Content-Type: multipart/mixed; boundary="-------------9909070109807" This is a multi-part message in MIME format. ---------------9909070109807 Content-Type: text/plain; name="99-326.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-326.keywords" holonomic constraints ---------------9909070109807 Content-Type: application/postscript; name="constraints.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="constraints.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: Constraints.dvi %%Pages: 13 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: Palatino-Roman %%EndComments %DVIPSCommandLine: dvips -t letter Constraints -o constraints.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 1999.09.06:2223 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 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h(nor)f(the)h(initial)f(velocity)g(exactly)270 3846 y(tangent)f(to)h Fn(\006)p Fp(.)36 b(Rubin)21 b(and)i(Ungar)g(also)g(consider)f(what)g (happens)i(if)e(initial)g(velocity)f(has)i(a)g(component)f(in)g(the)270 3995 y(dir)o(ection)17 b(normal)i(to)f Fn(\006)p Fp(.)27 b(In)18 b(this)g(case,)h(the)f(motion)f(in)h(the)g(normal)g(dir)o (ection)g(is)g(highly)g(oscillatory)-9 b(,)19 b(and)g(ther)o(e)e(is)270 4144 y(an)j(extra)f(potential)h(term,)f(depending)g(on)h(the)f(initial) g(condition,)g(in)h(the)f(Hamiltonian)g(for)h(the)f(limiting)g(motion) 270 4294 y(on)f Fn(\006)p Fp(.)28 b(A)19 b(mor)o(e)f(complete)f(r)o (esult)h(is)h(given)e(by)h(T)-7 b(akens)19 b([T].)g(Her)o(e)f(the)g (initial)g(conditions)h(ar)o(e)f(allowed)h(to)f(depend)270 4443 y(on)25 b Fj(\025)g Fp(in)f(such)h(a)g(way)g(that)f(the)g(initial) h(position)g(conver)o(ges)e(to)i(a)g(point)g(on)f Fn(\006)h Fp(and)g(the)f(initial)h(ener)o(gy)e(r)o(emains)270 4593 y(bounded.)k(\(W)-7 b(e)17 b(will)g(give)f(pr)o(ecise)i(assumptions)g (below)-7 b(.\))27 b(Once)17 b(again,)h(the)f(limiting)f(motion)h(on)h Fn(\006)f Fp(is)h(governed)270 4742 y(by)j(a)h(Hamiltonian)g(with)f(an) h(additional)h(potential.)34 b(T)-7 b(akens)22 b(noticed)g(that)f(a)h (non\255r)o(esonance)g(condition)g(on)g(the)270 4892 y(eigenvalues)e(of)i(the)e(Hessian)i(of)f(the)g(constraining)g (potential)g Fj(W)33 b Fp(along)22 b Fn(\006)f Fp(is)g(r)o(equir)o(ed)f (to)h(pr)o(ove)g(conver)o(gence.)270 5041 y(He)26 b(also)i(gave)f(an)g (example)f(showing)h(that)g(if)g(the)f(Hessian)h(of)h Fj(W)38 b Fp(does)28 b(have)e(an)i(eigenvalue)d(cr)o(ossing,)k(then)270 5191 y(ther)o(e)c(may)h(not)g(be)f(a)i(good)f(notion)g(of)h(limiting)e (motion.)46 b(In)26 b(his)g(example,)i(he)d(constr)o(ucts)i(two)f (sequences)g(of)270 5340 y(orbits,)j(each)f(one)g(conver)o(ging)e(to)i (an)g(limiting)e(orbit)h(on)h Fn(\006)p Fp(.)52 b(These)28 b(limiting)f(orbits)h(ar)o(e)f(identical)h(until)f(they)2020 5539 y(2)p eop %%Page: 3 3 3 2 bop 270 83 a Fp(hit)23 b(the)f(point)i(on)f Fn(\006)g Fp(wher)o(e)f(the)h(eigenvalues)f(cr)o(oss.)39 b(After)24 b(that,)g(they)e(ar)o(e)h(dif)o(fer)o(ent.)38 b(This)24 b(means)f(ther)o(e)f(is)i(no)270 232 y(dif)o(fer)o(ential)h(equation)h (on)g Fn(\006)f Fp(governing)g(the)h(limiting)e(motion.)46 b(For)26 b(another)g(discussion)h(of)f(the)f(question)h(of)270 382 y(r)o(ealizing)17 b(constraints)i(see)e([A2].)29 b(A)18 b(modern)f(survey)g(of)g(the)g(classical)i(mechanical)f(r)o (esults)f(that)g(emphasizes)i(the)270 531 y(systematic)h(use)f(of)h (weak)g(conver)o(gence)e(is)i(given)f(by)g(Bornemann)h(and)g(Sch)2658 528 y(\250)2647 531 y(utte)f([BS].)436 690 y(The)24 b(quantum)f(case)h (was)h(consider)o(ed)e(pr)o(eviously)g(by)h(T)-7 b(olar)24 b([T],)h(da)f(Costa)g([dC1,)i(dC2])e(and)h(in)e(the)h(path)270 839 y(integral)29 b(literatur)o(e)f(\(see)h(Anderson)h(and)f(Driver)g ([AD]\).)h(Related)e(work)h(can)h(also)g(be)e(found)h(in)g(Helf)o(fer)f (and)270 989 y(Sj)337 986 y(\250)329 989 y(ostrand)15 b([HS1])g([HS2],)h(who)d(obtained)h(WKB)g(expansions)g(for)h(the)e(gr)o (ound)g(state.)26 b(Ther)o(e)13 b(ar)o(e)h(r)o(eally)f(two)h(aspects) 270 1138 y(to)i(the)f(pr)o(oblem)h(of)g(r)o(ealizing)g(constraints:)27 b(a)17 b(lar)o(ge)f Fj(\025)g Fp(expansion)h(followed)f(by)g(an)g (averaging)g(pr)o(ocedur)o(e)g(to)g(deal)270 1288 y(with)h(highly)g (oscillatory)h(normal)f(motion.)27 b(Pr)o(evious)16 b(work)h(in)h (quantum)e(mechanics)i(concentrated)f(on)g(the)g(\002rst)270 1437 y(aspect.)28 b(Alr)o(eady)15 b(a)i(formal)f(lar)o(ge)f Fj(\025)h Fp(expansion)g(r)o(eveals)f(the)g(inter)o(esting)g(featur)o (e)g(that)g(the)g(limiting)f(Hamiltonian)270 1586 y(has)24 b(an)g(extra)f(potential)g(term)g(depending)g(on)h(scalar)g(and)g(the)f (mean)g(curvatur)o(es.)38 b(Since)24 b(the)e(mean)h(curvatur)o(e)270 1736 y(is)32 b(not)f(intrinsic,)k(this)c(potential)h(does)g(depend)g (on)f(the)g(imbedding)g(of)h Fn(\006)f Fp(in)h Fm(R)2861 1706 y Fl(n)p Fk(+)p Fl(m)3022 1736 y Fp(.)64 b(It)31 b(is)h(not)f(completely)270 1885 y(straightforwar)o(d)18 b(to)g(formulate)f(a)h(theor)o(em)e(in)i(the)f(quantum)g(case.)28 b(W)-7 b(e)17 b(have)g(chosen)h(a)g(formulation,)f(modelled)270 2035 y(on)i(the)f(classical)j(mechanical)e(theor)o(ems,)e(tracking)i(a) g(sequence)g(of)g(orbits)g(with)f(initial)h(positions)h(concentrating) 270 2184 y(on)25 b Fn(\006)f Fp(via)h(dilations)g(in)g(the)f(normal)h (dir)o(ection.)42 b(Actually)24 b(we)h(consider)g(the)f(equivalent)f (pr)o(oblem)h(of)h(tracking)270 2334 y(the)31 b(evolution)g(of)h Fj(H)947 2346 y Fl(\025)1022 2334 y Fp(conjugated)g(by)f(dilations,)36 b(acting)31 b(on)h(a)g(\002xed)g(initial)f(vector)-6 b(.)64 b(In)32 b(or)o(der)f(to)h(obtain)g(a)270 2483 y(limiting)27 b(orbit)h(we)h(must)e(assume)i(that)f(all)h(the)f (eigenvalues)g(of)h(the)e(Hessian)j(of)e(the)g(constraining)h (potential)270 2633 y Fj(W)42 b Fp(ar)o(e)30 b(constant)g(on)g Fn(\006)p Fp(.)59 b(In)29 b(fact)i(we)e(will)h(assume)g(that)g Fj(W)42 b Fp(is)30 b(exactly)g(quadratic.)59 b(When)30 b(the)f(eigenvalues)270 2782 y(ar)o(e)24 b(all)g(dif)o(fer)o(ent)f (\(and)i(non\255r)o(esonant\))f(we)f(ar)o(e)h(able)g(to)f(pr)o(ove)h (the)f(existence)g(of)h(a)g(limiting)f(motion)g(in)h Fj(L)3629 2752 y Fk(2)3666 2782 y Fn(\(\006\))p Fp(.)270 2931 y(Surprisingly)-9 b(,)18 b(for)f(very)f(symmetrical)h(r)o (estraining)f(potentials,)i(the)e(Hilbert)f(space)j(need)e(not)h(be)f Fj(L)3325 2901 y Fk(2)3362 2931 y Fn(\(\006\))p Fp(,)h(but)f(may)270 3081 y(be)j(the)g(space)i(of)f Fj(L)859 3051 y Fk(2)916 3081 y Fp(sections)g(of)g(a)g(vector)f(bundle)f(over)i Fn(\006)p Fp(.)436 3239 y(Our)f(formulation)h(of)g(the)f(quantum)f (theor)o(ems)h(invites)g(comparison)i(with)e(the)g(classical)i (mechanical)f(case.)270 3389 y(It)c(turns)h(out)f(that)h(extra)g (potentials)g(that)g(appear)h(in)f(the)f(two)h(cases)h(ar)o(e)f(quite)f (dif)o(fer)o(ent,)g(and)i(ther)o(e)e(is)h(no)g(obvious)270 3538 y(connection.)51 b(Upon)28 b(r)o(e\003ection,)g(the)f(r)o(eason)h (for)g(this)g(dif)o(fer)o(ence)f(is)g(clear)-6 b(.)53 b(If)27 b(we)g(have)g(a)h(sequence)f(of)h(initial)270 3688 y(quantum)18 b(states)h(whose)f(position)h(distribution)f(is)h (being)f(squeezed)h(to)f(lie)g(close)h(to)g Fn(\006)p Fp(,)f(then)g(by)g(the)g(uncertainty)270 3837 y(principle,)j(the)f (distribution)f(of)i(initial)f(momenta)g(will)g(be)g(spr)o(eading)h (out,)e(and)i(thus)f(the)g(initial)g(ener)o(gy)g(will)g(be)270 3986 y(unbounded.)46 b(However)-6 b(,)26 b(the)f(classical)j (mechanical)e(conver)o(gence)e(theor)o(ems)h(above)h(all)g(deal)g(with) g(bounded)270 4136 y(ener)o(gies.)h(As)21 b(Bornemann)e(and)h(Sch)1434 4133 y(\250)1423 4136 y(utte)f(point)h(out,)e(if)i(the)f(ener)o(gies)g (ar)o(e)g(unbounded,)f(it)h(may)h(happen)g(that)g(the)270 4285 y(weak)d(limit)g(of)h(the)f(orbits)h(does)g(not)f(even)g(lie)g(on) h Fn(\006)p Fp(.)27 b(However)-6 b(,)17 b(in)g(situations)h(wher)o(e)f (we)g(can)h(obtain)f(a)h(quantum)270 4435 y(mechanical)i(theor)o(em,)e (we)h(can)h(also)h(obtain)e(an)h(analogous)h(classical)g(theor)o(em)d (with)h(unbounded)g(ener)o(gies.)436 4593 y(The)h(pr)o(esent)f(note)g (contains)i(the)e(statements)g(of)h(our)g(r)o(esults.)27 b(Pr)o(oofs)20 b(will)g(be)f(given)f(in)i([FH].)270 5023 y Fq(Classical)30 b(mechanics:)39 b(b)s(ounded)28 b(energy)436 5340 y Fp(T)-7 b(o)23 b(give)e(a)i(pr)o(ecise)g(statement)f(of)h(our)f (r)o(esults)g(we)g(must)g(intr)o(oduce)g(some)g(notation.)37 b(The)22 b(normal)h(bundle)2020 5539 y(3)p eop %%Page: 4 4 4 3 bop 270 83 a Fp(to)19 b Fn(\006)h Fp(is)g(the)f(subset)h(of)g Fm(R)1028 53 y Fl(n)p Fk(+)p Fl(m)1207 83 y Fi(\002)e Fm(R)1344 53 y Fl(n)p Fk(+)p Fl(m)1525 83 y Fp(given)h(by)1450 319 y Fj(N)9 b Fn(\006)22 b(=)h Fi(f)p Fn(\()p Fj(\033)n(;)14 b(n)p Fn(\))24 b(:)f Fj(\033)j Fi(2)e Fn(\006)p Fj(;)j(n)c Fi(2)h Fj(N)2485 331 y Fl(\033)2529 319 y Fn(\006)p Fi(g)270 555 y Fp(Her)o(e)19 b Fj(N)528 567 y Fl(\033)573 555 y Fn(\006)h Fp(denotes)h(the)f(normal)g(space)i(to)e Fn(\006)h Fp(at)g Fj(\033)s Fp(,)f(identi\002ed)g(with)g(a)h(subspace)g (of)g Fm(R)2989 525 y Fl(n)p Fk(+)p Fl(m)3150 555 y Fp(.)31 b(Ther)o(e)20 b(is)h(a)g(natural)270 705 y(map)f(fr)o(om)g Fj(N)9 b Fn(\006)19 b Fp(into)g Fm(R)1001 674 y Fl(n)p Fk(+)p Fl(m)1182 705 y Fp(given)g(by)1715 854 y Fj(\023)24 b Fn(:)f(\()p Fj(\033)n(;)14 b(n)p Fn(\))24 b Fi(7!)f Fj(\033)f Fn(+)c Fj(n:)270 1056 y Fp(W)-7 b(e)23 b(now)g(\002x)g(a)h (suf)o(\002ciently)e(small)i Fj(\016)j Fp(so)d(that)f(this)h(map)g(is)g (a)g(dif)o(feomorphism)g(of)g Fj(N)9 b Fn(\006)2997 1068 y Fl(\016)3061 1056 y Fn(=)29 b Fi(f)p Fn(\()p Fj(\033)n(;)14 b(n)p Fn(\))29 b(:)g Fi(k)p Fj(n)p Fi(k)e Fj(<)h(\016)s Fi(g)270 1205 y Fp(onto)17 b(a)h(tubular)e(neigbour)o(hood)g(of)h Fn(\006)g Fp(in)h Fm(R)1593 1175 y Fl(n)p Fk(+)p Fl(m)1754 1205 y Fp(.)27 b(Then)17 b(we)g(can)g(pull)h(back)f(the)f(Euclidean)i (metric)e(fr)o(om)h Fm(R)3563 1175 y Fl(n)p Fk(+)p Fl(m)3741 1205 y Fp(to)270 1355 y Fj(N)9 b Fn(\006)406 1367 y Fl(\016)442 1355 y Fp(.)33 b(Since)21 b(we)g(ar)o(e)g(inter)o(ested)f(in)i(the)e (motion)h(close)g(to)g Fn(\006)h Fp(we)e(may)i(use)f Fj(N)9 b Fn(\006)2751 1367 y Fl(\016)2808 1355 y Fp(as)22 b(the)f(classical)i(con\002guration)270 1504 y(space.)29 b(This)21 b(will)e(be)g(convenient)f(in)i(what)f(follows.)436 1654 y(W)-7 b(e)23 b(will)h(want)g(to)f(decompose)i(vectors)e(in)h(the) g(cotangent)f(spaces)i(of)g Fj(N)9 b Fn(\006)2788 1666 y Fl(\016)2848 1654 y Fp(into)23 b(horizontal)i(and)f(vertical)270 1803 y(vectors,)18 b(so)h(we)g(now)f(explain)h(this)g(decomposition.)28 b(For)18 b(each)h(point)g Fj(\033)26 b Fi(2)e Fn(\006)p Fp(,)18 b(we)g(may)h(decompose)g Fj(T)3463 1815 y Fl(\033)3507 1803 y Fm(R)3561 1773 y Fl(n)p Fk(+)p Fl(m)3745 1803 y Fn(=)270 1952 y Fj(T)319 1964 y Fl(\033)363 1952 y Fn(\006)h Fi(\010)f Fj(N)594 1964 y Fl(\033)639 1952 y Fn(\006)j Fp(into)g(the)f(tangent)h(and)g(normal)h(space.)36 b(Using)23 b(the)e(natural)i(identi\002cation)e(of)i(all)f(tangent)f (spaces)270 2102 y(with)h Fm(R)507 2072 y Fl(n)p Fk(+)p Fl(m)668 2102 y Fp(,)i(we)f(may)g(r)o(egar)o(d)g(this)g(as)h(a)f (decomposition)g(of)h Fm(R)2264 2072 y Fl(n)p Fk(+)p Fl(m)2425 2102 y Fp(.)38 b(Let)23 b Fj(P)2683 2072 y Fl(T)2671 2122 y(\033)2758 2102 y Fp(and)h Fj(P)2980 2072 y Fl(N)2968 2122 y(\033)3066 2102 y Fp(be)e(the)h(corr)o (esponding)270 2251 y(orthogonal)16 b(pr)o(ojections.)28 b(Since)16 b(we)f(ar)o(e)h(thinking)f(of)i Fj(N)9 b Fn(\006)15 b Fp(as)i(an)g Fj(n)10 b Fn(+)g Fj(m)p Fp(\226dimensional)15 b(submanifold)h(of)h Fm(R)3575 2221 y Fl(n)p Fk(+)p Fl(m)3745 2251 y Fi(\002)270 2401 y Fm(R)324 2371 y Fl(n)p Fk(+)p Fl(m)485 2401 y Fp(,)h(we)g(can)g(identify)g Fj(T)1124 2416 y Fk(\()p Fl(\033)o(;n)p Fk(\))1277 2401 y Fj(N)9 b Fn(\006)18 b Fp(with)f(the)g Fj(n)d Fn(+)g Fj(m)p Fp(\226dimensional) 19 b(subspace)g(of)f Fm(R)2919 2371 y Fl(n)p Fk(+)p Fl(m)3094 2401 y Fi(\002)c Fm(R)3227 2371 y Fl(n)p Fk(+)p Fl(m)3406 2401 y Fp(given)j(by)h(all)270 2550 y(vectors)j(of)h(the)f(form)h Fn(\()p Fj(X)r(;)14 b(Y)k Fn(\))26 b(=)g(\()14 b(_)-37 b Fj(\033)s Fn(\(0\))p Fj(;)28 b Fn(_)-37 b Fj(n)p Fn(\(0\)\))p Fp(,)22 b(wher)o(e)f Fn(\()p Fj(\033)s Fn(\()p Fj(t)p Fn(\))p Fj(;)14 b(n)p Fn(\()p Fj(t)p Fn(\)\))23 b Fp(is)f(a)g(curve)f (in)g Fj(N)9 b Fn(\006)22 b Fp(passing)h(thr)o(ough)d Fn(\()p Fj(\033)n(;)14 b(n)p Fn(\))270 2700 y Fp(at)j(time)f Fj(t)23 b Fn(=)g(0)p Fp(.)k(For)17 b(such)f(a)i(tangent)e(vector)-6 b(,)17 b(the)f(decomposition)h(into)g(horizontal)g(and)g(vertical)g (vectors)g(is)g(given)270 2849 y(by)1468 2998 y Fn(\()p Fj(X)r(;)d(Y)k Fn(\))24 b(=)e(\()p Fj(X)r(;)14 b(P)2022 2964 y Fl(T)2010 3019 y(\033)2074 2998 y Fj(Y)19 b Fn(\))g(+)f(\(0)p Fj(;)c(P)2451 2964 y Fl(N)2439 3019 y(\033)2513 2998 y Fj(Y)19 b Fn(\))270 3200 y Fp(For)h(points)g Fn(\()p Fj(\033)n(;)14 b(n)p Fn(\))24 b Fi(2)g Fj(N)9 b Fn(\006)1081 3212 y Fl(\016)1117 3200 y Fp(,)19 b(the)g(pulled)h(back)f(inner)h(pr)o (oduct)f(on)h Fj(T)2371 3215 y Fk(\()p Fl(\033)o(;n)p Fk(\))2524 3200 y Fj(N)9 b Fn(\006)20 b Fp(is)g(given)f(by)1297 3436 y Fi(h)p Fn(\()p Fj(X)1430 3448 y Fk(1)1468 3436 y Fj(;)14 b(Y)1553 3448 y Fk(1)1590 3436 y Fn(\))p Fj(;)g Fn(\()p Fj(X)1760 3448 y Fk(2)1798 3436 y Fj(;)g(Y)1883 3448 y Fk(2)1920 3436 y Fn(\))p Fi(i)24 b Fn(=)e Fi(h)p Fj(X)2196 3448 y Fk(1)2252 3436 y Fn(+)c Fj(Y)2383 3448 y Fk(1)2421 3436 y Fj(;)c(X)2527 3448 y Fk(2)2582 3436 y Fn(+)k Fj(Y)2713 3448 y Fk(2)2750 3436 y Fi(i)270 3672 y Fp(wher)o(e)24 b(the)h(inner)g(pr)o(oduct)h(on)f(the)g(right)g(is)h (the)f(usual)g(Euclidean)g(inner)h(pr)o(oduct.)45 b(This)26 b(pr)o(ovides)f(an)h(identi\255)270 3822 y(\002cation)g(of)i Fj(T)702 3837 y Fk(\()p Fl(\033)o(;n)p Fk(\))855 3822 y Fj(N)9 b Fn(\006)26 b Fp(with)g Fj(T)1265 3792 y Fh(\003)1253 3849 y Fk(\()p Fl(\033)o(;n)p Fk(\))1407 3822 y Fj(N)9 b Fn(\006)26 b Fp(for)h Fn(\()p Fj(\033)n(;)14 b(n)p Fn(\))34 b Fi(2)g Fj(N)9 b Fn(\006)2151 3834 y Fl(\016)2187 3822 y Fp(.)49 b(Thus)27 b(we)g(may)f(decompose)h(cotangent)f(vectors) 270 3971 y(into)g(their)g(horizontal)i(and)f(vertical)f(parts)i(as)f (well.)49 b(W)-7 b(e)26 b(will)g(denote)g(by)g Fn(\()p Fj(\030)t(;)14 b(\021)s Fn(\))28 b Fp(the)e(horizontal)h(and)g (vertical)270 4121 y(components)20 b(of)g(a)g(vector)f(in)g Fj(T)1251 4091 y Fh(\003)1239 4147 y Fk(\()p Fl(\033)o(;n)p Fk(\))1393 4121 y Fj(N)9 b Fn(\006)p Fp(.)436 4270 y(W)-7 b(e)19 b(will)h(assume)g(that)f(the)g(constraining)h(potential)g(is)g (a)g Fj(C)2272 4240 y Fh(1)2362 4270 y Fp(function)f(of)h(the)g(form) 1621 4506 y Fj(W)12 b Fn(\()p Fj(\033)n(;)i(n)p Fn(\))24 b(=)2029 4450 y(1)p 2029 4487 42 4 v 2029 4563 a(2)2080 4506 y Fi(h)p Fj(n;)14 b(A)p Fn(\()p Fj(\033)s Fn(\))p Fj(n)p Fi(i)1182 b Fn(\(2)p Fj(:)p Fn(1\))270 4742 y Fp(wher)o(e)19 b(for)h(each)g Fj(\033)s Fp(,)g Fj(A)p Fn(\()p Fj(\033)s Fn(\))h Fp(is)f(a)g(positive)f(de\002nite)g(linear)h (transformation)h(on)e Fj(N)2748 4754 y Fl(\033)2793 4742 y Fn(\006)p Fp(.)436 4892 y(Since)h(we)f(expect)g(the)g(motion)f (in)i(the)f(normal)g(dir)o(ections)h(to)f(consist)h(of)g(rapid)h (harmonic)e(oscillations,)h(it)g(is)270 5041 y(natural)j(to)g(intr)o (oduce)f(action)h(variables)h(for)f(this)g(motion.)38 b(Ther)o(e)22 b(is)i(one)f(for)g(each)g(distinct)g(eigenvalue)f Fj(!)3651 5011 y Fk(2)3648 5062 y Fl(\013)3695 5041 y Fn(\()p Fj(\033)s Fn(\))270 5191 y Fp(of)e Fj(A)p Fn(\()p Fj(\033)s Fn(\))p Fp(.)30 b(Let)19 b Fj(P)769 5203 y Fl(\013)817 5191 y Fn(\()p Fj(\033)s Fn(\))h Fp(be)g(the)f(pr)o (ojection)g(onto)h(the)f(eigenspace)h(of)g Fj(!)2402 5160 y Fk(2)2399 5211 y Fl(\013)2446 5191 y Fn(\()p Fj(\033)s Fn(\))p Fp(.)30 b(This)20 b(pr)o(ojection)g(is)g(de\002ned)f(on)h Fj(N)3686 5203 y Fl(\033)3730 5191 y Fn(\006)p Fp(,)270 5340 y(which)f(we)g(may)g(think)g(of)h(as)g(the)f(range)h(of)f Fj(P)1685 5310 y Fl(N)1673 5361 y(\033)1768 5340 y Fp(in)g Fm(R)1910 5310 y Fl(n)p Fk(+)p Fl(m)2071 5340 y Fp(.)28 b(Thus)20 b(the)f(pr)o(ojection)g(is)h(de\002ned)f(on)g(vertical)g (vectors)2020 5539 y(4)p eop %%Page: 5 5 5 4 bop 270 83 a Fp(in)18 b Fj(T)406 98 y Fk(\()p Fl(\033)o(;n)p Fk(\))559 83 y Fj(N)9 b Fn(\006)18 b Fp(and,)h(via)f(the)g(natural)g (identi\002cation,)g(on)g(vertical)g(vectors)g(in)g Fj(T)2712 53 y Fh(\003)2700 110 y Fk(\()p Fl(\033)o(;n)p Fk(\))2854 83 y Fj(N)9 b Fn(\006)p Fp(.)27 b(W)l(ith)17 b(this)i(notation,)f(the) 270 232 y(corr)o(esponding)i(action)g(variable,)f(multiplied)g(by)g Fj(\025)1893 202 y Fk(2)1951 232 y Fp(for)h(notational)g(convenience,)e (is)i(given)f(by)1119 481 y Fj(I)1162 447 y Fl(\025)1155 502 y(\013)1206 481 y Fn(\()p Fj(\033)n(;)14 b(n;)g(\030)t(;)g(\021)s Fn(\))24 b(=)1789 425 y(1)p 1682 462 256 4 v 1682 538 a(2)p Fj(!)1776 550 y Fl(\013)1822 538 y Fn(\()p Fj(\033)s Fn(\))1947 481 y Fi(h)p Fj(\021)s(;)14 b(P)2113 493 y Fl(\013)2161 481 y Fj(\021)s Fi(i)19 b Fn(+)2349 425 y Fj(\025)2397 395 y Fk(4)2435 425 y Fj(!)2487 437 y Fl(\013)2534 425 y Fn(\()p Fj(\033)s Fn(\))p 2349 462 300 4 v 2478 538 a(2)2659 481 y Fi(h)p Fj(n;)14 b(P)2831 493 y Fl(\013)2878 481 y Fj(n)p Fi(i)679 b Fn(\(2)p Fj(:)p Fn(2\))270 730 y Fp(Notice)19 b(that)h(the)g(total)g(normal)g(ener)o (gy)f(is)h(given)g(by)1914 668 y Fg(P)2002 755 y Fl(\013)2063 730 y Fj(!)2115 742 y Fl(a)2155 730 y Fj(I)2191 742 y Fl(\013)2239 730 y Fp(.)29 b(The)20 b(following)g(is)g(a)h(version)f (of)g(the)g(theor)o(em)f(of)270 879 y(T)-7 b(akens)20 b(and)g(Rubin,)f(Ungar)-6 b(.)270 1079 y Ff(Theorem)26 b(2.1)41 b Fe(L)l(et)26 b(the)g(Hamiltonian)h Fj(H)1654 1091 y Fl(\025)1724 1079 y Fe(b)l(e)f(given)h(by)f(\(1.2\))h(wher)l(e)g Fj(W)38 b Fe(has)27 b(the)f(form)h(\(2.1\))g(and)g(satis\014es)436 1228 y(\(i\))j(The)h(eigenvalues)g Fj(!)1219 1198 y Fk(2)1216 1248 y Fl(\013)1263 1228 y Fn(\()p Fj(\033)s Fn(\))f Fe(of)h Fj(A)p Fn(\()p Fj(\033)s Fn(\))g Fe(have)g(c)l(onstant)e (multiplicity.)270 1377 y(Supp)l(ose)h(that)g Fn(\()p Fj(\033)835 1389 y Fl(\025)879 1377 y Fj(;)14 b(n)966 1389 y Fl(\025)1009 1377 y Fj(;)g(\030)1082 1389 y Fl(\025)1126 1377 y Fj(;)g(\021)1204 1389 y Fl(\025)1248 1377 y Fn(\))30 b Fe(ar)l(e)g(initial)h(c)l(onditions)g(in)e Fj(T)2254 1347 y Fh(\003)2292 1377 y Fj(N)9 b Fn(\006)2428 1389 y Fl(\016)2494 1377 y Fe(satisfying)436 1527 y(\(a\))30 b Fj(\033)623 1539 y Fl(\025)690 1527 y Fi(!)23 b Fj(\033)843 1539 y Fk(0)881 1527 y Fe(,)436 1676 y(\(b\))30 b Fj(\030)608 1688 y Fl(\025)675 1676 y Fi(!)23 b Fj(\030)817 1688 y Fk(0)854 1676 y Fe(,)436 1826 y(\(c\))30 b Fj(I)615 1796 y Fl(\025)608 1846 y(\013)658 1826 y Fn(\()p Fj(\033)737 1838 y Fl(\025)782 1826 y Fj(;)14 b(n)869 1838 y Fl(\025)912 1826 y Fj(;)g(\030)985 1838 y Fl(\025)1029 1826 y Fj(;)g(\021)1107 1838 y Fl(\025)1150 1826 y Fn(\))24 b Fi(!)f Fj(I)1355 1796 y Fk(0)1348 1846 y Fl(\013)1419 1826 y Fj(>)f Fn(0)p Fe(,)270 1975 y(as)30 b Fj(\025)23 b Fi(!)h(1)p Fe(.)38 b(L)l(et)29 b Fn(\()p Fj(\033)922 1987 y Fl(\025)967 1975 y Fn(\()p Fj(t)p Fn(\))p Fj(;)14 b(n)1148 1987 y Fl(\025)1191 1975 y Fn(\()p Fj(t)p Fn(\))p Fj(;)g(\030)1358 1987 y Fl(\025)1403 1975 y Fn(\()p Fj(t)p Fn(\))p Fj(;)g(\021)1575 1987 y Fl(\025)1619 1975 y Fn(\()p Fj(t)p Fn(\)\))30 b Fe(denote)g(the)g(subse)l(quent)e(orbit)j(in)e Fj(T)2948 1945 y Fh(\003)2986 1975 y Fj(N)9 b Fn(\006)3122 1987 y Fl(\016)3187 1975 y Fe(under)30 b(the)g(Hamil-)270 2125 y(tonian)g Fj(H)600 2137 y Fl(\025)644 2125 y Fe(.)436 2274 y(Supp)l(ose)37 b(that)f Fn(\()p Fj(\033)s Fn(\()p Fj(t)p Fn(\))p Fj(;)14 b(\030)t Fn(\()p Fj(t)p Fn(\)\))38 b Fe(is)f(the)f(orbit)h(in)f Fj(T)1966 2244 y Fh(\003)2003 2274 y Fn(\006)h Fe(with)f(initial)i(c)l(onditions)f Fn(\()p Fj(\033)3019 2286 y Fk(0)3057 2274 y Fj(;)14 b(\030)3130 2286 y Fk(0)3167 2274 y Fn(\))37 b Fe(governe)l(d)g(by)g (the)270 2423 y(Hamiltonian)1303 2573 y Fj(h)p Fn(\()p Fj(\033)n(;)14 b(\030)t Fn(\))25 b(=)1659 2517 y(1)p 1659 2554 42 4 v 1659 2630 a(2)1711 2573 y Fi(h)p Fj(\030)t(;)14 b(\030)t Fi(i)1892 2585 y Fl(\033)1956 2573 y Fn(+)k Fj(V)h Fn(\()p Fj(\033)s Fn(\))g(+)2322 2494 y Fg(X)2360 2668 y Fl(\013)2456 2573 y Fj(I)2499 2539 y Fk(0)2492 2593 y Fl(\013)2540 2573 y Fj(!)2592 2585 y Fl(\013)2639 2573 y Fn(\()p Fj(\033)s Fn(\))p Fj(:)436 2801 y Fe(Then)30 b(for)h(any)f Fj(T)k Fi(\025)23 b Fn(0)1367 3049 y(sup)1325 3119 y Fk(0)p Fh(\024)p Fl(t)p Fh(\024)p Fl(T)1549 3049 y Fi(j)p Fj(\033)1619 3061 y Fl(\025)1663 3049 y Fn(\()p Fj(t)p Fn(\))c Fi(\000)f Fj(\033)s Fn(\()p Fj(t)p Fn(\))p Fi(j)h Fn(+)f Fi(j)p Fj(\030)2187 3061 y Fl(\025)2231 3049 y Fn(\()p Fj(t)p Fn(\))h Fi(\000)f Fj(\030)t Fn(\()p Fj(t)p Fn(\))p Fi(j)24 b(!)f Fn(0)270 3298 y Fe(as)30 b Fj(\025)24 b Fi(!)f(1)p Fe(.)436 3497 y Fp(Implicit)12 b(in)g(this)g(statement)g(is)g(the)g(fact)g(that)g(the)g(appr)o (oximating)g(orbit)g(stays)g(in)g(the)g(tubular)g(neighbour)o(hood)270 3647 y(for)22 b Fn(0)j Fi(\024)g Fj(t)h Fi(\024)f Fj(T)12 b Fp(,)21 b(pr)o(ovided)g Fj(\025)h Fp(is)g(suf)o(\002ciently)e(lar)o (ge.)34 b(This)22 b(theor)o(em)e(is)i(actually)f(tr)o(ue)g(in)h(gr)o (eater)e(generality)-9 b(.)34 b(W)-7 b(e)270 3796 y(can)15 b(consider)g(smooth)f(r)o(estraining)h(potentials)g Fj(W)26 b Fp(wher)o(e)2090 3763 y Fk(1)p 2090 3777 34 4 v 2090 3825 a(2)2134 3796 y Fi(h)p Fj(n;)14 b(A)p Fn(\()p Fj(\033)s Fn(\))p Fj(n)p Fi(i)h Fp(is)g(the)f(\002rst)h(term)f(in)g(an)h (expansion.)27 b(If)14 b(we)270 3946 y(choose)k(our)g(tubular)g (neighbour)o(hood)e(so)j(that)f Fj(W)12 b Fn(\()p Fj(\033)19 b Fn(+)14 b Fj(n)p Fn(\))23 b Fi(\025)g Fj(c)p Fi(j)p Fj(n)p Fi(j)2343 3915 y Fk(2)2398 3946 y Fp(and)c(impose)f(the)g (non\255r)o(esonance)g(condition)270 4095 y Fj(!)322 4107 y Fl(\013)369 4095 y Fn(\()p Fj(\033)s Fn(\))24 b Fi(6)p Fn(=)f Fj(!)647 4107 y Fl(\014)691 4095 y Fn(\()p Fj(\033)s Fn(\))11 b(+)g Fj(!)944 4107 y Fl(\015)988 4095 y Fn(\()p Fj(\033)s Fn(\))17 b Fp(for)g(every)f(choice)h(of)g Fj(\013)p Fp(,)g Fj(\014)k Fp(and)c Fj(\015)22 b Fp(and)17 b(for)g(every)f Fj(\033)s Fp(,)h(then)f(the)g(same)h(conclusion)g (holds.)270 4244 y(This)22 b(theor)o(em)f(is)h(also)h(r)o(eally)e(a)i (local)f(theor)o(em:)31 b(if)22 b(we)f(impose)h(the)g(conditions)g(on)f Fj(W)34 b Fp(and)23 b(the)e(non\255r)o(esonance)270 4394 y(condition)g(locally)-9 b(,)23 b(and)f(take)f Fj(T)33 b Fp(to)21 b(be)g(the)g(time)f(wher)o(e)h(the)g(orbit)g(leaves)g(the)g (set)h(wher)o(e)e(all)i(the)f(conditions)h(ar)o(e)270 4543 y(tr)o(ue,)d(and)h(the)f(same)h(conclusion)g(holds)g(as)h(well.) 270 4892 y Fq(Classical)30 b(mechanics:)39 b(unb)s(ounded)27 b(energy)436 5191 y Fp(W)-7 b(e)30 b(now)f(describe)h(our)g(theor)o (ems)f(in)h(classical)i(mechanics)e(wher)o(e)f(the)h(initial)g(ener)o (gies)f(ar)o(e)h(diver)o(ging)270 5340 y(as)k(they)f(do)g(in)h(the)e (quantum)h(case.)69 b(Our)33 b(assumptions)i(will)e(imply)g(that)g(the) g(initial)g(value)g(of)g(the)g(action)2020 5539 y(5)p eop %%Page: 6 6 6 5 bop 270 83 a Fp(variables)32 b Fj(I)660 53 y Fl(\025)653 104 y(\013)735 83 y Fp(scales)h(like)e Fj(\025)1176 53 y Fk(2)1214 83 y Fj(I)1257 53 y Fk(0)1250 104 y Fl(\013)1297 83 y Fp(,)k(and)d(thus)f(the)g(initial)h(normal)f(ener)o(gy)g(diver)o (ges)g(like)g Fj(\025)3168 53 y Fk(2)3206 83 y Fp(.)63 b(Examining)32 b(the)270 232 y(ef)o(fective)15 b(Hamiltonian)i Fj(h)p Fn(\()p Fj(\033)n(;)d(\030)t Fn(\))k Fp(in)e(Theor)o(em)j(2.1)q (,)e(one)f(would)h(expect)f(ther)o(e)g(to)g(be)g(a)h(diver)o(ging)f Fj(\025)3312 202 y Fk(2)3363 170 y Fg(P)3451 257 y Fl(\013)3512 232 y Fj(I)3555 202 y Fk(0)3548 253 y Fl(\013)3596 232 y Fj(!)3648 244 y Fl(\013)3695 232 y Fn(\()p Fj(\033)s Fn(\))270 382 y Fp(potential)g(term)g(that)f(for)o(ces)i(the)e (particles)i(to)f(the)g(local)h(minima)e(of)2352 320 y Fg(P)2439 407 y Fl(\013)2501 382 y Fj(I)2544 352 y Fk(0)2537 402 y Fl(\013)2584 382 y Fj(!)2636 394 y Fl(\013)2683 382 y Fn(\()p Fj(\033)s Fn(\))p Fp(.)28 b(If)16 b(these)g(local)h (minima)e(\(called)270 531 y(mini\255wells)22 b(in)f([HS1,)i(HS2]\))g (exist,)f(we)f(would)h(expect)f(the)g(limiting)g(motion)h(to)f(take)h (place)g(ther)o(e.)34 b(W)-7 b(e)21 b(want)h(to)270 681 y(consider)27 b(the)f(case)h(wher)o(e)f(the)g(limiting)g(motion)g (takes)g(place)i(on)e(all)h(of)g Fn(\006)p Fp(.)49 b(Ther)o(efor)o(e)26 b(we)g(will)g(assume)h(that)270 830 y(ther)o(e)19 b(ar)o(e)g(no)h (mini\255wells,)f(i.e.,)g(the)g(fr)o(equencies)g Fj(!)1851 842 y Fl(\013)1918 830 y Fp(ar)o(e)g(constant.)436 980 y(The)i(\002rst)g(step)g(in)g(our)g(analysis)h(is)f(a)h(lar)o(ge)e Fj(\025)i Fp(expansion.)32 b(It)20 b(is)i(convenient)d(to)i(implement)e (this)i(expansion)270 1129 y(using)e(dilations)g(in)g(the)f(\002br)o(e) g(of)h(the)f(normal)i(bundle.)26 b(It)19 b(is)g(also)h(convenient)d(to) i(assume)g(that)g(ar)o(e)g(con\002guration)270 1279 y(space)h(is)g(all) g(of)f Fj(N)9 b Fn(\006)p Fp(.)28 b(This)20 b(makes)f(no)g(dif)o(fer)o (ence,)g(since)g(the)g(orbits)g(we)g(ar)o(e)g(considering)g(never)g (leave)f Fj(N)9 b Fn(\006)3616 1291 y Fl(\016)3653 1279 y Fp(.)28 b(W)-7 b(e)270 1428 y(de\002ne)15 b(a)h(symplectic)f (transformation)h(on)f Fj(T)1650 1398 y Fh(\003)1688 1428 y Fj(N)9 b Fn(\006)15 b Fp(by)g(dilating)g(the)g(normal)g (position)h(and)g(momentum)e(variables)270 1577 y(in)20 b(opposite)g(dir)o(ections.)28 b(Let)1482 1727 y Fj(d)1525 1739 y Fl(\025)1568 1727 y Fn(\()p Fj(\033)n(;)14 b(n;)g(\030)t(;)g (\021)s Fn(\))25 b(=)d(\()p Fj(\033)n(;)14 b(\025n;)g(\030)t(;)g(\025) 2408 1693 y Fh(\000)p Fk(1)2499 1727 y Fj(\021)s Fn(\))p Fj(:)270 1932 y Fp(Instead)g(of)h(the)f(original)h(Hamiltonian)f Fj(H)1563 1944 y Fl(\025)1620 1932 y Fp(we)g(may)h(now)f(consider)h (the)f(equivalent)f(Hamiltonian)h Fj(d)3395 1897 y Fh(\000)p Fk(1)3395 1957 y Fl(\025)3490 1932 y Fi(\016)6 b Fj(H)3607 1944 y Fl(\025)3656 1932 y Fi(\016)g Fj(d)3747 1944 y Fl(\025)3790 1932 y Fp(.)270 2082 y(A)20 b(lar)o(ge)g Fj(\025)g Fp(expansion)g(yields)1336 2326 y Fj(d)1379 2291 y Fh(\000)p Fk(1)1379 2351 y Fl(\025)1487 2326 y Fi(\016)e Fj(H)1616 2338 y Fl(\025)1678 2326 y Fi(\016)g Fj(d)1781 2338 y Fl(\025)1848 2326 y Fn(=)23 b Fj(H)2005 2338 y Fl(B)2080 2326 y Fn(+)18 b Fj(\025)2211 2292 y Fk(2)2249 2326 y Fj(H)2318 2338 y Fl(O)2393 2326 y Fn(+)g Fj(O)r Fn(\()p Fj(\025)2621 2292 y Fh(\000)p Fk(2)2711 2326 y Fn(\))270 2571 y Fp(wher)o(e)h Fj(H)576 2583 y Fl(O)652 2571 y Fp(is)h(the)f(harmonic)h(oscillator)g(Hamiltonian)1405 2815 y Fj(H)1474 2827 y Fl(O)1531 2815 y Fn(\()p Fj(\033)n(;)14 b(n;)g(\030)t(;)g(\021)s Fn(\))24 b(=)2007 2759 y(1)p 2007 2796 42 4 v 2007 2872 a(2)2058 2815 y Fi(h)p Fj(\021)s(;)14 b(\021)s Fi(i)20 b Fn(+)2360 2759 y(1)p 2360 2796 V 2360 2872 a(2)2411 2815 y Fi(h)p Fj(n;)14 b(An)p Fi(i)270 3059 y Fp(and)20 b Fj(H)492 3071 y Fl(B)569 3059 y Fp(is)g(the)f (bundle)g(Hamiltonian)g(given)g(by)1029 3304 y Fj(H)1098 3316 y Fl(B)1155 3304 y Fn(\()p Fj(\033)n(;)14 b(n;)g(\030)t(;)g(\021)s Fn(\))25 b(=)1631 3248 y(1)p 1631 3285 V 1631 3361 a(2)1683 3304 y Fi(h)p Fj(\030)d Fn(+)c Fj(f)9 b Fn(\()p Fj(\033)n(;)14 b(n;)g(\021)s Fn(\))p Fj(;)g(\030)24 b Fn(+)18 b Fj(f)9 b Fn(\()p Fj(\033)n(;)14 b(n;)g(\021)s Fn(\))p Fi(i)2722 3316 y Fl(\033)2786 3304 y Fn(+)k Fj(V)h Fn(\()p Fj(\033)s Fn(\))270 3548 y Fp(The)31 b(inner)g(pr)o(oduct)h Fi(h\001)p Fj(;)14 b Fi(\001i)1104 3560 y Fl(\033)1180 3548 y Fp(is)32 b(the)f(inner)g(pr)o(oduct)g(on)g Fj(T)2113 3518 y Fh(\003)2150 3548 y Fn(\006)h Fp(de\002ned)f(by)f(the)h(imbedding.)62 b(Her)o(e)31 b Fj(f)48 b Fi(2)41 b Fj(T)3713 3518 y Fh(\003)3701 3569 y Fl(\033)3750 3548 y Fn(\006)270 3698 y Fp(is)32 b(de\002ned)e(locally)i(as)g(follows.)63 b(Choose)32 b(a)f(local)h(co\255or)o(dinate)f(system)g Fj(x)41 b Fi(7!)f Fj(\033)s Fn(\()p Fj(x)p Fn(\))32 b Fp(for)g Fn(\006)f Fp(and)h(local)g(frame)270 3847 y Fj(n)320 3859 y Fk(1)357 3847 y Fn(\()p Fj(\033)s Fn(\))p Fj(;)14 b(:)g(:)g(:)h(;)f(n)707 3859 y Fl(m)769 3847 y Fn(\()p Fj(\033)s Fn(\))37 b Fp(for)f(the)e(normal)i(bundle.)74 b(Then)35 b(we)g(obtain)g(local)h(co\255or)o(dinates)g(for)g Fj(N)9 b Fn(\006)35 b Fp(by)g(sending)270 3997 y Fn(\()p Fj(x;)14 b(y)s Fn(\))32 b Fi(7!)f Fn(\()p Fj(\033)s Fn(\()p Fj(x)p Fn(\))p Fj(;)838 3934 y Fg(P)942 3997 y Fj(y)983 4009 y Fl(k)1023 3997 y Fj(n)1073 4009 y Fl(k)1114 3997 y Fn(\))p Fp(.)45 b(\(Her)o(e)25 b Fj(n)1484 4009 y Fl(k)1550 3997 y Fp(denotes)h Fj(n)1898 4009 y Fl(k)1938 3997 y Fn(\()p Fj(\033)s Fn(\()p Fj(x)p Fn(\)\))p Fp(.\))48 b(Associated)27 b(with)d(the)h(frame)h(is)g(the)f(connection)270 4146 y(one\255form.)j(This)21 b(is)f(the)f(antisymmetric)g(matrix)g (valued)g(one\255form)h(given)f(by)1701 4390 y Fj(b)1737 4402 y Fl(k)q(;l)1819 4390 y Fn([)p Fi(\001)p Fn(])k(=)g Fi(h)p Fj(n)2081 4402 y Fl(k)2122 4390 y Fj(;)14 b(dn)2252 4402 y Fl(l)2277 4390 y Fn([)p Fi(\001)p Fn(])p Fi(i)1261 b Fn(\(3)p Fj(:)p Fn(1\))270 4635 y Fp(\(Her)o(e)22 b(we)f(ar)o(e)i (thinking)e(of)i Fj(n)1206 4647 y Fl(k)1269 4635 y Fp(as)g(a)g(map)g (fr)o(om)f Fn(\006)g Fp(to)g Fm(R)2023 4605 y Fl(n)p Fk(+)p Fl(m)2184 4635 y Fp(,)h(and)g(the)e(inner)h(pr)o(oduct)h(is)f (in)g Fj(R)3253 4605 y Fl(n)p Fk(+)p Fl(m)3408 4635 y Fp(.\))37 b(W)l(ith)21 b(this)270 4784 y(notation,)1209 4934 y Fj(f)9 b Fn(\()p Fj(\033)s Fn(\()p Fj(x)p Fn(\))p Fj(;)1489 4855 y Fg(X)1624 4934 y Fj(y)1665 4946 y Fl(k)1706 4934 y Fj(n)1756 4946 y Fl(k)1797 4934 y Fj(;)1834 4855 y Fg(X)1967 4934 y Fj(r)2004 4946 y Fl(k)2046 4934 y Fj(dy)2130 4946 y Fl(k)2170 4934 y Fn(\)[)p Fi(\001)p Fn(])24 b(=)2383 4855 y Fg(X)2404 5034 y Fl(k)q(;l)2516 4934 y Fj(y)2557 4946 y Fl(k)2598 4934 y Fj(b)2634 4946 y Fl(k)q(;l)2715 4934 y Fn([)p Fi(\001)p Fn(])p Fj(r)2821 4946 y Fl(l)2847 4934 y Fj(:)436 5191 y Fp(Notice)d(that)h Fj(d)890 5155 y Fh(\000)p Fk(1)890 5216 y Fl(\025)999 5191 y Fi(\016)d Fj(I)1103 5160 y Fl(\025)1096 5211 y(\013)1166 5191 y Fi(\016)h Fj(d)1271 5203 y Fl(\025)1341 5191 y Fn(=)25 b Fj(\025)1479 5160 y Fh(\000)p Fk(2)1569 5191 y Fj(I)1612 5160 y Fl(\025)1605 5211 y(\013)1656 5191 y Fp(.)34 b(Our)22 b(assumptions)h(will)e(imply)h(that)g(this)g (quantity)f(conver)o(ges)g(on)270 5340 y(the)e(sequence)g(of)h(initial) g(conditions.)2020 5539 y(6)p eop %%Page: 7 7 7 6 bop 270 83 a Ff(Theorem)26 b(3.1)41 b Fe(L)l(et)26 b(the)g(Hamiltonian)h Fj(H)1654 95 y Fl(\025)1724 83 y Fe(b)l(e)f(given)h(by)f(\(1.2\))h(wher)l(e)g Fj(W)38 b Fe(has)27 b(the)f(form)h(\(2.1\))g(and)g(satis\014es)436 236 y(\(i\))j(The)h(eigenvalues)g Fj(!)1219 206 y Fk(2)1216 257 y Fl(\013)1292 236 y Fe(of)g Fj(A)p Fn(\()p Fj(\033)s Fn(\))g Fe(do)g(not)e(dep)l(end)i(on)f Fj(\033)s Fe(.)270 389 y(Supp)l(ose)g(that)g Fn(\()p Fj(\033)835 401 y Fl(\025)879 389 y Fj(;)14 b(n)966 401 y Fl(\025)1009 389 y Fj(;)g(\030)1082 401 y Fl(\025)1126 389 y Fj(;)g(\021)1204 401 y Fl(\025)1248 389 y Fn(\))30 b Fe(ar)l(e)g(initial)h(c)l(onditions)g(in)e Fj(T)2254 359 y Fh(\003)2292 389 y Fj(N)9 b Fn(\006)2428 401 y Fl(\016)2494 389 y Fe(satisfying)436 542 y(\(a\))30 b Fj(\033)623 554 y Fl(\025)690 542 y Fi(!)23 b Fj(\033)843 554 y Fk(0)881 542 y Fe(,)436 695 y(\(b\))30 b Fj(\030)608 707 y Fl(\025)675 695 y Fi(!)23 b Fj(\030)817 707 y Fk(0)854 695 y Fe(,)436 848 y(\(c\))30 b Fj(\025n)670 860 y Fl(\025)737 848 y Fi(!)23 b Fj(n)893 860 y Fk(0)930 848 y Fe(,)436 1001 y(\(d\))30 b Fj(\025)624 971 y Fh(\000)p Fk(1)714 1001 y Fj(\021)755 1013 y Fl(\025)821 1001 y Fi(!)24 b Fj(\021)969 1013 y Fk(0)1006 1001 y Fe(,)270 1155 y(as)30 b Fj(\025)23 b Fi(!)h(1)p Fe(.)38 b(L)l(et)29 b Fn(\()p Fj(\033)922 1167 y Fl(\025)967 1155 y Fn(\()p Fj(t)p Fn(\))p Fj(;)14 b(n)1148 1167 y Fl(\025)1191 1155 y Fn(\()p Fj(t)p Fn(\))p Fj(;)g(\030)1358 1167 y Fl(\025)1403 1155 y Fn(\()p Fj(t)p Fn(\))p Fj(;)g(\021)1575 1167 y Fl(\025)1619 1155 y Fn(\()p Fj(t)p Fn(\)\))30 b Fe(denote)g(the)g(subse)l(quent)e (orbit)j(in)e Fj(T)2948 1124 y Fh(\003)2986 1155 y Fj(N)9 b Fn(\006)3122 1167 y Fl(\016)3187 1155 y Fe(under)30 b(the)g(Hamil-)270 1304 y(tonian)g Fj(H)600 1316 y Fl(\025)644 1304 y Fe(.)436 1457 y(Supp)l(ose)38 b(that)f Fn(\()p Fj(\033)s Fn(\()p Fj(t)p Fn(\))p Fj(;)14 b(n)p Fn(\()p Fj(t)p Fn(\))p Fj(;)g(\030)t Fn(\()p Fj(t)p Fn(\))p Fj(;)g(\021)s Fn(\()p Fj(t)p Fn(\)\))41 b Fe(is)d(the)f(orbit)h(in)g Fj(T)2332 1427 y Fh(\003)2369 1457 y Fn(\006)g Fe(with)g(initial)g(c)l (onditions)h Fn(\()p Fj(\033)3390 1469 y Fk(0)3428 1457 y Fj(;)14 b(n)3515 1469 y Fk(0)3552 1457 y Fj(;)g(\030)3625 1469 y Fk(0)3662 1457 y Fj(;)g(\021)3740 1469 y Fk(0)3778 1457 y Fn(\))270 1607 y Fe(governe)l(d)31 b(by)f(the)g(Hamiltonian)h Fj(H)1410 1619 y Fl(B)1485 1607 y Fn(+)18 b Fj(\025)1616 1576 y Fk(2)1654 1607 y Fj(H)1723 1619 y Fl(O)1809 1607 y Fe(de\014ne)l(d)30 b(ab)l(ove.)436 1760 y(Then)g(for)h(any)f Fj(T)k Fi(\025)23 b Fn(0)700 2020 y(sup)657 2089 y Fk(0)p Fh(\024)p Fl(t)p Fh(\024)p Fl(T)881 2020 y Fi(j)p Fj(\033)951 2032 y Fl(\025)995 2020 y Fn(\()p Fj(t)p Fn(\))c Fi(\000)f Fj(\033)s Fn(\()p Fj(t)p Fn(\))p Fi(j)i Fn(+)e Fi(j)p Fj(\025n)1582 2032 y Fl(\025)1626 2020 y Fn(\()p Fj(t)p Fn(\))h Fi(\000)f Fj(n)p Fn(\()p Fj(t)p Fn(\))p Fi(j)h Fn(+)f Fi(j)p Fj(\030)2150 2032 y Fl(\025)2194 2020 y Fn(\()p Fj(t)p Fn(\))h Fi(\000)f Fj(\030)t Fn(\()p Fj(t)p Fn(\))p Fi(j)h Fn(+)f Fi(j)p Fj(\025)2720 1985 y Fh(\000)p Fk(1)2809 2020 y Fj(\021)2850 2032 y Fl(\025)2894 2020 y Fn(\()p Fj(t)p Fn(\))h Fi(\000)f Fj(\021)s Fn(\()p Fj(t)p Fn(\))p Fi(j)24 b(!)f Fn(0)270 2280 y Fe(as)30 b Fj(\025)24 b Fi(!)f(1)p Fe(.)436 2490 y Fp(This)g(theor)o(em)e(gives) g(a)i(satisfactory)h(description)e(of)h(the)e(limiting)g(motion)h(if)g (the)g(Poisson)h(bracket)e(of)i Fj(H)3753 2502 y Fl(B)270 2639 y Fp(and)f Fj(H)494 2651 y Fl(O)571 2639 y Fp(vanishes.)33 b(Then)21 b(the)f(\003ows)h(generated)g(by)f Fj(H)2013 2651 y Fl(B)2092 2639 y Fp(and)h Fj(H)2315 2651 y Fl(O)2393 2639 y Fp(commute)e(and)j(the)e(motion)h(is)g(given)f(by)h(the)270 2789 y(rapid)d(oscillations)g(generated)f(by)g Fj(\025)1402 2758 y Fk(2)1439 2789 y Fj(H)1508 2801 y Fl(O)1582 2789 y Fp(superimposed)h(on)f(the)f(\003ow)h(generated)g(by)f Fj(H)3043 2801 y Fl(B)3101 2789 y Fp(.)27 b(In)16 b(this)i(situation)e (we)270 2938 y(can)k(perform)g(averaging)g(by)f(simply)h(ignoring)f (the)g(oscillations.)436 3091 y(An)24 b(example)g(wher)o(e)f Fi(f)p Fj(H)1239 3103 y Fl(B)1296 3091 y Fj(;)14 b(H)1402 3103 y Fl(O)1458 3091 y Fi(g)24 b Fp(is)g(zer)o(o)g(is)g(when)g Fn(\006)g Fp(has)h(codimension)f(one,)g(or)-6 b(,)25 b(mor)o(e)f(generally)-9 b(,)24 b(if)h(the)270 3241 y(connection)19 b(form)g(vanishes.)28 b(Then)19 b Fj(H)1476 3253 y Fl(B)1557 3241 y Fn(=)1654 3208 y Fk(1)p 1654 3222 34 4 v 1654 3269 a(2)1697 3241 y Fi(h)p Fj(\030)t(;)14 b(\030)t Fi(i)k Fn(+)f Fj(V)i Fn(\()p Fj(\033)s Fn(\))h Fp(only)f(involves)g(variables) g(on)h Fj(T)3173 3210 y Fh(\003)3210 3241 y Fn(\006)p Fp(,)f(so)g(the)g(limiting)270 3390 y(motion)g(is)h(a)g(motion)f(on)h Fn(\006)g Fp(with)f(independent)g(oscillations)i(in)e(the)g(normal)h (variables.)436 3543 y(The)31 b(Poisson)h(bracket)e Fi(f)p Fj(H)1298 3555 y Fl(B)1355 3543 y Fj(;)14 b(H)1461 3555 y Fl(O)1517 3543 y Fi(g)31 b Fp(also)g(vanishes)h(if)f(all)g(the)g(fr)o (equencies)f Fj(!)2912 3555 y Fl(\013)2990 3543 y Fp(ar)o(e)h(equal.)62 b(In)30 b(this)h(case)270 3693 y(the)25 b(motion)g(generated)h(by)f Fj(H)1237 3705 y Fl(B)1320 3693 y Fp(need)h(not)f(only)h(involve)e(the) i(variables)g(on)g Fj(T)2778 3662 y Fh(\003)2815 3693 y Fn(\006)p Fp(.)46 b(It)26 b(can)g(be)f(thought)g(of)h(as)h(a)270 3842 y(generalized)h(minimal)f(coupling)g(type)g(\003ow)-7 b(.)51 b(\(See)28 b([GS])g(for)g(a)g(description)g(of)g(the)f(geometry) f(of)i(this)g(sort)g(of)270 3991 y(\003ow)-7 b(.\))44 b(The)25 b(\003ow)g(has)g(the)g(pr)o(operty)g(that)g(the)f (trajectories)i(in)e Fj(N)9 b Fn(\006)25 b Fp(ar)o(e)g(parallel)h (along)f(their)g(pr)o(ojections)g(onto)270 4141 y Fn(\006)p Fp(.)49 b(In)27 b(particular)-6 b(,)29 b Fi(j)p Fj(n)p Fi(j)980 4111 y Fk(2)1044 4141 y Fp(is)e(pr)o(eserved)g(by)f(this)h (motion.)49 b(In)26 b(certain)h(situations)g(ther)o(e)f(may)g(be)h (other)-6 b(,)28 b(angular)270 4290 y(momentum)23 b(like,)j(conserved)g (quantities)f(for)g Fj(H)1826 4302 y Fl(B)1884 4290 y Fp(,)h(which)f(would)g(show)g(up)h(as)g(adiabatic)g(invariants)g(for)g (the)270 4440 y(original)20 b(\003ow)-7 b(.)436 4593 y(In)19 b(general,)g(when)g(the)g(fr)o(equencies)g(ar)o(e)g(not)h(all)f (equal,)h(the)e(\003ows)i(generated)f(by)g Fj(H)3066 4605 y Fl(B)3143 4593 y Fp(and)h Fj(\025)3344 4563 y Fk(2)3382 4593 y Fj(H)3451 4605 y Fl(O)3527 4593 y Fp(interact,)270 4742 y(and)28 b(their)f(interaction)g(pr)o(oduces)g(an)h(extra)f (potential)h(term)e(in)i(the)f(limiting)f(Hamiltonian.)51 b(T)-7 b(o)27 b(describe)h(the)270 4892 y(limiting)18 b(motion,)h(we)g(must)g(perform)h(an)g(averaging.)28 b(This)20 b(r)o(equir)o(es)e(non\255r)o(esonance)i(conditions.)28 b(In)19 b(this)h(note,)270 5041 y(we)14 b(only)g(consider)h(the)f(extr) o(eme)f(case)i(wher)o(e)f(all)h(the)f(eigenvalues)f(ar)o(e)i(distinct.) 26 b(\(W)-7 b(e)14 b(will)g(tr)o(eat)g(the)g(mor)o(e)g(general)270 5191 y(case)25 b(wher)o(e)e(some,)h(but)f(not)h(all,)h(the)e(fr)o (equencies)h(ar)o(e)g(equal)f(in)h(our)g(forthcoming)f(paper)j([FH].\)) e(W)-7 b(e)23 b(have)h(the)270 5340 y(following)19 b(theor)o(em.)2020 5539 y(7)p eop %%Page: 8 8 8 7 bop 270 83 a Ff(Theorem)26 b(3.2)41 b Fe(L)l(et)26 b(the)g(Hamiltonian)h Fj(H)1654 95 y Fl(\025)1724 83 y Fe(b)l(e)f(given)h(by)f(\(1.2\))h(wher)l(e)g Fj(W)38 b Fe(has)27 b(the)f(form)h(\(2.1\))g(and)g(satis\014es)436 232 y(\(i\))j(The)h(eigenvalues)g Fj(!)1219 202 y Fk(2)1216 253 y(1)1256 232 y Fj(;)14 b(:)g(:)g(:)f(;)h(!)1495 202 y Fk(2)1492 253 y Fl(m)1585 232 y Fe(of)30 b Fj(A)p Fn(\()p Fj(\033)s Fn(\))h Fe(do)g(not)e(dep)l(end)i(on)f Fj(\033)s Fe(,)436 382 y(\(ii\))h(If)f Fj(j)e Fi(6)p Fn(=)22 b Fj(k)33 b Fe(then)c Fj(!)1133 394 y Fl(j)1187 382 y Fi(\006)18 b Fj(!)1322 394 y Fl(k)1385 382 y Fi(6)p Fn(=)23 b(0)p Fe(,)436 531 y(\(iii\))31 b(If)f Fj(j)e Fi(6)p Fn(=)23 b Fj(k)32 b Fe(and)f Fj(l)24 b Fi(6)p Fn(=)f Fj(m)29 b Fe(then)h Fj(!)1560 543 y Fl(j)1613 531 y Fi(\006)18 b Fj(!)1748 543 y Fl(k)1807 531 y Fi(\006)g Fj(!)1942 543 y Fl(l)1986 531 y Fi(\006)g Fj(!)2121 543 y Fl(m)2206 531 y Fi(6)p Fn(=)23 b(0)p Fe(.)270 681 y(Supp)l(ose)30 b(that)g Fn(\()p Fj(\033)835 693 y Fl(\025)879 681 y Fj(;)14 b(n)966 693 y Fl(\025)1009 681 y Fj(;)g(\030)1082 693 y Fl(\025)1126 681 y Fj(;)g(\021)1204 693 y Fl(\025)1248 681 y Fn(\))30 b Fe(ar)l(e)g(initial)h(c)l(onditions)g(in)e Fj(T)2254 651 y Fh(\003)2292 681 y Fj(N)9 b Fn(\006)2428 693 y Fl(\016)2494 681 y Fe(satisfying)436 830 y(\(a\))30 b Fj(\033)623 842 y Fl(\025)690 830 y Fi(!)23 b Fj(\033)843 842 y Fk(0)881 830 y Fe(,)436 980 y(\(b\))30 b Fj(\030)608 992 y Fl(\025)675 980 y Fi(!)23 b Fj(\030)817 992 y Fk(0)854 980 y Fe(,)436 1129 y(\(c\))30 b Fj(\025)620 1099 y Fh(\000)p Fk(2)709 1129 y Fj(I)752 1099 y Fl(\025)745 1153 y(k)796 1129 y Fn(\()p Fj(\033)875 1141 y Fl(\025)919 1129 y Fj(;)14 b(n)1006 1141 y Fl(\025)1049 1129 y Fj(;)g(\030)1122 1141 y Fl(\025)1166 1129 y Fj(;)g(\021)1244 1141 y Fl(\025)1288 1129 y Fn(\))23 b Fi(!)g Fj(I)1492 1099 y Fk(0)1485 1153 y Fl(k)1553 1129 y Fj(>)f Fn(0)p Fe(,)270 1279 y(as)30 b Fj(\025)23 b Fi(!)h(1)p Fe(.)38 b(L)l(et)29 b Fn(\()p Fj(\033)922 1291 y Fl(\025)967 1279 y Fn(\()p Fj(t)p Fn(\))p Fj(;)14 b(n)1148 1291 y Fl(\025)1191 1279 y Fn(\()p Fj(t)p Fn(\))p Fj(;)g(\030)1358 1291 y Fl(\025)1403 1279 y Fn(\()p Fj(t)p Fn(\))p Fj(;)g(\021)1575 1291 y Fl(\025)1619 1279 y Fn(\()p Fj(t)p Fn(\)\))30 b Fe(denote)g(the)g(subse)l(quent)e (orbit)j(in)e Fj(T)2948 1248 y Fh(\003)2986 1279 y Fj(N)9 b Fn(\006)3122 1291 y Fl(\016)3187 1279 y Fe(under)30 b(the)g(Hamil-)270 1428 y(tonian)g Fj(H)600 1440 y Fl(\025)644 1428 y Fe(.)436 1577 y(Supp)l(ose)37 b(that)f Fn(\()p Fj(\033)s Fn(\()p Fj(t)p Fn(\))p Fj(;)14 b(\030)t Fn(\()p Fj(t)p Fn(\)\))38 b Fe(is)f(the)f(orbit)h(in)f Fj(T)1966 1547 y Fh(\003)2003 1577 y Fn(\006)h Fe(with)f(initial)i(c)l(onditions) f Fn(\()p Fj(\033)3019 1589 y Fk(0)3057 1577 y Fj(;)14 b(\030)3130 1589 y Fk(0)3167 1577 y Fn(\))37 b Fe(governe)l(d)g(by)g (the)270 1727 y(Hamiltonian)1419 1876 y Fj(h)p Fn(\()p Fj(\033)n(;)14 b(\030)t Fn(\))24 b(=)1775 1820 y(1)p 1775 1857 42 4 v 1775 1933 a(2)1826 1876 y Fi(h)p Fj(\030)t(;)14 b(\030)t Fi(i)2007 1888 y Fl(\033)2071 1876 y Fn(+)k Fj(V)h Fn(\()p Fj(\033)s Fn(\))g(+)f Fj(V)2485 1888 y Fk(1)2523 1876 y Fn(\()p Fj(\033)s Fn(\))p Fj(;)270 2069 y Fe(wher)l(e)30 b Fj(V)552 2081 y Fk(1)620 2069 y Fe(is)g(describ)l(e) l(d)h(b)l(elow.)40 b(Then)30 b(for)h(any)f Fj(T)k Fi(\025)23 b Fn(0)1367 2284 y(sup)1325 2354 y Fk(0)p Fh(\024)p Fl(t)p Fh(\024)p Fl(T)1549 2284 y Fi(j)p Fj(\033)1619 2296 y Fl(\025)1663 2284 y Fn(\()p Fj(t)p Fn(\))c Fi(\000)f Fj(\033)s Fn(\()p Fj(t)p Fn(\))p Fi(j)h Fn(+)f Fi(j)p Fj(\030)2187 2296 y Fl(\025)2231 2284 y Fn(\()p Fj(t)p Fn(\))h Fi(\000)f Fj(\030)t Fn(\()p Fj(t)p Fn(\))p Fi(j)24 b(!)f Fn(0)270 2499 y Fe(as)30 b Fj(\025)24 b Fi(!)f(1)p Fe(.)436 2691 y Fp(T)-7 b(o)32 b(complete)f(the)f(statement)h(of)h(the) f(theor)o(em,)i(we)e(must)g(de\002ne)f(the)h(potential)h Fj(V)3125 2703 y Fk(1)3162 2691 y Fp(.)64 b(The)31 b(normalized)270 2841 y(eigenvectors)15 b(of)i Fj(A)p Fn(\()p Fj(\033)s Fn(\))g Fp(pr)o(ovide)f(an)h(orthonormal)f(frame)h Fj(n)2124 2853 y Fk(1)2161 2841 y Fn(\()p Fj(\033)s Fn(\))p Fj(;)d(:)g(:)g(:)g(;) g(n)2510 2853 y Fl(m)2573 2841 y Fn(\()p Fj(\033)s Fn(\))j Fp(for)g(the)f(normal)g(bundle.)26 b(\(Each)17 b Fj(n)3769 2853 y Fl(k)270 2990 y Fp(is)j(de\002ned)e(up)h(to)g(a)h(choice)f(of)g (sign.\))29 b(Let)19 b Fj(b)1606 3002 y Fl(k)q(;l)1706 2990 y Fp(be)g(the)f(associated)j(connection)d(one\255form)i(given)e (by)g(\(3.1\).)30 b(Then)1558 3205 y Fj(V)1606 3217 y Fk(1)1644 3205 y Fn(\()p Fj(\033)s Fn(\))24 b(=)1870 3126 y Fg(X)1891 3305 y Fl(k)q(;l)2013 3149 y Fj(I)2056 3119 y Fk(0)2049 3173 y Fl(k)2094 3149 y Fj(I)2137 3119 y Fk(0)2130 3173 y Fl(l)2174 3149 y Fj(!)2226 3161 y Fl(l)p 2013 3186 238 4 v 2086 3262 a Fj(!)2138 3274 y Fl(k)2261 3205 y Fi(k)p Fj(b)2339 3217 y Fl(k)q(;l)2420 3205 y Fi(k)2462 3171 y Fk(2)2499 3205 y Fj(:)1117 b Fn(\(3)p Fj(:)p Fn(2\))270 3461 y Fp(Notice)19 b(that)g(the)g(norm)h Fi(k)p Fj(b)1093 3473 y Fl(k)q(;l)1174 3461 y Fi(k)f Fp(is)h(insensitive)f(to)h(the)f(choice)g(of)h(signs)h(for)f(the)f (frame.)270 3780 y Fq(Quantum)27 b(mechanics)436 4079 y Fp(In)20 b(quantum)f(mechanics,)i(it)f(is)h(also)g(convenient)e(to)i (work)f(with)g(the)g(normal)g(bundle)g Fj(N)9 b Fn(\006)p Fp(.)30 b(If)20 b(we)g(consider)270 4228 y(initial)c(conditions)h(in)f Fj(L)1014 4198 y Fk(2)1051 4228 y Fn(\()p Fm(R)1137 4198 y Fl(n)p Fk(+)p Fl(m)1298 4228 y Fn(\))g Fp(that)h(ar)o(e)f(supported)h (near)f Fn(\006)h Fp(then,)f(to)g(a)g(good)h(appr)o(oximation)g(for)f (lar)o(ge)g Fj(\025)p Fp(,)i(the)270 4378 y(time)j(evolution)f(stays)j (near)f Fn(\006)p Fp(.)35 b(Thus)22 b(we)g(lose)g(nothing)f(by)g (inserting)h(Dirichlet)f(boundary)h(conditions)g(on)g(the)270 4527 y(boundary)f(of)h(the)f(tubular)f(neighbour)o(hood)g(of)i Fn(\006)p Fp(,)f(and)h(may)g(transfer)g(our)f(considerations)h(to)g Fj(L)3331 4497 y Fk(2)3367 4527 y Fn(\()p Fj(N)9 b Fn(\006)3535 4539 y Fl(\016)3572 4527 y Fj(;)14 b(d)p Fn(v)n(ol)o(\))p Fp(,)270 4676 y(wher)o(e)20 b Fj(d)p Fn(v)n(ol)h Fp(is)g(computed)g (using)g(the)f(pulled)h(back)g(metric.)32 b(If)21 b(we)g(extend)f(the)h (pulled)g(back)g(metric,)g(and)g(make)270 4826 y(a)g(suitable)e (de\002nition)g(of)i Fj(H)1142 4838 y Fl(\025)1206 4826 y Fp(in)f(the)f(complement)g(of)i Fj(N)9 b Fn(\006)2105 4838 y Fl(\016)2141 4826 y Fp(,)20 b(we)f(may)h(r)o(emove)f(the)h (boundary)g(condition.)29 b(Thus)270 4975 y(we)19 b(may)h(assume)g (that)f(that)h(the)f(Hamiltonian)g Fj(H)1821 4987 y Fl(\025)1885 4975 y Fp(acts)h(in)f Fj(L)2183 4945 y Fk(2)2220 4975 y Fn(\()p Fj(N)9 b Fn(\006)p Fj(;)14 b(d)p Fn(v)n(ol)o(\))p Fp(.)436 5125 y(W)-7 b(e)19 b(now)h(intr)o(oduce)e(the)h(gr)o(oup)h(of) g(dilations)g(in)g(the)f(normal)h(dir)o(ections)f(by)g(de\002ning)1517 5340 y Fn(\()p Fj(D)1618 5352 y Fl(\025)1662 5340 y Fj( )s Fn(\)\()p Fj(\033)n(;)14 b(n)p Fn(\))24 b(=)f Fj(\025)2107 5306 y Fl(m=)p Fk(2)2237 5340 y Fj( )s Fn(\()p Fj(\033)n(;)14 b(\025n)p Fn(\))p Fj(:)2020 5539 y Fp(8)p eop %%Page: 9 9 9 8 bop 270 83 a Fp(This)23 b(is)g(a)g(unitary)g(operator)g(fr)o(om)f Fj(L)1433 53 y Fk(2)1470 83 y Fn(\()p Fj(N)9 b Fn(\006)p Fj(;)14 b(d)p Fn(v)n(ol)1824 95 y Fl(\025)1868 83 y Fn(\))23 b Fp(to)f Fj(L)2071 53 y Fk(2)2108 83 y Fn(\()p Fj(N)9 b Fn(\006)p Fj(;)14 b(d)p Fn(v)n(ol)o(\))23 b Fp(wher)o(e)f Fj(d)p Fn(v)n(ol)2906 95 y Fl(\025)2972 83 y Fp(denotes)h(the)f(pulled) g(back)270 232 y(density)i Fj(d)p Fn(v)n(ol)702 244 y Fl(\025)745 232 y Fn(\()p Fj(\033)n(;)14 b(n)p Fn(\))32 b(=)e Fj(d)p Fn(v)n(ol)o(\()p Fj(\033)n(;)14 b(\025)1379 202 y Fh(\000)p Fk(1)1469 232 y Fj(n)p Fn(\))p Fp(.)43 b(Since)25 b(the)f(spaces)i Fj(L)2264 202 y Fk(2)2301 232 y Fn(\()p Fj(N)9 b Fn(\006)p Fj(;)14 b(d)p Fn(v)n(ol)2655 244 y Fl(\025)2699 232 y Fn(\))25 b Fp(depend)g(on)f Fj(\025)p Fp(,)i(and)g(we)e(want)g(to)270 382 y(deal)c(with)f(a)h (\002xed)f(Hilbert)g(space)i(as)f Fj(\025)k Fi(!)f(1)p Fp(,)c(we)h(perform)g(an)g(an)g(additional)g(unitary)g(transformation.) 29 b(Let)1294 611 y Fj(d)p Fn(v)n(ol)1444 623 y Fl(N)6 b Fk(\006)1577 611 y Fn(=)51 b(lim)1665 665 y Fl(\025)p Fh(!1)1850 611 y Fj(d)p Fn(v)n(ol)1999 623 y Fl(\025)2066 611 y Fn(=)23 b Fj(d)p Fn(v)n(ol)2303 623 y Fk(\006)2373 611 y Fi(\012)18 b Fj(d)p Fn(v)n(ol)2605 627 y Fd(R)2652 610 y Fc(n)p Fb(+)p Fc(m)270 840 y Fp(Then)27 b(the)f(quotient)g(of)h (densities)g Fj(d)p Fn(v)n(ol)1512 852 y Fl(N)6 b Fk(\006)1622 840 y Fj(=d)p Fn(v)n(ol)1813 852 y Fl(\025)1883 840 y Fp(is)27 b(a)h(function)e(on)h Fj(N)9 b Fn(\006)27 b Fp(and)g(we)f(may)h(de\002ne)g Fj(M)3427 852 y Fl(\025)3497 840 y Fp(to)f(be)h(the)270 989 y(operator)d(of)g(multiplication)f(by) 1314 918 y Fg(p)p 1397 918 494 4 v 71 x Fj(d)p Fn(v)n(ol)1546 1001 y Fl(N)6 b Fk(\006)1656 989 y Fj(=d)p Fn(v)n(ol)1847 1001 y Fl(\025)1891 989 y Fp(.)39 b(The)23 b(operator)i Fj(M)2509 1001 y Fl(\025)2575 989 y Fp(is)f(unitary)f(fr)o(om)g Fj(L)3184 959 y Fk(2)3221 989 y Fn(\()p Fj(N)9 b Fn(\006)p Fj(;)14 b(d)p Fn(v)n(ol)3575 1001 y Fl(N)6 b Fk(\006)3685 989 y Fn(\))24 b Fp(to)270 1139 y Fj(L)327 1109 y Fk(2)364 1139 y Fn(\()p Fj(N)9 b Fn(\006)p Fj(;)14 b(d)p Fn(v)n(ol)718 1151 y Fl(\025)761 1139 y Fn(\))p Fp(.)29 b(Let)1805 1288 y Fj(U)1862 1300 y Fl(\025)1928 1288 y Fn(=)23 b Fj(D)2085 1300 y Fl(\025)2128 1288 y Fj(M)2209 1300 y Fl(\025)2252 1288 y Fj(:)270 1487 y Fp(Notice)h(that)i(the)f(support)h (of)f(a)h(family)g(of)g(initial)f(conditions)h(of)f(the)g(form)h Fj(U)2731 1499 y Fl(\025)2774 1487 y Fj( )2828 1499 y Fk(0)2891 1487 y Fp(is)f(being)g(squeezed)h(close)f(to)270 1636 y Fn(\006)g Fp(as)g Fj(\025)31 b Fi(!)f(1)p Fp(.)43 b(W)-7 b(e)24 b(want)g(to)g(consider)h(a)g(such)g(a)g(sequence)f(of)h (initial)f(conditions.)43 b(Ther)o(efor)o(e)24 b(it)h(is)f(natural)h (to)270 1786 y(consider)20 b(the)f(conjugated)h(Hamiltonian)1765 1935 y Fj(L)1822 1947 y Fl(\025)1888 1935 y Fn(=)i Fj(U)2041 1901 y Fh(\003)2032 1956 y Fl(\025)2079 1935 y Fj(H)2148 1947 y Fl(\025)2192 1935 y Fj(U)2249 1947 y Fl(\025)2292 1935 y Fj(;)270 2134 y Fp(since)e(the)f(evolution)g(generated)g(by)h Fj(L)1480 2146 y Fl(\025)1543 2134 y Fp(acting)g(on)g Fj( )1939 2146 y Fk(0)1996 2134 y Fp(is)g(unitarily)g(equivalent)e(to)i (the)f(evolution)g(generated)h(by)270 2283 y Fj(H)339 2295 y Fl(\025)402 2283 y Fp(acting)g(on)g Fj(U)801 2295 y Fl(\025)844 2283 y Fj( )898 2295 y Fk(0)935 2283 y Fp(.)436 2433 y(As)h(a)f(\002rst)f(step)i(we)e(perform)h(a)g(lar)o(ge)g Fj(\025)g Fp(expansion.)28 b(This)21 b(yields)1531 2662 y Fj(L)1588 2674 y Fl(\025)1654 2662 y Fn(=)h Fj(H)1810 2674 y Fl(B)1886 2662 y Fn(+)c Fj(\025)2017 2627 y Fk(2)2055 2662 y Fj(H)2124 2674 y Fl(O)2199 2662 y Fn(+)g Fj(O)r Fn(\()p Fj(\025)2427 2627 y Fh(\000)p Fk(1)2517 2662 y Fn(\))270 2891 y Fp(wher)o(e)32 b Fj(H)589 2903 y Fl(O)678 2891 y Fp(and)h Fj(H)913 2903 y Fl(B)1003 2891 y Fp(ar)o(e)f(de\002ned) h(in)f(the)g(local)h(co\255or)o(dinates)g Fn(\()p Fj(x;)14 b(y)s Fn(\))34 b Fp(associated)g(to)e(some)h(local)g(frame)g(\(as)270 3040 y(intr)o(oduced)19 b(above\))g(as)i(follows.)29 b(Let)1460 3340 y Fj(D)1529 3352 y Fl(x)1594 3340 y Fn(=)1681 3149 y Fg(2)1681 3295 y(6)1681 3348 y(4)1755 3213 y Fj(D)1824 3225 y Fl(x)1862 3233 y Fb(1)1816 3301 y Fp(.)1816 3334 y(.)1816 3368 y(.)1750 3467 y Fj(D)1819 3479 y Fl(x)1857 3487 y Fc(n)1916 3149 y Fg(3)1916 3295 y(7)1916 3348 y(5)1985 3340 y Fj(;)93 b(D)2170 3352 y Fl(y)2232 3340 y Fn(=)2320 3149 y Fg(2)2320 3295 y(6)2320 3348 y(4)2400 3212 y Fj(D)2469 3224 y Fl(y)2503 3232 y Fb(1)2460 3303 y Fp(.)2460 3336 y(.)2460 3369 y(.)2389 3469 y Fj(D)2458 3481 y Fl(y)2492 3489 y Fc(m)2565 3149 y Fg(3)2565 3295 y(7)2565 3348 y(5)270 3644 y Fp(and)22 b(let)e Fj(H)32 b Fn(=)25 b([)p Fi(h)p Fj(@)5 b(=@)g(x)965 3656 y Fl(i)993 3644 y Fj(;)14 b(@)5 b(=@)g(x)1217 3656 y Fl(j)1251 3644 y Fi(i)p Fn(])21 b Fp(be)g(the)g(local)g(expr)o(ession)h(for)f(the)g (metric)f(on)i Fn(\006)p Fp(.)32 b(De\002ne)21 b Fj(h)k Fn(=)f(det)q(\()p Fj(H)7 b Fn(\))p Fp(.)33 b(De\002ne)270 3793 y(the)19 b(operator)i Fj(F)771 3805 y Fl(i)818 3793 y Fp(by)1571 3943 y Fj(F)1624 3955 y Fl(i)1675 3943 y Fn(=)1763 3864 y Fg(X)1784 4043 y Fl(k)q(;l)1896 3943 y Fj(y)1937 3955 y Fl(k)1978 3943 y Fj(b)2014 3955 y Fl(k)q(;l)2095 3943 y Fn([)p Fj(@)5 b(=@)g(x)2305 3955 y Fl(i)2332 3943 y Fn(])p Fj(D)2424 3955 y Fl(y)2458 3964 y Fc(l)2486 3943 y Fj(;)270 4178 y Fp(wher)o(e)19 b Fj(b)543 4190 y Fl(k)q(;l)644 4178 y Fp(is)h(the)f(connection)g(form) h(de\002ned)f(by)g(\(3.1\),)i(and)f(let)1815 4474 y Fj(F)35 b Fn(=)1991 4307 y Fg(2)1991 4457 y(4)2064 4351 y Fj(F)2117 4363 y Fk(1)2099 4431 y Fp(.)2099 4464 y(.)2099 4498 y(.)2060 4597 y Fj(F)2113 4609 y Fl(n)2172 4307 y Fg(3)2172 4457 y(5)2242 4474 y Fj(:)270 4763 y Fp(Then)1813 4912 y Fj(H)1882 4924 y Fl(O)1962 4912 y Fn(=)22 b Fj(D)2120 4878 y Fh(\003)2118 4933 y Fl(y)2158 4912 y Fj(D)2227 4924 y Fl(y)270 5111 y Fp(is)e(the)f(harmonic)h(oscillator)g (Hamiltonian)g(in)f(the)g(normal)h(variables,)g(and)1236 5340 y Fj(H)1305 5352 y Fl(B)1386 5340 y Fn(=)i(\()p Fj(D)1574 5352 y Fl(x)1635 5340 y Fn(+)c Fj(F)12 b Fn(\))1815 5306 y Fh(\003)1853 5340 y Fj(H)1929 5306 y Fh(\000)p Fk(1)2018 5340 y Fn(\()p Fj(D)2119 5352 y Fl(x)2179 5340 y Fn(+)18 b Fj(F)12 b Fn(\))19 b(+)f Fj(K)24 b Fn(+)18 b Fj(V)h Fn(\()p Fj(\033)s Fn(\))p Fj(:)2020 5539 y Fp(9)p eop %%Page: 10 10 10 9 bop 270 83 a Fp(Her)o(e)26 b(the)g(adjoints)i(ar)o(e)f(taken)f (with)g(r)o(espect)h(to)g Fj(d)p Fn(v)n(ol)1967 95 y Fl(N)6 b Fk(\006)2077 83 y Fp(,)29 b(given)c(locally)j(by)2719 12 y Fi(p)p 2788 12 48 4 v 71 x Fj(h)p Fi(j)p Fj(d)2902 53 y Fl(n)2947 83 y Fj(x)p Fi(j)p Fj(d)3060 53 y Fl(m)3124 83 y Fj(y)s Fi(j)p Fp(.)49 b(In)27 b(other)f(wor)o(ds,)270 232 y Fj(D)341 202 y Fh(\003)339 253 y Fl(y)402 232 y Fn(=)d Fi(\000)p Fj(D)626 202 y Fl(t)624 253 y(y)683 232 y Fp(and)d Fj(D)907 202 y Fh(\003)905 253 y Fl(x)970 232 y Fn(=)i Fi(\000)p Fn(\(1)p Fj(=)1238 161 y Fg(p)p 1320 161 33 4 v 1320 232 a Fn(\()p Fj(h)p Fn(\)\))p Fj(D)1535 202 y Fl(t)1533 253 y(x)1575 162 y Fi(p)p 1645 162 48 4 v 1645 232 a Fj(h)o Fp(.)29 b(The)19 b(additional)i(potential)f Fj(K)25 b Fp(is)20 b(given)f(by)1614 490 y Fj(K)29 b Fn(=)1811 434 y Fj(n)p Fn(\()p Fj(n)19 b Fi(\000)f Fn(1\))p 1811 471 308 4 v 1944 547 a(2)2128 490 y Fj(s)h Fi(\000)2279 434 y Fj(n)2329 404 y Fk(2)p 2279 471 88 4 v 2301 547 a Fn(4)2376 490 y Ff(h)2429 456 y Fk(2)270 733 y Fp(wher)o(e)d Fj(s)h Fp(is)h(the)f(scalar)h(curvatur)o(e)e(and)i Ff(h)f Fp(is)h(the)e(mean)h(curvatur)o(e)f(vector)-6 b(.)27 b(Notice)16 b(that)h(this)g(extra)g(potential)h(does)270 882 y(depend)i(on)g(the)f(imbedding)g(of)h Fn(\006)f Fp(in)h Fm(R)1507 852 y Fl(n)p Fk(+)p Fl(m)1668 882 y Fp(,)f(since)h(the)f(mean)h(curvatur)o(e)e(does.)270 1080 y Ff(Theorem)46 b(4.1)35 b Fe(L)l(et)42 b Fj(H)1074 1092 y Fl(\025)1160 1080 y Fe(b)l(e)h(the)f(Hamiltonian)i(\(1.2\),)j(c) l(onsider)l(e)l(d)d(to)e(b)l(e)h(acting)g(in)f Fj(L)3239 1050 y Fk(2)3276 1080 y Fn(\()p Fj(N)9 b Fn(\006)p Fj(;)14 b(d)p Fn(v)n(ol)o(\))p Fe(,)47 b(as)270 1230 y(explaine)l(d)31 b(ab)l(ove,)g(wher)l(e)g Fj(W)41 b Fe(has)31 b(the)f(form)g(\(2.1\))h (and)g(satis\014es)436 1379 y(\(i\))f(the)g(eigenvalues)h(of)g Fj(!)1285 1349 y Fk(2)1282 1400 y Fl(\013)1358 1379 y Fe(of)g Fj(A)p Fn(\()p Fj(\033)s Fn(\))g Fe(do)f(not)g(dep)l(end)h(on)e Fj(\033)s Fe(.)270 1529 y(L)l(et)g Fj(L)470 1541 y Fl(\025)536 1529 y Fn(=)23 b Fj(U)690 1498 y Fh(\003)681 1552 y Fl(\025)728 1529 y Fj(H)797 1541 y Fl(\025)840 1529 y Fj(U)897 1541 y Fl(\025)970 1529 y Fe(acting)31 b(in)e Fj(L)1377 1498 y Fk(2)1414 1529 y Fn(\()p Fj(N)9 b Fn(\006)p Fj(;)14 b(d)p Fn(v)n(ol)1768 1541 y Fl(N)6 b Fk(\006)1879 1529 y Fn(\))p Fe(.)39 b(Then)1393 1772 y Fp(s\255lim)1394 1827 y Fl(\025)p Fh(!1)1582 1679 y Fg(\020)1631 1772 y Fj(e)1670 1737 y Fh(\000)p Fl(itL)1816 1746 y Fc(\025)1878 1772 y Fi(\000)18 b Fj(e)2000 1737 y Fh(\000)p Fl(it)p Fk(\()p Fl(H)2180 1745 y Fc(B)2229 1737 y Fk(+)p Fl(\025)2319 1712 y Fb(2)2352 1737 y Fl(H)2406 1745 y Fc(O)2455 1737 y Fk(\))2486 1679 y Fg(\021)2558 1772 y Fn(=)23 b(0)436 2015 y Fp(Just)33 b(as)i(in)e(the)g(classical)i(case,)i(this)c(theor)o (em)f(pr)o(ovides)i(a)g(satisfactory)h(description)f(of)f(the)g(motion) g(if)270 2164 y Fn([)p Fj(H)362 2176 y Fl(B)419 2164 y Fj(;)14 b(H)525 2176 y Fl(O)582 2164 y Fn(])36 b(=)g(0)p Fp(,)30 b(so)f(that)g Fn(exp\()p Fi(\000)p Fj(it)p Fn(\()p Fj(H)1489 2176 y Fl(B)1569 2164 y Fn(+)24 b Fj(\025)1706 2134 y Fk(2)1743 2164 y Fj(H)1812 2176 y Fl(O)1869 2164 y Fn(\)\))37 b(=)e(exp\()p Fi(\000)p Fj(itH)2422 2176 y Fl(B)2479 2164 y Fn(\))14 b(exp\()p Fi(\000)p Fj(it\025)2856 2134 y Fk(2)2893 2164 y Fj(H)2962 2176 y Fl(O)3018 2164 y Fn(\))p Fp(.)56 b(As)30 b(befor)o(e,)g(this)e(will)270 2313 y(happen,)20 b(for)g(example,)g(if)g Fn(\006)f Fp(has)i (co\255dimension)f(one,)f(or)h(if)g(all)g(the)f(fr)o(equencies)g Fj(!)2843 2325 y Fl(\013)2910 2313 y Fp(ar)o(e)g(equal.)436 2463 y(If)e Fn(\006)g Fp(has)h(co\255dimension)f(one,)g(then)f(the)g (normal)i(bundle)d(is)j(trivial.)27 b(\(W)-7 b(e)17 b(ar)o(e)f (assuming)i(that)e Fn(\006)h Fp(is)h(compact.\))270 2612 y(Then)29 b(we)f(have)g Fj(L)862 2582 y Fk(2)899 2612 y Fn(\()p Fj(N)9 b Fn(\006)p Fj(;)14 b(d)p Fn(v)n(ol)1253 2624 y Fl(N)6 b Fk(\006)1364 2612 y Fn(\))36 b(=)g Fj(L)1590 2582 y Fk(2)1627 2612 y Fn(\(\006)p Fj(;)14 b(d)p Fn(v)n(ol)1905 2624 y Fk(\006)1957 2612 y Fn(\))24 b Fi(\012)f Fj(L)2158 2582 y Fk(2)2195 2612 y Fn(\()p Fm(R)p Fj(;)14 b(dy)s Fn(\))35 b Fp(and)29 b Fj(H)2703 2624 y Fl(B)2797 2612 y Fn(=)35 b Fj(h)2945 2624 y Fl(B)3026 2612 y Fi(\012)23 b Fj(I)36 b Fp(for)29 b(a)h(Schr)3545 2609 y(\250)3537 2612 y(odinger)270 2762 y(operator)h Fj(h)647 2774 y Fl(B)734 2762 y Fp(acting)e(in)h Fj(L)1132 2732 y Fk(2)1169 2762 y Fn(\(\006)p Fj(;)14 b(d)p Fn(v)n(ol)1447 2774 y Fk(\006)1499 2762 y Fn(\))p Fp(.)59 b(Since)30 b Fj(H)1892 2774 y Fl(O)1987 2762 y Fn(=)37 b Fj(I)32 b Fi(\012)24 b Fj(h)2294 2774 y Fl(O)2380 2762 y Fp(we)29 b(have)h(that)g Fn(exp)o(\()p Fi(\000)p Fj(it)p Fn(\()p Fj(H)3261 2774 y Fl(B)3343 2762 y Fn(+)24 b Fj(\025)3480 2732 y Fk(2)3517 2762 y Fj(H)3586 2774 y Fl(O)3643 2762 y Fn(\)\))38 b(=)270 2911 y(exp\()p Fi(\000)p Fj(ith)601 2923 y Fl(B)657 2911 y Fn(\))16 b Fi(\012)e Fn(exp\()p Fi(\000)p Fj(it\025)1115 2881 y Fk(2)1152 2911 y Fj(h)1200 2923 y Fl(O)1257 2911 y Fn(\))p Fp(.)28 b(This)19 b(can)f(be)g(interpr)o(eted)f(as)j(a)e (motion)g(in)g Fj(L)2714 2881 y Fk(2)2751 2911 y Fn(\(\006)p Fj(;)c(d)p Fn(v)n(ol)3029 2923 y Fk(\006)3081 2911 y Fn(\))19 b Fp(with)e(superimposed)270 3061 y(normal)j(oscillations.)436 3210 y(In)c(the)f(case)i(wher)o(e)e(the)h(fr)o(equencies)f Fj(!)1637 3222 y Fl(\013)1700 3210 y Fp(ar)o(e)h(all)h(equal,)f(the)g (normal)g(bundle)f(may)h(be)g(non\255trivial,)g(and)h(ther)o(e)270 3360 y(is)j(not)f(such)h(a)g(simple)f(tensor)h(pr)o(oduct)f (decomposition)h(of)g Fj(L)2191 3329 y Fk(2)2227 3360 y Fn(\()p Fj(N)9 b Fn(\006)p Fj(;)14 b(d)p Fn(v)n(ol)2582 3372 y Fl(N)6 b Fk(\006)2692 3360 y Fn(\))p Fp(.)28 b(However)-6 b(,)19 b(ther)o(e)f(is)i(a)g(geometric)270 3509 y(interpr)o(etation,)c (which)f(we)g(now)h(explain)g(brie\003y)-9 b(.)27 b(Consider)16 b(the)f(normal)h(frame)g(bundle)f Fj(F)d Fn(\006)k Fp(for)g Fn(\006)p Fp(.)27 b(Concr)o(etely)-9 b(,)270 3658 y(this)20 b(may)f(be)g(thought)g(of)h(as)h(a)f(subset)f(of)h Fm(R)1614 3628 y Fl(n)p Fk(+)p Fl(m)1793 3658 y Fi(\002)f Fm(R)1931 3628 y Fk(\()p Fl(n)p Fk(+)p Fl(m)p Fk(\))p Fl(m)2222 3658 y Fp(given)g(by)555 3901 y Fj(F)12 b Fn(\006)23 b(=)f Fi(f)p Fj(\033)n(;)14 b(n)964 3913 y Fk(1)1002 3901 y Fj(;)g(:)g(:)g(:)f(;)h(n)1236 3913 y Fl(m)1322 3901 y Fn(:)23 b Fj(\033)k Fi(2)c Fn(\006)79 b Fp(and)g Fj(n)1921 3913 y Fk(1)1958 3901 y Fj(;)14 b(:)g(:)g(:)g(;)g(n)2193 3913 y Fl(m)2335 3901 y Fp(is)20 b(an)g(orthonormal)g(basis)g(for)g Fj(N)3356 3913 y Fl(\033)3401 3901 y Fn(\006)p Fi(g)p Fj(:)270 4144 y Fp(The)32 b(bundle)f Fj(F)12 b Fn(\006)32 b Fp(is)h(a)g(principal)g(bundle)e(with)h(str)o(uctur)o(e)f(gr)o(oup)h (the)f(orthogonal)i(gr)o(oup)e Fj(O)r Fn(\()p Fj(m)p Fn(\))p Fp(.)67 b(Given)32 b(a)270 4294 y(r)o(epr)o(esentation)27 b(of)h Fj(O)r Fn(\()p Fj(m)p Fn(\))h Fp(on)e(a)i(vector)e(space)h Fj(V)47 b Fp(ther)o(e)26 b(is)j(an)f(associated)h(vector)e(bundle)f (with)h(\002br)o(e)g Fj(V)19 b Fp(.)52 b(For)270 4443 y(example,)16 b Fj(N)9 b Fn(\006)16 b Fp(is)h(the)e(vector)h(bundle)f (associated)i(to)f(the)g(identity)f(r)o(epr)o(esentation)g(of)i Fj(O)r Fn(\()p Fj(m)p Fn(\))g Fp(by)f Fj(m)10 b Fi(\002)g Fj(m)15 b Fp(matrices.)270 4593 y(It)k(turns)h(out)g(that)g Fj(L)897 4563 y Fk(2)934 4593 y Fn(\()p Fj(N)9 b Fn(\006)p Fj(;)14 b(d)p Fn(v)n(ol)1288 4605 y Fl(N)6 b Fk(\006)1398 4593 y Fn(\))21 b Fp(is)f(the)g(vector)f(bundle)h(with)f(\002br)o(e)g Fj(L)2577 4563 y Fk(2)2614 4593 y Fn(\()p Fm(R)2700 4563 y Fl(m)2769 4593 y Fj(;)14 b(d)2849 4563 y Fl(m)2912 4593 y Fj(y)s Fn(\))21 b Fp(associated)g(with)f(the)f(left)270 4742 y(r)o(egular)d(r)o(epr)o(esentation)g(of)h Fj(O)r Fn(\()p Fj(m)p Fn(\))p Fp(.)28 b(W)l(ith)16 b(this)g(interpr)o (etation,)g(the)g(operator)i Fj(D)2753 4754 y Fl(x)2805 4742 y Fn(+)11 b Fj(F)28 b Fp(is)17 b(a)g(covariant)g(derivative.)436 4892 y(For)24 b(some)h(initial)f(conditions)g(\(i.e.,)h(the)e(vector)h (on)g(which)g Fn(exp\()p Fi(\000)p Fj(itL)2644 4904 y Fl(\025)2686 4892 y Fn(\))h Fp(acts,\))h(the)d(limiting)g(motion)h(may) 270 5041 y(again)30 b(be)f(thought)g(of)h(as)g(taking)f(place)i(in)e Fj(L)1730 5011 y Fk(2)1767 5041 y Fn(\(\006)p Fj(;)14 b(d)p Fn(v)n(ol)2045 5053 y Fk(\006)2097 5041 y Fn(\))30 b Fp(with)f(superimposed)h(oscillations.)59 b(For)30 b(example,)270 5191 y(consider)19 b(the)f(subspace)h(of)g(functions)f (in)g Fj(L)1626 5160 y Fk(2)1663 5191 y Fn(\()p Fj(N)9 b Fn(\006)p Fj(;)14 b(d)p Fn(v)n(ol)2017 5203 y Fl(N)6 b Fk(\006)2127 5191 y Fn(\))19 b Fp(that)f(ar)o(e)h(radially)g (symmetric)f(in)g(the)g(\002br)o(e)f(variable)270 5340 y Fj(n)p Fp(.)28 b(This)20 b(subspace)g(does)g(have)f(a)g(tensor)h(pr)o (oduct)f(decomposition)g Fj(L)2429 5310 y Fk(2)2466 5340 y Fn(\(\006)p Fj(;)14 b(d)p Fn(v)n(ol)2744 5352 y Fk(\006)2796 5340 y Fn(\))k Fi(\012)f Fj(L)2985 5310 y Fk(2)2985 5364 y(radial)3155 5340 y Fn(\()p Fm(R)3241 5310 y Fl(m)3310 5340 y Fj(;)d(d)3390 5310 y Fl(m)3453 5340 y Fj(y)s Fn(\))p Fp(.)28 b(It)19 b(is)h(an)2001 5539 y(10)p eop %%Page: 11 11 11 10 bop 270 83 a Fp(invariant)17 b(subspace)i(for)f Fj(H)1128 95 y Fl(B)1185 83 y Fp(.)27 b(Furthermor)o(e,)16 b Fj(F)35 b Fn(=)23 b(0)17 b Fp(on)g(this)g(subspace,)i(so)f(the)f(r)o (estriction)f(of)i Fj(H)3302 95 y Fl(B)3377 83 y Fp(has)g(the)f(form) 270 232 y Fj(h)318 244 y Fl(B)396 232 y Fi(\012)j Fj(I)7 b Fp(.)39 b(Thus,)25 b(if)e Fj( )929 244 y Fk(0)990 232 y Fp(is)h(a)g(radial)h(function)d(in)i Fj(n)p Fp(,)g(then)e Fn(exp\()p Fi(\000)p Fj(itL)2381 244 y Fl(\025)2424 232 y Fn(\))p Fj( )2510 244 y Fk(0)2576 232 y Fn(=)28 b(exp\()p Fi(\000)p Fj(ith)3000 244 y Fl(B)3056 232 y Fn(\))21 b Fi(\012)f Fn(exp\()p Fi(\000)p Fj(it\025)3525 202 y Fk(2)3562 232 y Fj(h)3610 244 y Fl(O)3667 232 y Fn(\))p Fj( )3753 244 y Fk(0)3790 232 y Fp(.)270 382 y(As)h(above,)d(we)i (interpr)o(et)f(this)g(as)i(motion)e(in)g Fj(L)1735 352 y Fk(2)1772 382 y Fn(\(\006)p Fj(;)14 b(d)p Fn(v)n(ol)2051 394 y Fk(\006)2102 382 y Fn(\))20 b Fp(with)f(superimposed)i(normal)f (oscillations.)436 531 y(On)26 b(the)g(other)g(hand,)i(if)f Fj(N)9 b Fn(\006)26 b Fp(is)h(non\255trivial,)g(it)f(may)g(happen)i (that)e(the)g(limiting)f(motion)h(takes)g(place)h(on)270 681 y(a)e(space)g(of)g(sections)g(of)g(a)g(vector)e(bundle)h(over)g Fn(\006)p Fp(.)42 b(Instead)25 b(of)f(giving)g(mor)o(e)g(details)h (about)e(the)h(general)h(case,)270 830 y(we)d(of)o(fer)g(the)f (following)h(illustrative)f(example.)35 b(Instead)22 b(of)g(a)h(normal)f(bundle,)f(consider)i(the)e(M)3349 827 y(\250)3341 830 y(obius)h(band)g Fi(B)270 980 y Fp(de\002ned)f(by)g Fm(R)k Fi(\002)20 b Fm(R)14 b Fj(=)31 b Fi(\030)p Fp(,)21 b(wher)o(e)f Fn(\()p Fj(x;)14 b(y)s Fn(\))27 b Fi(\030)e Fn(\()p Fj(x)20 b Fn(+)f(1)p Fj(;)14 b Fi(\000)p Fj(y)s Fn(\))p Fp(.)33 b(This)22 b(an)g Fj(O)r Fn(\(1\))g Fp(bundle)e(over)h Fj(S)3053 950 y Fk(1)3112 980 y Fp(with)g(\002br)o(e)f Fm(R)p Fp(.)39 b(An)22 b Fj(L)3773 950 y Fk(2)270 1129 y Fp(function)i Fj( )29 b Fp(on)c Fi(B)i Fp(can)e(be)g(thought)f(of)h (as)h(a)f(function)g(on)g Fm(R)j Fi(\002)21 b Fm(R)31 b Fp(satisfying)26 b Fj( )s Fn(\()p Fj(x)c Fn(+)f(1)p Fj(;)14 b Fi(\000)p Fj(y)s Fn(\))30 b(=)h Fj( )s Fn(\()p Fj(x;)14 b(y)s Fn(\))p Fp(.)45 b(If)24 b(we)270 1279 y(decompose)c Fj( )s Fn(\()p Fj(x;)14 b(y)s Fn(\))p Fp(,)20 b(for)g(\002xed)f Fj(x)p Fp(,)h(into)g(odd)g(and)g(even)f(functions)g (of)h Fj(y)1440 1524 y( )s Fn(\()p Fj(x;)14 b(y)s Fn(\))23 b(=)g Fj( )1854 1536 y Fk(ev)n(en)1987 1524 y Fn(\()p Fj(x;)14 b(y)s Fn(\))19 b(+)f Fj( )2335 1536 y Fk(o)r(dd)2448 1524 y Fn(\()p Fj(x;)c(y)s Fn(\))270 1769 y Fp(then)26 b Fj( )506 1781 y Fk(ev)n(en)639 1769 y Fn(\()p Fj(x)d Fn(+)f(1)p Fj(;)14 b(y)s Fn(\))32 b(=)g Fj( )1166 1781 y Fk(ev)n(en)1299 1769 y Fn(\()p Fj(x;)14 b(y)s Fn(\))27 b Fp(and)g Fj( )1732 1781 y Fk(o)r(dd)1844 1769 y Fn(\()p Fj(x)c Fn(+)f(1)p Fj(;)14 b(y)s Fn(\))32 b(=)h Fi(\000)p Fj( )2437 1781 y Fk(o)r(dd)2549 1769 y Fn(\()p Fj(x;)14 b(y)s Fn(\))p Fp(.)48 b(\(Notice)26 b(that)g(these)g(ar)o(e)g (eigen\255)270 1919 y(functions)g(for)h(the)f(left)g(r)o(egular)g(r)o (epr)o(esentation)g(of)h Fj(O)r Fn(\(1\))g Fp(on)g Fj(L)2299 1889 y Fk(2)2336 1919 y Fn(\()p Fm(R)p Fn(\))p Fp(.\))55 b(Thus)27 b Fj( )2811 1931 y Fk(ev)n(en)2970 1919 y Fp(can)g(be)f (thought)f(of)i(as)h(an)270 2068 y Fj(L)327 2038 y Fk(2)364 2068 y Fn(\()p Fm(R)p Fj(;)14 b(dy)s Fn(\))24 b Fp(valued)18 b(function)g(on)g Fj(S)1363 2038 y Fk(1)1400 2068 y Fp(,)g(while)g Fj( )1706 2080 y Fk(o)r(dd)1837 2068 y Fp(can)g(be)g(thought)f(of)i(as) g(an)g Fj(L)2711 2038 y Fk(2)2747 2068 y Fn(\()p Fm(R)q Fj(;)14 b(dy)s Fn(\))24 b Fp(valued)18 b(section)g(of)h(a)f(line)270 2218 y(bundle)h(over)g Fj(S)768 2187 y Fk(1)825 2218 y Fp(\(which)g(happens)i(to)f(be)f Fi(B)j Fp(itself\).)28 b(In)19 b(this)h(way)g(we)f(obtain)g(the)h(decomposition)926 2463 y Fj(L)983 2429 y Fk(2)1019 2463 y Fn(\()p Fi(B)s Fn(\))j(=)g Fj(L)1309 2429 y Fk(2)1345 2463 y Fn(\()p Fj(S)1433 2429 y Fk(1)1471 2463 y Fj(;)14 b(dx)p 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Fl(x)1287 3007 y Fn(+)c Fj(V)h Fn(\()p Fj(x)p Fn(\))i Fp(acting)e(in)h Fj(L)1947 2977 y Fk(2)1984 3007 y Fn(\()p Fj(S)2072 2977 y Fk(1)2109 3007 y Fj(;)14 b(dx)p Fn(\))21 b Fp(and)f Fj(h)2490 3019 y Fh(\000)2569 3007 y Fn(=)j Fi(\000)p Fj(D)2793 2977 y Fk(2)2791 3028 y Fl(x)2850 3007 y Fn(+)18 b Fj(V)h Fn(\()p Fj(x)p Fn(\))i Fp(acting)f(in)f Fn(\000\()p Fj(S)3593 2977 y Fk(1)3630 3007 y Fj(;)14 b(dx)p Fn(\))p Fp(.)270 3157 y(Let)19 b Fj(h)449 3169 y Fk(0)510 3157 y Fn(=)j Fi(\000)p Fj(D)733 3126 y Fk(2)731 3177 y Fl(y)790 3157 y Fp(acting)e(in)f Fj(L)1168 3126 y Fk(2)1205 3157 y Fn(\()p Fm(R)p Fj(;)14 b(dy)s Fn(\))p Fp(,)26 b(with)19 b Fj(L)1730 3126 y Fk(2)1730 3177 y(ev)n(en)1862 3157 y Fn(\()p Fm(R)q Fj(;)14 b(dy)s Fn(\))25 b Fp(and)c Fj(L)2341 3126 y Fk(2)2341 3180 y(o)r(dd)2452 3157 y Fn(\()p Fm(R)q Fj(;)14 b(dy)s Fn(\))25 b Fp(as)c(invariant)f (subspaces.)29 b(Then)972 3402 y Fj(e)1011 3368 y Fh(\000)p Fl(it)p Fk(\()p Fl(H)1191 3376 y Fc(B)1241 3368 y Fk(+)p Fl(\025)1331 3343 y Fb(2)1363 3368 y Fl(H)1417 3376 y Fc(O)1467 3368 y Fk(\))1520 3402 y Fn(=)23 b Fj(e)1647 3368 y Fh(\000)p Fl(ith)p Fk(+)1837 3376 y Fb(+)1906 3402 y Fi(\012)18 b Fj(e)2028 3368 y Fh(\000)p Fl(it\025)2167 3343 y Fb(2)2200 3368 y Fl(h)2239 3376 y Fc(O)2324 3402 y Fi(\010)32 b Fj(e)2460 3368 y Fh(\000)p Fl(ith)p Fk(+)2650 3376 y Fa(\000)2722 3402 y Fi(\012)18 b Fj(e)2844 3368 y Fh(\000)p Fl(it\025)2983 3343 y Fb(2)3016 3368 y Fl(h)3055 3376 y Fc(O)270 3647 y Fp(So)25 b(if)g(the)f(initial)h(condition)f (happens)i(to)e(lie)h(in)f Fn(\000)e Fi(\012)f Fj(L)2006 3617 y Fk(2)2006 3671 y(o)r(dd)2117 3647 y Fp(,)26 b(then)e(we)g(would) g(think)g(of)h(the)g(limiting)e(motion)h(as)270 3797 y(taking)19 b(place)i(in)e Fn(\000)p Fp(,)h(with)f(superimposed)h (oscillations)h(in)e Fj(L)2149 3766 y Fk(2)2149 3820 y(o)r(dd)2261 3797 y Fp(.)436 3946 y(Finally)-9 b(,)21 b(we)g(consider)g(the)f(case)i(wher)o(e)e(the)g(eigenvalues)g Fj(!)2305 3958 y Fk(1)2342 3946 y Fj(;)14 b(:)g(:)g(:)f(;)h(!)2578 3958 y Fl(m)2662 3946 y Fp(of)21 b Fj(A)p Fn(\()p Fj(\033)s Fn(\))h Fp(ar)o(e)f(all)g(dif)o(fer)o(ent.)31 b(\(For)22 b(the)270 4095 y(case)c(when)e(some)h(eigenvalues,)f(but)g(not)g(all,)i (ar)o(e)f(the)f(same,)i(see)f([FH].\))g(For)g(simplicity)f(we)h(will)f (assume)i(that)e(we)270 4245 y(can)21 b(make)f(a)h(global)f(choice)g (of)h(sign)g(for)g(the)f(eigenvectors,)f(ther)o(eby)g(obtaining)h(a)h (global)f(frame)h Fj(n)3355 4257 y Fk(1)3392 4245 y Fj(;)14 b(:)g(:)g(:)f(;)h(n)3626 4257 y Fl(m)3710 4245 y Fp(for)270 4394 y Fj(N)9 b Fn(\006)p Fp(.)28 b(Then)20 b(the)f(bundle)g Fj(N)9 b Fn(\006)19 b Fp(is)h(trivial,)g(with)f(the)g(map)h(fr)o(om)g Fn(\006)e Fi(\002)g Fm(R)2417 4364 y Fl(m)2509 4394 y Fi(!)23 b Fj(N)9 b Fn(\006)20 b Fp(sending)g Fn(\()p Fj(\033)n(;)14 b(y)s Fn(\))24 b Fi(7!)f Fn(\()p Fj(\033)n(;)3503 4332 y Fg(P)3605 4394 y Fj(y)3646 4406 y Fl(k)3687 4394 y Fj(n)3737 4406 y Fl(k)3778 4394 y Fn(\))270 4544 y Fp(pr)o(oviding)c(the)g(trivialization.)29 b(Thus)20 b Fj(L)1517 4514 y Fk(2)1554 4544 y Fn(\()p Fj(N)9 b Fn(\006)p Fj(;)14 b(d)p Fn(v)n(ol)1908 4556 y Fl(N)6 b Fk(\006)2018 4544 y Fn(\))23 b(=)g Fj(L)2218 4514 y Fk(2)2255 4544 y Fn(\(\006)p Fj(;)14 b(d)p Fn(v)n(ol)2533 4556 y Fk(\006)2585 4544 y Fn(\))19 b Fi(\012)f Fj(L)2776 4514 y Fk(2)2812 4544 y Fn(\()p Fm(R)2899 4514 y Fl(m)2968 4544 y Fj(;)c(d)3048 4514 y Fl(m)3111 4544 y Fj(y)s Fn(\))p Fp(.)436 4693 y(In)19 b(this)h(situation,)f Fn([)p Fj(H)1114 4705 y Fl(B)1171 4693 y Fj(;)14 b(H)1277 4705 y Fl(O)1334 4693 y Fn(])23 b Fi(6)p Fn(=)f(0)p Fp(,)d(so)i(to)e(determine)g(the)g (limiting)f(motion)h(we)h(must)f(perform)h(a)g(quantum)270 4843 y(version)g(of)g(averaging.)28 b(W)-7 b(e)19 b(have)g(the)g (following)h(theor)o(em.)270 5041 y Ff(Theorem)46 b(4.2)35 b Fe(L)l(et)42 b Fj(H)1074 5053 y Fl(\025)1160 5041 y Fe(b)l(e)h(the)f(Hamiltonian)i(\(1.2\),)j(c)l(onsider)l(e)l(d)d(to)e(b) l(e)h(acting)g(in)f Fj(L)3239 5011 y Fk(2)3276 5041 y Fn(\()p Fj(N)9 b Fn(\006)p Fj(;)14 b(d)p Fn(v)n(ol)o(\))p Fe(,)47 b(as)270 5191 y(explaine)l(d)31 b(ab)l(ove,)g(wher)l(e)g Fj(W)41 b Fe(has)31 b(the)f(form)g(\(2.1\))h(and)g(satis\014es)436 5340 y(\(i\))f(the)g(eigenvalues)h(of)g Fj(!)1285 5310 y Fk(2)1282 5361 y(1)1321 5340 y Fj(;)14 b(:)g(:)g(:)g(;)g(!)1561 5310 y Fk(2)1558 5361 y Fl(m)1650 5340 y Fe(of)31 b Fj(A)p Fn(\()p Fj(\033)s Fn(\))g Fe(do)g(not)e(dep)l(end)i(on)f Fj(\033)s Fe(.)2001 5539 y Fp(11)p eop %%Page: 12 12 12 11 bop 436 83 a Fe(\(ii\))31 b(If)f Fj(j)e Fi(6)p Fn(=)22 b Fj(k)33 b Fe(then)c Fj(!)1133 95 y Fl(j)1187 83 y Fi(\006)18 b Fj(!)1322 95 y Fl(k)1385 83 y Fi(6)p Fn(=)23 b(0)p Fe(,)436 232 y(\(iii\))31 b(If)f Fj(j)e Fi(6)p Fn(=)23 b Fj(k)32 b Fe(and)f Fj(l)24 b Fi(6)p Fn(=)f Fj(m)29 b Fe(then)h Fj(!)1560 244 y Fl(j)1613 232 y Fi(\006)18 b Fj(!)1748 244 y Fl(k)1807 232 y Fi(\006)g Fj(!)1942 244 y Fl(l)1986 232 y Fi(\006)g Fj(!)2121 244 y Fl(m)2206 232 y Fi(6)p Fn(=)23 b(0)270 382 y Fe(L)l(et)29 b Fj(L)470 394 y Fl(\025)536 382 y Fn(=)23 b Fj(U)690 352 y Fh(\003)681 405 y Fl(\025)728 382 y Fj(H)797 394 y Fl(\025)840 382 y Fj(U)897 394 y Fl(\025)970 382 y Fe(acting)31 b(in)e Fj(L)1377 352 y Fk(2)1414 382 y Fn(\()p Fj(N)9 b Fn(\006)p Fj(;)14 b(d)p Fn(v)n(ol)1768 394 y Fl(N)6 b Fk(\006)1879 382 y Fn(\))23 b(=)g Fj(L)2079 352 y Fk(2)2115 382 y Fn(\(\006)p Fj(;)14 b(d)p Fn(v)n(ol)2394 394 y Fk(\006)2445 382 y Fn(\))19 b Fi(\012)f Fj(L)2636 352 y Fk(2)2673 382 y Fn(\()p Fm(R)2759 352 y Fl(m)2828 382 y Fj(;)c(d)2908 352 y Fl(m)2971 382 y Fj(y)s Fn(\))p Fe(.)39 b(L)l(et)1458 606 y Fj(h)1506 618 y Fl(B)1586 606 y Fn(=)22 b Fi(\000)p Fn(\001)1807 618 y Fk(\006)1877 606 y Fn(+)c Fj(K)24 b Fn(+)18 b Fj(V)h Fn(\()p Fj(\033)s Fn(\))g(+)g Fj(V)2470 618 y Fk(1)2507 606 y Fn(\()p Fj(\033)s Fn(\))270 830 y Fe(acting)40 b(in)f Fj(L)696 800 y Fk(2)733 830 y Fn(\(\006)p Fj(;)14 b(d)p Fn(v)n(ol)1011 842 y Fk(\006)1063 830 y Fn(\))p Fe(,)42 b(wher)l(e)e Fj(V)1454 842 y Fk(1)1531 830 y Fe(is)f(de\014ne)l(d)h(by)g(\(3.2\))g(ab)l(ove.) 68 b(L)l(et)39 b Fj(h)2742 842 y Fl(O)2837 830 y Fe(b)l(e)h(the)f (harmonic)i(oscil)t(lator)270 979 y(Hamiltonian)31 b(acting)f(in)g Fj(L)1157 949 y Fk(2)1193 979 y Fn(\()p Fm(R)1280 949 y Fl(m)1349 979 y Fj(;)14 b(d)1429 949 y Fl(m)1492 979 y Fj(y)s Fn(\))p Fe(.)39 b(Then)1337 1203 y Fp(s\255lim)1339 1259 y Fl(\025)p Fh(!1)1526 1111 y Fg(\020)1576 1203 y Fj(e)1615 1169 y Fh(\000)p Fl(itL)1761 1178 y Fc(\025)1822 1203 y Fi(\000)18 b Fj(e)1944 1169 y Fh(\000)p Fl(ith)2083 1177 y Fc(B)2155 1203 y Fi(\012)g Fj(e)2277 1169 y Fh(\000)p Fl(it\025)2416 1144 y Fb(2)2449 1169 y Fl(h)2488 1177 y Fc(O)2541 1111 y Fg(\021)2614 1203 y Fn(=)k(0)270 1604 y Fq(References)270 1903 y Fp([AD])e(L.)f(Anderson)h(and)f(B.K.)g (Driver)-6 b(,)19 b Fe(Finite)30 b(dimensional)h(appr)l(oximations)g (to)e(Wiener)g(me)l(asur)l(e)g(and)g(p)l(ath)270 2052 y(inte)l(gr)l(al)h(formulas)h(on)e(manifolds)p Fp(,)22 b(USCD)e(pr)o(eprint,)g(July)f(1998.)270 2202 y([A1])i(V)-10 b(.I.)19 b(Arnold,)h Fe(Mathematic)l(al)31 b(Metho)l(ds)g(of)g(Classic) l(al)h(Me)l(chanics,)21 b Fp(Springer)o(\255V)-9 b(erlag,)21 b(1978.)270 2351 y([A2])36 b(V)-10 b(.I.)34 b(Arnold,)39 b(V)-10 b(.V)g(.)34 b(Kozlov)i(and)f(A.I.)g(Neishtadt,)j Fe(Mathematic)l(al)44 b(Asp)l(e)l(cts)e(of)h(Classic)l(al)h(Me)l (chanics,)270 2501 y Fp(Encyclop\346dia)20 b(of)h(Mathemaitcal)e (sciences,)h(V)-9 b(olume)18 b(3,)i(Springer)o(\255V)-9 b(erlag,)21 b(1988.)270 2650 y([BS])28 b(Bornemann)e(and)h(Sch)1165 2647 y(\250)1154 2650 y(utte,)g Fe(Homo)l(genization)37 b(of)f(Hamiltonian)g(systems)f(with)h(a)g(str)l(ong)f(c)l(onstr)l (aining)270 2800 y(p)l(otential,)21 b Fp(Physica)f(D,)f Ff(102)g Fp(\(1997\))j(57\25577)270 2949 y([dC1])f(R.C.T)-6 b(.)20 b(da)g(Costa,)h Fe(Quantum)28 b(me)l(chanics)k(of)f(a)f(c)l (onstr)l(aine)l(d)g(p)l(article,)22 b Fp(Physical)f(Review)e(A,)h(V)-9 b(olume)19 b(23)270 3098 y(Number)f(4)i(\(1981\))i(1982\2261987.)270 3248 y([dC2])g(R.C.T)-6 b(.)19 b(da)i(Costa,)g Fe(Constr)l(aints)31 b(in)f(quantum)f(me)l(chanics,)23 b Fp(Physical)d(Review)f(A,)i(V)-9 b(olume)19 b(25)j(Number)d(6)270 3397 y(\(1982\))j(2893\2262900.)270 3547 y([FH])g(R.)g(Fr)o(oese)g(and)g(I.)g(Herbst,)g Fe(R)l(e)l(alizing) 32 b(holonomic)i(c)l(onstr)l(aints)d(in)h(classic)l(al)i(and)e(quantum) e(me)l(chanics,)270 3696 y Fp(in)20 b(pr)o(eparation.)270 3846 y([GS])29 b(V)-10 b(.)28 b(Guillemn)f(and)i(S.)g(Sternber)o(g,)g Fe(Symple)l(ctic)37 b(te)l(chniques)g(in)g(physics,)32 b Fp(Cambridge)c(University)g(Pr)o(ess,)270 3995 y(1984.)270 4144 y([HS1])21 b(B.)g(Helf)o(fer)f(and)h(J.)f(Sj)1132 4141 y(\250)1124 4144 y(ostrand,)j Fe(Puits)30 b(multiples)h(en)f(m)n (\023)-40 b(echanique)31 b(semi-classique)h(V,)3243 4123 y(\023)3230 4144 y(Etude)f(des)g(mini-)270 4294 y(puits,)22 b Fp(Curr)o(ent)e(topics)i(in)f(partial)i(dif)o(fer)o(ential)e (equations,)g(Kinokuniya)g(Company)h(Ltd.,)f(133\226186)j(\(V)-9 b(olume)21 b(in)270 4443 y(honour)e(of)h(S.)g(Mizohata\))270 4593 y([HS2])29 b(B.)f(Helf)o(fer)f(and)i(J.)f(Sj)1170 4590 y(\250)1162 4593 y(ostrand,)j Fe(Puits)36 b(multiples)h(en)f(m)n (\023)-40 b(echanique)38 b(semi-classique)f(V,)g(Cas)g(des)g(puits)270 4742 y(sous-vari)n(\023)-40 b(et)n(\023)g(es,)21 b Fp(Annales)f(de)g (l'Institut)e(Henri)h(Poincar)2003 4739 y(\264)1997 4742 y(e,)h(Physique)f(Th)2518 4739 y(\264)2512 4742 y(eorique,)g Ff(46)g Fp(no.)g(4)i(\(1987\))g(353\226372)270 4892 y([RU])c(H.)f (Rubin)g(and)h(P)-10 b(.)15 b(Ungar)-6 b(,)18 b Fe(Motion)28 b(under)e(a)h(str)l(ong)f(c)l(onstr)l(aining)g(for)l(c)l(e,)19 b Fp(Communications)d(on)g(pur)o(e)g(and)270 5041 y(applied)21 b(mathematics,)e Ff(10)g Fp(\(1957\))j(28\22642)270 5191 y([T)-7 b(a])18 b(F)-7 b(.)17 b(T)-7 b(akens,)18 b Fe(Motion)28 b(under)f(the)g(in\015uenc)l(e)g(of)h(a)f(str)l(ong)g(c)l(onstr)l (aining)g(for)l(c)l(e,)19 b Fp(in)e(Global)g(theory)g(of)h(dynam\255) 270 5340 y(ical)i(systems,)g(Springer)g(lectur)o(e)e(notes)i(in)g (mathmatics,)f Ff(819)g Fp(\(1980\))j(425\226445)2001 5539 y(12)p eop %%Page: 13 13 13 12 bop 270 83 a Fp([T)-7 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