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y(w)m(e)d(use)h(w)m(as)f(motiv)-5 b(ated)32 b(and)h(in)g(part)g(dev)m(elop)s(ed)h(in)e(the)i(con)m(text)h (of)e(our)g(study)h(of)f(a)g(class)g(of)g(nonlinear)-74 1553 y(Sc)m(hr\177)-49 b(odinger)35 b(equations)g(with)g(m)m(ultiple)d (nonlinear)h(b)s(ound)i(states)h(and)f(the)g(quan)m(tum)g(resonance)h (prob-)-74 1703 y(lem)30 b([60],)i([61)o(],)g([62],)g([63].)43 b(See)32 b(also)f([11)o(],[12])h(and)f([46].)43 b(Related)31 b(problems)g(are)g(also)g(considered)h(in)f([32])-74 1852 y(and)i([44)o(].)72 2001 y(W)-8 b(e)28 b(b)s(egin)e(with)g(a)h (linear)e(disp)s(ersiv)m(e)i(Hamiltonian)d(PDE)j(for)f(a)g(function)h Fq(u)p Ft(\()p Fq(x;)17 b(t)p Ft(\),)27 b Fq(x)i Fp(2)f Ft(I)-19 b Fo(R)3474 1959 y Fn(n)3548 2001 y Ft(and)27 b Fq(t)g(>)h Ft(0.)-74 2151 y(Supp)s(ose)j(that)f(this)f(system)i(has)f (spatially)e(lo)s(calized)g(and)i(time-p)s(erio)s(dic)c(solutions.)42 b(Suc)m(h)31 b(solutions)e(are)-74 2300 y(often)c(called)e Fr(b)-5 b(ound)27 b(states)p Ft(.)41 b(A)25 b(t)m(ypical)f(solution)f (to)h(suc)m(h)i(a)f(linear)e(system)i(consists)h(of)e(\(i\))f(a)i (non-deca)m(ying)-74 2450 y(part,)35 b(expressible)g(as)g(a)f(linear)f (com)m(bination)f(of)i(b)s(ound)h(states,)h(plus)e(\(ii\))f(a)h(part)g (whic)m(h)h(deca)m(ys)h(to)e(zero)-74 2599 y(in)e(suitable)g(norms)g (\(disp)s(ersion\).)43 b(This)32 b(pap)s(er)h(is)f(dev)m(oted)i(to)e (the)h(study)h(of)e(the)h(follo)m(wing)d(questions:)-74 2792 y Fo(\(1\))36 b Ft(Do)f(small)f(amplitude)g(spatially)g(lo)s (calized)g(and)j(time-p)s(erio)s(dic)32 b(solutions)k(p)s(ersist)g(for) g(t)m(ypical)f(non-)-74 2942 y(linear)c(and)i(Hamiltonian)c(p)s (erturbations?)-74 3091 y Fo(\(2\))j Ft(What)g(is)h(the)g(c)m(haracter) g(of)f(general)g(small)e(amplitude)h(solutions)h(to)g(the)h(p)s(erturb) s(ed)g(dynamics?)-74 3241 y Fo(\(3\))25 b Ft(Ho)m(w)i(are)f(the)h (structures)h(of)d(the)i(unp)s(erturb)s(ed)g(dynamics)f(manifested)f (in)h(the)g(p)s(erturb)s(ed)h(dynamics?)72 3434 y(Represen)m(tativ)m(e) 35 b(of)d(the)h(class)g(of)f(equations)g(of)h(in)m(terest)g(is)f(the)h (nonlinear)e(Klein-Gordon)f(equation:)983 3732 y Fq(@)1039 3691 y Fm(2)1034 3757 y Fn(t)1112 3732 y Fq(u)60 b Ft(=)1364 3636 y Fl(\020)1413 3732 y Ft(\001)23 b Fp(\000)f Fq(V)g Ft(\()p Fq(x)p Ft(\))g Fp(\000)h Fq(m)2033 3691 y Fm(2)2073 3636 y Fl(\021)2139 3732 y Fq(u)f Ft(+)g Fq(\025f)11 b Ft(\()p Fq(u)p Ft(\))p Fq(;)48 b(\025)28 b Fp(2)g Ft(I)-19 b Fo(R)856 b Ft(\(1.1\))-74 3927 y(with)32 b Fq(f)11 b Ft(\()p Fq(u)p Ft(\))32 b(real-v)-5 b(alued,)31 b(smo)s(oth)h(in)f(a) i(neigh)m(b)s(orho)s(o)s(d)e(of)h Fq(u)27 b Ft(=)h(0)k(and)h(ha)m(ving) f(an)h(expansion:)1542 4154 y Fq(f)11 b Ft(\()p Fq(u)p Ft(\))27 b(=)g Fq(u)1919 4113 y Fm(3)1980 4154 y Ft(+)22 b Fp(O)s Ft(\()p Fq(u)2254 4113 y Fm(4)2293 4154 y Ft(\))p Fq(:)1415 b Ft(\(1.2\))-74 4381 y(Here,)36 b Fq(u)30 b Ft(:)g(\()p Fq(x;)17 b(t)p Ft(\))31 b Fp(7!)f Fq(u)p Ft(\()p Fq(x;)17 b(t)p Ft(\))31 b Fp(2)g Ft(I)-19 b Fo(R)p Ft(,)34 b Fq(x)d Fp(2)g Ft(I)-18 b Fo(R)1539 4339 y Fm(3)1578 4381 y Ft(,)35 b(and)f Fq(t)d(>)f Ft(0.)49 b(W)-8 b(e)34 b(shall)f(restrict)h(our)h(large)e(time)g(asymptotic)-74 4531 y(analysis)g(to)f(the)i(case)g Fq(f)11 b Ft(\()p Fq(u)p Ft(\))28 b(=)g Fq(u)1168 4494 y Fm(3)1207 4531 y Ft(.)45 b(The)34 b(more)f(general)f(case)i(\(1.2\))f(can)g(b)s(e)h (treated)f(b)m(y)h(the)g(tec)m(hnique)g(of)-74 4680 y(this)e(pap)s(er)h (b)m(y)h(making)d(suitable)g(mo)s(di\014cation)f(of)i Fq(W)2011 4644 y Fn(s;p)2135 4680 y Ft(norms)h(used.)72 4829 y(W)-8 b(e)33 b(consider)g(\(1.1\))f(with)h(Cauc)m(h)m(y)h(data:) 1061 5057 y Fq(u)p Ft(\()p Fq(x;)17 b Ft(0\))60 b(=)g Fq(u)1593 5072 y Fm(0)1632 5057 y Ft(\()p Fq(x)p Ft(\))p Fq(;)49 b Ft(and)33 b Fq(@)2080 5072 y Fn(t)2110 5057 y Fq(u)p Ft(\()p Fq(x;)17 b Ft(0\))59 b(=)i Fq(u)2642 5072 y Fm(1)2680 5057 y Ft(\()p Fq(x)p Ft(\))p Fq(:)935 b Ft(\(1.3\))1926 5306 y(3)p eop %%Page: 4 4 4 3 bop -74 59 a Ft(Equation)32 b(\(1.1\))h(is)f(a)g(Hamiltonian)d (system)k(with)f(energy:)341 291 y Fp(E)9 b Ft([)p Fq(u;)17 b(@)581 306 y Fn(t)609 291 y Fq(u)p Ft(])60 b Fp(\021)900 223 y Ft(1)p 900 267 49 4 v 900 359 a(2)1008 173 y Fl(Z)1091 291 y Ft(\()p Fq(@)1180 306 y Fn(t)1210 291 y Fq(u)p Ft(\))1304 249 y Fm(2)1397 291 y Ft(+)55 b Fp(jr)p Fq(u)p Fp(j)1723 249 y Fm(2)1816 291 y Ft(+)f Fq(m)2031 249 y Fm(2)2071 291 y Fq(u)2127 249 y Fm(2)2221 291 y Ft(+)g Fq(V)22 b Ft(\()p Fq(x)p Ft(\))p Fq(u)2617 249 y Fm(2)2689 291 y Fq(dx)54 b Ft(+)h Fq(\025)3054 173 y Fl(Z)3186 291 y Fq(F)14 b Ft(\()p Fq(u)p Ft(\))31 b Fq(dx;)214 b Ft(\(1.4\))-74 507 y(where)34 b Fq(F)285 471 y Fk(0)308 507 y Ft(\()p Fq(u)p Ft(\))27 b(=)h Fq(f)11 b Ft(\()p Fq(u)p Ft(\))31 b(and)i Fq(F)14 b Ft(\(0\))27 b(=)g(0.)72 657 y(In)35 b(the)g(con)m(text)h(of)e(equations)h(of)f(t)m(yp)s(e)h (\(1.1\),)g(w)m(e)g(ha)m(v)m(e)h(found)f(the)g(follo)m(wing)c(answ)m (ers)37 b(to)d(questions)-74 806 y(\(1\),)e(\(2\))h(and)f(\(3\).)-74 955 y Fo(\(A1\))22 b Ft(In)i(a)f(small)e(op)s(en)i(neigh)m(b)s(orho)s (o)s(d)f(of)h(the)g(origin,)g(there)h(are)g(no)f(p)s(erio)s(dic)e(or)i (quasip)s(erio)s(dic)f(solutions;)-74 1105 y(Corollary)31 b(1.1.)-74 1254 y Fo(\(A2\))g Ft(All)g(solutions)h(in)g(this)g(neigh)m (b)s(orho)s(o)s(d)f(tend)i(to)g(zero)g(\(radiate\))e(as)i Fq(t)28 b Fp(!)f(1)p Ft(;)32 b(Theorem)h(1.1.)-74 1404 y Fo(\(A3\))c Ft(The)h(time)f(deca)m(y)i(of)e(solutions)g(is)g (anomalously)f(slo)m(w)2218 1368 y Fk(\003)2257 1404 y Ft(,)j Fr(i.e.)41 b Ft(a)30 b(rate)g(whic)m(h)g(is)f(slo)m(w)m(er)h (than)g(the)g(free)-74 1553 y(disp)s(ersiv)m(e)j(rate;)g(Theorem)g (1.1.)72 1784 y(Dynamical)25 b(systems)k(of)e(the)g(t)m(yp)s(e)i(w)m(e) f(analyze)f(app)s(ear)g(in)g(a)g(n)m(um)m(b)s(er)g(of)g(ph)m(ysical)g (settings.)42 b(Consider)-74 1933 y(a)37 b(nonlinear)f(medium)g(in)h (whic)m(h)h(w)m(a)m(v)m(es)i(can)e(propagate.)57 b(If)38 b(the)g(medium)e(has)i(lo)s(cal)d(inhomogeneities,)-74 2083 y(defects)c(or)f(impurities,)e(these)j(arise)f(in)f(the)i (mathematical)26 b(mo)s(del)j(as)h(a)f(spatially)f(dep)s(enden)m(t)k (co)s(e\016cien)m(t)-74 2232 y(in)46 b(the)h(equation)f(\()p Fr(e.g.)85 b Ft(lo)s(calized)44 b(p)s(oten)m(tial\).)84 b(Suc)m(h)47 b(p)s(erturbations)f(of)h(the)f(original)e(homogeneous)-74 2381 y(\(translation)31 b(in)m(v)-5 b(arian)m(t\))31 b(dynamics)i(in)m(tro)s(duce)f(new)i(mo)s(des)f(in)m(to)e(the)j(system) f(\()p Fr(impurity)j(mo)-5 b(des)p Ft(\))31 b(whic)m(h)-74 2531 y(can)i(trap)f(some)h(of)f(the)h(energy)g(and)g(a\013ect)g(the)g (time)e(ev)m(olution)h(of)g(the)h(system;)h(see)f([41],)g([35)o(],)g ([73].)72 2761 y(Let)41 b Fp(h)p Fq(K)7 b Fp(i)79 b Ft(=)h(\(1)55 b(+)f Fp(j)p Fq(K)7 b Fp(j)1076 2725 y Fm(2)1115 2761 y Ft(\))1163 2682 y Fj(1)p 1163 2694 31 4 v 1163 2735 a(2)1207 2761 y Ft(.)66 b(F)-8 b(or)39 b(the)h(nonlinear)f (Klein-Gordon)e(equation,)42 b(\(1.1\),)f(w)m(e)g(pro)m(v)m(e)g(the)-74 2911 y(follo)m(wing)30 b(result.)-74 3131 y Fo(Theorem)37 b(1.1.)49 b Fi(Let)33 b Fq(V)22 b Ft(\()p Fq(x)p Ft(\))32 b Fi(b)s(e)h(real-v)-5 b(alued)31 b(and)i(suc)m(h)h(that)-74 3321 y Fo(\(V1\))d Fi(F)-8 b(or)32 b Fq(\016)g(>)27 b Ft(5)33 b Fi(and)f Fp(j)p Fq(\013)q Fp(j)27 b(\024)h Ft(2)p Fi(,)33 b Fp(j)p Fq(@)1254 3285 y Fn(\013)1304 3321 y Fq(V)21 b Ft(\()p Fq(x)p Ft(\))p Fp(j)28 b(\024)g Fq(C)1744 3336 y Fn(\013)1793 3321 y Fp(h)p Fq(x)p Fp(i)1926 3285 y Fk(\000)p Fn(\016)2019 3321 y Fi(.)-74 3470 y Fo(\(V2\))j Ft(\()p Fp(\000)p Ft(\001)23 b(+)f(1\))590 3434 y Fk(\000)p Fm(1)701 3374 y Fl(\020)751 3470 y Ft(\()p Fq(x)g Fp(\001)g(r)p Ft(\))1037 3434 y Fn(l)1063 3470 y Fq(V)f Ft(\()p Fq(x)p Ft(\))1272 3374 y Fl(\021)1339 3470 y Ft(\()p Fp(\000)p Ft(\001)i(+)f(1\))1743 3434 y Fk(\000)p Fm(1)1869 3470 y Fi(is)32 b(b)s(ounded)i(on)e Fq(L)2567 3434 y Fm(2)2640 3470 y Fi(for)g Fp(j)p Fq(l)r Fp(j)27 b(\024)h Fq(N)3086 3485 y Fk(\003)3158 3470 y Fi(with)k Fq(N)3458 3485 y Fk(\003)3525 3470 y Fp(\025)d Ft(10)p Fi(.)-74 3620 y Fo(\(V3\))i Fi(Zero)i(is)f(not)g(a)g Fr(r)-5 b(esonanc)g(e)32 b Fi(of)g(the)h(op)s(erator)f Fp(\000)p Ft(\001)23 b(+)f Fq(V)f Fi(;)33 b(see)h([31)o(],)f([72].)72 3769 y(Assume)h(the)f(op)s(erator)1458 3919 y Fq(B)1537 3878 y Fm(2)1605 3919 y Ft(=)27 b Fp(\000)p Ft(\001)c(+)f Fq(V)g Ft(\()p Fq(x)p Ft(\))g(+)g Fq(m)2402 3878 y Fm(2)3773 3919 y Ft(\(1.5\))-74 4108 y Fi(has)33 b(con)m(tin)m(uous)h(sp)s (ectrum,)f Fq(\033)1097 4123 y Fn(cont)1235 4108 y Ft(\()p Fq(H)8 b Ft(\))28 b(=)g([)p Fq(m)1644 4072 y Fm(2)1683 4108 y Fq(;)17 b Fp(1)p Ft(\))p Fi(,)33 b(and)g(a)f(unique)i(strictly)e (p)s(ositiv)m(e)g(simple)f(eigen)m(v)-5 b(alue,)-74 4258 y Ft(\012)-4 4222 y Fm(2)64 4258 y Fq(<)27 b(m)252 4222 y Fm(2)325 4258 y Fi(with)32 b(asso)s(ciated)g(normalized)f (eigenfunction,)h Fq(')p Fi(:)1693 4474 y Fq(B)1772 4433 y Fm(2)1811 4474 y Fq(')c Ft(=)f(\012)2076 4433 y Fm(2)2116 4474 y Fq(':)1566 b Ft(\(1.6\))-74 4691 y Fi(Corresp)s(ondingly)-8 b(,)49 b(the)e(linear)e(Klein-Gordon)f(equation)i(\(1.1\),)j(with)d Fq(\025)51 b Ft(=)g(0)p 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3041 y Fq(;)49 b Ft(where)929 b(\(2.12\))1566 3215 y Fq(E)1638 3230 y Fm(0)1678 3215 y Ft(\()p Fq(t)p Ft(\))p Fq(f)126 b Ft(=)115 b(cos)q(\()p Fq(B)5 b(t)p Ft(\))p Fq(f)43 b(;)50 b Ft(and)925 b(\(2.13\))1574 3415 y Fq(E)1646 3430 y Fm(1)1686 3415 y Ft(\()p Fq(t)p Ft(\))p Fq(g)119 b Ft(=)2164 3348 y(sin\()p Fq(B)5 b(t)p Ft(\))p 2164 3392 311 4 v 2280 3483 a Fq(B)2484 3415 y(g)36 b(:)72 3648 y Ft(One)29 b(exp)s(ects)g(that)f(estimates)f(of)g (the)h(t)m(yp)s(e)h(app)s(earing)e(in)g(Theorem)h(2.1)f(and)h (Corollary)e(2.1)h(will)f(hold)-74 3797 y(as)j(w)m(ell)f(for)g Fq(E)453 3812 y Fm(0)492 3797 y Ft(\()p Fq(t)p Ft(\))h(and)g Fq(E)890 3812 y Fm(1)929 3797 y Ft(\()p Fq(t)p Ft(\))g(restricted)g(to) f Fo(P)1691 3812 y Fh(c)1731 3797 y Fq(L)1797 3761 y Fm(2)1837 3797 y Ft(,)h(the)g(con)m(tin)m(uous)g(sp)s(ectral)g(part)f (of)g Fq(H)35 b Ft(=)28 b Fq(B)3517 3761 y Fm(2)3557 3797 y Ft(.)42 b(One)29 b(can)-74 3947 y(relate)36 b(functions)h(of)g (the)g(op)s(erator)f Fq(H)8 b Ft(,)38 b(on)e(its)h(con)m(tin)m(uous)g (sp)s(ectral)g(part,)h(to)e(functions)h(of)g Fq(H)3597 3962 y Fm(0)3708 3947 y Ft(=)71 b Fq(B)3934 3911 y Fm(2)3929 3971 y(0)-74 4096 y Ft(using)32 b Fr(wave)i(op)-5 b(er)g(ators)p Ft(.)43 b(Let)1224 4246 y Fq(W)1316 4261 y Fm(+)1435 4246 y Ft(=)60 b(strong)c Fp(\000)71 b Ft(lim)2025 4300 y Fn(t)p Fk(!1)2241 4246 y Fq(e)2286 4204 y Fn(itH)2403 4246 y Fq(e)2448 4204 y Fk(\000)p Fn(itH)2610 4213 y Fj(0)2649 4246 y Fq(:)1049 b Ft(\(2.14\))-74 4442 y(W)-8 b(a)m(v)m(e)34 b(op)s(erators)e(relate)g Fq(H)972 4457 y Fm(0)1044 4442 y Ft(to)g Fq(H)40 b Ft(on)33 b Fo(P)1497 4457 y Fh(c)1537 4442 y Fq(L)1603 4406 y Fm(2)1675 4442 y Ft(via)f(the)h(in)m(tert)m(wining)e(prop)s(ert)m(y:)1339 4675 y Fq(H)67 b Ft(=)28 b Fq(W)1683 4690 y Fm(+)1775 4675 y Fq(H)1856 4690 y Fm(0)1927 4675 y Fq(W)2033 4634 y Fk(\003)2019 4699 y Fm(+)2176 4675 y Ft(on)33 b Fo(P)2389 4690 y Fh(c)2428 4675 y Fq(L)2494 4634 y Fm(2)2534 4675 y Fq(:)1164 b Ft(\(2.15\))-74 4907 y(K.)40 b(Y)-8 b(a)5 b(jima)39 b([72)o(])h(has)h(pro)m(v)m(ed)h(the)e Fq(W)1395 4871 y Fn(k)r(;p)1493 4907 y Ft(\(I)-18 b Fo(R)1631 4865 y Fn(n)1678 4907 y Ft(\))40 b(b)s(oundedness)i(of)e(w)m(a)m(v)m(e)i(op) s(erators)e(for)g(the)h(Sc)m(hr\177)-49 b(odinger)-74 5057 y(equation.)43 b(A)33 b(consequence)j(of)c(this)g(w)m(ork)h(is)g (the)g(follo)m(wing)c(result)k(for)f(spatial)f(dimension)g Fq(n)d Ft(=)f(3:)1901 5306 y(13)p eop %%Page: 14 14 14 13 bop -74 59 a Fo(Theorem)37 b(2.2.)49 b Fi(Let)41 b Fq(H)50 b Ft(=)42 b Fp(\000)p Ft(\001)28 b(+)g Fq(V)21 b Fi(,)43 b(where)g Fq(V)21 b Ft(\()p Fq(x)p Ft(\))41 b Fi(b)s(e)g(real-v)-5 b(alued)40 b(function)g(on)h Ft(I)-18 b Fo(R)3315 17 y Fm(3)3395 59 y Fi(satisfying)40 b(the)-74 208 y(follo)m(wing)26 b(h)m(yp)s(otheses,)32 b(whic)m(h)d(are)g (satis\014ed)g(b)m(y)h(smo)s(oth)d(and)i(su\016cien)m(tly)g(rapidly)f (deca)m(ying)h(p)s(oten)m(tials:)-74 358 y(F)-8 b(or)32 b(an)m(y)h Fp(j)p Fq(\013)q Fp(j)27 b(\024)h Fq(`)k Fi(there)i(is)e(a)g (constan)m(t)i Fq(C)1502 373 y Fn(\013)1583 358 y Fi(suc)m(h)g(that) 1253 472 y Fl(\014)1253 522 y(\014)1253 572 y(\014)1253 621 y(\014)1290 529 y Fq(@)1346 493 y Fn(\013)1397 529 y Fq(V)p 1290 573 185 4 v 1302 665 a(@)5 b(x)1413 636 y Fn(\013)1485 597 y Ft(\()p Fq(x)p Ft(\))1616 472 y Fl(\014)1616 522 y(\014)1616 572 y(\014)1616 621 y(\014)1704 597 y Fp(\024)61 b Fq(C)1912 612 y Fn(\013)1961 597 y Fp(h)p Fq(x)p Fp(i)2094 555 y Fk(\000)p Fn(\016)2220 597 y Fq(;)114 b(\016)31 b(>)d Ft(5)k Fq(:)-74 823 y Fi(Additionally)-8 b(,)42 b(assume)h(that)f Ft(0)g Fi(is)g(neither)h (an)f(eigen)m(v)-5 b(alue)42 b(nor)h(a)f Fr(r)-5 b(esonanc)g(e)41 b Fi(of)h Fq(H)8 b Fi(;)47 b(see)c([72],)i([31].)73 b(If)-74 973 y Ft(0)37 b Fp(\024)h Fq(k)s(;)17 b(k)279 937 y Fk(0)339 973 y Fp(\024)38 b Fq(l)i Fi(and)f Ft(1)e Fp(\024)g Fq(p;)17 b(p)1061 937 y Fk(0)1122 973 y Fp(\024)37 b(1)p Fi(,)j(then)e(there)h (exists)g(a)f(constan)m(t)h Fq(C)44 b(>)38 b Ft(0)f Fi(suc)m(h)j(that)e (for)g(an)m(y)h(Borel)-74 1122 y(function)32 b Fq(f)43 b Fi(on)33 b Fo(R)619 1086 y Fm(1)690 1122 y Fi(w)m(e)h(ha)m(v)m(e)143 1341 y Fq(C)220 1300 y Fk(\000)p Fm(1)347 1341 y Fp(jj)p Fq(f)11 b Ft(\()p Fq(H)581 1356 y Fm(0)619 1341 y Ft(\))p Fp(jj)713 1367 y Fn(B)s Fm(\()p Fn(W)873 1349 y Fg(k)q(;p)962 1367 y Fn(;W)1059 1349 y Fg(k)1093 1335 y Fb(0)1114 1349 y Fg(;p)1165 1335 y Fb(0)1191 1367 y Fm(\))1283 1341 y Fp(\024)61 b(jj)p Fq(f)11 b Ft(\()p Fq(H)d Ft(\))p Fo(P)1778 1356 y Fh(c)1816 1341 y Ft(\()p Fq(H)g Ft(\))p Fp(jj)2037 1367 y Fn(B)s Fm(\()p Fn(W)2197 1349 y Fg(k)q(;p)2285 1367 y Fn(;W)2382 1349 y Fg(k)2416 1335 y Fb(0)2437 1349 y Fg(;p)2488 1335 y Fb(0)2515 1367 y Fm(\))2607 1341 y Fp(\024)60 b Fq(C)40 b Fp(jj)p Fq(f)11 b Ft(\()p Fq(H)3088 1356 y Fm(0)3126 1341 y Ft(\))p Fp(jj)3220 1367 y Fn(B)s Fm(\()p Fn(W)3380 1349 y Fg(k)q(;p)3469 1367 y Fn(;W)3566 1349 y Fg(k)3600 1335 y Fb(0)3621 1349 y Fg(;p)3672 1335 y Fb(0)3698 1367 y Fm(\))3730 1341 y Fq(:)-74 1559 y Fi(Here,)34 b Fo(P)261 1574 y Fh(c)300 1559 y Ft(\()p Fq(H)8 b Ft(\))32 b 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Fq(E)1319 2468 y Fn(j)1356 2453 y Fi(.)-74 2863 y Fo(2.3.)75 b(Singular)37 b(resolv)m(en)m(ts)g(and)h(time)e(deca)m(y) -74 3077 y Ft(As)f(discussed)i(in)d(the)h(in)m(tro)s(duction,)f(the)i (damping)d(term)h(in)g(the)h(e\013ectiv)m(e)h(nonlinear)d(oscillator)f (\(1.26\),)-74 3227 y(is)45 b(related)h(to)f(the)h(ev)-5 b(aluation)44 b(of)i(a)f(singular)f(limit)f(of)i(the)h(resolv)m(en)m(t) h(as)f(the)g(eigen)m(v)-5 b(alue)45 b(parameter)-74 3376 y(approac)m(hes)i(the)g(con)m(tin)m(uous)g(sp)s(ectrum.)84 b(T)-8 b(o)46 b(ensure)i(that)e(the)g(correction)g(terms)g(to)g(the)g (nonlinear)-74 3526 y(oscillator)27 b(\(1.26\))i(can)h(b)s(e)g(treated) g(p)s(erturbativ)m(ely)-8 b(,)30 b(w)m(e)h(require)f(lo)s(cal)d(deca)m (y)k(estimates)e(for)g(the)h(op)s(erator)-74 3675 y Fq(e)-29 3639 y Fn(iB)s(t)81 3675 y Ft(\()p Fq(B)23 b Fp(\000)18 b Ft(\003)e Fp(\006)i Fq(i)p Ft(0\))610 3639 y Fk(\000)p Fm(1)704 3675 y Ft(,)31 b(where)h(\003)d(is)h(a)g(p)s(oin)m(t)f(in)h (the)g(in)m(terior)f(of)h(the)g(con)m(tin)m(uous)h(sp)s(ectrum)g(of)e Fq(B)36 b Ft(\(\003)27 b Fq(>)h(m)p Ft(\).)-74 3824 y(Suc)m(h)34 b(estimates)e(are)g(analogous)f(to)h(those)h(used)h(b)m(y)f(the)g (authors)f(in)g(recen)m(t)h(w)m(ork)h(on)e(a)g(time)f(dep)s(enden)m(t) -74 3974 y(approac)m(h)i(to)f(the)h(quan)m(tum)g(resonance)h(problem)e ([62)o(].)72 4123 y(W)-8 b(e)40 b(b)s(egin)e(with)h(a)f(prop)s (osition,)h(whic)m(h)h(is)e(essen)m(tially)h(a)f(restatemen)m(t)i(of)f (Theorem)g(2.3)g(for)f Fq(E)3799 4138 y Fn(j)3836 4123 y Ft(\()p Fq(t)p Ft(\),)-74 4273 y(and)33 b(is)f(pro)m(v)m(ed)i(the)f (same)f(w)m(a)m(y)-8 b(.)-74 4495 y Fo(Prop)s(osition)36 b(2.1.)49 b Fi(\()p Fq(W)899 4459 y Fn(s;p)1024 4495 y Fr(estimates)33 b(for)h Fq(e)1651 4459 y Fn(iB)s(t)1761 4495 y Fi(\))e(Assume)g(that)f Fq(V)22 b Ft(\()p Fq(x)p Ft(\))32 b Fi(satis\014es)g(the)g(h)m(yp)s(otheses)h(of)e(The-)-74 4645 y(orem)f(2.2)g(\()p Fq(n)e Ft(=)g(3)p Fi(\);)j(see)h([72)o(])f (for)f(h)m(yp)s(otheses)j(for)d(general)g(space)i(dimension,)e Fq(n)e Fp(\025)g Ft(3)p Fi(.)43 b(Let)31 b Fq(p)f Fi(and)h Fq(p)3702 4609 y Fk(0)3756 4645 y Fi(b)s(e)g(as)-74 4794 y(in)h(Theorem)h(2.1)f(and)h(let)f Fq(l)d Fp(\025)g Fq(s)e Fp(\025)h Ft(\()1330 4755 y Fm(1)p 1330 4771 36 4 v 1330 4829 a(2)1398 4794 y Fp(\000)1519 4755 y Fm(1)p 1507 4771 58 4 v 1507 4829 a Fn(p)1543 4810 y Fb(0)1575 4794 y Ft(\)\()p Fq(n)22 b Ft(+)g(2\))p Fi(,)33 b(where)h Fq(l)g Fi(is)e(as)h(in)f(Theorem)h(2.2.)43 b(Then,)995 5029 y Fp(k)32 b Fq(e)1122 4988 y Fn(iB)s(t)1265 5029 y Fo(P)1342 5044 y Fh(c)1415 5029 y Fq( )t Fp(k)1532 5044 y Fn(p)1568 5025 y Fb(0)1654 5029 y Fp(\024)60 b Fq(C)7 b Fp(j)p Fq(t)p Fp(j)1959 4978 y Fk(\000)p Fn(n)p Fm(\()2094 4951 y Fj(1)p 2094 4963 31 4 v 2094 5004 a(2)2135 4978 y Fk(\000)2211 4951 y Fj(1)p 2199 4963 55 4 v 2199 5009 a Fg(p)2231 4995 y Fb(0)2264 4978 y Fm(\))2328 5029 y Fp(k)p Fq( )t Fp(k)2495 5044 y Fn(s;p)2586 5029 y Fq(;)50 b(t)27 b Fp(6)p Ft(=)h(0)p Fq(;)820 b Ft(\(2.16\))1901 5306 y(14)p eop %%Page: 15 15 15 14 bop 72 59 a Ft(The)36 b(next)g(prop)s(osition)d(is)h(the)h ("singular")e(resolv)m(en)m(t)j(estimate)e(w)m(e)i(shall)d(require)i (in)f(section)h(7.)50 b(Let)-74 208 y Fq(\033)-19 223 y Fk(\003)21 208 y Ft(\()p Fq(n)p Ft(\))28 b(=)f(max)p Fp(f)528 169 y Fn(n)p 528 185 43 4 v 532 242 a Fm(2)580 208 y Fq(;)17 b Ft(2)22 b(+)830 169 y Fm(2)p Fn(n)p 803 185 133 4 v 803 242 a(n)p Fm(+2)946 208 y Fp(g)p Ft(.)-74 404 y Fo(Prop)s(osition)36 b(2.2.)49 b Fi(Assume)35 b(the)g(h)m(yp)s (otheses)i(of)d(Prop)s(osition)e(2.1.)49 b(Additionally)-8 b(,)32 b(assume)j(that)f Fq(V)22 b Ft(\()p Fq(x)p Ft(\))-74 553 y Fi(satis\014es)34 b(h)m(yp)s(othesis)h Fo(\(V2\))d Fi(of)h(Theorem)h(1.1.)45 b(Also,)34 b(let)e Fq(\033)h(>)c(\033)2389 568 y Fk(\003)2429 553 y Ft(\()p Fq(n)p Ft(\))p Fi(.)46 b(Then,)35 b(for)e(an)m(y)h(p)s(oin)m(t)f Ft(\003)g Fi(\()p Ft(\003)c Fq(>)g(m)p Fi(\))-74 703 y(in)j(the)h(con)m(tin)m(uous)g(sp)s (ectrum)g(of)f Fq(B)38 b Fi(w)m(e)c(ha)m(v)m(e)f(for)f Fq(l)e Ft(=)e(0)p Fq(;)17 b Ft(1)p Fq(;)g Ft(2)p Fi(:)388 909 y Fp(kh)p Fq(x)p Fp(i)571 868 y Fk(\000)p Fn(\033)705 909 y Fq(e)750 868 y Fn(iB)s(t)910 909 y Ft(\()p Fq(B)27 b Fp(\000)c Ft(\003)f(+)g Fq(i)p Ft(0\))1457 860 y Fk(\000)p Fn(l)1587 909 y Fo(P)1664 924 y Fh(c)1736 909 y Fp(h)p Fq(x)p Fp(i)1869 868 y Fk(\000)p Fn(\033)2003 909 y Fq( )t Fp(k)2120 924 y Fm(2)2275 909 y Fp(\024)116 b Fq(C)40 b Fp(h)p Fq(t)p Fp(i)2691 865 y Fk(\000)2778 838 y Fj(2)p Fg(n)p 2755 850 116 4 v 2755 891 a(n)p Fj(+2)2917 909 y Fp(k)p Fq( )t Fp(k)3084 924 y Fm(1)p Fn(;)p Fm(2)3178 909 y Fq(;)49 b(t)28 b(>)g Ft(0)p Fq(;)387 1083 y Fp(k)k(h)p Fq(x)p Fp(i)602 1042 y Fk(\000)p Fn(\033)736 1083 y Fq(e)781 1042 y Fn(iB)s(t)941 1083 y Ft(\()p Fq(B)27 b Fp(\000)c Ft(\003)f Fp(\000)g Fq(i)p Ft(0\))1489 1035 y Fk(\000)p Fn(l)1619 1083 y Fo(P)1696 1098 y Fh(c)1769 1083 y Fp(h)p Fq(x)p Fp(i)1902 1042 y Fk(\000)p Fn(\033)2003 1083 y Fq( )t Fp(k)2120 1098 y Fm(2)2275 1083 y Fp(\024)116 b Fq(C)40 b Fp(h)p Fq(t)p Fp(i)2691 1039 y Fk(\000)2778 1012 y Fj(2)p Fg(n)p 2755 1024 V 2755 1066 a(n)p Fj(+2)2917 1083 y Fp(k)p Fq( )t Fp(k)3084 1098 y Fm(1)p Fn(;)p Fm(2)3178 1083 y Fq(;)49 b(t)28 b(<)g Ft(0)p Fq(;)228 b Ft(\(2.17\))-74 1289 y Fr(pr)-5 b(o)g(of)p Ft(:)56 b(W)-8 b(e)39 b(pro)m(v)m(e)i(the)e (estimate)f(\(2.17\))h(for)f(the)h(case)h(of)f(the)g Fq(t)g(>)g Ft(0.)62 b(F)-8 b(or)39 b(the)g(case,)j Fq(t)d(<)f Ft(0,)j(the)e(same)-74 1439 y(argumen)m(t)32 b(with)h(simple)e(mo)s (di\014cations)f(applies.)72 1588 y(Let)46 b Fq(g)307 1603 y Fm(\001)420 1588 y Ft(=)k Fq(g)593 1603 y Fm(\001)656 1588 y Ft(\()p Fq(B)5 b Ft(\))45 b(denote)i(a)e(smo)s(oth)g(c)m (haracteristic)h(functions)g(of)f(an)h(op)s(en)f(in)m(terv)-5 b(al,)48 b(\001)e(whic)m(h)-74 1738 y(con)m(tains)34 b(\003)g(and)g(is)g(con)m(tained)g(in)f(the)i(con)m(tin)m(uous)g(sp)s (ectrum)f(of)g Fq(B)5 b Ft(.)48 b(Also,)34 b(let)p 3006 1685 51 4 v 33 w Fq(g)3057 1761 y Fm(\001)3150 1738 y Fp(\021)c Ft(1)23 b Fp(\000)h Fq(g)3477 1753 y Fm(\001)3540 1738 y Ft(.)48 b(W)-8 b(e)34 b(then)-74 1887 y(write,)f(for)f Fq(l)d Ft(=)f(1)p Fq(;)17 b Ft(2:)-39 2093 y Fq(e)6 2052 y Fn(iB)s(t)165 2093 y Ft(\()p Fq(B)27 b Fp(\000)c Ft(\003)f(+)g Fq(i")p Ft(\))709 2045 y Fk(\000)p Fn(l)806 2093 y Fo(P)883 2108 y Fh(c)1039 2093 y Ft(=)116 b Fq(e)1276 2052 y Fn(iB)s(t)1419 2093 y Fq(g)1466 2108 y Fm(\001)1529 2093 y Ft(\()p Fq(B)5 b Ft(\))49 b(\()p Fq(B)27 b Fp(\000)c Ft(\003)f(+)g Fq(i")p Ft(\))2277 2045 y Fk(\000)p Fn(l)2374 2093 y Fo(P)2451 2108 y Fh(c)2546 2093 y Ft(+)54 b Fq(e)2721 2052 y Fn(iB)s(t)p 2864 2040 V 2864 2093 a Fq(g)2914 2117 y Fm(\001)2977 2093 y Ft(\()p Fq(B)5 b Ft(\))50 b(\()o Fq(B)28 b Fp(\000)22 b Ft(\003)g(+)g Fq(i")p Ft(\))3725 2045 y Fk(\000)p Fn(l)3823 2093 y Fo(P)3900 2108 y Fh(c)1038 2268 y Fp(\021)116 b Fq(S)1297 2226 y Fn(")1291 2292 y Fm(1)1334 2268 y Ft(\()p Fq(t)p Ft(\))55 b(+)f Fq(S)1696 2226 y Fn(")1690 2292 y Fm(2)1733 2268 y Ft(\()p Fq(t)p Ft(\))p Fq(:)1854 b Ft(\(2.18\))72 2474 y(W)-8 b(e)33 b(no)m(w)h(estimate)d(the)i(op)s (erators)g Fq(S)1494 2489 y Fn(j)1558 2474 y Fp(\021)28 b Ft(lim)1799 2489 y Fn(")p Fk(!)p Fm(0)1958 2474 y Fq(S)2024 2438 y Fn(")2018 2498 y(j)2060 2474 y Ft(,)33 b Fq(j)h Ft(=)27 b(1)p Fq(;)17 b Ft(2)32 b(individually)-8 b(.)1523 2698 y Fr(Estimation)34 b(of)h Fq(S)2197 2713 y Fm(1)2236 2698 y Ft(\()p Fq(t)p Ft(\))p Fr(:)72 2885 y Ft(Consider)e(the)g(case)h Fq(l)c Ft(=)d(2;)33 b(the)g(case)g Fq(l)d Ft(=)e(1)k(is)g(simpler.)42 b(First,)32 b(w)m(e)h(note)g(that:)825 3091 y Fp(h)p Fq(x)p Fp(i)958 3050 y Fk(\000)p Fn(\033)1092 3091 y Fq(S)1158 3050 y Fn(")1152 3116 y Fm(1)1195 3091 y Ft(\()p Fq(t)p Ft(\))f Fp(h)p Fq(x)p Fp(i)1471 3050 y Fk(\000)p Fn(\033)666 3266 y Ft(=)115 b Fp(\000)p Fq(e)979 3225 y Fn(it)p Fm(\(\003)p Fk(\000)p Fn(i")p Fm(\))1298 3149 y Fl(Z)1381 3175 y Fk(1)1344 3337 y Fn(t)1505 3266 y Fq(ds)1651 3149 y Fl(Z)1734 3175 y Fk(1)1697 3337 y Fn(s)1858 3266 y Fq(d\034)43 b Fp(h)p Fq(x)p Fp(i)2127 3225 y Fk(\000)p Fn(\033)2262 3266 y Fq(e)2307 3225 y Fn(i\034)8 b Fm(\()p Fn(B)s Fk(\000)p Fm(\003+)p Fn(i")p Fm(\))2733 3266 y Fq(g)2780 3281 y Fm(\001)2843 3266 y Ft(\()p Fq(B)d Ft(\))p Fo(P)3075 3281 y Fh(c)3147 3266 y Fp(h)p Fq(x)p Fp(i)3280 3225 y Fk(\000)p Fn(\033)3725 3266 y Ft(\(2.19\))-74 3472 y(No)m(w)33 b(w)m(e)h(claim)c(that)i(the)h(op)s(erator)f(in)g(the) h(in)m(tegrand)f(satis\014es)i(the)f(estimate:)847 3700 y Fp(kh)p Fq(x)p Fp(i)1030 3659 y Fk(\000)p Fn(\033)1164 3700 y Fq(e)1209 3659 y Fn(i\034)8 b Fm(\()p Fn(B)s Fk(\000)p Fm(\003+)p Fn(i")p Fm(\))1635 3700 y Fq(g)1682 3715 y Fm(\001)1745 3700 y Ft(\()p Fq(B)d Ft(\))p Fo(P)1977 3715 y Fh(c)2049 3700 y Fp(h)p Fq(x)p Fp(i)2182 3659 y Fk(\000)p Fn(\033)2284 3700 y Fp(k)2334 3717 y Fk(B)r Fm(\()p Fn(L)2457 3698 y Fj(2)2492 3717 y Fm(\))2583 3700 y Fp(\024)61 b Fq(C)2841 3632 y(e)2886 3596 y Fk(\000)p Fn("\034)p 2841 3677 176 4 v 2844 3768 a Fp(h)p Fq(\034)11 b Fp(i)2975 3739 y Fn(r)3026 3700 y Fq(;)672 b Ft(\(2.20\))-74 3933 y(for)27 b Fq(r)j(<)e(\033)t Ft(.)42 b(The)28 b(desired)g (estimate)f(on)g Fq(S)1479 3897 y Fn(")1473 3957 y Fm(1)1543 3933 y Ft(then)h(follo)m(ws)e(b)m(y)i(use)h(of)e(\(2.20\))f(in)h (\(2.19\))g(and)g(that)h Fq(\033)j(>)d(\033)3773 3948 y Fk(\003)3813 3933 y Ft(\()p Fq(n)p Ft(\).)72 4082 y(It)j(is)f(simple) f(to)h(see)i(that)e(\(2.20\))g(is)g(exp)s(ected.)44 b(F)-8 b(or,)30 b(supp)s(ose)i(that)e(instead)h(of)f Fq(B)5 b Fo(P)3249 4097 y Fh(c)3319 4082 y Ft(w)m(e)32 b(had)e Fq(B)3722 4097 y Fm(0)3762 4082 y Ft(.)43 b(Let)-74 4232 y Fq(!)t Ft(\()p Fq(k)s Ft(\))c(=)275 4150 y Fp(p)p 358 4150 339 4 v 82 x Fq(m)443 4203 y Fm(2)505 4232 y Ft(+)22 b Fq(k)657 4203 y Fm(2)696 4232 y Ft(,)41 b(the)f(disp)s(ersion)f (relation)e(of)i Fq(B)1961 4247 y Fm(0)2001 4232 y Ft(.)63 b(Then,)42 b(setting)e Fq(L)f Ft(=)2929 4135 y Fl(\020)2979 4232 y Fq(i@)3063 4247 y Fn(k)3100 4257 y Fg(j)3137 4232 y Fq(!)3202 4135 y Fl(\021)3251 4158 y Fk(\000)p Fm(1)3362 4232 y Fq(@)3413 4247 y Fn(k)3450 4257 y Fg(j)3487 4232 y Ft(,)i(and)e(using)-74 4381 y(that)32 b Fp(jr)248 4396 y Fn(k)291 4381 y Fq(!)t Fp(j)27 b(6)p Ft(=)g(0)32 b(on)h(the)g(supp)s (ort)g(of)f Fq(g)1418 4396 y Fm(\001)1481 4381 y Ft(,)g(w)m(e)i(get)658 4587 y Fq(e)703 4546 y Fn(iB)780 4555 y Fj(0)815 4546 y Fn(t)845 4587 y Fq(g)892 4602 y Fm(\001)954 4587 y Ft(\()p Fq(B)1066 4602 y Fm(0)1106 4587 y Ft(\))p Fq(f)126 b Ft(=)1510 4470 y Fl(Z)1642 4587 y Fq(e)1687 4546 y Fn(i!)r Fm(\()p Fn(k)r Fm(\))p Fn(t)1913 4587 y Fq(e)1958 4546 y Fn(ik)r Fk(\001)p Fn(x)2084 4587 y Fq(g)2131 4602 y Fm(\001)2194 4587 y Ft(\()p Fq(k)s Ft(\))2345 4561 y(^)2324 4587 y Fq(f)10 b Ft(\()p Fq(k)s Ft(\))33 b Fq(dk)1318 4805 y Ft(=)116 b Fq(t)1545 4764 y Fk(\000)p Fn(r)1687 4688 y Fl(Z)1819 4805 y Fq(L)1885 4764 y Fn(r)1940 4709 y Fl(\020)1989 4805 y Fq(e)2034 4764 y Fn(i!)r Fm(\()p Fn(k)r Fm(\))p Fn(t)2228 4709 y Fl(\021)2327 4805 y Fq(e)2372 4764 y Fn(ik)r Fk(\001)p Fn(x)2530 4805 y Fq(g)2577 4820 y Fm(\001)2640 4805 y Ft(\()p Fq(k)s Ft(\))2791 4779 y(^)2770 4805 y Fq(f)11 b Ft(\()p Fq(k)s Ft(\))32 b Fq(dk)1318 5023 y Ft(=)116 b Fq(t)1545 4982 y Fk(\000)p Fn(r)1687 4906 y Fl(Z)1819 5023 y Fq(e)1864 4982 y Fn(i!)r Fm(\()p Fn(k)r Fm(\))p Fn(t)2074 4926 y Fl(\020)2124 5023 y Fq(L)2190 4982 y Fk(y)2226 4926 y Fl(\021)2275 4949 y Fn(r)2362 4926 y Fl(\020)2412 5023 y Fq(e)2457 4982 y Fn(ik)r Fk(\001)p Fn(x)2583 5023 y Fq(g)2630 5038 y Fm(\001)2693 5023 y Ft(\()p Fq(k)s Ft(\))2844 4996 y(^)2823 5023 y Fq(f)10 b Ft(\()p Fq(k)s Ft(\))3011 4926 y Fl(\021)3110 5023 y Fq(dk)s(:)483 b Ft(\(2.21\))1901 5306 y(15)p eop %%Page: 16 16 16 15 bop -74 59 a Ft(Estimation)20 b(of)i Fp(h)p Fq(x)p Fp(i)650 23 y Fk(\000)p Fn(\033)751 59 y Fq(e)796 23 y Fn(iB)873 32 y Fj(0)908 23 y Fn(t)938 59 y Fq(g)985 74 y Fm(\001)1048 59 y Ft(\()p Fq(B)1160 74 y Fm(0)1199 59 y Ft(\))p Fq(f)33 b Ft(in)21 b Fq(L)1487 23 y Fm(2)1549 59 y Ft(b)m(y)i(F)-8 b(ourier)21 b(transform)g(metho)s(ds,)j(using)e (that)g Fp(jr)p Fq(!)t Fp(j)e Ft(is)i(b)s(ounded)-74 208 y(a)m(w)m(a)m(y)34 b(from)d(zero)i(on)g(the)g(supp)s(ort)g(of)f Fq(g)1429 223 y Fm(\001)1491 208 y Ft(,)h(w)m(e)h(ha)m(v)m(e)g(that)e (if)f Fq(\033)h(>)c Ft(max)o Fp(f)2652 169 y Fn(n)p 2652 185 43 4 v 2656 242 a Fm(2)2705 208 y Fq(;)17 b Ft(2)p Fp(g)32 b Ft(and)g Fq(r)f(<)c(\033)t Ft(,)33 b(then)1048 457 y Fp(kh)p Fq(x)p Fp(i)1231 416 y Fk(\000)p Fn(\033)1333 457 y Fq(e)1378 416 y Fn(iB)1455 425 y Fj(0)1490 416 y Fn(t)1520 457 y Fq(g)1567 472 y Fm(\001)1629 457 y Ft(\()p Fq(B)1741 472 y Fm(0)1781 457 y Ft(\))p Fq(f)11 b Fp(k)1928 472 y Fm(2)2027 457 y Fp(\024)60 b Fq(C)7 b Fp(h)p Fq(t)p Fp(i)2354 416 y Fk(\000)p Fn(r)2447 457 y Fp(kh)p Fq(x)p Fp(i)2630 416 y Fn(\033)2677 457 y Fq(f)k Fp(k)2786 472 y Fm(2)2825 457 y Fq(:)873 b Ft(\(2.22\))-74 706 y(Use)28 b(of)f(this)g(estimate,)h(in)f(the)h(ab)s(o)m(v)m(e)g (expression)h(for)e Fq(S)2031 670 y Fn(")2025 731 y Fm(1)2094 706 y Ft(\(with)g Fq(B)33 b Ft(replaced)28 b(b)m(y)g Fq(B)3040 721 y Fm(0)3079 706 y Ft(\))g(giv)m(es,)h(for)e Fq(l)j Ft(=)d(0)p Fq(;)17 b Ft(1)p Fq(;)g Ft(2:)870 955 y Fp(kh)p Fq(x)p Fp(i)1053 914 y Fk(\000)p Fn(\033)1155 955 y Fq(e)1200 914 y Fn(iB)1277 923 y Fj(0)1312 914 y Fn(t)1358 955 y Ft(\()p Fq(B)1470 970 y Fm(0)1532 955 y Fp(\000)22 b Ft(\003)g(+)g Fq(i)p Ft(0\))1939 907 y Fk(\000)p Fn(l)2037 955 y Fp(h)p Fq(x)p Fp(i)2170 914 y Fk(\000)p Fn(\033)2271 955 y Fp(k)2321 972 y Fk(B)r Fm(\()p Fn(L)2444 953 y Fj(2)2479 972 y Fm(\))2538 955 y Fp(\024)28 b Fq(C)7 b Fp(h)p Fq(t)p Fp(i)2833 914 y Fk(\000)p Fn(r)r Fm(+)p Fn(l)3003 955 y Fq(:)695 b Ft(\(2.23\))72 1204 y(Tw)m(o)31 b(approac)m(hes)h(to)d(pro)m(ving)h(an)g(estimate)f (of)h(the)g(t)m(yp)s(e)h(\(2.22\))f(and)g(\(2.23\))f(for)h Fq(e)3205 1168 y Fn(iB)s(t)3315 1204 y Fq(g)3362 1219 y Fm(\001)3425 1204 y Ft(\()p Fq(B)5 b Ft(\))p Fo(P)3657 1219 y Fh(c)3727 1204 y Ft(are)30 b(as)-74 1354 y(follo)m(ws:)-74 1503 y(\(a\))36 b(One)h(can)f("map")f(the)i(estimate)e(for)h Fq(B)1590 1518 y Fm(0)1666 1503 y Ft(to)g(that)g(for)f Fq(B)5 b Fo(P)2312 1518 y Fh(c)2389 1503 y Ft(using)36 b(the)g(w)m(a)m(v)m(e)i(op)s(erator)e Fq(W)3550 1518 y Fm(+)3609 1503 y Ft(.)55 b(In)36 b(this)-74 1653 y(approac)m(h)23 b(w)m(e)h(w)m(ould)f(need)h(to)f(deriv)m(e)g(estimates)g(for)f Fq(W)2001 1668 y Fm(+)2060 1653 y Fq(g)2107 1668 y Fm(\001)2170 1653 y Ft(\()p Fq(B)2282 1668 y Fm(0)2322 1653 y Ft(\))g(in)h(w)m(eigh) m(ted)g Fq(L)2949 1617 y Fm(2)3012 1653 y Ft(spaces,)j(or)d(alternativ) m(ely)-74 1802 y(\(b\))j(One)h(can)g(use)g(the)g(approac)m(h)f(based)i (on)e(the)g("Mourre)h(estimate",)g(dev)m(elop)s(ed)g(in)e(quan)m(tum)i (scattering)-74 1952 y(theory)-8 b(.)44 b(This)31 b(approac)m(h)h(is)f (more)f(in)h(the)g(spirit)f(of)h(energy)h(estimates)f(for)g(partial)e (di\013eren)m(tial)h(equations)-74 2101 y(and)j(do)s(es)g(not)f (require)h(the)g(use)h(of)e(w)m(a)m(v)m(e)i(op)s(erators.)72 2250 y(Here,)29 b(w)m(e)f(shall)d(follo)m(w)g(the)i(latter)e(approac)m (h.)42 b(Time)26 b(deca)m(y)i(estimates)e(lik)m(e)g(\(2.22\))g(are)g(a) h(consequence)-74 2400 y(of)d(an)g(approac)m(h)g(to)g Fr(minimal)i(velo)-5 b(city)27 b(estimates)c Ft(of)h(Sigal)e(and)i (So\013er)g([58],)i(whic)m(h)f(w)m(ere)g(pro)m(v)m(ed)h(using)d(the)-74 2549 y(ideas)30 b(of)g(Mourre)h([43)o(];)g(see)h(also)d(P)m(erry)-8 b(,)32 b(Sigal)c(&)i(Simon)f([48)o(].)43 b(F)-8 b(or)30 b(our)g(application)d(to)j(the)h(the)f(op)s(erator)-74 2699 y Fq(e)-29 2663 y Fk(\000)p Fn(iB)s(t)136 2699 y Fq(g)183 2714 y Fm(\001)246 2699 y Ft(\()p Fq(B)5 b Ft(\),)29 b(w)m(e)g(shall)e(refer)i(to)e(sp)s(ecial)h(cases)h(of)f(results)g (stated)h(in)f(Skibsted)h([59)o(])f(and)h(Debi)m(\023)-46 b(evre,)29 b(Hislop)-74 2848 y(&)k(Sigal)d([20].)72 2998 y(W)-8 b(e)33 b(\014rst)g(in)m(tro)s(duce)g(some)f(de\014nitions)g(and) h(assumptions:.)-74 3147 y Fo(\(A0\))106 3297 y Ft(\(a\))g(Let)h Fq(N)40 b Fp(\025)30 b Ft(2)k(and)g Fq(g)986 3312 y Fm(\001)1078 3297 y Fp(2)c Fq(C)1251 3260 y Fk(1)1244 3321 y Fm(0)1326 3297 y Ft(,)k(b)s(e)g(a)f(smo)s(oth)g(c)m(haracteristic)g(function)h (whic)m(h)g(is)f(equal)h(to)f(one)h(on)-74 3446 y(the)f(op)s(en)g(in)m (terv)-5 b(al)31 b(\001.)102 3595 y(\(b\))f(Let)g Fq(H)37 b Ft(and)30 b Fq(A)g Ft(denote)h(self)e(adjoin)m(t)g(op)s(erators)h(on) f(a)h(Hilb)s(ert)e(space)j Fp(H)q Ft(.)43 b(Assume)30 b Fq(H)38 b Ft(is)29 b(b)s(ounded)-74 3745 y(b)s(elo)m(w.)43 b(Let)33 b Fp(D)h Ft(denote)f(the)f(domain)f(of)g Fq(A)h Ft(and)h Fp(D)s Ft(\()p Fq(H)8 b Ft(\))31 b(the)h(domain)e(of)i Fq(H)8 b Ft(.)43 b(Assume)33 b Fp(D)23 b(\\)f(D)s Ft(\()p Fq(H)8 b Ft(\))31 b(is)g(dense)-74 3894 y(in)h Fp(D)s Ft(\()p Fq(H)8 b Ft(\),)32 b(and)g(let)g Fp(h)p Fq(A)p Fp(i)27 b(\021)i Ft(\()p Fq(I)h Ft(+)22 b Fp(j)p Fq(A)p Fp(j)1296 3858 y Fm(2)1334 3894 y Ft(\))1382 3830 y Fj(1)p 1382 3842 31 4 v 1382 3884 a(2)1427 3894 y Ft(.)-74 4094 y Fo(\(A1\))31 b Ft(Let)i(ad)464 4051 y Fm(0)464 4118 y Fn(A)521 4094 y Ft(\()p Fq(H)8 b Ft(\))27 b(=)h Fq(H)8 b Ft(.)43 b(F)-8 b(or)32 b(1)27 b Fp(\024)h Fq(k)j Fp(\024)d Fq(N)1597 4109 y Fk(\003)1637 4094 y Ft(,)k(de\014ne)i(iterativ)m(ely)d (the)i(comm)m(utator)f(form)1181 4343 y Fq(i)1214 4301 y Fn(k)1257 4343 y Ft(ad)1360 4300 y Fn(k)1360 4367 y(A)1417 4343 y Ft(\()p Fq(H)8 b Ft(\))59 b(=)i Fq(i)1828 4246 y Fl(h)1899 4343 y Fq(i)1932 4301 y Fn(k)r Fk(\000)p Fm(1)2065 4343 y Ft(ad)2168 4300 y Fn(k)r Fk(\000)p Fm(1)2168 4367 y Fn(A)2301 4343 y Ft(\()p Fq(H)8 b Ft(\))32 b Fq(;)49 b(A)2680 4246 y Fl(i)3725 4343 y Ft(\(2.24\))-74 4592 y(on)37 b Fp(D)27 b(\\)f(D)s Ft(\()p Fq(H)8 b Ft(\).)55 b(Assume)37 b(that)g(for)f(1)f Fp(\024)g Fq(k)j Fp(\024)d Fq(N)1798 4607 y Fk(\003)1838 4592 y Ft(,)j Fq(i)1936 4556 y Fn(k)1979 4592 y Ft(ad)2081 4549 y Fn(k)2081 4616 y(A)2139 4592 y Ft(\()p Fq(H)8 b Ft(\))36 b(extends)i(to)f(a)f (symmetric)g(op)s(erator)g(with)-74 4741 y(domain)31 b Fp(D)s Ft(\()p Fq(H)8 b Ft(\),)32 b(where)h Fq(N)936 4756 y Fk(\003)1004 4741 y Ft(=)1107 4645 y Fl(h)1146 4741 y Fq(N)g Ft(+)1365 4702 y Fm(3)p 1365 4718 36 4 v 1365 4775 a(2)1410 4645 y Fl(i)1472 4741 y Ft(+)22 b(1.)-74 4940 y Fo(\(A2\))31 b Ft(F)-8 b(or)32 b Fp(j)p Fq(s)p Fp(j)27 b Fq(<)h Ft(1,)k Fq(e)747 4904 y Fn(iAs)889 4940 y Ft(:)27 b Fp(D)s Ft(\()p Fq(H)8 b Ft(\))60 b Fp(!)27 b(D)s Ft(\()p Fq(H)8 b Ft(\))32 b(and)g(sup)1988 4964 y Fk(j)p Fn(s)p Fk(j)p Fn(<)p Fm(1)2171 4940 y Fp(k)g Fq(H)8 b(e)2387 4904 y Fn(iAs)2501 4940 y Fq( )36 b Fp(k)2650 4955 y Fk(H)2774 4940 y Fq(<)60 b Fp(1)p Fq(;)33 b Ft(for)f(an)m(y)h Fq( )f Fp(2)c(D)s Ft(\()p Fq(H)8 b Ft(\).)1901 5306 y(16)p eop %%Page: 17 17 17 16 bop -74 59 a Fo(\(A3\))31 b Fr(Mourr)-5 b(e)36 b(estimate)p Ft(:)1204 208 y Fq(g)1251 223 y Fm(\001)1313 208 y Ft(\()p Fq(H)8 b Ft(\))32 b Fq(i)p Ft([)p Fq(H)r(;)17 b(A)p Ft(])33 b Fq(g)1877 223 y Fm(\001)1940 208 y Ft(\()p Fq(H)8 b Ft(\))59 b Fp(\025)29 b Fq(\022)35 b(g)2397 223 y Fm(\001)2460 208 y Ft(\()p Fq(H)8 b Ft(\))2625 167 y Fm(2)3725 208 y Ft(\(2.25\))-74 387 y(for)32 b(some)h Fq(\022)d(>)e Ft(0.)72 603 y(Under)34 b(the)f(ab)s(o)m(v)m(e)g 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y Fp(\000)55 b Ft(2)p Fq(m)2928 5016 y Fm(2)2967 5057 y Fq(B)3046 5016 y Fk(\000)p Fm(1)3141 5057 y Fq(:)557 b Ft(\(2.36\))1901 5306 y(18)p eop %%Page: 19 19 19 18 bop -74 59 a Ft(Substitution)32 b(in)m(to)g(\(2.34\))f(w)m(e)j (get:)248 313 y([)p Fq(A;)17 b(B)5 b Ft(])33 b Fq(B)610 272 y Fk(\000)p Fm(1)821 313 y Ft(=)1028 246 y(2)p 1023 290 59 4 v 1023 382 a Fq(\031)1141 196 y Fl(Z)1224 223 y Fk(1)1187 385 y Fm(0)1348 313 y Fq(w)1431 245 y Fj(1)p 1430 257 31 4 v 1430 298 a(2)1507 313 y Fq(B)38 b Ft(\()p Fq(B)1736 272 y Fm(2)1797 313 y Ft(+)22 b Fq(w)s Ft(\))2006 272 y Fk(\000)p Fm(2)2133 313 y Fq(dw)820 550 y Fp(\000)1028 482 y Ft(1)p 1023 526 59 4 v 1023 618 a Fq(\031)1141 433 y Fl(Z)1224 459 y Fk(1)1187 621 y Fm(0)1331 550 y Ft(\()p Fq(B)1448 509 y Fm(2)1510 550 y Ft(+)g Fq(w)s Ft(\))1719 509 y Fk(\000)p Fm(1)1861 453 y Fl(\020)1911 550 y Fq(x)h Fp(\001)e(r)p Fq(V)77 b Ft(+)22 b(2)p Fq(V)43 b Fp(\000)23 b Ft(2)p Fq(m)2736 509 y Fm(2)2775 453 y Fl(\021)2841 550 y Fq(B)2920 509 y 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Fp(j)p Fq(J)1153 2345 y Fm(2)1192 2330 y Fp(j)60 b(\024)h Fq(C)1511 2212 y Fl(Z)1594 2239 y Fk(1)1557 2401 y Fm(0)1718 2330 y Fq(w)1801 2261 y Fj(1)p 1801 2273 31 4 v 1801 2314 a(2)1878 2330 y Ft(\(\012)1986 2288 y Fm(2)2048 2330 y Ft(+)22 b Fq(w)s Ft(\))2257 2288 y Fk(\000)p Fm(2)2383 2330 y Fq(dw)62 b(<)e Fp(1)p Fq(:)896 b Ft(\(2.40\))-74 2569 y(The)34 b(higher)e(order)i(comm)m(utators)d(are)i(handled)g(in)f (a)h(similar)d(manner;)j(they)h(are)f(ev)m(en)h(simpler)d(b)s(ecause) -74 2718 y(eac)m(h)j(successiv)m(e)h(comm)m(utator)c(results)i(in)f(an) g(extra)h(factor)g(of)f(\()p Fq(B)2466 2682 y Fm(2)2527 2718 y Ft(+)22 b Fq(w)s Ft(\))2736 2682 y Fk(\000)p Fm(1)2830 2718 y Ft(.)72 2868 y(This)34 b(lea)m(v)m(es)h(us)g(with)e(v)m (eri\014cation)h(of)f(the)h(Mourre)h(estimate)e(\(A3\))g(for)g Fq(H)38 b Ft(=)29 b Fq(B)5 b Ft(.)48 b(Prop)s(osition)32 b(2.2)h(of)-74 3017 y([20])f(sa)m(ys)i(that)f(under)g(h)m(yp)s(otheses) i(on)e Fq(B)1517 2981 y Fm(2)1556 3017 y Ft(,)g(v)m(eri\014ed)g(in)f 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Fk(0\000)p Fm(1)699 4055 y Ft(=)k(1)p Fq(=)p Ft(2)32 b(and)h Fq(p)1220 4019 y Fk(0\000)p Fm(1)1355 4055 y Ft(+)22 b Fq(p)1502 4019 y Fk(\000)p Fm(1)1624 4055 y Ft(=)28 b(1.)43 b(Since)33 b Fq(\033)e(>)2302 4016 y Fn(n)p 2302 4032 43 4 v 2306 4089 a Fm(2)2355 4055 y Ft(,)h Fp(h)p Fq(x)p Fp(i)2547 4019 y Fk(\000)p Fn(\033)2677 4055 y Fp(2)c Fq(L)2837 4019 y Fn(q)2908 4055 y Ft(and)k(therefore:)163 4294 y Fp(kh)p Fq(x)p Fp(i)346 4253 y Fk(\000)p Fn(\033)480 4294 y Fq(S)546 4253 y Fn(")540 4319 y Fm(2)583 4294 y Ft(\()p Fq(t)p Ft(\))h Fp(h)p Fq(x)p Fp(i)860 4253 y Fk(\000)p Fn(\033)994 4294 y Fq(f)11 b Fp(k)1103 4309 y Fm(2)1202 4294 y Fp(\024)61 b Fq(C)7 b Fp(kh)p Fq(x)p Fp(i)1600 4253 y Fk(\000)p Fn(\033)1701 4294 y Fp(k)1751 4321 y Fk(B)r Fm(\()p Fn(L)1874 4302 y Fg(p)1906 4288 y Fb(0)1933 4321 y Fn(;L)2001 4302 y Fj(2)2035 4321 y Fm(\))2099 4294 y Fp(k)p Fq(e)2194 4253 y Fn(iB)2271 4262 y Fj(0)2306 4253 y Fn(t)2336 4294 y Ft(\()p Fq(B)2448 4309 y Fm(0)2509 4294 y Fp(\000)23 b 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Fp(k)1673 326 y Fk(B)r Fm(\()p Fn(W)1825 307 y Fj(1)p Fg(;p)1910 326 y Fn(;L)1978 307 y Fg(p)2010 293 y Fb(0)2036 326 y Fm(\))2128 299 y Fp(\024)61 b Fq(C)39 b Fp(j)p Fq(t)p Fp(j)2466 255 y Fk(\000)2554 228 y Fj(2)p Fg(n)p 2530 240 116 4 v 2530 282 a(n)p Fj(+2)2660 299 y Fq(;)1038 b Ft(\(2.42\))-74 540 y(where)34 b Fq(p)28 b Ft(=)f(2)22 b Fp(\000)h Ft(8)p Fq(=)p Ft(\()p Fq(n)f Ft(+)g(6\))32 b(and)g Fq(p)1230 503 y Fk(0)1281 540 y Ft(=)c(2\()p Fq(n)22 b Ft(+)g(2\))p Fq(=)p Ft(\()p Fq(n)g Fp(\000)g Ft(2\).)72 689 y(\(ii\))27 b Fp(k)p 281 636 51 4 v Fq(g)332 713 y Fm(\001)395 689 y Ft(\()p Fq(B)507 704 y Fm(0)546 689 y Ft(\)\()p Fq(B)696 704 y Fm(0)750 689 y Fp(\000)15 b Ft(\003)g(+)g Fq(i)p Ft(0\))1136 653 y Fk(\000)p Fm(1)1230 689 y Fq(B)1304 704 y Fm(0)1344 689 y Fp(k)1394 705 y Fk(B)r Fm(\()p Fn(L)1517 686 y Fg(p)1553 705 y Fm(\))1614 689 y Ft(is)28 b(b)s(ounded)i(b)s(ecause)p 2460 636 V 30 w Fq(g)2511 713 y Fm(\001)2573 689 y Ft(\()p Fq(\026)p Ft(\)\()p Fq(\026)15 b Fp(\000)g Ft(\003)g(+)g Fq(i)p Ft(0\))3206 653 y Fk(\000)p Fm(1)3299 689 y Fq(\026)29 b Ft(is)g(a)f(m)m(ultiplier)-74 839 y(on)33 b Fq(L)128 802 y Fn(p)200 839 y Ft([64].)-74 988 y(Using)f(the)h(b)s(oundedness)i (of)d Fq(W)1144 1003 y Fm(+)1236 988 y Ft(on)g Fq(W)1477 952 y Fm(2)p Fn(;p)1604 988 y Ft(and)h(that)f Fq(\033)g(>)27 b Ft(max)p Fp(f)2437 949 y Fn(n)p 2437 965 43 4 v 2441 1022 a Fm(2)2489 988 y Fq(;)17 b Ft(2)p Fp(g)p Ft(,)32 b(w)m(e)i(ha)m(v)m(e)895 1228 y Fp(k)p Fq(B)1019 1243 y Fm(0)1058 1228 y Fq(W)1164 1187 y Fk(\003)1150 1253 y Fm(+)1209 1228 y Fp(h)p Fq(x)p Fp(i)1342 1187 y Fk(\000)p Fn(\033)1476 1228 y Fq(f)11 b Fp(k)1585 1243 y Fn(p)1740 1228 y Fp(\024)116 b(k)p Fq(W)2089 1187 y Fk(\003)2075 1253 y Fm(+)2134 1228 y Fp(h)p Fq(x)p Fp(i)2267 1187 y Fk(\000)p Fn(\033)2401 1228 y Fq(f)11 b Fp(k)2510 1243 y Fm(1)p Fn(;p)1740 1403 y Fp(\024)116 b(k)p Fq(W)2089 1362 y Fk(\003)2075 1427 y Fm(+)2134 1403 y Fp(k)2184 1419 y Fk(B)r Fm(\()p Fn(W)2336 1401 y Fj(2)p Fg(;p)2421 1419 y Fm(\))2485 1403 y Fp(kh)p Fq(x)p Fp(i)2668 1362 y Fk(\000)p Fn(\033)2802 1403 y Fq(f)11 b Fp(k)2911 1418 y Fm(1)p Fn(;p)1740 1577 y Fp(\024)116 b Fq(C)7 b Fp(k)p Fq(f)k Fp(k)2169 1592 y Fm(1)p Fn(;)p Fm(2)2263 1577 y Fq(:)-74 1818 y Ft(Therefore,)1110 1967 y Fp(kh)p Fq(x)p Fp(i)1293 1926 y Fk(\000)p Fn(\033)1394 1967 y Fq(S)1460 1926 y Fn(")1454 1992 y Fm(2)1497 1967 y Fp(h)p Fq(x)p Fp(i)1630 1926 y Fk(\000)p Fn(\033)1732 1967 y Fq( )t Fp(k)1849 1982 y Fm(2)1948 1967 y Fp(\024)61 b Fq(C)7 b Fp(h)p Fq(t)p Fp(i)2276 1923 y Fk(\000)2363 1896 y Fj(2)p Fg(n)p 2340 1908 116 4 v 2340 1949 a(n)p Fj(+2)2502 1967 y Fp(k)p Fq( )t Fp(k)2669 1982 y Fm(1)p Fn(;)p Fm(2)2763 1967 y Fq(:)935 b Ft(\(2.43\))72 2167 y(W)-8 b(e)37 b(no)m(w)f(com)m (bine)g(the)g(estimates)g(of)f Fq(S)1620 2131 y Fn(")1614 2191 y(j)1657 2167 y Ft(\()p Fq(t)p Ft(\))p Fq(;)53 b(j)39 b Ft(=)33 b(1)p Fq(;)17 b Ft(2)35 b(to)h(complete)f(the)i(pro)s(of.)52 b(Com)m(bining)35 b(\(2.32\))-74 2316 y(and)e(\(2.43\))f(and)g(taking)g 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b(exactly)g(one)g(\(simple\))e(eigen)m(v)-5 b(alue)32 b(\012)1915 4861 y Fm(2)1988 4897 y Ft(satisfying)f(0)60 b Fq(<)g Ft(\012)2737 4861 y Fm(2)2837 4897 y Fq(<)g(m)3058 4861 y Fm(2)3098 4897 y Ft(,)33 b(with)f(corresp)s(onding)-74 5046 y(eigenfunction)g Fq(')c Fp(2)g Fq(L)777 5010 y Fm(2)816 5046 y Fq(;)50 b Fp(jj)p Fq(')p Fp(jj)1069 5061 y Fm(2)1134 5046 y Ft(=)28 b(1)p Fq(:)1901 5306 y Ft(21)p eop %%Page: 22 22 22 21 bop -74 59 a Fo(\(N\))31 b Fq(f)11 b Ft(\()p Fq(u)p Ft(\))60 b(=)g Fq(u)576 23 y Fm(3)669 59 y Ft(+)55 b Fq(f)848 74 y Fm(4)887 59 y Ft(\()p Fq(u)p Ft(\))p Fq(;)49 b(f)1143 74 y Fm(4)1183 59 y Ft(\()p Fq(u)p Ft(\))59 b(=)h Fp(O)s Ft(\()p Fq(u)1686 23 y Fm(4)1725 59 y Ft(\))33 b(and)f Fq(f)11 b Ft(\()p Fq(u)p Ft(\))32 b(is)g(smo)s(oth)g(in)g(a)g (neigh)m(b)s(orho)s(o)s(d)f(of)h Fq(u)c Ft(=)f(0.)72 208 y(These)48 b(h)m(yp)s(otheses)f(are)f(b)m(y)g(no)f(means)h(the)f (least)g(stringen)m(t,)k(but)d(are)f(su\016cien)m(t)i(for)d(the)i (presen)m(t)-74 358 y(purp)s(oses.)-74 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4616 y Fk(\000)p Fm(1)1015 4657 y Fq(e)1060 4616 y Fk(\000)p Fn(i)p Fm(\012)p Fn(t)1220 4657 y Fq(F)1283 4672 y Fm(2)1323 4657 y Ft(\()p Fq(a;)17 b(\021)1508 4616 y Fn(nr)1504 4681 y Fm(2)1588 4657 y Ft(\))115 b(=)1976 4549 y Fm(7)1934 4574 y Fl(X)1933 4756 y Fn(j)t Fm(=1)2055 4657 y Ft(\(2)p Fq(i)p Ft(\012\))2283 4616 y Fk(\000)p Fm(1)2378 4657 y Fq(e)2423 4616 y Fk(\000)p Fn(i)p Fm(\012)p Fn(t)2615 4657 y Fq(F)2678 4672 y Fm(2)2718 4657 y Ft(\()p Fq(a;)17 b(\021)2903 4616 y Fn(nr)2899 4681 y Fm(2)p Fn(j)2983 4657 y Ft(\))1741 4970 y(=)116 b(3)p Fq(\025)p Ft(\(2)p Fq(i)p Ft(\012\))2267 4928 y Fk(\000)p Fm(1)2361 4970 y Fq(a)2412 4928 y Fm(2)2452 4970 y Fq(e)2497 4928 y Fk(\000)p Fn(i)p Fm(\012)p Fn(t)2717 4862 y Fm(7)2674 4887 y Fl(X)2673 5069 y Fn(j)t Fm(=1)2812 4852 y Fl(Z)2912 4970 y Fq(')2976 4928 y Fm(3)3015 4970 y Fq(\021)3067 4928 y Fn(nr)3063 4994 y Fm(2)p Fn(j)3180 4970 y Fq(:)518 b Ft(\(4.38\))1901 5306 y(30)p eop %%Page: 31 31 31 30 bop -74 59 a Ft(Eac)m(h)43 b(of)f(the)g(sev)m(en)i(terms)e(con)m (tributing)f(to)h Fq(\021)1773 23 y Fn(nr)1769 83 y Fm(2)1896 59 y Ft(is)f(expressed)k(as)d(a)g(part)g(whic)m(h)g(is)g Fp(O)s Ft(\()p Fp(j)p Fq(A)p Fp(j)3536 23 y Fm(5)3575 59 y Ft(\))g(plus)g(an)-74 208 y(error)33 b(term)f(whic)m(h)h(is)f (estimated)g(in)f(magnitude)h(in)f(section)i(5.)43 b(The)34 b Fp(O)s Ft(\()p Fp(j)p Fq(A)p Fp(j)2824 172 y Fm(5)2863 208 y Ft(\))e(part)g(and)h(error)f(terms)h(are)-74 358 y(displa)m(y)m(ed)g(in)f(the)h(follo)m(wing)d(t)m(w)m(o)j(prop)s (ositions.)-74 634 y Fo(Prop)s(osition)j(4.6.)49 b Fi(Let)1205 783 y Fq(\032)p Ft(\()p Fq(\020)8 b Ft(\))27 b Fp(\021)h Ft(\()p Fo(P)1629 798 y Fh(c)1668 783 y Fq(')1732 742 y Fm(3)1772 783 y Fq(;)17 b(B)1895 742 y Fk(\000)p Fm(1)1989 783 y Ft(\()p Fq(B)27 b Fp(\000)c Fq(\020)8 b Ft(\))2317 742 y Fk(\000)p Fm(1)2410 783 y Fo(P)2487 798 y Fh(c)2527 783 y Fq(')2591 742 y Fm(3)2630 783 y Ft(\))p Fq(:)1030 b Ft(\(4.39\))-74 985 y Fi(Then,)853 1229 y Ft(\(2)p Fq(i)p Ft(\012\))1081 1188 y Fk(\000)p Fm(1)1175 1229 y Fq(e)1220 1188 y Fk(\000)p Fn(i)p Fm(\012)p Fn(t)1380 1229 y Fq(F)1443 1244 y Fm(2)1483 1229 y Ft(\()p Fq(a;)17 b(\021)1668 1188 y Fn(nr)1664 1254 y Fm(21)1748 1229 y Ft(\))694 1434 y(=)895 1366 y Fq(\025)952 1330 y Fm(2)p 895 1410 97 4 v 908 1502 a Ft(\012)1001 1434 y Fq(\032)p Ft(\(\012\))1197 1337 y Fl(h)1264 1366 y Ft(3)p 1247 1410 83 4 v 1247 1502 a(4)p Fq(i)1339 1434 y Fp(j)p Fq(A)p Fp(j)1468 1393 y Fm(2)1507 1434 y Fq(A)1580 1393 y Fm(3)1620 1434 y Fq(e)1665 1393 y Fm(2)p Fn(i)p Fm(\012)p Fn(t)1827 1434 y Ft(+)1952 1366 y(3)p 1935 1410 V 1935 1502 a(2)p Fq(i)2027 1434 y Fp(j)p Fq(A)p Fp(j)2156 1393 y Fm(4)2195 1434 y Fq(A)22 b Ft(+)2415 1366 y(3)p 2398 1410 V 2398 1502 a(4)p Fq(i)2490 1434 y Fp(j)p Fq(A)p Fp(j)2619 1393 y Fm(4)p 2658 1356 74 4 v 2658 1434 a Fq(Ae)2776 1393 y Fk(\000)p Fm(2)p Fn(i)p Fm(\012)p Fn(t)2971 1337 y Fl(i)3032 1434 y Ft(+)g Fq(E)3208 1393 y Fn(nr)3202 1458 y Fm(21)3725 1434 y Ft(\(4.40\))786 1947 y(\(2)p Fq(i)p Ft(\012\))1014 1906 y Fk(\000)p Fm(1)1109 1947 y Fq(e)1154 1906 y Fk(\000)p Fn(i)p Fm(\012)p Fn(t)1314 1947 y Fq(F)1377 1962 y Fm(2)1416 1947 y Ft(\()p Fq(a;)17 b(\021)1601 1906 y Fn(nr)1597 1971 y Fm(22)1682 1947 y Ft(\))628 2151 y(=)829 2084 y Fq(\025)886 2048 y Fm(2)p 829 2128 97 4 v 842 2220 a Ft(\012)935 2151 y Fq(\032)p Ft(\()p Fp(\000)p Ft(\012\))1208 2055 y Fl(h)1275 2084 y Ft(9)p 1258 2128 83 4 v 1258 2220 a(4)p Fq(i)1350 2151 y Fp(j)p Fq(A)p Fp(j)1479 2110 y Fm(4)1518 2151 y Fq(A)23 b Ft(+)1738 2084 y(9)p 1722 2128 V 1722 2220 a(2)p Fq(i)1814 2151 y Fp(j)p Fq(A)p Fp(j)1943 2110 y Fm(4)p 1982 2073 74 4 v 1982 2151 a Fq(Ae)2100 2110 y Fk(\000)p Fm(2)p Fn(i)p Fm(\012)p Fn(t)2317 2151 y Ft(+)2441 2084 y(9)p 2425 2128 83 4 v 2425 2220 a(4)p Fq(i)2517 2151 y Fp(j)p Fq(A)p Fp(j)2646 2110 y Fm(2)p 2685 2073 74 4 v 2685 2151 a Fq(A)2758 2093 y Fm(3)2797 2151 y Fq(e)2842 2110 y Fk(\000)p Fm(4)p Fn(i)p Fm(\012)p Fn(t)3037 2055 y Fl(i)3099 2151 y Ft(+)f Fq(E)3275 2110 y Fn(nr)3269 2176 y Fm(22)3725 2151 y Ft(\(4.41\))740 2664 y(\(2)p Fq(i)p Ft(\012\))968 2623 y Fk(\000)p Fm(1)1062 2664 y Fq(e)1107 2623 y Fk(\000)p Fn(i)p Fm(\012)p Fn(t)1267 2664 y Fq(F)1330 2679 y Fm(2)1370 2664 y Ft(\()p Fq(a;)17 b(\021)1555 2623 y Fn(nr)1551 2689 y Fm(23)1635 2664 y Ft(\))581 2869 y(=)782 2801 y Fq(\025)839 2765 y Fm(2)p 782 2846 97 4 v 795 2937 a Ft(\012)888 2869 y Fq(\032)p Ft(\()p Fp(\000)p Ft(3\012\))1210 2773 y Fl(h)1277 2801 y Ft(3)p 1260 2846 83 4 v 1260 2937 a(4)p Fq(i)1352 2869 y Fp(j)p Fq(A)p Fp(j)1481 2828 y Fm(4)p 1520 2791 74 4 v 1520 2869 a Fq(Ae)1638 2828 y Fk(\000)p Fm(2)p Fn(i)p Fm(\012)p Fn(t)1856 2869 y Ft(+)1964 2801 y(3)p 1964 2846 49 4 v 1964 2937 a(2)2022 2869 y Fp(j)p Fq(A)p Fp(j)2151 2828 y Fm(2)p 2190 2791 74 4 v 2190 2869 a Fq(A)2263 2810 y Fm(3)2303 2869 y Fq(e)2348 2828 y Fk(\000)p Fm(4)p Fn(i)p Fm(\012)p Fn(t)2565 2869 y Ft(+)2673 2801 y(3)p 2673 2846 49 4 v 2673 2937 a(4)p 2732 2791 74 4 v 2732 2869 a Fq(A)2805 2810 y Fm(5)2844 2869 y Fq(e)2889 2828 y Fk(\000)p 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4882 a Fq(A)1024 4824 y Fm(5)1096 4882 y Fq(e)1141 4841 y Fk(\000)p Fm(6)p Fn(i)p Fm(\012)p Fn(t)1369 4786 y Fl(h)1441 4882 y Fq(i\032)p Ft(\()p Fp(\000)p Ft(3\012\))g Fp(\000)1994 4815 y Ft(4)p 1994 4859 49 4 v 1994 4951 a(3)2052 4882 y Fq(i\032)p Ft(\(3\012)23 b(+)f Fq(i)p Ft(0\))2566 4786 y Fl(i)2660 4882 y Ft(+)54 b Fo(E)3725 5057 y Ft(\(4.48\))1901 5306 y(32)p eop %%Page: 33 33 33 32 bop -74 59 a Fi(where)1052 301 y Fo(E)83 b Ft(=)115 b(\(2)p Fq(i)p Ft(\012\))1628 260 y Fk(\000)p Fm(1)1723 301 y Fq(e)1768 260 y Fn(i)p Fm(\012)p Fn(t)1905 301 y Ft(3)p Fq(\025a)2079 184 y Fl(Z)2178 301 y Fq('\021)2294 260 y Fm(2)1209 518 y Ft(+)g(\(2)p Fq(i)p Ft(\012\))1628 477 y Fk(\000)p Fm(1)1723 518 y Fq(\025e)1825 477 y Fk(\000)p Fn(i)p Fm(\012)p Fn(t)2001 401 y Fl(Z)2101 518 y Fq('\021)2217 477 y Fm(3)1209 778 y Ft(+)1444 670 y Fm(7)1401 695 y Fl(X)1400 877 y Fn(j)t Fm(=1)1539 778 y Fq(E)1617 737 y Fn(nr)1611 803 y Fm(2)p Fn(j)1209 1031 y Ft(+)g(\(2)p Fq(i)p Ft(\012\))1628 990 y Fk(\000)p 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Fk(\000)p Fm(4)p Fn(i)p Fm(\012)p Fn(t)719 570 y Ft(+)975 502 y(1)p 920 546 159 4 v 920 638 a(4\012)1039 609 y Fm(2)1138 449 y Fl(\022)1241 502 y Ft(1)p 1241 546 49 4 v 1241 638 a(3)1300 570 y Fq(\013)1362 585 y Fm(03)1437 570 y Fq(\013)1499 585 y Fm(12)1628 570 y Ft(+)1769 502 y(1)p 1769 546 V 1769 638 a(2)1827 570 y Fq(\013)1889 585 y Fm(03)p 1964 517 63 4 v 1964 570 a Fq(\013)2027 585 y Fm(30)2134 449 y Fl(\023)p 2244 492 74 4 v 2244 570 a Fq(A)2317 511 y Fm(5)2389 570 y Fq(e)2434 529 y Fk(\000)p Fm(6)p Fn(i)p Fm(\012)p Fn(t)2629 570 y Fq(:)1069 b Ft(\(5.19\))-74 814 y(Referring)32 b(bac)m(k)i(to)e(\(5.8\),)g(w)m(e)h(see)h(w)m(e)g(m) m(ust)f(no)m(w)g(obtain)e(an)-74 1056 y Fr(Exp)-5 b(ansion)34 b(of)508 985 y Fl(R)563 1011 y Fn(t)547 1081 y Fm(0)642 1056 y Fq(O)717 1071 y Fm(5)756 1056 y Ft(\()p Fq(A)p Ft(\))f Fq(ds)p Ft(:)72 1251 y(F)-8 b(rom)31 b(\(5.10\))h(w)m(e)i(ha)m (v)m(e,)g(after)e(in)m(tegration)f(b)m(y)j(parts:)858 1386 y Fl(Z)941 1412 y Fn(t)904 1574 y Fm(0)1020 1503 y Fq(O)1095 1518 y Fm(5)1134 1503 y Ft(\()p Fq(A)p Ft(\))e Fq(ds)115 b Ft(=)h Fq(h)1775 1518 y Fm(3)p Fn(b)1845 1503 y Ft(\()p Fq(A)p Ft(\()p Fq(t)p Ft(\))p Fq(;)17 b(t)p Ft(\))54 b Fp(\000)h Fq(h)2426 1518 y Fm(3)p Fn(b)2496 1503 y Ft(\()p Fq(A)2607 1518 y Fm(0)2647 1503 y Fq(;)17 b Ft(0\))1527 1739 y(+)1719 1622 y Fl(Z)1802 1648 y Fn(t)1765 1811 y Fm(0)1881 1739 y Fp(O)1979 1643 y Fl(\020)2061 1739 y Fp(j)p Fq(A)p Fp(j)2190 1698 y Fm(4)2229 1739 y Ft(\()p Fp(j)p Fq(A)p Fp(j)2396 1698 y Fm(3)2490 1739 y Ft(+)54 b Fp(j)p Fo(E)p Fp(j)p Ft(\))2820 1643 y Fl(\021)2918 1739 y Fq(ds;)683 b Ft(\(5.20\))-74 1975 y(where)494 2211 y Fq(h)550 2226 y Fm(3)p Fn(b)619 2211 y Ft(\()p Fq(A;)17 b(t)p Ft(\))117 b(=)1174 2144 y Fq(\013)1236 2159 y Fm(50)p 1166 2188 153 4 v 1166 2279 a Ft(4)p Fq(i)p Ft(\012)1361 2211 y Fq(A)1434 2170 y Fm(5)1506 2211 y Fq(e)1551 2170 y Fm(4)p Fn(i)p Fm(\012)p Fn(t)1746 2211 y Ft(+)1894 2144 y Fq(\013)1956 2159 y Fm(41)p 1886 2188 V 1886 2279 a Ft(2)p Fq(i)p Ft(\012)2081 2211 y Fq(A)2154 2170 y Fm(4)p 2194 2133 74 4 v 2194 2211 a Fq(A)32 b(e)2344 2170 y Fm(2)p Fn(i)p Fm(\012)p Fn(t)2539 2211 y Fp(\000)2689 2144 y Fq(\013)2751 2159 y Fm(23)p 2681 2188 153 4 v 2681 2279 a Ft(2)p Fq(i)p Ft(\012)2876 2211 y Fq(A)2949 2170 y Fm(2)p 3021 2133 74 4 v 3021 2211 a Fq(A)3094 2153 y Fm(3)3166 2211 y Fq(e)3211 2170 y Fk(\000)p Fm(2)p Fn(i)p Fm(\012)p Fn(t)963 2423 y Fp(\000)1174 2356 y Fq(\013)1236 2371 y Fm(14)p 1166 2400 153 4 v 1166 2491 a Ft(4)p Fq(i)p Ft(\012)1361 2423 y Fq(A)p 1434 2345 74 4 v(A)1507 2365 y Fm(4)1579 2423 y Fq(e)1624 2382 y Fk(\000)p Fm(4)p Fn(i)p Fm(\012)p Fn(t)1874 2423 y Fp(\000)2024 2356 y Fq(\013)2086 2371 y Fm(05)p 2016 2400 153 4 v 2016 2491 a Ft(6)p Fq(i)p Ft(\012)p 2211 2345 74 4 v 2211 2423 a Fq(A)2284 2365 y Fm(5)2356 2423 y Fq(e)2401 2382 y Fk(\000)p Fm(6)p Fn(i)p Fm(\012)p Fn(t)2596 2423 y Fq(:)1102 b Ft(\(5.21\))-74 2659 y(Therefore)34 b(from)d(\(5.11\))h(and)h(our)f(computations)g(w)m(e)h(ha)m(v)m(e:)680 2895 y Fq(A)55 b Fp(\000)g Fq(h)996 2910 y Fm(3)1036 2895 y Ft(\()p Fq(A;)17 b(t)p Ft(\))54 b Fp(\000)h Fq(h)1506 2910 y Fm(3)p Fn(a)1583 2895 y Ft(\()p Fq(A;)17 b(t)p Ft(\))55 b Fp(\000)g Fq(h)2054 2910 y Fm(3)p Fn(b)2124 2895 y Ft(\()p Fq(A;)17 b(t)p Ft(\))680 3069 y(=)60 b Fq(A)889 3084 y Fm(0)983 3069 y Fp(\000)55 b Fq(h)1171 3084 y Fm(3)1211 3069 y Ft(\()p Fq(A)1322 3084 y Fm(0)1362 3069 y Fq(;)17 b Ft(0\))54 b Fp(\000)h Fq(h)1735 3084 y Fm(3)p Fn(a)1812 3069 y Ft(\()p Fq(A)1923 3084 y Fm(0)1962 3069 y Fq(;)17 b Ft(0\))54 b Fp(\000)h Fq(h)2335 3084 y Fm(3)p Fn(b)2405 3069 y Ft(\()p Fq(A)2516 3084 y Fm(0)2556 3069 y Fq(;)17 b Ft(0\))489 3259 y(+)647 3142 y Fl(Z)730 3168 y Fn(t)694 3331 y Fm(0)809 3259 y Fq(\013)871 3274 y Fm(21)979 3259 y Fp(j)p Fq(A)p Fp(j)1108 3218 y Fm(2)1146 3259 y Fq(A)55 b Ft(+)g Fq(\013)1467 3274 y Fm(32)1574 3259 y Fp(j)p Fq(A)p Fp(j)1703 3218 y Fm(4)1742 3259 y Fq(A)33 b(ds)489 3496 y Ft(+)742 3428 y(1)p 690 3472 153 4 v 690 3564 a(2)p Fq(i)p Ft(\012)902 3374 y 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Fp(j)p Fo(E)p Fp(j)p Ft(\))2890 3639 y Fl(\021)2988 3735 y Fq(ds:)613 b Ft(\(5.22\))72 3971 y(This)33 b(suggests)h(the)f(c)m(hange)h(of)e(v)-5 b(ariables:)1139 4207 y Fq(A)1212 4222 y Fm(1)1368 4207 y Fp(\021)116 b Fq(A)54 b Fp(\000)h Fq(H)1901 4222 y Fm(1)1941 4207 y Ft(\()p Fq(A;)17 b(t)p Ft(\))p Fq(;)114 b Ft(where)904 4382 y Fq(H)985 4397 y Fm(1)1024 4382 y Ft(\()p Fq(A;)17 b(t)p Ft(\))116 b(=)h Fq(h)1617 4397 y Fm(3)1656 4382 y Ft(\()p Fq(A;)17 b(t)p Ft(\))55 b(+)f Fq(h)2125 4397 y Fm(3)p Fn(a)2202 4382 y Ft(\()p Fq(A;)17 b(t)p Ft(\))55 b(+)f Fq(h)2671 4397 y Fm(3)p Fn(b)2741 4382 y Ft(\()p Fq(A;)17 b(t)p Ft(\))p Fq(;)729 b Ft(\(5.23\))-74 4618 y(whic)m(h)41 b(for)e(small)f Fp(j)p Fq(A)p Fp(j)p Ft(,)k(is)d(a)h(near-iden)m(tit)m(y)g(c)m(hange)h(of)f(v)-5 b(ariables.)65 b(Using)39 b(this)h(c)m(hange)h(of)f(v)-5 b(ariables)39 b(w)m(e)-74 4767 y(ha)m(v)m(e)34 b(that)e(\(5.22\))g(b)s (ecomes)417 5019 y Fq(A)490 5034 y Fm(1)645 5019 y Ft(=)115 b Fq(A)909 5034 y Fm(10)1039 5019 y Ft(+)1169 4901 y Fl(Z)1252 4928 y Fn(t)1215 5090 y Fm(0)1331 5019 y Fq(\013)1393 5034 y Fm(21)1500 5019 y Fp(j)p Fq(A)1601 5034 y Fm(1)1641 5019 y Fp(j)1669 4977 y Fm(2)1708 5019 y Fq(A)1781 5034 y Fm(1)1875 5019 y Ft(+)54 b Fq(\013)2067 5034 y Fm(32)2175 5019 y Fp(j)p Fq(A)2276 5034 y Fm(1)2315 5019 y Fp(j)2343 4977 y Fm(4)2382 5019 y Fq(A)2455 5034 y Fm(1)2527 5019 y Fq(ds)1901 5306 y Ft(37)p eop %%Page: 38 38 38 37 bop 645 91 a Ft(+)898 23 y(1)p 846 68 153 4 v 846 159 a(2)p Fq(i)p Ft(\012)1058 -30 y Fl(\022)1161 23 y Ft(3)p 1161 68 49 4 v 1161 159 a(2)1220 91 y Fp(j)p Fq(\013)1310 106 y Fm(03)1385 91 y Fp(j)1413 50 y Fm(2)1506 91 y Fp(\000)55 b Ft(2)p Fq(\013)1749 106 y Fm(30)1824 91 y Fq(\013)1886 106 y Fm(12)2015 91 y Ft(+)g(2)p Fp(j)p Fq(\013)2285 106 y Fm(12)2359 91 y Fp(j)2387 50 y Fm(2)2459 -30 y Fl(\023)2569 -26 y(Z)2652 0 y Fn(t)2615 162 y Fm(0)2731 91 y Fp(j)p Fq(A)2832 106 y Fm(1)2871 91 y Fp(j)2899 50 y Fm(4)2938 91 y Fq(A)3011 106 y Fm(1)3083 91 y Fq(ds)645 330 y Ft(+)836 213 y Fl(Z)919 240 y Fn(t)882 402 y Fm(0)998 234 y Fl(\020)1080 330 y Ft(2)p Fq(\013)1191 345 y Fm(21)1298 330 y Fp(j)p Fq(A)1399 345 y Fm(1)1438 330 y Fp(j)1466 289 y Fm(2)1538 330 y Fq(H)1619 345 y Fm(1)1658 330 y Ft(\()p Fq(A)1769 345 y Fm(1)1809 330 y Fq(;)17 b(s)p Ft(\))54 b(+)22 b Fq(A)2162 289 y Fm(2)2162 355 y(1)p 2234 252 89 4 v 2234 330 a Fq(H)2323 345 y Fm(1)2362 330 y Ft(\()p Fq(A)2473 345 y Fm(1)2513 330 y Fq(;)17 b(s)p Ft(\))2673 234 y Fl(\021)2771 330 y Fq(ds)55 b Ft(+)3053 213 y Fl(Z)3136 240 y Fn(t)3099 402 y Fm(0)3215 330 y Fo(E)3289 345 y Fh(1)3366 330 y Fq(ds)645 567 y Ft(+)836 450 y Fl(Z)919 476 y Fn(t)882 638 y Fm(0)998 567 y Fp(O)1097 471 y Fl(\020)1179 567 y Fp(j)p Fq(A)1280 582 y Fm(1)1319 567 y Fp(j)1347 526 y Fm(4)1386 567 y Ft(\()p Fp(j)p Fq(A)1525 582 y Fm(1)1564 567 y Fp(j)1592 526 y Fm(3)1686 567 y Ft(+)22 b Fp(j)p Fo(E)1886 582 y Fh(1)1930 567 y Fp(j)p Ft(\))2028 471 y Fl(\021)2133 567 y Ft(+)54 b Fp(O)2362 471 y Fl(\020)2444 567 y Fp(j)p Fq(A)2545 582 y Fm(1)2584 567 y Fp(j)2612 526 y Fm(2)2651 567 y Ft(\()p Fp(j)p Fq(A)2790 582 y Fm(1)2829 567 y Fp(j)2857 526 y Fm(5)2951 567 y Ft(+)22 b Fp(j)p Fo(E)3151 582 y Fh(1)3195 567 y Fp(j)p Ft(\))3261 471 y Fl(\021)3360 567 y Fq(ds:)241 b Ft(\(5.24\))72 762 y(W)-8 b(e)33 b(expand)h(the)f (terms)g(in)m(v)m(olving)e Fq(H)1517 777 y Fm(1)1556 762 y Ft(\()p Fq(A;)17 b(t)p Ft(\).)44 b(Using)32 b(in)m(tegration)f(b) m(y)i(parts,)g(as)g(ab)s(o)m(v)m(e,)g(w)m(e)h(obtain:)-1 972 y(2)p Fq(\013)110 987 y Fm(21)201 855 y Fl(Z)284 881 y Fn(t)247 1043 y Fm(0)363 972 y Fp(j)p Fq(A)464 987 y Fm(1)503 972 y Fp(j)531 931 y Fm(2)603 972 y Fq(H)684 987 y Fm(1)755 972 y Fq(ds)60 b Ft(=)28 b Fq(h)1072 987 y Fm(5)p Fn(a)1149 972 y Ft(\()p Fq(A)1260 987 y Fm(1)1299 972 y Fq(;)17 b(t)p Ft(\))55 b Fp(\000)g Fq(h)1659 987 y Fm(5)p Fn(a)1736 972 y Ft(\()p Fq(A)1847 987 y Fm(10)1922 972 y Fq(;)17 b Ft(0\))54 b(+)2237 855 y Fl(Z)2320 881 y Fn(t)2284 1043 y Fm(0)2399 972 y Fp(O)2498 875 y Fl(\020)2580 972 y 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1082 y Fp(\024)61 b Fq(C)7 b Fp(jj)p Fq(B)e(\021)t Fp(jj)2306 1097 y Fm(4)2344 1082 y Fq(:)1354 b Ft(\(7.24\))-74 1313 y(This)33 b(follo)m(ws)e(if)h(w)m(e)h(sho)m(w)h(that)e(the)h(op)s (erator)f Fq(@)1767 1328 y Fn(i)1796 1313 y Fq(B)1875 1277 y Fk(\000)p Fm(1)2002 1313 y Ft(is)g(b)s(ounded)h(on)g Fq(L)2700 1277 y Fn(p)2772 1313 y Ft(\(with)f Fq(p)c Ft(=)g(4\).)72 1551 y(T)-8 b(o)35 b(pro)m(v)m(e)g(the)g Fq(L)718 1515 y Fn(p)793 1551 y Ft(b)s(oundedness)h(of)e Fq(@)1531 1566 y Fn(i)1560 1551 y Fq(B)1639 1515 y Fk(\000)p Fm(1)1733 1551 y Ft(,)h(w)m(e)h(can)e(apply)g(the)h(results)g(on)f(the) h(w)m(a)m(v)m(e)h(op)s(erator,)e Fq(W)3934 1566 y Fk(\003)-74 1701 y Ft(in)e Fp(x)p Ft(2.2.)44 b(Indeed,)34 b(for)e(an)m(y)i Fq(g)d Fp(2)d Fq(L)1196 1665 y Fn(p)1236 1701 y Ft(,)33 b(w)m(e)h(ha)m(v)m(e)g(using)e(the)h(b)s(oundedness)i(of)e(the)g(w)m(a) m(v)m(e)h(op)s(erators)f(on)g Fq(W)3853 1665 y Fm(1)p Fn(;p)3947 1701 y Ft(,)-74 1850 y(for)f Fq(p)c Fp(\025)g Ft(1,)k(that)1120 2081 y Fp(k)p 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Fq(:)-74 2834 y Fo(Remark:)52 b Ft(W)-8 b(e)37 b(o\013er)g(here)h(an) f(alternativ)m(e)f(pro)s(of)g(whic)m(h)h(do)s(es)h(not)e(mak)m(e)h(use) h(of)f(the)g(w)m(a)m(v)m(e)i(op)s(erators,)-74 2984 y(and)f(therefore)g (applies)f(under)h(w)m(eak)m(er)i(h)m(yp)s(otheses)g(on)e Fq(V)21 b Ft(.)59 b(W)-8 b(e)38 b(b)s(egin)f(with)g(the)h(square)h(ro)s (ot)e(form)m(ula)-74 3133 y(see)d([49)o(])f(and)g(\(2.33\):)-74 3283 y(F)-8 b(or)32 b(an)m(y)h Fq( )f Fp(2)c(D)s Ft(\()p Fq(B)671 3246 y Fm(2)710 3283 y Ft(\):)1110 3432 y Fq(B)1189 3391 y Fk(\000)p Fm(1)1344 3432 y Ft(=)60 b Fq(\031)1539 3391 y Fk(\000)p Fm(1)1682 3315 y Fl(Z)1765 3341 y Fk(1)1728 3504 y Fm(0)1889 3432 y Fq(w)1962 3391 y Fk(\000)p Fm(1)p Fn(=)p Fm(2)2126 3432 y Ft(\()p Fq(B)2243 3391 y Fm(2)2305 3432 y Ft(+)22 b Fq(w)s Ft(\))2514 3391 y Fk(\000)p Fm(1)2640 3432 y Fq(dw)s(:)934 b Ft(\(7.25\))-74 3627 y(By)33 b(\(7.25\))f(and)h (the)g(second)h(resolv)m(en)m(t)f(form)m(ula)e(w)m(e)j(ha)m(v)m(e:)482 3858 y Fq(@)533 3873 y Fn(i)562 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Fj(2)2369 4661 y Fm(+)p Fn(w)r Fm(\))2541 4703 y Fq(dt)1098 b Ft(\(7.27\))-74 4907 y(Since)33 b(inf)22 b Fq(\033)t Ft(\()p Fq(B)491 4871 y Fm(2)531 4907 y Ft(\))28 b(=)f(\012)770 4871 y Fm(2)838 4907 y Fq(>)g Ft(0,)1424 5057 y Fp(k)p Fq(e)1519 5016 y Fk(\000)p Fn(t)p Fm(\()p Fn(B)1682 4992 y Fj(2)1718 5016 y Fk(\000)p Fm(\012)1824 4992 y Fj(2)1858 5016 y Fn(=)p Fm(2\))1960 5057 y Fp(k)2010 5072 y Fk(B)r Fm(\()p Fn(L)2133 5053 y Fg(p)2170 5072 y Fm(\))2262 5057 y Fp(\024)60 b Fq(C)1256 b Ft(\(7.28\))1901 5306 y(45)p eop %%Page: 46 46 46 45 bop -74 59 a Ft(and)33 b(therefore)722 208 y Fp(k)p Ft(\()p Fq(B)889 167 y Fm(2)951 208 y Ft(+)22 b Fq(w)s Ft(\))1160 167 y Fk(\000)p Fm(1)1254 208 y Fp(k)1304 224 y Fk(B)r Fm(\()p Fn(L)1427 205 y Fg(p)1463 224 y Fm(\))1555 208 y Fp(\024)60 b Fq(C)1818 91 y Fl(Z)1901 117 y Fk(1)1865 280 y Fm(0)1993 208 y Fq(e)2038 167 y Fk(\000)p Fn(t)p Fm(\(\012)2196 144 y Fj(2)2231 167 y Fn(=)p Fm(2+)p Fn(w)r Fm(\))2501 208 y Fp(\024)h Fq(C)2716 167 y Fk(0)2739 208 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Fk(\000)p Fm(1)p Fn(=)p Fm(2)3106 626 y Ft(\(1)h(+)g Fq(w)s Ft(\))3424 585 y Fk(\000)p Fm(1)3550 626 y Fq(dw)62 b(<)28 b Fp(1)p Fq(:)3725 776 y Ft(\(7.30\))72 1116 y(F)-8 b(rom)31 b(\(7.22\))h(w)m(e)i(see)g(that)e Fp(jj)p Fq(B)5 b(\021)t Fp(jj)1366 1131 y Fm(4)1436 1116 y Ft(m)m(ust)33 b(b)s(e)g(estimated.)43 b(Recall)31 b(that)1338 1335 y Fq(B)5 b(\021)64 b Ft(=)c Fq(B)5 b(\021)1792 1350 y Fm(1)1886 1335 y Ft(+)55 b Fq(B)5 b(\021)2144 1350 y Fm(2)2238 1335 y Ft(+)55 b Fq(B)5 b(\021)2496 1350 y Fm(3)2535 1335 y Fq(:)1163 b Ft(\(7.31\))-74 1553 y(By)33 b(\(7.4\))1149 1703 y Fq(B)5 b(\021)1276 1718 y Fm(1)1376 1703 y Ft(=)60 b Fq(E)1584 1718 y Fm(0)1623 1703 y Ft(\()p Fq(t)p Ft(\))p Fq(B)5 b Fo(P)1890 1718 y Fh(c)1930 1703 y Fq(u)1986 1718 y Fm(0)2080 1703 y Ft(+)55 b Fq(E)2283 1718 y Fm(1)2322 1703 y Ft(\()p Fq(t)p Ft(\))p Fq(B)5 b Fo(P)2589 1718 y Fh(c)2629 1703 y Fq(u)2685 1718 y Fm(1)2724 1703 y Fq(;)974 b Ft(\(7.32\))-74 1893 y(and)33 b(so)1297 2043 y Fp(jj)p Fq(B)5 b(\021)1480 2058 y Fm(1)1519 2043 y Ft(\()p Fq(t)p Ft(\))p Fp(jj)1686 2058 y Fm(4)1785 2043 y Fp(\024)60 b Fq(C)7 b Fp(h)p Fq(t)p Fp(i)2112 2001 y Fk(\000)p Fm(1)p Fn(=)p Fm(2)2310 2043 y Fp(k)p Fo(u)2422 2058 y Fm(0)2461 2043 y Fp(k)2511 2058 y Fh(X)2576 2043 y Ft(;)1122 b(\(7.33\))-74 2233 y(see)34 b(\(3.7\))e(for)g (de\014nition)f(of)i Fp(k)p Fo(u)1123 2248 y Fm(0)1162 2233 y Fp(k)1212 2248 y Fh(X)1277 2233 y Ft(.)72 2382 y(By)h(\(7.5\))1105 2547 y Fq(B)5 b(\021)1232 2562 y Fm(2)1332 2547 y Ft(=)60 b Fq(\025)1542 2430 y Fl(Z)1625 2456 y Fn(t)1588 2619 y Fm(0)1671 2547 y Fq(E)1743 2562 y Fm(1)1783 2547 y Ft(\()p Fq(t)22 b Fp(\000)h Fq(s)p Ft(\))32 b Fq(a)2145 2506 y Fm(3)2185 2547 y Ft(\()p Fq(s)p Ft(\))g Fq(B)5 b Fo(P)2495 2562 y Fh(c)2535 2547 y Fq(')2599 2506 y Fm(3)2671 2547 y Fq(ds;)930 b Ft(\(7.34\))-74 2738 y(whic)m(h)33 b(can)g(b)m(y)h(Theorem)e(2.3)h(b)s(e)f(estimated)g (as)158 2966 y Fp(jj)p Fq(B)5 b(\021)341 2981 y Fm(2)380 2966 y Ft(\()p Fq(t)p Ft(\))p Fp(jj)547 2981 y Fm(4)646 2966 y Fp(\024)61 b Fq(C)7 b Fp(j)p Fq(\025)p Fp(j)991 2849 y Fl(Z)1073 2875 y Fn(t)1036 3038 y Fm(0)1119 2966 y Fp(j)p Fq(t)22 b Fp(\000)g Fq(s)p Fp(j)1377 2925 y Fk(\000)p Fm(3)p Fn(=)p Fm(4)1574 2966 y Fp(j)p Fq(A)p Ft(\()p Fq(s)p Ft(\))p Fp(j)1825 2925 y Fm(3)1897 2966 y Fp(k)p Fo(P)2024 2981 y Fh(c)2063 2966 y Fq(')2127 2925 y Fm(3)2167 2966 y Fp(k)2217 2982 y Fm(4)p Fn(=)p Fm(3)2359 2966 y Fq(ds)60 b Fp(\024)h Fq(C)2724 2981 y Fn(')2774 2966 y Fp(j)p Fq(\025)p Fp(j)31 b Ft([)p Fq(A)p Ft(])3045 2925 y Fm(3)3045 2991 y(1)p Fn(=)p Fm(4)3188 2966 y Fp(h)p Fq(t)p Fp(i)3301 2925 y Fk(\000)p Fm(1)p Fn(=)p Fm(2)3466 2966 y Fq(:)232 b Ft(\(7.35\))-74 3185 y(The)34 b(estimation)c(of)i Fp(jj)p Fq(B)5 b(\021)898 3200 y Fm(3)937 3185 y Ft(\()p Fq(t)p Ft(\))p Fp(jj)1104 3200 y Fm(4)1176 3185 y Ft(is)32 b(more)g(in)m(v)m(olv)m(ed.)44 b(By)33 b(\(7.6\),)625 3419 y Fq(B)5 b(\021)752 3434 y Fm(3)851 3419 y Ft(=)61 b Fq(\025)1062 3302 y Fl(Z)1144 3328 y Fn(t)1107 3491 y Fm(0)1223 3419 y Ft(sin)16 b Fq(B)5 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Ft(+)54 b Fp(jj)p Fq(\021)3160 3825 y Fm(3)3199 3867 y Fp(jj)3255 3882 y Fm(1)p Fn(;)p Fm(4)p Fn(=)p Fm(3)3452 3770 y Fl(i)3540 3867 y Fq(ds:)61 b Ft(\(7.37\))-74 4085 y(The)30 b(follo)m(wing)c(prop)s(osition)h(pro)m (vides)j(estimates)e(for)g(the)i(in)m(tegrand.)42 b(It)28 b(is)h(pro)m(v)m(ed)h(in)e(the)h(same)g(manner)-74 4235 y(as)k(Prop)s(osition)e(7.2.)-74 4457 y Fo(Prop)s(osition)36 b(7.3.)373 4676 y Fp(j)p Fq(A)p Fp(j)502 4635 y Fm(2)574 4676 y Fp(jj)p Fq(')694 4635 y Fm(2)733 4676 y Fq(\021)t Fp(jj)841 4692 y Fm(1)p Fn(;)p Fm(4)p Fn(=)p Fm(3)1120 4676 y Fp(\024)116 b Fq(C)1383 4691 y Fn(')1465 4676 y Fp(j)p Fq(A)p Fp(j)1594 4635 y Fm(2)1683 4676 y Ft(\()o Fp(jj)p Fq(\021)t Fp(jj)1884 4691 y Fm(8)1977 4676 y Ft(+)54 b Fp(jj)p Fq(B)5 b(\021)2290 4691 y Fm(1)2330 4676 y Fp(jj)2386 4691 y Fm(4)2479 4676 y Ft(+)54 b Fp(jj)p Fq(B)5 b(\021)2792 4691 y Fm(2)2832 4676 y Fp(jj)2888 4691 y Fm(4)2981 4676 y Ft(+)55 b Fp(jj)p Fq(B)5 b(\021)3295 4691 y Fm(3)3334 4676 y Fp(jj)3390 4691 y Fm(4)3428 4676 y Ft(\))17 b Fq(;)413 4850 y Fp(j)p Fq(A)p Fp(j)32 b(jj)p Fq('\021)746 4809 y Fm(2)784 4850 y Fp(jj)840 4866 y Fm(1)p Fn(;)p Fm(4)p Fn(=)p Fm(3)1120 4850 y Fp(\024)116 b Fq(C)1383 4865 y Fn(')1465 4850 y Fp(j)p Fq(A)p Fp(j)1643 4754 y Fl(\020)1693 4850 y Fp(jj)p Fq(\021)t Fp(jj)1857 4809 y Fm(2)1857 4875 y(8)1949 4850 y Ft(+)55 b Fp(jj)p Fq(\021)t Fp(jj)2244 4865 y Fm(8)2314 4850 y Fp(jj)p Fq(B)5 b(\021)t Fp(jj)2557 4865 y Fm(4)2628 4754 y Fl(\021)638 5025 y Fp(jj)p Fq(\021)746 4984 y Fm(3)784 5025 y Fp(jj)840 5040 y Fm(1)p Fn(;)p Fm(4)p Fn(=)p Fm(3)1120 5025 y Fp(\024)116 b Ft(3)32 b Fp(jj)p Fq(\021)t Fp(jj)1558 4984 y Fm(2)1558 5049 y(8)1629 5025 y Fp(jj)p Fq(@)5 b(\021)t Fp(jj)1849 5040 y Fm(2)1887 5025 y Fq(:)1811 b Ft(\(7.38\))1901 5306 y(46)p eop %%Page: 47 47 47 46 bop 72 59 a Ft(An)m(ticipating)31 b(the)i(b)s(eha)m(vior)1273 307 y Fp(j)p Fq(A)p Ft(\()p Fq(t)p Ft(\))p Fp(j)27 b(\030)h Fq(t)1680 266 y Fk(\000)p Fm(1)p Fn(=)p Fm(4)1845 307 y Fq(;)50 b Fp(jj)p Fq(\021)t Ft(\()p Fq(t)p Ft(\))p Fp(jj)2197 322 y Fm(8)2262 307 y Fp(\030)61 b Fq(t)2435 266 y Fk(\000)p Fm(3)p Fn(=)p Fm(4)2600 307 y Fq(;)-74 556 y Ft(and)36 b(using)g(\(7.33\),)h(w)m(e)g(\014nd)g(that)f Fq(B)5 b(\021)1380 571 y Fm(3)1455 556 y Ft(is)36 b(driv)m(en)h(b)m(y)g (terms)f(whic)m(h)h(are)f(formally)d(of)j(order)g Fp(h)p Fq(t)p Fp(i)3596 520 y Fk(\000)p Fm(1)3690 556 y Ft(.)54 b(Esti-)-74 706 y(mation)37 b(in)h Fq(L)449 669 y Fm(4)528 706 y Ft(will)e(lead)i(to)h(con)m(v)m(olution)f(of)g Fp(h)p Fq(t)p Fp(i)1815 669 y Fk(\000)p Fm(1)1948 706 y Ft(with)h Fp(h)p Fq(t)p Fp(i)2290 669 y Fk(\000)p Fm(1)p Fn(=)p Fm(2)2493 706 y Ft(giving)e(a)i(rate)g(of)f Fp(h)p Fq(t)p Fp(i)3314 669 y Fk(\000)p Fm(1)p Fn(=)p Fm(2+)p Fn(\016)3560 678 y Fj(0)3599 706 y Ft(,)j(for)d(an)m(y)-74 855 y Fq(\016)-31 870 y Fm(0)36 855 y Fq(>)28 b Ft(0.)72 1104 y(F)-8 b(or)32 b(0)c Fp(\024)g Fq(t)g Fp(\024)g Fq(T)14 b Ft(,)32 b(with)g Fq(T)47 b Ft(\014xed)33 b(and)g(arbitrary)-8 b(,)32 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Fp(h)p Fq(t)p Fp(i)2907 1644 y Fk(\000)p Fm(1+)p Fn(\016)3083 1653 y Fj(0)3121 1588 y Fl(\021)3187 1685 y Fq(;)-74 1933 y Ft(where)34 b Fq(\016)251 1948 y Fm(0)323 1933 y Ft(is)e(p)s(ositiv)m(e)g(and)h(arbitrary)-8 b(.)42 b(Also,)459 2182 y Fp(j)p Fq(A)p Fp(j)32 b(jj)p Fq('\021)792 2141 y Fm(2)831 2182 y Fp(jj)887 2198 y Fm(1)p Fn(;)p Fm(4)p Fn(=)p Fm(3)1166 2182 y Fp(\024)84 b Fq(C)1397 2197 y Fn(')1479 2182 y Ft([)p Fq(A)p Ft(])1606 2198 y Fm(1)p Fn(=)p Fm(4)1733 2086 y Fl(\020)1815 2182 y Ft([)p Fq(\021)t Ft(])1921 2141 y Fm(2)1921 2207 y(8)p Fn(;)p Fm(3)p Fn(=)p Fm(4)2086 2182 y Fp(h)p Fq(t)p Fp(i)2199 2141 y Fk(\000)p Fm(7)p Fn(=)p Fm(4)2418 2182 y Ft(+)55 b([)p Fq(\021)t Ft(])2655 2198 y Fm(8)p Fn(;)p Fm(3)p Fn(=)p Fm(4)2820 2182 y Ft([)p Fq(B)5 b(\021)2974 2197 y Fm(1)3014 2182 y Ft(])3041 2197 y Fm(4)p Fn(;)p Fm(0)3135 2182 y Fp(h)p Fq(t)p Fp(i)3248 2141 y Fk(\000)p Fm(1)3375 2086 y Fl(\021)684 2356 y Fp(jj)p Fq(\021)792 2315 y Fm(3)831 2356 y Fp(jj)887 2372 y Fm(1)p Fn(;)p 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Fq(T)14 b Ft(:)-74 4678 y Fp(j)p Fq(A)p Fp(j)55 4637 y Fm(2)127 4678 y Fp(jj)p Fq(')247 4637 y Fm(2)285 4678 y Fq(\021)t Fp(jj)393 4694 y Fm(1)p Fn(;)p Fm(8)p Fn(=)p Fm(7)673 4678 y Fp(\024)116 b Fq(C)936 4693 y Fn(')1018 4678 y Ft([)p Fq(A)p Ft(])1145 4637 y Fm(2)1145 4703 y(1)p Fn(=)p Fm(4)1305 4582 y Fl(\020)1387 4678 y Ft([)p Fq(\021)t Ft(])1493 4694 y Fm(8)p Fn(;)p Fm(3)p Fn(=)p Fm(4)1712 4678 y Ft(+)55 b Fp(k)p Fo(u)1955 4693 y Fm(0)1994 4678 y Fp(k)2044 4693 y Fh(X)2164 4678 y Ft(+)g([)p Fq(A)p Ft(])2422 4637 y Fm(3)2422 4703 y(1)p Fn(=)p Fm(4)2587 4678 y Ft(+)f([)p Fq(B)5 b(\021)2871 4693 y Fm(3)2911 4678 y Ft(])2938 4694 y Fm(4)p Fn(;)p Fm(1)p Fn(=)p Fm(2)p Fk(\000)p Fn(\016)3184 4703 y Fj(0)3256 4582 y Fl(\021)3355 4678 y Fp(h)p Fq(t)p Fp(i)3468 4637 y Fk(\000)p Fm(1+)p Fn(\016)3644 4646 y Fj(0)3683 4678 y Fq(;)-34 4861 y Fp(j)p Fq(A)p Fp(j)32 b(jj)p Fq('\021)299 4820 y Fm(2)337 4861 y Fp(jj)393 4876 y Fm(1)p Fn(;)p Fm(8)p Fn(=)p Fm(7)673 4861 y Fp(\024)116 b Fq(C)936 4876 y Fn(')1018 4861 y Ft([)p Fq(A)p Ft(])1145 4876 y Fm(1)p Fn(=)p Fm(4)1305 4765 y Fl(\020)1387 4861 y Ft([)p Fq(\021)t Ft(])1493 4820 y Fm(2)1493 4886 y(8)p Fn(;)p Fm(3)p Fn(=)p Fm(4)1712 4861 y Ft(+)55 b Fp(fk)p Fo(u)2005 4876 y Fm(0)2044 4861 y Fp(k)2094 4876 y Fh(X)2214 4861 y Ft(+)f([)p Fq(B)5 b(\021)2498 4876 y Fm(3)2538 4861 y Ft(])2565 4876 y Fm(4)p Fn(;)p Fm(1)p Fn(=)p Fm(2)p Fk(\000)p Fn(\016)2811 4885 y Fj(0)2906 4861 y Ft(+)54 b([)p Fq(A)p Ft(])3163 4820 y Fm(3)3163 4886 y(1)p Fn(=)p Fm(4)3273 4861 y Fp(g)33 b Ft([)p Fq(\021)t Ft(])3462 4876 y Fm(8)p Fn(;)p Fm(3)p Fn(=)p Fm(4)3659 4765 y Fl(\021)3758 4861 y Fp(h)p Fq(t)p Fp(i)3871 4820 y Fk(\000)p Fm(1+)p Fn(\016)4047 4829 y Fj(0)4086 4861 y Fq(;)190 5045 y Fp(jj)p Fq(\021)298 5003 y Fm(3)337 5045 y Fp(jj)393 5060 y Fm(1)p Fn(;)p Fm(8)p Fn(=)p Fm(7)673 5045 y Fp(\024)116 b Fq(C)936 5060 y Fn(')1002 5045 y Ft(\()p Fp(jj)p Fq(\021)t Fp(jj)1204 5060 y Fm(1)p Fn(;)p Fm(2)1297 5045 y Ft(\))49 b([)p Fq(\021)t Ft(])1490 4994 y Fm(5)p Fn(=)p Fm(3)1490 5073 y(8)p Fn(;)p Fm(3)p Fn(=)p Fm(4)1687 5045 y Fp(h)p Fq(t)p Fp(i)1800 5003 y Fk(\000)p Fm(5)p Fn(=)p Fm(4)1965 5045 y Fq(:)1888 b Ft(\(7.41\))1901 5306 y(47)p eop %%Page: 48 48 48 47 bop -74 59 a Ft(Substitution)32 b(of)g(\(7.41\))g(in)m(to)g (\(7.17-7.20\))e(leads)j(to)f(an)g(estimate)g(for)g Fp(jj)p Fq(\021)2706 74 y Fm(3)2745 59 y Ft(\()p Fq(t)p Ft(\))p Fp(jj)2912 74 y Fm(8)2951 59 y Ft(.)43 b(F)-8 b(or)32 b(0)c Fp(\024)g Fq(t)g Fp(\024)g Fq(T)14 b Ft(:)181 308 y Fp(jj)p Fq(\021)285 323 y Fm(3)324 308 y Ft(\()p Fq(t)p Ft(\))p Fp(jj)491 323 y Fm(8)646 308 y Fp(\024)116 b(j)p Fq(\025)p Fp(j)31 b Fq(C)1053 323 y Fn(')1136 308 y Fp(h)p Fq(t)p Fp(i)1249 267 y Fk(\000)p Fm(1+)p Fn(\016)1425 276 y Fj(0)1513 308 y Ft(\()26 b([)p Fq(A)p Ft(])1704 267 y Fm(2)1704 332 y(1)p Fn(=)p Fm(4)1815 308 y Fp(f)p Ft([)p Fq(\021)t Ft(])1971 323 y Fm(8)p Fn(;)p Fm(3)p Fn(=)p Fm(4)2190 308 y Ft(+)54 b Fp(k)p Fo(u)2432 323 y Fm(0)2472 308 y Fp(k)2522 323 y Fh(X)2609 308 y Ft(+)h([)p Fq(A)p Ft(])2867 267 y Fm(3)2867 332 y(1)p Fn(=)p Fm(4)3032 308 y Ft(+)f([)p Fq(B)5 b(\021)3316 323 y Fm(3)3356 308 y Ft(])3383 323 y Fm(4)p Fn(;)p Fm(1)p Fn(=)p Fm(2)p Fk(\000)p Fn(\016)3629 332 y Fj(0)3669 308 y Fp(g)647 482 y Ft(+)83 b([)p Fq(A)p Ft(])933 498 y Fm(1)p Fn(=)p Fm(4)1076 482 y Fp(f)32 b Ft([)p Fq(\021)t Ft(])1264 441 y Fm(2)1264 507 y(8)p Fn(;)p Fm(3)p Fn(=)p Fm(4)1484 482 y Ft(+)1614 386 y Fl(h)1686 482 y Fp(k)p Fo(u)1798 497 y Fm(0)1838 482 y Fp(k)1888 497 y Fh(X)1975 482 y Ft(+)54 b([)p Fq(B)5 b(\021)2259 497 y Fm(3)2299 482 y Ft(])2326 498 y Fm(4)p Fn(;)p Fm(1)p Fn(=)p Fm(2)p Fk(\000)p Fn(\016)2572 507 y Fj(0)2667 482 y Ft(+)54 b([)p Fq(A)p Ft(])2924 441 y Fm(3)2924 507 y(1)p Fn(=)p Fm(4)3067 386 y Fl(i)3155 482 y Ft([)p Fq(\021)t Ft(])3261 498 y Fm(8)p Fn(;)p Fm(3)p Fn(=)p Fm(4)3459 482 y Fp(g)647 666 y Ft(+)116 b Fq(C)23 b Ft(\()p Fp(jj)p Fq(\021)t Fp(jj)1134 681 y Fm(1)p Fn(;)p Fm(2)1227 666 y Fq(;)17 b(')p Ft(\))49 b([)p Fq(\021)t Ft(])1528 615 y Fm(5)p Fn(=)p Fm(3)1528 695 y(8)p Fn(;)p Fm(3)p Fn(=)p Fm(4)1751 666 y Ft(\))17 b Fq(;)1892 b Ft(\(7.42\))-74 915 y(where)34 b Fq(\016)251 930 y Fm(0)323 915 y Ft(is)e(arbitary)-8 b(.)72 1064 y(Finally)30 b(w)m(e)k(can)f(no)m(w)g(estimate)f Fq(F)1384 1079 y Fm(2)1423 1064 y Ft(\()p Fq(a;)17 b(\021)1604 1079 y Fm(3)1644 1064 y Ft(\))32 b(using)g(\(4.18\))g(and)h(\(7.42\).) 43 b(Cho)s(osing)32 b Fq(\016)3230 1079 y Fm(0)3302 1064 y Ft(so)h(that)1167 1313 y Fp(\000)p Ft(1)22 b(+)g Fq(\016)1456 1328 y Fm(0)1556 1313 y Ft(=)60 b Fp(\000)p Ft(3)p Fq(=)p Ft(4)22 b Fp(\000)h Fq(\033)2093 1328 y Fm(0)2133 1313 y Fq(;)49 b Ft(with)32 b Fq(\033)2486 1328 y Fm(0)2554 1313 y Fq(>)27 b Ft(0)p Fq(;)-74 1562 y Ft(w)m(e)34 b(ha)m(v)m(e)-74 1795 y Fo(Prop)s(osition)i(7.4.)128 2044 y Fp(j)p Fq(F)219 2059 y Fm(2)259 2044 y Ft(\()p Fq(a;)17 b(\021)440 2059 y Fm(3)479 2044 y Ft(\))p Fp(j)148 b(\024)116 b Fq(C)956 2059 y Fn(')1006 2044 y Fp(j)p Fq(\025)p Fp(j)32 b Ft([)p Fq(A)p Ft(])1278 2003 y Fm(2)1278 2069 y(1)p Fn(=)p Fm(4)1437 1948 y Fl(\020)1519 2044 y Ft([)p Fq(A)p Ft(])1646 2003 y Fm(2)1646 2069 y(1)p Fn(=)p Fm(4)1756 2044 y Fp(f)h Ft([)p Fq(\021)t Ft(])1945 2060 y Fm(8)p Fn(;)p Fm(3)p Fn(=)p Fm(4)2164 2044 y Ft(+)55 b Fp(k)p Fo(u)2407 2059 y Fm(0)2446 2044 y Fp(j)2474 2059 y Fh(X)2594 2044 y Ft(+)g([)p Fq(A)p Ft(])2852 2003 y Fm(3)2852 2069 y(1)p Fn(=)p Fm(4)3016 2044 y Ft(+)g([)p Fq(B)5 b(\021)3301 2059 y Fm(3)3341 2044 y Ft(])3368 2060 y Fm(4)p Fn(;)p Fm(1)p Fn(=)p Fm(4+)p Fn(\033)3623 2069 y Fj(0)3695 2044 y Fp(g)694 2227 y Ft(+)83 b([)p Fq(A)p Ft(])980 2242 y Fm(1)p Fn(=)p Fm(4)1123 2227 y Fp(f)33 b Ft([)p Fq(\021)t Ft(])1312 2186 y Fm(2)1312 2251 y(8)p Fn(;)p Fm(3)p Fn(=)p Fm(4)1531 2227 y Ft(+)1661 2130 y Fl(h)1701 2227 y Fp(k)p Fo(u)1813 2242 y Fm(0)1852 2227 y Fp(k)1902 2242 y Fh(X)2022 2227 y Ft(+)55 b([)p Fq(A)p Ft(])2280 2186 y Fm(3)2280 2251 y(1)p Fn(=)p Fm(4)2445 2227 y Ft(+)f([)p Fq(B)5 b(\021)2729 2242 y Fm(3)2769 2227 y Ft(])2796 2242 y Fm(4)p Fn(;)p Fm(1)p Fn(=)p Fm(4+)p Fn(\033)3051 2251 y Fj(0)3123 2130 y Fl(i)3212 2227 y Ft([)p Fq(\021)t Ft(])3318 2242 y Fm(8)p Fn(;)p Fm(3)p Fn(=)p Fm(4)3482 2227 y Fp(g)694 2410 y Ft(+)158 b Fq(C)24 b Ft(\()p Fp(jj)p Fq(\021)t Fp(jj)1224 2425 y Fm(1)p Fn(;)p Fm(2)1317 2410 y Fq(;)17 b(')p Ft(\))48 b([)p Fq(\021)t Ft(])1617 2359 y Fm(5)p Fn(=)p Fm(3)1617 2439 y(8)p Fn(;)p Fm(3)p Fn(=)p Fm(4)1815 2314 y Fl(\021)1913 2410 y Fp(h)p Fq(t)p Fp(i)2026 2369 y Fk(\000)p Fm(5)p Fn(=)p Fm(4)p Fk(\000)p Fn(\033)2281 2378 y Fj(0)2321 2410 y Fq(:)1377 b Ft(\(7.43\))1072 2759 y Fr(Estimation)35 b(of)f Fq(F)1749 2774 y Fm(2)1789 2759 y Ft(\()p Fq(a;)17 b(\021)1974 2723 y Fn(nr)r Fm(1)1970 2784 y Fk(\003)2090 2759 y Ft(\))34 b Fr(and)h Fq(F)2415 2774 y Fm(2)2454 2759 y Ft(\()p Fq(a;)17 b(\021)2639 2723 y Fn(nr)r Fm(2)2635 2784 y Fk(\003)2755 2759 y Ft(\))72 3008 y(By)34 b(\(4.23\))532 3278 y Fq(\021)584 3236 y Fn(nr)r Fm(1)580 3302 y Fk(\003)815 3278 y Ft(=)116 b Fp(\000)1094 3210 y Fq(\025)p 1094 3254 57 4 v 1098 3346 a Ft(2)1161 3278 y Fq(A)1234 3236 y Fm(3)1234 3302 y(0)1306 3278 y Fq(B)1385 3236 y Fk(\000)p Fm(1)1480 3278 y Ft(\()p Fq(B)27 b Fp(\000)c Ft(3\012)f(+)g Fq(i)p Ft(0\))2078 3236 y Fk(\000)p Fm(1)2205 3278 y Fq(e)2250 3236 y Fn(iB)s(t)2393 3278 y Fo(P)2470 3293 y Fh(c)2542 3278 y Fq(')2606 3236 y Fm(3)2645 3278 y Fq(;)50 b Ft(and)532 3516 y Fq(\021)584 3475 y Fn(nr)r Fm(2)580 3540 y Fk(\003)815 3516 y Ft(=)83 b Fp(\000)1061 3448 y Ft(3)p Fq(\025)p 1061 3492 106 4 v 1090 3584 a Ft(2)1210 3516 y Fq(B)1289 3475 y Fk(\000)p Fm(1)1383 3516 y Ft(\()p Fq(B)28 b Fp(\000)22 b Ft(3\012)h(+)f Fq(i)p Ft(0\))1982 3475 y Fk(\000)p Fm(1)2125 3399 y Fl(Z)2208 3425 y Fn(t)2171 3587 y Fm(0)2254 3516 y Fq(e)2299 3475 y Fn(iB)s Fm(\()p Fn(t)p Fk(\000)p Fn(s)p Fm(\))2585 3516 y Fq(e)2630 3475 y Fn(i)p Fm(3\012)p Fn(s)2809 3516 y Fq(A)2882 3475 y Fm(2)2922 3516 y Fq(A)2995 3475 y Fk(0)3051 3516 y Fq(ds)p Fo(P)3225 3531 y Fh(c)3264 3516 y Fq(')3328 3475 y Fm(3)72 3765 y Ft(Estimation)31 b(using)h(Prop)s (osition)f(2.2)h(giv)m(es:)113 3914 y Fl(\014)113 3964 y(\014)113 4014 y(\014)p Fq(F)204 4029 y Fm(2)243 4014 y Ft(\()p Fq(a;)17 b(\021)428 3973 y Fn(nr)r Fm(1)424 4038 y Fk(\003)544 4014 y Ft(\))582 3914 y Fl(\014)582 3964 y(\014)582 4014 y(\014)116 b Ft(=)g(3)32 b Fp(j)p Fq(\025)p Fp(j)g(j)p Fq(a)p Fp(j)1251 3973 y Fm(2)1339 3889 y Fl(\014)1339 3939 y(\014)1339 3989 y(\014)1339 4039 y(\014)1367 3897 y(Z)1466 4014 y Fq(')1530 3973 y Fm(3)1570 4014 y Fq(\021)1622 3973 y Fn(nr)r Fm(1)1618 4038 y Fk(\003)1738 3889 y Fl(\014)1738 3939 y(\014)1738 3989 y(\014)1738 4039 y(\014)726 4246 y Ft(=)116 b Fq(C)7 b Fp(j)p Fq(\025)p Fp(j)1108 4205 y Fm(2)1179 4246 y Fp(j)p Fq(A)1280 4261 y Fm(0)1319 4246 y Fp(j)1347 4205 y Fm(3)1387 4246 y Fp(j)p Fq(A)p Fp(j)1516 4205 y Fm(2)1604 4122 y Fl(\014)1604 4172 y(\014)1604 4221 y(\014)1604 4271 y(\014)1664 4129 y(Z)1796 4246 y Fq(')1860 4205 y Fm(3)1948 4246 y Ft([)p Fq(B)e Ft(\()p Fq(B)28 b Fp(\000)22 b Ft(3\012)h(+)f Fq(i)p Ft(0\)])2680 4198 y Fk(\000)p Fm(1)2823 4246 y Fq(e)2868 4205 y Fn(iB)s(t)3011 4246 y Fo(P)3088 4261 y Fh(c)3161 4246 y Fq(')3225 4205 y Fm(3)3296 4122 y Fl(\014)3296 4172 y(\014)3296 4221 y(\014)3296 4271 y(\014)725 4441 y Fp(\024)116 b Fq(C)7 b Fp(j)p Fq(\025)p Fp(j)1108 4400 y Fm(2)1179 4441 y Fp(j)p Fq(A)1280 4456 y Fm(0)1319 4441 y Fp(j)1347 4400 y Fm(3)1387 4441 y Fp(j)p Fq(A)p Fp(j)1516 4400 y Fm(2)1587 4441 y Fp(k)32 b(h)p Fq(x)p Fp(i)1802 4400 y Fn(\033)1849 4441 y Fq(')1913 4400 y Fm(3)1985 4441 y Fp(k)2035 4456 y Fm(2)2107 4441 y Fp(kh)p Fq(x)p Fp(i)2290 4400 y Fk(\000)p Fn(\033)2391 4441 y Ft(\()p Fq(B)27 b Fp(\000)c Ft(3\012)f(+)g Fq(i)p Ft(0\))2989 4400 y Fk(\000)p Fm(1)3116 4441 y Fq(e)3161 4400 y Fn(iB)s(t)3304 4441 y Fo(P)3381 4456 y Fh(c)3421 4441 y Fq(B)3500 4400 y Fk(\000)p Fm(1)3594 4441 y Fq(')3658 4400 y Fm(3)3698 4441 y Fp(k)3748 4456 y Fm(2)725 4615 y Fp(\024)116 b Fq(C)988 4630 y Fn(')1070 4615 y Fp(j)p Fq(\025)p Fp(j)1183 4574 y Fm(2)1255 4615 y Fp(j)p Fq(A)1356 4630 y Fm(0)1395 4615 y Fp(j)1423 4574 y Fm(2)1495 4615 y Ft([)p Fq(A)p Ft(])1622 4574 y Fm(2)1622 4640 y(1)p Fn(=)p Fm(4)1764 4615 y Fp(h)p Fq(t)p Fp(i)1877 4574 y Fk(\000)1942 4547 y Fj(5)p 1942 4559 31 4 v 1942 4600 a(4)1982 4574 y Fm(+)p Fn(\016)2075 4615 y Fq(:)1623 b Ft(\(7.44\))-74 4864 y(Here,)49 b(w)m(e)c(ha)m(v)m(e)h(used)g(that)e (1)p Fq(=)p Ft(2)30 b(+)g(6)p Fq(=)p Ft(5)48 b(=)g(5)p Fq(=)p Ft(4)29 b(+)h Fq(\016)t Ft(,)48 b(for)c(some)g Fq(\016)52 b(>)c Ft(0.)80 b(The)45 b(constan)m(t)h Fq(C)3535 4879 y Fn(')3629 4864 y Ft(dep)s(ends)-74 5014 y(on)40 b Fp(kh)p Fq(x)p Fp(i)252 4977 y Fn(\033)299 5014 y Fq(B)378 4977 y Fk(\000)p Fm(1)472 5014 y Fp(h)p Fq(x)p Fp(i)605 4977 y Fk(\000)p Fn(\033)707 5014 y Fp(k)757 5030 y Fk(B)r Fm(\()p Fn(L)880 5011 y Fj(2)915 5030 y Fm(\))946 5014 y Ft(.)67 b(It)40 b(is)g(easy)i(to)e(c)m(hec)m(k)i(that)e(this)g(norm)g (is)f(b)s(ounded)i(if)f(w)m(e)h(replace)f Fq(B)3736 4977 y Fk(\000)p Fm(1)3871 5014 y Ft(b)m(y)1901 5306 y(48)p eop %%Page: 49 49 49 48 bop -74 59 a Fq(B)5 23 y Fk(\000)p Fm(2)100 59 y Ft(.)42 b(The)31 b(estimate)e(of)g(in)m(terest)i(is)e(reduced)i(to)f (this)f(case)i(using)e(the)i(Kato)e(square)i(ro)s(ot)e(form)m(ula)e (\(7.25\).)-74 208 y(Similarly)-8 b(,)29 b(w)m(e)k(ha)m(v)m(e)38 357 y Fl(\014)38 407 y(\014)38 457 y(\014)p Fq(F)129 472 y Fm(2)169 457 y Ft(\()p Fq(a;)17 b(\021)354 416 y Fn(nr)r Fm(2)350 481 y Fk(\003)469 457 y Ft(\))507 357 y Fl(\014)507 407 y(\014)507 457 y(\014)84 b Ft(=)116 b(3)p Fp(j)p Fq(\025)p Fp(j)32 b(j)p Fq(a)p Fp(j)1112 416 y Fm(2)1200 332 y Fl(\014)1200 382 y(\014)1200 432 y(\014)1200 482 y(\014)1227 340 y(Z)1327 457 y Fq(')1391 416 y Fm(3)1430 457 y Fq(\021)1482 416 y Fn(nr)r Fm(2)1478 481 y Fk(\003)1598 332 y Fl(\014)1598 382 y(\014)1598 432 y(\014)1598 482 y(\014)618 696 y Fp(\024)116 b Fq(C)39 b Fp(j)p Fq(\025)p Fp(j)1033 655 y Fm(2)1105 696 y Fp(j)p Fq(A)p Fp(j)1234 655 y Fm(2)1322 572 y Fl(\014)1322 622 y(\014)1322 672 y(\014)1322 721 y(\014)1382 579 y(Z)1514 696 y Fq(')1578 655 y Fm(3)1634 696 y Ft([)p Fq(B)5 b Ft(\()p Fq(B)27 b Fp(\000)c Ft(3\012)f(+)g Fq(i")p Ft(\)])2362 648 y Fk(\000)p Fm(1)2506 579 y Fl(Z)2589 606 y Fn(t)2552 768 y Fm(0)2668 696 y Fq(e)2713 655 y Fn(i)p Fm(\()p Fn(t)p Fk(\000)p Fn(s)p Fm(\)\()p Fn(B)s Fk(\000)p Fm(3\012\))3194 696 y Fq(A)3267 655 y Fm(2)3306 696 y Fq(A)3379 655 y Fk(0)3435 696 y Fq(ds)33 b Fo(P)3642 711 y Fh(c)3681 696 y Fq(')3745 655 y Fm(3)3817 572 y Fl(\014)3817 622 y(\014)3817 672 y(\014)3817 721 y(\014)618 936 y Fp(\024)116 b Fq(C)881 951 y Fn(')963 936 y Fp(j)p Fq(\025)p Fp(j)1076 895 y Fm(2)1148 936 y Fp(j)p Fq(A)p Fp(j)1277 895 y Fm(2)1365 819 y Fl(Z)1448 845 y Fn(t)1411 1008 y Fm(0)1477 936 y Fp(h)p Fq(t)22 b Fp(\000)h Fq(s)p Fp(i)1758 895 y Fk(\000)1823 868 y Fj(3)p 1823 880 31 4 v 1823 921 a(2)1863 895 y Fm(+)p Fn(\026)1997 936 y Fp(j)p Fq(A)p Fp(j)2126 895 y Fm(2)2197 936 y Fp(j)p Fq(A)2298 895 y Fk(0)2322 936 y Fp(j)32 b Fq(ds)618 1141 y Fp(\024)116 b Fq(C)881 1156 y Fn(')963 1141 y Fp(j)p Fq(\025)p Fp(j)1076 1100 y 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Fq(Q)230 309 y Fm(0)270 294 y Ft(\()p Fq(A;)17 b(\021)t Ft(\))59 b(=)h([)p Fq(\021)t Ft(])816 253 y Fm(3)816 319 y(8)p Fn(;)p Fm(3)p Fn(=)p Fm(4)1036 294 y Ft(+)54 b([)p Fq(A)p Ft(])1293 309 y Fm(1)p Fn(=)p Fm(4)1453 198 y Fl(\020)1535 294 y Ft(1)g(+)h Fq(C)1862 198 y Fl(\020)1912 294 y Ft([)p Fq(A)p Ft(])2039 309 y Fm(1)p Fn(=)p Fm(4)2149 294 y Fq(;)17 b Ft([)p Fq(\021)t Ft(])2299 309 y Fm(8)p Fn(;)p Fm(3)p Fn(=)p Fm(4)2464 294 y Fq(;)g Ft([)p Fq(B)5 b(\021)2662 309 y Fm(3)2701 294 y Ft(])2728 309 y Fm(4)p Fn(;)p Fm(1)p Fn(=)p Fm(4+)p Fn(\033)2983 318 y Fj(0)3023 294 y Fq(;)17 b Fp(k)p Fo(u)3179 309 y Fm(0)3218 294 y Fp(k)3268 309 y Fh(X)3366 198 y Fl(\021)32 b(\021)3725 294 y Ft(\(7.48\))72 581 y(It)h(no)m(w)g(remains)f(to)g(estimate)g Fp(jj)p Fq(\021)t Fp(jj)1419 596 y Fm(8)1457 581 y Ft(.)36 966 y Fp(jj)p Fq(\021)t Fp(jj)200 981 y Fm(8)321 966 y Fp(\024)116 b(jj)p Fq(\021)618 981 y Fm(1)657 966 y Fp(jj)713 981 y Fm(8)806 966 y Ft(+)55 b Fp(jj)p Fq(\021)1041 981 y Fm(2)1080 966 y Fp(jj)1136 981 y Fm(8)1229 966 y Ft(+)g Fp(jj)p Fq(\021)1464 981 y Fm(3)1502 966 y Fp(jj)1558 981 y Fm(8)321 1156 y Fp(\024)116 b(jj)p Fq(E)642 1171 y Fm(0)681 1156 y Ft(\()p Fq(t)p Ft(\))p Fo(P)869 1171 y Fh(c)909 1156 y Fq(u)965 1171 y Fm(0)1004 1156 y Fp(jj)1060 1171 y Fm(8)1153 1156 y Ft(+)55 b Fp(jj)p Fq(E)1412 1171 y Fm(1)1451 1156 y Ft(\()p Fq(t)p Ft(\))p Fo(P)1639 1171 y Fh(c)1679 1156 y Fq(u)1735 1171 y Fm(1)1774 1156 y Fp(jj)1830 1171 y Fm(8)1923 1156 y Ft(+)g Fp(j)p Fq(\025)p Fp(j)2184 1039 y Fl(Z)2265 1065 y Fn(t)2229 1228 y Fm(0)2312 1156 y Fp(jj)p Fq(E)2440 1171 y Fm(1)2479 1156 y Ft(\()p Fq(t)22 b Fp(\000)h Fq(s)p Ft(\))p Fq(a)2809 1115 y Fm(3)2848 1156 y Ft(\()p Fq(s)p Ft(\))p Fo(P)3047 1171 y Fh(c)3087 1156 y Fq(')3151 1115 y Fm(3)3190 1156 y Fp(jj)3246 1171 y Fm(8)3317 1156 y Fq(ds)55 b Ft(+)f Fp(jj)p Fq(\021)3703 1171 y Fm(3)3742 1156 y Fp(jj)3798 1171 y Fm(8)3837 1156 y Fq(:)-74 1391 y Ft(Using)32 b(Theorem)h(2.3)f(and)h(the)g(b)s(ound)g (\(7.42\))f(w)m(e)h(get)-74 1612 y Fo(Prop)s(osition)j(7.6.)388 1847 y Fp(jj)p Fq(\021)t Fp(jj)552 1862 y Fm(8)673 1847 y Fp(\024)116 b Fq(C)936 1862 y Fn(')1035 1751 y Fl(\020)1117 1847 y Fp(k)p Fo(u)1229 1862 y Fm(0)1268 1847 y Fp(k)1318 1862 y Fh(X)1438 1847 y Ft(+)55 b([)p Fq(A)p Ft(])1696 1806 y Fm(3)1696 1872 y(1)p Fn(=)p Fm(4)674 2022 y Ft(+)116 b([)p Fq(A)p Ft(])993 1981 y Fm(2)993 2046 y(1)p Fn(=)p Fm(4)1103 2022 y Fp(f)32 b Ft([)p Fq(\021)t Ft(])1291 2037 y Fm(8)p Fn(;)p Fm(3)p Fn(=)p Fm(4)1511 2022 y Ft(+)54 b Fp(k)p Fo(u)1753 2037 y Fm(0)1793 2022 y Fp(k)1843 2037 y Fh(X)1963 2022 y Ft(+)g([)p Fq(A)p Ft(])2220 1981 y Fm(3)2220 2046 y(1)p Fn(=)p Fm(4)2385 2022 y Ft(+)22 b([)p Fq(B)5 b(\021)2637 2037 y Fm(3)2677 2022 y Ft(])2704 2037 y Fm(4)p Fn(;)p Fm(1)p Fn(=)p Fm(4+)p Fn(\033)2959 2046 y Fj(0)2999 2022 y Fp(g)674 2196 y Ft(+)116 b([)p Fq(A)p Ft(])993 2212 y Fm(1)p Fn(=)p Fm(4)1136 2196 y Fp(f)32 b Ft([)p Fq(\021)t Ft(])1324 2155 y Fm(2)1324 2221 y(8)p Fn(;)p Fm(3)p Fn(=)p Fm(4)1543 2196 y Ft(+)1674 2100 y Fl(h)1713 2196 y Fp(k)p Fo(u)1825 2211 y Fm(0)1865 2196 y Fp(k)1915 2211 y Fh(X)2035 2196 y Ft(+)54 b([)p Fq(A)p Ft(])2292 2155 y Fm(3)2292 2221 y(1)p Fn(=)p Fm(4)2457 2196 y Ft(+)h([)p Fq(B)5 b(\021)2742 2211 y Fm(3)2781 2196 y Ft(])2808 2212 y Fm(4)p Fn(;)p Fm(1)p Fn(=)p Fm(4+)p Fn(\033)3063 2221 y Fj(0)3103 2100 y Fl(i)3191 2196 y Ft([)p Fq(\021)t Ft(])3297 2212 y Fm(8)p Fn(;)p Fm(3)p Fn(=)p Fm(4)3462 2196 y Fp(g)674 2380 y Ft(+)126 b Fq(C)23 b Ft(\()p Fp(jj)p Fq(\021)t Fp(jj)1171 2395 y Fm(1)p Fn(;)p Fm(2)1264 2380 y Fq(;)17 b(')p Ft(\))49 b([)p Fq(\021)t Ft(])1565 2329 y Fm(5)p Fn(=)p Fm(3)1565 2409 y(8)p Fn(;)p Fm(3)p Fn(=)p Fm(4)1762 2283 y Fl(\021)1861 2380 y Fp(h)p Fq(t)p Fp(i)1974 2339 y Fk(\000)p Fm(3)p Fn(=)p Fm(4)2138 2380 y Fq(:)1560 b Ft(\(7.49\))-74 2707 y(The)34 b(follo)m(wing)c(prop)s(osition)g(summarizes)i(our)g(lab)s (ors.)-74 2948 y Fo(Prop)s(osition)k(7.7.)49 b Fi(F)-8 b(or)32 b(an)m(y)h Fq(T)41 b(>)28 b Ft(0)p Fi(:)370 3184 y Ft([)p Fq(\021)t Ft(])476 3199 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Ft(,)e(whic)m(h)h(is)f(indep)s(enden)m (t)i(of)e Fq(T)14 b Ft(,)33 b(suc)m(h)h(that)e(for)g(all)f Fq(T)1677 457 y(M)10 b Ft(\()p Fq(T)k Ft(\))28 b Fp(\024)h Fq(M)2156 472 y Fk(\003)2195 457 y Fq(:)1503 b Ft(\(7.52\))-74 706 y(This)33 b(completes)f(the)h(pro)s(of)f(of)g(Theorem)h(1.1.)-74 1105 y Fs(8.)90 b(Summary)45 b(and)g(discussion)-74 1354 y Ft(W)-8 b(e)31 b(ha)m(v)m(e)g(considered)g(a)f(class)g(of)g (nonlinear)f(Klein-Gordon)e(equations,)k(\(1.1\),)f(whic)m(h)h(are)f(p) s(erturbations)-74 1503 y(of)35 b(a)h(linear)e(disp)s(ersiv)m(e)i (equation)f(whic)m(h)h(has)g(a)g(time)e(p)s(erio)s(dic)g(and)i (spatially)d(lo)s(calized)h(\(b)s(ound)h(state\))-74 1652 y(solution.)42 b(The)31 b(unp)s(erturb)s(ed)h(and)f(p)s(erturb)s (ed)h(dynamical)d(systems)j(are)f(Hamiltonian.)39 b(W)-8 b(e)31 b(ha)m(v)m(e)h(sho)m(wn)-74 1802 y(that)h(if)f(a)g(nonlinear)g (resonance)i(condition)d(\(1.8\))i(holds,)f(then)i(solutions)e(with)g (su\016cien)m(tly)i(small)d(initial)-74 1951 y(data)23 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y Fq(u)60 b Ft(=)g(0)p Fq(:)1266 b Ft(\(8.1\))-74 4197 y(Results)29 b(on)f(this)g(are)g(presen)m(ted)j(in)c Fp(x)p Ft(2.)42 b(Roughly)28 b(sp)s(eaking,)h(if)e(the)i(initial)24 b(data)k(has)h(a)f(su\016cien)m(t)h(n)m(um)m(b)s(er)-74 4346 y(of)37 b(deriv)-5 b(ativ)m(es)37 b(in)g Fq(L)717 4310 y Fn(p)757 4346 y Fq(;)54 b Ft(1)36 b Fq(<)f(p)h Fp(\024)h Ft(2,)h(then)g(the)g(solution)e(deca)m(ys)j(at)e(a)g(rate)g Fp(O)s Ft(\()p Fq(t)3005 4296 y Fk(\000)p Fn(n)p Fm(\()3140 4269 y Fj(1)p 3141 4281 31 4 v 3141 4322 a(2)3181 4296 y Fk(\000)3258 4269 y Fj(1)p 3246 4281 55 4 v 3246 4326 a Fg(p)3278 4312 y Fb(0)3310 4296 y Fm(\))3342 4346 y Ft(\))g(in)g Fq(L)3602 4310 y Fn(p)3638 4287 y Fb(0)3664 4346 y Ft(.)58 b(Here,)-74 4496 y Fq(p)-25 4460 y Fk(\000)p Fm(1)92 4496 y Ft(+)22 b(\()p Fq(p)277 4460 y Fk(0)300 4496 y Ft(\))338 4460 y Fk(\000)p Fm(1)460 4496 y Ft(=)27 b(1.)72 4645 y(\(1b\))33 b Fr(T)-7 b(r)i(anslation)33 b(invariant)h(nonline)-5 b(ar)34 b(Klein)g(Gor)-5 b(don)35 b(e)-5 b(quation)32 b Ft(\()p Fq(V)49 b Fp(\021)28 b Ft(0)33 b(and)f Fq(\025)c Fp(6)p Ft(=)f(0\):)-74 4795 y(F)-8 b(or)34 b(small)f(initial)e(conditions)j(it)g(has)h(b)s(een)h (sho)m(wn)g(that)f(solutions)e(deca)m(y)k(at)d(free)i(disp)s(ersiv)m(e) f(rates;)i(see)-74 4944 y([65])32 b(and)h(references)i(cited)d (therein.)1901 5306 y(51)p eop %%Page: 52 52 52 51 bop 72 59 a Ft(\(1c\))43 b Fr(Line)-5 b(ar)43 b(Klein-Gor)-5 b(don)43 b(e)-5 b(quation)43 b(with)h(a)f(p)-5 b(otential)44 b(having)e(a)i(b)-5 b(ound)43 b(state,)j(as)e(hyp)-5 b(othesize)g(d)-74 208 y Ft(\()p Fq(V)59 b Fp(6)p Ft(=)37 b(0)h(and)h Fq(\025)e Ft(=)h(0\):)54 b(By)39 b(the)g(sp)s(ectral)f (theorem,)i(a)e(t)m(ypical)f(solution)g(will)f(decomp)s(ose)j(in)m(to)e (a)h(linear)-74 407 y(sup)s(erp)s(osition)31 b(of)h(\(i\))f(a)h(b)s (ound)g(state)h(part,)f(of)g(the)h(form:)42 b Fq(R)2253 422 y Fm(0)2309 407 y Ft(cos)q(\(\012)p Fq(t)22 b Ft(+)f Fq(\032)2752 422 y Fm(0)2791 407 y Ft(\))p Fq(')p Ft(\()p Fq(x)p Ft(\),)33 b(and)f(\(ii\))e(a)i(part)g(whic)m(h)-74 557 y(disp)s(erses)i(to)e(zero)h(at)f(free)h(disp)s(ersiv)m(e)g(rates.) 72 706 y(\(1d\))c Fr(Nonline)-5 b(ar)31 b(Klein-Gor)-5 b(don)30 b(e)-5 b(quation)31 b(with)g(a)g(p)-5 b(otential)31 b(having)g(a)g(b)-5 b(ound)31 b(state,)h(as)f(hyp)-5 b(othesize)g(d)-74 856 y Ft(\()p Fq(V)65 b Fp(6)p Ft(=)43 b(0)e(and)h Fq(\025)h Fp(6)p Ft(=)f(0\):)62 b(Whereas)42 b(the)h(deca)m(ying)f(part)f(of)g(the)h(solution)e(in)h(the)h(ab)s(o)m (v)m(e)g(cases)h(tends)g(to)-74 1005 y(zero)34 b(at)g(a)g(free)g(disp)s (ersiv)m(e)h(rate,)f(Theorem)h(1.1)e(implies)f(that)h(the)i(deca)m(y)g (rate)f(is)g(anomalously)e(slo)m(w.)47 b(In)-74 1155 y(particular,)25 b(the)h(deca)m(y)h(rate)e(obtained)g(in)f Fq(L)1581 1118 y Fm(8)1646 1155 y Ft(is)h Fp(O)s Ft(\()p Fq(t)1892 1118 y Fk(\000)1957 1091 y Fj(1)p 1957 1103 31 4 v 1957 1144 a(4)2002 1155 y Ft(\),)i(while)d(the)i(free)g(disp)s (ersiv)m(e)f(rate)h(in)e Fq(L)3488 1118 y Fm(8)3553 1155 y Ft(is)h Fp(O)s Ft(\()p Fq(t)3799 1118 y Fk(\000)3864 1091 y Fj(9)p 3864 1103 V 3864 1144 a(8)3909 1155 y Ft(\).)-74 1304 y(This)37 b(slo)m(w)h(rate)f(of)g(deca)m(y)-8 b(,)39 b(due)f(to)f(the)h(nonlinear)e(resonan)m(t)i(in)m(teractions)e(giv)m(e) h(rise)g(to)g(a)g(long-liv)m(ed)e(or)-74 1453 y Fr(metastable)f(states) p Ft(.)-74 1603 y Fo(2.)50 b(Disp)s(ersiv)m(e)37 b(Hamiltonian)d (normal)j(form)72 1752 y Ft(In)43 b(a)e(\014nite)h(dimensional)d (Hamiltonian)f(system)43 b(of)e(one)h(degree)h(of)e(freedom,)j(the)f (general)e(normal)-74 1902 y(form)31 b(is:)752 2051 y Fq(A)825 2010 y Fk(0)877 2051 y Ft(=)c Fq(i)1030 1955 y Fl(\020)1080 2051 y Fq(c)1122 2066 y Fm(10)1218 2051 y Ft(+)22 b Fq(c)1358 2066 y Fm(21)1433 2051 y Fp(j)p Fq(A)p Fp(j)1562 2010 y Fm(2)1623 2051 y Ft(+)g Fq(c)1763 2066 y Fm(32)1838 2051 y Fp(j)p Fq(A)p Fp(j)1967 2010 y Fm(4)2028 2051 y Ft(+)g Fq(:::)g Ft(+)g Fq(c)2369 2066 y Fn(n)p Fm(+1)p Fn(;n)2569 2051 y Fp(j)p Fq(A)p Fp(j)2698 2010 y Fm(2)p Fn(n)2802 2051 y Ft(+)g Fq(:::)2981 1955 y Fl(\021)3047 2051 y Fq(A;)626 b Ft(\(8.2\))-74 2255 y(where)25 b Fq(c)241 2270 y Fn(n;n)p Fm(+1)465 2255 y Ft(are)f(real)f(n)m(um)m(b)s(ers.)42 b(In)24 b(the)h(presen)m(t)g (con)m(text)h(w)m(e)f(ha)m(v)m(e)g(the)g Fr(disp)-5 b(ersive)25 b(Hamiltonian)i(normal)-74 2404 y(form)760 2553 y Fq(A)833 2512 y Fk(0)884 2553 y Ft(=)988 2457 y Fl(\020)1037 2553 y Fq(k)1088 2568 y Fm(10)1185 2553 y Ft(+)22 b Fq(k)1334 2568 y Fm(21)1408 2553 y Fp(j)p Fq(A)p Fp(j)1537 2512 y Fm(2)1598 2553 y Ft(+)g Fq(k)1747 2568 y Fm(32)1822 2553 y Fp(j)p Fq(A)p Fp(j)1951 2512 y Fm(4)2012 2553 y Ft(+)g Fq(:::)g Ft(+)g Fq(k)2362 2568 y Fn(n)p Fm(+1)p Fn(;n)2562 2553 y Fp(j)p Fq(A)p Fp(j)2691 2512 y Fm(2)p Fn(n)2794 2553 y Ft(+)g Fq(:::)2973 2457 y Fl(\021)3040 2553 y Fq(A;)633 b Ft(\(8.3\))-74 2757 y(where)1437 2906 y Fq(k)1488 2921 y Fn(n)p Fm(+1)p Fn(;n)1715 2906 y Ft(=)27 b Fq(d)1869 2921 y Fn(n)p Fm(+1)p Fn(;n)2091 2906 y Ft(+)22 b Fq(ic)2264 2921 y Fn(n)p Fm(+1)p Fn(;n)-74 3110 y Ft(are,)52 b(in)47 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b(The)-74 4604 y(corresp)s(onding)33 b(deca)m(y)h(rate)e(w)m(ould)h (then)g(b)s(e)g(slo)m(w)m(er,)g(sp)s(eci\014cally)f Fp(O)s Ft(\()p Fp(j)p Fq(t)p Fp(j)2689 4559 y Fk(\000)2790 4532 y Fj(1)p 2753 4544 105 4 v 2753 4585 a(2)p Fg(n)2821 4593 y Fb(\003)2872 4604 y Ft(\).)72 4754 y(F)-8 b(rom)32 b(this)g(p)s(ersp)s(ectiv)m(e)i(one)f(ma)m(y)f(exp)s(ect)i(the)f (existence)h(of)f(breather)g(solutions)e(of)i(in)m(tegrable)e(non-)-74 4903 y(linear)39 b(\015o)m(ws)j(lik)m(e)d(the)i(sine-Gordon)f(equation) g(or)g(the)h(extremely)f(long-liv)m(ed)f(breather)i(-lik)m(e)e(states)i (of)-74 5052 y(the)34 b Fq(\036)153 5016 y Fm(4)226 5052 y Ft(-)f(mo)s(del)f(as)i(corresp)s(onding)f(to)h(the)g(case)g(where)h (to)e(all)f(orders)i(the)g(co)s(e\016cien)m(ts)h Fq(d)3377 5067 y Fn(n)p Fm(+1)p Fn(;n)3610 5052 y Ft(are)e(zero.)1901 5306 y(52)p eop %%Page: 53 53 53 52 bop -74 59 a Ft(Do)s(es)30 b(the)g(v)-5 b(anishing)29 b(of)g(all)f(suc)m(h)k(co)s(e\016cien)m(ts)f(ha)m(v)m(e)g(an)f(in)m (terpretation)f(in)g(terms)h(of)f(the)h(in\014nitely)f(man)m(y)-74 208 y(time-in)m(v)-5 b(arian)m(ts)30 b(for)i(the)h(in)m(tegrable)f (\015o)m(w?)-74 358 y Fo(3.)47 b(Multiple)25 b(b)s(ound)k(state)f (problems:)38 b Ft(Of)24 b(in)m(terest)h(are)f(problems)f(where)i(the)g (underlying)e(p)s(oten)m(tial,)-74 507 y Fq(V)f Ft(\()p Fq(x)p Ft(\),)36 b(supp)s(orts)g(more)e(than)i(one)f(b)s(ound)h(state.) 51 b(What)36 b(is)e(the)i(large)e(time)g(b)s(eha)m(vior)h(of)g(suc)m(h) h(systems?)-74 656 y(Our)23 b(analysis)g(and)g(the)g(ab)s(o)m(v)m(e)h (remarks)g(suggest)g(that)f(a)g(more)f(general)h(normal)e(form)h(could) g(b)s(e)i(dev)m(elop)s(ed,)-74 806 y(and)h(the)f(corresp)s(onding)h(\() f(in)f(general)h(slo)m(w)m(er\))h(deca)m(y)h(rates)e(an)m(ticipated.)40 b(Is)25 b(it)f(p)s(ossible)f(that)h(a)g(resonance)-74 955 y(among)45 b(the)h(m)m(ultiple)d("discrete)j(oscillators")e(can)i (b)s(e)g(arranged)g(so)g(that)f(one)h(gets)h(a)e(p)s(ersistence)i(of) -74 1105 y(nondeca)m(ying)33 b(solutions,)f(or)g(do)s(es)h(radiation)e (to)h(the)h(con)m(tin)m(uum)f(alw)m(a)m(ys)h(win)f(out?)-74 1254 y Fo(4.)48 b(Relation)29 b(to)i(cen)m(ter)g(manifold)f(theory:)41 b Ft(Our)27 b(program)e(of)i(decomp)s(osing)f(the)i(original)23 b(conserv)-5 b(a-)-74 1404 y(tiv)m(e)25 b(dynamical)e(system)j(in)m(to) f(a)f(\014nite)h(dimensional)d(dynamical)i(system)i(\(1.20\))e(whic)m (h)h(is)g(w)m(eakly)h(coupled)-74 1553 y(to)i(an)g(in\014nite)f (dimensional)f(dynamical)g(system)k(is)d(one)i(commonly)d(used)j(in)f (dissipativ)m(e)g(dynamical)e(sys-)-74 1703 y(tems.)52 b(There,)37 b(it)d(is)h(often)g(p)s(ossible)g(using)g(the)g(cen)m(ter)i (manifold)c(approac)m(h)i([14],)h([67],)g(to)f(construct)h(an)-74 1852 y(in)m(v)-5 b(arian)m(t)34 b(cen)m(ter-stable)i(manifold.)48 b(The)36 b(dynamics)f(in)g(a)g(neigh)m(b)s(orho)s(o)s(d)f(of)g(an)h (equilibrium)e(p)s(oin)m(t)h(are)-74 2001 y(c)m(haracterized)k(b)m(y)g (an)g(exp)s(onen)m(tially)e(fast)h(con)m(traction)g(on)g(to)g(the)h (stable-cen)m(ter)g(manifold.)55 b(The)38 b(con-)-74 2151 y(traction)33 b(is)h(exp)s(onen)m(tial)g(b)s(ecause)h(the)g(part)f (of)f(the)i(linearized)e(sp)s(ectrum)h(asso)s(ciated)g(with)g(the)h (in\014nite)-74 2300 y(dimensional)e(part)i(of)g(the)h(dynamics)f(is)g (con)m(tained)h(in)f(the)h(left)e(half)h(plane.)51 b(In)36 b(the)g(curren)m(t)h(con)m(text)f(of)-74 2450 y(conserv)-5 b(ativ)m(e)34 b(dynamical)d(systems,)j(the)f(linearization)c(ab)s(out)j (the)h(equilibrium)d(p)s(oin)m(t)i(\(here)h Fq(u)27 b Fp(\021)i Ft(0\))j(lies)-74 2599 y(on)25 b(the)g(imaginary)d(axis;)27 b(in)d(particular,)h(t)m(w)m(o)h(complex)e(conjugate)h(eigen)m(v)-5 b(alues)24 b Fp(\006)p Fq(i)p Ft(\012)i(and)f(con)m(tinous)g(sp)s(ec-) -74 2749 y(trum)34 b(from)f Fp(\006)p Fq(im)i Ft(to)f Fp(\006)p Fq(i)p Fp(1)p Ft(.)48 b(The)36 b(analogue)d(of)g(dissipation) g(is)g(the)i(mec)m(hanism)f(of)f(disp)s(ersiv)m(e)i(radiation)-74 2898 y(of)d(energy)-8 b(,)34 b(related)e(to)g(the)h(con)m(tin)m(uous)g (sp)s(ectrum,)h(and)e(the)h(asso)s(ciated)g(algebraic)e(time-deca)m(y) -8 b(.)72 3047 y(There)29 b(is)e(recen)m(t)i(w)m(ork)f(on)f(the)h (application)d(of)i(cen)m(ter)h(manifold)d(metho)s(ds)i(to)g(certain)g (sp)s(ecial)f(conser-)-74 3197 y(v)-5 b(ativ)m(e)30 b(dynamical)e (systems)j(of)e(nonlinear)f(Sc)m(hr\177)-49 b(odinger)30 b(t)m(yp)s(e;)i(see)f(the)f(cen)m(ter)h(manifold)c(analysis)i(of)g ([46])-74 3346 y(applied)d(to)h(problem)f(studied)i(b)m(y)g(the)g (authors)f(in)g([60)o(].)42 b(It)28 b(w)m(ould)f(b)s(e)g(of)g(in)m (terest)h(to)f(understand)h(whether)-74 3496 y(geometric)g(insigh)m(t)f (on)i(the)g(structure)h(of)f(the)g(phase)g(space)h(for)e(problems)g(of) h(the)g(t)m(yp)s(e)h(considered)f(in)f(this)-74 3645 y(pap)s(er)33 b(can)g(b)s(e)f(obtained)g(using)h(the)g(ideas)f(of)g(in) m(v)-5 b(arian)m(t)31 b(manifold)f(theory)-8 b(.)-74 3795 y Fo(5.)104 b(Systems)56 b(with)e(disorder:)75 b Ft(In)48 b(this)g(pap)s(er,)53 b(w)m(e)c(ha)m(v)m(e)h(seen)f(the)g (e\013ect)g(of)f(a)g(single)f(lo)s(calized)-74 3944 y(defect)34 b(on)f(the)g(w)m(a)m(v)m(e)i(propagation)c(dynamics)i(in)f(a)h (nonlinear)e(system.)45 b(Of)33 b(great)g(in)m(terest)g(w)m(ould)g(b)s (e)g(an)-74 4094 y(understanding)f(of)f(the)g(e\013ects)i(of)e(a)g (spatially)e(random)h(distribution)g(of)g(defects)j(mo)s(deled,)e(for)f (example,)-74 4243 y(b)m(y)47 b(random)e(p)s(oten)m(tial)f Fq(V)21 b Ft(\()p Fq(x)p Ft(\))46 b(on)g(the)g(lo)s(calization)c(of)k (energy)g(in)f(nonlinear)g(systems)i(suc)m(h)g(as)f(\(1.1\).)-74 4392 y(Related)32 b(questions)i(are)e(studied)h(in)f([22],)g([25],)h ([10)o(].)1901 5306 y(53)p eop %%Page: 54 54 54 53 bop -74 59 a Fs(References)-25 307 y Ft([1])49 b(S.)28 b(Agmon,)g(Lectures)h(on)e(Exp)s(onen)m(tial)h(Deca)m(y)g(of)f (Solutions)g(of)g(Second-Order)i(Elliptic)24 b(Equations:)127 456 y(Bounds)40 b(on)g(Eigenfunctions)f(of)g(N-Bo)s(dy)h(Sc)m(hr\177) -49 b(odinger)40 b(Op)s(erators,)h(Princeton)f(Univ)m(ersit)m(y)g (Press)127 606 y(\(1982\))-25 838 y([2])49 b(P)-8 b(.C.)33 b(Aic)m(helburg)f(&)g(R.)g(Beig,)g Fr(R)-5 b(adiation)34 b(damping)g(as)g(an)g(initial)h(value)f(pr)-5 b(oblem)p Ft(,)32 b(Ann.)h(Ph)m(ys.)h Fo(98)127 988 y Ft(\(1976\))d(264{283.)-25 1220 y([3])49 b(V.I.)60 b(Arnol'd,)67 b(Geometric)58 b(Metho)s(ds)j(in)f(the)g(Theory)i(of)d(Ordinary)h(Di\013eren)m(tial)d (Equations,)127 1369 y(Springer-V)-8 b(erlag,)30 b(New)k(Y)-8 b(ork)33 b(\(1983\))-25 1602 y([4])49 b(J.)23 b(Bourgain,)h Fr(Quasi-p)-5 b(erio)g(dic)25 b(solutions)h(of)g(Hamiltonian)g(p)-5 b(erturb)g(ations)26 b(of)g(2D)g(line)-5 b(ar)25 b(Schr\177)-50 b(odinger)127 1751 y(e)-5 b(quations)p Ft(,)32 b(preprin)m(t)g(1994.) -25 1984 y([5])49 b(P)-8 b(.)45 b(Brenner,)k Fr(On)c(the)i(existenc)-5 b(e)45 b(of)h(glob)-5 b(al)45 b(smo)-5 b(oth)46 b(solutions)f(of)h(c)-5 b(ertain)46 b(semiline)-5 b(ar)45 b(hyp)-5 b(erb)g(olic)127 2133 y(e)g(quations)p Ft(,)44 b(Math.)e(Z.)g Fo(167)h Ft(\(1979\))e(99-135;)k Fr(On)f(sc)-5 b(attering)43 b(and)g(everywher) -5 b(e)43 b(de\014ne)-5 b(d)43 b(sc)-5 b(attering)127 2283 y(op)g(er)g(ators)34 b(for)h(nonline)-5 b(ar)33 b(Klein-Gor)-5 b(don)34 b(e)-5 b(quations)p Ft(,)32 b(J.)h(Di\013.)e (Eqns.)j Fo(56)f Ft(\(1985\))e(310-344.)-25 2515 y([6])49 b(H.)d(Brezis,)k Fr(Perio)-5 b(dic)47 b(solutions)f(of)i(nonline)-5 b(ar)46 b(vibr)-5 b(ating)46 b(strings)h(and)g(duality)h(principles)p Ft(,)g(Bull.)127 2665 y(Amer.)32 b(Math.)h(So)s(c.)g(\(N.S.\))f Fo(8)h Ft(\(1983\))e(409-426.)-25 2897 y([7])49 b(L.)34 b(Bonilla)e(&)i(J.B.)h(Keller,)e Fr(Irr)-5 b(eversibility)36 b(and)g(nonr)-5 b(e)g(curr)g(enc)g(e)p Ft(,)34 b(J.)h(Stat.)f(Ph)m(ys.) i Fo(42)f Ft(\(1986\))e(1115-)127 3046 y(1125.)-25 3279 y([8])49 b(B.)42 b(Birnir,)i(H.P)-8 b(.)43 b(McKean)h(&)e(A.)h(W)-8 b(einstein,)45 b Fr(The)f(rigidity)g(of)g(sine-Gor)-5 b(don)43 b(br)-5 b(e)g(athers)p Ft(,)45 b(Comm.)127 3428 y(Pure)33 b(Appl.)f(Math.)h Fo(47)g Ft(\(1994\),)e(1043{1051.)-25 3661 y([9])49 b(B.)32 b(Birnir,)e Fr(Qualitative)k(analysis)g(of)f(r)-5 b(adiating)34 b(br)-5 b(e)g(athers)p Ft(,)32 b(Comm.)e(Pure)j(Appl.)f (Math.)g Fo(47)g Ft(\(1994\),)127 3810 y(103{119.)-74 4043 y([10])49 b(J.C.)g(Bronski,)k Fr(A)d(numeric)-5 b(al)49 b(study)h(of)f(solitary)h(wave)f(pr)-5 b(op)g(agation)48 b(in)i(a)f(disor)-5 b(der)g(e)g(d)49 b(me)-5 b(dium)p Ft(,)127 4192 y(preprin)m(t.)-74 4425 y([11])49 b(V.S.)36 b(Buslaev)g(&)f(G.S.)h(P)m(erel'man,)g Fr(Sc)-5 b(attering)38 b(for)f(the)h(nonline)-5 b(ar)36 b(Schr\177)-50 b(odinger)37 b(e)-5 b(quation:)50 b(states)127 4574 y(close)34 b(to)h(a)f(soliton)p Ft(,)f(St.)f(P)m(etersburg)j(Math.)e(J.)f Fo(4)h Ft(\(1993\))e (1111{1142.)-74 4807 y([12])49 b(V.S.)26 b(Buslaev)h(&)f(G.S.)h(P)m (erel'man,)g Fr(On)i(the)g(stability)g(of)g(solitary)g(waves)f(for)h (nonline)-5 b(ar)28 b(Schr\177)-50 b(odinger)127 4956 y(e)-5 b(quation)p Ft(,)32 b(preprin)m(t.)1901 5306 y(54)p eop %%Page: 55 55 55 54 bop -74 59 a Ft([13])49 b(D.K.)32 b(Campb)s(ell)e(&)j(M.)g(P)m (eyrard,)h Fr(Solitary)h(wave)f(c)-5 b(ol)5 b(lisions)33 b(r)-5 b(evisite)g(d)p Ft(,)32 b(Ph)m(ysica)i(D)e Fo(18)g Ft(\(1986\))g(47)-74 291 y([14])49 b(J.)32 b(Carr,)h(Applications)e(of) h(Cen)m(tre)i(Manifold)d(Theory)-8 b(,)34 b(Springer-V)-8 b(erlag,)30 b(New)k(Y)-8 b(ork)32 b(\(1981\))-74 524 y([15])49 b(E.)39 b(Co)s(ddington)f(&)h(N.)g(Levinson,)i(Theory)e(of)g (Ordinary)f(Di\013eren)m(tial)e(Equations,)41 b(McGra)m(w-Hill,)127 673 y(New)33 b(Y)-8 b(ork,)33 b(1955.)-74 906 y([16])49 b(J.M.)34 b(Coron,)g Fr(Perio)-5 b(de)35 b(minimale)g(p)-5 b(our)36 b(une)f(c)-5 b(or)g(de)36 b(vibr)-5 b(ante)35 b(de)h(longeur)f(in\014nie)p Ft(,)e(C.R.)h(Acad.)g(Sci.)127 1055 y Fo(A294)e Ft(\(1982\))f(127-129)-74 1287 y([17])49 b(W.)34 b(Craig)f(&)h(C.E.)h(W)-8 b(a)m(yne,)36 b Fr(Newton)-10 b('s)36 b(metho)-5 b(d)36 b(and)g(p)-5 b(erio)g(dic)35 b(solutions)h(of)g(nonline)-5 b(ar)36 b(wave)f(e)-5 b(qua-)127 1437 y(tions)p Ft(,)32 b(Comm.)f(Pure)j(Appl.)e(Math.)h Fo(46)g Ft(\(1993\),)e(1409{1498.)-74 1669 y([18])49 b(H.L.)32 b(Cycon,)i(R.G.)e(F)-8 b(ro)s(ese,)32 b(W.)h(Kirsc)m(h)f(&)g (B.)h(Simon,)e(Sc)m(hr\177)-49 b(odinger)32 b(Op)s(erators,)h (Springer-V)-8 b(erlag,)127 1819 y(Berlin)31 b(Heidelb)s(erg)h(New)h(Y) -8 b(ork,)33 b(1987)-74 2051 y([19])49 b(J.)e(Denzler,)52 b Fr(Nonp)-5 b(ersistenc)g(e)47 b(of)i(br)-5 b(e)g(ather)48 b(families)g(for)g(the)h(p)-5 b(erturb)g(e)g(d)49 b(Sine-Gor)-5 b(don)47 b(e)-5 b(quation)p Ft(,)127 2201 y(Comm)m(un.)32 b(Math.)h(Ph)m(ys.)h Fo(158)f Ft(\(1993\))e(397{430)-74 2433 y([20])49 b(S.)44 b(Debi)m(\023)-46 b(evre,)47 b(P)-8 b(.D.)44 b(Hislop)f(&)g(I.M.)i(Sigal,)g Fr(Sc)-5 b(attering)45 b(the)-5 b(ory)45 b(for)h(the)f(wave)g(e)-5 b(quation)45 b(on)g(non-)127 2583 y(c)-5 b(omp)g(act)34 b(manifolds)p Ft(,)d(Rev.)i(Math.)g(Ph)m(ys.)h Fo(4)f Ft(\(1992\))e(575{618.)-74 2815 y([21])49 b(G.)36 b(F)-8 b(ord,)38 b(M.)f(Kac)g(&)g(P)-8 b(.)37 b(Mazur,)i Fr(Statistic)-5 b(al)39 b(pr)-5 b(op)g(erties)38 b(of)h(assemblies)e(of)h(c)-5 b(ouple)g(d)39 b(oscil)5 b(lators)36 b Ft(J.)127 2964 y(Math.)d(Ph)m(ys.)h Fo(6)f Ft(\(1965\))e(504{515)-74 3197 y([22])49 b(J.)f(F)-8 b(r\177)-49 b(ohlic)m(h,)52 b(T.)d(Sp)s(encer)h(&)e(C.E.)i(W)-8 b(a)m(yne,)54 b Fr(L)-5 b(o)g(c)g(alization)49 b(in)g(disor)-5 b(der)g(e)g(d)49 b(nonline)-5 b(ar)48 b(dynamic)-5 b(al)127 3346 y(systems)p Ft(,)32 b(J.)h(Stat.)f(Ph)m(ys.)j Fo(42)d Ft(\(1986\))g(247{274.)-74 3579 y([23])49 b(J.)37 b(Ginibre)e(&)i(G.)g (V)-8 b(elo,)38 b Fr(Sc)-5 b(attering)39 b(the)-5 b(ory)39 b(in)g(the)g(ener)-5 b(gy)39 b(sp)-5 b(ac)g(e)38 b(for)h(a)g(class)f (of)h(nonline)-5 b(ar)38 b(wave)127 3728 y(e)-5 b(quations)p Ft(,)32 b(Comm.)f(Math.)i(Ph)m(ys.)i Fo(123)d Ft(\(1989\))g(535{573.) -74 3961 y([24])49 b(M.)35 b(Grillakis,)c(J.)k(Shatah)f(&)h(W.)f (Strauss,)i Fr(Stability)h(the)-5 b(ory)37 b(of)f(solitary)h(waves)f (in)g(the)h(pr)-5 b(esenc)g(e)35 b(of)127 4110 y(symmetry,)g(I)p Ft(,)d(J.)g(F)-8 b(unc.)33 b(Anal.)f Fo(74)h Ft(\(1987\))e(160{197.)-74 4343 y([25])49 b(S.A.)c(Gredeskul,)k(Y.S.)c(Kivshar,)j(A.)e(Sanc)m(hez) g(&)f(L.)g(V)-8 b(azquez,)50 b Fr(L)-5 b(o)g(c)g(alization)45 b(de)-5 b(c)g(ay)46 b(induc)-5 b(e)g(d)46 b(by)127 4492 y(str)-5 b(ong)34 b(nonline)-5 b(arity)35 b(in)f(disor)-5 b(der)g(e)g(d)34 b(systems)p Ft(,)e(Ph)m(ys.)j(Rev.)e(Lett.)g Fo(64)f Ft(\(1990\))g(1693{1696.)-74 4725 y([26])49 b(J.)22 b(Guc)m(k)m(enheimer)h(&)f(P)-8 b(.)22 b(Holmes,)h(Nonlinear)e (Oscillations,)g(Dynamical)f(Systems,)25 b(and)d(Bifurcations)127 4874 y(of)32 b(V)-8 b(ector)33 b(Fields,)e(Springer-V)-8 b(erlag)31 b(\(1983\))1901 5306 y(55)p eop %%Page: 56 56 56 55 bop -74 59 a Ft([27])49 b(C.)40 b(G)m(\023)-46 b(erard)39 b(&)h(I.M.)g(Sigal,)f Fr(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 Ft(,)40 b(Comm)m(un.)f(Math.)127 208 y(Ph)m(ys.)34 b Fo(145)f Ft(\(1992\))e(281{328.)-74 441 y([28])49 b(W.)33 b(Hunzik)m(er)g(&)g(I.M.)g(Sigal,)e Fr(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 Ft(,)127 590 y(ETH-Z)s(\177)-51 b(uric)m(h)31 b(preprin)m(t)i(\(1997\)) -74 822 y([29])49 b(W.)26 b(Hunzik)m(er,)i(I.M.)f(Sigal)d(&)i(A.)g (So\013er,)i Fr(Minimal)g(esc)-5 b(ap)g(e)28 b(velo)-5 b(cities)p Ft(,)27 b(ETH-Z)s(\177)-51 b(uric)m(h)25 b(preprin)m(t)h (\(1997\))-74 1055 y([30])49 b(V.)c(Jaksi)m(\023)-46 b(c)46 b(&)f(C.-A.)h(Pillet,)h Fr(On)f(a)g(mo)-5 b(del)46 b(for)h(quantum)f(friction.)h(I.)f(F)-7 b(ermi's)45 b(golden)h(rule)g (and)127 1204 y(dynamics)34 b(at)h(zer)-5 b(o)34 b(temp)-5 b(er)g(atur)g(e)p Ft(,)33 b(Ann.)g(Inst.)g(H.)g(P)m(oincar)m(\023)-46 b(e)33 b(Ph)m(ys.)h(Th)m(\023)-46 b(eor.)34 b Fo(62)e Ft(\(1995\))g(47{68.)-74 1437 y([31])49 b(A.)33 b(Jensen)i(&)e(T.)h (Kato,)e Fr(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 1586 y(functions)p Ft(,)c(Duk)m(e)h(Math.)g(J.)g Fo(46)f Ft(\(1979\),)g(583{611.)-74 1819 y([32])49 b(G.)39 b(Jona-Lasinio,)h(C.)g(Presilla)e(&)i(J.)g (Sj\177)-49 b(ostrand,)42 b Fr(On)f(Schr\177)-50 b(odinger)40 b(e)-5 b(quations)41 b(with)h(c)-5 b(onc)g(entr)g(ate)g(d)127 1968 y(nonline)g(arities)p Ft(,)31 b(Ann.)i(Ph)m(ys.)h Fo(240)f Ft(\(1995\))-74 2201 y([33])49 b(J.-L.)36 b(Journ)m(\023)-46 b(e,)39 b(A.)d(So\013er)h(&)g(C.D.)g(Sogge,)g Fr(De)-5 b(c)g(ay)39 b(estimates)f(for)g(Schr\177)-50 b(odinger)38 b(op)-5 b(er)g(ators)p Ft(,)37 b(Comm.)127 2350 y(Pure)c(Appl.)f(Math.) h Fo(44)g Ft(\(1991\))e(573{604.)-74 2583 y([34])49 b(S.)44 b(Kic)m(henassam)m(y)-8 b(,)48 b Fr(Br)-5 b(e)g(ather)45 b(solutions)g(of)h(nonline)-5 b(ar)44 b(wave)h(e)-5 b(quations)p Ft(,)46 b(Comm)m(un.)e(Pure)h(Appl.)127 2732 y(Math.)33 b Fo(44)f Ft(\(1991\))g(789)-74 2964 y([35])49 b(Y)-8 b(u.S.)33 b(Kivshar)f(&)g(B.A.)h(Malomed,)f(Rev.)h(Mo)s(d.)g(Ph)m(ys.)h Fo(61)f Ft(\(1989\))e(763)-74 3197 y([36])49 b(Klainerman,)28 b(S.)i Fr(R)-5 b(emark)32 b(on)g(the)g(asymptotic)g(b)-5 b(ehavior)32 b(of)g(the)g(Klein-Gor)-5 b(don)31 b(e)-5 b(quation)33 b(in)f Ft(I)-19 b Fo(R)3807 3155 y Fn(n)p Fm(+1)3944 3197 y Fr(,)127 3346 y Ft(Comm.)31 b(Pure)i(Appl.)g(Math.)g Fo(46)f Ft(\(1993\))g(137{144.)-74 3579 y([37])49 b(S.B.)32 b(Kuksin,)g Fr(Ne)-5 b(arly)35 b(inte)-5 b(gr)g(able)33 b(in\014nite-dimensional)f(Hamiltonian)h(systems)p Ft(,)f(Lecture)h (Notes)g(in)127 3728 y(Math.)g Fo(1556)p Ft(,)f(Springer)h(V)-8 b(erlag,)31 b(Berlin)g(-)h(Heidelb)s(erg)g(-)g(New)i(Y)-8 b(ork,)32 b(1993.)-74 3961 y([38])49 b(H.)34 b(Lam)m(b,)g Fr(On)i(a)h(p)-5 b(e)g(culiarity)36 b(of)g(the)h(wave-system)e(due)i (to)g(the)f(fr)-5 b(e)g(e)36 b(vibr)-5 b(ations)36 b(of)g(a)g (nucleusin)g(an)127 4110 y(extende)-5 b(d)34 b(me)-5 b(dium)p Ft(,)32 b(Pro)s(c.)g(London)h(Math.)g(So)s(c.)f Fo(32)h Ft(\(1900\))f(208-211)-74 4343 y([39])49 b(P)-8 b(.-L.)36 b(Lions,)g Fr(The)i(c)-5 b(onc)g(entr)g(ation)37 b(c)-5 b(omp)g(actness)37 b(principle)g(in)h(the)h(c)-5 b(alculus)38 b(of)g(variations,)g(I.)f(The)127 4492 y(lo)-5 b(c)g(al)5 b(ly)34 b(c)-5 b(omp)g(act)34 b(c)-5 b(ase)p Ft(,)32 b(Ann.)h(Inst.)h(Henri)e(P)m(oincar)m(\023)-46 b(e,)33 b(Analyse)g(nonlin)m(\023)-46 b(eaire,)30 b Fo(1)j Ft(\(1984\))e(109{145.)-74 4725 y([40])49 b(R.S.)26 b(MacKa)m(y)h(&)g (S.)f(Aubry)-8 b(,)29 b Fr(Pr)-5 b(o)g(of)29 b(of)g(existenc)-5 b(e)28 b(of)h(br)-5 b(e)g(athers)29 b(for)g(time-r)-5 b(eversible)28 b(or)h(Hamiltonian)127 4874 y(networks)34 b(of)g(we)-5 b(akly)35 b(c)-5 b(ouple)g(d)34 b(oscil)5 b(lators)p Ft(,)32 b(Nonlinearit)m(y)f Fo(7)h Ft(\(1994\))g(1623-1643.) 1901 5306 y(56)p eop %%Page: 57 57 57 56 bop -74 59 a Ft([41])49 b(D.W.)26 b(McLaughlin)f(&)h(A.C.)h (Scott,)h Fr(Perturb)-5 b(ation)30 b(analysis)e(of)h(\015uxon)g (dynamics)p Ft(,)e(Ph)m(ys.)h(Rev.)f Fo(A18)127 208 y Ft(\(1978\))k(1652)-74 441 y([42])49 b(D.W.)32 b(McLaughlin)g(&)g(J.)h (Shatah,)f Fr(Homo)-5 b(clinic)34 b(orbits)h(for)g(p)-5 b(de's)p Ft(,)31 b(preprin)m(t.)-74 673 y([43])49 b(E.)24 b(Mourre,)j Fr(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 Ft(,)26 b(Com-)127 822 y(m)m(un.)32 b(Math.)h(Ph)m(ys.)i Fo(78)d Ft(\(1981\))g(391)-74 1055 y([44])49 b(F.)f(Nier,)53 b Fr(The)c(dynamics)g(of)h(some)f(op)-5 b(en)49 b(quantum)h(systems)f(with)h(short-r)-5 b(ange)49 b(nonline)-5 b(arities)p Ft(,)127 1204 y(preprin)m(t)32 b(1997.)-74 1437 y([45])49 b(L.)38 b(Niren)m(b)s(erg,)i Fr(T)-7 b(opics)39 b(in)h(Nonline)-5 b(ar)40 b(F)-7 b(unctional)39 b(A)n(nalysis)p Ft(,)g(Couran)m(t)g(Institute)g(Lecture)g(Notes,)127 1586 y(1974.)-74 1819 y([46])49 b(C.-A.)41 b(Pillet)f(&)h(C.E.)i(W)-8 b(a)m(yne,)45 b Fr(Invariant)d(manifolds)f(for)i(a)g(class)f(of)h(disp) -5 b(ersive)42 b(,)j(Hamiltonian,)127 1968 y(p)-5 b(artial)34 b(di\013er)-5 b(ential)34 b(e)-5 b(quations)p Ft(,)32 b(to)h(app)s(ear)f(in)g(Journal)f(of)h(Di\013eren)m(tial)f(Equations.) -74 2201 y([47])49 b(R.M.)28 b(Pyk)m(e)i(&)e(I.M.)h(Sigal,)e Fr(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 2350 y(b)g(ounds)34 b(for)h(p)-5 b(erio)g(dic)33 b(solutions)i Ft(,)e(Duk)m(e)g(Math.)g(J.)g Fo(88)f Ft(\(1997\))-74 2583 y([48])49 b(P)-8 b(.)37 b(P)m(erry)-8 b(,)40 b(I.M.)e(Sigal)d(&)i (B.)g(Simon,)g Fr(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 Ft(,)38 b(Ann.)127 2732 y(Math)32 b Fo(114)h Ft(\(1981\))f(519{567)-74 2964 y([49])49 b(M.)41 b(Reed)h(&)e(B.)h(Simon,)h(Mo)s(dern)f(Metho)s (ds)h(of)f(Mathematical)e(Ph)m(ysics,)44 b(V)-8 b(olume)40 b(1,)j(F)-8 b(unctional)127 3114 y(Analysis,)32 b(Academic)g(Press)i (1972)-74 3346 y([50])49 b(M.)39 b(Reed)h(&)e(B.)h(Simon,)h(Mo)s(dern)f (Metho)s(ds)h(of)e(Mathematical)f(Ph)m(ysics,)43 b(V)-8 b(olume)37 b(4,)k(Analysis)d(of)127 3496 y(Op)s(erators,)32 b(Academic)g(Press)i(1978)-74 3728 y([51])49 b(H.A.)40 b(Rose)h(&)f(M.I.)g(W)-8 b(einstein,)42 b Fr(On)g(the)g(b)-5 b(ound)41 b(states)h(of)f(the)h(nonline)-5 b(ar)41 b(Schr\177)-50 b(odinger)40 b(e)-5 b(quation)127 3878 y(with)34 b(a)h(line)-5 b(ar)34 b(p)-5 b(otential)p Ft(,)33 b(Ph)m(ysica)g(D)f Fo(30)h Ft(\(1988\))e(207{218.)-74 4110 y([52])49 b(J.A.)30 b(Sanders)h(&)f(F.)g(V)-8 b(erh)m(ulst,)31 b(Av)m(eraging)f(Metho)s(ds) h(in)f(Nonlinear)e(Dynamical)g(Systems,)k(Applied)127 4260 y(Mathematical)e(Sciences)k Fo(59)f Ft(Springer-V)-8 b(erlag,)30 b(New)k(Y)-8 b(ork)33 b(\(1985\))-74 4492 y([53])49 b(L.S.)30 b(Sc)m(h)m(ulman,)g(C.R.)h(Do)s(ering)d(&)h(B.)i (Ga)m(v)m(eau,)g Fr(Line)-5 b(ar)32 b(de)-5 b(c)g(ay)32 b(in)g(multilevel)g(quantum)g(systems)p Ft(,)f(J.)127 4641 y(Ph)m(ys.)j(A:)f(Gen.)g Fo(24)f Ft(\(1991\))g(2053{2060.)-74 4874 y([54])49 b(H.)30 b(Segur)g(&)f(M.D.)h(Krusk)-5 b(al,)30 b Fr(Nonexistenc)-5 b(e)31 b(of)h(smal)5 b(l)32 b(amplitude)f(br)-5 b(e)g(ather)32 b(solutions)g(in)g Fq(\036)3615 4838 y Fm(4)3687 4874 y Fr(the)-5 b(ory)p Ft(,)127 5023 y(Ph)m(ys.)34 b(Rev.)f(Lett.,)g Fo(58)g Ft(\(1987\))e(747)1901 5306 y(57)p eop %%Page: 58 58 58 57 bop -74 59 a Ft([55])49 b(J.)26 b(Shatah)g(&)g(W.A.)h(Strauss,)h Fr(Br)-5 b(e)g(athers)28 b(as)h(homo)-5 b(clinic)27 b(ge)-5 b(ometric)28 b(wave)h(maps)p Ft(,)d(Ph)m(ysica)h(D)f(\(1996\))-74 288 y([56])49 b(I.M.)43 b(Sigal,)g Fr(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)127 438 y(quasip)-5 b(erio)g(dic)33 b(solutions)p Ft(,)f(Comm)m(un.)h (Math.)f(Ph)m(ys.)j Fo(153)e Ft(\(1993\))e(297)-74 668 y([57])49 b(I.M.)33 b(Sigal,)e Fr(Gener)-5 b(al)35 b(char)-5 b(acteristics)34 b(of)h(nonline)-5 b(ar)34 b(dynamics)p Ft(,)e(in)g(Sp)s(ectral)g(and)h(Scattering)f(The-)127 817 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,)127 966 y(Inc.)33 b(New)g(Y)-8 b(ork)33 b(-)f(Basel)g(-)h(Hong)f(Kong)g(1994)-74 1196 y([58])49 b(I.M.)33 b(Sigal)e(&)i(A.)g(So\013er,)g Fr(L)-5 b(o)g(c)g(al)35 b(de)-5 b(c)g(ay)35 b(and)g(velo)-5 b(city)35 b(b)-5 b(ounds)34 b(for)h(quantum)h(pr)-5 b(op)g(agation)p Ft(,)32 b(preprin)m(t,)127 1346 y(Princeton)g(1988)g (\(ftp://www.math.rutgers.edu/pub/so\013er\))-74 1575 y([59])49 b(E.)h(Skibsted,)55 b Fr(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 Ft(,)53 b(Comm)m(un.)c(Math.)127 1725 y(Ph)m(ys.)34 b Fo(142)f Ft(\(1991\))e(67{98)-74 1954 y([60])49 b(A.)24 b(So\013er)g(&)f(M.I.)i(W)-8 b(einstein,)25 b Fr(Multichannel)i(nonline)-5 b(ar)25 b(sc)-5 b(attering)27 b(the)-5 b(ory)27 b(for)g(noninte)-5 b(gr)g(able)25 b(e)-5 b(qua-)127 2104 y(tions)31 b(I&)f(II)p Ft(,)e(Comm)m(un.)h(Math.)g(Ph)m (ys.)h Fo(133)f Ft(\(1990\),)g(119-146;)f(J.)h(Di\013.)e(Eqns.)j Fo(98)p Ft(,)g(\(1992\),)f(376-390.)-74 2334 y([61])49 b(A.)26 b(So\013er)h(&)f(M.I.)h(W)-8 b(einstein,)28 b Fr(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 2483 y(of)27 b(emb)-5 b(e)g(dde)g(d)26 b(eigenvalues)p Ft(,)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 2632 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 2782 y(Ivrii,)c(L.Seco,)i(&)g(C.)g (Sulem)-74 3012 y([62])49 b(A.)30 b(So\013er)h(&)f(M.I.)i(W)-8 b(einstein,)30 b Fr(Time)i(dep)-5 b(endent)32 b(r)-5 b(esonanc)g(e)32 b(the)-5 b(ory)p Ft(,)31 b(to)f(app)s(ear)h(in)f (Geometric)f(and)127 3161 y(F)-8 b(unctional)30 b(Analysis.)-74 3391 y([63])49 b(A.)30 b(So\013er)g(&)f(M.I.)i(W)-8 b(einstein,)30 b Fr(The)i(lar)-5 b(ge)32 b(time)g(b)-5 b(ehavior)32 b(of)g(the)g(nonline)-5 b(ar)31 b(Schr\177)-50 b(odinger)31 b(e)-5 b(quation;)127 3540 y(Sele)g(ction)34 b(of)g(the)h(gr)-5 b(ound)35 b(state)g(and)f(instability)h(of)f(excite)-5 b(d)35 b(states)p Ft(,)d(in)g(preparation.)-74 3770 y([64])49 b(E.M.)29 b(Stein,)g(Singular)e(In)m(tegrals)i(and)f(Di\013eren)m (tiabilit)m(y)d(Prop)s(erties)k(of)f(F)-8 b(unctions,)29 b(Princeton)g(Uni-)127 3919 y(v)m(ersit)m(y)34 b(Press)g(\(1970\))-74 4149 y([65])49 b(W.A.)26 b(Strauss,)i(Nonlinear)d(W)-8 b(a)m(v)m(e)27 b(Equations,)h(CBMS)f(Regional)c(Conference)28 b(Series)f(in)e(Mathemat-)127 4299 y(ics)32 b(#7)h(AMS)g(\(1985\))-74 4528 y([66])49 b(D.M.)30 b(Stuart,)g Fr(Minimal)i(p)-5 b(erio)g(ds)31 b(for)h(solutions)g(of)h(some)e(classic)-5 b(al)31 b(\014eld)h(e)-5 b(quations)p Ft(,)30 b(SIAM)h(Journal)127 4678 y(of)h(Mathematical)e(Analysis,)j(to)f(app)s(ear.)-74 4907 y([67])49 b(A.)36 b(V)-8 b(anderbau)m(whede)39 b(&)d(G.)g(Io)s (oss)g Fr(Center)i(manifold)f(the)-5 b(ory)39 b(in)f(in\014nite)f (dimensions)p Ft(,)f(Dynamics)127 5057 y(Rep)s(orted)c Fo(2)h Ft(\(1990\))1901 5306 y(58)p eop %%Page: 59 59 59 58 bop -74 59 a Ft([68])49 b(C.E.)39 b(W)-8 b(a)m(yne,)42 b Fr(Perio)-5 b(dic)40 b(and)g(quasi-p)-5 b(erio)g(dic)39 b(solutions)h(of)h(nonline)-5 b(ar)39 b(wave)h(e)-5 b(quations)40 b(via)g(KAM)127 208 y(the)-5 b(ory)p Ft(,)33 b(Comm)m(un.)f(Math.)h(Ph) m(ys.)h Fo(127)f Ft(\(1990\))e(479)-74 441 y([69])49 b(A.)42 b(W)-8 b(einstein,)45 b Fr(Perio)-5 b(dic)43 b(nonline)-5 b(ar)43 b(waves)g(on)h(a)f(half-line)e Ft(Comm)m(un.)h (Math.)h(Ph)m(ys.)h Fo(99)f Ft(\(1985\))127 590 y(383;)32 b(Erratum)g Fo(107)g Ft(\(1986\))g(177.)-74 822 y([70])49 b(M.I.)33 b(W)-8 b(einstein,)33 b Fr(Lyapunov)i(stability)g(of)g(gr)-5 b(ound)35 b(states)g(of)g(nonline)-5 b(ar)34 b(disp)-5 b(ersive)34 b(evolution)g(e)-5 b(qua-)127 972 y(tions)p Ft(,)32 b(Comm)m(un.)g(Pure)h(Appl.)g(Math.)g Fo(39)f Ft(\(1986\))g(51{68.)-74 1204 y([71])49 b(V.)31 b(W)-8 b(eissk)m(opf)33 b(&)e(E.)h(Wigner,)f Fr(Ber)-5 b(e)g(chnung)32 b(der)i(nat)q(\177)-51 b(urlichen)33 b(Linienbr)-5 b(eite)33 b(auf)h(Grund)g(der)g(Dir)-5 b(ac-)127 1354 y(schen)34 b(Lichtthe)-5 b(orie)p Ft(,)32 b(Z.)g(Ph)m(ys.)j Fo(63)d Ft(\(1930\))g(54{73.)-74 1586 y([72])49 b(K.)d(Y)-8 b(a)5 b(jima,)48 b Fq(W)765 1550 y Fn(k)r(;p)862 1586 y Fr(-c)-5 b(ontinuity)48 b(of)g(wave)e(op)-5 b(er)g(ators)47 b(for)h(Schr\177)-50 b(odinger)46 b(op)-5 b(er)g(ators)p Ft(,)49 b(J.)e(Math.)g(So)s(c.)127 1736 y(Japan)32 b Fo(47)h Ft(\(1995\))e(551-581.)-74 1968 y([73])49 b(F.)32 b(Zhang,)g(Y.S.)h(Kivshar,)f(B.A.)h(Malomed)e(&) i(L.)f(V\023)-49 b(azquez,)34 b Fr(Kink)g(c)-5 b(aptur)g(e)35 b(by)g(a)g(lo)-5 b(c)g(al)34 b(impurity)h(in)127 2118 y(the)g(sine)f(Gor)-5 b(don)34 b(mo)-5 b(del)p Ft(,)32 b(Ph)m(ys.)j(Lett.)d Fo(A159)h Ft(\(1991\))e(318{322.)1901 5306 y(59)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF