Content-Type: multipart/mixed; boundary="-------------0105111133108" This is a multi-part message in MIME format. ---------------0105111133108 Content-Type: text/plain; name="01-176.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-176.keywords" QED, Metastable States, Resonances, Fermi's Golden Rule, Dilatation Analyticity ---------------0105111133108 Content-Type: application/postscript; name="mueck_mar01.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="mueck_mar01.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: mueck_mar01.dvi %%CreationDate: Fri May 11 10:51:26 2001 %%Pages: 29 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: C:\TEXMF\MIKTEX\BIN\DVIPS.EXE mueck_mar01.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.05.11:1049 %%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: special.pro %! 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5250 2550 75 75 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 345.00 165.00] PATmp PATsp ef gr PATusp gs col0 s gr % Ellipse n 6150 2550 75 75 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 405.00 165.00] PATmp PATsp ef gr PATusp gs col0 s gr % Polyline 15.000 slw n 2475 2550 m 8700 2550 l gs col0 s gr % Polyline 7.500 slw n 7050 2475 m 8700 2475 l 8700 2550 l 7050 2550 l cp gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 470.00 165.00] PATmp PATsp ef gr PATusp gs col0 s gr % Polyline n 6150 2550 m 6150 2850 l 8700 2850 l gs col0 s gr % Polyline n 7050 2550 m 7050 2700 l 8700 2700 l gs col0 s gr % Polyline n 3000 2550 m 3000 3300 l 8700 3300 l gs col0 s gr % Polyline n 5250 2550 m 5250 3000 l 8700 3000 l gs col0 s gr % Polyline n 3750 2550 m 3750 3150 l 8700 3150 l gs col0 s gr $F2psEnd rs end %%EndDocument @endspecial 1499 492 a Fu(E)1571 507 y Fs(j)2249 492 y 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FF(in)d(a)f(complex)g(neigh)n(b)r(ourho)r(o)r(d)g(of) g FB(\022)61 b FF(=)c(0,)d(for)227 1458 y(ev)n(ery)46 b(v)n(ector)g FB( )k FF(in)d(the)g(domain)g(of)g FB(w)1605 1470 y Fy(m;n)1729 1458 y FF(\()p FB(:)14 b(:)g(:)g FF(\).)96 b(This)47 b(is)f(ensured)h(b)n(y)g(a)f(suitable)h(UV-)227 1558 y(regularization)d(of)i(the)h(v)n(ector)e(p)r(oten)n(tial)h(as)f (discussed)h(in)g([BFS3].)93 b FB(\022)48 b FF(is)e(the)h(dilatation) 227 1657 y(parameter)26 b(\(see)i(Sect.)g(1.3\).)352 1758 y(Under)33 b(these)h(assumptions,)g(it)g(is)f(p)r(ossible)g(to)g (deduce)h(suitable)f(relativ)n(e)f(b)r(ounds)i(on)f(the)227 1857 y(p)r(erturbation:)227 2081 y FG(1.2)e(Lemma)43 b Fm(F)-7 b(or)33 b FB(m;)14 b(n)33 b FA(2)h Fw(N)1241 2093 y Fz(0)1278 2081 y Fm(,)h FF(1)e FA(\024)f FB(m)23 b FF(+)f FB(n)33 b FA(\024)g FF(2)p Fm(,)h FB(\022)i FA(2)d FB(U)2191 2096 y Fy(\031)r(=)p Fz(4)2336 2081 y FF(:=)2457 1989 y Fx(n)2519 2059 y FF(~)2512 2081 y FB(\022)26 b FA(2)d Fw(C)2748 1986 y Fx(\014)2748 2036 y(\014)2748 2085 y(\014)2789 2081 y FA(j)2819 2059 y FF(~)2812 2081 y FB(\022)r FA(j)h FB(<)e(\031)s(=)p FF(4)3134 1989 y Fx(o)3190 2081 y Fm(,)35 b(the)227 2206 y(op)r(erators)e FB(W)680 2218 y Fy(m;n)805 2206 y FF(\()p FB(\022)r FF(\))j Fm(are)e(de\014ned)h(on)g FA(D)r FF(\()p FB(H)1675 2218 y Fz(0)1713 2206 y FF(\))h(=)f FA(D)r FF(\()p FB(H)2048 2218 y Fz(el)2101 2206 y FF(\))24 b FA(\012)f(D)r FF(\()p FB(H)2412 2218 y Fz(f)2442 2206 y FF(\))35 b Fm(and)g(satisfy)g(the)h (relativ)n(e)227 2305 y(b)r(ound:)835 2337 y Fx(\015)835 2386 y(\015)881 2407 y FB(W)959 2419 y Fy(m;n)1084 2407 y FF(\()p FB(\022)r FF(\))p FA(j)p FB(H)1281 2419 y Fz(0)1337 2407 y FF(+)18 b FB(i)p FA(j)1472 2373 y Fq(\000)p Fz(1)1561 2337 y Fx(\015)1561 2386 y(\015)1607 2440 y Fq(H)1695 2407 y FA(\024)1783 2340 y Fx(\002)1818 2407 y FF(4\(1)f(+)h(\003)2092 2373 y Fz(2)2092 2428 y(0)2148 2407 y FF(+)g(\003)2289 2373 y Fz(2)2289 2428 y Fq(\000)p Fz(1)2378 2407 y FF(\))2410 2340 y Fx(\003)2444 2356 y Fz(\()p Fy(m)p Fz(+)p Fy(n)p Fz(\))p Fy(=)p Fz(2)2732 2407 y FB(:)461 b FF(\(17\))352 2572 y FG(Pro)s(of:)79 b FF(W)-7 b(e)28 b(refer)e(the)i(reader)f(to)g ([BFS3],)g(Lemma)h(I.1.)p 3309 2572 55 55 v 352 2744 a(An)37 b(immediate)h(consequence)e(is)g(the)i(follo)n(wing)e(theorem)g (whic)n(h)h(ensures)f(that)h(the)h(p)r(er-)227 2844 y(turb)r(ed)26 b(Hamiltonian)g(is)f(semib)r(ounded)h(and)f(has)h(the)f(same)g(domain)h (as)f(the)h(unp)r(erturb)r(ed)g(one.)227 3047 y FG(1.3)31 b(Theorem)44 b Fm(The)32 b(p)r(erturbation)g FB(W)1545 3059 y Fy(g)1616 3047 y Fm(is)g(de\014ned)g(on)g FA(D)r FF(\()p FB(H)2281 3059 y Fz(0)2319 3047 y FF(\))e(=)g FA(D)r FF(\()p FB(H)2643 3059 y Fz(el)2696 3047 y FF(\))22 b FA(\012)f(D)r FF(\()p FB(H)3003 3059 y Fz(f)3033 3047 y FF(\))p Fm(.)50 b(If)33 b FB(g)g(<)254 3114 y Fz(1)p 237 3128 67 4 v 237 3175 a(10)313 3147 y FF(\(1)25 b(+)e(\003)558 3117 y Fz(2)558 3167 y(0)620 3147 y FF(+)h(\003)767 3117 y Fz(2)767 3167 y Fq(\000)p Fz(1)855 3147 y FF(\))887 3117 y Fq(\000)p Fz(1)p Fy(=)p Fz(2)1044 3147 y Fm(,)39 b(then)e FB(H)1373 3159 y Fy(g)1449 3147 y FF(=)h FB(H)1621 3159 y Fz(0)1682 3147 y FF(+)24 b FB(W)1849 3159 y Fy(g)1925 3147 y Fm(is)36 b(self-adjoin)n(t)g(on)g FA(D)r FF(\()p FB(H)2744 3159 y Fy(g)2783 3147 y FF(\))i(=)g FA(D)r FF(\()p FB(H)3123 3159 y Fz(0)3161 3147 y FF(\))f Fm(and)227 3246 y(b)r(ounded)28 b(from)g(b)r(elo)n(w.)352 3347 y FG(Pro)s(of:)79 b FF(\(see)27 b([BFS3],)h(Corollary)d(I.2\).)p 3309 3347 55 55 v 227 3655 a FC(1.2)113 b(Surv)m(ey)38 b(of)f(the)h(Main)f(Result)227 3809 y FF(In)g([BFS3])f(it)h(w)n(as)f (pro)n(v)n(en)f(that,)k(for)d(su\016cien)n(tly)h(small)f FB(g)k(>)e FF(0,)g(the)f(excited)g(energy)e(lev)n(els)227 3909 y(disapp)r(ear)25 b(\(see)g(also)f(Thm.)i(1.10\))e(when)h (electrons)g(and)g(photons)f(are)h(minimally)g(coupled.)36 b(W)-7 b(e)227 4009 y(study)28 b(the)g(fate)g(of)f(the)h(corresp)r (onding)e(eigenstates,)h(stated)g(in)h(Sect.)g(2.)352 4109 y(In)36 b(Sect.)g(2.1)e(w)n(e)h(construct)h(metastable)f(states)g (close)g(to)g(the)h(unp)r(erturb)r(ed)g(eigenstates,)227 4209 y(i.e.,)h(w)n(e)d(explicitely)h(de\014ne)f(a)g(v)n(ector)g(\010)g (in)h(the)g(Hilb)r(ert)g(space)f FA(H)39 b FF(whic)n(h)34 b(appro)n(ximates)f(the)227 4308 y(unp)r(erturb)r(ed,)28 b(excited)f(eigenstate)f FB(')1454 4320 y Fy(j)1506 4308 y FA(\012)16 b FF(\012)27 b(with)g(energy)f FB(E)2188 4320 y Fy(j)2250 4308 y FF(of)h(the)g(decoupled)f(system)h(as)f(the)227 4408 y(coupling)h(constan)n(t)g FB(g)j FF(decreases.)352 4508 y(In)22 b(Sect.)h(2.2,)g(w)n(e)f(iden)n(tify)g(the)h(time-ev)n (olution)f(of)g(this)h(state)f(\010)g(to)g(b)r(e)h(exp)r(onen)n(tially) f(deca)n(y-)227 4608 y(ing,)30 b(up)g(to)f(small)g(errors.)40 b(W)-7 b(e)30 b(compare)e(this)i(result)f(with)h(predictions)f(from)g (time-dep)r(enden)n(t)227 4707 y(p)r(erturbation)c(theory)f(to)h (\014rst)g(order.)35 b(F)-7 b(ollo)n(wing)24 b(standard)g(exp)r (ositions)h(in)g(ph)n(ysics)g(textb)r(o)r(oks)227 4807 y(\(see,)38 b(e.g.,)g([Me],)h(Chap.)62 b(20\),)38 b(the)e(probabilit)n (y)f(p)r(er)h(unit)h(time,)i FB(p)2477 4819 y Fz(\010)p Fq(!)p Fy(\036)2630 4827 y Fo(0)2667 4807 y FF(,)f(of)e(transition)g (of)g(a)227 4907 y(p)r(erturb)r(ed,)31 b(excited)f(state)g(\010)g(in)n (to)g(the)h(ground)e(state)h FB(\036)2079 4919 y Fz(0)2146 4907 y FF(is)g(giv)n(en)g(b)n(y)f(the)i(matrix)e(elemen)n(t)h(of)227 5006 y(the)e(p)r(erturbation:)1084 5145 y FB(p)1126 5157 y Fz(\010)p Fq(!)p Fy(\036)1279 5165 y Fo(0)1339 5145 y FF(=)23 b(2)p FB(\031)1532 5050 y Fx(\014)1532 5100 y(\014)1532 5149 y(\014)1560 5145 y FA(h)14 b FB(\036)1655 5157 y Fz(0)1717 5145 y FA(j)g FB(W)1832 5157 y Fy(g)1884 5145 y FF(\010)g FA(i)1991 5173 y Fq(H)2056 5050 y Fx(\014)2056 5100 y(\014)2056 5149 y(\014)2083 5070 y Fz(2)2144 5145 y FF(=)22 b FA(O)2314 5078 y Fx(\000)2352 5145 y FB(g)2395 5111 y Fz(2)2431 5078 y Fx(\001)2483 5145 y FB(:)710 b FF(\(18\))227 5317 y(This)36 b(constan)n(t)g(transition)f(rate)g (implies)h(an)g(exp)r(onen)n(tial)g(deca)n(y)f(of)h(the)g(state)g(\010) g(under)g(the)227 5417 y(time-ev)n(olution)27 b(with)h(a)f(life-time)i FB(p)1402 5381 y Fq(\000)p Fz(1)1402 5442 y(\010)p Fq(!)p Fy(\036)1555 5450 y Fo(0)1591 5417 y FF(,)f(a)f(result)h(kno)n(wn)f(as) f FE(F)-6 b(ermi's)31 b(Golden)g(R)n(ule)p FF(.)352 5517 y(The)38 b(main)h(result)f(of)g(this)h(pap)r(er)f(\(Thm.)h(2.9\))f(sho) n(ws)f(the)i(w)n(eakness)e(of)i(the)g(\014rst)f(order)227 5617 y(appro)n(ximation)33 b(of)h(time)h(dep)r(enden)n(t)h(p)r (erturbation)e(theory)-7 b(.)57 b(It)34 b(states)h(that)f(the)h (time-deca)n(y)227 5717 y(la)n(w)21 b(of)h(the)g(constructed)g(v)n (ector)e(\010)i(can)f(b)r(e)h(divided)g(in)n(to)g(a)f(part)h(of)f(exp)r (onen)n(tial)h(deca)n(y)-7 b(,)22 b(whic)n(h)f(is)p eop %%Page: 5 5 5 4 bop 0 236 a Fv(1.2)92 b(Surv)m(ey)30 b(of)g(the)h(Main)f(Result) 2287 b FL(5)227 493 y FF(predicted)25 b(b)n(y)g(F)-7 b(ermi's)25 b(Golden)g(Rule,)h(and)f(an)g FE(err)l(or)i(term)e FF(\(\\bac)n(kground"\))e(whic)n(h)i(dominates)227 593 y(the)j(deca)n(y)f(for)g(large)f(times)i FB(t)23 b FA(\035)g FF(1:)827 719 y Fx(\014)827 769 y(\014)827 819 y(\014)855 815 y FA(h)14 b FF(\010)24 b FA(j)14 b FF(exp\()p FA(\000)p FB(iH)1344 827 y Fy(g)1382 815 y FB(t)p FF(\)\010)g FA(i)1550 843 y Fq(H)1615 719 y Fx(\014)1615 769 y(\014)1615 819 y(\014)1666 815 y FA(\024)93 b FB(C)20 b FF(exp\()p FA(\000)p FB(S)2178 827 y Fy(g)2216 815 y FB(t)p FF(\))1824 873 y Fx(|)p 1861 873 153 10 v 153 w({z)p 2088 873 V 153 w(})1754 952 y Fz(F)-5 b(ermi's)20 b(Golden)i(Rule)2363 815 y FF(+)2453 759 y FB(C)2512 771 y Fy(n)p 2453 796 105 4 v 2467 872 a FB(t)2497 848 y Fy(n)2567 815 y FB(g)2610 780 y Fz(4+)p Fy(\014)2443 907 y Fx(|)p 2480 907 74 10 v 74 w({z)p 2628 907 V 74 w(})2442 981 y Fz(error)f(term)2740 815 y FB(;)453 b FF(\(19\))227 1153 y(where)36 b(0)h FB(<)f(\014)42 b(<)37 b FF(1)p FB(=)p FF(9,)f FB(S)1083 1165 y Fy(g)1159 1153 y FF(=)h FA(O)1343 1086 y Fx(\000)1381 1153 y FB(g)1424 1123 y Fz(2)1461 1086 y Fx(\001)1499 1153 y FF(,)h FB(n)f FF(=)g(0)p FB(;)14 b FF(1)35 b(and)h FB(C)q(;)14 b(C)2231 1165 y Fy(n)2314 1153 y FB(<)37 b FA(1)f FF(are)f(suitable)h(constan)n(ts.)227 1253 y(W)-7 b(e)33 b(p)r(oin)n(t)g(out)g(that)f(this)h(result)g(is)f(an)g(impro)n (v)n(emen)n(t)g(on)g([BFS3],)i(where)e(the)h(time)g(deca)n(y)e(of)227 1352 y(the)d(eigenstate)f FB(')812 1364 y Fy(j)865 1352 y FA(\012)17 b FF(\012)28 b(itself)g(w)n(as)e(iden)n(ti\014ed)i(to)f(b) r(e)h(of)f(this)g(kind,)h(except)f(for)g(the)h(error)d(term;)227 1452 y(sp)r(eci\014cally)-7 b(,)26 b(Bac)n(h,)f(F)-7 b(r\177)-42 b(ohlic)n(h,)26 b(and)f(Sigal)g(pro)n(v)n(ed)f(that)i(the)g (error)e(term)h(is)h FA(O)r FF(\()p FB(t)2793 1422 y Fq(\000)p Fy(n)2891 1452 y FB(g)2934 1422 y Fz(4)2970 1452 y FF(\))g(\(but)h(here)227 1552 y(v)-5 b(alid)21 b(for)f(all)h FB(n)i FA(2)g Fw(N)p FF(\).)35 b(Some)21 b(ideas)f(for)g(estimating)h(metastable)f(states)g(used)h(here)g(are)e (b)r(orro)n(w)n(ed)227 1651 y(from)32 b(this)g(pap)r(er.)49 b(Our)31 b(aim)h(is)g(to)f(impro)n(v)n(e)g(the)h(order)e(of)i(the)g (error)e(term)i(\(in)h([BFS3)o(]\))g(in)f FB(g)s FF(,)227 1751 y(4)9 b(+)g FB(\014)27 b(>)c FF(4.)34 b(Our)23 b(impro)n(v)n(ed)e (estimate)i(expands)f(the)i(time)f(slot)g(in)g(whic)n(h)g(the)g(exp)r (onen)n(tial)f(deca)n(y)227 1851 y(dominates)34 b(the)h(error)e(term)h (and,)i(th)n(us,)g(in)f(whic)n(h)g(F)-7 b(ermi's)34 b(Golden)g(Rule)h (can)f(b)r(e)h(observ)n(ed,)227 1950 y(i.e.,)28 b(where)1344 2090 y FB(C)20 b FF(exp\()p FA(\000)p FB(S)1698 2102 y Fy(g)1736 2090 y FB(t)p FF(\))k FA(\035)1938 2034 y FB(C)1997 2046 y Fy(n)p 1938 2071 105 4 v 1952 2147 a FB(t)1982 2123 y Fy(n)2052 2090 y FB(g)2095 2056 y Fz(4+)p Fy(\014)2224 2090 y FB(:)969 b FF(\(20\))227 2266 y(This)28 b(is)f(ful\014lled)i(for)1159 2368 y FB(t)23 b FA(\024)g FB(S)1356 2334 y Fq(\000)p Fz(1)1351 2389 y Fy(g)1458 2301 y Fx(\002)1493 2368 y FF(const)18 b(+)g(\(4)g(+)g FB(\014)t FF(\))c(log)q(\()p FB(g)2239 2334 y Fq(\000)p Fz(1)2328 2368 y FF(\))2360 2301 y Fx(\003)2409 2368 y FB(:)784 b FF(\(21\))227 2520 y(Comparing)26 b(to)i([BFS3],)f(w)n(e)h (note)f(that)h(the)g(size)f(of)h(the)g(time)g(slot)f(is)h(enlarged)e(b) n(y)1396 2731 y FB(\014)t(S)1503 2697 y Fq(\000)p Fz(1)1498 2751 y Fy(g)1606 2731 y FF(log\()p FB(g)1788 2697 y Fq(\000)p Fz(1)1877 2731 y FF(\))1932 2680 y Fy(g)r Fq(!)p Fz(0)1932 2731 y FA(\000)-14 b(!)23 b(1)p FB(:)1021 b FF(\(22\))352 2916 y(Increasing)32 b(the)h(order)f(of)i(the)f(error)f(term)h (requires)f(a)h(re\014nemen)n(t)g(of)g(the)h(c)n(hoice)e(of)i(\010)f (as)227 3016 y(compared)f(to)i([BFS3)o(].)55 b(The)33 b(construction)g(of)g(the)h(state)f(\010)g(is)g(complicated,)i(and)e(w) n(e)g(dev)n(ote)227 3115 y(an)26 b(en)n(tire)g(section)g(to)h(this)f (task.)36 b(In)27 b(our)f(pap)r(er,)g(w)n(e)g(b)r(orro)n(w)f(an)h(idea) g(of)g([Hu)q(])g(for)g(constructing)227 3215 y(metastable)i(states.)40 b(Hunzik)n(er)28 b(pro)n(v)n(ed)e(for)i(a)h(quan)n(tum)f(system)g(with) h(isolated)f(eigen)n(v)-5 b(alues)27 b(of)227 3315 y(the)21 b(unp)r(erturb)r(ed)f(system)g(that)g(the)h(error)d(is)i(of)g(order)f FA(O)r FF(\()p FB(t)2110 3285 y Fq(\000)p Fy(n)2207 3315 y FB(g)2250 3285 y Fy(M)2324 3315 y FF(\),)j(for)d(all)h FB(n)j FA(2)g Fw(N)p FF(,)f(assuming)d(the)227 3414 y(existence)g(of)f (a)h FE(R)l(ayleigh-Schr\177)-42 b(odinger)24 b(exp)l(ansion)19 b FF(to)g(order)e FB(M)27 b FF(of)19 b(the)g(\(nonexisten)n(t\))g(p)r (erturb)r(ed)227 3514 y(eigenstate.)60 b(Moreo)n(v)n(er,)35 b(Hunzik)n(er)g(sho)n(w)n(ed)f(ho)n(w)h(metastable)g(states)h(of)f(a)g (quan)n(tum)h(system)227 3614 y(and)d FE(r)l(esonanc)l(es)f FF(of)h(the)g(in)n(teracting)e(system)h(are)g(related:)46 b(The)32 b(resonance)f(energies)g(are)h(the)227 3713 y(complex)d(eigen)n(v)-5 b(alues)28 b(of)h(the)h FE(dilate)l(d)j (Hamiltonian)d FB(H)2041 3725 y Fy(g)2080 3713 y FF(\()p FB(\022)r FF(\))g(\(de\014ned)g(in)f(Sect.)h(1.3\),)f(and)g(their)227 3813 y(imaginary)f(parts)h(determine)h(the)g(recipro)r(cal)e(life-time) i FB(S)2130 3825 y Fy(g)2199 3813 y FF(of)f(the)h(metastable)g(states.) 42 b(In)30 b(our)227 3912 y(case,)c(the)h(Ra)n(yleigh-Sc)n(hr\177)-42 b(odinger)23 b(expansion)j(to)g(\014rst)h(order)e(of)h(the)h (eigenstate)f(corresp)r(onding)227 4012 y(to)i(the)g(resonance)870 3991 y(~)851 4012 y FB(E)912 4024 y Fy(j)975 4012 y FF(of)f(the)h (dilated)g(op)r(erator)e FB(H)1893 4024 y Fy(g)1931 4012 y FF(\()p FB(\022)r FF(\),)j(Im\()p FB(\022)r FF(\))24 b FB(>)f FF(0,)k(app)r(ears)f(as)376 4249 y FB( )433 4206 y Fz(\(0\))430 4274 y Fy(\022)545 4249 y FF(=)633 4132 y Fx(\024)677 4249 y FB(P)12 b FF(\()p FB(\022)r FF(\))19 b FA(\000)p 949 4183 65 4 v 18 w FB(P)12 b FF(\()p FB(\022)r FF(\))1133 4157 y Fx(\020)p 1183 4183 V 92 x FB(P)g FF(\()p FB(\022)r FF(\))p FB(H)1422 4261 y Fy(g)1461 4249 y FF(\()p FB(\022)r FF(\))p 1566 4183 V FB(P)h FF(\()p FB(\022)r FF(\))19 b FA(\000)1859 4228 y FF(~)1839 4249 y FB(E)1900 4261 y Fy(j)1936 4157 y Fx(\021)1985 4174 y Fq(\000)p Fz(1)p 2088 4183 V 2088 4249 a FB(P)12 b FF(\()p FB(\022)r FF(\))p FB(W)2336 4261 y Fy(g)2376 4249 y FF(\()p FB(\022)r FF(\))p FB(P)g FF(\()p FB(\022)r FF(\))2651 4132 y Fx(\025)2710 4249 y FB(')2764 4261 y Fy(j)2799 4249 y FF(\()p FB(\022)r FF(\))20 b FA(\012)e FF(\012)149 b(\(23\))227 4484 y(\(as)29 b(deriv)n(ed)f(in)h([BFS1)o (]\),)h(where)e FB(P)12 b FF(\()p FB(\022)r FF(\))30 b(is)e(a)h(pro)5 b(jection)27 b(on)i(the)g(unp)r(erturb)r(ed)g (eigenspace)f(cor-)227 4583 y(resp)r(onding)d(to)g FB(E)808 4595 y Fy(j)869 4583 y FF(and)g(on)g(\\small)g(photon)g(energies")f (and)p 2190 4517 V 25 w FB(P)12 b FF(\()p FB(\022)r FF(\))24 b(=)f Fw(1)2522 4595 y Fq(H)2601 4583 y FA(\000)14 b FB(P)e FF(\()p FB(\022)r FF(\).)37 b(It)25 b(is)h(the)g(\014rst)227 4683 y(protot)n(yp)r(e)i(of)g(a)f(metastable)h(state.)38 b(But)29 b(the)f(transformation)e(bac)n(k)i(to)g(the)g(undilated)h (system)227 4794 y(is)24 b(problematic)f(b)r(ecause)g(lim)1179 4809 y Fz(Im)o(\()p Fy(\022)r Fz(\))p Fq(!)p Fz(0)1460 4794 y FB( )1517 4751 y Fz(\(0\))1514 4819 y Fy(\022)1630 4794 y FF(do)r(es)g(not)h(exist.)35 b(In)24 b(the)h(limit)f(Im\()p FB(\022)r FF(\))g FA(!)f FF(0,)h(the)h(sp)r(ec-)227 4894 y(trum)33 b(of)f FB(H)608 4906 y Fy(g)646 4894 y FF(\()p FB(\022)r FF(\))h(is)g(turned)f(bac)n(k)f(from)h(the)h(lo)n(w)n(er)d (half)i(plane)g(in)n(to)g(the)h(real)e(axis.)50 b(Th)n(us,)33 b(the)227 5002 y(resonance)628 4981 y(~)608 5002 y FB(E)669 5014 y Fy(j)732 5002 y FF(=)26 b FB(E)884 5014 y Fy(j)939 5002 y FF(+)20 b FA(O)1106 4935 y Fx(\000)1144 5002 y FB(g)1187 4972 y Fz(2)1224 4935 y Fx(\001)1282 5002 y FA(\000)g FB(i)p FA(O)1478 4935 y Fx(\000)1516 5002 y FB(g)1559 4972 y Fz(2)1595 4935 y Fx(\001)1664 5002 y FF(is)29 b(co)n(v)n(ered)f(b)n(y)i(a)g(sector)e(of)i(absolutely)f(con)n (tin)n(uous)227 5102 y(sp)r(ectrum)f(of)f FB(H)753 5114 y Fy(g)792 5102 y FF(\()p FB(\022)r FF(\),)h(whic)n(h)g(results)f(from) g(p)r(erturbing)g(the)h(branc)n(hes)e FB(E)2607 5114 y Fy(k)2666 5102 y FF(+)18 b FB(e)2788 5072 y Fq(\000)p Fy(\022)2877 5102 y FF([0)p FB(;)c FA(1)p FF(\),)28 b FB(k)e(<)c(j)5 b FF(,)227 5201 y(of)36 b(the)g(sp)r(ectrum)g(of)g FB(H)1024 5213 y Fz(0)1061 5201 y FF(\()p FB(\022)r FF(\))h(\(this)g (is)e(illustrated)h(in)g(Fig.)g(3;)k(for)35 b(sp)r(ectral)g(prop)r (erties)g(of)h(the)227 5301 y(dilated)30 b(Hamiltonian)f(see)g(also)f (Sect.)h(1.3)g(and)g(Fig.)g(4\).)42 b(Note)29 b(that)h(this)f(is)g(the) h(crucial)f(di\013er-)227 5401 y(ence)34 b(to)g([Hu)q(])g(in)g(the)h (construction)e(of)h(a)g(metastable)f(state)h(b)r(ecause)g(im)n(b)r (edded)h(eigen)n(v)-5 b(alues)227 5500 y(are)27 b(excluded)g(in)h(the)g (latter)g(pap)r(er.)352 5601 y(W)-7 b(e)23 b(brie\015y)f(sk)n(etc)n(h)g (our)g(approac)n(h)f(of)i(constructing)f(a)g(metastable)h(state.)35 b(Inspired)22 b(b)n(y)h([Hu],)227 5717 y(w)n(e)34 b(mo)r(dify)h(the)g (state)f FB( )1059 5673 y Fz(\(0\))1056 5742 y Fy(\022)1183 5717 y FF(so)f(that)i(it)g(b)r(ecomes)f(analytic)f(in)i FB(\022)i FF(\(i.e.,)f(w)n(e)e(require)g FE(dilatation)p eop %%Page: 6 6 6 5 bop 0 236 a FL(6)1423 b Fv(1)91 b(INTR)m(ODUCTION)30 b(AND)h(SUR)-10 b(VEY)30 b(OF)h(RESUL)-8 b(TS)221 1930 y @beginspecial @setspecial %%BeginDocument: fig-spectrumHgth.ps %!PS-Adobe-2.0 EPSF-2.0 %%Title: fig-spectrumHgth.pstex %%Creator: fig2dev Version 3.2 Patchlevel 3c %%CreationDate: Thu Mar 15 09:47:02 2001 %%For: mueck@coxeter.math.toronto.edu (Matthias Muck) %%BoundingBox: 0 0 375 185 %%Magnification: 1.0000 %%EndComments /MyAppDict 100 dict dup begin def /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save newpath 0 185 moveto 0 0 lineto 375 0 lineto 375 185 lineto closepath clip newpath -18.0 240.0 translate 1 -1 scale % This junk string is used by the show operators /PATsstr 1 string def /PATawidthshow { % cx cy cchar rx ry string % Loop over each character in the string { % cx cy cchar rx ry char % Show the character dup % cx cy cchar rx ry char char PATsstr dup 0 4 -1 roll put % cx cy cchar rx ry char (char) false charpath % cx cy cchar rx ry char /clip load PATdraw % Move past the character (charpath modified the % current point) currentpoint % cx cy cchar rx ry char x y newpath moveto % cx cy cchar rx ry char % Reposition by cx,cy if the character in the string is cchar 3 index eq { % cx cy cchar rx ry 4 index 4 index rmoveto } if % Reposition all characters by rx ry 2 copy rmoveto % cx cy cchar rx ry } forall pop pop pop pop pop % - currentpoint newpath moveto } bind def /PATcg { 7 dict dup begin /lw currentlinewidth def /lc currentlinecap def /lj currentlinejoin def /ml currentmiterlimit def /ds [ currentdash ] def /cc [ currentrgbcolor ] def /cm matrix currentmatrix def end } bind def % PATdraw - calculates the boundaries of the object and % fills it with the current pattern /PATdraw { % proc save exch PATpcalc % proc nw nh px py 5 -1 roll exec % nw nh px py newpath PATfill % - restore } bind def % PATfill - performs the tiling for the shape /PATfill { % nw nh px py PATfill - PATDict /CurrentPattern get dup begin setfont % Set the coordinate system to Pattern Space PatternGState PATsg % Set the color for uncolored pattezns PaintType 2 eq { PATDict /PColor get PATsc } if % Create the string for showing 3 index string % nw nh px py str % Loop for each of the pattern sources 0 1 Multi 1 sub { % nw nh px py str source % Move to the starting location 3 index 3 index % nw nh px py str source px py moveto % nw nh px py str source % For multiple sources, set the appropriate color Multi 1 ne { dup PC exch get PATsc } if % Set the appropriate string for the source 0 1 7 index 1 sub { 2 index exch 2 index put } for pop % Loop over the number of vertical cells 3 index % nw nh px py str nh { % nw nh px py str currentpoint % nw nh px py str cx cy 2 index oldshow % nw nh px py str cx cy YStep add moveto % nw nh px py str } repeat % nw nh px py str } for 5 { pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proe char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind def % PATmp - the makepattern equivalent /PATmp { % patdict patmtx PATmp patinstance exch dup length 7 add % We will add 6 new entries plus 1 FID dict copy % Create a new dictionary begin % Matrix to install when painting the pattern TilingType PATtcalc /PatternGState PATcg def PatternGState /cm 3 -1 roll put % Check for multi pattern sources (Level 1 fast color patterns) currentdict /Multi known not { /Multi 1 def } if % Font dictionary definitions /FontType 3 def % Create a dummy encoding vector /Encoding 256 array def 3 string 0 1 255 { Encoding exch dup 3 index cvs cvn put } for pop /FontMatrix matrix def /FontBBox BBox def /BuildChar { mark 3 1 roll % mark dict char exch begin Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata] PaintType 2 eq Multi 1 ne or { XStep 0 FontBBox aload pop setcachedevice } { XStep 0 setcharwidth } ifelse currentdict % mark [paintdata] dict /PaintProc load % mark [paintdata] dict paintproc end gsave false PATredef exec true PATredef grestore cleartomark % - } bind def currentdict end % newdict /foo exch % /foo newlict definefont % newfont } bind def % PATpcalc - calculates the starting point and width/height % of the tile fill for the shape /PATpcalc { % - PATpcalc nw nh px py PATDict /CurrentPattern get begin gsave % Set up the coordinate system to Pattern Space % and lock down pattern PatternGState /cm get setmatrix BBox aload pop pop pop translate % Determine the bounding box of the shape pathbbox % llx lly urx ury grestore % Determine (nw, nh) the # of cells to paint width and height PatHeight div ceiling % llx lly urx qh 4 1 roll % qh llx lly urx PatWidth div ceiling % qh llx lly qw 4 1 roll % qw qh llx lly PatHeight div floor % qw qh llx ph 4 1 roll % ph qw qh llx PatWidth div floor % ph qw qh pw 4 1 roll % pw ph qw qh 2 index sub cvi abs % pw ph qs qh-ph exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph % Determine the starting point of the pattern fill %(px, py) 4 2 roll % nw nh pw ph PatHeight mul % nw nh pw py exch % nw nh py pw PatWidth mul exch % nw nh px py end } bind def % Save the original routines so that we can use them later on /oldfill /fill load def /oldeofill /eofill load def /oldstroke /stroke load def /oldshow /show load def /oldashow /ashow load def /oldwidthshow /widthshow load def /oldawidthshow /awidthshow load def /oldkshow /kshow load def % These defs are necessary so that subsequent procs don't bind in % the originals /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def /PATredef { MyAppDict begin { /fill { /clip load PATdraw newpath } bind def /eofill { /eoclip load PATdraw newpath } bind def /stroke { PATstroke } bind def /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def /ashow { 0 0 null 6 3 roll PATawidthshow } bind def /widthshow { 0 0 3 -1 roll PATawidthshow } bind def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor if not available /setcmykcolor where { pop } { /setcmykcolor { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor - pop } bind def } ifelse /PATsc { % colorarray aload length % c1 ... cn length dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /PATsg { % dict begin lw setlinewidth lc setlinecap lj setlinejoin ml setmiterlimit ds aload pop setdash cc aload pop setrgbcolor cm setmatrix end } bind def /PATDict 3 dict def /PATsp { true PATredef PATDict begin /CurrentPattern exch def % If it's an uncolored pattern, save the color CurrentPattern /PaintType get 2 eq { /PColor exch def } if /CColor [ currentrgbcolor ] def end } bind def % PATstroke - stroke with the current pattern /PATstroke { countdictstack save mark { currentpoint strokepath moveto PATpcalc % proc nw nh px py clip newpath PATfill } stopped { (*** PATstroke Warning: Path is too complex, stroking with gray) = cleartomark restore countdictstack exch sub dup 0 gt { { end } repeat } { pop } ifelse gsave 0.5 setgray oldstroke grestore } { pop restore pop } ifelse newpath } bind def /PATtcalc { % modmtx tilingtype PATtcalc tilematrix % Note: tiling types 2 and 3 are not supported gsave exch concat % tilingtype matrix currentmatrix exch % cmtx tilingtype % Tiling type 1 and 3: constant spacing 2 ne { % Distort the pattern so that it occupies % an integral number of device pixels dup 4 get exch dup 5 get exch % tx ty cmtx XStep 0 dtransform round exch round exch % tx ty cmtx dx.x dx.y XStep div exch XStep div exch % tx ty cmtx a b 0 YStep dtransform round exch round exch % tx ty cmtx a b dy.x dy.y YStep div exch YStep div exch % tx ty cmtx a b c d 7 -3 roll astore % { a b c d tx ty } } if grestore } bind def /PATusp { false PATredef PATDict begin CColor PATsc end } bind def % this is the pattern fill program from the Second edition Reference Manual % with changes to call the above pattern fill % left30 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 16 true [ 32 0 0 -16 0 16 ] {} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P1 exch def /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def $F2psBegin %%Page: 1 1 10 setmiterlimit 0.06000 0.06000 sc % % Fig objects follow % % Polyline 7.500 slw n 2190 1695 m 3525 2400 l 2925 2400 l 2115 1620 l cp gs 0.65 setgray ef gr gs col0 s gr % Polyline [15 45] 45 sd n 2175 1650 m 5025 1650 l 4200 2100 l 2175 1650 l cp gs 0.95 setgray ef gr gs col0 s gr [] 0 sd % Polyline n 525 1275 m 2100 2400 l 1575 2400 l cp gs 0.65 setgray ef gr gs col0 s gr % Polyline n 1245 3718 m 1725 3975 l 1500 3975 l cp gs 0.65 setgray ef gr gs col0 s gr % Ellipse n 3825 1800 75 75 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 250.00 115.00] PATmp PATsp ef gr PATusp gs col0 s gr % Ellipse n 2100 1650 75 75 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 135.00 105.00] PATmp PATsp ef gr PATusp gs col0 s gr % Ellipse n 1200 3150 75 75 0 360 DrawEllipse gs col0 s gr % Ellipse n 1200 3675 75 75 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 75.00 240.00] PATmp PATsp ef gr PATusp gs col0 s gr % Polyline n 375 1275 m 6525 1275 l gs col0 s gr % Polyline gs clippath 3868 1714 m 3910 1756 l 4016 1648 l 3911 1713 l 3973 1606 l cp eoclip n 4342 1277 m 3900 1725 l gs col0 s gr gr % arrowhead n 3973 1606 m 3911 1713 l 4016 1648 l 3973 1606 l cp gs 0.00 setgray ef gr col0 s % Ellipse n 2400 1275 75 75 0 360 DrawEllipse gs col0 s gr % Polyline gs clippath 585 1170 m 585 1230 l 736 1230 l 616 1200 l 736 1170 l cp eoclip n 975 1200 m 600 1200 l gs col0 s gr gr % arrowhead n 736 1170 m 616 1200 l 736 1230 l 736 1170 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 2096 1555 m 2140 1596 l 2243 1485 l 2140 1553 l 2200 1444 l cp eoclip n 2400 1275 m 2129 1565 l gs col0 s gr gr % arrowhead n 2200 1444 m 2140 1553 l 2243 1485 l 2200 1444 l cp gs 0.00 setgray ef gr col0 s % Polyline [60] 0 sd n 2400 1275 m 3913 2380 l gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 160.00 85.00] PATmp PATsp ef gr PATusp gs col0 s gr [] 0 sd % Polyline n 6525 1275 m 6525 2400 l 6300 2400 l 5100 1275 l cp gs 0.65 setgray ef gr gs col0 s gr % Ellipse n 1050 1275 75 75 0 360 DrawEllipse gs col0 s gr % Ellipse n 525 1275 75 75 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 30.00 80.00] PATmp PATsp ef gr PATusp gs col0 s gr % Polyline [60] 0 sd n 1037 1295 m 2550 2400 l gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 69.13 86.33] PATmp PATsp ef gr PATusp gs col0 s gr [] 0 sd % Polyline [45 23 15 23] 0 sd gs clippath 3181 1960 m 3139 1918 l 3032 2024 l 3139 1961 l 3075 2067 l cp eoclip n 2550 2550 m 3150 1950 l gs col0 s gr gr [] 0 sd % arrowhead n 3075 2067 m 3139 1961 l 3032 2024 l 3075 2067 l cp gs 0.00 setgray ef gr col0 s % Arc gs clippath 3327 1630 m 3267 1640 l 3293 1789 l 3303 1666 l 3353 1779 l cp eoclip n 2831.2 1800.0 492.2 49.6 -17.7 arcn gs col0 s gr gr % arrowhead n 3353 1779 m 3303 1666 l 3293 1789 l 3353 1779 l cp gs 0.00 setgray ef gr col0 s % Polyline n 3849 1844 m 5175 2400 l 4575 2400 l cp gs 0.65 setgray ef gr gs col0 s gr % Polyline [60] 0 sd n 4350 1275 m 5863 2380 l gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 290.00 85.00] PATmp PATsp ef gr PATusp gs col0 s gr [] 0 sd % Polyline n 5100 1200 m 6525 1200 l 6525 1275 l 5100 1275 l cp gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 340.00 80.00] PATmp PATsp ef gr PATusp gs col0 s gr % Ellipse n 4350 1275 75 75 0 360 DrawEllipse gs col0 s gr % Polyline [60] 0 sd n 1187 3170 m 1575 3450 l gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 79.13 211.33] PATmp PATsp ef gr PATusp gs col0 s gr [] 0 sd % Polyline gs clippath 1020 3615 m 1080 3614 l 1078 3463 l 1050 3584 l 1018 3464 l cp eoclip n 1044 3154 m 1050 3600 l gs col0 s gr gr % arrowhead n 1018 3464 m 1050 3584 l 1078 3463 l 1018 3464 l cp gs 0.00 setgray ef gr col0 s $F2psEnd rs end %%EndDocument @endspecial 933 1812 a Fk(r)j(esonanc)g(e)32 b FH(with)g(sector)h(of)f (con)m(tin)m(uous)i(sp)s(ectrum)956 661 y(~)933 687 y Fu(E)1005 702 y Fs(k)1931 699 y FH(~)1908 724 y Fu(E)1980 739 y Fs(j)821 499 y Fu(:)17 b(:)g(:)260 b(E)1268 514 y Fs(k)2546 499 y FH(\006)-2095 b Fu(E)593 514 y Ft(0)2171 499 y Fu(E)2243 514 y Fs(j)1646 499 y Fu(:)17 b(:)g(:)560 b(:)17 b(:)g(:)244 474 y FH(~)221 499 y Fu(E)293 514 y Ft(0)1196 1324 y Fj(O)s FH(\(Im)o(\()p Fu(\022)s FH(\)\))933 1549 y(unp)s(erturb)s(ed)34 b(ernegy)g(lev)m(el)e(with)g(branc)m(h)h (of)g(con)m(tin)m(uous)g(sp)s(ectrum)0 2126 y FL(Figure)45 b(3:)70 b(Sp)s(ectrum)44 b(of)h(the)h(p)s(erturb)s(ed,)g(dilated)e (Hamiltonian)g Fr(H)2559 2140 y Fs(g)2598 2126 y FL(\()p Fr(\022)s FL(\),)49 b Fr(g)54 b(>)49 b FL(0,)h(Im)o(\()p Fr(\022)s FL(\))g Fr(>)g FL(0,)0 2239 y(resulting)44 b(from)h Fr(H)696 2253 y Ft(0)735 2239 y FL(\()p Fr(\022)s FL(\).)85 b(The)45 b(sectors)i(of)e(con)m(tin)m(uous)g(sp)s(ectrum)g (\014xed)g(at)h(the)f 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Fx(\000)1309 3814 y FB(e)1348 3780 y Fq(\000)p Fy(\022)1437 3814 y FB(H)1506 3826 y Fz(f)1554 3814 y FF(+)g FB(ig)1709 3780 y Fy(\015)1751 3747 y Fx(\001)1789 3760 y Fq(\000)p Fz(1)1892 3814 y FB(P)1945 3826 y Fy(j)1980 3814 y FF(\()p FB(\022)r FF(\))i(+)e(\()p FB(H)2289 3826 y Fz(0)2326 3814 y FF(\()p FB(\022)r FF(\))p FB(P)2484 3826 y Fy(>j)2572 3814 y FF(\()p FB(\022)r FF(\))i FA(\000)e FB(E)2841 3826 y Fy(j)2876 3814 y FF(\))2908 3773 y Fq(\000)p Fz(1)3011 3814 y FB(P)3064 3826 y Fy(>j)3151 3814 y FF(\()p FB(\022)r FF(\))p FB(:)227 3984 y FF(Here,)44 b FB(P)516 3996 y Fy(k)557 3984 y FF(\()p FB(\022)r FF(\))d(is)g(the)f(pro)5 b(jection)40 b(on)g(the)h(eigenspace)e(of)h FB(H)2242 3996 y Fz(0)2280 3984 y FF(\()p FB(\022)r FF(\))h(corresp)r(onding)e (to)h FB(E)3148 3996 y Fy(k)3230 3984 y FF(and)227 4093 y FB(P)280 4105 y Fy(>j)367 4093 y FF(\()p FB(\022)r FF(\))24 b(:=)f Fw(1)657 4105 y Fq(H)734 4093 y FA(\000)811 4031 y Fx(P)899 4052 y Fy(j)899 4118 y(k)q Fz(=0)1037 4093 y FB(P)1090 4105 y Fy(k)1132 4093 y FF(\()p FB(\022)r FF(\).)36 b(In)25 b(Sect.)g(2.1,)f(w)n(e)g(see)g(that)h(this)g(mo)r (di\014cation)f(of)g(the)h(resolv)n(en)n(t)227 4193 y(enables)32 b(us)h(to)g(de\014ne)g(a)f(state)h(with)g(the)g(required)f(prop)r (erties.)52 b(Ho)n(w)n(ev)n(er,)32 b(w)n(e)g(note)h(that)g(the)227 4305 y(analyticit)n(y)18 b(of)h(the)g(resolv)n(en)n(t)1191 4238 y Fx(\000)1229 4305 y FB(e)1268 4275 y Fq(\000)p Fy(\022)1357 4305 y FB(H)1426 4317 y Fz(f)1474 4305 y FF(+)f FB(E)1618 4317 y Fy(k)1678 4305 y FA(\000)g FB(E)1822 4317 y Fy(j)1876 4305 y FA(\000)g FB(ig)2031 4275 y Fy(\013)2077 4238 y Fx(\001)2115 4255 y Fq(\000)p Fz(1)2223 4305 y FF(in)h FB(\022)i FF(is)e(dep)r(enden)n(t)g(on)g(the)g(sp)r(ectral)227 4404 y(parameter)h FB(E)679 4416 y Fy(j)721 4404 y FA(\000)6 b FB(E)853 4416 y Fy(k)900 4404 y FF(+)g FB(ig)1043 4374 y Fy(\013)1111 4404 y FF(and)22 b(therefore)e(on)i(the)g(coupling)f (constan)n(t.)34 b(Once)21 b(more,)h(w)n(e)f(observ)n(e)227 4504 y(that)28 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b(Sigal)f(examine)g(the)h(time-deca)n(y)f(of)h(the)g (zero-order)d(Ra)n(yleigh-)227 5418 y(Sc)n(hr\177)-42 b(odinger)22 b(expansion)g(of)h(the)h(p)r(erturb)r(ed)g(eigenstates,)f (i.e.,)h(of)g(the)f(unp)r(erturb)r(ed)h(eigenstates)227 5517 y(themselv)n(es.)58 b(As)35 b(no)g(resolv)n(en)n(ts)e(en)n(ter)h (their)h(de\014nition)h(of)e(mestable)h(states,)h(the)g(dilatation)227 5617 y(parameter)g FB(\022)j FF(w)n(as)d(unrestricted,)i(while)f(in)g (our)g(case,)h(the)f(consideration)e(of)i(the)g(\014rst-order)227 5717 y(Ra)n(yleigh-Sc)n(hr\177)-42 b(odinger)25 b(expansion)j(allo)n (ws)f(us)i(to)f(\014nd)h(stronger)e(estimates)h(on)g(the)h(error)d (term)p eop %%Page: 7 7 7 6 bop 0 236 a Fv(1.3)92 b(Dilatation)30 b(Analyticit)m(y)g(of)g(the)h (System)1884 b FL(7)227 493 y FF(at)30 b(the)h(exp)r(ense)f(of)g FB(g)s FF(-dep)r(endence)f(of)h(the)h(dilatation)e(parameter.)43 b(W)-7 b(e)31 b(note)e(that)i(the)f(crucial)227 593 y(di\013erence)i(b) 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b(metastable)f(states,)h(as)f(w)n(ell)g(as,)h (for)f(the)h(deriv)-5 b(ation)30 b(of)h(sp)r(ectral)f(prop)r(erties)f (of)i(the)227 1518 y(p)r(erturb)r(ed)h(Hamiltonian)f(from)g(the)h(unp)r (erturb)r(ed)g(\(non-in)n(teracting\))f(one.)47 b(F)-7 b(or)31 b(estimates)g(of)227 1618 y(the)j(metastable)f(states)g(in)h (Sect.)f(2.2)g(w)n(e)g(need)g(an)g(analytic)g(con)n(tin)n(uation)g(of)g (the)h(matrix)f(el-)227 1717 y(emen)n(t)h FB(z)h FA(7!)669 1650 y Fx(\012)722 1717 y FB(')800 1647 y Fx(\014)800 1697 y(\014)842 1717 y FF(\()p FB(H)943 1729 y Fy(g)1000 1717 y FA(\000)18 b FB(z)t FF(\))1158 1687 y Fq(\000)p Fz(1)1247 1717 y FB( )1318 1650 y Fx(\013)1357 1750 y Fq(H)1455 1717 y FF(from)33 b(the)g(upp)r(er)h(half)f(plane)g(accross)e (the)i(real)f(axis)h(in)n(to)227 1817 y(the)27 b(lo)n(w)n(er)e(half)i (plane)f(for)h(suitable)f FE(dilatation)31 b(analytic)d FF(states)e FB(';)14 b( )26 b FA(2)d(H)t FF(.)37 b(This)27 b(enables)f(us)g(to)227 1917 y(deform)c(in)n(tegration)f(con)n(tours)g 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moveto % nw nh px py str source % For multiple sources, set the appropriate color Multi 1 ne { dup PC exch get PATsc } if % Set the appropriate string for the source 0 1 7 index 1 sub { 2 index exch 2 index put } for pop % Loop over the number of vertical cells 3 index % nw nh px py str nh { % nw nh px py str currentpoint % nw nh px py str cx cy 2 index show % nw nh px py str cx cy YStep add moveto % nw nh px py str } repeat % nw nh px py str } for 5 { pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proe char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind def % PATmp - the makepattern equivalent /PATmp { % patdict patmtx PATmp patinstance exch dup length 7 add % We will add 6 new entries plus 1 FID dict copy % Create a new dictionary begin % Matrix to install when painting the pattern TilingType PATtcalc /PatternGState PATcg def PatternGState /cm 3 -1 roll put % Check for multi pattern sources (Level 1 fast color patterns) currentdict /Multi known not { /Multi 1 def } if % Font dictionary definitions /FontType 3 def % Create a dummy encoding vector /Encoding 256 array def 3 string 0 1 255 { Encoding exch dup 3 index cvs cvn put } for pop /FontMatrix matrix def /FontBBox BBox def /BuildChar { mark 3 1 roll % mark dict char exch begin Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata] PaintType 2 eq Multi 1 ne or { XStep 0 FontBBox aload pop setcachedevice } { XStep 0 setcharwidth } ifelse currentdict % mark [paintdata] dict /PaintProc load % mark [paintdata] dict paintproc end gsave false PATredef exec true PATredef grestore cleartomark % - } bind def currentdict end % newdict /foo exch % /foo newlict definefont % newfont } bind def % PATpcalc - calculates the starting point and width/height % of the tile fill for the shape /PATpcalc { % - PATpcalc nw nh px py PATDict /CurrentPattern get begin gsave % Set up the coordinate system to Pattern Space % and lock down pattern PatternGState /cm get setmatrix BBox aload pop pop pop translate % Determine the bounding box of the shape pathbbox % llx lly urx ury grestore % Determine (nw, nh) the # of cells to paint width and height PatHeight div ceiling % llx lly urx qh 4 1 roll % qh llx lly urx PatWidth div ceiling % qh llx lly qw 4 1 roll % qw qh llx lly PatHeight div floor % qw qh llx ph 4 1 roll % ph qw qh llx PatWidth div floor % ph qw qh pw 4 1 roll % pw ph qw qh 2 index sub cvi abs % pw ph qs qh-ph exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph % Determine the starting point of the pattern fill %(px, py) 4 2 roll % nw nh pw ph PatHeight mul % nw nh pw py exch % nw nh py pw PatWidth mul exch % nw nh px py end } bind def % Save the original routines so that we can use them later on /oldfill /fill load def /oldeofill /eofill load def /oldstroke /stroke load def /oldshow /show load def /oldashow /ashow load def /oldwidthshow /widthshow load def /oldawidthshow /awidthshow load def /oldkshow /kshow load def % These defs are necessary so that subsequent procs don't bind in % the originals /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def /PATredef { MyAppDict begin { /fill { /clip load PATdraw newpath } bind def /eofill { /eoclip load PATdraw newpath } bind def /stroke { PATstroke } bind def /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def /ashow { 0 0 null 6 3 roll PATawidthshow } bind def /widthshow { 0 0 3 -1 roll PATawidthshow } bind def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor if not available /setcmykcolor where { pop } { /setcmykcolor { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor - pop } bind def } ifelse /PATsc { % colorarray aload length % c1 ... cn length dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /PATsg { % dict begin lw setlinewidth lc setlinecap lj setlinejoin ml setmiterlimit ds aload pop setdash cc aload pop setrgbcolor cm setmatrix end } bind def /PATDict 3 dict def /PATsp { true PATredef PATDict begin /CurrentPattern exch def % If it's an uncolored pattern, save the color CurrentPattern /PaintType get 2 eq { /PColor exch def } if /CColor [ currentrgbcolor ] def end } bind def % PATstroke - stroke with the current pattern /PATstroke { countdictstack save mark { currentpoint strokepath moveto PATpcalc % proc nw nh px py clip newpath PATfill } stopped { (*** PATstroke Warning: Path is too complex, stroking with gray) = cleartomark restore countdictstack exch sub dup 0 gt { { end } repeat } { pop } ifelse gsave 0.5 setgray oldstroke grestore } { pop restore pop } ifelse newpath } bind def 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3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n -1000 5137 m -1000 -1000 l 9780 -1000 l 9780 5137 l cp clip 0.06000 0.06000 sc % Arc 7.500 slw gs n 3300.0 2700.0 335.4 -26.6 63.4 arc gs col0 s gr gr % Ellipse n 3000 2550 75 75 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 195.00 165.00] PATmp PATsp ef gr PATusp gs col0 s gr % Ellipse n 3750 2550 75 75 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 245.00 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[0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 470.00 165.00] PATmp PATsp ef gr PATusp gs col0 s gr % Polyline n 8700 2550 m 8700 4050 l 7800 4050 l 7050 2550 l cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 470.00 170.00] PATmp PATsp ef gr PATusp gs col0 s gr % Ellipse n 7050 2550 75 75 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 465.00 165.00] PATmp PATsp ef gr PATusp gs col0 s gr $F2psEnd rs end %%EndDocument @endspecial 1161 492 a Fu(:)17 b(:)g(:)-490 b(E)858 507 y Ft(1)411 492 y Fu(E)483 507 y Ft(0)2978 766 y FH(~)2961 792 y Fu(S)336 1092 y FH(Im\()p Fu(\022)s FH(\))1499 492 y Fu(E)1571 507 y Fs(j)1949 492 y Fu(E)2021 507 y Fs(j)t Ft(+1)2474 492 y FH(\006)-333 b Fu(:)17 b(:)g(:)220 1556 y FL(Figure)30 b(4:)41 b(Sp)s(ectrum)29 b(of)i(the)f(unp)s(erturb) s(ed,)d(dilated)i(Hamiltonian)g Fr(H)2743 1570 y Ft(0)2782 1556 y FL(\()p Fr(\022)s FL(\),)i(Im)o(\()p Fr(\022)s FL(\))25 b Fr(>)g FL(0.)227 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PATawidthshow } bind def /widthshow { 0 0 3 -1 roll PATawidthshow } bind def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor if not available /setcmykcolor where { pop } { /setcmykcolor { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor - pop } bind def } ifelse /PATsc { % colorarray aload length % c1 ... cn length dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /PATsg { % dict begin lw setlinewidth lc setlinecap lj setlinejoin ml setmiterlimit ds aload pop setdash cc aload pop setrgbcolor cm setmatrix end } bind def /PATDict 3 dict def /PATsp { true PATredef PATDict begin /CurrentPattern exch def % If it's an uncolored pattern, save the color CurrentPattern /PaintType get 2 eq { /PColor exch def } if /CColor [ currentrgbcolor ] def end } bind def % PATstroke - stroke with the current pattern /PATstroke { countdictstack save mark { currentpoint strokepath moveto PATpcalc % proc nw nh px py clip newpath PATfill } stopped { (*** PATstroke Warning: Path is too complex, stroking with gray) = cleartomark restore countdictstack exch sub dup 0 gt { { end } repeat } { pop } ifelse gsave 0.5 setgray oldstroke grestore } { pop restore pop } ifelse newpath } bind def /PATtcalc { % modmtx tilingtype PATtcalc tilematrix % Note: tiling types 2 and 3 are not supported gsave exch concat % tilingtype matrix currentmatrix exch % cmtx tilingtype % Tiling type 1 and 3: constant spacing 2 ne { % Distort the pattern so that it occupies % an integral number of device pixels dup 4 get exch dup 5 get exch % tx ty cmtx XStep 0 dtransform round exch round exch % tx ty cmtx dx.x dx.y XStep div exch XStep div exch % tx ty cmtx a b 0 YStep dtransform round exch round exch % tx ty cmtx a b dy.x dy.y YStep div exch YStep div exch % tx ty cmtx a b c d 7 -3 roll astore % { a b c d tx ty } } if grestore } bind def /PATusp { false PATredef PATDict begin CColor PATsc end } bind def % right45 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 32 true [ 32 0 0 -32 0 32 ] {<010101010202020204040404080808081010101020202020 404040408080808001010101020202020404040408080808 101010102020202040404040808080800101010102020202 040404040808080810101010202020204040404080808080 010101010202020204040404080808081010101020202020 4040404080808080>} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P5 exch def 1.1111 1.1111 scale %restore scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /reencdict 12 dict def /ReEncode { reencdict begin /newcodesandnames exch def /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup /FID ne { dup /Encoding eq { exch dup length array copy newfont 3 1 roll put } { exch newfont 3 1 roll put } ifelse } { pop pop } ifelse } forall newfont /FontName newfontname put newcodesandnames aload pop 128 1 255 { newfont /Encoding get exch /.notdef put } for newcodesandnames length 2 idiv { newfont /Encoding get 3 1 roll put } repeat newfontname newfont definefont pop end } def /isovec [ 8#200 /grave 8#201 /acute 8#202 /circumflex 8#203 /tilde 8#204 /macron 8#205 /breve 8#206 /dotaccent 8#207 /dieresis 8#210 /ring 8#211 /cedilla 8#212 /hungarumlaut 8#213 /ogonek 8#214 /caron 8#220 /dotlessi 8#230 /oe 8#231 /OE 8#240 /space 8#241 /exclamdown 8#242 /cent 8#243 /sterling 8#244 /currency 8#245 /yen 8#246 /brokenbar 8#247 /section 8#250 /dieresis 8#251 /copyright 8#252 /ordfeminine 8#253 /guillemotleft 8#254 /logicalnot 8#255 /endash 8#256 /registered 8#257 /macron 8#260 /degree 8#261 /plusminus 8#262 /twosuperior 8#263 /threesuperior 8#264 /acute 8#265 /mu 8#266 /paragraph 8#267 /periodcentered 8#270 /cedilla 8#271 /onesuperior 8#272 /ordmasculine 8#273 /guillemotright 8#274 /onequarter 8#275 /onehalf 8#276 /threequarters 8#277 /questiondown 8#300 /Agrave 8#301 /Aacute 8#302 /Acircumflex 8#303 /Atilde 8#304 /Adieresis 8#305 /Aring 8#306 /AE 8#307 /Ccedilla 8#310 /Egrave 8#311 /Eacute 8#312 /Ecircumflex 8#313 /Edieresis 8#314 /Igrave 8#315 /Iacute 8#316 /Icircumflex 8#317 /Idieresis 8#320 /Eth 8#321 /Ntilde 8#322 /Ograve 8#323 /Oacute 8#324 /Ocircumflex 8#325 /Otilde 8#326 /Odieresis 8#327 /multiply 8#330 /Oslash 8#331 /Ugrave 8#332 /Uacute 8#333 /Ucircumflex 8#334 /Udieresis 8#335 /Yacute 8#336 /Thorn 8#337 /germandbls 8#340 /agrave 8#341 /aacute 8#342 /acircumflex 8#343 /atilde 8#344 /adieresis 8#345 /aring 8#346 /ae 8#347 /ccedilla 8#350 /egrave 8#351 /eacute 8#352 /ecircumflex 8#353 /edieresis 8#354 /igrave 8#355 /iacute 8#356 /icircumflex 8#357 /idieresis 8#360 /eth 8#361 /ntilde 8#362 /ograve 8#363 /oacute 8#364 /ocircumflex 8#365 /otilde 8#366 /odieresis 8#367 /divide 8#370 /oslash 8#371 /ugrave 8#372 /uacute 8#373 /ucircumflex 8#374 /udieresis 8#375 /yacute 8#376 /thorn 8#377 /ydieresis] def /Times-Roman /Times-Roman-iso isovec ReEncode /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n -1000 4958 m -1000 -1000 l 7922 -1000 l 7922 4958 l cp clip 0.06000 0.06000 sc 7.500 slw % Ellipse n 5250 2550 75 75 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 345.00 165.00] PATmp PATsp ef gr PATusp gs col0 s gr % Ellipse n 3000 2550 75 75 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 195.00 165.00] PATmp PATsp ef gr PATusp gs 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(the)h(p)r(olar)f(decomp)r(osition)g(of)g(the)h(normal)f(op)r(erator) 711 2723 y Fx(\000)749 2791 y FB(e)788 2756 y Fq(\000)p Fy(\022)872 2764 y Fn(g)911 2791 y FB(H)980 2803 y Fz(f)1027 2791 y FF(+)18 b FB(g)1153 2756 y Fy(\015)1196 2723 y Fx(\001)1234 2737 y Fq(\000)p Fz(1)1346 2791 y FF(=)1433 2720 y Fx(\014)1433 2770 y(\014)1461 2791 y FB(e)1500 2756 y Fq(\000)p Fy(\022)1584 2764 y Fn(g)1622 2791 y FB(H)1691 2803 y Fz(f)1739 2791 y FF(+)g FB(g)1865 2756 y Fy(\015)1907 2720 y Fx(\014)1907 2770 y(\014)1935 2737 y Fq(\000)p Fz(1)p Fy(=)p Fz(2)2105 2791 y FB(U)2185 2720 y Fx(\014)2185 2770 y(\014)2212 2791 y FB(e)2251 2756 y Fq(\000)p Fy(\022)2335 2764 y Fn(g)2373 2791 y FB(H)2442 2803 y Fz(f)2490 2791 y FF(+)g FB(g)2616 2756 y Fy(\015)2658 2720 y Fx(\014)2658 2770 y(\014)2686 2737 y Fq(\000)p Fz(1)p Fy(=)p Fz(2)2856 2791 y FB(;)337 b FF(\(60\))227 2994 y(where)27 b FB(U)37 b FF(is)27 b(a)g(unitary)h(op)r (erator.)35 b(Using)27 b(this)h(equation,)f(w)n(e)g(ha)n(v)n(e)261 3217 y FA(k)p FB(T)352 3229 y Fz(2)388 3217 y FF(\()p FB(\022)459 3229 y Fy(g)498 3217 y FF(\))p FA(k)572 3245 y Fq(H)720 3217 y FF(=)868 3122 y Fx(\015)868 3172 y(\015)868 3221 y(\015)914 3217 y FF(\()p FB(W)1024 3229 y Fy(g)1063 3217 y FF(\()p FB(\022)1134 3229 y Fy(g)1173 3217 y FF(\))19 b FA(\000)f FB(g)1350 3183 y Fy(\015)1392 3217 y FF(\))c FB(P)1491 3229 y Fy(j)1527 3217 y FF(\()p FB(\022)1598 3229 y Fy(g)1637 3217 y FF(\))1683 3150 y Fx(\000)1721 3217 y FB(e)1760 3183 y Fq(\000)p Fy(\022)1844 3191 y Fn(g)1882 3217 y FB(H)1951 3229 y Fz(f)1999 3217 y FF(+)k FB(g)2125 3183 y Fy(\015)2167 3150 y Fx(\001)2205 3163 y Fq(\000)p Fz(1)2308 3217 y FB(P)2361 3229 y Fy(j)2396 3217 y FF(\()p FB(\022)2467 3229 y Fy(g)2506 3217 y FF(\))p FB(W)2616 3229 y Fy(g)2655 3217 y FF(\()p FB(\022)2726 3229 y Fy(g)2765 3217 y FF(\))c FB(')2865 3229 y Fy(j)2901 3217 y FF(\()p FB(\022)2972 3229 y Fy(g)3010 3217 y FF(\))19 b FA(\012)f FF(\012)3204 3122 y Fx(\015)3204 3172 y(\015)3204 3221 y(\015)3250 3275 y Fq(H)720 3415 y FA(\024)83 b(k)o FF(\()q FB(W)1020 3427 y Fy(g)1059 3415 y FF(\()p FB(\022)1130 3427 y Fy(g)1169 3415 y FF(\))18 b FA(\000)g FB(g)1345 3380 y Fy(\015)1388 3415 y FF(\))c FB(P)1487 3427 y Fy(j)1522 3415 y FF(\()p FB(\022)1593 3427 y Fy(g)1632 3415 y FF(\))p FA(k)1706 3443 y Fq(H)1785 3319 y Fx(\015)1785 3369 y(\015)1785 3419 y(\015)1831 3344 y(\014)1831 3394 y(\014)1859 3415 y FB(e)1898 3380 y Fq(\000)p Fy(\022)1982 3388 y Fn(g)2020 3415 y FB(H)2089 3427 y Fz(f)2136 3415 y FF(+)k FB(g)2262 3380 y Fy(\015)2305 3344 y Fx(\014)2305 3394 y(\014)2332 3361 y Fq(\000)p Fz(1)p Fy(=)p Fz(2)2489 3319 y Fx(\015)2489 3369 y(\015)2489 3419 y(\015)2535 3473 y Fq(H)2614 3415 y FA(k)o FB(U)9 b FA(k)2762 3440 y Fq(H)2846 3415 y FA(\001)868 3612 y(\001)905 3517 y Fx(\015)905 3566 y(\015)905 3616 y(\015)951 3541 y(\014)951 3591 y(\014)979 3612 y FB(e)1018 3578 y Fq(\000)p Fy(\022)1102 3586 y Fn(g)1140 3612 y FB(H)1209 3624 y Fz(f)1257 3612 y FF(+)18 b FB(g)1383 3578 y Fy(\015)1425 3541 y Fx(\014)1425 3591 y(\014)1452 3558 y Fq(\000)p Fz(1)p Fy(=)p Fz(2)1622 3612 y FB(P)1675 3624 y Fy(j)1711 3612 y FF(\()p FB(\022)1782 3624 y Fy(g)1821 3612 y FF(\))p FB(W)1931 3624 y Fy(g)1970 3612 y FF(\()p FB(\022)2041 3624 y Fy(g)2080 3612 y FF(\))c FB(')2180 3624 y Fy(j)2215 3612 y FF(\()p FB(\022)2286 3624 y Fy(g)2325 3612 y FF(\))19 b FA(\012)f FF(\012)2519 3517 y Fx(\015)2519 3566 y(\015)2519 3616 y(\015)2565 3670 y Fq(H)2644 3612 y FB(:)227 3848 y FF(T)-7 b(o)28 b(obtain)740 3956 y Fx(\015)740 4005 y(\015)740 4055 y(\015)786 3981 y(\014)786 4030 y(\014)814 4051 y FB(e)853 4017 y Fq(\000)p Fy(\022)937 4025 y Fn(g)975 4051 y FB(H)1044 4063 y Fz(f)1091 4051 y FF(+)18 b FB(g)1217 4017 y Fy(\015)1260 3981 y Fx(\014)1260 4030 y(\014)1287 3997 y Fq(\000)p Fz(1)p Fy(=)p Fz(2)1457 4051 y FB(P)1510 4063 y Fy(j)1546 4051 y FF(\()p FB(\022)1617 4063 y Fy(g)1656 4051 y FF(\))p FB(W)1766 4063 y Fy(g)1805 4051 y FF(\()p FB(\022)1876 4063 y Fy(g)1915 4051 y FF(\))c FB(')2015 4063 y Fy(j)2050 4051 y FF(\()p FB(\022)2121 4063 y Fy(g)2160 4051 y FF(\))19 b FA(\012)f FF(\012)2354 3956 y Fx(\015)2354 4005 y(\015)2354 4055 y(\015)2400 4109 y Fq(H)2488 4051 y FF(=)23 b FA(O)2658 3984 y Fx(\000)2696 4051 y FB(g)2739 4017 y Fz(1)2776 3984 y Fx(\001)2828 4051 y FB(;)227 4304 y FF(w)n(e)k(refer)g(to)h(Lemma)f(2.8.)36 b(This)28 b(estimate)f(implies)h(that)g FA(k)p FB(T)2180 4316 y Fz(2)2216 4304 y FF(\()p FB(\022)2287 4316 y Fy(g)2326 4304 y FF(\))p FA(k)2400 4332 y Fq(H)2488 4304 y FF(=)23 b FA(O)2658 4237 y Fx(\000)2696 4304 y FB(g)2739 4274 y Fz(1+)2833 4247 y Fn(\015)p 2833 4261 34 3 v 2836 4294 a Fo(2)2881 4237 y Fx(\001)2919 4304 y FF(,)28 b(and)f(\014nally)312 4607 y FA(F)8 b FB(\036)p FF(\()p FB(\022)500 4619 y Fy(g)540 4607 y FF(\))18 b FA(\000)676 4499 y Fy(j)s Fq(\000)p Fz(1)674 4528 y Fx(X)673 4707 y Fy(k)q Fz(=0)808 4607 y FB(\030)844 4619 y Fy(k)885 4607 y FF(\()p FB(\022)956 4619 y Fy(g)995 4607 y FF(\))83 b(=)g FA(\000)1339 4499 y Fy(j)s Fq(\000)p Fz(1)1337 4528 y Fx(X)1337 4707 y Fy(k)q Fz(=0)1471 4607 y FB(T)1532 4573 y Fy(k)1520 4627 y Fz(1)1572 4607 y FF(\()p FB(\022)1643 4619 y Fy(g)1682 4607 y FF(\))18 b FA(\000)g FB(T)1864 4619 y Fz(2)1901 4607 y FF(\()p FB(\022)1972 4619 y Fy(g)2011 4607 y FF(\))h FA(\000)f FB(T)2194 4619 y Fz(3)2231 4607 y FF(\()p FB(\022)2302 4619 y Fy(g)2341 4607 y FF(\))23 b(=)f FA(O)2566 4515 y Fx(\020)2615 4607 y FB(g)2658 4573 y Fz(1+min)o Fq(f)2896 4546 y Fn(\015)p 2896 4560 V 2899 4593 a Fo(2)2940 4573 y Fy(;)p Fz(1)p Fq(\000)3054 4546 y Fn(\014)p 3054 4560 35 3 v 3057 4593 a Fo(2)3099 4573 y Fq(g)3138 4515 y Fx(\021)3201 4607 y FB(;)768 4903 y FA(F)8 b FB(\036)p FF(\()p FB(\022)956 4915 y Fy(g)995 4903 y FF(\))83 b(=)1260 4795 y Fy(j)s Fq(\000)p Fz(1)1258 4824 y Fx(X)1258 5002 y Fy(k)q Fz(=0)1393 4903 y FB(\030)1429 4915 y Fy(k)1470 4903 y FF(\()p FB(\022)1541 4915 y Fy(g)1580 4903 y FF(\))18 b(+)g FA(O)1796 4810 y Fx(\020)1845 4903 y FB(g)1888 4868 y Fz(1+min)o Fq(f)2126 4841 y Fn(\015)p 2126 4855 34 3 v 2129 4888 a Fo(2)2170 4868 y Fy(;)p Fz(1)p Fq(\000)2284 4841 y Fn(\014)p 2284 4855 35 3 v 2287 4888 a Fo(2)2329 4868 y Fq(g)2367 4810 y Fx(\021)2440 4903 y FF(=)23 b FA(O)2610 4810 y Fx(\020)2659 4903 y FB(g)2702 4868 y Fz(1+min)o Fq(f)2940 4841 y Fn(\015)p 2940 4855 34 3 v 2943 4888 a Fo(2)2984 4868 y Fy(;\013)p Fq(\000)3109 4841 y Fn(\014)p 3109 4855 35 3 v 3112 4888 a Fo(2)3154 4868 y Fq(g)3192 4810 y Fx(\021)3255 4903 y FB(:)227 5185 y FF(The)28 b(estimate)f(of)h(\(56\))f(is)h(similar.)p 3309 5185 55 55 v 227 5517 a FG(2.6)j(Remark)45 b Fm(Later,)28 b(w)n(e)g(c)n(ho)r(ose)f FB(\014)33 b Fm(smaller)27 b(than)i FB("=)p FF(2)23 b FB(<)h FF(1)p FB(=)p FF(9)j Fm(in)i(order)e(to)i(use) f(prop)r(erties)f(of)227 5617 y(the)j(dilated)f(sp)r(ectrum)g(of)g FB(H)1179 5629 y Fy(g)1217 5617 y Fm(.)41 b(T)-7 b(o)29 b(get)g(the)g(largest)f(p)r(ossible)g(exp)r(onen)n(t)h(in)g(\(57\))g (and)g(\(59\),)g(w)n(e)227 5717 y(c)n(ho)r(ose)d FB(\013)h Fm(as)f(large)f(as)h(p)r(ossible,)h(i.e.)37 b FB(\013)23 b FF(=)g FB(\014)t Fm(,)k(in)g(accordance)e(with)i(Lemma)g(2.4.)35 b(F)-7 b(urthermore,)p eop %%Page: 15 15 15 14 bop 0 236 a Fv(2.1)92 b(Construction)29 b(of)i(a)g(Metastable)g (State)1894 b FL(15)227 493 y Fm(w)n(e)27 b(c)n(ho)r(ose)g FB(\015)h FA(\025)22 b FB(\013)i FF(=)e FB(\014)t Fm(.)38 b(Then,)27 b(the)h(Eqs.)36 b(\(56\))28 b({)f(\(59\))g(read:)1096 738 y FF(\()p FB(\036)p FF(\()p FB(\022)1248 750 y Fy(g)1287 738 y FF(\))19 b FA(\000)f FB(')1475 750 y Fy(j)1510 738 y FF(\()p FB(\022)1581 750 y Fy(g)1620 738 y FF(\))h FA(\012)f FF(\012\))83 b(=)g FA(O)2159 646 y Fx(\020)2209 738 y FB(g)2252 704 y Fz(1)p Fq(\000)2346 677 y Fn(\015)p 2346 691 34 3 v 2349 724 a Fo(2)2394 646 y Fx(\021)2458 738 y FB(;)735 b FF(\(61\))1627 921 y FB(\030)1663 933 y Fy(k)1704 921 y FF(\()p FB(\022)1775 933 y Fy(g)1814 921 y FF(\))83 b(=)g FA(O)2159 829 y Fx(\020)2209 921 y FB(g)2252 887 y Fz(1+)2345 860 y Fn(\014)p 2345 874 35 3 v 2348 907 a Fo(2)2394 829 y Fx(\021)2458 921 y FB(;)735 b FF(\(62\))1131 1163 y FA(F)8 b FB(\036)p FF(\()p FB(\022)1319 1175 y Fy(g)1359 1163 y FF(\))18 b FA(\000)1495 1055 y Fy(j)s Fq(\000)p Fz(1)1493 1084 y Fx(X)1492 1263 y Fy(k)q Fz(=0)1627 1163 y FB(\030)1663 1175 y Fy(k)1704 1163 y FF(\()p FB(\022)1775 1175 y Fy(g)1814 1163 y FF(\))83 b(=)g FA(O)2159 1071 y Fx(\020)2209 1163 y FB(g)2252 1128 y Fz(1+)2345 1102 y Fn(\015)p 2345 1116 34 3 v 2348 1148 a Fo(2)2393 1071 y Fx(\021)2457 1163 y FB(;)736 b FF(\(63\))1587 1400 y FA(F)8 b FB(\036)p FF(\()p FB(\022)1775 1412 y Fy(g)1814 1400 y FF(\))83 b(=)g FA(O)2159 1307 y Fx(\020)2209 1400 y FB(g)2252 1365 y Fz(1+)2345 1339 y Fn(\014)p 2345 1353 35 3 v 2348 1385 a Fo(2)2394 1307 y Fx(\021)2458 1400 y FB(:)735 b FF(\(64\))227 1746 y FG(2.7)31 b(Lemma)43 b Fm(Let)32 b FF(0)d FB(<)g(\013)h FA(\024)f FB(\014)k(<)c FF(1)p FB(=)p FF(9)p Fm(,)i FB(#)1631 1758 y Fy(g)1702 1746 y Fm(as)f(de\014ned)i(in)g(\(50\),)g(and)f FB(\022)2605 1758 y Fy(g)2673 1746 y FF(:=)e FA(\006)p FB(i#)2933 1758 y Fy(g)3001 1746 y FF(=)g FA(O)3177 1679 y Fx(\000)3215 1746 y FB(g)3258 1716 y Fy(\014)3302 1679 y Fx(\001)3340 1746 y Fm(.)227 1846 y(Abbreviate)e FB(E)717 1858 y Fy(j)s(k)812 1846 y FF(:=)c FB(E)984 1858 y Fy(j)1038 1846 y FA(\000)18 b FB(E)1182 1858 y Fy(k)1246 1846 y FB(>)23 b FF(0)p Fm(.)36 b(Then,)28 b(w)n(e)f(ha)n(v)n(e)g(the)h (estimates:)573 1999 y Fx(\015)573 2049 y(\015)573 2099 y(\015)619 2094 y FB(P)672 2106 y Fy(k)713 2094 y FF(\()p FB(\022)784 2106 y Fy(g)823 2094 y FF(\))869 2027 y Fx(\000)907 2094 y FB(e)946 2060 y Fq(\000)p Fy(\022)1030 2068 y Fn(g)1068 2094 y FB(H)1137 2106 y Fz(f)1185 2094 y FA(\000)18 b FB(E)1329 2106 y Fy(j)s(k)1419 2094 y FA(\000)g FB(ig)1574 2060 y Fy(\013)1621 2027 y Fx(\001)1659 2040 y Fq(\000)p Fz(1)1762 2094 y FB(W)1840 2106 y Fy(g)1879 2094 y FF(\()p FB(\022)1950 2106 y Fy(g)1989 2094 y FF(\))c FB(')2089 2106 y Fy(j)2124 2094 y FF(\()p FB(\022)2195 2106 y Fy(g)2234 2094 y FF(\))19 b FA(\012)f FF(\012)2428 1999 y Fx(\015)2428 2049 y(\015)2428 2099 y(\015)2474 2153 y Fq(H)2562 2094 y FF(=)23 b FA(O)2732 2002 y Fx(\020)2782 2094 y FB(g)2825 2060 y Fz(1)p Fq(\000)2919 2033 y Fn(\014)p 2919 2047 V 2922 2080 a Fo(2)2968 2002 y Fx(\021)3216 2094 y FF(\(65\))227 2348 y Fm(and)285 2473 y Fx(\015)285 2523 y(\015)285 2573 y(\015)331 2569 y FB(P)384 2581 y Fy(k)425 2569 y FF(\()p FB(\022)496 2581 y Fy(g)535 2569 y FF(\))581 2501 y Fx(\000)619 2569 y FB(e)658 2534 y Fq(\000)p Fy(\022)742 2542 y Fn(g)780 2569 y FB(H)849 2581 y Fz(f)897 2569 y FA(\000)18 b FB(E)1041 2581 y Fy(j)s(k)1132 2569 y FA(\007)g FB(!)j FF(+)d FB(is)1439 2501 y Fx(\001)1476 2514 y Fq(\000)p Fz(1)1579 2501 y Fx(\000)1617 2569 y FB(e)1656 2534 y Fq(\000)p Fy(\022)1740 2542 y Fn(g)1778 2569 y FB(H)1847 2581 y Fz(f)1895 2569 y FA(\000)g FB(E)2039 2581 y Fy(j)s(k)2130 2569 y FA(\000)g FB(ig)2285 2534 y Fy(\013)2331 2501 y Fx(\001)2369 2514 y Fq(\000)p Fz(1)2472 2569 y FB(W)2550 2581 y Fy(g)2589 2569 y FF(\()p FB(\022)2660 2581 y Fy(g)2699 2569 y FF(\))c FB(')2799 2581 y Fy(j)2834 2569 y FF(\()p FB(\022)2905 2581 y Fy(g)2944 2569 y FF(\))19 b FA(\012)f FF(\012)3138 2473 y Fx(\015)3138 2523 y(\015)3138 2573 y(\015)3184 2627 y Fq(H)1632 2764 y FF(=)23 b FA(O)1802 2671 y Fx(\020)1851 2764 y FB(g)1894 2729 y Fz(1)p Fq(\000)1989 2703 y Fn(\014)p 1989 2717 V 1992 2749 a Fo(2)2038 2671 y Fx(\021)2101 2764 y FB(;)1092 b FF(\(66\))227 3015 y Fm(uniformly)28 b(in)g FB(\016)s(=)p FF(4)22 b FA(\024)g FB(!)k FA(\024)d FB(\016)s(=)p FF(2)j Fm(and)i FA(\000)p FF(const)13 b FB(g)1725 2984 y Fz(2)1785 3015 y FA(\024)22 b FB(s)h FA(\024)g FF(const)13 b FB(g)2268 2984 y Fz(2)2305 3015 y Fm(.)352 3133 y FG(Pro)s(of:)79 b FF(Note)27 b(that)562 3408 y FB(W)640 3420 y Fy(g)680 3408 y FF(\()p FB(\022)751 3420 y Fy(g)789 3408 y FF(\))14 b FB(')889 3420 y Fy(j)925 3408 y FF(\()p FB(\022)996 3420 y Fy(g)1035 3408 y FF(\))k FA(\012)g FF(\012)23 b(=)1339 3266 y Fx( )1405 3408 y FB(g)s(W)1526 3420 y Fz(1)p Fy(;)p Fz(0)1616 3408 y FF(\()p FB(\022)1687 3420 y Fy(g)1726 3408 y FF(\))18 b(+)1917 3329 y Fx(X)1860 3505 y Fy(m)p Fz(+)p Fy(n)p Fz(=2)2108 3408 y FB(g)2151 3374 y Fz(2)2188 3408 y FB(W)2266 3420 y Fy(m;n)2390 3408 y FF(\()p FB(\022)2461 3420 y Fy(g)2500 3408 y FF(\))2532 3266 y Fx(!)2612 3408 y FB(')2666 3420 y Fy(j)2701 3408 y FF(\()p FB(\022)2772 3420 y Fy(g)2811 3408 y FF(\))h FA(\012)f FF(\012)p FB(;)227 3707 y FF(b)r(ecause)27 b(of)719 3949 y FB(W)797 3961 y Fz(0)p Fy(;)p Fz(1)888 3949 y FF(\()p FB(\022)959 3961 y Fy(g)997 3949 y FF(\))14 b FB(')1097 3961 y Fy(j)1133 3949 y FF(\()p FB(\022)1204 3961 y Fy(g)1243 3949 y FF(\))k FA(\012)h FF(\012)k(=)1547 3836 y Fx(Z)1644 3949 y FB(dk)p 1687 3962 46 4 v 17 w(w)1806 3961 y Fz(0)p Fy(;)p Fz(1)1896 3949 y FF(\()p FB(k)p 1928 3962 V 4 w(;)14 b(\022)2051 3961 y Fy(g)2089 3949 y FF(\))g FB(')2189 3961 y Fy(j)2224 3949 y FF(\()p FB(\022)2295 3961 y Fy(g)2334 3949 y FF(\))19 b FA(\012)f FB(a)p FF(\()p FB(k)p 2544 3962 V 3 w FF(\))c(\012)23 b(=)g(0)p FB(:)227 4212 y FF(W)-7 b(e)28 b(start)f(with)h(Eq.)f(\(65\).)37 b(First,)28 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1468 y(\014)1798 1488 y FB(w)1857 1500 y Fz(1)p Fy(;)p Fz(0)1948 1488 y FF(\()p FB(k)p 1980 1501 46 4 v 3 w(;)14 b(\022)2102 1500 y Fy(g)2140 1488 y FF(\))g FB(')2240 1500 y Fy(j)2276 1488 y FF(\()p FB(\022)2347 1500 y Fy(g)2386 1488 y FF(\))2432 1421 y Fx(\013)2471 1522 y Fq(H)2528 1531 y Fo(el)p 1440 1544 1138 4 v 1619 1635 a FB(e)1658 1610 y Fq(\000)p Fy(\022)1742 1618 y Fn(g)1780 1635 y FB(!)s FF(\()1863 1613 y FB(~)1867 1635 y(k)s FF(\))19 b FA(\000)f FB(E)2108 1647 y Fy(j)s(k)2198 1635 y FA(\000)g FB(ig)2353 1611 y Fy(\013)2588 1563 y FB(a)p FF(\()p FB(k)p 2664 1576 46 4 v 3 w FF(\))2742 1529 y Fq(\003)2794 1563 y FF(\012)2854 1418 y Fx(\015)2854 1467 y(\015)2854 1517 y(\015)2854 1567 y(\015)2854 1617 y(\015)2901 1671 y Fq(F)2958 1421 y Fx(#)3007 1430 y Fz(2)310 1870 y FA(\024)83 b FB(g)501 1835 y Fz(2)538 1870 y FB(N)605 1882 y Fy(k)728 1870 y FF(max)659 1924 y Fy(l)p Fz(=1)p Fy(;:::)o(;N)916 1933 y Fn(k)966 1799 y Fx(\015)966 1849 y(\015)1012 1870 y FB(')1066 1835 y Fy(l)1066 1890 y(k)1107 1870 y FF(\()p FB(\022)1178 1882 y Fy(g)1217 1870 y FF(\))1249 1799 y Fx(\015)1249 1849 y(\015)1295 1816 y Fz(2)1295 1903 y Fq(H)1352 1912 y Fo(el)1416 1757 y Fx(Z)1513 1870 y FB(dk)p 1556 1883 V 1630 1724 a Fx(\014)1630 1774 y(\014)1630 1824 y(\014)1630 1874 y(\014)1630 1924 y(\014)1668 1728 y(\012)1721 1795 y FB(')1775 1765 y Fy(l)1775 1819 y(k)1816 1795 y FF(\()p 1848 1728 42 4 v FB(\022)1889 1807 y Fy(g)1928 1795 y FF(\))1984 1725 y Fx(\014)1984 1775 y(\014)2026 1795 y FB(w)2085 1807 y Fz(1)p Fy(;)p Fz(0)2175 1795 y FF(\()p FB(k)p 2207 1808 46 4 v 3 w(;)14 b(\022)2329 1807 y Fy(g)2368 1795 y FF(\))g FB(')2468 1807 y Fy(j)2503 1795 y FF(\()p FB(\022)2574 1807 y Fy(g)2613 1795 y FF(\))2659 1728 y Fx(\013)2698 1829 y Fq(H)2755 1838 y Fo(el)p 1668 1851 1138 4 v 1846 1942 a FB(e)1885 1917 y Fq(\000)p Fy(\022)1969 1925 y Fn(g)2007 1942 y FB(!)s FF(\()2090 1920 y FB(~)2094 1942 y(k)s FF(\))19 b FA(\000)f FB(E)2335 1954 y Fy(j)s(k)2425 1942 y FA(\000)g FB(ig)2580 1918 y 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FF(whic)n(h)28 b(pro)n(v)n(es)d(\(66\).)p 3309 5624 55 55 v eop %%Page: 18 18 18 17 bop 0 236 a FL(18)1565 b Fv(2)91 b(TIME-DECA)-8 b(Y)31 b(OF)g(MET)-8 b(AST)g(ABLE)30 b(ST)-8 b(A)g(TES)227 493 y FG(2.8)31 b(Lemma)43 b Fm(Let)28 b FB(\022)d FF(=)e FB(i#)g FA(2)g FB(U)1270 505 y Fy(#)1310 513 y Fn(g)1349 493 y Fm(,)28 b(i.e.,)f FA(j)p FB(#)p FA(j)d(\024)e FB(#)1811 505 y Fy(g)1878 493 y Fm(and)27 b FB(\015)h FA(\024)22 b FF(2)p Fm(.)37 b(Then,)28 b(w)n(e)f(ha)n(v)n(e)351 625 y Fx(\015)351 675 y(\015)351 724 y(\015)398 720 y FB(P)451 732 y Fy(j)486 720 y FF(\()p FB(\022)r FF(\))605 650 y Fx(\014)605 699 y(\014)633 720 y FB(e)672 686 y Fq(\000)p Fy(\022)761 720 y FB(H)830 732 y Fz(f)878 720 y FF(+)18 b FB(g)1004 686 y Fy(\015)1046 650 y Fx(\014)1046 699 y(\014)1074 666 y Fq(\000)p Fz(1)p Fy(=)p Fz(2)1244 720 y FB(W)1322 732 y Fy(g)1361 720 y FF(\()p FB(\022)r FF(\))1480 650 y Fx(\014)1480 699 y(\014)1508 720 y FB(e)1547 686 y Fq(\000)p Fy(\022)1637 720 y FB(H)1706 732 y Fz(f)1753 720 y FF(+)g FB(g)1879 686 y Fy(\015)1922 650 y Fx(\014)1922 699 y(\014)1949 666 y Fq(\000)p Fz(1)p Fy(=)p Fz(2)2119 720 y FB(P)2172 732 y Fy(j)2208 720 y FF(\()p FB(\022)r FF(\))2313 625 y Fx(\015)2313 675 y(\015)2313 724 y(\015)2360 778 y Fq(H)2508 720 y FF(=)82 b FA(O)2738 628 y Fx(\020)2787 720 y FB(g)2830 686 y Fz(1)p Fq(\000)2925 659 y Fn(\015)p 2925 673 34 3 v 2928 706 a Fo(2)2973 628 y Fx(\021)3036 720 y FB(;)157 b FF(\(70\))841 822 y Fx(\015)841 872 y(\015)841 922 y(\015)888 918 y FB(P)941 930 y Fy(j)976 918 y FF(\()p FB(\022)r FF(\))1095 847 y Fx(\014)1095 897 y(\014)1123 918 y FB(e)1162 883 y Fq(\000)p Fy(\022)1251 918 y FB(H)1320 930 y Fz(f)1368 918 y FF(+)18 b FB(g)1494 883 y Fy(\015)1536 847 y Fx(\014)1536 897 y(\014)1564 863 y Fq(\000)p Fz(1)p Fy(=)p Fz(2)1734 918 y FB(W)1812 930 y Fy(g)1851 918 y FF(\()p FB(\022)r FF(\))p FB(')2010 930 y Fy(j)2046 918 y FF(\()p FB(\022)r FF(\))i FA(\012)e FF(\012)2314 822 y Fx(\015)2314 872 y(\015)2314 922 y(\015)2360 976 y Fq(H)2508 918 y FF(=)82 b FA(O)2738 850 y Fx(\000)2776 918 y FB(g)2819 883 y Fz(1)2856 850 y Fx(\001)2907 918 y FB(;)286 b FF(\(71\))227 1147 y Fm(uniformly)28 b(in)g FB(#)23 b FA(2)g FF(\()p FA(\000)p FB(#)999 1159 y Fy(g)1038 1147 y FB(;)14 b FF(+)p FB(#)1189 1159 y Fy(g)1227 1147 y FF(\))p Fm(.)352 1251 y FG(Pro)s(of:)79 b FF(Recall)27 b(that)h FB(W)1198 1263 y Fy(g)1237 1251 y FF(\()p FB(\022)r FF(\))c(=)1453 1189 y Fx(P)1541 1276 y Fy(m)p Fz(+)p Fy(n)p Fz(=1)p Fy(;)p Fz(2)1847 1251 y FB(g)1890 1221 y Fy(m)p Fz(+)p Fy(n)2044 1251 y FB(W)2122 1263 y Fy(m;n)2247 1251 y FF(\()p FB(\022)r FF(\).)37 b(First,)28 b(w)n(e)f(deriv)n(e)g(a) g(lemma:)352 1356 y FG(Prop)s(osition)60 b FE(Ther)l(e)30 b(is)g(a)h(c)l(onstant)e FB(C)g(>)23 b FF(0)29 b FE(so)h(that)g(for)g FB(m)19 b FF(+)f FB(n)k FF(=)h(1)p FE(:)1008 1475 y Fx(\015)1008 1525 y(\015)1008 1575 y(\015)1054 1570 y FB(P)1107 1582 y Fy(j)1142 1570 y FF(\()p FB(\022)r FF(\))1261 1500 y Fx(\014)1261 1550 y(\014)1290 1570 y FB(e)1329 1536 y Fq(\000)p Fy(\022)1418 1570 y FB(H)1487 1582 y Fz(f)1535 1570 y FF(+)18 b FB(\026)1668 1500 y Fx(\014)1668 1550 y(\014)1696 1516 y Fq(\000)p Fz(1)p Fy(=)p Fz(2)1866 1570 y FB(W)1944 1582 y Fy(m;n)2068 1570 y FF(\()p FB(\022)r FF(\))2187 1500 y Fx(\014)2187 1550 y(\014)2215 1570 y FB(e)2254 1536 y Fq(\000)p Fy(\022)2343 1570 y FB(H)2412 1582 y Fz(f)2460 1570 y FF(+)g FB(\026)2593 1500 y Fx(\014)2593 1550 y(\014)2621 1516 y Fq(\000)p Fz(1)p Fy(=)p Fz(2)2791 1570 y FB(P)2844 1582 y Fy(j)2879 1570 y FF(\()p FB(\022)r FF(\))2984 1475 y Fx(\015)2984 1525 y(\015)2984 1575 y(\015)3031 1629 y Fq(H)1564 1885 y FA(\024)82 b FB(C)1776 1677 y Fx(v)1776 1723 y(u)1776 1773 y(u)1776 1823 y(t)p 1864 1677 719 4 v 1874 1799 a FF(2)1930 1706 y Fx(\020)2017 1762 y Fz(1+)p Fy(\026)p 1989 1780 181 4 v 1989 1827 a Fz(cos\()p Fy(#)p Fz(\))2180 1799 y FF(\003)2238 1770 y Fz(2)2238 1821 y Fq(\000)p Fz(1)2345 1799 y FF(+)18 b(\003)2486 1770 y Fz(2)2486 1821 y(0)2523 1706 y Fx(\021)p 1874 1866 699 4 v 2198 1942 a 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2431 y Fx(\015)2256 2481 y(\015)2256 2531 y(\015)2303 2585 y Fq(H)326 2747 y FF(=)473 2626 y Fx(\015)473 2676 y(\015)473 2726 y(\015)473 2776 y(\015)520 2634 y(Z)616 2676 y(\014)616 2726 y(\014)644 2747 y FB(e)683 2713 y Fq(\000)p Fy(\022)772 2747 y FB(H)841 2759 y Fz(f)889 2747 y FF(+)k FB(\026)1022 2676 y Fx(\014)1022 2726 y(\014)1050 2693 y Fq(\000)p Fz(1)p Fy(=)p Fz(2)1220 2747 y FB(P)1273 2759 y Fy(j)1308 2747 y FF(\()p FB(\022)r FF(\))p FB(w)1472 2759 y Fz(0)p Fy(;)p Fz(1)1563 2747 y FF(\()p FB(k)p 1595 2760 46 4 v 4 w(;)c(\022)r FF(\))p FB(P)1805 2759 y Fy(j)1840 2747 y FF(\()p FB(\022)r FF(\))1959 2651 y Fx(\014)1959 2701 y(\014)1959 2751 y(\014)1988 2747 y FB(e)2027 2713 y Fq(\000)p Fy(\022)2116 2747 y FF(\()p FB(H)2217 2759 y Fz(f)2265 2747 y FF(+)k FB(!)s FF(\()2431 2725 y FB(~)2435 2747 y(k)s FF(\)\))h(+)f FB(\026)2697 2651 y Fx(\014)2697 2701 y(\014)2697 2751 y(\014)2724 2672 y Fq(\000)p Fz(1)p Fy(=)p Fz(2)2895 2747 y FB(a)p FF(\()p FB(k)p 2971 2760 V 3 w FF(\))c FB( 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(in)h(\(73\))f(follo)n(ws)g(from)326 3710 y Fx(Z)423 3728 y(\015)423 3778 y(\015)423 3828 y(\015)469 3823 y FF(\()p FB(H)570 3835 y Fz(f)618 3823 y FF(+)18 b FB(!)s FF(\()784 3802 y FB(~)788 3823 y(k)s FF(\)\))898 3789 y Fq(\000)p Fz(1)p Fy(=)p Fz(2)1055 3823 y FB(a)p FF(\()p FB(k)p 1131 3836 V 3 w FF(\))p FB( )1266 3728 y Fx(\015)1266 3778 y(\015)1266 3828 y(\015)1312 3748 y Fz(2)1312 3882 y Fq(H)1392 3823 y FB(!)s FF(\()1475 3802 y FB(~)1479 3823 y(k)r FF(\))c FB(dk)p 1613 3836 V 409 4041 a FF(=)557 3928 y Fx(Z)654 3949 y(D)718 4041 y FF(\()p FB(H)819 4053 y Fz(f)867 4041 y FF(+)k FB(!)s FF(\()1033 4019 y FB(~)1037 4041 y(k)s FF(\)\))1147 4007 y Fq(\000)p Fz(1)p Fy(=)p Fz(2)1304 4041 y FB(a)p FF(\()p FB(k)p 1380 4054 V 3 w FF(\))p FB( )1539 3946 y Fx(\014)1539 3995 y(\014)1539 4045 y(\014)1581 4041 y FF(\()p FB(H)1682 4053 y Fz(f)1730 4041 y FF(+)g FB(!)s FF(\()1896 4019 y FB(~)1900 4041 y(k)s FF(\)\))2010 4007 y Fq(\000)p Fz(1)p Fy(=)p Fz(2)2166 4041 y FB(a)p FF(\()p FB(k)p 2242 4054 V 3 w FF(\))p FB( )2392 3949 y Fx(E)2442 4099 y Fq(H)2521 4041 y FB(!)s FF(\()2604 4019 y FB(~)2608 4041 y(k)s FF(\))c FB(dk)p 2743 4054 V 409 4259 a FF(=)557 4146 y Fx(Z)654 4167 y(D)718 4259 y FB(a)p FF(\()p FB(k)p 794 4272 V 3 w FF(\)\()p FB(H)973 4271 y Fz(f)1003 4259 y FB(P)1068 4225 y Fq(?)1056 4279 y Fz(\012)1124 4259 y FF(\))1156 4225 y Fq(\000)p Fz(1)p Fy(=)p Fz(2)1313 4259 y FB(P)1378 4225 y Fq(?)1366 4279 y Fz(\012)1447 4259 y FB( )1528 4163 y Fx(\014)1528 4213 y(\014)1528 4263 y(\014)1570 4259 y FB(a)p FF(\()p FB(k)p 1646 4272 V 3 w FF(\)\()p FB(H)1825 4271 y Fz(f)1855 4259 y FB(P)1920 4225 y Fq(?)1908 4279 y Fz(\012)1976 4259 y FF(\))2008 4225 y Fq(\000)p Fz(1)p Fy(=)p Fz(2)2164 4259 y FB(P)2229 4225 y Fq(?)2217 4279 y Fz(\012)2299 4259 y FB( )2370 4167 y Fx(E)2421 4317 y Fq(H)2500 4259 y FB(!)s FF(\()2583 4237 y FB(~)2587 4259 y(k)r FF(\))g FB(dk)p 2721 4272 V 27 w FF(=)2878 4188 y Fx(\015)2878 4238 y(\015)2924 4259 y FB(P)2989 4225 y Fq(?)2977 4279 y Fz(\012)3059 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ry char % Show the character dup % cx cy cchar rx ry char char PATsstr dup 0 4 -1 roll put % cx cy cchar rx ry char (char) false charpath % cx cy cchar rx ry char /clip load PATdraw % Move past the character (charpath modified the % current point) currentpoint % cx cy cchar rx ry char x y newpath moveto % cx cy cchar rx ry char % Reposition by cx,cy if the character in the string is cchar 3 index eq { % cx cy cchar rx ry 4 index 4 index rmoveto } if % Reposition all characters by rx ry 2 copy rmoveto % cx cy cchar rx ry } forall pop pop pop pop pop % - currentpoint newpath moveto } bind def /PATcg { 7 dict dup begin /lw currentlinewidth def /lc currentlinecap def /lj currentlinejoin def /ml currentmiterlimit def /ds [ currentdash ] def /cc [ currentrgbcolor ] def /cm matrix currentmatrix def end } bind def % PATdraw - calculates the boundaries of the object and % fills it with the current pattern /PATdraw { % proc save exch PATpcalc % proc nw nh px py 5 -1 roll exec % nw nh px py newpath PATfill % - restore } bind def % PATfill - performs the tiling for the shape /PATfill { % nw nh px py PATfill - PATDict /CurrentPattern get dup begin setfont % Set the coordinate system to Pattern Space PatternGState PATsg % Set the color for uncolored pattezns PaintType 2 eq { PATDict /PColor get PATsc } if % Create the string for showing 3 index string % nw nh px py str % Loop for each of the pattern sources 0 1 Multi 1 sub { % nw nh px py str source % Move to the starting location 3 index 3 index % nw nh px py str source px py moveto % nw nh px py str source % For multiple sources, set the appropriate color Multi 1 ne { dup PC exch get PATsc } if % Set the appropriate string for the source 0 1 7 index 1 sub { 2 index exch 2 index put } for pop % Loop over the number of vertical cells 3 index % nw nh px py str nh { % nw nh px py str currentpoint % nw nh px py str cx cy 2 index show % nw nh px py str cx cy YStep add moveto % nw nh px py str } repeat % nw nh px py str } for 5 { pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proe char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind def % PATmp - the makepattern equivalent /PATmp { % patdict patmtx PATmp patinstance exch dup length 7 add % We will add 6 new entries plus 1 FID dict copy % Create a new dictionary begin % Matrix to install when painting the pattern TilingType PATtcalc /PatternGState PATcg def PatternGState /cm 3 -1 roll put % Check for multi pattern sources (Level 1 fast color patterns) currentdict /Multi known not { /Multi 1 def } if % Font dictionary definitions /FontType 3 def % Create a dummy encoding vector /Encoding 256 array def 3 string 0 1 255 { Encoding exch dup 3 index cvs cvn put } for pop /FontMatrix matrix def /FontBBox BBox def /BuildChar { mark 3 1 roll % mark dict char exch begin Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata] PaintType 2 eq Multi 1 ne or { XStep 0 FontBBox aload pop setcachedevice } { XStep 0 setcharwidth } ifelse currentdict % mark [paintdata] dict /PaintProc load % mark [paintdata] dict paintproc end gsave false PATredef exec true PATredef grestore cleartomark % - } bind def currentdict end % newdict /foo exch % /foo newlict definefont % newfont } bind def % PATpcalc - calculates the starting point and width/height % of the tile fill for the shape /PATpcalc { % - PATpcalc nw nh px py PATDict /CurrentPattern get begin gsave % Set up the coordinate system to Pattern Space % and lock down pattern PatternGState /cm get setmatrix BBox aload pop pop pop translate % Determine the bounding box of the shape pathbbox % llx lly urx ury grestore % Determine (nw, nh) the # of cells to paint width and height PatHeight div ceiling % llx lly urx qh 4 1 roll % qh llx lly urx PatWidth div ceiling % qh llx lly qw 4 1 roll % qw qh llx lly PatHeight div floor % qw qh llx ph 4 1 roll % ph qw qh llx PatWidth div floor % ph qw qh pw 4 1 roll % pw ph qw qh 2 index sub cvi abs % pw ph qs qh-ph exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph % Determine the starting point of the pattern fill %(px, py) 4 2 roll % nw nh pw ph PatHeight mul % nw nh pw py exch % nw nh py pw PatWidth mul exch % nw nh px py end } bind def % Save the original routines so that we can use them later on /oldfill /fill load def /oldeofill /eofill load def /oldstroke /stroke load def /oldshow /show load def /oldashow /ashow load def /oldwidthshow /widthshow load def /oldawidthshow /awidthshow load def /oldkshow /kshow load def % These defs are necessary so that subsequent procs don't bind in % the originals /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def /PATredef { MyAppDict begin { /fill { /clip load PATdraw newpath } bind def /eofill { /eoclip load PATdraw newpath } bind def /stroke { PATstroke } bind def /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def /ashow { 0 0 null 6 3 roll PATawidthshow } bind def /widthshow { 0 0 3 -1 roll PATawidthshow } bind def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor if not available /setcmykcolor where { pop } { /setcmykcolor { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor - pop } bind def } ifelse /PATsc { % colorarray aload length % c1 ... cn length dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /PATsg { % dict begin lw setlinewidth lc setlinecap lj setlinejoin ml setmiterlimit ds aload pop setdash cc aload pop setrgbcolor cm setmatrix end } bind def /PATDict 3 dict def /PATsp { true PATredef PATDict begin /CurrentPattern exch def % If it's an uncolored pattern, save the color CurrentPattern /PaintType get 2 eq { /PColor exch def } if /CColor [ currentrgbcolor ] def end } bind def % PATstroke - stroke with the current pattern /PATstroke { countdictstack save mark { currentpoint strokepath moveto PATpcalc % proc nw nh px py clip newpath PATfill } stopped { (*** PATstroke Warning: Path is too complex, stroking with gray) = cleartomark restore countdictstack exch sub dup 0 gt { { end } repeat } { pop } ifelse gsave 0.5 setgray oldstroke grestore } { pop restore pop } ifelse newpath } bind def /PATtcalc { % modmtx tilingtype PATtcalc tilematrix % Note: tiling types 2 and 3 are not supported gsave exch concat % tilingtype matrix currentmatrix exch % cmtx tilingtype % Tiling type 1 and 3: constant spacing 2 ne { % Distort the pattern so that it occupies % an integral number of device pixels dup 4 get exch dup 5 get exch % tx ty cmtx XStep 0 dtransform round exch round exch % tx ty cmtx dx.x dx.y XStep div exch XStep div exch % tx ty cmtx a b 0 YStep dtransform round exch round exch % tx ty cmtx a b dy.x dy.y YStep div exch YStep div exch % tx ty cmtx a b c d 7 -3 roll astore % { a b c d tx ty } } if grestore } bind def /PATusp { false PATredef PATDict begin CColor PATsc end } bind def % this is the pattern fill program from the Second edition Reference Manual % with changes to call the above pattern fill % left30 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 16 true [ 32 0 0 -16 0 16 ] {} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P1 exch def % right45 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 32 true [ 32 0 0 -32 0 32 ] {<010101010202020204040404080808081010101020202020 404040408080808001010101020202020404040408080808 101010102020202040404040808080800101010102020202 040404040808080810101010202020204040404080808080 010101010202020204040404080808081010101020202020 4040404080808080>} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P5 exch def 1.1111 1.1111 scale %restore scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /reencdict 12 dict def /ReEncode { reencdict begin /newcodesandnames exch def /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup /FID ne { dup /Encoding eq { exch dup length array copy newfont 3 1 roll put } { exch newfont 3 1 roll put } ifelse } { pop pop } ifelse } forall newfont /FontName newfontname put newcodesandnames aload pop 128 1 255 { newfont /Encoding get exch /.notdef put } for newcodesandnames length 2 idiv { newfont /Encoding get 3 1 roll put } repeat newfontname newfont definefont pop end } def /isovec [ 8#200 /grave 8#201 /acute 8#202 /circumflex 8#203 /tilde 8#204 /macron 8#205 /breve 8#206 /dotaccent 8#207 /dieresis 8#210 /ring 8#211 /cedilla 8#212 /hungarumlaut 8#213 /ogonek 8#214 /caron 8#220 /dotlessi 8#230 /oe 8#231 /OE 8#240 /space 8#241 /exclamdown 8#242 /cent 8#243 /sterling 8#244 /currency 8#245 /yen 8#246 /brokenbar 8#247 /section 8#250 /dieresis 8#251 /copyright 8#252 /ordfeminine 8#253 /guillemotleft 8#254 /logicalnot 8#255 /endash 8#256 /registered 8#257 /macron 8#260 /degree 8#261 /plusminus 8#262 /twosuperior 8#263 /threesuperior 8#264 /acute 8#265 /mu 8#266 /paragraph 8#267 /periodcentered 8#270 /cedilla 8#271 /onesuperior 8#272 /ordmasculine 8#273 /guillemotright 8#274 /onequarter 8#275 /onehalf 8#276 /threequarters 8#277 /questiondown 8#300 /Agrave 8#301 /Aacute 8#302 /Acircumflex 8#303 /Atilde 8#304 /Adieresis 8#305 /Aring 8#306 /AE 8#307 /Ccedilla 8#310 /Egrave 8#311 /Eacute 8#312 /Ecircumflex 8#313 /Edieresis 8#314 /Igrave 8#315 /Iacute 8#316 /Icircumflex 8#317 /Idieresis 8#320 /Eth 8#321 /Ntilde 8#322 /Ograve 8#323 /Oacute 8#324 /Ocircumflex 8#325 /Otilde 8#326 /Odieresis 8#327 /multiply 8#330 /Oslash 8#331 /Ugrave 8#332 /Uacute 8#333 /Ucircumflex 8#334 /Udieresis 8#335 /Yacute 8#336 /Thorn 8#337 /germandbls 8#340 /agrave 8#341 /aacute 8#342 /acircumflex 8#343 /atilde 8#344 /adieresis 8#345 /aring 8#346 /ae 8#347 /ccedilla 8#350 /egrave 8#351 /eacute 8#352 /ecircumflex 8#353 /edieresis 8#354 /igrave 8#355 /iacute 8#356 /icircumflex 8#357 /idieresis 8#360 /eth 8#361 /ntilde 8#362 /ograve 8#363 /oacute 8#364 /ocircumflex 8#365 /otilde 8#366 /odieresis 8#367 /divide 8#370 /oslash 8#371 /ugrave 8#372 /uacute 8#373 /ucircumflex 8#374 /udieresis 8#375 /yacute 8#376 /thorn 8#377 /ydieresis] def /Times-Roman /Times-Roman-iso isovec ReEncode /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n -1000 5437 m -1000 -1000 l 7762 -1000 l 7762 5437 l cp clip 0.06000 0.06000 sc % Polyline 7.500 slw n 900 750 m 6300 750 l 6300 3225 l 900 3225 l cp gs col7 0.90 shd ef gr gs col7 s gr % Ellipse n 3600 1800 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 232.93 112.93] PATmp PATsp ef gr PATusp gs col0 s gr % Polyline n 600 1800 m 6600 1800 l gs col0 s gr % Polyline n 1500 1800 m 5700 1800 l 5700 3075 l 1500 3075 l cp gs col0 s gr % Polyline gs clippath 1530 2328 m 1500 2448 l 1470 2328 l 1470 2490 l 1530 2490 l cp clip n 1500 1800 m 1500 2475 l gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 100.00 120.00] PATmp PATsp ef gr PATusp gs col0 s gr gr % arrowhead n 1530 2328 m 1500 2448 l 1470 2328 l 1500 2328 l 1530 2328 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 3453 3045 m 3573 3075 l 3453 3105 l 3615 3105 l 3615 3045 l cp clip n 1500 3075 m 3600 3075 l gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 100.00 205.00] PATmp PATsp ef gr PATusp gs col0 s gr gr % arrowhead n 3453 3045 m 3573 3075 l 3453 3105 l 3453 3075 l 3453 3045 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 5670 2547 m 5700 2427 l 5730 2547 l 5730 2385 l 5670 2385 l cp clip n 5700 3075 m 5700 3000 l 5700 2775 l 5700 2550 l 5700 2475 l 5700 2400 l gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 380.00 160.00] PATmp PATsp ef gr PATusp gs col0 s gr gr % arrowhead n 5670 2547 m 5700 2427 l 5730 2547 l 5700 2547 l 5670 2547 l cp gs 0.00 setgray ef gr col0 s % Polyline n 3750 4425 m 6750 4425 l 2400 3375 l 3750 4425 l cp gs /PC [[1.00 1.00 1.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 160.00 225.00] PATmp PATsp ef gr PATusp gs col0 s gr % Polyline [60] 0 sd gs clippath 2661 3199 m 2566 3278 l 2613 3164 l 2517 3294 l 2565 3330 l cp clip n 3600 1875 m 2550 3300 l gs col0 s gr gr [] 0 sd % arrowhead n 2661 3199 m 2566 3278 l 2613 3164 l 2637 3182 l 2661 3199 l cp gs 0.00 setgray ef gr col0 s % Polyline 15.000 slw n 900 750 m 900 3225 l 6300 3225 l 6300 750 l gs col0 s gr 7.500 slw % Ellipse n 2475 3375 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 157.93 217.93] PATmp PATsp ef gr PATusp gs col0 s gr $F2psEnd rs end %%EndDocument @endspecial 1630 788 a Fu(E)1702 803 y Fs(j)505 788 y Fu(E)577 803 y Fs(j)636 788 y Fj(\000)23 b Fu(!)730 1538 y(E)802 1553 y Fs(j)861 1538 y Fj(\000)g Fu(!)i Fj(\000)e Fu(iS)1240 1553 y Fs(g)1593 525 y Fj(A)1673 489 y Fs(")1818 2100 y Fu(\033)t FH(\()p Fu(H)1996 2115 y Fs(g)2036 2100 y FH(\()p Fu(\022)s FH(\)\))1593 525 y Fj(A)1673 489 y Fs(")1293 1725 y Fk(r)-5 b(esonanc)g(e)2605 788 y Fu(E)2677 803 y Fs(j)2736 788 y FH(+)22 b Fu(!)2193 1538 y(E)2265 1553 y Fs(j)2324 1538 y FH(+)g Fu(!)j Fj(\000)e Fu(iS)2701 1553 y Fs(g)309 2452 y FL(Figure)29 b(6:)42 b(Deformation)30 b(of)h(the)f(in)m(tegration)h(con)m(tour)g(in)m(to)f (the)h(lo)m(w)m(er)f(half)g(plane.)227 2795 y FF(where)498 3066 y FB(M)579 3078 y Fq(\006)718 3066 y FF(:=)83 b FA(\000)986 2929 y Fy(\016)r(=)p Fz(2)975 2953 y Fx(Z)949 3195 y Fy(\016)r(=)p Fz(4)1080 3066 y FB(d!)1202 3010 y(F)1267 2980 y Fq(0)1290 3010 y FF(\()p FB(E)1383 3022 y Fy(j)1437 3010 y FF(+)18 b FB(!)s FF(\))p 1202 3047 406 4 v 1344 3123 a(2)p FB(\031)s(i)1653 2927 y Fy(S)1694 2935 y Fn(g)1631 2953 y Fx(Z)1638 3189 y Fz(0)1728 3066 y FB(ds)28 b FF(exp\()p FA(\000)p FB(it)p FF(\()p FB(E)2214 3078 y Fy(j)2267 3066 y FA(\006)18 b FB(!)j FA(\000)d FB(is)p FF(\)\))1085 3346 y FA(\001)1127 3254 y Fx(h)c(D)1244 3346 y FB(\036)p FF(\()p 1325 3279 42 4 v FB(\022)1367 3358 y Fy(g)1406 3346 y FF(\))1462 3251 y Fx(\014)1462 3301 y(\014)1462 3351 y(\014)1503 3346 y FF(\()q FB(H)1605 3358 y Fy(g)1643 3346 y FF(\()p FB(\022)1714 3358 y Fy(g)1753 3346 y FF(\))19 b FA(\000)f FB(E)1948 3358 y Fy(j)2002 3346 y FA(\007)g FB(!)j FF(+)d FB(is)p FF(\))2341 3305 y Fq(\000)p Fz(1)2444 3346 y FB(\036)p FF(\()p FB(\022)2564 3358 y Fy(g)2603 3346 y FF(\))2649 3254 y Fx(E)2700 3405 y Fq(H)1397 3542 y FA(\000)1480 3449 y Fx(D)1545 3542 y FB(\036)p FF(\()p FB(\022)1665 3554 y Fy(g)1704 3542 y FF(\))1760 3446 y Fx(\014)1760 3496 y(\014)1760 3546 y(\014)1802 3474 y(\000)1840 3542 y FB(H)1909 3554 y Fy(g)1948 3542 y FF(\()p 1980 3474 V FB(\022)2021 3554 y Fy(g)2060 3542 y FF(\))g FA(\000)h FB(E)2255 3554 y Fy(j)2308 3542 y FA(\007)f FB(!)j FF(+)d FB(is)2615 3474 y Fx(\001)2653 3488 y Fq(\000)p Fz(1)2756 3542 y FB(\036)p FF(\()p 2837 3474 V FB(\022)2879 3554 y Fy(g)2918 3542 y FF(\))2964 3449 y Fx(E)3014 3600 y Fq(H)3093 3449 y Fx(i)3133 3542 y FB(;)60 b FF(\(79\))374 3830 y FB(M)455 3845 y Fq(k)493 3830 y FF(\()p FB(\022)564 3842 y Fy(g)603 3830 y FF(\))83 b(:=)g FA(\000)986 3693 y Fy(\016)r(=)p Fz(2)975 3717 y Fx(Z)949 3959 y Fy(\016)r(=)p Fz(4)1080 3830 y FB(d!)1202 3774 y(F)1267 3744 y Fq(0)1290 3774 y FF(\()p FB(E)1383 3786 y Fy(j)1437 3774 y FF(+)18 b FB(!)s FF(\))p 1202 3811 406 4 v 1344 3887 a(2)p FB(\031)s(i)889 4128 y FA(\001)953 4015 y Fx(Z)907 4257 y Fq(I)t Fz(\()o Fy(!)r Fz(\))1077 4128 y FB(d\025)c(e)1221 4093 y Fq(\000)p Fy(it)p Fz(\()p Fy(\025)p Fq(\000)p Fy(iS)1502 4101 y Fn(g)1537 4093 y Fz(\))1581 4036 y Fx(D)1646 4128 y FB(\036)p FF(\()p 1727 4060 42 4 v FB(\022)1769 4140 y Fy(g)1808 4128 y FF(\))1864 4032 y Fx(\014)1864 4082 y(\014)1864 4132 y(\014)1905 4128 y FF(\()p FB(H)2006 4140 y Fy(g)2045 4128 y FF(\()p FB(\022)2116 4140 y Fy(g)2155 4128 y FF(\))19 b FA(\000)f FB(\025)h FF(+)f FB(iS)2519 4140 y Fy(g)2557 4128 y FF(\))2589 4086 y Fq(\000)p Fz(1)2692 4128 y FB(\036)p FF(\()p FB(\022)2812 4140 y Fy(g)2851 4128 y FF(\))2898 4036 y Fx(E)2948 4186 y Fq(H)3027 4128 y FB(:)166 b FF(\(80\))227 4420 y(Thm.)40 b(2.9)e(follo)n(ws)h(directly)g(from)g(Lemmata)f(2.11)g (and)h(2.12)f(b)r(elo)n(w)h(whic)n(h)g(state)g(that)h(the)227 4519 y(in)n(tegration)23 b(along)h(the)g(horizonzal)f(con)n(tour)g FA(\000)p FB(iS)1842 4531 y Fy(g)1892 4519 y FF(+)12 b FA(I)20 b FF(\()p FB(!)s FF(\))25 b(\(see)f(Fig.)g(6\))h(giv)n(es)e 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4399 y Fy(k)1296 4387 y FA(\000)g FB(E)1440 4399 y Fy(j)1494 4387 y FA(\000)g FB(ig)1649 4353 y Fy(\013)1695 4295 y Fx(\021)1745 4312 y Fq(\000)p Fz(1)1848 4387 y FB(P)1901 4399 y Fy(k)1942 4387 y FF(\()p 1974 4320 42 4 v FB(\022)2016 4399 y Fy(g)2054 4387 y FF(\))p FB(W)2164 4399 y Fy(g)2204 4387 y FF(\()p 2236 4320 V FB(\022)2277 4399 y Fy(g)2316 4387 y FF(\))c FB(')2416 4399 y Fy(j)2451 4387 y FF(\()p 2483 4320 V FB(\022)2525 4399 y Fy(g)2563 4387 y FF(\))19 b FA(\012)f FF(\012)2757 4292 y Fx(\015)2757 4342 y(\015)2757 4392 y(\015)2803 4445 y Fq(H)544 4582 y FA(\001)p FB(g)610 4548 y Fy(\013)657 4487 y Fx(\015)657 4537 y(\015)657 4587 y(\015)717 4515 y(\000)755 4582 y FB(e)794 4548 y Fq(\000)p Fy(\022)878 4556 y Fn(g)916 4582 y FB(H)985 4594 y Fz(f)1033 4582 y FF(+)g FB(E)1177 4594 y Fy(k)1237 4582 y FA(\000)g FB(E)1381 4594 y Fy(j)1434 4582 y FA(\007)g FB(!)j FF(+)d FB(is)1741 4515 y Fx(\001)1779 4528 y Fq(\000)p Fz(1)722 4765 y FA(\001)763 4698 y Fx(\000)801 4765 y FB(e)840 4731 y 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Fy(g)964 5125 y FF(\))1020 5029 y Fx(\014)1020 5079 y(\014)1020 5129 y(\014)1062 5125 y FF(\()p FB(H)1163 5137 y Fy(g)1202 5125 y FF(\()p FB(\022)1273 5137 y Fy(g)1311 5125 y FF(\))19 b FA(\000)f FB(E)1506 5137 y Fy(j)1560 5125 y FA(\007)g FB(!)j FF(+)d FB(is)p FF(\))1899 5083 y Fq(\000)p Fy(\027)t Fq(\000)p Fz(1)2091 5125 y FA(F)8 b FB(\036)p FF(\()p FB(\022)2279 5137 y Fy(g)2318 5125 y FF(\))2364 5032 y Fx(E)2415 5183 y Fq(H)2503 5125 y FF(=)23 b FA(O)2673 5057 y Fx(\000)2711 5125 y FB(g)2754 5090 y Fz(2+)p Fy(\014)2883 5057 y Fx(\001)227 5311 y FF(for)k FB(\015)h FF(=)23 b(1)k(and)g FB(\014)h(<)22 b FF(1)p FB(=)p FF(9.)36 b(W)-7 b(e)28 b(omit)f(the)h(estimate)g(of)575 5392 y Fx(D)640 5484 y FA(F)8 b FB(\036)p FF(\()p FB(\022)828 5496 y Fy(g)867 5484 y FF(\))923 5389 y Fx(\014)923 5438 y(\014)923 5488 y(\014)964 5417 y(\000)1002 5484 y FB(H)1071 5496 y Fy(g)1110 5484 y FF(\()p 1142 5416 V FB(\022)1184 5496 y Fy(g)1222 5484 y FF(\))19 b FA(\000)f FB(E)1417 5496 y Fy(j)1471 5484 y FA(\007)g FB(!)j FF(+)d FB(is)1778 5417 y Fx(\001)1816 5430 y Fq(\000)p Fy(\027)t Fq(\000)p Fz(1)2008 5484 y FA(F)8 b FB(\036)p FF(\()p 2157 5416 V FB(\022)2198 5496 y Fy(g)2237 5484 y FF(\))2283 5392 y Fx(E)2334 5542 y Fq(H)1284 5696 y FF(=)p 1371 5590 1811 4 v 1371 5604 a Fx(D)1436 5696 y FA(F)g FB(\036)p FF(\()p 1585 5628 42 4 v FB(\022)1627 5708 y Fy(g)1665 5696 y FF(\))1722 5600 y Fx(\014)1722 5650 y(\014)1722 5700 y(\014)1763 5696 y FF(\()p FB(H)1864 5708 y Fy(g)1903 5696 y FF(\()p FB(\022)1974 5708 y Fy(g)2013 5696 y FF(\))19 b FA(\000)f FB(E)2208 5708 y Fy(j)2261 5696 y FA(\007)g FB(!)j FA(\000)d FB(is)p FF(\))2600 5654 y Fq(\000)p Fy(\027)t Fq(\000)p Fz(1)2792 5696 y FA(F)8 b FB(\036)p FF(\()p FB(\022)2980 5708 y Fy(g)3020 5696 y FF(\))3066 5604 y Fx(E)3116 5754 y Fq(H)p eop %%Page: 25 25 25 24 bop 0 236 a Fv(2.2)92 b(Quasi-Exp)s(onen)m(tial)28 b(Time-Deca)m(y)k(La)m(w)1852 b FL(25)227 493 y FF(whic)n(h)28 b(is)f(analogous.)35 b(This)27 b(pro)n(v)n(es)f(\(90\))h(and)h 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string is cchar 3 index eq { % cx cy cchar rx ry 4 index 4 index rmoveto } if % Reposition all characters by rx ry 2 copy rmoveto % cx cy cchar rx ry } forall pop pop pop pop pop % - currentpoint newpath moveto } bind def /PATcg { 7 dict dup begin /lw currentlinewidth def /lc currentlinecap def /lj currentlinejoin def /ml currentmiterlimit def /ds [ currentdash ] def /cc [ currentrgbcolor ] def /cm matrix currentmatrix def end } bind def % PATdraw - calculates the boundaries of the object and % fills it with the current pattern /PATdraw { % proc save exch PATpcalc % proc nw nh px py 5 -1 roll exec % nw nh px py newpath PATfill % - restore } bind def % PATfill - performs the tiling for the shape /PATfill { % nw nh px py PATfill - PATDict /CurrentPattern get dup begin setfont % Set the coordinate system to Pattern Space PatternGState PATsg % Set the color for uncolored pattezns PaintType 2 eq { PATDict /PColor get PATsc } if % Create the string for showing 3 index string % nw nh px py str % Loop for each of the pattern sources 0 1 Multi 1 sub { % nw nh px py str source % Move to the starting location 3 index 3 index % nw nh px py str source px py moveto % nw nh px py str source % For multiple sources, set the appropriate color Multi 1 ne { dup PC exch get PATsc } if % Set the appropriate string for the source 0 1 7 index 1 sub { 2 index exch 2 index put } for pop % Loop over the number of vertical cells 3 index % nw nh px py str nh { % nw nh px py str currentpoint % nw nh px py str cx cy 2 index show % nw nh px py str cx cy YStep add moveto % nw nh px py str } repeat % nw nh px py str } for 5 { pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proe char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind def % PATmp - the makepattern equivalent /PATmp { % patdict patmtx PATmp patinstance exch dup length 7 add % We will add 6 new entries plus 1 FID dict copy % Create a new dictionary begin % Matrix to install when painting the pattern TilingType PATtcalc /PatternGState PATcg def PatternGState /cm 3 -1 roll put % Check for multi pattern sources (Level 1 fast color patterns) currentdict /Multi known not { /Multi 1 def } if % Font dictionary definitions /FontType 3 def % Create a dummy encoding vector /Encoding 256 array def 3 string 0 1 255 { Encoding exch dup 3 index cvs cvn put } for pop /FontMatrix matrix def /FontBBox BBox def /BuildChar { mark 3 1 roll % mark dict char exch begin Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata] PaintType 2 eq Multi 1 ne or { XStep 0 FontBBox aload pop setcachedevice } { XStep 0 setcharwidth } ifelse currentdict % mark [paintdata] dict /PaintProc load % mark [paintdata] dict paintproc end gsave false PATredef exec true PATredef grestore cleartomark % - } bind def currentdict end % newdict /foo exch % /foo newlict definefont % newfont } bind def % PATpcalc - calculates the starting point and width/height % of the tile fill for the shape /PATpcalc { % - PATpcalc nw nh px py PATDict /CurrentPattern get begin gsave % Set up the coordinate system to Pattern Space % and lock down pattern PatternGState /cm get setmatrix BBox aload pop pop pop translate % Determine the bounding box of the shape pathbbox % llx lly urx ury grestore % Determine (nw, nh) the # of cells to paint width and height PatHeight div ceiling % llx lly urx qh 4 1 roll % qh llx lly urx PatWidth div ceiling % qh llx lly qw 4 1 roll % qw qh llx lly PatHeight div floor % qw qh llx ph 4 1 roll % ph qw qh llx PatWidth div floor % ph qw qh pw 4 1 roll % pw ph qw qh 2 index sub cvi abs % pw ph qs qh-ph exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph % Determine the starting point of the pattern fill %(px, py) 4 2 roll % nw nh pw ph PatHeight mul % nw nh pw py exch % nw nh py pw PatWidth mul exch % nw nh px py end } bind def % Save the original routines so that we can use them later on /oldfill /fill load def /oldeofill /eofill load def /oldstroke /stroke load def /oldshow /show load def /oldashow /ashow load def /oldwidthshow /widthshow load def /oldawidthshow /awidthshow load def /oldkshow /kshow load def % These defs are necessary so that subsequent procs don't bind in % the originals /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def /PATredef { MyAppDict begin { /fill { /clip load PATdraw newpath } bind def /eofill { /eoclip load PATdraw newpath } bind def /stroke { PATstroke } bind def /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def /ashow { 0 0 null 6 3 roll PATawidthshow } bind def /widthshow { 0 0 3 -1 roll PATawidthshow } bind def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor if not available /setcmykcolor where { pop } { /setcmykcolor { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor - pop } bind def } ifelse /PATsc { % colorarray aload length % c1 ... cn length dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /PATsg { % dict begin lw setlinewidth lc setlinecap lj setlinejoin ml setmiterlimit ds aload pop setdash cc aload pop setrgbcolor cm setmatrix end } bind def /PATDict 3 dict def /PATsp { true PATredef PATDict begin /CurrentPattern exch def % If it's an uncolored pattern, save the color CurrentPattern /PaintType get 2 eq { /PColor exch def } if /CColor [ currentrgbcolor ] def end } bind def % PATstroke - stroke with the current pattern /PATstroke { countdictstack save mark { currentpoint strokepath moveto PATpcalc % proc nw nh px py clip newpath PATfill } stopped { (*** PATstroke Warning: Path is too complex, stroking with gray) = cleartomark restore countdictstack exch sub dup 0 gt { { end } repeat } { pop } ifelse gsave 0.5 setgray oldstroke grestore } { pop restore pop } ifelse newpath } bind def /PATtcalc { % modmtx tilingtype PATtcalc tilematrix % Note: tiling types 2 and 3 are not supported gsave exch concat % tilingtype matrix currentmatrix exch % cmtx tilingtype % Tiling type 1 and 3: constant spacing 2 ne { % Distort the pattern so that it occupies % an integral number of device pixels dup 4 get exch dup 5 get exch % tx ty cmtx XStep 0 dtransform round exch round exch % tx ty cmtx dx.x dx.y XStep div exch XStep div exch % tx ty cmtx a b 0 YStep dtransform round exch round exch % tx ty cmtx a b dy.x dy.y YStep div exch YStep div exch % tx ty cmtx a b c d 7 -3 roll astore % { a b c d tx ty } } if grestore } bind def /PATusp { false PATredef PATDict begin CColor PATsc end } bind def % this is the pattern fill program from the Second edition Reference Manual % with changes to call the above pattern fill % left30 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 16 true [ 32 0 0 -16 0 16 ] {} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P1 exch def 1.1111 1.1111 scale %restore scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /reencdict 12 dict def /ReEncode { reencdict begin /newcodesandnames exch def /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup /FID ne { dup /Encoding eq { exch dup length array copy newfont 3 1 roll put } { exch newfont 3 1 roll put } ifelse } { pop pop } ifelse } forall newfont /FontName newfontname put newcodesandnames aload pop 128 1 255 { newfont /Encoding get exch /.notdef put } for newcodesandnames length 2 idiv { newfont /Encoding get 3 1 roll put } repeat newfontname newfont definefont pop end } def /isovec [ 8#200 /grave 8#201 /acute 8#202 /circumflex 8#203 /tilde 8#204 /macron 8#205 /breve 8#206 /dotaccent 8#207 /dieresis 8#210 /ring 8#211 /cedilla 8#212 /hungarumlaut 8#213 /ogonek 8#214 /caron 8#220 /dotlessi 8#230 /oe 8#231 /OE 8#240 /space 8#241 /exclamdown 8#242 /cent 8#243 /sterling 8#244 /currency 8#245 /yen 8#246 /brokenbar 8#247 /section 8#250 /dieresis 8#251 /copyright 8#252 /ordfeminine 8#253 /guillemotleft 8#254 /logicalnot 8#255 /endash 8#256 /registered 8#257 /macron 8#260 /degree 8#261 /plusminus 8#262 /twosuperior 8#263 /threesuperior 8#264 /acute 8#265 /mu 8#266 /paragraph 8#267 /periodcentered 8#270 /cedilla 8#271 /onesuperior 8#272 /ordmasculine 8#273 /guillemotright 8#274 /onequarter 8#275 /onehalf 8#276 /threequarters 8#277 /questiondown 8#300 /Agrave 8#301 /Aacute 8#302 /Acircumflex 8#303 /Atilde 8#304 /Adieresis 8#305 /Aring 8#306 /AE 8#307 /Ccedilla 8#310 /Egrave 8#311 /Eacute 8#312 /Ecircumflex 8#313 /Edieresis 8#314 /Igrave 8#315 /Iacute 8#316 /Icircumflex 8#317 /Idieresis 8#320 /Eth 8#321 /Ntilde 8#322 /Ograve 8#323 /Oacute 8#324 /Ocircumflex 8#325 /Otilde 8#326 /Odieresis 8#327 /multiply 8#330 /Oslash 8#331 /Ugrave 8#332 /Uacute 8#333 /Ucircumflex 8#334 /Udieresis 8#335 /Yacute 8#336 /Thorn 8#337 /germandbls 8#340 /agrave 8#341 /aacute 8#342 /acircumflex 8#343 /atilde 8#344 /adieresis 8#345 /aring 8#346 /ae 8#347 /ccedilla 8#350 /egrave 8#351 /eacute 8#352 /ecircumflex 8#353 /edieresis 8#354 /igrave 8#355 /iacute 8#356 /icircumflex 8#357 /idieresis 8#360 /eth 8#361 /ntilde 8#362 /ograve 8#363 /oacute 8#364 /ocircumflex 8#365 /otilde 8#366 /odieresis 8#367 /divide 8#370 /oslash 8#371 /ugrave 8#372 /uacute 8#373 /ucircumflex 8#374 /udieresis 8#375 /yacute 8#376 /thorn 8#377 /ydieresis] def /Times-Roman /Times-Roman-iso isovec ReEncode /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n -1000 5212 m -1000 -1000 l 7720 -1000 l 7720 5212 l cp clip 0.06000 0.06000 sc 7.500 slw % Ellipse n 3600 1800 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 232.93 112.93] PATmp PATsp ef gr PATusp gs col0 s gr % Ellipse n 3750 2625 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 242.93 167.93] PATmp PATsp ef gr PATusp gs col0 s gr % Polyline n 600 1800 m 6600 1800 l gs col0 s gr % Polyline gs clippath 1530 2328 m 1500 2448 l 1470 2328 l 1470 2490 l 1530 2490 l cp clip n 1500 1800 m 1500 2475 l gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 100.00 120.00] PATmp PATsp ef gr PATusp gs col0 s gr gr % arrowhead n 1530 2328 m 1500 2448 l 1470 2328 l 1500 2328 l 1530 2328 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 5670 2547 m 5700 2427 l 5730 2547 l 5730 2385 l 5670 2385 l cp clip n 5700 3075 m 5700 3000 l 5700 2775 l 5700 2550 l 5700 2475 l 5700 2400 l gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 380.00 160.00] PATmp PATsp ef gr PATusp gs col0 s gr gr % arrowhead n 5670 2547 m 5700 2427 l 5730 2547 l 5700 2547 l 5670 2547 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 3453 3720 m 3573 3750 l 3453 3780 l 3615 3780 l 3615 3720 l cp clip n 1500 3750 m 3600 3750 l gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 100.00 250.00] PATmp PATsp ef gr PATusp gs col0 s gr gr % arrowhead n 3453 3720 m 3573 3750 l 3453 3780 l 3453 3750 l 3453 3720 l cp gs 0.00 setgray ef gr col0 s % Polyline n 1500 2475 m 1500 3150 l gs col0 s gr % Polyline n 5700 2400 m 5700 1800 l gs col0 s gr % Polyline [60] 0 sd n 1500 3150 m 1500 3750 l gs col0 s gr [] 0 sd % Polyline [60] 0 sd n 5700 3150 m 5700 3675 l gs col0 s gr [] 0 sd % Polyline n 3600 3750 m 5700 3750 l gs col0 s gr % Polyline gs clippath 3555 4053 m 3525 4173 l 3495 4053 l 3495 4215 l 3555 4215 l cp clip n 3525 3750 m 3525 4200 l gs col0 s gr gr % arrowhead n 3555 4053 m 3525 4173 l 3495 4053 l col0 s % Polyline n 3375 2250 m 4125 2250 l 4125 3000 l 3375 3000 l cp gs col0 s gr % Polyline gs clippath 4155 2553 m 4125 2673 l 4095 2553 l 4095 2715 l 4155 2715 l cp clip n 4125 2250 m 4125 2700 l gs col0 s gr gr % arrowhead n 4155 2553 m 4125 2673 l 4095 2553 l 4125 2553 l 4155 2553 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 3822 3030 m 3702 3000 l 3822 2970 l 3660 2970 l 3660 3030 l cp clip n 4125 3000 m 3675 3000 l gs col0 s gr gr % arrowhead n 3822 3030 m 3702 3000 l 3822 2970 l 3822 3000 l 3822 3030 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 3345 2697 m 3375 2577 l 3405 2697 l 3405 2535 l 3345 2535 l cp clip n 3375 3000 m 3375 2550 l gs col0 s gr gr % arrowhead n 3345 2697 m 3375 2577 l 3405 2697 l 3375 2697 l 3345 2697 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 3678 2220 m 3798 2250 l 3678 2280 l 3840 2280 l 3840 2220 l cp clip n 3375 2250 m 3825 2250 l gs col0 s gr gr % arrowhead n 3678 2220 m 3798 2250 l 3678 2280 l 3678 2250 l 3678 2220 l cp gs 0.00 setgray ef gr col0 s % Polyline [15 45] 45 sd n 2700 3150 m 3750 2625 l gs col0 s gr [] 0 sd $F2psEnd rs end %%EndDocument @endspecial 1668 480 a Fu(E)1740 495 y Fs(j)543 480 y Fu(E)615 495 y Fs(j)673 480 y Fj(\000)23 b Fu(!)1809 b(E)2715 495 y Fs(j)2774 480 y FH(+)22 b Fu(!)1780 1755 y(R)29 b Fj(!)e(1)1443 1567 y(\000)p Fu(iR)d FH(+)e Fj(I)i FH(\()p Fu(!)t FH(\))1240 1354 y(~)1218 1380 y Fu(E)1290 1395 y Fs(j)1354 1380 y FH(=)k Fu(E)1530 1395 y Fs(j)1588 1380 y FH(+)22 b(Re)q(\()p Fj(O)d FH(\()p Fu(g)2027 1343 y Ft(2)2066 1380 y FH(\)\))j Fj(\000)g Fu(iC)2366 1395 y Fs(S)2417 1380 y Fu(g)2468 1343 y Ft(2+)p Fs(")463 2032 y FL(Figure)30 b(7:)41 b(Applying)28 b Fi(Cauchy's)33 b(inte)-5 b(gr)g(al)34 b(formula)e FL(and)e Fi(r)-5 b(esidue)33 b(the)-5 b(or)g(em)p FL(.)227 2379 y FF(b)r(ecause)26 b FB(\020)j FA(\025)696 2347 y Fz(3)p 696 2361 34 4 v 696 2408 a(2)739 2379 y FB(")d FF(and)f FB(\014)j(<)22 b("=)p 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Fz(2)1714 853 y FF(\)\))i FA(\000)f FB(iC)1968 865 y Fy(S)2016 853 y FB(g)2059 829 y Fz(2+)p Fy(")2202 796 y FB(d\025)2293 601 y Fx(\014)2293 651 y(\014)2293 700 y(\014)2293 750 y(\014)2293 800 y(\014)2293 850 y(\014)2293 900 y(\014)2404 796 y FA(\024)117 b FF(lim)2552 849 y Fy(R)p Fq(!1)2776 683 y Fx(Z)2730 925 y Fq(I)t Fz(\()o Fy(!)r Fz(\))2910 740 y FB(e)2949 710 y Fq(\000)p Fy(Rt)p 2910 777 171 4 v 2963 853 a FB(R)3103 796 y(d\025)24 b FF(=)f(0)227 1099 y(and)299 1191 y Fx(\014)299 1240 y(\014)299 1290 y(\014)299 1340 y(\014)327 1311 y FF(exp\()p FA(\000)p FB(it)p FF(\()p FB(\025)c FA(\000)f FB(iS)872 1323 y Fy(g)910 1311 y FF(\)\))974 1215 y Fx(\014)974 1265 y(\014)974 1315 y(\014)1002 1369 y Fy(\025)p Fz(=)p Fy(E)1141 1377 y Fn(j)1172 1369 y Fz(+Re\()p Fq(O)r Fz(\()o Fy(g)1441 1352 y Fo(2)1474 1369 y Fz(\)\))p Fq(\000)p Fy(iC)1649 1377 y Fn(S)1691 1369 y Fy(g)1725 1352 y Fo(2+)p Fn(")1832 1191 y Fx(\014)1832 1240 y(\014)1832 1290 y(\014)1832 1340 y(\014)1883 1311 y FF(=)23 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1963 y Fy(g)2109 1951 y FF(\)\).)38 b(This)27 b(pro)n(v)n(es)f(the)i(lemma.)p 3309 1951 55 55 v 227 2298 a FD(A)134 b(The)45 b(Pull-Through)f(F)-11 b(orm)l(ula)227 2480 y FF(A)25 b(useful)f(to)r(ol)f(for)h(calculations) f(with)h(the)g(free)g(\014eld)g(Hamiltonian)g(and)g(creation)e(and)i (annihila-)227 2580 y(tion)j(op)r(erators)d(is)i(the)h(so)f(called)g FE(Pul)t(l-Thr)l(ough)k(F)-6 b(ormula)p FF(,)27 b(whic)n(h)f(can)g (easily)f(b)r(e)i(deriv)n(ed)f(from)227 2679 y(functional)i(calculus)f (\(see,)h(e.g.,)f([BFS1],)g(Lemma)h(A.1\):)227 2880 y FG(A.1)k(Lemma)d(\(Pull-Through)j(F)-8 b(orm)m(ula\))44 b Fm(Let)26 b FB(F)35 b FF(:)23 b Fw(R)2152 2850 y Fz(+)2230 2880 y FA(!)g Fw(C)k Fm(b)r(e)f(a)g(measurable)f(function,)227 2979 y(ob)r(eying)i FB(F)12 b FF(\()p FB(r)r FF(\))25 b(=)d FA(O)r FF(\()p FB(r)r FF(\))p Fm(.)39 b(Then)28 b FB(F)12 b FF(\()p FB(H)1432 2991 y Fz(f)1461 2979 y FF(\))28 b Fm(is)g(de\014ned)g(on)f FA(D)r FF(\()p FB(H)2173 2991 y Fz(f)2203 2979 y FF(\))h Fm(and)1199 3173 y FB(a)p FF(\()p FB(k)p 1275 3186 46 4 v 3 w FF(\))14 b FB(F)e FF(\()p FB(H)1533 3185 y Fz(f)1563 3173 y FF(\))23 b(=)g FB(F)12 b FF(\()p FB(H)1872 3185 y Fz(f)1920 3173 y FF(+)18 b FB(!)s FF(\()2086 3151 y FB(~)2090 3173 y(k)s FF(\)\))c FB(a)p FF(\()p FB(k)p 2290 3186 V 3 w FF(\))p FB(;)1161 3317 y(F)e FF(\()p FB(H)1327 3329 y Fz(f)1357 3317 y FF(\))i FB(a)p FF(\()p FB(k)p 1479 3330 V 3 w FF(\))1557 3283 y Fq(\003)1618 3317 y FF(=)23 b FB(a)p FF(\()p FB(k)p 1782 3330 V 3 w FF(\))1860 3283 y Fq(\003)1912 3317 y FB(F)12 b FF(\()p FB(H)2078 3329 y Fz(f)2126 3317 y FF(+)18 b FB(!)s FF(\()2292 3295 y FB(~)2296 3317 y(k)s FF(\)\))p FB(:)3174 3241 y FF(\(101\))267 3497 y FG(Ac)m(kno)m(wledgemen)m(ts)h FF(The)h(author)f(thanks)h(V.)g(Bac)n(h,)h(F.)f(Baldus,)h(J.)f(Lutgen,) i(M.)e(Sc)n(hneider,)227 3596 y(U.)30 b(Staude,)h(T.)e(W)-7 b(eth,)31 b(and)f(H.)g(Zenk)f(for)g(helpful)i(discussions.)42 b(He)30 b(is)f(esp)r(ecially)g(indebted)h(to)227 3696 y(V.)e(Bac)n(h)f(and)g(J.)h(Lutgen)f(for)g(careful)g(pro)r(ofreading.) 227 3971 y FD(References)227 4154 y FF([A)n(C])162 b Fa(J.)31 b(A)n(guilar,)f(J.M.)h(Combes.)d FE(A)h(Class)i(of)g(A)n (nalytic)f(Perturb)l(ations)g(for)h(One-b)l(o)l(dy)555 4253 y(Schr\177)-42 b(odinger)31 b(Hamiltonians.)f FF(Comm.)d(Math.)h (Ph)n(ys.,)f FG(22)p FF(,)g(269-279,)d(1971)227 4387 y([BFS1])81 b Fa(V.)43 b(Ba)n(ch,)j(J.)e(Fr)1209 4380 y(\177)1206 4387 y(ohlich,)h(I.M.)e(Sigal.)c FE(R)l(enormalization)i (Gr)l(oup)g(A)n(nalysis)g(of)555 4486 y(Sp)l(e)l(ctr)l(al)d(Pr)l (oblems)h(in)f(Quantum)e(Field)k(The)l(ory.)e FF(Adv.)f(in)g(Math.,)i FG(137)p FF(,)f(205-298,)555 4586 y(1998)227 4719 y([BFS2])81 b Fa(V.)30 b(Ba)n(ch,)g(J.)h(Fr)1167 4712 y(\177)1164 4719 y(ohlich,)f(I.M.)f(Sigal.)e FE(Quantum)g(Ele)l(ctr)l(o)l(dynamics) k(of)f(Con\014ne)l(d)555 4819 y(Non-R)l(elativistic)g(Particles.)g FF(Adv.)e(in)g(Math.,)f FG(137)p FF(,)g(299-395,)e(1998)227 4952 y([BFS3])81 b Fa(V.)27 b(Ba)n(ch,)i(J.)f(Fr)1160 4945 y(\177)1157 4952 y(ohlich,)g(I.M.)f(Sigal.)d FE(Sp)l(e)l(ctr)l(al) j(A)n(nalysis)g(for)h(Systems)f(of)h(A)n(toms)555 5052 y(and)d(Mole)l(cules)h(Couple)l(d)g(to)f(the)g(Quantize)l(d)g(R)l (adiation)g(Field.)g FF(Comm.)d(Math.)g(Ph)n(ys.,)555 5151 y FG(207)p FF(,)27 b(249-290,)d(1999)227 5284 y([BC])163 b Fa(E.)44 b(Balslev,)i(J.M.)e(Combes.)c FE(Sp)l(e)l(ctr)l(al)h(Pr)l (op)l(erties)h(of)g(Schr\177)-42 b(odinger)43 b(Op)l(er)l(ators)555 5384 y(with)30 b(Dilatation)h(A)n(nalytic)f(Potentials.)f FF(Comm.)f(Math.)g(Ph)n(ys.,)e FG(22)p FF(,)h(280-294,)e(1971)227 5517 y([CDG])94 b Fa(C.)55 b(Cohen-T)-7 b(annoudji,)61 b(J.)55 b(Dupont-R)n(oc,)61 b(G.)54 b(Gr)-6 b(ynber)n(g.)51 b FE(Photons)h(and)555 5617 y(A)n(toms)28 b({)h(Intr)l(o)l(duction)f (to)g(Quantum)f(Ele)l(ctr)l(o)l(dynamics.)h FF(John)e(Wiley)g(&)g (Sons,)g(New)555 5717 y(Y)-7 b(ork,)27 b(1991)p eop %%Page: 29 29 29 28 bop 0 236 a Fv(REFERENCES)2874 b FL(29)227 493 y FF([CFKS])57 b Fa(H.L.)24 b(Cycon,)j(R.G.)e(Fr)n(oese,)j(W.)d (Kirsch,)i(B.)e(Simon.)d FE(Schr\177)-42 b(odinger)27 b(Op)l(er)l(ators.)555 593 y FF(Springer-V)-7 b(erlag,)25 b(Berlin,)i(Heidelb)r(erg,)h(New)g(Y)-7 b(ork,)27 b(1987)227 725 y([HS1])132 b Fa(M.)29 b(H)764 718 y(\177)761 725 y(ubner,)g(H.)g(Spohn.)c FE(R)l(adiative)k(De)l(c)l(ay:)38 b(Nonp)l(erturb)l(ative)28 b(Appr)l(o)l(aches.)g FF(Rev.)555 825 y(Math.)g(Ph)n(ys.,)f FG(7)p FF(,)g(363-387,)d(1995)227 958 y([HS2])132 b Fa(M.)32 b(H)767 951 y(\177)764 958 y(ubner,)f(H.)g(Spohn.)d FE(Sp)l(e)l(ctr)l(al)i(Pr)l(op)l(erties)h(of)g (the)g(Spin-Boson)g(Hamiltonian.)555 1057 y FF(Ann.)d(Inst.)g(H.)g(P)n (oincare,)e FG(62)p FF(,)h(289-323,)d(1995)227 1190 y([Hu])174 b Fa(W.)38 b(Hunziker.)33 b FE(R)l(esonanc)l(es,)38 b(Metastable)g (States)d(and)i(Exp)l(onential)g(De)l(c)l(ay)f(L)l(aws)555 1290 y(in)30 b(Perturb)l(ation)g(The)l(ory.)f FF(Comm.)f(Math.)g(Ph)n (ys.,)e FG(132)p FF(,)h(177-188,)e(1990)227 1423 y([Ka])175 b Fa(T.)27 b(Ka)-6 b(to.)24 b FE(Grund)t(lehr)l(en)j(der)g (mathematischen)h(Wissenschaften)f(Bd.)h(132:)38 b(Pertur-)555 1522 y(b)l(ation)33 b(The)l(ory)h(of)g(Line)l(ar)f(Op)l(er)l(ators.)e FF(Springer-V)-7 b(erlag,)29 b(Berlin,)i(Heidelb)r(erg,)g(New)555 1622 y(Y)-7 b(ork,)27 b(1)821 1586 y Ft(st)917 1622 y FF(Edition,)h(1966)227 1755 y([Li])207 b Fa(E.H.)55 b(Lieb.)e FE(Bound)g(on)f(the)h(Maximum)f(Ne)l(gative)h(Ionization)h(of)f(A)n (toms)f(and)555 1854 y(Mole)l(cules.)29 b FF(Ph)n(ys.)e(Rev.)h(A,)g FG(29)p FF(,)f(3018-3028,)c(1984)227 1987 y([Me])169 b Fa(E.)31 b(Merzba)n(cher.)d FE(Quantum)g(Me)l(chanics.)i FF(John)d(Wiley)h(&)g(Sons,)f(Inc.,)h(New)g(Y)-7 b(ork,)555 2087 y(1964)227 2220 y([RS1])133 b Fa(M.)27 b(Reed,)i(B.)e(Simon.)c FE(Metho)l(ds)28 b(of)f(Mo)l(dern)h(Mathematic)l(al)g(Physics)g(I:)f(F) -6 b(unctional)555 2331 y(A)n(nalysis.)29 b FF(Academic)e(Press,)f(San) i(Diego,)f(2)1995 2295 y Ft(nd)2114 2331 y FF(Edition,)h(1980)227 2464 y([RS2])133 b Fa(M.)36 b(Reed,)i(B.)f(Simon.)31 b FE(Metho)l(ds)36 b(of)f(Mo)l(dern)g(Mathematic)l(al)h(Physics)g(II:)f (F)-6 b(ourier)555 2563 y(A)n(nalysis,)31 b(Self-A)l(djointness.)d FF(Academic)g(Press,)e(New)i(Y)-7 b(ork,)27 b(1)2624 2527 y Ft(st)2720 2563 y FF(Edition,)h(1975)227 2696 y([RS4])133 b Fa(M.)29 b(Reed,)i(B.)e(Simon.)c FE(Metho)l(ds)k(of)g(Mo) l(dern)g(Mathematic)l(al)i(Physics)e(IV:)g(A)n(nalysis)555 2796 y(of)i(Op)l(er)l(ators.)d FF(Academic)g(Press,)e(San)h(Diego,)g(1) 2141 2760 y Ft(st)2238 2796 y FF(Edition,)g(1978)227 2928 y([Sk])192 b Fa(E.)44 b(Skibsted.)d FE(Sp)l(e)l(ctr)l(al)h(A)n (nalysis)g(of)h FB(N)9 b FE(-Bo)l(dy)43 b(Systems)e(Couple)l(d)i(to)f (a)h(Bosonic)555 3028 y(Field.)30 b FF(Rev.)d(Math.)h(Ph)n(ys.,)f FG(10)p FF(\(7\),)g(989-1026,)d(1997)227 3161 y([Zi])208 b Fa(G.M.)38 b(Zishlin.)33 b FE(Discussion)k(of)g(the)g(Sp)l(e)l(ctrum) e(of)i(the)g(Schr\177)-42 b(odinger)38 b(Op)l(er)l(ator)e(for)555 3261 y(Systems)29 b(of)i(Many)f(Particles.)g FF(T)-7 b(r.)27 b(Mosk.)g(Math.)h(O.)f(-v)-5 b(a,)27 b FG(9)p FF(,)h(81-120,)d(1960)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0105111133108--