Content-Type: multipart/mixed; boundary="-------------9812041112978" This is a multi-part message in MIME format. ---------------9812041112978 Content-Type: text/plain; name="98-741.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="98-741.keywords" periodic potentials, phase space tunneling ---------------9812041112978 Content-Type: application/postscript; name="kam.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="kam.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: kam15.dvi %%Pages: 21 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips kam15.dvi %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1998.12.04:1501 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def 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80B3A3B812FEAAD800070180C8FCB3B3AA81A21704170C6D6D131E177E9238F001FE9238 F80FFF92B6FC7FA26D1680A26EECFE0017F86E14E06E91C7FC6E13F8020390C8FC31607D DE3A>II E end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: a4 %%EndSetup %%Page: 1 1 1 0 bop 800 1034 a FD(Quantum)55 b(T)-13 b(ransp)t(o)l(rt)55 b(on)f(KAM)g(T)-13 b(o)l(ri)1080 1329 y FC(Joachim)37 b(Asch)1755 1286 y FB(\003)2092 1329 y FC(Andreas)i(Knauf)2814 1286 y FB(y)1563 1563 y FC(Novemb)s(er)e(1998)1769 2038 y FA(Abstract)708 2209 y Fz(Although)h(quantum)g(tunneling)h(b)s(et)m (w)m(een)h(phase)e(space)g(to)m(ri)h(o)s(ccurs,)i(it)d(is)572 2322 y(supp)m(ressed)32 b(in)f(the)h(semiclassical)f(limit)f Fy(~)e Fx(&)f Fw(0)32 b Fz(fo)m(r)g(the)f(Schr\177)-45 b(odinger)33 b(equation)572 2435 y(of)d(a)g(pa)m(rticle)h(in)f Fy(R)1215 2402 y Fv(d)1291 2435 y Fz(under)h(the)f(in\015uence)h(of)g (a)f(smo)s(oth)h(p)s(erio)s(dic)f(p)s(otential.)708 2548 y(In)22 b(pa)m(rticula)m(r)f(this)g(implies)g(that)g(the)g (distribution)g(of)g(quantum)g(group)h(velo)s(ci-)572 2661 y(ties)f(nea)m(r)h(energy)h Fu(E)k Fz(converges)d(to)d(the)h (distribution)f(of)h(the)f(classical)g(asymptotic)572 2774 y(velo)s(cities)30 b(nea)m(r)h Fu(E)5 b Fz(,)31 b(up)f(to)h(a)f(term)g(of)h(the)g(o)m(rder)g Fx(O)s Fw(\(1)p Fu(=)2454 2697 y Fx(p)p 2531 2697 73 4 v 2531 2774 a Fu(E)6 b Fw(\))p Fz(.)328 3111 y Ft(1)158 b(Intro)t(duction)328 3333 y Fs(Consider)37 b(abstractly)g(a)g(self-adjoint)e(op)s(erato)m(r) i Fr(H)44 b Fs(on)37 b(its)g(domain)f Fr(D)s Fq(\()p Fr(H)8 b Fq(\))35 b Fs(in)i(a)f(Hilb)s(ert)328 3454 y(space)d Fp(H)q Fs(.)44 b(Then)32 b(fo)m(r)h Fr(")27 b Fp(\025)h Fq(0)33 b Fs(one)f(ma)m(y)h(call)f(a)g(pair)572 3657 y Fq(\()630 3631 y(~)610 3657 y Fr( )t(;)743 3632 y Fq(~)721 3657 y Fr(E)5 b Fq(\))28 b Fp(2)g Fr(D)s Fq(\()p Fr(H)8 b Fq(\))21 b Fp(\002)i Fo(R)109 b Fs(,)97 b Fp(k)1692 3631 y Fq(~)1672 3657 y Fr( )t Fp(k)28 b Fq(=)f(1)97 b Fs(,)h Fp(k)p Fq(\()p Fr(H)29 b Fp(\000)2512 3632 y Fq(~)2489 3657 y Fr(E)6 b Fq(\))2625 3631 y(~)2605 3657 y Fr( )t Fp(k)28 b(\024)g Fr(")328 3861 y Fs(an)40 b Fr(")p Fs(-)p Fn(quasimo)s(de)f Fs([1].)66 b(In)40 b(pa)m(rticula)m(r,) j(eigenfunctions)e Fr( )j Fs(with)c(eigenvalues)h Fr(E)46 b Fs(a)m(re)41 b Fq(0)p Fs(-)328 3981 y(quasimo)s(des.)474 4101 y(The)j(existence)i(of)e(an)g Fr(")p Fs(-quasimo)s(de)g Fq(\()1959 4075 y(~)1940 4101 y Fr( )s(;)2073 4076 y Fq(~)2050 4101 y Fr(E)6 b Fq(\))44 b Fs(implies)h(that)g(the)g(op)s (erato)m(r)f Fr(H)52 b Fs(has)328 4222 y(sp)s(ectrum)32 b Fr(\033)t Fq(\()p Fr(H)8 b Fq(\))29 b Fs(in)i Fq([)1138 4197 y(~)1115 4222 y Fr(E)24 b Fp(\000)18 b Fr(";)1418 4197 y Fq(~)1396 4222 y Fr(E)23 b Fq(+)18 b Fr(")p Fq(])p Fs(.)42 b(In)30 b(pa)m(rticula)m(r)h(w)m(e)g(a)m(re)g(sure)h(to)e (\014nd)h(an)f(eigenvalue)328 4342 y Fr(E)37 b Fs(in)31 b(an)g(interval)h Fq([)1049 4317 y(~)1027 4342 y Fr(E)25 b Fp(\000)20 b Fr(\026;)1346 4317 y Fq(~)1324 4342 y Fr(E)25 b Fq(+)19 b Fr(\026)p Fq(])31 b Fs(fo)m(r)h Fr(")27 b Fp(\024)h Fr(\026)p Fs(,)j(if)g(w)m(e)i(kno)m(w)f(that)f(the)h(sp)s (ectrum)g(in)g(that)328 4463 y(interval)g(is)h(purely)g(discrete.)p 328 4550 1296 4 v 439 4611 a Fm(\003)477 4641 y Fl(CPT-CNRS,)21 b(Luminy)g(Case)g(907,)h(F-13288)d(Ma)n(rseille)h(Cedex)h(9,)i(F)n (rance.)35 b(e-mail:)e(asch@cpt.univ-)328 4741 y(mrs.fr)443 4811 y Fm(y)477 4841 y Fl(Max-Planck-Institute)c(fo)n(r)i(Mathematics)e (in)i(the)g(Sciences,)g(Inselstr.)f(22{26,)g(D-04103)e(Leipzig,)328 4941 y(Germany)-7 b(.)38 b(e-mail:)e(knauf@mis.mpg.de)1922 5214 y Fs(1)p eop %%Page: 2 2 2 1 bop 474 631 a Fs(If)24 b(w)m(e)i(kno)m(w)f(in)g(addition)f(that)h Fr(E)30 b Fs(is)25 b(the)g(only)f(such)i(eigenvalue,)g(then,)h(after)e (cho)s(osing)328 751 y(an)32 b(app)m(rop)m(riate)h(phase)g(fo)m(r)g (its)g(no)m(rmalized)g(eigenfunction)h Fr( )t Fs(,)e(w)m(e)i(have)572 994 y Fp(k)642 967 y Fq(~)622 994 y Fr( )26 b Fp(\000)c Fr( )t Fp(k)28 b(\024)1070 926 y Fq(2)p Fr(")p 1070 971 95 4 v 1088 1062 a(\026)1174 994 y(:)2240 b Fs(\(1\))474 1247 y(Ho)m(w)m(ever,)41 b(due)d(to)g(nea)m(r-degeneracies)i(of)d Fr(\033)t Fq(\()p Fr(H)8 b Fq(\))37 b Fs(there)i(ma)m(y)f(b)s(e)f(no)h (eigenfunction)328 1368 y(of)32 b Fr(H)40 b Fs(nea)m(r)783 1341 y Fq(~)763 1368 y Fr( )t Fs(:)328 1535 y(The)d(standa)m(rd)h (example)f(is)g(that)g(of)g(the)h(Schr\177)-49 b(odinger)38 b(op)s(erato)m(r)f(on)g(the)g(line)h(with)f(the)328 1656 y(double)27 b(w)m(ell)g(p)s(otential)f Fr(V)c Fq(\()p Fr(q)t Fq(\))27 b(:=)h(\()p Fr(q)13 b Fp(\000)d Fq(1\))1811 1620 y Fk(2)1851 1656 y Fq(\()p Fr(q)j Fq(+)d(1\))2118 1620 y Fk(2)2158 1656 y Fs(.)41 b(Then)26 b(fo)m(r)h(energies)h Fr(E)34 b(<)27 b Fq(1)f Fs(one)h(ma)m(y)328 1776 y(construct)40 b Fo(~)797 1740 y Fj(1)871 1776 y Fs(-quasimo)s(des)e(lo)s(calized)g (in)g(one)h(o)m(r)f(the)h(other)f(w)m(ell,)j(whereas)e(all)f(eigen-)328 1897 y(functions)i(of)e Fr(H)941 1860 y Fi(~)1022 1897 y Fs(exhibit)h(pa)m(rit)m(y)-8 b(.)63 b(Here)40 b(the)g(nea)m (r-degeneracy)g(of)e(the)i(eigenenergies,)328 2017 y(which)e(is)f(of)g (o)m(rder)h Fp(O)s Fq(\(exp)q(\()p Fp(\000)p Fr(c=)p Fo(~)p Fq(\)\))p Fs(,)g(is)g(connected)h(with)e(tunneling)h(b)s(et)m(w) m(een)h(the)f(t)m(w)m(o)328 2137 y(comp)s(onents)33 b(of)f(the)i (energy)e(shell)i(of)e(the)h(classical)g(system)h(\(see,)f(e.g.)g (Lazutkin,)g([6]\).)474 2258 y(So)41 b(in)g(that)h(case)g(phase)f (space)h(tunneling)g(survives)f(the)h(semiclassical)g(limit,)i(and)328 2378 y(one)38 b(cannot)f(con\014ne)h(a)f(pa)m(rticle)i(fo)m(rever)f(in) f(a)g(w)m(ell.)59 b(As)38 b(a)g(physical)f(consequence)j(one)328 2498 y(ma)m(y)32 b(mention)h(the)g Fr(N)10 b(H)1216 2513 y Fk(3)1289 2498 y Fs(micro)m(w)m(ave)33 b(radiation.)328 2666 y(F)m(o)m(r)45 b(higher)g(degrees)g(of)f(freedom)h Fr(d)f Fs(these)h(energy)g(shell)g(comp)s(onents)g(generalize)g(to)328 2786 y(inva)m(riant)31 b(Lagrangian)g(to)m(ri)h(in)g(phase)g(space.)44 b(If)31 b(such)i(inva)m(riant)e(to)m(ri)i(exist)f(and)g(one)g(has)328 2907 y(some)h(control)g(over)f(the)h(bicha)m(racteristic)i(\015o)m(w)d (in)h(their)g(vicinit)m(y)-8 b(,)32 b(then)h(it)g(is)g(p)s(ossible)g (to)328 3027 y(construct)38 b Fr(")p Fs(-quasimo)s(des)f(of)g(high)g (accuracy)g(\()p Fr(")e Fq(=)g Fo(~)2320 2991 y Fv(N)2387 3027 y Fs(\))i(and)g(thus)g(to)g(extract)h(p)m(recise)328 3148 y(sp)s(ectral)33 b(info)m(rmations)g(in)g(the)g(semiclassical)h (limit)f Fo(~)27 b Fp(&)g Fq(0)p Fs(,)33 b([6].)474 3268 y(Ho)m(w)m(ever,)27 b(b)s(ecause)e(of)e(nea)m(r-degeneracies)i(in)f (the)g(sp)s(ectrum,)j(in)c(general)h(one)g(cannot)328 3388 y(dra)m(w)33 b(any)f(conclusion)h(concerning)g(the)h (semiclassical)g(eigen)p Fn(functions)p Fs(.)328 3556 y(In)26 b(recent)h(y)m(ea)m(rs)g(re\014ned)h(epitactic)f(metho)s(ds)g (allo)m(w)m(ed)g(to)f(p)m(ro)s(duce)h(semiconducto)m(rs)h(with)328 3676 y(p)s(erio)s(dic)41 b(sup)s(erlattices.)69 b(The)41 b(electrons)h(in)e(these)i(p)s(erio)s(dic)f(p)s(otentials)g(have)f(a)h (small)328 3797 y(e\013ective)f(value)e(of)g Fo(~)p Fs(,)i(leading)e (to)h(interesting)g(e\013ects)i(\(see)e([12]\).)61 b(In)38 b(this)i(context)f(it)328 3917 y(is)k(imp)s(o)m(rtant)f(to)h(kno)m(w)g (to)f(which)h(extent)g(one)g(ma)m(y)f(mo)s(del)g(the)h(electronic)h(b)s (ehavio)m(r)328 4038 y(classically)-8 b(.)474 4158 y(This)33 b(motivates)g(our)f(study)h(of)f(Schr\177)-49 b(odinger)33 b(op)s(erato)m(rs)572 4415 y Fr(H)661 4374 y Fi(~)730 4415 y Fq(=)28 b Fp(\000)921 4348 y Fo(~)975 4311 y Fk(2)p 921 4392 94 4 v 944 4483 a Fq(2)1025 4415 y(\001)22 b(+)g Fr(V)119 b Fs(on)98 b Fp(H)28 b Fq(:=)g Fr(L)1908 4374 y Fk(2)1948 4415 y Fq(\()p Fo(R)2052 4374 y Fv(d)2098 4415 y Fq(\))328 4657 y Fs(whose)33 b(p)s(otential)g Fr(V)49 b Fp(2)28 b Fr(C)1274 4621 y Fj(1)1349 4657 y Fq(\()p Fo(R)1453 4621 y Fv(d)1499 4657 y Fr(;)17 b Fo(R)5 b Fq(\))38 b Fs(is)33 b(p)s(erio)s(dic)g(w.r.t.)g(a)f(regula)m(r)h (lattice)h Fp(L)27 b(\032)h Fo(R)3316 4621 y Fv(d)3363 4657 y Fs(.)474 4777 y(W)m(e)33 b(ma)m(y)f(consider)i Fr(V)54 b Fs(as)33 b(a)f(function)g Fr(V)50 b Fq(:)27 b Fo(T)32 b Fp(!)27 b Fo(R)43 b Fs(on)33 b(the)g Fr(d)p Fs(-to)m(rus)f Fo(T)f Fq(:=)d Fo(R)3256 4741 y Fv(d)3303 4777 y Fr(=)p Fp(L)p Fs(.)1922 5214 y(2)p eop %%Page: 3 3 3 2 bop 474 631 a Fs(The)32 b Fp(L)p Fs(-inva)m(riance)g(and)g(the)h (Blo)s(ch)e(theo)m(rem)j(imply)e(that)g Fr(H)2704 595 y Fi(~)2778 631 y Fs(conjugates)h(unita)m(rily)328 751 y(to)f(the)i(direct)f(integral)g(of)f(the)h(op)s(erato)m(rs)572 923 y Fr(H)661 882 y Fi(~)703 923 y Fq(\()p Fr(k)s Fq(\))27 b(:=)1001 884 y Fk(1)p 1001 900 36 4 v 1001 958 a(2)1046 923 y Fq(\()p Fr(D)e Fq(+)d Fo(~)p Fr(k)s Fq(\))1434 887 y Fk(2)1495 923 y Fq(+)g Fr(V)120 b Fs(on)97 b Fr(L)2032 887 y Fk(2)2072 923 y Fq(\()p Fo(T)p Fq(\))198 b(\()p Fr(k)31 b Fp(2)d Fo(T)2686 887 y Fj(\003)2729 923 y Fq(\))p Fr(;)647 b Fs(\(2\))328 1095 y(acting)32 b(on)1526 1137 y Fh(Z)1625 1163 y Fj(\010)1581 1362 y Fi(T)1631 1343 y Fg(\003)1701 1272 y Fr(L)1767 1231 y Fk(2)1807 1272 y Fq(\()p Fo(T)p Fr(;)17 b(dq)6 b Fq(\))2163 1205 y Fr(dk)p 2100 1249 231 4 v 2100 1341 a Fq(v)m(ol)p Fo(T)2288 1312 y Fj(\003)2341 1272 y Fr(;)328 1491 y Fs(where)31 b Fp(L)663 1455 y Fj(\003)732 1491 y Fs(is)f(the)g(dual)g(lattice)g(with)h Fn(Brillouin)e(zone)h Fo(T)2309 1455 y Fj(\003)2380 1491 y Fq(:=)d Fo(R)2576 1455 y Fv(d)2623 1491 y Fr(=)p Fp(L)2741 1455 y Fj(\003)2809 1491 y Fs(and)j Fr(D)g Fq(:=)e Fp(\000)p Fr(i)p Fo(~)p Fp(r)i Fs(is)328 1611 y(the)k(momentum)g(op)s(erato)m(r.) 47 b(It)33 b(follo)m(ws)h(that)f(the)h(sp)s(ectrum)h(consists)g(of)e (bands.)47 b(Up)34 b(to)328 1732 y(measure)h(zero)f(sets)h(due)g(to)e (degeneracies,)j(the)f(eigenvalues)f Fr(E)2671 1696 y Fi(~)2665 1756 y Fv(n)2714 1732 y Fq(\()p Fr(k)s Fq(\))f Fs(a)m(re)i(analytic)e(in)h Fr(k)s Fs(,)328 1852 y(and)d(a)m(re)h (non-constant,)g(\(see,)h(e.g.,Thomas)e([10],)h(Wilco)m(x)f([13)o(],)h (and)g(Reed)f(and)h(Simon)328 1973 y([9]\).)42 b(Thus)29 b(the)h Fn(group)f(velo)s(cit)m(y)g Fo(~)1580 1936 y Fj(\000)p Fk(1)1674 1973 y Fp(r)1757 1988 y Fv(k)1799 1973 y Fr(E)1877 1936 y Fi(~)1871 1997 y Fv(n)1920 1973 y Fq(\()p Fr(k)s Fq(\))g Fs(vanishes)g(at)g(most)h(on)e(a)h(set)h(of)f (measure)328 2093 y(zero.)474 2213 y(On)41 b(the)g(other)g(hand)f(the)h (symmetry)g Fr(E)1974 2177 y Fi(~)1968 2238 y Fv(n)2017 2213 y Fq(\()p Fp(\000)p Fr(k)s Fq(\))g(=)g Fr(E)2460 2177 y Fi(~)2454 2238 y Fv(n)2503 2213 y Fq(\()p Fr(k)s Fq(\))f Fs(of)g(the)h(band)f(functions)328 2334 y(implies)g(in)g(the)h (non-degenerate)f(case)h(that)f(the)g(group)f(velo)s(cit)m(y)h (vanishes)g(fo)m(r)g Fr(k)j Fq(=)d(0)328 2454 y Fs(and)32 b(the)h(other)g Fq(2)960 2418 y Fv(d)1023 2454 y Fp(\000)22 b Fq(1)33 b Fs(\014xed)f(p)s(oints)h(of)f Fr(k)f Fp(7!)c(\000)p Fr(k)36 b Fs(on)d Fo(T)2385 2418 y Fj(\003)2427 2454 y Fs(.)474 2574 y(T)-8 b(o)26 b(see)h(ho)m(w)g(this)g(vanishing)f(of)g (the)i(group)e(velo)s(cit)m(y)g(is)h(connected)h(with)f(phase)g(space) 328 2695 y(tunneling,)34 b(w)m(e)g(consider)h(the)f(simplest)g(case)h (of)e Fr(d)28 b Fq(=)h(1)k Fs(dimension)h(\(fo)m(r)g Fr(d)28 b Fq(=)h(2)34 b Fs(see)g(also)328 2815 y([4]\).)474 2936 y(In)24 b(that)h(case)g(the)g(energy)g(shell)g Fq(\006)1674 2951 y Fv(E)1761 2936 y Fq(:=)j Fr(H)1981 2899 y Fj(\000)p Fk(1)2075 2936 y Fq(\()p Fr(E)6 b Fq(\))27 b Fp(\032)i(P)j Fs(of)24 b(the)h(Hamiltonian)g(function)572 3107 y Fr(H)8 b Fq(\()p Fr(p;)17 b(q)t Fq(\))26 b(:=)1044 3068 y Fk(1)p 1044 3085 36 4 v 1044 3142 a(2)1089 3107 y Fr(p)1138 3071 y Fk(2)1200 3107 y Fq(+)c Fr(V)f Fq(\()p Fr(q)t Fq(\))98 b Fs(on)32 b(the)h(phase)g(space)98 b Fp(P)37 b Fq(:=)27 b Fr(T)2774 3071 y Fj(\003)2813 3107 y Fo(T)p Fr(;)328 3279 y Fs(consists)j(fo)m(r)e(energies)i Fr(E)k(>)27 b(V)1423 3294 y Fk(max)1586 3279 y Fq(:=)h(max)1898 3294 y Fv(q)1953 3279 y Fr(V)22 b Fq(\()p Fr(q)t Fq(\))28 b Fs(of)g(t)m(w)m(o)h(comp)s(onents,)h(co)m(rresp)s(onding)328 3400 y(to)d(ballistic)h(motion)f(to)g(the)h(right)g(resp.)g(to)f(the)h (left.)42 b(These)28 b(comp)s(onents)g(a)m(re)g(p)s(ermuted)328 3520 y(b)m(y)k(the)i(time)f(reversal)g(transfo)m(rmation)g Fq(\()p Fr(p;)17 b(q)t Fq(\))27 b Fp(7!)g Fq(\()p Fp(\000)p Fr(p;)17 b(q)t Fq(\))33 b Fs(on)f Fp(P)8 b Fs(.)474 3641 y(As)36 b(the)g(eigenfunction)g Fr( )1416 3604 y Fi(~)1412 3665 y Fv(n)1459 3641 y Fq(\()p Fr(k)s Fq(\))f Fs(can)g(b)s(e)h(chosen) g(to)f(b)s(e)g(real)g(fo)m(r)h Fr(k)f Fq(=)d(0)p Fs(,)j(it)h(is)f (semic-)328 3761 y(lassically)i(equally)g(concentrated)i(on)d(b)s(oth)h (\(one-dimensional\))g(to)m(ri)h(co)m(rresp)s(onding)g(to)328 3881 y(the)33 b(energy)g Fr(E)h Fq(=)27 b Fr(E)1072 3845 y Fi(~)1066 3906 y Fv(n)1115 3881 y Fq(\(0\))p Fs(.)474 4002 y(The)45 b(vanishing)g(group)g(velo)s(cit)m(y)g(is)g(one)g (manifestation)h(of)e(that)i(fact.)81 b(Thus)45 b(fo)m(r)328 4122 y Fr(k)k Fq(=)d(0)p Fs(,)g(a)m(rbitra)m(rily)e(small)g(values)f (of)g Fo(~)g Fs(and)h(la)m(rge)f(times)i Fr(t)e Fs(the)h(quantum)g (evolution)328 4242 y Fq(exp)q(\()p Fp(\000)p Fr(iH)714 4206 y Fi(~)757 4242 y Fq(\()p Fr(k)s Fq(\))p Fr(t=)p Fo(~)p Fq(\))38 b Fs(and)h(the)g(classical)h(\015o)m(w)f Fq(\010)2090 4206 y Fv(t)2158 4242 y Fq(:)g Fp(P)47 b(!)38 b(P)47 b Fs(generated)40 b(b)m(y)f Fr(H)46 b Fs(b)s(ehave)328 4363 y(very)32 b(di\013erently)-8 b(.)474 4483 y(Ho)m(w)m(ever)37 b(w)m(e)f(a)m(rgue)g(that)f(fo)m(r)h(general)f(quasimomenta)h Fr(k)j Fs(in)c Fo(T)2801 4447 y Fj(\003)2879 4483 y Fn(phase)h(space)g (tun-)328 4604 y(neling)d(is)f(exceptional)h(in)g(the)g(limit)g Fo(~)27 b Fp(&)g Fq(0)p Fs(.)474 4724 y(Mo)m(re)d(sp)s(eci\014cally)-8 b(,)25 b(w)m(e)g(conjectured)f(in)g([2])f(that)h(the)f(quantum)h (distribution)g(of)f(group)328 4844 y(velo)s(cities)30 b(converges)g(in)g(the)g(semiclassical)h(limit)f(to)f(the)h(classical)h (one,)f(see)h(Conjecture)328 4965 y(5.1)h(b)s(elo)m(w.)1922 5214 y(3)p eop %%Page: 4 4 4 3 bop 474 631 a Fs(W)m(e)36 b(p)m(roved)g(this)h(in)f([2])g(fo)m(r)g (the)g(extreme)h(cases)g(of)f(p)s(otentials)g(leading)g(to)g(ergo)s (dic)328 751 y(motion,)50 b(and)c(fo)m(r)h(sepa)m(rable)g(p)s (otentials)g(\(which)h(a)m(re)f(the)g(only)f(kno)m(wn)h(examples)g(of) 328 872 y(p)s(erio)s(dic)32 b(p)s(otentials)h(leading)g(to)f (integrable)h(motion\).)474 992 y(Here)38 b(w)m(e)g(sho)m(w)f(a)f (simila)m(r)h(statement)h(fo)m(r)f(a)m(rbitra)m(ry)g(smo)s(oth)f(p)s (otentials)h(and)g(la)m(rge)328 1112 y(energies,)d(where)g(KAM)e(to)m (ri)h(a)m(re)g(kno)m(wn)h(to)e(dominate)h(the)g(phase)g(space)g (volume.)328 1280 y(After)k(p)m(resenting)g(the)g(strategy)f(in)g Fn(Sect.)h(2)p Fs(,)f(w)m(e)h(adapt)f(in)g Fn(Sect.)h(3)52 b Fs(Lazutkin's)36 b(results)328 1401 y(on)j(KAM-quasimo)s(des)h(to)g (the)g(p)m(resent)h(situation)f(of)g(a)f(family)g Fr(H)2809 1364 y Fi(~)2850 1401 y Fq(\()p Fr(k)s Fq(\))h Fs(of)f(di\013erential) 328 1521 y(op)s(erato)m(rs.)k Fn(Thm.)28 b(4.1)44 b Fs(contains)28 b(our)h(main)f(result.)43 b(It)28 b(states)h(that)g(fo)m(r)f(la)m(rge)g (energies)i Fr(E)328 1641 y Fs(a)h(p)m(rop)s(o)m(rtion)h Fq(1)20 b Fp(\000)g(O)s Fq(\(1)p Fr(=)1240 1557 y Fp(p)p 1323 1557 79 4 v 84 x Fr(E)6 b Fq(\))31 b Fs(of)g(the)h(eigenfunctions) h(is)f(semiclassically)g(concentrated)328 1762 y(nea)m(r)h(a)f(KAM)h (to)m(rus.)474 1882 y(This)24 b(then)g(leads)h(to)e(a)h(co)m(rresp)s (onding)g(statement)i(\()p Fn(Thm.)d(5.3)16 b Fs(\))24 b(fo)m(r)g(the)g(semiclassical)328 2002 y(distribution)33 b(of)f(group)h(velo)s(cities,)g(in)f(acco)m(rdance)i(with)f(the)g(ab)s (ove)f(conjecture.)474 2123 y(In)f(a)f(\014nal)h(section,)h(w)m(e)g (try)f(to)g(abstract)h(our)e(strategy)-8 b(.)44 b(W)m(e)31 b(a)m(rgue)g(that)g(a)f(mere)i Fn(exi-)328 2243 y(stence)g(p)m(ro)s(of) e Fs(fo)m(r)g(a)g(full)g(set)h(of)f Fo(~)1533 2207 y Fv(N)1600 2243 y Fs({quasimo)s(des)g(with)h(lo)s(calized)g(asymptotic)f (velo)s(cities)328 2364 y(could)h(imply)f(the)h(conjectured)h (classical)f(limit)f(of)g(the)i(distribution)f(of)f(group)g(velo)s (cities.)328 2602 y Ff(Ackno)m(wledgments.)52 b Fs(W)m(e)35 b(thank)h(Ruedi)g(Seiler)g(and)f(SFB)h(288,)g(TU)f(Berlin,)i(fo)m(r)e (hos-)328 2722 y(pitalit)m(y)-8 b(,)32 b(fo)m(r)h(which)g(J.A.)g(also)f (thanks)i(MPI)d(in)i(Leipzig.)328 3057 y Ft(2)158 b(Heuristics)328 3279 y Fs(Befo)m(re)40 b(w)m(e)g(turn)f(to)g(fo)m(rmal)g(statements)i (and)e(p)m(ro)s(ofs,)i(w)m(e)f(sho)m(rtly)g(describ)s(e)g(the)g(main) 328 3400 y(ideas,)33 b(sta)m(rting)g(with)g(the)g(follo)m(wing)g (observation.)474 3520 y(Tw)m(o)d(given)h(quasimo)s(des)f(asso)s (ciated)h(to)g(di\013erent)g(KAM)g(to)m(ri)f(give)g(rise)h(to)g (di\013erent)328 3641 y(exp)s(ectations)38 b(of)e(the)h(sub-p)m (rincipal)h(symb)s(ol)e Fo(~)p Fr(k)28 b Fp(\001)d Fr(D)39 b Fs(of)e(the)g(op)s(erato)m(r)g Fr(H)3068 3604 y Fi(~)3110 3641 y Fq(\()p Fr(k)s Fq(\))g Fs(de\014ned)328 3761 y(in)30 b(\(2\).)42 b(Thus)30 b(they)g(can)g(b)s(e)g(sepa)m(rated)h (energetically)g(b)m(y)e(va)m(rying)g(the)i(quasimomentum)328 3881 y Fr(k)s Fs(,)48 b(and)d(fo)m(r)g(t)m(ypical)g Fr(k)k Fs(in)c(the)g(Brillouin)g(zone)g Fo(T)2178 3845 y Fj(\003)2266 3881 y Fs(one)h(should)f(not)g(have)g(to)s(o)f(many)328 4002 y(nea)m(r-degeneracies)34 b(of)e(energies.)474 4122 y(Of)j(course)h(w)m(e)g(must)g(consider)g(scales)g(in)f(o)m(rder)h(to)f (mak)m(e)h(this)f(a)m(rgument)h(w)m(o)m(rk.)52 b(In)328 4242 y Fr(d)43 b Fs(dimensions)h(the)f(mean)g(spacing)h Fr(E)1757 4206 y Fi(~)1751 4267 y Fv(n)p Fk(+1)1888 4242 y Fq(\()p Fr(k)s Fq(\))29 b Fp(\000)h Fr(E)2232 4206 y Fi(~)2226 4267 y Fv(n)2274 4242 y Fq(\()p Fr(k)s Fq(\))43 b Fs(b)s(et)m(w)m(een)i(the)f(eigenvalues)f(of)328 4363 y Fr(H)417 4327 y Fi(~)459 4363 y Fq(\()p Fr(k)s Fq(\))36 b Fs(nea)m(r)h Fr(E)j(>)34 b(V)1111 4378 y Fk(min)1269 4363 y Fs(is)j(of)f(the)h(o)m(rder)g Fo(~)1942 4327 y Fv(d)1982 4363 y Fs(.)55 b(Thus)37 b(a)f(p)m(rio)m(ri)h(one)f(must)h (consider)h(in)e(a)328 4483 y(\014xed)27 b(energy)g(interval)g(ab)s (out)f Fo(~)1475 4447 y Fj(\000)p Fv(d)1597 4483 y Fs(quasimo)s(des)h (which)g(ma)m(y)g(lead)g(to)g(a)f(nea)m(r-degeneracy)328 4604 y(with)k(a)g(given)g(quasimo)s(de.)43 b(F)m(o)m(r)31 b Fo(~)1564 4567 y Fv(N)1631 4604 y Fs(-quasimo)s(des)f(w)m(e)h(need)g (an)e(energy)i(sepa)m(ration)f(of)g(at)328 4724 y(least)j Fo(~)600 4688 y Fv(N)667 4724 y Fs(.)44 b(So)32 b Fr(N)43 b Fs(should)33 b(b)s(e)g(la)m(rger)f(than)h Fr(d)p Fs(.)474 4844 y(Such)46 b(high)g(p)m(recision)g(KAM)g(quasimo)s(des)g(a)m(re)g (constructed)i(in)d(the)h(b)s(o)s(ok)f([6])h(b)m(y)328 4965 y(Lazutkin)c(\(see)g(also)g(the)g(a)m(rticle)h([11)o(])f(b)m(y)g (Thomas)f(and)g(W)m(assell)h(fo)m(r)g(related)g(results\))1922 5214 y(4)p eop %%Page: 5 5 5 4 bop 328 631 a Fs(W)m(e)38 b(apply)f(this)i(metho)s(d)f(after)g (some)h(straightfo)m(rw)m(a)m(rd)g(adaptation)e(to)h(our)g(family)f (\(2\))328 751 y(of)32 b(di\013erential)h(op)s(erato)m(rs.)474 872 y(An)g(imp)s(o)m(rtant)g(input)g(fo)m(r)g(that)f(construction)i (consists)g(in)f(the)g(re\014nement)h(of)f(KAM)328 992 y(theo)m(ry)27 b(p)m(resented)i(in)d(the)h(pap)s(er)g([7])f(b)m(y)h (P\177)-49 b(oschel.)42 b(Roughly)25 b(sp)s(eaking)i(one)g(uses)h(that) e(the)328 1112 y(deviation)33 b(of)h(the)h(Hamiltonian)e(function)h Fr(H)42 b Fs(from)34 b(an)f(integrable)i(one)f(vanishes)g(faster)328 1233 y(than)k(any)g(p)s(o)m(w)m(er)h(of)f(the)g(phase)h(space)g (distance)g(to)f(the)h(KAM)g(set.)61 b(In)38 b(pa)m(rticula)m(r)h(w)m (e)328 1353 y(ma)m(y)29 b(apply)g(p)s(erturbative)h(semiclassical)g (techniques)i(in)d(some)h Fo(~)2668 1317 y Fv(\013)2717 1353 y Fs(-neighb)s(o)m(rho)s(o)s(d)e(of)h(the)328 1474 y(set)k(of)g(KAM)f(to)m(ri.)474 1594 y(A)41 b(\014nal)g(rema)m(rk)h (concerns)g(the)f(phase)h(space)f(complement)h(of)f(the)g(KAM)g(set.)69 b(In)328 1714 y(general)33 b(w)m(e)g(do)g(not)f(have)g(any)g(info)m (rmation)h(over)f(individual)g(eigenfunctions)474 1835 y(and)h(eigenvalues)g(concentrating)g(semiclassically)h(in)e(that)h (region.)474 1955 y(In)e(pa)m(rticula)m(r)i(w)m(e)f(cannot)f(hop)s(e)h (to)f(lift)h(nea)m(r-degeneracies)h(b)s(et)m(w)m(een)g(such)f(eigenva-) 328 2076 y(lues)26 b(and)g(the)g(energies)h(of)f(the)g(KAM-quasimo)s (des)g(b)m(y)f(changing)h(the)g(quasi-momentum.)328 2196 y(Mo)m(reover,)36 b(if)f(a)f(quasimo)s(de)i(is)f(involved)f(in)i(such)f (a)g(nea)m(r-degeneracy)-8 b(,)37 b(there)f(need)g(not)328 2316 y(b)s(e)d(any)e(eigenfunction)j Fr( )1269 2280 y Fi(~)1265 2341 y Fv(n)1312 2316 y Fq(\()p Fr(k)s Fq(\))f Fs(nea)m(r)g(to)f(that)h(quasimo)s(de.)474 2437 y(Ho)m(w)m(ever,)e(w)m (e)f(can)f(apply)f(a)g(b)s(o)m(x)g(counting)h(p)m(rinciple.)44 b(W)m(e)28 b(kno)m(w)i(from)e(KAM)h(theo)m(ry)328 2557 y(that)j(fo)m(r)g(la)m(rge)g(energies)i Fr(E)k Fs(the)32 b(complement)h(of)f(the)g(KAM)h(set)g(is)f(of)f(relative)i(measure)328 2677 y Fp(O)s Fq(\(1)p Fr(=)546 2593 y Fp(p)p 629 2593 79 4 v 84 x Fr(E)6 b Fq(\))p Fs(.)474 2798 y(Then)34 b(a)g(W)m(eyl)g(a)m(rgument)h(implies)g(that)f(up)h(to)f(an)g (exceptional)g(set)i(of)d(relative)i(size)328 2918 y Fp(O)s Fq(\(1)p Fr(=)546 2834 y Fp(p)p 629 2834 V 84 x Fr(E)6 b Fq(\))34 b Fs(the)h(eigenvalues)g Fr(E)1508 2882 y Fi(~)1502 2943 y Fv(n)1550 2918 y Fq(\()p Fr(k)s Fq(\))f Fs(nea)m(r)h Fr(E)41 b Fs(a)m(re)34 b(w)m(ell-app)m(ro)m (ximated)h(b)m(y)g(KAM)f(quasi-)328 3039 y(mo)s(des.)474 3159 y(In)f(the)i(semiclassical)g(limit)f(these)h Fo(~)1797 3123 y Fv(N)1864 3159 y Fs(-quasimo)s(des)e Fq(\()2454 3133 y(~)2434 3159 y Fr( )t(;)2568 3134 y Fq(~)2545 3159 y Fr(E)6 b Fq(\))34 b Fs(a)m(re)g(t)m(ypically)f(energeti-)328 3279 y(cally)h(sepa)m(rated)j(in)d(the)i(sense)g(that)g(the)f(asso)s (ciated)h(intervals)f Fq([)2716 3254 y(~)2693 3279 y Fr(E)30 b Fp(\000)24 b Fo(~)2950 3243 y Fv(N)3017 3279 y Fr(;)3084 3254 y Fq(~)3061 3279 y Fr(E)30 b Fq(+)23 b Fo(~)3316 3243 y Fv(N)3383 3279 y Fq(])35 b Fs(a)m(re)328 3400 y(disjoint.)71 b(W)m(e)41 b(have)h(at)f(least)i(one)e(eigenvalue)h Fr(E)2189 3364 y Fi(~)2183 3424 y Fv(n)2231 3400 y Fq(\()p Fr(k)s Fq(\))g Fs(in)g(each)g(such)g(interval.)71 b(Thus)328 3520 y(only)37 b(an)g(exceptional)h(set)g(of)f(relative)h(p)m(rop)s(o)m (rtion)f Fp(O)s Fq(\(1)p Fr(=)2427 3436 y Fp(p)p 2510 3436 V 84 x Fr(E)6 b Fq(\))37 b Fs(of)g(these)i(intervals)f(ma)m(y)328 3641 y(contain)33 b(mo)m(re)g(than)f(one)h(eigenvalue.)474 3761 y(So)44 b(fo)m(r)f(t)m(ypical)g Fr(k)49 b Fp(2)e Fo(T)1355 3725 y Fj(\003)1441 3761 y Fs(most)d Fr(E)1761 3725 y Fi(~)1755 3786 y Fv(n)1803 3761 y Fq(\()p Fr(k)s Fq(\))f Fs(a)m(re)h(not)g(nea)m(r-degenerate,)j(and)c(thus)h(the)328 3881 y(co)m(rresp)s(onding)33 b(eigenfunctions)h Fr( )1596 3845 y Fi(~)1592 3906 y Fv(n)1639 3881 y Fq(\()p Fr(k)s Fq(\))f Fs(a)m(re)g(w)m(ell)g(app)m(ro)m(ximated)g(b)m(y)f(quasimo)s (des)3359 3855 y Fq(~)3340 3881 y Fr( )t Fs(.)328 4219 y Ft(3)158 b(KAM)52 b(Estimates)i(and)f(Quasimo)t(des)328 4441 y Fs(In)32 b(o)m(rder)h(to)g(apply)f(KAM)g(theo)m(ry)h(to)g Fr(H)40 b Fs(with)33 b(energies)h(in)572 4643 y Fr(I)h Fq(:=)28 b([\(1)22 b Fp(\000)h Fr(\016)t Fq(\))p Fr(E)6 b(;)17 b Fq(\(1)k(+)h Fr(\016)t Fq(\))p Fr(E)6 b Fq(])1821 b Fs(\(3\))328 4844 y(nea)m(r)51 b Fr(E)63 b(>)58 b Fq(0)p Fs(,)c(w)m(e)d(change)g(co)s(o)m(rdinates.)97 b(So)50 b(consider)h(the)g Fr(d)33 b Fp(\002)i Fr(d)50 b Fs(matrix)g Fr(L)58 b Fq(:=)328 4965 y(\()p Fr(`)407 4980 y Fk(1)446 4965 y Fr(;)17 b(:)g(:)g(:)f(;)h(`)706 4980 y Fv(d)746 4965 y Fq(\))p Fr(=)p Fq(\(2)p Fr(\031)t Fq(\))43 b Fs(of)h(a)g(basis)g Fq(\()p Fr(`)1591 4980 y Fk(1)1630 4965 y Fr(;)17 b(:)g(:)g(:)f(;)h(`) 1890 4980 y Fv(d)1930 4965 y Fq(\))44 b Fs(fo)m(r)g(the)h (con\014guration)e(space)i(lattice)g Fp(L)p Fs(,)1922 5214 y(5)p eop %%Page: 6 6 6 5 bop 328 631 a Fs(set)498 606 y Fq(^)483 631 y Fr(V)21 b Fq(\()p Fr(')p Fq(\))39 b(:=)g Fr(V)21 b Fq(\()p Fr(L')p Fq(\))p Fs(,)41 b(denote)f(b)m(y)1698 606 y Fq(^)1675 631 y Fp(P)47 b Fq(:=)39 b Fr(T)2004 595 y Fj(\003)2052 605 y Fq(^)2043 631 y Fo(T)k Fs(the)c(phase)h(space)g(over)f(the)g (standa)m(rd)328 751 y(to)m(rus)581 929 y Fq(^)572 955 y Fo(T)31 b Fq(:=)d Fo(R)862 914 y Fv(d)909 955 y Fr(=)p Fq(\(2)p Fr(\031)t Fo(Z)p Fq(\))1211 914 y Fv(d)328 1158 y Fs(and)k(de\014ne,)i(using)e(the)i(matrix)e Fr(M)38 b Fq(:=)28 b(\()p Fr(L)1868 1122 y Fv(t)1898 1158 y Fr(L)p Fq(\))2002 1122 y Fj(\000)p Fk(1)2097 1158 y Fs(,)k(the)h(Hamiltonian) 597 1336 y Fq(^)572 1362 y Fr(H)653 1377 y Fv(")717 1362 y Fq(:)794 1336 y(^)772 1362 y Fp(P)j(!)27 b Fo(R)109 b Fs(,)1324 1336 y Fq(^)1298 1362 y Fr(H)1379 1377 y Fv(")1416 1362 y Fq(\()p Fr(J)n(;)17 b(')p Fq(\))27 b(:=)1820 1322 y Fk(1)p 1820 1339 36 4 v 1820 1396 a(2)1865 1362 y Fq(\()p Fr(J)n(;)17 b(M)10 b(J)f Fq(\))23 b(+)f Fr(")2386 1336 y Fq(^)2371 1362 y Fr(V)f Fq(\()p Fr(')p Fq(\))p Fr(:)328 1565 y Fs(Then)32 b(fo)m(r)h(the)g(di\013eomo)m(rphism)572 1790 y Fp(M)692 1805 y Fv(E)779 1790 y Fq(:)27 b Fp(P)37 b(!)1089 1765 y Fq(^)1066 1790 y Fp(P)106 b Fs(,)98 b Fq(\()p Fr(p;)17 b(q)t Fq(\))27 b Fp(7!)g Fq(\()p Fr(J)n(;)17 b(')p Fq(\))27 b(:=)2130 1680 y Fh(\020)2190 1790 y Fr(L)2256 1749 y Fv(t)2286 1790 y Fr(p=)2384 1701 y Fp(p)p 2466 1701 79 4 v 2466 1790 a Fr(E)6 b(;)17 b(L)2654 1749 y Fj(\000)p Fk(1)2749 1790 y Fr(q)2796 1680 y Fh(\021)328 2015 y Fs(w)m(e)33 b(have)572 2219 y Fr(E)28 b Fp(\001)747 2194 y Fq(^)722 2219 y Fr(H)803 2234 y Fk(1)p Fv(=E)955 2219 y Fp(\016)22 b(M)1147 2234 y Fv(E)1234 2219 y Fq(=)27 b Fr(H)r(;)328 2437 y Fs(and)h(the)g(\015o)m(w)862 2412 y Fq(^)851 2437 y(\010)921 2401 y Fv(t)921 2461 y(")987 2437 y Fs(generated)g(b)m(y)1554 2412 y Fq(^)1528 2437 y Fr(H)1609 2452 y Fv(")1674 2437 y Fs(\(w.r.t.)h(the)f(standa)m(rd)h (symplectic)g(structure)h(on)3473 2412 y Fq(^)3450 2437 y Fp(P)9 b Fs(\))328 2557 y(is)33 b(conjugate)f(to)h(the)g(o)m(riginal) f(\015o)m(w,)h(up)g(to)f(a)g(change)h(of)f(time)h(scale:)583 2746 y Fq(^)572 2771 y(\010)642 2674 y Fj(p)p 701 2674 56 3 v 56 x Fv(E)t(t)642 2803 y Fk(1)p Fv(=E)808 2771 y Fp(\016)22 b(M)1000 2786 y Fv(E)1087 2771 y Fq(=)28 b Fp(M)1311 2786 y Fv(E)1392 2771 y Fp(\016)22 b Fq(\010)1534 2730 y Fv(t)1759 2771 y Fq(\()p Fr(t)28 b Fp(2)g Fo(R)5 b Fq(\))p Fr(:)339 2980 y Fq(^)328 3005 y(\010)398 2969 y Fv(t)398 3030 y(")468 3005 y Fs(b)s(ecomes)33 b(fully)f(integrable)h (fo)m(r)g(p)s(erturbation)g(pa)m(rameter)g Fr(")28 b Fq(=)f(0)p Fs(.)43 b(Namely)583 3183 y Fq(^)572 3209 y(\010)642 3167 y Fv(t)642 3233 y Fk(0)682 3209 y 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Fs(and)328 4521 y Fr(\034)39 b(>)28 b(d)21 b Fp(\000)i Fq(1)32 b Fs(w)m(e)i(consider)f(the)h(Diophantine)e(sets) 572 4724 y Fq(\012)642 4739 y Fv(\015)714 4724 y Fq(:=)845 4644 y Fh(\010)903 4724 y Fr(!)f Fp(2)d Fo(R)1155 4683 y Fv(d)1229 4724 y Fp(j)g(8)p Fr(k)j Fp(2)d Fo(Z)1585 4683 y Fv(d)1645 4724 y Fp(n)22 b(f)p Fq(0)p Fp(g)27 b Fq(:)h Fp(j)p Fr(!)d Fp(\001)d Fr(k)s Fp(j)27 b(\025)h Fr(\015)5 b Fp(k)p Fr(k)s Fp(k)2536 4683 y Fj(\000)p Fv(\034)2634 4644 y Fh(\011)2709 4724 y Fr(:)705 b Fs(\(5\))328 4928 y(These)33 b(a)m(re)g(asymptotically)f(of)g(full)h(measure)g(as)g Fr(\015)g Fp(&)27 b Fq(0)p Fs(.)1922 5214 y(6)p eop %%Page: 7 7 7 6 bop 474 631 a Fs(Denote)40 b(the)f(interval)g(of)g(new)g(energies)h (b)m(y)2129 606 y Fq(^)2117 631 y Fr(I)47 b Fq(:=)38 b([1)26 b Fp(\000)h Fr(\016)n(;)17 b Fq(1)27 b(+)f Fr(\016)t Fq(])p Fs(.)62 b(F)m(o)m(r)40 b Fr(")e Fq(=)h(0)f Fs(the)328 751 y(phase)33 b(space)g(region)1144 726 y Fq(^)1121 751 y Fp(P)1190 766 y Fv(")1255 751 y Fq(:=)1411 726 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974 y(denote)33 b(b)m(y)572 1144 y Fr(P)649 1103 y Fi(~)635 1169 y Fv(`)664 1150 y Fg(\003)704 1144 y Fq(\()p Fr(k)s Fq(\))195 b(\()p Fr(`)1108 1103 y Fj(\003)1175 1144 y Fp(2)28 b Fq(\003)1337 1103 y Fi(~)1337 1169 y Fv(I)1379 1144 y Fq(\()p Fr(k)s Fq(\)\))328 1315 y Fs(the)41 b(sp)s(ectral)g(p)m(rojecto)m(r)g(fo)m(r)f Fr(H)1486 1279 y Fi(~)1528 1315 y Fq(\()p Fr(k)s Fq(\))h Fs(and)f(the)g(interval)h Fq([)2443 1290 y(~)2421 1315 y Fr(E)2499 1279 y Fi(~)2493 1341 y Fv(`)2522 1322 y Fg(\003)2562 1315 y Fq(\()p Fr(k)s Fq(\))27 b Fp(\000)h Fo(~)2878 1279 y Fv(N)2945 1315 y Fr(;)3012 1290 y Fq(~)2989 1315 y Fr(E)3067 1279 y Fi(~)3061 1341 y Fv(`)3090 1322 y Fg(\003)3130 1315 y Fq(\()p Fr(k)s Fq(\))f(+)h Fo(~)3445 1279 y Fv(N)3512 1315 y Fq(])p Fs(.)328 1436 y(F)m(o)m(r)33 b(each)g Fr(`)749 1400 y Fj(\003)816 1436 y Fp(2)28 b Fq(\003)978 1400 y Fi(~)978 1461 y Fv(I)1021 1436 y Fq(\()p Fr(k)s Fq(\))k Fs(there)i(is)f(a)f(nea)m(rb)m(y)h(eigenvalue)572 1607 y Fr(E)650 1565 y Fi(~)644 1631 y Fv(n)692 1607 y Fq(\()p Fr(k)s Fq(\))130 b Fs(with)h Fp(j)p Fr(E)1363 1565 y Fi(~)1357 1631 y Fv(n)1405 1607 y Fq(\()p Fr(k)s Fq(\))22 b Fp(\000)1680 1581 y Fq(~)1657 1607 y Fr(E)1735 1565 y Fi(~)1729 1631 y Fv(`)1758 1612 y Fg(\003)1798 1607 y Fq(\()p Fr(k)s Fq(\))p Fp(j)27 b(\024)h Fo(~)2142 1565 y Fk(2)p Fv(N)2245 1607 y Fr(:)1120 b Fs(\(17\))328 1777 y(So)32 b(fo)m(r)h Fo(~)27 b Fr(<)h Fo(~)844 1792 y Fk(0)916 1777 y Fs(w)m(e)33 b(kno)m(w)h(in)e(pa)m(rticula)m(r)h(that) g Fq(dim)o(\()p Fr(P)2292 1741 y Fi(~)2278 1803 y Fv(`)2307 1784 y Fg(\003)2346 1777 y Fq(\()p Fr(k)s Fq(\)\))28 b Fp(\025)g Fq(1)p Fs(.)474 1898 y(But)46 b(since)h(the)f(quasimo)s(de) g(construction)h(is)g(only)e(based)h(on)g(the)g(KAM)g(pa)m(rt)g(of)328 2018 y(phase)31 b(space,)i(it)e(do)s(es)g(not)g(su\016ce)h(to)f(kno)m (w)g(that)g(the)h(quasimo)s(de)f(energies)3166 1993 y Fq(~)3143 2018 y Fr(E)3221 1982 y Fi(~)3215 2044 y Fv(`)3244 2025 y Fg(\003)3284 2018 y Fq(\()p Fr(k)s Fq(\))g Fs(a)m(re)328 2139 y(sepa)m(rated)40 b(from)e(each)h(other)g(to)f(ensure)i(that)e (the)h(eigenenergies)h(a)m(re)f(isolated.)62 b(Thus)328 2259 y(w)m(e)33 b(consider)h(the)f(subset)572 2430 y Fp(F)-7 b Fq(\003)705 2389 y Fi(~)705 2455 y Fv(I)747 2430 y Fq(\()p Fr(k)s Fq(\))27 b(:=)h Fp(f)p Fr(`)1126 2389 y Fj(\003)1193 2430 y Fp(2)g(G)-11 b Fq(\003)1403 2389 y Fi(~)1403 2455 y Fv(I)1446 2430 y Fq(\()p Fr(k)s Fq(\))27 b Fp(j)h Fq(dim)n(\()p Fr(P)1936 2389 y Fi(~)1922 2455 y Fv(`)1951 2436 y Fg(\003)1991 2430 y Fq(\()p Fr(k)s Fq(\)\))g(=)f(1)p Fp(g)195 b Fq(\()p Fr(k)31 b Fp(2)d Fo(T)2861 2389 y Fj(\003)2904 2430 y Fq(\))p Fr(:)423 b Fs(\(18\))328 2601 y(of)32 b(the)h(index)g(set)572 2772 y Fp(G)-11 b Fq(\003)688 2731 y Fi(~)688 2796 y Fv(I)731 2772 y Fq(\()p Fr(k)s Fq(\))27 b(:=)1019 2661 y Fh(n)1085 2772 y Fr(`)1126 2731 y Fj(\003)1193 2772 y Fp(2)h Fq(\003)1355 2731 y Fi(~)1355 2796 y Fv(I)1397 2772 y Fq(\()p Fr(k)s Fq(\))g Fp(j)f(j)1661 2746 y Fq(~)1638 2772 y Fr(E)1716 2731 y Fi(~)1710 2796 y Fv(`)1739 2777 y Fg(\003)1779 2772 y Fq(\()p Fr(k)s Fq(\))22 b Fp(\000)2054 2746 y Fq(~)2031 2772 y Fr(E)2109 2731 y Fi(~)2103 2796 y Fv(`)2132 2777 y Fg(0)2159 2772 y Fq(\()p Fr(k)s Fq(\))p Fp(j)27 b Fr(>)g Fq(2)p Fo(~)2550 2731 y Fv(N)2650 2772 y Fs(fo)m(r)32 b Fr(`)2832 2731 y Fj(0)2883 2772 y Fp(2)c Fq(\003)3045 2731 y Fi(~)3045 2796 y Fv(I)3087 2772 y Fq(\()p Fr(k)s Fq(\))p Fp(n)3267 2787 y Fj(f)p Fv(`)3331 2768 y Fg(\003)3367 2787 y Fj(g)3407 2661 y Fh(o)3490 2772 y Fr(;)328 2964 y Fs(W)m(e)33 b(obtain)f(a)g(map)572 3135 y Fp(I)626 3150 y Fv(k)696 3135 y Fq(:)c Fp(G)-11 b Fq(\003)867 3094 y Fi(~)867 3160 y Fv(I)910 3135 y Fq(\()p Fr(k)s Fq(\))28 b Fp(!)f Fo(N)328 3306 y Fs(b)m(y)32 b(setting)i Fp(I)812 3321 y Fv(k)855 3306 y Fq(\()p Fr(`)934 3270 y Fj(\003)973 3306 y Fq(\))28 b(:=)f Fr(n)33 b Fs(fo)m(r)g(some)g Fr(n)g Fs(meeting)g(\(17\).)43 b(This)32 b(map)h(is)f(one-to-one.)474 3427 y(Its)37 b(restriction)g(to)g Fp(F)-7 b Fq(\003)1305 3390 y Fi(~)1305 3452 y Fv(I)1347 3427 y Fq(\()p Fr(k)s Fq(\))36 b Fs(is)h(uniquely)f(de\014ned,)i(since)f(fo)m(r)g Fr(`)2753 3390 y Fj(\003)2826 3427 y Fp(2)e(F)-7 b Fq(\003)3060 3390 y Fi(~)3060 3452 y Fv(I)3102 3427 y Fq(\()p Fr(k)s Fq(\))72 b Fr(P)3381 3390 y Fi(~)3367 3452 y Fv(`)3396 3434 y Fg(\003)3436 3427 y Fq(\()p Fr(k)s Fq(\))328 3547 y Fs(is)29 b(the)h(one-dimensional)f(p)m(rojecto)m(r)h(fo)m(r)f(the)h (eigenfunction)f Fr( )2566 3511 y Fi(~)2562 3577 y Fj(I)2601 3589 y Fd(k)2640 3577 y Fk(\()p Fv(`)2696 3558 y Fg(\003)2732 3577 y Fk(\))2764 3547 y Fq(\()p Fr(k)s Fq(\))f Fs(of)h Fr(H)3119 3511 y Fi(~)3161 3547 y Fq(\()p Fr(k)s Fq(\))g Fs(whose)328 3693 y(eigenvalue)k Fr(E)855 3657 y Fi(~)849 3722 y Fj(I)888 3734 y Fd(k)926 3722 y Fk(\()p Fv(`)982 3703 y Fg(\003)1018 3722 y Fk(\))1050 3693 y Fq(\()p Fr(k)s Fq(\))f Fs(lies)h(in)g Fq([)1527 3668 y(~)1504 3693 y Fr(E)1582 3657 y Fi(~)1576 3719 y Fv(`)1605 3700 y Fg(\003)1645 3693 y Fq(\()p Fr(k)s Fq(\))22 b Fp(\000)h Fo(~)1951 3657 y Fv(N)2018 3693 y Fr(;)2084 3668 y Fq(~)2062 3693 y Fr(E)2140 3657 y Fi(~)2134 3719 y Fv(`)2163 3700 y Fg(\003)2203 3693 y Fq(\()p Fr(k)s Fq(\))f(+)g Fo(~)2507 3657 y Fv(N)2574 3693 y Fq(])p Fs(.)474 3829 y(The)37 b(index)g(set)h Fp(G)-11 b Fq(\003)1188 3793 y Fi(~)1188 3854 y Fv(I)1231 3829 y Fq(\()p Fr(k)s Fq(\))37 b Fs(of)g(the)h(sepa)m (rated)g(quasimo)s(des)g(ma)m(y)e(b)s(e)i(very)f(small.)57 b(F)m(o)m(r)328 3950 y(example)27 b(it)g(is)g(even)g(empt)m(y)h(fo)m(r) e Fr(k)31 b Fq(=)d(0)e Fs(in)h Fr(d)g Fq(=)h(1)e Fs(dimensions,)k(if)c Fo(~)i Fr(>)f Fq(0)g Fs(is)g(small)g(enough.)328 4070 y(Ho)m(w)m(ever,)34 b(its)f Fn(mean)g Fs(ca)m(rdinalit)m(y)572 4204 y Fh(\012)619 4285 y Fp(jG)-11 b Fq(\003)763 4244 y Fi(~)763 4309 y Fv(I)805 4285 y Fp(j)833 4204 y Fh(\013)908 4285 y Fq(:=)1038 4149 y Fh(Z)1094 4375 y Fi(T)1144 4356 y Fg(\003)1195 4285 y Fp(jG)g Fq(\003)1339 4244 y Fi(~)1339 4309 y Fv(I)1381 4285 y Fq(\()p Fr(k)s Fq(\))p Fp(j)1611 4217 y Fr(dk)p 1549 4262 231 4 v 1549 4353 a Fq(v)m(ol)o Fo(T)1736 4324 y Fj(\003)328 4503 y Fs(over)32 b(the)h(Brillouin)g (zone)g(turns)g(out)f(to)h(b)s(e)g(asymptotic)g(to)572 4593 y Fh(\012)619 4673 y Fp(jG)-11 b Fq(\003)763 4632 y Fi(~)763 4698 y Fv(I)805 4673 y Fp(j)833 4593 y Fh(\013)908 4673 y Fp(\030)28 b Fq(\(2)p Fr(\031)t Fo(~)p Fq(\))1251 4632 y Fj(\000)p Fv(d)1346 4673 y Fq(v)m(ol)o(\()p Fp(K)1584 4688 y Fv(I)1624 4673 y Fq(\))p Fr(;)328 4844 y Fs(with)40 b(the)h(KAM)f(subset)h Fp(K)1336 4859 y Fv(I)1416 4844 y Fs(as)e(de\014ned)i(in)f(\(6\).)65 b(This)39 b(is)i(the)f(reason)g (why)g(indices)h(in)328 4965 y Fp(F)-7 b Fq(\003)461 4929 y Fi(~)461 4990 y Fv(I)503 4965 y Fq(\()p Fr(k)s Fq(\))32 b Fs(a)m(re)h(abundant)g(on)f(the)h(average;)g(it)f(holds:) 1898 5214 y(12)p eop %%Page: 13 13 13 12 bop 328 631 a Ff(Theo)m(rem)35 b(4.1)49 b Fn(F)m(o)m(r)37 b Fr(`)1168 595 y Fj(\003)1240 631 y Fp(2)c(F)-7 b Fq(\003)1472 595 y Fi(~)1472 656 y Fv(I)1514 631 y Fq(\()p Fr(k)s Fq(\))17 b Fr(;)33 b(k)j Fp(2)e Fo(T)1971 595 y Fj(\003)2049 631 y Fn(and)i(a)f(suitable)h(choice)h(of)e(phase)h(of)f(the)328 751 y(eigenfunction)e Fr( )966 715 y Fi(~)962 781 y Fj(I)1001 793 y Fd(k)1040 781 y Fk(\()p Fv(`)1096 762 y Fg(\003)1132 781 y Fk(\))1163 751 y Fq(\()p Fr(k)s Fq(\))p Fn(,)572 979 y Fp(k)p Fr( )689 937 y Fi(~)685 1003 y Fj(I)724 1015 y Fd(k)762 1003 y Fk(\()p Fv(`)818 984 y Fg(\003)854 1003 y Fk(\))886 979 y Fq(\()p Fr(k)s Fq(\))22 b Fp(\000)1158 952 y Fq(~)1138 979 y Fr( )1205 937 y Fi(~)1201 1003 y Fv(`)1230 984 y Fg(\003)1270 979 y Fq(\()p Fr(k)s Fq(\))p Fp(k)27 b(\024)i Fq(2)p Fo(~)1686 937 y Fv(N)1752 979 y Fr(:)1613 b Fs(\(19\))328 1179 y Fn(F)m(o)m(r)38 b Fr(N)46 b(>)35 b(d)25 b Fq(+)g(2)37 b Fn(there)h(is)f(a)g Fr(\014)k(>)35 b Fq(0)i Fn(such)g(that)h(fo)m(r)f Fr(I)43 b Fq(:=)35 b([\(1)25 b Fp(\000)h Fr(\016)t Fq(\))p Fr(E)6 b(;)17 b Fq(\(1)24 b(+)h Fr(\016)t Fq(\))p Fr(E)6 b Fq(])37 b Fn(with)328 1299 y Fr(E)d(>)27 b(E)609 1314 y Fk(th)572 1500 y Fq(v)m(ol)o(\()p Fp(K)810 1515 y Fv(I)850 1500 y Fq(\))16 b Fp(\000)g Fq(v)m(ol)q(\()p Fp(K)1238 1459 y Fv(c)1237 1525 y(I)1277 1500 y Fq(\))g Fp(\000)g(O)1503 1515 y Fv(E)1563 1500 y Fq(\()p Fo(~)1655 1459 y Fv(\014)1702 1500 y Fq(\))28 b Fp(\024)g Fq(\(2)p Fr(\031)t Fo(~)p Fq(\))2111 1459 y Fv(d)2167 1419 y Fh(\012)2215 1500 y Fp(jF)-7 b Fq(\003)2376 1459 y Fi(~)2376 1525 y Fv(I)2417 1500 y Fp(j)2445 1419 y Fh(\013)2520 1500 y Fp(\024)28 b Fq(v)m(ol)o(\()p Fp(K)2863 1515 y Fv(I)2903 1500 y Fq(\))16 b(+)g Fp(O)3128 1515 y Fv(E)3188 1500 y Fq(\()p Fo(~)3280 1459 y Fv(\014)3327 1500 y Fq(\))p Fr(;)p Fs(\(20\))328 1701 y Fn(with)33 b Fp(K)612 1665 y Fv(c)611 1726 y(I)678 1701 y Fq(:=)28 b Fp(P)878 1716 y Fv(I)940 1701 y Fp(n)22 b(K)1088 1716 y Fv(I)1128 1701 y Fn(.)44 b(In)32 b(pa)m(rticula)m(r)572 1814 y Fh(\014)572 1874 y(\014)572 1934 y(\014)572 1994 y(\014)572 2054 y(\014)615 1915 y Fq(\(2)p Fr(\031)t Fo(~)p Fq(\))853 1879 y Fv(d)909 1834 y Fh(\012)956 1915 y Fp(jF)-7 b Fq(\003)1117 1879 y Fi(~)1117 1940 y Fv(I)1159 1915 y Fp(j)1187 1834 y Fh(\013)p 615 1966 619 4 v 770 2057 a Fq(v)m(ol)o(\()p Fp(P)1001 2072 y Fv(I)1041 2057 y Fq(\))1266 1989 y Fp(\000)23 b Fq(1)1415 1814 y Fh(\014)1415 1874 y(\014)1415 1934 y(\014)1415 1994 y(\014)1415 2054 y(\014)1475 1989 y Fp(\024)1580 1830 y Fh(r)p 1680 1830 163 4 v 1690 1921 a Fr(E)1762 1936 y Fk(th)p 1690 1966 144 4 v 1723 2057 a Fr(E)1865 1989 y Fq(+)f Fp(O)2042 2004 y Fv(E)2102 1989 y Fq(\()p Fo(~)2194 1948 y Fv(\014)2241 1989 y Fq(\))p Fr(:)1086 b Fs(\(21\))328 2295 y Ff(Rema)m(rk)36 b(4.2)49 b Fq(The)33 b(Liouville)d(measure)j(of)f(the)h(thic)m(k)m (ened)h(energy)g(shell)e(is)g(of)g(order)572 2496 y(v)m(ol)o(\()p Fp(P)803 2511 y Fv(I)844 2496 y Fq(\))27 b(=)h Fr(c)p Fq(\()p Fr(\016)t Fq(\))22 b Fp(\001)g Fr(E)1328 2455 y Fv(d=)p Fk(2)1461 2496 y Fp(\001)g Fq(\(1)f(+)h Fp(O)s Fq(\(1)p Fr(=E)6 b Fq(\)\))p Fr(:)1276 b Fs(\(22\))328 2720 y Ff(Pro)s(of.)43 b Fs(Estimate)31 b(\(19\))e(follo)m(ws)h(from)g (\(1\))f(and)h(Def.)g(\(18\),)g(since)g(the)h Fq(\()2971 2694 y(~)2951 2720 y Fr( )3018 2684 y Fi(~)3014 2746 y Fv(`)3043 2727 y Fg(\003)3083 2720 y Fq(\()p Fr(k)s Fq(\))p Fr(;)3280 2695 y Fq(~)3257 2720 y Fr(E)3335 2684 y Fi(~)3329 2746 y Fv(`)3358 2727 y Fg(\003)3398 2720 y Fq(\()p Fr(k)s Fq(\)\))328 2841 y Fs(a)m(re)i Fo(~)535 2804 y Fk(2)p Fv(N)637 2841 y Fs(-quasimo)s(des.)474 2961 y(The)g(upp)s(er)g(b)s(ound)f(in)g(\(20\))g(follo)m(ws)h(from)f (the)h(Lazutkin)g(result)h(\(8\))e(fo)m(r)h Fp(j)p Fq(\003)3243 2925 y Fi(~)3243 2986 y Fv(I)3284 2961 y Fq(\()p Fr(k)s Fq(\))p Fp(j)p Fs(.)474 3081 y(W)m(e)g(claim)g(that)572 3282 y Fq(\(2)p Fr(\031)t Fo(~)p Fq(\))810 3241 y Fv(d)866 3201 y Fh(\012)913 3282 y Fp(jG)-11 b Fq(\003)1057 3241 y Fi(~)1057 3307 y Fv(I)1100 3282 y Fp(j)1128 3201 y Fh(\013)1202 3282 y Fp(\025)28 b Fq(v)m(ol\()p Fp(K)1546 3297 y Fv(I)1586 3282 y Fq(\))22 b Fp(\000)g(O)s Fq(\()p Fo(~)1919 3241 y Fv(\014)1966 3282 y Fq(\))p Fr(:)1361 b Fs(\(23\))328 3483 y(By)32 b(\(8\))g(this)h(follo)m(ws)g(from)f(an)h (estimate)g(of)f(the)i(fo)m(rm)572 3683 y Fq(\(2)p Fr(\031)t Fo(~)p Fq(\))810 3642 y Fv(d)866 3603 y Fh(\012)913 3683 y Fp(j)p Fq(\003)1009 3642 y Fi(~)1009 3708 y Fv(I)1073 3683 y Fp(n)22 b(G)-11 b Fq(\003)1261 3642 y Fi(~)1261 3708 y Fv(I)1304 3683 y Fp(j)1332 3603 y Fh(\013)1406 3683 y Fq(=)28 b Fp(O)s Fq(\()p Fo(~)1684 3642 y Fv(\014)1731 3683 y Fq(\))p Fr(:)1596 b Fs(\(24\))328 3884 y(But)572 4028 y Fh(\012)619 4109 y Fp(j)p Fq(\003)715 4068 y Fi(~)715 4134 y Fv(I)779 4109 y Fp(n)22 b(G)-11 b Fq(\003)967 4068 y Fi(~)967 4134 y Fv(I)1009 4109 y Fp(j)1037 4028 y Fh(\013)1112 4109 y Fp(\024)1217 3973 y Fh(Z)1272 4199 y Fi(T)1322 4180 y Fg(\003)1460 4014 y Fh(X)1373 4238 y Fv(`)1402 4247 y Fb(1)1437 4238 y Fj(6)p Fk(=)p Fv(`)1521 4247 y Fb(2)1555 4238 y Fj(2)p Fk(\003)1651 4215 y Fc(~)1651 4261 y Fd(I)1706 4109 y Fr(\037)1784 3998 y Fh(\020)1866 4084 y Fq(~)1843 4109 y Fr(E)1921 4068 y Fi(~)1915 4134 y Fv(`)1944 4143 y Fb(1)1983 4109 y Fq(\()p Fr(k)s Fq(\))22 b Fp(\000)2258 4084 y Fq(~)2235 4109 y Fr(E)2313 4068 y Fi(~)2307 4134 y Fv(`)2336 4143 y Fb(2)2375 4109 y Fq(\()p Fr(k)s Fq(\))2505 3998 y Fh(\021)2654 4042 y Fr(dk)p 2591 4086 231 4 v 2591 4177 a Fq(v)m(ol)o Fo(T)2778 4149 y Fj(\003)2831 4109 y Fr(;)534 b Fs(\(25\))328 4441 y(where)34 b Fr(\037)p Fq(\()p Fr(x)p Fq(\))28 b(:=)f(1)33 b Fs(fo)m(r)f Fp(j)p Fr(x)p Fp(j)c(\024)g Fq(2)p Fo(~)1517 4405 y Fv(N)1616 4441 y Fs(and)k Fq(0)h Fs(otherwise.)474 4561 y(F)m(o)m(r)h Fr(E)k Fs(la)m(rge)33 b(and)f Fo(~)c Fr(<)f Fo(~)1392 4576 y Fk(0)572 4669 y Fh(\014)572 4729 y(\014)572 4789 y(\014)605 4784 y Fp(r)705 4673 y Fh(\020)787 4758 y Fq(~)764 4784 y Fr(E)842 4742 y Fi(~)836 4808 y Fv(`)865 4817 y Fb(1)904 4784 y Fq(\()p Fr(k)s Fq(\))22 b Fp(\000)1178 4758 y Fq(~)1156 4784 y Fr(E)1234 4742 y Fi(~)1228 4808 y Fv(`)1257 4817 y Fb(2)1295 4784 y Fq(\()p Fr(k)s Fq(\))1425 4673 y Fh(\021)1485 4669 y(\014)1485 4729 y(\014)1485 4789 y(\014)1601 4784 y Fp(\025)1772 4744 y Fk(1)p 1772 4761 36 4 v 1772 4818 a(2)1833 4784 y Fp(jr)17 b Fq(\()p Fr(E)2071 4799 y Fk(0)2110 4784 y Fq(\()p Fr(`)2189 4799 y Fk(1)2229 4784 y Fr(;)g(k)s Fq(\))k Fp(\000)i Fr(E)2558 4799 y Fk(0)2598 4784 y Fq(\()p Fr(`)2677 4799 y Fk(2)2716 4784 y Fr(;)17 b(k)s Fq(\)\))p Fp(j)1601 4964 y(\025)1772 4924 y Fk(1)p 1772 4941 V 1772 4998 a(4)1817 4964 y Fo(~)1871 4927 y Fk(2)1910 4964 y Fp(j)p Fr(`)1979 4979 y Fk(1)2040 4964 y Fp(\000)23 b Fr(`)2181 4979 y Fk(2)2220 4964 y Fp(j)28 b(\025)g Fr(cte:)p Fo(~)2584 4927 y Fk(2)1898 5214 y Fs(13)p eop %%Page: 14 14 14 13 bop 328 631 a Fs(unifo)m(rmly)26 b(fo)m(r)f(all)h Fr(k)31 b Fp(2)d Fo(T)1223 595 y Fj(\003)1291 631 y Fs(and)e Fr(`)1505 646 y Fk(1)1572 631 y Fp(6)p Fq(=)h Fr(`)1716 646 y Fk(2)1784 631 y Fp(2)h Fq(\003)1946 595 y Fi(~)1946 656 y Fv(I)1988 631 y Fs(.)41 b(Thus)26 b(b)m(y)g(the)g(implicit)g (function)g(theo)m(rem)328 751 y(the)33 b(set)h(of)e(quasimomenta)h Fr(k)d Fp(2)f Fo(T)1606 715 y Fj(\003)1681 751 y Fs(leading)j(to)h(a)f (degeneracy)595 925 y Fq(~)572 951 y Fr(E)650 909 y Fi(~)644 975 y Fv(`)673 984 y Fb(1)712 951 y Fq(\()p Fr(k)s Fq(\))27 b(=)996 925 y(~)973 951 y Fr(E)1051 909 y Fi(~)1045 975 y Fv(`)1074 984 y Fb(2)1113 951 y Fq(\()p Fr(k)s Fq(\))328 1150 y Fs(of)32 b(quasi-energies)i(fo)m(rms)f(a)f(hyp)s(ersurface,)h (and)572 1257 y Fh(Z)627 1483 y Fi(T)677 1464 y Fg(\003)728 1393 y Fr(\037)806 1282 y Fh(\020)888 1368 y Fq(~)865 1393 y Fr(E)943 1352 y Fi(~)937 1418 y Fv(`)966 1427 y Fb(1)1005 1393 y Fq(\()p Fr(k)s Fq(\))22 b Fp(\000)1280 1368 y Fq(~)1257 1393 y Fr(E)1335 1352 y Fi(~)1329 1418 y Fv(`)1358 1427 y Fb(2)1397 1393 y Fq(\()p Fr(k)s Fq(\))1527 1282 y Fh(\021)1676 1326 y Fr(dk)p 1613 1370 231 4 v 1613 1461 a Fq(v)m(ol)o Fo(T)1800 1432 y Fj(\003)1881 1393 y Fq(=)27 b Fp(O)s Fq(\()p Fo(~)2158 1352 y Fv(N)7 b Fj(\000)p Fk(2)2316 1393 y Fq(\))p Fr(:)328 1652 y Fs(Since)37 b Fp(j)p Fq(\003)674 1616 y Fi(~)674 1677 y Fv(I)715 1652 y Fp(j)f Fs(is)g(of)f(o)m(rder)i Fp(O)s Fq(\()p Fo(~)1405 1616 y Fj(\000)p Fv(d)1500 1652 y Fq(\))p Fs(,)g(the)f(r.h.s.)h(of)e(\(25\))h(is)g(thus)g(of)g(o)m(rder)g Fp(O)s Fq(\()p Fo(~)3162 1616 y Fj(\000)p Fk(2)p Fv(d)p Fk(+)p Fv(N)7 b Fj(\000)p Fk(2)3501 1652 y Fq(\))p Fs(.)328 1773 y(So)32 b(fo)m(r)h Fr(N)38 b(>)28 b(d)22 b Fq(+)g(2)f(+)h Fr(\014)39 b Fs(estimate)33 b(\(24\))f(holds)h(true,)g(implying)f (\(23\).)474 1893 y(W)m(e)h(estimate)h(the)f(numb)s(er)572 2012 y Fh(\012)619 2092 y Fp(jF)-7 b Fq(\003)780 2051 y Fi(~)780 2117 y Fv(I)822 2092 y Fp(j)850 2012 y Fh(\013)924 2092 y Fq(=)1028 2012 y Fh(\012)1075 2092 y Fp(jG)c Fq(\003)1219 2051 y Fi(~)1219 2117 y Fv(I)1261 2092 y Fp(j)1289 2012 y Fh(\013)1358 2092 y Fp(\000)1457 2012 y Fh(\012)1504 2092 y Fp(jG)g Fq(\003)1648 2051 y Fi(~)1648 2117 y Fv(I)1713 2092 y Fp(n)22 b(F)-7 b Fq(\003)1918 2051 y Fi(~)1918 2117 y Fv(I)1960 2092 y Fp(j)1988 2012 y Fh(\013)3392 2092 y Fs(\(26\))328 2292 y(from)32 b(b)s(elo)m(w)h(b)m(y)g(using)f (\(23\))g(and)h(the)g(relation)572 2406 y Fh(\014)572 2466 y(\014)605 2491 y Fp(G)-11 b Fq(\003)721 2450 y Fi(~)721 2515 y Fv(I)764 2491 y Fq(\()p Fr(k)s Fq(\))22 b 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h(l.h.s.)g(equals)f(the)h(numb)s(er)g(of)e(intervals)572 3408 y Fq([)622 3383 y(~)599 3408 y Fr(E)677 3367 y Fi(~)671 3433 y Fv(`)700 3414 y Fg(\003)740 3408 y Fq(\()p Fr(k)s Fq(\))22 b Fp(\000)h Fo(~)1046 3367 y Fv(N)1113 3408 y Fr(;)1179 3383 y Fq(~)1157 3408 y Fr(E)1235 3367 y Fi(~)1229 3433 y Fv(`)1258 3414 y Fg(\003)1298 3408 y Fq(\()p Fr(k)s Fq(\))f(+)g Fo(~)1602 3367 y Fv(N)1669 3408 y Fq(])98 b Fs(fo)m(r)f Fr(`)28 b Fp(2)g(G)-11 b Fq(\003)2279 3367 y Fi(~)2279 3433 y Fv(I)2322 3408 y Fq(\()p Fr(k)s Fq(\))328 3607 y Fs(containing)34 b(t)m(w)m(o)h(o)m(r)g (mo)m(re)g(eigenvalues)g Fr(E)1874 3571 y Fi(~)1868 3632 y Fv(n)1916 3607 y Fq(\()p Fr(k)s Fq(\))g Fs(\(counted)g(with)g (multiplicit)m(y\).)49 b(By)34 b(de\014-)328 3728 y(nition)e(of)h Fp(G)-11 b Fq(\003)818 3692 y Fi(~)818 3753 y Fv(I)860 3728 y Fq(\()p Fr(k)s Fq(\))33 b Fs(these)h(intervals)f(a)m(re)g (disjoint,)f(and)g(w)m(e)i(have)572 3927 y Fr(E)650 3886 y Fi(~)644 3952 y Fj(I)683 3964 y Fd(k)721 3952 y Fk(\()p Fv(`)777 3933 y Fg(\003)813 3952 y Fk(\))845 3927 y Fq(\()p Fr(k)s Fq(\))27 b Fp(2)i Fq([)1146 3902 y(~)1124 3927 y Fr(E)1202 3886 y Fi(~)1196 3952 y Fv(`)1225 3933 y Fg(\003)1265 3927 y Fq(\()p Fr(k)s Fq(\))22 b Fp(\000)h Fo(~)1571 3886 y Fv(N)1638 3927 y Fr(;)1704 3902 y Fq(~)1682 3927 y Fr(E)1760 3886 y Fi(~)1754 3952 y Fv(`)1783 3933 y Fg(\003)1822 3927 y Fq(\()p Fr(k)s Fq(\))g(+)f Fo(~)2127 3886 y Fv(N)2194 3927 y Fq(])p Fr(;)328 4126 y Fs(so)37 b(that)f(further)h(eigenvalues)g(must)g(b)s(e)g(indexed)g(b)m(y)f(an)g (integer)i(b)s(elonging)d(to)i(the)g(set)328 4247 y(which)c(app)s(ea)m (rs)g(on)f(the)i(r.h.s.)f(of)f(\(27\).)474 4367 y(The)h(W)m(eyl)f (estimate)572 4566 y Fq(\(2)p Fr(\031)t Fo(~)p Fq(\))810 4525 y Fv(d)866 4482 y Fh(\014)866 4541 y(\014)899 4566 y Fq(\004)964 4525 y Fi(~)964 4591 y Fv(I)1007 4566 y Fq(\()p Fr(k)s Fq(\))1137 4482 y Fh(\014)1137 4541 y(\014)1198 4566 y Fq(=)27 b(v)m(ol\()p Fp(P)1533 4581 y Fv(I)1573 4566 y Fq(\))22 b(+)g Fp(O)s Fq(\()p Fo(~)p Fq(\))195 b(\()p Fr(k)31 b Fp(2)d Fo(T)2415 4525 y Fj(\003)2458 4566 y Fq(\))328 4765 y Fs(is)33 b(unifo)m(rm)g(in)f Fr(k)s Fs(,)h(since)g(the)g(slop)s(e)g(of)f(the)h(band)g(functions)g (is)g(b)s(ounded)f(ab)s(ove)g(b)m(y)572 4965 y Fp(jr)683 4980 y Fv(k)725 4965 y Fr(E)803 4924 y Fi(~)797 4989 y Fv(n)846 4965 y Fq(\()p Fr(k)s Fq(\))p Fp(j)27 b(\024)h Fo(~)1190 4875 y Fh(p)p 1290 4875 676 4 v 90 x Fq(2\()p Fr(E)1455 4936 y Fi(~)1449 4989 y Fv(n)1497 4965 y Fq(\()p Fr(k)s Fq(\))22 b Fp(\000)g Fr(V)1805 4980 y Fk(min)1927 4965 y Fq(\))1898 5214 y Fs(14)p eop %%Page: 15 15 15 14 bop 328 631 a Fs(and)32 b(thus)h(of)g(o)m(rder)g Fo(~)f Fs(if)g Fr(E)1311 595 y Fi(~)1305 656 y Fv(n)1354 631 y Fq(\()p Fr(k)s Fq(\))c Fp(2)g Fr(I)40 b Fs(\(see)34 b([2],)e(Co)m(rr.)i(2.4\).)474 751 y(Thus)f(the)g(r.h.s.)g(of)f(\(27\)) g(is)h(b)s(ounded)g(ab)s(ove)f(b)m(y)572 870 y Fh(\014)572 930 y(\014)605 955 y Fq(\004)670 914 y Fi(~)670 979 y Fv(I)712 955 y Fq(\()p Fr(k)s Fq(\))23 b Fp(n)f(I)991 970 y Fv(k)1033 955 y Fq(\()p Fp(G)-11 b Fq(\003)1187 914 y Fi(~)1187 979 y Fv(I)1230 955 y Fq(\()p Fr(k)s Fq(\)\))1398 870 y Fh(\014)1398 930 y(\014)1459 955 y Fp(\024)28 b Fq(\(2)p Fr(\031)t Fo(~)p Fq(\))1802 914 y Fj(\000)p Fv(d)1897 955 y Fq(v)m(ol)o(\()p Fp(P)2128 970 y Fv(I)2169 955 y Fq(\))22 b Fp(\000)g(jG)-11 b Fq(\003)2472 914 y Fi(~)2472 979 y Fv(I)2515 955 y Fq(\()p Fr(k)s Fq(\))p Fp(j)21 b(\000)i(O)s Fq(\()p Fo(~)2968 914 y Fk(1)p Fj(\000)p Fv(d)3099 955 y Fq(\))p Fr(:)474 1158 y Fs(Inserting)28 b(that)g(upp)s(er)f(estimate)i(fo)m(r)e(\(27\))g(in)g (\(26\))g(and)g(using)h(\(23\))f(p)m(roves)h(the)g(lo)m(w)m(er)328 1279 y(b)s(ound)k(in)h(\(20\).)474 1399 y(Finally)-8 b(,)32 b(estimate)i(\(21\))e(follo)m(ws)h(from)f(\(20\))g(and)h(the)g (result)582 1590 y Fq(v)m(ol)o(\()p Fp(K)821 1554 y Fv(c)820 1615 y(I)860 1590 y Fq(\))p 582 1634 317 4 v 585 1726 a(v)m(ol\()p Fp(P)817 1741 y Fv(I)857 1726 y Fq(\))936 1657 y(=)27 b Fp(O)1138 1547 y Fh(\020)1198 1657 y Fq(1)p Fr(=)1296 1568 y Fp(p)p 1378 1568 79 4 v 1378 1657 a Fr(E)1456 1547 y Fh(\021)1532 1657 y Fr(;)328 1917 y Fs(see)34 b(\(7\).)2854 b Fa(2)328 2254 y Ft(5)158 b(Asymptotic)52 b(V)l(elo)t(cit)l(y)328 2476 y Fs(As)33 b(a)g(consequence)h(of)e (Birkho\013)7 b('s)33 b(Ergo)s(dic)f(Theo)m(rem)h(fo)m(r)g Fr(\025)p Fs({almost)f(all)h Fr(x)3077 2491 y Fk(0)3144 2476 y Fp(2)28 b(P)576 2742 y Fq(\026)-53 b Fr(v)623 2701 y Fj(\006)682 2742 y Fq(\()p Fr(x)775 2757 y Fk(0)814 2742 y Fq(\))28 b(:=)83 b(lim)1011 2804 y Fv(T)10 b Fj(!\0061)1295 2674 y Fq(1)p 1284 2719 71 4 v 1284 2810 a Fr(T)1381 2606 y Fh(Z)1481 2632 y Fv(T)1437 2832 y Fk(0)1553 2742 y Fr(p)p Fq(\()p Fr(t;)17 b(x)1774 2757 y Fk(0)1813 2742 y Fq(\))p Fr(dt)328 2998 y Fs(exist)41 b(and)g(a)m(re)g(equal)g(\()p Fr(\025)f Fs(denoting)h(the)h(Liouville)e(measure)i(on)e Fp(P)8 b Fs(\).)69 b(In)40 b(this)h(case)h(w)m(e)328 3118 y(set)k Fq(\026)-53 b Fr(v)t Fq(\()p Fr(x)629 3133 y Fk(0)669 3118 y Fq(\))42 b(:=)k(\026)-53 b Fr(v)945 3082 y Fj(\006)1004 3118 y Fq(\()p Fr(x)1097 3133 y Fk(0)1137 3118 y Fq(\))p Fs(,)43 b(and)e(otherwise)46 b Fq(\026)-53 b Fr(v)t Fq(\()p Fr(x)1998 3133 y Fk(0)2038 3118 y Fq(\))42 b(:=)g(0)p Fs(,)h(thus)f(de\014ning)g(the)f Fn(asymptotic)328 3239 y(velo)s(cit)m(y)576 3442 y Fq(\026)-53 b Fr(v)31 b Fq(:)d Fp(P)36 b(!)27 b Fo(R)1003 3401 y Fv(d)328 3645 y Fs(which)33 b(is)g(a)f Fr(\025)p Fs({measurable)h(phase)g(space)h (function.)474 3766 y(W)m(e)24 b(a)m(re)h(pa)m(rticula)m(rly)f (interested)i(in)d(the)i(energy)f(dep)s(endence)h(of)f(asymptotic)g (velo)s(cit)m(y)328 3886 y(and)32 b(thus)h(intro)s(duce)h(the)f Fn(energy-velo)s(cit)m(y)f(map)572 4090 y Fr(A)c Fq(:=)f(\()p Fr(H)r(;)21 b Fq(\026)-53 b Fr(v)t Fq(\))27 b(:)h Fp(P)36 b(!)28 b Fo(R)1438 4048 y Fv(d)p Fk(+1)1574 4090 y Fr(:)1791 b Fs(\(28\))328 4293 y Fr(A)33 b Fs(is)f Fr(\025)p Fs({measurable)h (and)g(generates)h(an)e(image)g(measure)i Fr(\027)g Fq(:=)28 b Fr(\025A)2816 4257 y Fj(\000)p Fk(1)2943 4293 y Fs(on)k Fo(R)3140 4257 y Fv(d)p Fk(+1)3277 4293 y Fs(.)474 4413 y(On)e(the)h(other)f(hand)g(\(see)h([2]\))f(fo)m(r)g(almost)g(all)g Fr(k)g Fp(2)e Fo(T)2415 4377 y Fj(\003)2488 4413 y Fs(the)j(op)s(erato) m(r)f(of)f(asymptotic)328 4534 y(velo)s(cit)m(y)576 4799 y Fq(\026)-53 b Fr(v)623 4758 y Fi(~)665 4799 y Fq(\()p Fr(k)s Fq(\))28 b(:=)56 b(lim)953 4861 y Fv(T)10 b Fj(!1)1183 4732 y Fq(1)p 1172 4776 V 1172 4867 a Fr(T)1269 4663 y Fh(Z)1369 4690 y Fv(T)1324 4889 y Fk(0)1440 4799 y Fr(e)1485 4758 y Fv(iH)1572 4735 y Fc(~)1611 4758 y Fk(\()p Fv(k)r Fk(\))q Fv(t)1734 4799 y Fq(\()p Fr(D)25 b Fq(+)d Fo(~)p Fr(k)s Fq(\))p Fr(e)2167 4758 y Fj(\000)p Fv(iH)2309 4735 y Fc(~)2348 4758 y Fk(\()p Fv(k)r Fk(\))p Fv(t)2503 4799 y Fr(dt:)1898 5214 y Fs(15)p eop %%Page: 16 16 16 15 bop 328 631 a Fs(exists)33 b(and)g(is)g(given)f(b)m(y)576 844 y Fq(\026)-53 b Fr(v)623 803 y Fi(~)665 844 y Fq(\()p Fr(k)s Fq(\))28 b(=)926 750 y Fh(X)1087 844 y Fr(P)1164 803 y Fi(~)1150 869 y Fv(m)1216 844 y Fq(\()p Fr(k)s Fq(\)\()p Fr(D)d Fq(+)d Fo(~)p Fr(k)s Fq(\))p Fr(P)1811 803 y Fi(~)1797 869 y Fv(m)1863 844 y Fq(\()p Fr(k)s Fq(\))28 b(=)2124 750 y Fh(X)2285 844 y Fo(~)2339 803 y Fj(\000)p Fk(1)2433 844 y Fp(r)2516 859 y Fv(k)2559 844 y Fr(E)2637 803 y Fi(~)2631 869 y Fv(m)2697 844 y Fq(\()p Fr(k)s Fq(\))p Fr(P)2904 803 y Fi(~)2890 869 y Fv(m)2957 844 y Fq(\()p Fr(k)s Fq(\))328 1070 y Fs(with)33 b(the)g(eigenp)m(rojections)h Fr(P)1450 1034 y Fi(~)1436 1095 y Fv(m)1503 1070 y Fq(\()p Fr(k)s Fq(\))e Fs(of)g Fr(H)1865 1034 y Fi(~)1907 1070 y Fq(\()p Fr(k)s Fq(\))p Fs(.)474 1191 y(The)h Fn(quantum)f(asymptotic)h(velo)s(cities)h Fs(a)m(re)f(de\014ned)g(b)m(y)576 1444 y Fq(\026)-53 b Fr(v)623 1403 y Fi(~)619 1469 y Fv(n)666 1444 y Fq(\()p Fr(k)s Fq(\))28 b(:=)954 1304 y Fh(\032)1070 1383 y Fo(~)1124 1347 y Fj(\000)p Fk(1)1219 1383 y Fp(r)1302 1398 y Fv(k)1344 1383 y Fr(E)1422 1347 y Fi(~)1416 1408 y Fv(n)1465 1383 y Fq(\()p Fr(k)s Fq(\))83 b Fs(,)32 b(gradient)h(exists)1308 1504 y Fq(0)321 b Fs(,)32 b(otherwise.)328 1711 y(W)m(e)42 b(equip)g(the)h Fn(semiclassical)g(phase)g(space)g Fp(P)2087 1675 y Fi(~)2173 1711 y Fq(:=)h Fo(N)f Fp(\002)29 b Fo(T)2589 1675 y Fj(\003)2674 1711 y Fs(with)43 b(the)f Fn(semiclassical)328 1831 y(measure)31 b Fr(\025)746 1795 y Fi(~)816 1831 y Fq(:=)d(\(2)p Fr(\031)t Fo(~)p Fq(\))1185 1795 y Fv(d)1224 1831 y Fr(\026)1283 1846 y Fk(1)1339 1831 y Fp(\002)17 b Fr(\026)1492 1846 y Fk(2)1531 1831 y Fs(,)31 b(where)g Fr(\026)1914 1846 y Fk(1)1983 1831 y Fs(denotes)g(counting)f(measure)h (on)e Fo(N)45 b Fs(and)30 b Fr(\026)3527 1846 y Fk(2)328 1952 y Fs(Haa)m(r)j(measure)g(on)g(the)g(Brillouin)f(zone)h Fo(T)1860 1916 y Fj(\003)1903 1952 y Fs(.)474 2072 y(In)51 b(o)m(rder)i(to)e(compa)m(re)h(classical)g(and)g(quantum)f(quantities,) 58 b(w)m(e)52 b(intro)s(duce)g(the)328 2193 y Fn(energy-velo)s(cit)m(y) 32 b(map)572 2394 y Fr(A)645 2353 y Fi(~)715 2394 y Fq(:)c Fp(P)847 2353 y Fi(~)917 2394 y Fp(!)g Fo(R)1111 2353 y Fv(d)p Fk(+1)1345 2394 y Fq(with)97 b Fr(A)1705 2353 y Fi(~)1748 2394 y Fq(\()p Fr(n;)17 b(k)s Fq(\))27 b(:=)h(\()p Fr(E)2254 2353 y Fi(~)2248 2419 y Fv(n)2296 2394 y Fq(\()p Fr(k)s Fq(\))p Fr(;)21 b Fq(\026)-53 b Fr(v)2521 2353 y Fi(~)2517 2419 y Fv(n)2564 2394 y Fq(\()p Fr(k)s Fq(\)\))328 2595 y Fs(and)32 b(the)h(image)g(measure)g Fr(\027)1357 2559 y Fi(~)1428 2595 y Fq(:=)28 b Fr(\025)1616 2559 y Fi(~)1658 2595 y Fq(\()p Fr(A)1769 2559 y Fi(~)1812 2595 y Fq(\))1850 2559 y Fj(\000)p Fk(1)1944 2595 y Fs(.)328 2763 y Ff(Example:)43 b Fs(F)m(o)m(r)34 b Fr(V)49 b Fp(\021)28 b Fq(0)33 b Fs(\(free)g(motion\))f Fr(\027)1871 2727 y Fi(~)1942 2763 y Fq(=)27 b Fr(\027)39 b Fs(indep)s(endent)34 b(of)e(the)h(value)f(of)g Fo(~)p Fs(.)328 2931 y(In)e([2])g(w)m(e)h (stated)g(the)f(follo)m(wing)g(conjecture,)i(which)f(w)m(e)g(p)m(roved) f(fo)m(r)g(smo)s(oth)g Fr(V)52 b Fs(leading)328 3051 y(to)31 b(integrable)g(resp.)g(to)g(ergo)s(dic)f(motion)h(\(see)h(also) e([5])h(fo)m(r)f(ergo)s(dic)h(motions)g(generated)328 3171 y(b)m(y)h(Coulombic)h(p)s(erio)s(dic)g Fr(V)21 b Fs(\):)328 3373 y Ff(Conjecture)35 b(5.1)49 b Fn(F)m(o)m(r)34 b(all)e Fp(L)p Fn({p)s(erio)s(dic)g(p)s(otentials)h Fr(V)49 b Fp(2)28 b Fr(C)2511 3336 y Fj(1)2586 3373 y Fq(\()p Fo(R)2690 3336 y Fv(d)2736 3373 y Fr(;)17 b Fo(R)5 b Fq(\))572 3574 y Fr(w)645 3533 y Fj(\003)689 3574 y Fp(\000)10 b Fq(lim)772 3637 y Fi(~)p Fj(&)p Fk(0)933 3574 y Fr(\027)987 3533 y Fi(~)1058 3574 y Fq(=)27 b Fr(\027)328 3831 y Fn(\(which)35 b(means)f Fq(lim)1057 3846 y Fi(~)p Fj(&)p Fk(0)1222 3750 y Fh(R)1269 3865 y Fi(R)1317 3846 y Fd(d)p Fb(+1)1451 3831 y Fr(f)11 b Fq(\()p Fr(x)p Fq(\))p Fr(d\027)1746 3795 y Fi(~)1788 3831 y Fq(\()p Fr(x)p Fq(\))31 b(=)2056 3750 y Fh(R)2104 3865 y Fi(R)2152 3846 y Fd(d)p Fb(+1)2285 3831 y Fr(f)11 b Fq(\()p Fr(x)p Fq(\))p Fr(d\027)6 b Fq(\()p Fr(x)p Fq(\))35 b Fn(fo)m(r)f(continuous)h(func-)328 3951 y(tions)e Fr(f)38 b Fp(2)28 b Fr(C)812 3915 y Fk(0)805 3976 y(0)852 3951 y Fq(\()p Fo(R)956 3915 y Fv(d)p Fk(+1)1092 3951 y Fr(;)17 b Fo(R)5 b Fq(\))38 b Fn(of)32 b(compact)i(supp)s(o)m (rt\).)328 4177 y Ff(Rema)m(rk)i(5.2)49 b Fq(One)30 b(ma)m(y)g(also)f (consider)h(the)g(stronger)g(conjecture)i(with)d(con)m(tin)m(uous)328 4297 y Fn(b)s(ounded)g Fq(test)h(functions)g Fr(f)11 b Fq(,)30 b(that)f(is)g(w)m(eak)i(con)m(v)m(ergence)h(in)c(the)i (language)e(of)h(proba-)328 4417 y(bilit)m(y)i(theory)-8 b(.)328 4643 y Fs(Here)27 b(w)m(e)g(obtain)f(a)g(statement)i(which)e (veri\014es)h(the)g(conjecture)g(in)g(the)f(high)g(energy)g(limit.)328 4763 y(T)-8 b(o)32 b(this)h(aim)f(w)m(e)i(intro)s(duce)f(the)g Fn(ballistic)g(scaling)572 4965 y Fr(f)620 4980 y Fv(E)679 4965 y Fq(\()p Fr(e;)17 b(v)t Fq(\))28 b(:=)f Fr(E)1131 4924 y Fj(\000)p Fv(d=)p Fk(2)1297 4965 y Fr(f)11 b Fq(\()p Fr(e=E)6 b(;)17 b(v)t(=)1710 4876 y Fp(p)p 1792 4876 79 4 v 1792 4965 a Fr(E)6 b Fq(\))195 b(\()p Fr(E)34 b(>)28 b Fq(0\))1898 5214 y Fs(16)p eop %%Page: 17 17 17 16 bop 328 631 a Fs(of)34 b(a)g(test)h(function)g Fr(f)41 b Fp(2)31 b Fr(C)1335 595 y Fk(0)1328 656 y(0)1374 631 y Fq(\()p Fo(R)1478 595 y Fv(d)p Fk(+1)1615 631 y Fr(;)17 b Fo(R)t Fq(\))p Fs(,)41 b(so)35 b(that)f Fr(f)2200 646 y Fk(1)2270 631 y Fq(=)c Fr(f)11 b Fs(.)49 b(W)m(e)34 b(notice)h(that)g(fo)m(r)f Fr(V)52 b Fp(\021)31 b Fq(0)328 751 y Fs(w)m(e)i(have)g Fr(\027)6 b Fq(\()p Fr(E)g(;)17 b(v)t Fq(\))27 b(=)h Fr(C)h Fp(\001)22 b Fr(\016)t Fq(\()p Fr(E)28 b Fp(\000)1562 712 y Fk(1)p 1562 728 36 4 v 1562 786 a(2)1608 751 y Fr(v)1659 715 y Fk(2)1698 751 y Fq(\))k Fs(so)h(that)572 877 y Fh(Z)627 1103 y Fi(R)675 1084 y Fd(d)p Fb(+1)809 1013 y Fr(f)857 1028 y Fv(E)917 1013 y Fq(\()p Fr(x)p Fq(\))p Fr(d\027)6 b Fq(\()p Fr(x)p Fq(\))28 b Fp(\021)1417 877 y Fh(Z)1473 1103 y Fi(R)1521 1084 y Fd(d)p Fb(+1)1655 1013 y Fr(f)11 b Fq(\()p Fr(x)p Fq(\))p Fr(d\027)6 b Fq(\()p Fr(x)p Fq(\))195 b(\()p Fr(E)34 b(>)27 b Fq(0\))p Fr(:)328 1263 y Fs(The)32 b(result)i(is)328 1466 y Ff(Theo)m(rem)h(5.3)49 b Fn(F)m(o)m(r)39 b(all)e Fr(f)46 b Fp(2)37 b Fr(C)1533 1430 y Fk(0)1572 1466 y Fq(\()p Fo(R)1676 1430 y Fv(d)p Fk(+1)1812 1466 y Fr(;)17 b Fo(R)5 b Fq(\))43 b Fn(with)38 b(compact)g(supp)s(o)m(rt)g(in)g Fo(R)3116 1430 y Fk(+)3206 1466 y Fp(\002)26 b Fo(R)3375 1430 y Fv(d)3459 1466 y Fn(w)m(e)328 1587 y(have)572 1823 y Fq(lim)17 b(sup)649 1905 y Fi(~)p Fj(&)p Fk(0)887 1678 y Fh(\014)887 1738 y(\014)887 1798 y(\014)887 1858 y(\014)921 1687 y(Z)976 1913 y Fi(R)1024 1894 y Fd(d)p Fb(+1)1158 1823 y Fr(f)1206 1838 y Fv(E)1266 1823 y Fq(\()p Fr(x)p Fq(\))p Fr(d\027)1502 1782 y Fi(~)1544 1823 y Fq(\()p Fr(x)p Fq(\))23 b Fp(\000)1797 1687 y Fh(Z)1853 1913 y Fi(R)1901 1894 y Fd(d)p Fb(+1)2035 1823 y Fr(f)2083 1838 y Fv(E)2142 1823 y Fq(\()p Fr(x)p Fq(\))p Fr(d\027)6 b Fq(\()p Fr(x)p Fq(\))2509 1678 y Fh(\014)2509 1738 y(\014)2509 1798 y(\014)2509 1858 y(\014)2571 1823 y Fq(=)28 b Fp(O)s Fq(\(1)p Fr(=)2893 1733 y Fp(p)p 2975 1733 79 4 v 2975 1823 a Fr(E)6 b Fq(\))p Fr(:)274 b Fs(\(29\))328 2094 y Ff(Pro)s(of.)58 b Fs(By)37 b(our)g(assumption)h(on)f Fr(f)48 b Fs(there)38 b(is)g(an)f(interval)g Fr(I)45 b Fs(of)37 b(the)h(fo)m(rm)f(\(3\))g(so)h(that)328 2214 y Fr(I)30 b Fp(\002)23 b Fo(R)567 2178 y Fv(d)645 2214 y Fs(strictly)34 b(contains)f(the)g(supp)s(o)m(rt)g(of)f Fr(f)1977 2229 y Fv(E)2037 2214 y Fs(.)474 2334 y(The)h(index)f(set)i (of)e(eigenenergies)i(in)f Fr(I)40 b Fs(splits)34 b(into)e(the)h (disjoint)g(union)572 2538 y Fq(\004)637 2497 y Fi(~)637 2562 y Fv(I)679 2538 y Fq(\()p Fr(k)s Fq(\))28 b(=)f(\004)1005 2497 y Fi(~)1005 2562 y Fk(1)1048 2538 y Fq(\()p Fr(k)s Fq(\))22 b Fp([)h Fq(\004)1354 2497 y Fi(~)1354 2562 y Fk(2)1396 2538 y Fq(\()p Fr(k)s Fq(\))98 b Fs(with)g Fq(\004)1961 2497 y Fi(~)1961 2562 y Fk(1)2003 2538 y Fq(\()p Fr(k)s Fq(\))28 b(:=)f Fp(I)2345 2553 y Fv(k)2388 2538 y Fq(\()p Fp(F)-7 b Fq(\003)2559 2497 y Fi(~)2559 2562 y Fv(I)2601 2538 y Fq(\()p Fr(k)s Fq(\)\))p Fr(:)328 2741 y Fs(By)32 b(\(21\),)g(the)h(volume)f(estimate)i(\(22\))e(and)h (injectivit)m(y)f(of)g Fp(I)2528 2756 y Fv(k)572 2945 y Fq(\(2)p Fr(\031)t Fo(~)p Fq(\))810 2904 y Fv(d)866 2864 y Fh(\012)913 2945 y Fp(j)p Fq(\004)1006 2904 y Fi(~)1006 2969 y Fk(2)1048 2945 y Fp(j)1076 2864 y Fh(\013)1151 2945 y Fq(=)27 b Fp(O)s Fq(\()p Fr(E)1452 2904 y Fk(\()p Fv(d)p Fj(\000)p Fk(1\))p Fv(=)p Fk(2)1709 2945 y Fq(\))22 b(+)g Fp(O)1946 2960 y Fv(E)2006 2945 y Fq(\()p Fo(~)2098 2904 y Fv(\014)2144 2945 y Fq(\))p Fr(;)328 3148 y Fs(so)33 b(that)572 3375 y Fq(\(2)p Fr(\031)t Fo(~)p Fq(\))810 3334 y Fv(d)866 3239 y Fh(Z)1045 3280 y(X)982 3504 y Fv(n)p Fj(2)p Fk(\004)1119 3481 y Fc(~)1119 3525 y Fb(2)1158 3504 y Fk(\()p Fv(k)r Fk(\))1268 3375 y Fr(f)1316 3390 y Fv(E)1376 3375 y Fq(\()p Fr(E)1492 3334 y Fi(~)1486 3400 y Fv(n)1534 3375 y Fq(\()p Fr(k)s Fq(\))p Fr(;)20 b Fq(\026)-52 b Fr(v)1759 3334 y Fi(~)1755 3400 y Fv(n)1802 3375 y Fq(\()p Fr(k)s Fq(\)\))p Fr(dk)30 b Fq(=)e Fp(O)s Fq(\(1)p Fr(=)2424 3286 y Fp(p)p 2506 3286 V 2506 3375 a Fr(E)6 b Fq(\))22 b(+)g Fp(O)2821 3390 y Fv(E)2881 3375 y Fq(\()p Fo(~)2973 3334 y Fv(\014)3020 3375 y Fq(\))p Fr(:)328 3718 y Fs(This)38 b(leads)h(to)g(a)f(contribution)h(of)f(o)m (rder)h Fp(O)s Fq(\(1)p Fr(=)2098 3634 y Fp(p)p 2181 3634 V 84 x Fr(E)6 b Fq(\))38 b Fs(to)h(\(29\),)g(so)g(that)g(w)m(e)g (need)g(only)328 3839 y(estimate)34 b(the)f(contribution)g(of)f Fq(\004)1562 3803 y Fi(~)1562 3863 y Fk(1)1605 3839 y Fs(.)43 b(By)32 b(\(8\))386 4059 y Fq(lim)382 4121 y Fi(~)p Fj(&)p Fk(0)526 4059 y Fq(\(2)p Fr(\031)t Fo(~)p Fq(\))764 4018 y Fv(d)820 3978 y Fh(\012)868 4059 y Fp(j)p Fq(\003)964 4018 y Fi(~)964 4083 y Fv(I)1027 4059 y Fp(n)22 b Fq(\003)1167 4018 y Fi(~)1167 4083 y Fk(1)1210 4059 y Fp(j)1238 3978 y Fh(\013)1312 4059 y Fq(=)28 b(0)97 b Fs(fo)m(r)g Fq(\003)1836 4018 y Fi(~)1836 4083 y Fk(1)1879 4059 y Fq(\()p Fr(k)s Fq(\))27 b(:=)h Fp(f)p Fr(`)2258 4018 y Fj(\003)2325 4059 y Fp(2)g Fq(\003)2487 4018 y Fi(~)2487 4083 y Fv(I)2529 4059 y Fq(\()p Fr(k)s Fq(\))g Fp(j)f Fo(~)p Fq(\()p Fr(`)2875 4018 y Fj(\003)2936 4059 y Fq(+)22 b Fr(k)s Fq(\))28 b Fp(2)g(J)3334 4018 y Fj(1)3316 4083 y Fv(\015)t(;E)3435 4059 y Fp(g)p Fr(:)328 4321 y Fs(So)k(it)h(su\016ces)h(to)f(consider)g(the)h(contribution)f(of)f (the)h(index)f(set)572 4525 y Fq(\004)637 4484 y Fi(~)637 4550 y Fk(1)p Fv(;)p Fk(1)731 4525 y Fq(\()p Fr(k)s Fq(\))c(:=)f Fp(I)1073 4540 y Fv(k)1116 4525 y Fq(\()p Fp(F)-7 b Fq(\003)1287 4484 y Fi(~)1287 4550 y Fv(I)1329 4525 y Fq(\()p Fr(k)s Fq(\))22 b Fp(\\)h Fq(\003)1638 4484 y Fi(~)1638 4550 y Fk(1)1680 4525 y Fq(\()p Fr(k)s Fq(\)\))28 b Fp(\032)g Fq(\004)2046 4484 y Fi(~)2046 4550 y Fk(1)2088 4525 y Fq(\()p Fr(k)s Fq(\))p Fr(:)328 4728 y Fs(The)k(result)i(\(29\))e(then) h(follo)m(ws)g(from)f(the)h(estimate)576 4932 y Fq(\026)-53 b Fr(v)623 4891 y Fi(~)619 4956 y Fv(n)666 4932 y Fq(\()p Fr(k)s Fq(\))28 b(=)f Fr(@)978 4947 y Fv(P)1037 4932 y Fr(K)7 b Fq(\()p Fo(~)p Fq(\()p Fr(`)1298 4891 y Fj(\003)1360 4932 y Fq(+)22 b Fr(k)s Fq(\)\))g(+)g Fp(O)s Fq(\()p Fo(~)p Fq(\))p Fr(:)1445 b Fs(\(30\))1898 5214 y(17)p eop %%Page: 18 18 18 17 bop 328 631 a Fs(fo)m(r)32 b Fr(`)510 595 y Fj(\003)578 631 y Fp(2)c(F)-7 b Fq(\003)805 595 y Fi(~)805 656 y Fv(I)847 631 y Fq(\()p Fr(k)s Fq(\))22 b Fp(\\)g Fq(\003)1155 595 y Fi(~)1155 656 y Fk(1)1197 631 y Fq(\()p Fr(k)s Fq(\))33 b Fs(and)f Fr(n)c Fq(:=)g Fp(I)1810 646 y Fv(k)1853 631 y Fq(\()p Fr(`)1932 595 y Fj(\003)1971 631 y Fq(\))k Fs(and)h(the)g(identit)m(y)576 815 y Fq(\026)-53 b Fr(v)t Fq(\()p Fr(x)p Fq(\))28 b(=)f Fr(@)936 830 y Fv(P)995 815 y Fr(K)7 b Fq(\()p Fr(P)14 b Fq(\))195 b(\()p Fr(P)41 b Fp(2)28 b(J)1755 773 y Fj(1)1830 815 y Fr(;)17 b(x)27 b Fp(2)h Fr(T)14 b Fq(\()p Fp(f)p Fr(P)g Fp(g)21 b(\002)i Fo(T)p Fq(\)\))328 998 y Fs(fo)m(r)32 b(the)i(group)e(velo)s(cit)m(y)g (on)h(the)g(KAM)g(to)m(ri)f(which)i(w)m(e)f(b)s(oth)f(p)m(rove)h(no)m (w.)474 1162 y(By)26 b(de\014nition)i(\(18\))e(of)g Fp(F)-7 b Fq(\003)1451 1126 y Fi(~)1451 1187 y Fv(I)1493 1162 y Fq(\()p Fr(k)s Fq(\))p Fs(,)28 b(the)f(eigenvalue)g Fr(E)2354 1126 y Fi(~)2348 1186 y Fv(n)2397 1162 y Fq(\()p Fr(k)s Fq(\))f Fs(is)h(non-degenerate)h(so)f(that)576 1345 y Fq(\026)-53 b Fr(v)623 1304 y Fi(~)619 1370 y Fv(n)666 1345 y Fq(\()p Fr(k)s Fq(\))28 b(=)927 1265 y Fh(\012)974 1345 y Fr( )1041 1304 y Fi(~)1037 1370 y Fv(n)1085 1345 y Fq(\()p Fr(k)s Fq(\))o Fr(;)21 b Fq(\026)-53 b Fr(v)1309 1304 y Fi(~)1351 1345 y Fq(\()p Fr(k)s Fq(\))p Fr( )1548 1304 y Fi(~)1544 1370 y Fv(n)1592 1345 y Fq(\()p Fr(k)s Fq(\))1722 1265 y Fh(\013)3392 1345 y Fs(\(31\))474 1529 y(F)m(o)m(r)29 b Fr(\036)e Fs(in)g(the)h(\()p Fr(k)s Fs({inva)m(riant\))f(domain)g(of)g Fr(H)2049 1493 y Fi(~)2091 1529 y Fq(\()p Fr(k)s Fq(\))g Fs(and)h Fr(E)33 b Fp(2)28 b Fo(R)39 b Fs(w)m(e)28 b(have)f(the)h(estimate)582 1690 y Fq(1)p 582 1734 49 4 v 582 1825 a(2)641 1757 y Fp(k)t Fq(\026)-53 b Fr(v)742 1716 y Fi(~)783 1757 y Fq(\()p Fr(k)s Fq(\))p Fr(\036)p Fp(k)1021 1716 y Fk(2)1088 1757 y Fp(\024)28 b(k)p Fq(\()p Fr(H)1370 1716 y Fi(~)1412 1757 y Fq(\()p Fr(k)s Fq(\))22 b Fp(\000)h Fr(E)6 b Fq(\))p Fr(\036)p Fp(kk)p Fr(\036)p Fp(k)21 b Fq(+)h Fp(k)p Fr(V)43 b Fp(\000)23 b Fr(E)6 b Fp(kk)p Fr(\036)p Fp(k)2701 1716 y Fk(2)2935 1757 y Fq(\()p Fr(k)30 b Fp(2)e Fo(T)3211 1716 y Fj(\003)3254 1757 y Fq(\))p Fr(:)474 1972 y Fs(It)33 b(follo)m(ws)f(from)h(Theo)m(rem)g(\(4.1\))f(that)572 2155 y Fp(k)t Fq(\026)-53 b Fr(v)673 2114 y Fi(~)715 2155 y Fq(\()p Fr(k)s Fq(\)\()p Fr( )950 2114 y Fi(~)946 2180 y Fj(I)985 2192 y Fd(k)1023 2180 y Fk(\()p Fv(`)1079 2161 y Fg(\003)1115 2180 y Fk(\))1147 2155 y Fq(\()p Fr(k)s Fq(\))22 b Fp(\000)1419 2129 y Fq(~)1399 2155 y Fr( )1466 2114 y Fi(~)1462 2180 y Fv(`)1491 2161 y Fg(\003)1531 2155 y Fq(\()p Fr(k)s Fq(\)\))p Fp(k)27 b Fq(=)h Fp(O)1979 2074 y Fh(\000)2024 2155 y Fo(~)2078 2114 y Fv(N)2145 2074 y Fh(\001)328 2339 y Fs(which)33 b(implies)g(fo)m(r)g(the)g(exp)s(ectation)339 2544 y Fq(\026)-52 b Fr(v)387 2503 y Fi(~)383 2569 y Fv(n)430 2544 y Fq(\()p Fr(k)s Fq(\))27 b(=)691 2463 y Fh(\012)738 2544 y Fr( )805 2503 y Fi(~)801 2569 y Fv(n)848 2544 y Fq(\()p Fr(k)s Fq(\))p Fr(;)17 b Fq(\()p Fr(D)25 b Fq(+)d Fo(~)p Fr(k)s Fq(\))p Fr( )1477 2503 y Fi(~)1473 2569 y Fv(n)1520 2544 y Fq(\()p Fr(k)s Fq(\))1650 2463 y Fh(\013)1725 2544 y Fq(=)1828 2434 y Fh(D)1909 2518 y Fq(~)1889 2544 y Fr( )1956 2503 y Fi(~)1952 2569 y Fv(`)1981 2550 y Fg(\003)2021 2544 y Fq(\()p Fr(k)s Fq(\))p Fr(;)17 b Fq(\()p Fr(D)25 b Fq(+)d Fo(~)p Fr(k)s Fq(\))2603 2518 y(~)2583 2544 y Fr( )2650 2503 y Fi(~)2646 2569 y Fv(`)2675 2550 y Fg(\003)2715 2544 y Fq(\()p Fr(k)s Fq(\))2845 2434 y Fh(E)2928 2544 y Fq(+)g Fp(O)3125 2463 y Fh(\000)3170 2544 y Fo(~)3224 2503 y Fv(N)3292 2463 y Fh(\001)3337 2544 y Fr(:)28 b Fs(\(32\))328 2750 y(By)k(construction) i(of)e(the)h(quasimo)s(des)572 2866 y Fh(D)653 2950 y Fq(~)633 2976 y Fr( )700 2935 y Fi(~)696 3001 y Fv(`)725 2982 y Fg(\003)765 2976 y Fq(\()p Fr(k)s Fq(\))p Fr(;)17 b Fq(\()p Fr(D)24 b Fq(+)f Fo(~)p Fr(k)s Fq(\))1346 2950 y(~)1327 2976 y Fr( )1394 2935 y Fi(~)1390 3001 y Fv(`)1419 2982 y Fg(\003)1459 2976 y Fq(\()p Fr(k)s Fq(\))1589 2866 y Fh(E)1677 2976 y Fq(=)1781 2841 y Fh(Z)1836 3066 y Fi(T)1901 2976 y Fr(@)1952 2991 y Fv(q)2007 2951 y Fq(~)1991 2976 y Fr(S)5 b Fq(\()p Fr(P)s(;)17 b(q)t Fq(\))32 b Fr(d\026)2431 2991 y Fv(P)2489 2976 y Fq(\()p Fr(q)t Fq(\))22 b(+)g Fp(O)e Fq(\()p Fo(~)t Fq(\))427 b Fs(\(33\))328 3229 y(fo)m(r)32 b Fr(P)42 b Fq(:=)27 b Fo(~)p Fq(\()p Fr(`)837 3193 y Fj(\003)898 3229 y Fq(+)c Fr(k)s Fq(\))p Fr(;)1149 3204 y Fq(~)1133 3229 y Fr(S)38 b Fs(as)32 b(de\014ned)i(in)e(\(14\),)g(and)572 3494 y Fr(d\026)682 3509 y Fv(P)740 3494 y Fq(\()p Fr(q)t Fq(\))27 b(:=)1121 3417 y Fr(@)1177 3380 y Fk(2)1172 3442 y Fv(q)r(P)1282 3391 y Fq(~)1265 3417 y Fr(S)6 b Fq(\()p Fr(P)s(;)17 b(q)t Fq(\))32 b Fr(dq)p 1031 3471 753 4 v 1031 3494 a Fh(R)1078 3608 y Fi(T)1143 3574 y Fr(@)1199 3540 y Fk(2)1194 3602 y Fv(q)1228 3583 y Fg(0)1251 3602 y Fv(P)1326 3549 y Fq(~)1309 3574 y Fr(S)6 b Fq(\()p Fr(P)s(;)17 b(q)1570 3545 y Fj(0)1593 3574 y Fq(\))32 b Fr(dq)1761 3545 y Fj(0)1794 3494 y Fr(:)328 3756 y Fs(Finally)e(from)g(the)i (Hamilton{Jacobi)e(equation,)h(since)g(the)g(classical)h(\015o)m(w)f (is)f(ergo)s(dic)h(on)328 3876 y(the)40 b(inva)m(riant)f(to)m(rus)i (indexed)f(b)m(y)g Fr(P)14 b Fs(,)40 b(and)g(since)g Fr(d\026)2280 3891 y Fv(P)2378 3876 y Fs(is)g(the)g(inva)m(riant)g (measure)g(in)g Fr(q)328 3996 y Fs(co)s(o)m(rdinates,)33 b(it)g(holds)542 4216 y Fq(\026)-52 b Fr(v)s Fq(\()p Fr(x)p Fq(\))28 b(=)852 4080 y Fh(Z)907 4306 y Fi(T)972 4216 y Fr(@)1023 4231 y Fv(q)1078 4191 y Fq(~)1062 4216 y Fr(S)6 b Fq(\()p Fr(P)s(;)17 b(q)t Fq(\))31 b Fr(d\026)1502 4231 y Fv(P)1560 4216 y Fq(\()p Fr(q)t Fq(\))d(=)f Fr(@)1865 4231 y Fv(P)1924 4216 y Fr(K)7 b Fq(\()p Fr(P)14 b Fq(\))195 b(\()p Fr(P)41 b Fp(2)28 b(J)2684 4175 y Fj(1)2759 4216 y Fr(;)17 b(x)27 b Fp(2)h Fr(T)14 b Fq(\()p Fp(f)p Fr(P)g Fp(g)21 b(\002)i Fo(T)p Fq(\)\))p Fr(:)328 4452 y Fs(Thus)32 b(\(30\))g(follo)m(ws)h(from)g(\(31\),)f(\(32\))g(and)g(\(33\).)1373 b Fa(2)328 4636 y Ff(Rema)m(rk)36 b(5.4)49 b Fq(Actually)24 b(w)m(e)i(ha)m(v)m(e)g(pro)m(v)m(en)h(in)d(addition)f(to)i(Theorem)g (3.1)g(that)3323 4610 y(~)3303 4636 y Fr( )3370 4600 y Fi(~)3366 4662 y Fv(`)3395 4643 y Fg(\003)3436 4636 y Fq(\()p Fr(k)s Fq(\))328 4756 y(lead)32 b(to)g(join)m(t)g(quasimo)s (des)g(of)g Fr(H)1602 4720 y Fi(~)1644 4756 y Fq(\()p Fr(k)s Fq(\))p Fr(;)20 b Fq(\026)-52 b Fr(v)1869 4720 y Fi(~)1911 4756 y Fq(\()p Fr(k)s Fq(\),)33 b(namely:)572 4940 y Fp(k)p Fq(\()t(\026)-53 b Fr(v)711 4899 y Fi(~)775 4940 y Fp(\000)22 b Fr(@)925 4955 y Fv(P)985 4940 y Fr(K)7 b Fq(\()p Fo(~)p Fq(\()p Fr(`)1246 4899 y Fj(\003)1307 4940 y Fq(+)22 b Fr(k)s Fq(\)\)\))1593 4914 y(~)1573 4940 y Fr( )1640 4899 y Fi(~)1636 4964 y Fv(`)1665 4946 y Fg(\003)1705 4940 y Fq(\()p Fr(k)s Fq(\))p Fp(k)28 b Fq(=)f Fp(O)20 b Fq(\()p Fo(~)t Fq(\))195 b(\()p Fr(k)31 b Fp(2)d Fo(T)2721 4899 y Fj(\003)2764 4940 y Fr(;)17 b(`)2849 4899 y Fj(\003)2915 4940 y Fp(2)28 b(F)-7 b Fq(\003)3142 4899 y Fi(~)3142 4964 y Fv(I)3184 4940 y Fq(\()p Fr(k)s Fq(\)\))p Fr(:)1898 5214 y Fs(18)p eop %%Page: 19 19 19 18 bop 328 631 a Ft(6)158 b(Bey)l(ond)53 b(KAM)328 853 y Fs(Theo)m(rem)31 b(5.3)e(gives)h(a)g(pa)m(rtial)g(answ)m(er)i(to) e(Conjecture)h(5.1,)f(based)h(on)f(the)g(KAM)h(region)328 974 y Fp(K)404 989 y Fv(I)472 974 y Fp(\032)f(P)648 989 y Fv(I)688 974 y Fs(.)45 b(But)33 b(what)g(happ)s(ens)h(in)f(the)h (complement)g Fp(K)2407 937 y Fv(c)2406 999 y(I)2463 974 y Fs(?)45 b(There)33 b(the)h(classical)g(dyna-)328 1094 y(mics)d(is)f(very)g(complicated)h(in)f(general,)g(since)i(one)e (ma)m(y)f(encounter)j(there)f(further)g(KAM)328 1214 y(to)m(ri)42 b(\(not)h(p)m(redicted)h(b)m(y)e(the)h(estimates\),)j (Canto)m(ri,)f(elliptic)e(and)f(hyp)s(erb)s(olic)g(p)s(erio)s(dic)328 1335 y(o)m(rbits,)33 b(la)m(rge)g(ergo)s(dic)g(comp)s(onents)g(etc.)474 1455 y(With)23 b(the)h(exception)g(of)e(the)i(elliptic)g(o)m(rbits,)i (there)e(is)f(no)g(direct)h(generalization)g(of)e(the)328 1575 y(ab)s(ove)35 b(KAM)i(metho)s(ds,)h(and)e(thus)h(it)g(seems)h(hop) s(eless)f(to)f(control)h(the)g(w)m(avefunctions)328 1696 y(semiclassically)50 b(supp)s(o)m(rted)f(in)g(that)g(region.)91 b(Ho)m(w)m(ever,)55 b(as)48 b(the)h(follo)m(wing)g(example)328 1816 y(sho)m(ws,)34 b(other)f(metho)s(ds)g(ma)m(y)f(w)m(o)m(rk.)328 1984 y Ff(Example.)59 b Fs(Consider)39 b Fr(d)e Fq(=)f(2)i Fs(dimensions.)60 b(As)39 b(sho)m(wn)g(in)f([2],)h(in)f(the)g(p)m (resence)i(of)e(at)328 2104 y(least)33 b(t)m(w)m(o)f(geometrically)h (distinct)g(KAM)f(to)m(ri)h(the)f(motion)g(on)g Fq(\006)2711 2119 y Fv(E)2803 2104 y Fs(is)h(ballistic)f(\()t Fq(\026)-53 b Fr(v)31 b Fp(6)p Fq(=)d(0)p Fs(\).)328 2225 y(This)33 b(is)g(caused)h(b)m(y)f(the)g(fact)g(that)g(these)h(to)m(ri)g(have)e (co)s(dimension)i(one)f(in)g Fq(\006)3121 2240 y Fv(E)3213 2225 y Fs(and)g(thus)328 2345 y(con\014ne)h(the)f(\015o)m(w)h(b)s(et)m (w)m(een)h(them.)46 b(W)m(e)33 b(denote)h(b)m(y)f Fp(R)2326 2360 y Fv(I)2394 2345 y Fp(\032)c(P)2569 2360 y Fv(I)2643 2345 y Fs(the)k(phase)h(space)g(region)328 2465 y(enclosed)g(b)m(y)f(t) m(w)m(o)h(nea)m(rb)m(y)f(KAM)h(to)m(ri)f(\(o)m(r)g(rather)h(families)f (of)g(such)g(to)m(ri)h(indexed)f(b)m(y)g(the)328 2586 y(energy)g(in)f Fr(I)8 b Fs(\).)474 2706 y(Using)25 b(microlo)s(cal)g (techniques,)j(Shnirelman)d(sho)m(w)m(ed)h(in)e([8])h(the)g(existence)h (of)e(a)h(la)m(rge)328 2827 y(numb)s(er)39 b(of)e(quasimo)s(des)i (concentrated)h(in)e Fp(R)2050 2842 y Fv(I)2090 2827 y Fs(,)i(see)f(also)f([3].)61 b(No)m(w)39 b(fo)m(r)f(la)m(rge)g(energy) 328 2947 y Fr(E)48 b Fs(the)42 b(va)m(riation)g(of)j Fq(\026)-53 b Fr(v)46 b Fs(w.r.t.)d(the)f(restriction)i(of)d(Liouville) g(measure)i(to)f Fp(R)3174 2962 y Fv(I)3256 2947 y Fs(is)h(small)328 3067 y(in)36 b(compa)m(rison)h(with)f Fr(E)6 b Fs(.)54 b(Thus)36 b(b)m(y)g(Ego)m(rov's)g(Theo)m(rem)h(the)f(ab)s(ove)g (quasimo)s(des)g(have)328 3188 y(group)c(velo)s(cities)h(nea)m(r)g(the) h(classical)j Fq(\026)-53 b Fr(v)36 b Fs(values)d(\(see)g(also)g([2],)f (Sect.)i(5\).)474 3308 y(Di\013erent)g(such)f(regions)g Fp(R)1476 3323 y Fv(I)1516 3308 y Fs(,)g(ho)m(w)m(ever,)g(have)f (di\013erent)i(classical)f(asymptotic)g(velo-)328 3428 y(cities.)57 b(Thus)36 b(one)h(should)g(b)s(e)f(able)h(to)f(apply)g (the)h(heuristics)i(develop)s(ed)d(in)h(Sect.)h(2)e(to)328 3549 y(that)d(case,)g(to)s(o)f(|)g Fn(without)h(explicitly)g(kno)m (wing)g(the)g(quasimo)s(des)p Fs(.)328 3716 y(When)41 b(trying)f(to)h(w)m(o)m(rk)h(on)e(this)i(kind)f(of)f(a)m(rguments,)k (one)d(is)g(led)g(to)g(the)g(pa)m(rado)m(xical)328 3837 y(conclusion)f(that)f(sometimes)i(it)e(is)h(mo)m(re)g(useful)g(to)f (kno)m(w)h(quasimo)s(des)f(\(with)h(certain)328 3957 y(additional)32 b(p)m(rop)s(erties\))i(of)e(an)g(op)s(erato)m(r)h(than) f(to)h(kno)m(w)g(its)g(eigenfunctions.)474 4078 y(T)-8 b(o)32 b(explain)g(this,)h(consider)h(the)f(algeb)m(ra)g(generated)g(b) m(y)572 4281 y Fp(f)p Fr(H)711 4240 y Fi(~)753 4281 y Fq(\()p Fr(k)s Fq(\))p Fr(;)20 b Fq(\026)-52 b Fr(v)978 4240 y Fi(~)974 4306 y Fk(1)1019 4281 y Fq(\()p Fr(k)s Fq(\))p Fr(;)17 b(:)g(:)g(:)f(;)21 b Fq(\026)-53 b Fr(v)1419 4240 y Fi(~)1415 4306 y Fv(d)1461 4281 y Fq(\()p Fr(k)s Fq(\))p Fp(g)p Fr(;)332 4484 y Fq(\026)g Fr(v)379 4448 y Fi(~)375 4509 y Fv(i)445 4484 y Fs(b)s(eing)23 b(the)h(comp)s(onents) h(of)e(the)h(op)s(erato)m(r)g(of)f(asymptotic)h(velo)s(cit)m(y)f({)h (which)g(commute)328 4605 y(with)32 b Fr(H)623 4569 y Fi(~)665 4605 y Fq(\()p Fr(k)s Fq(\))g Fs({)g(and)g(try)g(to)g(sho)m(w) h(the)g(existence)g(of)f Fn(joint)g Fs(quasimo)s(des.)43 b(Arguing)33 b(along)328 4725 y(the)c(lines)f(of)g(Sect.)h(2,)g(such)g (an)f(existence)h(p)m(ro)s(of)f(could)g(su\016ce)i(to)e(p)m(rove)g (Conjecture)h(5.1)328 4846 y(in)j(full)h(generalit)m(y)-8 b(.)1898 5214 y(19)p eop %%Page: 20 20 20 19 bop 328 631 a Ft(References)377 853 y Fs([1])49 b(Arnold,)27 b(V.I.:)40 b Fn(Mo)s(des)26 b(and)f(quasi-mo)s(des.)h Fs(F)m(unct.)h(Anal.)f(and)f(its)h(Appl.)g Ff(5)p Fs(,)h(94{101,)531 974 y(\(1972\))377 1172 y([2])49 b(Asch,)39 b(J.,)f(Knauf,)g(A.:)53 b Fn(Motion)36 b(in)h(P)m(erio)s(dic)g(P)m(otentials.)h Fs(Nonlinea)m(rit)m(y)g Ff(11)p Fs(,)g(175{)531 1293 y(200)31 b(\(1998\))377 1491 y([3])49 b(Colin)41 b(de)g(V)m(erdi)m (\022)-46 b(ere,)45 b(Y.:)61 b(KAM)42 b(et)f(limites)h (semi-classiques.)h(Dijon,)g(17-19)c(sep-)531 1612 y(temb)m(re)34 b(1997,)d(W)m(o)m(rkshop)j(on)e(KAM)h(in)f(Quantum)h(Mechanics.)377 1810 y([4])49 b(Dob)m(rokhotov,)31 b(S.,)h(Shafa)m(revich,)g(A.:)43 b Fn(`Momentum')32 b(tunneling)f(b)s(et)m(w)m(een)i(to)m(ri)f(and)531 1931 y(the)43 b(splitting)h(of)f(eigenvalues)g(of)g(the)h (Beltrami-Laplace)e(op)s(erato)m(r)i(on)f(Liouville)531 2051 y(surfaces.)33 b Fs(Institute)h(fo)m(r)e(Problems)i(in)e (Mechanics)i(Prep)m(rint)f Ff(599)f Fs(\(1997\))377 2250 y([5])49 b(Knauf,)37 b(A.:)51 b Fn(Coulombic)36 b(P)m(erio)s(dic)h(P)m (otentials:)52 b(The)36 b(Quantum)g(Case.)h Fs(Annals)g(of)531 2370 y(Physics)c Ff(191)p Fs(,)f(205{240)f(\(1989\))377 2569 y([6])49 b(Lazutkin,)26 b(V.)e(F.:)40 b Fn(KAM)24 b(theo)m(ry)g(and)g(semiclassical)i(app)m(ro)m(ximations)e(to)g (eigenfunc-)531 2689 y(tions.)37 b Fs(Ergebnisse)h(der)f(Mathematik)g (und)g(ihrer)g(Grenzgebiete,)j(V)m(ol.)d Ff(24)p Fs(.)g(Berlin;)531 2810 y(New)c(Y)-8 b(o)m(rk:)45 b(Sp)m(ringer)33 b(1993.)377 3008 y([7])49 b(P\177)-49 b(oschel,)32 b(J.:)43 b Fn(Integrabilit)m(y) 31 b(of)f(Hamiltonian)h(systems)i(on)e(Canto)m(r)h(sets.)g Fs(Commun.)531 3129 y(Pure)h(Appl.)g(Math.)f Ff(35)p Fs(,)g(653{696)f(\(1982\))377 3327 y([8])49 b(Shnirelman,)44 b(A.I.:)61 b Fn(On)41 b(the)h(Asymptotic)g(Prop)s(erties)g(of)f (Eigenfunctions)h(in)f(the)531 3448 y(Region)32 b(of)g(Chaotic)h (Motion.)f Fs(Addendum)i(to)e([6])377 3646 y([9])49 b(Reed,)35 b(M.,)g(Simon,)g(B.:)48 b Fn(Metho)s(ds)35 b(in)f(Mathematical)h (Physics,)h(V)m(ol.)f(IV:)f(Analysis)531 3767 y(of)e(Op)s(erato)m(rs.)h Fs(New)h(Y)-8 b(o)m(rk:)44 b(Academic)34 b(Press)g(1978)328 3965 y([10])49 b(Thomas,)36 b(L.E.:)51 b Fn(Time)36 b(dep)s(endent)i (app)m(roach)e(to)g(Scattering)h(from)f(impurities)i(in)531 4086 y(a)32 b(Crystal.)h Fs(Commun.)g(Math.)f(Phys.)h Ff(33)p Fs(,)f(335{343)f(\(1973\))328 4285 y([11])49 b(Thomas,)36 b(L.E.,)h(W)m(assell,)g(S.R.:)50 b Fn(Semiclassical)37 b(App)m(ro)m(ximation)f(fo)m(r)g(Schr\177)-49 b(odinger)531 4405 y(Op)s(erato)m(rs)47 b(at)g(High)g(Energy)-8 b(.)47 b Fs(In:)72 b Fn(Schr\177)-49 b(odinger)47 b(Op)s(erato)m(rs)p Fs(,)52 b(Lecture)c(Notes)g(in)531 4525 y(Physics)33 b Ff(403)p Fs(,)f(ed.)h(E.)g(Balslev.)f(New)i(Y)-8 b(o)m(rk:)44 b(Sp)m(ringer)33 b(1992)328 4724 y([12])49 b(W)m(eiss,)37 b(D.,)f(L)q(\177)-49 b(utjering,)36 b(G.,)g(Richter,)h(K.:)50 b Fn(Chaotic)36 b(Electron)g(Motion)f(in)g(Macros-)531 4844 y(copic)c(and)f(Mesoscopic)i(Antidot)f(Lattices.)h Fs(Chaos,)f(Solitons)g(&)g(F)m(ractals)h Ff(8)p Fs(,)f(1337)531 4965 y(\(1997\))1898 5214 y(20)p eop %%Page: 21 21 21 20 bop 328 631 a Fs([13])49 b(Wilco)m(x,)35 b(C.H.:)50 b Fn(Theo)m(ry)35 b(of)f(Blo)s(ch)h(w)m(aves.)h Fs(Journ.)f(d')h(Anal.) f(Math.)g Ff(33)p Fs(,)g(146{167)531 751 y(\(1978\))1898 5214 y(21)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9812041112978--