Content-Type: multipart/mixed; boundary="-------------0211110729461" This is a multi-part message in MIME format. ---------------0211110729461 Content-Type: text/plain; name="02-452.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-452.keywords" quantum and classical dynamics, long-range magnetic fields, asymptotic completeness ---------------0211110729461 Content-Type: application/postscript; name="preprint_CHS.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="preprint_CHS.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: preprint_CHS.dvi %%Pages: 95 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Helvetica %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips preprint_CHS.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.11.11:1339 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: 8r.enc % @@psencodingfile@{ % author = "S. 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f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g (.)141 b(42)296 3928 y(4.2)100 b(Construction)33 b(of)f(the)h(appro)m (ximate)e(ev)m(olution)65 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h (.)g(.)g(.)g(.)g(.)g(.)141 b(44)296 4048 y(4.3)100 b(Stating)31 b(the)i(main)e(result)i(of)f(the)h(pap)s(er)f(.)50 b(.)f(.)h(.)g(.)g(.) g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)141 b(45)150 4266 y Fv(5)90 b(Pro)s(of)37 b(of)h(the)f(existence)g(of)h (scattering)e(states)1377 b(46)296 4387 y Fz(5.1)100 b(Existence)34 b(of)e(\012)1140 4351 y Ft(d)1140 4411 y Fs(+)1252 4387 y Fz(.)50 b(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g (.)g(.)141 b(46)296 4507 y(5.2)100 b(Pro)m(ving)32 b(the)h(inclusion)e (Ran\(\012)1743 4471 y Ft(d)1743 4532 y Fs(+)1802 4507 y Fz(\))d Fr(\022)g(H)2057 4522 y Ft(E)2109 4534 y Fq(d)2175 4507 y Fz(.)49 b(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.) 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Fz(.)50 b(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g (.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)141 b(60)1960 5776 y(ii)p eop %%Page: 3 3 3 2 bop 296 407 a Fz(6.6)531 381 y(~)521 407 y Fu(\030)37 b Fz(is)32 b(not)g(lo)s(calized)e(on)j(the)g(negativ)m(e)g(axis)50 b(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g (.)g(.)141 b(62)296 527 y(6.7)109 b(~)-58 b Fu(\032)33 b Fz(is)f(lo)s(calized)e(b)s(elo)m(w)i Fu(\032)1426 542 y Ft(E)1508 527 y Fz(+)22 b Fu(\017)68 b Fz(.)50 b(.)g(.)g(.)g(.)g(.)g (.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.) g(.)141 b(63)296 648 y(6.8)100 b Fr(\000)p Fu(@)649 663 y Ft(t)679 648 y Fu(S)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))33 b(is)f(close)g(to)g Fu(H)39 b Fz(.)50 b(.)g(.)g(.)g(.)g(.)g(.) g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g (.)g(.)141 b(68)150 866 y Fv(7)90 b(Asymptotic)36 b(completeness:)49 b Fu(\015)1679 829 y Fs(2)1674 890 y(1)1740 866 y Fz(+)22 b Fu(\015)1894 829 y Fs(2)1889 890 y(2)1971 866 y Fv(is)37 b(in)m(tegrable)1142 b(70)296 986 y Fz(7.1)100 b(A)32 b(quan)m(tum)h(v)m(ersion)g(for)f(\(3.128\))50 b(.)g(.)g(.)g(.)g(.)f(.) h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)141 b(70)296 1106 y(7.2)100 b(A)32 b(propagation)f(estimate)h(for)g(the)37 b(^)-53 b Fu(\015)5 b Fz('s)101 b(.)50 b(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.) g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)141 b(74)521 1227 y(7.2.1)111 b(Starting)31 b(the)i(pro)s(of)f(of)g(Prop)s(osition)f (7.1)86 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.) 141 b(74)521 1347 y(7.2.2)111 b(Estimating)30 b Fu(R)q Fz(\()p Fu(\034)6 b(;)17 b(\026)p Fz(\).)51 b(.)f(.)g(.)g(.)g(.)g(.)g (.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.) g(.)141 b(75)521 1467 y(7.2.3)111 b(The)33 b(Heisen)m(b)s(erg)h(deriv) -5 b(ativ)m(e)32 b(of)g(\000\()p Fu(B)2261 1482 y Ft(\034)t(;\026)2362 1467 y Fz(\))82 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.) g(.)g(.)141 b(79)521 1588 y(7.2.4)111 b(A)32 b(propagation)f(estimate)h (for)g Fu(L)2086 1603 y Ft(\016)2175 1588 y Fz(.)49 b(.)h(.)g(.)g(.)g (.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)141 b(81)150 1806 y Fv(8)90 b(Asymptotic)36 b(completeness:)49 b(existence)37 b(of)g Fz(\012)2301 1821 y Fs(+)3713 1806 y Fv(82)150 2024 y(9)90 b(Appro)m(ximate)36 b(dynamics)i(for)f(negativ) m(e)g(times)1363 b(88)150 2242 y(10)34 b(Op)s(en)39 b(problems)2669 b(90)150 2460 y(11)72 b(Ac)m(kno)m(wledgmen)m(ts)2499 b(91)1947 5776 y Fz(iii)p eop %%Page: 1 4 1 3 bop 150 407 a Fw(1)161 b(In)l(tro)t(duction)150 655 y Fp(1.1)135 b(Tw)l(o-dimensional)47 b(purely)d(magnetic)j (Hamiltonians)150 840 y Fz(A)33 b(classical)e(particle)g(in)h(a)g (magnetic)f(\014eld)i(is)f(describ)s(ed)h(b)m(y)h(the)f(Hamiltonian) 1165 1099 y Fu(h)p Fz(\()p Fv(x)p Fu(;)17 b Fo(\030)r Fz(\))28 b(=)1595 1031 y(1)p 1595 1076 49 4 v 1595 1167 a(2)1653 1099 y(\()p Fo(\030)d Fr(\000)e Fv(a)p Fz(\()p Fv(x)p Fz(\)\))2094 1057 y Fs(2)2133 1099 y Fu(;)115 b Fz(\()p Fv(x)p Fu(;)17 b Fo(\030)r Fz(\))28 b Fr(2)g Fn(R)2695 1057 y Fs(2)p Ft(n)2783 1099 y Fu(:)814 b Fz(\(1.1\))150 1350 y(The)25 b(magnetic)e(\014eld)g Fv(B)p Fz(\()p Fv(x)p Fz(\))h(is)f(obtain)g(from)g(the)h(v)m(ector)h(p)s(oten)m(tial)e Fv(a)h Fz(b)m(y)h(exterior)f(di\013eren)m(tiation,)150 1470 y Fv(B)p Fz(\()p Fv(x)p Fz(\))37 b(=)f Fu(d)p Fv(a)p Fz(\()p Fv(x)p Fz(\).)60 b(In)39 b(this)e(pap)s(er)i(w)m(e)g(study)g (the)f(classical)f(and)h(quan)m(tum)g(dynamics)g(of)g(a)f(t)m(w)m(o)150 1590 y(dimensional)30 b(particle)h(in)h(a)g(magnetic)g(\014eld)g(of)g (the)h(form)912 1860 y Fv(B)p Fz(\()p Fv(x)p Fz(\))27 b(=)1268 1792 y Fu(b)p Fz(\()p Fu(\022)s Fz(\))p 1268 1837 166 4 v 1327 1928 a Fu(r)1443 1860 y(dx)1549 1875 y Fs(1)1611 1860 y Fr(^)c Fu(dx)1806 1875 y Fs(2)1845 1860 y Fu(;)114 b Fv(x)28 b Fz(=)g(\()p Fu(r)19 b Fz(cos)e Fu(\022)s(;)g(r)i Fz(sin)e Fu(\022)s Fz(\))27 b Fr(2)i Fn(R)2991 1819 y Fs(2)3036 1860 y Fu(:)561 b Fz(\(1.2\))150 2111 y(W)-8 b(e)33 b(are)g(in)m(terested)g(in)f(orbits)g(\()p Fv(x)p Fz(\()p Fu(t)p Fz(\))p Fu(;)17 b Fo(\030)r Fz(\()p Fu(t)p Fz(\)\))33 b(for)f(whic)m(h)1717 2331 y(lim)1702 2390 y Ft(t)p Fm(!1)1885 2331 y Fu(r)s Fz(\()p Fu(t)p Fz(\))27 b(=)h Fr(1)1350 b Fz(\(1.3\))150 2587 y(and)26 b(hence)h(in)e(scattering)h(theory)-8 b(.)42 b(The)26 b(deca)m(y)i(rate)d Fr(h)p Fv(x)p Fr(i)2270 2551 y Fm(\000)p Fs(1)2390 2587 y Fz(in)g(\(1.2\))h(seems)g(to)g(b)s(e)g(the)g(b)s (orderline)150 2708 y(rate)k(of)g(deca)m(y)i(for)d(whic)m(h)i(w)m(e)g (can)g(b)s(e)f(assured)i(of)d(\(1.3\))h(\(at)g(least)g(for)f(some)h (range)h(of)e(energies\).)150 2828 y(F)-8 b(or)39 b(if)f(w)m(e)j(tak)m (e)g Fv(B)p Fz(\()p Fv(x)p Fz(\))e(=)h(\()p Fu(b=r)1343 2792 y Ft(\015)1387 2828 y Fz(\))p Fu(dx)1531 2843 y Fs(1)1598 2828 y Fr(^)27 b Fu(dx)1797 2843 y Fs(2)1876 2828 y Fz(with)40 b Fu(b)g Fz(a)f(nonzero)i(constan)m(t)f(and)g(0)g Fu(<)f(\015)45 b(<)40 b Fz(1,)h(a)150 2948 y(v)m(ector)31 b(p)s(oten)m(tial)e(satisfying)g Fv(B)p Fz(\()p Fv(x)p Fz(\))e(=)h Fu(d)p Fv(a)p Fz(\()p Fv(x)p Fz(\))i(is)f(readily)g(found)i (and)f(leads)g(to)g(the)g(conserv)-5 b(ation)150 3069 y(la)m(ws)580 3238 y Fu(l)30 b Fz(=)753 3171 y Fu(@)5 b(h)p 753 3215 114 4 v 757 3306 a(@)g(\022)903 3238 y Fz(=)28 b(Const)q Fu(;)114 b(E)34 b Fz(=)27 b Fu(h)h Fz(=)1805 3171 y(1)p 1805 3215 49 4 v 1805 3306 a(2)1880 3098 y Fl(\022)1963 3171 y Fu(dr)p 1963 3215 98 4 v 1969 3306 a(dt)2071 3098 y Fl(\023)2144 3120 y Fs(2)2206 3238 y Fz(+)2314 3171 y(1)p 2314 3215 49 4 v 2314 3306 a(2)2389 3098 y Fl(\022)2480 3171 y Fu(l)p 2472 3215 47 4 v 2472 3306 a(r)2551 3238 y Fr(\000)23 b Fu(cr)2740 3197 y Fs(1)p Fm(\000)p Ft(\015)2874 3098 y Fl(\023)2947 3120 y Fs(2)3014 3238 y Fz(=)28 b(Const)q Fu(:)150 3480 y Fz(Here)h Fu(c)f Fz(=)610 3441 y Ft(b)p 560 3457 131 4 v 560 3515 a Fs(2)p Fm(\000)p Ft(\015)700 3480 y Fz(,)i Fu(l)h Fz(is)d(the)h(angular)e (momen)m(tum)h(and)h Fu(E)34 b Fz(is)29 b(the)g(energy)-8 b(.)43 b(It)29 b(follo)m(ws)e(that)i(all)e(orbits)150 3601 y(are)37 b(con\014ned)h(to)f(a)g(b)s(ounded)g(region)f(of)h(phase) h(space.)58 b(Indeed,)39 b(Miller)c(and)i(Simon)f(analyzed)150 3721 y(the)h(corresp)s(onding)f(quan)m(tum)g(Hamiltonian)c(and)k(sho)m (w)m(ed)j(that)c(its)h(sp)s(ectrum)g(is)g(pure)h(p)s(oin)m(t,)150 3842 y(dense)d(in)e([0)p Fu(;)17 b Fr(1)p Fz(\))32 b(\(for)g(more)f (details,)h(see)i([C-F-K-S,)d(Theorem)i(6.2]\).)296 3962 y(On)26 b(the)f(other)g(hand,)j(m)m(uc)m(h)d(w)m(ork)h(o)m(v)m(er)h (the)e(last)g(t)m(w)m(en)m(t)m(y)i(y)m(ears)g(has)e(b)s(een)h(done)g (in)e(analyzing)150 4082 y(the)38 b(quan)m(tum)g(problem)f(with)g Fr(j)p Fv(B)p Fz(\()p Fv(x)p Fz(\))p Fr(j)f Fz(=)h Fr(O)s Fz(\()p Fr(h)p Fv(x)p Fr(i)2029 4046 y Fm(\000)p Fs(1)p Fm(\000)p Ft(\017)2206 4082 y Fz(\))h(with)f Fu(\017)g(>)g Fz(0)g(\(in)g(an)m(y)i(dimension)d Fr(\025)h Fz(2\).)150 4203 y(W)-8 b(e)32 b(brie\015y)g(review)g(kno)m(wn)h(results)f(in)e (this)h(case.)45 b(Firstly)-8 b(,)30 b(the)i(existence)h(part)e(for)g (1)d Fu(<)f(\015)33 b(<)27 b Fr(1)150 4323 y Fz(is)h(co)m(v)m(ered)j(b) m(y)e(general)f(results)h(of)f(H\177)-49 b(ormander)28 b(\(see)i([H1]\))f(whic)m(h)g(hold)f(in)f(com)m(bination)g(with)h(a)150 4443 y(long-range)e(scalar)h(p)s(oten)m(tial.)40 b(The)29 b(comparison)d(dynamics)h(used)i(in)e([H1])h(to)f(construct)i(a)e(w)m (a)m(v)m(e)150 4564 y(op)s(erator)j(preserv)m(es)i(the)f(momen)m(tum)e (\(it)g(is)h(a)g(re\014ned)h(Dollard-t)m(yp)s(e)d(dynamics\).)42 b(Asymptotic)150 4684 y(completeness)f(w)m(as)g(pro)m(v)m(ed)h(b)m(y)f (H\177)-49 b(ormander)40 b(\(using)g(stationary)f(metho)s(ds\))h(in)g ([H2,)i(Chapter)150 4805 y(30].)78 b(In)45 b(addition)d(w)m(e)j(men)m (tion)e(here)i(the)g(w)m(ork)g(of)f(Rob)s(ert)g(\(see)h([R]\))f(whic)m (h)h(also)e(includes)150 4925 y(long-range)k(scalar)h(p)s(oten)m (tials.)91 b(The)50 b(w)m(a)m(v)m(e)h(op)s(erators)d(in)g([R])h(are)g (constructed)h(using)f(the)150 5045 y(stationary)39 b(mo)s(di\014er)f (of)h(Isozaki-Kitada)f(\(see)j([I-K])e(for)g(details\).)64 b(V)-8 b(ery)40 b(recen)m(tly)h(Roux)f(and)150 5166 y(Y)-8 b(afaev)23 b(revisited)g(this)f(problem)g(in)g([R-Y],)g(and)h(they)g (also)f(in)m(v)m(estigated)h(the)g(sp)s(ectral)g(prop)s(erties)150 5286 y(of)32 b(the)h(corresp)s(onding)g(scattering)f(matrix)f Fu(S)6 b Fz(.)296 5406 y(Secondly)-8 b(,)48 b(the)d(case)g Fu(\015)53 b(>)48 b Fz(3)p Fu(=)p Fz(2)43 b(w)m(as)j(further)e(in)m(v)m (estigated)h(b)m(y)g(Loss)g(and)g(Thaller)e(in)g([L-)150 5527 y(T1,2])32 b(for)e(purely)i(magnetic)e(Sc)m(hr\177)-49 b(odinger)32 b(and)f(Dirac)f(op)s(erators)i(where)g(they)h(pro)m(v)m(e) f(existence)1963 5776 y(1)p eop %%Page: 2 5 2 4 bop 150 407 a Fz(and)40 b(completeness)h(for)e(the)i(ordinary)e (M\034ller)g(op)s(erators)h(emplo)m(ying)e(Enss')k(time-dep)s(enden)m (t)150 527 y(approac)m(h.)51 b(Then)37 b(Nicoleau)c(and)j(Rob)s(ert)e (in)h([N-R])f(treat)h(the)h(Sc)m(hr\177)-49 b(odinger)35 b(problem)f(for)g Fu(\015)j(>)150 648 y Fz(3)p Fu(=)p 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(p)s(oten)m(tial\))d(do)s(es)j(not)e(app)s(ear)h(to)f(b)s(e)i(treated) 150 1490 y(in)33 b(the)i(literature.)46 b(Similar)30 b(problems)k(with)f(homogeneous)h(of)g(degree)h(zero)f(electric)g(p)s (oten)m(tial)150 1611 y(ha)m(v)m(e)47 b(b)s(een)g(considered)g(b)m(y)h (t)m(w)m(o)e(of)g(the)h(authors)f(in)f([He],)51 b([H-S];)46 b(see)h(also)e(a)h(related)g(w)m(ork)150 1731 y(of)39 b(Hassell,)i(Melrose)f(and)f(V)-8 b(asy)41 b(in)d([H-M-V].)i(In)g (those)g(cases,)j(the)d(Hamiltonian)35 b(is)40 b(roughly)150 1851 y Fu(H)d Fz(=)29 b Fr(\000)p Fz(\001)23 b(+)g Fu(V)e Fz(\()p Fv(x)p Fu(=)p Fr(j)p Fv(x)p Fr(j)p Fz(\),)33 b(where)i Fu(V)55 b Fz(is)32 b(de\014ned)j(on)f(the)f(unit)g(sphere.)48 b(The)34 b(generic)f(b)s(eha)m(vior)g(for)150 1972 y(the)28 b(classical)e(orbits)g(in)h(this)g(situation)e(is)i(that)g(they)i(are)e (ev)m(en)m(tually)h(trapp)s(ed)f(in)g(the)h(directions)150 2092 y(in)f(whic)m(h)i Fu(V)50 b Fz(has)28 b(lo)s(cal)e(extrema)i(\(in) g(the)g(quan)m(tum)h(case)g(the)f(lo)s(cal)e(maxima)g(and)i(saddle)g(p) s(oin)m(ts)150 2213 y(are)33 b(excluded\),)g(hence)h(v)m(ery)g(roughly) e(the)h(tra)5 b(jectories)33 b(are)g(asymptotically)d(straigh)m(t.)296 2333 y(The)42 b(b)s(eha)m(vior)f(for)f(the)i(t)m(w)m(o-dimensional)c (magnetic)i(case)i(with)e Fu(\015)47 b Fz(=)42 b(1)f(turns)g(out)g(to)g (b)s(e)150 2453 y(di\013eren)m(t)k(\(at)g(least)f(for)g(the)i(case)f (treated)h(in)e(this)g(pap)s(er\).)81 b(Assume)46 b(that)e(w)m(e)i(are) f(giv)m(en)g(a)150 2574 y(magnetic)36 b(\014eld)h(whic)m(h)h(is)f (homogeneous)g(of)g(degree)i Fr(\000)p Fz(1)e(outside)g(the)h(unit)f (disc,)i(i.e.)57 b(is)37 b(giv)m(en)150 2694 y(b)m(y)c Fu(r)332 2658 y Fm(\000)p Fs(1)426 2694 y Fu(b)p Fz(\()p Fu(\022)s Fz(\))f(for)f Fu(r)f Fr(\025)e Fz(1.)43 b(F)-8 b(or)31 b(the)h(classical)e(orbits)h(sta)m(ying)h(outside)g(the)g(unit) f(disc)g(\(for)g(all)f(large)150 2814 y(times\))i(and)g(with)g(energy)i Fu(E)g(>)27 b(E)1453 2829 y Ft(b)1520 2814 y Fz(where)1568 3004 y Fu(E)1640 3019 y Ft(b)1703 3004 y Fz(=)64 b(max)1806 3070 y Ft(\022)r Fm(2)p Fs([0)p Ft(;)p Fs(2)p Ft(\031)r Fs(])2077 3004 y Fu(b)2118 2962 y Fs(2)2158 3004 y Fz(\()p Fu(\022)s Fz(\))p Fu(=)p Fz(2)p Fu(;)1217 b Fz(\(1.4\))150 3243 y(w)m(e)32 b(ha)m(v)m(e)g(an)f(\\easy")h(Mourre)f(estimate)f (implying)e(that)j(their)f(radial)f(v)m(elo)s(cities)h(ev)m(en)m (tually)h(b)s(e-)150 3363 y(come)d(p)s(ositiv)m(e,)g(hence)h(these)g (orbits)e(mo)m(v)m(e)h(to)g(in\014nit)m(y)-8 b(.)41 b(\(W)-8 b(e)28 b(remark)g(that)f(when)i Fu(b)g Fz(is)e(constan)m(t,)150 3484 y Fu(E)222 3499 y Ft(b)290 3484 y Fz(equals)33 b(the)h(mobilit)m (y)c(edge)k(in)e(the)i(Miller-Simon)29 b(mo)s(del.\))44 b(A)33 b(more)f(detailed)h(analysis)f(un-)150 3604 y(der)h(the)g (additional)c(condition)i(that)h Fu(b)h Fz(is)f(strictly)f(negativ)m(e) i(\(a)f(similar)d(analysis)j(ma)m(y)g(b)s(e)g(done)150 3724 y(for)h Fu(b)e(>)e Fz(0\))34 b(sho)m(ws)h(that)f(the)g(asymptotic) f(orbits)g(are)h(logarithmic)c(spirals)j(and)h(not)g(asymptot-)150 3845 y(ically)i(straigh)m(t)h(as)h(in)f(the)h(p)s(oten)m(tial)e(case.) 60 b(Moreo)m(v)m(er,)41 b(w)m(e)e(can)f(ev)m(en)h(go)f(b)s(elo)m(w)f Fu(E)3384 3860 y Ft(b)3457 3845 y Fz(with)g(our)150 3965 y(considerations)32 b(if)g Fu(b)h Fz(is)f(not)g(constan)m(t.)296 4086 y(The)j(main)d(goal)h(of)g(the)i(pap)s(er)f(is)f(to)h(demonstrate) g(analogous)f(b)s(eha)m(vior)g(in)h(quan)m(tum)g(me-)150 4206 y(c)m(hanics.)52 b(In)35 b(a)g(follo)m(wing)e(pap)s(er)i([C-H-S])g (w)m(e)h(will)d(study)j(the)g(case)g(in)e(whic)m(h)i Fu(b)f Fz(has)h(zeros)g(and)150 4336 y(in)30 b(particular)f(the)j(zero) f(\015ux)h(case)1477 4256 y Fl(R)1543 4282 y Fs(2)p Ft(\031)1524 4370 y Fs(0)1642 4336 y Fu(b)p Fz(\()p Fu(\022)s Fz(\))p Fu(d\022)f Fz(=)d(0.)42 b(F)-8 b(or)31 b(the)g(latter)f(case)i(the)f (classical)f(scatter-)150 4456 y(ing)36 b(orbits)h(approac)m(h)h(a)f (direction)f(in)g(whic)m(h)i Fu(b)p Fz(\()p Fu(\022)s Fz(\))e(=)g(0.)57 b(When)39 b Fu(b)e Fz(c)m(hanges)i(sign)e(but)g(the)h (\015ux)150 4577 y(is)f(di\013eren)m(t)g(from)f(zero)h(b)s(oth)g(t)m (yp)s(es)i(of)d(b)s(eha)m(vior)h(ma)m(y)g(o)s(ccur:)53 b(some)37 b(tra)5 b(jectories)37 b(are)g(dra)m(wn)150 4697 y(to)m(w)m(ard)c(the)g(half-lines)d(de\014ned)k(b)m(y)g(the)f (zero)s(es)g(of)f Fu(b)h Fz(while)f(others)h(will)e(spiral.)296 4817 y(There)41 b(are)e(indications)f(of)h(somewhat)h(similar)c (results)k(in)e(dimensions)h(higher)g(than)g(t)m(w)m(o,)150 4938 y(although)31 b(the)i(geometry)g(and)g(analysis)e(are)i(more)f (complicated.)150 5222 y Fp(1.2)135 b(Classical)47 b(mec)l(hanics:)61 b(preliminaries)46 b(and)f(main)h(results)150 5407 y Fz(Our)40 b(system)h(consists)f(here)h(of)e(a)h(classical)e(particle)h (con\014ned)i(to)e(a)h(plane)f(and)h(sub)5 b(jected)42 b(to)150 5527 y(a)d(magnetic)f(\014eld)i Fv(B)f Fz(whic)m(h)h(is)e (assumed)j(to)e(b)s(e)g(homogeneous)h(of)f(degree)h Fr(\000)p Fz(1.)64 b(As)40 b(usual,)h Fv(B)1963 5776 y Fz(2)p eop %%Page: 3 6 3 5 bop 150 407 a Fz(is)40 b(\\orthogonal")f(to)i(the)g(plane)g(in)f (whic)m(h)h(the)h(particle)d(mo)m(v)m(es)j(so)f(it)f(has)i(only)e(one)h (nonzero)150 527 y(comp)s(onen)m(t)d(\(the)f(\\third")g(one\))g(whic)m (h)i(is)e(of)g(the)h(form)e Fu(B)5 b Fz(\()p Fv(x)p Fz(\))36 b(=)g Fu(b)p Fz(\()p Fu(\022)s Fz(\))p Fu(=r)s Fz(.)58 b(W)-8 b(e)38 b(assume)g(that)g Fu(b)150 648 y Fz(is)30 b(smo)s(oth)g(and)h(negativ)m(e.)43 b(The)31 b(asso)s(ciated)g(transv)m (erse)i(magnetic)c(v)m(ector)j(p)s(oten)m(tial)d(is)h Fv(a)p Fz(\()p Fv(x)p Fz(\))e(=)150 768 y(\()p Fr(\000)17 b Fz(sin\()p Fu(\022)s Fz(\))p Fu(;)g Fz(cos\()p Fu(\022)s Fz(\)\))p Fu(b)p Fz(\()p Fu(\022)s Fz(\).)296 888 y(The)34 b(corresp)s(onding)e(classical)f(Hamiltonian)e(function)j(is)g(\(in)g (p)s(olar)f(co)s(ordinates\))1217 1167 y Fu(h)p Fz(\()p Fu(r)m(;)17 b(\022)s Fz(;)g Fu(\032;)g(l)r Fz(\))28 b(:=)1820 1099 y(1)p 1820 1144 49 4 v 1820 1235 a(2)1879 1167 y Fu(\032)1929 1125 y Fs(2)1991 1167 y Fz(+)2098 1099 y(1)p 2098 1144 V 2098 1235 a(2)2174 1026 y Fl(\022)2265 1099 y Fu(l)p 2257 1144 47 4 v 2257 1235 a(r)2336 1167 y Fr(\000)23 b Fu(b)p Fz(\()p Fu(\022)s Fz(\))2601 1026 y Fl(\023)2674 1048 y Fs(2)2730 1167 y Fu(;)867 b Fz(\(1.5\))150 1428 y(where)30 b Fu(\032)e Fz(=)639 1389 y Fk(x)p 619 1405 83 4 v 619 1463 a Fm(j)p Fk(x)p Fm(j)726 1428 y Fr(\001)14 b Fz(\()p Fo(\030)i Fr(\000)e Fv(a)p Fz(\))30 b(is)e(the)h(radial)e(v)m (elo)s(cit)m(y)h(and)h Fu(l)h Fz(=)e Fu(x)2377 1443 y Fs(1)2417 1428 y Fu(p)2466 1443 y Fs(2)2519 1428 y Fr(\000)14 b Fu(x)2665 1443 y Fs(2)2706 1428 y Fu(p)2755 1443 y Fs(1)2823 1428 y Fz(is)28 b(the)h(canonical)e(angular)150 1549 y(momen)m(tum.)42 b(The)34 b(Hamilton)29 b(equations)k(for)f Fu(r)j Fz(and)e Fu(\022)j Fz(are)1135 1738 y Fu(dr)p 1135 1782 98 4 v 1141 1873 a(dt)1270 1805 y Fz(=)1384 1738 y Fu(@)5 b(h)p 1384 1782 114 4 v 1387 1873 a(@)g(\032)1535 1805 y Fz(=)27 b Fu(\032;)1840 1738 y(d\022)p 1840 1782 99 4 v 1846 1873 a(dt)1976 1805 y Fz(=)2090 1738 y Fu(@)5 b(h)p 2090 1782 114 4 v 2102 1873 a(@)g(l)2240 1805 y Fz(=)2354 1738 y(1)p 2354 1782 49 4 v 2355 1873 a Fu(r)2429 1664 y Fl(\022)2520 1738 y Fu(l)p 2513 1782 47 4 v 2513 1873 a(r)2591 1805 y Fr(\000)23 b Fu(b)2732 1664 y Fl(\023)2822 1805 y Fu(:)775 b Fz(\(1.6\))150 2061 y(The)34 b(Hamilton)29 b(equations)k(for)f Fu(\032)h Fz(and)g Fu(l)h Fz(are)1389 2250 y Fu(d\032)p 1389 2294 102 4 v 1397 2386 a(dt)1528 2317 y Fz(=)27 b Fr(\000)1718 2250 y Fu(@)5 b(h)p 1718 2294 114 4 v 1723 2386 a(@)g(r)1869 2317 y Fz(=)1973 2177 y Fl(\022)2064 2250 y Fu(l)p 2056 2294 47 4 v 2056 2386 a(r)2135 2317 y Fr(\000)23 b Fu(b)p Fz(\()p Fu(\022)s Fz(\))2400 2177 y Fl(\023)2527 2250 y Fu(l)p 2500 2294 87 4 v 2500 2386 a(r)2547 2357 y Fs(2)3624 2317 y Fz(\(1.7\))150 2574 y(and)1344 2659 y Fu(dl)p 1342 2703 86 4 v 1342 2795 a(dt)1465 2726 y Fz(=)28 b Fr(\000)1656 2659 y Fu(@)5 b(h)p 1656 2703 114 4 v 1660 2795 a(@)g(\022)1807 2726 y Fz(=)1910 2586 y Fl(\022)2001 2659 y Fu(l)p 1994 2703 47 4 v 1994 2795 a(r)2072 2726 y Fr(\000)23 b Fu(b)p Fz(\()p Fu(\022)s Fz(\))2337 2586 y Fl(\023)2428 2726 y Fu(b)2469 2685 y Fm(0)2492 2726 y Fz(\()p Fu(\022)s Fz(\))p Fu(:)981 b Fz(\(1.8\))150 2962 y(Let)33 b(us)g(in)m(tro)s(duce) g(the)g(transv)m(erse)h(v)m(elo)s(cit)m(y)f Fu(\030)f Fz(:=)2095 2923 y Ft(l)p 2089 2939 34 4 v 2089 2996 a(r)2155 2962 y Fr(\000)23 b Fu(b)p Fz(,)33 b(whic)m(h)g(ob)s(eys)h(the)f (equations)1452 3158 y Fu(d\022)p 1452 3202 99 4 v 1458 3294 a(dt)1588 3225 y Fz(=)1702 3158 y Fu(\030)p 1702 3202 48 4 v 1702 3294 a(r)1759 3225 y(;)1910 3158 y(d\030)p 1910 3202 99 4 v 1916 3294 a(dt)2046 3225 y Fz(=)27 b Fr(\000)2236 3158 y Fu(\030)h Fz(+)22 b Fu(b)p 2236 3202 210 4 v 2318 3294 a(r)2456 3225 y(\032:)1091 b Fz(\(1.9\))296 3461 y(Since)36 b Fu(h)f Fz(in)g(\(1.5\))g(do)s(es)h(not)f(dep)s(end)h (on)g Fu(t)p Fz(,)g(the)g(energy)g(is)f(conserv)m(ed;)40 b(that)35 b(is,)g(on)h(a)f(giv)m(en)150 3581 y(tra)5 b(jectory)33 b(one)g(has)1585 3701 y Fu(\032)1635 3660 y Fs(2)1675 3701 y Fz(\()p Fu(t)p Fz(\))22 b(+)g Fu(\030)1954 3660 y Fs(2)1993 3701 y Fz(\()p Fu(t)p Fz(\))28 b(=)g(2)p Fu(E)6 b(:)1186 b Fz(\(1.10\))150 3959 y Fv(1.2.1)112 b(An)38 b(attractiv)m(e)e(Lagrangian)i(manifold)150 4144 y Fz(W)-8 b(e)39 b(will)e(no)m(w)j(discuss)g(v)-5 b(arious)39 b(results)g(obtained)g(in)f(the)i(classical)d(framew)m(ork.)63 b(Apart)39 b(from)150 4264 y(their)h(o)m(wn)i(in)m(trinsic)d(in)m (terest)i(they)h(serv)m(e)h(as)d(motiv)-5 b(ation)38 b(for)i(our)h(main)e(result)i(in)f(quan)m(tum)150 4384 y(mec)m(hanics)33 b(\(see)h(Theorem)e(1.1)h(b)s(elo)m(w\).)296 4505 y(De\014ne)g(the)g(extended)i(con\014guration)d(space)1094 4709 y Fr(A)27 b Fz(:=)h Fr(f)p 1382 4654 56 4 v Fu(x)g Fz(=)f(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))28 b(:)55 b Fu(t)28 b(>)g Fz(0)p Fu(;)44 b(r)30 b(>)e Fz(0)p Fu(;)44 b(\022)31 b Fr(2)d Fn(T)p Fr(g)20 b Fu(;)150 4914 y Fz(and)33 b(consider)g(the)g(function)p 844 5037 57 4 v 844 5118 a Fu(h)q Fz(\()p 939 5063 56 4 v Fu(x;)p 1038 5063 52 4 v 17 w(\021)s Fz(\))28 b(:=)g Fu(\034)33 b Fz(+)22 b Fu(h)p Fz(\()p Fu(r)m(;)17 b(\022)s Fz(;)g Fu(\032;)g(l)r Fz(\))p Fu(;)p 2132 5063 56 4 v 212 w(x)28 b Fr(2)g(A)p Fu(;)p 2628 5063 52 4 v 212 w(\021)j Fz(=)d(\()p Fu(\034)6 b(;)17 b(\032;)g(l)r Fz(\))p Fu(:)150 5323 y Fz(W)-8 b(e)33 b(in)m(tro)s(duce)g(a)f(symplectic)g(form)f(on)i Fu(T)1750 5287 y Fm(\003)1789 5323 y Fr(A)f Fz(giv)m(en)h(b)m(y)967 5527 y Fu(d)p Fv(x)22 b Fr(^)h Fu(d)p Fo(\030)h Fz(+)e Fu(dt)g Fr(^)h Fu(d\034)39 b Fz(=)27 b Fu(d\022)e Fr(^)e Fu(dl)h Fz(+)e Fu(dr)j Fr(^)d Fu(d\032)g Fz(+)g Fu(dt)g Fr(^)h Fu(d\034)6 b(:)1963 5776 y Fz(3)p eop %%Page: 4 7 4 6 bop 150 407 a Fz(W)-8 b(e)46 b(construct)h(a)f(solution)e Fu(S)51 b Fz(\(de\014ned)d(on)d Fr(A)p Fz(\))h(to)f(the)h(equation)p 2786 326 57 4 v 46 w Fu(h)p Fz(\()p 2880 352 56 4 v Fu(x;)17 b Fr(r)p Fu(S)6 b Fz(\))50 b(=)g(0,)f(whic)m(h)d(is)150 527 y(nothing)32 b(but)h(the)g(Hamilton-Jacobi)28 b(equation)k(\(see)i (Section)f(2)f(for)g(details\))1421 706 y Fu(@)1472 721 y Ft(t)1502 706 y Fu(S)c Fz(+)22 b Fu(h)p 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y(precisely)-8 b(,)44 b(assume)f(that)e(\()p Fu(r)m(;)17 b(\022)s Fz(;)g Fu(\032;)g(l)r Fz(\))42 b(is)g(a)f (solution)f(for)i(the)g(sym)m(b)s(ol)f Fu(h)h Fz(with)g(energy)g Fu(E)50 b(>)43 b(E)3784 2094 y Ft(d)150 2199 y Fz(whic)m(h)j(exists)g (for)e(all)f Fu(t)50 b Fr(\025)f Fz(0.)81 b(Then)47 b(for)d(an)m(y)i Fu(\016)53 b(>)c Fz(0,)f(the)e(quan)m(tities)f Fu(E)37 b Fz(+)30 b Fu(@)3282 2214 y Ft(t)3312 2199 y Fu(S)6 b Fz(,)48 b Fu(\032)31 b Fr(\000)h Fu(@)3694 2214 y Ft(r)3732 2199 y Fu(S)6 b Fz(,)150 2320 y(\()p Fu(l)24 b Fr(\000)f Fu(@)392 2335 y Ft(\022)431 2320 y Fu(S)6 b Fz(\))p Fu(=r)35 b Fz(are)e(all)d Fr(O)s Fz(\()p Fu(t)1116 2284 y Fm(\000)p Fs(1+)p Ft(\016)1299 2320 y Fz(\))j(as)g Fu(t)28 b Fr(!)f(1)p Fz(.)43 b(Here)33 b(and)g(henceforth)h Fu(t)27 b Fr(!)h(1)k Fz(means)g Fu(t)c Fr(!)g Fz(+)p Fr(1)p Fz(.)296 2440 y(Ev)m(en)39 b(though)d(\(1.11\))g(ma)m(y)g(seem)h(somewhat)g (complicated)e(at)h(\014rst)h(glance,)h(w)m(e)f(obtain)f(in)150 2561 y(Section)30 b(2)g(that)785 2534 y(~)778 2561 y Fu(\022)s Fz(\()p Fu(t)p Fz(\))g(is)g(strictly)g(increasing)g(and)g (gro)m(ws)h(logarithmically)25 b(in)30 b(time.)41 b(This)31 b(allo)m(ws)150 2681 y(us)g(to)f(consider)h(the)g(radius)f(as)h(a)f (function)g(of)g(the)g(angle)g Fu(r)s Fz(\()2422 2655 y(~)2414 2681 y Fu(\022)s Fz(\))d(=)32 b(~)-53 b Fu(r)s Fz(\()p Fu(t)p Fz(\()2797 2655 y(~)2789 2681 y Fu(\022)s Fz(\)\))30 b(and)h(ev)m(en)m(tually)g(pro)m(v)m(e)150 2801 y(that)1490 2922 y(ln)15 b Fu(r)31 b Fz(=)c Fu(C)7 b Fz(\()p Fu(E)f Fz(\))2004 2895 y(~)1996 2922 y Fu(\022)26 b Fz(+)c Fu(R)q Fz(\()p Fu(E)6 b(;)2407 2895 y Fz(~)2400 2922 y Fu(\022)r Fz(\))1091 b(\(1.12\))150 3078 y(where)46 b Fu(C)7 b Fz(\()p Fu(E)f Fz(\))45 b(is)g(a)f(p)s(ositiv)m(e)h (increasing)f(function)h(and)g Fu(R)h Fz(is)e(2)p Fu(\031)t Fz(-p)s(erio)s(dic)f(in)3254 3052 y(~)3246 3078 y Fu(\022)s Fz(.)81 b(Moreo)m(v)m(er,)150 3199 y Fu(C)7 b Fz(\()p Fu(E)f Fz(\))36 b Fr(&)g Fz(0)i(when)h Fu(E)j Fr(&)36 b Fu(E)1222 3214 y Ft(d)1263 3199 y Fz(,)j(and)f(one)g(can)g(pro)m(v)m (e)h(that)e Fu(C)7 b Fz(\()p Fu(E)f Fz(\))p Fu(=)2656 3114 y Fr(p)p 2739 3114 127 4 v 85 x Fz(2)p Fu(E)43 b Fz(approac)m(hes)c(a)f(p)s(ositiv)m(e)150 3319 y(constan)m(t)e(when)g Fu(E)k Fz(increases)c(to)f(in\014nit)m(y)-8 b(.)49 b(With)34 b Fu(R)q Fz(\()p Fu(E)6 b(;)2316 3293 y Fz(~)2308 3319 y Fu(\022)s Fz(\))32 b(=)f Fu(R)q Fz(\()p Fu(E)6 b Fz(\))35 b(indep)s(enden)m(t)h(of)3473 3293 y(~)3465 3319 y Fu(\022)s Fz(,)g(\(1.12\))150 3439 y(is)c(the)h(equation)f(for)g(the)h(so-called) f(logarithmic)c(spiral.)296 3560 y(An)34 b(example)f(of)g(a)h(t)m (ypical)f(in)m(tegral)f(curv)m(e)j(\014rst)f(de\014ned)h(for)e(the)h (full)e(Hamiltonian)e(v)m(ector)150 3680 y(\014eld)i(and)h(then)g(pro)5 b(jected)34 b(to)e(the)h(rectangular)f(con\014guration)g(space)i(is)e (giv)m(en)g(b)s(elo)m(w.)925 5527 y @beginspecial 74 @llx 106 @lly 493 @urx 655 @ury 2551 @rwi 1984 @rhi @setspecial %%BeginDocument: Efinal401.eps %!PS-Adobe-3.0 EPSF-2.0 %%Title: b=-4.8-2sin, E=12.5, 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4326 4312 l S NP 4327 4297 m 4326 4304 l S NP 4327 4289 m 4327 4297 l S NP 4327 4281 m 4327 4289 l S NP 4328 4274 m 4327 4281 l S NP 4328 4266 m 4328 4274 l S NP 4328 4259 m 4328 4266 l S NP 4329 4251 m 4328 4259 l S NP 4329 4243 m 4329 4251 l S NP 4329 4236 m 4329 4243 l S NP 4329 4228 m 4329 4236 l S NP 4330 4220 m 4329 4228 l S NP 4330 4213 m 4330 4220 l S NP 4330 4205 m 4330 4213 l S NP 4330 4198 m 4330 4205 l S NP 4330 4190 m 4330 4198 l S NP 4331 4182 m 4330 4190 l S NP 4331 4175 m 4331 4182 l S NP 4331 4167 m 4331 4175 l S NP 4331 4159 m 4331 4167 l S NP 4331 4152 m 4331 4159 l S NP 4331 4144 m 4331 4152 l S NP 4331 4137 m 4331 4144 l S NP 4332 4129 m 4331 4137 l S NP 4332 4121 m 4332 4129 l S NP 4332 4114 m 4332 4121 l S NP 4332 4106 m 4332 4114 l S NP 4332 4098 m 4332 4106 l S NP 4332 4091 m 4332 4098 l S NP 4332 4083 m 4332 4091 l S NP 4332 4075 m 4332 4083 l S NP 4332 4068 m 4332 4075 l S NP 4332 4060 m 4332 4068 l S NP 4332 4053 m 4332 4060 l S NP 4332 4045 m 4332 4053 l S NP 4332 4037 m 4332 4045 l S NP 4332 4030 m 4332 4037 l S NP 4332 4022 m 4332 4030 l S NP 4332 4014 m 4332 4022 l S NP 4331 4007 m 4332 4014 l S NP 4331 3999 m 4331 4007 l S NP 4331 3991 m 4331 3999 l S NP 4331 3984 m 4331 3991 l S NP 4331 3976 m 4331 3984 l S NP 4331 3969 m 4331 3976 l S NP 4331 3961 m 4331 3969 l S NP 4331 3953 m 4331 3961 l S NP 4330 3946 m 4331 3953 l S NP 4330 3938 m 4330 3946 l S NP 4330 3930 m 4330 3938 l S NP 4330 3923 m 4330 3930 l S NP 4330 3915 m 4330 3923 l S NP 4329 3908 m 4330 3915 l S NP 4329 3900 m 4329 3908 l S NP 4329 3892 m 4329 3900 l S NP 4329 3885 m 4329 3892 l S NP 4328 3877 m 4329 3885 l S NP 4328 3869 m 4328 3877 l S NP 4328 3862 m 4328 3869 l S NP 4327 3854 m 4328 3862 l S NP 4327 3847 m 4327 3854 l S NP 4327 3839 m 4327 3847 l S NP 4326 3831 m 4327 3839 l S NP 4326 3824 m 4326 3831 l S NP 4326 3816 m 4326 3824 l S NP 4325 3808 m 4326 3816 l S NP 4325 3801 m 4325 3808 l S NP 4324 3793 m 4325 3801 l S NP 4324 3786 m 4324 3793 l S NP 4323 3778 m 4324 3786 l S NP 4323 3770 m 4323 3778 l S NP 4323 3763 m 4323 3770 l S NP 4322 3755 m 4323 3763 l S NP 4322 3748 m 4322 3755 l S NP 4321 3740 m 4322 3748 l S NP 4321 3733 m 4321 3740 l S NP 4320 3725 m 4321 3733 l S NP 4320 3717 m 4320 3725 l S NP 4319 3710 m 4320 3717 l S NP 4318 3702 m 4319 3710 l S NP 4318 3695 m 4318 3702 l S NP 4317 3687 m 4318 3695 l S NP 4317 3679 m 4317 3687 l S NP 4316 3672 m 4317 3679 l S NP 4316 3664 m 4316 3672 l S NP 4315 3657 m 4316 3664 l S NP 4314 3649 m 4315 3657 l S NP 4314 3642 m 4314 3649 l S NP 4313 3634 m 4314 3642 l S NP 4312 3627 m 4313 3634 l S NP 4312 3619 m 4312 3627 l S NP 4311 3611 m 4312 3619 l S NP 4310 3604 m 4311 3611 l S NP 4310 3596 m 4310 3604 l S NP 4309 3589 m 4310 3596 l S NP 4308 3581 m 4309 3589 l S NP 4307 3574 m 4308 3581 l S NP 4307 3566 m 4307 3574 l S NP 4306 3559 m 4307 3566 l S NP 4305 3551 m 4306 3559 l S NP 4304 3544 m 4305 3551 l S NP 4303 3536 m 4304 3544 l S NP 4303 3529 m 4303 3536 l S NP 4302 3521 m 4303 3529 l S NP 4301 3514 m 4302 3521 l S NP 4300 3506 m 4301 3514 l S NP 4299 3499 m 4300 3506 l S NP 4298 3491 m 4299 3499 l S NP 4297 3484 m 4298 3491 l S NP 4297 3476 m 4297 3484 l S NP 4296 3469 m 4297 3476 l S NP 4295 3461 m 4296 3469 l S NP 4294 3454 m 4295 3461 l S NP 4293 3446 m 4294 3454 l S NP 4292 3439 m 4293 3446 l S NP 4291 3431 m 4292 3439 l S NP 4290 3424 m 4291 3431 l S NP 4289 3416 m 4290 3424 l S NP 4288 3409 m 4289 3416 l S NP 4287 3401 m 4288 3409 l S NP 4286 3394 m 4287 3401 l S NP 4285 3387 m 4286 3394 l S NP 4284 3379 m 4285 3387 l S NP 4283 3372 m 4284 3379 l S NP 4282 3364 m 4283 3372 l S NP 4281 3357 m 4282 3364 l S NP 4279 3349 m 4281 3357 l S NP 4278 3342 m 4279 3349 l S NP 4277 3335 m 4278 3342 l S NP 4276 3327 m 4277 3335 l S NP 4275 3320 m 4276 3327 l S NP 4274 3313 m 4275 3320 l S NP 4273 3305 m 4274 3313 l S NP 4272 3298 m 4273 3305 l S NP 4270 3290 m 4272 3298 l S NP 4269 3283 m 4270 3290 l S NP 4268 3276 m 4269 3283 l S NP 4267 3268 m 4268 3276 l S NP 4265 3261 m 4267 3268 l S NP 4264 3254 m 4265 3261 l S NP 4263 3246 m 4264 3254 l S NP 4262 3239 m 4263 3246 l S NP 4260 3232 m 4262 3239 l S NP 4259 3224 m 4260 3232 l S NP 4258 3217 m 4259 3224 l S NP 4257 3210 m 4258 3217 l S NP 4255 3202 m 4257 3210 l S NP 4254 3195 m 4255 3202 l S NP 4253 3188 m 4254 3195 l S NP 4251 3180 m 4253 3188 l S NP 4250 3173 m 4251 3180 l S NP 4248 3166 m 4250 3173 l S NP 4247 3159 m 4248 3166 l S NP 4246 3151 m 4247 3159 l S NP 4244 3144 m 4246 3151 l S NP 4243 3137 m 4244 3144 l S NP 4241 3130 m 4243 3137 l S NP 4240 3122 m 4241 3130 l S NP 4238 3115 m 4240 3122 l S NP 4237 3108 m 4238 3115 l S NP 4235 3101 m 4237 3108 l S NP 4234 3093 m 4235 3101 l S NP 4232 3086 m 4234 3093 l S NP 4231 3079 m 4232 3086 l S NP 4229 3072 m 4231 3079 l S NP 4228 3065 m 4229 3072 l S NP 4226 3057 m 4228 3065 l S NP 4225 3050 m 4226 3057 l S NP 4223 3043 m 4225 3050 l S NP 4222 3036 m 4223 3043 l S NP 4220 3029 m 4222 3036 l S NP 4218 3022 m 4220 3029 l S NP 4217 3015 m 4218 3022 l S NP 4215 3007 m 4217 3015 l S NP 4214 3000 m 4215 3007 l S NP 4212 2993 m 4214 3000 l S NP 4210 2986 m 4212 2993 l S NP 4209 2979 m 4210 2986 l S NP 4207 2972 m 4209 2979 l S NP 4205 2965 m 4207 2972 l S NP 4204 2958 m 4205 2965 l S NP 4202 2951 m 4204 2958 l S NP 4200 2944 m 4202 2951 l S NP 4198 2937 m 4200 2944 l S NP 4197 2929 m 4198 2937 l S NP 4195 2922 m 4197 2929 l S NP 4193 2915 m 4195 2922 l S NP 4191 2908 m 4193 2915 l S NP 4190 2901 m 4191 2908 l S NP 4188 2894 m 4190 2901 l S NP 4186 2887 m 4188 2894 l S NP 4184 2880 m 4186 2887 l S NP 4182 2873 m 4184 2880 l S NP 4180 2866 m 4182 2873 l S NP 4179 2859 m 4180 2866 l S NP 4177 2852 m 4179 2859 l S NP 4175 2845 m 4177 2852 l S NP 4173 2839 m 4175 2845 l S NP 4171 2832 m 4173 2839 l S NP 4169 2825 m 4171 2832 l S NP 4167 2818 m 4169 2825 l S NP 4165 2811 m 4167 2818 l S NP 4163 2804 m 4165 2811 l S NP 4161 2797 m 4163 2804 l S NP 4159 2790 m 4161 2797 l S NP 4157 2783 m 4159 2790 l S NP 4155 2776 m 4157 2783 l S NP 4153 2770 m 4155 2776 l S NP 4151 2763 m 4153 2770 l S NP 4149 2756 m 4151 2763 l S NP 4147 2749 m 4149 2756 l S NP 4145 2742 m 4147 2749 l S NP 4143 2735 m 4145 2742 l S NP 4141 2729 m 4143 2735 l S NP 4139 2722 m 4141 2729 l S NP 4137 2715 m 4139 2722 l S NP 4135 2708 m 4137 2715 l S NP 4133 2701 m 4135 2708 l S NP 4131 2695 m 4133 2701 l S NP 4129 2688 m 4131 2695 l S NP 4126 2681 m 4129 2688 l S NP 4124 2675 m 4126 2681 l S NP 4122 2668 m 4124 2675 l S NP 4120 2661 m 4122 2668 l S NP 4118 2654 m 4120 2661 l S NP 4116 2648 m 4118 2654 l S NP 4113 2641 m 4116 2648 l S NP 4111 2634 m 4113 2641 l S NP 4109 2628 m 4111 2634 l S NP 4107 2621 m 4109 2628 l S NP 4104 2614 m 4107 2621 l S NP 4102 2608 m 4104 2614 l S NP 4100 2601 m 4102 2608 l S NP 4098 2595 m 4100 2601 l S NP 4095 2588 m 4098 2595 l S NP 4093 2581 m 4095 2588 l S NP 4091 2575 m 4093 2581 l S NP 4088 2568 m 4091 2575 l S NP 4086 2562 m 4088 2568 l S NP 4084 2555 m 4086 2562 l S NP 4081 2549 m 4084 2555 l S NP 4079 2542 m 4081 2549 l S NP 4077 2535 m 4079 2542 l S NP 4074 2529 m 4077 2535 l S NP 4072 2522 m 4074 2529 l S NP 4070 2516 m 4072 2522 l S NP 4067 2509 m 4070 2516 l S NP 4065 2503 m 4067 2509 l S NP 4062 2497 m 4065 2503 l S NP 4060 2490 m 4062 2497 l S NP 4057 2484 m 4060 2490 l S NP 4055 2477 m 4057 2484 l S NP 4052 2471 m 4055 2477 l S NP 4050 2464 m 4052 2471 l S NP 4048 2458 m 4050 2464 l S NP 4045 2452 m 4048 2458 l S NP 4043 2445 m 4045 2452 l S NP 4040 2439 m 4043 2445 l S NP 4037 2433 m 4040 2439 l S NP 4035 2426 m 4037 2433 l S NP 4032 2420 m 4035 2426 l S NP 4030 2414 m 4032 2420 l S NP 4027 2407 m 4030 2414 l S NP 4025 2401 m 4027 2407 l S NP 4022 2395 m 4025 2401 l S NP 4019 2389 m 4022 2395 l S NP 4017 2382 m 4019 2389 l S NP 4014 2376 m 4017 2382 l S NP 4012 2370 m 4014 2376 l S NP 4009 2364 m 4012 2370 l S NP 4006 2357 m 4009 2364 l S NP 4004 2351 m 4006 2357 l S NP 4001 2345 m 4004 2351 l S NP 3998 2339 m 4001 2345 l S NP 3996 2333 m 3998 2339 l S NP 3993 2327 m 3996 2333 l S NP 3990 2320 m 3993 2327 l S NP 3987 2314 m 3990 2320 l S NP 3985 2308 m 3987 2314 l S NP 3982 2302 m 3985 2308 l S NP 3979 2296 m 3982 2302 l S NP 3977 2290 m 3979 2296 l S NP 3974 2284 m 3977 2290 l S NP 3971 2278 m 3974 2284 l S NP 3968 2272 m 3971 2278 l S NP 3965 2266 m 3968 2272 l S NP 3963 2260 m 3965 2266 l S NP 3960 2254 m 3963 2260 l S NP 3957 2248 m 3960 2254 l S NP 3954 2242 m 3957 2248 l S NP 3951 2236 m 3954 2242 l S NP 3948 2230 m 3951 2236 l S NP 3946 2224 m 3948 2230 l S NP 3943 2218 m 3946 2224 l S NP 3940 2212 m 3943 2218 l S NP 3937 2206 m 3940 2212 l S NP 3934 2200 m 3937 2206 l S NP 3931 2195 m 3934 2200 l S NP 3928 2189 m 3931 2195 l S NP 3925 2183 m 3928 2189 l S NP 3922 2177 m 3925 2183 l S NP 3919 2171 m 3922 2177 l S NP 3916 2165 m 3919 2171 l S NP 3913 2160 m 3916 2165 l S NP 3910 2154 m 3913 2160 l S NP 3907 2148 m 3910 2154 l S NP 3904 2142 m 3907 2148 l S NP 3901 2137 m 3904 2142 l S NP 3898 2131 m 3901 2137 l S NP 3895 2125 m 3898 2131 l S NP 3892 2119 m 3895 2125 l S NP 3889 2114 m 3892 2119 l S NP 3886 2108 m 3889 2114 l S NP 3883 2102 m 3886 2108 l S NP 3880 2097 m 3883 2102 l S NP 3877 2091 m 3880 2097 l S NP 3874 2086 m 3877 2091 l S NP 3871 2080 m 3874 2086 l S NP 3868 2074 m 3871 2080 l S NP 3865 2069 m 3868 2074 l S NP 3862 2063 m 3865 2069 l S NP 3859 2058 m 3862 2063 l S NP 3855 2052 m 3859 2058 l S NP 3852 2047 m 3855 2052 l S NP 3849 2041 m 3852 2047 l S NP 3846 2036 m 3849 2041 l S NP 3843 2030 m 3846 2036 l S NP 3840 2025 m 3843 2030 l S NP 3836 2019 m 3840 2025 l S NP 3833 2014 m 3836 2019 l S NP 3830 2009 m 3833 2014 l S NP 3827 2003 m 3830 2009 l S NP 3824 1998 m 3827 2003 l S NP 3820 1992 m 3824 1998 l S NP 3817 1987 m 3820 1992 l S NP 3814 1982 m 3817 1987 l S NP 3811 1976 m 3814 1982 l S NP 3807 1971 m 3811 1976 l S NP 3804 1966 m 3807 1971 l S NP 3801 1960 m 3804 1966 l S NP 3797 1955 m 3801 1960 l S NP 3794 1950 m 3797 1955 l S NP 3791 1945 m 3794 1950 l S NP 3787 1940 m 3791 1945 l S NP 3784 1934 m 3787 1940 l S NP 3781 1929 m 3784 1934 l S NP 3777 1924 m 3781 1929 l S NP 3774 1919 m 3777 1924 l S NP 3771 1914 m 3774 1919 l S NP 3767 1908 m 3771 1914 l S NP 3764 1903 m 3767 1908 l S NP 3761 1898 m 3764 1903 l S NP 3757 1893 m 3761 1898 l S NP 3754 1888 m 3757 1893 l S NP 3750 1883 m 3754 1888 l S NP 3747 1878 m 3750 1883 l S NP 3744 1873 m 3747 1878 l S NP 3740 1868 m 3744 1873 l S NP 3737 1863 m 3740 1868 l S NP 3733 1858 m 3737 1863 l S NP 3730 1853 m 3733 1858 l S NP 3726 1848 m 3730 1853 l S NP 3723 1843 m 3726 1848 l S NP 3719 1838 m 3723 1843 l S NP 3716 1833 m 3719 1838 l S NP 3712 1829 m 3716 1833 l S NP 3709 1824 m 3712 1829 l S NP 3705 1819 m 3709 1824 l S NP 3702 1814 m 3705 1819 l S NP 3698 1809 m 3702 1814 l S NP 3695 1804 m 3698 1809 l S NP 3691 1800 m 3695 1804 l S NP 3688 1795 m 3691 1800 l S NP 3684 1790 m 3688 1795 l S NP 3681 1785 m 3684 1790 l S NP 3677 1781 m 3681 1785 l S NP 3673 1776 m 3677 1781 l S NP 3670 1771 m 3673 1776 l S NP 3666 1767 m 3670 1771 l S NP 3663 1762 m 3666 1767 l S NP 3659 1757 m 3663 1762 l S NP 3655 1753 m 3659 1757 l S NP 3652 1748 m 3655 1753 l S NP 3648 1743 m 3652 1748 l S NP 3645 1739 m 3648 1743 l S NP 3641 1734 m 3645 1739 l S NP 3637 1730 m 3641 1734 l S NP 3634 1725 m 3637 1730 l S NP 3630 1721 m 3634 1725 l S NP 3626 1716 m 3630 1721 l S NP 3623 1712 m 3626 1716 l S NP 3619 1707 m 3623 1712 l S NP 3615 1703 m 3619 1707 l S NP 3611 1698 m 3615 1703 l S NP 3608 1694 m 3611 1698 l S NP 3604 1690 m 3608 1694 l S NP 3600 1685 m 3604 1690 l S NP 3597 1681 m 3600 1685 l S NP 3593 1677 m 3597 1681 l S NP 3589 1672 m 3593 1677 l S NP 3585 1668 m 3589 1672 l S NP 3582 1664 m 3585 1668 l S NP 3578 1659 m 3582 1664 l S NP 3574 1655 m 3578 1659 l S NP 3570 1651 m 3574 1655 l S NP 3566 1647 m 3570 1651 l S NP 3563 1642 m 3566 1647 l S NP 3559 1638 m 3563 1642 l S NP 3555 1634 m 3559 1638 l S NP 3551 1630 m 3555 1634 l S NP 3547 1626 m 3551 1630 l S NP 3544 1622 m 3547 1626 l S NP 3540 1618 m 3544 1622 l S NP 3536 1614 m 3540 1618 l S NP 3532 1609 m 3536 1614 l S NP 3528 1605 m 3532 1609 l S NP 3524 1601 m 3528 1605 l S NP 3520 1597 m 3524 1601 l S NP 3517 1593 m 3520 1597 l S NP 3513 1589 m 3517 1593 l S NP 3509 1585 m 3513 1589 l S NP 3505 1581 m 3509 1585 l S NP 3501 1578 m 3505 1581 l S NP 3497 1574 m 3501 1578 l S NP 3493 1570 m 3497 1574 l S NP 3489 1566 m 3493 1570 l S NP 3485 1562 m 3489 1566 l S NP 3481 1558 m 3485 1562 l S NP 3478 1554 m 3481 1558 l S NP 3474 1551 m 3478 1554 l S NP 3470 1547 m 3474 1551 l S NP 3466 1543 m 3470 1547 l S NP 3462 1539 m 3466 1543 l S NP 3458 1536 m 3462 1539 l S NP 3454 1532 m 3458 1536 l S NP 3450 1528 m 3454 1532 l S NP 3446 1525 m 3450 1528 l S NP 3442 1521 m 3446 1525 l S NP 3438 1517 m 3442 1521 l S NP 3434 1514 m 3438 1517 l S NP 3430 1510 m 3434 1514 l S NP 3426 1506 m 3430 1510 l S NP 3422 1503 m 3426 1506 l S NP 3418 1499 m 3422 1503 l S NP 3414 1496 m 3418 1499 l S NP 3410 1492 m 3414 1496 l S NP 3406 1489 m 3410 1492 l S NP 3402 1485 m 3406 1489 l S NP 3398 1482 m 3402 1485 l S NP 3393 1478 m 3398 1482 l S NP 3389 1475 m 3393 1478 l S NP 3385 1472 m 3389 1475 l S NP 3381 1468 m 3385 1472 l S NP 3377 1465 m 3381 1468 l S NP 3373 1462 m 3377 1465 l S NP 3369 1458 m 3373 1462 l S NP 3365 1455 m 3369 1458 l S NP 3361 1452 m 3365 1455 l S NP 3357 1448 m 3361 1452 l S NP 3353 1445 m 3357 1448 l S NP 3348 1442 m 3353 1445 l S NP 3344 1439 m 3348 1442 l S NP 3340 1435 m 3344 1439 l S NP 3336 1432 m 3340 1435 l S NP 3332 1429 m 3336 1432 l S NP 3328 1426 m 3332 1429 l S NP 3324 1423 m 3328 1426 l S NP 3320 1420 m 3324 1423 l S NP 3315 1417 m 3320 1420 l S NP 3311 1414 m 3315 1417 l S NP 3307 1411 m 3311 1414 l S NP 3303 1408 m 3307 1411 l S NP 3299 1405 m 3303 1408 l S NP 3294 1402 m 3299 1405 l S NP 3290 1399 m 3294 1402 l S NP 3286 1396 m 3290 1399 l S NP 3282 1393 m 3286 1396 l S NP 3278 1390 m 3282 1393 l S NP 3273 1387 m 3278 1390 l S NP 3269 1384 m 3273 1387 l S NP 3265 1381 m 3269 1384 l S NP 3261 1378 m 3265 1381 l S NP 3257 1375 m 3261 1378 l S NP 3252 1373 m 3257 1375 l S NP 3248 1370 m 3252 1373 l S NP 3244 1367 m 3248 1370 l S NP 3240 1364 m 3244 1367 l S NP 3235 1362 m 3240 1364 l S NP 3231 1359 m 3235 1362 l S NP 3227 1356 m 3231 1359 l S NP 3223 1354 m 3227 1356 l S NP 3218 1351 m 3223 1354 l S NP 3214 1348 m 3218 1351 l S NP 3210 1346 m 3214 1348 l S NP 3206 1343 m 3210 1346 l S NP 3201 1341 m 3206 1343 l S NP 3197 1338 m 3201 1341 l S NP 3193 1336 m 3197 1338 l S NP 3188 1333 m 3193 1336 l S NP 3184 1331 m 3188 1333 l S NP 3180 1328 m 3184 1331 l S NP 3176 1326 m 3180 1328 l S NP 3171 1323 m 3176 1326 l S NP 3167 1321 m 3171 1323 l S NP 3163 1318 m 3167 1321 l S NP 3158 1316 m 3163 1318 l S NP 3154 1314 m 3158 1316 l S NP 3150 1311 m 3154 1314 l S NP 3145 1309 m 3150 1311 l S NP 3141 1307 m 3145 1309 l S NP 3137 1304 m 3141 1307 l S NP 3132 1302 m 3137 1304 l S NP 3128 1300 m 3132 1302 l S NP 3124 1298 m 3128 1300 l S NP 3119 1296 m 3124 1298 l S NP 3115 1293 m 3119 1296 l S NP 3111 1291 m 3115 1293 l S NP 3106 1289 m 3111 1291 l S NP 3102 1287 m 3106 1289 l S NP 3097 1285 m 3102 1287 l S NP 3093 1283 m 3097 1285 l S NP 3089 1281 m 3093 1283 l S NP 3084 1279 m 3089 1281 l S NP 3080 1277 m 3084 1279 l S NP 3076 1275 m 3080 1277 l S NP 3071 1273 m 3076 1275 l S NP 3067 1271 m 3071 1273 l S NP 3062 1269 m 3067 1271 l S NP 3058 1267 m 3062 1269 l S NP 3054 1265 m 3058 1267 l S NP 3049 1263 m 3054 1265 l S NP 3045 1261 m 3049 1263 l S NP 3040 1259 m 3045 1261 l S NP 3036 1257 m 3040 1259 l S NP 3031 1256 m 3036 1257 l S NP 3027 1254 m 3031 1256 l S NP 3023 1252 m 3027 1254 l S NP 3018 1250 m 3023 1252 l S NP 3014 1249 m 3018 1250 l S NP 3009 1247 m 3014 1249 l S NP 3005 1245 m 3009 1247 l S NP 3000 1244 m 3005 1245 l S NP 2996 1242 m 3000 1244 l S NP 2992 1240 m 2996 1242 l S NP 2987 1239 m 2992 1240 l S NP 2983 1237 m 2987 1239 l S NP 2978 1236 m 2983 1237 l S NP 2974 1234 m 2978 1236 l S NP 2969 1233 m 2974 1234 l S NP 2965 1231 m 2969 1233 l S NP 2960 1230 m 2965 1231 l S NP 2956 1228 m 2960 1230 l S NP 2952 1227 m 2956 1228 l S NP 2947 1225 m 2952 1227 l S NP 2943 1224 m 2947 1225 l S NP 2938 1222 m 2943 1224 l S NP 2934 1221 m 2938 1222 l S NP 2929 1220 m 2934 1221 l S NP 2925 1218 m 2929 1220 l S NP 2920 1217 m 2925 1218 l S NP 2916 1216 m 2920 1217 l S NP 2911 1215 m 2916 1216 l S NP 2907 1213 m 2911 1215 l S NP 2902 1212 m 2907 1213 l S NP 2898 1211 m 2902 1212 l S NP 2893 1210 m 2898 1211 l S NP 2889 1209 m 2893 1210 l S NP 2884 1207 m 2889 1209 l S NP 2880 1206 m 2884 1207 l S NP 2875 1205 m 2880 1206 l S NP 2871 1204 m 2875 1205 l S NP 2866 1203 m 2871 1204 l S NP 2862 1202 m 2866 1203 l S NP 2857 1201 m 2862 1202 l S NP 2853 1200 m 2857 1201 l S NP 2848 1199 m 2853 1200 l S NP 2844 1198 m 2848 1199 l S NP 2839 1197 m 2844 1198 l S NP 2835 1196 m 2839 1197 l S NP 2830 1195 m 2835 1196 l S NP 2826 1194 m 2830 1195 l S NP 2821 1194 m 2826 1194 l S NP 2817 1193 m 2821 1194 l S NP 2812 1192 m 2817 1193 l S NP 2808 1191 m 2812 1192 l S NP 2803 1190 m 2808 1191 l S NP 2799 1190 m 2803 1190 l S NP 2794 1189 m 2799 1190 l S NP 2790 1188 m 2794 1189 l S NP 2785 1187 m 2790 1188 l S NP 2780 1187 m 2785 1187 l S NP 2776 1186 m 2780 1187 l S NP 2771 1185 m 2776 1186 l S NP 2767 1185 m 2771 1185 l S NP 2762 1184 m 2767 1185 l S NP 2758 1184 m 2762 1184 l S NP 2753 1183 m 2758 1184 l S NP 2749 1183 m 2753 1183 l S NP 2744 1182 m 2749 1183 l S NP 2740 1181 m 2744 1182 l S NP 2735 1181 m 2740 1181 l S NP 2731 1181 m 2735 1181 l S NP 2726 1180 m 2731 1181 l S NP 2722 1180 m 2726 1180 l S NP 2717 1179 m 2722 1180 l S NP 2712 1179 m 2717 1179 l S NP 2708 1179 m 2712 1179 l S NP 2703 1178 m 2708 1179 l S NP 2699 1178 m 2703 1178 l S NP 2694 1178 m 2699 1178 l S NP 2690 1177 m 2694 1178 l S NP 2685 1177 m 2690 1177 l S NP 2681 1177 m 2685 1177 l S NP 2676 1177 m 2681 1177 l S NP 2672 1177 m 2676 1177 l S NP 2667 1176 m 2672 1177 l S NP 2663 1176 m 2667 1176 l S NP 2658 1176 m 2663 1176 l S NP 2653 1176 m 2658 1176 l S NP 2649 1176 m 2653 1176 l S NP 2644 1176 m 2649 1176 l S NP 2640 1176 m 2644 1176 l S NP 2635 1176 m 2640 1176 l S NP 2631 1176 m 2635 1176 l S NP 2626 1176 m 2631 1176 l S NP 2622 1176 m 2626 1176 l S NP 2617 1176 m 2622 1176 l S NP 2613 1176 m 2617 1176 l S NP 2608 1176 m 2613 1176 l S NP 2603 1176 m 2608 1176 l S NP 2599 1176 m 2603 1176 l S NP 2594 1176 m 2599 1176 l S NP 2590 1177 m 2594 1176 l S NP 2585 1177 m 2590 1177 l S NP 2581 1177 m 2585 1177 l S NP 2576 1177 m 2581 1177 l S NP 2572 1178 m 2576 1177 l S NP 2567 1178 m 2572 1178 l S NP 2563 1178 m 2567 1178 l S NP 2558 1178 m 2563 1178 l S NP 2553 1179 m 2558 1178 l S NP 2549 1179 m 2553 1179 l S NP 2544 1180 m 2549 1179 l S NP 2540 1180 m 2544 1180 l S NP 2535 1180 m 2540 1180 l S NP 2531 1181 m 2535 1180 l S NP 2526 1181 m 2531 1181 l S NP 2522 1182 m 2526 1181 l S NP 2517 1182 m 2522 1182 l S NP 2513 1183 m 2517 1182 l S NP 2508 1183 m 2513 1183 l S NP 2504 1184 m 2508 1183 l S NP 2499 1184 m 2504 1184 l S NP 2494 1185 m 2499 1184 l S NP 2490 1186 m 2494 1185 l S NP 2485 1186 m 2490 1186 l S NP 2481 1187 m 2485 1186 l S NP 2476 1188 m 2481 1187 l S NP 2472 1188 m 2476 1188 l S NP 2467 1189 m 2472 1188 l S NP 2463 1190 m 2467 1189 l S NP 2458 1191 m 2463 1190 l S NP 2454 1192 m 2458 1191 l S NP 2449 1192 m 2454 1192 l S NP 2445 1193 m 2449 1192 l S NP 2440 1194 m 2445 1193 l S NP 2436 1195 m 2440 1194 l S NP 2431 1196 m 2436 1195 l S NP 2427 1197 m 2431 1196 l S NP 2422 1198 m 2427 1197 l S NP 2418 1199 m 2422 1198 l S NP 2413 1200 m 2418 1199 l S NP 2409 1201 m 2413 1200 l S NP 2404 1202 m 2409 1201 l S NP 2400 1203 m 2404 1202 l S NP 2395 1204 m 2400 1203 l S NP 2391 1205 m 2395 1204 l S NP 2386 1206 m 2391 1205 l S NP 2382 1207 m 2386 1206 l S NP 2377 1208 m 2382 1207 l S NP 2373 1209 m 2377 1208 l S NP 2368 1211 m 2373 1209 l S NP 2364 1212 m 2368 1211 l S NP 2359 1213 m 2364 1212 l S NP 2355 1214 m 2359 1213 l S NP 2350 1215 m 2355 1214 l S NP 2346 1217 m 2350 1215 l S NP 2341 1218 m 2346 1217 l S NP 2337 1219 m 2341 1218 l S NP 2332 1221 m 2337 1219 l S NP 2328 1222 m 2332 1221 l S NP 2323 1224 m 2328 1222 l S NP 2319 1225 m 2323 1224 l S NP 2314 1226 m 2319 1225 l S NP 2310 1228 m 2314 1226 l S NP 2305 1229 m 2310 1228 l S NP 2301 1231 m 2305 1229 l S NP 2297 1232 m 2301 1231 l S NP 2292 1234 m 2297 1232 l S NP 2288 1235 m 2292 1234 l S NP 2283 1237 m 2288 1235 l S NP 2279 1239 m 2283 1237 l S NP 2274 1240 m 2279 1239 l S NP 2270 1242 m 2274 1240 l S NP 2265 1243 m 2270 1242 l S NP 2261 1245 m 2265 1243 l S NP 2257 1247 m 2261 1245 l S NP 2252 1249 m 2257 1247 l S NP 2248 1250 m 2252 1249 l S NP 2243 1252 m 2248 1250 l S NP 2239 1254 m 2243 1252 l S NP 2234 1256 m 2239 1254 l S NP 2230 1258 m 2234 1256 l S NP 2226 1259 m 2230 1258 l S NP 2221 1261 m 2226 1259 l S NP 2217 1263 m 2221 1261 l S NP 2212 1265 m 2217 1263 l S NP 2208 1267 m 2212 1265 l S NP 2204 1269 m 2208 1267 l S NP 2199 1271 m 2204 1269 l S NP 2195 1273 m 2199 1271 l S NP 2190 1275 m 2195 1273 l S NP 2186 1277 m 2190 1275 l S NP 2182 1279 m 2186 1277 l S NP 2177 1281 m 2182 1279 l S NP 2173 1283 m 2177 1281 l S NP 2169 1285 m 2173 1283 l S NP 2164 1287 m 2169 1285 l S NP 2160 1290 m 2164 1287 l S NP 2156 1292 m 2160 1290 l S NP 2151 1294 m 2156 1292 l S NP 2147 1296 m 2151 1294 l S NP 2142 1298 m 2147 1296 l S NP 2138 1301 m 2142 1298 l S NP 2134 1303 m 2138 1301 l S NP 2129 1305 m 2134 1303 l S NP 2125 1308 m 2129 1305 l S NP 2121 1310 m 2125 1308 l S NP 2117 1312 m 2121 1310 l S NP 2112 1315 m 2117 1312 l S NP 2108 1317 m 2112 1315 l S NP 2104 1320 m 2108 1317 l S NP 2099 1322 m 2104 1320 l S NP 2095 1324 m 2099 1322 l S NP 2091 1327 m 2095 1324 l S NP 2086 1329 m 2091 1327 l S NP 2082 1332 m 2086 1329 l S NP 2078 1334 m 2082 1332 l S NP 2074 1337 m 2078 1334 l S NP 2069 1340 m 2074 1337 l S NP 2065 1342 m 2069 1340 l S NP 2061 1345 m 2065 1342 l S NP 2056 1348 m 2061 1345 l S NP 2052 1350 m 2056 1348 l S NP 2048 1353 m 2052 1350 l S NP 2044 1356 m 2048 1353 l S NP 2039 1358 m 2044 1356 l S NP 2035 1361 m 2039 1358 l S NP 2031 1364 m 2035 1361 l S NP 2027 1367 m 2031 1364 l S NP 2023 1370 m 2027 1367 l S NP 2018 1372 m 2023 1370 l S NP 2014 1375 m 2018 1372 l S NP 2010 1378 m 2014 1375 l S NP 2006 1381 m 2010 1378 l S NP 2001 1384 m 2006 1381 l S NP 1997 1387 m 2001 1384 l S NP 1993 1390 m 1997 1387 l S NP 1989 1393 m 1993 1390 l S NP 1985 1396 m 1989 1393 l S NP 1981 1399 m 1985 1396 l S NP 1976 1402 m 1981 1399 l S NP 1972 1405 m 1976 1402 l S NP 1968 1408 m 1972 1405 l S NP 1964 1411 m 1968 1408 l S NP 1960 1414 m 1964 1411 l S NP 1956 1417 m 1960 1414 l S NP 1951 1421 m 1956 1417 l S NP 1947 1424 m 1951 1421 l S NP 1943 1427 m 1947 1424 l S NP 1939 1430 m 1943 1427 l S NP 1935 1433 m 1939 1430 l S NP 1931 1437 m 1935 1433 l S NP 1927 1440 m 1931 1437 l S NP 1923 1443 m 1927 1440 l S NP 1919 1447 m 1923 1443 l S NP 1915 1450 m 1919 1447 l S NP 1910 1453 m 1915 1450 l S NP 1906 1457 m 1910 1453 l S NP 1902 1460 m 1906 1457 l S NP 1898 1464 m 1902 1460 l S NP 1894 1467 m 1898 1464 l S NP 1890 1471 m 1894 1467 l S NP 1886 1474 m 1890 1471 l S NP 1882 1478 m 1886 1474 l S NP 1878 1481 m 1882 1478 l S NP 1874 1485 m 1878 1481 l S NP 1870 1488 m 1874 1485 l S NP 1866 1492 m 1870 1488 l S NP 1862 1495 m 1866 1492 l S NP 1858 1499 m 1862 1495 l S NP 1854 1503 m 1858 1499 l S NP 1850 1506 m 1854 1503 l S NP 1846 1510 m 1850 1506 l S NP 1842 1514 m 1846 1510 l S NP 1838 1518 m 1842 1514 l S NP 1834 1521 m 1838 1518 l S NP 1830 1525 m 1834 1521 l S NP 1826 1529 m 1830 1525 l S NP 1822 1533 m 1826 1529 l S NP 1818 1537 m 1822 1533 l S NP 1814 1540 m 1818 1537 l S NP 1811 1544 m 1814 1540 l S NP 1807 1548 m 1811 1544 l S NP 1803 1552 m 1807 1548 l S NP 1799 1556 m 1803 1552 l S NP 1795 1560 m 1799 1556 l S NP 1791 1564 m 1795 1560 l S NP 1787 1568 m 1791 1564 l S NP 1783 1572 m 1787 1568 l S NP 1779 1576 m 1783 1572 l S NP 1776 1580 m 1779 1576 l S NP 1772 1584 m 1776 1580 l S NP 1768 1588 m 1772 1584 l S NP 1764 1592 m 1768 1588 l S NP 1760 1596 m 1764 1592 l S NP 1756 1601 m 1760 1596 l S NP 1753 1605 m 1756 1601 l S NP 1749 1609 m 1753 1605 l S NP 1745 1613 m 1749 1609 l S NP 1741 1617 m 1745 1613 l S NP 1737 1622 m 1741 1617 l S NP 1734 1626 m 1737 1622 l S NP 1730 1630 m 1734 1626 l S NP 1726 1635 m 1730 1630 l S NP 1722 1639 m 1726 1635 l S NP 1719 1643 m 1722 1639 l S NP 1715 1648 m 1719 1643 l S NP 1711 1652 m 1715 1648 l S NP 1708 1656 m 1711 1652 l S NP 1704 1661 m 1708 1656 l S NP 1700 1665 m 1704 1661 l S NP 1696 1670 m 1700 1665 l S NP 1693 1674 m 1696 1670 l S NP 1689 1679 m 1693 1674 l S NP 1685 1683 m 1689 1679 l S NP 1682 1688 m 1685 1683 l S NP 1678 1692 m 1682 1688 l S NP 1675 1697 m 1678 1692 l S NP 1671 1702 m 1675 1697 l S NP 1667 1706 m 1671 1702 l S NP 1664 1711 m 1667 1706 l S NP 1660 1715 m 1664 1711 l S NP 1656 1720 m 1660 1715 l S NP 1653 1725 m 1656 1720 l S NP 1649 1729 m 1653 1725 l S NP 1646 1734 m 1649 1729 l S NP 1642 1739 m 1646 1734 l S NP 1639 1744 m 1642 1739 l S NP 1635 1749 m 1639 1744 l S NP 1632 1753 m 1635 1749 l S NP 1628 1758 m 1632 1753 l S NP 1625 1763 m 1628 1758 l S NP 1621 1768 m 1625 1763 l S NP 1618 1773 m 1621 1768 l S NP 1614 1778 m 1618 1773 l S NP 1611 1783 m 1614 1778 l S NP 1607 1787 m 1611 1783 l S NP 1604 1792 m 1607 1787 l S NP 1600 1797 m 1604 1792 l S NP 1597 1802 m 1600 1797 l S NP 1593 1807 m 1597 1802 l S NP 1590 1812 m 1593 1807 l S NP 1586 1817 m 1590 1812 l S NP 1583 1823 m 1586 1817 l S NP 1580 1828 m 1583 1823 l S NP 1576 1833 m 1580 1828 l S NP 1573 1838 m 1576 1833 l S NP 1570 1843 m 1573 1838 l S NP 1566 1848 m 1570 1843 l S NP 1563 1853 m 1566 1848 l S NP 1559 1858 m 1563 1853 l S NP 1556 1864 m 1559 1858 l S NP 1553 1869 m 1556 1864 l S NP 1550 1874 m 1553 1869 l S NP 1546 1879 m 1550 1874 l S NP 1543 1885 m 1546 1879 l S NP 1540 1890 m 1543 1885 l S NP 1536 1895 m 1540 1890 l S NP 1533 1901 m 1536 1895 l S NP 1530 1906 m 1533 1901 l S NP 1527 1911 m 1530 1906 l S NP 1523 1917 m 1527 1911 l S NP 1520 1922 m 1523 1917 l S NP 1517 1928 m 1520 1922 l S NP 1514 1933 m 1517 1928 l S NP 1511 1938 m 1514 1933 l S NP 1508 1944 m 1511 1938 l S NP 1504 1949 m 1508 1944 l S NP 1501 1955 m 1504 1949 l S NP 1498 1960 m 1501 1955 l S NP 1495 1966 m 1498 1960 l S NP 1492 1972 m 1495 1966 l S NP 1489 1977 m 1492 1972 l S NP 1486 1983 m 1489 1977 l S NP 1482 1988 m 1486 1983 l S NP 1479 1994 m 1482 1988 l S NP 1476 2000 m 1479 1994 l S NP 1473 2005 m 1476 2000 l S NP 1470 2011 m 1473 2005 l S NP 1467 2017 m 1470 2011 l S NP 1464 2022 m 1467 2017 l S NP 1461 2028 m 1464 2022 l S NP 1458 2034 m 1461 2028 l S NP 1455 2039 m 1458 2034 l S NP 1452 2045 m 1455 2039 l S NP 1449 2051 m 1452 2045 l S NP 1446 2057 m 1449 2051 l S NP 1443 2063 m 1446 2057 l S NP 1440 2068 m 1443 2063 l S NP 1437 2074 m 1440 2068 l S NP 1435 2080 m 1437 2074 l S NP 1432 2086 m 1435 2080 l S NP 1429 2092 m 1432 2086 l S NP 1426 2098 m 1429 2092 l S NP 1423 2104 m 1426 2098 l S NP 1420 2110 m 1423 2104 l S NP 1417 2116 m 1420 2110 l S NP 1415 2122 m 1417 2116 l S NP 1412 2128 m 1415 2122 l S NP 1409 2134 m 1412 2128 l S NP 1406 2140 m 1409 2134 l S NP 1403 2146 m 1406 2140 l S NP 1401 2152 m 1403 2146 l S NP 1398 2158 m 1401 2152 l S NP 1395 2164 m 1398 2158 l S NP 1392 2170 m 1395 2164 l S NP 1390 2176 m 1392 2170 l S NP 1387 2182 m 1390 2176 l S NP 1384 2188 m 1387 2182 l S NP 1381 2195 m 1384 2188 l S NP 1379 2201 m 1381 2195 l S NP 1376 2207 m 1379 2201 l S NP 1373 2213 m 1376 2207 l S NP 1371 2219 m 1373 2213 l S NP 1368 2226 m 1371 2219 l S NP 1366 2232 m 1368 2226 l S NP 1363 2238 m 1366 2232 l S NP 1360 2244 m 1363 2238 l S NP 1358 2251 m 1360 2244 l S NP 1355 2257 m 1358 2251 l S NP 1353 2263 m 1355 2257 l S NP 1350 2270 m 1353 2263 l S NP 1348 2276 m 1350 2270 l S NP 1345 2282 m 1348 2276 l S NP 1343 2289 m 1345 2282 l S NP 1340 2295 m 1343 2289 l S NP 1338 2301 m 1340 2295 l S NP 1335 2308 m 1338 2301 l S NP 1333 2314 m 1335 2308 l S NP 1330 2321 m 1333 2314 l S NP 1328 2327 m 1330 2321 l S NP 1325 2334 m 1328 2327 l S NP 1323 2340 m 1325 2334 l S NP 1321 2347 m 1323 2340 l S NP 1318 2353 m 1321 2347 l S NP 1316 2360 m 1318 2353 l S NP 1314 2366 m 1316 2360 l S NP 1311 2373 m 1314 2366 l S NP 1309 2379 m 1311 2373 l S NP 1307 2386 m 1309 2379 l S NP 1304 2392 m 1307 2386 l S NP 1302 2399 m 1304 2392 l S NP 1300 2406 m 1302 2399 l S NP 1297 2412 m 1300 2406 l S NP 1295 2419 m 1297 2412 l S NP 1293 2426 m 1295 2419 l S NP 1291 2432 m 1293 2426 l S NP 1288 2439 m 1291 2432 l S NP 1286 2446 m 1288 2439 l S NP 1284 2452 m 1286 2446 l S NP 1282 2459 m 1284 2452 l S NP 1280 2466 m 1282 2459 l S NP 1278 2472 m 1280 2466 l S NP 1276 2479 m 1278 2472 l S NP 1273 2486 m 1276 2479 l S NP 1271 2493 m 1273 2486 l S NP 1269 2500 m 1271 2493 l S NP 1267 2506 m 1269 2500 l S NP 1265 2513 m 1267 2506 l S NP 1263 2520 m 1265 2513 l S NP 1261 2527 m 1263 2520 l S NP 1259 2534 m 1261 2527 l S NP 1257 2540 m 1259 2534 l S NP 1255 2547 m 1257 2540 l S NP 1253 2554 m 1255 2547 l S NP 1251 2561 m 1253 2554 l S NP 1249 2568 m 1251 2561 l S NP 1247 2575 m 1249 2568 l S NP 1245 2582 m 1247 2575 l S NP 1243 2589 m 1245 2582 l S NP 1241 2596 m 1243 2589 l S NP 1239 2603 m 1241 2596 l S NP 1238 2610 m 1239 2603 l S NP 1236 2617 m 1238 2610 l S NP 1234 2623 m 1236 2617 l S NP 1232 2630 m 1234 2623 l S NP 1230 2637 m 1232 2630 l S NP 1228 2644 m 1230 2637 l S NP 1227 2652 m 1228 2644 l S NP 1225 2659 m 1227 2652 l S NP 1223 2666 m 1225 2659 l S NP 1221 2673 m 1223 2666 l S NP 1220 2680 m 1221 2673 l S NP 1218 2687 m 1220 2680 l S NP 1216 2694 m 1218 2687 l S NP 1215 2701 m 1216 2694 l S NP 1213 2708 m 1215 2701 l S NP 1211 2715 m 1213 2708 l S NP 1210 2722 m 1211 2715 l S NP 1208 2729 m 1210 2722 l S NP 1206 2737 m 1208 2729 l S NP 1205 2744 m 1206 2737 l S NP 1203 2751 m 1205 2744 l S NP 1202 2758 m 1203 2751 l S NP 1200 2765 m 1202 2758 l S NP 1198 2772 m 1200 2765 l S NP 1197 2780 m 1198 2772 l S NP 1195 2787 m 1197 2780 l S NP 1194 2794 m 1195 2787 l S NP 1192 2801 m 1194 2794 l S NP 1191 2808 m 1192 2801 l S NP 1189 2816 m 1191 2808 l S NP 1188 2823 m 1189 2816 l S NP 1187 2830 m 1188 2823 l S NP 1185 2837 m 1187 2830 l S NP 1184 2845 m 1185 2837 l S NP 1182 2852 m 1184 2845 l S NP 1181 2859 m 1182 2852 l S NP 1180 2866 m 1181 2859 l S NP 1178 2874 m 1180 2866 l S NP 1177 2881 m 1178 2874 l S NP 1176 2888 m 1177 2881 l S NP 1174 2896 m 1176 2888 l S NP 1173 2903 m 1174 2896 l S NP 1172 2910 m 1173 2903 l S NP 1171 2918 m 1172 2910 l S NP 1169 2925 m 1171 2918 l S NP 1168 2932 m 1169 2925 l S NP 1167 2940 m 1168 2932 l S NP 1166 2947 m 1167 2940 l S NP 1164 2954 m 1166 2947 l S NP 1163 2962 m 1164 2954 l S NP 1162 2969 m 1163 2962 l S NP 1161 2977 m 1162 2969 l S NP 1160 2984 m 1161 2977 l S NP 1159 2991 m 1160 2984 l S NP 1158 2999 m 1159 2991 l S NP 1157 3006 m 1158 2999 l S NP 1156 3014 m 1157 3006 l S NP 1154 3021 m 1156 3014 l S NP 1153 3029 m 1154 3021 l S NP 1152 3036 m 1153 3029 l S NP 1151 3043 m 1152 3036 l S NP 1150 3051 m 1151 3043 l S NP 1149 3058 m 1150 3051 l S NP 1148 3066 m 1149 3058 l S NP 1148 3073 m 1148 3066 l S NP 1147 3081 m 1148 3073 l S NP 1146 3088 m 1147 3081 l S NP 1145 3096 m 1146 3088 l S NP 1144 3103 m 1145 3096 l S NP 1143 3111 m 1144 3103 l S NP 1142 3118 m 1143 3111 l S NP 1141 3126 m 1142 3118 l S NP 1141 3133 m 1141 3126 l S NP 1140 3141 m 1141 3133 l S NP 1139 3148 m 1140 3141 l S NP 1138 3156 m 1139 3148 l S NP 1137 3163 m 1138 3156 l S NP 1137 3171 m 1137 3163 l S NP 1136 3178 m 1137 3171 l S NP 1135 3186 m 1136 3178 l S NP 1134 3193 m 1135 3186 l S NP 1134 3201 m 1134 3193 l S NP 1133 3209 m 1134 3201 l S NP 1132 3216 m 1133 3209 l S NP 1132 3224 m 1132 3216 l S NP 1131 3231 m 1132 3224 l S NP 1131 3239 m 1131 3231 l S NP 1130 3246 m 1131 3239 l S NP 1129 3254 m 1130 3246 l S NP 1129 3262 m 1129 3254 l S NP 1128 3269 m 1129 3262 l S NP 1128 3277 m 1128 3269 l S NP 1127 3284 m 1128 3277 l S NP 1127 3292 m 1127 3284 l S NP 1126 3299 m 1127 3292 l S NP 1126 3307 m 1126 3299 l S NP 1125 3315 m 1126 3307 l S NP 1125 3322 m 1125 3315 l S NP 1124 3330 m 1125 3322 l S NP 1124 3337 m 1124 3330 l S NP 1124 3345 m 1124 3337 l S NP 1123 3353 m 1124 3345 l S NP 1123 3360 m 1123 3353 l S NP 1123 3368 m 1123 3360 l S NP 1122 3376 m 1123 3368 l S NP 1122 3383 m 1122 3376 l S NP 1122 3391 m 1122 3383 l S NP 1121 3398 m 1122 3391 l S NP 1121 3406 m 1121 3398 l S NP 1121 3414 m 1121 3406 l S NP 1120 3421 m 1121 3414 l S NP 1120 3429 m 1120 3421 l S NP 1120 3437 m 1120 3429 l S NP 1120 3444 m 1120 3437 l S NP 1120 3452 m 1120 3444 l S NP 1119 3459 m 1120 3452 l S NP 1119 3467 m 1119 3459 l S NP 1119 3475 m 1119 3467 l S NP 1119 3482 m 1119 3475 l S NP 1119 3490 m 1119 3482 l S NP 1119 3498 m 1119 3490 l S NP 1119 3505 m 1119 3498 l S NP 1119 3513 m 1119 3505 l S NP 1119 3520 m 1119 3513 l S NP 1119 3528 m 1119 3520 l S NP 1119 3536 m 1119 3528 l S NP 1119 3543 m 1119 3536 l S NP 1119 3551 m 1119 3543 l S NP 1119 3559 m 1119 3551 l S NP 1119 3566 m 1119 3559 l S NP 1119 3574 m 1119 3566 l S NP 1119 3582 m 1119 3574 l S NP 1119 3589 m 1119 3582 l S NP 1119 3597 m 1119 3589 l S NP 1119 3604 m 1119 3597 l S NP 1119 3612 m 1119 3604 l S NP 1120 3620 m 1119 3612 l S NP 1120 3627 m 1120 3620 l S NP 1120 3635 m 1120 3627 l S NP 1120 3643 m 1120 3635 l S NP 1120 3650 m 1120 3643 l S NP 1121 3658 m 1120 3650 l S NP 1121 3666 m 1121 3658 l S NP 1121 3673 m 1121 3666 l S NP 1121 3681 m 1121 3673 l S NP 1122 3688 m 1121 3681 l S NP 1122 3696 m 1122 3688 l S NP 1122 3704 m 1122 3696 l S NP 1123 3711 m 1122 3704 l S NP 1123 3719 m 1123 3711 l S NP 1123 3726 m 1123 3719 l S NP 1124 3734 m 1123 3726 l S NP 1124 3742 m 1124 3734 l S NP 1125 3749 m 1124 3742 l S NP 1125 3757 m 1125 3749 l S NP 1125 3764 m 1125 3757 l S NP 1126 3772 m 1125 3764 l S NP 1126 3780 m 1126 3772 l S NP 1127 3787 m 1126 3780 l S NP 1127 3795 m 1127 3787 l S NP 1128 3802 m 1127 3795 l S NP 1128 3810 m 1128 3802 l S NP 1129 3818 m 1128 3810 l S NP 1130 3825 m 1129 3818 l S NP 1130 3833 m 1130 3825 l S NP 1131 3840 m 1130 3833 l S NP 1131 3848 m 1131 3840 l S NP 1132 3855 m 1131 3848 l S NP 1133 3863 m 1132 3855 l S NP 1133 3871 m 1133 3863 l S NP 1134 3878 m 1133 3871 l S NP 1135 3886 m 1134 3878 l S NP 1135 3893 m 1135 3886 l S NP 1136 3901 m 1135 3893 l S NP 1137 3908 m 1136 3901 l S NP 1138 3916 m 1137 3908 l S NP 1138 3923 m 1138 3916 l S NP 1139 3931 m 1138 3923 l S NP 1140 3938 m 1139 3931 l S NP 1141 3946 m 1140 3938 l S NP 1142 3953 m 1141 3946 l S NP 1143 3961 m 1142 3953 l S NP 1143 3968 m 1143 3961 l S NP 1144 3976 m 1143 3968 l S NP 1145 3983 m 1144 3976 l S NP 1146 3991 m 1145 3983 l S NP 1147 3998 m 1146 3991 l S NP 1148 4006 m 1147 3998 l S NP 1149 4013 m 1148 4006 l S NP 1150 4021 m 1149 4013 l S NP 1151 4028 m 1150 4021 l S NP 1152 4036 m 1151 4028 l S NP 1153 4043 m 1152 4036 l S NP 1154 4051 m 1153 4043 l S NP 1155 4058 m 1154 4051 l S NP 1156 4065 m 1155 4058 l S NP 1157 4073 m 1156 4065 l S NP 1158 4080 m 1157 4073 l S NP 1159 4088 m 1158 4080 l S NP 1160 4095 m 1159 4088 l S NP 1161 4102 m 1160 4095 l S NP 1163 4110 m 1161 4102 l S NP 1164 4117 m 1163 4110 l S NP 1165 4125 m 1164 4117 l S NP 1166 4132 m 1165 4125 l S NP 1167 4139 m 1166 4132 l S NP 1169 4147 m 1167 4139 l S NP 1170 4154 m 1169 4147 l S NP 1171 4161 m 1170 4154 l S NP 1172 4169 m 1171 4161 l S NP 1174 4176 m 1172 4169 l S NP 1175 4183 m 1174 4176 l S NP 1176 4191 m 1175 4183 l S NP 1178 4198 m 1176 4191 l S NP 1179 4205 m 1178 4198 l S NP 1180 4213 m 1179 4205 l S NP 1182 4220 m 1180 4213 l S NP 1183 4227 m 1182 4220 l S NP 1184 4234 m 1183 4227 l S NP 1186 4242 m 1184 4234 l S NP 1187 4249 m 1186 4242 l S NP 1189 4256 m 1187 4249 l S NP 1190 4263 m 1189 4256 l S NP 1192 4271 m 1190 4263 l S NP 1193 4278 m 1192 4271 l S NP 1195 4285 m 1193 4278 l S NP 1196 4292 m 1195 4285 l S NP 1198 4299 m 1196 4292 l S NP 1199 4307 m 1198 4299 l S NP 1201 4314 m 1199 4307 l S NP 1202 4321 m 1201 4314 l S NP 1204 4328 m 1202 4321 l S NP 1206 4335 m 1204 4328 l S NP 1207 4342 m 1206 4335 l S NP 1209 4349 m 1207 4342 l S NP 1210 4357 m 1209 4349 l S NP 1212 4364 m 1210 4357 l S NP 1214 4371 m 1212 4364 l S NP 1215 4378 m 1214 4371 l S NP 1217 4385 m 1215 4378 l S NP 1219 4392 m 1217 4385 l S NP 1221 4399 m 1219 4392 l S NP 1222 4406 m 1221 4399 l S NP 1224 4413 m 1222 4406 l S NP 1226 4420 m 1224 4413 l S NP 1228 4427 m 1226 4420 l S NP 1230 4434 m 1228 4427 l S NP 1231 4441 m 1230 4434 l S NP 1233 4448 m 1231 4441 l S NP 1235 4455 m 1233 4448 l S NP 1237 4462 m 1235 4455 l S NP 1239 4469 m 1237 4462 l S NP 1241 4476 m 1239 4469 l S NP 1243 4483 m 1241 4476 l S NP 1244 4490 m 1243 4483 l S NP 1246 4497 m 1244 4490 l S NP 1248 4504 m 1246 4497 l S NP 1250 4511 m 1248 4504 l S NP 1252 4517 m 1250 4511 l S NP 1254 4524 m 1252 4517 l S NP 1256 4531 m 1254 4524 l S NP 1258 4538 m 1256 4531 l S NP 1260 4545 m 1258 4538 l S NP 1262 4552 m 1260 4545 l S NP 1264 4559 m 1262 4552 l S NP 1267 4565 m 1264 4559 l S NP 1269 4572 m 1267 4565 l S NP 1271 4579 m 1269 4572 l S NP 1273 4586 m 1271 4579 l S NP 1275 4592 m 1273 4586 l S NP 1277 4599 m 1275 4592 l S NP 1279 4606 m 1277 4599 l S NP 1281 4613 m 1279 4606 l S NP 1284 4619 m 1281 4613 l S NP 1286 4626 m 1284 4619 l S NP 1288 4633 m 1286 4626 l S NP 1290 4639 m 1288 4633 l S NP 1292 4646 m 1290 4639 l S NP 1295 4653 m 1292 4646 l S NP 1297 4659 m 1295 4653 l S NP 1299 4666 m 1297 4659 l S NP 1301 4672 m 1299 4666 l S NP 1304 4679 m 1301 4672 l S NP 1306 4686 m 1304 4679 l S NP 1308 4692 m 1306 4686 l S NP 1311 4699 m 1308 4692 l S NP 1313 4705 m 1311 4699 l S NP 1316 4712 m 1313 4705 l S NP 1318 4718 m 1316 4712 l S NP 1320 4725 m 1318 4718 l S NP 1323 4731 m 1320 4725 l S NP 1325 4738 m 1323 4731 l S NP 1328 4744 m 1325 4738 l S NP 1330 4751 m 1328 4744 l S NP 1332 4757 m 1330 4751 l S NP 1335 4763 m 1332 4757 l S NP 1337 4770 m 1335 4763 l S NP 1340 4776 m 1337 4770 l S NP 1342 4782 m 1340 4776 l S NP 1345 4789 m 1342 4782 l S NP 1347 4795 m 1345 4789 l S NP 1350 4802 m 1347 4795 l S NP 1353 4808 m 1350 4802 l S NP 1355 4814 m 1353 4808 l S NP 1358 4820 m 1355 4814 l S NP 1360 4827 m 1358 4820 l S NP 1363 4833 m 1360 4827 l S NP 1366 4839 m 1363 4833 l S NP 1368 4845 m 1366 4839 l S NP 1371 4852 m 1368 4845 l S NP 1374 4858 m 1371 4852 l S NP 1376 4864 m 1374 4858 l S NP 1379 4870 m 1376 4864 l S NP 1382 4876 m 1379 4870 l S NP 1384 4882 m 1382 4876 l S NP 1387 4888 m 1384 4882 l S NP 1390 4895 m 1387 4888 l S NP 1393 4901 m 1390 4895 l S NP 1395 4907 m 1393 4901 l S NP 1398 4913 m 1395 4907 l S NP 1401 4919 m 1398 4913 l S NP 1404 4925 m 1401 4919 l S NP 1407 4931 m 1404 4925 l S NP 1409 4937 m 1407 4931 l S NP 1412 4943 m 1409 4937 l S NP 1415 4949 m 1412 4943 l S NP 1418 4955 m 1415 4949 l S NP 1421 4961 m 1418 4955 l S NP 1424 4966 m 1421 4961 l S NP 1427 4972 m 1424 4966 l S NP 1429 4978 m 1427 4972 l S NP 1432 4984 m 1429 4978 l S NP 1435 4990 m 1432 4984 l S NP 1438 4996 m 1435 4990 l S NP 1441 5001 m 1438 4996 l S NP 1444 5007 m 1441 5001 l S NP 1447 5013 m 1444 5007 l S NP 1450 5019 m 1447 5013 l S NP 1453 5025 m 1450 5019 l S NP 1456 5030 m 1453 5025 l S NP 1459 5036 m 1456 5030 l S NP 1462 5042 m 1459 5036 l S NP 1465 5047 m 1462 5042 l S NP 1468 5053 m 1465 5047 l S NP 1471 5059 m 1468 5053 l S NP 1474 5064 m 1471 5059 l S NP 1478 5070 m 1474 5064 l S NP 1481 5075 m 1478 5070 l S NP 1484 5081 m 1481 5075 l S NP 1487 5086 m 1484 5081 l S NP 1490 5092 m 1487 5086 l S NP 1493 5097 m 1490 5092 l S NP 1496 5103 m 1493 5097 l S NP 1500 5108 m 1496 5103 l S NP 1503 5114 m 1500 5108 l S NP 1506 5119 m 1503 5114 l S NP 1509 5125 m 1506 5119 l S NP 1512 5130 m 1509 5125 l S NP 1516 5135 m 1512 5130 l S NP 1519 5141 m 1516 5135 l S NP 1522 5146 m 1519 5141 l S NP 1525 5151 m 1522 5146 l S NP 1529 5157 m 1525 5151 l S NP 1532 5162 m 1529 5157 l S NP 1535 5167 m 1532 5162 l S NP 1538 5173 m 1535 5167 l S NP 1542 5178 m 1538 5173 l S NP 1545 5183 m 1542 5178 l S NP 1548 5188 m 1545 5183 l S NP 1552 5193 m 1548 5188 l S NP 1555 5199 m 1552 5193 l S NP 1559 5204 m 1555 5199 l S NP 1562 5209 m 1559 5204 l S NP 1565 5214 m 1562 5209 l S NP 1569 5219 m 1565 5214 l S NP 1572 5224 m 1569 5219 l S NP 1576 5229 m 1572 5224 l S NP 1579 5234 m 1576 5229 l S NP 1582 5239 m 1579 5234 l S NP 1586 5244 m 1582 5239 l S NP 1589 5249 m 1586 5244 l S NP 1593 5254 m 1589 5249 l S NP 1596 5259 m 1593 5254 l S NP 1600 5264 m 1596 5259 l S NP 1603 5269 m 1600 5264 l S NP 1607 5273 m 1603 5269 l S NP 1610 5278 m 1607 5273 l S NP 1614 5283 m 1610 5278 l S NP 1617 5288 m 1614 5283 l S NP 1621 5293 m 1617 5288 l S NP 1624 5297 m 1621 5293 l S NP 1628 5302 m 1624 5297 l S NP 1632 5307 m 1628 5302 l S NP 1635 5312 m 1632 5307 l S NP 1639 5316 m 1635 5312 l S NP 1642 5321 m 1639 5316 l S NP 1646 5325 m 1642 5321 l S NP 1650 5330 m 1646 5325 l S NP 1653 5335 m 1650 5330 l S NP 1657 5339 m 1653 5335 l S NP 1661 5344 m 1657 5339 l S NP 1664 5348 m 1661 5344 l S NP 1668 5353 m 1664 5348 l S NP 1672 5357 m 1668 5353 l S NP 1675 5362 m 1672 5357 l S NP 1679 5366 m 1675 5362 l S NP 1683 5371 m 1679 5366 l S NP 1686 5375 m 1683 5371 l S NP 1690 5379 m 1686 5375 l S NP 1694 5384 m 1690 5379 l S NP 1698 5388 m 1694 5384 l S NP 1701 5392 m 1698 5388 l S NP 1705 5397 m 1701 5392 l S NP 1709 5401 m 1705 5397 l S NP 1713 5405 m 1709 5401 l S NP 1716 5409 m 1713 5405 l S NP 1720 5414 m 1716 5409 l S NP 1724 5418 m 1720 5414 l S NP 1728 5422 m 1724 5418 l S NP 1732 5426 m 1728 5422 l S NP 1735 5430 m 1732 5426 l S NP 1739 5434 m 1735 5430 l S NP 1743 5438 m 1739 5434 l S NP 1747 5442 m 1743 5438 l S NP 1751 5447 m 1747 5442 l S NP 1755 5451 m 1751 5447 l S NP 1759 5455 m 1755 5451 l S NP 1762 5458 m 1759 5455 l S NP 1766 5462 m 1762 5458 l S NP 1770 5466 m 1766 5462 l S NP 1774 5470 m 1770 5466 l S NP 1778 5474 m 1774 5470 l S NP 1782 5478 m 1778 5474 l S NP 1786 5482 m 1782 5478 l S NP 1790 5486 m 1786 5482 l S NP 1794 5489 m 1790 5486 l S NP 1798 5493 m 1794 5489 l S NP 1802 5497 m 1798 5493 l S NP 1806 5501 m 1802 5497 l S NP 1810 5504 m 1806 5501 l S NP 1814 5508 m 1810 5504 l S NP 1818 5512 m 1814 5508 l S NP 1822 5515 m 1818 5512 l S NP 1826 5519 m 1822 5515 l S NP 1830 5522 m 1826 5519 l S NP 1834 5526 m 1830 5522 l S NP 1838 5530 m 1834 5526 l S NP 1842 5533 m 1838 5530 l S NP 1846 5537 m 1842 5533 l S NP 1850 5540 m 1846 5537 l S NP 1854 5543 m 1850 5540 l S NP 1858 5547 m 1854 5543 l S NP 1862 5550 m 1858 5547 l S NP 1866 5554 m 1862 5550 l S NP 1870 5557 m 1866 5554 l S NP 1874 5560 m 1870 5557 l S NP 1878 5564 m 1874 5560 l S NP 1883 5567 m 1878 5564 l S NP 1887 5570 m 1883 5567 l S NP 1891 5573 m 1887 5570 l S NP 1895 5577 m 1891 5573 l S NP 1899 5580 m 1895 5577 l S NP 1903 5583 m 1899 5580 l S NP 1907 5586 m 1903 5583 l S NP 1911 5589 m 1907 5586 l S NP 1916 5592 m 1911 5589 l S NP 1920 5595 m 1916 5592 l S NP 1924 5598 m 1920 5595 l S NP 1928 5601 m 1924 5598 l S NP 1932 5604 m 1928 5601 l S NP 1937 5607 m 1932 5604 l S NP 1941 5610 m 1937 5607 l S NP 1945 5613 m 1941 5610 l S NP 1949 5616 m 1945 5613 l S NP 1953 5619 m 1949 5616 l S NP 1958 5622 m 1953 5619 l S NP 1962 5625 m 1958 5622 l S NP 1966 5627 m 1962 5625 l S NP 1970 5630 m 1966 5627 l S NP 1974 5633 m 1970 5630 l S NP 1979 5636 m 1974 5633 l S NP 1983 5638 m 1979 5636 l S NP 1987 5641 m 1983 5638 l S NP 1991 5644 m 1987 5641 l S NP 1996 5646 m 1991 5644 l S NP 2000 5649 m 1996 5646 l S NP 2004 5651 m 2000 5649 l S NP 2009 5654 m 2004 5651 l S NP 2013 5656 m 2009 5654 l S NP 2017 5659 m 2013 5656 l S NP 2021 5661 m 2017 5659 l S NP 2026 5664 m 2021 5661 l S NP 2030 5666 m 2026 5664 l S NP 2034 5669 m 2030 5666 l S NP 2039 5671 m 2034 5669 l S NP 2043 5673 m 2039 5671 l S NP 2047 5676 m 2043 5673 l S NP 2052 5678 m 2047 5676 l S NP 2056 5680 m 2052 5678 l S NP 2060 5683 m 2056 5680 l S NP 2065 5685 m 2060 5683 l S NP 2069 5687 m 2065 5685 l S NP 2073 5689 m 2069 5687 l S NP 2078 5691 m 2073 5689 l S NP 2082 5694 m 2078 5691 l S NP 2087 5696 m 2082 5694 l S NP 2091 5698 m 2087 5696 l S NP 2095 5700 m 2091 5698 l S NP 2100 5702 m 2095 5700 l S NP 2104 5704 m 2100 5702 l S NP 2108 5706 m 2104 5704 l S NP 2113 5708 m 2108 5706 l S NP 2117 5710 m 2113 5708 l S NP 2122 5712 m 2117 5710 l S NP 2126 5713 m 2122 5712 l S NP 2130 5715 m 2126 5713 l S NP 2135 5717 m 2130 5715 l S NP 2139 5719 m 2135 5717 l S NP 2144 5721 m 2139 5719 l S NP 2148 5722 m 2144 5721 l S NP 2153 5724 m 2148 5722 l S NP 2157 5726 m 2153 5724 l S NP 2161 5728 m 2157 5726 l S NP 2166 5729 m 2161 5728 l S NP 2170 5731 m 2166 5729 l S NP 2175 5732 m 2170 5731 l S NP 2179 5734 m 2175 5732 l S NP 2184 5735 m 2179 5734 l S NP 2188 5737 m 2184 5735 l S NP 2193 5739 m 2188 5737 l S NP 2197 5740 m 2193 5739 l S NP 2202 5741 m 2197 5740 l S NP 2206 5743 m 2202 5741 l S NP 2210 5744 m 2206 5743 l S NP 2215 5746 m 2210 5744 l S NP 2219 5747 m 2215 5746 l S NP 2224 5748 m 2219 5747 l S NP 2228 5750 m 2224 5748 l S NP 2233 5751 m 2228 5750 l S NP 2237 5752 m 2233 5751 l S NP 2242 5753 m 2237 5752 l S NP 2246 5754 m 2242 5753 l S NP 2251 5756 m 2246 5754 l S NP 2255 5757 m 2251 5756 l S NP 2260 5758 m 2255 5757 l S NP 2264 5759 m 2260 5758 l S NP 2269 5760 m 2264 5759 l S NP 2273 5761 m 2269 5760 l S NP 2278 5762 m 2273 5761 l S NP 2282 5763 m 2278 5762 l S NP 2287 5764 m 2282 5763 l S NP 2291 5765 m 2287 5764 l S NP 2296 5766 m 2291 5765 l S NP 2300 5767 m 2296 5766 l S NP 2305 5767 m 2300 5767 l S NP 2309 5768 m 2305 5767 l S NP 2314 5769 m 2309 5768 l S NP 2318 5770 m 2314 5769 l S NP 2323 5771 m 2318 5770 l S NP 2327 5771 m 2323 5771 l S NP 2332 5772 m 2327 5771 l S NP 2337 5773 m 2332 5772 l S NP 2341 5773 m 2337 5773 l S NP 2346 5774 m 2341 5773 l S NP 2350 5774 m 2346 5774 l S NP 2355 5775 m 2350 5774 l S NP 2359 5776 m 2355 5775 l S NP 2364 5776 m 2359 5776 l S NP 2368 5777 m 2364 5776 l S NP 2373 5777 m 2368 5777 l S NP 2377 5777 m 2373 5777 l S NP 2382 5778 m 2377 5777 l S NP 2386 5778 m 2382 5778 l S NP 2391 5779 m 2386 5778 l S NP 2396 5779 m 2391 5779 l S NP 2400 5779 m 2396 5779 l S NP 2405 5779 m 2400 5779 l S NP 2409 5780 m 2405 5779 l S NP 2414 5780 m 2409 5780 l S NP 2418 5780 m 2414 5780 l S NP 2423 5780 m 2418 5780 l S NP 2427 5780 m 2423 5780 l S NP 2432 5780 m 2427 5780 l S NP 2436 5781 m 2432 5780 l S NP 2441 5781 m 2436 5781 l S NP 2445 5781 m 2441 5781 l S NP 2450 5781 m 2445 5781 l S NP 2455 5781 m 2450 5781 l S NP 2459 5781 m 2455 5781 l S NP 2464 5781 m 2459 5781 l S NP 2468 5780 m 2464 5781 l S NP 2473 5780 m 2468 5780 l S NP 2477 5780 m 2473 5780 l S NP 2482 5780 m 2477 5780 l S NP 2486 5780 m 2482 5780 l S NP 2491 5780 m 2486 5780 l S NP 2495 5779 m 2491 5780 l S NP 2500 5779 m 2495 5779 l S NP 2505 5779 m 2500 5779 l S NP 2509 5778 m 2505 5779 l S NP 2514 5778 m 2509 5778 l S NP 2518 5778 m 2514 5778 l S NP 2523 5777 m 2518 5778 l S NP 2527 5777 m 2523 5777 l S NP 2532 5776 m 2527 5777 l S NP 2536 5776 m 2532 5776 l S NP 2541 5775 m 2536 5776 l S NP 2545 5775 m 2541 5775 l S NP 2550 5774 m 2545 5775 l S NP 2554 5774 m 2550 5774 l S NP 2559 5773 m 2554 5774 l S NP 2564 5772 m 2559 5773 l S NP 2568 5772 m 2564 5772 l S NP 2573 5771 m 2568 5772 l S NP 2577 5770 m 2573 5771 l S NP 2582 5769 m 2577 5770 l S NP 2586 5769 m 2582 5769 l S NP 2591 5768 m 2586 5769 l S NP 2595 5767 m 2591 5768 l S NP 2600 5766 m 2595 5767 l S NP 2604 5765 m 2600 5766 l S NP 2609 5764 m 2604 5765 l S NP 2613 5763 m 2609 5764 l S NP 2618 5762 m 2613 5763 l S NP 2622 5761 m 2618 5762 l S NP 2627 5760 m 2622 5761 l S NP 2631 5759 m 2627 5760 l S NP 2636 5758 m 2631 5759 l S NP 2640 5757 m 2636 5758 l S NP 2645 5756 m 2640 5757 l S NP 2649 5755 m 2645 5756 l S NP 2654 5754 m 2649 5755 l S NP 2658 5753 m 2654 5754 l S NP 2663 5751 m 2658 5753 l S NP 2667 5750 m 2663 5751 l S NP 2672 5749 m 2667 5750 l S NP 2676 5747 m 2672 5749 l S NP 2681 5746 m 2676 5747 l S NP 2685 5745 m 2681 5746 l S NP 2690 5743 m 2685 5745 l S NP 2694 5742 m 2690 5743 l S NP 2699 5741 m 2694 5742 l S NP 2703 5739 m 2699 5741 l S NP 2707 5738 m 2703 5739 l S NP 2712 5736 m 2707 5738 l S NP 2716 5734 m 2712 5736 l S NP 2721 5733 m 2716 5734 l S NP 2725 5731 m 2721 5733 l S NP 2730 5730 m 2725 5731 l S NP 2734 5728 m 2730 5730 l S NP 2739 5726 m 2734 5728 l S NP 2743 5725 m 2739 5726 l S NP 2747 5723 m 2743 5725 l S NP 2752 5721 m 2747 5723 l S NP 2756 5719 m 2752 5721 l S NP 2761 5718 m 2756 5719 l S NP 2765 5716 m 2761 5718 l S NP 2770 5714 m 2765 5716 l S NP 2774 5712 m 2770 5714 l S NP 2778 5710 m 2774 5712 l S NP 2783 5708 m 2778 5710 l S NP 2787 5706 m 2783 5708 l S NP 2791 5704 m 2787 5706 l S NP 2796 5702 m 2791 5704 l S NP 2800 5700 m 2796 5702 l S NP 2805 5698 m 2800 5700 l S NP 2809 5696 m 2805 5698 l S NP 2813 5694 m 2809 5696 l S NP 2818 5692 m 2813 5694 l S NP 2822 5690 m 2818 5692 l S NP 2826 5688 m 2822 5690 l S NP 2831 5685 m 2826 5688 l S NP 2835 5683 m 2831 5685 l S NP 2839 5681 m 2835 5683 l S NP 2844 5679 m 2839 5681 l S NP 2848 5676 m 2844 5679 l S NP 2852 5674 m 2848 5676 l S NP 2857 5672 m 2852 5674 l S NP 2861 5669 m 2857 5672 l S NP 2865 5667 m 2861 5669 l S NP 2870 5664 m 2865 5667 l S NP 2874 5662 m 2870 5664 l S NP 2878 5659 m 2874 5662 l S NP 2883 5657 m 2878 5659 l S NP 2887 5654 m 2883 5657 l S NP 2891 5652 m 2887 5654 l S NP 2895 5649 m 2891 5652 l S NP 2900 5647 m 2895 5649 l S NP 2904 5644 m 2900 5647 l S NP 2908 5641 m 2904 5644 l S NP 2913 5639 m 2908 5641 l S NP 2917 5636 m 2913 5639 l S NP 2921 5633 m 2917 5636 l S NP 2925 5630 m 2921 5633 l S NP 2930 5628 m 2925 5630 l S NP 2934 5625 m 2930 5628 l S NP 2938 5622 m 2934 5625 l S NP 2942 5619 m 2938 5622 l S NP 2946 5616 m 2942 5619 l S NP 2951 5613 m 2946 5616 l S NP 2955 5610 m 2951 5613 l S NP 2959 5607 m 2955 5610 l S NP 2963 5604 m 2959 5607 l S NP 2967 5601 m 2963 5604 l S NP 2971 5598 m 2967 5601 l S NP 2976 5595 m 2971 5598 l S NP 2980 5592 m 2976 5595 l S NP 2984 5589 m 2980 5592 l S NP 2988 5586 m 2984 5589 l S NP 2992 5583 m 2988 5586 l S NP 2996 5580 m 2992 5583 l S NP 3001 5576 m 2996 5580 l S NP 3005 5573 m 3001 5576 l S NP 3009 5570 m 3005 5573 l S NP 3013 5567 m 3009 5570 l S NP 3017 5563 m 3013 5567 l S NP 3021 5560 m 3017 5563 l S NP 3025 5557 m 3021 5560 l S NP 3029 5553 m 3025 5557 l S NP 3033 5550 m 3029 5553 l S NP 3037 5547 m 3033 5550 l S NP 3041 5543 m 3037 5547 l S NP 3045 5540 m 3041 5543 l S NP 3049 5536 m 3045 5540 l S NP 3054 5533 m 3049 5536 l S NP 3058 5529 m 3054 5533 l S NP 3062 5525 m 3058 5529 l S NP 3066 5522 m 3062 5525 l S NP 3070 5518 m 3066 5522 l S NP 3074 5515 m 3070 5518 l S NP 3078 5511 m 3074 5515 l S NP 3082 5507 m 3078 5511 l S NP 3086 5504 m 3082 5507 l S NP 3089 5500 m 3086 5504 l S NP 3093 5496 m 3089 5500 l S NP 3097 5492 m 3093 5496 l S NP 3101 5489 m 3097 5492 l S NP 3105 5485 m 3101 5489 l S NP 3109 5481 m 3105 5485 l S NP 3113 5477 m 3109 5481 l S NP 3117 5473 m 3113 5477 l S NP 3121 5469 m 3117 5473 l S NP 3125 5465 m 3121 5469 l S NP 3129 5461 m 3125 5465 l S NP 3133 5457 m 3129 5461 l S NP 3136 5453 m 3133 5457 l S NP 3140 5449 m 3136 5453 l S NP 3144 5445 m 3140 5449 l S NP 3148 5441 m 3144 5445 l S NP 3152 5437 m 3148 5441 l S NP 3156 5433 m 3152 5437 l S NP 3160 5429 m 3156 5433 l S NP 3163 5425 m 3160 5429 l S NP 3167 5420 m 3163 5425 l S NP 3171 5416 m 3167 5420 l S NP 3175 5412 m 3171 5416 l S NP 3178 5408 m 3175 5412 l S NP 3182 5403 m 3178 5408 l S NP 3186 5399 m 3182 5403 l S NP 3190 5395 m 3186 5399 l S NP 3193 5390 m 3190 5395 l S NP 3197 5386 m 3193 5390 l S NP 3201 5382 m 3197 5386 l S NP 3205 5377 m 3201 5382 l S NP 3208 5373 m 3205 5377 l S NP 3212 5368 m 3208 5373 l S NP 3216 5364 m 3212 5368 l S NP 3219 5359 m 3216 5364 l S NP 3223 5355 m 3219 5359 l S NP 3227 5350 m 3223 5355 l S NP 3230 5346 m 3227 5350 l S NP 3234 5341 m 3230 5346 l S NP 3238 5337 m 3234 5341 l S NP 3241 5332 m 3238 5337 l S NP 3245 5327 m 3241 5332 l S NP 3248 5323 m 3245 5327 l S NP 3252 5318 m 3248 5323 l S NP 3256 5313 m 3252 5318 l S NP 3259 5309 m 3256 5313 l S NP 3263 5304 m 3259 5309 l S NP 3266 5299 m 3263 5304 l S NP 3270 5294 m 3266 5299 l S NP 3273 5289 m 3270 5294 l S NP 3277 5285 m 3273 5289 l S NP 3280 5280 m 3277 5285 l S NP 3284 5275 m 3280 5280 l S NP 3287 5270 m 3284 5275 l S NP 3291 5265 m 3287 5270 l S NP 3294 5260 m 3291 5265 l S NP 3298 5255 m 3294 5260 l S NP 3301 5250 m 3298 5255 l S NP 3305 5245 m 3301 5250 l S NP 3308 5240 m 3305 5245 l S NP 3311 5235 m 3308 5240 l S NP 3315 5230 m 3311 5235 l S NP 3318 5225 m 3315 5230 l S NP 3322 5220 m 3318 5225 l S NP 3325 5215 m 3322 5220 l S NP 3328 5209 m 3325 5215 l S NP 3332 5204 m 3328 5209 l S NP 3335 5199 m 3332 5204 l S NP 3338 5194 m 3335 5199 l S NP 3342 5189 m 3338 5194 l S NP 3345 5183 m 3342 5189 l S NP 3348 5178 m 3345 5183 l S NP 3352 5173 m 3348 5178 l S NP 3355 5168 m 3352 5173 l S NP 3358 5162 m 3355 5168 l S NP 3361 5157 m 3358 5162 l S NP 3365 5152 m 3361 5157 l S NP 3368 5146 m 3365 5152 l S NP 3371 5141 m 3368 5146 l S NP 3374 5135 m 3371 5141 l S NP 3377 5130 m 3374 5135 l S NP 3381 5124 m 3377 5130 l S NP 3384 5119 m 3381 5124 l S NP 3387 5113 m 3384 5119 l S NP 3390 5108 m 3387 5113 l S NP 3393 5102 m 3390 5108 l S NP 3396 5097 m 3393 5102 l S NP 3399 5091 m 3396 5097 l S NP 3402 5086 m 3399 5091 l S NP 3406 5080 m 3402 5086 l S NP 3409 5074 m 3406 5080 l S NP 3412 5069 m 3409 5074 l S NP 3415 5063 m 3412 5069 l S NP 3418 5057 m 3415 5063 l S NP 3421 5052 m 3418 5057 l S NP 3424 5046 m 3421 5052 l S NP 3427 5040 m 3424 5046 l S NP 3430 5035 m 3427 5040 l S NP 3433 5029 m 3430 5035 l S NP 3436 5023 m 3433 5029 l S NP 3439 5017 m 3436 5023 l S NP 3442 5011 m 3439 5017 l S NP 3445 5005 m 3442 5011 l S NP 3447 5000 m 3445 5005 l S NP 3450 4994 m 3447 5000 l S NP 3453 4988 m 3450 4994 l S NP 3456 4982 m 3453 4988 l S NP 3459 4976 m 3456 4982 l S NP 3462 4970 m 3459 4976 l S NP 3465 4964 m 3462 4970 l S NP 3467 4958 m 3465 4964 l S NP 3470 4952 m 3467 4958 l S NP 3473 4946 m 3470 4952 l S NP 3476 4940 m 3473 4946 l S NP 3479 4934 m 3476 4940 l S NP 3481 4928 m 3479 4934 l S NP 3484 4922 m 3481 4928 l S NP 3487 4916 m 3484 4922 l S NP 3490 4910 m 3487 4916 l S NP 3492 4903 m 3490 4910 l S NP 3495 4897 m 3492 4903 l S NP 3498 4891 m 3495 4897 l S NP 3500 4885 m 3498 4891 l S NP 3503 4879 m 3500 4885 l S NP 3506 4873 m 3503 4879 l S NP 3508 4866 m 3506 4873 l S NP 3511 4860 m 3508 4866 l S NP 3513 4854 m 3511 4860 l S NP 3516 4848 m 3513 4854 l S NP 3519 4841 m 3516 4848 l S NP 3521 4835 m 3519 4841 l S NP 3524 4829 m 3521 4835 l S NP 3526 4822 m 3524 4829 l S NP 3529 4816 m 3526 4822 l S NP 3531 4810 m 3529 4816 l S NP 3534 4803 m 3531 4810 l S NP 3536 4797 m 3534 4803 l S NP 3539 4790 m 3536 4797 l S NP 3541 4784 m 3539 4790 l S NP 3544 4777 m 3541 4784 l S NP 3546 4771 m 3544 4777 l S NP 3548 4765 m 3546 4771 l S NP 3551 4758 m 3548 4765 l S NP 3553 4752 m 3551 4758 l S NP 3556 4745 m 3553 4752 l S NP 3558 4739 m 3556 4745 l S NP 3560 4732 m 3558 4739 l S NP 3563 4725 m 3560 4732 l S NP 3565 4719 m 3563 4725 l S NP 3567 4712 m 3565 4719 l S NP 3570 4706 m 3567 4712 l S NP 3572 4699 m 3570 4706 l S NP 3574 4692 m 3572 4699 l S NP 3576 4686 m 3574 4692 l S NP 3579 4679 m 3576 4686 l S NP 3581 4672 m 3579 4679 l S NP 3583 4666 m 3581 4672 l S NP 3585 4659 m 3583 4666 l S NP 3587 4652 m 3585 4659 l S NP 3589 4646 m 3587 4652 l S NP 3592 4639 m 3589 4646 l S NP 3594 4632 m 3592 4639 l S NP 3596 4625 m 3594 4632 l S NP 3598 4619 m 3596 4625 l S NP 3600 4612 m 3598 4619 l S NP 3602 4605 m 3600 4612 l S NP 3604 4598 m 3602 4605 l S NP 3606 4591 m 3604 4598 l S NP 3608 4584 m 3606 4591 l S NP 3610 4578 m 3608 4584 l S NP 3612 4571 m 3610 4578 l S NP 3614 4564 m 3612 4571 l S NP 3616 4557 m 3614 4564 l S NP 3618 4550 m 3616 4557 l S NP 3620 4543 m 3618 4550 l S NP 3622 4536 m 3620 4543 l S NP 3624 4529 m 3622 4536 l S NP 3626 4522 m 3624 4529 l S NP 3628 4515 m 3626 4522 l S NP 3629 4508 m 3628 4515 l S NP 3631 4501 m 3629 4508 l S NP 3633 4494 m 3631 4501 l S NP 3635 4487 m 3633 4494 l S NP 3637 4480 m 3635 4487 l S NP 3638 4473 m 3637 4480 l S NP 3640 4466 m 3638 4473 l S NP 3642 4459 m 3640 4466 l S NP 3644 4452 m 3642 4459 l S NP 3645 4445 m 3644 4452 l S NP 3647 4438 m 3645 4445 l S NP 3649 4431 m 3647 4438 l S NP 3651 4424 m 3649 4431 l S NP 3652 4417 m 3651 4424 l S NP 3654 4410 m 3652 4417 l S NP 3655 4403 m 3654 4410 l S NP 3657 4395 m 3655 4403 l S NP 3659 4388 m 3657 4395 l S NP 3660 4381 m 3659 4388 l S NP 3662 4374 m 3660 4381 l S NP 3663 4367 m 3662 4374 l S NP 3665 4360 m 3663 4367 l S NP 3666 4352 m 3665 4360 l S NP 3668 4345 m 3666 4352 l S NP 3669 4338 m 3668 4345 l S NP 3671 4331 m 3669 4338 l S NP 3672 4324 m 3671 4331 l S NP 3674 4316 m 3672 4324 l S NP 3675 4309 m 3674 4316 l S NP 3677 4302 m 3675 4309 l S NP 3678 4294 m 3677 4302 l S NP 3679 4287 m 3678 4294 l S NP 3681 4280 m 3679 4287 l S NP 3682 4273 m 3681 4280 l S NP 3683 4265 m 3682 4273 l S NP 3685 4258 m 3683 4265 l S NP 3686 4251 m 3685 4258 l S NP 3687 4243 m 3686 4251 l S NP 3689 4236 m 3687 4243 l S NP 3690 4229 m 3689 4236 l S NP 3691 4221 m 3690 4229 l S NP 3692 4214 m 3691 4221 l S NP 3693 4207 m 3692 4214 l S NP 3695 4199 m 3693 4207 l S NP 3696 4192 m 3695 4199 l S NP 3697 4184 m 3696 4192 l S NP 3698 4177 m 3697 4184 l S NP 3699 4170 m 3698 4177 l S NP 3700 4162 m 3699 4170 l S NP 3701 4155 m 3700 4162 l S NP 3702 4147 m 3701 4155 l S NP 3703 4140 m 3702 4147 l S NP 3704 4133 m 3703 4140 l S NP 3706 4125 m 3704 4133 l S NP 3707 4118 m 3706 4125 l S NP 3708 4110 m 3707 4118 l S NP 3708 4103 m 3708 4110 l S NP 3709 4095 m 3708 4103 l S NP 3710 4088 m 3709 4095 l S NP 3711 4080 m 3710 4088 l S NP 3712 4073 m 3711 4080 l S NP 3713 4065 m 3712 4073 l S NP 3714 4058 m 3713 4065 l S NP 3715 4050 m 3714 4058 l S NP 3716 4043 m 3715 4050 l S NP 3716 4035 m 3716 4043 l S NP 3717 4028 m 3716 4035 l S NP 3718 4020 m 3717 4028 l S NP 3719 4013 m 3718 4020 l S NP 3720 4005 m 3719 4013 l S NP 3720 3998 m 3720 4005 l S NP 3721 3990 m 3720 3998 l S NP 3722 3983 m 3721 3990 l S NP 3722 3975 m 3722 3983 l S NP 3723 3968 m 3722 3975 l S NP 3724 3960 m 3723 3968 l S NP 3724 3952 m 3724 3960 l S NP 3725 3945 m 3724 3952 l S NP 3726 3937 m 3725 3945 l S NP 3726 3930 m 3726 3937 l S NP 3727 3922 m 3726 3930 l S NP 3727 3915 m 3727 3922 l S NP 3728 3907 m 3727 3915 l S NP 3728 3899 m 3728 3907 l S NP 3729 3892 m 3728 3899 l S NP 3729 3884 m 3729 3892 l S NP 3730 3877 m 3729 3884 l S NP 3730 3869 m 3730 3877 l S NP 3731 3861 m 3730 3869 l S NP 3731 3854 m 3731 3861 l S NP 3732 3846 m 3731 3854 l S NP 3732 3839 m 3732 3846 l S NP 3732 3831 m 3732 3839 l S NP 3733 3823 m 3732 3831 l S NP 3733 3816 m 3733 3823 l S NP 3733 3808 m 3733 3816 l S NP 3734 3800 m 3733 3808 l S NP 3734 3793 m 3734 3800 l S NP 3734 3785 m 3734 3793 l S NP 3734 3778 m 3734 3785 l S NP 3735 3770 m 3734 3778 l S NP 3735 3762 m 3735 3770 l S NP 3735 3755 m 3735 3762 l S NP 3735 3747 m 3735 3755 l S NP 3735 3739 m 3735 3747 l S NP 3736 3732 m 3735 3739 l S NP 3736 3724 m 3736 3732 l S NP 3736 3717 m 3736 3724 l S NP 3736 3709 m 3736 3717 l S NP 3736 3701 m 3736 3709 l S NP 3736 3694 m 3736 3701 l S NP 3736 3686 m 3736 3694 l S NP 3736 3678 m 3736 3686 l S NP 3736 3671 m 3736 3678 l S NP 3736 3663 m 3736 3671 l S NP 3736 3655 m 3736 3663 l S NP 3736 3648 m 3736 3655 l S NP 3736 3640 m 3736 3648 l S NP 3736 3633 m 3736 3640 l S NP 3736 3625 m 3736 3633 l S NP 3736 3617 m 3736 3625 l S NP 3735 3610 m 3736 3617 l S NP 3735 3602 m 3735 3610 l S NP 3735 3594 m 3735 3602 l S NP 3735 3587 m 3735 3594 l S NP 3735 3579 m 3735 3587 l S NP 3735 3572 m 3735 3579 l S NP 3734 3564 m 3735 3572 l S NP 3734 3556 m 3734 3564 l S NP 3734 3549 m 3734 3556 l S NP 3734 3541 m 3734 3549 l S NP 3733 3533 m 3734 3541 l S NP 3733 3526 m 3733 3533 l S NP 3733 3518 m 3733 3526 l S NP 3732 3511 m 3733 3518 l S NP 3732 3503 m 3732 3511 l S NP 3731 3495 m 3732 3503 l S NP 3731 3488 m 3731 3495 l S NP 3731 3480 m 3731 3488 l S NP 3730 3473 m 3731 3480 l S NP 3730 3465 m 3730 3473 l S NP 3729 3457 m 3730 3465 l S NP 3729 3450 m 3729 3457 l S NP 3728 3442 m 3729 3450 l S NP 3728 3435 m 3728 3442 l S NP 3727 3427 m 3728 3435 l S NP 3727 3419 m 3727 3427 l S NP 3726 3412 m 3727 3419 l S NP 3726 3404 m 3726 3412 l S NP 3725 3397 m 3726 3404 l S NP 3724 3389 m 3725 3397 l S NP 3724 3382 m 3724 3389 l S NP 3723 3374 m 3724 3382 l S NP 3722 3366 m 3723 3374 l S NP 3722 3359 m 3722 3366 l S NP 3721 3351 m 3722 3359 l S NP 3720 3344 m 3721 3351 l S NP 3719 3336 m 3720 3344 l S NP 3719 3329 m 3719 3336 l S NP 3718 3321 m 3719 3329 l S NP 3717 3314 m 3718 3321 l S NP 3716 3306 m 3717 3314 l S NP 3715 3299 m 3716 3306 l S NP 3715 3291 m 3715 3299 l S NP 3714 3284 m 3715 3291 l S NP 3713 3276 m 3714 3284 l S NP 3712 3269 m 3713 3276 l S NP 3711 3261 m 3712 3269 l S NP 3710 3254 m 3711 3261 l S NP 3709 3246 m 3710 3254 l S NP 3708 3239 m 3709 3246 l S NP 3707 3231 m 3708 3239 l S NP 3706 3224 m 3707 3231 l S NP 3705 3217 m 3706 3224 l S NP 3704 3209 m 3705 3217 l S NP 3703 3202 m 3704 3209 l S NP 3702 3194 m 3703 3202 l S NP 3701 3187 m 3702 3194 l S NP 3700 3179 m 3701 3187 l S NP 3699 3172 m 3700 3179 l S NP 3697 3165 m 3699 3172 l S NP 3696 3157 m 3697 3165 l S NP 3695 3150 m 3696 3157 l S NP 3694 3142 m 3695 3150 l S NP 3693 3135 m 3694 3142 l S NP 3691 3128 m 3693 3135 l S NP 3690 3120 m 3691 3128 l S NP 3689 3113 m 3690 3120 l S NP 3688 3106 m 3689 3113 l S NP 3686 3098 m 3688 3106 l S NP 3685 3091 m 3686 3098 l S NP 3684 3084 m 3685 3091 l S NP 3682 3077 m 3684 3084 l S NP 3681 3069 m 3682 3077 l S NP 3680 3062 m 3681 3069 l S NP 3678 3055 m 3680 3062 l S NP 3677 3047 m 3678 3055 l S NP 3675 3040 m 3677 3047 l S NP 3674 3033 m 3675 3040 l S NP 3673 3026 m 3674 3033 l S NP 3671 3019 m 3673 3026 l S NP 3670 3011 m 3671 3019 l S NP 3668 3004 m 3670 3011 l S NP 3667 2997 m 3668 3004 l S NP 3665 2990 m 3667 2997 l S NP 3663 2983 m 3665 2990 l S NP 3662 2975 m 3663 2983 l S NP 3660 2968 m 3662 2975 l S NP 3659 2961 m 3660 2968 l S NP 3657 2954 m 3659 2961 l S NP 3655 2947 m 3657 2954 l S NP 3654 2940 m 3655 2947 l S NP 3652 2933 m 3654 2940 l S NP 3650 2926 m 3652 2933 l S NP 3649 2918 m 3650 2926 l S NP 3647 2911 m 3649 2918 l S NP 3645 2904 m 3647 2911 l S NP 3643 2897 m 3645 2904 l S NP 3642 2890 m 3643 2897 l S NP 3640 2883 m 3642 2890 l S NP 3638 2876 m 3640 2883 l S NP 3636 2869 m 3638 2876 l S NP 3634 2862 m 3636 2869 l S NP 3632 2855 m 3634 2862 l S NP 3631 2848 m 3632 2855 l S NP 3629 2841 m 3631 2848 l S NP 3627 2834 m 3629 2841 l S NP 3625 2828 m 3627 2834 l S NP 3623 2821 m 3625 2828 l S NP 3621 2814 m 3623 2821 l S NP 3619 2807 m 3621 2814 l S NP 3617 2800 m 3619 2807 l S NP 3615 2793 m 3617 2800 l S NP 3613 2786 m 3615 2793 l S NP 3611 2779 m 3613 2786 l S NP 3609 2773 m 3611 2779 l S NP 3607 2766 m 3609 2773 l S NP 3605 2759 m 3607 2766 l S NP 3603 2752 m 3605 2759 l S NP 3601 2746 m 3603 2752 l S NP 3598 2739 m 3601 2746 l S NP 3596 2732 m 3598 2739 l S NP 3594 2725 m 3596 2732 l S NP 3592 2719 m 3594 2725 l S NP 3590 2712 m 3592 2719 l S NP 3588 2705 m 3590 2712 l S NP 3585 2699 m 3588 2705 l S NP 3583 2692 m 3585 2699 l S NP 3581 2685 m 3583 2692 l S NP 3579 2679 m 3581 2685 l S NP 3576 2672 m 3579 2679 l S NP 3574 2666 m 3576 2672 l S NP 3572 2659 m 3574 2666 l S NP 3569 2652 m 3572 2659 l S NP 3567 2646 m 3569 2652 l S NP 3565 2639 m 3567 2646 l S NP 3562 2633 m 3565 2639 l S NP 3560 2626 m 3562 2633 l S NP 3557 2620 m 3560 2626 l S NP 3555 2613 m 3557 2620 l S NP 3553 2607 m 3555 2613 l S NP 3550 2601 m 3553 2607 l S NP 3548 2594 m 3550 2601 l S NP 3545 2588 m 3548 2594 l S NP 3543 2581 m 3545 2588 l S NP 3540 2575 m 3543 2581 l S NP 3538 2569 m 3540 2575 l S NP 3535 2562 m 3538 2569 l S NP 3532 2556 m 3535 2562 l S NP 3530 2550 m 3532 2556 l S NP 3527 2544 m 3530 2550 l S NP 3525 2537 m 3527 2544 l S NP 3522 2531 m 3525 2537 l S NP 3519 2525 m 3522 2531 l S NP 3517 2519 m 3519 2525 l S NP 3514 2512 m 3517 2519 l S NP 3511 2506 m 3514 2512 l S NP 3509 2500 m 3511 2506 l S NP 3506 2494 m 3509 2500 l S NP 3503 2488 m 3506 2494 l S NP 3501 2482 m 3503 2488 l S NP 3498 2476 m 3501 2482 l S NP 3495 2470 m 3498 2476 l S NP 3492 2464 m 3495 2470 l S NP 3489 2458 m 3492 2464 l S NP 3487 2452 m 3489 2458 l S NP 3484 2446 m 3487 2452 l S NP 3481 2440 m 3484 2446 l S NP 3478 2434 m 3481 2440 l S NP 3475 2428 m 3478 2434 l S NP 3472 2422 m 3475 2428 l S NP 3469 2416 m 3472 2422 l S NP 3467 2410 m 3469 2416 l S NP 3464 2404 m 3467 2410 l S NP 3461 2398 m 3464 2404 l S NP 3458 2393 m 3461 2398 l S NP 3455 2387 m 3458 2393 l S NP 3452 2381 m 3455 2387 l S NP 3449 2375 m 3452 2381 l S NP 3446 2370 m 3449 2375 l S NP 3443 2364 m 3446 2370 l S NP 3440 2358 m 3443 2364 l S NP 3437 2353 m 3440 2358 l S NP 3434 2347 m 3437 2353 l S NP 3431 2341 m 3434 2347 l S NP 3427 2336 m 3431 2341 l S NP 3424 2330 m 3427 2336 l S NP 3421 2325 m 3424 2330 l S NP 3418 2319 m 3421 2325 l S NP 3415 2313 m 3418 2319 l S NP 3412 2308 m 3415 2313 l S NP 3409 2302 m 3412 2308 l S NP 3405 2297 m 3409 2302 l S NP 3402 2292 m 3405 2297 l S NP 3399 2286 m 3402 2292 l S NP 3396 2281 m 3399 2286 l S NP 3393 2275 m 3396 2281 l S NP 3389 2270 m 3393 2275 l S NP 3386 2265 m 3389 2270 l S NP 3383 2259 m 3386 2265 l S NP 3380 2254 m 3383 2259 l S NP 3376 2249 m 3380 2254 l S NP 3373 2244 m 3376 2249 l S NP 3370 2238 m 3373 2244 l S NP 3366 2233 m 3370 2238 l S NP 3363 2228 m 3366 2233 l S NP 3360 2223 m 3363 2228 l S NP 3356 2218 m 3360 2223 l S NP 3353 2213 m 3356 2218 l S NP 3349 2208 m 3353 2213 l S NP 3346 2203 m 3349 2208 l S NP 3343 2198 m 3346 2203 l S NP 3339 2193 m 3343 2198 l S NP 3336 2188 m 3339 2193 l S NP 3332 2183 m 3336 2188 l S NP 3329 2178 m 3332 2183 l S NP 3325 2173 m 3329 2178 l S NP 3322 2168 m 3325 2173 l S NP 3318 2163 m 3322 2168 l S NP 3315 2158 m 3318 2163 l S NP 3311 2153 m 3315 2158 l S NP 3308 2149 m 3311 2153 l S NP 3304 2144 m 3308 2149 l S NP 3301 2139 m 3304 2144 l S NP 3297 2134 m 3301 2139 l S NP 3293 2130 m 3297 2134 l S NP 3290 2125 m 3293 2130 l S NP 3286 2120 m 3290 2125 l S NP 3283 2116 m 3286 2120 l S NP 3279 2111 m 3283 2116 l S NP 3275 2107 m 3279 2111 l S NP 3272 2102 m 3275 2107 l S NP 3268 2098 m 3272 2102 l S NP 3264 2093 m 3268 2098 l S NP 3261 2089 m 3264 2093 l S NP 3257 2084 m 3261 2089 l S NP 3253 2080 m 3257 2084 l S NP 3250 2075 m 3253 2080 l S NP 3246 2071 m 3250 2075 l S NP 3242 2067 m 3246 2071 l S NP 3238 2062 m 3242 2067 l S NP 3235 2058 m 3238 2062 l S NP 3231 2054 m 3235 2058 l S NP 3227 2049 m 3231 2054 l S NP 3223 2045 m 3227 2049 l S NP 3219 2041 m 3223 2045 l S NP 3216 2037 m 3219 2041 l S NP 3212 2033 m 3216 2037 l S NP 3208 2029 m 3212 2033 l S NP 3204 2025 m 3208 2029 l S NP 3200 2021 m 3204 2025 l S NP 3196 2016 m 3200 2021 l S NP 3193 2012 m 3196 2016 l S NP 3189 2008 m 3193 2012 l S NP 3185 2005 m 3189 2008 l S NP 3181 2001 m 3185 2005 l S NP 3177 1997 m 3181 2001 l S NP 3173 1993 m 3177 1997 l S NP 3169 1989 m 3173 1993 l S NP 3165 1985 m 3169 1989 l S NP 3161 1981 m 3165 1985 l S NP 3157 1978 m 3161 1981 l S NP 3153 1974 m 3157 1978 l S NP 3149 1970 m 3153 1974 l S NP 3145 1967 m 3149 1970 l S NP 3141 1963 m 3145 1967 l S NP 3137 1959 m 3141 1963 l S NP 3133 1956 m 3137 1959 l S NP 3129 1952 m 3133 1956 l S NP 3125 1949 m 3129 1952 l S NP 3121 1945 m 3125 1949 l S NP 3117 1942 m 3121 1945 l S NP 3113 1938 m 3117 1942 l S NP 3109 1935 m 3113 1938 l S NP 3105 1931 m 3109 1935 l S NP 3101 1928 m 3105 1931 l S NP 3097 1924 m 3101 1928 l S NP 3093 1921 m 3097 1924 l S NP 3089 1918 m 3093 1921 l S NP 3085 1915 m 3089 1918 l S NP 3081 1911 m 3085 1915 l S NP 3076 1908 m 3081 1911 l S NP 3072 1905 m 3076 1908 l S NP 3068 1902 m 3072 1905 l S NP 3064 1899 m 3068 1902 l S NP 3060 1896 m 3064 1899 l S NP 3056 1893 m 3060 1896 l S NP 3051 1890 m 3056 1893 l S NP 3047 1886 m 3051 1890 l S NP 3043 1884 m 3047 1886 l S NP 3039 1881 m 3043 1884 l S NP 3035 1878 m 3039 1881 l S NP 3031 1875 m 3035 1878 l S NP 3026 1872 m 3031 1875 l S NP 3022 1869 m 3026 1872 l S NP 3018 1866 m 3022 1869 l S NP 3014 1863 m 3018 1866 l S NP 3009 1861 m 3014 1863 l S NP 3005 1858 m 3009 1861 l S NP 3001 1855 m 3005 1858 l S NP 2997 1853 m 3001 1855 l S NP 2992 1850 m 2997 1853 l S NP 2988 1847 m 2992 1850 l S NP 2984 1845 m 2988 1847 l S NP 2980 1842 m 2984 1845 l S NP 2975 1840 m 2980 1842 l S NP 2971 1837 m 2975 1840 l S NP 2967 1835 m 2971 1837 l S NP 2962 1832 m 2967 1835 l S NP 2958 1830 m 2962 1832 l S NP 2954 1828 m 2958 1830 l S NP 2949 1825 m 2954 1828 l S NP 2945 1823 m 2949 1825 l S NP 2941 1821 m 2945 1823 l S NP 2936 1819 m 2941 1821 l S NP 2932 1816 m 2936 1819 l S NP 2928 1814 m 2932 1816 l S NP 2923 1812 m 2928 1814 l S NP 2919 1810 m 2923 1812 l S NP 2915 1808 m 2919 1810 l S NP 2910 1806 m 2915 1808 l S NP 2906 1804 m 2910 1806 l S NP 2901 1802 m 2906 1804 l S NP 2897 1800 m 2901 1802 l S NP 2893 1798 m 2897 1800 l S NP 2888 1796 m 2893 1798 l S NP 2884 1794 m 2888 1796 l S NP 2879 1792 m 2884 1794 l S NP 2875 1790 m 2879 1792 l S NP 2871 1788 m 2875 1790 l S NP 2866 1787 m 2871 1788 l S NP 2862 1785 m 2866 1787 l S NP 2857 1783 m 2862 1785 l S NP 2853 1782 m 2857 1783 l S NP 2848 1780 m 2853 1782 l S NP 2844 1778 m 2848 1780 l S NP 2839 1777 m 2844 1778 l S NP 2835 1775 m 2839 1777 l S NP 2831 1774 m 2835 1775 l S NP 2826 1772 m 2831 1774 l S NP 2822 1771 m 2826 1772 l S NP 2817 1769 m 2822 1771 l S NP 2813 1768 m 2817 1769 l S NP 2808 1767 m 2813 1768 l S NP 2804 1765 m 2808 1767 l S NP 2799 1764 m 2804 1765 l S NP 2795 1763 m 2799 1764 l S NP 2790 1761 m 2795 1763 l S NP 2786 1760 m 2790 1761 l S NP 2781 1759 m 2786 1760 l S NP 2777 1758 m 2781 1759 l S NP 2772 1757 m 2777 1758 l S NP 2768 1756 m 2772 1757 l S NP 2763 1755 m 2768 1756 l S NP 2759 1754 m 2763 1755 l S NP 2754 1753 m 2759 1754 l S NP 2750 1752 m 2754 1753 l S NP 2745 1751 m 2750 1752 l S NP 2741 1750 m 2745 1751 l S NP 2736 1749 m 2741 1750 l S NP 2732 1748 m 2736 1749 l S NP 2727 1747 m 2732 1748 l S NP 2723 1746 m 2727 1747 l S NP 2718 1746 m 2723 1746 l S NP 2714 1745 m 2718 1746 l S NP 2709 1744 m 2714 1745 l S NP 2705 1744 m 2709 1744 l S NP 2700 1743 m 2705 1744 l S NP 2696 1742 m 2700 1743 l S NP 2691 1742 m 2696 1742 l S NP 2687 1741 m 2691 1742 l S NP 2682 1741 m 2687 1741 l S NP 2677 1740 m 2682 1741 l S NP 2673 1740 m 2677 1740 l S NP 2668 1740 m 2673 1740 l S NP 2664 1739 m 2668 1740 l S NP 2659 1739 m 2664 1739 l S NP 2655 1738 m 2659 1739 l S NP 2650 1738 m 2655 1738 l S NP 2646 1738 m 2650 1738 l S NP 2641 1738 m 2646 1738 l S NP 2637 1738 m 2641 1738 l S NP 2632 1737 m 2637 1738 l S NP 2627 1737 m 2632 1737 l S NP 2623 1737 m 2627 1737 l S NP 2618 1737 m 2623 1737 l S NP 2614 1737 m 2618 1737 l S NP 2609 1737 m 2614 1737 l S NP 2605 1737 m 2609 1737 l S NP 2600 1737 m 2605 1737 l S NP 2596 1737 m 2600 1737 l S NP 2591 1737 m 2596 1737 l S NP 2587 1738 m 2591 1737 l S NP 2582 1738 m 2587 1738 l S NP 2577 1738 m 2582 1738 l S NP 2573 1738 m 2577 1738 l S NP 2568 1739 m 2573 1738 l S NP 2564 1739 m 2568 1739 l S NP 2559 1739 m 2564 1739 l S NP 2555 1740 m 2559 1739 l S NP 2550 1740 m 2555 1740 l S NP 2546 1741 m 2550 1740 l S NP 2541 1741 m 2546 1741 l S NP 2537 1742 m 2541 1741 l S NP 2532 1742 m 2537 1742 l S NP 2528 1743 m 2532 1742 l S NP 2523 1743 m 2528 1743 l S NP 2519 1744 m 2523 1743 l S NP 2514 1745 m 2519 1744 l S NP 2509 1745 m 2514 1745 l S NP 2505 1746 m 2509 1745 l S NP 2500 1747 m 2505 1746 l S NP 2496 1748 m 2500 1747 l S NP 2491 1748 m 2496 1748 l S NP 2487 1749 m 2491 1748 l S NP 2482 1750 m 2487 1749 l S NP 2478 1751 m 2482 1750 l S NP 2473 1752 m 2478 1751 l S NP 2469 1753 m 2473 1752 l S NP 2464 1754 m 2469 1753 l S NP 2460 1755 m 2464 1754 l S NP 2455 1756 m 2460 1755 l S NP 2451 1757 m 2455 1756 l S NP 2446 1758 m 2451 1757 l S NP 2442 1760 m 2446 1758 l S NP 2437 1761 m 2442 1760 l S NP 2433 1762 m 2437 1761 l S NP 2428 1763 m 2433 1762 l S NP 2424 1765 m 2428 1763 l S NP 2419 1766 m 2424 1765 l S NP 2415 1767 m 2419 1766 l S NP 2410 1769 m 2415 1767 l S NP 2406 1770 m 2410 1769 l S NP 2402 1772 m 2406 1770 l S NP 2397 1773 m 2402 1772 l S NP 2393 1775 m 2397 1773 l S NP 2388 1776 m 2393 1775 l S NP 2384 1778 m 2388 1776 l S NP 2379 1779 m 2384 1778 l S NP 2375 1781 m 2379 1779 l S NP 2370 1783 m 2375 1781 l S NP 2366 1784 m 2370 1783 l S NP 2362 1786 m 2366 1784 l S NP 2357 1788 m 2362 1786 l S NP 2353 1790 m 2357 1788 l S NP 2348 1792 m 2353 1790 l S NP 2344 1794 m 2348 1792 l S NP 2340 1795 m 2344 1794 l S NP 2335 1797 m 2340 1795 l S NP 2331 1799 m 2335 1797 l S NP 2326 1801 m 2331 1799 l S NP 2322 1803 m 2326 1801 l S NP 2318 1806 m 2322 1803 l S NP 2313 1808 m 2318 1806 l S NP 2309 1810 m 2313 1808 l S NP 2305 1812 m 2309 1810 l S NP 2300 1814 m 2305 1812 l S NP 2296 1816 m 2300 1814 l S NP 2291 1819 m 2296 1816 l S NP 2287 1821 m 2291 1819 l S NP 2283 1823 m 2287 1821 l S NP 2278 1826 m 2283 1823 l S NP 2274 1828 m 2278 1826 l S NP 2270 1830 m 2274 1828 l S NP 2266 1833 m 2270 1830 l S NP 2261 1835 m 2266 1833 l S NP 2257 1838 m 2261 1835 l S NP 2253 1841 m 2257 1838 l S NP 2248 1843 m 2253 1841 l S NP 2244 1846 m 2248 1843 l S NP 2240 1848 m 2244 1846 l S NP 2236 1851 m 2240 1848 l S NP 2231 1854 m 2236 1851 l S NP 2227 1857 m 2231 1854 l S NP 2223 1859 m 2227 1857 l S NP 2219 1862 m 2223 1859 l S NP 2214 1865 m 2219 1862 l S NP 2210 1868 m 2214 1865 l S NP 2206 1871 m 2210 1868 l S NP 2202 1874 m 2206 1871 l S NP 2198 1877 m 2202 1874 l S NP 2193 1880 m 2198 1877 l S NP 2189 1883 m 2193 1880 l S NP 2185 1886 m 2189 1883 l S NP 2181 1889 m 2185 1886 l S NP 2177 1892 m 2181 1889 l S NP 2173 1895 m 2177 1892 l S NP 2169 1898 m 2173 1895 l S NP 2164 1902 m 2169 1898 l S NP 2160 1905 m 2164 1902 l S NP 2156 1908 m 2160 1905 l S NP 2152 1911 m 2156 1908 l S NP 2148 1915 m 2152 1911 l S NP 2144 1918 m 2148 1915 l S NP 2140 1922 m 2144 1918 l S NP 2136 1925 m 2140 1922 l S NP 2132 1929 m 2136 1925 l S NP 2128 1932 m 2132 1929 l S NP 2124 1936 m 2128 1932 l S NP 2120 1939 m 2124 1936 l S NP 2116 1943 m 2120 1939 l S NP 2112 1946 m 2116 1943 l S NP 2108 1950 m 2112 1946 l S NP 2104 1954 m 2108 1950 l S NP 2100 1958 m 2104 1954 l S NP 2096 1961 m 2100 1958 l S NP 2092 1965 m 2096 1961 l S NP 2088 1969 m 2092 1965 l S NP 2084 1973 m 2088 1969 l S NP 2080 1977 m 2084 1973 l S NP 2076 1981 m 2080 1977 l S NP 2072 1985 m 2076 1981 l S NP 2069 1989 m 2072 1985 l S NP 2065 1993 m 2069 1989 l S NP 2061 1997 m 2065 1993 l S NP 2057 2001 m 2061 1997 l S NP 2053 2005 m 2057 2001 l S NP 2049 2009 m 2053 2005 l S NP 2046 2013 m 2049 2009 l S NP 2042 2017 m 2046 2013 l S NP 2038 2022 m 2042 2017 l S NP 2034 2026 m 2038 2022 l S NP 2030 2030 m 2034 2026 l S NP 2027 2035 m 2030 2030 l S NP 2023 2039 m 2027 2035 l S NP 2019 2043 m 2023 2039 l S NP 2015 2048 m 2019 2043 l S NP 2012 2052 m 2015 2048 l S NP 2008 2057 m 2012 2052 l S NP 2004 2061 m 2008 2057 l S NP 2001 2066 m 2004 2061 l S NP 1997 2070 m 2001 2066 l S NP 1994 2075 m 1997 2070 l S NP 1990 2079 m 1994 2075 l S NP 1986 2084 m 1990 2079 l S NP 1983 2089 m 1986 2084 l S NP 1979 2094 m 1983 2089 l S NP 1976 2098 m 1979 2094 l S NP 1972 2103 m 1976 2098 l S NP 1969 2108 m 1972 2103 l S NP 1965 2113 m 1969 2108 l S NP 1961 2118 m 1965 2113 l S NP 1958 2122 m 1961 2118 l S NP 1955 2127 m 1958 2122 l S NP 1951 2132 m 1955 2127 l S NP 1948 2137 m 1951 2132 l S NP 1944 2142 m 1948 2137 l S NP 1941 2147 m 1944 2142 l S NP 1937 2152 m 1941 2147 l S NP 1934 2158 m 1937 2152 l S NP 1931 2163 m 1934 2158 l S NP 1927 2168 m 1931 2163 l S NP 1924 2173 m 1927 2168 l S NP 1921 2178 m 1924 2173 l S NP 1917 2183 m 1921 2178 l S NP 1914 2189 m 1917 2183 l S NP 1911 2194 m 1914 2189 l S NP 1907 2199 m 1911 2194 l S NP 1904 2205 m 1907 2199 l S NP 1901 2210 m 1904 2205 l S NP 1898 2215 m 1901 2210 l S NP 1895 2221 m 1898 2215 l S NP 1891 2226 m 1895 2221 l S NP 1888 2232 m 1891 2226 l S NP 1885 2237 m 1888 2232 l S NP 1882 2243 m 1885 2237 l S NP 1879 2248 m 1882 2243 l S NP 1876 2254 m 1879 2248 l S NP 1873 2260 m 1876 2254 l S NP 1870 2265 m 1873 2260 l S NP 1867 2271 m 1870 2265 l S NP 1864 2277 m 1867 2271 l S NP 1861 2282 m 1864 2277 l S NP 1858 2288 m 1861 2282 l S NP 1855 2294 m 1858 2288 l S NP 1852 2300 m 1855 2294 l S NP 1849 2306 m 1852 2300 l S NP 1846 2311 m 1849 2306 l S NP 1843 2317 m 1846 2311 l S NP 1840 2323 m 1843 2317 l S NP 1837 2329 m 1840 2323 l S NP 1834 2335 m 1837 2329 l S NP 1831 2341 m 1834 2335 l S NP 1829 2347 m 1831 2341 l S NP 1826 2353 m 1829 2347 l S NP 1823 2359 m 1826 2353 l S NP 1820 2365 m 1823 2359 l S NP 1818 2371 m 1820 2365 l S NP 1815 2377 m 1818 2371 l S NP 1812 2384 m 1815 2377 l S NP 1810 2390 m 1812 2384 l S NP 1807 2396 m 1810 2390 l S NP 1804 2402 m 1807 2396 l S NP 1802 2408 m 1804 2402 l S NP 1799 2415 m 1802 2408 l S NP 1796 2421 m 1799 2415 l S NP 1794 2427 m 1796 2421 l S NP 1791 2434 m 1794 2427 l S NP 1789 2440 m 1791 2434 l S NP 1786 2446 m 1789 2440 l S NP 1784 2453 m 1786 2446 l S NP 1781 2459 m 1784 2453 l S NP 1779 2466 m 1781 2459 l S NP 1777 2472 m 1779 2466 l S NP 1774 2479 m 1777 2472 l S NP 1772 2485 m 1774 2479 l S NP 1769 2492 m 1772 2485 l S NP 1767 2498 m 1769 2492 l S NP 1765 2505 m 1767 2498 l S NP 1762 2511 m 1765 2505 l S NP 1760 2518 m 1762 2511 l S NP 1758 2525 m 1760 2518 l S NP 1756 2531 m 1758 2525 l S NP 1753 2538 m 1756 2531 l S NP 1751 2545 m 1753 2538 l S NP 1749 2551 m 1751 2545 l S NP 1747 2558 m 1749 2551 l S NP 1745 2565 m 1747 2558 l S NP 1743 2572 m 1745 2565 l S NP 1741 2578 m 1743 2572 l S NP 1739 2585 m 1741 2578 l S NP 1737 2592 m 1739 2585 l S NP 1735 2599 m 1737 2592 l S NP 1733 2606 m 1735 2599 l S NP 1731 2613 m 1733 2606 l S NP 1729 2619 m 1731 2613 l S NP 1727 2626 m 1729 2619 l S NP 1725 2633 m 1727 2626 l S NP 1723 2640 m 1725 2633 l S NP 1721 2647 m 1723 2640 l S NP 1719 2654 m 1721 2647 l S NP 1717 2661 m 1719 2654 l S NP 1716 2668 m 1717 2661 l S NP 1714 2675 m 1716 2668 l S NP 1712 2682 m 1714 2675 l S NP 1710 2689 m 1712 2682 l S NP 1709 2696 m 1710 2689 l S NP 1707 2704 m 1709 2696 l S NP 1705 2711 m 1707 2704 l S NP 1704 2718 m 1705 2711 l S NP 1702 2725 m 1704 2718 l S NP 1700 2732 m 1702 2725 l S NP 1699 2739 m 1700 2732 l S NP 1697 2746 m 1699 2739 l S NP 1696 2754 m 1697 2746 l S NP 1694 2761 m 1696 2754 l S NP 1693 2768 m 1694 2761 l S NP 1691 2775 m 1693 2768 l S NP 1690 2782 m 1691 2775 l S NP 1688 2790 m 1690 2782 l S NP 1687 2797 m 1688 2790 l S NP 1686 2804 m 1687 2797 l S NP 1684 2812 m 1686 2804 l S NP 1683 2819 m 1684 2812 l S NP 1682 2826 m 1683 2819 l S NP 1680 2834 m 1682 2826 l S NP 1679 2841 m 1680 2834 l S NP 1678 2848 m 1679 2841 l S NP 1677 2856 m 1678 2848 l S NP 1676 2863 m 1677 2856 l S NP 1675 2870 m 1676 2863 l S NP 1673 2878 m 1675 2870 l S NP 1672 2885 m 1673 2878 l S NP 1671 2893 m 1672 2885 l S NP 1670 2900 m 1671 2893 l S NP 1669 2908 m 1670 2900 l S NP 1668 2915 m 1669 2908 l S NP 1667 2922 m 1668 2915 l S NP 1666 2930 m 1667 2922 l S NP 1665 2937 m 1666 2930 l S NP 1664 2945 m 1665 2937 l S NP 1663 2952 m 1664 2945 l S NP 1663 2960 m 1663 2952 l S NP 1662 2967 m 1663 2960 l S NP 1661 2975 m 1662 2967 l S NP 1660 2982 m 1661 2975 l S NP 1659 2990 m 1660 2982 l S NP 1659 2997 m 1659 2990 l S NP 1658 3005 m 1659 2997 l S NP 1657 3013 m 1658 3005 l S NP 1657 3020 m 1657 3013 l S NP 1656 3028 m 1657 3020 l S NP 1655 3035 m 1656 3028 l S NP 1655 3043 m 1655 3035 l S NP 1654 3050 m 1655 3043 l S NP 1654 3058 m 1654 3050 l S NP 1653 3066 m 1654 3058 l S NP 1653 3073 m 1653 3066 l S NP 1652 3081 m 1653 3073 l S NP 1652 3088 m 1652 3081 l S NP 1651 3096 m 1652 3088 l S NP 1651 3104 m 1651 3096 l S NP 1651 3111 m 1651 3104 l S NP 1650 3119 m 1651 3111 l S NP 1650 3126 m 1650 3119 l S NP 1650 3134 m 1650 3126 l S NP 1649 3142 m 1650 3134 l S NP 1649 3149 m 1649 3142 l S NP 1649 3157 m 1649 3149 l S NP 1649 3165 m 1649 3157 l S NP 1649 3172 m 1649 3165 l S NP 1648 3180 m 1649 3172 l S NP 1648 3187 m 1648 3180 l S NP 1648 3195 m 1648 3187 l S NP 1648 3203 m 1648 3195 l S NP 1648 3210 m 1648 3203 l S NP 1648 3218 m 1648 3210 l S NP 1648 3226 m 1648 3218 l S NP 1648 3233 m 1648 3226 l S NP 1648 3241 m 1648 3233 l S NP 1648 3248 m 1648 3241 l S NP 1648 3256 m 1648 3248 l S NP 1649 3264 m 1648 3256 l S NP 1649 3271 m 1649 3264 l S NP 1649 3279 m 1649 3271 l S NP 1649 3287 m 1649 3279 l S NP 1649 3294 m 1649 3287 l S NP 1650 3302 m 1649 3294 l S NP 1650 3310 m 1650 3302 l S NP 1650 3317 m 1650 3310 l S NP 1650 3325 m 1650 3317 l S NP 1651 3332 m 1650 3325 l S NP 1651 3340 m 1651 3332 l S NP 1652 3348 m 1651 3340 l S NP 1652 3355 m 1652 3348 l S NP 1653 3363 m 1652 3355 l S NP 1653 3370 m 1653 3363 l S NP 1654 3378 m 1653 3370 l S NP 1654 3386 m 1654 3378 l S NP 1655 3393 m 1654 3386 l S NP 1655 3401 m 1655 3393 l S NP 1656 3408 m 1655 3401 l S NP 1656 3416 m 1656 3408 l S NP 1657 3423 m 1656 3416 l S NP 1658 3431 m 1657 3423 l S NP 1658 3438 m 1658 3431 l S NP 1659 3446 m 1658 3438 l S NP 1660 3454 m 1659 3446 l S NP 1661 3461 m 1660 3454 l S NP 1662 3469 m 1661 3461 l S NP 1662 3476 m 1662 3469 l S NP 1663 3484 m 1662 3476 l S NP 1664 3491 m 1663 3484 l S NP 1665 3499 m 1664 3491 l S NP 1666 3506 m 1665 3499 l S NP 1667 3514 m 1666 3506 l S NP 1668 3521 m 1667 3514 l S NP 1669 3528 m 1668 3521 l S NP 1670 3536 m 1669 3528 l S NP 1671 3543 m 1670 3536 l S NP 1672 3551 m 1671 3543 l S NP 1673 3558 m 1672 3551 l S NP 1674 3566 m 1673 3558 l S NP 1675 3573 m 1674 3566 l S NP 1677 3580 m 1675 3573 l S NP 1678 3588 m 1677 3580 l S NP 1679 3595 m 1678 3588 l S NP 1680 3602 m 1679 3595 l S NP 1682 3610 m 1680 3602 l S NP 1683 3617 m 1682 3610 l S NP 1684 3624 m 1683 3617 l S NP 1685 3632 m 1684 3624 l S NP 1687 3639 m 1685 3632 l S NP 1688 3646 m 1687 3639 l S NP 1690 3653 m 1688 3646 l S NP 1691 3661 m 1690 3653 l S NP 1693 3668 m 1691 3661 l S NP 1694 3675 m 1693 3668 l S NP 1696 3682 m 1694 3675 l S NP 1697 3690 m 1696 3682 l S NP 1699 3697 m 1697 3690 l S NP 1700 3704 m 1699 3697 l S NP 1702 3711 m 1700 3704 l S NP 1703 3718 m 1702 3711 l S NP 1705 3725 m 1703 3718 l S NP 1707 3732 m 1705 3725 l S NP 1708 3739 m 1707 3732 l S NP 1710 3746 m 1708 3739 l S NP 1712 3754 m 1710 3746 l S NP 1714 3761 m 1712 3754 l S NP 1716 3768 m 1714 3761 l S NP 1717 3775 m 1716 3768 l S NP 1719 3782 m 1717 3775 l S NP 1721 3789 m 1719 3782 l S NP 1723 3795 m 1721 3789 l S NP 1725 3802 m 1723 3795 l S NP 1727 3809 m 1725 3802 l S NP 1729 3816 m 1727 3809 l S NP 1731 3823 m 1729 3816 l S NP 1733 3830 m 1731 3823 l S NP 1735 3837 m 1733 3830 l S NP 1737 3844 m 1735 3837 l S NP 1739 3850 m 1737 3844 l S NP 1741 3857 m 1739 3850 l S NP 1743 3864 m 1741 3857 l S NP 1745 3871 m 1743 3864 l S NP 1747 3877 m 1745 3871 l S NP 1749 3884 m 1747 3877 l S NP 1752 3891 m 1749 3884 l S NP 1754 3897 m 1752 3891 l S NP 1756 3904 m 1754 3897 l S NP 1758 3911 m 1756 3904 l S NP 1761 3917 m 1758 3911 l S NP 1763 3924 m 1761 3917 l S NP 1765 3930 m 1763 3924 l S NP 1768 3937 m 1765 3930 l S NP 1770 3943 m 1768 3937 l S NP 1772 3950 m 1770 3943 l S NP 1775 3956 m 1772 3950 l S NP 1777 3963 m 1775 3956 l S NP 1780 3969 m 1777 3963 l S NP 1782 3976 m 1780 3969 l S NP 1785 3982 m 1782 3976 l S NP 1787 3988 m 1785 3982 l S NP 1790 3995 m 1787 3988 l S NP 1792 4001 m 1790 3995 l S NP 1795 4007 m 1792 4001 l S NP 1798 4014 m 1795 4007 l S NP 1800 4020 m 1798 4014 l S NP 1803 4026 m 1800 4020 l S NP 1806 4032 m 1803 4026 l S NP 1808 4038 m 1806 4032 l S NP 1811 4045 m 1808 4038 l S NP 1814 4051 m 1811 4045 l S NP 1816 4057 m 1814 4051 l S NP 1819 4063 m 1816 4057 l S NP 1822 4069 m 1819 4063 l S NP 1825 4075 m 1822 4069 l S NP 1828 4081 m 1825 4075 l S NP 1831 4087 m 1828 4081 l S NP 1833 4093 m 1831 4087 l S NP 1836 4099 m 1833 4093 l S NP 1839 4104 m 1836 4099 l S NP 1842 4110 m 1839 4104 l S NP 1845 4116 m 1842 4110 l S NP 1848 4122 m 1845 4116 l S NP 1851 4128 m 1848 4122 l S NP 1854 4133 m 1851 4128 l S NP 1857 4139 m 1854 4133 l S NP 1860 4145 m 1857 4139 l S NP 1863 4150 m 1860 4145 l S NP 1866 4156 m 1863 4150 l S NP 1869 4162 m 1866 4156 l S NP 1872 4167 m 1869 4162 l S NP 1876 4173 m 1872 4167 l S NP 1879 4178 m 1876 4173 l S NP 1882 4184 m 1879 4178 l S NP 1885 4189 m 1882 4184 l S NP 1888 4194 m 1885 4189 l S NP 1892 4200 m 1888 4194 l S NP 1895 4205 m 1892 4200 l S NP 1898 4210 m 1895 4205 l S NP 1901 4216 m 1898 4210 l S NP 1905 4221 m 1901 4216 l S NP 1908 4226 m 1905 4221 l S NP 1911 4231 m 1908 4226 l S NP 1915 4237 m 1911 4231 l S NP 1918 4242 m 1915 4237 l S NP 1921 4247 m 1918 4242 l S NP 1925 4252 m 1921 4247 l S NP 1928 4257 m 1925 4252 l S NP 1932 4262 m 1928 4257 l S NP 1935 4267 m 1932 4262 l S NP 1939 4272 m 1935 4267 l S NP 1942 4277 m 1939 4272 l S NP 1946 4282 m 1942 4277 l S NP 1949 4286 m 1946 4282 l S NP 1953 4291 m 1949 4286 l S NP 1956 4296 m 1953 4291 l S NP 1960 4301 m 1956 4296 l S NP 1963 4305 m 1960 4301 l S NP 1967 4310 m 1963 4305 l S NP 1971 4315 m 1967 4310 l S NP 1974 4319 m 1971 4315 l S NP 1978 4324 m 1974 4319 l S NP 1981 4328 m 1978 4324 l S NP 1985 4333 m 1981 4328 l S NP 1989 4337 m 1985 4333 l S NP 1992 4342 m 1989 4337 l S NP 1996 4346 m 1992 4342 l S NP 2000 4351 m 1996 4346 l S NP 2004 4355 m 2000 4351 l S NP 2007 4359 m 2004 4355 l S NP 2011 4363 m 2007 4359 l S NP 2015 4368 m 2011 4363 l S NP 2019 4372 m 2015 4368 l S NP 2023 4376 m 2019 4372 l S NP 2026 4380 m 2023 4376 l S NP 2030 4384 m 2026 4380 l S NP 2034 4388 m 2030 4384 l S NP 2038 4392 m 2034 4388 l S NP 2042 4396 m 2038 4392 l S NP 2046 4400 m 2042 4396 l S NP 2050 4404 m 2046 4400 l S NP 2054 4408 m 2050 4404 l S NP 2058 4412 m 2054 4408 l S NP 2062 4415 m 2058 4412 l S NP 2065 4419 m 2062 4415 l S NP 2069 4423 m 2065 4419 l S NP 2073 4427 m 2069 4423 l S NP 2077 4430 m 2073 4427 l S NP 2081 4434 m 2077 4430 l S NP 2085 4437 m 2081 4434 l S NP 2090 4441 m 2085 4437 l S NP 2094 4444 m 2090 4441 l S NP 2098 4448 m 2094 4444 l S NP 2102 4451 m 2098 4448 l S NP 2106 4455 m 2102 4451 l S NP 2110 4458 m 2106 4455 l S NP 2114 4461 m 2110 4458 l S NP 2118 4464 m 2114 4461 l S NP 2122 4468 m 2118 4464 l S NP 2126 4471 m 2122 4468 l S NP 2130 4474 m 2126 4471 l S NP 2135 4477 m 2130 4474 l S NP 2139 4480 m 2135 4477 l S NP 2143 4483 m 2139 4480 l S NP 2147 4486 m 2143 4483 l S NP 2151 4489 m 2147 4486 l S NP 2156 4492 m 2151 4489 l S NP 2160 4495 m 2156 4492 l S NP 2164 4498 m 2160 4495 l S NP 2168 4500 m 2164 4498 l S NP 2172 4503 m 2168 4500 l S NP 2177 4506 m 2172 4503 l S NP 2181 4508 m 2177 4506 l S NP 2185 4511 m 2181 4508 l S NP 2190 4514 m 2185 4511 l S NP 2194 4516 m 2190 4514 l S NP 2198 4519 m 2194 4516 l S NP 2202 4521 m 2198 4519 l S NP 2207 4524 m 2202 4521 l S NP 2211 4526 m 2207 4524 l S NP 2215 4528 m 2211 4526 l S NP 2220 4531 m 2215 4528 l S NP 2224 4533 m 2220 4531 l S NP 2228 4535 m 2224 4533 l S NP 2233 4537 m 2228 4535 l S NP 2237 4539 m 2233 4537 l S NP 2242 4541 m 2237 4539 l S NP 2246 4543 m 2242 4541 l S NP 2250 4545 m 2246 4543 l S NP 2255 4547 m 2250 4545 l S NP 2259 4549 m 2255 4547 l S NP 2263 4551 m 2259 4549 l S NP 2268 4553 m 2263 4551 l S NP 2272 4555 m 2268 4553 l S NP 2277 4557 m 2272 4555 l S NP 2281 4558 m 2277 4557 l S NP 2286 4560 m 2281 4558 l S NP 2290 4562 m 2286 4560 l S NP 2295 4563 m 2290 4562 l S NP 2299 4565 m 2295 4563 l S NP 2303 4566 m 2299 4565 l S NP 2308 4568 m 2303 4566 l S NP 2312 4569 m 2308 4568 l S NP 2317 4570 m 2312 4569 l S NP 2321 4572 m 2317 4570 l S NP 2326 4573 m 2321 4572 l S NP 2330 4574 m 2326 4573 l S NP 2335 4575 m 2330 4574 l S NP 2339 4577 m 2335 4575 l S NP 2344 4578 m 2339 4577 l S NP 2348 4579 m 2344 4578 l S NP 2353 4580 m 2348 4579 l S NP 2357 4581 m 2353 4580 l S NP 2362 4582 m 2357 4581 l S NP 2366 4583 m 2362 4582 l S NP 2371 4583 m 2366 4583 l S NP 2375 4584 m 2371 4583 l S NP 2380 4585 m 2375 4584 l S NP 2384 4586 m 2380 4585 l S NP 2389 4586 m 2384 4586 l S NP 2393 4587 m 2389 4586 l S NP 2398 4588 m 2393 4587 l S NP 2402 4588 m 2398 4588 l S NP 2407 4589 m 2402 4588 l S NP 2412 4589 m 2407 4589 l S NP 2416 4590 m 2412 4589 l S NP 2421 4590 m 2416 4590 l S NP 2425 4590 m 2421 4590 l S NP 2430 4591 m 2425 4590 l S NP 2434 4591 m 2430 4591 l S NP 2439 4591 m 2434 4591 l S NP 2443 4591 m 2439 4591 l S NP 2448 4591 m 2443 4591 l S NP 2452 4591 m 2448 4591 l S NP 2457 4591 m 2452 4591 l S NP 2462 4591 m 2457 4591 l S NP 2466 4591 m 2462 4591 l S NP 2471 4591 m 2466 4591 l S NP 2475 4591 m 2471 4591 l S NP 2480 4591 m 2475 4591 l S NP 2484 4591 m 2480 4591 l S NP 2489 4591 m 2484 4591 l S NP 2493 4590 m 2489 4591 l S NP 2498 4590 m 2493 4590 l S NP 2502 4589 m 2498 4590 l S NP 2507 4589 m 2502 4589 l S NP 2511 4588 m 2507 4589 l S NP 2516 4588 m 2511 4588 l S NP 2521 4587 m 2516 4588 l S NP 2525 4587 m 2521 4587 l S NP 2530 4586 m 2525 4587 l S NP 2534 4585 m 2530 4586 l S NP 2539 4585 m 2534 4585 l S NP 2543 4584 m 2539 4585 l S NP 2548 4583 m 2543 4584 l S NP 2552 4582 m 2548 4583 l S NP 2557 4581 m 2552 4582 l S NP 2561 4580 m 2557 4581 l S NP 2566 4579 m 2561 4580 l S NP 2570 4578 m 2566 4579 l S NP 2575 4577 m 2570 4578 l S NP 2579 4576 m 2575 4577 l S NP 2584 4575 m 2579 4576 l S NP 2588 4574 m 2584 4575 l S NP 2593 4572 m 2588 4574 l S NP 2597 4571 m 2593 4572 l S NP 2602 4570 m 2597 4571 l S NP 2606 4568 m 2602 4570 l S NP 2611 4567 m 2606 4568 l S NP 2615 4565 m 2611 4567 l S NP 2619 4564 m 2615 4565 l S NP 2624 4562 m 2619 4564 l S NP 2628 4561 m 2624 4562 l S NP 2633 4559 m 2628 4561 l S NP 2637 4557 m 2633 4559 l S NP 2642 4555 m 2637 4557 l S NP 2646 4554 m 2642 4555 l S NP 2651 4552 m 2646 4554 l S NP 2655 4550 m 2651 4552 l S NP 2659 4548 m 2655 4550 l S NP 2664 4546 m 2659 4548 l S NP 2668 4544 m 2664 4546 l S NP 2672 4542 m 2668 4544 l S NP 2677 4540 m 2672 4542 l S NP 2681 4538 m 2677 4540 l S NP 2686 4536 m 2681 4538 l S NP 2690 4533 m 2686 4536 l S NP 2694 4531 m 2690 4533 l S NP 2699 4529 m 2694 4531 l S NP 2703 4527 m 2699 4529 l S NP 2707 4524 m 2703 4527 l S NP 2712 4522 m 2707 4524 l S NP 2716 4519 m 2712 4522 l S NP 2720 4517 m 2716 4519 l S NP 2724 4514 m 2720 4517 l S NP 2729 4512 m 2724 4514 l S NP 2733 4509 m 2729 4512 l S NP 2737 4506 m 2733 4509 l S NP 2742 4504 m 2737 4506 l S NP 2746 4501 m 2742 4504 l S NP 2750 4498 m 2746 4501 l S NP 2754 4495 m 2750 4498 l S NP 2758 4492 m 2754 4495 l S NP 2763 4489 m 2758 4492 l S NP 2767 4487 m 2763 4489 l S NP 2771 4484 m 2767 4487 l S NP 2775 4480 m 2771 4484 l S NP 2779 4477 m 2775 4480 l S NP 2783 4474 m 2779 4477 l S NP 2788 4471 m 2783 4474 l S NP 2792 4468 m 2788 4471 l S NP 2796 4465 m 2792 4468 l S NP 2800 4461 m 2796 4465 l S NP 2804 4458 m 2800 4461 l S NP 2808 4455 m 2804 4458 l S NP 2812 4451 m 2808 4455 l S NP 2816 4448 m 2812 4451 l S NP 2820 4444 m 2816 4448 l S NP 2824 4441 m 2820 4444 l S NP 2828 4437 m 2824 4441 l S NP 2832 4434 m 2828 4437 l S NP 2836 4430 m 2832 4434 l S NP 2840 4426 m 2836 4430 l S NP 2844 4423 m 2840 4426 l S NP 2848 4419 m 2844 4423 l S NP 2852 4415 m 2848 4419 l S NP 2856 4411 m 2852 4415 l S NP 2860 4407 m 2856 4411 l S NP 2864 4404 m 2860 4407 l S NP 2868 4400 m 2864 4404 l S NP 2872 4396 m 2868 4400 l S NP 2876 4392 m 2872 4396 l S NP 2879 4388 m 2876 4392 l S NP 2883 4383 m 2879 4388 l S NP 2887 4379 m 2883 4383 l S NP 2891 4375 m 2887 4379 l S NP 2895 4371 m 2891 4375 l S NP 2898 4367 m 2895 4371 l S NP 2902 4362 m 2898 4367 l S NP 2906 4358 m 2902 4362 l S NP 2910 4354 m 2906 4358 l S NP 2913 4349 m 2910 4354 l S NP 2917 4345 m 2913 4349 l S NP 2921 4340 m 2917 4345 l S NP 2925 4336 m 2921 4340 l S NP 2928 4331 m 2925 4336 l S NP 2932 4327 m 2928 4331 l S NP 2935 4322 m 2932 4327 l S NP 2939 4318 m 2935 4322 l S NP 2943 4313 m 2939 4318 l S NP 2946 4308 m 2943 4313 l S NP 2950 4303 m 2946 4308 l S NP 2953 4299 m 2950 4303 l S NP 2957 4294 m 2953 4299 l S NP 2960 4289 m 2957 4294 l S NP 2964 4284 m 2960 4289 l S NP 2967 4279 m 2964 4284 l S NP 2971 4274 m 2967 4279 l S NP 2974 4269 m 2971 4274 l S NP 2978 4264 m 2974 4269 l S NP 2981 4259 m 2978 4264 l S NP 2984 4254 m 2981 4259 l S NP 2988 4249 m 2984 4254 l S NP 2991 4244 m 2988 4249 l S NP 2994 4238 m 2991 4244 l S NP 2998 4233 m 2994 4238 l S NP 3001 4228 m 2998 4233 l S NP 3004 4223 m 3001 4228 l S NP 3008 4217 m 3004 4223 l S NP 3011 4212 m 3008 4217 l S NP 3014 4207 m 3011 4212 l S NP 3017 4201 m 3014 4207 l S NP 3020 4196 m 3017 4201 l S NP 3024 4190 m 3020 4196 l S NP 3027 4185 m 3024 4190 l S NP 3030 4179 m 3027 4185 l S NP 3033 4174 m 3030 4179 l S NP 3036 4168 m 3033 4174 l S NP 3039 4162 m 3036 4168 l S NP 3042 4157 m 3039 4162 l S NP 3045 4151 m 3042 4157 l S NP 3048 4145 m 3045 4151 l S NP 3051 4139 m 3048 4145 l S NP 3054 4134 m 3051 4139 l S NP 3057 4128 m 3054 4134 l S NP 3060 4122 m 3057 4128 l S NP 3063 4116 m 3060 4122 l S NP 3066 4110 m 3063 4116 l S NP 3069 4104 m 3066 4110 l S NP 3071 4098 m 3069 4104 l S NP 3074 4092 m 3071 4098 l S NP 3077 4086 m 3074 4092 l S NP 3080 4080 m 3077 4086 l S NP 3083 4074 m 3080 4080 l S NP 3085 4068 m 3083 4074 l S NP 3088 4062 m 3085 4068 l S NP 3091 4056 m 3088 4062 l S NP 3093 4049 m 3091 4056 l S NP 3096 4043 m 3093 4049 l S NP 3099 4037 m 3096 4043 l S NP 3101 4031 m 3099 4037 l S NP 3104 4024 m 3101 4031 l S NP 3106 4018 m 3104 4024 l S NP 3109 4012 m 3106 4018 l S NP 3111 4005 m 3109 4012 l S NP 3114 3999 m 3111 4005 l S NP 3116 3993 m 3114 3999 l S NP 3119 3986 m 3116 3993 l S NP 3121 3980 m 3119 3986 l S NP 3123 3973 m 3121 3980 l S NP 3126 3967 m 3123 3973 l S NP 3128 3960 m 3126 3967 l S NP 3131 3954 m 3128 3960 l S NP 3133 3947 m 3131 3954 l S NP 3135 3940 m 3133 3947 l S NP 3137 3934 m 3135 3940 l S NP 3140 3927 m 3137 3934 l S NP 3142 3920 m 3140 3927 l S NP 3144 3914 m 3142 3920 l S NP 3146 3907 m 3144 3914 l S NP 3148 3900 m 3146 3907 l S NP 3150 3893 m 3148 3900 l S NP 3152 3887 m 3150 3893 l S NP 3154 3880 m 3152 3887 l S NP 3156 3873 m 3154 3880 l S NP 3158 3866 m 3156 3873 l S NP 3160 3859 m 3158 3866 l S NP 3162 3852 m 3160 3859 l S NP 3164 3845 m 3162 3852 l S NP 3166 3838 m 3164 3845 l S NP 3168 3832 m 3166 3838 l S NP 3170 3825 m 3168 3832 l S NP 3172 3818 m 3170 3825 l S NP 3174 3811 m 3172 3818 l S NP 3175 3804 m 3174 3811 l S NP 3177 3796 m 3175 3804 l S NP 3179 3789 m 3177 3796 l S NP 3181 3782 m 3179 3789 l S NP 3182 3775 m 3181 3782 l S NP 3184 3768 m 3182 3775 l S NP 3185 3761 m 3184 3768 l S NP 3187 3754 m 3185 3761 l S NP 3189 3747 m 3187 3754 l S NP 3190 3739 m 3189 3747 l S NP 3192 3732 m 3190 3739 l S NP 3193 3725 m 3192 3732 l S NP 3195 3718 m 3193 3725 l S NP 3196 3711 m 3195 3718 l S NP 3198 3703 m 3196 3711 l S NP 3199 3696 m 3198 3703 l S NP 3200 3689 m 3199 3696 l S NP 3202 3681 m 3200 3689 l S NP 3203 3674 m 3202 3681 l S NP 3204 3667 m 3203 3674 l S NP 3205 3659 m 3204 3667 l S NP 3207 3652 m 3205 3659 l S NP 3208 3645 m 3207 3652 l S NP 3209 3637 m 3208 3645 l S NP 3210 3630 m 3209 3637 l S NP 3211 3623 m 3210 3630 l S NP 3212 3615 m 3211 3623 l S NP 3213 3608 m 3212 3615 l S NP 3214 3600 m 3213 3608 l S NP 3215 3593 m 3214 3600 l S NP 3216 3585 m 3215 3593 l S NP 3217 3578 m 3216 3585 l S NP 3218 3570 m 3217 3578 l S NP 3219 3563 m 3218 3570 l S NP 3220 3555 m 3219 3563 l S NP 3221 3548 m 3220 3555 l S NP 3222 3540 m 3221 3548 l S NP 3222 3533 m 3222 3540 l S NP 3223 3525 m 3222 3533 l S NP 3224 3518 m 3223 3525 l S NP 3225 3510 m 3224 3518 l S NP 3225 3503 m 3225 3510 l S NP 3226 3495 m 3225 3503 l S NP 3227 3488 m 3226 3495 l S NP 3227 3480 m 3227 3488 l S NP 3228 3472 m 3227 3480 l S NP 3228 3465 m 3228 3472 l S NP 3229 3457 m 3228 3465 l S NP 3229 3450 m 3229 3457 l S NP 3230 3442 m 3229 3450 l S NP 3230 3435 m 3230 3442 l S NP 3231 3427 m 3230 3435 l S NP 3231 3419 m 3231 3427 l S NP 3231 3412 m 3231 3419 l S NP 3231 3404 m 3231 3412 l S NP 3232 3396 m 3231 3404 l S NP 3232 3389 m 3232 3396 l S NP 3232 3381 m 3232 3389 l S NP 3232 3374 m 3232 3381 l S NP 3233 3366 m 3232 3374 l S NP 3233 3358 m 3233 3366 l S NP 3233 3351 m 3233 3358 l S NP 3233 3343 m 3233 3351 l S NP 3233 3335 m 3233 3343 l S NP 3233 3328 m 3233 3335 l S NP 3233 3320 m 3233 3328 l S NP 3233 3312 m 3233 3320 l S NP 3233 3305 m 3233 3312 l S NP 3233 3297 m 3233 3305 l S NP 3233 3290 m 3233 3297 l S NP 3232 3282 m 3233 3290 l S NP 3232 3274 m 3232 3282 l S NP 3232 3267 m 3232 3274 l S NP 3232 3259 m 3232 3267 l S NP 3231 3251 m 3232 3259 l S NP 3231 3244 m 3231 3251 l S NP 3231 3236 m 3231 3244 l S NP 3230 3229 m 3231 3236 l S NP 3230 3221 m 3230 3229 l S NP 3230 3213 m 3230 3221 l S NP 3229 3206 m 3230 3213 l S NP 3229 3198 m 3229 3206 l S NP 3228 3191 m 3229 3198 l S NP 3228 3183 m 3228 3191 l S NP 3227 3175 m 3228 3183 l S NP 3227 3168 m 3227 3175 l S NP 3226 3160 m 3227 3168 l S NP 3225 3153 m 3226 3160 l S NP 3225 3145 m 3225 3153 l S NP 3224 3138 m 3225 3145 l S NP 3223 3130 m 3224 3138 l S NP 3222 3123 m 3223 3130 l S NP 3222 3115 m 3222 3123 l S NP 3221 3108 m 3222 3115 l S NP 3220 3100 m 3221 3108 l S NP 3219 3093 m 3220 3100 l S NP 3218 3085 m 3219 3093 l S NP 3217 3078 m 3218 3085 l S NP 3216 3070 m 3217 3078 l S NP 3215 3063 m 3216 3070 l S NP 3214 3055 m 3215 3063 l S NP 3213 3048 m 3214 3055 l S NP 3212 3040 m 3213 3048 l S NP 3211 3033 m 3212 3040 l S NP 3210 3026 m 3211 3033 l S NP 3209 3018 m 3210 3026 l S NP 3207 3011 m 3209 3018 l S NP 3206 3004 m 3207 3011 l S NP 3205 2996 m 3206 3004 l S NP 3204 2989 m 3205 2996 l S NP 3202 2982 m 3204 2989 l S NP 3201 2974 m 3202 2982 l S NP 3200 2967 m 3201 2974 l S NP 3198 2960 m 3200 2967 l S NP 3197 2952 m 3198 2960 l S NP 3195 2945 m 3197 2952 l S NP 3194 2938 m 3195 2945 l S NP 3192 2931 m 3194 2938 l S NP 3191 2924 m 3192 2931 l S NP 3189 2916 m 3191 2924 l S NP 3188 2909 m 3189 2916 l S NP 3186 2902 m 3188 2909 l S NP 3184 2895 m 3186 2902 l S NP 3183 2888 m 3184 2895 l S NP 3181 2881 m 3183 2888 l S NP 3179 2874 m 3181 2881 l S NP 3177 2867 m 3179 2874 l S NP 3176 2860 m 3177 2867 l S NP 3174 2853 m 3176 2860 l S NP 3172 2846 m 3174 2853 l S NP 3170 2839 m 3172 2846 l S NP 3168 2832 m 3170 2839 l S NP 3166 2825 m 3168 2832 l S NP 3164 2818 m 3166 2825 l S NP 3162 2811 m 3164 2818 l S NP 3160 2804 m 3162 2811 l S NP 3158 2798 m 3160 2804 l S NP 3156 2791 m 3158 2798 l S NP 3154 2784 m 3156 2791 l S NP 3152 2777 m 3154 2784 l S NP 3150 2771 m 3152 2777 l S NP 3148 2764 m 3150 2771 l S NP 3145 2757 m 3148 2764 l S NP 3143 2750 m 3145 2757 l S NP 3141 2744 m 3143 2750 l S NP 3139 2737 m 3141 2744 l S NP 3136 2731 m 3139 2737 l S NP 3134 2724 m 3136 2731 l S NP 3132 2718 m 3134 2724 l S NP 3129 2711 m 3132 2718 l S NP 3127 2705 m 3129 2711 l S NP 3124 2698 m 3127 2705 l S NP 3122 2692 m 3124 2698 l S NP 3119 2685 m 3122 2692 l S NP 3117 2679 m 3119 2685 l S NP 3114 2673 m 3117 2679 l S NP 3112 2666 m 3114 2673 l S NP 3109 2660 m 3112 2666 l S NP 3107 2654 m 3109 2660 l S NP 3104 2648 m 3107 2654 l S NP 3101 2641 m 3104 2648 l S NP 3099 2635 m 3101 2641 l S NP 3096 2629 m 3099 2635 l S NP 3093 2623 m 3096 2629 l S NP 3090 2617 m 3093 2623 l S NP 3088 2611 m 3090 2617 l S NP 3085 2605 m 3088 2611 l S NP 3082 2599 m 3085 2605 l S NP 3079 2593 m 3082 2599 l S NP 3076 2587 m 3079 2593 l S NP 3073 2581 m 3076 2587 l S NP 3070 2576 m 3073 2581 l S NP 3067 2570 m 3070 2576 l S NP 3064 2564 m 3067 2570 l S NP 3061 2558 m 3064 2564 l S NP 3058 2553 m 3061 2558 l S NP 3055 2547 m 3058 2553 l S NP 3052 2541 m 3055 2547 l S NP 3049 2536 m 3052 2541 l S NP 3046 2530 m 3049 2536 l S NP 3043 2525 m 3046 2530 l S NP 3040 2519 m 3043 2525 l S NP 3036 2514 m 3040 2519 l S NP 3033 2508 m 3036 2514 l S NP 3030 2503 m 3033 2508 l S NP 3027 2498 m 3030 2503 l S NP 3023 2492 m 3027 2498 l S NP 3020 2487 m 3023 2492 l S NP 3017 2482 m 3020 2487 l S NP 3013 2477 m 3017 2482 l S NP 3010 2472 m 3013 2477 l S NP 3007 2467 m 3010 2472 l S NP 3003 2462 m 3007 2467 l S NP 3000 2457 m 3003 2462 l S NP 2996 2452 m 3000 2457 l S NP 2993 2447 m 2996 2452 l S NP 2989 2442 m 2993 2447 l S NP 2986 2437 m 2989 2442 l S NP 2982 2432 m 2986 2437 l S NP 2979 2427 m 2982 2432 l S NP 2975 2423 m 2979 2427 l S NP 2972 2418 m 2975 2423 l S NP 2968 2413 m 2972 2418 l S NP 2964 2409 m 2968 2413 l S NP 2961 2404 m 2964 2409 l S NP 2957 2400 m 2961 2404 l S NP 2953 2395 m 2957 2400 l S NP 2950 2391 m 2953 2395 l S NP 2946 2386 m 2950 2391 l S NP 2942 2382 m 2946 2386 l S NP 2938 2378 m 2942 2382 l S NP 2935 2374 m 2938 2378 l S NP 2931 2369 m 2935 2374 l S NP 2927 2365 m 2931 2369 l S NP 2923 2361 m 2927 2365 l S NP 2919 2357 m 2923 2361 l S NP 2916 2353 m 2919 2357 l S NP 2912 2349 m 2916 2353 l S NP 2908 2345 m 2912 2349 l S NP 2904 2341 m 2908 2345 l S NP 2900 2338 m 2904 2341 l S NP 2896 2334 m 2900 2338 l S NP 2892 2330 m 2896 2334 l S NP 2888 2326 m 2892 2330 l S NP 2884 2323 m 2888 2326 l S NP 2880 2319 m 2884 2323 l S NP 2876 2316 m 2880 2319 l S NP 2872 2312 m 2876 2316 l S NP 2868 2309 m 2872 2312 l S NP 2864 2305 m 2868 2309 l S NP 2860 2302 m 2864 2305 l S NP 2856 2299 m 2860 2302 l S NP 2851 2295 m 2856 2299 l S NP 2847 2292 m 2851 2295 l S NP 2843 2289 m 2847 2292 l S NP 2839 2286 m 2843 2289 l S NP 2835 2283 m 2839 2286 l S NP 2831 2280 m 2835 2283 l S NP 2827 2277 m 2831 2280 l S NP 2822 2274 m 2827 2277 l S NP 2818 2271 m 2822 2274 l S NP 2814 2268 m 2818 2271 l S NP 2810 2266 m 2814 2268 l S NP 2805 2263 m 2810 2266 l S NP 2801 2260 m 2805 2263 l S NP 2797 2258 m 2801 2260 l S NP 2793 2255 m 2797 2258 l S NP 2788 2253 m 2793 2255 l S NP 2784 2250 m 2788 2253 l S NP 2780 2248 m 2784 2250 l S NP 2775 2245 m 2780 2248 l S NP 2771 2243 m 2775 2245 l S NP 2767 2241 m 2771 2243 l S NP 2762 2239 m 2767 2241 l S NP 2758 2237 m 2762 2239 l S NP 2753 2235 m 2758 2237 l S NP 2749 2233 m 2753 2235 l S NP 2745 2231 m 2749 2233 l S NP 2740 2229 m 2745 2231 l S NP 2736 2227 m 2740 2229 l S NP 2731 2225 m 2736 2227 l S NP 2727 2223 m 2731 2225 l S NP 2723 2222 m 2727 2223 l S NP 2718 2220 m 2723 2222 l S NP 2714 2218 m 2718 2220 l S NP 2709 2217 m 2714 2218 l S NP 2705 2215 m 2709 2217 l S NP 2700 2214 m 2705 2215 l S NP 2696 2213 m 2700 2214 l S NP 2691 2211 m 2696 2213 l S NP 2687 2210 m 2691 2211 l S NP 2682 2209 m 2687 2210 l S NP 2678 2208 m 2682 2209 l S NP 2673 2207 m 2678 2208 l S NP 2669 2206 m 2673 2207 l S NP 2664 2205 m 2669 2206 l S NP 2660 2204 m 2664 2205 l S NP 2655 2203 m 2660 2204 l S NP 2651 2202 m 2655 2203 l S NP 2646 2201 m 2651 2202 l S NP 2642 2201 m 2646 2201 l S NP 2637 2200 m 2642 2201 l S NP 2633 2199 m 2637 2200 l S NP 2628 2199 m 2633 2199 l S NP 2624 2198 m 2628 2199 l S NP 2619 2198 m 2624 2198 l S NP 2615 2198 m 2619 2198 l S NP 2610 2197 m 2615 2198 l S NP 2606 2197 m 2610 2197 l S NP 2601 2197 m 2606 2197 l S NP 2596 2197 m 2601 2197 l S NP 2592 2197 m 2596 2197 l S NP 2587 2197 m 2592 2197 l S NP 2583 2197 m 2587 2197 l S NP 2578 2197 m 2583 2197 l S NP 2574 2197 m 2578 2197 l S NP 2569 2197 m 2574 2197 l S NP 2565 2197 m 2569 2197 l S NP 2560 2198 m 2565 2197 l S NP 2556 2198 m 2560 2198 l S NP 2551 2198 m 2556 2198 l S NP 2546 2199 m 2551 2198 l S NP 2542 2199 m 2546 2199 l S NP 2537 2200 m 2542 2199 l S NP 2533 2201 m 2537 2200 l S NP 2528 2201 m 2533 2201 l S NP 2524 2202 m 2528 2201 l S NP 2519 2203 m 2524 2202 l S NP 2515 2204 m 2519 2203 l S NP 2510 2205 m 2515 2204 l S NP 2506 2206 m 2510 2205 l S NP 2501 2207 m 2506 2206 l S NP 2497 2208 m 2501 2207 l S NP 2492 2209 m 2497 2208 l S NP 2488 2210 m 2492 2209 l S NP 2483 2212 m 2488 2210 l S NP 2479 2213 m 2483 2212 l S NP 2474 2214 m 2479 2213 l S NP 2470 2216 m 2474 2214 l S NP 2465 2217 m 2470 2216 l S NP 2461 2219 m 2465 2217 l S NP 2457 2220 m 2461 2219 l S NP 2452 2222 m 2457 2220 l S NP 2448 2224 m 2452 2222 l S NP 2443 2226 m 2448 2224 l S NP 2439 2228 m 2443 2226 l S NP 2434 2230 m 2439 2228 l S NP 2430 2231 m 2434 2230 l S NP 2426 2234 m 2430 2231 l S NP 2421 2236 m 2426 2234 l S NP 2417 2238 m 2421 2236 l S NP 2413 2240 m 2417 2238 l S NP 2408 2242 m 2413 2240 l S NP 2404 2245 m 2408 2242 l S NP 2400 2247 m 2404 2245 l S NP 2395 2249 m 2400 2247 l S NP 2391 2252 m 2395 2249 l S NP 2387 2254 m 2391 2252 l S NP 2383 2257 m 2387 2254 l S NP 2378 2260 m 2383 2257 l S NP 2374 2263 m 2378 2260 l S NP 2370 2265 m 2374 2263 l S NP 2366 2268 m 2370 2265 l S NP 2361 2271 m 2366 2268 l S NP 2357 2274 m 2361 2271 l S NP 2353 2277 m 2357 2274 l S NP 2349 2280 m 2353 2277 l S NP 2345 2283 m 2349 2280 l S NP 2341 2287 m 2345 2283 l S NP 2336 2290 m 2341 2287 l S NP 2332 2293 m 2336 2290 l S NP 2328 2297 m 2332 2293 l S NP 2324 2300 m 2328 2297 l S NP 2320 2303 m 2324 2300 l S NP 2316 2307 m 2320 2303 l S NP 2312 2311 m 2316 2307 l S NP 2308 2314 m 2312 2311 l S NP 2304 2318 m 2308 2314 l S NP 2300 2322 m 2304 2318 l S NP 2296 2326 m 2300 2322 l S NP 2292 2330 m 2296 2326 l S NP 2289 2334 m 2292 2330 l S NP 2285 2338 m 2289 2334 l S NP 2281 2342 m 2285 2338 l S NP 2277 2346 m 2281 2342 l S NP 2273 2350 m 2277 2346 l S NP 2269 2354 m 2273 2350 l S NP 2266 2358 m 2269 2354 l S NP 2262 2363 m 2266 2358 l S NP 2258 2367 m 2262 2363 l S NP 2255 2372 m 2258 2367 l S NP 2251 2376 m 2255 2372 l S NP 2247 2381 m 2251 2376 l S NP 2244 2386 m 2247 2381 l S NP 2240 2390 m 2244 2386 l S NP 2237 2395 m 2240 2390 l S NP 2233 2400 m 2237 2395 l S NP 2229 2405 m 2233 2400 l S NP 2226 2410 m 2229 2405 l S NP 2223 2415 m 2226 2410 l S NP 2219 2420 m 2223 2415 l S NP 2216 2425 m 2219 2420 l S NP 2212 2430 m 2216 2425 l S NP 2209 2435 m 2212 2430 l S NP 2206 2440 m 2209 2435 l S NP 2202 2445 m 2206 2440 l S NP 2199 2451 m 2202 2445 l S NP 2196 2456 m 2199 2451 l S NP 2193 2462 m 2196 2456 l S NP 2190 2467 m 2193 2462 l S NP 2187 2473 m 2190 2467 l S NP 2183 2478 m 2187 2473 l S NP 2180 2484 m 2183 2478 l S NP 2177 2490 m 2180 2484 l S NP 2174 2495 m 2177 2490 l S NP 2171 2501 m 2174 2495 l S NP 2169 2507 m 2171 2501 l S NP 2166 2513 m 2169 2507 l S NP 2163 2519 m 2166 2513 l S NP 2160 2525 m 2163 2519 l S NP 2157 2531 m 2160 2525 l S NP 2154 2537 m 2157 2531 l S NP 2152 2543 m 2154 2537 l S NP 2149 2549 m 2152 2543 l S NP 2146 2556 m 2149 2549 l S NP 2144 2562 m 2146 2556 l S NP 2141 2568 m 2144 2562 l S NP 2139 2574 m 2141 2568 l S NP 2136 2581 m 2139 2574 l S NP 2134 2587 m 2136 2581 l S NP 2131 2594 m 2134 2587 l S NP 2129 2600 m 2131 2594 l S NP 2127 2607 m 2129 2600 l S NP 2124 2613 m 2127 2607 l S NP 2122 2620 m 2124 2613 l S NP 2120 2627 m 2122 2620 l S NP 2118 2633 m 2120 2627 l S NP 2116 2640 m 2118 2633 l S NP 2113 2647 m 2116 2640 l S NP 2111 2654 m 2113 2647 l S NP 2109 2661 m 2111 2654 l S NP 2107 2667 m 2109 2661 l S NP 2106 2674 m 2107 2667 l S NP 2104 2681 m 2106 2674 l S NP 2102 2688 m 2104 2681 l S NP 2100 2695 m 2102 2688 l S NP 2098 2702 m 2100 2695 l S NP 2097 2709 m 2098 2702 l S NP 2095 2717 m 2097 2709 l S NP 2093 2724 m 2095 2717 l S NP 2092 2731 m 2093 2724 l S NP 2090 2738 m 2092 2731 l S NP 2089 2745 m 2090 2738 l S NP 2087 2753 m 2089 2745 l S NP 2086 2760 m 2087 2753 l S NP 2084 2767 m 2086 2760 l S NP 2083 2774 m 2084 2767 l S NP 2082 2782 m 2083 2774 l S NP 2081 2789 m 2082 2782 l S NP 2080 2796 m 2081 2789 l S NP 2078 2804 m 2080 2796 l S NP 2077 2811 m 2078 2804 l S NP 2076 2819 m 2077 2811 l S NP 2075 2826 m 2076 2819 l S NP 2074 2834 m 2075 2826 l S NP 2074 2841 m 2074 2834 l S NP 2073 2849 m 2074 2841 l S NP 2072 2856 m 2073 2849 l S NP 2071 2864 m 2072 2856 l S NP 2070 2871 m 2071 2864 l S NP 2070 2879 m 2070 2871 l S NP 2069 2886 m 2070 2879 l S NP 2069 2894 m 2069 2886 l S NP 2068 2902 m 2069 2894 l S NP 2068 2909 m 2068 2902 l S NP 2067 2917 m 2068 2909 l S NP 2067 2924 m 2067 2917 l S NP 2067 2932 m 2067 2924 l S NP 2066 2940 m 2067 2932 l S NP 2066 2947 m 2066 2940 l S NP 2066 2955 m 2066 2947 l S NP 2066 2963 m 2066 2955 l S NP 2066 2970 m 2066 2963 l S NP 2066 2978 m 2066 2970 l S NP 2066 2985 m 2066 2978 l S NP 2066 2993 m 2066 2985 l S NP 2066 3001 m 2066 2993 l S NP 2066 3008 m 2066 3001 l S NP 2066 3016 m 2066 3008 l S NP 2067 3024 m 2066 3016 l S NP 2067 3031 m 2067 3024 l S NP 2067 3039 m 2067 3031 l S NP 2068 3046 m 2067 3039 l S NP 2068 3054 m 2068 3046 l S NP 2069 3062 m 2068 3054 l S NP 2069 3069 m 2069 3062 l S NP 2070 3077 m 2069 3069 l S NP 2071 3084 m 2070 3077 l S NP 2071 3092 m 2071 3084 l S NP 2072 3099 m 2071 3092 l S NP 2073 3107 m 2072 3099 l S NP 2074 3114 m 2073 3107 l S NP 2075 3122 m 2074 3114 l S NP 2076 3129 m 2075 3122 l S NP 2077 3137 m 2076 3129 l S NP 2078 3144 m 2077 3137 l S NP 2079 3152 m 2078 3144 l S NP 2080 3159 m 2079 3152 l S NP 2081 3166 m 2080 3159 l S NP 2083 3174 m 2081 3166 l S NP 2084 3181 m 2083 3174 l S NP 2085 3188 m 2084 3181 l S NP 2087 3196 m 2085 3188 l S NP 2088 3203 m 2087 3196 l S NP 2090 3210 m 2088 3203 l S NP 2091 3217 m 2090 3210 l S NP 2093 3224 m 2091 3217 l S NP 2094 3231 m 2093 3224 l S NP 2096 3239 m 2094 3231 l S NP 2098 3246 m 2096 3239 l S NP 2099 3253 m 2098 3246 l S NP 2101 3260 m 2099 3253 l S NP 2103 3267 m 2101 3260 l S NP 2105 3274 m 2103 3267 l S NP 2107 3280 m 2105 3274 l S NP 2109 3287 m 2107 3280 l S NP 2111 3294 m 2109 3287 l S NP 2113 3301 m 2111 3294 l S NP 2115 3308 m 2113 3301 l S NP 2117 3314 m 2115 3308 l S NP 2120 3321 m 2117 3314 l S NP 2122 3328 m 2120 3321 l S NP 2124 3334 m 2122 3328 l S NP 2127 3341 m 2124 3334 l S NP 2129 3347 m 2127 3341 l S NP 2131 3354 m 2129 3347 l S NP 2134 3360 m 2131 3354 l S NP 2136 3367 m 2134 3360 l S NP 2139 3373 m 2136 3367 l S NP 2142 3379 m 2139 3373 l S NP 2144 3385 m 2142 3379 l S NP 2147 3391 m 2144 3385 l S NP 2150 3398 m 2147 3391 l S NP 2152 3404 m 2150 3398 l S NP 2155 3410 m 2152 3404 l S NP 2158 3416 m 2155 3410 l S NP 2161 3421 m 2158 3416 l S NP 2164 3427 m 2161 3421 l S NP 2167 3433 m 2164 3427 l S NP 2170 3439 m 2167 3433 l S NP 2173 3445 m 2170 3439 l S NP 2176 3450 m 2173 3445 l S NP 2179 3456 m 2176 3450 l S NP 2182 3461 m 2179 3456 l S NP 2185 3467 m 2182 3461 l S NP 2189 3472 m 2185 3467 l S NP 2192 3477 m 2189 3472 l S NP 2195 3483 m 2192 3477 l S NP 2198 3488 m 2195 3483 l S NP 2202 3493 m 2198 3488 l S NP 2205 3498 m 2202 3493 l S NP 2209 3503 m 2205 3498 l S NP 2212 3508 m 2209 3503 l S NP 2216 3513 m 2212 3508 l S NP 2219 3518 m 2216 3513 l S NP 2223 3522 m 2219 3518 l S NP 2226 3527 m 2223 3522 l S NP 2230 3532 m 2226 3527 l S NP 2234 3536 m 2230 3532 l S NP 2237 3541 m 2234 3536 l S NP 2241 3545 m 2237 3541 l S NP 2245 3550 m 2241 3545 l S NP 2248 3554 m 2245 3550 l S NP 2252 3558 m 2248 3554 l S NP 2256 3562 m 2252 3558 l S NP 2260 3566 m 2256 3562 l S NP 2264 3570 m 2260 3566 l S NP 2268 3574 m 2264 3570 l S NP 2272 3578 m 2268 3574 l S NP 2276 3582 m 2272 3578 l S NP 2280 3585 m 2276 3582 l S NP 2284 3589 m 2280 3585 l S NP 2288 3592 m 2284 3589 l S NP 2292 3596 m 2288 3592 l S NP 2296 3599 m 2292 3596 l S NP 2300 3603 m 2296 3599 l S NP 2304 3606 m 2300 3603 l S NP 2308 3609 m 2304 3606 l S NP 2312 3612 m 2308 3609 l S NP 2317 3615 m 2312 3612 l S NP 2321 3618 m 2317 3615 l S NP 2325 3621 m 2321 3618 l S NP 2329 3623 m 2325 3621 l S NP 2333 3626 m 2329 3623 l S NP 2338 3629 m 2333 3626 l S NP 2342 3631 m 2338 3629 l S NP 2346 3633 m 2342 3631 l S NP 2351 3636 m 2346 3633 l S NP 2355 3638 m 2351 3636 l S NP 2359 3640 m 2355 3638 l S NP 2364 3642 m 2359 3640 l S NP 2368 3644 m 2364 3642 l S NP 2373 3646 m 2368 3644 l S NP 2377 3648 m 2373 3646 l S NP 2381 3650 m 2377 3648 l S NP 2386 3651 m 2381 3650 l S NP 2390 3653 m 2386 3651 l S NP 2395 3654 m 2390 3653 l S NP 2399 3656 m 2395 3654 l S NP 2404 3657 m 2399 3656 l S NP 2408 3658 m 2404 3657 l S NP 2413 3660 m 2408 3658 l S NP 2417 3661 m 2413 3660 l S NP 2422 3662 m 2417 3661 l S NP 2426 3662 m 2422 3662 l S NP 2431 3663 m 2426 3662 l S NP 2435 3664 m 2431 3663 l S NP 2440 3665 m 2435 3664 l S NP 2444 3665 m 2440 3665 l S NP 2449 3666 m 2444 3665 l S NP 2453 3666 m 2449 3666 l S NP 2458 3666 m 2453 3666 l S NP 2462 3666 m 2458 3666 l S NP 2467 3666 m 2462 3666 l S NP 2472 3666 m 2467 3666 l S NP 2476 3666 m 2472 3666 l S NP 2481 3666 m 2476 3666 l S NP 2485 3666 m 2481 3666 l S NP 2490 3666 m 2485 3666 l S NP 2494 3665 m 2490 3666 l S NP 2499 3665 m 2494 3665 l S NP 2503 3664 m 2499 3665 l S NP 2508 3663 m 2503 3664 l S NP 2512 3662 m 2508 3663 l S NP 2517 3662 m 2512 3662 l S NP 2521 3661 m 2517 3662 l S NP 2526 3660 m 2521 3661 l S NP 2530 3658 m 2526 3660 l S NP 2535 3657 m 2530 3658 l S NP 2539 3656 m 2535 3657 l S NP 2544 3654 m 2539 3656 l S NP 2548 3653 m 2544 3654 l S NP 2553 3651 m 2548 3653 l S NP 2557 3650 m 2553 3651 l S NP 2562 3648 m 2557 3650 l S NP 2566 3646 m 2562 3648 l S NP 2570 3644 m 2566 3646 l S NP 2575 3642 m 2570 3644 l S NP 2579 3640 m 2575 3642 l S NP 2584 3638 m 2579 3640 l S NP 2588 3636 m 2584 3638 l S NP 2592 3633 m 2588 3636 l S NP 2596 3631 m 2592 3633 l S NP 2601 3628 m 2596 3631 l S NP 2605 3626 m 2601 3628 l S NP 2609 3623 m 2605 3626 l S NP 2614 3620 m 2609 3623 l S NP 2618 3617 m 2614 3620 l S NP 2622 3614 m 2618 3617 l S NP 2626 3611 m 2622 3614 l S NP 2630 3608 m 2626 3611 l S NP 2634 3605 m 2630 3608 l S NP 2638 3602 m 2634 3605 l S NP 2643 3598 m 2638 3602 l S NP 2647 3595 m 2643 3598 l S NP 2651 3591 m 2647 3595 l S NP 2655 3588 m 2651 3591 l S NP 2659 3584 m 2655 3588 l S NP 2663 3580 m 2659 3584 l S NP 2667 3576 m 2663 3580 l S NP 2670 3572 m 2667 3576 l S NP 2674 3568 m 2670 3572 l S NP 2678 3564 m 2674 3568 l S NP 2682 3560 m 2678 3564 l S NP 2686 3556 m 2682 3560 l S NP 2689 3552 m 2686 3556 l S NP 2693 3547 m 2689 3552 l S NP 2697 3543 m 2693 3547 l S NP 2701 3538 m 2697 3543 l S NP 2704 3534 m 2701 3538 l S NP 2708 3529 m 2704 3534 l S NP 2711 3524 m 2708 3529 l S NP 2715 3519 m 2711 3524 l S NP 2718 3515 m 2715 3519 l S NP 2722 3510 m 2718 3515 l S NP 2725 3505 m 2722 3510 l S NP 2729 3499 m 2725 3505 l S NP 2732 3494 m 2729 3499 l S NP 2735 3489 m 2732 3494 l S NP 2738 3484 m 2735 3489 l S NP 2742 3478 m 2738 3484 l S NP 2745 3473 m 2742 3478 l S NP 2748 3467 m 2745 3473 l S NP 2751 3462 m 2748 3467 l S NP 2754 3456 m 2751 3462 l S NP 2757 3450 m 2754 3456 l S NP 2760 3445 m 2757 3450 l S NP 2763 3439 m 2760 3445 l S NP 2766 3433 m 2763 3439 l S NP 2769 3427 m 2766 3433 l S NP 2772 3421 m 2769 3427 l S NP 2775 3415 m 2772 3421 l S NP 2777 3409 m 2775 3415 l S NP 2780 3403 m 2777 3409 l S NP 2783 3396 m 2780 3403 l S NP 2785 3390 m 2783 3396 l S NP 2788 3384 m 2785 3390 l S NP 2790 3377 m 2788 3384 l S NP 2793 3371 m 2790 3377 l S NP 2795 3365 m 2793 3371 l S NP 2797 3358 m 2795 3365 l S NP 2800 3351 m 2797 3358 l S NP 2802 3345 m 2800 3351 l S NP 2804 3338 m 2802 3345 l S NP 2806 3331 m 2804 3338 l S NP 2808 3325 m 2806 3331 l S NP 2810 3318 m 2808 3325 l S NP 2812 3311 m 2810 3318 l S NP 2814 3304 m 2812 3311 l S NP 2816 3297 m 2814 3304 l S NP 2818 3290 m 2816 3297 l S NP 2820 3283 m 2818 3290 l S NP 2822 3276 m 2820 3283 l S NP 2823 3269 m 2822 3276 l S NP 2825 3262 m 2823 3269 l S NP 2826 3255 m 2825 3262 l S NP 2828 3247 m 2826 3255 l S NP 2829 3240 m 2828 3247 l S NP 2831 3233 m 2829 3240 l S NP 2832 3226 m 2831 3233 l S NP 2834 3218 m 2832 3226 l S NP 2835 3211 m 2834 3218 l S NP 2836 3204 m 2835 3211 l S NP 2837 3196 m 2836 3204 l S NP 2838 3189 m 2837 3196 l S NP 2839 3181 m 2838 3189 l S NP 2840 3174 m 2839 3181 l S NP 2841 3166 m 2840 3174 l S NP 2842 3159 m 2841 3166 l S NP 2843 3151 m 2842 3159 l S NP 2843 3144 m 2843 3151 l S NP 2844 3136 m 2843 3144 l S NP 2845 3129 m 2844 3136 l S NP 2845 3121 m 2845 3129 l S NP 2846 3114 m 2845 3121 l S NP 2846 3106 m 2846 3114 l S NP 2846 3098 m 2846 3106 l S NP 2847 3091 m 2846 3098 l S NP 2847 3083 m 2847 3091 l S NP 2847 3075 m 2847 3083 l S NP 2847 3068 m 2847 3075 l S NP 2847 3060 m 2847 3068 l S NP 2847 3053 m 2847 3060 l S NP 2847 3045 m 2847 3053 l S NP 2847 3037 m 2847 3045 l S NP 2847 3030 m 2847 3037 l S NP 2847 3022 m 2847 3030 l S NP 2847 3014 m 2847 3022 l S NP 2846 3007 m 2847 3014 l S NP 2846 2999 m 2846 3007 l S NP 2845 2992 m 2846 2999 l S NP 2845 2984 m 2845 2992 l S NP 2844 2976 m 2845 2984 l S NP 2843 2969 m 2844 2976 l S NP 2843 2961 m 2843 2969 l S NP 2842 2954 m 2843 2961 l S NP 2841 2946 m 2842 2954 l S NP 2840 2939 m 2841 2946 l S NP 2839 2931 m 2840 2939 l S NP 2838 2924 m 2839 2931 l S NP 2837 2917 m 2838 2924 l S NP 2836 2909 m 2837 2917 l S NP 2835 2902 m 2836 2909 l S NP 2834 2894 m 2835 2902 l S NP 2832 2887 m 2834 2894 l S NP 2831 2880 m 2832 2887 l S NP 2829 2873 m 2831 2880 l S NP 2828 2865 m 2829 2873 l S NP 2826 2858 m 2828 2865 l S NP 2825 2851 m 2826 2858 l S NP 2823 2844 m 2825 2851 l S NP 2821 2837 m 2823 2844 l S NP 2819 2830 m 2821 2837 l S NP 2817 2823 m 2819 2830 l S NP 2816 2816 m 2817 2823 l S NP 2814 2809 m 2816 2816 l S NP 2812 2802 m 2814 2809 l S NP 2809 2796 m 2812 2802 l S NP 2807 2789 m 2809 2796 l S NP 2805 2782 m 2807 2789 l S NP 2803 2776 m 2805 2782 l S NP 2800 2769 m 2803 2776 l S NP 2798 2763 m 2800 2769 l S NP 2796 2756 m 2798 2763 l S NP 2793 2750 m 2796 2756 l S NP 2791 2743 m 2793 2750 l S NP 2788 2737 m 2791 2743 l S NP 2785 2731 m 2788 2737 l S NP 2783 2725 m 2785 2731 l S NP 2780 2719 m 2783 2725 l S NP 2777 2713 m 2780 2719 l S NP 2774 2707 m 2777 2713 l S NP 2771 2701 m 2774 2707 l S NP 2768 2695 m 2771 2701 l S NP 2765 2690 m 2768 2695 l S NP 2762 2684 m 2765 2690 l S NP 2759 2679 m 2762 2684 l S NP 2756 2673 m 2759 2679 l S NP 2752 2668 m 2756 2673 l S NP 2749 2662 m 2752 2668 l S NP 2746 2657 m 2749 2662 l S NP 2743 2652 m 2746 2657 l S NP 2739 2647 m 2743 2652 l S NP 2736 2642 m 2739 2647 l S NP 2732 2637 m 2736 2642 l S NP 2729 2633 m 2732 2637 l S NP 2725 2628 m 2729 2633 l S NP 2721 2623 m 2725 2628 l S NP 2718 2619 m 2721 2623 l S NP 2714 2614 m 2718 2619 l S NP 2710 2610 m 2714 2614 l S NP 2706 2606 m 2710 2610 l S NP 2703 2602 m 2706 2606 l S NP 2699 2598 m 2703 2602 l S NP 2695 2594 m 2699 2598 l S NP 2691 2590 m 2695 2594 l S NP 2687 2587 m 2691 2590 l S NP 2683 2583 m 2687 2587 l S NP 2679 2580 m 2683 2583 l S NP 2675 2576 m 2679 2580 l S NP 2670 2573 m 2675 2576 l S NP 2666 2570 m 2670 2573 l S NP 2662 2567 m 2666 2570 l S NP 2658 2564 m 2662 2567 l S NP 2654 2561 m 2658 2564 l S NP 2649 2559 m 2654 2561 l S NP 2645 2556 m 2649 2559 l S NP 2641 2554 m 2645 2556 l S NP 2637 2551 m 2641 2554 l S NP 2632 2549 m 2637 2551 l S NP 2628 2547 m 2632 2549 l S NP 2623 2545 m 2628 2547 l S NP 2619 2543 m 2623 2545 l S NP 2615 2542 m 2619 2543 l S NP 2610 2540 m 2615 2542 l S NP 2606 2539 m 2610 2540 l S NP 2601 2537 m 2606 2539 l S NP 2597 2536 m 2601 2537 l S NP 2592 2535 m 2597 2536 l S NP 2588 2534 m 2592 2535 l S NP 2583 2533 m 2588 2534 l S NP 2579 2533 m 2583 2533 l S NP 2574 2532 m 2579 2533 l S NP 2570 2532 m 2574 2532 l S NP 2565 2531 m 2570 2532 l S NP 2560 2531 m 2565 2531 l S NP 2556 2531 m 2560 2531 l S NP 2551 2531 m 2556 2531 l S NP 2547 2532 m 2551 2531 l S NP 2542 2532 m 2547 2532 l S NP 2538 2533 m 2542 2532 l S NP 2533 2533 m 2538 2533 l S NP 2529 2534 m 2533 2533 l S NP 2524 2535 m 2529 2534 l S NP 2520 2536 m 2524 2535 l S NP 2515 2537 m 2520 2536 l S NP 2511 2539 m 2515 2537 l S NP 2506 2540 m 2511 2539 l S NP 2502 2542 m 2506 2540 l S NP 2497 2544 m 2502 2542 l S NP 2493 2546 m 2497 2544 l S NP 2489 2548 m 2493 2546 l S NP 2484 2550 m 2489 2548 l S NP 2480 2552 m 2484 2550 l S NP 2476 2555 m 2480 2552 l S NP 2471 2557 m 2476 2555 l S NP 2467 2560 m 2471 2557 l S NP 2463 2563 m 2467 2560 l S NP 2459 2566 m 2463 2563 l S NP 2455 2569 m 2459 2566 l S NP 2451 2573 m 2455 2569 l S NP 2447 2576 m 2451 2573 l S NP 2443 2580 m 2447 2576 l S NP 2439 2583 m 2443 2580 l S NP 2435 2587 m 2439 2583 l S NP 2431 2591 m 2435 2587 l S NP 2427 2596 m 2431 2591 l S NP 2423 2600 m 2427 2596 l S NP 2420 2605 m 2423 2600 l S NP 2416 2609 m 2420 2605 l S NP 2413 2614 m 2416 2609 l S NP 2409 2619 m 2413 2614 l S NP 2406 2624 m 2409 2619 l S NP 2402 2629 m 2406 2624 l S NP 2399 2635 m 2402 2629 l S NP 2396 2640 m 2399 2635 l S NP 2393 2646 m 2396 2640 l S NP 2390 2651 m 2393 2646 l S NP 2387 2657 m 2390 2651 l S NP 2384 2663 m 2387 2657 l S NP 2381 2669 m 2384 2663 l S NP 2379 2676 m 2381 2669 l S NP 2376 2682 m 2379 2676 l S NP 2374 2688 m 2376 2682 l S NP 2372 2695 m 2374 2688 l S NP 2369 2702 m 2372 2695 l S NP 2367 2709 m 2369 2702 l S NP 2365 2715 m 2367 2709 l S NP 2363 2722 m 2365 2715 l S NP 2362 2729 m 2363 2722 l S NP 2360 2737 m 2362 2729 l S NP 2359 2744 m 2360 2737 l S NP 2357 2751 m 2359 2744 l S NP 2356 2758 m 2357 2751 l S NP 2355 2766 m 2356 2758 l S NP 2354 2773 m 2355 2766 l S NP 2353 2781 m 2354 2773 l S NP 2353 2788 m 2353 2781 l S NP 2352 2796 m 2353 2788 l S NP 2352 2804 m 2352 2796 l S NP 2351 2811 m 2352 2804 l S NP 2351 2819 m 2351 2811 l S NP 2351 2827 m 2351 2819 l S NP 2351 2834 m 2351 2827 l S NP 2352 2842 m 2351 2834 l S NP 2352 2849 m 2352 2842 l S NP 2353 2857 m 2352 2849 l S NP 2353 2864 m 2353 2857 l S NP 2354 2872 m 2353 2864 l S NP 2355 2879 m 2354 2872 l S NP 2356 2887 m 2355 2879 l S NP 2358 2894 m 2356 2887 l S NP 2359 2901 m 2358 2894 l S NP 2361 2909 m 2359 2901 l S NP 2363 2916 m 2361 2909 l S NP 2364 2923 m 2363 2916 l S NP 2366 2929 m 2364 2923 l S NP 2369 2936 m 2366 2929 l S NP 2371 2943 m 2369 2936 l S NP 2373 2949 m 2371 2943 l S NP 2376 2956 m 2373 2949 l S NP 2378 2962 m 2376 2956 l S NP 2381 2968 m 2378 2962 l S NP 2384 2974 m 2381 2968 l S NP 2387 2980 m 2384 2974 l S NP 2390 2985 m 2387 2980 l S NP 2393 2990 m 2390 2985 l S NP 2397 2996 m 2393 2990 l S NP 2400 3001 m 2397 2996 l S NP 2404 3005 m 2400 3001 l S NP 2407 3010 m 2404 3005 l S NP 2411 3014 m 2407 3010 l S NP 2415 3019 m 2411 3014 l S NP 2419 3022 m 2415 3019 l S NP 2423 3026 m 2419 3022 l S NP 2427 3030 m 2423 3026 l S NP 2431 3033 m 2427 3030 l S NP 2435 3036 m 2431 3033 l S NP 2439 3038 m 2435 3036 l S NP 2444 3041 m 2439 3038 l S NP 2448 3043 m 2444 3041 l S NP 2452 3045 m 2448 3043 l S NP 2457 3047 m 2452 3045 l S NP 2461 3048 m 2457 3047 l S NP 2466 3049 m 2461 3048 l S NP 2470 3050 m 2466 3049 l S NP 2475 3051 m 2470 3050 l S NP 2479 3051 m 2475 3051 l S NP 2484 3051 m 2479 3051 l S NP 2488 3051 m 2484 3051 l S NP 2493 3050 m 2488 3051 l S NP 2497 3050 m 2493 3050 l S NP 2502 3049 m 2497 3050 l S NP 2506 3047 m 2502 3049 l S NP 2511 3046 m 2506 3047 l S NP 2515 3044 m 2511 3046 l S NP 2520 3041 m 2515 3044 l S NP 2524 3039 m 2520 3041 l S NP 2528 3036 m 2524 3039 l S NP 2532 3033 m 2528 3036 l S NP 2536 3030 m 2532 3033 l S NP 2541 3026 m 2536 3030 l S NP 2544 3023 m 2541 3026 l S NP 2548 3019 m 2544 3023 l S NP 2552 3014 m 2548 3019 l S NP 2556 3010 m 2552 3014 l S NP 2559 3005 m 2556 3010 l S NP 2563 3000 m 2559 3005 l S NP 2566 2995 m 2563 3000 l S NP 2569 2989 m 2566 2995 l S NP 2572 2983 m 2569 2989 l S NP 2575 2978 m 2572 2983 l S NP 2578 2972 m 2575 2978 l S NP 2580 2965 m 2578 2972 l S NP 2583 2959 m 2580 2965 l S NP 2585 2952 m 2583 2959 l S NP 2587 2946 m 2585 2952 l S NP 2589 2939 m 2587 2946 l S NP 2591 2932 m 2589 2939 l S NP 2593 2924 m 2591 2932 l S NP 2594 2917 m 2593 2924 l S NP 2595 2910 m 2594 2917 l S NP 2596 2902 m 2595 2910 l S NP 2597 2895 m 2596 2902 l S NP 2598 2887 m 2597 2895 l S NP 2598 2880 m 2598 2887 l S NP 2599 2872 m 2598 2880 l S NP 2599 2865 m 2599 2872 l S NP 2598 2857 m 2599 2865 l S NP 2598 2849 m 2598 2857 l S NP 2597 2842 m 2598 2849 l S NP 2597 2834 m 2597 2842 l S NP 2596 2827 m 2597 2834 l S NP 2594 2819 m 2596 2827 l S NP 2593 2812 m 2594 2819 l S NP 2591 2805 m 2593 2812 l S NP 2589 2798 m 2591 2805 l S NP 2587 2791 m 2589 2798 l S NP 2585 2785 m 2587 2791 l S NP 2582 2778 m 2585 2785 l S NP 2580 2772 m 2582 2778 l S NP 2577 2766 m 2580 2772 l S NP 2574 2761 m 2577 2766 l S NP 2570 2756 m 2574 2761 l S NP 2567 2751 m 2570 2756 l S NP 2563 2746 m 2567 2751 l S NP 2560 2742 m 2563 2746 l S NP 2556 2738 m 2560 2742 l S NP 2552 2735 m 2556 2738 l S NP 2547 2732 m 2552 2735 l S NP 2543 2729 m 2547 2732 l S NP 2539 2727 m 2543 2729 l S NP 2534 2726 m 2539 2727 l S NP 2530 2725 m 2534 2726 l S NP 2525 2725 m 2530 2725 l S NP 2521 2725 m 2525 2725 l S NP 2516 2727 m 2521 2725 l S NP 2512 2728 m 2516 2727 l S NP 2507 2731 m 2512 2728 l S NP 2503 2734 m 2507 2731 l S NP 2500 2738 m 2503 2734 l S medium setlinewidth (b=\2614.8\2612sin, E=12.5, \(u,v\)=position) dup 0 stringbbox 6230 exch sub exch 2 idiv 2500 exch sub exch m show medium setlinewidth NP 500 692 m 500 6068 l S thin setlinewidth NP 474 692 m 500 692 l S NP 474 814 m 500 814 l S NP 449 936 m 500 936 l S NP 474 1058 m 500 1058 l S NP 474 1181 m 500 1181 l S NP 474 1303 m 500 1303 l S NP 474 1425 m 500 1425 l S NP 449 1547 m 500 1547 l S NP 474 1669 m 500 1669 l S NP 474 1791 m 500 1791 l S NP 474 1914 m 500 1914 l S NP 474 2036 m 500 2036 l S NP 449 2158 m 500 2158 l S NP 474 2280 m 500 2280 l S NP 474 2402 m 500 2402 l S NP 474 2525 m 500 2525 l S NP 474 2647 m 500 2647 l S NP 449 2769 m 500 2769 l S NP 474 2891 m 500 2891 l S NP 474 3013 m 500 3013 l S NP 474 3136 m 500 3136 l S NP 474 3258 m 500 3258 l S NP 449 3380 m 500 3380 l S NP 474 3502 m 500 3502 l S NP 474 3624 m 500 3624 l S NP 474 3746 m 500 3746 l S NP 474 3869 m 500 3869 l S NP 449 3991 m 500 3991 l S NP 474 4113 m 500 4113 l S NP 474 4235 m 500 4235 l S NP 474 4357 m 500 4357 l S NP 474 4480 m 500 4480 l S NP 449 4602 m 500 4602 l S NP 474 4724 m 500 4724 l S NP 474 4846 m 500 4846 l S NP 474 4968 m 500 4968 l S NP 474 5091 m 500 5091 l S NP 449 5213 m 500 5213 l S NP 474 5335 m 500 5335 l S NP 474 5457 m 500 5457 l S NP 474 5579 m 500 5579 l S NP 474 5701 m 500 5701 l S NP 449 5824 m 500 5824 l S NP 474 5946 m 500 5946 l S NP 474 6068 m 500 6068 l S medium setlinewidth NP 500 692 m 4500 692 l S thin setlinewidth NP 681 654 m 681 692 l S NP 863 673 m 863 692 l S NP 1045 673 m 1045 692 l S NP 1227 673 m 1227 692 l S NP 1409 673 m 1409 692 l S NP 1590 654 m 1590 692 l S NP 1772 673 m 1772 692 l S NP 1954 673 m 1954 692 l S NP 2136 673 m 2136 692 l S NP 2318 673 m 2318 692 l S NP 2500 654 m 2500 692 l S NP 2681 673 m 2681 692 l S NP 2863 673 m 2863 692 l S NP 3045 673 m 3045 692 l S NP 3227 673 m 3227 692 l S NP 3409 654 m 3409 692 l S NP 3590 673 m 3590 692 l S NP 3772 673 m 3772 692 l S NP 3954 673 m 3954 692 l S NP 4136 673 m 4136 692 l S NP 4318 654 m 4318 692 l S NP 4500 673 m 4500 692 l S medium setlinewidth (\26160) dup 0 stringbbox 2 idiv 936 exch sub exch 427 exch sub exch m show (\26140) dup 0 stringbbox 2 idiv 1547 exch sub exch 427 exch sub exch m show (\26120) dup 0 stringbbox 2 idiv 2158 exch sub exch 427 exch sub exch m show (0) dup 0 stringbbox 2 idiv 2769 exch sub exch 427 exch sub exch m show (20) dup 0 stringbbox 2 idiv 3380 exch sub exch 427 exch sub exch m show (40) dup 0 stringbbox 2 idiv 3991 exch sub exch 427 exch sub exch m show (60) dup 0 stringbbox 2 idiv 4602 exch sub exch 427 exch sub exch m show (80) dup 0 stringbbox 2 idiv 5213 exch sub exch 427 exch sub exch m show (100) dup 0 stringbbox 2 idiv 5824 exch sub exch 427 exch sub exch m show (v\(t\)) dup 0 stringbbox 2 idiv 3380 exch sub exch 259 exch sub exch m show (\261100) dup 0 stringbbox 637 exch sub exch 2 idiv 681 exch sub exch m show (\26150) dup 0 stringbbox 637 exch sub exch 2 idiv 1590 exch sub exch m show (0) dup 0 stringbbox 637 exch sub exch 2 idiv 2500 exch sub exch m show (50) dup 0 stringbbox 637 exch sub exch 2 idiv 3409 exch sub exch m show (100) dup 0 stringbbox 637 exch sub exch 2 idiv 4318 exch sub exch m show (u\(t\)) dup 0 stringbbox 477 exch sub exch 2 idiv 2500 exch sub exch m show boundarythick setlinewidth % The following draws a box around the plot, % if the variable drawborder is true drawborder { /bd boundarythick 2 idiv def [] 0 setdash NP bd bd m bd 6923 bd sub l 5000 bd sub 6923 bd sub l 5000 bd sub bd l bd bd l S } if % end of if to draw the border showpage grestore end %%EOF %%EndDocument @endspecial 1963 5776 a(4)p eop %%Page: 5 8 5 7 bop 150 407 a Fv(1.2.2)112 b(Classical)36 b(comparison)h(dynamics)g (and)i(asymptotic)d(completeness)150 592 y Fz(Motiv)-5 b(ated)32 b(b)m(y)i(the)f(ab)s(o)m(v)m(e)g(considerations,)f(w)m(e)i (in)m(tro)s(duce)376 855 y Fu(\015)427 870 y Fs(1)494 855 y Fz(=)27 b Fu(\032)c Fr(\000)g Fu(@)821 870 y Ft(r)859 855 y Fu(S;)45 b(\015)1042 870 y Fs(2)1108 855 y Fz(=)1222 788 y Fu(l)24 b Fr(\000)f Fu(@)1426 803 y Ft(\022)1466 788 y Fu(S)p 1222 833 310 4 v 1353 924 a(r)1541 855 y(;)45 b(h)1669 870 y Ft(a)1738 855 y Fz(=)28 b Fu(h)22 b Fr(\000)2030 788 y Fz(1)p 2030 833 49 4 v 2030 924 a(2)2105 775 y Fl(\000)2151 855 y Fu(\015)2207 814 y Fs(2)2202 880 y(1)2268 855 y Fz(+)g Fu(\015)2422 814 y Fs(2)2417 880 y(2)2462 775 y Fl(\001)2524 855 y Fu(;)p 2595 774 57 4 v 44 w(h)2651 870 y Ft(a)2693 855 y Fz(\()p 2731 801 56 4 v Fu(x;)p 2830 801 52 4 v 17 w(\021)t Fz(\))27 b(=)h Fu(\034)34 b Fz(+)22 b Fu(h)3281 870 y Ft(a)3322 855 y Fu(:)227 b Fz(\(1.13\))150 1106 y(It)34 b(is)f(easy)i(to)e(see)i(that)e(the)i (Hamilton)30 b(equations)k(for)g(\()p Fu(r)m(;)17 b(\022)s Fz(\))33 b(obtained)g(from)g Fu(h)3185 1121 y Ft(a)3260 1106 y Fz(coincide)g(with)150 1227 y(the)g(system)h(in)e(\(1.11\).)44 b(Moreo)m(v)m(er,)35 b(the)e(dynamics)g(of)p 2245 1146 57 4 v 33 w Fu(h)g Fz(and)p 2524 1146 V 33 w Fu(h)2580 1242 y Ft(a)2655 1227 y Fz(coincide)f(on)h Fr(L)p Fz(,)g(hence)h (recall-)150 1347 y(ing)h(the)h(result)g(of)f(Lemma)g(3.14)g(men)m (tioned)g(ab)s(o)m(v)m(e,)j(the)e(dynamics)g(generated)h(b)m(y)p 3414 1266 V 36 w Fu(h)3471 1362 y Ft(a)3548 1347 y Fz(should)150 1468 y(appro)m(ximate)32 b(the)h(real)e(one.)296 1588 y(T)-8 b(o)33 b(\014x)g(notation,)f(denote)h(b)m(y)h Fv(q)1505 1603 y Ft(a;t)1620 1588 y Fz(=)27 b(\()p Fu(\032)1811 1603 y Ft(a)1853 1588 y Fz(\()p Fu(t)p Fz(\))p Fu(;)17 b(l)2037 1603 y Ft(a)2079 1588 y Fz(\()p Fu(t)p Fz(\)\))33 b(the)g(\\momen)m(ta")e(generated)i(b)m(y)h Fu(h)3588 1603 y Ft(a)3630 1588 y Fz(;)f(the)150 1708 y(\015o)m(w)e(generated)g (b)m(y)h Fu(h)983 1723 y Ft(a)1055 1708 y Fz(is)e(denoted)h(with)f Fv(V)1822 1723 y Ft(a;t)1939 1708 y Fz(and)h(acts)g(as)f Fv(V)2528 1723 y Ft(a;t)2615 1708 y Fz(\()p Fu(r)2697 1723 y Fs(1)2736 1708 y Fu(;)17 b(\022)2825 1723 y Fs(1)2864 1708 y Fz(;)g Fv(q)2967 1723 y Ft(a;)p Fs(1)3064 1708 y Fz(\))27 b(=)h(\()p Fv(v)3330 1723 y Ft(t)3360 1708 y Fz(;)17 b Fv(q)3463 1723 y Ft(a;t)3549 1708 y Fz(\).)43 b(The)150 1829 y(\015o)m(w)23 b(generated)g(b)m(y)h Fu(h)f Fz(is)f(denoted)h(b)m(y)g Fv(V)1638 1844 y Ft(t)1690 1829 y Fz(and)g Fv(V)1955 1844 y Ft(t)1984 1829 y Fz(\()p Fu(r)s Fz(\(1\))p Fu(;)17 b(\022)s Fz(\(1\);)g Fu(\032)p Fz(\(1\))p Fu(;)g(l)r Fz(\(1\)\))26 b(=)h(\()p Fu(r)s Fz(\()p Fu(t)p Fz(\))p Fu(;)17 b(\022)s Fz(\()p Fu(t)p Fz(\);)g Fu(\032)p Fz(\()p Fu(t)p Fz(\))p Fu(;)g(l)r Fz(\()p Fu(t)p Fz(\)\))150 1949 y(giv)m(es)33 b(the)g(classical)e (solutions)h(of)g(the)h(true)g(dynamics.)296 2069 y(Let)k Fv(W)591 2084 y Ft(a;t)712 2069 y Fz(=)d Fv(V)909 2028 y Fm(\000)p Fs(1)907 2092 y Ft(a;t)1039 2069 y Fz(denote)j(the)g(in)m (v)m(erse)h(\015o)m(w)f(\(explicitly)-8 b(,)36 b(writing)f(the)i (equations)g(for)f Fv(V)3739 2084 y Ft(a;t)150 2190 y Fz(as)e Fu(d)p Fv(x)p Fu(=dt)29 b Fz(=)h Fv(F)p Fz(\()p Fu(t;)17 b Fv(x)p Fz(\),)35 b(the)f(equations)g(for)f Fv(W)1870 2205 y Ft(a;t)1990 2190 y Fz(are)h Fu(d)p Fv(z)p Fu(=ds)29 b Fz(=)h Fr(\000)p Fv(F)p Fz(\()p Fu(t)24 b Fz(+)e(1)h Fr(\000)h Fu(s;)17 b Fv(z)p Fz(\))33 b(where)i Fv(z)p Fz(\(1\))29 b(=)150 2310 y Fv(V)235 2325 y Ft(a;t)321 2310 y Fz(\()p Fv(x)p Fz(\))h(and)g Fv(W)789 2325 y Ft(a;t)875 2310 y Fz(\()p Fv(z)p Fz(\(1\)\))d(=)h Fv(z)p Fz(\()p Fu(t)p Fz(\))f(=)h Fv(x)p Fz(\).)42 b(The)31 b(ob)m(vious)f(in)m (terpretation)e(of)h Fv(W)3104 2325 y Ft(a;t)3220 2310 y Fz(is)h(that)f(it)g(giv)m(es)150 2431 y(bac)m(k)34 b(the)f(initial)28 b(conditions)k(used)i(in)e(computing)f Fv(V)2198 2446 y Ft(a;t)2284 2431 y Fz(.)296 2551 y(Classical)g (asymptotic)h(completeness)i(w)m(ould)e(b)s(e)h(the)g(existence)h(of) 1533 2771 y(\012)1603 2786 y Fs(+)1690 2771 y Fz(:=)43 b(lim)1821 2831 y Ft(t)p Fm(!1)2004 2771 y Fv(W)2120 2786 y Ft(a;t)2229 2771 y Fr(\016)22 b Fv(V)2386 2786 y Ft(t)2415 2771 y Fu(;)1134 b Fz(\(1.14\))150 3018 y(and)24 b(the)g(limit)c(represen)m(ts)26 b(the)d(initial)d(data)j(one)h(should) g(put)f(in)m(to)g(the)h(dynamics)f Fv(V)3303 3033 y Ft(a;t)3413 3018 y Fz(generated)150 3138 y(b)m(y)33 b Fu(h)341 3153 y Ft(a)416 3138 y Fz(in)f(order)g(to)h(get)f(a)g(go)s(o)s(d)g(appro)m (ximation)e(to)i(an)m(y)i(true)f(orbit)e Fv(V)2869 3153 y Ft(t)2898 3138 y Fz(.)296 3259 y(Although)e(w)m(e)i(do)e(not)h(pro)m (v)m(e)h(the)f(existence)h(of)e(the)h(ab)s(o)m(v)m(e)h(limit,)26 b(w)m(e)31 b(do)e(pro)m(v)m(e)i(for)e(energies)150 3379 y(larger)i(than)i Fu(E)726 3394 y Ft(d)799 3379 y Fz(the)g(existence)h (of)1460 3599 y(\005\012)1603 3614 y Fs(+)1690 3599 y Fz(:=)43 b(lim)1821 3659 y Ft(t)p Fm(!1)2004 3599 y Fz(\005)p Fv(W)2193 3614 y Ft(a;t)2302 3599 y Fr(\016)22 b Fv(V)2459 3614 y Ft(t)2488 3599 y Fu(;)1061 b Fz(\(1.15\))150 3851 y(where)44 b(\005)f(pro)5 b(jects)44 b(on)f(the)h(con\014guration)e (space)i(of)e(\()p Fu(r)m(;)17 b(\022)s Fz(\)'s.)75 b(Note)43 b(for)g(the)g(\015o)m(w)h Fv(V)3517 3866 y Ft(a;t)3646 3851 y Fz(that)150 3972 y(the)d(equations)g(for)f(the)h (con\014guration)f(space)i(part)f Fv(v)2251 3987 y Ft(t)2321 3972 y Fz(in)f(\(1.11\))g(are)h(completely)e(decoupled)150 4092 y(from)32 b(the)i(momen)m(ta)e Fv(q)1035 4107 y Ft(a;t)1122 4092 y Fz(,)i(and)g(that)f Fv(W)1702 4107 y Ft(a;t)1822 4092 y Fz(consequen)m(tly)i(enjo)m(ys)g(the)f(same)f (prop)s(ert)m(y)-8 b(.)47 b(In)34 b(fact)150 4213 y(denoting)28 b(the)g(in)m(v)m(erse)i(of)d Fv(v)1194 4228 y Ft(t)1252 4213 y Fz(b)m(y)i Fv(w)1464 4228 y Ft(t)1494 4213 y Fz(,)g(w)m(e)g (obtain)e Fv(w)2069 4228 y Ft(t)2126 4213 y Fz(=)g(\005)p Fv(W)2418 4228 y Ft(a;t)2533 4213 y Fz(simplifying)e(the)j(righ)m(t)f (hand)i(side)150 4333 y(of)i(\(1.15\).)42 b(W)-8 b(e)32 b(denote)g(b)m(y)h(\()p Fu(r)1276 4348 y Fs(+)1335 4333 y Fu(;)17 b(\022)1424 4348 y Fs(+)1483 4333 y Fz(\))31 b(the)h(limit)c(in)j(\(1.15\))g(and)g(call)f(the)i(en)m(tries)g(the)g (asymptotic)150 4453 y(radius)k(and)h(angle,)g(resp)s(ectiv)m(ely)-8 b(.)57 b(In)m(tuitiv)m(ely)-8 b(,)38 b(when)f(put)g(in)m(to)f(the)i (direct)e(\015o)m(w)i Fv(v)3364 4468 y Ft(t)3430 4453 y Fz(this)e(limit)150 4574 y(pro)m(vides)41 b(us)h(with)e(a)g(go)s(o)s (d)g(appro)m(ximation)e(of)i(the)h(con\014guration)f(space)i(comp)s (onen)m(t)e(of)h(the)150 4694 y(true)33 b(orbit)e(at)i(time)e Fu(t)p Fz(.)43 b(See)34 b(Prop)s(osition)d(3.13)h(for)g(details.)150 4983 y Fp(1.3)135 b(Quan)l(tum)45 b(mec)l(hanics:)61 b(preliminaries)47 b(and)d(main)i(results)150 5168 y Fz(Motiv)-5 b(ated)22 b(b)m(y)i(its)f(classical)e(coun)m(terpart,)26 b(w)m(e)e(c)m(ho)s(ose)g(a)f(magnetic)f(v)m(ector)i(p)s(oten)m(tial)d (\(w)m(e)j(denote)150 5288 y Fv(x)k Fz(=)f(\()p Fu(r)m(;)17 b(\022)s Fz(\)\))1070 5408 y Fv(a)p Fz(\()p Fv(x)p Fz(\))28 b(=)g(\()p Fr(\000)17 b Fz(sin)f Fu(\022)s(;)h Fz(cos)g Fu(\022)s Fz(\))p Fu(b)p Fz(\()p Fu(\022)s Fz(\))p Fu(m)2234 5423 y Fs(+)2294 5408 y Fz(\()p Fu(r)s Fz(\))28 b Fr(2)g Fu(C)2616 5367 y Fm(1)2690 5408 y Fz(\()p Fn(R)2794 5367 y Fs(2)2840 5408 y Fz(\))p Fu(;)671 b Fz(\(1.16\))1963 5776 y(5)p eop %%Page: 6 9 6 8 bop 150 407 a Fz(where)33 b(0)28 b Fr(\024)g Fu(m)698 422 y Fs(+)785 407 y Fr(\024)g Fz(1)j(is)h(a)f(smo)s(oth)g(cut-o\013)h (function)f(equal)h(to)f(zero)h(if)f Fu(r)f Fr(\024)3038 368 y Fs(1)p 3038 384 36 4 v 3038 441 a(4)3115 407 y Fz(and)i(equal)f(to)h(one)150 527 y(if)j Fu(r)g Fr(\025)443 488 y Fs(1)p 443 504 V 443 562 a(2)488 527 y Fz(.)53 b(Notice)35 b(that)h Fv(a)g Fz(is)f(homogeneous)h(of)f(degree)i(zero)f (outside)g(the)g(unit)f(disc)h(while)f(the)150 648 y(corresp)s(onding)d (magnetic)g(\014eld)g(is)g(homogeneous)h(of)f(degree)h Fr(\000)p Fz(1.)296 768 y(The)h(classical)d(Hamiltonian)e(from)i (\(1.5\))h(no)m(w)h(b)s(ecomes)g(an)g(op)s(erator)673 1032 y Fu(H)i Fz(=)902 965 y(1)p 902 1009 49 4 v 902 1101 a(2)961 1032 y(\()p Fv(p)22 b Fr(\000)h Fv(a)p Fz(\))1275 991 y Fs(2)1398 1032 y Fz(=)1567 965 y(1)p 1567 1009 V 1567 1101 a(2)1626 1032 y Fv(p)1688 991 y Fs(2)1749 1032 y Fr(\000)1859 965 y Fz(1)p 1859 1009 V 1859 1101 a(2)1918 1032 y(\()p Fv(p)f Fr(\001)g Fv(a)g Fz(+)g Fv(a)h Fr(\001)f Fv(p)p Fz(\))g(+)2621 965 y(1)p 2621 1009 V 2621 1101 a(2)2680 1032 y Fu(m)2765 991 y Fs(2)2765 1057 y(+)2824 1032 y Fu(b)2865 991 y Fs(2)1398 1295 y Fz(=)83 b Fr(\000)1644 1228 y Fz(1)p 1644 1272 V 1644 1363 a(2)1736 1228 y Fu(@)1792 1191 y Fs(2)p 1713 1272 144 4 v 1713 1363 a Fu(@)5 b(r)1816 1334 y Fs(2)1888 1295 y Fr(\000)2021 1228 y Fz(1)p 1998 1272 96 4 v 1998 1363 a(2)p Fu(r)2136 1228 y(@)p 2113 1272 104 4 v 2113 1363 a(@)g(r)2249 1295 y Fz(+)2357 1228 y(1)p 2357 1272 49 4 v 2357 1363 a(2)2432 1154 y Fl(\022)2515 1228 y Fu(L)p 2515 1272 67 4 v 2525 1363 a(r)2614 1295 y Fr(\000)22 b Fu(m)2798 1310 y Fs(+)2858 1295 y Fz(\()p Fu(r)s Fz(\))p Fu(b)p Fz(\()p Fu(\022)s Fz(\))3146 1154 y Fl(\023)3219 1177 y Fs(2)3275 1295 y Fu(;)274 b Fz(\(1.17\))150 1576 y(whic)m(h)37 b(is)g(essen)m(tially)f (self-adjoin)m(t)f(on)i Fu(C)1730 1540 y Fm(1)1723 1601 y Fs(0)1805 1576 y Fz(\()p Fn(R)1909 1540 y Fs(2)1954 1576 y Fz(\))g(with)f(the)i(domain)d Fu(H)2868 1540 y Fs(2)2907 1576 y Fz(\()p Fn(R)3011 1540 y Fs(2)3056 1576 y Fz(\))i(\(w)m(e)h(denoted)g(the)150 1697 y(angular)31 b(momen)m(tum)g Fu(L)d Fz(=)g Fr(\000)p Fu(i@)1373 1712 y Ft(\022)1413 1697 y Fz(\).)296 1817 y(W)-8 b(e)32 b(will)d(often)i (iden)m(tify)g Fu(L)1311 1781 y Fs(2)1351 1817 y Fz(\()p Fn(R)1454 1781 y Fs(2)1500 1817 y Fz(\))g(with)g Fu(L)1856 1781 y Fs(2)1896 1817 y Fz(\()p Fn(R)2000 1832 y Fs(+)2084 1817 y Fr(\002)20 b Fn(T)p Fz(\))34 b(through)e(the)f(unitary)g (transformation)1032 2037 y Fu(L)1098 1996 y Fs(2)1137 2037 y Fz(\()p Fn(R)1241 1996 y Fs(2)1287 2037 y Fz(\))c Fr(3)i Fu(f)11 b Fz(\()p Fu(r)m(;)17 b(\022)s Fz(\))27 b Fr(!)g Fu(r)1916 1996 y Fs(1)p Ft(=)p Fs(2)2026 2037 y Fu(f)11 b Fz(\()p Fu(r)m(;)17 b(\022)s Fz(\))27 b Fr(2)h Fu(L)2481 1996 y Fs(2)2521 2037 y Fz(\()p Fn(R)2625 2052 y Fs(+)2712 2037 y Fr(\002)23 b Fn(T)p Fz(\))p Fu(:)636 b Fz(\(1.18\))150 2257 y(As)33 b(an)g(op)s(erator)f(on)g Fu(L)1024 2221 y Fs(2)1064 2257 y Fz(\()p Fn(R)1168 2272 y Fs(+)1255 2257 y Fr(\002)23 b Fn(T)p Fz(\))35 b(the)e(Hamiltonian)c Fu(H)40 b Fz(tak)m(es)34 b(the)f(form)956 2551 y Fu(H)i Fz(=)27 b Fr(\000)1262 2484 y Fz(1)p 1262 2528 49 4 v 1262 2619 a(2)1355 2484 y Fu(@)1411 2447 y Fs(2)p 1331 2528 144 4 v 1331 2619 a Fu(@)5 b(r)1434 2591 y Fs(2)1507 2551 y Fr(\000)1659 2484 y Fz(1)p 1616 2528 135 4 v 1616 2619 a(8)p Fu(r)1712 2591 y Fs(2)1783 2551 y Fz(+)1891 2484 y(1)p 1891 2528 49 4 v 1891 2619 a(2)1966 2411 y Fl(\022)2040 2551 y Fr(\000)2134 2484 y Fu(i)p 2127 2528 47 4 v 2127 2619 a(r)2218 2484 y(@)p 2194 2528 106 4 v 2194 2619 a(@)g(\022)2331 2551 y Fr(\000)23 b Fu(m)2516 2566 y Fs(+)2575 2551 y Fz(\()p Fu(r)s Fz(\))p Fu(b)p Fz(\()p Fu(\022)s Fz(\))2863 2411 y Fl(\023)2936 2433 y Fs(2)2992 2551 y Fu(:)557 b Fz(\(1.19\))150 2823 y(W)-8 b(e)42 b(will)d(see)k(that)e(there)h(is)f(an)h(\\easy")g(Mourre)g (estimate)f(with)g(the)h(generator)f(of)g(dilations)150 2943 y Fu(A)28 b Fz(=)f(1)p Fu(=)p Fz(2\()p Fv(p)22 b Fr(\001)g Fv(x)g Fz(+)g Fv(x)g Fr(\001)g Fv(p)p Fz(\))33 b(as)g(conjugate)f(op)s(erator,)1405 3163 y Fu(i)p Fz([)p Fu(H)r(;)17 b(A)p Fz(])28 b Fr(\025)g Fu(H)i Fr(\000)23 b Fu(b)2077 3122 y Fs(2)2116 3163 y Fz(\()p Fu(\022)s Fz(\))p Fu(=)p Fz(2)f(+)g Fu(K)r(;)1006 b Fz(\(1.20\))150 3383 y(with)34 b Fu(K)41 b Fz(b)s(eing)33 b(relativ)m(ely)g(compact)h (to)g Fu(H)8 b Fz(,)34 b(see)h(\(2.2\))f(for)f(the)i(classical)d(coun)m (terpart.)49 b(Indeed,)150 3504 y(after)32 b(easy)i(computations)d (emplo)m(ying)h(p)s(olar)f(co)s(ordinates)h(one)g(obtains)1093 3724 y Fu(i)p Fz([)p Fv(p)22 b Fr(\001)g Fv(a)p Fu(;)17 b Fv(p)23 b Fr(\001)f Fv(x)p Fz(])27 b(=)h Fu(i)p Fz([)p Fv(p)23 b Fr(\001)f Fv(a)p Fu(;)17 b Fv(x)22 b Fr(\001)g Fv(p)p Fz(])28 b(=)f Fv(p)c Fr(\001)f Fz(\()p Fv(a)g Fz(+)g Fv(a)2782 3739 y Ft(c)2817 3724 y Fz(\))p Fu(;)150 3944 y Fz(where)33 b Fv(a)485 3959 y Ft(c)520 3944 y Fz(\()p Fv(x)p Fz(\))28 b(=)f(\(sin\()p Fu(\022)s Fz(\))p Fu(;)17 b Fr(\000)g Fz(cos\()p Fu(\022)s Fz(\)\))p Fu(b)p Fz(\()p Fu(\022)s Fz(\))p Fu(r)s(m)1795 3907 y Fm(0)1795 3968 y Fs(+)1855 3944 y Fz(\()p Fu(r)s Fz(\))31 b(is)g(smo)s(oth)g(and) h(compactly)f(supp)s(orted.)44 b(Then)685 4164 y Fu(i)p Fz([)p Fu(H)r(;)17 b(A)p Fz(])85 b(=)e Fv(p)1278 4122 y Fs(2)1340 4164 y Fr(\000)23 b Fz(\(1)p Fu(=)p Fz(2\))p Fv(p)e Fr(\001)h Fz(\()p Fv(a)h Fz(+)f Fv(a)2063 4179 y Ft(c)2098 4164 y Fz(\))g Fr(\000)h Fz(\(1)p Fu(=)p Fz(2\)\()p Fv(a)e Fz(+)h Fv(a)2746 4179 y Ft(c)2782 4164 y Fz(\))g Fr(\001)g Fv(p)g Fz(+)g Fv(a)h Fr(\001)e Fv(a)3254 4179 y Ft(c)1057 4368 y Fz(=)83 b Fu(H)30 b Fr(\000)1437 4301 y Fz(\()p Fv(a)22 b Fr(\000)h Fv(a)1705 4316 y Ft(c)1740 4301 y Fz(\))1778 4265 y Fs(2)p 1437 4345 381 4 v 1603 4436 a Fz(2)1849 4368 y(+)1957 4301 y(1)p 1957 4345 49 4 v 1957 4436 a(2)2016 4368 y(\()p Fv(p)f Fr(\000)h Fv(a)2292 4383 y Ft(c)2327 4368 y Fz(\))2365 4327 y Fs(2)1056 4616 y Fr(\025)83 b Fu(H)30 b Fr(\000)1437 4549 y Fz(\()p Fv(a)22 b Fr(\000)h Fv(a)1705 4564 y Ft(c)1740 4549 y Fz(\))1778 4512 y Fs(2)p 1437 4593 381 4 v 1603 4684 a Fz(2)1827 4616 y Fu(:)1722 b Fz(\(1.21\))150 4867 y(Th)m(us)34 b(w)m(e)g(obtain)d(\(1.20\).)296 4987 y(This)j(computation)f(indicates) g(that)h(ab)s(o)m(v)m(e)g Fu(E)2061 5002 y Ft(b)2126 4987 y Fz(=)c(max)16 b Fr(j)p Fu(b)p Fr(j)2527 4951 y Fs(2)2566 4987 y Fu(=)p Fz(2)34 b(things)f(are)h(somewhat)g(\\eas-)150 5108 y(ier",)j(just)g(as)f(they)i(are)f(in)e(the)i(classical)e(case)j (\(see)f(\(2.2\)\).)55 b(W)-8 b(e)37 b(will)e(see)i(in)f(Section)g(3)h (that)f(if)150 5228 y Fu(b)g(<)f Fz(0)i(is)g(not)g(constan)m(t,)i(w)m (e)f(can)g(go)e(b)s(elo)m(w)h Fu(E)1950 5243 y Ft(b)2022 5228 y Fz(do)m(wn)h(to)f(the)h(critical)c(energy)39 b Fu(E)3299 5243 y Ft(d)3376 5228 y Fz(b)m(y)f(using)f(a)150 5348 y(more)31 b(in)m(v)m(olv)m(ed)i(conjugate)f(op)s(erator;)f(notice) h(though)f(that)h Fu(E)2534 5363 y Ft(d)2603 5348 y Fz(=)27 b Fu(E)2778 5363 y Ft(b)2844 5348 y Fz(when)33 b Fu(b)g Fz(is)e(constan)m(t)i(\(see)150 5469 y(Prop)s(osition)e(2.10\).)43 b(F)-8 b(or)31 b(the)i(classical)e(v)m(ersion)j(of)e(this)g(conjugate)h (op)s(erator)e(see)j(\(2.33\).)1963 5776 y(6)p eop %%Page: 7 10 7 9 bop 296 407 a Fz(Hence,)35 b(according)d(to)h(Mourre)h(\(see)g ([M]\),)g(the)f(in)m(terv)-5 b(al)32 b(\()p Fu(E)2587 422 y Ft(d)2628 407 y Fu(;)17 b Fr(1)p Fz(\))32 b(is)h(a)f(subset)j(of) e(the)g(abso-)150 527 y(lutely)e(con)m(tin)m(uous)h(sp)s(ectrum)g(and)f (do)s(es)h(not)f(con)m(tain)g(singular)f(con)m(tin)m(uous)i(sp)s (ectrum.)44 b(P)m(ossi-)150 648 y(ble)34 b(em)m(b)s(edded)g(eigen)m(v) -5 b(alues)34 b(in)g(this)f(in)m(terv)-5 b(al)33 b(are)h(discrete)g (and)g(ma)m(y)g(at)f(most)h(accum)m(ulate)f(at)150 768 y Fu(E)222 783 y Ft(d)263 768 y Fz(.)296 888 y(W)-8 b(e)32 b(in)m(tro)s(duce)f(a)g(comparison)f(dynamics)i(roughly)e(generated)i (b)m(y)h(the)e(quan)m(tization)g(of)g(the)150 1009 y(sym)m(b)s(ol)j Fu(h)542 1024 y Ft(a)619 1009 y Fz(of)h(\(1.13\).)50 b(A)35 b(similar)d(appro)m(ximate)i(dynamics)h(w)m(as)h(used)h(in)d ([H-S].)h(This)g(t)m(yp)s(e)h(of)150 1129 y(dynamics)27 b(is)h(motiv)-5 b(ated)26 b(b)m(y)i(a)g(related)f(one)h(in)m(tro)s (duced)g(b)m(y)g(Y)-8 b(afaev)28 b(\([Y]\);)g(see)h(also)e([D-G1])f (and)150 1249 y([R-Y].)32 b(W)-8 b(e)33 b(can)g(de\014ne)h(a)e(family)e (of)i(isometries)862 1469 y Fu(L)928 1428 y Fs(2)968 1469 y Fz(\(\()p Fu(E)1116 1484 y Ft(d)1157 1469 y Fu(;)17 b Fr(1)p Fz(\))k Fr(\002)i Fn(T)p Fz(\))31 b Fr(3)d Fu(f)38 b Fr(7!)27 b Fu(U)1965 1484 y Fs(0)2005 1469 y Fz(\()p Fu(t)p Fz(\))p Fu(f)39 b Fr(2)28 b Fu(L)2363 1428 y Fs(2)2402 1469 y Fz(\()p Fn(R)2506 1484 y Fs(+)2594 1469 y Fr(\002)22 b Fn(T)p Fz(\))p Fu(;)48 b(t)28 b Fr(\025)g Fz(1)p Fu(;)150 1689 y Fz(b)m(y)33 b(\(see)h(\(3.56\))e(and)h(\(3.57\)\))526 1923 y([)p Fu(U)619 1938 y Fs(0)659 1923 y Fz(\()p Fu(t)p Fz(\))p Fu(f)11 b Fz(]\()p Fu(r)m(;)17 b(\022)s Fz(\))28 b(=)f Fu(e)1241 1882 y Ft(iS)t Fs(\()p Ft(t;r)n(;\022)r Fs(\))1501 1923 y Fu(J)1564 1872 y Fs(1)p Ft(=)p Fs(2)1555 1946 y Ft(t)1674 1923 y Fz(\()p Fu(r)m(;)17 b(\022)s Fz(\))p Fu(f)27 b Fz(\()p Fr(\000)p Fz(\()p Fu(@)2162 1938 y Ft(t)2193 1923 y Fu(S)6 b Fz(\)\(1)p Fu(;)17 b Fv(w)2509 1938 y Ft(t)2537 1923 y Fz(\()p Fu(r)m(;)g(\022)s Fz(\)\))p Fu(;)g(\022)2873 1938 y Fs(1)2913 1923 y Fz(\()p Fu(t;)g(r)m(;)g(\022)s Fz(\)\))f Fu(;)294 b Fz(\(1.22\))150 2143 y(where)25 b Fu(J)477 2158 y Ft(t)530 2143 y Fz(is)e(the)h (Jacobian)f(determinan)m(t)g(arising)f(from)g(the)i(v)-5 b(arious)23 b(c)m(hanges)i(of)e(v)-5 b(ariables)22 b(whic)m(h)150 2263 y(mak)m(es)33 b Fu(U)509 2278 y Fs(0)549 2263 y Fz(\()p Fu(t)p Fz(\))f(an)h(isometry)-8 b(.)296 2384 y(With)32 b(this)h(comparison)e(dynamics)i(w)m(e)g(ha)m(v)m(e)h (existence)h(of)d(the)h(direct)f(w)m(a)m(v)m(e)j(op)s(erator)d(\(de-) 150 2504 y(noted)d(b)m(y)h(\012)619 2468 y Ft(d)619 2529 y Fs(+)678 2504 y Fz(\))f(and)g(completeness)h(\(see)g(the)f(remark)f (after)h(the)g(statemen)m(t)h(of)e(Theorem)h(4.2)f(for)150 2625 y(a)k(more)g(general)g(result)h(including)d(short-range)j(p)s (erturbations\):)150 2828 y Fv(Theorem)49 b(1.1.)e FA(Denote)c(by)h Fr(H)1422 2843 y Ft(E)1474 2855 y Fq(d)1559 2828 y Fz(:=)h Fv(1)1763 2844 y Fs(\()p Ft(E)1842 2856 y Fq(d)1878 2844 y Ft(;)p Fm(1)p Fs(\))p Fm(n)p Ft(\033)2071 2852 y Fq(pp)2140 2844 y Fs(\()p Ft(H)5 b Fs(\))2262 2828 y Fz(\()p Fu(H)j Fz(\))p Fu(L)2493 2792 y Fs(2)2532 2828 y Fz(\()p Fn(R)2636 2843 y Fs(+)2730 2828 y Fr(\002)29 b Fn(T)p Fz(\))p Fu(:)48 b FA(Then)43 b(the)h(fol)5 b(lowing)150 2948 y(limits)34 b(exist)h(and)f(de\014ne)g(unitary)i(op)-5 b(er)g(ators)34 b(which)g(ar)-5 b(e)35 b(mutual)5 b(ly)35 b(inverse)833 3168 y Fz(\012)903 3127 y Ft(d)903 3193 y Fs(+)1046 3168 y Fz(=)83 b Fu(s)22 b Fr(\000)38 b Fz(lim)1372 3228 y Ft(t)p Fm(!1)1555 3168 y Fu(e)1600 3127 y Ft(itH)1717 3168 y Fu(U)1783 3183 y Fs(0)1823 3168 y Fz(\()p Fu(t)p Fz(\))28 b(:)g Fu(L)2083 3127 y Fs(2)2122 3168 y Fz(\(\()p Fu(E)2270 3183 y Ft(d)2311 3168 y Fu(;)17 b Fr(1)p Fz(\))k Fr(\002)i Fn(T)p Fz(\))31 b Fr(7!)c(H)2957 3183 y Ft(E)3009 3195 y Fq(d)3050 3168 y Fu(;)833 3358 y Fz(\012)903 3373 y Fs(+)1046 3358 y Fz(=)83 b Fu(s)22 b Fr(\000)38 b Fz(lim)1372 3418 y Ft(t)p Fm(!1)1555 3358 y Fu(U)1631 3317 y Fm(\003)1621 3382 y Fs(0)1672 3358 y Fz(\()p Fu(t)p Fz(\))p Fu(e)1828 3317 y Fm(\000)p Ft(itH)2027 3358 y Fz(:)28 b Fr(H)2166 3373 y Ft(E)2218 3385 y Fq(d)2286 3358 y Fr(7!)f Fu(L)2479 3317 y Fs(2)2519 3358 y Fz(\(\()p Fu(E)2667 3373 y Ft(d)2707 3358 y Fu(;)17 b Fr(1)p Fz(\))22 b Fr(\002)g Fn(T)p Fz(\))p Fu(:)438 b Fz(\(1.23\))296 3610 y(W)-8 b(e)33 b(ha)m(v)m(e)h(the)f (existence)h(of)e(the)h(asymptotic)f(observ)-5 b(ables)33 b(de\014ned)h(on)f Fr(H)3122 3625 y Ft(E)3174 3637 y Fq(d)3247 3610 y Fz(\(see)g(\(3.56\)\))1119 3830 y Fu(r)1163 3845 y Fs(+)1305 3830 y Fz(:=)83 b Fu(s:r)m(:)22 b Fr(\000)38 b Fz(lim)1754 3890 y Ft(t)p Fm(!1)1937 3830 y Fu(e)1982 3789 y Ft(itH)2099 3830 y Fu(M)27 b Fz(\()p Fu(r)2302 3845 y Fs(1)2342 3830 y Fz(\()p Fu(t;)17 b Fr(\001)p Fu(;)g Fr(\001)p Fz(\)\))e Fu(e)2695 3789 y Fm(\000)p Ft(itH)2867 3830 y Fu(;)1064 4024 y(e)1109 3983 y Ft(i\022)1166 3992 y Fj(+)1305 4024 y Fz(:=)83 b Fu(s:r)m(:)22 b Fr(\000)38 b Fz(lim)1754 4084 y Ft(t)p Fm(!1)1937 4024 y Fu(e)1982 3983 y Ft(itH)2099 4024 y Fu(M)2220 3943 y Fl(\000)2266 4024 y Fu(e)2311 3983 y Ft(i\022)2368 3992 y Fj(1)2403 3983 y Fs(\()p Ft(t;)p Fm(\001)p Ft(;)p Fm(\001)p Fs(\))2566 3943 y Fl(\001)2628 4024 y Fu(e)2673 3983 y Fm(\000)p Ft(itH)2845 4024 y Fu(;)704 b Fz(\(1.24\))150 4277 y(where)36 b(the)f(notation)f Fu(M)10 b Fz(\()p Fr(\001)p Fz(\))35 b(signi\014es)f(m)m(ultiplication)c(op)s(erator)35 b(and)g Fu(s:r)m(:)24 b Fr(\000)g Fz(lim)32 b(means)j(strong)150 4397 y(resolv)m(en)m(t)42 b(limit.)63 b(These)43 b(op)s(erators)d(can)h (b)s(e)f(expressed)k(in)39 b(terms)i(of)f(the)h(w)m(a)m(v)m(e)h(op)s (erators)e(of)150 4517 y(Theorem)f(1.1;)j(they)e(represen)m(t)h(quan)m (tum)e(analogs)f(of)g(the)i(classical)d(asymptotic)h(radius)h(and)150 4638 y(angle,)32 b Fu(r)476 4653 y Fs(+)567 4638 y Fz(and)h Fu(\022)802 4653 y Fs(+)862 4638 y Fz(,)f(discussed)i(previously)-8 b(.)44 b(F)-8 b(or)32 b(details,)f(see)j(Theorem)f(4.2.)296 4758 y(In)e(the)g(case)g(where)h Fu(b)c(<)f Fz(0)j(do)s(es)h(not)f(dep) s(end)i(on)e Fu(\022)j Fz(w)m(e)f(ha)m(v)m(e)f(the)g(result)f Fu(E)3125 4773 y Ft(d)3194 4758 y Fz(=)d Fu(E)3369 4773 y Ft(b)3431 4758 y Fz(=)h Fu(b)3576 4722 y Fs(2)3616 4758 y Fu(=)p Fz(2)i(in)150 4878 y(\(1.23\))i(and)g(\(1.24\),)g(cf.)44 b(the)33 b(Miller-Simon)c(result,)j(and)h(the)g(form)m(ula)e(for)h Fu(U)3054 4893 y Fs(0)3093 4878 y Fz(\()p Fu(t)p Fz(\))h(reads)512 5165 y([)p Fu(U)605 5180 y Fs(0)645 5165 y Fz(\()p Fu(t)p Fz(\))p Fu(f)11 b Fz(]\()p Fu(r)m(;)17 b(\022)s Fz(\))27 b(=)h(exp)1347 5025 y Fl(\032)1422 5165 y Fu(i)1465 5098 y(r)1512 5062 y Fs(2)p 1465 5142 87 4 v 1466 5233 a Fz(2)p Fu(t)1584 5165 y Fr(\000)22 b Fu(i)1726 5098 y(b)1767 5062 y Fs(2)1807 5098 y Fu(t)p 1726 5142 117 4 v 1760 5233 a Fz(2)1852 5025 y Fl(\033)1944 5019 y(r)p 2043 5019 95 4 v 2067 5098 a Fu(r)p 2053 5142 75 4 v 2053 5233 a(t)2088 5205 y Fs(2)2138 5165 y Fu(f)2213 5025 y Fl(\022)2315 5098 y Fu(r)2362 5062 y Fs(2)p 2297 5142 124 4 v 2297 5233 a Fz(2)p Fu(t)2381 5205 y Fs(2)2452 5165 y Fz(+)2560 5098 y Fu(b)2601 5062 y Fs(2)p 2560 5142 81 4 v 2576 5233 a Fz(2)2651 5165 y Fu(;)17 b(\022)25 b Fz(+)d Fu(b)2920 5098 y(t)p 2914 5142 47 4 v 2914 5233 a(r)2988 5165 y Fz(ln)o(\()p Fu(t)p Fz(\))3180 5025 y Fl(\023)3270 5165 y Fu(:)279 b Fz(\(1.25\))150 5460 y(Moreo)m(v)m(er)34 b(in)e(this)g(case)i(the)f(term)f Fu(R)q Fz(\()p Fu(E)6 b(;)1732 5434 y Fz(~)1725 5460 y Fu(\022)s Fz(\))32 b(in)g(\(1.12\))g (is)g(indeed)h(constan)m(t)g(in)3161 5434 y(~)3153 5460 y Fu(\022)s Fz(.)1963 5776 y(7)p eop %%Page: 8 11 8 10 bop 296 407 a Fz(Heuristically)-8 b(,)48 b(our)e(comparison)f (dynamics)h(mo)m(v)m(es)i(the)e(supp)s(ort)h(of)f(an)g(initial)d(state) j(of)150 527 y(su\016cien)m(tly)e(high)e(energy)j(along)d(the)h(in)m (tegral)f(curv)m(es)j(of)e(\(1.11\))g(whic)m(h)h(are)f(spirals)f(mo)m (ving)150 648 y(coun)m(ter-clo)s(c)m(kwise)23 b(to)e(in\014nit)m(y)h (with)f(the)i(radius)e(prop)s(ortional)e(to)j Fu(t)g Fz(and)g(the)h(angle)e(prop)s(ortional)150 768 y(to)41 b(ln)16 b Fu(t)p Fz(,)45 b(cf.)71 b(\(1.12\))41 b(\(see)i(also)e (\(3.65\))g(and)h(\(3.66\)\).)70 b(\(Clearly)41 b(this)h(picture)g(is)f (con\014rmed)h(b)m(y)150 888 y(\(1.25\))f(in)g(the)h(constan)m(t)g Fu(b)g Fz(case.\))71 b(Asymptotic)42 b(completeness)g(means)g(that)f (an)m(y)h(state)g(with)150 1009 y(high)37 b(enough)i(energy)g(can)f(b)s (e)h(though)m(t)f(of)f(\(asymptotically)f(in)i(time\))e(as)i(a)g(sup)s (erp)s(osition)f(of)150 1129 y(translates)32 b(along)g(these)h (logarithmic)c(spirals.)150 1418 y Fp(1.4)135 b(Gauge)52 b(co)l(v)-7 b(ariance)53 b(of)f(the)g(w)l(a)l(v)l(e)h(op)t(erators)g (and)f(unitarit)l(y)h(of)456 1567 y(the)46 b FF(S)p Fp(-matrix)150 1752 y Fz(Using)31 b(an)g(argumen)m(t)g(similar)d(to)j(the)h(one)f (whic)m(h)h(led)f(to)g(our)g(\\outgoing")f(w)m(a)m(v)m(e)j(op)s (erators)e(\(see)150 1872 y(\(1.23\)\),)g(w)m(e)h(can)f(also)f(giv)m(e) h(an)g(appro)m(ximate)f(dynamics)h(at)g(negativ)m(e)g(times)g(\(see)h (Section)f(9)f(for)150 1993 y(details\),)f(and)g(consequen)m(tly)i (de\014ne)f(some)f(\\incoming")d(w)m(a)m(v)m(e)31 b(op)s(erators)e (\(denoted)h(b)m(y)f(\012)3579 1957 y Ft(d)3579 2018 y Fm(\000)3668 1993 y Fz(and)150 2113 y(\012)220 2128 y Fm(\000)279 2113 y Fz(\).)44 b(It)31 b(can)h(b)s(e)g(sho)m(wn)h(that) e(\012)1377 2077 y Ft(d)1377 2138 y Fm(\000)1468 2113 y Fz(maps)g(unitarily)f Fu(L)2186 2077 y Fs(2)2225 2113 y Fz(\(\()p Fu(E)2373 2128 y Ft(d)2414 2113 y Fu(;)17 b Fr(1)p Fz(\))i Fr(\002)i Fn(T)p Fz(\))35 b(on)m(to)c Fr(H)3151 2128 y Ft(E)3203 2140 y Fq(d)3275 2113 y Fz(and)g(\012)3533 2128 y Fm(\000)3624 2113 y Fz(is)g(its)150 2234 y(in)m(v)m(erse.)296 2354 y(The)43 b(non-trivial)38 b(fact)k(that)f(needs)i(to)e(b)s(e)h (sho)m(wn)h(\(see)f(Prop)s(osition)e(9.2\))h(is)g(that)g(the)h Fu(E)3784 2369 y Ft(d)150 2474 y Fz(for)i(negativ)m(e)g(times)g(is)g (the)g(same)h(as)f(that)g(for)g(p)s(ositiv)m(e)g(times.)78 b(One)44 b(migh)m(t)f(feel)h(that)g(this)150 2595 y(follo)m(ws)31 b(from)h(time)f(rev)m(ersal)i(in)m(v)-5 b(ariance.)43 b(But)32 b(it)g(do)s(es)h(not)f(seem)h(to.)43 b(In)33 b(fact)g(time)e(in)m(v)-5 b(ariance)150 2715 y(is)40 b(di\013eren)m(t)h(with)g(a)f(magnetic)g(\014eld.)68 b(If)41 b(the)g(time)f(rev)m(ersed)j(orbit)d(is)g Fv(x)2982 2730 y Ft(r)3020 2715 y Fz(\()p Fu(t)p Fz(\))i(:=)g Fv(x)p Fz(\()p Fr(\000)p Fu(t)p Fz(\),)h(then)150 2836 y(the)37 b(time)f(rev)m(ersed)j(v)m(elo)s(cit)m(y)d(is)h Fv(v)1459 2851 y Ft(r)1497 2836 y Fz(\()p Fu(t)p Fz(\))e(:=)1816 2835 y(_)1781 2836 y Fv(x)1840 2851 y Ft(r)1878 2836 y Fz(\()p Fu(t)p Fz(\))g(=)g Fr(\000)2228 2835 y Fz(_)2212 2836 y Fv(x)p Fz(\()p Fr(\000)p Fu(t)p Fz(\).)57 b(If)37 b(the)g(force)g(is)f(giv)m(en)h(only)g(b)m(y)g(a)150 2956 y(scalar)h(p)s(oten)m(tial,)h(then)g(\()p Fv(x)1205 2971 y Ft(r)1243 2956 y Fu(;)17 b Fv(v)1346 2971 y Ft(r)1384 2956 y Fz(\))39 b(satisfy)g(Newton's)h(equations.)62 b(But)39 b(if)f(there)h(is)g(a)f(magnetic)150 3076 y(\014eld)32 b Fv(B)p Fz(\()p Fv(x)p Fz(\))h(this)f(also)f(needs)k(to)d(b)s(e)h(c)m (hanged)g(to)f Fr(\000)p Fv(B)p Fz(\()p Fv(x)p Fz(\).)296 3197 y(Hence)j(the)f(S-matrix)e(de\014ned)j(as)1619 3116 y Fl(\000)1665 3197 y Fz(\012)1735 3161 y Ft(d)1735 3221 y Fs(+)1794 3116 y Fl(\001)1840 3138 y Fm(\003)1896 3197 y Fz(\012)1966 3161 y Ft(d)1966 3221 y Fm(\000)2059 3197 y Fz(is)f(unitary)f(on)g Fu(L)2706 3161 y Fs(2)2746 3197 y Fz(\(\()p Fu(E)2894 3212 y Ft(d)2935 3197 y Fu(;)17 b Fr(1)p Fz(\))22 b Fr(\002)h Fn(T)p Fz(\).)50 b(Our)34 b(w)m(a)m(v)m(e)150 3317 y(op)s(erators)g(ha)m(v)m(e)h(a)f(simple)f (transformation)f(la)m(w)i(under)h(a)f(time)e(indep)s(enden)m(t)k (gauge)e(transfor-)150 3437 y(mation.)42 b(If)1042 3558 y Fv(a)28 b Fr(!)f Fv(a)c Fz(+)f Fr(r)p Fu(f)5 b(;)39 b(H)c Fr(!)28 b Fu(e)1917 3517 y Ft(if)1986 3558 y Fu(H)8 b(e)2120 3517 y Fm(\000)p Ft(if)2267 3558 y Fz(and)22 b Fu(S)33 b Fr(!)28 b Fu(S)g Fz(+)22 b Fu(f)5 b(;)150 3732 y Fz(then)1710 3853 y(\012)1780 3811 y Ft(d)1780 3877 y Fm(\006)1867 3853 y Fr(!)27 b Fu(e)2039 3811 y Ft(if)2109 3853 y Fz(\012)2179 3811 y Ft(d)2179 3877 y Fm(\006)2238 3853 y Fu(:)150 4027 y Fz(It)33 b(follo)m(ws)e(that)h (the)h(S-matrix)e(is)h(gauge)h(in)m(v)-5 b(arian)m(t.)296 4147 y(In)30 b(the)h(gauge)e(w)m(e)i(use)g(here)f(there)h(is)e(no)h (asymptotic)f(momen)m(tum)f(and)i(w)m(e)h(conjecture)g(that)150 4268 y(there)g(do)s(es)g(not)f(exist)g(a)g(gauge)g(where)h(an)f (asymptotic)g(momen)m(tum)e(exists.)44 b(This)30 b(w)m(ould)g(mean)150 4388 y(that)35 b(no)h(momen)m(tum)e(preserving)i(appro)m(ximate)f (dynamics)h(is)f(a)m(v)-5 b(ailable.)50 b(This)35 b(partially)e(mo-)150 4508 y(tiv)-5 b(ates)31 b(our)g(c)m(hoice)g(of)g(appro)m(ximate)f (dynamics)h(\(1.22\))g(whic)m(h)h(is)e(not)h(momen)m(tum)f(preserving.) 150 4629 y(F)-8 b(or)34 b(a)h(comparativ)m(e)f(discussion)i(of)e(v)-5 b(arious)35 b(t)m(yp)s(es)h(of)f(w)m(a)m(v)m(e)i(op)s(erators)e(as)g (used)h(in)e([H1],)i([I-K])150 4749 y(and)d([Y],)g(see)g([D-G1].)1963 5776 y(8)p eop %%Page: 9 12 9 11 bop 150 407 a Fw(2)161 b(Classical)50 b(mec)l(hanics:)68 b(the)47 b(globally)j(attracting)f(p)t(eri-)392 589 y(o)t(dic)54 b(orbit)150 808 y Fz(Our)28 b(main)f(in)m(terest)i(in)e(this)h(section) g(is)g(to)g(determine)g(the)h(so)f(called)f(\\scattering)h(energies")h (and)150 929 y(to)i(study)h(the)g(long)d(time)h(b)s(eha)m(vior)h(of)f (an)m(y)i(classical)e(orbit)g(corresp)s(onding)h(to)f(suc)m(h)j(an)e (energy)-8 b(.)150 1049 y(W)g(e)31 b(will)d(see)j(that)f Fu(E)934 1064 y Ft(b)996 1049 y Fz(=)e(max)16 b Fr(j)p Fu(b)p Fr(j)1395 1013 y Fs(2)1434 1049 y Fu(=)p Fz(2)30 b(is)g(a)g(threshold)g(ab)s(o)m(v)m(e)i(whic)m(h)e(things)g(are)h (\\simpler")d(while)150 1170 y(b)s(elo)m(w)k(it)g(they)h(are)g(\\more)f (complicated".)296 1290 y(W)-8 b(e)48 b(assume)g(that)g(the)g(orbits)f (sta)m(y)h(a)m(w)m(a)m(y)h(from)e(the)h(origin)d(at)i(all)f(p)s(ositiv) m(e)h(times,)j(i.e.)150 1410 y Fu(r)s Fz(\()p Fu(t)p Fz(\))35 b Fr(6)p Fz(=)h(0)g(for)h Fu(t)f Fr(\025)g Fz(0.)57 b(W)-8 b(e)37 b(assume)h(that)f Fu(b)f(<)f Fz(0;)k(in)e(this)f(section) i(w)m(e)g(tak)m(e)g(the)f(magnetic)f(\014eld)150 1531 y Fu(B)5 b Fz(\()p Fv(x)p Fz(\))28 b(=)f Fu(b)p Fz(\()p Fu(\022)s Fz(\))p Fu(=r)36 b Fz(for)c(all)e Fv(x)e Fr(6)p Fz(=)g(0,)k(cf.)44 b(Subsection)33 b(1.2.)296 1651 y(Let)h(us)f (\014rst)h(notice)f(that)g(w)m(e)h(alw)m(a)m(ys)g(ha)m(v)m(e)g(a)f (maximal)d(v)m(elo)s(cit)m(y)j(b)s(ound)g(as)h(an)f(immediate)150 1772 y(consequence)j(of)c(energy)h(conserv)-5 b(ation)33 b(\(1.10\))f(whic)m(h)h(giv)m(es)g Fr(j)15 b Fz(_)-42 b Fu(r)r Fr(j)27 b(\024)2720 1687 y(p)p 2803 1687 127 4 v 85 x Fz(2)p Fu(E)38 b Fz(and)33 b(yields)1458 1999 y Fu(r)s(=t)27 b Fr(\024)1721 1910 y(p)p 1804 1910 V 89 x Fz(2)p Fu(E)h Fz(+)22 b Fu(\017;)115 b(t)28 b Fr(\025)g Fu(T)2457 2014 y Ft(\017)2490 1999 y Fu(:)1107 b Fz(\(2.1\))150 2287 y Fp(2.1)135 b Fi(E)40 b(>)33 b(E)794 2305 y Fh(b)879 2287 y Fp(leads)45 b(to)h(an)f(\\easy")h(classical)h(Mourre)d(estimate) 150 2472 y Fz(Next)d(w)m(e)h(shall)d(sho)m(w)i(that)f(if)g Fu(E)47 b(>)40 b(E)1657 2487 y Ft(b)1732 2472 y Fz(then)h(not)g(only)e (do)s(es)i(the)g(particle)e(lea)m(v)m(e)i(the)g(origin,)150 2592 y(but)34 b(w)m(e)h(also)e(ha)m(v)m(e)j(a)e(p)s(ositiv)m(e)f(lo)m (w)m(er)h(b)s(ound)g(for)g(its)f(v)m(elo)s(cit)m(y)-8 b(.)48 b(More)34 b(precisely)-8 b(,)35 b(w)m(e)g(sho)m(w)g(that)165 2713 y(_)-42 b Fu(r)s Fz(\()p Fu(t)p Fz(\))27 b(=)h Fu(\032)p Fz(\()p Fu(t)p Fz(\))g Fr(\025)g Fu(\032)783 2728 y Fs(0)851 2713 y Fu(>)f Fz(0)32 b(if)g Fu(t)h Fz(is)f(su\016cien)m(tly)h(large)e (\(see)j(\(2.4\)\).)296 2833 y(T)-8 b(o)33 b(ac)m(hiev)m(e)h(that,)e (compute)h(the)g(time)e(deriv)-5 b(ativ)m(e)32 b(of)g Fu(\032r)k Fz(using)c(\(1.6\){\(1.10\))f(and)i(get)733 3042 y Fu(d)p 716 3086 86 4 v 716 3178 a(dt)812 3109 y Fz(\()p Fu(\032r)s Fz(\))27 b(=)47 b(_)-46 b Fu(\032r)25 b Fz(+)d Fu(\032)1383 3068 y Fs(2)1450 3109 y Fz(=)28 b Fu(\030)5 b Fz(\()p Fu(b)22 b Fz(+)g Fu(\030)5 b Fz(\))21 b(+)h(2)p Fu(E)28 b Fr(\000)23 b Fu(\030)2303 3068 y Fs(2)2369 3109 y Fz(=)28 b(2)p Fu(E)g 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Fu(;)117 b(t)28 b Fr(\025)g Fu(T)8 b(:)150 5286 y Fv(Pro)s(of.)78 b Fz(W)-8 b(e)45 b(already)e(kno)m(w)j(that)e(for)f Fu(t)i Fz(large)e(enough,)48 b(the)c(radial)e(v)m(elo)s(cit)m(y)i Fu(\032)h Fz(is)e(nonzero,)150 5407 y(hence)j Fr(j)p Fu(\030)5 b Fz(\()p Fu(t)p Fz(\))p Fr(j)46 b Fu(<)817 5322 y Fr(p)p 900 5322 127 4 v 85 x Fz(2)p Fu(E)6 b Fz(,)47 b(for)d(ev)m(ery)i Fu(t)i Fr(\025)g Fu(T)1796 5422 y Fs(1)1835 5407 y Fz(.)79 b(Fix)43 b Fu(\017)48 b(>)f Fz(0)d(and)g(assume)h(that)f(either)g Fu(\030)5 b Fz(\()p Fu(T)3634 5422 y Fs(1)3673 5407 y Fz(\))48 b Fr(2)150 5527 y Fz(\(max)16 b Fr(j)p Fu(b)p Fr(j)29 b Fz(+)f Fu(\017;)699 5443 y Fr(p)p 782 5443 V 84 x Fz(2)p Fu(E)6 b Fz(\))42 b(or)g Fu(\030)5 b Fz(\()p Fu(T)1261 5542 y Fs(1)1300 5527 y Fz(\))44 b Fr(2)g Fz(\()p Fr(\000)1607 5443 y(p)p 1690 5443 V 84 x Fz(2)p Fu(E)6 b(;)17 b Fz(min)e Fr(j)p Fu(b)p Fr(j)28 b(\000)i Fu(\017)p Fz(\).)72 b(W)-8 b(e)42 b(\014rst)h(pro)m(v)m(e)g(that)f(there)h (exists)1963 5776 y(9)p eop %%Page: 10 13 10 12 bop 150 407 a Fu(T)207 422 y Fs(2)274 407 y Fu(>)28 b(T)435 422 y Fs(1)504 407 y Fz(suc)m(h)j(that)e Fu(\030)5 b Fz(\()p Fu(T)1072 422 y Fs(2)1111 407 y Fz(\))28 b Fr(2)g Fz([min)15 b Fr(j)p Fu(b)p Fr(j)h(\000)g Fu(\017;)h Fz(max)g Fr(j)p Fu(b)p Fr(j)f Fz(+)g Fu(\017)p Fz(].)42 b(Indeed,)32 b(if)c(w)m(e)j(assume)f(the)g(con)m(trary)g(it)150 527 y(means)k(that)f(either)h Fu(\030)5 b Fz(\()p Fu(t)p Fz(\))29 b Fr(\025)h Fz(max)16 b Fr(j)p Fu(b)p Fr(j)23 b Fz(+)g Fu(\017)34 b Fz(or)f Fu(\030)5 b Fz(\()p Fu(t)p Fz(\))29 b Fr(\024)i Fz(min)15 b Fr(j)p Fu(b)p Fr(j)22 b(\000)i Fu(\017)34 b Fz(for)f(ev)m(ery)j Fu(t)30 b Fr(\025)g Fu(T)3251 542 y Fs(1)3290 527 y Fz(.)47 b(In)34 b(the)g(\014rst)150 648 y(situation,)d(\(2.3\))h(implies)e(that)j Fu(\030)5 b Fz(\()p Fu(t)p Fz(\))32 b(is)g(decreasing)h(and)1224 843 y Fu(d\030)p 1224 887 99 4 v 1230 978 a(dt)1332 910 y Fz(\()p Fu(t)p Fz(\))28 b Fr(\024)g(\000)1667 843 y Fu(\017)p 1663 887 47 4 v 1663 978 a(r)1720 820 y Fl(p)p 1820 820 508 4 v 90 x Fz(2)p Fu(E)g Fr(\000)22 b Fu(\030)2116 881 y Fs(2)2155 910 y Fz(\()p Fu(T)2250 925 y Fs(1)2290 910 y Fz(\))p Fu(;)114 b(t)28 b Fr(\025)g Fu(T)2694 925 y Fs(1)2734 910 y Fu(;)150 1154 y Fz(while)36 b(in)f(the)i(second)h (situation,)e(\(2.3\))g(implies)e(that)i Fu(\030)5 b Fz(\()p Fu(t)p Fz(\))36 b(is)g(increasing)g(and)h(if)e(w)m(e)i (de\014ne)h Fu(M)150 1275 y Fz(as)33 b(the)g(maxim)m(um)d(b)s(et)m(w)m (een)35 b Fu(\030)1318 1238 y Fs(2)1357 1275 y Fz(\()p Fu(T)1452 1290 y Fs(1)1491 1275 y Fz(\))e(and)f(\(min)15 b Fr(j)p Fu(b)p Fr(j)22 b(\000)h Fu(\017)p Fz(\))2264 1238 y Fs(2)2336 1275 y Fz(then)1348 1470 y Fu(d\030)p 1348 1514 99 4 v 1354 1605 a(dt)1457 1537 y Fz(\()p Fu(t)p Fz(\))k Fr(\025)1714 1470 y Fu(\017)p 1710 1514 47 4 v 1710 1605 a(r)1767 1452 y Fr(p)p 1850 1452 354 4 v 85 x Fz(2)p Fu(E)h Fr(\000)23 b Fu(M)10 b(;)115 b(t)28 b Fr(\025)g Fu(T)2570 1552 y Fs(1)2609 1537 y Fu(:)150 1781 y Fz(Using)j(\(2.1\),)h(w)m(e)h(see)g(that)e(in)g(b)s(oth)h(cases) h(the)f(v)-5 b(ariation)30 b(in)h Fu(\030)36 b Fz(is)31 b(logarithmic)e(in)i Fu(t)h Fz(so)g Fu(\030)k Fz(cannot)150 1902 y(b)s(e)d(b)s(ounded.)296 2022 y(Th)m(us)g(in)e(an)m(y)h(case)g(w) m(e)g(can)g(\014nd)g Fu(T)1614 2037 y Fs(2)1685 2022 y Fz(with)f Fu(\030)5 b Fz(\()p Fu(T)2049 2037 y Fs(2)2088 2022 y Fz(\))27 b Fr(2)i Fz([min)15 b Fr(j)p Fu(b)p Fr(j)k(\000)h Fu(\017;)d Fz(max)g Fr(j)p Fu(b)p Fr(j)i Fz(+)h Fu(\017)p Fz(].)43 b(Let)32 b(us)g(pro)m(v)m(e)150 2142 y(that)42 b Fu(\030)5 b Fz(\()p Fu(t)p Fz(\))42 b(will)e(sta)m(y)k(in)e(the)h (same)f(closed)h(in)m(terv)-5 b(al)41 b(for)h(all)e(times)i Fu(t)j Fr(\025)g Fu(T)3033 2157 y Fs(2)3072 2142 y Fz(.)74 b(Assume)43 b(on)g(the)150 2263 y(con)m(trary)28 b(that)g(there)g (exists)g Fu(T)1307 2278 y Fs(3)1375 2263 y Fu(>)f(T)1535 2278 y Fs(2)1603 2263 y Fz(suc)m(h)i(that)e Fu(\030)5 b Fz(\()p Fu(T)2167 2278 y Fs(3)2206 2263 y Fz(\))28 b Fu(>)f Fz(max)16 b Fr(j)p Fu(b)p Fr(j)c Fz(+)g Fu(\017)p Fz(.)42 b(Then)29 b(the)f(in)m(termediate)150 2383 y(v)-5 b(alue)32 b(theorem)g(giv)m(es)h(exactly)g(one)g(p)s(oin)m(t)f Fu(T)1847 2398 y Fs(4)1914 2383 y Fr(2)c Fz([)p Fu(T)2092 2398 y Fs(2)2132 2383 y Fu(;)17 b(T)2233 2398 y Fs(3)2272 2383 y Fz(\))33 b(where)812 2596 y Fu(\030)5 b Fz(\()p Fu(T)955 2611 y Fs(4)994 2596 y Fz(\))28 b(=)f(max)17 b Fr(j)p Fu(b)p Fr(j)22 b Fz(+)g Fu(\017;)114 b Fz(max)16 b Fr(j)p Fu(b)p Fr(j)22 b Fz(+)g Fu(\017)28 b(<)g(\030)5 b Fz(\()p Fu(t)p Fz(\))p Fu(;)114 b(T)2702 2611 y Fs(4)2769 2596 y Fu(<)28 b(t)f(<)h(T)3096 2611 y Fs(3)3136 2596 y Fu(:)150 2809 y Fz(The)38 b(mean)e(v)-5 b(alue)36 b(theorem)g(w)m (ould)h(lead)f(to)h(the)g(existence)h(of)e Fu(\034)46 b Fr(2)35 b Fz(\()p Fu(T)2871 2824 y Fs(4)2911 2809 y Fu(;)17 b(T)3012 2824 y Fs(3)3051 2809 y Fz(\))37 b(where)3422 2766 y Ft(d\030)p 3422 2786 70 4 v 3426 2844 a(dt)3502 2809 y Fz(\()p Fu(\034)11 b Fz(\))35 b Fu(>)f Fz(0)150 2942 y(whic)m(h)26 b(con)m(tradicts)f(\(2.3\))f(whic)m(h)i(sa)m(ys)g (that)1826 2898 y Ft(d\030)p 1826 2919 V 1830 2976 a(dt)1906 2942 y Fz(\()p Fu(\034)11 b Fz(\))25 b(has)h(the)f(same)g(sign)f(as)i Fr(\000)p Fu(\030)5 b Fz(\()p Fu(\034)11 b Fz(\))c(+)g Fr(j)p Fu(b)p Fr(j)26 b Fu(<)i Fr(\000)p Fu(\017)h(<)e Fz(0.)296 3062 y(In)37 b(a)f(similar)d(w)m(a)m(y)-8 b(,)39 b(one)d(pro)m(v)m(es)i(that)e Fu(\030)5 b Fz(\()p Fu(t)p Fz(\))36 b(cannot)h(b)s(ecome)f(less)h(than)f(min)15 b Fr(j)p Fu(b)p Fr(j)25 b(\000)g Fu(\017)36 b Fz(for)g(an)m(y)150 3183 y Fu(t)28 b Fr(\025)g Fu(T)375 3198 y Fs(2)447 3183 y Fz(and)33 b(w)m(e)g(are)g(done.)p 3754 3183 4 66 v 3758 3120 59 4 v 3758 3183 V 3816 3183 4 66 v 296 3421 a(W)-8 b(e)29 b(ma)m(y)f(no)m(w)h(conclude)g(that)f(w)m(e)h(also)f(ha)m (v)m(e)i(a)e(minimal)c(v)m(elo)s(cit)m(y)k(b)s(ound,)h(that)f(is)g(for) g(ev)m(ery)150 3542 y(su\016cien)m(tly)44 b(small)d Fu(\017)p Fz(,)46 b(there)e(exists)h Fu(T)59 b Fz(=)46 b Fu(T)1863 3557 y Ft(\017)1939 3542 y Fz(large)c(enough)i(suc)m(h)h(that)e (\(remem)m(b)s(er)g(that)g(w)m(e)150 3662 y(imp)s(osed)32 b(the)h(condition)e Fu(E)i(>)28 b(E)1410 3677 y Ft(b)1445 3662 y Fz(\))720 3886 y Fu(\032)p Fz(\()p Fu(E)6 b(;)17 b(t)p Fz(\))28 b(=)1134 3796 y Fl(p)p 1234 3796 569 4 v 90 x Fz(2)p Fu(E)g Fr(\000)23 b Fu(\030)1531 3857 y Fs(2)1570 3886 y Fz(\()p Fu(E)6 b(;)17 b(t)p Fz(\))27 b Fr(\025)1935 3796 y Fl(p)p 2035 3796 819 4 v 90 x Fz(2)p Fu(E)h Fr(\000)22 b Fz(\(max)17 b Fr(j)p Fu(b)p Fr(j)k Fz(+)h Fu(\017)p Fz(\))2813 3857 y Fs(2)2853 3886 y Fu(;)114 b(t)28 b Fr(\025)h Fu(T)8 b(:)369 b Fz(\(2.4\))296 4099 y(Another)39 b(imp)s(ortan)m(t)d(consequence)41 b(of)d(Lemma)f(2.1)g (is)h(that)g(the)g(transv)m(erse)j(v)m(elo)s(cit)m(y)c Fu(\030)5 b Fz(\()p Fu(t)p Fz(\))150 4219 y(ev)m(en)m(tually)34 b(will)e(b)s(e)i(p)s(ositiv)m(e.)46 b(Using)34 b(this)f(fact)h(w)m(e)h (ma)m(y)e(express)j(the)e(time)f(v)-5 b(ariable)32 b(from)h(the)150 4340 y(\014rst)26 b(equation)e(in)h(\(1.9\))f(as)h(a)g(function)g(of)f Fu(\022)k Fz(and)d(in)m(tro)s(duce)g(it)f(in)h(\(2.3\),)h(obtaining)d (one)i(equation)150 4460 y(for)32 b Fu(\030)5 b Fz(\()p Fu(\022)s Fz(\):)664 4664 y Fu(d\030)p 663 4708 99 4 v 663 4799 a(d\022)800 4731 y Fz(=)27 b Fr(\000)990 4664 y Fu(b)p Fz(\()p Fu(\022)s Fz(\))c(+)f Fu(\030)5 b Fz(\()p Fu(\022)s Fz(\))p 990 4708 458 4 v 1134 4799 a Fu(\030)g Fz(\()p Fu(\022)s Fz(\))1458 4641 y Fl(p)p 1557 4641 460 4 v 1557 4731 a Fz(2)p Fu(E)28 b Fr(\000)23 b Fu(\030)1854 4702 y Fs(2)1893 4731 y Fz(\()p Fu(\022)s Fz(\))p Fu(;)114 b(\022)31 b Fr(\025)d Fu(\022)2384 4746 y Fs(0)2424 4731 y Fu(;)114 b Fz(0)28 b Fu(<)f(\030)5 b Fz(\()p Fu(\022)2876 4746 y Fs(0)2916 4731 y Fz(\))27 b Fu(<)3085 4642 y Fr(p)p 3168 4642 127 4 v 89 x Fz(2)p Fu(E)6 b(:)302 b Fz(\(2.5\))296 4958 y(More)29 b(generally)-8 b(,)28 b(motiv)-5 b(ated)26 b(b)m(y)k(these)f(considerations)f(w)m(e)h(transform)e(\(1.7\))h(and)g (\(1.9\))g(in)m(to)150 5079 y(a)k(system)i(of)e(equations)h(with)f Fu(\022)k Fz(as)c(v)-5 b(ariable)31 b(and)i Fu(E)h(>)27 b Fz(0:)578 5209 y Fl(8)578 5298 y(<)578 5478 y(:)708 5283 y Fu(@)759 5298 y Ft(\022)798 5283 y Fu(\032)h Fz(=)g Fu(b)23 b Fz(+)f Fu(\030)708 5409 y(@)759 5424 y Ft(\022)798 5409 y Fu(\030)33 b Fz(=)27 b Fr(\000)1064 5366 y Ft(b)p Fs(+)p Ft(\030)p 1064 5386 119 4 v 1107 5444 a(\030)1193 5409 y Fu(\032)708 5541 y(\030)756 5505 y Fs(2)817 5541 y Fz(+)22 b Fu(\032)965 5505 y Fs(2)1032 5541 y Fz(=)28 b(2)p Fu(E)1331 5413 y(;)142 b(\032)p Fz(\()p Fu(\022)1633 5428 y Fs(0)1673 5413 y Fz(\))28 b Fr(2)g Fz(\()p Fr(\000)1948 5324 y(p)p 2031 5324 127 4 v 89 x Fz(2)p Fu(E)6 b(;)2202 5324 y Fr(p)p 2285 5324 V 89 x Fz(2)p Fu(E)f Fz(\))28 b(and)g Fu(\030)5 b Fz(\()p Fu(\022)2793 5428 y Fs(0)2832 5413 y Fz(\))28 b Fr(2)g Fz(\(0)p Fu(;)3123 5324 y Fr(p)p 3205 5324 V 3205 5413 a Fz(2)p Fu(E)6 b Fz(\))p Fu(:)227 b Fz(\(2.6\))1939 5776 y(10)p eop %%Page: 11 14 11 13 bop 296 407 a Fz(Notice)43 b(that)f(\(2.6\))g(is)h(deriv)m(ed)g (under)h(the)f(assumption)f(that)g Fu(\030)50 b(>)44 b Fz(0,)i(while)41 b(no)i(sign)f(as-)150 527 y(sumption)26 b(on)h Fu(\032)h Fz(is)f(imp)s(osed.)40 b(Although)27 b(\(2.6\))g(ha)m(v)m(e)h(maximal)c(solutions)i(that)h(are)h(not)f (globally)150 648 y(de\014ned)42 b(\(dep)s(ending)f(on)g(the)h(initial) 37 b(conditions\),)42 b(w)m(e)g(shall)d(only)i(b)s(e)g(in)m(terested)h (in)e(globally)150 768 y(de\014ned)c(in)d(fact)h(p)s(erio)s(dic)f (solutions.)47 b(As)35 b(w)m(e)g(will)d(see)k(shortly)e(the)g(ab)s(o)m (v)m(e)h(system)h(ma)m(y)e(admit)150 888 y(p)s(erio)s(dic)d(solutions)g (ev)m(en)k(for)d(some)g(energies)h Fu(E)h Fr(\024)28 b Fu(E)2199 903 y Ft(b)2234 888 y Fz(.)296 1009 y(Clearly)k(the)h (single)f(equation)979 1229 y Fu(@)1030 1244 y Ft(\022)1069 1229 y Fu(\032)c Fz(=)g Fu(b)23 b Fz(+)1413 1137 y Fl(p)p 1512 1137 339 4 v 1512 1229 a Fz(2)p Fu(E)28 b Fr(\000)23 b Fu(\032)1811 1200 y Fs(2)1851 1229 y Fu(;)141 b(\032)p Fz(\()p Fu(\022)2152 1244 y Fs(0)2193 1229 y Fz(\))27 b Fr(2)h Fz(\()p Fr(\000)2467 1140 y(p)p 2551 1140 127 4 v 2551 1229 a Fz(2)p Fu(E)6 b(;)2722 1140 y Fr(p)p 2804 1140 V 2804 1229 a Fz(2)p Fu(E)g Fz(\))p Fu(;)628 b Fz(\(2.7\))150 1449 y(is)30 b(equiv)-5 b(alen)m(t)31 b(to)f(\(2.6\))g(if)f(w)m(e)j(only)e(consider)h(solutions)f(to)g (\(2.7\))g(with)g Fu(\030)i Fz(:=)3055 1362 y Fl(p)p 3155 1362 339 4 v 87 x Fz(2)p Fu(E)c Fr(\000)22 b Fu(\032)3453 1420 y Fs(2)3524 1449 y Fz(strictly)150 1569 y(p)s(ositiv)m(e.)150 1858 y Fp(2.2)135 b(Existence)46 b(of)g(a)f(p)t(erio)t(dic)g(solution)g (to)h(\(2.5\))150 2043 y Fz(With)30 b(the)g(standing)g(h)m(yp)s (otheses)j Fu(E)h(>)27 b(E)1735 2058 y Ft(b)1800 2043 y Fz(and)k Fu(b)d(<)f Fz(0)j(w)m(e)i(lo)s(ok)d(at)h(\(2.5\))g(with)f (the)i(initial)c(angle)150 2163 y Fu(\022)195 2178 y Fs(0)269 2163 y Fz(c)m(hosen)35 b(to)f(b)s(e)g(zero.)49 b(F)-8 b(or)33 b(an)m(y)h Fu(a)d Fr(2)f Fz(\(0)p Fu(;)1756 2079 y Fr(p)p 1839 2079 127 4 v 84 x Fz(2)p Fu(E)6 b Fz(\),)34 b(denote)h(b)m(y)g Fu(\030)5 b Fz(\()p Fu(E)h(;)17 b(a)p Fz(;)g Fr(\001)p Fz(\))32 b(the)j(maximal)30 b(solution)150 2283 y(to)i(\(2.5\))g(with)g(sp)s(eci\014ed)i(initial)29 b(data)j Fu(\030)5 b Fz(\()p Fu(E)h(;)17 b(a)p Fz(;)g(0\))26 b(=)i Fu(a)p Fz(.)150 2487 y Fv(Lemma)44 b(2.2.)h FA(F)-7 b(or)39 b(every)h Fu(a)e Fr(2)g Fz([min)15 b Fr(j)p Fu(b)p Fr(j)p Fu(=)p Fz(2)p Fu(;)1861 2410 y Fr(p)p 1943 2410 305 4 v 1943 2487 a Fu(E)28 b Fz(+)22 b Fu(E)2213 2502 y Ft(b)2248 2487 y Fz(])p FA(,)41 b(the)g(maximal)d(solution)i(is)g (glob)-5 b(al)40 b(and)150 2607 y(stays)35 b(in)g(the)g(same)f (interval)g(for)h(al)5 b(l)34 b Fu(\022)d Fr(2)d Fn(R)1813 2622 y Fs(+)1878 2607 y FA(.)150 2811 y Fv(Pro)s(of.)51 b Fz(W)-8 b(e)36 b(reason)g(as)f(in)g(Lemma)f(2.1.)51 b(Let)36 b(us)f(assume)h(on)f(the)h(con)m(trary)g(that)f(there)h (exists)150 2931 y Fu(\022)195 2946 y Fs(1)269 2931 y Fu(>)f Fz(0)h(suc)m(h)i(that)f(either)f(0)f Fu(<)f(\030)5 b Fz(\()p Fu(E)h(;)17 b(a)p Fz(;)g Fu(\022)1727 2946 y Fs(1)1766 2931 y Fz(\))35 b Fu(<)f Fz(min)15 b Fr(j)p Fu(b)p Fr(j)p Fu(=)p Fz(2)36 b(or)h Fu(\030)5 b Fz(\()p Fu(E)h(;)17 b(a)p Fz(;)g Fu(\022)2831 2946 y Fs(1)2869 2931 y Fz(\))35 b Fu(>)3052 2854 y Fr(p)p 3135 2854 V 77 x Fu(E)29 b Fz(+)22 b Fu(E)3406 2946 y Ft(b)3440 2931 y Fz(.)56 b(Denote)150 3051 y(b)m(y)44 b Fu(\022)341 3066 y Fs(2)424 3051 y Fz(the)g(largest)f(argumen)m(t)g(in)f([0)p Fu(;)17 b(\022)1665 3066 y Fs(1)1705 3051 y Fz(\))43 b(where)i(w)m(e)f(ha)m(v)m(e)h(that)e Fu(\030)5 b Fz(\()p Fu(E)h(;)17 b(a)p Fz(;)g Fu(\022)3039 3066 y Fs(2)3078 3051 y Fz(\))46 b(=)g(min)15 b Fr(j)p Fu(b)p Fr(j)p Fu(=)p Fz(2)42 b(\(or)150 3095 y Fr(p)p 233 3095 V 77 x Fu(E)28 b Fz(+)22 b Fu(E)503 3187 y Ft(b)570 3172 y Fz(resp)s(ectiv)m(ely\).)45 b(Then)34 b(\(2.5\))e(implies)e(that)1087 3374 y Fu(@)5 b(\030)p 1087 3418 106 4 v 1087 3510 a(@)g(\022)1202 3441 y Fz(\()p Fu(E)h(;)17 b(a)p Fz(;)g Fu(x)p Fz(\))28 b Fu(>)f Fz(0)g(\()p Fu(<)h Fz(0\))p Fu(;)211 b Fz(\()p Fr(8)p Fz(\))28 b Fu(\022)2428 3456 y Fs(2)2496 3441 y Fu(<)g(x)g(<)f(\022)2831 3456 y Fs(1)2871 3441 y Fu(;)150 3692 y Fz(whic)m(h)33 b(clearly)f(con)m(tradicts)h Fu(\030)5 b Fz(\()p Fu(E)h(;)17 b(a)p Fz(;)g Fu(\022)1591 3707 y Fs(1)1629 3692 y Fz(\))28 b Fu(<)g Fz(\()p Fu(>)p Fz(\))p Fu(\030)5 b Fz(\()p Fu(E)h(;)17 b(a)p Fz(;)g Fu(\022)2299 3707 y Fs(2)2337 3692 y Fz(\).)p 3754 3692 4 66 v 3758 3630 59 4 v 3758 3692 V 3816 3692 4 66 v 296 3931 a(W)-8 b(e)33 b(ma)m(y)g(no)m(w)g(conclude)g(that)f(the)h(mapping)842 4151 y Fu(g)t Fz(\()p Fu(E)6 b Fz(;)17 b Fr(\001)p Fz(\))26 b(:)i([min)15 b Fr(j)p Fu(b)p Fr(j)p Fu(=)p Fz(2)p Fu(;)1645 4059 y Fl(p)p 1744 4059 305 4 v 1744 4151 a Fu(E)28 b Fz(+)22 b Fu(E)2014 4166 y Ft(b)2048 4151 y Fz(])28 b Fr(7!)f Fz([min)16 b Fr(j)p Fu(b)p Fr(j)p Fu(=)p Fz(2)p Fu(;)2676 4059 y Fl(p)p 2774 4059 V 2774 4151 a Fu(E)29 b Fz(+)22 b Fu(E)3045 4166 y Ft(b)3079 4151 y Fz(])p Fu(;)1549 4371 y(g)t Fz(\()p Fu(E)6 b Fz(;)17 b Fu(a)p Fz(\))26 b(=)i Fu(\030)5 b Fz(\()p Fu(E)h(;)17 b(a)p Fz(;)g(2)p Fu(\031)t Fz(\))150 4545 y(has)33 b(at)f(least)g(one)h (\014xed)h(p)s(oin)m(t)e(so)g(there)i(exists)f(at)f(least)g(one)h(2)p Fu(\031)t Fz(-p)s(erio)s(dic)d(solution.)150 4665 y Fv(Remark.)61 b Fz(An)39 b(analysis)f(similar)d(to)j(the)h(one)g(w)m(e)h(did)e(b)s (efore)h(sho)m(ws)h(that)e(for)g(an)m(y)h(c)m(hoice)g(of)150 4786 y Fu(a)201 4801 y Fs(1)290 4786 y Fr(2)49 b Fz(\(0)p Fu(;)17 b Fz(min)e Fr(j)p Fu(b)p Fr(j)p Fz(])45 b(and)g Fu(a)1137 4801 y Fs(2)1226 4786 y Fr(2)k Fz([max)16 b Fr(j)p Fu(b)p Fr(j)p Fu(;)1707 4701 y Fr(p)p 1790 4701 127 4 v 85 x Fz(2)p Fu(E)6 b Fz(\))45 b(w)m(e)h(again)d(ha)m(v)m(e)k (that)e(the)g(in)m(terv)-5 b(al)44 b([)p Fu(a)3513 4801 y Fs(1)3553 4786 y Fu(;)17 b(a)3648 4801 y Fs(2)3687 4786 y Fz(])45 b(is)150 4906 y(left)36 b(in)m(v)-5 b(arian)m(t)35 b(b)m(y)j(the)g(ab)s(o)m(v)m(e)f(mapping)f Fu(g)t Fz(.)55 b(In)37 b(fact,)h(the)f(next)h(result)f(sa)m(ys)h(there)g(is)e(only)h (one)150 5027 y(p)s(erio)s(dic)23 b(solution)g(whic)m(h)j(do)s(es)f (not)g(get)f(outside)h([min)15 b Fr(j)p Fu(b)p Fr(j)p Fu(;)i Fz(max)f Fr(j)p Fu(b)p Fr(j)p Fz(].)41 b(Moreo)m(v)m(er,)28 b(all)22 b(other)j(global)150 5147 y(\(non-p)s(erio)s(dic\))30 b(solutions)i(will)e(b)s(e)j(dra)m(wn)g(to)g(the)g(p)s(erio)s(dic)e (one.)296 5267 y(Denote)40 b(b)m(y)h Fu(\030)824 5282 y Ft(E)922 5267 y Fz(an)m(y)g(2)p Fu(\031)t Fz(-p)s(erio)s(dic)c (solution)h(to)h(the)h(equation)g(\(2.5\),)h(with)e(the)h(conditions) 150 5388 y Fu(\030)193 5403 y Ft(E)252 5388 y Fz(\()p Fu(\022)s Fz(\))28 b Fr(2)g Fz(\(0)p Fu(;)629 5303 y Fr(p)p 711 5303 V 711 5388 a Fz(2)p Fu(E)6 b Fz(\))33 b(and)f Fu(\022)f Fr(2)d Fz([0)p Fu(;)17 b Fr(1)p Fz(\).)1939 5776 y(11)p eop %%Page: 12 15 12 14 bop 150 407 a Fv(Lemma)37 b(2.3.)42 b FA(Ther)-5 b(e)34 b(is)h(exactly)g(one)f(such)h(solution,)f(and)g(it)h(ob)-5 b(eys)1292 627 y Fz(Ran\()p Fu(\030)1548 642 y Ft(E)1607 627 y Fz(\))28 b Fr(\022)g Fu(I)35 b Fz(:=)28 b([min)15 b Fr(j)p Fu(b)p Fr(j)p Fu(;)i Fz(max)f Fr(j)p Fu(b)p Fr(j)p Fz(])p Fu(:)150 847 y Fv(Pro)s(of.)87 b Fz(The)48 b(existence)h(of)d(a)h(solution)f Fu(\030)1833 862 y Ft(E)1939 847 y Fz(with)h(v)-5 b(alues)47 b(in)f Fu(I)55 b Fz(follo)m(ws)46 b(from)g(the)h(previous)150 967 y(considerations.)55 b(Consider)37 b(an)m(y)g(other)g(solution)e(to)h(the)h(same)f(equation) g(\(whic)m(h)h(ma)m(y)f(not)h(b)s(e)150 1088 y(p)s(erio)s(dic\),)j (starting)f(at)h Fu(\030)5 b Fz(\(0\))39 b Fr(2)h Fz(\(0)p Fu(;)1545 1003 y Fr(p)p 1628 1003 127 4 v 85 x Fz(2)p Fu(E)6 b Fz(\).)65 b(Assume)41 b(that)e Fu(\030)5 b Fz(\(0\))40 b Fr(6)p Fz(=)g Fu(\030)2844 1103 y Ft(E)2903 1088 y Fz(\(0\),)h(otherwise)g Fu(\030)5 b Fz(\()p Fu(\022)s Fz(\))39 b(=)150 1208 y Fu(\030)193 1223 y Ft(E)252 1208 y Fz(\()p Fu(\022)s Fz(\))33 b(ev)m(erywhere.)296 1328 y(If)j Fu(\032)447 1343 y Ft(E)507 1328 y Fz(\()p Fu(\022)s Fz(\))c(=)771 1243 y Fl(p)p 871 1243 475 4 v 85 x Fz(2)p Fu(E)c Fr(\000)23 b Fu(\030)1168 1294 y Fs(2)1163 1355 y Ft(E)1222 1328 y Fz(\()p Fu(\022)s Fz(\))35 b(and)h Fu(\032)c Fz(=)1765 1243 y Fl(p)p 1864 1243 460 4 v 1864 1328 a Fz(2)p Fu(E)c Fr(\000)23 b Fu(\030)2161 1300 y Fs(2)2200 1328 y Fz(\()p Fu(\022)s Fz(\))35 b(then)h(in)m(tro)s(duce)g Fu(F)46 b Fz(=)32 b Fu(\032)24 b Fr(\000)h Fu(\032)3464 1343 y Ft(E)3524 1328 y Fz(.)52 b(Since)150 1449 y(equation)32 b(\(2.5\))g(can)h(b)s(e)g(rewritten)g(as)1375 1651 y Fu(d)p 1351 1695 99 4 v 1351 1786 a(d\022)1460 1628 y Fl(p)p 1559 1628 460 4 v 1559 1718 a Fz(2)p Fu(E)28 b Fr(\000)23 b Fu(\030)1856 1689 y Fs(2)1895 1718 y Fz(\()p Fu(\022)s Fz(\))k(=)h Fu(b)p Fz(\()p Fu(\022)s Fz(\))23 b(+)f Fu(\030)5 b Fz(\()p Fu(\022)s Fz(\))p Fu(;)989 b Fz(\(2.8\))150 1956 y(w)m(e)34 b(get)958 2053 y Fu(dF)p 958 2097 128 4 v 972 2188 a(d\022)1123 2120 y Fz(=)28 b Fu(\030)f Fr(\000)22 b Fu(\030)1439 2135 y Ft(E)1526 2120 y Fz(=)28 b Fr(\000)p Fu(hF)14 b Fz(;)212 b Fu(h)28 b Fz(=)2281 2053 y Fu(\030)2324 2068 y Ft(E)2406 2053 y Fr(\000)22 b Fu(\030)p 2276 2097 282 4 v 2276 2188 a(\032)g Fr(\000)h Fu(\032)2498 2203 y Ft(E)2596 2120 y Fz(=)2709 2053 y Fu(\032)g Fz(+)f Fu(\032)2930 2068 y Ft(E)p 2709 2097 281 4 v 2714 2188 a Fu(\030)27 b Fz(+)22 b Fu(\030)2925 2203 y Ft(E)2999 2120 y Fu(:)598 b Fz(\(2.9\))150 2339 y(Notice)35 b(that)g(w)m(e)i(used)g(the)f(energy)g(conserv)-5 b(ation)36 b(in)f(order)g(to)g(obtain)g(the)h(second)h(expression)150 2459 y(for)32 b Fu(h)p Fz(.)296 2580 y(Then)i(b)m(y)g(in)m(tegration:) 1282 2863 y Fu(F)14 b Fz(\()p Fu(\022)s Fz(\))27 b(=)g Fu(F)14 b Fz(\(0\))j(exp)1997 2722 y Fl(\024)2050 2863 y Fr(\000)2144 2727 y Fl(Z)2243 2753 y Ft(\022)2199 2953 y Fs(0)2299 2863 y Fu(h)p Fz(\()p Fu(\036)p Fz(\))p Fu(d\036)2598 2722 y Fl(\025)2666 2863 y Fu(:)883 b Fz(\(2.10\))296 3134 y(Next)34 b(using)e(the)h(b)s(ound)888 3424 y Fu(h)27 b Fr(\025)1248 3357 y Fu(\032)1298 3372 y Ft(E)p 1087 3401 433 4 v 1087 3421 a Fr(p)p 1170 3421 127 4 v 84 x Fz(2)p Fu(E)h Fz(+)22 b Fu(\030)1460 3520 y Ft(E)1556 3424 y Fr(\025)1672 3270 y Fl(p)p 1771 3270 549 4 v 1771 3355 a Fz(2)p Fu(E)28 b Fr(\000)23 b Fz(max)16 b Fu(\030)2266 3321 y Fs(2)2261 3382 y Ft(E)p 1672 3401 649 4 v 1866 3505 a Fz(2)1915 3421 y Fr(p)p 1998 3421 127 4 v 84 x Fz(2)p Fu(E)2358 3424 y Fz(=:)27 b Fu(C)7 b Fz(\()p Fu(E)f(;)17 b(b)p Fz(\))28 b(\()p Fu(>)g Fz(0\))p Fu(;)150 3694 y Fz(w)m(e)34 b(get)1012 3814 y Fr(j)p Fu(F)14 b Fz(\()p Fu(\022)s Fz(\))p Fr(j)27 b(\024)h(j)p Fu(F)14 b Fz(\(0\))p Fr(j)j Fz(exp)o([)p Fr(\000)p Fu(C)7 b Fz(\()p Fu(E)f(;)17 b(b)p Fz(\))p Fu(\022)s Fz(])p Fu(;)213 b Fz(\()p Fr(8)p Fz(\))17 b Fu(\022)31 b Fr(\025)d Fz(0)p Fu(;)150 3989 y Fz(and)33 b(conclude)g(that)1521 4109 y(lim)1500 4172 y Ft(\022)r Fm(!1)1693 4109 y Fr(j)p Fu(\030)5 b Fz(\()p Fu(\022)s Fz(\))21 b Fr(\000)i Fu(\030)2057 4124 y Ft(E)2116 4109 y Fz(\()p Fu(\022)s Fz(\))p Fr(j)k Fz(=)h(0)p Fu(:)1101 b Fz(\(2.11\))150 4313 y(In)30 b(particular,)f(this)g(will)f(pro)m(v)m (e)j(the)f(lemma.)41 b(Indeed,)32 b(if)c(there)j(w)m(ere)g(a)f (di\013eren)m(t)g(p)s(erio)s(dic)e(orbit)160 4407 y(~)150 4434 y Fu(\030)193 4449 y Ft(E)285 4434 y Fz(w)m(e)33 b(w)m(ould)g(ha)m(v)m(e)860 4654 y Fr(j)p Fu(\030)931 4669 y Ft(E)990 4654 y Fz(\(2)p Fu(n\031)t Fz(\))21 b Fr(\000)1363 4627 y Fz(~)1353 4654 y Fu(\030)1396 4669 y Ft(E)1455 4654 y Fz(\(2)p Fu(n\031)t Fz(\))p Fr(j)27 b Fz(=)h Fr(j)p Fu(\030)1927 4669 y Ft(E)1986 4654 y Fz(\(0\))22 b Fr(\000)2242 4627 y Fz(~)2232 4654 y Fu(\030)2275 4669 y Ft(E)2334 4654 y Fz(\(0\))p Fr(j)27 b Fu(>)h Fz(0)p Fu(;)114 b Fr(8)p Fu(n)28 b Fr(2)g Fn(N)150 4874 y Fz(and)33 b(this)f(w)m(ould)g(con)m(tradict)h(\(2.11\).)p 3754 4874 4 66 v 3758 4811 59 4 v 3758 4874 V 3816 4874 4 66 v 1939 5776 a(12)p eop %%Page: 13 16 13 15 bop 150 407 a Fp(2.3)135 b(P)l(erio)t(dic)46 b(solutions)g(b)t (elo)l(w)f Fi(E)2017 425 y Fh(b)150 592 y Fz(This)37 b(is)g(the)h(\014rst)f(place)g(where)h(w)m(e)h(are)e(going)f(to)g(allo) m(w)g Fu(E)43 b Fz(to)37 b(go)g(b)s(elo)m(w)g Fu(E)3086 607 y Ft(b)3120 592 y Fz(.)57 b(First)36 b(w)m(e)j(pro)m(v)m(e)150 712 y(that)c(there)i(exists)f Fu(E)961 727 y Ft(d)1034 712 y Fr(2)d Fz([min)16 b Fr(j)p Fu(b)p Fr(j)1437 676 y Fs(2)1476 712 y Fu(=)p Fz(2)p Fu(;)h(E)1690 727 y Ft(b)1724 712 y Fz(])35 b(suc)m(h)i(that)f(\(2.6\))f(admits)f(global)f(p)s(erio)s (dic)h(solutions)150 832 y(for)e(ev)m(ery)i Fu(E)g(>)28 b(E)838 847 y Ft(d)878 832 y Fz(.)44 b(If)32 b Fu(b)h Fz(is)f(not)h(constan)m(t)g(then)h(w)m(e)f(sho)m(w)h(that)e Fu(E)2676 847 y Ft(d)2745 832 y Fr(2)c Fz(\(min)15 b Fr(j)p Fu(b)p Fr(j)3153 796 y Fs(2)3192 832 y Fu(=)p Fz(2)p Fu(;)i(E)3406 847 y Ft(b)3440 832 y Fz(\).)296 953 y(Fix)27 b Fu(E)538 968 y Fs(0)606 953 y Fu(>)g Fz(0)h(and)g (consider)g(\(2.7\))g(with)f(the)i(notation)d(extended)k(to)e(include)f (the)i(energy)g(and)150 1073 y(initial)g(v)-5 b(alue)32 b(of)g Fu(\032)1160 1282 y(@)1211 1297 y Ft(\022)1250 1282 y Fu(\032)p Fz(\()p Fu(E)1410 1297 y Fs(0)1450 1282 y Fu(;)17 b(a;)g(\022)s Fz(\))28 b(=)f Fu(b)p Fz(\()p Fu(\022)s Fz(\))c(+)2092 1192 y Fl(p)p 2191 1192 747 4 v 2191 1282 a Fz(2)p Fu(E)2312 1297 y Fs(0)2374 1282 y Fr(\000)g Fu(\032)2524 1254 y Fs(2)2563 1282 y Fz(\()p Fu(E)2673 1297 y Fs(0)2713 1282 y Fu(;)17 b(a;)g(\022)s Fz(\))p Fu(;)1160 1441 y(\032)p Fz(\()p Fu(E)1320 1456 y Fs(0)1360 1441 y Fu(;)g(a;)g Fz(0\))27 b(=)g Fu(a)h Fr(2)g Fz(\()p Fr(\000)2004 1349 y Fl(p)p 2104 1349 161 4 v 92 x Fz(2)p Fu(E)2225 1456 y Fs(0)2265 1441 y Fu(;)2309 1349 y Fl(p)p 2408 1349 V 2408 1441 a Fz(2)p Fu(E)2529 1456 y Fs(0)2568 1441 y Fz(\))p Fu(;)943 b Fz(\(2.12\))150 1650 y(or)32 b(equiv)-5 b(alen)m(tly)874 1922 y Fu(\032)p Fz(\()p Fu(E)1034 1937 y Fs(0)1073 1922 y Fu(;)17 b(a;)g(\022)s Fz(\))28 b(=)f Fu(a)c Fz(+)1601 1786 y Fl(Z)1700 1813 y Ft(\022)1656 2012 y Fs(0)1739 1922 y Fz(\()p Fu(b)p Fz(\()p Fu(\036)p Fz(\))g(+)2073 1832 y Fl(p)p 2172 1832 756 4 v 2172 1922 a Fz(2)p Fu(E)2293 1937 y Fs(0)2355 1922 y Fr(\000)f Fu(\032)2504 1893 y Fs(2)2544 1922 y Fz(\()p Fu(E)2654 1937 y Fs(0)2694 1922 y Fu(;)17 b(a;)g(\036)p Fz(\))o(\))p Fu(d\036:)150 2184 y Fv(Prop)s(osition)40 b(2.4.)j FA(Assume)38 b(that)h(for)f(some)f Fu(E)2030 2199 y Fs(0)2103 2184 y Fu(>)d Fz(0)k FA(the)g(e)-5 b(quation)37 b(\(2.12\))h(admits)f(a)h(glob)-5 b(al,)150 2304 y Fu(C)227 2268 y Fs(1)301 2304 y FA(and)35 b Fz(2)p Fu(\031)t FA(-p)-5 b(erio)g(dic)33 b(solution)i(denote)-5 b(d)34 b(by)h Fu(\032)1900 2319 y Ft(E)1952 2328 y Fj(0)1991 2304 y FA(.)44 b(Then)34 b(it)h(satis\014es)1189 2549 y Fu(c)27 b Fz(:=)h(inf)1431 2612 y Ft(\022)1524 2433 y Fl(q)p 1623 2433 548 4 v 1623 2549 a Fz(2)p Fu(E)1744 2564 y Fs(0)1806 2549 y Fr(\000)23 b Fu(\032)1956 2515 y Fs(2)1956 2576 y Ft(E)2008 2585 y Fj(0)2046 2549 y Fz(\()p Fu(\022)s Fz(\))28 b Fr(\025)g Fz(min)15 b Fr(j)p Fu(b)p Fr(j)28 b Fu(>)f Fz(0)p Fu(:)790 b Fz(\(2.13\))150 2788 y FA(If)34 b(in)h(addition)f(we)g(have)1166 3057 y Fu(I)8 b Fz(\()p Fu(E)1327 3072 y Fs(0)1366 3057 y Fu(;)17 b(\032)1460 3072 y Ft(E)1512 3081 y Fj(0)1550 3057 y Fz(\))28 b(:=)1747 2921 y Fl(Z)1846 2947 y Fs(2)p Ft(\031)1802 3147 y Fs(0)2146 2989 y Fu(\032)2196 3004 y Ft(E)2248 3013 y Fj(0)p 1955 3034 523 4 v 1955 3054 a Fl(q)p 2055 3054 423 4 v 110 x Fz(2)p Fu(E)2176 3179 y Fs(0)2237 3164 y Fr(\000)23 b Fu(\032)2387 3130 y Fs(2)2387 3191 y Ft(E)2439 3200 y Fj(0)2488 3057 y Fu(d')k(>)h Fz(0)p Fu(;)766 b Fz(\(2.14\))150 3401 y FA(then)43 b(ther)-5 b(e)43 b(exists)f Fu(\017)h(>)f Fz(0)h FA(such)g(that)g(for)f(every)h Fu(E)49 b Fr(2)43 b Fz(\()p Fu(E)2398 3416 y Fs(0)2465 3401 y Fr(\000)29 b Fu(\017;)17 b(E)2726 3416 y Fs(0)2794 3401 y Fz(+)28 b Fu(\017)p Fz(\))43 b FA(we)f(have)g(a)h(p)-5 b(erio)g(dic)150 3521 y(solution)35 b Fu(\032)569 3536 y Ft(E)663 3521 y FA(ob)-5 b(eying)35 b Fu(I)8 b Fz(\()p Fu(E)e(;)17 b(\032)1271 3536 y Ft(E)1330 3521 y Fz(\))28 b Fu(>)f Fz(0)35 b FA(as)f(in)h(\(2.14\).)150 3716 y Fv(Pro)s(of.)43 b Fz(Before)32 b(starting)f(the)h(pro)s(of,)f(let)g(us)h(explain)f(the) h(meaning)e(of)h(\(2.14\).)43 b(Assuming)31 b(that)150 3836 y(w)m(e)j(ha)m(v)m(e)g Fu(\032)569 3851 y Ft(E)628 3836 y Fz(,)f(de\014ne)h Fu(\030)1013 3851 y Ft(E)1100 3836 y Fz(:=)1230 3751 y Fl(p)p 1330 3751 359 4 v 85 x Fz(2)p Fu(E)28 b Fr(\000)23 b Fu(\032)1629 3802 y Fs(2)1629 3863 y Ft(E)1688 3836 y Fz(.)44 b(Consider)33 b(the)g(initial)c(v)-5 b(alue)32 b(problem)f(\()p Fu(t)d Fr(\025)g Fz(1\):)947 4056 y Fu(d)t Fz(~)-53 b Fu(r)p 947 4100 98 4 v 953 4191 a(dt)1082 4123 y Fz(=)28 b Fu(\032)1236 4138 y Ft(E)1296 4123 y Fz(\()1342 4097 y(~)1334 4123 y Fu(\022)s Fz(\))p Fu(;)1571 4056 y(d)1630 4029 y Fz(~)1622 4056 y Fu(\022)p 1571 4100 99 4 v 1577 4191 a(dt)1708 4123 y Fz(=)1821 4056 y Fu(\030)1864 4071 y Ft(E)1923 4056 y Fz(\()1969 4029 y(~)1961 4056 y Fu(\022)s Fz(\))p 1821 4100 227 4 v 1915 4191 a(~)-53 b Fu(r)2057 4123 y(;)114 b Fz(\()t(~)-53 b Fu(r)s Fz(\(1\))p Fu(;)2459 4097 y Fz(~)2452 4123 y Fu(\022)s Fz(\(1\)\))27 b(=)g(\(1)p Fu(;)17 b Fz(0\))p Fu(:)538 b Fz(\(2.15\))150 4358 y(It)41 b(is)g(easy)i(to)e(c)m(hec)m(k) j(that)d(at)g(least)g(for)g Fu(t)g Fz(close)h(to)f(1)g(the)h(ab)s(o)m (v)m(e)g(system)g(admits)e(a)h(solution)150 4478 y(\()t(~)-53 b Fu(r)s(;)286 4452 y Fz(~)279 4478 y Fu(\022)r Fz(\))38 b(whic)m(h)g(also)e(solv)m(es)j(the)e(Hamilton)e(equations)j (\(1.6\){\(1.9\),)f(hence)i(it)d(corresp)s(onds)j(to)e(a)150 4599 y(real)32 b(orbit)f(at)h(energy)i Fu(E)6 b Fz(.)43 b(W)-8 b(e)33 b(notice)g(that)f(the)h(ab)s(o)m(v)m(e)g(system)h(giv)m (es)1111 4887 y(~)-53 b Fu(r)s Fz(\()p Fu(t)p Fz(\))28 b(=)j(~)-53 b Fu(r)s Fz(\()1489 4860 y(~)1481 4887 y Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))28 b(=)f(exp)q Fr(f)2008 4751 y Fl(Z)2113 4760 y Fs(~)2108 4777 y Ft(\022)q Fs(\()p Ft(t)p Fs(\))2063 4976 y(0)2227 4887 y Fz(\()p Fu(\032)2315 4902 y Ft(E)2375 4887 y Fu(=\030)2467 4902 y Ft(E)2526 4887 y Fz(\)\()p Fu(')p Fz(\))p Fu(d')p Fr(g)707 b Fz(\(2.16\))150 5166 y(hence)38 b(the)e(solution)f(is)h(global,)f(Ran\()1611 5140 y(~)1603 5166 y Fu(\022)s Fz(\))f(=)g([0)p Fu(;)17 b Fr(1)p Fz(\))35 b(and)41 b(~)-53 b Fu(r)r Fz(\()p Fu(t)p Fz(\))37 b(increases)g(\\in)e(mean")h(after)g(eac)m(h)150 5286 y(complete)26 b(rev)m(olution)f(around)h(the)h(origin)d(and)j (escap)s(es)h(to)e(in\014nit)m(y)-8 b(.)41 b(It)26 b(is)g(imp)s(ortan)m (t)e(to)i(remark)150 5407 y(though)31 b(that)k(~)-53 b Fu(r)r Fz(\()p Fu(t)p Fz(\))31 b(ma)m(y)g(not)g(b)s(e)g(strictly)f (increasing)g(as)i(a)e(function)h(of)f(time)g(as)h(it)f(w)m(as)i(the)f (case)150 5527 y(for)h(energies)h(ab)s(o)m(v)m(e)g Fu(E)1016 5542 y Ft(b)1051 5527 y Fz(.)1939 5776 y(13)p eop %%Page: 14 17 14 16 bop 296 407 a Fz(No)m(w)42 b(let)f(us)h(start)f(the)h(pro)s(of)e (of)h(the)h(prop)s(osition.)67 b(If)41 b Fu(\032)2522 422 y Ft(E)2574 431 y Fj(0)2654 407 y Fz(is)g(p)s(erio)s(dic)f(and)h Fu(C)3419 371 y Fs(1)3459 407 y Fz(,)i(then)f(it)150 527 y(cannot)36 b(equal)f Fr(\006)810 450 y(p)p 894 450 161 4 v 894 527 a Fz(2)p Fu(E)1015 542 y Fs(0)1090 527 y Fz(at)g(an)m(y)h(p)s(oin)m(t)f Fu(\022)1702 542 y Fs(1)1778 527 y Fz(since)h(otherwise)g Fu(\022)2502 542 y Fs(1)2577 527 y Fz(w)m(ould)g(b)s(e)g(an)f(extrem)m(um)h(p)s(oin)m(t)150 648 y(and)d(th)m(us)h(0)29 b(=)f Fu(@)788 663 y Ft(\022)828 648 y Fu(\032)878 663 y Ft(E)930 672 y Fj(0)969 648 y Fz(\()p Fu(\022)1052 663 y Fs(1)1091 648 y Fz(\))h(=)f Fu(b)p Fz(\()p Fu(\022)1386 663 y Fs(1)1427 648 y Fz(\))33 b(whic)m(h)g(con)m(tradicts)h Fu(b)29 b(<)f Fz(0)33 b(ev)m(erywhere.)48 b(Next,)35 b(let)d(us)i(sho)m(w)150 799 y(that)g Fu(c)d Fr(\025)h Fz(min)15 b Fr(j)p Fu(b)p Fr(j)p Fz(.)49 b(Indeed,)37 b(assume)e(that)f(the)h(minim)m(um)c(of)j Fu(\030)2560 814 y Ft(E)2612 823 y Fj(0)2682 799 y Fz(:=)2816 689 y Fl(q)p 2915 689 423 4 v 2915 799 a Fz(2)p Fu(E)3036 814 y Fs(0)3098 799 y Fr(\000)22 b Fu(\032)3247 765 y Fs(2)3247 826 y Ft(E)3299 835 y Fj(0)3373 799 y Fz(is)34 b(attained)150 957 y(at)39 b(some)g(p)s(oin)m(t)f Fu(\022)833 972 y Fs(1)873 957 y Fz(.)63 b(Then)40 b Fu(\032)1274 972 y Ft(E)1326 981 y Fj(0)1365 957 y Fz(\()p Fu(\022)1448 972 y Fs(1)1488 957 y Fz(\))p Fu(@)1577 972 y Ft(\022)1616 957 y Fu(\032)1666 972 y Ft(E)1718 981 y Fj(0)1757 957 y Fz(\()p Fu(\022)1840 972 y Fs(1)1880 957 y Fz(\))f(=)g(0.)63 b(If)39 b Fu(\032)2365 972 y Ft(E)2417 981 y Fj(0)2455 957 y Fz(\()p Fu(\022)2538 972 y Fs(1)2578 957 y Fz(\))g(=)g(0)g(then)g Fu(\030)3129 972 y Ft(E)3181 981 y Fj(0)3258 957 y Fz(=)3373 880 y Fr(p)p 3456 880 161 4 v 77 x Fz(2)p Fu(E)3577 972 y Fs(0)3616 957 y Fz(,)i Fu(\032)3734 972 y Ft(E)3786 981 y Fj(0)150 1078 y Fz(v)-5 b(anishes)33 b(iden)m(tically)d(and)j Fu(b)28 b Fz(=)g Fr(\000)1444 1001 y(p)p 1527 1001 V 77 x Fz(2)p Fu(E)1648 1093 y Fs(0)1688 1078 y Fz(.)43 b(Ob)m(viously)32 b Fu(c)c Fz(=)g(min)15 b Fr(j)p Fu(b)p Fr(j)32 b Fz(in)g(this)g(case.)44 b(If)32 b Fu(\032)3393 1093 y Ft(E)3445 1102 y Fj(0)3484 1078 y Fz(\()p Fu(\022)3567 1093 y Fs(1)3607 1078 y Fz(\))c Fr(6)p Fz(=)f(0)150 1198 y(then)33 b Fu(@)423 1213 y Ft(\022)463 1198 y Fu(\032)513 1213 y Ft(E)565 1222 y Fj(0)603 1198 y Fz(\()p Fu(\022)686 1213 y Fs(1)726 1198 y Fz(\))28 b(=)f(0)33 b(th)m(us)g Fu(\030)1234 1213 y Ft(E)1286 1222 y Fj(0)1324 1198 y Fz(\()p Fu(\022)1407 1213 y Fs(1)1447 1198 y Fz(\))28 b(=)f Fu(c)h Fz(=)g Fr(\000)p Fu(b)p Fz(\()p Fu(\022)1991 1213 y Fs(1)2031 1198 y Fz(\))33 b(and)g(\(2.13\))e(follo)m(ws.)296 1318 y(Next,)47 b(denote)c(b)m(y)h Fu(a)1096 1333 y Fs(0)1181 1318 y Fz(=)h Fu(\032)1352 1333 y Ft(E)1404 1342 y Fj(0)1443 1318 y Fz(\(0\).)74 b(General)42 b(results)h(in)f(O.D.E.)h(ensure)i (the)e(existence)h(of)150 1439 y Fu(\016)193 1454 y Fs(1)233 1439 y Fu(;)17 b(\016)320 1454 y Fs(2)402 1439 y Fu(>)43 b Fz(0)e(suc)m(h)i(that)e Fu(\032)p Fz(\()p Fu(E)6 b(;)17 b(a;)g(\022)s Fz(\))42 b(exists)g(and)g(is)f(smo)s(oth)f(in)h(all)f (argumen)m(ts)h(for)g Fu(E)49 b Fr(2)44 b Fz(\()p Fu(E)3680 1454 y Fs(0)3748 1439 y Fr(\000)150 1559 y Fu(\016)193 1574 y Fs(1)233 1559 y Fu(;)17 b(E)349 1574 y Fs(0)416 1559 y Fz(+)28 b Fu(\016)563 1574 y Fs(1)602 1559 y Fz(\),)44 b Fu(a)e Fr(2)h Fz(\()p Fu(a)1002 1574 y Fs(0)1069 1559 y Fr(\000)29 b Fu(\016)1218 1574 y Fs(2)1257 1559 y Fu(;)17 b(a)1352 1574 y Fs(0)1420 1559 y Fz(+)27 b Fu(\016)1566 1574 y Fs(2)1606 1559 y Fz(\))41 b(and)g Fu(\022)46 b Fr(2)c Fz([0)p Fu(;)17 b Fz(2)p Fu(\031)t Fz(].)69 b(W)-8 b(e)41 b(in)m(tend)h(to)e(apply)h(the)h(implicit)150 1680 y(function)30 b(theorem)h(to)f(the)h(equation)g Fu(\032)p Fz(\()p Fu(E)6 b(;)17 b(a;)g Fz(2)p Fu(\031)t Fz(\))27 b(=)h Fu(a)p Fz(;)j(di\013eren)m(tiating)e(\(2.12\))h(with)g (resp)s(ect)i(to)150 1800 y Fu(a)h Fz(w)m(e)g(get)1087 1938 y Fu(@)1138 1953 y Ft(\022)1177 1938 y Fz(\()p Fu(@)1266 1953 y Ft(a)1308 1938 y Fu(\032)p Fz(\)\()p Fu(E)6 b(;)17 b(a;)g(\022)s Fz(\))28 b(=)g Fr(\000)2150 1870 y Fu(\032)p 1956 1915 438 4 v 1956 1935 a Fl(p)p 2056 1935 339 4 v 86 x Fz(2)p Fu(E)g Fr(\000)22 b Fu(\032)2354 1992 y Fs(2)2404 1938 y Fu(@)2455 1953 y Ft(a)2497 1938 y Fu(\032)p Fz(\()p Fu(E)6 b(;)17 b(a;)g(\022)s Fz(\))150 2174 y(whic)m(h)33 b(b)m(y)h(in)m(tegration)c(yields)1139 2376 y Fu(@)1190 2391 y Ft(a)1232 2376 y Fu(\032)p Fz(\()p Fu(E)1392 2391 y Fs(0)1432 2376 y Fu(;)17 b(a)1527 2391 y Fs(0)1566 2376 y Fu(;)g Fz(2)p Fu(\031)t Fz(\))27 b(=)h(exp)18 b(\()p Fr(\000)p Fu(I)8 b Fz(\()p Fu(E)2329 2391 y Fs(0)2368 2376 y Fu(;)17 b(\032)2462 2391 y Ft(E)2514 2400 y Fj(0)2553 2376 y Fz(\)\))28 b Fu(<)f Fz(1)p Fu(:)296 2577 y Fz(Then)39 b(the)f(implicit)c(function)k(theorem)f(giv)m(es)h(a)g(smo)s(oth)f (solution)f Fu(a)p Fz(\()p Fu(E)6 b Fz(\))38 b(to)f(the)h(equation)150 2698 y Fu(\032)p Fz(\()p Fu(E)6 b(;)17 b(a)p Fz(\()p Fu(E)6 b Fz(\))p Fu(;)17 b Fz(2)p Fu(\031)t Fz(\))44 b(=)h Fu(a)p Fz(\()p Fu(E)6 b Fz(\),)46 b(in)41 b(a)i(small)d(op)s(en)j (in)m(terv)-5 b(al)42 b(cen)m(tered)i(at)f Fu(E)2886 2713 y Fs(0)2925 2698 y Fz(.)74 b(The)44 b(prop)s(osition)c(is)150 2818 y(concluded)33 b(b)m(y)h(putting)e Fu(\032)1138 2833 y Ft(E)1197 2818 y Fz(\()p Fu(\022)s Fz(\))c(=)g Fu(\032)p Fz(\()p Fu(E)6 b(;)17 b(a)p Fz(\()p Fu(E)6 b Fz(\))p Fu(;)17 b(\022)s Fz(\).)p 3754 2818 4 66 v 3758 2755 59 4 v 3758 2818 V 3816 2818 4 66 v 296 3056 a(F)-8 b(or)36 b(ev)m(ery)j Fu(E)809 3071 y Fs(0)883 3056 y Fu(>)c Fz(0)h(denote)i(b)m(y)f Fu(P)14 b Fz(\()p Fu(E)1724 3071 y Fs(0)1763 3056 y Fz(\))37 b(the)g(statemen)m(t)g (\\there)h(is)e(a)g Fu(C)3034 3020 y Fs(1)3110 3056 y Fz(p)s(erio)s(dic)f(solution)150 3177 y(to)d(\(2.12\))g(satisfying)g (\(2.14\)".)42 b(Then)34 b(de\014ne)1036 3378 y Fr(E)i Fz(:=)27 b Fr(f)p Fu(E)1377 3393 y Fs(0)1444 3378 y Fu(>)h Fz(0)f(:)56 b Fu(P)14 b Fz(\()p Fu(E)1894 3393 y Fs(0)1933 3378 y Fz(\))22 b(is)g(true)p Fr(g)p Fu(;)114 b(E)2517 3393 y Ft(d)2586 3378 y Fz(:=)27 b(inf)c Fr(E)9 b Fu(:)636 b Fz(\(2.17\))150 3580 y Fv(Lemma)40 b(2.5.)k FA(F)-7 b(or)36 b(every)i Fu(E)h(>)32 b(E)1504 3595 y Ft(b)1576 3580 y FA(let)38 b Fu(\030)1759 3595 y Ft(E)1856 3580 y FA(b)-5 b(e)37 b(the)g(p)-5 b(erio)g(dic)37 b(solution)g(to)h (\(2.5\))f(and)g(let)g Fu(\032)3629 3595 y Ft(E)3722 3580 y Fz(:=)150 3615 y Fl(p)p 250 3615 351 4 v 85 x Fz(2)p Fu(E)28 b Fr(\000)22 b Fu(\030)546 3666 y Fs(2)541 3727 y Ft(E)640 3700 y Fz(\()p Fu(>)40 b Fz(0\))p FA(.)64 b(Then)41 b Fu(\032)1286 3715 y Ft(E)1387 3700 y FA(satis\014es)g(the)g (b)-5 b(ound)41 b Fu(I)8 b Fz(\()p Fu(E)e(;)17 b(\032)2477 3715 y Ft(E)2537 3700 y Fz(\))40 b Fu(>)f Fz(0)j FA(as)f(in)g (\(2.14\).)63 b(Mor)-5 b(e)g(over)150 3821 y Fz(\()p Fu(\032)238 3836 y Ft(E)298 3821 y Fu(;)17 b(\030)385 3836 y Ft(E)444 3821 y Fz(\))35 b FA(is)f(the)h(unique)g(p)-5 b(erio)g(dic)34 b(solution)g(to)h(\(2.6\))g(which)f(satis\014es)g (\(2.14\).)150 4008 y Fv(Pro)s(of)p Fz(.)63 b(The)40 b(facts)f(that)g Fu(\032)1224 4023 y Ft(E)1323 4008 y Fz(satis\014es)h(\(2.14\))e(and)h(that)g(\()p Fu(\032)2479 4023 y Ft(E)2539 4008 y Fu(;)17 b(\030)2626 4023 y Ft(E)2685 4008 y Fz(\))39 b(is)f(a)h(solution)f(to)g(\(2.6\))h(are)150 4129 y(trivial;)47 b(w)m(e)e(only)e(need)i(to)f(pro)m(v)m(e)h(that)f (there)g(are)g(no)g(other)g(solutions)f(to)g(\(2.6\))g(whic)m(h)i(also) 150 4249 y(satis\014es)33 b(\(2.14\).)296 4370 y(Assume)24 b(that)e(\()9 b(~)-58 b Fu(\032)937 4385 y Ft(E)997 4370 y Fu(;)1051 4343 y Fz(~)1041 4370 y Fu(\030)1084 4385 y Ft(E)1143 4370 y Fz(\))22 b(is)g(suc)m(h)i(a)f(solution.)38 b(First,)33 b(~)-58 b Fu(\032)2283 4385 y Ft(E)2365 4370 y Fz(cannot)23 b(b)s(e)g(strictly)f(negativ)m(e)h(b)s(ecause)150 4490 y(in)30 b(that)h(case)g(\(2.14\))g(cannot)g(b)s(e)g(ful\014lled.) 40 b(Second,)33 b(if)38 b(~)-58 b Fu(\032)2286 4505 y Ft(E)2346 4490 y Fz(\()p Fu(\022)s Fz(\))31 b(is)g(strictly)f(p)s (ositiv)m(e,)g(then)i(w)m(e)g(can)150 4654 y(write)37 b(~)-58 b Fu(\032)445 4669 y Ft(E)533 4654 y Fz(=)636 4531 y Fl(q)p 736 4531 V 123 x Fz(2)p Fu(E)28 b Fr(\000)995 4628 y Fz(~)984 4654 y Fu(\030)1032 4620 y Fs(2)1027 4681 y Ft(E)1115 4654 y Fz(and)g(hence)1577 4628 y(~)1567 4654 y Fu(\030)1610 4669 y Ft(E)1697 4654 y Fz(satis\014es)i(\(2.5\).) 41 b(Whence)30 b(b)m(y)f(Lemma)e(2.3)h(w)m(e)h(conclude)150 4807 y(that)373 4781 y(~)363 4807 y Fu(\030)406 4822 y Ft(E)495 4807 y Fz(=)g Fu(\030)643 4822 y Ft(E)736 4807 y Fz(and)43 b(~)-58 b Fu(\032)977 4822 y Ft(E)1067 4807 y Fz(=)29 b Fu(\032)1222 4822 y Ft(E)1282 4807 y Fz(.)47 b(Third)34 b(if)42 b(~)-58 b Fu(\032)1775 4822 y Ft(E)1835 4807 y Fz(\()p Fu(\022)1918 4822 y Fs(1)1958 4807 y Fz(\))29 b(=)h(0)j(w)m(e)i(ha)m(v)m(e)2595 4781 y(~)2584 4807 y Fu(\030)2627 4822 y Ft(E)2687 4807 y Fz(\()p Fu(\022)2770 4822 y Fs(1)2809 4807 y Fz(\))30 b(=)2983 4723 y Fr(p)p 3066 4723 127 4 v 84 x Fz(2)p Fu(E)39 b Fz(and)34 b(hence)i(the)150 4928 y(\014rst)d(equation)g(of)f (\(2.6\))h(leads)g(to)f Fu(@)1510 4943 y Ft(\022)1558 4928 y Fz(~)-57 b Fu(\032)1600 4943 y Ft(E)1659 4928 y Fz(\()p Fu(\022)1742 4943 y Fs(1)1782 4928 y Fz(\))28 b Fr(\025)1954 4843 y(p)p 2037 4843 V 85 x Fz(2)p Fu(E)g Fr(\000)23 b Fz(max)16 b Fr(j)p Fu(b)p Fr(j)28 b Fu(>)g Fz(0.)45 b(F)-8 b(rom)31 b(the)i(p)s(erio)s(dicit)m(y)e(of)159 5048 y(~)-58 b Fu(\032)200 5063 y Ft(E)260 5048 y Fz(\()p Fu(\022)s Fz(\))32 b(w)m(e)i(then)f(conclude)g(that)f(indeed)h(this)f (function)g(cannot)h(ha)m(v)m(e)g(zeros.)45 b(W)-8 b(e)33 b(ha)m(v)m(e)g(reduced)150 5168 y(to)f(the)h(previous)g(case.)p 3754 5168 4 66 v 3758 5106 59 4 v 3758 5168 V 3816 5168 4 66 v 296 5407 a(Ho)m(w)m(ev)m(er,)g(it)28 b(is)h(not)g(y)m(et)h (clear)f(whether)i(w)m(e)f(can)f(construct)i(p)s(erio)s(dic)c (solutions)i(whic)m(h)g(ob)s(ey)150 5527 y(\(2.14\))j(for)g(all)e (energies)j(in)f(\()p Fu(E)1309 5542 y Ft(d)1350 5527 y Fu(;)17 b Fr(1)p Fz(\).)43 b(This)32 b(question)h(will)e(b)s(e)h (answ)m(ered)j(shortly)-8 b(.)1939 5776 y(14)p eop %%Page: 15 18 15 17 bop 296 407 a Fz(But)40 b(\014rst)f(let)g(us)g(pro)m(v)m(e)i(a)e (few)g(v)m(ery)i(in)m(teresting)e(facts)g(ab)s(out)g Fu(\032)2804 422 y Ft(E)2864 407 y Fz(\()p Fu(\022)s Fz(\).)63 b(W)-8 b(e)40 b(start)f(with)f(its)150 527 y(energy)c(dep)s(endence:)150 731 y Fv(Prop)s(osition)k(2.6.)43 b FA(L)-5 b(et)37 b Fu(E)1217 746 y Fs(0)1288 731 y Fu(>)32 b Fz(0)k FA(and)h Fu(\032)1723 746 y Ft(E)1775 755 y Fj(0)1850 731 y FA(b)-5 b(e)37 b(as)g(in)f(Pr)-5 b(op)g(osition)36 b(2.4)h(\(ob)-5 b(eying)36 b(also)g(\(2.14\)\).)150 851 y(Denote)e(by)h Fu(\032)662 866 y Ft(E)722 851 y Fz(\()p Fu(\022)s Fz(\))g FA(the)f(smo)-5 b(oth)35 b(and)f(p)-5 b(erio)g(dic)34 b(solution)g(to)h(\(2.12\))f(c)-5 b(onstructe)g(d)35 b(in)f(the)h(pr)-5 b(o)g(of)34 b(of)150 971 y(Pr)-5 b(op)g(osition)31 b(2.4)g(for)h(every)g Fu(E)h Fr(2)c Fz(\()p Fu(E)1544 986 y Fs(0)1598 971 y Fr(\000)15 b Fu(\017;)i(E)1845 986 y Fs(0)1901 971 y Fz(+)e Fu(\017)p Fz(\))p FA(.)44 b(De\014ne)31 b Fu(\030)2491 986 y Ft(E)2578 971 y Fz(=)2682 886 y Fl(p)p 2781 886 359 4 v 2781 971 a Fz(2)p Fu(E)d Fr(\000)23 b Fu(\032)3080 937 y Fs(2)3080 998 y Ft(E)3167 971 y Fz(\()p Fu(>)28 b Fz(0\))p FA(.)43 b(We)32 b(then)150 1092 y(have)1644 1212 y Fz(inf)1686 1275 y Ft(\022)1779 1212 y Fu(@)1830 1227 y Ft(E)1890 1212 y Fu(\032)1940 1227 y Ft(E)2000 1212 y Fz(\()p Fu(\022)s Fz(\))27 b Fu(>)h Fz(0)p Fu(;)1245 b Fz(\(2.18\))1515 1431 y(sup)1571 1513 y Ft(\022)1678 1431 y Fu(@)1734 1390 y Fs(2)1729 1456 y Ft(E)1789 1431 y Fu(\032)1839 1446 y Ft(E)1899 1431 y Fz(\()p Fu(\022)s Fz(\))28 b Fu(<)g Fz(0)99 b(and)1116 b(\(2.19\))1634 1696 y(inf)1676 1758 y Ft(\022)1769 1696 y Fu(@)1820 1711 y Ft(E)1890 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y(and)33 b(ob)m(viously)1500 4077 y Fu(\032)1550 4092 y Ft(E)1610 4077 y Fu(@)1661 4092 y Ft(E)1721 4077 y Fu(\032)1771 4092 y Ft(E)1858 4077 y Fz(=)28 b(1)22 b Fr(\000)h Fu(\030)2176 4092 y Ft(E)2235 4077 y Fu(@)2286 4092 y Ft(E)2346 4077 y Fu(\030)2389 4092 y Ft(E)2448 4077 y Fu(:)1101 b Fz(\(2.24\))150 4252 y(No)m(w)33 b(let)e(us)h(pro)m (v)m(e)h(the)g(monotonicit)m(y)d(of)h Fu(\032)1816 4267 y Ft(E)1908 4252 y Fz(in)g(\(2.18\).)43 b(Di\013eren)m(tiate)30 b(\(2.23\))i(with)f(resp)s(ect)i(to)150 4372 y Fu(E)39 b Fz(and)32 b(get)h Fu(@)664 4387 y Ft(E)724 4372 y Fu(\030)767 4387 y Ft(E)854 4372 y Fz(=)27 b Fu(@)1008 4387 y Ft(\022)1048 4372 y Fz(\()p Fu(@)1137 4387 y Ft(E)1197 4372 y Fu(\032)1247 4387 y Ft(E)1307 4372 y Fz(\),)32 b(then)i(in)m(tro)s(duce)e(this)g(in) g(\(2.24\))g(to)g(obtain)975 4637 y Fu(@)1026 4652 y Ft(\022)1066 4637 y Fz(\()p Fu(@)1155 4652 y Ft(E)1215 4637 y Fu(\032)1265 4652 y Ft(E)1325 4637 y Fz(\))27 b(=)h Fr(\000)1581 4569 y Fu(\032)1631 4584 y Ft(E)p 1581 4614 V 1585 4705 a Fu(\030)1628 4720 y Ft(E)1701 4637 y Fz(\()p Fu(@)1790 4652 y Ft(E)1850 4637 y Fu(\032)1900 4652 y Ft(E)1960 4637 y Fz(\))22 b(+)2155 4569 y(1)p 2128 4614 103 4 v 2128 4705 a Fu(\030)2171 4720 y Ft(E)2268 4637 y Fz(=)28 b Fr(\000)p Fu(h)p Fz(\()p Fu(@)2594 4652 y Ft(E)2654 4637 y Fu(\032)2704 4652 y Ft(E)2764 4637 y Fz(\))22 b(+)g Fu(g)t(:)576 b Fz(\(2.25\))150 4929 y(Notice)28 b(that)h(according)f(to)g(\(2.14\))g(w)m(e)i(ha)m(v)m(e)g (that)2057 4849 y Fl(R)2123 4875 y Fs(2)p Ft(\031)2104 4964 y Fs(0)2222 4929 y Fu(h)p Fz(\()p Fu(\022)s Fz(\))p Fu(d\022)h(>)c Fz(0.)42 b(Th)m(us)30 b Fu(g)i Fz(is)d(strictly)f(p)s (ositiv)m(e)150 5049 y(and)39 b(the)g(equation)f(has)h(the)h(same)e (form)g(as)h(in)e(\(2.21\).)62 b(It)38 b(follo)m(ws)g(according)g(to)g (\(2.22\))g(that)150 5170 y Fu(@)201 5185 y Ft(E)261 5170 y Fu(\032)311 5185 y Ft(E)399 5170 y Fu(>)27 b Fz(0.)296 5290 y(The)34 b(conca)m(vit)m(y)f(in)f(\(2.19\))g(is)g(sho)m(wn)i (using)e(the)h(same)g(idea.)43 b(W)-8 b(rite)840 5510 y Fu(\032)890 5469 y Fs(3)890 5535 y Ft(E)950 5510 y Fz(\()p Fu(@)1044 5469 y Fs(2)1039 5535 y Ft(E)1099 5510 y Fu(\032)1149 5525 y Ft(E)1209 5510 y Fz(\))28 b(=)f Fr(\000)p Fz(\(1)c Fr(\000)f Fu(\030)1707 5525 y Ft(E)1767 5510 y Fu(@)1818 5525 y Ft(E)1878 5510 y Fu(\030)1921 5525 y Ft(E)1980 5510 y Fz(\))2018 5469 y Fs(2)2079 5510 y Fr(\000)h Fu(\032)2229 5469 y Fs(2)2229 5535 y Ft(E)2289 5510 y Fz([\()p Fu(@)2405 5525 y Ft(E)2465 5510 y Fu(\030)2508 5525 y Ft(E)2567 5510 y Fz(\))2605 5469 y Fs(2)2667 5510 y Fz(+)f Fu(\030)2808 5525 y Ft(E)2867 5510 y Fu(@)2923 5469 y Fs(2)2918 5535 y Ft(E)2978 5510 y Fu(\030)3021 5525 y Ft(E)3080 5510 y Fz(])p Fu(;)442 b Fz(\(2.26\))1939 5776 y(15)p eop %%Page: 16 19 16 18 bop 150 407 a Fz(then)33 b(isolate)e Fu(@)732 371 y Fs(2)727 432 y Ft(E)787 407 y Fu(\030)830 422 y Ft(E)890 407 y Fz(:)941 667 y Fu(@)997 626 y Fs(2)992 692 y Ft(E)1052 667 y Fu(\030)1095 682 y Ft(E)1182 667 y Fz(=)d Fr(\000)1373 600 y Fu(\032)1423 615 y Ft(E)p 1373 644 110 4 v 1377 735 a Fu(\030)1420 750 y Ft(E)1515 667 y Fr(\001)22 b Fz(\()p Fu(@)1659 626 y Fs(2)1654 692 y Ft(E)1714 667 y Fu(\032)1764 682 y Ft(E)1824 667 y Fz(\))g Fr(\000)2020 600 y Fz(1)p 1993 644 103 4 v 1993 735 a Fu(\030)2036 750 y Ft(E)2122 586 y Fl(\002)2164 667 y Fz(\()p Fu(@)2253 682 y Ft(E)2313 667 y Fu(\032)2363 682 y Ft(E)2423 667 y Fz(\))2461 626 y Fs(2)2522 667 y Fz(+)g(\()p Fu(@)2709 682 y Ft(E)2769 667 y Fu(\030)2812 682 y Ft(E)2871 667 y Fz(\))2909 626 y Fs(2)2949 586 y Fl(\003)3007 667 y Fu(:)542 b Fz(\(2.27\))296 896 y(Di\013eren)m(tiate)32 b(\(2.23\))h(t)m(wice)g(with)g(resp)s(ect)i(to)e Fu(E)39 b Fz(to)33 b(get)g Fu(@)2522 860 y Fs(2)2517 921 y Ft(E)2577 896 y Fu(\030)2620 911 y Ft(E)2708 896 y Fz(=)c Fu(@)2864 911 y Ft(\022)2904 896 y Fz(\()p Fu(@)2998 860 y Fs(2)2993 921 y Ft(E)3053 896 y Fu(\032)3103 911 y Ft(E)3162 896 y Fz(\))34 b(and)f(in)m(tro)s(duce)150 1016 y(it)f(in)f(\(2.27\):)1414 1257 y Fu(@)1465 1272 y Ft(\022)1505 1257 y Fz(\()p Fu(@)1599 1216 y Fs(2)1594 1282 y Ft(E)1654 1257 y Fu(\032)1704 1272 y Ft(E)1764 1257 y Fz(\))c(=)h Fr(\000)p Fu(h)p Fz(\()p Fu(@)2160 1216 y Fs(2)2155 1282 y Ft(E)2215 1257 y Fu(\032)2265 1272 y Ft(E)2325 1257 y Fz(\))22 b(+)g Fu(g)t(:)150 1432 y Fz(Since)32 b(w)m(e)g(again)e(ha)m(v)m(e)j(the)e (in)m(tegral)f(condition)g(on)i Fu(h)f Fz(while)g Fu(g)j Fz(no)m(w)e(is)f(strictly)g(negativ)m(e,)h(\(2.22\))150 1552 y(implies)e Fu(@)537 1516 y Fs(2)532 1577 y Ft(E)592 1552 y Fu(\032)642 1567 y Ft(E)730 1552 y Fu(<)d Fz(0)33 b(and)f(w)m(e)i(are)f(done.)296 1672 y(Finally)-8 b(,)30 b(let)i(us)h(pro)m(v)m(e)h(\(2.20\).)43 b(Using)32 b Fu(\030)1813 1687 y Ft(E)1900 1672 y Fz(=)2004 1587 y Fl(p)p 2103 1587 359 4 v 2103 1672 a Fz(2)p Fu(E)c Fr(\000)23 b Fu(\032)2402 1638 y Fs(2)2402 1699 y Ft(E)2494 1672 y Fz(w)m(e)34 b(ha)m(v)m(e)1281 1892 y Fu(@)1332 1907 y Ft(E)1392 1892 y Fz(\()p Fu(\032)1480 1907 y Ft(E)1540 1892 y Fu(=\030)1632 1907 y Ft(E)1691 1892 y Fz(\))28 b(=)f(\(2)p Fu(E)6 b(@)2076 1907 y 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Fz(and)c Fu(E)2160 3642 y Fs(0)2248 3627 y Fr(62)49 b(E)9 b Fz(.)80 b(Since)44 b Fu(E)2870 3642 y Ft(d)2956 3627 y Fz(is)g(the)h(in\014m)m(um)e(of)i Fr(E)150 3747 y Fz(then)f(there)g(m)m(ust)f(exist)h Fu(E)1209 3762 y Fs(1)1294 3747 y Fr(2)i(E)52 b Fz(so)43 b(that)g Fu(E)1935 3762 y Fs(1)2020 3747 y Fu(<)j(E)2214 3762 y Fs(0)2254 3747 y Fz(.)75 b(It)43 b(follo)m(ws)f(that)h(some)g Fu(\032)3330 3762 y Ft(E)3382 3771 y Fj(1)3421 3747 y Fz(\()p Fu(\022)s Fz(\))g(exists)150 3867 y(and)e(satis\014es)h(\(2.14\).)69 b(Then)42 b(w)m(e)g(can)g(apply)f(the)g(lo)s(cal)e(construction)j(of)e (Prop)s(osition)g(2.4)h(in)150 3988 y(order)j(to)f(obtain)g(similar)e (solutions)h(for)i(sligh)m(tly)e(larger)h(energies.)77 b(In)44 b(fact,)j(w)m(e)e(claim)c(that)150 4108 y(this)36 b(construction)i(can)f(b)s(e)g(con)m(tin)m(ued)g(up)h(to)e Fu(E)2017 4123 y Fs(0)2093 4108 y Fz(and)h(cannot)g(stop)h(b)s(efore.) 56 b(Indeed,)39 b(assume)150 4229 y(there)45 b(exists)g Fu(E)764 4244 y Fs(2)850 4229 y Fr(2)j Fz(\()p Fu(E)1074 4244 y Fs(1)1114 4229 y Fu(;)17 b Fr(1)p Fz(\))43 b(suc)m(h)i(that)f (the)h(branc)m(h)g(coming)d(from)h Fu(\032)2946 4244 y Ft(E)2998 4253 y Fj(1)3081 4229 y Fz(stops)h(at)g Fu(E)3545 4244 y Fs(2)3585 4229 y Fz(.)78 b(W)-8 b(e)150 4349 y(kno)m(w)34 b(that)g Fu(\032)668 4364 y Ft(E)727 4349 y Fz(\()p Fu(\022)s Fz(\))g(is)e(increasing)h(as)g(a)g(function)g(of)g Fu(E)39 b Fz(on)34 b(\()p Fu(E)2495 4364 y Fs(1)2534 4349 y Fu(;)17 b(E)2650 4364 y Fs(2)2689 4349 y Fz(\))34 b(and)f(b)s(ounded)h(from)e (ab)s(o)m(v)m(e)150 4469 y(b)m(y)293 4392 y Fr(p)p 376 4392 161 4 v 77 x Fz(2)p Fu(E)497 4484 y Fs(0)537 4469 y Fz(,)42 b(hence)g(admits)d(a)h(limit)d Fu(\032)1593 4484 y Ft(E)1645 4493 y Fj(2)1684 4469 y Fz(\()p Fu(\022)s Fz(\).)67 b(W)-8 b(e)41 b(conclude)g(that)f Fu(\032)2756 4484 y Ft(E)2808 4493 y Fj(2)2887 4469 y Fz(is)g(p)s(erio)s(dic,)h (solv)m(es)h(the)150 4590 y(in)m(tegral)31 b(v)m(ersion)i(of)f (\(2.12\))g(with)g Fu(a)c Fz(=)g Fu(\032)1684 4605 y Ft(E)1736 4614 y Fj(2)1774 4590 y Fz(\(0\),)33 b(th)m(us)g(it)f(is)g Fu(C)2446 4554 y Fs(1)2485 4590 y Fz(.)296 4742 y(Then)43 b(de\014ne)f Fu(\030)893 4757 y Ft(E)945 4766 y Fj(2)1026 4742 y Fz(=)1145 4631 y Fl(q)p 1244 4631 423 4 v 1244 4742 a Fz(2)p Fu(E)1365 4757 y Fs(2)1427 4742 y Fr(\000)23 b Fu(\032)1577 4707 y Fs(2)1577 4769 y Ft(E)1629 4778 y Fj(2)1709 4742 y Fz(and)41 b(notice)g(that)g(according)g(to)g (\(2.13\))g(it)f(cannot)i(b)s(e)150 4889 y(zero.)54 b(Hence)38 b Fu(\032)748 4904 y Ft(E)800 4913 y Fj(2)875 4889 y Fz(and)e Fu(\030)1111 4904 y Ft(E)1163 4913 y Fj(2)1237 4889 y Fz(are)g(smo)s(oth)f(functions)h(of)g Fu(\022)j Fz(whic)m(h)e(solv)m(e)f(\(2.6\).)54 b(Then)37 b(they)g(also)150 5010 y(ob)s(ey)d(\(2.14\))g(b)s(ecause)h(\(2.20\))e(ensures)j(that)e Fu(\032)1920 5025 y Ft(E)1980 5010 y Fu(=\030)2072 5025 y Ft(E)2165 5010 y Fz(is)g(increasing)f(in)g Fu(E)40 b Fz(at)34 b(\014xed)h Fu(\022)s Fz(.)48 b(It)34 b(follo)m(ws)150 5130 y(that)27 b(the)h(construction)g(of)f(Prop)s(osition)f(2.4)h(can)h (b)s(e)f(rep)s(eated)i(at)e Fu(E)2732 5145 y Fs(2)2771 5130 y Fz(.)42 b(Hence)29 b Fu(\032)3175 5145 y Ft(E)3262 5130 y Fz(do)s(es)f(not)g(stop)150 5250 y(there.)44 b(Th)m(us)34 b(w)m(e)g(m)m(ust)f(ha)m(v)m(e)h Fu(E)1367 5265 y Fs(0)1434 5250 y Fr(2)28 b(E)41 b Fz(and)33 b(therefore)g Fr(E)j Fz(=)28 b(\()p Fu(E)2524 5265 y Ft(d)2564 5250 y Fu(;)17 b Fr(1)p Fz(\).)296 5371 y(No)m(w)29 b(let)e(us)i(pro)m(v)m(e)g(the)f (uniqueness)i(of)e(suc)m(h)h(a)f(solution.)40 b(Assume)29 b(that)f(\()9 b(~)-58 b Fu(\032)3135 5386 y Ft(E)3195 5371 y Fu(;)3248 5345 y Fz(~)3239 5371 y Fu(\030)3282 5386 y Ft(E)3341 5371 y Fz(\))27 b(is)h(another)150 5491 y(solution)46 b(to)i(\(2.6\))g(ob)s(eying)f(\(2.14\).)89 b(Then)49 b(using)f(the)g(lo)s(cal)e(construction)i(of)g(Prop)s (osition)1939 5776 y(16)p eop %%Page: 17 20 17 19 bop 150 407 a Fz(2.4)41 b(together)g(with)g(the)h(monotonicit)m (y)e(in)g(\(2.20\))h(w)m(e)h(can)g(uniquely)f(construct)i(the)e (solution)150 527 y(\\branc)m(hes")46 b(originating)41 b(from)i Fu(\032)1461 542 y Ft(E)1566 527 y Fz(and)53 b(~)-58 b Fu(\032)1817 542 y Ft(E)1922 527 y Fz(as)44 b(functions)h(of)f Fu(E)50 b Fz(un)m(til)44 b(w)m(e)h(reac)m(h)g(an)g (energy)150 648 y Fu(E)228 611 y Fm(0)282 648 y Fu(>)31 b(E)461 663 y Ft(b)530 648 y Fz(where)36 b(w)m(e)g(kno)m(w)f(that)g Fu(\032)1480 663 y Ft(E)1536 644 y Ff(0)1593 648 y Fz(=)40 b(~)-58 b Fu(\032)1750 663 y Ft(E)1806 644 y Ff(0)1866 648 y Fz(due)36 b(to)e(Lemma)f(2.5.)49 b(Going)33 b(bac)m(kw)m(ards)j (from)e Fu(E)3802 611 y Fm(0)150 768 y Fz(using)28 b(the)i(uniqueness)g (part)f(of)f(the)h(implicit)c(function)k(theorem,)g(cf.)42 b(the)29 b(pro)s(of)f(of)h(Prop)s(osition)150 888 y(2.4,)j(w)m(e)i (conclude)f(that)41 b(~)-58 b Fu(\032)1140 903 y Ft(E)1228 888 y Fz(=)27 b Fu(\032)1381 903 y Ft(E)1474 888 y Fz(and)33 b(w)m(e)g(are)g(done.)p 3754 888 4 66 v 3758 826 59 4 v 3758 888 V 3816 888 4 66 v 150 1127 a Fv(Remark)p Fz(.)42 b(The)30 b(ab)s(o)m(v)m(e)f(uniqueness)i(pro)s(of)d(w)m(orks)i(only)f (for)f(solutions)g(whic)m(h)h(w)m(e)h(a)f(priori)e(kno)m(w)150 1247 y(ob)s(ey)j(\(2.14\).)41 b(If)29 b(w)m(e)i(drop)e(\(2.14\))f(in)h (Corollary)e(2.7)i(uniqueness)i(is)e(no)g(longer)f(true,)j(see)f (Section)150 1368 y(9.)296 1488 y(W)-8 b(e)36 b(kno)m(w)g(b)m(y)g(no)m (w)f(that)g(if)f Fu(E)k(>)32 b(E)1663 1503 y Ft(b)1732 1488 y Fz(then)k Fu(\032)2007 1503 y Ft(E)2102 1488 y Fz(is)e(p)s(ositiv)m(e.)50 b(F)-8 b(rom)34 b(Prop)s(osition)f(2.4)i(w)m (e)h(see)150 1608 y(that)27 b(if)f Fu(\032)490 1623 y Ft(E)577 1608 y Fz(is)h(non-negativ)m(e)g(\(but)g(not)g(iden)m(tically) e(zero\),)k(w)m(e)f(can)f(still)e(go)i(lo)m(w)m(er)g(in)f(energy)i (with)150 1729 y(our)g(construction.)43 b(Hence)29 b(w)m(e)h(giv)m(e)e (here)h(another)g(de\014nition)e(related)h(to)g(the)h(energy)h(for)d (whic)m(h)150 1849 y Fu(\032)200 1864 y Ft(E)292 1849 y Fz(can)33 b(b)s(ecome)g(zero:)1242 1970 y Fu(E)1314 1985 y Ft(c)1376 1970 y Fz(:=)28 b(inf)6 b Fr(f)p Fu(E)1747 1985 y Fs(0)1814 1970 y Fu(>)28 b Fz(0)f(:)56 b(inf)2118 2032 y Ft(\022)2211 1970 y Fu(\032)2261 1985 y Ft(E)2313 1994 y Fj(0)2352 1970 y Fz(\()p Fu(\022)s Fz(\))28 b Fu(>)f Fz(0)p Fr(g)p Fu(:)843 b Fz(\(2.28\))150 2162 y(Ho)m(w)m(ev)m(er,)30 b(if)24 b Fu(b)j Fz(is)e(a)h(negativ)m(e)g (constan)m(t,)i(then)f(it)d(is)i(easy)h(to)e(see)i(that)f Fu(E)2820 2177 y Ft(d)2888 2162 y Fz(=)i Fu(E)3064 2177 y Ft(c)3127 2162 y Fz(=)f Fu(E)3302 2177 y Ft(b)3337 2162 y Fz(.)41 b(Indeed,)29 b(in)150 2282 y(this)k(case)i(w)m(e)f(ha)m (v)m(e)h Fu(\030)962 2297 y Ft(E)1014 2306 y Fj(0)1082 2282 y Fz(=)29 b Fr(\000)p Fu(b)i(>)e Fz(0)k(indep)s(enden)m(t)i(of)e Fu(\022)k Fz(and)d Fu(E)2534 2297 y Fs(0)2573 2282 y Fz(.)47 b(Moreo)m(v)m(er,)35 b Fu(\032)3153 2297 y Ft(E)3205 2306 y Fj(0)3274 2282 y Fz(=)3379 2203 y Fr(p)p 3462 2203 364 4 v 79 x Fz(2)p Fu(E)3583 2297 y Fs(0)3644 2282 y Fr(\000)23 b Fu(b)3785 2253 y Fs(2)150 2402 y Fz(and)33 b(the)g(result)f(follo)m(ws.)296 2523 y(More)d(in)m(teresting)f(is)g (the)h(situation)e(in)g(whic)m(h)i Fu(b)g Fz(is)f(not)g(constan)m(t.)43 b(W)-8 b(e)29 b(start)f(with)g(a)g(lemma:)150 2702 y Fv(Lemma)37 b(2.8.)42 b FA(If)34 b Fu(b)i FA(is)e(not)h(c)-5 b(onstant)35 b(then)f Fu(E)1895 2717 y Ft(d)1964 2702 y Fu(<)27 b(E)2139 2717 y Ft(c)2202 2702 y Fu(<)g(E)2377 2717 y Ft(b)2412 2702 y FA(.)150 2882 y Fv(Pro)s(of)p Fz(.)53 b(Since)36 b Fu(\032)810 2897 y Ft(E)870 2882 y Fz(\()p Fu(\022)s Fz(\))g(is)f(increasing)g(in)g Fu(E)42 b Fz(and)36 b(is)f(b)s(ounded)i(from)d(b)s(elo)m(w)i(b)m(y)h(zero)f(if) f Fu(E)k(>)33 b(E)3763 2897 y Ft(c)3798 2882 y Fz(,)150 3002 y(w)m(e)h(can)e(de\014ne)i(the)f(functions)764 3194 y Fu(\032)814 3209 y Ft(c)849 3194 y Fz(\()p Fu(\022)s Fz(\))28 b(=)64 b(lim)1104 3257 y Ft(E)t Fm(&)p Ft(E)1283 3265 y Fq(c)1330 3194 y Fu(\032)1380 3209 y Ft(E)1440 3194 y Fz(\()p Fu(\022)s Fz(\))p Fu(;)115 b(\030)1749 3209 y Ft(c)1783 3194 y Fz(\()p Fu(\022)s Fz(\))28 b(=)64 b(lim)2038 3257 y Ft(E)t Fm(&)p Ft(E)2217 3265 y Fq(c)2265 3194 y Fu(\030)2308 3209 y Ft(E)2367 3194 y Fz(\()p Fu(\022)s Fz(\))28 b(=)2622 3104 y Fl(p)p 2722 3104 463 4 v 90 x Fz(2)p Fu(E)g Fr(\000)22 b Fu(\032)3020 3166 y Fs(2)3020 3219 y Ft(c)3060 3194 y Fz(\()p Fu(\022)s Fz(\))p Fu(:)365 b Fz(\(2.29\))150 3435 y(Notice)38 b(that)h Fu(\032)730 3450 y Ft(c)804 3435 y Fz(and)g Fu(\030)1043 3450 y Ft(c)1116 3435 y Fz(are)f(smo)s(oth)g(and)h(p)s(erio)s(dic)e(solutions)h(to)h (\(2.6\))f(where)i Fu(E)k Fz(=)38 b Fu(E)3580 3450 y Ft(c)3615 3435 y Fz(.)62 b(By)150 3555 y(\(2.13\))42 b(w)m(e)j(ha)m(v)m(e)f Fu(\030)875 3570 y Ft(c)909 3555 y Fz(\()p Fu(\022)s Fz(\))i Fu(>)g Fz(0)d(for)f(all)f Fu(\022)47 b Fz(while)42 b Fu(\032)2005 3570 y Ft(c)2040 3555 y Fz(\()p Fu(\022)2123 3570 y Fs(1)2163 3555 y Fz(\))k(=)f(0)e (for)g(some)g Fu(\022)2920 3570 y Fs(1)2960 3555 y Fz(.)75 b(But)43 b Fu(\032)3316 3570 y Ft(c)3394 3555 y Fz(cannot)h(b)s(e)150 3676 y(iden)m(tically)30 b(zero,)k(otherwise)f(\(2.12\))e(w)m(ould)i (read)g(as)1632 3850 y Fu(d)p 1608 3894 99 4 v 1608 3986 a(d\022)1717 3917 y(\032)1767 3932 y Ft(c)1829 3917 y Fz(=)1933 3826 y Fl(p)p 2033 3826 156 4 v 91 x Fz(2)p Fu(E)2154 3932 y Ft(c)2210 3917 y Fz(+)22 b Fu(b:)1200 b Fz(\(2.30\))150 4135 y(Since)37 b(the)g(righ)m(t)e(hand)i(side)f(is)g (not)h(iden)m(tically)d(zero)j(this)f(leads)g(to)h(a)f(con)m (tradiction.)54 b(In)37 b(par-)150 4255 y(ticular,)31 b(this)h(means)1683 4299 y Fl(Z)1782 4326 y Fs(2)p Ft(\031)1738 4525 y Fs(0)1891 4368 y Fu(\032)1941 4383 y Ft(c)p 1891 4412 86 4 v 1895 4503 a Fu(\030)1938 4518 y Ft(c)1986 4435 y Fu(d\022)f(>)c Fz(0)p Fu(;)1284 b Fz(\(2.31\))150 4644 y(and)33 b(according)f(to)g(Prop)s(osition)f(2.4)h(it)f(follo)m (ws)h(that)g Fu(E)2281 4659 y Ft(d)2349 4644 y Fu(<)c(E)2525 4659 y Ft(c)2560 4644 y Fz(.)296 4765 y(No)m(w)h(let)f(us)h(pro)m(v)m (e)g(that)g Fu(E)1310 4780 y Ft(c)1372 4765 y Fu(<)f(E)1548 4780 y Ft(b)1582 4765 y Fz(.)42 b(Assume)30 b(on)e(the)h(con)m(trary)g (that)f Fu(E)2969 4780 y Ft(c)3031 4765 y Fz(=)g Fu(E)3207 4780 y Ft(b)3241 4765 y Fz(.)42 b(Pic)m(k)29 b Fu(\022)3569 4780 y Fs(1)3637 4765 y Fz(suc)m(h)150 4885 y(that)j Fu(\032)411 4900 y Ft(c)446 4885 y Fz(\()p Fu(\022)529 4900 y Fs(1)569 4885 y Fz(\))c(=)f(0)33 b(and)f Fu(\030)1052 4900 y Ft(c)1087 4885 y Fz(\()p Fu(\022)1170 4900 y Fs(1)1210 4885 y Fz(\))27 b(=)1379 4808 y Fr(p)p 1462 4808 156 4 v 77 x Fz(2)p Fu(E)1583 4900 y Ft(b)1617 4885 y Fz(.)43 b(Since)33 b Fu(b)23 b Fz(+)2104 4808 y Fr(p)p 2187 4808 V 77 x Fz(2)p Fu(E)2308 4900 y Ft(b)2370 4885 y Fr(\025)28 b Fz(0)k(w)m(e)i(learn)e(from)f(\(2.5\))h(that)1049 5059 y Fu(d)p 1025 5104 99 4 v 1025 5195 a(d\022)1134 5127 y Fz(\()p Fu(\030)1215 5142 y Ft(c)1272 5127 y Fr(\000)1371 5035 y Fl(p)p 1471 5035 156 4 v 92 x Fz(2)p Fu(E)1592 5142 y Ft(b)1626 5127 y Fz(\))c(=)f Fr(\000)p Fu(h)p Fz(\()p Fu(\030)2009 5142 y Ft(c)2066 5127 y Fr(\000)2166 5035 y Fl(p)p 2266 5035 V 92 x Fz(2)p Fu(E)2387 5142 y Ft(b)2421 5127 y Fz(\))22 b(+)g Fu(g)t Fz(;)44 b Fu(g)31 b Fr(\024)d Fz(0)p Fu(:)150 5344 y Fz(In)33 b(particular)1405 5441 y Fu(d)p 1380 5486 99 4 v 1380 5577 a(d\022)1506 5398 y Fl(n)1572 5509 y Fu(e)1617 5414 y Fg(R)1676 5468 y Ft(hd\036)1799 5509 y Fz(\()p Fu(\030)1880 5524 y Ft(c)1937 5509 y Fr(\000)2036 5417 y Fl(p)p 2136 5417 156 4 v 92 x Fz(2)p Fu(E)2257 5524 y Ft(b)2291 5509 y Fz(\))2329 5398 y Fl(o)2423 5509 y Fr(\024)c Fz(0)p Fu(;)1939 5776 y Fz(17)p eop %%Page: 18 21 18 20 bop 150 407 a Fz(whic)m(h)34 b(sho)m(ws)h(that)e Fu(\030)966 422 y Ft(c)1000 407 y Fz(\()p Fu(\022)s Fz(\))c Fr(\025)1259 330 y(p)p 1342 330 156 4 v 77 x Fz(2)p Fu(E)1463 422 y Ft(b)1531 407 y Fz(\(and)k(therefore)h Fu(\030)2212 422 y Ft(c)2247 407 y Fz(\()p Fu(\022)s Fz(\))29 b(=)2504 330 y Fr(p)p 2587 330 V 77 x Fz(2)p Fu(E)2708 422 y Ft(b)2743 407 y Fz(\))k(for)g Fu(\022)f(<)c(\022)3190 422 y Fs(1)3230 407 y Fz(.)46 b(This)33 b(implies)150 527 y(that)46 b Fu(\032)425 542 y Ft(c)460 527 y Fz(\()p Fu(\022)s Fz(\))51 b(=)g(0)46 b(for)g Fu(\022)54 b(<)d(\022)1291 542 y Fs(1)1331 527 y Fz(,)f(and)c(the)h(p)s(erio)s(dicit)m(y)d(giv)m(es)j(that)f Fu(\032)2822 542 y Ft(c)2857 527 y Fz(\()p Fu(\022)s Fz(\))g(is)g(iden)m(tically)e(zero,)150 648 y(con)m(tradicting)31 b(\(2.31\).)p 3754 648 4 66 v 3758 585 59 4 v 3758 648 V 3816 648 4 66 v 296 886 a(No)m(w)j(let)e(us)h(fo)s(cus)g(on)g Fu(E)1243 901 y Ft(d)1284 886 y Fz(,)f(again)g(in)g(the)h(case)h(when)g Fu(b)f Fz(is)f(not)h(constan)m(t.)45 b(Reasoning)32 b(as)h(w)m(e)150 1006 y(did)i(for)h Fu(E)546 1021 y Ft(c)581 1006 y Fz(,)g(there)h (exist)f(t)m(w)m(o)h(smo)s(oth)e(and)h(p)s(erio)s(dic)e(functions)i Fu(\032)2711 1021 y Ft(d)2788 1006 y Fz(and)g Fu(\030)3024 1021 y Ft(d)3100 1006 y Fz(whic)m(h)g(solv)m(e)g(\(2.6\))150 1127 y(with)c Fu(E)i Fz(=)27 b Fu(E)653 1142 y Ft(d)727 1127 y Fz(and)1390 1302 y Fu(\030)1433 1317 y Ft(d)1501 1302 y Fr(\025)h Fu(c)g(>)f Fz(0)p Fu(;)1969 1167 y Fl(Z)2069 1193 y Fs(2)p Ft(\031)2025 1392 y Fs(0)2178 1235 y Fu(\032)2228 1250 y Ft(d)p 2178 1279 91 4 v 2182 1371 a Fu(\030)2225 1386 y Ft(d)2279 1302 y Fu(d\022)j Fz(=)e(0)p Fu(:)991 b Fz(\(2.32\))150 1524 y(It)38 b(is)f(in)m(teresting)h(what)g(happ)s (ens)h(when)g(the)f(energy)h(decreases)h(from)d Fu(E)2974 1539 y Ft(c)3047 1524 y Fz(to)g Fu(E)3243 1539 y Ft(d)3284 1524 y Fz(.)59 b(A)m(t)38 b Fu(E)3588 1539 y Ft(c)3623 1524 y Fz(,)i(the)150 1644 y(radial)j(v)m(elo)s(cit)m(y)i(is)g(zero)h (somewhere)h(but)f(still)d(non-negativ)m(e;)51 b(then)46 b(for)f Fu(E)56 b Fr(2)50 b Fz(\()p Fu(E)3415 1659 y Ft(d)3456 1644 y Fu(;)17 b(E)3572 1659 y Ft(c)3606 1644 y Fz(\))46 b(the)150 1764 y(v)m(elo)s(cit)m(y)35 b Fu(\032)563 1779 y Ft(E)657 1764 y Fz(attains)f(more)h(and)g(more)f(negativ)m(e)h (v)-5 b(alues)35 b(and)g(the)g(in)m(tegral)e(in)h(\(2.14\))g(b)s (ecomes)150 1885 y(smaller)d(and)h(smaller)f(when)i Fu(E)1339 1900 y Ft(d)1413 1885 y Fz(is)f(approac)m(hed.)150 2088 y Fv(Lemma)37 b(2.9.)42 b FA(If)34 b Fu(b)i FA(is)e(not)h(c)-5 b(onstant)35 b(then)1405 2317 y Fz(1)p 1405 2361 49 4 v 1405 2452 a(2)1480 2244 y Fl(\024)1533 2384 y Fr(\000)1650 2317 y Fz(1)p 1620 2361 108 4 v 1620 2452 a(2)p Fu(\031)1754 2248 y Fl(Z)1854 2275 y Fs(2)p Ft(\031)1810 2474 y Fs(0)1953 2384 y Fu(b)p Fz(\()p Fu(\022)s Fz(\))p Fu(d\022)2217 2244 y Fl(\025)2270 2258 y Fs(2)2337 2384 y Fu(<)28 b(E)2513 2399 y Ft(d)2553 2384 y Fu(:)150 2680 y Fv(Pro)s(of)p Fz(.)43 b(Since)33 b(w)m(e)h(ha)m(v)m(e)f Fu(@)1166 2695 y Ft(\022)1206 2680 y Fu(\032)1256 2695 y Ft(d)1324 2680 y Fz(=)28 b Fu(b)23 b Fz(+)1590 2595 y Fl(p)p 1689 2595 374 4 v 1689 2680 a Fz(2)p Fu(E)1810 2695 y Ft(d)1873 2680 y Fr(\000)f Fu(\032)2022 2645 y Fs(2)2022 2707 y Ft(d)2063 2680 y Fz(,)33 b(then)g(b)m(y)h(in)m(tegration)c(w)m(e)k(get) 1207 2970 y(0)27 b(=)1387 2835 y Fl(Z)1487 2861 y Fs(2)p Ft(\031)1442 3060 y Fs(0)1585 2970 y Fu(b)p Fz(\()p Fu(\022)s Fz(\))p Fu(d\022)f Fz(+)1970 2835 y Fl(Z)2070 2861 y Fs(2)p Ft(\031)2025 3060 y Fs(0)2169 2851 y Fl(q)p 2268 2851 V 2268 2970 a Fz(2)p Fu(E)2389 2985 y Ft(d)2452 2970 y Fr(\000)c Fu(\032)2601 2936 y Fs(2)2601 2998 y Ft(d)2642 2970 y Fu(d\022)s(:)150 3265 y Fz(Since)33 b Fu(\032)455 3280 y Ft(d)528 3265 y Fz(is)f(not)h(iden)m(tically)d (zero)j(w)m(e)h(get)e Fr(\000)1877 3185 y Fl(R)1944 3211 y Fs(2)p Ft(\031)1924 3299 y Fs(0)2043 3265 y Fu(b)p Fz(\()p Fu(\022)s Fz(\))p Fu(d\022)f(<)c Fz(2)p Fu(\031)2546 3188 y Fr(p)p 2629 3188 162 4 v 77 x Fz(2)p Fu(E)2750 3280 y Ft(d)2823 3265 y Fz(and)33 b(w)m(e)g(are)g(done.)p 3754 3265 4 66 v 3758 3202 59 4 v 3758 3265 V 3816 3265 4 66 v 296 3503 a(No)m(w)g(let)f(us)i(summarize)d(what)i(w)m(e)g (obtained)f(in)g(the)h(last)f(few)h(lemmas:)150 3707 y Fv(Prop)s(osition)38 b(2.10.)43 b FA(If)37 b Fu(b)g FA(is)g(a)g(ne)-5 b(gative)36 b(c)-5 b(onstant)37 b(then)f Fu(E)2467 3722 y Ft(d)2540 3707 y Fz(=)31 b Fu(E)2719 3722 y Ft(c)2786 3707 y Fz(=)g Fu(E)2965 3722 y Ft(b)3031 3707 y Fz(=)h Fu(b)3180 3671 y Fs(2)3220 3707 y Fu(=)p Fz(2)p FA(.)50 b(If)37 b Fu(b)g FA(is)g(not)150 3827 y(c)-5 b(onstant,)34 b(then)691 4056 y Fz(min)15 b Fr(j)p Fu(b)p Fr(j)967 4020 y Fs(2)p 691 4100 316 4 v 824 4191 a Fz(2)1044 4123 y Fu(<)1157 4056 y Fz(1)p 1157 4100 49 4 v 1157 4191 a(2)1233 3983 y Fl(\024)1285 4123 y Fr(\000)1402 4056 y Fz(1)p 1372 4100 108 4 v 1372 4191 a(2)p Fu(\031)1507 3987 y Fl(Z)1607 4014 y Fs(2)p Ft(\031)1562 4213 y Fs(0)1705 4123 y Fu(b)p Fz(\()p Fu(\022)s Fz(\))p Fu(d\022)1969 3983 y Fl(\025)2022 3997 y Fs(2)2090 4123 y Fu(<)27 b(E)2265 4138 y Ft(d)2333 4123 y Fu(<)h(E)2509 4138 y Ft(c)2572 4123 y Fu(<)f(E)2747 4138 y Ft(b)2809 4123 y Fz(=)2923 4056 y(max)16 b Fr(j)p Fu(b)p Fr(j)3218 4020 y Fs(2)p 2923 4100 335 4 v 3066 4191 a Fz(2)3267 4123 y Fu(:)150 4448 y Fp(2.4)135 b(A)45 b(classical)h(Mourre)f (estimate)i(ab)t(o)l(v)l(e)f Fi(E)2610 4466 y Fh(d)150 4633 y Fz(W)-8 b(e)37 b(no)m(w)g(in)m(v)m(estigate)g(the)g(long)e(term) h(b)s(eha)m(vior)h(of)f(an)g(arbitrary)g(classical)f(orbit,)h (de\014ned)i(for)150 4754 y(ev)m(ery)31 b Fu(t)d Fr(\025)g Fz(0,)h(corresp)s(onding)g(to)g(an)g(energy)h Fu(E)j(>)28 b(E)2137 4769 y Ft(d)2178 4754 y Fz(.)42 b(W)-8 b(e)29 b(will)e(see)j(that)f(the)g(p)s(erio)s(dic)e(solution)150 4874 y(pro)m(vided)33 b(b)m(y)h(Corollary)c(2.7)j(generates)g(an)g (attractor.)296 4994 y(Consider)38 b(an)f(orbit)f(\()p Fu(r)s Fz(\()p Fu(t)p Fz(\))p Fu(;)17 b(\022)s Fz(\()p Fu(t)p Fz(\)\))37 b(solving)f(\(1.6\){\(1.9\))g(and)h(ob)s(eying)g (\(1.10\))f(with)h Fu(E)42 b(>)35 b(E)3757 5009 y Ft(d)3798 4994 y Fz(.)150 5115 y(W)-8 b(e)33 b(use)h(the)f(notation)e(\()p Fu(\032)1133 5130 y Ft(E)1193 5115 y Fu(;)17 b(\030)1280 5130 y Ft(E)1338 5115 y Fz(\))33 b(for)f(the)h(p)s(erio)s(dic)e (solution)g(pro)m(vided)i(b)m(y)g(Corollary)e(2.7.)296 5235 y(F)-8 b(or)32 b(ev)m(ery)j Fu(C)f(>)28 b Fz(0)k(de\014ne)954 5455 y Fu(A)1027 5470 y Ft(C)1086 5455 y Fz(\()p Fu(t)p Fz(\))c(=)g Fr(f)p Fu(@)1430 5470 y Ft(E)1490 5455 y Fu(\032)1540 5470 y Ft(E)1600 5455 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))22 b(+)g Fu(C)7 b Fz([)p Fu(\032)p Fz(\()p Fu(t)p Fz(\))22 b Fr(\000)h Fu(\032)2392 5470 y Ft(E)2452 5455 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\)])p Fr(g)f(\001)g Fu(r)s Fz(\()p Fu(t)p Fz(\))p Fu(:)555 b Fz(\(2.33\))1939 5776 y(18)p eop %%Page: 19 22 19 21 bop 150 407 a Fv(Prop)s(osition)31 b(2.11.)38 b FA(F)-7 b(or)30 b(every)h Fu(\017)d(>)f Fz(0)p FA(,)32 b(ther)-5 b(e)31 b(exists)f Fu(T)42 b(>)27 b Fz(1)k FA(lar)-5 b(ge)30 b(enough)g(\(dep)-5 b(ending)29 b(on)i(the)150 527 y(p)-5 b(articular)35 b(orbit\))g(such)f(that)1319 744 y Fu(\032)p Fz(\()p Fu(t)p Fz(\))23 b Fr(\000)f Fu(\032)1652 759 y Ft(E)1712 744 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))28 b Fr(\025)g(\000)p Fu(\017;)117 b Fr(8)p Fu(t)28 b Fr(\025)h Fu(T)8 b(:)920 b Fz(\(2.34\))150 961 y Fv(Pro)s(of)p Fz(.)59 b(W)-8 b(e)38 b(\014rst)g(compute)g(the)g(deriv)-5 b(ativ)m(e)37 b(of)h Fu(A)2103 976 y Ft(C)2200 961 y Fz(with)f(resp)s(ect)i(to)e Fu(t)p Fz(;)k(using)d(the)g(Hamilton)150 1082 y(equations)33 b(w)m(e)g(get)g(\(also)f(use)h(\(2.24\)\))519 1281 y Fu(d)p 501 1325 86 4 v 501 1416 a(dt)597 1348 y(r)s(@)695 1363 y Ft(E)755 1348 y Fu(\032)805 1363 y Ft(E)892 1348 y Fz(=)28 b Fu(\030)5 b(@)1095 1363 y Ft(E)1154 1348 y Fu(\030)1197 1363 y Ft(E)1279 1348 y Fz(+)22 b Fu(\032@)1478 1363 y Ft(E)1538 1348 y Fu(\032)1588 1363 y Ft(E)1676 1348 y Fz(=)27 b(1)22 b(+)g(\()p Fu(\030)27 b Fr(\000)22 b Fu(\030)2198 1363 y Ft(E)2257 1348 y Fz(\))p Fu(@)2346 1363 y Ft(E)2406 1348 y Fu(\030)2449 1363 y Ft(E)2531 1348 y Fz(+)g(\()p Fu(\032)g Fr(\000)h Fu(\032)2889 1363 y Ft(E)2949 1348 y Fz(\))p Fu(@)3038 1363 y Ft(E)3098 1348 y Fu(\032)3148 1363 y Ft(E)3208 1348 y Fu(;)341 b Fz(\(2.35\))150 1590 y(and)522 1764 y Fu(d)p 504 1809 V 504 1900 a(dt)600 1832 y(r)s Fz(\()p Fu(\032)22 b Fr(\000)h Fu(\032)907 1847 y Ft(E)967 1832 y Fz(\))k(=)h(\()p Fu(\030)e Fr(\000)d Fu(\030)1386 1847 y Ft(E)1445 1832 y Fz(\))p Fu(\030)k Fz(+)22 b(\()p Fu(\032)g Fr(\000)h Fu(\032)1911 1847 y Ft(E)1971 1832 y Fz(\))p Fu(\032)k Fz(=)2200 1764 y(1)p 2200 1809 49 4 v 2200 1900 a(2)2259 1832 y(\()p Fu(\032)22 b Fr(\000)h Fu(\032)2519 1847 y Ft(E)2579 1832 y Fz(\))2617 1791 y Fs(2)2678 1832 y Fz(+)2786 1764 y(1)p 2786 1809 V 2786 1900 a(2)2845 1832 y(\()p Fu(\030)j Fr(\000)d Fu(\030)3095 1847 y Ft(E)3154 1832 y Fz(\))3192 1791 y Fs(2)3576 1832 y Fz(\(2.36\))150 2074 y(where)34 b(the)f(last)f(equalit)m(y)g(is)g(a)g(consequence)k(of)c(energy)i (conserv)-5 b(ation)32 b(and)h(the)g(iden)m(tit)m(y)221 2330 y(\()p Fu(\030)27 b Fr(\000)c Fu(\030)472 2345 y Ft(E)531 2330 y Fz(\))p Fu(\030)j Fz(+)c(\()p Fu(\032)h Fr(\000)f Fu(\032)996 2345 y Ft(E)1056 2330 y Fz(\))p Fu(\032)28 b Fz(=)1286 2263 y(1)p 1286 2307 V 1286 2398 a(2)1344 2330 y(\()p Fu(\032)1432 2289 y Fs(2)1494 2330 y Fz(+)22 b Fu(\030)1640 2289 y Fs(2)1679 2330 y Fz(\))g Fr(\000)1849 2263 y Fz(1)p 1849 2307 V 1849 2398 a(2)1907 2330 y(\()p Fu(\032)1995 2289 y Fs(2)1995 2355 y Ft(E)2077 2330 y Fz(+)g Fu(\030)2223 2289 y Fs(2)2218 2355 y Ft(E)2278 2330 y Fz(\))g(+)2446 2263 y(1)p 2446 2307 V 2446 2398 a(2)2504 2330 y(\()p Fu(\032)h Fr(\000)f Fu(\032)2764 2345 y Ft(E)2824 2330 y Fz(\))2862 2289 y Fs(2)2924 2330 y Fz(+)3032 2263 y(1)p 3032 2307 V 3032 2398 a(2)3091 2330 y(\()p Fu(\030)k Fr(\000)d Fu(\030)3341 2345 y Ft(E)3400 2330 y Fz(\))3438 2289 y Fs(2)3477 2330 y Fu(:)72 b Fz(\(2.37\))150 2578 y(Com)m(bining)31 b(\(2.33\),)h(\(2.35\))g(and)g(\(2.36\))g(w)m(e) i(get)354 2777 y Fu(dA)478 2792 y Ft(C)p 354 2821 183 4 v 403 2913 a Fu(dt)575 2844 y Fz(=)27 b(1)22 b(+)g(\()p Fu(\030)27 b Fr(\000)c Fu(\030)1098 2859 y Ft(E)1157 2844 y Fz(\))p Fu(@)1246 2859 y Ft(E)1306 2844 y Fu(\030)1349 2859 y Ft(E)1430 2844 y Fz(+)f(\()p Fu(\032)g Fr(\000)h Fu(\032)1788 2859 y Ft(E)1848 2844 y Fz(\))p Fu(@)1937 2859 y Ft(E)1997 2844 y Fu(\032)2047 2859 y Ft(E)2129 2844 y Fz(+)2237 2777 y Fu(C)p 2237 2821 77 4 v 2251 2913 a Fz(2)2346 2844 y Fr(\001)f Fz([\()p Fu(\032)g Fr(\000)h Fu(\032)2683 2859 y Ft(E)2743 2844 y Fz(\))2781 2803 y Fs(2)2842 2844 y Fz(+)f(\()p Fu(\030)27 b Fr(\000)c Fu(\030)3191 2859 y Ft(E)3250 2844 y Fz(\))3288 2803 y Fs(2)3327 2844 y Fz(])p Fu(:)195 b Fz(\(2.38\))150 3092 y(Since)39 b Fu(@)462 3107 y Ft(E)522 3092 y Fu(\030)565 3107 y Ft(E)663 3092 y Fz(and)h Fu(@)911 3107 y Ft(E)971 3092 y Fu(\032)1021 3107 y Ft(E)1119 3092 y Fz(are)f(b)s(ounded,)j(it)c (is)g(easy)j(to)d(see)i(that)f(there)h(exists)g Fu(C)7 b Fz(\()p Fu(E)f Fz(\))38 b Fu(>)h Fz(0)f(\(i.e.)150 3213 y(only)32 b(dep)s(ending)h(on)f Fu(E)6 b Fz(\))33 b(suc)m(h)h(that)e(if)g Fu(C)i Fr(\025)28 b Fu(C)7 b Fz(\()p Fu(E)f Fz(\))1564 3412 y Fu(dA)1688 3427 y Ft(C)p 1564 3456 183 4 v 1612 3547 a Fu(dt)1784 3479 y Fr(\025)29 b Fz(1)p Fu(=)p Fz(2)p Fu(;)113 b(t)28 b Fr(\025)g Fz(1)p Fu(:)1155 b Fz(\(2.39\))150 3727 y(Then)34 b(in)m(tegrating)d(\(2.33\)) g(w)m(e)j(obtain)e(that)g(for)g Fu(T)46 b Fz(large)32 b(enough:)1564 3986 y Fu(A)1637 4001 y Ft(C)1696 3986 y Fz(\()p Fu(t)p Fz(\))c Fr(\025)1957 3918 y Fu(t)p 1950 3963 49 4 v 1950 4054 a Fz(4)2009 3986 y Fu(;)114 b(t)28 b Fr(\025)g Fu(T)8 b(:)1166 b Fz(\(2.40\))150 4233 y(Using)32 b(\(2.33\))g(w)m(e)i(\014nally)d(get)1057 4495 y Fu(\032)p Fz(\()p Fu(t)p Fz(\))22 b Fr(\000)h Fu(\032)1390 4510 y Ft(E)1450 4495 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))f(+)1829 4428 y(1)p 1815 4472 77 4 v 1815 4563 a Fu(C)1902 4495 y(@)1953 4510 y Ft(E)2013 4495 y Fu(\032)2063 4510 y Ft(E)2123 4495 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))28 b Fu(>)f Fz(0)p Fu(;)114 b(t)28 b Fr(\025)g Fu(T)150 4737 y Fz(and)33 b(the)g(pro)s(of)e(follo)m(ws)h(b)m(y)h(c)m(ho)s (osing)f Fu(C)40 b Fz(large)31 b(enough.)p 3754 4737 4 66 v 3758 4675 59 4 v 3758 4737 V 3816 4737 4 66 v 150 5188 a Fw(3)161 b(Propagation)54 b(estimates)f(in)h(classical)i (mec)l(hanics)150 5407 y Fz(In)38 b(this)f(section)h(w)m(e)h(w)m(ork)f (with)f(classical)f(orbits)h(that)h(are)g(de\014ned)h(for)e(all)e (times)i Fu(t)f Fr(\025)h Fz(0.)59 b(W)-8 b(e)150 5527 y(w)m(an)m(t)32 b(to)f(sho)m(w)i(that)e(the)h(radial)e(and)h(transv)m (erse)j(v)m(elo)s(cit)m(y)d Fu(\032)p Fz(\()p Fu(t)p Fz(\))h(and)g Fu(\030)5 b Fz(\()p Fu(t)p Fz(\))30 b(corresp)s(onding)i (to)f(an)1939 5776 y(19)p eop %%Page: 20 23 20 22 bop 150 407 a Fz(arbitrary)34 b(orbit)g(\()p Fu(r)s Fz(\()p Fu(t)p Fz(\))p Fu(;)17 b(\022)s Fz(\()p Fu(t)p Fz(\)\))34 b(at)h(energy)h Fu(E)i(>)31 b(E)2004 422 y Ft(d)2045 407 y Fz(,)36 b(will)c(get)j(arbitrarily)d(close)j(to)g Fu(\032)3338 422 y Ft(E)3398 407 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))g(and)150 527 y Fu(\030)193 542 y Ft(E)252 527 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))e(for)f (large)f(enough)i(times.)296 648 y(Let)c(us)h(in)m(tro)s(duce)f(four)f (real-v)-5 b(alued)28 b(cut-o\013)g(functions)h Fu(F)41 b Fr(2)28 b Fu(C)2705 611 y Fm(1)2780 648 y Fz(\()p Fn(R)5 b Fz(\);)36 b(they)30 b(dep)s(end)g(on)f(real)150 768 y(parameters)k Fu(a)27 b(<)h(b)g(<)g(c)f(<)h(d)p Fz(:)1131 1104 y Fu(F)1194 1119 y Fs(+)1281 1104 y Fz(=)g Fu(F)1462 1057 y Ft(a;b)1448 1126 y Fs(+)1553 1104 y Fz(;)1131 1249 y Fu(F)1194 1264 y Fs(+)1254 1249 y Fz(\()p Fu(x)p Fz(\))g(=)f(0)h(for)f Fu(x)h(<)f(a;)45 b(F)2109 1264 y Fs(+)2168 1249 y Fz(\()p Fu(x)p Fz(\))28 b(=)g(1)f(for)g Fu(x)h(>)g(b;)718 b Fz(\(3.1\))1131 1404 y Fu(F)1208 1363 y Fm(0)1194 1428 y Fs(+)1254 1404 y Fz(\()p Fu(x)p Fz(\))28 b Fr(\025)g Fz(0)p Fu(;)1638 1316 y Fl(p)p 1738 1316 123 4 v 88 x Fu(F)1801 1419 y Fs(+)1860 1404 y Fu(;)1904 1316 y Fl(p)p 2003 1316 293 4 v 2003 1404 a Fz(1)22 b Fr(\000)h Fu(F)2237 1419 y Fs(+)2296 1404 y Fu(;)2340 1317 y Fl(p)p 2439 1317 123 4 v 2439 1404 a Fu(F)2516 1369 y Fm(0)2502 1426 y Fs(+)2589 1404 y Fr(2)28 b Fu(C)2760 1363 y Fm(1)2835 1404 y Fz(\()p Fn(R)5 b Fz(\))p Fu(:)1529 1740 y(F)1592 1755 y Fm(\000)1679 1740 y Fz(=)27 b Fu(F)1859 1693 y Ft(c;d)1845 1762 y Fm(\000)1977 1740 y Fz(=)h(1)22 b Fr(\000)h Fu(F)2329 1693 y Ft(c;d)2315 1762 y Fs(+)2419 1740 y Fu(:)1178 b Fz(\(3.2\))1449 2033 y Fu(F)1512 2048 y Fs(+)p Fm(\000)1654 2033 y Fz(=)28 b Fu(F)1835 1986 y Ft(a;b;c;d)1821 2055 y Fs(+)p Fm(\000)2060 2033 y Fz(=)f Fu(F)2240 1986 y Ft(a;b)2226 2055 y Fs(+)2331 2033 y Fu(F)2408 1986 y Ft(c;d)2394 2055 y Fm(\000)2499 2033 y Fu(:)1098 b Fz(\(3.3\))1159 2352 y Fu(F)1222 2367 y Fs(++)1336 2352 y Fz(\()p Fu(x)p Fz(\))28 b(=)g Fu(F)1676 2304 y Ft(a;b;c;d)1662 2373 y Fs(++)1873 2352 y Fz(\()p Fu(x)p Fz(\))g(=)2136 2216 y Fl(Z)2235 2242 y Ft(x)2191 2442 y(a)2296 2352 y Fu(F)2373 2304 y Ft(a;b;c;d)2359 2373 y Fs(+)p Fm(\000)2570 2352 y Fz(\()p Fu(s)p Fz(\))p Fu(ds:)808 b Fz(\(3.4\))150 2672 y Fp(3.1)135 b(A)45 b(minimal)h(v)l(elo)t(cit)l(y)g(b)t(ound:)59 b(the)46 b(particle)g(lea)l(v)l(es)h(the)e(origin)150 2856 y Fv(Prop)s(osition)k (3.1.)f FA(Ther)-5 b(e)44 b(exists)h Fu(d)i(>)h Fz(0)d FA(smal)5 b(l)44 b(enough)h(such)h(that)f(for)h(every)f Fu(c)i(<)g(d)f FA(and)150 2977 y Fu(F)213 2992 y Fm(\000)300 2977 y Fz(=)27 b Fu(F)480 2930 y Ft(c;d)466 2999 y Fm(\000)606 2977 y FA(we)34 b(have)1560 3097 y Fz(lim)1545 3157 y Ft(t)p Fm(!1)1728 3097 y Fr(j)p Fu(F)1819 3112 y Fm(\000)1878 3097 y Fz(\()p Fu(r)s Fz(\()p Fu(t)p Fz(\))p Fu(=t)p Fz(\))p Fr(j)27 b Fz(=)g(0)p Fu(:)1194 b Fz(\(3.5\))150 3302 y FA(In)33 b(p)-5 b(articular,)35 b(\014xing)e Fu(c)28 b Fz(=)f Fu(d=)p Fz(2)33 b FA(implies)g(the)i(existenc)-5 b(e)33 b(of)h Fu(T)47 b FA(lar)-5 b(ge)34 b(enough)g(such)g(that)g Fu(r)s Fz(\()p Fu(t)p Fz(\))p Fu(=t)28 b Fr(\025)150 3423 y Fu(d=)p Fz(2)34 b FA(for)h(al)5 b(l)34 b Fu(t)28 b Fr(\025)g Fu(T)14 b FA(.)150 3622 y Fv(Pro)s(of)p Fz(.)43 b(The)32 b(metho)s(ds)f(w)m(e)h(use)g(here)g(will)c(from)i(no)m(w)i(on) f(b)s(e)g(emplo)m(y)m(ed)g(throughout)g(the)h(pap)s(er)150 3743 y(for)g(b)s(oth)g(classical)f(and)i(quan)m(tum)g(mec)m(hanics.)44 b(W)-8 b(e)33 b(\014rst)g(need)g(the)g(follo)m(wing)d(lemma:)150 3943 y Fv(Lemma)41 b(3.2.)i FA(Ther)-5 b(e)37 b(exists)h(a)f Fu(d)1459 3958 y Fs(1)1531 3943 y Fu(>)c Fz(0)38 b FA(smal)5 b(l)37 b(enough)g(such)g(that)i(for)e(al)5 b(l)38 b Fu(c)3093 3958 y Fs(1)3165 3943 y Fu(<)33 b(d)3325 3958 y Fs(1)3402 3943 y FA(and)k Fu(F)3657 3958 y Fm(\000)3749 3943 y Fz(=)150 4063 y Fu(F)227 4016 y Ft(c)258 4025 y Fj(1)292 4016 y Ft(;d)348 4025 y Fj(1)213 4085 y Fm(\000)421 4063 y FA(we)e(have)1375 4102 y Fl(Z)1475 4128 y Fm(1)1430 4328 y Fs(1)1576 4170 y Fz(1)p 1576 4215 49 4 v 1583 4306 a Fu(t)1635 4238 y Fr(j)p Fu(F)1740 4197 y Fm(0)1726 4262 y(\000)1785 4238 y Fr(j)p Fz(\()p Fu(A)1924 4253 y Ft(C)1982 4238 y Fu(=t)p Fz(\))p Fu(dt)28 b Fr(\024)g Fz(Const)q Fu(;)1024 b Fz(\(3.6\))150 4457 y FA(wher)-5 b(e)34 b(the)h(ab)-5 b(ove)34 b(c)-5 b(onstant)35 b(do)-5 b(es)34 b(not)h(dep)-5 b(end)34 b(on)g(the)h(p)-5 b(articular)35 b(orbit.)44 b(Mor)-5 b(e)g(over,)1540 4673 y Fu(F)1617 4622 y Ft(c)1648 4631 y Fj(1)1682 4622 y Ft(=)p Fs(3)p Ft(;c)1803 4631 y Fj(1)1837 4622 y Ft(=)p Fs(2)1603 4695 y Fm(\000)1912 4673 y Fz(\()p Fu(A)2023 4688 y Ft(C)2082 4673 y Fu(=t)p Fz(\))28 b Fr(!)f Fz(0)p Fu(;)1189 b Fz(\(3.7\))150 4889 y FA(but)35 b(the)g(r)-5 b(ate)35 b(of)g(c)-5 b(onver)g(genc)g(e) 33 b(dep)-5 b(ends)34 b(on)g(the)h(p)-5 b(articular)35 b(orbit.)150 5089 y Fv(Pro)s(of.)44 b Fz(W)-8 b(e)33 b(start)g(with)f(\(3.6\).)44 b(Di\013eren)m(tiate)32 b(the)h(b)s(ounded)g(propagation)f(observ)-5 b(able)33 b(\010\()p Fu(t)p Fz(\))28 b(=)150 5209 y Fu(F)213 5224 y Fm(\000)272 5209 y Fz(\()p Fu(A)383 5224 y Ft(C)442 5209 y Fz(\()p Fu(t)p Fz(\))p Fu(=t)p Fz(\))33 b(and)g(get)f(\(the)h (deriv)-5 b(ativ)m(e)32 b(of)g Fu(F)1887 5224 y Fm(\000)1979 5209 y Fz(is)g(negativ)m(e\))904 5482 y Fu(@)955 5497 y Ft(t)985 5482 y Fz(\010\()p Fu(t)p Fz(\))c(=)g Fr(\000j)p Fu(F)1480 5441 y Fm(0)1466 5507 y(\000)1525 5482 y Fr(j)p Fz(\()p Fu(A)1664 5497 y Ft(C)1723 5482 y Fz(\()p Fu(t)p Fz(\))p Fu(=t)p Fz(\))22 b Fr(\001)2028 5342 y Fl(\022)2111 5415 y Fz(1)p 2111 5459 V 2118 5551 a Fu(t)2170 5482 y(@)2221 5497 y Ft(t)2251 5482 y Fu(A)2324 5497 y Ft(C)2383 5482 y Fz(\()p Fu(t)p Fz(\))h Fr(\000)2639 5415 y Fz(1)p 2626 5459 75 4 v 2626 5551 a Fu(t)2661 5522 y Fs(2)2711 5482 y Fu(A)2784 5497 y Ft(C)2843 5482 y Fz(\()p Fu(t)p Fz(\))2954 5342 y Fl(\023)3044 5482 y Fu(;)1939 5776 y Fz(20)p eop %%Page: 21 24 21 23 bop 150 407 a Fz(or)32 b(equiv)-5 b(alen)m(tly)32 b(\(here)h Fu(t)28 b Fr(\025)h Fz(1\))884 593 y(1)p 866 637 85 4 v 866 728 a(2)p Fu(t)960 660 y Fr(j)p Fu(F)1065 619 y Fm(0)1051 685 y(\000)1110 660 y Fr(j)p Fz(\()p Fu(A)1249 675 y Ft(C)1308 660 y Fz(\()p Fu(t)p Fz(\))p Fu(=t)p Fz(\))e Fr(\024)i Fu(@)1725 675 y Ft(t)1755 660 y Fu(A)1828 675 y Ft(C)1887 660 y Fz(\()p Fu(t)p Fz(\))2008 593 y(1)p 2008 637 49 4 v 2015 728 a Fu(t)2067 660 y Fr(j)p Fu(F)2172 619 y Fm(0)2158 685 y(\000)2217 660 y Fr(j)p Fz(\()p Fu(A)2356 675 y Ft(C)2415 660 y Fz(\()p Fu(t)p Fz(\))p Fu(=t)p Fz(\))e(=)h Fr(\000)p Fu(@)2907 675 y Ft(t)2937 660 y Fz(\010\()p Fu(t)p Fz(\))1374 957 y(+)p Fr(j)p Fu(F)1555 916 y Fm(0)1541 982 y(\000)1599 957 y Fr(j)p Fz(\()p Fu(A)1738 972 y Ft(C)1797 957 y Fz(\()p Fu(t)p Fz(\))p Fu(=t)p Fz(\))22 b Fr(\001)g Fz(\()p Fu(A)2213 972 y Ft(C)2272 957 y Fz(\()p Fu(t)p Fz(\))p Fu(=t)p Fz(\))2515 889 y(1)p 2515 934 V 2522 1025 a Fu(t)2574 957 y(;)150 1157 y Fz(where)37 b(the)g(\014rst)g(inequalit)m(y)e(is)g (giv)m(en)i(b)m(y)g(\(2.39\).)53 b(Since)37 b Fu(F)2419 1172 y Fm(\000)2514 1157 y Fz(is)f(supp)s(orted)h(in)e(\()p Fr(\0001)p Fu(;)17 b(d)3505 1172 y Fs(1)3544 1157 y Fz(],)37 b(then)150 1278 y Fr(j)p Fu(F)255 1242 y Fm(0)241 1302 y(\000)300 1278 y Fr(j)p Fz(\()p Fu(A)439 1293 y Ft(C)498 1278 y Fz(\()p Fu(t)p Fz(\))p Fu(=t)p Fz(\))22 b Fr(\001)g Fz(\()p Fu(A)914 1293 y Ft(C)973 1278 y Fz(\()p Fu(t)p Fz(\))p Fu(=t)p Fz(\))28 b Fr(\024)g(j)p Fu(F)1444 1242 y Fm(0)1430 1302 y(\000)1489 1278 y Fr(j)p Fz(\()p Fu(A)1628 1293 y Ft(C)1687 1278 y Fz(\()p Fu(t)p Fz(\))p Fu(=t)p Fz(\))22 b Fr(\001)g Fu(d)2043 1293 y Fs(1)2114 1278 y Fz(hence)1191 1410 y Fl(\022)1274 1483 y Fz(1)p 1274 1528 V 1274 1619 a(2)1355 1551 y Fr(\000)h Fu(d)1506 1566 y Fs(1)1545 1410 y Fl(\023)1645 1483 y Fz(1)p 1645 1528 V 1652 1619 a Fu(t)1703 1551 y Fr(j)p Fu(F)1808 1509 y Fm(0)1794 1575 y(\000)1853 1551 y Fr(j)p Fz(\()p Fu(A)1992 1566 y Ft(C)2051 1551 y Fz(\()p Fu(t)p Fz(\))p Fu(=t)p Fz(\))28 b Fr(\024)g(\000)p Fu(@)2545 1566 y Ft(t)2576 1551 y Fz(\010\()p Fu(t)p Fz(\))p Fu(:)150 1817 y Fz(Because)34 b(\010)28 b Fr(\024)g Fz(1,)33 b(c)m(ho)s(osing)f Fu(d)1282 1832 y Fs(1)1349 1817 y Fr(2)c Fz(\(0)p Fu(;)17 b Fz(1)p Fu(=)p Fz(2\))31 b(and)i(after)f(in)m(tegration)f(\(3.6\))h (follo)m(ws.)296 1937 y(A)k(few)g(w)m(ords)g(ab)s(out)f(\(3.7\).)51 b(W)-8 b(e)36 b(could)f(sa)m(y)h(that)f(due)h(to)f(\(2.40\))g(the)h (limit)c(follo)m(ws)i(easily)-8 b(.)150 2058 y(But)34 b(w)m(e)h(use)g(another)f(argumen)m(t)g(whic)m(h)h(is)e(based)i(on)f (\(3.6\))g(and)g(can)g(b)s(e)g(generalized)g(later)f(on)150 2178 y(to)k(quan)m(tum)h(mec)m(hanics.)58 b(W)-8 b(e)38 b(pro)m(v)m(e)g(t)m(w)m(o)g(things:)53 b(\014rst,)39 b(that)e(the)h(limit)c(exists;)41 b(second,)f(the)150 2298 y(limit)29 b(equals)k(zero.)296 2419 y(No)m(w)f(consider)e Fu(F)971 2368 y Ft(c)1002 2377 y Fj(1)1036 2368 y Ft(=)p Fs(3)p Ft(;c)1157 2377 y Fj(1)1192 2368 y Ft(=)p Fs(2)957 2440 y Fm(\000)1266 2419 y Fz(\()p Fu(A)1377 2434 y Ft(C)1437 2419 y Fz(\()p Fu(t)p Fz(\))p Fu(=t)p Fz(\).)42 b(T)-8 b(o)31 b(pro)m(v)m(e)h(the)f(existence)h(of)e(a)h(limit)c(when)32 b Fu(t)27 b Fr(!)h(1)i Fz(w)m(e)150 2539 y(emplo)m(y)38 b(a)g(Co)s(ok)h(t)m(yp)s(e)g(argumen)m(t,)h(that)f(is)f(w)m(e)h(sho)m (w)h(the)f(absolute)f(in)m(tegrabilit)m(y)e(of)i(its)g(time)150 2659 y(deriv)-5 b(ativ)m(e:)1232 2780 y Fr(j)p Fu(@)1311 2795 y Ft(t)1341 2780 y Fu(F)1418 2729 y Ft(c)1449 2738 y Fj(1)1483 2729 y Ft(=)p Fs(3)p Ft(;c)1604 2738 y Fj(1)1638 2729 y Ft(=)p Fs(2)1404 2802 y Fm(\000)1713 2780 y Fz(\()p Fu(A)1824 2795 y Ft(C)1883 2780 y Fz(\()p Fu(t)p Fz(\))p Fu(=t)p Fz(\))p Fr(j)28 b(2)g Fu(L)2332 2739 y Fs(1)2372 2780 y Fz(\(\(1)p Fu(;)17 b Fr(1)p Fz(\)\))p Fu(:)150 2949 y Fz(But)33 b(w)m(e)g(see)h(that)e(on)g(the)h(giv)m(en)g(orbit,)f (the)g(ab)s(o)m(v)m(e)i(deriv)-5 b(ativ)m(e)32 b(ma)m(y)g(b)s(e)h(b)s (ounded)g(b)m(y)g(\(w)m(e)h(omit)150 3070 y(the)f(sup)s(erscripts\)) 1502 3235 y(Const)23 b Fr(\001)f(j)p Fu(F)1929 3193 y Fm(0)1915 3259 y(\000)1974 3235 y Fr(j)p Fz(\()p Fu(A)2113 3250 y Ft(C)2171 3235 y Fz(\()p Fu(t)p Fz(\))p Fu(=t)p Fz(\))2414 3167 y(1)p 2414 3212 V 2421 3303 a Fu(t)150 3430 y Fz(where)31 b(the)f(constan)m(t)g(dep)s(ends)i(on)d(the)h (orbit;)g(here)g(w)m(e)h(used)g(the)f(b)s(oundedness)h(of)f(the)g(supp) s(ort)150 3550 y(of)i Fu(F)338 3514 y Fm(0)324 3575 y(\000)383 3550 y Fz(,)h(and)g(that)f Fu(@)895 3565 y Ft(t)926 3550 y Fu(A)999 3565 y Ft(C)1090 3550 y Fz(is)h(b)s(ounded.)44 b(No)m(w)34 b(since)f Fu(c)2129 3565 y Fs(1)2168 3550 y Fu(=)p Fz(2)28 b Fu(<)f(d)2448 3565 y Fs(1)2487 3550 y Fz(,)33 b(the)g(in)m(tegrabilit)m(y)d(is)j(ensured)h(b)m(y)150 3670 y(\(3.6\).)296 3791 y(No)m(w)i(let)f(us)h(pro)m(v)m(e)h(that)e (the)h(limit)c(is)i(zero.)53 b(W)-8 b(e)36 b(remark)f(\014rst)h(that)f (mimic)m(king)d(the)k(pro)s(of)150 3911 y(of)c(\(3.6\))g(in)g(Lemma)f (3.2)h(w)m(e)i(can)f(also)e(sho)m(w)j(that)1348 4035 y Fl(Z)1447 4061 y Fm(1)1403 4261 y Fs(1)1548 4103 y Fz(1)p 1548 4148 V 1555 4239 a Fu(t)1607 4171 y Fr(j)p Fu(F)1712 4130 y Fm(0)1698 4195 y Fs(++)1812 4171 y Fr(j)p Fz(\()p Fu(A)1951 4186 y Ft(C)2010 4171 y Fu(=t)p Fz(\))p Fu(dt)27 b Fr(\024)i Fz(Const)p Fu(;)997 b Fz(\(3.8\))150 4458 y(where)36 b Fu(F)497 4473 y Fs(++)643 4458 y Fz(=)c Fu(F)828 4407 y Ft(c)859 4416 y Fj(1)892 4407 y Ft(=)p Fs(4)p Ft(;c)1013 4416 y Fj(1)1048 4407 y Ft(=)p Fs(3)p Ft(;c)1169 4416 y Fj(1)1203 4407 y Ft(=)p Fs(2)p Ft(;c)1324 4416 y Fj(1)814 4480 y Fs(++)1397 4458 y Fz(is)j(lik)m(e)f(in)g (\(3.4\).)50 b(W)-8 b(e)36 b(see)g(that)f Fu(F)2695 4422 y Fm(0)2681 4483 y Fs(++)2826 4458 y Fz(=)d(1)j(on)f(\()p Fu(c)3235 4473 y Fs(1)3275 4458 y Fu(=)p Fz(3)p Fu(;)17 b(c)3459 4473 y Fs(1)3497 4458 y Fu(=)p Fz(2\))35 b(and)150 4579 y(has)e(compact)f(supp)s(ort.)44 b(W)-8 b(e)33 b(then)g(can)g (write)1093 4788 y Fu(F)1156 4803 y Fm(\000)1215 4788 y Fz(\()p Fu(A)1326 4803 y Ft(C)1385 4788 y Fz(\()p Fu(t)p Fz(\))p Fu(=t)p Fz(\))p Fu(F)1695 4746 y Fm(0)1681 4812 y Fs(++)1795 4788 y Fz(\()p Fu(A)1906 4803 y Ft(C)1965 4788 y Fz(\()p Fu(t)p Fz(\))p Fu(=t)p Fz(\))28 b(=)g Fu(F)2393 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5776 a(21)p eop %%Page: 22 25 22 24 bop 296 407 a Fz(W)-8 b(e)33 b(are)g(no)m(w)g(ready)g(to)g (\014nish)f(the)h(pro)s(of)f(of)g(\(3.5\).)43 b(W)-8 b(rite)774 640 y Fu(F)851 593 y Ft(c;d)837 662 y Fm(\000)941 640 y Fz(\()p Fu(r)s(=t)p Fz(\))28 b(=)f Fu(F)1356 593 y Ft(c;d)1342 662 y Fm(\000)1447 640 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fr(f)p Fu(F)1781 589 y Ft(c)1812 598 y Fj(1)1845 589 y Ft(=)p Fs(3)p Ft(;c)1966 598 y Fj(1)2000 589 y Ft(=)p Fs(2)1767 662 y Fm(\000)2075 640 y Fz(\()p Fu(A)2186 655 y Ft(C)2245 640 y Fu(=t)p Fz(\))22 b(+)g Fu(F)2564 589 y Ft(c)2595 598 y Fj(1)2629 589 y Ft(=)p Fs(3)p Ft(;c)2750 598 y Fj(1)2784 589 y Ft(=)p Fs(2)2550 662 y(+)2859 640 y Fz(\()p Fu(A)2970 655 y Ft(C)3029 640 y Fu(=t)p Fz(\))p Fr(g)150 860 y Fz(where)40 b(w)m(e)h(c)m(ho)s(ose)f(0)f Fu(<)g(c)g(<)f(d)h(<<)g(c)1626 875 y Fs(1)1665 860 y Fz(.)64 b(The)40 b(con)m(tribution)e(from)g(the)i(\014rst)g(term)e(is)h (handled)150 981 y(b)m(y)34 b(\(3.7\);)g(the)g(second)h(will)c(ev)m(en) 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Fu(F)3366 2206 y Fm(\000)p Ft(d;)p Fm(\000)p Ft(d=)p Fs(2)p Ft(;d=)p Fs(2)p Ft(;d)3352 2279 y Fs(++)150 2378 y FA(\(se)-5 b(e)34 b(\(3.4\)\))1376 2413 y Fl(Z)1476 2440 y Fm(1)1432 2639 y Fs(1)1577 2481 y Fz(1)p 1577 2526 49 4 v 1584 2617 a Fu(t)1636 2549 y Fr(j)p Fu(F)1741 2508 y Fm(0)1727 2574 y Fs(++)1841 2549 y Fr(j)p Fz(\()p Fu(\030)5 b Fz(\()p Fu(t)p Fz(\)\))p Fu(dt)26 b Fr(\024)j Fz(Const)p Fu(;)978 b Fz(\(3.10\))150 2776 y FA(Mor)-5 b(e)g(over,)38 b(\(se)-5 b(e)36 b(\(3.3\)\))h(if)g Fz(0)32 b Fu(<)g(d)1437 2791 y Fs(1)1509 2776 y Fu(<)g(d=)p Fz(2)k FA(is)h(also)g(su\016ciently)h(smal)5 b(l)36 b(then)h(for)g(any) h Fu(F)3497 2791 y Fs(+)p Fm(\000)3648 2776 y FA(with)150 2896 y(supp)-5 b(ort)35 b(in)g Fz([)p Fr(\000)p Fu(d)770 2911 y Fs(1)809 2896 y Fu(;)17 b(d)904 2911 y Fs(1)943 2896 y Fz(])35 b FA(we)f(have)1602 3017 y Fz(lim)1587 3076 y Ft(t)p Fm(!1)1770 3017 y Fu(F)1833 3032 y Fs(+)p Fm(\000)1947 3017 y Fz(\()p Fu(\030)5 b Fz(\()p Fu(t)p Fz(\)\))27 b(=)h(0)p Fu(:)1187 b Fz(\(3.11\))150 3252 y Fv(Pro)s(of.)42 b Fz(W)-8 b(e)27 b(start)h(with)f(\(3.10\).)41 b(Di\013eren)m(tiate)26 b(the)i(b)s(ounded)g(propagation)d(observ)-5 b(able)28 b(\010\()p Fu(t)p Fz(\))g(=)150 3373 y Fu(F)213 3388 y Fs(++)327 3373 y Fz(\()p Fu(\030)5 b Fz(\()p Fu(t)p Fz(\)\))32 b(and)h(get)689 3648 y Fu(@)740 3663 y Ft(t)771 3648 y Fz(\010\()p Fu(t)p Fz(\))28 b(=)f Fu(F)1160 3607 y Fm(0)1146 3672 y Fs(++)1260 3648 y Fz(\()p Fu(\030)5 b Fz(\()p Fu(t)p Fz(\)\))1505 3580 y Fu(d\030)p 1505 3625 99 4 v 1511 3716 a(dt)1641 3648 y Fz(=)27 b Fr(\000)p Fu(F)1898 3607 y Fm(0)1884 3672 y Fs(++)1999 3648 y Fz(\()p Fu(\030)5 b Fz(\()p Fu(t)p Fz(\)\))2244 3580 y Fu(\030)g Fz(\()p Fu(t)p Fz(\))p Fu(\032)p 2244 3625 209 4 v 2324 3716 a(r)2484 3648 y Fz(+)22 b Fu(F)2659 3607 y Fm(0)2645 3672 y Fs(++)2759 3648 y Fz(\()p Fu(\030)5 b Fz(\()p Fu(t)p Fz(\)\))3004 3580 y(\()p Fr(\000)p Fu(b)p Fz(\))p Fu(\032)p 3004 3625 246 4 v 3103 3716 a(r)3258 3648 y(:)291 b Fz(\(3.12\))296 3893 y(A)33 b(consequence)j(of)c(Prop)s(osition)f (2.11)g(is)i(the)g(inequalit)m(y)1407 4113 y(lim)17 b(inf)1459 4173 y Ft(t)p Fm(!1)1677 4113 y Fz([)p Fu(\032)p Fz(\()p Fu(t)p Fz(\))23 b Fr(\000)f Fu(\032)2037 4128 y Ft(E)2097 4113 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\)])28 b Fr(\025)g Fz(0)p Fu(;)150 4366 y Fz(and)i(b)m(y)h(\(2.13\))f(there)h (exists)f Fu(\017)f(>)e Fz(0)j(\(only)f(dep)s(ending)i(on)f(the)g(lo)s (cation)e(of)i Fu(E)6 b Fz(\))30 b(and)g Fu(T)42 b(>)27 b Fz(1)j(large)150 4486 y(enough)j(\(dep)s(ending)g(on)f(the)h(orbit\)) f(suc)m(h)i(that)1416 4716 y Fu(\032)p Fz(\()p Fu(t)p Fz(\))28 b Fr(\025)g(\000)1787 4627 y(p)p 1871 4627 127 4 v 1871 4716 a Fz(2)p Fu(E)g Fz(+)22 b Fu(\017;)114 b(t)28 b Fr(\025)h Fu(T)8 b(:)1017 b Fz(\(3.13\))150 4936 y(But)32 b Fu(F)420 4900 y Fm(0)406 4961 y Fs(++)520 4936 y Fz(\()p Fu(\030)5 b Fz(\()p Fu(t)p Fz(\)\))32 b(is)f(zero)h(unless)h Fr(j)p Fu(\030)5 b Fr(j)31 b Fz(is)g(less)i (than)f Fu(d)p Fz(.)42 b(Energy)34 b(conserv)-5 b(ation)31 b(will)f(force)i Fu(\032)g Fz(to)g(sta)m(y)150 5066 y(either)h(ab)s(o)m (v)m(e)703 4984 y Fr(p)p 786 4984 339 4 v 82 x Fz(2)p Fu(E)28 b Fr(\000)23 b Fu(d)1086 5037 y Fs(2)1158 5066 y Fz(or)32 b(b)s(elo)m(w)h Fr(\000)1631 4984 y(p)p 1714 4984 V 82 x Fz(2)p Fu(E)28 b Fr(\000)23 b Fu(d)2014 5037 y Fs(2)2053 5066 y Fz(.)44 b(If)33 b(w)m(e)h(c)m(ho)s(ose)g Fu(d)e Fz(small)f(enough)i(\(dep)s(ending)150 5186 y(on)f Fu(\017)h Fz(in)f(\(3.13\)\))g(w)m(e)i(can)e(rule)h(out)f(the)h(second) h(alternativ)m(e)e(so)g(w)m(e)i(ha)m(v)m(e)779 5476 y Fu(F)856 5435 y Fm(0)842 5501 y Fs(++)957 5476 y Fz(\()p Fu(\030)5 b Fz(\()p Fu(t)p Fz(\)\))1202 5409 y(\()p Fr(\000)p Fu(b)p Fz(\))p Fu(\032)p 1202 5453 246 4 v 1300 5545 a(r)1484 5476 y Fr(\025)28 b Fu(F)1666 5435 y Fm(0)1652 5501 y Fs(++)1766 5476 y Fz(\()p Fu(\030)5 b Fz(\()p Fu(t)p Fz(\)\))2011 5409 y(\(min)14 b Fr(j)p Fu(b)p Fr(j)p Fz(\))2362 5327 y Fr(p)p 2445 5327 339 4 v 82 x Fz(2)p Fu(E)28 b Fr(\000)23 b Fu(d)2745 5380 y Fs(2)p 2011 5453 774 4 v 2374 5545 a Fu(r)2794 5476 y(;)114 b(t)28 b Fr(\025)g Fu(T)8 b(:)1939 5776 y Fz(22)p eop %%Page: 23 26 23 25 bop 150 407 a Fz(In)m(tro)s(ducing)28 b(this)h(in)f(\(3.12\))g(w) m(e)h(obtain)f(another)h Fu(T)42 b Fz(large)28 b(enough)h(suc)m(h)h (that)f(with)f Fu(C)3420 422 y Fs(1)3487 407 y Fu(>)f Fz(0)i(and)150 527 y Fu(t)f Fr(\025)g Fu(T)449 780 y(@)500 795 y Ft(t)530 780 y Fz(\010\()p Fu(t)p Fz(\))h Fr(\025)855 713 y Fz(1)p 855 757 49 4 v 856 849 a Fu(r)913 780 y(F)990 739 y Fm(0)976 805 y Fs(++)1091 780 y Fz(\()p Fu(\030)5 b Fz(\()p Fu(t)p Fz(\)\))21 b Fr(\001)h Fz(\()p Fr(\000)p Fu(d)1563 691 y Fr(p)p 1646 691 127 4 v 89 x Fz(2)p Fu(E)28 b Fz(+)22 b(\(min)15 b Fr(j)p Fu(b)p Fr(j)p Fz(\))2245 693 y Fr(p)p 2328 693 339 4 v 87 x Fz(2)p Fu(E)28 b Fr(\000)22 b Fu(d)2627 751 y Fs(2)2667 780 y Fz(\))27 b Fr(\025)h Fu(C)2907 795 y Fs(1)2969 780 y Fr(\001)3028 713 y Fz(1)p 3028 757 49 4 v 3035 849 a Fu(t)3087 780 y(F)3164 739 y Fm(0)3150 805 y Fs(++)3264 780 y Fz(\()p Fu(\030)5 b Fz(\()p Fu(t)p Fz(\)\))p Fu(;)150 1031 y Fz(where)37 b(in)d(the)i(last)e(inequalit)m(y)h(w)m(e)h(used)g(the)g(maximal)c(v)m (elo)s(cit)m(y)j(b)s(ound)h(\(2.1\))f(and)g(w)m(e)h(to)s(ok)f Fu(d)150 1152 y Fz(small)i(enough)i(in)g(order)g(to)g(get)g(a)f(p)s (ositiv)m(e)h(lo)m(w)m(er)g(b)s(ound.)63 b(W)-8 b(e)40 b(then)g(in)m(tegrate)e(and)h(get)h(the)150 1272 y(result.)296 1392 y(The)32 b(pro)s(of)d(of)h(the)h(limit)c(in)i(\(3.11\))h(is)g (similar)d(to)j(the)h(one)f(w)m(e)i(ga)m(v)m(e)f(for)f(\(3.7\).)42 b(First,)30 b(with)g(a)150 1513 y(Co)s(ok)35 b(t)m(yp)s(e)g(argumen)m (t)g(w)m(e)g(reduce)h(the)f(existence)h(of)f(the)g(limit)c(to)j(the)h (absolute)f(in)m(tegrabilit)m(y)150 1633 y(of)1222 1778 y Fr(j)p Fu(F)1327 1737 y Fm(0)1313 1803 y Fs(+)p Fm(\000)1427 1778 y Fz(\()p Fu(\030)5 b Fz(\()p Fu(t)p Fz(\)\))1672 1711 y Fu(d\030)p 1672 1755 99 4 v 1678 1846 a(dt)1780 1778 y Fr(j)27 b(\024)h Fz(Const)2200 1711 y(1)p 2200 1755 49 4 v 2207 1846 a Fu(t)2259 1778 y Fr(j)p Fu(F)2364 1737 y Fm(0)2350 1803 y Fs(+)p Fm(\000)2463 1778 y Fz(\()p Fu(\030)5 b Fz(\()p Fu(t)p Fz(\)\))p Fr(j)p Fu(:)150 1977 y Fz(W)-8 b(e)33 b(can)g(write)1305 2097 y Fu(F)1382 2056 y Fm(0)1368 2122 y Fs(+)p Fm(\000)1482 2097 y Fz(\()p Fu(\030)5 b Fz(\()p Fu(t)p Fz(\)\))p Fu(F)1794 2056 y Fm(0)1780 2122 y Fs(++)1893 2097 y Fz(\()p Fu(\030)g Fz(\()p Fu(t)p Fz(\)\))27 b(=)h Fu(F)2336 2056 y Fm(0)2322 2122 y Fs(+)p Fm(\000)2436 2097 y Fz(\()p Fu(\030)5 b Fz(\()p Fu(t)p Fz(\)\))150 2271 y(b)s(ecause)46 b(of)e(their)g(supp)s (ort)g(prop)s(erties,)k(so)c(the)h(in)m(tegrabilit)m(y)d(problem)h(can) i(b)s(e)f(reduced)i(to)150 2392 y(\(3.10\).)296 2512 y(Second,)34 b(notice)e(that)h(w)m(e)g(also)f(ma)m(y)g(write)1305 2732 y Fu(F)1368 2747 y Fs(+)p Fm(\000)1482 2732 y Fz(\()p Fu(\030)5 b Fz(\()p Fu(t)p Fz(\)\))p Fu(F)1794 2691 y Fm(0)1780 2757 y Fs(++)1893 2732 y Fz(\()p Fu(\030)g Fz(\()p Fu(t)p Fz(\)\))27 b(=)h Fu(F)2322 2747 y Fs(+)p Fm(\000)2436 2732 y Fz(\()p Fu(\030)5 b Fz(\()p Fu(t)p Fz(\)\))150 2952 y(and)39 b(rep)s(eat)g(the)h(sub-sequence)i(argumen)m (t)d(w)m(e)h(\014rst)f(emplo)m(y)m(ed)h(righ)m(t)e(after)h(\(3.9\).)62 b(It)39 b(follo)m(ws)150 3073 y(that)32 b(the)h(limit)c(m)m(ust)k(b)s (e)g(zero.)p 3754 3073 4 66 v 3758 3010 59 4 v 3758 3073 V 3816 3073 4 66 v 150 3480 a Fp(3.3)135 b Fi(\030)51 b Fp(is)45 b(p)t(ositiv)l(e)i(for)e(large)h(times)150 3664 y Fz(W)-8 b(e)32 b(sa)m(w)g(in)f(the)h(previous)g(subsection)g (that)g Fu(\030)k Fz(could)31 b(not)g(return)h(to)f(zero)h(for)f(large) f(times;)h(no)m(w)150 3785 y(w)m(e)j(pro)m(v)m(e)f(that)g Fu(\030)k Fz(b)s(ecomes)c(p)s(ositiv)m(e.)150 3988 y Fv(Lemma)44 b(3.4.)g FA(If)c Fu(d)931 4003 y Fs(1)1010 3988 y FA(is)g(the)g(same)f(as)h(in)f(\(3.11\),)i(c)-5 b(onsider)39 b Fr(\000)p Fu(d)2644 4003 y Fs(1)2721 3988 y Fu(<)e(c)2876 4003 y Fs(2)2952 3988 y Fu(<)g(d)3116 4003 y Fs(2)3193 3988 y Fu(<)g Fz(0)p FA(.)60 b(Then)39 b(for)150 4108 y Fu(F)213 4123 y Fm(\000)300 4108 y Fz(=)27 b Fu(F)480 4061 y Ft(c)511 4070 y Fj(2)545 4061 y Ft(;d)601 4070 y Fj(2)466 4130 y Fm(\000)675 4108 y FA(we)34 b(have)1630 4229 y Fz(lim)1614 4289 y Ft(t)p Fm(!1)1797 4229 y Fu(F)1860 4244 y Fm(\000)1919 4229 y Fz(\()p Fu(\030)5 b Fz(\()p Fu(t)p Fz(\)\))28 b(=)f(0)p Fu(:)1215 b Fz(\(3.14\))150 4459 y Fv(Pro)s(of.)62 b Fz(The)39 b(existence)h(of)f(the)g(ab)s(o)m(v) m(e)g(limit)c(again)j(follo)m(ws)f(after)i(using)f(a)g(Co)s(ok)h (argumen)m(t;)150 4579 y(notice)d(that)g Fu(F)733 4543 y Fm(0)719 4604 y(\000)814 4579 y Fz(is)f(supp)s(orted)i(in)f(\()p Fr(\000)p Fu(d)1661 4594 y Fs(1)1700 4579 y Fu(;)17 b(d)1795 4594 y Fs(1)1834 4579 y Fz(\))36 b(so)g(w)m(e)h(can)g(apply)f(the)g (propagation)e(estimate)i(in)150 4700 y(\(3.10\).)296 4820 y(W)-8 b(e)49 b(no)m(w)g(pro)m(v)m(e)g(that)f(the)h(limit)c(is)j (zero.)91 b(W)-8 b(e)48 b(kno)m(w)i(from)d(energy)i(conserv)-5 b(ation)48 b(that)150 4941 y Fu(\032)p Fz(\()p Fu(t)p Fz(\))g Fr(2)g Fz([)p Fr(\000)577 4856 y(p)p 661 4856 127 4 v 661 4941 a Fz(2)p Fu(E)6 b(;)832 4856 y Fr(p)p 915 4856 V 85 x Fz(2)p Fu(E)f Fz(].)79 b(Fix)44 b(some)g Fu(a)k(<)g(b)g(<)g Fr(\000)2130 4856 y(p)p 2214 4856 V 2214 4941 a Fz(2)p Fu(E)i Fz(and)2586 4856 y Fr(p)p 2669 4856 V 85 x Fz(2)p Fu(E)k(<)48 b(c)f(<)h(d)c Fz(and)h(consider)150 5061 y Fu(F)213 5076 y Fs(+)p Fm(\000)355 5061 y Fz(=)27 b Fu(F)535 5014 y Ft(a;b;c;d)521 5083 y Fs(+)p Fm(\000)765 5061 y Fz(\(see)34 b(\(3.3\)\);)e(w)m(e)h(ha)m(v)m(e)1677 5281 y Fu(F)1740 5296 y Fs(+)p Fm(\000)1854 5281 y Fz(\()p Fu(\032)p Fz(\()p Fu(t)p Fz(\)\))28 b(=)f(1)p Fu(:)1278 b Fz(\(3.15\))1939 5776 y(23)p eop %%Page: 24 27 24 26 bop 150 407 a Fz(Consider)29 b(also)f(that)g(particular)f Fu(F)1461 422 y Fs(++)1603 407 y Fz(whose)j(deriv)-5 b(ativ)m(e)28 b(giv)m(es)h(bac)m(k)h(the)f(ab)s(o)m(v)m(e)g Fu(F)3288 422 y Fs(+)p Fm(\000)3402 407 y Fz(.)42 b(W)-8 b(e)29 b(then)150 527 y(pro)m(v)m(e)34 b(the)f(propagation)e(estimate:) 1224 633 y Fl(Z)1324 659 y Fm(1)1280 858 y Fs(1)1425 701 y Fz(1)p 1425 745 49 4 v 1432 837 a Fu(t)1484 768 y(F)1561 727 y Fm(0)1547 793 y Fs(++)1661 768 y Fz(\()p Fu(\032)p Fz(\()p Fu(t)p Fz(\)\))p Fu(F)1961 783 y Fm(\000)2021 768 y Fz(\()p Fu(\030)5 b Fz(\()p Fu(t)p Fz(\)\))p Fu(dt)27 b Fr(\024)h Fz(Const)q Fu(;)825 b Fz(\(3.16\))150 1017 y(and)34 b(b)s(ecause)h(of)e(\(3.15\))g(w)m(e)i(conclude)f(that)g Fu(F)1921 1032 y Fm(\000)1980 1017 y Fz(\()p Fu(\030)5 b Fz(\()p Fu(t)p Fz(\)\))33 b(admits)g(a)g(div)m(ergen)m(t)i(sequence)i (of)c(times)150 1137 y(along)e(whic)m(h)i(it)f(tends)h(to)g(zero)g(and) f(this)h(pro)m(v)m(es)h(the)f(lemma.)296 1258 y(W)-8 b(e)34 b(no)m(w)g(pro)m(v)m(e)h(\(3.16\).)46 b(Consider)34 b(the)g(observ)-5 b(able)33 b(\010\()p Fu(t)p Fz(\))d(=)f Fu(F)2687 1273 y Fs(++)2801 1258 y Fz(\()p Fu(\032)p Fz(\()p Fu(t)p Fz(\)\))p Fu(F)3101 1273 y Fm(\000)3161 1258 y Fz(\()p Fu(\030)5 b Fz(\()p Fu(t)p Fz(\)\))32 b(and)i(com-)150 1378 y(pute)f(its)f(time)f(deriv)-5 b(ativ)m(e:)1147 1623 y Fu(@)1198 1638 y Ft(t)1228 1623 y Fz(\010\()p Fu(t)p Fz(\))28 b(=)1550 1556 y Fu(\030)f Fr(\001)22 b Fz(\()p Fu(\030)k Fz(+)c Fu(b)p Fz(\))p 1550 1601 405 4 v 1729 1692 a Fu(r)1965 1623 y(F)2042 1582 y Fm(0)2028 1648 y Fs(++)2142 1623 y Fu(F)2205 1638 y Fm(\000)2286 1623 y Fz(+)g Fu(F)2447 1638 y Fs(++)2561 1623 y Fu(F)2638 1582 y Fm(0)2624 1648 y(\000)2693 1556 y Fu(d\030)p 2693 1601 99 4 v 2699 1692 a(dt)2801 1623 y(:)150 1845 y Fz(No)m(w)33 b(notice)f(that)h(on)f(the)h(supp)s(ort)g (of)f Fu(F)1709 1860 y Fm(\000)1801 1845 y Fz(w)m(e)i(ha)m(v)m(e)f Fu(\030)g(<)27 b(d)2399 1860 y Fs(2)2466 1845 y Fu(<)g Fz(0)33 b(hence)663 2030 y Fu(\030)26 b Fr(\001)c Fz(\()p Fu(\030)27 b Fz(+)22 b Fu(b)p Fz(\))p 663 2074 405 4 v 842 2165 a Fu(r)1077 2097 y(F)1154 2056 y Fm(0)1140 2122 y Fs(++)1254 2097 y Fu(F)1317 2112 y Fm(\000)1404 2097 y Fr(\025)1519 2030 y(j)p Fu(d)1598 2045 y Fs(2)1637 2030 y Fr(j)1665 1993 y Fs(2)1726 2030 y Fz(+)g Fr(j)p Fu(d)1903 2045 y Fs(2)1942 2030 y Fr(j)17 b Fz(min)d Fr(j)p Fu(b)p Fr(j)p 1519 2074 744 4 v 1867 2165 a Fu(r)2272 2097 y(F)2349 2056 y Fm(0)2335 2122 y Fs(++)2449 2097 y Fu(F)2512 2112 y Fm(\000)2599 2097 y Fr(\025)28 b Fz(Const)2964 2030 y(1)p 2964 2074 49 4 v 2971 2165 a Fu(t)3023 2097 y(F)3100 2056 y Fm(0)3086 2122 y Fs(++)3200 2097 y Fu(F)3263 2112 y Fm(\000)150 2318 y Fz(where)34 b(in)e(the)h(second)g(inequalit)m (y)f(w)m(e)i(used)f(the)g(maximal)d(v)m(elo)s(cit)m(y)i(b)s(ound.)44 b(Finally)-8 b(,)1121 2558 y(Const)1381 2491 y(1)p 1381 2535 V 1388 2627 a Fu(t)1439 2558 y(F)1516 2517 y Fm(0)1502 2583 y Fs(++)1616 2558 y Fu(F)1679 2573 y Fm(\000)1766 2558 y Fr(\024)28 b Fu(F)1934 2573 y Fs(++)2049 2558 y Fr(j)p Fu(F)2154 2517 y Fm(0)2140 2583 y(\000)2198 2558 y Fr(j)22 b(\001)g(j)2336 2491 y Fu(d\030)p 2336 2535 99 4 v 2342 2627 a(dt)2444 2558 y Fr(j)g Fz(+)g Fu(@)2643 2573 y Ft(t)2673 2558 y Fz(\010\()p Fu(t)p Fz(\))150 2780 y(and)33 b(the)g(in)m(tegrabilit)m(y)d(of)i(the)h(righ)m (t)f(hand)g(side)h(\014nishes)h(the)f(pro)s(of.)p 3754 2780 4 66 v 3758 2717 59 4 v 3758 2780 V 3816 2780 4 66 v 150 3018 a Fv(Remark)p Fz(.)53 b(If)36 b(w)m(e)h(put)g(together)f (the)g(previous)h(t)m(w)m(o)f(lemmas,)f(w)m(e)j(obtain)c(that)i Fu(\030)5 b Fz(\()p Fu(t)p Fz(\))33 b Fr(\025)h Fu(d)3594 3033 y Fs(1)3667 3018 y Fu(>)f Fz(0)150 3139 y(for)f(all)f Fu(t)c Fr(\025)i Fu(T)46 b Fz(where)34 b Fu(T)46 b Fz(is)32 b(large)f(enough.)150 3424 y Fp(3.4)135 b Fi(\032)46 b Fp(gets)f(trapp)t(ed)g(near)g Fi(\032)1708 3442 y Fh(E)1824 3424 y Fp(and)f Fi(\030)51 b Fp(near)45 b Fi(\030)2537 3442 y Fh(E)150 3609 y Fz(W)-8 b(e)25 b(already)f(kno)m(w)h(from)e (energy)i(conserv)-5 b(ation)25 b(and)f(Prop)s(osition)f(2.11)g(that)h (the)h(radial)d(v)m(elo)s(cit)m(y)150 3729 y(is)32 b(lo)s(calized)e (somewhere)k(in)e(the)h(in)m(terv)-5 b(al)1238 3925 y Fu(\032)p Fz(\()p Fu(t)p Fz(\))28 b Fr(2)g Fz([)p Fu(\032)1598 3940 y Ft(E)1658 3925 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))23 b Fr(\000)f Fu(\017;)2098 3836 y Fr(p)p 2181 3836 127 4 v 89 x Fz(2)p Fu(E)6 b Fz(])p Fu(;)114 b(t)28 b Fr(\025)h Fu(T)8 b(;)839 b Fz(\(3.17\))150 4121 y(where)45 b Fu(T)58 b Fz(is)44 b(large)f(enough)i(and)f(dep)s(ends)i(on)e(ev)m (erything.)79 b(W)-8 b(e)45 b(no)m(w)g(in)m(tend)f(to)g(sho)m(w)h(that) 150 4241 y Fu(\032)p Fz(\()p Fu(t)p Fz(\))35 b(can)g(only)f(sp)s(end)h (a)g(\014nite)f(amoun)m(t)g(of)g(time)f(outside)i(an)f(in)m(terv)-5 b(al)33 b(of)i(width)f Fu(\017)h Fz(cen)m(tered)h(at)150 4362 y Fu(\032)200 4377 y Ft(E)260 4362 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\).)296 4482 y(F)-8 b(or)32 b(ev)m(ery)j Fu(E)807 4446 y Fm(0)858 4482 y Fr(\025)28 b Fu(E)6 b Fz(,)32 b(de\014ne)1436 4678 y Fu(B)1510 4693 y Ft(E)1566 4674 y Ff(0)1592 4678 y Fz(\()p Fu(t)p Fz(\))c(:=)f Fu(\032)p Fz(\()p Fu(t)p Fz(\))c Fr(\000)g Fu(\032)2195 4693 y Ft(E)2251 4674 y Ff(0)2277 4678 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))p Fu(:)150 4888 y Fv(Prop)s(osition)36 b(3.5.)41 b FA(Denote)34 b(by)i Fu(F)1497 4903 y Fs(+)1583 4888 y Fz(=)28 b Fu(F)1764 4837 y Ft(\017=)p Fs(2)p Ft(;\017)1750 4909 y Fs(+)1915 4888 y FA(.)45 b(Then)1586 5083 y Fz(lim)1571 5143 y Ft(t)p Fm(!1)1754 5083 y Fu(F)1817 5098 y Fs(+)1876 5083 y Fz(\()p Fu(B)1988 5098 y Ft(E)2048 5083 y Fz(\()p Fu(t)p Fz(\)\))28 b(=)f(0)p Fu(:)150 5306 y FA(Mor)-5 b(e)g(over,)34 b(ther)-5 b(e)35 b(exists)g Fu(T)48 b FA(lar)-5 b(ge)35 b(enough)f(such)h(that)1358 5502 y Fr(j)p Fu(\032)p Fz(\()p Fu(t)p Fz(\))22 b Fr(\000)h Fu(\032)1719 5517 y Ft(E)1778 5502 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))p Fr(j)28 b(\024)g Fu(\017;)117 b(t)28 b Fr(\025)g Fu(T)8 b(:)959 b Fz(\(3.18\))1939 5776 y(24)p eop %%Page: 25 28 25 27 bop 150 407 a Fv(Pro)s(of)p Fz(.)68 b(Notice)40 b(that)h Fu(F)1114 422 y Fs(+)1173 407 y Fz(\()p Fu(B)1285 422 y Ft(E)1344 407 y Fz(\()p Fu(t)p Fz(\)\))g(is)f(zero)i(unless)f Fu(\032)p Fz(\()p Fu(t)p Fz(\))h Fr(2)g Fz([)p Fu(\032)2539 422 y Ft(E)2599 407 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))28 b(+)g Fu(\017=)p Fz(2)p Fu(;)3147 322 y Fr(p)p 3229 322 127 4 v 3229 407 a Fz(2)p Fu(E)6 b Fz(].)68 b(W)-8 b(e)41 b(no)m(w)150 527 y(try)f(to)g(break)h(this)f(in)m(terv)-5 b(al)39 b(in)m(to)g(smaller)f(pieces)j(whic)m(h)g(can)f(b)s(e)h(more)e (easily)h(treated)g(using)150 648 y(propagation)31 b(estimates.)150 827 y Fv(Lemma)41 b(3.6.)j FA(De\014ne)37 b Fu(\017)1119 842 y Fs(0)1193 827 y Fz(:=)1581 787 y Ft(\017)p 1340 804 510 4 v 1340 861 a Fs(4)12 b(sup)1494 878 y Fq(\022)1529 861 y Fs(\()p Ft(@)1593 872 y Fq(E)1645 861 y Ft(\032)1681 872 y Fq(E)1733 861 y Fs(\()p Ft(\022)r Fs(\)\))1860 827 y FA(.)55 b(Then)38 b(ther)-5 b(e)38 b(exists)g Fu(M)49 b FA(lar)-5 b(ge)38 b(enough)f(such)i(that)150 961 y(uniformly)c(in)f Fu(\022)d Fr(2)d Fz([0)p Fu(;)17 b Fz(2)p Fu(\031)t Fz(])34 b FA(we)h(have)924 1163 y Fz([)p Fu(\032)1001 1178 y Ft(E)1061 1163 y Fz(\()p Fu(\022)s Fz(\))22 b(+)g Fu(\017=)p Fz(2)p Fu(;)1486 1073 y Fr(p)p 1569 1073 127 4 v 90 x Fz(2)p Fu(E)5 b Fz(])28 b Fr(\032)g(f)p Fu(\032)1955 1178 y Ft(E)2011 1159 y Ff(0)2037 1163 y Fz(\()p Fu(\022)s Fz(\))g(:)56 b Fu(E)2350 1122 y Fm(0)2401 1163 y Fr(2)28 b Fz([)p Fu(E)g Fz(+)22 b Fu(\017)2759 1178 y Fs(0)2799 1163 y Fu(;)17 b(M)10 b Fz(])p Fr(g)p Fu(:)525 b Fz(\(3.19\))150 1354 y FA(If)30 b Fu(N)39 b Fr(\025)28 b Fz(1)i FA(is)h(an)f(inte)-5 b(ger,)32 b(de\014ne)e Fu(E)1493 1318 y Fm(0)1487 1379 y Fs(0)1554 1354 y Fz(:=)e Fu(E)19 b Fz(+)13 b Fu(\017)1904 1369 y Fs(0)1975 1354 y FA(and)30 b Fu(E)2238 1318 y Fm(0)2232 1380 y Ft(k)2303 1354 y Fz(:=)d Fu(E)2511 1318 y Fm(0)2505 1379 y Fs(0)2558 1354 y Fz(+)2670 1315 y Ft(k)p 2657 1331 64 4 v 2657 1389 a(N)2731 1354 y Fz(\()p Fu(M)d Fr(\000)13 b Fu(E)3055 1318 y Fm(0)3049 1379 y Fs(0)3089 1354 y Fz(\))p FA(.)43 b(Then)30 b(for)h(every)150 1475 y Fu(\017)189 1490 y Fs(2)256 1475 y Fu(>)d Fz(0)35 b FA(ther)-5 b(e)35 b(exists)f Fu(N)45 b FA(lar)-5 b(ge)35 b(enough)f(such)h(that)742 1753 y Fr(f)p Fu(\032)842 1768 y Ft(E)898 1749 y Ff(0)924 1753 y Fz(\()p Fu(\022)s Fz(\))27 b(:)56 b Fu(E)1236 1712 y Fm(0)1287 1753 y Fr(2)28 b Fz([)p Fu(E)g Fz(+)22 b Fu(\017)1645 1768 y Fs(0)1685 1753 y Fu(;)17 b(M)10 b Fz(])p Fr(g)28 b(\032)2076 1629 y Ft(N)2052 1659 y Fl([)2043 1871 y Ft(k)r Fs(=1)2172 1753 y Fz([)p Fu(\032)2249 1770 y Ft(E)2305 1747 y Ff(0)2301 1793 y Fq(k)2343 1753 y Fz(\()p Fu(\022)s Fz(\))23 b Fr(\000)f Fu(\017)2628 1768 y Fs(2)2668 1753 y Fu(;)17 b(\032)2762 1770 y Ft(E)2818 1747 y Ff(0)2814 1793 y Fq(k)2856 1753 y Fz(\()p Fu(\022)s Fz(\))22 b(+)g Fu(\017)3139 1768 y Fs(2)3179 1753 y Fz(])p Fu(:)343 b Fz(\(3.20\))150 2035 y Fv(Pro)s(of)p Fz(.)52 b(W)-8 b(e)36 b(kno)m(w)g(that)f Fu(\030)1186 2050 y Ft(E)1281 2035 y Fz(cannot)g(exceed)j(max)16 b Fr(j)p Fu(b)p Fr(j)35 b Fz(for)g(all)e(energies,)k(hence)g(energy)f (conser-)150 2155 y(v)-5 b(ation)33 b(giv)m(es)i Fu(\032)738 2170 y Ft(E)794 2152 y Ff(0)850 2155 y Fr(\030)958 2071 y(p)p 1041 2071 151 4 v 84 x Fz(2)p Fu(E)1168 2127 y Fm(0)1225 2155 y Fz(when)h Fu(E)1560 2119 y Fm(0)1613 2155 y Fr(!)30 b(1)p Fz(.)48 b(This)34 b(is)g(ho)m(w)h(w)m(e)g(get)f (the)h(existence)h(of)d Fu(M)10 b Fz(.)49 b(No)m(w)150 2276 y(since)33 b Fu(\032)439 2291 y Ft(E)531 2276 y Fz(increases)g(with)f Fu(E)6 b Fz(,)32 b(in)g(order)g(to)g(get)h (\(3.19\))e(w)m(e)i(only)f(need)h(to)f(v)m(erify)h(that)f(uniformly)150 2396 y(in)g Fu(\022)1530 2517 y(\032)1580 2533 y Ft(E)1636 2510 y Ff(0)1632 2554 y Fj(0)1670 2517 y Fz(\()p Fu(\022)s Fz(\))c Fr(\024)g Fu(\032)1977 2532 y Ft(E)2037 2517 y Fz(\()p Fu(\022)s Fz(\))22 b(+)g Fu(\017=)p Fz(2)p Fu(:)150 2679 y Fz(The)34 b(conca)m(vit)m(y)f(in)f(energy)h(and)g(the)g (de\014nition)f(of)g Fu(\017)2151 2694 y Fs(0)2223 2679 y Fz(then)h(giv)m(es)1022 2870 y Fu(\032)1072 2887 y Ft(E)1128 2864 y Ff(0)1124 2908 y Fj(0)1163 2870 y Fz(\()p Fu(\022)s Fz(\))22 b Fr(\000)h Fu(\032)1459 2885 y Ft(E)1519 2870 y Fz(\()p Fu(\022)s Fz(\))k Fr(\024)i Fz(\()p Fu(E)1892 2829 y Fm(0)1886 2895 y Fs(0)1947 2870 y Fr(\000)23 b Fu(E)6 b Fz(\))22 b Fr(\001)g Fz(\()p Fu(@)2324 2885 y Ft(E)2384 2870 y Fu(\032)2434 2885 y Ft(E)2494 2870 y Fz(\()p Fu(\022)s Fz(\)\))28 b Fr(\024)g Fu(\017=)p Fz(4)p Fu(;)150 3062 y Fz(and)33 b(\(3.19\))e(is)i(pro)m(v)m(ed.)296 3182 y(As)g(for)f(\(3.20\),)g(it)g(is)g(su\016cien)m(t)i(that)e (uniformly)e(in)i Fu(\022)983 3374 y(\032)1033 3390 y Ft(E)1089 3367 y Ff(0)1085 3414 y Fq(k)q Fj(+1)1205 3374 y Fz(\()p Fu(\022)s Fz(\))27 b Fr(\024)i Fu(\032)1512 3390 y Ft(E)1568 3367 y Ff(0)1564 3414 y Fq(k)1606 3374 y Fz(\()p Fu(\022)s Fz(\))22 b(+)g Fu(\017)1889 3389 y Fs(2)1929 3374 y Fu(;)114 b(k)31 b Fr(2)d(f)p Fz(0)p Fu(;)17 b Fz(1)p Fu(;)g(:)g(:)g(:)e(;)i(N)32 b Fr(\000)23 b Fz(1)p Fr(g)p Fu(:)150 3575 y Fz(Since)31 b Fu(\032)453 3591 y Ft(E)509 3568 y Ff(0)505 3615 y Fq(k)q Fj(+1)624 3575 y Fz(\()p Fu(\022)s Fz(\))19 b Fr(\000)g Fu(\032)913 3591 y Ft(E)969 3568 y Ff(0)965 3615 y Fq(k)1007 3575 y Fz(\()p Fu(\022)s Fz(\))28 b Fr(\024)g Fz(Const)20 b Fr(\001)e Fz(\()p Fu(E)1695 3539 y Fm(0)1689 3601 y Ft(k)r Fs(+1)1841 3575 y Fr(\000)h Fu(E)2015 3539 y Fm(0)2009 3601 y Ft(k)2052 3575 y Fz(\))27 b(=)h(Const)q Fu(=)-5 b(N)10 b Fz(,)31 b(w)m(e)h(can)f(mak)m(e)g(this)f(di\013erence)150 3695 y(smaller)h(than)h Fu(\017)753 3710 y Fs(2)826 3695 y Fz(and)g(w)m(e)i(are)e(done.)p 3754 3695 4 66 v 3758 3633 59 4 v 3758 3695 V 3816 3695 4 66 v 296 3934 a(The)j(next)g(step)g (is)f(to)f(rule)h(out)g(the)g(p)s(ossibilit)m(y)e(for)h Fu(\032)p Fz(\()p Fu(t)p Fz(\))i(to)e(b)s(e)i(lo)s(cated)e(in)g(small)f (in)m(terv)-5 b(als)150 4054 y(lik)m(e)32 b(those)h(in)f(\(3.20\).)296 4175 y(F)-8 b(or)32 b(ev)m(ery)j Fu(\017)768 4190 y Fs(2)835 4175 y Fu(>)28 b Fz(0)k(consider)1498 4366 y Fu(F)1561 4381 y Fs(+)p Fm(\000)1703 4366 y Fz(=)c Fu(F)1884 4319 y Fm(\000)p Fs(5)p Ft(\017)2003 4328 y Fj(2)2037 4319 y Ft(;)p Fm(\000)p Fs(4)p Ft(\017)2176 4328 y Fj(2)2209 4319 y Ft(;)p Fs(4)p Ft(\017)2293 4328 y Fj(2)2327 4319 y Ft(;)p Fs(5)p Ft(\017)2411 4328 y Fj(2)1870 4388 y Fs(+)p Fm(\000)2450 4366 y Fu(:)150 4558 y Fz(Denote)g(b)m(y)h Fu(F)674 4573 y Fs(++)816 4558 y Fz(precisely)f(that)g(function)f(of)g (the)i(t)m(yp)s(e)f(\(3.4\))g(whose)h(deriv)-5 b(ativ)m(e)27 b(giv)m(es)i(bac)m(k)g(the)150 4678 y(ab)s(o)m(v)m(e)k Fu(F)489 4693 y Fs(+)p Fm(\000)603 4678 y Fz(.)150 4857 y Fv(Lemma)k(3.7.)42 b FA(Uniformly)35 b(in)f Fu(E)1425 4821 y Fm(0)1476 4857 y Fr(2)28 b Fz([)p Fu(E)h Fz(+)22 b Fu(\017)1835 4872 y Fs(0)1874 4857 y Fu(;)17 b(M)10 b Fz(])36 b FA(ther)-5 b(e)35 b(exists)f Fu(\017)2633 4872 y Fs(2)2708 4857 y FA(smal)5 b(l)34 b(enough)g(such)h(that)927 4964 y Fl(Z)1027 4990 y Fm(1)983 5189 y Fs(1)1128 5032 y Fz(1)p 1128 5077 49 4 v 1135 5168 a Fu(t)1187 5100 y(F)1264 5058 y Fm(0)1250 5124 y Fs(++)1364 5100 y Fz(\()p Fu(B)1476 5115 y Ft(E)1532 5096 y Ff(0)1558 5100 y Fz(\()p Fu(t)p Fz(\)\))p Fu(dt)28 b Fz(=)1924 4964 y Fl(Z)2024 4990 y Fm(1)1979 5189 y Fs(1)2125 5032 y Fz(1)p 2125 5077 V 2132 5168 a Fu(t)2184 5100 y(F)2247 5115 y Fs(+)p Fm(\000)2361 5100 y Fz(\()p Fu(B)2473 5115 y Ft(E)2529 5096 y Ff(0)2555 5100 y Fz(\()p Fu(t)p Fz(\)\))p Fu(dt)f(<)h Fr(1)p Fu(:)528 b Fz(\(3.21\))150 5337 y FA(Mor)-5 b(e)g(over,)1315 5457 y Fz(lim)1299 5517 y Ft(t)p Fm(!1)1483 5457 y Fu(F)1560 5410 y Fm(\000)p Fs(4)p Ft(\017)1679 5419 y Fj(2)1713 5410 y Ft(;)p Fm(\000)p Fs(3)p Ft(\017)1852 5419 y Fj(2)1885 5410 y Ft(;)p Fs(3)p Ft(\017)1969 5419 y Fj(2)2003 5410 y Ft(;)p Fs(4)p Ft(\017)2087 5419 y Fj(2)1546 5479 y Fs(+)p Fm(\000)2126 5457 y Fz(\()p Fu(B)2238 5472 y Ft(E)2294 5453 y Ff(0)2320 5457 y Fz(\()p Fu(t)p Fz(\)\))27 b(=)h(0)p Fu(:)900 b Fz(\(3.22\))1939 5776 y(25)p eop %%Page: 26 29 26 28 bop 150 407 a Fv(Pro)s(of)p Fz(.)40 b(W)-8 b(e)22 b(only)f(pro)m(v)m(e)j(\(3.21\),)f(since)f(the)h(limit)18 b(follo)m(ws)j(from)f(the)j(usual)e(Co)s(ok)h(t)m(yp)s(e)h(argumen)m (t.)296 527 y(W)-8 b(e)36 b(in)m(tro)s(duce)f(the)h(b)s(ounded)g (observ)-5 b(able)35 b(\010\()p Fu(t)p Fz(\))e(=)e Fr(\000)p Fu(F)2413 542 y Fs(++)2528 527 y Fz(\()p Fu(B)2640 542 y Ft(E)2696 523 y Ff(0)2722 527 y Fz(\()p Fu(t)p Fz(\)\))k(and)g (compute)h(its)e(time)150 648 y(deriv)-5 b(ativ)m(e:)1089 798 y Fu(@)1140 813 y Ft(t)1170 798 y Fz(\010\()p Fu(t)p Fz(\))29 b(=)e Fu(F)1546 813 y Fs(+)p Fm(\000)1660 798 y Fz(\()p Fu(B)1772 813 y Ft(E)1828 794 y Ff(0)1854 798 y Fz(\()p Fu(t)p Fz(\)\))2013 731 y Fu(\030)g Fr(\001)22 b Fz(\()p Fu(\030)2214 746 y Ft(E)2270 727 y Ff(0)2295 731 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))g Fr(\000)h Fu(\030)5 b Fz(\()p Fu(t)p Fz(\)\))p 2013 775 836 4 v 2408 866 a Fu(r)2859 798 y(:)690 b Fz(\(3.23\))150 1003 y(Since)35 b(w)m(e)i(kno)m(w)f(that)f Fu(\030)5 b Fz(\()p Fu(t)p Fz(\))32 b Fr(\025)h Fu(d)1377 1018 y Fs(1)1449 1003 y Fu(>)f Fz(0)j(after)g(some)g(time,)g(w)m(e)h(can)g(write)f (using)g(energy)h(conser-)150 1124 y(v)-5 b(ation)31 b(that)676 1378 y Fu(\030)719 1393 y Ft(E)775 1374 y Ff(0)800 1378 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))23 b Fr(\000)f Fu(\030)5 b Fz(\()p Fu(t)p Fz(\))83 b(=)1568 1311 y(2)p Fu(E)1695 1274 y Fm(0)1740 1311 y Fr(\000)22 b Fu(\032)1889 1274 y Fs(2)1889 1337 y Ft(E)1945 1318 y Ff(0)1972 1311 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))g Fr(\000)h Fz(2)p Fu(E)28 b Fz(+)22 b Fu(\032)2626 1274 y Fs(2)2665 1311 y Fz(\()p Fu(t)p Fz(\))p 1568 1355 1209 4 v 1853 1446 a Fu(\030)1896 1461 y Ft(E)1952 1442 y Ff(0)1977 1446 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))h(+)f Fu(\030)5 b Fz(\()p Fu(t)p Fz(\))1399 1647 y(=)1674 1579 y(2\()p Fu(E)1839 1543 y Fm(0)1884 1579 y Fr(\000)23 b Fu(E)6 b Fz(\))p 1568 1624 639 4 v 1568 1715 a Fu(\030)1611 1730 y Ft(E)1667 1711 y Ff(0)1692 1715 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))22 b(+)g Fu(\030)5 b Fz(\()p Fu(t)p Fz(\))2238 1647 y(+)2346 1579 y Fu(\032)p Fz(\()p Fu(t)p Fz(\))23 b(+)f Fu(\032)2678 1594 y Ft(E)2734 1575 y Ff(0)2760 1579 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))p 2346 1624 649 4 v 2351 1715 a Fu(\030)2394 1730 y Ft(E)2450 1711 y Ff(0)2476 1715 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))g(+)g Fu(\030)5 b Fz(\()p Fu(t)p Fz(\))3005 1647 y Fu(B)3079 1662 y Ft(E)3135 1643 y Ff(0)3161 1647 y Fz(\()p Fu(t)p Fz(\))p Fu(:)277 b Fz(\(3.24\))150 1921 y(Since)33 b Fu(E)483 1885 y Fm(0)528 1921 y Fr(\000)23 b Fu(E)34 b Fr(\025)28 b Fu(\017)878 1936 y Fs(0)918 1921 y Fz(,)k(w)m(e)i(get)e(that)h(for)f Fu(t)c Fr(\025)g Fu(T)1869 1936 y Fs(1)1941 1921 y Fz(w)m(e)34 b(ha)m(v)m(e)684 2141 y Fu(F)747 2156 y Fs(+)p Fm(\000)861 2141 y Fz(\()p Fu(B)973 2156 y Ft(E)1029 2137 y Ff(0)1055 2141 y Fz(\()p Fu(t)p Fz(\)\))22 b Fr(\001)g Fz(\()p Fu(\030)1357 2156 y Ft(E)1413 2137 y Ff(0)1439 2141 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))g Fr(\000)h Fu(\030)5 b Fz(\()p Fu(t)p Fz(\)\))27 b Fr(\025)h Fz(\()p Fu(C)2233 2156 y Fs(1)2272 2141 y Fu(\017)2311 2156 y Fs(0)2373 2141 y Fr(\000)23 b Fu(C)2543 2156 y Fs(2)2582 2141 y Fu(\017)2621 2156 y Fs(2)2661 2141 y Fz(\))f Fr(\001)g Fu(F)2834 2156 y Fs(+)p Fm(\000)2948 2141 y Fz(\()p Fu(B)3060 2156 y Ft(E)3116 2137 y Ff(0)3142 2141 y Fz(\()p Fu(t)p Fz(\)\))285 b(\(3.25\))150 2361 y(where)39 b Fu(C)507 2376 y Fs(1)584 2361 y Fz(and)f Fu(C)849 2376 y Fs(2)926 2361 y Fz(are)f(p)s(ositiv)m (e)g(constan)m(ts)j(not)d(dep)s(ending)h(on)g Fu(\017)2729 2376 y Fs(2)2769 2361 y Fz(.)58 b(Moreo)m(v)m(er,)41 b(they)e(can)f(b)s(e)150 2481 y(uniformly)f(c)m(hosen)j(if)e Fu(E)1093 2445 y Fm(0)1155 2481 y Fz(is)g(restricted)h(to)g(compact)f (sets.)63 b(Hence)40 b(if)e Fu(\017)2899 2496 y Fs(2)2977 2481 y Fz(is)h(small)d(enough,)41 b(w)m(e)150 2602 y(ha)m(v)m(e)34 b(the)f(desired)g(p)s(ositivit)m(y)e(and)i(w)m(e)g(can)g(in)m(tegrate)f (in)g(\(3.23\))g(th)m(us)i(yielding)c(\(3.21\).)p 3754 2602 4 66 v 3758 2539 59 4 v 3758 2602 V 3816 2602 4 66 v 296 2840 a(No)m(w)43 b(let)f(us)h(get)g(bac)m(k)h(to)e(the)h(prop) s(osition.)71 b(With)42 b Fu(\017)2413 2855 y Fs(2)2495 2840 y Fz(pro)m(vided)h(b)m(y)h(the)f(ab)s(o)m(v)m(e)g(lemma,)150 2960 y(construct)32 b(the)e(in)m(terv)-5 b(als)30 b(from)f(\(3.20\))h (and)g(see)i(that)e(there)h(exists)g(a)g(constan)m(t)g Fu(c)d(>)f Fz(0)j(suc)m(h)i(that)869 3273 y Fu(c)22 b Fr(\001)g Fu(\037)1044 3301 y Fs([)p Ft(\032)1100 3312 y Fq(E)1151 3301 y Fs(\()p Ft(\022)r Fs(\)+)p Ft(\017=)p Fs(2)p Ft(;)1414 3245 y Fm(p)p 1473 3245 91 3 v 56 x Fs(2)p Ft(E)t Fs(])1588 3273 y Fz(\()p Fu(x)p Fz(\))28 b Fr(\024)1892 3149 y Ft(N)1852 3179 y Fl(X)1860 3391 y Ft(k)r Fs(=1)2012 3273 y Fu(F)2089 3226 y Fm(\000)p Fs(2)p Ft(\017)2208 3235 y Fj(2)2242 3226 y Ft(;)p Fm(\000)p Ft(\017)2346 3235 y Fj(2)2380 3226 y Ft(;\017)2429 3235 y Fj(2)2463 3226 y Ft(;)p Fs(2)p Ft(\017)2547 3235 y Fj(2)2075 3295 y Fs(+)p Fm(\000)2585 3273 y Fz(\()p Fu(x)22 b Fr(\000)h Fu(\032)2850 3289 y Ft(E)2906 3267 y Ff(0)2902 3313 y Fq(k)2944 3273 y Fz(\()p Fu(\022)s Fz(\)\))470 b(\(3.26\))150 3578 y(whic)m(h)33 b(leads)g(to)814 3866 y Fu(F)877 3881 y Fs(+)937 3866 y Fz(\()p Fu(B)1049 3881 y Ft(E)1108 3866 y Fz(\()p Fu(t)p Fz(\)\))28 b Fr(\024)1400 3798 y Fz(1)p 1400 3843 49 4 v 1403 3934 a Fu(c)1481 3866 y Fr(\001)22 b Fu(F)1594 3881 y Fs(+)1653 3866 y Fz(\()p Fu(B)1765 3881 y Ft(E)1825 3866 y Fz(\()p Fu(t)p Fz(\)\))g Fr(\001)2086 3741 y Ft(N)2046 3771 y Fl(X)2053 3983 y Ft(k)r Fs(=1)2206 3866 y Fu(F)2283 3818 y Fm(\000)p Fs(2)p Ft(\017)2402 3827 y Fj(2)2436 3818 y Ft(;)p Fm(\000)p Ft(\017)2540 3827 y Fj(2)2574 3818 y Ft(;\017)2623 3827 y Fj(2)2656 3818 y Ft(;)p Fs(2)p Ft(\017)2740 3827 y Fj(2)2269 3887 y Fs(+)p Fm(\000)2778 3866 y Fz(\()p Fu(B)2890 3882 y Ft(E)2946 3859 y Ff(0)2942 3905 y Fq(k)2985 3866 y Fz(\()p Fu(t)p Fz(\)\))p Fu(:)415 b Fz(\(3.27\))150 4176 y(Then)34 b(the)f(use)g(of)f(\(3.22\))g(\014nishes)i(the)f(pro)s (of.)p 3754 4176 4 66 v 3758 4113 59 4 v 3758 4176 V 3816 4176 4 66 v 150 4414 a Fv(Remark)p Fz(.)41 b(An)25 b(immediate)e(consequence)28 b(of)d(the)g(ab)s(o)m(v)m(e)h(prop)s (osition)e(and)h(energy)h(conserv)-5 b(ation)150 4535 y(is)34 b(a)h(sharp)g(lo)s(calization)c(for)k Fu(\030)5 b Fz(,)35 b(to)s(o.)49 b(Namely)-8 b(,)35 b(for)f(ev)m(ery)j Fu(\017)32 b(>)f Fz(0)k(there)h(exists)f Fu(T)49 b Fz(large)34 b(enough)150 4655 y(suc)m(h)g(that)1364 4775 y Fr(j)p Fu(\030)5 b Fz(\()p Fu(t)p Fz(\))21 b Fr(\000)i Fu(\030)1715 4790 y Ft(E)1774 4775 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))p Fr(j)28 b(\024)g Fu(\017;)114 b(t)28 b Fr(\025)h Fu(T)8 b(:)965 b Fz(\(3.28\))150 5064 y Fp(3.5)135 b(The)45 b(eik)l(onal)h(and)f(Hamilton-Jacobi)i(equation)150 5249 y Fz(Let)33 b(\()p Fu(\032)413 5264 y Ft(E)473 5249 y Fu(;)17 b(\030)560 5264 y Ft(E)619 5249 y Fz(\))32 b(b)s(e)h(as)f(in)g (Corollary)f(2.7.)43 b(De\014ne)2000 5224 y(~)1983 5249 y Fu(S)34 b Fz(:)28 b(\()p Fu(E)2242 5264 y Ft(d)2282 5249 y Fu(;)17 b Fr(1)p Fz(\))22 b Fr(\002)g Fz(\(0)p Fu(;)17 b Fr(1)p Fz(\))k Fr(\002)i Fn(R)39 b Fr(7!)27 b Fn(R)43 b Fz(b)m(y)1573 5444 y(~)1556 5469 y Fu(S)6 b Fz(\()p Fu(E)g(;)17 b(r)m(;)g(\022)s Fz(\))27 b(:=)h Fu(r)s(\032)2208 5484 y Ft(E)2268 5469 y Fz(\()p Fu(\022)s Fz(\))p Fu(:)1157 b Fz(\(3.29\))1939 5776 y(26)p eop %%Page: 27 30 27 29 bop 150 407 a Fz(By)33 b(direct)f(computation)g(w)m(e)h(can)g (sho)m(w)h(that)1941 382 y(~)1924 407 y Fu(S)k Fz(solv)m(es)c(the)f (eik)m(onal)e(equation)1322 620 y(\()p Fu(@)1411 635 y Ft(r)1466 595 y Fz(~)1450 620 y Fu(S)6 b Fz(\))1554 579 y Fs(2)1615 620 y Fz(+)22 b(\()p Fu(@)1802 635 y Ft(\022)1858 595 y Fz(~)1842 620 y Fu(S)5 b(=r)25 b Fr(\000)e Fu(b)p Fz(\()p Fu(\022)s Fz(\)\))2328 579 y Fs(2)2395 620 y Fz(=)28 b(2)p Fu(E)6 b(:)296 833 y Fz(De\014ne)33 b Fu(r)s Fz(\()p Fu(E)6 b(;)17 b(t;)g(\022)s Fz(\))28 b(:=)f Fu(t=)p Fz(\()p Fu(@)1301 848 y Ft(E)1361 833 y Fu(\032)1411 848 y Ft(E)1471 833 y Fz(\)\()p Fu(\022)s Fz(\))h Fu(>)f Fz(0.)44 b(W)-8 b(e)32 b(then)i(ha)m(v)m(e)150 1030 y Fv(Lemma)j(3.8.)42 b FA(F)-7 b(or)34 b(al)5 b(l)34 b Fu(t)28 b(>)g Fz(0)p FA(,)34 b Fu(E)g(>)28 b(E)1646 1045 y Ft(d)1721 1030 y FA(and)34 b Fu(\022)d Fr(2)d Fn(R)618 1243 y Fu(@)669 1258 y Ft(E)729 1243 y Fu(r)s Fz(\()p Fu(E)6 b(;)17 b(t;)g(\022)s Fz(\))27 b Fu(>)g Fz(0)p Fu(;)147 b Fz(lim)1424 1305 y Ft(E)t Fm(!1)1637 1243 y Fu(r)s Fz(\()p Fu(E)6 b(;)17 b(t;)g(\022)s Fz(\))27 b(=)g Fr(1)p Fu(;)155 b Fz(lim)2382 1306 y Ft(E)t Fm(&)p Ft(E)2561 1318 y Fq(d)2613 1243 y Fu(r)s Fz(\()p Fu(E)6 b(;)17 b(t;)g(\022)s Fz(\))27 b(=)g(0)p Fu(:)385 b Fz(\(3.30\))150 1505 y Fv(Pro)s(of.)43 b Fz(Clearly)-8 b(,)32 b Fu(r)s Fz(\()p Fu(E)6 b(;)17 b(t;)g(\022)s Fz(\))32 b(increases)h(with)f Fu(E)39 b Fz(b)s(ecause)33 b(of)f(the)h(prop)s(erties)g(of)f Fu(\032)3257 1520 y Ft(E)3317 1505 y Fz(,)g(see)i(Prop)s(o-)150 1625 y(sition)d(2.6.)43 b(The)33 b(only)e(non)m(trivial)f(thing)i(is)f (pro)m(ving)h(that)g(Ran)c Fu(r)s Fz(\()p Fr(\001)p Fu(;)17 b(t;)g(\022)s Fz(\))26 b(=)i(\(0)p Fu(;)17 b Fr(1)p Fz(\).)42 b(But)33 b(this)e(is)150 1746 y(equiv)-5 b(alen)m(t)32 b(to)h(pro)m(ving)f(that)1132 1959 y(lim)1093 2021 y Ft(E)t Fm(&)p Ft(E)1272 2033 y Fq(d)1324 1959 y Fu(@)1375 1974 y Ft(E)1435 1959 y Fu(\032)1485 1974 y Ft(E)1545 1959 y Fz(\()p Fu(\022)s Fz(\))27 b(=)h(+)p Fr(1)p Fu(;)144 b Fz(lim)2117 2021 y Ft(E)t Fm(!1)2330 1959 y Fu(@)2381 1974 y Ft(E)2441 1959 y Fu(\032)2491 1974 y Ft(E)2551 1959 y Fz(\()p Fu(\022)s Fz(\))27 b(=)h(0)p Fu(:)694 b Fz(\(3.31\))150 2220 y(F)-8 b(or,)32 b(let)g(us)h(in)m(tro)s(duce)g (\(2.25\))e(in)m(to)h(\(2.22\))g(and)h(obtain)777 2501 y Fu(@)828 2516 y Ft(E)888 2501 y Fu(\032)938 2516 y Ft(E)998 2501 y Fz(\()p Fu(\022)s Fz(\))27 b(=)1253 2366 y Fl(Z)1353 2392 y Fm(1)1308 2591 y Fs(0)1632 2434 y Fz(1)p 1454 2478 406 4 v 1454 2570 a Fu(\030)1497 2585 y Ft(E)1556 2570 y Fz(\()p Fu(\022)e Fr(\000)e Fu(\036)p Fz(\))1886 2501 y(exp)2052 2361 y Fl(\032)2127 2501 y Fr(\000)2221 2366 y Fl(Z)2320 2392 y Ft(\036)2276 2591 y Fs(0)2393 2434 y Fu(\032)2443 2449 y Ft(E)p 2393 2478 110 4 v 2397 2570 a Fu(\030)2440 2585 y Ft(E)2513 2501 y Fz(\()p Fu(\022)i Fr(\000)e Fu(\036)2779 2460 y Fm(0)2802 2501 y Fz(\))p Fu(d\036)2949 2460 y Fm(0)2971 2361 y Fl(\033)3063 2501 y Fu(d\036:)377 b Fz(\(3.32\))296 2772 y(Since)32 b Fu(\030)593 2787 y Ft(E)652 2772 y Fz(\()p Fu(\022)s Fz(\))g(is)f(con)m(tin)m(uous)h(on)g(b)s(oth)f(argumen)m(ts)h (and)f(strictly)g(p)s(ositiv)m(e,)g(for)g(an)m(y)h Fu(E)3541 2787 y Fs(1)3609 2772 y Fu(>)27 b(E)3784 2787 y Ft(d)150 2892 y Fz(w)m(e)34 b(can)e(\014nd)i Fu(c)27 b(>)h Fz(0)k(suc)m(h)i (that)1576 3127 y(inf)1467 3194 y Ft(E)t Fm(2)p Fs([)p Ft(E)1642 3206 y Fq(d)1677 3194 y Ft(;E)1749 3203 y Fj(1)1783 3194 y Fs(])1921 3127 y Fz(inf)1917 3190 y Ft(\022)r Fm(2)p Fe(T)2158 3060 y Fz(1)p 2070 3104 227 4 v 2070 3196 a Fu(\030)2113 3211 y Ft(E)2172 3196 y Fz(\()p Fu(\022)s Fz(\))2334 3127 y Fr(\025)28 b Fu(c:)296 3396 y Fz(Cho)s(ose)34 b Fu(M)43 b Fz(to)32 b(b)s(e)h(arbitrarily)d(large)h(and)i(see)h(that)e (for)g Fu(E)i Fr(2)28 b Fz([)p Fu(E)2739 3411 y Ft(d)2779 3396 y Fu(;)17 b(E)2895 3411 y Fs(1)2935 3396 y Fz(])32 b(w)m(e)i(ha)m(v)m(e)938 3677 y Fu(@)989 3692 y Ft(E)1049 3677 y Fu(\032)1099 3692 y Ft(E)1159 3677 y Fz(\()p Fu(\022)s Fz(\))28 b Fr(\025)g Fu(c)22 b Fr(\001)1530 3542 y Fl(Z)1629 3568 y Ft(M)1585 3767 y Fs(0)1725 3677 y Fz(exp)1891 3537 y Fl(\032)1965 3677 y Fr(\000)2059 3542 y Fl(Z)2159 3568 y Ft(\036)2115 3767 y Fs(0)2232 3610 y Fu(\032)2282 3625 y Ft(E)p 2232 3655 110 4 v 2236 3746 a Fu(\030)2279 3761 y Ft(E)2352 3677 y Fz(\()p Fu(\022)j Fr(\000)d Fu(\036)2617 3636 y Fm(0)2641 3677 y Fz(\))p Fu(d\036)2788 3636 y Fm(0)2810 3537 y Fl(\033)2902 3677 y Fu(d\036:)538 b Fz(\(3.33\))150 3948 y(Hence)34 b(w)m(e)g(get)e(\(see)i(\(2.32\)\))804 4229 y(lim)17 b(inf)832 4292 y Ft(E)t Fm(&)p Ft(E)1011 4304 y Fq(d)1091 4229 y Fu(@)1142 4244 y Ft(E)1202 4229 y Fu(\032)1252 4244 y Ft(E)1312 4229 y Fz(\()p Fu(\022)s Fz(\))28 b Fr(\025)g Fu(c)22 b Fr(\001)1683 4094 y Fl(Z)1782 4120 y Ft(M)1738 4319 y Fs(0)1878 4229 y Fz(exp)2044 4089 y Fl(\032)2118 4229 y Fr(\000)2212 4094 y Fl(Z)2312 4120 y Ft(\036)2268 4319 y Fs(0)2385 4162 y Fu(\032)2435 4177 y Ft(d)p 2385 4206 91 4 v 2389 4297 a Fu(\030)2432 4312 y Ft(d)2485 4229 y Fz(\()p Fu(\022)k Fr(\000)c Fu(\036)2751 4188 y Fm(0)2774 4229 y Fz(\))p Fu(d\036)2921 4188 y Fm(0)2944 4089 y Fl(\033)3035 4229 y Fu(d\036:)150 4500 y Fz(Because)37 b(the)f(in)m(tegral)e(in)h(\(2.32\))g(is)h(zero,)h(w)m (e)f(get)g(the)g(existence)h(of)f(another)f(constan)m(t)i Fu(c)3611 4463 y Fm(0)3667 4500 y Fu(>)c Fz(0)150 4620 y(suc)m(h)h(that)e(for)g(ev)m(ery)j Fu(\022)h Fz(and)c Fu(\036)1305 4896 y Fz(exp)1470 4755 y Fl(\032)1545 4896 y Fr(\000)1639 4760 y Fl(Z)1739 4786 y Ft(\036)1694 4986 y Fs(0)1811 4828 y Fu(\032)1861 4843 y Ft(d)p 1811 4873 V 1815 4964 a Fu(\030)1858 4979 y Ft(d)1912 4896 y Fz(\()p Fu(\022)25 b Fr(\000)e Fu(\036)2178 4855 y Fm(0)2201 4896 y Fz(\))p Fu(d\036)2348 4855 y Fm(0)2371 4755 y Fl(\033)2473 4896 y Fr(\025)28 b Fu(c)2620 4855 y Fm(0)2643 4896 y Fu(;)150 5166 y Fz(and)33 b(since)g Fu(M)43 b Fz(w)m(as)33 b(arbitrary)-8 b(,)32 b(w)m(e)i(get)e(lim)1779 5181 y Ft(E)t Fm(&)p Ft(E)1958 5193 y Fq(d)2013 5166 y Fu(@)2064 5181 y Ft(E)2124 5166 y Fu(\032)2174 5181 y Ft(E)2234 5166 y Fz(\()p Fu(\022)s Fz(\))c(=)g Fr(1)p Fz(.)296 5286 y(The)j(other)f(limit)c(is)k(pro)m(v)m(en)h(using)f (\(3.32\))f(again.)41 b(When)31 b Fu(E)36 b Fz(b)s(ecomes)30 b(large,)f(w)m(e)i(kno)m(w)g(that)150 5407 y Fu(\030)193 5422 y Ft(E)289 5407 y Fz(is)36 b(trapp)s(ed)g(inside)g([min)15 b Fr(j)p Fu(b)p Fr(j)p Fu(;)i Fz(max)f Fr(j)p Fu(b)p Fr(j)p Fz(])36 b(while)g Fu(\032)2055 5422 y Ft(E)2149 5407 y Fr(\030)2261 5322 y(p)p 2344 5322 127 4 v 85 x Fz(2)p Fu(E)6 b Fz(.)55 b(It)37 b(follo)m(ws)e(that)h(the)h(righ)m(t)f (hand)150 5527 y(side)d(in)e(\(3.32\))h(can)h(b)s(e)g(b)s(ounded)g(b)m (y)h(a)e(constan)m(t)h(times)f Fu(E)2400 5491 y Fm(\000)p Fs(1)p Ft(=)p Fs(2)2598 5527 y Fz(and)g(w)m(e)i(are)e(done.)p 3754 5527 4 66 v 3758 5465 59 4 v 3758 5527 V 3816 5527 4 66 v 1939 5776 a(27)p eop %%Page: 28 31 28 30 bop 296 407 a Fz(No)m(w)33 b(consider)g(the)g(partial)e(Legendre) i(transform)f(of)2375 382 y(~)2358 407 y Fu(S)6 b Fz(:)411 649 y Fu(S)34 b Fz(:)28 b(\(0)p Fu(;)17 b Fr(1)p Fz(\))k Fr(\002)h Fz(\(0)p Fu(;)17 b Fr(1)p Fz(\))22 b Fr(\002)g Fn(R)39 b Fr(7!)27 b Fn(R)5 b Fu(;)120 b(S)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))28 b(:=)53 b(sup)2291 730 y Ft(E)t Fm(\025)p Ft(E)2454 742 y Fq(d)2506 538 y Fl(n)2589 623 y Fz(~)2573 649 y Fu(S)5 b Fz(\()p Fu(E)h(;)17 b(r)m(;)g(\022)s Fz(\))22 b Fr(\000)h Fu(tE)3204 538 y Fl(o)3287 649 y Fu(:)262 b Fz(\(3.34\))296 929 y(The)34 b(previous)f(lemma)d(stated)j(that)1513 1149 y(Ran)27 b Fu(r)s Fz(\()p Fr(\001)p Fu(;)17 b(t;)g(\022)s Fz(\))27 b(=)g(\(0)p Fu(;)17 b Fr(1)p Fz(\))p Fu(:)1113 b Fz(\(3.35\))296 1369 y(Since)47 b Fu(@)616 1384 y Ft(E)676 1369 y Fu(r)s Fz(\()p Fr(\001)p Fu(;)17 b(t;)g(\022)s Fz(\))50 b Fu(>)i Fz(0,)e(one)c(can)h(express)i Fu(E)j Fz(as)47 b(a)f(function)g Fu(E)6 b Fz(\()p Fu(t;)17 b Fr(\001)p Fu(;)g(\022)s Fz(\))46 b(of)g Fu(r)54 b Fr(2)e Fz(\(0)p Fu(;)17 b Fr(1)p Fz(\),)150 1489 y(ob)s(eying)741 1730 y Fu(E)6 b Fz(\()p Fu(t;)17 b Fr(\001)p Fu(;)g(\022)s Fz(\))27 b(:)h(\(0)p Fu(;)17 b Fr(1)p Fz(\))26 b Fr(7!)i Fz(\()p Fu(E)1709 1745 y Ft(d)1749 1730 y Fu(;)17 b Fr(1)p Fz(\))p Fu(;)211 b(t)28 b Fz(=)g(\()p Fu(@)2425 1745 y Ft(E)2501 1705 y Fz(~)2485 1730 y Fu(S)6 b Fz(\)\()p Fu(E)g Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))p Fu(;)g(r)m(;)g(\022)s Fz(\))p Fu(:)341 b Fz(\(3.36\))296 1904 y(With)32 b(this)h(de\014nition,)e(the)i(function)f Fu(S)39 b Fz(in)31 b(\(3.34\))h(b)s(ecomes)1238 2145 y Fu(S)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))27 b(=)g Fu(r)s(\032)1819 2161 y Ft(E)t Fs(\()p Ft(t;r)n(;\022)r Fs(\))2063 2145 y Fz(\()p Fu(\022)s Fz(\))c Fr(\000)f Fu(tE)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))p Fu(:)839 b Fz(\(3.37\))296 2319 y(This)24 b(will)e(b)s(e)i(our)g (Hamilton-Jacobi)c(function.)40 b(Using)23 b(\(2.8\),)j(\(3.29\))d(and) h(\(3.34\))f(one)h(readily)150 2440 y(v)m(eri\014es)34 b(the)f(follo)m(wing)d(iden)m(tities:)1469 2660 y Fu(@)1520 2675 y Ft(t)1550 2660 y Fu(S)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))28 b(=)f Fr(\000)p Fu(E)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))p Fu(;)1071 b Fz(\(3.38\))1478 2880 y Fu(@)1529 2895 y Ft(r)1567 2880 y Fu(S)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))27 b(=)h Fu(\032)2102 2895 y Ft(E)t Fs(\()p Ft(t;r)n(;\022)r Fs(\))2346 2880 y Fz(\()p Fu(\022)s Fz(\))p Fu(;)1079 b Fz(\(3.39\))1288 3054 y Fu(@)1339 3069 y Ft(\022)1378 3054 y Fu(S)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))28 b(=)f Fu(r)s Fz([)p Fu(b)p Fz(\()p Fu(\022)s Fz(\))c(+)f Fu(\030)2266 3070 y Ft(E)t Fs(\()p Ft(t;r)n(;\022)r Fs(\))2509 3054 y Fz(\()p Fu(\022)s Fz(\)])p Fu(:)889 b Fz(\(3.40\))296 3229 y(Therefore,)34 b Fu(S)k Fz(is)33 b Fu(C)1039 3192 y Fm(1)1146 3229 y Fz(on)f(its)g(domain)f(and)i(ob)s(eys)g(the)g (Hamilton-Jacobi)c(equation)1215 3517 y(0)f(=)f Fu(@)1446 3532 y Ft(t)1476 3517 y Fu(S)h Fz(+)1672 3449 y(1)p 1672 3494 49 4 v 1672 3585 a(2)1748 3376 y Fl(\022)1831 3449 y Fu(@)1882 3464 y Ft(\022)1921 3449 y Fu(S)p 1831 3494 157 4 v 1886 3585 a(r)2019 3517 y Fr(\000)23 b Fu(b)2160 3376 y Fl(\023)2234 3399 y Fs(2)2295 3517 y Fz(+)2403 3449 y(1)p 2403 3494 49 4 v 2403 3585 a(2)2462 3517 y(\()p Fu(@)2551 3532 y Ft(r)2589 3517 y Fu(S)6 b Fz(\))2693 3476 y Fs(2)2733 3517 y Fu(:)816 b Fz(\(3.41\))150 3794 y Fv(Prop)s(osition)41 b(3.9.)j FA(The)39 b(function)g Fu(S)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))39 b FA(is)g(homo)-5 b(gene)g(ous)38 b(of)h(de)-5 b(gr)g(e)g(e)39 b Fz(1)g FA(in)g(the)g(variables)150 3914 y Fz(\()p Fu(r)m(;)17 b(t)p Fz(\))p FA(,)42 b(i.e.)62 b Fu(S)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))38 b(=)h Fu(tS)6 b Fz(\(1)p Fu(;)17 b(r)s(=t;)g(\022)s Fz(\))p FA(.)61 b(Supp)-5 b(ose)40 b Fu(K)46 b Fr(\032)39 b Fz(\(0)p Fu(;)17 b Fr(1)p Fz(\))40 b FA(is)g(c)-5 b(omp)g(act.)62 b(Then)40 b(for)g(every)150 4035 y Fu(n)28 b Fr(\025)g Fz(0)35 b FA(and)f Fu(m)28 b Fr(\025)g Fz(0)35 b FA(we)f(c)-5 b(an)35 b(\014nd)f(a)h(c)-5 b(onstant)34 b Fu(C)h Fz(=)27 b Fu(C)2185 4050 y Ft(m;n;K)2433 4035 y FA(such)34 b(that)1121 4306 y Fz(sup)1177 4388 y Ft(\022)1314 4306 y Fz(sup)1284 4392 y Ft(r)r(=t)p Fm(2)p Ft(K)1507 4306 y Fr(j)p Fu(@)1591 4265 y Ft(n)1586 4331 y(r)1638 4306 y Fu(@)1694 4265 y Ft(m)1689 4331 y(\022)1762 4306 y Fu(S)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))p Fr(j)27 b(\024)2334 4239 y Fu(C)p 2286 4283 173 4 v 2286 4374 a(t)2321 4346 y Ft(n)p Fm(\000)p Fs(1)2468 4306 y Fu(;)117 b(t)28 b(>)f Fz(1)p Fu(:)722 b Fz(\(3.42\))150 4596 y Fv(Pro)s(of.)79 b Fz(Let)44 b(us)h(see)h(wh)m(y)g Fu(S)k Fz(has)44 b(the)h(stated)g(homogeneit)m(y)f(prop)s(ert)m(y)-8 b(.)80 b(First,)46 b(notice)e(that)150 4716 y Fu(E)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))43 b(only)f(dep)s(ends)i(on)f Fu(r)s(=t)f Fz(and)h Fu(\022)j Fz(since)d(it)f(w)m(as)i(obtained)e (from)g(the)h(equation)f Fu(t=r)48 b Fz(=)150 4836 y Fu(@)201 4851 y Ft(E)261 4836 y Fu(\032)311 4851 y Ft(E)371 4836 y Fz(\()p Fu(E)6 b(;)17 b(\022)s Fz(\))32 b(\(see)i(\(3.36\)\).)43 b(Second,)33 b(apply)g(\(3.37\).)296 4957 y(In)43 b(order)f(to)g(pro)m (v)m(e)h(\(3.42\),)h(w)m(e)f(use)g(the)f(scaling)f(prop)s(ert)m(y)i (and)f(the)g(fact)g(that)g Fu(E)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))150 5077 y(is)41 b(restricted)i(to)e(some)h (compact)f(in)g(\()p Fu(E)1715 5092 y Ft(d)1756 5077 y Fu(;)17 b Fr(1)p Fz(\))41 b(only)g(dep)s(ending)h(on)g Fu(K)7 b Fz(.)71 b(F)-8 b(urther)42 b(details)f(are)150 5198 y(omitted.)p 3754 5198 4 66 v 3758 5135 59 4 v 3758 5198 V 3816 5198 4 66 v 1939 5776 a(28)p eop %%Page: 29 32 29 31 bop 150 407 a Fp(3.6)135 b(A)45 b(priori)g(lo)t(calization)i(for) e Fi(E)7 b FF(\()p Fi(t;)20 b(r)s FF(\()p Fi(t)p FF(\))p Fi(;)g(\022)s FF(\()p Fi(t)p FF(\)\))150 592 y Fz(Consider)35 b(again)f(a)h(tra)5 b(jectory)35 b(corresp)s(onding)g(to)g(an)g(energy) h Fu(E)h(>)32 b(E)2855 607 y Ft(d)2896 592 y Fz(;)k(w)m(e)g(kno)m(w)g (that)f(\(3.18\))150 712 y(and)48 b(\(3.28\))f(hold)f(and)i(w)m(e)h(no) m(w)f(w)m(an)m(t)g(a)g(similar)c(lo)s(calization)g(for)j(the)h (di\013erence)g(b)s(et)m(w)m(een)150 832 y Fu(E)6 b Fz(\()p Fu(t;)17 b(r)s Fz(\()p Fu(t)p Fz(\))p Fu(;)g(\022)s Fz(\()p Fu(t)p Fz(\)\))32 b(and)h Fu(E)6 b Fz(.)296 953 y(W)-8 b(e)48 b(again)e(use)i(propagation)e(estimates)h(whic)m(h)h(can)g(and)f (will)e(b)s(e)j(generalized)f(later)f(to)150 1073 y(quan)m(tum)31 b(mec)m(hanics.)44 b(F)-8 b(or)30 b(simplifying)e(notation,)i(replace)h Fu(E)6 b Fz(\()p Fu(t;)17 b(r)s Fz(\()p Fu(t)p Fz(\))p Fu(;)g(\022)s Fz(\()p Fu(t)p Fz(\)\))31 b(b)m(y)h Fu(E)3309 1088 y Ft(t)3339 1073 y Fz(.)43 b(W)-8 b(e)32 b(tak)m(e)g Fu(\017)150 1193 y Fz(in)g(\(3.18\))g(and)g(\(3.28\))g(v)m(ery)i (small.)41 b(De\014ne)1359 1468 y Fu(D)s Fz(\()p Fu(t)p Fz(\))28 b(:=)f(1)22 b Fr(\000)1893 1401 y Fu(r)s Fz(\()p Fu(t)p Fz(\))p 1893 1445 158 4 v 1954 1537 a Fu(t)2083 1468 y Fr(\001)g Fu(@)2184 1483 y Ft(E)2244 1468 y Fu(\032)2294 1483 y Ft(E)2354 1468 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))p Fu(:)960 b Fz(\(3.43\))150 1714 y(Then)34 b(w)m(e)f(ha)m(v)m (e)150 1917 y Fv(Prop)s(osition)e(3.10.)40 b FA(L)-5 b(et)31 b Fu(F)1248 1932 y Fs(+)p Fm(\000)1390 1917 y Fz(=)d Fu(F)1571 1866 y Ft(\017)1600 1875 y Fj(1)1634 1866 y Ft(=)p Fs(2)p Ft(;\017)1753 1875 y Fj(1)1787 1866 y Ft(;c;d)1557 1939 y Fs(+)p Fm(\000)1897 1917 y FA(,)k Fu(\017)1998 1932 y Fs(1)2066 1917 y Fu(>)27 b Fz(0)p FA(,)32 b(and)f(c)-5 b(onsider)31 b(the)g(c)-5 b(orr)g(esp)g(onding)30 b Fu(F)3681 1932 y Fs(++)3795 1917 y FA(.)150 2038 y(Then)k(we)h(have:) 1000 2048 y Fl(Z)1099 2075 y Fm(1)1055 2274 y Fs(1)1201 2116 y Fz(1)p 1201 2161 49 4 v 1208 2252 a Fu(t)1259 2184 y(F)1336 2143 y Fm(0)1322 2208 y Fs(++)1436 2184 y Fz(\()p Fu(D)s Fz(\()p Fu(t)p Fz(\)\))p Fu(dt)28 b Fz(=)1924 2048 y Fl(Z)2024 2075 y Fm(1)1979 2274 y Fs(1)2125 2116 y Fz(1)p 2125 2161 V 2132 2252 a Fu(t)2184 2184 y(F)2247 2199 y Fs(+)p Fm(\000)2361 2184 y Fz(\()p Fu(D)s Fz(\()p Fu(t)p Fz(\)\))p Fu(dt)f(<)h Fr(1)p Fu(:)600 b Fz(\(3.44\))150 2405 y FA(Mor)-5 b(e)g(over,)34 b(ther)-5 b(e)35 b(exists)g Fu(T)48 b FA(lar)-5 b(ge)35 b(enough)f(such)h(that) 1222 2625 y Fr(j)p Fu(E)28 b Fz(+)22 b Fu(@)1499 2640 y Ft(t)1529 2625 y Fu(S)6 b Fz(\()p Fu(t;)17 b(r)s Fz(\()p Fu(t)p Fz(\))p Fu(;)g(\022)s Fz(\()p Fu(t)p Fz(\)\))p Fr(j)26 b(\024)j Fu(\017)2310 2640 y Fs(1)2349 2625 y Fu(;)117 b(t)28 b Fr(\025)g Fu(T)8 b(:)823 b Fz(\(3.45\))150 2845 y Fv(Pro)s(of)p Fz(.)43 b(Di\013eren)m(tiating)30 b Fu(D)35 b Fz(and)e(using)f(the)h(iden)m(tit)m(y)g(1)22 b Fr(\000)g Fu(\032)2451 2860 y Ft(E)2511 2845 y Fu(@)2562 2860 y Ft(E)2622 2845 y Fu(\032)2672 2860 y Ft(E)2760 2845 y Fz(=)27 b Fu(\030)2906 2860 y Ft(E)2966 2845 y Fu(@)3017 2860 y Ft(E)3077 2845 y Fu(\030)3120 2860 y Ft(E)3211 2845 y Fz(w)m(e)34 b(obtain:)531 3042 y Fu(dD)p 531 3086 135 4 v 555 3177 a(dt)703 3109 y Fz(=)27 b Fr(\000)893 3042 y Fu(D)p 893 3086 84 4 v 918 3177 a(t)1010 3109 y Fr(\000)1119 3042 y Fz(1)p 1119 3086 49 4 v 1126 3177 a Fu(t)1200 3109 y Fr(\001)22 b Fz([\()p Fu(\032)g Fr(\000)h Fu(\032)1537 3124 y Ft(E)1597 3109 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\)\))f Fr(\001)g Fu(@)1993 3124 y Ft(E)2053 3109 y Fu(\032)2103 3124 y Ft(E)2185 3109 y Fz(+)g(\()p Fu(\030)27 b Fr(\000)22 b Fu(\030)2533 3124 y Ft(E)2592 3109 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\)\))h Fr(\001)e Fu(@)2988 3124 y Ft(E)3049 3109 y Fu(\030)3092 3124 y Ft(E)3151 3109 y Fz(])p Fu(:)371 b Fz(\(3.46\))150 3360 y(Consider)32 b(the)g(b)s(ounded)g(propagation)e(observ)-5 b(able)31 b(\010\()p Fu(t)p Fz(\))d(=)g Fr(\000)p Fu(F)2587 3375 y Fs(++)2701 3360 y Fz(\()p Fu(D)s Fz(\()p Fu(t)p Fz(\)\),)k(di\013eren)m(tiate)e(it)h(and)150 3481 y(use)j(\(3.46\),)d (\(3.18\))h(and)h(\(3.28\):)681 3750 y(\010)751 3709 y Fm(0)803 3750 y Fz(=)27 b Fr(\000)p Fu(F)1046 3765 y Fs(+)p Fm(\000)1161 3750 y Fz(\()p Fu(D)s Fz(\))22 b Fr(\001)1402 3683 y Fu(dD)p 1402 3727 135 4 v 1427 3818 a(dt)1574 3750 y Fr(\025)1689 3683 y Fz(1)p 1689 3727 49 4 v 1696 3818 a Fu(t)1748 3750 y(F)1811 3765 y Fs(+)p Fm(\000)1925 3750 y Fz(\()p Fu(D)s Fz(\))g Fr(\001)g Fz(\()p Fu(\017)2234 3765 y Fs(1)2274 3750 y Fu(=)p Fz(2)f Fr(\000)i Fz(Const)g Fr(\001)f Fu(\017)p Fz(\))p Fu(;)114 b(t)28 b Fr(\025)g Fu(T)8 b(:)296 3995 y Fz(Hence)38 b(if)d Fu(\017)i Fz(is)f(small)d(enough,)38 b(w)m(e)g(ha)m(v)m(e)f(the) g(desired)g(p)s(ositivit)m(y)e(and)h(w)m(e)h(can)g(in)m(tegrate)f(in) 150 4116 y(order)d(to)f(get)h(\(3.44\).)42 b(No)m(w)33 b(using)g(the)g(usual)f(pro)s(cedure,)i(w)m(e)f(can)g(pro)m(v)m(e)h (that)1598 4336 y(lim)1582 4396 y Ft(t)p Fm(!1)1765 4336 y Fu(F)1828 4351 y Fs(+)p Fm(\000)1942 4336 y Fz(\()p Fu(D)s Fz(\()p Fu(t)p Fz(\)\))28 b(=)f(0)150 4583 y(and)34 b(b)s(ecause)i Fu(D)h Fz(is)d(b)s(ounded)h(and)f(the)h(ab)s(o)m(v)m(e)g (limit)30 b(did)k(not)g(dep)s(end)h(on)g(the)f(c)m(hoice)h(of)f Fu(c)g Fz(and)150 4703 y Fu(d)e Fz(in)g Fu(F)410 4718 y Fs(+)p Fm(\000)524 4703 y Fz(,)h(w)m(e)g(can)g(write)1524 4834 y(lim)1508 4894 y Ft(t)p Fm(!1)1692 4834 y Fu(F)1769 4783 y Ft(\017)1798 4792 y Fj(1)1831 4783 y Ft(=)p Fs(2)p Ft(;\017)1950 4792 y Fj(1)1755 4856 y Fs(+)1989 4834 y Fz(\()p Fu(D)s Fz(\()p Fu(t)p Fz(\)\))27 b(=)h(0)p Fu(:)296 5035 y Fz(Reasoning)35 b(in)f(a)h(v)m(ery)h(similar)c(w)m(a)m (y)-8 b(,)36 b(w)m(e)h(can)e(also)f(pro)m(v)m(e)i(that)f(for)f Fu(a;)17 b(b)32 b(<)g Fr(\000)p Fu(\017)3273 5050 y Fs(1)3348 5035 y Fz(and)j Fu(F)3603 5050 y Fs(+)p Fm(\000)3749 5035 y Fz(=)150 5166 y Fu(F)227 5115 y Ft(a;b;)p Fm(\000)p Ft(\017)418 5124 y Fj(1)452 5115 y Ft(;)p Fm(\000)p Ft(\017)556 5124 y Fj(1)589 5115 y Ft(=)p Fs(2)213 5188 y(+)p Fm(\000)696 5166 y Fz(w)m(e)f(ha)m(v)m(e)270 5305 y Fl(Z)370 5331 y Fm(1)325 5530 y Fs(1)471 5373 y Fz(1)p 471 5418 V 478 5509 a Fu(t)530 5440 y(F)593 5455 y Fs(+)p Fm(\000)707 5440 y Fz(\()p Fu(D)s Fz(\()p Fu(t)p Fz(\)\))p Fu(dt)27 b(<)h Fr(1)p Fu(;)129 b Fz(lim)1435 5500 y Ft(t)p Fm(!1)1619 5440 y Fu(F)1682 5455 y Fs(+)p Fm(\000)1796 5440 y Fz(\()p Fu(D)s Fz(\()p Fu(t)p Fz(\)\))27 b(=)h(0)p Fu(;)129 b Fz(lim)2388 5500 y Ft(t)p Fm(!1)2571 5440 y Fu(F)2648 5390 y Fm(\000)p Ft(\017)2732 5399 y Fj(1)2766 5390 y Ft(;)p Fm(\000)p Ft(\017)2870 5399 y Fj(1)2903 5390 y Ft(=)p Fs(2)2634 5462 y Fm(\000)2978 5440 y Fz(\()p Fu(D)s Fz(\()p Fu(t)p Fz(\)\))27 b(=)h(0)p Fu(:)120 b Fz(\(3.47\))1939 5776 y(29)p eop %%Page: 30 33 30 32 bop 150 407 a Fz(W)-8 b(e)33 b(conclude)g(that)1556 527 y Fr(j)p Fu(D)s Fz(\()p Fu(t)p Fz(\))p Fr(j)27 b(\024)h Fu(\017)1978 542 y Fs(1)2018 527 y Fu(;)114 b(t)28 b Fr(\025)g Fu(T)8 b(:)1157 b Fz(\(3.48\))296 702 y(Using)33 b(the)g(equation)f(de\014ning)g Fu(E)6 b Fz(\()p Fu(t;)17 b(r)s Fz(\()p Fu(t)p Fz(\))p Fu(;)g(\022)s Fz(\()p Fu(t)p Fz(\)\))32 b(and)h(\(3.43\))f(w)m(e)i(obtain)1034 976 y Fu(t=r)s Fz(\()p Fu(t)p Fz(\))27 b(=)h Fu(@)1458 991 y Ft(E)1518 976 y Fu(\032)1568 991 y Ft(E)1620 999 y Fq(t)1652 976 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))p Fu(;)114 b(D)31 b Fz(=)c(1)22 b Fr(\000)2436 909 y Fu(@)2487 924 y Ft(E)2547 909 y Fu(\032)2597 924 y Ft(E)2657 909 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))p 2424 954 481 4 v 2424 1045 a Fu(@)2475 1060 y Ft(E)2535 1045 y Fu(\032)2585 1060 y Ft(E)2637 1068 y Fq(t)2669 1045 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))2914 976 y Fu(:)635 b Fz(\(3.49\))150 1247 y(Then)1329 1405 y Fu(D)s Fz(\()p Fu(t)p Fz(\))28 b(=)1665 1334 y Fu(@)1721 1298 y Fs(2)1716 1359 y Ft(E)1776 1334 y Fu(\032)1826 1350 y Ft(x)p Fs(\()p Ft(t)p Fs(\))1951 1334 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))p 1665 1382 521 4 v 1686 1473 a Fu(@)1737 1488 y Ft(E)1797 1473 y Fu(\032)1847 1488 y Ft(E)1899 1496 y Fq(t)1931 1473 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))2218 1405 y Fr(\001)22 b Fz(\()p Fu(E)2378 1420 y Ft(t)2430 1405 y Fr(\000)h Fu(E)6 b Fz(\))930 b(\(3.50\))150 1635 y(where)34 b Fu(x)p Fz(\()p Fu(t)p Fz(\))g(is)f(somewhere)h(b)s(et)m(w) m(een)h Fu(E)k Fz(and)33 b Fu(E)1975 1650 y Ft(t)2005 1635 y Fz(.)45 b(Because)35 b(of)e(the)g(minimal)c(and)k(maximal)d(v)m (e-)150 1756 y(lo)s(cit)m(y)25 b(cut-o\013s)h(in)f(\(2.1\))g(and)i (\(3.5\),)f(w)m(e)h(kno)m(w)h(that)d Fu(E)2183 1771 y Ft(t)2239 1756 y Fz(th)m(us)i Fu(x)p Fz(\()p Fu(t)p Fz(\))g(v)-5 b(aries)25 b(in)g(a)h(time)f(indep)s(enden)m(t)150 1876 y(compact)32 b(in)m(terv)-5 b(al)31 b(pro)m(vided)j Fu(t)e Fz(is)g(large)g(enough.)44 b(This)32 b(means)h(that)f(w)m(e)i(can)f (write)1256 2096 y Fr(j)p Fu(E)1356 2111 y Ft(t)1407 2096 y Fr(\000)23 b Fu(E)6 b Fr(j)27 b(\024)i Fz(Const)23 b Fr(\001)e(j)p Fu(D)s Fz(\()p Fu(t)p Fz(\))p Fr(j)p Fu(;)114 b(t)28 b Fr(\025)g Fu(T)8 b(;)150 2316 y Fz(hence)34 b(\(3.45\))e(follo)m(ws)f(and)i(the)g(prop)s(osition)d(is)j(pro)m(v)m (en.)p 3754 2316 4 66 v 3758 2254 59 4 v 3758 2316 V 3816 2316 4 66 v 150 2723 a Fp(3.7)135 b(The)45 b(classical)h (comparison)f(dynamics)150 2908 y Fz(W)-8 b(e)36 b(already)f(kno)m(w)i (that)e(giv)m(en)h(an)f(energy)i Fu(E)i(>)32 b(E)2142 2923 y Ft(d)2183 2908 y Fz(,)k(the)g(momen)m(ta)f(of)g(an)m(y)h(real)f (orbit)f(corre-)150 3028 y(sp)s(onding)c(to)g Fu(E)36 b Fz(tend)31 b(to)f(get)g(closer)h(and)f(closer)g(to)g(the)h(p)s(erio)s (dic)e(ones.)43 b(W)-8 b(e)31 b(w)m(ould)f(also)g(lik)m(e)f(to)150 3149 y(ha)m(v)m(e)36 b(a)f(similar)d(prop)s(ert)m(y)j(for)g(the)g(tra)5 b(jectory)36 b(itself,)e(but)h(w)m(e)h(\014rst)g(need)g(a)e(comparison) g(orbit,)150 3269 y(whic)m(h)f(is)f(constructed)i(in)e(what)h(follo)m (ws.)296 3389 y(F)-8 b(or)32 b(ev)m(ery)j Fu(\022)774 3404 y Fs(1)841 3389 y Fr(2)28 b Fn(R)44 b Fz(and)32 b(ev)m(ery)j Fu(E)e(>)28 b(E)1768 3404 y Ft(d)1809 3389 y Fz(,)k(denote)i(b)m(y)f(\(see)h(Lemma)d(3.8\):)1286 3654 y Fu(r)1330 3669 y Fs(1)1397 3654 y Fz(:=)d Fu(r)s Fz(\()p Fu(E)6 b(;)17 b Fz(1)p Fu(;)g(\022)1873 3669 y Fs(1)1911 3654 y Fz(\))28 b(=)2257 3586 y(1)p 2090 3631 382 4 v 2090 3722 a Fu(@)2141 3737 y Ft(E)2201 3722 y Fu(\032)2251 3737 y Ft(E)2311 3722 y Fz(\()p Fu(\022)2394 3737 y Fs(1)2434 3722 y Fz(\))2510 3654 y Fu(>)f Fz(0)p Fu(;)887 b Fz(\(3.51\))296 3924 y(Consider)33 b(the)g(follo)m(wing)d (system)k(of)e(equations)1356 4055 y Fl(\()1515 4109 y Ft(d)s Fs(~)-38 b Ft(r)p 1515 4126 71 4 v 1519 4183 a(dt)1595 4149 y Fz(\()p Fu(E)6 b(;)17 b(t)p Fz(\))28 b(=)g Fu(\032)2010 4164 y Ft(E)2069 4149 y Fz(\()2115 4122 y(~)2107 4149 y Fu(\022)t Fz(\()p Fu(E)6 b(;)17 b(t)p Fz(\)\))1488 4255 y Ft(d)1530 4238 y Fs(~)1524 4255 y Ft(\022)p 1488 4271 72 4 v 1493 4329 a(dt)1569 4294 y Fz(\()p Fu(E)6 b(;)17 b(t)p Fz(\))27 b(=)1943 4247 y Ft(\030)1974 4258 y Fq(E)2026 4247 y Fs(\()2059 4229 y(~)2053 4247 y Ft(\022)r Fs(\()p Ft(E)t(;t)p Fs(\)\))p 1943 4271 328 4 v 2015 4329 a(~)-38 b Ft(r)r Fs(\()p Ft(E)t(;t)p Fs(\))2592 4226 y Fu(;)957 b Fz(\(3.52\))150 4551 y(with)48 b(~)-53 b Fu(r)s Fz(\()p Fu(E)6 b(;)17 b Fz(1\))47 b(=)g Fu(r)892 4566 y Fs(1)976 4551 y Fz(and)1185 4524 y(~)1178 4551 y Fu(\022)s Fz(\()p Fu(E)6 b(;)17 b Fz(1\))47 b(=)g Fu(\022)1688 4566 y Fs(1)1728 4551 y Fz(.)79 b(Notice)44 b(that)g(in)g(rectangular)g(co)s(ordinates)f (this)h(is)150 4671 y(equiv)-5 b(alen)m(t)32 b(to)879 4867 y Fu(d)935 4866 y Fz(~)930 4867 y Fv(x)p 879 4912 110 4 v 891 5003 a Fu(dt)1027 4935 y Fz(=)27 b Fr(r)1230 4910 y Fz(~)1213 4935 y Fu(S)6 b Fz(\()p Fu(E)g(;)1444 4934 y Fz(~)1439 4935 y Fv(x)p Fz(\))22 b Fr(\000)h Fv(a)p Fz(\()1755 4934 y(~)1750 4935 y Fv(x)p Fz(\))p Fu(;)1994 4934 y Fz(~)1989 4935 y Fv(x)p Fz(\(1\))k(=)h Fu(r)2348 4950 y Fs(1)2387 4935 y Fz(\(cos)q(\()p Fu(\022)2639 4950 y Fs(1)2678 4935 y Fz(\))p Fu(;)17 b Fz(sin\()p Fu(\022)2963 4950 y Fs(1)3003 4935 y Fz(\)\))p Fu(:)470 b Fz(\(3.53\))296 5186 y(With)32 b(reference)i(to)f(\(3.36\))e(w)m(e)j (ha)m(v)m(e:)150 5389 y Fv(Lemma)j(3.11.)42 b FA(F)-7 b(or)34 b(al)5 b(l)34 b Fu(t)28 b Fr(\025)g Fz(1)35 b FA(we)g(have)f Fu(E)6 b Fz(\()p Fu(t;)21 b Fz(~)-53 b Fu(r)r Fz(\()p Fu(E)6 b(;)17 b(t)p Fz(\))p Fu(;)2287 5363 y Fz(~)2279 5389 y Fu(\022)s Fz(\()p Fu(E)6 b(;)17 b(t)p Fz(\)\))28 b(=)f Fu(E)34 b Fz(=)27 b Fu(E)6 b Fz(\(1)p Fu(;)17 b(r)3191 5404 y Fs(1)3230 5389 y Fu(;)g(\022)3319 5404 y Fs(1)3359 5389 y Fz(\))p Fu(:)1939 5776 y Fz(30)p eop %%Page: 31 34 31 33 bop 150 407 a Fv(Pro)s(of.)43 b Fz(It)33 b(su\016ces)i(to)d(sho)m (w)i(the)f(iden)m(tit)m(y)j(~)-53 b Fu(r)s Fz(\()p Fu(E)6 b(;)17 b(t)p Fz(\))27 b(=)h Fu(r)s Fz(\()p Fu(E)6 b(;)17 b(t;)2527 381 y Fz(~)2520 407 y Fu(\022)r Fz(\()p Fu(E)6 b(;)17 b(t)p Fz(\)\))p Fu(:)296 527 y Fz(Di\013eren)m(tiating)25 b(the)i(equation)f Fu(r)s Fz(\()p Fu(E)6 b(;)17 b(t;)1787 501 y Fz(~)1780 527 y Fu(\022)s Fz(\()p Fu(E)6 b(;)17 b(t)p Fz(\)\))27 b(=)2438 488 y Ft(t)p 2240 504 422 4 v 2240 570 a(@)2277 581 y Fq(E)2329 570 y Ft(\032)2365 581 y Fq(E)2416 570 y Fs(\()2449 553 y(~)2443 570 y Ft(\022)s Fs(\()p Ft(E)t(;t)p Fs(\)\))2698 527 y Fz(with)g(resp)s(ect)h(to)e Fu(t)h Fz(and)g(using)150 670 y(\(2.24\))32 b(one)h(obtains)503 882 y Fu(dr)p 503 926 98 4 v 509 1017 a(dt)610 949 y Fz(\()p Fu(E)6 b(;)17 b(t;)857 923 y Fz(~)849 949 y Fu(\022)s Fz(\()p Fu(E)6 b(;)17 b(t)p Fz(\)\))83 b(=)1684 882 y(1)p 1420 926 578 4 v 1420 1030 a Fu(@)1471 1045 y Ft(E)1531 1030 y Fu(\032)1581 1045 y Ft(E)1641 1030 y Fz(\()1687 1004 y(~)1679 1030 y Fu(\022)s Fz(\()p Fu(E)6 b(;)17 b(t)p Fz(\)\))2030 949 y Fr(\000)2139 882 y Fu(t@)2225 897 y Ft(E)2286 882 y Fz(\()p Fu(@)2375 897 y Ft(\022)2414 882 y Fu(\032)2464 897 y Ft(E)2524 882 y Fz(\))p 2139 926 423 4 v 2193 1017 a([)p Fu(@)2271 1032 y Ft(E)2331 1017 y Fu(\032)2381 1032 y Ft(E)2441 1017 y Fz(])2468 989 y Fs(2)2572 949 y Fz(\()p Fu(E)6 b(;)2739 923 y Fz(~)2732 949 y Fu(\022)s Fz(\()p Fu(E)g(;)17 b(t)p Fz(\)\))22 b Fr(\001)3132 882 y Fu(d)3191 855 y Fz(~)3183 882 y Fu(\022)p 3132 926 99 4 v 3138 1017 a(dt)3241 949 y Fz(\()p Fu(E)6 b(;)17 b(t)p Fz(\))1251 1231 y(=)1506 1163 y(1)p 1420 1208 221 4 v 1420 1299 a Fu(@)1471 1314 y Ft(E)1531 1299 y Fu(\032)1581 1314 y Ft(E)1667 1120 y Fl(h)1714 1231 y Fz(1)22 b Fr(\000)h Fu(\030)1928 1246 y Ft(E)1987 1231 y Fz(\()p Fu(@)2076 1246 y Ft(E)2136 1231 y Fu(\030)2179 1246 y Ft(E)2238 1231 y Fz(\))2286 1163 y Fu(r)p 2286 1208 47 4 v 2290 1299 a Fz(~)-53 b Fu(r)2343 1120 y Fl(i)2417 1231 y Fz(=)28 b Fu(\032)2571 1246 y Ft(E)2653 1231 y Fr(\000)2763 1163 y Fu(\030)2806 1178 y Ft(E)2865 1163 y Fz(\()p Fu(@)2954 1178 y Ft(E)3014 1163 y Fu(\030)3057 1178 y Ft(E)3116 1163 y Fz(\)\()p Fu(r)d Fr(\000)i Fz(~)-54 b Fu(r)s Fz(\))p 2763 1208 683 4 v 2932 1299 a(\()p Fu(@)3021 1314 y Ft(E)3081 1299 y Fu(\032)3131 1314 y Ft(E)3191 1299 y Fz(\))t(~)h Fu(r)3455 1231 y(;)150 1493 y Fz(or)32 b(using)g(\(3.52\))g(again)882 1687 y Fu(d)p 865 1732 86 4 v 865 1823 a(dt)977 1644 y Fl(h)1024 1754 y Fu(r)s Fz(\()p Fu(E)6 b(;)17 b(t;)1317 1728 y Fz(~)1310 1754 y Fu(\022)s Fz(\()p Fu(E)6 b(;)17 b(t)p Fz(\)\))k Fr(\000)27 b Fz(~)-53 b Fu(r)s Fz(\()p Fu(E)6 b(;)17 b(t)p Fz(\))2030 1644 y Fl(i)2160 1754 y Fz(=)83 b Fr(\000)2406 1687 y Fu(\030)2449 1702 y Ft(E)2508 1687 y Fz(\()p Fu(@)2597 1702 y Ft(E)2657 1687 y Fu(\030)2700 1702 y Ft(E)2759 1687 y Fz(\))p 2406 1732 392 4 v 2430 1823 a(\()p Fu(@)2519 1838 y Ft(E)2579 1823 y Fu(\032)2629 1838 y Ft(E)2689 1823 y Fz(\))t(~)-53 b Fu(r)2807 1754 y Fz([)p Fu(r)25 b Fr(\000)i Fz(~)-53 b Fu(r)s Fz(])p Fu(;)1031 1973 y(r)s Fz(\()p Fu(E)6 b(;)17 b Fz(1)p Fu(;)1337 1947 y Fz(~)1331 1973 y Fu(\022)r Fz(\()p Fu(E)6 b(;)17 b Fz(1\)\))k Fr(\000)27 b Fz(~)-53 b Fu(r)r Fz(\()p Fu(E)6 b(;)17 b Fz(1\))83 b(=)g Fu(r)2363 1988 y Fs(1)2424 1973 y Fr(\000)23 b Fu(r)2568 1988 y Fs(1)2635 1973 y Fz(=)k(0)p Fu(:)150 2180 y Fz(Solving)34 b(this)h(initial)d(v)-5 b(alue)35 b(problem)f(sho)m(ws)j(that)f(the)g(di\013erence)g(m)m(ust)g(b)s(e)f (zero)h(at)f(all)f(times,)150 2300 y(therefore)f(the)g(iden)m(tit)m(y)g (follo)m(ws.)p 3754 2300 4 66 v 3758 2238 59 4 v 3758 2300 V 3816 2300 4 66 v 296 2539 a(W)-8 b(e)33 b(de\014ne)h(the)f (mapping:)279 2865 y Fv(v)338 2880 y Ft(t)395 2865 y Fz(:)28 b Fn(R)516 2880 y Fs(+)603 2865 y Fr(\002)23 b Fn(T)31 b Fr(7!)c Fn(R)990 2880 y Fs(+)1077 2865 y Fr(\002)c Fn(T)p Fu(;)117 b Fv(v)1443 2880 y Ft(t)1473 2865 y Fz(\()p Fu(r)1555 2880 y Fs(1)1595 2865 y Fu(;)17 b(\022)1684 2880 y Fs(1)1723 2865 y Fz(\))28 b(:=)1919 2755 y Fl(\020)1983 2865 y Fz(~)-53 b Fu(r)s Fz(\()p Fu(E)6 b Fz(\(1)p Fu(;)17 b(r)2317 2880 y Fs(1)2355 2865 y Fu(;)g(\022)2444 2880 y Fs(1)2484 2865 y Fz(\))p Fu(;)g(t)p Fz(\))p Fu(;)2690 2839 y Fz(~)2683 2865 y Fu(\022)s Fz(\()p Fu(E)6 b Fz(\(1)p Fu(;)17 b(r)3022 2880 y Fs(1)3060 2865 y Fu(;)g(\022)3149 2880 y Fs(1)3189 2865 y Fz(\))p Fu(;)g(t)p Fz(\))3344 2755 y Fl(\021)3420 2865 y Fu(:)129 b Fz(\(3.54\))296 3099 y(Notice)25 b(\014rst)g(that)g(the)g(system)h(of)f(equations)g (de\014ning)g Fv(v)2421 3114 y Ft(t)2476 3099 y Fz(in)f(\(3.52\))g(can) h(b)s(e)g(more)g(compactly)150 3220 y(written)32 b(as)h(\(see)h (\(3.38\),)e(\(3.39\),)g(\(3.40\))f(and)i(Lemma)e(3.11\))966 3416 y Fu(d)p Fv(v)1076 3431 y Ft(t)p 966 3460 140 4 v 993 3552 a Fu(dt)1143 3483 y Fz(=)1247 3343 y Fl(\022)1320 3483 y Fz(\()p Fu(@)1409 3498 y Ft(r)1447 3483 y Fu(S)6 b Fz(\)\()p Fu(t;)17 b Fv(v)1727 3498 y Ft(t)1757 3483 y Fz(\))p Fu(;)g Fz(\()1905 3416 y(1)p 1887 3460 87 4 v 1887 3552 a Fu(r)1934 3523 y Fs(2)1982 3483 y Fu(@)2033 3498 y Ft(\022)2073 3483 y Fu(S)28 b Fr(\000)2273 3416 y Fu(b)p 2270 3460 47 4 v 2270 3552 a(r)2327 3483 y Fz(\)\()p Fu(t;)17 b Fv(v)2541 3498 y Ft(t)2571 3483 y Fz(\))2609 3343 y Fl(\023)2710 3483 y Fz(=)27 b Fv(X)p Fz(\()p Fu(t;)17 b Fv(v)3074 3498 y Ft(t)3103 3483 y Fz(\))p Fu(;)956 3686 y Fv(v)1015 3701 y Fs(1)1082 3686 y Fz(=)28 b(\()p Fu(r)1268 3701 y Fs(1)1307 3686 y Fu(;)17 b(\022)1396 3701 y Fs(1)1436 3686 y Fz(\))27 b Fr(2)h Fn(R)1661 3701 y Fs(+)1749 3686 y Fr(\002)22 b Fn(T)p Fu(;)1638 b Fz(\(3.55\))150 3892 y(where)34 b(w)m(e)f(in)m(tro)s(duced)792 4131 y Fv(X)p Fz(\()p Fu(t;)17 b Fv(x)p Fz(\))28 b(=)1222 3991 y Fl(\022)1296 4131 y Fz(\()p Fu(@)1385 4146 y Ft(r)1423 4131 y Fu(S)6 b Fz(\)\()p Fu(t;)17 b Fv(x)p Fz(\))p Fu(;)g Fz(\()1851 4064 y(1)p 1833 4108 87 4 v 1833 4200 a Fu(r)1880 4171 y Fs(2)1928 4131 y Fu(@)1979 4146 y Ft(\022)2019 4131 y Fu(S)28 b Fr(\000)2219 4064 y Fu(b)p 2216 4108 47 4 v 2216 4200 a(r)2273 4131 y Fz(\)\()p Fu(t;)17 b Fv(x)p Fz(\))2525 3991 y Fl(\023)2615 4131 y Fu(;)114 b Fv(x)28 b Fz(=)f(\()p Fu(r)m(;)17 b(\022)s Fz(\))p Fu(:)296 4389 y Fz(Then)34 b Fv(v)610 4404 y Ft(t)672 4389 y Fz(admits)e(an)g(in)m(v)m(erse)i(denoted)g(b)m(y)f Fv(w)2036 4404 y Ft(t)2066 4389 y Fz(:)745 4630 y Fv(w)826 4645 y Ft(t)883 4630 y Fz(:)28 b Fn(R)1004 4645 y Fs(+)1091 4630 y Fr(\002)22 b Fn(T)31 b Fr(7!)d Fn(R)1478 4645 y Fs(+)1565 4630 y Fr(\002)23 b Fn(T)p Fu(;)117 b Fv(w)1953 4645 y Ft(t)1983 4630 y Fz(\()p Fu(r)m(;)17 b(\022)s Fz(\))27 b(=)h(\()p Fu(r)2405 4645 y Fs(1)2444 4630 y Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))p Fu(;)g(\022)2821 4645 y Fs(1)2861 4630 y Fz(\()p Fu(t;)g(r)m(;)g(\022)s Fz(\)\))f Fu(:)346 b Fz(\(3.56\))150 4799 y(If)33 b(w)m(e)g(denote)g(b) m(y)h Fv(u)903 4814 y Ft(\034)946 4799 y Fz(\()p Fu(r)m(;)17 b(\022)s Fz(\))33 b(the)g(solution)e(to)h(the)h(equation)675 4987 y Fu(d)p Fv(u)788 5002 y Ft(\034)p 675 5032 157 4 v 701 5123 a Fu(d\034)869 5055 y Fz(=)27 b Fr(\000)p Fv(X)p Fz(\()p Fu(t)c Fr(\000)f Fu(\034)34 b Fz(+)22 b(1)p Fu(;)17 b Fv(u)1658 5070 y Ft(\034)1701 5055 y Fz(\))p Fu(;)114 b Fv(u)1942 5070 y Fs(1)2009 5055 y Fz(=)28 b(\()p Fu(r)m(;)17 b(\022)s Fz(\))28 b Fr(2)g Fn(R)2510 5070 y Fs(+)2597 5055 y Fr(\002)22 b Fn(T)p Fu(;)118 b(\034)39 b Fr(2)28 b Fz(\(1)p Fu(;)17 b(t)p Fz(\))p Fu(;)266 b Fz(\(3.57\))150 5286 y(w)m(e)34 b(ha)m(v)m(e)g Fv(w)600 5301 y Ft(t)657 5286 y Fz(=)27 b Fv(u)822 5301 y Ft(t)852 5286 y Fz(.)296 5407 y(W)-8 b(e)23 b(p)s(oin)m(t)e(out)h (that)g Fv(v)1121 5422 y Ft(t)1151 5407 y Fz(\()p Fu(r)1233 5422 y Fs(1)1273 5407 y Fu(;)17 b(\022)1362 5422 y Fs(1)1401 5407 y Fz(\))22 b(also)g(solv)m(es)h(t)m(w)m(o)g(of)e(the)i(Hamilton)d (equations)i(\(corresp)s(onding)150 5527 y(to)35 b(the)h (con\014guration)e(space\))j(for)e(the)h(sym)m(b)s(ol)e Fu(h)2053 5542 y Ft(a)2130 5527 y Fz(w)m(e)j(in)m(tro)s(duced)e(in)g (\(1.13\).)51 b(The)37 b(other)e(t)m(w)m(o)1939 5776 y(31)p eop %%Page: 32 35 32 34 bop 150 407 a Fz(equations)35 b(giv)m(e)g(a)g(solution)f(w)m(e)i (denoted)g(b)m(y)g Fv(q)1962 422 y Ft(a;t)2080 407 y Fz(=)c(\()p Fu(\032)2276 422 y Ft(a)2318 407 y Fz(\()p Fu(t)p Fz(\))p Fu(;)17 b(l)2502 422 y Ft(a)2543 407 y Fz(\()p Fu(t)p Fz(\)\))35 b(in)g(Subsection)g(1.2,)h(and)f(w)m(e)150 527 y(de\014ned)c(the)f(total)d(direct)j(\015o)m(w)g(to)f(b)s(e)g Fv(V)1681 542 y Ft(a;t)1795 527 y Fz(=)f(\()p Fv(v)1996 542 y Ft(t)2026 527 y Fz(;)17 b Fv(q)2129 542 y Ft(a;t)2215 527 y Fz(\))29 b(corresp)s(onding)h(to)f(a)g(set)h(of)f(initial)c(data) 150 648 y(\()p Fu(r)232 663 y Fs(1)271 648 y Fu(;)17 b(\022)360 663 y Fs(1)400 648 y Fz(;)g Fu(\032)494 663 y Ft(a)536 648 y Fz(\(1\))p Fu(;)g(l)734 663 y Ft(a)775 648 y Fz(\(1\)\).)48 b(The)35 b(in)m(v)m(erse)h(\015o)m(w)f(denoted)g (b)m(y)g Fv(W)2368 663 y Ft(a;t)2489 648 y Fz(has)f(as)h(the)g (\\con\014guration)e(space)150 768 y(part")f(the)h(\015o)m(w)g Fv(w)862 783 y Ft(t)924 768 y Fz(describ)s(ed)h(ab)s(o)m(v)m(e.)296 888 y(In)28 b(order)g(to)g(get)f(an)h(idea)f(ab)s(out)g(ho)m(w)i Fv(w)1821 903 y Ft(t)1878 888 y Fz(dep)s(ends)g(on)f(the)g(initial)c (conditions,)k(w)m(e)h(\014rst)f(lo)s(ok)150 1009 y(at)f(the)h(case)g (when)g Fu(b)p Fz(\()p Fu(\022)s Fz(\))h(=)e Fu(b)h(<)g Fz(0)f(is)g(a)g(negativ)m(e)g(constan)m(t.)43 b(W)-8 b(e)28 b(then)g(kno)m(w)g(that)f Fu(E)3308 1024 y Ft(d)3377 1009 y Fz(=)g Fu(E)3552 1024 y Ft(c)3615 1009 y Fz(=)g Fu(E)3790 1024 y Ft(b)150 1129 y Fz(and)33 b(the)g(p)s(erio)s(dic)e (solution)g(is)h Fu(\030)1392 1144 y Ft(E)1451 1129 y Fz(\()p Fu(\022)s Fz(\))c(=)f Fr(\000)p Fu(b)i(>)e Fz(0,)33 b(whic)m(h)g(from)e(\(3.29\))h(and)h(\(3.34\))f(leads)g(to)1002 1383 y(~)985 1408 y Fu(S)6 b Fz(\()p Fu(E)g(;)17 b(r)m(;)g(\022)s Fz(\))28 b(=)f Fu(r)1560 1321 y Fr(p)p 1643 1321 330 4 v 87 x Fz(2)p Fu(E)h Fr(\000)22 b Fu(b)1932 1380 y Fs(2)1972 1408 y Fu(;)115 b(S)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))27 b(=)2609 1341 y Fu(r)2656 1305 y Fs(2)p 2609 1385 87 4 v 2610 1477 a Fz(2)p Fu(t)2727 1408 y Fr(\000)2836 1341 y Fu(b)2877 1305 y Fs(2)2917 1341 y Fu(t)p 2836 1385 117 4 v 2870 1477 a Fz(2)2962 1408 y Fu(:)587 b Fz(\(3.58\))296 1659 y(The)34 b(energy)f(function)g Fu(E)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))32 b(in)g(\(3.36\))g(is)1014 1938 y Fu(E)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))28 b(=)1540 1871 y Fu(r)1587 1835 y Fs(2)p 1521 1916 124 4 v 1521 2007 a Fz(2)p Fu(t)1605 1978 y Fs(2)1677 1938 y Fz(+)1785 1871 y Fu(b)1826 1835 y Fs(2)p 1785 1916 81 4 v 1801 2007 a Fz(2)1876 1938 y Fu(;)114 b(E)6 b Fz(\(1)p Fu(;)17 b(r)2270 1953 y Fs(1)2309 1938 y Fu(;)g(\022)2398 1953 y Fs(1)2437 1938 y Fz(\))28 b(=)2617 1871 y Fu(r)2664 1835 y Fs(2)2661 1896 y(1)p 2617 1916 87 4 v 2635 2007 a Fz(2)2735 1938 y(+)2843 1871 y Fu(b)2884 1835 y Fs(2)p 2843 1916 81 4 v 2859 2007 a Fz(2)2934 1938 y Fu(:)615 b Fz(\(3.59\))296 2189 y(W)-8 b(e)33 b(no)m(w)h(can)e(explicitly)f (solv)m(e)i(\(3.52\))f(obtaining)987 2459 y Fu(r)s Fz(\()p Fu(t;)17 b(r)1195 2474 y Fs(1)1234 2459 y Fu(;)g(\022)1323 2474 y Fs(1)1363 2459 y Fz(\))27 b(=)h Fu(t)g(r)1639 2474 y Fs(1)1678 2459 y Fu(;)115 b(\022)s Fz(\()p Fu(t;)17 b(r)2029 2474 y Fs(1)2068 2459 y Fu(;)g(\022)2157 2474 y Fs(1)2197 2459 y Fz(\))27 b(=)h Fr(\000)2474 2391 y Fu(b)p 2453 2436 84 4 v 2453 2527 a(r)2497 2542 y Fs(1)2563 2459 y Fz(ln)o(\()p Fu(t)p Fz(\))23 b(+)f Fu(\022)2921 2474 y Fs(1)2961 2459 y Fu(;)588 b Fz(\(3.60\))150 2719 y(while)32 b(the)h(in)m(v)m(erse)h(\015o)m(w)f(is)1124 2956 y Fu(r)1168 2971 y Fs(1)1207 2956 y Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))28 b(=)1637 2888 y Fu(r)p 1637 2933 47 4 v 1643 3024 a(t)1693 2956 y(;)114 b(\022)1879 2971 y Fs(1)1919 2956 y Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))28 b(=)f Fu(\022)f Fz(+)c Fu(b)2564 2888 y(t)p 2558 2933 V 2558 3024 a(r)2631 2956 y Fz(ln\()p Fu(t)p Fz(\))p Fu(:)725 b Fz(\(3.61\))150 3201 y(or)32 b(in)g(a)g(more)g (compact)g(form)1369 3474 y Fv(w)1450 3489 y Ft(t)1480 3474 y Fz(\()p Fu(r)m(;)17 b(\022)s Fz(\))27 b(=)1820 3334 y Fl(\022)1903 3407 y Fu(r)p 1903 3452 V 1909 3543 a(t)1960 3474 y(;)17 b(\022)25 b Fz(+)d Fu(b)2229 3407 y(t)p 2223 3452 V 2223 3543 a(r)2296 3474 y Fz(ln\()p Fu(t)p Fz(\))2489 3334 y Fl(\023)2579 3474 y Fu(:)970 b Fz(\(3.62\))296 3706 y(The)34 b(Jacobian)e(matrix)f(\(with)h(resp)s (ect)i(to)e(the)h(initial)c(conditions\))i(is)h(then)1339 3980 y Fv(w)1422 3939 y Fm(0)1445 3995 y Ft(t)1474 3980 y Fz(\()p Fu(r)m(;)17 b(\022)s Fz(\))28 b(=)1815 3840 y Fl(\022)2134 3880 y Fs(1)p 2134 3896 36 4 v 2139 3953 a Ft(t)2429 3919 y Fz(0)1929 4039 y Fr(\000)p Fu(b)2080 4000 y Ft(t)p 2057 4017 69 4 v 2057 4074 a(r)2091 4055 y Fj(2)2153 4039 y Fz(ln\()p Fu(t)p Fz(\))83 b(1)2519 3840 y Fl(\023)2609 3980 y Fu(:)940 b Fz(\(3.63\))296 4207 y(The)35 b(remaining)d(part)i(of)f(this)h(subsection)h(is)f(dev)m (oted)h(to)f(sho)m(wing)g(a)g(similar)d(b)s(eha)m(vior)i(for)150 4327 y(the)d(in)m(v)m(erse)h(\015o)m(w)g(ev)m(en)g(when)g Fu(b)p Fz(\()p Fu(\022)s Fz(\))f(is)f(not)h(constan)m(t.)43 b(That)30 b(is,)g(w)m(e)h(lo)s(ok)d(again)g(at)i(the)g(Jacobian)150 4448 y(matrix)g Fv(w)549 4412 y Fm(0)572 4463 y Ft(t)601 4448 y Fz(\()p Fu(r)m(;)17 b(\022)s Fz(\))32 b(and)f(pro)m(v)m(e)i (that)e(giv)m(en)g(an)m(y)h(small)d Fu(\016)j(>)27 b Fz(0,)32 b(then)g(b)m(y)g(p)s(erforming)d(a)i(deriv)-5 b(ativ)m(e)150 4568 y(with)34 b(resp)s(ect)h(to)f Fu(r)j Fz(w)m(e)e(in)m(tro)s(duce)f(a)g(deca)m(y)i(of)e(order)g Fu(t)2251 4532 y Ft(\016)r Fm(\000)p Fs(1)2413 4568 y Fz(while)g(di\013eren)m(tiating)e(with)i(resp)s(ect)150 4689 y(to)j Fu(\022)k Fz(w)m(e)d(in)m(tro)s(duce)f(a)h(gro)m(wth)f(of)g (at)g(most)g Fu(t)1884 4652 y Ft(\016)1922 4689 y Fz(.)58 b(These)39 b(estimates)e(\(and)h(extensions)g(to)f(higher)150 4809 y(order)c(deriv)-5 b(ativ)m(es\))32 b(will)e(pla)m(y)j(an)f(imp)s (ortan)m(t)f(role)h(in)f(the)i(rest)h(of)e(the)h(pap)s(er.)296 4929 y(W)-8 b(e)42 b(\014rst)g(study)g(the)g(direct)f(\015o)m(w)g (\(3.52\),)i(since)f(w)m(e)g(in)m(tend)f(to)g(use)h(the)g(in)m(v)m (erse)g(function)150 5050 y(theorem.)h(An)33 b(immediate)d(consequence) 36 b(of)c(Lemma)f(3.11)h(is)962 5306 y(~)-53 b Fu(r)r Fz(\()p Fu(E)6 b Fz(\(1)p Fu(;)17 b(r)1295 5321 y Fs(1)1334 5306 y Fu(;)g(\022)1423 5321 y Fs(1)1463 5306 y Fz(\))p Fu(;)g(t)p Fz(\))27 b(=)2352 5238 y Fu(t)p 1759 5283 1222 4 v 1759 5387 a(@)1810 5402 y Ft(E)1870 5387 y Fu(\032)1920 5403 y Ft(E)t Fs(\(1)p Ft(;r)2090 5412 y Fj(1)2124 5403 y Ft(;\022)2177 5412 y Fj(1)2211 5403 y Fs(\))2243 5387 y Fz(\()2289 5361 y(~)2281 5387 y Fu(\022)s Fz(\()p Fu(E)6 b Fz(\(1)p Fu(;)17 b(r)2620 5402 y Fs(1)2659 5387 y Fu(;)g(\022)2748 5402 y Fs(1)2788 5387 y Fz(\))p Fu(;)g(t)p Fz(\)\))2990 5306 y Fu(:)559 b Fz(\(3.64\))1939 5776 y(32)p eop %%Page: 33 36 33 35 bop 150 407 a Fz(Moreo)m(v)m(er,)28 b(if)d(w)m(e)h(imp)s(ose)e (the)i(condition)e(that)h Fu(E)6 b Fz(\(1)p Fu(;)17 b(r)2178 422 y Fs(1)2217 407 y Fu(;)g(\022)2306 422 y Fs(1)2345 407 y Fz(\))28 b Fr(2)g Fu(K)33 b Fz(where)26 b Fu(K)32 b Fz(is)25 b(a)g(compact)g(subset)150 527 y(of)34 b(\()p Fu(E)373 542 y Ft(d)414 527 y Fu(;)17 b Fr(1)p Fz(\),)34 b(then)h(there)h(exist)f(t)m(w)m(o)g(p)s(ositiv)m(e)f(constan)m(ts)i Fu(C)2419 542 y Fs(1)2458 527 y Fz(\()p Fu(K)7 b Fz(\))31 b Fu(<)g(C)2832 542 y Fs(2)2872 527 y Fz(\()p Fu(K)7 b Fz(\))34 b(indep)s(enden)m(t)i(of)e Fu(\022)3785 542 y Fs(1)150 648 y Fz(suc)m(h)g(that)1116 768 y Fu(t)22 b Fr(\001)g Fu(C)1293 783 y Fs(1)1360 768 y Fr(\024)32 b Fz(~)-53 b Fu(r)s Fz(\()p Fu(E)6 b Fz(\(1)p Fu(;)17 b(r)1803 783 y Fs(1)1841 768 y Fu(;)g(\022)1930 783 y Fs(1)1970 768 y Fz(\))p Fu(;)g(t)p Fz(\))28 b Fr(\024)g Fu(t)22 b Fr(\001)g Fu(C)2435 783 y Fs(2)2474 768 y Fu(;)114 b(t)28 b Fr(\025)g Fz(1)p Fu(:)717 b Fz(\(3.65\))296 942 y(Kno)m(wing)23 b(that)g(the)h(p)s(erio)s(dic)e(solution)g Fu(\030)1825 957 y Ft(E)1907 942 y Fz(is)h(alw)m(a)m(ys)h(trapp)s(ed)g (b)s(et)m(w)m(een)h(min)15 b Fr(j)p Fu(b)p Fr(j)23 b Fz(and)h(max)16 b Fr(j)p Fu(b)p Fr(j)p Fz(,)150 1063 y(using)24 b(\(3.65\))g(and)h(\(3.52\))e(w)m(e)j(get)e(that)h(if)e Fu(E)6 b Fz(\(1)p Fu(;)17 b(r)1954 1078 y Fs(1)1993 1063 y Fu(;)g(\022)2082 1078 y Fs(1)2121 1063 y Fz(\))28 b Fr(2)g Fu(K)k Fz(then)25 b(there)g(exist)g(t)m(w)m(o)g(other)g(p)s (ositiv)m(e)150 1183 y(constan)m(ts)34 b Fu(C)652 1198 y Fs(3)691 1183 y Fz(\()p Fu(K)7 b Fz(\))28 b Fu(<)f(C)1058 1198 y Fs(4)1098 1183 y Fz(\()p Fu(K)7 b Fz(\))32 b(indep)s(enden)m(t)i (of)e Fu(\022)2005 1198 y Fs(1)2077 1183 y Fz(suc)m(h)i(that)1157 1403 y Fu(C)1227 1418 y Fs(3)1288 1403 y Fr(\001)22 b Fz(ln)o(\()p Fu(t)p Fz(\))28 b Fr(\024)1671 1377 y Fz(~)1663 1403 y Fu(\022)s Fz(\()p Fu(t;)17 b(r)1872 1418 y Fs(1)1912 1403 y Fu(;)g(\022)2001 1418 y Fs(1)2040 1403 y Fz(\))22 b Fr(\000)h Fu(\022)2245 1418 y Fs(1)2312 1403 y Fr(\024)29 b Fu(C)2488 1418 y Fs(4)2549 1403 y Fr(\001)22 b Fz(ln)o(\()p Fu(t)p Fz(\))p Fu(:)758 b Fz(\(3.66\))150 1623 y(In)34 b(a)f(similar)e(w)m(a)m(y)-8 b(,)35 b(imp)s(osing)c(the)j(condition)e Fu(E)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))29 b Fr(2)h Fu(K)41 b Fz(w)m(e)35 b(ha)m(v)m(e)g(the)f(p)s(ositiv)m(e)f (constan)m(ts)150 1743 y Fu(C)i(<)27 b(C)435 1707 y Fm(0)491 1743 y Fz(only)32 b(dep)s(ending)h(on)f Fu(K)40 b Fz(suc)m(h)34 b(that)1222 1981 y Fu(C)29 b Fr(\001)1381 1914 y Fu(r)p 1381 1958 47 4 v 1387 2049 a(t)1466 1981 y Fr(\024)f Fu(r)1615 1996 y Fs(1)1654 1981 y Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))28 b Fr(\024)g Fu(C)2152 1940 y Fm(0)2197 1981 y Fr(\001)2257 1914 y Fu(r)p 2257 1958 V 2263 2049 a(t)2314 1981 y(;)1235 b Fz(\(3.67\))1222 2161 y Fr(\000)p Fu(C)1376 2120 y Fm(0)1422 2161 y Fr(\001)22 b Fz(ln)o(\()p Fu(t)p Fz(\))28 b Fr(\024)g Fu(\022)1842 2176 y Fs(1)1882 2161 y Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))22 b Fr(\000)h Fu(\022)31 b Fr(\024)d(\000)p Fu(C)h Fr(\001)22 b Fz(ln\()p Fu(t)p Fz(\))p Fu(:)150 2381 y Fz(These)36 b(estimates)d(giv)m(e)h (some)g(information)d(ab)s(out)i(the)h(lo)s(cation)e(of)h Fu(r)2820 2396 y Fs(1)2894 2381 y Fz(and)h Fu(\022)3130 2396 y Fs(1)3203 2381 y Fz(as)h(functions)e(of)150 2501 y Fu(t)p Fz(,)g Fu(r)i Fz(and)e Fu(\022)s Fz(.)43 b(It)33 b(remains)f(to)g(study)i(ho)m(w)f(their)f(deriv)-5 b(ativ)m(es)32 b(with)h(resp)s(ect)h(to)e Fu(r)j Fz(and)e Fu(\022)i Fz(b)s(eha)m(v)m(e.)150 2761 y Fv(3.7.1)112 b(Dep)s(endence)39 b(of)f(the)f(direct)f(\015o)m(w)i(on)g Fu(r)2199 2776 y Fs(1)2275 2761 y Fv(and)h Fu(\022)2537 2776 y Fs(1)150 2946 y Fz(W)-8 b(e)33 b(start)g(b)m(y)g(lo)s(oking)d(at)j(the)g (Jacobian)e(determinan)m(t)i(for)f(the)h(direct)f(\015o)m(w)1167 3224 y Fu(J)1221 3239 y Fk(v)1264 3247 y Fq(t)1323 3224 y Fz(=)c Fu(J)1481 3239 y Fk(v)1524 3247 y Fq(t)1556 3224 y Fz(\()p Fu(t;)17 b(r)1717 3239 y Fs(1)1756 3224 y Fu(;)g(\022)1845 3239 y Fs(1)1885 3224 y Fz(\))27 b(=)2054 3079 y Fl(\014)2054 3139 y(\014)2054 3199 y(\014)2054 3258 y(\014)2156 3158 y Fu(@)2207 3173 y Ft(r)2239 3182 y Fj(1)2283 3158 y Fz(~)-54 b Fu(r)198 b(@)2571 3173 y Ft(\022)2604 3182 y Fj(1)2648 3158 y Fz(~)-54 b Fu(r)2129 3287 y(@)2180 3302 y Ft(r)2212 3311 y Fj(1)2258 3261 y Fz(~)2251 3287 y Fu(\022)198 b(@)2545 3302 y Ft(\022)2578 3311 y Fj(1)2625 3261 y Fz(~)2617 3287 y Fu(\022)2731 3079 y Fl(\014)2731 3139 y(\014)2731 3199 y(\014)2731 3258 y(\014)2781 3224 y Fu(:)768 b Fz(\(3.68\))150 3460 y(W)-8 b(e)36 b(shall)e(pro)m(v)m(e)j(that)f(it)e(gro)m(ws)j(precisely) f(lik)m(e)f Fu(t)p Fz(.)53 b(With)35 b(the)h(notations)f(from)f (\(3.55\))h(w)m(e)i(ha)m(v)m(e)150 3608 y Fu(J)204 3623 y Fk(v)247 3631 y Fq(t)313 3608 y Fz(=)e Fu(det)572 3497 y Fl(\020)644 3569 y Ft(@)t Fk(v)728 3577 y Fq(t)p 641 3585 119 4 v 641 3642 a Ft(@)t Fk(v)725 3651 y Fj(1)769 3497 y Fl(\021)865 3608 y Fz(whic)m(h)i(according)f(to)h(general)f (results)g(ob)s(eys)i(the)f(equation)f(\(w)m(e)i(denote)f(b)m(y)150 3756 y Fu(E)222 3771 y Ft(t)280 3756 y Fz(:=)27 b Fu(E)6 b Fz(\()p Fu(t;)17 b Fv(v)664 3771 y Ft(t)694 3756 y Fz(\))27 b(=)h Fr(\000)p Fu(@)991 3771 y Ft(t)1022 3756 y Fu(S)6 b Fz(\()p Fu(t;)17 b Fv(v)1264 3771 y Ft(t)1293 3756 y Fz(\)\):)593 3967 y Fu(d)p 575 4012 86 4 v 575 4103 a(dt)671 4035 y(J)725 4050 y Fk(v)768 4058 y Fq(t)828 4035 y Fz(=)27 b(\()p Fr(r)1052 4050 y Fk(x)1122 4035 y Fr(\001)21 b Fv(X)p Fz(\)\()p Fu(t;)c Fv(v)1470 4050 y Ft(t)1500 4035 y Fz(\))p Fu(J)1592 4050 y Fk(v)1635 4058 y Fq(t)1749 4035 y Fz(=)1908 3894 y Fl(\024)1961 4035 y Fu(@)2017 3994 y Fs(2)2012 4059 y Ft(r)2057 4035 y Fu(S)6 b Fz(\()p Fu(t;)17 b Fv(v)2299 4050 y Ft(t)2329 4035 y Fz(\))22 b(+)2497 3967 y(1)p 2497 4012 49 4 v 2502 4103 a(~)-53 b Fu(r)2572 3894 y Fl(\022)2655 3967 y Fu(@)2711 3931 y Fs(2)2706 3993 y Ft(\022)2752 3967 y Fu(S)p 2655 4012 163 4 v 2718 4103 a Fz(~)f Fu(r)2850 4035 y Fr(\000)22 b Fu(b)2990 3994 y Fm(0)3014 4035 y Fz(\()3060 4009 y(~)3052 4035 y Fu(\022)s Fz(\))3138 3894 y Fl(\023\025)3281 4035 y Fu(J)3335 4050 y Fk(v)3378 4058 y Fq(t)1749 4307 y Fz(=)1908 4167 y Fl(\024)1961 4307 y Fu(@)2017 4266 y Fs(2)2012 4332 y Ft(r)2057 4307 y Fu(S)6 b Fz(\()p Fu(t;)17 b Fv(v)2299 4322 y Ft(t)2329 4307 y Fz(\))22 b(+)2497 4240 y(1)p 2497 4284 49 4 v 2502 4376 a(~)-53 b Fu(r)2556 4307 y(@)2613 4317 y Fs(~)2607 4335 y Ft(\022)2646 4307 y Fz([)p Fu(\030)2716 4322 y Ft(E)2768 4330 y Fq(t)2800 4307 y Fz(\()2846 4281 y(~)2838 4307 y Fu(\022)s Fz(\)])2951 4167 y Fl(\025)3020 4307 y Fu(J)3074 4322 y Fk(v)3117 4330 y Fq(t)3149 4307 y Fu(:)400 b Fz(\(3.69\))296 4584 y(W)-8 b(e)33 b(in)m(tegrate)f (\(3.69\))g(and)h(obtain)535 4867 y Fu(J)589 4882 y Fk(v)632 4890 y Fq(t)830 4867 y Fz(=)27 b(exp)1099 4727 y Fl(\032)1174 4732 y(Z)1273 4758 y Ft(t)1229 4957 y Fs(1)1319 4727 y Fl(\024)1372 4867 y Fu(@)1428 4826 y Fs(2)1423 4892 y Ft(r)1468 4867 y Fu(S)6 b Fz(\()p Fu(\034)g(;)17 b Fv(v)1723 4882 y Ft(\034)1766 4867 y Fz(\))22 b(+)1934 4800 y(1)p 1934 4845 V 1939 4936 a(~)-53 b Fu(r)2010 4727 y Fl(\022)2093 4800 y Fu(@)2149 4764 y Fs(2)2144 4826 y Ft(\022)2189 4800 y Fu(S)p 2093 4845 163 4 v 2155 4936 a Fz(~)g Fu(r)2287 4867 y Fr(\000)23 b Fu(b)2428 4826 y Fm(0)2451 4867 y Fz(\()2497 4841 y(~)2489 4867 y Fu(\022)t Fz(\))2576 4727 y Fl(\023\025)2718 4867 y Fu(d\034)2822 4727 y Fl(\033)830 5145 y Fz(=)k(exp)1099 5005 y Fl(\032)1174 5010 y(Z)1273 5036 y Ft(t)1229 5235 y Fs(1)1319 5005 y Fl(\024)1372 5145 y Fu(@)1428 5104 y Fs(2)1423 5170 y Ft(r)1468 5145 y Fu(S)6 b Fz(\()p Fu(\034)g(;)17 b Fv(v)1723 5160 y Ft(\034)1766 5145 y Fz(\))22 b(+)1934 5078 y(1)p 1934 5123 49 4 v 1939 5214 a(~)-53 b Fu(r)2010 5035 y Fl(\020)2069 5145 y Fz(\()p Fu(@)2158 5160 y Ft(E)2218 5145 y Fu(\030)2261 5160 y Ft(E)2313 5168 y Fq(t)2344 5145 y Fz(\)\()2428 5119 y(~)2420 5145 y Fu(\022)s Fz(\))23 b Fr(\001)e Fz(\()p Fu(@)2667 5160 y Ft(\022)2707 5145 y Fu(E)6 b Fz(\)\()p Fu(t;)21 b Fz(~)-53 b Fu(r)r(;)3038 5119 y Fz(~)3030 5145 y Fu(\022)s Fz(\))3116 5035 y Fl(\021)3176 5005 y(\025)3245 5145 y Fu(d\034)3349 5005 y Fl(\033)830 5424 y Fr(\002)17 b Fz(exp)1089 5283 y Fl(\032)1164 5288 y(Z)1264 5314 y Ft(t)1219 5513 y Fs(1)1320 5356 y Fz(1)p 1320 5401 V 1325 5492 a(~)-53 b Fu(r)1379 5424 y Fz(\()p Fu(@)1468 5439 y Ft(\022)1507 5424 y Fu(\030)1550 5439 y Ft(E)1602 5447 y Fq(t)1633 5424 y Fz(\)\()1717 5397 y(~)1709 5424 y Fu(\022)t Fz(\))p Fu(d\034)1900 5283 y Fl(\033)1991 5424 y Fu(:)1558 b Fz(\(3.70\))1939 5776 y(33)p eop %%Page: 34 37 34 36 bop 296 407 a Fz(W)-8 b(e)24 b(ma)m(y)e(simplify)e(the)k(second)g (exp)s(onen)m(tial)e(realizing)e(that)j(one)g(can)g(express)i Fu(t)e Fz(as)g Fu(t)p Fz(\()3491 381 y(~)3483 407 y Fu(\022)s(;)17 b(r)3619 422 y Fs(1)3658 407 y Fu(;)g(\022)3747 422 y Fs(1)3787 407 y Fz(\))150 527 y(b)m(y)43 b(in)m(v)m(erting)e(the)h (function)1290 501 y(~)1282 527 y Fu(\022)s Fz(\()p Fu(t;)17 b(r)1491 542 y Fs(1)1530 527 y Fu(;)g(\022)1619 542 y Fs(1)1659 527 y Fz(\))42 b(giv)m(en)g(b)m(y)g(the)h(direct)e(\015o)m (w.)72 b(Consequen)m(tly)44 b(w)m(e)f(ma)m(y)150 648 y(in)m(tro)s(duce)1303 745 y(~)1303 768 y(~)-53 b Fu(r)s Fz(\()1392 742 y(~)1384 768 y Fu(\022)r(;)17 b(r)1519 783 y Fs(1)1558 768 y Fu(;)g(\022)1647 783 y Fs(1)1687 768 y Fz(\))28 b(=)k(~)-54 b Fu(r)s Fz(\()p Fu(t)p Fz(\()2022 742 y(~)2014 768 y Fu(\022)s(;)17 b(r)2150 783 y Fs(1)2189 768 y Fu(;)g(\022)2278 783 y Fs(1)2318 768 y Fz(\))p Fu(;)g(r)2444 783 y Fs(1)2483 768 y Fu(;)g(\022)2572 783 y Fs(1)2611 768 y Fz(\))p Fu(;)900 b Fz(\(3.71\))150 932 y(and)33 b(\(3.52\))e(reduces)k(to)d(\(denote)h(b)m(y)h Fu(E)1648 947 y Fs(1)1715 932 y Fz(=)28 b Fu(E)6 b Fz(\(1)p Fu(;)17 b(r)2072 947 y Fs(1)2111 932 y Fu(;)g(\022)2200 947 y Fs(1)2239 932 y Fz(\)\))1289 1130 y Fu(@)1350 1107 y Fz(~)1350 1130 y(~)-54 b Fu(r)p 1288 1175 106 4 v 1288 1266 a(@)5 b(\022)1403 1198 y Fz(\()p Fu(\022)s(;)17 b(r)1577 1213 y Fs(1)1616 1198 y Fu(;)g(\022)1705 1213 y Fs(1)1745 1198 y Fz(\))27 b(=)1924 1130 y Fu(\032)1974 1145 y Ft(E)2026 1154 y Fj(1)2065 1130 y Fz(\()p Fu(\022)s Fz(\))p 1924 1175 265 4 v 1928 1266 a Fu(\030)1971 1281 y Ft(E)2023 1290 y Fj(1)2061 1266 y Fz(\()p Fu(\022)s Fz(\))2221 1198 y Fr(\001)2275 1175 y Fz(~)2275 1198 y(~)-53 b Fu(r)s Fz(\()p Fu(\022)s(;)17 b(r)2492 1213 y Fs(1)2530 1198 y Fu(;)g(\022)2619 1213 y Fs(1)2659 1198 y Fz(\))879 b(\(3.72\))150 1444 y(whic)m(h)33 b(in)f(turn)h(leads) f(to)1205 1661 y(~)1205 1684 y(~)-54 b Fu(r)s Fz(\()p Fu(\022)s(;)17 b(r)1421 1699 y Fs(1)1460 1684 y Fu(;)g(\022)1549 1699 y Fs(1)1589 1684 y Fz(\))27 b(=)h Fu(r)1802 1699 y Fs(1)1858 1684 y Fz(exp)2024 1543 y Fl(\032)2098 1548 y(Z)2198 1575 y Ft(\022)2154 1774 y(\022)2187 1783 y Fj(1)2264 1617 y Fu(\032)2314 1632 y Ft(E)2366 1641 y Fj(1)2404 1617 y Fz(\()p Fu(\036)p Fz(\))p 2264 1661 275 4 v 2268 1752 a Fu(\030)2311 1767 y Ft(E)2363 1776 y Fj(1)2400 1752 y Fz(\()p Fu(\036)p Fz(\))2548 1684 y Fu(d\036)2657 1543 y Fl(\033)2748 1684 y Fu(:)801 b Fz(\(3.73\))296 1942 y(With)34 b(the)g(same)g(c)m(hange)g(of)g(v)-5 b(ariables)32 b(and)i(k)m(eeping)h(in)e(mind)g(that)g Fu(E)3001 1957 y Ft(t)3061 1942 y Fz(=)d Fu(E)3239 1957 y Fs(1)3312 1942 y Fz(\(see)35 b(Lemma)150 2062 y(3.11\):)774 2134 y Fl(Z)874 2161 y Ft(t)830 2360 y Fs(1)930 2202 y Fz(1)p 930 2247 49 4 v 935 2338 a(~)-53 b Fu(r)989 2270 y Fz(\()p Fu(@)1078 2285 y Ft(\022)1117 2270 y Fu(\030)1160 2285 y Ft(E)1212 2293 y Fq(t)1244 2270 y Fz(\)\()1328 2244 y(~)1320 2270 y Fu(\022)s Fz(\))p Fu(d\034)39 b Fz(=)1641 2134 y Fl(Z)1746 2143 y Fs(~)1741 2161 y Ft(\022)1696 2360 y(\022)1729 2369 y Fj(1)1848 2202 y Fz(1)p 1806 2247 134 4 v 1806 2338 a Fu(\030)1849 2353 y Ft(E)1901 2362 y Fj(1)1949 2270 y Fz(\()p Fu(@)2038 2285 y Ft(\036)2085 2270 y Fu(\030)2128 2285 y Ft(E)2180 2294 y Fj(1)2218 2270 y Fz(\)\()p Fu(\036)p Fz(\))p Fu(d\036)27 b Fz(=)g(ln)2727 2100 y Fl(")2813 2202 y Fu(\030)2856 2217 y Ft(E)2908 2226 y Fj(1)2947 2202 y Fz(\()2993 2176 y(~)2985 2202 y Fu(\022)s Fz(\))p 2795 2247 294 4 v 2795 2338 a Fu(\030)2838 2353 y Ft(E)2890 2362 y Fj(1)2928 2338 y Fz(\()p Fu(\022)3011 2353 y Fs(1)3051 2338 y Fz(\))3099 2100 y Fl(#)3174 2270 y Fu(;)375 b Fz(\(3.74\))150 2521 y(whic)m(h)33 b(in)m(tro)s(duced)g (in)f(\(3.70\))f(yields)375 2790 y Fu(J)429 2805 y Fk(v)472 2813 y Fq(t)531 2790 y Fz(=)663 2723 y Fu(\030)706 2738 y Ft(E)758 2747 y Fj(1)796 2723 y Fz(\()842 2696 y(~)834 2723 y Fu(\022)s Fz(\))p 645 2767 V 645 2858 a Fu(\030)688 2873 y Ft(E)740 2882 y Fj(1)778 2858 y Fz(\()p Fu(\022)861 2873 y Fs(1)901 2858 y Fz(\))965 2790 y(exp)1131 2650 y Fl(\032)1206 2654 y(Z)1305 2681 y Ft(t)1261 2880 y Fs(1)1352 2650 y Fl(\022)1425 2790 y Fu(@)1481 2749 y Fs(2)1476 2815 y Ft(r)1521 2790 y Fu(S)6 b Fz(\()p Fu(\034)g(;)17 b Fv(v)1776 2805 y Ft(\034)1819 2790 y Fz(\))22 b(+)1987 2723 y(1)p 1987 2767 49 4 v 1992 2858 a(~)-53 b Fu(r)2046 2790 y Fz(\()p Fu(@)2135 2805 y Ft(E)2195 2790 y Fu(\030)2238 2805 y Ft(E)2290 2814 y Fj(1)2328 2790 y Fz(\()2374 2764 y(~)2366 2790 y Fu(\022)s Fz(\))22 b Fr(\001)g Fz(\()p Fu(@)2613 2805 y Ft(\022)2652 2790 y Fu(E)6 b Fz(\)\()p Fu(\034)g(;)17 b Fv(v)2957 2805 y Ft(\034)3000 2790 y Fz(\))3038 2650 y Fl(\023)3128 2790 y Fu(d\034)3232 2650 y Fl(\033)3324 2790 y Fu(:)225 b Fz(\(3.75\))296 3043 y(W)-8 b(e)48 b(no)m(w)g(treat)g(the)g(in)m(tegral)e(in)g(\(3.75\).)88 b(W)-8 b(e)48 b(compute)f(using)h(\(3.39\),)i(\(3.52\))d(and)h(the)150 3164 y(iden)m(tit)m(y)32 b Fu(@)561 3179 y Ft(E)621 3164 y Fu(\032)671 3179 y Ft(E)t Fs(\()p Ft(t;r)n(;\022)r Fs(\))916 3164 y Fz(\()p Fu(\022)s Fz(\))27 b(=)h Fu(t=r)s Fz(:)1050 3440 y Fu(@)1106 3399 y Fs(2)1101 3465 y Ft(r)1146 3440 y Fu(S)6 b Fz(\()p Fu(\034)g(;)17 b Fv(v)1401 3455 y Ft(\034)1444 3440 y Fz(\))22 b(+)1612 3372 y(1)p 1612 3417 V 1617 3508 a(~)-53 b Fu(r)1671 3440 y Fz(\()p Fu(@)1760 3455 y Ft(E)1820 3440 y Fu(\030)1863 3455 y Ft(E)1915 3463 y Fq(\034)1957 3440 y Fz(\()2003 3414 y(~)1995 3440 y Fu(\022)s Fz(\))22 b Fr(\001)g Fz(\()p Fu(@)2242 3455 y Ft(\022)2281 3440 y Fu(E)6 b Fz(\)\()p Fu(\034)g(;)17 b Fv(v)2586 3455 y Ft(\034)2629 3440 y Fz(\))22 b Fr(\000)2799 3372 y Fu(\032)2849 3387 y Ft(E)2901 3395 y Fq(\034)2944 3372 y Fz(\()2990 3346 y(~)2982 3372 y Fu(\022)s Fz(\))p 2799 3417 270 4 v 2915 3508 a(~)-54 b Fu(r)891 3701 y Fz(=)83 b Fr(\000)1140 3634 y Fu(d)1199 3608 y Fz(~)1191 3634 y Fu(\022)p 1137 3678 105 4 v 1137 3770 a(d\034)1268 3561 y Fl(\024)1503 3634 y Fz(1)p 1330 3678 394 4 v 1330 3770 a Fu(\030)1373 3785 y Ft(E)1425 3793 y Fq(\034)1468 3770 y Fu(@)1524 3735 y Fs(2)1519 3797 y Ft(E)1579 3770 y Fu(\032)1629 3785 y Ft(E)1681 3793 y Fq(\034)1734 3701 y Fr(f)p Fz(\()p Fu(@)1873 3716 y Ft(E)1933 3701 y Fu(\032)1983 3716 y Ft(E)2035 3724 y Fq(\034)2078 3701 y Fz(\))2116 3660 y Fs(2)2177 3701 y Fz(+)22 b(\()p Fu(@)2364 3716 y Ft(E)2424 3701 y Fu(\030)2467 3716 y Ft(E)2519 3724 y Fq(\034)2562 3701 y Fz(\))2600 3660 y Fs(2)2639 3701 y Fr(g)g Fz(+)2819 3634 y Fu(\032)2869 3649 y Ft(E)2921 3657 y Fq(\034)p 2819 3678 146 4 v 2823 3770 a Fu(\030)2866 3785 y Ft(E)2918 3793 y Fq(\034)2974 3561 y Fl(\025)3043 3701 y Fz(\()3089 3675 y(~)3081 3701 y Fu(\022)s Fz(\))891 3990 y(=)83 b Fr(\000)1140 3922 y Fu(d)1199 3896 y Fz(~)1191 3922 y Fu(\022)p 1137 3967 105 4 v 1137 4058 a(d\034)1273 3990 y Fr(\001)22 b Fu(@)1374 4005 y Ft(\022)1430 3990 y Fz(ln)16 b Fr(j)p Fu(@)1612 3948 y Fs(2)1607 4014 y Ft(E)1667 3990 y Fu(\032)1717 4005 y Ft(E)1769 4013 y Fq(\034)1812 3990 y Fr(j)1736 b Fz(\(3.76\))150 4217 y(where)34 b(the)f(last)f(equalit)m(y)g(comes)h(from)e(\(2.27\).)43 b(W)-8 b(e)33 b(conclude)g(that)848 4362 y Fl(Z)947 4388 y Fs(2)p Ft(\031)903 4588 y Fs(0)1046 4357 y Fl(\024)1281 4430 y Fz(1)p 1109 4475 394 4 v 1109 4566 a Fu(\030)1152 4581 y Ft(E)1204 4589 y Fq(\034)1246 4566 y Fu(@)1302 4532 y Fs(2)1297 4593 y Ft(E)1357 4566 y Fu(\032)1407 4581 y Ft(E)1459 4589 y Fq(\034)1512 4498 y Fr(f)p Fz(\()p Fu(@)1651 4513 y Ft(E)1711 4498 y Fu(\032)1761 4513 y Ft(E)1813 4521 y Fq(\034)1856 4498 y Fz(\))1894 4456 y Fs(2)1956 4498 y Fz(+)22 b(\()p Fu(@)2143 4513 y Ft(E)2203 4498 y Fu(\030)2246 4513 y Ft(E)2298 4521 y Fq(\034)2340 4498 y Fz(\))2378 4456 y Fs(2)2417 4498 y Fr(g)g Fz(+)2597 4430 y Fu(\032)2647 4445 y Ft(E)2699 4453 y Fq(\034)p 2597 4475 146 4 v 2601 4566 a Fu(\030)2644 4581 y Ft(E)2696 4589 y Fq(\034)2752 4357 y Fl(\025)2822 4498 y Fu(d)2881 4471 y Fz(~)2873 4498 y Fu(\022)30 b Fz(=)e(0)p Fu(:)296 4720 y Fz(W)-8 b(e)33 b(also)f(compute)h(\(using)f(\(3.52\))g(again\)) 1341 4833 y Fl(\()1421 4867 y(Z)1521 4894 y Ft(t)1477 5093 y Fs(1)1577 4936 y Fu(\032)1627 4951 y Ft(E)1679 4959 y Fq(\034)1722 4936 y Fz(\()1768 4909 y(~)1760 4936 y Fu(\022)s Fz(\()p Fu(\034)11 b Fz(\)\))p 1577 4980 399 4 v 1693 5071 a(~)-54 b Fu(r)s Fz(\()p Fu(\034)11 b Fz(\))1986 4833 y Fl(\))2082 5003 y Fu(d\034)39 b Fz(=)28 b(ln)2437 4936 y(~)-54 b Fu(r)s Fz(\()p Fu(t)p Fz(\))p 2426 4980 172 4 v 2430 5071 a(~)h Fu(r)r Fz(\(1\))2607 5003 y Fu(:)296 5286 y Fz(Consequen)m(tly)40 b(w)m(e)e(conclude)f(that) g(there)g(exists)h Fu(C)k Fz(=)35 b Fu(C)7 b Fz(\()p Fu(r)2584 5301 y Fs(1)2623 5286 y Fu(;)17 b(\022)2712 5301 y Fs(1)2752 5286 y Fz(\))35 b Fu(>)g Fz(0)i(suc)m(h)h(that)f (\(see)h(also)150 5407 y(\(3.65\)\))1362 5527 y Fu(C)1439 5486 y Fm(\000)p Fs(1)1555 5527 y Fr(\001)22 b Fu(t)28 b Fr(\024)g Fu(J)1827 5542 y Fk(v)1870 5550 y Fq(t)1902 5527 y Fz(\()p Fu(t;)17 b(r)2063 5542 y Fs(1)2102 5527 y Fu(;)g(\022)2191 5542 y Fs(1)2231 5527 y Fz(\))28 b Fr(\024)g Fu(C)h Fr(\001)22 b Fu(t:)963 b Fz(\(3.77\))1939 5776 y(34)p eop %%Page: 35 38 35 37 bop 296 407 a Fz(W)-8 b(e)28 b(no)m(w)f(estimate)f(the)i (individual)c(deriv)-5 b(ativ)m(es)27 b(of)j(~)-53 b Fu(r)s Fz(\()p Fu(t;)17 b(r)2449 422 y Fs(1)2488 407 y Fu(;)g(\022)2577 422 y Fs(1)2617 407 y Fz(\))26 b(and)2873 381 y(~)2865 407 y Fu(\022)t Fz(\()p Fu(t;)17 b(r)3075 422 y Fs(1)3114 407 y Fu(;)g(\022)3203 422 y Fs(1)3242 407 y Fz(\))27 b(with)g(resp)s(ect)150 527 y(to)33 b Fu(r)314 542 y Fs(1)386 527 y Fz(and)h Fu(\022)622 542 y Fs(1)661 527 y Fz(.)46 b(In)33 b(conjunction)g(with)g(\(3.77\))f (those)i(will)d(b)s(e)i(useful)g(for)g(estimating)e(deriv)-5 b(ativ)m(es)150 648 y(of)32 b(the)h(in)m(v)m(erse)h(\015o)m(w.)296 768 y(Using)41 b(\(3.71\))e(in)h(the)h(second)h(equation)e(of)g (\(3.52\))g(after)h(separating)f(v)-5 b(ariables)39 b(and)i(in)m(te-) 150 888 y(grating)31 b(w)m(e)j(get)e(\(remem)m(b)s(er)g(the)h (abbreviation)f Fu(E)2092 903 y Fs(1)2159 888 y Fz(=)c Fu(E)6 b Fz(\(1)p Fu(;)17 b(r)2516 903 y Fs(1)2554 888 y Fu(;)g(\022)2643 903 y Fs(1)2683 888 y Fz(\)\))1319 1007 y Fl(Z)1424 1016 y Fs(~)1419 1033 y Ft(\022)q Fs(\()p Ft(t;r)1557 1042 y Fj(1)1592 1033 y Ft(;\022)1645 1042 y Fj(1)1679 1033 y Fs(\))1374 1232 y Ft(\022)1407 1241 y Fj(1)1742 1052 y Fz(~)1742 1075 y(~)-53 b Fu(r)s Fz(\()p Fu(\036;)17 b(r)1969 1090 y Fs(1)2007 1075 y Fu(;)g(\022)2096 1090 y Fs(1)2136 1075 y Fz(\))p 1738 1120 436 4 v 1822 1211 a Fu(\030)1865 1226 y Ft(E)1917 1235 y Fj(1)1955 1211 y Fz(\()p Fu(\036)p Fz(\))2184 1143 y Fu(d\036)27 b Fz(=)g Fu(t)c Fr(\000)f Fz(1)p Fu(:)920 b Fz(\(3.78\))296 1369 y(Di\013eren)m(tiating)30 b(with)i(resp)s(ect)i(to)f Fu(r)1665 1384 y Fs(1)1736 1369 y Fz(w)m(e)h(obtain)1060 1550 y Fu(@)1124 1524 y Fz(~)1116 1550 y Fu(\022)p 1042 1595 141 4 v 1042 1686 a(@)5 b(r)1142 1701 y Fs(1)1214 1617 y Fr(\001)1279 1527 y Fz(~)1279 1550 y(~)-54 b Fu(r)s Fz(\()1367 1524 y(~)1359 1550 y Fu(\022)s(;)17 b(r)1495 1565 y Fs(1)1534 1550 y Fu(;)g(\022)1623 1565 y Fs(1)1663 1550 y Fz(\))p 1274 1595 427 4 v 1359 1699 a Fu(\030)1402 1714 y Ft(E)1454 1723 y Fj(1)1492 1699 y Fz(\()1538 1673 y(~)1530 1699 y Fu(\022)s Fz(\))1738 1617 y(=)28 b Fr(\000)1936 1482 y Fl(Z)2041 1491 y Fs(~)2035 1508 y Ft(\022)1991 1707 y(\022)2024 1716 y Fj(1)2091 1617 y Fu(@)2142 1632 y Ft(r)2174 1641 y Fj(1)2230 1477 y Fl(\024)2297 1527 y Fz(~)2297 1550 y(~)-54 b Fu(r)s Fz(\()p Fu(\036;)17 b(r)2523 1565 y Fs(1)2562 1550 y Fu(;)g(\022)2651 1565 y Fs(1)2690 1550 y Fz(\))p 2292 1595 436 4 v 2377 1686 a Fu(\030)2420 1701 y Ft(E)2472 1710 y Fj(1)2510 1686 y Fz(\()p Fu(\036)p Fz(\))2738 1477 y Fl(\025)2807 1617 y Fu(d\036:)633 b Fz(\(3.79\))296 1856 y(Let)44 b(us)g(pro)m(v)m(e)g(a) f(v)m(ery)i(rough)e(estimate)g(of)f(the)i(ab)s(o)m(v)m(e)g(deriv)-5 b(ativ)m(e)43 b(\(here)h(and)f(b)s(elo)m(w)h(the)150 1977 y(constan)m(ts)34 b(are)f(uniformly)d(b)s(ounded)j(on)g(compact)f (energy)i(in)m(terv)-5 b(als)31 b Fu(K)k Fr(\032)28 b Fz(\()p Fu(E)3148 1992 y Ft(d)3189 1977 y Fu(;)17 b Fr(1)p Fz(\)\):)703 2060 y Fl(\014)703 2119 y(\014)703 2179 y(\014)703 2239 y(\014)703 2299 y(\014)764 2167 y Fu(@)828 2140 y Fz(~)820 2167 y Fu(\022)p 746 2211 141 4 v 746 2302 a(@)5 b(r)846 2317 y Fs(1)896 2060 y Fl(\014)896 2119 y(\014)896 2179 y(\014)896 2239 y(\014)896 2299 y(\014)957 2234 y Fr(\024)28 b Fz(Const)23 b Fr(\001)f Fz([1)g(+)g(\()1626 2208 y(~)1618 2234 y Fu(\022)j Fr(\000)e Fu(\022)1833 2249 y Fs(1)1873 2234 y Fz(\))1911 2193 y Fs(2)1950 2234 y Fz(])28 b Fr(\024)g Fz(Const)23 b Fr(\001)f Fz([1)g(+)g(ln)2709 2191 y Fs(2)2749 2234 y Fz(\()p Fu(t)p Fz(\)];)114 b Fu(t)28 b Fr(\025)g Fz(1)p Fu(:)304 b Fz(\(3.80\))296 2491 y(Indeed,)40 b(di\013eren)m(tiating)35 b(the)j(in)m(tegrand)f(on)g(the)h(righ)m(t)e(hand)h(side)h(of)e (\(3.79\))h(with)g(resp)s(ect)150 2612 y(to)32 b Fu(r)313 2627 y Fs(1)385 2612 y Fz(w)m(e)i(get)e(\(see)i(\(3.73\)\):)1271 2761 y(~)1271 2784 y(~)-53 b Fu(r)s Fz(\()p Fu(\036;)17 b(r)1498 2799 y Fs(1)1537 2784 y Fu(;)g(\022)1626 2799 y Fs(1)1665 2784 y Fz(\))p 1267 2828 436 4 v 1352 2920 a Fu(\030)1395 2935 y Ft(E)1447 2944 y Fj(1)1485 2920 y Fz(\()p Fu(\036)p Fz(\))1730 2711 y Fl(\032)1804 2851 y Fz(1)p Fu(=r)1946 2866 y Fs(1)2007 2851 y Fz(+)2105 2716 y Fl(Z)2205 2742 y Ft(\036)2161 2941 y(\022)2194 2950 y Fj(1)2268 2851 y Fu(@)2319 2866 y Ft(r)2351 2875 y Fj(1)2400 2784 y Fu(\032)2450 2799 y Ft(E)2502 2808 y Fj(1)2541 2784 y Fz(\()p Fu(\021)t Fz(\))p 2400 2828 269 4 v 2404 2920 a Fu(\030)2447 2935 y Ft(E)2499 2944 y Fj(1)2537 2920 y Fz(\()p Fu(\021)t Fz(\))2678 2851 y Fu(d\021)2781 2711 y Fl(\033)1097 3140 y Fr(\000)1271 3050 y Fz(~)1271 3073 y(~)-53 b Fu(r)s Fz(\()p Fu(\036;)17 b(r)1498 3088 y Fs(1)1537 3073 y Fu(;)g(\022)1626 3088 y Fs(1)1665 3073 y Fz(\))p 1267 3117 436 4 v 1352 3209 a Fu(\030)1400 3174 y Fs(2)1395 3236 y Ft(E)1447 3245 y Fj(1)1485 3209 y Fz(\()p Fu(\036)p Fz(\))1735 3140 y Fr(\001)22 b Fz(\()p Fu(@)1874 3155 y Ft(E)1934 3140 y Fu(\030)1977 3155 y Ft(E)2029 3164 y Fj(1)2067 3140 y Fz(\)\()p Fu(\036)p Fz(\))g Fr(\001)g Fz(\()p Fu(@)2400 3155 y Ft(r)2438 3140 y Fu(E)6 b Fz(\)\(1)p Fu(;)17 b(r)2729 3155 y Fs(1)2768 3140 y Fu(;)g(\022)2857 3155 y Fs(1)2896 3140 y Fz(\))p Fu(:)615 b Fz(\(3.81\))150 3392 y(This)33 b(together)g(with)f(the)h(fact)f(that)1556 3369 y(~)1556 3392 y(~)-54 b Fu(r)36 b Fz(increases)d(with)f Fu(\036)h Fz(o)m(v)m(er)g(a)g(p)s(erio)s(d,)e(leads)i(us)g(to)852 3482 y Fl(\014)852 3541 y(\014)852 3601 y(\014)852 3661 y(\014)885 3626 y Fu(@)936 3641 y Ft(r)968 3650 y Fj(1)1024 3486 y Fl(\024)1091 3536 y Fz(~)1091 3559 y(~)-54 b Fu(r)s Fz(\()p Fu(\036;)17 b(r)1317 3574 y Fs(1)1356 3559 y Fu(;)g(\022)1445 3574 y Fs(1)1485 3559 y Fz(\))p 1086 3603 V 1171 3695 a Fu(\030)1214 3710 y Ft(E)1266 3719 y Fj(1)1304 3695 y Fz(\()p Fu(\036)p Fz(\))1532 3486 y Fl(\025)1585 3482 y(\014)1585 3541 y(\014)1585 3601 y(\014)1585 3661 y(\014)1646 3626 y Fr(\024)28 b Fz(Const)23 b Fr(\001)2077 3603 y Fz(~)2077 3626 y(~)-53 b Fu(r)s Fz(\()2166 3600 y(~)2158 3626 y Fu(\022)s(;)17 b(r)2294 3641 y Fs(1)2333 3626 y Fu(;)g(\022)2422 3641 y Fs(1)2461 3626 y Fz(\))g([1)22 b(+)g(\()p Fu(\036)g Fr(\000)g Fu(\022)2974 3641 y Fs(1)3014 3626 y Fz(\)])17 b Fu(;)150 3853 y Fz(whic)m(h)33 b(in)m(tro)s(duced)g(in)f(\(3.79\))f(and)i(using)f(\(3.66\))g(leads)h (to)f(the)h(desired)g(estimate)f(in)g(\(3.80\).)296 3974 y(One)h(can)f(iterate)g(this)g(pro)s(cedure)h(b)m(y)g(p)s(erforming)d (higher)i(order)h(deriv)-5 b(ativ)m(es)32 b(with)g(resp)s(ect)150 4094 y(to)23 b Fu(\022)305 4109 y Fs(1)368 4094 y Fz(and)g Fu(r)592 4109 y Fs(1)655 4094 y Fz(in)g(\(3.79\))f(and)h(isolating)e (the)j(term)e(ha)m(ving)h(all)f(the)h(deriv)-5 b(ativ)m(es)24 b(acting)e(on)3460 4068 y(~)3452 4094 y Fu(\022)s Fz(.)41 b(Hence)150 4215 y(reasoning)35 b(b)m(y)h(induction)e(\(w)m(e)j(skip)e (the)h(details\),)f(one)g(obtains)g(logarithmic)c(t)m(yp)s(e)37 b(b)s(ounds)e(for)150 4335 y(all)30 b(deriv)-5 b(ativ)m(es)33 b(of)890 4309 y(~)882 4335 y Fu(\022)s Fz(:)234 4393 y Fl(\014)234 4453 y(\014)234 4512 y(\014)234 4572 y(\014)234 4632 y(\014)354 4500 y Fu(@)410 4464 y Ft(i)447 4474 y Fz(~)439 4500 y Fu(\022)p 277 4544 289 4 v 277 4647 a(@)5 b(r)380 4600 y Ft(j)377 4672 y Fs(1)417 4647 y Fu(@)g(\022)521 4613 y Ft(k)518 4672 y Fs(1)575 4393 y Fl(\014)575 4453 y(\014)575 4512 y(\014)575 4572 y(\014)575 4632 y(\014)625 4567 y Fz(\()p Fu(t;)17 b(r)786 4582 y Fs(1)825 4567 y Fu(;)g(\022)914 4582 y Fs(1)954 4567 y Fz(\))27 b Fr(\024)h Fz(Const)23 b Fr(\001)f Fz([1)g(+)g(\()1688 4541 y(~)1680 4567 y Fu(\022)j Fr(\000)e Fu(\022)1895 4582 y Fs(1)1935 4567 y Fz(\))1973 4526 y Ft(i)p Fs(+1)2091 4567 y Fz(])28 b Fr(\024)g Fz(Const)23 b Fr(\001)f Fz([1)g(+)g(ln)2850 4524 y Ft(i)p Fs(+1)2969 4567 y Fz(\()p Fu(t)p Fz(\)])p Fu(;)114 b(t)28 b Fr(\025)g Fz(1)p Fu(;)84 b Fz(\(3.82\))150 4824 y(for)32 b Fu(i;)17 b(j;)g(k)31 b Fr(2)d(f)p Fz(0)p Fu(;)17 b Fz(1)p Fu(;)g Fz(2)p Fu(;)g Fz(3)p Fr(g)p Fz(,)31 b Fu(i)c(>)h Fz(0,)k Fu(j)d Fz(+)22 b Fu(k)30 b Fz(=)e Fu(i)p Fz(.)296 4945 y(Let)34 b(us)h(no)m(w)f(in)m(v)m(estigate)g(the)g (deriv)-5 b(ativ)m(es)34 b(of)j(~)-53 b Fu(r)s Fz(.)46 b(Using)34 b(\(3.71\),)f(\(3.73\),)h(\(3.66\),)f(\(3.82\))g(and)150 5065 y(reasoning)f(b)m(y)i(induction)d(one)i(infers:)904 5145 y Fl(\014)904 5205 y(\014)904 5265 y(\014)904 5325 y(\014)1025 5223 y Fu(@)1081 5186 y Ft(i)1115 5223 y Fz(~)-53 b Fu(r)p 947 5267 V 947 5370 a(@)5 b(r)1050 5323 y Ft(j)1047 5395 y Fs(1)1088 5370 y Fu(@)g(\022)1192 5336 y Ft(k)1189 5395 y Fs(1)1245 5145 y Fl(\014)1245 5205 y(\014)1245 5265 y(\014)1245 5325 y(\014)1295 5290 y Fz(\()p Fu(t;)17 b(r)1456 5305 y Fs(1)1495 5290 y Fu(;)g(\022)1584 5305 y Fs(1)1624 5290 y Fz(\))28 b Fr(\024)g Fz(Const)23 b Fr(\001)f Fu(t)p Fz([1)g(+)g(ln)2429 5247 y Ft(i)p Fs(+1)2547 5290 y Fz(\()p Fu(t)p Fz(\)])p Fu(;)115 b(t)28 b Fr(\025)g Fz(1)p Fu(;)505 b Fz(\(3.83\))150 5527 y(for)32 b Fu(i;)17 b(j;)g(k)31 b Fr(2)d(f)p Fz(0)p Fu(;)17 b Fz(1)p Fu(;)g Fz(2)p Fu(;)g Fz(3)p Fr(g)p Fz(,)31 b Fu(i)c(>)h Fz(0,)k Fu(j)d Fz(+)22 b Fu(k)30 b Fz(=)e Fu(i)p Fz(.)1939 5776 y(35)p eop %%Page: 36 39 36 38 bop 150 407 a Fv(3.7.2)112 b(Dep)s(endence)39 b(of)f(the)f(in)m (v)m(erse)g(\015o)m(w)h(on)f Fu(r)j Fv(and)f Fu(\022)150 592 y Fz(The)j(follo)m(wing)d(lemma)g(\(and)i(similar)d(b)s(ounds)k (for)f(some)g(higher)g(order)g(deriv)-5 b(ativ)m(es;)46 b(see)d(the)150 712 y(remark)i(after)f(the)h(lemma\))e(will)f(b)s(e)k (imp)s(ortan)m(t)c(in)j(the)g(next)h(sections.)80 b(It)45 b(giv)m(es)h(a)e(precise)150 832 y(meaning)f(to)g(the)i(statemen)m(t)f (sa)m(ying)g(that)g(when)h(one)f(di\013eren)m(tiates)g(the)g(in)m(v)m (erse)i(\015o)m(w)e(with)150 953 y(resp)s(ect)33 b(to)f Fu(r)i Fz(one)e(gains)f(a)h(deca)m(y)h(of)e(almost)f Fu(t)1917 917 y Fm(\000)p Fs(1)2044 953 y Fz(while)h(di\013eren)m (tiating)f(with)h(resp)s(ect)i(to)f Fu(\022)j Fz(one)150 1073 y(gets)e(bac)m(k)h(something)d(almost)g(b)s(ounded.)296 1193 y(Giv)m(en)i(a)f(compact)g(energy)i(in)m(terv)-5 b(al)31 b Fu(K)k Fr(\032)28 b Fz(\()p Fu(E)2047 1208 y Ft(d)2088 1193 y Fu(;)17 b Fr(1)p Fz(\))32 b(w)m(e)h(in)m(tro)s(duce) g(for)f Fu(t)c Fr(\025)g Fz(1)k(the)h(sets)986 1413 y Fr(A)1066 1428 y Ft(t)1095 1413 y Fz(\()p Fu(K)7 b Fz(\))28 b(=)g Fr(f)p Fz(\()p Fu(r)m(;)17 b(\022)s Fz(\))27 b Fr(2)h Fz(\(0)p Fu(;)17 b Fr(1)p Fz(\))k Fr(\002)i Fn(T)31 b Fz(:)50 b Fu(E)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))28 b Fr(2)g Fu(K)7 b Fr(g)p Fu(:)587 b Fz(\(3.84\))150 1632 y(W)-8 b(e)33 b(recall,)e(cf.)44 b(\(3.56\))32 b(and)g(\(3.57\),) 819 1872 y Fu(E)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))28 b(=)f Fu(E)6 b Fz(\(1)p Fu(;)17 b Fv(w)1606 1887 y Ft(t)1635 1872 y Fz(\()p Fu(r)m(;)g(\022)s Fz(\)\);)114 b Fv(w)2104 1887 y Ft(t)2162 1872 y Fz(=)27 b(\()p Fu(r)2347 1887 y Fs(1)2387 1872 y Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))p Fu(;)g(\022)2764 1887 y Fs(1)2803 1872 y Fz(\()p Fu(t;)g(r)m(;)g(\022)s Fz(\)\))p Fu(;)420 b Fz(\(3.85\))1404 2046 y Fu(r)30 b Fz(=)i(~)-53 b Fu(r)r Fz(\()p Fu(t;)17 b(r)1789 2061 y Fs(1)1829 2046 y Fz(\()p Fu(t;)g(r)m(;)g(\022)s Fz(\))p Fu(;)g(\022)2206 2061 y Fs(1)2245 2046 y Fz(\()p Fu(t;)g(r)m(;)g(\022)s Fz(\)\))1005 b(\(3.86\))150 2220 y(and)1389 2341 y Fu(\022)31 b Fz(=)1576 2315 y(~)1568 2341 y Fu(\022)s Fz(\()p Fu(t;)17 b(r)1777 2356 y Fs(1)1817 2341 y Fz(\()p Fu(t;)g(r)m(;)g(\022)s Fz(\))p Fu(;)g(\022)2194 2356 y Fs(1)2233 2341 y Fz(\()p Fu(t;)g(r)m(;)g(\022)s Fz(\)\))p Fu(:)990 b Fz(\(3.87\))150 2543 y Fv(Lemma)44 b(3.12.)h FA(Fix)40 b Fu(\016)h Fr(2)e Fz(\(0)p Fu(;)17 b Fz(1\))p FA(.)60 b(Then)40 b(for)g(any)g(c)-5 b(omp)g(act)40 b(interval)g Fu(K)45 b Fr(\032)38 b Fz(\()p Fu(E)3216 2558 y Ft(d)3257 2543 y Fu(;)17 b Fr(1)p Fz(\))39 b FA(and)h(any)150 2664 y(initial)34 b(data)h(for)g(the)g(inverse)f(\015ow)g(lo)-5 b(c)g(ate)g(d)35 b(in)f Fr(A)2028 2679 y Ft(t)2057 2664 y Fz(\()p Fu(K)7 b Fz(\))35 b FA(we)g(have)720 2826 y Fl(\014)720 2886 y(\014)720 2945 y(\014)720 3005 y(\014)763 2903 y Fu(@)5 b(\022)864 2918 y Fs(1)p 763 2947 142 4 v 782 3039 a Fu(@)g(r)915 2826 y Fl(\014)915 2886 y(\014)915 2945 y(\014)915 3005 y(\014)976 2970 y Fz(=)1080 2796 y Fl(\014)1080 2856 y(\014)1080 2915 y(\014)1080 2975 y(\014)1080 3035 y(\014)1113 2970 y Fr(\000)p Fu(J)1253 2929 y Fm(\000)p Fs(1)1244 2995 y Fk(v)1287 3003 y Fq(t)1375 2903 y Fu(@)1439 2877 y Fz(~)1431 2903 y Fu(\022)p 1358 2947 141 4 v 1358 3039 a(@)g(r)1458 3054 y Fs(1)1508 2796 y Fl(\014)1508 2856 y(\014)1508 2915 y(\014)1508 2975 y(\014)1508 3035 y(\014)1569 2970 y Fr(\024)28 b Fu(C)h Fr(\001)22 b Fu(t)1858 2929 y Ft(\016)r Fm(\000)p Fs(1)1986 2970 y Fu(;)2129 2826 y Fl(\014)2129 2886 y(\014)2129 2945 y(\014)2129 3005 y(\014)2173 2903 y Fu(@)5 b(\022)2274 2918 y Fs(1)p 2173 2947 142 4 v 2191 3039 a Fu(@)g(\022)2324 2826 y Fl(\014)2324 2886 y(\014)2324 2945 y(\014)2324 3005 y(\014)2385 2970 y Fz(=)2489 2826 y Fl(\014)2489 2886 y(\014)2489 2945 y(\014)2489 3005 y(\014)2522 2970 y Fu(J)2585 2929 y Fm(\000)p Fs(1)2576 2995 y Fk(v)2619 3003 y Fq(t)2708 2903 y Fu(@)10 b Fz(~)-54 b Fu(r)p 2689 2947 141 4 v 2689 3039 a(@)5 b(r)2789 3054 y Fs(1)2840 2826 y Fl(\014)2840 2886 y(\014)2840 2945 y(\014)2840 3005 y(\014)2901 2970 y Fr(\024)28 b Fu(C)h Fr(\001)22 b Fu(t)3190 2929 y Ft(\016)3228 2970 y Fu(;)321 b Fz(\(3.88\))150 3271 y FA(and)720 3309 y Fl(\014)720 3369 y(\014)720 3429 y(\014)720 3488 y(\014)763 3386 y Fu(@)5 b(r)863 3401 y Fs(1)p 763 3431 V 782 3522 a Fu(@)g(r)914 3309 y Fl(\014)914 3369 y(\014)914 3429 y(\014)914 3488 y(\014)974 3454 y Fz(=)1078 3279 y Fl(\014)1078 3339 y(\014)1078 3399 y(\014)1078 3459 y(\014)1078 3518 y(\014)1111 3454 y Fu(J)1174 3412 y Fm(\000)p Fs(1)1165 3478 y Fk(v)1208 3486 y Fq(t)1297 3386 y Fu(@)1361 3360 y Fz(~)1353 3386 y Fu(\022)p 1279 3431 142 4 v 1279 3522 a(@)g(\022)1380 3537 y Fs(1)1430 3279 y Fl(\014)1430 3339 y(\014)1430 3399 y(\014)1430 3459 y(\014)1430 3518 y(\014)1491 3454 y Fr(\024)28 b Fu(C)i Fr(\001)21 b Fu(t)1780 3412 y Ft(\016)r Fm(\000)p Fs(1)1909 3454 y Fu(;)2052 3309 y Fl(\014)2052 3369 y(\014)2052 3429 y(\014)2052 3488 y(\014)2095 3386 y Fu(@)5 b(r)2195 3401 y Fs(1)p 2095 3431 141 4 v 2113 3522 a Fu(@)g(\022)2245 3309 y Fl(\014)2245 3369 y(\014)2245 3429 y(\014)2245 3488 y(\014)2306 3454 y Fz(=)2410 3309 y Fl(\014)2410 3369 y(\014)2410 3429 y(\014)2410 3488 y(\014)2443 3454 y Fr(\000)p Fu(J)2583 3412 y Fm(\000)p Fs(1)2574 3478 y Fk(v)2617 3486 y Fq(t)2707 3386 y Fu(@)10 b Fz(~)-54 b Fu(r)p 2688 3431 142 4 v 2688 3522 a(@)5 b(\022)2789 3537 y Fs(1)2840 3309 y Fl(\014)2840 3369 y(\014)2840 3429 y(\014)2840 3488 y(\014)2901 3454 y Fr(\024)28 b Fu(C)h Fr(\001)22 b Fu(t)3190 3412 y Ft(\016)3228 3454 y Fu(;)321 b Fz(\(3.89\))150 3709 y FA(wher)-5 b(e)34 b Fu(C)h(>)27 b Fz(0)35 b FA(only)g(dep)-5 b(ends)33 b(on)i Fu(K)42 b FA(and)34 b Fu(\016)t FA(.)150 3912 y Fv(Pro)s(of.)70 b Fz(The)42 b(result)g(comes)f(after)h(a)f(straigh)m (tforw)m(ard)g(application)e(of)i(the)h(in)m(v)m(erse)g(function)150 4032 y(theorem)32 b(and)h(b)m(y)h(using)e(\(3.77\),)g(\(3.82\),)g (\(3.83\);)g(w)m(e)h(replace)g(the)g(logarithms)d(b)m(y)j Fu(t)3332 3996 y Ft(\016)r(=)p Fs(2)3441 4032 y Fz(.)p 3754 4032 4 66 v 3758 3970 59 4 v 3758 4032 V 3816 4032 4 66 v 296 4271 a(W)-8 b(e)39 b(will)e(later)g(on)i(need)h(similar)35 b(b)s(ounds)k(for)f(some)h(higher)f(order)h(deriv)-5 b(ativ)m(es;)42 b(those)d(are)150 4391 y(obtained)48 b(along)f(the)i(same)g(line,)i(that)e(is)f(b)m(y)i(applying)d(the)i(in) m(v)m(erse)h(function)e(theorem)h(in)150 4512 y(conjunction)32 b(with)h(the)g(b)s(ounds)g(obtained)f(for)g(deriv)-5 b(ativ)m(es)33 b(of)f(the)h(direct)f(\015o)m(w.)150 4800 y Fp(3.8)135 b(In)l(v)l(estigating)48 b(classical)e(asymptotic)g (completeness)150 4985 y Fz(W)-8 b(e)40 b(no)m(w)g(are)f(in)g(a)g(p)s (osition)f(to)h(pro)m(v)m(e)h(\(1.15\),)h(whic)m(h)f(as)f(w)m(e)i(ha)m (v)m(e)f(already)f(stated)h(is)f(only)g(a)150 5105 y(\014rst)33 b(step)h(in)d(sho)m(wing)i(classical)e(asymptotic)h(completeness.)150 5308 y Fv(Prop)s(osition)j(3.13.)41 b FA(Consider)33 b(an)h(arbitr)-5 b(ary)35 b(classic)-5 b(al)33 b(orbit)h(de\014ne)-5 b(d)33 b(for)h(al)5 b(l)34 b(p)-5 b(ositive)33 b(times:)1451 5527 y Fv(V)1536 5542 y Ft(t)1593 5527 y Fz(=)27 b(\()p Fu(r)s Fz(\()p Fu(t)p Fz(\))p Fu(;)17 b(\022)s Fz(\()p Fu(t)p Fz(\);)g Fu(\032)p Fz(\()p Fu(t)p Fz(\))p Fu(;)g(l)r Fz(\()p Fu(t)p Fz(\)\))1939 5776 y(36)p eop %%Page: 37 40 37 39 bop 150 407 a FA(c)-5 b(orr)g(esp)g(onding)46 b(to)i(an)f(ener)-5 b(gy)47 b Fu(E)57 b(>)51 b(E)1708 422 y Ft(d)1749 407 y FA(.)82 b(F)-7 b(or)47 b(such)g(an)g(orbit)h(the)f(asymptotic)g(r)-5 b(adius)48 b(and)150 527 y(angle)c(de\014ne)-5 b(d)43 b(as)h(entries)g(of)g(the)g(limit)g(\(1.15\),)i(and)e(denote)-5 b(d)43 b Fu(r)2698 542 y Fs(+)2802 527 y FA(and)h Fu(\022)3046 542 y Fs(+)3149 527 y FA(r)-5 b(esp)g(e)g(ctively,)46 b(ex-)150 648 y(ist.)70 b(Mor)-5 b(e)g(over)43 b(the)g(ener)-5 b(gy)43 b(of)g(the)g(orbit)h(is)f(r)-5 b(elate)g(d)43 b(to)g(the)g(asymptotic)g(quantities)g(by)h Fu(E)49 b Fz(=)150 768 y Fr(\000)p Fu(@)278 783 y Ft(t)309 768 y Fu(S)6 b Fz(\(1)p Fu(;)17 b(r)550 783 y Fs(+)608 768 y Fu(;)g(\022)697 783 y Fs(+)756 768 y Fz(\))p FA(.)150 971 y Fv(Pro)s(of.)43 b Fz(W)-8 b(e)33 b(start)g(b)m(y)g(\014xing)g (further)f(notation.)43 b(Denote)32 b(b)m(y)i(\(see)f(\(1.5\))f(and)h (\(1.13\)\))1098 1191 y Fv(F)p Fz(\()p Fu(r)m(;)17 b(\022)s(;)g(\032;)g (l)r Fz(\))84 b(=)f(\()p Fu(@)1878 1206 y Ft(\032)1918 1191 y Fu(h;)17 b(@)2069 1206 y Ft(l)2096 1191 y Fu(h;)g Fr(\000)p Fu(@)2324 1206 y Ft(r)2363 1191 y Fu(h;)g Fr(\000)p Fu(@)2591 1206 y Ft(\022)2630 1191 y Fu(h)p Fz(\))p Fu(;)1057 1337 y Fv(F)1127 1352 y Ft(a)1169 1337 y Fz(\()p Fu(r)m(;)g(\022)s(;)g (\032;)g(l)r Fz(\))83 b(=)g(\()p Fu(@)1878 1352 y Ft(\032)1918 1337 y Fu(h)1974 1352 y Ft(a)2016 1337 y Fu(;)17 b(@)2111 1352 y Ft(l)2137 1337 y Fu(h)2193 1352 y Ft(a)2235 1337 y Fu(;)g Fr(\000)p Fu(@)2407 1352 y Ft(r)2446 1337 y Fu(h)2502 1352 y Ft(a)2544 1337 y Fu(;)g Fr(\000)p Fu(@)2716 1352 y Ft(\022)2755 1337 y Fu(h)2811 1352 y Ft(a)2853 1337 y Fz(\))p Fu(:)658 b Fz(\(3.90\))296 1557 y(W)-8 b(e)47 b(no)m(w)f(explicitly)f(need)i(the)f(dep)s(endence)j(on)c(the)i (initial)42 b(conditions.)83 b(F)-8 b(or)45 b(example,)150 1677 y Fv(V)235 1692 y Ft(t)264 1677 y Fz(\()p Fv(x)p Fz(\))33 b(means)f(the)h(particular)e(orbit)h(whic)m(h)h(equals)g Fv(x)f Fz(at)h Fu(t)27 b Fz(=)h(1,)k(i.e.)43 b Fv(V)2900 1692 y Fs(1)2967 1677 y Fz(=)28 b Fv(x)g Fr(2)g Fn(R)3318 1641 y Fs(4)3363 1677 y Fz(.)296 1797 y(The)37 b(Hamilton)32 b(equations)k(for)f(the)h(true)g(and)f(comparison)f(dynamics)i(ma)m(y)f (b)s(e)g(written)h(in)150 1918 y(the)d(form)940 1995 y Fu(d)p Fv(V)1076 2010 y Ft(t)p 940 2040 165 4 v 979 2131 a Fu(dt)1115 2063 y Fz(\()p Fv(x)p Fz(\))27 b(=)h Fv(F)p Fz(\()p Fv(V)1574 2078 y Ft(t)1604 2063 y Fz(\()p Fv(x)p Fz(\)\))p Fu(;)2025 1995 y(d)p Fv(V)2161 2010 y Ft(a;t)p 2025 2040 223 4 v 2093 2131 a Fu(dt)2257 2063 y Fz(\()p Fv(x)p Fz(\))g(=)g Fv(F)2594 2078 y Ft(a)2636 2063 y Fz(\()p Fv(V)2759 2078 y Ft(a;t)2845 2063 y Fz(\()p Fv(x)p Fz(\)\))p Fu(:)531 b Fz(\(3.91\))296 2262 y(Since)33 b Fv(W)667 2277 y Ft(a;t)786 2262 y Fz(denotes)h(the)f(in)m(v)m(erse)h (for)e Fv(V)1864 2277 y Ft(a;t)1950 2262 y Fz(,)h(w)m(e)g(ha)m(v)m(e) 327 2535 y(0)27 b(=)535 2467 y Fu(d)p 517 2512 86 4 v 517 2603 a(dt)629 2535 y Fz(\()p Fv(W)783 2550 y Ft(a;t)870 2535 y Fz(\()p Fv(V)993 2550 y Ft(a;t)1079 2535 y Fz(\()p Fv(x)p Fz(\)\)\))h(=)1431 2467 y Fu(d)p Fv(W)1598 2482 y Ft(a;t)p 1431 2512 254 4 v 1515 2603 a Fu(dt)1711 2535 y Fz(\()p Fv(V)1834 2550 y Ft(a;t)1920 2535 y Fz(\()p Fv(x)p Fz(\)\))22 b(+)g([)p Fv(W)2358 2494 y Fm(0)2381 2550 y Ft(a;t)2468 2535 y Fz(\()p Fv(V)2591 2550 y Ft(a;t)2677 2535 y Fz(\()p Fv(x)p Fz(\)\)])p Fv(F)2947 2550 y Ft(a)2989 2535 y Fz(\()p Fv(V)3112 2550 y Ft(a;t)3199 2535 y Fz(\()p Fv(x)p Fz(\)\))p Fu(;)177 b Fz(\(3.92\))150 2786 y(where)34 b Fv(W)550 2750 y Fm(0)572 2801 y Ft(a;t)692 2786 y Fz(denotes)f(the)g (total)f(deriv)-5 b(ativ)m(e)32 b(of)g(the)h(v)m(ector)g(v)-5 b(alued)33 b(function)f Fv(y)d Fr(7!)e Fv(W)3484 2801 y Ft(a;t)3571 2786 y Fz(\()p Fv(y)r Fz(\).)296 2906 y(W)-8 b(e)29 b(shall)d(pro)m(v)m(e)j(the)g(existence)g(of)e(the)i(limit)24 b(\(1.15\))j(b)m(y)i(a)f(Co)s(ok)g(t)m(yp)s(e)g(argumen)m(t,)h(that)f (is)f(b)m(y)150 3027 y(sho)m(wing)j(that)g(the)g(time)f(deriv)-5 b(ativ)m(e)29 b(is)g(in)g Fu(L)1829 2990 y Fs(1)1869 3027 y Fz(\(\(1)p Fu(;)17 b Fr(1)p Fz(\)\).)42 b(Hence)31 b(w)m(e)g(c)m(ho)s(ose)f(an)g(initial)c(condition)150 3147 y Fv(y)34 b Fz(for)e(the)h(true)g(orbit)f(at)g(energy)i Fu(E)f(>)28 b(E)1717 3162 y Ft(d)1790 3147 y Fz(and)33 b(compute)467 3343 y Fu(d)p 449 3388 86 4 v 449 3479 a(dt)562 3411 y Fz(\(\005)p Fv(W)789 3426 y Ft(a;t)875 3411 y Fz(\()p Fv(V)998 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Fz(0)k(for)f(large)g Fu(t)p Fz(.)48 b(According)34 b(to)150 967 y(\(3.18\))j(and)i(\(3.28\),)f Fu(L)p Fz(\()p Fu(t)p Fz(\))g Fr(!)f Fz(0)g(as)i Fu(t)e Fr(!)g(1)p Fz(,)i(so)f(in)m(tegrating)f(\(3.103\))g(giv)m(es)h Fu(L)p Fz(\()p Fu(t)p Fz(\))g(=)f Fr(O)s Fz(\()p Fu(t)3577 931 y Fm(\000)p Fs(2+)p Ft(\016)3760 967 y Fz(\).)150 1088 y(Then)d(in)m(tegrating)d(\(3.103\))g(again)g(with)i(this)f(new)h (information)d(giv)m(es)j(\(3.99\).)296 1208 y(T)-8 b(o)43 b(bridge)f(the)i(gap)e(b)s(et)m(w)m(een)j(\()p Fu(\032)1606 1223 y Ft(E)1666 1208 y Fu(;)17 b(\030)1753 1223 y Ft(E)1812 1208 y Fz(\))42 b(and)h(\()p Fu(\032)2180 1223 y Ft(E)2232 1231 y Fq(t)2264 1208 y Fu(;)17 b(\030)2351 1223 y Ft(E)2403 1231 y Fq(t)2434 1208 y Fz(\))43 b(\(remem)m(b)s(er)f(our)h(notation)e Fu(E)3674 1223 y Ft(t)3749 1208 y Fz(=)150 1328 y Fr(\000)p Fu(@)278 1343 y Ft(t)309 1328 y Fu(S)6 b Fz(\()p Fu(t;)17 b(r)s Fz(\()p Fu(t)p Fz(\))p Fu(;)g(\022)s Fz(\()p Fu(t)p Fz(\)\)\),)41 b(w)m(e)h(mak)m(e)e(a)h(T)-8 b(a)m(ylor)40 b(expansion)h(and)f(k)m(eep)i(only)e(terms)h(up)f(to)h(the)f(\014rst) 150 1449 y(order)33 b(in)f Fu(E)c Fr(\000)22 b Fu(E)790 1464 y Ft(t)820 1449 y Fz(.)44 b(Th)m(us)34 b(w)m(e)f(de\014ne)825 1669 y(^)-53 b Fu(\015)872 1684 y Fs(1)911 1669 y Fz(\()p Fu(t)p Fz(\))83 b(:=)g Fu(\032)p Fz(\()p Fu(t)p Fz(\))23 b Fr(\000)g(f)o Fu(\032)1674 1684 y Ft(E)1726 1692 y Fq(t)1759 1669 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))f(+)g Fu(@)2165 1684 y Ft(E)2225 1669 y Fu(\032)2275 1684 y Ft(E)2327 1692 y Fq(t)2359 1669 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))g Fr(\001)g Fz(\()p Fu(E)28 b Fr(\000)23 b Fu(E)2976 1684 y Ft(t)3006 1669 y Fz(\))p Fr(g)16 b Fu(;)825 1814 y Fz(^)-53 b Fu(\015)872 1829 y Fs(2)911 1814 y Fz(\()p Fu(t)p Fz(\))83 b(:=)g Fu(\030)5 b Fz(\()p Fu(t)p Fz(\))22 b Fr(\000)h(f)p Fu(\030)1665 1829 y Ft(E)1717 1837 y Fq(t)1748 1814 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))f(+)g Fu(@)2154 1829 y Ft(E)2214 1814 y Fu(\030)2257 1829 y Ft(E)2309 1837 y Fq(t)2341 1814 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))g Fr(\001)g Fz(\()p Fu(E)28 b Fr(\000)23 b Fu(E)2958 1829 y Ft(t)2987 1814 y Fz(\))p Fr(g)17 b Fu(:)408 b Fz(\(3.104\))150 2034 y(W)-8 b(e)30 b(will)d(also)i(need)i(to)e(sho)m(w)i(that)e Fu(E)22 b Fr(\000)16 b Fu(E)1736 2049 y Ft(t)1796 2034 y Fz(is)29 b(small)e(so)j(w)m(e)h(in)m(tro)s(duce)e(a)h(third)e Fu(\015)35 b Fz(and)30 b(a)f(third)k(^)-53 b Fu(\015)5 b Fz(:)981 2254 y Fu(\015)1032 2269 y Fs(3)1071 2254 y Fz(\()p Fu(t)p Fz(\))28 b(=)j(^)-52 b Fu(\015)1365 2269 y Fs(3)1404 2254 y Fz(\()p Fu(t)p Fz(\))27 b(:=)h Fu(E)g Fz(+)22 b Fu(@)1922 2269 y Ft(t)1952 2254 y Fu(S)6 b Fz(\()p Fu(t;)17 b(r)s Fz(\()p Fu(t)p Fz(\))p Fu(;)g(\022)s Fz(\()p Fu(t)p Fz(\)\))27 b(=)h Fu(E)g Fr(\000)23 b Fu(E)2937 2269 y Ft(t)2967 2254 y Fu(:)533 b Fz(\(3.105\))150 2474 y(W)-8 b(e)33 b(can)g(then)g(rewrite)g(\(3.104\))e(as)333 2694 y(^)-53 b Fu(\015)380 2709 y Fs(1)419 2694 y Fz(\()p Fu(t)p Fz(\))28 b(=)g Fu(\015)713 2709 y Fs(1)752 2694 y Fz(\()p Fu(t)p Fz(\))22 b 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Fu(\015)1226 3390 y Ft(j)1262 3375 y Fz(\()p Fu(t)p Fz(\))p Fr(j)p Fu(;)44 b Fr(j)p Fu(\015)1551 3390 y Ft(k)1593 3375 y Fz(\()p Fu(t)p Fz(\))p Fr(j)28 b Fz(:)55 b Fu(j;)17 b(k)31 b Fr(2)d(f)p Fz(1)p Fu(;)17 b Fz(2)p Fu(;)g Fz(3)p Fr(gg)26 b(\024)i Fu(\017;)115 b(t)28 b Fr(\025)g Fu(T)8 b(:)468 b Fz(\(3.107\))296 3595 y(W)-8 b(e)33 b(split)f(the)h(pro)s(of)e(of)h(Lemma)g(3.14)g(in)m (to)f(sev)m(eral)j(pieces:)150 3715 y Fv(I)p Fz(.)28 b(W)-8 b(e)29 b(start)f(with)g(a)g(few)h(constrain)m(ts)f(w)m(e)i(ha)m (v)m(e)f(on)f(the)h Fu(\015)5 b Fz('s)29 b(coming)e(from)g(energy)i (conserv)-5 b(ation.)150 3836 y(The)34 b(\014rst)f(one)f(is)200 4056 y(2)p Fu(E)h Fz(=)28 b Fu(\032)508 4014 y Fs(2)548 4056 y Fz(\()p Fu(t)p Fz(\))s(+)s Fu(\030)789 4014 y Fs(2)828 4056 y Fz(\()p Fu(t)p Fz(\))g(=)g(\()p Fu(\015)1160 4071 y Fs(1)1202 4056 y Fz(+)s Fu(\032)1331 4071 y Ft(E)1383 4079 y Fq(t)1415 4056 y Fz(\))1453 4014 y Fs(2)1496 4056 y Fz(+)s(\()p Fu(\015)1664 4071 y Fs(2)1707 4056 y Fz(+)s Fu(\030)1829 4071 y Ft(E)1881 4079 y Fq(t)1913 4056 y Fz(\))1951 4014 y Fs(2)2018 4056 y Fz(=)f(2)p Fu(E)2242 4071 y Ft(t)2275 4056 y Fz(+)s(2)p Fu(\015)2454 4071 y Fs(1)2497 4056 y Fr(\001)s Fu(\032)2578 4071 y Ft(E)2630 4079 y Fq(t)2666 4056 y Fz(+)s(2)p Fu(\015)2845 4071 y Fs(2)2887 4056 y Fr(\001)s Fu(\030)2961 4071 y Ft(E)3013 4079 y Fq(t)3048 4056 y Fz(+)s(\()p Fu(\015)3221 4014 y Fs(2)3216 4080 y(1)3264 4056 y Fz(+)s Fu(\015)3399 4014 y Fs(2)3394 4080 y(2)3439 4056 y Fz(\))50 b(\(3.108\))150 4276 y(or)32 b(equiv)-5 b(alen)m(tly)560 4496 y(2)p Fu(\015)660 4511 y Fs(3)699 4496 y Fz(\()p Fu(t)p Fz(\))28 b(=)f(2)p Fu(\015)1041 4511 y Fs(1)1080 4496 y Fz(\()p Fu(t)p Fz(\))c Fr(\001)e Fu(\032)1313 4511 y Ft(E)1365 4519 y Fq(t)1397 4496 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))i(+)f(2)p Fu(\015)1853 4511 y Fs(2)1892 4496 y Fz(\()p Fu(t)p Fz(\))g Fr(\001)g Fu(\030)2118 4511 y Ft(E)2170 4519 y Fq(t)2201 4496 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))g(+)g Fu(\015)2612 4455 y Fs(2)2607 4520 y(1)2652 4496 y Fz(\()p Fu(t)p Fz(\))g(+)g Fu(\015)2939 4455 y Fs(2)2934 4520 y(2)2978 4496 y Fz(\()p Fu(t)p Fz(\))p Fu(:)411 b Fz(\(3.109\))296 4716 y(Rewriting)46 b(\(3.109\))h(with)j(^)-52 b Fu(\015)5 b Fz('s)48 b(w)m(e)g(get)g(a)f(linear)e(dep)s(endence)50 b(\(up)e(to)f(quadratic)g(terms\))150 4836 y(b)s(et)m(w)m(een)i(^)-53 b Fu(\015)588 4851 y Fs(1)670 4836 y Fz(and)47 b(^)-53 b Fu(\015)921 4851 y Fs(2)960 4836 y Fz(,)46 b(similar)40 b(to)j(\(3.100\);)k(w)m(e)d(again)e(emplo)m(y)g(the)i(iden)m(tit)m(y)f (1)29 b Fr(\000)h Fu(\032)3423 4851 y Ft(E)3483 4836 y Fu(@)3534 4851 y Ft(E)3594 4836 y Fu(\032)3644 4851 y Ft(E)3749 4836 y Fz(=)150 4956 y Fu(\030)193 4971 y Ft(E)252 4956 y Fu(@)303 4971 y Ft(E)363 4956 y Fu(\030)406 4971 y Ft(E)465 4956 y Fz(:)1034 5077 y(2)t(^)-53 b Fu(\015)1134 5092 y Fs(1)1173 5077 y Fz(\()p Fu(t)p Fz(\))22 b Fr(\001)g Fu(\032)1406 5092 y Ft(E)1458 5100 y Fq(t)1490 5077 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))h(+)f(2)t(^)-53 b Fu(\015)1946 5092 y Fs(2)1984 5077 y Fz(\()p Fu(t)p Fz(\))23 b Fr(\001)e Fu(\030)2210 5092 y Ft(E)2262 5100 y Fq(t)2294 5077 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))28 b(=)f Fr(O)s Fz(\()t(^)-53 b Fu(\015)2836 5036 y Fs(2)2876 5077 y Fz(\))p Fu(:)586 b Fz(\(3.110\))150 5251 y Fv(I)s(I)p Fz(.)32 b(W)-8 b(e)33 b(con)m(tin)m(ue)g(with)g(the)g(time)e(deriv)-5 b(ativ)m(e)32 b(of)k(^)-53 b Fu(\015)2076 5266 y Fs(3)2115 5251 y Fz(.)43 b(The)34 b(k)m(ey)g(equation)e(is)1561 5471 y Fu(t=r)s Fz(\()p Fu(t)p Fz(\))27 b(=)h Fu(@)1985 5486 y Ft(E)2045 5471 y Fu(\032)2095 5486 y Ft(E)2147 5494 y Fq(t)2179 5471 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))1113 b(\(3.111\))1939 5776 y(39)p eop %%Page: 40 43 40 42 bop 150 407 a Fz(P)m(erforming)31 b(the)i(deriv)-5 b(ativ)m(e)32 b(w)m(e)i(get)958 599 y(1)p Fu(=r)24 b Fr(\000)f Fz(\()p Fu(t=r)1393 558 y Fs(2)1432 599 y Fz(\))p Fu(\032)28 b Fz(=)f Fr(\000)p Fz(\()p Fu(@)1822 558 y Fs(2)1817 624 y Ft(E)1878 599 y Fu(\032)1928 614 y Ft(E)1980 622 y Fq(t)2012 599 y Fz(\))22 b Fr(\001)g Fz(\()p Fu(@)2211 614 y Ft(t)2245 599 y Fz(^)-53 b Fu(\015)2292 614 y Fs(3)2331 599 y Fz(\))22 b(+)g(\()p Fu(@)2578 614 y Ft(E)2638 599 y Fu(\030)2681 614 y Ft(E)2733 622 y Fq(t)2764 599 y Fu(=r)s Fz(\))g Fr(\001)g Fu(\030)514 b Fz(\(3.112\))150 792 y(and)33 b(using)f(again)f(1)22 b Fr(\000)h Fu(\032)1076 807 y Ft(E)1136 792 y Fu(@)1187 807 y Ft(E)1247 792 y Fu(\032)1297 807 y Ft(E)1384 792 y Fz(=)28 b Fu(\030)1531 807 y Ft(E)1590 792 y Fu(@)1641 807 y Ft(E)1701 792 y Fu(\030)1744 807 y Ft(E)1836 792 y Fz(w)m(e)33 b(are)g(led)f(to)927 1023 y Fu(@)978 1038 y Ft(t)1011 1023 y Fz(^)-52 b Fu(\015)1059 1038 y Fs(3)1125 1023 y Fz(=)28 b Fu(@)1280 1038 y Ft(t)1310 1023 y Fu(\015)1361 1038 y Fs(3)1428 1023 y Fz(=)1663 956 y(1)p 1541 1001 292 4 v 1541 1092 a Fu(r)s(@)1644 1058 y Fs(2)1639 1119 y Ft(E)1699 1092 y Fu(\032)1749 1107 y Ft(E)1801 1115 y Fq(t)1843 1023 y Fr(f)p Fz(\()p Fu(@)1982 1038 y Ft(E)2042 1023 y Fu(\032)2092 1038 y Ft(E)2144 1046 y Fq(t)2176 1023 y Fz(\))22 b Fr(\001)g Fu(\015)2337 1038 y Fs(1)2398 1023 y Fz(+)g(\()p Fu(@)2585 1038 y Ft(E)2645 1023 y Fu(\030)2688 1038 y Ft(E)2740 1046 y Fq(t)2772 1023 y Fz(\))g Fr(\001)f Fu(\015)2932 1038 y Fs(2)2971 1023 y Fr(g)p Fu(;)479 b Fz(\(3.113\))150 1274 y(and)33 b(\014nally)-8 b(,)31 b(using)h(\(3.111\))g(in)g(order)g (to)h(get)f(rid)g(of)g Fu(r)s Fz(\()p Fu(t)p Fz(\))g(on)h(the)g(righ)m (t)f(hand)g(side:)200 1516 y Fu(@)251 1531 y Ft(t)285 1516 y Fz(^)-53 b Fu(\015)332 1531 y Fs(3)399 1516 y Fz(=)530 1449 y Fu(@)581 1464 y Ft(E)641 1449 y Fu(\032)691 1464 y Ft(E)743 1472 y Fq(t)p 512 1493 281 4 v 512 1584 a Fu(t@)603 1550 y Fs(2)598 1611 y Ft(E)658 1584 y Fu(\032)708 1599 y Ft(E)760 1607 y Fq(t)803 1516 y Fr(f)p Fz(\()p Fu(@)942 1531 y Ft(E)1001 1516 y Fu(\032)1051 1531 y Ft(E)1103 1539 y Fq(t)1135 1516 y Fz(\))22 b Fr(\001)k Fz(^)-53 b Fu(\015)1296 1531 y Fs(1)1357 1516 y Fz(+)22 b(\()p Fu(@)1544 1531 y Ft(E)1604 1516 y Fu(\030)1647 1531 y Ft(E)1699 1539 y Fq(t)1731 1516 y Fz(\))g Fr(\001)j Fz(^)-53 b Fu(\015)1891 1531 y Fs(2)1930 1516 y Fr(g)22 b Fz(+)2128 1449 y Fu(@)2179 1464 y Ft(E)2239 1449 y Fu(\032)2289 1464 y Ft(E)2341 1472 y Fq(t)p 2110 1493 V 2110 1584 a Fu(t@)2201 1550 y Fs(2)2196 1611 y Ft(E)2257 1584 y Fu(\032)2307 1599 y Ft(E)2359 1607 y Fq(t)2401 1516 y Fz([\()p Fu(@)2517 1531 y Ft(E)2577 1516 y Fu(\032)2627 1531 y Ft(E)2679 1539 y Fq(t)2711 1516 y Fz(\))2749 1475 y Fs(2)2810 1516 y Fz(+)g(\()p Fu(@)2997 1531 y Ft(E)3057 1516 y Fu(\030)3100 1531 y Ft(E)3152 1539 y Fq(t)3184 1516 y Fz(\))3222 1475 y Fs(2)3261 1516 y Fz(])g Fr(\001)k Fz(^)-53 b Fu(\015)3411 1531 y Fs(3)3450 1516 y Fu(:)50 b Fz(\(3.114\))150 1761 y Fv(I)s(I)s(I)p Fz(.)31 b(Next)h(comes)g(the)g (time)e(deriv)-5 b(ativ)m(e)32 b(of)i(^)-52 b Fu(\015)1860 1776 y Fs(1)1899 1761 y Fz(.)43 b(Compute)31 b(\014rst)i(the)f(deriv)-5 b(ativ)m(e)31 b(of)g Fu(\015)3365 1776 y Fs(1)3435 1761 y Fz(using)h(the)150 1882 y(Hamilton)e(equations)j(and)f(\(2.6\):)752 2129 y Fu(@)803 2144 y Ft(t)833 2129 y Fu(\015)884 2144 y Fs(1)923 2129 y Fz(\()p Fu(t)p Fz(\))84 b(=)1288 2062 y Fu(\030)5 b Fz(\()p Fu(t)p Fz(\))p 1288 2106 159 4 v 1288 2197 a Fu(r)s Fz(\()p Fu(t)p Fz(\))1456 2129 y([)p Fu(b)p Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))23 b(+)f Fu(\030)5 b Fz(\()p Fu(t)p Fz(\)])1117 2398 y Fr(\000)1288 2330 y Fu(\030)g Fz(\()p Fu(t)p Fz(\))p 1288 2375 V 1288 2466 a Fu(r)s Fz(\()p Fu(t)p Fz(\))1456 2398 y([)p Fu(b)p Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))23 b(+)f Fu(\030)1923 2413 y Ft(E)1975 2421 y Fq(t)2006 2398 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\)])h(+)f Fu(@)2440 2413 y Ft(E)2500 2398 y Fu(\032)2550 2413 y Ft(E)2602 2421 y Fq(t)2634 2398 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))g Fr(\001)g Fu(@)2992 2413 y Ft(t)3022 2398 y Fu(\015)3073 2413 y Fs(3)3112 2398 y Fz(\()p Fu(t)p Fz(\))1118 2666 y(=)1288 2599 y Fu(\030)5 b Fz(\()p Fu(t)p Fz(\))p 1288 2643 V 1288 2735 a Fu(r)s Fz(\()p Fu(t)p Fz(\))1456 2666 y Fu(\015)1507 2681 y Fs(2)1546 2666 y Fz(\()p Fu(t)p Fz(\))22 b(+)g Fu(@)1828 2681 y Ft(E)1888 2666 y Fu(\032)1938 2681 y Ft(E)1990 2689 y Fq(t)2022 2666 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))h Fr(\001)f Fu(@)2381 2681 y Ft(t)2411 2666 y Fu(\015)2462 2681 y Fs(3)2501 2666 y Fz(\()p Fu(t)p Fz(\))p Fu(:)888 b Fz(\(3.115\))150 2915 y(W)-8 b(e)33 b(then)g(obtain)f(\(using)g(\(2.6\),)g(\(1.9\),)g(and)h(\(3.111\)\)) 1085 3107 y Fu(@)1136 3122 y Ft(t)1170 3107 y Fz(^)-52 b Fu(\015)1218 3122 y Fs(1)1340 3107 y Fz(=)84 b Fu(@)1551 3122 y Ft(t)1581 3107 y Fu(\015)1632 3122 y Fs(1)1693 3107 y Fr(\000)23 b Fz(\()p Fu(@)1882 3122 y Ft(E)1942 3107 y Fu(\030)1985 3122 y Ft(E)2037 3130 y Fq(t)2068 3107 y Fz(\))g Fr(\001)e Fz(\()p Fu(\030)5 b(=r)s Fz(\))21 b Fr(\001)h Fu(\015)2520 3122 y Fs(3)1340 3253 y Fr(\000)83 b Fz(\()p Fu(@)1589 3268 y Ft(E)1649 3253 y Fu(\032)1699 3268 y Ft(E)1751 3276 y Fq(t)1783 3253 y Fz(\))22 b Fr(\001)g Fu(@)1944 3268 y Ft(t)1974 3253 y Fu(\015)2025 3268 y Fs(3)2086 3253 y Fz(+)g Fu(\015)2235 3268 y Fs(3)2297 3253 y Fr(\001)f Fz(\()p Fu(@)2440 3211 y Fs(2)2435 3277 y Ft(E)2495 3253 y Fu(\032)2545 3268 y Ft(E)2597 3276 y Fq(t)2630 3253 y Fz(\))p Fu(@)2719 3268 y Ft(t)2749 3253 y Fu(\015)2800 3268 y Fs(3)1340 3398 y Fz(=)84 b(\()p Fu(\030)1581 3413 y Ft(E)1633 3421 y Fq(t)1664 3398 y Fu(=r)s Fz(\))22 b Fr(\001)j Fz(^)-52 b Fu(\015)1921 3413 y Fs(2)1982 3398 y Fz(+)22 b Fr(O)s Fz(\()t(^)-53 b Fu(\015)2256 3357 y Fs(2)2295 3398 y Fu(=t)p Fz(\))1340 3543 y(=)84 b Fr(\000)p Fz([)p Fu(\032)1654 3558 y Ft(E)1706 3566 y Fq(t)1761 3543 y Fr(\001)22 b Fz(\()p Fu(@)1900 3558 y Ft(E)1960 3543 y Fu(\032)2010 3558 y Ft(E)2062 3566 y Fq(t)2094 3543 y Fz(\))p Fu(=t)p Fz(])g Fr(\001)j Fz(^)-52 b Fu(\015)2366 3558 y Fs(1)2427 3543 y Fz(+)22 b Fr(O)s Fz(\()t(^)-53 b Fu(\015)2701 3502 y Fs(2)2741 3543 y Fu(=t)p Fz(\))p Fu(;)637 b Fz(\(3.116\))150 3736 y(where)34 b(the)f(last)f(equalit)m(y)g(came)g(from)f(\(3.110\),)h (\(3.111\),)g(and)h(\(3.114\).)296 3856 y(De\014ne)1455 3996 y Fu(f)1503 4011 y Fs(1)1543 3996 y Fz(\()p Fu(t)p Fz(\))28 b(:=)1920 3929 y(1)p 1822 3973 246 4 v 1822 4064 a Fu(@)1873 4079 y Ft(E)1933 4064 y Fu(\032)1983 4079 y Ft(E)2035 4087 y Fq(t)2077 3996 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))g Fu(>)g Fz(0)p Fu(:)1007 b Fz(\(3.117\))150 4213 y(W)-8 b(e)33 b(see)h(that)e(its)g(time)f(deriv)-5 b(ativ)m(e)32 b(giv)m(es)h(\(use)h(\(3.111\))d(and)i(\(3.114\)\):)821 4405 y Fu(@)872 4420 y Ft(t)902 4405 y Fu(f)950 4420 y Fs(1)1072 4405 y Fz(=)83 b(\()p Fu(@)1325 4364 y Fs(2)1320 4430 y Ft(E)1380 4405 y Fu(\032)1430 4420 y Ft(E)1482 4428 y Fq(t)1514 4405 y Fz(\))p Fu(=)p Fz(\()p Fu(@)1690 4420 y Ft(E)1750 4405 y Fu(\032)1800 4420 y Ft(E)1852 4428 y Fq(t)1884 4405 y Fz(\))1922 4364 y Fs(2)1984 4405 y Fr(\001)21 b Fz(\()p Fu(@)2122 4420 y Ft(t)2156 4405 y Fz(^)-52 b Fu(\015)2204 4420 y Fs(3)2243 4405 y Fz(\))22 b Fr(\000)g Fz(\()p Fu(f)2488 4420 y Fs(1)2528 4405 y Fu(=t)p Fz(\))g Fr(\001)g Fz(\()p Fu(@)2811 4420 y Ft(E)2871 4405 y Fu(\030)2914 4420 y Ft(E)2966 4428 y Fq(t)2997 4405 y Fz(\))g Fr(\001)g Fu(\030)1072 4551 y Fz(=)83 b Fr(\000)p Fu(f)1356 4566 y Fs(1)1418 4551 y Fr(\001)22 b Fz(\(1)g Fr(\000)h Fu(\032)1727 4566 y Ft(E)1779 4574 y Fq(t)1811 4551 y Fu(@)1862 4566 y Ft(E)1922 4551 y Fu(\032)1972 4566 y Ft(E)2024 4574 y Fq(t)2056 4551 y Fz(\))p Fu(=t)f Fz(+)g Fr(O)s Fz(\()t(^)-53 b Fu(\015)5 b(=t)p Fz(\))p Fu(:)904 b Fz(\(3.118\))150 4743 y(Com)m(bining)31 b(this)h(with)g(\(3.116\))g(w)m(e)h(get)1243 4936 y Fu(@)1294 4951 y Ft(t)1324 4936 y Fz(\()p Fu(f)1410 4951 y Fs(1)1453 4936 y Fz(^)-53 b Fu(\015)1500 4951 y Fs(1)1539 4936 y Fz(\))28 b(=)f Fr(\000)p Fz(\(1)p Fu(=t)p Fz(\)\()p Fu(f)2080 4951 y Fs(1)2123 4936 y Fz(^)-52 b Fu(\015)2171 4951 y Fs(1)2210 4936 y Fz(\))22 b(+)g Fr(O)s Fz(\()t(^)-53 b Fu(\015)2544 4895 y Fs(2)2583 4936 y Fu(=t)p Fz(\))p Fu(:)795 b Fz(\(3.119\))150 5129 y Fv(IV)p Fz(.)33 b(No)m(w)i(w)m(e)g (deal)e(with)h(the)g(time)f(deriv)-5 b(ativ)m(e)33 b(of)k(^)-52 b Fu(\015)2142 5144 y Fs(2)2181 5129 y Fz(.)47 b(The)35 b(computations)e(are)h(more)f(in)m(v)m(olv)m(ed)150 5249 y(and)g(w)m(e)g(only)f(giv)m(e)h(the)g(relev)-5 b(an)m(t)32 b(equation:)1153 5485 y Fu(@)1204 5500 y Ft(t)1238 5485 y Fz(^)-52 b Fu(\015)1286 5500 y Fs(2)1352 5485 y Fz(=)28 b Fr(\000)1543 5418 y Fu(b)23 b Fz(+)f Fu(\030)1748 5433 y Ft(E)1800 5441 y Fq(t)p 1543 5462 289 4 v 1664 5554 a Fu(r)1845 5485 y Fz(^)-53 b Fu(\015)1892 5500 y Fs(1)1953 5485 y Fz(+)2061 5418 y Fu(b\032)2152 5433 y Ft(E)2204 5441 y Fq(t)p 2061 5462 176 4 v 2062 5554 a Fu(r)s(\030)2152 5569 y Ft(E)2204 5577 y Fq(t)2251 5485 y Fz(^)g Fu(\015)2298 5500 y Fs(2)2359 5485 y Fz(+)22 b Fr(O)s Fz(\()t(^)-53 b Fu(\015)2633 5444 y Fs(2)2673 5485 y Fu(=t)p Fz(\))p Fu(:)705 b Fz(\(3.120\))1939 5776 y(40)p eop %%Page: 41 44 41 43 bop 150 407 a Fz(W)-8 b(e)44 b(remark)g(that)f(the)i(r.h.s.)78 b(of)43 b(the)i(ab)s(o)m(v)m(e)f(equation)g(con)m(tains)j(^)-52 b Fu(\015)2809 422 y Fs(3)2892 407 y Fz(only)43 b(in)g(the)h(quadratic) 150 527 y(remainder.)f(Then)33 b(notice)1475 605 y Fu(b\032)1566 620 y Ft(E)1618 628 y Fq(t)p 1475 649 176 4 v 1476 740 a Fu(r)s(\030)1566 755 y Ft(E)1618 763 y Fq(t)1688 672 y Fz(=)1802 605 y Fu(b)23 b Fz(+)f Fu(\030)2007 620 y Ft(E)2059 628 y Fq(t)p 1802 649 289 4 v 1859 740 a Fu(r)s(\030)1949 755 y Ft(E)2001 763 y Fq(t)2100 672 y Fu(\032)2150 687 y Ft(E)2202 695 y Fq(t)2256 672 y Fr(\000)2366 605 y Fu(\032)2416 620 y Ft(E)2468 628 y Fq(t)p 2366 649 135 4 v 2409 740 a Fu(r)150 889 y Fz(and)1191 1034 y Fu(@)1242 1049 y Ft(t)1272 1034 y Fz([)p Fu(\030)1342 1049 y Ft(E)1394 1057 y Fq(t)1426 1034 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\)])28 b(=)f Fr(\000)1906 967 y Fu(b)c Fz(+)f Fu(\030)2111 982 y Ft(E)2163 990 y Fq(t)p 1907 1011 289 4 v 2027 1103 a Fu(r)2205 1034 y(\032)2255 1049 y Ft(E)2307 1057 y Fq(t)2361 1034 y Fz(+)g Fr(O)s Fz(\()t(^)-53 b Fu(\015)5 b(=t)p Fz(\))p Fu(:)150 1228 y Fz(W)-8 b(e)33 b(in)m(tro)s(duce)g(the)g (second)g(in)m(tegrating)e(factor)1430 1435 y Fu(f)1478 1450 y Fs(2)1517 1435 y Fz(\()p Fu(t)p Fz(\))d(:=)g(\()p Fu(\030)1868 1450 y Ft(E)1920 1458 y Fq(t)1951 1435 y Fu(=@)2051 1450 y Ft(E)2111 1435 y Fu(\032)2161 1450 y Ft(E)2213 1458 y Fq(t)2245 1435 y Fz(\)\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))p Fu(:)982 b Fz(\(3.121\))150 1642 y(Then)492 1873 y Fu(@)543 1888 y Ft(t)573 1873 y Fz(\()p Fu(f)659 1888 y Fs(2)702 1873 y Fz(^)-52 b Fu(\015)750 1888 y Fs(2)789 1873 y Fz(\))27 b(=)h Fr(\000)1045 1805 y Fu(b)23 b Fz(+)f Fu(\030)1250 1820 y Ft(E)1302 1828 y Fq(t)p 1045 1850 V 1172 1941 a Fu(t)1343 1873 y Fz(\()p Fu(@)1432 1888 y Ft(E)1492 1873 y Fu(\032)1542 1888 y Ft(E)1594 1896 y Fq(t)1627 1873 y Fz(\))p Fu(\030)1708 1888 y Ft(E)1760 1896 y Fq(t)1813 1873 y Fr(\001)g Fz(\()p Fu(f)1949 1888 y Fs(1)1992 1873 y Fz(^)-53 b Fu(\015)2039 1888 y Fs(1)2078 1873 y Fz(\))22 b Fr(\000)h Fz(\(1)p Fu(=t)p Fz(\)\()p Fu(f)2533 1888 y Fs(2)2576 1873 y Fz(^)-53 b Fu(\015)2623 1888 y Fs(2)2662 1873 y Fz(\))22 b(+)g Fr(O)s Fz(\()t(^)-53 b Fu(\015)2996 1832 y Fs(2)3036 1873 y Fu(=t)p Fz(\))p Fu(:)342 b Fz(\(3.122\))296 2111 y(No)m(w)33 b(w)m(e)h(are)f(ready)g(to)f(rewrite)h(\(3.114\))e(in)h(a)g (more)g(con)m(v)m(enien)m(t)j(form.)42 b(De\014ne)1345 2317 y Fu(f)1393 2332 y Fs(3)1433 2317 y Fz(\()p Fu(t)p Fz(\))28 b(=)f Fr(\000)p Fz(\()p Fu(@)1846 2276 y Fs(2)1841 2342 y Ft(E)1902 2317 y Fu(\032)1952 2332 y Ft(E)2004 2340 y Fq(t)2036 2317 y Fu(=@)2136 2332 y Ft(E)2196 2317 y Fu(\032)2246 2332 y Ft(E)2298 2340 y Fq(t)2330 2317 y Fz(\)\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))p Fu(:)897 b Fz(\(3.123\))150 2524 y(Using)48 b(the)h(iden)m(tit)m(y)f(\()p Fu(@)1089 2539 y Ft(E)1149 2524 y Fu(\030)1192 2539 y Ft(E)1244 2547 y Fq(t)1275 2524 y Fz(\))1313 2488 y Fs(2)1385 2524 y Fz(+)33 b(\()p Fu(@)1583 2539 y Ft(E)1643 2524 y Fu(\032)1693 2539 y Ft(E)1745 2547 y Fq(t)1777 2524 y Fz(\))1815 2488 y Fs(2)1909 2524 y Fz(=)54 b Fr(\000)p Fu(\032)2166 2539 y Ft(E)2218 2547 y Fq(t)2251 2524 y Fu(@)2307 2488 y Fs(2)2302 2549 y Ft(E)2362 2524 y Fu(\032)2412 2539 y Ft(E)2464 2547 y Fq(t)2529 2524 y Fr(\000)33 b Fu(\030)2682 2539 y Ft(E)2734 2547 y Fq(t)2765 2524 y Fu(@)2821 2488 y Fs(2)2816 2549 y Ft(E)2877 2524 y Fu(\030)2920 2539 y Ft(E)2972 2547 y Fq(t)3051 2524 y Fz(together)49 b(with)f(the)150 2644 y(\\linear")31 b(dep)s(endence)j(\(3.110\))e(w)m (e)i(obtain)870 2851 y Fu(@)921 2866 y Ft(t)951 2851 y Fz(\()p Fu(f)1037 2866 y Fs(3)1080 2851 y Fz(^)-52 b Fu(\015)1128 2866 y Fs(3)1167 2851 y Fz(\))27 b(=)h([)p Fu(a)1414 2866 y Fs(31)1489 2851 y Fz(\()p Fu(t)p Fz(\))p Fu(=t)p Fz(])22 b Fr(\001)g Fz(\()p Fu(f)1869 2866 y Fs(1)1912 2851 y Fz(^)-53 b Fu(\015)1959 2866 y Fs(1)1999 2851 y Fz(\))22 b Fr(\000)g Fz(\(1)p Fu(=t)p Fz(\)\()p Fu(f)2453 2866 y Fs(3)2496 2851 y Fz(^)-53 b Fu(\015)2543 2866 y Fs(3)2582 2851 y Fz(\))22 b(+)g Fr(O)s Fz(\()t(^)-53 b Fu(\015)2916 2810 y Fs(2)2956 2851 y Fu(=t)p Fz(\))p Fu(;)422 b Fz(\(3.124\))150 3058 y(where)34 b Fu(a)483 3073 y Fs(31)558 3058 y Fz(\()p Fu(t)p Fz(\))e(is)g(a)h(b)s(ounded)g (scalar.)150 3178 y Fv(V)p Fz(.)42 b(W)-8 b(e)43 b(are)g(no)m(w)g (ready)h(to)e(giv)m(e)h(a)f(di\013eren)m(tial)f(inequalit)m(y)g(in)m(v) m(olving)h(all)e(three)47 b(^)-52 b Fu(\015)5 b Fz('s.)74 b(First,)150 3298 y(rewrite)33 b(\(3.119\),)e(\(3.122\))h(and)h (\(3.124\))e(in)h(a)g(more)g(compact)g(form:)768 3595 y Fu(@)819 3610 y Ft(t)850 3595 y Fz(\()p Fu(f)936 3610 y Ft(j)976 3595 y Fz(^)-53 b Fu(\015)1023 3610 y Ft(j)1059 3595 y Fz(\))28 b(=)1283 3471 y Fs(3)1228 3500 y Fl(X)1236 3713 y Ft(k)r Fs(=1)1372 3595 y Fz([)p Fu(a)1450 3610 y Ft(j)t(k)1526 3595 y Fz(\()p Fu(t)p Fz(\))p Fu(=t)p Fz(])22 b Fr(\001)g Fz(\()p Fu(f)1906 3610 y Ft(k)1952 3595 y Fz(^)-53 b Fu(\015)1999 3610 y Ft(k)2042 3595 y Fz(\))22 b(+)g Fr(O)s Fz(\()t(^)-53 b Fu(\015)2376 3554 y Fs(2)2415 3595 y Fu(=t)p Fz(\))p Fu(;)114 b(j)34 b Fr(2)28 b(f)p Fz(1)p Fu(;)17 b Fz(2)p Fu(;)g Fz(3)p Fr(g)p Fu(;)319 b Fz(\(3.125\))150 3892 y(where)34 b Fu(a)483 3907 y Ft(j)t(j)581 3892 y Fz(=)29 b Fr(\000)p Fz(1)34 b(for)e(all)f Fu(j)6 b Fz(,)34 b(and)f Fu(a)1479 3907 y Fs(12)1583 3892 y Fz(=)c Fu(a)1739 3907 y Fs(13)1842 3892 y Fz(=)g Fu(a)1998 3907 y Fs(23)2102 3892 y Fz(=)g Fu(a)2258 3907 y Fs(32)2361 3892 y Fz(=)g(0.)45 b(In)34 b(particular,)e(the)h(matrix)f Fr(f)p Fu(a)p Fr(g)150 4012 y Fz(is)f(lo)m(w)m(er)h(triangular.)41 b(Notice)31 b(also)g(that)h(w)m(e)h(ha)m(v)m(e)g(b)s(een)f(using)g(the)g(fact)f (that)h(when)h(the)f(energy)150 4133 y(is)f(lo)s(calized)d(around)j Fu(E)j(>)28 b(E)1254 4148 y Ft(d)1294 4133 y Fz(,)k(one)f(ma)m(y)g(get) g(upp)s(er)g(and)h(lo)m(w)m(er)f(b)s(ounds)g(for)g Fu(f)3153 4148 y Ft(j)3190 4133 y Fz('s)g(uniform)e(in)i Fu(t)p Fz(;)150 4253 y(there)i(exist)g(upp)s(er)g(b)s(ounds)h(for)e Fu(a)1449 4268 y Ft(j)t(k)1524 4253 y Fz('s)h(to)s(o.)296 4373 y(De\014ne)g(the)g(Liapuno)m(v-t)m(yp)s(e)g(function)1212 4580 y Fu(L)1278 4595 y Ft(C)1365 4580 y Fz(:=)28 b Fu(C)h Fr(\001)21 b Fz(\()p Fu(f)1730 4595 y Fs(1)1773 4580 y Fz(^)-52 b Fu(\015)1821 4595 y Fs(1)1860 4580 y Fz(\))1898 4539 y Fs(2)1959 4580 y Fz(+)22 b(\()p Fu(f)2143 4595 y Fs(2)2187 4580 y Fz(^)-53 b Fu(\015)2234 4595 y Fs(2)2273 4580 y Fz(\))2311 4539 y Fs(2)2372 4580 y Fz(+)22 b(\()p Fu(f)2556 4595 y Fs(3)2600 4580 y Fz(^)-53 b Fu(\015)2647 4595 y Fs(3)2686 4580 y Fz(\))2724 4539 y Fs(2)3527 4580 y Fz(\(3.126\))150 4787 y(where)37 b Fu(C)j(>)32 b Fz(0)k(is)f(a)g(v)m (ery)i(large)e(p)s(ositiv)m(e)g(constan)m(t)h(only)f(dep)s(ending)h(on) g(the)g(energy)-8 b(.)53 b(No)m(w)37 b(let)150 4907 y(us)c(see)h(ho)m (w)f(w)m(e)h(c)m(ho)s(ose)f Fu(C)7 b Fz(.)44 b(Compute)200 5114 y Fu(@)251 5129 y Ft(t)281 5114 y Fu(L)347 5129 y Ft(C)434 5114 y Fz(=)28 b Fr(\000)p Fz(\(2)p Fu(=t)p Fz(\))p Fu(L)890 5129 y Ft(C)958 5114 y Fz(+)9 b(\(2)p Fu(=t)p Fz(\)[)p Fu(a)1330 5129 y Fs(21)1414 5114 y Fr(\001)g Fz(\()p Fu(f)1537 5129 y Fs(2)1579 5114 y Fz(^)-52 b Fu(\015)1627 5129 y Fs(2)1666 5114 y Fz(\)\()p Fu(f)1790 5129 y Fs(1)1833 5114 y Fz(^)f Fu(\015)1880 5129 y Fs(1)1919 5114 y Fz(\)])9 b(+)g(\(2)p Fu(=t)p Fz(\)[)p Fu(a)2365 5129 y Fs(31)2448 5114 y Fr(\001)g Fz(\()p Fu(f)2571 5129 y Fs(3)2614 5114 y Fz(^)-53 b Fu(\015)2661 5129 y Fs(3)2700 5114 y Fz(\)\()p Fu(f)2824 5129 y Fs(1)2867 5114 y Fz(^)h Fu(\015)2915 5129 y Fs(1)2954 5114 y Fz(\)])9 b(+)g Fr(O)s Fz(\()t(^)-53 b Fu(\015)3289 5073 y Fs(3)3328 5114 y Fu(=t)p Fz(\))p Fu(:)50 b Fz(\(3.127\))150 5321 y(W)-8 b(e)33 b(see)h(that)e(the)h(cross)h(terms)e(can)h(b)s(e)g(b)s (ounded)g(b)m(y)476 5527 y(2)p Fr(j)p Fz(\()p Fu(f)639 5542 y Ft(j)679 5527 y Fz(^)-53 b Fu(\015)726 5542 y Ft(j)762 5527 y Fz(\)\()p Fu(f)886 5542 y Fs(1)929 5527 y Fz(^)h Fu(\015)977 5542 y Fs(1)1016 5527 y Fz(\))p Fr(j)27 b(\024)h Fz(\(1)p Fu(=)1350 5438 y Fr(p)p 1433 5438 77 4 v 89 x Fu(C)7 b Fz(\)[\()p Fu(f)1661 5542 y Ft(j)1701 5527 y Fz(^)-53 b Fu(\015)1748 5542 y Ft(j)1784 5527 y Fz(\))1822 5486 y Fs(2)1884 5527 y Fz(+)22 b Fu(C)7 b Fz(\()p Fu(f)2145 5542 y Fs(1)2188 5527 y Fz(^)-53 b Fu(\015)2235 5542 y Fs(1)2274 5527 y Fz(\))2312 5486 y Fs(2)2352 5527 y Fz(])27 b Fr(\024)i Fz(\()p Fu(L)2616 5542 y Ft(C)2675 5527 y Fu(=)2724 5438 y Fr(p)p 2807 5438 V 89 x Fu(C)7 b Fz(\))p Fu(;)114 b(j)34 b Fr(2)28 b(f)p Fz(2)p Fu(;)17 b Fz(3)p Fr(g)p Fu(:)1939 5776 y Fz(41)p eop %%Page: 42 45 42 44 bop 150 407 a Fz(W)-8 b(e)37 b(conclude)f(that)g(for)g(ev)m(ery)i Fu(\016)f(>)d Fz(0,)j(w)m(e)g(can)f(c)m(ho)s(ose)h Fu(C)7 b Fz(\()p Fu(\016)t Fz(\))36 b(su\016cien)m(tly)h(large)e(suc)m(h)j (that)d(for)150 527 y(and)e(for)f(some)g(large)g(constan)m(t)h Fu(K)1448 542 y Ft(\016)1518 527 y Fz(\(w)m(e)h(write)e Fu(L)2015 542 y Ft(\016)2086 527 y Fz(instead)h(of)f Fu(L)2600 542 y Ft(C)2650 554 y Fq(\016)2689 527 y Fz(\):)1306 776 y Fu(@)1357 791 y Ft(t)1387 776 y Fu(L)1453 791 y Ft(\016)1519 776 y Fr(\024)c(\000)1711 709 y Fz(2)22 b Fr(\000)h Fu(\016)t(=)p Fz(2)p 1711 753 316 4 v 1851 844 a Fu(t)2036 776 y(L)2102 791 y Ft(\016)2163 776 y Fz(+)f Fu(K)2344 791 y Ft(\016)2382 776 y Fu(L)2448 725 y Fs(3)p Ft(=)p Fs(2)2448 804 y Ft(\016)2558 776 y Fu(=t:)858 b Fz(\(3.128\))150 999 y(Moreo)m(v)m(er,)32 b(due)e(to)e(the)i(a)f (priori)e(smallness)i(of)g(the)k(^)-53 b Fu(\015)5 b Fz('s)30 b(\(see)h(\(3.107\)\))d(w)m(e)i(get)f(that)h(giv)m(en)f Fu(\016)j(>)27 b Fz(0,)150 1120 y(there)33 b(is)f Fu(T)47 b Fz(large)31 b(enough)i(suc)m(h)h(that:)1499 1312 y Fu(K)1582 1327 y Ft(\016)1620 1312 y Fu(L)1686 1261 y Fs(1)p Ft(=)p Fs(2)1686 1339 y Ft(\016)1824 1312 y Fr(\024)28 b Fu(\016)t(=)p Fz(2)p Fu(;)114 b(t)28 b Fr(\025)g Fu(T)8 b(:)150 1504 y Fz(Com)m(bining)24 b(this)i(with)g(\(3.128\))f(w)m(e)i (\014nally)e(obtain)g(that)h(for)g(ev)m(ery)i Fu(\016)j(>)d Fz(0)e(there)h(is)e Fu(T)40 b Fz(su\016cien)m(tly)150 1624 y(large)31 b(suc)m(h)j(that)1307 1788 y Fu(@)1358 1803 y Ft(t)1388 1788 y Fu(L)1454 1803 y Ft(\016)1492 1788 y Fz(\()p Fu(t)p Fz(\))28 b Fr(\024)g(\000)1823 1721 y Fz(2)23 b Fr(\000)f Fu(\016)p 1823 1765 218 4 v 1915 1857 a(t)2051 1788 y(L)2117 1803 y Ft(\016)2155 1788 y Fz(\()p Fu(t)p Fz(\))p Fu(;)115 b(t)28 b Fr(\025)g Fu(T)8 b(:)859 b Fz(\(3.129\))296 1982 y(W)-8 b(e)34 b(are)g(\014nally)e(in)h(p)s(osition)e(to)i(end)i(the)e(pro)s(of)g(of)g (Lemma)f(3.14.)46 b(Indeed,)35 b(\(3.129\))d(implies)150 2102 y(that)g Fu(t)396 2066 y Fs(2)p Fm(\000)p Ft(\016)525 2102 y Fu(L)591 2117 y Ft(\016)629 2102 y Fz(\()p Fu(t)p Fz(\))g(decreases)j(if)d Fu(t)c Fr(\025)g Fu(T)14 b Fz(,)32 b(hence)1065 2294 y Fr(j)t Fz(^)-53 b Fu(\015)1144 2309 y Ft(j)1179 2294 y Fr(j)p Fz(\()p Fu(t)p Fz(\))28 b Fr(\024)g Fz(Const)23 b Fr(\001)f Fu(t)1808 2253 y Ft(\016)r Fm(\000)p Fs(1)1936 2294 y Fu(;)114 b(j)34 b Fr(2)28 b(f)p Fz(1)p Fu(;)17 b Fz(2)p Fu(;)g Fz(3)p Fr(g)p Fu(;)43 b(t)28 b Fr(\025)g Fu(T)8 b(;)150 2486 y Fz(and)33 b(b)m(y)g(in)m(tro)s (ducing)e(this)i(in)e(\(3.106\))h(and)h(the)g(pro)s(of)e(is)i (complete.)p 3754 2486 4 66 v 3758 2424 59 4 v 3758 2486 V 3816 2486 4 66 v 150 2933 a Fw(4)161 b(The)53 b(main)i(result)150 3152 y Fz(F)-8 b(rom)31 b(no)m(w)j(on)e(w)m(e)i(deal)e(with)h(quan)m (tum)g(mec)m(hanics.)44 b(This)33 b(section)g(con)m(tains)g(the)g(form) m(ulation)150 3273 y(of)f(our)h(main)d(theorem.)150 3557 y Fp(4.1)135 b(A)45 b(Mourre)f(estimate)j(ab)t(o)l(v)l(e)f Fi(E)2096 3575 y Fh(d)150 3742 y Fz(W)-8 b(e)42 b(kno)m(w)g(from)e (\(1.20\))h(and)g(\(1.21\))f(that)h(the)h(generator)f(of)g(dilations)e (is)i(a)g(go)s(o)s(d)f(conjugate)150 3862 y(op)s(erator)g(for)g(states) h(with)f(energy)h(lo)s(calization)c(ab)s(o)m(v)m(e)k Fu(E)2424 3877 y Ft(b)2458 3862 y Fz(.)67 b(W)-8 b(e)41 b(ha)m(v)m(e)h(already)e(encoun)m(tered)150 3983 y(this)f(in)g(the)h (classical)f(case)h(\(see)h(\(2.2\)\).)64 b(No)m(w)41 b(the)f(natural)e(question)j(is)e(whether)i(or)e(not)h(w)m(e)150 4103 y(can)f(giv)m(e)f(a)g(quan)m(tum)h(coun)m(terpart)g(to)g(the)f (quan)m(tit)m(y)i(w)m(e)f(de\014ned)h(in)e(\(2.33\).)60 b(The)40 b(answ)m(er)f(is)150 4224 y(p)s(ositiv)m(e)32 b(and)h(stated)g(in)f(what)g(follo)m(ws.)296 4344 y(W)-8 b(e)45 b(in)m(tro)s(duce)p 921 4289 47 4 v 44 w Fu(r)968 4359 y Fs(1)1007 4344 y Fz(\()p Fv(x)p Fz(\))j(:=)f Fu(F)1417 4293 y Fs(1)p Ft(=)p Fs(2)p Ft(;)p Fs(1)1403 4366 y(+)1582 4344 y Fz(\()p Fr(j)p Fv(x)p Fr(j)p Fz(\))29 b Fr(\001)h(j)p Fv(x)p Fr(j)p Fz(.)78 b(F)-8 b(or)43 b(ev)m(ery)j Fu(C)55 b(>)47 b Fz(0)d(and)g Fu(E)54 b(>)47 b(E)3398 4359 y Ft(d)3483 4344 y Fz(de\014ne)e(a)150 4464 y(\\rotated")32 b(generator)h(of)f(dilations:)589 4699 y Fu(A)662 4714 y Ft(C)721 4699 y Fz(\()p Fu(E)6 b Fz(\))28 b(:=)1043 4632 y Fu(C)p 1043 4676 77 4 v 1057 4768 a Fz(2)1130 4699 y Fu(e)1175 4658 y Fm(\000)p Ft(i)p 1254 4621 34 3 v(r)1288 4667 y Fj(1)1343 4658 y Fs([)p Ft(@)1400 4669 y Fq(E)1451 4658 y Ft(\032)1487 4669 y Fq(E)1539 4658 y Fs(\()p Ft(\022)r Fs(\))p Ft(=C)5 b Fm(\000)p Ft(\032)1809 4669 y Fq(E)1861 4658 y Fs(\()p Ft(\022)r Fs(\)])1992 4699 y Fz([)p Fv(p)22 b Fr(\001)g Fv(x)g Fz(+)g Fv(x)g Fr(\001)g Fv(p)p Fz(])17 b Fu(e)2614 4658 y Ft(i)p 2638 4621 V(r)2672 4667 y Fj(1)2726 4658 y Fs([)p Ft(@)2783 4669 y Fq(E)2835 4658 y Ft(\032)2871 4669 y Fq(E)2923 4658 y Fs(\()p Ft(\022)r Fs(\))p Ft(=C)5 b Fm(\000)p Ft(\032)3193 4669 y Fq(E)3245 4658 y Fs(\()p Ft(\022)r Fs(\)])3359 4699 y Fu(;)238 b Fz(\(4.1\))150 4922 y(where)34 b Fu(\032)482 4937 y Ft(E)574 4922 y Fz(is)e(the)h(p)s(erio)s(dic)e (solution)g(giv)m(en)i(b)m(y)h(Corollary)c(\(2.7\).)150 5102 y Fv(Prop)s(osition)52 b(4.1.)d FA(Fix)e Fu(E)58 b(>)52 b(E)1511 5117 y Ft(d)1552 5102 y FA(.)83 b(Then)48 b(for)f(every)h(smal)5 b(l)47 b(enough)h Fu(\017)k(>)g Fz(0)p FA(,)e(ther)-5 b(e)48 b(exists)150 5222 y Fu(C)35 b Fz(=)27 b Fu(C)7 b Fz(\()p Fu(\017;)17 b(E)6 b Fz(\))35 b FA(lar)-5 b(ge)34 b(enough)h(and)f(a)h(c)-5 b(omp)g(act)34 b(op)-5 b(er)g(ator)34 b Fu(K)42 b FA(such)35 b(that)615 5459 y Fv(1)671 5474 y Fs([)p Ft(E)t Fm(\000)p Ft(\017;E)t Fs(+)p Ft(\017)p Fs(])1012 5459 y Fz(\()p Fu(H)8 b Fz(\))27 b Fu(i)p Fz([)p Fu(H)r(;)17 b(A)1464 5474 y Ft(C)1523 5459 y Fz(\()p Fu(E)6 b Fz(\)])28 b Fv(1)1788 5474 y Fs([)p Ft(E)t Fm(\000)p Ft(\017;E)t Fs(+)p Ft(\017)p Fs(])2129 5459 y Fz(\()p Fu(H)8 b Fz(\))27 b Fr(\025)2436 5391 y Fz(1)p 2436 5436 49 4 v 2436 5527 a(2)2517 5459 y Fr(\001)22 b Fv(1)2623 5474 y Fs([)p Ft(E)t Fm(\000)p Ft(\017;E)t Fs(+)p Ft(\017)p Fs(])2963 5459 y Fz(\()p Fu(H)8 b Fz(\))22 b(+)g Fu(K)r(:)1939 5776 y Fz(42)p eop %%Page: 43 46 43 45 bop 150 407 a Fv(Pro)s(of)p Fz(.)65 b(W)-8 b(e)41 b(can)f(compute)g(the)g(comm)m(utator)e(b)s(et)m(w)m(een)k Fu(H)48 b Fz(and)40 b Fu(A)2794 422 y Ft(C)2853 407 y Fz(\()p Fu(E)6 b Fz(\))40 b(b)m(y)h(reading)e(o\013)h(the)150 527 y(classical)22 b(computations)h(w)m(e)i(ha)m(v)m(e)h(done)e(in)g (\(2.35\){\(2.37\))e(and)i(making)f(ev)m(erything)i(symmetric.)296 648 y(W)-8 b(e)43 b(\014rst)g(need)g(a)f(few)h(de\014nitions.)72 b(F)-8 b(or)42 b Fu(a)i Fz(=)g(1)p Fu(=)p Fz(2)e(and)g Fu(b)j Fz(=)f(1)e(w)m(e)h(de\014ne)h(a)e(regularized)150 768 y(\\mo)s(dulus")31 b(b)m(y:)1138 955 y(\026)-53 b Fu(r)r Fz(\()p Fv(x)p Fz(\))28 b(=)1447 819 y Fl(Z)1546 845 y Fm(j)p Fk(x)p Fm(j)1502 1045 y Fs(0)1649 955 y Fu(F)1712 970 y Fs(+)1771 955 y Fz(\()p Fu(s)p Fz(\))p Fu(ds)22 b Fz(+)2110 819 y Fl(Z)2209 845 y Fs(1)2165 1045 y(0)2249 955 y Fz([1)g Fr(\000)h Fu(F)2510 970 y Fs(+)2569 955 y Fz(\()p Fu(s)p Fz(\)])p Fu(ds;)782 b Fz(\(4.2\))150 1165 y(\(notice)32 b(that)h(for)f Fr(j)p Fv(x)p Fr(j)27 b(\025)h Fz(1)k(w)m(e)i(ha)m(v)m(e)k(\026)-53 b Fu(r)r Fz(\()p Fv(x)p Fz(\))28 b(=)g Fr(j)p Fv(x)p Fr(j)p Fz(\).)296 1286 y(The)34 b(radial)c(v)m(elo)s(cit)m(y)j Fu(\032)f Fz(and)h(its)f(b)s(ounded)h(v)m(ersion)43 b(~)-58 b Fu(\032)32 b Fz(are)h(giv)m(en)g(b)m(y)1133 1546 y Fu(\032)28 b Fz(:=)1352 1478 y(1)p 1352 1523 49 4 v 1352 1614 a(2)1411 1546 y Fr(f)p Fz(\()p Fv(p)22 b Fr(\000)h Fv(a)p Fz(\))f Fr(\001)g Fz(\()p Fr(r)t Fz(\026)-53 b Fu(r)s Fz(\))22 b(+)g(\()p Fr(r)t Fz(\026)-53 b Fu(r)r Fz(\))22 b Fr(\001)g Fz(\()p Fv(p)h Fr(\000)f Fv(a)p Fz(\))p Fr(g)p Fu(:)783 b Fz(\(4.3\))150 1739 y(It)33 b(is)f(easy)h(to)g(see)g(that)g Fu(\032)f Fz(is)h(essen)m(tially)f (self-adjoin)m(t)f(on)h Fu(C)2419 1703 y Fm(1)2412 1764 y Fs(0)2494 1739 y Fz(\()p Fn(R)2598 1703 y Fs(2)2643 1739 y Fz(\).)296 1859 y(Using)h(p)s(olar)e(co)s(ordinates)h(in)f Fu(L)1520 1823 y Fs(2)1560 1859 y Fz(\()p Fn(R)1664 1823 y Fs(2)1710 1859 y Fz(\))h(and)h(\(4.2\),)f(w)m(e)h(ha)m(v)m(e)903 2098 y Fu(\032)83 b Fz(=)g Fr(\000)1290 2031 y Fu(i)p 1282 2075 V 1282 2166 a Fz(2)1358 1958 y Fl(\032)1466 2031 y Fu(@)p 1442 2075 104 4 v 1442 2166 a(@)5 b(r)1578 2098 y Fr(\001)22 b Fu(F)1691 2113 y Fs(+)1750 2098 y Fz(\()p Fu(r)s Fz(\))g(+)g Fu(F)2056 2113 y Fs(+)2115 2098 y Fz(\()p Fu(r)s Fz(\))g Fr(\001)2343 2031 y Fu(@)p 2319 2075 V 2319 2166 a(@)5 b(r)2455 2098 y Fz(+)2563 2031 y(1)p 2563 2075 49 4 v 2564 2166 a Fu(r)2622 2098 y(F)2685 2113 y Fs(+)2744 2098 y Fz(\()p Fu(r)s Fz(\))2867 1958 y Fl(\033)2969 2098 y Fz(=)1036 2370 y(=)83 b Fr(\000)p Fu(i)1305 2282 y Fl(p)p 1405 2282 123 4 v 88 x Fu(F)1468 2385 y Fs(+)1544 2230 y Fl(\022)1651 2303 y Fu(@)p 1627 2348 104 4 v 1627 2439 a(@)5 b(r)1763 2370 y Fz(+)1894 2303 y(1)p 1871 2348 96 4 v 1871 2439 a(2)p Fu(r)1976 2230 y Fl(\023)2066 2282 y(p)p 2166 2282 123 4 v 88 x Fu(F)2229 2385 y Fs(+)2288 2370 y Fz(;)45 b Fu(r)30 b Fz(=)d Fr(j)p Fv(x)p Fr(j)p Fu(:)945 b Fz(\(4.4\))296 2614 y(As)33 b(an)g(op)s(erator)f(on)g Fu(L)1170 2577 y Fs(2)1210 2614 y Fz(\()p Fn(R)1314 2629 y Fs(+)1401 2614 y Fr(\002)23 b Fn(T)p Fz(\))36 b(the)d(radial)d(momen)m(tum)h(tak) m(es)j(the)f(form:)1561 2804 y Fu(\032)28 b Fz(=)g Fr(\000)p Fu(i)1853 2716 y Fl(p)p 1953 2716 V 88 x Fu(F)2016 2819 y Fs(+)2075 2804 y Fu(@)2126 2819 y Ft(r)2165 2716 y Fl(p)p 2264 2716 V 2264 2804 a Fu(F)2327 2819 y Fs(+)2387 2804 y Fu(:)1210 b Fz(\(4.5\))150 2986 y(Moreo)m(v)m(er)37 b(there)g(exists)f(a)g(smo)s(oth)f(function)g Fu(m)2009 3001 y Fs(0)2049 2986 y Fz(\()p Fu(r)s Fz(\))g(supp)s(orted)h(a)m(w)m (a)m(y)h(from)e(zero)h(and)g(with)f(a)150 3106 y(deca)m(y)f(of)e(at)g (least)g(order)h Fu(r)1181 3070 y Fm(\000)p Fs(2)1308 3106 y Fz(suc)m(h)h(that)1471 3288 y Fu(\032)1521 3246 y Fs(2)1588 3288 y Fz(=)28 b Fr(\000)p Fu(F)1832 3303 y Fs(+)1891 3288 y Fu(@)1947 3246 y Fs(2)1942 3312 y Ft(r)1988 3288 y Fu(F)2051 3303 y Fs(+)2132 3288 y Fz(+)22 b Fu(m)2315 3303 y Fs(0)2355 3288 y Fz(\()p Fu(r)s Fz(\))p Fu(:)1119 b Fz(\(4.6\))296 3469 y(The)34 b(transv)m(erse)h(v)m(elo)s (cit)m(y)d Fu(\030)37 b Fz(and)c(its)f(b)s(ounded)h(coun)m(terpart)2662 3443 y(~)2651 3469 y Fu(\030)k Fz(are)c(giv)m(en)g(b)m(y)1551 3693 y Fu(\030)f Fz(:=)1767 3626 y Fu(L)p 1767 3670 67 4 v 1781 3762 a Fz(\026)-53 b Fu(r)1865 3693 y Fr(\000)23 b Fu(m)2050 3708 y Fs(+)2109 3693 y Fz(\()p Fu(r)s Fz(\))p Fu(b)p Fz(\()p Fu(\022)s Fz(\))p Fu(:)1200 b Fz(\(4.7\))150 3906 y(Finally)35 b(w)m(e)k(note)f(\(see)h(Lemma)d(6.1)i(for)f(related) h(b)s(ounds\))g(that)g(for)f(ev)m(ery)j Fu(\014)i(>)37 b Fz(0)g(there)i(exist)150 4026 y Fu(C)r(;)17 b(N)37 b Fr(\025)29 b Fz(1)j(suc)m(h)i(that)e(for)g(all)f Fu(z)h Fr(2)c Fn(C)48 b Fr(n)22 b Fn(R)583 4284 y Fz(max)p Fr(fjj)t Fz(\026)-53 b Fu(r)918 4243 y Ft(\014)964 4284 y Fz(\()p Fu(H)29 b Fr(\000)23 b Fu(z)t Fz(\))1299 4243 y Fm(\000)p Fs(1)1394 4284 y Fz(\()t(\026)-53 b Fu(r)s Fz(\))1517 4243 y Fm(\000)p Ft(\014)1619 4284 y Fr(jj)p Fu(;)17 b Fr(jj)t Fz(\026)-53 b Fu(r)1822 4243 y Ft(\014)1867 4284 y Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(z)t Fz(\))2164 4243 y Fm(\000)p Fs(1)2259 4284 y Fz(\()t(\026)-53 b Fu(r)r Fz(\))2381 4243 y Fm(\000)p Ft(\014)2483 4284 y Fr(jjg)27 b(\024)2819 4217 y Fu(C)p 2731 4261 253 4 v 2731 4353 a Fr(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)3011 4144 y Fl(\022)3157 4217 y Fr(h)p Fu(z)t Fr(i)p 3094 4261 V 3094 4353 a(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)3357 4144 y Fl(\023)3430 4166 y Ft(N)3514 4284 y Fz(;)83 b(\(4.8\))778 4503 y Fr(h)p Fu(z)t Fr(i)28 b Fz(=)g(\(1)22 b(+)g Fr(j)p Fu(z)t Fr(j)1349 4462 y Fs(2)1388 4503 y Fz(\))1426 4462 y Fs(1)p Ft(=)p Fs(2)1536 4503 y Fu(:)150 4685 y Fz(W)-8 b(e)33 b(shall)e(use)j(\(4.8\))e(in)g(the)h(con)m(text)g(of)f (estimating)f(comm)m(utators.)296 4805 y(The)45 b(v)-5 b(arious)43 b(cut-o\013s)h(will)e(generate)i(through)g(comm)m(utation)d (sev)m(eral)k(terms)f(whic)m(h)g(are)150 4925 y(relativ)m(ely)39 b(compact)h(to)g Fu(H)8 b Fz(,)41 b(while)f Fu(\032)1578 4889 y Fs(2)1645 4925 y Fz(+)27 b Fu(\030)1796 4889 y Fs(2)1875 4925 y Fz(no)m(w)41 b(equals)g(2)p Fu(H)47 b Fz(\(up)40 b(to)g(relativ)m(ely)f(compact)h(re-)150 5046 y(mainders\))32 b(and)g(not)h(2)p Fu(E)38 b Fz(as)33 b(in)f(the)h(classical)e(case.)44 b(Hence)34 b(w)m(e)g(can)f(write)484 5272 y Fu(i)p Fz([)p Fu(H)r(;)17 b(A)744 5287 y Ft(C)803 5272 y Fz(\()p Fu(E)6 b Fz(\)])83 b(=)g(1)22 b(+)g Fu(H)30 b Fr(\000)22 b Fu(E)28 b Fz(+)1813 5204 y(1)p 1813 5249 49 4 v 1813 5340 a(2)1889 5272 y([\()p Fu(\030)e Fr(\000)d Fu(\030)2166 5287 y Ft(E)2225 5272 y Fz(\))p Fu(@)2314 5287 y Ft(E)2374 5272 y Fu(\030)2417 5287 y Ft(E)2498 5272 y Fz(+)f(\()p Fu(\032)h Fr(\000)f Fu(\032)2856 5287 y Ft(E)2916 5272 y Fz(\))p Fu(@)3005 5287 y Ft(E)3065 5272 y Fu(\032)3115 5287 y Ft(E)3197 5272 y Fz(+)g Fu(h:c:)p Fz(])1067 5509 y(+)1236 5441 y Fu(C)p 1236 5486 77 4 v 1250 5577 a Fz(2)1339 5428 y Fl(\002)1381 5509 y Fz(\()p Fu(\030)k Fr(\000)d Fu(\030)1631 5524 y Ft(E)1690 5509 y Fz(\))1728 5468 y Fs(2)1790 5509 y Fz(+)f(\()p Fu(\032)g Fr(\000)h Fu(\032)2148 5524 y Ft(E)2208 5509 y Fz(\))2246 5468 y Fs(2)2285 5428 y Fl(\003)2349 5509 y Fz(+)f Fu(K)2530 5524 y Fs(1)2569 5509 y Fu(;)1028 b Fz(\(4.9\))1939 5776 y(43)p eop %%Page: 44 47 44 46 bop 150 407 a Fz(where)41 b Fu(h:c:)e Fz(means)h(hermitian)d (conjugate)j(and)f Fu(K)2115 422 y Fs(1)2194 407 y Fz(is)g(a)g(relativ) m(ely)f(compact)h(remainder.)64 b(In)150 527 y(the)33 b(form)e(sense)k(w)m(e)e(ha)m(v)m(e)h(the)f(inequalit)m(y)747 791 y Fr(\000)834 723 y Fz(1)p 834 768 49 4 v 834 859 a(2)910 791 y([\()p Fu(\030)26 b Fr(\000)d Fu(\030)1187 806 y Ft(E)1246 791 y Fz(\))p Fu(@)1335 806 y Ft(E)1395 791 y Fu(\030)1438 806 y Ft(E)1520 791 y Fz(+)f Fu(h:c:)p Fz(])28 b Fr(\024)1940 723 y Fu(C)2017 687 y Fs(1)p Ft(=)p Fs(2)p 1940 768 187 4 v 2009 859 a Fz(2)2137 791 y(\()p Fu(\030)e Fr(\000)d Fu(\030)2387 806 y Ft(E)2446 791 y Fz(\))2484 750 y Fs(2)2546 791 y Fz(+)2747 723 y(1)p 2654 768 236 4 v 2654 859 a(2)p Fu(C)2780 830 y Fs(1)p Ft(=)p Fs(2)2899 791 y Fz(\()p Fu(@)2988 806 y Ft(E)3048 791 y Fu(\030)3091 806 y Ft(E)3151 791 y Fz(\))3189 750 y Fs(2)150 1025 y Fz(while)36 b(a)h(similar)e(one)i(holds)g(for)g(\()p Fu(\032)25 b Fr(\000)h Fu(\032)1684 1040 y Ft(E)1744 1025 y Fz(\))p Fu(@)1833 1040 y Ft(E)1893 1025 y Fu(\032)1943 1040 y Ft(E)2003 1025 y Fz(.)58 b(W)-8 b(e)38 b(ha)m(v)m(e)g(\()p Fu(H)33 b Fr(\000)26 b Fu(E)6 b Fz(\))36 b Fr(\025)g(\000)p Fu(\017)i Fz(when)g(restricted)150 1145 y(to)45 b(the)h(range)g(of)f Fv(1)922 1161 y Fs([)p Ft(E)t Fm(\000)p Ft(\017;E)t Fs(+)p Ft(\017)p Fs(])1262 1145 y Fz(\()p Fu(H)8 b Fz(\);)52 b(then)46 b(c)m(ho)s(osing)f Fu(C)52 b Fz(large)45 b(enough)h(w)m(e)g (obtain)e(the)i(desired)150 1265 y(p)s(ositivit)m(y)31 b(and)i(the)g(prop)s(osition)d(is)j(pro)m(v)m(en.)p 3754 1265 4 66 v 3758 1203 59 4 v 3758 1265 V 3816 1265 4 66 v 296 1504 a(Under)g(the)f(conditions)e(w)m(e)j(imp)s(osed,)e(1)p Fu(=)t Fz(\026)-53 b Fu(r)1920 1468 y Ft(\013)2000 1504 y Fz(with)31 b Fu(\013)d(>)g Fz(1)p Fu(=)p Fz(2)j(is)g(a)g(lo)s(cally)e (smo)s(oth)h(p)s(erturba-)150 1624 y(tion)i(for)h Fu(H)8 b Fz(,)33 b(cf.)47 b(Prop)s(osition)31 b(4.1,)j([M])g(and)f([R-S].)g(W) -8 b(e)34 b(ha)m(v)m(e)h(that)e(for)g(an)m(y)h(state)g(lik)m(e)f(the)h (one)150 1745 y(in)e(\(6.1\):)679 1876 y Fl(Z)778 1902 y Fm(1)734 2101 y Fs(1)853 2011 y Fr(h)p Fu( )t Fz(\()p Fu(t)p Fz(\))p Fu(;)17 b Fz(\()t(\026)-53 b Fu(r)r Fz(\))1236 1970 y Fm(\000)p Fs(2)p Ft(\013)1376 2011 y Fu( )t Fz(\()p Fu(t)p Fz(\))p Fr(i)p Fu(dt)27 b Fz(=)1810 1876 y Fl(Z)1909 1902 y Fm(1)1865 2101 y Fs(1)2001 2011 y Fr(jj)p Fz(\()t(\026)-53 b Fu(r)r Fz(\))2179 1970 y Fm(\000)p Ft(\013)2283 2011 y Fu( )t Fz(\()p Fu(t)p Fz(\))p Fr(jj)2517 1970 y Fs(2)2556 2011 y Fu(dt)27 b Fr(\024)i Fz(Const)f Fr(jj)p Fu( )t Fr(jj)3231 1970 y Fs(2)3269 2011 y Fu(:)280 b Fz(\(4.10\))150 2231 y(W)-8 b(e)35 b(also)f(learn)h(that)f(the)i(p)s(oin)m(t)e(sp)s (ectrum)h(of)g Fu(H)42 b Fz(in)34 b(\()p Fu(E)2297 2246 y Ft(d)2338 2231 y Fu(;)17 b Fr(1)p Fz(\))34 b(is)g(discrete)i(with)f (eigen)m(v)-5 b(alues)34 b(of)150 2351 y(\014nite)e(m)m(ultiplicit)m(y) -8 b(.)150 2638 y Fp(4.2)135 b(Construction)46 b(of)f(the)g(appro)l (ximate)i(ev)l(olution)150 2822 y Fz(F)-8 b(rom)31 b(no)m(w)i(on)g(w)m (e)g(abbreviate)g(\(0)p Fu(;)17 b Fr(1)p Fz(\))31 b(b)m(y)j Fn(R)1870 2837 y Fs(+)1935 2822 y Fz(.)43 b(De\014ne)34 b(for)e Fu(t)27 b Fr(\025)i Fz(1)j(the)h(op)s(erator)1146 3025 y Fu(U)1212 3040 y Fs(0)1251 3025 y Fz(\()p Fu(t)p Fz(\))28 b(:)g Fu(L)1511 2984 y Fs(2)1551 3025 y Fz(\(\()p Fu(E)1699 3040 y Ft(d)1739 3025 y Fu(;)17 b Fr(1)p Fz(\))22 b Fr(\002)g Fn(T)p Fz(\))31 b Fr(7!)c Fu(L)2367 2984 y Fs(2)2407 3025 y Fz(\()p Fn(R)2511 3040 y Fs(+)2598 3025 y Fr(\002)c Fn(T)p Fz(\))p Fu(;)750 b Fz(\(4.11\))150 3228 y(where)34 b(\(see)f(also)f(\(3.56\)\))308 3445 y([)p Fu(U)401 3460 y Fs(0)440 3445 y Fz(\()p Fu(t)p Fz(\))p Fu(f)11 b Fz(]\()p Fu(r)m(;)17 b(\022)s Fz(\))28 b(:=)g(exp)q Fr(f)p Fu(iS)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))p Fr(g)p Fu(J)1704 3394 y Fs(1)p Ft(=)p Fs(2)1695 3467 y Ft(t)1813 3445 y Fz(\()p Fu(r)m(;)g(\022)s Fz(\))23 b Fr(\001)e Fu(f)28 b Fz(\()p Fr(\000)p Fu(@)2336 3460 y Ft(t)2366 3445 y Fu(S)6 b Fz(\(1)p Fu(;)17 b Fv(w)2644 3460 y Ft(t)2673 3445 y Fz(\()p Fu(r)m(;)g(\022)s Fz(\)\))p Fu(;)g(\022)3009 3460 y Fs(1)3049 3445 y Fz(\()p Fu(t;)g(r)m(;)g(\022)s Fz(\)\))f Fu(:)158 b Fz(\(4.12\))150 3648 y(Here)34 b Fu(J)435 3663 y Ft(t)498 3648 y Fz(is)e(a)h(Jacobian)g(determinan)m(t)g (whic)m(h)g(assures)i(that)e Fu(U)2528 3663 y Fs(0)2568 3648 y Fz(\()p Fu(t)p Fz(\))g(is)g(unitary)-8 b(.)45 b(More)34 b(precisely)-8 b(,)150 3768 y(it)35 b(equals)i(the)g(pro)s (duct)f(b)s(et)m(w)m(een)i(the)f(Jacobian)f Fu(J)2110 3783 y Fk(w)2169 3791 y Fq(t)2237 3768 y Fz(of)f(the)i(in)m(v)m(erse)h (\015o)m(w,)g(and)e(the)h(Jacobian)150 3888 y(of)32 b(the)h (transformation)e Fn(R)1159 3903 y Fs(+)1252 3888 y Fr(3)d Fz(\()p Fu(r)m(;)17 b(\022)s Fz(\))28 b Fr(7!)f Fz(\()p Fr(\000)p Fu(@)1876 3903 y Ft(t)1907 3888 y Fu(S)6 b Fz(\(1)p Fu(;)17 b(r)m(;)g(\022)s Fz(\))p Fu(;)g(\022)s Fz(\))27 b Fr(2)h Fz(\()p Fu(E)2636 3903 y Ft(d)2676 3888 y Fu(;)17 b Fr(1)p Fz(\).)43 b(W)-8 b(e)33 b(also)f(in)m(tro)s (duce)1083 4091 y Fu(W)14 b Fz(\()p Fu(t)p Fz(\))27 b(:)h Fu(L)1448 4050 y Fs(2)1488 4091 y Fz(\(\()p Fu(E)1636 4106 y Ft(d)1676 4091 y Fu(;)17 b Fr(1)p Fz(\))22 b Fr(\002)g Fn(T)p Fz(\))31 b Fr(7!)c Fu(L)2304 4050 y Fs(2)2344 4091 y Fz(\()p Fn(R)2448 4106 y Fs(+)2535 4091 y Fr(\002)c Fn(T)p Fz(\))p Fu(;)1083 4237 y Fz([)p Fu(W)14 b Fz(\()p Fu(t)p Fz(\))p Fu(f)d Fz(]\()p Fu(r)m(;)17 b(\022)s Fz(\))27 b(:=)h(exp)q Fr(f\000)p Fu(iS)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))p Fr(g)p Fz([)p Fu(U)2586 4252 y Fs(0)2625 4237 y Fz(\()p Fu(t)p Fz(\))p Fu(f)11 b Fz(]\()p Fu(r)m(;)17 b(\022)s Fz(\))p Fu(:)518 b Fz(\(4.13\))150 4440 y(F)-8 b(or)32 b Fu(f)38 b Fr(2)28 b Fu(C)582 4403 y Fm(1)575 4464 y Fs(0)657 4440 y Fz(\(\()p Fu(E)805 4455 y Ft(d)845 4440 y Fu(;)17 b Fr(1)p Fz(\))22 b Fr(\002)g Fn(T)p Fz(\))36 b(w)m(e)e(ha)m(v)m(e)g(that)e Fu(W)14 b Fz(\()p Fu(t)p Fz(\))p Fu(f)43 b Fz(is)32 b(strongly)h(di\013eren)m(tiable)e(and)1393 4692 y Fr(\000)p Fu(i)1532 4624 y(d)p 1513 4669 86 4 v 1513 4760 a(dt)1610 4692 y(W)14 b Fz(\()p Fu(t)p Fz(\))p Fu(f)38 b Fz(=)28 b Fr(\000)p Fu(B)5 b Fz(\()p Fu(t)p Fz(\))p Fu(W)14 b Fz(\()p Fu(t)p Fz(\))p Fu(f)5 b(:)995 b Fz(\(4.14\))150 4919 y(Moreo)m(v)m(er,)1109 5122 y Fu(B)5 b Fz(\()p Fu(t)p Fz(\))p Fu(W)14 b Fz(\()p Fu(t)p Fz(\))p Fu(f)39 b Fr(2)28 b Fu(C)1774 5081 y Fm(1)1767 5147 y Fs(0)1848 5122 y Fz(\()p Fn(R)1952 5137 y Fs(+)2039 5122 y Fr(\002)23 b Fn(T)p Fz(\))p Fu(;)1309 b Fz(\(4.15\))628 5325 y(2)p Fu(B)5 b Fz(\()p Fu(t)p Fz(\))83 b(=)g Fr(\000)p Fu(i)p Fz(\()p Fu(@)1308 5340 y Ft(r)1347 5325 y Fu(S)6 b Fz(\)\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))p Fu(@)1790 5340 y Ft(r)1851 5325 y Fr(\000)22 b Fu(i)2000 5184 y Fl(\022)2083 5257 y Fu(@)2134 5272 y Ft(\022)2174 5257 y Fu(S)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))p 2083 5302 445 4 v 2282 5393 a Fu(r)2560 5325 y Fr(\000)22 b Fu(b)p Fz(\()p Fu(\022)s Fz(\))2824 5184 y Fl(\023)2925 5257 y Fz(1)p 2925 5302 49 4 v 2926 5393 a Fu(r)2984 5325 y(@)3035 5340 y Ft(\022)3096 5325 y Fz(+)g Fu(h:c:)950 5527 y Fz(=)83 b(\()p Fr(r)1230 5542 y Fk(x)1277 5527 y Fu(S)6 b Fz(\()p Fu(t;)17 b Fv(x)p Fz(\))22 b Fr(\000)h Fv(a)p Fz(\()p Fv(x)p Fz(\)\))f Fr(\001)g Fv(p)h Fz(+)f Fu(h:c:;)1236 b Fz(\(4.16\))1939 5776 y(44)p eop %%Page: 45 48 45 47 bop 150 407 a Fz(where)35 b(h.c.)46 b(means)33 b(hermitian)f(conjugate.)46 b(The)34 b(computation)e(is)h(fairly)e (standard,)j(and)g(relies)150 527 y(on)e(\(3.55\))g(and)h(\(3.69\).)296 648 y(W)-8 b(e)40 b(no)m(w)f(w)m(an)m(t)h(to)f(determine)f(the)i (\\generator)f(of)f(the)i(free)f(ev)m(olution",)g(i.e.)63 b(to)38 b(describ)s(e)150 768 y(the)33 b(strong)g(time)e(deriv)-5 b(ativ)m(e)32 b(of)g Fu(U)1463 783 y Fs(0)1503 768 y Fz(\()p Fu(t)p Fz(\).)296 888 y(Consider)e(the)g(follo)m(wing)d (symmetric)h(op)s(erators)i(on)f Fu(L)2372 852 y Fs(2)2412 888 y Fz(\()p Fn(R)2516 903 y Fs(+)2596 888 y Fr(\002)16 b Fn(T)p Fz(\))33 b(\(de\014ned)e(on)e(smo)s(oth)g(and)150 1009 y(compactly)j(supp)s(orted)h(functions\))1208 1249 y Fu(\015)1259 1264 y Fs(1)1326 1249 y Fz(=)28 b Fu(\032)22 b Fr(\000)h Fu(@)1653 1264 y Ft(r)1691 1249 y Fu(S)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))28 b(=)f Fu(\032)c Fr(\000)f Fu(\032)2398 1265 y Ft(E)t Fs(\()p Ft(t;r)n(;\022)r Fs(\))2643 1249 y Fz(\()p Fu(\022)s Fz(\))809 b(\(4.17\))150 1424 y(and)974 1577 y Fu(\015)1025 1592 y Fs(2)1092 1577 y Fz(=)27 b Fu(\030)g Fr(\000)1365 1436 y Fl(\022)1448 1509 y Fu(@)1499 1524 y Ft(\022)1538 1509 y Fu(S)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))p 1448 1554 445 4 v 1647 1645 a Fu(r)1924 1577 y Fr(\000)23 b Fu(b)p Fz(\()p Fu(\022)s Fz(\))2189 1436 y Fl(\023)2290 1577 y Fz(=)28 b Fu(\030)f Fr(\000)22 b Fu(\030)2606 1592 y Ft(E)t Fs(\()p Ft(t;r)n(;\022)r Fs(\))2850 1577 y Fz(\()p Fu(\022)s Fz(\))p Fu(:)575 b Fz(\(4.18\))296 1808 y(F)-8 b(rom)31 b(\(3.41\))h(and)h(\(4.14\))f(one)h(infers)f(that)g(for)g(an)m (y)i Fu(f)k Fr(2)28 b Fu(C)2550 1772 y Fm(1)2543 1833 y Fs(0)2625 1808 y Fz(\(\()p Fu(E)2773 1823 y Ft(d)2813 1808 y Fu(;)17 b Fr(1)p Fz(\))22 b Fr(\002)g Fn(T)p Fz(\))36 b(the)d(mapping)1248 2028 y(\(1)p Fu(;)17 b Fr(1)p Fz(\))27 b Fr(3)h Fu(t)g Fr(7!)f Fu(U)1894 2043 y Fs(0)1934 2028 y Fz(\()p Fu(t)p Fz(\))p Fu(f)39 b Fr(2)28 b Fu(L)2292 1987 y Fs(2)2331 2028 y Fz(\()p Fn(R)2435 2043 y Fs(+)2523 2028 y Fr(\002)22 b Fn(T)p Fz(\))150 2248 y(is)32 b(di\013eren)m (tiable)f(and)i(if)e Fu(t)i Fz(is)f(large)g(enough)h(then)752 2522 y Fu(i)841 2454 y(d)p 823 2499 86 4 v 823 2590 a(dt)919 2522 y(U)985 2537 y Fs(0)1025 2522 y Fz(\()p Fu(t)p Fz(\))p Fu(f)38 b Fz(=)28 b Fu(H)1407 2537 y Fs(0)1446 2522 y Fz(\()p Fu(t)p Fz(\))p Fu(U)1623 2537 y Fs(0)1663 2522 y Fz(\()p Fu(t)p Fz(\))p Fu(f)5 b(;)114 b(H)2049 2537 y Fs(0)2089 2522 y Fz(\()p Fu(t)p Fz(\))27 b(:=)h Fu(H)i Fz(+)2620 2454 y(1)p 2577 2499 135 4 v 2577 2590 a(8)p Fu(r)2673 2561 y Fs(2)2744 2522 y Fr(\000)2853 2454 y Fu(\015)2909 2418 y Fs(2)2904 2479 y(1)p 2853 2499 96 4 v 2877 2590 a Fz(2)2981 2522 y Fr(\000)3090 2454 y Fu(\015)3146 2418 y Fs(2)3141 2479 y(2)p 3090 2499 V 3114 2590 a Fz(2)3196 2522 y Fu(:)353 b Fz(\(4.19\))150 2821 y Fp(4.3)135 b(Stating)46 b(the)f(main)h(result)f(of)h(the)f(pap)t (er)150 3006 y Fz(W)-8 b(e)33 b(are)g(\014nally)e(prepared)i(to)g(giv)m (e)f(the)h(main)e(theorem.)150 3209 y Fv(Theorem)c(4.2.)35 b FA(The)26 b(limits)g(in)g(\(1.23\))g(exist)h(and)f(de\014ne)f(mutual) 5 b(ly)28 b(inverse)d(unitary)j(op)-5 b(er)g(ators.)150 3329 y(Sp)g(el)5 b(le)-5 b(d)34 b(out:)296 3450 y Fz(I)50 b FA(\(Existenc)-5 b(e)49 b(of)g(sc)-5 b(attering)49 b(states.\))88 b(F)-7 b(or)49 b(every)g Fu(\036)g FA(in)g Fu(L)2629 3414 y Fs(2)2669 3450 y Fz(\(\()p Fu(E)2817 3465 y Ft(d)2858 3450 y Fu(;)17 b Fr(1)p Fz(\))32 b Fr(\002)h Fn(T)p Fz(\))53 b FA(ther)-5 b(e)49 b(exists)150 3570 y Fu( )32 b Fz(=)27 b(\012)418 3534 y Ft(d)418 3595 y Fs(+)478 3570 y Fu(\036)34 b FA(in)h(the)g(sp)-5 b(ac)g(e)34 b Fr(H)1190 3585 y Ft(E)1242 3597 y Fq(d)1310 3570 y Fz(=)27 b Fv(1)1469 3586 y Fs(\()p Ft(E)1548 3598 y Fq(d)1585 3586 y Ft(;)p Fm(1)p Fs(\))p Fm(n)p Ft(\033)1778 3594 y Fq(pp)1846 3586 y Fs(\()p Ft(H)5 b Fs(\))1968 3570 y Fz(\()p Fu(H)j Fz(\))p Fu(L)2199 3534 y Fs(2)2238 3570 y Fz(\()p Fn(R)2342 3585 y Fs(+)2429 3570 y Fr(\002)23 b Fn(T)p Fz(\))38 b FA(such)d(that)1352 3799 y Fz(lim)1337 3859 y Ft(t)p Fm(!1)1520 3714 y Fl(\015)1520 3774 y(\015)1575 3799 y Fu(e)1620 3758 y Fm(\000)p Ft(itH)1792 3799 y Fz(\012)1862 3758 y Ft(d)1862 3824 y Fs(+)1922 3799 y Fu(\036)22 b Fr(\000)g Fu(U)2167 3814 y Fs(0)2207 3799 y Fz(\()p Fu(t)p Fz(\))p Fu(\036)2376 3714 y Fl(\015)2376 3774 y(\015)2459 3799 y Fz(=)27 b(0;)296 4051 y(I)s(I)42 b FA(\(Asymptotic)h(c)-5 b(ompleteness.\))64 b(F)-7 b(or)42 b(every)g Fu( )k FA(in)41 b(the)h Fr(H)2596 4066 y Ft(E)2648 4078 y Fq(d)2689 4051 y FA(,)h(ther)-5 b(e)42 b(exists)g Fu(\036)f Fz(=)g(\012)3572 4066 y Fs(+)3631 4051 y Fu( )46 b FA(in)150 4172 y Fu(L)216 4136 y Fs(2)256 4172 y Fz(\(\()p Fu(E)404 4187 y Ft(d)444 4172 y Fu(;)17 b Fr(1)p Fz(\))22 b Fr(\002)g Fn(T)p Fz(\))39 b FA(such)34 b(that)1343 4392 y Fz(lim)1328 4451 y Ft(t)p Fm(!1)1511 4307 y Fl(\015)1511 4367 y(\015)1566 4392 y Fu(e)1611 4351 y Fm(\000)p Ft(itH)1783 4392 y Fu( )26 b Fr(\000)d Fu(U)2038 4407 y Fs(0)2077 4392 y Fz(\()p Fu(t)p Fz(\)\012)2258 4407 y Fs(+)2318 4392 y Fu( )2385 4307 y Fl(\015)2385 4367 y(\015)2468 4392 y Fz(=)28 b(0;)296 4644 y(I)s(I)s(I)k FA(\(Asymptotic)g (observables.\))43 b(The)31 b(asymptotic)h(r)-5 b(adius)31 b(and)h(angle)f(\(mo)-5 b(d)31 b Fz(2)p Fu(\031)t FA(\))g(as)h (de\014ne)-5 b(d)150 4764 y(in)49 b(\(1.24\))g(exist)g(as)g(op)-5 b(er)g(ators)49 b(on)g Fr(H)1649 4779 y Ft(E)1701 4791 y Fq(d)1741 4764 y FA(;)57 b(mor)-5 b(e)g(over)48 b(\()p Fu(M)10 b Fz(\()p Fr(\001)p Fz(\))50 b FA(denotes)f(multiplic)-5 b(ation)48 b(by)i(the)150 4885 y(ar)-5 b(gument,)35 b(se)-5 b(e)34 b(L)-5 b(emma)34 b(3.8)g(for)h(the)g(de\014nition)f(of)g Fu(r)s Fz(\()p Fu(E)6 b(;)17 b(t;)g(\022)s Fz(\))p FA(\):)381 5105 y Fu(e)426 5064 y Ft(i\022)483 5073 y Fj(+)566 5105 y Fz(=)27 b(\012)739 5064 y Ft(d)739 5129 y Fs(+)799 5105 y Fu(M)10 b Fz(\()p Fu(e)986 5064 y Ft(i\022)1050 5105 y Fz(\)\012)1158 5120 y Fs(+)1217 5105 y Fu(;)45 b(r)1333 5120 y Fs(+)1419 5105 y Fz(=)28 b(\012)1593 5064 y Ft(d)1593 5129 y Fs(+)1652 5105 y Fu(M)10 b Fz(\()p Fu(r)s Fz(\()p Fu(E)c(;)17 b Fz(1)p Fu(;)g(\022)s Fz(\)\)\012)2288 5120 y Fs(+)2347 5105 y Fu(;)45 b(H)8 b Fv(1)2564 5120 y Fm(H)2624 5131 y Fq(E)2669 5148 y(d)2740 5105 y Fz(=)28 b(\012)2914 5064 y Ft(d)2914 5129 y Fs(+)2973 5105 y Fu(M)10 b Fz(\()p Fu(E)c Fz(\)\012)3301 5120 y Fs(+)3362 5105 y Fv(1)3418 5120 y Fm(H)3478 5131 y Fq(E)3523 5148 y(d)3567 5105 y Fu(:)1939 5776 y Fz(45)p eop %%Page: 46 49 46 48 bop 150 407 a Fv(Remark.)42 b Fz(W)-8 b(e)32 b(ma)m(y)e(readily)g (add)h(some)g(short)g(range)g(magnetic)f(and)h(scalar)f(p)s (erturbations)g(to)150 527 y Fu(H)8 b Fz(,)29 b(and)g(the)h(ab)s(o)m(v) m(e)g(theorem)e(remains)h(true.)42 b(Namely)-8 b(,)29 b(w)m(e)h(de\014ne)g(\(see)g(also)e(\(1.16\))h(and)g(\(1.17\)\))385 786 y Fu(H)466 801 y Ft(s)530 786 y Fz(:=)671 718 y(1)p 671 763 49 4 v 671 854 a(2)730 786 y(\()p Fv(p)22 b Fr(\000)h Fv(a)f Fr(\000)h Fv(a)1182 801 y Ft(s)1219 786 y Fz(\))1257 744 y Fs(2)1319 786 y Fz(+)f Fu(V)1474 801 y Ft(s)1538 786 y Fz(=)28 b Fu(H)h Fr(\000)1862 718 y Fz(1)p 1862 763 V 1862 854 a(2)1921 786 y(\()p Fv(p)22 b Fr(\000)h Fv(a)p Fz(\))f Fr(\001)g Fv(a)2361 801 y Ft(s)2420 786 y Fr(\000)2530 718 y Fz(1)p 2530 763 V 2530 854 a(2)2589 786 y Fv(a)2643 801 y Ft(s)2702 786 y Fr(\001)g Fz(\()p Fv(p)g Fr(\000)h Fv(a)p Fz(\))f(+)3196 718 y(1)p 3196 763 V 3196 854 a(2)3255 786 y Fv(a)3309 744 y Fs(2)3309 810 y Ft(s)3371 786 y Fz(+)g Fu(V)3526 801 y Ft(s)3563 786 y Fu(;)150 1035 y Fz(where)40 b Fv(a)492 1050 y Ft(s)568 1035 y Fz(is)e Fu(C)749 999 y Fs(1)789 1035 y Fz(,)i Fu(V)913 1050 y Ft(s)989 1035 y Fz(is)e(relativ)m(ely)g(b)s(ounded)h (with)g(resp)s(ect)h(to)e Fr(\000)p Fz(\001)i(with)e(b)s(ound)h(less)g (than)g(1)150 1155 y(and,)33 b(in)f(addition,)e(for)i(some)h Fu(\017)28 b(>)g Fz(0)k(and)g(as)h Fr(j)p Fv(x)p Fr(j)27 b(!)h(1)p Fz(,)1227 1369 y Fu(V)1284 1384 y Ft(s)1321 1369 y Fz(\()p Fv(x)p Fz(\))p Fu(;)44 b Fv(a)1581 1384 y Ft(s)1618 1369 y Fz(\()p Fv(x)p Fz(\))28 b(=)f Fr(O)s Fz(\()p Fr(j)p Fv(x)p Fr(j)2119 1328 y Fm(\000)p Fs(\(1+)p Ft(\017)p Fs(\))2352 1369 y Fz(\);)1227 1514 y Fu(D)1311 1473 y Ft(\013)1360 1514 y Fu(V)1417 1529 y Ft(s)1454 1514 y Fz(\()p Fv(x)p Fz(\))p Fu(;)44 b(D)1744 1473 y Ft(\013)1793 1514 y Fv(a)1847 1529 y Ft(s)1885 1514 y Fz(\()p Fv(x)p Fz(\))27 b(=)h Fr(O)s Fz(\()p Fr(j)p Fv(x)p Fr(j)2386 1473 y Fm(\000)p Fs(2)2480 1514 y Fz(\);)44 b Fr(j)p Fu(\013)q Fr(j)27 b Fz(=)g(2)p Fu(:)150 1728 y Fz(Then)37 b(b)m(y)g(standard)g(Mourre)f(theory)h(and)f(the)h(theory) g(of)e(smo)s(oth)h(p)s(erturbations,)g(cf.)55 b(\(4.10\),)150 1849 y(one)31 b(\014rst)g(constructs)i(the)e(relativ)m(e)f(w)m(a)m(v)m (e)j(op)s(erator)d(for)g(the)i(pair)d(\()p Fu(H)r(;)17 b(H)2889 1864 y Ft(s)2926 1849 y Fz(\))30 b(.)43 b(Next)32 b(in)m(v)m(oking)f(the)150 1969 y(stated)g(theorem)f(\(for)g Fu(H)8 b Fz(\))30 b(and)g(the)h(c)m(hain)g(rule)f(for)g(w)m(a)m(v)m(e)i (op)s(erators)e(one)h(deduces)h(the)f(theorem)150 2089 y(with)h Fu(H)40 b Fz(replaced)33 b(b)m(y)h Fu(H)1095 2104 y Ft(s)1131 2089 y Fz(.)150 2421 y Fw(5)161 b(Pro)t(of)54 b(of)g(the)f(existence)f(of)i(scattering)f(states)150 2640 y Fz(Our)33 b(pro)s(of)f(of)g(the)h(\014rst)g(statemen)m(t)h(of)e (Theorem)h(4.2)f(is)h(divided)f(in)m(to)g(t)m(w)m(o)h(parts.)45 b(Cho)s(ose)33 b(an)m(y)150 2761 y Fu(\036)27 b Fr(2)h Fu(C)406 2724 y Fm(1)399 2785 y Fs(0)481 2761 y Fz(\(\()p Fu(E)629 2776 y Ft(d)670 2761 y Fu(;)17 b Fr(1)p Fz(\))k Fr(\002)i Fn(T)p Fz(\))36 b(and)c(denote)i(its)e(supp)s(ort)h(b)m(y)g Fu(I)8 b Fz(.)296 2881 y(Firstly)-8 b(,)31 b(w)m(e)j(sho)m(w)g(that)e (the)h(follo)m(wing)d(limit)f(exists)k(in)f Fu(L)2490 2845 y Fs(2)2530 2881 y Fz(\()p Fn(R)2634 2896 y Fs(+)2721 2881 y Fr(\002)23 b Fn(T)p Fz(\):)1491 3095 y(\012)1561 3054 y Ft(d)1561 3120 y Fs(+)1621 3095 y Fu(\036)k Fz(:=)43 b(lim)1837 3155 y Ft(t)p Fm(!1)2020 3095 y Fu(e)2065 3054 y Ft(iH)5 b(t)2182 3095 y Fu(U)2248 3110 y Fs(0)2288 3095 y Fz(\()p Fu(t)p Fz(\))p Fu(\036:)1140 b Fz(\(5.1\))296 3336 y(Secondly)-8 b(,)33 b(w)m(e)h(sho)m(w)g(that)e(this)g(limit)d(b)s (elongs)j(to)h Fr(H)2298 3351 y Ft(E)2350 3363 y Fq(d)2390 3336 y Fz(.)150 3624 y Fp(5.1)135 b(Existence)46 b(of)g FF(\012)1299 3580 y Fh(d)1299 3653 y Fx(+)150 3808 y Fz(Emplo)m(ying)h(the)i(usual)f(Co)s(ok)g(argumen)m(t,)53 b(w)m(e)c(can)g(reduce)h(the)e(ab)s(o)m(v)m(e)i(limit)45 b(to)j(the)h(\\time)150 3929 y(in)m(tegrabilit)m(y")30 b(of)i(the)h(\\p)s(erturbation")e(\(see)j(\(4.19\)\):)708 4066 y Fl(Z)808 4093 y Fm(1)763 4292 y Fs(1)899 4057 y Fl(\015)899 4117 y(\015)899 4177 y(\015)899 4237 y(\015)954 4061 y(\022)1061 4134 y Fu(@)1117 4098 y Fs(2)p 1038 4179 144 4 v 1038 4270 a Fu(@)5 b(r)1141 4241 y Fs(2)1213 4202 y Fz(+)1339 4134 y(1)p 1321 4179 87 4 v 1321 4270 a Fu(r)1368 4241 y Fs(2)1451 4134 y Fu(@)1507 4098 y Fs(2)p 1427 4179 145 4 v 1427 4270 a Fu(@)g(\022)1531 4241 y Fs(2)1603 4202 y Fz(+)1755 4134 y(1)p 1711 4179 135 4 v 1711 4270 a(4)p Fu(r)1807 4241 y Fs(2)1856 4061 y Fl(\023)1946 4202 y Fz([)p Fu(W)14 b Fz(\()p Fu(t)p Fz(\))p Fu(\036)p Fz(]\()p Fu(r)m(;)j(\022)s Fz(\))2484 4057 y Fl(\015)2484 4117 y(\015)2484 4177 y(\015)2484 4237 y(\015)2539 4301 y Ft(L)2587 4282 y Fj(2)2622 4301 y Fs(\()p Fe(R)2697 4310 y Fj(+)2748 4301 y Fm(\002)p Fe(T)p Fs(\))2923 4202 y Fu(dt)28 b(<)f Fr(1)p Fu(:)357 b Fz(\(5.2\))296 4494 y(In)29 b(order)f(to)g(get)g(an)g(idea)g(ab)s (out)g(wh)m(y)h(\(5.2\))f(should)g(hold,)g(w)m(e)i(again)c(lo)s(ok)h (at)h(the)h(case)g(when)150 4614 y Fu(b)p Fz(\()p Fu(\022)s Fz(\))41 b(=)f Fu(b)g(<)g Fz(0)g(is)f(a)h(negativ)m(e)g(constan)m(t.)66 b(F)-8 b(rom)38 b(the)i(de\014nition)f(\(4.13\))g(w)m(e)i(get)f(using)g (\(3.58\),)150 4735 y(\(3.59\),)32 b(\(3.62\))g(and)g(\(3.63\))g(that) 977 5015 y([)p Fu(W)14 b Fz(\()p Fu(t)p Fz(\))p Fu(\036)p Fz(]\()p Fu(r)m(;)j(\022)s Fz(\))27 b(=)1646 4869 y Fl(r)p 1745 4869 95 4 v 1769 4948 a Fu(r)p 1755 4992 75 4 v 1755 5084 a(t)1790 5055 y Fs(2)1840 5015 y Fu(\036)1915 4875 y Fl(\022)2016 4948 y Fu(r)2063 4912 y Fs(2)p 1998 4992 124 4 v 1998 5084 a Fz(2)p Fu(t)2082 5055 y Fs(2)2153 5015 y Fz(+)2261 4948 y Fu(b)2302 4912 y Fs(2)p 2261 4992 81 4 v 2277 5084 a Fz(2)2352 5015 y Fu(;)17 b(\022)25 b Fz(+)d Fu(b)2621 4948 y(t)p 2615 4992 47 4 v 2615 5084 a(r)2689 5015 y Fz(ln)o(\()p Fu(t)p Fz(\))2881 4875 y Fl(\023)2971 5015 y Fu(;)626 b Fz(\(5.3\))150 5286 y(in)32 b(accordance)h(with)g(the)g(form)m(ula)d(\(1.25\))i(for)g Fu(U)2007 5301 y Fs(0)2047 5286 y Fz(\()p Fu(t)p Fz(\))p Fu(f)11 b Fz(.)296 5407 y(W)-8 b(e)30 b(ma)m(y)g(no)m(w)g(see)h(that)f (on)f(the)h(supp)s(ort)g(of)g Fu(W)14 b Fz(\()p Fu(t)p Fz(\))p Fu(\036)p Fz(,)29 b Fu(r)k Fz(and)d Fu(t)f Fz(are)h(e\013ectiv) m(ely)h(prop)s(ortional;)150 5527 y(when)c(w)m(e)g(lo)s(ok)e(at)h(the)g (in)m(tegrand)g(in)f(\(5.2\),)i(w)m(e)g(realize)e(that)g(b)m(y)i (di\013eren)m(tiating)d(with)i(resp)s(ect)h(to)1939 5776 y(46)p eop %%Page: 47 50 47 49 bop 150 407 a Fu(r)32 b Fz(w)m(e)e(obtain)e(an)h(extra)h(deca)m (y)h(of)d(order)i(either)f Fu(t)1982 371 y Fm(\000)p Fs(1)2106 407 y Fz(or)f Fu(t)2256 371 y Fm(\000)p Fs(1)2368 407 y Fz(ln)o(\()p Fu(t)p Fz(\).)42 b(Tw)m(o)30 b(deriv)-5 b(ativ)m(es)30 b(will)d(mak)m(e)i(it)150 527 y(in)m(tegrable)i(in)f Fu(t)p Fz(.)44 b(As)32 b(for)f(the)h(deriv)-5 b(ativ)m(es)32 b(with)f(resp)s(ect)i(to)e Fu(\022)s Fz(,)h(w)m(e)h(see)g(that)e(they)h (are)g(b)s(ounded;)150 648 y(but)h(1)p Fu(=r)474 611 y Fs(2)545 648 y Fz(is)f(transformed)g(in)m(to)g(1)p Fu(=t)1519 611 y Fs(2)1591 648 y Fz(so)h(\(5.2\))f(follo)m(ws.)296 768 y(A)27 b(similar)c(argumen)m(t)j(can)h(b)s(e)f(carried)h(out)f (when)h Fu(b)g Fz(is)f(not)h(constan)m(t.)42 b(Compared)26 b(to)g(Lemma)150 888 y(3.12)h(w)m(e)i(need)f(to)f(go)h(further,)h(i.e.) 41 b(to)27 b(in)m(v)m(estigate)h(what)g(happ)s(ens)g(when)h(w)m(e)g (di\013eren)m(tiate)e(t)m(wice)150 1009 y(with)32 b(resp)s(ect)i(to)e Fu(r)j Fz(and)e Fu(\022)s Fz(.)44 b(T)-8 b(aking)32 b(t)m(w)m(o)h (deriv)-5 b(ativ)m(es)33 b(in)f(\(3.86\))f(and)i(\(3.87\))f(w)m(e)h (get)953 1221 y Fu(@)10 b Fz(~)-54 b Fu(r)p 934 1265 141 4 v 934 1356 a(@)5 b(r)1034 1371 y Fs(1)1095 1221 y Fu(@)1151 1184 y Fs(2)1191 1221 y Fu(r)1235 1236 y Fs(1)p 1095 1265 180 4 v 1113 1356 a Fu(@)g(r)1216 1327 y Fs(2)1306 1288 y Fz(+)1433 1221 y Fu(@)10 b Fz(~)-54 b Fu(r)p 1414 1265 142 4 v 1414 1356 a(@)5 b(\022)1515 1371 y Fs(1)1576 1221 y Fu(@)1632 1184 y Fs(2)1672 1221 y Fu(\022)1717 1236 y Fs(1)p 1576 1265 182 4 v 1595 1356 a Fu(@)g(r)1698 1327 y Fs(2)1795 1288 y Fz(=)28 b Fu(A)1972 1303 y Fs(1)2011 1288 y Fu(;)924 1565 y(A)997 1580 y Fs(1)1065 1565 y Fz(=)f Fr(\000)1255 1498 y Fu(@)1311 1462 y Fs(2)1356 1498 y Fz(~)-53 b Fu(r)p 1255 1542 144 4 v 1255 1634 a(@)5 b(r)1358 1599 y Fs(2)1355 1658 y(1)1425 1425 y Fl(\022)1508 1498 y Fu(@)g(r)1608 1513 y Fs(1)p 1508 1542 141 4 v 1527 1634 a Fu(@)g(r)1659 1425 y Fl(\023)1732 1447 y Fs(2)1794 1565 y Fr(\000)22 b Fz(2)2022 1498 y Fu(@)2078 1462 y Fs(2)2122 1498 y Fz(~)-53 b Fu(r)p 1952 1542 283 4 v 1952 1634 a(@)5 b(r)2052 1649 y Fs(1)2092 1634 y Fu(@)g(\022)2193 1649 y Fs(1)2254 1498 y Fu(@)g(r)2354 1513 y Fs(1)p 2254 1542 141 4 v 2272 1634 a Fu(@)g(r)2414 1498 y(@)g(\022)2515 1513 y Fs(1)p 2414 1542 142 4 v 2433 1634 a Fu(@)g(r)2588 1565 y Fr(\000)2699 1498 y Fu(@)2755 1462 y Fs(2)2799 1498 y Fz(~)-53 b Fu(r)p 2698 1542 145 4 v 2698 1634 a(@)5 b(\022)2802 1599 y Fs(2)2799 1658 y(1)2869 1425 y Fl(\022)2952 1498 y Fu(@)g(\022)3053 1513 y Fs(1)p 2952 1542 142 4 v 2971 1634 a Fu(@)g(r)3104 1425 y Fl(\023)3177 1447 y Fs(2)3624 1565 y Fz(\(5.4\))150 1837 y(and)929 2037 y Fu(@)993 2011 y Fz(~)985 2037 y Fu(\022)p 912 2082 141 4 v 912 2173 a(@)g(r)1012 2188 y Fs(1)1072 2037 y Fu(@)1128 2001 y Fs(2)1168 2037 y Fu(r)1212 2052 y Fs(1)p 1072 2082 180 4 v 1090 2173 a Fu(@)g(r)1193 2144 y Fs(2)1284 2105 y Fz(+)1410 2037 y Fu(@)1474 2011 y Fz(~)1466 2037 y Fu(\022)p 1392 2082 142 4 v 1392 2173 a(@)g(\022)1493 2188 y Fs(1)1553 2037 y Fu(@)1609 2001 y Fs(2)1650 2037 y Fu(\022)1695 2052 y Fs(1)p 1553 2082 182 4 v 1573 2173 a Fu(@)g(r)1676 2144 y Fs(2)1772 2105 y Fz(=)28 b Fu(A)1949 2120 y Fs(2)1988 2105 y Fu(;)902 2382 y(A)975 2397 y Fs(2)1042 2382 y Fz(=)g Fr(\000)1233 2315 y Fu(@)1289 2279 y Fs(2)1337 2289 y Fz(~)1329 2315 y Fu(\022)p 1233 2359 145 4 v 1234 2451 a(@)5 b(r)1337 2416 y Fs(2)1334 2475 y(1)1404 2242 y Fl(\022)1487 2315 y Fu(@)g(r)1587 2330 y Fs(1)p 1487 2359 141 4 v 1506 2451 a Fu(@)g(r)1638 2242 y Fl(\023)1711 2264 y Fs(2)1773 2382 y Fr(\000)22 b Fz(2)2000 2315 y Fu(@)2056 2279 y Fs(2)2104 2289 y Fz(~)2096 2315 y Fu(\022)p 1931 2359 283 4 v 1931 2451 a(@)5 b(r)2031 2466 y Fs(1)2071 2451 y Fu(@)g(\022)2172 2466 y Fs(1)2233 2315 y Fu(@)g(r)2333 2330 y Fs(1)p 2233 2359 141 4 v 2251 2451 a Fu(@)g(r)2393 2315 y(@)g(\022)2494 2330 y Fs(1)p 2393 2359 142 4 v 2412 2451 a Fu(@)g(r)2567 2382 y Fr(\000)2677 2315 y Fu(@)2733 2279 y Fs(2)2781 2289 y Fz(~)2773 2315 y Fu(\022)p 2677 2359 145 4 v 2677 2451 a(@)g(\022)2781 2416 y Fs(2)2778 2475 y(1)2848 2242 y Fl(\022)2931 2315 y Fu(@)g(\022)3032 2330 y Fs(1)p 2931 2359 142 4 v 2950 2451 a Fu(@)g(r)3083 2242 y Fl(\023)3156 2264 y Fs(2)3212 2382 y Fu(:)385 b Fz(\(5.5\))150 2654 y(W)-8 b(e)33 b(then)g(ha)m(v)m(e)1259 2769 y Fu(@)1315 2733 y Fs(2)1355 2769 y Fu(r)1399 2784 y Fs(1)p 1259 2814 180 4 v 1277 2905 a Fu(@)5 b(r)1380 2876 y Fs(2)1476 2837 y Fz(=)28 b Fu(J)1643 2796 y Fm(\000)p Fs(1)1634 2861 y Fk(v)1677 2869 y Fq(t)1754 2666 y Fl( )1861 2769 y Fu(@)1925 2743 y Fz(~)1917 2769 y Fu(\022)p 1843 2814 142 4 v 1843 2905 a(@)5 b(\022)1944 2920 y Fs(1)2017 2837 y Fr(\001)22 b Fu(A)2140 2852 y Fs(1)2201 2837 y Fr(\000)2330 2769 y Fu(@)10 b Fz(~)-54 b Fu(r)p 2311 2814 V 2311 2905 a(@)5 b(\022)2412 2920 y Fs(1)2485 2837 y Fr(\001)22 b Fu(A)2608 2852 y Fs(2)2647 2666 y Fl(!)3624 2837 y Fz(\(5.6\))150 3093 y(and)1200 3208 y Fu(@)1256 3172 y Fs(2)1296 3208 y Fu(\022)1341 3223 y Fs(1)p 1200 3252 182 4 v 1219 3344 a Fu(@)5 b(r)1322 3315 y Fs(2)1419 3275 y Fz(=)27 b Fu(J)1585 3234 y Fm(\000)p Fs(1)1576 3300 y Fk(v)1619 3308 y Fq(t)1696 3105 y Fl( )1775 3275 y Fr(\000)1880 3208 y Fu(@)1944 3182 y Fz(~)1936 3208 y Fu(\022)p 1862 3252 141 4 v 1862 3344 a(@)5 b(r)1962 3359 y Fs(1)2035 3275 y Fr(\001)22 b Fu(A)2158 3290 y Fs(1)2220 3275 y Fz(+)2346 3208 y Fu(@)10 b Fz(~)-54 b Fu(r)p 2328 3252 V 2328 3344 a(@)5 b(r)2428 3359 y Fs(1)2500 3275 y Fr(\001)22 b Fu(A)2623 3290 y Fs(2)2662 3105 y Fl(!)2758 3275 y Fu(:)839 b Fz(\(5.7\))150 3537 y(Using)29 b(\(3.82\),)g(\(3.83\),)g(\(3.88\))f(and)h(\(3.89\))f(in)h (\(5.4\))f(and)h(\(5.5\))g(w)m(e)h(get)f(up)g(to)g(additional)d (factors)150 3657 y(of)40 b Fu(t)304 3621 y Ft(\016)383 3657 y Fz(\(remem)m(b)s(er)g(that)g Fu(\016)45 b Fz(is)40 b(arbitrarily)e(small;)k(these)g(terms)f(app)s(ear)f(when)i(w)m(e)g (replace)f(the)150 3778 y(logarithms\))1480 3898 y Fu(A)1553 3913 y Fs(1)1620 3898 y Fr(\030)28 b Fu(t)1760 3857 y Fm(\000)p Fs(1)1855 3898 y Fu(;)211 b(A)2166 3913 y Fs(2)2234 3898 y Fr(\030)28 b Fu(t)2374 3857 y Fm(\000)p Fs(2)2468 3898 y Fu(;)150 4072 y Fz(whic)m(h)33 b(leads)g(to)1617 4160 y Fu(@)1673 4123 y Fs(2)1713 4160 y Fu(r)1757 4175 y Fs(1)p 1617 4204 180 4 v 1635 4295 a Fu(@)5 b(r)1738 4267 y Fs(2)1806 4227 y Fu(;)1888 4160 y(@)1944 4123 y Fs(2)1984 4160 y Fu(\022)2029 4175 y Fs(1)p 1888 4204 182 4 v 1907 4295 a Fu(@)g(r)2010 4267 y Fs(2)2107 4227 y Fr(\030)28 b Fu(t)2247 4186 y Fm(\000)p Fs(2)2341 4227 y Fu(:)1256 b Fz(\(5.8\))150 4440 y(Similarly)-8 b(,)29 b(one)j(ma)m(y)h(sho)m(w)g(that)g(\(again)e(forgetting)g(ab)s(out)h (factors)h(of)f Fu(t)2915 4404 y Ft(\016)2953 4440 y Fz(\))1042 4652 y Fu(@)1098 4616 y Fs(2)1138 4652 y Fu(r)1182 4667 y Fs(1)p 1042 4696 180 4 v 1059 4787 a Fu(@)5 b(\022)1163 4759 y Fs(2)1231 4719 y Fu(;)1313 4652 y(@)1369 4616 y Fs(2)1409 4652 y Fu(\022)1454 4667 y Fs(1)p 1313 4696 182 4 v 1331 4787 a Fu(@)g(\022)1435 4759 y Fs(2)1532 4719 y Fr(\030)28 b Fz(Const)q Fu(;)2150 4652 y(@)2206 4616 y Fs(2)2246 4652 y Fu(r)2290 4667 y Fs(1)p 2135 4696 209 4 v 2135 4787 a Fu(@)5 b(\022)s(@)g(r)2354 4719 y(;)2449 4652 y(@)2505 4616 y Fs(2)2545 4652 y Fu(\022)2590 4667 y Fs(1)p 2435 4696 V 2435 4787 a Fu(@)g(\022)s(@)g(r)2682 4719 y Fr(\030)28 b Fu(t)2822 4678 y Fm(\000)p Fs(1)2916 4719 y Fu(:)681 b Fz(\(5.9\))150 4965 y(These)31 b(estimates)d(and)h (Lemma)e(3.12)i(lead)f(to)g(the)h(conclusion)g(that)f(the)h(\014rst)h (and)f(second)h(order)150 5085 y(deriv)-5 b(ativ)m(es)43 b(of)g(the)h(Jacobian)e(corresp)s(onding)h(to)g(the)h(in)m(v)m(erse)h (\015o)m(w)e Fu(J)2912 5100 y Fk(w)2971 5108 y Fq(t)3003 5085 y Fz(\()p Fu(r)m(;)17 b(\022)s Fz(\))46 b(=)g Fu(J)3443 5049 y Fm(\000)p Fs(1)3434 5110 y Fk(v)3477 5118 y Fq(t)3537 5085 y Fz(\()p Fu(r)3619 5100 y Fs(1)3658 5085 y Fu(;)17 b(\022)3747 5100 y Fs(1)3787 5085 y Fz(\))1939 5776 y(47)p eop %%Page: 48 51 48 50 bop 150 407 a Fz(b)s(eha)m(v)m(e)34 b(lik)m(e)1080 638 y([)p Fu(@)1158 653 y Ft(r)1196 638 y Fu(J)1259 597 y Fs(1)p Ft(=)p Fs(2)1250 662 y Fk(w)1309 670 y Fq(t)1369 638 y Fz(])p Fu(J)1459 597 y Fm(\000)p Fs(1)p Ft(=)p Fs(2)1450 662 y Fk(w)1509 670 y Fq(t)1652 638 y Fz(=)28 b Fr(\000)1843 570 y Fz(1)p 1843 615 49 4 v 1843 706 a(2)1902 638 y Fu(@)1953 653 y Ft(r)1991 638 y Fz(\(ln)16 b Fu(J)2181 653 y Fk(v)2224 661 y Fq(t)2278 638 y Fr(\016)22 b Fv(w)2431 653 y Ft(t)2461 638 y Fz(\))27 b Fr(\030)i Fu(t)2667 597 y Fm(\000)p Fs(1)2761 638 y Fu(;)788 b Fz(\(5.10\))1080 830 y([)p Fu(@)1158 845 y Ft(\022)1197 830 y Fu(J)1260 789 y Fs(1)p Ft(=)p Fs(2)1251 855 y Fk(w)1310 863 y Fq(t)1371 830 y Fz(])p Fu(J)1461 789 y Fm(\000)p Fs(1)p Ft(=)p Fs(2)1452 855 y Fk(w)1511 863 y Fq(t)1653 830 y Fr(\030)29 b Fz(Const)p Fu(;)1541 b Fz(\(5.11\))1080 1028 y Fu(@)1131 1043 y Ft(r)1169 1028 y Fz([[)p Fu(@)1274 1043 y Ft(r)1313 1028 y Fu(J)1376 987 y Fs(1)p Ft(=)p Fs(2)1367 1053 y Fk(w)1426 1061 y Fq(t)1486 1028 y Fz(])p Fu(J)1576 987 y Fm(\000)p Fs(1)p Ft(=)p Fs(2)1567 1053 y Fk(w)1626 1061 y Fq(t)1741 1028 y Fz(])28 b(=)f Fr(\000)1986 961 y Fz(1)p 1986 1005 V 1986 1096 a(2)2046 1028 y Fu(@)2102 987 y Fs(2)2097 1053 y Ft(r)2142 1028 y Fz(\(ln)16 b Fu(J)2332 1043 y Fk(v)2375 1051 y Fq(t)2429 1028 y Fr(\016)22 b Fv(w)2582 1043 y Ft(t)2611 1028 y Fz(\))28 b Fr(\030)g Fu(t)2817 987 y Fm(\000)p Fs(2)2912 1028 y Fu(;)637 b Fz(\(5.12\))1080 1261 y Fu(@)1131 1276 y Ft(\022)1170 1261 y Fz([[)p Fu(@)1275 1276 y Ft(\022)1315 1261 y Fu(J)1378 1220 y Fs(1)p Ft(=)p Fs(2)1369 1286 y Fk(w)1428 1294 y Fq(t)1488 1261 y Fz(])p Fu(J)1578 1220 y Fm(\000)p Fs(1)p Ft(=)p Fs(2)1569 1286 y Fk(w)1628 1294 y Fq(t)1743 1261 y Fz(])28 b(=)g Fr(\000)1989 1194 y Fz(1)p 1989 1238 V 1989 1330 a(2)2048 1261 y Fu(@)2104 1220 y Fs(2)2099 1286 y Ft(\022)2144 1261 y Fz(\(ln)16 b Fu(J)2334 1276 y Fk(v)2377 1284 y Fq(t)2431 1261 y Fr(\016)22 b Fv(w)2584 1276 y Ft(t)2614 1261 y Fz(\))27 b Fr(\030)h Fz(Const)q Fu(;)515 b Fz(\(5.13\))1016 1561 y Fu(@)1067 1576 y Ft(\022)1106 1561 y Fz([[)p Fu(@)1211 1576 y Ft(r)1250 1561 y Fu(J)1313 1519 y Fs(1)p Ft(=)p Fs(2)1304 1585 y Fk(w)1363 1593 y Fq(t)1423 1561 y Fz(])p Fu(J)1513 1519 y Fm(\000)p Fs(1)p Ft(=)p Fs(2)1504 1585 y Fk(w)1563 1593 y Fq(t)1678 1561 y Fz(])28 b(=)f Fr(\000)1923 1493 y Fz(1)p 1923 1538 V 1923 1629 a(2)1983 1561 y Fu(@)2034 1576 y Ft(\022)2073 1561 y Fu(@)2124 1576 y Ft(r)2163 1561 y Fz(\(ln)15 b Fu(J)2352 1576 y Fk(v)2395 1584 y Fq(t)2449 1561 y Fr(\016)22 b Fv(w)2602 1576 y Ft(t)2632 1561 y Fz(\))28 b Fr(\030)g Fu(t)2838 1519 y Fm(\000)p Fs(1)2932 1561 y Fu(:)617 b Fz(\(5.14\))296 1770 y(Up)33 b(to)f(factors)h(of)f Fu(t)1039 1734 y Ft(\016)1110 1770 y Fz(w)m(e)h(conclude)g(from)f (Lemma)f(3.12,)h(\(5.8\))g(-)g(\(\(5.14\))g(that)g(indeed)420 1896 y Fl(\015)420 1956 y(\015)476 1981 y Fu(@)532 1940 y Fs(2)527 2006 y Ft(r)572 1981 y Fz([)p Fu(W)14 b Fz(\()p Fu(t)p Fz(\))p Fu(\036)p Fz(]\()p Fu(r)m(;)j(\022)s Fz(\))1110 1896 y Fl(\015)1110 1956 y(\015)1182 1981 y Fu(;)1253 1896 y Fl(\015)1253 1956 y(\015)1308 1981 y Fu(r)1355 1940 y Fm(\000)p Fs(2)1449 1981 y Fu(@)1505 1940 y Fs(2)1500 2006 y Ft(\022)1546 1981 y Fz([)p Fu(W)d Fz(\()p Fu(t)p Fz(\))p Fu(\036)p Fz(]\()p Fu(r)m(;)j(\022)s Fz(\))2084 1896 y Fl(\015)2084 1956 y(\015)2155 1981 y Fu(;)2227 1896 y Fl(\015)2227 1956 y(\015)2282 1981 y Fu(r)2329 1940 y Fm(\000)p Fs(2)2423 1981 y Fz([)p Fu(W)d Fz(\()p Fu(t)p Fz(\))p Fu(\036)p Fz(]\()p Fu(r)m(;)j(\022)s Fz(\))2961 1896 y Fl(\015)2961 1956 y(\015)3044 1981 y Fr(\030)28 b Fu(t)3184 1940 y Fm(\000)p Fs(2)3278 1981 y Fu(;)271 b Fz(\(5.15\))150 2193 y(whic)m(h)33 b(\014nally)e(leads)i(to)f (\(5.2\).)p 3754 2193 4 66 v 3758 2130 59 4 v 3758 2193 V 3816 2193 4 66 v 150 2598 a Fp(5.2)135 b(Pro)l(ving)46 b(the)f(inclusion)g FF(Ran\(\012)2097 2555 y Fh(d)2097 2628 y Fx(+)2165 2598 y FF(\))32 b Fd(\022)i(H)2471 2616 y Fh(E)2532 2628 y Fc(d)150 2783 y Fz(W)-8 b(e)36 b(\014rst)g(sho)m(w)h (that)e(Ran\(\012)1267 2747 y Ft(d)1267 2808 y Fs(+)1327 2783 y Fz(\))d Fr(\022)i Fz(Ran\()p Fv(1)1777 2799 y Fs(\()p Ft(E)1856 2811 y Fq(d)1892 2799 y Ft(;)p Fm(1)p Fs(\))2014 2783 y Fz(\()p Fu(H)8 b Fz(\)\).)52 b(The)36 b(inclusion)e(is)h(based)i(on)f(the)g(in)m(ter-)150 2903 y(t)m(wining)c(form)m(ula)1548 3024 y Fu(H)8 b Fz(\012)1707 2983 y Ft(d)1707 3048 y Fs(+)1766 3024 y Fu(\036)27 b Fz(=)g(\012)2024 2983 y Ft(d)2024 3048 y Fs(+)2084 3024 y Fu(M)10 b Fz(\()p Fu(E)c Fz(\))p Fu(\036;)1149 b Fz(\(5.16\))150 3195 y(where)31 b Fu(\036)d Fr(2)g Fu(C)686 3158 y Fm(1)679 3219 y Fs(0)761 3195 y Fz(\(\()p Fu(E)909 3210 y Ft(d)949 3195 y Fu(;)17 b Fr(1)p Fz(\))g Fr(\002)g Fn(T)p Fz(\).)46 b(This)31 b(form)m(ula)d(implies)g(a)i(similar)d(in)m(tert)m(wining)i (for)g(resolv)m(en)m(ts)150 3315 y(and)k(through)f(functional)f (calculus)h(for)g(an)m(y)i(real)d(and)i(b)s(ounded)g(function:)1471 3526 y Fu(f)11 b Fz(\()p Fu(H)d Fz(\)\012)1765 3485 y Ft(d)1765 3551 y Fs(+)1851 3526 y Fz(=)27 b(\012)2024 3485 y Ft(d)2024 3551 y Fs(+)2084 3526 y Fu(M)10 b Fz(\()p Fu(f)h Fz(\()p Fu(E)6 b Fz(\)\))p Fu(;)1072 b Fz(\(5.17\))150 3738 y(whic)m(h)33 b(yields)f(the)h(result.)296 3858 y(F)-8 b(or)32 b(ev)m(ery)j Fu( )c Fr(2)d Fz(Dom)o(\()p Fu(H)8 b Fz(\))32 b(and)g Fu(t)c Fr(\025)g Fz(1)33 b(w)m(e)g(ha)m(v)m (e)479 4214 y Fr(h)p Fu(H)8 b( )t(;)17 b(e)763 4172 y Ft(itH)879 4214 y Fu(U)945 4229 y Fs(0)985 4214 y Fz(\()p Fu(t)p Fz(\))p Fu(\036)p Fr(i)27 b Fz(=)1323 4078 y Fl(Z)1440 4214 y Fu(dr)1554 4078 y Fl(Z)1670 4214 y Fu(d\022)p 1796 4127 373 4 v 30 w(H)8 b(e)1930 4185 y Fm(\000)p Ft(itH)2102 4214 y Fu( )t Fz(\()p Fu(r)m(;)17 b(\022)s Fz(\))p Fu(e)2423 4172 y Ft(iS)t Fs(\()p Ft(t;r)n(;\022)r Fs(\))2682 4214 y Fz([)p Fu(W)d Fz(\()p Fu(t)p Fz(\))p Fu(\036)p Fz(]\()p Fu(r)m(;)j(\022)s Fz(\))p Fu(:)329 b Fz(\(5.18\))150 4473 y(W)-8 b(e)33 b(mo)m(v)m(e)g Fu(H)40 b Fz(to)32 b(the)h(righ)m(t)f(and)h(use)g(\(4.19\):)792 4744 y Fr(h)p Fu(H)8 b( )t(;)17 b(e)1076 4703 y Ft(itH)1192 4744 y Fu(U)1258 4759 y Fs(0)1298 4744 y Fz(\()p Fu(t)p Fz(\))p Fu(\036)p Fr(i)27 b Fz(=)h Fr(h)p Fu(e)1721 4703 y Fm(\000)p Ft(itH)1892 4744 y Fu( )t(;)2003 4604 y Fl(\022)2076 4744 y Fr(\000)2207 4677 y Fz(1)p 2163 4721 135 4 v 2163 4812 a(8)p Fu(r)2259 4784 y Fs(2)2331 4744 y Fz(+)2439 4677 y Fu(\015)2495 4641 y Fs(2)2490 4701 y(1)p 2439 4721 96 4 v 2462 4812 a Fz(2)2566 4744 y(+)2674 4677 y Fu(\015)2730 4641 y Fs(2)2725 4701 y(2)p 2674 4721 V 2698 4812 a Fz(2)2780 4604 y Fl(\023)2870 4744 y Fu(U)2936 4759 y Fs(0)2975 4744 y Fz(\()p Fu(t)p Fz(\))p Fu(\036)p Fr(i)1539 5025 y Fz(+)p Fr(h)p Fu(e)1699 4984 y Fm(\000)p Ft(itH)1870 5025 y Fu( )t(;)17 b(i@)2065 5040 y Ft(t)2096 5025 y Fu(U)2162 5040 y Fs(0)2201 5025 y Fz(\()p Fu(t)p Fz(\))p Fu(\036)p Fr(i)p Fu(:)150 5195 y Fz(When)26 b(w)m(e)h (di\013eren)m(tiate)d Fu(iU)1197 5210 y Fs(0)1237 5195 y Fz(\()p Fu(t)p Fz(\))p Fu(\036)h Fz(with)g(resp)s(ect)i(to)e Fu(t)p Fz(,)i(w)m(e)f(get)g(t)m(w)m(o)g(terms:)40 b(the)25 b(\014rst)h(one)g(con)m(tains)150 5316 y(the)33 b(expression)806 5527 y Fr(\000)p Fz([)p Fu(@)961 5542 y Ft(t)992 5527 y Fu(S)6 b Fz(]\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))p Fu(e)1418 5486 y Ft(iS)t Fs(\()p Ft(t;r)n(;\022)r Fs(\))1677 5527 y Fz([)p Fu(W)d Fz(\()p Fu(t)p Fz(\))p Fu(\036)p Fz(]\()p Fu(r)m(;)j(\022)s Fz(\))27 b(=)h([)p Fu(U)2439 5542 y Fs(0)2478 5527 y Fz(\()p Fu(t)p Fz(\))p Fu(M)10 b Fz(\()p Fu(E)c Fz(\))p Fu(\036)p Fz(]\()p Fu(r)m(;)17 b(\022)s Fz(\))p Fu(;)1939 5776 y Fz(48)p eop %%Page: 49 52 49 51 bop 150 407 a Fz(cf.)44 b(\(3.38\);)32 b(the)h(second)g(one)g (con)m(tains)1218 614 y Fu(i@)1302 629 y Ft(t)1333 614 y Fz([)p Fu(W)14 b Fz(\()p Fu(t)p Fz(\))p Fu(\036)p Fz(]\()p Fu(r)m(;)j(\022)s Fz(\))27 b(=)h Fu(B)5 b Fz(\()p Fu(t)p Fz(\)[)p Fu(W)14 b Fz(\()p Fu(t)p Fz(\))p Fu(\036)p Fz(]\()p Fu(r)m(;)j(\022)s Fz(\))p Fu(;)150 820 y Fz(cf.)44 b(\(4.14\).)296 941 y(W)-8 b(e)33 b(then)g(obtain)691 1147 y Fr(h)p Fu(H)8 b( )t(;)17 b(e)975 1106 y Ft(itH)1091 1147 y Fu(U)1157 1162 y Fs(0)1197 1147 y Fz(\()p Fu(t)p Fz(\))p Fu(\036)p Fr(i)27 b Fz(=)g Fr(h)p Fu( )t(;)17 b(e)1730 1106 y Ft(itH)1847 1147 y Fu(U)1913 1162 y Fs(0)1952 1147 y Fz(\()p Fu(t)p Fz(\))p Fu(M)10 b Fz(\()p Fu(E)c Fz(\))p Fu(\036)p Fr(i)691 1352 y Fz(+)p Fr(h)p Fu(e)851 1311 y Fm(\000)p Ft(itH)1022 1352 y Fu( )t(;)1133 1211 y Fl(\022)1206 1352 y Fr(\000)1337 1284 y Fz(1)p 1293 1329 135 4 v 1293 1420 a(8)p Fu(r)1389 1391 y Fs(2)1461 1352 y Fz(+)1569 1284 y Fu(\015)1625 1248 y Fs(2)1620 1309 y(1)p 1569 1329 96 4 v 1592 1420 a Fz(2)1696 1352 y(+)1804 1284 y Fu(\015)1860 1248 y Fs(2)1855 1309 y(2)p 1804 1329 V 1828 1420 a Fz(2)1910 1211 y Fl(\023)2000 1352 y Fu(U)2066 1367 y Fs(0)2105 1352 y Fz(\()p Fu(t)p Fz(\))p Fu(\036)p Fr(i)22 b Fz(+)g Fr(h)p Fu(e)2517 1311 y Fm(\000)p Ft(itH)2689 1352 y Fu( )t(;)17 b(e)2845 1311 y Ft(iS)2919 1352 y Fu(B)5 b Fz(\()p Fu(t)p Fz(\))p Fu(W)14 b Fz(\()p Fu(t)p Fz(\))p Fu(\036)p Fr(i)p Fu(:)296 1610 y Fz(No)m(w)41 b(the)g(second)h(and)e (the)h(third)e(term)h(on)g(the)h(righ)m(t)f(hand)g(side)h(of)e(the)i (ab)s(o)m(v)m(e)g(equalit)m(y)150 1731 y(con)m(v)m(erge)c(to)e(zero)h (as)g Fu(t)d Fr(!)f(1)p Fz(.)52 b(This)36 b(follo)m(ws)e(readily)h (from)f(the)i(estimates)f(of)g(Subsection)h(7.1)150 1851 y(together)d(with)f(the)h(iden)m(tit)m(y)529 1977 y Fl(\000)575 2058 y Fu(\015)631 2017 y Fs(2)626 2082 y(1)692 2058 y Fz(+)22 b Fu(\015)846 2017 y Fs(2)841 2082 y(2)886 1977 y Fl(\001)948 2058 y Fu(e)993 2017 y Ft(iS)t Fs(\()p Ft(t;r)n(;\022)r Fs(\))1252 2058 y Fz([)p Fu(W)14 b Fz(\()p Fu(t)p Fz(\))p Fu(\036)p Fz(]\()p Fu(r)m(;)j(\022)s Fz(\))28 b(=)f Fr(\000)p Fu(e)2043 2017 y Ft(iS)t Fs(\()p Ft(t;r)n(;\022)r Fs(\))2320 1977 y Fl(\000)2365 2058 y Fu(@)2421 2017 y Fs(2)2416 2082 y Ft(r)2484 2058 y Fz(+)22 b Fu(r)2629 2017 y Fm(\000)p Fs(2)2723 2058 y Fu(@)2779 2017 y Fs(2)2774 2082 y Ft(\022)2819 1977 y Fl(\001)2881 2058 y Fz([)p Fu(W)14 b Fz(\()p Fu(t)p Fz(\))p Fu(\036)p Fz(]\()p Fu(r)m(;)j(\022)s Fz(\))p Fu(:)296 2265 y Fz(In)33 b(conclusion)1359 2385 y Fr(h)p Fu(H)8 b( )t(;)17 b Fz(\012)1668 2344 y Ft(d)1668 2410 y Fs(+)1727 2385 y Fu(\036)p Fr(i)27 b Fz(=)g Fr(h)p Fu( )t(;)17 b Fz(\012)2174 2344 y Ft(d)2174 2410 y Fs(+)2233 2385 y Fu(M)10 b Fz(\()p Fu(E)c Fz(\))p Fu(\036)p Fr(i)p Fu(;)150 2554 y Fz(yielding)31 b(\(5.16\).)296 2674 y(Clearly)h(\012) 705 2638 y Ft(d)705 2699 y Fs(+)765 2674 y Fu(M)10 b Fz(\()p Fu(E)c Fz(\))p Fu(\036)28 b Fr(?)g(H)1298 2689 y Ft(pp)1373 2674 y Fz(.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)g(pro)m(v)m(ed)i (Ran)o(\(\012)2436 2638 y Ft(d)2436 2699 y Fs(+)2496 2674 y Fz(\))27 b Fr(\022)h(H)2750 2689 y Ft(E)2802 2701 y Fq(d)2843 2674 y Fz(.)p 3754 2674 4 66 v 3758 2611 59 4 v 3758 2674 V 3816 2674 4 66 v 296 2913 a(W)-8 b(e)43 b(ha)m(v)m(e)h(no)m(w)g(established)f(I)f(of)h(Theorem)g(4.2.)73 b(The)44 b(remaining)c(part)i(of)h(the)g(pap)s(er)f(is)150 3033 y(dev)m(oted)c(to)f(pro)m(ving)f(I)s(I.)56 b(Giv)m(en)36 b(these)i(results,)g(I)s(I)s(I)f(of)f(Theorem)h(4.2)f(follo)m(ws)g (easily)-8 b(.)55 b(F)-8 b(or)36 b(the)150 3153 y(last)c(iden)m(tit)m (y)g Fu(H)8 b Fv(1)840 3168 y Fm(H)900 3179 y Fq(E)945 3197 y(d)1017 3153 y Fz(=)27 b(\012)1190 3117 y Ft(d)1190 3178 y Fs(+)1250 3153 y Fu(M)10 b Fz(\()p Fu(E)c Fz(\)\012)1578 3168 y Fs(+)1638 3153 y Fv(1)1694 3168 y Fm(H)1754 3179 y Fq(E)1799 3197 y(d)1876 3153 y Fz(w)m(e)34 b(in)m(v)m(ok)m(e)f (\(5.16\).)150 3484 y Fw(6)161 b(Asymptotic)40 b(completeness:)63 b(propagation)41 b(estimates)392 3667 y(in)54 b(quan)l(tum)f(mec)l (hanics)150 3886 y Fz(W)-8 b(e)33 b(no)m(w)h(start)f(the)h(pro)s(of)e (of)h(lo)s(cal)e(asymptotic)h(completeness.)46 b(In)33 b(this)g(section)g(w)m(e)h(are)f(going)150 4006 y(to)c(pro)m(v)m(e)i(v) -5 b(arious)29 b(propagation)f(estimates)i(for)f(states)h Fu( )t Fz(\()p Fu(t)p Fz(\))g(of)f(the)h(form)f(\(6.1\),)h(v)m(ery)h (similar)26 b(to)150 4127 y(the)33 b(ones)g(w)m(e)h(had)f(for)f(the)h (classical)e(problem.)296 4247 y(The)f(metho)s(d)e(w)m(e)i(use)g(is)e (to)g(sho)m(w)i(that)f(for)f(large)g Fu(t)p Fz(,)h Fu( )t Fz(\()p Fu(t)p Fz(\))g(is)f(\\lo)s(calized)f(near)i(the)g(attractor") 150 4367 y(in)35 b(the)g(sense)i(that)e Fu( )t Fz(\()p Fu(t)p Fz(\))e Fr(\030)g Fz(\005)1298 4331 y Ft(N)1298 4392 y(i)p Fs(=1)1416 4367 y Fu(F)1479 4382 y Ft(i)1508 4367 y Fz(\()p Fu(A)1619 4382 y Ft(i)1647 4367 y Fz(\()p Fu(t)p Fz(\)\))p Fu( )t Fz(\()p Fu(t)p Fz(\))i(with)g(eac)m(h)h Fu(F)2519 4382 y Ft(i)2548 4367 y Fz(\()p Fu(A)2659 4382 y Ft(i)2687 4367 y Fz(\()p Fu(t)p Fz(\)\))f(a)g(\\cut-o\013)g (function")f(of)150 4488 y(some)41 b(observ)-5 b(able.)71 b(It)42 b(will)d(b)s(e)j(crucial)e(that)h(the)h(Heisen)m(b)s(erg)h (deriv)-5 b(ativ)m(e)41 b(of)g(the)h(pro)s(duct)f(of)150 4608 y(cut-o\013)35 b(functions)g(is)g(essen)m(tially)g(p)s(ositiv)m (e.)51 b(F)-8 b(or)35 b(further)g(motiv)-5 b(ation)33 b(see)j(the)g(remarks)g(at)f(the)150 4728 y(b)s(eginning)c(of)h(the)h (pro)s(of)f(of)g(Lemma)f(8.2.)150 5015 y Fp(6.1)135 b(Preliminaries)47 b(and)e(the)g(Hel\013er-Sj\177)-67 b(ostrand)47 b(form)l(ula)150 5200 y Fz(W)-8 b(e)48 b(start)g(with)f(v)-5 b(arious)47 b(de\014nitions)g(and)h(preliminary)d(tec)m(hnical)i(results.)90 b(Let)47 b Fu(H)56 b Fz(b)s(e)48 b(the)150 5320 y(Hamiltonian)29 b(in)m(tro)s(duced)k(in)e(\(1.17\).)43 b(W)-8 b(e)33 b(are)g(going)e(to)h(study)i(states)f(of)f(the)h(form)1541 5527 y Fu( )t Fz(\()p Fu(t)p Fz(\))28 b(=)g Fu(e)1896 5486 y Fm(\000)p Ft(itH)2067 5527 y Fu(f)2115 5542 y Ft(E)2175 5527 y Fz(\()p Fu(H)8 b Fz(\))p Fu( )t(;)1190 b Fz(\(6.1\))1939 5776 y(49)p eop %%Page: 50 53 50 52 bop 150 407 a Fz(where)39 b(~)-50 b Fu(\017)35 b(>)g Fz(0)h(is)h(v)m(ery)h(small,)d Fu(f)1362 422 y Ft(E)1457 407 y Fr(2)g Fu(C)1635 371 y Fm(1)1628 431 y Fs(0)1710 407 y Fz(\(\()p Fu(E)c Fr(\000)26 b Fz(~)-50 b Fu(\017)q(=)p Fz(2)p Fu(;)17 b(E)30 b Fz(+)25 b(~)-49 b Fu(\017=)p Fz(2\)\),)37 b(0)e Fr(\024)g Fu(f)2897 422 y Ft(E)2992 407 y Fr(\024)g Fz(1)i(and)f Fu(f)3431 422 y Ft(E)3526 407 y Fz(=)f(1)h(on)150 527 y(\()p Fu(E)28 b Fr(\000)23 b Fz(~)-49 b Fu(\017=)p Fz(4)p Fu(;)17 b(E)28 b Fz(+)22 b(~)-49 b Fu(\017=)p Fz(4\))32 b(for)g Fu(E)i Fr(2)28 b Fz(\()p Fu(E)1433 542 y Ft(d)1473 527 y Fu(;)17 b Fr(1)p Fz(\))p Fr(n)p Fu(\033)1760 542 y Ft(pp)1835 527 y Fz(\()p Fu(H)8 b Fz(\).)43 b(The)34 b(state)f Fu( )e Fr(2)d Fu(L)2764 491 y Fs(2)2804 527 y Fz(\()p Fn(R)2908 491 y Fs(2)2953 527 y Fz(\).)296 648 y(W)-8 b(e)38 b(will)c(later)i (need)i(t)m(w)m(o)g(other)f(cut-o\013)g(functions.)56 b(Namely)-8 b(,)37 b(consider)h Fu(f)3166 663 y Fs(1)p Ft(;E)3315 648 y Fr(2)e Fu(C)3494 611 y Fm(1)3487 672 y Fs(0)3568 648 y Fz(\(\()p Fu(E)c Fr(\000)150 768 y Fz(4)q(~)-50 b Fu(\017=)p Fz(5)p Fu(;)17 b(E)34 b Fz(+)29 b(4)q(~)-50 b Fu(\017=)p Fz(5\)\),)44 b(0)g Fr(\024)g Fu(f)1186 783 y Fs(1)p Ft(;E)1345 768 y Fr(\024)g Fz(1)e(and)g(equal)g (to)g(1)g(on)g(\()p Fu(E)35 b Fr(\000)29 b Fz(3)q(~)-50 b Fu(\017=)p Fz(4)p Fu(;)17 b(E)34 b Fz(+)28 b(3)q(~)-50 b Fu(\017)q(=)p Fz(4\).)71 b(Moreo)m(v)m(er,)150 888 y(consider)550 862 y(~)529 888 y Fu(f)577 903 y Ft(E)664 888 y Fr(2)28 b Fu(C)835 852 y Fm(1)828 913 y Fs(0)910 888 y Fz(\(\()p Fu(E)e Fr(\000)c Fz(~)-50 b Fu(\017)q(;)17 b(E)26 b Fz(+)21 b(~)-50 b Fu(\017)q Fz(\)\),)31 b(0)d Fr(\024)1837 862 y Fz(~)1816 888 y Fu(f)1864 903 y Ft(E)1951 888 y Fr(\024)h Fz(1)i(and)h(equal)f(to)h(1)f(on)h(\()p Fu(E)26 b Fr(\000)21 b Fz(7)q(~)-50 b Fu(\017=)p Fz(8)p Fu(;)17 b(E)26 b Fz(+)20 b(7)q(~)-50 b Fu(\017=)p Fz(8\).)150 1009 y(W)-8 b(e)33 b(then)g(ha)m(v)m(e)h(the)f(prop)s(ert)m(y)g(that) 1367 1229 y Fu(f)1415 1244 y Fs(1)p Ft(;E)1529 1229 y Fu(f)1577 1244 y Ft(E)1665 1229 y Fz(=)27 b Fu(f)1816 1244 y Ft(E)1876 1229 y Fu(;)2038 1202 y Fz(~)2017 1229 y Fu(f)2065 1244 y Ft(E)2125 1229 y Fu(f)2173 1244 y Fs(1)p Ft(;E)2315 1229 y Fz(=)h Fu(f)2467 1244 y Fs(1)p Ft(;E)2581 1229 y Fu(:)150 1449 y Fz(Let)33 b(us)g(de\014ne)h(the)f (regularized)e(radial)g(and)h(transv)m(erse)j(v)m(elo)s(cities)d(b)m(y) 1112 1669 y(~)-58 b Fu(\032)28 b Fz(:=)1333 1642 y(~)1312 1669 y Fu(f)1360 1684 y Ft(E)1419 1669 y Fz(\()p Fu(H)8 b Fz(\))p Fu(\032)1655 1642 y Fz(~)1634 1669 y Fu(f)1682 1684 y Ft(E)1742 1669 y Fz(\()p Fu(H)g Fz(\))p Fu(;)2058 1642 y Fz(~)2047 1669 y Fu(\030)33 b Fz(:=)2274 1642 y(~)2253 1669 y Fu(f)2301 1684 y Ft(E)2361 1669 y Fz(\()p Fu(H)8 b Fz(\))p Fu(\030)2594 1642 y Fz(~)2574 1669 y Fu(f)2622 1684 y Ft(E)2680 1669 y Fz(\()p Fu(H)g Fz(\))p Fu(:)752 b Fz(\(6.2\))296 1889 y(Let)40 b(us)g(recall)f(\(see)h([D-G2,) g(App)s(endix)g(C])h(or)e([M\034]\))h(that)f(for)g(all)f(b)s(ounded)i (\(or)f(p)s(ossibly)150 2009 y(un)m(b)s(ounded\))44 b(self-adjoin)m(t)d (op)s(erators)h Fu(A)h Fz(and)g Fu(B)k Fz(\(on)c(the)g(same)g(Hilb)s (ert)e(space\))i(and)g(for)f(all)150 2130 y Fu(f)c Fr(2)28 b Fu(C)407 2093 y Fm(1)400 2154 y Fs(0)482 2130 y Fz(\()p Fn(R)5 b Fz(\))38 b(w)m(e)c(ma)m(y)e(represen)m(t)j(\()p 1476 2048 57 4 v Fu(@)j Fz(denotes)c(di\013eren)m(tiation)d(with)h (resp)s(ect)i(to)p 3222 2075 50 4 v 32 w Fu(z)5 b Fz(\):)440 2398 y([)p Fu(A;)17 b(f)11 b Fz(\()p Fu(B)5 b Fz(\)])28 b(=)f Fr(\000)1048 2331 y Fz(1)p 1043 2375 59 4 v 1043 2466 a Fu(\031)1129 2262 y Fl(Z)1185 2488 y Fe(C)p 1253 2317 57 4 v 1253 2398 a Fu(@)1332 2372 y Fz(~)1310 2398 y Fu(f)11 b Fz(\()p Fu(z)t Fz(\)\()p Fu(B)28 b Fr(\000)23 b Fu(z)t Fz(\))1821 2357 y Fm(\000)p Fs(1)1916 2398 y Fz([)p Fu(A;)17 b(B)5 b Fz(]\()p Fu(B)27 b Fr(\000)c Fu(z)t Fz(\))2492 2357 y Fm(\000)p Fs(1)2587 2398 y Fu(dxdy)t Fz(;)43 b Fu(z)32 b Fz(=)c Fu(x)22 b Fz(+)g Fu(iy)t(;)290 b Fz(\(6.3\))150 2688 y(where)453 2662 y(~)432 2688 y Fu(f)43 b Fz(is)32 b(a)g(smo)s(oth)g(compactly)g(supp)s(orted)h(almost) e(analytic)g(extension)j(of)e Fu(f)11 b Fz(.)296 2809 y(W)-8 b(e)36 b(shall)e(need)j(this)e(and)g(other)h(comm)m(utator)e (form)m(ulas)g(for)h(functions)g(\000)h(of)f(the)g(t)m(yp)s(e)i Fu(F)3766 2824 y Fs(+)150 2929 y Fz(or)32 b(t)m(yp)s(e)i Fu(F)552 2944 y Fs(++)666 2929 y Fz(.)43 b(An)33 b(almost)e(analytic)g (extension)j(of)e(\000)g(can)h(b)s(e)g(constructed)h(ob)s(eying)1010 3149 y Fr(j)p 1038 3068 V Fu(@)1100 3124 y Fz(~)1094 3149 y(\000\()p Fu(z)t Fz(\))p Fr(j)28 b(\024)g Fu(C)1511 3164 y Ft(k)1554 3149 y Fr(h)p Fu(z)t Fr(i)1681 3108 y Fm(\000)p Fs(1)p Fm(\000)p Ft(k)1869 3149 y Fr(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)2122 3108 y Ft(k)2164 3149 y Fz(;)115 b Fu(z)32 b Fr(2)c Fn(C)20 b Fu(;)120 b(k)31 b Fr(2)d Fn(N)9 b Fu(:)665 b Fz(\(6.4\))150 3369 y(Then,)753 3632 y([)p Fu(A;)17 b Fz(\000\()p Fu(B)5 b Fz(\)])83 b(=)g Fr(\000)1474 3565 y Fz(1)p 1469 3609 59 4 v 1469 3700 a Fu(\031)1555 3496 y Fl(Z)1610 3722 y Fe(C)p 1679 3551 57 4 v 1679 3632 a Fu(@)1742 3607 y Fz(~)1736 3632 y(\000\()p Fu(z)t Fz(\)\()p Fu(B)27 b Fr(\000)c Fu(z)t Fz(\))2248 3591 y Fm(\000)p Fs(1)2343 3632 y Fz([)p Fu(A;)17 b(B)5 b Fz(]\()p Fu(B)28 b Fr(\000)22 b Fu(z)t Fz(\))2919 3591 y Fm(\000)p Fs(1)3014 3632 y Fu(dxdy)1223 3887 y Fz(=)1392 3819 y(1)p 1392 3864 49 4 v 1392 3955 a(2)1450 3887 y Fr(f)p Fz(\000)1561 3846 y Fm(0)1585 3887 y Fz(\()p Fu(B)5 b Fz(\)[)p Fu(A;)17 b(B)5 b Fz(])22 b(+)g([)p Fu(A;)17 b(B)5 b Fz(]\000)2421 3846 y Fm(0)2444 3887 y Fz(\()p Fu(B)g Fz(\))p Fr(g)22 b Fz(+)g Fu(R)2843 3902 y Fs(1)2883 3887 y Fu(;)714 b Fz(\(6.5\))150 4132 y(where)720 4376 y Fu(R)794 4391 y Fs(1)862 4376 y Fz(=)27 b Fr(\000)1082 4308 y Fz(1)p 1052 4353 108 4 v 1052 4444 a(2)p Fu(\031)1187 4240 y Fl(Z)1242 4466 y Fe(C)p 1311 4295 57 4 v 1311 4376 a Fu(@)1374 4350 y Fz(~)1368 4376 y(\000\()p Fu(z)t Fz(\)\()p Fu(B)h Fr(\000)22 b Fu(z)t Fz(\))1880 4335 y Fm(\000)p Fs(1)3624 4376 y Fz(\(6.6\))720 4583 y Fr(\002)814 4503 y Fl(\010)872 4583 y Fz(\()p Fu(B)28 b Fr(\000)22 b Fu(z)t Fz(\))1198 4542 y Fm(\000)p Fs(1)1293 4583 y Fz([)p Fu(B)5 b(;)17 b Fz([)p Fu(A;)g(B)5 b Fz(]])23 b Fr(\000)f Fz([)p Fu(B)5 b(;)17 b Fz([)p Fu(A;)g(B)5 b Fz(]]\()p Fu(B)28 b Fr(\000)22 b Fu(z)t Fz(\))2595 4542 y Fm(\000)p Fs(1)2690 4503 y Fl(\011)2765 4583 y Fz(\()p Fu(B)27 b Fr(\000)c Fu(z)t Fz(\))3091 4542 y Fm(\000)p Fs(1)3186 4583 y Fu(dxdy)t(:)150 4803 y Fz(Also,)1034 4924 y([)p Fu(A;)17 b Fz(\000\()p Fu(B)5 b Fz(\)])27 b(=)1552 4835 y Fr(p)p 1635 4835 85 4 v 89 x Fz(\000)1696 4895 y Fm(0)1719 4924 y Fz(\()p Fu(B)5 b Fz(\)[)p Fu(A;)17 b(B)5 b Fz(])2124 4835 y Fr(p)p 2207 4835 V 89 x Fz(\000)2268 4895 y Fm(0)2292 4924 y Fz(\()p Fu(B)g Fz(\))22 b(+)g Fu(R)2641 4939 y Fs(1)2703 4924 y Fz(+)g Fu(R)2875 4939 y Fs(2)2914 4924 y Fu(;)683 b Fz(\(6.7\))150 5098 y(where)540 5458 y Fu(R)614 5473 y Fs(2)681 5458 y Fz(=)795 5391 y(1)p 795 5435 49 4 v 795 5526 a(2)870 5348 y Fl(h)917 5458 y Fz([)p Fu(A;)17 b(B)5 b Fz(])p Fu(;)1211 5368 y Fl(p)p 1311 5368 240 4 v 90 x Fz(\000)1372 5429 y Fm(0)1395 5458 y Fz(\()p Fu(B)g Fz(\))1550 5348 y Fl(i)1614 5368 y(p)p 1713 5368 V 1713 5458 a Fz(\000)1774 5429 y Fm(0)1797 5458 y Fz(\()p Fu(B)g Fz(\))23 b Fr(\000)2084 5391 y Fz(1)p 2084 5435 49 4 v 2084 5526 a(2)2143 5368 y Fl(p)p 2243 5368 240 4 v 90 x Fz(\000)2304 5429 y Fm(0)2327 5458 y Fz(\()p Fu(B)5 b Fz(\))2498 5348 y Fl(h)2545 5458 y Fz([)p Fu(A;)17 b(B)5 b Fz(])p Fu(;)2839 5368 y Fl(p)p 2939 5368 V 90 x Fz(\000)3000 5429 y Fm(0)3023 5458 y Fz(\()p Fu(B)g Fz(\))3178 5348 y Fl(i)3242 5458 y Fu(:)355 b Fz(\(6.8\))1939 5776 y(50)p eop %%Page: 51 54 51 53 bop 150 407 a Fp(6.2)135 b(Non-comm)l(utativit)l(y)48 b(and)c(\\in)l(tegrable)k(remainders")150 592 y Fz(T)-8 b(o)31 b(go)f(from)g(classical)f(to)i(quan)m(tum)g(mec)m(hanics)g(w)m (e)h(ha)m(v)m(e)h(to)d(deal)g(with)h(\\errors")g(coming)e(from)150 712 y(the)d(non-comm)m(utativit)m(y)d(of)i(certain)g(op)s(erators.)41 b(W)-8 b(e)26 b(will)d(use)j(the)g(follo)m(wing)d(tec)m(hnical)i (lemmas)150 832 y(to)k(sho)m(w)h(that)f(these)h(\\errors")f(are)g (small.)40 b(In)29 b(order)g(to)g(simplify)d(notation,)j(w)m(e)h(will)c (often)j(write)150 953 y Fu(r)35 b Fz(instead)e(of)f(the)h(regularized) e(mo)s(dulus)p 1737 898 47 4 v 32 w Fu(r)k Fz(\(see)f(\(4.2\)\).)150 1127 y Fv(Lemma)j(6.1.)42 b FA(Supp)-5 b(ose)35 b Fu(F)s(;)17 b(G)27 b Fr(2)i Fu(C)1521 1090 y Fm(1)1514 1151 y Fs(0)1595 1127 y Fz(\()p Fn(R)1699 1142 y Fs(+)1764 1127 y Fz(\))35 b FA(and)g Fu(F)42 b Fz(=)28 b(1)34 b FA(on)h(a)g(neighb)-5 b(orho)g(o)g(d)33 b(of)i(the)g(supp)-5 b(ort)35 b(of)150 1247 y Fu(G)p FA(.)43 b(L)-5 b(et)32 b Fu(B)k FA(b)-5 b(e)30 b(one)h(of)f(the)h(op)-5 b(er)g(ators)31 b Fu(H)8 b FA(,)31 b Fu(A)1782 1262 y Ft(C)1841 1247 y Fu(=t)p FA(,)41 b Fz(~)-58 b Fu(\032)31 b FA(or)2200 1221 y Fz(~)2190 1247 y Fu(\030)t FA(.)44 b(Then)30 b(for)h(every)g(inte)-5 b(ger)30 b Fu(N)38 b Fr(\025)29 b Fz(1)p FA(,)i(ther)-5 b(e)150 1367 y(exists)35 b Fu(C)f(>)28 b Fz(0)34 b FA(such)h(that)760 1607 y Fr(jj)p Fz(\(1)21 b Fr(\000)i Fu(F)14 b Fz(\()p Fu(r)s(=t)p Fz(\)\)\()p Fu(B)26 b Fr(\000)d Fu(z)t Fz(\))1671 1566 y Fm(\000)p Fs(1)1766 1607 y Fu(G)p Fz(\()p Fu(r)s(=t)p Fz(\))p Fr(jj)j(\024)2336 1540 y Fu(C)p 2248 1584 253 4 v 2248 1675 a Fr(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)2527 1467 y Fl(\022)2673 1540 y Fr(h)p Fu(z)t Fr(i)p 2610 1584 V 2610 1675 a(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)2873 1467 y Fl(\023)2947 1489 y Ft(N)3030 1607 y Fu(t)3065 1566 y Fm(\000)p Ft(N)3188 1607 y Fu(:)150 1844 y Fv(Pro)s(of.)45 b Fz(One)34 b(can)f(\014nd)h(a)f(function)f Fu(G)1621 1859 y Fs(1)1694 1844 y Fz(suc)m(h)i(that)f Fu(G)2203 1859 y Fs(1)2243 1844 y Fu(G)28 b Fz(=)h Fu(G)k Fz(and)g Fu(F)14 b(G)2907 1859 y Fs(1)2975 1844 y Fz(=)29 b Fu(G)3157 1859 y Fs(1)3196 1844 y Fz(.)45 b(Then)35 b(for)d(an)m(y)150 1965 y Fu(N)38 b Fr(2)28 b Fn(N)48 b Fz(w)m(e)33 b(ma)m(y)g(write)f (\(abbreviating)f(b)s(elo)m(w)i Fu(F)41 b Fz(=)28 b Fu(F)14 b Fz(\()p Fu(r)s(=t)p Fz(\),)31 b Fu(G)d Fz(=)g Fu(G)p Fz(\()p Fu(r)s(=t)p Fz(\))k(and)g Fu(G)3291 1980 y Fs(1)3358 1965 y Fz(=)c Fu(G)3539 1980 y Fs(1)3578 1965 y Fz(\()p Fu(r)s(=t)p Fz(\)\))1015 2150 y(\(1)22 b Fr(\000)h Fu(F)14 b Fz(\)\()p Fu(B)27 b Fr(\000)22 b Fu(z)t Fz(\))1664 2109 y Fm(\000)p Fs(1)1759 2150 y Fu(G)28 b Fz(=)f(\(1)22 b Fr(\000)h Fu(F)14 b Fz(\)\()p Fu(B)27 b Fr(\000)22 b Fu(z)t Fz(\))2616 2109 y Fm(\000)p Fs(1)2711 2150 y Fu(G)2788 2109 y Ft(N)2788 2175 y Fs(1)2856 2150 y Fu(G:)150 2336 y Fz(Due)33 b(to)f(the)h(supp)s(ort)g(conditions)e(w)m(e)j(ha)m(v) m(e)952 2521 y(\(1)22 b Fr(\000)g Fu(F)14 b Fz(\)\()p Fu(B)27 b Fr(\000)c Fu(z)t Fz(\))1601 2480 y Fm(\000)p Fs(1)1696 2521 y Fu(G)k Fz(=)h(\(1)22 b Fr(\000)g Fu(F)14 b Fz(\))p Fu(ad)2329 2480 y Ft(N)2329 2546 y(G)2384 2555 y Fj(1)2423 2521 y Fz(\(\()p Fu(B)27 b Fr(\000)22 b Fu(z)t Fz(\))2786 2480 y Fm(\000)p Fs(1)2881 2521 y Fz(\))p Fu(G;)601 b Fz(\(6.9\))150 2716 y(where)34 b Fu(ad)534 2680 y Fs(0)534 2741 y Ft(G)589 2750 y Fj(1)627 2716 y Fz(\()p Fu(B)5 b Fz(\))28 b(=)g Fu(B)37 b Fz(and)c Fu(ad)1317 2680 y Ft(k)1317 2741 y(G)1372 2750 y Fj(1)1411 2716 y Fz(\()p Fu(B)5 b Fz(\))27 b(=)h([)p Fu(ad)1826 2675 y Ft(k)r Fm(\000)p Fs(1)1826 2743 y Ft(G)1881 2752 y Fj(1)1959 2716 y Fz(\()p Fu(B)5 b Fz(\))p Fu(;)17 b(G)2235 2731 y Fs(1)2274 2716 y Fz(])32 b(for)g Fu(k)f Fr(\025)d Fz(1.)44 b(Then)1000 2930 y Fu(ad)1102 2889 y Ft(N)1102 2955 y(G)1157 2964 y Fj(1)1196 2930 y Fz(\(\()p Fu(B)27 b Fr(\000)c Fu(z)t Fz(\))1560 2889 y Fm(\000)p Fs(1)1655 2930 y Fz(\))k(=)1971 2836 y Fl(X)1824 3048 y Ft(k)1861 3057 y Fj(1)1895 3048 y Fs(+)p Fm(\001\001\001)o Fs(+)p Ft(k)2101 3056 y Fq(n)2143 3048 y Fs(=)p Ft(N)2278 2930 y Fu(C)2348 2945 y Ft(k)2385 2954 y Fj(1)2419 2945 y Ft(;)p Fm(\001\001\001)10 b Ft(;k)2566 2953 y Fq(n)3576 2930 y Fz(\(6.10\))1000 3190 y(\()p Fu(B)28 b Fr(\000)22 b Fu(z)t Fz(\))1326 3149 y Fm(\000)p Fs(1)1421 3190 y Fu(ad)1523 3146 y Ft(k)1560 3155 y Fj(1)1523 3217 y Ft(G)1578 3226 y Fj(1)1617 3190 y Fz(\()p Fu(B)5 b Fz(\))17 b Fr(\001)g(\001)g (\001)d Fz(\()p Fu(B)27 b Fr(\000)c Fu(z)t Fz(\))2247 3149 y Fm(\000)p Fs(1)2342 3190 y Fu(ad)2444 3147 y Ft(k)2481 3155 y Fq(n)2444 3217 y Ft(G)2499 3226 y Fj(1)2538 3190 y Fz(\()p Fu(B)5 b Fz(\)\()p Fu(B)27 b Fr(\000)c Fu(z)t Fz(\))3019 3149 y Fm(\000)p Fs(1)3114 3190 y Fu(:)150 3375 y Fz(It)29 b(is)g(then)h(easy)g(to)f(see)h(that)f(eac)m(h)h(comm)m (utator)e(brings)h(a)g(deca)m(y)h(of)f(order)g Fu(t)3047 3339 y Fm(\000)p Fs(1)3142 3375 y Fz(;)h(more)f(precisely)-8 b(,)150 3496 y(if)31 b Fu(B)38 b Fz(is)32 b(either)41 b(~)-57 b Fu(\032)32 b Fz(or)938 3470 y(~)928 3496 y Fu(\030)37 b Fz(then)1434 3616 y Fr(jj)p Fu(ad)1592 3562 y Ft(k)1629 3572 y Fq(j)1592 3643 y Ft(G)1647 3652 y Fj(1)1685 3616 y Fz(\()p Fu(B)5 b Fz(\))p Fr(jj)27 b(\024)h Fz(Const)23 b Fr(\001)f Fu(t)2385 3575 y Fm(\000)p Ft(k)2477 3585 y Fq(j)2514 3616 y Fu(:)150 3776 y Fz(If)33 b Fu(B)f Fz(=)c Fu(H)40 b Fz(then)512 3991 y Fr(jj)p Fu(ad)670 3937 y Ft(k)707 3947 y Fq(j)670 4018 y Ft(G)725 4027 y Fj(1)762 3991 y Fz(\()p Fu(H)8 b Fz(\))22 b Fr(\001)g(h)p Fu(H)8 b Fr(i)1166 3950 y Fm(\000)p Fs(1)1259 3991 y Fr(jj)27 b(\024)h Fz(Const)23 b Fr(\001)f Fu(t)1804 3950 y Fm(\000)p Ft(k)1896 3960 y Fq(j)1933 3991 y Fu(;)114 b Fr(jjh)p Fu(H)8 b Fr(i)p Fz(\()p Fu(H)28 b Fr(\000)23 b Fu(z)t Fz(\))2631 3950 y Fm(\000)p Fs(1)2726 3991 y Fr(jj)k(\024)h Fz(Const)3236 3924 y Fr(h)p Fu(z)t Fr(i)p 3173 3969 V 3173 4060 a(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)3436 3991 y Fu(:)150 4233 y Fz(If)33 b Fu(B)f Fz(=)c Fu(A)531 4248 y Ft(C)590 4233 y Fu(=t)33 b Fz(then)g(the)g(only)f(nonzero)h(con) m(tribution)f(comes)g(from)g Fu(k)2799 4248 y Ft(j)2863 4233 y Fz(=)27 b(1)32 b(whic)m(h)h(giv)m(es)1299 4466 y Fu(ad)1401 4425 y Fs(1)1401 4490 y Ft(G)1456 4499 y Fj(1)1494 4466 y Fz(\()p Fu(A)1605 4481 y Ft(C)1664 4466 y Fu(=t)p Fz(\))28 b(=)1928 4398 y(1)p 1928 4443 49 4 v 1935 4534 a Fu(t)2008 4466 y Fr(\001)2068 4398 y Fu(r)p 2068 4443 47 4 v 2074 4534 a(t)2125 4466 y(G)2202 4425 y Fm(0)2202 4490 y Fs(1)2241 4466 y Fz(\()p Fu(r)s(=t)p Fz(\))f Fr(\030)2591 4398 y Fz(1)p 2591 4443 49 4 v 2598 4534 a Fu(t)2649 4466 y(:)150 4677 y Fz(W)-8 b(e)42 b(then)h(sum)f(up)g (the)h(con)m(tributions)e(coming)g(from)f(eac)m(h)j(comm)m(utator)e (and)h(conclude)h(the)150 4797 y(lemma.)p 3754 4797 4 66 v 3758 4735 59 4 v 3758 4797 V 3816 4797 4 66 v 150 5036 a Fv(Remark)p Fz(.)51 b(Com)m(bining)34 b(Lemma)g(6.1)h(with)g (\(6.3\))f(w)m(e)j(obtain)d(for)h(ev)m(ery)i(function)e Fu(f)43 b Fr(2)33 b Fu(C)3603 5000 y Fm(1)3596 5060 y Fs(0)3677 5036 y Fz(\()p Fn(R)5 b Fz(\))150 5156 y(and)33 b Fu(B)k Fz(an)m(y)d(of)e(the)h(op)s(erators)f Fu(H)8 b Fz(,)32 b Fu(A)1567 5171 y Ft(C)1626 5156 y Fu(=t)p Fz(,)42 b(~)-58 b Fu(\032)33 b Fz(or)1982 5130 y(~)1972 5156 y Fu(\030)5 b Fz(:)1178 5342 y Fr(jj)p Fz(\(1)21 b Fr(\000)i Fu(F)14 b Fz(\()p Fu(r)s(=t)p Fz(\)\))p Fu(f)d Fz(\()p Fu(B)5 b Fz(\))p Fu(G)p Fz(\()p Fu(r)s(=t)p Fz(\))p Fr(jj)25 b Fz(=)j Fr(O)s Fz(\()p Fu(t)2602 5300 y Fm(\0001)2732 5342 y Fz(\))p Fu(:)779 b Fz(\(6.11\))296 5527 y(W)-8 b(e)33 b(con)m(tin)m(ue)g(with)g(another)f(lo)s(calization)d(result)j (needed)j(later.)1939 5776 y(51)p eop %%Page: 52 55 52 54 bop 150 407 a Fv(Lemma)34 b(6.2.)40 b FA(Assume)32 b Fu(F)1182 422 y Fs(+)p Fm(\000)1328 407 y FA(has)g(supp)-5 b(ort)32 b(in)g Fn(R)2025 422 y Fs(+)2123 407 y FA(and)f(c)-5 b(onsider)31 b Fu(F)s(;)17 b(G;)g(L)28 b Fr(2)g Fu(C)3187 371 y Fm(1)3180 431 y Fs(0)3261 407 y Fz(\()p Fn(R)5 b Fz(\))39 b FA(such)32 b(that)150 527 y Fu(F)46 b FA(e)-5 b(quals)32 b Fz(1)g FA(on)g(a)g(neighb)-5 b(orho)g(o)g(d)31 b(of)h(the)g(supp)-5 b(ort)32 b(of)g Fu(G)p FA(.)44 b(Assume)32 b(that)h Fu(B)38 b FA(and)31 b Fu(D)k FA(ar)-5 b(e)32 b(either)h Fu(H)8 b FA(,)159 648 y Fz(~)-58 b Fu(\032)35 b FA(or)371 621 y Fz(~)361 648 y Fu(\030)t FA(.)45 b(Then)924 768 y Fr(jj)p Fz(\(1)21 b Fr(\000)i Fu(F)14 b Fz(\()p Fu(B)5 b Fz(\)\))21 b Fr(\001)h Fu(L)p Fz(\()p Fu(D)s Fz(\))g Fr(\001)g Fu(G)p Fz(\()p Fu(B)5 b Fz(\))22 b Fr(\001)g Fu(F)2194 783 y Fs(+)p Fm(\000)2308 768 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fr(jj)27 b Fz(=)g Fr(O)s Fz(\()p Fu(t)2856 727 y Fm(\0001)2986 768 y Fz(\))p Fu(:)525 b Fz(\(6.12\))150 971 y Fv(Pro)s(of.)43 b Fz(Clearly)32 b(if)f Fu(D)f Fz(=)e Fu(B)38 b Fz(there)33 b(is)f(nothing)f(to)h(b)s(e) h(pro)m(v)m(en.)45 b(First,)32 b(assume)h(that)f Fu(B)h Fz(=)j(~)-58 b Fu(\032)33 b Fz(and)150 1092 y Fu(D)38 b Fz(=)390 1065 y(~)380 1092 y Fu(\030)t Fz(;)h(w)m(e)f(can)f(\014nd)h (a)e(function)h Fu(G)1573 1107 y Fs(1)1649 1092 y Fz(as)g(in)f(the)i (previous)f(lemma)e(suc)m(h)j(that)f Fu(G)3292 1107 y Fs(1)3331 1092 y Fu(G)e Fz(=)g Fu(G)i Fz(and)150 1212 y Fu(F)14 b(G)304 1227 y Fs(1)371 1212 y Fz(=)27 b Fu(G)551 1227 y Fs(1)591 1212 y Fz(.)43 b(W)-8 b(e)33 b(then)g(see)h(that)425 1432 y(\(1)22 b Fr(\000)h Fu(F)14 b Fz(\()p Fu(B)5 b Fz(\)\))21 b Fr(\001)h Fu(L)p Fz(\()p Fu(D)s Fz(\))g Fr(\001)g Fu(G)p Fz(\()p Fu(B)5 b Fz(\))28 b(=)f(\(1)22 b Fr(\000)h Fu(F)14 b Fz(\()p Fu(B)5 b Fz(\)\))22 b Fr(\001)f Fu(ad)2288 1391 y Ft(N)2288 1457 y(G)2343 1466 y Fj(1)2378 1457 y Fs(\()p Ft(B)s Fs(\))2493 1432 y Fz(\()p Fu(L)p Fz(\()p Fu(D)s Fz(\)\))h Fr(\001)g Fu(G)p Fz(\()p Fu(B)5 b Fz(\))p Fu(;)114 b(N)39 b Fr(2)28 b Fn(N)9 b Fu(:)150 1674 y Fz(Lo)s(oking)22 b(at)g(\(4.5\))g(and)h(\(4.7\))g(w)m(e)h(see)g (that)e(ev)m(ery)j(time)c(w)m(e)j(comm)m(ute)32 b(~)-58 b Fu(\032)23 b Fz(with)3035 1647 y(~)3025 1674 y Fu(\030)k Fz(w)m(e)d(gain)e(an)h(\\extra)150 1794 y(1)p Fu(=)p 248 1739 47 4 v(r)32 b Fz(deca)m(y".)43 b(The)31 b(same)e(thing)g(is)g (true)h(when)h(w)m(e)f(comm)m(ute)f Fu(G)2555 1809 y Fs(1)2595 1794 y Fz(\()9 b(~)-58 b Fu(\032)p Fz(\))29 b(with)h Fu(L)p Fz(\()3084 1768 y(~)3074 1794 y Fu(\030)t Fz(\))g(via)f(the)h(Hel\013er-)150 1914 y(Sj\177)-49 b(ostrand)38 b(form)m(ula;)f(hence)i Fu(N)48 b Fz(comm)m(utators)36 b(pro)m(vide)i(us)f(with)g Fu(N)48 b Fz(extra)38 b(factors)f(of)g(1)p Fu(=)p 3605 1860 V(r)r Fz(.)58 b(In)150 2035 y(order)24 b(to)f(transform)f(them)h(in)m(to)g(1)p Fu(=t)g Fz(w)m(e)i(emplo)m(y)e (the)h(previous)g(lemma:)36 b(m)m(ultiply)21 b(eac)m(h)k(1)p Fu(=r)g Fz(with)150 2169 y(1)i(=)h Fu(F)407 2118 y Fs(\(1\))393 2191 y(+)p Fm(\000)507 2169 y Fz(\()p Fu(r)s(=t)p Fz(\))8 b(+)g(1)g Fr(\000)g Fu(F)1025 2118 y Fs(\(1\))1011 2191 y(+)p Fm(\000)1126 2169 y Fz(\()p Fu(r)s(=t)p Fz(\))25 b(where)i Fu(F)1710 2118 y Fs(\(1\))1696 2191 y(+)p Fm(\000)1836 2169 y Fz(has)f(the)g(prop)s(ert)m(y)h(that)e(\(1)8 b Fr(\000)g Fu(F)3017 2118 y Fs(\(1\))3003 2191 y(+)p Fm(\000)3118 2169 y Fz(\))p Fu(F)3219 2184 y Fs(+)p Fm(\000)3361 2169 y Fz(=)27 b(0)f(and)f(its)150 2289 y(supp)s(ort)33 b(is)g(included)f (in)h Fn(R)1180 2304 y Fs(+)1246 2289 y Fz(.)44 b(The)34 b(previous)f(lemma)e(ensures)k(that)e(the)g(resulting)f(cross)i(terms) 150 2420 y(con)m(taining)h(at)h(least)g(one)1175 2381 y Fs(1)p 1175 2397 36 4 v 1176 2417 34 3 v 1176 2454 a Ft(r)1220 2420 y Fz(\(1)25 b Fr(\000)g Fu(F)1511 2369 y Fs(\(1\))1497 2442 y(+)p Fm(\000)1611 2420 y Fz(\))36 b(are)h(of)e(order)i Fr(O)s Fz(\()p Fu(t)2380 2384 y Fm(\0001)2510 2420 y Fz(\))f(while)f(\(1)p Fu(=)p 2978 2365 47 4 v(r)r Fz(\))25 b Fr(\001)f Fu(F)3216 2369 y Fs(\(1\))3202 2442 y(+)p Fm(\000)3316 2420 y Fz(\()p Fu(r)s(=t)p Fz(\))34 b Fr(\030)g Fu(t)3703 2384 y Fm(\000)p Fs(1)3798 2420 y Fz(.)150 2540 y(W)-8 b(e)33 b(then)g(get)643 2760 y(\(1)22 b Fr(\000)g Fu(F)14 b Fz(\()9 b(~)-58 b Fu(\032)p Fz(\)\))22 b Fr(\001)g Fu(ad)1266 2719 y Ft(N)1266 2785 y(G)1321 2794 y Fj(1)1356 2785 y Fs(\()6 b(~)-41 b Ft(\032)p Fs(\))1451 2760 y Fz(\()p Fu(L)p Fz(\()1603 2734 y(~)1593 2760 y Fu(\030)5 b Fz(\)\))22 b Fr(\001)f Fu(G)p Fz(\()9 b(~)-58 b Fu(\032)q Fz(\))22 b Fr(\001)f Fu(F)2126 2775 y Fs(+)p Fm(\000)2241 2760 y Fz(\()p Fu(r)s(=t)p Fz(\))27 b(=)g Fr(O)s Fz(\()p Fu(t)2733 2719 y Fm(\000)p Ft(N)2856 2760 y Fz(\))p Fu(;)114 b(N)38 b Fr(\025)28 b Fz(1)p Fu(:)150 2980 y Fz(Finally)-8 b(,)31 b(let)h(us)i(notice)f (that)g(w)m(e)h(can)g(follo)m(w)d(the)j(same)f(argumen)m(t)g(for)g(all) e(p)s(ossible)h(c)m(hoices)i(for)150 3101 y Fu(B)40 b Fz(and)35 b Fu(D)s Fz(,)h(since)f(ev)m(ery)i(time)d(w)m(e)i(comm)m(ute) e(an)m(y)i(t)m(w)m(o)g(of)f(the)g(op)s(erators)g Fu(H)8 b Fz(,)44 b(~)-58 b Fu(\032)35 b Fz(and)3428 3075 y(~)3418 3101 y Fu(\030)k Fz(w)m(e)d(gain)150 3221 y(the)d(extra)g(1)p Fu(=r)i Fz(deca)m(y)f(and)e(w)m(e)i(can)f(rep)s(eat)g(the)g(same)f(pro) s(cedure)i(as)e(b)s(efore.)p 3754 3221 4 66 v 3758 3159 59 4 v 3758 3221 V 3816 3221 4 66 v 296 3460 a(The)24 b(last)d(tec)m(hnical)h(result)g(w)m(e)i(presen)m(t)g(here)f(is)f(a)g (quan)m(tum)h(v)m(ersion)g(of)f(energy)h(conserv)-5 b(ation.)150 3580 y(Consider)24 b Fu(f)596 3595 y Fs(1)635 3580 y Fz(\()9 b(~)-58 b Fu(\032)p Fz(\),)25 b Fu(f)861 3595 y Fs(2)901 3580 y Fz(\()949 3554 y(~)939 3580 y Fu(\030)t Fz(\))e(and)h Fu(f)1276 3595 y Ft(E)1335 3580 y Fz(\()p Fu(H)8 b Fz(\))23 b(where)h Fu(f)1843 3595 y Fs(1)1906 3580 y Fz(is)e(supp)s(orted)i(in)f([)p Fu(\032)2625 3595 y Fs(0)2667 3580 y Fr(\000)s Fu(\017)2786 3595 y Fs(1)2827 3580 y Fu(;)17 b(\032)2921 3595 y Fs(0)2963 3580 y Fz(+)s Fu(\017)3081 3595 y Fs(1)3121 3580 y Fz(],)25 b Fu(f)3248 3595 y Fs(2)3310 3580 y Fz(is)e(supp)s(orted)150 3700 y(in)32 b([)p Fu(\030)334 3715 y Fs(0)395 3700 y Fr(\000)23 b Fu(\017)534 3715 y Fs(1)574 3700 y Fu(;)17 b(\030)661 3715 y Fs(0)721 3700 y Fz(+)22 b Fu(\017)858 3715 y Fs(1)898 3700 y Fz(])33 b(and)g Fu(f)1196 3715 y Fs(1)p Ft(;E)1343 3700 y Fz(is)f(de\014ned)i(righ)m(t)d(after)i(\(6.1\).)150 3904 y Fv(Lemma)42 b(6.3.)h FA(Assume)c Fu(F)1200 3919 y Fs(+)p Fm(\000)1352 3904 y FA(is)f(supp)-5 b(orte)g(d)39 b(on)f Fn(R)2108 3919 y Fs(+)2211 3904 y FA(and)g(assume)g(that)g Fr(j)p Fu(\032)3029 3868 y Fs(2)3029 3929 y(0)3094 3904 y Fz(+)24 b Fu(\030)3242 3868 y Fs(2)3237 3929 y(0)3306 3904 y Fr(\000)h Fz(2)p Fu(E)6 b Fr(j)34 b Fu(>)g Fz(3)q(~)-50 b Fu(\017)p FA(.)150 4024 y(Then)34 b(ther)-5 b(e)35 b(exists)f Fu(\017)952 4039 y Fs(1)1020 4024 y Fu(>)27 b Fz(0)35 b FA(smal)5 b(l)34 b(enough)g(such)h(that)g(for)g(every)g Fu(f)2680 4039 y Fs(1)2754 4024 y FA(and)f Fu(f)2991 4039 y Fs(2)3066 4024 y FA(as)g(ab)-5 b(ove)34 b(we)h(have)1016 4244 y Fr(jj)p Fu(f)1120 4259 y Fs(1)1159 4244 y Fz(\()9 b(~)-58 b Fu(\032)p Fz(\))22 b Fr(\001)g Fu(f)1405 4259 y Fs(2)1445 4244 y Fz(\()1493 4218 y(~)1483 4244 y Fu(\030)t Fz(\))g Fr(\001)g Fu(f)1688 4259 y Fs(1)p Ft(;E)1802 4244 y Fz(\()p Fu(H)8 b Fz(\))22 b Fr(\001)g Fu(F)2102 4259 y Fs(+)p Fm(\000)2216 4244 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fr(jj)k Fz(=)i Fr(O)s Fz(\()p Fu(t)2764 4203 y Fm(\0001)2894 4244 y Fz(\))p Fu(:)617 b Fz(\(6.13\))150 4464 y Fv(Pro)s(of.)55 b Fz(Assume)38 b(without)e(loss)g(that)g Fu(\032)1695 4428 y Fs(2)1695 4489 y(0)1760 4464 y Fz(+)24 b Fu(\030)1908 4428 y Fs(2)1903 4489 y(0)1972 4464 y Fr(\000)h Fz(2)p Fu(E)41 b(>)34 b Fz(3)q(~)-50 b Fu(\017)p Fz(.)55 b(Cho)s(ose)38 b Fu(\036)c Fr(2)h Fu(L)3119 4428 y Fs(2)3158 4464 y Fz(\()p Fn(R)3262 4428 y Fs(2)3308 4464 y Fz(\))h(and)h(de\014ne)150 4585 y Fu(\036)p Fz(\()p Fu(t;)17 b(\017)364 4600 y Fs(1)403 4585 y Fz(\))31 b(=)f Fu(f)626 4600 y Fs(1)665 4585 y Fz(\()9 b(~)-58 b Fu(\032)q Fz(\))23 b Fr(\001)g Fu(f)914 4600 y Fs(2)953 4585 y Fz(\()1001 4558 y(~)991 4585 y Fu(\030)5 b Fz(\))23 b Fr(\001)g Fu(f)1199 4600 y Fs(1)p Ft(;E)1313 4585 y Fz(\()p Fu(H)8 b Fz(\))23 b Fr(\001)g Fu(F)1615 4600 y Fs(+)p Fm(\000)1729 4585 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu(\036)p Fz(.)47 b(Let)35 b(us)f(pro)m(v)m(e)i(that)e(for)f(su\016cien)m(tly)i (small)d Fu(\017)3785 4600 y Fs(1)150 4705 y Fz(w)m(e)i(ma)m(y)e(write) 674 4925 y Fr(h)p Fu(\036)p Fz(\()p Fu(t;)17 b(\017)927 4940 y Fs(1)966 4925 y Fz(\))p Fu(;)g Fz(\()9 b(~)-58 b Fu(\032)1136 4884 y Fs(2)1198 4925 y Fz(+)1306 4899 y(~)1296 4925 y Fu(\030)1344 4884 y Fs(2)1405 4925 y Fr(\000)22 b Fz(2)p Fu(H)8 b Fz(\))p Fu(\036)p Fz(\()p Fu(t;)17 b(\017)1894 4940 y Fs(1)1933 4925 y Fz(\))p Fr(i)27 b(\025)i Fz(~)-50 b Fu(\017)q Fr(jj)p Fu(\036)p Fz(\()p Fu(t;)17 b(\017)2452 4940 y Fs(1)2490 4925 y Fz(\))p Fr(jj)2584 4884 y Fs(2)2645 4925 y Fz(+)22 b Fr(O)s Fz(\()p Fu(t)2898 4884 y Fm(\0001)3028 4925 y Fz(\))p Fr(jj)p Fu(\036)p Fr(jj)3236 4884 y Fs(2)3274 4925 y Fu(:)275 b Fz(\(6.14\))150 5145 y(Indeed,)35 b(consider)f(the)g (exp)s(ectation)f(in)g(the)h(left)e(hand)i(side)f(of)g(the)h(ab)s(o)m (v)m(e)g(inequalit)m(y)-8 b(.)45 b(W)-8 b(e)34 b(can)150 5265 y(put)g(near)43 b(~)-58 b Fu(\032)598 5229 y Fs(2)671 5265 y Fz(some)34 b(function)1320 5239 y(~)1300 5265 y Fu(f)1348 5280 y Fs(1)1388 5265 y Fz(\()9 b(~)-58 b Fu(\032)p Fz(\))34 b(whic)m(h)g(equals)g(1)f(on)h(the)g(supp)s(ort)g (of)g Fu(f)3036 5280 y Fs(1)3109 5265 y Fz(and)g(is)f(supp)s(orted)150 5386 y(on)j([)p Fu(\032)366 5401 y Fs(0)430 5386 y Fr(\000)26 b Fz(2)p Fu(\017)621 5401 y Fs(1)660 5386 y Fu(;)17 b(\032)754 5401 y Fs(0)818 5386 y Fz(+)25 b(2)p Fu(\017)1007 5401 y Fs(1)1046 5386 y Fz(];)38 b(then)f(put)f(near)1777 5360 y(~)1767 5386 y Fu(\030)1815 5350 y Fs(2)1890 5386 y Fz(another)g(function)2656 5360 y(~)2637 5386 y Fu(f)2685 5401 y Fs(2)2724 5386 y Fz(\()2772 5360 y(~)2762 5386 y Fu(\030)t Fz(\))g(whic)m(h)h(equals)g(1)e(on)i(the)1939 5776 y(52)p eop %%Page: 53 56 53 55 bop 150 407 a Fz(supp)s(ort)38 b(of)f Fu(f)680 422 y Fs(2)757 407 y Fz(and)g(is)g(supp)s(orted)h(on)g([)p Fu(\030)1728 422 y Fs(0)1792 407 y Fr(\000)26 b Fz(2)p Fu(\017)1983 422 y Fs(1)2023 407 y Fu(;)17 b(\030)2110 422 y Fs(0)2174 407 y Fz(+)25 b(2)p Fu(\017)2363 422 y Fs(1)2403 407 y Fz(].)58 b(According)37 b(to)g(the)h(previous)g(t)m (w)m(o)150 527 y(lemmas)31 b(w)m(e)i(ha)m(v)m(e)h(that)522 700 y(max)p Fr(fjj)p Fz([1)21 b Fr(\000)1026 674 y Fz(~)1006 700 y Fu(f)1054 715 y Fs(2)1094 700 y Fz(\()1142 674 y(~)1132 700 y Fu(\030)t Fz(\)])p Fu(\036)p Fz(\()p Fu(t;)c(\017)1458 715 y Fs(1)1498 700 y Fz(\))p Fr(jj)p Fu(;)43 b Fr(jj)p Fz([1)22 b Fr(\000)1935 674 y Fz(~)1915 700 y Fu(f)1963 715 y Fs(1)2003 700 y Fz(\()9 b(~)-58 b Fu(\032)p Fz(\)])p Fu(\036)p Fz(\()p Fu(t;)17 b(\017)2370 715 y Fs(1)2410 700 y Fz(\))p Fr(jjg)26 b(\024)i(O)s Fz(\()p Fu(t)2840 659 y Fm(\0001)2970 700 y Fz(\))p Fr(jj)p Fu(\036)p Fr(jj)p Fu(:)371 b Fz(\(6.15\))150 873 y(In)39 b(a)g(similar)d(w)m(a)m(y)-8 b(,)42 b(w)m(e)e(ma)m(y)f(put)1492 847 y(~)1471 873 y Fu(f)1519 888 y Ft(E)1579 873 y Fz(\()p Fu(H)8 b Fz(\))38 b(near)h(2)p Fu(H)47 b Fz(at)38 b(the)i(exp)s(ense)h(of)e(another)g Fr(O)s Fz(\()p Fu(t)3489 837 y Fm(\0001)3619 873 y Fz(\))p Fr(jj)p Fu(\036)p Fr(jj)150 994 y Fz(error.)k(Then)34 b(w)m(e)g(see)f(that)g(in)f(the)h(form)e(sense)j(w)m(e)g(ha)m(v)m(e)649 1167 y(~)-58 b Fu(\032)690 1126 y Fs(2)749 1140 y Fz(~)730 1167 y Fu(f)778 1182 y Fs(1)817 1167 y Fz(\()9 b(~)-58 b Fu(\032)q Fz(\))22 b(+)1074 1140 y(~)1064 1167 y Fu(\030)1112 1126 y Fs(2)1170 1140 y Fz(~)1151 1167 y Fu(f)1199 1182 y Fs(2)1238 1167 y Fz(\()1286 1140 y(~)1276 1167 y Fu(\030)t Fz(\))g Fr(\000)h Fz(2)p Fu(H)1642 1140 y Fz(~)1621 1167 y Fu(f)1669 1182 y Ft(E)1728 1167 y Fz(\()p Fu(H)8 b Fz(\))82 b Fr(\025)i Fz(\()p Fr(j)p Fu(\032)2252 1182 y Fs(0)2291 1167 y Fr(j)22 b(\000)h Fz(2)p Fu(\017)2529 1182 y Fs(1)2568 1167 y Fz(\))2606 1126 y Fs(2)2665 1140 y Fz(~)2646 1167 y Fu(f)2694 1182 y Fs(1)2733 1167 y Fz(\()9 b(~)-58 b Fu(\032)q Fz(\))22 b(+)g(\()p Fr(j)p Fu(\030)3089 1182 y Fs(0)3127 1167 y Fr(j)1975 1312 y(\000)84 b Fz(2)p Fu(\017)2224 1327 y Fs(1)2264 1312 y Fz(\))2302 1271 y Fs(2)2360 1286 y Fz(~)2341 1312 y Fu(f)2389 1327 y Fs(2)2428 1312 y Fz(\()2476 1286 y(~)2466 1312 y Fu(\030)5 b Fz(\))22 b Fr(\000)h Fz(2\()p Fu(E)k Fz(+)c(~)-50 b Fu(\017)q Fz(\))3057 1286 y(~)3036 1312 y Fu(f)3084 1327 y Ft(E)3143 1312 y Fz(\()p Fu(H)8 b Fz(\))p Fu(:)241 b Fz(\(6.16\))150 1485 y(When)39 b(w)m(e)f(tak)m(e)h(the)f(exp)s (ectation)g(of)f(the)h(righ)m(t)f(hand)h(side)g(of)f(\(6.16\))g(on)h Fu(\036)p Fz(\()p Fu(t;)17 b(\017)3247 1500 y Fs(1)3287 1485 y Fz(\))37 b(w)m(e)i(can)f(get)150 1605 y(rid)31 b(of)g(the)i(cut-o\013s)942 1579 y(~)922 1605 y Fu(f)970 1620 y Fs(1)1010 1605 y Fz(,)1088 1579 y(~)1069 1605 y Fu(f)1117 1620 y Fs(2)1188 1605 y Fz(and)1398 1579 y(~)1377 1605 y Fu(f)1425 1620 y Ft(E)1517 1605 y Fz(at)e(the)h(price)g (of)f(another)h Fr(O)s Fz(\()p Fu(t)2662 1569 y Fm(\0001)2792 1605 y Fz(\))p Fr(jj)p Fu(\036)p Fr(jj)e Fz(error.)43 b(Then)33 b(if)e Fu(\017)3688 1620 y Fs(1)3759 1605 y Fz(is)150 1726 y(small)f(enough)j(w)m(e)h(ha)m(v)m(e)1093 1899 y(\()p Fr(j)p Fu(\032)1209 1914 y Fs(0)1249 1899 y Fr(j)22 b(\000)g Fz(2)p Fu(\017)1486 1914 y Fs(1)1526 1899 y Fz(\))1564 1858 y Fs(2)1625 1899 y Fz(+)g(\()p Fr(j)p Fu(\030)1832 1914 y Fs(0)1871 1899 y Fr(j)g(\000)g Fz(2)p Fu(\017)2108 1914 y Fs(1)2148 1899 y Fz(\))2186 1858 y Fs(2)2248 1899 y Fr(\000)g Fz(2\()p Fu(E)28 b Fz(+)23 b(~)-50 b Fu(\017)p Fz(\))28 b Fr(\025)h Fz(~)-50 b Fu(\017)150 2072 y Fz(and)33 b(\(6.14\))e(follo)m(ws.)43 b(No)m(w)33 b(let)f(us)h(pro)m(v)m(e)h(that)335 2291 y Fr(h)p Fu(\036)p Fz(\()p Fu(t;)17 b(\017)588 2306 y Fs(1)627 2291 y Fz(\))p Fu(;)g Fz(\()9 b(~)-58 b Fu(\032)797 2250 y Fs(2)859 2291 y Fz(+)967 2265 y(~)957 2291 y Fu(\030)1005 2250 y Fs(2)1066 2291 y Fr(\000)22 b Fz(2)p Fu(H)8 b Fz(\))p Fu(\036)p Fz(\()p Fu(t;)17 b(\017)1555 2306 y Fs(1)1594 2291 y Fz(\))p Fr(i)27 b(\024)1819 2224 y Fz(~)-50 b Fu(\017)p 1813 2268 49 4 v 1813 2360 a Fz(2)1872 2291 y Fr(jj)p Fu(\036)p Fz(\()p Fu(t;)17 b(\017)2142 2306 y Fs(1)2181 2291 y Fz(\))p Fr(jj)2275 2250 y Fs(2)2336 2291 y Fz(+)22 b Fr(O)s Fz(\()p Fu(t)2589 2250 y Fm(\0001)2719 2291 y Fz(\))p Fr(jj)p Fu(\036)p Fr(jj)2927 2250 y Fs(2)2964 2291 y Fu(;)115 b(t)28 b Fr(\025)g Fu(T)3332 2306 y Fs(~)-36 b Ft(\017)3364 2291 y Fu(;)185 b Fz(\(6.17\))150 2495 y(whic)m(h)36 b(together)g(with)f(\(6.14\))g(implies)f(\(6.13\).)52 b(Indeed,)38 b(from)c(\(4.5\),)i(\(4.7\))f(and)h(\(1.19\))f(w)m(e)i (see)150 2616 y(that)873 2624 y Fl(\015)873 2684 y(\015)873 2744 y(\015)873 2803 y(\015)928 2769 y Fz([)9 b(~)-58 b Fu(\032)1005 2727 y Fs(2)1067 2769 y Fz(+)1175 2742 y(~)1165 2769 y Fu(\030)1213 2727 y Fs(2)1274 2769 y Fr(\000)23 b Fz(2)p Fu(H)8 b Fz(])p Fu(\036)p Fz(\()p Fu(t;)17 b(\017)1753 2784 y Fs(1)1792 2769 y Fz(\))22 b Fr(\000)2004 2701 y Fz(1)p 1961 2746 135 4 v 1961 2837 a(4)p 2010 2782 47 4 v Fu(r)2057 2800 y Fs(2)2106 2769 y Fu(\036)p Fz(\()p Fu(t;)17 b(\017)2320 2784 y Fs(1)2359 2769 y Fz(\))2397 2624 y Fl(\015)2397 2684 y(\015)2397 2744 y(\015)2397 2803 y(\015)2480 2769 y Fz(=)28 b Fr(O)s Fz(\()p Fu(t)2739 2727 y Fm(\0001)2869 2769 y Fz(\))p Fr(jj)p Fu(\036)p Fr(jj)p Fu(:)150 2975 y Fz(Then)34 b(if)d Fu(t)i Fz(is)f(large)f(enough)i(w)m(e)h(ha)m(v)m(e)875 3059 y Fl(\034)950 3199 y Fu(\036)p Fz(\()p Fu(t;)17 b(\017)1164 3214 y Fs(1)1203 3199 y Fz(\))p Fu(;)1338 3132 y Fz(1)p 1295 3176 135 4 v 1295 3268 a(4)p 1344 3213 47 4 v Fu(r)1390 3231 y Fs(2)1440 3199 y Fu(\036)p Fz(\()p Fu(t;)g(\017)1654 3214 y Fs(1)1693 3199 y Fz(\))1731 3059 y Fl(\035)1834 3199 y Fr(\024)1954 3132 y Fz(~)-50 b Fu(\017)p 1949 3176 49 4 v 1949 3268 a Fz(2)2007 3199 y Fr(jj)p Fu(\036)p Fz(\()p Fu(t;)17 b(\017)2277 3214 y Fs(1)2316 3199 y Fz(\))p Fr(jj)2410 3158 y Fs(2)2471 3199 y Fz(+)22 b Fr(O)s Fz(\()p Fu(t)2724 3158 y Fm(\0001)2854 3199 y Fz(\))p Fr(jj)p Fu(\036)p Fr(jj)3062 3158 y Fs(2)150 3424 y Fz(and)33 b(w)m(e)g(are)g(done.)p 3754 3545 4 66 v 3758 3482 59 4 v 3758 3545 V 3816 3545 4 66 v 150 3945 a Fp(6.3)135 b(A)45 b(maximal)h(v)l(elo)t(cit)l(y)h(b)t(ound)150 4129 y Fz(The)34 b(notation)d Fr(h\001i)847 4144 y Ft(t)909 4129 y Fz(will)f(b)s(e)k(used)g(to)e(signify)g(the)h(exp)s(ectation)g (in)f(a)h(state)g(lik)m(e)f(\(6.1\))h(at)f(time)g Fu(t)p Fz(.)150 4250 y(W)-8 b(e)30 b(will)d(often)i(sligh)m(tly)e(abuse)j (notation)e(b)m(y)i(abbreviating)e(the)i(notation)i(\026)-53 b Fu(r)32 b Fz(for)c(the)i(function)f(in)150 4370 y(\(4.2\))36 b(as)g Fu(r)s Fz(.)55 b(F)-8 b(or)35 b(example,)i(w)m(e)g(use)h(the)e (notation)f Fr(h)p Fu(F)2232 4334 y Fm(0)2272 4370 y Fz(\()p Fu(r)s(=t)p Fz(\))o Fr(i)2517 4399 y Ft(t)2583 4370 y Fz(in)g(the)i(in)m(tegral)e(in)g(Lemma)g(6.4)150 4490 y(stated)h(b)s(elo)m(w)g(for)f(the)h(exp)s(ectation)g(of)g(the)g (op)s(erator)f(of)g(m)m(ultiplication)c(b)m(y)37 b Fv(x)c Fr(!)g Fu(F)3444 4454 y Fm(0)3483 4490 y Fz(\()5 b(\026)-54 b Fu(r)s Fz(\()p Fv(x)p Fz(\))p Fu(=t)p Fz(\))150 4611 y(in)32 b(the)h(state)g Fu( )t Fz(\()p Fu(t)p Fz(\).)296 4731 y(A)g(standard)g(computation)e(will)f(sho)m(w)k(that)1618 4904 y Fr(jj)9 b Fz(~)-58 b Fu(\032)p Fr(jj)27 b(\024)1912 4819 y(p)p 1995 4819 336 4 v 85 x Fz(2)p Fu(E)h Fz(+)22 b(2)q(~)-50 b Fu(\017:)1219 b Fz(\(6.18\))150 5077 y(Indeed)39 b(the)e(result)g(follo)m(ws)f(from)g(the)h(de\014nition)f(\(6.2\),)i (the)g(fact)e(that)h Fr(jr)p Fu(r)s Fz(\()p Fv(x)p Fz(\))p Fr(j)e(\024)h Fz(1)g(and)i(the)150 5198 y(Cauc)m(h)m(y-Sc)m(h)m(w)m (arz)e(inequalit)m(y:)527 5415 y Fr(jj)9 b Fz(~)-58 b Fu(\032)p Fr(jj)27 b Fz(=)59 b(sup)820 5501 y Fm(jj)p Ft(\036)p Fm(jj)p Fs(=1)1047 5415 y Fr(jh)p Fu(\036;)24 b Fz(~)-56 b Fu(\032)o(\036)p Fr(ij)27 b(\024)60 b Fz(sup)1522 5501 y Fm(jj)p Ft(\036)p Fm(jj)p Fs(=1)1749 5286 y Fl(q)p 1849 5286 919 4 v 129 x Fr(h)1909 5389 y Fz(~)1888 5415 y Fu(f)1936 5430 y Ft(E)1995 5415 y Fz(\()p Fu(H)8 b Fz(\))p Fu(\036;)17 b Fz(2)p Fu(H)2419 5389 y Fz(~)2400 5415 y Fu(f)2448 5430 y Ft(E)2506 5415 y Fz(\()p Fu(H)8 b Fz(\))p Fu(\036)p Fr(i)26 b(\024)j Fz(\(2)p Fu(E)e Fz(+)22 b(2)q(~)-50 b Fu(\017)q Fz(\))3311 5374 y Fs(1)p Ft(=)p Fs(2)3421 5415 y Fu(:)1939 5776 y Fz(53)p eop %%Page: 54 57 54 56 bop 150 407 a Fv(Lemma)33 b(6.4.)38 b FA(L)-5 b(et)32 b Fu(K)j Fz(=)1141 326 y Fr(p)p 1224 326 296 4 v 81 x Fz(2)p Fu(E)28 b Fz(+)22 b(1)p FA(,)32 b Fu(a)c Fz(=)f Fu(K)7 b FA(,)33 b Fu(b)28 b Fz(=)f Fu(K)21 b Fz(+)14 b(1)31 b FA(and)g(let)h Fu(F)44 b FA(denote)31 b(either)g Fu(F)3435 422 y Fs(+)3526 407 y FA(or)g Fu(F)3711 422 y Fs(++)150 527 y FA(\(se)-5 b(e)34 b(\(3.1\){\(3.4\)\).)43 b(Then)1326 662 y Fl(Z)1425 689 y Fm(1)1381 888 y Fs(1)1517 798 y Fu(t)1552 757 y Fm(\000)p Fs(1)1663 798 y Fr(h)o Fu(F)1778 757 y Fm(0)1818 798 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fr(i)2063 827 y Ft(t)2110 798 y Fu(dt)27 b Fr(\024)h Fu(C)7 b Fr(jj)p Fu( )t Fr(jj)2584 757 y Fs(2)2622 798 y Fu(:)150 1065 y Fv(Pro)s(of.)43 b Fz(Consider)33 b(the)g(observ)-5 b(able)1349 1285 y(\010\()p Fu(t)p Fz(\))28 b(=)f Fr(\000)1760 1259 y Fz(~)1738 1285 y Fu(f)1786 1300 y Ft(E)1846 1285 y Fz(\()p Fu(H)8 b Fz(\))p Fu(F)30 b Fz(\()p Fu(r)s(=t)p Fz(\))2349 1259 y(~)2327 1285 y Fu(f)2375 1300 y Ft(E)2435 1285 y Fz(\()p Fu(H)8 b Fz(\))p Fu(:)949 b Fz(\(6.19\))150 1505 y(By)33 b(di\013eren)m(tiating)e(with)h(resp)s(ect)i(to)e Fu(t)h Fz(w)m(e)g(get)671 1747 y Fu(@)722 1762 y Ft(t)769 1747 y Fr(h)o Fz(\010\()p Fu(t)p Fz(\))p Fr(i)1028 1776 y Ft(t)1085 1747 y Fz(=)28 b Fu(t)1224 1706 y Fm(\000)p Fs(1)1335 1747 y Fr(h)p Fu(r)s(=tF)1582 1706 y Fm(0)1621 1747 y Fz(\()p Fu(r)s(=t)p Fz(\))o Fr(i)1866 1776 y Ft(t)1918 1747 y Fr(\000)2017 1636 y Fl(D)2100 1721 y Fz(~)2078 1747 y Fu(f)2126 1762 y Ft(E)2186 1747 y Fz(\()p Fu(H)8 b Fz(\))p Fu(i)17 b Fz([)o Fu(H)r(;)g(F)30 b Fz(\()p Fu(r)s(=t)p Fz(\)])2919 1721 y(~)2898 1747 y Fu(f)2946 1762 y Ft(E)3005 1747 y Fz(\()p Fu(H)8 b Fz(\))3170 1636 y Fl(E)3231 1816 y Ft(t)3277 1747 y Fu(:)272 b Fz(\(6.20\))150 1993 y(The)34 b(ab)s(o)m(v)m(e)f(comm)m(utator)e(ma)m(y)h(b)s(e)h (written)g(as:)1079 2333 y Fu(i)17 b Fz([)p Fu(H)r(;)g(F)30 b Fz(\()p Fu(r)s(=t)p Fz(\))o(])e(=)g(\(2)p Fu(t)p Fz(\))1901 2292 y Fm(\000)p Fs(1)2011 2333 y Fr(f)p Fv(p)23 b Fr(\001)e Fz(\()p Fr(r)p Fu(r)s Fz(\))p Fu(F)2478 2292 y Fm(0)2518 2333 y Fz(\()o Fu(r)s(=t)p Fz(\))h(+)g(h)p Fu(:)p Fz(c)p Fu(:)q Fr(g)1079 2489 y Fz(=)27 b(\(2)p Fu(t)p Fz(\))1342 2447 y Fm(\000)p Fs(1)1436 2399 y Fr(p)p 1519 2399 100 4 v 90 x Fu(F)1596 2460 y Fm(0)1636 2489 y Fz(\()p Fu(r)s(=t)p Fz(\)\))16 b Fr(f)32 b Fv(p)23 b Fr(\001)f(r)p Fu(r)i Fz(+)e(h)p Fu(:)p Fz(c)p Fu(:)q Fr(g)2582 2399 y(p)p 2665 2399 V 90 x Fu(F)2742 2460 y Fm(0)2781 2489 y Fz(\()p Fu(r)s(=t)p Fz(\))1079 2644 y(=)27 b Fu(t)1217 2603 y Fm(\000)p Fs(1)1312 2555 y Fr(p)p 1395 2555 V 89 x Fu(F)1472 2615 y Fm(0)1511 2644 y Fz(\()p Fu(r)s(=t)p Fz(\))16 b Fu(\032)1784 2555 y Fr(p)p 1868 2555 V 1868 2644 a Fu(F)1945 2615 y Fm(0)1984 2644 y Fz(\()p Fu(r)s(=t)p Fz(\))h Fu(:)1341 b Fz(\(6.21\))296 2864 y(By)33 b(\(6.3\))471 2997 y Fl(\015)471 3057 y(\015)471 3116 y(\015)527 3001 y(h)595 3085 y Fz(~)574 3111 y Fu(f)622 3126 y Ft(E)681 3111 y Fz(\()p Fu(H)8 b Fz(\))p Fu(;)890 3022 y Fr(p)p 972 3022 V 972 3111 a Fu(F)1049 3083 y Fm(0)1089 3111 y Fz(\()p Fu(r)s(=t)p Fz(\))1296 3001 y Fl(i)1343 2997 y(\015)1343 3057 y(\015)1343 3116 y(\015)1426 3111 y Fr(\024)28 b Fu(C)1652 2997 y Fl(\015)1652 3057 y(\015)1652 3116 y(\015)1707 3111 y Fz(\()p Fu(H)i Fr(\000)23 b Fu(i)p Fz(\))2027 3070 y Fm(\000)p Fs(1)2121 3111 y Fu(i)p Fz([)p Fu(H)r(;)2308 3022 y Fr(p)p 2391 3022 V 89 x Fu(F)2468 3083 y Fm(0)2508 3111 y Fz(\()p Fu(r)s(=t)p Fz(\)]\()p Fu(H)29 b Fr(\000)23 b Fu(i)p Fz(\))3061 3070 y Fm(\000)p Fs(1)3155 2997 y Fl(\015)3155 3057 y(\015)3155 3116 y(\015)3227 3111 y Fu(:)322 b Fz(\(6.22\))296 3376 y(In)m(tro)s(ducing)43 b(\(6.21\))f(in)g(\(6.20\),)j(comm)m(uting)2111 3350 y(~)2090 3376 y Fu(f)2138 3391 y Ft(E)2197 3376 y Fz(\()p Fu(H)8 b Fz(\))43 b(and)2605 3292 y Fr(p)p 2688 3292 V 84 x Fu(F)2765 3348 y Fm(0)2804 3376 y Fz(\()p Fu(r)s(=t)p Fz(\))f(\(using)h(\(6.3\)\),)i(and)150 3497 y(in)m(v)m(oking)32 b(that)h(\()p Fu(r)s(=t)p Fz(\))p Fu(F)1036 3461 y Fm(0)1075 3497 y Fz(\()p Fu(r)s(=t)p Fz(\))27 b Fr(\025)h Fu(K)7 b(F)1581 3461 y Fm(0)1621 3497 y Fz(\()p Fu(r)s(=t)p Fz(\))o(,)33 b(w)m(e)h(obtain)754 3744 y Fu(@)805 3759 y Ft(t)852 3744 y Fr(h)p Fz(\010\()p Fu(t)p Fz(\))p Fr(i)1111 3773 y Ft(t)1168 3744 y Fr(\025)28 b Fu(t)1308 3703 y Fm(\000)p Fs(1)1420 3634 y Fl(D)1481 3655 y Fr(p)p 1564 3655 V 89 x Fu(F)1641 3715 y Fm(0)1680 3744 y Fz(\()p Fu(r)s(=t)p Fz(\))16 b(\()p Fu(K)29 b Fr(\000)j Fz(~)-58 b Fu(\032)p Fz(\))2241 3655 y Fr(p)p 2324 3655 V 89 x Fu(F)2401 3715 y Fm(0)2441 3744 y Fz(\()p Fu(r)s(=t)p Fz(\))2648 3634 y Fl(E)2708 3813 y Ft(t)2760 3744 y Fz(+)22 b Fu(R)2932 3759 y Fs(1)2972 3744 y Fz(\()p Fu(t)p Fz(;)17 b Fu( )t Fz(\))p Fu(;)355 b Fz(\(6.23\))150 3996 y(where)34 b(the)f(remainder)e Fu(R)1134 4011 y Fs(1)1174 3996 y Fz(\()p Fu(t)p Fz(;)17 b Fu( )t Fz(\))32 b(ob)s(eys)1500 4216 y Fr(j)p Fu(R)1602 4231 y Fs(1)1642 4216 y Fz(\()p Fu(t)p Fz(;)17 b Fu( )t Fz(\))p Fr(j)27 b(\024)h Fu(C)7 b(t)2136 4175 y Fm(\000)p Fs(2)2230 4216 y Fr(jj)p Fu( )t Fr(jj)2409 4175 y Fs(2)2448 4216 y Fu(:)1101 b Fz(\(6.24\))296 4436 y(With)32 b(our)h(c)m(hoice)g(of)g Fu(K)7 b Fz(,)33 b Fu(K)c Fr(\000)j Fz(~)-58 b Fu(\032)28 b Fr(\025)1670 4355 y(p)p 1753 4355 296 4 v 81 x Fz(2)p Fu(E)g Fz(+)22 b(1)g Fr(\000)2170 4355 y(p)p 2253 4355 336 4 v 81 x Fz(2)p Fu(E)28 b Fz(+)22 b(2)p Fu(\017)28 b(>)g Fz(0)33 b(\(for)f Fu(\017)c(<)g Fz(1)p Fu(=)p Fz(2\),)k(cf.)45 b(\(6.18\).)150 4556 y(Applied)36 b(to)g(the)h(righ)m(t)f(hand)h(side)f (of)g(\(6.23\))g(w)m(e)i(obtain)d(after)h(an)h(in)m(tegration)e(that)h (for)g(ev)m(ery)150 4677 y Fu(T)41 b(>)28 b Fz(1)734 4707 y Fl(Z)833 4733 y Ft(T)789 4932 y Fs(1)905 4842 y Fu(t)940 4801 y Fm(\000)p Fs(1)1051 4842 y Fr(h)p Fu(F)1167 4801 y Fm(0)1206 4842 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fr(i)1452 4871 y Ft(t)1498 4842 y Fu(dt)f Fr(\024)i(h)o Fz(\010\()p Fu(T)14 b Fz(\))p Fr(i)2011 4871 y Ft(T)2088 4842 y Fr(\000)23 b(h)p Fz(\010\(1\))p Fr(i)2460 4871 y Fs(1)2522 4842 y Fr(\000)2622 4707 y Fl(Z)2721 4733 y Ft(T)2677 4932 y Fs(1)2793 4842 y Fu(R)2867 4857 y Fs(1)2907 4842 y Fz(\()p Fu(t)p Fz(;)17 b Fu( )t Fz(\))p Fu(dt:)334 b Fz(\(6.25\))296 5069 y(Finally)-8 b(,)35 b(using)h(the)g(fact)g(that)g Fr(jj)p Fz(\010\()p Fu(T)14 b Fz(\))p Fr(jj)33 b(\024)h Fu(C)43 b Fz(uniformly)34 b(in)h Fu(T)48 b(>)34 b Fz(1)h(and)i (\(6.24\),)f(the)h(righ)m(t)150 5189 y(hand)d(side)f(of)g(\(6.25\))g (ma)m(y)g(b)s(e)h(estimated)e(indep)s(enden)m(tly)j(of)d Fu(T)43 b(>)29 b Fz(1,)34 b(and)f(the)h(lemma)d(follo)m(ws.)p 3754 5310 4 66 v 3758 5247 59 4 v 3758 5310 V 3816 5310 4 66 v 1939 5776 a(54)p eop %%Page: 55 58 55 57 bop 150 407 a Fv(Corollary)41 b(6.5.)j FA(L)-5 b(et)38 b Fu(C)i(>)33 b(K)e Fz(+)24 b(1)38 b FA(\(with)f Fu(K)45 b FA(as)38 b(in)f(L)-5 b(emma)37 b(6.4\))h(and)f(supp)-5 b(ose)37 b Fu(F)3364 422 y Fs(1)3437 407 y Fr(2)c Fu(L)3602 371 y Fm(1)3677 407 y Fz(\()p Fn(R)5 b Fz(\))150 527 y FA(with)35 b Fz(supp)q(\()p Fu(F)664 542 y Fs(1)703 527 y Fz(\))28 b Fr(\022)g Fz([)p Fu(K)h Fz(+)22 b(1)p Fu(;)17 b(C)7 b Fz(])p FA(.)45 b(Then)1138 653 y Fl(Z)1238 679 y Fm(1)1194 879 y Fs(1)1329 789 y Fu(t)1364 747 y Fm(\000)p Fs(1)1459 789 y Fr(jj)p Fu(F)1578 804 y Fs(1)1617 789 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu( )t Fz(\()p Fu(t)p Fz(\))p Fr(jj)2058 747 y Fs(2)2096 789 y Fu(dt)28 b Fr(\024)g Fz(Const)g Fr(jj)p Fu( )t Fr(jj)2771 747 y Fs(2)2810 789 y Fu(:)150 1052 y Fv(Pro)s(of.)42 b Fz(Using)30 b(the)h(inputs)f Fu(a)e Fz(=)f Fu(K)7 b Fz(,)31 b Fu(b)d Fz(=)g Fu(K)c Fz(+)17 b(1,)31 b Fu(c)c Fz(=)h Fu(C)37 b Fz(and)30 b Fu(d)e Fz(=)f Fu(C)d Fz(+)17 b(1)30 b(in)f(\(3.3\))h(and)g(\(3.4\))g(w) m(e)150 1172 y(infer)i(that)586 1091 y Fl(p)p 686 1091 178 4 v 81 x Fu(F)763 1138 y Fm(0)749 1194 y Fs(++)891 1172 y Fu(F)954 1187 y Fs(1)1021 1172 y Fz(=)27 b Fu(F)1187 1187 y Fs(1)1227 1172 y Fu(:)33 b Fz(Then)961 1399 y Fr(jj)p Fu(F)1080 1414 y Fs(1)1119 1399 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu( )t Fz(\()p Fu(t)p Fz(\))p Fr(jj)1560 1358 y Fs(2)1682 1399 y Fr(\024)83 b(jj)p Fu(F)1961 1414 y Fs(1)2000 1399 y Fr(jj)2056 1358 y Fs(2)2056 1424 y Fm(1)2152 1399 y Fr(\001)22 b(jj)2258 1313 y Fl(p)p 2357 1313 V 2357 1399 a Fu(F)2434 1365 y Fm(0)2420 1421 y Fs(++)2534 1399 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu( )t Fz(\()p Fu(t)p Fz(\))p Fr(jj)2975 1358 y Fs(2)1683 1545 y Fz(=)83 b Fr(jj)p Fu(F)1961 1560 y Fs(1)2000 1545 y Fr(jj)2056 1503 y Fs(2)2056 1569 y Fm(1)2147 1464 y Fl(\012)2194 1545 y Fu(F)2271 1503 y Fm(0)2257 1569 y Fs(++)2388 1545 y Fz(\()o Fu(r)s(=t)p Fz(\))2594 1464 y Fl(\013)2641 1584 y Ft(t)2687 1545 y Fu(;)862 b Fz(\(6.26\))150 1755 y(and)33 b(w)m(e)g(ma)m(y)g(use)g(Lemma)e(6.4)i(to)f(conclude)h(the)g (estimate.)p 3754 1755 4 66 v 3758 1693 59 4 v 3758 1755 V 3816 1755 4 66 v 150 2069 a Fv(Corollary)54 b(6.6.)49 b FA(L)-5 b(et)48 b Fu(C)59 b(>)52 b(K)39 b Fz(+)32 b(1)47 b FA(and)h(supp)-5 b(ose)47 b Fu(F)2288 2084 y Fs(1)2380 2069 y Fr(2)52 b Fu(C)2575 2033 y Fm(1)2568 2093 y Fs(0)2650 2069 y Fz(\()p Fn(R)5 b Fz(\))53 b FA(is)48 b(r)-5 b(e)g(al-value)g(d) 47 b(and)h(that)150 2189 y Fz(supp)q(\()p Fu(F)452 2204 y Fs(1)492 2189 y Fz(\))27 b Fr(\022)h Fz([)p Fu(K)i Fz(+)22 b(1)p Fu(;)17 b(C)7 b Fz(])p FA(.)44 b(Then)1492 2400 y Fz(lim)1477 2460 y Ft(t)p Fm(!1)1660 2400 y Fr(jj)p Fu(F)1779 2415 y Fs(1)1835 2400 y Fz(\()o Fu(r)s(=t)p Fz(\))17 b Fu( )t Fz(\()p Fu(t)p Fz(\))p Fr(jj)27 b Fz(=)g(0)p Fu(:)150 2647 y Fv(Pro)s(of.)76 b Fz(W)-8 b(e)43 b(will)e(pro)m(v)m(e)k (that)e Fu(\021)t Fz(\()p Fu(t)p Fz(\))j(:=)g Fr(jj)p Fu(F)1875 2662 y Fs(1)1930 2647 y Fz(\()p Fu(r)s(=t)p Fz(\))16 b Fu( )t Fz(\()p Fu(t)p Fz(\))p Fr(jj)2387 2611 y Fs(2)2470 2647 y Fz(go)s(es)43 b(to)g(zero)h(as)f Fu(t)j Fr(!)g(1)p Fz(.)75 b(F)-8 b(rom)150 2768 y(Corollary)31 b(6.5)h(w)m(e)i(kno)m(w)f(that)g(there)g(exists)g(a)g(sequence)i(\()p Fu(t)2425 2783 y Ft(n)2472 2768 y Fz(\))2510 2783 y Ft(n)p Fm(\025)p Fs(1)2680 2768 y Fz(with)d Fu(t)2937 2783 y Ft(n)3012 2768 y Fr(!)27 b(1)32 b Fz(suc)m(h)i(that)1703 2978 y(lim)1679 3038 y Ft(n)p Fm(!1)1879 2978 y Fu(\021)t Fz(\()p Fu(t)2004 2993 y Ft(n)2051 2978 y Fz(\))28 b(=)f(0)p Fu(:)296 3216 y Fz(Hence)k(w)m(e)f(only)e(need)i(to)f(pro)m(v)m(e)h (that)f Fu(\021)j Fz(has)e(a)e(limit)e(at)i(in\014nit)m(y;)i(b)m(y)g(a) f(standard)g(Co)s(ok)g(t)m(yp)s(e)150 3336 y(argumen)m(t,)j(this)h(w)m (ould)f(b)s(e)h(true)g(if)e Fu(\021)1568 3300 y Fm(0)1619 3336 y Fr(2)d Fu(L)1779 3300 y Fs(1)1819 3336 y Fz(\(\(1)p Fu(;)17 b Fr(1)p Fz(\)\).)296 3457 y(W)-8 b(e)33 b(compute)934 3667 y Fu(\021)986 3626 y Fm(0)1009 3667 y Fz(\()p Fu(t)p Fz(\))28 b(=)g Fr(\000)p Fz(2)p Fu(t)1413 3626 y Fm(\000)p Fs(1)1524 3667 y Fr(h)p Fu(r)s(=t)p Fz(\()p Fu(F)1795 3682 y Fs(1)1834 3667 y Fu(F)1911 3626 y Fm(0)1897 3692 y Fs(1)1937 3667 y Fz(\)\()p Fu(r)s(=t)p Fz(\))p Fr(i)2220 3696 y Ft(t)2272 3667 y Fz(+)22 b Fr(h)p Fu(i)p Fz([)p Fu(H)r(;)17 b(F)2673 3626 y Fs(2)2659 3692 y(1)2712 3667 y Fz(\()p Fu(r)s(=t)p Fz(\)])p Fr(i)2985 3682 y Ft(t)3014 3667 y Fu(:)296 3878 y Fz(The)34 b(\014rst)f(term)f(ma)m(y)g(b)s(e)h (estimated)f(b)m(y)781 4004 y Fl(\014)781 4064 y(\014)815 4089 y Fr(\000)p Fz(2)p Fu(t)976 4048 y Fm(\000)p Fs(1)1087 4089 y Fr(h)p Fu(r)s(=t)p Fz(\()p Fu(F)1358 4104 y Fs(1)1397 4089 y Fu(F)1474 4048 y Fm(0)1460 4113 y Fs(1)1500 4089 y Fz(\)\()p Fu(r)s(=t)p Fz(\))p Fr(i)1783 4118 y Ft(t)1812 4004 y Fl(\014)1812 4064 y(\014)1873 4089 y Fr(\024)c Fz(2)p Fu(C)7 b(t)2139 4048 y Fm(\000)p Fs(1)2234 4089 y Fr(jj)2290 3999 y Fl(p)p 2389 3999 261 4 v 2389 4089 a Fr(j)p Fu(F)2480 4104 y Fs(1)2519 4089 y Fu(F)2596 4054 y Fm(0)2582 4113 y Fs(1)2622 4089 y Fr(j)o Fz(\()p Fu(r)s(=t)p Fz(\)\))p Fu( )t Fz(\()p Fu(t)p Fz(\))p Fr(jj)3128 4048 y Fs(2)3167 4089 y Fu(;)150 4299 y Fz(whic)m(h)33 b(is)f(in)g Fu(L)707 4263 y Fs(1)779 4299 y Fz(b)m(y)i(Corollary)d (6.5.)296 4420 y(The)48 b(second)g(term)e(can)h(b)s(e)f(rewritten)h (using)g(that)2379 4393 y(~)2358 4420 y Fu(f)2406 4435 y Ft(E)2466 4420 y Fz(\()p Fu(H)8 b Fz(\))p Fu( )55 b Fz(=)c Fu( )f Fz(and)d(a)f(computation)150 4540 y(similar)29 b(to)k(\(6.21\))1198 4725 y(~)1177 4751 y Fu(f)1225 4766 y Ft(E)1285 4751 y Fz(\()p Fu(H)8 b Fz(\))p Fu(i)1500 4670 y Fl(\002)1541 4751 y Fu(H)r(;)17 b(F)1745 4710 y Fs(2)1731 4775 y(1)1800 4751 y Fz(\()p Fu(r)s(=t)p Fz(\))2007 4670 y Fl(\003)2086 4725 y Fz(~)2065 4751 y Fu(f)2113 4766 y Ft(E)2173 4751 y Fz(\()p Fu(H)8 b Fz(\))1018 4911 y(=)83 b Fu(t)1212 4870 y Fm(\000)p Fs(1)1328 4885 y Fz(~)1307 4911 y Fu(f)1355 4926 y Ft(E)1414 4911 y Fz(\()p Fu(H)8 b Fz(\))p Fu(F)1642 4926 y Fs(1)1681 4911 y Fv(p)23 b Fr(\001)f Fz(\()p Fr(r)p Fu(r)s Fz(\))p Fu(F)2099 4870 y Fm(0)2085 4936 y Fs(1)2145 4885 y Fz(~)2124 4911 y Fu(f)2172 4926 y Ft(E)2231 4911 y Fz(\()p Fu(H)8 b Fz(\))22 b(+)g(h)p Fu(:)p Fz(c)p Fu(:)1018 5057 y Fz(=)83 b Fu(t)1212 5015 y Fm(\000)p Fs(1)1307 5057 y Fu(F)1370 5072 y Fs(1)1431 5030 y Fz(~)1409 5057 y Fu(f)1457 5072 y Ft(E)1517 5057 y Fz(\()p Fu(H)8 b Fz(\))p Fv(p)22 b Fr(\001)g Fz(\()p Fr(r)p Fu(r)s Fz(\))2042 5030 y(~)2022 5057 y Fu(f)2070 5072 y Ft(E)2129 5057 y Fz(\()p Fu(H)8 b Fz(\))p Fu(F)2371 5015 y Fm(0)2357 5081 y Fs(1)2418 5057 y Fz(+)22 b(h)p Fu(:)p Fz(c)p Fu(:)h Fz(+)f Fu(R)2862 5072 y Fs(2)2902 5057 y Fz(\()p Fu(t)p Fz(\))p Fu(:)536 b Fz(\(6.27\))150 5267 y(Using)32 b(\(6.3\))g(one)g(can)h(sho)m(w)g (that)f Fr(jj)p Fu(R)1596 5282 y Fs(2)1635 5267 y Fz(\()p Fu(t)p Fz(\))p Fr(jj)c Fz(=)f Fr(O)s Fz(\()p Fu(t)2088 5231 y Fm(\000)p Fs(2)2183 5267 y Fz(\),)32 b(cf.)44 b(\(6.22\),)31 b(and)i(hence)g(in)m(tegrable.)43 b(Let)150 5388 y(us)33 b(lo)s(ok)f(at)1041 5508 y Fu(t)1076 5467 y Fm(\000)p Fs(1)1187 5397 y Fl(D)1248 5508 y Fu(F)1311 5523 y Fs(1)1367 5508 y Fz(\()p Fu(r)s(=t)p Fz(\))1611 5482 y(~)1590 5508 y Fu(f)1638 5523 y Ft(E)1698 5508 y Fz(\()p Fu(H)8 b Fz(\))p Fv(p)22 b Fr(\001)g Fz(\()p Fr(r)p Fu(r)s Fz(\))2223 5482 y(~)2203 5508 y Fu(f)2251 5523 y Ft(E)2310 5508 y Fz(\()p Fu(H)8 b Fz(\))p Fu(F)2552 5467 y Fm(0)2538 5533 y Fs(1)2593 5508 y Fz(\()p Fu(r)s(=t)p Fz(\))2800 5397 y Fl(E)2861 5577 y Ft(t)2907 5508 y Fu(:)1939 5776 y Fz(55)p eop %%Page: 56 59 56 58 bop 150 407 a Fz(T)-8 b(aking)32 b(the)h(mo)s(dulus)f(w)m(e)h (get)g(an)f(upp)s(er)h(b)s(ound)g(for)f(it)g(of)g(the)h(form)678 575 y(1)p 661 620 85 4 v 661 711 a(2)p Fu(t)755 642 y Fr(jj)831 616 y Fz(~)811 642 y Fu(f)859 657 y Ft(E)918 642 y Fz(\()p Fu(H)8 b Fz(\))p Fv(p)22 b Fr(\001)f Fz(\()p Fr(r)p Fu(r)s Fz(\))1443 616 y(~)1422 642 y Fu(f)1470 657 y Ft(E)1530 642 y Fz(\()p Fu(H)8 b Fz(\))p Fr(jj)1768 532 y Fl(\020)1826 642 y Fr(k)o Fu(F)1938 657 y Fs(1)1995 642 y Fz(\()o Fu(r)s(=t)p Fz(\))17 b Fu( )t Fz(\()p Fu(t)p Fz(\))p Fr(k)2446 594 y Fs(2)2507 642 y Fz(+)22 b Fr(k)p Fu(F)2732 601 y Fm(0)2718 667 y Fs(1)2774 642 y Fz(\()p Fu(r)s(=t)p Fz(\))16 b Fu( )t Fz(\()p Fu(t)p Fz(\))p Fr(k)3225 591 y Fs(2)3265 532 y Fl(\021)3576 642 y Fz(\(6.28\))150 874 y(whic)m(h)33 b(is)f(in)m(tegrable)g(due)h(to)f(Corollary)f(6.5.)43 b(In)33 b(conclusion,)f Fu(\021)2582 838 y Fm(0)2638 874 y Fz(is)g Fu(L)2802 838 y Fs(1)2874 874 y Fz(and)h(w)m(e)g(are)g (done.)p 3754 874 4 66 v 3758 812 59 4 v 3758 874 V 3816 874 4 66 v 150 1176 a Fv(Lemma)27 b(6.7.)34 b FA(Supp)-5 b(ose)26 b Fu(C)35 b(>)27 b Fz(1)f FA(and)g Fu(F)1635 1191 y Fs(2)1702 1176 y Fr(2)j Fu(L)1863 1140 y Fm(1)1938 1176 y Fz(\()p Fn(R)5 b Fz(\))32 b FA(with)26 b Fz(supp)q(\()p Fu(F)2617 1191 y Fs(2)2657 1176 y Fz(\))h Fr(\032)i Fz(\()p Fu(C)r(;)17 b Fr(1)p Fz(\))25 b FA(and)h Fr(jj)p Fu(F)3445 1191 y Fs(2)3483 1176 y Fr(jj)3539 1191 y Fm(1)3641 1176 y Fr(\024)i Fz(1)p FA(.)150 1296 y(Then)34 b(for)h(al)5 b(l)34 b Fu( )39 b FA(in)c(the)g(Schwartz)f(sp)-5 b(ac)g(e)34 b Fr(S)7 b Fz(\()p Fn(R)1920 1260 y Fs(2)1966 1296 y Fz(\))1301 1548 y(sup)1316 1629 y Ft(t)p Fm(\025)p Fs(1)1464 1548 y Fr(jj)p Fu(F)1583 1563 y Fs(2)1622 1548 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu( )t Fz(\()p Fu(t)p Fz(\))p Fr(jj)2063 1507 y Fs(2)2129 1548 y Fr(\024)2244 1481 y Fz(Const\()p Fu( )t Fz(\))p 2244 1525 393 4 v 2383 1616 a Fu(C)2460 1588 y Fs(2)2647 1548 y Fu(:)150 1804 y Fv(Pro)s(of.)43 b Fz(First,)32 b(notice)g(that)h(due)g(to)f(the)h(supp)s(ort)g (condition)e(w)m(e)j(ha)m(v)m(e)852 2000 y Fu(C)929 1959 y Fs(2)968 2000 y Fu(t)1003 1959 y Fs(2)1043 2000 y Fr(jj)p Fu(F)1162 2015 y Fs(2)1201 2000 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu( )t Fz(\()p Fu(t)p Fz(\))p Fr(jj)1642 1959 y Fs(2)1708 2000 y Fr(\024)28 b(jj)p Fu(r)s(F)1979 2015 y Fs(2)2017 2000 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu( )t Fz(\()p Fu(t)p Fz(\))p Fr(jj)2458 1959 y Fs(2)2524 2000 y Fr(\024)h(jj)e(j)p Fu(x)p Fr(j)p Fu( )t Fz(\()p Fu(t)p Fz(\))p Fr(jj)3058 1959 y Fs(2)3096 2000 y Fu(:)296 2197 y Fz(Second,)46 b(since)d Fu( )k Fr(2)e(S)7 b Fz(\()p Fn(R)1308 2161 y Fs(2)1353 2197 y Fz(\))42 b(w)m(e)h(ha)m(v)m(e)h(that) e Fu( )t Fz(\()p Fu(t)p Fz(\))i(=)f Fu(e)2428 2161 y Fm(\000)p Ft(itH)2600 2197 y Fu(f)2648 2212 y Ft(E)2708 2197 y Fz(\()p Fu(H)8 b Fz(\))p Fu( )45 b Fz(is)d(in)f(the)i(domain)d (of)150 2317 y(m)m(ultiplication)d(with)42 b(an)m(y)h(p)s(o)m(w)m(er)g (of)e Fr(j)p Fu(x)p Fr(j)h Fz(and)g Fu(A)g Fz(\(the)g(dilation)d (generator,)45 b(cf.)72 b(\(1.20\)\).)e(By)150 2438 y(in)m(tegrating)31 b(the)i(second)h(order)e(deriv)-5 b(ativ)m(e)33 b(\(from)e Fu(t)d Fz(=)f(0\))33 b(w)m(e)g(get)g(the)g(estimate,)f(cf.)43 b(\(1.21\),)1069 2635 y Fu(e)1114 2593 y Ft(itH)1231 2635 y Fv(x)1290 2593 y Fs(2)1330 2635 y Fu(e)1375 2593 y Fm(\000)p Ft(itH)1575 2635 y Fr(\024)28 b Fv(x)1739 2593 y Fs(2)1800 2635 y Fz(+)22 b(2)p Fu(tA)h Fz(+)f(Const)h Fr(\001)f Fz(\()p Fu(H)29 b Fz(+)22 b(1\))p Fu(t)2866 2593 y Fs(2)150 2831 y Fz(whic)m(h)33 b(leads)g(to)645 3028 y Fr(jj)27 b(j)p Fu(x)p Fr(j)p Fu( )t Fz(\()p Fu(t)p Fz(\))p Fr(jj)1073 2987 y Fs(2)1195 3028 y Fr(\024)83 b(h)p Fu(f)1442 3043 y Ft(E)1502 3028 y Fz(\()p Fu(H)8 b Fz(\))p Fu( )t(;)17 b Fv(x)1837 2987 y Fs(2)1875 3028 y Fu(f)1923 3043 y Ft(E)1983 3028 y Fz(\()p Fu(H)8 b Fz(\))p Fu( )t Fr(i)21 b Fz(+)h(2)p Fu(t)p Fr(h)p Fu(f)2544 3043 y Ft(E)2604 3028 y Fz(\()p Fu(H)8 b Fz(\))p Fu( )t(;)17 b(Af)3001 3043 y Ft(E)3060 3028 y Fz(\()p Fu(H)8 b Fz(\))p Fu( )t Fr(i)1196 3173 y Fz(+)83 b(Const)23 b Fr(\001)f Fz(\()p Fu(E)28 b Fz(+)22 b(1\))p Fu(t)2035 3132 y Fs(2)2075 3173 y Fr(jj)p Fu(f)2179 3188 y Ft(E)2238 3173 y Fz(\()p Fu(H)8 b Fz(\))p Fu( )t Fr(jj)2526 3132 y Fs(2)2564 3173 y Fu(:)150 3370 y Fz(No)m(w)33 b(com)m(bine)f(the)h(t)m(w)m(o)g (estimates)g(and)f(the)h(lemma)e(follo)m(ws.)p 3754 3370 4 66 v 3758 3308 59 4 v 3758 3370 V 3816 3370 4 66 v 150 3672 a Fv(Prop)s(osition)39 b(6.8.)44 b FA(Consider)37 b(the)g(function)h Fu(F)2009 3687 y Fs(+)2106 3672 y FA(in)g(\(3.1\))f(with)g Fu(a)d Fz(=)f Fu(K)e Fz(+)25 b(1)37 b FA(and)h Fu(b)33 b Fz(=)g Fu(K)f Fz(+)24 b(2)150 3792 y FA(\(and)34 b(with)h Fu(K)g Fz(=)812 3712 y Fr(p)p 895 3712 296 4 v 80 x Fz(2)p Fu(E)28 b Fz(+)22 b(1)p FA(\).)45 b(Then)34 b(for)g(al)5 b(l)35 b(states)g Fu( )t Fz(\()p Fu(t)p Fz(\))g FA(as)f(in)h(\(6.1\))f(we)g(have)1483 3989 y Fz(lim)1467 4049 y Ft(t)p Fm(!1)1650 3989 y Fr(jj)p Fu(F)1769 4004 y Fs(+)1844 3989 y Fz(\()p Fu(r)s(=t)p Fz(\))16 b Fu( )t Fz(\()p Fu(t)p Fz(\))p Fr(jj)28 b Fz(=)f(0)p Fu(:)150 4218 y Fv(Pro)s(of.)42 b Fz(What)29 b(follo)m(ws)f(is)h(an)g Fu("=)p Fz(3)f(-)g(argumen)m(t.)42 b(Start)29 b(b)m(y)h(\014xing)f Fu(")e(>)h Fz(0)h(\(this)f(is)h(not)g(the)g Fu(\017)h Fz(used)150 4338 y(to)i(sp)s(ecify)h(the)g(state)g(\(6.1\)\).)43 b(There)34 b(are)f(three)g(steps:)296 4459 y(I.)g(Cho)s(ose)g Fu( )793 4474 y Ft(")858 4459 y Fr(2)28 b(S)7 b Fz(\()p Fn(R)1124 4423 y Fs(2)1169 4459 y Fz(\))33 b(suc)m(h)h(that)e Fr(jj)p Fu( )26 b Fr(\000)c Fu( )1978 4474 y Ft(")2016 4459 y Fr(jj)27 b Fu(<)g("=)p Fz(3.)296 4579 y(I)s(I.)34 b(Denote)g(b)m(y)g Fv(1)958 4594 y Ft(\024)1037 4579 y Fz(the)g(c)m(haracteristic)f(function)h(for)f(\()p Fu(\024;)17 b Fr(1)p Fz(\).)46 b(Emplo)m(ying)32 b(Lemma)g(6.7,)i(w)m (e)150 4700 y(\014nd)f Fu(C)415 4715 y Ft(")479 4700 y Fu(>)28 b(K)h Fz(+)22 b(2)32 b(suc)m(h)i(that)1419 4896 y(sup)1434 4977 y Ft(t)p Fm(\025)p Fs(1)1582 4896 y Fr(jj)p Fv(1)1694 4911 y Ft(C)1744 4919 y Fq(")1782 4896 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu( )2052 4911 y Ft(")2088 4896 y Fz(\()p Fu(t)p Fz(\))p Fr(jj)27 b Fu(<)h("=)p Fz(3)p Fu(:)296 5157 y Fz(I)s(I)s(I.)38 b(Consider)h Fu(F)948 5172 y Fs(+)p Fm(\000)1100 5157 y Fz(in)e(\(3.3\))h(with)g (the)g(same)g Fu(a)g Fz(and)g Fu(b)h Fz(as)f(ab)s(o)m(v)m(e,)i(and)f (with)e Fu(c)g Fz(=)g Fu(C)3593 5172 y Ft(")3668 5157 y Fz(and)150 5278 y Fu(d)27 b Fz(=)h Fu(c)22 b Fz(+)g(1.)43 b(Then)34 b(b)m(y)f(Corollary)e(6.6)h(there)i(exists)f Fu(T)2167 5293 y Ft(")2232 5278 y Fu(>)27 b Fz(1)33 b(suc)m(h)h(that) 1391 5474 y(sup)1387 5556 y Ft(t)p Fm(\025)p Ft(T)1508 5564 y Fq(")1559 5474 y Fr(k)o Fu(F)1671 5489 y Fs(+)p Fm(\000)1802 5474 y Fz(\()p Fu(r)s(=t)p Fz(\))16 b Fu( )2088 5489 y Ft(")2125 5474 y Fz(\()p Fu(t)p Fz(\))p Fr(k)28 b Fu(<)g("=)p Fz(3)p Fu(:)1939 5776 y Fz(56)p eop %%Page: 57 60 57 59 bop 296 407 a Fz(W)-8 b(e)33 b(decomp)s(ose)1038 599 y Fu(F)1101 614 y Fs(+)1177 599 y Fz(\()p Fu(r)s(=t)p Fz(\))16 b Fu( )t Fz(\()p Fu(t)p Fz(\))83 b(=)g Fu(F)1883 614 y Fs(+)1959 599 y Fz(\()p Fu(r)s(=t)p Fz(\))16 b(\()p Fu( )t Fz(\()p Fu(t)p Fz(\))22 b Fr(\000)h Fu( )2583 614 y Ft(")2620 599 y Fz(\()p Fu(t)p Fz(\)\))1661 744 y(+)83 b([)p Fu(F)1910 759 y Fs(+)1986 744 y Fz(\()p Fu(r)s(=t)p Fz(\))22 b Fr(\000)h Fv(1)2371 759 y Ft(C)2421 767 y Fq(")2475 744 y Fz(\()p Fu(r)s(=t)p Fz(\)])16 b Fu( )2788 759 y Ft(")2825 744 y Fz(\()p Fu(t)p Fz(\))1661 889 y(+)83 b Fv(1)1876 904 y Ft(C)1926 912 y Fq(")1981 889 y Fz(\()p Fu(r)s(=t)p Fz(\))16 b Fu( )2267 904 y Ft(")2304 889 y Fz(\()p Fu(t)p Fz(\))p Fu(;)1134 b Fz(\(6.29\))150 1081 y(use)34 b(the)f(triangle)d(inequalit)m(y)-8 b(,)32 b(and)g(then)i(the)f(estimate)949 1274 y Fr(k)p Fz([)p Fu(F)1089 1289 y Fs(+)1165 1274 y Fz(\()p Fu(r)s(=t)p Fz(\))22 b Fr(\000)g Fv(1)1549 1289 y Ft(C)1599 1297 y Fq(")1654 1274 y Fz(\()p Fu(r)s(=t)p Fz(\))o(])17 b Fu( )1967 1289 y Ft(")2004 1274 y Fz(\()p Fu(t)p Fz(\))p Fr(k)28 b(\024)g(k)p Fu(F)2411 1289 y Fs(+)p Fm(\000)2541 1274 y Fz(\()p Fu(r)s(=t)p Fz(\))17 b Fu( )2828 1289 y Ft(")2865 1274 y Fz(\()p Fu(t)p Fz(\))p Fr(k)150 1466 y Fz(and)33 b(the)g(estimates)f(from)f(I,)i(I)s(I)g(and)f(I)s(I)s(I,)h (yielding)1484 1658 y(sup)1480 1739 y Ft(t)p Fm(\025)p Ft(T)1601 1747 y Fq(")1651 1658 y Fr(k)p Fu(F)1764 1673 y Fs(+)1840 1658 y Fz(\()p Fu(r)s(=t)p Fz(\))16 b Fu( )t Fz(\()p Fu(t)p Fz(\))p Fr(k)28 b Fu(<)f(";)150 1909 y Fz(and)33 b(therefore)g(the)g(prop)s(osition.)p 3754 1909 4 66 v 3758 1847 59 4 v 3758 1909 V 3816 1909 4 66 v 296 2148 a(As)g(a)g(consequence)i(of)d(Prop)s(osition)f(6.8)h(w)m (e)i(de\014ne)947 2340 y Fu(F)1024 2299 y Ft(M)s(:v)r(:b:)1010 2364 y Fm(\000)1252 2340 y Fz(:=)28 b Fu(F)1460 2293 y Ft(K)5 b Fs(+1)p Ft(;K)g Fs(+2)1446 2362 y Fm(\000)1792 2340 y Fu(;)114 b( )1996 2355 y Fs(1)2036 2340 y Fz(\()p Fu(t)p Fz(\))28 b(:=)g Fu(F)2383 2299 y Ft(M)s(:v)r(:b:)2369 2364 y Fm(\000)2600 2340 y Fz(\()p Fu(r)s(=t)p Fz(\))16 b Fu( )t Fz(\()p Fu(t)p Fz(\))p Fu(;)548 b Fz(\(6.30\))150 2532 y(and)33 b(notice)f(that)g(lim)974 2547 y Ft(t)p Fm(!1)1145 2532 y Fz(\()p Fu( )1246 2547 y Fs(1)1285 2532 y Fz(\()p Fu(t)p Fz(\))23 b Fr(\000)f Fu( )t Fz(\()p Fu(t)p Fz(\)\))28 b(=)f(0.)44 b(W)-8 b(e)33 b(rewrite)f(this)g(as)1712 2724 y Fu( )1775 2739 y Fs(1)1814 2724 y Fz(\()p Fu(t)p Fz(\))c Fr(\030)g Fu( )t Fz(\()p Fu(t)p Fz(\))p Fu(:)150 3009 y Fp(6.4)135 b(A)45 b(minimal)h(v)l(elo)t(cit)l(y)g(b)t(ound)150 3193 y Fz(W)-8 b(e)34 b(follo)m(w)e(the)i(same)f(strategy)h(as)g(in)e (the)i(classical)e(case)j(and)e(w)m(e)i(use)f(almost)e(the)i(same)f (tec)m(h-)150 3314 y(nique)38 b(\(with)f(some)g(complications)e(due)j (to)g(non-comm)m(utativit)m(y\).)56 b(Since)38 b(w)m(e)g(ha)m(v)m(e)h (the)f(max-)150 3434 y(imal)h(v)m(elo)s(cit)m(y)j(b)s(ound,)i(w)m(e)f (can)f(de\014ne)h(a)f(regularized)f(conjugate)h(op)s(erator)f(as)h (\(here)3578 3409 y(~)3556 3434 y Fu(F)3619 3449 y Fm(\000)3722 3434 y Fz(:=)150 3555 y Fu(F)227 3507 y Ft(K)5 b Fs(+3)p Ft(;K)g Fs(+4)213 3576 y Fm(\000)559 3555 y Fz(,)33 b Fu(K)i Fz(=)840 3474 y Fr(p)p 923 3474 296 4 v 81 x Fz(2)p Fu(E)28 b Fz(+)22 b(1\))1047 3730 y(~)1021 3756 y Fu(A)1094 3771 y Ft(C)1154 3756 y Fz(\()p Fu(E)6 b Fz(\))27 b(=)1461 3730 y(~)1439 3756 y Fu(F)1502 3771 y Fm(\000)1561 3756 y Fz(\()p Fu(r)s(=t)p Fz(\))1789 3729 y(~)1768 3756 y Fu(f)1816 3771 y Ft(E)1875 3756 y Fz(\()p Fu(H)8 b Fz(\))p Fu(A)2113 3771 y Ft(C)2172 3756 y Fz(\()p Fu(E)e Fz(\))2347 3729 y(~)2326 3756 y Fu(f)2374 3771 y Ft(E)2433 3756 y Fz(\()p Fu(H)i Fz(\))2620 3730 y(~)2598 3756 y Fu(F)2661 3771 y Fm(\000)2720 3756 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu(:)622 b Fz(\(6.31\))150 3948 y(Clearly)32 b(this)g(op)s(erator)g(is) g(b)s(ounded)h(and)g(gro)m(ws)g(at)f(most)g(linearly)f(in)h Fu(t)p Fz(:)1531 4140 y Fr(jj)1612 4115 y Fz(~)1587 4140 y Fu(A)1660 4155 y Ft(C)1718 4140 y Fz(\()p Fu(E)6 b Fz(\))p Fr(jj)27 b(\024)h Fz(Const)23 b Fr(\001)f Fu(t:)1132 b Fz(\(6.32\))150 4332 y(W)-8 b(e)33 b(start)g(with)f(the)h(quan)m(tum) g(equiv)-5 b(alen)m(t)32 b(of)g(Lemma)f(3.2:)150 4511 y Fv(Lemma)e(6.9.)37 b FA(F)-7 b(or)28 b(every)g Fu(F)1240 4526 y Fs(+)p Fm(\000)1382 4511 y Fz(=)g Fu(F)1563 4460 y Ft(a;b;d=)p Fs(2)p Ft(;d)1549 4533 y Fs(+)p Fm(\000)1865 4511 y FA(denote)g(by)h Fu(F)2354 4526 y Fs(++)2496 4511 y FA(exactly)g(that)g(function)f(as)g(in)h(\(3.4\))150 4632 y(whose)39 b(derivative)g(gives)f(b)-5 b(ack)39 b Fu(F)1416 4647 y Fs(+)p Fm(\000)1530 4632 y FA(.)59 b(Then)39 b(ther)-5 b(e)40 b(exists)f Fu(d)d(>)h Fz(0)i FA(smal)5 b(l)39 b(enough)g(such)g(that)h(we)150 4752 y(have)34 b(\(se)-5 b(e)34 b(\(6.30\)\))948 4859 y Fl(Z)1048 4885 y Fm(1)1003 5085 y Fs(1)1139 4880 y Fl(\015)1139 4940 y(\015)1139 5000 y(\015)1194 4908 y(p)p 1294 4908 178 4 v 87 x Fu(F)1371 4960 y Fm(0)1357 5016 y Fs(++)1471 4995 y Fz(\()1535 4970 y(~)1509 4995 y Fu(A)1582 5010 y Ft(C)1641 4995 y Fz(\()p Fu(E)6 b Fz(\))p Fu(=t)p Fz(\))p Fu( )1980 5010 y Fs(1)2020 4995 y Fz(\()p Fu(t)p Fz(\))2131 4880 y Fl(\015)2131 4940 y(\015)2131 5000 y(\015)2186 4906 y Fs(2)2242 4995 y Fu(dt)28 b Fr(\024)g Fz(Const)23 b Fr(\001)f(jj)p Fu( )t Fr(jj)2962 4954 y Fs(2)3000 4995 y Fu(:)549 b Fz(\(6.33\))150 5265 y FA(Mor)-5 b(e)g(over,)34 b(for)h Fu(F)827 5280 y Fm(\000)914 5265 y Fz(=)28 b Fu(F)1095 5214 y Ft(d=)p Fs(4)p Ft(;d=)p Fs(2)1081 5287 y Fm(\000)1367 5265 y FA(we)34 b(have)1417 5457 y Fz(lim)1401 5517 y Ft(t)p Fm(!1)1584 5457 y Fu(F)1647 5472 y Fm(\000)1707 5457 y Fz(\()1770 5432 y(~)1745 5457 y Fu(A)1818 5472 y Ft(C)1877 5457 y Fz(\()p Fu(E)6 b Fz(\))p Fu(=t)p Fz(\))p Fu( )2216 5472 y Fs(1)2256 5457 y Fz(\()p Fu(t)p Fz(\))27 b(=)h(0)p Fu(:)1002 b Fz(\(6.34\))1939 5776 y(57)p eop %%Page: 58 61 58 60 bop 150 407 a Fv(Pro)s(of)p Fz(.)43 b(De\014ne)33 b(the)g(b)s(ounded)h(observ)-5 b(able)1035 605 y(\010\()p Fu(t)p Fz(\))29 b(=)e Fu(F)1425 564 y Ft(M)s(:v)r(:b:)1411 629 y Fm(\000)1626 605 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu(F)1896 620 y Fs(++)2009 605 y Fz(\()2073 580 y(~)2047 605 y Fu(A)2120 620 y Ft(C)2180 605 y Fz(\()p Fu(E)6 b Fz(\))p Fu(=t)p Fz(\))p Fu(F)2533 564 y Ft(M)s(:v)r(:b:)2519 629 y Fm(\000)2733 605 y Fz(\()p Fu(r)s(=t)p Fz(\))150 803 y(and)33 b(di\013eren)m(tiate)f Fr(h)p Fz(\010\()p Fu(t)p Fz(\))p Fr(i)1141 818 y Ft( )r Fs(\()p Ft(t)p Fs(\))1306 803 y Fz(with)g(resp)s(ect)i(to)e Fu(t)h Fz(and)g(get:)971 1035 y Fu(@)1022 1050 y Ft(t)1052 1035 y Fr(h)p Fz(\010\()p Fu(t)p Fz(\))p Fr(i)1311 1051 y Ft( )r Fs(\()p Ft(t)p Fs(\))1471 1035 y Fz(=)28 b Fu(R)1649 1050 y Fs(1)1689 1035 y Fz(\()p Fu(t)p Fz(\))22 b(+)1920 925 y Fl(D)1981 1035 y Fu(D)2062 1050 y Ft(H)2129 1035 y Fz([)2182 1010 y(~)2156 1035 y Fu(A)2229 1050 y Ft(C)2288 1035 y Fz(\()p Fu(E)6 b Fz(\))p Fu(=t)p Fz(])2553 925 y Fl(E)2614 1043 y Fr(p)p 2697 1043 142 4 v 61 x Ft(F)2752 1081 y Ff(0)2742 1125 y Fj(++)2839 1104 y Ft( )2885 1113 y Fj(1)2920 1104 y Fs(\()p Ft(t)p Fs(\))3576 1035 y Fz(\(6.35\))150 1294 y(where)1327 1414 y Fu(D)1408 1429 y Ft(H)1475 1414 y Fu(X)i Fz(\()p Fu(t)p Fz(\))27 b(:=)h Fu(@)1884 1429 y Ft(t)1914 1414 y Fu(X)8 b Fz(\()p Fu(t)p Fz(\))22 b(+)g Fu(i)p Fz([)p Fu(H)r(;)17 b(X)8 b Fz(\()p Fu(t)p Fz(\)])928 b(\(6.36\))150 1579 y(denotes)32 b(the)f(Heisen)m(b)s(erg)g(deriv)-5 b(ativ)m(e;)31 b(w)m(e)h(also)d(emplo)m(y)m(ed)i(\(6.4\).)43 b(The)31 b(ab)s(o)m(v)m(e)h(remainder)d Fu(R)3674 1594 y Fs(1)3714 1579 y Fz(\()p Fu(t)p Fz(\))150 1699 y(can)k(b)s(e)g (treated)g(with)f(the)h(same)f(metho)s(ds)h(as)f(b)s(efore)h(and)g(sho) m(wn)h(to)e(ob)s(ey)h(the)g(estimate)1360 1807 y Fl(Z)1459 1833 y Fm(1)1415 2033 y Fs(1)1551 1943 y Fr(j)p Fu(R)1653 1958 y Fs(1)1692 1943 y Fz(\()p Fu(t)p Fz(\))p Fr(j)p Fu(dt)27 b Fr(\024)h Fz(Const)23 b Fr(\001)f(jj)p Fu( )t Fr(jj)2550 1902 y Fs(2)2588 1943 y Fu(:)150 2210 y Fz(P)m(erforming)28 b(the)i(Heisen)m(b)s(erg)h(deriv)-5 b(ativ)m(e)29 b(of)1889 2185 y(~)1863 2210 y Fu(A)1936 2225 y Ft(C)1996 2210 y Fz(\()p Fu(E)6 b Fz(\))p Fu(=t)p Fz(,)30 b(w)m(e)g(obtain)f(sev)m (eral)h(terms)g(\(see)h(\(6.31\)\):)513 2408 y Fu(D)594 2423 y Ft(H)661 2408 y Fz([)714 2383 y(~)688 2408 y Fu(A)761 2423 y Ft(C)820 2408 y Fz(\()p Fu(E)6 b Fz(\))p Fu(=t)p Fz(])28 b(=)f Fu(R)1290 2423 y Fs(2)1330 2408 y Fz(\()p Fu(t)p Fz(\))22 b(+)1584 2383 y(~)1561 2408 y Fu(F)1624 2423 y Fm(\000)1684 2408 y Fz(\()p Fu(r)s(=t)p Fz(\))1911 2381 y(~)1891 2408 y Fu(f)1939 2423 y Ft(E)1998 2408 y Fz(\()p Fu(H)8 b Fz(\))p Fr(f)p Fu(D)2294 2423 y Ft(H)2360 2408 y Fz([)p Fu(A)2460 2423 y Ft(C)2520 2408 y Fz(\()p Fu(E)e Fz(\))p Fu(=t)p Fz(])p Fr(g)2856 2381 y Fz(~)2835 2408 y Fu(f)2883 2423 y Ft(E)2942 2408 y Fz(\()p Fu(H)i Fz(\))3129 2383 y(~)3107 2408 y Fu(F)3170 2423 y Fm(\000)3229 2408 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu(:)150 2606 y Fz(Using)32 b(Prop)s(osition)f(4.1)h(w)m(e)i(can)f(write)605 2843 y Fu(D)686 2858 y Ft(H)753 2843 y Fz([)806 2817 y(~)780 2843 y Fu(A)853 2858 y Ft(C)912 2843 y Fz(\()p Fu(E)6 b Fz(\))p Fu(=t)p Fz(])28 b Fr(\025)g Fu(R)1384 2858 y Fs(3)1424 2843 y Fz(\()p Fu(t)p Fz(\))22 b(+)1683 2775 y(1)p 1665 2820 85 4 v 1665 2911 a(3)p Fu(t)1781 2817 y Fz(~)1759 2843 y Fu(F)1822 2858 y Fm(\000)1881 2843 y Fz(\()p Fu(r)s(=t)p Fz(\))2109 2816 y(~)2088 2843 y Fu(f)2136 2858 y Ft(E)2196 2843 y Fz(\()p Fu(H)8 b Fz(\))2361 2801 y Fs(2)2422 2817 y Fz(~)2399 2843 y Fu(F)2462 2858 y Fm(\000)2522 2843 y Fz(\()p Fu(r)s(=t)p Fz(\))21 b Fr(\000)2860 2775 y Fz(1)p 2860 2820 49 4 v 2867 2911 a Fu(t)2919 2843 y Fz([)2971 2817 y(~)2946 2843 y Fu(A)3019 2858 y Ft(C)3078 2843 y Fz(\()p Fu(E)6 b Fz(\))p Fu(=t)p Fz(])p Fu(:)150 3071 y Fz(The)34 b(remainder)d Fu(R)885 3086 y Fs(3)925 3071 y Fz(\()p Fu(t)p Fz(\))h(will)f(also)g(b)s(e)i(in) m(tegrable)f(in)f(the)i(sense)i(that)1126 3185 y Fl(Z)1225 3211 y Fm(1)1181 3410 y Fs(1)1317 3320 y Fr(jh)p Fu(R)1458 3335 y Fs(3)1497 3320 y Fz(\()p Fu(t)p Fz(\))p Fr(i)1647 3294 y(p)p 1730 3294 142 4 v 61 x Ft(F)1785 3332 y Ff(0)1775 3376 y Fj(++)1872 3355 y Ft( )1918 3364 y Fj(1)1952 3355 y Fs(\()p Ft(t)p Fs(\))2037 3320 y Fr(j)p Fu(dt)27 b Fr(\024)h Fz(Const)23 b Fr(\001)f(jj)p Fu( )t Fr(jj)2784 3279 y Fs(2)2822 3320 y Fu(:)150 3565 y Fz(Due)33 b(to)f(Lemmas)f(6.1)i (and)f(6.2)g(w)m(e)i(ma)m(y)e(write)660 3763 y Fr(h)721 3738 y Fz(~)699 3763 y Fu(F)762 3778 y Fm(\000)821 3763 y Fz(\()p Fu(r)s(=t)p Fz(\))1049 3737 y(~)1028 3763 y Fu(f)1076 3778 y Ft(E)1135 3763 y Fz(\()p Fu(H)8 b Fz(\))1300 3722 y Fs(2)1361 3738 y Fz(~)1339 3763 y Fu(F)1402 3778 y Fm(\000)1461 3763 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fr(i)1707 3737 y(p)p 1790 3737 V 61 x Ft(F)1845 3775 y Ff(0)1835 3819 y Fj(++)1932 3798 y Ft( )1978 3807 y Fj(1)2012 3798 y Fs(\()p Ft(t)p Fs(\))2124 3763 y Fz(=)28 b Fr(jj)2284 3677 y Fl(p)p 2383 3677 178 4 v 2383 3763 a Fu(F)2460 3729 y Fm(0)2446 3785 y Fs(++)2560 3763 y Fz(\()2624 3738 y(~)2598 3763 y Fu(A)2671 3778 y Ft(C)2730 3763 y Fz(\()p Fu(E)6 b Fz(\))p Fu(=t)p Fz(\))p Fu( )3069 3778 y Fs(1)3109 3763 y Fz(\()p Fu(t)p Fz(\))p Fr(jj)3276 3722 y Fs(2)1666 4002 y Fz(+)p Fr(O)s Fz(\()p Fu(t)1897 3961 y Fm(\0001)2027 4002 y Fz(\))p Fr(jj)p Fu( )t Fr(jj)2244 3961 y Fs(2)2282 4002 y Fu(:)150 4167 y Fz(Putting)32 b(ev)m(erything)i(bac)m(k)f(in)m(to)f(\(6.35\))g(w)m(e)h(obtain)679 4410 y Fu(@)730 4425 y Ft(t)760 4410 y Fr(h)p Fz(\010\()p Fu(t)p Fz(\))p Fr(i)1019 4425 y Ft( )r Fs(\()p Ft(t)p Fs(\))1180 4410 y Fr(\025)28 b Fu(R)1359 4425 y Fs(4)1399 4410 y Fz(\()p Fu(t)p Fz(\))22 b(+)1640 4342 y(1)p 1640 4387 49 4 v 1647 4478 a Fu(t)1715 4295 y Fl(\015)1715 4355 y(\015)1715 4414 y(\015)1771 4323 y(p)p 1870 4323 178 4 v 1870 4410 a Fu(F)1947 4375 y Fm(0)1933 4431 y Fs(++)2047 4410 y Fz(\()2111 4384 y(~)2085 4410 y Fu(A)2158 4425 y Ft(C)2218 4410 y Fz(\()p Fu(E)6 b Fz(\))p Fu(=t)p Fz(\))p Fu( )2557 4425 y Fs(1)2596 4410 y Fz(\()p Fu(t)p Fz(\))2707 4295 y Fl(\015)2707 4355 y(\015)2707 4414 y(\015)2763 4321 y Fs(2)2824 4410 y Fr(\001)22 b Fz(\(1)p Fu(=)p Fz(3)g Fr(\000)g Fu(d)p Fz(\))p Fu(:)150 4638 y Fz(W)-8 b(e)29 b(then)h(in)m(tegrate)e(and)h(\(6.33\))f(follo)m(ws.) 41 b(The)30 b(pro)s(of)e(of)g(\(6.34\))g(uses)i(the)f(same)g(strategy)h (as)f(the)150 4759 y(one)k(emplo)m(y)m(ed)g(in)e(Lemma)h(3.2)g(or)g (Corollary)f(6.6.)p 3754 4759 4 66 v 3758 4696 59 4 v 3758 4759 V 3816 4759 4 66 v 296 4997 a(De\014ne)i(\(here)h Fu(d)e Fz(is)g(the)h(one)g(giv)m(en)g(b)m(y)g(Lemma)e(6.9\))417 5209 y Fu(F)494 5168 y Ft(dil)480 5233 y Fs(+)580 5209 y Fz(\()644 5184 y(~)618 5209 y Fu(A)691 5224 y Ft(C)750 5209 y Fz(\()p Fu(E)6 b Fz(\))p Fu(=t)p Fz(\))28 b(:=)g Fu(F)1262 5158 y Ft(d=)p Fs(8)p Ft(;d=)p Fs(4)1248 5231 y(+)1499 5209 y Fz(\()1562 5184 y(~)1537 5209 y Fu(A)1610 5224 y Ft(C)1669 5209 y Fz(\()p Fu(E)6 b Fz(\))p Fu(=t)p Fz(\))p Fu(;)114 b( )2149 5224 y Fs(2)2189 5209 y Fz(\()p Fu(t)p Fz(\))28 b(:=)f Fu(F)2535 5168 y Ft(dil)2521 5233 y Fs(+)2621 5209 y Fz(\()2685 5184 y(~)2659 5209 y Fu(A)2732 5224 y Ft(C)2792 5209 y Fz(\()p Fu(E)6 b Fz(\))p Fu(=t)p Fz(\))p Fu( )3131 5224 y Fs(1)3170 5209 y Fz(\()p Fu(t)p Fz(\))p Fu(:)268 b Fz(\(6.37\))150 5407 y(Clearly)-8 b(,)36 b(\(6.34\))g(implies)d(that)j Fu( )1410 5422 y Fs(2)1450 5407 y Fz(\()p Fu(t)p Fz(\))e Fr(\030)h Fu( )1770 5422 y Fs(1)1809 5407 y Fz(\()p Fu(t)p Fz(\))f Fr(\030)h Fu( )t Fz(\()p Fu(t)p Fz(\))h(when)h Fu(t)d Fr(!)g(1)p Fz(.)54 b(W)-8 b(e)36 b(no)m(w)h(are)f(ready)h(to)150 5527 y(pro)m(v)m(e)d(a)e(minimal)c(v)m(elo)s(cit)m(y)33 b(b)s(ound:)1939 5776 y(58)p eop %%Page: 59 62 59 61 bop 150 408 a Fv(Prop)s(osition)42 b(6.10.)j FA(F)-7 b(or)39 b Fu(d)1275 423 y Fs(1)1351 408 y Fu(>)e Fz(0)i FA(de\014ne)g Fu(F)1904 423 y Fm(\000)1963 408 y Fz(\()p Fu(r)s(=t)p Fz(\))e(:=)g Fu(F)2424 357 y Ft(d)2460 366 y Fj(1)2495 357 y Ft(=)p Fs(2)p Ft(;d)2621 366 y Fj(1)2410 430 y Fm(\000)2660 408 y Fz(\()p Fu(r)s(=t)p Fz(\))p FA(.)59 b(Then)39 b(ther)-5 b(e)40 b(exists)g Fu(d)3786 423 y Fs(1)150 528 y FA(smal)5 b(l)34 b(enough)g(such)h(that)1331 649 y Fr(jj)p Fu(F)1450 664 y Fm(\000)1509 649 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu( )1779 664 y Fs(2)1818 649 y Fz(\()p Fu(t)p Fz(\))p Fr(jj)27 b Fz(=)h Fr(O)s Fz(\()p Fu(t)2271 608 y Fm(\0001)2401 649 y Fz(\))p Fr(jj)p Fu( )t Fr(jj)p Fu(:)931 b Fz(\(6.38\))150 852 y Fv(Pro)s(of)p Fz(.)43 b(The)34 b(strategy)f(is)f(pro)m(ving)g(that)h(for)f Fu(d)1932 867 y Fs(1)2003 852 y Fz(small)f(enough)i(w)m(e)g(ha)m(v)m(e) 1187 1072 y Fr(jj)p Fu(F)1306 1087 y Fm(\000)1364 1072 y Fz(\()p Fu(r)s(=t)p Fz(\))22 b Fr(\001)g Fu(F)1720 1031 y Ft(dil)1706 1097 y Fs(+)1806 1072 y Fz(\()1869 1047 y(~)1844 1072 y Fu(A)1917 1087 y Ft(C)1976 1072 y Fz(\()p Fu(E)6 b Fz(\))p Fu(=t)p Fz(\))p Fr(jj)27 b Fz(=)g Fr(O)s Fz(\()p Fu(t)2593 1031 y Fm(\0001)2723 1072 y Fz(\))p Fu(:)788 b Fz(\(6.39\))150 1303 y(Since)24 b(sup)542 1326 y Ft(t)p Fm(\025)p Fs(1)679 1303 y Fr(jj)760 1278 y Fz(~)735 1303 y Fu(A)808 1318 y Ft(C)866 1303 y Fz(\()p Fu(E)6 b Fz(\))p Fu(=t)p Fr(jj)27 b Fz(=)h Fu(M)38 b(<)28 b Fr(1)p Fz(,)d(w)m(e)f(can)g(replace)g Fu(F)2382 1267 y Ft(dil)2368 1328 y Fs(+)2491 1303 y Fz(with)f(some)g(compactly)g(supp)s(orted)150 1447 y(function)36 b Fu(F)613 1396 y Ft(d=)p Fs(8)p Ft(;d=)p Fs(4)p Ft(;M)s(;M)7 b Fs(+1)599 1468 y(+)p Fm(\000)1162 1447 y Fz(suc)m(h)38 b(that)e Fu(F)1678 1410 y Ft(dil)1664 1471 y Fs(+)1764 1447 y Fz(\()1828 1421 y(~)1802 1447 y Fu(A)1875 1462 y Ft(C)1934 1447 y Fz(\()p Fu(E)6 b Fz(\))p Fu(=t)p Fz(\))35 b(=)f Fu(F)2418 1462 y Fs(+)p Fm(\000)2532 1447 y Fz(\()2596 1421 y(~)2570 1447 y Fu(A)2643 1462 y Ft(C)2702 1447 y Fz(\()p Fu(E)6 b Fz(\))p Fu(=t)p Fz(\).)56 b(Using)36 b(the)h(Hel\013er-)150 1567 y(Sj\177)-49 b(ostrand)33 b(form)m(ula)d(w)m(e)k(get)992 1970 y Fu(F)1055 1985 y Fm(\000)1114 1970 y Fz(\()p Fu(r)s(=t)p Fz(\))22 b Fr(\001)g Fu(F)1470 1929 y Ft(dil)1456 1994 y Fs(+)1573 1799 y Fl( )1687 1877 y Fz(~)1661 1902 y Fu(A)1734 1917 y Ft(C)1794 1902 y Fz(\()p Fu(E)6 b Fz(\))p 1661 1947 287 4 v 1787 2038 a Fu(t)1958 1799 y Fl(!)833 2332 y Fz(=)83 b Fr(\000)1084 2264 y Fz(1)p 1079 2309 59 4 v 1079 2400 a Fu(\031)1165 2196 y Fl(Z)1221 2422 y Fe(C)p 1289 2251 57 4 v 1289 2332 a Fu(@)1368 2306 y Fz(~)1346 2332 y Fu(F)1409 2347 y Fs(+)p Fm(\000)1523 2332 y Fz(\()p Fu(z)t Fz(\))1665 2127 y Fl(8)1665 2217 y(<)1665 2396 y(:)1754 2332 y Fu(F)1817 2347 y Fm(\000)1876 2332 y Fz(\()p Fu(r)s(=t)p Fz(\))22 b Fr(\001)2155 2161 y Fl( )2269 2239 y Fz(~)2244 2264 y Fu(A)2317 2279 y Ft(C)2376 2264 y Fz(\()p Fu(E)6 b Fz(\))p 2244 2309 287 4 v 2369 2400 a Fu(t)2562 2332 y Fr(\000)22 b Fu(z)2710 2161 y Fl(!)2790 2184 y Fm(\000)p Fs(1)2884 2127 y Fl(9)2884 2217 y(=)2884 2396 y(;)2990 2332 y Fu(dxdy)t(:)350 b Fz(\(6.40\))150 2709 y(De\014ne)478 2660 y(~)478 2683 y(~)452 2709 y Fu(A)525 2724 y Ft(C)584 2709 y Fz(\()p Fu(E)6 b Fz(\))28 b(=)f Fu(F)946 2661 y Ft(d)982 2670 y Fj(1)1017 2661 y Ft(;)p Fs(2)p Ft(d)1108 2670 y Fj(1)932 2730 y Fm(\000)1147 2709 y Fz(\()p Fu(r)s(=t)p Fz(\))1379 2683 y(~)1354 2709 y Fu(A)1427 2724 y Ft(C)1486 2709 y Fz(\()p Fu(E)6 b Fz(\))p Fu(F)1717 2661 y Ft(d)1753 2670 y Fj(1)1787 2661 y Ft(;)p Fs(2)p Ft(d)1878 2670 y Fj(1)1703 2730 y Fm(\000)1917 2709 y Fz(\()p Fu(r)s(=t)p Fz(\))32 b(and)g(notice)g(that)g(for)g (su\016cien)m(tly)h(small)d Fu(d)3786 2724 y Fs(1)150 2829 y Fz(and)j(some)f Fu(T)46 b Fz(large)32 b(enough)h(w)m(e)g(ha)m(v) m(e)1525 3071 y(sup)1533 3152 y Ft(t)p Fm(\025)p Ft(T)1688 3071 y Fr(jj)1769 3022 y Fz(~)1769 3046 y(~)1744 3071 y Fu(A)1817 3086 y Ft(C)1876 3071 y Fz(\()p Fu(E)6 b Fz(\))p Fu(=t)p Fr(jj)27 b(\024)h Fu(d=)p Fz(9)150 3396 y(whic)m(h)33 b(implies)c(that)j Fu(F)1047 3345 y Ft(d=)p Fs(8)p Ft(;d=)p Fs(4)p Ft(;M)s(;M)7 b Fs(+1)1033 3417 y(+)p Fm(\000)1559 3396 y Fz(\()1623 3347 y(~)1623 3371 y(~)1597 3396 y Fu(A)1670 3411 y Ft(C)1729 3396 y Fz(\()p Fu(E)f Fz(\))p Fu(=t)p Fz(\))28 b(=)f(0)32 b(for)g(all)e Fu(t)d Fr(\025)i Fu(T)14 b Fz(.)43 b(Reasoning)31 b(as)h(in)f(Lemma)150 3516 y(6.1)h(w)m(e)i(ma)m(y)e(write)g(that)h(for)f Fu(N)38 b Fr(\025)28 b Fz(1)836 3758 y Fr(jj)p Fu(F)969 3707 y Ft(d)1005 3716 y Fj(1)1039 3707 y Ft(=)p Fs(2)p Ft(;d)1165 3716 y Fj(1)955 3780 y Fm(\000)1204 3758 y Fz(\()p Fu(r)s(=t)p Fz(\))1428 3647 y Fl(h)1474 3758 y Fz(\()1538 3733 y(~)1512 3758 y Fu(A)1585 3773 y Ft(C)1645 3758 y Fz(\()p Fu(E)6 b Fz(\))p Fu(=t)22 b Fr(\000)g Fu(z)t Fz(\))2091 3717 y Fm(\000)p Fs(1)2208 3758 y Fr(\000)h Fz(\()2372 3710 y(~)2372 3733 y(~)2346 3758 y Fu(A)2419 3773 y Ft(C)2478 3758 y Fz(\()p Fu(E)6 b Fz(\))p Fu(=t)22 b Fr(\000)h Fu(z)t Fz(\))2925 3717 y Fm(\000)p Fs(1)3020 3647 y Fl(i)3084 3758 y Fr(jj)1366 4094 y(\024)1481 4027 y Fz(Const)q(\()p Fu(N)10 b Fz(\))p 1481 4071 415 4 v 1542 4163 a Fr(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)1795 4134 y Fs(2)1922 3954 y Fl(\022)2068 4027 y Fr(h)p Fu(z)t Fr(i)p 2005 4071 253 4 v 2005 4163 a(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)2268 3954 y Fl(\023)2341 3976 y Ft(N)2425 4094 y Fu(t)2460 4053 y Fm(\000)p Ft(N)2582 4094 y Fu(:)150 4326 y Fz(Put)33 b(this)f(bac)m(k)i(in)m(to)e(\(6.40\))f(and)i(the)g(pro)s(of)f(is)g (\014nished.)p 3754 4326 4 66 v 3758 4263 59 4 v 3758 4326 V 3816 4326 4 66 v 296 4564 a(De\014ne)h(with)g(the)g Fu(d)1040 4579 y Fs(1)1111 4564 y Fz(pro)m(vided)g(b)m(y)h(the)f (previous)g(prop)s(osition)761 4784 y Fu(F)838 4743 y Ft(m:v)r(:b:)824 4809 y Fs(+)1030 4784 y Fz(\()p Fu(r)s(=t)p Fz(\))27 b(:=)g Fu(F)1471 4733 y Ft(d)1507 4742 y Fj(1)1542 4733 y Ft(=)p Fs(3)p Ft(;d)1668 4742 y Fj(1)1703 4733 y Ft(=)p Fs(2)1457 4806 y(+)1778 4784 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu(;)114 b( )2189 4799 y Fs(3)2228 4784 y Fz(\()p Fu(t)p Fz(\))28 b(:=)g Fu(F)2575 4743 y Ft(m:v)r(:b:)2561 4809 y Fs(+)2767 4784 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu( )3037 4799 y Fs(2)3076 4784 y Fz(\()p Fu(t)p Fz(\))p Fu(:)362 b Fz(\(6.41\))150 5004 y(Then)34 b(w)m(e)f(ha)m(v)m(e)h(sho)m(wn)g Fu( )1132 5019 y Fs(3)1172 5004 y Fz(\()p Fu(t)p Fz(\))28 b Fr(\030)g Fu( )1479 5019 y Fs(2)1519 5004 y Fz(\()p Fu(t)p Fz(\))g Fr(\030)g Fu( )1826 5019 y Fs(1)1866 5004 y Fz(\()p Fu(t)p Fz(\))f Fr(\030)h Fu( )t Fz(\()p Fu(t)p Fz(\).)1939 5776 y(59)p eop %%Page: 60 63 60 62 bop 150 407 a Fp(6.5)146 b FF(~)-70 b Fi(\032)46 b Fp(is)f(lo)t(calized)h(ab)t(o)l(v)l(e)g Fi(\032)1687 425 y Fh(E)1784 407 y Fd(\000)26 b Fi(\017)150 592 y Fz(W)-8 b(e)33 b(no)m(w)g(form)m(ulate)e(Prop)s(osition)g(2.11)h(in)g (terms)g(of)g(quan)m(tum)h(mec)m(hanics.)150 790 y Fv(Prop)s(osition)j (6.11.)41 b FA(Fix)35 b Fz(~)-49 b Fu(\017)1240 805 y Fs(2)1307 790 y Fu(>)28 b Fz(0)p FA(.)44 b(Consider)34 b Fu(F)2012 805 y Fm(\000)2099 790 y Fz(=)27 b Fu(F)2279 742 y Fm(\000)p Fs(2)q(~)-36 b Ft(\017)2398 751 y Fj(2)2432 742 y Ft(;)p Fm(\000)q Fs(~)g Ft(\017)2536 751 y Fj(2)2265 811 y Fm(\000)2574 790 y FA(.)45 b(Then)34 b(we)g(have)976 1003 y Fr(k)p Fu(F)1089 1018 y Fm(\000)1148 1003 y Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)1408 1018 y Ft(E)1490 1003 y Fz(+)f(\()p Fu(@)1677 1018 y Ft(E)1737 1003 y Fu(\032)1787 1018 y Ft(E)1847 1003 y Fz(\))p Fu(=C)7 b Fz(\))p Fu( )2112 1018 y Fs(3)2152 1003 y Fz(\()p Fu(t)p Fz(\))p Fr(k)27 b Fz(=)h Fr(O)s Fz(\()p Fu(t)2599 962 y Fm(\0001)2729 1003 y Fz(\))f Fr(jj)p Fu( )t Fr(jj)p Fu(:)576 b Fz(\(6.42\))150 1217 y Fv(Pro)s(of.)61 b Fz(If)38 b(w)m(e)i(lo)s(ok)d(at)h Fu(F)1186 1181 y Ft(dil)1172 1242 y Fs(+)1311 1217 y Fz(w)m(e)h(in)m(tro)s(duced)g(in)f(\(6.37\))f (w)m(e)j(can)f(write)f(\(w)m(e)h(drop)g(the)g(energy)150 1338 y(dep)s(endence\))1029 1493 y Fu(F)1106 1452 y Ft(dil)1092 1518 y Fs(+)1192 1493 y Fz(\()1256 1468 y(~)1230 1493 y Fu(A)1303 1508 y Ft(C)1362 1493 y Fu(=t)p Fz(\))28 b(=)g Fu(F)1740 1405 y Fq(d)p 1702 1417 109 3 v 1702 1459 a Fj(16)p Fq(C)1821 1433 y Ft(;)1873 1405 y Fq(d)p 1850 1417 78 3 v 1850 1459 a Fj(8)p Fq(C)1679 1515 y Fs(+)1959 1382 y Fl(\020)2044 1468 y Fz(~)2018 1493 y Fu(A)2091 1508 y Ft(C)2151 1493 y Fu(=)p Fz(\()p Fu(tC)7 b Fz(\))2388 1382 y Fl(\021)2463 1493 y Fu(F)2540 1452 y Ft(dil)2526 1518 y Fs(+)2627 1493 y Fz(\()2690 1468 y(~)2665 1493 y Fu(A)2738 1508 y Ft(C)2797 1493 y Fu(=t)p Fz(\))p Fu(:)150 1687 y Fz(De\014ne)1419 1807 y(\001)p Fu(\032)29 b Fz(:=)36 b(~)-58 b Fu(\032)23 b Fr(\000)f Fu(\032)1931 1822 y Ft(E)2013 1807 y Fz(+)g(\()p Fu(@)2200 1822 y Ft(E)2260 1807 y Fu(\032)2310 1822 y Ft(E)2370 1807 y Fz(\))p Fu(=C)r(:)1020 b Fz(\(6.43\))150 1979 y(The)34 b(prop)s(osition)c(w)m(ould)j(then)g(b)s(e)g(implied)d(b)m(y)j (the)g(estimate)345 2107 y Fl(\015)345 2167 y(\015)345 2227 y(\015)400 2222 y Fu(F)463 2237 y Fm(\000)522 2222 y Fz(\(\001)p Fu(\032)p Fz(\))p Fu(F)855 2134 y Fq(d)p 816 2146 109 3 v 816 2187 a Fj(16)p Fq(C)935 2161 y Ft(;)987 2134 y Fq(d)p 964 2146 78 3 v 964 2187 a Fj(8)p Fq(C)792 2244 y Fs(+)1073 2111 y Fl(\020)1158 2197 y Fz(~)1132 2222 y Fu(A)1205 2237 y Ft(C)1265 2222 y Fu(=)p Fz(\()p Fu(tC)7 b Fz(\))1502 2111 y Fl(\021)1578 2222 y Fu(F)1655 2181 y Ft(m:v)r(:b:)1641 2246 y Fs(+)1847 2222 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu(F)2131 2181 y Ft(M)s(:v)r(:b)2117 2246 y Fm(\000)2311 2222 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu(f)2566 2237 y Fs(1)p Ft(;E)2680 2222 y Fz(\()p Fu(H)h Fz(\))2845 2107 y Fl(\015)2845 2167 y(\015)2845 2227 y(\015)2928 2222 y Fz(=)27 b Fr(O)s Fz(\()p Fu(t)3186 2181 y Fm(\0001)3316 2222 y Fz(\))p Fu(:)195 b Fz(\(6.44\))150 2457 y(The)34 b(in)m(terpretation)e(of)h(the)g(ab)s(o)m(v)m(e)h (estimate)e(is)h(that)g(as)g(in)f(the)i(classical)d(situation,)41 b(~)-58 b Fu(\032)23 b Fr(\000)g Fu(\032)3667 2472 y Ft(E)3749 2457 y Fz(+)150 2578 y(\()p Fu(@)239 2593 y Ft(E)299 2578 y Fu(\032)349 2593 y Ft(E)409 2578 y Fz(\))p Fu(=C)7 b Fz(\))30 b(and)h Fu(A)902 2593 y Ft(C)993 2578 y Fz(tend)g(to)g(ha)m(v)m(e)h(the)g(same)e(sign.)43 b(The)32 b(strategy)f(of)g(the)g(pro)s(of)f(is)h(somewhat)150 2698 y(similar)36 b(to)i(the)i(one)f(w)m(e)h(ha)m(v)m(e)g(used)g(for)e (\(6.39\).)62 b(As)40 b(a)e(general)g(remark,)j(w)m(e)f(will)c(often)j (write)150 2819 y Fr(O)s Fz(\()p Fu(t)305 2782 y Fm(\0001)435 2819 y Fz(\))26 b(instead)g(of)f(terms)h(con)m(taining)f(comm)m (utators)g(of)g(the)i(t)m(yp)s(e)g(w)m(e)g(encoun)m(tered)h(in)d(Lemma) 150 2939 y(6.2.)296 3059 y(If)32 b Fu(F)456 3074 y Fs(+)p Fm(\000)602 3059 y Fz(is)g(supp)s(orted)h(on)f Fn(R)1359 3074 y Fs(+)1456 3059 y Fz(and)g(equals)h(1)e(on)h(the)h(supp)s(ort)f (of)g Fu(F)2872 3023 y Ft(m:v)r(:b:)2858 3084 y Fs(+)3064 3059 y Fu(F)3141 3023 y Ft(M)s(:v)r(:b)3127 3084 y Fm(\000)3354 3059 y Fz(then)h(de\014ne)176 3165 y(~)176 3189 y(~)150 3214 y Fu(A)223 3229 y Ft(C)315 3214 y Fz(as)454 3428 y Fu(f)502 3443 y Fs(1)p Ft(;E)616 3428 y Fz(\()p Fu(H)8 b Fz(\))p Fu(F)844 3443 y Fs(+)p Fm(\000)958 3428 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu(F)1242 3377 y Fm(\000)q Fs(~)-36 b Ft(\017)1326 3386 y Fj(2)1359 3377 y Ft(=)p Fs(2)p Ft(;)p Fm(\000)q Fs(~)g Ft(\017)1533 3386 y Fj(2)1567 3377 y Ft(=)p Fs(4)1228 3449 y Fm(\000)1642 3428 y Fz(\(\001)p Fu(\032)p Fz(\))23 b Fr(\001)1947 3402 y Fz(~)1921 3428 y Fu(A)1994 3443 y Ft(C)2076 3428 y Fr(\001)f Fu(F)2203 3377 y Fm(\000)q Fs(~)-36 b Ft(\017)2287 3386 y Fj(2)2320 3377 y Ft(=)p Fs(2)p Ft(;)p Fm(\000)q Fs(~)g Ft(\017)2494 3386 y Fj(2)2528 3377 y Ft(=)p Fs(4)2189 3449 y Fm(\000)2603 3428 y Fz(\(\001)p Fu(\032)p Fz(\))p Fu(F)2873 3443 y Fs(+)p Fm(\000)2988 3428 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu(f)3243 3443 y Fs(1)p Ft(;E)3357 3428 y Fz(\()p Fu(H)8 b Fz(\))150 3673 y(and)30 b(let)g(us)h(pro)m(v)m(e)h(that)e(the)g(sp)s (ectrum)h(of)1793 3624 y(~)1793 3647 y(~)1767 3673 y Fu(A)1840 3688 y Ft(C)1899 3673 y Fu(=)p Fz(\()p Fu(C)7 b(t)p Fz(\))30 b(tends)i(to)e(b)s(e)g(negativ)m(e.)43 b(Indeed,)33 b(using)d(the)150 3793 y(expressions)k(in)e(\(6.31\),)g (\(4.1\))g(and)h(\(2.33\),)f(w)m(e)h(\014rst)g(get)g(that)f(for)g Fu(t)c Fr(\025)g Fz(1)426 4005 y(~)400 4030 y Fu(A)473 4045 y Ft(C)532 4030 y Fu(=)p Fz(\()p Fu(C)7 b(t)p Fz(\))28 b(=)922 4005 y(~)900 4030 y Fu(F)963 4045 y Fm(\000)1022 4030 y Fz(\()p Fu(r)s(=t)p Fz(\))1250 4004 y(~)1229 4030 y Fu(f)1277 4045 y Ft(E)1337 4030 y Fz(\()p Fu(H)8 b Fz(\))21 b Fr(\001)h Fz(\()p Fu(\032)g Fr(\000)h Fu(\032)1833 4045 y Ft(E)1915 4030 y Fz(+)f Fu(@)2064 4045 y Ft(E)2124 4030 y Fu(\032)2174 4045 y Ft(E)2234 4030 y Fu(=C)7 b Fz(\))2408 3962 y Fu(r)p 2408 4007 47 4 v 2414 4098 a(t)2486 4030 y Fr(\001)2558 4004 y Fz(~)2536 4030 y Fu(f)2584 4045 y Ft(E)2644 4030 y Fz(\()p Fu(H)h Fz(\))2830 4005 y(~)2809 4030 y Fu(F)2872 4045 y Fm(\000)2931 4030 y Fz(\()p Fu(r)s(=t)p Fz(\))21 b(+)h Fr(O)s Fz(\(1)p Fu(=t)p Fz(\))p Fu(:)150 4314 y Fz(Because)32 b(of)d(the)i(presence)h(of)e Fu(f)1340 4329 y Fs(1)p Ft(;E)1454 4314 y Fz(\()p Fu(H)8 b Fz(\))30 b(in)f(the)i(de\014nition)e(of)2491 4266 y(~)2491 4289 y(~)2466 4314 y Fu(A)2539 4329 y Ft(C)2628 4314 y Fz(w)m(e)i(can)g(use)g(an)f(argumen)m(t)g(lik)m(e)150 4435 y(in)24 b(Lemma)f(6.2)g(b)m(y)j(enlarging)c(a)i(bit)g(the)g(supp)s (ort)h(of)f Fu(f)2166 4450 y Fs(1)p Ft(;E)2305 4435 y Fz(and)g(put)h(it)e(near)2976 4409 y(~)2955 4435 y Fu(f)3003 4450 y Ft(E)3062 4435 y Fz(\()p Fu(H)8 b Fz(\),)26 b(transforming)171 4529 y(~)150 4555 y Fu(f)198 4570 y Ft(E)258 4555 y Fz(\()p Fu(H)8 b Fz(\))21 b Fr(\001)h Fz(\()p Fu(\032)g Fr(\000)h Fu(\032)754 4570 y Ft(E)836 4555 y Fr(\000)g Fu(@)987 4570 y Ft(E)1047 4555 y Fu(\032)1097 4570 y Ft(E)1157 4555 y Fu(=C)7 b Fz(\))21 b Fr(\001)1414 4529 y Fz(~)1392 4555 y Fu(f)1440 4570 y Ft(E)1500 4555 y Fz(\()p Fu(H)8 b Fz(\))32 b(in)m(to)g(\001)p Fu(\032)22 b Fz(+)g Fr(O)s Fz(\()p Fu(t)2301 4519 y Fm(\0001)2431 4555 y Fz(\).)296 4676 y(Then)34 b(w)m(e)f(do)f(the)g(same)g(thing)f(with)h(the)h (cut-o\013)e(in)h(\001)p Fu(\032)g Fz(th)m(us)h(making)e(the)h (\\leading)f(term")150 4796 y(negativ)m(e)i(in)f(form)f(sense;)j(w)m(e) g(\014nally)d(get)1426 4983 y(~)1426 5006 y(~)1400 5032 y Fu(A)1473 5047 y Ft(C)1532 5032 y Fu(=)p Fz(\()p Fu(C)7 b(t)p Fz(\))28 b Fr(\024)g(O)s Fz(\()p Fu(t)2057 4990 y Fm(\000)p Fs(1)2152 5032 y Fz(\))p Fu(;)114 b(t)28 b Fr(\025)g Fz(1)p Fu(:)150 5245 y Fz(It)33 b(follo)m(ws)e(that)h (there)i(exist)e Fu(T)47 b Fz(large)31 b(enough)i(suc)m(h)h(that)f(for) f(ev)m(ery)i Fu(t)28 b Fr(\025)g Fu(T)41 b Fz(=)28 b Fu(T)14 b Fz(\()p Fu(C)7 b Fz(\):)1075 5460 y(~)1075 5483 y(~)1050 5509 y Fu(A)1123 5524 y Ft(C)1182 5509 y Fu(=)p Fz(\()p Fu(C)g(t)p Fz(\))27 b Fr(\024)1623 5441 y Fu(d)p 1561 5486 175 4 v 1561 5577 a Fz(16)p Fu(C)1746 5509 y(;)114 b(F)2012 5421 y Fq(d)p 1974 5433 109 3 v 1974 5474 a Fj(16)p Fq(C)2092 5448 y Ft(;)2145 5421 y Fq(d)p 2122 5433 78 3 v 2122 5474 a Fj(8)p Fq(C)1950 5530 y Fs(+)2230 5398 y Fl(\020)2316 5460 y Fz(~)2316 5483 y(~)2290 5509 y Fu(A)2363 5524 y Ft(C)2422 5509 y Fu(=)p Fz(\()p Fu(C)7 b(t)p Fz(\))2659 5398 y Fl(\021)2746 5509 y Fz(=)28 b(0)p Fu(:)650 b Fz(\(6.45\))1939 5776 y(60)p eop %%Page: 61 64 61 63 bop 150 429 a Fz(Then)32 b(reasoning)f(as)h(in)e(Prop)s(osition)g (6.10,)h(w)m(e)h(can)f(insert)g Fu(F)2537 342 y Fq(d)p 2499 354 109 3 v 2499 395 a Fj(16)p Fq(C)2617 369 y Ft(;)2670 342 y Fq(d)p 2647 354 78 3 v 2647 395 a Fj(8)p Fq(C)2475 451 y Fs(+)2756 319 y Fl(\020)2841 381 y Fz(~)2841 404 y(~)2815 429 y Fu(A)2888 444 y Ft(C)2947 429 y Fu(=)p Fz(\()p Fu(C)7 b(t)p Fz(\))3184 319 y Fl(\021)3275 429 y Fz(in)30 b(\(6.44\))h(and)150 572 y(up)36 b(to)g(the)g(use)h(of)e (Hel\013er-Sj\177)-49 b(ostrand)36 b(form)m(ula)e(and)i(of)f(v)-5 b(arious)36 b(comm)m(utator)e(estimates)i(as)g(in)150 692 y(Lemma)31 b(6.2)h(w)m(e)i(obtain)d(the)i(result.)p 3754 812 4 66 v 3758 750 59 4 v 3758 812 V 3816 812 4 66 v 296 1051 a(A)g(consequence)j(of)c(the)h(ab)s(o)m(v)m(e)g(prop)s (osition)e(is)150 1205 y Fv(Corollary)j(6.12.)39 b FA(Fix)32 b Fz(~)-49 b Fu(\017)1125 1220 y Fs(2)1192 1205 y Fu(>)28 b Fz(0)j FA(as)h(given)f(by)h(the)g(pr)-5 b(evious)32 b(pr)-5 b(op)g(osition.)42 b(Then)32 b(cho)-5 b(osing)30 b Fu(C)7 b Fz(\()p Fu(E)f Fz(\))150 1325 y FA(lar)-5 b(ge)34 b(enough)h(we)f(have)1099 1395 y Fl(\015)1099 1455 y(\015)1099 1515 y(\015)1155 1510 y Fu(F)1232 1463 y Fm(\000)p Fs(9)q(~)-36 b Ft(\017)1351 1472 y Fj(2)1385 1463 y Ft(;)p Fm(\000)p Fs(8)q(~)g Ft(\017)1524 1472 y Fj(2)1218 1531 y Fm(\000)1562 1510 y Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)1822 1525 y Ft(E)1882 1510 y Fz(\))p Fu( )1983 1525 y Fs(3)2023 1510 y Fz(\()p Fu(t)p Fz(\))2134 1395 y Fl(\015)2134 1455 y(\015)2134 1515 y(\015)2217 1510 y Fz(=)k Fr(O)s Fz(\()p Fu(t)2475 1469 y Fm(\0001)2605 1510 y Fz(\))h Fr(jj)p Fu( )t Fr(jj)p Fu(:)699 b Fz(\(6.46\))150 1694 y Fv(Pro)s(of.)67 b Fz(The)41 b(in)m(terpretation)f(is)g(again)f(simple:)57 b(if)40 b Fu(C)47 b Fz(is)40 b(large)f(then)50 b(~)-58 b Fu(\032)28 b Fr(\000)g Fu(\032)3118 1709 y Ft(E)3219 1694 y Fz(cannot)40 b(b)s(e)h(to)s(o)150 1815 y(small,)30 b(due)j(to)f(the)h(previous)g (prop)s(osition.)42 b(If)f(~)-58 b Fu(\032)33 b Fz(had)f(comm)m(uted)g (with)g Fu(@)2958 1830 y Ft(E)3019 1815 y Fu(\032)3069 1830 y Ft(E)3161 1815 y Fz(then)h(this)f(w)m(ould)150 1935 y(ha)m(v)m(e)41 b(b)s(een)g(automatic.)63 b(But)39 b(ev)m(en)j(if)c(they)j(do)f(not)f(comm)m(ute,)i(their)e(comm)m(utator) f(b)s(ecomes)150 2055 y(small)30 b(in)i(time)f(on)i(the)g(particular)e (cut-o\013s)h(whic)m(h)h(build)f(the)h(state)g Fu( )2838 2070 y Fs(3)2877 2055 y Fz(\()p Fu(t)p Fz(\).)296 2176 y(Cho)s(ose)h(a)e(function)g Fu(F)1162 2191 y Fs(+)p Fm(\000)1309 2176 y Fz(supp)s(orted)h(on)g Fn(R)1968 2191 y Fs(+)2066 2176 y Fz(whic)m(h)g(satis\014es)777 2338 y Fu(F)840 2353 y Fs(+)p Fm(\000)954 2338 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu(F)1238 2297 y Ft(m:v)r(:b:)1224 2363 y Fs(+)1429 2338 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu(F)1713 2297 y Ft(M)s(:v)r(:b)1699 2363 y Fm(\000)1894 2338 y Fz(\()p Fu(r)s(=t)p Fz(\))27 b(=)g Fu(F)2308 2297 y Ft(m:v)r(:b:)2294 2363 y Fs(+)2500 2338 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu(F)2784 2297 y Ft(M)s(:v)r(:b)2770 2363 y Fm(\000)2965 2338 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu(:)296 2501 y Fz(Since)45 b(w)m(e)g(ha)m(v)m(e)h(already)e(pro)m(v)m(en)h(\(6.42\),)i(the)d (corollary)f(w)m(ould)h(follo)m(w)f(if)g(w)m(e)i(can)g(pro)m(v)m(e)150 2621 y(\(remem)m(b)s(er)32 b(the)h(notation)e(\001)p Fu(\032)e Fz(=)36 b(~)-58 b Fu(\032)23 b Fr(\000)f Fu(\032)1686 2636 y Ft(E)1768 2621 y Fz(+)g Fu(@)1917 2636 y Ft(E)1977 2621 y Fu(\032)2027 2636 y Ft(E)2087 2621 y Fu(=C)7 b Fz(\))522 2697 y Fl(\015)522 2756 y(\015)522 2816 y(\015)577 2811 y Fu(F)654 2764 y Fm(\000)p Fs(9)q(~)-36 b Ft(\017)773 2773 y Fj(2)807 2764 y Ft(;)p Fm(\000)p Fs(8)q(~)g Ft(\017)946 2773 y Fj(2)640 2833 y Fm(\000)984 2811 y Fz(\()9 b(~)-58 b Fu(\032)23 b Fr(\000)f Fu(\032)1244 2826 y Ft(E)1304 2811 y Fz(\))p Fu(F)1419 2764 y Fm(\000)p Fs(3)q(~)-36 b Ft(\017)1538 2773 y Fj(2)1572 2764 y Ft(;)p Fm(\000)p Fs(2)q(~)g Ft(\017)1711 2773 y Fj(2)1405 2833 y Fs(+)1749 2811 y Fz(\(\001)p Fu(\032)p Fz(\))p Fu(F)2019 2826 y Fs(+)p Fm(\000)2134 2811 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu(f)2389 2826 y Fs(1)p Ft(;E)2503 2811 y Fz(\()p Fu(H)8 b Fz(\))2668 2697 y Fl(\015)2668 2756 y(\015)2668 2816 y(\015)2750 2811 y Fz(=)28 b Fr(O)s Fz(\()p Fu(t)3009 2770 y Fm(\0001)3139 2811 y Fz(\))p Fu(:)372 b Fz(\(6.47\))296 3018 y(Denote)33 b(b)m(y)g Fu(G)843 3033 y Fs(1)915 3018 y Fz(the)g(function)f Fu(F)1542 2971 y Fm(\000)p Fs(5)q(~)-36 b Ft(\017)1661 2980 y Fj(2)1695 2971 y Ft(;)p Fm(\000)p Fs(4)q(~)g Ft(\017)1834 2980 y Fj(2)1528 3040 y Fs(+)1905 3018 y Fz(and)33 b(observ)m(e)h(that)1443 3189 y Fu(G)1520 3204 y Fs(1)1559 3189 y Fu(F)1636 3142 y Fm(\000)p Fs(3)q(~)-36 b Ft(\017)1755 3151 y Fj(2)1789 3142 y Ft(;)p Fm(\000)p Fs(2)q(~)g Ft(\017)1928 3151 y Fj(2)1622 3211 y Fs(+)1994 3189 y Fz(=)28 b Fu(F)2175 3142 y Fm(\000)p Fs(3)q(~)-36 b Ft(\017)2294 3151 y Fj(2)2328 3142 y Ft(;)p Fm(\000)p Fs(2)q(~)g Ft(\017)2467 3151 y Fj(2)2161 3211 y Fs(+)2505 3189 y Fu(:)150 3352 y Fz(F)-8 b(or)32 b(ev)m(ery)i Fu(\036)28 b Fr(2)g Fu(L)828 3315 y Fs(2)867 3352 y Fz(\()p Fn(R)971 3315 y Fs(2)1017 3352 y Fz(\))k(de\014ne)741 3514 y(\011\()p Fu(t)p Fz(\))c(=)f Fu(F)1136 3467 y Fm(\000)p Fs(9)q(~)-36 b Ft(\017)1255 3476 y Fj(2)1289 3467 y Ft(;)p Fm(\000)p Fs(8)q(~)g Ft(\017)1428 3476 y Fj(2)1122 3536 y Fm(\000)1466 3514 y Fz(\()9 b(~)-58 b Fu(\032)23 b Fr(\000)f Fu(\032)1726 3529 y Ft(E)1786 3514 y Fz(\))p Fu(F)1901 3467 y Fm(\000)p Fs(3)q(~)-36 b Ft(\017)2020 3476 y Fj(2)2054 3467 y Ft(;)p Fm(\000)p Fs(2)q(~)g Ft(\017)2193 3476 y Fj(2)1887 3536 y Fs(+)2231 3514 y Fz(\(\001)p Fu(\032)p Fz(\))p Fu(F)2501 3529 y Fs(+)p Fm(\000)2616 3514 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu(f)2871 3529 y Fs(1)p Ft(;E)2985 3514 y Fz(\()p Fu(H)8 b Fz(\))p Fu(\036:)150 3677 y Fz(Then)34 b(w)m(e)f(ha)m(v)m(e) 410 3839 y(~)-50 b Fu(\017)448 3854 y Fs(2)488 3839 y Fr(k)p Fz(\011\()p Fu(t)p Fz(\))p Fr(k)775 3798 y Fs(2)897 3839 y Fr(\024)83 b(h\000)p Fz(7)q(~)-50 b Fu(\017)1261 3854 y Fs(2)1323 3839 y Fr(\000)23 b Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)1683 3854 y Ft(E)1743 3839 y Fz(\))p Fr(i)1820 3868 y Fs(\011\()p Ft(t)p Fs(\))3576 3839 y Fz(\(6.48\))898 3985 y(=)83 b Fr(h\000)p Fz(7)q(~)-50 b Fu(\017)1261 4000 y Fs(2)1323 3985 y Fr(\000)23 b Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)1683 4000 y Ft(E)1765 3985 y Fz(+)f Fu(@)1914 4000 y Ft(E)1974 3985 y Fu(\032)2024 4000 y Ft(E)2084 3985 y Fu(=C)7 b Fz(\))22 b(+)g Fu(@)2419 4000 y Ft(E)2479 3985 y Fu(\032)2529 4000 y Ft(E)2589 3985 y Fu(=C)7 b Fr(i)2753 4014 y Fs(\011\()p Ft(t)p Fs(\))898 4147 y Fz(=)83 b Fr(h\000)p Fz(7)q(~)-50 b Fu(\017)1261 4162 y Fs(2)1323 4147 y Fr(\000)23 b Fz(\(\001)p Fu(\032)p Fz(\))g Fr(\001)e Fu(G)1779 4162 y Fs(1)1819 4147 y Fz(\(\001)p Fu(\032)p Fz(\))p Fr(i)2065 4176 y Fs(\011\()p Ft(t)p Fs(\))2227 4147 y Fz(+)h Fr(h)o Fu(@)2414 4162 y Ft(E)2474 4147 y Fu(\032)2524 4162 y Ft(E)2584 4147 y Fu(=C)7 b Fr(i)2749 4176 y Fs(\011\()p Ft(t)p Fs(\))2910 4147 y Fz(+)22 b Fr(O)s Fz(\()p Fu(t)3163 4106 y Fm(\0001)3293 4147 y Fz(\))p Fr(jj)p Fu(\036)p Fr(jj)3501 4106 y Fs(2)3539 4147 y Fu(:)150 4320 y Fz(In)39 b(the)f(ab)s(o)m(v)m(e)i(second)f(line)e(w)m(e)i(put)g Fu(G)1655 4335 y Fs(1)1733 4320 y Fz(near)f(\001)p Fu(\032)h Fz(at)f(the)h(exp)s(ense)h(of)e(a)g(usual)g Fr(O)s Fz(\()p Fu(t)3413 4284 y Fm(\0001)3543 4320 y Fz(\))g(error)150 4440 y(coming)i(from)g(the)h(comm)m(utations.)68 b(Then)43 b(on)e(the)g(supp)s(ort)h(of)f Fu(G)2757 4455 y Fs(1)2837 4440 y Fz(w)m(e)i(ha)m(v)m(e)f Fr(\000)p Fz(7)q(~)-50 b Fu(\017)3388 4455 y Fs(2)3456 4440 y Fr(\000)29 b Fz(\(\001)p Fu(\032)p Fz(\))f Fr(\001)150 4561 y Fu(G)227 4576 y Fs(1)266 4561 y Fz(\(\001)p Fu(\032)p Fz(\))40 b Fr(\024)g Fz(0)f(in)g(the)h(form)e(sense.)66 b(Finally)-8 b(,)38 b(if)g Fu(C)47 b Fz(is)39 b(large)f(enough)i(w)m(e)g(get)g Fr(h)o Fu(@)3234 4576 y Ft(E)3294 4561 y Fu(\032)3344 4576 y Ft(E)3404 4561 y Fu(=C)7 b Fr(i)3569 4590 y Fs(\011\()p Ft(t)p Fs(\))3748 4561 y Fr(\024)150 4693 y Fz(\()q(~)-50 b Fu(\017)227 4708 y Fs(2)267 4693 y Fu(=)p Fz(2\))p Fr(jj)p Fz(\011\()p Fu(t)p Fz(\))p Fr(jj)702 4657 y Fs(2)772 4693 y Fz(and)33 b(w)m(e)g(are)g(done.)p 3754 4693 V 3758 4631 59 4 v 3758 4693 V 3816 4693 4 66 v 296 4932 a(Inspired)g(b)m(y)h(the)f(last)f(t)m(w)m(o)h(results,)g(w)m(e)h (de\014ne)470 5095 y Fu(F)547 5053 y Fs(\001)606 5062 y Fj(1)533 5119 y Fs(+)644 5095 y Fz(\(\001)p Fu(\032)p Fz(\))28 b(:=)g Fu(F)1087 5047 y Fm(\000)p Fs(3)q(~)-36 b Ft(\017)1206 5056 y Fj(2)1240 5047 y Ft(;)p Fm(\000)p Fs(2)q(~)g Ft(\017)1379 5056 y Fj(2)1073 5116 y Fs(+)1417 5095 y Fz(\(\001)p Fu(\032)p Fz(\))p Fu(;)114 b(F)1842 5053 y Fs(\001)1901 5062 y Fj(2)1828 5119 y Fs(+)1940 5095 y Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)2200 5110 y Ft(E)2260 5095 y Fz(\))k(:=)h Fu(F)2533 5047 y Fm(\000)p Fs(10)q(~)-36 b Ft(\017)2687 5056 y Fj(2)2721 5047 y Ft(;)p Fm(\000)p Fs(9)q(~)g Ft(\017)2860 5056 y Fj(2)2519 5116 y Fs(+)2898 5095 y Fz(\()9 b(~)-58 b Fu(\032)23 b Fr(\000)f Fu(\032)3158 5110 y Ft(E)3218 5095 y Fz(\))320 b(\(6.49\))150 5257 y(and)1237 5377 y Fu( )1300 5392 y Fs(4)1340 5377 y Fz(\()p Fu(t)p Fz(\))28 b(=)g Fu(F)1660 5336 y Fs(\001)1719 5345 y Fj(2)1646 5402 y Fs(+)1757 5377 y Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)2017 5392 y Ft(E)2077 5377 y Fz(\))p Fu(F)2192 5336 y Fs(\001)2251 5345 y Fj(1)2178 5402 y Fs(+)2289 5377 y Fz(\(\001)p Fu(\032)p Fz(\))p Fu( )2559 5392 y Fs(3)2599 5377 y Fz(\()p Fu(t)p Fz(\))p Fu(:)839 b Fz(\(6.50\))150 5527 y(Then)34 b(w)m(e)f(ha)m(v)m(e)h(sho)m(wn)g Fu( )1132 5542 y Fs(4)1172 5527 y Fz(\()p Fu(t)p Fz(\))28 b Fr(\030)g Fu( )1479 5542 y Fs(3)1519 5527 y Fz(\()p Fu(t)p Fz(\))g Fr(\030)g(\001)17 b(\001)g(\001)26 b(\030)i Fu( )t Fz(\()p Fu(t)p Fz(\).)1939 5776 y(61)p eop %%Page: 62 65 62 64 bop 150 418 a Fp(6.6)469 387 y FF(~)456 418 y Fi(\030)51 b Fp(is)45 b(not)g(lo)t(calized)h(on)f(the)g(negativ)l(e)i(axis)150 603 y Fz(W)-8 b(e)40 b(no)m(w)g(giv)m(e)g(the)g(quan)m(tum)g(equiv)-5 b(alen)m(t)39 b(of)g(Lemma)f(3.3.)65 b(Remem)m(b)s(er)39 b(that)g(one)h(imp)s(ortan)m(t)150 723 y(ingredien)m(t)30 b(of)g(the)i(pro)s(of)e(w)m(as)h(\(3.13\))f(whic)m(h)h(said)g(that)f (the)h(radial)e(v)m(elo)s(cit)m(y)i(sta)m(y)m(ed)h(a)m(w)m(a)m(y)g (from)150 844 y Fr(\000)227 759 y(p)p 311 759 127 4 v 311 844 a Fz(2)p Fu(E)5 b Fz(.)44 b(That)33 b(is)f(wh)m(y)i(w)m(e)f (start)g(with)f(a)g(preliminary)e(result:)150 1030 y Fv(Lemma)i(6.13.)38 b FA(Assume)30 b(that)i Fz(~)-50 b Fu(\017)1403 1045 y Fs(2)1474 1030 y FA(entering)30 b(the)g(de\014nition)g(of)g Fu( )2614 1045 y Fs(4)2654 1030 y Fz(\()p Fu(t)p Fz(\))g FA(is)g(very)h(smal)5 b(l.)42 b(Mor)-5 b(e)g(over,)150 1150 y(assume)34 b(that)h Fu(\032)740 1165 y Ft(E)823 1150 y Fz(+)921 1066 y Fr(p)p 1004 1066 V 84 x Fz(2)p Fu(E)e Fr(\025)28 b Fz(20)q(~)-50 b Fu(\017)1400 1165 y Fs(2)1440 1150 y FA(.)44 b(We)35 b(then)g(have)694 1257 y Fl(\015)694 1317 y(\015)694 1377 y(\015)749 1372 y Fu(F)826 1324 y Fs(3)q(~)-36 b Ft(\017)890 1333 y Fj(2)924 1324 y Ft(;)p Fs(4)q(~)g Ft(\017)1008 1333 y Fj(2)812 1393 y Fm(\000)1046 1372 y Fz(\()9 b(~)-58 b Fu(\032)23 b Fz(+)1255 1282 y Fr(p)p 1338 1282 V 90 x Fz(2)p Fu(E)5 b Fz(\))p Fu(F)1579 1330 y Fs(\001)1638 1339 y Fj(2)1565 1396 y Fs(+)1677 1372 y Fz(\()k(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)1937 1387 y Ft(E)1996 1372 y Fz(\))p Fu(F)2097 1387 y Fs(+)p Fm(\000)2212 1372 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu(f)2467 1387 y Fs(1)p Ft(;E)2581 1372 y Fz(\()p Fu(H)8 b Fz(\))2746 1257 y Fl(\015)2746 1317 y(\015)2746 1377 y(\015)2828 1372 y Fz(=)28 b Fr(O)s Fz(\()p Fu(t)3087 1330 y Fm(\0001)3216 1372 y Fz(\))p Fu(:)295 b Fz(\(6.51\))150 1599 y Fv(Pro)s(of.)43 b Fz(The)33 b(in)m(terpretation)d(of)h(this)h (lemma)d(is)i(easy)-8 b(.)44 b(Since)32 b Fu( )2586 1614 y Fs(4)2626 1599 y Fz(\()p Fu(t)p Fz(\))g(is)f(lo)s(calized)e(on)j(the) g(region)150 1719 y(where)37 b(~)-58 b Fu(\032)27 b Fz(is)f(essen)m (tially)h(larger)e(than)i Fu(\032)1599 1734 y Ft(E)1659 1719 y Fz(,)h(and)f(since)g(the)g(p)s(erio)s(dic)e(solution)g Fu(\032)3074 1734 y Ft(E)3161 1719 y Fz(is)h(strictly)g(larger)150 1839 y(than)33 b Fr(\000)455 1755 y(p)p 538 1755 V 84 x Fz(2)p Fu(E)6 b Fz(,)33 b(then)g(the)g(same)f(m)m(ust)h(b)s(e)g(true) g(for)41 b(~)-58 b Fu(\032)p Fz(.)296 1960 y(The)39 b(strategy)g(w)m(e) g(follo)m(w)e(is)g(similar)e(to)j(the)g(one)h(w)m(e)g(used)g(b)s (efore.)60 b(Fix)37 b Fu(\036)g Fr(2)h Fu(L)3403 1924 y Fs(2)3443 1960 y Fz(\()p Fn(R)3546 1924 y Fs(2)3592 1960 y Fz(\))g(and)150 2080 y(de\014ne)774 2201 y(\011\()p Fu(t)p Fz(\))28 b(:=)g Fu(F)1197 2153 y Fs(3)q(~)-36 b Ft(\017)1261 2162 y Fj(2)1295 2153 y Ft(;)p Fs(4)q(~)g Ft(\017)1379 2162 y Fj(2)1183 2222 y Fm(\000)1417 2201 y Fz(\()9 b(~)-58 b Fu(\032)22 b Fz(+)1625 2111 y Fr(p)p 1708 2111 V 90 x Fz(2)p Fu(E)6 b Fz(\))p Fu(F)1950 2160 y Fs(\001)2009 2169 y Fj(2)1936 2225 y Fs(+)2047 2201 y Fz(\()j(~)-58 b Fu(\032)23 b Fr(\000)f Fu(\032)2307 2216 y Ft(E)2367 2201 y Fz(\))p Fu(F)2468 2216 y Fs(+)p Fm(\000)2582 2201 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu(f)2837 2216 y Fs(1)p Ft(;E)2951 2201 y Fz(\()p Fu(H)8 b Fz(\))p Fu(\036:)150 2378 y Fz(De\014ne)33 b Fu(G)529 2393 y Fs(2)596 2378 y Fz(:=)28 b Fu(F)804 2330 y Fm(\000)p Fs(11)q(~)-36 b Ft(\017)958 2339 y Fj(2)992 2330 y Ft(;)p Fm(\000)p Fs(10)q(~)g Ft(\017)1166 2339 y Fj(2)790 2399 y Fs(+)1237 2378 y Fz(and)33 b(notice)f(that)1103 2618 y Fu(F)1180 2577 y Fs(\001)1239 2586 y Fj(2)1166 2643 y Fs(+)1277 2618 y Fz(\()9 b(~)-58 b Fu(\032)23 b Fr(\000)f Fu(\032)1537 2633 y Ft(E)1597 2618 y Fz(\))g Fr(\001)g Fu(G)1784 2633 y Fs(2)1824 2618 y Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)2084 2633 y Ft(E)2144 2618 y Fz(\))k(=)h Fu(F)2390 2577 y Fs(\001)2449 2586 y Fj(2)2376 2643 y Fs(+)2487 2618 y Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)2747 2633 y Ft(E)2807 2618 y Fz(\))p Fu(:)296 2784 y Fz(Then)34 b(w)m(e)g(ha)m(v)m(e)491 2984 y(~)-50 b Fu(\017)529 2999 y Fs(2)569 2984 y Fr(k)p Fz(\011\()p Fu(t)p Fz(\))p Fr(k)856 2943 y Fs(2)978 2984 y Fr(\024)1139 2873 y Fl(D)1199 2984 y Fz(5)q(~)g Fu(\017)1287 2999 y Fs(2)1349 2984 y Fr(\000)23 b Fz(\()9 b(~)-58 b Fu(\032)22 b Fz(+)1657 2895 y Fr(p)p 1740 2895 V 89 x Fz(2)p Fu(E)6 b Fz(\))1905 2873 y Fl(E)1966 3053 y Fs(\011\()p Ft(t)p Fs(\))3576 2984 y Fz(\(6.52\))979 3217 y(=)1139 3106 y Fl(D)1199 3217 y Fz(5)q(~)-50 b Fu(\017)1287 3232 y Fs(2)1349 3217 y Fr(\000)23 b Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)1709 3232 y Ft(E)1769 3217 y Fz(\))f Fr(\000)g Fz(\()1966 3128 y Fr(p)p 2049 3128 V 89 x Fz(2)p Fu(E)28 b Fz(+)22 b Fu(\032)2346 3232 y Ft(E)2406 3217 y Fz(\))2444 3106 y Fl(E)2505 3286 y Fs(\011\()p Ft(t)p Fs(\))978 3420 y Fr(\024)84 b(h)o Fz(5)q(~)-50 b Fu(\017)1265 3435 y Fs(2)1327 3420 y Fr(\000)23 b Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)1687 3435 y Ft(E)1747 3420 y Fz(\))f Fr(\001)f Fu(G)1933 3435 y Fs(2)1973 3420 y Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)2233 3435 y Ft(E)2293 3420 y Fz(\))f Fr(\000)g Fz(20)q(~)-50 b Fu(\017)2589 3435 y Fs(2)2629 3420 y Fr(i)2668 3449 y Fs(\011\()p Ft(t)p Fs(\))2829 3420 y Fz(+)22 b Fr(O)s Fz(\()p Fu(t)3082 3379 y Fm(\0001)3212 3420 y Fz(\))p Fr(jj)p Fu(\036)p Fr(jj)3420 3379 y Fs(2)3458 3420 y Fu(;)150 3645 y Fz(where)j(w)m(e)f(put)g(a)f Fu(G)876 3660 y Fs(2)938 3645 y Fz(near)33 b(~)-58 b Fu(\032)t Fr(\000)s Fu(\032)1330 3660 y Ft(E)1414 3645 y Fz(at)23 b(the)g(usual)g(exp)s(ense)j(and)d(replaced)h Fr(\000)p Fz(\()2954 3561 y Fr(p)p 3037 3561 V 84 x Fz(2)p Fu(E)9 b Fz(+)s Fu(\032)3296 3660 y Ft(E)3357 3645 y Fz(\))23 b(b)m(y)h Fr(\000)p Fz(20)q(~)-50 b Fu(\017)3758 3660 y Fs(2)3798 3645 y Fz(.)150 3766 y(It)33 b(follo)m(ws)e(that)h(the)h (\014rst)g(term)f(in)g(the)h(second)h(line)d(is)h(negativ)m(e)h(and)g (the)g(pro)s(of)f(follo)m(ws.)p 3754 3766 4 66 v 3758 3703 59 4 v 3758 3766 V 3816 3766 4 66 v 296 4004 a(De\014ne)h(a)g(new) g(\\ev)m(olv)m(ed")h(state)1335 4204 y Fu( )1398 4219 y Fs(5)1438 4204 y Fz(\()p Fu(t)p Fz(\))28 b(=)f Fu(F)1757 4157 y Fs(~)-35 b Ft(\017)1786 4166 y Fj(2)1820 4157 y Ft(;)p Fs(2)q(~)f Ft(\017)1904 4166 y Fj(2)1743 4226 y Fs(+)1942 4204 y Fz(\()9 b(~)-58 b Fu(\032)23 b Fz(+)2151 4115 y Fr(p)p 2234 4115 127 4 v 89 x Fz(2)p Fu(E)6 b Fz(\))p Fu( )2462 4219 y Fs(4)2501 4204 y Fz(\()p Fu(t)p Fz(\))p Fu(:)937 b Fz(\(6.53\))150 4404 y(Then)34 b(w)m(e)f(ha)m(v)m(e) h(sho)m(wn)g Fu( )1132 4419 y Fs(5)1172 4404 y Fz(\()p Fu(t)p Fz(\))28 b Fr(\030)g Fu( )1479 4419 y Fs(4)1519 4404 y Fz(\()p Fu(t)p Fz(\))g Fr(\030)g(\001)17 b(\001)g(\001)26 b(\030)i Fu( )t Fz(\()p Fu(t)p Fz(\).)296 4524 y(W)-8 b(e)33 b(\014nally)f(giv)m(e)g(the)h(equiv)-5 b(alen)m(t)33 b(of)f(Lemma)f(3.3)h(in)g(quan)m(tum)h(mec)m(hanics:)150 4710 y Fv(Lemma)k(6.14.)42 b FA(Ther)-5 b(e)34 b(exists)h Fu(d)1417 4725 y Fs(2)1484 4710 y Fu(>)27 b Fz(0)35 b FA(smal)5 b(l)34 b(enough)g(so)h(that)g(for)f Fu(F)2817 4659 y Fm(\000)p Ft(d)2908 4668 y Fj(2)2943 4659 y Ft(;)p Fm(\000)p Ft(d)3054 4668 y Fj(2)3088 4659 y Ft(=)p Fs(2)p Ft(;d)3214 4668 y Fj(2)3249 4659 y Ft(=)p Fs(2)p Ft(;d)3375 4668 y Fj(2)2803 4732 y Fs(++)1083 4829 y Fl(Z)1183 4855 y Fm(1)1139 5054 y Fs(1)1284 4897 y Fz(1)p 1284 4941 49 4 v 1291 5033 a Fu(t)1343 4964 y Fr(jj)1399 4878 y Fl(p)p 1498 4878 178 4 v 1498 4964 a Fu(F)1575 4930 y Fm(0)1561 4986 y Fs(++)1675 4964 y Fz(\()1723 4938 y(~)1713 4964 y Fu(\030)t Fz(\))p Fu( )1861 4979 y Fs(5)1901 4964 y Fz(\()p Fu(t)p Fz(\))p Fr(jj)2068 4923 y Fs(2)2107 4964 y Fu(dt)28 b Fr(\024)g Fz(Const)23 b Fr(\001)f(jj)p Fu( )t Fr(jj)2827 4923 y Fs(2)2865 4964 y Fu(:)684 b Fz(\(6.54\))150 5217 y FA(Mor)-5 b(e)g(over,)36 b(if)g Fz(0)31 b Fu(<)f(d)943 5232 y Fs(3)1013 5217 y Fu(<)g(d)1170 5232 y Fs(2)1209 5217 y Fu(=)p Fz(2)36 b FA(is)g(also)g(su\016ciently)g (smal)5 b(l)36 b(then)g(for)g(any)g Fu(F)3030 5232 y Fs(+)p Fm(\000)3181 5217 y FA(with)g(supp)-5 b(ort)36 b(in)150 5337 y Fz([)p Fr(\000)p Fu(d)305 5352 y Fs(3)345 5337 y Fu(;)17 b(d)440 5352 y Fs(3)479 5337 y Fz(])35 b FA(we)f(have)1551 5457 y Fz(lim)1535 5517 y Ft(t)p Fm(!1)1718 5457 y Fu(F)1781 5472 y Fs(+)p Fm(\000)1895 5457 y Fz(\()1943 5431 y(~)1933 5457 y Fu(\030)5 b Fz(\))p Fu( )2082 5472 y Fs(5)2122 5457 y Fz(\()p Fu(t)p Fz(\))27 b(=)h(0)p Fu(:)1136 b Fz(\(6.55\))1939 5776 y(62)p eop %%Page: 63 66 63 65 bop 150 407 a Fv(Pro)s(of.)70 b Fz(Using)41 b(all)e(the)j (cut-o\013s)f(en)m(tering)h(in)f Fu( )2052 422 y Fs(5)2091 407 y Fz(\()p Fu(t)p Fz(\))h(w)m(e)g(de\014ne)h(a)e(b)s(ounded)h(and)g (symmetric)150 527 y(propagation)27 b(observ)-5 b(able)29 b(\010\()p Fu(t)p Fz(\))h(in)e(suc)m(h)i(a)e(w)m(a)m(y)i(that)f(when)h (taking)e(the)h(exp)s(ectation)g(on)g(a)g(state)150 648 y(lik)m(e)j(in)g(\(6.1\))g(w)m(e)h(get)1441 768 y Fr(h)p Fz(\010\()p Fu(t)p Fz(\))p Fr(i)1700 783 y Ft( )r Fs(\()p Ft(t)p Fs(\))1861 768 y Fz(=)27 b Fr(h)p Fu(F)2066 783 y Fs(++)2180 768 y Fz(\()2228 742 y(~)2218 768 y Fu(\030)t Fz(\))p Fr(i)2342 783 y Ft( )2388 792 y Fj(5)2423 783 y Fs(\()p Ft(t)p Fs(\))2507 768 y Fu(:)150 942 y Fz(When)e(w)m(e)f (di\013eren)m(tiate)f(suc)m(h)i(an)f(exp)s(ectation)g(with)f(resp)s (ect)i(to)e Fu(t)h Fz(w)m(e)h(are)e(led)h(to)f(the)h(Heisen)m(b)s(erg) 150 1063 y(deriv)-5 b(ativ)m(e)35 b(of)h(eac)m(h)h(cut-o\013)e (function;)i(in)e(the)i(pro)s(cess)g(w)m(e)g(obtain)d(a)i(n)m(um)m(b)s (er)g(of)g(terms)f(whic)m(h)150 1183 y(can)e(b)s(e)g(regroup)s(ed)g(as) 765 1424 y Fu(@)816 1439 y Ft(t)847 1424 y Fr(h)p Fz(\010\()p Fu(t)p Fz(\))p Fr(i)1106 1439 y Ft( )r Fs(\()p Ft(t)p Fs(\))1266 1424 y Fz(=)28 b Fr(h)p Fu(D)1490 1439 y Ft(H)1557 1424 y Fz(\010\()p Fu(t)p Fz(\))p Fr(i)1777 1439 y Ft( )r Fs(\()p Ft(t)p Fs(\))1938 1424 y Fz(=)2041 1313 y Fl(D)2102 1424 y Fu(D)2183 1439 y Ft(H)2261 1398 y Fz(~)2250 1424 y Fu(\030)2298 1313 y Fl(E)2359 1432 y Fr(p)p 2442 1432 142 4 v 61 x Ft(F)2497 1470 y Ff(0)2487 1514 y Fj(++)2584 1493 y Fs(\()2618 1475 y(~)2611 1493 y Ft(\030)s Fs(\))p Ft( )2718 1502 y Fj(5)2753 1493 y Fs(\()p Ft(t)p Fs(\))2860 1424 y Fz(+)22 b Fu(R)3032 1439 y Fs(1)3071 1424 y Fz(\()p Fu(t)p Fz(\))p Fu(:)367 b Fz(\(6.56\))150 1664 y(As)49 b(usual,)j Fu(R)686 1679 y Fs(1)726 1664 y Fz(\()p Fu(t)p Fz(\))c(is)g(just)h(a)f(remainder)f(whic)m(h)i(can)f(b)s(e)h(sho)m(wn)h (\(based)f(on)f(the)h(previously)150 1785 y(obtained)32 b(propagation)f(estimates\))h(to)g(b)s(eha)m(v)m(e)j(lik)m(e)1360 1920 y Fl(Z)1459 1946 y Fm(1)1415 2145 y Fs(1)1551 2055 y Fr(j)p Fu(R)1653 2070 y Fs(1)1692 2055 y Fz(\()p Fu(t)p Fz(\))p Fr(j)p Fu(dt)27 b Fr(\024)h Fz(Const)23 b Fr(\001)f(jj)p Fu( )t Fr(jj)2550 2014 y Fs(2)2588 2055 y Fu(:)150 2328 y Fz(The)29 b(equation)f(\(6.56\))f(is)h(the)h(quan)m(tum)f(equiv)-5 b(alen)m(t)28 b(of)g(\(3.12\).)41 b(When)29 b(w)m(e)g(p)s(erform)e(the) i(Heisen-)150 2448 y(b)s(erg)41 b(deriv)-5 b(ativ)m(e)41 b(of)964 2422 y(~)953 2448 y Fu(\030)5 b Fz(,)43 b(w)m(e)f(obtain)e(t)m (w)m(o)i(leading)e(terms)h(\(i.e.)68 b(b)s(eha)m(ving)42 b(lik)m(e)e(1)p Fu(=t)h Fz(due)h(to)f(the)150 2569 y(v)-5 b(arious)42 b(cut-o\013s\).)75 b(Because)45 b(of)e(the)g(same)g (cut-o\013s)g(whic)m(h)h(build)e Fu( )2853 2584 y Fs(5)2893 2569 y Fz(\()p Fu(t)p Fz(\))h(w)m(e)h(can)f(essen)m(tially)150 2689 y(rep)s(eat)c(the)g(pro)s(of)f(of)h(Lemma)e(3.3;)42 b(the)d(non-comm)m(utativit)m(y)e(is)i(b)m(ypassed)i(b)m(y)f(putting)e (other)150 2810 y(cut-o\013s)32 b(with)f(larger)g(\(or)h(smaller\))d (supp)s(ort)k(near)f(the)g(relev)-5 b(an)m(t)32 b(op)s(erators.)43 b(Let)32 b(us)g(only)g(men-)150 2930 y(tion)f(that)h(here)h(is)f(the)h (place)f(where)i(Lemma)d(6.3)g(comes)i(in)m(to)e(pla)m(y)i(and)f (forces)42 b(~)-58 b Fu(\032)33 b Fz(to)f(sta)m(y)h(near)150 2966 y Fr(p)p 233 2966 127 4 v 84 x Fz(2)p Fu(E)38 b Fz(and)33 b(hence)h(to)e(b)s(e)h(p)s(ositiv)m(e.)43 b(F)-8 b(urther)32 b(details)g(are)h(omitted.)p 3754 3050 4 66 v 3758 2988 59 4 v 3758 3050 V 3816 3050 4 66 v 296 3289 a(The)d(next)h(step)f(is)e(pro)m(ving)h(a)g(quan)m(tum)h(equiv)-5 b(alen)m(t)29 b(of)f(Lemma)g(3.4.)42 b(One)30 b(can)f(see)i(that)e(the) 150 3409 y(classical)24 b(computations)g(w)m(e)i(did)f(there)g(can)h (easily)e(b)s(e)i(translated)e(in)m(to)h(the)g(quan)m(tum)h(language,) 150 3530 y(as)34 b(it)e(w)m(as)i(the)g(case)g(with)f(the)h(previous)g (lemma.)43 b(That)34 b(is)f(wh)m(y)h(w)m(e)h(only)e(form)m(ulate)e(the) j(result)150 3650 y(in)f(terms)g(of)g(adding)f(a)i(new)g(cut-o\013)f (on)g(our)g(state:)46 b(if)32 b Fu(d)2290 3665 y Fs(1)2358 3650 y Fu(>)d Fz(0)k(is)g(the)h(one)g(obtained)f(in)f(Lemma)150 3770 y(3.3)g(then)1476 4011 y Fu( )1539 4026 y Fs(6)1579 4011 y Fz(\()p Fu(t)p Fz(\))c(=)g Fu(F)1899 3960 y Ft(d)1935 3969 y Fj(1)1969 3960 y Ft(=)p Fs(2)p Ft(;d)2095 3969 y Fj(1)1885 4033 y Fs(+)2134 4011 y Fz(\()2182 3985 y(~)2172 4011 y Fu(\030)5 b Fz(\))p Fu( )2321 4026 y Fs(5)2360 4011 y Fz(\()p Fu(t)p Fz(\))p Fu(:)1078 b Fz(\(6.57\))150 4185 y(W)-8 b(e)33 b(ha)m(v)m(e)h(sho)m(wn)g Fu( )902 4200 y Fs(6)942 4185 y Fz(\()p Fu(t)p Fz(\))27 b Fr(\030)i Fu( )1249 4200 y Fs(5)1288 4185 y Fz(\()p Fu(t)p Fz(\))f Fr(\030)g(\001)17 b(\001)g(\001)26 b(\030)i Fu( )t Fz(\()p Fu(t)p Fz(\).)150 4474 y Fp(6.7)146 b FF(~)-70 b Fi(\032)46 b Fp(is)f(lo)t(calized)h(b)t(elo)l(w)f Fi(\032)1688 4492 y Fh(E)1786 4474 y FF(+)26 b Fi(\017)150 4659 y Fz(Remem)m(b)s(er)36 b(that)h Fu( )922 4674 y Fs(6)962 4659 y Fz(\()p Fu(t)p Fz(\))g(\(and)g(already)g Fu( )1754 4674 y Fs(4)1793 4659 y Fz(\()p Fu(t)p Fz(\))g(in)g(\(6.50\)\))f(con)m(tains)h(a)f(lo)s (calization)d(of)46 b(~)-58 b Fu(\032)37 b Fz(ab)s(o)m(v)m(e)150 4779 y Fu(\032)200 4794 y Ft(E)282 4779 y Fr(\000)23 b Fz(9)q(~)-50 b Fu(\017)470 4794 y Fs(2)509 4779 y Fz(.)44 b(W)-8 b(e)32 b(no)m(w)i(w)m(an)m(t)f(to)f(pro)m(v)m(e)i(the)f(analog)e (of)h(Prop)s(osition)f(3.5)h(in)g(quan)m(tum)h(mec)m(hanics,)150 4900 y(whic)m(h)g(w)m(ould)g(pro)m(vide)f(us)i(with)e(an)g(upp)s(er)h (b)s(ound)g(for)41 b(~)-58 b Fu(\032)33 b Fz(of)f(the)h(form)f Fu(\032)2915 4915 y Ft(E)2997 4900 y Fz(+)22 b Fu(\017)p Fz(.)296 5020 y(W)-8 b(e)30 b(\014rst)g(start)f(with)f(a)h(propagation) f(estimate)g(of)h(the)g(same)g(t)m(yp)s(e)i(as)e(the)g(one)h(in)e (\(3.21\);)i(w)m(e)150 5141 y(emplo)m(y)i(the)h(notations)f(in)m(tro)s (duced)g(in)g(Lemma)f(3.6.)150 5344 y Fv(Lemma)42 b(6.15.)i FA(L)-5 b(et)39 b Fu(\017)c Fz(:=)f(9)q(~)-50 b Fu(\017)1297 5359 y Fs(2)1376 5344 y FA(and)38 b(de\014ne)f Fu(\017)1895 5359 y Fs(0)1974 5344 y FA(as)h(in)h(L)-5 b(emma)37 b(3.6.)56 b(F)-7 b(or)38 b(every)g Fu(\017)3276 5359 y Fs(2)3351 5344 y Fu(>)c Fz(0)39 b FA(denote)150 5464 y(by)32 b Fu(F)337 5479 y Fs(+)p Fm(\000)479 5464 y Fz(=)27 b Fu(F)659 5417 y Fm(\000)p Fs(10)p Ft(\017)813 5426 y Fj(2)848 5417 y Ft(;)p Fm(\000)p Fs(9)p Ft(\017)987 5426 y Fj(2)1020 5417 y Ft(;)p Fs(9)p Ft(\017)1104 5426 y Fj(2)1138 5417 y Ft(;)p Fs(10)p Ft(\017)1257 5426 y Fj(2)645 5486 y Fs(+)p Fm(\000)1328 5464 y FA(and)k(by)h Fu(F)1701 5479 y Fs(++)1847 5464 y FA(pr)-5 b(e)g(cisely)31 b(that)h(function)g(as)f (in)h(\(3.4\))f(for)g(which)g(we)1939 5776 y Fz(63)p eop %%Page: 64 67 64 66 bop 150 407 a FA(have)40 b Fu(F)457 371 y Fm(0)443 431 y Fs(++)597 407 y Fz(=)f Fu(F)775 422 y Fs(+)p Fm(\000)889 407 y FA(.)63 b(F)-7 b(or)41 b(every)g Fu(E)1513 371 y Fm(0)1575 407 y Fr(2)f Fz([)p Fu(E)33 b Fz(+)26 b Fu(\017)1954 422 y Fs(0)1994 407 y Fu(;)17 b(M)10 b Fz(])42 b FA(de\014ne)e Fu(B)2575 422 y Ft(E)2631 403 y Ff(0)2657 407 y Fz(\()p Fu(t)p Fz(\))f(:=)49 b(~)-58 b Fu(\032)27 b Fr(\000)g Fu(\032)3181 422 y Ft(E)3237 403 y Ff(0)3263 407 y FA(.)64 b(Then)40 b(ther)-5 b(e)150 527 y(exists)35 b Fu(\017)456 542 y Fs(2)523 527 y Fu(>)28 b Fz(0)34 b FA(smal)5 b(l)34 b(enough)g(such)h(that)g(uniformly)g(in)g Fu(E)2364 491 y Fm(0)2422 527 y FA(we)f(have)921 657 y Fl(Z)1020 683 y Fm(1)976 882 y Fs(1)1122 725 y Fz(1)p 1122 769 49 4 v 1129 861 a Fu(t)1197 678 y Fl(\015)1197 738 y(\015)1197 797 y(\015)1252 706 y(p)p 1352 706 178 4 v 86 x Fu(F)1429 758 y Fm(0)1415 814 y Fs(++)1529 792 y Fz(\()p Fu(B)1641 807 y Ft(E)1697 788 y Ff(0)1723 792 y Fz(\()p Fu(t)p Fz(\)\))22 b Fr(\001)g Fu( )2007 807 y Fs(6)2047 792 y Fz(\()p Fu(t)p Fz(\))2158 678 y Fl(\015)2158 738 y(\015)2158 797 y(\015)2213 704 y Fs(2)2269 792 y Fu(dt)28 b Fr(\024)g Fz(Const)23 b Fr(\001)f(jj)p Fu( )t Fr(jj)2989 751 y Fs(2)3027 792 y Fu(:)522 b Fz(\(6.58\))150 1059 y FA(and)1221 1180 y Fz(lim)1206 1240 y Ft(t)p Fm(!1)1389 1180 y Fu(F)1466 1133 y Fm(\000)p Fs(8)p Ft(\017)1585 1142 y Fj(2)1619 1133 y Ft(;)p Fs(7)p Ft(\017)1703 1142 y Fj(2)1737 1133 y Ft(;)p Fs(7)p Ft(\017)1821 1142 y Fj(2)1855 1133 y Ft(;)p Fs(8)p Ft(\017)1939 1142 y Fj(2)1452 1202 y Fs(+)p Fm(\000)1977 1180 y Fz(\()p Fu(B)2089 1195 y Ft(E)2145 1176 y Ff(0)2171 1180 y Fz(\()p Fu(t)p Fz(\)\))28 b Fu( )2411 1195 y Fs(6)2451 1180 y Fz(\()p Fu(t)p Fz(\))g(=)f(0)p Fu(:)807 b Fz(\(6.59\))150 1416 y Fv(Pro)s(of.)64 b Fz(W)-8 b(e)40 b(only)f(pro)m(v)m(e)i(\(6.58\).)64 b(Remem)m(b)s(er)39 b(that)g Fu(\017)2277 1431 y Fs(0)2357 1416 y Fz(is)g(\014xed)i(and)e (prop)s(ortional)e(to)j(~)-49 b Fu(\017)3635 1431 y Fs(2)3714 1416 y Fz(w)m(e)150 1536 y(obtained)37 b(in)g(the)h(previous)g (subsection.)59 b(Using)37 b(all)e(the)j(cut-o\013s)g(en)m(tering)g Fu( )3153 1551 y Fs(6)3192 1536 y Fz(\()p Fu(t)p Fz(\))g(w)m(e)g (de\014ne)h(a)150 1656 y(b)s(ounded)d(and)g(symmetric)e(propagation)g (observ)-5 b(able)36 b(\010\()p Fu(t)p Fz(\))g(in)e(suc)m(h)j(a)e(w)m (a)m(y)i(that)e(when)i(taking)150 1777 y(the)c(exp)s(ectation)g(on)f(a) g(state)h(lik)m(e)f(in)g(\(6.1\))g(w)m(e)i(get)1273 1997 y Fr(h)p Fz(\010\()p Fu(t)p Fz(\))p Fr(i)1532 2012 y Ft( )r Fs(\()p Ft(t)p Fs(\))1693 1997 y Fz(=)27 b Fr(\000h)p Fu(F)1975 2012 y Fs(++)2090 1997 y Fz(\(\()p Fu(B)2240 2012 y Ft(E)2296 1993 y Ff(0)2322 1997 y Fz(\()p Fu(t)p Fz(\)\))p Fr(i)2510 2012 y Ft( )2556 2021 y Fj(6)2590 2012 y Fs(\()p Ft(t)p Fs(\))2675 1997 y Fu(:)150 2217 y Fz(Di\013eren)m(tiate)34 b(with)i(resp)s(ect)h(to)e Fu(t)h Fz(as)g(in)f(the)h(classical)e(case)j(\(see)g(\(3.23\)\))e(and)h (notice)f(that)h(the)150 2337 y(\\in)m(teresting")c(term)g(is)g(going)f (to)h(b)s(e)1235 2468 y Fl(D)1306 2553 y Fz(~)1296 2579 y Fu(\030)26 b Fr(\001)c Fz(\()p Fu(\030)1496 2594 y Ft(E)1552 2575 y Ff(0)1600 2579 y Fr(\000)1709 2553 y Fz(~)1699 2579 y Fu(\030)5 b Fz(\))1785 2468 y Fl(E)1846 2648 y Ft(r)1880 2628 y Ff(\000)p Fj(1)p Fq(=)p Fj(2)2024 2587 y Fr(p)p 2107 2587 142 4 v 61 x Ft(F)2162 2625 y Ff(0)2152 2669 y Fj(++)2249 2648 y Fs(\()p Ft(B)2329 2667 y Fq(E)2377 2653 y Ff(0)2404 2648 y Fs(\()p Ft(t)p Fs(\)\))p Fm(\001)p Ft( )2576 2657 y Fj(6)2612 2648 y Fs(\()p Ft(t)p Fs(\))2713 2579 y Fu(:)150 2865 y Fz(W)-8 b(e)33 b(then)h(p)s(erform)e(the)i(same)e(manipulations)e(as)k(w)m(e)g (did)e(in)h(order)g(to)f(get)h(\(3.24\);)g(b)s(efore)g(that)150 2986 y(w)m(e)i(ha)m(v)m(e)g(to)f(giv)m(e)g(a)g(prop)s(er)g(sense)h(to)f (the)g(in)m(v)m(erse)i(of)2228 2959 y(~)2218 2986 y Fu(\030)28 b Fz(+)23 b Fu(\030)2431 3001 y Ft(E)2487 2982 y Ff(0)2512 2986 y Fz(.)48 b(This)34 b(can)g(b)s(e)g(done)h(b)s(ecause)g Fu( )3785 3001 y Fs(6)150 3106 y Fz(con)m(tains)30 b(the)h(lo)s (calization)c(of)1331 3080 y(~)1320 3106 y Fu(\030)35 b Fz(on)c(the)f(p)s(ositiv)m(e)g(semi-axis)f(\(see)i(\(6.57\)\).)42 b(Indeed,)33 b(since)e Fu(\030)3648 3121 y Ft(E)3704 3102 y Ff(0)3759 3106 y Fz(is)150 3226 y(strictly)h(p)s(ositiv)m(e,)g (it)f(is)i(enough)g(to)f(use)1204 3374 y Fl(\020)1263 3485 y Fu(F)1340 3434 y Fs(0)p Ft(;d)1431 3443 y Fj(1)1466 3434 y Ft(=)p Fs(4)1326 3506 y(+)1540 3485 y Fz(\()1588 3458 y(~)1578 3485 y Fu(\030)5 b Fz(\))22 b Fr(\001)1746 3458 y Fz(~)1736 3485 y Fu(\030)k Fz(+)c Fu(\030)1946 3500 y Ft(E)2002 3481 y Ff(0)2028 3374 y Fl(\021)2087 3396 y Fm(\000)p Fs(1)2209 3485 y Fr(\025)28 b Fz(Const)h Fu(>)e Fz(0)p Fu(:)150 3727 y Fz(More)33 b(precisely)-8 b(,)33 b(w)m(e)g(consider)g(the)g(pro)s(duct)271 3968 y(\()p Fu(\030)352 3983 y Ft(E)408 3964 y Ff(0)455 3968 y Fr(\000)565 3942 y Fz(~)555 3968 y Fu(\030)t Fz(\))22 b Fr(\001)712 3858 y Fl(\020)772 3968 y Fu(F)849 3918 y Fs(0)p Ft(;d)940 3927 y Fj(1)974 3918 y Ft(=)p Fs(4)835 3990 y(+)1049 3968 y Fz(\()1097 3942 y(~)1087 3968 y Fu(\030)t Fz(\))g Fr(\001)1254 3942 y Fz(~)1244 3968 y Fu(\030)27 b Fz(+)22 b Fu(\030)1455 3983 y Ft(E)1511 3964 y Ff(0)1536 3858 y Fl(\021)1623 3968 y Fz(=)28 b(2\()p Fu(E)1892 3927 y Fm(0)1937 3968 y Fr(\000)23 b Fu(H)8 b Fz(\))21 b(+)h(\()9 b(~)-58 b Fu(\032)23 b Fz(+)f Fu(\032)2542 3983 y Ft(E)2598 3964 y Ff(0)2624 3968 y Fz(\))g Fr(\001)g Fu(B)2808 3983 y Ft(E)2864 3964 y Ff(0)2890 3968 y Fz(\()p Fu(t)p Fz(\))g(+)g(small)n(\()p Fu(t)p Fz(\))121 b(\(6.60\))150 4216 y(where)41 b(small)m(\()p Fu(t)p Fz(\))f(means)f(that)g(after)g (taking)g(the)g(exp)s(ectation)h(it)e(will)f(con)m(v)m(erge)k(to)e (zero.)64 b(Re-)150 4336 y(mem)m(b)s(er)33 b(that)h Fu(H)41 b Fz(is)34 b(lo)s(calized)d(in)i(a)h(v)m(ery)h(narro)m(w)g(in)m(terv)-5 b(al)32 b(around)i Fu(E)40 b Fz(\(of)33 b(width)i(~)-50 b Fu(\017)p Fz(,)35 b(see)g(\(6.1\)\))150 4457 y(Hence)g(reasoning)f (as)g(in)f(the)i(classical)d(case)j(w)m(e)g(ev)m(en)m(tually)g(obtain)e (the)h(desired)h(p)s(ositivit)m(y)d(b)m(y)150 4577 y(making)f Fu(\017)533 4592 y Fs(2)605 4577 y Fz(su\016cien)m(tly)j(small)c(and)j (w)m(e)g(can)g(in)m(tegrate)f(in)g(the)h(usual)f(w)m(a)m(y)-8 b(.)p 3754 4577 4 66 v 3758 4514 59 4 v 3758 4577 V 3816 4577 4 66 v 296 4815 a(W)g(e)33 b(no)m(w)h(giv)m(e)e(the)h(quan)m(tum)g (v)m(ersion)g(of)f(Prop)s(osition)f(3.5:)150 5019 y Fv(Prop)s(osition) 36 b(6.16.)41 b FA(L)-5 b(et)36 b Fu(\017)28 b Fz(=)f(9)q(~)-50 b Fu(\017)1454 5034 y Fs(2)1494 5019 y FA(.)45 b(Then)1359 5239 y Fz(lim)1343 5299 y Ft(t)p Fm(!1)1527 5239 y Fu(F)1604 5192 y Fs(10)p Ft(\017;)p Fs(11)p Ft(\017)1590 5261 y Fs(+)1825 5239 y Fz(\()9 b(~)-58 b Fu(\032)23 b Fr(\000)f Fu(\032)2085 5254 y Ft(E)2145 5239 y Fz(\))28 b Fu( )2274 5254 y Fs(6)2314 5239 y Fz(\()p Fu(t)p Fz(\))f(=)h(0)p Fu(:)944 b Fz(\(6.61\))1939 5776 y(64)p eop %%Page: 65 68 65 67 bop 150 407 a Fv(Pro)s(of.)45 b Fz(What)34 b(w)m(e)g(need)g (\014rst)g(is)f(to)g(restate)h(\(3.26\))e(and)i(\(3.27\))e(in)h(terms)g (of)g(op)s(erators.)45 b(Lo)s(ok)150 527 y(\014rst)h(at)g(the)h(righ)m (t)e(hand)h(side)g(of)g(\(3.26\).)83 b(W)-8 b(e)46 b(w)m(an)m(t)h(to)e (get)h(rid)g(of)f(the)i Fu(\022)i Fz(dep)s(endence)f(b)m(y)150 648 y(in)m(tro)s(ducing)39 b(a)i(partition)d(of)i(unit)m(y)h(in)f(the)h (angular)e(v)-5 b(ariable.)66 b(So)40 b(w)m(e)i(can)f(write)f(that)g (there)150 768 y(exists)33 b Fu(J)42 b Fz(large)31 b(enough)i(suc)m(h)h (that)318 1068 y Fu(F)395 1021 y Fm(\000)p Fs(2)p Ft(\017)514 1030 y Fj(2)548 1021 y Ft(;)p Fm(\000)p Ft(\017)652 1030 y Fj(2)686 1021 y Ft(;\017)735 1030 y Fj(2)768 1021 y Ft(;)p Fs(2)p Ft(\017)852 1030 y Fj(2)381 1090 y Fs(+)p Fm(\000)890 1068 y Fz(\()p Fu(x)23 b Fr(\000)g Fu(\032)1156 1084 y Ft(E)1212 1061 y Ff(0)1208 1108 y Fq(k)1250 1068 y Fz(\()p Fu(\022)s Fz(\)\))28 b Fr(\024)1594 944 y Ft(J)1545 973 y Fl(X)1555 1183 y Ft(j)t Fs(=1)1705 1068 y Fu(F)1782 1021 y Fm(\000)p Fs(3)p Ft(\017)1901 1030 y Fj(2)1935 1021 y Ft(;)p Fm(\000)p Fs(2)p Ft(\017)2074 1030 y Fj(2)2108 1021 y Ft(;)p Fs(2)p Ft(\017)2192 1030 y Fj(2)2226 1021 y Ft(;)p Fs(3)p Ft(\017)2310 1030 y Fj(2)1768 1090 y Fs(+)p Fm(\000)2348 1068 y Fz(\()p Fu(x)23 b Fr(\000)f Fu(\032)2613 1084 y Ft(E)2669 1061 y Ff(0)2665 1108 y Fq(k)2708 1068 y Fz(\(2)p Fu(\031)t(j)6 b(=J)j Fz(\)\))21 b Fr(\001)h Fu(\037)3220 1083 y Ft(j)3257 1068 y Fz(\()p Fu(\022)s Fz(\))p Fu(;)168 b Fz(\(6.62\))150 1401 y(where)35 b Fu(\037)494 1416 y Ft(j)564 1401 y Fz(are)e(functions)h(of)f(the)h Fu(F)1493 1416 y Fs(+)p Fm(\000)1640 1401 y Fz(t)m(yp)s(e,)1888 1326 y Fl(P)1993 1353 y Ft(J)1993 1430 y(j)t Fs(=1)2136 1401 y Fu(\037)2197 1416 y Ft(j)2234 1401 y Fz(\()p Fu(\022)s Fz(\))29 b(=)g(1)34 b(and)f(the)h(supp)s(ort)g(of)f(eac)m(h)h Fu(\037)3689 1416 y Ft(j)3759 1401 y Fz(is)150 1521 y(sharply)e(lo)s (calized)d(around)j(2)p Fu(\031)t(j)6 b(=J)j Fz(.)43 b(Notice)31 b(that)h(b)m(y)h(\014xing)e(the)h(angle)f(to)h Fu(\022)3060 1536 y Ft(j)3124 1521 y Fz(=)c(2)p Fu(\031)t(j)6 b(=J)40 b Fz(w)m(e)33 b(had)150 1642 y(to)27 b(enlarge)g(the)h(region)f (where)i Fu(F)1386 1657 y Fs(+)p Fm(\000)1527 1642 y Fz(equals)f(1.)42 b(Then)28 b(w)m(e)h(can)f(rewrite)f(again)g(\(3.26\)) f(in)h(the)h(form)200 1947 y Fu(F)277 1896 y Ft(\017=)p Fs(2)p Ft(;\017)263 1969 y Fs(+)428 1947 y Fz(\()p Fu(x)22 b Fr(\000)g Fu(\032)692 1962 y Ft(E)752 1947 y Fz(\()p Fu(\022)s Fz(\)\))27 b Fr(\024)1087 1823 y Ft(N)1046 1853 y Fl(X)1054 2065 y Ft(k)r Fs(=1)1257 1823 y Ft(J)1207 1853 y Fl(X)1218 2063 y Ft(j)t Fs(=1)1367 1885 y Fr(p)p 1451 1885 98 4 v 1451 1947 a Fu(\037)1512 1962 y Ft(j)1548 1947 y Fz(\()p Fu(\022)s Fz(\))22 b Fr(\001)f Fu(F)1820 1900 y Fm(\000)p Fs(3)p Ft(\017)1939 1909 y Fj(2)1973 1900 y Ft(;)p Fm(\000)p Fs(2)p Ft(\017)2112 1909 y Fj(2)2146 1900 y Ft(;)p Fs(2)p Ft(\017)2230 1909 y Fj(2)2264 1900 y Ft(;)p Fs(3)p Ft(\017)2348 1909 y Fj(2)1806 1969 y Fs(+)p Fm(\000)2386 1947 y Fz(\()p Fu(x)h Fr(\000)g Fu(\032)2650 1964 y Ft(E)2706 1941 y Ff(0)2702 1987 y Fq(k)2744 1947 y Fz(\(2)p Fu(\031)t(j)6 b(=J)j Fz(\)\))21 b Fr(\001)3194 1885 y(p)p 3277 1885 V 62 x Fu(\037)3338 1962 y Ft(j)3375 1947 y Fz(\()p Fu(\022)s Fz(\))p Fu(:)50 b Fz(\(6.63\))296 2256 y(Let)33 b(us)g(no)m(w)h(replace)e Fu(x)h Fz(b)m(y)42 b(~)-57 b Fu(\032)32 b Fz(and)h(see)h(what)f(happ)s(ens:)150 2453 y Fv(Lemma)44 b(6.17.)h FA(De\014ne)39 b Fz(\011\()p Fu(t)p Fz(\))f(:=)g Fu(F)1585 2406 y Fs(10)p Ft(\017;)p Fs(11)p Ft(\017)1571 2475 y Fs(+)1806 2453 y Fz(\()9 b(~)-58 b Fu(\032)27 b Fr(\000)g Fu(\032)2075 2468 y Ft(E)2134 2453 y Fz(\))38 b Fu( )2273 2468 y Fs(6)2313 2453 y Fz(\()p Fu(t)p Fz(\))p FA(.)61 b(Denote)40 b(by)g Fu(\022)3033 2468 y Ft(j)3108 2453 y Fz(:=)e(2)p Fu(\031)t(j)6 b(=J)j FA(.)61 b(Then)150 2573 y(for)43 b Fu(J)52 b FA(lar)-5 b(ge)43 b(enough)g(\(or)f(e)-5 b(quivalently,)45 b(for)e(a)g (su\016ciently)g(\014ne)g(angular)g(p)-5 b(artition\))43 b(and)f(for)150 2694 y Fu(t)28 b(>)f(T)14 b Fz(\()p Fu(J)9 b Fz(\))35 b FA(we)g(have)348 2875 y Ft(N)308 2905 y Fl(X)315 3117 y Ft(k)r Fs(=1)518 2875 y Ft(J)468 2905 y Fl(X)479 3115 y Ft(j)t Fs(=1)629 2919 y Fl(\012)676 2999 y Fu(F)753 2952 y Fm(\000)p Fs(3)p Ft(\017)872 2961 y Fj(2)906 2952 y Ft(;)p Fm(\000)p Fs(2)p Ft(\017)1045 2961 y Fj(2)1079 2952 y Ft(;)p Fs(2)p Ft(\017)1163 2961 y Fj(2)1197 2952 y Ft(;)p Fs(3)p Ft(\017)1281 2961 y Fj(2)739 3021 y Fs(+)p Fm(\000)1319 2999 y Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)1579 3016 y Ft(E)1635 2993 y Ff(0)1631 3039 y Fq(k)1673 2999 y Fz(\()p Fu(\022)1756 3014 y Ft(j)1793 2999 y Fz(\)\))1869 2919 y Fl(\013)1916 3005 y Fm(p)p 1975 3005 77 3 v 38 x Ft(\037)2019 3053 y Fq(j)2051 3043 y Fs(\011\()p Ft(t)p Fs(\))2218 2999 y Fr(\025)2333 2932 y Fz(1)p 2333 2977 49 4 v 2333 3068 a(2)2392 2999 y Fr(jj)p Fz(\011\()p Fu(t)p Fz(\))p Fr(jj)2691 2958 y Fs(2)2751 2999 y Fr(\000)g(O)s Fz(\()p Fu(t)3006 2958 y Fm(\0001)3136 2999 y Fz(\))p Fr(jj)p Fu( )t Fr(jj)3353 2958 y Fs(2)3391 2999 y Fu(:)158 b Fz(\(6.64\))150 3329 y Fv(Pro)s(of.)47 b Fz(First,)33 b(w)m(e)i(apply)e(\(6.63\))g(but)h (with)g Fu(\022)f Fz(=)c Fu(\022)2095 3344 y Ft(j)2166 3329 y Fz(and)34 b Fu(F)2434 3281 y Fs(6)p Ft(\017;)p Fs(7)p Ft(\017)2420 3350 y Fs(+)2585 3329 y Fz(\()9 b(~)-58 b Fu(\032)23 b Fr(\000)g Fu(\032)2846 3344 y Ft(E)2906 3329 y Fz(\()p Fu(\022)2989 3344 y Ft(j)3026 3329 y Fz(\)\))34 b(on)g(the)g(left)f(hand)150 3449 y(side)f(\(w)m(e)g(can)g(do)g(this)f (b)s(ecause)i(ev)m(erything)g(comm)m(utes)e(since)h(w)m(e)h(\014xed)f (the)h(angle\).)42 b(Then)33 b(the)150 3569 y(left)f(hand)h(side)f(of)g (\(6.64\))g(will)e(b)s(e)j(larger)f(than)1055 3750 y Ft(J)1005 3780 y Fl(X)1016 3990 y Ft(j)t Fs(=1)1166 3794 y Fl(\012)1213 3875 y Fz(\011\()p Fu(t)p Fz(\))p Fu(;)1444 3812 y Fr(p)p 1526 3812 98 4 v 1526 3875 a Fu(\037)1587 3890 y Ft(j)1646 3875 y Fr(\001)22 b Fu(F)1773 3828 y Fs(6)p Ft(\017;)p Fs(7)p Ft(\017)1759 3897 y Fs(+)1924 3875 y Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)2184 3890 y Ft(E)2244 3875 y Fz(\()p Fu(\022)2327 3890 y Ft(j)2364 3875 y Fz(\)\))f Fr(\001)2512 3812 y(p)p 2595 3812 V 63 x Fu(\037)2656 3890 y Ft(j)2692 3875 y Fz(\011\()p Fu(t)p Fz(\))2879 3794 y Fl(\013)2943 3875 y Fu(:)296 4204 y Fz(Remem)m(b)s(er)32 b(that)f(\011\()p Fu(t)p Fz(\))h(liv)m(es)f(in)g(the)h(range)g(of)f Fu(F)2162 4157 y Fs(10)p Ft(\017;)p Fs(11)p Ft(\017)2148 4226 y Fs(+)2383 4204 y Fz(\()9 b(~)-58 b Fu(\032)21 b Fr(\000)g Fu(\032)2640 4219 y Ft(E)2700 4204 y Fz(\))31 b(hence)i(w)m(e)g(can)f (rewrite)g(the)150 4324 y(ab)s(o)m(v)m(e)h(sum)g(as)846 4481 y Ft(J)797 4511 y Fl(X)807 4720 y Ft(j)t Fs(=1)957 4524 y Fl(\012)1004 4605 y Fz(\011\()p Fu(t)p Fz(\))p Fu(;)1235 4542 y Fr(p)p 1318 4542 V 63 x Fu(\037)1379 4620 y Ft(j)1438 4605 y Fr(\001)22 b Fu(F)1565 4558 y Fs(8)p Ft(\017;)p Fs(9)p Ft(\017)1551 4627 y Fs(+)1716 4605 y Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)1976 4620 y Ft(E)2036 4605 y Fz(\))f Fr(\001)f Fu(F)2222 4558 y Fs(6)p Ft(\017;)p Fs(7)p Ft(\017)2208 4627 y Fs(+)2374 4605 y Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)2634 4620 y Ft(E)2693 4605 y Fz(\()p Fu(\022)2776 4620 y Ft(j)2813 4605 y Fz(\)\))f Fr(\001)2961 4542 y(p)p 3044 4542 V 63 x Fu(\037)3105 4620 y Ft(j)3142 4605 y Fz(\011\()p Fu(t)p Fz(\))3357 4524 y Fl(\013)638 4861 y Fz(+)83 b Fr(O)s Fz(\()p Fu(t)952 4819 y Fm(\0001)1082 4861 y Fz(\))p Fr(jj)p Fu( )t Fr(jj)1299 4819 y Fs(2)1337 4861 y Fu(;)2212 b Fz(\(6.65\))150 5073 y(since)39 b(the)h(error)e(in)m(tro)s(duced)h(b) m(y)h(the)f(comm)m(utation)e(with)h Fu(\037)2513 5088 y Ft(j)2589 5073 y Fz(and)h(the)g(other)g Fu(F)3283 5088 y Fs(+)3381 5073 y Fz(is)f(of)h(order)150 5194 y Fu(t)185 5158 y Fm(\0001)346 5194 y Fz(due)33 b(to)e(the)h(cut-o\013s)g(whic)m (h)g(build)e(\011.)43 b(The)33 b(next)f(step)h(is)e(pro)m(ving)g(that)h (w)m(e)g(can)g(get)g(rid)e(of)150 5314 y Fu(F)227 5267 y Fs(6)p Ft(\017;)p Fs(7)p Ft(\017)213 5336 y Fs(+)378 5314 y Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)638 5329 y Ft(E)698 5314 y Fz(\()p Fu(\022)781 5329 y Ft(j)818 5314 y Fz(\)\))32 b(if)g(the)h(angular)e(partition)f(is)j(\014ne)g (enough.)44 b(In)33 b(fact,)f(w)m(e)i(sho)m(w)f(that)682 5442 y Fl(\015)682 5502 y(\015)737 5527 y Fu(F)814 5480 y Fs(8)p Ft(\017;)p Fs(9)p Ft(\017)800 5549 y Fs(+)966 5527 y Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)1226 5542 y Ft(E)1285 5527 y Fz(\))g Fr(\001)e Fu(F)1472 5480 y Fs(6)p Ft(\017;)p Fs(7)p Ft(\017)1458 5549 y Fm(\000)1623 5527 y Fz(\()9 b(~)-58 b Fu(\032)23 b Fr(\000)f Fu(\032)1883 5542 y Ft(E)1943 5527 y Fz(\()p Fu(\022)2026 5542 y Ft(j)2063 5527 y Fz(\)\))g Fr(\001)2211 5464 y(p)p 2294 5464 V 63 x Fu(\037)2355 5542 y Ft(j)2392 5527 y Fz(\011\()p Fu(t)p Fz(\))2579 5442 y Fl(\015)2579 5502 y(\015)2662 5527 y Fz(=)27 b Fr(O)s Fz(\()p Fu(t)2920 5486 y Fm(\0001)3050 5527 y Fz(\))p Fr(jj)p Fu( )t Fr(jj)p Fu(:)282 b Fz(\(6.66\))1939 5776 y(65)p eop %%Page: 66 69 66 68 bop 150 407 a Fz(Indeed,)34 b(denote)f(b)m(y)861 609 y(\011)937 624 y Fs(1)976 609 y Fz(\()p Fu(t)p Fz(\))28 b(:=)g Fu(F)1323 562 y Fs(8)p Ft(\017;)p Fs(9)p Ft(\017)1309 631 y Fs(+)1474 609 y Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)1734 624 y Ft(E)1794 609 y Fz(\))f Fr(\001)g Fu(F)1981 562 y Fs(6)p Ft(\017;)p Fs(7)p Ft(\017)1967 631 y Fm(\000)2132 609 y Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)2392 624 y Ft(E)2452 609 y Fz(\()p Fu(\022)2535 624 y Ft(j)2571 609 y Fz(\)\))f Fr(\001)2719 546 y(p)p 2802 546 98 4 v 63 x Fu(\037)2863 624 y Ft(j)2900 609 y Fz(\011\()p Fu(t)p Fz(\))p Fu(:)150 811 y Fz(W)-8 b(e)33 b(then)g(ha)m(v)m(e)574 1013 y(8)p Fu(\017)p Fr(jj)p Fz(\011)794 1028 y Fs(1)833 1013 y Fz(\()p Fu(t)p Fz(\))p Fr(jj)1000 972 y Fs(2)1122 1013 y Fr(\024)83 b(h)9 b Fz(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)1543 1028 y Ft(E)1603 1013 y Fr(i)1642 1042 y Fs(\011)1697 1051 y Fj(1)1731 1042 y Fs(\()p Ft(t)p Fs(\))1123 1176 y Fz(=)83 b Fr(h)9 b Fz(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)1543 1191 y Ft(E)1603 1176 y Fz(\()p Fu(\022)1686 1191 y Ft(j)1723 1176 y Fz(\))p Fr(i)1800 1208 y Fs(\011)1855 1217 y Fj(1)1889 1208 y Fs(\()p Ft(t)p Fs(\))1996 1176 y Fz(+)f Fu(o)p Fz(\(1\))f Fr(\001)h(jj)p Fz(\011)2469 1191 y Fs(1)2508 1176 y Fz(\()p Fu(t)p Fz(\))p Fr(jj)2675 1134 y Fs(2)2736 1176 y Fz(+)g Fr(O)s Fz(\()p Fu(t)2989 1134 y Fm(\0001)3119 1176 y Fz(\))p Fr(jj)p Fu( )t Fr(jj)3336 1134 y Fs(2)3374 1176 y Fu(;)175 b Fz(\(6.67\))150 1389 y(where)34 b(w)m(e)g(used)g (that)e Fu(\032)1060 1404 y Ft(E)1120 1389 y Fz(\()p Fu(\022)s Fz(\))22 b Fr(\000)h Fu(\032)1416 1404 y Ft(E)1476 1389 y Fz(\()p Fu(\022)1559 1404 y Ft(j)1596 1389 y Fz(\))32 b(can)h(b)s(e)g(made)g(as)f(small)f(as)i(w)m(e)g(w)m(an)m(t)h(if)e Fu(\037)3267 1404 y Ft(j)3336 1389 y Fz(has)h(a)f(sharp)150 1509 y(supp)s(ort;)37 b(simply)d(put)48 b(~)-61 b Fu(\037)1100 1524 y Ft(j)1137 1509 y Fz(\()p Fu(\022)s Fz(\))35 b(near)h(it)e(where) 48 b(~)-60 b Fu(\037)1962 1524 y Ft(j)2034 1509 y Fz(has)36 b(a)f(sligh)m(tly)e(larger)i(supp)s(ort)g(than)h Fu(\037)3574 1524 y Ft(j)3611 1509 y Fz(.)52 b(W)-8 b(e)150 1646 y(then)36 b(put)f(near)44 b(~)-58 b Fu(\032)24 b Fr(\000)g Fu(\032)1000 1661 y Ft(E)1060 1646 y Fz(\()p Fu(\022)1143 1661 y Ft(j)1180 1646 y Fz(\))35 b(a)f(factor)h(of)f Fu(F)1808 1595 y Fs(29)p Ft(\017=)p Fs(4)p Ft(;)p Fs(31)p Ft(\017=)p Fs(4)1794 1668 y Fm(\000)2171 1646 y Fz(\()9 b(~)-58 b Fu(\032)24 b Fr(\000)g Fu(\032)2434 1661 y Ft(E)2494 1646 y Fz(\()p Fu(\022)2577 1661 y Ft(j)2614 1646 y Fz(\)\);)36 b(the)g(price)e(w)m(e) i(pa)m(y)g(is)e(again)150 1766 y(of)e(order)h Fr(O)s Fz(\()p Fu(t)671 1730 y Fm(\0001)801 1766 y Fz(\))p Fr(jj)p Fu( )t Fr(jj)1018 1730 y Fs(2)1056 1766 y Fz(.)43 b(Hence)716 2089 y(8)p Fu(\017)p Fr(jj)p Fz(\011)936 2104 y Fs(1)975 2089 y Fz(\()p Fu(t)p Fz(\))p Fr(jj)1142 2048 y Fs(2)1264 2089 y Fr(\024)1425 1978 y Fl(D)1486 2089 y Fz([)9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)1735 2104 y Ft(E)1795 2089 y Fz(\()p Fu(\022)1878 2104 y Ft(j)1915 2089 y Fz(\)])f Fr(\001)g Fu(F)2129 2038 y Fs(29)p Ft(\017=)p Fs(4)p Ft(;)p Fs(31)p Ft(\017=)p Fs(4)2115 2111 y Fm(\000)2491 2089 y Fz(\()9 b(~)-58 b Fu(\032)23 b Fr(\000)f Fu(\032)2751 2104 y Ft(E)2811 2089 y Fz(\()p Fu(\022)2894 2104 y Ft(j)2931 2089 y Fz(\)\))3007 1978 y Fl(E)3068 2158 y Fs(\011)3123 2167 y Fj(1)3157 2158 y Fs(\()p Ft(t)p Fs(\))1265 2292 y Fz(+)84 b Fu(o)p Fz(\(1\))21 b Fr(\001)h(jj)p Fz(\011)1800 2307 y Fs(1)1839 2292 y Fz(\()p Fu(t)p Fz(\))p Fr(jj)2006 2250 y Fs(2)2067 2292 y Fz(+)g Fr(O)s Fz(\()p Fu(t)2320 2250 y Fm(\0001)2450 2292 y Fz(\))p Fr(jj)p Fz(\011\()p Fu(t)p Fz(\))p Fr(jj)2787 2250 y Fs(2)1264 2437 y Fr(\024)84 b Fz(\(31)p Fu(\017=)p Fz(4)21 b(+)h Fu(o)p Fz(\(1\)\))g Fr(\001)g(jj)p Fz(\011)2231 2452 y Fs(1)2269 2437 y Fz(\()p Fu(t)p Fz(\))p Fr(jj)2436 2396 y Fs(2)2497 2437 y Fz(+)g Fr(O)s Fz(\()p Fu(t)2750 2396 y Fm(\0001)2880 2437 y Fz(\))p Fr(jj)p Fu( )t Fr(jj)3097 2396 y Fs(2)3576 2437 y Fz(\(6.68\))150 2639 y(whic)m(h)38 b(ends)g(the)g(pro)s(of)e(of)g (\(6.66\).)57 b(No)m(w)38 b(go)e(bac)m(k)i(to)f(\(6.65\))g(and)g (replace)g Fu(F)3152 2592 y Fs(6)p Ft(\017;)p Fs(7)p Ft(\017)3138 2661 y Fs(+)3303 2639 y Fz(\()9 b(~)-58 b Fu(\032)26 b Fr(\000)f Fu(\032)3569 2654 y Ft(E)3629 2639 y Fz(\()p Fu(\022)3712 2654 y Ft(j)3749 2639 y Fz(\)\))150 2759 y(b)m(y)35 b(1.)46 b(Finally)-8 b(,)31 b(replace)j Fu(F)1170 2712 y Fs(8)p Ft(\017;)p Fs(9)p Ft(\017)1156 2781 y Fs(+)1321 2759 y Fz(\()9 b(~)-58 b Fu(\032)23 b Fr(\000)h Fu(\032)1583 2774 y Ft(E)1643 2759 y Fz(\))33 b(b)m(y)i(1)e(b)s(ecause)i(\011)f(is)f(in)g(the)h(range)g(of)f Fu(F)3244 2712 y Fs(10)p Ft(\017;)p Fs(11)p Ft(\017)3230 2781 y Fs(+)3466 2759 y Fz(\()9 b(~)-58 b Fu(\032)23 b Fr(\000)g Fu(\032)3727 2774 y Ft(E)3787 2759 y Fz(\))150 2880 y(and)33 b(the)g(lemma)d(is)i(pro)m(v)m(en.)p 3754 2880 4 66 v 3758 2817 59 4 v 3758 2880 V 3816 2880 4 66 v 296 3118 a(W)-8 b(e)28 b(no)m(w)f(con)m(tin)m(ue)h(the)g(pro)s(of) e(of)g(\(6.61\).)41 b(Consider)28 b(separately)f(eac)m(h)h(term)f(in)f (the)h(left)g(hand)150 3238 y(side)33 b(of)f(\(6.64\).)42 b(Denote)33 b(b)m(y)h(\(at)e Fu(k)j Fz(and)e Fu(j)39 b Fz(\014xed\))951 3487 y(\011)1027 3502 y Fs(2)1067 3487 y Fz(\()p Fu(t)p Fz(\))27 b(:=)1336 3361 y Fl(q)p 1436 3361 644 4 v 126 x Fu(F)1513 3440 y Fm(\000)p Fs(3)p Ft(\017)1632 3449 y Fj(2)1666 3440 y Ft(;)p Fm(\000)p Fs(2)p Ft(\017)1805 3449 y Fj(2)1839 3440 y Ft(;)p Fs(2)p Ft(\017)1923 3449 y Fj(2)1957 3440 y Ft(;)p Fs(3)p Ft(\017)2041 3449 y Fj(2)1499 3509 y Fs(+)p Fm(\000)2079 3487 y Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)2339 3503 y Ft(E)2395 3480 y Ff(0)2391 3527 y Fq(k)2433 3487 y Fz(\()p Fu(\022)2516 3502 y Ft(j)2553 3487 y Fz(\)\))2629 3424 y Fr(p)p 2712 3424 98 4 v 63 x Fu(\037)2773 3502 y Ft(j)2810 3487 y Fz(\011\()p Fu(t)p Fz(\))p Fu(:)296 3689 y Fz(Notice)32 b(that)h(if)e(w)m(e)j(pro)m(v)m(e)g(that)751 3891 y Fr(jj)p Fz(\011)883 3906 y Fs(2)922 3891 y Fz(\()p Fu(t)p Fz(\))p Fr(jj)1089 3850 y Fs(2)1155 3891 y Fr(\024)1260 3810 y Fl(\012)1308 3891 y Fu(F)1385 3844 y Fm(\000)p Fs(7)p Ft(\017)1504 3853 y Fj(2)1537 3844 y Ft(;)p Fm(\000)p Fs(6)p Ft(\017)1676 3853 y Fj(2)1710 3844 y Ft(;)p Fs(6)p Ft(\017)1794 3853 y Fj(2)1828 3844 y Ft(;)p Fs(7)p Ft(\017)1912 3853 y Fj(2)1371 3913 y Fs(+)p Fm(\000)1951 3891 y Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)2211 3907 y Ft(E)2267 3884 y Ff(0)2263 3931 y Fq(k)2305 3891 y Fz(\))2343 3810 y Fl(\013)2390 3935 y Fs(\011)2445 3944 y Fj(2)2479 3935 y Fs(\()p Ft(t)p Fs(\))2586 3891 y Fz(+)f Fr(O)s Fz(\()p Fu(t)2839 3850 y Fm(\0001)2969 3891 y Fz(\))p Fr(jj)p Fu( )t Fr(jj)3186 3850 y Fs(2)3576 3891 y Fz(\(6.69\))150 4116 y(then)32 b(w)m(e)g(can)f(apply)g(\(6.59\))f(\(up)i(to)f(another)g (small)e(enlargemen)m(t)h(of)h(the)g(supp)s(ort\).)44 b(The)32 b(pro)s(of)150 4237 y(of)g(\(6.69\))g(is)g(v)m(ery)i(similar) 29 b(to)k(what)f(w)m(e)i(did)e(in)g(\(6.66\).)43 b(Namely)-8 b(,)31 b(w)m(e)j(sho)m(w)g(that)1122 4354 y Fl(\015)1122 4414 y(\015)1178 4439 y Fu(F)1255 4391 y Fs(6)p Ft(\017)1319 4400 y Fj(2)1353 4391 y Ft(;)p Fs(7)p Ft(\017)1437 4400 y Fj(2)1241 4460 y Fs(+)1475 4439 y Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)1735 4455 y Ft(E)1791 4432 y Ff(0)1787 4478 y Fq(k)1829 4439 y Fz(\))f Fr(\001)g Fz(\011)2015 4454 y Fs(2)2054 4439 y Fz(\()p Fu(t)p Fz(\))2165 4354 y Fl(\015)2165 4414 y(\015)2249 4439 y Fz(=)27 b Fr(O)s Fz(\()p Fu(t)2507 4397 y Fm(\0001)2637 4439 y Fz(\))p Fr(jj)p Fu( )t Fr(jj)722 b Fz(\(6.70\))150 4641 y(and)1054 4676 y Fl(\015)1054 4736 y(\015)1109 4761 y Fu(F)1186 4714 y Fm(\000)p Fs(7)p Ft(\017)1305 4723 y Fj(2)1339 4714 y Ft(;)p Fm(\000)p Fs(6)p Ft(\017)1478 4723 y Fj(2)1172 4783 y Fm(\000)1516 4761 y Fz(\()9 b(~)-58 b Fu(\032)23 b Fr(\000)f Fu(\032)1776 4777 y Ft(E)1832 4754 y Ff(0)1828 4801 y Fq(k)1871 4761 y Fz(\))g Fr(\001)g Fz(\011)2057 4776 y Fs(2)2096 4761 y Fz(\()p Fu(t)p Fz(\))2207 4676 y Fl(\015)2207 4736 y(\015)2290 4761 y Fz(=)27 b Fr(O)s Fz(\()p Fu(t)2548 4720 y Fm(\0001)2678 4761 y Fz(\))p Fr(jj)p Fu( )t Fr(jj)p Fu(;)654 b Fz(\(6.71\))150 4928 y(whic)m(h)38 b(is)f(true)g(pro)m(vided)h(the)g(angular)e (partition)f(is)i(\014ne)h(enough.)59 b(Hence)38 b(the)g(prop)s (osition)d(is)150 5048 y(pro)m(v)m(en.)p 3754 5048 4 66 v 3758 4985 59 4 v 3758 5048 V 3816 5048 4 66 v 296 5286 a(The)48 b(ab)s(o)m(v)m(e)f(result)f(sa)m(ys)i(that)e(w)m(e)h(can) g(put)f(another)h(cut-o\013)f(on)g Fu( )t Fz(\()p Fu(t)p Fz(\))g(\(remem)m(b)s(er)g(that)150 5407 y Fu(\017)28 b Fz(=)g(9)q(~)-50 b Fu(\017)409 5422 y Fs(2)448 5407 y Fz(\):)1383 5527 y Fu(F)1460 5480 y Fs(12)p Ft(\017;)p Fs(13)p Ft(\017)1446 5549 y Fm(\000)1682 5527 y Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)1942 5542 y Ft(E)2002 5527 y Fz(\))p Fu( )2103 5542 y Fs(6)2143 5527 y Fz(\()p Fu(t)p Fz(\))k Fr(\030)i Fu( )t Fz(\()p Fu(t)p Fz(\))p Fu(:)984 b Fz(\(6.72\))1939 5776 y(66)p eop %%Page: 67 70 67 69 bop 150 407 a Fz(F)-8 b(or)44 b(further)h(purp)s(oses,)k(w)m(e)d (are)f(forced)g(to)f(replace)h(it)f(with)g(another)h(cut-o\013)f(in)m (v)m(olving)g Fu(\032)3743 422 y Ft(E)3799 403 y Ff(0)150 527 y Fz(with)37 b Fu(E)455 491 y Fm(0)515 527 y Fz(=)f Fu(E)c Fz(+)26 b(2)p Fu(\017)921 542 y Fs(0)960 527 y Fz(;)41 b(remem)m(b)s(er)c(that)g Fu(\017)1743 542 y Fs(0)1821 527 y Fz(in)g(Lemma)f(3.6)i(and)f(Lemma)g(6.15)g(w)m(as)h (small)e(and)150 648 y(prop)s(ortional)30 b(to)i Fu(\017)p Fz(.)44 b(W)-8 b(e)33 b(see)h(that)780 855 y(~)-58 b Fu(\032)23 b Fr(\000)g Fu(\032)994 870 y Ft(E)1081 855 y Fr(\024)28 b Fz(13)p Fu(\017)98 b Fz(and)106 b(~)-58 b Fu(\032)23 b Fr(\000)g Fu(\032)1898 870 y Ft(E)1954 851 y Ff(0)2007 855 y Fr(\025)29 b Fz(30)p Fu(\017)22 b Fz(+)g(2)p Fu(\017)2458 870 y Fs(0)2497 855 y Fz(\(sup)2591 937 y Ft(\022)2699 855 y Fu(@)2750 870 y Ft(E)2810 855 y Fu(\032)2860 870 y Ft(E)2920 855 y Fz(\))27 b(:=)h Fu(\021)3164 870 y Fs(1)3576 855 y Fz(\(6.73\))150 1117 y(are)33 b(classically)d(incompatible,)g(th)m(us)k(w)m(e)f(can)g (rewrite)g(\(6.72\))f(as)1342 1324 y Fu(F)1419 1277 y Ft(\021)1454 1286 y Fj(1)1489 1277 y Ft(;\021)1544 1286 y Fj(1)1578 1277 y Fs(+)p Ft(\017)1662 1286 y Fj(2)1405 1346 y Fm(\000)1701 1324 y Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)1961 1339 y Ft(E)2017 1320 y Ff(0)2043 1324 y Fz(\))p Fu( )2144 1339 y Fs(6)2184 1324 y Fz(\()p Fu(t)p Fz(\))k Fr(\030)i Fu( )t Fz(\()p Fu(t)p Fz(\))p Fu(;)943 b Fz(\(6.74\))150 1531 y(up)36 b(to)f(a)h Fr(O)s Fz(\()p Fu(t)656 1495 y Fm(\0001)785 1531 y Fz(\))p Fr(jj)p Fu( )t Fr(jj)f Fz(error.)52 b(The)37 b(reason)f(for)f(doing)g(this)g (replacemen)m(t)h(will)d(app)s(ear)j(clear)f(in)150 1652 y(the)e(pro)s(of)f(of)g(Prop)s(osition)f(8.1.)296 1772 y(Since)e(w)m(e)h(w)m(ould)e(still)e(lik)m(e)i(to)g(ha)m(v)m(e)i(an)f (explicit)e(upp)s(er)i(b)s(ound)g(for)37 b(~)-58 b Fu(\032)14 b Fr(\000)g Fu(\032)3058 1787 y Ft(E)3118 1772 y Fz(,)30 b(w)m(e)f(again)e(notice)150 1893 y(that)1083 2013 y(~)-58 b Fu(\032)23 b Fr(\000)g Fu(\032)1297 2028 y Ft(E)1384 2013 y Fr(\025)28 b Fz(2)p Fu(\021)1586 2028 y Fs(1)1648 2013 y Fz(+)22 b Fu(\017)1785 2028 y Fs(2)1922 2013 y Fz(and)107 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)2399 2028 y Ft(E)2455 2009 y Ff(0)2509 2013 y Fr(\024)28 b Fu(\021)2662 2028 y Fs(1)2724 2013 y Fz(+)22 b Fu(\017)2861 2028 y Fs(2)3576 2013 y Fz(\(6.75\))150 2182 y(are)39 b(classically)e(incompatible,)g(th)m(us)j(w)m(e)g(can)f(put)g Fu(F)2211 2135 y Fs(2)p Ft(\021)2281 2144 y Fj(1)2316 2135 y Fs(+)p Ft(\017)2400 2144 y Fj(2)2434 2135 y Ft(;)p Fs(2)p Ft(\021)2524 2144 y Fj(1)2559 2135 y Fs(+2)p Ft(\017)2678 2144 y Fj(2)2197 2204 y Fm(\000)2716 2182 y Fz(\()9 b(~)-58 b Fu(\032)27 b Fr(\000)f Fu(\032)2984 2197 y Ft(E)3044 2182 y Fz(\))39 b(on)g(the)g(left)f(hand)150 2302 y(side)33 b(of)f(\(6.74\))g(at)g(the)h(exp)s(ense)h(of)f(a)f Fr(O)s Fz(\()p Fu(t)1737 2266 y Fm(\0001)1867 2302 y Fz(\))p Fr(jj)p Fu( )t Fr(jj)f Fz(error.)296 2423 y(No)m(w)i(de\014ne)841 2630 y Fu( )904 2645 y Fs(7)944 2630 y Fz(\()p Fu(t)p Fz(\))28 b(:=)g Fu(F)1291 2583 y Fs(2)p Ft(\021)1361 2592 y Fj(1)1396 2583 y Fs(+)p Ft(\017)1480 2592 y Fj(2)1514 2583 y Ft(;)p Fs(2)p Ft(\021)1604 2592 y Fj(1)1639 2583 y Fs(+2)p Ft(\017)1758 2592 y Fj(2)1277 2652 y Fm(\000)1796 2630 y Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)2056 2645 y Ft(E)2116 2630 y Fz(\))p Fu(F)2231 2583 y Ft(\021)2266 2592 y Fj(1)2301 2583 y Ft(;\021)2356 2592 y Fj(1)2390 2583 y Fs(+)p Ft(\017)2474 2592 y Fj(2)2217 2652 y Fm(\000)2512 2630 y Fz(\()9 b(~)-58 b Fu(\032)23 b Fr(\000)f Fu(\032)2772 2645 y Ft(E)2828 2626 y Ff(0)2855 2630 y Fz(\))p Fu( )2956 2645 y Fs(6)2995 2630 y Fz(\()p Fu(t)p Fz(\))p Fu(:)443 b Fz(\(6.76\))150 2837 y(W)-8 b(e)33 b(ha)m(v)m(e)h(sho)m(wn)g Fu( )902 2852 y Fs(7)942 2837 y Fz(\()p Fu(t)p Fz(\))27 b Fr(\030)i Fu( )1249 2852 y Fs(6)1288 2837 y Fz(\()p Fu(t)p Fz(\))f Fr(\030)g(\001)17 b(\001)g(\001)26 b(\030)i Fu( )t Fz(\()p Fu(t)p Fz(\).)296 2958 y(Then)k(the)g(new)g(state)f(has) g(the)h(prop)s(ert)m(y)g(that)39 b(~)-58 b Fu(\032)32 b Fz(is)e(e\013ectiv)m(ely)i(lo)s(calized)c(in)j(a)f(v)m(ery)j(narro)m (w)150 3078 y(band)40 b(around)g Fu(\032)789 3093 y Ft(E)849 3078 y Fz(\()p Fu(\022)s Fz(\))g(\(see)h(also)e(\(6.50\)\).)64 b(W)-8 b(e)40 b(are)g(no)m(w)h(prepared)f(to)g(sho)m(w)h(a)e(similar)e (strong)150 3198 y(lo)s(calization)29 b(for)827 3172 y(~)817 3198 y Fu(\030)37 b Fz(around)32 b Fu(\030)1270 3213 y Ft(E)1329 3198 y Fz(.)150 3391 y Fv(Corollary)h(6.18.)38 b FA(L)-5 b(et)32 b Fu(\021)1126 3406 y Fs(2)1193 3391 y Fz(:=)c(2)p Fu(\021)1421 3406 y Fs(1)1473 3391 y Fz(+)14 b(2)p Fu(\017)1651 3406 y Fs(2)1690 3391 y FA(.)44 b(Then)30 b(ther)-5 b(e)31 b(exists)g(a)g(c)-5 b(onstant)30 b Fu(M)39 b(>)27 b Fz(0)k FA(lar)-5 b(ge)31 b(enough)150 3511 y(so)k(that)930 3719 y Fr(jj)p Fu(F)1063 3662 y Fm(\000)p Fs(\()p Ft(M)7 b Fs(+1\))1337 3621 y Fm(p)p 1395 3621 71 3 v 1395 3662 a Ft(\021)1430 3671 y Fj(2)1465 3662 y Ft(;)p Fm(\000)p Ft(M)1615 3621 y Fm(p)p 1674 3621 V 41 x Ft(\021)1709 3671 y Fj(2)1049 3740 y Fm(\000)1748 3719 y Fz(\()1796 3692 y(~)1786 3719 y Fu(\030)26 b Fr(\000)d Fu(\030)1998 3734 y Ft(E)2057 3719 y Fz(\))f Fr(\001)g Fu( )2230 3734 y Fs(7)2270 3719 y Fz(\()p Fu(t)p Fz(\))p Fr(jj)27 b Fz(=)g Fr(O)s Fz(\()p Fu(t)2722 3677 y Fm(\0001)2852 3719 y Fz(\))22 b Fr(\001)g(jj)p Fu( )t Fr(jj)p Fu(;)930 3887 y Fr(jj)p Fu(F)1063 3830 y Ft(M)1138 3789 y Fm(p)p 1195 3789 V 1195 3830 a Ft(\021)1230 3839 y Fj(2)1265 3830 y Ft(;)p Fs(\()p Ft(M)7 b Fs(+1\))1504 3789 y Fm(p)p 1564 3789 V 1564 3830 a Ft(\021)1599 3839 y Fj(2)1049 3908 y Fs(+)1638 3887 y Fz(\()1686 3860 y(~)1676 3887 y Fu(\030)26 b Fr(\000)d Fu(\030)1888 3902 y Ft(E)1947 3887 y Fz(\))f Fr(\001)g Fu( )2120 3902 y Fs(7)2160 3887 y Fz(\()p Fu(t)p Fz(\))p Fr(jj)27 b Fz(=)h Fr(O)s Fz(\()p Fu(t)2613 3845 y Fm(\0001)2743 3887 y Fz(\))22 b Fr(\001)f(jj)p Fu( )t Fr(jj)p Fu(:)518 b Fz(\(6.77\))150 4094 y Fv(Pro)s(of)p Fz(.)44 b(W)-8 b(e)33 b(only)f(consider)h(the)h(\014rst)f(norm)f(in)g (\(6.77\),)g(the)h(other)g(one)g(b)s(eing)f(analogous.)43 b(The)150 4214 y(idea)35 b(is)g(to)g(use)i(energy)f(conserv)-5 b(ation)35 b(together)h(with)f(the)h(already)f(kno)m(wn)i(lo)s (calization)32 b(for)43 b(~)-57 b Fu(\032)p Fz(.)150 4335 y(De\014ne)33 b(as)g(usual)1103 4455 y(\011\()p Fu(t)p Fz(\))28 b(:=)g Fu(F)1526 4399 y Fm(\000)p Fs(\()p Ft(M)7 b Fs(+1\))1800 4357 y Fm(p)p 1859 4357 V 42 x Ft(\021)1894 4408 y Fj(2)1929 4399 y Ft(;)p Fm(\000)p Ft(M)2079 4357 y Fm(p)p 2137 4357 V 2137 4399 a Ft(\021)2172 4408 y Fj(2)1512 4477 y Fm(\000)2211 4455 y Fz(\()2259 4429 y(~)2249 4455 y Fu(\030)27 b Fr(\000)22 b Fu(\030)2461 4470 y Ft(E)2521 4455 y Fz(\))g Fr(\001)g Fu( )2694 4470 y Fs(7)2733 4455 y Fz(\()p Fu(t)p Fz(\))p Fu(:)150 4624 y Fz(Then)647 4831 y Fu(M)751 4790 y Fs(2)791 4831 y Fu(\021)839 4846 y Fs(2)879 4831 y Fr(jj)p Fz(\011\()p Fu(t)p Fz(\))p Fr(jj)1178 4790 y Fs(2)1299 4831 y Fr(\024)1460 4721 y Fl(D)1520 4831 y Fz(\()1568 4805 y(~)1558 4831 y Fu(\030)27 b Fr(\000)c Fu(\030)1771 4846 y Ft(E)1830 4831 y Fz(\))1868 4790 y Fs(2)1907 4721 y Fl(E)1968 4900 y Ft( )r Fs(\()p Ft(t)p Fs(\))3576 4831 y Fz(\(6.78\))1300 5064 y(=)1460 4954 y Fl(D)1520 5064 y Fr(\000)1607 5038 y Fz(~)1597 5064 y Fu(\030)1645 5023 y Fs(2)1707 5064 y Fr(\000)1817 5038 y Fz(~)1807 5064 y Fu(\030)j Fr(\001)c Fz(\()p Fu(\030)2007 5079 y Ft(E)2088 5064 y Fr(\000)2198 5038 y Fz(~)2188 5064 y Fu(\030)t Fz(\))g Fr(\000)h Fz(\()p Fu(\030)2476 5079 y Ft(E)2557 5064 y Fr(\000)2667 5038 y Fz(~)2657 5064 y Fu(\030)t Fz(\))f Fr(\001)2824 5038 y Fz(~)2814 5064 y Fu(\030)27 b Fz(+)22 b Fu(\030)3030 5023 y Fs(2)3025 5089 y Ft(E)3084 4954 y Fl(E)3145 5133 y Fs(\011\()p Ft(t)p Fs(\))3301 5064 y Fu(:)150 5320 y Fz(Since)33 b(w)m(e)g(kno)m(w)h(that)e(on)h(the)g(supp)s(ort)g(of)f (the)h(cut-o\013s)g(in)e(\011\()p Fu(t)p Fz(\))i(w)m(e)h(ha)m(v)m(e) 1159 5501 y(~)1149 5527 y Fu(\030)1197 5486 y Fs(2)1258 5527 y Fr(\000)23 b Fu(\030)1406 5486 y Fs(2)1401 5552 y Ft(E)1488 5527 y Fz(=)k(2\()p Fu(H)j Fr(\000)22 b Fu(E)6 b Fz(\))22 b Fr(\000)32 b Fz(~)-58 b Fu(\032)2176 5486 y Fs(2)2238 5527 y Fz(+)22 b Fu(\032)2386 5486 y Fs(2)2386 5552 y Ft(E)2474 5527 y Fr(\030)28 b Fz(~)-49 b Fu(\017)22 b Fz(+)g Fu(\021)2786 5542 y Fs(2)1939 5776 y Fz(67)p eop %%Page: 68 71 68 70 bop 150 407 a Fz(w)m(e)34 b(can)e(rewrite)h(up)g(to)f(the)h (usual)f(errors)h(in)m(tro)s(duced)g(b)m(y)h(comm)m(utations)d (\(6.78\))g(as)842 614 y Fu(M)946 573 y Fs(2)986 614 y Fu(\021)1034 629 y Fs(2)1074 614 y Fr(jj)p Fz(\011\()p Fu(t)p Fz(\))p Fr(jj)1373 573 y Fs(2)1494 614 y Fr(\024)83 b Fz(Const)23 b Fr(\001)f Fu(\021)2024 629 y Fs(2)2064 614 y Fr(jj)p Fz(\011\()p Fu(t)p Fz(\))p Fr(jj)2363 573 y Fs(2)2423 614 y Fz(+)g Fr(O)s Fz(\()p Fu(t)2676 573 y Fm(\0001)2806 614 y Fz(\))g Fr(\001)g(jj)p Fu( )t Fr(jj)3095 573 y Fs(2)1494 786 y Fr(\000)1654 676 y Fl(D)1726 760 y Fz(~)1715 786 y Fu(\030)27 b Fr(\001)22 b Fz(\()p Fu(\030)1916 801 y Ft(E)1997 786 y Fr(\000)2107 760 y Fz(~)2097 786 y Fu(\030)t Fz(\))g(+)g(\()p Fu(\030)2383 801 y Ft(E)2464 786 y Fr(\000)2574 760 y Fz(~)2564 786 y Fu(\030)t Fz(\))h Fr(\001)2732 760 y Fz(~)2721 786 y Fu(\030)2769 676 y Fl(E)2830 855 y Fs(\011\()p Ft(t)p Fs(\))2986 786 y Fu(:)563 b Fz(\(6.79\))150 1059 y(No)m(w)31 b(remem)m(b)s(er)f(\(see)h (\(6.57\)\))f(that)g Fu( )1606 1074 y Fs(7)1676 1059 y Fz(has)h(a)f(lo)s(calization)c(for)2599 1033 y(~)2589 1059 y Fu(\030)35 b Fz(ab)s(o)m(v)m(e)c(zero;)h(so)e(w)m(e)i(can)e (write)160 1153 y(~)150 1180 y Fu(\030)42 b Fz(=)c Fu(F)426 1144 y Fs(2)465 1180 y Fz(\()513 1153 y(~)503 1180 y Fu(\030)t Fz(\))h(where)h Fu(F)52 b Fz(is)38 b(a)g(smo)s(oth)g(v)m (ersion)h(of)f(the)h(square)g(ro)s(ot)f(on)g(the)h(p)s(ositiv)m(e)f (semi-axis.)150 1300 y(Hence)484 1507 y Fr(\000)578 1396 y Fl(D)649 1481 y Fz(~)639 1507 y Fu(\030)26 b Fr(\001)c Fz(\()p Fu(\030)839 1522 y Ft(E)921 1507 y Fr(\000)1030 1481 y Fz(~)1020 1507 y Fu(\030)5 b Fz(\))22 b(+)g(\()p Fu(\030)1307 1522 y Ft(E)1388 1507 y Fr(\000)1498 1481 y Fz(~)1488 1507 y Fu(\030)t Fz(\))g Fr(\001)1655 1481 y Fz(~)1645 1507 y Fu(\030)1692 1396 y Fl(E)1753 1576 y Fs(\011\()p Ft(t)p Fs(\))1976 1507 y Fz(=)82 b(2)2200 1396 y Fl(D)2261 1507 y Fu(F)14 b Fz(\()2386 1481 y(~)2376 1507 y Fu(\030)t Fz(\))22 b Fr(\001)g Fz(\()2581 1481 y(~)2571 1507 y Fu(\030)k Fr(\000)d Fu(\030)2783 1522 y Ft(E)2842 1507 y Fz(\))f Fr(\001)g Fu(F)14 b Fz(\()3077 1481 y(~)3067 1507 y Fu(\030)t Fz(\))3152 1396 y Fl(E)3213 1576 y Fs(\011\()p Ft(t)p Fs(\))3576 1507 y Fz(\(6.80\))1976 1710 y(+)82 b Fr(O)s Fz(\()p Fu(t)2289 1669 y Fm(\000)p Fs(1)2384 1710 y Fz(\))p Fr(jj)p Fz(\011\()p Fu(t)p Fz(\))p Fr(jj)2721 1669 y Fs(2)2781 1710 y Fz(+)22 b Fr(O)s Fz(\()p Fu(t)3034 1669 y Fm(\0001)3164 1710 y Fz(\))g Fr(\001)g(jj)p Fu( )t Fr(jj)3453 1669 y Fs(2)150 1917 y Fz(where)45 b(the)e(term)g Fr(O)s Fz(\()p Fu(t)1020 1881 y Fm(\000)p Fs(1)1115 1917 y Fz(\))p Fr(jj)p Fz(\011\()p Fu(t)p Fz(\))p Fr(jj)1452 1881 y Fs(2)1533 1917 y Fz(comes)g(from)g(comm)m(uting)e Fu(F)14 b Fz(\()2715 1890 y(~)2705 1917 y Fu(\030)t Fz(\))43 b(with)g Fu(\030)3109 1932 y Ft(E)3168 1917 y Fz(.)76 b(But)44 b(no)m(w)g(the)150 2037 y(exp)s(ectation)30 b(on)h(the)f(righ)m(t)g(hand)g(side)h(of)f(\(6.80\))f(is)h(essen)m (tially)g(negativ)m(e)g(b)s(ecause)i(of)e(the)h(extra)150 2157 y(cut-o\013)h(on)h(\011\()p Fu(t)p Fz(\).)43 b(Therefore)34 b(\(6.79\))e(b)s(ecomes)481 2364 y Fu(M)585 2323 y Fs(2)626 2364 y Fu(\021)674 2379 y Fs(2)713 2364 y Fr(jj)p Fz(\011\()p Fu(t)p Fz(\))p Fr(jj)1012 2323 y Fs(2)1078 2364 y Fr(\024)c Fz(\(Const)23 b Fr(\001)f Fu(\021)1591 2379 y Fs(2)1653 2364 y Fz(+)g Fr(O)s Fz(\()p Fu(t)1906 2323 y Fm(\000)p Fs(1)2000 2364 y Fz(\)\))g Fr(\001)g(jj)p Fz(\011\()p Fu(t)p Fz(\))p Fr(jj)2447 2323 y Fs(2)2507 2364 y Fz(+)g Fr(O)s Fz(\()p Fu(t)2760 2323 y Fm(\0001)2890 2364 y Fz(\))g Fr(\001)g(jj)p Fu( )t Fr(jj)3179 2323 y Fs(2)3217 2364 y Fu(;)332 b Fz(\(6.81\))150 2571 y(hence)34 b(c)m(ho)s(osing)e Fu(M)43 b Fz(large)32 b(enough)h(the)g(corollary)e(is)h(pro)m(v)m(en.)p 3754 2571 4 66 v 3758 2509 59 4 v 3758 2571 V 3816 2571 4 66 v 296 2810 a(With)g(the)h Fu(M)44 b Fz(pro)m(vided)33 b(b)m(y)g(the)g(ab)s(o)m(v)m(e)h(corollary)c(de\014ne)939 3030 y Fu(F)1016 2989 y Ft(M)1002 3055 y Fs(+)p Fm(\000)1199 3030 y Fz(:=)83 b Fu(F)1462 2974 y Fm(\000)p Fs(\()p Ft(M)7 b Fs(+2\))1736 2933 y Fm(p)p 1795 2933 71 3 v 41 x Ft(\021)1830 2983 y Fj(2)1865 2974 y Ft(;)p Fm(\000)p Fs(\()p Ft(M)g Fs(+1\))2159 2933 y Fm(p)p 2218 2933 V 41 x Ft(\021)2253 2983 y Fj(2)2288 2974 y Ft(;)p Fs(\()p Ft(M)g Fs(+1\))2527 2933 y Fm(p)p 2587 2933 V 2587 2974 a Ft(\021)2622 2983 y Fj(2)2657 2974 y Ft(;)p Fs(\()p Ft(M)g Fs(+2\))2896 2933 y Fm(p)p 2955 2933 V 41 x Ft(\021)2990 2983 y Fj(2)1448 3052 y Fs(+)p Fm(\000)3029 3030 y Fu(;)902 3188 y( )965 3203 y Fs(8)1005 3188 y Fz(\()p Fu(t)p Fz(\))83 b(:=)g Fu(F)1462 3146 y Ft(M)1448 3212 y Fs(+)p Fm(\000)1562 3188 y Fz(\()1610 3161 y(~)1600 3188 y Fu(\030)27 b Fr(\000)22 b Fu(\030)1812 3203 y Ft(E)1871 3188 y Fz(\))g Fr(\001)g Fu( )2044 3203 y Fs(7)2084 3188 y Fz(\()p Fu(t)p Fz(\))p Fu(:)1354 b Fz(\(6.82\))150 3395 y(W)-8 b(e)33 b(ha)m(v)m(e)h(sho)m(wn) g Fu( )902 3410 y Fs(8)942 3395 y Fz(\()p Fu(t)p Fz(\))27 b Fr(\030)i Fu( )1249 3410 y Fs(7)1288 3395 y Fz(\()p Fu(t)p Fz(\))f Fr(\030)g(\001)17 b(\001)g(\001)26 b(\030)i Fu( )t Fz(\()p Fu(t)p Fz(\).)150 3681 y Fp(6.8)135 b Fd(\000)p Fi(@)611 3699 y Fh(t)645 3681 y Fi(S)7 b FF(\()p Fi(t;)20 b(r)m(;)g(\022)s FF(\))43 b Fp(is)i(close)g(to)h Fi(H)150 3866 y Fz(The)40 b(last)f(quan)m(tit)m(y)h(whic)m(h)g(w)m(e)g (w)m(ould)f(lik)m(e)g(to)g(kno)m(w)h(that)f(is)g(a)g(priori)e(small)g (on)j(our)f(state)g(is)150 3987 y Fu(H)30 b Fz(+)22 b Fu(@)410 4002 y Ft(t)440 3987 y Fu(S)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\).)43 b(W)-8 b(e)33 b(\014rst)g(giv)m(e)g(a)f (quan)m(tum)h(v)m(ersion)g(for)f(Prop)s(osition)f(3.10.)150 4179 y Fv(Prop)s(osition)k(6.19.)42 b FA(T)-7 b(ake)34 b Fu(F)1341 4143 y Ft(v)r(:b:)1485 4179 y FA(to)h(b)-5 b(e)35 b(of)f(the)h(typ)-5 b(e)35 b(\(3.3\))f(with)g(supp)-5 b(ort)35 b(on)g Fn(R)3207 4194 y Fs(+)3307 4179 y FA(and)f(e)-5 b(qual)35 b(to)150 4299 y Fz(1)d FA(on)h(the)g(supp)-5 b(ort)32 b(of)h(the)f(minimal)g(and)g(maximal)f(velo)-5 b(city)33 b(cut-o\013s.)44 b(A)n(bbr)-5 b(eviate)32 b Fr(\000)p Fu(@)3441 4314 y Ft(t)3471 4299 y Fu(S)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))150 4420 y FA(by)35 b Fu(E)349 4435 y Ft(t)414 4420 y FA(and)f(de\014ne)757 4670 y Fu(D)s Fz(\()p Fu(t)p Fz(\))28 b(:=)g(1)21 b Fr(\000)1291 4603 y Fu(r)p 1291 4647 47 4 v 1297 4739 a(t)1348 4670 y(@)1399 4685 y Ft(E)1459 4670 y Fu(\032)1509 4685 y Ft(E)1569 4670 y Fz(\()p Fu(\022)s Fz(\))h Fr(\001)g Fu(F)1842 4629 y Ft(v)r(:b:)1952 4670 y Fz(\()p Fu(r)s(=t)p Fz(\))27 b(=)g(1)22 b Fr(\000)2482 4603 y Fu(@)2533 4618 y Ft(E)2593 4603 y Fu(\032)2643 4618 y Ft(E)p 2470 4647 246 4 v 2470 4739 a Fu(@)2521 4754 y Ft(E)2581 4739 y Fu(\032)2631 4754 y Ft(E)2683 4762 y Fq(t)2747 4670 y Fr(\001)g Fu(F)2874 4629 y Ft(v)r(:b:)2984 4670 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu(:)150 4932 y FA(Denote)39 b(by)h Fu(c)e Fz(:=)f(1)26 b(+)f(sup)1165 4955 y Ft(t)p Fm(\025)p Fs(1)1302 4932 y Fr(jj)p Fu(D)s Fz(\()p Fu(t)p Fz(\))p Fr(jj)36 b Fu(<)h Fr(1)p FA(.)59 b(F)-7 b(or)40 b Fz(0)d Fu(<)g(\017)2373 4947 y Fs(3)2450 4932 y Fu(<)g Fz(1)p Fu(=)p Fz(3)i FA(denote)h(by)g Fu(F)3261 4947 y Fs(+)p Fm(\000)3415 4932 y FA(either)g(the)150 5065 y(function)g Fu(F)616 5017 y Fs(2)p Ft(\017)680 5026 y Fj(3)714 5017 y Ft(;)p Fs(3)p Ft(\017)798 5026 y Fj(3)832 5017 y Ft(;c;c)p Fs(+1)602 5086 y(+)p Fm(\000)1067 5065 y FA(or)h Fu(F)1276 5017 y Fm(\000)p Ft(c)p Fm(\000)p Fs(1)p Ft(;)p Fm(\000)p Ft(c;)p Fm(\000)p Fs(3)p Ft(\017)1697 5026 y Fj(3)1729 5017 y Ft(;)p Fm(\000)p Fs(2)p Ft(\017)1868 5026 y Fj(3)1262 5086 y Fs(+)p Fm(\000)1906 5065 y FA(.)61 b(Denote)40 b(by)g Fu(F)2533 5080 y Fs(++)2688 5065 y FA(its)g(asso)-5 b(ciate)g(d)39 b(\(3.4\)-typ)-5 b(e)40 b(of)150 5185 y(function.)k(Then)34 b(if)h Fu(\017)961 5200 y Fs(3)1036 5185 y FA(is)f(su\016ciently)h(smal)5 b(l)34 b(we)h(have)974 5302 y Fl(Z)1073 5328 y Fm(1)1029 5527 y Fs(1)1175 5370 y Fz(1)p 1175 5414 49 4 v 1182 5506 a Fu(t)1233 5437 y Fr(jj)1289 5349 y Fl(p)p 1388 5349 178 4 v 1388 5437 a Fu(F)1451 5452 y Fs(+)p Fm(\000)1565 5437 y Fz(\()p Fu(D)s Fz(\()p Fu(t)p Fz(\)\))22 b Fr(\001)g Fu( )1971 5452 y Fs(8)2011 5437 y Fz(\()p Fu(t)p Fz(\))p Fr(jj)2178 5396 y Fs(2)2217 5437 y Fu(dt)27 b Fr(\024)h Fz(Const)23 b Fr(\001)f(jj)p Fu( )t Fr(jj)2936 5396 y Fs(2)2974 5437 y Fu(:)575 b Fz(\(6.83\))1939 5776 y(68)p eop %%Page: 69 72 69 71 bop 150 407 a Fv(Pro)s(of)p Fz(.)48 b(The)35 b(pro)s(of)e(is)h (similar)d(in)i(spirit)g(to)h(the)g(one)g(w)m(e)i(ga)m(v)m(e)f(in)e (the)h(classical)f(case.)49 b(Assume)150 527 y(\014rst)44 b(that)g(w)m(e)g(w)m(ork)h(with)e Fu(F)1299 480 y Fs(2)p Ft(\017)1363 489 y Fj(3)1397 480 y Ft(;)p Fs(3)p Ft(\017)1481 489 y Fj(3)1515 480 y Ft(;c;c)p Fs(+1)1285 549 y(+)p Fm(\000)1710 527 y Fz(.)77 b(De\014ne)44 b(a)f(b)s(ounded)i(and)f (symmetric)f(propagation)150 648 y(observ)-5 b(able)30 b(\010\()p Fu(t)p Fz(\))h(in)f(suc)m(h)i(a)e(w)m(a)m(y)h(that)f(when)i (taking)d(the)i(exp)s(ectation)f(on)h(a)f(state)g(lik)m(e)g(in)f (\(6.1\))150 768 y(w)m(e)34 b(get)1328 888 y Fr(h)p Fz(\010\()p Fu(t)p Fz(\))p Fr(i)1587 904 y Ft( )r Fs(\()p Ft(t)p Fs(\))1748 888 y Fz(=)28 b Fr(\000h)p Fu(F)2031 903 y Fs(++)2145 888 y Fz(\()p Fu(D)s Fz(\()p Fu(t)p Fz(\)\))p Fr(i)2455 904 y Ft( )2501 913 y Fj(8)2535 904 y Fs(\()p Ft(t)p Fs(\))2620 888 y Fu(:)150 1060 y Fz(Di\013eren)m(tiate)34 b(with)i(resp)s(ect)h(to)e Fu(t)h Fz(as)g(in)f(the)h(classical)e(case)j (\(see)g(\(3.46\)\))e(and)h(notice)f(that)h(the)150 1180 y(\\in)m(teresting")c(term)g(is)g(going)f(to)h(b)s(e)452 1305 y Fl(\034)536 1378 y Fz(1)p 536 1423 49 4 v 543 1514 a Fu(t)617 1446 y Fr(\001)22 b Fz([)p Fu(D)s Fz(\()p Fu(t)p Fz(\))g(+)g Fr(O)s Fz(\()p Fu(\017)p Fz(\)])1233 1305 y Fl(\035)1308 1483 y Fr(p)p 1391 1483 142 4 v 62 x Ft(F)1446 1522 y Ff(0)1436 1565 y Fj(++)1533 1545 y Fs(\()p Ft(D)r Fs(\()p Ft(t)p Fs(\)\))p Fm(\001)p Ft( )1792 1554 y Fj(8)1829 1545 y Fs(\()p Ft(t)p Fs(\))1941 1446 y Fr(\025)2056 1378 y Fz(2)p Fu(\017)2144 1393 y Fs(3)2206 1378 y Fr(\000)g(O)s Fz(\()p Fu(\017)p Fz(\))p 2056 1423 447 4 v 2262 1514 a Fu(t)2513 1446 y Fr(jj)2569 1359 y Fl(p)p 2668 1359 178 4 v 2668 1446 a Fu(F)2745 1411 y Fm(0)2731 1467 y Fs(++)2845 1446 y Fz(\()p Fu(D)s Fz(\()p Fu(t)p Fz(\)\))g Fr(\001)f Fu( )3250 1461 y Fs(8)3290 1446 y Fz(\()p Fu(t)p Fz(\))p Fr(jj)3457 1405 y Fs(2)3496 1446 y Fu(;)150 1754 y Fz(where)32 b(w)m(e)g(emplo)m(y)m(ed)f(the)g(p)s (ositivit)m(y)f(of)g Fu(D)k Fz(on)c(the)i(supp)s(ort)f(of)f Fu(F)2669 1718 y Fm(0)2655 1778 y Fs(++)2800 1754 y Fz(together)h(with) f(the)i(small-)150 1886 y(ness)39 b(of)46 b(~)-58 b Fu(\032)25 b Fr(\000)h Fu(\032)706 1901 y Ft(E)803 1886 y Fz(and)1008 1860 y(~)998 1886 y Fu(\030)k Fr(\000)c Fu(\030)1217 1901 y Ft(E)1276 1886 y Fz(.)57 b(Then)39 b(w)m(e)f(in)m(tegrate)f(and) h(get)f(the)h(result)f(in)g(\(6.83\).)57 b(The)38 b(case)150 2006 y(where)28 b Fu(F)489 2021 y Fs(+)p Fm(\000)630 2006 y Fz(is)f(supp)s(orted)h(on)f(the)g(negativ)m(e)g(axis)g(is)g (similar,)d(only)j(the)g(propagation)f(observ)-5 b(able)150 2127 y(has)33 b(to)f(b)s(e)h(tak)m(en)h(with)e(an)g(opp)s(osite)g (sign.)43 b(The)34 b(prop)s(osition)c(is)j(pro)m(v)m(en.)p 3754 2127 4 66 v 3758 2064 59 4 v 3758 2127 V 3816 2127 4 66 v 296 2365 a(W)-8 b(e)35 b(are)f(no)m(w)h(ready)f(to)g(add)g(a)g (new)h(cut-o\013)f(to)f(our)h(state.)49 b(If)34 b Fu(\017)2724 2380 y Fs(3)2798 2365 y Fz(is)f(as)h(giv)m(en)h(b)m(y)g(the)f(ab)s(o)m (v)m(e)150 2486 y(prop)s(osition,)d(de\014ne)947 2699 y Fu(F)1010 2714 y Ft(D)1102 2699 y Fz(:=)d Fu(F)1310 2652 y Fm(\000)p Fs(4)p Ft(\017)1429 2661 y Fj(3)1462 2652 y Ft(;)p Fm(\000)p Fs(3)p Ft(\017)1601 2661 y Fj(3)1635 2652 y Ft(;)p Fs(3)p Ft(\017)1719 2661 y Fj(3)1753 2652 y Ft(;)p Fs(4)p Ft(\017)1837 2661 y Fj(3)1296 2721 y Fs(+)p Fm(\000)1876 2699 y Fu(;)114 b( )2080 2714 y Fs(9)2120 2699 y Fz(\()p Fu(t)p Fz(\))27 b(:=)h Fu(F)2452 2714 y Ft(D)2516 2699 y Fz(\()p Fu(D)s Fz(\()p Fu(t)p Fz(\)\))p Fu( )2850 2714 y Fs(8)2890 2699 y Fz(\()p Fu(t)p Fz(\))p Fu(:)548 b Fz(\(6.84\))150 2913 y(W)-8 b(e)33 b(ha)m(v)m(e)h(sho)m(wn)g Fu( )902 2928 y Fs(9)942 2913 y Fz(\()p Fu(t)p Fz(\))27 b Fr(\030)i Fu( )1249 2928 y Fs(8)1288 2913 y Fz(\()p Fu(t)p Fz(\))f Fr(\030)g(\001)17 b(\001)g(\001)26 b(\030)i Fu( )t Fz(\()p Fu(t)p Fz(\).)296 3033 y(Finally)-8 b(,)26 b(w)m(e)j(can)f(sho)m(w)g(that)g(on)f(our)h(state)g Fu(H)35 b Fz(can)28 b(b)s(e)f(w)m(ell)g(appro)m(ximated)g(b)m(y)i Fr(\000)p Fu(@)3414 3048 y Ft(t)3444 3033 y Fu(S)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\).)150 3154 y(W)-8 b(e)33 b(form)m(ulate)e(this)h(as)h(a)f(prop)s(osition.)150 3352 y Fv(Prop)s(osition)49 b(6.20.)f FA(If)c Fu(M)59 b(>)47 b Fz(0)e FA(c)-5 b(onsider)44 b Fu(F)1987 3367 y Fs(+)p Fm(\000)2149 3352 y Fz(=)j Fu(F)2349 3304 y Fm(\000)p Fs(2)p Ft(M)7 b(\017)2543 3313 y Fj(3)2577 3304 y Ft(;)p Fm(\000)p Ft(M)g(\017)2756 3313 y Fj(3)2789 3304 y Ft(;M)g(\017)2913 3313 y Fj(3)2947 3304 y Ft(;)p Fs(2)p Ft(M)g(\017)3106 3313 y Fj(3)2335 3373 y Fs(+)p Fm(\000)3144 3352 y FA(.)76 b(Denote)45 b(again)150 3472 y Fr(\000)p Fu(@)278 3487 y Ft(t)309 3472 y Fu(S)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))34 b FA(by)h Fu(E)896 3487 y Ft(t)961 3472 y FA(and)f(de\014ne)g(\()p Fu(F)1551 3436 y Ft(v)r(:b:)1696 3472 y FA(is)g(as)h(in)g(the)f(pr)-5 b(evious)35 b(pr)-5 b(op)g(osition\))1163 3686 y Fz(~)-53 b Fu(\015)1210 3701 y Fs(3)1277 3686 y Fz(:=)28 b Fu(H)h Fr(\001)1589 3659 y Fz(~)1568 3686 y Fu(f)1616 3701 y Ft(E)1676 3686 y Fz(\()p Fu(H)8 b Fz(\))21 b(+)h Fu(@)2011 3701 y Ft(t)2041 3686 y Fu(S)6 b Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))p Fu(F)2472 3644 y Ft(v)r(:b:)2582 3686 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu(:)760 b Fz(\(6.85\))150 3899 y FA(Then)34 b(ther)-5 b(e)35 b(exists)f Fu(M)46 b FA(lar)-5 b(ge)34 b(enough)h(such)f(that)1161 4113 y Fr(jj)p Fz(\()p Fv(1)22 b Fr(\000)h Fu(F)1496 4128 y Fs(+)p Fm(\000)1610 4113 y Fz(\()t(~)-53 b Fu(\015)1699 4128 y Fs(3)1738 4113 y Fz(\)\))p Fu( )1877 4128 y Fs(9)1916 4113 y Fz(\()p Fu(t)p Fz(\))p Fr(jj)27 b Fz(=)h Fr(O)s Fz(\()p Fu(t)2369 4072 y Fm(\0001)2499 4113 y Fz(\))22 b Fr(\001)g(jj)p Fu( )t Fr(jj)p Fu(:)761 b Fz(\(6.86\))150 4326 y Fv(Pro)s(of)p Fz(.)46 b(The)34 b(argumen)m(t)g(relies)e(on)i(the)g(classical)d (analog)h(that)i(can)f(b)s(e)h(traced)g(bac)m(k)g(to)g(\(3.48\),)150 4447 y(\(3.50\))e(and)g(\(3.45\).)43 b(De\014ne)1419 4567 y(\011\()p Fu(t)p Fz(\))28 b(:=)f Fu(F)1841 4520 y Ft(M)7 b(\017)1945 4529 y Fj(3)1979 4520 y Ft(;)p Fs(2)p Ft(M)g(\017)2138 4529 y Fj(3)1827 4589 y Fs(+)2176 4567 y Fz(\()t(~)-53 b Fu(\015)2265 4582 y Fs(3)2304 4567 y Fz(\))p Fu( )2405 4582 y Fs(9)2445 4567 y Fz(\()p Fu(t)p Fz(\))150 4739 y(and)33 b(compute)722 4952 y Fu(M)10 b(\017)865 4967 y Fs(3)906 4952 y Fr(jj)p Fz(\011\()p Fu(t)p Fz(\))p Fr(jj)1205 4911 y Fs(2)1271 4952 y Fr(\024)28 b(h)t Fz(~)-53 b Fu(\015)1466 4967 y Fs(3)1505 4952 y Fr(i)1544 4968 y Fs(\011\()p Ft(t)p Fs(\))1710 4952 y Fr(\024)29 b(h)p Fu(E)e Fr(\000)c Fu(E)2126 4967 y Ft(t)2178 4952 y Fz(+)g(~)-50 b Fu(\017)p Fr(i)2354 4968 y Fs(\011\()p Ft(t)p Fs(\))2516 4952 y Fz(+)22 b Fr(O)s Fz(\()p Fu(t)2769 4911 y Fm(\0001)2899 4952 y Fz(\))g Fr(\001)g(jj)p Fu( )t Fr(jj)3188 4911 y Fs(2)3226 4952 y Fu(:)150 5166 y Fz(But)40 b(then)g Fr(j)p Fu(E)33 b Fr(\000)27 b Fu(E)889 5181 y Ft(t)919 5166 y Fr(j)39 b Fz(can)h(b)s(e)g(b)s(ounded)g(b)m(y)h Fr(j)p Fu(D)s Fr(j)d Fz(times)h(a)g(\(big\))g(constan)m(t)h(so)g(it)f (can)h(b)s(e)f(made)150 5286 y(smaller)26 b(than)i(a)f(constan)m(t)i (times)e Fu(\017)1465 5301 y Fs(3)1505 5286 y Fz(.)41 b(Hence)30 b(if)c(w)m(e)j(tak)m(e)g Fu(M)38 b Fz(large)27 b(enough)h(w)m(e)h(obtain)e Fr(jj)p Fz(\011\()p Fu(t)p Fz(\))p Fr(jj)f Fz(=)150 5407 y Fr(O)s Fz(\()p Fu(t)305 5371 y Fm(\0001)435 5407 y Fz(\))p Fr(\001jj)p Fu( )t Fr(jj)p Fz(.)38 b(W)-8 b(e)22 b(then)h(follo)m(w)d(the)i(same)f (argumen)m(t)h(\\on)f(the)h(other)g(side")g(and)g(the)g(prop)s(osition) 150 5527 y(is)32 b(pro)m(v)m(en.)p 3754 5527 V 3758 5465 59 4 v 3758 5527 V 3816 5527 4 66 v 1939 5776 a(69)p eop %%Page: 70 73 70 72 bop 296 407 a Fz(No)m(w)34 b(let)f(us)h(in)m(tro)s(duce)f (another)h(cut-o\013)e(on)i(our)f(state.)46 b(If)33 b Fu(\017)2613 422 y Fs(3)2686 407 y Fz(and)h Fu(M)44 b Fz(are)33 b(as)h(giv)m(en)f(ab)s(o)m(v)m(e,)150 527 y(de\014ne)837 648 y Fu(F)900 663 y Ft(E)987 648 y Fz(:=)28 b Fu(F)1195 600 y Fm(\000)p Fs(3)p Ft(M)7 b(\017)1389 609 y Fj(3)1422 600 y Ft(;)p Fm(\000)p Fs(2)p Ft(M)g(\017)1636 609 y Fj(3)1670 600 y Ft(;)p Fs(2)p Ft(M)g(\017)1829 609 y Fj(3)1863 600 y Ft(;)p Fs(3)p Ft(M)g(\017)2022 609 y Fj(3)1181 669 y Fs(+)p Fm(\000)2060 648 y Fu(;)114 b( )2264 663 y Fs(10)2339 648 y Fz(\()p Fu(t)p Fz(\))28 b(:=)g Fu(F)2672 663 y Ft(E)2732 648 y Fz(\()t(~)-53 b Fu(\015)2821 663 y Fs(3)2860 648 y Fz(\))p Fu( )2961 663 y Fs(9)3000 648 y Fz(\()p Fu(t)p Fz(\))p Fu(:)438 b Fz(\(6.87\))150 822 y(W)-8 b(e)33 b(ha)m(v)m(e)h(sho)m(wn)g Fu( )902 837 y Fs(10)977 822 y Fz(\()p Fu(t)p Fz(\))28 b Fr(\030)g Fu( )1284 837 y Fs(9)1324 822 y Fz(\()p Fu(t)p Fz(\))f Fr(\030)i(\001)17 b(\001)g(\001)26 b(\030)i Fu( )t Fz(\()p Fu(t)p Fz(\).)150 1155 y Fw(7)161 b(Asymptotic)53 b(completeness:)68 b Fb(\015)2383 1103 y Fz(2)2375 1190 y(1)2468 1155 y Fa(+)32 b Fb(\015)2684 1103 y Fz(2)2676 1190 y(2)2790 1155 y Fw(is)55 b(in)l(tegrable)150 1374 y Fz(If)35 b(w)m(e)h(lo)s(ok)e (bac)m(k)i(at)e(\(4.19\))h(w)m(e)h(see)g(that)f(it)f(w)m(ould)g(b)s(e)i (go)s(o)s(d)e(to)g(kno)m(w)i(that)f(when)h(applied)e(on)150 1494 y(a)k(state)i(lik)m(e)e(the)h(one)g(w)m(e)h(ha)m(v)m(e)g(in)e (\(6.1\),)i(the)f(\\p)s(erturbation")e Fu(\015)2701 1458 y Fs(2)2696 1519 y(1)2767 1494 y Fz(+)26 b Fu(\015)2925 1458 y Fs(2)2920 1519 y(2)3003 1494 y Fz(deca)m(ys)41 b(at)d(least)h(lik)m(e)150 1615 y Fu(t)185 1578 y Fm(\000)p Fs(1)p Fm(\000)p Ft(\016)368 1615 y Fz(.)52 b(This)36 b(w)m(ould)g(mean)f(that)g(a)g(Co)s(ok)h(t)m(yp)s(e)g(argumen)m(t)g (for)f(the)h(existence)h(of)e(\012)3367 1630 y Fs(+)3462 1615 y Fz(could)g(b)s(e)150 1735 y(p)s(ossible.)296 1855 y(W)-8 b(e)40 b(\014rst)f(in)m(tro)s(duce)g(a)g(regularized)f(v)m (ersion)i(of)e(our)h(gammas.)61 b(De\014ne)40 b(\(see)g(Prop)s (ositions)150 1976 y(6.19)32 b(and)h(6.20)f(for)g(v)-5 b(arious)31 b(notations\))1342 2196 y(~)-53 b Fu(\015)1389 2211 y Fs(1)1511 2196 y Fz(:=)92 b(~)-58 b Fu(\032)23 b Fr(\000)f Fu(\032)1919 2211 y Ft(E)1971 2219 y Fq(t)2003 2196 y Fz(\()p Fu(\022)s Fz(\))h Fr(\001)e Fu(F)2276 2155 y Ft(v)r(:b:)2386 2196 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu(;)1004 b Fz(\(7.1\))1342 2341 y(~)-53 b Fu(\015)1389 2356 y Fs(2)1511 2341 y Fz(:=)1707 2315 y(~)1697 2341 y Fu(\030)27 b Fr(\000)c Fu(\030)1910 2356 y Ft(E)1962 2364 y Fq(t)1993 2341 y Fz(\()p Fu(\022)s Fz(\))f Fr(\001)g Fu(F)2266 2300 y Ft(v)r(:b:)2376 2341 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu(:)1014 b Fz(\(7.2\))296 2561 y(The)26 b(main)d(result)i(of)f (this)h(section)g(will)d(b)s(e)k(a)e(quan)m(tum)h(equiv)-5 b(alen)m(t)25 b(of)f(Lemma)g(3.14.)40 b(W)-8 b(e)25 b(will)150 2681 y(try)k(to)g(follo)m(w)e(the)i(same)g(steps)h(as)f(w)m(e)h(did)e (in)g(the)h(classical)f(case)i(since)f(there)g(is)g(a)f(close)h (analogy)150 2802 y(with)k(that)g(situation.)44 b(Of)33 b(course,)i(here)f(the)g(tec)m(hnique)g(is)f(more)g(in)m(v)m(olv)m(ed)h (since)f(w)m(e)i(no)m(w)f(ha)m(v)m(e)150 2922 y(to)c(deal)f(with)h (non-comm)m(utativit)m(y)-8 b(.)40 b(But)31 b(still,)d(the)j(main)d (idea)i(is)f(the)i(same:)42 b(\014nd)31 b(a)f(Liapuno)m(v-)150 3042 y(t)m(yp)s(e)43 b(function)f(of)f(the)47 b(~)-53 b Fu(\015)5 b Fz('s)43 b(whose)g(Heisen)m(b)s(erg)g(deriv)-5 b(ativ)m(e)42 b(ob)s(eys)h(a)f(certain)g(inequalit)m(y)f(\(see)150 3163 y(\(3.128\))32 b(for)g(the)h(classical)e(coun)m(terpart\).)296 3283 y(In)g(classical)f(mec)m(hanics)h(it)f(w)m(as)i(v)m(ery)g(easy)g (to)e(go)h(from)e(\(3.128\))h(to)h(\(3.129\))e(b)s(ecause)k(ev)m(ery-) 150 3404 y(thing)f(comm)m(utes.)43 b(In)33 b(quan)m(tum)g(mec)m(hanics) g(w)m(e)g(ha)m(v)m(e)h(to)f(b)s(e)f(more)g(careful)g(with)g (remainders.)150 3692 y Fp(7.1)135 b(A)45 b(quan)l(tum)g(v)l(ersion)h (for)f(\(3.128\))150 3877 y Fz(As)29 b(w)m(e)g(ha)m(v)m(e)h(said)e(b)s (efore,)h(w)m(e)h(closely)d(follo)m(w)g(the)i(steps)h(w)m(e)f(to)s(ok)f (in)f(the)i(pro)s(of)f(of)g(Lemma)e(3.14.)150 3998 y(Remem)m(b)s(er)32 b(the)h(\\third")f(gamma)e(w)m(e)k(de\014ned)g(in)d(\(6.85\))h(whic)m (h)h(should)g(replace)f(\(3.105\).)296 4118 y(W)-8 b(e)32 b(no)m(w)f(in)m(tro)s(duce)g(the)h(triplet)d(of)h(quan)m(tum)35 b(^)-52 b Fu(\015)5 b Fz('s)31 b(\(see)h(\(3.106\)\).)42 b(In)31 b(order)g(to)g(simplify)e(the)150 4238 y(writing)i(w)m(e)j (adopt)e(the)h(same)f(notation)g(for)g(them)g(as)h(in)f(the)h (classical)e(case.)44 b(De\014ne)232 4497 y(^)-52 b Fu(\015)280 4512 y Fs(1)346 4497 y Fz(:=)32 b(~)-53 b Fu(\015)528 4512 y Fs(1)589 4497 y Fr(\000)699 4430 y Fz(1)p 699 4474 49 4 v 699 4566 a(2)757 4497 y([\()p Fu(@)873 4512 y Ft(E)934 4497 y Fu(\032)984 4512 y Ft(E)1036 4520 y Fq(t)1068 4497 y Fz(\))22 b Fr(\001)j Fz(~)-53 b Fu(\015)1228 4512 y Fs(3)1290 4497 y Fz(+)22 b Fu(h:c:)p Fz(])p Fu(;)118 b Fz(^)-53 b Fu(\015)1759 4512 y Fs(2)1826 4497 y Fz(:=)31 b(~)-52 b Fu(\015)2008 4512 y Fs(2)2069 4497 y Fr(\000)2178 4430 y Fz(1)p 2178 4474 V 2178 4566 a(2)2237 4497 y([\()p Fu(@)2353 4512 y Ft(E)2413 4497 y Fu(\030)2456 4512 y Ft(E)2508 4520 y Fq(t)2540 4497 y Fz(\))22 b Fr(\001)j Fz(~)-52 b Fu(\015)2701 4512 y Fs(3)2762 4497 y Fz(+)22 b Fu(h:c:)p Fz(])p Fu(;)118 b Fz(^)-53 b Fu(\015)3231 4512 y Fs(3)3298 4497 y Fz(:=)31 b(~)-52 b Fu(\015)3480 4512 y Fs(3)3519 4497 y Fu(;)78 b Fz(\(7.3\))150 4743 y(where)34 b(as)g(usual,)f Fu(h:c:)h Fz(means)f(hermitian)e(conjugate.) 46 b(Notice)33 b(that)g(the)g(a)g(priori)e(smallness)i(w)m(e)150 4863 y(established)40 b(for)f(the)k(~)-53 b Fu(\015)5 b Fz('s)40 b(in)f(the)h(previous)g(section)g(can)f(b)s(e)h(easily)f (transferred)h(to)j(^)-53 b Fu(\015)5 b Fz('s)40 b(at)g(the)150 4983 y(exp)s(ense)d(of)e(in)m(tro)s(ducing)f(some)h(other)h (cut-o\013s.)52 b(That)35 b(is,)h(w)m(e)g(can)g(pro)m(v)m(e)h(the)e (existence)i(of)e Fu(M)150 5104 y Fz(large)c(enough)i(so)g(that)g(for)f (small)e Fu(\017)e(>)f Fz(0)33 b(w)m(e)g(ha)m(v)m(e)421 5324 y Fr(jj)p Fz([)p Fv(1)21 b Fr(\000)i Fu(F)758 5277 y Fm(\000)p Fs(2)p Ft(M)7 b(\017;)p Fm(\000)p Ft(M)g(\017;M)g(\017;)p Fs(2)p Ft(M)g(\017)744 5346 y Fs(+)p Fm(\000)1414 5324 y Fz(\()t(^)-53 b Fu(\015)1503 5339 y Ft(j)1539 5324 y Fz(\)])22 b Fr(\001)g Fu( )1739 5339 y Fs(10)1814 5324 y Fz(\()p Fu(t)p Fz(\))p Fr(jj)28 b Fz(=)f Fr(O)s Fz(\()p Fu(t)2267 5283 y Fm(\0001)2397 5324 y Fz(\))22 b Fr(\001)g(jj)p Fu( )t Fr(jj)p Fu(;)113 b(j)33 b Fr(2)c(f)p Fz(1)p Fu(;)17 b Fz(2)p Fu(;)g Fz(3)p Fr(g)p Fu(:)268 b Fz(\(7.4\))1939 5776 y(70)p eop %%Page: 71 74 71 73 bop 296 407 a Fz(Let)41 b(us)g(de\014ne)h(the)g(state)f(whic)m(h) g(also)f(tak)m(es)i(in)m(to)d(accoun)m(t)j(the)f(smallness)f(on)k(^)-52 b Fu(\015)3458 422 y Fs(1)3537 407 y Fz(and)45 b(^)-53 b Fu(\015)3786 422 y Fs(2)150 527 y Fz(\(remem)m(b)s(er)32 b(that)k(^)-52 b Fu(\015)906 542 y Fs(3)972 527 y Fz(=)31 b(~)-52 b Fu(\015)1127 542 y Fs(3)1166 527 y Fz(\):)835 697 y(^)813 722 y Fu(F)41 b Fz(:=)28 b Fu(F)1125 675 y Fm(\000)p Fs(4)p Ft(M)7 b(\017;)p Fm(\000)p Fs(3)p Ft(M)g(\017;)p Fs(3)p Ft(M)g(\017;)p Fs(4)p Ft(M)g(\017)1111 744 y Fs(+)p Fm(\000)1852 722 y Fu(;)114 b( )2056 737 y Fs(11)2131 722 y Fz(\()p Fu(t)p Fz(\))28 b(:=)2423 697 y(^)2401 722 y Fu(F)13 b Fz(\()t(^)-53 b Fu(\015)2566 737 y Fs(1)2605 722 y Fz(\))2665 697 y(^)2643 722 y Fu(F)14 b Fz(\()t(^)-53 b Fu(\015)2809 737 y Fs(2)2848 722 y Fz(\))p Fu( )2949 737 y Fs(10)3024 722 y Fz(\()p Fu(t)p Fz(\))p Fu(:)462 b Fz(\(7.5\))150 917 y(W)-8 b(e)33 b(ha)m(v)m(e)h(sho) m(wn)g Fu( )902 932 y Fs(11)977 917 y Fz(\()p Fu(t)p Fz(\))28 b Fr(\030)g Fu( )1284 932 y Fs(10)1359 917 y Fz(\()p Fu(t)p Fz(\))g Fr(\030)g(\001)17 b(\001)g(\001)26 b(\030)i Fu( )t Fz(\()p Fu(t)p Fz(\).)296 1037 y(W)-8 b(e)38 b(no)m(w)f(are)g(in)m(terested)h(in)e(writing)f(do)m(wn)j(the)f (Heisen)m(b)s(erg)h(deriv)-5 b(ativ)m(es)37 b(for)f(the)41 b(^)-53 b Fu(\015)5 b Fz('s)38 b(\(see)150 1158 y(\(6.36\)\).)k(As)32 b(a)f(general)g(rule,)g(p)s(erforming)e(the)j(Heisen)m(b)s(erg)g(deriv) -5 b(ativ)m(e)30 b(generates)j(\(up)e(to)g(some)150 1278 y(comm)m(utators\))38 b(the)h(same)g(result)g(as)h(if)d(the)j (computation)d(w)m(as)j(done)g(in)e(classical)g(mec)m(hanics)150 1398 y(b)m(y)33 b(p)s(erforming)e(the)i(time)e(deriv)-5 b(ativ)m(e)32 b(on)h(a)f(giv)m(en)h(classical)e(orbit.)150 1519 y Fv(Remark)p Fz(.)62 b(Since)39 b(at)f(the)i(end)f(w)m(e)h(will)d (apply)h(ev)m(erything)i(on)f(states)h(con)m(taining)d(all)g(\(elev)m (en)150 1639 y(b)m(y)43 b(no)m(w,)i(at)d(least\))g(cut-o\013s,)i(w)m(e) f(mak)m(e)g(the)f(con)m(v)m(en)m(tion)h(of)f(denoting)g(b)m(y)h Fr(O)s Fz(\()p Fu(t)3249 1603 y Fm(\000)p Ft(n)3351 1639 y Fz(\))f(an)m(y)g(term)150 1759 y(whic)m(h)i(con)m(tains)f(a)g(deca)m (y)i(of)e(order)h(1)p Fu(=r)1740 1723 y Ft(n)1786 1759 y Fz(,)i Fu(n)g Fr(\025)h Fz(1.)75 b(T)m(ypically)43 b(suc)m(h)i(terms)e(will)e(arise)i(from)150 1880 y(v)-5 b(arious)36 b(comm)m(utators)f(or)h(conserv)-5 b(ation)36 b(la)m(ws.)54 b(This)37 b(means)f(that)g(w)m(e)h(can)g(often)f(neglect) h(the)150 2000 y(non-comm)m(utativit)m(y)-8 b(.)150 2239 y Fv(Ia)p Fz(.)43 b(W)-8 b(e)30 b(start)f(with)g(the)h(constrain)m(ts)g (w)m(e)h(ha)m(v)m(e)g(on)e(the)34 b(^)-53 b Fu(\015)5 b Fz('s)30 b(coming)e(from)g(energy)j(conserv)-5 b(ation.)150 2359 y(The)34 b(\014rst)f(one)f(is)524 2554 y(2)p Fu(H)683 2528 y Fz(~)662 2554 y Fu(f)710 2569 y Ft(E)769 2554 y Fz(\()p Fu(H)8 b Fz(\))83 b(=)92 b(~)-58 b Fu(\032)1226 2513 y Fs(2)1288 2554 y Fz(+)1396 2528 y(~)1386 2554 y Fu(\030)1434 2513 y Fs(2)1495 2554 y Fz(+)22 b Fr(O)s Fz(\()p Fu(t)1748 2513 y Fm(\000)p Fs(2)1842 2554 y Fz(\))28 b(=)f(\()t(~)-53 b Fu(\015)2100 2569 y Fs(1)2161 2554 y Fz(+)22 b Fu(\032)2309 2569 y Ft(E)2361 2577 y Fq(t)2394 2554 y Fz(\))2432 2513 y Fs(2)2493 2554 y Fz(+)g(\()t(~)-53 b Fu(\015)2680 2569 y Fs(2)2741 2554 y Fz(+)22 b Fu(\030)2882 2569 y Ft(E)2934 2577 y Fq(t)2966 2554 y Fz(\))3004 2513 y Fs(2)3065 2554 y Fz(+)g Fr(O)s Fz(\()p Fu(t)3318 2513 y Fm(\000)p Fs(2)3413 2554 y Fz(\))1017 2699 y(=)83 b(2)p Fu(E)1297 2714 y Ft(t)1348 2699 y Fz(+)22 b(\()t(~)-53 b Fu(\015)1535 2714 y Fs(1)1597 2699 y Fr(\001)21 b Fu(\032)1696 2714 y Ft(E)1748 2722 y Fq(t)1803 2699 y Fz(+)k(~)-52 b Fu(\015)1952 2714 y Fs(2)2013 2699 y Fr(\001)22 b Fu(\030)2106 2714 y Ft(E)2158 2722 y Fq(t)2211 2699 y Fz(+)g Fu(h:c:)p Fz(\))h(+)f Fr(O)s Fz(\()t(~)-53 b Fu(\015)2796 2658 y Fs(2)2791 2724 y(1)2835 2699 y Fu(;)20 b Fz(~)-52 b Fu(\015)2935 2658 y Fs(2)2930 2724 y(2)2974 2699 y Fu(;)17 b(t)3053 2658 y Fm(\000)p Fs(2)3147 2699 y Fz(\))p Fu(:)412 b Fz(\(7.6\))150 2894 y(Let)32 b(us)g(commen)m(t)e(a)h(bit)g(these)i (equalities)d(whic)m(h)i(migh)m(t)d(lo)s(ok)i(strange)g(at)g(\014rst)h (sigh)m(t.)43 b(First,)31 b(as)150 3014 y(w)m(e)e(men)m(tioned)f(in)g (the)h(ab)s(o)m(v)m(e)g(remark,)g(the)g(cut-o\013s)f(en)m(tering)g(the) 33 b(^)-53 b Fu(\015)5 b Fz('s)29 b(w)m(ere)h(neglected)f(b)s(ecause) 150 3135 y(they)k(only)f(con)m(tribute)g(with)g Fr(O)s Fz(\()p Fu(t)1428 3099 y Fm(\0001)1557 3135 y Fz(\))g(when)i(faced)e (with)g(the)h(\\thinner")e(cut-o\013s)h(building)e(our)150 3255 y(state.)50 b(Second,)36 b(the)f(\014rst)g(equalit)m(y)f(con)m (tains)g Fr(O)s Fz(\()p Fu(t)2078 3219 y Fm(\000)p Fs(2)2173 3255 y Fz(\))g(whic)m(h)h(tak)m(es)h(in)m(to)e(accoun)m(t)h(\(see)h (\(1.19\)\))150 3375 y(that)30 b(\\2)p Fu(H)24 b Fr(\000)17 b Fu(\032)706 3339 y Fs(2)763 3375 y Fr(\000)g Fu(\030)905 3339 y Fs(2)972 3375 y Fr(\030)28 b Fz(1)p Fu(=)p 1175 3321 47 4 v(r)1221 3339 y Fs(2)1261 3375 y Fz(".)42 b(If)30 b Fu(A)g Fz(and)g Fu(B)35 b Fz(are)30 b(b)s(ounded)h(op)s(erators,)f(b) m(y)h Fr(O)s Fz(\()p Fu(A;)17 b(B)5 b Fz(\))30 b(w)m(e)h(denote)150 3496 y(an)m(y)k(\014nite)e(linear)f(com)m(bination)g(of)i(pro)s(ducts)g (of)g(b)s(ounded)g(op)s(erators)g(con)m(taining)f(either)g Fu(A)h Fz(or)150 3616 y Fu(B)47 b Fz(as)42 b(factors.)71 b(The)43 b(op)s(erators)e(en)m(tering)h(these)h(pro)s(ducts)f(will)e (ha)m(v)m(e)j(\\go)s(o)s(d")d(comm)m(utation)150 3737 y(prop)s(erties.)296 3857 y(An)33 b(equiv)-5 b(alen)m(t)33 b(expression)g(is)969 4052 y(2)t(~)-53 b Fu(\015)1069 4067 y Fs(3)1136 4052 y Fz(=)28 b(\()t(~)-53 b Fu(\015)1329 4067 y Fs(1)1390 4052 y Fr(\001)21 b Fu(\032)1489 4067 y Ft(E)1541 4075 y Fq(t)1596 4052 y Fz(+)k(~)-52 b Fu(\015)1745 4067 y Fs(2)1806 4052 y Fr(\001)22 b Fu(\030)1899 4067 y Ft(E)1951 4075 y Fq(t)2004 4052 y Fz(+)g Fu(h:c:)p Fz(\))h(+)f Fr(O)s Fz(\()t(~)-53 b Fu(\015)2589 4011 y Fs(2)2584 4076 y(1)2628 4052 y Fu(;)20 b Fz(~)-52 b Fu(\015)2728 4011 y Fs(2)2723 4076 y(2)2767 4052 y Fu(;)17 b(t)2846 4011 y Fm(\000)p Fs(2)2940 4052 y Fz(\))p Fu(:)619 b Fz(\(7.7\))296 4247 y(Rewriting)39 b(\(7.7\))h(with)k(^)-53 b Fu(\015)5 b Fz('s)41 b(w)m(e)g(again)e(get)i(an)f(\\almost")e(linear) h(dep)s(endence)k(b)s(et)m(w)m(een)j(^)-53 b Fu(\015)3786 4262 y Fs(1)150 4367 y Fz(and)36 b(^)-52 b Fu(\015)391 4382 y Fs(2)430 4367 y Fz(;)32 b(one)h(of)f(the)h(ingredien)m(ts)g(is)f (the)h(iden)m(tit)m(y)f(1)22 b Fr(\000)h Fu(\032)2291 4382 y Ft(E)2351 4367 y Fu(@)2402 4382 y Ft(E)2462 4367 y Fu(\032)2512 4382 y Ft(E)2599 4367 y Fz(=)28 b Fu(\030)2746 4382 y Ft(E)2805 4367 y Fu(@)2856 4382 y Ft(E)2916 4367 y Fu(\030)2959 4382 y Ft(E)3018 4367 y Fz(:)973 4562 y(2)t(^)-53 b Fu(\015)1073 4577 y Fs(1)1134 4562 y Fr(\001)22 b Fu(\032)1234 4577 y Ft(E)1286 4585 y Fq(t)1340 4562 y Fz(+)g(2)t(^)-53 b Fu(\015)1538 4577 y Fs(2)1599 4562 y Fr(\001)22 b Fu(\030)1692 4577 y Ft(E)1744 4585 y Fq(t)1803 4562 y Fz(=)27 b Fr(O)s Fz(\()t(^)-53 b Fu(\015)2082 4521 y Fs(2)2077 4586 y Ft(j)2122 4562 y Fu(;)17 b(t)2201 4521 y Fm(\000)p Fs(1)2295 4562 y Fz(\))p Fu(;)114 b(j)34 b Fr(2)28 b(f)p Fz(1)p Fu(;)17 b Fz(2)p Fu(;)g Fz(3)p Fr(g)p Fu(:)620 b Fz(\(7.8\))150 4757 y(where)34 b(w)m(e)f(c)m(hose)h (to)f(drop)f(the)h(symmetric)f(form)f(at)i(the)g(exp)s(ense)h(of)e(an)h (extra)g Fr(O)s Fz(\()p Fu(t)3358 4720 y Fm(\000)p Fs(1)3452 4757 y Fz(\))g(error.)150 4995 y Fv(Ib)p Fz(.)48 b(Before)35 b(starting)e(to)h(compute)g(v)-5 b(arious)33 b(Heisen)m(b)s(erg)i (deriv)-5 b(ativ)m(es)34 b(w)m(e)i(ha)m(v)m(e)f(to)f(sp)s(end)h(some) 150 5115 y(time)30 b(studying)i(the)h(comm)m(utation)c(prop)s(erties)j (of)f(our)h Fu(\015)5 b Fz('s.)44 b(T)-8 b(ec)m(hnically)31 b(the)h(follo)m(wing)d(formal)150 5236 y(iden)m(tit)m(y)j(will)f(b)s(e) h(v)m(ery)i(imp)s(ortan)m(t:)877 5482 y Fu(i)927 5342 y Fl(\024)980 5482 y Fr(\000)p Fu(i@)1141 5497 y Ft(r)1202 5482 y Fr(\000)23 b Fu(@)1353 5497 y Ft(r)1391 5482 y Fu(S;)17 b Fr(\000)1590 5415 y Fu(i)p 1582 5459 V 1582 5551 a(r)1639 5482 y(@)1690 5497 y Ft(\022)1752 5482 y Fr(\000)1862 5415 y Fu(@)1913 5430 y Ft(\022)1952 5415 y Fu(S)p 1862 5459 157 4 v 1917 5551 a(r)2028 5342 y Fl(\025)2108 5482 y Fz(=)28 b Fr(\000)2299 5415 y Fz(1)p 2299 5459 49 4 v 2300 5551 a Fu(r)2375 5342 y Fl(\022)2448 5482 y Fr(\000)2542 5415 y Fu(i)p 2535 5459 47 4 v 2535 5551 a(r)2592 5482 y(@)2643 5497 y Ft(\022)2705 5482 y Fr(\000)2814 5415 y Fu(@)2865 5430 y Ft(\022)2905 5415 y Fu(S)p 2814 5459 157 4 v 2869 5551 a(r)2981 5342 y Fl(\023)3070 5482 y Fu(:)1939 5776 y Fz(71)p eop %%Page: 72 75 72 74 bop 150 407 a Fz(It)33 b(implies)1052 527 y Fu(i)p Fz([)t(~)-53 b Fu(\015)1163 542 y Fs(1)1203 527 y Fu(;)20 b Fz(~)-52 b Fu(\015)1298 542 y Fs(2)1336 527 y Fz(])28 b Fr(')g(\000)p Fu(r)1620 486 y Fm(\000)p Fs(1)1735 527 y Fz(~)-69 b Fu(\015)1766 542 y Fs(2)1827 527 y Fz(+)22 b Fr(O)2024 446 y Fl(\000)2069 527 y Fu(t)2104 486 y Fm(\000)p Fs(2)2199 446 y Fl(\001)2272 527 y Fz(=)28 b Fr(O)2474 446 y Fl(\000)2524 527 y Fz(~)-53 b Fu(\015)5 b(=t;)17 b(t)2739 486 y Fm(\000)p Fs(2)2833 446 y Fl(\001)2896 527 y Fu(:)701 b Fz(\(7.9\))150 691 y(As)43 b(a)f(consequence)k(of)c (\(7.9\),)i(b)m(y)g(comm)m(uting)g(~)-53 b Fu(\015)2062 706 y Fs(1)2144 691 y Fz(or)46 b(~)-53 b Fu(\015)2324 706 y Fs(2)2406 691 y Fz(with)45 b(~)-52 b Fu(\015)2689 706 y Fs(3)2770 691 y Fz(expressed)45 b(as)e(in)f(\(7.7\))g(w)m(e)150 812 y(either)37 b(gain)f(an)h(extra)g(1)p Fu(=t)g Fz(and)g(k)m(eep)i (the)f(n)m(um)m(b)s(er)f(of)k(~)-53 b Fu(\015)2333 827 y Ft(i)2361 812 y Fz('s)37 b(as)h(b)s(efore)f(or)g(w)m(e)h(gain)e(a)h (1)p Fu(=t)3592 775 y Fs(2)3668 812 y Fz(and)150 932 y(lo)m(w)m(er)c(the)g(n)m(um)m(b)s(er)g(of)i(~)-52 b Fu(\015)1090 947 y Ft(i)1118 932 y Fz('s)33 b(b)m(y)g(one.)44 b(Hence)34 b(w)m(e)f(can)g(write)1132 1128 y Fu(i)p Fz([)t(~)-53 b Fu(\015)1243 1143 y Ft(j)1280 1128 y Fu(;)20 b Fz(~)-52 b Fu(\015)1375 1143 y Fs(3)1413 1128 y Fz(])28 b(=)g Fr(O)s Fz(\()t(~)-53 b Fu(\015)5 b(=t;)20 b Fz(~)-52 b Fu(\015)1932 1087 y Fs(2)1971 1128 y Fu(=t;)17 b(t)2134 1087 y Fm(\000)p Fs(3)2228 1128 y Fz(\))p Fu(;)114 b(j)34 b Fr(2)28 b(f)p Fz(1)p Fu(;)17 b Fz(2)p Fr(g)p Fu(:)732 b Fz(\(7.10\))150 1324 y(And)33 b(th)m(us)449 1520 y Fu(i)p Fz([)t(~)-53 b Fu(\015)560 1535 y Ft(j)596 1520 y Fu(;)20 b Fz(~)-52 b Fu(\015)691 1535 y Ft(k)733 1520 y Fz(])28 b(=)f Fr(O)990 1439 y Fl(\000)1039 1520 y Fz(~)-52 b Fu(\015)5 b(=t;)17 b(t)1255 1479 y Fm(\000)p Fs(2)1349 1439 y Fl(\001)1411 1520 y Fu(;)115 b(i)p Fz([)t(^)-53 b Fu(\015)1664 1535 y Ft(j)1700 1520 y Fu(;)20 b Fz(^)-52 b Fu(\015)1795 1535 y Ft(k)1837 1520 y Fz(])28 b(=)f Fr(O)2094 1439 y Fl(\000)2143 1520 y Fz(^)-52 b Fu(\015)5 b(=t;)17 b(t)2359 1479 y Fm(\000)p Fs(2)2453 1439 y Fl(\001)2515 1520 y Fu(;)114 b(j;)17 b(k)31 b Fr(2)e(f)p Fz(1)p Fu(;)17 b Fz(2)p Fu(;)g Fz(3)p Fr(g)p Fu(:)297 b Fz(\(7.11\))150 1834 y Fv(I)s(I)p Fz(.)29 b(W)-8 b(e)30 b(con)m(tin)m(ue)g(with)f(the)h (Heisen)m(b)s(erg)g(deriv)-5 b(ativ)m(e)29 b(of)k(^)-53 b Fu(\015)2325 1849 y Fs(3)2364 1834 y Fz(.)42 b(W)-8 b(rite)29 b(the)h(equation)f(whic)m(h)h(de\014nes)150 1955 y Fu(E)222 1970 y Ft(t)1672 2075 y Fu(t=r)g Fz(=)e Fu(@)1985 2090 y Ft(E)2045 2075 y Fu(\032)2095 2090 y Ft(E)2147 2098 y Fq(t)2179 2075 y Fz(\()p Fu(\022)s Fz(\))1273 b(\(7.12\))150 2239 y(and)36 b(compute)g(the)g(Heisen)m(b)s(erg)h (deriv)-5 b(ativ)m(e)35 b(on)h(b)s(oth)g(sides)g(\(notice)g(that)f Fu(D)3097 2254 y Ft(H)3168 2239 y Fz(^)-52 b Fu(\015)3216 2254 y Fs(3)3288 2239 y Fz(=)33 b Fr(\000)p Fu(D)3555 2254 y Ft(H)3623 2239 y Fu(E)3695 2254 y Ft(t)3749 2239 y Fz(+)150 2359 y Fr(O)s Fz(\()p Fu(t)305 2323 y Fm(\0001)435 2359 y Fz(\)\):)378 2555 y(1)p Fu(=r)25 b Fr(\000)d Fz(\()p Fu(t=r)813 2514 y Fs(2)852 2555 y Fz(\))9 b(~)-58 b Fu(\032)23 b Fz(+)f Fr(O)s Fz(\()p Fu(t)1216 2514 y Fm(\000)p Fs(2)1310 2555 y Fz(\))28 b(=)g Fr(\000)p Fz(\()p Fu(@)1651 2514 y Fs(2)1646 2580 y Ft(E)1706 2555 y Fu(\032)1756 2570 y Ft(E)1808 2578 y Fq(t)1840 2555 y Fz(\))22 b Fr(\001)g Fz(\()p Fu(D)2069 2570 y Ft(H)2140 2555 y Fz(^)-53 b Fu(\015)2187 2570 y Fs(3)2226 2555 y Fz(\))22 b(+)g(\()p Fu(@)2473 2570 y Ft(E)2533 2555 y Fu(\030)2576 2570 y Ft(E)2628 2578 y Fq(t)2660 2555 y Fu(=r)s Fz(\))f Fr(\001)2875 2529 y Fz(~)2865 2555 y Fu(\030)27 b Fz(+)22 b Fr(O)s Fz(\()p Fu(t)3188 2514 y Fm(\000)p Fs(2)3282 2555 y Fz(\))p Fu(:)229 b Fz(\(7.13\))150 2752 y(Emplo)m(ying)31 b(1)22 b Fr(\000)h Fu(\032)860 2767 y Ft(E)920 2752 y Fu(@)971 2767 y Ft(E)1031 2752 y Fu(\032)1081 2767 y Ft(E)1168 2752 y Fz(=)28 b Fu(\030)1315 2767 y Ft(E)1374 2752 y Fu(@)1425 2767 y Ft(E)1485 2752 y Fu(\030)1528 2767 y Ft(E)1620 2752 y Fz(again)j(w)m(e)i(are)g(led)f(to)656 2986 y Fu(D)737 3001 y Ft(H)808 2986 y Fz(^)-53 b Fu(\015)855 3001 y Fs(3)922 2986 y Fz(=)27 b Fu(D)1106 3001 y Ft(H)1177 2986 y Fz(~)-53 b Fu(\015)1224 3001 y Fs(3)1291 2986 y Fz(=)1526 2919 y(1)p 1405 2964 292 4 v 1405 3055 a Fu(r)s(@)1508 3020 y Fs(2)1503 3082 y Ft(E)1562 3055 y Fu(\032)1612 3070 y Ft(E)1664 3078 y Fq(t)1706 2986 y Fr(f)p Fz(\()p Fu(@)1845 3001 y Ft(E)1905 2986 y Fu(\032)1955 3001 y Ft(E)2007 3009 y Fq(t)2039 2986 y Fz(\))22 b Fr(\001)k Fz(~)-53 b Fu(\015)2200 3001 y Fs(1)2261 2986 y Fz(+)22 b(\()p Fu(@)2448 3001 y Ft(E)2508 2986 y Fu(\030)2551 3001 y Ft(E)2603 3009 y Fq(t)2635 2986 y Fz(\))g Fr(\001)j Fz(~)-52 b Fu(\015)2796 3001 y Fs(2)2835 2986 y Fr(g)22 b Fz(+)g Fr(O)s Fz(\()p Fu(t)3160 2945 y Fm(\000)p Fs(2)3254 2986 y Fz(\))p Fu(;)257 b Fz(\(7.14\))150 3240 y(and)33 b(\014nally)-8 b(,)31 b(using)h(\(7.12\))g(in)g(order)h(to)f(get)g(rid) g(of)g Fu(r)j Fz(on)e(the)g(righ)m(t)f(hand)h(side:)604 3486 y Fu(D)685 3501 y Ft(H)756 3486 y Fz(^)-53 b Fu(\015)803 3501 y Fs(3)925 3486 y Fz(=)1112 3418 y Fu(@)1163 3433 y Ft(E)1223 3418 y Fu(\032)1273 3433 y Ft(E)1325 3441 y Fq(t)p 1094 3463 281 4 v 1094 3554 a Fu(t@)1185 3520 y Fs(2)1180 3581 y Ft(E)1241 3554 y Fu(\032)1291 3569 y Ft(E)1343 3577 y Fq(t)1385 3486 y Fr(f)p Fz(\()p Fu(@)1524 3501 y Ft(E)1583 3486 y Fu(\032)1633 3501 y Ft(E)1685 3509 y Fq(t)1718 3486 y Fz(\))22 b Fr(\001)j Fz(^)-53 b Fu(\015)1878 3501 y Fs(1)1940 3486 y Fz(+)22 b(\()p Fu(@)2127 3501 y Ft(E)2187 3486 y Fu(\030)2230 3501 y Ft(E)2282 3509 y Fq(t)2313 3486 y Fz(\))g Fr(\001)k Fz(^)-53 b Fu(\015)2474 3501 y Fs(2)2513 3486 y Fr(g)22 b Fz(+)2711 3418 y Fu(@)2762 3433 y Ft(E)2822 3418 y Fu(\032)2872 3433 y Ft(E)2924 3441 y Fq(t)p 2693 3463 V 2693 3554 a Fu(t@)2784 3520 y Fs(2)2779 3581 y Ft(E)2839 3554 y Fu(\032)2889 3569 y Ft(E)2941 3577 y Fq(t)2983 3486 y Fz([\()p Fu(@)3099 3501 y Ft(E)3159 3486 y Fu(\032)3209 3501 y Ft(E)3261 3509 y Fq(t)3293 3486 y Fz(\))3331 3445 y Fs(2)925 3698 y Fz(+)83 b(\()p Fu(@)1173 3713 y Ft(E)1233 3698 y Fu(\030)1276 3713 y Ft(E)1328 3721 y Fq(t)1360 3698 y Fz(\))1398 3657 y Fs(2)1437 3698 y Fz(])22 b Fr(\001)k Fz(^)-53 b Fu(\015)1587 3713 y Fs(3)1648 3698 y Fz(+)22 b Fr(O)s Fz(\()p Fu(t)1901 3657 y Fm(\000)p Fs(2)1996 3698 y Fz(\))p Fu(:)1515 b Fz(\(7.15\))150 4013 y Fv(I)s(I)s(I)p Fz(.)32 b(Next)h(comes)g(the)g(Heisen)m(b)s(erg)h(deriv)-5 b(ativ)m(e)32 b(of)j(^)-52 b Fu(\015)2141 4028 y Fs(1)2180 4013 y Fz(:)530 4209 y Fu(D)611 4224 y Ft(H)682 4209 y Fz(^)g Fu(\015)730 4224 y Fs(1)852 4209 y Fz(=)83 b Fu(D)1092 4224 y Ft(H)1162 4209 y Fz(~)-52 b Fu(\015)1210 4224 y Fs(1)1271 4209 y Fr(\000)23 b Fz(\()p Fu(@)1460 4224 y Ft(E)1520 4209 y Fu(\030)1563 4224 y Ft(E)1615 4232 y Fq(t)1646 4209 y Fu(=r)s Fz(\))f Fr(\001)1862 4182 y Fz(~)1851 4209 y Fu(\030)27 b Fr(\001)e Fz(^)-52 b Fu(\015)2022 4224 y Fs(3)2083 4209 y Fr(\000)23 b Fz(\()p Fu(@)2272 4224 y Ft(E)2332 4209 y Fu(\032)2382 4224 y Ft(E)2434 4232 y Fq(t)2466 4209 y Fz(\))f Fr(\001)g Fu(D)2657 4224 y Ft(H)2727 4209 y Fz(~)-52 b Fu(\015)2775 4224 y Fs(3)2836 4209 y Fz(+)22 b Fr(O)s Fz(\()t(^)-53 b Fu(\015)3110 4167 y Fs(2)3149 4209 y Fu(=t;)17 b(t)3312 4167 y Fm(\000)p Fs(2)3407 4209 y Fz(\))852 4354 y(=)83 b(\()p Fu(\030)1092 4369 y Ft(E)1144 4377 y Fq(t)1175 4354 y Fu(=r)s Fz(\))21 b Fr(\001)26 b Fz(^)-53 b Fu(\015)1431 4369 y Fs(2)1492 4354 y Fz(+)22 b Fr(O)s Fz(\()t(^)-53 b Fu(\015)1766 4313 y Fs(2)1806 4354 y Fu(=t;)17 b(t)1969 4313 y Fm(\000)p Fs(2)2063 4354 y Fz(\))852 4499 y(=)83 b Fr(\000)p Fz([)p Fu(\032)1165 4514 y Ft(E)1217 4522 y Fq(t)1271 4499 y Fr(\001)22 b Fz(\()p Fu(@)1410 4514 y Ft(E)1470 4499 y Fu(\032)1520 4514 y Ft(E)1572 4522 y Fq(t)1604 4499 y Fz(\))p Fu(=t)p Fz(])g Fr(\001)k Fz(^)-53 b Fu(\015)1876 4514 y Fs(1)1937 4499 y Fz(+)22 b Fr(O)s Fz(\()t(^)-53 b Fu(\015)2211 4458 y Fs(2)2251 4499 y Fu(=t;)17 b(t)2414 4458 y Fm(\000)p Fs(2)2508 4499 y Fz(\))p Fu(;)1003 b Fz(\(7.16\))150 4695 y(where)34 b(the)f(last)f(equalit)m(y)g(came)g (from)f(\(7.8\))i(and)f(\(7.12\).)43 b(As)33 b(in)f(the)h(classical)e (case,)i(de\014ne)1485 4936 y Fu(f)1533 4951 y Fs(1)1572 4936 y Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))28 b(:=)2126 4868 y(1)p 2028 4913 246 4 v 2028 5004 a Fu(@)2079 5019 y Ft(E)2139 5004 y Fu(\032)2189 5019 y Ft(E)2241 5027 y Fq(t)2311 4936 y Fu(>)g Fz(0)p Fu(:)1085 b Fz(\(7.17\))150 5186 y(W)-8 b(e)33 b(see)h(that)e(its)g(Heisen)m(b)s(erg)h(deriv)-5 b(ativ)m(e)33 b(giv)m(es)g(\(use)g(\(7.12\))f(and)h(\(7.14\)\):)550 5382 y Fu(D)631 5397 y Ft(H)698 5382 y Fu(f)746 5397 y Fs(1)869 5382 y Fz(=)82 b(\()p Fu(@)1121 5341 y Fs(2)1116 5407 y Ft(E)1176 5382 y Fu(\032)1226 5397 y Ft(E)1278 5405 y Fq(t)1311 5382 y Fz(\))p Fu(=)p Fz(\()p Fu(@)1487 5397 y Ft(E)1546 5382 y Fu(\032)1596 5397 y Ft(E)1648 5405 y Fq(t)1680 5382 y Fz(\))1718 5341 y Fs(2)1780 5382 y Fr(\001)22 b Fz(\()p Fu(D)1949 5397 y Ft(H)2019 5382 y Fz(^)-52 b Fu(\015)2067 5397 y Fs(3)2106 5382 y Fz(\))22 b Fr(\000)h Fz(\()p Fu(f)2352 5397 y Fs(1)2391 5382 y Fu(=t)p Fz(\))f Fr(\001)g Fz(\()p Fu(@)2674 5397 y Ft(E)2734 5382 y Fu(\030)2777 5397 y Ft(E)2829 5405 y Fq(t)2860 5382 y Fz(\))g Fr(\001)2980 5356 y Fz(~)2970 5382 y Fu(\030)27 b Fz(+)22 b Fr(O)s Fz(\()p Fu(t)3293 5341 y Fm(\000)p Fs(2)3387 5382 y Fz(\))869 5527 y(=)82 b Fr(\000)p Fu(f)1152 5542 y Fs(1)1215 5527 y Fr(\001)21 b Fz(\(1)h Fr(\000)h Fu(\032)1523 5542 y Ft(E)1575 5550 y Fq(t)1607 5527 y Fu(@)1658 5542 y Ft(E)1718 5527 y Fu(\032)1768 5542 y Ft(E)1820 5550 y Fq(t)1852 5527 y Fz(\))p Fu(=t)f Fz(+)g Fr(O)s Fz(\()t(^)-53 b Fu(\015)5 b(=t;)17 b(t)2433 5486 y Fm(\000)p Fs(2)2527 5527 y Fz(\))p Fu(:)984 b Fz(\(7.18\))1939 5776 y(72)p eop %%Page: 73 76 73 75 bop 150 407 a Fz(Com)m(bining)31 b(this)h(with)g(\(7.16\))g(w)m (e)i(obtain)1122 627 y Fu(D)1203 642 y Ft(H)1271 627 y Fz(\()p Fu(f)1357 642 y Fs(1)1400 627 y Fz(^)-53 b Fu(\015)1447 642 y Fs(1)1486 627 y Fz(\))28 b(=)f Fr(\000)p Fz(\(1)p Fu(=t)p Fz(\)\()p Fu(f)2027 642 y Fs(1)2070 627 y Fz(^)-52 b Fu(\015)2118 642 y Fs(1)2157 627 y Fz(\))22 b(+)g Fr(O)s Fz(\()t(^)-53 b Fu(\015)2491 586 y Fs(2)2530 627 y Fu(=t;)17 b(t)2693 586 y Fm(\000)p Fs(2)2788 627 y Fz(\))p Fu(:)723 b Fz(\(7.19\))150 965 y Fv(IV)p Fz(.)32 b(No)m(w)h(w)m(e)h(compute)e(the)h(Heisen)m(b)s(erg)h(deriv)-5 b(ativ)m(e)32 b(of)k(^)-53 b Fu(\015)2371 980 y Fs(2)2410 965 y Fz(:)1033 1229 y Fu(D)1114 1244 y Ft(H)1185 1229 y Fz(^)g Fu(\015)1232 1244 y Fs(2)1299 1229 y Fz(=)28 b Fr(\000)1490 1161 y Fu(b)23 b Fz(+)f Fu(\030)1695 1176 y Ft(E)1747 1184 y Fq(t)p 1490 1206 289 4 v 1611 1297 a Fu(r)1792 1229 y Fz(^)-53 b Fu(\015)1839 1244 y Fs(1)1900 1229 y Fz(+)2008 1161 y Fu(b\032)2099 1176 y Ft(E)2151 1184 y Fq(t)p 2008 1206 176 4 v 2009 1297 a Fu(r)s(\030)2099 1312 y Ft(E)2151 1320 y Fq(t)2197 1229 y Fz(^)h Fu(\015)2245 1244 y Fs(2)2306 1229 y Fz(+)22 b Fr(O)s Fz(\()t(^)-53 b Fu(\015)2580 1188 y Fs(2)2619 1229 y Fu(=t;)17 b(t)2782 1188 y Fm(\000)p Fs(2)2877 1229 y Fz(\))p Fu(:)634 b Fz(\(7.20\))150 1498 y(W)-8 b(e)44 b(remark)g(that)f(the)i(r.h.s.)78 b(of)43 b(the)i(ab)s(o)m(v)m(e)f(equation)g(con)m(tains)j(^)-52 b Fu(\015)2809 1513 y Fs(3)2892 1498 y Fz(only)43 b(in)g(the)h (quadratic)150 1618 y(remainder.)f(Then)1202 1763 y Fu(D)1283 1778 y Ft(H)1350 1763 y Fz([)p Fu(\030)1420 1778 y Ft(E)1472 1786 y Fq(t)1504 1763 y Fz(])28 b(=)f Fr(\000)1749 1695 y Fu(b)c Fz(+)f Fu(\030)1954 1710 y Ft(E)2006 1718 y Fq(t)p 1750 1740 289 4 v 1870 1831 a Fu(r)2047 1763 y(\032)2097 1778 y Ft(E)2149 1786 y Fq(t)2204 1763 y Fz(+)g Fr(O)s Fz(\()t(^)-53 b Fu(\015)5 b(=t;)17 b(t)2641 1722 y Fm(\000)p Fs(2)2735 1763 y Fz(\))150 1963 y(and)33 b(with)f(the)h(in)m(tegrating) e(factor)1473 2183 y Fu(f)1521 2198 y Fs(2)1560 2183 y Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))28 b(:=)f Fu(\030)2049 2198 y Ft(E)2101 2206 y Fq(t)2133 2183 y Fu(=)p Fz(\()p Fu(@)2271 2198 y Ft(E)2330 2183 y Fu(\032)2380 2198 y Ft(E)2432 2206 y Fq(t)2465 2183 y Fz(\))1073 b(\(7.21\))150 2403 y(w)m(e)34 b(obtain)396 2647 y Fu(D)477 2662 y Ft(H)545 2647 y Fz(\()p Fu(f)631 2662 y Fs(2)674 2647 y Fz(^)-53 b Fu(\015)721 2662 y Fs(2)760 2647 y Fz(\))28 b(=)f Fr(\000)1016 2580 y Fu(b)c Fz(+)f Fu(\030)1221 2595 y Ft(E)1273 2603 y Fq(t)p 1017 2624 V 1143 2715 a Fu(t)1315 2647 y Fz(\()p Fu(@)1404 2662 y Ft(E)1464 2647 y Fu(\032)1514 2662 y Ft(E)1566 2670 y Fq(t)1598 2647 y Fz(\))p Fu(\030)1679 2662 y Ft(E)1731 2670 y Fq(t)1784 2647 y Fr(\001)g Fz(\()p Fu(f)1920 2662 y Fs(1)1963 2647 y Fz(^)-52 b Fu(\015)2011 2662 y Fs(1)2050 2647 y Fz(\))22 b Fr(\000)g Fz(\(1)p Fu(=t)p Fz(\)\()p Fu(f)2504 2662 y Fs(2)2547 2647 y Fz(^)-53 b Fu(\015)2594 2662 y Fs(2)2633 2647 y Fz(\))23 b(+)f Fr(O)s Fz(\()t(^)-53 b Fu(\015)2968 2606 y Fs(2)3007 2647 y Fu(=t;)17 b(t)3170 2606 y Fm(\000)p Fs(2)3264 2647 y Fz(\))p Fu(:)247 b Fz(\(7.22\))296 2898 y(No)m(w)33 b(w)m(e)h(are)f(ready)g(to)f(rewrite)h(\(7.15\))f(in)f(a)i(more)f(con)m (v)m(enien)m(t)i(form.)42 b(De\014ne)1374 3118 y Fu(f)1422 3133 y Fs(3)1462 3118 y Fz(\()p Fu(t;)17 b(r)m(;)g(\022)s Fz(\))27 b(=)h Fr(\000)p Fz(\()p Fu(@)2052 3077 y Fs(2)2047 3143 y Ft(E)2108 3118 y Fu(\032)2158 3133 y Ft(E)2210 3141 y Fq(t)2242 3118 y Fu(=@)2342 3133 y Ft(E)2402 3118 y Fu(\032)2452 3133 y Ft(E)2504 3141 y Fq(t)2536 3118 y Fz(\))p Fu(:)975 b Fz(\(7.23\))150 3338 y(Using)48 b(the)h(iden)m(tit)m(y)f(\()p Fu(@)1089 3353 y Ft(E)1149 3338 y Fu(\030)1192 3353 y Ft(E)1244 3361 y Fq(t)1275 3338 y Fz(\))1313 3302 y Fs(2)1385 3338 y Fz(+)33 b(\()p Fu(@)1583 3353 y Ft(E)1643 3338 y Fu(\032)1693 3353 y Ft(E)1745 3361 y Fq(t)1777 3338 y Fz(\))1815 3302 y Fs(2)1909 3338 y Fz(=)54 b Fr(\000)p Fu(\032)2166 3353 y Ft(E)2218 3361 y Fq(t)2251 3338 y Fu(@)2307 3302 y Fs(2)2302 3363 y Ft(E)2362 3338 y Fu(\032)2412 3353 y Ft(E)2464 3361 y Fq(t)2529 3338 y Fr(\000)33 b Fu(\030)2682 3353 y Ft(E)2734 3361 y Fq(t)2765 3338 y Fu(@)2821 3302 y Fs(2)2816 3363 y Ft(E)2877 3338 y Fu(\030)2920 3353 y Ft(E)2972 3361 y Fq(t)3051 3338 y Fz(together)49 b(with)f(the)150 3458 y(\\linear")31 b(dep)s(endence)j(\(7.8\))f(w)m(e)g(obtain)750 3678 y Fu(D)831 3693 y Ft(H)898 3678 y Fz(\()p Fu(f)984 3693 y Fs(3)1027 3678 y Fz(^)-52 b Fu(\015)1075 3693 y Fs(3)1114 3678 y Fz(\))27 b(=)h([)p Fu(a)1361 3693 y Fs(31)1436 3678 y Fz(\()p Fu(t)p Fz(\))p Fu(=t)p Fz(])22 b Fr(\001)g Fz(\()p Fu(f)1816 3693 y Fs(1)1859 3678 y Fz(^)-53 b Fu(\015)1906 3693 y Fs(1)1945 3678 y Fz(\))23 b Fr(\000)f Fz(\(1)p Fu(=t)p Fz(\)\()p Fu(f)2400 3693 y Fs(3)2443 3678 y Fz(^)-53 b Fu(\015)2490 3693 y Fs(3)2529 3678 y Fz(\))22 b(+)g Fr(O)s Fz(\()t(^)-53 b Fu(\015)2863 3637 y Fs(2)2903 3678 y Fu(=t;)17 b(t)3066 3637 y Fm(\000)p Fs(2)3160 3678 y Fz(\))p Fu(;)351 b Fz(\(7.24\))150 3899 y(where)34 b Fu(a)483 3914 y Fs(31)590 3899 y Fz(is)e(an)h(op)s(erator) f(uniformly)e(b)s(ounded)j(in)f(time.)150 4019 y Fv(V)p Fz(.)c(Let)g(us)g(no)m(w)h(giv)m(e)f(a)g(di\013eren)m(tial)e(inequalit) m(y)h(in)m(v)m(olving)f(all)g(three)33 b(^)-53 b Fu(\015)5 b Fz('s.)43 b(First,)28 b(rewrite)g(\(7.19\),)150 4139 y(\(7.22\))k(and)g(\(7.24\))g(as:)524 4449 y Fu(D)605 4464 y Ft(H)672 4449 y Fz(\()p Fu(f)758 4464 y Ft(j)798 4449 y Fz(^)-53 b Fu(\015)845 4464 y Ft(j)881 4449 y Fz(\))28 b(=)1105 4325 y Fs(3)1051 4355 y Fl(X)1058 4567 y Ft(k)r Fs(=1)1195 4449 y Fz([)p Fu(a)1273 4464 y Ft(j)t(k)1348 4449 y Fz(\()p Fu(t)p Fz(\))p Fu(=t)p Fz(])22 b Fr(\001)g Fz(\()p Fu(f)1728 4464 y Ft(k)1774 4449 y Fz(^)-52 b Fu(\015)1822 4464 y Ft(k)1864 4449 y Fz(\))22 b(+)g Fr(O)s Fz(\()t(^)-53 b Fu(\015)2198 4408 y Fs(2)2238 4449 y Fu(=t;)17 b(t)2401 4408 y Fm(\000)p Fs(2)2495 4449 y Fz(\))p Fu(;)114 b(j)34 b Fr(2)28 b(f)p Fz(1)p Fu(;)17 b Fz(2)p Fu(;)g Fz(3)p Fr(g)p Fu(;)372 b Fz(\(7.25\))150 4754 y(where)37 b Fu(a)486 4769 y Ft(j)t(j)587 4754 y Fz(=)32 b Fr(\000)p Fz(1)k(for)e(all)g Fu(j)6 b Fz(,)35 b(and)h Fu(a)1499 4769 y Fs(12)1606 4754 y Fz(=)c Fu(a)1765 4769 y Fs(13)1872 4754 y Fz(=)h Fu(a)2032 4769 y Fs(23)2139 4754 y Fz(=)f Fu(a)2298 4769 y Fs(32)2405 4754 y Fz(=)g(0.)52 b(As)36 b(in)e(the)i(classical)e(case,)j(the)150 4874 y(matrix)c Fr(f)p Fu(a)p Fr(g)g Fz(is)h(lo)m(w)m(er)g(triangular.)46 b(Notice)33 b(again)g(that)h(when)h(the)f(energy)i(is)d(lo)s(calized)f (around)150 4995 y Fu(E)k(>)30 b(E)436 5010 y Ft(d)477 4995 y Fz(,)35 b(w)m(e)g(ha)m(v)m(e)g(upp)s(er)g(and)f(lo)m(w)m(er)g(b) s(ounds)h(for)e Fu(f)2177 5010 y Ft(j)2214 4995 y Fz('s)i(uniform)d(in) h Fu(t)p Fz(;)i(there)g(exist)f(uniform)f(in)150 5115 y(time)e(upp)s(er)i(b)s(ounds)h(for)e Fu(a)1191 5130 y Ft(j)t(k)1266 5115 y Fz('s,)h(to)s(o.)296 5236 y(De\014ne)g(the)g (Liapuno)m(v-t)m(yp)s(e)g(function)f(of)k(^)-52 b Fu(\015)5 b Fz('s)1196 5456 y Fu(L)1262 5471 y Ft(C)1349 5456 y Fz(:=)28 b Fu(C)h Fr(\001)21 b Fz(\()t(^)-53 b Fu(\015)1717 5471 y Fs(1)1756 5456 y Fu(f)1815 5414 y Fs(2)1804 5480 y(1)1858 5456 y Fz(^)h Fu(\015)1906 5471 y Fs(1)1945 5456 y Fz(\))22 b(+)j(^)-52 b Fu(\015)2154 5471 y Fs(2)2193 5456 y Fu(f)2252 5414 y Fs(2)2241 5480 y(2)2294 5456 y Fz(^)g Fu(\015)2342 5471 y Fs(2)2403 5456 y Fz(+)26 b(^)-53 b Fu(\015)2552 5471 y Fs(3)2591 5456 y Fu(f)2650 5414 y Fs(2)2639 5480 y(3)2693 5456 y Fz(^)g Fu(\015)2740 5471 y Fs(3)3576 5456 y Fz(\(7.26\))1939 5776 y(73)p eop %%Page: 74 77 74 76 bop 150 407 a Fz(where)27 b Fu(C)34 b(>)28 b Fz(0)d(is)g(a)g(v)m (ery)i(large)d(p)s(ositiv)m(e)h(constan)m(t)i(only)d(dep)s(ending)i(on) f(the)h(energy)h(lo)s(calization.)150 527 y(No)m(w)33 b(let)f(us)h(see)h(ho)m(w)f(w)m(e)h(c)m(ho)s(ose)f Fu(C)7 b Fz(.)44 b(Compute)856 747 y Fu(D)937 762 y Ft(H)1004 747 y Fu(L)1070 762 y Ft(C)1213 747 y Fz(=)83 b Fr(\000)p Fz(\(2)p Fu(=t)p Fz(\))p Fu(L)1724 762 y Ft(C)1805 747 y Fz(+)22 b(\(2)p Fu(=t)p Fz(\))g Fr(\001)g Fz([)p Fu(a)2262 762 y Fs(21)2359 747 y Fr(\001)g Fz(\()p Fu(f)2495 762 y Fs(2)2538 747 y Fz(^)-53 b Fu(\015)2585 762 y Fs(2)2625 747 y Fz(\)\()p Fu(f)2749 762 y Fs(1)2792 747 y Fz(^)g Fu(\015)2839 762 y Fs(1)2878 747 y Fz(\)])1213 892 y(+)83 b(\(2)p Fu(=t)p Fz(\))21 b Fr(\001)h Fz([)p Fu(a)1730 907 y Fs(31)1827 892 y Fr(\001)g Fz(\()p Fu(f)1963 907 y Fs(3)2006 892 y Fz(^)-52 b Fu(\015)2054 907 y Fs(3)2093 892 y Fz(\)\()p Fu(f)2217 907 y Fs(1)2260 892 y Fz(^)f Fu(\015)2307 907 y Fs(1)2346 892 y Fz(\)])22 b(+)g Fr(O)s Fz(\()t(^)-53 b Fu(\015)2707 851 y Fs(3)2747 892 y Fu(=t;)20 b Fz(^)-52 b Fu(\015)5 b(=t)3015 851 y Fs(2)3054 892 y Fz(\))p Fu(:)457 b Fz(\(7.27\))150 1112 y(W)-8 b(e)33 b(see)h(that)e(the)h(cross)h(terms)e(can)h(b)s(e)g(b)s(ounded)g(in)f (the)h(form)e(sense)k(b)m(y)e(\()p Fu(j)h Fr(2)28 b(f)p Fz(2)p Fu(;)17 b Fz(3)p Fr(g)p Fz(\):)438 1377 y(2)p Fr(jh)p Fz(\()p Fu(f)640 1392 y Ft(j)680 1377 y Fz(^)-53 b Fu(\015)727 1392 y Ft(j)763 1377 y Fz(\)\()p Fu(f)887 1392 y Fs(1)930 1377 y Fz(^)h Fu(\015)978 1392 y Fs(1)1017 1377 y Fz(\))p Fr(i)1094 1392 y Ft(')1143 1377 y Fr(j)28 b(\024)1370 1310 y Fz(1)p 1314 1354 160 4 v 1314 1374 a Fr(p)p 1397 1374 77 4 v 84 x Fu(C)1484 1377 y Fz([)p Fr(h)p Fz(\()p Fu(f)1636 1392 y Ft(j)1676 1377 y Fz(^)-53 b Fu(\015)1723 1392 y Ft(j)1759 1377 y Fz(\))1797 1336 y Fm(\003)1837 1377 y Fz(\()p Fu(f)1923 1392 y Ft(j)1963 1377 y Fz(^)g Fu(\015)2010 1392 y Ft(j)2046 1377 y Fz(\))22 b(+)g Fu(C)7 b Fz(\()p Fu(f)2367 1392 y Fs(1)2410 1377 y Fz(^)-52 b Fu(\015)2458 1392 y Fs(1)2497 1377 y Fz(\))2535 1336 y Fm(\003)2574 1377 y Fz(\()p Fu(f)2660 1392 y Fs(1)2703 1377 y Fz(^)g Fu(\015)2751 1392 y Fs(1)2790 1377 y Fz(\))p Fr(i)2867 1392 y Ft(')2917 1377 y Fz(])27 b Fr(\024)3142 1310 y Fz(1)p 3086 1354 160 4 v 3086 1374 a Fr(p)p 3169 1374 77 4 v 84 x Fu(C)3256 1377 y Fr(h)p Fu(L)3361 1392 y Ft(C)3421 1377 y Fr(i)3460 1392 y Ft(')3510 1377 y Fu(:)150 1652 y Fz(Moreo)m(v)m(er,)37 b(the)f(a)f(priori)e(smallness)h (of)h(the)k(^)-53 b Fu(\015)6 b Fz('s)35 b(stated)h(in)e(\(7.5\))h (enable)g(us)h(to)e(b)s(ound)i(in)e(form)150 1773 y(sense)e(an)m(y)f (pro)s(duct)g(of)f(three)h(gammas)e(with)h(\(up)h(to)f(a)g(constan)m (t\))h Fu(\017L)2805 1788 y Ft(C)2865 1773 y Fz(.)43 b(W)-8 b(e)30 b(conclude)h(that)f(for)150 1893 y(ev)m(ery)35 b Fu(\016)e(>)28 b Fz(0,)33 b(w)m(e)h(can)g(c)m(ho)s(ose)g Fu(C)7 b Fz(\()p Fu(\016)t Fz(\))32 b(su\016cien)m(tly)i(large)e(suc)m (h)j(that)e(in)f(the)i(form)e(sense)i(w)m(e)h(ha)m(v)m(e)150 2013 y Fu(L)216 2028 y Ft(C)266 2040 y Fq(\016)1244 2190 y Fu(D)1325 2205 y Ft(H)1392 2190 y Fu(L)1458 2205 y Ft(C)1508 2217 y Fq(\016)1574 2190 y Fr(\024)28 b(\000)1766 2123 y Fz(2)23 b Fr(\000)f Fu(\016)t(=)p Fz(2)p 1766 2168 316 4 v 1907 2259 a Fu(t)2092 2190 y(L)2158 2205 y Ft(C)2208 2217 y Fq(\016)2269 2190 y Fz(+)g Fr(O)s Fz(\()t(^)-53 b Fu(\015)5 b(=t)2627 2149 y Fs(2)2666 2190 y Fz(\))p Fu(:)845 b Fz(\(7.28\))150 2390 y(As)33 b(in)f(the)h(classical)e(case,)j(w)m(e)f(abuse)h(notation)d(and)i (write)f Fu(L)2495 2405 y Ft(\016)2566 2390 y Fz(instead)g(of)g Fu(L)3079 2405 y Ft(C)3129 2417 y Fq(\016)3168 2390 y Fz(.)150 2679 y Fp(7.2)135 b(A)45 b(propagation)h(estimate)h(for)e(the) 50 b FF(^)-64 b Fi(\015)6 b Fp('s)150 2864 y Fz(The)48 b(next)g(step)g(is)e(pro)m(ving)h(that)g(when)h(restricted)g(to)e (states)i(lik)m(e)f Fu( )2925 2879 y Fs(11)3000 2864 y Fz(\()p Fu(t)p Fz(\))g(\(see)h(\(7.5\)\),)i(our)150 2984 y(gammas)31 b(deca)m(y)j(b)s(etter)f(than)f Fu(t)1353 2948 y Fm(\000)p Fs(1)p Ft(=)p Fs(2)1551 2984 y Fz(as)h(is)f(the)h (case)g(in)f(the)h(classical)e(situation.)150 3188 y Fv(Prop)s(osition)37 b(7.1.)42 b FA(Fix)35 b(a)h(smal)5 b(l)35 b Fz(0)29 b Fu(<)h(\016)j(<<)c Fz(2)36 b FA(in)f(\(7.28\).)47 b(Then)35 b(ther)-5 b(e)36 b(exists)f(\(a)h(su\016ciently)150 3308 y(smal)5 b(l\))34 b Fu(\017)488 3323 y Fs(4)555 3308 y Fu(>)28 b Fz(0)35 b FA(so)f(that)h(the)g(fol)5 b(lowing)34 b(estimate)g(holds:)1297 3528 y Fz(lim)1281 3588 y Ft(t)p Fm(!1)1465 3528 y Fr(jj)p Fv(1)1577 3543 y Fs(\(1)p Ft(;)p Fm(1)p Fs(\))1777 3447 y Fl(\000)1823 3528 y Fu(t)1858 3487 y Fs(1+)p Ft(\017)1977 3496 y Fj(4)2015 3528 y Fu(L)2081 3543 y Ft(\016)2120 3447 y Fl(\001)2182 3528 y Fu( )2245 3543 y Fs(11)2320 3528 y Fz(\()p Fu(t)p Fz(\))p Fr(jj)27 b Fz(=)h(0)p Fu(:)296 3775 y Fz(The)34 b(pro)s(of)e(is)g(long)f(and)i(complicated,)e(so)h(w)m(e)i(split)d(it)h (in)m(to)g(sev)m(eral)h(parts.)150 4035 y Fv(7.2.1)112 b(Starting)37 b(the)g(pro)s(of)h(of)g(Prop)s(osition)d(7.1)150 4219 y Fz(Let)f(\000)h(b)s(e)f(a)g(cut-o\013)g(function)f(of)h(the)h(t) m(yp)s(e)g Fu(F)1897 4234 y Fs(+)1990 4219 y Fz(with)f Fu(a)d Fz(=)f(1)p Fu(=)p Fz(2)k(and)g Fu(b)d Fz(=)f(1.)48 b(Then)36 b(Prop)s(osition)150 4340 y(7.1)c(follo)m(ws)f(if)h(w)m(e)h (can)g(sho)m(w)h(that)1326 4579 y(lim)1310 4639 y Ft(t)p Fm(!1)1493 4464 y Fl(\015)1493 4524 y(\015)1493 4584 y(\015)1549 4490 y Fr(p)p 1632 4490 61 4 v 89 x Fz(\000)1709 4498 y Fl(\000)1755 4579 y Fu(t)1790 4538 y Fs(1+)p Ft(\017)1909 4547 y Fj(4)1947 4579 y Fu(L)2013 4594 y Ft(\016)2052 4498 y Fl(\001)2114 4579 y Fu( )2177 4594 y Fs(11)2252 4579 y Fz(\()p Fu(t)p Fz(\))2363 4464 y Fl(\015)2363 4524 y(\015)2363 4584 y(\015)2419 4490 y Fs(2)2486 4579 y Fz(=)27 b(0)p Fu(:)911 b Fz(\(7.29\))150 4826 y(Equiv)-5 b(alen)m(tly)d(,)32 b(in)m(tro)s(ducing)1233 5046 y Fu(F)14 b Fz(\()p Fu(t)p Fz(\))28 b(:=)1579 4965 y Fl(\012)1626 5046 y Fu( )1689 5061 y Fs(11)1765 5046 y Fz(\()p Fu(t)p Fz(\))p Fu(;)17 b Fz(\000)1998 4965 y Fl(\000)2043 5046 y Fu(t)2078 5004 y Fs(1+)p Ft(\017)2197 5013 y Fj(4)2235 5046 y Fu(L)2301 5061 y Ft(\016)2339 4965 y Fl(\001)2402 5046 y Fu( )2465 5061 y Fs(11)2540 5046 y Fz(\()p Fu(t)p Fz(\))2651 4965 y Fl(\013)2715 5046 y Fu(;)150 5266 y Fz(w)m(e)34 b(need)f(to)f(pro)m(v)m(e)i(that)f Fu(F)14 b Fz(\()p Fu(t)p Fz(\))27 b Fr(!)g Fz(0)33 b(when)g Fu(t)28 b Fr(!)g(1)p Fz(.)43 b(W)-8 b(e)33 b(de\014ne)g(for)g Fu(t;)17 b(\026)27 b(>)g Fz(1)425 5486 y Fu(F)14 b Fz(\()p Fu(t;)j(\026)p Fz(\))26 b(:=)873 5405 y Fl(\012)920 5486 y Fu( )983 5501 y Fs(11)1058 5486 y Fz(\()p Fu(t)p Fz(\))p Fu(;)17 b Fz(\000)1291 5405 y Fl(\000)1336 5486 y Fu(\026)1395 5444 y Fm(\000)p Fs(1)1489 5486 y Fu(t)1524 5444 y Fs(1+2)p Ft(\017)1678 5453 y Fj(4)1717 5486 y Fu(L)1783 5501 y Ft(\016)1821 5405 y Fl(\001)1884 5486 y Fu( )1947 5501 y Fs(11)2022 5486 y Fz(\()p Fu(t)p Fz(\))2133 5405 y Fl(\013)2197 5486 y Fz(;)114 b Fu(B)5 b Fz(\()p Fu(t;)17 b(\026)p Fz(\))27 b(:=)h Fu(\026)2848 5444 y Fm(\000)p Fs(1)2942 5486 y Fu(t)2977 5444 y Fs(1+2)p Ft(\017)3131 5453 y Fj(4)3170 5486 y Fu(L)3236 5501 y Ft(\016)3274 5486 y Fu(:)275 b Fz(\(7.30\))1939 5776 y(74)p eop %%Page: 75 78 75 77 bop 150 407 a Fz(Assume)46 b(for)e(the)h(momen)m(t)f(t)m(w)m(o)i (prop)s(erties)f(\(for)f(a)g(similar)e(pro)s(cedure)k(see)g([D-G2,)g (Section)150 527 y(6.13]\):)150 648 y(\()p Fu(A)p Fz(\):)d(F)-8 b(or)32 b(ev)m(ery)j Fu(a)28 b(>)f Fz(0,)32 b(there)i(exists)f Fu(t)1646 663 y Ft(a)1720 648 y Fz(indep)s(enden)m(t)h(of)e Fu(\026)c(>)f Fz(1)33 b(suc)m(h)h(that)e(for)g(all)e Fu(t)e Fr(\025)h Fu(t)3575 663 y Ft(a)1462 860 y Fu(F)14 b Fz(\()p Fu(t;)j(\026)p Fz(\))27 b Fr(\024)h Fu(F)14 b Fz(\()p Fu(t)2035 875 y Ft(a)2077 860 y Fu(;)j(\026)p Fz(\))k(+)h Fu(a=)p Fz(2)p Fu(;)150 1073 y Fz(and)150 1193 y(\()p Fu(B)5 b Fz(\):)43 b(There)34 b(exists)g Fu(\026)986 1208 y Ft(a)1055 1193 y Fu(>)27 b Fz(1)32 b(so)h(that)g(whenev)m(er)i Fu(\026)27 b Fr(\025)h Fu(\026)2249 1208 y Ft(a)1668 1406 y Fu(F)14 b Fz(\()p Fu(t)1818 1421 y Ft(a)1860 1406 y Fu(;)j(\026)p Fz(\))27 b Fu(<)g(a=)p Fz(2)p Fu(:)296 1618 y Fz(Giv)m(en)53 b(\()p Fu(A)p Fz(\))g(and)f(\()p Fu(B)5 b Fz(\),)58 b(Prop)s(osition)51 b(7.1)h(readily)g(follo)m(ws)f (b)m(y)j(observing)f(that)f Fu(F)14 b Fz(\()p Fu(t)p Fz(\))62 b(=)150 1752 y Fu(F)14 b Fz(\()p Fu(t;)j(t)379 1716 y Ft(\017)408 1725 y Fj(4)446 1752 y Fz(\).)43 b(Indeed)34 b(for)e(an)m(y)i(giv)m(en)e Fu(a)c(>)g Fz(0)k(w)m(e)i(ha)m(v)m(e)g(for) e(all)e Fu(t)e Fr(\025)g Fz(max)p Fr(f)p Fu(t)2811 1767 y Ft(a)2853 1752 y Fu(;)17 b(\026)2956 1701 y Fs(1)p Ft(=\017)3055 1710 y Fj(4)2956 1762 y Ft(a)3093 1752 y Fr(g)1309 1965 y Fz(0)28 b Fr(\024)g Fu(F)14 b Fz(\()p Fu(t)p Fz(\))27 b Fr(\024)i Fu(F)14 b Fz(\()p Fu(t)1962 1980 y Ft(a)2003 1965 y Fu(;)j(t)2082 1924 y Ft(\017)2111 1933 y Fj(4)2149 1965 y Fz(\))22 b(+)g Fu(a=)p Fz(2)28 b Fu(<)f(a:)296 2177 y Fz(Hence)44 b(what)e(remains)f(to)g(b)s(e)h(pro) m(v)m(ed)i(is)d(\()p Fu(A)p Fz(\))h(and)g(\()p Fu(B)5 b Fz(\).)71 b(Notice)42 b(that)f(\()p Fu(B)5 b Fz(\))42 b(immediately)150 2298 y(follo)m(ws)31 b(from)h(\()p Fu(A)p Fz(\))g(and)h(the)g(fact)f(that)h(\000)f(is)g(supp)s(orted)i(a)m (w)m(a)m(y)f(from)f(zero,)h(yielding)1377 2510 y(s)22 b Fr(\000)47 b Fz(lim)1537 2570 y Ft(\026)p Fm(!1)1737 2510 y Fz(\000)1815 2430 y Fl(\000)1860 2510 y Fu(\026)1919 2469 y Fm(\000)p Fs(1)2013 2510 y Fu(t)2048 2469 y Fs(1+2)p Ft(\017)2202 2478 y Fj(4)2048 2535 y Ft(a)2241 2510 y Fu(L)2307 2525 y Ft(\016)2346 2430 y Fl(\001)2419 2510 y Fz(=)27 b(0)p Fu(:)296 2768 y Fz(As)33 b(for)f(\()p Fu(A)p Fz(\))h(w)m(e)h(write)1203 3044 y Fu(F)14 b Fz(\()p Fu(t;)j(\026)p Fz(\))27 b(=)h Fu(F)14 b Fz(\()p Fu(t)1775 3059 y Fs(0)1814 3044 y Fu(;)j(\026)p Fz(\))22 b(+)2075 2908 y Fl(Z)2174 2934 y Ft(t)2130 3134 y(t)2155 3143 y Fj(0)2220 3044 y Fu(@)2271 3059 y Ft(\034)2315 3044 y Fu(F)14 b Fz(\()p Fu(\034)6 b(;)17 b(\026)p Fz(\))27 b Fu(d\034)6 b(:)150 3322 y Fz(The)34 b(idea)d(is)i(to)f(\014nd)h(a)f (function)g Fu(\021)g Fr(2)c Fu(L)1672 3286 y Fs(1)1712 3322 y Fz(\(\(1)p Fu(;)17 b Fr(1)p Fz(\)\))31 b(suc)m(h)j(that)1536 3535 y(sup)1544 3614 y Ft(\026>)p Fs(1)1700 3535 y Fu(@)1751 3550 y Ft(\034)1795 3535 y Fu(F)14 b Fz(\()p Fu(\034)6 b(;)17 b(\026)p Fz(\))26 b Fr(\024)i Fu(\021)t Fz(\()p Fu(\034)11 b Fz(\))p Fu(:)1138 b Fz(\(7.31\))296 3812 y(By)45 b(taking)f(the)h(partial)e(deriv)-5 b(ativ)m(e)44 b(with)g(resp)s(ect)i(to)e Fu(\034)56 b Fz(in)44 b Fu(F)14 b Fz(\()p Fu(\034)6 b(;)17 b(\026)p Fz(\),)47 b(w)m(e)e(obtain)f(v)-5 b(arious)150 3932 y(terms)35 b(con)m(taining)f(the)h(Heisen)m(b)s(erg)h (deriv)-5 b(ativ)m(e)35 b(of)f(the)i(cut-o\013)e(factors)h(in)g Fu( )3112 3947 y Fs(11)3187 3932 y Fz(\()p Fu(\034)11 b Fz(\))35 b(\(their)f(total)150 4053 y(sum)f(is)f(denoted)i(b)m(y)f Fu(R)q Fz(\()p Fu(\034)6 b(;)17 b(\026)p Fz(\)\))32 b(and)h(one)g(term) f(with)h(the)g(Heisen)m(b)s(erg)g(deriv)-5 b(ativ)m(e)33 b(of)f(\000)17 b(\()o Fu(B)5 b Fz(\()p Fu(t;)17 b(\026)p Fz(\)\))150 4173 y(\(see)37 b(\(7.30\)\).)53 b(Remem)m(b)s(er)35 b(that)h(the)g(Heisen)m(b)s(erg)h(deriv)-5 b(ativ)m(e)35 b(of)h(a)g(time)e(dep)s(enden)m(t)k(family)33 b(of)150 4294 y(op)s(erators)f Fu(A)p Fz(\()p Fu(t)p Fz(\))h(is)f(denoted)i(b)m (y)f Fu(D)1481 4309 y Ft(H)1548 4294 y Fu(A)p Fz(\()p Fu(t)p Fz(\))g(and)g(it)e(means)i Fu(@)2402 4309 y Ft(t)2432 4294 y Fu(A)p Fz(\()p Fu(t)p Fz(\))23 b(+)f Fu(i)p Fz([)p Fu(H)r(;)17 b(A)p Fz(\()p Fu(t)p Fz(\)].)150 4552 y Fv(7.2.2)112 b(Estimating)35 b Fu(R)q Fz(\()p Fu(\034)6 b(;)17 b(\026)p Fz(\))p Fv(.)150 4737 y Fz(Let)31 b(us)h(\014rst)f(deal)f(with)h Fu(R)q Fz(\()p Fu(\034)6 b(;)17 b(\026)p Fz(\),)30 b(i.e.)43 b(the)31 b(terms)g(coming)e(up)j(b)m(y)g(p)s(erforming)c(the)k(H-deriv) -5 b(ativ)m(e)150 4857 y(of)37 b(the)g(v)-5 b(arious)36 b(cut-o\013s)h(building)e Fu( )1572 4872 y Fs(11)1647 4857 y Fz(\()p Fu(t)p Fz(\).)57 b(T)-8 b(ak)m(e)38 b(\014rst)g(the)f (term)g(generated)h(b)m(y)g(the)f(maximal)150 4978 y(v)m(elo)s(cit)m(y) 32 b(cut-o\013)h Fu(F)880 4993 y Fm(\000)939 4978 y Fz(\()p Fu(r)s(=\034)11 b Fz(\))27 b(=)h Fu(F)1372 4942 y Ft(M)s(:v)r(:b:)1358 5002 y Fm(\000)1572 4978 y Fz(\()p Fu(r)s(=\034)11 b Fz(\);)33 b(it)e(ma)m(y)i(b)s(e)f(written)h(as)552 5174 y(2)p 549 5218 54 4 v 549 5310 a Fu(\034)613 5241 y Fr(<)17 b(h)o Fu( )803 5256 y Fs(11)878 5241 y Fz(\()p Fu(\034)11 b Fz(\))p Fu(;)17 b Fz(\000)g(\()p Fu(B)1241 5256 y Ft(\034)t(;\026) 1342 5241 y Fz(\))22 b Fr(\001)g Fz(\(other)67 b(cut)22 b Fr(\000)h Fz(o\013s\))p Fr(\001)1330 b Fz(\(7.32\))539 5489 y Fr(\002)643 5403 y Fl(p)p 743 5403 200 4 v 86 x Fr(\000)p Fu(F)897 5455 y Fm(0)883 5511 y(\000)943 5489 y Fz(\()p Fu(r)s(=\034)11 b Fz(\)\()p Fu(r)s(=\034)33 b Fr(\000)22 b Fu(\032)p Fz(\))1564 5403 y Fl(p)p 1664 5403 V 86 x Fr(\000)p Fu(F)1818 5455 y Fm(0)1804 5511 y(\000)1864 5489 y Fz(\()p Fu(r)s(=\034)11 b Fz(\))p Fu( )t Fz(\()p Fu(\034)g Fz(\))2285 5379 y Fl(E)2362 5489 y Fz(;)45 b Fu(B)2508 5504 y Ft(\034)t(;\026)2637 5489 y Fz(:=)27 b Fu(B)5 b Fz(\()p Fu(\034)h(;)17 b(\026)p Fz(\))28 b(=)3214 5422 y Fu(\034)3267 5386 y Fs(1+2)p Ft(\017)3421 5395 y Fj(4)p 3214 5466 247 4 v 3308 5558 a Fu(\026)3470 5489 y(L)3536 5504 y Ft(\016)3575 5489 y Fu(:)1939 5776 y Fz(75)p eop %%Page: 76 79 76 78 bop 150 407 a Fz(In)38 b(order)g(to)f(b)s(e)h(able)g(to)f(use)i (Lemma)d(6.4,)j(w)m(e)g(need)g(to)e(put)h(the)g(left)f(factor)3177 325 y Fl(p)p 3276 325 200 4 v 3276 407 a Fr(\000)p Fu(F)3430 372 y Fm(0)3416 429 y(\000)3514 407 y Fz(next)h(to)150 527 y(the)31 b Fu( )t Fz(\()p Fu(t)p Fz(\))f(in)f(the)i(\014rst)g(en)m (try)g(through)f(rep)s(eated)h(comm)m(utations.)41 b(When)31 b(comm)m(uting)e(with)h(the)150 648 y(\\old")h(cut-o\013s)i(one)g (gains)e(a)i(deca)m(y)h(of)e(1)p Fu(=\034)11 b Fz(,)33 b(so)f(the)h(remainders)f(are)h(in)m(tegrable.)296 768 y(Hence)h(the)e(only)f(problematic)f(terms)i(could)g(arise)f(from)g (the)h(comm)m(utation)e(with)i(\000.)43 b(Since)150 888 y(w)m(e)d(use)h(the)e(\014rst)h(equalit)m(y)f(in)f(\(6.5\),)j(w)m(e)f (are)g(motiv)-5 b(ated)37 b(to)i(study)i(v)-5 b(arious)38 b(comm)m(utators)g(of)150 1009 y Fu(B)5 b Fz(\()p Fu(\034)h(;)17 b(\026)p Fz(\).)150 1212 y Fv(Lemma)37 b(7.2.)42 b FA(F)-7 b(or)34 b(every)h Fu(G)27 b Fr(2)h Fu(C)1478 1176 y Fm(1)1471 1237 y Fs(0)1553 1212 y Fz(\()p Fn(R)1657 1227 y Fs(+)1722 1212 y Fz(\))35 b FA(we)f(have)862 1414 y Fl(\015)862 1473 y(\015)918 1418 y(\002)959 1498 y Fu(G)p Fz(\()p Fu(r)s(=t)p Fz(\))p Fu(;)17 b Fz(\()p Fu(B)1399 1513 y Ft(t;\026)1512 1498 y Fr(\000)23 b Fu(z)t Fz(\))1699 1457 y Fm(\000)p Fs(1)1794 1418 y Fl(\003)1835 1414 y(\015)1835 1473 y(\015)1918 1498 y Fr(\024)28 b Fz(Const)2311 1431 y Fr(h)p Fu(z)t Fr(i)2438 1395 y Fs(1)p Ft(=)p Fs(2)p 2283 1475 293 4 v 2283 1567 a Fr(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)2536 1538 y Fs(2)2608 1498 y Fr(\001)2667 1431 y Fu(t)2702 1395 y Fm(\000)p Fs(\(1)p Fm(\000)p Fs(2)p Ft(\017)2938 1404 y Fj(4)2974 1395 y Fs(\))p Ft(=)p Fs(2)p 2667 1475 409 4 v 2787 1567 a Fu(\026)2846 1538 y Fs(1)p Ft(=)p Fs(2)3086 1498 y Fu(:)150 1769 y Fv(Pro)s(of.)43 b Fz(W)-8 b(e)33 b(rely)g(on)f(the)h(iden)m(tit)m(y)668 1908 y Fl(\002)710 1989 y Fu(G)p Fz(\()p Fu(r)s(=t)p Fz(\))p Fu(;)17 b Fz(\()p Fu(B)1150 2004 y Ft(t;\026)1263 1989 y Fr(\000)23 b Fu(z)t Fz(\))1450 1948 y Fm(\000)p Fs(1)1545 1908 y Fl(\003)1614 1989 y Fz(=)k(\()p Fu(B)1829 2004 y Ft(t;\026)1943 1989 y Fr(\000)c Fu(z)t Fz(\))2130 1948 y Fm(\000)p Fs(1)2225 1989 y Fz([)p Fu(B)2326 2004 y Ft(t;\026)2418 1989 y Fu(;)17 b(G)p Fz(\()p Fu(r)s(=t)p Fz(\)]\()p Fu(B)2885 2004 y Ft(t;\026)2998 1989 y Fr(\000)23 b Fu(z)t Fz(\))3185 1948 y Fm(\000)p Fs(1)3280 1989 y Fu(;)150 2209 y Fz(where)901 2463 y([)p Fu(B)1002 2478 y Ft(t;\026)1094 2463 y Fu(;)17 b(G)p Fz(])82 b(=)1493 2396 y Fu(t)1528 2359 y Fs(1+2)p Ft(\017)1682 2368 y Fj(4)p 1493 2440 228 4 v 1578 2531 a Fu(\026)1748 2382 y Fl(\010)1806 2463 y Fu(C)1876 2478 y Ft(\016)1917 2463 y Fz(^)-52 b Fu(\015)1965 2478 y Fs(1)2004 2463 y Fz([)p Fu(f)2090 2422 y Fs(2)2079 2488 y(1)2133 2463 y Fz(^)f Fu(\015)2180 2478 y Fs(1)2219 2463 y Fu(;)17 b(G)p Fz(])22 b(+)g Fu(C)2557 2478 y Ft(\016)2595 2463 y Fz([)t(^)-53 b Fu(\015)2673 2478 y Fs(1)2712 2463 y Fu(;)17 b(G)p Fz(])p Fu(f)2919 2422 y Fs(2)2908 2488 y(1)2961 2463 y Fz(^)-52 b Fu(\015)3009 2478 y Fs(1)1324 2761 y Fz(+)1548 2637 y Fs(3)1493 2667 y Fl(X)1504 2877 y Ft(j)t Fs(=2)1654 2680 y Fl(\000)1700 2761 y Fz([)t(^)f Fu(\015)1778 2776 y Ft(j)1814 2761 y Fu(;)17 b(G)p Fz(])p Fu(f)2021 2720 y Fs(2)2010 2786 y Ft(j)2063 2761 y Fz(^)-52 b Fu(\015)2111 2776 y Ft(j)2169 2761 y Fz(+)26 b(^)-53 b Fu(\015)2318 2776 y Ft(j)2354 2761 y Fz([)p Fu(f)2440 2720 y Fs(2)2429 2786 y Ft(j)2483 2761 y Fz(^)g Fu(\015)2530 2776 y Ft(j)2566 2761 y Fu(;)17 b(G)p Fz(])2714 2680 y Fl(\001)2760 2591 y(\))2857 2761 y Fu(:)692 b Fz(\(7.33\))150 3082 y(Since)25 b(the)44 b(^)-69 b Fu(\015)607 3097 y Ft(j)644 3082 y Fz('s)24 b(are)h(essen)m(tially)f(\014rst)h(order)f(deriv)-5 b(ativ)m(es,)26 b(when)g(w)m(e)f(comm)m(ute)f(them)g(with)h Fu(G)p Fz(\()p Fu(r)s(=t)p Fz(\))150 3202 y(w)m(e)34 b(gain)d(a)h(factor)g(of)g(1)p Fu(=t)p Fz(.)44 b(Hence)34 b(the)f(lemma)d(w)m(ould)i(follo)m(w)f(from)h(the)h(estimate)941 3440 y Fu(t)986 3367 y Fj(1+2)p Fq(\017)1119 3382 y Fj(4)p 986 3388 169 3 v 1055 3429 a(2)p 941 3485 228 4 v 970 3576 a Fu(\026)1029 3547 y Fs(1)p Ft(=)p Fs(2)1195 3423 y Fl(\015)1195 3483 y(\015)1254 3508 y Fz(^)-53 b Fu(\015)1301 3523 y Ft(j)1337 3508 y Fz(\()p Fu(B)1449 3523 y Ft(t;\026)1563 3508 y Fr(\000)23 b Fu(z)t Fz(\))1750 3466 y Fm(\000)p Fs(1)1845 3423 y Fl(\015)1845 3483 y(\015)1928 3508 y Fr(\024)28 b Fz(Const)23 b Fr(\001)2372 3440 y(h)p Fu(z)t Fr(i)2499 3404 y Fs(1)p Ft(=)p Fs(2)p 2365 3485 253 4 v 2365 3576 a Fr(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)2627 3508 y Fz(;)45 b Fu(j)33 b Fz(=)28 b(1)p Fu(;)17 b Fz(2)p Fu(:)531 b Fz(\(7.34\))150 3788 y(Clearly)30 b(\(7.34\))g(follo)m(ws)g (from)f(the)i(quadratic)g(estimate)f(\(notice)g(that)35 b(^)-53 b Fu(\015)2893 3752 y Fs(2)2888 3812 y Ft(j)2960 3788 y Fr(\024)28 b Fz(Const)20 b Fr(\001)e Fu(L)3446 3803 y Ft(\016)3515 3788 y Fz(in)30 b(form)150 3908 y(sense\))471 4067 y Fl(\015)471 4126 y(\015)530 4151 y Fz(^)-53 b Fu(\015)577 4166 y Ft(j)613 4151 y Fz(\()p Fu(B)725 4166 y Ft(t;\026)839 4151 y Fr(\000)23 b Fu(z)t Fz(\))1026 4110 y Fm(\000)p Fs(1)1121 4151 y Fu(\036)1179 4067 y Fl(\015)1179 4126 y(\015)1234 4093 y Fs(2)1356 4151 y Fr(\024)84 b Fz(Const)23 b Fr(\001)1933 4084 y Fu(\026)p 1849 4128 228 4 v 1849 4220 a(t)1884 4191 y Fs(1+2)p Ft(\017)2038 4200 y Fj(4)2087 4151 y Fr(h)p Fz(\()p Fu(B)2238 4166 y Ft(t;\026)2351 4151 y Fr(\000)g Fu(z)t Fz(\))2538 4110 y Fm(\000)p Fs(1)2633 4151 y Fu(\036;)17 b(B)2809 4166 y Ft(t;\026)2900 4151 y Fz(\()p Fu(B)3012 4166 y Ft(t;\026)3126 4151 y Fr(\000)23 b Fu(z)t Fz(\))3313 4110 y Fm(\000)p Fs(1)3408 4151 y Fu(\036)p Fr(i)1356 4397 y(\024)84 b Fz(Const)23 b Fr(\001)1933 4330 y Fu(\026)p 1849 4374 V 1849 4466 a(t)1884 4437 y Fs(1+2)p Ft(\017)2038 4446 y Fj(4)2103 4257 y Fl(\032)2290 4330 y Fz(1)p 2188 4374 253 4 v 2188 4466 a Fr(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)2473 4397 y Fz(+)2675 4330 y Fr(j)p Fu(z)t Fr(j)p 2581 4374 293 4 v 2581 4466 a(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)2834 4437 y Fs(2)2883 4257 y Fl(\033)2975 4397 y Fr(jj)p Fu(\036)p Fr(jj)3145 4356 y Fs(2)3182 4397 y Fu(:)367 b Fz(\(7.35\))p 3754 4666 4 66 v 3758 4603 59 4 v 3758 4666 V 3816 4666 4 66 v 150 4987 a Fv(Corollary)37 b(7.3.)42 b FA(Under)35 b(the)g(c)-5 b(onditions)34 b(of)g(L)-5 b(emma)34 b(7.2)756 5183 y Fl(\015)756 5243 y(\015)811 5268 y Fz([)p Fu(B)912 5283 y Ft(t;\026)1004 5268 y Fu(;)17 b(G)p Fz(\()p Fu(r)s(=t)p Fz(\)]])f(\()p Fu(B)1514 5283 y Ft(t;\026)1628 5268 y Fr(\000)22 b Fu(z)t Fz(\))1814 5227 y Fm(\000)p Fs(1)1909 5183 y Fl(\015)1909 5243 y(\015)1992 5268 y Fr(\024)29 b Fz(Const)22 b Fr(\001)2437 5200 y(h)p Fu(z)t Fr(i)2564 5164 y Fs(1)p Ft(=)p Fs(2)p 2429 5245 253 4 v 2429 5336 a Fr(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)2714 5268 y(\001)2774 5200 y Fu(t)2809 5164 y Fm(\000)p Fs(\(1)p Fm(\000)p Fs(2)p Ft(\017)3045 5173 y Fj(4)3080 5164 y Fs(\))p Ft(=)p Fs(2)p 2774 5245 409 4 v 2894 5336 a Fu(\026)2953 5307 y Fs(1)p Ft(=)p Fs(2)3192 5268 y Fu(:)1939 5776 y Fz(76)p eop %%Page: 77 80 77 79 bop 150 407 a FA(Pr)-5 b(o)g(of)p Fz(.)41 b(W)-8 b(e)25 b(comm)m(ute)f(the)i(\\free")i(^)-52 b Fu(\015)5 b Fz('s)25 b(in)f(\(7.33\))h(to)f(the)i(righ)m(t.)40 b(These)27 b(comm)m(utations)c(in)m(tro)s(duce)150 540 y(an)41 b(extra)h(deca)m(y)g(of)f Fu(t)989 504 y Fm(\000)p Fs(1)1084 540 y Fz(,)i(hence)f(the)g(corresp)s(onding)f(terms)g(are)g (b)s(ounded)h(b)m(y)g Fu(C)3334 501 y Ft(t)3359 477 y Ff(\000)p Fj(\(1)p Ff(\000)p Fj(2)p Fq(\017)3565 492 y Fj(4)3601 477 y(\))p 3334 517 296 4 v 3370 574 a Ft(\026)p Fm(j=)p Fs(\()p Ft(z)s Fs(\))p Fm(j)3639 540 y Fz(.)70 b(In)150 674 y(addition)31 b(w)m(e)i(use)h(\(7.34\).)p 3754 674 4 66 v 3758 611 59 4 v 3758 674 V 3816 674 4 66 v 150 993 a Fv(Lemma)j(7.4.)42 b FA(Supp)-5 b(ose)35 b Fu(F)s(;)17 b(G)27 b Fr(2)i Fu(C)1521 957 y Fm(1)1514 1017 y Fs(0)1595 993 y Fz(\()p Fn(R)1699 1008 y Fs(+)1764 993 y Fz(\))35 b FA(and)g Fu(F)42 b Fz(=)28 b(1)34 b FA(on)h(a)g(neighb)-5 b(orho)g(o)g(d)33 b(of)i(the)g(supp)-5 b(ort)35 b(of)150 1113 y Fu(G)p FA(.)48 b(Then)35 b(for)h(every)g(inte) -5 b(ger)36 b Fu(N)k Fr(\025)30 b Fz(1)p FA(,)36 b(ther)-5 b(e)36 b(exists)g(a)g(c)-5 b(onstant)36 b Fu(C)g(>)30 b Fz(0)36 b FA(indep)-5 b(endent)34 b(of)i Fu(\026)30 b(>)f Fz(1)150 1233 y FA(such)35 b(that)560 1417 y Fl(\015)560 1477 y(\015)615 1502 y Fz(\(1)22 b Fr(\000)h Fu(F)14 b Fz(\()p Fu(r)s(=t)p Fz(\)\)\()p Fu(B)1258 1517 y Ft(t;\026)1371 1502 y Fr(\000)22 b Fu(z)t Fz(\))1557 1461 y Fm(\000)p Fs(1)1652 1502 y Fu(G)p Fz(\()p Fu(r)s(=t)p Fz(\))1936 1417 y Fl(\015)1936 1477 y(\015)2019 1502 y Fr(\024)2222 1434 y Fu(C)p 2134 1479 253 4 v 2134 1570 a Fr(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)2413 1361 y Fl(\022)2560 1434 y Fr(h)p Fu(z)t Fr(i)p 2497 1479 V 2497 1570 a(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)2760 1361 y Fl(\023)2833 1383 y Ft(N)2917 1502 y Fu(t)2952 1461 y Fm(\000)p Ft(N)7 b Fs(\(1)p Fm(\000)p Fs(2)p Ft(\017)3251 1470 y Fj(4)3286 1461 y Fs(\))p Ft(=)p Fs(2)3388 1502 y Fu(:)150 1770 y Fv(Pro)s(of.)45 b Fz(One)34 b(can)f(\014nd)h(a)f(function)f Fu(G)1621 1785 y Fs(1)1694 1770 y Fz(suc)m(h)i(that)f Fu(G)2203 1785 y Fs(1)2243 1770 y Fu(G)28 b Fz(=)h Fu(G)k Fz(and)g Fu(F)14 b(G)2907 1785 y Fs(1)2975 1770 y Fz(=)29 b Fu(G)3157 1785 y Fs(1)3196 1770 y Fz(.)45 b(Then)35 b(for)d(an)m(y)150 1890 y Fu(N)39 b Fr(2)29 b Fn(N)48 b Fz(w)m(e)34 b(ma)m(y)f(write)g(\(abbreviating)f (b)s(elo)m(w)h Fu(F)42 b Fz(=)29 b Fu(F)14 b Fz(\()p Fu(r)s(=t)p Fz(\),)32 b Fu(G)d Fz(=)f Fu(G)p Fz(\()p Fu(r)s(=t)p Fz(\),)33 b Fu(G)3139 1905 y Fs(1)3207 1890 y Fz(=)c Fu(G)3389 1905 y Fs(1)3428 1890 y Fz(\()p Fu(r)s(=t)p Fz(\))k(and)150 2011 y Fu(B)g Fz(=)27 b Fu(B)434 2026 y Ft(t;\026)526 2011 y Fz(\))1015 2131 y(\(1)22 b Fr(\000)h Fu(F)14 b Fz(\)\()p Fu(B)27 b Fr(\000)22 b Fu(z)t Fz(\))1664 2090 y Fm(\000)p Fs(1)1759 2131 y Fu(G)28 b Fz(=)f(\(1)22 b Fr(\000)h Fu(F)14 b Fz(\)\()p Fu(B)27 b Fr(\000)22 b Fu(z)t Fz(\))2616 2090 y Fm(\000)p Fs(1)2711 2131 y Fu(G)2788 2090 y Ft(N)2788 2156 y Fs(1)2856 2131 y Fu(G:)296 2304 y Fz(Due)33 b(to)f(the)h(supp)s(ort)g(conditions)f(w)m(e)h(ha)m(v) m(e)952 2521 y(\(1)22 b Fr(\000)g Fu(F)14 b Fz(\)\()p Fu(B)27 b Fr(\000)c Fu(z)t Fz(\))1601 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b(ma)m(y)f(b)s(ound)370 3265 y Fr(jj)p Fu(ad)528 3224 y Ft(k)528 3289 y(G)583 3298 y Fj(1)621 3265 y Fz(\()p Fu(B)5 b Fz(\)\()p Fu(B)27 b Fr(\000)c Fu(z)t Fz(\))1102 3224 y Fm(\000)p Fs(1)1197 3265 y Fr(jj)k(\024)h Fu(C)1455 3280 y Ft(k)1507 3197 y Fu(t)1542 3161 y Fm(\000)p Ft(k)r Fs(+1+2)p Ft(\017)1845 3170 y Fj(4)p 1507 3242 377 4 v 1540 3333 a Fu(\026)p Fr(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)1921 3265 y(\024)g Fu(C)2096 3280 y Ft(k)2161 3265 y Fr(\001)2229 3197 y(h)p Fu(z)t Fr(i)2356 3161 y Fs(1)p Ft(=)p Fs(2)p 2221 3242 253 4 v 2221 3333 a Fr(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)2506 3265 y(\001)2566 3197 y Fu(t)2601 3161 y Fm(\000)p Ft(k)r Fs(+1+2)p Ft(\017)2904 3170 y Fj(4)p 2566 3242 377 4 v 2669 3333 a Fu(\026)2728 3304 y Fs(1)p Ft(=)p Fs(2)2952 3265 y Fu(;)114 b(k)31 b Fr(\025)d Fz(2)p Fu(:)220 b Fz(\(7.37\))296 3537 y(W)-8 b(e)32 b(no)m(w)h(in)m(v)m(estigate)e(the)h Fu(N)10 b Fz(-th)32 b(order)g(comm)m(utator)e(in)h(\(7.36\).)43 b(W)-8 b(e)32 b(simplify)d(notation)h(b)m(y)150 3658 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y Ft(t;\026)2793 4159 y Fr(\000)g Fu(z)t Fz(\))2979 4117 y Fm(\000)p Fs(1)3074 4159 y Fu(:)150 4375 y Fz(When)43 b(w)m(e)h(estimate)d(the)i(norm)e(of) h(eac)m(h)h(term)f(in)f(the)i(ab)s(o)m(v)m(e)g(sum,)i(w)m(e)e(mak)m(e)g (a)f(distinction)150 4496 y(b)s(et)m(w)m(een)d(the)f(factors)f(with)g Fu(k)i Fz(=)d(1)h(and)g(those)h(with)f Fu(k)i(>)d Fz(1.)57 b(Cho)s(ose)38 b(a)f(term)g(with)g(the)g(total)150 4616 y(n)m(um)m(b)s(er)43 b(of)g(factors)g(to)f(b)s(e)h Fu(n)j Fr(\024)g Fu(N)53 b Fz(and)43 b(assume)g(that)g(w)m(e)h(ha)m(v)m(e)g Fu(n)2811 4631 y Fs(1)2894 4616 y Fz(factors)f(of)f Fu(ad)3445 4580 y Fs(1)3527 4616 y Fz(and)h Fu(n)3785 4631 y Fs(2)150 4736 y Fz(factors)31 b(with)g Fu(k)g Fr(\025)d Fz(2;)k(clearly)e Fu(n)1350 4751 y Fs(1)1410 4736 y Fz(+)19 b Fu(n)1563 4751 y Fs(2)1630 4736 y Fz(=)28 b Fu(n)j Fz(and)h Fu(n)2070 4751 y Fs(2)2137 4736 y Fr(\024)c Fz(\()p Fu(N)j Fr(\000)20 b Fu(n)2544 4751 y Fs(1)2583 4736 y Fz(\))p Fu(=)p Fz(2.)43 b(W)-8 b(e)32 b(use)g(Corollary)e(7.3)g(for)150 4857 y(the)h Fu(n)374 4872 y Fs(1)443 4857 y Fz(factors)g(and)f(the)g (second)i(inequalit)m(y)d(in)g(\(7.37\))h(for)f(the)i(remaining)d (factors)i(obtaining)e(a)150 4977 y(b)s(ound)33 b(of)f(the)h(form)e (\(uniformly)g(in)g Fu(\026)d(>)f Fz(1\):)888 5193 y Fu(C)p 800 5237 253 4 v 800 5328 a Fr(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)1101 5193 y(h)p Fu(z)t Fr(i)1228 5157 y Ft(n=)p Fs(2)p 1073 5237 300 4 v 1073 5328 a Fr(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)1326 5300 y Ft(n)1405 5260 y Fr(\001)22 b Fu(t)1490 5219 y Fm(\000)p Ft(l)1566 5228 y Fj(1)1601 5219 y Fs(+1+2)p Ft(\017)1810 5228 y Fj(4)1865 5260 y Fr(\001)17 b(\001)g(\001)e Fu(t)2033 5219 y Fm(\000)p Ft(l)2109 5227 y Fq(n)2147 5242 y Fj(2)2186 5219 y Fs(+1+2)p Ft(\017)2395 5228 y Fj(4)2456 5260 y Fr(\001)22 b Fu(t)2541 5219 y Fm(\000)p Ft(n)2639 5228 y Fj(1)2673 5219 y Fs(\(1)p Fm(\000)p Fs(2)p Ft(\017)2854 5228 y Fj(4)2889 5219 y Fs(\))p Ft(=)p Fs(2)2991 5260 y Fu(;)558 b Fz(\(7.39\))150 5527 y(where)34 b Fu(l)461 5542 y Fs(1)522 5527 y Fz(+)22 b Fr(\001)17 b(\001)g(\001)k Fz(+)h Fu(l)886 5542 y Ft(n)929 5551 y Fj(2)995 5527 y Fz(=)28 b Fu(N)k Fr(\000)23 b Fu(n)1367 5542 y Fs(1)1439 5527 y Fz(and)33 b(eac)m(h)g Fu(l)d Fr(\025)e Fz(2.)1939 5776 y(77)p eop %%Page: 78 81 78 80 bop 296 407 a Fz(Since)35 b(there)f(at)g(at)g(most)f(\()p Fu(N)h Fr(\000)24 b Fu(n)1594 422 y Fs(1)1633 407 y Fz(\))p Fu(=)p Fz(2)34 b(factors)g(of)f(the)i(form)e Fu(t)2671 371 y Fm(\000)p Ft(l)q Fs(+1+2)p Ft(\017)2957 380 y Fj(4)3029 407 y Fz(w)m(e)i(ma)m(y)f(b)s(ound)h(the)150 527 y(time)c(dep)s (endence)k(in)d(\(7.39\))g(b)m(y)911 738 y Fu(t)946 697 y Fm(\000)p Ft(N)7 b Fs(+)p Ft(n)1162 706 y Fj(1)1223 738 y Fr(\001)22 b Fu(t)1308 697 y Fs(\()p Ft(N)7 b Fm(\000)p Ft(n)1496 706 y Fj(1)1531 697 y Fs(\)\(1+2)p Ft(\017)1739 706 y Fj(4)1774 697 y Fs(\))p Ft(=)p Fs(2)1899 738 y Fr(\001)21 b Fu(t)1983 697 y Fm(\000)p Ft(n)2081 706 y Fj(1)2116 697 y Fs(\(1)p Fm(\000)p Fs(2)p Ft(\017)2297 706 y Fj(4)2332 697 y Fs(\))p Ft(=)p Fs(2)2462 738 y Fz(=)27 b Fu(t)2600 697 y Fm(\000)p Ft(N)7 b Fs(\(1)p Fm(\000)p Fs(2)p Ft(\017)2899 706 y Fj(4)2935 697 y Fs(\))p Ft(=)p Fs(2)3037 738 y Fu(:)150 948 y Fz(As)33 b(for)f(the)h Fu(z)k Fz(dep)s(endence)e(of)e(the)g(b)s(ound)f(w)m(e)i(notice)e(that) 1337 1162 y Fr(h)p Fu(z)t Fr(i)1464 1126 y Ft(n=)p Fs(2)p 1309 1207 300 4 v 1309 1298 a Fr(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)1562 1269 y Ft(n)1647 1230 y Fr(\024)1752 1089 y Fl(\022)1898 1162 y Fr(h)p Fu(z)t Fr(i)p 1835 1207 253 4 v 1835 1298 a(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)2098 1089 y Fl(\023)2171 1112 y Ft(N)2255 1230 y Fu(;)114 b(n)28 b Fr(\024)g Fu(N)150 1492 y Fz(and)33 b(the)g(lemma)d(follo)m(ws.)p 3754 1492 4 66 v 3758 1430 59 4 v 3758 1492 V 3816 1492 4 66 v 296 1731 a(No)m(w)h(let)f(us)h(go)f (bac)m(k)i(to)e(\(7.32\))f(and)i(see)g(what)g(happ)s(ens)h(when)f(w)m (e)h(comm)m(ute)3306 1649 y Fl(p)p 3405 1649 200 4 v 3405 1731 a Fr(\000)p Fu(F)3559 1696 y Fm(0)3545 1752 y(\000)3635 1731 y Fz(with)150 1851 y(\000\()p Fu(B)323 1866 y Ft(\034)t(;\026)424 1851 y Fz(\).)47 b(W)-8 b(e)34 b(use)h(the)f(form)m(ula)e(\(6.5\):)46 b(b)m(y)34 b(in)m(tro)s(ducing)f (the)h(estimate)f(from)g(Lemma)f(7.2)h(with)150 1971 y Fu(G)28 b Fz(=)358 1890 y Fl(p)p 458 1890 V 81 x Fr(\000)p Fu(F)612 1937 y Fm(0)598 1993 y(\000)688 1971 y Fz(in)i(that)g(form)m (ula)f(w)m(e)j(get)e(that)h(the)g(comm)m(utator)e(brings)h(an)g(extra)h (deca)m(y)h(\(to)f(the)150 2092 y(already)25 b(existing)h(1)p Fu(=\034)36 b Fz(in)25 b(fron)m(t)h(of)f(the)i(scalar)e(pro)s(duct\))h (of)f(order)h Fu(t)2677 2056 y Fm(\000)p Fs(1)p Ft(=)p Fs(2+)p Ft(\017)2921 2065 y Fj(4)2960 2092 y Fz(,)i(uniformly)23 b(in)i Fu(\026)j(>)f Fz(1;)150 2212 y(notice)37 b(that)g(the)h(in)m (tegral)e(with)h(resp)s(ect)i(to)e Fu(z)43 b Fz(is)37 b(also)f(absolutely)h(con)m(v)m(ergen)m(t)j(\(put)d Fu(k)j Fz(=)35 b(2)j(in)150 2332 y(\(6.4\)\).)61 b(Finally)-8 b(,)37 b(apply)h(Lemma)f(6.4)h(and)h(w)m(e)g(are)f(done)h(with)f(all)f (the)i(con)m(tributions)f(coming)150 2453 y(from)31 b(the)i(Heisen)m(b) s(erg)h(deriv)-5 b(ativ)m(e)32 b(of)g(the)h(maximal)c(v)m(elo)s(cit)m (y)k(cut-o\013.)296 2573 y(But)d(there)f(are)g(some)g(other)h (cut-o\013s)f(whic)m(h)g(ha)m(v)m(e)i(to)e(b)s(e)g(di\013eren)m (tiated.)42 b(T)-8 b(ak)m(e)30 b(for)f(instance)150 2694 y(the)i(con)m(tribution)e(coming)f(from)h(the)i(Heisen)m(b)s(erg)g (deriv)-5 b(ativ)m(e)29 b(of)h Fu(F)2720 2650 y Fs(\001)2779 2659 y Fj(2)2706 2715 y Fs(+)2817 2694 y Fz(\()9 b(~)-58 b Fu(\032)18 b Fr(\000)f Fu(\032)3067 2709 y Ft(E)3127 2694 y Fz(\))30 b(\(see)h(\(6.43\))f(and)150 2814 y(\(6.50\)\).)39 b(Denote)23 b(for)f(simplicit)m(y)d Fu(F)1479 2770 y Fs(\001)1538 2779 y Fj(2)1465 2836 y Fs(+)1599 2814 y Fz(with)j Fu(F)1874 2829 y Fs(+)1933 2814 y Fz(.)40 b(Then)24 b Fu(D)2326 2829 y Ft(H)2393 2814 y Fu(F)2456 2829 y Fs(+)2537 2814 y Fz(will)d(ha)m(v)m(e)i(only)f(one)h(\\dangerous")150 2934 y(term)33 b(with)h(a)f(deca)m(y)j(of)d(just)h(1)p Fu(=r)i Fz(but)e(this)g(one)g(will)d(also)i(con)m(tain)h Fu(F)2804 2898 y Fm(0)2790 2959 y Fs(+)2883 2934 y Fz(whic)m(h)g(is)f (supp)s(orted)i(on)150 3055 y(the)28 b(classical)f(forbidden)h(region)f (\(see)i(\(6.46\)\))e(hence)j(in)m(tegrable.)41 b(The)29 b(go)s(o)s(d)e(thing)g(here)i(is)e(that)150 3175 y(w)m(e)34 b(do)e(not)h(ha)m(v)m(e)g(to)g(comm)m(ute)f(an)m(ything)g(with)g (\000\()p Fu(B)2168 3190 y Ft(\034)t(;\026)2270 3175 y Fz(\).)296 3296 y(There)f(is)f(a)g(third)f(t)m(yp)s(e)i(of)e(terms)h (in)f Fu(R)q Fz(\()p Fu(\034)6 b(;)17 b(\026)p Fz(\),)30 b(coming)e(for)i(instance)g(from)e(the)j(H-deriv)-5 b(ativ)m(e)150 3416 y(of)40 b Fu(F)332 3431 y Fs(+)392 3416 y Fz(\()440 3390 y(~)430 3416 y Fu(\030)t Fz(\))h(in)f(\(6.57\).)67 b(In)41 b(this)g(case)g(w)m(e)h(again)e(ha)m(v)m(e)i(to)e(comm)m(ute) 2774 3334 y Fl(p)p 2874 3334 123 4 v 82 x Fu(F)2951 3382 y Fm(0)2937 3438 y Fs(+)2996 3416 y Fz(\()3044 3390 y(~)3034 3416 y Fu(\030)5 b Fz(\))40 b(with)h(\000\()p Fu(B)3564 3431 y Ft(\034)t(;\026)3665 3416 y Fz(\))g(in)150 3536 y(order)32 b(to)g(apply)g(Lemma)f(6.14.)43 b(If)32 b Fu(F)1557 3500 y Ft(v)r(:b:)1699 3536 y Fz(is)g(lik)m(e)g(in)f(Prop)s (osition)g(6.19,)g(then)i(it)f(su\016ces)i(to)e(sho)m(w)150 3657 y(the)h(b)s(ound)997 3755 y Fl(\015)997 3815 y(\015)997 3875 y(\015)997 3934 y(\015)1052 3764 y(Z)1107 3990 y Fe(C)p 1176 3818 57 4 v 1176 3900 a Fu(@)1239 3874 y Fz(~)1233 3900 y(\000\()p Fu(z)t Fz(\))1436 3789 y Fl(h)1483 3813 y(p)p 1583 3813 123 4 v 87 x Fu(F)1660 3865 y Fm(0)1646 3921 y Fs(+)1705 3900 y Fz(\()1753 3873 y(~)1743 3900 y Fu(\030)t Fz(\))p Fu(;)17 b Fz(\()p Fu(B)1984 3915 y Ft(\034)t(;\026)2107 3900 y Fr(\000)23 b Fu(z)t Fz(\))2294 3858 y Fm(\000)p Fs(1)2389 3789 y Fl(i)2453 3900 y Fu(F)2530 3858 y Ft(v)r(:b:)2640 3900 y Fz(\()p Fu(r)s(=\034)11 b Fz(\))p Fu(dxdy)3074 3755 y Fl(\015)3074 3815 y(\015)3074 3875 y(\015)3074 3934 y(\015)997 4118 y Fr(\024)28 b Fu(C)7 b(\034)1232 4077 y Fm(\000)p Fs(1)p Ft(=)p Fs(2+)p Ft(\017)1476 4086 y Fj(4)1515 4118 y Fu(;)2034 b Fz(\(7.40\))150 4329 y(uniformly)30 b(in)i Fu(\026)c(>)f Fz(1.)43 b(T)-8 b(o)33 b(pro)m(v)m(e)h(\(7.40\))e(w)m(e)h(expand)h(the)f(comm)m(utator) 507 4456 y Fl(h)554 4481 y(p)p 653 4481 V 653 4567 a Fu(F)730 4533 y Fm(0)716 4589 y Fs(+)776 4567 y Fz(\()824 4541 y(~)814 4567 y Fu(\030)t Fz(\))p Fu(;)17 b Fz(\()p Fu(B)1055 4582 y Ft(\034)t(;\026)1178 4567 y Fr(\000)23 b Fu(z)t Fz(\))1365 4526 y Fm(\000)p Fs(1)1460 4456 y Fl(i)1534 4567 y Fz(=)28 b Fr(\000)p Fz(\()p Fu(B)1827 4582 y Ft(\034)t(;\026)1951 4567 y Fr(\000)23 b Fu(z)t Fz(\))2138 4526 y Fm(\000)p Fs(1)2249 4456 y Fl(h)2296 4481 y(p)p 2396 4481 V 86 x Fu(F)2473 4533 y Fm(0)2459 4589 y Fs(+)2518 4567 y Fz(\()2566 4541 y(~)2556 4567 y Fu(\030)t Fz(\))p Fu(;)17 b(B)2759 4582 y Ft(\034)t(;\026)2860 4456 y Fl(i)2924 4567 y Fz(\()p Fu(B)3036 4582 y Ft(\034)t(;\026)3160 4567 y Fr(\000)22 b Fu(z)t Fz(\))3346 4526 y Fm(\000)p Fs(1)3441 4567 y Fu(;)150 4805 y Fz(and)29 b(substitute)g(in)m(to)f (the)h(in)m(tegral)e(in)h(\(7.40\).)42 b(As)29 b(b)s(efore)g(w)m(e)h(v) m(erify)f(the)g(absolute)f(in)m(tegrabilit)m(y)150 4925 y(of)h(the)i(in)m(tegral)d(b)m(y)i(pro)m(viding)f(a)h(b)s(ound)g(for)f (the)h(in)m(tegrand)g(that)f(exhibits)h(appropriate)f Fu(z)t Fz(-)h(and)150 5046 y Fu(\034)11 b Fz(-deca)m(y)-8 b(.)296 5166 y(The)40 b(argumen)m(t)f(closely)g(follo)m(ws)f(the)i(one) f(w)m(e)i(used)f(in)e(the)i(pro)s(of)e(Lemma)g(7.2,)j(with)e(just)150 5286 y(one)g(notable)e(di\013erence:)57 b(when)39 b(comm)m(uting)1947 5205 y Fl(p)p 2046 5205 V 2046 5286 a Fu(F)2123 5252 y Fm(0)2109 5308 y Fs(+)2169 5286 y Fz(\()2217 5260 y(~)2207 5286 y Fu(\030)t Fz(\))f(with)g Fu(B)2632 5301 y Ft(\034)t(;\026)2772 5286 y Fz(w)m(e)i(do)e(not)g(automatically)150 5407 y(get)g(a)f(1)p Fu(=\034)49 b Fz(but)38 b(rather)g(a)g(1)p Fu(=r)i Fz(deca)m(y;)i(ho)m (w)m(ev)m(er)e(the)e(presence)i(of)e Fu(F)2792 5371 y Ft(v)r(:b:)2939 5407 y Fz(in)f(the)i(in)m(tegrand)e(will)150 5527 y(transform)32 b(1)p Fu(=r)i Fz(in)m(to)e(1)p Fu(=\034)11 b Fz(.)1939 5776 y(78)p eop %%Page: 79 82 79 81 bop 296 407 a Fz(No)m(w)30 b(let)f(us)h(giv)m(e)f(details.)41 b(W)-8 b(e)30 b(pic)m(k)g(a)f(smo)s(oth)f(function)h Fu(G)2561 422 y Fs(+)p Fm(\000)2704 407 y Fz(of)g(the)h(t)m(yp)s(e)g Fu(F)3256 422 y Fs(+)p Fm(\000)3399 407 y Fz(supp)s(orted)150 527 y(on)42 b Fn(R)361 542 y Fs(+)468 527 y Fz(and)g(equal)g(to)g(1)f (on)h(the)h(supp)s(ort)f(of)g(the)g(function)g Fu(F)2613 491 y Ft(v)r(:b:)2723 527 y Fz(.)71 b(Using)42 b(Lemmas)f(7.4)h(and)150 672 y(6.1)36 b(w)m(e)i(ma)m(y)e(put)h(a)g(factor)f Fu(G)1303 687 y Fs(+)p Fm(\000)1417 672 y Fz(\()p Fu(r)s(=\034)11 b Fz(\))36 b(next)i(to)e(the)h(comm)m(utator)2752 561 y Fl(h)2799 590 y(p)p 2898 590 123 4 v 2898 672 a Fu(F)2975 638 y Fm(0)2961 694 y Fs(+)3020 672 y Fz(\()3068 646 y(~)3058 672 y Fu(\030)5 b Fz(\))p Fu(;)17 b(B)3262 687 y Ft(\034)t(;\026)3363 561 y Fl(i)3446 672 y Fz(since)38 b(the)150 824 y(comm)m(utators)33 b(with)g(\()p Fu(B)1075 839 y Ft(\034)t(;\026)1199 824 y Fr(\000)24 b Fu(z)t Fz(\))1387 788 y Fm(\000)p Fs(1)1515 824 y Fz(induced)35 b(b)m(y)f(this)g(op)s(eration)e(pro)s(duce)i(in)m(tegrable)f(terms)h (in)150 944 y(agreemen)m(t)f(with)f(\(7.40\).)296 1092 y(Computing)806 982 y Fl(h)853 1011 y(p)p 952 1011 V 952 1092 a Fu(F)1029 1058 y Fm(0)1015 1114 y Fs(+)1075 1092 y Fz(\()1123 1066 y(~)1113 1092 y Fu(\030)t Fz(\))p Fu(;)17 b(B)1316 1107 y Ft(\034)t(;\026)1417 982 y Fl(i)1497 1092 y Fz(as)32 b(in)g(\(7.33\),)g(w)m(e)i(ha)m(v)m(e)g(to)e(deal)g (with)370 1327 y Fu(\034)423 1291 y Fs(1+2)p Ft(\017)577 1300 y Fj(4)p 370 1371 247 4 v 464 1463 a Fu(\026)626 1394 y Fz(\()p Fu(B)738 1409 y Ft(\034)t(;\026)861 1394 y Fr(\000)23 b Fu(z)t Fz(\))1048 1353 y Fm(\000)p Fs(1)1147 1394 y Fz(~)-53 b Fu(\015)1194 1409 y Ft(j)1247 1284 y Fl(h)1298 1394 y Fz(~)g Fu(\015)1345 1409 y Ft(j)1381 1394 y Fu(;)1425 1308 y Fl(p)p 1524 1308 123 4 v 1524 1394 a Fu(F)1601 1360 y Fm(0)1587 1416 y Fs(+)1647 1394 y Fz(\()1695 1368 y(~)1685 1394 y Fu(\030)t Fz(\))1770 1284 y Fl(i)1833 1394 y Fu(G)1910 1409 y Fs(+)p Fm(\000)2024 1394 y Fz(\()p Fu(r)s(=\034)11 b Fz(\)\()p Fu(B)2361 1409 y Ft(\034)t(;\026)2485 1394 y Fr(\000)22 b Fu(z)t Fz(\))2671 1353 y Fm(\000)p Fs(1)2766 1394 y Fz(;)45 b Fu(j)33 b Fr(2)28 b(f)p Fz(1)p Fu(;)17 b Fz(2)p Fu(;)g Fz(3)p Fr(g)p Fu(:)209 b Fz(\(7.41\))150 1648 y(\(and)33 b(a)f(similar)e(expression)k(with)i(~)-52 b Fu(\015)1524 1663 y Ft(j)1592 1648 y Fz(to)33 b(the)g(righ)m(t\).)44 b(By)33 b(coupling)e(one)j(resolv)m(en)m(t)f(with)52 b(~)-68 b Fu(\015)3599 1663 y Ft(j)3668 1648 y Fz(and)150 1768 y(estimating)30 b(the)j(norm)f(as)h(in)f(\(7.34\),)g(w)m(e)h(see)h (that)e(\(7.40\))g(follo)m(ws)f(from)1411 1884 y Fl(\015)1411 1944 y(\015)1411 2003 y(\015)1466 1888 y(h)1517 1998 y Fz(~)-53 b Fu(\015)1564 2013 y Ft(j)1600 1998 y Fu(;)1644 1912 y Fl(p)p 1743 1912 V 1743 1998 a Fu(F)1820 1964 y Fm(0)1806 2020 y Fs(+)1866 1998 y Fz(\()1914 1972 y(~)1904 1998 y Fu(\030)t Fz(\))1989 1888 y Fl(i)2053 1998 y Fu(r)2100 1884 y Fl(\015)2100 1944 y(\015)2100 2003 y(\015)2182 1998 y Fr(\024)28 b Fz(Const)q Fu(;)1012 b Fz(\(7.42\))150 2247 y(\(this)33 b(is)g(just)g(a)g(consequence)k(of)32 b(the)i(fact)f(that)g Fu(i)p Fz([)t(~)-53 b Fu(\015)2101 2262 y Ft(j)2138 2247 y Fu(;)2192 2220 y Fz(~)2182 2247 y Fu(\030)t Fz(])33 b(brings)g(an)g(extra)h(1)p Fu(=r)i Fz(factor\).)45 b(Finally)-8 b(,)150 2367 y(1)p Fu(=r)35 b Fz(is)d(transformed)g(in)m(to)g(1)p Fu(=\034)43 b Fz(b)m(y)34 b(the)f(factor)f Fu(G)2011 2382 y Fs(+)p Fm(\000)2125 2367 y Fz(\()p Fu(r)s(=\034)11 b Fz(\))32 b(and)h(w)m(e)g(are)g(done.) 296 2487 y(W)-8 b(e)27 b(therefore)g(conclude)f(that)g Fr(j)p Fu(R)q Fz(\()p Fu(\034)6 b(;)17 b(\026)p Fz(\))p Fr(j)25 b Fz(is)h(in)m(tegrable)f(in)h Fu(\034)37 b Fz(uniformly)24 b(in)i Fu(\026)h(>)h Fz(1,)f(cf.)42 b(\(7.31\).)150 2745 y Fv(7.2.3)112 b(The)38 b(Heisen)m(b)s(erg)f(deriv)-6 b(ativ)m(e)36 b(of)i Fz(\000\()p Fu(B)2119 2760 y Ft(\034)t(;\026)2220 2745 y Fz(\))150 2929 y(W)-8 b(e)35 b(con)m(tin)m(ue)h(the)f(v)m (eri\014cation)f(of)h(\(7.31\))f(\(for)g(some)g Fu(\021)i Fr(2)c Fu(L)2470 2893 y Fs(1)2509 2929 y Fz(\))j(b)m(y)h(considering)e (the)h(remaining)150 3050 y(con)m(tribution)i(from)g(the)i(Heisen)m(b)s (erg)g(deriv)-5 b(ativ)m(e)37 b(of)h(\000)17 b(\()p Fu(B)2382 3065 y Ft(\034)t(;\026)2483 3050 y Fz(\).)60 b(F)-8 b(ormally)35 b(using)j(\(6.7\))g(with)g Fu(A)150 3170 y Fz(giv)m(en)33 b(b)m(y)g Fu(@)591 3185 y Ft(\034)657 3170 y Fz(+)22 b Fu(iH)8 b Fz(,)32 b(w)m(e)i(get)876 3373 y Fr(h)p Fu(D)996 3388 y Ft(H)1063 3373 y Fz(\000\()p Fu(B)1236 3388 y Ft(\034)t(;\026)1338 3373 y Fz(\))p Fr(i)1414 3405 y Ft( )1460 3414 y Fj(11)1557 3373 y Fz(=)28 b Fu(\026)1720 3332 y Fm(\000)p Fs(1)1814 3373 y Fu(\034)1867 3332 y Fs(1+2)p Ft(\017)2021 3341 y Fj(4)2077 3373 y Fr(h)o Fu(C)2185 3388 y Ft(\034)2228 3373 y Fr(i)2267 3402 y Fs(\011)2322 3410 y Fq(\026)2390 3373 y Fz(+)22 b Fr(h)p Fu(R)2601 3388 y Fs(1)2640 3373 y Fr(i)2679 3388 y Ft( )2725 3397 y Fj(11)2816 3373 y Fz(+)g Fr(h)p Fu(R)3027 3388 y Fs(2)3067 3373 y Fr(i)3106 3388 y Ft( )3152 3397 y Fj(11)3221 3373 y Fz(;)876 3580 y Fu(C)946 3595 y Ft(\034)1017 3580 y Fz(=)28 b(\()o(1)22 b(+)g(2)p Fu(\017)1415 3595 y Fs(4)1455 3580 y Fz(\))17 b Fu(\034)1563 3539 y Fm(\000)p Fs(1)1657 3580 y Fu(L)1723 3595 y Ft(\016)1784 3580 y Fz(+)22 b Fu(D)1963 3595 y Ft(H)2030 3580 y Fu(L)2096 3595 y Ft(\016)2134 3580 y Fu(;)114 b Fz(\011)2351 3595 y Ft(\026)2426 3580 y Fz(=)2529 3461 y Fl(q)p 2629 3461 336 4 v 119 x Fz(\000)2690 3551 y Fm(0)2713 3580 y Fz(\()p Fu(B)2825 3595 y Ft(\034)t(;\026)2926 3580 y Fz(\))p Fu( )3027 3595 y Fs(11)3102 3580 y Fu(:)447 b Fz(\(7.43\))150 3797 y(W)-8 b(e)32 b(no)m(w)h(concen)m(trate)g(on)f(the)g(term)f(in)m (v)m(olving)g(the)h(\\\014rst)g(comm)m(utator")e Fu(C)3077 3812 y Ft(\034)3120 3797 y Fz(.)43 b(If)32 b(w)m(e)h(could)e(use)150 3917 y(\(7.28\))39 b(then)i(b)m(y)g(c)m(ho)s(osing)e Fu(\017)1254 3932 y Fs(4)1334 3917 y Fz(small)f(enough,)k(the)f (dangerous)f(term)g(whic)m(h)g(only)g(deca)m(ys)i(lik)m(e)150 4038 y(1)p Fu(=\034)i Fz(b)s(ecomes)34 b(negativ)m(e)g(so)f(w)m(e)i (can)e(discard)g(it.)45 b(Remem)m(b)s(er)33 b(that)g(\(7.28\))g(w)m(as) h(deriv)m(ed)g(ha)m(ving)150 4158 y(in)f(mind)f(that)i(b)m(y)g(sligh)m (tly)e(enlarging)g(the)i(supp)s(orts)g(of)f(the)h(v)-5 b(arious)33 b(cut-o\013s)h(building)d Fu( )3605 4173 y Fs(11)3714 4158 y Fz(w)m(e)150 4278 y(ma)m(y)39 b(put)g(them)f(an)m (ywhere)j(w)m(e)f(w)m(an)m(t)g(at)f(the)g(exp)s(ense)i(of)d Fr(O)s Fz(\()p Fu(\034)2603 4242 y Fm(\0001)2734 4278 y Fz(\))p Fr(jj)p Fu( )t Fr(jj)f Fz(errors.)63 b(Hence)40 b(if)e(w)m(e)150 4399 y(pro)m(v)m(e)e(that)e(w)m(e)i(can)f(comm)m(ute)f (the)h(old)e(cut-o\013s)i(o)m(v)m(er)2269 4315 y Fl(p)p 2368 4315 V 2368 4399 a Fz(\000)2429 4370 y Fm(0)2452 4399 y Fz(\()p Fu(B)2564 4414 y Ft(\034)t(;\026)2666 4399 y Fz(\))f(in)g(the)h(same)f(w)m(a)m(y)-8 b(,)37 b(then)e(w)m(e)150 4519 y(are)28 b(done.)42 b(But)29 b(this)e(is)h(essen)m(tially)f(con)m(tained)i(in)e(Lemmas)g(7.2)h(and)g (7.4)f(and)h(then)h(in)e(the)i(pro)s(of)150 4640 y(follo)m(wing)h (\(7.40\).)43 b(W)-8 b(e)33 b(giv)m(e)f(no)h(other)f(details.)296 4760 y(No)m(w)45 b(let)d(us)j(go)e(bac)m(k)h(to)g(the)g(in)m(v)m (estigation)e(of)h Fu(C)2282 4775 y Ft(\034)2369 4760 y Fz(in)f(\(7.43\).)76 b(Namely)-8 b(,)46 b(w)m(e)e(treat)g(\\the)150 4880 y(quan)m(tum)36 b(errors")g(in)m(tro)s(duced)g(b)m(y)h Fr(O)s Fz(\()t(^)-53 b Fu(\015)5 b(=\034)1802 4844 y Fs(2)1842 4880 y Fz(\))35 b(in)g(the)i(righ)m(t)e(hand)h(side)g(of)f (\(7.28\).)52 b(In)37 b(this)e(case)150 5001 y(w)m(e)28 b(can)g(use)g(the)f(b)s(ound)h Fr(jj)t Fz(^)-53 b Fu(\015)1190 5016 y Ft(j)1225 5001 y Fz(\011)1301 5016 y Ft(\026)1348 5001 y Fz(\()p Fu(\034)11 b Fz(\))p Fr(jj)27 b(\024)h Fz(Const)h Fu(\026)2002 4965 y Fs(1)p Ft(=)p Fs(2)2111 5001 y Fu(\034)2164 4965 y Fm(\000)p Fs(1)p Ft(=)p Fs(2)p Fm(\000)p Ft(\017)2408 4974 y Fj(4)2475 5001 y Fz(whic)m(h)f(put)f(bac) m(k)h(in)m(to)f(\(7.43\))f(leads)150 5121 y(to)32 b(a)h(con)m (tribution)e(of)h(order)1030 5324 y Fu(\026)1089 5283 y Fm(\000)p Fs(1)1183 5324 y Fu(\034)1236 5283 y Fs(1+2)p Ft(\017)1390 5292 y Fj(4)1451 5324 y Fr(\001)22 b Fu(\026)1560 5283 y Fs(1)p Ft(=)p Fs(2)1669 5324 y Fu(\034)1722 5283 y Fm(\000)p Fs(1)p Ft(=)p Fs(2)p Fm(\000)p Ft(\017)1966 5292 y Fj(4)2028 5324 y Fr(\001)g Fu(\034)2131 5283 y Fm(\000)p Fs(2)2253 5324 y Fr(\024)28 b Fu(\026)2417 5283 y Fm(\000)p Fs(1)p Ft(=)p Fs(2)2582 5324 y Fu(\034)2635 5283 y Fm(\000)p Fs(3)p Ft(=)p Fs(2+)p Ft(\017)2879 5292 y Fj(4)2918 5324 y Fu(;)150 5527 y Fz(whic)m(h)33 b(clearly)f(implies)e (uniform)h(in)m(tegrabilit)m(y)-8 b(.)1939 5776 y(79)p eop %%Page: 80 83 80 82 bop 296 407 a Fz(W)-8 b(e)29 b(no)m(w)g(treat)f(the)g(last)f(t)m (w)m(o)i(terms)f(on)g(the)h(righ)m(t)e(hand)h(side)h(of)e(\(7.43\).)41 b(Since)29 b(they)g(in)m(v)m(olv)m(e)150 527 y(a)j(comm)m(utator)f(b)s (et)m(w)m(een)k Fu(L)1224 542 y Ft(\016)1295 527 y Fz(and)d Fu(D)1565 542 y Ft(H)1633 527 y Fu(L)1699 542 y Ft(\016)1737 527 y Fz(,)h(w)m(e)g(are)g(motiv)-5 b(ated)31 b(to)h(write)g(it)g(as) 721 832 y([)p Fu(L)814 847 y Ft(\016)852 832 y Fu(;)17 b(D)977 847 y Ft(H)1044 832 y Fu(L)1110 847 y Ft(\016)1148 832 y Fz(])28 b Fr(')1400 707 y Fs(3)1345 737 y Fl(X)1308 949 y Ft(j;k)r(;l)q Fs(=1)1543 832 y Fu(A)1616 847 y Ft(j)t(k)r(l)1717 832 y Fz(~)-53 b Fu(\015)1764 847 y Ft(j)1804 832 y Fz(~)g Fu(\015)1851 847 y Ft(k)1897 832 y Fz(~)g Fu(\015)1944 847 y Ft(l)1992 832 y Fz(+)2161 707 y Fs(3)2106 737 y Fl(X)2090 949 y Ft(j;k)r Fs(=1)2283 832 y Fu(A)2356 847 y Ft(j)t(k)2435 832 y Fz(~)h Fu(\015)2483 847 y Ft(j)2522 832 y Fz(~)g Fu(\015)2570 847 y Ft(k)2634 832 y Fz(+)2786 707 y Fs(3)2732 737 y Fl(X)2743 947 y Ft(j)t Fs(=1)2893 832 y Fu(A)2966 847 y Ft(j)3006 832 y Fz(~)f Fu(\015)3053 847 y Ft(j)3111 832 y Fz(+)22 b Fu(A)p Fz(;)721 1089 y Fu(A)794 1104 y Ft(j)t(k)r(l)918 1089 y Fz(=)28 b Fr(O)s Fz(\()p Fu(\034)1195 1048 y Fm(\000)p Fs(2)1290 1089 y Fz(\))p Fu(;)114 b(A)1542 1104 y Ft(j)t(k)1645 1089 y Fz(=)27 b Fr(O)s Fz(\()p Fu(\034)1921 1048 y Fm(\000)p Fs(3)2016 1089 y Fz(\))p Fu(;)114 b(A)2268 1104 y Ft(j)2333 1089 y Fz(=)27 b Fr(O)s Fz(\()p Fu(\034)2609 1048 y Fm(\000)p Fs(4)2704 1089 y Fz(\))p Fu(;)114 b(A)28 b Fz(=)g Fr(O)s Fz(\()p Fu(\034)3261 1048 y Fm(\000)p Fs(5)3355 1089 y Fz(\))p Fu(;)156 b Fz(\(7.44\))150 1309 y(whic)m(h)33 b(is)f(obtained)g(b)m(y)i(rep)s(eatedly)f(applying)e(\(7.11\).)43 b(With)32 b(this)g(form)m(ula)f(w)m(e)i(can)g(no)m(w)g(pro)m(v)m(e)150 1513 y Fv(Lemma)k(7.5.)42 b FA(The)34 b(r)-5 b(emainder)34 b Fu(R)1502 1528 y Fs(1)1577 1513 y FA(in)g(\(7.43\))g(ob)-5 b(eys)1336 1733 y Fz(sup)1343 1812 y Ft(\026>)p Fs(1)1499 1733 y Fr(jh)p Fu(R)1640 1748 y Fs(1)1679 1733 y Fr(i)1718 1748 y Ft( )1764 1757 y Fj(1)1803 1733 y Fr(j)27 b(\024)h Fz(Const)h Fu(\034)2294 1692 y Fm(\000)p Fs(3)p Ft(=)p Fs(2+3)p Ft(\017)2573 1701 y Fj(4)2612 1733 y Fu(:)937 b Fz(\(7.45\))150 2017 y Fv(Pro)s(of.)43 b Fz(Lo)s(oking)32 b(at)g(\(6.6\),)g(w)m(e)i(see)f(that)g(the)g(relev)-5 b(an)m(t)32 b(quan)m(tit)m(y)h(to)g(b)s(ound)f(is)452 2229 y Fu(\034)505 2193 y Fs(2+4)p Ft(\017)659 2202 y Fj(4)p 452 2274 247 4 v 526 2365 a Fu(\026)585 2336 y Fs(2)725 2161 y Fl(Z)780 2387 y Fe(C)849 2297 y Fr(j)p 877 2215 57 4 v Fu(@)940 2271 y Fz(~)934 2297 y(\000)p Fr(j)p Fz(\()p Fu(z)t Fz(\))1165 2212 y Fl(\015)1165 2272 y(\015)1220 2297 y Fz(\()p Fu(B)1332 2312 y Ft(\034)t(;\026)1455 2297 y Fr(\000)23 b Fu(z)t Fz(\))1642 2255 y Fm(\000)p Fs(1)1737 2297 y Fz([)p Fu(L)1830 2312 y Ft(\016)1868 2297 y Fu(;)17 b(D)1993 2312 y Ft(H)2060 2297 y Fu(L)2126 2312 y Ft(\016)2165 2297 y Fz(]\()p Fu(B)2304 2312 y Ft(\034)t(;\026)2427 2297 y Fr(\000)23 b Fu(z)t Fz(\))2614 2255 y Fm(\000)p Fs(2)2709 2297 y Fu( )2772 2312 y Fs(11)2847 2297 y Fz(\()p Fu(\034)11 b Fz(\))2976 2212 y Fl(\015)2976 2272 y(\015)3048 2297 y Fu(dxdy)t Fz(;)292 b(\(7.46\))150 2569 y(and)44 b(a)g(similar)d(expression)k(with)f(the)g(p)s(o)m(w)m (ers)i(of)e(resolv)m(en)m(ts)h(in)m(terc)m(hanged)g(\(whic)m(h)g(ma)m (y)f(b)s(e)150 2690 y(treated)33 b(similarly\).)296 2810 y(W)-8 b(e)36 b(no)m(w)g(insert)f(eac)m(h)h(term)e(from)g(\(7.44\))g (in)m(to)h(\(7.46\))f(and)h(c)m(hec)m(k)i(the)f(deca)m(y)g(in)f Fu(\034)11 b Fz(.)51 b(Let)35 b(us)150 2930 y(start)e(with)915 3018 y Fu(\034)968 2981 y Fs(2+4)p Ft(\017)1122 2990 y Fj(4)p 915 3062 247 4 v 989 3153 a Fu(\026)1048 3125 y Fs(2)1187 3000 y Fl(\015)1187 3060 y(\015)1243 3085 y Fz(\()p Fu(B)1355 3100 y Ft(\034)t(;\026)1478 3085 y Fr(\000)23 b Fu(z)t Fz(\))1665 3044 y Fm(\000)p Fs(1)1760 3085 y Fu(A)1833 3100 y Ft(j)t(k)r(l)1933 3085 y Fz(^)-52 b Fu(\015)1981 3100 y Ft(j)2021 3085 y Fz(^)f Fu(\015)2068 3100 y Ft(k)2114 3085 y Fz(^)g Fu(\015)2161 3100 y Ft(l)2187 3085 y Fz(\()p Fu(B)2299 3100 y Ft(\034)t(;\026)2422 3085 y Fr(\000)23 b Fu(z)t Fz(\))2609 3044 y Fm(\000)p Fs(2)2704 3085 y Fu( )2767 3100 y Fs(11)2842 3085 y Fz(\()p Fu(\034)11 b Fz(\))2971 3000 y Fl(\015)2971 3060 y(\015)3043 3085 y Fz(;)150 3304 y(w)m(e)34 b(claim)c(there)j(is)f(a)g(uniform)f (upp)s(er)i(b)s(ound)g(of)f(the)h(form)1254 3585 y Fu(C)1355 3517 y Fr(h)27 b Fu(z)t Fr(i)1509 3481 y Fs(1)p Ft(=)p Fs(2+)p Ft(m)p 1341 3562 410 4 v 1341 3653 a Fr(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)1594 3624 y Fs(2+)p Ft(m)1760 3585 y Fu(\034)1813 3544 y Fm(\000)p Fs(3)p Ft(=)p Fs(2+3)p Ft(\017)2092 3553 y Fj(4)2132 3585 y Fu(\026)2191 3544 y Fm(\000)p Fs(1)p Ft(=)p Fs(2)2356 3585 y Fu(;)44 b(m)28 b Fr(\025)g Fz(1)p Fu(;)855 b Fz(\(7.47\))150 3861 y(whic)m(h)26 b(together)g(with)f(\(6.4\))g(yields)g(absolutely)g(in)m(tegrabilit)m (y)d(in)j Fu(z)30 b Fz(and)c(agreemen)m(t)g(with)f(\(7.45\).)150 3981 y(The)34 b(other)e(terms)h(from)e(\(7.44\))h(will)e(ob)s(ey)j(the) g(same)g(b)s(ound.)296 4101 y(First,)c(notice)f(that)g(w)m(e)i(ma)m(y)e (rewrite)g(the)h(middle)e(term)h(as)k(^)-53 b Fu(\015)2596 4116 y Ft(j)2633 4101 y Fu(A)2706 4116 y Ft(j)t(k)r(l)2806 4101 y Fz(^)h Fu(\015)2854 4116 y Ft(k)2899 4101 y Fz(^)g Fu(\015)2947 4116 y Ft(l)3001 4101 y Fz(since)29 b(the)g(comm)m(uta-) 150 4222 y(tor)k(b)s(et)m(w)m(een)i Fu(A)758 4237 y Ft(j)t(k)r(l)888 4222 y Fz(and)i(^)-52 b Fu(\015)1130 4237 y Ft(j)1199 4222 y Fz(b)s(eha)m(v)m(es)35 b(lik)m(e)e Fu(\034)1796 4186 y Fm(\000)p Fs(3)1924 4222 y Fz(and)g(therefore)h(ma)m(y)f(b)s(e)g (treated)h(along)e(with)h(the)150 4342 y(terms)39 b Fu(A)501 4357 y Ft(j)t(k)579 4342 y Fz(^)-52 b Fu(\015)627 4357 y Ft(j)666 4342 y Fz(^)g Fu(\015)714 4357 y Ft(k)795 4342 y Fz(from)37 b(\(7.44\).)61 b(Bounding)38 b(the)h(factor)j(^)-53 b Fu(\015)2328 4357 y Ft(j)2403 4342 y Fz(b)m(y)39 b(use)h(of)e (\(7.34\))g(yields)g(the)h(upp)s(er)150 4463 y(b)s(ounds)648 4717 y Fu(C)735 4650 y(\034)788 4613 y Fs(4)p Ft(\017)852 4622 y Fj(4)p 735 4694 157 4 v 764 4785 a Fu(\026)823 4756 y Fs(2)918 4632 y Fl(\015)918 4692 y(\015)973 4717 y Fz(\()p Fu(B)1085 4732 y Ft(\034)t(;\026)1208 4717 y Fr(\000)23 b Fu(z)t Fz(\))1395 4676 y Fm(\000)p Fs(1)1494 4717 y Fz(^)-53 b Fu(\015)1541 4732 y Ft(j)1577 4632 y Fl(\015)1577 4692 y(\015)1655 4717 y Fr(\001)1704 4602 y Fl(\015)1704 4662 y(\015)1704 4722 y(\015)1763 4717 y Fz(^)h Fu(\015)1811 4732 y Ft(k)1857 4717 y Fz(^)f Fu(\015)1904 4732 y Ft(l)1943 4692 y Fz(^)1930 4717 y(\011\()p Fu(\034)11 b Fz(\))2135 4602 y Fl(\015)2135 4662 y(\015)2135 4722 y(\015)648 4991 y Fr(\024)28 b Fz(Const)23 b Fr(\001)1085 4924 y Fu(\034)1138 4888 y Fs(3)p Ft(\017)1202 4897 y Fj(4)1237 4888 y Fm(\000)p Fs(1)p Ft(=)p Fs(2)p 1085 4968 317 4 v 1159 5060 a Fu(\026)1218 5031 y Fs(3)p Ft(=)p Fs(2)1430 4924 y Fr(h)p Fu(z)t Fr(i)1557 4888 y Fs(1)p Ft(=)p Fs(2)p 1422 4968 253 4 v 1422 5060 a Fr(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)1701 4877 y Fl(\015)1701 4936 y(\015)1701 4996 y(\015)1760 4991 y Fz(^)-52 b Fu(\015)1808 5006 y Ft(k)1853 4991 y Fz(^)g Fu(\015)1901 5006 y Ft(l)1940 4966 y Fz(^)1926 4991 y(\011\()p Fu(\034)11 b Fz(\))2131 4877 y Fl(\015)2131 4936 y(\015)2131 4996 y(\015)2204 4991 y Fu(;)2358 4966 y Fz(^)2345 4991 y(\011\()p Fu(\034)g Fz(\))28 b(=)f(\()p Fu(B)2793 5006 y Ft(\034)t(;\026)2917 4991 y Fr(\000)22 b Fu(z)t Fz(\))3103 4950 y Fm(\000)p Fs(2)3198 4991 y Fu( )3261 5006 y Fs(11)3337 4991 y Fz(\()p Fu(\034)11 b Fz(\))p Fu(:)150 5260 y Fz(Next,)34 b(write)346 5480 y(^)-53 b Fu(\015)393 5495 y Ft(k)439 5480 y Fz(^)g Fu(\015)486 5495 y Ft(l)526 5455 y Fz(^)512 5480 y(\011\()p Fu(\034)11 b Fz(\))28 b(=)j(^)-52 b Fu(\015)900 5495 y Ft(k)942 5480 y Fz(\()p Fu(B)1054 5495 y Ft(\034)t(;\026)1177 5480 y Fr(\000)23 b Fu(z)t Fz(\))1364 5439 y Fm(\000)p Fs(1)1463 5480 y Fz(^)-53 b Fu(\015)1510 5495 y Ft(l)1535 5480 y Fz(\()p Fu(B)1647 5495 y Ft(\034)t(;\026)1771 5480 y Fr(\000)23 b Fu(z)t Fz(\))1958 5439 y Fm(\000)p Fs(1)2053 5480 y Fu( )2116 5495 y Fs(11)2191 5480 y Fz(\()p Fu(\034)11 b Fz(\))22 b(+)k(^)-53 b Fu(\015)2491 5495 y Ft(k)2533 5480 y Fz(\()p Fu(B)2645 5495 y Ft(\034)t(;\026)2769 5480 y Fr(\000)23 b Fu(z)t Fz(\))2956 5439 y Fm(\000)p Fs(1)3051 5480 y Fz([)p Fu(B)3152 5495 y Ft(\034)t(;\026)3253 5480 y Fu(;)d Fz(^)-52 b Fu(\015)3348 5495 y Ft(l)3373 5480 y Fz(])3414 5455 y(^)3400 5480 y(\011\()p Fu(\034)11 b Fz(\))p Fu(:)1939 5776 y Fz(80)p eop %%Page: 81 84 81 83 bop 150 407 a Fz(Substituting)32 b(w)m(e)h(ha)m(v)m(e)h(to)e (estimate)936 620 y Fu(\034)989 584 y Fs(3)p Ft(\017)1053 593 y Fj(4)1088 584 y Fm(\000)p Fs(1)p Ft(=)p Fs(2)p 936 665 317 4 v 1010 756 a Fu(\026)1069 727 y Fs(3)p Ft(=)p Fs(2)1280 620 y Fr(h)p Fu(z)t Fr(i)1407 584 y Fs(1)p Ft(=)p Fs(2)p 1272 665 253 4 v 1272 756 a Fr(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)1552 603 y Fl(\015)1552 663 y(\015)1611 687 y Fz(^)-53 b Fu(\015)1658 702 y Ft(k)1700 687 y Fz(\()p Fu(B)1812 702 y Ft(\034)t(;\026)1936 687 y Fr(\000)23 b Fu(z)t Fz(\))2123 646 y Fm(\000)p Fs(1)2218 603 y Fl(\015)2218 663 y(\015)2295 687 y Fr(\001)2345 603 y Fl(\015)2345 663 y(\015)2404 687 y Fz(^)-53 b Fu(\015)2451 702 y Ft(l)2477 687 y Fz(\()p Fu(B)2589 702 y Ft(\034)t(;\026)2712 687 y Fr(\000)23 b Fu(z)t Fz(\))2899 646 y Fm(\000)p Fs(1)2994 603 y Fl(\015)2994 663 y(\015)3576 687 y Fz(\(7.48\))150 958 y(and)933 1052 y Fu(\034)986 1016 y Fs(3)p Ft(\017)1050 1025 y Fj(4)1085 1016 y Fm(\000)p Fs(1)p Ft(=)p Fs(2)p 933 1097 317 4 v 1007 1188 a Fu(\026)1066 1159 y Fs(3)p Ft(=)p Fs(2)1278 1052 y Fr(h)p Fu(z)t Fr(i)1405 1016 y Fs(1)p Ft(=)p Fs(2)p 1270 1097 253 4 v 1270 1188 a Fr(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)1549 1035 y Fl(\015)1549 1095 y(\015)1608 1119 y Fz(^)-52 b Fu(\015)1656 1134 y Ft(k)1698 1119 y Fz(\()p Fu(B)1810 1134 y Ft(\034)t(;\026)1933 1119 y Fr(\000)23 b Fu(z)t Fz(\))2120 1078 y Fm(\000)p Fs(1)2215 1035 y Fl(\015)2215 1095 y(\015)2293 1119 y Fr(\001)2342 1005 y Fl(\015)2342 1065 y(\015)2342 1124 y(\015)2398 1119 y Fz([)p Fu(B)2499 1134 y Ft(\034)t(;\026)2600 1119 y Fu(;)d Fz(^)-52 b Fu(\015)2695 1134 y Ft(l)2720 1119 y Fz(])2761 1094 y(^)2747 1119 y(\011\()p Fu(\034)11 b Fz(\))2952 1005 y Fl(\015)2952 1065 y(\015)2952 1124 y(\015)3025 1119 y Fu(:)524 b Fz(\(7.49\))150 1350 y(In)m(tro)s(ducing) 32 b(\(7.34\))g(in)g(\(7.48\))g(w)m(e)h(get)g(the)g(b)s(ound)1639 1569 y(2)p 1579 1613 169 4 v 1579 1704 a Fu(\026)1638 1675 y Fs(1)p Ft(=)p Fs(2)1757 1636 y Fu(\034)1810 1595 y Fm(\000)p Fs(3)p Ft(=)p Fs(2+)p Ft(\017)2054 1604 y Fj(4)2131 1569 y Fr(h)p Fu(z)t Fr(i)2258 1532 y Fs(3)p Ft(=)p Fs(2)p 2104 1613 293 4 v 2104 1704 a Fr(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)2357 1675 y Fs(3)150 1912 y Fz(whic)m(h)g(clearly)f (is)g(of)g(the)h(form)e(\(7.47\).)296 2032 y(Let)i(us)g(fo)s(cus)g(on)g (\(7.49\).)42 b(In)m(tro)s(ducing)33 b(again)e(\(7.34\))h(w)m(e)h(get) 1252 2173 y Fr(p)p 1335 2173 49 4 v 82 x Fz(2)p 1252 2300 132 4 v 1288 2391 a Fu(\026)1394 2323 y(\034)1447 2282 y Fm(\000)p Fs(1+2)p Ft(\017)1656 2291 y Fj(4)1787 2255 y Fr(h)p Fu(z)t Fr(i)p 1705 2300 293 4 v 1705 2391 a(j=)p Fz(\()p Fu(z)t Fz(\))p Fr(j)1958 2362 y Fs(2)2024 2208 y Fl(\015)2024 2268 y(\015)2024 2328 y(\015)2079 2323 y Fz([)p Fu(B)2180 2338 y Ft(\034)t(;\026)2281 2323 y Fu(;)21 b Fz(^)-53 b Fu(\015)2376 2338 y Ft(l)2402 2323 y Fz(])2443 2298 y(^)2429 2323 y(\011\()p Fu(\034)11 b Fz(\))2634 2208 y Fl(\015)2634 2268 y(\015)2634 2328 y(\015)2706 2323 y Fu(:)150 2599 y Fz(Computing)31 b(the)i(comm)m (utator)f(as)g(in)g(\(7.33\))g(yields)999 2878 y([)p Fu(B)1100 2893 y Ft(\034)t(;\026)1202 2878 y Fu(;)20 b Fz(^)-52 b Fu(\015)1297 2893 y Ft(l)1322 2878 y Fz(])28 b(=)1490 2811 y Fu(\034)1543 2774 y Fs(1+2)p Ft(\017)1697 2783 y Fj(4)p 1490 2855 247 4 v 1584 2946 a Fu(\026)1763 2797 y Fl(\010)1821 2878 y Fr(O)s Fz(\()p Fu(r)1988 2837 y Fm(\000)p Fs(1)2082 2878 y Fz(\))t(^)-53 b Fu(\015)2171 2893 y Ft(i)2221 2878 y Fz(+)22 b Fr(O)s Fz(\()p Fu(r)2486 2837 y Fm(\000)p Fs(2)2580 2878 y Fz(\))2618 2797 y Fl(\011)2693 2878 y Fu(;)33 b(i)28 b Fr(6)p Fz(=)f Fu(l)r(:)150 3148 y Fz(W)-8 b(e)30 b(substitute)g(and)g(use)h(the)f(minimal)25 b(v)m(elo)s(cit)m(y)k(cut-o\013)g(from)g Fu( )2613 3163 y Fs(11)2717 3148 y Fz(to)h(transform)e(1)p Fu(=r)k Fz(in)m(to)d(1)p Fu(=\034)11 b Fz(,)150 3269 y(cf.)45 b(Lemma)31 b(7.4.)44 b(F)-8 b(or)32 b(the)h(\014rst)h(term)e(w)m(e)i(then)g(use)f(\(7.34\))g (again.)43 b(In)33 b(conclusion,)f(b)s(oth)h(terms)150 3389 y(con)m(tribute)j(with)e(a)i(b)s(ound)f(of)g(the)h(form)e (\(7.47\))g(and)i(w)m(e)g(ma)m(y)f(deduce)i(that)e(the)h(con)m (tribution)150 3509 y(coming)k(from)g(the)i(\014rst)g(term)f(on)g(the)h (righ)m(t)f(hand)h(side)f(of)g(\(7.44\))g(to)g Fr(h)p Fu(R)3038 3524 y Fs(1)3077 3509 y Fr(i)3116 3524 y Ft( )3162 3533 y Fj(1)3242 3509 y Fz(b)s(eha)m(v)m(es)j(as)e(in)150 3630 y(\(7.45\).)296 3750 y(As)37 b(for)f(the)h(con)m(tribution)e(to)h Fr(h)p Fu(R)1564 3765 y Fs(1)1603 3750 y Fr(i)1642 3765 y Ft( )1688 3774 y Fj(1)1763 3750 y Fz(coming)e(from)h(the)i(second)h (term)d(on)i(the)f(righ)m(t)g(hand)150 3871 y(side)30 b(of)f(\(7.44\))g(the)h(situation)f(is)g(no)m(w)h(b)s(etter)h(since)f (w)m(e)h(trade)f(one)j(^)-52 b Fu(\015)2739 3886 y Ft(j)2804 3871 y Fz(with)30 b(an)g(extra)g(1)p Fu(=\034)41 b Fz(deca)m(y)150 3991 y(in)32 b Fu(A)337 4006 y Ft(j)t(k)412 3991 y Fz(;)h(similarly)28 b(for)k(the)h(remaining)e(terms.)43 b(Details)31 b(are)i(omitted.)p 3754 3991 4 66 v 3758 3928 59 4 v 3758 3991 V 3816 3991 4 66 v 150 4229 a(There)c(is)e(a)g(completely)f(similar)f(b)s(ound)i (for)g Fu(R)1910 4244 y Fs(2)1978 4229 y Fz(as)g(for)g Fu(R)2310 4244 y Fs(1)2377 4229 y Fz(in)g(\(7.45\).)41 b(W)-8 b(e)28 b(skip)g(the)g(pro)s(of)e(whic)m(h)150 4350 y(is)i(similar.)39 b(W)-8 b(e)29 b(conclude)g(that)g(\(7.31\))f (holds)g(for)g(an)h(in)m(tegrable)e Fu(\021)t Fz(.)42 b(Prop)s(osition)27 b(7.1)h(is)g(pro)m(v)m(en.)150 4610 y Fv(7.2.4)112 b(A)37 b(propagation)h(estimate)e(for)i Fu(L)1926 4625 y Ft(\016)150 4794 y Fv(Lemma)i(7.6.)j FA(Consider)37 b(a)g(function)g(of)g(the)g(typ)-5 b(e)38 b Fu(F)2209 4809 y Fm(\000)2305 4794 y FA(with)g Fu(a)32 b Fz(=)g(1)37 b FA(and)g Fu(b)c Fz(=)f(2)37 b FA(and)g(denote)g(it)150 4915 y(by)f Fz(\000)339 4930 y Fs(1)379 4915 y FA(.)49 b(Ther)-5 b(e)35 b(exist)h Fu(\017)1006 4930 y Fs(4)1077 4915 y Fu(>)30 b Fz(0)36 b FA(smal)5 b(l)35 b(enough)h(so)g(that)h (with)f Fu(B)2475 4930 y Ft(t)2535 4915 y Fz(:=)30 b Fu(t)2703 4879 y Fs(1+)p Ft(\017)2822 4888 y Fj(4)2861 4915 y Fu(L)2927 4930 y Ft(\016)3001 4915 y FA(and)36 b Fu( )3255 4930 y Fs(11)3330 4915 y Fz(\()p Fu(t)p Fz(\))g FA(given)g(by)150 5035 y(\(7.5\))e(we)h(have)1029 5071 y Fl(Z)1128 5097 y Fm(1)1084 5296 y Fs(1)1230 5139 y Fz(1)p 1230 5183 49 4 v 1237 5275 a Fu(t)1305 5092 y Fl(\015)1305 5151 y(\015)1305 5211 y(\015)1360 5118 y(p)p 1460 5118 178 4 v 88 x Fr(\000)p Fz(\000)1598 5172 y Fm(0)1598 5231 y Fs(1)1638 5206 y Fz(\()p Fu(B)1750 5221 y Ft(t)1780 5206 y Fz(\))p Fu( )1881 5221 y Fs(11)1956 5206 y Fz(\()p Fu(t)p Fz(\))2067 5092 y Fl(\015)2067 5151 y(\015)2067 5211 y(\015)2122 5118 y Fs(2)2206 5206 y Fu(dt)27 b Fr(\024)i Fz(Const)f Fr(jj)p Fu( )t Fr(jj)2881 5165 y Fs(2)2919 5206 y Fu(:)1939 5776 y Fz(81)p eop %%Page: 82 85 82 84 bop 150 407 a Fv(Pro)s(of.)44 b Fz(W)-8 b(e)33 b(pro)s(ceed)g(as)g(in)f(the)h(previous)g(section)g(b)m(y)h (constructing)f(a)f(b)s(ounded)i(propagation)150 527 y(observ)-5 b(able.)44 b(T)-8 b(o)32 b(simplify)e(notation,)i(w)m(e)h (only)f(giv)m(e)h(its)f(exp)s(ectation)h(on)f Fu( )t Fz(\()p Fu(t)p Fz(\))h(whic)m(h)g(equals:)1342 747 y Fu(V)21 b Fz(\()p Fu(t)p Fz(\))28 b(:=)g Fr(h)p Fu( )1792 762 y Fs(11)1867 747 y Fz(\()p Fu(t)p Fz(\))p Fu(;)17 b Fz(\000)2083 762 y Fs(1)2122 747 y Fz(\()p Fu(B)2234 762 y Ft(t)2264 747 y Fz(\))p Fu( )2365 762 y Fs(11)2440 747 y Fz(\()p Fu(t)p Fz(\))p Fr(i)f Fu(:)943 b Fz(\(7.50\))150 967 y(Di\013eren)m(tiating)30 b Fu(V)22 b Fz(\()p Fu(t)p Fz(\))32 b(w)m(e)i(get)487 1215 y Fu(@)538 1230 y Ft(t)568 1215 y Fu(V)21 b Fz(\()p Fu(t)p Fz(\))28 b(=)g Fr(\000)983 1104 y Fl(D)1044 1126 y(p)p 1143 1126 178 4 v 1143 1215 a Fr(\000)p Fz(\000)1281 1180 y Fm(0)1281 1239 y Fs(1)1321 1215 y Fz(\()p Fu(B)1433 1230 y Ft(t)1463 1215 y Fz(\))p Fu( )1564 1230 y Fs(11)1639 1215 y Fz(\()p Fu(t)p Fz(\))p Fu(;)17 b Fz(\()p Fu(D)1913 1230 y Ft(H)1980 1215 y Fu(B)2054 1230 y Ft(t)2084 1215 y Fz(\))2138 1126 y Fl(p)p 2238 1126 V 89 x Fr(\000)p Fz(\000)2376 1180 y Fm(0)2376 1239 y Fs(1)2416 1215 y Fz(\()p Fu(B)2528 1230 y Ft(t)2557 1215 y Fz(\))p Fu( )2658 1230 y Fs(11)2734 1215 y Fz(\()p Fu(t)p Fz(\))2845 1104 y Fl(E)2928 1215 y Fz(+)22 b Fu(R)q Fz(\()p Fu(t)p Fz(\))p Fu(;)337 b Fz(\(7.51\))150 1462 y(where)33 b Fu(R)g Fz(con)m(tains)f(remainders)f(of)g(the)i(t)m(yp)s (e)f Fu(R)1988 1477 y Fs(1)2060 1462 y Fz(and)g Fu(R)2323 1477 y Fs(2)2394 1462 y Fz(as)g(in)f(\(7.43\))g(together)h(with)g (Heisen-)150 1582 y(b)s(erg)23 b(deriv)-5 b(ativ)m(es)24 b(of)f(the)h(other)g(cut-o\013)f(functions)g(whic)m(h)i(build)d Fu( )2626 1597 y Fs(11)2701 1582 y Fz(\()p Fu(t)p Fz(\).)41 b(Using)23 b(v)-5 b(arious)22 b(previous)150 1703 y(estimates,)32 b(cf.)44 b(the)33 b(pro)s(of)f(of)g(Prop)s(osition)f(7.1,)h(w)m(e)h(ma) m(y)g(pro)m(v)m(e)h(that)1379 1832 y Fl(Z)1479 1859 y Fm(1)1434 2058 y Fs(1)1570 1968 y Fr(j)p Fu(R)q Fz(\()p Fu(t)p Fz(\))p Fr(j)44 b Fu(dt)27 b Fr(\024)h Fz(Const)h Fr(jj)p Fu( )t Fr(jj)2531 1927 y Fs(2)2569 1968 y Fu(;)150 2235 y Fz(hence)34 b(w)m(e)g(only)e(ha)m(v)m(e)i(to)e(deal)g(with)g (the)h(\014rst)g(term.)43 b(Firstly)-8 b(,)31 b(rewrite)i(it)e(as)711 2507 y Fr(\000)p Fu(t)823 2466 y Fs(1+)p Ft(\017)942 2475 y Fj(4)997 2366 y Fl(\034)1072 2507 y Fz(\(1)22 b(+)g Fu(\017)1318 2522 y Fs(4)1358 2507 y Fz(\))1422 2439 y Fu(L)1488 2454 y Ft(\016)p 1422 2484 105 4 v 1457 2575 a Fu(t)1559 2507 y Fz(+)g Fu(D)1738 2522 y Ft(H)1805 2507 y Fu(L)1871 2522 y Ft(\016)1909 2366 y Fl(\035)1984 2606 y Fs(\011)2059 2507 y Fu(;)45 b Fz(\011)27 b(:=)2365 2418 y Fl(p)p 2465 2418 178 4 v 89 x Fr(\000)p Fz(\000)2603 2472 y Fm(0)2603 2531 y Fs(1)2642 2507 y Fz(\()p Fu(B)2754 2522 y Ft(t)2784 2507 y Fz(\))p Fu( )2885 2522 y Fs(11)2960 2507 y Fz(\()p Fu(t)p Fz(\))p Fu(:)478 b Fz(\(7.52\))150 2788 y(Up)34 b(to)f(an)g(in)m(tegrable)g(remainder)f(this)h(term)g(is)g (\\p)s(ositiv)m(e",)g(in)f(fact)i(\(see)g(the)g(pro)s(of)f(of)g(Prop)s (o-)150 2909 y(sition)e(7.1\):)1148 3186 y Fr(\000)p Fu(t)1260 3145 y Fs(1+)p Ft(\017)1379 3154 y Fj(4)1435 3045 y Fl(\034)1510 3186 y Fz(\()o(1)22 b(+)g Fu(\017)1755 3201 y Fs(4)1795 3186 y Fz(\))1860 3119 y Fu(L)1926 3134 y Ft(\016)p 1860 3163 105 4 v 1894 3254 a Fu(t)1996 3186 y Fz(+)g Fu(D)2175 3201 y Ft(H)2242 3186 y Fu(L)2308 3201 y Ft(\016)2347 3045 y Fl(\035)2421 3285 y Fs(\011)988 3455 y Fr(\025)1158 3387 y Fz(1)g Fr(\000)h Fu(\016)j Fr(\000)c Fu(\017)1536 3402 y Fs(4)p 1158 3432 419 4 v 1349 3523 a Fu(t)1603 3455 y Fr(h)o Fu(B)1715 3470 y Ft(t)1745 3455 y Fr(i)1784 3484 y Fs(\011)1865 3455 y Fz(+)g Fr(O)s Fz(\()p Fu(t)2118 3413 y Fm(\000)p Fs(3)p Ft(=)p Fs(2)2283 3455 y Fz(\))p Fr(jj)p Fu( )t Fr(jj)2500 3413 y Fs(2)988 3693 y Fr(\025)1158 3625 y Fz(1)g Fr(\000)h Fu(\016)j Fr(\000)c Fu(\017)1536 3640 y Fs(4)p 1158 3670 V 1349 3761 a Fu(t)1603 3578 y Fl(\015)1603 3638 y(\015)1603 3698 y(\015)1658 3604 y(p)p 1758 3604 178 4 v 89 x Fr(\000)p Fz(\000)1896 3658 y Fm(0)1896 3717 y Fs(1)1935 3693 y Fz(\()p Fu(B)2047 3708 y Ft(t)2077 3693 y Fz(\))p Fu( )2178 3708 y Fs(11)2253 3578 y Fl(\015)2253 3638 y(\015)2253 3698 y(\015)2309 3604 y Fs(2)2370 3693 y Fz(+)g Fr(O)s Fz(\()p Fu(t)2623 3652 y Fm(\000)p Fs(3)p Ft(=)p Fs(2)2788 3693 y Fz(\))p Fr(jj)p Fu( )t Fr(jj)3005 3652 y Fs(2)3043 3693 y Fu(;)506 b Fz(\(7.53\))150 3944 y(where)34 b(the)f(last)f (inequalit)m(y)f(comes)i(from)e(the)i(fact)g(that)f(\000)2388 3907 y Fm(0)2388 3968 y Fs(1)2460 3944 y Fz(is)g(supp)s(orted)h(in)f ([1)p Fu(;)17 b Fz(2].)p 3754 3944 4 66 v 3758 3881 59 4 v 3758 3944 V 3816 3944 4 66 v 150 4395 a Fw(8)161 b(Asymptotic)53 b(completeness:)68 b(existence)53 b(of)h Fa(\012)3275 4416 y Fz(+)150 4614 y(W)-8 b(e)34 b(will)d(no)m(w)i(pro)m (v)m(e)i(statemen)m(t)f(I)s(I)f(of)g(Theorem)g(4.2.)45 b(By)34 b(a)f(co)m(v)m(ering)h(argumen)m(t)f(it)f(su\016ces)j(to)150 4734 y(sho)m(w)150 4937 y Fv(Prop)s(osition)k(8.1.)44 b FA(Assume)38 b(we)g(have)f Fu(E)j(>)33 b(E)2012 4952 y Ft(d)2053 4937 y FA(,)38 b Fu(E)i Fr(62)34 b Fu(\033)2388 4952 y Ft(pp)2463 4937 y Fz(\()p Fu(H)8 b Fz(\))p FA(.)54 b(Supp)-5 b(ose)37 b(that)i Fu( )t Fz(\()p Fu(t)p Fz(\))f FA(is)g(as)f(in)150 5058 y(\(6.1\))d(with)i Fz(~)-50 b Fu(\017)28 b(>)f Fz(0)35 b FA(very)g(smal)5 b(l.)44 b(Then)34 b(ther)-5 b(e)35 b(exists)f(a)h(ve)-5 b(ctor)35 b Fu(\036)27 b Fr(2)h Fu(L)2753 5022 y Fs(2)2793 5058 y Fz(\(\()p Fu(E)2941 5073 y Ft(d)2981 5058 y Fu(;)17 b Fr(1)p Fz(\))22 b Fr(\002)h Fn(T)p Fz(\))38 b FA(such)c(that)1471 5278 y Fz(lim)1455 5338 y Ft(t)p Fm(!1)1639 5278 y Fr(k)o Fu(U)1754 5293 y Fs(0)1794 5278 y Fz(\()p Fu(t)p Fz(\))p Fu(\036)22 b Fr(\000)h Fu( )t Fz(\()p Fu(t)p Fz(\))p Fr(k)k Fz(=)h(0)p Fu(:)1939 5776 y Fz(82)p eop %%Page: 83 86 83 85 bop 150 407 a Fv(Pro)s(of.)70 b Fz(The)43 b(prop)s(osition)d(is)h (an)h(easy)g(consequence)j(of)c(the)h(unitarit)m(y)f(of)g Fu(U)3189 422 y Fs(0)3229 407 y Fz(\()p Fu(t)p Fz(\))g(\(since)h(this) 150 527 y(op)s(erator)29 b(is)g(essen)m(tially)g(a)g(c)m(hange)i(of)e (v)-5 b(ariables\))28 b(and)h(of)h(the)f(existence)j(of)d(the)h(follo)m (wing)c(limit:)1572 768 y Fu(\036)h Fz(:=)43 b(lim)1788 828 y Ft(t)p Fm(!1)1971 768 y Fu(U)2047 727 y Fm(\003)2037 793 y Fs(0)2087 768 y Fz(\()p Fu(t)p Fz(\))p Fu( )t Fz(\()p Fu(t)p Fz(\))p Fu(:)1221 b Fz(\(8.1\))150 973 y(The)31 b(pro)s(of)d(of)i(\(8.1\))f(is)g(complicated)f(and)i(uses)h(the)f (propagation)e(estimates)i(w)m(e)g(ha)m(v)m(e)h(obtained)150 1093 y(so)37 b(far.)54 b(Because)38 b(of)e(the)h(v)-5 b(arious)36 b(lo)s(calization)c(prop)s(erties)37 b(w)m(e)g(ha)m(v)m(e)h (already)e(pro)m(v)m(en,)j(w)m(e)f(can)150 1214 y(replace)33 b Fu( )t Fz(\()p Fu(t)p Fz(\))f(b)m(y)1409 1334 y Fu( )1472 1349 y Fs(12)1547 1334 y Fz(\()p Fu(t)p Fz(\))c(:=)g(\000)1878 1349 y Fs(1)1917 1334 y Fz(\()p Fu(t)1990 1293 y Fs(1+)p Ft(\017)2109 1302 y Fj(4)2148 1334 y Fu(L)2214 1349 y Ft(\016)2252 1334 y Fz(\))p Fu( )2353 1349 y Fs(11)2428 1334 y Fz(\()p Fu(t)p Fz(\))p Fu(:)1058 b Fz(\(8.2\))150 1507 y(since)46 b(the)g(di\013erence)g(b)s(et)m(w)m(een)i(them)d(tends) i(to)e(zero)h(in)e(time.)81 b(There)47 b(are)f(\014fteen)g(cut-o\013s) 150 1627 y(whic)m(h)38 b(build)e Fu( )751 1642 y Fs(12)827 1627 y Fz(\()p Fu(t)p Fz(\))h(and)h(w)m(e)g(w)m(ould)g(lik)m(e)f(to)g (k)m(eep)i(trac)m(k)g(of)e(their)g(Heisen)m(b)s(erg)h(deriv)-5 b(ativ)m(es)38 b(in)150 1748 y(a)h(more)f(e\016cien)m(t)h(w)m(a)m(y)-8 b(.)64 b(Moreo)m(v)m(er,)42 b(they)d(split)f(in)m(to)g(t)m(w)m(o)h (categories:)56 b(the)40 b(\014rst)f(one)g(con)m(tains)150 1868 y(those)28 b(cut-o\013s)f(in)m(tro)s(duced)g(through)g(a)g(w)m (eak)h(propagation)d(estimate)h(\(pro)m(v)m(en)j(b)m(y)f(constructing) 150 1988 y(a)44 b(b)s(ounded)h(propagation)d(observ)-5 b(able)45 b(whose)g(Heisen)m(b)s(erg)g(deriv)-5 b(ativ)m(e)44 b(has)h(a)f(sign\))f(and)i(the)150 2109 y(second)40 b(one)f(con)m (tains)g(the)g(cut-o\013s)g(whose)h(complemen)m(tary)e(lo)s (calizations)d(are)k(in)f(classically)150 2229 y(forbidden)32 b(regions.)296 2350 y(Let)38 b(us)h(lo)s(ok)d(bac)m(k)j(for)f(the)g (cut-o\013s)g(b)s(elonging)e(to)h(the)i(\014rst)f(category)-8 b(.)60 b(In)38 b(order)g(to)f(k)m(eep)150 2470 y(trac)m(k)27 b(of)e(them)g(more)g(easily)-8 b(,)27 b(w)m(e)g(in)m(tro)s(duce)e (uni\014ed)h(notations)f(for)h(them)f(as)h(follo)m(ws:)39 b Fu(F)3473 2485 y Fs(1)3540 2470 y Fz(:=)28 b Fu(F)3734 2485 y Ft(D)3798 2470 y Fz(,)150 2590 y Fu(A)223 2605 y Fs(1)295 2590 y Fz(=)33 b Fu(r)s(=t)i Fz(\(see)h(\(6.84\)\),)f Fu(F)1181 2605 y Fs(2)1254 2590 y Fz(:=)d Fu(F)1466 2554 y Ft(dil)1452 2615 y Fs(+)1552 2590 y Fz(,)k Fu(A)1688 2605 y Fs(2)1760 2590 y Fz(:=)1922 2565 y(~)1896 2590 y Fu(A)1969 2605 y Ft(C)2028 2590 y Fz(\()p Fu(E)6 b Fz(\))p Fu(=t)35 b Fz(\(see)i(\(6.37\)\),)e Fu(F)2913 2605 y Fs(3)2985 2590 y Fz(:=)e Fu(F)3198 2539 y Ft(d)3234 2548 y Fj(1)3269 2539 y Ft(=)p Fs(2)p Ft(;d)3395 2548 y Fj(1)3184 2612 y Fs(+)3434 2590 y Fz(,)j Fu(A)3570 2605 y Fs(3)3642 2590 y Fz(:=)3788 2564 y(~)3778 2590 y Fu(\030)150 2711 y Fz(\(see)j(\(6.57\)\),)g Fu(F)768 2726 y Fs(4)845 2711 y Fz(:=)e Fu(F)1062 2663 y Ft(\021)1097 2672 y Fj(1)1132 2663 y Ft(;\021)1187 2672 y Fj(1)1222 2663 y Fs(+)p Ft(\017)1306 2672 y Fj(2)1048 2732 y Fm(\000)1344 2711 y Fz(,)i Fu(A)1483 2726 y Fs(4)1560 2711 y Fz(=)46 b(~)-58 b Fu(\032)27 b Fr(\000)f Fu(\032)1903 2726 y Ft(E)1959 2707 y Ff(0)2024 2711 y Fz(\(see)39 b(\(6.76\)\),)g Fu(F)2642 2726 y Fs(5)2719 2711 y Fz(:=)e Fu(F)2936 2675 y Ft(M)s(:v)r(:b:)2922 2735 y Fm(\000)3136 2711 y Fz(,)j Fu(A)3276 2726 y Fs(5)3353 2711 y Fz(:=)d Fu(r)s(=t)h Fz(\(see)150 2831 y(\(6.30\)\),)32 b(and)h Fu(F)750 2846 y Fs(6)817 2831 y Fz(:=)27 b(\000)1008 2846 y Fs(1)1048 2831 y Fz(,)32 b Fu(A)1180 2846 y Fs(6)1248 2831 y Fz(:=)27 b Fu(t)1413 2795 y Fs(1+)p Ft(\017)1532 2804 y Fj(4)1571 2831 y Fu(L)1637 2846 y Ft(\016)1708 2831 y Fz(\(see)33 b(\(8.2\)\).)296 2951 y(W)-8 b(e)46 b(do)e(the)i(same)f(thing)f(with)g (the)i(second)g(category:)68 b Fu(F)2589 2966 y Fs(7)2678 2951 y Fz(:=)48 b Fu(F)2906 2915 y Ft(m:v)r(:b:)2892 2976 y Fs(+)3098 2951 y Fz(,)g Fu(A)3246 2966 y Fs(7)3335 2951 y Fz(:=)g Fu(r)s(=t)d Fz(\(see)150 3080 y(\(6.41\)\),)d Fu(F)570 3095 y Fs(8)650 3080 y Fz(:=)f Fu(F)871 3037 y Fs(\001)930 3046 y Fj(2)857 3102 y Fs(+)968 3080 y Fz(,)h Fu(A)1110 3095 y Fs(8)1191 3080 y Fz(:=)49 b(~)-58 b Fu(\032)28 b Fr(\000)g Fu(\032)1567 3095 y Ft(E)1667 3080 y Fz(\(see)42 b(\(6.50\)\),)f Fu(F)2290 3095 y Fs(9)2371 3080 y Fz(:=)f Fu(F)2591 3037 y Fs(\001)2650 3046 y Fj(1)2577 3102 y Fs(+)2689 3080 y Fz(,)i Fu(A)2831 3095 y Fs(9)2911 3080 y Fz(:=)50 b(~)-58 b Fu(\032)28 b Fr(\000)g Fu(\032)3288 3095 y Ft(E)3375 3080 y Fz(+)f Fu(@)3529 3095 y Ft(E)3589 3080 y Fu(\032)3639 3095 y Ft(E)3699 3080 y Fu(=C)150 3210 y Fz(\(see)40 b(again)d(\(6.50\)\),)j Fu(F)1036 3225 y Fs(10)1150 3210 y Fz(:=)e Fu(F)1368 3162 y Ft(\017)1397 3171 y Fj(2)1431 3162 y Ft(;)p Fs(2)p Ft(\017)1515 3171 y Fj(2)1354 3231 y Fs(+)1553 3210 y Fz(,)j Fu(A)1694 3225 y Fs(10)1807 3210 y Fz(:=)47 b(~)-58 b Fu(\032)27 b Fz(+)2128 3125 y Fr(p)p 2211 3125 127 4 v 85 x Fz(2)p Fu(E)44 b Fz(\(see)c(\(6.53\)\),)g Fu(F)2996 3225 y Fs(11)3109 3210 y Fz(:=)f Fu(F)3328 3162 y Fs(2)p Ft(\021)3398 3171 y Fj(1)3433 3162 y Fs(+)p Ft(\017)3517 3171 y Fj(2)3551 3162 y Ft(;\021)3606 3171 y Fj(1)3640 3162 y Fs(+2)p Ft(\017)3759 3171 y Fj(2)3314 3231 y Fm(\000)3798 3210 y Fz(,)150 3339 y Fu(A)223 3354 y Fs(11)341 3339 y Fz(=)52 b(~)-58 b Fu(\032)29 b Fr(\000)f Fu(\032)694 3354 y Ft(E)754 3339 y Fz(,)44 b Fu(F)888 3354 y Fs(12)1006 3339 y Fz(:=)f Fu(F)1229 3303 y Ft(M)1215 3364 y Fs(+)p Fm(\000)1329 3339 y Fz(,)h Fu(A)1473 3354 y Fs(12)1591 3339 y Fz(:=)1747 3313 y(~)1737 3339 y Fu(\030)33 b Fr(\000)c Fu(\030)1962 3354 y Ft(E)2062 3339 y Fz(\(see)43 b(\(6.82\)\),)g Fu(F)2688 3354 y Fs(13)2806 3339 y Fz(:=)g Fu(F)3015 3354 y Ft(E)3075 3339 y Fz(,)h Fu(A)3219 3354 y Fs(13)3337 3339 y Fz(:=)64 b(~)-70 b Fu(\015)3534 3354 y Fs(3)3616 3339 y Fz(=)47 b(^)-53 b Fu(\015)3786 3354 y Fs(3)150 3473 y Fz(\(see)35 b(\(6.87\)\),)f Fu(F)759 3488 y Fs(14)864 3473 y Fz(:=)1019 3448 y(^)997 3473 y Fu(F)13 b Fz(,)35 b Fu(A)1208 3488 y Fs(14)1312 3473 y Fz(:=)51 b(^)-70 b Fu(\015)1496 3488 y Fs(1)1569 3473 y Fz(\(see)35 b(\(7.5\)\))f(and)g(\014nally)f Fu(F)2590 3488 y Fs(15)2694 3473 y Fz(:=)2850 3448 y(^)2827 3473 y Fu(F)14 b Fz(,)35 b Fu(A)3039 3488 y Fs(15)3143 3473 y Fz(:=)51 b(^)-70 b Fu(\015)3327 3488 y Fs(2)3400 3473 y Fz(\(see)35 b(again)150 3594 y(\(7.5\)\).)296 3714 y(W)-8 b(e)38 b(remark)e(that)h(the)h(order)f(of)f(the)i(ab)s(o)m (v)m(e)g(cut-o\013s)f(is)f(not)h(imp)s(ortan)m(t)e(when)j(applied)e(on) 150 3834 y Fu( )t Fz(\()p Fu(t)p Fz(\))24 b(in)f(\(6.1\),)j(since)e(ev) m(ery)i(comm)m(utation)c(will)f(at)j(least)f(b)s(e)i(of)e(order)h Fr(O)s Fz(\()p Fu(t)2908 3798 y Fm(\000)p Fs(1)p Ft(=)p Fs(2+)p Ft(\017)3152 3807 y Fj(4)3187 3798 y Ft(=)p Fs(2)3262 3834 y Fz(\).)41 b(W)-8 b(e)24 b(rewrite)150 3955 y(then)964 4075 y Fu( )1027 4090 y Ft(f)1073 4075 y Fz(\()p Fu(t)p Fz(\))k(:=)g Fu(F)1406 4090 y Fs(1)1445 4075 y Fz(\()p Fu(A)1556 4090 y Fs(1)1596 4075 y Fz(\()p Fu(t)p Fz(\)\))22 b Fr(\001)g Fu(F)1880 4090 y Fs(2)1919 4075 y Fz(\()p Fu(A)2030 4090 y Fs(2)2070 4075 y Fz(\))g Fr(\001)g Fu(:::)g Fr(\001)g Fu(F)2396 4090 y Fs(15)2471 4075 y Fz(\()p Fu(A)2582 4090 y Fs(15)2656 4075 y Fz(\()p Fu(t)p Fz(\)\))p Fu( )t Fz(\()p Fu(t)p Fz(\))p Fu(:)614 b Fz(\(8.3\))296 4248 y(De\014ne)49 b Fu(Q)p Fz(\()p Fu(t;)17 b(')p Fz(\))55 b(=)g Fr(h)p Fu(U)1201 4263 y Fs(0)1241 4248 y Fz(\()p Fu(t)p Fz(\))p Fu(';)17 b( )1523 4263 y Ft(f)1568 4248 y Fz(\()p Fu(t)p Fz(\))p Fr(i)p Fz(,)53 b(where)d Fu(')k Fr(2)i Fu(C)2413 4212 y Fm(1)2406 4272 y Fs(0)2487 4248 y Fz(\(\()p Fu(E)2635 4263 y Ft(d)2676 4248 y Fu(;)17 b Fr(1)p Fz(\))32 b Fr(\002)i Fn(T)p Fz(\).)95 b(Since)49 b Fu( )3558 4263 y Ft(f)3603 4248 y Fz(\()p Fu(t)p Fz(\))34 b Fr(\000)150 4368 y Fu( )t Fz(\()p Fu(t)p Fz(\))55 b Fr(!)f Fz(0,)f(the)48 b(existence)j(of)d(the)g(limit)d(in)j(\(8.1\))g (is)g(equiv)-5 b(alen)m(t)48 b(to)h(the)f(existence)i(of)f(the)150 4489 y(limit)35 b(lim)525 4504 y Ft(t)p Fm(!1)712 4489 y Fu(U)788 4452 y Fm(\003)778 4513 y Fs(0)828 4489 y Fz(\()p Fu(t)p Fz(\))p Fu( )1002 4504 y Ft(f)1048 4489 y Fz(\()p Fu(t)p Fz(\).)62 b(Moreo)m(v)m(er,)42 b(com)m(bining)37 b(the)j(Cauc)m(h)m(y)g(criterion)e(for)g(the)h(existence)150 4609 y(of)44 b(a)f(limit)e(with)j(the)g(Co)s(ok)h(argumen)m(t)e(and)i (the)f(Riesz)g(represen)m(tation)h(theorem)f(for)g(linear)150 4729 y(functionals)36 b(on)h Fu(L)857 4693 y Fs(2)897 4729 y Fz(,)h(w)m(e)g(see)g(that)f(this)f(limit)e(exists)k(if)d(for)i (ev)m(ery)i Fu(")c(>)g Fz(0,)j(there)f(exists)h Fu(T)3593 4744 y Ft(")3665 4729 y Fu(>)d Fz(1)150 4850 y(suc)m(h)f(that)e(for)g (all)f Fu(t)901 4865 y Fs(2)968 4850 y Fr(\025)d Fu(t)1108 4865 y Fs(1)1176 4850 y Fu(>)f(T)1336 4865 y Ft(")1406 4850 y Fz(and)32 b(all)f Fu(')c Fr(2)i Fu(C)1994 4814 y Fm(1)1987 4874 y Fs(0)2068 4850 y Fz(\(\()p Fu(E)2216 4865 y Ft(d)2257 4850 y Fu(;)17 b Fr(1)p Fz(\))k Fr(\002)i Fn(T)p Fz(\),)36 b(w)m(e)d(ha)m(v)m(e)1092 4993 y Fl(Z)1192 5020 y Ft(t)1217 5029 y Fj(2)1147 5219 y Ft(t)1172 5228 y Fj(1)1273 5129 y Fr(j)o Fu(@)1351 5144 y Ft(t)1381 5129 y Fu(Q)p Fz(\()p Fu(t;)17 b(')p Fz(\))p Fr(j)44 b Fu(dt)28 b Fr(\024)g Fu(")f Fr(jj)p Fu(')p Fr(jj)2217 5146 y Ft(L)2265 5127 y Fj(2)2299 5146 y Fs(\(\()p Ft(E)2405 5158 y Fq(d)2441 5146 y Ft(;)p Fm(1)p Fs(\))p Fm(\002)p Fe(T)p Fs(\))2690 5129 y Fu(:)907 b Fz(\(8.4\))296 5407 y(Pro)m(ving)44 b(\(8.4\))g(will)d(b)s(e)k(the)f(task)h(in)e(the)i (remaining)c(of)j(this)g(subsection.)79 b(W)-8 b(e)44 b(start)g(b)m(y)150 5527 y(expanding)28 b(the)h(deriv)-5 b(ativ)m(e)28 b(of)f Fu(Q)i Fz(with)e(resp)s(ect)j(to)e Fu(t)p Fz(.)42 b(P)m(erforming)27 b(the)h(deriv)-5 b(ativ)m(e)28 b(with)g(resp)s(ect)1939 5776 y(83)p eop %%Page: 84 87 84 86 bop 150 407 a Fz(to)36 b Fu(t)h Fz(and)f(using)g(\(4.19\))g(w)m (e)h(get)g(\(b)m(y)g Fu(d)1625 422 y Ft(H)1692 407 y Fu( )1755 422 y Ft(f)1837 407 y Fz(w)m(e)g(denote)h(the)e(Heisen)m(b)s (erg)i(deriv)-5 b(ativ)m(e)36 b(acting)f(on)150 527 y(the)e(cut-o\013s) g(in)e Fu( )839 542 y Ft(f)885 527 y Fz(\))293 795 y Fu(@)344 810 y Ft(t)374 795 y Fu(Q)p Fz(\()p Fu(t;)17 b(')p Fz(\))28 b(=)819 728 y Fu(i)p 812 772 49 4 v 812 864 a Fz(8)887 655 y Fl(\034)962 795 y Fu(U)1028 810 y Fs(0)1067 795 y Fz(\()p Fu(t)p Fz(\))p Fu(';)1315 728 y Fz(1)p 1296 772 87 4 v 1296 864 a Fu(r)1343 835 y Fs(2)1392 795 y Fu( )1455 810 y Ft(f)1501 795 y Fz(\()p Fu(t)p Fz(\))1612 655 y Fl(\035)1709 795 y Fr(\000)1826 728 y Fu(i)p 1818 772 49 4 v 1818 864 a Fz(2)1877 795 y Fr(h)p Fu(\015)1972 754 y Fs(2)2011 795 y Fu(U)2077 810 y Fs(0)2117 795 y Fz(\()p Fu(t)p Fz(\))p Fu(';)17 b( )2399 810 y Ft(f)2444 795 y Fz(\()p Fu(t)p Fz(\))p Fr(i)22 b Fz(+)g Fr(h)p Fu(U)2819 810 y Fs(0)2859 795 y Fz(\()p Fu(t)p Fz(\))p Fu(';)17 b(d)3129 810 y Ft(H)3196 795 y Fu( )3259 810 y Ft(f)3304 795 y Fz(\()p Fu(t)p Fz(\))p Fr(i)p Fu(:)143 b Fz(\(8.5\))296 1060 y(The)38 b(\014rst)f(term)e(on)i(the)f(righ)m(t)g (hand)h(side)f(of)g(\(8.5\))g(is)g(clearly)f(b)s(ounded)i(up)g(to)f(a)g (constan)m(t)150 1181 y(b)m(y)j Fr(jj)p Fu(')p Fr(jj)p Fu(=t)551 1145 y Fs(2)589 1181 y Fz(,)h(b)s(ecause)g Fu(F)1086 1196 y Fs(1)1164 1181 y Fz(implies)c(in)h(particular)g(that)h Fu(r)s(=t)g Fz(is)g(b)s(ounded)h(and)g(a)m(w)m(a)m(y)g(from)f(zero)150 1301 y(at)d(the)g(same)g(time.)50 b(Hence)36 b(after)f(in)m(tegration)e (w)m(e)j(get)f(an)g(estimate)g(as)g(in)f(\(8.4\).)51 b(The)36 b(second)150 1422 y(term)f(is)g(tec)m(hnically)g(more)g (complicated:)48 b(b)m(y)37 b(v)-5 b(arious)34 b(comm)m(utations)g(in)m (v)m(olving)h(the)h(cut-o\013s)150 1542 y(in)c Fu( )327 1557 y Ft(f)373 1542 y Fz(\()p Fu(t)p Fz(\))g(\(w)m(e)i(skip)f(the)g (details\))e(one)i(can)g(pro)m(v)m(e)g(that)1372 1750 y Fu(\015)1428 1709 y Fs(2)1467 1750 y Fu( )1530 1765 y Ft(f)1576 1750 y Fz(\()p Fu(t)p Fz(\))28 b(=)j(~)-53 b Fu(\015)1874 1709 y Fs(2)1914 1750 y Fu( )1977 1765 y Ft(f)2022 1750 y Fz(\()p Fu(t)p Fz(\))23 b(+)f Fr(O)s Fz(\()p Fu(t)2409 1709 y Fm(\0001)2538 1750 y Fz(\))p Fu(:)1021 b Fz(\(8.6\))150 1958 y(In)m(v)m(oking)30 b(\(8.6\),)f(the)g (second)i(term)d(in)g(\(8.5\))h(b)s(ecomes)g Fr(\000)p Fu(i=)p Fz(2)p Fr(h)p Fu(U)2544 1973 y Fs(0)2584 1958 y Fz(\()p Fu(t)p Fz(\))p Fu(';)20 b Fz(~)-52 b Fu(\015)2859 1922 y Fs(2)2898 1958 y Fu( )2961 1973 y Ft(f)3007 1958 y Fz(\()p Fu(t)p Fz(\))p Fr(i)p Fu(:)29 b Fz(Because)h(of)f(the)150 2078 y(presence)39 b(of)d Fu(F)723 2093 y Fs(6)763 2078 y Fz(\()p Fu(A)874 2093 y Fs(6)913 2078 y Fz(\))h(in)f Fu( )1169 2093 y Ft(f)1215 2078 y Fz(\()p Fu(t)p Fz(\),)i(w)m(e)g(c)m (ho)s(ose)f(t)m(w)m(o)h(functions)f(\000)2527 2093 y Fs(2)2603 2078 y Fz(and)g(\000)2858 2093 y Fs(3)2934 2078 y Fz(with)f(a)h(sligh)m(tly)e(wider)150 2199 y(supp)s(ort)e(than)g (\000)800 2214 y Fs(1)871 2199 y Fz(and)g(decomp)s(ose)860 2406 y(~)-53 b Fu(\015)912 2365 y Fs(2)907 2431 y(1)952 2406 y Fu( )1015 2421 y Ft(f)1060 2406 y Fz(\()p Fu(t)p Fz(\))28 b(=)j(~)-52 b Fu(\015)1354 2421 y Fs(1)1393 2406 y Fz(\000)1454 2421 y Fs(3)1493 2406 y Fz(\()p Fu(t)1566 2365 y Fs(1+)p Ft(\017)1685 2374 y Fj(4)1724 2406 y Fu(L)1790 2421 y Ft(\016)1828 2406 y Fz(\))t(~)f Fu(\015)1917 2421 y Fs(1)1956 2406 y Fz(\000)2017 2421 y Fs(2)2056 2406 y Fz(\()p Fu(t)2129 2365 y Fs(1+)p Ft(\017)2248 2374 y Fj(4)2287 2406 y Fu(L)2353 2421 y Ft(\016)2391 2406 y Fz(\))p Fu( )2492 2421 y Ft(f)2538 2406 y Fz(\()p Fu(t)p Fz(\))22 b(+)g Fr(O)s Fz(\()p Fu(t)2924 2365 y Fm(\0001)3054 2406 y Fz(\))p Fu(:)150 2614 y Fz(Using)32 b(\(7.3\))h(w)m(e)g(can)g (easily)f(sho)m(w)i(that)f(there)g(is)g(a)f(constan)m(t)i(indep)s (enden)m(t)g(of)e(time)f(so)i(that)g(in)150 2735 y(the)i(form)e(sense) 40 b(~)-52 b Fu(\015)861 2699 y Fs(2)856 2759 y(1)931 2735 y Fr(\024)32 b Fz(Const)24 b Fr(\001)f Fu(L)1430 2750 y Ft(\016)1503 2735 y Fz(whic)m(h)35 b(means)g(that)f Fr(jj)t Fz(~)-53 b Fu(\015)2405 2750 y Fs(1)2444 2735 y Fz(\000)2505 2750 y Fs(3)2544 2735 y Fz(\()p Fu(t)2617 2699 y Fs(1+)p Ft(\017)2736 2708 y Fj(4)2775 2735 y Fu(L)2841 2750 y Ft(\016)2879 2735 y Fz(\))p Fr(jj)30 b(\024)i Fz(Const)24 b Fr(\001)f Fu(t)3471 2699 y Fm(\000)p Fs(1)p Ft(=)p Fs(2)p Fm(\000)p Ft(\017)3715 2708 y Fj(4)3750 2699 y Ft(=)p Fs(2)150 2855 y Fz(and)33 b(similarly)d(for)36 b(~)-52 b Fu(\015)941 2870 y Fs(1)980 2855 y Fz(\000)1041 2870 y Fs(2)1080 2855 y Fz(\()p Fu(t)1153 2819 y Fs(1+)p Ft(\017)1272 2828 y Fj(4)1311 2855 y Fu(L)1377 2870 y Ft(\016)1415 2855 y Fz(\);)34 b(w)m(e)g(conclude)g(that)f(the)g (\014rst)h(term)f(on)g(the)h(righ)m(t)e(hand)i(side)150 2976 y(is)e Fr(O)s Fz(\()p Fu(t)403 2939 y Fm(\000)p Fs(1)p Fm(\000)p Ft(\017)577 2948 y Fj(4)616 2976 y Fz(\).)43 b(After)33 b(in)m(tegration)e(w)m(e)i(get)g(a)f(b)s(ound)h(in)f (agreemen)m(t)g(with)h(\(8.4\).)296 3096 y(W)-8 b(e)36 b(no)m(w)g(lo)s(ok)e(at)h(the)g(third)g(term)g(on)g(the)h(righ)m(t)e (hand)i(side)f(of)g(\(8.5\),)g(the)h(one)f(con)m(taining)150 3216 y(the)e(Heisen)m(b)s(erg)g(deriv)-5 b(ativ)m(es)33 b(of)f(all)f(cut-o\013)h(functions.)43 b(Acting)32 b(with)g Fu(d)2901 3231 y Ft(H)3001 3216 y Fz(on)g(\(8.3\))g(w)m(e)i(get)646 3514 y Fu(d)697 3529 y Ft(H)764 3514 y Fu( )827 3529 y Ft(f)873 3514 y Fz(\()p Fu(t)p Fz(\))28 b(=)1152 3390 y Fs(15)1115 3420 y Fl(X)1126 3630 y Ft(j)t Fs(=1)1276 3514 y Fu(F)1339 3529 y Fs(1)1378 3514 y Fz(\()p Fu(A)1489 3529 y Fs(1)1529 3514 y Fz(\()p Fu(t)p Fz(\)\))p Fu(:::)p Fr(f)p Fu(D)1890 3529 y Ft(H)1957 3514 y Fu(F)2020 3529 y Ft(j)2057 3514 y Fz(\()p Fu(A)2168 3529 y Ft(j)2205 3514 y Fz(\()p Fu(t)p Fz(\)\))p Fr(g)p Fu(:::F)2548 3529 y Fs(15)2623 3514 y Fz(\()p Fu(A)2734 3529 y Fs(15)2808 3514 y Fz(\()p Fu(t)p Fz(\)\))p Fu( )t Fz(\()p Fu(t)p Fz(\))p Fu(:)462 b Fz(\(8.7\))150 3817 y(Here)36 b Fu(D)464 3832 y Ft(H)566 3817 y Fz(denotes)g(as)f(usual)g(the)g(Heisen)m(b)s (erg)h(deriv)-5 b(ativ)m(e.)50 b(W)-8 b(e)35 b(k)m(eep)i(in)d(mind)g (that)g(all)f(of)i(the)150 3938 y Fu(F)227 3902 y Fm(0)213 3962 y Ft(j)250 3938 y Fz('s)g(of)f(the)h(t)m(yp)s(e)h Fu(F)932 3902 y Fm(0)918 3962 y Fs(+)1011 3938 y Fz(or)f Fu(F)1210 3902 y Fm(0)1196 3962 y(\000)1289 3938 y Fz(ha)m(v)m(e)h (de\014nite)f(signs.)49 b(The)36 b(functions)f(of)f(t)m(yp)s(e)h Fu(F)3185 3902 y Fm(0)3171 3962 y Fs(+)p Fm(\000)3320 3938 y Fz(do)f(not)h(ha)m(v)m(e)150 4058 y(this)28 b(prop)s(ert)m(y)g (but)h(can)f(b)s(e)g(rewritten)g(as)g Fu(F)1816 4022 y Fm(0)1802 4083 y Fs(+)p Fm(\000)1944 4058 y Fz(=)f Fu(g)2094 4073 y Fs(+)2166 4058 y Fr(\000)13 b Fu(g)2303 4073 y Fm(\000)2362 4058 y Fz(,)29 b(where)g(the)f(terms)g(are)g (non-negativ)m(e)150 4179 y(and)37 b(ha)m(v)m(e)g(non-o)m(v)m (erlapping)f(supp)s(orts.)56 b(T)-8 b(o)37 b(\014x)g(a)f(uniform)f (notation)g(let)g(us)j(write)e(in)g(general)150 4299 y Fu(F)227 4263 y Fm(0)213 4324 y Ft(j)278 4299 y Fz(=)27 b Fu(g)428 4314 y Ft(j)t Fs(+)541 4299 y Fr(\000)c Fu(g)688 4314 y Ft(j)t Fm(\000)779 4299 y Fz(.)44 b(Then)33 b(writing)1102 4559 y Fu(S)1162 4574 y Ft(j)1199 4559 y Fz(\()p Fu(t)p Fz(\))28 b(:=)1468 4440 y Fl(q)p 1568 4440 435 4 v 119 x Fu(g)1615 4574 y Ft(j)t Fs(+)1706 4559 y Fz(\()p Fu(A)1817 4574 y Ft(j)1854 4559 y Fz(\()p Fu(t)p Fz(\)\))16 b Fr(f)p Fu(D)2150 4574 y Ft(H)2217 4559 y Fu(A)2290 4574 y Ft(j)2327 4559 y Fz(\()p Fu(t)p Fz(\))p Fr(g)2504 4440 y Fl(q)p 2604 4440 V 119 x Fu(g)2651 4574 y Ft(j)t Fs(+)2742 4559 y Fz(\()p Fu(A)2853 4574 y Ft(j)2890 4559 y Fz(\()p Fu(t)p Fz(\)\))1102 4775 y(+)1178 4657 y Fl(q)p 1278 4657 V 118 x Fu(g)1325 4790 y Ft(j)t Fm(\000)1416 4775 y Fz(\()p Fu(A)1527 4790 y Ft(j)1563 4775 y Fz(\()p Fu(t)p Fz(\)\))h Fr(f\000)p Fu(D)1937 4790 y Ft(H)2005 4775 y Fu(A)2078 4790 y Ft(j)2114 4775 y Fz(\()p Fu(t)p Fz(\))p Fr(g)2292 4657 y Fl(q)p 2391 4657 V 2391 4775 a Fu(g)2438 4790 y Ft(j)t Fm(\000)2530 4775 y Fz(\()p Fu(A)2641 4790 y Ft(j)2677 4775 y Fz(\()p Fu(t)p Fz(\)\))798 b(\(8.8\))150 5020 y(one)24 b(obtains)g(\(w)m(e)h(can)f(put)g Fu(f)1215 5035 y Fs(1)p Ft(;E)1354 5020 y Fz(on)f(the)i(righ)m(t)e(hand)h(side)g (since)h Fu(f)2568 5035 y Fs(1)p Ft(;E)2682 5020 y Fu(f)2730 5035 y Ft(E)2818 5020 y Fz(=)i Fu(f)2969 5035 y Ft(E)3053 5020 y Fz(and)d Fu(f)3282 5035 y Fs(1)p Ft(;E)3418 4994 y Fz(~)3396 5020 y Fu(f)3444 5035 y Ft(E)3532 5020 y Fz(=)j Fu(f)3683 5035 y Fs(1)p Ft(;E)3798 5020 y Fz(;)150 5141 y(see)34 b(also)d(\(6.1\)\))430 5439 y Fu(d)481 5454 y Ft(H)548 5439 y Fu( )611 5454 y Ft(f)657 5439 y Fz(\()p Fu(t)p Fz(\))c(=)h Fu(f)947 5454 y Fs(1)p Ft(;E)1062 5439 y Fz(\()p Fu(H)8 b Fz(\))1279 5314 y Fs(15)1243 5344 y Fl(X)1253 5554 y Ft(j)t Fs(=1)1403 5439 y Fu(F)1466 5454 y Fs(1)1506 5439 y Fz(\()p Fu(A)1617 5454 y Fs(1)1656 5439 y Fz(\()p Fu(t)p Fz(\)\))p Fu(:::S)1946 5454 y Ft(j)1983 5439 y Fz(\()p Fu(t)p Fz(\))p Fu(:::F)2238 5454 y Fs(15)2313 5439 y Fz(\()p Fu(A)2424 5454 y Fs(15)2499 5439 y Fz(\()p Fu(t)p Fz(\)\))p Fu( )t Fz(\()p Fu(t)p Fz(\))22 b(+)g Fr(O)s Fz(\()p Fu(t)3101 5398 y Fm(\000)p Fs(1)p Fm(\000)p Ft(\017)3275 5407 y Fj(4)3314 5439 y Fz(\))p Fu(:)245 b Fz(\(8.9\))1939 5776 y(84)p eop %%Page: 85 88 85 87 bop 296 407 a Fz(Also)32 b(notice)h(that)f(for)g(ev)m(ery)i Fu(j)39 b Fz(w)m(e)33 b(ha)m(v)m(e)822 627 y Fr(k)p Fu(f)920 642 y Fs(1)p Ft(;E)1034 627 y Fz(\()p Fu(H)8 b Fz(\))p Fu(F)1262 642 y Fs(1)1301 627 y Fz(\()p Fu(A)1412 642 y Fs(1)1452 627 y Fz(\()p Fu(t)p Fz(\)\))p Fu(:::S)1742 642 y Ft(j)1778 627 y Fz(\()p Fu(t)p Fz(\))p Fu(:::F)2033 642 y Fs(15)2109 627 y Fz(\()p Fu(A)2220 642 y Fs(15)2295 627 y Fz(\()p Fu(t)p Fz(\)\))p Fr(k)27 b(\024)h Fz(Const)q Fu(=t:)589 b Fz(\(8.10\))150 847 y(W)-8 b(e)33 b(then)g(ha)m(v)m(e)548 1067 y Fr(h)p Fu(U)653 1082 y Fs(0)693 1067 y Fz(\()p Fu(t)p Fz(\))p Fu(';)17 b(d)963 1082 y Ft(H)1030 1067 y Fu( )1093 1082 y Ft(f)1138 1067 y Fz(\()p Fu(t)p Fz(\))p Fr(i)28 b Fz(=)2184 b(\(8.11\))585 1181 y Fs(15)548 1211 y Fl(X)559 1421 y Ft(j)t Fs(=1)709 1195 y Fl(D)770 1213 y(p)p 870 1213 103 4 v 92 x Fu(F)933 1320 y Fs(1)972 1305 y Fz(\()p Fu(A)1083 1320 y Fs(1)1123 1305 y Fz(\))p Fu(:::)1242 1220 y Fl(p)p 1341 1220 190 4 v 1341 1305 a Fu(F)1404 1320 y Ft(j)t Fm(\000)p Fs(1)1531 1305 y Fz(\()p Fu(A)1642 1320 y Ft(j)t Fm(\000)p Fs(1)1769 1305 y Fz(\))1807 1220 y Fl(p)p 1907 1220 V 85 x Fu(F)1970 1320 y Ft(j)t Fs(+1)2096 1305 y Fz(\()p Fu(A)2207 1320 y Ft(j)t Fs(+1)2334 1305 y Fz(\))p Fu(:::)2453 1213 y Fl(p)p 2553 1213 138 4 v 92 x Fu(F)2616 1320 y Fs(15)2691 1305 y Fz(\()p Fu(A)2802 1320 y Fs(15)2877 1305 y Fz(\))p Fu(f)2963 1320 y Fs(1)p Ft(;E)3077 1305 y Fz(\()p Fu(H)8 b Fz(\))p Fu(U)3308 1320 y Fs(0)3347 1305 y Fz(\()p Fu(t)p Fz(\))p Fu(')27 b(;)558 1591 y(S)618 1606 y Ft(j)655 1591 y Fz(\()p Fu(t)p Fz(\))766 1499 y Fl(p)p 866 1499 103 4 v 92 x Fu(F)929 1606 y Fs(1)968 1591 y Fz(\()p Fu(A)1079 1606 y Fs(1)1119 1591 y Fz(\))p Fu(:::)1238 1506 y Fl(p)p 1338 1506 190 4 v 85 x Fu(F)1401 1606 y Ft(j)t Fm(\000)p Fs(1)1527 1591 y Fz(\()p Fu(A)1638 1606 y Ft(j)t Fm(\000)p Fs(1)1765 1591 y Fz(\))1803 1506 y Fl(p)p 1903 1506 V 85 x Fu(F)1966 1606 y Ft(j)t Fs(+1)2093 1591 y Fz(\()p Fu(A)2204 1606 y Ft(j)t Fs(+1)2330 1591 y Fz(\))p Fu(:::)2449 1499 y Fl(p)p 2549 1499 138 4 v 92 x Fu(F)2612 1606 y Fs(15)2687 1591 y Fz(\()p Fu(A)2798 1606 y Fs(15)2873 1591 y Fz(\))p Fu( )t Fz(\()p Fu(t)p Fz(\))3089 1481 y Fl(E)548 1773 y Fz(+)p Fr(O)s Fz(\()p Fu(t)779 1732 y Fm(\000)p Fs(1)p Fm(\000)p Ft(\017)953 1741 y Fj(4)992 1773 y Fz(\))p Fr(jj)p Fu(')p Fr(jj)p Fu(;)150 1993 y Fz(where)d(w)m(e)g(used)g(the)g(fact)f(that)f(b)m(y)i (comm)m(uting)d(an)m(y)j(t)m(w)m(o)g(cut-o\013)e(functions)h(w)m(e)h (get)f(an)g(in)m(tegrable)150 2113 y(con)m(tribution.)296 2234 y(The)32 b(next)f(step)g(is)f(to)h(see)g(that)f(from)g Fu(S)1783 2249 y Fs(8)1853 2234 y Fz(up)g(to)g Fu(S)2168 2249 y Fs(15)2274 2234 y Fz(w)m(e)h(ha)m(v)m(e)h(a)e Fr(O)s Fz(\()p Fu(t)2872 2198 y Fm(\0001)3002 2234 y Fz(\))18 b Fr(\001)f(jj)p Fu(\036)p Fr(jj)29 b Fz(con)m(tribution)150 2354 y(b)s(ecause)46 b(the)e(supp)s(orts)h(of)f Fu(g)1283 2369 y Ft(j)t Fm(\006)1418 2354 y Fz(en)m(tering)g(them)g(are)g(lo)s (calized)e(in)h(the)i(classically)d(forbidden)150 2475 y(regions.)h(Hence)34 b(w)m(e)g(can)e(rewrite)h(\(8.11\))f(as)548 2695 y Fr(h)p Fu(U)653 2710 y Fs(0)693 2695 y Fz(\()p Fu(t)p Fz(\))p Fu(';)17 b(d)963 2710 y Ft(H)1030 2695 y Fu( )1093 2710 y Ft(f)1138 2695 y Fz(\()p Fu(t)p Fz(\))p Fr(i)28 b Fz(=)2184 b(\(8.12\))603 2808 y Fs(6)548 2838 y Fl(X)559 3048 y Ft(j)t Fs(=1)709 2822 y Fl(D)770 2841 y(p)p 870 2841 103 4 v 92 x Fu(F)933 2948 y Fs(1)972 2933 y Fz(\()p Fu(A)1083 2948 y Fs(1)1123 2933 y Fz(\))p Fu(:::)1242 2848 y Fl(p)p 1341 2848 190 4 v 1341 2933 a Fu(F)1404 2948 y Ft(j)t Fm(\000)p Fs(1)1531 2933 y Fz(\()p Fu(A)1642 2948 y Ft(j)t Fm(\000)p Fs(1)1769 2933 y Fz(\))1807 2848 y Fl(p)p 1907 2848 V 85 x Fu(F)1970 2948 y Ft(j)t Fs(+1)2096 2933 y Fz(\()p Fu(A)2207 2948 y Ft(j)t Fs(+1)2334 2933 y Fz(\))p Fu(:::)2453 2841 y Fl(p)p 2553 2841 138 4 v 92 x Fu(F)2616 2948 y Fs(15)2691 2933 y Fz(\()p Fu(A)2802 2948 y Fs(15)2877 2933 y Fz(\))p Fu(f)2963 2948 y Fs(1)p Ft(;E)3077 2933 y Fz(\()p Fu(H)8 b Fz(\))p Fu(U)3308 2948 y Fs(0)3347 2933 y Fz(\()p Fu(t)p Fz(\))p Fu(')27 b(;)558 3219 y(S)618 3234 y Ft(j)655 3219 y Fz(\()p Fu(t)p Fz(\))766 3127 y Fl(p)p 866 3127 103 4 v 92 x Fu(F)929 3234 y Fs(1)968 3219 y Fz(\()p Fu(A)1079 3234 y Fs(1)1119 3219 y Fz(\))p Fu(:::)1238 3134 y Fl(p)p 1338 3134 190 4 v 85 x Fu(F)1401 3234 y Ft(j)t Fm(\000)p Fs(1)1527 3219 y Fz(\()p Fu(A)1638 3234 y Ft(j)t Fm(\000)p Fs(1)1765 3219 y Fz(\))1803 3134 y Fl(p)p 1903 3134 V 85 x Fu(F)1966 3234 y Ft(j)t Fs(+1)2093 3219 y Fz(\()p Fu(A)2204 3234 y Ft(j)t Fs(+1)2330 3219 y Fz(\))p Fu(:::)2449 3127 y Fl(p)p 2549 3127 138 4 v 92 x Fu(F)2612 3234 y Fs(15)2687 3219 y Fz(\()p Fu(A)2798 3234 y Fs(15)2873 3219 y Fz(\))p Fu( )t Fz(\()p Fu(t)p Fz(\))3089 3108 y Fl(E)548 3401 y Fz(+)p Fr(O)s Fz(\()p Fu(t)779 3360 y Fm(\000)p Fs(1)p Fm(\000)p Ft(\017)953 3369 y Fj(4)992 3401 y Fz(\))p Fr(jj)p Fu(')p Fr(jj)p Fu(;)150 3621 y Fz(where)34 b(no)m(w)f(the)g(sum)g(only)f(runs)h(o)m(v) m(er)h(the)f(\014rst)g(six)f(cut-o\013s.)296 3741 y(The)c(next)g(lemma) d(pla)m(ys)i(a)f(crucial)g(role)g(in)g(what)h(follo)m(ws.)41 b(F)-8 b(or)26 b Fu( )32 b Fr(2)c Fu(L)2953 3705 y Fs(2)2992 3741 y Fz(\()p Fn(R)3096 3756 y Fs(+)3172 3741 y Fr(\002)11 b Fn(T)p Fz(\))30 b(\(as)d(ab)s(o)m(v)m(e\))150 3862 y(and)33 b Fu(j)g Fr(2)28 b(f)p Fz(1)p Fu(;)17 b(:::;)g Fz(15)p Fr(g)32 b Fz(w)m(e)h(de\014ne)376 4094 y Fu( )439 4109 y Ft(t;j)548 4094 y Fz(:=)28 b Fu(:::)760 4009 y Fl(p)p 860 4009 190 4 v 85 x Fu(F)923 4109 y Ft(j)t Fm(\000)p Fs(1)1050 4094 y Fz(\()p Fu(A)1161 4109 y Ft(j)t Fm(\000)p Fs(1)1287 4094 y Fz(\))1325 4014 y Fl(p)p 1425 4014 397 4 v 80 x Fu(g)1472 4109 y Ft(j)t Fs(+)1585 4094 y Fz(+)22 b Fu(g)1730 4109 y Ft(j)t Fm(\000)1821 4009 y Fl(p)p 1921 4009 190 4 v 85 x Fu(F)1984 4109 y Ft(j)t Fs(+1)2111 4094 y Fz(\()p Fu(A)2222 4109 y Ft(j)t Fs(+1)2349 4094 y Fz(\))p Fu(:::)2468 4002 y Fl(p)p 2567 4002 138 4 v 2567 4094 a Fu(F)2630 4109 y Fs(15)2705 4094 y Fz(\()p Fu(A)2816 4109 y Fs(15)2891 4094 y Fz(\))p Fu(f)2977 4109 y Fs(1)p Ft(;E)3092 4094 y Fz(\()p Fu(H)8 b Fz(\))p Fu( )t(:)225 b Fz(\(8.13\))150 4314 y Fv(Lemma)51 b(8.2.)e FA(Ther)-5 b(e)45 b(exist)h(two)g(c)-5 b(onstants)46 b Fz(0)j Fu(<)f(c)h(<)g(C)55 b(<)49 b Fr(1)d FA(such)g(that)g(for)h(al) 5 b(l)45 b Fu( )t(;)17 b(\036)49 b Fr(2)150 4435 y Fu(L)216 4398 y Fs(2)256 4435 y Fz(\()p Fn(R)360 4450 y Fs(+)447 4435 y Fr(\002)22 b Fn(T)p Fz(\))39 b FA(and)34 b Fu(j)g Fr(2)28 b(f)p Fz(1)p Fu(;)17 b(:::;)g Fz(6)p Fr(g)33 b FA(one)i(has)f(\(for)h Fu(\017)2037 4450 y Fs(4)2104 4435 y Fu(>)28 b Fz(0)34 b FA(smal)5 b(l\))746 4610 y Fu(c)p 746 4655 42 4 v 749 4746 a(t)798 4678 y Fr(jj)p Fu( )917 4693 y Ft(t;j)998 4678 y Fr(jj)1054 4637 y Fs(2)1115 4678 y Fr(\000)22 b(O)s Fz(\()p Fu(t)1369 4637 y Fm(\000)p Fs(1)p Fm(\000)p Ft(\017)1543 4646 y Fj(4)1582 4678 y Fz(\))p Fr(jj)p Fu( )t Fr(jj)1799 4637 y Fs(2)1865 4678 y Fr(\024)1634 b Fz(\(8.14\))736 4783 y Fl(D)796 4802 y(p)p 896 4802 103 4 v 92 x Fu(F)959 4909 y Fs(1)999 4894 y Fz(\()p Fu(A)1110 4909 y Fs(1)1149 4894 y Fz(\))p Fu(:::)1268 4809 y Fl(p)p 1368 4809 190 4 v 85 x Fu(F)1431 4909 y Ft(j)t Fm(\000)p Fs(1)1558 4894 y Fz(\()p Fu(A)1669 4909 y Ft(j)t Fm(\000)p Fs(1)1796 4894 y Fz(\))1834 4809 y Fl(p)p 1933 4809 V 1933 4894 a Fu(F)1996 4909 y Ft(j)t Fs(+1)2123 4894 y Fz(\()p Fu(A)2234 4909 y Ft(j)t Fs(+1)2361 4894 y Fz(\))p Fu(:::)2480 4802 y Fl(p)p 2580 4802 138 4 v 92 x Fu(F)2643 4909 y Fs(15)2718 4894 y Fz(\()p Fu(A)2829 4909 y Fs(15)2903 4894 y Fz(\))p Fu(f)2989 4909 y Fs(1)p Ft(;E)3104 4894 y Fz(\()p Fu(H)8 b Fz(\))p Fu( )30 b(;)736 5066 y(S)796 5081 y Ft(j)832 5066 y Fz(\()p Fu(t)p Fz(\))746 5150 y Fl(p)p 845 5150 103 4 v 845 5242 a Fu(F)908 5257 y Fs(1)948 5242 y Fz(\()p Fu(A)1059 5257 y Fs(1)1098 5242 y Fz(\))p Fu(:::)1217 5157 y Fl(p)p 1317 5157 190 4 v 85 x Fu(F)1380 5257 y Ft(j)t Fm(\000)p Fs(1)1507 5242 y Fz(\()p Fu(A)1618 5257 y Ft(j)t Fm(\000)p Fs(1)1745 5242 y Fz(\))1783 5157 y Fl(p)p 1882 5157 V 1882 5242 a Fu(F)1945 5257 y Ft(j)t Fs(+1)2072 5242 y Fz(\()p Fu(A)2183 5257 y Ft(j)t Fs(+1)2310 5242 y Fz(\))p Fu(:::)2429 5150 y Fl(p)p 2529 5150 138 4 v 92 x Fu(F)2592 5257 y Fs(15)2667 5242 y Fz(\()p Fu(A)2778 5257 y Fs(15)2852 5242 y Fz(\))p Fu(f)2938 5257 y Fs(1)p Ft(;E)3053 5242 y Fz(\()p Fu(H)8 b Fz(\))p Fu( )3285 5132 y Fl(E)1939 5776 y Fz(85)p eop %%Page: 86 89 86 88 bop 150 407 a FA(and)746 512 y Fl(\014)746 572 y(\014)746 632 y(\014)779 516 y(D)840 535 y(p)p 940 535 103 4 v 92 x Fu(F)1003 642 y Fs(1)1042 627 y Fz(\()p Fu(A)1153 642 y Fs(1)1193 627 y Fz(\))p Fu(:::)1312 542 y Fl(p)p 1412 542 190 4 v 85 x Fu(F)1475 642 y Ft(j)t Fm(\000)p Fs(1)1602 627 y Fz(\()p Fu(A)1713 642 y Ft(j)t Fm(\000)p Fs(1)1839 627 y Fz(\))1877 542 y Fl(p)p 1977 542 V 85 x Fu(F)2040 642 y Ft(j)t Fs(+1)2167 627 y Fz(\()p Fu(A)2278 642 y Ft(j)t Fs(+1)2404 627 y Fz(\))p Fu(:::)2523 535 y Fl(p)p 2623 535 138 4 v 92 x Fu(F)2686 642 y Fs(15)2761 627 y Fz(\()p Fu(A)2872 642 y Fs(15)2947 627 y Fz(\))p Fu(f)3044 586 y Fs(1)3033 651 y Ft(E)3093 627 y Fz(\()p Fu(H)8 b Fz(\))p Fu(\036)35 b(;)746 800 y(S)806 815 y Ft(j)843 800 y Fz(\()p Fu(t)p Fz(\))766 883 y Fl(p)p 866 883 103 4 v 92 x Fu(F)929 990 y Fs(1)968 975 y Fz(\()p Fu(A)1079 990 y Fs(1)1119 975 y Fz(\))p Fu(:::)1238 890 y Fl(p)p 1338 890 190 4 v 85 x Fu(F)1401 990 y Ft(j)t Fm(\000)p Fs(1)1527 975 y Fz(\()p Fu(A)1638 990 y Ft(j)t Fm(\000)p Fs(1)1765 975 y Fz(\))1803 890 y Fl(p)p 1903 890 V 85 x Fu(F)1966 990 y Ft(j)t Fs(+1)2093 975 y Fz(\()p Fu(A)2204 990 y Ft(j)t Fs(+1)2330 975 y Fz(\))p Fu(:::)2449 883 y Fl(p)p 2549 883 138 4 v 92 x Fu(F)2612 990 y Fs(15)2687 975 y Fz(\()p Fu(A)2798 990 y Fs(15)2873 975 y Fz(\))p Fu(f)2970 934 y Fs(1)2959 1000 y Ft(E)3018 975 y Fz(\()p Fu(H)8 b Fz(\))p Fu( )3250 865 y Fl(E)3311 861 y(\014)3311 920 y(\014)3311 980 y(\014)746 1209 y Fr(\024)861 1141 y Fu(C)p 861 1186 77 4 v 882 1277 a(t)948 1209 y Fr(jj)p Fu( )1067 1224 y Ft(t;j)1148 1209 y Fr(jj)28 b(jj)p Fu(\036)1346 1224 y Ft(t;j)1426 1209 y Fr(jj)22 b Fz(+)g Fr(O)s Fz(\()p Fu(t)1757 1167 y Fm(\000)p Fs(1)p Fm(\000)p Ft(\017)1931 1176 y Fj(4)1969 1209 y Fz(\))p Fr(jj)p Fu( )t Fr(jj)27 b(jj)p Fu(\036)p Fr(jj)p Fu(:)1166 b Fz(\(8.15\))150 1460 y Fv(Pro)s(of.)73 b Fz(The)44 b(\014rst)f(estimate)e(sa)m(ys)j (that)f(eac)m(h)g Fu(S)2076 1475 y Ft(j)2112 1460 y Fz(,)j Fu(j)k Fr(2)45 b(f)p Fz(1)p Fu(;)17 b(:)g(:)g(:)e(;)i Fz(6)p Fr(g)42 b Fz(is)g(\\p)s(ositiv)m(e".)73 b(That)42 b(is,)150 1580 y(up)s(on)32 b(restricting)f(the)i(Heisen)m(b)s(erg)g (deriv)-5 b(ativ)m(es)32 b(of)f(ev)m(ery)j(observ)-5 b(able)33 b(to)e(states)i(con)m(taining)e(all)150 1700 y(other)e(cut-o\013s)g(w)m(e)g(alw)m(a)m(ys)h(get)f(the)g(plus)f(sign.) 42 b(W)-8 b(e)29 b(used)h(this)e(sort)h(of)f(\\de\014nite)h(sign")f (prop)s(ert)m(y)150 1821 y(whenev)m(er)47 b(w)m(e)f(had)g(to)e(obtain)g (a)h(w)m(eak)h(propagation)d(estimate;)51 b(the)45 b(only)f(truly)h (imp)s(ortan)m(t)150 1941 y(fact)32 b(here)g(is)f(that)h(all)d(\014v)m (e)k Fu(S)1232 1956 y Ft(j)1269 1941 y Fz('s)f(are)g(sim)m(ultaneously) e(p)s(ositiv)m(e.)43 b(Before)32 b(v)m(erifying)f(the)h(prop)s(ert)m(y) 150 2061 y(for)38 b(eac)m(h)i(term,)f(w)m(e)h(w)m(ould)e(lik)m(e)g(to)g (giv)m(e)h(a)f(simple)f(explanation)h(to)g(this)g(apparen)m(tly)h (striking)150 2182 y(coincidence.)77 b(Consider)45 b(the)f(function)f Fu(x)k Fz(:)g(\(1)p Fu(;)17 b Fr(1)p Fz(\))46 b Fr(7!)g Fn(R)5 b Fz(,)53 b Fu(x)p Fz(\()p Fu(t)p Fz(\))47 b(=)g Fr(\006)p Fz(10)p Fu(t)3056 2146 y Fm(\000)p Fs(1)3194 2182 y Fz(.)77 b(W)-8 b(e)44 b(see)h(that)150 2302 y(\()p Fu(dx=dt)p Fz(\))38 b(=)h Fr(\000)p Fu(x=t)h Fz(th)m(us)g(b)s(oth)f Fu(F)1411 2255 y Fm(\000)p Fs(2)p Ft(;)p Fm(\000)p Fs(1)1397 2324 y(+)1615 2302 y Fz(\()p Fu(x)p Fz(\()p Fu(t)p Fz(\)\))g(and)g Fu(F)2169 2255 y Fs(1)p Ft(;)p Fs(2)2155 2324 y Fm(\000)2263 2302 y Fz(\()p Fu(x)p Fz(\()p Fu(t)p Fz(\)\))h(are)f(increasing)f(with) h Fu(t)p Fz(.)63 b(This)39 b(is)150 2423 y(the)30 b(phenomenon)g(b)s (ehind)f(the)h(\\p)s(ositivit)m(y")d(in)i(the)g(case)i(when)f Fu(j)k Fz(=)27 b(1)i(\(see)i(b)s(elo)m(w\).)42 b(As)30 b(a)f(\014nal)150 2543 y(remark,)36 b(let)e(us)i(notice)f(that)g(these) i(six)e(cut-o\013s)g(are)g(c)m(hosen)i(in)e(suc)m(h)h(a)f(w)m(a)m(y)i (that)e(the)h(gro)m(wth)150 2663 y(of)e(their)g(appro)m(ximate)g(c)m (haracteristic)h(functions)f(indicates)g(the)i(tendency)g(of)f(a)f(tra) 5 b(jectory)35 b(to)150 2784 y(b)s(e)e(dra)m(wn)g(to)f(the)h(spiraling) e(attractor.)296 2904 y(In)f(fact)g(if)e(w)m(e)j(in)m(terpret)f(the)g (pro)s(duct)g(of)f(cut-o\013)g(functions)h(as)f(an)h(appro)m(ximate)f (c)m(haracter-)150 3025 y(istic)35 b(function)h(of)h(the)f(attractor,)h (the)g(appro)m(ximate)f(p)s(ositivit)m(y)f(of)h(its)g(Heisen)m(b)s(erg) h(deriv)-5 b(ativ)m(e)150 3145 y(indicates)32 b(the)h(increasing)f (probabilit)m(y)e(that)j Fu( )t Fz(\()p Fu(t)p Fz(\))f(is)g(\\in)g(the) h(attractor".)296 3265 y Fu(j)38 b Fz(=)31 b(1:)47 b(Since)35 b Fu(F)924 3280 y Fs(1)998 3265 y Fz(is)g(a)f Fu(F)1245 3280 y Fs(+)p Fm(\000)1394 3265 y Fz(function,)g(w)m(e)i(write)f(the)g (deriv)-5 b(ativ)m(e)34 b Fu(F)2899 3229 y Fm(0)2885 3290 y Fs(1)2956 3265 y Fz(=)d Fu(g)3110 3280 y Fs(+)3192 3265 y Fr(\000)24 b Fu(g)3340 3280 y Fm(\000)3399 3265 y Fz(.)50 b(W)-8 b(e)35 b(lo)s(ok)150 3386 y(at)k(the)h(op)s(erators) 890 3325 y Fr(p)p 973 3325 106 4 v 61 x Fu(g)1020 3401 y Fs(+)1079 3386 y Fz(\()p Fu(r)s(=t)p Fz(\))26 b Fr(\001)h Fu(D)1448 3401 y Ft(H)1515 3386 y Fu(D)s Fz(\()p Fu(t)p Fz(\))g Fr(\001)1792 3325 y(p)p 1875 3325 V 61 x Fu(g)1922 3401 y Fs(+)1980 3386 y Fz(\()p Fu(r)s(=t)p Fz(\))40 b(and)f Fr(\000)2500 3325 y(p)p 2584 3325 V 2584 3386 a Fu(g)2631 3401 y Fm(\000)2690 3386 y Fz(\()p Fu(r)s(=t)p Fz(\))26 b Fr(\001)h Fu(D)3059 3401 y Ft(H)3126 3386 y Fu(D)s Fz(\()p Fu(t)p Fz(\))g Fr(\001)3403 3325 y(p)p 3486 3325 V 61 x Fu(g)3533 3401 y Fm(\000)3591 3386 y Fz(\()p Fu(r)s(=t)p Fz(\).)150 3506 y(F)-8 b(rom)36 b(the)i(pro)s(of)e (of)h(Prop)s(osition)f(6.19)g(it)h(follo)m(ws)f(that)h(these)i(are)e (essen)m(tially)g(p)s(ositiv)m(e)g(when)150 3626 y(the)c(other)g (cut-o\013s)f(are)h(tak)m(en)h(in)m(to)d(accoun)m(t.)296 3747 y Fu(j)j Fz(=)28 b(2:)43 b Fu(F)656 3762 y Fs(2)728 3747 y Fz(is)32 b(of)g Fu(F)1000 3762 y Fs(+)1092 3747 y Fz(t)m(yp)s(e)h(and)804 3967 y Fu(S)864 3982 y Fs(2)903 3967 y Fz(\()p Fu(t)p Fz(\))28 b(=)1146 3880 y Fl(p)p 1245 3880 123 4 v 1245 3967 a Fu(F)1322 3933 y Fm(0)1308 3989 y Fs(+)1368 3967 y Fz(\()1431 3942 y(~)1406 3967 y Fu(A)1479 3982 y Ft(C)1538 3967 y Fz(\()p Fu(E)6 b Fz(\))p Fu(=t)p Fz(\))22 b Fr(\001)g Fu(D)1967 3982 y Ft(H)2060 3942 y Fz(~)2034 3967 y Fu(A)2107 3982 y Ft(C)2166 3967 y Fz(\()p Fu(E)6 b Fz(\))p Fu(=t)22 b Fr(\001)2476 3880 y Fl(p)p 2576 3880 V 87 x Fu(F)2653 3933 y Fm(0)2639 3989 y Fs(+)2698 3967 y Fz(\()2761 3942 y(~)2736 3967 y Fu(A)2809 3982 y Ft(C)2868 3967 y Fz(\()p Fu(E)6 b Fz(\))p Fu(=t)p Fz(\))p Fu(:)150 4187 y Fz(F)-8 b(or)32 b(its)g(\\p)s(ositivit)m(y")e(go)j(bac)m(k)g(to)f(Lemma)g(6.9.)296 4307 y Fu(j)i Fz(=)28 b(3:)43 b Fu(F)656 4322 y Fs(3)728 4307 y Fz(is)32 b(again)f(of)h Fu(F)1260 4322 y Fs(+)1352 4307 y Fz(t)m(yp)s(e)h(and)1275 4527 y Fu(S)1335 4542 y Fs(3)1374 4527 y Fz(\()p Fu(t)p Fz(\))28 b(:=)1643 4441 y Fl(p)p 1743 4441 V 86 x Fu(F)1820 4493 y Fm(0)1806 4549 y Fs(+)1865 4527 y Fz(\()1913 4501 y(~)1903 4527 y Fu(\030)5 b Fz(\))22 b Fr(\001)g Fu(D)2142 4542 y Ft(H)2219 4501 y Fz(~)2209 4527 y Fu(\030)k Fr(\001)2328 4441 y Fl(p)p 2428 4441 V 86 x Fu(F)2505 4493 y Fm(0)2491 4549 y Fs(+)2550 4527 y Fz(\()2598 4501 y(~)2588 4527 y Fu(\030)t Fz(\))p Fu(:)150 4747 y Fz(See)33 b(for)f(details)g(Lemma)f(6.14.)296 4868 y Fu(j)j Fz(=)28 b(4:)43 b Fu(F)656 4883 y Fs(4)728 4868 y Fz(is)32 b(of)g Fu(F)1000 4883 y Fm(\000)1092 4868 y Fz(t)m(yp)s(e)h(and)785 5088 y Fu(S)845 5103 y Fs(4)884 5088 y Fz(\()p Fu(t)p Fz(\))28 b(=)f Fr(\000)1203 5001 y Fl(p)p 1303 5001 200 4 v 87 x Fr(\000)p Fu(F)1457 5053 y Fm(0)1443 5109 y(\000)1503 5088 y Fz(\()9 b(~)-58 b Fu(\032)23 b Fr(\000)f Fu(\032)1763 5103 y Ft(E)1819 5084 y Ff(0)1845 5088 y Fz(\))h Fr(\001)e Fu(D)2036 5103 y Ft(H)2104 5088 y Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)2364 5103 y Ft(E)2420 5084 y Ff(0)2446 5088 y Fz(\))2484 5001 y Fl(p)p 2583 5001 V 2583 5088 a Fr(\000)p Fu(F)2737 5053 y Fm(0)2723 5109 y(\000)2783 5088 y Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)3043 5103 y Ft(E)3099 5084 y Ff(0)3125 5088 y Fz(\))p Fu(:)150 5308 y Fz(W)-8 b(e)33 b(treated)g(suc)m(h)g(terms)g(in)e(Lemma)g(6.15;)h(w)m (e)i(see)f(that)f Fu(D)2432 5323 y Ft(H)2499 5308 y Fz(\()9 b(~)-58 b Fu(\032)22 b Fr(\000)h Fu(\032)2759 5323 y Ft(E)2815 5304 y Ff(0)2841 5308 y Fz(\))32 b(is)g(\\almost")e Fu(E)e Fr(\000)22 b Fu(E)3698 5271 y Fm(0)3749 5308 y Fz(=)150 5428 y Fr(\000)p Fz(2)p Fu(\017)315 5443 y Fs(0)383 5428 y Fu(<)27 b Fz(0)33 b(if)e Fu(\017)696 5443 y Fs(2)768 5428 y Fz(is)h(small)f(enough.)1939 5776 y(86)p eop %%Page: 87 90 87 89 bop 296 407 a Fu(j)34 b Fz(=)28 b(5:)43 b Fu(F)656 422 y Fs(5)728 407 y Fz(is)32 b(of)g Fu(F)1000 422 y Fm(\000)1092 407 y Fz(t)m(yp)s(e)h(\(the)g(maximal)c(v)m(elo)s(cit)m(y) k(cut-o\013)7 b(\),)32 b(so)1086 627 y Fu(S)1146 642 y Fs(5)1185 627 y Fz(\()p Fu(t)p Fz(\))c(=)g Fr(\000)1505 540 y Fl(p)p 1605 540 200 4 v 87 x Fr(\000)p Fu(F)1759 593 y Fm(0)1745 649 y(\000)1805 627 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu(i)p Fz([)p Fu(H)r(;)17 b(r)s(=t)p Fz(])2357 540 y Fl(p)p 2456 540 V 2456 627 a Fr(\000)p Fu(F)2610 593 y Fm(0)2596 649 y(\000)2655 627 y Fz(\()p Fu(r)s(=t)p Fz(\))p Fu(:)150 847 y Fz(Its)33 b(\\p)s(ositivit)m(y")e(comes)i(from)e (the)i(considerations)f(w)m(e)i(made)e(in)g(the)h(pro)s(of)e(of)h (Lemma)g(6.4.)296 967 y Fu(j)i Fz(=)28 b(6:)43 b Fu(F)656 982 y Fs(6)728 967 y Fz(is)32 b(again)f(of)h Fu(F)1260 982 y Fm(\000)1352 967 y Fz(t)m(yp)s(e)h(and)746 1187 y Fu(S)806 1202 y Fs(6)846 1187 y Fz(\()p Fu(t)p Fz(\))27 b(:=)h Fr(\000)1192 1101 y Fl(p)p 1292 1101 V 86 x Fr(\000)p Fu(F)1446 1153 y Fm(0)1432 1209 y(\000)1492 1187 y Fz(\()p Fu(t)1565 1146 y Fs(1+)p Ft(\017)1684 1155 y Fj(4)1722 1187 y Fu(L)1788 1202 y Ft(\016)1827 1187 y Fz(\))22 b Fr(\001)g Fu(D)2018 1202 y Ft(H)2085 1187 y Fz(\()p Fu(t)2158 1146 y Fs(1+)p Ft(\017)2277 1155 y Fj(4)2315 1187 y Fu(L)2381 1202 y Ft(\016)2420 1187 y Fz(\))g Fr(\001)2530 1101 y Fl(p)p 2629 1101 V 2629 1187 a Fr(\000)p Fu(F)2783 1153 y Fm(0)2769 1209 y(\000)2829 1187 y Fz(\()p Fu(t)2902 1146 y Fs(1+)p Ft(\017)3021 1155 y Fj(4)3059 1187 y Fu(L)3125 1202 y Ft(\016)3164 1187 y Fz(\))p Fu(:)150 1407 y Fz(See)33 b(for)f(details)g(Lemma)f(7.6.)296 1528 y(Ab)s(out)49 b(\(8.15\):)74 b(these)49 b(estimates)f(are)h(b)s(oundedness)h(prop)s (erties)e(whic)m(h)h(easily)f(ma)m(y)g(b)s(e)150 1648 y(deduced)34 b(from)e(the)h(ab)s(o)m(v)m(e)g(considerations)f(\(see)i (also)e(\(8.10\)\).)p 3754 1648 4 66 v 3758 1585 59 4 v 3758 1648 V 3816 1648 4 66 v 150 1886 a Fv(Completion)53 b(of)i(the)g(pro)s(of)g(of)g(Prop)s(osition)e(8.1)48 b Fz(In)m(tro)s(ducing)g(\(8.15\))f(in)g(\(8.12\))g(and)150 2007 y(applying)31 b(the)i(Cauc)m(h)m(y-Sc)m(h)m(w)m(arz)k(inequalit)m (y)31 b(w)m(e)j(obtain)589 2227 y Fr(jh)p Fu(U)722 2242 y Fs(0)761 2227 y Fz(\()p Fu(t)p Fz(\))p Fu(';)17 b(d)1031 2242 y Ft(H)1098 2227 y Fu( )1161 2242 y Ft(f)1206 2227 y Fz(\()p Fu(t)p Fz(\))p Fr(ij)28 b(\024)2087 b Fz(\(8.16\))589 2489 y Fu(C)682 2318 y Fl( )771 2421 y Fz(1)p 771 2466 49 4 v 778 2557 a Fu(t)901 2364 y Fs(6)847 2394 y Fl(X)857 2604 y Ft(j)t Fs(=1)1007 2489 y Fr(jj)p Fz(\()p Fu(U)1167 2504 y Fs(0)1206 2489 y Fz(\()p Fu(t)p Fz(\))p Fu(')p Fz(\))1419 2504 y Ft(t;j)1501 2489 y Fr(jj)1557 2448 y Fs(2)1595 2318 y Fl(!)1674 2338 y Fs(1)p Ft(=)p Fs(2)1801 2318 y Fl( )1890 2421 y Fz(1)p 1890 2466 V 1897 2557 a Fu(t)2019 2364 y Fs(6)1965 2394 y Fl(X)1976 2604 y Ft(j)t Fs(=1)2126 2489 y Fr(jj)p Fz(\()p Fu(e)2265 2448 y Fm(\000)p Ft(itH)2436 2489 y Fu( )t Fz(\))2541 2504 y Ft(t;j)2622 2489 y Fr(jj)2678 2448 y Fs(2)2717 2318 y Fl(!)2796 2338 y Fs(1)p Ft(=)p Fs(2)2928 2489 y Fz(+)22 b Fu(C)7 b(t)3138 2448 y Fm(\000)p Fs(1)p Fm(\000)p Ft(\017)3312 2457 y Fj(4)3351 2489 y Fr(jj)p Fu(')p Fr(jj)p Fu(:)296 2810 y Fz(Recall)40 b(that)h(w)m(e)h(had)g(to)e(lo)s(ok)h(at)f(an)i(in) m(tegral)d(as)j(in)e(\(8.4\).)69 b(The)42 b(prop)s(osition)e(w)m(ould)h (b)s(e)150 2930 y(concluded)33 b(if)f(w)m(e)h(could)f(pro)m(v)m(e)i (that)461 3122 y Fl(Z)561 3149 y Ft(t)586 3158 y Fj(2)517 3348 y Ft(t)542 3357 y Fj(1)642 3088 y Fl( )730 3191 y Fz(1)p 730 3235 V 737 3326 a Fu(t)860 3134 y Fs(6)806 3163 y Fl(X)816 3373 y Ft(j)t Fs(=1)966 3258 y Fr(jj)p Fz(\()p Fu(U)1126 3273 y Fs(0)1165 3258 y Fz(\()p Fu(t)p Fz(\))p Fu(')p Fz(\))1378 3273 y Ft(t;j)1460 3258 y Fr(jj)1516 3217 y Fs(2)1555 3088 y Fl(!)1633 3107 y Fs(1)p Ft(=)p Fs(2)1760 3088 y Fl( )1849 3191 y Fz(1)p 1849 3235 V 1856 3326 a Fu(t)1979 3134 y Fs(6)1924 3163 y Fl(X)1935 3373 y Ft(j)t Fs(=1)2085 3258 y Fr(jj)p Fz(\()p Fu(e)2224 3217 y Fm(\000)p Ft(itH)2395 3258 y Fu( )t Fz(\))2500 3273 y Ft(t;j)2581 3258 y Fr(jj)2637 3217 y Fs(2)2676 3088 y Fl(!)2755 3107 y Fs(1)p Ft(=)p Fs(2)2882 3258 y Fu(dt)27 b Fr(\024)h Fu(")p Fr(jj)p Fu(')p Fr(jj)p Fu(:)227 b Fz(\(8.17\))150 3573 y(This)33 b(will)d(b)s(e)j(ac)m(hiev)m (ed)h(as)e(so)s(on)h(as)g(w)m(e)g(obtain)f(the)h(next)g(t)m(w)m(o)h (estimates)1120 3723 y Fl(Z)1219 3749 y Fm(1)1175 3949 y Fs(1)1321 3791 y Fz(1)p 1321 3836 V 1328 3927 a Fu(t)1450 3734 y Fs(6)1396 3764 y Fl(X)1407 3974 y Ft(j)t Fs(=1)1557 3859 y Fr(jj)p Fz(\()p Fu(U)1717 3874 y Fs(0)1756 3859 y Fz(\()p Fu(t)p Fz(\))p Fu(')p Fz(\))1969 3874 y Ft(t;j)2050 3859 y Fr(jj)2106 3817 y Fs(2)2145 3859 y Fu(dt)27 b Fr(\024)i Fz(Const)f Fr(jj)p Fu(')p Fr(jj)2817 3817 y Fs(2)3576 3859 y Fz(\(8.18\))150 4174 y(and)1103 4224 y Fl(Z)1203 4250 y Fm(1)1159 4449 y Fs(1)1304 4292 y Fz(1)p 1304 4337 V 1311 4428 a Fu(t)1434 4235 y Fs(6)1379 4265 y Fl(X)1390 4475 y Ft(j)t Fs(=1)1540 4360 y Fr(jj)p Fz(\()p Fu(e)1679 4318 y Fm(\000)p Ft(itH)1850 4360 y Fu( )t Fz(\))1955 4375 y Ft(t;j)2037 4360 y Fr(jj)2093 4318 y Fs(2)2131 4360 y Fu(dt)g Fr(\024)g Fz(Const)h Fr(jj)p Fu( )t Fr(jj)2807 4318 y Fs(2)2845 4360 y Fu(:)704 b Fz(\(8.19\))296 4635 y(Since)30 b(\(8.19\))f(follo)m(ws)g(from)f(the) i(propagation)e(estimates)i(w)m(e)g(ha)m(v)m(e)h(obtained)e(so)h(far,)g (w)m(e)h(are)150 4755 y(only)i(left)g(with)g(pro)m(ving)g(\(8.18\).)46 b(This)33 b(is)g(the)h(place)g(where)g(the)g(sim)m(ultaneous)f(\\p)s (ositivit)m(y")e(of)150 4875 y Fu(S)210 4890 y Ft(j)247 4875 y Fz(,)h Fu(j)i Fr(2)28 b(f)p Fz(1)p Fu(;)17 b(:::;)g Fz(6)p Fr(g)31 b Fz(from)h(\(8.14\))f(pla)m(ys)i(a)g(cen)m(tral)f (role.)296 4996 y(No)m(w)40 b(let)f(us)h(pro)m(v)m(e)g(\(8.18\).)63 b(W)-8 b(e)40 b(in)m(tro)s(duce)f(the)h(function)e(\(remem)m(b)s(er)h (that)g(the)h(order)f(of)150 5116 y(cut-o\013s)33 b(do)s(es)g(not)f (coun)m(t\))503 5380 y Fu(V)22 b Fz(\()p Fu(t;)17 b(')p Fz(\))27 b(=)932 5266 y Fl(\015)932 5325 y(\015)932 5385 y(\015)987 5380 y Fu(f)1035 5395 y Fs(1)p Ft(;E)1150 5380 y Fz(\()p Fu(H)8 b Fz(\))1315 5288 y Fl(p)p 1414 5288 138 4 v 1414 5380 a Fu(F)1477 5395 y Fs(15)1552 5380 y Fz(\()p Fu(A)1663 5395 y Fs(15)1737 5380 y Fz(\()p Fu(t)p Fz(\)\))p Fu(:::)1967 5288 y Fl(p)p 2067 5288 103 4 v 92 x Fu(F)2130 5395 y Fs(6)2170 5380 y Fz(\()p Fu(A)2281 5395 y Fs(6)2320 5380 y Fz(\()p Fu(t)p Fz(\)\))p Fu(:::)2550 5288 y Fl(p)p 2650 5288 V 92 x Fu(F)2713 5395 y Fs(1)2753 5380 y Fz(\()p Fu(A)2864 5395 y Fs(1)2903 5380 y Fz(\()p Fu(t)p Fz(\)\))p Fu(U)3118 5395 y Fs(0)3158 5380 y Fz(\()p Fu(t)p Fz(\))p Fu(')3333 5266 y Fl(\015)3333 5325 y(\015)3333 5385 y(\015)3389 5292 y Fs(2)3445 5380 y Fu(:)1939 5776 y Fz(87)p eop %%Page: 88 91 88 90 bop 296 407 a Fz(Di\013eren)m(tiate)34 b Fu(V)21 b Fz(\()p Fu(t;)c(')p Fz(\))35 b(with)f(resp)s(ect)i(to)e Fu(t)p Fz(.)51 b(When)35 b(the)h(deriv)-5 b(ativ)m(e)34 b(acts)h(on)g Fu(U)3300 422 y Fs(0)3339 407 y Fz(\()p Fu(t)p Fz(\))p Fu(')g Fz(w)m(e)h(get)150 539 y Fu(:)17 b(:)g(:)281 462 y Fr(p)p 364 462 103 4 v 77 x Fu(F)427 554 y Fs(1)467 539 y Fz(\()p Fu(A)578 554 y Fs(1)617 539 y Fz(\()p Fu(t)p Fz(\)\))22 b Fr(\001)g Fz(\()p Fu(H)30 b Fz(+)1129 500 y Fs(1)p 1095 516 104 4 v 1095 574 a(8)p Ft(r)1164 555 y Fj(2)1231 539 y Fr(\000)1340 496 y Ft(\015)1380 472 y Fj(2)p 1340 516 75 4 v 1360 574 a Fs(2)1425 539 y Fz(\))p Fu(U)1529 554 y Fs(0)1569 539 y Fz(\()p Fu(t)p Fz(\))p Fu(')p Fz(;)i(see)i(\(4.19\).)43 b(Using)32 b(that)1170 743 y Fu(f)1218 758 y Fs(1)p Ft(;E)1332 743 y Fz(\()p Fu(H)8 b Fz(\))1497 652 y Fl(p)p 1596 652 103 4 v 1596 743 a Fu(F)1659 758 y Fs(1)1699 743 y Fz(\()p Fu(r)s(=t)p Fz(\))1923 663 y Fl(\010)1984 743 y Fz(~)-53 b Fu(\015)2036 702 y Fs(2)2098 743 y Fr(\000)23 b Fu(\015)2254 702 y Fs(2)2293 663 y Fl(\011)2379 743 y Fz(=)k Fr(O)s Fz(\()p Fu(t)2637 702 y Fm(\0001)2767 743 y Fz(\))150 926 y(and)1411 1046 y(~)-53 b Fu(\015)1463 1005 y Fs(2)1502 954 y Fl(p)p 1602 954 V 92 x Fu(F)1665 1061 y Fs(6)1705 1046 y Fz(\()p Fu(A)1816 1061 y Fs(6)1855 1046 y Fz(\()p Fu(t)p Fz(\)\))28 b(=)f Fr(O)s Fz(\()p Fu(t)2290 1005 y Fm(\000)p Fs(1)p Fm(\000)p Ft(\017)2464 1014 y Fj(4)2503 1046 y Fz(\))p Fu(;)150 1204 y Fz(w)m(e)34 b(get)803 1471 y Fu(@)854 1486 y Ft(t)884 1471 y Fu(V)21 b Fz(\()p Fu(t;)c(')p Fz(\))28 b(=)f(2)1414 1346 y Fs(15)1378 1376 y Fl(X)1388 1586 y Ft(j)t Fs(=1)1538 1471 y Fr(<)1627 1360 y Fl(D)1688 1471 y Fu(:::)1786 1360 y Fl(n)1852 1471 y Fu(D)1933 1486 y Ft(H)2000 1386 y Fl(p)p 2100 1386 100 4 v 85 x Fu(F)2163 1486 y Ft(j)2199 1471 y Fz(\()p Fu(A)2310 1486 y Ft(j)2347 1471 y Fz(\()p Fu(t)p Fz(\)\))2496 1360 y Fl(o)2579 1471 y Fu(:::f)2708 1486 y Fs(1)p Ft(;E)2823 1471 y Fz(\()p Fu(H)8 b Fz(\))p Fu(U)3054 1486 y Fs(0)3093 1471 y Fz(\()p Fu(t)p Fz(\))p Fu(';)813 1757 y(:::)894 1671 y Fl(p)p 994 1671 V 86 x Fu(F)1057 1772 y Ft(j)1093 1757 y Fz(\()p Fu(A)1204 1772 y Ft(j)1241 1757 y Fz(\()p Fu(t)p Fz(\)\))p Fu(:::f)1519 1772 y Fs(1)p Ft(;E)1634 1757 y Fz(\()p Fu(H)g Fz(\))p Fu(U)1865 1772 y Fs(0)1904 1757 y Fz(\()p Fu(t)p Fz(\))p Fu(')2079 1646 y Fl(E)2162 1757 y Fz(+)22 b Fr(O)s Fz(\()p Fu(t)2415 1716 y Fm(\000)p Fs(1)p Fm(\000)p Ft(\017)2589 1725 y Fj(4)2627 1757 y Fz(\))p Fr(jj)p Fu(')p Fr(jj)2841 1716 y Fs(2)2879 1757 y Fu(:)670 b Fz(\(8.20\))296 1966 y(Rearranging)21 b(the)i(ab)s(o)m(v)m 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3754 2762 4 66 v 3758 2700 59 4 v 3758 2762 V 3816 2762 4 66 v 296 3001 a(W)-8 b(e)39 b(ha)m(v)m(e)h(therefore)e(pro)m(v)m(en)i (b)s(oth)e(I)g(and)h(I)s(I)f(of)g(Theorem)g(4.2,)i(or)e(equiv)-5 b(alen)m(tly)d(,)38 b(that)h(the)150 3121 y(limits)31 b(de\014ned)36 b(in)e(\(1.23\))g(exist)h(and)f(de\014ne)i(unitary)e(op) s(erators)g(whic)m(h)h(are)g(m)m(utually)d(in)m(v)m(erse.)150 3241 y(As)h(for)f(I)s(I)s(I)h(w)m(e)g(refer)g(to)f(the)h(discussion)g (at)g(the)g(end)g(of)f(section)h(5.)p 3754 3241 V 3758 3179 59 4 v 3758 3241 V 3816 3241 4 66 v 150 3687 a Fw(9)161 b(Appro)l(ximate)53 b(dynamics)g(for)h(negativ)l(e)f(times)150 3906 y Fz(The)36 b(\014rst)g(issue)g(w)m(e)h(w)m(an)m(t)f(to)f(explain) f(in)h(this)g(section)h(is)e(the)i(b)s(eha)m(vior)f(of)g(our)h(system)g (in)f(the)150 4026 y(distan)m(t)i(past.)56 b(The)38 b(appro)m(ximate)e (dynamics)h Fu(U)2020 4041 y Fs(0)2059 4026 y Fz(\()p Fu(t)p Fz(\))g(in)f(\(4.11\))g(and)h(\(4.12\))f(only)h(mak)m(es)g (sense)150 4147 y(for)29 b(p)s(ositiv)m(e)g(times)g(and)g(sho)m(ws)j (that)d(in)g(the)h(distan)m(t)f(future)h(ev)m(ery)i(scattering)d(state) h(will)d(spiral)150 4267 y(a)m(w)m(a)m(y)34 b(from)d(the)i(origin.)296 4387 y(F)-8 b(or)44 b(negativ)m(e)g(times,)j(the)d(picture)h(is)e(rev)m (ersed.)81 b(Our)44 b(task)h(is)f(to)g(\014nd)h(an)f(appro)m(ximate)150 4508 y(dynamics)c(with)g(the)h(usual)g(spiraling)d(feature)j(in)e(the)i (distan)m(t)g(past,)h(and)f(whic)m(h)g(sho)m(ws)h(ho)m(w)150 4628 y(the)e(particle)e(is)i(dra)m(wn)g(to)f(the)i(origin.)62 b(In)40 b(other)g(w)m(ords,)j(instead)c(of)h(lo)s(oking)d(for)i (attracting)150 4748 y(p)s(erio)s(dic)32 b(solutions)g(at)h(p)s(ositiv) m(e)g(times,)g(w)m(e)h(no)m(w)g(searc)m(h)h(for)e(attracting)f(p)s (erio)s(dic)g(solutions)g(at)150 4869 y(negativ)m(e)h(times.)296 4989 y(W)-8 b(e)32 b(are)f(therefore)h(in)m(terested)h(in)e(obtaining)e Fu(C)2101 4953 y Fs(1)2172 4989 y Fz(and)i(p)s(erio)s(dic)f(solutions)g (to)h(the)h(system)g(of)150 5110 y(equations)377 5209 y Fl(8)377 5298 y(<)377 5478 y(:)507 5283 y Fu(@)558 5298 y Ft(\022)598 5283 y Fu(\032)c Fz(=)f Fu(b)c Fz(+)f Fu(\030)507 5409 y(@)558 5424 y Ft(\022)598 5409 y Fu(\030)32 b Fz(=)27 b Fr(\000)863 5366 y Ft(b)p Fs(+)p Ft(\030)p 863 5386 119 4 v 906 5444 a(\030)992 5409 y Fu(\032)507 5541 y(\030)555 5505 y Fs(2)616 5541 y Fz(+)22 b Fu(\032)764 5505 y Fs(2)832 5541 y Fz(=)27 b(2)p Fu(E)1130 5413 y(;)114 b(\022)31 b Fr(\025)d Fz(0)p Fu(;)44 b(\032)p Fz(\(0\))28 b Fr(2)g Fz(\()p Fr(\000)1984 5324 y(p)p 2068 5324 127 4 v 2068 5413 a Fz(2)p Fu(E)5 b(;)2238 5324 y Fr(p)p 2321 5324 V 89 x Fz(2)p Fu(E)h Fz(\))28 b(and)f Fu(\030)5 b Fz(\(0\))27 b Fr(2)h Fz(\(0)p Fu(;)3123 5324 y Fr(p)p 3206 5324 V 89 x Fz(2)p Fu(E)5 b Fz(\))p Fu(;)227 b Fz(\(9.1\))1939 5776 y(88)p eop %%Page: 89 92 89 91 bop 150 407 a Fz(with)32 b(the)h(supplemen)m(tary)g(condition) 1638 532 y Fl(Z)1738 558 y Fs(2)p Ft(\031)1693 757 y Fs(0)1847 600 y Fu(\032)p 1847 645 51 4 v 1848 736 a(\030)1907 668 y Fz(\()p Fu(\022)s Fz(\))p Fu(d\022)d(<)e Fz(0)p Fu(:)1287 b Fz(\(9.2\))150 921 y(Let)36 b(us)h(explain)e(the)i(meaning) d(of)i(\(9.2\).)53 b(Assume)37 b(that)f(w)m(e)h(ha)m(v)m(e)g(suc)m(h)h (solutions)d(and)h(denote)150 1042 y(them)c(b)m(y)i Fu(\032)585 1057 y Ft(e)654 1042 y Fz(and)f Fu(\030)887 1057 y Ft(e)924 1042 y Fz(.)43 b(Consider)33 b(the)g(initial)c(v)-5 b(alue)32 b(problem)f(\()p Fu(t)d Fr(\025)g(\000)p Fz(1\):)893 1248 y Fu(d)t Fz(~)-53 b Fu(r)p 893 1293 98 4 v 899 1384 a(dt)1028 1316 y Fz(=)27 b Fu(\032)1181 1331 y Ft(e)1218 1316 y Fz(\()1264 1289 y(~)1256 1316 y Fu(\022)s Fz(\))p Fu(;)1494 1248 y(d)1553 1222 y Fz(~)1545 1248 y Fu(\022)p 1494 1293 99 4 v 1500 1384 a(dt)1630 1316 y Fz(=)1744 1248 y Fu(\030)1787 1263 y Ft(e)1823 1248 y Fz(\()1869 1222 y(~)1861 1248 y Fu(\022)s Fz(\))p 1744 1293 204 4 v 1826 1384 a(~)-53 b Fu(r)1957 1316 y(;)114 b Fz(\()t(~)-53 b Fu(r)s Fz(\()p Fr(\000)p Fz(1\))p Fu(;)2437 1289 y Fz(~)2429 1316 y Fu(\022)s Fz(\()p Fr(\000)p Fz(1\)\))28 b(=)f(\(1)p Fu(;)17 b Fz(0\))p Fu(:)531 b Fz(\(9.3\))150 1542 y(It)37 b(is)g(easy)h(to)f(c)m(hec)m(k)i(that)e(at)g(least)f(for)h Fu(t)g Fz(close)g(to)g Fr(\000)p Fz(1)g(the)h(ab)s(o)m(v)m(e)g(system)g (admits)e(a)g(solution)150 1663 y(\()t(~)-53 b Fu(r)s(;)286 1637 y Fz(~)279 1663 y Fu(\022)r Fz(\))36 b(whic)m(h)h(also)e(solv)m (es)i(the)g(Hamilton)c(equations,)38 b(th)m(us)f(it)e(corresp)s(onds)i (to)f(a)g(real)f(orbit)g(at)150 1783 y(energy)f Fu(E)6 b Fz(.)43 b(W)-8 b(e)33 b(notice)f(that)h(the)g(ab)s(o)m(v)m(e)g (system)h(giv)m(es)1134 2063 y(~)-53 b Fu(r)r Fz(\()p Fu(t)p Fz(\))28 b(=)k(~)-53 b Fu(r)r Fz(\()1511 2037 y(~)1503 2063 y Fu(\022)t Fz(\()p Fu(t)p Fz(\)\))27 b(=)h(exp)q Fr(f)2031 1928 y Fl(Z)2136 1936 y Fs(~)2130 1954 y Ft(\022)r Fs(\()p Ft(t)p Fs(\))2086 2153 y(0)2250 2063 y Fz(\()p Fu(\032)2338 2078 y Ft(e)2375 2063 y Fu(=\030)2467 2078 y Ft(e)2503 2063 y Fz(\)\()p Fu(')p Fz(\))p Fu(d')p Fr(g)778 b Fz(\(9.4\))150 2335 y(th)m(us)47 b(Ran)o(\()598 2308 y(~)590 2335 y Fu(\022)t Fz(\))i(=)h([0)p Fu(;)17 b Fr(1)p Fz(\))45 b(and)50 b(~)-53 b Fu(r)s Fz(\()p Fu(t)p Fz(\))45 b(decreases)j(\\in)d(mean")g(after)g(eac)m(h)i(complete)e(rev)m (olution)150 2455 y(around)39 b(the)g(origin)e(and)i(collapses)g(to)f (it)g(in)h(a)f(\014nite)h(amoun)m(t)f(of)h(time.)61 b(Nev)m(ertheless,) 43 b(going)150 2576 y(bac)m(kw)m(ards)35 b(in)d Fu(t)g Fz(w)m(e)i(see)g(the)f(spiraling)d(b)s(eha)m(vior)i(again.)296 2696 y(No)m(w)h(let)e(us)i(in)m(v)m(estigate)f(the)g(existence)i(of)d (suc)m(h)j(solutions.)42 b(De\014ne)32 b Fu(b)2967 2711 y Fs(1)3007 2696 y Fz(\()p Fu(\022)s Fz(\))c(:=)g Fu(b)p Fz(\()p Fr(\000)p Fu(\022)s Fz(\).)44 b(Then)150 2816 y Fu(b)191 2831 y Fs(1)270 2816 y Fz(is)38 b(negativ)m(e,)i(p)s(erio)s (dic)e(and)g(if)g Fu(E)44 b(>)38 b(E)1767 2831 y Ft(d)1808 2816 y Fz(\()p Fu(b)1887 2831 y Fs(1)1927 2816 y Fz(\))h(then)g(\(see)h (Corollary)d(2.7\))h(w)m(e)i(ha)m(v)m(e)g(a)e(unique)150 2937 y(p)s(erio)s(dic)31 b(solution)g(to)369 3156 y Fl(8)369 3246 y(<)369 3425 y(:)499 3231 y Fu(@)550 3246 y Ft(\022)590 3231 y Fu(\032)d Fz(=)f Fu(b)812 3246 y Fs(1)874 3231 y Fz(+)c Fu(\030)499 3357 y(@)550 3372 y Ft(\022)590 3357 y Fu(\030)32 b Fz(=)c Fr(\000)856 3313 y Ft(b)886 3322 y Fj(1)921 3313 y Fs(+)p Ft(\030)p 856 3334 154 4 v 916 3391 a(\030)1019 3357 y Fu(\032)499 3488 y(\030)547 3452 y Fs(2)608 3488 y Fz(+)22 b Fu(\032)756 3452 y Fs(2)824 3488 y Fz(=)27 b(2)p Fu(E)1138 3360 y(;)114 b(\022)31 b Fr(\025)d Fz(0)p Fu(;)44 b(\032)p Fz(\(0\))28 b Fr(2)g Fz(\()p Fr(\000)1992 3271 y(p)p 2075 3271 127 4 v 89 x Fz(2)p Fu(E)6 b(;)2246 3271 y Fr(p)p 2329 3271 V 89 x Fz(2)p Fu(E)g Fz(\))27 b(and)h Fu(\030)5 b Fz(\(0\))27 b Fr(2)h Fz(\(0)p Fu(;)3131 3271 y Fr(p)p 3213 3271 V 3213 3360 a Fz(2)p Fu(E)6 b Fz(\))p Fu(;)219 b Fz(\(9.5\))150 3683 y(with)32 b(the)h(supplemen)m(tary)g(condition)1638 3808 y Fl(Z)1738 3834 y Fs(2)p Ft(\031)1693 4033 y Fs(0)1847 3876 y Fu(\032)p 1847 3920 51 4 v 1848 4012 a(\030)1907 3943 y Fz(\()p Fu(\022)s Fz(\))p Fu(d\022)d(>)e Fz(0)p Fu(:)1287 b Fz(\(9.6\))296 4192 y(The)35 b(in)m(timate)c(connection)i (b)s(et)m(w)m(een)i(attractiv)m(e)e(solutions)g(at)g(p)s(ositiv)m(e)f (times)g(and)i(attrac-)150 4312 y(tiv)m(e)f(solutions)e(at)h(negativ)m (e)h(times)f(is)g(giv)m(en)h(b)m(y)g(the)g(follo)m(wing)d(prop)s (osition:)150 4499 y Fv(Prop)s(osition)41 b(9.1.)j FA(Assume)39 b(that)h Fz(\()p Fu(\032)1639 4514 y Ft(E)1698 4499 y Fu(;)17 b(\030)1785 4514 y Ft(E)1844 4499 y Fz(\))39 b FA(is)g(a)g(p)-5 b(erio)g(dic)38 b(solution)h(which)f(solves)g (\(9.5\))h(and)150 4620 y(ob)-5 b(eys)42 b(\(9.6\).)67 b(Denote)42 b(by)h Fu(\032)1249 4635 y Ft(e)1286 4620 y Fz(\()p Fu(\022)s Fz(\))f(:=)g Fr(\000)p Fu(\032)1724 4635 y Ft(E)1784 4620 y Fz(\()p Fr(\000)p Fu(\022)s Fz(\))h FA(and)f Fu(\030)2268 4635 y Ft(e)2305 4620 y Fz(\()p Fu(\022)s Fz(\))g(=)g Fu(\030)2632 4635 y Ft(E)2691 4620 y Fz(\()p Fr(\000)p Fu(\022)s Fz(\))p FA(.)68 b(Then)42 b Fz(\()p Fu(\032)3340 4635 y Ft(e)3377 4620 y Fu(;)17 b(\030)3464 4635 y Ft(e)3500 4620 y Fz(\))43 b FA(solves)150 4740 y(\(9.1\))f(and)g(ob)-5 b(eys)42 b(\(9.2\).)67 b(R)-5 b(e)g(cipr)g(o)g(c)g(al)5 b(ly,)44 b(assume)e(that)g Fz(\()p Fu(\032)2391 4755 y Ft(e)2429 4740 y Fu(;)17 b(\030)2516 4755 y Ft(e)2552 4740 y Fz(\))42 b FA(is)h(a)f(p)-5 b(erio)g(dic)42 b(solution)g(which)150 4861 y(solves)36 b(\(9.1\))g(and)h(ob)-5 b(eys)36 b(\(9.2\).)51 b(Denote)36 b(by)h Fu(\032)1934 4876 y Ft(E)1994 4861 y Fz(\()p Fu(\022)s Fz(\))31 b(:=)h Fr(\000)p Fu(\032)2411 4876 y Ft(e)2449 4861 y Fz(\()p Fr(\000)p Fu(\022)s Fz(\))37 b FA(and)g Fu(\030)2922 4876 y Ft(E)2981 4861 y Fz(\()p Fu(\022)s Fz(\))31 b(=)h Fu(\030)3287 4876 y Ft(e)3323 4861 y Fz(\()p Fr(\000)p Fu(\022)s Fz(\))p FA(.)52 b(Then)150 4981 y Fz(\()p Fu(\032)238 4996 y Ft(E)298 4981 y Fu(;)17 b(\030)385 4996 y Ft(E)444 4981 y Fz(\))35 b FA(solves)f(\(9.5\))g(and)g(ob)-5 b(eys)34 b(\(9.6\).)150 5168 y Fv(Pro)s(of)p Fz(.)43 b(Simple)31 b(computation.)p 3754 5168 4 66 v 3758 5106 59 4 v 3758 5168 V 3816 5168 4 66 v 296 5407 a(The)k(next)g(imp)s (ortan)m(t)d(thing)h(is)h(kno)m(wing)g(the)h(energy)g(range)f(for)f (whic)m(h)i(w)m(e)g(can)f(construct)150 5527 y(attracting)28 b(solutions)h(for)g(negativ)m(e)h(times.)42 b(Denote)30 b(b)m(y)g Fu(E)2375 5491 y Fm(0)2369 5553 y Ft(d)2410 5527 y Fz(\()p Fu(b)p Fz(\))g(the)g(in\014m)m(um)e(of)i(all)d(energies) j(for)1939 5776 y(89)p eop %%Page: 90 93 90 92 bop 150 407 a Fz(whic)m(h)39 b(\(9.1\))f(has)i(a)e(solution)f(ob) s(eying)h(\(9.2\).)62 b(It)39 b(is)f(ob)m(vious)h(\(using)f(the)h(ab)s (o)m(v)m(e)h(prop)s(osition\))150 527 y(that)34 b Fu(E)441 491 y Fm(0)435 553 y Ft(d)476 527 y Fz(\()p Fu(b)p Fz(\))c(=)h Fu(E)802 542 y Ft(d)842 527 y Fz(\()p Fu(b)921 542 y Fs(1)961 527 y Fz(\).)49 b(But)34 b(is)g(it)f(true)h(that)g Fu(E)1967 491 y Fm(0)1961 553 y Ft(d)2002 527 y Fz(\()p Fu(b)p Fz(\))d(=)f Fu(E)2328 542 y Ft(d)2369 527 y Fz(\()p Fu(b)p Fz(\)?)48 b(In)35 b(other)f(w)m(ords,)i(can)e(w)m(e)i(pro)m(v)m (e)150 648 y(that)h(\\)p Fu(E)487 663 y Ft(d)527 648 y Fz(")g(corresp)s(onding)g(to)f Fu(b)p Fz(\()p Fu(\022)s Fz(\))i(equals)f(the)g(\\)p Fu(E)2163 663 y Ft(d)2204 648 y Fz(")f(asso)s(ciated)h(to)g Fu(b)p Fz(\()p Fr(\000)p Fu(\022)s Fz(\)?)57 b(The)38 b(answ)m(er)g(is)150 768 y(a\016rmativ)m(e:)150 967 y Fv(Prop)s(osition)e(9.2.)41 b FA(The)35 b(critic)-5 b(al)34 b(ener)-5 b(gies)34 b(ar)-5 b(e)35 b(e)-5 b(qual:)44 b Fu(E)2397 982 y Ft(d)2438 967 y Fz(\()p Fu(b)2517 982 y Fs(1)2557 967 y Fz(\))27 b(=)h Fu(E)2804 930 y Fm(0)2798 992 y Ft(d)2838 967 y Fz(\()p Fu(b)p Fz(\))g(=)g Fu(E)3159 982 y Ft(d)3200 967 y Fz(\()p Fu(b)p Fz(\))p FA(.)150 1165 y Fv(Pro)s(of)p Fz(.)66 b(Assume)40 b Fu(E)955 1180 y Ft(d)996 1165 y Fz(\()p Fu(b)p Fz(\))h Fu(<)f(E)1342 1180 y Ft(d)1383 1165 y Fz(\()p Fu(b)1462 1180 y Fs(1)1502 1165 y Fz(\).)65 b(Then)42 b(consider)e(the)g(critical)e(solution)g(\()p Fu(\032)3254 1180 y Fs(1)3294 1165 y Fu(;)17 b(\030)3381 1180 y Fs(1)3420 1165 y Fz(\))40 b(to)f(\(9.5\))150 1295 y(corresp)s(onding)c(to)f Fu(E)972 1310 y Ft(d)1013 1295 y Fz(\()p Fu(b)1092 1310 y Fs(1)1131 1295 y Fz(\))h(and)g(satisfying) 1833 1215 y Fl(R)1899 1241 y Fs(2)p Ft(\031)1880 1330 y Fs(0)2008 1251 y Ft(\032)2044 1260 y Fj(1)p 2008 1272 71 4 v 2011 1329 a Ft(\030)2042 1338 y Fj(1)2089 1295 y Fz(\()p Fu(\022)s Fz(\))p Fu(d\022)f Fz(=)d(0.)50 b(Then)35 b(de\014ne)h Fu(\032)p Fz(\()p Fu(\022)s Fz(\))c(:=)f Fr(\000)p Fu(\032)3583 1310 y Fs(1)3623 1295 y Fz(\()p Fr(\000)p Fu(\022)s Fz(\))150 1425 y(and)f Fu(\030)5 b Fz(\()p Fu(\022)s Fz(\))27 b(:=)h Fu(\030)710 1440 y Fs(1)749 1425 y Fz(\()p Fr(\000)p Fu(\022)s Fz(\),)j(and)f(notice)g (that)f(they)i(solv)m(e)g(\(9.1\))e(but)h(with)g(a)g(zero)g(in)m (tegral)e(condition.)150 1546 y(If)39 b(w)m(e)h(lo)s(ok)e(at)h(its)f (asso)s(ciated)h(\\real)f(orbit")g(\()p Fu(r)s Fz(\()p Fu(t)p Fz(\))p Fu(;)17 b(\022)s Fz(\()p Fu(t)p Fz(\)\))39 b(w)m(e)h(see)g(that)f(it)f(exists)i(for)f(all)e Fu(t)i Fr(\025)g Fz(0)150 1683 y(since)33 b Fu(r)s Fz(\()p Fu(t)p Fz(\))27 b(=)h(exp)q Fr(f)877 1603 y Fl(R)943 1629 y Ft(\022)r Fs(\()p Ft(t)p Fs(\))924 1717 y(0)1063 1683 y Fz(\()p Fu(\032=\030)5 b Fz(\)\()p Fu(')p Fz(\))p Fu(d')p Fr(g)31 b Fz(is)h(b)s(ounded)h(from)e(b)s(elo)m(w)i(and)g(ab)s(o)m(v)m (e.)44 b(Then)34 b(consider)1321 1897 y Fu(A)p Fz(\()p Fu(t)p Fz(\))28 b(=)g([)p Fu(\032)p Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\))22 b Fr(\000)h Fu(\032)2121 1912 y Ft(d)2162 1897 y Fz(\()p Fu(\022)s Fz(\()p Fu(t)p Fz(\)\)])f Fr(\001)g Fu(r)s Fz(\()p Fu(t)p Fz(\))150 2112 y(and)34 b(see)h(that)f Fu(A)786 2075 y Fm(0)810 2112 y Fz(\()p Fu(t)p Fz(\))c Fr(\025)h Fu(E)1131 2127 y Ft(d)1172 2112 y Fz(\()p Fu(b)1251 2127 y Fs(1)1291 2112 y Fz(\))23 b Fr(\000)g Fu(E)1524 2127 y Ft(d)1565 2112 y Fz(\()p Fu(b)p Fz(\))31 b Fu(>)f Fz(0)k(for)f Fu(t)e Fr(\025)g Fz(0,)j(whic)m(h)h(con)m(tradicts)f(its)g(b)s(oundedness.)150 2232 y(Then)g Fu(E)477 2247 y Ft(d)517 2232 y Fz(\()p Fu(b)596 2247 y Fs(1)636 2232 y Fz(\))28 b Fu(<)f(E)877 2247 y Ft(d)918 2232 y Fz(\()p Fu(b)p Fz(\))33 b(is)f(con)m(tradicted)h (b)m(y)g(a)g(similar)c(argumen)m(t)j(and)h(w)m(e)h(are)e(done.)p 3754 2232 4 66 v 3758 2169 59 4 v 3758 2232 V 3816 2232 4 66 v 150 2470 a Fv(Remark)p Fz(.)80 b(F)-8 b(or)44 b(ev)m(ery)j Fu(E)55 b(>)49 b(E)1426 2485 y Ft(d)1511 2470 y Fz(there)d(are)f(exactly)g(t)m(w)m(o)h(branc)m(hes)g(of)f Fu(C)3110 2434 y Fs(1)3194 2470 y Fz(p)s(erio)s(dic)f(orbits)150 2591 y(whic)m(h)30 b(solv)m(e)f(\(9.1\),)g(b)s(oth)g(ha)m(ving)g Fu(\030)j(>)c Fz(0)g(but)i(their)e(in)m(tegral)g(condition)f(is)i(with) f(opp)s(osite)h(signs.)150 2711 y(While)39 b(the)h(\\propagating")e Fu(\032)1314 2726 y Ft(E)1414 2711 y Fz(is)h(increasing)g(and)h(conca)m (v)m(e)i(in)d(energy)-8 b(,)42 b(the)f(\\collapsing")c Fu(\032)3788 2726 y Ft(e)150 2832 y Fz(is)42 b(decreasing)i(and)f(con)m (v)m(ex)i(in)d(energy)-8 b(.)76 b(Then)44 b(notice)e(that)h Fu(r)32 b Fr(\001)d Fu(\032)2735 2847 y Ft(e)2772 2832 y Fz(\()p Fu(E)6 b(;)17 b(\022)s Fz(\))43 b(solv)m(es)h(the)f(eik)m (onal)150 2952 y(equation)36 b(and)h(pro)m(vides)g(a)f(Hamilton-Jacobi) d(function)j(for)g(negativ)m(e)h(times)e Fu(t)g(<)f Fz(0.)55 b(W)-8 b(e)37 b(then)150 3072 y(can)29 b(de\014ne)h(a)e(direct)g(and)h (in)m(v)m(erse)h(\015o)m(w)f(and)g(\014nally)e(construct)j(the)f(appro) m(ximate)e(dynamics)h(for)150 3193 y(negativ)m(e)33 b(times)e(using)i (the)g(same)f(ideas)h(as)f(in)g(the)h(case)h(of)e Fu(U)2493 3208 y Fs(0)2532 3193 y Fz(.)44 b(W)-8 b(e)33 b(giv)m(e)f(no)h(further) g(details.)150 3525 y Fw(10)161 b(Op)t(en)53 b(problems)150 3744 y Fz(W)-8 b(e)23 b(men)m(tion)e(t)m(w)m(o)i(related)f(problems)g (concerning)h(dynamics)f(and)g(sp)s(ectral)g(theory)h(for)f(magnetic) 150 3864 y(\014elds)33 b(considered)g(in)f(this)g(pap)s(er:)296 3984 y(\()p Fu(a)p Fz(\).)49 b(Dynamics)33 b(b)s(elo)m(w)h Fu(E)1302 3999 y Ft(d)1343 3984 y Fz(:)46 b(w)m(e)36 b(write)e Fu(\032)c Fz(=)h Fu(!)20 b Fz(cos\()p Fu(')p Fz(\),)35 b Fu(\030)g Fz(=)30 b Fu(!)20 b Fz(sin\()p Fu(')p Fz(\))33 b(with)h Fu(!)g Fz(=)3396 3900 y Fr(p)p 3479 3900 127 4 v 84 x Fz(2)p Fu(E)6 b Fz(,)35 b(and)150 4105 y(in)m(tro)s(duce)i(a)f(new)i(time)d Fu(\034)48 b Fz(with)36 b Fu(d\034)11 b(=dt)35 b Fz(=)f(1)p Fu(=r)s Fz(\()p Fu(t)p Fz(\).)55 b(The)38 b(v)-5 b(ariables)35 b(\()p Fu(';)17 b(\022)s Fz(\))37 b(mo)m(v)m(e)g(on)f(a)h(torus)g Fn(T)150 4225 y Fz(according)32 b(to)g(the)h(di\013eren)m(tial)e (equation)1185 4416 y Fu(d)p 1158 4461 105 4 v 1158 4552 a(d\034)1272 4483 y Fz(\()p Fu(';)17 b(\022)s Fz(\))27 b(=)h(\()p Fr(\000)p Fu(b)p Fz(\()p Fu(\022)s Fz(\))23 b Fr(\000)g Fu(!)c Fz(sin\()p Fu(')p Fz(\))p Fu(;)e(!)j Fz(sin)o(\()p Fu(')p Fz(\)\))p Fu(:)150 4729 y Fz(F)-8 b(or)32 b(0)27 b Fu(<)h(E)33 b(<)28 b(E)786 4744 y Ft(d)859 4729 y Fz(it)k(turns)h(out)f(that)h Fu(')p Fz(\()p Fu(\034)11 b Fz(\))28 b Fr(!)f(1)32 b Fz(and)h Fu(\022)s Fz(\()p Fu(\034)11 b Fz(\))28 b Fr(!)f(1)33 b Fz(as)f Fu(t)c Fr(!)f(1)p Fz(.)44 b(W)-8 b(e)33 b(can)f(write)1427 4952 y Fu(r)s Fz(\()p Fu(\034)11 b Fz(\))28 b(=)g Fu(r)s Fz(\(0\))p Fu(e)1952 4857 y Fg(R)1998 4878 y Fq(\034)1985 4935 y Fj(0)2049 4911 y Ft(!)14 b Fs(cos\()p Ft(')p Fs(\()p Ft(\034)2340 4887 y Ff(0)2364 4911 y Fs(\)\))p Ft(d\034)2493 4887 y Ff(0)2520 4952 y Fu(:)150 5166 y Fz(Are)36 b(there)g(an)m(y)g (orbits)e(with)h(energy)i Fu(E)h(<)32 b(E)1888 5181 y Ft(d)1964 5166 y Fz(for)j(whic)m(h)h Fu(r)s Fz(\()p Fu(\034)11 b Fz(\))32 b Fr(!)g(1)j Fz(as)g Fu(\034)44 b Fr(!)32 b(1)p Fz(?)51 b(There)37 b(are)150 5286 y(none)c(if)e Fu(b)p Fz(\()p Fu(\022)s Fz(\))i(is)g(constan)m(t)g(and)g(nonzero.)296 5407 y(\()p Fu(b)p Fz(\).)43 b(W)-8 b(e)28 b(kno)m(w)h(that)f(the)h(sp) s(ectrum)f(of)g Fu(H)35 b Fz(is)28 b([0)p Fu(;)17 b Fr(1)p Fz(\).)41 b(What)28 b(is)g(the)g(nature)g(of)g(the)h(sp)s(ectrum)150 5527 y(in)j([0)p Fu(;)17 b(E)456 5542 y Ft(d)496 5527 y Fz(]?)44 b(F)-8 b(or)31 b(the)i(constan)m(t)h Fu(b)f Fz(case,)g(it)f(is)g(pure)h(p)s(oin)m(t.)1939 5776 y(90)p eop %%Page: 91 94 91 93 bop 150 407 a Fw(11)215 b(Ac)l(kno)l(wledgmen)l(ts)150 626 y Fz(This)38 b(w)m(ork)g(w)m(as)h(completed)e(during)g(a)g(sta)m(y) h(at)g(Institut)f(Mittag-Le\017er.)58 b(The)38 b(authors)g(w)m(an)m(t) 150 746 y(to)32 b(express)j(their)d(gratitude)g(for)g(in)m(vitations)f (and)h(\014nancial)g(supp)s(ort.)150 1079 y Fw(References)150 1298 y Fz([C-F-K-S])48 b(Cycon,)c(H.L.,)f(F)-8 b(ro)s(ese,)43 b(R.G.,)g(Kirsc)m(h,)g(W.,)g(Simon,)f(B.:)60 b(Sc)m(hr\177)-49 b(odinger)41 b(op)s(erators)594 1418 y(with)29 b(application)e(to)h (quan)m(tum)i(mec)m(hanics)f(and)g(global)e(geometry)-8 b(.)29 b(Berlin,)f(Heidel-)594 1539 y(b)s(erg,)33 b(New)g(Y)-8 b(ork:)44 b(Springer-V)-8 b(erlag)31 b(1987)150 1742 y([C-H-S])127 b(Cornean,)38 b(H.,)f(Herbst,)h(I.,)f(Skibsted,)h(E.:)51 b(Scattering)35 b(theory)i(for)e(a)h(class)g(of)g(long-)594 1863 y(range)d(magnetic)e(\014elds.)i(In)g(preparation.)150 2066 y([D-G1])156 b(Derezi)s(\023)-51 b(nski,)30 b(J.,)h(G)m(\023)-46 b(erard,)31 b(C.:)43 b(Long-range)29 b(scattering)i(in)e(the)i(p)s (osition)e(represen)m(ta-)594 2186 y(tion.)j(J.)h(Math.)g(Ph)m(ys.)h Fv(38)f Fz(no.)f(8,)h(3925-3942)d(\(1997\))150 2390 y([D-G2])156 b(Derezi)s(\023)-51 b(nski,)43 b(J.,)g(G)m(\023)-46 b(erard,)43 b(C.:)62 b(Scattering)40 b(theory)i(of)f(classical)f(and)h(quan)m(tum)g (N-)594 2510 y(particle)32 b(systems.)i(Berlin:)42 b(Springer-V)-8 b(erlag)31 b(1997)150 2714 y([E])324 b(Enss,)53 b(V.:)73 b(Quan)m(tum)47 b(scattering)g(with)f(long-range)g(magnetic)g (\014elds.)i(Op)s(erator)594 2834 y(calculus)39 b(and)g(sp)s(ectral)g (theory)g(\(Lam)m(brec)m(h)m(t,)j(1991\))37 b(61{70,)j(Op)s(er.)f (Theory)h(Adv.)594 2954 y(Appl.)33 b Fv(57)f Fz(Birkh\177)-49 b(auser,)33 b(Basel)g(1992)150 3158 y([He])274 b(Herbst,)49 b(I.:)67 b(Sp)s(ectral)44 b(and)h(scattering)f(theory)h(for)f(Sc)m (hr\177)-49 b(odinger)44 b(op)s(erators)g(with)594 3278 y(p)s(oten)m(tials)32 b(indep)s(enden)m(t)i(of)e Fr(j)p Fv(x)p Fr(j)p Fz(.)g(Amer.)g(J.)h(Math.)g Fv(113)f Fz(no.)h(3,)f (509-565)f(\(1991\))150 3482 y([H1])268 b(H\177)-49 b(ormander,)32 b(L.:)43 b(The)32 b(existence)h(of)e(w)m(a)m(v)m(e)i(op)s(erators)f(in) e(scattering)i(theory)-8 b(.)32 b(Math.)594 3602 y(Z.)h Fv(146)f Fz(no.)h(1,)f(69-91)f(\(1976\))150 3805 y([H2])268 b(H\177)-49 b(ormander,)56 b(L.:)81 b(The)53 b(analysis)e(of)g(linear)e (partial)h(di\013eren)m(tial)f(op)s(erators)i Fu(I)8 b(V)22 b Fz(.)594 3926 y(Berlin:)43 b(Springer-V)-8 b(erlag)30 b(1985)150 4129 y([H-M-V])89 b(Hassell,)38 b(A.,)h(Melrose,)g(R.,)f(V) -8 b(asy)g(,)39 b(A.:)53 b(Sp)s(ectral)36 b(and)i(scattering)f(theory)h (for)e(sym-)594 4249 y(b)s(olic)55 b(p)s(oten)m(tials)h(of)g(order)g (zero.)h(T)-8 b(o)57 b(app)s(ear)g(in)e(Adv)-5 b(ances)58 b(in)e(Mathematics.)594 4370 y(h)m (ttp://au.arXiv.org/abs/math.AP/0112163)150 4573 y([H-S])230 b(Herbst,)41 b(I.,)f(Skibsted,)h(E.:)56 b(Quan)m(tum)38 b(scattering)g(for)g(p)s(oten)m(tials)f(indep)s(enden)m(t)i(of)594 4694 y Fr(j)p Fv(x)p Fr(j)p Fz(:)44 b(asymptotic)33 b(completeness)g (for)g(high)f(and)h(lo)m(w)g(energies.)g(MaPh)m(ySto)i(preprin)m(t)594 4814 y(no.)e(7,)f(Marc)m(h)i(2002)150 5017 y([I-K])246 b(Isozaki,)53 b(H.,)g(Kitada,)e(H.:)75 b(Mo)s(di\014ed)48 b(w)m(a)m(v)m(e)i(op)s(erators)f(with)f(time-indep)s(enden)m(t)594 5138 y(mo)s(di\014ers.)32 b(J.)h(F)-8 b(ac.)32 b(Sci.)g(Univ.)h(T)-8 b(oky)m(o)33 b(Sect.)h(IA)f(Math.)g Fv(32)f Fz(no.)h(1,)f(77-104)f (\(1985\))150 5341 y([L-T1])177 b(Loss,)37 b(M.,)f(Thaller,)f(B.:)49 b(Scattering)35 b(of)g(particles)f(b)m(y)i(long-range)e(magnetic)g (\014elds.)594 5462 y(Ann.)g(Ph)m(ysics)g Fv(176)e Fz(no.)h(1,)f (159-180)f(\(1987\))1939 5776 y(91)p eop %%Page: 92 95 92 94 bop 150 407 a Fz([L-T2])177 b(Loss,)38 b(M.,)g(Thaller,)e(B.:)51 b(Short-range)36 b(scattering)g(in)f(long-range)g(magnetic)g(\014elds:) 594 527 y(the)e(relativistic)e(case.)i(J.)g(Di\013eren)m(tial)d (Equations)j Fv(73)f Fz(no.)h(2,)f(225-236)f(\(1988\))150 731 y([M])301 b(Mourre,)31 b(E.:)43 b(Absence)31 b(of)e(singular)f(con) m(tin)m(uous)j(sp)s(ectrum)f(for)f(certain)g(self-adjoin)m(t)594 851 y(op)s(erators.)k(Comm)m(un.)f(Math.)h(Ph)m(ys.)h Fv(78)f Fz(no.3,)f(391-408)f(\(1981\))150 1054 y([M\034])252 b(M\034ller,)43 b(J.S.:)63 b(An)43 b(abstract)f(radiation)e(condition)g (and)i(applications)e(to)i Fu(N)10 b Fz(-b)s(o)s(dy)594 1175 y(systems.)35 b(Rev.)e(Math.)g(Ph)m(ys.)h Fv(12)f Fz(no.5,)f(767-803)f(\(2000\))150 1378 y([N-R])212 b(Nicoleau,)43 b(F.,)h(Rob)s(ert,)f(D.:)61 b(Th)m(\022)-46 b(eorie)42 b(de)g(la)e(di\013usion)h(quan)m(tique)h(p)s(our)g(des)g(p)s(er-)594 1499 y(turbations)37 b(\022)-49 b(a)37 b(longue)g(et)h(courte)g(p)s (ort)m(\022)-46 b(ee)37 b(du)h(c)m(hamp)f(magn)m(\023)-46 b(etique.)37 b(Ann.)h(F)-8 b(ac.)37 b(Sci.)594 1619 y(T)-8 b(oulouse)33 b(Math.)g(\(5\))f Fv(12)h Fz(no.)f(2,)h(185-194)d (\(1991\))150 1822 y([R])318 b(Rob)s(ert,)45 b(D.:)62 b(On)43 b(scattering)f(theory)h(for)e(long)g(range)i(p)s(erturbations)f (of)f(Laplace)594 1943 y(op)s(erators.)33 b(J.)g(Anal.)e(Math.)i Fv(59)p Fz(,)g(189-203)e(\(1992\))150 2146 y([R-S])231 b(Reed,)41 b(M.,)f(Simon,)e(B.:)55 b(Metho)s(ds)39 b(of)e(mo)s(dern)h (mathematical)d(ph)m(ysics)40 b Fv(I)s(I)s(I)p Fz(.)d(Scat-)594 2267 y(tering)32 b(theory)-8 b(.)33 b(New)h(Y)-8 b(ork,)33 b(London:)43 b(Academic)32 b(Press)i(1979)150 2470 y([R-Y])212 b(Roux,)44 b(P)-8 b(.,)44 b(Y)-8 b(afaev,)43 b(D.:)61 b(The)42 b(scattering)f(matrix)f(for)g(the)i(Sc)m(hr\177)-49 b(odinger)42 b(op)s(erator)594 2590 y(with)32 b(a)g(long-range)f (electromagnetic)g(p)s(oten)m(tial.)g(mp)p 2654 2590 30 4 v 34 w(arc)i(preprin)m(t)f(arc)m(hiv)m(e)h(\(2002\))594 2711 y(h)m(ttp://www.ma.utexas.edu/mp)p 1937 2711 V 38 w(arc-bin/mpa?yn=02-364)150 2914 y([Y])317 b(Y)-8 b(afaev,)40 b(D.:)53 b(W)-8 b(a)m(v)m(e)39 b(op)s(erators)f(for)f(the)i(Sc)m (hr\177)-49 b(odinger)38 b(op)s(erators.)g(T)-8 b(eor.)38 b(Mat.)g(Fiz.)594 3034 y Fv(45)p Fz(,)33 b(224-234)e(\(1980\))1939 5776 y(92)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0211110729461--