Content-Type: multipart/mixed; boundary="-------------0203250657127" This is a multi-part message in MIME format. ---------------0203250657127 Content-Type: text/plain; name="02-146.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-146.keywords" shrodinger equation, Ehrenfest time, coherent states, semiclassical limit ---------------0203250657127 Content-Type: application/postscript; name="DeRo200302.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="DeRo200302.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: DeRo200302.dvi %%Pages: 25 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Times-Roman Times-Italic Times-Bold %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o DeRo200302.ps DeRo200302.dvi %DVIPSParameters: dpi=1200, compressed %DVIPSSource: TeX output 2002.03.22:1712 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict 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Fo(\))p Fp(;)1125 9495 y Fm(\017)83 b Fp(If)38 b Fn(t)46 b Fm(\031)h Fn(T)23 b Fo(\(\026)-83 b Fn(h)o Fo(\))37 b Fp(then)g Fn(\026)2526 9520 y Fj(\026)-68 b Fk(h;t)2746 9495 y Fm(!)47 b Fn(g)6 b Fo(\()p Fn(X)13 b Fo(\))p Fn(d)p Fo(\006)3531 9520 y Fj(0)3604 9495 y Fo(\()p Fn(X)g Fo(\))p Fp(,)37 b(where)g Fo(\006)4527 9520 y Fj(0)4639 9495 y Fp(is)g(the)h(0-ener)m (gy)f(curv)n(e)g(\(Fig.)51 b(2\),)2617 9858 y Fo(\006)2737 9883 y Fj(0)2858 9858 y Fo(=)46 b Fm(f)p Fn(X)59 b Fo(=)46 b(\()p Fn(q)6 b(;)28 b(p)p Fo(\))p Fn(;)119 b(p)4105 9790 y Fj(2)4179 9858 y Fn(=)p Fo(2)37 b(+)g Fn(V)g Fo(\()p Fn(q)6 b Fo(\))45 b(=)h(0)p Fm(g)1291 10222 y Fp(and)i Fn(g)55 b Fp(is)48 b(a)g(smooth)g(positi)l(v)n(e)g(density)g(supported) g(on)h(the)f(unstable)g(part)g(of)h Fo(\006)6011 10247 y Fj(0)6134 10222 y Fp(joining)1291 10421 y Fo(\()p Fn(q)1430 10446 y Fl(\000)1542 10421 y Fn(;)28 b Fo(0\))41 b Fp(to)g Fo(\()p Fn(q)2114 10446 y Fj(+)2224 10421 y Fn(;)28 b Fo(0\))p Fp(.)3699 10919 y(3)p eop %%Page: 4 4 4 3 bop 1125 1455 a Fp(Here)44 b(and)h(in)f(what)h(follo)l(ws,)g Fn(t)51 b(<<)h(T)23 b Fo(\(\026)-83 b Fn(h)p Fo(\))44 b Fp(means)g(that,)h(for)g(all)f Fn(\017)53 b(>)f Fo(0)p Fp(,)45 b Fn(t)51 b Fm(\024)h Fo(\(1)39 b Fm(\000)g Fn(\017)p Fo(\))p Fn(T)23 b Fo(\(\026)-83 b Fn(h)p Fo(\))p Fp(;)876 1654 y(the)52 b(con)-7 b(v)n(er)m(gence)51 b(as)h Fo(\026)-83 b Fn(h)65 b Fm(!)h Fo(0)53 b Fp(is)e(of)i(course)f(not)g(uniform)h (with)g(respect)e(to)h Fn(\017)p Fp(.)84 b(It)53 b(is)f(clear)f(that) 876 1853 y(such)46 b(results)g(can)f(only)i(be)f(obtained)g(by)h (controlling)g(the)f(error)h(term)g(in)f(the)h(Egoro)n(v)f(theorem)876 2052 y(for)g(long)g(enough)g(times.)64 b(In)46 b(particular)-7 b(,)47 b(to)f(observ)n(e)f(the)h(transition)g(in)g(the)g(beha)m(viour)f (of)i Fn(\026)6431 2077 y Fj(\026)-68 b Fk(h;t)876 2252 y Fp(at)45 b Fn(T)23 b Fo(\(\026)-83 b Fn(h)o Fo(\))p Fp(,)45 b(one)f(needs)g(such)g(control)h(up)g(to)g(times)f Fn(\014)9 b Fm(j)28 b Fo(log)i(\026)-83 b Fn(h)p Fm(j)p Fp(,)45 b(for)g Fn(\014)61 b(>)5149 2186 y Fj(1)p 5075 2213 215 7 v 5075 2309 a(2)p Fk(!)5225 2326 y Fh(0)5310 2252 y Fp(.)f(W)-13 b(e)44 b(shall)h(combine)876 2451 y(techniques)62 b(de)l(v)n(eloped)f(in)i([BR)o(])g(with)g(precise)e (estimates)h(on)g(the)h(\003o)l(w)f(of)h Fn(H)76 b Fp(close)62 b(to)h(the)876 2650 y(zero-ener)m(gy)41 b(curv)n(e)g(to)h(control)g (the)f(error)h(for)g(times)g(up)f(to)4515 2585 y Fj(2)p 4441 2612 V 4441 2708 a(3)p Fk(!)4591 2725 y Fh(0)4675 2650 y Fm(j)28 b Fo(log)i(\026)-83 b Fn(h)p Fm(j)42 b Fp(\(Proposition)g(3.2\).)1125 2850 y(Intuiti)l(v)n(ely)-11 b(,)60 b(it)c(is)g(not)h(too)f(dif)l(\002cult)h(to)g(con)-7 b(vince)55 b(oneself)h(that)h(a)e(coherent)h(state)g(located)876 3049 y(initially)41 b(in)f Fn(q)52 b Fo(=)46 b(0)g(=)h Fn(p)39 b Fp(and)h(which)g(therefore)g(has)f(a)h(microlocal)g(support)g (in)h(phase)e(space)f(that)876 3248 y(is)45 b(concentrated)f(in)i(a)f (re)n(gion)g(of)g(size)3173 3107 y Fm(p)p 3311 3107 96 7 v 141 x Fo(\026)-83 b Fn(h)45 b Fp(around)h(this)f(point,)h(will)f(e) n(xplore)g(the)g(macroscopic)876 3447 y(re)n(gion)63 b(of)h(phase)f(space)f(that)i(e)n(xtends)e(up)i(to)g(the)f(nearest)g (neighbours)g Fn(q)5416 3472 y Fl(\006)5592 3447 y Fp(at)h(times)f Fn(t)87 b(>>)970 3581 y Fj(1)p 896 3608 215 7 v 896 3704 a(2)p Fk(!)1046 3721 y Fh(0)1130 3647 y Fm(j)28 b Fo(log)i(\026)-83 b Fn(h)p Fm(j)p Fp(.)51 b(This)41 b(intuition)h(will)g(be)f(further)h (e)n(xplained)e(belo)l(w)-11 b(,)41 b(b)m(ut)g(it)g(is)g(corroborated)g (by)h(the)876 3846 y(second)49 b(statement)g(abo)n(v)n(e.)73 b(W)-13 b(e)49 b(remark)h(that)f(pre)l(vious)h(ef)l(forts)g(to)f (estimate)g(the)h(e)l(v)m(olution)f(of)876 4045 y(coherent)43 b(states)f(\(in)h(particular)g(Combescure-Robert)g([CR])g(and)g (Hagedorn-Jo)n(ye)f([HJ]\))i(ha)m(v)n(e)876 4244 y(concentrated)61 b(on)h(constructing)h(good)f(semi-classical)e Fn(L)4398 4184 y Fj(2)4473 4244 y Fp(-approximations)i(of)h(the)f(e)l(v)m(olv)n (ed)876 4444 y(state.)48 b(This)35 b(yields)g(much)g(more)g(detailed)g (information)h(on)f(the)f(state)h(than)g(is)f(needed)h(here)f(since)876 4643 y(we)57 b(only)g(study)g(the)g(e)n(xpectation)f(v)l(alues)g(of)h (W)-13 b(e)n(yl-quantized)56 b(classical)g(observ)l(ables.)96 b(As)57 b(a)876 4842 y(counterpart,)37 b(the)g(a)m(v)l(ailable)e (results)i(on)f Fn(L)3365 4782 y Fj(2)3440 4842 y Fp(-estimates)g(are)g (v)l(alid)h(only)g(up)f(to)h(times)5945 4777 y Fj(1)p 5871 4804 V 5871 4899 a(6)p Fk(!)6021 4916 y Fh(0)6105 4842 y Fm(j)28 b Fo(log)i(\026)-83 b Fn(h)p Fm(j)p Fp(,)876 5041 y(during)51 b(which)f(the)h(microlocal)f(support)h(of)g(the)f (initial)h(state)f(does)f(not)i(reach)f(a)g(macroscopic)876 5241 y(size,)42 b(and)g(the)n(y)g(can)h(therefore)g(not)g(be)f(used)g (to)h(obtain)g(the)g(abo)n(v)n(e)e(results.)54 b(This)43 b(restriction)g(on)876 5440 y(the)49 b(time)g(is)f(related)h(to)g(the)g (observ)l(ation)f(that,)i(in)f([CR])g([HJ],)i(only)e(the)f(linearized)h (dynamics)876 5639 y(close)42 b(to)h Fn(X)1570 5664 y Fj(0)1693 5639 y Fo(=)48 b(\(0)p Fn(;)28 b Fo(0\))42 b Fp(is)h(used)f(in)h(the)g(analysis.)54 b(F)n(or)43 b(the)g(longer)g(times)f(we)h(e)n(xplore)f(here,)h(this)876 5838 y(is)e(insuf)l(\002cient.)1125 6038 y(A)46 b(similar)g(transition) g(in)f(the)h(beha)m(viour)f(of)h Fn(\026)3884 6063 y Fj(\026)-68 b Fk(h;\017;t)4199 6038 y Fp(\()p Fn(\017)54 b Fo(=)4558 5896 y Fm(p)p 4697 5896 96 7 v 4697 6038 a Fo(\026)-83 b Fn(h)o Fp(\))46 b(at)g Fn(T)23 b Fo(\(\026)-83 b Fn(h)o Fo(\))45 b Fp(w)n(as)h(sho)l(wn)f(in)h(the)876 6237 y(conte)n(xt)51 b(of)i(a)e(simple)h(chaotic)g(Hamiltonian)g (system)f(on)h(the)g(tw)n(o-torus)h(in)f([BDB1][BDB2].)876 6436 y(In)d(that)h(case)d(the)i Fn(\026)2048 6461 y Fj(\026)-68 b Fk(h;\017;t)2366 6436 y Fp(con)-7 b(v)n(er)m(ge)49 b(to)g(the)g(Lebesgue)g(measure)f(for)i Fn(T)23 b Fo(\(\026)-83 b Fn(h)o Fo(\))60 b Fn(<<)g(t)g(<<)h Fo(3)p Fn(T)23 b Fo(\(\026)-83 b Fn(h)o Fo(\))p Fp(.)876 6635 y(Roughly)59 b(speaking,)i(this)e(is)f(due)h(to)f(the)h(f)n(act)g(that)f(in)h(that)g (case,)i(the)e(trajectories)f(starting)h(a)876 6835 y(distance)1476 6693 y Fm(p)p 1615 6693 V 1615 6835 a Fo(\026)-83 b Fn(h)56 b Fp(a)n(w)n(ay)f(from)i(the)f(origin)h(e)n(xplore)f(all)g(of)h(phase)e (space)g(on)i(the)f(time)g(scale)f(con-)876 7034 y(sidered,)47 b(due)g(to)g(the)g(e)n(xponential)g(mixing)g(of)g(the)g(dynamics.)67 b(In)48 b(the)f(present)f(situation,)i(such)876 7233 y(trajectories)41 b(con)-7 b(v)n(er)m(ge)41 b(either)h(to)f Fn(q)2957 7258 y Fj(+)3109 7233 y Fp(or)h(to)g Fn(q)3534 7258 y Fl(\000)3646 7233 y Fp(,)f(leading)g(to)h(the)f(second)g(result) h(abo)n(v)n(e.)1125 7432 y(F)n(or)j(coherent)f(states)f(we)h(ha)m(v)n (e)g Fn(")52 b Fm(\030)3348 7367 y Fj(\026)-68 b Fk(h)p 3348 7394 76 7 v 3355 7490 a(")3443 7432 y Fp(.)60 b(F)n(or)45 b(a)f(deeper)g(understanding)h(of)g(the)f(phenomena)876 7632 y(described)51 b(in)i(this)f(paper)g(it)h(is)f(useful)g(to)g (consider)g(a)g(weak)n(er)g(localization)g(condition)g(on)h(the)876 7831 y(state)42 b Fn(\021)1305 7856 y Fk(")1419 7831 y Fp(and)h(to)g(w)n(ork)g(with)g(the)g(tw)n(o)g(scale)f(paramaters)g Fo(\026)-83 b Fn(h;)28 b(")43 b Fp(independently)-11 b(.)55 b(By)43 b(considering)876 8030 y(initial)37 b(states)f(in)g (which)h Fn(")46 b(<<)2762 7965 y Fj(\026)-68 b Fk(h)p 2762 7992 V 2769 8087 a(")2857 8030 y Fp(,)37 b(we)g(will)g(sho)l(w)f (\(Theorem)h(2.3\))f(that)h(on)f(the)h(same)f(time)h(scale)876 8229 y(as)f(the)g(one)g(considered)g(abo)n(v)n(e)f(\()p Fe(i.e)n(.)49 b Fn(t)d Fm(\024)3431 8164 y Fj(2)p 3357 8191 215 7 v 3357 8287 a(3)p Fk(!)3507 8304 y Fh(0)3591 8229 y Fm(j)28 b Fo(log)i(\026)-83 b Fn(h)p Fm(j)p Fp(\),)37 b(the)g(state)f(e)n(xplores)f(basically)h(the)g(entire)876 8429 y(zero-ener)m(gy)41 b(surf)n(ace,)g(as)g(we)g(no)l(w)g(e)n (xplain.)1125 8628 y(W)-13 b(e)36 b(consider)h(an)f(arbitrary)i(\(not)f (necessarily)f(Gaussian\))g(pro\002le)h Fn(')g Fp(in)g(the)g(Schw)n (artz)g(space)876 8827 y Fm(S)12 b Fo(\(I)-37 b(R\))41 b Fp(such)h(that)1943 8693 y Fi(R)2021 8886 y Fj(I)-30 b(R)2171 8827 y Fm(j)p Fn(')p Fo(\()p Fn(q)6 b Fo(\))p Fm(j)2582 8767 y Fj(2)2656 8827 y Fn(dq)53 b Fo(=)47 b(1)42 b Fp(and)g(de\002ne)g(the)g(follo)l(wing)g(normalized)g(state,)f (localized)876 9026 y(at)g(the)h(origin)g Fo(\(0)p Fn(;)28 b Fo(0\))40 b Fp(of)i(the)g(phase)e(space,)3021 9470 y Fn(\021)3103 9495 y Fk(")3174 9470 y Fo(\()p Fn(q)6 b Fo(\))46 b(=)3727 9358 y(1)p 3625 9432 287 7 v 3625 9587 a Fn(")3702 9539 y Fj(1)p Fk(=)p Fj(2)3931 9470 y Fn(')4068 9286 y Fi(\020)4186 9358 y Fn(q)p 4186 9432 81 7 v 4187 9584 a(")4286 9286 y Fi(\021)4413 9470 y Fn(:)1952 b Fp(\(4\))876 9880 y(Hence)1143 10292 y Fn(<)1362 10251 y Fo(^)1318 10292 y Fn(A)p Fo(\()p Fn(t)p Fo(\))p Fn(\021)1715 10317 y Fk(")1786 10292 y Fn(;)28 b(\021)1942 10317 y Fk(")2059 10292 y Fn(>)p Fo(=)2363 10066 y Fi(Z)2456 10444 y Fj(I)-30 b(R)2570 10405 y Fh(2)2671 10292 y Fn(A)p Fo(\()p Fn(X)13 b Fo(\))p Fn(d\026)3263 10317 y Fj(\026)-68 b Fk(h)n(;";t)3537 10292 y Fo(\()p Fn(X)13 b Fo(\))45 b(=)4038 10066 y Fi(Z)4131 10444 y Fj(I)-30 b(R)4245 10405 y Fh(2)4346 10292 y Fn(A)p Fo(\(\026)-83 b Fn(h)n(;)28 b(t;)g(X)13 b Fo(\))p Fn(W)5211 10317 y Fk(')5333 10108 y Fi(\020)5452 10180 y Fn(q)p 5452 10254 V 5453 10406 a(")5552 10292 y(;)5655 10180 y(")p 5646 10254 96 7 v 5646 10406 a Fo(\026)-83 b Fn(h)5762 10292 y(p)5846 10108 y Fi(\021)6013 10180 y Fn(dX)p 5992 10254 280 7 v 5992 10406 a Fo(2)p Fn(\031)6 b Fo(\026)-83 b Fn(h)6291 10292 y(;)3699 10919 y Fp(4)p eop %%Page: 5 5 5 4 bop 876 1455 a Fp(where)33 b Fn(W)1472 1480 y Fk(')1601 1455 y Fp(is)h(the)f(W)-7 b(igner)34 b(function)h(of)f Fn(')f Fp(\(computed)h(with)g Fo(\026)-83 b Fn(h)45 b Fo(=)i(1)p Fp(\))34 b(and)f(where,)i(by)e(de\002nition,)876 1676 y Fn(A)p Fo(\(\026)-83 b Fn(h)o(;)28 b(t;)g(X)13 b Fo(\))43 b Fp(is)i(the)g(W)-13 b(e)n(yl-symbol)44 b(of)3192 1634 y Fo(^)3148 1676 y Fn(A)o Fo(\()p Fn(t)p Fo(\))p Fp(.)61 b(More)44 b(generally)-11 b(,)45 b(we)g(could)g(also)f (consider)h(semi-)876 1875 y(classical)36 b(measures)h Fn(\026)2232 1900 y Fj(\026)-68 b Fk(h;";t)2546 1875 y Fp(de\002ned)38 b(by)g(a)g(density)f(matrix)54 b Fo(^)-98 b Fn(\032)4464 1900 y Fj(\026)-68 b Fk(h;\017)4681 1875 y Fp(with)38 b Fn(\032)5100 1900 y Fj(\026)-68 b Fk(h;\017)5279 1875 y Fo(\()p Fn(X)13 b Fo(\))45 b(=)h Fn(\032)p Fo(\()5951 1801 y Fk(q)p 5950 1837 66 7 v 5951 1933 a(")6035 1875 y Fn(;)6136 1810 y Fk(")p 6129 1837 76 7 v 6129 1933 a Fj(\026)-68 b Fk(h)6225 1875 y Fn(p)p Fo(\))37 b Fp(for)876 2097 y(some)k Fn(\032)46 b Fm(2)g(S)12 b Fo(\(I)-37 b(R)1880 2024 y Fj(2)1954 2097 y Fo(\))41 b Fp(as)g(follo)l(ws)1301 2322 y Fi(Z)1394 2700 y Fj(I)-30 b(R)1508 2661 y Fh(2)1608 2548 y Fn(A)p Fo(\()p Fn(X)13 b Fo(\))p Fn(d\026)2200 2573 y Fj(\026)-68 b Fk(h)o(;";t)2475 2548 y Fo(\()p Fn(X)13 b Fo(\))45 b(=)h(T)-14 b(r[)3237 2506 y(^)3193 2548 y Fn(A)p Fo(\()p Fn(t)p Fo(\))15 b(^)-98 b Fn(\032)3594 2573 y Fj(\026)-68 b Fk(h)o(;")3779 2548 y Fo(])47 b(=)4047 2322 y Fi(Z)4139 2700 y Fj(I)-30 b(R)4253 2661 y Fh(2)4354 2548 y Fn(A)p Fo(\(\026)-83 b Fn(h)o(;)28 b(t;)g(X)13 b Fo(\))p Fn(\032)5177 2364 y Fi(\020)5294 2436 y Fn(q)p 5294 2510 81 7 v 5295 2662 a(")5394 2548 y(;)5497 2436 y(")p 5488 2510 96 7 v 5488 2662 a Fo(\026)-83 b Fn(h)5603 2548 y(p)5687 2364 y Fi(\021)5855 2436 y Fn(dX)p 5834 2510 280 7 v 5834 2662 a Fo(2)p Fn(\031)6 b Fo(\026)-83 b Fn(h)6133 2548 y(:)1125 2996 y Fp(Note)35 b(that)h(for)g(the)f (states)g(in)g(\(4\))h(the)g(W)-7 b(igner)35 b(function)h(has)f(a)g (spread)g Fo(\001)p Fn(q)42 b Fp(in)35 b(the)h Fn(q)6 b Fp(-direction)876 3195 y(gi)l(v)n(en)41 b(by)g Fo(\001)p Fn(q)53 b Fm(\030)46 b Fn(")p Fp(,)c(and)f(a)g(spread)g Fo(\001)p Fn(p)46 b Fm(\030)3422 3130 y Fj(\026)-68 b Fk(h)p 3422 3157 76 7 v 3429 3252 a(")3560 3195 y Fp(in)41 b(the)h Fn(p)p Fp(-direction.)51 b(De\002ning)2528 3663 y Fo(\001)46 b(=)h(max)p Fm(f)p Fo(\001)p Fn(q)6 b(;)28 b Fo(\001)p Fn(p)p Fm(g)46 b 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b(study)f(will)h(be)g(done)f(in)h(Section)f(4.)1125 8214 y(T)-13 b(o)42 b(understand)f(intuiti)l(v)n(ely)g(what)h(happens,)e (one)h(can)g(assume)2938 8652 y Fn(H)13 b Fo(\()p Fn(X)g Fo(\))46 b(=)3611 8540 y(1)p 3611 8614 84 7 v 3611 8766 a(2)3714 8652 y(\()p Fn(p)3863 8584 y Fj(2)3973 8652 y Fm(\000)37 b Fn(!)4248 8584 y Fj(2)4242 8693 y(0)4323 8652 y Fn(q)4403 8584 y Fj(2)4478 8652 y Fo(\))876 9096 y Fp(for)42 b Fn(X)53 b Fp(in)42 b(a)f(small)g(neighbourhood)g(of)h Fo(\(0)p Fn(;)28 b Fo(0\))p Fp(.)50 b(No)l(w)-11 b(,)40 b(if)i Fm(j)p Fn(X)13 b Fm(j)46 b(\024)g Fo(\001)p Fp(,)41 b(then)g Fm(j)p Fo(\010)5388 9036 y Fk(t)5388 9142 y(H)5514 9096 y Fo(\()p Fn(X)13 b Fo(\))p Fm(j)45 b(\024)h Fn(e)6138 9036 y Fk(!)6222 9053 y Fh(0)6287 9036 y Fl(j)p Fk(t)p Fl(j)6425 9096 y Fo(\001)p Fp(,)876 9295 y(so)41 b(as)g(long)h(as)2399 9567 y Fm(j)p Fn(t)p Fm(j)k Fn(<<)h(T)2999 9592 y Fj(\001)3162 9567 y Fo(=)3405 9455 y(1)p 3357 9529 178 7 v 3357 9681 a Fn(!)3460 9706 y Fj(0)3555 9567 y Fm(j)28 b Fo(log)i(\001)p Fm(j)47 b(\024)4386 9455 y Fo(1)p 4297 9529 261 7 v 4297 9681 a(2)p Fn(!)4483 9706 y Fj(0)4578 9567 y Fm(j)28 b Fo(log)i(\026)-83 b Fn(h)o Fm(j)p Fn(;)876 9954 y Fm(j)p Fo(\010)1042 9894 y Fk(t)1042 10000 y(H)1168 9954 y Fo(\()p Fn(X)13 b Fo(\))p Fm(j)45 b(!)h Fo(0)c Fp(as)e Fo(\001)47 b Fm(!)f Fo(0)p Fp(,)41 b(so)g(we)g(e)n(xpect)g(that)2929 10340 y Fm(h)p Fn(\021)3076 10365 y Fk(")3148 10340 y Fn(;)3265 10298 y Fo(^)3222 10340 y Fn(A)o Fo(\()p Fn(t)p Fo(\))p Fn(\021)3618 10365 y Fk(")3688 10340 y Fm(i)46 b(!)g Fn(A)p Fo(\(0)p Fn(;)28 b Fo(0\))p Fn(;)3699 10919 y Fp(5)p eop %%Page: 6 6 6 5 bop 876 1455 a Fp(which)62 b(is)g(indeed)f(correct)h(\(Theorem)h (2.1\(i\)\).)113 b(T)-13 b(o)63 b(understand)e(what)h(happens)f(for)i (slightly)876 1675 y(longer)44 b(times)g Fn(t)50 b Fo(=)g(\(1)39 b(+)f Fn(\014)9 b Fo(\))2579 1594 y Fl(j)23 b Fj(log)i(\001)p Fl(j)p 2579 1637 406 7 v 2707 1732 a Fk(!)2791 1749 y Fh(0)3004 1675 y Fp(,)43 b(one)h(has)f(to)h(remark)g(\002rst)g(that)g 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3035 y Fo(=)47 b Fm(f)p Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\))p Fn(;)72 b(H)13 b Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\))45 b Fn(<)i Fo(0)p Fn(;)73 b(q)53 b(>)46 b Fo(0)p Fm(g)p Fp(,)41 b Fo(\012)3891 3060 y Fk(I)9 b(I)4081 3035 y Fo(=)46 b Fm(f)p Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\))p Fn(;)73 b(H)13 b Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\))45 b Fn(>)h Fo(0)p Fn(;)74 b(p)46 b(>)g Fo(0)p Fm(g)p Fp(,)1125 3361 y Fm(\017)83 b Fo(\012)1411 3386 y Fk(I)9 b(I)g(I)1669 3361 y Fo(=)46 b Fm(f)p Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\))p Fn(;)73 b(H)13 b Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\))45 b Fn(<)h Fo(0)p Fn(;)74 b(q)52 b(<)47 b Fo(0)p Fm(g)p Fp(,)41 b Fo(\012)4027 3386 y Fk(I)9 b(V)4256 3361 y Fo(=)46 b Fm(f)p Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\))p Fn(;)73 b(H)13 b Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\))45 b Fn(>)h Fo(0)p Fn(;)74 b(p)46 b(<)g Fo(0)p Fm(g)p Fp(.)876 3708 y(As)65 b(long)h(as)e Fn(\014)100 b Fm(\024)90 b Fo(1)p Fp(,)71 b(the)66 b(e)n(xtra)e(time)i Fn(\014)3446 3627 y Fl(j)23 b Fj(log)i(\001)p Fl(j)p 3446 3669 V 3574 3765 a Fk(!)3658 3782 y Fh(0)3936 3708 y Fp(serv)n(es)64 b(those)h(points)h (to)f(reach)g(a)g(distance)876 3939 y Fo(\001)1014 3879 y Fk(\014)1169 3939 y Fp(from)i(either)e Fn(q)2075 3964 y Fj(+)2251 3939 y Fp(or)h Fn(q)2529 3964 y Fl(\000)2641 3939 y Fp(.)123 b(F)n(or)65 b(longer)h(times)f Fn(\014)99 b Fm(\025)91 b Fo(1)p Fp(,)71 b(the)n(y)65 b(in)g(turn)h(start)g(lea)m (ving)f(the)876 4138 y(neighbourhood)e(of)g Fn(q)2219 4163 y Fl(\006)2332 4138 y Fp(,)k(reaching)c(a)g(macroscopic)f (distance)g(from)h(those)g(points)g(at)g(a)f(time)876 4338 y(roughly)42 b(equal)f(to)h Fo(3)p Fn(T)2186 4363 y Fj(\001)2303 4338 y Fp(.)51 b(This)42 b(yields)f(for)3058 4688 y Fn(T)3155 4713 y Fj(\001)3318 4688 y Fn(<<)47 b Fm(j)p Fn(t)p Fm(j)f Fn(<<)g Fo(3)p Fn(T)4305 4713 y Fj(\001)876 5038 y Fp(that)1784 5237 y Fm(h)p Fn(\021)1931 5262 y Fk(")2002 5237 y Fn(;)2119 5195 y Fo(^)2076 5237 y Fn(A)o Fo(\()p Fn(t)p Fo(\))p Fn(\021)2472 5262 y Fk(")2543 5237 y Fm(i)f(!)h Fo(\()p Fn(\013)3036 5262 y Fk(I)3149 5237 y Fo(+)37 b Fn(\013)3421 5262 y Fk(I)9 b(I)3566 5237 y Fo(\))p Fn(A)p Fo(\()p Fn(q)3895 5262 y Fj(+)4004 5237 y Fo(\))36 b(+)h(\()p Fn(\013)4442 5262 y Fk(I)9 b(I)g(I)4691 5237 y Fo(+)37 b Fn(\013)4963 5262 y Fk(I)9 b(V)5146 5237 y Fo(\))p Fn(A)p Fo(\()p Fn(q)5475 5262 y Fl(\000)5586 5237 y Fo(\))p Fn(;)876 5527 y Fp(where)41 b Fn(\013)1429 5552 y Fk(i)1526 5527 y Fp(is)g(the)h(fraction)g(of)f (the)h(total)g(mass)e(of)i(the)f(state)g(li)l(ving)h(in)g Fo(\012)5101 5552 y Fk(i)5156 5527 y Fp(:)2495 5956 y Fn(\013)2601 5981 y Fk(i)2703 5956 y Fo(=)85 b(lim)2878 6063 y Fj(\001)p Fl(!)p Fj(0)3332 5844 y Fo(1)p 3234 5918 280 7 v 3234 6070 a(2)p Fn(\031)6 b Fo(\026)-83 b Fn(h)3561 5730 y Fi(Z)46 b(Z)3791 6107 y Fj(\012)3886 6124 y Ff(i)3975 5956 y Fn(\032)p Fo(\()4146 5844 y Fn(q)p 4146 5918 81 7 v 4147 6070 a(")4245 5956 y(;)4348 5844 y(")p 4339 5918 96 7 v 4339 6070 a Fo(\026)-83 b Fn(h)4454 5956 y(p)p Fo(\))p Fn(dq)6 b(dp:)1426 b Fp(\(9\))876 6432 y(Of)50 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currentpoint translate 1 0.59677 div 1 0.59677 div scale neg exch neg exch translate 906 3248 a 1785 3613 a Fp(Figure)42 b(2:)51 b(The)42 b(phase)f(portrait)h(for)g(a)f(quadruple)h(well)f(potential.)1125 4159 y(Let)53 b(us)g(denote)f(by)h Fn(X)2452 4184 y Fk(j)2589 4159 y Fo(=)67 b(\()p Fn(q)2924 4184 y Fk(j)2994 4159 y Fn(;)28 b Fo(0\))52 b Fp(the)h(abo)n(v)n(e)e(non-de)n(generate)h (critical)h(points)g(of)g Fn(H)67 b Fp(for)876 4358 y Fn(j)56 b Fo(=)46 b(0)p Fn(;)28 b Fm(\006)p Fp(.)51 b(W)-13 b(e)41 b(are)g(interested)g(in)h(a)f(neighborhood)h(in)g(the)f(phase)g (space)f(of)i(the)f(curv)n(e)1880 4723 y Fo(\006)2000 4655 y Fl(0)2000 4764 y Fj(0)2121 4723 y Fo(=)46 b Fm(f)p Fn(X)59 b Fo(=)46 b(\()p Fn(q)6 b(;)28 b(p)p Fo(\))p Fn(;)118 b(p)3367 4655 y Fj(2)3441 4723 y Fn(=)p Fo(2)37 b(+)g Fn(V)g Fo(\()p Fn(q)6 b Fo(\))46 b(=)g(0)p Fn(;)74 b(q)4652 4748 y Fl(\000)4810 4723 y Fm(\024)46 b Fn(q)53 b Fm(\024)46 b Fn(q)5361 4748 y Fj(+)5471 4723 y Fm(g)p Fn(:)876 5089 y Fp(Ob)n(viously)g(\(see)h(Fig.)68 b(2\))47 b Fo(\006)2563 5028 y Fl(0)2563 5130 y Fj(0)2637 5089 y Fm(nf)p Fn(X)2941 5114 y Fj(0)3015 5089 y Fn(;)28 b(X)3227 5114 y Fl(\000)3339 5089 y Fn(;)g(X)3551 5114 y Fj(+)3660 5089 y Fm(g)47 b Fp(is)g(split)g(into)h(4)f(branches)f Fn(\015)5440 5114 y Fj(+)p Fk(;)p Fj(+)5691 5089 y Fn(;)i(\015)5871 5114 y Fl(\000)p Fk(;)p Fl(\000)6126 5089 y Fn(;)f(\015)6305 5114 y Fj(+)p Fk(;)p Fl(\000)6558 5089 y Fn(;)876 5288 y(\015)962 5313 y Fl(\000)p Fk(;)p Fj(+)1215 5288 y Fp(,)34 b(where)d(the)h(\002rst)h(inde)n(x)e(refers)h(to)g(the)g(sign)g(of)g (the)g Fn(q)6 b Fp(-component)32 b(and)g(the)g(second)f(inde)n(x)g(to) 876 5487 y(the)38 b(sign)h(of)g(the)f Fn(p)p Fp(-component.)50 b(Moreo)n(v)n(er)37 b Fo(\006)3628 5512 y Fk(ust)3874 5487 y Fo(:=)46 b Fn(\015)4181 5512 y Fj(+)p Fk(;)p Fj(+)4459 5487 y Fm([)26 b Fn(\015)4682 5512 y Fl(\000)p Fk(;)p Fl(\000)4976 5487 y Fp(and)38 b Fo(\006)5374 5512 y Fk(st)5541 5487 y Fo(:=)47 b Fn(\015)5849 5512 y Fl(\000)p Fk(;)p Fj(+)6128 5487 y Fm([)26 b Fn(\015)6351 5512 y Fj(+)p Fk(;)p Fl(\000)876 5687 y Fp(are)42 b(respecti)l(v)n(ely)g(the)g (unstable)g(component)h(and)f(the)h(stable)f(component)g(of)h(the)g (ener)m(gy)f(curv)n(e)876 5886 y Fo(\006)996 5825 y Fl(0)996 5927 y Fj(0)1070 5886 y Fp(.)50 b(In)37 b(particular)g(we)g(ha)m(v)n(e) 129 b Fo(lim)2586 5985 y Fk(t)p Fl(!)p Fj(+)p Fl(1)3030 5886 y Fo(\010)3150 5817 y Fk(t)3150 5927 y(H)3276 5886 y Fo(\()p Fn(X)13 b Fo(\))45 b(=)h Fn(X)3915 5911 y Fj(0)4026 5886 y Fp(for)38 b Fn(X)59 b Fm(2)45 b Fo(\006)4730 5911 y Fk(st)4888 5886 y Fp(and)130 b Fo(lim)5165 5985 y Fk(t)p Fl(!)p Fj(+)p Fl(1)5610 5886 y Fo(\010)5730 5817 y Fk(t)5730 5927 y(H)5856 5886 y Fo(\()p Fn(X)13 b Fo(\))44 b(=)j Fn(X)6495 5911 y Fj(+)876 6146 y Fp(\(respecti)l(v)n(ely)40 b Fn(X)1906 6171 y Fl(\000)2018 6146 y Fp(\))i(for)g Fn(X)59 b Fm(2)46 b Fn(\015)2790 6171 y Fj(+)p Fk(;)p Fj(+)3083 6146 y Fp(\(respecti)l(v)n(ely)40 b Fn(X)59 b Fm(2)46 b Fn(\015)4415 6171 y Fl(\000)p Fk(;)p Fl(\000)4670 6146 y Fp(\).)1125 6345 y(No)l(w)41 b(we)g(can)g(formulate)h(our)g (\002rst)g(main)g(result.)876 6677 y Fu(Theor)m(em)f(2.1)83 b Fe(Let)64 b(us)g(assume)f(that)i Fp(\(H1\))g Fe(and)f Fp(\(H2\))h Fe(hold)g(and)f(let)h(us)f(intr)-7 b(oduce)63 b Fn(T)6213 6591 y Fj(\()p Fk(j)7 b Fj(\))6190 6726 y(\001)6475 6677 y Fo(=)896 6830 y Fl(j)23 b Fj(log)i(\001)p Fl(j)p 896 6872 406 7 v 1026 6968 a Fk(!)1110 6985 y Ff(j)1321 6910 y Fn(;)69 b Fo(\()p Fn(j)56 b Fo(=)46 b(0)p Fm(\006)p Fo(\))p Fe(.)51 b(Let)41 b Fn(!)53 b Fo(=)46 b(max)p Fm(f)p Fn(!)3249 6935 y Fj(+)3360 6910 y Fn(;)28 b(!)3537 6935 y Fj(0)3611 6910 y Fn(;)g(!)3788 6935 y Fl(\000)3901 6910 y Fm(g)p Fe(.)51 b(Let)41 b Fn(A)46 b Fm(2)g(S)12 b Fo(\(I)-37 b(R)4981 6838 y Fj(2)5055 6910 y Fo(\))p Fe(,)41 b Fn(\017)46 b(>)h Fo(0)41 b Fe(and)h(suppose)2944 7402 y Fm(j)p Fn(t)p Fm(j)k(\024)3392 7290 y Fo(2)p 3338 7364 193 7 v 3338 7516 a(3)p Fn(!)3550 7402 y Fo(\(1)36 b Fm(\000)h Fn(\017)p Fo(\))p Fm(j)28 b Fo(log)i(\026)-83 b Fn(h)p Fm(j)p Fn(:)1792 b Fp(\(10\))1051 7872 y Fe(\(i\))84 b(Let)42 b Fo(0)k Fn(<)g(\016)1923 7897 y Fj(1)2044 7872 y Fn(<)g Fo(1)41 b Fe(and)h(suppose)e(in)i(addition)g(to)g(\(10\))g (that)g Fm(j)p Fn(t)p Fm(j)k(\024)h Fo(\(1)36 b Fm(\000)h Fn(\016)5563 7897 y Fj(1)5637 7872 y Fo(\))p Fn(T)5822 7786 y Fj(\(0\))5799 7921 y(\001)6000 7872 y Fe(.)51 b(Then)2943 8261 y Fo(lim)2905 8368 y Fj(\001)p Fl(!)p Fj(0)3259 8261 y Fn(<)3478 8219 y Fo(^)3434 8261 y Fn(A)o Fo(\()p Fn(t)p Fo(\))p Fn(\021)3830 8286 y Fk(")3901 8261 y Fn(;)28 b(\021)4057 8286 y Fk(")4174 8261 y Fn(>)p Fo(=)47 b Fn(A)p Fo(\()p Fn(X)4807 8286 y Fj(0)4880 8261 y Fo(\))p Fn(:)1337 b Fp(\(11\))1005 8749 y Fe(\(ii\))84 b(Let)42 b Fo(0)k Fn(<)g(\016)1923 8774 y Fj(1)1997 8749 y Fn(;)28 b(\016)2145 8774 y Fj(2)2261 8749 y Fe(be)41 b(small)h(enough)f(and)h(suppose)e(in)i(addition)g(to)g(\(10\))g(that) 2103 9139 y Fo(\(1)36 b(+)h Fn(\016)2527 9164 y Fj(1)2602 9139 y Fo(\))p Fn(T)2787 9053 y Fj(\(0\))2764 9188 y(\001)3011 9139 y Fm(\024)46 b Fn(t)g Fm(\024)g Fn(T)3587 9053 y Fj(\(0\))3564 9188 y(\001)3803 9139 y Fo(+)37 b(\(2)f Fm(\000)h Fn(\016)4393 9164 y Fj(2)4468 9139 y Fo(\))28 b(min)p Fm(f)p Fn(T)5040 9053 y Fj(\()p Fl(\000)p Fj(\))5017 9188 y(\001)5256 9139 y Fn(;)g(T)5450 9053 y Fj(\(+\))5427 9188 y(\001)5663 9139 y Fm(g)p Fn(:)536 b Fp(\(12\))1291 9505 y Fe(Then)1582 9870 y Fo(lim)1543 9977 y Fj(\001)p Fl(!)p Fj(0)1897 9870 y Fn(<)2116 9828 y Fo(^)2072 9870 y Fn(A)p Fo(\()p Fn(t)p Fo(\))p Fn(\021)2469 9895 y Fk(")2539 9870 y Fn(;)28 b(\021)2695 9895 y Fk(")2813 9870 y Fn(>)p Fo(=)46 b(\()p Fn(\013)3288 9895 y Fk(I)3401 9870 y Fo(+)37 b Fn(\013)3673 9895 y Fk(I)9 b(I)3817 9870 y Fo(\))p Fn(A)p Fo(\()p Fn(X)4210 9895 y Fj(+)4319 9870 y Fo(\))36 b(+)h(\()p Fn(\013)4757 9895 y Fk(I)9 b(I)g(I)5006 9870 y Fo(+)37 b Fn(\013)5278 9895 y Fk(I)9 b(V)5462 9870 y Fo(\))p Fn(A)p Fo(\()p Fn(X)5855 9895 y Fl(\000)5965 9870 y Fo(\))p Fn(:)252 b Fp(\(13\))3699 10919 y(7)p eop %%Page: 8 8 8 7 bop 1125 1455 a Fp(The)53 b(restriction)g(\(10\))h(on)f(the)g (times)g(comes)f(from)h(the)g(control)h(on)f(the)g(error)g(term)g(in)h (the)876 1654 y(Egoro)n(v)59 b(theorem)f(\(Proposition)i(3.2\).)101 b(The)59 b(tw)n(o)g(parts)f(of)h(the)f(theorem)g(correspond)g(to)h(tw)n (o)876 1853 y(dif)l(ferent)k(re)n(gimes)f(in)h(the)g(beha)m(viour)g(of) g(the)g(right-hand)g(side)g(of)g(\(7\).)116 b(F)n(or)63 b(part)g(\(ii\))h(to)f(be)876 2052 y(non-tri)l(vial,)42 b(we)f(need)g(of)h(course)f(that)2473 2304 y Fo(1)p 2426 2378 178 7 v 2426 2530 a Fn(!)2529 2555 y Fj(0)2623 2416 y Fm(j)28 b Fo(log)i(\001)p Fm(j)47 b Fn(<)3420 2304 y Fo(2)p 3365 2378 193 7 v 3365 2530 a(3)p Fn(!)3577 2416 y Fm(j)28 b Fo(log)i(\026)-83 b Fn(h)p Fm(j)p Fn(;)69 b Fp(or)42 b Fo(\001)47 b Fn(>)f Fo(\026)-83 b Fn(h)4806 2278 y Fh(2)p Ff(!)4937 2303 y Fh(0)p 4806 2316 195 7 v 4836 2382 a(3)p Ff(!)5029 2416 y Fn(:)876 2789 y Fp(This)38 b(can)f(be)g(seen)g(as)g(a)g(restriction)h(on)g Fo(\001)p Fp(,)g(gi)l(v)n(en)f Fn(!)3927 2814 y Fj(0)4002 2789 y Fn(;)28 b(!)4179 2814 y Fl(\006)4291 2789 y Fp(,)38 b(or)g(as)f(a)g(restriction)h(on)f(the)h Fn(!)6092 2814 y Fk(j)6162 2789 y Fp(,)g(gi)l(v)n(en)876 3008 y Fo(\001)p Fp(.)68 b(F)n(or)47 b(e)n(xample,)g(for)h(coherent)f(states,)g(where)f Fo(\001)57 b(=)4201 2866 y Fm(p)p 4339 2866 96 7 v 142 x Fo(\026)-83 b Fn(h)p Fp(,)48 b(this)f(yields)g Fo(4)p Fn(!)5442 3033 y Fj(0)5573 3008 y Fn(>)56 b Fo(3)p Fn(!)6 b Fp(,)49 b(which)e(is)876 3207 y(satis\002ed)41 b(in)g(particular)h (when)f Fn(!)2814 3232 y Fl(\006)2973 3207 y Fo(=)46 b Fn(!)3251 3232 y Fj(0)3326 3207 y Fp(.)1125 3406 y(As)h(e)n(xplained) f(in)i(the)f(introduction,)j(these)c(results)h(can)g(be)g(understood)h (by)f(recalling)g(that)876 3606 y(the)41 b(initial)g(state)g(li)l(v)n (es)f(in)h(a)f(re)n(gion)h(of)g(phase)f(space)g(of)h(linear)g(size)g Fo(\001)g Fp(around)g(the)g(origin.)51 b(This)876 3805 y(re)n(gion)62 b(is)g(stretched)g(to)h(linear)f(size)g Fn(L)3261 3830 y Fk(t)3404 3805 y Fo(=)85 b(e)3692 3745 y Fk(!)3776 3762 y Fh(0)3840 3745 y Fk(t)3898 3805 y Fo(\001)63 b Fp(in)g(a)f(time)g Fn(t)h Fp(due)f(the)g(instability)h(of) g(the)876 4025 y(dynamics)43 b(around)h Fn(X)2199 4050 y Fj(0)2324 4025 y Fo(=)51 b(0)p Fp(.)58 b(As)43 b(a)h(result,)g(for)g (times)g(shorter)g(than)4984 3945 y Fl(j)23 b Fj(log)i(\001)p Fl(j)p 4984 3987 406 7 v 5112 4082 a Fk(!)5196 4099 y Fh(0)5409 4025 y Fp(,)44 b(this)g(size)f(tends)g(to)876 4224 y(0)49 b(as)f Fo(\001)60 b Fm(!)g Fo(0)49 b Fp(and)f(the)h(e)l(v)m (olv)n(ed)e(state)i(therefore)g(still)g(concentrates)e(around)i Fn(X)5681 4249 y Fj(0)5755 4224 y Fp(.)73 b(But)49 b(for)h(the)876 4424 y(longer)40 b(times)f(considered)g(in)g(the)h(second)e(part)i(of)f (the)h(Theorem)g(this)f(is)g(no)h(longer)f(correct)h(and)876 4623 y(the)57 b(trajectories)f(starting)h(at)g(a)f(distance)g Fo(\001)h Fp(no)l(w)g(reach)f(the)h(neighbouring)g(barrier)g(top)g Fn(X)6451 4648 y Fl(\006)6563 4623 y Fp(,)876 4822 y(leading)41 b(to)h(the)f(second)g(result.)1125 5021 y(It)56 b(is)g(also)g(possible) f(to)h(describe)g(in)g(detail)g(what)g(happens)f(on)h(the)g(transition) g(time)h(scale)876 5221 y(when)39 b Fn(t)g Fp(is)g(of)h(order)2153 5155 y Fj(1)p 2111 5182 149 7 v 2111 5278 a Fk(!)2195 5295 y Fh(0)2280 5221 y Fm(j)28 b Fo(log)i(\001)p Fm(j)p Fp(.)51 b(As)38 b(e)n(xpected,)g(the)h(state)g(then)g(stretches)g(out)g (along)g(the)h(unsta-)876 5420 y(ble)h(manifold)h(of)g Fn(X)2079 5445 y Fj(0)2153 5420 y Fp(:)876 5723 y Fu(Theor)m(em)f(2.2) 83 b Fe(Let)55 b(us)f(assume)g Fp(\(H1\))i Fe(and)f Fp(\(H2\))p Fe(.)92 b(Suppose)54 b(that)h(the)g(times)g Fn(t)g Fe(satisfy)-9 b(,)57 b(in)e(ad-)876 6002 y(dition)h(to)g(\(10\),)98 b Fo(lim)1881 6109 y Fj(\001)p Fl(!)p Fj(0)2217 6002 y Fo(\001e)2429 5933 y Fk(!)2513 5950 y Fh(0)2577 5933 y Fk(t)2682 6002 y Fo(=)46 b Fn(\024)72 b(>)g Fo(0)56 b Fe(and)g(that)71 b Fo(lim)3984 6104 y Fk(")p Fl(!)p Fj(0)4328 5890 y Fo(\026)-83 b Fn(h)p 4293 5964 167 7 v 4293 6116 a(")4370 6068 y Fk(\026)4525 6002 y Fo(=)46 b Fn(c)56 b Fe(for)f(some)g Fn(c)72 b(>)h Fo(0)56 b Fe(and)f(some)876 6267 y Fn(\026)46 b Fm(2)p Fo(]1)p Fn(;)28 b Fo(+)p Fm(1)p Fo([)p Fe(.)51 b(Then)42 b(ther)-6 b(e)40 b(e)m(xists)g(a)i(positive)f (smooth)g(density)g Fn(g)4722 6292 y Fk(\024;c;\026)5069 6267 y Fe(on)g Fo(\006)5396 6292 y Fk(ust)5637 6267 y Fe(suc)n(h)g(that)1952 6661 y Fo(lim)1914 6768 y Fj(\001)p Fl(!)p Fj(0)2268 6661 y Fn(<)2487 6619 y Fo(^)2443 6661 y Fn(A)o Fo(\()p Fn(t)p Fo(\))p Fn(\021)2839 6686 y Fk(")2910 6661 y Fn(;)28 b(\021)3066 6686 y Fk(")3183 6661 y Fn(>)p Fo(=)3488 6435 y Fi(Z)3580 6812 y Fj(\006)3675 6829 y Ff(ust)3892 6661 y Fn(A)p Fo(\()p Fn(X)13 b Fo(\))p Fn(g)4377 6686 y Fk(\024;c;\026)4680 6661 y Fo(\()p Fn(X)g Fo(\))p Fn(d)p Fo(\006)5167 6686 y Fj(0)5241 6661 y Fo(\()p Fn(X)g Fo(\))p Fn(;)876 7097 y Fe(for)55 b(e)n(very)f(observable)g Fn(A)71 b Fm(2)h Fn(C)2832 7037 y Fl(1)2820 7138 y Fj(0)2972 7097 y Fo(\(I)-37 b(R)3182 7025 y Fj(2)3256 7097 y Fo(\))55 b Fe(suc)n(h)g(that)h(supp)p Fo(\()p Fn(A)p Fo(\))70 b Fm(\032)i(f)p Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\);)98 b Fn(q)5557 7122 y Fl(\000)5741 7097 y Fn(<)72 b(q)78 b(<)73 b(q)6370 7122 y Fj(+)6480 7097 y Fm(g)p Fe(.)876 7296 y(Her)-6 b(e)32 b Fn(d)p Fo(\006)1441 7321 y Fj(0)1516 7296 y Fo(\()p Fn(X)13 b Fo(\))31 b Fe(is)i(the)g(\003ow-in)-7 b(variant)33 b(Liouville)h(measur)-6 b(e)31 b(on)i Fo(\006)4707 7321 y Fj(0)4782 7296 y Fe(.)48 b(Mor)-6 b(eo)n(ver)32 b(we)g(can)h(compute)876 7495 y Fn(g)955 7520 y Fk(\024;c;\026)1260 7495 y Fe(,)41 b(in)h(position)f(coor)-6 b(dinates)41 b Fn(q)6 b Fe(,)41 b(by)g(the)h(following)g(formulae:)1608 7799 y Fn(g)1687 7824 y Fk(\024;c;\026)1992 7799 y Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\()p Fn(q)6 b Fo(\)\))164 b(=)i Fn(!)3132 7824 y Fj(0)3207 7799 y Fn(`)p Fo(\()p Fn(q)6 b Fo(\))p Fn(G)3617 7824 y Fk(\024;c;\026)3921 7799 y Fo(\()p Fn(`)p Fo(\()p Fn(q)g Fo(\)\))p Fn(;)211 b Fo(where)2290 8203 y Fn(`)p Fo(\()p Fn(q)6 b Fo(\))165 b(=)3074 8091 y Fm(j)p Fn(q)6 b Fm(j)p 3049 8165 222 7 v 3049 8198 a(p)p 3188 8198 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4725 m gs 1 -1 sc (q-3) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 10350 4725 m gs 1 -1 sc (q3) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 4395 3570 m gs 1 -1 sc (V) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 4350 6165 m gs 1 -1 sc (-v) col0 sh gr $F2psEnd rs %%EndDocument @endspecial 906 5642 a /End PSfrag 906 5642 a 906 3467 a /Hide PSfrag 906 3467 a -464 3580 a Fp(PSfrag)42 b(replacements)p -464 3643 1370 7 v 906 3649 a /Unhide PSfrag 906 3649 a 794 3802 a { 794 3802 a Fc(q)794 3802 y } 0/Place PSfrag 794 3802 a 717 4048 a { 717 4048 a Fc(V)717 4048 y } 1/Place PSfrag 717 4048 a 581 4197 a { 581 4197 a Fc(q)685 4233 y Fm(\000)p Fo(3)581 4197 y } 2/Place PSfrag 581 4197 a 581 4397 a { 581 4397 a Fc(q)685 4433 y Fm(\000)p Fo(2)581 4397 y } 3/Place PSfrag 581 4397 a 581 4596 a { 581 4596 a Fc(q)685 4632 y Fm(\000)p Fo(1)581 4596 y } 4/Place PSfrag 581 4596 a 710 4798 a { 710 4798 a Fc(q)814 4834 y Fo(0)710 4798 y } 5/Place PSfrag 710 4798 a 710 4998 a { 710 4998 a Fc(q)814 5034 y 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Fo(\()p Fn(X)5116 4463 y Fk(j)e Fj(\()p Fk(\034)f(;i)p Fj(\))5442 4433 y Fo(\))p Fn(;)1291 4916 y Fe(wher)-6 b(e)40 b(the)i Fn(\013)2083 4941 y Fk(i)2180 4916 y Fe(ar)-6 b(e)40 b(de\002ned)h(in)h(\(9\).)1125 5253 y Fp(Theorem)32 b(2.4)f(can)g(be)g(paraphrased)g(by)h(saying)f (that)g(\223most)h(of)f(the)h(time\224)f(the)h(semi-classical)876 5452 y(measures)43 b Fn(\026)1638 5477 y Fj(\026)-68 b Fk(h;\017;t)1950 5452 y Fp(con)-7 b(v)n(er)m(ge)43 b(to)h(a)g(sum)f(of)i(at)e(most)h(four)g(Dirac)g(delta-measures)f(on)h (the)f(critical)876 5652 y(points)e Fn(X)1461 5677 y Fk(j)1531 5652 y Fp(.)51 b(The)42 b(set)f Fn(I)54 b Fp(is)41 b(in)h(f)n(act)g(a)f(\002nite)h(union)g(of)g(interv)l(als.)1125 5889 y(Theorem)j(2.3)e(sho)l(ws)g(the)h(occurrence)g(of)g(interv)l(als) g(of)g(time)h(of)f(length)h Fo(2)p Fn(T)5656 5803 y Fj(\(0\))5633 5938 y(\001)5878 5889 y Fp(centered)f(at)876 6128 y(e)l(v)n(en)54 b(multiples)h(of)g Fn(T)2224 6042 y Fj(\(0\))2201 6177 y(\001)2402 6128 y Fp(.)91 b(First,)59 b(as)54 b(in)h(part)g(\(i\))g (of)h(Theorem)f(2.1)f(one)h(sees)e(that)i(for)h(times)e Fn(t)876 6367 y Fp(between)h Fm(\000)p Fn(T)1734 6280 y Fj(\(0\))1711 6416 y(\001)1969 6367 y Fp(and)h Fn(T)2385 6280 y Fj(\(0\))2362 6416 y(\001)2620 6367 y Fp(\()p Fe(i.e)n(.)95 b Fn(`)73 b Fo(=)h(0)p Fp(\))56 b(the)h(particle)f(stays) f(close)h(to)g Fo(0)p Fp(.)96 b(Ne)n(xt,)58 b(as)e(will)h(be)876 6566 y(seen)36 b(in)g(Section)h(4)g(\(see)e(also)h(the)h(informal)g (discussion)f(in)g(the)h(introduction\),)h(the)e(time)h(needed)876 6765 y(to)50 b(go)h(with)f(the)h(classical)e(dynamics)g(along)i(the)f (ener)m(gy)g(curv)n(e)g Fo(\006)4879 6795 y Fj(\001)4988 6762 y Fh(2)5123 6765 y Fo(:=)63 b Fm(fj)p Fn(H)13 b Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\))p Fm(j)62 b Fo(=)h(\001)6447 6705 y Fj(2)6521 6765 y Fm(g)876 6964 y Fp(from)c(some)f(neighborhood)h (of)g Fn(X)2982 6989 y Fj(0)3115 6964 y Fp(to)f(a)h(neighborhood)f(of)h Fn(X)4750 6989 y Fj(1)4883 6964 y Fp(or)g Fn(X)5218 6989 y Fl(\000)p Fj(1)5454 6964 y Fp(is)f(slightly)h(longer)876 7193 y(than)48 b Fn(T)1330 7107 y Fj(\(0\))1307 7242 y(\001)1509 7193 y Fp(.)72 b(Once)48 b(the)g(particle)h(has)f(come)g (close)g(to)g(one)h(of)g(these)e(tw)n(o)i(points,)h(it)f(stays)f(ag)o (ain)876 7432 y(a)h(period)h Fo(2)p Fn(T)1676 7346 y Fj(\(0\))1653 7481 y(\001)1904 7432 y Fp(close)e(to)i(them,)h(before)e (e)n(xploring)h(further)g(barrier)g(tops)f(\()p Fn(`)61 b(>)g Fo(1)p Fp(\))50 b(starting)g(at)876 7671 y Fn(t)74 b Fo(=)h(3)p Fn(T)1417 7585 y Fj(\(0\))1394 7720 y(\001)1595 7671 y Fp(.)97 b(In)58 b(general,)h(it)e(will)h(therefore)f(approach)f (the)h Fn(`)p Fp(-th)g(top)g(a)n(w)n(ay)f(from)i Fo(0)f Fp(at)g(times)876 7910 y(slightly)42 b(longer)f(than)h Fo(\(2)p Fn(`)37 b Fo(+)g(1\))p Fn(T)2897 7823 y Fj(\(0\))2874 7959 y(\001)3075 7910 y Fp(.)51 b(This)41 b(helps)g(to)h(e)n(xplain)f (the)g(formulas)h(\(16\)-\(17\).)1125 8109 y(No)l(w)-11 b(,)46 b(as)f(pointed)h(out)g(in)g(the)g(introduction,)h(under)e(the)h (h)o(ypotheses)f(of)h(Theorem)g(2.3,)g(we)876 8308 y(can)f(control)i (the)e(quantum)i(dynamics)e(only)h(up)g(to)g(times)4455 8243 y Fj(2)p 4381 8270 215 7 v 4381 8365 a(3)p Fk(!)4531 8382 y Fh(0)4615 8308 y Fm(j)28 b Fo(log)i(\026)-83 b Fn(h)p Fm(j)p Fp(.)64 b(So,)47 b(for)g(the)e(particle)h(to)876 8507 y(reach)41 b(the)g Fn(`)p Fp(-th)i(top)e(in)h(this)f(time,)h(one)f (needs)3268 8888 y Fo(\001)3426 8760 y Fh(3\(2)p Ff(`)p Fh(+1\))p 3426 8794 395 7 v 3595 8860 a(2)3895 8888 y Fm(\025)46 b Fo(\026)-83 b Fn(h;)876 9226 y Fp(which)50 b(e)n(xplains)f(the)h(origin)h(of)g(the)f(condition)g(on)h Fn(k)56 b Fp(and)50 b Fn(m)g Fp(in)g(the)g(h)o(ypotheses)f(of)i (Theorem)876 9425 y(2.3.)57 b(In)43 b(particular)-7 b(,)45 b(if)f(we)f(choose)f Fo(\001)51 b(=)f Fn(")g Fo(=)3640 9283 y Fm(p)p 3779 9283 96 7 v 3779 9425 a Fo(\026)-83 b Fn(h)43 b Fp(\()p Fe(i.e)n(.)56 b Fn(m)50 b Fo(=)g(1)p Fp(\),)44 b(as)f(for)h(coherent)f(states,)g(then)876 9624 y(the)50 b(w)n(a)m(v)n(efunction)g(only)h(visits)f(the)h(sites)f Fn(X)3561 9649 y Fl(\006)3723 9624 y Fp(in)h(the)f(allo)l(wed)g(time)h (scale)f(and)g(Theorem)h(2.3)876 9823 y(reduces)41 b(to)g(Theorem)h (2.1.)51 b(Only)41 b(if)h Fn(m)k(>)g Fo(1)c Fp(do)g(we)f(obtain)g(ne)l (w)g(results.)1125 10023 y(Let)33 b(us)f(\002nally)h(remark)g(that,)h (when)f Fn(m)46 b(>)g Fo(1)p Fp(,)34 b(we)e(ha)m(v)n(e)g(tw)n(o)h(re)n (gimes)f(:)47 b(either)32 b Fo(\001)47 b(=)6049 9957 y Fj(\026)-68 b Fk(h)p 6049 9984 76 7 v 6059 10080 a(\017)6191 10023 y Fn(<<)47 b(\017)p Fp(,)876 10222 y(where)66 b(the)g(spread)f (of)h(the)g(W)-7 b(igner)67 b(function)f(is)g(lar)m(ger)g(in)h(the)f Fn(q)6 b Fp(-direction)66 b(than)g(in)h(the)f Fn(p)p Fp(-)876 10421 y(direction)55 b(or)g Fo(\001)70 b(=)2142 10356 y Fj(\026)-68 b Fk(h)p 2142 10383 V 2152 10478 a(\017)2308 10421 y Fn(>>)71 b(\017)p Fp(,)58 b(where)c(the)g(spread)g (of)h(the)g(W)-7 b(igner)55 b(function)g(is)f(lar)m(ger)h(in)g(the)3657 10919 y(11)p eop %%Page: 12 12 12 11 bop 876 1455 a Fn(p)p Fp(-direction)41 b(than)f(in)h(the)f Fn(q)6 b Fp(-direction.)51 b(As)40 b(is)h(easily)f(check)n(ed,)f(in)i (the)f(\002rst)h(re)n(gime)f(only)h Fn(\013)6285 1480 y Fk(I)9 b(;I)g(I)g(I)876 1654 y Fp(are)52 b(dif)l(ferent)h(from)g (zero)f(so)g(that)h(the)f(quantum)g(particle)h(ne)l(v)n(er)e(actually)h (visits)g(the)g(sites)g Fn(X)6541 1679 y Fk(`)876 1853 y Fp(and)47 b Fn(X)1301 1878 y Fl(\000)p Fk(`)1516 1853 y Fp(\()p Fn(`)58 b(>)g Fo(1)p Fp(\))48 b(on)g(the)f(time)h(scale)f (considered.)68 b(The)48 b(most)g(interesting)f(case)g(is)g(therefore) 876 2052 y(the)41 b(second)g(re)n(gime.)1125 2252 y(In)36 b(conclusion,)g(our)h(result)f(in)g(Theorem)g(2.3)g(sho)l(ws)f(that)h (the)g(initial)g(state,)g(essentially)g(sup-)876 2451 y(ported)c(in)f(four)i(small)e(neighborhoods)g Fn(C)3329 2476 y Fk(i)3416 2451 y Fp(near)g Fn(X)3871 2476 y Fj(0)3977 2451 y Fp(located)g(inside)g Fo(\012)5036 2476 y Fk(i)5091 2451 y Fp(,)i(for)g Fn(i)46 b Fo(=)g Fn(I)13 b(;)28 b(I)13 b(I)g(;)28 b(I)13 b(I)g(I)g(;)28 b(I)13 b(V)36 b Fp(,)876 2650 y(is)58 b(carried)g(by)g(the)g(time)h(e)l(v)m(olution)f(into)g Fo(4)h Fp(dif)l(ferent)f(small)g(neighbourhoods)h(of)f(one)g(of)h(the) 876 2850 y(critical)45 b(points)f Fn(X)1970 2875 y Fk(j)2040 2850 y Fp(.)60 b(The)45 b(components)f Fn(C)3411 2875 y Fk(I)3531 2850 y Fp(and)g Fn(C)3934 2875 y Fk(I)9 b(I)g(I)4191 2850 y Fp(are)44 b(oscillating)g(between)g Fn(X)5910 2875 y Fj(0)6029 2850 y Fp(and)g Fn(X)6451 2875 y Fl(\006)6563 2850 y Fp(,)876 3049 y(whereas)c Fn(C)1580 3074 y Fk(I)9 b(I)1765 3049 y Fp(and)41 b Fn(C)2165 3074 y Fk(I)9 b(V)2388 3049 y Fp(are)41 b(mo)n(ving)g(in)g(the)g(clockwise)f(direction)i (along)f(the)g Fo(0)g Fp(ener)m(gy)g(curv)n(e.)876 3248 y(A)j(ne)l(w)g(macroscopic)f(distrib)m(ution)i(of)f(the)h(total)f(mass) f(of)i(the)f(initial)h(state)e(occurs)h(at)g(each)f(ap-)876 3447 y(proximate)56 b(period)g(of)g(time)g Fo(\(2)p Fn(`)48 b Fo(+)f(1\))p Fn(T)3316 3472 y Fj(\001)3489 3447 y Fp(\(transition)57 b(times\).)94 b(Actually)-11 b(,)59 b(if)d(the)g(potential)g(is)876 3647 y(con\002ning,)44 b(as)f(in)h(Fig.)58 b(3,)44 b(for)g(e)n(xample,) f(we)g(can)h(easily)f(follo)l(w)h(the)f(e)l(v)m(olution)h(of)g Fn(\026)5925 3672 y Fj(\026)-68 b Fk(h;";t)6245 3647 y Fp(when)876 3846 y(the)41 b(state)g(mak)n(es)g(loops)h(around)f Fo(\006)2954 3871 y Fj(0)3029 3846 y Fp(.)1125 4045 y(As)57 b(in)i(Theorem)f(2.2,)j(we)c(could)h(also)g(describe)f(what)h(happens)f (at)h(the)f(transition)i(times)876 4271 y Fo(\(2)p Fn(`)29 b Fo(+)g(1\))p Fn(T)1548 4185 y Fj(\(0\))1525 4320 y(\001)1765 4271 y Fp(in)40 b(terms)f(of)h(absolutely)e(continuous)i(measures)e(on) h(unstable)g(curv)n(es,)f(assuming)876 4470 y(the)j(condition)81 b Fo(lim)1789 4577 y Fj(\001)p Fl(!)p Fj(0)2124 4470 y Fo(\001)2262 4402 y Fj(2)p Fk(`)p Fj(+1)2561 4470 y Fo(e)2635 4402 y Fk(!)2719 4419 y Fh(0)2783 4402 y Fk(t)2888 4470 y Fo(=)46 b Fn(\024)g(>)g Fo(0)p Fp(.)876 5024 y Fq(3)239 b(Reduction)59 b(to)g(a)g(classical)i(estimate)876 5395 y Fp(Our)g(goal)f(in)h(this)f(section)g(is)g(to)g(pro)n(v)n(e)g (\(7\),)65 b(or)-7 b(,)66 b(more)60 b(precisely)-11 b(,)65 b(Proposition)c(3.7.)107 b(Let)61 b(us)876 5594 y(assume)42 b(that)i(\(H1\))g(and)f(either)h(\(H3\),)g(\(H4\))g(or)g(\(H5\))g(are)f (satis\002ed,)g(together)g(with)h(\(15\).)58 b(Let)44 b Fn(\037)876 5793 y Fp(be)c(a)f(smooth)h(function)g(with)g(compact)g (support)g(such)f(that)h Fn(\037)p Fo(\()p Fn(u)p Fo(\))45 b(=)h(1)40 b Fp(for)h Fm(j)p Fn(u)p Fm(j)46 b(\024)g Fo(2)p Fp(.)51 b(In)40 b(order)g(to)876 5993 y(localize)h(close)g(to)g (the)h(ener)m(gy)f(0,)g(let)h(us)f(introduce)g(the)h(observ)l(able)2825 6421 y Fi(c)2789 6463 y Fn(B)2915 6488 y Fk(E)3073 6463 y Fo(:=)k Fn(\037)3426 6179 y Fi( )3621 6308 y Fo(^)3577 6350 y Fn(H)p 3577 6425 152 7 v 3587 6577 a(E)3749 6179 y Fi(!)3951 6421 y Fo(^)3908 6463 y Fn(A)o(\037)4164 6179 y Fi( )4359 6308 y Fo(^)4315 6350 y Fn(H)p 4315 6425 V 4325 6577 a(E)4487 6179 y Fi(!)4646 6463 y Fn(:)876 6961 y Fp(\277From)60 b(no)l(w)g(on)g(it)g(is)f(understood)h(that)g Fn(A)80 b Fm(2)f(S)12 b Fo(\(I)-37 b(R)4084 6889 y Fj(2)4159 6961 y Fo(\))p Fp(,)63 b(unless)c(we)h(w)n(ork)g(under)g(h)o(ypothesis) 876 7161 y(\(H5\),)41 b(in)h(which)f(case)g Fn(A)k Fm(2)h Fn(C)2676 7100 y Fl(1)2664 7202 y Fj(0)2816 7161 y Fo(\(I)-37 b(R)3026 7088 y Fj(2)3101 7161 y Fo(\))p Fp(.)50 b(Then)42 b(we)f(ha)m(v)n(e)f(\002rst)i(of)g(all:)876 7461 y Fu(Pr)m(oposition)h (3.1)82 b Fe(F)-17 b(or)42 b(e)n(very)e Fn(a)46 b(>)i Fo(0)p Fe(,)41 b(e)n(very)f Fn(N)66 b(>)47 b Fo(0)p Fe(,)41 b(ther)-6 b(e)41 b(e)m(xist)g Fn(C)4955 7486 y Fj(1)5077 7461 y Fn(>)47 b Fo(0)p Fe(,)41 b Fn(C)5538 7486 y Fj(2)5659 7461 y Fn(>)48 b Fo(0)p Fe(,)41 b(suc)n(h)g(that)876 7661 y(for)g(all)h Fn(t)k Fm(2)g Fo(I)-37 b(R)1627 7927 y Fi(\014)1627 8027 y(\014)1627 8126 y(\014)1682 8118 y Fn(<)1901 8076 y Fo(^)1857 8118 y Fn(A)p Fo(\()p Fn(t)p Fo(\))p Fn(\021)2254 8143 y Fk(")2325 8118 y Fn(;)28 b(\021)2481 8143 y Fk(")2598 8118 y Fn(>)46 b Fm(\000)h Fn(<)3160 8076 y Fi(c)3124 8118 y Fn(B)3250 8143 y Fk(E)3362 8118 y Fo(\()p Fn(t)p Fo(\))p Fn(\021)3634 8143 y Fk(")3705 8118 y Fn(;)28 b(\021)3861 8143 y Fk(")3978 8118 y Fn(>)4107 7927 y Fi(\014)4107 8027 y(\014)4107 8126 y(\014)4209 8118 y Fm(\024)46 b Fn(C)4503 8143 y Fj(1)4615 8006 y Fo(\026)-83 b Fn(h)p 4597 8080 133 7 v 4597 8232 a(E)4786 8118 y Fo(+)37 b Fn(C)5071 8143 y Fj(2)5173 7884 y Fi(\022)5381 8006 y Fo(\001)p 5315 8080 271 7 v 5315 8113 a Fm(p)p 5453 8113 133 7 v 140 x Fn(E)5605 7884 y Fi(\023)5728 7918 y Fk(N)6328 8118 y Fp(\(18\))876 8573 y Fe(for)k(e)n(very)f Fn(E)55 b Fm(\024)46 b Fn(a)p Fe(,)41 b Fo(\001)47 b Fm(\024)f Fn(a)2482 8433 y Fm(p)p 2620 8433 V 140 x Fn(E)9 b Fe(.)1125 8874 y Fp(The)66 b(follo)l(wing)g(result)g(then)g(sho)l(ws) e(that)i(the)g(main)g(contrib)m(ution)g(to)g Fn(<)5734 8832 y Fo(^)5690 8874 y Fn(A)o Fo(\()p Fn(t)p Fo(\))p Fn(\021)6086 8899 y Fk(")6157 8874 y Fn(;)28 b(\021)6313 8899 y Fk(")6475 8874 y Fn(>)876 9073 y Fp(comes)41 b(from)h(the)f(e)l 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10348 y Fi(c)2196 10390 y Fn(B)2322 10415 y Fk(E)2433 10390 y Fo(\()p Fn(t)p Fo(\))36 b Fm(\000)h Fn(O)5 b(p)3041 10322 y Fk(w)3041 10431 y Fj(\026)-68 b Fk(h)3148 10390 y Fn(B)3274 10415 y Fk(E)6 b(;)p Fj(0)3491 10390 y Fo(\()p Fn(t)p Fo(\)\))p Fn(\021)3828 10415 y Fk(")3899 10390 y Fn(;)28 b(\021)4055 10415 y Fk(")4172 10390 y Fn(>)4301 10199 y Fi(\014)4301 10299 y(\014)4301 10399 y(\014)4403 10390 y Fm(\024)46 b Fn(C)4697 10415 y Fj(1)4771 10390 y Fo(\026)-83 b Fn(h)4867 10316 y Fk(s)4975 10390 y Fo(+)37 b Fn(C)5260 10415 y Fj(2)5372 10278 y Fo(\026)-83 b Fn(h)5468 10204 y Fj(2)p 5354 10352 207 7 v 5354 10504 a Fn(E)5487 10456 y Fj(3)6328 10390 y Fp(\(19\))3657 10919 y(12)p eop %%Page: 13 13 13 12 bop 876 1455 a Fe(for)41 b(e)n(very)f Fm(j)p Fn(t)p Fm(j)46 b(\024)1903 1389 y Fj(2)p Fl(\000)p Fj(3)p Fk(\016)p 1903 1416 301 7 v 1977 1512 a Fj(3)p Fk(!)2224 1455 y Fm(j)28 b Fo(log)r(\(\026)-83 b Fn(h)o Fo(\))p Fm(j)p Fe(,)41 b Fo(\026)-83 b Fn(h)46 b Fm(\024)g Fn(a)p Fe(,)41 b Fo(\026)-83 b Fn(h)3450 1380 y Fj(2)3570 1455 y Fm(\024)47 b Fn(aE)3967 1394 y Fj(3)4040 1455 y Fe(.)876 1784 y Fp(It)42 b(is)f(not)h(dif)l(\002cult)g(to)g(see)e(that)i(Proposition)g (3.1)f(results)g(directly)h(from)g(the)g(follo)l(wing)g(lemma.)876 2113 y Fu(Lemma)g(3.3)82 b Fe(F)-17 b(or)41 b(e)n(very)f Fn(N)64 b(>)47 b Fo(0)p Fe(,)41 b(e)n(very)e Fn(a)46 b(>)g Fo(0)c Fe(ther)-6 b(e)41 b(e)m(xist)f Fn(C)4633 2138 y Fj(1)4753 2113 y Fn(>)47 b Fo(0)p Fe(,)41 b Fn(C)5214 2138 y Fj(2)5334 2113 y Fn(>)46 b Fo(0)c Fe(suc)n(h)e(that)2241 2327 y Fi(\015)2241 2427 y(\015)2241 2526 y(\015)2241 2626 y(\015)2241 2725 y(\015)2333 2618 y Fn(\037)2465 2334 y Fi( )2659 2463 y Fo(^)2616 2505 y Fn(H)p 2616 2579 152 7 v 2626 2731 a(E)2787 2334 y Fi(!)2946 2618 y Fn(\021)3028 2643 y Fk(")3137 2618 y Fm(\000)d Fn(\021)3385 2643 y Fk(")3456 2327 y Fi(\015)3456 2427 y(\015)3456 2526 y(\015)3456 2626 y(\015)3456 2725 y(\015)3595 2618 y Fm(\024)46 b Fn(C)3889 2643 y Fj(1)4001 2505 y Fo(\026)-83 b Fn(h)p 3983 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4188 V 4397 4340 a(E)4558 3992 y Fi(\023)q(\023)4803 4027 y Fj(2)4914 4226 y Fo(+)g Fn(O)5 b Fo(\()5314 4114 y(\026)-83 b Fn(h)p 5297 4188 133 7 v 5297 4340 a(E)5448 4226 y Fo(\))p Fn(:)876 4726 y Fp(But,)41 b(using)g(the)h(W)-7 b(igner)41 b(function)h(of)g Fn(\021)3234 4751 y Fk(")3347 4726 y Fp(we)f(ha)m(v)n(e)988 5216 y Fn(<)46 b(O)5 b(p)1379 5148 y Fk(w)1379 5257 y Fj(\026)-68 b Fk(h)1513 4982 y Fi(\022)1635 5216 y Fo(1)37 b Fm(\000)g Fn(\037)2053 4982 y Fi(\022)2195 5104 y Fn(H)p 2195 5178 152 7 v 2232 5330 a(")2366 4982 y Fi(\023)q(\023)2611 5017 y Fj(2)2713 5216 y Fn(\021)2795 5241 y Fk(")2866 5216 y Fn(;)28 b(\021)3022 5241 y Fk(")3140 5216 y Fn(>)p Fo(=)3444 4990 y Fi(Z)3536 5368 y Fj(I)-30 b(R)3651 5329 y Fh(2)3751 4982 y Fi(\022)3874 5216 y Fo(1)37 b Fm(\000)g Fn(\037)4292 4982 y Fi(\022)4433 5104 y Fn(H)13 b Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\))p 4433 5178 519 7 v 4626 5330 a Fn(E)4971 4982 y Fi(\023)q(\023)5216 5017 y Fj(2)5318 5216 y Fn(\032)5432 5032 y Fi(\020)5551 5104 y Fn(q)p 5551 5178 81 7 v 5552 5330 a(")5651 5216 y(;)g(p)5837 5104 y(")p 5829 5178 96 7 v 5829 5330 a Fo(\026)-83 b Fn(h)5943 5032 y Fi(\021)6090 5104 y Fn(dq)6 b(dp)p 6090 5178 337 7 v 6119 5330 a Fo(2)p Fn(\031)g Fo(\026)-83 b Fn(h)6447 5216 y(;)876 5666 y Fp(so)41 b(that)h(we)f(ha)m(v)n(e)f(to)i(estimate)2261 6074 y Fn(F)23 b Fo(\()p Fn(E)10 b(;)28 b Fo(\026)-83 b Fn(h)n Fo(\))46 b(:=)3089 5848 y Fi(Z)3182 6225 y Fj([)o Fl(j)p Fk(H)9 b Fj(\()p Fk(q)t(;p)p Fj(\))p Fl(j\025)p Fk(E)d Fj(])3974 5883 y Fi(\014)3974 5983 y(\014)3974 6082 y(\014)4029 6074 y Fn(\032)4143 5890 y Fi(\020)4262 5962 y Fn(q)p 4262 6036 81 7 v 4263 6188 a(")4362 6074 y(;)28 b(p)4548 5962 y(")p 4540 6036 96 7 v 4540 6188 a Fo(\026)-83 b Fn(h)4654 5890 y Fi(\021)4754 5883 y(\014)4754 5983 y(\014)4754 6082 y(\014)4837 6074 y Fn(dq)6 b(dp:)876 6558 y Fp(Because)40 b Fn(\032)h Fp(is)g(f)n(ast)h(decreasing)e(at)i Fm(1)f Fp(we)g(ha)m(v)n(e)g(for)h(e)l(v)n(ery)e Fn(\022)51 b Fm(2)p Fo(]0)p Fn(;)28 b Fo(1[)p Fp(,)2151 6779 y Fi(Z)101 b(Z)2437 7156 y Fj([)o Fl(j)p Fk(q)t Fl(j)p Fj(+)p Fl(j)p Fk(p)p Fl(j\025)p Fj(\001)3078 7123 y Ff(\022)3148 7156 y Fj(])3221 6814 y Fi(\014)3221 6914 y(\014)3221 7013 y(\014)3277 7005 y Fn(\032)3391 6821 y Fi(\020)3509 6893 y Fn(q)p 3509 6967 81 7 v 3510 7119 a(")3609 7005 y(;)28 b(p)3796 6893 y(")p 3787 6967 96 7 v 3787 7119 a Fo(\026)-83 b Fn(h)3902 6821 y Fi(\021)4001 6814 y(\014)4001 6914 y(\014)4001 7013 y(\014)4084 7005 y Fn(dq)6 b(dp)47 b Fo(=)f Fn(O)5 b Fo(\(\001)4977 6936 y Fj(+)p Fl(1)5219 7005 y Fo(\))p Fn(:)876 7478 y Fp(But)41 b(close)g(to)h(0)f(we)g(can)g (use)g(the)h(T)-13 b(aylor)42 b(formula:)2288 7912 y Fn(H)13 b Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\))45 b(=)3047 7800 y(1)p 3047 7874 84 7 v 3047 8026 a(2)3178 7778 y Fi(\000)3254 7912 y Fn(p)3338 7844 y Fj(2)3449 7912 y Fm(\000)37 b Fn(!)3724 7844 y Fj(2)3718 7953 y(0)3799 7912 y Fn(q)3879 7844 y Fj(2)3953 7778 y Fi(\001)4066 7912 y Fo(+)g Fn(O)5 b Fo(\()p Fm(j)p Fn(q)h Fm(j)4601 7844 y Fj(3)4712 7912 y Fo(+)37 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y(\014)5161 8951 y Fn(dq)6 b(dp)37 b Fo(+)g Fn(O)5 b Fo(\(\001)6035 8882 y Fj(+)p Fl(1)6277 8951 y Fo(\))p Fn(:)876 9586 y Fp(The)34 b(conclusion)f(of)h(the)g (lemma)g(will)g(therefore)g(be)f(a)h(consequence)e(of)i(the)g(follo)l (wing)g(estimate:)876 9785 y(for)42 b(e)l(v)n(ery)f Fn(\032)g Fp(in)h(the)f(Schw)n(artz)h(space)e Fm(S)12 b Fo(\(I)-37 b(R)3448 9713 y Fj(2)3522 9785 y Fo(\))42 b Fp(and)f(e)l(v)n(ery)g Fn(N)64 b Fm(\025)46 b Fo(1)c Fp(we)f(ha)m(v)n(e)1561 10297 y Fo(\(2)p Fn(\031)6 b Fo(\026)-83 b Fn(h)o Fo(\))1970 10228 y Fl(\000)p Fj(1)2176 10071 y Fi(Z)101 b(Z)2462 10460 y Fo([)2508 10448 y Fl(j)p Fk(p)2615 10415 y Fh(2)2680 10448 y Fl(\000)p Fk(!)2872 10408 y Fh(2)2868 10488 y(0)2937 10448 y Fk(q)3002 10415 y Fh(2)3067 10448 y Fl(j\025)p Fk(E)3313 10460 y Fo(])3396 10106 y Fi(\014)3396 10205 y(\014)3396 10305 y(\014)3451 10297 y Fn(\032)3565 10112 y Fi(\020)3684 10184 y Fn(q)p 3684 10258 81 7 v 3685 10410 a(")3784 10297 y(;)28 b(p)3970 10184 y(")p 3962 10258 96 7 v 3962 10410 a Fo(\026)-83 b Fn(h)4077 10112 y Fi(\021)4176 10106 y(\014)4176 10205 y(\014)4176 10305 y(\014)4259 10297 y Fn(dq)6 b(dp)46 b Fo(=)h Fn(O)4976 10013 y Fi( )5107 10062 y(\022)5315 10184 y Fo(\001)p 5249 10258 271 7 v 5249 10292 a Fm(p)p 5388 10292 133 7 v 5388 10432 a Fn(E)5540 10062 y Fi(\023)5662 10097 y Fk(N)5788 10013 y Fi(!)6328 10297 y Fp(\(21\))3657 10919 y(13)p eop %%Page: 14 14 14 13 bop 876 1455 a Fp(T)-13 b(o)50 b(pro)n(v)n(e)f(this,)j(we)d (start)h(by)g(the)g(change)g(of)g(v)l(ariables)f Fn(q)68 b Fo(=)62 b Fn("u)p Fp(,)52 b Fn(p)62 b Fo(=)5212 1389 y Fj(\026)-68 b Fk(h)p 5212 1416 76 7 v 5219 1512 a(")5307 1455 y Fn(v)57 b Fp(and)50 b(we)f(introduce)914 1639 y Fo(~)876 1681 y Fn(E)93 b Fo(:=)1406 1616 y Fk(E)p 1371 1643 174 7 v 1371 1739 a Fj(\001)1480 1706 y Fh(2)1565 1681 y Fp(.)112 b(Assuming)62 b Fo(\001)84 b(=)g Fn(\017)63 b Fp(\(the)f(case)f(where)g Fo(\001)84 b(=)h(\026)-83 b Fn(h)o(=\017)63 b Fp(is)e(treated)h(similarly\),)67 b(it)c(is)876 1908 y(not)f(dif)l(\002cult)h(to)f(see)f(that)h(we)f(ha)m (v)n(e)g(to)h(estimate)f(for)4235 1866 y Fo(~)4197 1908 y Fn(E)93 b Fm(!)84 b Fo(+)p Fm(1)p Fp(,)67 b(the)62 b(follo)l(wing)g(inte)n(grals)876 2107 y(depending)41 b(on)h(the)f(parameter)g Fn(c)46 b Fm(2)p Fo(]0)p Fn(;)28 b(c)3270 2132 y Fj(0)3344 2107 y Fo([)p Fp(,)42 b Fn(c)3546 2132 y Fj(0)3666 2107 y Fn(>)k Fo(0)p Fp(.)2467 2558 y Fn(I)13 b Fo(\()2656 2516 y(~)2618 2558 y Fn(E)8 b(;)28 b(c)p Fo(\))46 b(=)3181 2332 y Fi(Z)3273 2709 y Fl(j)p Fk(c)3371 2676 y Fh(2)3437 2709 y Fk(p)3505 2676 y Fh(2)3570 2709 y Fl(\000)p Fk(q)3739 2676 y Fh(2)3804 2709 y Fl(j\025)3976 2680 y Fj(~)3947 2709 y Fk(E)4087 2558 y Fm(j)p Fn(\032)p Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\))p Fm(j)p Fn(dq)6 b(dp:)876 3094 y Fp(Considering)55 b(the)g(tw)n(o)h(dif)l(ferent)g(cases)d Fn(q)3412 3033 y Fj(2)3559 3094 y Fm(\025)3809 2999 y Fj(~)3779 3028 y Fk(E)3882 2978 y Fh(2)p 3779 3055 169 7 v 3831 3151 a Fj(4)4023 3094 y Fp(and)i Fn(q)4398 3033 y Fj(2)4545 3094 y Fm(\024)4795 2999 y Fj(~)4765 3028 y Fk(E)4868 2978 y Fh(2)p 4765 3055 V 4817 3151 a Fj(4)5009 3094 y Fp(we)g(get)h(easily)-11 b(,)57 b(for)f(e)l(v)n(ery)876 3293 y Fn(N)65 b Fm(\025)46 b Fo(1)p Fp(,)3072 3492 y Fn(I)13 b Fo(\()3261 3450 y(~)3223 3492 y Fn(E)c(;)28 b(c)p Fo(\))45 b(=)h Fn(O)5 b Fo(\()4020 3450 y(~)3983 3492 y Fn(E)4116 3423 y Fl(\000)p Fk(N)4344 3492 y Fo(\))876 3791 y Fp(uniformly)45 b(in)f Fn(c)51 b Fm(2)p Fo(]0)p Fn(;)28 b(c)2266 3816 y Fj(0)2340 3791 y Fo([)p Fp(.)58 b(This)44 b(implies)g(\(21\),)h(and)e(ends)h(the)g(proof)g(of)g(the)g (Lemma,)h(which)e(in)876 3990 y(turn)f(implies)f(Proposition)i(3.1.)p 6404 4023 93 133 v 1125 4289 a(The)k(proof)h(of)g(Proposition)f(3.2)g (will)h(be)f(obtained)g(by)g(adapting)g(the)g(methods)g(of)g([BR])g(to) 876 4488 y(Hamiltonions)38 b(depending)f(on)h(the)g(e)n(xtra)f (parameter)h Fn(")p Fp(.)50 b(The)38 b(application)g(of)g(the)g (constructions)876 4688 y(of)51 b([BR)o(])g(is)f(relati)l(v)n(ely)g 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b(;\034)f(;R)6070 5817 y Fe(with)46 b(the)876 6016 y(following)37 b(pr)-7 b(operties.)48 b(F)-17 b(or)36 b(e)n(very)f Fn(E)10 b Fe(,)36 b Fo(0)46 b Fn(<)g(E)56 b Fm(\024)46 b Fn(E)3961 6041 y Fj(1)4035 6016 y Fe(,)37 b(ther)-6 b(e)36 b(e)m(xists)f Fn(\016)4962 6041 y Fk(E)5111 6016 y Fe(satisfying)72 b Fo(lim)5784 6123 y Fk(E)6 b Fl(!)p Fj(0)6114 6016 y Fn(\016)6188 6041 y Fk(E)6346 6016 y Fo(=)46 b(0)876 6281 y Fe(and)c(suc)n(h)e(that)i(for)f(e)n(very) f Fn(X)13 b Fe(,)40 b Fm(j)p Fn(H)13 b Fo(\()p Fn(X)g Fo(\))p Fm(j)46 b(\024)g Fn(E)10 b Fe(,)40 b(we)h(have)2308 6505 y Fi(\014)2308 6605 y(\014)2364 6646 y Fn(@)2461 6566 y Fk(\015)2452 6695 y(X)2578 6646 y Fo(\010)2698 6577 y Fk(t)2698 6687 y(H)2824 6646 y Fo(\()p Fn(X)13 b Fo(\))3105 6505 y Fi(\014)3105 6605 y(\014)3205 6646 y Fm(\024)46 b Fn(C)3499 6671 y Fk(\015)7 b(;\034)f(;R)3859 6646 y Fo(exp[\()p Fn(!)43 b Fo(+)37 b Fn(\016)4610 6671 y Fk(E)4722 6646 y Fo(\))p Fm(j)p Fn(\015)9 b Fm(jj)p Fn(t)p Fm(j)p Fo(])1156 b Fp(\(22\))876 7043 y Fe(for)41 b(all)h Fm(j)p Fn(t)p Fm(j)k(\024)h Fn(\034)19 b Fm(j)28 b Fo(log)h Fn(E)10 b Fm(j)41 b Fe(and,)g(if)h(\(H5\))g(holds,)f(pr)-7 b(o)n(vided)40 b Fm(j)p Fo(\010)4381 6983 y Fk(t)4431 6933 y Fg(0)4381 7089 y Fk(H)4507 7043 y Fo(\()p Fn(X)13 b Fo(\))p Fm(j)45 b Fn(<)h(R)q(;)28 b Fo(0)47 b Fm(\024)f Fn(t)5620 6983 y Fl(0)5713 7043 y Fm(\024)g Fn(t)p Fe(.)876 7375 y Fu(Pr)m(oof)j(of)g(Pr)m(oposition)h(3.2)p Fp(:)66 b(Using)48 b(Lemma)h(3.4)g(and)f(w)n(orking)i(as)e(in)h([BR])g(we)f (shall)g(obtain)876 7575 y(Proposition)j(3.2.)77 b(F)n(or)51 b(that)f(purpose)g(we)g(\002rst)h(easily)f(pro)n(v)n(e)f(the)i(follo)l (wing)g(estimates,)g(using)876 7774 y(Lemma)42 b(3.4,)1771 8171 y Fm(j)p Fn(@)1914 8091 y Fk(\015)1905 8220 y(X)2031 8171 y Fn(B)2157 8196 y Fk(E)6 b(;)p Fj(0)2375 8171 y Fo(\(\010)2560 8102 y Fk(t)2560 8212 y(H)2686 8171 y Fo(\()p Fn(X)13 b Fo(\)\))p Fm(j)44 b(\024)i Fn(C)3416 8196 y Fj(0)p Fk(;\015)3635 7987 y Fi(\020)3734 8171 y Fn(E)3867 8102 y Fl(j)p Fk(\015)7 b 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Fp(be)h(one)g(of)h(the)f(critical)h(points)f(appearing) 876 5710 y(in)62 b(the)f(h)o(ypotheses.)110 b(Then)62 b(we)f(ha)m(v)n(e)f(the)i(follo)l(wing)g(reduction)f(to)h(a)f(normal)h (form)g(for)h(the)876 5909 y(Hamiltonian)42 b Fn(H)55 b Fp(\(see)41 b([CV)o(,)g(HS)q(])g(for)i(a)e(proof\).)876 6241 y Fu(Pr)m(oposition)i(3.5)82 b Fe(Ther)-6 b(e)59 b(e)m(xists)g Fn(\032)80 b(>)h Fo(0)60 b Fe(,)k(a)c(smooth)g (symplectic)f(dif)m(feomorphism)g Fn(F)6168 6266 y Fk(j)6298 6241 y Fe(fr)-7 b(om)876 6458 y Fn(D)1013 6483 y Fk(\032;)1121 6500 y Ff(j)1238 6458 y Fo(=])p Fn(q)1487 6483 y Fk(j)1584 6458 y Fm(\000)26 b Fn(\032;)i(q)1973 6483 y Fk(j)2069 6458 y Fo(+)e Fn(\032)p Fo([)p Fm(\002)p Fo(])g Fm(\000)g Fn(\032;)i(\032)p Fo([)40 b Fe(into)f(a)g(neighborhood)4495 6416 y Fo(~)4433 6458 y Fn(D)4570 6483 y Fk(j)4679 6458 y Fe(of)g(the)f(origin)h(\(0,)g(0\))g(in)g Fo(I)-37 b(R)6242 6385 y Fj(2)6355 6458 y Fe(and)876 6657 y(a)41 b(smooth)h(function)f Fn(f)2188 6682 y Fk(j)2305 6657 y Fo(:)46 b 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Fj(0)3922 2687 y Fo(=)4128 2622 y Fj(1)p 4117 2649 88 7 v 4117 2744 a Fk(!)4225 2687 y Fp(;)h(the)e(reason)f(is)h(that)5480 2622 y Fj(1)p 5469 2649 V 5469 2744 a Fk(!)5577 2687 y Fm(j)28 b Fo(log)r(\()p Fn(E)10 b Fo(\))p Fm(j)35 b Fp(is)g(less)876 2886 y(than)41 b(the)g(time)g(needed)g(to)g(tra)m(v)n(el)f(accross)g(an)n(y)g(one)h (box)g Fn(D)4394 2916 y Fk(\032=)p Fj(2)p Fk(;j)4749 2886 y Fp(on)g(a)f(path)i(of)f(ener)m(gy)g Fm(\006)p Fn(E)10 b Fp(,)40 b(for)876 3086 y Fn(E)48 b Fp(small)39 b(enough)f(\(since)h Fn(!)52 b Fm(\025)47 b Fn(!)2843 3111 y Fk(j)2913 3086 y Fp(\).)j(W)-13 b(e)38 b(\002rst)i(gi)l(v)n(e)e (the)h(proof)g(under)g(conditions)g(\(H3\))g(or)g(\(H4\).)876 3285 y(Let)49 b Fn(\032)59 b(>)g Fo(0)49 b Fp(be)f(\002x)n(ed)h(and)f (small)g(enough)h(so)f(that)h(the)f(box)n(es)89 b Fm(f)p Fn(D)4869 3310 y Fk(\032;j)5048 3285 y Fn(;)74 b Fm(\000)p Fn(N)5430 3310 y Fl(\000)5589 3285 y Fm(\024)46 b Fn(j)56 b Fm(\024)46 b Fn(N)6196 3310 y Fj(+)6307 3285 y Fm(g)i Fp(do)876 3484 y(not)53 b(intersect.)85 b(In)54 b(what)f(follo)l(ws)g (we)f(al)n(w)n(ays)h(assume)f Fn(t)68 b Fm(\025)f Fo(0)p Fp(,)56 b(so)c Fn(t)i Fp(will)f(be)g(increasing,)i(and)876 3683 y Fn(E)71 b(>)62 b Fo(0)p Fp(.)76 b(\(The)50 b(other)g(cases)f (are)g(similar)h(and)g(will)g(not)h(be)e(gi)l(v)n(en)g(in)i(detail.\)) 76 b(As)49 b(a)h(result,)i(the)876 3883 y(classical)32 b(paths)h(are)h(oriented)g(by)f(the)h(time)g(and)f(the)h(notion)g(of)g (consecuti)l(v)n(e)e(box)n(es)h(is)g(relati)l(v)n(e)g(to)876 4082 y(this)43 b(ordering.)57 b(Since)43 b Fn(E)58 b(>)49 b Fo(0)p Fp(,)43 b(the)h(classical)d(path)i(mo)n(v)n(es)f(from)i(\(the) f(neighbourhood)h(of\))g(one)876 4281 y(critical)53 b(point)f Fn(X)1921 4306 y Fk(j)2043 4281 y Fp(to)h(the)g(ne)n(xt.)83 b(F)n(or)53 b(con\002ning)g(potentials)f(\(condition)h(\(H3\)\),)i (after)e(visiting)876 4480 y(\(a)d(neighbourhood)g(of\))g Fn(X)2491 4505 y Fk(N)2596 4522 y Fh(+)2698 4480 y Fp(,)h(the)f(path)g (is)f(re\003ected)h(by)f(the)h(turning)h(point)f Fn(q)5625 4505 y Fk(N)5730 4522 y Fh(+)5824 4505 y Fj(+1)6000 4480 y Fp(,)h(turns)f(to)876 4680 y(the)40 b(left)g(to)g(visit)g Fn(X)2013 4705 y Fk(N)2118 4722 y Fh(+)2259 4680 y Fp(ag)o(ain)g(and)f (then)h(continues)f(on)h(to)l(w)n(ards)g Fn(X)4843 4705 y Fk(N)4948 4722 y Fh(+)5041 4705 y Fl(\000)p Fj(1)5219 4680 y Fp(,)g(and)g(so)f(on;)i(the)e(same)876 4879 y(for)46 b Fn(X)1253 4904 y Fl(\000)p Fk(N)1462 4921 y Fg(\000)1559 4904 y Fj(1)1679 4879 y Fp(where,)f(arri)l(ving)h(from)g(the)f(left,)h (the)f(particle)h(goes)e(after)h(re\003ection)h(to)f(the)g(right.)876 5078 y(Under)c(condition)h(\(H4\),)g(the)f(particle)h(just)f(k)n(eeps)g (going)h(in)f(one)h(direction.)1125 5278 y(First)k(of)g(all,)g(it)g(is) f(clear)g(that)g(there)h(e)n(xists)e(a)h(time)h Fn(T)4225 5303 y Fj(1)4345 5278 y Fp(such)f(that,)h(for)g(e)l(v)n(ery)f Fo(0)53 b Fm(\024)h Fn(E)62 b Fm(\024)54 b Fn(E)6489 5303 y Fj(1)6563 5278 y Fp(,)876 5477 y(the)65 b(time)g Fn(t)1564 5502 y Fj(1)p Fk(;j)1740 5477 y Fo(\()p Fn(E)10 b Fo(\))63 b Fp(it)j(tak)n(es)e(to)i(tra)m(v)n(el)e(along)h Fm(j)p Fn(H)13 b Fo(\()p Fn(X)g Fo(\))p Fm(j)89 b Fo(=)h Fn(E)75 b Fp(between)64 b(\(the)h(edges)f(of\))i(an)n(y)876 5676 y(tw)n(o)49 b(consecuti)l(v)n(e)d(box)n(es)i Fn(D)2568 5706 y Fk(\032=)p Fj(2)p Fk(;j)2929 5676 y Fp(and)g Fn(D)3354 5706 y Fk(\032=)p Fj(2)p Fk(;j)7 b Fj(+1)3884 5676 y Fp(satis\002es)47 b Fm(j)p Fn(t)4564 5701 y Fj(1)p Fk(;j)4740 5676 y Fo(\()p Fn(E)10 b Fo(\))p Fm(j)57 b(\024)j Fn(T)5392 5701 y Fj(1)5466 5676 y Fp(,)50 b(and)e(is)g(therefore)876 5875 y(uniformly)h(bounded)f(for)g Fo(0)58 b Fm(\024)g Fn(E)67 b Fm(\024)58 b Fn(E)3277 5900 y Fj(1)3351 5875 y Fp(.)70 b(It)48 b(is)f(clearly)h(suf)l(\002cient)g(to)g(pro)n(v)n(e)f (the)g(result)h(for)g(all)876 6075 y Fn(t)e Fm(\025)g Fn(T)1254 6100 y Fj(1)1329 6075 y Fp(.)51 b(W)-13 b(e)40 b(shall)i(consider)f(the)g(follo)l(wing)h(tw)n(o)g(cases.)876 6274 y Fu(1)p Fp(.)55 b Fn(X)80 b(=)-102 b Fm(2)49 b([)1526 6299 y Fk(j)p 1595 6140 450 7 v 1595 6274 a Fn(D)1732 6304 y Fk(\032=)p Fj(2)p Fk(;j)2045 6274 y Fp(.)56 b(In)43 b(other)g(w)n(ords,)g(the)g(initial)g(condition)g(is)g(situated)f(some) l(where)g(between)876 6473 y(tw)n(o)h(consecuti)l(v)n(e)d(box)n(es)h Fn(D)2549 6503 y Fk(\032=)p Fj(2)p Fk(;j)2846 6520 y Fh(0)2963 6473 y Fp(and)h Fn(D)3382 6503 y Fk(\032=)p Fj(2)p Fk(;j)3679 6520 y Fh(0)3745 6503 y Fj(+1)3921 6473 y Fp(.)54 b(Then)42 b(there)g(e)n(xists)g Fn(t)5253 6498 y Fj(2)5327 6473 y Fo(\()p Fn(E)10 b Fo(\))46 b Fm(\024)i Fn(t)5873 6498 y Fj(1)5947 6473 y Fo(\()p Fn(E)10 b Fo(\))46 b Fm(\024)i Fn(T)6530 6498 y Fj(1)876 6722 y Fp(\(we)41 b(drop)g(the)g(inde)n(x)g Fn(j)2231 6747 y Fj(0)2306 6722 y Fp(\))g(such)f(that)i Fo(\010)3158 6636 y Fk(t)3208 6653 y Fh(2)3272 6636 y Fj(\()p Fk(E)6 b Fj(\))3158 6771 y Fk(H)3488 6722 y Fo(\()p Fn(X)13 b Fo(\))40 b Fp(belongs)h(to)g(the)g(edge)f(of)i Fn(D)5452 6752 y Fk(\032=)p Fj(2)p Fk(;j)5749 6769 y Fh(0)5815 6752 y Fj(+1)5991 6722 y Fp(.)51 b(In)41 b(other)876 6921 y(w)n(ords,)63 b(it)c(tak)n(es)g(a)g(\002nite)g(time)g(less)f (than)h Fn(T)3572 6946 y Fj(1)3706 6921 y Fp(for)h(the)e(particle)h(to) h(reach)e(the)h(ne)n(xt)g(box.)103 b(Let)876 7121 y(us)51 b(furthermore)i(remark)f(that)f(the)h(particle)g(is)f(entering)h(the)f (small)h(box)f Fn(D)5408 7151 y Fk(\032=)p Fj(2)p Fk(;j)5705 7168 y Fh(0)5771 7151 y Fj(+1)5999 7121 y Fp(at)h Fn(X)6309 7146 y Fk(j)6364 7163 y Fh(0)6428 7146 y Fj(+1)876 7320 y Fp(after)45 b(a)f(time)h Fn(t)1743 7345 y Fj(2)1818 7320 y Fo(\()p Fn(E)10 b Fo(\))50 b Fm(\024)i Fn(T)2409 7345 y Fj(1)2528 7320 y Fp(and)45 b(that)f(it)h(will)g(then)g(stay)f (there)h(for)g(an)f(amount)h(of)g(time)g(of)g(order)876 7551 y Fn(!)985 7480 y Fl(\000)p Fj(1)979 7598 y Fk(j)1034 7615 y Fh(0)1164 7551 y Fm(j)28 b Fo(log)h Fn(E)10 b Fm(j)70 b(\025)g Fn(\034)2000 7576 y Fj(0)2074 7551 y Fm(j)28 b Fo(log)i Fn(E)10 b Fm(j)p Fp(.)88 b(Writing)55 b Fo(\010)3390 7491 y Fk(t)3390 7597 y(H)3516 7551 y Fo(\()p Fn(X)13 b Fo(\))68 b(=)i(\010)4184 7465 y Fk(t)p 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10162 y Fk(t)1462 10268 y(H)1588 10222 y Fo(\()p Fn(X)13 b Fo(\))80 b Fm(2)i Fn(D)2279 10247 y Fk(\032;j)2442 10264 y Fh(0)2516 10222 y Fp(,)64 b(for)e(all)e Fo(0)82 b Fm(\024)g Fn(t)f Fm(\024)h Fn(\034)3904 10247 y Fj(0)3978 10222 y Fm(j)28 b Fo(log)i Fn(E)10 b Fm(j)p Fp(.)107 b(This)61 b(means)f(that)h(at)f(the)h (initial)876 10421 y(v)l(alue)47 b Fn(X)60 b Fp(the)48 b(trajectory)f(is)h(just)f(\223entering\224)g(the)h(box)f Fn(D)4280 10451 y Fk(\032=)p Fj(2)p Fk(;j)4577 10468 y Fh(0)4699 10421 y Fp(and)g(therefore)h(spends)f(at)g(least)3657 10919 y(16)p eop %%Page: 17 17 17 16 bop 876 1455 a Fp(a)60 b(time)i Fn(\034)1440 1480 y Fj(0)1514 1455 y Fm(j)28 b Fo(log)i Fn(E)10 b Fm(j)60 b Fp(before)g(lea)m(ving)h(ag)o(ain.)109 b(In)61 b(that)g(case)e(one)i (can)f(apply)h(the)f(normal)i(form)876 1654 y(directly)e(during)g(the)g (entire)g(time)g(span)f(and)g(obtain)h(the)g(result)f(as)g(abo)n(v)n (e.)104 b(If,)65 b(on)60 b(the)f(other)876 1853 y(hand,)46 b(there)g(e)n(xists)f Fo(0)54 b 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Fg(0)2614 3349 y Fk(H)2740 3303 y Fo(\()p Fn(X)13 b Fo(\))49 b Fp(enters)h(a)f(box)i Fn(D)4077 3333 y Fk(\032=)p Fj(2)p Fk(;j)4374 3350 y Fh(0)4448 3303 y Fp(,)h(for)e(some)g Fn(j)5254 3328 y Fj(0)5329 3303 y Fp(.)76 b(Under)51 b(conditions)876 3502 y(\(H3\))c(or)f(\(H4\),)h(this)f(is)g(true)h(for)g(all)f Fn(X)58 b Fp(with)47 b Fm(j)p Fn(H)13 b Fo(\()p Fn(X)g Fo(\))p Fm(j)54 b(\024)h Fn(E)4445 3527 y Fj(1)4519 3502 y Fp(.)65 b(In)47 b(the)f(case)e(of)j(condition)f(\(H5\),)876 3701 y(this)34 b(is)g(still)h(true)f(for)h(all)f Fn(X)59 b Fo(=)46 b(\()p Fn(q)6 b(;)28 b(p)p Fo(\))33 b Fp(with)i Fn(q)3476 3726 y Fk(N)3581 3743 y Fg(\000)3679 3726 y Fl(\000)p Fj(1)3867 3701 y Fo(+)10 b Fn(\032)47 b(<)f(q)52 b(<)46 b(q)4689 3726 y Fk(N)4794 3743 y Fh(+)4888 3726 y Fj(+1)5075 3701 y Fm(\000)10 b Fn(\032)p Fp(,)35 b(b)m(ut)f(not)h (necessarily)876 3901 y(otherwise,)41 b(since)g(some)g(trajectories)g (escape)f(to)i(in\002nity)-11 b(.)53 b(Let)42 b(us)f(therefore)h (consider)f(the)h(case)876 4100 y(where)i Fn(q)1400 4125 y Fk(N)1505 4142 y Fh(+)1599 4125 y Fj(+1)1827 4100 y Fn(<)52 b(q)58 b(<)52 b(R)46 b Fp(and)e(call)g(the)h(point)g Fn(X)57 b Fp(\223outgoing\224)44 b(if)h(the)g(trajectory)g(starting)f (at)h Fn(X)876 4299 y Fp(does)e(not)g(meet)g(the)h(box)f Fn(D)2521 4329 y Fk(\032=)p Fj(2)p Fk(;N)2868 4346 y Fh(+)3015 4299 y Fp(in)g(the)h(future)g(or)f(lea)m(v)n(es)f(it)i (before)f Fn(\034)5184 4324 y Fj(0)5258 4299 y Fm(j)28 b Fo(log)i Fn(E)10 b Fm(j)p Fp(.)56 b(The)44 b(crucial)876 4498 y(observ)l(ation)i(is)h(no)l(w)g(that,)g(in)g(that)g(case,)g(it)g (tak)n(es)g(at)f(most)h(a)g(\002nite)g(time)g Fn(T)5376 4523 y Fk(R)5532 4498 y Fp(to)g(reach)f Fn(q)62 b Fo(=)57 b Fn(R)q Fp(,)876 4698 y(uniformly)48 b(in)f Fn(E)10 b Fp(;)49 b(for)f(such)e(initial)h(conditions)g(the)g(estimate)f(is)h (tri)l(vial.)68 b(The)47 b(same)f(reasoning)876 4897 y(w)n(orks)c(for)g(outgoing)g(initial)g(conditions)f(with)h Fn(q)52 b(<)46 b(q)4063 4922 y Fk(N)4168 4939 y Fg(\000)4267 4922 y Fl(\000)p Fj(1)4482 4897 y Fm(\000)37 b Fn(\032)p Fp(.)p 6404 4930 93 133 v 1125 5185 a(Until)42 b(no)l(w)f(we)g(ha)m(v)n (e)f(pro)n(v)n(ed)h(that)h(for)g(some)f Fn(s)46 b(>)g Fo(0)p Fp(,)41 b(for)h(e)l(v)n(ery)f Fn(N)65 b Fm(\025)46 b Fo(1)41 b Fp(we)g(ha)m(v)n(e)1041 5667 y Fn(<)1260 5625 y Fo(^)1216 5667 y Fn(A)p Fo(\()p Fn(t)p Fo(\))p Fn(\021)1613 5692 y Fk(")1683 5667 y Fn(;)28 b(\021)1839 5692 y Fk(")1957 5667 y Fn(>)p Fo(=)p Fn(<)46 b(O)5 b(p)2606 5599 y Fk(w)2606 5708 y Fj(\026)-68 b Fk(h)2713 5667 y Fo([)p Fn(B)2885 5692 y Fk(E)6 b(;)p Fj(0)3103 5667 y Fo(\()p Fn(t)p Fo(\)])p Fn(\021)3421 5692 y Fk(")3491 5667 y Fn(;)28 b(\021)3647 5692 y Fk(")3765 5667 y Fn(>)46 b Fo(+)p Fn(O)4228 5383 y Fi( )4359 5667 y Fo(\026)-83 b Fn(h)4455 5593 y Fk(s)4563 5667 y Fo(+)4767 5555 y(\026)g Fn(h)4863 5481 y Fj(2)p 4749 5629 207 7 v 4749 5781 a Fn(E)4882 5733 y Fj(3)5012 5667 y Fo(+)5217 5555 y(\026)g Fn(h)p 5198 5629 133 7 v 5198 5781 a(E)5387 5667 y Fo(+)5553 5433 y Fi(\022)5762 5555 y Fo(\001)p 5696 5629 271 7 v 5696 5662 a Fm(p)p 5834 5662 133 7 v 141 x Fn(E)5986 5433 y Fi(\023)6108 5468 y Fk(N)6234 5383 y Fi(!)6393 5667 y Fn(;)876 6144 y Fp(where)1052 6543 y Fn(<)46 b(O)5 b(p)1443 6474 y Fk(w)1443 6584 y Fj(\026)-68 b Fk(h)1550 6543 y Fo([)p Fn(B)1722 6568 y Fk(E)6 b(;)p Fj(0)1939 6543 y Fo(\()p Fn(t)p Fo(\)])p Fn(\021)2257 6568 y Fk(")2328 6543 y Fn(;)28 b(\021)2484 6568 y Fk(")2601 6543 y Fn(>)p Fo(=)2906 6317 y Fi(Z)46 b(Z)3137 6694 y Fj(I)-30 b(R)3251 6656 y Fh(2)3351 6543 y Fn(\037)3455 6474 y Fj(2)3557 6309 y Fi(\022)3700 6430 y Fn(H)13 b Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\))p 3700 6505 519 7 v 3893 6657 a Fn(E)4238 6309 y Fi(\023)4388 6543 y Fn(A)p Fo(\(\010)4698 6474 y Fk(t)4698 6584 y(H)4823 6543 y Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\)\))p Fn(\032)5370 6359 y Fi(\020)5486 6430 y Fn(q)p 5486 6505 81 7 v 5487 6657 a(")5586 6543 y(;)g(p)5773 6430 y(")p 5764 6505 96 7 v 5764 6657 a Fo(\026)-83 b Fn(h)5879 6359 y Fi(\021)6026 6430 y Fn(dq)6 b(dp)p 6026 6505 337 7 v 6054 6657 a Fo(2)p Fn(\031)g Fo(\026)-83 b Fn(h)6382 6543 y(:)6328 6830 y Fp(\(25\))876 7030 y(The)33 b(\002nal)g(step)g(is)g(to)g(pro)n(v)n(e)f(that)h(only)g(ener)m(gies)g Fn(E)42 b Fp(between)32 b Fo(\001)4591 6969 y Fj(2+)p Fk(\033)4836 6986 y Fh(1)4943 7030 y Fp(and)g Fo(\001)5353 6969 y Fj(2)p Fl(\000)p Fk(\033)5600 6986 y Fh(2)5707 7030 y Fp(\()p Fn(\033)5857 7055 y Fk(j)5973 7030 y Fn(>)46 b Fo(0)p Fn(;)74 b(j)56 b Fo(=)876 7229 y(1)p Fn(;)74 b Fo(2)p Fp(,)41 b(arbitrary)h(small\))g(contrib)m(ute)f(in)h(\(25\).) 876 7540 y Fu(Lemma)g(3.6)82 b Fe(F)-17 b(or)41 b(e)n(very)f Fn(\033)2519 7565 y Fj(1)2639 7540 y Fn(>)47 b Fo(0)p Fe(,)41 b Fn(\033)3076 7565 y Fj(2)3196 7540 y Fn(>)46 b Fo(0)p Fe(:)1716 7956 y Fo(\026)-83 b Fn(h)1812 7881 y Fl(\000)p Fj(1)2018 7730 y Fi(Z)46 b(Z)2248 8107 y Fj([)p Fl(j)p Fk(H)9 b Fj(\()p Fk(q)t(;p)p Fj(\))p Fl(j\024)p Fj(\001)2971 8070 y Fh(2+)p Ff(\033)3181 8095 y Fh(1)3255 8107 y Fj(])3328 7765 y Fi(\014)3328 7864 y(\014)3328 7964 y(\014)3384 7956 y Fn(\032)3498 7771 y Fi(\020)3616 7843 y Fn(q)p 3616 7917 81 7 v 3617 8070 a(")3716 7956 y(;)28 b(p)3903 7843 y(")p 3894 7917 96 7 v 3894 8070 a Fo(\026)-83 b Fn(h)4009 7771 y Fi(\021)4108 7765 y(\014)4108 7864 y(\014)4108 7964 y(\014)4191 7956 y Fn(dq)6 b(dp)47 b Fo(=)f Fn(O)5 b Fo(\(\001)5084 7887 y Fk(\033)5161 7904 y Fh(1)5225 7887 y Fk(=)p Fj(2)5368 7956 y Fo(\))1753 8428 y(\026)-83 b Fn(h)1849 8354 y Fl(\000)p Fj(1)2055 8202 y Fi(Z)46 b(Z)2285 8580 y Fj([)p Fl(j)p Fk(H)9 b Fj(\()p Fk(q)t(;p)p Fj(\))p Fl(j\025)p Fj(\001)3008 8542 y Fh(2)p Fg(\000)p Ff(\033)3223 8567 y Fh(2)3296 8580 y Fj(])3370 8237 y Fi(\014)3370 8337 y(\014)3370 8437 y(\014)3425 8428 y Fn(\032)3539 8244 y Fi(\020)3658 8316 y Fn(q)p 3658 8390 81 7 v 3659 8542 a(")3758 8428 y(;)28 b(p)3944 8316 y(")p 3936 8390 96 7 v 3936 8542 a Fo(\026)-83 b Fn(h)4051 8244 y Fi(\021)4150 8237 y(\014)4150 8337 y(\014)4150 8437 y(\014)4233 8428 y Fn(dq)6 b(dp)46 b Fo(=)h Fn(O)5 b Fo(\(\001)5126 8360 y Fj(+)p Fl(1)5368 8428 y Fo(\))876 8879 y Fu(Pr)m(oof)p Fp(:)102 b(The)66 b(second)g(estimate)f(w)n(as)h(already)g(pro)n(v)n(ed)g(in)g(\(21\).) 126 b(Let)67 b(us)f(estimate)f(here)i(the)876 9078 y(inte)n(gral)2217 9358 y Fn(I)13 b Fo(\()p Fn(E)d(;)28 b Fo(\026)-83 b Fn(h)n Fo(\))46 b(=)2955 9132 y Fi(Z)h(Z)3186 9510 y Fj([)o Fl(j)p Fk(H)9 b Fj(\()p Fk(q)t(;p)p Fj(\))p Fl(j\024)p Fk(E)d Fj(])3978 9167 y Fi(\014)3978 9267 y(\014)3978 9367 y(\014)4033 9358 y Fn(\032)4147 9174 y Fi(\020)4266 9246 y Fn(q)p 4266 9320 81 7 v 4267 9472 a(")4366 9358 y(;)28 b(p)4552 9246 y(")p 4544 9320 96 7 v 4544 9472 a Fo(\026)-83 b Fn(h)4659 9174 y Fi(\021)4758 9167 y(\014)4758 9267 y(\014)4758 9367 y(\014)4861 9246 y Fn(dq)6 b(dp)p 4861 9320 337 7 v 4981 9472 a Fo(\026)-83 b Fn(h)5217 9358 y(:)876 9780 y Fp(Because)51 b Fn(\032)i Fp(is)f(f)n(ast)h (decreasing)f(it)h(is)f(suf)l(\002cient)h(to)g(inte)n(grate)f(in)h Fm(j)p Fn(q)6 b Fm(j)67 b(\024)g Fo(\001)5381 9720 y Fk(\022)5510 9780 y Fp(for)53 b(some)f Fo(2)p Fn(=)p Fo(3)67 b Fn(<)876 9979 y(\022)51 b(<)46 b Fo(1)p Fp(.)51 b(But)41 b(in)h(a)f(domain)h(where)f Fn(q)48 b Fp(is)41 b(small,)g(we)g(can)g(use)g(the)g(T)-13 b(aylor)42 b(formula)g(to)g (obtain)2563 10390 y Fn(H)13 b Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\))45 b(=)3322 10278 y(1)p 3322 10352 84 7 v 3322 10504 a(2)3425 10390 y(\()p Fn(p)3574 10322 y Fj(2)3685 10390 y Fm(\000)37 b Fn(!)3960 10322 y Fj(2)3954 10431 y(0)4034 10390 y Fn(q)4114 10322 y Fj(2)4189 10390 y Fo(\))f(+)i Fn(O)5 b Fo(\()p Fn(q)4734 10322 y Fj(3)4807 10390 y Fo(\))p Fn(;)3657 10919 y Fp(17)p eop %%Page: 18 18 18 17 bop 876 1455 a Fp(so)41 b(that)h(it)f(is)h(enough)f(to)h (estimate)1863 1897 y Fn(I)1936 1922 y Fj(0)2010 1897 y Fo(\()p Fn(E)10 b(;)28 b Fo(\026)-83 b Fn(h)n Fo(\))46 b(=)2662 1671 y Fi(Z)h(Z)2893 2061 y Fo([)2939 2049 y Fl(j)p Fk(p)3046 2016 y Fh(2)3112 2049 y Fl(\000)p Fk(!)3304 2009 y Fh(2)3300 2089 y(0)3368 2049 y Fk(q)3433 2016 y Fh(2)3498 2049 y Fl(j\024)p Fj(2\()p Fk(E)6 b Fj(+\001)4073 2016 y Fh(3)p Ff(\022)4198 2049 y Fj(\))4250 2061 y Fo(])4332 1707 y Fi(\014)4332 1806 y(\014)4332 1906 y(\014)4387 1897 y Fn(\032)4501 1713 y Fi(\020)4620 1785 y Fn(q)p 4620 1859 81 7 v 4621 2011 a(")4720 1897 y(;)28 b(p)4906 1785 y(")p 4898 1859 96 7 v 4898 2011 a Fo(\026)-83 b Fn(h)5013 1713 y Fi(\021)5112 1707 y(\014)5112 1806 y(\014)5112 1906 y(\014)5215 1785 y Fn(dq)6 b(dp)p 5215 1859 337 7 v 5335 2011 a Fo(\026)-83 b Fn(h)5571 1897 y(:)711 b Fp(\(26\))876 2395 y(Assume)61 b(no)l(w)h Fo(\001)85 b(=)f Fn(\017)p Fp(.)114 b(The)62 b(other)h(case)e(is)h(treated)f (similarly)-11 b(.)114 b(W)-13 b(e)62 b(change)f(v)l(ariables)g(by)876 2595 y(putting)42 b Fn(q)52 b Fo(=)46 b Fn("u;)28 b(p)46 b Fo(=)2261 2529 y Fj(\026)-68 b Fk(h)p 2261 2556 76 7 v 2268 2652 a(")2356 2595 y Fn(v)48 b Fp(and)42 b(we)f(introduce)3708 2553 y Fo(~)3669 2595 y Fn(E)56 b Fo(=)4078 2529 y Fk(E)p 4043 2556 174 7 v 4043 2652 a Fj(\001)4152 2619 y Fh(2)4237 2595 y Fp(,)41 b Fn(c)46 b Fo(=)4682 2529 y Fj(\026)-68 b Fk(h)p 4633 2556 V 4633 2652 a Fj(\001)4742 2619 y Fh(2)4827 2595 y Fp(,)41 b(so)g(that)2160 3057 y Fn(I)2233 3082 y Fj(0)2307 3057 y Fo(\()p Fn(E)10 b(;)28 b Fo(\026)-83 b Fn(h)n Fo(\))46 b Fm(\024)2959 2831 y Fi(Z)h(Z)3190 3221 y Fo([)3236 3208 y Fl(j)p Fk(c)3334 3175 y Fh(2)3400 3208 y Fk(v)3470 3175 y Fh(2)3535 3208 y Fl(\000)p Fk(!)3727 3168 y Fh(2)3723 3248 y(0)3791 3208 y Fk(u)3870 3175 y Fh(2)3935 3208 y Fl(j\024)p Fj(3)4174 3179 y(~)4144 3208 y Fk(E)4248 3220 y Fo(])4330 3057 y Fm(j)p Fn(\032)p Fo(\()p Fn(u;)28 b(v)6 b Fo(\))p Fm(j)27 b Fn(dudv)6 b(:)876 3594 y Fp(Splitting)58 b(the)g(domain)f(of)h(the)f(inte)n(gral) g Fn(I)3412 3619 y Fj(0)3487 3594 y Fo(\()p Fn(E)10 b(;)28 b Fo(\026)-83 b Fn(h)n Fo(\))57 b Fp(into)h Fm(f)p Fn(u)4469 3534 y Fj(2)4619 3594 y Fm(\024)76 b Fo(4)4945 3552 y(~)4907 3594 y Fn(E)9 b Fm(g)57 b Fp(and)g Fm(f)p Fn(u)5654 3534 y Fj(2)5804 3594 y Fn(>)76 b Fo(4)6131 3552 y(~)6092 3594 y Fn(E)9 b Fm(g)58 b Fp(and)876 3794 y(using)41 b Fo(0)46 b Fn(<)h(c)f Fm(\024)g Fo(1)41 b Fp(\(Uncertainty)h (Principle!\))53 b(we)41 b(get)3038 4207 y Fn(I)3111 4232 y Fj(0)3185 4207 y Fo(\()p Fn(E)10 b(;)28 b Fo(\026)-83 b Fn(h)o Fo(\))45 b(=)i Fn(O)5 b Fo(\()4035 4022 y Fi(p)p 4200 4022 133 7 v 4238 4165 a Fo(~)4200 4207 y Fn(E)k Fo(\))p Fn(;)1885 b Fp(\(27\))876 4569 y(which)41 b(concludes)g(the)g (proof)h(of)g(Lemma)g(3.6.)p 6404 4602 93 133 v 1125 4866 a(The)c(conclusion)f(of)g(the)h(w)n(ork)g(done)f(so)g(f)n(ar)i (can)d(be)i(formulated)g(in)g(the)f(follo)l(wing)h(proposi-)876 5065 y(tion,)44 b(which)g(sho)l(ws)e(that)i(the)g(study)f(of)h(the)g (long-time)g(quantum)g(a)m(v)n(erages)e(considered)h(in)h(this)876 5265 y(paper)d(can)g(be)g(reduced)g(to)h(a)f(purely)h(classical)e (long-time)i(analysis.)876 5594 y Fu(Pr)m(oposition)h(3.7)82 b Fe(F)-17 b(or)42 b(e)n(very)e Fo(0)48 b Fn(<)g(\033)3102 5619 y Fj(1)3176 5594 y Fn(;)28 b(\033)3345 5619 y Fj(2)3420 5594 y Fn(;)g(\033)3589 5619 y Fj(3)3711 5594 y Fn(<)47 b Fo(1)p Fe(,)c Fn(\016)53 b(>)48 b Fo(0)p Fe(,)42 b Fo(\001)4664 5619 y Fj(0)4787 5594 y Fn(>)48 b Fo(0)42 b Fe(ther)-6 b(e)42 b(e)m(xists)f Fn(s)47 b(>)h Fo(0)43 b Fe(suc)n(h)876 5793 y(that)1470 6039 y Fn(<)1689 5997 y Fo(^)1645 6039 y Fn(A)p Fo(\()p Fn(t)p Fo(\))p Fn(\021)2042 6064 y Fk(")2112 6039 y Fn(;)28 b(\021)2268 6064 y Fk(")2386 6039 y Fn(>)p Fo(=)2690 5813 y Fi(Z)46 b(Z)2921 6190 y Fj(\012)3016 6207 y Ff(\033)3094 6190 y Fj(\(\001\))3343 6039 y Fn(A)p Fo(\(\010)3653 5970 y Fk(t)3653 6080 y(H)3778 6039 y Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\)\))p Fn(\032)4325 5855 y Fi(\020)4442 5927 y Fn(q)p 4442 6001 81 7 v 4443 6153 a(")4542 6039 y(;)g(p)4728 5927 y(")p 4720 6001 96 7 v 4720 6153 a Fo(\026)-83 b Fn(h)4835 5855 y Fi(\021)4982 5927 y Fn(dq)6 b(dp)p 4982 6001 337 7 v 5010 6153 a Fo(2)p Fn(\031)g Fo(\026)-83 b Fn(h)5375 6039 y Fo(+)37 b Fn(O)5 b Fo(\(\001)5876 5970 y Fk(s)5946 6039 y Fo(\))876 6479 y Fe(uniformly)41 b(for)g Fm(j)p Fn(t)p Fm(j)47 b(\024)2192 6414 y Fj(2)p Fl(\000)p Fj(3)p Fk(\016)p 2192 6441 301 7 v 2265 6537 a Fj(3)p Fk(!)2513 6479 y Fm(j)28 b Fo(log)r(\(\026)-83 b Fn(h)o Fo(\))p Fm(j)p Fe(,)41 b Fo(\001)46 b Fm(\024)h Fo(\001)3653 6504 y Fj(0)3727 6479 y Fe(.)k(Her)-6 b(e)1203 6863 y Fo(\012)1323 6888 y Fk(\033)1413 6863 y Fo(\(\001\))45 b(=)1902 6729 y Fi(\010)1998 6863 y Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\))45 b Fm(2)h Fo(I)-37 b(R)2713 6791 y Fj(2)2788 6863 y Fn(;)28 b Fo(\001)3000 6795 y Fj(2+)p Fk(\033)3245 6812 y Fh(1)3364 6863 y Fm(\024)46 b(j)p Fn(H)13 b Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\))p Fm(j)46 b(\024)g Fo(\001)4509 6795 y Fj(2)p Fl(\000)p Fk(\033)4756 6812 y Fh(2)4829 6863 y Fn(;)74 b Fm(j)p Fn(q)6 b Fm(j)38 b Fo(+)f Fm(j)p Fn(p)p Fm(j)46 b(\024)g Fo(\001)5860 6795 y Fj(1)p Fl(\000)p Fk(\033)6107 6812 y Fh(3)6180 6729 y Fi(\011)876 7225 y Fe(and)c Fn(!)52 b Fo(=)46 b(max)p Fm(f)p Fn(!)1992 7250 y Fk(j)2063 7225 y Fm(g)p Fe(.)876 7785 y Fq(4)239 b(Contr)l(olling)58 b(the)i(classical)h(\003o)n(w)876 8156 y Fp(No)l(w)48 b(we)f(can)h(proceed)g(to)g(the)g(proof)h(of)f (Theorems)g(2.1-2.2-2.3-2.4.)70 b(In)49 b(vie)l(w)e(of)i(the)f(results) 876 8355 y(of)60 b(the)f(pre)l(vious)h(section)f(\(Proposition)h(3.7\)) g(it)g(remains)f(to)h(study)f(carefully)h(the)f(long)h(time)876 8554 y(beha)m(viour)41 b(of)h(the)f(classical)f(trajectories)i(for)g (initial)g(conditions)f(near)g(the)h(critical)g(point)f Fn(X)6378 8579 y Fj(0)6452 8554 y Fp(.)876 8754 y Fu(Pr)m(oof)e(of)g (Theor)m(em)f(2.1)h(\(i\))p Fp(:)51 b(Let)39 b(us)f(recall)g(that)h Fn(X)3945 8779 y Fj(0)4065 8754 y Fo(=)46 b(\(0)p Fn(;)28 b Fo(0\))p Fp(.)49 b(Using)39 b(the)f(T)-13 b(aylor)39 b(formula)h(we)876 8953 y(ha)m(v)n(e,)g(for)i(e)l(v)n(ery)e Fn(X)59 b Fm(2)46 b Fo(\012)2379 8978 y Fk(\033)2468 8953 y Fo(\(\001\))1633 9315 y Fm(j)p Fn(A)p Fo(\(\010)1989 9246 y Fk(t)1989 9356 y(H)2115 9315 y Fo(\()p Fn(X)13 b Fo(\)\))35 b Fm(\000)i Fn(A)p Fo(\(\010)2972 9246 y Fk(t)2972 9356 y(H)3097 9315 y Fo(\()p Fn(X)3300 9340 y Fj(0)3373 9315 y Fo(\)\))p Fm(j)46 b(\024)g Fn(C)12 b Fo(\001)4039 9246 y Fj(1)p 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Fj(1)p Fl(\000)p Fk(\033)4389 10370 y Fh(3)4490 10421 y Fo(exp)o([\()p Fn(!)4957 10446 y Fj(0)5069 10421 y Fo(+)37 b Fn(\021)5317 10446 y Fj(\001)5435 10421 y Fo(\))p Fm(j)p Fn(t)p Fm(j)p Fo(])p Fn(:)584 b Fp(\(28\))3657 10919 y(18)p eop %%Page: 19 19 19 18 bop 876 1455 a Fp(where)52 b Fn(\021)1416 1480 y Fj(\001)1599 1455 y Fm(!)66 b Fo(0)52 b Fp(as)g Fo(\001)66 b Fm(!)f Fo(0)p Fp(.)83 b(Since)52 b Fn(\033)3316 1480 y Fj(3)3443 1455 y Fp(can)f(be)h(chosen)f(as)h(small)g(as)f(desired)h (the)g(result)g(\(i\))876 1654 y(follo)l(ws)41 b(easily)g(from)i (Proposition)f(3.7.)p 6404 1687 93 133 v 876 1932 a Fu(Pr)m(oof)37 b(of)f(Theor)m(em)g(2.1)f(\(ii\))p Fp(:)50 b(W)-13 b(e)36 b(de\002ne)g Fo(\012)3516 1872 y Fk(i)3516 1973 y(\033)3605 1932 y Fo(\(\001\))46 b(=)g(\012)4214 1957 y Fk(i)4286 1932 y Fm(\\)17 b Fo(\012)4534 1957 y Fk(\033)4623 1932 y Fo(\(\001\))36 b Fp(where)g Fn(i)46 b Fo(=)g Fn(I)13 b(;)28 b(I)13 b(I)g(;)28 b(I)13 b(I)g(I)g(;)28 b(I)13 b(V)876 2131 y Fp(\(Fig.)124 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b(the)h(positi)l(v)n(e)e(numbers)i Fn(\033)6489 4336 y Fj(1)6563 4311 y Fp(,)876 4510 y Fn(\033)971 4535 y Fj(2)1045 4510 y Fp(,)46 b Fn(\033)1228 4535 y Fj(3)1347 4510 y Fp(small)f(enough,)h(and)f(times)g Fn(t)g Fp(in)g(the)g(abo)n(v) n(e)f(windo)l(w)-11 b(,)45 b(the)g(\003o)l(w)g(carries)g Fo(\012)5782 4450 y Fk(I)9 b(;I)g(I)5782 4551 y(\033)6033 4510 y Fo(\(\001\))45 b Fp(into)876 4709 y(a)f(small)g(neighborhood)h (of)g Fn(X)2686 4734 y Fj(+)2840 4709 y Fp(and)f Fo(\012)3244 4649 y Fk(I)9 b(I)g(I)g(;I)g(V)3244 4750 y(\033)3670 4709 y Fo(\(\001\))44 b Fp(into)h(a)f(neighborhood)h(of)f Fn(X)5690 4734 y Fl(\000)5802 4709 y Fp(,)h(as)e Fo(\001)i Fp(tends)876 4909 y(to)d Fo(0)p Fp(.)51 b(More)41 b(precisely)-11 b(,)40 b(we)i(shall)f(pro)n(v)n(e)f(that)3091 5221 y Fo(lim)3052 5328 y Fj(\001)p Fl(!)p Fj(0)3387 5221 y Fo(\010)3507 5153 y Fk(t)3507 5262 y(H)3633 5221 y Fo(\()p Fn(X)13 b Fo(\))45 b(=)h Fn(X)4272 5246 y Fj(+)4382 5221 y Fn(;)1900 b Fp(\(30\))876 5615 y(uniformly)49 b(for)f(all)f Fn(X)70 b Fm(2)57 b Fo(\012)2538 5555 y Fk(I)9 b(;I)g(I)2538 5656 y(\033)2790 5615 y Fo(\(\001\))47 b Fp(and)g(for)h(all)g Fn(t)f Fp(in)h(the)g(windo)l(w)f(\(12\),)i(and)f(a)f(similar)h(state-) 876 5814 y(ment)43 b(for)h Fo(\012)1608 5754 y Fk(I)9 b(I)g(I)g(;I)g(V)1608 5855 y(\033)2035 5814 y Fo(\(\001\))p Fp(.)56 b(This,)43 b(together)h(with)f(the)h(de\002nition)g(of)f(the)h Fn(\013)5168 5839 y Fk(i)5266 5814 y Fp(\(see)f(\(9\)\))h(and)f(Propo-) 876 6013 y(sition)f(3.7)f(clearly)g(implies)g(\(29\).)1125 6213 y(Assumption)d Fu(\(H2\))h Fp(is)f(a)g(special)f(case)g(of)i Fu(\(H3\))g Fp(\(with)g Fn(j)56 b Fo(=)46 b(0)p Fn(;)28 b Fo(+)p Fn(;)g Fm(\000)p Fp(\),)39 b(and)f(it)g(will)h(be)f(con)-7 b(v)n(e-)876 6412 y(nient)43 b(for)g(later)f(reference)h(in)f(the)h (proof)g(of)g(Theorem)g(2.3)f(to)h(consider)f(here)h(assumption)f Fu(\(H3\))876 6611 y Fp(with)47 b(its)f(associated)g(notations.)66 b(W)-13 b(e)46 b(need)g(to)h(study)f(classical)g(paths)g(starting)h (from)g(the)g(small)876 6810 y(neighborhoods)e Fo(\012)2028 6750 y Fk(i)2028 6851 y(\033)2117 6810 y Fo(\(\001\))39 b(+)g Fn(X)2730 6835 y Fk(j)2844 6810 y Fo(\()p Fn(i)52 b Fo(=)h Fn(I)13 b(;)28 b(I)13 b(I)g(;)28 b(I)13 b(I)g(I)g(;)28 b(I)13 b(V)36 b Fo(\))44 b Fp(of)h Fn(X)4587 6835 y Fk(j)4709 6810 y Fo(=)52 b(\()p Fn(q)5029 6835 y Fk(j)5099 6810 y Fn(;)28 b Fo(0\))p Fp(.)60 b(W)-13 b(e)44 b(deal)g(\002rst)h(with)876 7010 y(the)i(case)e Fn(i)57 b Fo(=)f Fn(I)13 b(I)g Fp(.)67 b(Let)47 b Fn(\013)57 b Fm(2)f Fo([0)p Fn(;)28 b Fo(1)p Fn(=)p Fo(2])p Fp(.)67 b(An)n(y)47 b(point)g Fn(X)69 b Fo(=)56 b(\()p Fn(q)6 b(;)28 b(p)p Fo(\))45 b Fp(belonging)i(to)g Fo(\012)5774 6949 y Fk(I)9 b(I)5774 7051 y(\033)5918 7010 y Fo(\(\001\))41 b(+)g Fn(X)6535 7035 y Fk(j)876 7209 y Fp(can)g(then)g(be)h(labeled)f(by)g(a)g(real)h Fn(x)3232 7522 y(q)52 b Fo(=)47 b Fn(q)3608 7547 y Fk(j)3715 7522 y Fo(+)37 b Fn(xE)4109 7453 y Fk(\013)4202 7522 y Fn(;)2080 b Fp(\(31\))876 7834 y(where)69 b Fn(E)108 b Fo(=)99 b Fn(H)13 b Fo(\()p Fn(X)g Fo(\))98 b Fn(>)h Fo(0)p Fp(.)136 b(W)-13 b(e)69 b(can)h(therefore)g(consider)f Fn(x)h Fp(and)g Fn(E)79 b Fp(as)69 b(coordinates)g(on)876 8034 y Fo(\012)996 7973 y Fk(I)9 b(I)996 8075 y(\033)1140 8034 y Fo(\(\001\))36 b(+)h Fn(X)1748 8059 y Fk(j)1818 8034 y Fp(.)51 b(Remark)41 b(that)h(clearly)2141 8451 y Fm(j)p Fn(q)h Fm(\000)37 b Fn(q)2544 8476 y Fk(j)2615 8451 y Fm(j)46 b(\024)g Fo(\001)3020 8383 y Fj(1)p Fl(\000)p Fk(\033)3267 8400 y Fh(3)3387 8451 y Fm(\024)g Fn(H)13 b Fo(\()p Fn(X)g Fo(\))4014 8338 y Fh(1)p 4013 8357 57 7 v 4013 8423 a(2)4089 8383 y Fl(\000)p Fk(\021)4274 8451 y Fn(;)69 b(\021)53 b Fo(=)4719 8329 y Fn(\033)4814 8354 y Fj(3)4925 8329 y Fo(+)5111 8261 y Fk(\033)5188 8278 y Fh(1)p 5111 8291 142 7 v 5149 8387 a Fj(2)p 4719 8413 554 7 v 4768 8565 a Fo(1)37 b(+)g Fn(\033)5149 8590 y Fj(1)5293 8451 y Fn(:)876 8861 y Fp(Consequently)-11 b(,)40 b(if)i(we)f(choose)g Fn(\013)47 b Fo(=)3090 8795 y Fj(1)p 3090 8822 67 7 v 3090 8918 a(2)3213 8861 y Fm(\000)37 b Fn(\021)6 b Fp(,)42 b Fn(x)f Fp(belongs)g(to)h Fo([)p Fm(\000)p Fo(1)p Fn(;)28 b Fo(1])p Fp(.)1125 9060 y(F)n(or)46 b(simplicity)g(of)g(notation,)g(we)g(shall)f(assume)f(in)i(the)g(follo) l(wing)g Fn(j)64 b Fm(6)p Fo(=)54 b Fn(N)5617 9085 y Fj(+)5727 9060 y Fp(,)47 b(b)m(ut)e(the)h(case)876 9259 y Fn(j)56 b Fo(=)46 b Fn(N)1308 9284 y Fj(+)1460 9259 y Fp(is)c(treated)f(in)h(the)f(same)g(w)n(ay)-11 b(.)1125 9458 y(W)e(e)50 b(need)h(an)g(estimate)g(on)g(the)g(time)g Fn(\034)3461 9483 y Fk(E)3573 9458 y Fo(\()p Fn(\013)q(;)28 b(x)p Fo(\))50 b Fp(it)h(tak)n(es)g(to)g(go)h(from)g Fo(\()p Fn(q)5474 9483 y Fk(j)5543 9458 y Fn(;)5617 9318 y Fm(p)p 5755 9318 216 7 v 140 x Fo(2)p Fn(E)10 b Fo(\))50 b Fp(to)i Fo(\()p Fn(q)69 b Fo(=)876 9675 y Fn(q)950 9700 y Fk(j)1058 9675 y Fo(+)37 b Fn(xE)1452 9615 y Fk(\013)1546 9675 y Fn(;)28 b(p)47 b Fo(=)1928 9536 y Fi(p)p 2094 9536 1479 7 v 139 x Fo(2\()p Fn(E)e Fm(\000)37 b Fn(V)g Fo(\()p Fn(q)2849 9700 y Fk(j)2956 9675 y Fo(+)g Fn(xE)3350 9627 y Fk(\013)3443 9675 y Fo(\)\))42 b Fp(along)g(the)g(curv)n(e)g Fm(f)p Fn(H)61 b Fo(=)48 b Fn(E)10 b Fm(g)p Fp(.)53 b(A)42 b(simple)g(compu-)876 9874 y(tation)g(sho)l(ws)e(it)i(satis\002es,)e (for)i(an)n(y)f Fn(\013)47 b Fm(2)f Fo([0)p Fn(;)28 b Fo(1)p Fn(=)p Fo(2])p Fp(,)41 b(when)g Fn(E)55 b Fm(&)46 b Fo(0)p Fp(,)1462 10317 y Fn(\034)1535 10342 y Fk(E)1647 10317 y Fo(\()p Fn(\013)q(;)28 b(x)p Fo(\))45 b(=)h Fm(j)2347 10091 y Fi(Z)2513 10132 y Fk(q)2574 10149 y Ff(j)2636 10132 y Fj(+)p Fk(xE)2916 10081 y Ff(\013)2439 10468 y Fk(q)2500 10485 y Ff(j)3502 10204 y Fn(du)p 3057 10278 1072 7 v 3057 10312 a Fi(p)p 3223 10312 906 7 v 141 x Fo(2\()p Fn(E)g Fm(\000)37 b Fn(V)f Fo(\()p Fn(u)p Fo(\)\))4148 10317 y Fm(j)46 b Fo(=)4436 10204 y(1)37 b Fm(\000)g Fo(2)p Fn(\013)p 4436 10278 476 7 v 4545 10430 a Fo(2)p Fn(!)4731 10455 y Fk(j)4931 10317 y Fm(j)28 b Fo(log)i Fn(E)10 b Fm(j)36 b Fo(+)i Fn(O)5 b Fo(\(1\))p Fn(;)308 b Fp(\(32\))3657 10919 y(19)p eop %%Page: 20 20 20 19 bop 876 1455 a Fp(uniformly)42 b(for)g Fn(x)f Fp(in)h(a)f (bounded)g(set.)50 b(In)41 b(particular)-7 b(,)42 b(if)f Fn(\013)47 b Fo(=)g(0)p Fp(,)40 b(this)i(sho)l(ws)e(that)h(it)h(tak)n (es)e(a)h(time)2218 1893 y Fn(T)2315 1918 y Fk(j;j)7 b Fj(+1)2647 1893 y Fo(\()p Fn(E)j Fo(\))44 b Fm(\021)i Fo(\()3301 1780 y(1)p 3214 1854 257 7 v 3214 2007 a(2)p Fn(!)3400 2032 y Fk(j)3527 1893 y Fo(+)3884 1780 y(1)p 3713 1854 425 7 v 3713 2007 a(2)p Fn(!)3899 2032 y Fk(j)7 b Fj(+1)4157 1893 y Fo(\))p Fm(j)28 b Fo(log)i Fn(E)10 b Fm(j)36 b Fo(+)h Fn(O)5 b Fo(\(1\))876 2391 y Fp(to)58 b(go)h(from)g Fo(\()p Fn(q)1808 2416 y Fk(j)1877 2391 y Fn(;)1951 2250 y Fm(p)p 2090 2250 216 7 v 2090 2391 a Fo(2)p Fn(E)9 b Fo(\))58 b Fp(to)g Fo(\()p Fn(q)2754 2416 y Fk(j)7 b Fj(+1)2992 2391 y Fn(;)3066 2250 y Fm(p)p 3204 2250 V 141 x Fo(2)p Fn(E)i Fo(\))p Fp(.)101 b(Consequently)-11 b(,)62 b(for)c(e)l(v)n(ery)g Fn(\016)5386 2330 y Fl(0)5510 2391 y Fn(>)77 b Fo(0)p Fp(,)62 b(there)c(e)n(xist)876 2590 y Fn(\033)971 2615 y Fj(1)1045 2590 y Fn(;)28 b(\033)1214 2615 y Fj(2)1330 2590 y Fp(such)41 b(that,)g(for)h(all)g Fo(\001)2589 2530 y Fj(2+)p Fk(\033)2834 2547 y Fh(1)2953 2590 y Fm(\024)47 b Fn(E)55 b Fm(\024)46 b Fo(\001)3620 2530 y Fj(2)p Fl(\000)p Fk(\033)3867 2547 y Fh(2)3940 2590 y Fp(,)1223 2978 y Fn(T)1343 2892 y Fj(\()p Fk(j)7 b Fj(\))1320 3027 y(\001)1554 2978 y Fo(+)37 b Fn(T)1840 2892 y Fj(\()p Fk(j)7 b Fj(+1\))1817 3027 y(\001)2219 2978 y Fm(\000)37 b Fn(\016)2465 2910 y Fl(0)2512 2978 y Fm(j)28 b Fo(log)i(\001)p Fm(j)46 b(\024)h Fn(T)3331 3003 y Fk(j;j)7 b Fj(+1)3663 2978 y Fo(\()p Fn(E)j Fo(\))44 b Fm(\024)i Fn(T)4265 2892 y Fj(\()p Fk(j)7 b Fj(\))4242 3027 y(\001)4476 2978 y Fo(+)37 b Fn(T)4762 2892 y Fj(\()p Fk(j)7 b Fj(+1\))4739 3027 y(\001)5141 2978 y Fo(+)37 b Fn(\016)5387 2910 y Fl(0)5434 2978 y Fm(j)28 b Fo(log)i(\001)p Fm(j)p Fn(:)348 b Fp(\(33\))876 3343 y(On)41 b(the)h(other)f(hand,)g (if)h Fn(\013)47 b Fo(=)2644 3278 y Fj(1)p 2644 3305 67 7 v 2644 3401 a(2)2767 3343 y Fm(\000)37 b Fn(\021)6 b Fp(,)42 b(we)f(\002nd)2685 3767 y Fn(\034)2758 3792 y Fk(E)2870 3767 y Fo(\()p Fn(x;)28 b(\013)q Fo(\))44 b(=)3558 3655 y Fn(\021)p 3516 3729 174 7 v 3516 3881 a(!)3619 3906 y Fk(j)3709 3767 y Fm(j)28 b Fo(log)i Fn(E)10 b Fm(j)36 b Fo(+)h Fn(O)5 b Fo(\(1\))p Fn(:)1531 b Fp(\(34\))876 4251 y(As)44 b(a)h(result,)h(if)f Fn(X)65 b Fo(=)53 b(\()p Fn(q)6 b(;)28 b(p)p Fo(\))51 b Fm(2)i Fo(\012)2915 4191 y Fk(I)9 b(I)2915 4292 y(\033)3059 4251 y Fo(\(\001\))39 b(+)g Fn(X)3672 4276 y Fk(j)3787 4251 y Fp(with)45 b Fn(E)62 b Fo(=)53 b Fn(H)13 b Fo(\()p Fn(X)g Fo(\))44 b Fp(and)h Fn(x)g Fp(de\002ned)g(as)f(in)h(\(31\),)876 4450 y(then)1290 4816 y Fo(\010)1410 4726 y Fk(T)1488 4743 y Ff(j)o(;j)5 b Fh(+1)1778 4726 y Fj(\()p Fk(E)h Fj(\))1410 4864 y Fk(H)1994 4816 y Fo(\()p Fn(X)13 b Fo(\))45 b(=)h(\010)2615 4726 y Fk(T)2693 4743 y Ff(j)o(;j)5 b Fh(+1)2983 4726 y Fj(\()p Fk(E)h Fj(\)+)p Fk(\034)3353 4743 y Ff(E)3450 4726 y Fj(\()p Fk(x;\013)p Fj(\))2615 4864 y Fk(H)3763 4816 y Fo(\()p Fn(q)3902 4841 y Fk(j)3972 4816 y Fn(;)4046 4667 y Fm(p)p 4184 4667 216 7 v 149 x Fo(2)p Fn(E)j Fo(\))46 b(=)g(\010)4805 4729 y Fk(\034)4866 4746 y Ff(E)4963 4729 y Fj(\()p Fk(x;\013)p Fj(\))4805 4864 y Fk(H)5276 4816 y Fo(\()p Fn(q)5415 4841 y Fk(j)7 b Fj(+1)5653 4816 y Fn(;)5727 4667 y Fm(p)p 5865 4667 V 149 x Fo(2)p Fn(E)i Fo(\))p Fn(:)876 5181 y Fp(Using)52 b(no)l(w)f(a)h(normal)g(form)h(around)f Fn(X)3332 5206 y Fk(j)7 b Fj(+1)3622 5181 y Fp(as)51 b(in)h(the)g(proof)h(of)f(Lemma)g (3.4,)i(together)e(with)876 5380 y(\(33\))42 b(and)g(\(34\),)g(we)f (easily)g(conclude)g(that,)h(gi)l(v)n(en)f Fo(0)46 b Fn(<)h(\033)4284 5320 y Fl(0)4278 5421 y Fj(3)4399 5380 y Fn(<)g Fo(1)p Fp(,)41 b(there)h(e)n(xist)f Fo(0)46 b Fn(<)h(\033)5868 5405 y Fj(1)5942 5380 y Fn(;)28 b(\033)6111 5405 y Fj(2)6186 5380 y Fn(;)g(\033)6355 5405 y Fj(3)6475 5380 y Fn(<)876 5579 y Fo(1)41 b Fp(small)h(enough)f(so)g(that)1897 6021 y Fo(\010)2017 5921 y Fk(T)2113 5860 y Fh(\()p Ff(j)5 b Fh(\))2095 5964 y(\001)2266 5921 y Fj(+)p Fk(T)2464 5860 y Fh(\()p Ff(j)g Fh(+1\))2446 5964 y(\001)2017 6070 y Fk(H)2766 6021 y Fo(\(\012)2951 5953 y Fk(I)k(I)2951 6062 y(\033)3094 6021 y Fo(\(\001\))37 b(+)g Fn(X)3703 6046 y Fk(j)3772 6021 y Fo(\))46 b Fm(\032)g Fo(\012)4178 5953 y Fk(I)9 b(I)4178 6065 y(\033)4255 6082 y Fh(1)4320 6065 y Fk(;\033)4436 6082 y Fh(2)4501 6065 y Fk(;\033)4621 6025 y Fg(0)4617 6105 y Fh(3)4691 6021 y Fo(\(\001\))36 b(+)h Fn(X)5299 6046 y Fk(j)7 b Fj(+1)5537 6021 y Fn(:)745 b Fp(\(35\))876 6386 y(No)l(w)-11 b(,)41 b(for)h Fn(t)f Fp(in)h(the)f(windo)l(w)2276 6752 y Fo(\(1)c(+)g Fn(\016)2701 6777 y Fj(1)2775 6752 y Fo(\))p Fn(T)2960 6665 y Fj(\()p Fk(j)7 b Fj(\))2937 6801 y(\001)3179 6752 y Fm(\024)47 b Fn(t)f Fm(\024)g Fn(T)3756 6665 y Fj(\()p Fk(j)7 b Fj(\))3733 6801 y(\001)3967 6752 y Fo(+)37 b(\(2)f Fm(\000)h Fn(\016)4557 6777 y Fj(2)4632 6752 y Fo(\))p Fn(T)4817 6665 y Fj(\()p Fk(j)7 b Fj(+1\))4794 6801 y(\001)5158 6752 y Fn(;)1124 b Fp(\(36\))876 7157 y(we)41 b(write)h Fn(t)k Fo(=)g Fn(T)1895 7070 y Fj(\()p Fk(j)7 b Fj(\))1872 7205 y(\001)2106 7157 y Fo(+)37 b Fn(T)2392 7070 y Fj(\()p Fk(j)7 b Fj(+1\))2369 7205 y(\001)2771 7157 y Fo(+)37 b Fn(t)2997 7096 y Fl(0)3085 7157 y Fp(so)k(that)2312 7561 y Fm(\000)p Fn(T)2561 7475 y Fj(\()p Fk(j)7 b Fj(+1\))2538 7610 y(\001)2940 7561 y Fo(+)37 b Fn(\016)3180 7586 y Fj(1)3254 7561 y Fn(T)3374 7475 y Fj(\()p Fk(j)7 b Fj(\))3351 7610 y(\001)3594 7561 y Fm(\024)46 b Fn(t)3829 7493 y Fl(0)3922 7561 y Fm(\024)g Fo(\(1)37 b Fm(\000)g Fn(\016)4522 7586 y Fj(2)4596 7561 y Fo(\))p Fn(T)4781 7475 y Fj(\()p Fk(j)7 b Fj(+1\))4758 7610 y(\001)5123 7561 y Fn(:)876 7927 y Fp(Hence,)40 b(as)h(in)h(part)f(\(i\),)h(a)f(direct)h (application)f(of)h(the)f(normal)h(form)h(around)e Fn(X)5528 7952 y Fk(j)7 b Fj(+1)5807 7927 y Fp(sho)l(ws)41 b(that)1898 8322 y Fo(lim)1860 8429 y Fj(\001)p Fl(!)p Fj(0)2195 8322 y Fo(\010)2315 8253 y Fk(t)2365 8203 y Fg(0)2315 8363 y Fk(H)2441 8322 y Fo(\()p Fn(X)2657 8253 y Fl(0)2702 8322 y Fo(\))46 b(=)g Fn(X)3126 8347 y Fk(j)7 b Fj(+1)3364 8322 y Fn(;)194 b Fm(8)p Fn(X)3847 8253 y Fl(0)3939 8322 y Fm(2)46 b Fo(\012)4216 8253 y Fk(I)9 b(I)4216 8366 y(\033)4293 8383 y Fh(1)4358 8366 y Fk(;\033)4474 8383 y Fh(2)4539 8366 y Fk(;\033)4659 8326 y Fg(0)4655 8406 y Fh(3)4729 8322 y Fo(\(\001\))36 b(+)h Fn(X)5337 8347 y Fk(j)7 b Fj(+1)5575 8322 y Fn(;)876 8753 y Fp(pro)n(vided)54 b Fn(\033)1619 8693 y Fl(0)1613 8794 y Fj(3)1741 8753 y Fp(is)g(small)f(enough.)88 b(Combining)55 b(this)f(with)g(\(35\),)j (we)c(ha)m(v)n(e)g(\002nally)h(pro)n(v)n(en)f(that)876 8952 y(there)g(e)n(xist)g Fo(0)68 b Fn(<)h(\033)2070 8977 y Fj(1)2144 8952 y Fn(;)28 b(\033)2313 8977 y Fj(2)2387 8952 y Fn(;)g(\033)2556 8977 y Fj(3)2699 8952 y Fn(<)68 b Fo(1)54 b Fp(so)f(that,)j(for)e(all)g Fn(t)f Fp(in)h(the)f(windo)l(w) h(\(36\))g(and)f(for)h(all)g Fn(X)81 b Fm(2)876 9151 y Fo(\012)996 9091 y Fk(I)9 b(I)1140 9151 y Fo(\()p Fn(\033)d Fo(\))p Fp(,)40 b(one)h(has)3027 9351 y Fo(lim)2988 9458 y Fj(\001)p Fl(!)p Fj(0)3323 9351 y Fo(\010)3443 9282 y Fk(t)3443 9392 y(H)3569 9351 y Fo(\()p Fn(X)13 b Fo(\))45 b(=)h Fn(X)4208 9376 y Fk(j)7 b Fj(+1)4446 9351 y Fn(:)1125 9730 y Fp(This,)46 b(applied)f(to)g Fn(j)63 b Fo(=)53 b(0)p Fp(,)45 b(implies)h(\(30\).)62 b(F)n(or)45 b Fo(\012)4007 9670 y Fk(I)9 b(V)4007 9771 y(\033)4084 9788 y Fh(1)4149 9771 y Fk(;\033)4265 9788 y Fh(2)4330 9771 y Fk(;\033)4459 9730 y Fo(\(\001\))45 b Fp(the)g(reasoning)g(is)g(completely)876 9930 y(analogous.)1125 10129 y(T)-13 b(o)42 b(treat)f(the)h(case)e (where)i Fn(i)k Fo(=)h Fn(I)13 b Fp(,)41 b(we)g(introduce,)g(for)i Fn(E)55 b(<)46 b Fo(0)p Fp(,)c Fn(q)4923 10058 y Fj(+)4917 10175 y Fk(j)5033 10129 y Fo(\()p Fn(E)10 b Fo(\))40 b Fp(which)h(is)h(the)f(\(right\))876 10328 y(turning)i(point,)g Fe(i.e)n(.)52 b Fp(the)43 b(smallest)f Fn(q)53 b(>)c(q)3290 10353 y Fk(j)3402 10328 y Fp(such)42 b(that)g Fn(V)37 b Fo(\()p Fn(q)6 b Fo(\))47 b(=)h Fn(E)10 b Fp(.)53 b(Then)43 b Fn(\034)5293 10353 y Fk(E)5405 10328 y Fo(\()p Fn(\013)q(;)28 b(x)p Fo(\))41 b Fp(denotes)h(the)3657 10919 y(20)p eop %%Page: 21 21 21 20 bop 876 1489 a Fp(time)32 b(to)f(go)h(from)g Fo(\()p Fn(q)2060 1418 y Fj(+)2054 1536 y Fk(j)2170 1489 y Fo(\()p Fn(E)10 b Fo(\))p Fn(;)28 b Fo(0\))h Fp(to)j Fo(\()p Fn(q)52 b Fo(=)46 b Fn(q)3291 1418 y Fj(+)3285 1536 y Fk(j)3402 1489 y Fo(\()p Fn(E)10 b Fo(\)+)p Fn(x)p Fm(j)p Fn(E)g Fm(j)4114 1429 y Fk(\013)4206 1489 y Fn(;)28 b(p)45 b Fo(=)4585 1302 y Fi(q)p 4751 1302 1750 7 v 187 x Fo(2\()p Fn(E)g Fm(\000)37 b Fn(V)g Fo(\()p Fn(q)5512 1418 y Fj(+)5506 1536 y Fk(j)5622 1489 y Fo(\()p Fn(E)10 b Fo(\))35 b(+)i Fn(x)p Fm(j)p Fn(E)10 b Fm(j)6406 1441 y Fk(\013)6500 1489 y Fo(\))p Fn(;)59 b Fo(\()p Fn(x)46 b(>)876 1735 y Fo(0\))41 b Fp(along)g(the)h(curv)n(e)f Fm(f)p Fn(H)59 b Fo(=)47 b Fn(E)10 b Fm(g)40 b Fp(and)i(we)f(ha)m(v)n(e)f(when)h Fn(E)55 b Fm(\045)46 b Fo(0)1108 2215 y Fn(\034)1181 2240 y Fk(E)1292 2215 y Fo(\()p Fn(\013)q(;)28 b(x)p Fo(\))45 b(=)1918 1989 y Fi(Z)2084 2030 y Fk(q)2149 1976 y Fh(+)2145 2072 y Ff(j)2243 2030 y Fj(\()p Fk(E)6 b Fj(\)+)p Fk(x)p Fl(j)p Fk(E)g Fl(j)2808 1980 y Ff(\013)2011 2366 y Fk(q)2076 2313 y Fh(+)2072 2408 y Ff(j)2169 2366 y Fj(\()p Fk(E)g Fj(\))3396 2103 y Fn(du)p 2951 2177 1072 7 v 2951 2210 a Fi(p)p 3117 2210 906 7 v 142 x Fo(2\()p Fn(E)45 b Fm(\000)37 b Fn(V)g Fo(\()p Fn(u)p Fo(\)\))4088 2215 y(=)4283 2103 y(1)g Fm(\000)g Fo(2)p Fn(\013)p 4283 2177 476 7 v 4393 2329 a Fo(2)p Fn(!)4579 2354 y Fk(j)4779 2215 y Fm(j)28 b Fo(log)r(\()p Fm(\000)p Fn(E)10 b Fo(\))p Fm(j)36 b Fo(+)h Fn(O)5 b Fo(\(1\))p Fn(:)230 b Fp(\(37\))876 2682 y(The)56 b(reasoning)f(is)h(then)g(ag)o(ain)g(similar)g(as)f(abo)n (v)n(e)f(and)i(the)g(case)e Fn(i)73 b Fo(=)g Fn(I)13 b(I)g(I)69 b Fp(is)56 b(treated)g(analo-)876 2881 y(gously)-11 b(.)p 6445 2915 93 133 v 876 3364 a Fu(Pr)m(oof)42 b(of)g(Theor)m(em)f (2.2)p Fp(:)51 b(W)-13 b(e)41 b(ha)m(v)n(e)f(to)i(estimate)f(the)g (inte)n(gral)1711 3770 y Fn(J)1803 3795 y Fj(\001)1921 3770 y Fo(\()p Fn(t;)28 b Fo(\026)-83 b Fn(h)o Fo(\))45 b(=)i(\(2)p Fn(\031)6 b Fo(\026)-83 b Fn(h)n Fo(\))2909 3702 y Fl(\000)p Fj(1)3115 3544 y Fi(Z)46 b(Z)3346 3921 y Fj(\012)3441 3938 y Ff(\033)3519 3921 y Fj(\(\001\))3768 3770 y Fn(A)p Fo(\(\010)4078 3702 y Fk(t)4078 3811 y(H)4203 3770 y Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\)\))p Fn(\032)4750 3586 y Fi(\020)4867 3658 y Fn(q)p 4867 3732 81 7 v 4868 3884 a(")4967 3770 y(;)g(p)5153 3658 y(")p 5145 3732 96 7 v 5145 3884 a Fo(\026)-83 b Fn(h)5260 3586 y Fi(\021)5387 3770 y Fn(dq)6 b(dp;)559 b Fp(\(38\))876 4242 y(for)49 b Fn(A)59 b Fm(2)g Fn(C)1603 4182 y Fl(1)1591 4283 y Fj(0)1744 4242 y Fo(\(I)-37 b(R)1953 4170 y Fj(2)2028 4242 y Fo(\))48 b Fp(such)g(that)h(supp)p Fo(\()p Fn(A)p Fo(\))58 b Fm(\032)h(f)p Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\);)86 b Fn(q)4264 4267 y Fl(\000)4436 4242 y Fn(<)59 b(q)66 b(<)59 b(q)5026 4267 y Fj(+)5137 4242 y Fm(g)p Fp(.)72 b(It)50 b(is)e(not)h(dif)l(\002cult)g(to)876 4441 y(see)41 b(that)h(we)g(can)f(al)n(w)n(ays)h(assume)f(that)h Fn(A)g Fp(is)g(supported)g(in)g(a)g(small)f(neighbourhood)i(of)f(one)g (of)876 4640 y(the)f(four)h(branches)f(of)h(the)f(zero-ener)m(gy)h (curv)n(e)f(close)f(to)i Fn(X)4425 4665 y Fj(0)4499 4640 y Fp(.)1125 4840 y(In)54 b(a)f(\002rst)i(step)e(we)g(shall)g(assume)g (that)h(the)f(support)h(of)g Fn(A)f Fp(is)h(contained)f(in)h Fn(D)5900 4870 y Fk(\032=)p Fj(2)p Fk(;)p Fj(0)6271 4840 y Fp(for)h Fn(\032)876 5039 y Fp(small)48 b(enough.)71 b(W)-13 b(e)47 b(shall)h(therefore)h(be)e(able)h(to)h(compute)f Fn(J)4580 5064 y Fj(\001)4697 5039 y Fo(\()p Fn(t;)28 b Fo(\026)-83 b Fn(h)o Fo(\))48 b Fp(using)g(the)g(normal)h(form)876 5238 y(\(Proposition)g(3.5\).)72 b(F)n(or)48 b(simplicity)h(we)f(erase) f(in)i(what)f(follo)l(ws)g(the)g(inde)n(x)g(0)h(\()p Fn(F)81 b Fo(=)59 b Fn(F)6169 5263 y Fj(0)6244 5238 y Fn(;)28 b(f)76 b Fo(=)876 5437 y Fn(f)957 5462 y Fj(0)1032 5437 y Fo(\))p Fp(.)51 b(W)-13 b(e)41 b(will)i(use)e(the)g(notation)i Fn(J)62 b Fm(\031)47 b Fn(M)60 b Fp(to)42 b(mean)g(that)g(the)f(dif)l (ference)h Fn(J)53 b Fm(\000)38 b Fn(M)60 b Fp(is)41 b Fn(O)5 b Fo(\(\001)6234 5377 y Fk(s)6304 5437 y Fo(\))42 b Fp(for)876 5637 y(some)47 b Fn(s)57 b(>)h Fo(0)p Fp(.)70 b Fn(F)g Fp(is)47 b(a)h(symplectic)f(transform)h(so)g(we)f(get,)h (after)g(a)g(slight)f(modi\002cation)i(of)f(the)876 5836 y Fn(\033)6 b Fp(')-9 b(s,)989 6294 y Fn(J)1081 6319 y Fj(\001)1198 6294 y Fo(\()p Fn(t;)28 b Fo(\026)-83 b Fn(h)o Fo(\))46 b Fm(\031)g Fo(\(2)p Fn(\031)6 b Fo(\026)-83 b Fn(h)o Fo(\))2187 6226 y Fl(\000)p Fj(1)2393 6068 y Fi(Z)46 b(Z)2637 6416 y Fj(~)2623 6445 y(\012)2718 6462 y Ff(\033)2797 6445 y Fj(\(\001\))3090 6252 y Fo(~)3046 6294 y Fn(A)o Fo(\(\010)3355 6208 y Fk(t=f)3551 6158 y Fg(0)3596 6208 y Fj(\()p Fk(H)9 b Fl(\016)p Fk(F)3936 6158 y Fg(\000)p Fh(1)4090 6208 y Fj(\()4178 6179 y(~)4142 6208 y Fk(X)g Fj(\))3355 6343 y Fk(K)4320 6294 y Fo(\()j(~)-95 b Fn(q)6 b(;)42 b Fo(~)-97 b Fn(p)o Fo(\)\))p Fn(\032)4866 6010 y Fi( )5016 6182 y Fn(q)6 b Fo(\()5208 6140 y(~)5161 6182 y Fn(X)12 b 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Fk(\033)3422 7106 y Fh(2)3541 7157 y Fm(\024)47 b(j)12 b Fo(~)-95 b Fn(q)20 b Fo(~)-97 b Fn(p)o Fm(j)47 b(\024)f Fo(\001)4332 7089 y Fj(2)p Fl(\000)p Fk(\033)4579 7106 y Fh(1)4652 7157 y Fn(;)28 b Fm(j)12 b Fo(~)-95 b Fn(q)6 b Fm(j)37 b Fo(+)g Fm(j)14 b Fo(~)-97 b Fn(p)p Fm(j)46 b(\024)h Fo(\001)5637 7089 y Fj(1)p Fl(\000)p Fk(\033)5884 7106 y Fh(3)5957 7157 y Fm(g)p Fn(:)876 7484 y Fp(But)1212 7442 y Fo(~)1165 7484 y Fn(X)61 b Fp(is)48 b Fo(\001)1661 7424 y Fj(1)p Fl(\000)p Fk(\033)1908 7441 y Fh(3)1982 7484 y Fp(-close)g(to)h(zero)g(so)f(it)h(is)f(not)h(dif)l(\002cult)h (to)f(pro)n(v)n(e,)g(applying)g(Proposition)g(3.5,)876 7684 y(that)42 b(we)f(are)g(allo)l(wed)g(to)h(linearize)f Fn(F)3123 7623 y Fl(\000)p Fj(1)3342 7684 y Fp(so)h(that)f(we)g(ha)m(v) n(e)2101 8053 y Fn(F)2231 7985 y Fl(\000)p Fj(1)2409 8053 y Fo(\()2522 8011 y(~)2474 8053 y Fn(X)12 b Fo(\))46 b(=)3000 7941 y(1)p 2930 8015 222 7 v 2930 8048 a Fm(p)p 3069 8048 84 7 v 3069 8186 a Fo(2)3172 8053 y(\()12 b(~)-95 b Fn(q)42 b Fo(+)37 b Fn(!)3622 8078 y Fj(0)3711 8053 y Fo(~)-97 b Fn(p;)28 b Fm(\000)4065 7941 y Fo(~)-96 b Fn(q)p 4004 8015 178 7 v 4004 8167 a(!)4107 8192 y Fj(0)4238 8053 y Fo(+)51 b(~)-97 b Fn(p)p Fo(\))36 b(+)h Fn(O)5 b Fo(\()p Fm(j)5045 8011 y Fo(~)4998 8053 y Fn(X)12 b Fm(j)5194 7985 y Fj(2)5268 8053 y Fo(\))p Fn(:)876 8505 y Fp(W)-13 b(e)62 b(also)g(ha)m(v)n(e)g Fn(f)1956 8444 y Fl(0)2003 8505 y Fo(\()p Fn(H)j Fm(\016)53 b Fn(F)2537 8444 y Fl(\000)p Fj(1)2715 8505 y Fo(\()2828 8463 y(~)2780 8505 y Fn(X)12 b Fo(\)\))85 b(=)h Fn(f)3459 8444 y Fl(0)3506 8505 y Fo(\(0\))52 b(+)h Fn(O)5 b Fo(\()p Fm(j)4242 8463 y Fo(~)4196 8505 y Fn(X)11 b Fm(j)p Fo(\))p Fp(.)115 b(So)63 b(we)g(get,)k(using)c(the)g(e)n(xplicit)876 8704 y(e)n(xpression)40 b(for)i(the)g(\003o)l(w)f(of)h(the)f(Hamiltonian)h Fn(K)12 b Fp(,)41 b Fn(J)4061 8729 y Fj(\001)4179 8704 y Fo(\()p Fn(t;)28 b Fo(\026)-83 b Fn(h)o Fo(\))45 b Fm(\031)i Fn(J)4867 8644 y Fj(0)4851 8750 y(\001)4968 8704 y Fo(\()p Fn(t;)28 b Fo(\026)-83 b Fn(h)o Fo(\))p Fp(,)41 b(where)1696 9120 y Fn(J)1804 9051 y Fj(0)1788 9161 y(\001)1905 9120 y Fo(\()p Fn(t;)28 b Fo(\026)-83 b Fn(h)o Fo(\))46 b(=)2603 9007 y(1)p 2505 9082 280 7 v 2505 9234 a(2)p Fn(\031)6 b Fo(\026)-83 b Fn(h)2832 8894 y Fi(Z)46 b(Z)3077 9242 y Fj(~)3062 9271 y(\012)3157 9288 y Ff(\033)3236 9271 y Fj(\(\001\))3529 9078 y Fo(~)3485 9120 y Fn(A)o Fo(\(e)3748 9051 y Fk(t!)3882 9068 y Fh(0)3967 9120 y Fo(~)-95 b Fn(q)6 b(;)28 b Fo(e)4183 9051 y Fl(\000)p Fk(t!)4421 9068 y Fh(0)4507 9120 y Fo(~)-97 b Fn(p)p Fo(\))2507 9607 y Fm(\002)p Fn(\032)2750 9373 y Fi(\022)3000 9495 y Fo(1)p 2892 9569 299 7 v 2892 9602 a Fm(p)p 3030 9602 84 7 v 138 x Fo(2)p Fn(")3211 9607 y Fo(\()12 b(~)-95 b Fn(q)42 b Fo(+)37 b Fn(!)3661 9632 y Fj(0)3750 9607 y Fo(~)-97 b Fn(p)p Fo(\))p Fn(;)55 b Fo(\()p Fm(\000)4261 9495 y Fo(~)-96 b Fn(q)p 4200 9569 178 7 v 4200 9721 a(!)4303 9746 y Fj(0)4434 9607 y Fo(+)51 b(~)-97 b Fn(p)p Fo(\))4888 9495 y Fn(")p 4769 9569 318 7 v 4769 9602 a Fm(p)p 4906 9602 84 7 v 4906 9740 a Fo(2\026)-83 b Fn(h)5105 9373 y Fi(\023)5255 9607 y Fn(d)12 b Fo(~)-95 b Fn(q)6 b(d)14 b Fo(~)-97 b Fn(p:)691 b Fp(\(40\))876 10023 y(No)l(w)59 b(the)h(result)f(of)h(the)f(computation)h(of)g(the)g (limit)g(for)g Fo(\001)g Fp(going)g(to)g(0)f(of)h(the)f(last)h(inte)n (gral)876 10222 y(depends)47 b(on)h(the)g(relati)l(v)n(e)f(size)g(of)h Fn(")h Fp(and)f Fo(\026)-83 b Fn(h)o Fp(.)71 b(There)48 b(are)f(three)h(dif)l(ferent)h(cases)d(to)i(check:)64 b Fn(")58 b Fm(\031)876 10280 y(p)p 1014 10280 96 7 v 141 x Fo(\026)-83 b Fn(h)p Fp(,)41 b Fn(")47 b Fo(=)f Fn(o)p Fo(\()1637 10280 y Fm(p)p 1775 10280 V 141 x Fo(\026)-83 b Fn(h)p Fo(\))p Fp(,)2019 10280 y Fm(p)p 2157 10280 V 141 x Fo(\026)g Fn(h)46 b Fo(=)g Fn(o)p Fo(\()p Fn(")p Fo(\))p Fp(.)3657 10919 y(21)p eop %%Page: 22 22 22 21 bop 1125 1516 a Fp(Let)49 b(us)f(be)n(gin)g(with)h(the)f (symmetric)h(case)e(when)63 b Fo(lim)4093 1619 y Fk(")p Fl(!)p Fj(0)4430 1404 y Fo(\026)-83 b Fn(h)p 4402 1478 152 7 v 4402 1630 a(")4479 1582 y Fj(2)4633 1516 y Fo(=)59 b Fn(c)p Fp(,)50 b Fo(0)59 b Fn(<)g(c)g(<)g Fo(+)p Fm(1)p Fp(.)73 b(No)l(w)48 b(we)876 1798 y(compute)42 b(the)f(limit)i(with)f (the)f(change)g(of)h(v)l(ariables:)64 b Fo(~)-95 b Fn(q)52 b Fo(=)47 b(e)4433 1738 y Fl(\000)p Fk(t!)4671 1755 y Fh(0)4744 1798 y Fn(u)p Fp(,)55 b Fo(~)-97 b Fn(p)46 b Fo(=)h(e)5302 1738 y Fl(\000)p Fk(t!)5540 1755 y Fh(0)5612 1798 y Fn(v)5744 1733 y Fj(\026)-68 b Fk(h)p 5719 1760 128 7 v 5719 1855 a(")5782 1822 y Fh(2)5908 1798 y Fp(and)41 b(obtain)876 1997 y(under)h(the)f(e)n(xtra)g(condition)57 b Fo(lim)2580 2100 y Fk(")p Fl(!)p Fj(0)2869 1997 y Fn(")p Fo(e)3020 1929 y Fk(t!)3154 1946 y Fh(0)3273 1997 y Fo(=)46 b Fn(\024)p Fp(:)943 2540 y Fo(lim)905 2647 y Fj(\001)p Fl(!)p Fj(0)1240 2540 y Fn(J)1348 2472 y Fj(0)1332 2581 y(\001)1449 2540 y Fo(\()p Fn(t;)28 b Fo(\026)-83 b Fn(h)o Fo(\))166 b(=)2424 2428 y(1)p 2289 2502 354 7 v 2289 2654 a(2)p Fn(\031)6 b(\024)2569 2606 y Fj(2)2690 2314 y Fi(Z)47 b(Z)2921 2691 y Fj(I)-30 b(R)3035 2653 y Fh(2)3180 2498 y Fo(~)3136 2540 y Fn(A)o Fo(\()p Fn(u;)28 b Fo(0\))p Fn(\032)3756 2306 y Fi(\022)4014 2428 y Fo(1)p 3897 2502 318 7 v 3897 2535 a Fm(p)p 4036 2535 84 7 v 4036 2672 a Fo(2)p Fn(\024)4234 2540 y Fo(\()p Fn(u)37 b Fo(+)g Fn(c!)4772 2565 y Fj(0)4846 2540 y Fn(v)6 b Fo(\))p Fn(;)5208 2428 y Fo(1)p 5091 2502 318 7 v 5091 2535 a Fm(p)p 5230 2535 84 7 v 5230 2672 a Fo(2)p Fn(\024)5428 2540 y Fo(\()p Fn(v)43 b Fm(\000)5879 2428 y Fn(u)p 5802 2502 250 7 v 5802 2654 a(c!)5977 2679 y Fj(0)6072 2306 y Fi(\023)6222 2540 y Fn(dudv)1974 2990 y Fo(=)2269 2764 y Fi(Z)2361 3142 y Fj(I)-30 b(R)2555 2948 y Fo(~)2511 2990 y Fn(A)p Fo(\()p Fn(u;)28 b Fo(0\))p Fn(G)3149 3015 y Fk(\024;c;)p Fj(2)3438 2990 y Fo(\()p Fn(u)p Fo(\))p Fn(du;)876 3437 y Fp(where)59 b(we)f(used)h(\(14\).)104 b(Let)59 b(us)g(remark)g(that)h (close)e(to)h(zero,)k(in)c(the)g(phase)f(space,)k(we)d(ha)m(v)n(e)876 3637 y Fm(f)14 b Fo(~)-97 b Fn(p)83 b Fo(=)h(0)p Fm(g)61 b Fp(if)h(and)g(only)g(if)g Fm(f)p Fn(p)83 b Fo(=)g Fn(p)p Fo(\()p Fn(q)6 b Fo(\))83 b(=)g Fm(\006)3731 3495 y Fi(p)p 3898 3495 556 7 v 3898 3637 a Fm(\000)p Fo(2)p Fn(V)36 b Fo(\()p Fn(q)6 b Fo(\))p Fm(g)61 b Fp(where)h Fm(\006)83 b Fo(=)h(+)62 b Fp(if)g Fn(q)89 b(>)84 b Fo(0)62 b Fp(and)876 3836 y Fm(\006)72 b Fo(=)h Fm(\000)56 b Fp(if)g Fn(q)78 b(<)73 b Fo(0)p Fp(.)94 b(So)56 b(to)f(come)h(back)f(to)h(the)f(curv)n (e)g Fo(\006)4379 3861 y Fj(0)4454 3836 y Fp(,)k(let)c(us)h(try)g(to)g (compute)f(the)h(right)876 4035 y(hand)51 b(side)f(with)h(the)g(change) f(of)h(v)l(ariable)g Fn(u)64 b Fo(=)76 b(~)-96 b Fn(q)6 b Fo(\()p Fn(q)g(;)28 b(p)p Fo(\()p Fn(q)6 b Fo(\)\))p Fp(.)78 b(F)n(or)51 b(that)g(purpose,)h(we)f(compute)896 4160 y Fk(d)10 b Fj(~)-76 b Fk(q)p 896 4196 135 7 v 896 4292 a(dq)1050 4234 y Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\()p Fn(q)6 b Fo(\)\))86 b(=)1952 4160 y Fk(@)17 b Fj(~)-76 b Fk(q)p 1952 4196 144 7 v 1952 4292 a(@)7 b(q)2169 4234 y Fo(+)54 b Fn(p)2436 4174 y Fl(0)2482 4234 y Fo(\()p Fn(q)6 b Fo(\))2714 4160 y Fk(@)17 b Fj(~)-76 b Fk(q)p 2711 4196 147 7 v 2711 4292 a(@)7 b(p)2877 4234 y Fp(.)120 b(By)64 b(taking)h(the)f(\002rst)h Fn(X)13 b Fp(-deri)l(v)l(ati)l(v)n (e)63 b(in)i(the)f(normal)h(form)876 4434 y(e)n(xpression)40 b(we)h(ha)m(v)n(e)2402 4799 y Fn(f)2501 4730 y Fl(0)2548 4799 y Fo(\()p Fn(H)13 b Fo(\()p Fn(X)g Fo(\)\))p Fn(H)3261 4730 y Fl(0)3306 4799 y Fo(\()p Fn(X)g Fo(\))45 b(=)i Fn(F)3938 4730 y Fl(0)3984 4799 y Fo(\()p Fn(X)13 b Fo(\))4265 4730 y Fl(\003)4340 4799 y Fn(K)4493 4730 y Fl(0)4540 4799 y Fo(\()p Fn(F)23 b Fo(\()p Fn(X)13 b Fo(\)\))1247 b Fp(\(41\))876 5164 y(where)41 b Fn(M)1502 5104 y Fl(\003)1620 5164 y Fp(denotes)g(the)g(transpose)g(of)h(the)f(matrix)h Fn(M)18 b Fp(.)51 b(So)42 b(we)f(get)h(\()p Fn(f)5104 5104 y Fl(0)5151 5164 y Fo(\(0\))j(=)h Fn(!)5693 5093 y Fl(\000)p Fj(1)5687 5209 y(0)5872 5164 y Fp(\))2382 5448 y Fi(\022)2587 5580 y Fn(V)2721 5520 y Fl(0)2768 5580 y Fo(\()p Fn(q)6 b Fo(\))2740 5780 y Fn(p)3060 5448 y Fi(\023)3228 5682 y Fo(=)46 b Fn(!)3506 5707 y Fj(0)3609 5398 y Fi( )3845 5486 y Fk(@)17 b Fj(~)-76 b Fk(q)p 3845 5522 144 7 v 3845 5618 a(@)7 b(q)4196 5486 y(@)19 b Fj(~)-78 b Fk(p)p 4196 5522 147 7 v 4198 5618 a(@)7 b(q)3845 5721 y(@)17 b Fj(~)-76 b Fk(q)p 3843 5757 V 3843 5852 a(@)7 b(p)4196 5721 y(@)19 b Fj(~)-78 b Fk(p)p 4196 5757 V 4196 5852 a(@)7 b(p)4445 5398 y Fi(!)4604 5448 y(\022)4824 5580 y Fo(~)-97 b Fn(p)4824 5780 y Fo(~)h Fn(q)4976 5448 y Fi(\023)6328 5682 y Fp(\(42\))876 6185 y(A)41 b(little)h(algebraic)f (computation)h(gi)l(v)n(es)2750 6519 y Fn(d)12 b Fo(~)-95 b Fn(q)p 2750 6593 167 7 v 2750 6745 a(dq)2937 6631 y Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\()p Fn(q)6 b Fo(\)\))43 b(=)k Fm(\006)4155 6519 y Fn(!)4258 6544 y Fj(0)p 3883 6593 722 7 v 3883 6626 a Fi(p)p 4049 6626 556 7 v 142 x Fm(\000)p Fo(2)p Fn(V)37 b Fo(\()p Fn(q)6 b Fo(\))4636 6631 y(~)-95 b Fn(q)6 b(;)1578 b Fp(\(43\))876 7115 y(leading)41 b(to)2530 7458 y Fo(~)-96 b Fn(q)52 b Fo(=)47 b(exp)3100 7174 y Fi( )3232 7458 y Fm(\006)p Fn(!)3464 7483 y Fj(0)3566 7232 y Fi(Z)3732 7273 y Fk(q)3658 7609 y(q)3719 7626 y Fh(0)4131 7346 y Fn(du)p 3853 7420 737 7 v 3853 7453 a Fi(p)p 4019 7453 571 7 v 142 x Fm(\000)p Fo(2)p Fn(V)37 b Fo(\()p Fn(u)p Fo(\))4609 7174 y Fi(!)4801 7458 y Fo(~)-116 b Fn(q)4842 7483 y Fj(0)4917 7458 y Fn(:)1365 b Fp(\(44\))876 7906 y(W)-13 b(e)56 b(shall)g(compute)69 b Fo(~)-95 b Fn(q)6 b Fo(\()p Fn(q)g(;)28 b(p)p Fo(\()p Fn(q)6 b Fo(\)\))54 b Fp(more)j(e)n(xplicitly)g(by)g(going)g(to)g(the)f(limit)i Fn(q)5489 7931 y Fj(0)5638 7906 y Fm(!)74 b Fo(0)p Fp(.)97 b(W)-13 b(e)56 b(can)876 8106 y(easily)41 b(compute)1938 8032 y Fk(d)10 b Fj(~)-76 b Fk(q)p 1938 8067 135 7 v 1938 8163 a(dq)2092 8106 y Fo(\(0)p Fn(;)28 b Fo(0\))p Fp(;)41 b(hence)g(using)g(the)g(T)-13 b(aylor)42 b(formula)g(we)g(ha)m (v)n(e)3074 8588 y Fo(~)-116 b Fn(q)3115 8613 y Fj(0)3236 8588 y Fo(=)3500 8475 y(1)p 3431 8549 222 7 v 3431 8583 a Fm(p)p 3570 8583 84 7 v 3570 8720 a Fo(2)3673 8588 y Fn(q)3747 8613 y Fj(0)3858 8588 y Fo(+)37 b Fn(O)5 b Fo(\()p Fn(q)4301 8519 y Fj(2)4295 8629 y(0)4375 8588 y Fo(\))1888 b Fp(\(45\))876 9038 y(so)41 b(that)1821 9347 y Fo(~)-95 b Fn(q)52 b Fo(=)2199 9235 y(1)p 2130 9309 222 7 v 2130 9342 a Fm(p)p 2269 9342 84 7 v 2269 9479 a Fo(2)2372 9347 y Fm(j)p Fn(q)6 b Fm(j)28 b Fo(exp)2853 9063 y Fi( )2984 9347 y Fm(\006)3141 9121 y Fi(Z)3307 9162 y Fk(q)3233 9498 y Fj(0)3408 9063 y Fi( )3839 9235 y Fn(!)3942 9260 y Fj(0)p 3560 9309 737 7 v 3560 9342 a Fi(p)p 3726 9342 571 7 v 142 x Fm(\000)p Fo(2)p Fn(V)37 b Fo(\()p Fn(u)p Fo(\))4353 9347 y Fm(\000)4545 9235 y Fo(1)p 4539 9309 96 7 v 4539 9461 a Fn(u)4654 9063 y Fi(!)4813 9347 y Fn(du)4994 9063 y Fi(!)5172 9347 y Fm(\021)46 b Fn(`)p Fo(\()p Fn(q)6 b Fo(\))p Fn(:)656 b Fp(\(46\))876 9784 y(This)42 b(\002nally)g(yields)1628 10238 y Fo(lim)1590 10345 y Fj(\001)p Fl(!)p Fj(0)1925 10238 y Fn(J)2033 10169 y Fj(0)2017 10279 y(\001)2134 10238 y Fo(\()p Fn(t;)28 b Fo(\026)-83 b Fn(h)o Fo(\))46 b(=)g Fn(!)2817 10263 y Fj(0)2920 10012 y Fi(Z)3086 10053 y Fk(q)3147 10070 y Fh(+)3012 10389 y Fk(q)3073 10406 y Fg(\000)3276 10238 y Fn(A)p Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\()p Fn(q)6 b Fo(\)\))p Fn(`)p Fo(\()p Fn(q)g Fo(\))p Fn(G)4389 10263 y Fk(\024;c;)p Fj(2)4676 10238 y Fo(\()p Fn(`)p Fo(\()p Fn(q)g Fo(\)\))5381 10125 y Fn(dq)p 5105 10199 722 7 v 5105 10233 a Fi(p)p 5270 10233 556 7 v 5270 10375 a Fm(\000)p Fo(2)p Fn(V)37 b Fo(\()p Fn(q)6 b Fo(\))5845 10238 y Fn(:)437 b Fp(\(47\))3657 10919 y(22)p eop %%Page: 23 23 23 22 bop 876 1455 a Fp(This)40 b(pro)n(v)n(es)f(Theorem)h(2.2)f(in)h (the)g(symmetric)g(case)f Fn(\017)4121 1394 y Fj(2)4242 1455 y Fm(\030)46 b Fo(\026)-83 b Fn(h)40 b Fp(when)f Fn(A)h Fp(is)f(supported)h(in)g(a)g(small)876 1654 y(neighborhood)45 b(of)h Fo(0)p Fp(.)61 b(F)n(or)45 b(the)g(tw)n(o)h(other)f(cases)e(the) i(computation)g(is)g(almost)g(the)g(same)f(and)h(is)876 1853 y(left)d(to)f(the)h(reader)-9 b(.)1125 2052 y(Let)53 b(us)g(no)l(w)g(assume)f(that)h Fn(A)g Fp(is)g(supported)g(outside)g(a) g(small)f(disc)h Fn(D)5359 2082 y Fk(\032=)p Fj(4)p Fk(;)p Fj(0)5677 2052 y Fp(.)85 b(The)54 b(general)876 2252 y(case)47 b(will)h(follo)l(w)g(using)g(a)g(partition)g(of)h(unity)-11 b(.)71 b(It)48 b(is)g(enough)f(to)i(assume)d(that)i Fn(A)g Fp(is)g(supported)876 2451 y(in)i Fm(f)p Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\);)88 b Fn(p)61 b Fm(\025)h Fo(0)p Fn(;)89 b(q)2268 2476 y Fl(\000)2423 2451 y Fo(+)43 b Fn(\016)68 b(<)62 b(q)67 b(<)62 b(q)3334 2476 y Fj(+)3487 2451 y Fm(\000)44 b Fn(\016)m(;)28 b Fo(\001)3943 2391 y Fj(2+)p Fk(\033)4188 2408 y Fh(1)4322 2451 y Fm(\024)62 b(j)p Fn(H)13 b Fo(\()p Fn(q)6 b(;)28 b(p)p Fo(\))p Fm(j)61 b(\024)h Fo(\001)5514 2391 y Fj(2)p Fl(\000)p Fk(\033)5761 2408 y Fh(2)5834 2451 y Fm(g)p Fp(,)51 b(for)g(some)876 2650 y Fn(\016)h(>)46 b Fo(0)37 b Fp(and)f Fn(\033)1668 2675 y Fk(j)1784 2650 y Fn(>)46 b Fo(0)p Fp(.)j(By)36 b(the)g(same)g(ar)m(gument)g(already)g(used)g(in)g(the)g(proof)h(of)g (Theorem)f(2.1)g(\(ii\),)876 2868 y(there)49 b(e)n(xists)f(a)h (\002nite)h(time)f Fn(T)2637 2893 y Fj(1)2772 2868 y Fn(>)61 b Fo(0)49 b Fp(depending)g(only)h(on)f Fn(\016)55 b Fp(such)49 b(that)g Fn(A)42 b Fm(\016)h Fo(\010)5589 2794 y Fk(T)5667 2811 y Fh(1)5589 2916 y Fk(H)5790 2868 y Fp(is)49 b(supported)876 3086 y(in)44 b Fn(D)1186 3116 y Fk(\032=)p Fj(2)p Fk(;)p Fj(0)1503 3086 y Fp(.)59 b(So)44 b(we)f(can)g(apply)h(the)g(result)g(pro)n(v)n(ed)f(in)h(the)g(\002rst)g (step)f(to)h Fn(A)5238 3111 y Fj(1)5363 3086 y Fo(:=)50 b Fn(A)39 b Fm(\016)f Fo(\010)5993 3012 y Fk(T)6071 3029 y Fh(1)5993 3135 y Fk(H)6145 3086 y Fp(.)58 b(More)876 3285 y(e)n(xplicitely)41 b(we)g(ha)m(v)n(e)1585 3693 y Fn(J)1677 3718 y Fj(\001)1795 3693 y Fo(\()p Fn(t;)28 b Fo(\026)-83 b Fn(h)o Fo(\))45 b(=)i(\(2)p Fn(\031)6 b Fo(\026)-83 b Fn(h)n Fo(\))2783 3624 y Fl(\000)p Fj(1)2989 3467 y Fi(Z)46 b(Z)3220 3844 y Fj(\012)3315 3861 y Ff(\033)3393 3844 y Fj(\(\001\))3642 3693 y Fn(A)3767 3718 y Fj(1)3841 3693 y Fo(\(\010)4026 3619 y Fk(t)p Fj(+)p Fk(T)4256 3636 y Fh(1)4026 3741 y Fk(H)4329 3693 y Fo(\()p Fn(q)6 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(parametrize)h(the)f(curv)n(e)g Fo(\006)4571 5017 y Fj(0)4689 4992 y Fp(with)h(the)g(time)f(of)h(the)g(classical)876 5191 y(\003o)l(w)j(\(this)g(is)g(possible)f(because)g Fn(A)g Fp(is)h(supported)g(outside)f(the)h(equilibrium)h(points.)68 b(Then)47 b(we)876 5390 y(translate)35 b(in)i(time)f(by)g Fn(T)2270 5415 y Fj(1)2380 5390 y Fp(and)g(remark)g(that)g(an)g(e)n (xplicit)f(computation)h(gi)l(v)n(es)f(easily)h Fn(`)p Fo(\()p Fn(q)6049 5415 y Fk(s)p Fj(+)p Fk(T)6292 5432 y Fh(1)6365 5390 y Fo(\))45 b(=)876 5590 y(e)950 5529 y Fk(!)1034 5546 y Fh(0)1098 5529 y Fk(T)1176 5546 y Fh(1)1250 5590 y Fn(`)p Fo(\()p Fn(q)1458 5615 y Fk(s)1529 5590 y Fo(\))p Fp(.)50 b(Hence)41 b(we)g(get)g(\002nally)1462 5776 y Fi(Z)1554 6154 y Fj(\006)1649 6171 y Fh(unst)1908 6002 y Fn(A)2033 6027 y Fj(1)2107 6002 y Fo(\()p Fn(X)13 b Fo(\))p Fn(g)2467 6027 y Fk(\024)2545 6044 y Fh(1)2609 6027 y Fk(;c;\026)2836 6002 y Fo(\()p Fn(X)g Fo(\))p Fn(d)p Fo(\006)3323 6027 y Fj(0)3396 6002 y Fo(\()p Fn(X)g Fo(\))45 b(=)3897 5776 y Fi(Z)3990 6154 y Fj(\006)4085 6171 y Fh(unst)4344 6002 y Fn(A)p Fo(\()p Fn(X)13 b Fo(\))p Fn(g)4829 6027 y Fk(\024;c;\026)5132 6002 y Fo(\()p Fn(X)g Fo(\))p Fn(d)p Fo(\006)5619 6027 y Fj(0)5693 6002 y Fo(\()p Fn(X)g Fo(\))p Fn(;)308 b Fp(\(50\))876 6429 y(which)41 b(is)h(the)f(e)n(xpected)f(formula)i(to)g(end)f(the)h(proof)g(of)g (Theorem)g(2.2.)p 6404 6462 93 133 v 1125 6713 a Fu(Pr)m(oof)60 b(of)g(Theor)m(em)f(2.3)p Fp(:)87 b(This)60 b(Theorem)g(is)f(an)h(e)n (xtension)e(of)i(Theorem)g(2.1)f(\(ii\).)106 b(W)-13 b(e)876 6912 y(obtain)56 b(the)h(proof)g(in)f(this)g(more)h(general)f (case)f(easily)h(from)h(the)f(proof)h(of)g(Theorem)g(2.1)f(\(ii\))876 7111 y(where)45 b(the)g(general)h(mechanism)e(for)i(multiple)h(wells)d (has)h(already)g(been)g(described.)63 b(W)-13 b(e)44 b(only)876 7311 y(sk)n(etch)d(the)g(ar)m(gument)h(leading)f(to)h (\(16\))g(and)g(\(17\).)1125 7537 y(Fix)34 b Fo(0)47 b Fn(<)f(`)g Fm(\024)g Fn(N)2108 7562 y Fj(+)2219 7537 y Fp(.)j(Remembering)33 b(that)h Fn(T)3687 7450 y Fj(\()p Fk(j)7 b Fj(\))3664 7585 y(\001)3907 7537 y Fo(=)47 b Fn(T)4203 7450 y Fj(\(0\))4180 7585 y(\001)4415 7537 y Fp(in)34 b(this)g(case)f(and)g(iterating)i(equation)876 7736 y(\(35\))51 b Fn(`)f Fp(times,)h(one)f(pro)n(v)n(es)e(easily)i (that,)h(gi)l(v)n(en)f Fo(0)62 b Fn(<)f(\033)4181 7676 y Fl(0)4175 7777 y Fj(3)4312 7736 y Fn(<)g Fo(1)p Fp(,)52 b(there)e(e)n(xist)f Fo(0)62 b Fn(<)g(\033)5853 7761 y Fj(1)5927 7736 y Fn(;)28 b(\033)6096 7761 y Fj(2)6170 7736 y Fn(;)g(\033)6339 7761 y Fj(3)6475 7736 y Fn(<)876 7935 y Fp(small)41 b(enough,)g(so)g(that)2381 8339 y Fo(\010)2501 8239 y Fj(2)p Fk(`T)2719 8179 y Fh(\(0\))2701 8282 y(\001)2501 8388 y Fk(H)2882 8339 y Fo(\(\012)3067 8271 y Fk(I)9 b(I)3067 8381 y(\033)3211 8339 y Fo(\(\001\)\))45 b Fm(\032)h Fo(\012)3884 8271 y Fk(I)9 b(I)3884 8381 y(\033)3961 8398 y Fh(1)4026 8381 y Fk(;\033)4142 8398 y Fh(2)4207 8381 y Fk(;\033)4327 8347 y Fg(0)4381 8339 y Fo(\(\001\))36 b(+)h Fn(X)4989 8364 y Fk(`)5053 8339 y Fn(:)876 8667 y Fp(\277From)56 b(this)g(one)g(concludes)e(as)i (before)f(that)h(there)g(e)n(xist)f Fo(0)73 b Fn(<)f(\033)4862 8692 y Fj(1)4937 8667 y Fn(;)28 b(\033)5106 8692 y Fj(2)5180 8667 y Fn(;)g(\033)5349 8692 y Fj(3)5496 8667 y Fn(<)56 b Fp(so)f(that)h(for)g(all)876 8867 y Fn(X)j Fm(2)45 b Fo(\012)1349 8806 y Fk(I)9 b(I)1349 8908 y(\033)1493 8867 y Fo(\(\001\))p Fp(,)3114 9066 y Fo(lim)3075 9173 y Fj(\001)p Fl(!)p Fj(0)3411 9066 y Fo(\010)3531 8997 y Fk(t)3531 9107 y(H)3656 9066 y Fo(\()p Fn(X)k Fo(\))45 b(=)i Fn(X)4296 9091 y Fk(`)4359 9066 y Fn(;)876 9397 y Fp(for)42 b(all)g Fn(t)f Fp(such)g(that)2249 9725 y Fo(\(2)p Fn(`)c Fm(\000)g Fo(1)f(+)i Fn(\016)3029 9750 y Fj(1)3103 9725 y Fo(\))p Fn(T)3288 9638 y Fj(\(0\))3265 9773 y(\001)3512 9725 y Fm(\024)46 b Fn(t)g Fm(\024)g Fo(\(2)p Fn(`)37 b Fo(+)g(1)g Fm(\000)g Fn(\016)4748 9750 y Fj(2)4822 9725 y Fo(\))p Fn(T)5007 9638 y Fj(\(0\))4984 9773 y(\001)5185 9725 y Fn(:)876 10053 y Fp(This)42 b(in)f(turn)h (implies)g(as)f(before)g(that,)g(for)h(such)f Fn(t)p Fp(,)2819 10380 y Fo(lim)2781 10488 y Fj(\001)p Fl(!)p Fj(0)3116 10380 y Fn(J)3224 10312 y Fk(I)9 b(I)3208 10421 y Fj(\001)3368 10380 y Fo(\()p Fn(t;)28 b Fo(\026)-83 b Fn(h)o Fo(\))46 b(=)g Fn(\013)4054 10405 y Fk(I)9 b(I)4198 10380 y Fn(A)p Fo(\()p Fn(X)4526 10405 y Fk(`)4589 10380 y Fo(\))p Fn(:)3657 10919 y Fp(23)p eop %%Page: 24 24 24 23 bop 876 1455 a Fp(Similarly)-11 b(,)2728 1654 y Fo(lim)2690 1761 y Fj(\001)p Fl(!)p Fj(0)3025 1654 y Fn(J)3133 1585 y Fk(I)9 b(V)3117 1695 y Fj(\001)3316 1654 y Fo(\()p Fn(t;)28 b Fo(\026)-83 b Fn(h)o Fo(\))46 b(=)g Fn(\013)4002 1679 y Fk(I)9 b(V)4186 1654 y Fn(A)p Fo(\()p Fn(X)4514 1679 y Fl(\000)p Fk(`)4680 1654 y Fo(\))p Fn(:)876 2034 y Fp(F)n(or)46 b(the)f(component)h Fo(\012)2302 1973 y Fk(I)2378 2034 y Fp(,)g(one)f(has)g(to)h(remark)g(that)f(a)h (trajectory)f Fo(\010)4924 1973 y Fk(t)4924 2079 y(H)5050 2034 y Fo(\()p Fn(X)13 b Fo(\))p Fp(,)45 b(with)h Fn(X)66 b Fm(2)54 b Fo(\012)6248 1973 y Fk(I)6248 2075 y(\033)6337 2034 y Fo(\(\001\))876 2269 y Fp(will,)42 b(after)f(visiting)h Fn(X)2241 2294 y Fj(+)2393 2269 y Fp(for)g Fn(t)g Fp(between)f Fo(\(1)c(+)g Fn(\016)3750 2294 y Fj(1)3824 2269 y Fo(\))p Fn(T)4009 2183 y Fj(\(0\))3986 2318 y(\001)4229 2269 y Fp(and)k Fo(\(3)c Fm(\000)g Fn(\016)4935 2294 y Fj(2)5009 2269 y Fo(\))p Fn(T)5194 2183 y Fj(\(0\))5171 2318 y(\001)5372 2269 y Fp(,)42 b(return)g(to)g(a)f(neigh-)876 2508 y(bourhood)k(of)f Fn(X)1877 2533 y Fj(0)1995 2508 y Fp(during)h(the)f(period)h Fo([\(3)38 b(+)i Fn(\016)3664 2533 y Fj(1)3738 2508 y Fo(\))p Fn(T)3923 2422 y Fj(\(0\))3900 2557 y(\001)4101 2508 y Fn(;)28 b Fo(\(5)38 b Fm(\000)h Fn(\016)4603 2533 y Fj(2)4677 2508 y Fo(\))p Fn(T)4862 2422 y Fj(\(0\))4839 2557 y(\001)5040 2508 y Fo(])45 b Fe(etc.)p Fp(.)58 b(This)44 b(e)n(xplains)g(the)876 2707 y(dif)l(ference)d(between)g(\(16\))h(and)f (\(17\).)p 6404 2741 93 133 v 1125 3243 a Fu(Pr)m(oof)51 b(of)f(Theor)m(em)g(2.4)p Fp(:)68 b(This)50 b(is)g(an)g(easy)f (consequence)f(of)i(the)g(preceding)g(ar)m(guments,)876 3442 y(where)44 b(one)g(has)f(to)i(follo)l(w)f(the)g(four)h(components) f(of)g(the)g(initial)h(W)-7 b(igner)45 b(function)f(separately)876 3641 y(in)e(their)f(journe)n(y)h(from)g(one)f Fn(X)2722 3666 y Fk(j)2833 3641 y Fp(to)h(another)-9 b(.)1125 3840 y(Indeed,)64 b(this)d(time,)j(a)d(trajectory)f Fn(X)94 b Fm(2)80 b Fo(\012)3759 3780 y Fk(I)9 b(I)3759 3881 y(\033)3903 3840 y Fo(\(\001\))p Fp(,)65 b(will)60 b(be)g(close)g(to)h Fn(X)5543 3865 y Fj(1)5677 3840 y Fp(for)g Fn(t)f Fp(between)876 4072 y Fn(T)996 3986 y Fj(\(0\))973 4121 y(\001)1229 4072 y Fo(+)55 b Fn(\016)1487 4097 y Fj(1)1561 4072 y Fm(j)28 b Fo(log)i(\001)p Fm(j)66 b Fp(and)f Fn(T)2552 3986 y Fj(\(0\))2529 4121 y(\001)2785 4072 y Fo(+)55 b(2)p Fn(T)3172 3986 y Fj(\(1\))3149 4121 y(\001)3405 4072 y Fm(\000)g Fn(\016)3663 4097 y Fj(2)3737 4072 y Fm(j)28 b Fo(log)i(\001)p Fm(j)p Fp(,)72 b(then)65 b(it)h(will)g(be)f (close)g(to)h Fn(X)6146 4097 y Fj(2)6285 4072 y Fp(for)g Fn(t)876 4311 y Fp(between)56 b Fn(T)1606 4224 y Fj(\(0\))1583 4359 y(\001)1833 4311 y Fo(+)48 b(2)p Fn(T)2213 4224 y Fj(\(1\))2190 4359 y(\001)2440 4311 y Fo(+)h Fn(\016)2692 4336 y Fj(1)2766 4311 y Fm(j)28 b Fo(log)i(\001)p Fm(j)57 b Fp(and)g Fn(T)3740 4224 y Fj(\(0\))3717 4359 y(\001)3967 4311 y Fo(+)48 b(2)p Fn(T)4347 4224 y Fj(\(1\))4324 4359 y(\001)4574 4311 y Fo(+)h(2)p Fn(T)4955 4224 y Fj(\(2\))4932 4359 y(\001)5181 4311 y Fm(\000)g Fn(\016)5433 4336 y Fj(2)5507 4311 y Fm(j)28 b Fo(log)i(\001)p Fm(j)p Fp(,)61 b Fe(etc.)p Fp(.)97 b(A)876 4510 y(similar)56 b(statement)f(holds)g (for)h Fn(X)84 b Fm(2)72 b Fo(\012)3274 4450 y Fk(i)3274 4551 y(\033)3363 4510 y Fo(\(\001\))p Fp(,)58 b(with)e Fn(i)72 b Fo(=)g Fn(I)13 b(;)28 b(I)13 b(I)g(I)g(;)28 b(I)13 b(V)37 b Fp(,)58 b(b)m(ut)d(contrary)h(to)g(what)876 4709 y(happens)42 b(in)h(the)g(pre)l(vious)f(theorem,)h(with)g(dif)l (ferent)h(time-interv)l(als.)55 b(T)-13 b(aking)43 b(the)g (intersection)876 4908 y(of)38 b(the)f(interv)l(als)h(where)f(the)g (dynamics)g(is)h(controlled)g(yields)f(the)g(set)h Fn(I)50 b Fp(of)38 b(the)g(statement)f(of)h(the)876 5108 y(theorem.)p 6404 5141 V 876 5768 a Fq(Refer)l(ences)876 6139 y Fp([BB])277 b(Berry)-11 b(,)34 b(M.)d(V)-21 b(.,)32 b(Balasz)f(N.L.,)i Fe(Evolution)g(of)g(semi-classical)e(quantum)h(states)g(in)g(phase)1485 6338 y(space)p Fp(,)40 b(J.)g(Ph)o(ys.)i(A)f Fu(12)p Fp(,)g(625-642)h(\(1979\))876 6670 y([BGP])176 b(D.)44 b(Bamb)m(usi,)g(S.)h(Graf)l(\002)g(S.,)g(T)-12 b(.)45 b(P)n(aul,)f Fe(Long)i(time)e(semi-classical)g(appr)-7 b(oximation)44 b(of)1485 6869 y(quantum)c(\003ows:)51 b(a)41 b(pr)-7 b(oof)40 b(of)h(the)f(Ehr)-6 b(enfest)40 b(time)p Fp(,)g(Asymptot.)g(Anal.)g(21)h(\(1999\),)g(no.)1485 7069 y(2,)g(149\226160.)876 7401 y([BDB1])83 b(F)-13 b(.)50 b(Bonechi,)i(S.)e(De)g(Bi)2907 7400 y(\036)2898 7401 y(evre,)i Fe(Exponential)f(mixing)f(and)h Fo(ln)28 b(\026)-83 b Fn(h)50 b Fe(time)g(scales)f(in)i(quan-)1485 7600 y(tized)61 b(hyperbolic)g(maps)g(on)h(the)f(torus,)66 b Fp(mp)4198 7625 y Fl(\000)4310 7600 y Fp(arc)61 b(99-381)h(\(1999\),) 67 b(Comm.)61 b(Math.)1485 7799 y(Ph)o(ys.)41 b Fu(211)g Fp(3,)g(659-686)h(\(2000\).)876 8131 y([BDB2])83 b(F)-13 b(.)49 b(Bonechi,)h(S.)f(De)f(Bi)2901 8130 y(\036)2892 8131 y(evre,)j Fe(Contr)-7 b(olling)48 b(str)-7 b(ong)49 b(scarring)e(for)i(quantized)g(er)-6 b(godic)1485 8331 y(tor)n(al)41 b(automorphisms)p Fp(,)f(mp)3166 8356 y Fl(\000)3278 8331 y Fp(arc)h(02-81\(2002\),)h(Duk)n(e)g(Math.)e (Journal,)h(to)h(appear)-9 b(.)876 8663 y([BR])277 b(A.)45 b(Bouzouina,)h(D.)g(Robert.)f Fe(Uniform)h(semi-classical)f(estimates)g (for)h(the)g(pr)-7 b(opa)n(ga-)1485 8862 y(tion)42 b(of)f(quantum)h (observables,)d Fp(Duk)n(e)j(Math.)e(Journal)h(\(to)h(appear\).)876 9194 y([CP1])213 b(Y)-21 b(.)34 b(Colin)i(de)f(V)-18 b(erdi)2626 9193 y(\036)2617 9194 y(ere,)35 b(B.)g(P)n(arisse.)3663 9158 y Fe(\264)3630 9194 y(Equilibr)-6 b(e)35 b(instable)g(en)h(r)5114 9195 y(\264)5105 9194 y(egime)e(semi-classique)g(:)1485 9393 y(I-Concentr)n(ation)104 b(micr)-7 b(olocale)n(,)119 b Fp(Comm.)104 b(P)n(artial)h(Dif)l(ferential)g(Equations,)120 b(19)1485 9593 y(\(9&10\),)41 b(1535-1563)h(\(1994\).)876 9925 y([CP2])213 b(Y)-21 b(.)57 b(Colin)i(de)f(V)-18 b(erdi)2695 9924 y(\036)2686 9925 y(ere,)62 b(B.)57 b(P)n(arisse.)g Fe(Singular)i(Bohr)m(-Sommerfeld)f(Rules,)k Fp(Comm.)1485 10124 y(Math.)40 b(Ph)o(ys.)h(205,)g(459-500)h(\(1999\).)3657 10919 y(24)p eop %%Page: 25 25 25 24 bop 876 1455 a Fp([CC])277 b(Casati,)81 b(G.,)i(Chirik)n(o)n(v) -11 b(,)83 b(B.,)f(The)75 b(le)n(g)o(ac)n(y)e(of)j(chaos)e(in)h (quantum)g(mechanics,)81 b(in)1485 1654 y(\223Quantum)70 b(Chaos\224,)77 b(Ed.)71 b(G.)g(Casati)f(and)g(B.)h(Chirik)n(o)n(v)-11 b(,)78 b(Cambridge)71 b(Uni)l(v)n(ersity)1485 1853 y(Press)41 b(\(1995\).)876 2185 y([CV])268 b(Y)-21 b(.)41 b(Colin)i(de)f(V)-18 b(erdi)2647 2184 y(\036)2638 2185 y(ere,)42 b(J.)g(V)-18 b(e)n(y)-11 b(,)41 b Fe(Le)i(lemme)e(de)i(Mor)n(se)f(isoc)n(hor)-6 b(e)n(,)41 b Fp(T)-13 b(opology)i(,)43 b(v)m(ol.)f(18,)1485 2385 y(283-293)g(\(1979\).)876 2717 y([CR])277 b(M.)59 b(Combescure)g(M.,)64 b(D.)59 b(Robert.)h Fe(Semi-classical)f(spr)-6 b(eading)60 b(of)g(quantum)g(wave)1485 2916 y(pac)m(k)n(ets)42 b(and)i(applications)g(near)f(unstable)g(\002xed)g(point)h(of)g(the)f (classical)g(\003ow)p Fp(,)h(Note)1485 3115 y(CRAS)g(P)n(aris,)g (t.323,)h(S)2904 3114 y(\264)2895 3115 y(erie)g(I,)f(pp.)g(871-876)h (\(1996\))g(and)g(Asymptotic)f(Analysis)g(14,)1485 3314 y(pp.)d(377-404)h(\(1997\).)876 3647 y([DR])268 b(M.)61 b(Dauge,)66 b(D.)61 b(Robert,)66 b Fe(W)-15 b(e)-5 b(yl')e(s)62 b(formula)g(for)g(a)g(class)f(of)h(pseudodif)m(fer)-6 b(ential)61 b(op-)1485 3846 y(er)n(ator)n(s)c(with)h(ne)-7 b(gative)58 b(or)-6 b(der)57 b(on)i Fn(L)3736 3786 y Fj(2)3810 3846 y Fo(\(I)-37 b(R)4020 3773 y Fk(n)4110 3846 y Fo(\))p Fe(,)62 b Fp(Lecture)d(Notes)e(in)i(Math.)e(No.)h(1259,) 1485 4045 y(90-122,)41 b(Springer)m(-V)-18 b(erlag)42 b(\(1986\).)876 4377 y([DB])268 b(S.)81 b(De)h(Bi)2142 4376 y(\036)2133 4377 y(evre,)91 b Fe(Chaos,)g(quantization)83 b(and)f(the)g(classical)f(limit)i(on)f(the)g(torus)p Fp(,)1485 4576 y(\223Proceedings)h(of)h(the)g(XIVth)g(w)n(orkshop)g(on) g(geometric)g(methods)g(in)g(ph)o(ysics,)1485 4776 y(Bialo)l(wieza)40 b(July)h(1995\224)h(\226)f(Polish)h(Scienti\002c)g(Publishers)f(PWN)h (\(1998\).)876 5108 y([F])407 b(G.B.)46 b(F)n(olland)h(,)h Fe(Harmonic)f(Analysis)g(in)g(Phase)g(Space)n(.)f Fp(Annals)h(of)h (Math.)e(Studies,)1485 5307 y(122)p 1485 5334 250 7 v(,)41 b(Princeton)h(Uni)l(v)n(ersity)f(Press)g(\(1989\).)876 5639 y([HJ])314 b(G.)67 b(Hagedorn)h(,)74 b(A.)68 b(Jo)n(ye,)74 b Fe(Exponentially)69 b(Accur)n(ate)e(Semi-classical)g(Dynamics:)1485 5838 y(Pr)-7 b(opa)n(gation,)66 b(Localization,)g(Ehr)-6 b(enfest)61 b(T)-9 b(imes,)65 b(Scattering)n(,)h(and)c(Mor)-6 b(e)61 b(Gener)n(al)1485 6038 y(States)p Fp(,)40 b(Ann.)h(Henri)h (Poincar)3269 6037 y(\264)3260 6038 y(e)f Fu(1)p Fp(,)g(no.)g(5,)h (837\226883)f(\(2000\).)876 6370 y([HS])287 b(B.)64 b(Helf)l(fer)-7 b(,)70 b(J.)64 b(Sj)2605 6369 y(\250)2591 6370 y(ostrand,)71 b Fe(Semi-classical)64 b(analysis)g(for)g(Harper')-7 b(s)64 b(equation)h(III,)1485 6569 y Fp(Bull.)41 b(Soc.)g(Math.)f (France)i(117)f(\(4\))h(m)3776 6568 y(\264)3767 6569 y(emoire)g(No)g(39)f(\(1989\).)876 6901 y([O)-7 b(TH])165 b(O'Connor)-7 b(,)56 b(P)-18 b(.W)j(.)51 b(,)56 b(T)-13 b(omso)n(vic,)54 b(S.,)i(Heller)-7 b(,)55 b(E.J.,)g Fe(Semi-classical)d (dynamics)g(in)i(the)1485 7100 y(str)-7 b(ongly)33 b(c)n(haotic)g(r)-6 b(e)f(gime:)48 b(br)-6 b(eaking)33 b(the)h(lo)n(g)h(time)f(barrier)p Fp(,)g(Ph)o(ysica)g Fu(55)g Fp(D,)g(340-357)1485 7299 y(\(1992\).)876 7632 y([R])388 b(D.)53 b(Robert,)j Fe(A)m(utour)e(de)f (l')i(Appr)-7 b(oximation)53 b(Semi-classique)p Fp(,)j Fu(PM)e(68)p Fp(,)i(Birkh)6228 7631 y(\250)6219 7632 y(auser)-7 b(,)1485 7831 y(1987.)876 8163 y([TH])278 b(T)-13 b(omso)n(vic,)41 b(S.)g(,)h(Heller)-7 b(,)42 b(E.J.,)e Fe(Semi-classical)i(dynamics)f(of)h(c)n(haotic)f(motion:)53 b(une)m(x-)1485 8362 y(pected)41 b(long-time)g(accur)n(acy)p Fp(,)f(Ph)o(ys.)h(Re)l(v)-11 b(.)40 b(Lett.)i Fu(67)f Fp(6,)g(664)h(\(1991\).)876 8694 y([T])398 b(J.T)-13 b(oth,)45 b Fe(On)g(the)h(small-scale)e(mass)h(concentr)n(ation)f(of)i (modes)e Fp(Comm.)h(Math.)g(Ph)o(ys.)1485 8894 y(206,)c(409-428)h (\(1999\).)3657 10919 y(25)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0203250657127--