Content-Type: multipart/mixed; boundary="-------------0106050327473" This is a multi-part message in MIME format. ---------------0106050327473 Content-Type: text/plain; name="01-207.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-207.keywords" quantum chaos, quantum maps ---------------0106050327473 Content-Type: application/postscript; name="notesamsf2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="notesamsf2.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: notesamsf2.dvi %%Pages: 58 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o notesamsf2.ps notesamsf2.dvi %DVIPSParameters: dpi=1200, compressed %DVIPSSource: TeX output 2001.06.05:0944 %%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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b(a)g(short)h(in)-5 b(tro)5 b(duction)62 b(to)g(Hamiltonian)758 8933 y(mec)-5 b(hanics,)43 b(explain)c(in)g(whic)-5 b(h)39 b(sense)g(a)g(billiard)h(\035o)-5 b(w)39 b(is)g(a)g(Hamiltonian)g(\035o)-5 b(w)39 b(and)g(sho)-5 b(w)40 b(that)758 9132 y(the)49 b(most)h(regular)g(Hamiltonian)g(\035o) -5 b(ws)51 b(p)5 b(ossible)51 b(are)e(the)g(so-called)i(completely)e (in)-5 b(tegrable)758 9332 y(ones.)74 b(I)53 b(will)h(\034nally)g(try)f (to)h(con)-5 b(vince)53 b(y)-5 b(ou)54 b(that)f(the)h(notion)g(of)f (unpredictabilit)-5 b(y)54 b(is)h(rather)758 9531 y(w)-5 b(ell)55 b(captured)g(in)h(the)f(de\034nition)h(of)f(\020mixing\021.) 1057 9804 y Fx(Exer)-5 b(cise)62 b FC(4.3)p Fx(.)83 b FC(Use)47 b(the)g(W)-14 b(eyl)46 b(la)-5 b(w)48 b(to)e(deriv)-5 b(e)47 b(an)h(asymptotic)f(expression)g(for)g(the)g Fw(\025)6645 9829 y Ft(n)758 10003 y FC(as)55 b(a)h(function)f(of)g Fw(n)p FC(.)74 b(Wh)-5 b(y)55 b(can)g(the)g(result)h(not)f(b)5 b(e)55 b(used)h(to)f(study)g Fw(P)23 b Fy(\()p Fw(s)p Fy(\))p FC(?)2653 10342 y FD(5.)96 b(Hamiltonian)62 b(mec)-5 b(hanics)1057 10641 y FC(I)77 b(hop)5 b(e)77 b(y)-5 b(ou)77 b(remem)-5 b(b)5 b(er)78 b(Newton's)e(second)i(la)-5 b(w)77 b(from)h(y)-5 b(our)77 b(high)h(sc)-5 b(ho)5 b(ol)78 b(da)-5 b(ys.)140 b(It)758 10840 y(sa)-5 b(ys)70 b(that)e(the)g(force)h (exp)5 b(erienced)68 b(b)-5 b(y)69 b(a)g(p)5 b(oin)-5 b(t)69 b(particle)g(is)g(equal)g(to)f(its)i(mass)g(times)f(its)p eop %%Page: 10 10 10 9 bop 758 878 a FA(10)2127 b(STEPHAN)58 b(DE)h(BI\310VRE)758 1259 y FC(acceleration:)2240 1593 y Fw(ma)46 b Fy(=)g Fw(F)5 b(;)248 b FC(or)222 b Fw(m)3637 1481 y(d)3723 1420 y Fv(2)3797 1481 y Fw(q)p 3637 1555 241 7 v 3647 1707 a(dt)3793 1659 y Fv(2)3897 1593 y Fy(\()p Fw(t)p Fy(\))46 b(=)g Fw(F)23 b Fy(\()p Fw(q)6 b Fy(\()p Fw(t)p Fy(\))p Fw(;)57 b Fy(_)-75 b Fw(q)t Fy(\()p Fw(t)p Fy(\)\))p Fw(;)-4468 b FC(\(5.1\))758 1928 y(where)79 b Fw(t)87 b Fu(2)f Fs(R)123 b Fu(7!)87 b Fw(q)6 b Fy(\()p Fw(t)p Fy(\))86 b Fu(2)h Fs(R)2755 1868 y Fv(3)2918 1928 y FC(is)80 b(the)g(particle)f(tra)9 b(jectory)78 b(and)i Fw(F)110 b Fy(:)87 b Fs(R)5664 1868 y Fv(3)5800 1928 y Fu(\002)54 b Fs(R)6106 1868 y Fv(3)6276 1928 y Fu(!)87 b Fs(R)6652 1868 y Fv(3)758 2128 y FC(the)54 b Fz(for)-8 b(c)g(e)p FC(.)73 b(F)-14 b(rom)56 b(a)e(ph)-5 b(ysical)56 b(p)5 b(oin)-5 b(t)55 b(of)f(view)g(\020force\021)68 b(is)56 b(a)e(v)-5 b(ery)54 b(complicated)h(concept,)f(but)758 2327 y(mathematically)-14 b(,)47 b(it)e(is)h(just)f(a)g(function)h (from)f Fs(R)3863 2267 y Fv(6)3992 2327 y FC(to)g Fs(R)4307 2267 y Fv(3)4391 2327 y FC(.)70 b(Similarly)-14 b(,)49 b(from)c(a)g(mathematical)758 2526 y(p)5 b(oin)-5 b(t)46 b(of)h(view,)h(Newton's)e(second)h(la)-5 b(w)47 b(is)g(just)g(a)f (second)h(order)g(di\033eren)-5 b(tial)47 b(equation.)70 b(One)758 2725 y(sa)-5 b(ys)64 b(the)e(force)h Fw(F)86 b FC(is)64 b Fz(c)-8 b(onservative)64 b FC(if)f(there)f(exists)i(a)f (function)g Fw(V)96 b Fy(:)59 b Fs(R)5416 2665 y Fv(3)5558 2725 y Fu(!)g Fs(R)27 b FC(,)75 b(called)63 b(the)758 2925 y Fz(p)-8 b(otential)p FC(,)66 b(so)e(that)f Fw(F)23 b Fy(\()p Fw(q)6 b(;)28 b(v)6 b Fy(\))59 b(=)h Fu(\000r)p Fw(V)37 b Fy(\()p Fw(q)6 b Fy(\))p FC(.)98 b(In)64 b(particular,)i(a)d (conserv)-9 b(ativ)k(e)63 b(force)g(do)5 b(es)63 b(not)758 3124 y(dep)5 b(end)54 b(on)h(the)f(v)-5 b(elo)5 b(cit)-5 b(y)53 b(of)h(the)g(particle.)74 b(This)55 b(terminology)f(comes)h (from)f(the)g(fact)g(that,)758 3323 y(in)h(that)g(case,)g(the)g Fz(ener)-8 b(gy)p FC(,)56 b(de\034ned)g(b)-5 b(y)2414 3661 y Fw(E)56 b Fy(:)46 b(\()p Fw(q)6 b(;)28 b(v)6 b Fy(\))46 b Fu(2)f Fs(R)3380 3592 y Fv(6)3510 3661 y Fu(7!)3742 3549 y Fy(1)p 3742 3623 84 7 v 3742 3775 a(2)3845 3661 y Fw(mv)4077 3592 y Fv(2)4189 3661 y Fy(+)37 b Fw(V)f Fy(\()p Fw(q)6 b Fy(\))46 b Fu(2)f Fs(R)27 b Fw(;)758 3976 y FC(is)56 b(a)f Fz(c)-8 b(onstant)60 b(of)f(the)h(motion)p FC(,)55 b Fz(i.e.)3064 4210 y Fw(d)p 3061 4284 147 7 v 3061 4436 a(dt)3228 4322 y(E)10 b Fy(\()p Fw(q)c Fy(\()p Fw(t)p Fy(\))p Fw(;)56 b Fy(_)-74 b Fw(q)s Fy(\()p Fw(t)p Fy(\)\))45 b(=)h(0)p Fw(:)-3693 b FC(\(5.2\))1057 4655 y Fx(Exer)-5 b(cise)62 b FC(5.1)p Fx(.)83 b FC(Pro)-5 b(v)g(e)64 b(that)g(ev)-5 b(ery)64 b(solution)h(of)f(\(5.1\))f (satis\034es)j(\(5.2\))d(if)h(the)g(force)g(is)758 4854 y(conserv)-9 b(ativ)k(e.)1057 5131 y(There)74 b(is)h(an)f(elegan)-5 b(t)75 b(w)-5 b(a)g(y)75 b(to)f(rewrite)f(\(5.1\),)79 b(called)74 b(the)g(Hamiltonian)h(formalism.)758 5344 y(In)-5 b(tro)5 b(ducing)60 b(the)f Fz(line)-8 b(ar)64 b(momentum)c Fw(p)p Fy(\()p Fw(t)p Fy(\))53 b(=)h Fw(m)3967 5270 y Ft(dq)p 3967 5306 135 7 v 3974 5401 a(dt)4121 5344 y Fy(\()p Fw(t)p Fy(\))p FC(,)60 b(it)g(is)g(clear)g(that)f(the)h (second)g(order)758 5543 y(di\033eren)-5 b(tial)55 b(equation)g (\(5.1\))g(is)g(equiv)-9 b(alen)k(t)55 b(to)g(the)g(set)g(of)g(coupled) h(\034rst)f(order)h(equations)2503 5908 y Fy(_)-77 b Fw(q)6 b Fy(\()p Fw(t)p Fy(\))45 b(=)2983 5795 y Fw(p)p Fy(\()p Fw(t)p Fy(\))p 2983 5870 273 7 v 3046 6022 a Fw(m)3275 5908 y(;)393 b Fy(_)-79 b Fw(p)p Fy(\()p Fw(t)p Fy(\))45 b(=)h Fu(\000r)p Fw(V)37 b Fy(\()p Fw(q)6 b Fy(\()p Fw(t)p Fy(\)\))p Fw(:)-4264 b FC(\(5.3\))758 6222 y(In)-5 b(tro)5 b(ducing)55 b(furthermore)g(the)g Fz(Hamiltonian)2473 6588 y Fw(H)k Fy(:)47 b(\()p Fw(q)6 b(;)28 b(p)p Fy(\))44 b Fu(2)i Fs(R)3455 6520 y Fv(6)3585 6588 y Fu(7!)3852 6476 y Fw(p)3936 6416 y Fv(2)p 3817 6550 229 7 v 3817 6702 a Fy(2)p Fw(m)4102 6588 y Fy(+)37 b Fw(V)g Fy(\()p Fw(q)6 b Fy(\))46 b Fu(2)f Fs(R)27 b Fw(;)-4225 b FC(\(5.4\))758 6913 y(equation)55 b(\(5.3\))f(can)h(b)5 b(e)55 b(rewritten)2044 7269 y Fy(_)-77 b Fw(q)6 b Fy(\()p Fw(t)p Fy(\))45 b(=)2524 7156 y Fw(@)9 b(H)p 2524 7230 249 7 v 2558 7383 a(@)g(p)2792 7269 y Fy(\()p Fw(q)d Fy(\()p Fw(t)p Fy(\))p Fw(;)28 b(p)p Fy(\()p Fw(t)p Fy(\)\))p Fw(;)224 b Fy(_)-79 b Fw(p)p Fy(\()p Fw(t)p Fy(\))45 b(=)h Fu(\000)4420 7156 y Fw(@)9 b(H)p 4420 7230 V 4456 7383 a(@)g(q)4689 7269 y Fy(\()p Fw(q)d Fy(\()p Fw(t)p Fy(\))p Fw(;)28 b(p)p Fy(\()p Fw(t)p Fy(\)\))p Fw(:)-4725 b FC(\(5.5\))758 7616 y(That)66 b(all)h(this)g(rewriting)g(is)g(of)f (an)-5 b(y)67 b(use)h(can)e(not)h(b)5 b(e)66 b(clear)h(to)f(y)-5 b(ou)67 b(y)-5 b(et,)69 b(but)e(there)f(is)h(an)758 7815 y(in)-5 b(triguing)59 b(elegance)f(to)h(it)f(that)g(can)g(b)5 b(e)59 b(felt)e(as)i(follo)-5 b(ws.)85 b(Let)58 b Fw(f)69 b Fy(:)52 b Fs(R)5248 7755 y Fv(6)5383 7815 y Fu(!)g Fs(R)94 b FC(and)59 b(compute)758 8014 y(\()p Fw(x)p Fy(\()p Fw(t)p Fy(\))44 b(=)j(\()p Fw(q)6 b Fy(\()p Fw(t)p Fy(\))p Fw(;)28 b(p)p Fy(\()p Fw(t)p Fy(\)\))p FC(\))1736 8257 y Fw(d)p 1734 8331 147 7 v 1734 8483 a(dt)1900 8370 y(f)18 b Fy(\()p Fw(x)p Fy(\()p Fw(t)p Fy(\)\))164 b(=)2893 8257 y Fw(@)9 b(f)p 2893 8331 197 7 v 2903 8483 a(@)g(q)3110 8370 y Fy(\()p Fw(x)p Fy(\()p Fw(t)p Fy(\)\))29 b(_)-75 b Fw(q)t Fy(\()p Fw(t)p Fy(\))36 b(+)4015 8257 y Fw(@)9 b(f)p 4015 8331 V 4023 8483 a(@)g(p)4232 8370 y Fy(\()p Fw(x)p Fy(\()p Fw(t)p Fy(\)\))30 b(_)-76 b Fw(p)m Fy(\()p Fw(t)p Fy(\))2578 8810 y(=)2893 8697 y Fw(@)9 b(f)p 2893 8772 V 2903 8924 a(@)g(q)3110 8810 y Fy(\()p Fw(x)p Fy(\()p Fw(t)p Fy(\)\))3545 8697 y Fw(@)g(H)p 3545 8772 249 7 v 3577 8924 a(@)g(p)3812 8810 y Fy(\()p Fw(x)p Fy(\()p Fw(t)p Fy(\)\))35 b Fu(\000)4448 8697 y Fw(@)9 b(f)p 4448 8772 197 7 v 4456 8924 a(@)g(p)4664 8810 y Fy(\()p Fw(x)p Fy(\()p Fw(t)p Fy(\)\))5099 8697 y Fw(@)g(H)p 5099 8772 249 7 v 5133 8924 a(@)g(q)5366 8810 y Fy(\()p Fw(x)p Fy(\()p Fw(t)p Fy(\)\))2578 9147 y(=)166 b Fu(f)p Fw(f)9 b(;)28 b(H)13 b Fu(g)p Fw(;)-2642 b FC(\(5.6\))758 9416 y(where)63 b(I)g(in)-5 b(tro)5 b(duced,)65 b(for)e(an)-5 b(y)64 b(pair)f(of)g(functions)h Fw(f)9 b(;)28 b(g)65 b Fu(2)59 b Fw(C)4718 9356 y Fr(1)4859 9416 y Fy(\()p Fs(R)5046 9356 y Fv(6)5130 9416 y Fy(\))p FC(,)64 b(the)f Fz(Poisson)j(br)-8 b(acket)758 9615 y FC(of)55 b Fw(f)73 b FC(and)56 b Fw(g)61 b Fz(via)2285 9961 y Fu(f)p Fw(f)9 b(;)28 b(g)6 b Fu(g)p Fy(\()p Fw(x)p Fy(\))45 b(=)3165 9849 y Fw(@)9 b(f)p 3165 9923 197 7 v 3175 10075 a(@)g(q)3382 9961 y Fy(\()p Fw(x)p Fy(\))3627 9849 y Fw(@)g(g)p 3627 9923 183 7 v 3628 10075 a(@)g(p)3828 9961 y Fy(\()p Fw(x)p Fy(\))36 b Fu(\000)4275 9849 y Fw(@)9 b(f)p 4275 9923 197 7 v 4283 10075 a(@)g(p)4491 9961 y Fy(\()p Fw(x)p Fy(\))4736 9849 y Fw(@)g(g)p 4736 9923 183 7 v 4739 10075 a(@)g(q)4938 9961 y Fy(\()p Fw(x)p Fy(\))p Fw(:)-4451 b FC(\(5.7\))758 10308 y(Note)54 b(that)h(I)g(use)h(the)f(shorthand) 2379 10581 y Fw(@)9 b(f)p 2379 10656 197 7 v 2389 10808 a(@)g(q)2596 10694 y Fy(\()p Fw(x)p Fy(\))2841 10581 y Fw(@)g(g)p 2841 10656 183 7 v 2842 10808 a(@)g(p)3042 10694 y Fy(\()p Fw(x)p Fy(\))45 b(=)3575 10486 y Fv(3)3488 10536 y Fp(X)3500 10890 y Ft(i)p Fv(=1)3790 10581 y Fw(@)9 b(f)p 3775 10656 227 7 v 3775 10808 a(@)g(q)3946 10833 y Ft(i)4022 10694 y Fy(\()p Fw(x)p Fy(\))4293 10581 y Fw(@)g(g)p 4267 10656 237 7 v 4267 10808 a(@)g(p)4448 10833 y Ft(i)4522 10694 y Fy(\()p Fw(x)p Fy(\))p Fw(;)82 b FC(etc.)p eop %%Page: 11 11 11 10 bop 2362 861 a FA(QUANTUM)58 b(CHA)l(OS:)g(A)h(BRIEF)f(FIRST)g (VISIT)1472 b(11)758 1259 y FC(and)50 b(that)f(the)g(ab)5 b(o)-5 b(v)g(e)50 b(de\034nition)g(immediately)g(generalizes)g(to)f Fs(R)4992 1199 y Fv(2)p Ft(n)5157 1259 y FC(.)72 b(The)50 b(P)-5 b(oisson)51 b(brac)-5 b(k)g(et)758 1458 y(then)76 b(de\034nes)i(a)f(pro)5 b(duct)77 b(on)g Fw(C)2912 1398 y Fr(1)3052 1458 y Fy(\()p Fs(R)3239 1398 y Fv(2)q Ft(n)3405 1458 y Fy(\))g FC(that)f(has)h(the)g(follo)-5 b(wing)77 b(v)-5 b(ery)77 b(imp)5 b(ortan)-5 b(t)77 b(and)758 1658 y(easily)55 b(c)-5 b(hec)g(k)g(ed)56 b(prop)5 b(erties:)3263 1982 y Fu(f)p Fw(f)k(;)28 b(g)6 b Fu(g)46 b Fy(=)g Fu(\000f)p Fw(g)6 b(;)28 b(f)18 b Fu(g)167 b FC(\(An)-5 b(ti-symmetry\))-5090 b(\(5.8\))1495 2231 y Fu(ff)p Fw(f)9 b(;)28 b(g)6 b Fu(g)p Fw(;)28 b(h)p Fu(g)36 b Fy(+)h Fu(ff)p Fw(g)6 b(;)28 b(h)p Fu(g)p Fw(;)g(f)18 b Fu(g)37 b Fy(+)g Fu(ff)p Fw(h;)28 b(f)18 b Fu(g)p Fw(;)28 b(g)6 b Fu(g)45 b Fy(=)i(0)166 b FC(\(Jacobi)55 b(iden)-5 b(tit)g(y\))-5089 b(\(5.9\))2836 2480 y Fu(f)p Fw(f)9 b(;)28 b(g)6 b(h)p Fu(g)46 b Fy(=)g Fu(f)p Fw(f)9 b(;)28 b(g)6 b Fu(g)p Fw(h)37 b Fy(+)g Fw(g)6 b Fu(f)p Fw(f)j(;)28 b(h)p Fu(g)165 b FC(\(Deriv)-9 b(ation\))-5090 b(\(5.10\))1314 2731 y Fu(f)p Fw(f)9 b(;)28 b(g)6 b Fu(g)46 b Fy(=)h(0)p Fw(;)28 b Fu(8)p Fw(g)52 b Fu(2)46 b Fw(C)2619 2663 y Fr(1)2759 2731 y Fy(\()p Fs(R)2946 2663 y Fv(2)q Ft(n)3112 2731 y Fy(\))g Fu(\))g Fw(f)73 b FC(is)56 b(a)g(constan)-5 b(t)55 b (\(Non-degeneracy\))-5090 b(\(5.11\))1057 3061 y Fx(Exer)-5 b(cise)62 b FC(5.2)p Fx(.)83 b FC(\(i\))55 b(Chec)-5 b(k,)55 b(through)g(direct)g(calculations,)h(the)f(ab)5 b(o)-5 b(v)g(e)55 b(prop)5 b(erties.)1057 3261 y(\(ii\))38 b(Let)h Fu(H)i FC(b)5 b(e)39 b(a)g(Hilb)5 b(ert)39 b(space)g(and)h(let) f Fu(L)p Fy(\()p Fu(H)r Fy(\))f FC(b)5 b(e)39 b(the)g(set)g(of)g(b)5 b(ounded)40 b(linear)g(op)5 b(erators)758 3460 y(on)55 b Fu(H)r FC(.)74 b(De\034ne)55 b(on)h Fu(L)p Fy(\()p Fu(H)r Fy(\))d FC(the)i(follo)-5 b(wing)56 b(pro)5 b(duct)55 b(\(called)g(the)g(comm)-5 b(utator\):)1599 3784 y Fy([)p Fu(\001)p Fw(;)28 b Fu(\001)p Fy(])46 b(:)h(\()p Fw(A;)28 b(B)8 b Fy(\))45 b Fu(2)g(L)p Fy(\()p Fu(H)r Fy(\))36 b Fu(\002)h(L)p Fy(\()p Fu(H)r Fy(\))44 b Fu(7!)j Fy([)p Fw(A;)28 b(B)8 b Fy(])46 b(=)g Fw(AB)f Fu(\000)37 b Fw(B)8 b(A)45 b Fu(2)h(L)p Fy(\()p Fu(H)r Fy(\))p Fw(:)758 4108 y FC(Pro)-5 b(v)g(e)58 b(that)f(the)g(comm)-5 b(utator)58 b(has)g(prop)5 b(erties)58 b(completely)f(analogous)h(to)g(the)f(four)g (prop-)758 4307 y(erties)e(listed)g(ab)5 b(o)-5 b(v)g(e)56 b(for)f(the)g(P)-5 b(oisson)57 b(brac)-5 b(k)g(et.)1057 4638 y(In)68 b(conclusion,)k(the)67 b(P)-5 b(oisson)70 b(brac)-5 b(k)g(et)68 b(has)g(some)h(nice)f(prop)5 b(erties)67 b(and)i(they)e(ha)-5 b(v)g(e)68 b(a)758 4837 y(surprising)57 b(similarit)-5 b(y)57 b(to)e(the)h(prop)5 b(erties)56 b(of)f(the)g(comm)-5 b(utator.)76 b(This)57 b(is)f(quite)f(in)-5 b(triguing,)758 5036 y(and)55 b(w)-5 b(e)56 b(will)f(see)h(in)f(the)g (next)g(sections)h(that)e(it)h(has)h(some)g(imp)5 b(ortan)-5 b(t)56 b(consequences.)1057 5235 y(F)-14 b(or)54 b(no)-5 b(w,)55 b(let)f(me)g(p)5 b(oin)-5 b(t)55 b(out)f(ho)-5 b(w)55 b(the)e(P)-5 b(oisson)56 b(brac)-5 b(k)g(et)54 b(can)h(b)5 b(e)53 b(used)i(to)f(de\034ne)h(some)758 5435 y(asso)5 b(ciated)64 b(structures.)99 b(Giv)-5 b(en)64 b(an)-5 b(y)64 b Fw(g)j Fu(2)60 b Fw(C)3683 5374 y Fr(1)3823 5435 y Fy(\()p Fs(R)4011 5374 y Fv(2)p Ft(n)4176 5435 y Fy(\))p FC(,)66 b(one)e(can)f(\034rst)i(of)e(all)h(construct)g(the) 758 5634 y Fz(Hamiltonian)c(ve)-8 b(ctor)60 b(\034eld)c Fw(X)2697 5659 y Ft(g)2829 5634 y FC(b)-5 b(y)1936 6050 y Fw(X)2074 6075 y Ft(g)2197 6050 y Fy(:)46 b Fw(x)g Fu(2)g Fs(R)2710 5982 y Fv(2)p Ft(n)2922 6050 y Fu(7!)g Fw(X)3272 6075 y Ft(g)3349 6050 y Fy(\()p Fw(x)p Fy(\))f(=)h(\()3879 5938 y Fw(@)9 b(g)p 3879 6012 183 7 v 3880 6164 a(@)g(p)4081 6050 y Fy(\()p Fw(x)p Fy(\))p Fw(;)28 b Fu(\000)4529 5938 y Fw(@)9 b(g)p 4529 6012 V 4532 6164 a(@)g(q)4731 6050 y Fy(\()p Fw(x)p Fy(\)\))44 b Fu(2)i Fs(R)5345 5982 y Fv(2)p Ft(n)5511 6050 y Fw(;)758 6452 y FC(or,)55 b(for)g(those)h (familiar)f(with)g(the)g(rudimen)-5 b(ts)57 b(of)e(di\033eren)-5 b(tial)56 b(geometry)-14 b(,)54 b(b)-5 b(y)2962 6854 y Fw(X)3100 6879 y Ft(g)3223 6854 y Fy(=)3419 6741 y Fw(@)9 b(g)p 3419 6816 V 3420 6968 a(@)g(p)3653 6741 y(@)p 3641 6816 178 7 v 3641 6968 a(@)g(q)3875 6854 y Fu(\000)4061 6741 y Fw(@)g(g)p 4061 6816 183 7 v 4064 6968 a(@)g(q)4298 6741 y(@)p 4284 6816 181 7 v 4284 6968 a(@)g(p)4484 6854 y(:)758 7256 y FC(One)55 b(can)g(then)h(consider)g (the)e(asso)5 b(ciated)56 b(system)f(of)g(\034rst)h(order)f(di\033eren) -5 b(tial)56 b(equations)1647 7657 y Fy(_)-75 b Fw(x)p Fy(\()p Fw(t)p Fy(\))45 b(=)i Fw(X)2262 7682 y Ft(g)2338 7657 y Fy(\()p Fw(x)p Fy(\()p Fw(t)p Fy(\)\))220 b FC(or)86 b Fy(_)-77 b Fw(q)6 b Fy(\()p Fw(t)p Fy(\))46 b(=)3687 7545 y Fw(@)9 b(g)p 3687 7619 183 7 v 3688 7771 a(@)g(p)3889 7657 y Fy(\()p Fw(x)p Fy(\()p Fw(t)p Fy(\)\))p Fw(;)225 b Fy(_)-79 b Fw(p)p Fy(\()p Fw(t)p Fy(\))45 b(=)h Fu(\000)5185 7545 y Fw(@)9 b(g)p 5185 7619 V 5188 7771 a(@)g(q)5388 7657 y Fy(\()p Fw(x)p Fy(\()p Fw(t)p Fy(\)\))p Fw(;)-5091 b 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y(solutions)e(for)f(all)h Fw(t)46 b Fu(2)g Fs(R)27 b FC(.)82 b(F)-14 b(or)54 b(precise)g(tec)-5 b(hnical)53 b(conditions)h(guaran)-5 b(teeing)54 b(this,)g(y)-5 b(ou)54 b(ma)-5 b(y)758 9719 y(consult)55 b([)p FD(AM)q FC(],)h(for)f(example.)1057 9918 y(W)-14 b(e)54 b(will)i(need)f(a)h (few)e(prop)5 b(erties)56 b(of)f(Hamiltonian)g(\035o)-5 b(ws)56 b(that)f(I)g(shall)i(not)e(pro)-5 b(v)g(e:)1057 10248 y Fx(Pr)g(oposition)61 b FC(5.3)p Fx(.)83 b FC(Let)54 b Fw(f)9 b(;)28 b(g)6 b(;)28 b(h)46 b Fu(2)g Fw(C)3521 10188 y Fr(1)3661 10248 y Fy(\()p Fs(R)3848 10188 y Fv(2)q Ft(n)4014 10248 y Fy(\))p FC(.)74 b(W)-14 b(e)54 b(ha)-5 b(v)g(e:)1057 10448 y(\(i\))54 b(the)h(Hamiltonian)h(\035o)-5 b(ws)56 b(preserv)-5 b(e)55 b(the)g(P)-5 b(oisson)57 b(brac)-5 b(k)g(et,)55 b Fz(i.e.)74 b FC(for)55 b(all)h Fw(t)46 b Fu(2)g Fs(R)27 b FC(,)2638 10772 y Fu(f)p Fw(f)55 b Fu(\016)37 b Fy(\010)3097 10692 y Ft(g)3097 10813 y(t)3174 10772 y Fw(;)28 b(h)36 b Fu(\016)h Fy(\010)3620 10692 y Ft(g)3620 10813 y(t)3697 10772 y Fu(g)47 b Fy(=)f Fu(f)p Fw(f)9 b(;)28 b(h)p Fu(g)36 b(\016)h Fy(\010)4704 10692 y Ft(g)4704 10813 y(t)4781 10772 y Fy(;)-4069 b FC(\(5.13\))p eop %%Page: 12 12 12 11 bop 758 878 a FA(12)2127 b(STEPHAN)58 b(DE)h(BI\310VRE)1057 1259 y FC(\(ii\))65 b(the)h(Hamiltonian)g(\035o)-5 b(ws)67 b(preserv)-5 b(e)66 b(the)g(v)-5 b(olume,)69 b Fz(i.e.)p FC(,)g(for)d(all)h Fw(t)d Fu(2)g Fs(R)102 b FC(and)66 b(for)g(all)758 1458 y(Borel)54 b(subsets)j Fw(B)64 b FC(of)55 b Fs(R)2275 1398 y Fv(2)p Ft(n)2440 1458 y FC(,)h(one)f(has) 3063 1617 y Fp(Z)3155 1994 y Ft(B)3297 1843 y Fw(dx)47 b Fy(=)3700 1617 y Fp(Z)3792 1994 y Fv(\010)3887 1938 y Fo(g)3887 2028 y(t)3957 1994 y Fv(\()p Ft(B)6 b Fv(\))4202 1843 y Fw(dx:)1057 2314 y FC(\(iii\))54 b(If)h Fu(f)p Fw(f)9 b(;)28 b(g)6 b Fu(g)46 b Fy(=)h(0)p FC(,)55 b(then,)g Fu(8)p Fw(s;)28 b(t)46 b Fu(2)g Fs(R)27 b Fw(;)h Fy(\010)3614 2235 y Ft(f)3614 2355 y(t)3746 2314 y Fu(\016)37 b Fy(\010)3986 2254 y Ft(g)3986 2355 y(s)4109 2314 y Fy(=)46 b(\010)4404 2254 y Ft(g)4404 2355 y(s)4518 2314 y Fu(\016)37 b Fy(\010)4758 2235 y Ft(f)4758 2355 y(t)4844 2314 y FC(.)1057 2636 y(It)51 b(is)i(v)-5 b(ery)51 b(instructiv)-5 b(e)52 b(to)g(compute)g(a) g(few)f(simple)i(Hamiltonian)g(\035o)-5 b(ws,)53 b(so)g(I)f(urge)g(y)-5 b(ou)758 2835 y(strongly)46 b(to)g(do)h(the)f(follo)-5 b(wing)47 b(exercise,)h(the)e(results)h(of)g(whic)-5 b(h)47 b(I)f(will)h(use)g(extensiv)-5 b(ely)46 b(later)758 3035 y(on.)1057 3356 y Fx(Exer)-5 b(cise)62 b FC(5.4)p Fx(.)83 b FC(\(i\))55 b(Let)f Fw(H)13 b Fy(\()p Fw(x)p Fy(\))46 b(=)3367 3282 y Ft(p)3435 3232 y Fn(2)p 3342 3318 185 7 v 3342 3413 a Fv(2)p Ft(m)3546 3356 y FC(.)74 b(Compute)55 b Fy(\010)4505 3296 y Ft(H)4505 3397 y(t)4686 3356 y FC(explicitly)-14 b(.)1057 3608 y(\(ii\))54 b(Compute)i Fy(\010)2173 3547 y Ft(H)2173 3649 y(t)2354 3608 y FC(for)f Fw(H)13 b Fy(\()p Fw(x)p Fy(\))46 b(=)3250 3534 y Ft(p)3318 3484 y Fn(2)p 3225 3570 V 3225 3665 a Fv(2)p Ft(m)3466 3608 y Fy(+)3652 3542 y Fv(1)p 3652 3569 67 7 v 3652 3665 a(2)3738 3608 y Fw(m!)3993 3547 y Fv(2)4067 3608 y Fw(q)4147 3547 y Fv(2)4222 3608 y Fw(;)83 b Fy(\()p Fw(x)46 b Fy(=)g(\()p Fw(q)6 b(;)28 b(p)p Fy(\))44 b Fu(2)i Fs(R)5424 3547 y Fv(2)5508 3608 y Fw(;)28 b(!)52 b(>)46 b Fy(0\))p FC(.)1057 3817 y(\(iii\))65 b(Let)h Fw(L)e Fy(:)g Fs(R)2109 3757 y Fv(6)2257 3817 y Fu(!)g Fs(R)2610 3757 y Fv(3)2760 3817 y FC(b)5 b(e)66 b(de\034ned)h(b)-5 b(y)66 b Fw(L)p Fy(\()p Fw(x)p Fy(\))e(=)g Fw(x)44 b Fu(^)g Fw(p)p FC(.)106 b(This)67 b(v)-5 b(ector)66 b(is)g(called)h(the) 758 4016 y Fz(angular)62 b(momentum)c FC(in)h(ph)-5 b(ysics)59 b(texts.)80 b(Compute)58 b Fu(f)p Fw(L)4398 4041 y Ft(i)4454 4016 y Fw(;)28 b(L)4641 4041 y Ft(j)4710 4016 y Fu(g)58 b FC(for)g Fw(i;)28 b(j)60 b Fy(=)51 b(1)p Fw(;)28 b Fy(2)p Fw(;)g Fy(3)p FC(.)81 b(Compute)758 4233 y(also)50 b Fy(\010)1205 4160 y Ft(L)1296 4177 y Fo(i)1205 4274 y Ft(t)1406 4233 y FC(and)g(in)-5 b(terpret)49 b(the)g(result)h (geometrically)-14 b(.)71 b(Explain)50 b(the)f(statemen)-5 b(t,)50 b(found)g(in)g(all)758 4433 y(ph)-5 b(ysics)63 b(texts)e(on)i(classical)g(mec)-5 b(hanics,)65 b(that)c(\020angular)i (momen)-5 b(tum)64 b(is)e(the)g(generator)g(of)758 4632 y(rotations.\021)1057 4831 y(\(iv\))54 b(Let)g Fw(g)6 b Fy(\()p Fw(x)p Fy(\))45 b(=)i Fw(q)c Fu(\001)37 b Fw(p)p FC(.)73 b(Compute)56 b Fy(\010)3446 4751 y Ft(g)3446 4872 y(t)3523 4831 y FC(.)1057 5030 y(\(v\))68 b(Let)g Fw(g)76 b FC(b)5 b(e)69 b(a)g(homogeneous)i(quadratic)e(p)5 b(olynomial)70 b(in)f(the)g(v)-9 b(ariables)70 b Fw(q)6 b(;)28 b(p)69 b Fu(2)g Fs(R)27 b FC(,)758 5230 y Fz(i.e.)p FC(:)2081 5505 y Fw(g)6 b Fy(\()p Fw(x)p Fy(\))45 b(=)2631 5393 y(1)p 2631 5467 84 7 v 2631 5619 a(2)2734 5505 y Fw(\013)q(q)2921 5437 y Fv(2)3033 5505 y Fy(+)37 b Fw(\014)9 b(q)d(p)36 b Fy(+)3688 5393 y(1)p 3688 5467 V 3688 5619 a(2)3791 5505 y Fw(\015)9 b(p)3970 5437 y Fv(2)4044 5505 y Fw(;)360 b Fy(\()p Fw(\013)q(;)28 b(\014)9 b(;)28 b(\015)54 b Fu(2)46 b Fs(R)26 b Fy(\))p Fw(:)758 5846 y FC(Sho)-5 b(w)56 b(that,)f(for)g(eac)-5 b(h)55 b Fw(t)p FC(,)h Fy(\010)2511 5766 y Ft(g)2511 5887 y(t)2643 5846 y FC(is)g(a)f(linear)h (map)g(on)f Fs(R)4132 5786 y Fv(2)4271 5846 y FC(giv)-5 b(en)56 b(explicitly)e(b)-5 b(y)2396 6254 y Fy(\010)2516 6174 y Ft(g)2516 6294 y(t)2640 6254 y Fy(=)46 b(exp)27 b Fw(B)8 b(t;)360 b(B)55 b Fy(=)4052 6020 y Fp(\022)4324 6152 y Fw(\014)310 b(\015)4257 6352 y Fu(\000)p Fw(\013)167 b Fu(\000)p Fw(\014)4974 6020 y Fp(\023)758 6667 y FC(and)55 b(that)g(det)p Fy(\010)1791 6588 y Ft(g)1791 6708 y(t)1914 6667 y Fy(=)46 b(1)p FC(.)74 b(Compare)56 b(this)f(to)g(what)g(y)-5 b(ou)56 b(found)g(in)f(\(i\)-\(iv\).)1057 6867 y(\(vi\))f(Compute)h Fy(\010)2214 6787 y Ft(p)2282 6804 y Fo(i)2214 6908 y Ft(t)2343 6867 y Fw(;)28 b Fy(\010)2537 6787 y Ft(q)2598 6804 y Fo(i)2537 6908 y Ft(t)2659 6867 y FC(,)55 b(for)g Fw(i)47 b Fy(=)f(1)28 b Fw(:)g(:)g(:)f(n)p FC(.)1057 7085 y(\(vii\))35 b(Let)h Fw(f)1762 7110 y Fv(1)1837 7085 y Fw(;)28 b(f)1992 7110 y Fv(2)2113 7085 y Fu(2)46 b Fw(C)2401 7025 y Fr(1)2541 7085 y Fy(\()p Fs(R)2728 7025 y Fv(2)q Ft(n)2894 7085 y Fy(\))p FC(.)67 b(Sho)-5 b(w)38 b(that)e Fy(\010)3953 7005 y Ft(f)4018 7022 y Fn(1)3953 7126 y Ft(t)4137 7085 y Fy(=)46 b(\010)4432 7005 y Ft(f)4497 7022 y Fn(2)4432 7126 y Ft(t)4607 7085 y FC(for)37 b(all)g Fw(t)46 b Fu(2)g Fs(R)73 b FC(i\033)36 b Fw(f)5747 7110 y Fv(1)5822 7085 y Fu(\000)p Fw(f)6032 7110 y Fv(2)6153 7085 y Fy(=)47 b Fw(c)f Fu(2)f Fs(R)758 7285 y FC(i\033)55 b Fw(X)1094 7310 y Ft(f)1159 7327 y Fn(1)1277 7285 y Fy(=)47 b Fw(X)1591 7310 y Ft(f)1656 7327 y Fn(2)1728 7285 y FC(.)1057 7606 y(F)-14 b(or)69 b(further)g(reference,)i(let)e(me)g(note)g(that)f(a)h(di\033eomorphism) i Fy(\010)e FC(of)g Fs(R)5841 7546 y Fv(2)p Ft(n)6075 7606 y FC(that)g(pre-)758 7805 y(serv)-5 b(es)73 b(the)g(P)-5 b(oisson)74 b(brac)-5 b(k)g(et)73 b(as)g(in)h(\(5.13\))e(is)h(called)g (a)g Fz(c)-8 b(anonic)g(al)76 b(tr)-8 b(ansformation)72 b FC(or)h(a)758 8005 y Fz(symple)-8 b(ctomorphism)p FC(.)131 b(The)75 b(linear)g(canonical)g(transformations)h(will)f(pla)-5 b(y)75 b(a)g(particular)758 8204 y(role)55 b(further)g(on)g(in)h(these) f(notes)h(and)f(are)g(the)g(sub)9 b(ject)56 b(of)f(the)g(follo)-5 b(wing)56 b(exercise.)1057 8526 y Fx(Exer)-5 b(cise)62 b FC(5.5)p Fx(.)83 b FC(\(i\))53 b(Let)g Fw(A)g FC(b)5 b(e)54 b(a)f(t)-5 b(w)g(o)54 b(b)-5 b(y)54 b(t)-5 b(w)g(o)54 b(matrix)g(and)g Fy(\010)5094 8551 y Ft(A)5256 8526 y FC(the)f(asso)5 b(ciated)54 b(linear)758 8725 y(map)66 b(of)f Fs(R)1459 8665 y Fv(2)1542 8725 y FC(.)105 b(Sho)-5 b(w)66 b(that)f Fy(\010)2631 8750 y Ft(A)2804 8725 y FC(is)h(a)g(canonical)f(transformation)h(if)f(and)h(only)g(if)f(det)p Fw(A)d Fy(=)h(1)758 8924 y FC(\()p Fz(i.e.)73 b FC(i\033)55 b Fw(A)46 b Fu(2)55 b FC(SL)p Fy(\(2)p Fw(;)28 b Fs(R)e Fy(\))p FC(\).)1057 9123 y(\(ii\))39 b(Sho)-5 b(w)41 b(that,)i(if)d(T)-14 b(r)p Fw(A)45 b(>)i Fy(2)p FC(,)c(and)e(if)f Fw(A)45 b Fu(2)h FC(SL)p Fy(\(2)p Fw(;)28 b Fs(R)e Fy(\))p FC(,)52 b(then)40 b(there)g(exists)g(a)g(homogeneous)758 9323 y(quadratic)56 b(p)5 b(olynomial)58 b Fw(g)63 b FC(so)58 b(that)e Fy(\010)3191 9348 y Ft(A)3348 9323 y Fy(=)50 b(\010)3647 9243 y Ft(g)3647 9367 y Fv(1)3724 9323 y FC(.)79 b FD(Hin)-5 b(t:)77 b FC(Diagonalize)57 b Fw(A)p FC(,)g(then)g(use)h(\(iv\))e(and)758 9522 y(\(v\))e(of)h (Exercise)g(5.4.)1057 9721 y(\(iii\))62 b(Sho)-5 b(w)64 b(that,)g(if)f(T)-14 b(r)p Fw(A)59 b(<)g Fu(\000)p Fy(2)p FC(,)65 b(suc)-5 b(h)64 b(a)f(p)5 b(olynomial)63 b(do)5 b(es)64 b(not)e(exist.)97 b FD(Hin)-5 b(t:)89 b FC(Sho)-5 b(w)758 9920 y(that,)54 b(if)i(T)-14 b(r)p Fw(B)54 b Fy(=)46 b(0)p FC(,)56 b(then)f(T)-14 b(r)55 b Fy(exp)28 b Fw(B)54 b Fu(\025)46 b(\000)p Fy(2)p FC(.)1057 10120 y(\(iv\))54 b(What)g(happ)5 b(ens)57 b(if)e Fu(\000)p Fy(2)46 b Fu(\024)102 b FC(T)-14 b(r)55 b Fw(A)46 b Fu(\024)g Fy(2)p FC(?)1057 10441 y(Hamiltonians)59 b(in)h(ph)-5 b(ysical)60 b(problems)h(do)e(not)g(alw)-5 b(a)g(ys)60 b(ha)-5 b(v)g(e)60 b(the)f(form)g(\(5.4\))f(b)5 b(ecause)758 10641 y(forces)61 b(are)h(not)g(alw)-5 b(a)g(ys)62 b(of)g(the)f(form)h Fw(F)80 b Fy(=)57 b Fu(\000r)p Fw(V)37 b FC(.)94 b(An)62 b(imp)5 b(ortan)-5 b(t)61 b(exception)g(is)i(the)e(case)758 10840 y(of)70 b(a)g(particle)g(of)h(mass)g Fw(m)g FC(and)g(c)-5 b(harge)70 b Fw(e)h FC(mo)-5 b(ving)71 b(in)g(a)g Fz(magnetic)j (\034eld)d Fw(B)80 b Fy(:)72 b Fs(R)6090 10780 y Fv(3)6245 10840 y Fu(!)g Fs(R)6606 10780 y Fv(3)6689 10840 y FC(.)p eop %%Page: 13 13 13 12 bop 2362 861 a FA(QUANTUM)58 b(CHA)l(OS:)g(A)h(BRIEF)f(FIRST)g (VISIT)1472 b(13)758 1259 y FC(Newton's)55 b(second)g(la)-5 b(w)56 b(then)f(reads)2860 1535 y Fw(m)12 b Fy(\177)-95 b Fw(q)5 b Fy(\()p Fw(t)p Fy(\))46 b(=)g Fw(e)31 b Fy(_)-77 b Fw(q)6 b Fy(\()p Fw(t)p Fy(\))36 b Fu(^)h Fw(B)8 b Fy(\()p Fw(q)e Fy(\()p Fw(t)p Fy(\)\))p Fw(;)-3849 b FC(\(5.14\))758 1820 y(where)51 b(the)g(force)g Fw(F)23 b Fy(\()p Fw(q)6 b(;)28 b(v)6 b Fy(\))45 b(=)i Fw(ev)36 b Fu(^)29 b Fw(B)8 b Fy(\()p Fw(q)e Fy(\))51 b FC(is)h(called)f(the)g (Loren)-5 b(tz)51 b(force.)72 b(The)52 b(question)f(arises)758 2019 y(whether)57 b(this)i(second)g(order)f(equation)g(can)g(b)5 b(e)58 b(p)5 b(oured)59 b(in)-5 b(to)58 b(Hamiltonian)h(form.)83 b(This)59 b(is)758 2219 y(indeed)70 b(the)g(case.)120 b(First,)74 b(w)-5 b(e)71 b(need)f(to)g(recall)h(that)e (electromagnetic)h(theory)f(teac)-5 b(hes)71 b(us)758 2418 y(that)56 b(a)h(magnetic)g(\034eld)g(is)g(a)g(div)-5 b(ergence)57 b(free)f(v)-5 b(ector)56 b(\034eld)i(\()p Fz(i.e.)78 b FC(div)p Fw(B)57 b Fy(=)49 b(0)p FC(\))56 b(so)i(that)e(there)758 2617 y(exists)62 b(a)g(v)-5 b(ector)61 b(\034eld)i Fw(A)57 b Fy(:)g Fs(R)2648 2557 y Fv(3)2789 2617 y Fu(!)h Fs(R)3136 2557 y Fv(3)3281 2617 y FC(for)k(whic)-5 b(h)63 b Fw(B)j Fy(=)57 b Fw(cur)5 b(l)s(A)57 b Fy(=)g Fu(r)42 b(^)f Fw(A)p FC(.)94 b Fw(A)62 b FC(is)h(called)f(the)758 2816 y Fz(ve)-8 b(ctor)61 b(p)-8 b(otential)p FC(.)78 b(It)57 b(then)f(follo)-5 b(ws)58 b(from)e(a)h(straigh)-5 b(tforw)g(ard)58 b(and)f(instructiv)-5 b(e)56 b(calculation)758 3016 y(that,)e(if)i Fw(x)p Fy(\()p Fw(t)p Fy(\))45 b(=)h(\()p Fw(q)6 b Fy(\()p Fw(t)p Fy(\))p Fw(;)28 b(p)p Fy(\()p Fw(t)p Fy(\)\))52 b FC(is)k(a)f(\035o)-5 b(w)56 b(line)g(of)f(the)g (Hamiltonian)g(\035o)-5 b(w)56 b(generated)f(b)-5 b(y)2954 3397 y Fw(H)13 b Fy(\()p Fw(x)p Fy(\))46 b(=)3571 3285 y(\()p Fw(p)36 b Fu(\000)h Fw(eA)p Fy(\()p Fw(q)6 b Fy(\)\))4399 3225 y Fv(2)p 3571 3359 902 7 v 3907 3511 a Fy(2)p Fw(m)4492 3397 y(;)758 3729 y FC(then)55 b Fw(q)6 b Fy(\()p Fw(t)p Fy(\))54 b FC(satis\034es)j(\(5.14\).)1057 4031 y Fx(Exer)-5 b(cise)62 b FC(5.6)p Fx(.)83 b FC(\(i\))55 b(Pro)-5 b(v)g(e)55 b(this)h(last)f(statemen)-5 b(t.)1057 4231 y(\(ii\)*)59 b(Let)f Fw(A)p Fy(\()p Fw(x)p Fy(\))52 b(=)i(\(0)p Fw(;)28 b(B)8 b(q)2737 4256 y Fv(1)2811 4231 y Fw(;)28 b Fy(0\))59 b FC(for)h(some)g Fw(B)i(>)53 b Fy(0)p FC(.)87 b(Sho)-5 b(w)61 b Fw(B)8 b Fy(\()p Fw(q)e Fy(\))52 b(=)i(\(0)p Fw(;)28 b Fy(0)p Fw(;)g(B)8 b Fy(\))59 b FC(and)h(pro)-5 b(v)g(e)758 4430 y(that)54 b(the)h(solutions)i(of)e(\(5.14\))f(describ) 5 b(e)56 b(a)f(helicoidal)h(motion)f(along)h(the)f(v)-5 b(ertical)55 b(axis.)1057 4732 y(A)68 b(crucial)i(notion)f(in)h(all)f (what)g(follo)-5 b(ws)70 b(is)g(the)f(one)h(of)f Fz(c)-8 b(onstant)72 b(of)g(the)h(motion)p FC(.)116 b(A)758 4932 y(function)56 b Fw(f)75 b FC(is)58 b(said)f(to)g(b)5 b(e)56 b(a)h(constan)-5 b(t)57 b(of)f(the)h(motion)g(for)f(a)h (Hamiltonian)g(\035o)-5 b(w)57 b Fy(\010)6180 4852 y Ft(g)6180 4973 y(t)6314 4932 y FC(if)g(and)758 5131 y(only)e(if)2524 5302 y Fw(d)p 2521 5376 147 7 v 2521 5528 a(dt)2688 5414 y(f)g Fu(\016)37 b Fy(\010)3064 5334 y Ft(g)3064 5455 y(t)3141 5414 y Fy(\()p Fw(x)p Fy(\))45 b(=)h(0)p Fw(;)83 b Fu(8)p Fw(t)47 b Fu(2)e Fs(R)27 b Fw(;)h(x)55 b Fu(2)46 b Fs(R)4780 5345 y Fv(2)p Ft(n)4945 5414 y Fw(:)758 5700 y FC(Since,)55 b(as)h(w)-5 b(e)55 b(already)h(c)-5 b(hec)g(k)g(ed,)2707 5941 y Fw(d)p 2704 6015 V 2704 6167 a(dt)2871 6053 y(f)55 b Fu(\016)37 b Fy(\010)3247 5973 y Ft(g)3247 6094 y(t)3324 6053 y Fy(\()p Fw(x)p Fy(\))45 b(=)h Fu(f)p Fw(f)9 b(;)28 b(g)6 b Fu(g)p Fy(\(\010)4369 5973 y Ft(g)4369 6094 y(t)4446 6053 y Fy(\()p Fw(x)p Fy(\)\))p Fw(;)-4024 b FC(\(5.15\))758 6390 y Fw(f)73 b FC(is)56 b(a)f(constan)-5 b(t)56 b(of)f(the)g(motion)g (for)h Fy(\010)3299 6310 y Ft(g)3299 6431 y(t)3431 6390 y FC(i\033)f(it)g(has)h(v)-9 b(anishing)57 b(P)-5 b(oisson)57 b(brac)-5 b(k)g(et)55 b(with)g Fw(g)6 b FC(.)1057 6692 y Fx(Exer)-5 b(cise)62 b FC(5.7)p Fx(.)83 b FC(Sho)-5 b(w)54 b(that)f(the)g(set)h(of)f(constan)-5 b(ts)55 b(of)e(the)g (motion)h(of)f(a)h(giv)-5 b(en)54 b(Hamil-)758 6892 y(tonian)65 b(\035o)-5 b(w)66 b Fy(\010)1760 6812 y Ft(g)1760 6932 y(t)1902 6892 y FC(is)g(closed)g(under)f(the)g(P)-5 b(oisson)67 b(brac)-5 b(k)g(et)65 b(\()p Fz(i.e.)103 b FC(if)65 b Fw(f)5200 6917 y Fv(1)5275 6892 y Fw(;)28 b(f)5430 6917 y Fv(2)5570 6892 y FC(are)65 b(constan)-5 b(ts)66 b(of)758 7091 y(the)55 b(motion,)g(then)g(so)h(is)g Fu(f)p Fw(f)2565 7116 y Fv(1)2640 7091 y Fw(;)28 b(f)2795 7116 y Fv(2)2869 7091 y Fu(g)p FC(\).)1057 7393 y(Note)59 b(that,)i(clearly)-14 b(,)62 b Fu(f)p Fw(g)6 b(;)28 b(g)6 b Fu(g)55 b Fy(=)g(0)60 b FC(so)h(that)f Fw(g)67 b FC(is)61 b(alw)-5 b(a)g(ys)61 b(a)g(constan)-5 b(t)60 b(of)h(the)f(motion)h(for)758 7593 y(the)55 b(\035o)-5 b(w)55 b(it)g(generates.)74 b(As)55 b(a)h(result,)f(the)g(lev)-5 b(el)55 b(surfaces)h(of)f Fw(g)6 b FC(:)2500 7868 y Fy(\006)2620 7893 y Ft(c)2733 7868 y Fy(=)47 b Fu(f)p Fw(x)f Fu(2)g Fs(R)3413 7800 y Fv(2)p Ft(n)3624 7868 y Fu(j)g Fw(g)6 b Fy(\()p Fw(x)p Fy(\))46 b(=)g Fw(c)p Fu(g)55 b Fy(\()p Fw(c)46 b Fu(2)g Fs(R)27 b Fy(\))758 8144 y FC(are)43 b(in)-5 b(v)c(arian)k(t)44 b(surfaces)h(for)e(the)g(\035o)-5 b(w)45 b Fy(\010)3293 8064 y Ft(g)3293 8185 y(t)3370 8144 y FC(.)70 b(In)44 b(particular,)i(giv)-5 b(en)44 b(an)g(initial)g(p)5 b(oin)-5 b(t)44 b Fw(x)f FC(in)h Fz(phase)758 8343 y(sp)-8 b(ac)g(e)67 b Fs(R)1322 8283 y Fv(2)q Ft(n)1488 8343 y FC(,)72 b(the)c(curv)-5 b(e)68 b Fw(t)g Fu(7!)g Fy(\010)2843 8263 y Ft(g)2843 8384 y(t)2921 8343 y Fy(\()p Fw(x)p Fy(\))f FC(do)5 b(es)69 b(not)f(w)-5 b(ander)69 b(throughout)g(all)f(of)h(phase)g(space,)758 8542 y(but)60 b(sta)-5 b(ys)62 b(on)f(the)f Fy(2)p Fw(n)40 b Fu(\000)h Fy(1)p FC(-dimensional)63 b(surface)e Fy(\006)4153 8567 y Ft(c)4281 8542 y FC(with)f Fw(c)55 b Fy(=)h Fw(g)6 b Fy(\()p Fw(x)p Fy(\))p FC(.)89 b(This)62 b(constitutes)e(a)758 8741 y(simpli\034cation)g(with)f(resp)5 b(ect)58 b(to)h(the)g(case)g (of)g(an)g(arbitrary)g(\(non-Hamiltonian\))g(\035o)-5 b(w,)61 b(for)758 8941 y(whic)-5 b(h)69 b(suc)-5 b(h)69 b(a)g(priori)f(information)h(is)g(not)f(a)-5 b(v)c(ailable.)114 b(The)68 b(situation)g(further)g(simpli\034es)758 9140 y(if)c(there)f(are)h(sev)-5 b(eral)64 b(constan)-5 b(ts)65 b(of)f(the)f(motion)i Fw(f)4081 9165 y Fv(1)4155 9140 y Fw(;)28 b(f)4310 9165 y Fv(2)4385 9140 y Fw(;)g(:)g(:)g(:)f(f)4761 9165 y Ft(k)4843 9140 y FC(,)66 b(since)f(the)f(motion)g(is)g(then)758 9339 y(con\034ned)55 b(to)g(one)h(of)f(the)g(surfaces)1629 9615 y Fy(\006)1749 9640 y Ft(c)1863 9615 y Fy(=)46 b Fu(f)p Fw(x)g Fu(2)g Fs(R)2542 9546 y Fv(2)p Ft(n)2754 9615 y Fu(j)g Fw(f)2927 9640 y Ft(i)2982 9615 y Fy(\()p Fw(x)p Fy(\))g(=)g Fw(c)3500 9640 y Ft(i)3555 9615 y Fw(;)28 b(i)46 b Fy(=)g(1)28 b Fw(:)g(:)g(:)g(k)5 b Fu(g)46 b(\021)h Fw(F)4766 9546 y Fr(\000)p Fv(1)4944 9615 y Fy(\()p Fw(c)p Fy(\))54 b(\()p Fw(c)46 b Fu(2)f Fs(R)5662 9546 y Ft(k)5753 9615 y Fy(\))p Fw(:)758 9890 y FC(whic)-5 b(h)56 b(are)f Fy(2)p Fw(n)36 b Fu(\000)h Fw(k)61 b FC(dimensional)c (smo)5 b(oth)56 b(surfaces)g(if)f(the)g(Jacobian)h(of)f(the)g(map)2099 10166 y Fw(F)69 b Fy(:)46 b Fw(x)g Fu(2)g Fs(R)2787 10097 y Fv(2)q Ft(n)2999 10166 y Fu(!)g Fw(F)23 b Fy(\()p Fw(x)p Fy(\))45 b(=)i(\()p Fw(f)3933 10191 y Fv(1)4007 10166 y Fy(\()p Fw(x)p Fy(\))p Fw(;)28 b(:)g(:)g(:)e(f)4607 10191 y Ft(k)4689 10166 y Fy(\()p Fw(x)p Fy(\)\))45 b Fu(2)h Fs(R)5304 10097 y Ft(k)758 10441 y FC(has)61 b(rank)f Fw(k)5 b FC(.)89 b(This)61 b(reduces)g(the)e(set)i(of)f(di\033eren)-5 b(tial)60 b(equations)g(determining)h(the)f(\035o)-5 b(w)60 b(ef-)758 10641 y(fectiv)-5 b(ely)63 b(from)h(a)h(problem)g(in)g Fy(2)p Fw(n)f FC(v)-9 b(ariables)64 b(to)g(one)h(in)g Fy(2)p Fw(n)42 b Fu(\000)i Fw(k)70 b FC(v)-9 b(ariables.)101 b(One)65 b(ma)-5 b(y)65 b(b)5 b(e)758 10840 y(tempted)42 b(to)g(think)g(that)g(the)h(optimal)g(case)f(is)i(the)e(one)h(where)f Fw(k)52 b Fy(=)46 b(2)p Fw(n)12 b Fu(\000)g Fy(1)42 b FC(\(Wh)-5 b(y?\),)44 b(but)f(this)p eop %%Page: 14 14 14 13 bop 758 878 a FA(14)2127 b(STEPHAN)58 b(DE)h(BI\310VRE)758 1259 y FC(is)j(asking)g(a)f(bit)g(to)5 b(o)61 b(m)-5 b(uc)g(h.)94 b(There)61 b(are)g(only)g(a)h(few)f(in)-5 b(teresting)61 b(Hamiltonians)h(in)g(appli-)758 1458 y(cations)d(that)f(ha)-5 b(v)g(e)60 b(so)f(man)-5 b(y)60 b(constan)-5 b(ts)60 b(of)f(the)f(motion.)86 b(The)59 b(most)g(in)-5 b(teresting)60 b(systems)758 1658 y(ha)-5 b(ving)70 b(a)g(large)g(n)-5 b(um)g(b)5 b(er)72 b(of)d(constan)-5 b(ts)71 b(of)f(the)f(motion)i(turn)f(out)g(to)f(b)5 b(e)70 b(the)g Fz(c)-8 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v 3313 5260 a Fw(\026)3413 5285 y Ft(E)3525 5260 y Fy(\()p Fw(B)8 b Fy(\))4197 5146 y(=)46 b Fw(\026)4472 5171 y Ft(E)4584 5146 y Fy(\()p Fw(A)p Fy(\))p Fw(:)758 5485 y FC(In)62 b(fact,)g(this)h(de\034nition)f (can)g(b)5 b(e)61 b(found)i(in)f(all)g(b)5 b(o)g(oks)61 b(on)h(dynamical)h(systems)f(under)h(the)758 5685 y(term)55 b Fz(mixing)p FC(:)1057 5994 y Fx(Definition)61 b FC(9.2)p Fx(.)83 b FC(The)63 b(Hamiltonian)h(\035o)-5 b(w)64 b Fy(\010)4130 5934 y Ft(H)4130 6035 y(t)4319 5994 y FC(is)g(said)g(to)f (b)5 b(e)62 b Fz(mixing)i FC(with)f(resp)5 b(ect)758 6193 y(to)55 b(the)f(Liouville)i(measure)g Fw(\026)2674 6218 y Ft(E)2841 6193 y FC(on)g Fy(\006)3192 6218 y Ft(E)3359 6193 y FC(i\033)f(for)g(all)h(measurable)g(sets)g Fw(A;)28 b(B)54 b Fu(\032)46 b Fy(\006)5909 6218 y Ft(E)6076 6193 y FC(one)56 b(has)2433 6481 y Fy(lim)2391 6580 y Ft(t)p Fr(!1)2733 6481 y Fw(\026)2833 6506 y Ft(E)2946 6481 y Fy(\(\010)3131 6412 y Ft(H)3131 6522 y(t)3256 6481 y Fy(\()p Fw(B)8 b Fy(\))36 b Fu(\\)h 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y(its)61 b(\034lamen)-5 b(ts)63 b(end)f(up)g(wrapping)g(all)g(around)g Fy(\006)3935 8855 y Ft(E)4047 8830 y FC(.)92 b(No)-5 b(w)62 b Fz(if)f FC(the)g(motion)h(is)g(exp)5 b(onen)-5 b(tially)758 9030 y(unstable,)81 b(then)75 b(the)f(set)h Fw(B)84 b FC(will)75 b(b)5 b(e)75 b(stretc)-5 b(hed)75 b(exp)5 b(onen)-5 b(tially)75 b(fast)g(in)g(some)h(directions)758 9229 y(at)71 b(least)i(and)f(so)h (there)e(is)i(hop)5 b(e)72 b(that)g(the)g(required)g(spreading)h(happ)5 b(ens)73 b(exp)5 b(onen)-5 b(tially)758 9428 y(fast.)86 b(Although)59 b(this)h(in)-5 b(tuition)60 b(is)g(partially)f(corrob)5 b(orated)58 b(b)-5 b(y)60 b(theorems,)h(I)e(should)i(w)-5 b(arn)758 9627 y(y)g(ou)63 b(that)f(the)h(link)f(b)5 b(et)-5 b(w)g(een)63 b(exp)5 b(onen)-5 b(tial)63 b(instabilit)-5 b(y)63 b(\(for)f(whic)-5 b(h)64 b(the)e(tec)-5 b(hnical)63 b(term)g(is)758 9827 y(h)-5 b(yp)5 b(erb)g(olicit)-5 b(y\))57 b(and)i(exp)5 b(onen)-5 b(tial)58 b(mixing)g(is)h (considerably)g(more)f(subtle)h(than)f(the)g(ab)5 b(o)-5 b(v)g(e)758 10026 y(discussion)57 b(ma)-5 b(y)56 b(lead)f(y)-5 b(ou)56 b(to)f(b)5 b(eliev)-5 b(e)54 b(\(see)h([)p FD(Ba1)q FC(][)p FD(Ba2)p FC(][)p FD(D)p FC(])h(for)f(details\).)1057 10225 y(De\034nition)h(9.2)h(can)g(b)5 b(e)57 b(rewritten)e(in)j(terms) f(of)f(the)h(c)-5 b(haracteristic)56 b(functions)h Fw(\037)6335 10250 y Ft(A)6443 10225 y Fw(;)28 b(\037)6621 10250 y Ft(B)758 10424 y FC(of)55 b Fw(A)g FC(and)g Fw(B)64 b FC(as)56 b(follo)-5 b(ws:)1536 10755 y Fy(lim)1494 10855 y Ft(t)p Fr(!1)1837 10529 y Fp(Z)1929 10906 y Fq(T)2008 10873 y Fn(2)2107 10755 y Fw(\037)2211 10780 y Ft(B)2362 10755 y Fu(\016)37 b Fy(\010)2602 10686 y Ft(H)2602 10796 y(t)2728 10755 y Fy(\()p Fw(x)p Fy(\))p Fw(\037)3057 10780 y Ft(A)3164 10755 y Fy(\()p Fw(x)p Fy(\))p Fw(d\026)3575 10780 y Ft(E)3733 10755 y Fy(=)3908 10529 y Fp(Z)4001 10906 y Fq(T)4080 10873 y Fn(2)4179 10755 y Fw(\037)4283 10780 y Ft(B)4397 10755 y Fy(\()p Fw(x)p Fy(\))p Fw(d\026)4808 10780 y Ft(E)4947 10529 y Fp(Z)5040 10906 y Fq(T)5119 10873 y Fn(2)5218 10755 y Fw(\037)5322 10780 y Ft(A)5430 10755 y Fy(\()p Fw(x)p Fy(\))p Fw(d\026)5841 10780 y Ft(E)5953 10755 y Fw(:)p eop %%Page: 23 23 23 22 bop 2362 861 a FA(QUANTUM)58 b(CHA)l(OS:)g(A)h(BRIEF)f(FIRST)g (VISIT)1472 b(23)758 1259 y FC(A)55 b(simple)h(densit)-5 b(y)56 b(argumen)-5 b(t)55 b(then)h(sho)-5 b(ws)56 b(that)f(mixing)h (is)g(equiv)-9 b(alen)k(t)54 b(to:)920 1838 y Fu(8)p Fw(f)9 b(;)28 b(g)52 b Fu(2)46 b Fw(L)1577 1770 y Fv(2)1651 1838 y Fy(\(\006)1836 1863 y Ft(E)1948 1838 y Fw(;)28 b(d\026)2208 1863 y Ft(E)2321 1838 y Fy(\))p Fw(;)69 b Fy(lim)2460 1938 y Ft(t)p Fr(!1)2801 1612 y Fp(Z)2894 1989 y Fq(T)2973 1956 y Fn(2)3044 1838 y Fy(\()p Fw(f)55 b Fu(\016)37 b Fy(\010)3485 1770 y Ft(H)3485 1879 y(t)3611 1838 y Fy(\)\()p Fw(x)p Fy(\))p Fw(g)6 b Fy(\()p Fw(x)p Fy(\))p Fw(d\026)4397 1863 y Ft(E)4553 1838 y Fy(=)4729 1612 y Fp(Z)4821 1989 y Fq(T)4900 1956 y Fn(2)4999 1838 y Fw(f)18 b Fy(\()p Fw(x)p Fy(\))p Fw(d\026)5509 1863 y Ft(E)5649 1612 y Fp(Z)5741 1989 y Fq(T)5820 1956 y Fn(2)5919 1838 y Fw(g)6 b Fy(\()p Fw(x)p Fy(\))p Fw(d\026)6415 1863 y Ft(E)6527 1838 y Fw(:)758 1527 y FC(\(9.2\))758 2225 y(Requiring)55 b(a)h(\035o)-5 b(w)55 b(to)g(b)5 b(e)55 b(\(exp)5 b(onen)-5 b(tially\))54 b(mixing)i(is)g(a)f(v)-5 b(ery)55 b(strong)h(condition)f(and)h(there)758 2425 y(are)f(not)h(man)-5 b(y)56 b(Hamiltonian)g(systems)h(that)e(ha)-5 b(v)g(e)56 b(b)5 b(een)56 b(pro)-5 b(v)g(en)56 b(to)g(b)5 b(e)55 b(mixing.)76 b(F)-14 b(or)57 b(some)758 2624 y(billiard)g(\035o) -5 b(ws,)58 b(and)f(in)g(particular)f(for)h(the)f(Sinai)h(billiard,)h (this)f(is)g(kno)-5 b(wn)57 b(to)f(b)5 b(e)56 b(the)g(case,)758 2823 y(but)68 b(the)g(pro)5 b(of)68 b(is)g(far)g(from)h(easy)-14 b(.)112 b(In)69 b(fact,)h(it)e(is)h(to)f(some)h(exten)-5 b(t)67 b(the)h(absence)g(of)g(large)758 3022 y(classes)61 b(of)e(examples)i(of)e(Hamiltonian)h(\035o)-5 b(ws)61 b(that)f(are)f(\(exp)5 b(onen)-5 b(tially\))59 b(mixing)h(that)g(has) 758 3222 y(spurned)f(the)f(in)-5 b(terest)58 b(in)g(the)g(study)g(of)g (discrete)g(Hamiltonian)g(dynamical)g(systems)h(\(and)758 3421 y(their)c(quan)-5 b(tum)56 b(equiv)-9 b(alen)k(ts\),)54 b(to)h(whic)-5 b(h)56 b(I)f(will)h(come)f(in)h(the)f(next)f(section.) 1057 3734 y Fx(Exer)-5 b(cise)62 b FC(9.3)p Fx(.)83 b FC(*)40 b(Sho)-5 b(w)42 b(that)d(a)i(Hamiltonian)f(system)h(that)f(is)h (mixing)g(on)f Fy(\006)6053 3759 y Ft(E)6206 3734 y FC(can)g(not)758 3933 y(ha)-5 b(v)g(e)44 b(a)h(smo)5 b(oth)44 b(constan)-5 b(t)44 b(of)g(the)g(motion)h(\(functionally)e(indep)5 b(enden)-5 b(t)45 b(of)f Fw(H)57 b FC(itself)13 b(\).)70 b FD(Hin)-5 b(t:)758 4133 y FC(Y)-14 b(ou)66 b(could)h(sho)-5 b(w)67 b(that)f(if)g(the)g(\035o)-5 b(w)67 b(is)g(mixing)g(on)f Fy(\006)4273 4158 y Ft(E)4385 4133 y FC(,)k(then)c(an)-5 b(y)66 b(measurable)i(in)-5 b(v)c(arian)k(t)758 4332 y(subset)78 b(of)g Fy(\006)1621 4357 y Ft(E)1810 4332 y FC(is)h(of)e(Liouville)h(measure)g Fy(0)g FC(or)f Fy(1)p FC(.)142 b(W)-14 b(ould)78 b(y)-5 b(ou)78 b(agree)f(to)g(claim)h(that)f (a)758 4531 y(Hamiltonian)55 b(system)h(is)g(more)f(predictable)g(if)g (it)g(has)i(more)e(constan)-5 b(ts)56 b(of)f(the)g(motion?)758 4845 y FD(F)-16 b(urther)59 b(reading:)72 b FC(The)51 b(statistical)h(prop)5 b(erties)51 b(of)g(dynamical)h(systems)h(are)e (the)g(sub)9 b(ject)758 5044 y(of)57 b(ergo)5 b(dic)57 b(theory)-14 b(.)80 b(Here)57 b(are)g(some)h(of)f(the)g(man)-5 b(y)59 b(go)5 b(o)g(d)57 b(in)-5 b(tro)5 b(ductions)58 b(to)f(this)h(b)5 b(eautiful)758 5243 y(\034eld:)74 b([)p FD(CFS)p FC(])56 b([)p FD(P)p FC(])g([)p FD(W)p FC(].)1743 5654 y FD(10.)96 b(Hyp)5 b(erb)g(olic)63 b(automorphisms)h(of)h(the)f (t)-5 b(w)g(o-torus)1057 5953 y FC(A)68 b Fz(c)-8 b(ontinuous)72 b(dynamic)-8 b(al)72 b(system)d FC(is)g(a)g(collection)g Fy(\()o(\010)4717 5978 y Ft(t)4776 5953 y Fy(\))4840 6002 y Ft(t)p Fr(2)p Fq(R)5141 5953 y FC(of)g(measure-preserving)758 6152 y(transformations)50 b(of)f(a)h(probabilit)-5 b(y)50 b(space)g Fy(\()p Fw(M)9 b(;)28 b(\026)p Fy(\))48 b FC(suc)-5 b(h)51 b(that)e Fy(\010)4911 6177 y Ft(t)4995 6152 y Fu(\016)25 b Fy(\010)5223 6177 y Ft(s)5340 6152 y Fy(=)46 b(\010)5635 6177 y Ft(t)p Fv(+)p Ft(s)5858 6152 y Fw(;)28 b Fy(\010)6052 6177 y Fv(0)6173 6152 y Fy(=)101 b FC(id)6542 6177 y Ft(M)6689 6152 y FC(.)758 6352 y(Similarly)-14 b(,)67 b(one)e(obtains)g(a)f Fz(discr)-8 b(ete)68 b(dynamic)-8 b(al)67 b(system)d FC(when)h(the)f(collection)f(is)i(indexed)758 6551 y(b)-5 b(y)59 b Fw(t)54 b Fu(2)f Fs(Z)77 b FC(so)60 b(that)f Fy(\010)2143 6576 y Ft(t)2255 6551 y Fy(=)53 b(\(\010)2622 6576 y Fv(1)2696 6551 y Fy(\))2761 6491 y Ft(t)2879 6551 y FC(and)60 b(hence)f(a)h(discrete)f(dynamical)h (system)g(is)g(completely)758 6750 y(determined)e(b)-5 b(y)60 b(a)e(single)i(in)-5 b(v)g(ertible)59 b(measure)g(preserving)g (map)h Fy(\010)51 b(=)i(\010)5506 6775 y Fv(1)5639 6750 y FC(that)58 b(is)h(iterated)758 6949 y(to)74 b(obtain)g(the)g (dynamics.)131 b(W)-14 b(e)74 b(ha)-5 b(v)g(e)75 b(so)g(far)f(b)5 b(een)73 b(studying)i(con)-5 b(tin)g(uous)76 b(Hamiltonian)758 7149 y(dynamical)55 b(systems)2170 7014 y Fp(\000)2247 7149 y Fy(\010)2367 7088 y Ft(H)2367 7190 y(t)2492 7014 y Fp(\001)2568 7215 y Ft(t)p Fr(2)p Fq(R)2855 7149 y FC(on)g(energy)f(surfaces)i Fy(\(\006)4425 7174 y Ft(E)4536 7149 y Fw(;)28 b(\026)4710 7174 y Ft(E)4822 7149 y Fy(\))55 b FC(and)g(no)-5 b(w)55 b(turn)g(to)g(discrete)758 7355 y(Hamiltonian)g(dynamical)h(systems)g(on)g(the)e(t)-5 b(w)g(o-torus.)1057 7554 y(First,)53 b(let)e(us)i(consider)g(the)e(t)-5 b(w)g(o-torus)53 b Fs(T)3726 7494 y Fv(2)3855 7554 y Fy(=)47 b Fs(R)4153 7494 y Fv(2)4237 7554 y Fw(=)p Fs(Z)4433 7494 y Fv(2)4568 7554 y FC(as)53 b(the)e(quotien)-5 b(t)52 b(of)f(phase)i(space)758 7753 y Fs(R)881 7693 y Fv(2)1026 7753 y FC(b)-5 b(y)62 b Fs(Z)1376 7693 y Fv(2)1522 7753 y FC(and)g(recall)g(that)g(a)g(smo)5 b(oth)62 b(function)g Fw(f)75 b Fu(2)57 b Fw(C)4518 7693 y Fr(1)4659 7753 y Fy(\()p Fs(T)4836 7693 y Fv(2)4919 7753 y Fy(\))62 b FC(on)g(the)g(torus)g(is)h(nothing)758 7953 y(but)69 b(a)g(smo)5 b(oth)70 b(p)5 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b(in)-5 b(v)c(arian)k(t)56 b(under)g(translations)g(b)-5 b(y)55 b Fs(Z)3964 9878 y Fv(2)4048 9938 y FC(,)g Fz(i.e.)2225 10232 y Fy(\010)2345 10164 y Ft(H)2345 10273 y(t)2471 10232 y Fy(\()p Fw(x)36 b Fy(+)h Fw(n)p Fy(\))45 b(=)i(\010)3339 10257 y Ft(t)3397 10232 y Fy(\()p Fw(x)p Fy(\))36 b(+)h Fw(n;)83 b Fu(8)p Fw(x)46 b Fu(2)g Fs(R)4566 10164 y Fv(2)4649 10232 y Fw(;)28 b(n)46 b Fu(2)g Fs(Z)5139 10164 y Fv(2)5222 10232 y Fw(:)758 10527 y FC(As)j(a)g(result,)i Fy(\010)1753 10466 y Ft(H)1753 10568 y(t)1929 10527 y FC(de\034nes)f(a)f(map)h(on)g (the)f(torus)g(that)g(I)g(shall)i(denote)e(b)-5 b(y)49 b(the)g(same)h(sym)-5 b(b)5 b(ol.)1057 10840 y Fx(Exer)-5 b(cise)62 b FC(10.1)p Fx(.)83 b FC(Pro)-5 b(v)g(e)55 b(all)h(the)f(claims)h(made)g(in)f(the)g(previous)h(paragraph.)p eop %%Page: 24 24 24 23 bop 758 878 a FA(24)2127 b(STEPHAN)58 b(DE)h(BI\310VRE)1057 1259 y FC(Of)j(course,)k(since)d(w)-5 b(e)63 b(are)g(in)h(t)-5 b(w)g(o)63 b(dimensions,)k(suc)-5 b(h)64 b(\035o)-5 b(ws)64 b(are)f(automatically)g(com-)758 1458 y(pletely)49 b(in)-5 b(tegrable.)73 b(The)51 b(situation)f(is)i(v)-5 b(ery)50 b(di\033eren)-5 b(t)50 b(when)h(w)-5 b(e)51 b(consider)g Fz(discr)-8 b(ete)56 b(Hamil-)758 1658 y(tonian)68 b(dynamic)-8 b(al)67 b(systems)p FC(.)101 b(Those)65 b(are)g(de\034ned)g(as)g(the)f (iterates)g Fy(\010)5376 1597 y Ft(t)5434 1658 y Fy(\()p Fw(t)d Fu(2)h Fs(Z)8 b Fy(\))74 b FC(of)64 b(a)g(\034xed)758 1857 y(canonical)58 b(transformation)h Fy(\010)f FC(of)h(the)e(torus,)j (the)e(latter)g(b)5 b(eing)58 b(a)g(smo)5 b(oth)59 b(in)-5 b(v)g(ertible)59 b(map)758 2056 y Fy(\010)46 b(:)g Fs(T)1129 1996 y Fv(2)1258 2056 y Fu(!)g Fs(T)1583 1996 y Fv(2)1722 2056 y FC(suc)-5 b(h)57 b(that,)d(for)h(all)h Fw(f)9 b(;)28 b(g)52 b Fu(2)46 b Fw(C)3570 1996 y Fr(1)3710 2056 y Fy(\()p Fs(T)3887 1996 y Fv(2)3971 2056 y Fy(\))p FC(,)2778 2347 y Fu(f)p Fw(f)55 b Fu(\016)37 b Fy(\010)p Fw(;)28 b(g)43 b Fu(\016)37 b Fy(\010)p Fu(g)46 b Fy(=)g Fu(f)p Fw(f)9 b(;)28 b(g)6 b Fu(g)37 b(\016)g Fy(\010)p Fw(:)758 2638 y FC(The)61 b(examples)g(I)g(will)g(b)5 b(e)61 b(in)-5 b(terested)60 b(in)i(are)f(of)f(the)h(follo)-5 b(wing)61 b(v)-5 b(ery)61 b(simple)h(form.)91 b(Let)60 b Fw(A)758 2837 y FC(b)5 b(e)46 b(a)i(t)-5 b(w)g(o)47 b(b)-5 b(y)47 b(t)-5 b(w)g(o)48 b(matrix)e(and)i(consider)g(the)f(asso) 5 b(ciated)47 b(linear)g(map)h Fw(x)e Fu(2)g Fs(R)5719 2777 y Fv(2)5848 2837 y Fu(7!)g Fw(Ax)g Fu(2)g Fs(R)6606 2777 y Fv(2)6689 2837 y FC(.)758 3037 y(Y)-14 b(ou)79 b(will)g(undoubtedly)g(ha)-5 b(v)g(e)80 b(no)f(trouble)g(c)-5 b(hec)g(king)79 b(that)g(this)g(de\034nes)h(a)f(map)g(on)h(the)758 3236 y(t)-5 b(w)g(o-torus,)67 b Fz(i.e.)100 b Fy([)p Fw(Ax)p Fy(])61 b(=)g([)p Fw(Ax)2712 3176 y Fr(0)2758 3236 y Fy(])j FC(if)g Fy([)p Fw(x)p Fy(])d(=)g([)p Fw(x)3608 3176 y Fr(0)3655 3236 y Fy(])p FC(,)66 b(if)e(and)h(only)f(if)g Fw(A)f FC(has)i(in)-5 b(teger)64 b(en)-5 b(tries.)101 b(On)758 3435 y(the)67 b(other)h(hand,)k(w)-5 b(e)68 b(sa)-5 b(w)69 b(\(Exercise)e(5.5\))g(that)h Fw(A)f 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4743 y(discrete)67 b(Hamiltonian)h(dynamical)g (system)f(generated)g(b)-5 b(y)68 b Fy(\010)4851 4768 y Ft(A)5027 4743 y FC(is)g(exp)5 b(onen)-5 b(tially)67 b(mixing)758 4942 y(on)80 b Fs(T)1126 4882 y Fv(2)1210 4942 y FC(.)149 b(More)81 b(precisely)-14 b(,)87 b(there)80 b(exists)g(a)h(constan)-5 b(t)81 b Fw(C)4514 4967 y Ft(A)4702 4942 y FC(and)g Fw(\015)97 b(>)89 b Fy(0)80 b FC(so)h(that,)87 b(for)80 b(all)758 5141 y Fw(f)9 b(;)28 b(g)52 b Fu(2)45 b Fw(C)1340 5081 y Fv(1)1415 5141 y Fy(\()p Fs(T)1592 5081 y Fv(2)1675 5141 y Fy(\))55 b FC(one)h(has)764 5501 y Fp(?)764 5600 y(?)764 5700 y(?)764 5799 y(?)874 5515 y(Z)967 5893 y Fq(T)1046 5859 y Fn(2)1117 5741 y Fy(\()p Fw(f)f Fu(\016)37 b Fy(\010)1558 5673 y Ft(t)1558 5782 y(A)1666 5741 y Fy(\)\()p Fw(x)p Fy(\))p Fw(g)6 b Fy(\()p Fw(x)p Fy(\))p Fw(dx)35 b Fu(\000)2648 5507 y Fp(\022)2770 5515 y(Z)2862 5893 y Fq(T)2941 5859 y Fn(2)3041 5741 y Fw(f)18 b Fy(\()p Fw(x)p Fy(\))p Fw(dx)3546 5507 y Fp(\023)27 b(\022)3817 5515 y(Z)3910 5893 y Fq(T)3989 5859 y Fn(2)4088 5741 y Fw(g)6 b Fy(\()p Fw(x)p Fy(\))p Fw(dx)4579 5507 y Fp(\023)4701 5501 y(?)4701 5600 y(?)4701 5700 y(?)4701 5799 y(?)4857 5741 y Fu(\024)47 b Fw(C)5152 5766 y Ft(A)5306 5741 y Fu(k)f Fw(f)64 b Fu(k)5663 5766 y Fv(2)p Ft(;)p Fv(1)5843 5741 y Fu(k)46 b Fw(g)53 b Fu(k)6187 5766 y Fv(2)p Ft(;)p Fv(1)6413 5741 y Fw(e)6490 5673 y Fr(\000)p Ft(\015)7 b(t)758 5416 y FC(\(10.1\))758 6122 y(Here)2119 6387 y Fu(k)46 b Fw(g)52 b Fu(k)2462 6318 y Fv(2)2462 6428 y(2)p Ft(;)p Fv(1)2642 6387 y Fy(=)2818 6161 y Fp(Z)2910 6538 y Fq(T)2989 6505 y Fn(2)3107 6387 y Fu(j)46 b Fw(g)6 b Fy(\()p Fw(x)p Fy(\))45 b Fu(j)3600 6318 y Fv(2)3721 6387 y Fw(dx)37 b Fy(+)4105 6161 y Fp(Z)4197 6538 y Fq(T)4276 6505 y Fn(2)4394 6387 y Fu(j)46 b(r)p Fw(g)6 b Fy(\()p Fw(x)p Fy(\))46 b Fu(j)5026 6318 y Fv(2)5147 6387 y Fw(dx:)1057 6802 y FC(T)-14 b(o)61 b(simplify)i(the)e(notations) h(I)f(ha)-5 b(v)g(e)62 b(simply)h(written)e Fw(x)56 b Fu(2)h Fs(T)5038 6742 y Fv(2)5183 6802 y FC(rather)k(than)h Fy([)p Fw(x)p Fy(])56 b Fu(2)h Fs(T)6606 6742 y Fv(2)6689 6802 y FC(.)758 7001 y(The)h(deca)-5 b(y)57 b(rate)h Fw(\015)67 b FC(in)58 b(the)g(exp)5 b(onen)-5 b(tial)57 b(is)i(directly)e(related)h(to)f(the)h(matrix)f Fw(A)h FC(as)g(follo)-5 b(ws.)758 7200 y(The)66 b(h)-5 b(yp)5 b(otheses)68 b(on)f Fw(A)f FC(imply)h(that)f Fw(A)h FC(has)g(t)-5 b(w)g(o)67 b(eigen)-5 b(v)g(ectors)67 b Fw(v)5093 7225 y Fr(\006)5273 7200 y FC(and)g(t)-5 b(w)g(o)67 b(eigen)-5 b(v)c(alues)758 7400 y Fw(\025)855 7339 y Fr(\006)p Fv(1)1088 7400 y FC(with)55 b Fw(\025)46 b(>)g Fy(1)p FC(.)74 b(One)56 b(has)g Fw(\015)f Fy(=)46 b(ln)29 b Fw(\025)p FC(.)758 7599 y FD(Pro)5 b(of:)101 b FC(The)68 b(pro)5 b(of)68 b(of)g(this)h(w)-5 b(ell)68 b(kno)-5 b(wn)69 b(result)f(is)h(tak)-5 b(en)68 b(from)h([)p FD(DB)p FC(].)113 b(The)68 b(essence)h(of)758 7798 y(the)76 b(pro)5 b(of)76 b(is)i(con)-5 b(tained)77 b(in)g(its)g(last)g(few)f(lines,)83 b(whic)-5 b(h)77 b(y)-5 b(ou)77 b(migh)-5 b(t)77 b(wish)h(to)e(read)h(\034rst.)758 7997 y(I)e(start)h(ho)-5 b(w)g(ev)g(er)76 b(with)f(some)h (preliminaries.)137 b(W)-14 b(e)75 b(can)h(tak)-5 b(e)75 b Fw(v)5009 8022 y Fr(\006)5197 7997 y FC(to)g(b)5 b(e)75 b(normalized,)82 b(so)758 8197 y(that)74 b Fw(v)1217 8222 y Fr(\006)1409 8197 y Fy(=)79 b(\(cos)28 b Fw(\022)2010 8222 y Fr(\006)2122 8197 y Fw(;)g Fy(sin)g Fw(\022)2505 8222 y Fr(\006)2617 8197 y Fy(\))75 b FC(and)h Fw(s)3178 8222 y Fr(\006)3369 8197 y Fy(=)j(tan)28 b Fw(\022)3923 8222 y Fr(\006)4110 8197 y FC(for)75 b(some)h(angles)f Fw(\022)5416 8222 y Fr(\006)5528 8197 y FC(.)134 b(Note)74 b(for)h(later)758 8396 y(purp)5 b(oses)54 b(that)f(the)g(slop)5 b(es)54 b Fw(s)2649 8421 y Fr(\006)2814 8396 y FC(are)f(quadratic)g (irrationals,)i(whic)-5 b(h)54 b(means)g(they)f(are)g(zero)5 b(es)758 8595 y(of)55 b(a)g(quadratic)g(p)5 b(olynomial)56 b(with)f(in)-5 b(teger)55 b(co)5 b(e\036cien)-5 b(ts)55 b(\(Pro)-5 b(v)g(e)55 b(this!\).)1057 8794 y(It)60 b(will)h(ob)-5 b(viously)62 b(b)5 b(e)61 b(enough)g(to)f(sho)-5 b(w)62 b(\(10.1\))f(for)f Fw(f)9 b(;)28 b(g)67 b FC(of)61 b(mean)g(zero,)h(so) g(from)f(no)-5 b(w)758 8994 y(on)60 b(I)f(w)-5 b(ork)60 b(in)g(the)g(subspace)h(of)e Fw(L)3016 8933 y Fv(2)3091 8994 y Fy(\()p Fs(T)3268 8933 y Fv(2)3352 8994 y Fy(\))g FC(orthogonal)h(to)f(the)h(constan)-5 b(t)60 b(functions.)87 b(De\034ne)758 9193 y(there,)54 b(on)i(the)f(ob)-5 b(vious)56 b(domain)2027 9560 y Fu(f)p Fw(\036)47 b Fu(2)e Fw(L)2525 9491 y Fv(2)2600 9560 y Fy(\()p Fs(T)2777 9491 y Fv(2)2861 9560 y Fy(\))g Fu(j)3064 9334 y Fp(Z)3156 9711 y Fq(T)3235 9678 y Fn(2)3334 9560 y Fw(\036)p Fy(\()p Fw(x)p Fy(\))p Fw(dx)h Fy(=)g(0)p Fw(;)83 b(D)4409 9585 y Fr(\006)4522 9560 y Fw(\036)46 b Fu(2)f Fw(L)4936 9491 y Fv(2)5011 9560 y Fy(\()p Fs(T)5188 9491 y Fv(2)5272 9560 y Fy(\))p Fu(g)p Fw(;)758 9934 y FC(the)55 b(self-adjoin)-5 b(t)56 b(op)5 b(erators)2079 10294 y Fw(D)2216 10319 y Fr(\006)2374 10294 y Fy(=)2649 10181 y(1)p 2570 10256 241 7 v 2570 10408 a(2)p Fw(\031)h(i)2858 10294 y Fy(\()p Fw(v)3003 10319 y Fr(\006)3115 10294 y Fw(;)28 b Fu(r)p Fy(\))46 b(=)3712 10181 y(1)p 3633 10256 V 3633 10408 a(2)p Fw(\031)6 b(i)3894 10294 y Fy(\(cos)28 b Fw(\022)4287 10319 y Fr(\006)4399 10294 y Fw(@)4487 10319 y Ft(q)4597 10294 y Fy(+)37 b(sin)29 b Fw(\022)5073 10319 y Fr(\006)5185 10294 y Fw(@)5273 10319 y Ft(p)5350 10294 y Fy(\))758 10641 y FC(with)47 b(eigenfunctions)g Fw(\037)2308 10666 y Ft(n)2399 10641 y Fy(\()p Fw(x)p Fy(\))e(=)h(exp)27 b Fw(i)p Fy(2)p Fw(\031)6 b Fy(\()p Fw(n)3531 10666 y Fv(2)3605 10641 y Fw(q)27 b Fu(\000)21 b Fw(n)3956 10666 y Fv(1)4030 10641 y Fw(p)p Fy(\))p FC(:)69 b Fw(D)4431 10666 y Fr(\006)4544 10641 y Fw(\037)4648 10666 y Ft(n)4784 10641 y Fy(=)46 b Fu(h)p Fw(v)5104 10666 y Fr(\006)5216 10641 y Fw(;)28 b(n)p Fu(i)p Fw(\037)5559 10666 y Ft(n)5649 10641 y FC(.)71 b(Here,)48 b Fu(h)p Fw(a;)28 b(b)p Fu(i)45 b Fy(=)758 10840 y Fw(a)846 10865 y Fv(1)920 10840 y Fw(b)991 10865 y Fv(2)1107 10840 y Fu(\000)d Fw(a)1366 10865 y Fv(2)1440 10840 y Fw(b)1511 10865 y Fv(1)1586 10840 y FC(.)94 b(The)62 b(irrationalit)-5 b(y)62 b(of)g Fw(s)3278 10865 y Fr(\006)3452 10840 y FC(guaran)-5 b(tees)63 b(that)f(the)f(eigen)-5 b(v)c(alues)63 b Fu(h)p Fw(v)5956 10865 y Fr(\006)6068 10840 y Fw(;)28 b(n)p Fu(i)61 b FC(are)h(a)p eop %%Page: 25 25 25 24 bop 2362 861 a FA(QUANTUM)58 b(CHA)l(OS:)g(A)h(BRIEF)f(FIRST)g (VISIT)1472 b(25)758 1259 y FC(coun)-5 b(table)55 b(dense)g(subset)g (of)g Fs(R)90 b FC(and)55 b(that)f Fy(0)h FC(is)g(not)g(an)g(eigen)-5 b(v)c(alue)54 b(\(since)h Fw(n)45 b Fu(6)p Fy(=)i(0)p FC(\).)73 b(Indeed,)2710 1583 y Fu(h)p Fw(v)2855 1608 y Fr(\006)2967 1583 y Fw(;)28 b(n)p Fu(i)45 b Fy(=)i(cos)28 b Fw(\022)3755 1608 y Fr(\006)3867 1583 y Fy(\()p Fw(n)4032 1608 y Fv(2)4142 1583 y Fu(\000)37 b Fw(s)4386 1608 y Fr(\006)4498 1583 y Fw(n)4598 1608 y Fv(1)4672 1583 y Fy(\))p Fw(;)758 1906 y FC(and)58 b Fu(f)d FC(F)-14 b(rac)q Fy(\()p Fw(s)1681 1931 y Fr(\006)1792 1906 y Fw(n)1892 1931 y Fv(1)1966 1906 y Fy(\))51 b Fu(j)f Fw(n)2278 1931 y Fv(1)2403 1906 y Fu(2)g Fs(Z)9 b Fu(g)67 b FC(is)59 b(dense)f(in)g Fy([0)p Fw(;)28 b Fy(1])58 b FC(since)h Fw(s)4526 1931 y Fr(\006)4696 1906 y FC(is)f(irrational.)82 b(\(See)58 b(App)5 b(endix)758 2105 y(A,)55 b(if)g(y)-5 b(ou)55 b(did)h(not)f(kno)-5 b(w)56 b(this)g(last)f(fact.\))73 b(Also)2578 2429 y Fw(D)2715 2454 y Fr(\006)2827 2429 y Fy(\()p Fw(f)55 b Fu(\016)37 b Fy(\010)3268 2360 y Ft(t)3268 2470 y(A)3376 2429 y Fy(\))46 b(=)g Fw(e)3739 2360 y Fr(\006)p Ft(\015)7 b(t)3978 2429 y Fy(\()p Fw(D)4180 2454 y Fr(\006)4293 2429 y Fw(f)18 b Fy(\))36 b Fu(\016)h Fy(\010)4733 2360 y Ft(t)4733 2470 y(A)4841 2429 y Fw(;)-4129 b FC(\(10.2\))758 2752 y(and)2625 3076 y Fw(D)2767 3005 y Fr(\000)p Fv(1)2762 3117 y Fr(\006)2946 3076 y Fw(g)52 b Fy(=)3320 2918 y Fp(X)3252 3281 y Ft(n)p Fr(2)p Fq(Z)3508 3248 y Fn(2)p Fi(\003)3655 3076 y Fw(g)3734 3101 y Ft(n)3825 3076 y Fy(\()p Fu(h)p Fw(v)4035 3101 y Fr(\006)4146 3076 y Fw(;)28 b(n)p Fu(i)p Fy(\))4450 3007 y Fr(\000)p 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Ft(T)3676 10534 y Fp(X)3664 10890 y Ft(t)3714 10856 y Fi(0)3759 10890 y Fv(=1)3955 10691 y Fw(\016)4029 10738 y Ft(A)4129 10705 y Fi(\000)p Fo(t)4273 10738 y Ft(n;A)4494 10705 y Fi(\000)p Fo(t)4631 10681 y Fi(0)4685 10738 y Ft(n)4775 10691 y Fw(;)p eop %%Page: 27 27 27 26 bop 2362 861 a FA(QUANTUM)58 b(CHA)l(OS:)g(A)h(BRIEF)f(FIRST)g (VISIT)1472 b(27)758 1259 y FC(from)78 b(whic)-5 b(h)79 b(the)e(result)i(follo)-5 b(ws)79 b(for)f Fw(f)18 b Fy(\()p Fw(x)p Fy(\))83 b(=)h Fw(\037)4034 1284 y Ft(n)4125 1259 y Fy(\()p Fw(x)p Fy(\))77 b FC(since)h Fw(A)4981 1199 y Fr(\000)p Ft(t)5143 1259 y Fw(n)84 b Fy(=)h Fw(A)5666 1199 y Fr(\000)p Ft(t)5820 1149 y Fi(0)5872 1259 y Fw(n)78 b FC(i\033)g Fw(t)84 b Fy(=)h Fw(t)6689 1199 y Fr(0)758 1458 y FC(\(Remem)-5 b(b)5 b(er)59 b(that)f Fu(j)52 b FC(T)-14 b(r)p Fw(A)52 b Fu(j)p Fw(>)h Fy(2)p FC(\).)84 b(This)60 b(implies)g(immediately)f(the)f(result)i(for)e(trigonomet-) 758 1658 y(ric)k(p)5 b(olynomials)62 b(and)h(a)f(standard)h(appro)-5 b(ximation)63 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b(p)46 b Fu(2)g Fs(Q)38 b FC(.)1057 2984 y(\(ii\))54 b(Sho)-5 b(w)56 b(that)2462 3246 y Fy(#)p Fu(f)p Fw(x)46 b Fu(2)g Fs(T)3094 3178 y Fv(2)3223 3246 y Fu(j)h(9)p Fw(t)f Fu(\024)g Fw(T)14 b(;)28 b(A)3999 3178 y Ft(t)4057 3246 y Fw(x)46 b Fy(=)g Fw(x)p Fu(g)g(\030)g Fw(e)4849 3178 y Ft(\015)7 b(T)758 3536 y FC(as)55 b Fw(T)70 b Fu(!)46 b(1)p FC(.)1057 3735 y(\(iii\))52 b(Sho)-5 b(w)53 b(that,)g(if)f Fw(x)2460 3760 y Fv(0)2580 3735 y Fu(2)46 b Fs(T)2850 3675 y Fv(2)2986 3735 y FC(is)53 b(a)g(p)5 b(erio)g(dic)52 b(p)5 b(oin)-5 b(t)53 b(for)f Fy(\010)4722 3760 y Ft(A)4830 3735 y FC(,)h(then)g(\(10.6\))f(can)g (not)h(hold.)1057 4066 y(In)41 b(conclusion,)k(h)-5 b(yp)5 b(erb)g(olic)41 b(toral)g(automorphisms)i(are)f(p)5 b(erfect)39 b(examples)j(of)f(\020c)-5 b(haotic\021)758 4265 y(Hamiltonian)46 b(dynamical)h(systems)g(that)e(ha)-5 b(v)g(e)47 b(the)f(added)g(virtue) g(of)g(b)5 b(eing)46 b(extremely)f(sim-)758 4464 y(ple,)69 b(making)e(the)f(analysis)h(of)g(their)f(dynamical)h(prop)5 b(erties)66 b(\(exp)5 b(onen)-5 b(tial)65 b(mixing,)70 b(p)5 b(eri-)758 4664 y(o)g(dic)54 b(orbits,)h Fw(:)28 b(:)g(:)81 b FC(\))54 b(quite)g(straigh)-5 b(tforw)g(ard,)55 b(as)g(the)f(previous)h(pages)g(sho)-5 b(w)g(ed.)75 b(They)54 b(are)g(in)758 4863 y(particular)64 b(m)-5 b(uc)g(h)65 b(simpler)g(than)f(c)-5 b(haotic)64 b(billiard)h(\035o)-5 b(ws.)101 b(Although)64 b(they)f(do)h(not)g(mo)5 b(del)758 5062 y(real)67 b(life)h(ph)-5 b(ysical)68 b(systems,)k(they)66 b(are)i(considered)g(useful)g(to)-5 b(y-mo)5 b(dels)68 b(for)g(in)-5 b(v)g(estigating)758 5261 y(c)g(haos-related)74 b(problems,)79 b(and)74 b(in)g(particular)g(quan)-5 b(tum)74 b(c)-5 b(haos.)130 b(There)73 b(is)h(nev)-5 b(ertheless)758 5461 y(a)72 b(problem:)107 b(to)72 b(talk)f(ab)5 b(out)71 b(quan)-5 b(tum)73 b(c)-5 b(haos,)77 b(w)-5 b(e)72 b(need)g(a)f(sp)5 b(ectral)72 b(problem)g(naturally)758 5660 y(related)60 b(to)h(the)f(toral)h(automorphisms)i(that)d(can)h(pla)-5 b(y)61 b(the)g(role)g(pla)-5 b(y)g(ed)62 b(b)-5 b(y)61 b(the)f(Diric)-5 b(hlet)758 5859 y(Laplacian)51 b(in)g(the)f(case)h(of) f(billiards.)74 b(There)51 b(exists)f(indeed)i(suc)-5 b(h)51 b(a)g(sp)5 b(ectral)51 b(problem,)h(as)f(I)758 6058 y(will)c(sho)-5 b(w)49 b(in)f(section)g(13,)h(but)e(to)h (understand)g(its)g(origin)g(and)g(to)f(see)h(wh)-5 b(y)47 b(it)h(is)g(the)f(natural)758 6258 y(analog)e(of)g(the)g(Diric)-5 b(hlet)46 b(Laplacian,)h(y)-5 b(ou)46 b(need)f(to)g(understand)i(some)f (notions)g(of)f(quan)-5 b(tum)758 6457 y(mec)g(hanics.)106 b(This)66 b(is)g(the)g(sub)9 b(ject)65 b(of)h(the)f(next)g(t)-5 b(w)g(o)66 b(sections)g(in)g(whic)-5 b(h)66 b(I)g(will)g(explain)f(a) 758 6656 y(set)60 b(of)f(ideas)i(coming)g(from)f(quan)-5 b(tum)61 b(mec)-5 b(hanics)61 b(that)e(p)5 b(ermit)60 b(to)g(relate)f(a)h(Hamiltonian)758 6855 y(system)g(to)h(certain)f(sp)5 b(ectral)60 b(problems,)j(generalizing)e(in)g(this)g(manner)g(the)f (link)h(b)5 b(et)-5 b(w)g(een)758 7055 y(Diric)g(hlet)55 b(Laplacians)h(and)g(the)f(corresp)5 b(onding)56 b(billiard)g(\035o)-5 b(ws.)2732 7552 y FD(11.)96 b(Quan)-5 b(tum)64 b(mec)-5 b(hanics)1057 7851 y(11.1.)96 b(W)-16 b(a)-5 b(v)g(e)48 b(pac)-5 b(k)g(ets)47 b(and)h(the)g(Sc)-5 b(hr\366)5 b(dinger)47 b(equation.)83 b FC(In)41 b(this)h(section)f(I)h(wish)758 8050 y(to)63 b(in)-5 b(tro)5 b(duce)64 b(the)f(basic)i(ingredien)-5 b(ts)65 b(of)e(\(non-relativistic\))g(quan)-5 b(tum)65 b(mec)-5 b(hanics.)100 b(I)64 b(will)758 8250 y(not)i(b)5 b(e)66 b(rigorous)h(in)g(m)-5 b(y)66 b(discussion,)71 b(but)c(I)f(will)h(instead)f(try)g(to)g(bring)h(out)f(some)h(of)f(the) 758 8449 y(main)g(ideas)h(that)f(underly)g(the)f(theory)h(and)g (explain)g(its)h(mathematical)f(structure.)105 b(The)758 8648 y(quan)-5 b(tum)49 b(mec)-5 b(hanics)49 b(that)f(w)-5 b(e)49 b(use)f(to)5 b(da)-5 b(y)48 b(w)-5 b(as)50 b(b)5 b(orn)48 b(in)h(the)f(1920's)h(after)f(ab)5 b(out)48 b(a)g(quarter)758 8847 y(of)f(a)h(cen)-5 b(tury)48 b(of)g(hard)g(w)-5 b(ork)48 b(b)-5 b(y)48 b(some)h(of)f(the)f(b)5 b(est)48 b(ph)-5 b(ysicists)49 b(of)f(the)g(da)-5 b(y)-14 b(.)71 b(They)48 b(had)g(come)758 9047 y(to)72 b(realize)h(that)f(the)h(v)-5 b(ery)72 b(successful)i(description)f(of)g(nature)g(based)h(on)f(the)f (ideas)i(that)758 9246 y(I)68 b(brie\035y)g(presen)-5 b(ted)69 b(in)g(section)f(5)h(w)-5 b(as)69 b(inadequate)f(for)g(the)g (understanding)h(of)f(ph)-5 b(ysical)758 9445 y(phenomena)47 b(on)g(the)g(atomic)f(scale.)72 b(The)46 b(predictions)i(of)e (classical)i(mec)-5 b(hanics)48 b(w)-5 b(ere)46 b(simply)758 9644 y(not)h(in)h(agreemen)-5 b(t)48 b(with)g(a)g(certain)f(n)-5 b(um)g(b)5 b(er)49 b(of)f(exp)5 b(erimen)-5 b(ts)47 b(p)5 b(erformed)48 b(at)f(the)h(b)5 b(eginning)758 9844 y(of)70 b(the)h(cen)-5 b(tury)-14 b(.)120 b(It)70 b(to)5 b(ok)70 b(some)h(time)f(to)h(dev)-5 b(elop)71 b(the)f(new)h(theory)-14 b(,)74 b(esp)5 b(ecially)70 b(since)i(it)758 10043 y(turned)60 b(out)g(to)f(require)h(a)g(radically)g(new)g(approac)-5 b(h,)62 b(ha)-5 b(ving)61 b(a)f(n)-5 b(um)g(b)5 b(er)61 b(of)f(philosophical)758 10242 y(and)40 b(epistemological)g (implications)h(that)e(remain)h(puzzling)h(to)e(this)h(da)-5 b(y)-14 b(.)69 b(I)40 b(shall)h(not)e(go)h(in)-5 b(to)758 10441 y(an)g(y)46 b(of)g(this)h(but)g(instead)f(con)-5 b(ten)g(t)47 b(m)-5 b(yself)47 b(in)f(this)h(section)f(to)g(presen)-5 b(t)47 b(quan)-5 b(tum)47 b(mec)-5 b(hanics)758 10641 y(as)45 b(a)g(\020\034nished\021)61 b(pro)5 b(duct.)70 b(By)44 b(this)h(I)g(mean)h(that)e(I)h(will)g(not)g(motiv)-9 b(ate)44 b(its)h(form)-5 b(ulation)45 b(with)758 10840 y(exp)5 b(erimen)-5 b(tal)40 b(evidence,)i(as)f(is)g(often)f(done)h(in) f(ph)-5 b(ysics)42 b(b)5 b(o)g(oks,)43 b(nor)e(base)f(it)h(on)f(an)h (axiomatic)p eop %%Page: 28 28 28 27 bop 758 878 a FA(28)2127 b(STEPHAN)58 b(DE)h(BI\310VRE)758 1259 y FC(framew)-5 b(ork,)54 b(as)h(mathematicians)g(sometimes)g(lik) -5 b(e)54 b(to)g(do.)74 b(Instead,)55 b(I)f(will)h(simply)g(explain)758 1458 y(its)g(structure)g(and)h(analyze)f(\(some)g(of)13 b(\))55 b(its)g(predictions.)1057 1658 y(W)-14 b(e)42 b(ha)-5 b(v)g(e)44 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b(eha)-5 b(viour)57 b(of)g(the)g(solutions) i(to)e(\(11.4\).)79 b(This)58 b(will)g(bring)g(out)f(some)758 8034 y(striking)e(similarities)h(b)5 b(et)-5 b(w)g(een)54 b(quan)-5 b(tum)56 b(mec)-5 b(hanics)56 b(and)f(the)f(Hamiltonian)i (form)-5 b(ulation)758 8233 y(of)72 b(classical)i(mec)-5 b(hanics)74 b(and)f(allo)-5 b(w)73 b(me)g(to)f(sho)-5 b(w)74 b(ho)-5 b(w)73 b(these)g(ideas)g(naturally)f(lead)h(one)758 8432 y(to)68 b(asso)5 b(ciate)68 b(a)g(Hamiltonian)g(\035o)-5 b(w)69 b(on)g Fs(R)3529 8372 y Fv(2)p Ft(n)3763 8432 y FC(to)f(certain)f(sp)5 b(ectral)68 b(problems)i(on)e Fw(L)6263 8372 y Fv(2)6338 8432 y Fy(\()p Fs(R)6525 8372 y Ft(n)6625 8432 y Fy(\))p FC(,)758 8631 y(thereb)-5 b(y)70 b(generalizing)h(the)f(link)h(b)5 b(et)-5 b(w)g(een)70 b Fu(\000)p Fy(\001)3869 8656 y Ft(D)4060 8631 y FC(on)h Fy(\012)g FC(and)g(the)f(corresp)5 b(onding)71 b(billiard)758 8831 y(\035o)-5 b(w.)71 b(W)-14 b(e)46 b(will)g(then)g(\034nally)g(b)5 b(e)46 b(ready)g(to)f(return)h(to)g(the)g(h)-5 b(yp)5 b(erb)g(olic)45 b(toral)h(automorphisms)758 9030 y(and)55 b(the)g(sp)5 b(ectral)55 b(problem)h(to)f(whic)-5 b(h)56 b(they)f(are)g(naturally)g(asso)5 b(ciated.)1057 9720 y FD(11.2.)96 b(The)43 b(Ehrenfest)i(equations.)83 b FC(Let)36 b(us)h(start)g(b)-5 b(y)37 b(establishing)h(that)e(the)h(Sc) -5 b(hr\366)5 b(dinger)758 9919 y(equation)64 b(is)h(compatible)f(with) g(the)g(probabilistic)h(in)-5 b(terpretation)64 b(of)h(the)e(w)-5 b(a)g(v)g(e)65 b(function,)758 10118 y(giv)-5 b(en)55 b(in)h(\(11.1\)-\(11.2\).)72 b(Indeed,)56 b(if)f Fw( )3254 10143 y Ft(t)3368 10118 y FC(is)h(a)f(solution)i(of)e(\(11.4\),)f(then) 2309 10570 y Fw(d)p 2292 10644 147 7 v 2292 10796 a(dt)2505 10682 y Fu(k)46 b Fw( )2742 10707 y Ft(t)2847 10682 y Fu(k)2930 10613 y Fv(2)3004 10682 y Fy(=)3216 10570 y Fw(d)p 3199 10644 V 3199 10796 a(dt)3393 10448 y Fp(\022)3516 10456 y(Z)3608 10833 y Fq(R)3685 10800 y Fn(3)3811 10682 y Fu(j)g Fw( )4011 10707 y Ft(t)4116 10682 y Fu(j)4162 10613 y Fv(2)4283 10682 y Fy(\()p Fw(y)6 b Fy(\))p Fw(d)4586 10613 y Fv(3)4660 10682 y Fw(y)4747 10448 y Fp(\023)4916 10682 y Fy(=)46 b(0)p Fw(:)-4462 b FC(\(11.6\))p eop %%Page: 30 30 30 29 bop 758 878 a FA(30)2127 b(STEPHAN)58 b(DE)h(BI\310VRE)758 1259 y FC(Equation)52 b(\(11.6\))f(follo)-5 b(ws)54 b(from)e(a)g (simple)i(\(formal\))e(computation)g(up)5 b(on)53 b(using)g(\(11.4\))f (and)758 1458 y(a)58 b(partial)h(in)-5 b(tegration.)85 b(Hence,)59 b(if)f Fu(k)52 b Fw( )3300 1483 y Fv(0)3427 1458 y Fu(k)p Fy(=)g(1)p FC(,)60 b(then)e Fu(k)52 b Fw( )4504 1483 y Ft(t)4615 1458 y Fu(k)p Fy(=)g(1)p Fw(;)j Fu(8)p Fw(t)e Fu(2)e Fs(R)27 b FC(,)69 b(so)59 b(that)f(w)-5 b(e)59 b(can)758 1658 y(indeed)c(in)-5 b(terpret)55 b Fu(j)47 b Fw( )2172 1683 y Ft(t)2276 1658 y Fu(j)2322 1597 y Fv(2)2443 1658 y Fy(\()p Fw(y)6 b Fy(\))55 b FC(as)h(a)f (probabilit)-5 b(y)56 b(densit)-5 b(y)56 b(at)f(all)g(times)h Fw(t)p FC(.)1057 1857 y(Next,)d(let)i(me)h(rewrite)e(\(11.4\))h(as)h (follo)-5 b(ws:)3210 2147 y Fw(i)p Fh(~)p Fw(@)3445 2172 y Ft(t)3503 2147 y Fw( )3611 2172 y Ft(t)3716 2147 y Fy(=)46 b Fw(H)13 b( )4150 2172 y Ft(t)4209 2147 y Fw(;)-3497 b FC(\(11.7\))758 2437 y(where)55 b(I)g(in)-5 b(tro)5 b(duced)55 b(the)g(linear)h(partial)f(di\033eren)-5 b(tial)56 b(op)5 b(erator)1642 2825 y Fy(\()p Fw(H)13 b(\036)p Fy(\)\()p Fw(y)6 b Fy(\))45 b(=)2552 2713 y(1)p 2479 2787 229 7 v 2479 2939 a(2)p Fw(m)2728 2825 y Fy(\()p Fw(P)2923 2756 y Fv(2)2997 2825 y Fw(\036)p Fy(\)\()p Fw(y)6 b Fy(\))36 b(+)h Fw(V)g Fy(\()p Fw(y)6 b Fy(\))p Fw(\036)p Fy(\()p Fw(y)g Fy(\))45 b(=)h(\()4564 2713 y Fw(P)4694 2652 y Fv(2)p 4551 2787 V 4551 2939 a Fy(2)p Fw(m)4837 2825 y Fy(+)37 b Fw(V)g Fy(\()p Fw(Q)p Fy(\)\))p Fw(\036)p Fy(\()p Fw(y)6 b Fy(\))p Fw(;)-5067 b FC(\(11.8\))758 3161 y(where)2473 3496 y Fw(P)2603 3428 y Fv(2)2724 3496 y Fy(=)46 b Fw(P)3029 3428 y Fv(2)3006 3537 y(1)3140 3496 y Fy(+)37 b Fw(P)3436 3428 y Fv(2)3413 3537 y(2)3547 3496 y Fy(+)g Fw(P)3843 3428 y Fv(2)3820 3537 y(3)3917 3496 y Fw(;)194 b(P)4264 3521 y Ft(j)4380 3496 y Fy(=)4575 3384 y Fh(~)p 4575 3458 90 7 v 4591 3610 a Fw(i)4766 3384 y(@)p 4705 3458 249 7 v 4705 3610 a(@)9 b(y)4883 3635 y Ft(j)4973 3496 y Fw(:)-4261 b FC(\(11.9\))758 3890 y(One)82 b(calls)h Fw(H)96 b FC(the)82 b Fz(Hamiltonian)h FC(\(op)5 b(erator\))81 b(or)h(the)g Fz(Schr)-8 b(\366)g(dinger)85 b(op)-8 b(er)g(ator)81 b FC(and)i Fw(P)114 b Fy(=)758 4089 y(\()p Fw(P)930 4114 y Fv(1)1003 4089 y Fw(;)28 b(P)1184 4114 y Fv(2)1258 4089 y Fw(;)g(P)1439 4114 y Fv(3)1513 4089 y Fy(\))66 b FC(the)g Fz(momentum)g FC(op)5 b(erators.)106 b(Note)65 b(that)h(I)g(am)g(not)g(sh)-5 b(ying)68 b(a)-5 b(w)g(a)g(y)66 b(from)h(the)758 4288 y(widespread)54 b(abuse)h(of)e(notation)h(that)f(consists)i(of)e(using) i(the)e(same)i(letter)d Fw(H)68 b FC(to)53 b(designate)758 4515 y(the)69 b(function)g(on)h(phase)g(space)f Fw(H)13 b Fy(\()p Fw(x)p Fy(\))69 b(=)3589 4441 y Ft(p)3657 4391 y Fn(2)p 3564 4477 185 7 v 3564 4572 a Fv(2)p Ft(m)3814 4515 y Fy(+)47 b Fw(V)36 b Fy(\()p Fw(q)6 b Fy(\))69 b FC(and)h(the)f(op)5 b(erator)69 b(in)g(\(11.8\);)76 b(y)-5 b(ou)758 4714 y(b)5 b(etter)54 b(try)h(not)h(to)f(b)5 b(e)55 b(b)5 b(othered)56 b(b)-5 b(y)56 b(it.)75 b(Let)54 b(me)i(no)-5 b(w)57 b(justify)e(this)h(analogy)g(with)f(classical)758 4913 y(mec)-5 b(hanics,)49 b(b)5 b(oth)45 b(in)i(notation)e(and)i(in)f (terminology)-14 b(.)71 b(F)-14 b(or)47 b(that)e(purp)5 b(ose,)49 b(I)d(will)g(sho)-5 b(w)47 b(that,)758 5113 y(if)55 b Fw( )1018 5138 y Ft(t)1132 5113 y FC(is)h(a)f(solution)h(to)f (\(11.4\),)g(then)2590 5377 y Fw(d)p 2560 5451 147 7 v 2560 5603 a(dt)2772 5489 y(<)46 b(Q)3078 5514 y Ft(j)3195 5489 y Fw(>)3324 5514 y Ft( )3412 5531 y Fo(t)3649 5489 y Fy(=)3870 5424 y Fv(1)p 3844 5451 118 7 v 3844 5547 a Ft(m)4028 5489 y Fw(<)g(P)4310 5514 y Ft(j)4426 5489 y Fw(>)4555 5514 y Ft( )4643 5531 y Fo(t)4706 5489 y Fw(;)-3994 b FC(\(11.10\))2614 5785 y Fw(d)p 2584 5859 147 7 v 2584 6011 a(dt)2797 5897 y(<)46 b(P)3079 5922 y Ft(j)3195 5897 y Fw(>)3324 5922 y Ft( )3412 5939 y Fo(t)3641 5897 y Fy(=)h Fu(\000)f Fw(<)4198 5832 y Ft(@)7 b(V)p 4187 5859 208 7 v 4187 5955 a(@)g(y)4332 5972 y Fo(j)4460 5897 y Fw(>)4589 5922 y Ft( )4677 5939 y Fo(t)4741 5897 y Fw(;)-4029 b FC(\(11.11\))758 6233 y(where)2344 6569 y Fw(<)2548 6456 y(@)9 b(V)p 2540 6531 249 7 v 2540 6683 a(@)g(y)2718 6708 y Ft(j)2854 6569 y Fw(>)2983 6594 y Ft( )3071 6611 y Fo(t)3135 6569 y Fy(=)3310 6343 y Fp(Z)3402 6720 y Fq(R)3479 6687 y Fn(3)3616 6456 y Fw(@)g(V)p 3607 6531 V 3607 6683 a(@)g(y)3785 6708 y Ft(j)3876 6569 y Fy(\()p Fw(y)d Fy(\))45 b Fu(j)h Fw( )4338 6594 y Ft(t)4443 6569 y Fu(j)4489 6500 y Fv(2)4610 6569 y Fy(\()p Fw(y)6 b Fy(\))p Fw(d)4913 6500 y Fv(3)4987 6569 y Fw(y)g(;)-4362 b FC(\(11.12\))758 6952 y(and)2050 7287 y Fw(<)47 b(P)2333 7312 y Ft(j)2448 7287 y Fw(>)2577 7312 y Ft( )2665 7329 y Fo(t)2729 7287 y Fu(\021)f Fy(\()p Fw( )3077 7312 y Ft(t)3135 7287 y Fw(;)28 b(P)3316 7312 y Ft(j)3386 7287 y Fw( )3494 7312 y Ft(t)3552 7287 y Fy(\))46 b(=)3838 7061 y Fp(Z)3930 7439 y Fq(R)4007 7406 y Fn(3)p 4115 7152 167 7 v 4115 7287 a Fw( )4223 7312 y Ft(t)4282 7287 y Fy(\()p Fw(y)6 b Fy(\))4519 7175 y Fh(~)p 4519 7249 90 7 v 4535 7401 a Fw(i)4648 7175 y(@)j( )4853 7200 y Ft(t)p 4648 7249 264 7 v 4656 7401 a Fw(@)g(y)4834 7426 y Ft(j)4932 7287 y Fy(\()p Fw(y)d Fy(\))p Fw(d)5235 7219 y Fv(3)5309 7287 y Fw(y)g(:)-4684 b FC(\(11.13\))758 7681 y(Equations)75 b(\(11.10-11.11\))g(are)g(called)h Fz(the)i(Ehr)-8 b(enfest)78 b(e)-8 b(quations)p FC(.)134 b(Comparing)76 b(them)g(to)758 7880 y(\(5.3\),)54 b(the)h(similarities) h(are)f(striking.)74 b(If)55 b(one)g(in)-5 b(terprets)55 b Fw(<)46 b(P)4779 7905 y Ft(j)4895 7880 y Fw(>)5024 7905 y Ft( )5112 7922 y Fo(t)5230 7880 y FC(as)56 b(the)f(co)5 b(ordinates)55 b(of)758 8080 y(the)h(mean)i(momen)-5 b(tum)58 b(of)e(the)h(particle,)g(then)g(the)f(quan)-5 b(tum)58 b(mec)-5 b(hanical)58 b(mean)f(p)5 b(osition)758 8279 y(and)65 b(momen)-5 b(tum)67 b(satisfy)e(equations)g(that)g(are)g (v)-5 b(ery)64 b(close)i(to)e(the)h(classical)h(equations)g(of)758 8478 y(motion.)74 b(The)55 b(di\033erence)g(is)h(that)e(in)i(general) 2701 8846 y Fw(<)2905 8733 y(@)9 b(V)p 2897 8808 249 7 v 2897 8960 a(@)g(y)3075 8985 y Ft(j)3211 8846 y Fw(>)3340 8871 y Ft( )3428 8888 y Fo(t)3492 8846 y Fu(6)p Fy(=)3696 8733 y Fw(@)g(V)p 3687 8808 V 3687 8960 a(@)g(y)3865 8985 y Ft(j)3956 8846 y Fy(\()p Fw(<)45 b(Q)i(>)4502 8871 y Ft( )4590 8888 y Fo(t)4653 8846 y Fy(\))p Fw(;)-4006 b FC(\(11.14\))758 9241 y(so)64 b(that)f(\(11.10-11.11\))h(are)f(not)h (ordinary)g(di\033eren)-5 b(tial)64 b(equations)f(satis\034ed)i(b)-5 b(y)64 b Fw(t)c Fu(2)g Fs(R)96 b Fu(!)758 9440 y Fy(\()p Fw(<)53 b(Q)h(>)1319 9465 y Ft( )1407 9482 y Fo(t)1470 9440 y Fw(;)28 b(<)53 b(P)77 b(>)2039 9465 y Ft( )2127 9482 y Fo(t)2190 9440 y Fy(\))53 b Fu(2)g Fs(R)2595 9380 y Fv(6)2679 9440 y FC(.)87 b(Here)59 b(I)g(use)i(the)e(short)h(hand)g (notation)g Fw(Q)53 b Fy(=)h(\()p Fw(Q)5991 9465 y Fv(1)6065 9440 y Fw(;)28 b(Q)6270 9465 y Fv(2)6345 9440 y Fw(;)g(Q)6550 9465 y Fv(3)6625 9440 y Fy(\))p FC(,)758 9639 y(and)52 b(similarly)i(for)e Fw(P)23 b FC(.)72 b(Equalit)-5 b(y)52 b(is)h(obtained)f(in)h(\(11.14\))e(for)h(arbitrary)g Fw( )5577 9664 y Ft(t)5688 9639 y FC(only)g(if)g(grad)6521 9664 y Ft(y)6601 9639 y Fw(V)758 9838 y FC(is)j(linear)g(in)h(the)e(co) 5 b(ordinates)55 b Fw(y)2827 9863 y Ft(j)2897 9838 y FC(,)h(suc)-5 b(h)56 b(as)f(when)g Fw(V)92 b FC(is)55 b(the)g(harmonic)h(oscillator)f(p)5 b(oten)-5 b(tial)778 9977 y Fv(1)p 778 10004 67 7 v 778 10100 a(2)864 10043 y Fw(m!)1119 9983 y Fv(2)1193 10043 y Fw(q)1273 9983 y Fv(2)1348 10043 y FC(.)76 b(A)-5 b(t)56 b(an)-5 b(y)56 b(rate,)g(equations)g(\(11.10-11.11\),)g(apart)g(from)g(suggesting)h(a) f(further)g(in-)758 10242 y(terpretation)47 b(of)h(the)h(w)-5 b(a)g(v)g(e)49 b(function)f(through)h(the)f(in)-5 b(tro)5 b(duction)49 b(of)f(the)g(mean)h(momen)-5 b(tum)758 10441 y Fw(<)46 b(P)1040 10466 y Ft(j)1157 10441 y Fw(>)1286 10466 y Ft( )1374 10483 y Fo(t)1437 10441 y FC(,)56 b(also)g(giv)-5 b(e)55 b(a)h(\034rst)g(indication)g(on)g(the)f(p)5 b(ossibilit)-5 b(y)57 b(of)e(describing)i(the)e(motion)h(of)758 10641 y(a)f(p)5 b(oin)-5 b(t)55 b(particle)f(through)h(the)g(use)g(of)g(the)f (Sc)-5 b(hr\366)5 b(dinger)56 b(equation.)73 b(The)55 b(Ehrenfest)f(equa-)758 10840 y(tions)40 b(suggest)h(the)f(existence)g (of)g(a)g(natural)h(link)f(b)5 b(et)-5 b(w)g(een)40 b(the)g(solutions)i (of)e(the)g(Sc)-5 b(hr\366)5 b(dinger)p eop %%Page: 31 31 31 30 bop 2362 861 a FA(QUANTUM)58 b(CHA)l(OS:)g(A)h(BRIEF)f(FIRST)g (VISIT)1472 b(31)758 1259 y FC(equation)49 b(and)h(of)f(the)h (Hamiltonian)g(equations)f(of)h(motion,)h(that)e(w)-5 b(e)50 b(shall)g(further)g(explore)758 1458 y(b)5 b(elo)-5 b(w.)1057 1658 y(Let)48 b(me)h(no)-5 b(w)49 b(pro)-5 b(v)g(e)50 b(\(11.10-11.11\).)71 b(I)49 b(will)g(only)g(giv)-5 b(e)49 b(a)g(formal)g(pro)5 b(of,)50 b(but)f(if)g(y)-5 b(ou)49 b(sup-)758 1857 y(p)5 b(ose)48 b(throughout)h(that)f Fw( )2447 1882 y Ft(t)2555 1857 y FC(b)5 b(elongs)49 b(to)f(the)g(Sc)-5 b(h)g(w)g(artz)50 b(space)f Fu(S)12 b Fy(\()p Fs(R)5056 1797 y Fv(3)5140 1857 y Fy(\))p FC(,)50 b(then)e(the)g(calculation)758 2056 y(b)5 b(elo)-5 b(w)71 b(is)g(completely)f(justi\034ed.)121 b(Recall)71 b(that)f Fw( )78 b Fu(2)72 b(S)12 b Fy(\()p Fs(R)4580 1996 y Fv(3)4664 2056 y Fy(\))70 b FC(if)h Fw( )78 b Fu(2)72 b Fw(C)5467 1996 y Fr(1)5607 2056 y Fy(\()p Fs(R)5795 1996 y Fv(3)5878 2056 y Fy(\))e FC(and)i Fw( )6 b FC(,)75 b(as)758 2255 y(w)-5 b(ell)59 b(as)h(all)g(its)f(deriv)-9 b(ativ)k(es,)60 b(decreases)g(faster)f(than)h(an)-5 b(y)59 b(in)-5 b(v)g(erse)60 b(p)5 b(o)-5 b(w)g(er)60 b(of)f Fu(jj)p Fw(y)6 b Fu(jj)p FC(.)87 b(T)-5 b(ypical)758 2455 y(examples)55 b(are)g Fw( )6 b Fy(\()p Fw(y)g Fy(\))46 b(=)g Fw(p)p Fy(\()p Fw(y)6 b Fy(\))28 b(exp)f Fu(\000jj)p Fw(y)6 b Fu(jj)3312 2394 y Fv(2)3442 2455 y FC(where)55 b Fw(p)p Fy(\()p Fw(y)6 b Fy(\))55 b FC(is)h(a)f(p)5 b(olynomial.)1799 2711 y Fw(d)p 1769 2785 147 7 v 1769 2937 a(dt)1982 2824 y(<)46 b(Q)2288 2849 y Ft(j)2404 2824 y Fw(>)2533 2849 y Ft( )2621 2866 y Fo(t)2685 2824 y Fy(=)2860 2598 y Fp(Z)3054 2824 y Fw(y)3135 2849 y Ft(j)p 3205 2688 314 7 v 3205 2824 a Fw(@)3293 2849 y Ft(t)3352 2824 y Fw( )3460 2849 y Ft(t)3518 2824 y Fy(\()p Fw(y)6 b Fy(\))p Fw( )3843 2849 y Ft(t)3902 2824 y Fy(\()p Fw(y)g Fy(\))p Fw(d)4205 2755 y Fv(3)4279 2824 y Fw(y)43 b Fy(+)4569 2598 y Fp(Z)4763 2824 y Fw(y)4844 2849 y Ft(j)p 4914 2688 167 7 v 4914 2824 a Fw( )5022 2849 y Ft(t)5081 2824 y Fy(\()p Fw(y)6 b Fy(\)\()p Fw(@)5451 2849 y Ft(t)5509 2824 y Fw( )5617 2849 y Ft(t)5675 2824 y Fy(\)\()p Fw(y)g Fy(\))p Fw(d)6043 2755 y Fv(3)6117 2824 y Fw(y)g(:)-5492 b FC(\(11.15\))758 3191 y(No)-5 b(w)55 b(use)h(\(11.4\))e(to)h(replace)g Fw(@)2756 3216 y Ft(t)2815 3191 y Fw( )2923 3216 y Ft(t)2981 3191 y FC(:)1165 3461 y Fw(d)p 1135 3535 147 7 v 1135 3687 a(dt)1347 3573 y(<)46 b(Q)1653 3598 y Ft(j)1770 3573 y Fw(>)1899 3598 y Ft( )1987 3615 y Fo(t)2318 3573 y Fy(=)2714 3347 y Fp(Z)2908 3573 y Fw(y)2989 3598 y Ft(j)3087 3339 y Fp(\024)3175 3573 y Fu(\000)p Fw(i)3450 3461 y Fh(~)p 3381 3535 229 7 v 3381 3687 a Fy(2)p Fw(m)3630 3573 y Fy(\(\001)p 3833 3438 167 7 v Fw( )3941 3598 y Ft(t)3999 3573 y Fy(\()p Fw(y)6 b Fy(\))37 b(+)4455 3461 y Fw(i)p 4439 3535 90 7 v 4439 3687 a Fh(~)4548 3573 y Fw(V)g Fy(\()p Fw(y)6 b Fy(\))p 4899 3438 167 7 v Fw( )5007 3598 y Ft(t)5065 3573 y Fy(\()p Fw(y)g Fy(\))5282 3339 y Fp(\025)5397 3573 y Fw( )5505 3598 y Ft(t)5564 3573 y Fy(\()p Fw(y)g Fy(\))p Fw(d)5867 3504 y Fv(3)5941 3573 y Fw(y)2714 4038 y Fy(+)2871 3812 y Fp(Z)3065 4038 y Fw(y)3146 4063 y Ft(j)p 3216 3903 V 3216 4038 a Fw( )3324 4063 y Ft(t)3383 4038 y Fy(\()p Fw(y)g Fy(\))3628 3804 y Fp(\024)3715 4038 y Fw(i)3861 3925 y Fh(~)p 3792 4000 229 7 v 3792 4152 a Fy(2)p Fw(m)4040 4038 y Fy(\(\001)p Fw( )4351 4063 y Ft(t)4410 4038 y Fy(\)\()p Fw(y)g Fy(\))36 b Fu(\000)4930 3925 y Fw(i)p 4914 4000 90 7 v 4914 4152 a Fh(~)5024 4038 y Fw(V)g Fy(\()p Fw(y)6 b Fy(\))p Fw( )5482 4063 y Ft(t)5540 4038 y Fy(\()p Fw(y)g 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Fy(=)166 b Fw(p)2796 10072 y Ft(j)2895 10003 y Fp(b)2865 10047 y Fw( )7 b Fy(\()p Fw(p)p Fy(\))p Fw(;)758 10329 y FC(it)55 b(is)h(then)f(easy)g(to)g(see)g(that)g(w)-5 b(e)55 b(can)h(rewrite)e(\(11.13\))h(as)2635 10697 y Fw(<)47 b(P)2918 10722 y Ft(j)3033 10697 y Fw(>)3162 10722 y Ft( )3250 10739 y Fo(t)3314 10697 y Fy(=)3489 10471 y Fp(Z)3683 10697 y Fw(p)3767 10722 y Ft(j)3882 10697 y Fu(j)4004 10653 y Fp(b)3975 10697 y Fw( )4083 10722 y Ft(t)4141 10697 y Fy(\()p Fw(p)p Fy(\))e Fu(j)4446 10628 y Fv(2)4567 10697 y Fw(d)4653 10628 y Fv(3)4728 10697 y Fw(p:)-4100 b FC(\(11.18\))p eop %%Page: 32 32 32 31 bop 758 878 a FA(32)2127 b(STEPHAN)58 b(DE)h(BI\310VRE)758 1259 y FC(This)46 b(strongly)g(suggests)g(that)f(w)-5 b(e)46 b(should)h(in)-5 b(terpret)45 b Fu(j)4300 1215 y Fp(b)4270 1259 y Fw( )4378 1284 y Ft(t)4483 1259 y Fu(j)4529 1199 y Fv(2)4650 1259 y Fy(\()p Fw(p)p Fy(\))f FC(as)i(the)g(probabilit)-5 b(y)46 b(densit)-5 b(y)758 1458 y(of)51 b(linear)i(momen)-5 b(tum,)54 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Fu(j)2002 7914 y Fp(b)1972 7958 y Fw( )2080 7983 y Ft(t)2196 7958 y Fu(j)2242 7898 y Fv(2)2373 7958 y Fy(\()p Fw(p)p Fy(\))p FC(,)g(so)h(that)e(the)g (notion)h(of)g(tra)9 b(jectory)60 b(disapp)5 b(ears)63 b(in)f(quan)-5 b(tum)758 8157 y(mec)g(hanics.)69 b(In)38 b(view)f(of)h(the)g(suggestiv)-5 b(e)38 b(form)g(of)g(\(11.8\),)j(I)d (supp)5 b(ose)39 b(y)-5 b(ou)38 b(w)-5 b(on't)38 b(b)5 b(e)38 b(surprised)758 8357 y(if)55 b(I)g(no)-5 b(w)56 b(tell)f(y)-5 b(ou)55 b(that)g(one)g(in)-5 b(terprets)2988 8665 y Fw(<)47 b(H)59 b(>)3490 8690 y Ft( )3578 8707 y Fo(t)3642 8665 y Fu(\021)46 b Fy(\()p Fw( )3990 8690 y Ft(t)4048 8665 y Fw(;)28 b(H)13 b( )4381 8690 y Ft(t)4440 8665 y Fy(\))-3747 b FC(\(11.20\))758 8973 y(as)78 b(the)f(mean)h (energy)f(of)g(the)g(particle)g(at)h(time)f Fw(t)p FC(.)140 b(Here)77 b(also,)84 b(there)77 b(is)h(a)g(probabilit)-5 b(y)758 9172 y(measure)50 b(hidden)g(in)f(the)g(bac)-5 b(kground,)51 b(but)f(it)e(is)i(harder)g(\(in)f(fact)f(imp)5 b(ossible\))51 b(to)d(compute)758 9371 y(in)53 b(general,)g(and)g(I)g (will)g(come)g(bac)-5 b(k)53 b(to)f(this)h(p)5 b(oin)-5 b(t)53 b(later)f(on.)73 b(A)53 b(calculation)f(similar)i(to)f(the)758 9570 y(one)i(leading)h(to)f(\(11.10-11.11\))g(sho)-5 b(ws)3224 9852 y Fw(d)p 3194 9927 147 7 v 3194 10079 a(dt)3407 9965 y(<)46 b(H)60 b(>)3909 9990 y Ft( )3997 10007 y Fo(t)4060 9965 y Fy(=)46 b(0)-3560 b FC(\(11.21\))758 10319 y(if)59 b Fw( )1022 10344 y Ft(t)1140 10319 y FC(is)h(a)g (solution)g(to)f(the)g(Sc)-5 b(hr\366)5 b(dinger)60 b(equation.)86 b(Y)-14 b(ou)60 b(will)f(recognize)g(here)h(the)f(la)-5 b(w)60 b(of)758 10519 y(conserv)-9 b(ation)55 b(of)g(energy)-14 b(.)1057 10840 y Fx(Exer)-5 b(cise)62 b FC(11.1)p Fx(.)83 b FC(Pro)-5 b(v)g(e)55 b(\(11.21\).)p eop %%Page: 33 33 33 32 bop 2362 861 a FA(QUANTUM)58 b(CHA)l(OS:)g(A)h(BRIEF)f(FIRST)g (VISIT)1472 b(33)1057 1259 y FC(Quan)-5 b(tities)38 b(suc)-5 b(h)38 b(as)g(p)5 b(osition,)42 b(momen)-5 b(tum,)42 b(and)c(energy)f(are)h(examples)f(of)h(what)f(ph)-5 b(ysi-)758 1458 y(cists)59 b(call)f(\020observ)-9 b(ables\021.)85 b(In)59 b(classical)g(mec)-5 b(hanics,)61 b(observ)-9 b(ables)59 b(are)g(represen)-5 b(ted)58 b(b)-5 b(y)59 b(real)758 1658 y(v)-9 b(alued)55 b(functions)h Fw(f)74 b FC(on)56 b(phase)g(space.)76 b(As)56 b(w)-5 b(e)55 b(sa)-5 b(w)57 b(on)f(some)g(examples,)g(in)g(quan)-5 b(tum)57 b(me-)758 1857 y(c)-5 b(hanics)70 b(they)f(are)g(represen)-5 b(ted)70 b(b)-5 b(y)70 b(self-adjoin)-5 b(t)70 b(op)5 b(erators)69 b Fw(F)92 b FC(on)70 b Fw(L)5344 1797 y Fv(2)5418 1857 y Fy(\()p Fs(R)5606 1797 y Fv(3)5689 1857 y Fy(\))p FC(.)116 b(Recall)70 b(that)758 2056 y(self-adjoin)-5 b(t)56 b(op)5 b(erators)55 b(ha)-5 b(v)g(e)55 b(the)g(prop)5 b(ert)-5 b(y)55 b(that)3090 2360 y Fy(\()p Fw(\036;)28 b(F)23 b( )6 b Fy(\))45 b(=)h(\()p Fw(F)23 b(\036;)28 b( )6 b Fy(\))-3646 b FC(\(11.22\))758 2673 y(whic)-5 b(h)55 b(implies)h(that)e(their)g(diagonal)h(matrix)g(elemen)-5 b(ts)55 b(are)f(real.)74 b(Note)53 b(that)h(\(11.22\))g(do)5 b(es)758 2872 y(indeed)56 b(hold)g(for)g Fw(Q)2039 2897 y Ft(j)2109 2872 y Fw(;)28 b(P)2290 2897 y Ft(j)2359 2872 y FC(,)56 b(and)h(for)e Fw(H)13 b FC(,)57 b(as)f(y)-5 b(ou)56 b(can)g(c)-5 b(hec)g(k)56 b(easily)g(with)f(a)h(few)f(more)h (partial)758 3072 y(in)-5 b(tegrations)75 b(as)g(in)g(the)f(previous)h (subsection.)133 b(This)75 b(allo)-5 b(ws)76 b(for)e(the)g(in)-5 b(terpretation)75 b(of)758 3271 y(the)65 b(matrix)g(elemen)-5 b(t)66 b Fy(\()p Fw( )2401 3296 y Ft(t)2459 3271 y Fw(;)28 b(F)23 b( )2771 3296 y Ft(t)2830 3271 y Fy(\))65 b FC(as)h(the)f(mean)i (v)-9 b(alue)65 b(of)g(the)h(corresp)5 b(onding)66 b(observ)-9 b(able.)758 3470 y(As)61 b(a)g(last)h(example,)g(the)f(op)5 b(erators)61 b(corresp)5 b(onding)62 b(to)f(the)g(comp)5 b(onen)-5 b(ts)62 b(of)f(the)g(angular)758 3669 y(momen)-5 b(tum)56 b(v)-5 b(ector)55 b(are,)g(to)g(nob)5 b(o)g(dy's)55 b(surprise,)i(I)e(hop)5 b(e,)1770 3973 y Fw(L)1883 3998 y Fv(1)2003 3973 y Fy(=)46 b Fw(Q)2309 3998 y Fv(2)2384 3973 y Fw(P)2491 3998 y Fv(3)2602 3973 y Fu(\000)37 b Fw(Q)2899 3998 y Fv(3)2974 3973 y Fw(P)3081 3998 y Fv(2)3155 3973 y Fw(;)83 b(L)3397 3998 y Fv(2)3518 3973 y Fy(=)46 b Fw(Q)3824 3998 y Fv(3)3899 3973 y Fw(P)4006 3998 y Fv(1)4117 3973 y Fu(\000)37 b Fw(Q)4414 3998 y Fv(1)4489 3973 y Fw(P)4596 3998 y Fv(3)4670 3973 y Fw(;)83 b(L)4912 3998 y Fv(3)5033 3973 y Fy(=)46 b Fw(Q)5339 3998 y Fv(1)5414 3973 y Fw(P)5521 3998 y Fv(2)5632 3973 y Fu(\000)37 b Fw(Q)5929 3998 y Fv(2)6003 3973 y Fw(P)6110 3998 y Fv(1)6184 3973 y Fw(:)-5472 b FC(\(11.23\))758 4287 y(There)61 b(is)h(a)g(b)5 b(eautiful)61 b(analog)h(to)g(\(5.15\))e(in)j(quan)-5 b(tum)62 b(mec)-5 b(hanics.)94 b(Supp)5 b(ose)62 b Fw(F)85 b FC(is)62 b(a)g(self-)758 4486 y(adjoin)-5 b(t)55 b(op)5 b(erator)55 b(represen)-5 b(ting)56 b(some)g(ph)-5 b(ysical)56 b(observ)-9 b(able)55 b(and)h(compute)1897 4754 y Fw(d)p 1867 4829 147 7 v 1867 4981 a(dt)2034 4867 y Fy(\()p Fw( )2207 4892 y Ft(t)2265 4867 y Fw(;)28 b(F)23 b( 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b(are)g(the)f (so-called)i Fz(c)-8 b(anonic)g(al)88 b(c)-8 b(ommutation)87 b(r)-8 b(elations)86 b FC(\(CCR\).)g(Compute)g(also)758 9724 y Fy([)p Fw(L)917 9749 y Fv(1)991 9724 y Fw(;)28 b(L)1178 9749 y Fv(2)1253 9724 y Fy(])p Fw(;)g Fy([)p Fw(Q)1550 9664 y Fv(2)1550 9767 y Ft(j)1624 9724 y Fw(;)g(P)1828 9664 y Fv(2)1805 9771 y Ft(k)1902 9724 y Fy(])49 b FC(and)h(compare)f (the)f(result)h(to)g Fu(f)p Fw(L)4100 9749 y Fv(1)4174 9724 y Fw(;)28 b(L)4361 9749 y Fv(2)4436 9724 y Fu(g)p Fw(;)g Fu(f)p Fw(q)4756 9664 y Fv(2)4750 9767 y Ft(j)4830 9724 y Fw(;)g(p)4988 9664 y Fv(2)4988 9771 y Ft(k)5069 9724 y Fu(g)p FC(.)72 b(Do)5 b(es)48 b(this)i(lo)5 b(ok)48 b(in)-5 b(ter-)758 9923 y(esting?)1057 10242 y(In)59 b(section)g(5,)h(I)f(p)5 b(oin)-5 b(ted)59 b(out)g(that)g(Hamilton's)g (equations)g(of)g(motion)g(can)g(b)5 b(e)59 b(gener-)758 10441 y(alized)e(b)-5 b(y)57 b(replacing)g Fw(H)71 b FC(with)57 b(an)-5 b(y)57 b(function)g Fw(g)63 b FC(on)58 b(phase)f(space)h Fs(R)5137 10381 y Fv(2)p Ft(n)5302 10441 y FC(,)g(whic)-5 b(h)57 b(leads)h(to)f(the)758 10641 y(notion)62 b(of)g(a)g(Hamiltonian)g(\035o)-5 b(w)62 b(on)g Fs(R)3295 10580 y Fv(2)q Ft(n)3461 10641 y FC(.)94 b(What)61 b(w)-5 b(ould)63 b(b)5 b(e)62 b(the)f(corresp)5 b(onding)63 b(general-)758 10840 y(ization)55 b(of)g(the)g(Sc)-5 b(hr\366)5 b(dinger)56 b(equation)f(\(11.7\)?)73 b(I)55 b(start)g(with)g(a)h(de\034nition.)p eop %%Page: 34 34 34 33 bop 758 878 a FA(34)2127 b(STEPHAN)58 b(DE)h(BI\310VRE)1057 1259 y Fx(Definition)i FC(11.3)p Fx(.)83 b FC(A)69 b(map)h Fw(t)f Fu(2)h Fs(R)105 b Fu(!)69 b Fw(U)3863 1284 y Ft(t)3992 1259 y Fu(2)g(B)5 b Fy(\()p Fw(L)4464 1199 y Fv(2)4538 1259 y Fy(\()p Fs(R)4726 1199 y Fv(3)4809 1259 y Fy(\)\))69 b FC(is)g(called)h(a)f Fz(unitary)j(one-)758 1458 y(p)-8 b(ar)g(ameter)58 b(gr)-8 b(oup)55 b FC(i\033)2329 1743 y Fy(\()p Fw(i)p Fy(\))344 b Fw(U)2973 1768 y Fv(0)3392 1743 y Fy(=)46 b(id)3706 1773 y Ft(L)3797 1740 y Fn(2)3862 1773 y Fv(\()p Fq(R)3991 1740 y Fn(3)4062 1773 y Fv(\))4122 1743 y Fw(;)2272 2012 y Fy(\()p Fw(ii)p Fy(\))166 b Fu(k)46 b Fw(U)2924 2037 y Ft(t)2983 2012 y Fw( )52 b Fu(k)166 b Fy(=)p Fu(k)46 b Fw( )53 b Fu(k)378 b(8)p Fw( )52 b Fu(2)46 b Fw(L)4794 1943 y Fv(2)4869 2012 y Fy(\()p Fs(R)5056 1943 y Fv(3)5140 2012 y Fy(\))p Fw(;)2215 2263 y Fy(\()p Fw(iii)p Fy(\))254 b Fw(U)2883 2288 y Ft(t)p Fv(+)p Ft(t)3085 2255 y Fi(0)3392 2263 y Fy(=)46 b Fw(U)3680 2288 y Ft(t)3776 2263 y Fu(\016)37 b Fw(U)4009 2288 y Ft(t)4059 2255 y Fi(0)4445 2263 y Fu(8)p Fw(t;)28 b(t)4731 2195 y Fr(0)4823 2263 y Fu(2)46 b Fs(R)5103 2195 y Fv(2)5187 2263 y Fw(:)1057 2571 y FC(I)51 b(will)h(no)-5 b(w)53 b(argue)e(that)g(this)i(is)f(the)f (prop)5 b(er)52 b(analog)g(of)g(the)f(classical)h(notion)g(of)g(Hamil-) 758 2770 y(tonian)65 b(\035o)-5 b(w.)103 b(T)-14 b(o)65 b(see)g(this,)j(remark)d(that)f(for)h(an)-5 b(y)65 b(self-adjoin)-5 b(t)66 b(op)5 b(erator)64 b Fw(G)h FC(on)g Fw(L)6263 2710 y Fv(2)6338 2770 y Fy(\()p Fs(R)6525 2710 y Ft(n)6625 2770 y Fy(\))p FC(,)758 2970 y(w)-5 b(e)55 b(can)g(write)3220 3254 y Fw(i)p Fh(~)p Fw(@)3455 3279 y Ft(t)3514 3254 y Fw( )3622 3279 y Ft(t)3726 3254 y Fy(=)47 b Fw(G )4141 3279 y Ft(t)4199 3254 y Fw(;)-3487 b FC(\(11.27\))758 3549 y(in)67 b(analogy)h(with)f(\(11.7\).)109 b(F)-14 b(rom)68 b(this,)j(it)c(follo)-5 b(ws)68 b(as)g(ab)5 b(o)-5 b(v)g(e)67 b(that,)j(if)d Fw(F)90 b FC(is)68 b(a)g(self-adjoin) -5 b(t)758 3748 y(op)5 b(erator,)54 b(then)2779 4007 y Fw(d)p 2749 4081 147 7 v 2749 4234 a(dt)2961 4120 y(<)47 b(F)69 b(>)3442 4145 y Ft( )3530 4162 y Fo(t)3593 4120 y Fy(=)p Fw(<)3917 4007 y Fy([)p Fw(F)5 b(;)28 b(G)p Fy(])p 3917 4081 408 7 v 4048 4234 a Fw(i)p Fh(~)4391 4120 y Fw(>)4520 4145 y Ft( )4608 4162 y Fo(t)4718 4120 y Fw(:)-4006 b FC(\(11.28\))758 4460 y(It)55 b(is)g(tempting)h(to)e (solv)-5 b(e)56 b(\(11.27\))f(formally)g(b)-5 b(y)56 b(writing)2923 4844 y Fw( )3031 4869 y Ft(t)3136 4844 y Fy(=)3311 4610 y Fp(\022)3433 4844 y Fy(exp)27 b Fu(\000)3880 4732 y Fw(i)p 3863 4806 90 7 v 3863 4958 a Fh(~)3973 4844 y Fw(Gt)4164 4610 y Fp(\023)4314 4844 y Fw( )4422 4869 y Fv(0)4496 4844 y Fw(;)-3784 b FC(\(11.29\))758 5219 y(where)2567 5595 y Fy(exp)27 b Fu(\000)3014 5483 y Fw(i)p 2997 5557 V 2997 5709 a Fh(~)3107 5595 y Fw(Gt)46 b Fy(=)3578 5387 y Fr(1)3524 5437 y Fp(X)3519 5789 y Ft(n)p Fv(=0)3848 5483 y Fy(1)p 3817 5557 146 7 v 3817 5709 a Fw(n)p Fy(!)4010 5361 y Fp(\022)4152 5483 y Fu(\000)p Fw(it)p 4152 5557 247 7 v 4231 5709 a Fh(~)4419 5361 y Fp(\023)4541 5395 y Ft(n)4659 5595 y Fw(G)4790 5526 y Ft(n)4880 5595 y Fw(:)-4168 b FC(\(11.30\))758 6050 y(W)-14 b(riting)59 b Fw(U)1502 5990 y Ft(G)1484 6091 y(t)1667 6050 y Fy(=)53 b(exp)27 b Fu(\000)2293 5984 y Ft(i)p 2279 6011 74 7 v 2279 6107 a Fq(~)2373 6050 y Fw(Gt)p FC(,)60 b(a)f(further)g(formal)h(calculation)f(sho)-5 b(ws)61 b(that)e Fw(U)5720 5990 y Ft(G)5702 6091 y(t)5891 6050 y FC(is)h(a)f(unitary)758 6249 y(one-parameter)66 b(group.)108 b(All)66 b(of)g(this)h(can)g(b)5 b(e)66 b(turned)g(in)-5 b(to)67 b(prop)5 b(erly)66 b(form)-5 b(ulated)67 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b(:)g(:)g(:)f(;)h(p)4541 10524 y Ft(j)4647 10499 y Fy(+)37 b Fw(t;)28 b(:)g(:)g(:)f(;)h(p)5326 10524 y Ft(n)5416 10499 y Fy(\))758 10840 y FC(\(and)55 b(to)g(compare)g(all)h(this)g(to)f(Exercise)f(5.4)i(\(iii\)\).)p eop %%Page: 35 35 35 34 bop 2362 861 a FA(QUANTUM)58 b(CHA)l(OS:)g(A)h(BRIEF)f(FIRST)g (VISIT)1472 b(35)1057 1259 y Fx(Exer)-5 b(cise)62 b FC(11.4)p Fx(.)83 b FC(Sho)-5 b(w)56 b(that,)f(for)g Fw(b)46 b Fu(2)g Fs(R)3775 1199 y Ft(n)3874 1259 y Fw(;)28 b( )4056 1284 y Fv(0)4177 1259 y Fu(2)46 b Fw(L)4447 1199 y Fv(2)4521 1259 y Fy(\()p Fs(R)4709 1199 y Ft(n)4808 1259 y Fy(\))p FC(,)2558 1613 y Fy([\(exp)27 b Fu(\000)3116 1500 y Fw(i)p 3099 1574 90 7 v 3099 1726 a Fh(~)3209 1613 y Fw(b)37 b Fu(\001)g Fw(P)23 b Fy(\))p Fw( )3703 1638 y Fv(0)3777 1613 y Fy(]\()p Fw(y)6 b Fy(\))46 b(=)g Fw( )4369 1638 y Fv(0)4444 1613 y Fy(\()p Fw(y)d Fu(\000)37 b Fw(b)p Fy(\))-4177 b FC(\(11.31\))758 1931 y(and)2610 2243 y Fy([\(exp)3039 2131 y Fw(i)p 3022 2205 V 3022 2357 a Fh(~)3132 2243 y 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Fy(=)46 b Fw(E)3926 7649 y Ft(n)4016 7624 y Fy(\()p Fh(~)p Fy(\))p Fw(')4345 7556 y Fq(~)4345 7665 y Ft(n)4434 7624 y Fw(:)758 7897 y FC(In)37 b(terms)g(of)f(the)h (eigen)-5 b(v)c(alues)37 b Fw(E)2792 7922 y Ft(n)2882 7897 y Fy(\()p Fh(~)p Fy(\))f FC(and)h(eigenfunctions)g Fw(')4617 7837 y Fq(~)4617 7938 y Ft(n)4744 7897 y FC(the)g(solutions)h (of)e(the)h(Sc)-5 b(hr\366)5 b(dinger)758 8096 y(equation)55 b(can)g(then)g(b)5 b(e)55 b(written)2704 8408 y Fw( )2812 8433 y Ft(t)2916 8408 y Fy(=)3092 8250 y Fp(X)3171 8599 y Ft(n)3332 8408 y Fy(\()p Fw(')3506 8339 y Fq(~)3506 8449 y Ft(n)3595 8408 y Fw(;)28 b( )6 b Fy(\))p Fw(e)3925 8339 y Fr(\000)4049 8281 y Fo(itE)4224 8298 y(n)4305 8281 y Fn(\()p Ff(~)p Fn(\))p 4049 8313 412 6 v 4222 8379 a Ff(~)4544 8408 y Fw(')4653 8339 y Fq(~)4653 8449 y Ft(n)4743 8408 y Fw(:)758 8820 y FC(In)41 b(this)g(sense,)k(studying) d(the)e(solutions)i(of)f(the)g(Sc)-5 b(hr\366)5 b(dinger)41 b(equation)g(reduces)g(to)g(studying)758 9019 y(the)65 b(eigenfunctions)h(and)h(eigen)-5 b(v)c(alues)66 b(of)g(the)f(Sc)-5 b(hr\366)5 b(dinger)67 b(op)5 b(erator.)105 b(Except)65 b(for)h(some)758 9218 y(v)-5 b(ery)71 b(simple)h(cases)g(\(suc)-5 b(h)72 b(as)g Fw(V)37 b Fy(\()p Fw(q)6 b Fy(\))72 b(=)3418 9153 y Fv(1)p 3418 9180 67 7 v 3418 9276 a(2)3504 9218 y Fw(m!)3759 9158 y Fv(2)3834 9218 y Fw(q)3914 9158 y Fv(2)3988 9218 y FC(\),)j(it)c(is)i(ho)-5 b(w)g(ev)g(er)71 b(hard)h(to)f(obtain)h(m)-5 b(uc)g(h)758 9418 y(precise)37 b(information)g(ab)5 b(out)37 b(those)g(ob)9 b(jects.)68 b(As)37 b(p)5 b(oin)-5 b(ted)37 b(out)g(b)5 b(efore,)40 b(one)d(approac)-5 b(h)38 b(is)f(then)758 9617 y(to)j(study)g(the)g (problem)h(asymptotically)f(as)h Fh(~)f FC(tends)g(to)g(zero)g(and)h (to)f(analyze)g(the)f(prop)5 b(erties)758 9816 y(of)64 b(the)f Fw(E)1373 9841 y Ft(n)1463 9816 y Fy(\()p Fh(~)p Fy(\))p Fw(;)28 b(')1866 9756 y Fq(~)1866 9857 y Ft(n)2019 9816 y FC(in)65 b(this)f(limit,)j(called)d(the)g Fz(semi-classic)-8 b(al)65 b FC(regime.)100 b(It)64 b(turns)h(out)f(that,)758 10015 y(for)46 b(small)h Fh(~)p FC(,)h(the)e(prop)5 b(erties)46 b(of)g(those)g(quan)-5 b(tities)47 b(are)f(closely)g(related)g(to)g (prop)5 b(erties)46 b(of)g(the)758 10242 y(Hamiltonian)62 b(\035o)-5 b(w)63 b(generated)f(b)-5 b(y)63 b Fw(H)13 b Fy(\()p Fw(x)p Fy(\))58 b(=)3740 10168 y Ft(p)3808 10118 y Fn(2)p 3715 10204 185 7 v 3715 10299 a Fv(2)p Ft(m)3961 10242 y Fy(+)42 b Fw(V)36 b Fy(\()p Fw(q)6 b Fy(\))62 b FC(on)h(the)f(energy)g(surface)g Fw(H)13 b Fy(\()p Fw(x)p Fy(\))58 b(=)758 10441 y Fw(E)881 10466 y Ft(n)971 10441 y Fy(\()p Fh(~)p Fy(\))p FC(.)71 b(There)52 b(is)g(no)g(w)-5 b(a)g(y)52 b(I)g(can)g(go)f(in)-5 b(to)52 b(this)h(here,)f(or)g(ev)-5 b(en)51 b(to)h(giv)-5 b(e)51 b(a)h(simple)h(example)f(of)758 10641 y(this)g(phenomenon)h(in)f(some)g (detail,)g(but)g(I)g(will)g(illustrate)f(one)h(asp)5 b(ect)51 b(of)h(it)f(in)h(the)f(con)-5 b(text)758 10840 y(of)40 b(the)h(h)-5 b(yp)5 b(erb)g(olic)41 b(toral)g(automorphisms)i (b)-5 b(y)41 b(pro)-5 b(ving)42 b(the)e(Sc)-5 b(hnirelman)43 b(theorem)e(\(section)p eop %%Page: 36 36 36 35 bop 758 878 a FA(36)2127 b(STEPHAN)58 b(DE)h(BI\310VRE)758 1259 y FC(14\).)107 b(F)-14 b(or)67 b(more)g(information)g(on)g(the)f (enormous)i(\034eld)f(that)f(semi-classical)j(analysis)f(has)758 1458 y(b)5 b(ecome)55 b(o)-5 b(v)g(er)55 b(the)g(last)g(thirt)-5 b(y)55 b(y)-5 b(ears)56 b(y)-5 b(ou)55 b(ma)-5 b(y)56 b(consult)g([)p FD(H)p FC(][)p FD(I)q FC(][)p FD(R)p FC(])g(\(as)f(a)g(start!\).)1057 1801 y FD(11.5.)96 b(Billiards)90 b(again.)82 b FC(W)-14 b(e)81 b(ha)-5 b(v)g(e)82 b(seen,)88 b(in)82 b(section)f(6,)88 b(that)81 b(billiard)h(\035o)-5 b(ws)82 b(are)758 2000 y(Hamiltonian)63 b(systems.)100 b(In)63 b(the)g(previous)h(subsections,)k(I)63 b(explained)h(ho)-5 b(w)64 b(quan)-5 b(tum)64 b(me-)758 2227 y(c)-5 b(hanics)55 b(asso)5 b(ciates)55 b(to)g(eac)-5 b(h)55 b(Hamiltonian)f(of)h(the)f(t) -5 b(yp)5 b(e)54 b Fw(H)13 b Fy(\()p Fw(x)p Fy(\))46 b(=)5115 2153 y Ft(p)5183 2102 y Fn(2)p 5090 2188 185 7 v 5090 2284 a Fv(2)p Ft(m)5330 2227 y Fy(+)35 b Fw(V)i Fy(\()p Fw(q)6 b Fy(\))54 b FC(on)h Fs(R)6245 2166 y Fv(3)6364 2227 y Fu(\002)36 b Fs(R)6652 2166 y Fv(3)758 2470 y FC(a)61 b(self-adjoin)-5 b(t)62 b(op)5 b(erator)61 b Fw(H)70 b Fy(=)57 b Fu(\000)3003 2404 y Fq(~)3077 2354 y Fn(2)p 2981 2431 V 2981 2527 a Fv(2)p Ft(m)3185 2470 y Fy(\001)41 b(+)g Fw(V)98 b FC(on)62 b Fw(L)4079 2409 y Fv(2)4154 2470 y Fy(\()p Fs(R)4341 2409 y Ft(n)4440 2470 y Fy(\))p FC(.)92 b(Applying)62 b(this)g(precept)f(to)g(the)758 2669 y(family)54 b(of)h(Hamiltonians)g Fw(H)2610 2694 y Ft(\017)2729 2669 y FC(in)-5 b(tro)5 b(duced)55 b(in)g(section)g(6,)g (yields)g(the)f(family)h(of)g(self-adjoin)-5 b(t)758 2887 y(op)5 b(erators)54 b Fw(H)1630 2912 y Ft(\017)1740 2887 y Fy(=)46 b Fu(\000)2087 2821 y Fq(~)2161 2771 y Fn(2)p 2064 2848 V 2064 2944 a Fv(2)p Ft(m)2268 2887 y Fy(\001)36 b(+)g Fw(V)2704 2912 y Ft(\017)2822 2887 y FC(on)55 b Fw(L)3165 2826 y Fv(2)3239 2887 y Fy(\()p Fw(D)3441 2912 y Ft(a)p Fv(+)p Ft(\017)3679 2887 y Fy(\))p FC(.)73 b(As)55 b(in)g(the)f(previous)h(subsection,)h(it)e(can)h(b)5 b(e)758 3096 y(sho)-5 b(wn)56 b([)p FD(RS)q FC(])f(that)g(there)f (exists)i(an)f(orthonormal)h(basis)h Fw(')4599 3035 y Ft(\017)4599 3137 y(n)4744 3096 y FC(of)e Fw(L)5046 3035 y Fv(2)5121 3096 y Fy(\()p Fw(D)5323 3121 y Ft(a)p Fv(+)p Ft(\017)5560 3096 y Fy(\))g FC(suc)-5 b(h)57 b(that)3131 3386 y Fw(H)3269 3411 y Ft(\017)3333 3386 y Fw(')3442 3318 y Ft(\017)3442 3428 y(n)3578 3386 y Fy(=)46 b Fw(E)3876 3411 y Ft(n)3966 3386 y Fy(\()p Fw(\017)p Fy(\))p Fw(')4272 3318 y Ft(\017)4272 3428 y(n)758 3677 y FC(for)55 b(a)g(sequence)1534 3968 y Fy(0)46 b Fu(\024)g Fw(E)1961 3993 y Fv(0)2035 3968 y Fy(\()p Fw(\017)p Fy(\))g Fu(\024)g Fw(E)2576 3993 y Fv(1)2650 3968 y Fy(\()p Fw(\017)p Fy(\))g Fu(\024)g(\001)28 b(\001)g(\001)46 b(\024)g Fw(E)3606 3993 y Ft(n)3696 3968 y Fy(\()p Fw(\017)p Fy(\))g Fu(\024)g Fw(:)28 b(:)g(:)g(;)418 b Fy(lim)4741 4068 y Ft(n)p Fr(!1)5116 3968 y Fw(E)5239 3993 y Ft(n)5329 3968 y Fy(\()p Fw(\017)p Fy(\))46 b(=)g Fu(1)p Fw(:)758 4361 y FC(Moreo)-5 b(v)g(er,)64 b Fy(lim)1773 4386 y Ft(\017)p Fr(!)p Fv(0)2063 4361 y Fw(E)2186 4386 y Ft(n)2276 4361 y Fy(\()p Fw(\017)p Fy(\))57 b(=)2760 4296 y Fq(~)2834 4246 y Fn(2)p 2737 4323 V 2737 4419 a Fv(2)p Ft(m)2941 4361 y Fw(\025)3038 4386 y Ft(n)3128 4361 y FC(,)64 b(where)e Fw(\025)3822 4386 y Ft(n)3975 4361 y FC(is)h(the)f Fw(n)p FC(-th)g(eigen)-5 b(v)c(alue)62 b(of)g(the)g(Diric)-5 b(hlet)758 4561 y(Laplacian)49 b(on)g Fw(D)1871 4586 y Ft(a)1952 4561 y FC(.)71 b(These)49 b(results)h(generalize,)g(as)f(in)g(the)g(classical)h(case,)g(in)f(a)g (natural)g(w)-5 b(a)g(y)758 4760 y(to)61 b(more)g(general)g(billiard)i (shap)5 b(es.)92 b(W)-14 b(e)61 b(w)-5 b(on't)62 b(need)f(these)g (results)h(and)g(I)f(will)h(not)f(pro)-5 b(v)g(e)758 4959 y(them,)75 b(but)c(they)g(sho)-5 b(w)72 b(ho)-5 b(w)72 b(the)f(Diric)-5 b(hlet)71 b(Laplacian)h(mak)-5 b(es)72 b(its)f(en)-5 b(try)71 b(in)-5 b(to)72 b(quan)-5 b(tum)758 5159 y(mec)g(hanics)41 b(and)f(wh)-5 b(y)40 b(the)g(question)g(ask)-5 b(ed)40 b(in)g(section)g(2)g(is)h(natural)f (from)g(this)g(p)5 b(oin)-5 b(t)40 b(of)g(view.)758 5358 y(A)f(momen)-5 b(t's)41 b(re\035ection)e(also)h(allo)-5 b(ws)41 b(to)e(conclude)h(that)f(the)g(limit)h Fw(\025)5105 5383 y Ft(n)5242 5358 y Fu(!)46 b(1)39 b FC(discussed)j(there)758 5557 y(corresp)5 b(onds)56 b(precisely)f(to)g(the)g(limit)g Fh(~)46 b Fu(!)g Fy(0)56 b FC(of)f(the)g(previous)h(subsection.)1057 5756 y(In)65 b(the)f(follo)-5 b(wing)66 b(t)-5 b(w)g(o)65 b(sections)h(I)f(will)g(use)g(the)g(ideas)g(dev)-5 b(elop)5 b(ed)65 b(here)g(to)g(construct)758 5956 y(a)52 b(sp)5 b(ectral)52 b(problem)h(that)f(is)g(naturally)h(related)e(to)h(the)g(h) -5 b(yp)5 b(erb)g(olic)52 b(toral)g(automorphisms)758 6155 y(in)57 b(the)f(same)i(w)-5 b(a)g(y)57 b(as)g(the)g(Sc)-5 b(hr\366)5 b(dinger)57 b(op)5 b(erator)56 b(is)i(link)-5 b(ed)57 b(to)f(a)h(Hamiltonian)g(\035o)-5 b(w.)79 b(The)758 6354 y(last)61 b(section)g(of)g(these)f(notes)h(will)h(then)f(b)5 b(e)60 b(used)i(to)e(study)i(this)f(sp)5 b(ectral)61 b(problem)g(in)h(the)758 6553 y(semi-classical)57 b(regime.)758 6753 y FD(F)-16 b(urther)100 b(reading:)137 b FC(If)87 b(this)g(section)g(has)h(aroused)g(y)-5 b(our)87 b(in)-5 b(terest)87 b(and)h(y)-5 b(ou)87 b(w)-5 b(an)g(t)88 b(to)758 6952 y(kno)-5 b(w)63 b(more,)i(y)-5 b(ou)63 b(will)g(just)f(ha)-5 b(v)g(e)64 b(to)e(learn)h(quan)-5 b(tum)63 b(mec)-5 b(hanics.)98 b(Ho)-5 b(w)g(ev)g(er,)64 b(b)5 b(e)63 b(w)-5 b(arned)758 7151 y(that)56 b(quan)-5 b(tum)57 b(mec)-5 b(hanics)58 b(can)e(not)h(really)f(b)5 b(e)56 b(learned)h(without)g(reading)g(ph)-5 b(ysics)58 b(b)5 b(o)g(oks:)758 7350 y([)p FD(CTDL)p FC(][)p FD(LL2)p FC(][)p FD(M)p FC(])61 b(are)e(some)i(t)-5 b(ypical)60 b(examples.)88 b(This)61 b(is)f(a)g(tough)g(job)g(for)g (mathemati-)758 7550 y(cally)49 b(trained)g(p)5 b(eople)49 b(since)h(ph)-5 b(ysicists)51 b(don't)e(sh)-5 b(y)51 b(a)-5 b(w)g(a)g(y)50 b(from)f(a)h(priori)f(illegal)h(op)5 b(erations:)758 7749 y(for)54 b(them)h(all)g(limits)h(ma)-5 b(y)55 b(b)5 b(e)55 b(in)-5 b(terc)g(hanged,)56 b(all)f(series)h(con)-5 b(v)g(erge,)55 b(all)g(functions)g(are)g(in\034n-)758 7948 y(itely)e(di\033eren)-5 b(tiable,)55 b(or)f(so)g(it)g(seems,)h (and)g(this)f(tends)h(to)e(scare)h(mathematicians.)74 b(F)-14 b(ormal)758 8147 y(computations)52 b(are)f(nev)-5 b(ertheless)52 b(a)g(go)5 b(o)g(d)51 b(\034rst)i(approac)-5 b(h)52 b(to)f(an)-5 b(y)52 b(sub)9 b(ject)52 b(and)g(should)h(not)758 8347 y(b)5 b(e)65 b(despised.)105 b(They)64 b(often)h(allo)-5 b(w)66 b(y)-5 b(ou)65 b(to)g(feel)g(what)g(ough)-5 b(t)66 b(to)e(b)5 b(e)65 b(true)g(and)h(to)f(appreci-)758 8546 y(ate)58 b(the)g(b)5 b(eaut)-5 b(y)58 b(of)h(a)g(sub)9 b(ject)59 b(b)5 b(efore)58 b(getting)g(in)-5 b(to)59 b(the)f(nitt)-5 b(y-gritt)g(y)58 b(of)g(actually)h(pro)-5 b(ving)758 8745 y(things.)100 b(The)64 b(mathematics)h(underpinning)g (\(non-relativistic\))f(quan)-5 b(tum)64 b(mec)-5 b(hanics)66 b(has)758 8944 y(b)5 b(een)37 b(extensiv)-5 b(ely)37 b(dev)-5 b(elop)5 b(ed)37 b(and)h(can)g(b)5 b(e)37 b(learned)h(in)g([)p FD(RS)p FC(][)p FD(CFKS)p FC(][)p FD(HS)q FC(],)j(for)c(example.)68 b(In)758 9144 y(addition,)54 b(the)f(analogies)h(b)5 b(et)-5 b(w)g(een)52 b(symplectic)h(geometry)g(and)g(the)g(theory)g(of) g(self-adjoin)-5 b(t)758 9343 y(and)50 b(unitary)f(op)5 b(erators)49 b(on)h(Hilb)5 b(ert)48 b(space)i(ha)-5 b(v)g(e)50 b(spurred)g(on)g(m)-5 b(uc)g(h)51 b(b)5 b(eautiful)49 b(mathemat-)758 9542 y(ics)j(that)e(has)j(often)d(nothing)i(to)f(do)h (with)f(quan)-5 b(tum)52 b(mec)-5 b(hanics)53 b(as)f(suc)-5 b(h.)74 b(T)-14 b(o)51 b(giv)-5 b(e)52 b(just)f(one)758 9741 y(example,)j(y)-5 b(ou)54 b(can)f(appreciate)g(this)h(b)-5 b(y)54 b(reading)g(the)g(in)-5 b(tro)5 b(ductions)54 b(to)f([)p FD(K)p FC(][)p FD(V)p FC(])h(whic)-5 b(h)54 b(are)758 9941 y(b)5 b(o)g(oks)55 b(on)g(the)g(represen)-5 b(tation)55 b(theory)g(of)g(Lie)g(groups.)2817 10342 y FD(12.)96 b(W)-16 b(eyl)63 b(quan)-5 b(tization)1057 10641 y(12.1.)96 b(In)-5 b(tro)5 b(duction.)84 b FC(It)49 b(is)h(clear)g(from)f(the)h(previous)g(section)f(that)g(there)g(are)h (some)758 10840 y(rather)71 b(amazing)h(structural)g(similarities)h(b)5 b(et)-5 b(w)g(een)72 b(classical)g(and)h(quan)-5 b(tum)72 b(mec)-5 b(hanics.)p eop %%Page: 37 37 37 36 bop 2362 861 a FA(QUANTUM)58 b(CHA)l(OS:)g(A)h(BRIEF)f(FIRST)g (VISIT)1472 b(37)758 1259 y FC(The)81 b(mathematics)i(of)e(classical)i (mec)-5 b(hanics)83 b(deals)g(with)e(functions)i(on)f Fs(R)5797 1199 y Fv(2)p Ft(n)6044 1259 y FC(and)h(their)758 1458 y(P)-5 b(oisson)63 b(brac)-5 b(k)g(ets,)63 b(and)e(with)g (Hamiltonian)h(\035o)-5 b(ws.)92 b(The)62 b(mathematics)f(of)g(quan)-5 b(tum)62 b(me-)758 1658 y(c)-5 b(hanics)56 b(with)g(op)5 b(erators)56 b(on)g Fw(L)2794 1597 y Fv(2)2868 1658 y Fy(\()p Fs(R)3056 1597 y Ft(n)3155 1658 y Fy(\))f FC(and)i(their)e (comm)-5 b(utators,)57 b(and)f(with)g(unitary)f(\035o)-5 b(ws.)758 1857 y(W)-14 b(e)62 b(ha)-5 b(v)g(e)63 b(noticed)g(that)f (certain)h(op)5 b(erators)62 b(seem)h(to)g(\020naturally\021)77 b(corresp)5 b(ond)63 b(to)g(certain)758 2056 y(simple)77 b(functions)f(and)h(that)e(this)h(corresp)5 b(ondence)76 b(maps)h(P)-5 b(oisson)78 b(brac)-5 b(k)g(ets)76 b(in)-5 b(to)77 b(com-)758 2255 y(m)-5 b(utators)69 b(\(see)g(Exercise)g (11.2\).)115 b(An)69 b(ob)-5 b(vious)70 b(\(mathematical\))f(question)g (that)f(presen)-5 b(ts)758 2455 y(itself)60 b(\(and)g(that)g(w)-5 b(as)61 b(ask)-5 b(ed)61 b(b)-5 b(y)60 b(the)g(British)g(ph)-5 b(ysicist)62 b(P)-14 b(.A.M.)60 b(Dirac)g(in)h(the)f(early)g(da)-5 b(ys)758 2654 y(of)58 b(quan)-5 b(tum)58 b(mec)-5 b(hanics\))59 b(is)g(whether)f(this)g(corresp)5 b(ondence)58 b(can)h(b)5 b(e)57 b(extended)h(\(uniquely)758 2853 y(p)5 b(erhaps\))66 b(to)g(a)h(linear)g(map)g(asso)5 b(ciating)67 b(to)f(eac)-5 b(h)67 b(function)f Fw(f)85 b FC(in)67 b(a)f(suitable)h(\(preferably) 758 3052 y(large\))j(v)-5 b(ector)69 b(space)i(of)f(functions)h(on)g (phase)g(space)g 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Fy(\))4858 4581 y Fr(\003)4934 4650 y Fw(:)758 4901 y FC(If)58 b(suc)-5 b(h)60 b(a)f(map)g(existed)f(and)h(w) -5 b(ere)59 b(unique,)h(it)e(w)-5 b(ould)60 b(asso)5 b(ciate)58 b(to)h Fz(e)-8 b(ach)58 b FC(real)h(function)g(of)758 5101 y(the)70 b(v)-5 b(ector)69 b(space)i(considered)g(a)f(self-adjoin) -5 b(t)71 b(op)5 b(erator)70 b(and)h(hence)f(a)g(unitary)h(\035o)-5 b(w)70 b(\(b)-5 b(y)758 5300 y(exp)5 b(onen)-5 b(tiation\))54 b(in)i(a)f(v)-5 b(ery)55 b(natural)g(w)-5 b(a)g(y)-14 b(.)75 b(If,)55 b(in)g(addition,)1670 5693 y Fy(\()p Fw(D)5 b Fy(\))54 b Fw(O)5 b(p)55 b Fy([)2333 5580 y(\()p Fw(p)36 b Fu(\000)h Fw(A)p Fy(\()p Fw(q)6 b Fy(\)\))3084 5520 y Fv(2)p 2333 5654 824 7 v 2630 5806 a Fy(2)p Fw(m)3213 5693 y Fy(+)37 b Fw(V)g Fy(\()p Fw(q)6 b Fy(\)])46 b(=)4083 5580 y(1)p 4010 5654 229 7 v 4010 5806 a(2)p Fw(m)4259 5693 y Fy(\()p Fw(P)59 b Fu(\000)37 b Fw(A)p Fy(\()p Fw(Q)p Fy(\)\))5107 5624 y Fv(2)5217 5693 y Fy(+)g Fw(V)f Fy(\()p Fw(Q)p Fy(\))p Fw(:)758 6024 y FC(then)66 b(it)h(w)-5 b(ould)67 b(neatly)g(generalize)f(the)g(prescriptions)i(of)f(quan)-5 b(tum)67 b(mec)-5 b(hanics)68 b(and)f(giv)-5 b(e)758 6224 y(them)55 b(a)g(sound)i(mathematical)e(underpinning.)1057 6423 y(W)-14 b(e)57 b(will)h(in)-5 b(v)g(estigate)59 b(the)e(\(non\)existence)g(of)h(suc)-5 b(h)59 b(a)f(map)h Fw(O)5 b(p)57 b FC(in)h(this)g(section.)82 b(Note)758 6622 y(\034rst)49 b(that)f(w)-5 b(e)48 b(can)h(not)f(b)5 b(e)49 b(to)5 b(o)47 b(na\357v)-5 b(e.)71 b(One)49 b(can)g(not)f(just)h (replace)f(in)h(an)-5 b(y)49 b(function)f Fw(f)18 b Fy(\()p Fw(q)6 b(;)28 b(p)p Fy(\))p FC(,)758 6821 y Fw(q)47 b FC(par)41 b Fw(Q)h FC(and)f Fw(p)g FC(par)h Fw(P)63 b FC(b)5 b(ecause,)44 b(for)d(example)g Fw(q)6 b(p)46 b Fy(=)g Fw(pq)6 b FC(,)44 b(but)d Fw(QP)69 b Fu(6)p Fy(=)47 b Fw(P)23 b(Q)p FC(.)69 b(This)42 b(is)f(called)h(the)758 7021 y(\020ordering)51 b(problem\021)65 b(and)51 b(an)-5 b(y)51 b(c)-5 b(hoice)50 b(for)g(the)g(map)h Fw(O)5 b(p)49 b FC(is)i(sometimes)h(said)f(to)f(corresp)5 b(ond)758 7220 y(to)53 b(a)h(c)-5 b(hoice)54 b(of)f(ordering.)74 b(Note)53 b(also)h(that,)g Fz(if)g FC(a)f(map)i Fw(O)5 b(p)53 b FC(exists)g(satisfying)i(\(A\)-\(C\),)c(then)758 7419 y(the)k(canonical)g(comm)-5 b(utation)56 b(relations)g(\(see)f (Exercise)f(11.2\))h(imply)h(that)2363 7801 y Fw(O)5 b(p)54 b Fy(1)46 b(=)g Fw(O)5 b(p)54 b Fu(f)p Fw(q)3364 7826 y Fv(1)3439 7801 y Fw(;)28 b(p)3597 7826 y Fv(1)3671 7801 y Fu(g)46 b Fy(=)3995 7689 y([)p Fw(Q)4172 7714 y Fv(1)4247 7689 y Fw(;)28 b(P)4428 7714 y Fv(1)4502 7689 y Fy(])p 3995 7763 553 7 v 4198 7915 a Fw(i)p Fh(~)4614 7801 y Fy(=)102 b(id)4983 7831 y Ft(L)5074 7798 y Fn(2)5139 7831 y Fv(\()p Fq(R)5268 7798 y Fo(n)5356 7831 y Fv(\))5416 7801 y Fw(:)-4704 b FC(\(12.2\))758 8143 y(F)-14 b(urthermore,)48 b(under)f(some)g(additional)g(tec)-5 b(hnical)46 b(conditions,)j(one)d (can)g(sho)-5 b(w)47 b(\(see)f(Prop)5 b(o-)758 8342 y(sition)51 b(12.1)f(and)h(Exercise)f(12.6)g(\(i\)\))f(that,)i(if)f Fw(f)69 b FC(is)51 b(a)f(function)h(of)f Fw(q)56 b FC(or)51 b Fw(p)f FC(alone,)h(then)f(a)h(map)758 8542 y(satisfying)k(\(A\))f (and)i(\(B\))e(also)i(satis\034es)2146 8828 y Fy(\()p Fw(E)10 b Fy(\))331 b Fw(O)5 b(p)54 b(f)18 b Fy(\()p Fw(q)6 b Fy(\))45 b(=)i Fw(f)18 b Fy(\()p Fw(Q)p Fy(\))p Fw(;)193 b(O)5 b(p)54 b(f)18 b Fy(\()p Fw(p)p Fy(\))45 b(=)h Fw(f)18 b Fy(\()p Fw(P)23 b Fy(\))p Fw(:)758 9114 y FC(This)49 b(is)h(lo)5 b(oking)49 b(prett)-5 b(y)48 b(go)5 b(o)g(d,)49 b(b)5 b(ecause)49 b(the)g(conditions)h(\(A\)-\(C\))c (imply)k(therefore)e(\(D\),)g(at)758 9314 y(least)58 b(for)g Fw(A)p Fy(\()p Fw(q)6 b Fy(\))49 b(=)i(0)p FC(!)82 b(So)59 b(if)f Fw(O)5 b(p)57 b FC(existed)g(and)i(if)f(it)g(w)-5 b(ere)58 b(unique,)h(then)f(it)g(w)-5 b(ould)59 b(go)f(a)g(long)758 9513 y(w)-5 b(a)g(y)55 b(in)h(\020explaining\021)70 b(in)56 b(mathematical)f(terms)h(the)e(origin)i(of)f(the)g(Sc)-5 b(hr\366)5 b(dinger)56 b(equation.)1057 9846 y FD(12.2.)96 b(Quadratic)56 b(Hamiltonians.)82 b FC(T)-14 b(o)49 b(simplify)h(the)g 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1607 y Fw(O)5 b(p)54 b(q)2331 1538 y Fv(2)2452 1607 y Fy(=)46 b Fw(Q)2758 1538 y Fv(2)2833 1607 y Fw(;)28 b(O)5 b(p)54 b(p)3261 1538 y Fv(2)3381 1607 y Fy(=)46 b Fw(P)3686 1538 y Fv(2)3760 1607 y Fw(;)28 b(O)5 b(p)54 b(q)6 b(p)46 b Fy(=)4509 1495 y(1)p 4509 1569 84 7 v 4509 1721 a(2)4612 1607 y(\()p Fw(P)23 b(Q)36 b Fy(+)h Fw(QP)23 b Fy(\))p Fw(:)-4754 b FC(\(12.3\))1057 1962 y(Note)54 b(that)g(condition)i (\(C\))e(is)i(here)f(to)g(b)5 b(e)55 b(in)-5 b(terpreted)55 b(as)h(sa)-5 b(ying)56 b(that)1969 2232 y Fu(8)p Fw(f)64 b Fu(2)46 b(P)2478 2257 y Fv(2)2553 2232 y Fw(;)28 b Fu(8)p Fw(';)g( )52 b Fu(2)46 b(S)12 b Fy(\()p Fs(R)26 b Fy(\))p Fw(;)i Fy(\()p Fw(';)g(O)5 b(p)62 b(f)18 b( )6 b Fy(\))46 b(=)g(\()p Fw(O)5 b(p)5053 2188 y Fy(\026)5018 2232 y Fw(f)18 b(';)28 b( )6 b Fy(\))p Fw(:)758 2502 y FD(Pro)f(of:)72 b FC(Existence)49 b(do)5 b(es)50 b(not)g(p)5 b(ose)49 b(a)h(problem)g(since)g(it)g(is)g(easily)g(c)-5 b(hec)g(k)g(ed)50 b(that)f(the)g(c)-5 b(hoice)758 2701 y Fw(O)5 b(p)76 b(q)88 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Fy(\()p Fw(Q)36 b Fu(\000)i Fw(a)p Fy(\))p Fw(T)3233 7895 y Fv(1)3306 7870 y Fw(\016)3380 7895 y Ft(a)3507 7870 y Fy(=)46 b Fw(T)3779 7895 y Fv(1)3853 7870 y Fy(\()p Fw(Q)37 b Fu(\000)g Fw(a)p Fy(\))p Fw(\016)4479 7895 y Ft(a)4605 7870 y Fy(=)46 b(0)p Fw(;)758 8140 y FC(from)j(whic)-5 b(h)51 b(one)e(concludes)i(that)e(the)g(temp)5 b(ered)49 b(distribution)i Fw(T)5014 8165 y Fv(1)5088 8140 y Fw(\016)5162 8165 y Ft(a)5292 8140 y FC(is)f(necessarily)g(a)g(m)-5 b(ul-)758 8339 y(tiple)73 b(of)f Fw(\016)1434 8364 y Ft(a)1514 8339 y FC(,)78 b Fz(i.e.)127 b FC(that)73 b Fw(T)2469 8364 y Fv(1)2543 8339 y Fw(\016)2617 8364 y Ft(a)2773 8339 y Fy(=)j Fw(c)3050 8364 y Fv(1)3124 8339 y Fy(\()p Fw(a)p Fy(\))p Fw(\016)3416 8364 y Ft(a)3568 8339 y FC(for)d(some)h Fw(c)4346 8364 y Fv(1)4420 8339 y Fy(\()p Fw(a)p Fy(\))h Fu(2)g Fs(C)42 b FC(.)136 b(Consequen)-5 b(tly)-14 b(,)78 b(for)73 b(all)758 8539 y Fw( )52 b Fu(2)46 b(S)12 b Fy(\()p Fs(R)26 b Fy(\))p FC(,)p 2095 8611 415 7 v 2095 8755 a Fy(\()p 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y(particular)55 b(that)2046 1803 y Fw(O)5 b(p)54 b(q)2390 1828 y Ft(j)2460 1803 y Fw(p)2544 1828 y Ft(k)2671 1803 y Fy(=)2866 1691 y(1)p 2866 1765 84 7 v 2866 1917 a(2)2969 1803 y(\()p Fw(Q)3165 1828 y Ft(j)3235 1803 y Fw(P)3342 1828 y Ft(k)3460 1803 y Fy(+)37 b Fw(P)3733 1828 y Ft(k)3815 1803 y Fw(Q)3946 1828 y Ft(j)4016 1803 y Fy(\))p Fw(;)28 b Fu(8)p Fw(j;)g(k)52 b Fu(2)46 b(f)p Fy(1)p Fw(;)28 b(:)g(:)g(:)f(;)h(n)p Fu(g)p Fw(:)1057 2151 y FC(These)49 b(results)i(are)e(rather)g(encouraging.)72 b(In)50 b(particular,)h(in)f (view)f(of)g(what)g(w)-5 b(e)50 b(learned)758 2350 y(in)45 b(the)f(previous)h(section,)i(w)-5 b(e)44 b(exp)5 b(ect)43 b(there)h(to)g(b)5 b(e)45 b(a)f(natural)h(corresp)5 b(ondence)44 b(b)5 b(et)-5 b(w)g(een)45 b(the)758 2550 y(linear)55 b(Hamiltonian)g(\035o)-5 b(ws)56 b(generated)e(b)-5 b(y)55 b(homogeneous)h(quadratic)f(p)5 b(olynomials)56 b Fw(g)6 b Fy(\()p Fw(x)p Fy(\))45 b(=)778 2684 y Fv(1)p 778 2711 67 7 v 778 2806 a(2)864 2749 y Fw(\013)q(q)1051 2689 y 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3704 y(in)e(section)h(13.)69 b(A)40 b(closed)g(expression)h(for)g(the)f (op)5 b(erators)40 b Fy(exp)27 b Fu(\000)4855 3639 y Ft(i)p 4842 3666 74 7 v 4842 3762 a Fq(~)4936 3704 y Fy(\()p Fw(O)5 b(p)38 b(g)6 b Fy(\))41 b Fw(t)f FC(is)h(rather)f (di\036cult)758 3904 y(to)55 b(obtain)g([)p FD(F)p FC(],)h(but)f(some)h (sp)5 b(ecial)56 b(cases)f(are)g(easily)h(computed:)1057 4206 y Fx(Exer)-5 b(cise)62 b FC(12.4)p Fx(.)83 b FC(Sho)-5 b(w)56 b(that,)f(for)g(all)g Fw( )e Fu(2)45 b Fw(L)4038 4146 y Fv(2)4113 4206 y Fy(\()p Fs(R)26 b Fy(\))p FC(,)2671 4347 y Fp(\022)2793 4581 y Fy(exp)h Fu(\000)3240 4469 y Fw(i)p 3223 4543 90 7 v 3223 4695 a Fh(~)3333 4581 y Fw(Q)3464 4513 y Fv(2)3539 4581 y Fw(t)55 b( )3768 4347 y Fp(\023)3918 4581 y Fy(\()p Fw(y)6 b Fy(\))166 b(=)g Fw(e)4673 4513 y Fr(\000)4808 4468 y Fo(i)p 4797 4486 66 6 v 4797 4553 a Ff(~)4883 4513 y Ft(y)4954 4463 y Fn(2)5019 4513 y Ft(t)5133 4581 y Fw( )6 b Fy(\()p Fw(y)g Fy(\);)-4752 b FC(\(12.4\))2649 4812 y Fp(\022)2771 5046 y 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b Fy(\))64 b FC(in)-5 b(v)c(arian)k(t.)1057 6190 y FD(12.3.)96 b(The)67 b(Gro)5 b(enew)-5 b(old-V)-16 b(an)67 b(Ho)-5 b(v)g(e)67 b(theorem.)85 b FC(Encouraged)58 b(b)-5 b(y)58 b(the)f(result)i(of)758 6390 y(the)h(previous)h(subsection,)h(one)f(ma)-5 b(y)60 b(hop)5 b(e)61 b(to)f(extend)f(the)h(map)h Fw(O)5 b(p)59 b FC(to)h(the)g(v)-5 b(ector)60 b(space)758 6589 y Fu(P)79 b FC(of)65 b(all)h(p)5 b(olynomials)66 b(in)g Fw(q)71 b FC(and)66 b Fw(p)p FC(.)103 b(This)66 b(can)f(ho)-5 b(w)g(ev)g(er)66 b Fz(not)g FC(b)5 b(e)65 b(done,)j(as)e(the)f(follo)-5 b(wing)758 6788 y(theorem)59 b(sho)-5 b(ws.)90 b(\(There)59 b(exist)g(sev)-5 b(eral)60 b(v)-5 b(ersions)62 b(of)d(this)i(result,)g (see)f(for)g(example)f([)p FD(AM)q FC(])758 6988 y(or)c([)p FD(F)p FC(].\))1057 7290 y Fx(Theorem)61 b FC(12.5)p Fx(.)83 b Fz(\(Gr)-8 b(o)g(enewold-V)-13 b(an)51 b(Hove\))e(Ther)-8 b(e)49 b(do)-8 b(es)48 b(not)h(exist)g(a)g(map)f Fw(O)5 b(p)45 b Fy(:)h Fu(P)60 b(!)758 7489 y(L)p Fy(\()p Fu(S)12 b Fy(\()p Fs(R)25 b 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FC(and)h(deduce)g Fw(O)5 b(p)54 b(q)6 b(p)4017 8529 y Fv(2)4146 8589 y FC(from)56 b(it.)1057 8788 y(\(iv\))e(Chec)-5 b(k)55 b(that)2774 9072 y Fu(f)p Fw(q)6 b(p)3021 9004 y Fv(2)3095 9072 y Fw(;)28 b(q)3249 9004 y Fv(2)3324 9072 y Fw(p)p Fu(g)36 b Fy(+)3713 8960 y(1)p 3713 9034 84 7 v 3713 9186 a(3)3816 9072 y Fu(f)p Fw(q)3979 9004 y Fv(3)4054 9072 y Fw(;)28 b(p)4212 9004 y Fv(3)4285 9072 y Fu(g)46 b Fy(=)h(0)p Fw(:)1057 9369 y FC(\(v\))54 b(Compute)2704 9653 y Fy([)p Fw(O)5 b(p)55 b(q)6 b(p)3185 9584 y Fv(2)3259 9653 y Fw(;)28 b(O)5 b(p)54 b(q)3683 9584 y Fv(2)3757 9653 y Fw(p)p Fy(])37 b(+)4110 9541 y(1)p 4110 9615 V 4110 9767 a(3)4213 9653 y([)p Fw(Q)4390 9584 y Fv(3)4465 9653 y Fw(;)28 b(P)4669 9584 y Fv(3)4742 9653 y Fy(])758 9939 y FC(and)55 b(sho)-5 b(w)57 b(y)-5 b(ou)55 b(obtain)h(a)f(con)-5 b(tradiction)55 b(implying)i(the)d(theorem.)1057 10242 y(F)-14 b(aced)46 b(with)f(this)h(obstacle,)i(m)-5 b(uc)g(h)47 b(e\033ort)d(has)j(b)5 b(een)45 b(exp)5 b(ended)45 b(to)g(\034nd)i(a)e (set)h(of)f(natural)758 10441 y(conditions)k(in)g(the)f(spirit)g(of)h (\(A\)-\(E\))d(\(but)i(w)-5 b(eak)g(er,)50 b(of)e(course\))g(that)g(w) -5 b(ould)49 b(guaran)-5 b(tee)49 b(the)758 10641 y(existence)55 b(of)i(a)f(\(unique\))g(map)i Fw(O)5 b(p)55 b FC(satisfying)i(them.)78 b(An)-5 b(y)56 b(suc)-5 b(h)58 b(map)f(is)g(usually)h(called)e(a)758 10840 y(\020quan)-5 b(tization\021,)47 b(a)e(notion)g(for)g(whic)-5 b(h)45 b(no)g(generally)g(agreed)g(up)5 b(on)45 b(formal)g (de\034nition)g(exists.)p eop %%Page: 40 40 40 39 bop 758 878 a FA(40)2127 b(STEPHAN)58 b(DE)h(BI\310VRE)758 1259 y FC(The)g(oldest,)i(and)f(still)g(in)g(man)-5 b(y)60 b(w)-5 b(a)g(ys)61 b(the)e(most)h(satisfactory)f(suc)-5 b(h)61 b(quan)-5 b(tization,)60 b(is)h(the)758 1458 y(W)-14 b(eyl)60 b(quan)-5 b(tization)61 b(that)g(I)g(shall)h(no)-5 b(w)62 b(describ)5 b(e)62 b(since)f(it)g(is)h(an)g(imp)5 b(ortan)-5 b(t)61 b(to)5 b(ol)61 b(in)g(semi-)758 1658 y(classical)d(analysis)h(and)f(since)h(w)-5 b(e)58 b(will)g(need)f(it)h (in)g(the)g(form)-5 b(ulation)58 b(and)g(the)g(pro)5 b(of)57 b(of)h(the)758 1857 y(Sc)-5 b(hnirelman)56 b(theorem.)1057 2160 y FD(12.4.)96 b(W)-16 b(eyl)58 b(quan)-5 b(tization)56 b(for)i(Sc)-5 b(h)g(w)g(artz)56 b(functions.)83 b FC(I)50 b(will)h(con)-5 b(tin)g(ue)51 b(to)e(w)-5 b(ork)758 2359 y(on)51 b Fs(R)1107 2299 y Fv(2)1191 2359 y FC(,)h(the)f(extension)g (to)f Fs(R)2622 2299 y Fv(2)q Ft(n)2839 2359 y FC(b)5 b(eing)51 b(immediate.)73 b(Let)51 b Fw(f)64 b Fu(2)46 b(S)12 b Fy(\()p Fs(R)5053 2299 y Fv(2)5137 2359 y Fy(\))51 b FC(\(the)f(Sc)-5 b(h)g(w)g(artz)52 b(space,)758 2572 y(de\034ned)j(previously\))h(and)f(de\034ne)h(the)f Fz(symple)-8 b(ctic)59 b(F)-13 b(ourier)61 b(tr)-8 b(ansform)5466 2528 y Fy(~)5430 2572 y Fw(f)74 b FC(of)55 b Fw(f)73 b FC(as)56 b(follo)-5 b(ws:)2645 2885 y Fy(~)2609 2929 y Fw(f)18 b Fy(\()p Fw(a)p Fy(\))45 b(=)3146 2703 y Fp(Z)3239 3081 y Fq(R)3316 3047 y Fn(2)3423 2929 y Fw(f)18 b Fy(\()p Fw(x)p Fy(\))28 b(exp)4092 2817 y Fw(i)p 4076 2891 90 7 v 4076 3043 a Fh(~)4185 2929 y Fu(h)p Fw(a;)g(x)p Fu(i)4637 2817 y Fw(dx)p 4592 2891 274 7 v 4592 3043 a Fy(2)p Fw(\031)6 b Fh(~)758 2929 y FC(\(12.7\))758 3293 y(Recall)67 b(that)g Fu(h)p Fw(a;)28 b(x)p Fu(i)65 b Fy(=)i Fw(a)2378 3318 y Fv(1)2452 3293 y Fw(p)44 b Fu(\000)i Fw(a)2843 3318 y Fv(2)2917 3293 y Fw(q)6 b FC(.)110 b(This)69 b(is)f(of)f(course)h (just)f(the)g(F)-14 b(ourier)69 b(transform)e(in)h(a)758 3506 y(sligh)-5 b(tly)70 b(funn)-5 b(y)71 b(notation.)117 b(Note)69 b(that)h(the)f(function)4399 3462 y Fy(~)4363 3506 y Fw(f)88 b FC(dep)5 b(ends)71 b(on)f Fh(~)p FC(,)k(but)c(I)g (shall)g(not)758 3705 y(explicitly)54 b(indicate)h(this)h(dep)5 b(endence.)73 b(The)56 b(F)-14 b(ourier)56 b(in)-5 b(v)g(ersion)56 b(form)-5 b(ula)56 b(no)-5 b(w)56 b(yields:)2544 4054 y Fw(f)18 b Fy(\()p Fw(x)p Fy(\))46 b(=)3089 3828 y Fp(Z)3181 4205 y Fq(R)3258 4172 y Fn(2)3402 4010 y Fy(~)3366 4054 y Fw(f)18 b Fy(\()p Fw(a)p Fy(\))28 b(exp)e Fu(\000)4156 3941 y Fw(i)p 4140 4016 90 7 v 4140 4168 a Fh(~)4250 4054 y Fu(h)p Fw(a;)i(x)p Fu(i)4705 3941 y Fw(da)p 4657 4016 274 7 v 4657 4168 a Fy(2)p Fw(\031)6 b Fh(~)758 4054 y FC(\(12.8\))1057 4477 y Fx(Exer)-5 b(cise)62 b FC(12.7)p Fx(.)83 b FC(\(i\))55 b(Sho)-5 b(w)56 b(that)3344 4433 y Fy(~)3308 4477 y Fw(f)74 b FC(b)5 b(elongs)56 b(to)e Fu(S)12 b Fy(\()p Fs(R)4562 4417 y Fv(2)4645 4477 y Fy(\))p FC(.)74 b(\(ii\))55 b(Sho)-5 b(w)56 b(\(12.8\).)1057 4777 y(The)51 b(idea)i(of)e(W)-14 b(eyl)52 b(w)-5 b(as)52 b(to)g(apply)g(to)g(this)g(form)-5 b(ula)53 b(the)e(simple)i(rule:)73 b(\020replace)52 b Fw(q)58 b FC(b)-5 b(y)52 b Fw(Q)758 4976 y FC(and)j Fw(p)g FC(b)-5 b(y)56 b Fw(P)23 b FC(\021)69 b(to)55 b(yield)2157 5325 y Fw(O)5 b(p)2373 5256 y Ft(W)2578 5325 y Fw(f)64 b Fy(=)2899 5099 y Fp(Z)2991 5476 y Fq(R)3068 5443 y Fn(2)3211 5281 y Fy(~)3175 5325 y Fw(f)19 b Fy(\()p Fw(a)p Fy(\))28 b(exp)e Fu(\000)3966 5212 y Fw(i)p 3950 5286 90 7 v 3950 5439 a Fh(~)4059 5325 y Fy(\()p Fw(a)4212 5350 y Fv(1)4286 5325 y Fw(P)60 b Fu(\000)37 b Fw(a)4707 5350 y Fv(2)4781 5325 y Fw(Q)p Fy(\))5046 5212 y Fw(da)p 4997 5286 274 7 v 4997 5439 a Fy(2)p Fw(\031)6 b Fh(~)5290 5325 y Fw(:)-4578 b FC(\(12.9\))758 5679 y(Let)54 b(us)i(mak)-5 b(e)56 b(sense)g(of)f(this)g(form)-5 b(ula.)75 b(It)55 b(is)h(con)-5 b(v)g(enien)g(t)56 b(to)f(in)-5 b(tro)5 b(duce)2710 5989 y Fw(U)18 b Fy(\()p Fw(a)p Fy(\))45 b(=)h(exp)28 b Fu(\000)3726 5877 y Fw(i)p 3710 5951 90 7 v 3710 6103 a Fh(~)3820 5989 y Fy(\()p Fw(a)3973 6014 y Fv(1)4046 5989 y Fw(P)60 b Fu(\000)37 b Fw(a)4467 6014 y Fv(2)4541 5989 y Fw(Q)p Fy(\))p Fw(:)-4025 b FC(\(12.10\))758 6316 y(These)48 b(are)f(called)h(the)f Fz(phase)53 b(sp)-8 b(ac)g(e)51 b(tr)-8 b(anslation)53 b(op)-8 b(er)g(ators)p FC(.)70 b(W)-14 b(e)48 b(already)f(studied)i Fw(U)18 b Fy(\()p Fw(a)6440 6341 y Fv(1)6514 6316 y Fw(;)28 b Fy(0\))758 6515 y FC(and)49 b Fw(U)18 b Fy(\(0)p Fw(;)28 b(a)1515 6540 y Fv(2)1589 6515 y Fy(\))48 b FC(in)g(Exercise)g(11.4,)i (and)f(the)f(results)h(there,)h(together)d(with)h(those)h(of)f 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Fw(a)4821 7698 y Fv(1)4895 7673 y Fw(@)4983 7698 y Ft(y)5100 7673 y Fy(+)5286 7561 y Fw(ia)5431 7586 y Fv(2)p 5286 7635 220 7 v 5351 7787 a Fh(~)5525 7673 y Fw(y)5612 7439 y Fp(\023)5763 7673 y Fw( )6 b Fy(\()p Fw(y)g(;)28 b(s)p Fy(\))p Fw(;)-5534 b FC(\(12.11\))758 8044 y(with)47 b(initial)h(condition)g Fw( )6 b Fy(\(0)p Fw(;)28 b(y)6 b Fy(\))46 b(=)g Fw( )6 b Fy(\()p Fw(y)g Fy(\))p FC(.)71 b(This)49 b(equation)e(is)h(easily)g(solv)-5 b(ed)49 b(\(b)-5 b(y)47 b(the)h(metho)5 b(d)758 8243 y(of)55 b(c)-5 b(haracteristics,)55 b(for)g(example\))g(to)g(yield)1895 8595 y Fw( )6 b Fy(\()p Fw(s;)28 b(y)6 b Fy(\))45 b(=)i(exp)o(\()p Fu(\000)3124 8482 y Fw(i)p 3066 8557 173 7 v 3066 8709 a Fy(2)p Fh(~)3259 8595 y Fw(s)3337 8526 y Fv(2)3411 8595 y Fw(a)3499 8620 y Fv(1)3573 8595 y Fw(a)3661 8620 y Fv(2)3735 8595 y Fy(\))83 b(exp\()4237 8482 y Fw(i)p 4222 8557 90 7 v 4222 8709 a Fh(~)4331 8595 y Fw(sa)4497 8620 y Fv(2)4571 8595 y Fw(y)6 b Fy(\))55 b Fw( )6 b Fy(\()p Fw(y)43 b Fu(\000)37 b Fw(sa)5413 8620 y Fv(1)5487 8595 y Fy(\))p Fw(:)758 8922 y FC(On)55 b(the)g(other)g(hand,)h (\(12.11\))f(is)h(solv)-5 b(ed)56 b(formally)f(b)-5 b(y)56 b(the)e(expression)2392 9273 y Fw( )6 b Fy(\()p Fw(s;)28 b(y)6 b Fy(\))46 b(=)g([exp)28 b Fu(\000)3589 9161 y Fw(i)p 3573 9235 V 3573 9387 a Fh(~)3682 9273 y Fw(s)p Fy(\()p Fw(a)3913 9298 y Fv(1)3987 9273 y Fw(P)59 b Fu(\000)38 b Fw(a)4408 9298 y Fv(2)4482 9273 y Fw(Q)p Fy(\)])p Fw( )6 b Fy(\()p Fw(y)g Fy(\))p Fw(:)758 9600 y FC(This)56 b(leads)g(one)f(to) g Fz(de\034ne)h FC(\()p Fw(s)46 b Fy(=)g(1)p FC(\))856 10161 y Fw(U)18 b Fy(\()p Fw(a)p Fy(\))p Fw( )6 b Fy(\()p Fw(y)g Fy(\))46 b(=)g([exp)27 b Fu(\000)2250 10049 y Fw(i)p 2233 10123 V 2233 10275 a Fh(~)2343 10161 y Fy(\()p Fw(a)2496 10186 y Fv(1)2570 10161 y Fw(P)59 b Fu(\000)37 b Fw(a)2990 10186 y Fv(2)3065 10161 y Fw(Q)p Fy(\))p Fw( )6 b Fy(])p Fw( )g Fy(\()p Fw(y)g Fy(\))46 b(=)g(\(exp)27 b Fu(\000)4526 10049 y Fw(i)p 4468 10123 173 7 v 4468 10275 a Fy(2)p Fh(~)4660 10161 y Fw(a)4748 10186 y Fv(1)4823 10161 y Fw(a)4911 10186 y Fv(2)4985 10161 y Fy(\)\(exp)5432 10049 y Fw(i)p 5415 10123 90 7 v 5415 10275 a Fh(~)5525 10161 y Fw(a)5613 10186 y Fv(2)5687 10161 y Fw(y)6 b Fy(\))56 b Fw( )6 b Fy(\()p Fw(y)43 b Fu(\000)37 b Fw(a)6452 10186 y Fv(1)6526 10161 y Fy(\))p Fw(:)758 9854 y FC(\(12.12\))1057 10527 y Fx(Exer)-5 b(cise)62 b FC(12.8)p Fx(.)83 b FC(Pro)-5 b(v)g(e)55 b(that,)g(for)g(all)h Fw(a)46 b Fu(2)g Fs(R)4052 10467 y Fv(2)4136 10527 y FC(,)1847 10798 y Fw(U)18 b Fy(\()p Fu(\000)p Fw(a)p Fy(\))p Fw(P)23 b(U)18 b Fy(\()p Fw(a)p Fy(\))45 b(=)h Fw(P)60 b Fu(\000)37 b Fw(a)3445 10823 y Fv(2)3519 10798 y Fw(;)360 b(U)18 b Fy(\()p Fu(\000)p Fw(a)p Fy(\))p Fw(QU)g Fy(\()p Fw(a)p Fy(\))45 b(=)h Fw(Q)37 b Fu(\000)g Fw(a)5525 10823 y Fv(1)5600 10798 y Fw(:)p eop %%Page: 41 41 41 40 bop 2362 861 a FA(QUANTUM)58 b(CHA)l(OS:)g(A)h(BRIEF)f(FIRST)g (VISIT)1472 b(41)1057 1259 y Fx(Lemma)62 b FC(12.9)p Fx(.)83 b Fz(L)-8 b(et)88 b Fw(f)117 b Fu(2)97 b(S)12 b Fy(\()p 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Fy(exp)27 b Fu(\000)p Fy(2)p Fw(\031)6 b(i)p Fy(\()p Fw(n)4321 8258 y Fv(1)4395 8233 y Fw(p)36 b Fu(\000)h Fw(n)4781 8258 y Fv(2)4855 8233 y Fw(q)6 b Fy(\))p Fw(;)-4288 b FC(\(12.18\))758 8691 y(and)55 b(so,)h(in)g(complete)f(agreemen)-5 b(t)55 b(with)g(\(12.9\))g(w)-5 b(e)55 b(de\034ne)1855 9010 y Fw(O)5 b(p)2071 8942 y Ft(W)2276 9010 y Fw(f)65 b Fy(=)2634 8852 y Fp(X)2597 9216 y Ft(n)p Fr(2)p Fq(Z)2853 9183 y Fn(2)2939 9010 y Fw(f)3020 9035 y Ft(n)3138 9010 y Fy(exp)27 b Fu(\000)p Fy(2)p Fw(\031)6 b(i)p Fy(\()p Fw(n)3954 9035 y Fv(1)4028 9010 y Fw(P)60 b Fu(\000)37 b Fw(n)4461 9035 y Fv(2)4535 9010 y Fw(Q)p Fy(\))46 b(=)4989 8852 y Fp(X)4952 9216 y Ft(n)p Fr(2)p Fq(Z)5208 9183 y Fn(2)5293 9010 y Fw(f)5374 9035 y Ft(n)5465 9010 y Fw(U)18 b Fy(\(2)p Fw(\031)6 b Fh(~)p Fw(n)p Fy(\))p Fw(:)-5388 b FC(\(12.19\))758 9459 y(Clearly)55 b(this)g(sum)i(con)-5 b(v)g(erges)55 b(in)h(op)5 b(erator)55 b(norm)h(and)2888 9768 y Fu(k)47 b Fw(O)5 b(p)3234 9700 y Ft(W)3439 9768 y Fw(f)64 b Fu(k\024)3879 9610 y Fp(X)3842 9974 y Ft(n)p Fr(2)p Fq(Z)4098 9941 y Fn(2)4202 9768 y Fu(j)47 b Fw(f)4376 9793 y Ft(n)4512 9768 y Fu(j)p Fw(;)758 10242 y FC(so)42 b(that)f Fw(O)5 b(p)1510 10182 y Ft(W)1701 10242 y FC(de\034nes)42 b(a)g(linear)g(map)g(from)g Fw(C)3670 10182 y Fr(1)3810 10242 y Fy(\()p Fs(T)3987 10182 y Fv(2)4071 10242 y Fy(\))f FC(to)g(the)g(b)5 b(ounded)42 b(op)5 b(erators)42 b(on)g Fw(L)6354 10182 y Fv(2)6428 10242 y Fy(\()p Fs(R)27 b Fy(\))p FC(.)758 10441 y(Note)38 b(also)j(that)e(it)g(is)h(totally)f (ob)-5 b(vious)41 b(that)e(\(C\))f(and)j(\(E\))d(are)i(satis\034ed!)69 b(The)40 b(further)f(crucial)758 10641 y(prop)5 b(erties)45 b(of)h(the)g(W)-14 b(eyl)45 b(quan)-5 b(tization)46 b(that)f(w)-5 b(e)46 b(shall)h(need)f(are)g(collected)f(in)h(the)g(follo)-5 b(wing)758 10840 y(prop)5 b(osition.)p eop %%Page: 46 46 46 45 bop 758 878 a FA(46)2127 b(STEPHAN)58 b(DE)h(BI\310VRE)1057 1259 y Fx(Pr)-5 b(oposition)61 b FC(12.22)p Fx(.)83 b FC(Let)57 b Fw(f)9 b(;)28 b(g)57 b Fu(2)50 b Fw(C)3529 1199 y Fr(1)3669 1259 y Fy(\()p Fs(T)3847 1199 y Fv(2)3930 1259 y Fy(\))p FC(.)82 b(Then)58 b(there)f(exists)h(a)g(constan)-5 b(t)59 b Fw(C)69 b FC(\(de-)758 1458 y(p)5 b(ending)55 b(on)h Fw(g)61 b FC(and)56 b Fw(f)18 b FC(\))55 b(suc)-5 b(h)57 b(that)2430 1734 y Fy(\()p Fw(i)p Fy(\))45 b Fu(k)h Fw(O)5 b(p)3007 1666 y Ft(W)3212 1734 y Fw(f)74 b(O)5 b(p)3583 1666 y Ft(W)3788 1734 y Fw(g)43 b Fu(\000)37 b Fw(O)5 b(p)4292 1666 y Ft(W)4497 1734 y Fw(f)18 b(g)53 b Fu(k)166 b(\024)g Fw(C)12 b Fh(~)-4735 b FC(\(12.20\))2000 2094 y Fy(\()p Fw(ii)p Fy(\))46 b Fu(k)2439 1982 y Fy([)p Fw(O)5 b(p)2701 1921 y Ft(W)2907 1982 y Fw(f)k(;)28 b(O)5 b(p)3287 1921 y Ft(W)3491 1982 y Fw(g)h Fy(])p 2439 2056 1184 7 v 2958 2208 a Fw(i)p Fh(~)3680 2094 y Fu(\000)37 b Fw(O)5 b(p)4062 2026 y Ft(W)4267 2094 y Fu(f)p Fw(f)k(;)28 b(g)6 b Fu(g)46 b(k)166 b(\024)g Fw(C)12 b Fh(~)-4735 b FC(\(12.21\))758 2448 y FD(Pro)5 b(of:)75 b FC(\(i\))55 b(One)g(has,)h(using)h(\(12.14\),)811 2766 y Fw(O)5 b(p)1027 2698 y Ft(W)1232 2766 y Fw(f)18 b(g)43 b Fu(\000)37 b Fw(O)5 b(p)1835 2698 y Ft(W)2040 2766 y Fw(f)74 b(O)5 b(p)2411 2698 y Ft(W)2616 2766 y Fw(g)52 b Fy(=)3038 2609 y Fp(X)2923 2972 y Ft(n;m)p Fr(2)p Fq(Z)3336 2939 y Fn(2)3421 2766 y Fw(f)3502 2791 y Ft(n)3593 2766 y Fw(g)3672 2791 y Ft(m)3826 2582 y Fp(\020)3925 2766 y Fw(e)4061 2653 y Fo(i)p 4022 2671 123 6 v 4022 2738 a Fn(2)p Ff(~)4165 2698 y Fr(h)p Ft(n;m)p Fr(i)p Fv(\(2)p Ft(\031)t Fq(~)p Fv(\))4838 2648 y Fn(2)4947 2766 y Fu(\000)37 b Fy(1)5196 2582 y Fp(\021)5323 2766 y Fw(U)18 b Fy(\(2)p Fw(\031)6 b Fh(~)p Fw(n)p Fy(\))p Fw(U)18 b Fy(\(2)p Fw(\031)6 b Fh(~)p Fw(m)p Fy(\))p Fw(:)758 3212 y FC(The)55 b(simple)h(estimate)3084 3411 y Fu(j)46 b Fy(exp)28 b Fw(is)37 b Fu(\000)g Fy(1)46 b Fu(j\024j)g Fw(s)g Fu(j)p Fw(;)758 3645 y FC(then)55 b(allo)-5 b(ws)56 b(to)f(conclude)g(that) 1342 3914 y Fu(k)46 b Fw(O)5 b(p)1687 3846 y Ft(W)1892 3914 y Fw(f)18 b(g)43 b Fu(\000)37 b Fw(O)5 b(p)2495 3846 y Ft(W)2701 3914 y Fw(f)73 b(O)5 b(p)3071 3846 y Ft(W)3276 3914 y Fw(g)53 b Fu(k)3491 3944 y Ft(L)3582 3911 y Fn(2)3647 3944 y Fv(\()p Fq(R)p Fv(\))3843 3914 y Fu(\024)4133 3757 y Fp(X)4018 4120 y Ft(n;m)p Fr(2)p Fq(Z)4431 4087 y Fn(2)4535 3914 y Fu(j)46 b Fw(f)4708 3939 y Ft(n)4799 3914 y Fw(g)4878 3939 y Ft(m)5050 3914 y Fu(jj)h Fw(n)f Fu(jj)g Fw(m)g Fu(j)g Fy(2)p Fw(\031)5941 3846 y Fv(2)6015 3914 y Fh(~)p Fw(:)758 4381 y FC(No)-5 b(w,)55 b(since)h Fw(f)9 b(;)28 b(g)52 b Fu(2)45 b Fw(C)2171 4320 y Fr(1)2312 4381 y Fy(\()p Fs(T)2489 4320 y Fv(2)2572 4381 y Fy(\))p FC(,)56 b(one)f(easily)g(sho)-5 b(ws)57 b(that)3251 4525 y Fp(X)3214 4889 y Ft(n)p Fr(2)p Fq(Z)3470 4855 y Fn(2)3574 4683 y Fu(j)47 b Fw(f)3748 4708 y Ft(n)3884 4683 y Fu(jj)g Fw(n)f Fu(j)4215 4614 y Ft(`)758 5125 y FC(con)-5 b(v)g(erges)51 b(for)g(an)-5 b(y)52 b Fw(`)46 b(>)g Fy(0)p FC(,)52 b(from)g(whic)-5 b(h)51 b(the)g(result)h(follo)-5 b(ws.)73 b(P)-5 b(art)51 b(\(ii\))f(is)i(pro)-5 b(v)g(ed)52 b(similarly)-14 b(.)p 6614 5324 7 113 v 6621 5219 100 7 v 6621 5324 V 6720 5324 7 113 v 1057 5622 a Fx(Exer)-5 b(cise)62 b FC(12.23)p Fx(.)83 b FC(*)62 b(\(i\))g(Clearly)-14 b(,)64 b Fw(C)3495 5562 y Fr(1)3635 5622 y Fy(\()p Fs(T)3813 5562 y Fv(2)3896 5622 y Fy(\))e FC(is)h(a)g(comm)-5 b(utativ)g(e)62 b(algebra)h(under)g(m)-5 b(ul-)758 5821 y(tiplication)67 b(of)h(functions.)111 b(Sho)-5 b(w)68 b(that)f Fw(O)5 b(p)3620 5761 y Ft(W)3770 5821 y Fy(\()p Fw(C)3966 5761 y Fr(1)4106 5821 y Fy(\()p Fs(T)4283 5761 y Fv(2)4367 5821 y Fy(\)\))66 b FC(is)j(also)f(an)g(algebra)g(under)g(m)-5 b(ul-)758 6020 y(tiplication)72 b(of)f(op)5 b(erators,)77 b Fz(i.e.)124 b FC(that)71 b(for)h(all)g Fw(f)9 b(;)28 b(g)80 b Fu(2)74 b Fw(C)4462 5960 y Fr(1)4602 6020 y Fy(\()p Fs(T)4780 5960 y Fv(2)4863 6020 y Fy(\))p FC(,)i(there)c (exists)g(\(a)g(unique\))758 6220 y Fw(h)57 b Fu(2)g Fw(C)1210 6159 y Fr(1)1350 6220 y Fy(\()p Fs(T)1527 6159 y Fv(2)1611 6220 y Fy(\))62 b FC(suc)-5 b(h)63 b(that)e Fy(\()p Fw(O)5 b(p)2766 6159 y Ft(W)2978 6220 y Fw(f)18 b 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Fw(C)4086 6956 y Fr(1)4227 7017 y Fy(\()p Fs(T)4404 6956 y Fv(2)4488 7017 y Fy(\))p Fu(g)52 b FC(do)5 b(es)54 b(not)f(separate)g(the)g(p)5 b(oin)-5 b(ts)758 7216 y(of)62 b Fw(L)1067 7156 y Fv(2)1142 7216 y Fy(\()p Fs(R)26 b Fy(\))72 b FC(in)63 b(the)f(sense)i(of)f(Lemma)g(12.13)g(and)g(that)f (the)h(conclusions)h(of)f(Exercise)f(12.16)758 7415 y(also)56 b(no)f(longer)h(hold)g(if)f Fu(S)12 b Fy(\()p Fs(R)2641 7355 y Fv(2)2725 7415 y Fy(\))55 b FC(is)h(replaced)f(b)-5 b(y)55 b Fw(C)4028 7355 y Fr(1)4169 7415 y Fy(\()p Fs(T)4346 7355 y Fv(2)4429 7415 y Fy(\))p FC(.)1057 7713 y Fx(Exer)-5 b(cise)62 b FC(12.24)p Fx(.)83 b FC(Chec)-5 b(k)61 b(that)g(the)h (Egoro)-5 b(v)61 b(theorem)h(\(Theorem)f(12.17\))g(still)h(holds)758 7912 y(for)55 b Fw(f)64 b Fu(2)46 b Fw(C)1445 7852 y Fr(1)1585 7912 y Fy(\()p Fs(T)1762 7852 y Fv(2)1846 7912 y Fy(\))p Fw(;)28 b(A)45 b Fu(2)h FC(SL)55 b Fy(\(2)p Fw(;)28 b Fs(Z)8 b Fy(\))p FC(.)758 8210 y FD(F)-16 b(urther)72 b(reading:)87 b FC(The)62 b(links)h(b)5 b(et)-5 b(w)g(een)62 b(sym)-5 b(b)5 b(ol)63 b(calculus)g(and)g(quan)-5 b(tization,)64 b(and)f(the)758 8409 y(analysis)72 b(b)5 b(ehind)73 b(it,)i(has)e(b)5 b(een)71 b(extensiv)-5 b(ely)71 b(explained)h(in)g(man)-5 b(y)72 b(places)g(from)g(di\033eren)-5 b(t)758 8608 y(p)5 b(oin)-5 b(ts)47 b(of)f(view.)71 b(F)-14 b(or)47 b(a)g(nice)g(o)-5 b(v)g(erview)46 b(of)g(and)i(comparison)f(b)5 b(et)-5 b(w)g(een)47 b(the)f(di\033eren)-5 b(t)47 b(quan)-5 b(ti-)758 8807 y(zations)44 b(most)g(commonly)g(encoun)-5 b(tered)44 b(\(W)-14 b(eyl,)45 b(Wic)-5 b(k,)46 b(an)-5 b(ti-Wic)g(k,)46 b(K)-5 b(ohn-Niren)g(b)5 b(erg\))45 b(and)758 9007 y(their)54 b(merits,)i(y)-5 b(ou)55 b(ma)-5 b(y)55 b(consult)g([)p FD(F)q FC(].)74 b(The)54 b(detailed)h(analysis)h(of)f(the)f(con)-5 b(trol)55 b(of)g(the)f(error)758 9206 y(terms)k(in)g Fh(~)g FC(is)g(giv)-5 b(en)58 b(in)g([)p FD(R)q FC(].)82 b(The)57 b(lecture)h(notes)g(of)f(Prof.)82 b(Egoro)-5 b(v)58 b(and)g(Prof.)81 b(V)-14 b(asilevski)758 9405 y(in)55 b(this)h(v)-5 b(olume)56 b(also)g(deal)f(in)h(part)f(with)g (certain)g(asp)5 b(ects)55 b(of)g(these)g(issues.)2187 9744 y FD(13.)96 b(Quan)-5 b(tum)64 b(mec)-5 b(hanics)63 b(on)h(the)h(torus)1057 10043 y(13.1.)96 b(The)56 b(Hilb)5 b(ert)54 b(spaces.)84 b FC(W)-14 b(e)47 b(no)-5 b(w)48 b(disp)5 b(ose)49 b(of)e(all)h(the)f(ingredien)-5 b(ts)49 b(to)e(describ)5 b(e)758 10242 y(the)62 b(sp)5 b(ectral)62 b(problem)h(naturally)f(asso)5 b(ciated)62 b(to)g(the)g(h)-5 b(yp)5 b(erb)g(olic)62 b(toral)g(automorphisms.)758 10441 y(First,)69 b(w)-5 b(e)66 b(need)g(an)g(appropriate)h(Hilb)5 b(ert)65 b(space)h Fu(H)4200 10466 y Fq(~)4282 10441 y FC(.)106 b(When)66 b(the)f(classical)i(phase)g(space)758 10641 y(is)56 b Fs(R)1048 10580 y Fv(2)1132 10641 y FC(,)g(the)f (corresp)5 b(onding)57 b(Hilb)5 b(ert)55 b(space)h(used)g(in)g(quan)-5 b(tum)57 b(mec)-5 b(hanics)57 b(is)f Fw(L)5950 10580 y Fv(2)6025 10641 y Fy(\()p Fs(R)26 b Fy(\))65 b FC(as)56 b(w)-5 b(e)758 10840 y(sa)g(w)67 b(in)f(the)g(preceding)g(sections.)107 b(I)66 b(will)h(use)f(the)g(insigh)-5 b(ts)68 b(gained)e(there)g(to)f (construct)h(a)p eop %%Page: 47 47 47 46 bop 2362 861 a FA(QUANTUM)58 b(CHA)l(OS:)g(A)h(BRIEF)f(FIRST)g (VISIT)1472 b(47)758 1259 y FC(suitable)54 b(Hilb)5 b(ert)53 b(space)h Fu(H)2528 1284 y Fq(~)2664 1259 y FC(when)h(the)e(phase)i (space)f(is)h Fs(T)4563 1199 y Fv(2)4646 1259 y FC(.)73 b(I)54 b(w)-5 b(an)g(t)55 b(to)e(use)i(the)f(op)5 b(erators)758 1458 y Fw(O)g(p)974 1398 y Ft(W)1175 1458 y Fw(f)18 b FC(,)53 b(for)f Fw(f)70 b FC(b)5 b(elonging)52 b(to)f Fw(C)2853 1398 y Fr(1)2994 1458 y Fy(\()p Fs(T)3171 1398 y Fv(2)3254 1458 y Fy(\))h FC(as)g(quan)-5 b(tum)53 b(observ)-9 b(ables)52 b(on)g Fu(H)5506 1483 y Fq(~)5588 1458 y FC(.)73 b(Exercise)51 b(12.23)758 1658 y(\(ii\))60 b(giv)-5 b(es)62 b(some)g(indications)g(as)g(to)f(wh)-5 b(y)61 b(c)-5 b(ho)5 b(osing)63 b Fu(H)4317 1683 y Fq(~)4460 1658 y FC(equal)e(to)g Fw(L)5226 1597 y Fv(2)5301 1658 y Fy(\()p Fs(R)26 b Fy(\))70 b FC(is)62 b(not)f(the)g(righ)-5 b(t)758 1857 y(idea:)95 b Fw(L)1307 1797 y Fv(2)1381 1857 y Fy(\()p Fs(R)27 b Fy(\))74 b FC(is)67 b(\020to)5 b(o)65 b(big\021)81 b(for)65 b(this)i(purp)5 b(ose.)106 b(In)66 b(more)g(tec)-5 b(hnical)66 b(terms,)j(one)d(w)-5 b(an)g(ts)67 b(to)758 2056 y(\034nd)e(a)g(Hilb)5 b(ert)63 b(space)i(on)g(whic)-5 b(h)66 b(the)e(family)g Fw(O)5 b(p)4033 1996 y Ft(W)4211 1922 y Fp(\000)4287 2056 y Fw(C)4418 1996 y Fr(1)4558 2056 y Fy(\()p Fs(T)4736 1996 y Fv(2)4819 2056 y Fy(\))4884 1922 y Fp(\001)5024 2056 y FC(acts)65 b(irreducibly)-14 b(.)102 b(Recall)758 2255 y(from)62 b(Exercise)h(12.16)f(that)g(this)h (means)h(that)e(the)g(space)h Fu(H)4724 2280 y Fq(~)4869 2255 y FC(m)-5 b(ust)63 b(not)g(ha)-5 b(v)g(e)63 b(an)-5 b(y)63 b(closed)758 2455 y(subspaces)76 b(that)e(are)g(in)-5 b(v)c(arian)k(t)75 b(under)g Fw(O)5 b(p)3633 2394 y Ft(W)3810 2320 y Fp(\000)3886 2455 y Fw(C)4017 2394 y Fr(1)4158 2455 y Fy(\()p Fs(T)4335 2394 y Fv(2)4418 2455 y Fy(\))4483 2320 y Fp(\001)4559 2455 y FC(.)131 b(Since)75 b(y)-5 b(ou)75 b(undoubtedly)g(did)758 2654 y(Exercise)55 b(12.23,)g(y)-5 b(ou)56 b(probably)f(noticed)g(that)1989 2937 y Fy([)p Fw(U)18 b Fy(\()p Fw(n)p Fy(\))p Fw(;)28 b(O)5 b(p)2686 2868 y Ft(W)2891 2937 y Fw(f)18 b Fy(])46 b(=)g(0)p Fw(;)360 b Fu(8)p Fw(f)65 b Fu(2)46 b Fw(C)4272 2868 y Fr(1)4412 2937 y Fy(\()p Fs(T)4589 2868 y Fv(2)4673 2937 y Fy(\))p Fw(;)82 b Fu(8)p Fw(n)46 b Fu(2)g Fs(Z)5374 2868 y Fv(2)5457 2937 y Fw(:)758 3220 y FC(This)63 b(suggests)g(w)-5 b(e)62 b(should)i(b)5 b(e)62 b(lo)5 b(oking)62 b(for)g(a)g(Hilb)5 b(ert)62 b(space)g(on)h(whic)-5 b(h)63 b(all)f(the)g Fw(U)18 b Fy(\()p Fw(n)p Fy(\))62 b FC(act)758 3419 y(as)67 b(m)-5 b(ultiples)69 b(of)e(the)g(iden)-5 b(tit)g(y)67 b(op)5 b(erator!)109 b(It)67 b(is)g(ob)-5 b(vious)69 b(that)d(suc)-5 b(h)69 b(a)e(space)g(can)h(not)f(b)5 b(e)758 3618 y(constructed)39 b(using)j Fw(L)2171 3558 y Fv(2)2245 3618 y FC(-functions,)i(but)c(it)g(is)h(easy)f(to)g(see)g (that)g(the)f Fw(U)18 b Fy(\()p Fw(a)p Fy(\))p Fw(;)28 b(a)45 b Fu(2)h Fs(R)5884 3558 y Fv(2)5968 3618 y FC(,)d(originally)758 3817 y(de\034ned)55 b(on)h Fu(S)12 b Fy(\()p Fs(R)27 b Fy(\))p FC(,)64 b(extend)54 b(con)-5 b(tin)g(uously)57 b(to)e Fu(S)3856 3757 y Fr(0)3902 3817 y Fy(\()p Fs(R)27 b Fy(\))p FC(.)1057 4124 y Fx(Exer)-5 b(cise)62 b FC(13.1)p Fx(.)83 b FC(Pro)-5 b(v)g(e)55 b(this.)75 b FD(Hin)-5 b(t:)73 b FC(Remem)-5 b(b)5 b(er)56 b(Exercise)f(12.2.)1057 4431 y(With)f(this)i(in)g(mind,)g(let)f(us)h(therefore)e(de\034ne)2074 4714 y Fu(H)2214 4739 y Fq(~)2342 4714 y Fy(=)46 b Fu(f)p Fw( )52 b Fu(2)46 b(S)3030 4645 y Fr(0)3077 4714 y Fy(\()p Fs(R)26 b Fy(\))55 b Fu(j)46 b Fw(U)18 b Fy(\(1)p Fw(;)28 b Fy(0\))p Fw( )52 b Fy(=)46 b Fw( )53 b Fy(=)46 b Fw(U)18 b Fy(\(0)p Fw(;)28 b Fy(1\))p Fw( )6 b Fu(g)p Fw(;)-4634 b FC(\(13.1\))758 4997 y(so)55 b(that)g(for)g(all)h Fw( )c Fu(2)46 b(H)2263 5022 y Fq(~)2345 4997 y FC(,)55 b(one)h(has)3071 5299 y Fw(U)18 b Fy(\()p Fw(n)p Fy(\))p Fw( )51 b Fy(=)c Fw(e)3903 5185 y Fo(i)p 3864 5203 123 6 v 3864 5270 a 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b(;)249 b(U)18 b Fy(\(0)p Fw(;)28 b Fy(1\))p Fw( )52 b Fy(=)46 b Fw(e)4784 10771 y Ft(i\024)4909 10788 y Fn(2)4982 10840 y Fw( )6 b(:)p eop %%Page: 48 48 48 47 bop 758 878 a FA(48)2127 b(STEPHAN)58 b(DE)h(BI\310VRE)758 1259 y FC(Study)52 b(the)h(space)g Fu(H)2104 1284 y Fq(~)2186 1259 y Fy(\()p Fw(\024)p Fy(\))f FC(of)g(temp)5 b(ered)53 b(distributions)h(satisfying)f(these)g(conditions,)h(along)758 1458 y(the)h(lines)h(of)f(the)g(ab)5 b(o)-5 b(v)g(e)55 b(lemma.)1057 1760 y(F)-14 b(rom)70 b(no)-5 b(w)69 b(on,)k(I)c(will)h (alw)-5 b(a)g(ys)70 b(supp)5 b(ose)70 b(that)e Fh(~)h FC(is)h(c)-5 b(hosen)70 b(so)g(that)e(there)h(exists)g(an)758 1959 y(in)-5 b(teger)51 b Fw(N)69 b FC(for)51 b(whic)-5 b(h)52 b Fy(2)p Fw(\031)6 b Fh(~)p Fw(N)64 b Fy(=)47 b(1)k FC(and)g(whenev)-5 b(er)51 b(in)h(one)f(form)-5 b(ula)52 b(b)5 b(oth)51 b Fh(~)g FC(and)h Fw(N)69 b FC(app)5 b(ear,)758 2159 y(they)54 b(will)i(b)5 b(e)55 b(related)f(b)-5 b(y)56 b(this)g(equalit)-5 b(y)-14 b(.)1057 2468 y FD(13.2.)96 b(The)58 b(W)-16 b(eyl)58 b(calculus.)81 b FC(W)-14 b(e)50 b(are)g(no)-5 b(w)50 b(in)h(a)f(p)5 b(osition)51 b(to)e(dev)-5 b(elop)51 b(an)f(appropri-)758 2667 y(ate)k(W)-14 b(eyl)55 b(calculus)h(on)g Fu(H)2453 2692 y Fq(~)2535 2667 y FC(,)f(thanks)h(to) e(the)h(follo)-5 b(wing)56 b(remark:)1057 2969 y Fx(Lemma)62 b FC(13.4)p Fx(.)83 b Fz(\(i\))59 b Fw(U)18 b Fy(\()p Fw(a)p Fy(\))p Fu(H)2841 2994 y Fq(~)2969 2969 y Fu(\032)46 b(H)3284 2994 y Fq(~)3426 2969 y Fz(i\033)59 b Fw(a)46 b Fy(=)3985 2904 y Ft(n)p 3967 2931 118 7 v 3967 3027 a(N)4104 2969 y Fw(;)28 b(n)46 b Fu(2)g Fs(Z)4593 2909 y Fv(2)4677 2969 y Fy(;)1057 3183 y Fz(\(ii\))59 b(If)g Fw(f)64 b Fu(2)46 b Fw(C)1961 3123 y Fr(1)2101 3183 y Fy(\()p Fs(T)2278 3123 y Fv(2)2362 3183 y Fy(\))p Fz(,)59 b(then)h Fw(O)5 b(p)3122 3123 y Ft(W)3331 3183 y Fw(f)78 b Fu(H)3630 3208 y Fq(~)3758 3183 y Fu(\032)46 b(H)4073 3208 y Fq(~)4155 3183 y Fz(.)758 3485 y FD(Pro)5 b(of:)66 b FC(Let)37 b Fw(a)46 b Fu(2)f Fs(R)2032 3425 y Fv(2)2116 3485 y 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y Fx(Lemma)62 b FC(13.5)p Fx(.)83 b Fz(Ther)-8 b(e)55 b(exists)g(a)g(Hilb)-8 b(ert)56 b(sp)-8 b(ac)g(e)55 b(structur)-8 b(e)55 b Fu(h\001)p Fw(;)28 b Fu(\001i)5130 4609 y Fr(H)5244 4626 y Ff(~)5381 4584 y Fz(on)55 b Fu(H)5754 4609 y Fq(~)5891 4584 y Fz(\025)g(unique)h(up)758 4783 y(to)j(a)g(glob)-8 b(al)60 b(normalization)g(factor)f(\025)h(so)f(that)g Fw(U)18 b Fy(\()4029 4718 y Ft(n)p 4011 4745 118 7 v 4011 4840 a(N)4149 4783 y Fy(\))59 b Fz(is)g(unitary)g(for)g(e)-8 b(ach)60 b Fw(n)45 b Fu(2)h Fs(Z)6088 4723 y Fv(2)6171 4783 y Fz(.)758 5085 y FD(Pro)5 b(of:)75 b FC(One)56 b(has)2050 5396 y Fw(U)18 b Fy(\()2301 5284 y(1)p 2266 5358 152 7 v 2266 5510 a Fw(N)2438 5396 y(;)28 b Fy(0\))p Fw(e)2737 5421 y Ft(j)2853 5396 y Fy(=)46 b Fw(e)3105 5421 y Ft(j)7 b Fv(+1)3343 5396 y Fw(;)360 b(U)18 b Fy(\(0)p Fw(;)4156 5284 y Fy(1)p 4122 5358 V 4122 5510 a Fw(N)4293 5396 y Fy(\))p Fw(e)4435 5421 y Ft(j)4551 5396 y Fy(=)47 b Fw(e)4804 5327 y Fr(\000)p Ft(i)p Fv(2)p Ft(\031)5145 5274 y Fo(j)p 5123 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Ft(j)2091 10837 y Fw(;)28 b(O)5 b(p)2381 10769 y Ft(W)2586 10837 y Fw(f)73 b(')2849 10751 y Fv(\()p Ft(N)12 b Fv(\))2849 10884 y Ft(j)3078 10837 y Fu(i)3143 10862 y Fr(H)3257 10879 y Ff(~)3385 10837 y Fy(=)46 b Fu(h)p Fw(')3734 10751 y Fv(\()p Ft(N)12 b Fv(\))3734 10884 y Ft(j)3963 10837 y Fw(;)28 b(O)5 b(p)4253 10769 y Ft(W)4504 10837 y Fw(<)47 b(f)64 b(>)4954 10862 y Ft(T)5160 10837 y Fw(')5269 10751 y Fv(\()p Ft(N)12 b Fv(\))5269 10884 y Ft(j)5499 10837 y Fu(i)5564 10862 y Fr(H)5678 10879 y Ff(~)5759 10837 y Fw(;)p eop %%Page: 53 53 53 52 bop 2362 861 a FA(QUANTUM)58 b(CHA)l(OS:)g(A)h(BRIEF)f(FIRST)g (VISIT)1472 b(53)758 1259 y FC(where)67 b(I)h(used)g(the)f(notation)g Fw(<)g(f)85 b(>)3234 1284 y Ft(T)3406 1259 y FC(in)-5 b(tro)5 b(duced)68 b(in)g(Prop)5 b(osition)68 b(10.6)g(as)g(w)-5 b(ell)68 b(as)g(the)758 1458 y(Egoro)-5 b(v)60 b(theorem.)89 b(Lemma)60 b(14.3,)i(together)d(with)h(Lemma)h(13.9)g(\(i\))e(and)i (\(iii\))f(then)g(imply)758 1658 y(that,)54 b(for)i(ev)-5 b(ery)54 b Fw(T)69 b Fu(2)46 b Fs(N)31 b FC(,)64 b(there)55 b(exists)g(constan)-5 b(ts)57 b Fw(C)4162 1597 y Fr(0)4150 1703 y Ft(T)4254 1658 y Fw(;)28 b(C)4447 1683 y Ft(T)4551 1658 y FC(,)55 b(indep)5 b(enden)-5 b(t)56 b(of)f Fw(N)18 b FC(,)56 b(so)g(that)2528 1984 y Fy(1)p 2494 2058 152 7 v 2494 2210 a Fw(N)2754 1889 y Ft(N)2693 1939 y Fp(X)2698 2292 y Ft(j)7 b Fv(=1)2979 2097 y Fu(j)46 b(h)p Fw(')3245 2010 y Fv(\()p Ft(N)12 b Fv(\))3245 2143 y Ft(j)3474 2097 y Fw(;)28 b(O)5 b(p)3764 2028 y Ft(W)3969 2097 y Fw(f)74 b(')4233 2010 y Fv(\()p Ft(N)12 b Fv(\))4233 2143 y Ft(j)4462 2097 y Fu(i)4527 2122 y Fr(H)4641 2139 y Ff(~)4769 2097 y Fu(j)4815 2028 y Fv(2)4890 2097 y Fu(\024)1675 2798 y(\024)2024 2686 y Fy(1)p 1990 2760 V 1990 2912 a Fw(N)2250 2590 y Ft(N)2189 2640 y Fp(X)2194 2994 y Ft(j)7 b Fv(=1)2429 2798 y Fu(h)p Fw(')2603 2712 y Fv(\()p Ft(N)12 b Fv(\))2603 2844 y Ft(j)2832 2798 y Fw(;)2906 2664 y Fp(\000)2982 2798 y Fw(O)5 b(p)3198 2729 y Ft(W)3449 2798 y Fw(<)46 b(f)65 b(>)3899 2823 y Ft(T)4003 2664 y Fp(\001)4080 2691 y Fr(\003)4184 2664 y Fp(\000)4260 2798 y Fw(O)5 b(p)4476 2729 y Ft(W)4727 2798 y Fw(<)46 b(f)65 b(>)5177 2823 y Ft(T)5281 2664 y Fp(\001)5385 2798 y Fw(')5494 2712 y Fv(\()p Ft(N)12 b Fv(\))5494 2844 y Ft(j)5723 2798 y Fu(i)5788 2823 y Fr(H)5902 2840 y Ff(~)1675 3403 y Fu(\024)2024 3291 y Fy(1)p 1990 3365 V 1990 3517 a Fw(N)2250 3196 y Ft(N)2189 3245 y Fp(X)2194 3599 y Ft(j)7 b Fv(=1)2429 3403 y Fu(h)p Fw(')2603 3317 y Fv(\()p Ft(N)12 b Fv(\))2603 3449 y Ft(j)2832 3403 y Fw(;)2906 3269 y Fp(\000)2982 3403 y Fw(O)5 b(p)3198 3335 y Ft(W)p 3403 3268 555 7 v 3403 3403 a Fw(<)46 b(f)65 b(>)3853 3428 y Ft(T)4003 3403 y Fw(<)47 b(f)64 b(>)4453 3428 y Ft(T)4558 3269 y Fp(\001)4662 3403 y Fw(')4771 3317 y Fv(\()p Ft(N)12 b Fv(\))4771 3449 y Ft(j)5000 3403 y Fu(i)5065 3428 y Fr(H)5179 3445 y Ff(~)5297 3403 y Fy(+)5483 3291 y Fw(C)5614 3231 y Fr(0)5602 3337 y Ft(T)p 5483 3365 224 7 v 5519 3517 a Fw(N)1675 3931 y Fu(\024)1970 3705 y Fp(Z)2062 4082 y Fq(T)2141 4049 y Fn(2)p 2241 3786 825 7 v 2241 3931 a Fw(<)46 b(f)64 b(>)2690 3956 y Ft(T)2841 3931 y Fy(\()p Fw(x)p Fy(\))45 b Fw(<)h(f)65 b(>)3561 3956 y Ft(T)3712 3931 y Fy(\()p Fw(x)p Fy(\))p Fw(dx)36 b Fy(+)4340 3818 y Fw(C)4459 3843 y Ft(T)p 4340 3892 224 7 v 4376 4045 a Fw(N)758 4286 y FC(Since)53 b(w)-5 b(e)53 b(don't)h(kno)-5 b(w)53 b(a)g(priori)h(if)f(the)g(limit)h(as)f Fw(N)65 b Fu(!)46 b(1)53 b FC(of)g(the)g(left)g(hand)h(side)f(exists,)h(w)-5 b(e)758 4485 y(tak)g(e)54 b(the)h(limsup)i(of)e(b)5 b(oth)55 b(sides)h(with)g(resp)5 b(ect)54 b(to)h Fw(N)74 b FC(to)55 b(obtain)1383 4911 y Fy(lim)28 b(sup)1446 5050 y Ft(N)12 b Fr(!1)1973 4798 y Fy(1)p 1939 4873 152 7 v 1939 5025 a Fw(N)2199 4703 y Ft(N)2138 4753 y Fp(X)2143 5107 y Ft(j)7 b Fv(=1)2424 4911 y Fu(j)46 b(h)p Fw(')2690 4825 y Fv(\()p Ft(N)12 b Fv(\))2690 4957 y Ft(j)2919 4911 y Fw(;)28 b(O)5 b(p)3209 4842 y Ft(W)3414 4911 y Fw(f)74 b(')3678 4825 y Fv(\()p Ft(N)12 b Fv(\))3678 4957 y Ft(j)3907 4911 y Fu(i)3972 4936 y Fr(H)4086 4953 y Ff(~)4214 4911 y Fu(j)4260 4842 y Fv(2)4335 4911 y Fu(\024)4510 4685 y Fp(Z)4602 5062 y Fq(T)4681 5029 y Fn(2)4799 4911 y Fu(j)p Fw(<)46 b(f)64 b(>)5294 4936 y Ft(T)5445 4911 y Fy(\()p Fw(x)p Fy(\))46 b Fu(j)5762 4842 y Fv(2)5882 4911 y Fw(dx:)758 5350 y FC(Remarking)55 b(that)f(the)g(left)g(hand)h (side)g(do)5 b(es)55 b(not)f(dep)5 b(end)55 b(on)g Fw(T)23 b FC(,)55 b(and)g(using)h(the)e(ergo)5 b(dicit)-5 b(y)758 5549 y(of)45 b Fw(A)g FC(as)h(expressed)g(in)f(Prop)5 b(osition)46 b(10.8,)i(\(14.2\))c(follo)-5 b(ws)47 b(immediately)-14 b(.)70 b(Lemma)46 b(14.5)f(then)758 5748 y(implies)65 b(the)e(existence)g(of)g(a)h(set)g Fw(E)10 b Fy(\()p Fw(N)f(;)28 b(f)18 b Fy(\))59 b Fu(\032)h(f)p Fy(1)p Fw(;)28 b(:)g(:)g(:)f(;)h(N)18 b Fu(g)65 b FC(ha)-5 b(ving)64 b(prop)5 b(erties)64 b(\(i\))f(and)h(\(ii\))758 5948 y(of)52 b(the)f(theorem.)73 b(This)53 b(basically)g(ends)g(the)e(pro)5 b(of,)53 b(except)e(for)h(the)g(dep)5 b(endence)52 b(of)g(the)f(set)758 6147 y Fw(E)10 b Fy(\()p Fw(N)f(;)28 b(f)18 b Fy(\))57 b FC(on)j(the)e(function)h Fw(f)18 b FC(.)85 b(The)59 b(latter)f(can)i(b)5 b(e)58 b(eliminated)i(b)-5 b(y)59 b(using)h(the)f(same)g(tric)-5 b(ks)758 6346 y(as)50 b(those)h(used)g(to)f(pro)-5 b(v)g(e)50 b(Lemma)h(14.5.)72 b(I)51 b(will)f(merely)g(sk)-5 b(etc)g(h)51 b(the)f(argumen)-5 b(t,)51 b(lea)-5 b(ving)51 b(y)-5 b(ou)758 6545 y(to)64 b(\034ll)h(in)g(the)g(details.)102 b(In)-5 b(tro)5 b(duce)65 b Fu(G)10 b FC(,)67 b(the)d(set)h(of)f(all)h(trigonometric)f(p)5 b(olynomials)66 b(on)f Fs(T)6652 6485 y Fv(2)758 6745 y FC(with)57 b(rational)g(co)5 b(e\036cien)-5 b(ts.)79 b(This)58 b(is)g(a)f(coun)-5 b(table)58 b(set)f(the)g(elemen)-5 b(ts)58 b(of)f(whic)-5 b(h)57 b(w)-5 b(e)58 b(denote)758 6944 y(b)-5 b(y)55 b Fw(f)1069 6969 y Ft(`)1133 6944 y Fw(;)28 b(`)47 b Fu(2)e Fs(N)31 b FC(.)83 b(Applying)55 b(the)g(argumen)-5 b(ts)57 b(of)e(the)g(\034rst)g(part)g(of)g(the)g (pro)5 b(of)55 b(to)2436 7257 y Fy(1)p 2402 7331 V 2402 7483 a Fw(N)2662 7162 y Ft(N)2601 7212 y Fp(X)2606 7565 y Ft(j)7 b Fv(=1)2943 7162 y Ft(L)2869 7212 y Fp(X)2877 7569 y Ft(`)p Fv(=1)3155 7370 y Fu(j)46 b(h)p Fw(')3421 7283 y Fv(\()p Ft(N)12 b Fv(\))3421 7416 y Ft(j)3650 7370 y Fw(;)28 b(O)5 b(p)3940 7301 y Ft(W)4145 7370 y Fw(f)4226 7395 y Ft(`)4345 7370 y Fw(')4454 7283 y Fv(\()p Ft(N)12 b Fv(\))4454 7416 y Ft(j)4684 7370 y Fu(i)4749 7395 y Fr(H)4863 7412 y Ff(~)4990 7370 y Fu(j)5036 7301 y Fv(2)758 7819 y FC(one)39 b(sho)-5 b(ws)41 b(there)e(exist)g(sets)g Fw(E)10 b Fy(\()p Fw(N)f(;)28 b(L)p Fy(\))45 b Fu(\032)h(f)p Fy(1)p Fw(;)28 b(:)g(:)g(:)f(;)h(N)18 b Fu(g)40 b FC(for)f(whic)-5 b(h)40 b(\(i\))f(and)h(\(ii\))e(of)i(the)f(theorem)758 8018 y(hold)49 b(with)f Fw(f)65 b Fy(=)46 b Fw(f)1893 8043 y Ft(`)2006 8018 y FC(for)i(eac)-5 b(h)49 b Fy(1)d Fu(\024)g Fw(`)g Fu(\024)h Fw(L)p FC(.)71 b(One)49 b(can)g(moreo)-5 b(v)g(er)49 b(c)-5 b(ho)5 b(ose)49 b(the)f Fw(E)10 b Fy(\()p Fw(N)f(;)28 b(L)p Fy(\))47 b FC(so)i(that)758 8217 y Fw(E)10 b Fy(\()p Fw(N)f(;)28 b(L)p Fy(\))44 b 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Fw(N)18 b Fy(\))45 b(=)h Fw(E)10 b Fy(\()p Fw(N)f(;)28 b(L)4209 9400 y Ft(N)4334 9375 y Fy(\))p Fw(:)758 9647 y FC(Since)55 b Fw(L)1304 9672 y Ft(N)1476 9647 y Fu(!)46 b(1)56 b FC(as)g Fw(N)64 b Fu(!)46 b(1)p FC(,)56 b(the)f(theorem)g(follo)-5 b(ws)56 b(from)f(the)g(observ)-9 b(ation)55 b(that)1923 9944 y Fu(8)p Fw(f)64 b Fu(2)46 b Fw(C)2448 9875 y Fr(1)2588 9944 y Fy(\()p Fs(T)2766 9875 y Fv(2)2849 9944 y Fy(\))p Fw(;)28 b Fu(8)p Fw(\017)46 b(>)h Fy(0)p Fw(;)28 b Fu(9)p Fw(`)46 b Fu(2)g Fs(N)31 b Fw(;)4133 9786 y Fp(X)4087 10149 y Ft(n)p Fr(2)p Fq(Z)4343 10116 y Fn(2)4456 9944 y Fu(j)46 b Fw(f)4629 9969 y Ft(n)4756 9944 y Fu(\000)37 b Fw(f)5003 9969 y Ft(`;n)5235 9944 y Fu(j\024)47 b Fw(\017;)758 10379 y FC(together)54 b(with)h(the)g(observ)-9 b(ation)55 b(that)2767 10675 y Fu(k)46 b Fw(O)5 b(p)3112 10607 y Ft(W)3318 10675 y Fw(f)64 b Fu(k)3546 10700 y Fr(H)3660 10717 y Ff(~)3742 10675 y Fu(\024)3954 10517 y Fp(X)3917 10881 y Ft(n)p Fr(2)p Fq(Z)4173 10848 y Fn(2)4277 10675 y Fu(j)46 b Fw(f)4450 10700 y Ft(n)4587 10675 y Fu(j)h Fw(:)p eop %%Page: 54 54 54 53 bop 758 878 a FA(54)2127 b(STEPHAN)58 b(DE)h(BI\310VRE)p 6614 1259 7 113 v 6621 1154 100 7 v 6621 1259 V 6720 1259 7 113 v 1057 1458 a FC(The)64 b(three)g(essen)-5 b(tial)65 b(ingredien)-5 b(ts)66 b(of)e(this)h(pro)5 b(of)64 b(\(and)g(of)h(all)f(existing)h(pro)5 b(ofs)64 b(of)h(the)758 1658 y(Sc)-5 b(hnirelman)66 b(theorem\))f(are)g(clearly) g(some)g(basic)h(semi-classical)h(estimates,)h(the)d(Egoro)-5 b(v)758 1857 y(theorem)56 b(and)g(the)g(ergo)5 b(dicit)-5 b(y)56 b(of)g Fw(A)p FC(.)76 b(In)57 b(particular,)f(the)g(result)h(do) 5 b(es)56 b(not)g(dep)5 b(end)56 b(on)h(an)-5 b(y)758 2056 y(stronger)46 b(dynamical)h(prop)5 b(erties,)48 b(suc)-5 b(h)47 b(as)f(the)g(\(exp)5 b(onen)-5 b(tial\))45 b(mixing)i(for)f(example,)i(of)e(the)758 2255 y(underlying)53 b(dynamics.)74 b(As)52 b(suc)-5 b(h,)55 b(the)d(Sc)-5 b(hnirelman)54 b(theorem)f(is)g(clearly)f(only)h(a)f(\034rst)h(step)758 2455 y(in)61 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5382 y(A)57 b(particularly)h(in)-5 b(teresting)58 b(p)5 b(ossibilit)-5 b(y)59 b(w)-5 b(ould)59 b(b)5 b(e)58 b(the)f(existence)g(of)h(suc)-5 b(h)59 b(sequences)758 5582 y(where)2917 5921 y Fw(\026)p Fy(\()p Fw(f)18 b Fy(\))46 b(=)3506 5809 y(1)p 3487 5883 121 7 v 3487 6035 a Fw(T)3655 5714 y Ft(T)18 b Fr(\000)p Fv(1)3668 5764 y Fp(X)3679 6115 y Ft(t)p Fv(=0)3949 5921 y Fw(f)g Fy(\()p Fw(A)4238 5853 y Ft(t)4295 5921 y Fw(x)4390 5946 y Fv(0)4465 5921 y Fy(\);)758 6317 y FC(here)64 b Fw(x)1222 6342 y Fv(0)1362 6317 y FC(is)h(a)g(p)5 b(erio)g(dic)65 b(p)5 b(oin)-5 b(t)65 b(of)g(the)f(dynamics)i(generated)e(b)-5 b(y)66 b Fw(A)p FC(,)h(of)d(p)5 b(erio)g(d)65 b Fw(T)23 b FC(,)67 b(so)f(that)758 6517 y Fw(A)883 6456 y Ft(T)987 6517 y Fw(x)1082 6542 y Fv(0)1212 6517 y Fy(=)55 b Fw(x)1491 6542 y Fv(0)1565 6517 y FC(.)91 b(In)61 b(other)f(w)-5 b(ords,)64 b Fw(\026)d FC(is)g(the)f(natural)h Fw(A)p FC(-in)-5 b(v)c(arian)k(t)62 b(measure)f(supp)5 b(orted)61 b(on)g(a)758 6716 y(p)5 b(erio)g(dic)67 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10840 y(holds)63 b(b)5 b(oth)62 b(for)g(the)g(h)-5 b(yp)5 b(erb)g(olic)62 b(toral)g(automorphisms)j (and)e(their)f(p)5 b(erturbations\))62 b(I)g(am)p eop %%Page: 55 55 55 54 bop 2362 861 a FA(QUANTUM)58 b(CHA)l(OS:)g(A)h(BRIEF)f(FIRST)g (VISIT)1472 b(55)758 1259 y FC(a)-5 b(w)g(are)58 b(of)g(is)h(a)g(W)-14 b(eyl-t)-5 b(yp)5 b(e)57 b(la)-5 b(w)58 b(sa)-5 b(ying)59 b(that)f(the)g(eigen)-5 b(v)c(alues)58 b(equidistribute)h(on)f(the)g (unit)758 1458 y(circle,)d Fz(i.e.)2413 1736 y Fy(lim)2337 1843 y Ft(N)12 b Fr(!1)2801 1623 y Fy(1)p 2767 1697 152 7 v 2767 1849 a Fw(N)2939 1736 y(])p Fu(f)p Fy(1)45 b Fu(\024)i Fw(j)56 b Fu(\024)46 b Fw(N)64 b Fu(j)47 b Fw(\022)4063 1649 y Fv(\()p Ft(N)12 b Fv(\))4058 1782 y Ft(j)4338 1736 y Fu(2)46 b Fw(I)13 b Fu(g)46 b Fy(=)p Fu(j)g Fw(I)60 b Fu(j)p Fw(;)758 2044 y FC(for)38 b(an)-5 b(y)40 b(in)-5 b(terv)c(al)38 b Fw(I)52 b FC(on)40 b(the)e(unit)h (circle)g([)p FD(BDB1)p FC(],)k(but)c(this)g(do)5 b(es)40 b(of)e(course)i(not)e(sa)-5 b(y)40 b(an)-5 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10840 y Fw(\034)831 10865 y Fv(0)942 10840 y Fy(+)37 b Fw(m)55 b FC(from)g(whic)-5 b(h)56 b(the)f(lemma)h(easily)f(follo)-5 b(ws.)p 6614 10840 7 113 v 6621 10734 100 7 v 6621 10840 V 6720 10840 7 113 v eop %%Page: 56 56 56 55 bop 758 878 a FA(56)2127 b(STEPHAN)58 b(DE)h(BI\310VRE)1057 1259 y FC(Note)46 b(that)h(in)h(this)g(case)f(eac)-5 b(h)48 b(orbit)g(is)g(a)g(closed)g(one-dimensional)h(submanifold)g(of)f (the)758 1458 y(torus.)74 b(A)-5 b(t)55 b(the)g(other)f(extreme,)h(one) g(has)h(the)f(follo)-5 b(wing)56 b(result:)1057 1787 y Fx(Lemma)62 b FC(A.2)p Fx(.)83 b Fz(If)68 b(for)g(al)8 b(l)70 b Fw(m)63 b Fu(2)f Fs(Z)3264 1727 y Ft(n)p Fr(\003)3431 1787 y Fz(,)71 b Fw(!)50 b Fu(\001)44 b Fw(m)62 b Fu(6)p Fy(=)h(0)p Fz(,)71 b(then,)h(for)c(al)8 b(l)70 b Fw(f)81 b Fu(2)62 b Fw(C)12 b Fy(\()p Fs(T)5997 1727 y Ft(n)6096 1787 y Fy(\))p Fz(,)71 b(for)d(al)8 b(l)758 1987 y Fw(\034)831 2012 y Fv(0)951 1987 y Fu(2)46 b Fs(T)1220 1926 y Ft(n)1320 1987 y Fz(,)59 b(one)h(has)2419 2353 y Fy(lim)2354 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b(considered?)69 b(Can)39 b(y)-5 b(ou)38 b(pro)-5 b(v)g(e)39 b(y)-5 b(our)39 b(conjectures?)1057 9360 y Fx(Exer)-5 b(cise)62 b FC(A.6)p Fx(.)82 b FC(*)43 b(Study)f(the)g(motion)h(in)g(a)f(square)h(billiard.)70 b(Pro)-5 b(v)g(e)43 b(that,)i(if)d(the)g(slop)5 b(e)758 9559 y(of)49 b(the)h(tra)9 b(jectory)48 b(with)h(resp)5 b(ect)49 b(to)h(one)f(of)h(the)f(sides)i(of)f(the)f(billiard)i(is)f (rational,)h(then)f(the)758 9758 y(tra)9 b(jectory)53 b(is)j(p)5 b(erio)g(dic.)74 b(Sho)-5 b(w)56 b(that,)e(if)i(the)e(slop)5 b(e)56 b(is)g(irrational,)g(then)2195 10194 y Fy(lim)2130 10301 y Ft(T)18 b Fr(!1)2557 10082 y Fy(1)p 2539 10156 V 2539 10308 a Fw(T)2707 9968 y Fp(Z)2873 10009 y Ft(T)2799 10346 y Fv(0)3005 10194 y Fw(f)g Fy(\()p Fw(\015)9 b Fy(\()p Fw(t)p Fy(\)\))54 b Fw(dt)46 b Fy(=)3941 9968 y Fp(Z)4107 10009 y Fv(1)4033 10346 y(0)4209 9968 y Fp(Z)4375 10009 y Fv(1)4301 10346 y(0)4477 10194 y Fw(f)18 b Fy(\()p Fw(x;)28 b(y)6 b Fy(\))p Fw(dxdy)g(:)758 10599 y FC(A)-5 b(dapt)44 b(the)f(ab)5 b(o)-5 b(v)g(e)45 b(statemen)-5 b(ts)44 b(to)g(the)f(case)h(of)g(a)g(rectangular)g(billiard)h(with)f (sides)h(of)f(length)758 10798 y Fw(a)55 b FC(and)h Fw(b)p FC(.)74 b FD(Hin)-5 b(t:)73 b FC(Y)-14 b(ou)55 b(ma)-5 b(y)56 b(wish)g(to)f(consult)g([)p FD(CFS)q FC(])g(or)g([)p FD(T)p FC(])h(for)f(inspiration.)p eop %%Page: 57 57 57 56 bop 2362 861 a FA(QUANTUM)58 b(CHA)l(OS:)g(A)h(BRIEF)f(FIRST)g (VISIT)1472 b(57)3298 1259 y FD(References)758 1525 y FE([Arf)10 b(])350 b(Arfk)l(en)48 b(G.,)f Fe(Mathematic)-7 b(al)49 b(metho)-7 b(ds)50 b(for)h(physicists,)c FE(A)l(cademic)e (Press)h(\(1985\).)758 1691 y([A])458 b(Arnold,)49 b(V.,)h Fe(Mathematic)-7 b(al)51 b(metho)-7 b(ds)52 b(in)h(classic)-7 b(al)53 b(me)-7 b(chanics,)47 b FE(New-Y)-12 b(ork-Berlin,)48 b(Springer-)1400 1857 y(V)-12 b(erlag)45 b(\(1973\))758 2023 y([AM])329 b(Abraham)47 b(R.,)g(Marsden)g(J.,)h Fe(F)-11 b(oundations)51 b(of)g(me)-7 b(chanics,)44 b FE(Benjamin-Cummings)h(\(1978\))758 2189 y([B])464 b(Berry)65 b(M.,)70 b Fe(Semiclassic)-7 b(al)67 b(me)-7 b(chanics)65 b(of)i(r)-7 b(e)g(gular)67 b(and)h(irr)-7 b(e)g(gular)67 b(motion)p FE(,)i(in)64 b(\020Chaotic)f(b)t(e-)1400 2355 y(ha)l(viour)71 b(of)g(deterministic)d(systems\021,)76 b(Les)70 b(Houc)l(hes)h(Lectures)f(v)l(ol)g(XXXVI,)g(ed.)i(G.)f(Io)t (oss,)1400 2521 y(R.H.G.)47 b(Helleman)d(and)k(R.)f(Stora,)f(pp.)i (171-271,)d(Amsterdam)g(North-Holland)f(\(1983\).)758 2687 y([Ba1])322 b(Baladi,)44 b(V.,)h Fe(Positive)j(tr)-7 b(ansfer)49 b(op)-7 b(er)g(ators)48 b(and)h(de)-7 b(c)g(ay)48 b(of)g(c)-7 b(orr)g(elations,)45 b FE(W)-12 b(orld)44 b(Scien)l(ti\034c,)g(Sin-)1400 2853 y(gap)t(ore)i(\(2000\).)758 3019 y([Ba2])322 b(Baladi,)61 b(V.)e Fe(De)-7 b(c)g(ay)61 b(of)h(c)-7 b(orr)g(elations-Minic)g(ourse-Handout)61 b(notes)p FE(,)i(1999)57 b(AMS)j(Summer)f(Re-)1400 3185 y(searc)l(h)47 b(Institute,)f(Seattle)f(W)-12 b(ashington)46 b(\020Smo)t(oth)g(ergo)t(dic)f(theory)i(and)g(applications\021.)758 3351 y([BonDB])107 b(Bonec)l(hi)66 b(F.,)72 b(De)65 b(Bi\350vre)h(S.,) 72 b Fe(Exp)-7 b(onential)68 b(mixing)f(and)i Fl(j)j Fd(ln)23 b Fc(~)71 b Fl(j)f Fe(time)e(sc)-7 b(ales)69 b(in)g(quantize)-7 b(d)1400 3517 y(hyp)g(erb)g(olic)50 b(maps)g(on)h(the)g(torus,)d FE(Comm)l(un.)f(Math.)g(Ph)l(ys,)h Fb(211)p FE(,)g(3,)f(659-686)d(\(2000\).)758 3683 y([BDB1])185 b(Bouzouina)56 b(A.,)j(De)e(Bi\350vre)e(S.,)60 b Fe(Equip)-7 b(artition)60 b(of)f(the)h(eigenfunctions)f(of)h(quantize)-7 b(d)60 b(er)-7 b(go)g(dic)1400 3849 y(maps)51 b(on)g(the)g(torus)p FE(,)d(Comm)l(un.Math.Ph)l(ys.)g Fb(178)p FE(,)g(83-105)d(\(1996\).)758 4015 y([BDB2])185 b(Bouzouina)58 b(A.,)j(De)d(Bi\350vre)g(S.,)k Fe(Equidistribution)g(des)f(valeurs)h(pr)-7 b(opr)g(es)61 b(et)h(er)-7 b(go)g(dicit\351)59 b(semi-)1400 4181 y(classique)k(de)f (symple)-7 b(ctomorphismes)60 b(du)j(tor)-7 b(e)62 b(quanti\034\351s)11 b FE(,)62 b(C.)e(R.)f(A)l(cad.)g(Sci.)g(P)l(aris,)i(S\351rie)d(I)1400 4347 y Fb(t.326)p FE(,)49 b(1021-1024)44 b(\(1998\).)758 4514 y([BGS])275 b(Bohigas)54 b(O.,)k(Giannoni)c(M.-J.,)60 b(Sc)l(hmit)55 b(C.,)j Fe(Char)-7 b(acterization)58 b(of)h(chaotic)f (quantum)h(sp)-7 b(e)g(ctr)g(a)1400 4680 y(and)51 b(universality)h(of)e (level)h(\035uctuation)h(laws)p FE(,)47 b(Ph)l(ys.)h(Rev.)f(Lett.)g Fb(52)p FE(,)h(1-4)e(\(1984\).)758 4846 y([Bo])393 b(Bouzouina)59 b(A.,)64 b Fe(Comp)-7 b(ortement)62 b(semi-classique)g(de)h(symple)-7 b(ctomorphismes)62 b(du)h(tor)-7 b(e)63 b(quan-)1400 5012 y(ti\034\351s)p FE(,)48 b(do)t(ctoral)d(dissertation,)f(Univ)l (ersit\351)h(P)l(aris)g(9)i(\(1997\).)758 5178 y([BT])362 b(Berry)51 b(M.,)h(T)-12 b(ab)t(or)50 b(M.,)h Fe(L)-7 b(evel)54 b(clustering)g(in)g(the)g(r)-7 b(e)g(gular)54 b(sp)-7 b(e)g(ctrum)p FE(,)50 b(Pro)t(c.)g(Ro)l(y)-12 b(.)51 b(So)t(c.)f(A)g Fb(356)p FE(,)1400 5344 y(375-394)45 b(\(1977\).)758 5510 y([CdV])278 b(Colin)58 b(de)h(V)-12 b(erdi\350re)58 b(Y.,)63 b Fe(Er)-7 b(go)g(dicit\351)59 b(et)j(fonctions)g(pr)-7 b(opr)g(es)62 b(du)h(L)-7 b(aplacien)p FE(,)61 b(Comm)l(un.)e(Math.)1400 5676 y(Ph)l(ys.)48 b Fb(102)p FE(,)h(497-502)44 b(\(1985\).)758 5842 y([CFKS])182 b(Cycon)40 b(H.L.,)h(F)-12 b(ro)t(ese)38 b(R.G.,)j(Kirsc)l(h)d(W.,)j (Simon)d(B.,)j Fe(Schr)-7 b(\366)g(dinger)42 b(Op)-7 b(er)g(ators)p FE(,)41 b(Springer-V)-12 b(erlag)1400 6008 y(\(1987\).)758 6174 y([CFS])292 b(Cornfeld)47 b(I.P)-12 b(.,)47 b(F)-12 b(omin)46 b(S.V.,)i(Sinai)e(Y.G.,)h Fe(Er)-7 b(go)g(dic)49 b(the)-7 b(ory)p FE(,)47 b(Springer-V)-12 b(erlag)44 b(\(1982\).)758 6340 y([CR])358 b(Com)l(b)t(escure)67 b(M.,)75 b(Rob)t(ert)67 b(D.,)74 b Fe(Distribution)c(of)h(matrix)f (elements)g(and)h(level)f(sp)-7 b(acings)70 b(for)1400 6506 y(classic)-7 b(al)7 b(ly)52 b(chaotic)e(systems)p FE(,)e(Ann.)f(Inst.)g(H.)g(P)l(oincar\351)e Fb(61)p FE(,4)j(\(1994\),)d (443-483.)758 6672 y([CTDL])164 b(Cohen-T)-12 b(annoudji)59 b(C.,)j(Diu)c(B.,)k(Lalo)t(\353)56 b(F.,)62 b(Quan)l(tum)d(mec)l (hanics,)h(v)l(olumes)d(I)h(and)h(I)t(I,)f(John)1400 6838 y(Wiley)46 b(\(1977\).)758 7004 y([D])456 b(Dolgop)l(y)l(at)40 b(D.,)i Fe(On)k(de)-7 b(c)g(ay)44 b(of)h(c)-7 b(orr)g(elations)45 b(in)g(A)l(nosov)h(\035ows,)41 b FE(Annals)f(of)h(Math.)g Fb(147)p FE(,)j(357-390)1400 7170 y(\(1998\).)758 7336 y([DB])356 b(De)50 b(Bi\350vre)f(S.,)j Fe(Chaos,)i(quantization)h(and)e (the)h(classic)-7 b(al)54 b(limit)g(on)g(the)f(torus)p FE(,)g(Pro)t(ceedings)48 b(of)1400 7502 y(the)h(XIVth)f(W)-12 b(orkshop)49 b(on)g(Geometrical)44 b(Metho)t(d)49 b(in)f(Ph)l(ysics)g (\025)g(Bialo)l(wieza)d(\(1995\),)i(mp)6443 7520 y FB(\000)6547 7502 y FE(arc)1400 7668 y(96-191,)e(P)l(olish)h(Scien)l(ti\034c)f (Publishers)h(PWN)g(\(1998\).)758 7834 y([DEGI])198 b(Degli)43 b(Esp)t(osti,)h(M.,)j(Gra\036,)e(S.,)h(Isola,)e(S.,)i Fe(Sto)-7 b(chastic)49 b(pr)-7 b(op)g(erties)48 b(of)h(the)h(quantum)g (A)l(rnold)g(c)-7 b(at)1400 8000 y(in)51 b(the)g(classic)-7 b(al)51 b(limit)p FE(,)c(Comm.)f(Math.)h(Ph)l(ys)h Fb(167)g FE(,)f(471-509)d(\(1995\).)758 8166 y([F])472 b(F)-12 b(olland)53 b(G.,)k Fe(Harmonic)f(analysis)i(in)g(phase)f(sp)-7 b(ac)g(e)p FE(,)55 b(Princeton)e(Univ)l(ersit)l(y)g(Press,)i(Princeton) 1400 8333 y(\(1988\).)758 8499 y([G])453 b(Goldstein,)45 b(H.,)i Fe(Classic)-7 b(al)52 b(me)-7 b(chanics)p FE(,)45 b(A)l(ddison-W)-12 b(esley)45 b(\(1959\).)758 8665 y([GMW])179 b(Guhr)57 b(T.,)i(M\374ller-Gro)t(eling)53 b(H.,)58 b(W)-12 b(eidenm)l(\374ller)54 b(H.,)59 b Fe(R)-7 b(andom)59 b(matrix)g(the)-7 b(ories)58 b(in)i(quantum)1400 8831 y(physics:)66 b(c)-7 b(ommon)49 b(c)-7 b(onc)g(epts)p FE(,)46 b(Ph)l(ys.)h(Rep.)h Fb(299)p FE(,)g(189-425)c(\(1998\).)758 8997 y([GL])365 b(G\351rard)66 b(P)-12 b(.,)72 b(Leic)l(h)l(tnam)65 b(E.,)71 b Fe(Er)-7 b(go)g(dic)67 b(pr)-7 b(op)g(erties)67 b(of)i(the)f(eigenfunctions)g(for)h(the)f(Dirichlet)1400 9163 y(pr)-7 b(oblem)p FE(,)46 b(Duk)l(e.)i(Math.)f(J.)h Fb(71)g FE(\(1993\),)d(559-607.)758 9329 y([H])458 b(Hel\033er)44 b(B.,)i Fe(Semi-classic)-7 b(al)48 b(analysis)i(for)e(the)h(Schr)-7 b(\366)g(dinger)47 b(op)-7 b(er)g(ator)48 b(and)h(applic)-7 b(ations)p FE(,)45 b(LNM)1400 9495 y(1336,)h(Springer)h(V)-12 b(erlag)45 b(\(1988\).)758 9661 y([Ho])387 b(H\366rmander,)42 b(L.,)g Fe(The)j(analysis)h(of)f(line)-7 b(ar)45 b(p)-7 b(artial)45 b(di\033er)-7 b(ential)45 b(op)-7 b(er)g(ators)45 b(IV)p FE(,)c(Springer-V)-12 b(erlag)1400 9827 y(\(1985\).)758 9993 y([HB])358 b(Hanna)l(y)38 b(J.H.,)i(Berry)e(M.V.,)i Fe(Quantization)j(of)f(line)-7 b(ar)42 b(maps-F)-11 b(r)k(esnel)42 b(di\033r)-7 b(action)41 b(by)h(a)h(p)-7 b(erio)g(dic)1400 10159 y(gr)g(ating)p FE(,)47 b(Ph)l(ysica)f(D)h Fb(1)p FE(,)g(267)f(\(1980\).)758 10325 y([HS])380 b(Hislop)45 b(P)-12 b(.,)48 b(Sigal)d(I.M.,)i(In)l(tro)t(duction)g(to)f(sp)t (ectral)f(theory)-12 b(,)47 b(Springer-V)-12 b(erlag)45 b(\(1996\).)758 10491 y([I])513 b(Ivrii)52 b(V,)h Fe(Micr)-7 b(olo)g(c)g(al)55 b(analysis)j(and)e(pr)-7 b(e)g(cise)55 b(sp)-7 b(e)g(ctr)g(al)56 b(asymptotics)p FE(,)f(Springer)d(Monographs) h(in)1400 10657 y(Mathematics.)45 b(Springer-V)-12 b(erlag,)45 b(Berlin,)g(1998.)p eop %%Page: 58 58 58 57 bop 758 878 a FA(58)2127 b(STEPHAN)58 b(DE)h(BI\310VRE)758 1259 y FE([Ke])391 b(Keating)53 b(J.,)58 b Fe(The)g(c)-7 b(at)58 b(maps:)80 b(quantum)59 b(me)-7 b(chanics)56 b(and)i(classic)-7 b(al)58 b(motion,)e FE(Nonlinearit)l(y)d Fb(4)p FE(,)1400 1425 y(309\025341)44 b(\(1991\).)758 1591 y([Kh])376 b(Khinc)l(hin)43 b(A.,)i Fe(Continue)-7 b(d)47 b(F)-11 b(r)k(actions,)44 b FE(The)f(Univ)l(ersit)l(y)g(of)g (Chicago)e(Press,)j(Chicago)d(and)j(Lon-)1400 1757 y(don)k(\(1964\).) 758 1923 y([K])454 b(Kirillo)l(v)44 b(A.,)j Fe(Elements)k(of)f(the)h (the)-7 b(ory)51 b(of)f(r)-7 b(epr)g(esentations)p FE(,)47 b(Springer)f(\(1976\).)758 2089 y([KR1])279 b(Kurlb)t(erg,)68 b(P)-12 b(.,)69 b(Rudnic)l(k,)h(Z.)65 b Fe(He)-7 b(cke)64 b(the)-7 b(ory)67 b(and)g(e)-7 b(quidistribution)67 b(for)f(the)h (quantization)h(of)1400 2255 y(line)-7 b(ar)51 b(maps)g(of)f(the)h (torus)p FE(,)d(preprin)l(t)f(1999,)f(c)l(hao-dyn/9901031,)e(Duk)l(e)j (Math.)h(J.,)f(to)g(app)t(ear.)758 2421 y([KR2])279 b(Kurlb)t(erg,)42 b(P)-12 b(.,)44 b(Rudnic)l(k,)f(Z.)g Fe(On)k(quantum)f(er)-7 b(go)g(dicity)45 b(for)g(line)-7 b(ar)46 b(maps)g(of)g(the)g(torus)p FE(,)e(preprin)l(t)1400 2587 y(1999,)i(math.NT/9910145.)758 2753 y([LL1])317 b(Landau)48 b(L.D.,)f(Lifshitz)f(E.M.,)h Fe(Me)-7 b(chanics)p FE(,)46 b(P)l(ergamon)f(Press,)h(Oxford)h (\(1960\).)758 2919 y([LL2])317 b(Landau)70 b(L.D.,)75 b(Lifshitz)68 b(E.M.,)75 b Fe(Quantum)e(me)-7 b(chanics,)74 b(nonr)-7 b(elativistic)71 b(the)-7 b(ory)p FE(,)75 b(P)l(ergamon)1400 3086 y(Press,)47 b(Oxford)g(\(1965\).)758 3252 y([M])435 b(Messiah)46 b(A.,)h(Quan)l(tum)g(mec)l(hanics,)e(North)i(Holland,)e (Amsterdam)g(\(1961\).)758 3418 y([Ma])364 b(Marklof)48 b(J.,)j(P)l(air)c(correlation)f(densities)h(of)h(inhomogeneous)f (quadratic)h(forms,)h(preprin)l(t)g(ma)l(y)1400 3584 y(2000.)758 3750 y([MO])325 b(de)55 b(Matos)f(M.B.,)j(Ozorio)c(de)h (Almeida)e(A.M.,)57 b Fe(Quantization)j(of)d(A)l(nosov)h(maps,)e FE(Ann.)f(Ph)l(ys.)1400 3916 y Fb(237)p FE(,)49 b(46-65)c(\(1995\).)758 4082 y([P])468 b(P)l(etersen)47 b(K.,)g Fe(Er)-7 b(go)g(dic)48 b(the)-7 b(ory)p FE(,)47 b(Cam)l(bridge)f(UP)-12 b(,)47 b(\(1983\).)758 4248 y([R])460 b(Rob)t(ert)47 b(D.,)g Fe(A)l(utour)52 b(de)f(l'appr)-7 b(oximation)49 b(semi-classique)p FE(,)d(Birkh\344user)g(\(1987\).)758 4414 y([RS])382 b(Reed)63 b(M.,)68 b(Simon)62 b(B.)i Fe(A)h(c)-7 b(ourse)66 b(in)g(mathematic)-7 b(al)64 b(physics)p FE(,)j(v)l(olume)62 b(IV,)h(A)l(cademic)e(Press,)1400 4580 y(London)48 b(\(1978\).)758 4746 y([S1])415 b(Sarnak)48 b(P)-12 b(.,)47 b Fe(A)l(rithmetic)j (quantum)i(chaos)p FE(,)47 b(Israel)e(Math.)i(Conf.)g(Pro)t(c.)f Fb(8)p FE(,)i(183-236)c(\(1995\).)758 4912 y([S2])415 b(Sarnak)54 b(P)-12 b(.,)56 b Fe(Quantum)i(chaos,)g(symmetry)f(and)g (zeta)g(functions.)g(L)-7 b(e)g(ctur)g(es)56 b(I)h(and)g(II)p FE(,)c(Curren)l(t)1400 5078 y(dev)l(elopmen)l(ts)35 b(in)g (mathematics,)g(1997)f(\(Cam)l(bridge,)j(MA\),)f(127\025159,)f(In)l(t.) h(Press,)h(Boston,)g(MA)1400 5244 y(\(1999\).)758 5410 y([Sc])423 b(Sc)l(hnirelman)56 b(A.,)k Fe(Er)-7 b(go)g(dic)59 b(pr)-7 b(op)g(erties)59 b(of)h(eigenfunctions)p FE(,)f(Usp.)e(Math.)h (Nauk.)f Fb(29)p FE(,)k(181-182)1400 5576 y(\(1974\).)758 5742 y([T])462 b(T)-12 b(abac)l(hnik)l(o)l(v)48 b(S.,)f Fe(Bil)7 b(liar)-7 b(ds)p FE(,)48 b(P)l(anoramas)d(et)i(Syn)l (th\350ses)g(1,)g(SMF)h(\(1995\).)758 5908 y([V])458 b(V)-12 b(ogan)47 b(D.,)g Fe(Unitary)52 b(r)-7 b(epr)g(esentations)50 b(of)h(r)-7 b(e)g(ductive)50 b(Lie)h(gr)-7 b(oups)p FE(,)47 b(Princeton)f(Univ)l(ersit)l(y)g(Press)1400 6074 y(\(1987\).)758 6240 y([W])419 b(W)-12 b(alters)68 b(P)-12 b(.,)76 b Fe(A)l(n)c(intr)-7 b(o)g(duction)71 b(to)i(er)-7 b(go)g(dic)69 b(the)-7 b(ory)p FE(,)75 b(Graduate)69 b(T)-12 b(exts)69 b(in)g(Mathematics)e(79,)1400 6406 y(Springer)47 b(V)-12 b(erlag)45 b(\(1982\).)758 6572 y([Z1])407 b(Zelditc)l(h)42 b(S.,)i Fe(Uniform)h(distribution)i(of)f(the)h(eigenfunctions)e(on)i(c) -7 b(omp)g(act)45 b(hyp)-7 b(erb)g(olic)44 b(surfac)-7 b(es)p FE(,)1400 6739 y(Duk)l(e)48 b(Math.)f(J)h Fb(55)f FE(\(1987\),)f(919-941.)758 6905 y([Z2])407 b(Zelditc)l(h)56 b(S.,)k Fe(Quantum)i(tr)-7 b(ansition)60 b(amplitudes)h(for)f(er)-7 b(go)g(dic)58 b(and)i(for)g(c)-7 b(ompletely)59 b(inte)-7 b(gr)g(able)1400 7071 y(systems)p FE(,)48 b(J.)g(F)-12 b(unct.)48 b(Anal)e Fb(94)h FE(\(1990\),)f(415-436.)758 7237 y([Z3])407 b(Zelditc)l(h)46 b(S.,)i Fe(Quantum)k(mixing)p FE(,)46 b(J.)i(F)-12 b(unct)47 b(.)h(Anal)d Fb(140)j FE(\(1996\),)e(68-86.)758 7403 y([ZZ])392 b(Zelditc)l(h)33 b(S.,)38 b(Zw)l(orski)33 b(M.,)38 b Fe(Er)-7 b(go)g(dicity)37 b(of)i(eigenfunctions)e(for)i(er)-7 b(go)g(dic)37 b(bil)7 b(liar)-7 b(ds)p FE(,)37 b(Comm.)c(Math.)1400 7569 y(Ph)l(ys.)48 b Fb(175)p FE(,)h(3,)d(673-682)f(\(1996\).)1057 7880 y FF(UFR)i(de)f(Ma)-12 b(th\351ma)g(tiques)46 b(et)g(UMR)i(A)l(GA)-12 b(T,)46 b(Universit\351)g(des)g(Sciences)g(et)h(Technologies)758 8046 y(de)53 b(Lille,)g(59655)j(Villeneuve)d(d'Ascq)i(Cedex)d(France) 1057 8212 y Fe(E-mail)e(addr)-7 b(ess)11 b FE(:)62 b Fa(debievre@agat.univ-li)o(ll)o(e1.)o(fr)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0106050327473--