% ps file: A. Soffer and M.I. 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b(indicate)f(brie\015y)g(in)g(section)h (4)f(the)h(mo)s(di\014cations)e(required)i(in)e(the)i(analysis.)72 3549 y(View)m(ed)i(in)d(terms)h(of)g(the)h(co)s(ordinates)e(of)h(the)h (unp)s(erturb)s(ed)g(dynamical)e(system,)i(the)g(p)s(erturbation)-74 3698 y Fn(W)14 b Fo(\()p Fn(t)p Fo(\))38 b(has)h(the)g(e\013ect)g(of)f (coupling)f(the)i(b)s(ound)g(state)g(to)f(the)h(con)m(tin)m(uum)f(mo)s (des.)61 b(The)40 b(mec)m(hanism)d(of)-74 3848 y(instabilit)m(y)23 b(is)i(a)g(slo)m(w)g(transfer)g(of)g(energy)h(from)e(the)i(b)s(ound)f (state)h(part)f(of)f(the)i(solution)e(to)g(the)i(con)m(tin)m(uum)-74 3997 y(mo)s(des)g(and)g(an)h(asso)s(ciated)f(deca)m(y)h(due)g(to)f (radiation)e(of)i(energy)h(to)f(in\014nit)m(y)-8 b(.)41 b(W)-8 b(e)27 b(dev)m(elop)f(a)g(theory)h(whic)m(h)-74 4147 y(yields)37 b(a)g(detailed)f(picture)i(of)f(the)g(in)m(termediate) f(and)i(long)e(time)g(b)s(eha)m(vior)h(of)g(solutions)f(to)h(the)h (initial)-74 4296 y(v)-5 b(alue)31 b(problem.)42 b(Our)31 b(main)f(results)i(are)g(presen)m(ted)h(in)e(Theorems)h(2.1)g(and)f (2.2.)43 b(A)31 b(consequence)k(is)c(that)-74 4445 y(the)37 b(solution)f(of)g(the)h(initial)d(v)-5 b(alue)36 b(problem)g(for)g(the) h(p)s(erturb)s(ed)h(dynamical)d(system)j(with)e(initial)d(data)-74 4595 y(giv)m(en)25 b(b)m(y)h(the)f(time-p)s(erio)s(dic)c(b)s(ound)k (state)g(of)g(the)g(unp)s(erturb)s(ed)h(problem,)f(is)f(c)m (haracterized)i(b)m(y)g(transien)m(t)-74 4744 y(exp)s(onen)m(tial)33 b(deca)m(y)h Fl(\030)29 b Fn(e)872 4708 y Fi(\000)p Fm(\000)p Fi(j)p Fk(t)p Fi(j)1073 4744 y Fo(where)34 b(0)29 b Fn(<)f Fo(\000)g(=)h Fl(O)s Fo(\()p Fl(jjj)p Fn(W)14 b Fl(jjj)2125 4708 y Fm(2)2162 4744 y Fo(\))p Fn(;)33 b Fo(for)f Fl(jjj)p Fn(W)14 b Fl(jjj)31 b Fo(small,)g(and)i(b)m(y)h(algebraic)d(deca)m(y) -74 4894 y(as)k Fl(j)p Fn(t)p Fl(j)30 b(!)h(1)p Fo(.)48 b(W)-8 b(e)35 b(deriv)m(e)g(explicit)e(form)m(ulae)g(for)h(the)h Fj(lifetime)p Fo(,)f Fn(\034)42 b Fl(\021)32 b Fo(\000)2636 4858 y Fi(\000)p Fm(1)2764 4894 y Fo(\(eqn.)50 b(\(2.9\)\))34 b(and)g(the)h Fj(sp)-5 b(e)g(ctr)g(al)-74 5043 y(line)34 b(shift)f Fo(\(eqn.)44 b(\(4.21\)\).)1926 5306 y(2)p eop %%Page: 3 3 3 2 bop 72 59 a Fo(The)30 b(approac)m(h)f(w)m(e)h(tak)m(e)g(is)e (closely)h(related)f(to)g(our)h(w)m(ork)h(on)e(quan)m(tum)i(resonances) g([37],)g([38)o(])f(and)g(on)-74 208 y(the)36 b(radiation)e(damping)g (of)i(lo)s(calized)d(states)k(for)e(nonlinear)g(w)m(a)m(v)m(e)i (equations)f([39].)53 b(W)-8 b(e)36 b(no)m(w)h(illustrate)-74 358 y(the)c(main)e(idea)h(in)g(the)h(curren)m(t)g(con)m(text.)45 b(Consider)33 b(a)g(linear)e(Sc)m(hr\177)-49 b(odinger)32 b(equation:)734 607 y Fn(i@)818 622 y Fk(t)848 607 y Fn(\036)p Fo(\()p Fn(t;)17 b(x)p Fo(\))60 b(=)g(\()p Fl(\000)p Fo(\001)23 b(+)f Fn(V)g Fo(\()p Fn(x)p Fo(\)\))16 b Fn(\036)p Fo(\()p Fn(t;)h(x)p Fo(\))55 b(+)g(cos\()p Fn(\026t)p Fo(\))32 b Fn(\014)6 b Fo(\()p Fn(x)p Fo(\))p Fn(\036)p Fo(\()p Fn(t;)17 b(x)p Fo(\))p Fn(:)607 b Fo(\(1.3\))-74 856 y(W)-8 b(e)39 b(assume)g(that)g Fl(\000)p Fo(\001)27 b(+)f Fn(V)21 b Fo(\()p Fn(x)p Fo(\))39 b(has)g(exactly)g(one)g(eigen)m (v)-5 b(alue)38 b(at)g Fn(\025)2557 871 y Fm(0)2634 856 y Fn(<)g Fo(0,)i(with)e(corresp)s(onding)h(eigen-)-74 1005 y(function)46 b Fn( )385 1020 y Fm(0)425 1005 y Fo(\()p Fn(x)p Fo(\).)84 b(Therefore,)51 b(the)c(unp)s(erturb)s(ed)g (\()p Fn(\014)56 b Fl(\021)51 b Fo(0\))46 b(problem)f(has)i(a)f (spatially)e(lo)s(calized)g(and)-74 1155 y(time-p)s(erio)s(dic)39 b(solution,)k Fn( )1003 1170 y Fm(0)1043 1155 y Fo(\()p Fn(x)p Fo(\))p Fn(e)1219 1118 y Fi(\000)p Fk(i\025)1339 1127 y Fh(0)1374 1118 y Fk(t)1404 1155 y Fo(.)72 b(W)-8 b(e)42 b(assume)h(that)f(the)g(con)m(tin)m(uous)h(sp)s(ectrum)f(of)g Fn(H)3560 1170 y Fm(0)3641 1155 y Fo(is)g(equal)-74 1304 y(to)f(the)h(real)e(nonnegativ)m(e)i(half-line.)67 b(The)43 b(p)s(erturbation)d Fn(W)14 b Fo(\()p Fn(x;)j(t)p Fo(\))43 b(=)f Fn(\014)6 b Fo(\()p Fn(x)p Fo(\))17 b(cos\()p Fn(\026t)p Fo(\))42 b(is)e(assumed)j(to)e(b)s(e)-74 1453 y(real-v)-5 b(alued,)34 b(to)h(deca)m(y)i(su\016cien)m(tly)f(rapidly)e(as)h Fl(j)p Fn(x)p Fl(j)d(!)g(1)p Fo(,)j(and)h(to)e(b)s(e)i(small)d(in)h(an) h(appropriate)g(norm;)-74 1603 y(see)f(section)f(3.)-74 1802 y(W)-8 b(e)39 b(no)m(w)g(ask)f(the)h(follo)m(wing)d(question:)55 b Fj(Do)-5 b(es)39 b(the)h(b)-5 b(ound)40 b(state)g(solution)g(of)g (the)g(unp)-5 b(erturb)g(e)g(d)40 b(pr)-5 b(oblem)-74 1952 y(p)g(ersist)35 b(under)f(smal)5 b(l)34 b(p)-5 b(erturb)g(ations) 35 b(of)g(this)g(typ)-5 b(e?)72 2101 y Fo(As)29 b(indicated)e(ab)s(o)m (v)m(e,)i(w)m(e)g(\014nd)g(that)e(generic)h(p)s(erturbations)g(lead)f (to)g(a)h(slo)m(w)g(radiativ)m(e)e(deca)m(y)j(of)f(solu-)-74 2250 y(tions.)41 b(T)-8 b(o)25 b(illustrate)e(the)j(mec)m(hanism)f(of)f (deca)m(y)-8 b(,)29 b(w)m(e)d(\014rst)g(observ)m(e)h(that)e(it)f(is)h (natural)f(in)h(the)h(unp)s(erturb)s(ed)-74 2400 y(case)34 b(to)e(decomp)s(ose)h(the)g(solution)e(as:)932 2649 y Fn(\036)p Fo(\()p Fn(t;)17 b(x)p Fo(\))60 b(=)g Fn(a)p Fo(\()p Fn(t)p Fo(\))p Fn( )1621 2664 y Fm(0)1716 2649 y Fo(+)55 b Fn(\036)1905 2664 y Fk(d)1945 2649 y Fo(\()p Fn(t;)17 b(x)p Fo(\))p Fn(;)82 b Fo(\()p Fn( )2365 2664 y Fm(0)2405 2649 y Fn(;)17 b(\036)2507 2664 y Fk(d)2547 2649 y Fo(\()p Fn(t)p Fo(\)\))60 b(=)g(0)p Fn(:)805 b Fo(\(1.4\))-74 2898 y(Then,)34 b(w)m(e)g(ha)m(v)m(e)f(that)1216 3147 y Fn(i@)1300 3162 y Fk(t)1330 3147 y Fn(a)p Fo(\()p Fn(t)p Fo(\))116 b(=)g Fn(\025)1857 3162 y Fm(0)1929 3147 y Fn(a)p Fo(\()p Fn(t)p Fo(\))p Fn(;)1151 3321 y(i\036)1242 3336 y Fk(d)1282 3321 y Fo(\()p Fn(t;)17 b(x)p Fo(\))116 b(=)g(\()p Fl(\000)p Fo(\001)23 b(+)f Fn(V)f Fo(\()p Fn(x)p Fo(\)\))49 b Fn(\036)2471 3336 y Fk(d)2512 3321 y Fo(\()p Fn(t;)17 b(x)p Fo(\))p Fn(;)1024 b Fo(\(1.5\))-74 3571 y(with)32 b(initial)d(conditions)580 3820 y Fn(a)p Fo(\(0\))61 b(=)f(\()p Fn( )1054 3835 y Fm(0)1093 3820 y Fn(;)17 b(\036)p Fo(\(0\)\))f Fn(;)82 b(\036)1541 3835 y Fk(d)1581 3820 y Fo(\(0\))60 b(=)g Fg(P)1979 3835 y Ff(c)2019 3820 y Fn(\036)p Fo(\(0\))f Fl(\021)i Fn(\036)p Fo(\(0\))54 b 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Fo(\()p Fn(t;)17 b Fl(\001)p Fo(\))p Fl(k)769 4532 y Fk(p)836 4517 y Fl(!)27 b Fo(0,)32 b(\()p Fn(p)c(>)f Fo(2\))33 b(or)f(for)g(some)g Fn(\033)g(>)27 b Fo(0,)33 b Fl(kh)p Fn(x)p Fl(i)2403 4481 y Fi(\000)p Fk(\033)2504 4517 y Fn(\036)2562 4532 y Fk(d)2603 4517 y Fo(\()p Fn(t;)17 b Fl(\001)p Fo(\))p Fl(k)2836 4534 y Fk(L)2884 4515 y Fh(2)2949 4517 y Fl(!)28 b Fo(0.)72 4666 y(Ho)m(w)37 b(is)e(the)h(dynamics)g(e\013ected)h(b)m(y)g(a)e(small)e(spatially)h (lo)s(calized)g(and)h(time-p)s(erio)s(dic)e(p)s(erturbation)-74 4816 y(\()p Fn(\014)41 b Fl(6)p Fo(=)35 b(0\)?)57 b(Substitution)36 b(of)g(\(1.4\))h(in)m(to)f(\(1.3\),)i(leads)f(to)f(a)h Fj(we)-5 b(akly)39 b(c)-5 b(ouple)g(d)36 b Fo(dynamical)f(system)j(for) f Fn(a)p Fo(\()p Fn(t)p Fo(\))-74 4965 y(and)d Fn(\036)175 4980 y Fk(d)215 4965 y Fo(\()p Fn(t)p Fo(\).)48 b(This)34 b(system)h(is)f(deriv)m(ed)g(and)h(studied)f(in)f(sections)i(4-6.)46 b(A)34 b(simpli\014cation)d(of)i(the)i(resulting)1926 5306 y(3)p eop %%Page: 4 4 4 3 bop -74 59 a Fo(system)34 b(whic)m(h)f(w)m(e)g(shall)f(use)h(to)f (illustrate)f(the)i(main)e(idea)h(is)g(the)h(follo)m(wing:)919 308 y Fn(i@)1003 323 y Fk(t)1034 308 y Fn(a)p Fo(\()p Fn(t)p Fo(\))116 b(=)f Fn(\025)1560 323 y Fm(0)1632 308 y Fn(a)p Fo(\()p Fn(t)p Fo(\))55 b(+)f(\()33 b Fn(\014)6 b( )2174 323 y Fm(0)2246 308 y Fn(;)49 b(\036)2380 323 y Fk(d)2420 308 y Fo(\()p Fn(t)p Fo(\))33 b(\))49 b(cos)q(\()p Fn(\026t)p Fo(\))32 b Fn(;)872 482 y(i@)956 497 y Fk(t)987 482 y Fn(\036)1045 497 y Fk(d)1085 482 y Fo(\()p Fn(t)p Fo(\))116 b(=)f Fl(\000)p Fo(\001)p Fn(\036)1719 497 y Fk(d)1760 482 y Fo(\()p Fn(t)p Fo(\))55 b(+)f Fn(a)p Fo(\()p Fn(t)p Fo(\))c(cos\()p Fn(\026t)p Fo(\))33 b Fn(\014)6 b( )2725 497 y Fm(0)2764 482 y Fn(:)982 b Fo(\(1.7\))72 731 y(The)35 b(system)g(\(1.7\))e(can)i(view)m(ed)g(as)f(an)f (in\014nite)g(dimensional)f(dynamical)g(system)j(consisting)e(of)g(t)m (w)m(o)-74 881 y(comp)s(onen)m(ts:)46 b(\(i\))32 b(a)h(\014nite)h 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y(the)c(origin)e(of)h(this)g(e\013ect.)72 3620 y(Viewing)j Fn(\014)41 b Fo(as)36 b(small,)f(w)m(e)h(set)h Fn(a)p Fo(\()p Fn(t)p Fo(\))69 b(=)f Fn(e)1680 3584 y Fi(\000)p Fk(i\025)1800 3593 y Fh(0)1835 3584 y Fk(t)1900 3620 y Fn(A)p Fo(\()p Fn(t)p Fo(\),)37 b(where)g Fn(A)p Fo(\()p Fn(t)p Fo(\))f(is)f("slo)m(wly)g(v)-5 b(arying",)36 b(\()p Fn(@)3615 3635 y Fk(t)3645 3620 y Fn(A)p Fo(\()p Fn(t)p Fo(\))69 b(=)-74 3770 y Fl(O)s Fo(\()p Fn(\014)6 b Fo(\)\).)43 b(In)33 b(terms)f(of)h(the)g(amplitude,)d Fn(A)p Fo(\()p Fn(t)p Fo(\),)j(the)g(system)h(\(1.7\))e(b)s(ecomes:)914 4019 y Fn(@)965 4034 y Fk(t)1028 4019 y Fn(A)p Fo(\()p Fn(t)p Fo(\))116 b(=)f Fl(\000)p Fn(i)50 b Fo(\()33 b Fn(\014)6 b( )1874 4034 y Fm(0)1946 4019 y Fn(;)49 b(\036)2080 4034 y Fk(d)2120 4019 y Fo(\()p Fn(t)p Fo(\))33 b(\))49 b(cos)17 b Fn(\026t)32 b(e)2669 3978 y Fk(i\025)2734 3987 y Fh(0)2769 3978 y Fk(t)2799 4019 y Fn(;)947 b Fo(\(1.9\))1000 4193 y Fn(i@)1084 4208 y Fk(t)1114 4193 y Fn(\036)1172 4208 y Fk(d)1328 4193 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b(w)m(a)m(v)m(e-\014eld)g Fn(\036)595 4906 y Fk(d)636 4891 y Fo(.)42 b(The)30 b(c)m(haracter)g(of)f(solutions)g (to)f(\(1.9-1.10\))g(can)i(b)s(e)f(deduced)i(b)m(y)g(\014rst)e(solving) g(\(1.10\))-74 5040 y(for)44 b Fn(\036)145 5055 y Fk(d)229 5040 y Fo(and)g(then)h(substituting)e(the)i(result)f(in)m(to)f (\(1.9\).)77 b(The)45 b(result)f(is)g(a)g(reduction)g(to)f(an)h Fj(or)-5 b(dinary)1926 5306 y Fo(4)p eop %%Page: 5 5 5 4 bop -74 59 a Fj(di\013er)-5 b(ential)39 b(e)-5 b(quation)37 b Fo(for)g Fn(A)p Fo(\()p Fn(t)p Fo(\).)59 b(This)37 b(is)g(reminiscen)m(t)g(of)h(the)g(cen)m(ter)h(manifold)34 b(approac)m(h,)40 b(commonly)-74 208 y(used)f(in)f(dissipativ)m(e)f (systems)j(\(see,)h(for)c(example,)j([3)o(],)g([41]\))e(and)g(recen)m (tly)h(applied)e(in)h(a)g(Hamiltonian)-74 358 y(con)m(text)c([16].)72 507 y(The)28 b(asymptotic)d(b)s(eha)m(vior)i(of)e(the)i(ordinary)f (di\013eren)m(tial)f(equation)h(go)m(v)m(erning)g Fn(A)p Fo(\()p Fn(t)p Fo(\))h(can)f(b)s(e)h(deduced)-74 656 y(b)m(y)i(a)f(asymptotic)f(ev)-5 b(aluation)27 b(of)g(the)i(righ)m(t)e (hand)h(side)h(of)e(\(1.9\),)i(considering)e(carefully)g(the)i(con)m (tribution)-74 806 y(to)j Fn(\036)103 821 y Fk(d)176 806 y Fo(coming)f(from)g(a)i(neigh)m(b)s(orho)s(o)s(d)e(of)h(the)h (resonan)m(t)h(energy)-8 b(,)33 b Fn(\025)2496 821 y Fm(0)2558 806 y Fo(+)22 b Fn(\026)p Fo(.)383 1045 y Fl(\000)p Fn(i)50 b Fo(\()32 b Fn(\014)6 b( )737 1060 y Fm(0)809 1045 y Fn(;)49 b(\036)943 1060 y Fk(d)983 1045 y Fo(\()p Fn(t)p Fo(\))33 b(\))49 b(cos)17 b Fn(\026t)33 b(e)1533 1004 y Fk(i\025)1598 1013 y Fh(0)1633 1004 y Fk(t)224 1236 y Fo(=)115 b Fl(\000)510 1169 y Fn(i)p 502 1213 49 4 v 502 1304 a Fo(4)611 1115 y Fe(\022)704 1236 y Fn(\014)6 b( )828 1251 y Fm(0)900 1236 y Fn(;)976 1119 y Fe(Z)1059 1145 y Fk(t)1023 1308 y Fm(0)1138 1236 y Fn(e)1183 1195 y Fk(i)p Fm(\001\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))1438 1236 y Fn(e)1483 1195 y Fi(\000)p Fk(i)p Fm(\()p Fk(\025)1630 1204 y Fh(0)1666 1195 y Fm(+)p Fk(\026)p Fm(\))p 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y(terms)33 b(of)f(the)h(F)-8 b(ourier)31 b(transform,)h Fl(F)10 b Fo([)p Fn(z)t Fo(]\()p Fn(\030)5 b Fo(\))60 b(=)1794 2920 y Fe(R)1898 2991 y Fn(e)1943 2955 y Fi(\000)p Fk(i\030)s Fi(\001)p Fk(x)2152 2991 y Fn(z)t Fo(\()p Fn(x)p Fo(\))34 b Fn(dx)p Fo(,)e(w)m(e)i(\014nd:)257 3263 y(\000)60 b(=)549 3196 y(1)p 524 3240 98 4 v 524 3332 a(16)664 3263 y(\()p Fn(\025)759 3278 y Fm(0)821 3263 y Fo(+)22 b Fn(\026)p Fo(\))1016 3222 y Fi(\000)1081 3195 y Fh(1)p 1080 3207 31 4 v 1080 3248 a(2)1174 3117 y Fe( )1240 3139 y(\014)1240 3189 y(\014)1240 3238 y(\014)1240 3288 y(\014)1267 3263 y Fl(F)10 b Fo([)p Fn(\014)c( )1500 3278 y Fm(0)1539 3263 y Fo(])1583 3142 y Fe(\022)1644 3158 y(q)p 1727 3158 276 4 v 105 x Fn(\025)1784 3278 y Fm(0)1846 3263 y Fo(+)22 b Fn(\026)2002 3142 y Fe(\023)2063 3139 y(\014)2063 3189 y(\014)2063 3238 y(\014)2063 3288 y(\014)2091 3165 y Fm(2)2185 3263 y Fo(+)2316 3139 y Fe(\014)2316 3189 y(\014)2316 3238 y(\014)2316 3288 y(\014)2343 3263 y Fl(F)10 b Fo([)p Fn(\014)c( )2576 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Fn(W)-74 4134 y Fo(under)34 b(consideration.)-74 4283 y Fj(Emer)-5 b(genc)g(e)34 b(of)g(irr)-5 b(eversible)34 b(b)-5 b(ehavior)34 b(in)h(a)f(time-r)-5 b(eversible)34 b(system)72 4432 y Fo(Th)m(us,)40 b(the)e(original)c(dynamical)h (system,)k(\(1.3\),)f(whic)m(h)f(is)g(Hamiltonian)c(and,)39 b(in)d(particular,)h(time-)-74 4582 y(rev)m(ersible)27 b(\()g(in)m(v)-5 b(arian)m(t)25 b(under)i(the)g(transformation)d Fn(\036)p Fo(\()p Fn(t;)17 b(x)p Fo(\))28 b Fl(7!)p 2335 4503 58 4 v 27 w Fn(\036)p Fo(\()p Fl(\000)p Fn(t;)17 b(x)p Fo(\))27 b(\),)h(is)e(equiv)-5 b(alen)m(t)26 b(to)g(a)h (dynamical)-74 4731 y(system)34 b(with)f(some)g Fj(app)-5 b(ar)g(ent)33 b Fo(time-asymmetry)-8 b(.)1873 4695 y Fm(1)1957 4731 y Fo(This)34 b(is)f(related)g(to)f(the)i Fn(")h Fj(-)h(pr)-5 b(escription)32 b Fo(discussed)-74 4881 y(in)k(section)h(4;)i(see,)g(in)d(particular,)g(Prop)s(osition)f (4.2.)55 b(A)m(t)38 b(the)f(heart)g(of)f(this)g(apparen)m(t)i(parado)m (x)f(is)f(that)p -74 4965 1620 4 v 38 5027 a Fd(1)76 5057 y Fq(This)30 b(is)h(related)e(to)i(a)f("parado)n(x")e(raised)h(in) i(the)g(late)f(nineteen)n(th)h(cen)n(tury)f(b)n(y)h(Zermelo)e(and)i (others)e(concerning)h(the)1926 5306 y Fo(5)p eop %%Page: 6 6 6 5 bop -74 59 a Fo(the)40 b(energy)h(of)e(the)h(oscillator,)f(whic)m (h)h(is)f(coupled)h(to)f(the)h(con)m(tin)m(uum,)h(is)e(propagated)h (out)f(to)g(in\014nit)m(y)-74 208 y(b)s(ecause)34 b(the)f(singular)e (limit:)1241 420 y(lim)1257 480 y Fk(")p Fi(#)p Fm(0)1393 420 y Fn(e)1438 379 y Fi(\000)p Fk(iH)1575 388 y Fh(0)1610 379 y Fk(t)1656 420 y Fo(\()p Fl(\000)p Fo(\001)23 b Fl(\000)g Fo(\()p Fn(\025)2070 435 y Fm(0)2131 420 y Fo(+)f Fn(\026)p Fo(\))g Fl(\007)h Fn(i")p Fo(\))2564 372 y Fi(\000)p Fm(1)3725 420 y Fo(\(1.15\))-74 647 y(satis\014es)41 b(a)f(lo)s(cal)e(energy)k(deca)m(y)g(estimate)d(as)i Fn(t)g Fl(!)f(\0061)p Fo(;)45 b(see)c Fg(\(H4\))p Fo(.)66 b(Th)m(us,)44 b(for)c Fn(t)h(<)f Fo(0,)j(in)c(\(1.12\))h(w)m(e)-74 796 y(replace)33 b(\000)f(b)m(y)i Fl(\000)p Fo(\000.)-74 985 y Fj(Normal)h(F)-7 b(orms)33 b(for)i(c)-5 b(ouple)g(d)34 b(oscil)5 b(lator)34 b(-)h(wave)f(systems)72 1134 y Fo(The)42 b(damp)s(ed)f(oscillator)d(ab)s(o)m(v)m(e)k(is)e(a)h(normal)e(form)h (equation,)j(whic)m(h)e(illustrates)e(a)i(more)f(general)-74 1284 y(phenomenon.)45 b(Dynamical)30 b(systems)35 b(lik)m(e)d(\(1.3\))g (can)h(b)s(e)g(view)m(ed)h(as)f(a)g(Hamiltonian)c(systems)34 b(consisting)-74 1433 y(of)28 b(t)m(w)m(o)i(subsystems:)43 b(a)29 b(single)e(\(or,)j(in)d(general,)i(m)m(ultiple\))e(degree)i(of)f (freedom)h(oscillator)c(and)k(an)g(in\014nite)-74 1582 y(dimensional)h(radiating)g(w)m(a)m(v)m(e)35 b(\014eld.)-74 1732 y(\(I\))41 b(In)h(\(1.3\))e(the)h(discrete)h(oscillator)c (frequency)44 b(do)s(es)d(not)g(lie)e(in)i(the)g(con)m(tin)m(uous)h(sp) s(ectrum)f(of)g(w)m(a)m(v)m(e)-74 1881 y(frequencies.)j(Ho)m(w)m(ev)m (er,)34 b(the)c(time)f(p)s(erio)s(dicit)m(y)g(of)h(the)h(p)s(oten)m (tial)d(generates)k(frequencies)g(in)e(the)g(solution)-74 2031 y(whic)m(h)j(resonate)h(with)e(con)m(tin)m(uum)g(mo)s(des.)-74 2180 y(\(I)s(I\))24 b(In)g([37],)i([38)o(],)g(quan)m(tum)f(resonance)g (t)m(yp)s(e)g(problems)e(are)h(considered)h(in)e(whic)m(h)i(the)f (discrete)h(oscillator)-74 2330 y(frequency)34 b(coincides)d(with)g(a)g (frequency)i(of)e(con)m(tin)m(uum)g(mo)s(des.)43 b(Coupling)30 b(of)g(the)i(t)m(w)m(o)g(o)s(ccurs)g(due)g(to)f(a)-74 2479 y(time-indep)s(enden)m(t)h(lo)s(calized)e(p)s(oten)m(tial.)-74 2629 y(\(I)s(I)s(I\))25 b(In)g(our)f(study)i(of)f(radiation)d(damping)h (of)i("breather-lik)m(e")e(structures)k([39])d(w)m(e)i(\014nd,)h(as)e (in)f(\(1.3\),)i(that)-74 2778 y(the)41 b(discrete)g(oscillator)c (frequency)42 b(do)s(es)f(not)f(lie)f(in)g(the)i(con)m(tin)m(uous)g(sp) s(ectrum)f(of)g(w)m(a)m(v)m(e)i(frequencies.)-74 2927 y(Here,)37 b(the)f(coupling)e(is)h(due)h(to)f(a)g(nonlinear)f(p)s(oten) m(tial)g(whic)m(h)i(generates)h(frequencies)g(whic)m(h)f(lie)e(in)g (the)-74 3077 y(con)m(tin)m(uous)f(sp)s(ectrum.)72 3226 y(In)g(an)g(isolated)e(single)h(degree)h(of)f(freedom)g(Hamiltonian)d (system,)34 b(the)f(general)f(normal)e(form)i(is:)525 3438 y Fn(A)598 3397 y Fi(0)681 3438 y Fo(=)61 b Fn(i)868 3342 y Fe(\020)917 3438 y Fn(c)959 3453 y Fm(10)1088 3438 y Fo(+)55 b Fn(c)1261 3453 y Fm(21)1336 3438 y Fl(j)p Fn(A)p Fl(j)1465 3397 y Fm(2)1558 3438 y Fo(+)g Fn(c)1731 3453 y Fm(32)1805 3438 y Fl(j)p Fn(A)p Fl(j)1934 3397 y Fm(4)2028 3438 y Fo(+)f Fn(:::)h Fo(+)g Fn(c)2467 3453 y Fk(n)p Fm(+1)p Fk(;n)2699 3438 y Fl(j)p Fn(A)p Fl(j)2828 3397 y Fm(2)p Fk(n)2964 3438 y Fo(+)g Fn(:::)3176 3342 y Fe(\021)3275 3438 y Fn(A;)350 b Fo(\(1.16\))-74 3650 y(where)34 b(the)f(constan)m(ts)h Fn(c)850 3665 y Fk(n)p Fm(+1)p Fk(;n)1082 3650 y Fo(are)f Fg(real)f Fo(n)m(um)m(b)s(ers.)-74 3800 y Fj(How)c(is)f(this)h(normal)e(form)i(alter)-5 b(e)g(d)27 b(as)g(a)h(r)-5 b(esult)28 b(of)f(c)-5 b(oupling)27 b(of)g(the)h(oscil)5 b(lator)27 b(to)h(an)f(in\014nite)g(dimensional) -74 3949 y(r)-5 b(adiative)34 b(wave)g(e)-5 b(quation)35 b(?)72 4098 y Fo(In)27 b(the)h([37)o(],)g([38],)g([39)o(])f(and)g(in)f (the)h(presen)m(t)h(w)m(ork,)h(w)m(e)f(\014nd)f(the)g(more)f(general)g Fj(disp)-5 b(ersive)28 b(or)i(r)-5 b(adiative)-74 4248 y(Hamiltonian)34 b(normal)g(form)p Fo(:)532 4460 y Fn(A)605 4419 y Fi(0)689 4460 y Fo(=)825 4363 y Fe(\020)875 4460 y Fn(k)926 4475 y Fm(10)1055 4460 y Fo(+)54 b Fn(k)1236 4475 y Fm(21)1311 4460 y Fl(j)p Fn(A)p Fl(j)1440 4419 y Fm(2)1533 4460 y Fo(+)h Fn(k)1715 4475 y Fm(32)1789 4460 y Fl(j)p Fn(A)p Fl(j)1918 4419 y Fm(4)2012 4460 y Fo(+)f Fn(:::)h Fo(+)g Fn(k)2460 4475 y Fk(n)p Fm(+1)p Fk(;n)2692 4460 y Fl(j)p Fn(A)p Fl(j)2821 4419 y Fm(2)p Fk(n)2957 4460 y Fo(+)f Fn(:::)3168 4363 y Fe(\021)3267 4460 y Fn(A;)358 b Fo(\(1.17\))p -74 4567 1620 4 v -74 4658 a Fq(irrev)n(ersibilit)n(y)28 b(of)h(macroscopically)e(observ)n (ed)h(b)r(eha)n(vior)h(in)h(ph)n(ysical)e(systems)h(and)h(the)g(rev)n (ersibilit)n(y)e(of)h(the)h(microscopic)-74 4758 y(equations)23 b(of)g(motion.)35 b(This)23 b(w)n(as)f(addressed)h(in)g(the)h(con)n (text)f(of)g(classical)f(systems)g(via)h(a)g(statistical)g(approac)n(h) e(in)j(the)f(w)n(ork)-74 4858 y(of)31 b(Maxw)n(ell)f(and)h(Thomson)f (and)h(culminated)g(in)g(the)g(w)n(ork)f(of)h(Boltzmann.)46 b(In)31 b(quan)n(tum)g(systems,)h(for)e(example)g(in)i(the)-74 4957 y(quan)n(tum)c(measuremen)n(t)f(theory)-7 b(,)28 b(extensions)f(of)h(these)g(ideas)f(apply)-7 b(.)38 b(The)28 b(source)f(of)h(the)g(emergence)f(of)h(irrev)n(ersibilit)n(y)-7 b(,)26 b(is)-74 5057 y(the)i(coupling)f(of)h(systems)f(with)h(few)g (degrees)e(of)i(freedom)f(to)g(systems)g(with)i(man)n(y)-7 b(.)36 b(F)-7 b(or)27 b(a)g(discussion,)g(see)g([20)o(].)1926 5306 y Fo(6)p eop %%Page: 7 7 7 6 bop -74 59 a Fo(with)32 b(complex)g(co)s(e\016cien)m(ts;)1358 208 y Fn(k)1409 223 y Fk(n)p Fm(+1)p Fk(;n)1669 208 y Fo(=)60 b Fn(d)1856 223 y Fk(n)p Fm(+1)p Fk(;n)2110 208 y Fo(+)54 b Fn(ic)2315 223 y Fk(n)p Fm(+1)p Fk(;n)2515 208 y Fn(;)1183 b Fo(\(1.18\))-74 412 y(where)40 b Fn(d)265 427 y Fk(n)p Fm(+1)p Fk(;n)503 412 y Fo(and)e Fn(c)740 427 y Fk(n)p Fm(+1)p Fk(;n)978 412 y Fo(are)h(real)e(n)m(um)m(b)s(ers.) 62 b(Non-zero)39 b Fj(r)-5 b(e)g(al)40 b(p)-5 b(arts)38 b Fo(of)g(the)h(normal)d(form)i(co)s(e\016cien)m(ts)-74 561 y(arise)31 b(due)g(to)g(coupling)e(of)i(discrete)g(and)h(con)m(tin) m(uum)e(mo)s(des.)43 b(In)31 b(the)h(linear)d(problems:)42 b(\(I\))31 b(and)g(\(I)s(I\),)g(w)m(e)-74 710 y(alw)m(a)m(ys)40 b(ha)m(v)m(e)g Fn(d)527 725 y Fm(10)640 710 y Fl(\021)f(\000)p Fo(\000)g Fl(\024)g Fo(0.)62 b(In)39 b(\(I)s(I)s(I\))g(w)m(e)h(ha)m(v)m (e)g Fn(k)1973 725 y Fm(10)2086 710 y Fo(=)e Fn(ic)2275 725 y Fm(10)2389 710 y Fo(and,)j(dep)s(ending)e(on)g(the)g(details)f (of)g(the)-74 860 y(nonlinear)c(in)m(teraction,)h(w)m(e)i(ma)m(y)e(ha)m (v)m(e:)50 b Fn(k)1558 875 y Fm(21)1665 860 y Fo(=)33 b Fn(ic)1849 875 y Fm(21)1959 860 y Fo(and)j Fn(k)2203 875 y Fm(32)2310 860 y Fo(=)c Fl(\000)p Fo(\000)25 b(+)f Fn(ic)2756 875 y Fm(32)2866 860 y Fo([39],)36 b Fn(k)3132 875 y Fm(21)3239 860 y Fo(=)c Fl(\000)p Fo(\000)25 b(+)f Fn(ic)3685 875 y Fm(21)3795 860 y Fo([26],)-74 1009 y(or)37 b(other)g(p)s(ossibilities.)54 b(In)38 b(eac)m(h)g(case)g(there's)g(a)f (di\013eren)m(t)h(expression)g(for)f(\000,)h(the)g(\014rst)f(non)m (trivial)e Fj(r)-5 b(e)g(al)-74 1159 y Fo(con)m(tribution)25 b(to)h(a)g(normal)e(form)h(co)s(e\016cien)m(t,)j(but)e(the)h(essence)h (of)e(its)f(calculation)f(is)i(as)g(describ)s(ed)h(ab)s(o)m(v)m(e.)-74 1308 y(Generically)k(one)i(has)f Fl(\000)p Fo(\000)d Fn(<)e Fo(0,)32 b(and)h(it)f(is)g(natural)f(to)h(conjecture)i(that)e (the)h(\014rst)g(non-zero)g(\000)27 b(=)h Fn(d)3703 1323 y Fk(n)3746 1331 y Fc(\003)3781 1323 y Fm(+1)p Fk(;n)3934 1331 y Fc(\003)-74 1458 y Fo(is)k(strictly)g(negativ)m(e.)-74 1707 y Fj(Conne)-5 b(ction)34 b(with)g(p)-5 b(ersistenc)g(e)34 b(the)-5 b(ory)35 b(of)g(p)-5 b(erio)g(dic)34 b(solutions:)72 1906 y Fo(The)g(condition)e(\000)c Fn(>)g Fo(0)k(\(see)i(also)e Fg(\(H6\))p Fo(\))g(can)h(b)s(e)g(seen)h(to)e(arise)h(in)f(another)h (manner.)44 b(Motiv)-5 b(ated)32 b(b)m(y)-74 2055 y(the)j(p)s (erturbativ)m(e)f(approac)m(h)h(to)f(constructing)g(p)s(erio)s(dic)f (solutions)g(of)h(ordinary)f(di\013eren)m(tial)f(equations,)-74 2205 y(it)g(is)g(natural)f(to)h(seek)j(a)d(solution)f(of)h(\(1.3\))g (in)g(the)h(form:)891 2454 y Fn(\036)p Fo(\()p Fn(t)p Fo(\))60 b(=)g Fn(e)1301 2413 y Fi(\000)p Fk(i\025)1421 2422 y Fh(0)1456 2413 y Fk(t)1485 2454 y Fn( )1548 2469 y Fm(0)1643 2454 y Fo(+)54 b Fn( )1836 2469 y Fm(1)1931 2454 y Fo(+)g Fn( )2124 2469 y Fm(2)2219 2454 y Fo(+)g Fn(:::)h Fo(+)g Fn( )2679 2469 y Fk(j)2770 2454 y Fo(+)g Fn(:::;)716 b Fo(\(1.19\))-74 2703 y(where)34 b Fn( )271 2718 y Fk(j)340 2703 y Fo(is)e(formally)e(of)i(order)h Fn(\014)1250 2667 y Fk(j)1286 2703 y Fo(.)44 b(Substitution)31 b(of)h(the)h(ansatz)h(\(1.19\))d(in)m(to)h(\(1.3\))g(yields)g(that:) 1589 2952 y Fn( )1652 2967 y Fm(1)1752 2952 y Fo(=)60 b Fn( )1955 2911 y Fk(a)1951 2977 y Fm(1)2051 2952 y Fo(+)54 b Fn( )2248 2911 y Fk(b)2244 2977 y Fm(1)2284 2952 y Fn(;)1414 b Fo(\(1.20\))-74 3201 y(where)1095 3450 y Fn(i@)1179 3465 y Fk(t)1210 3450 y Fn( )1277 3409 y Fk(a)1273 3475 y Fm(1)1434 3450 y Fo(=)115 b Fn(H)1706 3465 y Fm(0)1778 3450 y Fn( )1845 3409 y Fk(a)1841 3475 y Fm(1)1941 3450 y Fo(+)2082 3383 y(1)p 2082 3427 49 4 v 2082 3518 a(2)2173 3450 y Fn(e)2218 3409 y Fi(\000)p Fk(i)p Fm(\()p Fk(\025)2365 3418 y Fh(0)2400 3409 y Fi(\000)p Fk(\026)p Fm(\))p Fk(t)2555 3450 y Fn(\014)6 b( )2679 3465 y Fm(0)2718 3450 y Fn(;)1101 3683 y(i@)1185 3698 y Fk(t)1216 3683 y Fn( )1283 3642 y Fk(b)1279 3708 y Fm(1)1434 3683 y Fo(=)115 b Fn(H)1706 3698 y Fm(0)1778 3683 y Fn( )1845 3642 y Fk(b)1841 3708 y Fm(1)1936 3683 y Fo(+)2076 3616 y(1)p 2076 3660 V 2076 3752 a(2)2167 3683 y Fn(e)2212 3642 y Fi(\000)p Fk(i)p Fm(\()p Fk(\025)2359 3651 y Fh(0)2395 3642 y Fm(+)p Fk(\026)p Fm(\))p Fk(t)2582 3683 y Fn(\014)6 b( )2706 3698 y Fm(0)2745 3683 y Fn(:)953 b Fo(\(1.21\))-74 3932 y(These)34 b(equations)f(can)g(b)s(e)g(solv)m (ed)g(b)m(y)h(setting:)1068 4181 y Fn( )1135 4140 y Fk(a)1131 4206 y Fm(1)1237 4181 y Fo(=)60 b Fn(e)1418 4140 y Fi(\000)p Fk(i)p Fm(\()p Fk(\025)1565 4149 y Fh(0)1600 4140 y Fi(\000)p Fk(\026)p Fm(\))p Fk(t)1787 4181 y Fo(\011)1863 4140 y Fk(a)1905 4181 y Fn(;)49 b( )2048 4140 y Fk(b)2044 4206 y Fm(1)2144 4181 y Fo(=)60 b Fn(e)2325 4140 y Fi(\000)p Fk(i)p Fm(\()p Fk(\025)2472 4149 y Fh(0)2508 4140 y Fm(+)p Fk(\026)p Fm(\))p Fk(t)2694 4181 y Fo(\011)2770 4140 y Fk(b)2805 4181 y Fn(:)893 b Fo(\(1.22\))-74 4430 y(F)-8 b(or)32 b Fn( )168 4394 y Fk(a)164 4455 y Fm(1)242 4430 y Fo(w)m(e)i(obtain)d(the)i(equation:)1197 4695 y(\()f Fn(H)1348 4710 y Fm(0)1442 4695 y Fl(\000)55 b Fo(\()p Fn(\025)1669 4710 y Fm(0)1731 4695 y Fl(\000)22 b Fn(\026)p Fo(\))33 b(\))49 b(\011)2123 4653 y Fk(a)2280 4695 y Fo(=)2481 4627 y(1)p 2481 4671 V 2481 4763 a(2)2540 4695 y Fn(\014)6 b( )2664 4710 y Fm(0)1206 4928 y Fo(\()32 b Fn(H)1357 4943 y Fm(0)1451 4928 y Fl(\000)55 b Fo(\()p Fn(\025)1678 4943 y Fm(0)1740 4928 y Fo(+)22 b Fn(\026)p Fo(\))32 b(\))49 b(\011)2130 4887 y Fk(b)2280 4928 y Fo(=)2481 4860 y(1)p 2481 4905 V 2481 4996 a(2)2540 4928 y Fn(\014)6 b( )2664 4943 y Fm(0)3725 4928 y Fo(\(1.23\))1926 5306 y(7)p eop %%Page: 8 8 8 7 bop -74 59 a Fo(The)24 b(\014rst)g(equation)f(of)g(\(1.23\))f(has)i (a)f(lo)s(calized)e(solution)h(b)s(ecause)i Fn(\025)2466 74 y Fm(0)2509 59 y Fl(\000)s Fn(\026)k(<)f Fo(0)c(and)g(lies)g(in)f (the)i(resolv)m(en)m(t)g(set)-74 208 y(of)35 b Fn(H)121 223 y Fm(0)160 208 y Fo(.)51 b(Note)35 b(ho)m(w)m(ev)m(er,)j(that)d Fn(\025)1151 223 y Fm(0)1214 208 y Fo(+)24 b Fn(\026)31 b(>)h Fo(0,)j(and)g(therefore)h(the)g(second)g(equation)f(can)g(ha)m(v) m(e)h(a)f(lo)s(calized)-74 358 y(solution)41 b(only)g(if)g(the)i(pro)5 b(jection)42 b(of)g Fn(\014)6 b( )1523 373 y Fm(0)1604 358 y Fo(on)m(to)42 b(the)h(generalized)e(eigenfunction)h(asso)s (ciated)g(with)f(the)-74 507 y(con)m(tin)m(uum)c(energy)i Fn(\025)787 522 y Fm(0)851 507 y Fo(+)26 b Fn(\026)36 b Fo(v)-5 b(anishes.)59 b(This)37 b(is)g(equiv)-5 b(alen)m(t)37 b(to)g(\000)f(=)g(0,)i(where)g(\000)g(is)f(de\014ned)h(b)m(y)g (\(1.13\);)-74 656 y(See)g(also)f(\(1.14\))f(or,)j(more)e(generally)-8 b(,)37 b Fg(\(H6\))p Fo(.)57 b(Generically)-8 b(,)37 b(\000)f Fn(>)f Fo(0.)58 b(The)38 b(results)g(of)f(this)g(pap)s(er)h (sho)m(w,)-74 806 y(in)32 b(particular,)e(that)i(this)g(obstruction)g (to)g(solv)-5 b(abilit)m(y)29 b(in)j(the)g(class)h(of)e(quasip)s(erio)s (dic)g(functions,)h(implies)-74 955 y(radiativ)m(e)g(deca)m(y)i(of)e (solutions)f(of)h(the)h(initial)c(v)-5 b(alue)32 b(problem.)72 1171 y(Finally)-8 b(,)44 b(w)m(e)h(discuss)g(v)-5 b(arious)43 b(con)m(texts,)48 b(where)d(problems)e(in)m(v)m(olving)f(time-dep)s (enden)m(t)i(p)s(oten)m(tials)-74 1321 y(ha)m(v)m(e)g(b)s(een)f (studied.)73 b(Equations)43 b(with)f(time)f(dep)s(enden)m(t)j(p)s(oten) m(tials)d(arise)h(in)g(mathematical)d(ph)m(ysics)-74 1470 y(and)29 b(are)g(imp)s(ortan)m(t)d(in)i(applications,)g Fj(e.g.)41 b Fo(electron)29 b(microscop)m(y)g(and)g(solid)e(state)i (devices.)43 b(In)30 b(quan)m(tum)-74 1620 y(ph)m(ysics)h(their)e (analysis)g(dates)i(bac)m(k)g(to)e(the)h(early)f(da)m(ys)i(of)e(the)i (theory)-8 b(,)30 b(where)h(the)f(pro)s(cess)h(of)f(ionization)-74 1769 y(b)m(y)j(external)e(\014eld)g(\(ligh)m(t)f(w)m(a)m(v)m(es\))k(w)m (as)f(studied)e(and)h(later)f(n)m(uclear)g(spin)h(resonance)h(and)e(c)m (harge)h(transfer)-74 1919 y(problems;)44 b(see)e(e.g.)68 b([19)o(],)43 b([7].)67 b(The)42 b(problem)d(of)h(ionization)e(and)j (more)f(generally)f(of)h(excitation)g(of)g(a)-74 2068 y(molecule)35 b(b)m(y)i(radiation)d(is)h(also)g(imp)s(ortan)m(t)g(in)g (laser)h(optics)g(and)g(c)m(hemistry)-8 b(.)54 b(Recen)m(t)37 b(mo)s(dels)e(of)h(c)m(haos)-74 2217 y([14])43 b(and)g(sto)s(c)m (hastic)h(resonances)h(also)d(in)m(v)m(olv)m(e)h(time)f(dep)s(enden)m (t)j(p)s(oten)m(tial)d(mo)s(dels,)i Fj(e.g.)75 b Fo([42].)g(The)-74 2367 y(expression)33 b(\(1.13\))e(\(see)i(also)e Fg(\(H6\))p Fo(\))f(is)h(the)h(analogue)f(of)g(the)h(F)-8 b(ermi)30 b(Golden)h(Rule,)g(whic)m(h)i(arises)e(in)g(the)-74 2516 y(study)39 b(of)e(sp)s(on)m(taneous)i(emission,)f(autoionization)d ([7],)k([19)o(],)h([29)o(].)59 b(In)38 b(the)g(con)m(text)h(of)f(the)g (problem)e(of)-74 2666 y(ionization)h(of)h(an)h(atom)f(b)m(y)i(a)f (time-dep)s(enden)m(t)h(electric)e(\014eld)h(a)g(heuristic)g(deriv)-5 b(ation)37 b(is)i(presen)m(ted)j(in)-74 2815 y(section)33 b(42)f(of)g([19].)72 2965 y(Equations)46 b(of)e(t)m(yp)s(e)i(\(1.1\))f (and)g(\(1.2\))g(are)g(also)f(relev)-5 b(an)m(t)45 b(in)f(the)i(study)g (of)f(coheren)m(t)h(solutions)e(of)-74 3114 y(nonlinear)31 b(systems.)45 b(F)-8 b(or)32 b(example,)g(a)g(p)s(erio)s(dic)e(or)i (quasip)s(erio)s(dic)f(solution)g(can)i(b)s(e)f(view)m(ed)i(as)f(a)f(b) s(ound)-74 3264 y(state)23 b(of)f(an)h(equation)f(of)g(this)g(t)m(yp)s (e)i(with)e(a)h(self-consisten)m(t)f(\(nonlinear\))f(time-dep)s(enden)m (t)i(p)s(oten)m(tial.)39 b(This)-74 3413 y(p)s(oin)m(t)30 b(of)g(view)h(w)m(as)g(\014rst)g(exploited)f(b)m(y)h(Sigal)e([33],)i ([24)o(])g(who)g(used)g(the)g(metho)s(ds)g(of)f(dilation)d(analyticit)m (y)-74 3562 y(to)e(relate)f(the)h(generic)g(structural)g(instabilit)m (y)e(of)h(quasi-p)s(erio)s(dic)f(solutions)h(of)g(certain)h(nonlinear)e (systems)-74 3712 y(to)38 b(the)g(generic)h(instabilit)m(y)c(of)j(an)g (em)m(b)s(edded)h(eigen)m(v)-5 b(alue)38 b(in)f(the)i(con)m(tin)m(uous) g(sp)s(ectrum)f(of)g(a)g(Flo)s(quet)-74 3861 y(op)s(erator.)k(This)32 b(approac)m(h)f(yields)g(the)h(sp)s(ectral)f(line)e(shift)i(and)g(the)h (F)-8 b(ermi)29 b(golden)i(rule,)g(whic)m(h)g(giv)m(e)g(the)-74 4011 y(lo)s(cation)d(of)i(the)h Fj(r)-5 b(esonanc)g(e)28 b Fo(to)i(whic)m(h)h(the)g(em)m(b)s(edded)g(eigen)m(v)-5 b(alue)30 b(has)h(b)s(een)g(p)s(erturb)s(ed.)43 b(The)31 b(approac)m(h)-74 4160 y(w)m(e)j(ha)m(v)m(e)g(tak)m(en)f(to)g(the)g (non-p)s(ersistence)h(question)f(is)f(to)g(sho)m(w)i(b)m(y)g(a)e(study) i(of)e(the)h(large)f(time)f(b)s(eha)m(vior)-74 4310 y(of)h(the)h (initial)c(v)-5 b(alue)32 b(problem)f(that)i(solutions)e(deca)m(y)j (slo)m(wly)e(as)h Fn(t)28 b Fl(!)f(1)p Fo(.)72 4459 y(The)f(analysis)e (of)h(time)e(dep)s(enden)m(t)k(resonan)m(t)f(problems)e(in)g(the)h(ph)m (ysics)i(literature)c(is)i(based)g(on)g(formal)-74 4608 y(time)31 b(dep)s(enden)m(t)j(p)s(erturbation)d(theory;)i(see)h([19)o (],)f([7],)f(where)h(Dyson)g(series)g(to)f(\014rst)g(order)h(is)e (used,)j(and)-74 4758 y([30])e(where)i(Dirac's)e(p)s(erturbation)f (theory)j(is)e(applied.)42 b(This)33 b(analysis)e(is)i(limited)c(in)j (man)m(y)g(resp)s(ects.)45 b(In)-74 4907 y([19],)32 b([7])h(it)e(is)h (assumed,)i(among)d(other)i(things,)f(that)g(the)h(relev)-5 b(an)m(t)32 b(time)g(scale)g(of)g(the)h(appro)m(ximation)d(is)-74 5057 y(smaller)k(than)i Fn(")543 5021 y Fi(\000)p Fm(1)673 5057 y Fo(where)i Fn(")d Fo(is)h(the)g(p)s(erturbation)f(size,)j(while) d(w)m(e)i(kno)m(w)g(that)f(the)h(lifetime,)d(is)h(of)h(order)1926 5306 y(8)p eop %%Page: 9 9 9 8 bop -74 59 a Fn(")-28 23 y Fi(\000)p Fm(2)66 59 y Fo(;)33 b(It)f(is)g(also)g(not)h(clear)f(ho)m(w)h(to)f(get)h(the)g(sp)s (ectral)f(line)f(shift)h(from)g(this)g(approac)m(h.)72 208 y(Problems)41 b(in)m(v)m(olving)f(time)h(dep)s(enden)m(t)i(p)s (oten)m(tials)d(ha)m(v)m(e)j(receiv)m(ed)g(considerable)e(atten)m(tion) g(in)f(the)-74 358 y(mathematical)h(ph)m(ysics)46 b(literature)d (during)h(the)h(last)f(t)m(w)m(en)m(t)m(y)j(y)m(ears,)h(and)d(a)f(n)m (um)m(b)s(er)h(of)f(di\013eren)m(t)g(ap-)-74 507 y(proac)m(hes,)35 b(based)f(on)f(scattering)g(theory)h(tec)m(hniques,)h(ha)m(v)m(e)f(b)s (een)g(dev)m(elop)s(ed.)47 b(In)33 b([9])g(a)g(kind)g(of)g(Flo)s(quet) -74 656 y(theory)41 b(is)e(used)i(to)f(analyze)f(the)i(case)g(of)e (time-dep)s(enden)m(t)h(p)s(oten)m(tials)f(deca)m(ying)h(lik)m(e)f Fn(L)3385 620 y Fm(1)3465 656 y Fo(functions)h(in)-74 806 y(time.)f(The)25 b(approac)m(h)g(has)f(also)f(b)s(een)i(used)g(to)e (treat)h(some)g(problems)f(in)m(v)m(olving)g(time-p)s(erio)s(dic)d(p)s (oten)m(tials)-74 955 y(and)33 b(long)e(range)i(p)s(oten)m(tials)e ([17],)h([43],)h([10)o(].)72 1105 y(Another)23 b(t)m(yp)s(e)h(of)e (time-dep)s(enden)m(t)h(problem)e(is)h(the)h(c)m(harge-transfer)g (problems.)40 b(These)24 b(are)f(problems)-74 1254 y(in)36 b(whic)m(h)h(the)h(p)s(oten)m(tial)d Fn(W)50 b Fo(is)37 b(a)f(sum)h(of)f(terms)h(of)f(the)h(t)m(yp)s(e:)53 b Fn(W)14 b Fo(\()p Fn(x)25 b Fl(\000)h Fn(v)t(t)p Fo(\).)56 b(In)37 b([45])g(a)f(c)m(harge)h(transfer)-74 1404 y(problem)23 b(is)h(mapp)s(ed)f(in)h(a)g(similar)c(w)m(a)m(y)26 b(on)m(to)e(a)g (three-b)s(o)s(dy)g(t)m(yp)s(e)h(problem)e(whic)m(h)i(is)e(treated)i(b) m(y)g(resolv)m(en)m(t)-74 1553 y(metho)s(ds;)40 b(In)e([6])f(a)g (geometric)f(approac)m(h)i(to)f(scattering)h(w)m(as)g(used)h(to)e (study)h(the)g(general)f(b)s(eha)m(vior)g(of)-74 1703 y(scattering)32 b(states)i(with)e(time-dep)s(enden)m(t)h(p)s(oten)m (tials.)72 1852 y(F)-8 b(or)32 b(p)s(oten)m(tials)f(deca)m(ying)i(slo)m (wly)e(in)h(time)f(whic)m(h)i(are)f(p)s(erturbations)g(of)f(man)m(y)i (b)s(o)s(dy)f(hamiltonians,)-74 2001 y(whic)m(h)k(app)s(ear)g(in)e(the) i(analysis)f(of)g(N-b)s(o)s(dy)g(long)g(range)g(scattering,)h (phase-space)h(metho)s(ds)f(w)m(ere)h(used)-74 2151 y(in)e([35],)i (based)f(on)g(the)g(existence)h(of)f(an)f(asymptotic)h(energy)g(op)s (erator)f(in)g(suc)m(h)j(cases.)54 b(The)37 b(A)m(C)g(Stark)-74 2300 y(e\013ect)e(w)m(as)h(treated)e(in)g([44])g(b)m(y)h(metho)s(ds)f (exploiting)f(the)h(sp)s(ecial)g(form)f(of)h(the)g(p)s(oten)m(tial.)47 b(Also)34 b(related)-74 2450 y(to)c(our)g(study)i(is)e(researc)m(h)i (concerning)f(the)f(w)m(a)m(v)m(e)j(equation)d(with)g(a)g(time)f(p)s (erio)s(dic)g(p)s(oten)m(tial)g(or)h(the)g(free)-74 2599 y(w)m(a)m(v)m(e)41 b(equation)e(in)g(the)g(exterior)g(of)g(a)g(p)s (erio)s(dically)d(oscillating)g(obstacle;)42 b(see,)g(for)d(example,)h ([4],)h([40],)-74 2749 y(and)33 b(those)g(cited)g(therein.)72 2898 y(W)-8 b(e)37 b(consider)g(a)f(Hamiltonian,)d Fn(H)1386 2913 y Fm(0)1425 2898 y Fo(,)38 b(with)d(one)i(b)s(ound)f(state,)i(p)s (erturb)s(ed)f(b)m(y)g(a)f(small)e(and)i(lo)s(calized)-74 3047 y(p)s(oten)m(tial,)25 b(time-p)s(erio)s(dic)c(p)s(oten)m(tial)h (of)i(the)h(simple)e(form)g(giv)m(en)i(in)e(Hyp)s(othesis)i Fg(\(H5\))p Fo(.)40 b(This)24 b(assumption)-74 3197 y(is)e(made)h(to)f (simplify)e(computations)i(but)h(the)g(general)g(metho)s(d)f(applies)g (to)g(problems)g(with)h(more)f(than)h(one)-74 3346 y(b)s(ound)k(state)h (and)f(to)g(more)f(general)h(time)f(-)g(p)s(erio)s(dic)g(or)h(quasi-p)s (erio)s(dic)e(p)s(erturbations.)41 b(An)27 b(application)-74 3496 y(curren)m(tly)k(under)f(in)m(v)m(estigation)f(in)m(v)m(olv)m(es)h (the)h(structural)e(instabilit)m(y)f(and)i(meta-stabilit)m(y)d(of)i ("breather-)-74 3645 y(lik)m(e")38 b(mo)s(des)h(of)f(certain)g(linear)g (w)m(a)m(v)m(e-guide)i(problems)e(related)g(to)h("M-soliton")d (solutions)i(of)g(in)m(tegral)-74 3795 y(nonlinear)31 b(\015o)m(ws)j([21].)72 3944 y(In)d(this)f(pap)s(er)g(w)m(e)h(treat)f (a)g(mo)s(del)f(of)g(the)i(ionization)c(problem)i([19],)i([7].)42 b(By)31 b(treating)e(the)i(pro)s(cess)g(of)-74 4094 y(ionization)c(as)i (a)g(resonance)i(caused)g(b)m(y)f(coupling)e(of)h(the)h(p)s(oin)m(t)f (sp)s(ectrum)g(to)g(the)h(con)m(tin)m(uous)g(sp)s(ectrum,)-74 4243 y(w)m(e)j(apply)e(the)h(time)e(dep)s(enden)m(t)k(resonance)f (theory)-8 b(,)32 b(recen)m(tly)g(dev)m(elop)s(ed)h(in)e([37)o(],)h ([38])g(to)f(\014nd)h(the)g(large)-74 4392 y(time)26 b(b)s(eha)m(vior)h(of)g(the)h(solution.)40 b(In)28 b(particular,)e(w)m (e)j(sho)m(w)f(that)f(under)h(our)g(conditions,)f(the)h(b)s(ound)f (state)-74 4542 y(is)38 b(alw)m(a)m(ys)g(disin)m(tegrating)e(and)j (\014nd)f(the)h(lifetime,)d Fn(\034)48 b Fl(\030)38 b Fo(\000)2187 4506 y Fi(\000)p Fm(1)2281 4542 y Fo(,)i(and)e(the)g(sp)s (ectral)g(line)f(shift,)i(\(4.21\);)h(see)-74 4691 y(the)34 b(remark)f(follo)m(wing)d(Theorem)k(2.2.)45 b(W)-8 b(e)34 b(also)e(pro)m(v)m(e)i(lo)s(cal)d(deca)m(y)k(estimates)e(for)g(the)g (solution,)f(whic)m(h)-74 4841 y(are)h(of)f(general)g(in)m(terest)h(to) f(nonlinear)f(and)i(other)g(problems.)1926 5306 y(9)p eop %%Page: 10 10 10 9 bop -74 59 a Fg(Notations)36 b(and)j(terminology:)-74 208 y Fo(Throughout)33 b(this)f(pap)s(er)h(w)m(e)h(will)c(denote)j(a)f (generic)h(constan)m(t)g(b)m(y)h Fn(C)7 b Fo(,)33 b Fn(D)s Fo(,)f(etc.)-74 358 y Fl(h)p Fn(x)p Fl(i)60 b Fo(=)g(\(1)22 b(+)g Fl(j)p Fn(x)p Fl(j)573 321 y Fm(2)612 358 y Fo(\))660 278 y Fh(1)p 660 290 31 4 v 660 331 a(2)705 358 y Fo(.)-74 507 y Fl(L)p Fo(\()p Fl(H)q Fo(\))27 b(=)65 b(the)33 b(space)h(of)e(b)s(ounded)h(linear)e(op)s(erators)i(on)f Fl(H)-74 656 y Fo(F)-8 b(unctions)32 b(of)g(self-adjoin)m(t)f(op)s (erators)i(are)f(de\014ned)i(via)e(the)h(sp)s(ectral)g(theorem)f([28].) -74 906 y Fg(Ac)m(kno)m(wledgemen)m(ts:)59 b Fo(The)42 b(authors)f(wish)g(to)g(thank)g(Eduard)h(Kirr)e(for)g(helpful)g(commen) m(ts)h(on)f(the)-74 1055 y(man)m(uscript.)49 b(AS)35 b(and)g(MIW)g(are)g(supp)s(orted)g(in)f(part)g(b)m(y)i(gran)m(ts)f (from)e(the)i(National)d(Science)k(F)-8 b(ounda-)-74 1204 y(tion.)-74 1504 y Fp(2.)46 b(General)f(form)l(ulation)i(and)d (main)i(results.)-74 1752 y Fo(Consider)33 b(the)g(general)f(system) 1337 1901 y Fn(i@)1421 1916 y Fk(t)1452 1901 y Fn(\036)p Fo(\()p Fn(t)p Fo(\))60 b(=)g(\()p Fn(H)1936 1916 y Fm(0)1997 1901 y Fo(+)22 b Fn(W)14 b Fo(\()p Fn(t)p Fo(\)\))j Fn(\036)p Fo(\()p Fn(t)p Fo(\))p Fn(;)1210 b Fo(\(2.1\))-74 2105 y(Here,)34 b Fn(\036)p Fo(\()p Fn(t)p Fo(\))e(denotes)i(a)e(function)g (of)g(time,)f Fn(t)p Fo(,)i(with)f(v)-5 b(alues)33 b(in)f(a)g(complex)g (Hilb)s(ert)f(space)j Fl(H)q Fo(.)-74 2254 y Fg(Hyp)s(otheses)k(on)g Fn(H)765 2269 y Fm(0)804 2254 y Fg(:)p -74 2287 910 4 v -74 2404 a(\(H1\))28 b Fn(H)267 2419 y Fm(0)335 2404 y Fo(is)h(self-adjoin)m(t)e(on)i Fl(H)h Fo(and)f(b)s(oth)g Fn(H)1668 2419 y Fm(0)1736 2404 y Fo(and)g Fn(W)14 b Fo(\()p Fn(t)p Fo(\))p Fn(;)46 b(t)28 b Fl(2)g Fo(I)-15 b(R)2461 2362 y Fm(1)2500 2404 y Fn(;)30 b Fo(are)f(densely)h (de\014ned)g(on)f(a)g(subspace)-74 2553 y Fl(D)35 b Fo(of)d Fl(H)q Fo(.)-74 2703 y(The)i(norm)d(on)i Fl(H)g Fo(is)g(denoted)g(b)m (y)h Fl(k)22 b(\001)f(k)p Fo(,)33 b(and)f(the)h(inner)g(pro)s(duct)g (of)f Fn(f)5 b(;)17 b(g)31 b Fl(2)d(H)q Fo(,)k(b)m(y)i(\()p Fn(f)5 b(;)17 b(g)t Fo(\))o(.)-74 2852 y Fg(\(H2\))39 b Fo(The)j(sp)s(ectrum)f(of)f Fn(H)1040 2867 y Fm(0)1120 2852 y Fo(is)g(assumed)h(to)f(consist)h(of)f(an)g(absolutely)g(con)m (tin)m(uous)h(part,)i Fn(\033)3615 2867 y Fm(con)n(t)3751 2852 y Fo(\()p Fn(H)3870 2867 y Fm(0)3909 2852 y Fo(\),)-74 3002 y(with)38 b(asso)s(ciated)g(sp)s(ectral)h(pro)5 b(jection)38 b Fg(P)1543 3017 y Ff(c)1621 3002 y Fo(and)h(a)f(single)f (isolated)g(eigen)m(v)-5 b(alue)38 b Fn(\025)3074 3017 y Fm(0)3152 3002 y Fo(with)g(corresp)s(onding)-74 3151 y(normalized)31 b(eigenstate,)i Fn( )971 3166 y Fm(0)1010 3151 y Fo(,)g Fj(i.e.)1398 3400 y Fn(H)1479 3415 y Fm(0)1518 3400 y Fn( )1581 3415 y Fm(0)1681 3400 y Fo(=)60 b Fn(\025)1874 3415 y Fm(0)1914 3400 y Fn( )1977 3415 y Fm(0)2016 3400 y Fn(;)50 b Fl(k)p Fn( )2206 3415 y Fm(0)2245 3400 y Fl(k)28 b Fo(=)f(1)p Fn(:)1271 b Fo(\(2.2\))72 3649 y(The)39 b(manner)f(in)f(whic)m(h)i(w)m(e)g(shall)d(measure)j(the)f(deca)m(y)h (of)f(solutions)f(is)g(t)m(ypically)g(in)g(a)h(lo)s(cal)d(deca)m(y)-74 3799 y(sense,)40 b Fj(e.g.)57 b Fo(for)36 b(the)i(scalar)e(Sc)m(hr\177) -49 b(odinger)38 b(equation)f(w)m(e)h(measure)f(lo)s(cal)e(deca)m(y)k (using)d(the)i(norms:)52 b Fn(f)46 b Fl(7!)-74 3948 y(kh)p Fn(x)p Fl(i)109 3912 y Fi(\000)p Fk(\033)211 3948 y Fn(f)11 b Fl(k)320 3965 y Fk(L)368 3946 y Fh(2)406 3948 y Fo(,)37 b(where)h Fn(\033)g(>)d Fo(0.)55 b(So)36 b(that)g(our)h(theory)g (applies)e(to)h(a)h(class)f(of)g(general)g(systems)i(\(in)m(v)m (olving,)-74 4097 y(for)h(example,)h(v)m(ector)g(equations)f(with)g (matrix)e(op)s(erators\),)k(w)m(e)f(assume)f(the)g(existence)i(of)e (self-adjoin)m(t)-74 4247 y("w)m(eigh)m(ts",)33 b Fn(w)468 4262 y Fi(\000)559 4247 y Fo(and)g Fn(w)819 4262 y Fm(+)910 4247 y Fo(suc)m(h)h(that)105 4396 y(\(i\))d Fn(w)310 4411 y Fm(+)402 4396 y Fo(is)h(de\014ned)i(on)e(a)h(dense)h(subspace)g (of)e Fl(H)i Fo(and)f(on)f(whic)m(h)h Fn(w)2629 4411 y Fm(+)2716 4396 y Fl(\025)28 b Fn(cI)8 b Fo(,)65 b Fn(c)27 b(>)h Fo(0.)105 4546 y(\(ii\))j Fn(w)338 4561 y Fi(\000)424 4546 y Fl(2)d(L)p Fo(\()p Fl(H)q Fo(\))105 4695 y(\(iii\))i Fn(w)365 4710 y Fi(\000)456 4695 y Fn(w)526 4710 y Fm(+)585 4695 y Fg(P)662 4710 y Ff(c)762 4695 y Fo(=)60 b Fg(P)975 4710 y Ff(c)1075 4695 y Fo(=)g Fg(P)1288 4710 y Ff(c)1360 4695 y Fn(w)1430 4710 y Fi(\000)1522 4695 y Fn(w)1592 4710 y Fm(+)72 4845 y Fo(In)34 b(the)h(scalar)e(case)h Fn(w)920 4860 y Fm(+)1013 4845 y Fo(and)g Fn(w)1274 4860 y Fi(\000)1366 4845 y Fo(corresp)s(ond)h(to)e(m)m(ultiplication)c(b)m (y)35 b Fl(h)p Fn(x)p Fl(i)2881 4808 y Fk(\033)2961 4845 y Fo(and)f Fl(h)p Fn(x)p Fl(i)3285 4808 y Fi(\000)p Fk(\033)3387 4845 y Fo(,)g(resp)s(ectiv)m(ely;)-74 4994 y(see)g(section)f(3.)1901 5306 y(10)p eop %%Page: 11 11 11 10 bop 72 59 a Fo(The)33 b(follo)m(wing)c(h)m(yp)s(othesis)k(and)f (Corollary)f(ensure)i(that)f(the)g(equation)g(satis\014es)g(su\016cien) m(tly)h(strong)-74 208 y(disp)s(ersiv)m(e)g(time-deca)m(y)f(estimates.) -74 358 y Fg(\(H3\))g Fo(Lo)s(cal)g(deca)m(y)i(estimates)e(on)g Fn(e)1335 321 y Fi(\000)p Fk(iH)1472 330 y Fh(0)1507 321 y Fk(t)p 190 390 1347 4 v 1537 358 a Fo(:)45 b(Let)34 b Fn(r)e Fl(\025)e Fo(2)22 b(+)h Fn(")p Fo(,)33 b(where)i Fn(")29 b(>)g Fo(0.)46 b(There)35 b(exist)f Fn(w)3414 373 y Fm(+)3506 358 y Fo(and)g Fn(w)3767 373 y Fi(\000)3826 358 y Fo(,)g(as)-74 507 y(ab)s(o)m(v)m(e,)g(suc)m(h)g(that)e(for)g(all) f Fn(f)11 b Fo(:)1101 656 y(\()p Fg(a)p Fo(\))65 b Fl(k)p Fn(w)1416 671 y Fi(\000)1475 656 y Fn(e)1520 615 y Fi(\000)p Fk(iH)1657 624 y Fh(0)1692 615 y Fk(t)1721 656 y Fg(P)1798 671 y Ff(c)1838 656 y Fn(f)11 b Fl(k)27 b(\024)i Fn(C)39 b Fl(h)p Fn(t)p Fl(i)2302 615 y Fi(\000)p Fk(r)r Fm(+1)2485 656 y Fl(k)p Fn(w)2605 671 y Fm(+)2663 656 y Fn(f)11 b Fl(k)p Fn(;)974 b Fo(\(2.3\))-74 860 y(and)618 1009 y(\()p Fg(b)p Fo(\))65 b Fl(k)p Fn(w)941 1024 y Fi(\000)1000 1009 y Fn(e)1045 968 y Fi(\000)p Fk(iH)1182 977 y Fh(0)1217 968 y Fk(t)1246 1009 y Fo(\()p Fn(H)1365 1024 y Fm(0)1427 1009 y Fl(\000)22 b Fn(\025)1583 1024 y Fm(0)1645 1009 y Fl(\000)g Fn(\026)g Fl(\000)h Fn(i)p Fo(0\))2045 968 y Fi(\000)p Fm(1)2172 1009 y Fg(P)2249 1024 y Ff(c)2289 1009 y Fn(f)11 b Fl(k)27 b(\024)60 b Fn(C)7 b Fl(h)p Fn(t)p Fl(i)2752 968 y Fi(\000)p Fk(r)r Fm(+1)2968 1009 y Fl(k)p Fn(w)3088 1024 y Fm(+)3146 1009 y Fn(f)k Fl(k)p Fn(:)491 b Fo(\(2.4\))-74 1213 y(\(1\))35 b(T)m(ypically)f(one)h(is)g (not)f("handed")i(the)f(singular)f(estimate)g(\(2.4\).)50 b(It)35 b(can)g(ho)m(w)m(ev)m(er)j(b)s(e)d(pro)m(v)m(ed)h(using)-74 1362 y(\(2.3\))c(and)h(the)g(follo)m(wing:)-74 1561 y Fg(\(b)32 1576 y Fm(\001)95 1561 y Fg(\))44 b Fo(Let)g(\001)h(denote)g (an)f(op)s(en)g(in)m(terv)-5 b(al)43 b(con)m(taining)g Fn(\025)2122 1576 y Fm(0)2191 1561 y Fo(+)30 b Fn(\026)44 b Fo(and)g(con)m(tained)h(in)e(the)i(in)m(terior)e(of)g(the)-74 1711 y(con)m(tin)m(uous)h(sp)s(ectrum)f(of)g Fn(H)1064 1726 y Fm(0)1103 1711 y Fo(.)75 b(Let)43 b Fg(g)1446 1726 y Ff(\001)1561 1711 y Fo(denote)h(a)f(smo)s(oth)f(function)g(of)h (compact)f(supp)s(ort)i(whic)m(h)f(is)-74 1860 y(iden)m(tically)31 b(one)h(on)h(\001.)44 b(Then,)33 b(for)g(all)d Fn(f)43 b Fo(suc)m(h)34 b(that)f Fn(w)2023 1875 y Fm(+)2081 1860 y Fn(f)39 b Fl(2)28 b(H)1194 2109 y(k)p Fn(w)1314 2124 y Fi(\000)1373 2109 y Fn(e)1418 2068 y Fi(\000)p Fk(iH)1555 2077 y Fh(0)1589 2068 y Fk(t)1619 2109 y Fg(g)1675 2124 y Ff(\001)1780 2109 y Fn(f)11 b Fl(k)27 b(\024)h Fn(C)2091 2124 y Fm(\001)2186 2109 y Fl(h)p Fn(t)p Fl(i)2299 2068 y Fi(\000)p Fk(r)2392 2109 y Fl(k)p Fn(w)2512 2124 y Fm(+)2570 2109 y Fn(f)11 b Fl(k)p Fn(:)1067 b Fo(\(2.5\))-74 2558 y(\(2\))35 b(T)-8 b(o)35 b(pro)m(v)m(e)i(that)e(\(2.4\))f(is)h (implied)e(b)m(y)j(estimates)f(\(2.3\))g(and)g(\(2.5\),)g(one)h(can)f (follo)m(w)f(v)m(ery)j(closely)d(the)-74 2707 y(approac)m(h)f(tak)m(en) h(in)d([37],)i([38)o(].)44 b(In)33 b(particular,)e(see)i(Prop)s (osition)e(2.1)h(and)h(App)s(endix)g(A)g(of)f([38].)43 b(In)33 b([38],)-74 2857 y(the)i(pro)s(ofs)f(are)g(carried)g(out)g(in)f (for)h(a)g(particular)e(c)m(hoice)j(of)f Fl(H)q Fn(;)51 b(w)2495 2872 y Fm(+)2554 2857 y Fn(;)34 b Fo(and)g Fn(w)2876 2872 y Fi(\000)2935 2857 y Fo(,)h(but)f(as)h(indicated)e(in)g(the)-74 3006 y(remark)g(follo)m(wing)e(the)j(main)d(theorem)i(of)g([38],)h(the) f(pro)s(of)g(can)g(b)s(e)h(adapted)g(to)f(a)g(more)f(general)h(setting) -74 3155 y(and)39 b(is)f(virtually)e(unc)m(hanged;)43 b(one)c(need)g(only)f(replace)h Fn(L)2189 3119 y Fm(2)2267 3155 y Fo(b)m(y)g(a)f(general)g(Hilb)s(ert)f(space)j Fl(H)q Fo(,)g Fl(h)p Fn(x)p Fl(i)3731 3119 y Fi(\000)p Fk(\033)3871 3155 y Fo(b)m(y)-74 3305 y Fn(w)-4 3320 y Fi(\000)55 3305 y Fo(,)33 b(and)f Fl(h)p Fn(x)p Fl(i)437 3269 y Fk(\033)516 3305 y Fo(b)m(y)i Fn(w)722 3320 y Fm(+)781 3305 y Fo(.)43 b(This)33 b(requires)g(h)m(yp)s(othesis:)-74 3454 y Fg(\(H4\))e Fo(There)j(is)e(a)h(c)m(hoice)g(of)f(real)f(n)m(um)m (b)s(er)i Fn(c)g Fo(suc)m(h)h(that)1461 3703 y Fl(k)p Fn(w)1581 3718 y Fm(+)1640 3703 y Fo(\()p Fn(H)1759 3718 y Fm(0)1820 3703 y Fo(+)22 b Fn(c)p Fo(\))1998 3662 y Fi(\000)p Fm(1)2092 3703 y Fn(w)2162 3718 y Fi(\000)2221 3703 y Fl(k)2271 3719 y Fi(L)p Fm(\()p Fi(H)p Fm(\))3773 3703 y Fo(\(2.6\))-74 3952 y(can)33 b(b)s(e)g(made)f(su\016cien)m(tly)h (small.)-74 4102 y(In)g(the)h(scalar)e(case)i Fn(w)771 4117 y Fm(+)862 4102 y Fo(and)f Fn(w)1122 4117 y Fi(\000)1214 4102 y Fo(corresp)s(ond)h(to)e(m)m(ultiplication)c(b)m(y)34 b Fl(h)p Fn(x)p Fl(i)2725 4066 y Fk(\033)2805 4102 y Fo(and)f Fl(h)p Fn(x)p Fl(i)3128 4066 y Fi(\000)p Fk(\033)3229 4102 y Fo(,)g(resp)s(ectiv)m(ely;)i(see)-74 4251 y(section)e(3.)-74 4401 y Fg(\(H5\))e(Hyp)s(otheses)39 b(on)e(the)h(p)s(erturbation)f Fn(W)14 b Fo(\()p Fn(t)p Fo(\))p Fg(:)p 189 4439 1862 4 v 72 4550 a Fo(W)-8 b(e)33 b(consider)g(time-dep)s(enden)m(t)g (symmetric)f(p)s(erturbations)g(of)g(the)h(form)1300 4799 y Fn(W)14 b Fo(\()p Fn(t)p Fo(\))28 b(=)f(cos)17 b Fn(\026t)33 b(\014)6 b(;)49 b Fo(with)32 b Fn(\014)2342 4758 y Fi(\003)2409 4799 y Fo(=)27 b Fn(\014)6 b(:)1173 b Fo(\(2.7\))-74 5048 y(In)33 b(man)m(y)g(applications,)d Fn(\014)6 b Fo(,)32 b(is)g(a)h(spatially)d(lo)s(calized)h(scalar)g(or)i (matrix)e(function.)1901 5306 y(11)p eop %%Page: 12 12 12 11 bop 72 59 a Fo(T)-8 b(o)33 b(measure)g(the)g(size)g(of)f(the)h(p) s(erturbation)f Fn(W)14 b Fo(,)32 b(w)m(e)h(in)m(tro)s(duce)g(the)g (norm)956 319 y Fl(jjj)p Fn(W)14 b Fl(jjj)58 b(\021)1426 222 y Fe(\020)1475 319 y Fl(k)p Fn(w)1595 334 y Fm(+)1682 319 y Fn(\014)6 b Fl(k)1793 277 y Fm(2)1793 343 y Fi(L)p Fm(\()p Fi(H)p Fm(\))2014 319 y Fo(+)55 b(\()p Fl(k)p Fn(w)2303 334 y Fm(+)2389 319 y Fn(\014)6 b(w)2520 334 y Fm(+)2578 319 y Fl(k)2628 277 y Fm(2)2628 343 y Fi(L)p Fm(\()p Fi(H)p Fm(\))2796 222 y Fe(\021)2856 218 y Fh(1)p 2856 230 31 4 v 2856 271 a(2)2917 319 y Fn(:)829 b Fo(\(2.8\))72 568 y(Our)35 b(\014nal)f(h)m(yp)s(othesis)i(is)e(a)g(resonance)i (condition)d(whic)m(h)i(sa)m(ys)h(that)f Fn(\026)23 b Fo(+)g Fn(\025)2979 583 y Fm(0)3084 568 y Fl(2)32 b Fn(\033)3237 583 y Fm(con)n(t)3373 568 y Fo(\()p Fn(H)3492 583 y Fm(0)3531 568 y Fo(\))i(and)h(that)-74 717 y(there)e(is)g(non)m(trivial)d (coupling.)-74 867 y Fg(\(H6\))h Fo(Resonance)j(condition)p 189 880 875 4 v 31 w(There)g(exists)g Fn(\022)1693 882 y Fm(0)1760 867 y Fn(>)28 b Fo(0,)k(indep)s(enden)m(t)i(of)e Fn(W)14 b Fo(,)32 b(suc)m(h)i(that)844 1116 y(\000)60 b Fl(\021)1113 1048 y Fn(\031)p 1113 1092 59 4 v 1118 1184 a Fo(4)1231 1116 y(\()p Fn(\014)6 b( )1393 1131 y Fm(0)1432 1116 y Fn(;)45 b(\016)t Fo(\()p Fn(H)1670 1131 y Fm(0)1731 1116 y Fl(\000)23 b Fn(\025)1888 1131 y Fm(0)1949 1116 y Fl(\000)g Fn(\026)p Fo(\))p Fn(\014)6 b( )2270 1131 y Fm(0)2309 1116 y Fo(\))28 b Fl(\025)g Fn(\022)2525 1131 y Fm(0)2565 1116 y Fl(jjj)p Fn(W)14 b Fl(jjj)2839 1074 y Fm(2)2903 1116 y Fn(>)28 b Fo(0)717 b(\(2.9\))-74 1365 y Fg(Remark:)42 b Fo(That)30 b(the)g(expression)i (in)d(\(2.9\))g(and)h(the)g(principal)e(v)-5 b(alue)29 b(term)h(in)f(\(2.13\))g(b)s(elo)m(w)h(are)g(\014nite)f(is)-74 1514 y(a)j(consequence)k([2])d(of)f(the)h(lo)s(cal)d(deca)m(y)k (estimate)e Fg(\(H3\))f Fo(and)i(the)g(represen)m(tation:)234 1667 y Fe(\020)316 1763 y Fn(\014)6 b( )440 1778 y Fm(0)480 1763 y Fn(;)49 b Fo(\()p Fn(H)675 1778 y Fm(0)736 1763 y Fl(\000)23 b Fn(\025)f Fl(\000)g Fn(\026)g Fl(\000)h Fn(i)p Fo(0\))1315 1722 y Fi(\000)p Fm(1)1442 1763 y Fn(\014)6 b( )1566 1778 y Fm(0)1638 1667 y Fe(\021)1747 1763 y Fo(=)60 b Fn(i)1966 1646 y Fe(Z)2049 1672 y Fi(1)2012 1835 y Fm(0)2173 1667 y Fe(\020)2255 1763 y Fn(\014)6 b( )2379 1778 y Fm(0)2418 1763 y Fn(;)17 b(e)2507 1722 y Fi(\000)p Fk(i)p Fm(\()p Fk(H)2671 1731 y Fh(0)2706 1722 y Fi(\000)p Fk(\025)2802 1731 y Fh(0)2837 1722 y Fi(\000)p Fk(\026)p Fm(\))p Fk(s)2998 1763 y Fn(\014)6 b( )3122 1778 y Fm(0)3194 1667 y Fe(\021)3293 1763 y Fn(ds:)308 b Fo(\(2.10\))72 2012 y(Our)33 b(main)e(result)h(is)g(the)h (follo)m(wing:)-74 2245 y Fg(Theorem)k(2.1.)49 b Fb(Let)34 b Fn(H)881 2260 y Fm(0)954 2245 y Fb(satisfy)f(h)m(yp)s(otheses)j Fg(\(H1\)-\(H6\))c Fb(and)i(assume)g Fn(w)2889 2260 y Fm(+)2948 2245 y Fn(\036)3006 2260 y Fm(0)3074 2245 y Fl(2)c(H)q Fb(.)47 b(Then,)35 b(if)d Fl(jjj)p Fn(W)14 b Fl(jjj)-74 2394 y Fb(is)32 b(su\016cien)m(tly)i(small,)c(the)j (solution)e(of)h(\(2.1\))g(satis\014es)h(the)g(lo)s(cal)e(deca)m(y)j (estimate:)1111 2643 y Fl(k)p Fn(w)1231 2658 y Fi(\000)1322 2643 y Fn(\036)p Fo(\()p Fn(t)p Fo(\))p Fl(k)27 b(\024)h Fn(C)7 b Fl(h)p Fn(t)p Fl(i)1863 2602 y Fi(\000)p Fk(r)r Fm(+1)2046 2643 y Fl(k)p Fn(w)2166 2658 y Fm(+)2257 2643 y Fn(\036)2315 2658 y Fm(0)2354 2643 y Fl(k)p Fn(;)82 b(t)28 b Fl(2)g Fo(I)-15 b(R)p Fn(:)936 b Fo(\(2.11\))-74 2892 y Fg(Remarks:)84 b Fo(In)35 b(the)h(pro)s(of)e(w)m(e)i(actually)e (use)i(a)f(less)g(stringen)m(t)h(prop)s(ert)m(y)g(than)f Fg(\(H6\))p Fo(.)50 b(Rather,)36 b(w)m(e)g(use)-74 3042 y(that:)-74 3191 y Fg(\(H6'\))c Fl(jjj)p Fn(W)14 b Fl(jjj)495 3155 y Fm(3)564 3191 y Fo(\000)625 3155 y Fi(\000)p Fm(1)752 3191 y Fo(is)32 b(su\016cien)m(tly)h(small.)72 3341 y(Under)43 b(the)f(same)g(h)m(yp)s(otheses)i(as)e(Theorem)g(2.1,)i(w)m(e)f(obtain) d(more)h(detailed)g(information)e(on)i(the)-74 3490 y(b)s(eha)m(vior)32 b(of)h Fn(\036)p Fo(\()p Fn(t)p Fo(\):)-74 3747 y Fg(Theorem)k(2.2.)951 3996 y Fn(\036)p Fo(\()p Fn(x;)17 b(t)p Fo(\))83 b(=)g Fn(a)p Fo(\()p Fn(t)p Fo(\))p Fn( )1686 4011 y Fm(0)1748 3996 y Fo(+)22 b Fn(\036)1904 4011 y Fk(d)1944 3996 y Fo(\()p Fn(t)p Fo(\))p Fn(;)82 b Fo(\()p Fn( )2265 4011 y Fm(0)2337 3996 y Fn(;)49 b(\036)2471 4011 y Fk(d)2512 3996 y Fo(\()p Fn(t)p Fo(\)\))60 b(=)g(0)p Fn(;)1056 4171 y(a)p Fo(\()p Fn(t)p Fo(\))84 b(=)f Fn(a)p Fo(\(0\))32 b Fn(e)1714 4130 y Fi(\000)p Fm(\000)p Fk(t)1875 4171 y Fn(e)1920 4130 y Fi(\000)p Fk(i!)2043 4139 y Fh(1)2078 4130 y Fm(\()p Fk(t)p Fm(\))2217 4171 y Fo(+)55 b Fn(R)2422 4186 y Fk(a)2463 4171 y Fo(\()p Fn(t)p Fo(\))1009 4345 y Fn(\036)1067 4360 y Fk(d)1108 4345 y Fo(\()p Fn(t)p Fo(\))83 b(=)g Fn(e)1506 4304 y Fi(\000)p Fk(iH)1643 4313 y Fh(0)1677 4304 y Fk(t)1739 4345 y Fg(P)1816 4360 y Ff(c)1856 4345 y Fn(\036)1914 4360 y Fm(0)2008 4345 y Fo(+)2151 4319 y(~)2139 4345 y Fn(\036)o Fo(\()p Fn(t)p Fo(\))p Fn(:)-74 4594 y Fb(F)-8 b(or)32 b(an)m(y)h(\014xed)h Fn(T)578 4609 y Fm(0)645 4594 y Fn(>)28 b Fo(0)p Fb(,)k(w)m(e)i(ha)m(v) m(e)1244 4843 y Fl(j)o Fn(R)1345 4858 y Fk(a)1387 4843 y Fo(\()p Fn(t)p Fo(\))p Fl(j)60 b(\024)h Fn(C)39 b Fl(jjj)p Fn(W)14 b Fl(jjj)p Fn(;)47 b Fl(j)p Fn(t)p Fl(j)27 b(\024)i Fn(T)2462 4858 y Fm(0)2501 4843 y Fo(\000)2562 4802 y Fi(\000)p Fm(1)3725 4843 y Fo(\(2.12\))1901 5306 y(12)p eop %%Page: 13 13 13 12 bop -74 59 a Fb(Moreo)m(v)m(er,)550 284 y Fl(j)p Fn(R)652 299 y Fk(a)693 284 y Fo(\()p Fn(t)p Fo(\))p Fl(j)83 b Fo(=)g Fl(O)s Fo(\()p Fl(h)p Fn(t)p Fl(i)1307 243 y Fi(\000)p Fk(r)r Fm(+1)1490 284 y Fo(\))p Fn(;)49 b Fl(j)p Fn(t)p Fl(j)27 b(!)h(1)588 474 y Fn(!)649 489 y Fm(1)689 474 y Fo(\()p Fn(t)p Fo(\))115 b(=)h Fn(\025)1164 489 y Fm(0)1235 474 y Fn(t)55 b Fl(\000)1467 406 y Fo(1)p 1467 450 49 4 v 1467 542 a(4)1543 377 y Fe(\020)1592 474 y Fn(\014)6 b( )1716 489 y Fm(0)1756 474 y Fn(;)17 b Fo(\()p Fn(H)1919 489 y Fm(0)1980 474 y Fl(\000)22 b Fn(\025)2136 489 y Fm(0)2198 474 y Fo(+)g Fn(\026)p Fo(\))2393 433 y Fi(\000)p Fm(1)2487 474 y Fg(P)2564 489 y Ff(c)2604 474 y Fn(\014)6 b( )2728 489 y Fm(0)2767 377 y Fe(\021)2866 474 y Fn(t)1107 707 y Fl(\000)1227 639 y Fo(1)p 1227 684 V 1227 775 a(4)1302 611 y Fe(\020)1351 707 y Fn(\014)g( )1475 722 y Fm(0)1515 707 y Fn(;)49 b Fo(P)p Fn(:)p Fo(V)q Fn(:)q Fo(\()p Fn(H)1905 722 y Fm(0)1967 707 y Fl(\000)22 b Fn(\025)2123 722 y Fm(0)2185 707 y Fl(\000)g Fn(\026)p Fo(\))2381 666 y Fi(\000)p Fm(1)2508 707 y Fg(P)2585 722 y Ff(c)2625 707 y Fn(\014)6 b( )2749 722 y Fm(0)2788 611 y Fe(\021)2887 707 y Fn(t)55 b Fo(+)f Fn(M)10 b Fo(\()p Fn(t)p Fo(\))p Fn(;)376 b Fo(\(2.13\))-74 952 y Fb(where)40 b Fn(M)10 b Fo(\()p Fn(t)p Fo(\))38 b Fb(is)g(of)g(order)g Fl(jjj)p Fn(W)14 b Fl(jjj)1222 916 y Fm(2)1297 952 y Fb(and)39 b(is)e(oscillatory)f(in)i Fn(t)p Fb(.)60 b(F)-8 b(urthermore,)2925 925 y Fo(~)2912 952 y Fn(\036)37 b Fo(=)g Fn(\036)3178 967 y Fm(1)3243 952 y Fo(+)26 b Fn(\036)3403 967 y Fm(2)3480 952 y Fb(is)38 b(explicitly)-74 1101 y(giv)m(en)g(in)e(\(4.9\),)i(with)f Fl(k)p Fn(w)917 1116 y Fi(\000)988 1075 y Fo(~)975 1101 y Fn(\036)p Fo(\()p Fn(t)p Fo(\))p Fl(k)73 b Fo(=)g Fl(O)s Fo(\()p Fl(h)p Fn(t)p Fl(i)1649 1065 y Fi(\000)p Fk(r)r Fm(+1)1831 1101 y Fo(\))37 b Fb(as)h Fl(j)p Fn(t)p Fl(j)d(!)g(1)p Fb(,)j(and)g(therefore,)h(b)m(y)f Fg(\(H3\))e Fb(same)h(holds)-74 1251 y Fl(k)p Fn(w)46 1266 y Fi(\000)105 1251 y Fn(\036)163 1266 y Fk(d)203 1251 y Fo(\()p Fn(t)p Fo(\))p Fl(k)60 b Fo(=)g Fl(O)s Fo(\()p Fl(h)p Fn(t)p Fl(i)793 1214 y Fi(\000)p Fk(r)r Fm(+1)976 1251 y Fo(\))32 b Fb(as)h Fl(j)p Fn(t)p Fl(j)27 b(!)h(1)p Fb(.)-74 1524 y Fg(Remark:)59 b Fo(Supp)s(ose)41 b(the)g(initial)c(data)j(is)g(giv)m(en)h(b)m(y)g (the)g(b)s(ound)g(state)g(of)f(the)g(unp)s(erturb)s(ed)i(problem,)-74 1673 y Fj(i.e.)61 b Fn(\036)p Fo(\()p Fn(x;)17 b Fo(0\))37 b(=)g Fn( )617 1688 y Fm(0)657 1673 y Fo(\()p Fn(x)p Fo(\),)k Fn(a)p Fo(\(0\))c(=)h(1,)g Fn(\036)1355 1688 y Fk(d)1395 1673 y Fo(\(0\))f(=)h(0.)61 b(Then,)41 b(from)c(the)i (expansion)g(of)f(the)h(solution)e(w)m(e)i(ha)m(v)m(e)-74 1823 y(that)33 b(for)g(0)28 b Fl(\024)i Fn(t)f Fl(\024)g Fo(\000)703 1787 y Fi(\000)p Fm(1)830 1823 y Fo(that)k(the)h(pro)5 b(jection)33 b(of)g(the)g(solution)f(on)h Fn( )2524 1838 y Fm(0)2597 1823 y Fo(is)g(of)g(order)g Fn(e)3108 1787 y Fi(\000)p Fm(\000)p Fk(t)3237 1823 y Fo(,)g(with)g(an)g(error)g(of) -74 1972 y(order)i Fl(jjj)p Fn(W)14 b Fl(jjj)p Fo(.)46 b(Hence)35 b(it)f(is)f(natural)g(to)h(view)h(the)g(state)f Fn( )2184 1987 y Fm(0)2224 1972 y Fn(e)2269 1936 y Fi(\000)p Fk(i\025)2389 1945 y Fh(0)2424 1936 y Fk(t)2488 1972 y Fo(as)g(a)g Fj(metastable)i(state)p Fo(,)f(with)f(lifetime)-74 2122 y Fn(\034)39 b Fo(=)28 b(\000)172 2085 y Fi(\000)p Fm(1)294 2122 y Fl(\030)g(jjj)p Fn(W)14 b Fl(jjj)673 2085 y Fi(\000)p Fm(2)765 2122 y Fo(.)-74 2418 y Fp(3.)90 b(An)45 b(example:)61 b(the)45 b(Sc)l(hr\177)-67 b(odinger)45 b(equation)-74 2666 y Fo(In)25 b(this)g(section)g(w)m(e)h(apply)f(the)g (results)h(of)e(Theorems)i(2.1)e(and)h(2.2)g(to)g(the)g(sp)s(ecial)f (case)i(of)e(the)i(Sc)m(hr\177)-49 b(odinger)-74 2815 y(equation)27 b(with)g(a)g(time-p)s(erio)s(dic)d(and)k(spatially)d(lo)s (calized)g(p)s(erturbing)i(p)s(oten)m(tial,)g(equation)g(\(1.3\).)41 b(Mo)s(d-)-74 2965 y(els)j(of)g(the)h(sort)g(considered)g(in)f(this)g (example)g(o)s(ccur)g(in)g(the)h(study)h(of)e(ionization)d(of)j(an)h (atom)e(b)m(y)i(a)-74 3114 y(time-v)-5 b(arying)30 b(electric)i (\014eld;)h(see)g([19],)g([7].)72 3264 y(W)-8 b(e)33 b(b)s(egin)f(b)m(y)i(indicating)c(assumptions)i(on)h Fn(V)21 b Fo(\()p Fn(x)p Fo(\))33 b(and)g Fn(\014)6 b Fo(\()p Fn(x)p Fo(\))32 b(under)i(whic)m(h)f Fg(\(H1\)-\(H6\))d Fo(hold.)72 3413 y(W)-8 b(e)43 b(tak)m(e)g Fl(H)h Fo(=)g Fn(L)785 3377 y Fm(2)825 3413 y Fo(\(I)-15 b(R)955 3371 y Fk(n)1002 3413 y Fo(\),)44 b(and)e Fn(H)1391 3428 y Fm(0)1516 3413 y Fl(\021)86 b(\000)p Fo(\001)71 b(+)f Fn(V)22 b Fo(\()p Fn(x)p Fo(\),)45 b(where)e Fn(V)21 b Fo(\()p Fn(x)p Fo(\))43 b(and)f Fn(\014)6 b Fo(\()p Fn(x)p Fo(\))41 b(are)h(real-v)-5 b(alued,)-74 3562 y(su\016cien)m(tly) 41 b(di\013eren)m(tiable)e(and)h(deca)m(ying)g(rapidly)f(as)h Fn(x)h Fl(2)g Fo(I)-15 b(R)2383 3521 y Fk(n)2470 3562 y Fo(tends)41 b(to)f(in\014nit)m(y)-8 b(.)65 b(Th)m(us,)44 b Fn(H)3618 3577 y Fm(0)3697 3562 y Fo(is)c(self-)-74 3712 y(adjoin)m(t,)d(and)g Fn(H)564 3727 y Fm(0)640 3712 y Fo(and)h Fn(W)14 b Fo(\()p Fn(t)p Fo(\))36 b(are)h(densely)h (de\014ned)h(op)s(erators)d(in)g Fn(L)2564 3676 y Fm(2)2604 3712 y Fo(.)57 b(Moreo)m(v)m(er,)39 b(w)m(e)f(assume)g(that)f Fn(H)3935 3727 y Fm(0)-74 3861 y Fo(has)h(exactly)g(one)f(negativ)m(e)h (eigen)m(v)-5 b(alue)36 b(and)i(the)g(absolutely)e(con)m(tin)m(uous)i (sp)s(ectrum)g(of)f Fn(H)3446 3876 y Fm(0)3522 3861 y Fo(is)g(equal)g(to)-74 4011 y(the)c(p)s(ositiv)m(e)f(half-line.)72 4160 y(With)38 b(regard)h(h)m(yp)s(othesis)g Fg(\(H3\))p Fo(,)g(w)m(e)g(tak)m(e)g(the)g(w)m(eigh)m(ts)g(used)h(to)e(measure)h (lo)s(cal)d(energy)j(deca)m(y)h(to)-74 4310 y(b)s(e)32 b Fn(w)128 4325 y Fi(\006)214 4310 y Fl(\021)d(h)p Fn(x)p Fl(i)453 4273 y Fi(\006)p Fk(\033)554 4310 y Fo(.)43 b(There)33 b(is)e(a)h(go)s(o)s(d)e(deal)h(of)g(literature)g(on)g(lo)s (cal)f(energy)i(deca)m(y)h(estimates)e(for)h Fn(e)3629 4273 y Fi(\000)p Fk(iH)3766 4282 y Fh(0)3800 4273 y Fk(t)3830 4310 y Fg(P)3907 4325 y Ff(c)3947 4310 y Fo(.)-74 4459 y(These)j(results)f(require)g(su\016cien)m(t)h(regularit)m(y)d(and)i (deca)m(y)h(of)e(the)h(p)s(oten)m(tial)d Fn(V)22 b Fo(\()p Fn(x)p Fo(\).)47 b(W)-8 b(e)34 b(refer)f(the)h(reader)-74 4608 y(to)e([13],)h([15)o(])g(and)g([23)o(];)g(see)h(also)d([27],)i ([31)o(].)72 4758 y(The)h(large)d(time)g(expansion)i(of)e(the)i(op)s (erator:)43 b Fl(h)p Fn(x)p Fl(i)2028 4722 y Fi(\000)p Fk(\033)2129 4758 y Fn(e)2174 4722 y Fi(\000)p Fk(iH)2311 4731 y Fh(0)2346 4722 y Fk(t)2376 4758 y Fg(P)2453 4773 y Ff(c)2493 4758 y Fl(h)p Fn(x)p Fl(i)2626 4722 y Fi(\000)p Fk(\033)2760 4758 y Fo(in)31 b Fn(L)2939 4722 y Fm(2)3011 4758 y Fo(con)m(tains)h(a)g Fj(non-generic)-74 4907 y Fo(leading)c(order)h(term)g(of)g(order)g Fn(t)1134 4871 y Fi(\000)1199 4843 y Fh(1)p 1199 4855 31 4 v 1199 4897 a(2)1244 4907 y Fo(.)42 b(More)30 b(rapid)e(deca)m(y)j(as)f Fn(t)e Fl(!)f(1)i Fo(is)g(ensured)h(pro)m(vided)g(w)m(e)h(assume)e(the) -74 5057 y(follo)m(wing)h(condition)h(concerning)i(the)g(b)s(eha)m (vior)f(of)g(the)h(resolv)m(en)m(t)h(of)e Fn(H)2701 5072 y Fm(0)2773 5057 y Fo(near)h(zero)g(energy:)1901 5306 y(13)p eop %%Page: 14 14 14 13 bop 72 59 a Fg(\(Z\))42 b Fo(Zero)f(is)h(not)f(a)h(resonance)h (of)f(the)g(op)s(erator)f Fl(\000)p Fo(\001)29 b(+)g Fn(V)21 b Fo(\()p Fn(x)p Fo(\).)72 b(F)-8 b(or)41 b(a)g(precise)i (de\014nition)e(of)g Fj(zer)-5 b(o)-74 208 y(ener)g(gy)35 b(r)-5 b(esonanc)g(e)31 b Fo(in)h(this)g(con)m(text)i(and)e(a)h (detailed)e(discussion)i(see)h([13])e(and)h([23)o(].)-74 358 y Fg(Remark:)43 b Fo(The)34 b(condition)d Fg(\(Z\))h Fo(holds)g(generically)f(in)h(the)h(space)h(of)e(p)s(oten)m(tials.)72 507 y(Hyp)s(othesis)i Fg(\(H3a\))e Fo(is)g(a)g(consequence)k(of:)-74 764 y Fg(Prop)s(osition)g(3.1.)49 b Fb(Assume)41 b(that)f Fn(V)22 b Fo(\()p Fn(x)p Fo(\))41 b Fb(is)f(su\016cien)m(tly)h(regular) e(and)i(deca)m(ys)h(su\016cien)m(tly)f(rapidly)f(as)-74 914 y Fl(j)p Fn(x)p Fl(j)33 b(!)f(1)p Fb(;)37 b(see)f([13],)h([15)o(],) g([23)o(].)53 b(Additionally)-8 b(,)33 b(assume)j(condition)e Fg(\(Z\))p Fb(.)i(Then,)h(w)m(e)g(ha)m(v)m(e)g(the)f(follo)m(wing)-74 1063 y(lo)s(cal)30 b(energy)k(deca)m(y)g(estimate.)-74 1213 y(\(a\))e(Assume)i Fn(n)28 b Fl(\025)g Fo(3)k Fb(and)h Fn(\033)e(>)d Fo(1)p Fb(.)43 b(Then,)34 b(there)f(exists)h Fn(r)c(>)e Fo(2)k Fb(suc)m(h)i(that)1000 1462 y Fl(kh)p Fn(x)p Fl(i)1183 1421 y Fi(\000)p Fk(\033)1317 1462 y Fn(e)1362 1421 y Fi(\000)p Fk(iH)1499 1430 y Fh(0)1533 1421 y Fk(t)1563 1462 y Fg(P)1640 1477 y Ff(c)1680 1462 y Fn(f)43 b Fl(k)1821 1477 y Fm(2)1920 1462 y Fl(\024)61 b Fn(C)40 b Fl(h)p Fn(t)p Fl(i)2281 1421 y Fi(\000)p Fk(r)r Fm(+1)2496 1462 y Fl(kh)p Fn(x)p Fl(i)2679 1421 y Fk(\033)2725 1462 y Fn(f)11 b Fl(k)2834 1477 y Fm(2)2873 1462 y Fn(:)873 b Fo(\(3.1\))-74 1711 y Fb(\(b\))33 b(Assume)g Fn(n)28 b Fo(=)g(1)p Fb(.)43 b(Then,)34 b(w)m(e)f(get)g(\(3.1\))f(with) g Fl(h)p Fn(t)p Fl(i)1915 1675 y Fi(\000)p Fk(r)r Fm(+1)2125 1711 y Fo(=)c Fl(h)p Fn(t)p Fl(i)2342 1674 y Fi(\000)2407 1647 y Fh(3)p 2407 1659 31 4 v 2407 1700 a(2)2451 1711 y Fb(.)-74 1860 y(\(c\))33 b(Assume)g Fn(n)28 b Fo(=)g(2)p Fb(.)43 b(Then,)34 b(w)m(e)f(get)g(\(3.1\))f(with)g Fl(h)p Fn(t)p Fl(i)1904 1824 y Fi(\000)p Fk(r)r Fm(+1)2119 1860 y Fb(replaced)h(b)m(y)h Fl(h)p Fn(t)17 b Fo(log)2856 1817 y Fm(2)2912 1860 y Fn(t)p Fl(i)2986 1824 y Fi(\000)p Fm(1)3081 1860 y Fb(.)72 2167 y Fo(Hyp)s(othesis)34 b Fg(\(H3b)825 2182 y Fm(\001)887 2167 y Fg(\))e Fo(is)g(a)h(consequence) i(of:)-74 2425 y Fg(Prop)s(osition)h(3.2.)49 b Fb(Assume)25 b(the)g(ab)s(o)m(v)m(e)g(h)m(yp)s(otheses)i(on)e Fn(H)2234 2440 y Fm(0)2297 2425 y Fb(and)g(assume)g(that)f Fo(\001)h Fb(denotes)h(an)e(op)s(en)h(and)-74 2574 y(b)s(ounded)36 b(subin)m(terv)-5 b(al)34 b(of)g(the)h(con)m(tin)m(uous)g(sp)s(ectrum)g (of)g Fn(H)2224 2589 y Fm(0)2263 2574 y Fb(.)49 b(Let)35 b Fn(g)2563 2589 y Fm(\001)2661 2574 y Fb(denote)g(a)f(smo)s(oth)g(c)m (haracteristic)-74 2724 y(function,)e(as)h(de\014ned)h(in)e(section)h (2.)43 b(Then,)34 b(for)e Fo(2)27 b Fn(<)h(r)i(<)e(\033)t Fb(,)950 2973 y Fl(kh)p Fn(x)p Fl(i)1133 2932 y Fi(\000)p Fk(\033)1267 2973 y Fn(e)1312 2932 y Fi(\000)p Fk(iH)1449 2941 y Fh(0)1484 2932 y Fk(t)1513 2973 y Fn(g)1560 2988 y Fm(\001)1623 2973 y Fo(\()p Fn(H)1742 2988 y Fm(0)1781 2973 y Fo(\))p Fn(f)44 b Fl(k)1961 2988 y Fm(2)2060 2973 y Fl(\024)61 b Fn(C)39 b Fl(h)p Fn(t)p Fl(i)2420 2932 y Fi(\000)p Fk(r)2545 2973 y Fl(kh)p Fn(x)p Fl(i)2728 2932 y Fk(\033)2775 2973 y Fn(f)11 b Fl(k)2884 2988 y Fm(2)2923 2973 y Fn(:)823 b Fo(\(3.2\))-74 3230 y Fj(pr)-5 b(o)g(of:)67 b Fo(If)45 b Fn(H)425 3245 y Fm(0)509 3230 y Fo(is)g(replaced)g(b)m(y)h(the)f(Laplacian,)h(this)f(estimate)f(is)g (straigh)m(tforw)m(ard)h(since)g Fn(g)3546 3245 y Fm(\001)3609 3230 y Fo(\()p Fn(H)3728 3245 y Fm(0)3767 3230 y Fo(\))p Fn(f)55 b Fo(is)-74 3380 y(sp)s(ectrally)38 b(supp)s(orted)i(a)m(w)m(a) m(y)h(from)c(p)s(oin)m(ts)i(of)f(stationary)h(phase.)63 b(An)40 b(approac)m(h)f(to)g(this)f(estimate)g(for)-74 3529 y Fn(H)7 3544 y Fm(0)46 3529 y Fo(,)52 b(a)c(v)-5 b(ariable)46 b(co)s(e\016cien)m(t)i(op)s(erator)f(is)h(review)m(ed)h (in)e(App)s(endix)h(A)g(and)g(is)g(based)g(on)g(the)g Fj(Mourr)-5 b(e)-74 3678 y(estimate)p Fo(,)32 b(where)i(the)f(op)s (erator)f Fn(A)h Fo(can)g(b)s(e)f(tak)m(en)i(to)e(b)s(e:)44 b Fn(A)28 b Fo(=)f(\()p Fn(x)c Fl(\001)f Fn(p)g Fo(+)g Fn(p)g Fl(\001)g Fn(x)p Fo(\))p Fn(=)p Fo(2,)32 b(where)i Fn(p)28 b Fo(=)f Fl(\000)p Fn(i)p Fl(r)p Fo(.)72 3828 y(W)-8 b(e)33 b(tak)m(e)h(the)f(time-dep)s(enden)m(t)f(p)s(erturbation) g(to)g(b)s(e)h(of)f(the)h(form:)1448 4077 y Fn(W)14 b Fo(\()p Fn(x;)j(t)p Fo(\))60 b(=)g(cos)17 b Fn(\026t)32 b(\014)6 b Fo(\()p Fn(x)p Fo(\))p Fn(;)1321 b Fo(\(3.3\))-74 4326 y(with)33 b Fn(\014)38 b Fo(su\016cien)m(tly)33 b(di\013eren)m(tiable)f(and)h(rapidly)e(deca)m(ying)j(in)e Fn(x)p Fo(,)h Fj(e.g.)44 b Fl(h)p Fn(x)p Fl(i)2797 4290 y Fm(2)p Fk(\033)2879 4326 y Fn(\014)6 b Fo(\()p Fn(x)p Fo(\))28 b Fl(2)g Fn(L)3259 4290 y Fm(2)3299 4326 y Fo(.)44 b(Th)m(us,)35 b Fg(\(H5\))c Fo(is)-74 4475 y(satis\014ed)f(as)g(w)m (ell.)42 b(Finally)-8 b(,)27 b(the)j(resonance)h(condition)d Fg(\(H6\))p Fo(,)h(holds)g(generically)f(in)h(the)h(p)s(oten)m(tial)e Fn(V)21 b Fo(\()p Fn(x)p Fo(\).)72 4625 y(Therefore,)35 b(our)f(main)d(theorems)j(Theorems)g(2.1)f(and)h(Theorems)g(2.2)f(on)g (the)h(structural)f(instabilit)m(y)-74 4774 y(of)f(the)h(unp)s(erturb)s (ed)h(b)s(ound)f(state,)g(and)f(large)g(time)f(b)s(eha)m(vior)h(of)g (suc)m(h)j(systems)f(apply)-8 b(.)1901 5306 y(14)p eop %%Page: 15 15 15 14 bop -74 59 a Fp(4.)90 b(Decomp)t(osition)47 b(and)d(deriv)-7 b(ation)46 b(of)f(the)g(disp)t(ersiv)l(e)h(normal)g(form)-74 307 y Fo(As)30 b(in)f([37],)h([38],)g(w)m(e)h(b)s(egin)e(b)m(y)i (deriving)e(a)g(decomp)s(osition)f(of)h(the)h(solution,)f Fn(\036)p Fo(\()p Fn(t)p Fo(\))h(,)g(whic)m(h)g(will)e(facilitate)-74 456 y(the)33 b(study)h(of)e(its)g(large)g(time)f(b)s(eha)m(vior.)43 b(Let)1489 700 y Fn(\036)p Fo(\()p Fn(t)p Fo(\))28 b(=)f Fn(a)p Fo(\()p Fn(t)p Fo(\))p Fn( )2014 715 y Fm(0)2076 700 y Fo(+)22 b Fn(\036)2232 715 y Fk(d)2273 700 y Fo(\()p Fn(t)p Fo(\))p Fn(;)1362 b Fo(\(4.1\))-74 944 y(with)32 b(the)h(orthogonalit)m(y)e(condition)1477 1188 y(\()p Fn( )1578 1203 y Fm(0)1618 1188 y Fn(;)17 b(\036)1720 1203 y Fk(d)1760 1188 y Fo(\))28 b(=)f(0)65 b(for)32 b(all)63 b Fn(t:)1351 b Fo(\(4.2\))-74 1432 y(Note)33 b(therefore)g(that)f Fn(\036)840 1447 y Fk(d)941 1432 y Fo(=)60 b Fg(P)1154 1447 y Ff(c)1194 1432 y Fn(\036)1252 1447 y Fk(d)1324 1432 y Fo(since)33 b Fn(H)1644 1447 y Fm(0)1716 1432 y Fo(has)g(b)s(een)g(assumed)h(to)e(ha)m(v)m(e)i(only) e(one)h(eigen)m(v)-5 b(alue.)-74 1582 y Fg(Remark:)42 b Fo(The)32 b(required)f(mo)s(di\014cations)d(in)i(our)h(analysis)f (for)g(the)h(case)g(where)h Fn(W)14 b Fo(\()p Fn(t)p Fo(\))30 b(has)h(more)f(general)-74 1731 y(p)s(erio)s(dic)25 b(or)i(quasi-p)s(erio)s(dic)e(time)g(dep)s(endence,)31 b(and)c(when)h Fn(H)2305 1746 y Fm(0)2371 1731 y Fo(has)f(more)f(than)h (one)g(b)s(ound)g(state)g(can)h(b)s(e)-74 1881 y(seen)h(at)f(this)g(p)s (oin)m(t.)41 b(If)28 b Fn(H)899 1896 y Fm(0)966 1881 y Fo(has)g Fn(M)39 b Fo(discrete)29 b(eigen)m(v)-5 b(alues,)29 b(then)f(the)h(decomp)s(osition)d(\(4.1\))h(is)h(replaced)g(b)m(y)-74 2030 y(a)34 b(sum)g(o)m(v)m(er)i(the)e Fn(M)45 b Fo(discrete)35 b(mo)s(des)f(with)g(amplitudes)f Fn(a)2175 2045 y Fk(j)2212 2030 y Fo(\()p Fn(t)p Fo(\))p Fn(;)51 b(j)36 b Fo(=)30 b(0)p Fn(;)17 b(:::;)g(M)34 b Fl(\000)24 b Fo(1,)34 b(plus)g(the)h(con) m(tin)m(uous)-74 2179 y(sp)s(ectral)45 b(part.)81 b(If)45 b Fn(W)14 b Fo(\()p Fn(t)p Fo(\))44 b(has)i(more)e(general)h(p)s(erio)s (dic)e(or)i(quasi-p)s(erio)s(dic)e(time)g(dep)s(endence,)51 b(in)44 b(the)-74 2329 y(analysis)34 b(of)g(the)h(coupled)f(equations)h (for)f Fn(a)1578 2344 y Fk(j)1614 2329 y Fo(\()p Fn(t)p Fo(\))h(and)f Fn(\036)2009 2344 y Fk(d)2050 2329 y Fo(\()p Fn(t)p Fo(\))g(\(see)h(b)s(elo)m(w\),)g(w)m(e)h(use)f(an)f(expansion)h (of)f Fn(W)14 b Fo(\()p Fn(t)p Fo(\))-74 2478 y(in)32 b(terms)g(of)h(op)s(erators)f(of)g(the)h(form)e(\(2.7\).)72 2676 y(W)-8 b(e)33 b(pro)s(ceed)h(b)m(y)f(\014rst)g(inserting)f (\(4.1\))g(in)m(to)g(\(2.1\),)g(whic)m(h)h(yields)f(the)h(equation:)806 2920 y Fn(i@)890 2935 y Fk(t)920 2920 y Fn(a)p Fo(\()p Fn(t)p Fo(\))p Fn( )1145 2935 y Fm(0)1240 2920 y Fo(+)54 b Fn(i@)1454 2935 y Fk(t)1485 2920 y Fn(\036)1543 2935 y Fk(d)1583 2920 y Fo(\()p Fn(t)p Fo(\))116 b(=)f Fn(\025)2058 2935 y Fm(0)2097 2920 y Fn(a)p Fo(\()p Fn(t)p Fo(\))p Fn( )2322 2935 y Fm(0)2417 2920 y Fo(+)22 b Fn(H)2596 2935 y Fm(0)2636 2920 y Fn(\036)2694 2935 y Fk(d)2734 2920 y Fo(\()p Fn(t)p Fo(\))1810 3095 y(+)115 b Fn(a)p Fo(\()p Fn(t)p Fo(\))p Fn(W)14 b Fo(\()p Fn(t)p Fo(\))p Fn( )2443 3110 y Fm(0)2538 3095 y Fo(+)54 b Fn(W)14 b Fo(\()p Fn(t)p Fo(\))p Fn(\036)2943 3110 y Fk(d)2983 3095 y Fo(\()p Fn(t)p Fo(\))679 b(\(4.3\))72 3339 y(T)-8 b(aking)33 b(the)g(inner)f(pro)s(duct)h(of)f(\(4.3\))g(with)g Fn( )1810 3354 y Fm(0)1882 3339 y Fo(yields)h(the)g(follo)m(wing)d (equation)i(for)g Fn(a)p Fo(\()p Fn(t)p Fo(\):)745 3583 y Fn(i@)829 3598 y Fk(t)859 3583 y Fn(a)116 b Fo(=)g Fn(\025)1275 3598 y Fm(0)1314 3583 y Fn(a)p Fo(\()p Fn(t)p Fo(\))55 b(+)f(\()p Fn( )1762 3598 y Fm(0)1802 3583 y Fn(;)17 b(W)d Fo(\()p Fn(t)p Fo(\))p Fn( )2126 3598 y Fm(0)2166 3583 y Fo(\))i Fn(a)p Fo(\()p Fn(t)p Fo(\))55 b(+)g(\()o Fn( )2668 3598 y Fm(0)2708 3583 y Fn(;)17 b(W)d Fo(\()p Fn(t)p Fo(\))p Fn(\036)3027 3598 y Fk(d)3067 3583 y Fo(\))j Fn(;)624 b Fo(\(4.4\))735 3757 y Fn(a)p Fo(\(0\))115 b(=)h(\()p Fn( )1319 3772 y Fm(0)1358 3757 y Fn(;)17 b(\036)p Fo(\(0\)\))-74 4001 y(In)33 b(deriving)f(\(4.4\))g (w)m(e)h(ha)m(v)m(e)h(used)g(that)e Fn( )1523 4016 y Fm(0)1596 4001 y Fo(is)g(normalized)e(and)j(the)g(relation)1646 4245 y(\()p Fn( )1747 4260 y Fm(0)1786 4245 y Fn(;)17 b(@)1881 4260 y Fk(t)1911 4245 y Fn(\036)1969 4260 y Fk(d)2009 4245 y Fo(\))28 b(=)g(0)p Fn(;)1518 b Fo(\(4.5\))-74 4489 y(a)32 b(consequence)k(of)c(\(4.2\).)72 4638 y(Applying)g Fg(P)572 4653 y Ff(c)645 4638 y Fo(to)g(\(4.3\),)g(w)m(e)h(obtain)f(an) g(equation)h(for)f Fn(\036)2212 4653 y Fk(d)2252 4638 y Fo(:)692 4882 y Fn(i@)776 4897 y Fk(t)806 4882 y Fn(\036)864 4897 y Fk(d)905 4882 y Fo(\()p Fn(t)p Fo(\))115 b(=)h Fn(H)1404 4897 y Fm(0)1443 4882 y Fn(\036)1501 4897 y Fk(d)1541 4882 y Fo(\()p Fn(t)p Fo(\))55 b(+)f Fg(P)1914 4897 y Ff(c)1954 4882 y Fn(W)14 b Fo(\()p Fn(t)p Fo(\))p Fn(\036)2229 4897 y Fk(d)2269 4882 y Fo(\()p Fn(t)p Fo(\))55 b(+)g Fn(a)p Fo(\()p Fn(t)p Fo(\))p Fg(P)2805 4897 y Ff(c)2845 4882 y Fn(W)14 b Fo(\()p Fn(t)p Fo(\))p Fn( )3125 4897 y Fm(0)3164 4882 y Fn(;)582 b Fo(\(4.6\))793 5057 y Fn(\036)851 5072 y Fk(d)891 5057 y Fo(\(0\))115 b(=)h Fg(P)1400 5072 y Ff(c)1440 5057 y Fn(\036)p Fo(\(0\))1901 5306 y(15)p eop %%Page: 16 16 16 15 bop -74 59 a Fo(Since)30 b(w)m(e)i(are)e(after)g(a)g(slo)m(w)g (resonan)m(t)h(deca)m(y)h(phenomenon,)f(it)e(will)f(pro)m(v)m(e)k(adv) -5 b(an)m(tageous)31 b(to)e(extract)i(the)-74 208 y(fast)i(oscillatory) d(b)s(eha)m(vior)i(of)g Fn(a)p Fo(\()p Fn(t)p Fo(\).)44 b(W)-8 b(e)33 b(therefore)g(de\014ne:)1609 415 y Fn(A)p Fo(\()p Fn(t)p Fo(\))28 b Fl(\021)h Fn(e)1972 374 y Fk(i\025)2037 383 y Fh(0)2072 374 y Fk(t)2101 415 y Fn(a)p Fo(\()p Fn(t)p Fo(\))p Fn(:)1483 b Fo(\(4.7\))-74 622 y(Then,)34 b(\(4.4\))e(reads)835 828 y Fn(@)886 843 y Fk(t)917 828 y Fn(A)60 b Fo(=)g Fl(\000)p Fn(iA)17 b Fo(\()p Fn( )1487 843 y Fm(0)1527 828 y Fn(;)g(W)d Fo(\()p Fn(t)p Fo(\))p Fn( )1851 843 y Fm(0)1891 828 y Fo(\))54 b Fl(\000)h Fn(ie)2193 787 y Fk(i\025)2258 796 y Fh(0)2294 787 y Fk(t)2372 828 y Fo(\()p Fn( )2473 843 y Fm(0)2513 828 y Fn(;)17 b(W)d Fo(\()p Fn(t)p Fo(\))p Fn(\036)2832 843 y Fk(d)2872 828 y Fo(\()p Fn(t)p Fo(\)\))i Fn(:)709 b Fo(\(4.8\))-74 1035 y(Solving)31 b(\(4.6\))h(b)m(y)i(Duhamel's)d(form)m (ula)g(w)m(e)j(ha)m(v)m(e)753 1257 y Fn(\036)811 1272 y Fk(d)851 1257 y Fo(\()p Fn(t)p Fo(\))84 b(=)g Fn(e)1251 1216 y Fi(\000)p Fk(iH)1388 1225 y Fh(0)1423 1216 y Fk(t)1452 1257 y Fn(\036)1510 1272 y Fk(d)1551 1257 y Fo(\(0\))54 b Fl(\000)h Fn(i)1912 1140 y Fe(Z)1995 1167 y Fk(t)1958 1329 y Fm(0)2041 1257 y Fn(e)2086 1216 y Fi(\000)p Fk(iH)2223 1225 y Fh(0)2258 1216 y Fm(\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))2430 1257 y Fg(P)2507 1272 y Ff(c)2547 1257 y Fn(W)14 b Fo(\()p Fn(s)p Fo(\))p Fn(a)p Fo(\()p Fn(s)p Fo(\))p Fn( )3011 1272 y Fm(0)3050 1257 y Fn(ds)1046 1494 y Fl(\000)83 b Fn(i)1256 1377 y Fe(Z)1339 1403 y Fk(t)1302 1565 y Fm(0)1385 1494 y Fn(e)1430 1453 y Fi(\000)p Fk(iH)1567 1462 y Fh(0)1602 1453 y Fm(\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))1774 1494 y Fg(P)1851 1509 y Ff(c)1891 1494 y Fn(W)14 b Fo(\()p Fn(s)p Fo(\))p Fn(\036)2177 1509 y Fk(d)2216 1494 y Fo(\()p Fn(s)p Fo(\))33 b Fn(ds)1046 1668 y Fl(\021)83 b Fn(\036)1264 1683 y Fm(0)1303 1668 y Fo(\()p Fn(t)p Fo(\))55 b(+)g Fn(\036)1658 1683 y Fm(1)1697 1668 y Fo(\()p Fn(t)p Fo(\))f(+)h Fn(\036)2051 1683 y Fm(2)2090 1668 y Fo(\()p Fn(t)p Fo(\))p Fn(:)1545 b Fo(\(4.9\))-74 1875 y(By)38 b(standard)g(metho)s(ds,)g(the)g(system)g(\(4.8\)-\(4.9\))e (for)h Fn(A)p Fo(\()p Fn(t)p Fo(\))h(and)f Fn(\036)2522 1890 y Fk(d)2562 1875 y Fo(\()p Fn(t)p Fo(\))f(=)g Fn(\036)p Fo(\()p Fn(t)p Fo(\))25 b Fl(\000)h Fn(e)3163 1839 y Fi(\000)p Fk(i\025)3283 1848 y Fh(0)3318 1839 y Fk(t)3385 1875 y Fn(A)p Fo(\()p Fn(t)p Fo(\))37 b Fn( )3669 1890 y Fm(0)3746 1875 y Fo(has)h(a)-74 2024 y(global)30 b(solution)h(in)h Fn(t)h 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Fe(X)2455 2936 y Fk(j)t Fm(=0)2594 2837 y Fo(\()p Fn( )2695 2852 y Fm(0)2735 2837 y Fn(;)17 b(W)d Fo(\()p Fn(t)p Fo(\))p Fn(\036)3054 2852 y Fk(j)3090 2837 y Fo(\))i Fn(:)554 b Fo(\(4.10\))72 3098 y(W)-8 b(e)33 b(next)g(giv)m(e)f(a)g(detailed)f(expansion)h(of)g (the)h(sum)f(in)f(\(4.10\).)43 b(It)32 b(is)f(in)h(the)g Fn(j)i Fo(=)27 b(1)32 b(term)g(that)g(the)g(k)m(ey)-74 3247 y(resonance)40 b(is)d(found.)61 b(This)38 b(mak)m(es)h(it)e(p)s (ossible)g(to)h(\014nd)h(a)e(normal)g(form)f(for)i(\(4.10\))f(in)h (whic)m(h)g Fj(internal)-74 3397 y(damping)24 b Fo(in)h(the)h(system)h (is)e(made)g(explicit.)39 b(This)26 b(damping)e(is)h(resp)s(onsible)h (for)f(the)h(energy)g(transfer)g(from)-74 3546 y(the)33 b(discrete)g(to)g(con)m(tin)m(uum)f(mo)s(des)g(of)g(the)h(system)h(and) f(the)g(asso)s(ciated)f(radiativ)m(e)f(deca)m(y)j(of)f(solutions.)-74 3755 y Fg(Prop)s(osition)j(4.1.)425 3962 y Fl(\000)p Fn(ie)580 3921 y Fk(i\025)645 3930 y Fh(0)681 3921 y Fk(t)727 3962 y Fo(cos)17 b Fn(\026t)g 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4152 y Fn(ds)70 4389 y(R)144 4404 y Fm(2)266 4389 y Fo(=)443 4322 y Fn(i)p 435 4366 V 435 4457 a Fo(2)494 4389 y Fn(e)539 4348 y Fi(\000)p Fk(i\026t)707 4389 y Fo(cos)17 b Fn(\026t)32 b(A)p Fo(\()p Fn(t)p Fo(\))1214 4293 y Fe(\020)1263 4389 y Fn(\014)6 b( )1387 4404 y Fm(0)1427 4389 y Fn(;)17 b Fo(\()p Fn(H)1590 4404 y Fm(0)1651 4389 y Fl(\000)22 b Fn(\025)1807 4404 y Fm(0)1869 4389 y Fl(\000)h Fn(\026)e Fl(\000)i Fn(i)p Fo(0\))2269 4348 y Fi(\000)p Fm(1)2363 4389 y Fg(P)2440 4404 y Ff(c)2480 4389 y Fn(\014)6 b( )2604 4404 y Fm(0)2644 4293 y Fe(\021)70 4623 y Fn(R)144 4638 y Fm(3)266 4623 y Fo(=)83 b Fl(\000)520 4556 y Fn(i)p 512 4600 V 512 4692 a Fo(2)571 4623 y Fn(A)p Fo(\(0\))p Fn(e)814 4582 y Fk(i\025)879 4591 y Fh(0)914 4582 y Fk(t)960 4623 y Fo(cos)18 b Fn(\026t)1218 4527 y Fe(\020)1268 4623 y Fn(\014)6 b( )1392 4638 y Fm(0)1431 4623 y Fn(;)17 b(e)1520 4582 y Fi(\000)p Fk(iH)1657 4591 y Fh(0)1692 4582 y Fk(t)1721 4623 y Fo(\()p Fn(H)1840 4638 y Fm(0)1902 4623 y Fl(\000)22 b Fn(\025)2058 4638 y Fm(0)2120 4623 y Fl(\000)g Fn(\026)g Fl(\000)h Fn(i)p Fo(0\))2520 4582 y Fi(\000)p Fm(1)2614 4623 y Fg(P)2691 4638 y Ff(c)2731 4623 y Fn(\014)6 b( )2855 4638 y Fm(0)2894 4527 y Fe(\021)70 4858 y Fn(R)144 4873 y Fm(4)266 4858 y Fo(=)83 b Fl(\000)520 4790 y Fn(i)p 512 4834 V 512 4926 a Fo(2)571 4858 y Fn(e)616 4816 y Fk(i\025)681 4825 y Fh(0)716 4816 y Fk(t)763 4858 y Fo(cos)17 b Fn(\026t)1021 4740 y Fe(Z)1103 4767 y Fk(t)1067 4929 y Fm(0)1150 4761 y Fe(\020)1199 4858 y Fn(\014)6 b( )1323 4873 y Fm(0)1363 4858 y Fn(;)17 b(e)1452 4816 y Fi(\000)p Fk(iH)1589 4825 y Fh(0)1623 4816 y Fm(\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))1795 4858 y Fo(\()p Fn(H)1914 4873 y Fm(0)1975 4858 y Fl(\000)23 b Fn(\025)2132 4873 y Fm(0)2194 4858 y Fl(\000)f Fn(\026)g Fl(\000)h Fn(i)p Fo(0\))2594 4816 y Fi(\000)p Fm(1)2688 4858 y Fn(e)2733 4816 y Fi(\000)p Fk(i)p Fm(\()p Fk(\025)2880 4825 y Fh(0)2915 4816 y Fm(+)p Fk(\026)p Fm(\))p Fk(s)3077 4858 y Fn(@)3128 4873 y Fk(s)3165 4858 y Fn(A)p Fo(\()p Fn(s)p Fo(\))p Fg(P)3437 4873 y Ff(c)3477 4858 y Fn(\014)6 b( )3601 4873 y Fm(0)3640 4761 y Fe(\021)3707 4858 y Fn(ds:)3725 5032 y Fo(\(4.11\))1901 5306 y(16)p eop %%Page: 17 17 17 16 bop -74 59 a Fo(Pro)s(of)32 b(of)g(Prop)s(osition)f(4.1)p -74 91 1028 4 v 72 208 a(Using)i(that)f(cos)17 b Fn(\026t)28 b Fo(=)940 169 y Fm(1)p 940 185 36 4 v 940 242 a(2)986 208 y Fo(\()p Fn(e)1069 172 y Fk(i\026t)1187 208 y Fo(+)22 b Fn(e)1330 172 y Fi(\000)p Fk(i\026t)1481 208 y Fo(\))33 b(and)f(the)h(de\014nition)f Fn(A)p Fo(\()p Fn(t)p Fo(\))c(=)f Fn(e)2703 172 y Fk(i\025)2768 181 y Fh(0)2803 172 y Fk(t)2833 208 y Fn(a)p Fo(\()p Fn(t)p Fo(\),)33 b(w)m(e)g(get)g(from)e(\(4.9\)) 889 482 y Fn(\036)947 497 y Fm(1)986 482 y Fo(\()p Fn(t)p Fo(\))d(=)83 b Fl(\000)1377 443 y Fk(i)p 1371 459 V 1371 516 a Fm(2)1500 365 y Fe(Z)1583 391 y Fk(t)1546 554 y Fm(0)1629 482 y Fn(e)1674 441 y Fi(\000)p Fk(iH)1811 450 y Fh(0)1846 441 y Fm(\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))2018 482 y Fn(e)2063 441 y Fi(\000)p Fk(i)p Fm(\()p Fk(\025)2210 450 y Fh(0)2245 441 y Fi(\000)p Fk(\026)p Fm(\))p Fk(s)2407 482 y Fn(A)p Fo(\()p Fn(s)p Fo(\))p Fg(P)2679 497 y Ff(c)2718 482 y Fn(\014)6 b( )2842 497 y Fm(0)2914 482 y Fn(ds)1284 719 y Fl(\000)1377 679 y Fk(i)p 1371 695 V 1371 753 a Fm(2)1500 601 y Fe(Z)1583 628 y Fk(t)1546 790 y Fm(0)1629 719 y Fn(e)1674 677 y Fi(\000)p Fk(iH)1811 686 y Fh(0)1846 677 y Fm(\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))2018 719 y Fn(e)2063 677 y Fi(\000)p Fk(i)p Fm(\()p Fk(\025)2210 686 y Fh(0)2245 677 y Fm(+)p Fk(\026)p Fm(\))p Fk(s)2407 719 y Fn(A)p Fo(\()p Fn(s)p Fo(\))p Fg(P)2679 734 y Ff(c)2718 719 y Fn(\014)g( )2842 734 y Fm(0)2882 719 y Fn(ds)1312 893 y Fl(\021)143 b Fn(K)1615 908 y Fm(0)1677 893 y Fo(+)22 b Fn(K)1858 908 y Fm(res)1949 893 y Fn(:)1749 b Fo(\(4.12\))-74 1142 y(Therefore,)617 1391 y Fl(\000)p Fn(ie)772 1350 y Fk(i\025)837 1359 y Fh(0)873 1350 y Fk(t)951 1391 y Fo(cos)18 b Fn(\026t)e Fo(\()p Fn(\014)6 b( )1371 1406 y Fm(0)1410 1391 y Fn(;)17 b(\036)1512 1406 y Fm(1)1551 1391 y Fo(\()p Fn(t)p Fo(\)\))116 b(=)f Fl(\000)p Fn(ie)2162 1350 y Fk(i\025)2227 1359 y Fh(0)2263 1350 y Fk(t)2309 1391 y Fo(cos)18 b Fn(\026t)e Fo(\()p Fn(\014)6 b( )2729 1406 y Fm(0)2768 1391 y Fn(;)17 b(K)2895 1406 y Fm(0)2957 1391 y Fo(+)22 b Fn(K)3138 1406 y Fm(res)3229 1391 y Fo(\))1816 1565 y(=)115 b Fn(R)2081 1580 y Fm(1)2176 1565 y Fl(\000)55 b Fn(ie)2386 1524 y Fk(i\025)2451 1533 y Fh(0)2486 1524 y Fk(t)2532 1565 y Fo(cos)18 b Fn(\026t)e Fo(\()p Fn(\014)6 b( )2952 1580 y Fm(0)2991 1565 y Fn(;)17 b(K)3118 1580 y Fm(res)3209 1565 y Fo(\))478 b(\(4.13\))-74 1814 y(The)36 b(terms)g Fn(R)478 1829 y Fk(j)515 1814 y Fn(;)52 b(j)38 b Fo(=)32 b(2)p Fn(;)17 b Fo(3)p Fn(;)g Fo(4)34 b(in)h(\(4.11\))g(come)g(from)f (an)h(expansion)h(of)f(the)h(last)e(term)h(in)g(\(4.13\).)51 b(This)35 b(is)-74 1964 y(the)h(source)h(of)e(the)g(k)m(ey)i(resonan)m (t)g(con)m(tribution.)51 b(W)-8 b(e)36 b(no)m(w)g(pro)s(ceed)g(with)f (a)h(careful)e(ev)-5 b(aluation)34 b(of)h(this)-74 2113 y(term.)72 2263 y(W)-8 b(e)33 b(\014rst)g(regularize)f Fn(K)969 2278 y Fm(res)1060 2263 y Fo(.)43 b(F)-8 b(or)32 b Fn(t)c(>)f Fo(0,)33 b(let)f Fn(")g Fo(b)s(e)h(p)s(ositiv)m(e)f(and)g (arbitrary)g(and)g(de\014ne:)840 2523 y Fn(K)930 2482 y Fk(")923 2547 y Fm(res)1074 2523 y Fo(=)60 b Fl(\000)1305 2455 y Fn(i)p 1297 2500 49 4 v 1297 2591 a Fo(2)1373 2406 y Fe(Z)1456 2432 y Fk(t)1419 2594 y Fm(0)1502 2523 y Fn(e)1547 2482 y Fi(\000)p Fk(iH)1684 2491 y Fh(0)1718 2482 y Fm(\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))1891 2523 y Fn(e)1936 2482 y Fi(\000)p Fk(i)p Fm(\()p Fk(\025)2083 2491 y Fh(0)2118 2482 y Fm(+)p Fk(\026)p Fi(\000)p Fk(i")p Fm(\))p Fk(s)2391 2523 y Fn(A)p Fo(\()p Fn(s)p Fo(\))32 b Fg(P)2695 2538 y Ff(c)2768 2523 y Fn(\014)6 b( )2892 2538 y Fm(0)2964 2523 y Fn(ds)664 b Fo(\(4.14\))-74 2772 y(Note)27 b(that)g Fn(K)445 2787 y Fm(res)564 2772 y Fo(=)g(lim)803 2787 y Fk(")p Fi(!)p Fm(0)962 2772 y Fn(K)1052 2736 y Fk(")1045 2796 y Fm(res)1136 2772 y Fo(.)42 b(In)m(tegration)26 b(of)h(the)h(expression)g(in)e(\(4.14\))h(b)m(y)h(parts,)h(and)e (letting)e Fn(")i Fo(tend)-74 2921 y(to)32 b(zero)h(from)f(ab)s(o)m(v)m (e)h(giv)m(es)-74 3179 y Fg(Prop)s(osition)j(4.2.)49 b Fb(The)33 b(follo)m(wing)d(expansion)j(of)f Fn(K)2014 3194 y Fm(res)2138 3179 y Fb(is)g(v)-5 b(alid)31 b(in)g Fl(S)2652 3142 y Fi(0)2676 3179 y Fb(:)489 3438 y Fn(K)572 3453 y Fm(res)746 3438 y Fo(=)83 b Fl(\000)992 3370 y Fo(1)p 992 3414 V 992 3506 a(2)1084 3438 y Fn(e)1129 3396 y Fi(\000)p Fk(i)p Fm(\()p Fk(\025)1276 3405 y Fh(0)1311 3396 y Fm(+)p Fk(\026)p Fm(\))p Fk(t)1465 3438 y Fn(A)p Fo(\()p Fn(t)p Fo(\)\()p Fn(H)1768 3453 y Fm(0)1830 3438 y Fl(\000)23 b Fn(\025)f Fl(\000)g Fn(\026)g Fl(\000)h Fn(i)p Fo(0\))2409 3396 y Fi(\000)p Fm(1)2503 3438 y Fg(P)2580 3453 y Ff(c)2620 3438 y Fn(\014)6 b( )2744 3453 y Fm(0)746 3671 y Fo(+)915 3603 y(1)p 915 3647 V 915 3739 a(2)974 3671 y Fn(A)p Fo(\(0\))32 b Fn(e)1249 3630 y Fi(\000)p Fk(iH)1386 3639 y Fh(0)1421 3630 y Fk(t)1450 3671 y Fo(\()p Fn(H)1569 3686 y Fm(0)1631 3671 y Fl(\000)22 b Fn(\025)g Fl(\000)h Fn(\026)f Fl(\000)g Fn(i)p Fo(0\))2209 3630 y Fi(\000)p Fm(1)2304 3671 y Fg(P)2381 3686 y Ff(c)2421 3671 y Fn(\014)6 b( )2545 3686 y Fm(0)746 3904 y Fo(+)915 3837 y(1)p 915 3881 V 915 3972 a(2)990 3787 y Fe(Z)1073 3813 y Fk(t)1036 3975 y Fm(0)1120 3904 y Fn(e)1165 3863 y Fi(\000)p Fk(iH)1302 3872 y Fh(0)1336 3863 y Fm(\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))1508 3904 y Fo(\()p Fn(H)1627 3919 y Fm(0)1689 3904 y Fl(\000)22 b Fn(\025)g Fl(\000)h Fn(\026)f Fl(\000)g Fn(i)p Fo(0\))2267 3863 y Fi(\000)p Fm(1)2362 3904 y Fn(e)2407 3863 y Fi(\000)p Fk(i)p Fm(\()p Fk(\025)2554 3872 y Fh(0)2589 3863 y Fm(+)p Fk(\026)p Fm(\))p Fk(s)2751 3904 y Fn(@)2802 3919 y Fk(s)2839 3904 y Fn(A)p Fo(\()p Fn(s)p Fo(\))p Fg(P)3111 3919 y Ff(c)3151 3904 y Fn(\014)6 b( )3275 3919 y Fm(0)3314 3904 y Fn(ds)314 b Fo(\(4.15\))-74 4161 y(The)36 b(c)m(hoice)g(of)f(regularization,)f Fl(\000)p Fn(i")p Fo(,)i(in)f(\(4.14\))g(ensures)i(that)e(the)h(latter) 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Fn(t)p Fo(\))p Fn(A)p Fo(\()p Fn(t)p Fo(\))55 b(+)f Fn(R)2146 323 y Fm(1)p Fk(a)2224 308 y Fo(\()p Fn(t)p Fo(\))g(+)h Fn(R)2594 323 y Fm(1)p Fk(b)2664 308 y Fo(\()p Fn(t)p Fo(\))p Fn(;)923 b Fo(\(4.16\))1081 482 y Fn(R)1155 497 y Fm(2)1311 482 y Fo(=)115 b(\()p Fn(\032)1590 497 y Fm(2)1630 482 y Fo(\()p Fn(t)p Fo(\))55 b Fl(\000)g Fo(\000\))p Fn(A)p Fo(\()p Fn(t)p Fo(\))p Fn(;)1487 b Fo(\(4.17\))-74 731 y Fb(where)178 980 y Fn(\032)228 995 y Fm(1)268 980 y Fo(\()p Fn(t)p Fo(\))116 b(=)672 913 y Fn(i)p 665 957 49 4 v 665 1049 a Fo(4)723 980 y(\(1)22 b(+)g Fn(e)975 939 y Fm(2)p Fk(i\026t)1107 980 y Fo(\))1162 884 y Fe(\020)1211 980 y Fn(\014)6 b( )1335 995 y Fm(0)1375 980 y Fn(;)17 b Fo(\()p Fn(H)1538 995 y Fm(0)1599 980 y Fl(\000)22 b Fn(\025)1755 995 y Fm(0)1817 980 y Fo(+)g Fn(\026)p Fo(\))2012 939 y Fi(\000)p Fm(1)2106 980 y Fg(P)2183 995 y Ff(c)2222 980 y Fn(\014)6 b( )2346 995 y Fm(0)2386 884 y Fe(\021)178 1214 y Fn(\032)228 1229 y Fm(2)268 1214 y Fo(\()p Fn(t)p Fo(\))116 b(=)705 1147 y Fn(i)p 697 1191 V 697 1283 a Fo(4)756 1214 y(\(1)22 b(+)g Fn(e)1008 1173 y Fi(\000)p Fm(2)p Fk(i\026t)1194 1214 y Fo(\))1249 1118 y Fe(\020)1298 1214 y Fn(\014)6 b( )1422 1229 y Fm(0)1462 1214 y Fn(;)17 b Fo(P)p Fn(:)p Fo(V)q Fn(:)33 b Fo(\()p Fn(H)1852 1229 y Fm(0)1914 1214 y Fl(\000)22 b Fn(\025)2070 1229 y Fm(0)2132 1214 y Fl(\000)g Fn(\026)p Fo(\))2328 1173 y Fi(\000)p Fm(1)2455 1214 y Fg(P)2532 1229 y Ff(c)2572 1214 y Fn(\014)6 b( )2696 1229 y Fm(0)2735 1118 y Fe(\021)494 1426 y Fl(\000)665 1359 y Fn(\031)p 665 1403 59 4 v 670 1495 a Fo(4)734 1426 y Fn(e)779 1385 y Fi(\000)p Fm(2)p Fk(i\026t)998 1426 y Fo(\()p Fn(\014)g( )1160 1441 y Fm(0)1199 1426 y Fn(;)17 b(\016)t Fo(\()p Fn(H)1409 1441 y Fm(0)1470 1426 y Fl(\000)23 b Fn(\025)1627 1441 y Fm(0)1688 1426 y Fl(\000)g Fn(\026)p Fo(\))p Fg(P)1962 1441 y Ff(c)2001 1426 y Fn(\014)6 b( )2125 1441 y Fm(0)2165 1426 y Fo(\))318 1638 y(\000)116 b(=)697 1571 y Fn(\031)p 697 1615 V 702 1706 a Fo(4)783 1638 y(\()p Fn(\014)6 b( )945 1653 y Fm(0)984 1638 y Fn(;)17 b(\016)t Fo(\()p Fn(H)1194 1653 y Fm(0)1255 1638 y Fl(\000)23 b Fn(\025)1412 1653 y Fm(0)1473 1638 y Fl(\000)g Fn(\026)p Fo(\))p Fg(P)1747 1653 y Ff(c)1786 1638 y Fn(\014)6 b( )1910 1653 y Fm(0)1950 1638 y Fo(\))116 1872 y Fn(R)190 1887 y Fm(1)p Fk(a)268 1872 y Fo(\()p Fn(t)p Fo(\))116 b(=)g Fl(\000)782 1805 y Fn(i)p 774 1849 49 4 v 774 1941 a Fo(2)833 1872 y Fn(A)p Fo(\(0\))p Fn(e)1076 1831 y Fk(i\025)1141 1840 y Fh(0)1176 1831 y Fk(t)1222 1872 y Fo(cos)18 b Fn(\026t)1513 1776 y Fe(\020)1562 1872 y Fn(\014)6 b( )1686 1887 y Fm(0)1726 1872 y Fn(;)49 b(e)1847 1831 y Fi(\000)p Fk(iH)1984 1840 y Fh(0)2019 1831 y Fk(t)2081 1872 y Fo(\()p Fn(H)2200 1887 y Fm(0)2261 1872 y Fl(\000)23 b Fn(\025)2418 1887 y Fm(0)2479 1872 y Fo(+)f Fn(\026)p Fo(\))2674 1831 y Fi(\000)p Fm(1)2768 1872 y Fg(P)2845 1887 y Ff(c)2885 1872 y Fn(\014)6 b( )3009 1887 y Fm(0)3049 1776 y Fe(\021)124 2106 y Fn(R)198 2121 y Fm(1)p Fk(b)268 2106 y Fo(\()p Fn(t)p Fo(\))116 b(=)714 2038 y(1)p 697 2082 83 4 v 697 2174 a(2)p Fn(i)789 2106 y(e)834 2065 y Fk(i\025)899 2074 y Fh(0)934 2065 y Fk(t)1013 2106 y Fo(cos)17 b Fn(\026t)1303 1984 y Fe(\022)1364 2106 y Fn(\014)6 b( )1488 2121 y Fm(0)1528 2106 y Fn(;)1572 1989 y Fe(Z)1655 2015 y Fk(t)1618 2177 y Fm(0)1733 2106 y Fn(e)1778 2065 y Fi(\000)p Fk(iH)1915 2074 y Fh(0)1950 2065 y Fm(\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))2122 2106 y Fn(e)2167 2065 y Fi(\000)p Fk(i)p Fm(\()p Fk(\025)2314 2074 y Fh(0)2349 2065 y Fi(\000)p Fk(\026)p Fm(\))p Fk(s)2511 2106 y Fn(@)2562 2121 y Fk(s)2599 2106 y Fn(A)p Fo(\()p Fn(s)p Fo(\)\()p Fn(H)2913 2121 y Fm(0)2974 2106 y Fl(\000)23 b Fn(\025)f Fo(+)g Fn(\026)p Fo(\))3348 2065 y Fi(\000)p Fm(1)3442 2106 y Fg(P)3519 2121 y Ff(c)3559 2106 y Fn(\014)6 b( )3683 2121 y Fm(0)3722 1984 y Fe(\023)-74 2363 y Fo(Pro)s(of.)p -74 2376 262 4 v 32 w(T)-8 b(o)33 b(obtain)e(\(4.17\))h(w)m(e)i(use)f (the)g(w)m(ell)f(kno)m(wn)i(distributional)29 b(iden)m(tit)m(y)1279 2612 y(\()p Fn(x)22 b Fl(\000)h Fn(i)p Fo(0\))1614 2571 y Fi(\000)p Fm(1)1736 2612 y Fo(=)k Fn(P)s(:V)5 b(:)33 b(x)2109 2571 y Fi(\000)p Fm(1)2226 2612 y Fo(+)22 b Fn(i\031)t(\016)t Fo(\()p Fn(x)p Fo(\))p Fn(:)1104 b Fo(\(4.18\))-74 2861 y(Equation)32 b(\(4.16\))g(follo)m(ws)g(b)m(y)h (in)m(tegration)e(b)m(y)i(parts.)174 b([])72 3011 y(Com)m(bining)31 b(equation)i(\(4.10\))e(with)i(the)g(previous)g(prop)s(ositions,)e(w)m (e)j(ha)m(v)m(e)-74 3268 y Fg(Prop)s(osition)i(4.4.)1089 3417 y Fn(@)1140 3432 y Fk(t)1170 3417 y Fn(A)p Fo(\()p Fn(t)p Fo(\))60 b(=)g(\()33 b Fl(\000)p Fo(\000)55 b(+)g Fn(\032)p Fo(\()p Fn(t)p Fo(\))32 b(\))50 b Fn(A)p Fo(\()p Fn(t)p Fo(\))k(+)h Fn(E)6 b Fo(\()p Fn(t)p Fo(\))p Fn(;)914 b Fo(\(4.19\))-74 3621 y Fb(where)105 3870 y Fn(E)6 b Fo(\()p Fn(t)p Fo(\))60 b(=)g Fn(R)564 3885 y Fm(1)p Fk(a)664 3870 y Fo(+)22 b Fn(R)836 3885 y Fm(1)p Fk(b)928 3870 y Fo(+)g Fn(R)1100 3885 y Fm(3)1162 3870 y Fo(+)g Fn(R)1334 3885 y Fm(4)1395 3870 y Fl(\000)h Fn(ie)1573 3829 y Fk(i\025)1638 3838 y Fh(0)1673 3829 y Fk(t)1720 3870 y Fo(cos)17 b Fn(\026t)g Fo(\()o Fn(\014)6 b( )2139 3885 y Fm(0)2179 3870 y Fn(;)17 b(\036)2281 3885 y Fm(0)2320 3870 y Fo(\()p Fn(t)p Fo(\)\))38 b Fl(\000)23 b Fn(ie)2685 3829 y Fk(i\025)2750 3838 y Fh(0)2785 3829 y Fk(t)2832 3870 y Fo(cos)17 b Fn(\026t)49 b Fo(\()p Fn(\014)6 b( )3284 3885 y Fm(0)3323 3870 y Fn(;)17 b(\036)3425 3885 y Fm(2)3464 3870 y Fo(\))g Fn(:)179 b Fo(\(4.20\))-74 4119 y Fb(Here,)34 b Fn(\036)242 4134 y Fm(0)313 4119 y Fb(and)f Fn(\036)561 4134 y Fm(2)633 4119 y Fb(are)f(giv)m(en)h(in)f(\(4.9\),)g Fn(R)1498 4134 y Fm(1)p Fk(a)1607 4119 y Fb(and)h Fn(R)1871 4134 y Fm(1)p Fk(b)1974 4119 y Fb(in)e(\(4.18\),)h Fn(R)2470 4134 y Fm(3)2542 4119 y Fb(and)h Fn(R)2806 4134 y Fm(4)2878 4119 y Fb(in)f(\(4.11\),)g(and)716 4368 y Fn(\032)p Fo(\()p Fn(t)p Fo(\))116 b Fl(\021)g(\000)p Fn(i)17 b Fo(cos)h Fn(\026t)e Fo(\()p Fn( )1672 4383 y Fm(0)1712 4368 y Fn(;)h(\014)6 b( )1880 4383 y Fm(0)1919 4368 y Fo(\))55 b(+)g Fn(\032)2193 4383 y Fm(1)2232 4368 y Fo(\()p Fn(t)p Fo(\))23 b(+)f Fn(\032)2514 4383 y Fm(2)2553 4368 y Fo(\()p Fn(t)p Fo(\))994 4559 y(=)1204 4491 y Fn(i)p 1196 4536 49 4 v 1196 4627 a Fo(4)1272 4463 y Fe(\020)1321 4559 y Fn(\014)6 b( )1445 4574 y Fm(0)1485 4559 y Fn(;)17 b Fo(\()p Fn(H)1648 4574 y Fm(0)1709 4559 y Fl(\000)22 b Fn(\025)1865 4574 y Fm(0)1927 4559 y Fo(+)g Fn(\026)p Fo(\))2122 4518 y Fi(\000)p Fm(1)2216 4559 y Fg(P)2293 4574 y Ff(c)2333 4559 y Fn(\014)6 b( )2457 4574 y Fm(0)2496 4463 y Fe(\021)994 4793 y Fo(+)1204 4726 y Fn(i)p 1196 4770 V 1196 4861 a Fo(4)1272 4697 y Fe(\020)1321 4793 y Fn(\014)g( )1445 4808 y Fm(0)1485 4793 y Fn(;)17 b Fo(P)p Fn(:)p Fo(V)q Fn(:)33 b Fo(\()p Fn(H)1875 4808 y Fm(0)1936 4793 y Fl(\000)23 b Fn(\025)2093 4808 y Fm(0)2154 4793 y Fl(\000)g Fn(\026)p Fo(\))2351 4752 y Fi(\000)p Fm(1)2445 4793 y Fg(P)2522 4808 y Ff(c)2562 4793 y Fn(\014)6 b( )2686 4808 y Fm(0)2725 4697 y Fe(\021)2830 4793 y Fo(+)54 b Fn(m)p Fo(\()p Fn(t)p Fo(\))p Fn(;)542 b Fo(\(4.21\))-74 5042 y Fb(where)34 b Fn(m)p Fo(\()p Fn(t)p Fo(\))f Fb(is)f(a)g(b)s (ounded)i(and)e(oscillatory)f(function)h(of)g(zero)h(mean.)1901 5306 y Fo(18)p eop %%Page: 19 19 19 18 bop -74 59 a Fg(Remarks:)-74 208 y Fo(\(1\))23 b(The)i(p)s(oin)m(t)d(of)h(\(4.19\))g(is)g(that)g(the)h(damping)e(co)s (e\016cien)m(t)i(\(\000)j Fn(>)h Fo(0\),)d(whic)m(h)f(arises)f(due)h (to)f(the)h(coupling)e(of)-74 358 y(the)29 b(discrete)f(b)s(ound)h (state)f(to)g(the)g(con)m(tin)m(uum)g(mo)s(des)g(b)m(y)h(the)f(p)s (erio)s(dic)f(p)s(erturbation,)h(is)g(made)f(explicit.)-74 507 y(This)34 b(facilitates)d(a)j(direct)f(study)i(of)f(the)g(transfer) g(of)f(energy)i(from)d(the)i(b)s(ound)g(state)h(to)e(the)h(con)m(tin)m (uum)-74 656 y(and)f(the)g(asso)s(ciated)f(radiativ)m(e)g(deca)m(y)i (of)e(solutions.)-74 806 y(\(2\))44 b(The)g(leading)f(order)g(part)h (of)f(equation)h(\(4.19\))f(is)g(the)i(analogue)d(of)i(the)g(disp)s (ersiv)m(e)g(normal)e(form)-74 955 y(deriv)m(ed)34 b(in)d([39])i(for)f (a)g(class)h(of)f(nonlinear)f(disp)s(ersiv)m(e)i(w)m(a)m(v)m(e)i (equations.)72 1176 y(In)e(the)g(next)h(section)f(w)m(e)g(estimate)f (the)h(remainder)f(terms)g(in)g(\(4.9\))g(and)h(\(4.19\).)-74 1468 y Fp(5.)46 b(ODE)f(Estimates)-74 1716 y Fo(Equations)33 b(\(4.9\))f(and)h(\(4.19\))f(comprise)h(a)f(dynamical)f(system)j(go)m (v)m(erning)f Fn(\036)2873 1731 y Fk(d)2913 1716 y Fo(\()p Fn(t)p Fo(\))g(and)g Fn(a)p Fo(\()p Fn(t)p Fo(\),)g(the)g(solution)-74 1865 y(of)48 b(whic)m(h)h(is)e(equiv)-5 b(alen)m(t)48 b(to)g(the)h(original)c(equation)j(\(1.1\).)89 b(Our)49 b(aim)d(in)h(this)h(decomp)s(osition)e(is)i(to)-74 2015 y(separate)29 b(the)g(disp)s(ersiv)m(e)g(\(p)s(olynomially)24 b(time-deca)m(ying\))i(part)i Fn(\036)2452 2030 y Fk(d)2493 2015 y Fo(\()p Fn(t)p Fo(\))g(from)f(the)h(solution,)g(as)h(w)m(ell)e (as)h(the)-74 2164 y(transien)m(t)39 b(\(exp)s(onen)m(tially)e (time-deca)m(ying\))g(part.)61 b(In)38 b(this)g(and)h(in)f(the)g(follo) m(wing)e(section)j(w)m(e)g(deriv)m(e)g(a)-74 2313 y(coupled)27 b(system)g(of)f(estimates)g(for)g Fn(A)p Fo(\()p Fn(t)p Fo(\))h(and)g Fn(\036)1719 2328 y Fk(d)1759 2313 y Fo(\()p Fn(t)p Fo(\),)h(w)m(e)f(sho)m(w)h(that)e Fn(A)p Fo(\()p Fn(t)p Fo(\))h(deca)m(ys)h(in)e(time,)h(pro)m(vided)g Fn(\036)3823 2328 y Fk(d)3863 2313 y Fo(\()p Fn(t)p Fo(\))-74 2463 y(is)k(disp)s(ersiv)m(ely)h(deca)m(ying)h(and)f(vice-v)m(ersa.)44 b(Then,)33 b(w)m(e)g(exploit)e(the)h(assumption)f(that)h(the)g(p)s (oten)m(tial,)e Fn(W)-74 2612 y Fo(is)i(small)e(to)j(close)f(the)h (resulting)f(inequalities,)f(and)h(pro)m(v)m(e)i(the)f(deca)m(y)h(of)e (b)s(oth)h Fn(A)p Fo(\()p Fn(t)p Fo(\))f(and)h Fn(\036)3439 2627 y Fk(d)3479 2612 y Fo(\()p Fn(t)p Fo(\).)72 2762 y(A)m(t)d(this)f(p)s(oin)m(t,)h(w)m(e)g(state)g(a)g(simple)e(Lemma)g (whic)m(h)i(w)m(e)g(shall)f(use)h(in)f(a)g(n)m(um)m(b)s(er)h(of)f (places)h(in)f(this)g(and)-74 2911 y(in)j(the)h(next)g(section.)-74 3111 y Fg(Lemma)k(5.1.)49 b Fb(Let)33 b Fn(\013)28 b(>)g Fo(1)p Fb(.)976 3192 y Fe(Z)1059 3219 y Fk(t)1022 3381 y Fm(0)1121 3309 y Fl(h)p Fn(t)22 b Fl(\000)g Fn(s)p Fl(i)1401 3268 y Fi(\000)p Fk(\013)1538 3309 y Fl(h)p Fn(s)p Fl(i)1662 3268 y Fi(\000)p Fk(\014)1796 3309 y Fn(ds)60 b Fl(\024)28 b Fn(C)2128 3324 y Fk(\013;\014)2272 3309 y Fl(h)p Fn(t)p Fl(i)2385 3268 y Fi(\000)12 b Fm(min)n(\()p Fk(\014)s(;\014)s Fm(+)p Fk(\013)p Fi(\000)p Fm(1\))3773 3309 y Fo(\(5.1\))-74 3509 y(Pro)s(of:)p -74 3522 262 4 v 26 w(The)28 b(b)s(ound)f(is)f(obtained)g(b)m(y)i(viewing)e(the)h (in)m(tegral)e(as)i(decomp)s(osed)h(in)m(to)e(a)g(part)h(o)m(v)m(er)h ([0)p Fn(;)17 b(t=)p Fo(2])26 b(and)-74 3658 y(the)36 b(part)f(o)m(v)m(er)h([)p Fn(t=)p Fo(2)p Fn(;)17 b(t)p Fo(].)51 b(W)-8 b(e)35 b(estimate)f(the)i(in)m(tegral)d(o)m(v)m(er)k ([0)p Fn(;)17 b(t=)p Fo(2])34 b(b)m(y)i(b)s(ounding)f Fl(h)p Fn(t)23 b Fl(\000)i Fn(s)p Fl(i)3339 3622 y Fi(\000)p Fk(\013)3478 3658 y Fo(b)m(y)36 b(its)e(v)-5 b(alue)-74 3808 y(at)37 b Fn(t=)p Fo(2)h(and)f(explicitly)f(computing)g(the)i (remaining)d(in)m(tegral.)57 b(The)39 b(in)m(tegral)d(o)m(v)m(er)i([)p Fn(t=)p Fo(2)p Fn(;)17 b(t)p Fo(])38 b(is)f(computed)-74 3957 y(b)m(y)j(b)s(ounding)e Fl(h)p Fn(s)p Fl(i)629 3921 y Fi(\000)p Fk(\014)769 3957 y Fo(b)m(y)i(its)e(v)-5 b(alue)39 b(at)f Fn(t=)p Fo(2)h(and)g(again)e(computing)h(explicitly)f (the)i(remaining)d(in)m(tegral.)-74 4107 y(Putting)c(the)h(t)m(w)m(o)g (estimates)g(together)f(yields)h(the)g(lemma.)72 4256 y(W)-8 b(e)27 b(no)m(w)h(turn)e(to)g(the)h(estimate)f(for)g Fn(A)p Fo(\()p Fn(t)p Fo(\))h(in)e(terms)i(of)f(the)h(disp)s(ersiv)m(e) g(norm)f(of)g Fn(\036)3150 4271 y Fk(d)3190 4256 y Fo(\()p Fn(t)p Fo(\))h(and)f(lo)s(cal)e(deca)m(y)-74 4406 y(estimates)32 b(for)g Fn(e)549 4369 y Fi(\000)p Fk(iH)686 4378 y Fh(0)721 4369 y Fk(t)751 4406 y Fg(P)828 4421 y Ff(c)868 4406 y Fo(\()p Fn(H)987 4421 y Fm(0)1026 4406 y Fo(\).)-74 4605 y Fg(Prop)s(osition)k(5.1.)49 b Fn(A)p Fo(\()p Fn(t)p Fo(\))p Fb(,)33 b(the)g(solution)e(of)h(\(4.19\),)g(can)h(b)s(e)f (expanded)j(as:)1132 4804 y Fn(A)p Fo(\()p Fn(t)p Fo(\))116 b(=)f Fn(e)1668 4697 y Fe(R)1723 4717 y Fa(t)1707 4793 y Fh(0)1763 4759 y Fk(\032)p Fm(\()p Fk(s)p Fm(\))p Fk(ds)1977 4707 y Fe(\020)2026 4804 y Fn(e)2071 4763 y Fi(\000)p Fm(\000)p Fk(t)2200 4804 y Fn(A)p Fo(\(0\))22 b(+)g Fn(R)2592 4819 y Fk(A)2649 4804 y Fo(\()p Fn(t)p Fo(\))2760 4707 y Fe(\021)3773 4804 y Fo(\(5.2\))1074 5019 y Fn(R)1148 5034 y Fk(A)1205 5019 y Fo(\()p Fn(t)p Fo(\))116 b(=)1623 4901 y Fe(Z)1706 4928 y Fk(t)1669 5090 y Fm(0)1785 5019 y Fn(e)1830 4977 y Fi(\000)p Fm(\000\()p Fk(t)p Fi(\000)p Fk(\034)8 b Fm(\))2163 4993 y Fo(~)2140 5019 y Fn(E)e Fo(\()p Fn(\034)11 b Fo(\))33 b Fn(d\034)6 b(;)1267 b Fo(\(5.3\))1901 5306 y(19)p eop %%Page: 20 20 20 19 bop -74 59 a Fb(where)231 33 y Fo(~)208 59 y Fn(E)6 b Fo(\()p Fn(t)p Fo(\))33 b Fb(is)f(giv)m(en)g(in)g(\(4.20\))g(and)h (\(5.9\).)43 b(F)-8 b(or)31 b(an)m(y)j Fn(\013)28 b(>)g Fo(1)p Fb(,)k Fn(R)2374 74 y Fk(A)2431 59 y Fo(\()p Fn(t)p Fo(\))h Fb(satis\014es)g(the)g(estimates:)225 300 y Fo(sup)119 384 y Fm(2\000)198 365 y Fc(\000)p Fa(\013)291 384 y Fi(\024)p Fk(t)p Fi(\024)p 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1771 y(Then,)234 1746 y(~)208 1771 y Fn(A)33 b Fo(satis\014es)g(the)g(equation)1459 2013 y Fn(@)1510 2028 y Fk(t)1566 1988 y Fo(~)1541 2013 y Fn(A)83 b Fo(=)g Fl(\000)p Fo(\000)2020 1988 y(~)1994 2013 y Fn(A)22 b Fo(+)2210 1988 y(~)2187 2013 y Fn(E)6 b Fo(\()p Fn(t)p Fo(\))1397 b(\(5.8\))1447 2162 y(~)1425 2187 y Fn(E)6 b Fo(\()p Fn(t)p Fo(\))83 b(=)g Fn(e)1901 2143 y Fi(\000)1967 2080 y Fe(R)2023 2101 y Fa(t)2007 2176 y Fh(0)2062 2143 y Fk(\032)p Fm(\()p Fk(s)p Fm(\))p Fk(ds)2259 2187 y Fn(E)6 b Fo(\()p Fn(t)p Fo(\))p Fn(:)1298 b Fo(\(5.9\))-74 2428 y(Solving)31 b(\(5.8\))h(w)m(e)i(get)1064 2660 y(~)1038 2685 y Fn(A)p Fo(\()p Fn(t)p Fo(\))117 b(=)f Fn(e)1576 2644 y Fi(\000)p Fm(\000)p Fk(t)1730 2660 y Fo(~)1705 2685 y Fn(A)p Fo(\(0\))54 b(+)2088 2568 y Fe(Z)2171 2594 y Fk(t)2134 2757 y Fm(0)2217 2685 y Fn(e)2262 2644 y Fi(\000)p Fm(\000\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))2556 2660 y Fo(~)2533 2685 y Fn(E)6 b Fo(\()p Fn(s)p Fo(\))32 b Fn(ds)863 b Fo(\(5.10\))1338 2881 y Fl(\021)116 b 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5306 y(20)p eop %%Page: 21 21 21 20 bop 72 59 a Fo(F)-8 b(rom)31 b(\(5.10\))h(w)m(e)i(ha)m(v)m(e)g (for)e(an)m(y)h Fn(M)39 b(>)27 b Fo(0:)514 314 y Fl(j)567 289 y Fo(~)542 314 y Fn(A)p Fo(\()p Fn(t)p Fo(\))p Fl(j)115 b(\024)84 b(j)p Fn(A)p Fo(\(0\))p Fl(j)p Fn(e)1329 273 y Fi(\000)p Fm(\000)p Fk(t)1479 314 y Fo(+)1577 197 y Fe(Z)1660 223 y Fk(M)1623 385 y Fm(0)1755 314 y Fn(e)1800 273 y Fi(\000)p Fm(\000\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))2072 314 y Fl(j)2122 289 y Fo(~)2100 314 y Fn(E)5 b Fo(\()p Fn(s)p Fo(\))p Fl(j)p Fn(ds)22 b Fo(+)2544 197 y Fe(Z)2627 223 y Fk(t)2590 385 y(M)2685 314 y Fn(e)2730 273 y Fi(\000)p Fm(\000\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))3002 314 y Fl(j)3052 289 y Fo(~)3030 314 y Fn(E)5 b Fo(\()p Fn(s)p Fo(\))p Fl(j)32 b Fn(ds)870 505 y Fo(=)116 b Fl(j)p Fn(A)p Fo(\(0\))p Fl(j)p Fn(e)1361 464 y Fi(\000)p Fm(\000)p Fk(t)1544 505 y Fo(+)54 b Fn(I)1717 520 y Fm(1)1757 505 y Fo(\()p Fn(t)p Fo(\))h(+)f Fn(I)2096 520 y Fm(2)2136 505 y Fo(\()p Fn(t)p Fo(\))p Fn(:)1451 b 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1568 a(2)1771 1542 y Fm(\000)p Fk(t)1867 1583 y Fl(\001)1917 1466 y Fe(Z)2000 1493 y Fk(M)1963 1655 y Fm(0)2096 1583 y Fn(e)2141 1542 y Fi(\000)p Fm(\000\()2277 1515 y Fh(1)p 2277 1527 V 2277 1568 a(2)2318 1542 y Fk(t)p Fi(\000)p Fk(s)p Fm(\))2495 1583 y Fn(ds)54 b Fl(\001)109 b Fo(sup)2696 1667 y Fm(0)p Fi(\024)p Fk(\034)8 b Fi(\024)p Fm(\000)2924 1648 y Fc(\000)p Fa(\013)3033 1583 y Fl(j)3083 1558 y Fo(~)3061 1583 y Fn(E)d Fo(\()p Fn(\034)11 b Fo(\))p Fl(j)1187 1819 y(\024)228 b Fo(sup)1380 1903 y Fm(2\000)1459 1884 y Fc(\000)p Fa(\013)1552 1903 y Fi(\024)p Fk(\034)8 b Fi(\024)p Fk(T)1751 1819 y Fo(\()p Fl(h)p Fn(t)p Fl(i)1902 1778 y Fk(r)r Fi(\000)p Fm(1)2030 1819 y Fn(e)2075 1778 y Fi(\000)p Fm(\000)2187 1751 y Fa(t)p 2184 1763 V 2184 1804 a Fh(2)2229 1819 y Fo(\))22 b Fl(\001)g Fn(C)7 b Fo(\000)2477 1778 y Fi(\000)p Fm(1)2593 1819 y Fl(\001)108 b Fo(sup)2643 1903 y Fm(0)p Fi(\024)p Fk(\034)8 b Fi(\024)p Fm(\000)2871 1884 y Fc(\000)p Fa(\013)2980 1819 y Fl(j)3030 1794 y Fo(~)3008 1819 y Fn(E)d Fo(\()p Fn(\034)11 b Fo(\))p Fl(j)1187 2055 y(\024)116 b Fn(C)7 b(e)1502 2014 y Fi(\000)p Fm(\000)1601 1990 y Fc(\000)p Fa(\016)1823 2055 y Fo(sup)1736 2139 y Fm(0)p Fi(\024)p Fk(\034)h Fi(\024)p Fm(\000)1964 2120 y Fc(\000)p Fa(\013)2073 2055 y Fl(j)2124 2030 y Fo(~)2101 2055 y Fn(E)e Fo(\()p Fn(\034)11 b Fo(\))p Fl(j)p Fn(;)1362 b Fo(\(5.14\))-74 2322 y(for)32 b(some)h Fn(\016)e(>)d Fo(0.)43 b(Therefore,)1008 2557 y(sup)896 2641 y Fm(2\000)975 2622 y Fc(\000)p Fa(\013)1068 2641 y Fi(\024)p Fk(\034)8 b Fi(\024)p Fk(T)1284 2460 y Fe(\020)1333 2557 y Fl(h)p Fn(t)p Fl(i)1446 2516 y Fk(r)r Fi(\000)p Fm(1)1607 2557 y Fn(I)1650 2572 y Fm(1)1689 2557 y Fo(\()p Fn(t)p Fo(\))1800 2460 y Fe(\021)1910 2557 y Fl(\024)61 b Fn(C)7 b(e)2170 2516 y Fi(\000)p Fm(\000)2269 2492 y Fc(\000)p Fa(\016)2491 2557 y Fo(sup)2404 2641 y Fm(0)p Fi(\024)p Fk(\034)h Fi(\024)p Fm(\000)2632 2622 y Fc(\000)p Fa(\013)2741 2557 y Fl(j)2792 2532 y Fo(~)2769 2557 y Fn(E)e Fo(\()p Fn(\034)11 b Fo(\))p 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-74 59 a Fg(Lemma)37 b(6.1.)49 b Fb(F)-8 b(or)32 b(an)m(y)h Fn(\021)f Fl(2)c Fo([0)p Fn(;)17 b(r)24 b Fl(\000)f Fo(1])687 187 y Fe(\015)687 237 y(\015)687 287 y(\015)687 337 y(\015)733 195 y(Z)816 221 y Fk(t)780 383 y Fm(0)863 312 y Fn(w)933 327 y Fi(\000)991 312 y Fn(e)1036 271 y Fi(\000)p Fk(iH)1173 280 y Fh(0)1208 271 y Fm(\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))1380 312 y Fg(P)1457 327 y Ff(c)1497 312 y Fn(f)11 b Fo(\()p Fn(s)p Fo(\))p Fn(ds)1775 187 y Fe(\015)1775 237 y(\015)1775 287 y(\015)1775 337 y(\015)1881 312 y Fl(\024)60 b Fn(C)7 b Fl(h)p Fn(t)p Fl(i)2208 271 y Fi(\000)p Fk(\021)2353 312 y Fo(sup)2321 390 y Fm(0)p Fi(\024)p Fk(\034)h Fi(\024)p Fk(t)2547 312 y Fo(\()p Fl(h)p Fn(\034)j Fl(i)2716 271 y Fk(\021)2758 312 y Fl(k)p Fn(w)2878 327 y Fm(+)2937 312 y Fn(f)g Fo(\()p Fn(\034)g Fo(\))p Fl(k)p Fo(\))560 b(\(6.1\))-74 575 y Fb(and)94 688 y Fe(\015)94 738 y(\015)94 788 y(\015)94 838 y(\015)140 696 y(Z)223 722 y Fk(t)186 884 y Fm(0)270 813 y Fn(w)340 828 y Fi(\000)398 813 y 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b(Namely)-8 b(,)-74 1244 y(using)32 b(that)h Fn(r)d(>)e Fo(2,)225 1368 y Fe(\015)225 1417 y(\015)225 1467 y(\015)225 1517 y(\015)271 1375 y(Z)354 1401 y Fk(t)317 1564 y Fm(0)400 1492 y Fn(w)470 1507 y Fi(\000)529 1492 y Fn(e)574 1451 y Fi(\000)p Fk(iH)711 1460 y Fh(0)746 1451 y Fm(\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))918 1492 y Fg(P)995 1507 y Ff(c)1035 1492 y Fn(f)11 b Fo(\()p Fn(s)p Fo(\))32 b Fn(ds)1345 1368 y Fe(\015)1345 1417 y(\015)1345 1467 y(\015)1345 1517 y(\015)1506 1492 y Fl(\024)1699 1375 y Fe(Z)1782 1401 y Fk(t)1745 1564 y Fm(0)1828 1492 y Fl(k)p Fn(w)1948 1507 y Fi(\000)2007 1492 y Fn(e)2052 1451 y Fi(\000)p Fk(iH)2189 1460 y Fh(0)2224 1451 y Fm(\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))2428 1492 y Fg(P)2505 1507 y Ff(c)2545 1492 y Fn(w)2615 1507 y Fi(\000)2674 1492 y Fl(k)2724 1508 y Fi(L)p Fm(\()p Fi(H)p Fm(\))2891 1492 y Fl(h)p Fn(s)p Fl(i)3015 1451 y Fi(\000)p Fk(\021)3144 1492 y Fn(ds)1699 1675 y Fl(\001)48 b Fo(sup)1744 1752 y Fm(0)p Fi(\024)p Fk(\034)8 b Fi(\024)p Fk(t)1969 1675 y Fo(\()p Fl(h)p Fn(\034)j Fl(i)2138 1634 y Fk(\021)2180 1675 y Fl(k)p Fn(w)2300 1690 y Fm(+)2359 1675 y Fn(f)g Fo(\()p Fn(\034)g Fo(\))p Fl(k)p Fo(\))1506 1937 y Fl(\024)116 b Fn(C)1793 1820 y Fe(Z)1876 1846 y Fk(t)1839 2008 y Fm(0)1905 1937 y Fl(h)p Fn(t)22 b Fl(\000)h Fn(s)p Fl(i)2186 1896 y Fi(\000)p Fk(r)r Fm(+1)2369 1937 y Fl(h)p Fn(s)p Fl(i)2493 1896 y Fi(\000)p Fk(\021)2621 1937 y Fn(ds)80 b Fo(sup)2767 2014 y Fm(0)p Fi(\024)p Fk(\034)8 b Fi(\024)p Fk(t)2993 1937 y Fo(\()p Fl(h)p Fn(\034)j Fl(i)3162 1896 y Fk(\021)3204 1937 y Fl(k)p Fn(w)3324 1952 y Fm(+)3383 1937 y Fn(f)g Fo(\()p Fn(\034)g Fo(\))p Fl(k)p Fo(\))1506 2148 y Fl(\024)83 b Fn(C)7 b Fl(h)p Fn(t)p Fl(i)1856 2107 y Fi(\000)p Fk(\021)2001 2148 y Fo(sup)1970 2226 y Fm(0)p Fi(\024)p Fk(\034)h Fi(\024)p Fk(t)2196 2148 y Fo(\()p Fl(h)p Fn(\034)j Fl(i)2365 2107 y Fk(\021)2423 2148 y Fl(k)p Fn(w)2543 2163 y Fm(+)2602 2148 y Fn(f)g Fo(\()p Fn(\034)g Fo(\))p Fl(k)p Fo(\))-74 2417 y(whic)m(h)32 b(pro)m(v)m(es)g(\(6.1\).)43 b(The)31 b(pro)s(of)f(of)h(\(6.2\))f(is)h (iden)m(tical,)e(and)i(uses)i(the)e(singular)e(lo)s(cal)g(deca)m(y)j (estimate)e(of)-74 2566 y Fg(\(H3\))p Fo(.)72 2716 y(W)-8 b(e)33 b(no)m(w)h(de\014ne)f(the)g(norms)1354 2865 y([)p Fn(A)p Fo(])1481 2880 y Fk(\013)1530 2865 y Fo(\()p Fn(T)14 b Fo(\))60 b(=)104 b(sup)1873 2944 y Fm(0)p Fi(\024)p Fk(\034)8 b Fi(\024)p Fk(T)2108 2865 y Fl(h)p Fn(\034)j Fl(i)2239 2824 y Fk(\013)2289 2865 y Fl(j)p Fn(A)p Fo(\()p Fn(\034)g Fo(\))p Fl(j)1226 b Fo(\(6.3\))-74 3090 y(and)1178 3239 y([)p Fn(\036)1263 3254 y Fk(d)1304 3239 y Fo(])1331 3254 y Fk(LD)r(;\013)1508 3239 y Fo(\()p Fn(T)14 b Fo(\))59 b(=)104 b(sup)1850 3318 y Fm(0)p Fi(\024)p Fk(\034)8 b Fi(\024)p Fk(T)2085 3239 y Fl(h)p Fn(\034)j Fl(i)2216 3198 y Fk(\013)2266 3239 y Fl(k)p Fn(w)2386 3254 y Fi(\000)2444 3239 y Fn(\036)2502 3254 y Fk(d)2543 3239 y Fo(\()p Fn(\034)g Fo(\))p Fl(k)1051 b Fo(\(6.4\))-74 3464 y(Then)34 b(w)m(e)f(ha)m(v)m(e) -74 3709 y Fg(Prop)s(osition)j(6.1.)49 b Fb(F)-8 b(or)32 b(an)m(y)h Fn(T)41 b(>)28 b Fo(0)p Fb(,)796 3946 y Fo([)p Fn(\036)881 3961 y Fk(d)921 3946 y Fo(])948 3961 y Fk(LD)r(;r)r Fi(\000)p Fm(1)1204 3946 y Fo(\()p Fn(T)14 b Fo(\))27 b Fl(\024)h Fn(C)c Fo(\()32 b Fl(k)p Fn(w)1767 3961 y Fm(+)1826 3946 y Fn(\036)1884 3961 y Fk(d)1924 3946 y Fo(\(0\))p Fl(k)54 b Fo(+)h Fl(jjj)p Fn(W)14 b Fl(jjj)30 b Fo([)p Fn(A)p Fo(])2715 3961 y Fk(r)r Fi(\000)p Fm(1)2843 3946 y Fo(\()p Fn(T)14 b Fo(\))32 b(\))17 b Fn(:)669 b Fo(\(6.5\))-74 4191 y(Pro)s(of.)p -74 4204 V 31 w(F)-8 b(rom)30 b(equation)h(\(4.9\))f(w)m(e)j(get,)e(using)g(the)h(assumed)g (lo)s(cal)d(deca)m(y)k(estimate)d(for)h Fn(e)3325 4155 y Fi(\000)p Fk(iH)3462 4164 y Fh(0)3497 4155 y Fk(t)3558 4191 y Fo(and)g(\(6.1\),)281 4596 y Fl(k)o Fn(w)400 4611 y Fi(\000)459 4596 y Fn(\036)517 4611 y Fk(d)557 4596 y Fo(\()p Fn(t)p Fo(\))p Fl(k)116 b(\024)1070 4488 y Fm(2)1028 4513 y Fe(X)1027 4695 y Fk(j)t Fm(=0)1199 4596 y Fl(k)p Fn(w)1319 4611 y Fi(\000)1377 4596 y Fn(\036)1435 4611 y Fk(j)1471 4596 y Fo(\()p Fn(t)p Fo(\))p Fl(k)834 4834 y(\024)g Fn(C)7 b Fl(h)p Fn(t)p Fl(i)1217 4793 y Fi(\000)p Fk(r)r Fm(+1)1416 4834 y Fl(k)p Fn(w)1536 4849 y Fm(+)1595 4834 y Fn(\036)1653 4849 y Fk(d)1693 4834 y Fo(\(0\))p Fl(k)22 b Fo(+)54 b Fn(C)7 b Fl(h)p Fn(t)p Fl(i)2210 4793 y Fi(\000)p Fk(r)r Fm(+1)2425 4834 y Fo([)p Fn(A)p Fo(])2552 4849 y Fk(r)r Fi(\000)p Fm(1)2681 4834 y Fo(\()p Fn(t)p Fo(\))77 b(sup)2841 4911 y Fm(0)p Fi(\024)p Fk(s)p Fi(\024)p Fk(t)3061 4834 y Fl(k)p Fn(w)3181 4849 y Fm(+)3239 4834 y Fn(W)14 b Fo(\()p Fn(s)p Fo(\))p Fn( )3530 4849 y Fm(0)3570 4834 y Fl(k)835 5048 y Fo(+)116 b Fn(C)39 b Fl(jjj)p Fn(W)14 b Fl(jjj)31 b(h)p Fn(t)p Fl(i)1554 5007 y Fi(\000)p Fk(r)r Fm(+1)1753 5048 y Fo([)p Fn(\036)1838 5063 y Fk(d)1878 5048 y Fo(])1905 5077 y Fk(LD)r(;r)r Fi(\000)p Fm(1)2177 5048 y Fo(\()p Fn(t)p Fo(\))p Fn(:)1458 b Fo(\(6.6\))1901 5306 y(22)p eop %%Page: 23 23 23 22 bop 72 59 a Fo(Since)28 b Fl(k)p Fn(w)442 74 y Fm(+)501 59 y Fn(W)14 b Fo(\()p Fn(s)p Fo(\))p Fn( )792 74 y Fm(0)831 59 y Fl(k)28 b(\024)g(jjj)p Fn(W)14 b Fl(jjj)25 b(k)p Fn( )1426 74 y Fm(0)1466 59 y Fl(k)55 b Fo(=)g Fl(jjj)p Fn(W)14 b Fl(jjj)26 b Fo(and)h Fl(jjj)p Fn(W)14 b Fl(jjj)26 b Fo(is)h(assumed)h(to)g(b)s(e)g(small,)e(m)m(ultiplying) -74 208 y(b)s(oth)32 b(sides)i(of)e(this)g(last)g(equation)g(b)m(y)i Fl(h)p Fn(t)p Fl(i)1524 172 y Fk(r)r Fi(\000)p Fm(1)1684 208 y Fo(and)f(taking)e(suprem)m(um)i(o)m(v)m(er)h Fn(t)28 b Fl(\024)g Fn(T)46 b Fo(yields)32 b(\(6.5\).)206 b([])72 358 y(W)-8 b(e)33 b(no)m(w)h(estimate)d Fn(E)6 b Fo(\()p Fn(t)p Fo(\).)-74 615 y Fg(Prop)s(osition)36 b(6.2.)49 b Fb(Let)33 b Fn(T)41 b(>)27 b Fo(0)p Fb(.)44 b(F)-8 b(or)31 b(an)m(y)j Fn(\021)d Fl(2)d Fo([0)p Fn(;)17 b(r)25 b Fl(\000)d Fo(1])p Fb(:)371 864 y Fo([)p Fn(E)6 b Fo(])503 879 y Fk(\021)545 864 y Fo(\()p Fn(T)14 b Fo(\))60 b Fl(\024)g Fn(C)983 768 y Fe(\020)1065 864 y Fl(jjj)p Fn(W)14 b Fl(jjj)1339 823 y Fm(2)1409 864 y Fl(j)p Fn(A)p Fo(\(0\))p Fl(j)53 b Fo(+)i Fl(jjj)p Fn(W)14 b Fl(jjj)30 b(k)p Fn(w)2271 879 y Fm(+)2330 864 y Fn(\036)2388 879 y Fk(d)2428 864 y Fo(\(0\))p Fl(k)54 b Fo(+)h Fl(jjj)p Fn(W)14 b Fl(jjj)3062 823 y Fm(3)3131 864 y Fo([)p Fn(A)p Fo(])3258 879 y Fk(\021)3301 864 y Fo(\()p Fn(T)g Fo(\))3480 768 y Fe(\021)3773 864 y Fo(\(6.7\))-74 1121 y(Pro)s(of.)p -74 1134 262 4 v 72 1271 a Fn(E)6 b Fo(\()p Fn(t)p Fo(\))32 b(is)f(de\014ned)h(in)f(\(5.6\))g(and)g(\(4.19\).)42 b(F)-8 b(rom)30 b(these)j(equations)e(it)g(is)f(seen)j(that)e(w)m(e)h (need)h(to)e(estimate)-74 1420 y(the)i(follo)m(wing)d(terms:)253 1819 y(\()p Fn(\013)353 1834 y Fm(1)393 1819 y Fo(\))83 b Fl(j)p Fn(R)616 1834 y Fm(1)p Fk(a)692 1819 y Fl(j)28 b Fo(=)f Fl(j)p Fn(A)p Fo(\(0\))p Fl(j)1153 1719 y Fe(\014)1153 1769 y(\014)1153 1819 y(\014)1181 1722 y(\020)1231 1819 y Fn(\014)6 b( )1355 1834 y Fm(0)1394 1819 y Fn(;)17 b(e)1483 1778 y Fi(\000)p Fk(iH)1620 1787 y Fh(0)1655 1778 y Fk(t)1684 1819 y Fo(\()p Fn(H)1803 1834 y Fm(0)1865 1819 y Fl(\000)22 b Fn(\025)2021 1834 y Fm(0)2083 1819 y Fo(+)g Fn(\026)p Fo(\))2278 1778 y Fi(\000)p Fm(1)2372 1819 y Fg(P)2449 1834 y Ff(c)2489 1819 y Fn(\014)6 b( )2613 1834 y Fm(0)2652 1722 y Fe(\021)2702 1719 y(\014)2702 1769 y(\014)2702 1819 y(\014)253 2033 y Fo(\()p Fn(\013)353 2048 y Fm(2)393 2033 y Fo(\))83 b Fl(j)p Fn(R)616 2048 y Fm(1)p Fk(b)685 2033 y Fl(j)27 b Fo(=)h Fl(j)p Fn(A)p Fo(\(0\))p Fl(j)1115 1909 y Fe(\014)1114 1959 y(\014)1114 2009 y(\014)1114 2058 y(\014)1141 1916 y(Z)1224 1943 y Fk(t)1187 2105 y Fm(0)1271 1937 y Fe(\020)1320 2033 y Fn(\014)6 b( )1444 2048 y Fm(0)1484 2033 y Fn(;)17 b(e)1573 1992 y Fi(\000)p Fk(iH)1710 2001 y Fh(0)1744 1992 y Fm(\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))1916 2033 y Fn(e)1961 1992 y Fk(i)p Fm(\()p Fk(\026)p Fi(\000)p Fk(\025)2150 2001 y Fh(0)2186 1992 y Fm(\))p Fk(s)2250 2033 y Fo(\()p Fn(H)2369 2048 y Fm(0)2430 2033 y Fl(\000)23 b Fn(\025)2587 2048 y Fm(0)2649 2033 y Fo(+)f Fn(\026)p Fo(\))2844 1992 y Fi(\000)p Fm(1)2937 2033 y Fn(@)2988 2048 y Fk(s)3026 2033 y Fn(A)p Fo(\()p Fn(s)p Fo(\))p Fg(P)3298 2048 y Ff(c)3338 2033 y Fn(\014)6 b( )3462 2048 y Fm(0)3501 1937 y Fe(\021)3567 2033 y Fn(ds)3664 1909 y Fe(\014)3664 1959 y(\014)3664 2009 y(\014)3664 2058 y(\014)253 2273 y Fo(\()p Fn(\013)353 2288 y Fm(3)393 2273 y Fo(\))83 b Fl(j)p Fn(R)616 2288 y Fm(3)655 2273 y Fl(j)27 b Fo(=)824 2206 y(1)p 824 2250 49 4 v 824 2341 a(2)882 2273 y Fl(j)p Fn(A)p Fo(\(0\))p Fl(j)1185 2173 y Fe(\014)1185 2223 y(\014)1185 2273 y(\014)o Fo(\()p Fn(\014)6 b( )1374 2288 y Fm(0)1414 2273 y Fn(;)17 b(e)1503 2232 y Fi(\000)p Fk(iH)1640 2241 y Fh(0)1674 2232 y Fk(t)1704 2273 y Fo(\()p Fn(H)1823 2288 y Fm(0)1884 2273 y Fl(\000)23 b Fn(\025)2041 2288 y Fm(0)2102 2273 y Fl(\000)g Fn(\026)f Fl(\000)g Fn(i)p Fo(0\))2502 2232 y Fi(\000)p Fm(1)2597 2273 y Fg(P)2674 2288 y Ff(c)2714 2273 y Fn(\014)6 b( )2838 2288 y Fm(0)2877 2273 y Fo(\))2915 2173 y Fe(\014)2915 2223 y(\014)2915 2273 y(\014)253 2506 y Fo(\()p Fn(\013)353 2521 y Fm(4)393 2506 y Fo(\))83 b Fl(j)p Fn(R)616 2521 y Fm(4)655 2506 y Fl(j)27 b Fo(=)824 2439 y(1)p 824 2483 V 824 2575 a(2)899 2382 y Fe(\014)899 2432 y(\014)899 2481 y(\014)899 2531 y(\014)927 2389 y(Z)1010 2415 y Fk(t)973 2578 y Fm(0)1039 2506 y Fo(\()p Fn(\014)6 b( )1201 2521 y Fm(0)1241 2506 y Fn(;)17 b(e)1330 2465 y Fi(\000)p Fk(iH)1467 2474 y Fh(0)1501 2465 y Fm(\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))1673 2506 y Fo(\()p Fn(H)1792 2521 y Fm(0)1854 2506 y Fl(\000)22 b Fn(\025)2010 2521 y Fm(0)2072 2506 y Fl(\000)g Fn(\026)g Fl(\000)h Fn(i)p Fo(0\))2472 2465 y Fi(\000)p Fm(1)2566 2506 y Fn(e)2611 2465 y Fi(\000)p Fk(i)p Fm(\()p Fk(\025)2758 2474 y Fh(0)2793 2465 y Fm(+)p Fk(\026)p Fm(\))p Fk(s)2955 2506 y Fn(@)3006 2521 y Fk(s)3043 2506 y Fn(A)p Fo(\()p Fn(s)p Fo(\))p Fg(P)3315 2521 y Ff(c)3355 2506 y Fn(\014)6 b( )3479 2521 y Fm(0)3519 2506 y Fo(\))32 b Fn(ds)3686 2382 y Fe(\014)3686 2432 y(\014)3686 2481 y(\014)3686 2531 y(\014)253 2701 y Fo(\()p Fn(\013)353 2716 y Fm(5)393 2701 y Fo(\))83 b Fl(j)p Fo(\()p Fn(\014)6 b( )704 2716 y Fm(0)743 2701 y Fn(;)17 b(\036)845 2716 y Fm(0)883 2701 y Fo(\))p Fl(j)28 b Fo(=)f Fl(j)p Fo(\()p Fn(\014)6 b( )1270 2716 y Fm(0)1309 2701 y Fn(;)17 b(e)1398 2660 y Fi(\000)p Fk(iH)1535 2669 y Fh(0)1570 2660 y Fk(t)1599 2701 y Fn(\036)1657 2716 y Fk(d)1698 2701 y Fo(\(0\)\))p Fl(j)253 2891 y Fo(\()p Fn(\013)353 2906 y Fm(6)393 2891 y Fo(\))83 b Fl(j)p Fo(\()p Fn(\014)6 b( )704 2906 y Fm(0)743 2891 y Fn(;)17 b(\036)845 2906 y Fm(2)883 2891 y Fo(\()p Fn(t)p Fo(\)\))p Fl(j)28 b Fo(=)1191 2766 y Fe(\014)1191 2816 y(\014)1191 2866 y(\014)1191 2915 y(\014)1219 2773 y(Z)1302 2800 y Fk(t)1265 2962 y Fm(0)1332 2891 y Fo(\()p Fn(\014)6 b( )1494 2906 y Fm(0)1533 2891 y Fn(;)17 b(e)1622 2849 y Fi(\000)p Fk(iH)1759 2858 y Fh(0)1793 2849 y Fm(\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))1982 2891 y Fo(cos)g Fn(\026s)32 b Fg(P)2343 2906 y Ff(c)2416 2891 y Fn(\014)6 b(\036)2535 2906 y Fk(d)2575 2891 y Fo(\()p Fn(s)p Fo(\))32 b Fn(ds)p Fo(\))2864 2766 y Fe(\014)2864 2816 y(\014)2864 2866 y(\014)2864 2915 y(\014)2908 2891 y Fn(:)-74 3148 y Fo(The)i(estimates) e(of)g(\()p Fn(\013)767 3163 y Fk(j)803 3148 y Fo(\))h(rep)s(eatedly)g (use)g(Lemma)f(6.1.)43 b(Let)32 b Fn(\021)g Fl(2)c Fo([0)p Fn(;)17 b(r)24 b Fl(\000)f Fo(1].)-74 3297 y(Estimation)31 b(of)h(\()p Fn(\013)638 3312 y Fm(1)677 3297 y Fo(\):)p -74 3336 817 4 v 532 3696 a Fl(j)p Fn(R)634 3711 y Fm(1)p Fk(a)710 3696 y Fl(j)83 b(\024)h(j)p Fn(A)p Fo(\(0\))p Fl(j)1253 3600 y Fe(\020)1301 3696 y Fn(w)1371 3711 y Fm(+)1430 3696 y Fn(\014)6 b( )1554 3711 y Fm(0)1593 3696 y Fn(;)49 b(w)1739 3711 y Fi(\000)1798 3696 y Fn(e)1843 3655 y Fi(\000)p Fk(iH)1980 3664 y Fh(0)2015 3655 y Fk(t)2044 3696 y Fo(\()p Fn(H)2163 3711 y Fm(0)2225 3696 y Fl(\000)22 b Fn(\025)2381 3711 y Fm(0)2443 3696 y Fo(+)g Fn(\026)p Fo(\))2638 3655 y Fi(\000)p Fm(1)2732 3696 y Fg(P)2809 3711 y Ff(c)2849 3696 y Fn(w)2919 3711 y Fi(\000)3010 3696 y Fn(w)3080 3711 y Fm(+)3139 3696 y Fn(\014)6 b( )3263 3711 y Fm(0)3302 3600 y Fe(\021)821 3870 y Fl(\024)84 b Fn(C)7 b Fl(j)p Fn(A)p Fo(\(0\))p Fl(j)31 b(jjj)p Fn(W)14 b Fl(jjj)1618 3829 y Fm(2)1688 3870 y Fl(h)p Fn(t)p Fl(i)1801 3829 y Fi(\000)p Fk(r)r Fm(+1)3773 3870 y Fo(\(6.8\))-74 4119 y(b)m(y)32 b(lo)s(cal)c(deca)m(y)33 b(estimates)d(and)h(our)g (assumptions)f(on)h Fn(H)2112 4134 y Fm(0)2151 4119 y Fo(,)g(since)h Fn(\025)2504 4134 y Fm(0)2561 4119 y Fl(\000)19 b Fn(\026)31 b Fo(is)f(not)h(in)f(the)h(sp)s(ectrum)g(of)g Fn(H)3908 4134 y Fm(0)3947 4119 y Fo(.)-74 4319 y(Estimation)g(of)h(\() p Fn(\013)638 4334 y Fm(2)677 4319 y Fo(\))p -74 4357 789 4 v 72 4468 a(F)-8 b(rom)31 b(\(5.6\))i(w)m(e)g(ha)m(v)m(e)h(that) 1211 4717 y Fl(j)p Fn(@)1290 4732 y Fk(s)1327 4717 y Fn(A)p Fo(\()p Fn(s)p Fo(\))p Fl(j)27 b(\024)h Fn(C)7 b Fl(jjj)p Fn(W)14 b Fl(jjj)30 b(j)p Fn(A)p Fo(\()p Fn(s)p Fo(\))p Fl(j)22 b Fo(+)g Fl(j)p Fn(E)6 b Fo(\()p Fn(s)p Fo(\))p Fl(j)1083 b Fo(\(6.9\))-74 4966 y(since)33 b Fn(\032)g Fo(and)g(\000)f(are,)h(resp)s(ectiv)m(ely)-8 b(,)33 b(linear)e(and)i(quadratic)f(in)g Fl(jjj)p Fn(W)14 b Fl(jjj)p Fo(.)1901 5306 y(23)p eop %%Page: 24 24 24 23 bop 72 59 a Fo(Applying)32 b(Lemma)f(6.1)h(to)h(\()p Fn(\013)1220 74 y Fm(2)1259 59 y Fo(\))g(w)m(e)g(then)g(get)108 283 y Fl(j)p Fn(R)210 298 y Fm(1)p Fk(b)280 283 y Fl(j)115 b(\024)h(j)p Fn(A)p Fo(\(0\))p Fl(j)887 158 y Fe(\014)886 208 y(\014)886 258 y(\014)886 307 y(\014)914 165 y(Z)997 192 y Fk(t)960 354 y Fm(0)1043 186 y Fe(\020)1092 283 y Fn(w)1162 298 y Fm(+)1221 283 y Fn(\014)6 b( )1345 298 y Fm(0)1385 283 y Fn(;)49 b(w)1531 298 y Fi(\000)1590 283 y Fn(e)1635 241 y Fi(\000)p Fk(iH)1772 250 y Fh(0)1806 241 y Fm(\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))1978 283 y Fo(\()p Fn(H)2097 298 y Fm(0)2159 283 y Fl(\000)22 b Fn(\025)2315 298 y Fm(0)2377 283 y Fo(+)g Fn(\026)p Fo(\))2572 241 y Fi(\000)p Fm(1)2666 283 y Fg(P)2743 298 y Ff(c)2783 283 y Fn(w)2853 298 y Fi(\000)2944 283 y Fn(@)2995 298 y Fk(s)3032 283 y Fn(A)p Fo(\()p Fn(s)p Fo(\))33 b Fn(w)3330 298 y Fm(+)3388 283 y Fn(\014)6 b( )3512 298 y Fm(0)3552 186 y Fe(\021)3651 283 y Fn(ds)3748 158 y Fe(\014)3748 208 y(\014)3748 258 y(\014)3748 307 y(\014)423 474 y Fl(\024)116 b Fn(C)7 b Fl(jjj)p Fn(W)14 b Fl(jjj)967 433 y Fm(2)1037 474 y Fl(h)p Fn(t)p Fl(i)1150 433 y Fi(\000)p Fk(\021)1263 474 y Fo(\()p Fl(jjj)p Fn(W)g Fl(jjj)30 b Fo([)p Fn(A)p Fo(])1732 489 y Fk(\021)1774 474 y Fo(\()p Fn(t)p Fo(\))55 b(+)g([)p Fn(E)6 b Fo(])2203 489 y Fk(\021)2245 474 y Fo(\()p Fn(t)p Fo(\)\))16 b Fn(:)1288 b Fo(\(6.10\))-74 683 y(Estimation)31 b(of)h(\()p Fn(\013)638 698 y Fm(3)677 683 y Fo(\))p -74 721 789 4 v 44 w(Since)43 b Fn(\025)1081 698 y Fm(0)1151 683 y Fo(+)29 b Fn(\026)46 b Fl(2)h Fn(\033)t Fo(\()p Fn(H)1652 698 y Fm(0)1692 683 y Fo(\),)f(w)m(e)f(use)f(\(2.4\))f(together)h(with) g(the)g(pro)s(cedure)g(used)h(to)-74 832 y(estimate)32 b(\()p Fn(\013)417 847 y Fm(2)456 832 y Fo(\).)44 b(The)33 b(result)g(is:)1287 1131 y Fl(j)p Fn(R)1389 1146 y Fm(3)1428 1131 y Fl(j)28 b(\024)g Fn(C)7 b Fl(j)p Fn(A)p Fo(\(0\))p Fl(j)31 b(jjj)p Fn(W)14 b Fl(jjj)2225 1090 y Fm(2)2290 1131 y Fl(h)p Fn(t)p Fl(i)2403 1090 y Fi(\000)p Fk(r)r Fm(+1)2586 1131 y Fn(:)1112 b Fo(\(6.11\))-74 1355 y(Estimation)31 b(of)h(\()p Fn(\013)638 1370 y Fm(4)677 1355 y Fo(\))p -74 1393 V 72 1505 a(This)h(term)f(is)g(estimated)g(in)g(a)g(manner)h (similar)c(to)j Fn(\013)2120 1520 y Fm(2)2192 1505 y Fo(and)h Fn(\013)2444 1520 y Fm(3)2483 1505 y Fo(:)943 1803 y Fl(j)p Fn(R)1045 1818 y Fm(4)1084 1803 y Fl(j)27 b(\024)i Fn(c)p Fl(jjj)p Fn(W)14 b Fl(jjj)1561 1762 y Fm(2)1614 1803 y Fl(fjjj)p Fn(W)g Fl(jjj)30 b Fo([)p Fn(A)p Fo(])2095 1818 y Fk(\021)2138 1803 y Fo(\()p Fn(t)p Fo(\))22 b(+)g([)p Fn(E)6 b Fo(])2501 1818 y Fk(\021)2543 1803 y Fo(\()p Fn(t)p Fo(\))p Fl(g)17 b(h)p Fn(t)p Fl(i)2834 1762 y Fi(\000)p Fk(\021)2930 1803 y Fn(:)768 b Fo(\(6.12\))-74 2028 y(Estimation)31 b(of)h(\()p Fn(\013)638 2043 y Fm(5)677 2028 y Fo(\):)p -74 2066 817 4 v 72 2177 a(Since,)h(b)m(y)h (de\014nition,)d Fn(\036)1008 2192 y Fk(\013)1058 2177 y Fo(\(0\))c(=)g Fg(P)1390 2192 y Ff(c)1430 2177 y Fn(\036)1488 2192 y Fk(d)1529 2177 y Fo(\(0\))32 b(w)m(e)h(can)g(apply)f(lo)s(cal)f (deca)m(y)j(estimates)e(for)g Fn(e)3401 2141 y Fi(\000)p Fk(iH)3538 2150 y Fh(0)3573 2141 y Fk(t)3635 2177 y Fo(to)g(get)986 2385 y Fl(j)p Fo(\()p Fn(\014)6 b( )1176 2400 y Fm(0)1215 2385 y Fn(;)17 b(\036)1317 2400 y Fm(0)1356 2385 y Fo(\()p Fn(t)p Fo(\)\))p Fl(j)60 b(\024)h Fn(C)7 b Fl(jjj)p Fn(W)14 b Fl(jjj)30 b(h)p Fn(t)p Fl(i)2225 2344 y Fi(\000)p Fk(r)r Fm(+1)2435 2385 y Fl(k)p Fn(w)2555 2400 y Fm(+)2614 2385 y Fn(\036)2672 2400 y Fk(d)2712 2385 y Fo(\(0\))p Fl(k)p Fn(:)811 b Fo(\(6.13\))-74 2632 y(Estimation)31 b(of)h(\()p Fn(\013)638 2647 y Fm(6)677 2632 y Fo(\))p -74 2670 789 4 v 72 2781 a(Applying)g(Lemma)f(6.1)h(as)h(b)s(efore)g(w)m(e)h(get,)e (for)g(0)c Fl(\024)g Fn(t)g Fl(\024)g Fn(T)14 b Fo(,)1017 2990 y Fl(j)p Fo(\()p Fn(\014)6 b( )1207 3005 y Fm(0)1246 2990 y Fn(;)17 b(\036)1348 3005 y Fm(2)1387 2990 y Fo(\()p Fn(t)p Fo(\)\))p Fl(j)60 b(\024)28 b Fn(C)7 b Fl(jjj)p Fn(W)14 b Fl(jjj)2080 2949 y Fm(2)2150 2990 y Fl(h)p Fn(t)p Fl(i)2263 2949 y Fi(\000)p Fk(\021)2387 2990 y Fo([)p Fn(\036)2472 3005 y Fk(d)2512 2990 y Fo(])2539 3005 y Fk(LD)r(;\021)2709 2990 y Fo(\()p Fn(T)g Fo(\))p Fn(:)842 b Fo(\(6.14\))-74 3198 y(Using)32 b(Prop)s(osition)f(6.1)h(to) h(estimate)e([)p Fn(\036)1477 3213 y Fk(d)1518 3198 y Fo(])1545 3213 y Fk(LD)r(;\021)1714 3198 y Fo(\()p Fn(t)p Fo(\))i(in)f(\(6.14\),)f(w)m(e)j(get)572 3406 y Fl(j)p Fo(\()p Fn(\014)6 b( )762 3421 y Fm(0)801 3406 y Fn(;)17 b(\036)903 3421 y Fm(2)942 3406 y Fo(\()p Fn(t)p Fo(\)\))p Fl(j)28 b(\024)g Fn(C)7 b Fl(jjj)p Fn(W)14 b Fl(jjj)1603 3365 y Fm(2)1672 3406 y Fl(h)p Fn(t)p Fl(i)1785 3365 y Fi(\000)p Fk(\021)1898 3406 y Fl(f)33 b(k)p Fn(w)2101 3421 y Fm(+)2159 3406 y Fn(\036)2217 3421 y Fk(d)2258 3406 y Fo(\(0\))p Fl(k)54 b Fo(+)g Fl(jjj)p Fn(W)14 b Fl(jjj)30 b Fo([)p Fn(A)p Fo(])3048 3421 y Fk(\021)3091 3406 y Fo(\()p Fn(t)p Fo(\))i Fl(g)17 b Fn(:)397 b Fo(\(6.15\))72 3615 y(Finally)-8 b(,)33 b(com)m(bining)g(the)i(ab)s(o)m(v)m(e)h (estimates)e(for)g(\()p Fn(\013)2028 3630 y Fk(j)2065 3615 y Fo(\))p Fn(;)51 b Fo(1)31 b Fl(\024)h Fn(j)37 b Fl(\024)32 b Fo(6,)j(w)m(e)h(can)f(no)m(w)g(b)s(ound)g([)p Fn(E)6 b Fo(])3634 3630 y Fk(\021)3676 3615 y Fo(\()p Fn(T)14 b Fo(\))34 b(for)-74 3764 y(an)m(y)f Fn(\021)f Fl(2)c Fo([0)p Fn(;)17 b(r)24 b Fl(\000)f Fo(1])32 b(as)h(follo)m(ws:) 99 4122 y([)p Fn(E)6 b Fo(])231 4137 y Fk(\021)273 4122 y Fo(\()p Fn(t)p Fo(\))28 b Fl(\024)g Fn(C)610 4026 y Fe(n)666 4122 y Fl(jjj)p Fn(W)14 b Fl(jjj)940 4081 y Fm(2)1009 4122 y Fl(j)p Fn(A)p Fo(\(0\))p Fl(j)22 b Fo(+)g Fl(jjj)p Fn(W)14 b Fl(jjj)30 b(k)p Fn(w)1807 4137 y Fm(+)1865 4122 y Fn(\036)1923 4137 y Fk(d)1964 4122 y Fo(\(0\))p Fl(k)21 b Fo(+)h Fl(jjj)p Fn(W)14 b Fl(jjj)2532 4081 y Fm(2)2602 4122 y Fo([)p Fn(E)6 b Fo(])2734 4137 y Fk(\021)2776 4122 y Fo(\()p Fn(T)14 b Fo(\))22 b(+)g Fl(jjj)p Fn(W)14 b Fl(jjj)3317 4081 y Fm(3)3386 4122 y Fo([)p Fn(A)p Fo(])3513 4137 y Fk(\021)3556 4122 y Fo(\()p Fn(T)g Fo(\))3703 4026 y Fe(o)3774 4122 y Fn(:)3725 4271 y Fo(\(6.16\))-74 4421 y(Since)33 b Fl(jjj)p Fn(W)14 b Fl(jjj)30 b Fo(assumed)j(to)g(b)s (e)f(small,)f(Prop)s(osition)g(6.2)h(follo)m(ws.)172 b([])72 4608 y(W)-8 b(e)41 b(can)g(no)m(w)g(complete)f(the)h(pro)s(ofs) f(of)g(our)g(main)f(results,)k(Theorem)e(2.1)f(and)g(Theorem)h(2.2.)67 b(T)-8 b(o)-74 4758 y(pro)m(v)m(e)41 b(the)g(assertions)f(concerning)g (the)g(in\014nite)f(time)g(b)s(eha)m(vior,)i(the)g(k)m(ey)g(is)f(to)f (establish)h(lo)s(cal)d(deca)m(y)-74 4907 y(of)i Fn(\036)102 4922 y Fk(d)142 4907 y Fo(,)i(in)e(particular,)g(the)h(uniform)d(b)s (oundedness)42 b(of)c([)p Fn(\036)2145 4922 y Fk(d)2186 4907 y Fo(])2213 4922 y Fk(LD)r(;r)r Fi(\000)p Fm(1)2468 4907 y Fo(\()p Fn(T)14 b Fo(\).)64 b(This)39 b(will)e(follo)m(w)g (directly)i(from)-74 5057 y(Prop)s(osition)31 b(6.1)h(if)g(w)m(e)h(pro) m(v)m(e)h(the)f(uniform)e(b)s(oundedness)k([)p Fn(A)p Fo(])2338 5072 y Fk(r)r Fi(\000)p Fm(1)2466 5057 y Fo(\()p Fn(T)14 b Fo(\),)32 b(or)h(equiv)-5 b(alen)m(tly)31 b([)3386 5032 y(~)3360 5057 y Fn(A)q Fo(])3461 5072 y Fk(r)r Fi(\000)p Fm(1)3589 5057 y Fo(\()p Fn(T)14 b Fo(\).)1901 5306 y(24)p eop %%Page: 25 25 25 24 bop -74 59 a Fg(Prop)s(osition)36 b(6.3.)49 b Fb(There)25 b(exists)g(a)f(constan)m(t)h Fn(C)1817 74 y Fi(\003)1856 59 y Fb(,)h(dep)s(ending)e(on)g Fl(k)p Fn(\036)2604 74 y Fm(0)2643 59 y Fl(k)p Fn(;)17 b Fl(k)p Fn(w)2857 74 y Fm(+)2915 59 y Fn(\036)2973 74 y Fm(0)3012 59 y Fl(k)p Fn(;)g Fl(jjj)p Fn(W)d Fl(jjj)p Fn(;)j Fo(\000)k Fb(and)j Fn(r)s Fb(,)h(suc)m(h)-74 208 y(that)32 b(for)h(an)m(y)g Fn(T)41 b(>)28 b Fo(0)1628 358 y([)p Fn(A)p Fo(])1755 373 y Fk(r)r Fi(\000)p Fm(1)1883 358 y Fo(\()p Fn(T)14 b Fo(\))28 b Fl(\024)g Fn(C)2233 373 y Fi(\003)3725 358 y Fo(\(6.17\))-74 607 y(Pro)s(of.)p -74 620 262 4 v 72 756 a(W)-8 b(e)36 b(b)s(egin)f(with)g(the)g(expansion)h(of)f Fn(A)p Fo(\()p Fn(t)p Fo(\))g(giv)m(en)h(in)e(Prop)s(osition)g(5.1.)51 b(Multiplying)33 b(\(5.2\))i(b)m(y)h Fl(h)p Fn(t)p Fl(i)3819 720 y Fk(r)r Fi(\000)p Fm(1)3947 756 y Fo(,)-74 906 y(and)d(taking)f (the)h(suprem)m(um)f(o)m(v)m(er)i(0)27 b Fl(\024)i Fn(t)e Fl(\024)i Fn(T)46 b Fo(w)m(e)33 b(ha)m(v)m(e:)297 1175 y([)p Fn(A)p Fo(])424 1190 y Fk(r)r Fi(\000)p Fm(1)553 1175 y Fo(\()p Fn(T)14 b Fo(\))60 b Fl(\024)897 1029 y Fe( )963 1175 y Fl(j)p Fn(A)p Fo(\(0\))p Fl(j)32 b Fo(\000)1310 1134 y Fi(\000)p Fk(r)r Fm(+1)1547 1175 y Fo(+)159 b(sup)1678 1259 y Fm(0)p Fi(\024)p Fk(\034)8 b Fi(\024)p Fm(2\000)1941 1240 y Fc(\000)p Fa(\013)2050 1175 y Fl(j)p Fn(R)2152 1190 y Fk(A)2209 1175 y Fo(\()p Fn(\034)j Fo(\))p Fl(j)55 b Fo(+)166 b(sup)2551 1259 y Fm(2\000)2630 1240 y Fc(\000)p Fa(\013)2723 1259 y Fi(\024)p Fk(\034)8 b Fi(\024)p Fk(T)2939 1175 y Fl(j)p Fn(R)3041 1190 y Fk(A)3098 1175 y Fo(\()p Fn(\034)j Fo(\))p Fl(j)3288 1029 y Fe(!)3725 1175 y Fo(\(6.18\))-74 1454 y(The)34 b(righ)m(t)d(hand)i(side)g(of)f(\(6.18\))g(is)g(estimated)g (using)g(Prop)s(osition)f(5.1.)-74 1696 y([)p Fn(A)p Fo(])53 1711 y Fk(r)r Fi(\000)p Fm(1)182 1696 y Fo(\()p Fn(T)14 b Fo(\))49 b Fl(\024)504 1600 y Fe(\020)554 1696 y Fl(j)p Fn(A)p Fo(\(0\))p Fl(j)32 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272 4 v 2142 3635 a Fn(\022)2187 3650 y Fm(0)2330 3567 y Fn(:)1368 b Fo(\(6.21\))-74 3826 y(Therefore,)34 b(for)e Fl(jjj)p Fn(W)14 b Fl(jjj)30 b Fo(su\016cien)m(tly)j(small)1628 4068 y([)p Fn(A)p Fo(])1755 4083 y Fk(r)r Fi(\000)p Fm(1)1883 4068 y Fo(\()p Fn(T)14 b Fo(\))28 b Fl(\024)g Fn(C)2233 4083 y Fi(\003)3725 4068 y Fo(\(6.22\))-74 4310 y(\(Alternativ)m(ely)-8 b(,)36 b(w)m(e)i(can)f(obtain)e(\(6.22\))h(b)m(y)h(imp)s(osing)d(that)i Fl(jjj)p Fn(W)14 b Fl(jjj)2549 4273 y Fm(3)2586 4310 y Fo(\000)2647 4273 y Fi(\000)p Fm(1)2778 4310 y Fo(b)s(e)36 b(su\016cien)m(tly)h(small.\))53 b(Here,)-74 4459 y Fn(C)-4 4474 y Fi(\003)68 4459 y Fo(dep)s(ends)34 b(on)f Fl(k)p Fn(\036)689 4474 y Fm(0)728 4459 y Fl(k)p Fn(;)17 b Fl(k)p Fn(w)942 4474 y Fm(+)1000 4459 y Fn(\036)1058 4474 y Fm(0)1097 4459 y Fl(k)p Fn(;)g Fl(jjj)p Fn(W)d Fl(jjj)p Fn(;)j Fo(\000)29 b(and)k Fn(r)s Fo(.)72 4608 y(This)h(completes)f(the) h(pro)s(of)e(of)h(Prop)s(osition)e(6.3)i(and)h(therewith)f(the)h Fn(t)29 b Fl(!)g(1)j Fo(asymptotics)h(asserted)-74 4758 y(in)f(Theorems)h(2.1)f(and)h(2.2)-74 4907 y Fg(Remark:)47 b Fo(This)35 b(estimation)e(pro)s(cedure)i(di\013ers)g(from)e(one)i (used)h(in)e([38],)h(where)h(the)f(cases)h Fn(r)26 b Fl(\000)f Fo(1)34 b(large)-74 5057 y(and)f Fn(r)25 b Fl(\000)d Fo(1)33 b(small)d(are)i(treated)h(di\013eren)m(tly)-8 b(.)1901 5306 y(25)p eop %%Page: 26 26 26 25 bop 72 59 a Fo(It)33 b(remains)f(to)g(pro)m(v)m(e)i(the)f(in)m (termediate)e(time)h(estimate)f(\(2.12\).)43 b(The)34 b(ingredien)m(ts)e(are)h(con)m(tained)g(in)-74 208 y(\(6.7\))f(and)h (its)f(pro)s(of.)43 b(First,)31 b(b)m(y)j(\(5.3\))1198 454 y Fn(R)1272 469 y Fk(A)1329 454 y Fo(\()p Fn(t)p Fo(\))60 b Fl(\024)28 b Fn(C)1731 337 y Fe(Z)1814 364 y Fk(t)1777 526 y Fm(0)1893 454 y Fn(e)1938 413 y Fi(\000)p Fm(\000\()p Fk(t)p Fi(\000)p Fk(\034)8 b Fm(\))2265 454 y Fl(j)o Fn(E)e Fo(\()p Fn(\034)11 b Fo(\))p Fl(j)50 b Fn(d\034)6 b(:)1022 b Fo(\(6.23\))-74 695 y(Let)40 b Fn(T)165 710 y Fm(0)244 695 y Fo(denote)g(an)f(arbitrary)f(\014xed)j (p)s(ositiv)m(e)e(n)m(um)m(b)s(er.)64 b(W)-8 b(e)40 b(estimate)e (\(6.23\))h(for)g Fn(t)g Fl(2)h Fo([0)p Fn(;)17 b(T)3526 710 y Fm(0)3565 695 y Fo(\000)3626 659 y Fi(\000)p Fm(1)3720 695 y Fo(].)64 b(W)-8 b(e)-74 844 y(b)s(ound)42 b(the)g(exp)s(onen)m (tial)f(in)f(the)i(in)m(tegrand)f(b)m(y)i(one)e(\(explicit)f(in)m (tegration)g(w)m(ould)h(giv)m(e)h(something)e(of)-74 994 y(order)47 b(\000)256 957 y Fi(\000)p Fm(1)350 994 y Fo(\),)j(and)d(b)s(ound)g Fl(j)p Fn(E)6 b Fo(\()p Fn(\034)11 b Fo(\))p Fl(j)46 b Fo(b)m(y)i(estimating)c(the)k(expressions)g(\()p Fn(\013)2739 1009 y Fk(j)2775 994 y Fo(\))p Fn(;)64 b(j)57 b Fo(=)52 b(1)p Fn(;)17 b(:::;)g Fo(6)46 b(in)g(the)h(pro)s(of)-74 1143 y(of)41 b(Prop)s(osition)e(6.2.)69 b(First,)42 b(the)g(estimates)e (of)h(Prop)s(osition)e(6.2)i(for)f Fn(\013)2727 1158 y Fm(1)2767 1143 y Fn(;)17 b(\013)2873 1158 y Fm(3)2953 1143 y Fo(and)41 b Fn(\013)3213 1158 y Fm(5)3294 1143 y Fo(are)g(useful)g(as)g(is.)-74 1292 y(In)m(tegration)32 b(of)g(the)h(b)s(ounds)g(\(6.8\),)g(\(6.11\))e(and)i(\(6.13\))f(giv)m (es:)836 1422 y Fe(Z)919 1448 y Fk(t)882 1610 y Fm(0)998 1539 y Fn(e)1043 1498 y Fi(\000)p Fm(\000\()p Fk(t)p Fi(\000)p Fk(\034)8 b Fm(\))1353 1539 y Fo(\()p Fn(\013)1453 1554 y Fk(j)1489 1539 y Fo(\))61 b Fl(\024)f Fn(C)40 b Fl(jjj)p Fn(W)14 b Fl(jjj)30 b(k)p Fn(w)2259 1554 y Fm(+)2317 1539 y Fn(\036)p Fo(\(0\))p Fl(k)p Fn(;)49 b(j)34 b Fo(=)27 b(1)p Fn(;)17 b Fo(3)p Fn(;)g Fo(5)p Fn(:)660 b Fo(\(6.24\))-74 1770 y(T)-8 b(o)35 b(estimate)f(the)i(con)m (tributions)e(of)g(\()p Fn(\013)1445 1785 y Fm(2)1485 1770 y Fo(\))h(and)g(\()p Fn(\013)1850 1785 y Fm(4)1889 1770 y Fo(\),)h(\014rst)f(observ)m(e)i(that)e(b)m(y)h(\(6.9\))e(and)h (Prop)s(osition)e(6.2)-74 1919 y(with)f Fn(\021)g Fo(=)27 b(0)1270 2069 y Fl(j)p Fn(@)1349 2084 y Fk(s)1386 2069 y Fn(A)p Fo(\()p Fn(s)p Fo(\))p Fl(j)59 b(\024)i Fn(C)39 b Fl(jjj)p Fn(W)14 b Fl(jjj)31 b(k)p Fn(w)2340 2084 y Fm(+)2398 2069 y Fn(\036)p Fo(\(0\))p Fl(k)1094 b Fo(\(6.25\))-74 2264 y(Therefore,)34 b(using)e(lo)s(cal)e(deca)m(y)k(estimates)f(w)m(e) g(ha)m(v)m(e:)824 2388 y Fe(Z)907 2414 y Fk(t)870 2577 y Fm(0)986 2505 y Fn(e)1031 2464 y Fi(\000)p Fm(\000\()p Fk(t)p Fi(\000)p Fk(\034)8 b Fm(\))1341 2505 y Fo(\()p Fn(\013)1441 2520 y Fk(j)1477 2505 y Fo(\))33 b Fn(d\034)94 b Fl(\024)116 b Fn(C)39 b(T)2094 2520 y Fm(0)2134 2505 y Fo(\000)2195 2464 y Fi(\000)p Fm(1)2322 2505 y Fl(jjj)p Fn(W)14 b Fl(jjj)2596 2464 y Fm(3)2665 2505 y Fl(k)p Fn(w)2785 2520 y Fm(+)2844 2505 y Fn(\036)p Fo(\(0\))p Fl(k)1735 2679 y(\024)116 b(jjj)p Fn(W)14 b Fl(jjj)30 b(k)p Fn(w)2352 2694 y Fm(+)2411 2679 y Fn(\036)p Fo(\(0\))p Fl(k)p Fn(;)48 b(j)34 b Fo(=)27 b(2)p Fn(;)17 b Fo(4)p Fn(:)-74 2910 y Fo(Finally)-8 b(,)30 b(w)m(e)k(come)e(to)g(the)h(con)m (tribution)f(of)g(\()p Fn(\013)1724 2925 y Fm(6)1763 2910 y Fo(\).)44 b(W)-8 b(e)33 b(rewrite)f(\()p Fn(\013)2470 2925 y Fm(6)2510 2910 y Fo(\))g(as)h(follo)m(ws.)430 3157 y(\()p Fn(\013)530 3172 y Fm(6)569 3157 y Fo(\))116 b(=)914 3032 y Fe(\014)914 3082 y(\014)914 3132 y(\014)914 3182 y(\014)942 3039 y(Z)1025 3066 y Fk(t)988 3228 y Fm(0)1054 3157 y Fo(\()p Fn(\014)38 b(e)1230 3115 y Fk(iH)1312 3124 y Fh(0)1347 3115 y Fm(\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))1519 3157 y Fg(P)1596 3172 y Ff(c)1636 3157 y Fn(\014)6 b( )1760 3172 y Fm(0)1800 3157 y Fn(;)17 b Fo(cos)g Fn(\026s)32 b(\036)2186 3172 y Fk(d)2226 3157 y Fo(\()p Fn(s)p Fo(\))g Fn(ds)p Fo(\))2515 3032 y Fe(\014)2515 3082 y(\014)2515 3132 y(\014)2515 3182 y(\014)723 3396 y Fo(=)914 3272 y Fe(\014)914 3322 y(\014)914 3371 y(\014)914 3421 y(\014)942 3279 y(Z)1025 3305 y Fk(t)988 3468 y Fm(0)1054 3396 y Fo(\()p Fn(w)1162 3411 y Fm(+)1221 3396 y Fn(\014)6 b(w)1352 3411 y Fm(+)1433 3396 y Fl(\001)21 b Fn(w)1552 3411 y Fi(\000)1611 3396 y Fn(e)1656 3355 y Fk(iH)1738 3364 y Fh(0)1773 3355 y Fm(\()p Fk(t)p Fi(\000)p Fk(s)p Fm(\))1945 3396 y Fg(P)2022 3411 y Ff(c)2062 3396 y Fn(w)2132 3411 y Fi(\000)2213 3396 y Fl(\001)h Fn(w)2333 3411 y Fm(+)2391 3396 y 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Fl(jjj)p Fn(W)14 b Fl(jjj)3101 3756 y Fm(2)3160 3792 y Fl(k)p Fn(w)3280 3807 y Fm(+)3339 3792 y Fn(\036)3397 3807 y Fk(d)3437 3792 y Fo(\(0\))p Fl(k)21 b(h)p Fn(\034)11 b Fl(i)3764 3756 y Fi(\000)p Fk(r)r Fm(+1)3947 3792 y Fo(.)-74 3941 y(Multiplication)44 b(of)k(this)f(b)s(ound)h(b)m(y)h Fn(e)1422 3905 y Fi(\000)p Fm(\000\()p Fk(t)p Fi(\000)p Fk(\034)8 b Fm(\))1747 3941 y Fo(and)48 b(in)m(tegration)f(with)g(resp) s(ect)i(to)f Fn(t)g Fo(giv)m(es)g(the)h(b)s(ound)-74 4091 y Fn(C)39 b Fl(jjj)p Fn(W)14 b Fl(jjj)309 4054 y Fm(2)346 4091 y Fl(k)p Fn(w)466 4106 y Fm(+)525 4091 y Fn(\036)583 4106 y Fk(d)623 4091 y Fo(\(0\))p Fl(k)p Fo(.)43 b(T)-8 b(o)32 b(assess)i(the)e(con)m(tributions)f(from)g Fn(\036)2337 4106 y Fm(1)2397 4091 y Fo(+)21 b Fn(\036)2552 4106 y Fm(2)2591 4091 y Fo(,)32 b(note)g(that)g(lo)s(cal)e(deca)m(y)j (estimates)-74 4240 y Fg(\(H3\))e Fo(imply)1159 4389 y Fl(k)p Fn(w)1279 4404 y Fi(\000)1337 4389 y Fo(\()p Fn(\036)1433 4404 y Fm(1)1494 4389 y Fo(+)22 b Fn(\036)1650 4404 y Fm(2)1690 4389 y Fo(\))p Fl(k)59 b(\024)i Fn(C)40 b Fl(jjj)p Fn(W)14 b Fl(jjj)30 b(k)p Fn(w)2509 4404 y Fm(+)2567 4389 y Fn(\036)2625 4404 y Fm(0)2665 4389 y Fl(k)p Fn(:)983 b Fo(\(6.27\))-74 4585 y(Putting)32 b(together)h(the)g (con)m(tributions)f(from)f Fn(\036)1720 4600 y Fm(0)1792 4585 y Fo(and)i(from)e Fn(\036)2270 4600 y Fm(1)2331 4585 y Fo(+)22 b Fn(\036)2487 4600 y Fm(2)2526 4585 y Fo(,)33 b(w)m(e)h(ha)m(v)m(e:)567 4709 y Fe(Z)650 4735 y Fk(t)613 4897 y Fm(0)729 4826 y Fn(e)774 4785 y Fi(\000)p Fm(\000\()p Fk(t)p Fi(\000)p Fk(\034)8 b Fm(\))1084 4826 y Fo(\()p Fn(\013)1184 4841 y Fm(6)1224 4826 y Fo(\))32 b Fn(d\034)71 b Fl(\024)61 b Fn(C)1690 4730 y Fe(\020)1772 4826 y Fl(jjj)p Fn(W)14 b Fl(jjj)2046 4785 y Fm(2)2115 4826 y Fl(k)p Fn(w)2235 4841 y Fm(+)2294 4826 y Fn(\036)2352 4841 y Fk(d)2392 4826 y Fo(\(0\))p Fl(k)54 b Fo(+)h(\000)2813 4785 y Fi(\000)p Fm(1)2940 4826 y Fl(jjj)p Fn(W)14 b Fl(jjj)3214 4785 y Fm(3)3283 4730 y Fe(\021)3725 4826 y Fo(\(6.28\))-74 5057 y(The)34 b(ab)s(o)m(v)m(e)f(estimates)f(and)h Fg(\(H6\))e Fo(imply)g(\(2.12\).)43 b(This)32 b(concludes)i(the)f(pro)s (of)f(of)g(Theorem)h(2.2.)1901 5306 y(26)p eop %%Page: 27 27 27 26 bop -74 59 a Fp(7.)46 b(App)t(endix)d(A:)i(General)h(approac)l(h) f(to)g(lo)t(cal)h(deca)l(y)f(estimates)-74 394 y Fo(Hyp)s(othesis)e Fg(\(H3\))e Fo(for)h(our)g(main)f(theorem)h(is)g(one)g(requiring)g (that)g(our)g(unp)s(erturb)s(ed)h(op)s(erator,)i Fn(H)3908 409 y Fm(0)3947 394 y Fo(,)-74 544 y(satisfy)25 b(a)g(suitable)f(lo)s (cal)e(deca)m(y)27 b(estimates.)40 b(It)25 b(is)g(remark)m(ed)g(follo)m (wing)e(this)h(h)m(yp)s(othesis)i(that,)h(in)d(practice,)-74 693 y(the)29 b(v)m(eri\014cation)e(of)g(\(2.4\))h(using)f(\(2.5\).)42 b(In)28 b(this)g(section)g(w)m(e)h(giv)m(e)f(an)g(outline)e(to)i(a)g(v) m(ery)h(general)e(approac)m(h)-74 843 y(to)i(obtaining)e(estimates)h (of)h(this)f(t)m(yp)s(e,)j(based)f(on)f(a)g(tec)m(hnique)h(originating) c(in)i(the)h(w)m(ork)h(of)f(Mourre)g([22];)-74 992 y(see)36 b(also)d([25].)49 b(In)35 b(the)g(follo)m(wing)c(general)j(discussion)h (w)m(e)h(shall)d(let)g Fn(H)42 b Fo(denote)35 b(self-adjoin)m(t)e(op)s (erator)h(on)-74 1142 y(a)39 b(Hilb)s(ert)f(space,)k Fl(H)q Fo(,)f(k)m(eeping)f(in)f(mind)f(that)h(our)g(application)e(is)i (to)g(the)g(unp)s(erturb)s(ed)i(op)s(erator)e Fn(H)3908 1157 y Fm(0)3947 1142 y Fo(.)-74 1291 y(Let)h Fn(E)46 b Fl(2)40 b Fn(\033)t Fo(\()p Fn(H)8 b Fo(\),)41 b(and)e(assume)h(that) g(an)f(op)s(erator)g Fn(A)h Fo(can)g(b)s(e)g(found)g(suc)m(h)h(that)e Fn(A)h Fo(is)f(self-adjoin)m(t)f(and)-74 1441 y Fl(D)s Fo(\()p Fn(A)p Fo(\))17 b Fl(\\)h(H)31 b Fo(is)f(dense)i(in)d Fl(H)q Fo(.)43 b(Let)31 b(\001)f(denote)h(an)f(op)s(en)h(in)m(terv)-5 b(al)29 b(with)h(compact)g(closure.)43 b(W)-8 b(e)30 b(shall)f(use)i(the)-74 1590 y(notation:)1259 1739 y(ad)1362 1697 y Fk(n)1362 1764 y(A)1419 1739 y Fo(\()p Fn(H)8 b Fo(\))59 b(=)h([)p Fl(\001)17 b(\001)g(\001)e Fo([)p Fn(H)r(;)i(A)p Fo(])p Fn(;)g(A)p Fo(])p Fn(;)g Fl(\001)g(\001)g(\001)e Fn(A)p Fo(])p Fn(;)1132 b Fo(\(7.1\))-74 1934 y(for)32 b(the)h(n-fold)e(comm)m(utator.)72 2083 y(Assume)j(the)f(t)m(w)m(o)g (conditions:)-74 2233 y Fg(\(M1\))f Fo(The)i(op)s(erators)465 2461 y Fn(g)512 2476 y Fm(\001)574 2461 y Fo(\()p Fn(H)8 b Fo(\))32 b([)p Fn(H)r(;)17 b(A)p Fo(])p Fn(;)49 b Fo([)p Fn(H)r(;)17 b(A)p Fo(])33 b Fn(g)1435 2476 y Fm(\001)1498 2461 y Fo(\()p Fn(H)8 b Fo(\))p Fn(;)48 b Fo(and)33 b Fn(g)1975 2476 y Fm(\001)2038 2461 y Fo(\()p Fn(H)8 b Fo(\))32 b(ad)2338 2418 y Fk(n)2338 2486 y(A)2395 2461 y Fo(\()p Fn(H)8 b Fo(\))32 b Fn(g)2639 2476 y Fm(\001)2701 2461 y Fo(\()p Fn(H)8 b Fo(\))p Fn(;)81 b Fo(2)28 b Fl(\024)g Fn(n)g Fl(\024)g Fn(N)348 b Fo(\(7.2\))-74 2689 y(can)33 b(all)d(b)s(e)j(extended)i(to)d(a)g(b)s(ounded)i(op)s(erator)e(on)g Fl(H)q Fo(.)-74 2926 y Fg(\(M2\))g Fj(Mourr)-5 b(e)36 b(estimate)p Fo(:)1082 3154 y Fn(g)1129 3169 y Fm(\001)1192 3154 y Fo(\()p Fn(H)8 b Fo(\))32 b Fn(i)p Fo([)p Fn(H)r(;)17 b(A)p Fo(])33 b Fn(g)1756 3169 y Fm(\001)1818 3154 y Fo(\()p Fn(H)8 b Fo(\))60 b Fl(\025)28 b Fn(\022)36 b(g)2276 3169 y Fm(\001)2339 3154 y Fo(\()p Fn(H)8 b Fo(\))2504 3113 y Fm(2)2597 3154 y Fo(+)55 b Fn(K)962 b Fo(\(7.3\))-74 3383 y(for)32 b(some)h Fn(\022)d(>)e Fo(0)k(and)h(compact)f(op)s (erator,)g Fn(K)7 b Fo(.)-74 3532 y Fg(Remark:)61 b Fo(The)43 b(theorems)e(b)s(elo)m(w)h(w)m(ere)h(orginally)38 b(pro)m(v)m(ed)43 b(under)g(more)d(restrictiv)m(e)i(conditions)f(than)-74 3681 y Fg(\(M1\))30 b Fo([5],)h([8],)g([22],)g([25],)g([34)o(],)h([36)o (].)43 b(The)32 b(more)e(general)g(results)h(stated)g(here)g(can)g(b)s (e)g(pro)m(v)m(ed)h(using)e(the)-74 3831 y(approac)m(h)j(to)f(v)m(elo)s (cit)m(y)h(b)s(ounds)g(in)f([12)o(].)-74 4152 y Fg(Theorem)37 b(7.1.)49 b Fb(\(Mourre;)33 b(see)h([5])f(Theorem)g(4.9,)f([2])g(Lemma) f(5.4\))-74 4302 y(Assume)26 b(conditions)e Fg(\(M1\)-\(M2\))p Fb(,)j(with)d(N=2.)41 b(Then,)28 b(in)d(the)g(in)m(terv)-5 b(al)24 b Fo(\001)p Fb(,)j Fn(H)33 b Fb(can)25 b(only)g(ha)m(v)m(e)h (absolutely)-74 4451 y(con)m(tin)m(uous)f(sp)s(ectrum)g(with)f (\014nitely)g(man)m(y)g(eigen)m(v)-5 b(alues)24 b(of)g(\014nite)g(m)m (ultiplicit)m(y)-8 b(.)37 b(Moreo)m(v)m(er,)28 b(the)c(op)s(erator)1304 4679 y Fl(h)p Fn(A)p Fl(i)1455 4638 y Fi(\000)p Fm(1)1582 4679 y Fn(g)1629 4694 y Fm(\001)1691 4679 y Fo(\()p Fn(H)8 b Fo(\))32 b(\()p Fn(H)e Fl(\000)22 b Fn(z)t Fo(\))2223 4638 y Fi(\000)p Fm(1)2351 4679 y Fl(h)p Fn(A)p Fl(i)2502 4638 y Fi(\000)p Fm(1)3773 4679 y Fo(\(7.4\))-74 4907 y Fb(is)32 b(uniformly)e(b)s(ounded)j(in)e Fn(z)t Fb(,)i(as)g(an)f(op)s (erator)f(on)i Fl(H)q Fb(.)43 b(If)32 b Fn(K)j Fo(=)28 b(0)p Fb(,)k(then)g(there)h(are)g(no)f(eigen)m(v)-5 b(alues)32 b(in)f(the)-74 5057 y(in)m(terv)-5 b(al)31 b Fo(\001)p Fb(.)1901 5306 y Fo(27)p eop %%Page: 28 28 28 27 bop -74 59 a Fg(Theorem)37 b(7.2.)49 b Fb(\(Sigal-So\013er;)30 b(see)k([34],[8],)f([36)o(],)g([11],)f([12]\))-74 208 y(Assume)i(conditions)d Fg(\(M1\)-\(M2\))h Fb(with)g Fn(N)38 b Fl(\025)29 b Fo(2)j Fb(and)h Fn(K)h Fo(=)28 b(0)p Fb(.)43 b(Then,)34 b(for)e(all)f Fn(")c(>)g Fo(0)616 476 y Fl(jj)32 b Fn(F)797 355 y Fe(\022)868 409 y Fn(A)p 868 453 74 4 v 887 545 a(t)979 476 y(<)27 b(\022)1130 355 y Fe(\023)1241 476 y Fn(e)1286 435 y Fi(\000)p Fk(iH)5 b(t)1490 476 y Fn(g)1537 491 y Fm(\001)1600 476 y Fo(\()p Fn(H)j Fo(\))p Fn( )36 b Fl(jj)1920 491 y Fm(2)2019 476 y Fl(\024)60 b Fn(C)7 b Fl(h)p Fn(t)p Fl(i)2346 435 y Fi(\000)p Fk(N)g Fm(+1+)p Fk(")2678 476 y Fl(k)32 b(h)p Fn(A)p Fl(i)2911 435 y Fk(N)7 b Fi(\000)p Fm(1)3068 476 y Fn( )37 b Fl(k)3218 491 y Fm(2)3257 476 y Fn(;)489 b Fo(\(7.5\))-74 725 y Fb(and)33 b(therefore)825 875 y Fl(jj)e(h)p Fn(A)p Fl(i)1063 834 y Fi(\000)p Fk(\033)1197 875 y Fn(e)1242 834 y Fi(\000)p Fk(iH)5 b(t)1447 875 y Fn(g)1494 890 y Fm(\001)1556 875 y Fo(\()p Fn(H)j Fo(\))p Fn( )36 b Fl(jj)1876 890 y Fm(2)1975 875 y Fl(\024)61 b Fn(C)39 b Fl(h)p Fn(t)p Fl(i)2335 834 y Fi(\000)p Fk(\033)2469 875 y Fl(k)33 b(h)p Fn(A)p Fl(i)2703 834 y Fk(N)7 b Fi(\000)p Fm(1)2860 875 y Fn( )36 b Fl(k)3009 890 y Fm(2)3048 875 y Fn(;)698 b Fo(\(7.6\))-74 1078 y Fb(for)32 b Fn(\033)g(<)c(N)33 b Fl(\000)23 b Fo(1)p Fb(.)44 b(Here,)34 b Fn(F)46 b Fb(is)32 b(a)h(smo)s(othed)f(out)h(c)m(haracteristic)g(function,)f(and) h Fn(F)3052 982 y Fe(\020)3112 1039 y Fk(A)p 3112 1055 53 4 v 3126 1113 a(t)3202 1078 y Fn(<)28 b(\022)3354 982 y Fe(\021)3436 1078 y Fb(is)33 b(de\014ned)h(b)m(y)-74 1228 y(the)f(sp)s(ectral)f(theorem.)-74 1485 y Fg(Remark:)58 b Fn(\033)43 b Fo(is)c(required)i(to)e(b)s(e)h(smaller)e(than)1851 1446 y Fk(N)p 1851 1462 64 4 v 1864 1519 a Fm(2)1963 1485 y Fo(in)h([8],)j([34)o(])e(and)g(smaller)e(than)i(that)f(in)g ([36].)65 b(The)-74 1635 y(ab)s(o)m(v)m(e)34 b(b)s(ound)e Fn(\033)g(<)c(N)k Fl(\000)23 b Fo(1)32 b(is)g(pro)m(v)m(ed)i(in)e ([12];)g(see)i(Theorem)f(1.1)f(and)h(the)g(remark)f(b)s(elo)m(w)h(it.) 72 1784 y(Let)g(\001)328 1799 y Fm(1)400 1784 y Fo(denote)h(an)e(op)s (en)h(in)m(terv)-5 b(al)31 b(con)m(taining)g(the)i(closure)g(of)f (\001.)-74 2091 y Fg(Corollary)k(7.1.)49 b Fb(Assume)37 b(that)f Fl(h)p Fn(x)p Fl(i)1360 2055 y Fi(\000)p Fk(\033)1497 2091 y Fn(g)1544 2106 y Fm(\001)1603 2115 y Fh(1)1641 2091 y Fo(\()p Fn(H)8 b Fo(\))p Fl(h)p Fn(A)p Fl(i)1957 2055 y Fk(\033)2039 2091 y Fb(is)36 b(b)s(ounded.)54 b(Then,)38 b(in)d(the)i(ab)s(o)m(v)m(e)g(theorems)f(w)m(e)-74 2241 y(can)d(replace)g(the)g(w)m(eigh)m(t)f Fl(h)p Fn(A)p Fl(i)1063 2204 y Fi(\000)p Fk(\033)1197 2241 y Fb(b)m(y)i Fl(h)p Fn(x)p Fl(i)1466 2204 y Fi(\000)p Fk(\033)1567 2241 y Fb(.)72 2598 y Fo(The)h(strategy)g(for)e(using)g(the)i(ab)s(o)m (v)m(e)f(results)g(to)g(pro)m(v)m(e)h(lo)s(cal)c(deca)m(y)36 b(estimates)d(lik)m(e)g(that)h(in)f Fg(\(H4\))f Fo(is)-74 2747 y(as)h(follo)m(ws.)42 b(Then)305 2996 y Fl(k)32 b(h)p Fn(x)p Fl(i)520 2955 y Fi(\000)p Fk(\033)654 2996 y Fn(e)699 2955 y Fi(\000)p Fk(iH)5 b(t)871 2996 y Fn(g)918 3011 y Fm(\001)981 2996 y Fo(\()p Fn(H)j Fo(\))p Fn( )36 b Fl(k)1295 3011 y Fm(2)1450 2996 y Fo(=)84 b Fl(k)33 b(h)p Fn(x)p Fl(i)1826 2955 y Fi(\000)p Fk(\033)1927 2996 y Fn(g)1974 3011 y Fm(\001)2033 3020 y Fh(1)2071 2996 y Fo(\()p Fn(H)8 b Fo(\))p Fn(e)2281 2955 y Fi(\000)p Fk(iH)d(t)2453 2996 y Fn(g)2500 3011 y Fm(\001)2562 2996 y Fo(\()p Fn(H)j Fo(\))p Fn( )t Fl(k)2844 3011 y Fm(2)1450 3170 y Fo(=)84 b Fl(k)33 b(h)p Fn(x)p Fl(i)1826 3129 y Fi(\000)p Fk(\033)1927 3170 y Fn(g)1974 3185 y Fm(\001)2033 3194 y Fh(1)2071 3170 y Fo(\()p Fn(H)8 b Fo(\))p Fl(h)p Fn(A)p Fl(i)2387 3129 y Fk(\033)2455 3170 y Fl(\001)22 b(h)p Fn(A)p Fl(i)2656 3129 y Fi(\000)p Fk(\033)2790 3170 y Fn(e)2835 3129 y Fi(\000)p Fk(iH)5 b(t)3007 3170 y Fn(g)3054 3185 y Fm(\001)3117 3170 y Fo(\()p Fn(H)j Fo(\))p Fn( )36 b Fl(k)1450 3345 y(\024)116 b(k)32 b(h)p Fn(x)p Fl(i)1858 3304 y Fi(\000)p Fk(\033)1960 3345 y Fn(g)2007 3360 y Fm(\001)2066 3369 y Fh(1)2104 3345 y Fo(\()p Fn(H)8 b Fo(\))p Fl(h)p Fn(A)p Fl(i)2420 3304 y Fk(\033)2466 3345 y Fl(k)22 b(\001)g(k)32 b(h)p Fn(A)p Fl(i)2821 3304 y Fi(\000)p Fk(\033)2955 3345 y Fn(e)3000 3304 y Fi(\000)p Fk(iH)5 b(t)3172 3345 y Fn(g)3219 3360 y Fm(\001)3281 3345 y Fo(\()p Fn(H)j Fo(\))p Fn( )36 b Fl(k)1450 3519 y(\024)116 b Fn(C)39 b Fl(k)32 b(h)p Fn(A)p Fl(i)1985 3478 y Fi(\000)p Fk(\033)2119 3519 y Fn(e)2164 3478 y Fi(\000)p Fk(iH)5 b(t)2336 3519 y Fn(g)2383 3534 y Fm(\001)2446 3519 y Fo(\()p Fn(H)j Fo(\))p Fn( )36 b Fl(k)1450 3713 y(\024)116 b Fn(C)1713 3728 y Fm(1)1752 3713 y Fl(k)32 b Fn(F)1928 3592 y Fe(\022)1999 3645 y Fn(A)p 1999 3690 74 4 v 2018 3781 a(t)2109 3713 y(<)c(\022)2261 3592 y Fe(\023)2371 3713 y Fl(h)p Fn(A)p Fl(i)2522 3672 y Fi(\000)p Fk(\033)2656 3713 y Fn(e)2701 3672 y Fi(\000)p Fk(iH)5 b(t)2873 3713 y Fn(g)2920 3728 y Fm(\001)2983 3713 y Fo(\()p Fn(H)j Fo(\))p Fn( )36 b Fl(k)3297 3728 y Fm(2)1643 3956 y Fo(+)c Fn(C)1821 3971 y Fm(2)1860 3956 y Fl(k)h Fn(F)2036 3835 y Fe(\022)2107 3889 y Fn(A)p 2107 3933 V 2126 4025 a(t)2218 3956 y Fl(\025)28 b Fn(\022)2404 3835 y Fe(\023)2481 3956 y Fl(h)p Fn(A)p Fl(i)2632 3915 y Fi(\000)p Fk(\033)2766 3956 y Fn(e)2811 3915 y Fi(\000)p Fk(iH)5 b(t)2983 3956 y Fn(g)3030 3971 y Fm(\001)3093 3956 y Fo(\()p Fn(H)j Fo(\))p Fn( )36 b Fl(k)3407 3971 y Fm(2)3773 3956 y Fo(\(7.7\))-74 4214 y(Theorem)e(7.2)e(is)h(used)h (to)f(obtain)f(the)i(deca)m(y)g(of)f(the)g(\014rst)h(term)f(on)g(the)g (righ)m(t)g(hand)g(side)g(of)g(\(7.7\),)g(while)-74 4363 y(w)m(e)h(can)f(replace)f Fn(A)h Fo(b)m(y)g Fn(\022)s(t)g Fo(in)f(the)h(second)h(term.)-74 4663 y Fp(References)-25 4911 y Fo([1])49 b(L.)32 b(Alan)g(&)g(J.H.)h(Eb)s(erley)-8 b(,)33 b Fj(Optic)-5 b(al)34 b(R)-5 b(esonanc)g(e)34 b(and)g(Two)h(L)-5 b(evel)34 b(A)n(toms)p Fo(,)e(Do)m(v)m(er)h(1987.) 1901 5306 y(28)p eop %%Page: 29 29 29 28 bop -25 59 a Fo([2])49 b(V.)42 b(Bac)m(h,)k(J.)c(F)-8 b(r\177)-49 b(ohlic)m(h,)44 b(I.M.)f(Sigal)d(&)i(A.)h(So\013er,)i Fj(Positive)f(c)-5 b(ommutators)43 b(and)g(the)h(sp)-5 b(e)g(ctrum)44 b(of)127 208 y(Pauli-Fierz)34 b(Hamiltonians)g(of)g (atoms)h(and)f(mole)-5 b(cules)p Fo(,)31 b(1997)h(preprin)m(t)-25 441 y([3])49 b(J.)32 b(Carr,)h(Applications)e(of)h(Cen)m(ter)i (Manifold)d(Theory)-8 b(,)34 b(Springer-V)-8 b(erlag,)30 b(New)k(Y)-8 b(ork)32 b(\(1981\))-25 673 y([4])49 b(J.)33 b(Co)s(op)s(er,)g(W.)g(Strauss)i Fj(Time)f(p)-5 b(erio)g(dic)34 b(sc)-5 b(attering)36 b(of)f(symmetric)g(hyp)-5 b(erb)g(olic)34 b(systems)p Fo(,)f(J.)g(Math.)127 822 y(Anal.)e(Appl.)i Fg(122)f Fo(\(1987\))g(444{452.)-25 1055 y([5])49 b(H.L.)31 b(Cycon,)h(R.G.)f(F)-8 b(ro)s(ese,)31 b(W.)g(Kirsc)m(h)g(&)g(B.)g (Simon,)f Fj(Schr\177)-50 b(odinger)32 b(Op)-5 b(er)g(ators)32 b(with)i(Applic)-5 b(ations)127 1204 y(to)31 b(Quantum)g(Me)-5 b(chanics)30 b(and)h(Glob)-5 b(al)30 b(Ge)-5 b(ometry)p Fo(,)30 b(Springer-V)-8 b(erlag)26 b(Berlin-Heidelb)s(erg-New)h(Y)-8 b(ork)127 1354 y(1987.)-25 1586 y([6])49 b(V.)25 b(Enss)i(&)e(K.)h(V)-8 b(eselic,)26 b Fj(Bound)i(states)g(and)g(pr)-5 b(op)g(agation)27 b(states)i(for)f(time)g(dep)-5 b(endent)27 b(hamiltonians)p Fo(,)127 1736 y(Ann.)33 b(Inst.)g(H.)g(P)m(oincar)m(\023)-46 b(e)33 b Fg(A39)f Fo(\(1983\))f(159{191.)-25 1968 y([7])49 b(A.)32 b(Galindo)f(&)h(P)-8 b(.)33 b(P)m(ascual,)g(Quan)m(tum)f(Mec)m (hanics)i(I)s(I,)f(Springer)f(1991)-25 2201 y([8])49 b(C.)40 b(G)m(\023)-46 b(erard)39 b(&)h(I.M.)g(Sigal,)f Fj(Sp)-5 b(ac)g(e-time)40 b(pictur)-5 b(e)42 b(of)f(semiclassic)-5 b(al)39 b(r)-5 b(esonanc)g(es)p Fo(,)40 b(Comm)m(un.)f(Math.)127 2350 y(Ph)m(ys.)34 b Fg(145)f Fo(\(1992\))e(281{328.)-25 2583 y([9])49 b(J.S.)38 b(Ho)m(wland,)h Fj(Stationary)g(sc)-5 b(attering)40 b(the)-5 b(ory)40 b(for)f(time-dep)-5 b(endent)38 b(Hamiltonians)p Fo(,)g(Math.)g(Ann.)127 2732 y Fg(207)32 b Fo(\(1974\))g(315-335)-74 2964 y([10])49 b(J.S.)42 b(Ho)m(wland,)i Fj(Sc)-5 b(attering)43 b(the)-5 b(ory)44 b(for)f(Hamiltonians)g(p)-5 b(erio)g(dic)42 b(in)h(time)p Fo(,)h(Indiana)e(Math.)g(J.)g Fg(28)127 3114 y Fo(\(1979\))31 b(471-494)-74 3346 y([11])49 b(W.)33 b(Hunzik)m(er)g(&)g(I.M.)g(Sigal,) e Fj(Time)j(dep)-5 b(endent)34 b(sc)-5 b(attering)35 b(the)-5 b(ory)35 b(for)g(N-b)-5 b(o)g(dy)35 b(quantum)g(systems)p Fo(,)127 3496 y(ETH-Z)s(\177)-51 b(uric)m(h)31 b(preprin)m(t)i (\(1997\))-74 3728 y([12])49 b(W.)32 b(Hunzik)m(er,)i(I.M.)f(Sigal)e(&) h(A.)h(So\013er,)g Fj(Minimal)h(esc)-5 b(ap)g(e)34 b(velo)-5 b(cities)p Fo(,)32 b(1998)f(preprin)m(t.)-74 3961 y([13])49 b(A.)33 b(Jensen)i(&)e(T.)h(Kato,)e Fj(Sp)-5 b(e)g(ctr)g(al)35 b(pr)-5 b(op)g(erties)35 b(of)h(Schr\177)-50 b(odinger)34 b(op)-5 b(er)g(ators)35 b(and)g(time-de)-5 b(c)g(ay)34 b(of)i(wave)127 4110 y(functions)p Fo(,)c(Duk)m(e)h(Math.)g(J.)g Fg(46)f Fo(\(1979\),)g(583{611.)-74 4343 y([14])49 b(H.R.)d(Jauslin)e (&)i(J.L.)g(Leb)s(o)m(witz,)j Fj(Gener)-5 b(alize)g(d)46 b(Flo)-5 b(quet)47 b(op)-5 b(er)g(ator)47 b(for)g(quasip)-5 b(erio)g(dic)g(al)5 b(ly)46 b(driven)127 4492 y(quantum)33 b(systems)p Fo(,)f(in)e(\\Mathematical)f(Ph)m(ysics)k(X",)e(K.)g(Sc)m (hm)s(\177)-51 b(udgen)31 b(\(ed.\))h(Springer)e(\(1991\),)h(and)127 4641 y(cited)h(ref.)-74 4874 y([15])49 b(J.-L.)29 b(Journ)m(\023)-46 b(e,)31 b(A.)f(So\013er)g(&)g(C.)g(Sogge,)h Fn(L)1702 4838 y Fk(p)1769 4874 y Fl(!)d Fn(L)1963 4838 y Fk(p)1999 4814 y Fc(0)2057 4874 y Fj(estimates)k(for)g(the)h(time)f(dep)-5 b(endent)31 b(Schr\177)-50 b(odinger)127 5023 y(e)-5 b(quation)p Fo(,)32 b(Bull.)f(AMS,)i Fg(23)g Fo(\(1990\))1901 5306 y(29)p eop %%Page: 30 30 30 29 bop -74 59 a Fo([16])49 b(C.-A.)e(Pillet)f(&)h(C.E.)i(W)-8 b(a)m(yne,)53 b Fj(Invariant)47 b(manifolds)h(for)g(a)h(class)e(of)i (disp)-5 b(ersive,)51 b(Hamiltonian)127 208 y(p)-5 b(artial)34 b(di\013er)-5 b(ential)34 b(e)-5 b(quations)p Fo(,)32 b(J.)h(Di\013.)e(Eq.)j Fg(141)e Fo(\(1997\))g(310{326)-74 441 y([17])49 b(H.)28 b(Kitada)e(&)i(K.)g(Y)-8 b(a)5 b(jima,)27 b Fj(A)k(sc)-5 b(attering)31 b(the)-5 b(ory)31 b(for)f(time)h(dep)-5 b(endent)29 b(long)h(r)-5 b(ange)31 b(p)-5 b(otentials)p Fo(,)28 b(Duk)m(e)127 590 y(Math.)33 b(J.)f Fg(49)h Fo(\(1982\))e(341-376)-74 822 y([18])49 b(H.)32 b(Lam)m(b,)g Fj(On)h(a)i(p)-5 b(e)g(culiarity)34 b(of)g(the)g(wave-system)g(due)g(to)h(the)f(fr)-5 b(e)g(e)34 b(vibr)-5 b(ations)34 b(of)g(a)g(nucleus)g(in)g(an)127 972 y(extende)-5 b(d)34 b(me)-5 b(dium)p Fo(,)32 b(Pro)s(c.)g(London)h (Math.)g(So)s(c.)f Fg(32)h Fo(\(1900\))f(208-211)-74 1204 y([19])49 b(L.D.)32 b(Landau)h(&)g(E.M.)h(Lifshitz,)e(Quan)m(tum)g (Mec)m(hanics:)46 b(Non)33 b(-relativistic)d(Theory)-8 b(,)34 b(V)-8 b(olume)32 b(3)g(of)127 1354 y Fj(Course)i(of)h(The)-5 b(or)g(etic)g(al)34 b(Physics)p Fo(,)e(P)m(ergamon)g(Press,)i(Oxford)f (1965)-74 1586 y([20])49 b(Leb)s(o)m(witz,)34 b(Jo)s(el)f(L.)h Fj(Macr)-5 b(osc)g(opic)35 b(laws,)h(micr)-5 b(osc)g(opic)34 b(dynamics,)i(time's)f(arr)-5 b(ow)36 b(and)f(Boltzmann)-10 b('s)127 1736 y(entr)-5 b(opy)p Fo(,)32 b(Ph)m(ys.)j(A)d(194)g (\(1993\),)g(no.)g(1-4,)g(1{27.)-74 1968 y([21])49 b(P)-8 b(.D.)31 b(Miller,)f(A.)i(So\013er)g(&)f(M.I.)i(W)-8 b(einstein,)32 b Fj(Structur)-5 b(al)35 b(instability)f(and)f (metastability)h(of)g(br)-5 b(e)g(ather)127 2118 y(states)35 b(of)f(soliton)h(wave)f(guides)p Fo(,)e(in)g(preparation.)-74 2350 y([22])49 b(E.)24 b(Mourre,)j Fj(A)n(bsenc)-5 b(e)26 b(of)i(singular)e(c)-5 b(ontinuous)28 b(sp)-5 b(e)g(ctrum)27 b(for)g(c)-5 b(ertain)27 b(self-adjoint)f(op)-5 b(er)g(ators)p Fo(,)26 b(Com-)127 2500 y(m)m(un.)32 b(Math.)h(Ph)m(ys.)i Fg(78)d Fo(\(1981\))g(391)-74 2732 y([23])49 b(M.)34 b(Murata,)h Fj(R)-5 b(ate)36 b(of)g(de)-5 b(c)g(ay)36 b(of)g(lo)-5 b(c)g(al)36 b(ener)-5 b(gy)36 b(and)g(sp)-5 b(e)g(ctr)g(al)36 b(pr)-5 b(op)g(erties)35 b(of)h(el)5 b(liptic)36 b(op)-5 b(er)g(ators)p Fo(,)34 b(Jpn.)127 2881 y(J.)e(Math.)h Fg(6)g Fo(\(1980\))e(77{127)-74 3114 y([24])49 b(R.M.)28 b(Pyk)m(e)i(&)e(I.M.)h(Sigal,)e Fj(Nonline)-5 b(ar)31 b(wave)f(e)-5 b(quations:)42 b(Constr)-5 b(aints)30 b(on)h(p)-5 b(erio)g(ds)30 b(and)g(exp)-5 b(onential)127 3263 y(b)g(ounds)34 b(for)h(p)-5 b(erio)g(dic)33 b(solutions)i Fo(,)e(Duk)m(e)g(Math.)g(J.)g Fg(88)f Fo(\(1997\))-74 3496 y([25])49 b(P)-8 b(.)37 b(P)m(erry)-8 b(,)40 b(I.M.)e(Sigal)d(&)i (B.)g(Simon,)g Fj(Sp)-5 b(e)g(ctr)g(al)39 b(analysis)f(of)h(N-b)-5 b(o)g(dy)40 b(Schr\177)-50 b(odinger)37 b(op)-5 b(er)g(ators)p Fo(,)38 b(Ann.)127 3645 y(Math.)33 b Fg(114)f Fo(\(1981\))g(519{567.) -74 3878 y([26])49 b(R.)32 b(Pyk)m(e,)j(A.)d(So\013er)h(&)f(M.I.)i(W)-8 b(einstein,)32 b Fj(Stability)j(of)g(Kinks)p Fo(,)d(in)g(progress.)-74 4110 y([27])49 b(J.)25 b(Rauc)m(h,)j Fj(L)-5 b(o)g(c)g(al)28 b(de)-5 b(c)g(ay)28 b(of)g(sc)-5 b(attering)28 b(solutions)g(to)g (Schr\177)-50 b(odinger's)27 b(e)-5 b(quation)p Fo(,)27 b(Comm.)d(Math.)i(Ph)m(ys.)127 4260 y Fg(61)32 b Fo(\(1978\),)g (149{168.)-74 4492 y([28])49 b(M.)37 b(Reed)g(&)g(B.)f(Simon,)h(Metho)s (ds)g(in)f(Mo)s(dern)h(Mathematical)e(Ph)m(ysics,)k(I.)e(F)-8 b(unctional)35 b(Analysis,)127 4641 y(Academic)d(Press)i(1972.)-74 4874 y([29])49 b(M.)27 b(Reed)h(&)f(B.)g(Simon,)g(Metho)s(ds)h(in)e(Mo) s(dern)i(Mathematical)d(Ph)m(ysics,)30 b(IV.)d(Analysis)g(of)f(Op)s (erator,)127 5023 y(Academic)32 b(Press)i(1978.)1901 5306 y(30)p eop %%Page: 31 31 31 30 bop -74 59 a Fo([30])49 b(J.J.)33 b(Sakurai,)f(Adv)-5 b(anced)33 b(Quan)m(tum)g(Mec)m(hanics,)h(Addison)e(W)-8 b(esley)34 b(\(1967\))-74 291 y([31])49 b(T.)35 b(Sc)m(hon)m(b)s(ek,)j Fj(De)-5 b(c)g(ay)36 b(of)h(solutions)f(of)h(Schr\177)-50 b(odinger)36 b(e)-5 b(quations)p Fo(,)35 b(Duk)m(e)h(Math.)f(J.)g(46)f (\(1979\),)h(203{)127 441 y(213.)-74 673 y([32])49 b(I.M.)43 b(Sigal,)g Fj(Nonline)-5 b(ar)43 b(wave)g(and)h(Schr\177)-50 b(odinger)42 b(e)-5 b(quations)44 b(I.)f(Instability)h(of)g(time-p)-5 b(erio)g(dic)42 b(and)127 822 y(quasip)-5 b(erio)g(dic)33 b(solutions)p Fo(,)f(Comm)m(un.)h(Math.)f(Ph)m(ys.)j Fg(153)e Fo(\(1993\))e(297)-74 1055 y([33])49 b(I.M.)33 b(Sigal,)e Fj(Gener)-5 b(al)35 b(char)-5 b(acteristics)34 b(of)h(nonline)-5 b(ar)34 b(dynamics)p Fo(,)e(in)g(Sp)s(ectral)g(and)h (Scattering)f(The-)127 1204 y(ory;)43 b(Pro)s(ceedings)d(of)f(the)h(T) -8 b(aniguc)m(hi)38 b(in)m(ternational)f(w)m(orkshop,)43 b(ed.)d(M.)g(Ik)-5 b(a)m(w)m(a,)42 b(Marcel)d(Dekk)m(er,)127 1354 y(Inc.)33 b(New)g(Y)-8 b(ork)33 b(-)f(Basel)g(-)h(Hong)f(Kong)g (1994)-74 1586 y([34])49 b(I.M.)31 b(Sigal)d(&)i(A.-So\013er,)h Fj(L)-5 b(o)g(c)g(al)32 b(de)-5 b(c)g(ay)32 b(and)h(velo)-5 b(city)32 b(b)-5 b(ounds)33 b(for)f(Quantum)h(pr)-5 b(op)g(agation)p Fo(,)30 b(preprin)m(t,)127 1736 y(Princeton)i(1988)g (\(ftp://www.math.rutgers.edu/pub/so\013er\))-74 1968 y([35])49 b(I.M.)c(Sigal)e(&)i(A.-So\013er,)j Fj(Asymptotic)f (Completeness)e(for)h(Short)g(R)-5 b(ange)46 b(Many)g(Bo)-5 b(dy)46 b(Systems)p Fo(,)127 2118 y(Ann.)33 b(Math.)g Fg(126)f Fo(\(1987\))g(35{108.)-74 2350 y([36])49 b(E.)h(Skibsted,)55 b Fj(Pr)-5 b(op)g(agation)50 b(estimates)g(for)g(N-b)-5 b(o)g(dy)51 b(Schr\177)-50 b(odinger)50 b(op)-5 b(er)g(ators)p Fo(,)53 b(Comm)m(un.)c(Math.)127 2500 y(Ph)m(ys.)34 b Fg(142)f Fo(\(1991\))e(67{98)-74 2732 y([37])49 b(A.)26 b(So\013er)h(&)f(M.I.)h(W)-8 b(einstein,)28 b Fj(Dynamic)g(the)-5 b(ory)29 b(of)g(quantum)h(r)-5 b(esonanc)g(es)28 b(and)g(p)-5 b(erturb)g(ation)30 b(the)-5 b(ory)127 2881 y(of)27 b(emb)-5 b(e)g(dde)g(d)26 b(eigenvalues)p Fo(,)f(in)f(Pro)s(ceedings)i(of)e (Conference)j(on)d(P)m(artial)f(Di\013eren)m(tial)g(Equations)i(and)127 3031 y(Applications,)33 b(Univ)m(ersit)m(y)i(of)f(T)-8 b(oron)m(to,)35 b(June,)h(1995,)e(CRM)h(Lecture)g(Notes,)h(Eds.)g(P)-8 b(.)34 b(Greiner,)h(V.)127 3180 y(Ivrii,)c(L.Seco,)i(&)g(C.)g(Sulem)-74 3413 y([38])49 b(A.)30 b(So\013er)h(&)f(M.I.)i(W)-8 b(einstein,)30 b Fj(Time)i(dep)-5 b(endent)32 b(r)-5 b(esonanc)g(e)32 b(the)-5 b(ory)p Fo(,)31 b(to)f(app)s(ear)h(in)f(Geometric)f(and)127 3562 y(F)-8 b(unctional)30 b(Analysis.)-74 3795 y([39])49 b(A.)32 b(So\013er)g(&)g(M.I.)h(W)-8 b(einstein,)32 b Fj(R)-5 b(esonanc)g(es,)33 b(r)-5 b(adiation)34 b(damping)f(and)h (instability)g(in)h(Hamiltonian)127 3944 y(nonline)-5 b(ar)33 b(wave)h(e)-5 b(quations)p Fo(,)33 b(to)f(app)s(ear)g(in)g(In)m (v)m(en)m(tiones)j(Mathematicae.)-74 4177 y([40])49 b(B.)40 b(V)-8 b(ain)m(b)s(erg)39 b Fj(Sc)-5 b(attering)42 b(of)f(waves)g(in)h (a)f(me)-5 b(dium)41 b(dep)-5 b(ending)40 b(p)-5 b(erio)g(dic)g(al)5 b(ly)41 b(on)h(time)p Fo(,)f(Asterisque)127 4326 y Fg(210)32 b Fo(\(1992\))g(327{340)-74 4558 y([41])49 b(A.)34 b(V)-8 b(anderbau)m(whede)36 b(&)d(G.)h(Io)s(oss,)g Fj(Center)i(manifold)e (the)-5 b(ory)36 b(in)g(in\014nite)f(dimensions)p Fo(,)e(Dynamics)127 4708 y(Rep)s(orted)f Fg(2)h Fo(\(1990\))-74 4940 y([42])49 b(J.M.G.)32 b(Vilar)f(&)h(J.M.)i(Rubi,)e(Ph)m(ys.)i(Rev.)f(Lett.)g Fg(77)g Fo(\(1996\))e(2863)h(and)g(cited)h(ref.)1901 5306 y(31)p eop %%Page: 32 32 32 31 bop -74 59 a Fo([43])49 b(K.)33 b(Y)-8 b(a)5 b(jima,)32 b Fj(Sc)-5 b(attering)36 b(the)-5 b(ory)36 b(for)g(Schr\177)-50 b(odinger)35 b(op)-5 b(er)g(ators)35 b(with)h(p)-5 b(otentials)35 b(p)-5 b(erio)g(dic)35 b(in)h(time)p Fo(,)e(J.)127 208 y(Math.)f(So)s(c.)f(Japan)h Fg(29)f Fo(\(1977\))g(729-743)-74 441 y([44])49 b(K.)30 b(Y)-8 b(a)5 b(jima,)29 b Fj(R)-5 b(esonanc)g(es)31 b(for)i(the)f(A)n(C-Stark)h(e\013e)-5 b(ct)p Fo(,)30 b(Comm)m(un.)g(Math.)h(Ph)m(ys.)h Fg(78)e Fo(\(1982\))f(331{352.)-74 673 y([45])49 b(K.)28 b(Y)-8 b(a)5 b(jima,)27 b Fj(A)32 b(multichannel)e(sc)-5 b(attering)30 b(the)-5 b(ory)32 b(for)f(some)f(time)h(dep)-5 b(endent)30 b(hamiltonians,)g(char)-5 b(ge)127 822 y(tr)g(ansfer)34 b(pr)-5 b(oblem)p Fo(,)32 b(Comm)m(un.)g(Math.)h(Ph)m(ys.)h Fg(75)f Fo(\(1980\))e(153{178.)1901 5306 y(32)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF