Content-Type: multipart/mixed; boundary="-------------0111182052493" This is a multi-part message in MIME format. ---------------0111182052493 Content-Type: text/plain; name="01-425.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-425.keywords" Schr\"odinger operators, imbedded singular continuous spectrum, wave operators, decaying potentials ---------------0111182052493 Content-Type: application/postscript; name="sc1.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="sc1.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.82 Copyright 1998 Radical Eye Software %%Title: sc1.dvi %%Pages: 30 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips sc1 -o sc1.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.11.16:1407 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IE 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IE N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IE 256 array N 2 string 0 1 255{IE S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! 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(completeness)h(is)f(that)g(the)h(dynamics)f(of)g(the)h(p)s(erturb)s (ed)149 498 y(op)s(erator)27 b(can)g(b)s(e)g(divided)g(in)m(to)f(t)m(w) m(o)i(w)m(ell-understo)s(o)s(d)e(parts:)41 b(scattering)27 b(states)h(tra)m(v)m(eling)e(to)149 614 y(in\014nit)m(y)e(in)g(a)g(w)m (a)m(y)i(similar)21 b(to)k(the)g(free)f(ev)m(olution,)i(and)e(b)s(ound) h(states)h(whic)m(h)f(remain)e(con\014ned)149 731 y(in)29 b(a)g(certain)g(sense)i(for)e(all)e(times.)42 b(On)29 b(the)h(other)f(hand,)i(the)f(singular)d(con)m(tin)m(uous)k(sp)s (ectrum)149 847 y(is)42 b(less)h(understo)s(o)s(d,)j(and)c(corresp)s (onds)i(to)e(an)h(in)m(termediate)e(situation:)62 b(states)43 b(whic)m(h)g(do)149 963 y(not)g(remain)e(b)s(ounded)i(in)e(time,)j(but) e(tra)m(v)m(el)h(slo)m(w)m(er,)i(ma)m(y)d(exhibit)g(recurren)m(t)i (phenomena)149 1079 y(and)33 b(do)f(not)f(lo)s(ok)g(lik)m(e)h (scattering)f(states)i(\(see,)h(for)d(example,)h([12])g(and)g (discussion)g(in)g([4)o(])g(for)149 1196 y(more)39 b(detailed)g (explanation)f(and)i(further)g(references\).)67 b(A)39 b(signi\014can)m(t)g(e\013ort)h(in)f(quan)m(tum)149 1312 y(mec)m(hanical)c(scattering)h(theory)h(w)m(as)h(dev)m(oted)f(to)f(pro) m(ving)g(asymptotic)g(completeness,)i(and)149 1428 y(th)m(us)k(absence) h(of)d(singular)f(con)m(tin)m(uous)j(sp)s(ectrum,)h(in)d(a)g(wide)h(v) -5 b(ariet)m(y)41 b(of)f(situations)g(\(see,)149 1544 y(e.g.)j([6])29 b(for)g(a)g(review)h(of)f(some)g(examples\).)42 b(Ho)m(w)m(ev)m(er,)33 b(to)c(the)g(b)s(est)h(of)f(our)g(kno)m(wledge,) i(there)149 1660 y(are)h(no)f(examples)g(of)f(Sc)m(hr\177)-49 b(odinger)31 b(op)s(erators)g(for)g(whic)m(h)g(w)m(a)m(v)m(e)i(op)s (erators)e(exist)h(but)f(are)g(not)149 1777 y(asymptotically)c (complete)i(due)h(to)f(the)h(presence)h(of)e(singular)f(con)m(tin)m (uous)i(sp)s(ectrum.)43 b(In)30 b(the)149 1893 y(problem)42 b(that)h(w)m(e)h(consider,)i(a)d(classical)f(result)g(is)h(the)h (existence)g(of)f(w)m(a)m(v)m(e)i(op)s(erators)d(for)149 2009 y Fr(V)50 b Fp(2)29 b Fr(L)417 2024 y Fo(1)456 2009 y Fr(;)k Ft(and)g(in)f(this)h(class)g(w)m(a)m(v)m(e)h(op)s(erators)f (are)g(also)f(asymptotically)e(complete.)44 b(Ho)m(w)m(ev)m(er)149 2125 y(recen)m(tly)f(in)e([4],)j(the)e(existence)h(of)e(\(mo)s (di\014ed\))f(w)m(a)m(v)m(e)k(op)s(erators)d(for)g Fr(V)65 b Fp(2)43 b Fr(L)3171 2140 y Fq(p)3211 2125 y Fr(;)f(p)h(<)g Ft(2)e(w)m(as)149 2244 y(established.)50 b(Moreo)m(v)m(er,)36 b(usual)e(M\177)-49 b(oller)33 b(w)m(a)m(v)m(e)k(op)s(erators)d (\(1.2\))g(exist)h(if)e(lim)3083 2259 y Fq(x)p Fm(!1)3285 2164 y Fj(R)3352 2190 y Fq(x)3332 2279 y Fo(0)3412 2244 y Fr(V)22 b Ft(\()p Fr(y)t Ft(\))17 b Fr(dy)149 2362 y Ft(exists.)42 b(Therefore,)28 b(an)e(example)f(of)g(\(conditionally)d (in)m(tegrable\))j(p)s(oten)m(tial)f(in)g Fr(L)3199 2326 y Fq(p)3239 2362 y Fr(;)i(p)i(<)f Ft(2)e(with)149 2478 y(im)m(b)s(edded)33 b(singular)f(con)m(tin)m(uous)h(sp)s(ectrum)h(w)m (ould)f(also)f(pro)m(vide)h(the)h(\014rst)f(kno)m(wn)h(example)149 2594 y(of)26 b(a)g(Sc)m(hr\177)-49 b(odinger)27 b(op)s(erator)f(for)f (whic)m(h)i(w)m(a)m(v)m(e)h(op)s(erators)e(exist)h(but)g(fail)d(to)i(b) s(e)g(asymptotically)149 2710 y(complete)32 b(due)i(to)e(the)h (presence)i(of)d(the)h(singular)e(con)m(tin)m(uous)i(sp)s(ectrum.)249 2827 y(Our)g(main)d(goal)h(in)h(this)g(pap)s(er)h(is)f(to)g(pro)m(v)m (e)i(the)f(follo)m(wing)149 3010 y FA(Theorem)f(1.1.)38 b Fi(Given)30 b(any)g(p)-5 b(ositive)30 b(function)g Fr(h)p Ft(\()p Fr(x)p Ft(\))h Fi(tending)f(to)g(in\014nity)h(as)f Fr(x)g Fi(gr)-5 b(ows,)31 b(ther)-5 b(e)149 3141 y(exist)44 b(p)-5 b(otentials)44 b Fr(V)21 b Ft(\()p Fr(x)p Ft(\))44 b Fi(such)g(that)h Fp(j)p Fr(V)21 b Ft(\()p Fr(x)p Ft(\))p Fp(j)44 b(\024)1967 3093 y Fq(h)p Fo(\()p Fq(x)p Fo(\))p 1967 3118 136 4 v 1970 3175 a(1+)p Fq(x)2156 3141 y Fi(and)g(the)g(op) -5 b(er)g(ator)44 b Fr(H)3000 3156 y Fq(V)3104 3141 y Fi(has)g(non-empty)149 3260 y(singular)35 b(c)-5 b(ontinuous)34 b(sp)-5 b(e)g(ctrum.)249 3443 y Ft(It)32 b(is)e(easy)j(to)e(see)h(that) g(without)f(loss)g(of)g(generalit)m(y)f(one)i(can)g(assume)g(that)f Fr(h)p Ft(\()p Fr(x)p Ft(\))h(is)e(mono-)149 3559 y(tone.)82 b(F)-8 b(or)45 b(the)h(rest)f(of)g(the)h(pap)s(er,)j(w)m(e)d(are)f (going)f(to)h(op)s(erate)g(under)i(this)d(assumption,)149 3676 y(whic)m(h)39 b(is)f(con)m(v)m(enien)m(t)i(for)d(tec)m(hnical)h (reasons.)61 b(W)-8 b(e)39 b(are)f(going)f(to)h(pro)m(vide)g(a)g (fairly)e(explicit)149 3792 y(construction)30 b(of)f(suc)m(h)i(p)s (oten)m(tials.)41 b(As)30 b(a)f(corollary)e(of)i(suc)m(h)i (construction)e(w)m(e)i(are)e(also)f(going)149 3908 y(to)33 b(pro)m(v)m(e)149 4092 y FA(Theorem)f(1.2.)38 b Fi(Given)30 b(any)g(p)-5 b(ositive)30 b(function)g Fr(h)p Ft(\()p Fr(x)p Ft(\))h Fi(tending)f(to)g(in\014nity)h(as)f Fr(x)g Fi(gr)-5 b(ows,)31 b(ther)-5 b(e)149 4222 y(exist)32 b(p)-5 b(otentials)31 b Fr(V)21 b Ft(\()p Fr(x)p Ft(\))32 b Fi(such)g(that)g Fp(j)p Fr(V)21 b Ft(\()p Fr(x)p Ft(\))p Fp(j)28 b(\024)1872 4175 y Fq(h)p Fo(\()p Fq(x)p Fo(\))p 1872 4199 V 1875 4257 a(1+)p Fq(x)2017 4222 y Fr(;)k Fi(and)f(the)g(wave)g(op)-5 b(er)g(ators)31 b(for)h(the)f(op)-5 b(er)g(ator)149 4341 y Fr(H)230 4356 y Fq(V)326 4341 y Fi(exist)35 b(but)g(ar)-5 b(e)35 b(not)g(asymptotic)-5 b(al)5 b(ly)34 b(c)-5 b(omplete.)249 4525 y Ft(Our)30 b(examples)g(can)g(b)s(e)g(easily)f(generalized)g(to)h(an)m(y)g (dimension)f(b)m(y)i(lo)s(oking)c(at)j(spherically)149 4641 y(symmetric)i(p)s(oten)m(tials.)249 4757 y(Let)45 b(us)g(outline)e(brie\015y)h(the)h(idea)f(of)g(construction.)79 b(Con)m(trolling)42 b(singular)h(con)m(tin)m(uous)149 4873 y(sp)s(ectrum)48 b(is)e(hard)h(b)s(ecause)i(unlik)m(e)d(p)s(oin)m (t)h(sp)s(ectrum,)k(one)c(has)g(to)g(con)m(trol)f(in)h(a)f(certain)149 4990 y(sense)h(an)d(uncoun)m(table)h(set)g(of)f(energies,)k(and)d (there)g(is)f(no)g(simple)f(criteria)g(\(lik)m(e)g Fr(L)3497 4953 y Fo(2)3537 4990 y Ft(\))h(for)149 5106 y(what)e(b)s(eha)m(vior)f (of)f(solutions)g(implies)f(existence)j(of)f(singular)e(con)m(tin)m (uous)j(comp)s(onen)m(t.)69 b(It)149 5222 y(turns)40 b(out)f(that)g(one)g(can)h(reduce)g(matters)f(to)g(k)m(eeping)g(trac)m (k)h(on)f(eac)m(h)h(step)g(of)f(only)f(\014nite)149 5338 y(n)m(um)m(b)s(er)31 b(of)f(energies,)h(but)f(the)h(price)f(one)g(has)h (to)f(pa)m(y)h(is)e(ha)m(ving)h(to)g(con)m(trol)f(rather)i(precisely) 149 5455 y(norms)38 b(of)f(the)i(solutions)d(at)i(these)h(energies.)60 b(The)39 b(con)m(trol)e(has)h(to)g(b)s(e)g(m)m(uc)m(h)g(more)f(precise) p eop %%Page: 4 4 4 3 bop 149 66 a Fn(4)1324 b(ALEXANDER)23 b(KISELEV)149 266 y Ft(than)40 b(in)e(constructions)i(of)e(Nab)s(ok)m(o)i(and)f (Simon.)62 b(A)m(t)39 b(the)h(heart)f(of)g(the)g(construction)h(is)e(a) 149 382 y(lemma)h(whic)m(h)i(allo)m(ws)f(instead)h(of)f(one)h(im)m(b)s (edded)g(eigen)m(v)-5 b(alue)40 b(in)g(Wigner-v)m(on)g(Neumann-)149 498 y(t)m(yp)s(e)29 b(construction)f(to)f(get)h(t)m(w)m(o,)h(eac)m(h)g (ha)m(ving)e(an)g(eigenfunction)g(\(normalized)f(at)h(the)h(origin\)) 149 614 y(with)j(the)g Fr(L)602 578 y Fo(2)672 614 y Ft(norm)f(squared)i(exactly)f(t)m(wice)g(the)g Fr(L)2100 578 y Fo(2)2171 614 y Ft(norm)e(squared)k(of)d(the)h(eigenfunction)f (in)149 731 y(a)36 b(usual)f(Wigner-v)m(on)h(Neumann)f(construction)h (in)f(a)h(certain)f(asymptotic)g(regime.)52 b(The)36 b Fr(L)3696 694 y Fo(2)149 847 y Ft(norm)g(squared)i(of)e(the)h (eigenfunction)f(is)g(the)g(in)m(v)m(erse)i(of)e(the)h(p)s(oin)m(t)f(w) m(eigh)m(t)h(the)g(measure)f Fr(\026)149 963 y Ft(assigns)d(to)f(a)g (giv)m(en)h(eigen)m(v)-5 b(alue.)43 b(Th)m(us)34 b(the)f(lemma)d(will)g (allo)m(w)h(to)i(\\spread)g(out")f(ev)m(enly)i(the)149 1079 y(w)m(eigh)m(t)g(of)e(one)i(p)s(oin)m(t)e(mass)h(to)f(t)m(w)m(o)i (nearb)m(y)g(p)s(oin)m(ts.)45 b(This)33 b(allo)m(ws)f(to)g(carry)i(out) f(Can)m(tor-lik)m(e)149 1196 y(construction,)28 b(building)23 b(a)i(sequence)k(of)c(p)s(oten)m(tials)f(whic)m(h)i(ha)m(v)m(e)h(p)s (oin)m(t)e(measure)g(comp)s(onen)m(ts)149 1312 y(appro)m(ximating)31 b(singular)g(con)m(tin)m(uous)i(measure.)44 b(In)33 b(the)g(limit,)c (one)k(gets)g(the)g(result.)249 1428 y(The)25 b(metho)s(ds)f(of)f(this) h(pap)s(er)g(can)g(b)s(e)h(generalized)e(to)h(the)g(discrete)h(Sc)m (hr\177)-49 b(odinger)24 b(op)s(erators,)149 1544 y(b)m(y)35 b(using)e(the)h(discrete)h(analog)c(of)j(the)g(Pr)s(\177)-51 b(ufer)33 b(transform)f(\(see,)j(e.g.)47 b([10]\).)f(The)35 b(results)f(are)149 1660 y(parallel)c(to)j(the)g(con)m(tin)m(uous)g (case.)44 b(W)-8 b(e)33 b(plan)f(to)g(address)i(these)g(applications)c (elsewhere.)1400 1971 y(2.)49 b Fs(A)38 b(Splitting)f(Lemma)249 2145 y Ft(F)-8 b(or)32 b(the)i(rest)g(of)f(the)g(pap)s(er,)h(let)e(us)i (\014x)g(for)f(simplicit)m(y)d(some)j(b)s(oundary)h(condition)e(at)h (the)149 2261 y(origin,)28 b(for)h(example)g(Diric)m(hlet.)40 b(Let)29 b(us)h(denote)g(b)m(y)g Fr(\026)f Ft(the)h(sp)s(ectral)e (measure)i(corresp)s(onding)149 2378 y(to)35 b(this)g(b)s(oundary)h (condition.)50 b(Throughout)35 b(the)h(pap)s(er,)g(w)m(e)g(are)f(going) f(to)h(use)h(notation)e Fr(C)149 2494 y Ft(for)i(univ)m(ersal)g (constan)m(ts)h(\(not)f(necessarily)h(the)f(same\),)h(more)e(precisely) -8 b(,)38 b(for)d(the)i(constan)m(ts)149 2610 y(whic)m(h)28 b(do)f(not)f(dep)s(end)i(in)f(an)m(y)g(w)m(a)m(y)h(on)f(the)g(step)h (of)f(the)g(inductiv)m(e)g(construction.)42 b(A)27 b(starting)149 2726 y(p)s(oin)m(t)32 b(is)g(the)h(follo)m(wing)d(elemen)m(tary)j (observ)-5 b(ation.)149 2923 y FA(Lemma)34 b(2.1.)40 b Fi(L)-5 b(et)32 b Fr(E)38 b Fi(b)-5 b(e)32 b(an)g(eigenvalue)f(of)g Fr(H)1946 2938 y Fq(V)2007 2923 y Fr(;)h Fi(and)g Fr(u)p Ft(\()p Fr(x;)17 b(E)6 b Ft(\))32 b Fi(b)-5 b(e)31 b(the)i(c)-5 b(orr)g(esp)g(onding)30 b(eigen-)149 3040 y(function)35 b(normalize)-5 b(d)33 b(by)i(the)g(c)-5 b(ondition)34 b Fr(u)1798 3004 y Fm(0)1821 3040 y Ft(\(0)p Fr(;)17 b(E)6 b Ft(\))27 b(=)h(1)p Fr(:)34 b Fi(Then)g Fr(\026)p Ft(\()p Fr(E)6 b Ft(\))28 b(=)f Fp(k)p Fr(u)p Ft(\()p Fr(x;)17 b(E)6 b Ft(\))p Fp(k)3316 2999 y Fm(\000)p Fo(2)3316 3073 y Fq(L)3364 3054 y Fk(2)3398 3073 y Fo(\()p Fh(R)3473 3054 y Fk(+)3524 3073 y Fo(\))3556 3040 y Fr(:)149 3253 y Fi(Pr)-5 b(o)g(of.)42 b Ft(Denote)21 b Fr(v)t Ft(\()p Fr(x;)c(E)6 b Ft(\))22 b(the)g(solution)e(satisfying)h(orthogonal)f(b)s (oundary)i(condition)e Fr(v)t Ft(\()p Fr(x;)d(E)6 b Ft(\))28 b(=)149 3370 y(1)p Fr(;)k(v)308 3333 y Fm(0)331 3370 y Ft(\()p Fr(x;)17 b(E)6 b Ft(\))27 b(=)h(0)p Fr(:)j Ft(Recall)f(that)h(the)h(W)-8 b(eyl)32 b Fr(m)p Ft(-function)f(is)f (de\014ned)j(for)e Fr(z)i Fp(2)28 b Fg(C)46 b Fp(n)19 b Fg(R)43 b Ft(as)31 b(a)g(unique)149 3486 y(complex)i(n)m(um)m(b)s(er) h Fr(m)p Ft(\()p Fr(z)t Ft(\))g(suc)m(h)h(that)e Fr(v)t Ft(\()p Fr(x;)17 b(z)t Ft(\))23 b(+)f Fr(m)p Ft(\()p Fr(z)t Ft(\))p Fr(u)p Ft(\()p Fr(x;)17 b(z)t Ft(\))31 b Fp(2)e Fr(L)2643 3450 y Fo(2)2716 3486 y Ft(\(assuming)j(w)m(e)j(are) e(in)g(the)149 3602 y(limit)c(p)s(oin)m(t)j(case\).)45 b(W)-8 b(e)33 b(ha)m(v)m(e)g(\(see)h(e.g.)44 b([5],)32 b(Chapter)i(9,)e(Section)g(3\))149 3860 y(\(2.1\))669 b Fr(m)p Ft(\()p Fr(z)t Ft(\))23 b Fp(\000)g Fr(m)p Ft(\()p Fr(z)1520 3875 y Fo(0)1560 3860 y Ft(\))28 b(=)1729 3724 y Fj(Z)1733 4008 y Fh(R)1845 3719 y Fj(\022)2008 3792 y Ft(1)p 1929 3837 207 4 v 1929 3928 a Fr(t)22 b Fp(\000)h Fr(z)2167 3860 y Fp(\000)2374 3792 y Ft(1)p 2277 3837 242 4 v 2277 3928 a Fr(t)f Fp(\000)h Fr(z)2479 3943 y Fo(0)2529 3719 y Fj(\023)2619 3860 y Fr(d\026)p Ft(\()p Fr(t)p Ft(\))p Fr(;)149 4181 y Ft(where)34 b Fr(z)476 4196 y Fo(0)549 4181 y Ft(is)e(some)g(\014xed)i(complex)e(n)m(um)m(b)s (er.)44 b(F)-8 b(rom)31 b(\(2.1\))h(it)f(follo)m(ws)h(that)957 4377 y Fr(\026)p Ft(\()p Fr(E)6 b Ft(\))28 b(=)f(lim)1302 4437 y Fq(\017)p Fm(!)p Fo(0)1453 4377 y Fr(\017)p Fp(j)p Fr(m)p Ft(\()p Fr(E)i Ft(+)22 b Fr(i\017)p Ft(\))p Fp(j)28 b Ft(=)f(lim)2112 4437 y Fq(\017)p Fm(!)p Fo(0)2263 4377 y Fr(\017)17 b Ft(Im)33 b Fr(m)p Ft(\()p Fr(E)28 b Ft(+)22 b Fr(i\017)p Ft(\))p Fr(:)149 4605 y Ft(On)33 b(the)g(other)g(hand,)621 4801 y(Im)g Fr(m)p Ft(\()p Fr(E)c Ft(+)22 b Fr(i\017)p Ft(\))28 b(=)f Fr(\017)p Fp(k)p Fr(v)t Ft(\()p Fr(x;)17 b(E)29 b Ft(+)22 b Fr(i\017)p Ft(\))g(+)g Fr(m)p Ft(\()p Fr(E)29 b Ft(+)22 b Fr(i\017)p Ft(\))p Fr(u)p Ft(\()p Fr(x;)17 b(E)28 b Ft(+)22 b Fr(i\017)p Ft(\))p Fp(k)3023 4760 y Fo(2)3023 4827 y Fq(L)3071 4808 y Fk(2)3106 4827 y Fo(\()p Fh(R)3181 4808 y Fk(+)3232 4827 y Fo(\))149 5007 y Ft(\(see,)37 b(e.g.)51 b([5]\).)g(Multiplying)32 b(b)s(oth)j(sides)g(of)g(the)g(ab)s(o)m(v)m(e)h(equalit)m(y)f(b)m(y)h Fr(\017)f Ft(and)g(passing)g(to)g(the)149 5123 y(limit)29 b Fr(\017)g Fp(!)e Ft(0)p Fr(;)32 b Ft(w)m(e)i(get)1384 5279 y Fr(\026)p Ft(\()p Fr(E)6 b Ft(\))27 b(=)h Fr(\026)p Ft(\()p Fr(E)6 b Ft(\))1941 5237 y Fo(2)1980 5279 y Fp(k)p Fr(u)p Ft(\()p Fr(x;)17 b(E)6 b Ft(\))p Fp(k)2389 5237 y Fo(2)2389 5304 y Fq(L)2437 5285 y Fk(2)2475 5279 y Fr(;)149 5455 y Ft(whic)m(h)34 b(is)e(exactly)h(the)g(claim)d(of)i(the) h(lemma.)1778 b Ff(\003)p eop %%Page: 5 5 5 4 bop 1220 66 a Fn(SINGULAR)31 b(CONTINUOUS)g(SPECTR)n(UM)1033 b(5)249 266 y Ft(Giv)m(en)28 b(the)h(energy)g Fr(E)k Ft(=)28 b Fr(k)1261 230 y Fo(2)1300 266 y Fr(;)h Ft(it)e(will)e(b)s(e)k (con)m(v)m(enien)m(t)h(to)d(in)m(tro)s(duce)h(Pr)s(\177)-51 b(ufer)28 b(v)-5 b(ariables)27 b Fr(R)i Ft(and)149 382 y Fr(\022)s(;)k(R)332 346 y Fo(2)399 382 y Ft(=)28 b(\()p Fr(u)597 346 y Fm(0)620 382 y Ft(\))658 346 y Fo(2)719 382 y Ft(+)22 b Fr(k)871 346 y Fo(2)911 382 y Fr(u)967 346 y Fo(2)1038 382 y Ft(and)33 b Fr(\022)e Ft(=)c(tan)1548 346 y Fm(\000)p Fo(1)1643 382 y Ft(\()p Fr(k)s(u=u)1896 346 y Fm(0)1918 382 y Ft(\))p Fr(:)32 b Ft(Then)i(it)d(is)i(easy)g(to)f (see)i(that)1120 593 y(\(log)17 b Fr(R)q Ft(\()p Fr(x;)g(k)s Ft(\))1605 552 y Fo(2)1644 593 y Ft(\))1682 552 y Fm(0)1733 593 y Ft(=)1849 526 y(1)p 1847 570 55 4 v 1847 661 a Fr(k)1911 593 y(V)k Ft(\()p Fr(x)p Ft(\))c(sin)f(2)p Fr(\022)s Ft(\()p Fr(x;)h(k)s Ft(\))-2450 b(\(2.2\))1381 830 y Fr(\022)s Ft(\()p Fr(x;)17 b(k)s Ft(\))1658 789 y Fm(0)1709 830 y Ft(=)28 b Fr(k)d Fp(\000)2001 763 y Ft(1)p 1999 807 V 1999 899 a Fr(k)2063 830 y(V)c Ft(\()p Fr(x)p Ft(\)\(sin)c Fr(\022)s Ft(\))2533 789 y Fo(2)2572 830 y Fr(:)-2450 b Ft(\(2.3\))249 1023 y(There)45 b(are)f(t)m(w)m(o)g (essen)m(tial)g(new)h(ideas)f(whic)m(h)g(pla)m(y)g(imp)s(ortan)m(t)d (role)i(in)g(the)i(result:)66 b(the)149 1139 y(Splitting)31 b(Lemma,)h(and)h(a)g(relation)e(b)s(et)m(w)m(een)k Fr(L)2000 1103 y Fo(2)2073 1139 y Ft(norm)e(of)f Fr(R)i Ft(and)g(deriv)-5 b(ativ)m(e)32 b(of)h(the)g(Pr)s(\177)-51 b(ufer)149 1255 y(angle)33 b Fr(\022)k Ft(in)32 b(energy)-8 b(.)47 b(The)35 b(purp)s(ose)f(of)f(this)g(section)h(is)f(to)g(pro)m(v)m(e)h(the)g (\014rst)g(k)m(ey)h(lemma)c(whic)m(h)149 1371 y(is)37 b(at)f(the)h(heart)f(of)h(construction,)g(the)g(Splitting)d(Lemma.)54 b(T)-8 b(o)37 b(clarify)e(its)h(meaning,)g(let)g(us)149 1487 y(form)m(ulate)31 b(\014rst)i(a)f(simpli\014ed)e(v)m(ersion)j(of)f (this)g(result.)43 b(Fix)31 b(an)i(arbitrary)e(p)s(ositiv)m(e)h(energy) h Fr(k)3697 1502 y Fo(0)149 1604 y Ft(\(w)m(e)38 b(are)f(going)f(to)g (w)m(ork)i(with)f(v)-5 b(alues)36 b(of)h(quasimomen)m(tum)e Fr(k)40 b Ft(throughout)d(the)g(pap)s(er,)i(but)149 1720 y(still)f(call)g(those)i(\\energies")g(for)f(simplicit)m(y\).)63 b(Fix)39 b(a)g(parameter)h Fr(g)t(;)f Ft(whic)m(h)h(w)m(e)h(think)e(of) h(as)149 1839 y(large.)57 b(De\014ne)41 b(~)-52 b Fr(g)39 b Ft(=)c(exp)q(\()p Fr(g)1182 1803 y Fo(3)p Fq(=)p Fo(4)1291 1839 y Ft(\))p Fr(:)i Ft(Fix)g(another)g(parameter,)h Fr(f)5 b(;)38 b Ft(whic)m(h)g(w)m(e)g(think)f(of)g(as)g(small.)149 1955 y(\(It)29 b(is)e(useful)i(to)e(k)m(eep)j(in)e(mind)f(that)h(in)f (the)i(actual)e(construction,)i(w)m(e)g(will)d(b)s(e)j(able)e(to)h(c)m (ho)s(ose)149 2072 y Fr(g)45 b Ft(as)d(large)e(as)h(w)m(e)i(w)m(an)m(t) f(and)f(then)h(after)f(that)g(to)g(c)m(ho)s(ose)i Fr(f)52 b Ft(as)41 b(small)e(as)j(w)m(e)g(w)m(an)m(t\).)71 b(Let)149 2188 y Fr(\016)t(k)31 b Ft(=)h(~)-53 b Fr(g)433 2152 y Fm(\000)p Fo(1)527 2188 y Fr(f)5 b(;)33 b Ft(and)f(denote)i Fr(k)1195 2203 y Fo(1)1234 2188 y Fr(;)f(k)1345 2203 y Fo(2)1384 2188 y Fr(;)g(k)1495 2203 y Fo(2)1562 2188 y Fr(>)27 b(k)1716 2203 y Fo(1)1756 2188 y Fr(;)32 b Ft(the)h(ends)h(of)f(the)g(in)m(terv)-5 b(al)31 b(of)h(length)g Fr(\016)t(k)k Ft(cen)m(tered)149 2304 y(at)e Fr(k)321 2319 y Fo(0)360 2304 y Fr(:)f Ft(Denote)h Fr(\022)801 2319 y Fq(i)829 2304 y Ft(\()p Fr(x)p Ft(\))c(=)f Fr(\022)s Ft(\()p Fr(x;)17 b(k)1331 2319 y Fq(i)1359 2304 y Ft(\))33 b(and)h Fr(\016)t(\022)e Ft(=)d Fr(\022)1895 2319 y Fo(2)1958 2304 y Fp(\000)23 b Fr(\022)2103 2319 y Fo(1)2143 2304 y Fr(:)34 b Ft(F)-8 b(urthermore,)33 b(de\014ne)h(a)g(small)d(angle)h Fr(\013)149 2420 y Ft(b)m(y)i(a)e(condition)f Fr(f)d Ft(sin)16 b Fr(\013)28 b Ft(=)g Fr(\016)t(k)s(:)33 b Ft(W)-8 b(e)33 b(are)f(going)f(to)h(use)i(notation)1403 2583 y Fr(x)1458 2598 y Fq(\014)1533 2583 y Ft(=)28 b(min)n Fp(f)p Fr(x)g Ft(:)g Fr(\016)t(\022)s Ft(\()p Fr(x)p Ft(\))g(=)g Fr(\014)6 b Fp(g)p Fr(:)149 2745 y Ft(De\014ne)34 b(a)e(p)s(oten)m(tial)f Fr(V)21 b Ft(\()p Fr(x)p Ft(\))33 b(on)g([0)p Fr(;)17 b Fp(1)p Ft(\))31 b(b)m(y)149 3029 y(\(2.4\))297 b Fr(V)22 b Ft(\()p Fr(x)p Ft(\))28 b(=)988 2825 y Fj(8)988 2915 y(<)988 3094 y(:)1118 2909 y Fp(\000)p Ft(2)p Fr(f)11 b(k)1354 2924 y Fo(0)1410 2909 y Ft(sin\()p Fr(\022)1613 2924 y Fo(1)1675 2909 y Ft(+)22 b Fr(\022)1818 2924 y Fo(2)1858 2909 y Ft(\))p Fr(;)737 b Ft(0)28 b Fp(\024)g Fr(x)g Fp(\024)g Fr(x)3085 2924 y Fq(\031)r Fm(\000)p Fq(\013)1118 3025 y Ft(2)p Fr(f)11 b(k)1277 3040 y Fo(0)1333 3025 y Ft(sin)o(\()p Fr(\022)1535 3040 y Fo(1)1597 3025 y Ft(+)22 b Fr(\022)1740 3040 y Fo(2)1780 3025 y Ft(\))p Fr(;)815 b(x)2715 3040 y Fq(\031)r Fm(\000)p Fq(\013)2890 3025 y Fp(\024)29 b Fr(x)f Fp(\024)g Fr(f)3243 2989 y Fm(\000)p Fo(1)3337 3025 y Fr(g)1118 3146 y Fp(\000)p Ft(min)1375 3065 y Fj(\000)1421 3146 y Fr(\016)t(k)s(;)1595 3102 y Fq(g)p 1575 3123 76 4 v 1575 3181 a Fo(2)p Fq(x)1660 3065 y Fj(\001)1733 3106 y Fq(k)1770 3115 y Fk(0)p 1733 3123 72 4 v 1751 3181 a Fo(4)1814 3146 y Ft(\(sin)16 b(2)p Fr(\022)2082 3161 y Fo(1)2144 3146 y Ft(+)22 b(sin)16 b(2)p Fr(\022)2472 3161 y Fo(2)2512 3146 y Ft(\))p Fr(;)83 b(x)28 b(>)g(f)2906 3110 y Fm(\000)p Fo(1)3000 3146 y Fr(g)t(:)149 3307 y Ft(The)45 b(de\014nition)e(of)g Fr(V)65 b Ft(in)m(v)m(olv)m(es)44 b(the)h(Pr)s(\177)-51 b(ufer)43 b(angles)g Fr(\022)2269 3322 y Fq(i)2298 3307 y Fr(;)g Ft(and)h Fr(\022)2614 3322 y Fq(i)2686 3307 y Ft(in)f(their)g(turn)h (dep)s(end)h(on)149 3424 y Fr(V)67 b Ft(through)45 b(\(2.3\).)82 b(This)45 b(am)m(biguit)m(y)f(is)h(easily)f(resolv)m(ed,)50 b(ho)m(w)m(ev)m(er,)h(b)m(y)46 b(substituting)f(the)149 3540 y(expression)34 b(\(2.4\))e(for)g Fr(V)54 b Ft(in)m(to)32 b(\(2.3\))f(and)i(solving)e(the)i(resulting)e(nonlinear)g(system)j(for) e Fr(\022)3575 3555 y Fq(i)3603 3540 y Fr(:)h Ft(It)149 3656 y(is)23 b(not)g(di\016cult)e(to)i(see)h(from)e(\(2.3\))g(that)h (this)f(resulting)g(system)i(has)f(unique)g(piecewise)h(smo)s(oth)149 3772 y(global)34 b(solution)g(b)m(y)i(standard)g(ODE)g(existence)h(and) e(uniqueness)j(theorem.)52 b(The)37 b(functions)149 3889 y Fr(\022)194 3904 y Fq(i)223 3889 y Ft(\()p Fr(x)p Ft(\))h(are)f(smo)s (oth)f(apart)h(from)f(three)i(p)s(oin)m(ts)e(where)j Fr(V)59 b Ft(is)36 b(not)h(smo)s(oth,)h(and)f(where)i Fr(\022)3549 3852 y Fm(0)3546 3913 y Fq(i)3611 3889 y Ft(\(or)149 4005 y Fr(\022)197 3969 y Fm(00)194 4029 y Fq(i)240 4005 y Ft(\))c(ma)m(y)g(jump.)51 b(W)-8 b(e)35 b(can)g(then)h(de\014ne)h Fr(V)56 b Ft(in)35 b(terms)g(of)f(these)j Fr(\022)2601 4020 y Fq(i)2664 4005 y Ft(b)m(y)h(\(2.4\))c(recip)s(e,)i (and)f(then)149 4121 y(solutions)g(of)42 b(\(2.3\))35 b(will)e(coincide)i(with)h Fr(\022)1752 4136 y Fq(i)1816 4121 y Ft(b)m(y)g(uniqueness.)55 b(F)-8 b(or)35 b(the)h(rest)g(of)f (the)h(pap)s(er,)h(w)m(e)149 4237 y(are)42 b(going)f(to)g(consider)h (on)g(eac)m(h)g(step)h Fr(V)63 b Ft(de\014ned)44 b(in)d(terms)g(of)h (Pr)s(\177)-51 b(ufer)41 b(angles)g(at)h(a)f(\014nite)149 4354 y(n)m(um)m(b)s(er)34 b(of)e(energies)h(without)g(further)g (explanation.)43 b(W)-8 b(e)33 b(are)g(going)e(to)i(ignore)f(for)g(no)m (w)i(the)149 4470 y(issue)29 b(whether)g Fr(V)50 b Ft(is)28 b(w)m(ell-de\014ned)g(in)f(a)h(sense)i(that)d Fr(x)2171 4485 y Fq(\031)r Fm(\000)p Fq(\013)2347 4470 y Fr(<)g(f)2509 4434 y Fm(\000)p Fo(1)2603 4470 y Fr(g)t Ft(;)i(this)f(will)e(b)s(e)i (v)m(eri\014ed)h(later)149 4586 y(for)k(all)e Fr(g)36 b Ft(large)31 b(enough.)45 b(W)-8 b(e)34 b(notice)e(that)h(the)g (constan)m(t)h Fr(f)43 b Ft(in)33 b(\(2.4\))f(can)h(b)s(e)g(replaced)g (b)m(y)h(an)149 4702 y(appropriate)39 b(slo)m(wly)f(v)-5 b(arying)39 b(function)g(with)f(no)h(di\013erence)h(for)f(the)g (result.)64 b(W)-8 b(e)39 b(c)m(ho)s(ose)h(a)149 4818 y(particular)31 b(represen)m(tation)j(\(2.4\))e(to)g(simplify)e(tec)m (hnicalities.)42 b(Then)34 b(w)m(e)f(ha)m(v)m(e)149 4996 y FA(Lemma)43 b(2.2.)h Fi(Assume)c(that)f Fr(g)k Fi(is)c(su\016ciently) g(lar)-5 b(ge,)40 b(and)e Fr(f)50 b Fi(is)39 b(su\016ciently)h(smal)5 b(l.)57 b(Then)149 5112 y(for)35 b(a)g(p)-5 b(otential)34 b Fr(V)22 b Ft(\()p Fr(x)p Ft(\))35 b Fi(given)f(by)44 b Ft(\(2.4\))o Fi(,)35 b(b)-5 b(oth)35 b Fr(k)1946 5127 y Fo(1)2020 5112 y Fi(and)f Fr(k)2260 5127 y Fo(2)2334 5112 y Fi(ar)-5 b(e)35 b(eigenvalues.)43 b(Mor)-5 b(e)g(over,)149 5283 y Ft(\(2.5\))666 b Fp(k)p Fr(R)q Ft(\()p Fr(x;)17 b(k)1329 5298 y Fq(i)1357 5283 y Ft(\))p Fp(k)1445 5242 y Fo(2)1445 5309 y Fq(L)1493 5290 y Fk(2)1527 5309 y Fo(\()p Fh(R)1602 5290 y Fk(+)1654 5309 y Fo(\))1713 5283 y Ft(=)27 b(2)p Fr(f)1924 5242 y Fm(\000)p Fo(1)2018 5283 y Ft(\(1)22 b(+)g Fr(O)s Ft(\()p Fr(f)11 b(e)2445 5242 y Fq(g)2484 5283 y Fr(;)20 b Ft(~)-52 b Fr(g)2579 5242 y Fm(\000)p Fo(1)p Fq(=)p Fo(2)2743 5283 y Fr(g)t Ft(\)\))149 5455 y Fi(pr)-5 b(ovide)g(d)34 b(that)i Fr(R)q Ft(\()p Fr(x;)17 b(k)997 5470 y Fq(i)1025 5455 y Ft(\))34 b Fi(ar)-5 b(e)35 b(normalize)-5 b(d)33 b(by)i(a)g(c)-5 b(ondition)34 b Fr(R)q Ft(\(0)p Fr(;)17 b(k)2652 5470 y Fq(i)2679 5455 y Ft(\))28 b(=)g(1)p Fr(;)34 b(i)28 b Ft(=)g(1)p Fr(;)17 b Ft(2)p Fr(:)p eop %%Page: 6 6 6 5 bop 149 66 a Fn(6)1324 b(ALEXANDER)23 b(KISELEV)249 266 y Ft(W)-8 b(e)29 b(are)g(not)f(going)g(to)g(pro)m(v)m(e)i(Lemma)d (2.2,)i(but)g(directly)f(a)h(more)f(adv)-5 b(anced)29 b(v)m(ersion)g(whic)m(h)149 382 y(is)43 b(going)f(to)h(b)s(e)h(needed)h (in)e(the)h(construction.)76 b(The)44 b(di\013erence)g(is)f(that)g(w)m (e)i(will)c(need)j(to)149 498 y(handle)37 b(not)g(just)h(a)f(pair)f(of) g(eigen)m(v)-5 b(alues,)39 b(but)e(2)2020 462 y Fq(n)p Fm(\000)p Fo(1)2194 498 y Ft(pairs)f(of)h(eigen)m(v)-5 b(alues)37 b(on)g(the)h Fr(n)p Ft(th)f(step)149 614 y(sim)m (ultaneously)-8 b(.)54 b(Ho)m(w)m(ev)m(er,)40 b(w)m(e)d(are)f(going)f (to)h(use)i(this)e(more)f(transparen)m(t)j(form)m(ulation)33 b(to)149 731 y(mak)m(e)47 b(a)f(few)g(clarifying)e(remarks.)85 b(First)45 b(of)h(all,)h(observ)m(e)i(that)d Fp(j)p Fr(V)21 b Ft(\()p Fr(x)p Ft(\))p Fp(j)51 b(\024)g Fr(h)p Ft(\()p Fr(x)p Ft(\))p Fr(=)p Ft(\()p Fr(x)32 b Ft(+)f(1\))149 847 y(pro)m(vided)i(that)g Fr(f)43 b Ft(is)32 b(su\016cien)m(tly)h (small.)42 b(Indeed,)34 b(w)m(e)f(basically)e(need)j(to)e(ensure)i (that)1348 1009 y(2)p Fr(k)1448 1024 y Fo(0)1487 1009 y Fr(f)k Fp(\024)28 b Fr(h)p Ft(\()p Fr(f)1831 968 y Fm(\000)p Fo(1)1925 1009 y Fr(g)t Ft(\))p Fr(=)p Ft(\()p Fr(f)2160 968 y Fm(\000)p Fo(1)2253 1009 y Fr(g)e Ft(+)c(1\))p Fr(;)149 1171 y Ft(or)31 b(2)p Fr(k)367 1186 y Fo(0)406 1171 y Ft(\()p Fr(g)21 b Ft(+)c Fr(f)11 b Ft(\))28 b Fp(\024)g Fr(h)p Ft(\()p Fr(f)988 1135 y Fm(\000)p Fo(1)1082 1171 y Fr(g)t Ft(\))p Fr(:)i Ft(This)h(clearly)e(holds)h(if)g(for)g (\014xed)h Fr(g)j Ft(w)m(e)d(tak)m(e)h Fr(f)41 b Ft(su\016cien)m(tly)31 b(small.)149 1287 y(Next,)j(consider)f(a)f(Wigner-v)m(on)h(Neumann-t)m (yp)s(e)g(p)s(oten)m(tial)e Fr(V)2535 1302 y Fo(0)2574 1287 y Ft(\()p Fr(x)p Ft(\))i(de\014ned)h(b)m(y)847 1511 y Fr(V)904 1526 y Fo(0)943 1511 y Ft(\()p Fr(x)p Ft(\))28 b(=)1206 1371 y Fj(\032)1322 1449 y Fp(\000)p Ft(2)p Fr(k)1499 1464 y Fo(0)1538 1449 y Fr(f)g Ft(sin)16 b(2)p Fr(\022)1844 1464 y Fo(0)1884 1449 y Fr(;)503 b Ft(0)27 b Fp(\024)i Fr(x)f Fp(\024)g Fr(f)2843 1413 y Fm(\000)p Fo(1)2937 1449 y Fr(g)1322 1570 y Fp(\000)p Ft(min)1578 1489 y Fj(\000)1624 1570 y Fr(\016)t(k)s(;)1798 1526 y Fq(g)p 1779 1547 76 4 v 1779 1604 a Fo(2)p Fq(x)1864 1489 y Fj(\001)1936 1530 y Fq(k)1973 1539 y Fk(0)p 1936 1547 72 4 v 1954 1604 a Fo(2)2034 1570 y Ft(sin)16 b(2)p Fr(\022)2264 1585 y Fo(0)2304 1570 y Fr(;)83 b(x)28 b(>)g(f)2660 1534 y Fm(\000)p Fo(1)2754 1570 y Fr(g)t(:)149 1735 y Ft(It)g(is)e(fairly)f(straigh)m(tforw)m(ard)i(to)g(c)m(hec)m(k)i(using) f(\(2.2\))o(,)h(\(2.3\))d(that)h Fr(V)2640 1750 y Fo(0)2706 1735 y Ft(leads)g(to)g(an)g(eigen)m(v)-5 b(alue)26 b(at)149 1851 y Fr(k)200 1866 y Fo(0)262 1851 y Ft(with)d Fp(k)p Fr(R)q Ft(\()p Fr(x;)17 b(k)788 1866 y Fo(0)827 1851 y Ft(\))p Fp(k)915 1815 y Fo(2)915 1882 y Fq(L)963 1863 y Fk(2)997 1882 y Fo(\()p Fh(R)1072 1863 y Fk(+)1123 1882 y Fo(\))1183 1851 y Ft(=)27 b Fr(f)1345 1815 y Fm(\000)p Fo(1)1439 1851 y Ft(\(1)r(+)r Fr(O)s Ft(\()p Fr(f)11 b(g)t(;)17 b(e)1921 1815 y Fm(\000)p Fq(g)2014 1851 y Ft(\)\))p Fr(:)23 b Ft(Moreo)m(v)m(er,)j(one)d(cannot)g(get)g(a)f (smaller)f(\(in)149 1981 y(the)28 b(main)c(term\))i(norm)g(for)g(an)h (eigen)m(v)-5 b(alue)26 b(at)g Fr(k)1960 1996 y Fo(0)2026 1981 y Ft(with)g(a)h(p)s(oten)m(tial)e Fr(V)48 b Ft(satisfying)26 b Fp(j)p Fr(V)21 b Ft(\()p Fr(x)p Ft(\))p Fp(j)27 b(\024)i Fr(f)5 b(:)149 2097 y Ft(By)45 b(Lemma)e(2.1,)k(this)d(means)g(that)g (w)m(e)h(cannot)f(ha)m(v)m(e)i(a)e(p)s(oin)m(t)f(mass)h(bigger)f(than)i Fr(f)11 b Ft(\(1)29 b(+)149 2214 y Fr(O)s Ft(\()p Fr(f)11 b(g)t(;)17 b(e)464 2178 y Fm(\000)p Fq(g)558 2214 y Ft(\)\))32 b(in)f(the)i(sp)s(ectral)f(measure)h(for)f(suc)m(h)i(p)s(oten)m(tials.) 42 b(Th)m(us)34 b(Lemma)d(2.2)h(pro)m(vides)h(a)149 2330 y(w)m(a)m(y)-8 b(,)40 b(using)d(p)s(oten)m(tial)e(satisfying)h(the)i (same)f(upp)s(er)h(b)s(ound)f(as)g Fr(V)2679 2345 y Fo(0)2719 2330 y Ft(,)h(to)f(split)f(the)h(eigen)m(v)-5 b(alue)149 2446 y Fr(k)200 2461 y Fo(0)279 2446 y Ft(in)m(to)39 b(t)m(w)m(o)i(nearb)m(y)g(eigen)m(v)-5 b(alues)39 b(with)h(equal)f(w)m (eigh)m(ts)i(without)e(loss)h(of)f(the)h(total)f(w)m(eigh)m(t)149 2562 y(\(in)46 b(the)i(asymptotic)e(regime\).)85 b(The)48 b(fact)f(that)f(it)g(can)h(b)s(e)g(done)h(is)e(not)h(surprising,)j(but) 149 2679 y(the)39 b(construction)g(is)e(not)i(trivial.)57 b(F)-8 b(or)38 b(example,)h(taking)f(just)g Fr(V)59 b Ft(=)2856 2639 y Fo(1)p 2856 2656 36 4 v 2856 2713 a(2)2902 2679 y Ft(\()p Fr(V)2997 2694 y Fo(1)3062 2679 y Ft(+)26 b Fr(V)3221 2694 y Fo(2)3260 2679 y Ft(\))p Fr(;)38 b Ft(where)i Fr(V)3708 2694 y Fq(i)149 2795 y Ft(is)i(giv)m(en)g(b)m(y)h (the)f(same)g(expression)h(as)f Fr(V)1762 2810 y Fo(0)1843 2795 y Ft(at)g Fr(k)2023 2810 y Fq(i)2051 2795 y Fr(;)g Ft(seems)g(to)g(lead)f(to)h(a)f(loss)h(of)f(a)h(constan)m(t)149 2911 y(factor)35 b(in)f(the)h(total)e(mass)h(of)h(t)m(w)m(o)g(eigen)m (v)-5 b(alues)35 b(and)f(th)m(us)i(do)s(es)f(not)g(w)m(ork)g(for)f(the) h(Splitting)149 3027 y(Lemma.)54 b(Moreo)m(v)m(er,)39 b(one)e(cannot)g(hop)s(e)f(for)g(a)h(co)s(e\016cien)m(t)g(smaller)d (than)j(2)f(in)g(fron)m(t)g(of)g Fr(f)3642 2991 y Fm(\000)p Fo(1)149 3144 y Ft(in)48 b(\(2.5\))o(.)90 b(As)48 b(the)h(construction) f(b)s(elo)m(w)g(will)e(sho)m(w,)53 b(that)47 b(w)m(ould)h(ha)m(v)m(e)i (led)d(b)m(y)i(splitting)149 3260 y(to)34 b(sp)s(ectral)g(measures)i (giving)c(arbitrary)i(large)f(w)m(eigh)m(t)h(to)g(a)g(\014xed)i (\014nite)e(in)m(terv)-5 b(al.)47 b(But)35 b(the)149 3385 y(sp)s(ectral)26 b(measure)f Fr(\026)h Ft(is)e(w)m(ell)h(kno)m(wn) i(to)e(satisfy)1958 3305 y Fj(R)2067 3337 y Fq(d\026)p Fo(\()p Fq(t)p Fo(\))p 2051 3362 190 4 v 2051 3419 a(1+)p Fm(j)p Fq(t)p Fm(j)2206 3400 y Fk(2)2279 3385 y Fr(<)i(C)r(;)e Ft(with)g Fr(C)33 b Ft(uniform)23 b(for)i(uniformly)149 3513 y(b)s(ounded)34 b Fr(V)5 b(:)33 b Ft(Therefore,)g(the)g(factor)g (2)f(in)g(the)h(Splitting)d(Lemma)h(is)h(optimal.)249 3629 y(After)f(these)h(preliminary)c(remarks,)k(let)e(us)h(consider)g (the)h(general)e(case.)44 b(On)31 b(the)g Fr(n)p Ft(th)g(step)149 3758 y(of)h(our)h(construction,)f(w)m(e)i(are)e(going)f(to)h(lo)s(ok)f (at)h(2)2083 3722 y Fq(n)2162 3758 y Ft(energies)h Fr(k)2585 3707 y Fo(\()p Fq(n)p Fo(\))2582 3783 y Fq(j)2687 3758 y Fr(;)g(j)g Ft(=)28 b(1)p Fr(;)17 b(:)g(:)g(:)e Ft(2)3196 3722 y Fq(n)3243 3758 y Fr(;)33 b Ft(ordered)g(in)149 3874 y(increasing)g(order.)45 b(W)-8 b(e)33 b(are)g(going)f(to)h (assume)g(that)g(for)f(ev)m(ery)j Fr(n;)f Ft(all)d(these)j(energies)f (lie)f(in)g(a)149 3990 y(\014xed)i(compact)e(in)m(terv)-5 b(al)32 b(a)m(w)m(a)m(y)i(from)d(zero;)i(moreo)m(v)m(er,)g(w)m(e)h(are) e(going)f(to)i(assume)g(that)149 4165 y(\(2.6\))916 b(2max)1496 4180 y Fq(j;j)1578 4161 y Fe(0)1603 4165 y Fp(j)p Fr(k)1685 4114 y Fo(\()p Fq(n)p Fo(\))1682 4190 y Fq(j)1809 4165 y Fp(\000)23 b Fr(k)1963 4114 y Fo(\()p Fq(n)p Fo(\))1960 4193 y Fq(j)1993 4174 y Fe(0)2065 4165 y Fp(j)k(\024)h Ft(min)o(\()p Fr(k)2480 4114 y Fo(\()p Fq(n)p Fo(\))2477 4190 y Fq(j)2582 4165 y Ft(\))149 4329 y(for)39 b(ev)m(ery)h Fr(j;)f(n:)g Ft(This)g(assumption)f(is)g(made)g(for)g(tec)m(hnical)g (con)m(v)m(enience)j(and)e(will)d(b)s(e)j(easy)149 4458 y(to)i(satisfy)f(on)h(eac)m(h)h(step)f(of)f(the)i(inductiv)m(e)e (construction.)68 b(W)-8 b(e)41 b(denote)h Fr(\022)3069 4407 y Fo(\()p Fq(n)p Fo(\))3066 4483 y Fq(j)3212 4458 y Ft(=)g Fr(\022)s Ft(\()p Fr(x;)17 b(k)3569 4407 y Fo(\()p Fq(n)p Fo(\))3566 4483 y Fq(j)3671 4458 y Ft(\))p Fr(;)149 4605 y(R)224 4554 y Fo(\()p Fq(n)p Fo(\))223 4631 y Fq(j)354 4605 y Ft(=)28 b Fr(R)q Ft(\()p Fr(x;)17 b(k)724 4554 y Fo(\()p Fq(n)p Fo(\))721 4631 y Fq(j)826 4605 y Ft(\))p Fr(;)29 b(\016)t(k)1021 4554 y Fo(\()p Fq(n)p Fo(\))1018 4631 y Fq(j)1150 4605 y Ft(=)f Fr(k)1308 4554 y Fo(\()p Fq(n)p Fo(\))1305 4631 y(2)p Fq(j)1425 4605 y Fp(\000)15 b Fr(k)1571 4554 y Fo(\()p Fq(n)p Fo(\))1568 4631 y(2)p Fq(j)t Fm(\000)p Fo(1)1730 4605 y Fr(;)29 b Ft(and)g Fr(\016)t(\022)2067 4554 y Fo(\()p Fq(n)p Fo(\))2064 4631 y Fq(j)2197 4605 y Ft(=)f Fr(\022)2349 4554 y Fo(\()p Fq(n)p Fo(\))2346 4631 y(2)p Fq(j)2466 4605 y Fp(\000)15 b Fr(\022)2606 4554 y Fo(\()p Fq(n)p Fo(\))2603 4631 y(2)p Fq(j)t Fm(\000)p Fo(1)2766 4605 y Fr(:)29 b Ft(Then)h Fr(V)3152 4569 y Fo(\()p Fq(n)p Fo(\))3253 4605 y Ft(\()p Fr(x)p Ft(\))g(is)e(going)149 4756 y(to)33 b(b)s(e)f(de\014ned)i(b)m(y) g Fr(V)951 4720 y Fo(\()p Fq(n)p Fo(\))1053 4756 y Ft(\()p Fr(x)p Ft(\))28 b(=)1316 4681 y Fj(P)1421 4707 y Fo(2)1456 4684 y Fl(n)p Fe(\000)p Fk(1)1421 4785 y Fq(j)t Fo(=1)1598 4756 y Fr(V)1677 4705 y Fo(\()p Fq(n)p Fo(\))1655 4781 y Fq(j)1778 4756 y Ft(\()p Fr(x)p Ft(\))p Fr(;)33 b Ft(where)149 4881 y(\(2.7\))149 5226 y Fr(V)228 5175 y Fo(\()p Fq(n)p Fo(\))206 5251 y Fq(j)330 5226 y Ft(\()p Fr(x)p Ft(\))28 b(=)592 4932 y Fj(8)592 5022 y(>)592 5051 y(>)592 5081 y(>)592 5111 y(<)592 5291 y(>)592 5320 y(>)592 5350 y(>)592 5380 y(:)722 5015 y Fp(\000)p Ft(2)p Fr(f)907 4964 y Fo(\()p Fq(n)p Fo(\))896 5040 y Fq(j)1009 5015 y Fr(k)1063 4964 y Fo(\()p Fq(n)p Fm(\000)p Fo(1\))1060 5040 y Fq(j)1272 5015 y Ft(sin\()p Fr(\022)1478 4964 y Fo(\()p Fq(n)p Fo(\))1475 5040 y(2)p Fq(j)t Fm(\000)p Fo(1)1659 5015 y Ft(+)22 b Fr(\022)1805 4964 y Fo(\()p Fq(n)p Fo(\))1802 5040 y(2)p Fq(j)1907 5015 y Ft(\))p Fr(;)746 b Ft(0)28 b Fp(\024)g Fr(x)g Fp(\024)g Fr(x)3143 4964 y Fo(\()p Fq(n)p Fo(\))3143 5069 y Fq(j;\031)r Fm(\000)p Fq(\013)3335 5034 y Fk(\()p Fl(n)p Fk(\))3335 5091 y Fl(j)722 5202 y Ft(2)p Fr(f)830 5151 y Fo(\()p Fq(n)p Fo(\))819 5227 y Fq(j)932 5202 y Fr(k)986 5151 y Fo(\()p Fq(n)p Fm(\000)p Fo(1\))983 5227 y Fq(j)1195 5202 y Ft(sin)o(\()p Fr(\022)1400 5151 y Fo(\()p Fq(n)p Fo(\))1397 5227 y(2)p Fq(j)t Fm(\000)p Fo(1)1582 5202 y Ft(+)22 b Fr(\022)1728 5151 y Fo(\()p Fq(n)p Fo(\))1725 5227 y(2)p Fq(j)1830 5202 y Ft(\))p Fr(;)823 b(x)2773 5151 y Fo(\()p Fq(n)p Fo(\))2773 5256 y Fq(j;\031)r Fm(\000)p Fq(\013)2965 5221 y Fk(\()p Fl(n)p Fk(\))2965 5278 y Fl(j)3087 5202 y Fp(\024)29 b Fr(x)f Fp(\024)g Ft(\()p Fr(f)3478 5151 y Fo(\()p Fq(n)p Fo(\))3467 5227 y Fq(j)3579 5202 y Ft(\))3617 5165 y Fm(\000)p Fo(1)3712 5202 y Fr(g)722 5423 y Fp(\000)p Ft(min)979 5312 y Fj(\020)1039 5423 y Fr(\016)t(k)1140 5372 y Fo(\()p Fq(n)p Fo(\))1137 5449 y Fq(j)1241 5423 y Fr(;)1295 5379 y Fq(g)1329 5387 y Fl(n)p 1295 5400 77 4 v 1296 5457 a Fo(2)p Fq(x)1381 5312 y Fj(\021)1467 5360 y Fq(k)1506 5325 y Fk(\()p Fl(n)p Fe(\000)p Fk(1\))1504 5382 y Fl(j)p 1467 5400 208 4 v 1553 5457 a Fo(4)1685 5423 y Ft(\(sin)16 b(2)p Fr(\022)1956 5372 y Fo(\()p Fq(n)p Fo(\))1953 5449 y(2)p Fq(j)t Fm(\000)p Fo(1)2137 5423 y Ft(+)22 b(sin)17 b(2)p Fr(\022)2469 5372 y Fo(\()p Fq(n)p Fo(\))2466 5449 y(2)p Fq(j)2570 5423 y Ft(\))p Fr(;)83 b(x)29 b(>)e Ft(\()p Fr(f)3002 5372 y Fo(\()p Fq(n)p Fo(\))2991 5449 y Fq(j)3104 5423 y Ft(\))3142 5387 y Fm(\000)p Fo(1)3236 5423 y Fr(g)t(:)p eop %%Page: 7 7 7 6 bop 1220 66 a Fn(SINGULAR)31 b(CONTINUOUS)g(SPECTR)n(UM)1033 b(7)149 283 y Ft(Here)34 b Fr(\013)443 232 y Fo(\()p Fq(n)p Fo(\))442 309 y Fq(j)577 283 y Ft(is)e(de\014ned)i(b)m(y)f Fr(f)1205 232 y Fo(\()p Fq(n)p Fo(\))1194 309 y Fq(j)1324 283 y Ft(sin)16 b Fr(\013)1523 232 y Fo(\()p Fq(n)p Fo(\))1522 309 y Fq(j)1652 283 y Ft(=)28 b Fr(\016)t(k)1857 232 y Fo(\()p Fq(n)p Fo(\))1854 309 y Fq(j)1959 283 y Fr(;)k Ft(and)1325 504 y Fr(x)1380 453 y Fo(\()p Fq(n)p Fo(\))1380 532 y Fq(j;\014)1510 504 y Ft(=)27 b(min)o Fp(f)p Fr(x)h Ft(:)g Fr(\016)t(\022)2059 453 y Fo(\()p Fq(n)p Fo(\))2056 530 y Fq(j)2161 504 y Ft(\()p Fr(x)p Ft(\))g(=)f Fr(\014)6 b Fp(g)p Fr(:)149 695 y Ft(Notice)34 b(that)f(the)h(v)-5 b(alue)34 b(of)f Fr(g)1251 710 y Fq(n)1331 695 y Ft(is)g(going)g(to)g (b)s(e)h(the)g(same)g(for)f(all)f Fr(j:)i Ft(Also,)f(as)h(b)s(efore,)g (w)m(e)h(will)149 824 y(denote)45 b(~)-52 b Fr(g)519 839 y Fq(n)606 824 y Ft(=)41 b(exp)q(\()p Fr(g)961 773 y Fo(3)p Fq(=)p Fo(4)957 834 y Fq(n)1070 824 y Ft(\))p Fr(:)g Ft(The)g(parameters)f Fr(g)1943 839 y Fq(n)1990 824 y Fr(;)h(f)2117 773 y Fo(\()p Fq(n)p Fo(\))2106 849 y Fq(j)2258 824 y Ft(and)g Fr(\016)t(k)2557 773 y Fo(\()p Fq(n)p Fo(\))2554 849 y Fq(j)2699 824 y Ft(are)g(to)f(b)s(e)g(c)m (hosen)i(on)f(the)149 971 y Fr(n)p Ft(th)32 b(step)h(of)e (construction.)43 b(The)32 b(energies)g Fr(k)1864 920 y Fo(\()p Fq(n)p Fm(\000)p Fo(1\))1861 997 y Fq(j)2088 971 y Ft(are)f(giv)m(en)h(\(from)e(the)i(previous)g(step\).)44 b(On)149 1119 y(the)38 b Fr(n)p Ft(th)g(step,)i(eac)m(h)e Fr(k)1028 1068 y Fo(\()p Fq(n)p Fm(\000)p Fo(1\))1025 1144 y Fq(j)1257 1119 y Ft(splits)f(in)m(to)g(t)m(w)m(o)h(eigen)m(v)-5 b(alues)37 b Fr(k)2473 1068 y Fo(\()p Fq(n)p Fo(\))2470 1144 y(2)p Fq(j)t Fm(\000)p Fo(1)2632 1119 y Fr(;)17 b(k)2730 1068 y Fo(\()p Fq(n)p Fo(\))2727 1144 y(2)p Fq(j)2832 1119 y Ft(;)37 b(that)g(is,)h(the)g(in)m(terv)-5 b(al)149 1266 y(\()p Fr(k)241 1215 y Fo(\()p Fq(n)p Fo(\))238 1292 y(2)p Fq(j)t Fm(\000)p Fo(1)400 1266 y Fr(;)17 b(k)498 1215 y Fo(\()p Fq(n)p Fo(\))495 1292 y(2)p Fq(j)600 1266 y Ft(\))43 b(is)h(cen)m(tered)h(at)f Fr(k)1374 1215 y Fo(\()p Fq(n)p Fm(\000)p Fo(1\))1371 1292 y Fq(j)1609 1266 y Ft(and)g(has)g(length)f Fr(\016)t(k)2405 1215 y Fo(\()p Fq(n)p Fo(\))2402 1292 y Fq(j)2507 1266 y Fr(:)h Ft(Throughout)g(the)g(pap)s(er,)j(w)m(e)149 1414 y(will)35 b(assume)i(the)f(relationship)f Fr(f)1445 1363 y Fo(\()p Fq(n)p Fo(\))1434 1439 y Fq(j)1581 1414 y Ft(=)j(~)-53 b Fr(g)1738 1429 y Fq(n)1785 1414 y Fr(\016)t(k)1886 1363 y Fo(\()p Fq(n)p Fo(\))1883 1439 y Fq(j)1988 1414 y Fr(:)36 b Ft(In)h(the)g(actual)e(construction,)j Fr(V)3315 1378 y Fo(\()p Fq(n)p Fo(\))3417 1414 y Ft(\()p Fr(x)p Ft(\))e(will)149 1561 y(b)s(e)42 b(giv)m(en)f(b)m(y)i(\(2.7\))d(only)h (to)g(the)g(righ)m(t)f(of)h(some)g(v)-5 b(alue)40 b(of)g Fr(x)j Ft(=)f Fr(x)2734 1576 y Fq(n)2781 1561 y Fr(;)f Ft(and)g Fr(R)q Ft(\()p Fr(x)3215 1576 y Fq(n)3263 1561 y Fr(;)17 b(k)3361 1510 y Fo(\()p Fq(n)p Fo(\))3358 1587 y Fq(j)3462 1561 y Ft(\))42 b Fp(6)p Ft(=)g(1)p Fr(:)149 1694 y Ft(Ho)m(w)m(ev)m(er)f(in)d(this)g(section)g(w)m(e)h(will)d (consider)j Fr(V)1998 1658 y Fo(\()p Fq(n)p Fo(\))2100 1694 y Ft(\()p Fr(x)p Ft(\))g(de\014ned)h(b)m(y)g(\(2.7\))e(on)g([0)p Fr(;)17 b Fp(1)p Ft(\))37 b(and)i(w)m(e)149 1828 y(will)33 b(assume)j(that)e Fr(R)q Ft(\(0)p Fr(;)17 b(k)s Ft(\))32 b(=)f(1)k(and)g Fr(\022)1650 1777 y Fo(\()p Fq(n)p Fo(\))1647 1853 y Fq(j)1752 1828 y Ft(\(0)p Fr(;)17 b(k)s Ft(\))31 b(=)h(0)i(for)h(all)e Fr(k)s(:)i Ft(Let)g(us)g(in)m(tro)s(duce)g(one)h (more)149 1975 y(parameter)31 b(that)f(w)m(e)i(need,)g Fr(a)1269 1990 y Fq(n)1344 1975 y Ft(=)27 b(min)1610 1990 y Fq(j;j)1692 1971 y Fe(0)1717 1975 y Fp(j)p Fr(k)1799 1924 y Fo(\()p Fq(n)p Fm(\000)p Fo(1\))1796 2001 y Fq(j)2009 1975 y Fp(\000)19 b Fr(k)2159 1924 y Fo(\()p Fq(n)p Fm(\000)p Fo(1\))2156 2004 y Fq(j)2189 1985 y Fe(0)2351 1975 y Fp(j)30 b Ft(\(essen)m(tially)-8 b(,)31 b(in)f(the)h(construction)149 2126 y(pro)s(cess)38 b(w)m(e)f(will)c(ha)m(v)m(e)38 b Fr(a)1107 2141 y Fq(n)1188 2126 y Ft(=)33 b(min)1460 2141 y Fq(j)1496 2126 y Ft(\()p Fr(\016)t(k)1635 2075 y Fo(\()p Fq(n)p Fm(\000)p Fo(1\))1632 2151 y Fq(j)1827 2126 y Ft(\)\).)54 b(T)-8 b(o)36 b(k)m(eep)h(notation)e(compact,)i (henceforth)g(in)149 2247 y(this)c(section)f(w)m(e)i(are)f(going)e(to)h (omit)f(the)i(index)g(\\)p Fr(n)p Ft(")f(for)g(most)g(v)-5 b(ariables,)32 b(including)f Fr(V)54 b Ft(\(but)149 2363 y(with)33 b(an)f(exception)h(of)f Fr(k)s Ft('s,)i(where)f(mem)m(b)s (ers)g(of)f(t)m(w)m(o)h(di\013eren)m(t)g(steps)h(participate)d (explicitly)149 2479 y(in)h(a)h(construction\).)149 2672 y FA(Lemma)38 b(2.3)f Ft(\(Splitting)30 b(Lemma\))p FA(.)40 b Fi(L)-5 b(et)36 b Fr(V)21 b Ft(\()p Fr(x)p Ft(\))35 b Fi(b)-5 b(e)35 b(given)f(by)44 b Ft(\(2.7\))o Fi(,)35 b(and)f(assume)g(that)149 2969 y Ft(\(2.8\))221 b(4)637 2874 y Fj(X)692 3084 y Fq(j)797 2969 y Fr(f)845 2984 y Fq(j)881 2969 y Fr(k)935 2918 y Fo(\()p Fq(n)p Fm(\000)p Fo(1\))932 2994 y Fq(j)1155 2969 y Fr(<)1269 2901 y Ft(1)p 1269 2946 49 4 v 1269 3037 a(3)1328 2969 y Fr(a;)99 b(g)31 b(>>)d Ft(1)p Fr(;)66 b Ft(2)1954 2928 y Fq(n)2001 2969 y Fr(g)t(a)2103 2928 y Fm(\000)p Fo(2)2213 2844 y(2)2248 2821 y Fl(n)p Fe(\000)p Fk(1)2219 2874 y Fj(X)2230 3084 y Fq(j)t Fo(=1)2386 2969 y Fr(f)2434 2984 y Fq(j)2498 2969 y Fr(<<)28 b Ft(1)p Fr(;)99 b(\016)t(k)2951 2984 y Fq(j)3015 2969 y Fr(<)3153 2901 y Ft(1)p 3129 2946 98 4 v 3129 3037 a(12)3236 2969 y Fr(a;)149 3267 y Fi(for)35 b Fr(j)f Ft(=)27 b(1)p Fr(;)17 b(:)g(:)g(:)f(;)h Ft(2)799 3231 y Fq(n)p Fm(\000)p Fo(1)935 3267 y Fr(:)35 b Fi(Then)g(for)f(any)h Fr(l)30 b Ft(=)d(1)p Fr(;)17 b(:)g(:)g(:)f(;)h Ft(2)2073 3231 y Fq(n)p Fm(\000)p Fo(1)2210 3267 y Fr(;)149 3558 y Ft(\(2.9\))289 b Fp(k)p Fr(R)q Ft(\()p Fr(x;)17 b(k)955 3507 y Fo(\()p Fq(n)p Fo(\))952 3590 y(2\()p Fq(l)q Fm(\000)p Fo(1\)+)p Fq(i)1237 3558 y Ft(\))p Fp(k)1325 3517 y Fo(2)1325 3584 y Fq(L)1373 3565 y Fk(2)1408 3584 y Fo(\()p Fh(R)1483 3565 y Fk(+)1534 3584 y Fo(\))1593 3558 y Ft(=)28 b(2)p Fr(f)1805 3517 y Fm(\000)p Fo(1)1794 3586 y Fq(l)1915 3388 y Fj( )1994 3558 y Ft(1)22 b(+)g Fr(O)2257 3388 y Fj( )2335 3558 y Ft(2)2384 3517 y Fq(n)2431 3558 y Fr(e)2476 3517 y Fq(g)2517 3558 y Fr(a)2568 3517 y Fm(\000)p Fo(2)2679 3434 y(2)2714 3410 y Fl(n)p Fe(\000)p Fk(1)2685 3464 y Fj(X)2696 3674 y Fq(j)t Fo(=1)2851 3558 y Fr(f)2899 3573 y Fq(j)2936 3558 y Fr(;)e Ft(~)-52 b Fr(g)3031 3517 y Fm(\000)p Fo(1)p Fq(=)p Fo(2)3195 3558 y Fr(g)3246 3388 y Fj(!)o(!)3420 3558 y Fr(;)149 3847 y Fi(wher)-5 b(e)35 b Fr(i)28 b Ft(=)f(1)p Fr(;)17 b Ft(2)p Fr(:)149 4040 y Fi(R)-5 b(emark.)43 b Ft(The)27 b(condition)e Fr(g)31 b(>>)c Ft(1)g(means)f(that)g Fr(g)k Ft(needs)e(to)e(b)s(e)h (greater)f(than)h(some)f(univ)m(ersal)149 4156 y(constan)m(t,)33 b(the)e(v)-5 b(alue)30 b(of)g(whic)m(h)i(can)f(b)s(e)g(deriv)m(ed)h (from)d(the)i(pro)s(of.)43 b(The)31 b(same)g(in)m(terpretation)149 4273 y(applies)36 b(to)h(the)g(other)g(condition)e(in)h(\(2.8\))g(in)m (v)m(olving)g(\\)p Fr(<<)p Ft(".)55 b(Notice)37 b(also)e(that)i(the)g (fourth)149 4389 y(condition)d(basically)f(follo)m(ws)g(from)h(the)h (\014rst)g(t)m(w)m(o)g(and)g Fr(\016)t(k)2385 4404 y Fq(j)2453 4389 y Ft(=)f(~)-52 b Fr(g)2611 4353 y Fm(\000)p Fo(1)2704 4389 y Fr(f)2752 4404 y Fq(j)2789 4389 y Ft(,)35 b(but)g(w)m(e)h(state)f(it)f(sepa-)149 4505 y(rately)f(for)f(con)m(v)m (enience.)249 4738 y(W)-8 b(e)46 b(are)g(going)e(to)i(\014x)g Fr(l)53 b Ft(=)d(1)p Fr(;)45 b(i)51 b Ft(=)f(1)45 b(and)h(pro)m(v)m(e)h (the)f(result)g(for)f(this)g(case;)54 b(other)46 b Fr(l)r(;)17 b(i)149 4866 y Ft(are)39 b(treated)g(in)f(exactly)h(the)g(same)g(w)m(a) m(y)-8 b(.)62 b(W)-8 b(rite)38 b Fr(V)60 b Ft(=)38 b Fr(V)2357 4881 y Fo(1)2422 4866 y Ft(+)27 b Fr(W)m(;)39 b Ft(where)g Fr(V)3024 4881 y Fo(1)3102 4866 y Ft(=)f Fr(V)3294 4815 y Fo(\()p Fq(n)p Fo(\))3273 4890 y(1)3435 4866 y Ft(and)g Fr(W)149 4982 y Ft(is)i(the)g(rest)h(of)e(the)h(p)s (oten)m(tial.)64 b(Direct)39 b(substitution)g(of)46 b(\(2.7\))40 b(in)m(to)f(\(2.2\))o(,)j(\(2.3\))d(and)h(some)149 5099 y(trigonometry)31 b(leads)i(to)f(the)h(follo)m(wing)d(equations)j(v)-5 b(alid)31 b(for)h Fr(x)c(<)f(f)2748 5058 y Fm(\000)p Fo(1)2737 5124 y(1)2842 5099 y Fr(g)k Ft(:)149 5374 y(\(2.10\))154 b(\(log)17 b Fr(R)809 5332 y Fo(2)848 5374 y Ft(\()p Fr(x;)g(k)1039 5323 y Fo(\()p Fq(n)p Fo(\))1036 5398 y(1)1141 5374 y Ft(\))1179 5332 y Fm(0)1230 5374 y Ft(=)28 b Fp(\000)1421 5306 y Fr(k)1475 5255 y Fo(\()p Fq(n)p Fm(\000)p Fo(1\))1472 5331 y(1)p 1421 5351 247 4 v 1466 5463 a Fr(k)1520 5412 y Fo(\()p Fq(n)p Fo(\))1517 5488 y(1)1698 5347 y Ft(~)1677 5374 y Fr(f)1725 5389 y Fo(1)1765 5374 y Ft(\()p Fr(x)p Ft(\)\(cos)17 b Fr(\016)t(\022)2173 5389 y Fo(1)2235 5374 y Fp(\000)23 b Ft(cos\()p Fr(\022)2548 5389 y Fo(2)2610 5374 y Ft(+)f(3)p Fr(\022)2802 5389 y Fo(1)2842 5374 y Ft(\)\))g(+)3048 5306 y Fr(W)14 b Ft(\()p Fr(x)p Ft(\))p 3048 5351 237 4 v 3089 5463 a Fr(k)3143 5412 y Fo(\()p Fq(n)p Fo(\))3140 5488 y(1)3312 5374 y Ft(sin)i(2)p Fr(\022)3542 5389 y Fo(1)p eop %%Page: 8 8 8 7 bop 149 66 a Fn(8)1324 b(ALEXANDER)23 b(KISELEV)338 357 y Ft(\()p Fr(\016)t(\022)468 372 y Fo(1)508 357 y Ft(\))546 316 y Fm(0)652 357 y Ft(=)83 b Fr(\016)t(k)909 372 y Fo(1)965 187 y Fj( )1044 357 y Ft(1)22 b Fp(\000)1300 290 y Fr(V)f Ft(\()p Fr(x)p Ft(\))p 1224 334 361 4 v 1224 447 a(2)p Fr(k)1327 396 y Fo(\()p Fq(n)p Fo(\))1324 471 y(1)1429 447 y Fr(k)1483 396 y Fo(\()p Fq(n)p Fo(\))1480 471 y(2)1595 357 y Ft(\(1)h(+)g(cos)17 b(2)p Fr(\022)2043 372 y Fo(2)2083 357 y Ft(\))2121 187 y Fj(!)2222 357 y Ft(+)-2149 b(\(2.11\))811 693 y(+)904 523 y Fj( )992 626 y Fr(k)1046 575 y Fo(\()p Fq(n)p Fm(\000)p Fo(1\))1043 650 y(1)p 992 670 247 4 v 1038 783 a Fr(k)1092 732 y Fo(\()p Fq(n)p Fo(\))1089 807 y(1)1270 667 y Ft(~)1249 693 y Fr(f)1297 708 y Fo(1)1336 693 y Ft(\()p Fr(x)p Ft(\)\(1)22 b Fp(\000)h Ft(cos)17 b(2\()p Fr(\022)1955 708 y Fo(1)2017 693 y Ft(+)22 b Fr(\022)2160 708 y Fo(2)2200 693 y Ft(\)\))g Fp(\000)2407 626 y Fr(W)14 b Ft(\()p Fr(x)p Ft(\))p 2407 670 237 4 v 2448 783 a Fr(k)2502 732 y Fo(\()p Fq(n)p Fo(\))2499 807 y(1)2671 693 y Ft(sin\()p Fr(\022)2874 708 y Fo(1)2936 693 y Ft(+)22 b Fr(\022)3079 708 y Fo(2)3118 693 y Ft(\))3156 523 y Fj(!)3252 693 y Ft(sin)16 b Fr(\016)t(\022)3480 708 y Fo(1)3520 693 y Ft(;)149 987 y(here)390 960 y(~)369 987 y Fr(f)417 1002 y Fo(1)456 987 y Ft(\()p Fr(x)p Ft(\))42 b(=)f Fp(\006)p Fr(f)871 1002 y Fo(1)952 987 y Ft(with)f(a)g(c)m(hange)h(from)f(+)g(to) g Fp(\000)h Ft(at)g Fr(x)g Ft(=)g Fr(x)2597 1002 y Fo(1)p Fq(;\031)r Fm(\000)p Fq(\013)2795 1011 y Fk(1)2835 987 y Fr(:)f Ft(F)-8 b(or)40 b(the)h(rest)g(of)f(this)149 1103 y(section,)i(w)m(e)f(are)e(also)g(going)f(to)h(omit)f(the)i(index) g(\\1")e(in)h Fr(x)2468 1118 y Fo(1)p Fq(;\014)2610 1103 y Ft(and)h(write)f(simply)f Fr(x)3437 1118 y Fq(\014)3525 1103 y Ft(for)h Fr(x)149 1219 y Ft(suc)m(h)34 b(that)f Fr(\016)t(\022)673 1234 y Fo(1)713 1219 y Ft(\()p Fr(x)806 1234 y Fq(\014)853 1219 y Ft(\))28 b(=)f Fr(\014)6 b(:)149 1452 y Fi(R)-5 b(emark.)64 b Ft(Equations)39 b(\(2.10\))g(and)h (\(2.11\))e(are)i(fairly)d(complicated,)j(but)g(not)f(all)f(terms)h (are)149 1568 y(of)g(equal)f(imp)s(ortance.)60 b(A)m(t)39 b(the)g(heart)g(of)f(the)h(matter)e(is)i(the)g(follo)m(wing)c(simpler)i (nonlinear)149 1684 y(dynamical)31 b(system:)1355 1872 y(\(log)17 b Fr(R)1611 1831 y Fo(2)1650 1872 y Ft(\))1688 1831 y Fm(0)1739 1872 y Ft(=)28 b Fp(\000)1942 1846 y Ft(~)1920 1872 y Fr(f)1968 1887 y Fo(1)2008 1872 y Ft(\()p Fr(x)p Ft(\))17 b(cos)g Fr(\016)t(\022)2395 1887 y Fo(1)149 1872 y Ft(\(2.12\))1285 2027 y(\()p Fr(\016)t(\022)1415 2042 y Fo(1)1454 2027 y Ft(\))1492 1986 y Fm(0)1543 2027 y Ft(=)28 b Fr(\016)t(k)1745 2042 y Fo(1)1806 2027 y Ft(+)1925 2001 y(~)1904 2027 y Fr(f)1952 2042 y Fo(1)1992 2027 y Ft(\()p Fr(x)p Ft(\))17 b(sin)f Fr(\016)t(\022)2368 2042 y Fo(1)2408 2027 y Fr(:)149 2212 y Ft(In)44 b(some)f(sense,)48 b(w)m(e)c(will)d(sho)m(w)j(that)f(the)g(rest)h(of)f(terms)g(in)f (\(2.10\))o(,)k(\(2.11\))d(pro)s(duce)g(only)149 2328 y(small)33 b(corrections)j(to)e(the)i(b)s(eha)m(vior)f(of)f(this)h (system.)52 b(It)36 b(is)e(quite)h(instructiv)m(e)h(\(and)f(tec)m(hni-) 149 2444 y(cally)30 b(simple\))g(to)g(pro)m(v)m(e)i(a)f(v)m(ersion)h (of)e(the)i(Splitting)c(Lemma)i(for)g(the)h(system)h(\(2.12\),)f(but)g (to)149 2560 y(sa)m(v)m(e)k(the)e(space,)g(w)m(e)h(are)f(going)e(to)h (treat)g(directly)g(the)h(system)h(\(2.10\))o(,)f(\(2.11\))o(.)249 2793 y(F)-8 b(rom)31 b(\(2.11\))h(and)h Fr(\016)t(\022)1069 2808 y Fo(1)1108 2793 y Ft(\(0\))28 b(=)f(0)p Fr(;)33 b Ft(w)m(e)g(can)g(write)149 3088 y(\(2.13\))701 b Fr(\016)t(\022)1192 3103 y Fo(1)1259 3088 y Ft(=)28 b Fr(\016)t(k)1461 3103 y Fo(1)1568 2933 y Fq(x)1516 2953 y Fj(Z)1526 3234 y Fo(0)1633 3088 y Fr(e)1678 2990 y Fd(R)1725 3010 y Fl(x)1711 3068 y(y)1776 3043 y Fq(\030)s Fo(\()p Fq(s)p Fo(\))11 b Fq(ds)1981 3088 y Ft(\(1)22 b(+)g Fr(O)s Ft(\()p Fp(j)p Fr(V)f Ft(\()p Fr(y)t Ft(\))p Fp(j)p Ft(\)\))c Fr(dy)t(;)149 3390 y Ft(where)149 3640 y(\(2.14\))173 b Fr(\030)5 b Ft(\()p Fr(y)t Ft(\))26 b(=)888 3573 y Fr(k)942 3522 y Fo(\()p Fq(n)p Fm(\000)p Fo(1\))939 3597 y(1)p 888 3617 247 4 v 933 3730 a Fr(k)987 3679 y Fo(\()p Fq(n)p Fo(\))984 3755 y(1)1160 3470 y Fj( )1260 3614 y Ft(~)1239 3640 y Fr(f)1287 3655 y Fo(1)1327 3640 y Ft(\()p Fr(y)t Ft(\)\(1)21 b Fp(\000)h Ft(cos)17 b(2\()p Fr(\022)1941 3655 y Fo(1)2003 3640 y Ft(+)22 b Fr(\022)2146 3655 y Fo(2)2186 3640 y Ft(\)\))g Fp(\000)2400 3573 y Fr(W)14 b Ft(\()p Fr(y)t Ft(\))p 2394 3617 V 2394 3730 a Fr(k)2448 3679 y Fo(\()p Fq(n)p Fm(\000)p Fo(1\))2445 3755 y(1)2666 3640 y Ft(sin\()p Fr(\022)2869 3655 y Fo(1)2931 3640 y Ft(+)22 b Fr(\022)3074 3655 y Fo(2)3114 3640 y Ft(\))3152 3470 y Fj(!)3257 3573 y Ft(sin)17 b Fr(\016)t(\022)3486 3588 y Fo(1)p 3257 3617 269 4 v 3326 3709 a Fr(\016)t(\022)3418 3724 y Fo(1)3536 3640 y Fr(:)249 3911 y Ft(W)-8 b(e)33 b(need)h(the)f(follo)m(wing)d(tec)m(hnical)i(lemma)149 4101 y FA(Lemma)41 b(2.4.)j Fi(Under)38 b(the)g(assumptions)f(of)g(the) h(Splitting)g(L)-5 b(emma,)38 b(we)f(have)g(for)h Fr(y)f(<)c(x)h Fp(\024)149 4217 y Fr(f)208 4176 y Fm(\000)p Fo(1)197 4241 y(1)302 4217 y Fr(g)149 4512 y Ft(\(2.15\))899 4357 y Fq(x)847 4376 y Fj(Z)856 4658 y Fq(y)963 4512 y Fr(\030)5 b Ft(\()p Fr(s)p Ft(\))17 b Fr(ds)26 b Ft(=)1429 4357 y Fq(x)1377 4376 y Fj(Z)1386 4658 y Fq(y)1514 4486 y Ft(~)1493 4512 y Fr(f)1541 4527 y Fo(1)1580 4512 y Ft(\()p Fr(s)p Ft(\))1712 4445 y(sin)17 b Fr(\016)t(\022)1941 4460 y Fo(1)p 1712 4489 V 1780 4580 a Fr(\016)t(\022)1872 4595 y Fo(1)2007 4512 y Fr(ds)p Ft(\(1)22 b(+)g Fr(O)s Ft(\()p Fr(\016)t(k)2525 4527 y Fo(1)2563 4512 y Ft(\)\))g(+)g Fr(O)s Ft(\()p Fp(E)2928 4527 y Fq(n)2974 4512 y Ft(\))p Fr(;)149 4836 y Fi(wher)-5 b(e)35 b Fp(E)478 4851 y Fq(n)552 4836 y Ft(=)27 b(2)704 4800 y Fq(n)751 4836 y Ft(\()789 4761 y Fj(P)894 4865 y Fq(j)947 4836 y Fr(f)995 4851 y Fq(j)1032 4836 y Ft(\))p Fr(g)t(a)1172 4800 y Fm(\000)p Fo(2)1266 4836 y Fr(:)149 5029 y Fi(Pr)-5 b(o)g(of.)42 b Ft(Let)32 b(us)i(consider)e(one)h(of)f(the)h(terms)g(en)m(tering)g (in)m(to)e Fr(\030)5 b(;)149 5319 y Ft(\(2.16\))1364 5164 y Fq(x)1312 5184 y Fj(Z)1321 5465 y Fq(y)1428 5319 y Fr(V)1485 5334 y Fq(j)1521 5319 y Ft(\()p Fr(s)p Ft(\))17 b(sin)o(\()p Fr(\022)1862 5334 y Fo(1)1924 5319 y Ft(+)22 b Fr(\022)2067 5334 y Fo(2)2107 5319 y Ft(\))2155 5252 y(sin)17 b Fr(\016)t(\022)2384 5267 y Fo(1)p 2155 5296 V 2223 5388 a Fr(\016)t(\022)2315 5403 y Fo(1)2450 5319 y Fr(ds;)p eop %%Page: 9 9 9 8 bop 1220 66 a Fn(SINGULAR)31 b(CONTINUOUS)g(SPECTR)n(UM)1033 b(9)149 275 y Ft(where)50 b(2)k Fp(\024)g Fr(j)60 b Fp(\024)55 b Ft(2)962 239 y Fq(n)p Fm(\000)p Fo(1)1147 275 y Ft(is)47 b(arbitrary)-8 b(.)89 b(Recall)47 b(that)h Fr(V)2345 290 y Fq(j)2381 275 y Ft(\()p Fr(s)p Ft(\))54 b(=)2708 249 y(~)2687 275 y Fr(f)2735 290 y Fq(j)2771 275 y Ft(\()p Fr(s)p Ft(\))17 b(sin\()p Fr(\022)3113 290 y Fo(2)p Fq(j)t Fm(\000)p Fo(1)3308 275 y Ft(+)32 b Fr(\022)3461 290 y Fo(2)p Fq(j)3534 275 y Ft(\))47 b(for)149 404 y Fr(x)36 b Fp(\024)g Ft(\()p Fr(f)439 419 y Fq(j)475 404 y Ft(\))513 367 y Fm(\000)p Fo(1)608 404 y Fr(g)t(;)g Ft(where)1029 377 y(~)1008 404 y Fr(f)1056 419 y Fq(j)1128 404 y Ft(=)f Fp(\006)p Fr(f)1364 419 y Fq(j)1438 404 y Ft(with)i(only)f(at)h(most)f (one)i(jump)e(\(w)m(e)i(do)f(not)g(assume)g(apri-)149 520 y(ori)44 b(that)h Fr(x)587 535 y Fq(j;\031)r Fm(\000)p Fq(\013)779 545 y Fl(j)865 520 y Fr(<)j Ft(\()p Fr(f)1075 535 y Fq(j)1112 520 y Ft(\))1150 484 y Fm(\000)p Fo(1)1244 520 y Fr(g)t(;)d Ft(but)g(will)e(sho)m(w)j(this)f(b)s(elo)m(w\).)81 b(F)-8 b(or)44 b Fr(x)50 b(>)f Ft(\()p Fr(f)3105 535 y Fq(j)3141 520 y Ft(\))3179 484 y Fm(\000)p Fo(1)3273 520 y Fr(g)t(;)c(V)3453 535 y Fq(j)3489 520 y Ft(\()p Fr(s)p Ft(\))k(=)171 637 y(~)149 664 y Fr(f)197 679 y Fq(j)234 664 y Ft(\()p Fr(s)p Ft(\)\(sin\(2)p Fr(\022)646 679 y Fo(2)p Fq(j)t Fm(\000)p Fo(1)808 664 y Ft(\))22 b(+)h(sin)o(\(2)p Fr(\022)1218 679 y Fo(2)p Fq(j)1290 664 y Ft(\)\))p Fr(;)33 b Ft(where)i Fp(j)1758 637 y Ft(~)1737 664 y Fr(f)1785 679 y Fq(j)1821 664 y Ft(\()p Fr(s)p Ft(\))p Fp(j)28 b(\024)h Ft(\()p Fr(k)2197 613 y Fo(\()p Fq(n)p Fm(\000)p Fo(1\))2194 689 y Fq(j)2389 664 y Fr(g)t Ft(\))p Fr(=)p Ft(2)p Fr(s;)j Fp(j)2730 637 y Ft(~)2709 664 y Fr(f)2768 627 y Fm(0)2757 688 y Fq(j)2794 664 y Ft(\()p Fr(s)p Ft(\))p Fp(j)c(\024)h Ft(\()p Fr(k)3170 613 y Fo(\()p Fq(n)p Fm(\000)p Fo(1\))3167 689 y Fq(j)3362 664 y Fr(g)t Ft(\))p Fr(=)p Ft(\(2)p Fr(s)3633 627 y Fo(2)3671 664 y Ft(\))p Fr(:)149 785 y Ft(Notice)k(that)f(b)m(y)h(assumption)f(\(2.8\))g(w)m(e)i(ha)m(v)m(e) 149 985 y(\(2.17\))50 b Fp(j)p Ft(\()p Fr(\022)560 1000 y Fo(2)p Fq(j)t Fm(\000)p Fo(1)730 985 y Ft(+)8 b Fr(\022)859 1000 y Fo(2)p Fq(j)940 985 y Fp(\000)g Fr(\022)1070 1000 y Fo(1)1119 985 y Fp(\000)g Fr(\022)1249 1000 y Fo(2)1290 985 y Ft(\))1328 944 y Fm(0)1351 985 y Fp(j)28 b(\025)1522 917 y Fr(a)p 1522 962 52 4 v 1523 1053 a Ft(2)1583 985 y Fr(;)83 b Fp(j)p Ft(\(2)p Fr(\022)1853 1000 y Fo(2)p Fq(j)t Fm(\000)p Fo(1)2023 985 y Fp(\000)8 b Fr(\022)2153 1000 y Fo(1)2202 985 y Fp(\000)g Fr(\022)2332 1000 y Fo(2)2373 985 y Ft(\))2411 944 y Fm(0)2434 985 y Fp(j)28 b(\025)2605 917 y Fr(a)p 2605 962 V 2606 1053 a Ft(2)2666 985 y Fr(;)83 b Fp(j)p Ft(\(2)p Fr(\022)2936 1000 y Fo(2)p Fq(j)3016 985 y Fp(\000)8 b Fr(\022)3146 1000 y Fo(1)3195 985 y Fp(\000)g Fr(\022)3325 1000 y Fo(2)3366 985 y Ft(\))3404 944 y Fm(0)3427 985 y Fp(j)28 b(\025)3598 917 y Fr(a)p 3598 962 V 3599 1053 a Ft(2)3659 985 y Fr(:)149 1192 y Ft(Substituting)38 b(the)g(expression)i(for)d Fr(V)1574 1207 y Fq(j)1648 1192 y Ft(in)m(to)h(\(2.16\))f(and)h(using)g(form)m (ula)e(for)i(the)g(pro)s(duct)g(of)149 1309 y(sines,)c(w)m(e)f(reduce)h (matters)e(to)h(estimation)d(of)i(the)h(in)m(tegrals)f(of)g(t)m(yp)s(e) 960 1587 y Fr(I)1003 1602 y Fm(\006)1062 1587 y Ft(\()p Fr(x;)17 b(y)t Ft(\))27 b(=)1472 1431 y Fq(x)1419 1451 y Fj(Z)1428 1732 y Fq(y)1557 1560 y Ft(~)1536 1587 y Fr(f)1584 1602 y Fq(j)1637 1587 y Ft(cos\(2)p Fr(\022)1899 1602 y Fo(2)p Fq(j)1994 1587 y Fp(\006)22 b Ft(\()p Fr(\022)2176 1602 y Fo(1)2238 1587 y Ft(+)g Fr(\022)2381 1602 y Fo(2)2421 1587 y Ft(\)\))2507 1519 y(sin)16 b Fr(\016)t(\022)2735 1534 y Fo(1)p 2507 1564 269 4 v 2575 1655 a Fr(\016)t(\022)2667 1670 y Fo(1)2802 1587 y Fr(ds:)149 1889 y Ft(Consider)33 b(the)g(\\)p Fp(\000)p Ft(")g(case,)h(and)e(in)m(tegrate)g(b)m(y)i (parts)f(in)m(tegrating)1123 2061 y(cos\(2)p Fr(\022)1385 2076 y Fo(2)p Fq(j)1479 2061 y Fp(\000)23 b Ft(\()p Fr(\022)1662 2076 y Fo(1)1724 2061 y Ft(+)f Fr(\022)1867 2076 y Fo(2)1907 2061 y Ft(\)\)\(2)p Fr(\022)2115 2076 y Fo(2)p Fq(j)2209 2061 y Fp(\000)h Ft(\()p Fr(\022)2392 2076 y Fo(1)2454 2061 y Ft(+)f Fr(\022)2597 2076 y Fo(2)2636 2061 y Ft(\)\))2712 2020 y Fm(0)2736 2061 y Fr(:)149 2234 y Ft(Observing)35 b(that)g Fp(j)p Fr(\022)905 2198 y Fm(00)902 2259 y Fq(j)947 2234 y Ft(\()p Fr(s)p Ft(\))p Fp(j)30 b(\024)i Fr(C)7 b Ft(\()p Fp(j)p Fr(V)21 b Ft(\()p Fr(s)p Ft(\))p Fp(j)i Ft(+)g Fp(j)p Fr(V)1835 2198 y Fm(0)1859 2234 y Ft(\()p Fr(s)p Ft(\))p Fp(j)p Ft(\))34 b(for)g(an)m(y)h Fr(j;)g Fp(j)p Ft(\()p Fr(\016)t(\022)2678 2249 y Fo(1)2717 2234 y Ft(\))2755 2198 y Fm(0)2779 2234 y Ft(\()p Fr(s)p Ft(\))p Fp(j)30 b(\024)i Fr(C)7 b Ft(\()p Fp(j)p Fr(V)21 b Ft(\()p Fr(s)p Ft(\))p Fp(j)i Ft(+)g Fr(\016)t(k)3659 2249 y Fo(1)3698 2234 y Ft(\))149 2354 y(and)k(using)f(the)i(inequalit)m(y)e (\(2.17\))g(for)g(the)h(deriv)-5 b(ativ)m(e)27 b(of)f(the)h(argumen)m (t)f(of)g(cosine,)i(w)m(e)g(obtain)149 2648 y Fp(j)p Fr(I)220 2663 y Fm(\000)279 2648 y Ft(\()p Fr(x;)17 b(y)t Ft(\))p Fp(j)27 b(\024)h Fr(C)7 b(a)794 2607 y Fm(\000)p Fo(1)905 2448 y Fj(0)905 2627 y(@)992 2648 y Fr(f)1040 2663 y Fq(j)1099 2648 y Ft(+)1249 2493 y Fq(x)1197 2513 y Fj(Z)1206 2794 y Fq(y)1313 2538 y Fj(\020)1373 2648 y Fp(j)1421 2622 y Ft(~)1401 2648 y Fr(f)1460 2607 y Fm(0)1449 2673 y Fq(j)1485 2648 y Ft(\()p Fr(s)p Ft(\))p Fp(j)21 b Ft(+)h Fp(j)1803 2622 y Ft(~)1782 2648 y Fr(f)1830 2663 y Fq(j)1867 2648 y Ft(\()p Fr(s)p Ft(\))p Fp(j)p Ft(\()p Fp(j)p Fr(V)e Ft(\()p Fr(s)p Ft(\))p Fp(j)i Ft(+)g Fr(\016)t(k)2528 2663 y Fo(1)2567 2648 y Ft(\))g(+)g Fp(j)2774 2622 y Ft(~)2753 2648 y Fr(f)2801 2663 y Fq(j)2838 2648 y Ft(\()p Fr(s)p Ft(\))p Fp(j)p Ft(\()p Fp(j)p 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b(for)j(all)d Fr(y)h(<)c(x)h Fp(\024)h Fr(f)1601 3182 y Fm(\000)p Fo(1)1590 3248 y(1)1695 3224 y Fr(g)813 3517 y Fp(j)p Fr(I)884 3532 y Fm(\000)942 3517 y Ft(\()p Fr(x;)17 b(y)t Ft(\))p Fp(j)27 b(\024)h Fr(C)1423 3317 y Fj(0)1423 3496 y(@)1510 3517 y Fr(f)1558 3532 y Fq(j)1594 3517 y Fr(a)1645 3476 y Fm(\000)p Fo(1)1762 3517 y Ft(+)22 b Fr(a)1911 3476 y Fm(\000)p Fo(2)2074 3361 y Fq(x)2022 3381 y Fj(Z)2031 3662 y Fq(y)2138 3517 y Fp(j)2187 3490 y Ft(~)2166 3517 y Fr(f)2214 3532 y Fq(j)2251 3517 y Ft(\()p Fr(s)p Ft(\))p Fp(j)2418 3422 y Fj(X)2478 3634 y Fq(l)2577 3517 y Fp(j)2626 3490 y Ft(~)2605 3517 y Fr(f)2653 3532 y Fq(l)2679 3517 y Ft(\()p Fr(s)p Ft(\))p Fp(j)17 b Fr(ds)2943 3317 y Fj(1)2943 3496 y(A)3046 3517 y Fr(:)149 3819 y Ft(The)40 b(estimate)e(on)h Fr(I)938 3834 y Fo(+)1036 3819 y Ft(is)f(similar,)g (but)h(in)m(v)m(olv)m(es)h(a)e(constan)m(t)i(indep)s(enden)m(t)g(of)e Fr(n)i Ft(instead)e(of)149 3950 y Fr(a)47 b Ft(\(arising)e(from)g(the)i (minimal)c(p)s(ossible)i(size)i(of)f Fr(k)2192 3899 y Fo(\()p Fq(n)p Fo(\))2189 3975 y Fq(j)2294 3950 y Ft(\).)85 b(The)48 b(con)m(tribution)d(of)h(the)h(term)171 4063 y(~)149 4089 y Fr(f)197 4104 y Fo(1)237 4089 y Ft(\()p Fr(s)p Ft(\))17 b(cos\(2\()p Fr(\022)676 4104 y Fo(1)731 4089 y Ft(+)e Fr(\022)867 4104 y Fo(2)907 4089 y Ft(\)\))29 b(in)g(\(2.14\))f(is)h(b)s(ounded)h(similarly)c(to)j Fr(I)2446 4104 y Fo(+)2505 4089 y Fr(:)g Ft(Summing)e(up)j(all)d(b)s (ounds,)k(w)m(e)149 4205 y(obtain)h(at)g(most)149 4474 y(\(2.18\))619 b Fr(C)1111 4274 y Fj(0)1111 4453 y(@)1198 4379 y(X)1254 4589 y Fq(j)1359 4474 y Fr(f)1407 4489 y Fq(j)1444 4474 y Fr(a)1495 4433 y Fm(\000)p Fo(1)1611 4474 y Ft(+)22 b Fr(a)1760 4433 y Fm(\000)p Fo(2)1923 4319 y Fq(x)1871 4338 y Fj(Z)1880 4620 y Fq(y)1988 4304 y Fj( )2066 4379 y(X)2122 4589 y Fq(j)2227 4474 y Fp(j)2276 4448 y Ft(~)2255 4474 y Fr(f)2303 4489 y Fq(j)2339 4474 y Ft(\()p Fr(s)p Ft(\))p Fp(j)2489 4304 y Fj(!)2568 4326 y Fo(2)2640 4474 y Fr(ds)2737 4274 y Fj(1)2737 4453 y(A)2841 4474 y Fr(:)149 4799 y Ft(W)-8 b(e)39 b(can)f(compute)f(explicitly)f (that)i Fp(k)1623 4773 y Ft(~)1602 4799 y Fr(f)1650 4814 y Fq(j)1686 4799 y Fp(k)1736 4763 y Fo(2)1736 4830 y Fq(L)1784 4811 y Fk(2)1859 4799 y Fp(\024)f Ft(2)p Fr(f)2070 4814 y Fq(j)2107 4799 y Fr(g)t(;)g Ft(and)h(therefore)g(the)g(total)f (b)s(ound)g(do)s(es)149 4918 y(not)c(exceed)h Fr(C)7 b(a)762 4881 y Fm(\000)p Fo(2)856 4918 y Ft(2)905 4881 y Fq(n)952 4918 y Ft(\()990 4843 y Fj(P)1095 4947 y Fq(j)1148 4918 y Fr(f)1196 4933 y Fq(j)1233 4918 y Ft(\))p Fr(g)t(:)31 b Ft(Notice)h(that)g(the)h(\014rst)g(term)e(in)h(\(2.18\))f(is)h (included)g(in)m(to)f(this)149 5037 y(error)36 b(since)h Fr(a)f Ft(is)f(less)i(than)f(a)f(\014xed)i(constan)m(t)g(and)f Fr(g)h(>)c Ft(1)p Fr(:)j Ft(This)g(is)g(exactly)g(the)h(error)e(term) 149 5153 y(in)d(\(2.15\);)g(the)h(origin)e(of)h(the)h(main)e(term)h(is) g(clear.)1560 b Ff(\003)149 5338 y Fi(R)-5 b(emark.)41 b Ft(The)23 b(error)f Fr(O)s Ft(\()p Fr(\016)t(k)1179 5353 y Fo(1)1217 5338 y Ft(\))f(in)g(the)i(main)d(term)h(of)28 b(\(2.15\))21 b(giv)m(es)h(after)g(in)m(tegration)e Fr(O)s Ft(\()p Fr(g)t(\016)t(k)3568 5353 y Fo(1)3605 5338 y Ft(\))28 b(=)149 5455 y Fr(O)s Ft(\()s(~)-52 b Fr(g)316 5418 y Fm(\000)p Fo(1)409 5455 y Fr(g)t(f)508 5470 y Fo(1)547 5455 y Ft(\))33 b(whic)m(h)g(can)f(also)g(b)s(e)h(subsumed)h (in)m(to)e Fp(E)2106 5470 y Fq(n)2152 5455 y Fr(:)p eop %%Page: 10 10 10 9 bop 149 66 a Fn(10)1286 b(ALEXANDER)23 b(KISELEV)249 266 y Ft(Next)44 b(w)m(e)h(sho)m(w)f(that,)j(pro)m(vided)c(that)h(the)g (error)f(terms)g(in)g(\(2.15\))g(are)g(small,)h(w)m(e)h(ha)m(v)m(e)149 383 y Fr(x)204 398 y Fq(\031)r Fm(\000)p Fq(\013)347 407 y Fk(1)414 383 y Fr(<)28 b(f)577 342 y Fm(\000)p Fo(1)566 407 y(1)671 383 y Fr(g)t(;)k Ft(and)h(th)m(us)g(our)g(p)s (oten)m(tial)d(is)j(w)m(ell)e(de\014ned.)149 542 y FA(Lemma)38 b(2.5.)j Fi(F)-7 b(or)34 b(su\016ciently)h(lar)-5 b(ge)35 b Fr(g)j Fi(and)c(for)h(su\016ciently)g(smal)5 b(l)34 b Fp(E)2906 557 y Fq(n)2952 542 y Fr(;)h Fi(we)g(have)149 699 y Ft(\(2.19\))870 b Fr(x)1324 714 y Fq(\031)r Fm(\000)p Fq(\013)1467 723 y Fk(1)1534 699 y Fp(\024)28 b Fr(\031)t(f)1757 658 y Fm(\000)p Fo(1)1746 724 y(1)1867 699 y Ft(log\(2)s(~)-52 b Fr(g)s Ft(\)\(1)22 b(+)g Fr(C)7 b Fp(E)2505 714 y Fq(n)2552 699 y Ft(\))p Fr(:)149 858 y Fi(R)-5 b(emark.)44 b Ft(Similarly)26 b(to)31 b(Lemma)e(2.3,)h(b)m(y)i(su\016cien)m(tly)f(large)e(\(or)h (small\))e(w)m(e)k(mean)e(that)g(there)149 974 y(exist)k(univ)m(ersal)e (constan)m(ts)i(suc)m(h)h(that)d(if)g(the)h(v)-5 b(alue)32 b(of)h Fr(g)j Ft(\(resp)s(ectiv)m(ely)e Fp(E)2985 989 y Fq(n)3031 974 y Ft(\))f(exceeds)i(\(is)e(less)149 1091 y(than\))g(these)h(constan)m(ts,)g(the)f(result)f(holds.)149 1257 y Fi(Pr)-5 b(o)g(of.)42 b Ft(Let)32 b(us)i(\014rst)f(estimate)e (from)h(ab)s(o)m(v)m(e)h Fr(x)1901 1272 y Fq(\031)r(=)p Fo(2)2019 1257 y Fr(:)g Ft(On)f(the)h(in)m(terv)-5 b(al)32 b([0)p Fr(;)17 b(x)2937 1272 y Fq(\031)r(=)p Fo(2)3054 1257 y Ft(])33 b(w)m(e)g(ha)m(v)m(e)1431 1414 y(2)p Fr(=\031)f Fp(\024)c Ft(\(sin)16 b Fr(\016)t(\022)1987 1429 y Fo(1)2027 1414 y Ft(\))p Fr(=\016)t(\022)2206 1429 y Fo(1)2273 1414 y Fp(\024)28 b Ft(1)p Fr(:)149 1570 y Ft(Th)m(us,)35 b(according)d(to)g(\(2.13\))o(,)h(Lemma)e(2.4)h(and)h(the)g(remark)f (after)g(its)g(pro)s(of,)g(w)m(e)i(ha)m(v)m(e)149 1781 y(\(2.20\))660 b Fr(\016)t(\022)1151 1796 y Fo(1)1191 1781 y Ft(\()p Fr(x)p Ft(\))28 b Fp(\025)g Fr(e)1514 1713 y Fk(2)p 1510 1725 38 3 v 1510 1766 a Fl(\031)1558 1740 y Fq(f)1592 1749 y Fk(1)1627 1740 y Fq(x)1681 1714 y Fr(\031)t(\016)t(k)1838 1729 y Fo(1)p 1681 1758 197 4 v 1710 1849 a Ft(2)p Fr(f)1807 1864 y Fo(1)1887 1781 y Ft(\(1)21 b Fp(\000)i Fr(e)2140 1740 y Fm(\000)2209 1713 y Fk(2)p 2205 1725 38 3 v 2205 1766 a Fl(\031)2253 1740 y Fq(f)2287 1749 y Fk(1)2322 1740 y Fq(x)2365 1781 y Ft(\)\(1)f Fp(\000)h Fr(C)7 b Fp(E)2742 1796 y Fq(n)2788 1781 y Ft(\))149 1997 y(for)34 b Fr(x)d(<)f Ft(\()p Fr(f)578 2012 y Fo(1)618 1997 y Ft(\))656 1960 y Fm(\000)p Fo(1)750 1997 y Fr(g)38 b Ft(and)c(suc)m(h)i(that)e Fr(\016)t(\022)1553 2012 y Fo(1)1623 1997 y Fp(2)d Ft([0)p Fr(;)17 b(\031)t(=)p Ft(2])p Fr(:)33 b Ft(It)i(is)f(clear)f(from)g(\(2.20\))h(that)g(if)f Fr(g)k Ft(is)d(large)149 2127 y(enough,)e(then)g Fr(x)787 2142 y Fq(\031)r(=)p Fo(2)933 2127 y Fr(<)27 b(f)1095 2086 y Fm(\000)p Fo(1)1084 2151 y(1)1189 2127 y Fr(g)t(:)k Ft(Indeed,)i(recall)c(that)i Fr(\016)t(k)2207 2142 y Fo(1)2246 2127 y Fr(=f)2343 2142 y Fo(1)2410 2127 y Ft(=)d Fr(e)2559 2091 y Fm(\000)p Fq(g)2650 2067 y Fk(3)p Fl(=)p Fk(4)2751 2127 y Fr(:)j Ft(If)g Fr(f)2964 2086 y Fm(\000)p Fo(1)2953 2151 y(1)3058 2127 y Fr(g)j Ft(w)m(ere)e(less)g(than)149 2243 y Fr(x)204 2258 y Fq(\031)r(=)p Fo(2)323 2243 y Fr(;)g Ft(\(2.20\))g(w)m(ould)g(imply)f(that)980 2439 y Fr(\016)t(\022)1072 2454 y Fo(1)1112 2439 y Ft(\()p Fr(f)1209 2398 y Fm(\000)p Fo(1)1198 2463 y(1)1303 2439 y Fr(g)t Ft(\))c Fp(\025)h Fr(e)1583 2370 y Fk(2)p 1579 2382 V 1579 2424 a Fl(\031)1627 2398 y Fq(g)r Fm(\000)p Fq(g)1754 2374 y Fk(3)p Fl(=)p Fk(4)1865 2371 y Fr(\031)p 1865 2416 59 4 v 1870 2507 a Ft(2)1951 2328 y Fj(\020)2010 2439 y Ft(1)22 b Fp(\000)h Fr(e)2226 2398 y Fm(\000)2295 2370 y Fk(2)p 2291 2382 38 3 v 2291 2424 a Fl(\031)2339 2398 y Fq(g)2379 2328 y Fj(\021)2455 2439 y Ft(\(1)f Fp(\000)g Fr(C)7 b Fp(E)2793 2454 y Fq(n)2840 2439 y Ft(\))p Fr(;)149 2625 y Ft(an)31 b(ob)m(vious)g(con)m(tradiction)e(for) i Fr(g)i Ft(large)d(enough)h(and)g(su\016cien)m(tly)g(small)e Fp(E)3005 2640 y Fq(n)3051 2625 y Fr(:)i Ft(Moreo)m(v)m(er,)i(pro-)149 2741 y(vided)k(that)g Fp(E)685 2756 y Fq(n)768 2741 y Ft(is)f(small,)g(w)m(e)h(can)g(estimate)f(\(assuming)g(also)f Fr(x)2603 2757 y Fq(\031)r(=)p Fo(2)2756 2741 y Fp(\025)2878 2702 y Fq(\031)p 2878 2718 43 4 v 2882 2776 a Fo(2)2931 2741 y Ft(\()p Fr(f)3017 2756 y Fo(1)3056 2741 y Ft(\))3094 2705 y Fm(\000)p Fo(1)3205 2741 y Ft(log)17 b(2)p Fr(;)36 b Ft(or)h(else)149 2859 y(w)m(e)d(ha)m(v)m(e)g(a)e(nice)h(upp)s(er)g(b) s(ound\))1426 3033 y Fr(e)1485 2965 y Fk(2)p 1481 2977 38 3 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3701 y Ft(=)27 b Fr(\031)d Fp(\000)c Fr(\016)t(\022)1086 3716 y Fo(1)1126 3701 y Fr(;)p 1184 3646 56 4 v 31 w(x)29 b Ft(=)e Fr(x)1426 3716 y Fq(\031)1493 3701 y Fp(\000)20 b Fr(x)32 b Ft(the)g(equation)f(\(2.11\))g (transforms)g(in)m(to)f(the)i(equation)149 3823 y(for)p 293 3742 96 4 v 27 w Fr(\016)t(\022)388 3838 y Fo(1)454 3823 y Ft(of)26 b(the)h(same)g(form)e(as)i(\(2.11\))o(,)h(with)e(the)h (same)g(initial)c(condition)i(and)i(same)f(estimates)149 3945 y(a)m(v)-5 b(ailable.)50 b(In)35 b(particular,)g(the)h(p)s(oin)m (t)p 1630 3890 56 4 v 34 w Fr(x)1685 3961 y Fq(\031)r(=)p Fo(2)1838 3945 y Ft(suc)m(h)h(that)p 2275 3864 96 4 v 35 w Fr(\016)t(\022)2370 3960 y Fo(1)2410 3945 y Ft(\()p 2448 3890 56 4 v Fr(x)2503 3961 y Fq(\031)r(=)p Fo(2)2621 3945 y Ft(\))32 b(=)g Fr(\031)t(=)p Ft(2)j(satis\014es)h(the)g(same)149 4075 y(b)s(ound)48 b(as)g Fr(x)655 4090 y Fq(\031)r(=)p Fo(2)774 4075 y Fr(:)f Ft(But)p 1057 3994 96 4 v 48 w Fr(\016)t(\022)1153 4090 y Fo(1)1192 4075 y Ft(\()p 1230 4020 56 4 v Fr(x)p Ft(\))54 b(=)f Fr(\031)37 b Fp(\000)c Fr(\016)t(\022)1800 4090 y Fo(1)1840 4075 y Ft(\()p Fr(x)1933 4090 y Fq(\031)2013 4075 y Fp(\000)g Fr(x)p Ft(\))p Fr(;)48 b Ft(and)g(th)m(us)p 2726 4020 V 49 w Fr(x)2781 4090 y Fq(\031)r(=)p Fo(2)2952 4075 y Ft(=)54 b Fr(x)3137 4090 y Fq(\031)3217 4075 y Fp(\000)33 b Fr(x)3382 4090 y Fq(\031)r(=)p Fo(2)3500 4075 y Fr(:)48 b Ft(But)149 4191 y Fr(x)204 4206 y Fq(\031)r Fm(\000)p Fq(\013)347 4215 y Fk(1)415 4191 y Fp(\024)28 b Fr(x)575 4206 y Fq(\031)622 4191 y Fr(;)33 b Ft(and)g(the)g(ab)s(o)m(v)m(e)g(observ)-5 b(ation)33 b(together)g(with)f(\(2.21\))g(\014nishes)h(the)g(pro)s(of)f (of)h(the)149 4307 y(lemma.)3199 b Ff(\003)249 4473 y Ft(Next,)34 b(w)m(e)f(pro)m(v)m(e)h(the)f(follo)m(wing)d(essen)m(tial) 149 4632 y FA(Lemma)41 b(2.6)g Ft(\(Small)32 b(Angle)j(Lemma\))p FA(.)42 b Fi(L)-5 b(et)37 b Fr(\015)43 b Fi(b)-5 b(e)37 b(a)g(smal)5 b(l)37 b(angle,)g(in)g(p)-5 b(articular)37 b(such)h(that)149 4749 y Fr(\015)33 b(<)28 b(g)388 4713 y Fm(\000)p Fo(1)481 4749 y Fr(:)35 b Fi(Denote)f Fr(x)933 4764 y Fq(\015)1013 4749 y Fi(the)h(smal)5 b(lest)34 b(value)h(of)f Fr(x)h Fi(wher)-5 b(e)35 b Fr(\016)t(\022)2375 4764 y Fo(1)2415 4749 y Ft(\()p Fr(x)p Ft(\))28 b(=)f Fr(\015)5 b(:)35 b Fi(Then)149 4962 y Ft(\(2.22\))641 b Fr(R)q Ft(\()p Fr(x)1208 4977 y Fq(\015)1253 4962 y Fr(;)17 b(k)1351 4911 y Fo(\()p Fq(n)p Fo(\))1348 4987 y(1)1453 4962 y Ft(\))1491 4921 y Fo(2)1558 4962 y Ft(=)1803 4895 y Fr(\016)t(k)1901 4910 y Fo(1)p 1672 4939 401 4 v 1672 5030 a Fr(\015)5 b(f)1776 5045 y Fo(1)1837 5030 y Ft(+)22 b Fr(\016)t(k)2033 5045 y Fo(1)2099 4881 y Fj(\000)2145 4962 y Ft(1)f(+)h Fr(O)s Ft(\()p Fr(\015)2485 4921 y Fo(2)2524 4962 y Fr(g)t(;)17 b Fp(E)2672 4977 y Fq(n)2718 4962 y Ft(\))2756 4881 y Fj(\001)2818 4962 y Fr(:)149 5167 y Fi(Mor)-5 b(e)g(over,)149 5367 y Ft(\(2.23\))387 b Fp(k)p Fr(R)q Ft(\()p Fr(x;)17 b(k)1102 5316 y Fo(\()p Fq(n)p Fo(\))1099 5391 y(1)1204 5367 y Ft(\))p Fp(k)1292 5326 y Fo(2)1292 5393 y Fq(L)1340 5374 y Fk(2)1375 5393 y Fo(\(0)p Fq(;x)1497 5401 y Fl(\015)1537 5393 y Fo(\))1596 5367 y Ft(=)27 b Fr(f)1758 5326 y Fm(\000)p Fo(1)1747 5391 y(1)2070 5299 y Ft(1)p 1862 5344 464 4 v 1862 5435 a(1)22 b(+)k(~)-53 b Fr(g)2082 5406 y Fm(\000)p Fo(1)2176 5435 y Fr(\015)2232 5406 y Fm(\000)p Fo(1)2353 5286 y Fj(\000)2399 5367 y Ft(1)21 b(+)h Fr(O)s Ft(\()p Fr(\015)2739 5326 y Fo(2)2778 5367 y Fr(g)t(;)17 b Fp(E)2926 5382 y Fq(n)2972 5367 y Ft(\))3010 5286 y Fj(\001)3072 5367 y Fr(:)p eop %%Page: 11 11 11 10 bop 1220 66 a Fn(SINGULAR)31 b(CONTINUOUS)g(SPECTR)n(UM)994 b(11)149 266 y Fi(Pr)-5 b(o)g(of.)42 b Ft(Notice)g(that)g(on)g([0)p Fr(;)17 b(x)1305 281 y Fq(\015)1350 266 y Ft(])p Fr(;)42 b Ft(w)m(e)i(ha)m(v)m(e)g(\(sin)16 b Fr(\016)t(\022)2101 281 y Fo(1)2141 266 y Ft(\))p Fr(=\016)t(\022)2320 281 y Fo(1)2404 266 y Ft(=)44 b(1)29 b(+)g Fr(O)s Ft(\()p Fr(\015)2879 230 y Fo(2)2917 266 y Ft(\))p Fr(:)43 b Ft(Ob)s(esrv)m(e)h(that)e(b)m(y)149 386 y(Lemma)g(2.5)g(w)m(e)i(ha)m(v) m(e)g(that)e Fr(x)1340 401 y Fq(\015)1430 386 y Fr(<)j(f)1610 345 y Fm(\000)p Fo(1)1599 410 y(1)1704 386 y Fr(g)t(:)d Ft(By)j(\(2.13\))o(,)g(Lemma)c(2.4)i(and)g(the)g(de\014nition)e(of)149 502 y(p)s(oten)m(tial)31 b(w)m(e)j(ha)m(v)m(e)g(that)871 788 y Fr(\016)t(k)969 803 y Fo(1)1057 623 y Fq(x)1097 631 y Fl(\015)1025 652 y Fj(Z)1035 933 y Fo(0)1141 788 y Fr(e)1186 747 y Fq(f)1220 756 y Fk(1)1255 747 y Fo(\()p Fq(x)1322 755 y Fl(\015)1362 747 y Fm(\000)p Fq(y)r Fo(\)\(1+)p Fq(O)r Fo(\()p Fq(\015)1721 723 y Fk(2)1757 747 y Fo(\)\)+)p Fq(O)r Fo(\()p Fm(E)1986 755 y Fl(n)2029 747 y Fo(\))2061 788 y Ft(\(1)22 b(+)g Fr(O)s Ft(\()p Fp(j)p Fr(V)e Ft(\()p Fr(y)t Ft(\))p Fp(j)p Ft(\))d Fr(dy)29 b Ft(=)e Fr(\015)5 b(:)149 1076 y Ft(Th)m(us,)35 b(w)m(e)e(obtain)149 1247 y(\(2.24\))600 b Fr(e)1044 1205 y Fq(f)1078 1214 y Fk(1)1113 1205 y Fq(x)1153 1213 y Fl(\015)1196 1247 y Ft(\(1)22 b Fp(\000)h Fr(e)1450 1205 y Fm(\000)p Fq(f)1539 1214 y Fk(1)1573 1205 y Fq(x)1613 1213 y Fl(\015)1657 1247 y Ft(\))p Fr(f)1754 1205 y Fm(\000)p Fo(1)1743 1271 y(1)1848 1247 y Fr(\016)t(k)1946 1262 y Fo(1)1985 1247 y Ft(\(1)f(+)g Fr(O)s Ft(\()p Fp(E)2361 1262 y Fq(n)2407 1247 y Fr(;)17 b(\015)2507 1205 y Fo(2)2546 1247 y Fr(g)t Ft(\)\))27 b(=)g Fr(\015)5 b(:)149 1418 y Ft(W)-8 b(e)35 b(included)e(the)h Fr(O)s Ft(\()p Fp(j)p Fr(V)21 b Ft(\()p Fr(y)t Ft(\))p Fp(j)p Ft(\))32 b(error)i(in)m(to)f Fp(E)1819 1433 y Fq(n)1866 1418 y Fr(;)g Ft(since)i(w)m(e)g(can)f(assume)g(freely)g (that)g Fr(a)c(<)f Ft(1)34 b(and)149 1534 y Fr(g)d(>)d Ft(1)p Fr(:)k Ft(Th)m(us)149 1761 y(\(2.25\))759 b Fr(e)1203 1720 y Fq(f)1237 1729 y Fk(1)1271 1720 y Fq(x)1311 1728 y Fl(\015)1383 1761 y Ft(=)1486 1620 y Fj(\022)1569 1693 y Fr(\015)5 b(f)1673 1708 y Fo(1)p 1569 1738 144 4 v 1572 1829 a Fr(\016)t(k)1670 1844 y Fo(1)1745 1761 y Ft(+)22 b(1)1892 1620 y Fj(\023)1982 1680 y(\000)2028 1761 y Ft(1)f(+)h Fr(O)s Ft(\()p Fr(\015)2368 1720 y Fo(2)2407 1761 y Fr(g)t(;)17 b Fp(E)2555 1776 y Fq(n)2601 1761 y Ft(\))2639 1680 y Fj(\001)2701 1761 y Fr(:)149 2002 y Ft(No)m(w)34 b(in)d(the)i(equation)g(\(2.10\))f(for)g(the)h (amplitude,)d(w)m(e)k(ha)m(v)m(e)g(for)e Fr(x)c(<)g(f)2902 1961 y Fm(\000)p Fo(1)2891 2027 y(1)2996 2002 y Fr(g)149 2285 y Ft(\(2.26\))472 b(log\()p Fr(R)q Ft(\()p Fr(x;)17 b(k)1301 2234 y Fo(\()p Fq(n)p Fo(\))1298 2309 y(1)1403 2285 y Ft(\))1441 2243 y Fo(2)1480 2285 y Ft(\))28 b(=)f Fp(\000)1795 2129 y Fq(x)1743 2149 y Fj(Z)1753 2430 y Fo(0)1881 2258 y Ft(~)1859 2285 y Fr(f)1907 2300 y Fo(1)1947 2285 y Ft(\()p Fr(y)t Ft(\))17 b(cos)f Fr(\016)t(\022)2330 2300 y Fo(1)2370 2285 y Ft(\()p Fr(y)t Ft(\))h Fr(dy)23 b Ft(+)f Fr(O)s Ft(\()p Fp(E)2904 2300 y Fq(n)2950 2285 y Ft(\))p Fr(;)149 2578 y Ft(b)m(y)29 b(an)f(estimate)f(directly)g (analogous)f(to)i(the)g(estimates)f(of)h(Lemma)e(2.4)h(\(but)h (easier\).)42 b(There-)149 2694 y(fore,)33 b(for)f Fr(x)c Fp(\024)g Fr(x)761 2709 y Fq(\015)806 2694 y Fr(;)33 b Ft(w)m(e)g(ha)m(v)m(e)149 2886 y(\(2.27\))792 b Fr(R)q Ft(\()p Fr(x;)17 b(k)1457 2835 y Fo(\()p Fq(n)p Fo(\))1454 2910 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Ft(+)22 b Fr(f)2231 4427 y Fo(1)2287 4412 y Ft(sin)2447 4385 y(~)2423 4412 y Fr(\016)t(\022)2519 4427 y Fo(1)2558 4412 y Ft(\()p Fr(x)p Ft(\))-2540 b(\(2.29\))149 4592 y(for)34 b(all)d Fr(x:)k Ft(Assume)f(furthermore)g(that)f(the)h (initial)d(conditions)h(for)2760 4567 y(~)2739 4592 y Fr(R)q Ft(,)2898 4566 y(~)2875 4592 y Fr(\016)t(\022)2970 4607 y Fo(1)3043 4592 y Ft(at)h Fr(x)e Ft(=)e(0)34 b(are)g(the)149 4708 y(same)f(as)g(for)f Fr(R)q(;)g(\016)t(\022)s(:)h Ft(Then)h(the)f(follo)m(wing)d(lemma)g(holds.)149 4891 y FA(Lemma)38 b(2.7.)j Fi(F)-7 b(or)34 b(every)h Fr(x)28 b Fp(\024)g Fr(x)1445 4906 y Fq(\031)r Fm(\000)p Fq(\013)1588 4915 y Fk(1)1628 4891 y Fr(;)35 b Fi(we)f(have)1360 5070 y Fp(j)1410 5044 y Ft(~)1388 5070 y Fr(\016)t(\022)1482 5085 y Fo(1)1522 5070 y Ft(\()p Fr(x)p Ft(\))22 b Fp(\000)h Fr(\016)t(\022)1867 5085 y Fo(1)1907 5070 y Ft(\()p Fr(x)p Ft(\))p Fp(j)k(\024)i Fr(C)11 b Ft(~)-53 b Fr(g)2327 5029 y Fm(\000)p Fo(1)2438 5045 y Ft(~)2420 5070 y Fp(E)2473 5085 y Fq(n)149 5070 y Ft(\(2.30\))1221 5208 y(~)1200 5233 y Fr(R)1275 5192 y Fo(2)1314 5233 y Ft(\()p Fr(x)p Ft(\))p Fr(=R)1569 5192 y Fo(2)1609 5233 y Ft(\()p Fr(x;)17 b(k)1800 5182 y Fo(\()p Fq(n)p Fo(\))1797 5258 y(1)1902 5233 y Ft(\))27 b(=)h(1)22 b(+)g Fr(O)s Ft(\()2373 5208 y(~)2356 5233 y Fp(E)2409 5248 y Fq(n)2455 5233 y Ft(\))p Fr(;)-2371 b Ft(\(2.31\))149 5413 y Fi(wher)-5 b(e)442 5387 y Ft(~)425 5413 y Fp(E)478 5428 y Fq(n)552 5413 y Ft(=)27 b(2)704 5376 y Fq(n)751 5413 y Fr(e)796 5376 y Fq(g)837 5413 y Fr(a)888 5376 y Fm(\000)p Fo(2)999 5338 y Fj(P)1104 5442 y Fq(j)1157 5413 y Fr(f)1205 5428 y Fq(j)1242 5413 y Fr(:)p eop %%Page: 12 12 12 11 bop 149 66 a Fn(12)1286 b(ALEXANDER)23 b(KISELEV)149 275 y Fi(Pr)-5 b(o)g(of.)42 b Ft(Denote)36 b Fr(\016)t(\022)881 239 y Fm(\003)921 275 y Ft(\()p Fr(x)p Ft(\))e(=)1219 249 y(~)1196 275 y Fr(\016)t(\022)1291 290 y Fo(1)1331 275 y Ft(\()p Fr(x)p Ft(\))25 b Fp(\000)g Fr(\016)t(\022)1681 290 y Fo(1)1721 275 y Ft(\()p Fr(x)p Ft(\))p Fr(:)37 b Ft(F)-8 b(rom)35 b(\(2.11\))o(,)i(\(2.29\))f(it)g(follo)m(ws)f(that)h (for)g Fr(x)f Fp(\024)149 392 y Fr(f)208 351 y Fm(\000)p Fo(1)197 417 y(1)302 392 y Fr(g)t(;)632 666 y Ft(\()p Fr(\016)t(\022)765 625 y Fm(\003)804 666 y Ft(\))842 625 y Fm(0)866 666 y Ft(\()p Fr(x)p Ft(\))83 b(=)g Fr(\016)t(k)1337 681 y Fo(1)1376 666 y Fr(O)s Ft(\()p Fp(j)p Fr(V)20 b Ft(\()p Fr(x)p Ft(\))p Fp(j)p Ft(\))i(+)g Fr(f)1962 681 y Fo(1)2012 599 y Ft(sin)2171 573 y(~)2148 599 y Fr(\016)t(\022)2243 614 y Fo(1)2305 599 y Fp(\000)h Ft(sin)16 b Fr(\016)t(\022)2633 614 y Fo(1)p 2012 644 661 4 v 2275 735 a Fr(\016)t(\022)2370 706 y Fm(\003)2683 666 y Fr(\016)t(\022)2778 625 y Fm(\003)2839 666 y Ft(+)1239 776 y Fj( )1318 947 y Fr(f)1366 962 y Fo(1)1422 947 y Ft(cos)h(2\()p Fr(\022)1701 962 y Fo(1)1763 947 y Ft(+)22 b Fr(\022)1906 962 y Fo(2)1946 947 y Ft(\))g(+)2114 879 y Fr(W)14 b Ft(\()p Fr(x)p Ft(\))p 2114 924 237 4 v 2154 1036 a Fr(k)2208 985 y Fo(\()p Fq(n)p Fo(\))2205 1061 y(1)2377 947 y Ft(sin\()p Fr(\022)2580 962 y Fo(1)2642 947 y Ft(+)22 b Fr(\022)2785 962 y Fo(2)2825 947 y Ft(\))2863 776 y Fj(!)2958 947 y Ft(sin)17 b Fr(\016)t(\022)3187 962 y Fo(1)3226 947 y Fr(:)149 1230 y Ft(Therefore,)149 1346 y(\(2.32\))149 1574 y Fr(\016)t(\022)244 1533 y Fm(\003)284 1574 y Ft(\()p Fr(x)p Ft(\))28 b(=)599 1418 y Fq(x)547 1438 y Fj(Z)557 1719 y Fo(0)663 1574 y Fr(e)708 1475 y Fd(R)755 1496 y Fl(x)741 1553 y(y)806 1529 y Fq(\021)r Fo(\()p Fq(s)p Fo(\))13 b Fq(ds)1032 1403 y Fj( )1111 1574 y Fr(\016)t(k)1209 1589 y Fo(1)1248 1574 y Fr(O)s Ft(\()p Fp(j)p Fr(V)21 b Ft(\()p Fr(y)t Ft(\))p Fp(j)p Ft(\))g(+)1783 1403 y Fj( )1861 1574 y Fr(f)1909 1589 y Fo(1)1966 1574 y Ft(cos)c(2\()p Fr(\022)2245 1589 y Fo(1)2307 1574 y Ft(+)22 b Fr(\022)2450 1589 y Fo(2)2489 1574 y Ft(\))h(+)2658 1506 y Fr(W)14 b Ft(\()p Fr(x)p Ft(\))p 2658 1551 V 2698 1663 a Fr(k)2752 1613 y Fo(\()p Fq(n)p Fo(\))2749 1688 y(1)2921 1574 y Ft(sin\()p Fr(\022)3124 1589 y Fo(1)3186 1574 y Ft(+)22 b Fr(\022)3329 1589 y Fo(2)3369 1574 y Ft(\))3407 1403 y Fj(!)3502 1574 y Ft(sin)16 b Fr(\016)t(\022)3730 1589 y Fo(1)3770 1403 y Fj(!)3866 1574 y Fr(;)149 1888 y Ft(where)1258 2074 y Fr(\021)t Ft(\()p Fr(s)p Ft(\))28 b(=)f(2)p Fr(f)1660 2089 y Fo(1)1709 2007 y Ft(sin)17 b Fr(\016)t(\022)1941 1971 y Fm(\003)p 1709 2051 271 4 v 1778 2143 a Fr(\016)t(\022)1873 2114 y Fm(\003)2007 2074 y Ft(cos)q(\()p Fr(\016)t(\022)2268 2089 y Fo(1)2329 2074 y Ft(+)2451 2048 y(~)2427 2074 y Fr(\016)t(\022)2523 2089 y Fo(1)2562 2074 y Ft(\))p Fr(:)149 2287 y Ft(No)m(w)32 b(all)d(oscillatory)g(terms)i(in)f (\(2.32\))g(\(that)h(is,)g(all)e(terms)i(except)i(the)e Fr(\016)t(k)2979 2302 y Fo(1)3018 2287 y Fr(O)s Ft(\()p Fp(j)p Fr(V)21 b Fp(j)p Ft(\))30 b(term\))h(are)149 2403 y(estimated)37 b(b)m(y)h(in)m(tegration)d(b)m(y)j(parts)f(similarly)d (to)i(Lemma)g(2.4.)56 b(F)-8 b(or)37 b(example,)g(for)g(an)m(y)g(of)149 2519 y(the)c(2)366 2483 y Fq(n)p Fm(\000)p Fo(1)525 2519 y Fp(\000)23 b Ft(1)32 b(terms)h(en)m(tering)g(in)m(to)e Fr(W)14 b Ft(\()p Fr(y)t Ft(\))p Fr(;)32 b Ft(w)m(e)h(ha)m(v)m(e)h(for) e Fr(x)c Fp(\024)g Fr(x)2605 2534 y Fq(\031)r Fm(\000)p Fq(\013)2748 2543 y Fk(1)1994 2634 y Fj(\014)1994 2694 y(\014)1994 2754 y(\014)1994 2813 y(\014)1994 2873 y(\014)1994 2933 y(\014)2096 2683 y Fq(x)2044 2703 y Fj(Z)2054 2984 y Fo(0)2160 2838 y Fr(e)2205 2740 y Fd(R)2253 2761 y Fl(x)2239 2818 y(y)2303 2794 y Fq(\021)r Fo(\()p Fq(s)p Fo(\))13 b Fq(ds)2513 2838 y Fr(V)2570 2853 y Fq(j)2607 2838 y Ft(\()p Fr(y)t Ft(\))k(sin)n(\()p Fr(\022)2953 2853 y Fo(1)3015 2838 y Ft(+)22 b Fr(\022)3158 2853 y Fo(2)3198 2838 y Ft(\))17 b Fr(dy)3356 2634 y Fj(\014)3356 2694 y(\014)3356 2754 y(\014)3356 2813 y(\014)3356 2873 y(\014)3356 2933 y(\014)3415 2838 y Fp(\024)-3343 b Ft(\(2.33\))199 3236 y Fr(C)7 b(a)327 3194 y Fm(\000)p Fo(1)425 3236 y Ft(~)-53 b Fr(g)488 3035 y Fj(0)488 3215 y(@)575 3236 y Fr(f)623 3251 y Fq(j)682 3236 y Ft(+)832 3080 y Fq(x)780 3100 y Fj(Z)790 3381 y Fo(0)896 3125 y Fj(\020)956 3236 y Fp(j)1005 3209 y Ft(~)984 3236 y Fr(f)1043 3194 y Fm(0)1032 3260 y Fq(j)1068 3236 y Ft(\()p Fr(y)t Ft(\))p Fp(j)21 b Ft(+)h Fp(j)p Fr(\021)t Ft(\()p Fr(y)t Ft(\))p Fp(jj)1626 3209 y Ft(~)1607 3236 y Fr(f)1655 3251 y Fq(j)1689 3236 y Ft(\()p Fr(y)t Ft(\))p Fp(j)f Ft(+)h Fp(j)2013 3209 y Ft(~)1992 3236 y Fr(f)2040 3251 y Fq(j)2076 3236 y Ft(\()p Fr(y)t Ft(\))p Fp(j)p Fr(O)s Ft(\()p Fp(j)p Fr(V)e Ft(\()p Fr(y)t Ft(\))p Fp(j)g Ft(+)i Fp(j)p Fr(V)2834 3194 y Fm(0)2857 3236 y Ft(\()p Fr(y)t Ft(\))p Fp(j)p Ft(\))p Fr(a)3102 3194 y Fm(\000)p Fo(1)3195 3125 y Fj(\021)3288 3236 y Fr(dy)3391 3035 y Fj(1)3391 3215 y(A)3493 3236 y Fr(;)149 3550 y Ft(where)149 3874 y(\(2.34\))784 3848 y(~)763 3874 y Fr(f)811 3889 y Fq(j)847 3874 y Ft(\()p Fr(y)t Ft(\))27 b(=)1106 3640 y Fj(8)1106 3730 y(>)1106 3760 y(<)1106 3939 y(>)1106 3969 y(:)1236 3729 y Fp(\000)p Fr(f)1361 3744 y Fq(j)1398 3729 y Fr(;)737 b Ft(0)27 b Fp(\024)h Fr(x)h Fp(\024)f Fr(x)2587 3744 y Fq(j;\031)r Fm(\000)p Fq(\013)2779 3754 y Fl(j)1236 3846 y Fr(f)1284 3861 y Fq(j)1320 3846 y Fr(;)815 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Ft(functions)d(from)f(the)h(t)m(w)m(o)g(jumps)g(of)3646 4311 y(~)3624 4338 y Fr(f)3672 4353 y Fq(j)3709 4338 y Ft(;)149 4454 y(those)c(jumps)f(con)m(tribute)g(to)g(the)h(\014rst,)i (o\013-in)m(tegral)39 b(term)j(in)f(the)h(estimate.)72 b(Notice)41 b(that)149 4574 y Fp(j)p Fr(\021)t Ft(\()p Fr(x)p Ft(\))p Fp(j)p Fr(;)17 b Fp(j)p Fr(V)k Ft(\()p Fr(x)p Ft(\))p Fp(j)p Fr(;)c Fp(j)p Fr(V)847 4538 y Fm(0)870 4574 y Ft(\()p Fr(x)p Ft(\))p Fp(j)30 b(\024)g Fr(C)1259 4499 y Fj(P)1364 4603 y Fq(j)1439 4548 y Ft(~)1418 4574 y Fr(f)1466 4589 y Fq(j)1502 4574 y Ft(\()p Fr(x)p Ft(\))p Fr(;)k Ft(where)h Fr(C)41 b Ft(is)33 b(a)h(univ)m(ersal)f(constan)m(t.) 48 b(Th)m(us)35 b(the)f(total)149 4699 y(b)s(ound)f(in)f(\(2.33\))g(do) s(es)h(not)f(exceed)1093 5018 y Fr(C)7 b(a)1221 4977 y Fm(\000)p Fo(1)1319 5018 y Ft(~)-52 b Fr(g)1383 4818 y Fj(0)1383 4997 y(@)1470 5018 y Fr(f)1518 5033 y Fq(j)1577 5018 y Ft(+)22 b Fr(a)1726 4977 y Fm(\000)p Fo(1)1873 4863 y Fm(1)1837 4883 y Fj(Z)1847 5164 y Fo(0)1974 4992 y Ft(~)1953 5018 y Fr(f)2001 5033 y Fq(j)2038 5018 y Ft(\()p Fr(y)t Ft(\))2183 4923 y Fj(X)2242 5136 y Fq(l)2363 4992 y Ft(~)2342 5018 y Fr(f)2390 5033 y Fq(l)2416 5018 y Ft(\()p Fr(y)t Ft(\))17 b Fr(dy)2664 4818 y Fj(1)2664 4997 y(A)2765 5018 y Fr(:)149 5338 y Ft(The)53 b(term)e Fr(f)669 5353 y Fo(1)725 5338 y Ft(cos)18 b(2\()p Fr(\022)1005 5353 y Fo(1)1079 5338 y Ft(+)35 b Fr(\022)1235 5353 y Fo(2)1275 5338 y Ft(\))52 b(is)f(estimated)g(similary)e(\(but)j(giv)m (es)g(smaller)d(error\),)57 b(while)149 5455 y(the)37 b(term)e Fr(\016)t(k)655 5470 y Fo(1)694 5455 y Fr(O)s Ft(\()p Fp(j)p Fr(V)20 b Ft(\()p Fr(y)t Ft(\))p Fp(j)p Ft(\))35 b(in)m(tegrated)g(o)m(v)m(er)i([0)p Fr(;)17 b(x)1997 5470 y Fq(\031)r Fm(\000)p Fq(\013)2140 5479 y Fk(1)2179 5455 y Ft(])36 b(with)f(the)i(exp)s(onen)m(tial)e(giv)m(es) h(at)g(most)p eop %%Page: 13 13 13 12 bop 1220 66 a Fn(SINGULAR)31 b(CONTINUOUS)g(SPECTR)n(UM)994 b(13)149 266 y Ft(\()p Fr(f)235 281 y Fo(1)275 266 y Ft(\))313 230 y Fm(\000)p Fo(1)407 266 y Fr(\016)t(k)505 281 y Fo(1)548 266 y Ft(~)-53 b Fr(g)611 191 y Fj(P)716 295 y Fq(j)769 266 y Fr(f)817 281 y Fq(j)887 266 y Ft(b)m(y)33 b(Lemma)e(2.5.)43 b(Summing)31 b(together)i(all)d(errors)j(w)m(e)h (obtain)914 572 y Fp(j)p Fr(\016)t(\022)1037 531 y Fm(\003)1076 572 y Fp(j)28 b(\024)g Fr(C)11 b Ft(~)-53 b Fr(g)1381 372 y Fj(0)1381 551 y(@)1468 572 y Fr(a)1519 531 y Fm(\000)p Fo(1)1630 477 y Fj(X)1691 689 y Fq(l)1791 572 y Fr(f)1839 587 y Fq(l)1887 572 y Ft(+)22 b Fr(a)2036 531 y Fm(\000)p Fo(2)2184 416 y Fm(1)2147 436 y Fj(Z)2157 718 y Fo(0)2263 402 y Fj( )2342 477 y(X)2398 687 y Fq(j)2524 546 y Ft(~)2502 572 y Fr(f)2550 587 y Fq(j)2587 402 y Fj(!)2666 424 y Fo(2)2739 572 y Fr(dy)2842 372 y Fj(1)2842 551 y(A)2944 572 y Fr(;)149 857 y Ft(whic)m(h)34 b(leads)e(to)149 1025 y(\(2.35\))976 b Fp(j)p Fr(\016)t(\022)1498 984 y Fm(\003)1538 1025 y Ft(\()p Fr(x)p Ft(\))p Fp(j)27 b(\024)h Fr(C)7 b Ft(2)1955 984 y Fq(n)2006 1025 y Ft(~)-53 b Fr(g)s(g)t(a)2154 984 y Fm(\000)p Fo(2)2265 930 y Fj(X)2321 1140 y Fq(j)2425 1025 y Fr(f)2473 1040 y Fq(j)149 1302 y Ft(for)32 b Fr(x)d Fp(\024)f Fr(x)542 1317 y Fq(\031)r Fm(\000)p Fq(\013)685 1326 y Fk(1)724 1302 y Fr(:)33 b Ft(This)f(pro)m(v)m(es)j(\(2.30\))o(,)e(since)g Fr(e)1901 1266 y Fq(g)1969 1302 y Fp(\025)28 b Fr(C)7 b(g)t Ft(\()s(~)-52 b Fr(g)2291 1266 y Fo(2)2329 1302 y Ft(\))32 b(for)g(some)h(univ)m (ersal)f Fr(C)39 b Ft(and)33 b(all)e Fr(g)t(:)249 1418 y Ft(By)k(\(2.10\))o(,)d(\(2.28\))g(and)h(an)f(argumen)m(t)h(parallel)c (to)k(Lemma)e(2.4,)541 1697 y(log)16 b Fr(R)758 1656 y Fo(2)798 1697 y Ft(\()p Fr(x;)h(k)989 1646 y Fo(\()p Fq(n)p Fo(\))986 1722 y(1)1091 1697 y Ft(\))22 b Fp(\000)g Ft(log)1414 1672 y(~)1393 1697 y Fr(R)1468 1656 y Fo(2)1508 1697 y Ft(\()p Fr(x)p Ft(\))28 b(=)f Fp(\000)1916 1542 y Fq(x)1864 1562 y Fj(Z)1874 1843 y Fo(0)1980 1697 y Fr(f)2028 1712 y Fo(1)2068 1697 y Ft(\(cos)17 b Fr(\016)t(\022)2345 1712 y Fo(1)2407 1697 y Fp(\000)23 b Ft(cos)2677 1671 y(~)2654 1697 y Fr(\016)t(\022)2749 1712 y Fo(1)2788 1697 y Ft(\))17 b Fr(dy)25 b Ft(+)d Fr(O)s Ft(\()p Fp(E)3234 1712 y Fq(n)3280 1697 y Ft(\))p Fr(:)149 1988 y Ft(Using)34 b(\(2.30\))e(and)g(Lemma)g(2.5)g(w)m(e)h(obtain)f(for)g Fr(x)c Fp(\024)g Fr(x)2241 2003 y Fq(\031)r Fm(\000)p Fq(\013)2384 2012 y Fk(1)149 2173 y Ft(\(2.36\))884 b Fr(R)1358 2132 y Fo(2)1397 2173 y Ft(\()p Fr(x;)17 b(k)1588 2123 y Fo(\()p Fq(n)p Fo(\))1585 2198 y(1)1690 2173 y Ft(\))p Fr(=)1798 2148 y Ft(~)1777 2173 y Fr(R)1852 2132 y Fo(2)1891 2173 y Ft(\()p Fr(x)p Ft(\))28 b(=)g(1)22 b(+)g Fr(O)s Ft(\()2464 2148 y(~)2439 2173 y Fp(E)2492 2188 y Fq(n)2538 2173 y Ft(\))p Fr(;)149 2350 y Ft(whic)m(h)36 b(\014nishes)h(the)f(pro)s(of)e(\(w)m(e)j(used)g(that)e(eviden)m(tly)h Fp(E)2306 2365 y Fq(n)2385 2350 y Fr(<)2519 2325 y Ft(~)2493 2350 y Fp(E)2546 2365 y Fq(n)2628 2350 y Ft(and)g(so)g(the)g(former)e (error)h(is)149 2469 y(absorb)s(ed)g(b)m(y)h(the)f(latter\).)48 b(Also)34 b(on)g(the)h(last)f(step)h(w)m(e)g(generously)g(exc)m(hanged) 40 b(~)-52 b Fr(g)3297 2433 y Fm(\000)p Fo(1)3425 2469 y Ft(for)34 b Fr(g)3627 2433 y Fo(3)p Fq(=)p Fo(4)149 2585 y Ft(to)f(arriv)m(e)f(at)g(an)h(error)f(in)g(\(2.36\))o(.)2231 b Ff(\003)249 2765 y Ft(No)m(w)33 b(w)m(e)h(are)e(ready)i(to)e(pro)s (ceed)h(with)g(the)g(pro)s(of)e(of)h(Lemma)g(2.3.)149 2944 y Fi(Pr)-5 b(o)g(of)35 b(of)g(the)g(Splitting)f(L)-5 b(emma.)41 b Ft(The)33 b(structure)g(of)e(the)i(eigenfunctions)e (obtained)h(in)f(Split-)149 3061 y(ting)h(Lemma)f(is)h(illustrated)f(b) m(y)i(Figure)f(1.)249 3189 y(1.)52 b(Consider)36 b(\014rst)g(the)g(in)m (terv)-5 b(al)34 b([0)p Fr(;)17 b(x)1692 3205 y Fq(\031)r(=)p Fo(2)1810 3189 y Ft(])p Fr(:)36 b Ft(The)g(estimate)f(for)g(the)h(norm) e(of)i Fr(R)q Ft(\()p Fr(x;)17 b(k)3458 3138 y Fo(\()p Fq(n)p Fo(\))3455 3213 y(1)3560 3189 y Ft(\))35 b(on)149 3305 y(the)c(in)m(terv)-5 b(al)29 b([0)p Fr(;)17 b(x)840 3320 y Fq(\015)885 3305 y Ft(])30 b(for)g(some)h(small)d Fr(\015)35 b Ft(is)30 b(pro)m(vided)h(b)m(y)h(Lemma)d(2.6.)42 b(By)32 b(\(2.22\))e(and)h(\(2.26\))o(,)149 3431 y(the)37 b(norm)f(on)g(the)h(in)m(terv)-5 b(al)35 b([)p Fr(x)1329 3446 y Fq(\015)1374 3431 y Fr(;)17 b(x)1473 3447 y Fq(\031)r(=)p Fo(2)1591 3431 y Ft(])36 b(do)s(es)h(not)f(exceed)j(\()2419 3388 y Fq(\015)t(f)2493 3397 y Fk(1)p 2419 3408 109 4 v 2421 3466 a Fq(\016)r(k)2491 3475 y Fk(1)2562 3431 y Ft(+)25 b(1\))2750 3395 y Fm(\000)p Fo(1)2844 3431 y Fr(x)2899 3447 y Fq(\031)r(=)p Fo(2)3017 3431 y Ft(\(1)f(+)h Fr(O)s Ft(\()p Fr(\015)3401 3395 y Fo(2)3439 3431 y Fr(g)t(;)17 b Fp(E)3587 3446 y Fq(n)3633 3431 y Ft(\)\))p Fr(:)149 3569 y Ft(Cho)s(osing)32 b Fr(\015)h Ft(=)e(~)-52 b Fr(g)814 3533 y Fm(\000)p Fo(1)p Fq(=)p Fo(2)978 3569 y Fr(;)32 b Ft(for)g(example,)h(and)f(using)h(the)g(b)s(ound)f(\(2.21\))g(on)h Fr(x)2982 3585 y Fq(\031)r(=)p Fo(2)3100 3569 y Fr(;)f Ft(w)m(e)i(\014nd)f(that)1014 3762 y Fp(k)p Fr(R)q Ft(\()p Fr(x;)17 b(k)1330 3711 y Fo(\()p Fq(n)p Fo(\))1327 3787 y(1)1432 3762 y Ft(\))p Fp(k)1520 3721 y Fo(2)1520 3788 y Fq(L)1568 3769 y Fk(2)1602 3788 y Fo([)p Fq(x)1662 3796 y Fl(\015)1701 3788 y Fq(;x)1761 3802 y Fl(\031)r(=)p Fk(2)1865 3788 y Fo(])1916 3762 y Ft(=)28 b Fr(f)2079 3721 y Fm(\000)p Fo(1)2068 3787 y(1)2173 3762 y Fr(O)s Ft(\()s(~)-52 b Fr(g)2340 3721 y Fm(\000)p Fo(1)p Fq(=)p Fo(2)2503 3762 y Fr(g)2554 3721 y Fo(3)p Fq(=)p Fo(4)2663 3762 y Fr(;)17 b Fp(E)2760 3777 y Fq(n)2807 3762 y Ft(\))p Fr(:)149 3944 y Ft(Th)m(us,)35 b(altogether,)149 4124 y(\(2.37\))569 b Fp(k)p Fr(R)q Ft(\()p Fr(x;)17 b(k)1284 4073 y Fo(\()p Fq(n)p Fo(\))1281 4149 y(1)1386 4124 y Ft(\))p Fp(k)1474 4083 y Fo(2)1474 4150 y Fq(L)1522 4131 y Fk(2)1557 4150 y Fo([0)p Fq(;x)1672 4164 y Fl(\031)r(=)p Fk(2)1775 4150 y Fo(])1827 4124 y Ft(=)27 b Fr(f)1989 4083 y Fm(\000)p Fo(1)1978 4149 y(1)2083 4124 y Ft(\(1)22 b(+)g Fr(O)s Ft(\()p Fp(E)2459 4139 y Fq(n)2505 4124 y Fr(;)e Ft(~)-52 b Fr(g)2600 4083 y Fm(\000)p Fo(1)p Fq(=)p Fo(2)2764 4124 y Fr(g)t Ft(\)\))p Fr(:)249 4312 y Ft(2.)41 b(Next,)28 b(w)m(e)f(estimate)e(norm)f(on)i([)p Fr(x)1605 4327 y Fq(\031)r(=)p Fo(2)1723 4312 y Fr(;)17 b(x)1822 4327 y Fq(\031)r Fm(\000)p Fq(\013)1965 4336 y Fk(1)2004 4312 y Ft(])p Fr(:)26 b Ft(Consider)g(an)g(auxiliary)d (simpli\014ed)h(system)149 4428 y(\(2.28\),)32 b(\(2.29\))g(of)g(Lemma) f(2.7.)44 b(By)34 b(\(2.30\))o(,)f(w)m(e)g(ha)m(v)m(e)622 4579 y(~)599 4606 y Fr(\016)t(\022)694 4621 y Fo(1)733 4606 y Ft(\()p Fr(x)826 4621 y Fq(\031)r(=)p Fo(2)944 4606 y Ft(\))28 b(=)g Fr(\031)t(=)p Ft(2)21 b(+)h Fr(O)s Ft(\()s(~)-52 b Fr(g)1557 4564 y Fm(\000)p Fo(1)1675 4580 y Ft(~)1650 4606 y Fp(E)1703 4621 y Fq(n)1750 4606 y Ft(\))p Fr(;)1904 4579 y Ft(~)1881 4606 y Fr(\016)t(\022)1976 4621 y Fo(1)2016 4606 y Ft(\()p Fr(x)2109 4621 y Fq(\031)r Fm(\000)p Fq(\013)2252 4630 y Fk(1)2291 4606 y Ft(\))27 b(=)h Fr(\031)e Fp(\000)d Fr(\013)2703 4621 y Fo(1)2764 4606 y Ft(+)f Fr(O)s Ft(\()s(~)-52 b Fr(g)3029 4564 y Fm(\000)p Fo(1)3147 4580 y Ft(~)3122 4606 y Fp(E)3175 4621 y Fq(n)3222 4606 y Ft(\))p Fr(:)149 4790 y Ft(Denote)40 b(~)-55 b Fr(x)540 4805 y Fq(\014)617 4790 y Ft(=)29 b(min)o Fp(f)p Fr(x)p Fp(j)1041 4764 y Ft(~)1018 4790 y Fr(\016)t(\022)1113 4805 y Fo(1)1152 4790 y Ft(\()p Fr(x)p Ft(\))h(=)f Fr(\014)6 b Fp(g)p Fr(:)33 b Ft(Notice)g(that)g(the) h(system)h(\(2.28\))o(,)f(\(2.29\))f(is)g(symmetric)149 4906 y(in)f(a)h(sense)h(that)e(for)g Fr(x)c Fp(\024)35 b Ft(~)-55 b Fr(x)1199 4921 y Fq(\031)1246 4906 y Fr(;)866 5052 y Ft(~)843 5078 y Fr(\016)t(\022)938 5093 y Fo(1)978 5078 y Ft(\()p Fr(x)p Ft(\))28 b(=)f Fr(\031)f Fp(\000)1444 5052 y Ft(~)1421 5078 y Fr(\016)t(\022)1516 5093 y Fo(1)1556 5078 y Ft(\()6 b(~)-55 b Fr(x)1649 5093 y Fq(\031)1718 5078 y Fp(\000)23 b Fr(x)p Ft(\))50 b(and)2189 5053 y(~)2168 5078 y Fr(R)2243 5037 y Fo(2)2282 5078 y Ft(\()p Fr(x)p Ft(\))28 b(=)2566 5053 y(~)2545 5078 y Fr(R)2620 5037 y Fo(2)2659 5078 y Ft(\()6 b(~)-55 b Fr(x)2752 5093 y Fq(\031)2822 5078 y Fp(\000)23 b Fr(x)p Ft(\))p Fr(:)149 5247 y Ft(Observ)m(e)35 b(also)d(that)g Fr(\013)993 5262 y Fo(1)1060 5247 y Ft(=)f(~)-52 b Fr(g)1215 5206 y Fm(\000)p Fo(1)1211 5272 y(1)1308 5247 y Ft(\(1)22 b(+)g Fr(O)s Ft(\()s(~)-52 b Fr(g)1682 5211 y Fm(\000)p Fo(2)1775 5247 y Ft(\)\))p Fr(:)32 b Ft(Th)m(us)j(the)e(norm)149 5424 y(\(2.38\))542 b Fp(k)1012 5398 y Ft(~)991 5424 y Fr(R)q Fp(k)1116 5383 y Fo(2)1116 5449 y Fq(L)1164 5431 y Fk(2)1198 5449 y Fo([)p Fq(x)1258 5463 y Fl(\031)r(=)p Fk(2)1362 5449 y Fq(;x)1422 5458 y Fl(\031)r Fe(\000)p Fl(\013)1548 5473 y Fk(1)1586 5449 y Fo(])1637 5424 y Ft(=)28 b Fp(k)1812 5398 y Ft(~)1791 5424 y Fr(R)q Fp(k)1916 5383 y Fo(2)1916 5449 y Fq(L)1964 5431 y Fk(2)1998 5449 y Fo([)t(~)-39 b Fq(x)2058 5473 y Fl(\013)2098 5488 y Fk(1)2132 5473 y(\(1+)p Fl(O)r Fk(\()2315 5460 y(~)2306 5473 y Fe(E)2339 5481 y Fl(n)2381 5473 y Fk(\)\))2433 5449 y Fq(;)t Fo(~)g Fq(x)2493 5473 y Fl(\031)r(=)p Fk(2\(1+)p Fl(O)r Fk(\()2776 5460 y(~)2767 5473 y Fe(E)2800 5481 y Fl(n)2842 5473 y Fk(\)\))2894 5449 y Fo(])2918 5424 y Fr(:)p eop %%Page: 14 14 14 13 bop 149 66 a Fn(14)1286 b(ALEXANDER)23 b(KISELEV)149 266 y Ft(By)45 b(iden)m(tical)d(argumen)m(ts,)47 b(Lemma)c(2.5)h(and)g (Lemma)e(2.6)i(\(Small)d(Angle)j(Lemma\))e(apply)149 382 y(to)e(the)g(system)h(\(2.28\))o(,)g(\(2.29\))e(\(with)h(the)g (only)f(di\013erence)h(that)g(the)g(conclusion)f(do)s(es)h(not)149 498 y(in)m(v)m(olv)m(e)31 b(the)h(error)e Fp(E)931 513 y Fq(n)1008 498 y Ft(coming)g(from)f(oscillatory)f(terms)j(whic)m(h)g (are)g(absen)m(t)h(in)e(the)h(simpli\014ed)149 614 y(system\).)43 b(Therefore,)30 b(the)d(considerations)g(close)h(to)f(the)g(\014rst)h (part)f(of)g(the)h(pro)s(of)e(ab)s(o)m(v)m(e)i(apply)149 739 y(to)33 b(the)g(norm)e(of)824 714 y(~)802 739 y Fr(R)j Ft(and)f(giv)m(e)149 899 y(\(2.39\))658 b Fp(k)1128 874 y Ft(~)1107 899 y Fr(R)q Fp(k)1232 858 y Fo(2)1232 925 y Fq(L)1280 906 y Fk(2)1314 925 y Fo([0)p Fq(;)t Fo(~)-39 b Fq(x)1429 948 y Fl(\031)r(=)p Fk(2\(1+)p Fl(O)r Fk(\()1712 935 y(~)1703 948 y Fe(E)1736 956 y Fl(n)1778 948 y Fk(\)\))1830 925 y Fo(])1881 899 y Ft(=)28 b Fr(f)2044 858 y Fm(\000)p Fo(1)2033 924 y(1)2138 899 y Ft(\(1)22 b(+)g Fr(O)s Ft(\()s(~)-52 b Fr(g)2512 858 y Fm(\000)p Fo(1)p Fq(=)p Fo(2)2675 899 y Fr(g)t Ft(\)\))p Fr(:)149 1085 y Ft(By)35 b(\(2.23\))d(of)g(Lemma)f (2.6,)h(w)m(e)i(also)d(ha)m(v)m(e)149 1291 y(\(2.40\))600 b Fp(k)1070 1266 y Ft(~)1049 1291 y Fr(R)q Fp(k)1174 1250 y Fo(2)1174 1317 y Fq(L)1222 1298 y Fk(2)1256 1317 y Fo([0)p Fq(;)t Fo(~)-39 b Fq(x)1371 1341 y Fl(\013)1411 1356 y Fk(1)1445 1341 y(\(1+)p Fl(O)r Fk(\()1628 1328 y(~)1619 1341 y Fe(E)1652 1349 y Fl(n)1694 1341 y Fk(\)\))1746 1317 y Fo(])1798 1291 y Ft(=)1911 1224 y(1)p 1911 1268 49 4 v 1911 1360 a(2)1970 1291 y Fr(f)2029 1250 y Fm(\000)p Fo(1)2018 1316 y(1)2123 1291 y Ft(\(1)22 b(+)g Fr(O)s Ft(\()2463 1266 y(~)2446 1291 y Fp(E)2499 1306 y Fq(n)2545 1291 y Fr(;)e Ft(~)-52 b Fr(g)2640 1250 y Fm(\000)p Fo(2)2733 1291 y Fr(g)t Ft(\)\))p Fr(:)149 1501 y Ft(Therefore,)27 b(the)e(norm)e(in)h(\(2.38\))f(is)h(equal)g(to)1856 1461 y Fo(1)p 1856 1478 36 4 v 1856 1535 a(2)1901 1501 y Fr(f)1960 1459 y Fm(\000)p Fo(1)1949 1525 y(1)2054 1501 y Ft(\(1)5 b(+)g Fr(O)s Ft(\()2359 1476 y(~)2343 1501 y Fp(E)2396 1516 y Fq(n)2441 1501 y Fr(;)20 b Ft(~)-52 b Fr(g)2536 1465 y Fm(\000)p Fo(1)p Fq(=)p Fo(2)2700 1501 y Fr(g)t Ft(\)\))p Fr(:)23 b Ft(No)m(w)i(b)m(y)g(the)g(estimate)149 1618 y(\(2.31\))32 b(of)g(Lemma)g(2.7)g(w)m(e)h(ha)m(v)m(e)149 1824 y(\(2.41\))450 b Fp(k)p Fr(R)q Ft(\()p Fr(x;)17 b(k)1165 1773 y Fo(\()p Fq(n)p Fo(\))1162 1849 y(1)1267 1824 y Ft(\))p Fp(k)1355 1783 y Fo(2)1355 1850 y Fq(L)1403 1831 y Fk(2)1438 1850 y Fo([)p Fq(x)1498 1864 y Fl(\031)r(=)p Fk(2)1601 1850 y Fq(;x)1661 1859 y Fl(\031)r Fe(\000)p Fl(\013)1787 1874 y Fk(1)1826 1850 y Fo(])1877 1824 y Ft(=)1990 1757 y(1)p 1990 1801 49 4 v 1990 1893 a(2)2049 1824 y Fr(f)2108 1783 y Fm(\000)p Fo(1)2097 1849 y(1)2202 1824 y Ft(\(1)22 b(+)g Fr(O)s Ft(\()2542 1799 y(~)2525 1824 y Fp(E)2578 1839 y Fq(n)2624 1824 y Fr(;)e Ft(~)-52 b Fr(g)2719 1783 y Fm(\000)p Fo(1)p Fq(=)p Fo(2)2883 1824 y Fr(g)t Ft(\)\))p Fr(:)149 2011 y Ft(In)33 b(addition,)e(b)m(y)j (Lemma)d(2.6)h(and)h(symmetry)870 2187 y(~)849 2212 y Fr(R)924 2171 y Fo(2)964 2212 y Ft(\()p Fr(x)1057 2227 y Fq(\031)r Fm(\000)p Fq(\013)1200 2236 y Fk(1)1239 2212 y Ft(\))28 b(=)1429 2187 y(~)1408 2212 y Fr(R)1483 2171 y Fo(2)1523 2212 y Ft(\()p Fr(x)1616 2239 y Fq(\013)1661 2248 y Fk(1)1696 2239 y Fo(\(1+)p Fq(O)r Fo(\()1919 2222 y(~)1896 2239 y Fm(E)1933 2247 y Fl(n)1976 2239 y Fo(\)\))2035 2212 y Ft(\))g(=)2214 2145 y(1)p 2214 2189 V 2214 2280 a(2)2273 2212 y(\(1)22 b(+)g Fr(O)s Ft(\()2613 2187 y(~)2596 2212 y Fp(E)2649 2227 y Fq(n)2695 2212 y Fr(;)e Ft(~)-52 b Fr(g)2790 2171 y Fm(\000)p Fo(2)2883 2212 y Fr(g)t Ft(\)\))p Fr(:)149 2399 y Ft(By)34 b(Lemma)d(2.7,)h(this)g(implies)149 2600 y(\(2.42\))749 b Fr(R)q Ft(\()p Fr(x)1316 2615 y Fq(\031)r Fm(\000)p Fq(\013)1459 2624 y Fk(1)1498 2600 y Fr(;)17 b(k)1596 2549 y Fo(\()p Fq(n)p Fo(\))1593 2624 y(1)1697 2600 y Ft(\))1735 2559 y Fo(2)1802 2600 y Ft(=)1916 2532 y(1)p 1916 2577 V 1916 2668 a(2)1975 2600 y(\(1)22 b(+)g Fr(O)s Ft(\()2314 2575 y(~)2298 2600 y Fp(E)2351 2615 y Fq(n)2396 2600 y Fr(;)f Ft(~)-53 b Fr(g)2491 2559 y Fm(\000)p Fo(2)2585 2600 y Fr(g)t Ft(\)\))p Fr(:)249 2802 y Ft(3.)75 b(Next)45 b(step)f(is)f(the)g(consideration)g(of)g([)p Fr(x)1966 2817 y Fq(\031)r Fm(\000)p Fq(\013)2109 2826 y Fk(1)2148 2802 y Fr(;)17 b(f)2251 2761 y Fm(\000)p Fo(1)2240 2826 y(1)2345 2802 y Fr(g)t Ft(])42 b(in)m(terv)-5 b(al.)75 b(On)43 b(this)g(in)m(terv)-5 b(al,)45 b(in)149 2918 y(the)d(idealized)d(system)j(\(2.12\))f(the)g(angle)f Fr(\016)t(\022)1889 2933 y Fo(1)1970 2918 y Ft(sta)m(ys)i(constan)m(t,) i(and)e(since)f(it)f(is)g(v)m(ery)j(close)149 3034 y(to)d Fr(\031)t(;)h Ft(this)e(allo)m(ws)h(for)f(a)h(consisten)m(t)i(deca)m(y) f(in)f Fr(R)h Ft(equation.)66 b(The)41 b(main)e(goal)g(therefore)h(is) 149 3151 y(to)h(con)m(trol)f(the)h(additional)d(p)s(erturbation)i (terms,)j(whic)m(h)e(are)g(either)f(small)f(or)h(oscillating.)149 3267 y(De\014ne)34 b Fr(\016)p 499 3186 V 4 w(\022)547 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Ft(~)2081 5303 y Fr(f)2129 5318 y Fq(j)2165 5303 y Ft(\()p Fr(y)t Ft(\))p Fp(j)p Fr(O)s Ft(\()p Fp(j)p Fr(V)d Ft(\()p Fr(y)t Ft(\))p Fp(j)i Ft(+)h Fp(j)p Fr(V)2923 5262 y Fm(0)2946 5303 y Ft(\()p Fr(y)t Ft(\))p Fp(j)p Ft(\))p Fr(a)3191 5262 y Fm(\000)p Fo(1)3284 5193 y Fj(\021)3377 5303 y Fr(dy)3480 5073 y Fj(1)3480 5249 y(C)3480 5312 y(A)3582 5303 y Fr(;)p eop %%Page: 16 16 16 15 bop 149 66 a Fn(16)1286 b(ALEXANDER)23 b(KISELEV)149 275 y Ft(where)458 249 y(~)436 275 y Fr(f)484 290 y Fq(j)559 275 y Ft(is)37 b(giv)m(en)g(b)m(y)j(\(2.34\))d(as)h(b)s(efore.)59 b(Similarly)34 b(to)j(the)h(computation)e(in)h(Lemma)f(2.7,)149 391 y(the)d(total)e(b)s(ound)i(in)f(\(2.47\))g(is)1023 706 y Fr(C)7 b(e)1145 665 y Fq(g)1202 476 y Fj(0)1202 651 y(B)1202 715 y(@)1289 706 y Fr(f)1337 721 y Fq(j)1374 706 y Fr(a)1425 665 y Fm(\000)p Fo(1)1542 706 y Ft(+)22 b Fr(a)1691 665 y Fm(\000)p Fo(2)1891 551 y Fm(1)1854 570 y Fj(Z)1780 852 y Fq(x)1820 861 y Fl(\031)r Fe(\000)p Fl(\013)1946 876 y Fk(1)2044 680 y Ft(~)2023 706 y Fr(f)2071 721 y Fq(j)2108 706 y Ft(\()p Fr(y)t Ft(\))2253 611 y Fj(X)2312 824 y Fq(l)2433 680 y Ft(~)2412 706 y Fr(f)2460 721 y Fq(l)2486 706 y Ft(\()p Fr(y)t Ft(\))17 b Fr(dy)2734 476 y Fj(1)2734 651 y(C)2734 715 y(A)2835 706 y Fr(:)149 1021 y Ft(The)34 b(term)f Fr(\016)t(k)682 1036 y Fo(1)737 1021 y Ft(cos)18 b(2\()p Fr(\022)1017 1036 y Fo(1)1079 1021 y Ft(+)k Fr(\022)1222 1036 y Fo(2)1262 1021 y Ft(\))32 b(is)h(estimated)f(similary)e(\(but)j(giv)m(es)h(smaller)d(error\),)h (while)h(the)149 1138 y(term)k Fr(\016)t(k)485 1153 y Fo(1)524 1138 y Fr(O)s Ft(\()p Fp(j)p Fr(V)21 b Ft(\()p Fr(y)t Ft(\))p Fp(j)p Ft(\))36 b(in)m(tegrated)h(o)m(v)m(er)i([)p Fr(x)1740 1153 y Fq(\031)r Fm(\000)p Fq(\013)1883 1162 y Fk(1)1922 1138 y Fr(;)17 b(f)2025 1097 y Fm(\000)p Fo(1)2014 1163 y(1)2119 1138 y Fr(g)t Ft(])37 b(with)g(the)h(exp)s (onen)m(tial)f(giv)m(es)g(at)h(most)149 1254 y(\()p Fr(f)235 1269 y Fo(1)275 1254 y Ft(\))313 1218 y Fm(\000)p Fo(1)407 1254 y Fr(\016)t(k)505 1269 y Fo(1)544 1254 y Fr(e)589 1218 y Fq(g)646 1180 y Fj(P)751 1283 y Fq(j)804 1254 y Fr(f)852 1269 y Fq(j)889 1254 y Fr(:)33 b Ft(Summing)d(together)j (all)e(errors)h(w)m(e)i(obtain)149 1457 y(\(2.48\))893 b Fp(j)p Fr(\016)p 1367 1376 49 4 v 4 w(\022)s Ft(\()p Fr(x)p Ft(\))p Fp(j)27 b(\024)h Fr(C)7 b Ft(2)1832 1416 y Fq(n)1883 1457 y Ft(~)-53 b Fr(g)1930 1416 y Fm(\000)p Fo(1)2024 1457 y Fr(g)t(e)2120 1416 y Fq(g)2159 1457 y Fr(a)2210 1416 y Fm(\000)p Fo(2)2321 1362 y Fj(X)2377 1572 y Fq(j)2482 1457 y Fr(f)2530 1472 y Fq(j)2567 1457 y Fr(:)149 1720 y Ft(No)m(w)34 b(notice)e(that)164 1944 y(\(log)17 b Fr(R)q Ft(\()p Fr(x;)g(k)611 1893 y Fo(\()p Fq(n)p Fo(\))608 1968 y(1)713 1944 y Ft(\))751 1903 y Fo(2)790 1944 y Ft(\))828 1903 y Fm(0)879 1944 y Ft(=)993 1876 y Fr(k)1047 1825 y Fo(\()p Fq(n)p Fm(\000)p Fo(1\))1044 1901 y(1)p 993 1921 247 4 v 1038 2034 a Fr(k)1092 1983 y Fo(\()p Fq(n)p Fo(\))1089 2058 y(1)1249 1944 y Fr(f)1297 1959 y Fo(1)1336 1944 y Ft(\()p Fp(\000)g Ft(cos)q(\()p Fr(\016)p 1684 1863 49 4 v 4 w(\022)1732 1959 y Fo(1)1794 1944 y Ft(+)22 b Fr(\013)1954 1959 y Fo(1)1993 1944 y Ft(\))g Fp(\000)h Ft(cos\()p Fr(\022)2366 1959 y Fo(2)2428 1944 y Ft(+)f(3)p Fr(\022)2620 1959 y Fo(1)2660 1944 y Ft(\)\))g(+)2866 1876 y Fr(W)14 b Ft(\()p Fr(x)p Ft(\))p 2866 1921 237 4 v 2907 2034 a Fr(k)2961 1983 y Fo(\()p Fq(n)p Fo(\))2958 2058 y(1)3130 1944 y Ft(sin)i(2)p Fr(\022)s Ft(\()p Fr(x;)h(k)3554 1893 y Fo(\()p Fq(n)p Fo(\))3551 1968 y(1)3656 1944 y Ft(\))p Fr(:)149 2189 y Ft(Using)34 b(\(2.42\))f(and)g(estimates)g(parallel)d(to)j(the)g(ones)h(considered) g(in)e(Lemma)g(2.4)h(w)m(e)h(\014nd)f(for)149 2305 y Fr(x)204 2320 y Fq(\031)r Fm(\000)p Fq(\013)347 2329 y Fk(1)414 2305 y Fp(\024)c Fr(x)f Fp(\024)g Ft(\()p Fr(f)794 2320 y Fo(1)833 2305 y Ft(\))871 2269 y Fm(\000)p Fo(1)966 2305 y Fr(g)35 b Ft(that)937 2517 y Fr(R)1012 2476 y Fo(2)1051 2517 y Ft(\()p Fr(x;)17 b(k)1242 2466 y Fo(\()p Fq(n)p Fo(\))1239 2541 y(1)1344 2517 y Ft(\))28 b(=)1524 2450 y(1)p 1524 2494 49 4 v 1524 2585 a(2)1582 2517 y Fr(e)1627 2476 y Fm(\000)p Fq(f)1716 2485 y Fk(1)1751 2476 y Fo(\()p Fq(x)p Fm(\000)p Fq(x)1913 2485 y Fl(\031)r Fe(\000)p Fl(\013)2039 2500 y Fk(1)2077 2476 y Fo(\))2126 2406 y Fj(\020)2185 2517 y Ft(1)22 b(+)g Fr(O)s Ft(\()2487 2492 y(~)2470 2517 y Fp(E)2523 2532 y Fq(n)2569 2517 y Fr(;)17 b(g)7 b Ft(~)-52 b Fr(g)2715 2476 y Fm(\000)p Fo(1)2807 2517 y Ft(\))2845 2406 y Fj(\021)2921 2517 y Fr(;)149 2732 y Ft(where)30 b(as)e(b)s(efore)848 2707 y(~)830 2732 y Fp(E)883 2747 y Fq(n)957 2732 y Ft(=)g(2)1110 2696 y Fq(n)1157 2732 y Fr(e)1202 2696 y Fq(g)1242 2732 y Fr(a)1293 2696 y Fm(\000)p Fo(2)1404 2658 y Fj(P)1509 2762 y Fq(j)1562 2732 y Fr(f)1610 2747 y Fq(j)1675 2732 y Ft(\(w)m(e)h(are)f(b)s(eing)g(generous)h(here,)g(replacing)e Fr(g)3364 2696 y Fo(2)3406 2732 y Ft(~)-52 b Fr(g)3454 2696 y Fm(\000)p Fo(1)3565 2707 y Ft(~)3548 2732 y Fp(E)3601 2747 y Fq(n)3660 2732 y Ft(+)149 2874 y Fp(E)202 2889 y Fq(n)271 2874 y Ft(b)m(y)413 2849 y(~)396 2874 y Fp(E)449 2889 y Fq(n)495 2874 y Ft(\).)40 b(W)-8 b(e)22 b(also)f(absorb)s(ed)h (the)h(error)e Fp(E)1784 2889 y Fq(n)1853 2874 y Ft(coming)f(from)h (estimates)g(of)g(oscillatory)f(in)m(tegrals)149 2999 y(parallel)30 b(to)j(Lemma)e(2.4)h(in)m(to)1336 2973 y(~)1319 2999 y Fp(E)1372 3014 y Fq(n)1418 2999 y Fr(:)h Ft(In)g(particular,)149 3205 y(\(2.49\))421 b Fp(k)p Fr(R)q Ft(\()p Fr(x;)17 b(k)1136 3154 y Fo(\()p Fq(n)p Fo(\))1133 3230 y(1)1237 3205 y Ft(\))p Fp(k)1325 3164 y Fo(2)1325 3231 y Fq(L)1373 3212 y Fk(2)1408 3231 y Fo([)p Fq(x)1468 3240 y Fl(\031)r Fe(\000)p Fl(\013)1594 3255 y Fk(1)1632 3231 y Fq(;)p Fo(\()p Fq(f)1713 3240 y Fk(1)1747 3231 y Fo(\))1774 3212 y Fe(\000)p Fk(1)1857 3231 y Fq(g)r Fo(])1945 3205 y Ft(=)2058 3138 y(1)p 2058 3182 V 2058 3274 a(2)2117 3205 y Fr(f)2176 3164 y Fm(\000)p Fo(1)2165 3230 y(1)2287 3095 y Fj(\020)2346 3205 y Ft(1)22 b(+)g Fr(O)s Ft(\()2648 3180 y(~)2631 3205 y Fp(E)2684 3220 y Fq(n)2730 3205 y Fr(;)17 b(g)7 b Ft(~)-52 b Fr(g)2876 3164 y Fm(\000)p Fo(1)2968 3205 y Ft(\))3006 3095 y Fj(\021)149 3399 y Ft(and,)33 b(giv)m(en)g(the)g(upp)s(er)g(b)s(ound)g(on)f Fr(x)1559 3414 y Fq(\031)r Fm(\000)p Fq(\013)1702 3423 y Fk(1)1774 3399 y Ft(of)g(Lemma)f(2.5,)149 3579 y(\(2.50\))656 b Fr(R)1130 3538 y Fo(2)1169 3579 y Ft(\()p Fr(f)1266 3538 y Fm(\000)p Fo(1)1255 3604 y(1)1360 3579 y Fr(g)t(;)17 b(k)1509 3528 y Fo(\()p Fq(n)p Fo(\))1506 3604 y(1)1610 3579 y Ft(\))28 b Fp(\024)g Fr(e)1826 3538 y Fm(\000)p Fq(g)r(=)p Fo(2)1992 3579 y Ft(\(1)21 b(+)h Fr(C)7 b Ft(\()2331 3554 y(~)2313 3579 y Fp(E)2366 3594 y Fq(n)2435 3579 y Ft(+)25 b(~)-52 b Fr(g)2584 3538 y Fm(\000)p Fo(1)2678 3579 y Fr(g)t Ft(\)\))p Fr(:)249 3744 y Ft(4.)43 b(Finally)-8 b(,)30 b(w)m(e)k(consider)f(the)g(solution)e(on)h([)p Fr(f)2001 3703 y Fm(\000)p Fo(1)1990 3768 y(1)2095 3744 y Fr(g)t(;)17 b Fp(1)p Ft(\))p Fr(:)32 b Ft(Here,)900 3993 y Fr(V)957 4008 y Fo(1)996 3993 y Ft(\()p Fr(x)p Ft(\))c(=)g Fp(\000)p Ft(min)1516 3882 y Fj(\020)1575 3993 y Fr(\016)t(k)1673 4008 y Fo(1)1712 3993 y 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4467 y Fr(;)1730 4399 y(g)p 1703 4444 105 4 v 1703 4535 a Ft(2)p Fr(x)1817 4356 y Fj(\021)1893 4467 y Ft(\(1)22 b(+)g(cos)17 b(2)p Fr(\016)t(\022)2388 4482 y Fo(1)2450 4467 y Fp(\000)23 b Ft(cos)17 b(4)p Fr(\022)2791 4482 y Fo(1)2853 4467 y Fp(\000)22 b Ft(cos)c(2\()p Fr(\022)3232 4482 y Fo(1)3293 4467 y Ft(+)k Fr(\022)3436 4482 y Fo(2)3476 4467 y Ft(\)\))g(+)3053 4760 y(+)3139 4693 y Fr(W)14 b Ft(\()p Fr(x)p Ft(\))p 3139 4737 237 4 v 3179 4850 a Fr(k)3233 4799 y Fo(\()p Fq(n)p Fo(\))3230 4874 y(1)3402 4760 y Ft(sin)j(2)p Fr(\022)3633 4775 y Fo(1)252 4760 y Ft(\(2.51\))387 5080 y(\()p Fr(\016)t(\022)517 5095 y Fo(1)556 5080 y Ft(\))594 5039 y Fm(0)645 5080 y Ft(=)28 b Fr(\016)t(k)847 5095 y Fo(1)886 5080 y Ft(\(1)22 b(+)g Fr(O)s Ft(\()p Fr(\016)t(k)1307 5095 y Fo(1)1345 5080 y Ft(\)\))g(+)1551 5012 y Fr(k)1605 4962 y Fo(\()p Fq(n)p Fm(\000)p Fo(1\))1602 5037 y(1)p 1551 5057 247 4 v 1572 5170 a Ft(8)p Fr(k)1675 5119 y Fo(\()p Fq(n)p Fo(\))1672 5194 y(1)1807 5080 y Ft(min)1986 4969 y Fj(\020)2046 5080 y Fr(\016)t(k)2144 5095 y Fo(1)2183 5080 y Fr(;)2264 5012 y(g)p 2237 5057 105 4 v 2237 5148 a Ft(2)p Fr(x)2351 4969 y Fj(\021)2427 5080 y Ft(sin)16 b(2)p Fr(\016)t(\022)2704 5095 y Fo(1)2744 5080 y Ft(\(1)22 b Fp(\000)g Ft(cos)c(2\()p Fr(\022)3232 5095 y Fo(1)3293 5080 y Ft(+)k Fr(\022)3436 5095 y Fo(2)3476 5080 y Ft(\)\))g(+)2553 5374 y(+)2639 5306 y Fr(W)14 b Ft(\()p Fr(x)p Ft(\))p 2639 5351 237 4 v 2655 5463 a(2)p Fr(k)2758 5412 y Fo(\()p Fq(n)p Fo(\))2755 5488 y(1)2886 5374 y Ft(\(cos)j(2)p Fr(\022)3165 5389 y Fo(2)3227 5374 y Fp(\000)22 b Ft(cos)17 b(2)p Fr(\022)3567 5389 y Fo(1)3607 5374 y Ft(\))p Fr(:)-3420 b Ft(\(2.52\))p eop %%Page: 17 17 17 16 bop 1220 66 a Fn(SINGULAR)31 b(CONTINUOUS)g(SPECTR)n(UM)994 b(17)149 266 y Ft(First)38 b(of)f(all,)h(an)m(y)g(con)m(tribution)f(to) h(the)h(solution)d Fr(R)j Ft(of)f(the)g(equation)g(\(2.51\))g(coming)e (from)149 382 y(the)j(last)e(term)g(in)m(v)m(olving)g Fr(W)51 b Ft(or)38 b(t)m(w)m(o)g(last)f(summands)h(in)f(brac)m(k)m(ets) j(in)d(the)h(\014rst)h(term)e(is)g(of)149 498 y(the)c(order)g Fr(O)s Ft(\()p Fp(E)741 513 y Fq(n)787 498 y Ft(\))f(b)m(y)i(the)f (same)f(estimates)g(as)h(b)s(efore)g(\(in)f(Lemma)f(2.4\).)43 b(Therefore,)149 761 y(\(2.53\))732 693 y Fr(R)q Ft(\()p Fr(x;)17 b(k)998 642 y Fo(\()p Fq(n)p Fo(\))995 718 y(1)1100 693 y Ft(\))1138 657 y Fo(2)p 658 738 594 4 v 658 850 a Fr(R)q Ft(\()p Fr(f)830 809 y Fm(\000)p Fo(1)819 875 y(1)924 850 y Fr(g)t(;)g(k)1073 800 y Fo(\()p Fq(n)p Fo(\))1070 875 y(1)1174 850 y Ft(\))1212 822 y Fo(2)1289 761 y Ft(=)27 b Fr(e)1437 690 y Fm(\000)1502 663 y Fk(1)p 1502 675 31 3 v 1502 716 a(8)1543 690 y Fo(min)n(\()p Fq(\016)r(k)1757 699 y Fk(1)1793 690 y Fq(;)1839 659 y Fl(g)p 1822 675 66 3 v 1822 716 a Fk(2)p Fl(x)1898 690 y Fo(\))1937 637 y Fd(R)1984 657 y Fl(x)1970 725 y(f)2006 701 y Fe(\000)p Fk(1)2000 750 y(1)2088 725 y Fl(g)2125 690 y Fo(\(1+cos)12 b(2)p Fq(\016)r(\022)2449 699 y Fk(1)2485 690 y Fo(\))g Fq(dy)r Fo(\(1+)p Fq(O)r Fo(\()p Fq(\016)r(k)2867 699 y Fk(1)2903 690 y Fo(\)\))2962 761 y Ft(\(1)22 b(+)g Fr(O)s Ft(\()p Fp(E)3338 776 y Fq(n)3384 761 y Ft(\)\))p Fr(:)149 1029 y Ft(F)-8 b(urthermore,)32 b(same)h(estimates)f(sho)m(w)i(that)e(for)g(an)m(y)h Fr(x)c(>)e(f)2447 988 y Fm(\000)p Fo(1)2436 1053 y(1)2541 1029 y Fr(g)t(;)149 1317 y Ft(\(2.54\))1189 1161 y Fq(x)1137 1181 y Fj(Z)1085 1481 y Fq(f)1126 1451 y Fe(\000)p Fk(1)1119 1502 y(1)1209 1481 y Fq(g)1284 1317 y Fr(W)14 b Ft(\()p Fr(y)t Ft(\)\(cos)h(2)p Fr(\022)1795 1332 y Fo(2)1857 1317 y Fp(\000)23 b Ft(cos)17 b(2)p Fr(\022)2198 1332 y Fo(1)2238 1317 y Ft(\))g Fr(dy)30 b Ft(=)d Fr(O)s Ft(\()p Fp(E)2694 1332 y Fq(n)2740 1317 y Ft(\))149 1666 y(as)33 b(w)m(ell.)43 b(Consider)33 b(a)f(sequence)k(of)c(p)s(oin)m(ts)g Fr(y)1850 1681 y Fq(n)1896 1666 y Fr(;)h Ft(where)h Fr(y)2286 1681 y Fo(1)2352 1666 y Ft(=)28 b Fr(f)2515 1625 y Fm(\000)p Fo(1)2504 1690 y(1)2609 1666 y Fr(g)t(;)k Ft(and)1157 1843 y Fr(y)1205 1858 y Fq(n)1280 1843 y Ft(=)27 b(min)o Fp(f)p Fr(y)j Ft(:)e Fr(\016)t(\022)1821 1858 y Fo(1)1861 1843 y Ft(\()p Fr(y)t Ft(\))f(=)g Fr(\031)t(n)22 b Ft(+)g Fr(\016)t(\022)2448 1858 y Fo(1)2488 1843 y Ft(\()p Fr(y)2574 1858 y Fo(1)2613 1843 y Ft(\))p Fp(g)p Fr(:)149 2021 y Ft(According)32 b(to)f(\(2.52\))o(,)h(\(2.54\))e(and)i(\(2.6\))o(,)g (w)m(e)h(ha)m(v)m(e)f Fr(y)2180 2036 y Fq(n)2247 2021 y Fp(\000)20 b Fr(y)2392 2036 y Fq(n)p Fm(\000)p Fo(1)2556 2021 y Fp(\024)28 b Ft(\(2)p Fr(\031)c Ft(+)19 b Fr(O)s Ft(\()p Fp(E)3091 2036 y Fq(n)3137 2021 y Ft(\)\)\()p Fr(\016)t(k)3349 2036 y Fo(1)3388 2021 y Ft(\))3426 1985 y Fm(\000)p Fo(1)3520 2021 y Fr(;)32 b Ft(and)149 2137 y(in)27 b(eac)m(h)i(\()p Fr(y)559 2152 y Fq(n)p Fm(\000)p Fo(1)695 2137 y Fr(;)17 b(y)787 2152 y Fq(n)833 2137 y Ft(\))28 b(there)g(is)f(an)h(in)m(terv)-5 b(al)26 b Fr(I)1757 2152 y Fq(n)1831 2137 y Ft(of)h(length)g(at)g(least)g(\()p Fr(\031)t(=)p Ft(4)12 b(+)g Fr(O)s Ft(\()p Fp(E)3031 2152 y Fq(n)3076 2137 y Ft(\)\)\()p Fr(\016)t(k)3288 2152 y Fo(1)3327 2137 y Ft(\))3365 2101 y Fm(\000)p Fo(1)3487 2137 y Ft(where)149 2253 y(cos)18 b(2)p Fr(\016)t(\022)438 2268 y Fo(1)505 2253 y Fr(>)27 b Ft(0)p Fr(:)i Ft(Let)f(us)h(assume)g (for)e(simplicit)m(y)f(that)i(the)h(error)f(terms)g(in)g(the)g(ab)s(o)m (v)m(e)h(estimates)149 2370 y(do)i(not)g(consume)g(more)g(than)f(half)g (of)h(the)g(principal)e(terms.)42 b(Then)33 b(from)c(\(2.51\))h(w)m(e)i (see)g(that)149 2486 y(on)h(\()p Fr(y)371 2501 y Fq(n)417 2486 y Fr(;)17 b(y)509 2501 y Fq(n)p Fo(+1)646 2486 y Ft(\))149 2685 y(\(2.55\))773 b Fr(R)q Ft(\()p Fr(y)t(;)17 b(k)1435 2634 y Fo(\()p Fq(n)p Fo(\))1432 2709 y(1)1536 2685 y Ft(\))1574 2643 y Fo(2)1641 2685 y Fp(\024)28 b Fr(C)7 b(R)q Ft(\()p Fr(y)1984 2700 y Fo(1)2023 2685 y Fr(;)17 b(k)2121 2634 y Fo(\()p Fq(n)p Fo(\))2118 2709 y(1)2223 2685 y Ft(\))2261 2643 y Fo(2)2300 2685 y Fr(e)2345 2642 y Fm(\000)2412 2592 y Fd(P)2487 2612 y Fl(n)p Fe(\000)p Fk(1)2487 2664 y Fl(j)s Fk(=1)2619 2642 y Fq(b)2649 2652 y Fl(j)2686 2685 y Fr(;)149 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y(>)592 761 y(<)592 941 y(>)592 970 y(>)592 1000 y(>)592 1030 y(:)722 665 y Fp(\000)p Ft(2)p Fr(f)907 614 y Fo(\()p Fq(n)p Fo(\))896 690 y Fq(j)1009 665 y Fr(k)1063 614 y Fo(\()p Fq(n)p Fm(\000)p Fo(1\))1060 690 y Fq(j)1272 665 y Ft(sin\()p Fr(\022)1478 614 y Fo(\()p Fq(n)p Fo(\))1475 690 y(2)p Fq(j)t Fm(\000)p Fo(1)1659 665 y Ft(+)22 b Fr(\022)1805 614 y Fo(\()p Fq(n)p Fo(\))1802 690 y(2)p Fq(j)1907 665 y Ft(\))p Fr(;)746 b(x)2773 680 y Fq(n)2849 665 y Fp(\024)28 b Fr(x)g Fp(\024)g Fr(x)3197 614 y Fo(\()p Fq(n)p Fo(\))3197 719 y Fq(j;\031)r Fm(\000)p Fq(\013)3389 684 y Fk(\()p Fl(n)p Fk(\))3389 741 y Fl(j)722 852 y Ft(2)p Fr(f)830 801 y Fo(\()p Fq(n)p Fo(\))819 877 y Fq(j)932 852 y Fr(k)986 801 y Fo(\()p Fq(n)p Fm(\000)p Fo(1\))983 877 y Fq(j)1195 852 y Ft(sin)o(\()p Fr(\022)1400 801 y Fo(\()p Fq(n)p Fo(\))1397 877 y(2)p Fq(j)t Fm(\000)p Fo(1)1582 852 y Ft(+)22 b Fr(\022)1728 801 y Fo(\()p Fq(n)p Fo(\))1725 877 y(2)p Fq(j)1830 852 y Ft(\))p Fr(;)823 b(x)2773 801 y Fo(\()p Fq(n)p 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Fq(j)3326 1073 y Ft(\))3364 1037 y Fm(\000)p Fo(1)3458 1073 y Fr(g)3505 1088 y Fq(n)3552 1073 y Fr(:)149 1310 y Ft(It)48 b(is)f(assumed)h(that)f Fr(\016)t(\022)1110 1259 y Fo(\()p Fq(n)p Fo(\))1107 1335 y Fq(j)1212 1310 y Ft(\()p Fr(x)1305 1325 y Fq(n)1352 1310 y Ft(\))53 b Fr(<)g(\031)t(=)p Ft(2;)47 b Fr(x)1858 1259 y Fo(\()p Fq(n)p Fo(\))1858 1364 y Fq(j;\031)r Fm(\000)p Fq(\013)2050 1329 y Fk(\()p Fl(n)p Fk(\))2050 1386 y Fl(j)2191 1310 y Ft(is)g(then)h(de\014ned)h(as)e(a)g(minimal)d(v)-5 b(alue)149 1505 y(of)47 b Fr(x)53 b(>)f(x)566 1520 y Fq(n)661 1505 y Ft(where)c Fr(\016)t(\022)1052 1454 y Fo(\()p Fq(n)p Fo(\))1049 1530 y Fq(j)1154 1505 y Ft(\()p Fr(x)p Ft(\))f(reac)m(hes)i Fr(\031)36 b Fp(\000)d Fr(\013)1950 1454 y Fo(\()p Fq(n)p Fo(\))1949 1530 y Fq(j)2051 1505 y Fr(:)47 b Ft(W)-8 b(e)48 b(adopt)f(this)g(de\014nition)f(in)g(order)h (to)149 1621 y(a)m(v)m(oid)38 b(making)e(to)s(o)h(man)m(y)g (assumptions)h(on)f(what)h(happ)s(ens)h(for)e Fr(x)f Fp(\024)h Fr(x)2958 1636 y Fq(n)3006 1621 y Fr(;)g Ft(although)g(in)g (the)149 1737 y(construction)k(pro)s(cess)h(that)f(app)s(ears)g(in)f (the)h(follo)m(wing)d(sections)j(it)f(is)g(straigh)m(tforw)m(ard)g(to) 149 1866 y(c)m(hec)m(k)50 b(that)d Fr(\016)t(\022)748 1815 y Fo(\()p Fq(n)p Fo(\))745 1891 y Fq(j)850 1866 y Ft(\()p Fr(x)p Ft(\))g(do)s(es)h(not)f(reac)m(h)h Fr(\031)36 b Fp(\000)c Fr(\013)1986 1815 y Fo(\()p Fq(n)p Fo(\))1985 1891 y Fq(j)2135 1866 y Ft(for)47 b Fr(x)53 b Fp(\024)f Fr(x)2591 1881 y Fq(n)2639 1866 y Fr(:)47 b Ft(Similarly)-8 b(,)47 b(in)f(the)i(actual)149 2013 y(construction)40 b(w)m(e)f(will)e(alw)m(a)m(ys)j(ha)m(v)m(e)g Fr(x)1660 2028 y Fq(n)1746 2013 y Fr(<<)e Ft(\()p Fr(f)2033 1962 y Fo(\()p Fq(n)p Fo(\))2022 2039 y Fq(j)2134 2013 y Ft(\))2172 1977 y Fm(\000)p Fo(1)2267 2013 y Fr(g)2314 2028 y Fq(n)2360 2013 y Fr(:)h Ft(As)h(usual,)g Fr(\016)t(k)2968 1962 y Fo(\()p Fq(n)p Fo(\))2965 2039 y Fq(j)3108 2013 y Ft(=)i(~)-52 b Fr(g)3274 1977 y Fm(\000)p Fo(1)3270 2038 y Fq(n)3367 2013 y Fr(f)3426 1962 y Fo(\()p Fq(n)p Fo(\))3415 2039 y Fq(j)3528 2013 y Fr(:)39 b Ft(F)-8 b(or)149 2129 y(simplicit)m(y)g(,) 30 b(w)m(e)k(are)f(also)e(going)g(to)i(assume)g(that)149 2352 y(\(3.2\))918 b Fp(k)p Fr(V)1396 2311 y Fo(\()p Fq(n)p Fo(\))1498 2352 y Fp(k)1548 2368 y Fq(L)1596 2349 y Fe(1)1661 2368 y Fo(\()p Fh(R)1736 2349 y Fk(+)1787 2368 y Fo(\))1846 2352 y Fp(\024)1962 2284 y Ft(1)p 1962 2329 49 4 v 1962 2420 a(2)2020 2352 y(min)2183 2367 y Fq(j)2219 2352 y Fp(f)p Ft(\()p Fr(k)2361 2301 y Fo(\()p Fq(n)p Fo(\))2358 2377 y Fq(j)2463 2352 y Ft(\))2501 2311 y Fo(2)2541 2352 y Fp(g)p Fr(:)149 2566 y Ft(W)-8 b(e)33 b(also)f(supp)s(ose)i(that)e(the)h(condition)e(\(2.8\))h(of)h (the)g(Splitting)d(Lemma)h(holds:)149 2852 y(\(3.3\))134 b(4)550 2758 y Fj(X)605 2968 y Fq(j)710 2852 y Fr(f)769 2801 y Fo(\()p Fq(n)p Fo(\))758 2878 y Fq(j)871 2852 y Fr(k)925 2801 y Fo(\()p Fq(n)p Fm(\000)p Fo(1\))922 2878 y Fq(j)1144 2852 y Fr(<)1258 2785 y Ft(1)p 1258 2829 V 1258 2921 a(3)1317 2852 y Fr(a)1368 2867 y Fq(n)1415 2852 y Fr(;)100 b(g)1589 2867 y Fq(n)1663 2852 y Fr(>>)27 b Ft(1)p Fr(;)67 b Ft(2)2034 2811 y Fq(n)2080 2852 y Fr(g)2127 2867 y Fq(n)2174 2852 y Fr(a)2225 2811 y Fm(\000)p Fo(2)2225 2877 y Fq(n)2336 2728 y Fo(2)2371 2704 y Fl(n)p Fe(\000)p Fk(1)2342 2758 y Fj(X)2353 2968 y Fq(j)t Fo(=1)2509 2852 y Fr(f)2568 2801 y Fo(\()p Fq(n)p Fo(\))2557 2878 y Fq(j)2697 2852 y Fr(<<)28 b Ft(1)p Fr(;)99 b(\016)t(k)3153 2801 y Fo(\()p Fq(n)p Fo(\))3150 2878 y Fq(j)3283 2852 y Fr(<)3421 2785 y Ft(1)p 3396 2829 98 4 v 3396 2921 a(12)3504 2852 y Fr(a)3555 2867 y Fq(n)149 3148 y Ft(for)44 b(all)e Fr(j:)j Ft(The)g(p)s(oten)m(tial)d Fr(V)1283 3112 y Fo(\()p Fq(n)p Fo(\))1385 3148 y Ft(\()p Fr(x)p Ft(\))i(on)g(\(0)p Fr(;)17 b(x)1893 3163 y Fq(n)1940 3148 y Ft(\))44 b(comes)g(from)f(the)i(previous)g(steps,)j(but)c(for) 149 3264 y(no)m(w)f(w)m(e)g(mak)m(e)f(no)g(more)f(sp)s(eci\014c)i (assumptions)f(ab)s(out)f(it.)71 b(Recall)41 b(our)g(notation)g Fr(a)3479 3279 y Fq(n)p Fo(+1)3660 3264 y Ft(=)149 3395 y(min)312 3410 y Fq(j;j)394 3391 y Fe(0)419 3395 y Fp(j)p Fr(k)501 3344 y Fo(\()p Fq(n)p Fo(\))498 3421 y Fq(j)625 3395 y Fp(\000)22 b Fr(k)778 3344 y Fo(\()p Fq(n)p Fo(\))775 3423 y Fq(j)808 3404 y Fe(0)880 3395 y Fp(j)p Fr(:)33 b Ft(Let)38 b(~)-55 b Fr(x)1197 3410 y Fq(n)1277 3395 y Ft(b)s(e)33 b(suc)m(h)h(that)e(for)g(an)m(y)i Fr(x)28 b Fp(\025)34 b Ft(~)-55 b Fr(x)2418 3410 y Fq(n)2465 3395 y Fr(;)149 3639 y Ft(\(3.4\))935 b Fp(j)p Fr(V)1392 3598 y Fo(\()p Fq(n)p Fo(\))1494 3639 y Ft(\()p Fr(x)p Ft(\))p Fp(j)27 b(\024)1795 3572 y Ft(1)p 1795 3616 49 4 v 1795 3707 a(8)1854 3639 y Fr(a)1905 3654 y Fq(n)p Fo(+1)2042 3639 y Ft(min)2205 3654 y Fq(j)2241 3639 y Fp(f)p Ft(\()p Fr(k)2383 3588 y Fo(\()p Fq(n)p Fo(\))2380 3665 y Fq(j)2485 3639 y Ft(\))p Fp(g)p Fr(:)149 3879 y Ft(Also,)33 b(tak)m(e)39 b(~)-55 b Fr(x)663 3894 y Fq(n)738 3879 y Fr(>)28 b(g)889 3894 y Fq(n)935 3879 y Ft(max)1117 3894 y Fq(j)1153 3879 y Fp(f)p Ft(\()p Fr(f)1300 3829 y Fo(\()p Fq(n)p Fo(\))1289 3905 y Fq(j)1402 3879 y Ft(\))1440 3843 y Fm(\000)p Fo(1)1534 3879 y Fp(g)k Ft(for)g(con)m(v)m(enience.)149 4083 y FA(Lemma)38 b(3.1.)j Fi(Assume)35 b(that)g Fr(V)1404 4047 y Fo(\()p Fq(n)p Fo(\))1506 4083 y Ft(\()p Fr(x)p Ft(\))g Fi(is)g(given)f(by)44 b Ft(\(3.1\))34 b Fi(for)h Fr(x)28 b(>)f(x)2799 4098 y Fq(n)2847 4083 y Fr(:)35 b Fi(Then)149 4370 y Ft(\(3.5\))1345 4303 y Fr(@)5 b(\022)p 1342 4347 111 4 v 1342 4438 a(@)g(k)1463 4370 y Ft(\()p Fr(x;)17 b(k)1654 4319 y Fo(\()p Fq(n)p Fo(\))1651 4396 y Fq(j)1756 4370 y Ft(\))28 b Fp(\025)g Fr(P)1990 4385 y Fq(n)2053 4230 y Fj(\022)2160 4303 y Fr(x)p 2137 4347 103 4 v 2143 4438 a Ft(~)-55 b Fr(x)2192 4453 y Fq(n)2249 4230 y Fj(\023)2322 4252 y Fq(\014)s Fo(\()p Fq(k)2431 4217 y Fk(\()p Fl(n)p Fk(\))2429 4274 y Fl(j)2522 4252 y Fo(\))149 4671 y Fi(for)35 b(any)g Fr(x)28 b Fp(\025)g Ft(2)6 b(~)-55 b Fr(x)784 4686 y Fq(n)831 4671 y Fr(:)35 b Fi(Her)-5 b(e)35 b Fr(\014)6 b Ft(\()p Fr(k)s Ft(\))28 b(=)1458 4609 y Fq(k)1497 4573 y Fk(\()p Fl(n)p Fe(\000)p Fk(1\))1495 4631 y Fl(j)1665 4609 y Fq(g)1699 4617 y Fl(n)p 1458 4649 284 4 v 1545 4706 a Fo(16)p Fq(k)1752 4671 y Fr(:)34 b Fi(Mor)-5 b(e)g(over,)35 b(if)g Fr(k)j Fi(is)c(such)h(that)149 4909 y Ft(\(3.6\))1191 b Fp(j)p Fr(k)25 b Fp(\000)d Fr(k)1798 4858 y Fo(\()p Fq(n)p Fo(\))1795 4934 y Fq(j)1900 4909 y Fp(j)27 b(\024)2071 4841 y Ft(1)p 2071 4886 49 4 v 2071 4977 a(4)2129 4909 y Fr(a)2180 4924 y Fq(n)p Fo(+1)2318 4909 y Fr(;)149 5118 y Fi(then)149 5363 y Ft(\(3.7\))1269 5296 y Fr(@)5 b(\022)p 1266 5340 111 4 v 1266 5432 a(@)g(k)1387 5363 y Ft(\()p Fr(x;)17 b(k)s Ft(\))28 b Fp(\024)g Fr(D)1830 5378 y Fq(n)1893 5193 y Fj( )1972 5363 y Ft(1)22 b(+)2141 5223 y Fj(\022)2248 5296 y Fr(x)p 2224 5340 103 4 v 2230 5432 a Ft(~)-55 b Fr(x)2279 5447 y Fq(n)2337 5223 y Fj(\023)2410 5245 y Fq(\014)s Fo(\()p Fq(k)r Fo(\))2551 5193 y Fj(!)p eop %%Page: 19 19 19 18 bop 1220 66 a Fn(SINGULAR)31 b(CONTINUOUS)g(SPECTR)n(UM)994 b(19)149 269 y Fi(for)42 b(any)f Fr(x:)h Fi(Her)-5 b(e)42 b Fr(D)950 284 y Fq(n)997 269 y Fr(;)f(P)1128 284 y Fq(n)1217 269 y Fi(ar)-5 b(e)41 b(p)-5 b(ositive)41 b(c)-5 b(onstants)41 b(de\014ne)-5 b(d)40 b(only)h(in)h(terms)f(of)g Fr(V)3341 232 y Fo(\()p Fq(n)p Fo(\))3442 269 y Ft(\()p Fr(x)p Ft(\))h Fi(for)149 403 y Fr(x)29 b Fp(\024)f Fr(x)393 418 y Fq(n)440 403 y Fr(;)35 b(x)557 418 y Fq(n)604 403 y Fr(;)g Fp(f)p Fr(f)775 352 y Fo(\()p Fq(n)p Fo(\))764 428 y Fq(j)877 403 y Fp(g)p Fr(;)g(a)1040 418 y Fq(n)p Fo(+1)1212 403 y Fi(and)f Fr(g)1448 418 y Fq(n)1495 403 y Fi(.)149 608 y(Pr)-5 b(o)g(of.)42 b Ft(It)30 b(is)g(clear)g(that)h (the)g(v)-5 b(alue)30 b(of)36 b(~)-55 b Fr(x)1668 623 y Fq(n)1745 608 y Ft(dep)s(ends)33 b(only)d(on)g Fr(x)2521 623 y Fq(n)2569 608 y Fr(;)g(f)2685 557 y Fo(\()p Fq(n)p Fo(\))2674 634 y Fq(j)2787 608 y Fr(;)g(a)2895 623 y Fq(n)p Fo(+1)3063 608 y Ft(and)h Fr(g)3298 623 y Fq(n)3344 608 y Fr(;)g Ft(and)g(can)149 724 y(b)s(e)i(easily)f(estimated)g(in)g (terms)g(of)g(these)i(quan)m(tities:)900 908 y(~)-55 b Fr(x)949 923 y Fq(n)1024 908 y Fp(\024)28 b Fr(C)7 b Ft(max)1404 797 y Fj(\020)1464 908 y Ft(2)1513 866 y Fq(n)1560 908 y Fr(g)1607 923 y Fq(n)1653 908 y Fr(a)1704 866 y Fm(\000)p Fo(1)1704 932 y Fq(n)p Fo(+1)1842 908 y Fr(;)17 b(x)1941 923 y Fq(n)2010 908 y Ft(+)22 b Fr(g)2155 923 y Fq(n)2202 908 y Ft(max)2383 923 y Fq(j)2420 908 y Fp(f)p Ft(\()p Fr(f)2567 857 y Fo(\()p Fq(n)p Fo(\))2556 933 y Fq(j)2668 908 y Ft(\))2706 866 y Fm(\000)p Fo(1)2800 908 y Ft(\))p Fp(g)2888 797 y Fj(\021)2964 908 y Fr(:)149 1096 y Ft(Notice)33 b(that)f(from)f(\(2.3\))h(it)g(follo)m(ws)f(that) 376 1185 y Fj(\022)462 1258 y Fr(@)5 b(\022)p 459 1302 111 4 v 459 1394 a(@)g(k)580 1325 y Ft(\()p Fr(x;)17 b(k)s Ft(\))809 1185 y Fj(\023)882 1207 y Fm(0)933 1325 y Ft(=)28 b(1)22 b(+)1238 1258 y(1)p 1216 1302 94 4 v 1216 1394 a Fr(k)1270 1365 y Fo(2)1319 1325 y Fr(V)g Ft(\()p Fr(x)p Ft(\)\(sin)16 b Fr(\022)s Ft(\()p Fr(x;)h(k)s Ft(\)\))2018 1284 y Fo(2)2080 1325 y Fp(\000)2192 1258 y Ft(1)p 2190 1302 55 4 v 2190 1394 a Fr(k)2254 1325 y(V)k Ft(\()p Fr(x)p Ft(\))c(sin)g(2)p Fr(\022)s Ft(\()p Fr(x;)g(k)s Ft(\))2960 1185 y Fj(\022)3046 1258 y Fr(@)5 b(\022)p 3043 1302 111 4 v 3043 1394 a(@)g(k)3164 1325 y Ft(\()p Fr(x;)17 b(k)s Ft(\))3393 1185 y Fj(\023)3483 1325 y Fr(:)149 1555 y Ft(Along)29 b(with)f(the)i(b)s(oundary)g (condition)1686 1515 y Fq(@)t(\022)p 1684 1532 80 4 v 1684 1589 a(@)t(k)1774 1555 y Ft(\(0)p Fr(;)17 b(k)s Ft(\))27 b(=)h(0)h(\(whic)m(h)g(follo)m(ws)f(from)g(the)i(fact)f(that)g (w)m(e)149 1672 y(consider)40 b(solutions)e(satisfying)g(\014xed)i(b)s (oundary)f(condition)f(for)g(all)f Fr(k)s Ft(\))i(this)f(implies)f(in)h (our)149 1788 y(setting)149 2051 y(\(3.8\))568 1983 y Fr(@)5 b(\022)p 565 2028 111 4 v 565 2119 a(@)g(k)686 2051 y Ft(\()p Fr(x;)17 b(k)s Ft(\))28 b(=)1099 1895 y Fq(x)1047 1915 y Fj(Z)1057 2196 y Fo(0)1163 2051 y Fr(e)1208 2006 y Fm(\000)1275 1979 y Fk(1)p 1273 1991 34 3 v 1273 2032 a Fl(k)1329 1952 y Fd(R)1376 1973 y Fl(x)1362 2030 y(y)1426 2006 y Fq(V)1483 1982 y Fk(\()p Fl(n)p Fk(\))1574 2006 y Fo(\()p Fq(s)p Fo(\))12 b(sin)o(\(2)p Fq(\022)r Fo(\()p Fq(s;k)r Fo(\)\))h Fq(ds)2132 1910 y Fj(\022)2205 2051 y Ft(1)22 b(+)2406 1983 y(1)p 2384 2028 94 4 v 2384 2119 a Fr(k)2438 2090 y Fo(2)2487 2051 y Fr(V)2566 2010 y Fo(\()p Fq(n)p Fo(\))2668 2051 y Ft(\()p Fr(y)t Ft(\)\(sin)15 b Fr(\022)s Ft(\))3055 2010 y Fo(2)3095 1910 y Fj(\023)3201 2051 y Fr(dy)t(:)149 2330 y Ft(F)-8 b(or)35 b(an)m(y)i Fr(k)i Ft(satisfying)c(the)h(assumption)f(\(3.6\))h (of)f(the)h(lemma)e(w)m(e)j(ha)m(v)m(e)g(for)k(~)-55 b Fr(x)3155 2345 y Fq(n)3236 2330 y Fp(\024)34 b Fr(y)i Fp(\024)e Fr(x)i Ft(b)m(y)149 2446 y(an)d(estimate)f(parallel)e(to)i (that)g(in)g(Lemma)f(2.4)h(and)h(using)h(\(3.4\))e(and)g(\(3.3\))149 2563 y(\(3.9\))202 2634 y Fq(x)149 2654 y Fj(Z)158 2935 y Fq(y)266 2790 y Fr(V)344 2749 y Fo(\()p Fq(n)p Fo(\))446 2790 y Ft(\()p Fr(y)t Ft(\))17 b(sin)e(2)p Fr(\022)s Ft(\()p Fr(y)t(;)i(k)s Ft(\))g Fr(dy)29 b Ft(=)e Fp(\000)1384 2713 y Fr(k)1438 2662 y Fo(\()p Fq(n)p Fm(\000)p Fo(1\))1435 2738 y Fq(j)1631 2713 y Fr(g)1678 2728 y Fq(n)p 1384 2767 340 4 v 1506 2858 a Ft(16)1803 2634 y Fq(x)1751 2654 y Fj(Z)1760 2935 y Fq(y)1877 2713 y Ft(cos)q(\(2)p Fr(\022)s Ft(\()p Fr(s;)17 b(k)s Ft(\))22 b Fp(\000)g Ft(2)p Fr(\022)2581 2662 y Fo(\()p Fq(n)p Fo(\))2578 2738 y Fq(j)2683 2713 y Ft(\()p Fr(s)p Ft(\)\))p 1877 2767 966 4 v 2337 2858 a Fr(s)2869 2790 y(ds)p Ft(+)p Fr(O)s Ft(\(2)3207 2749 y Fq(n)3253 2790 y Fr(a)3304 2749 y Fm(\000)p Fo(2)3304 2815 y Fq(n)p Fo(+1)3441 2790 y Fr(g)3488 2805 y Fq(n)3552 2695 y Fj(X)3607 2905 y Fq(j)3712 2790 y Fr(f)3771 2739 y Fo(\()p Fq(n)p Fo(\))3760 2815 y Fq(j)3873 2790 y Ft(\))p Fr(:)149 3082 y Ft(Equations)33 b(\(3.8\),)f(\(3.9\))g(together)h(imply)e(a)h(v)m(ery)i(rough)f (estimate)531 3283 y Fr(@)5 b(\022)p 528 3327 111 4 v 528 3419 a(@)g(k)649 3350 y Ft(\()p Fr(x;)17 b(k)s Ft(\))28 b Fp(\024)g Fr(C)13 b Ft(~)-55 b Fr(x)1143 3365 y Fq(n)1190 3350 y Fr(e)1247 3241 y Fk(1)p 1246 3253 34 3 v 1246 3294 a Fl(k)1312 3199 y Fk(~)-34 b Fl(x)1343 3207 y(n)1316 3214 y Fd(R)1318 3331 y Fk(0)1390 3268 y Fm(j)p Fq(V)1467 3244 y Fk(\()p Fl(n)p Fk(\))1557 3268 y Fo(\()p Fq(y)r Fo(\))p Fm(j)13 b Fq(dy)1781 3350 y Ft(+)22 b Fr(C)7 b(e)2001 3308 y Fq(C)e Fo(2)2091 3284 y Fl(n)2133 3308 y Fq(a)2170 3278 y Fe(\000)p Fk(2)2170 3329 y Fl(n)p Fk(+1)2290 3308 y Fq(g)2324 3316 y Fl(n)2378 3258 y Fd(P)2453 3329 y Fl(j)2497 3308 y Fq(f)2538 3273 y Fk(\()p Fl(n)p Fk(\))2531 3330 y Fl(j)2702 3195 y Fq(x)2650 3215 y Fj(Z)2640 3497 y Fo(~)-39 b Fq(x)2676 3505 y Fl(n)2766 3350 y Fr(e)2811 3309 y Fq(\014)s Fo(\()p Fq(k)r Fo(\))12 b(log)q(\()p Fq(x=y)r Fo(\))3238 3350 y Fr(dy)t(;)149 3634 y Ft(where)34 b Fr(C)40 b Ft(is)32 b(a)g(univ)m(ersal)h(constan)m(t.)44 b(Similarly)-8 b(,)825 3829 y Fr(@)5 b(\022)p 822 3874 111 4 v 822 3965 a(@)g(k)943 3897 y Ft(\()p Fr(x;)17 b(k)1134 3846 y Fo(\()p Fq(n)p Fo(\))1131 3922 y Fq(j)1236 3897 y Ft(\))28 b Fp(\025)g Fr(C)1477 3912 y Fo(1)1516 3897 y Fr(e)1561 3854 y Fm(\000)p Fq(C)5 b Fo(2)1706 3831 y Fl(n)1749 3854 y Fq(a)1786 3824 y Fe(\000)p Fk(2)1786 3875 y Fl(n)p Fk(+1)1906 3854 y Fq(g)1940 3862 y Fl(n)1994 3804 y Fd(P)2068 3875 y Fl(j)2113 3854 y Fq(f)2154 3819 y Fk(\()p Fl(n)p Fk(\))2147 3876 y Fl(j)2317 3741 y Fq(x)2265 3761 y Fj(Z)2256 4044 y Fo(~)-39 b Fq(x)2292 4052 y Fl(n)2381 3897 y Fr(e)2426 3854 y Fq(\014)s Fo(\()p Fq(k)2535 3819 y Fk(\()p Fl(n)p Fk(\))2533 3876 y Fl(j)2626 3854 y Fo(\))12 b(log)q(\()p Fq(x=y)r Fo(\))2944 3897 y Fr(dy)t(:)149 4180 y Ft(Notice)33 b(that)958 4227 y Fq(x)905 4247 y Fj(Z)896 4530 y Fo(~)-39 b Fq(x)932 4538 y Fl(n)1022 4383 y Fr(e)1067 4342 y Fq(\014)s Fo(\()p Fq(k)r Fo(\))12 b(log)q(\()p Fq(x=y)r Fo(\))1494 4383 y Fr(dy)31 b Ft(=)1873 4315 y(~)-55 b Fr(x)1922 4330 y Fq(n)p 1737 4360 362 4 v 1737 4451 a Fr(\014)6 b Ft(\()p Fr(k)s Ft(\))22 b Fp(\000)g Ft(1)2125 4212 y Fj( )2204 4242 y(\022)2310 4315 y Fr(x)p 2287 4360 103 4 v 2293 4451 a Ft(~)-55 b Fr(x)2342 4466 y Fq(n)2399 4242 y Fj(\023)2473 4265 y Fq(\014)s Fo(\()p Fq(k)r Fo(\))2635 4383 y Fp(\000)2768 4315 y Fr(x)p 2745 4360 V 2751 4451 a Ft(~)g Fr(x)2800 4466 y Fq(n)2857 4212 y Fj(!)2953 4383 y Fr(:)149 4651 y Ft(The)28 b(parameter)f Fr(\014)6 b Ft(\()p Fr(k)s Ft(\))26 b(is)h(large)f(b)s(ecause)i(of)e Fr(g)1855 4666 y Fq(n)1902 4651 y Fr(;)h Ft(and)g(w)m(e)h(can)f(assume)h(for)e (simplicit)m(y)e Fr(\014)6 b Ft(\()p Fr(k)s Ft(\))28 b Fr(>)f Ft(2)149 4785 y(for)j(all)e Fr(k)33 b Ft(of)d(in)m(terest)h (\(lying)d(in)i(a)g(compact)f(set)i(where)h(all)c Fr(k)2441 4734 y Fo(\()p Fq(n)p Fo(\))2438 4811 y Fq(j)2573 4785 y Ft(lie\).)41 b(Then)31 b(w)m(e)g(get)f(the)h(result)149 4902 y(of)i(the)g(lemma.)41 b(W)-8 b(e)33 b(can)g(tak)m(e,)g(for)f (example,)993 5164 y Fr(D)1074 5179 y Fq(n)1149 5164 y Ft(=)27 b Fr(C)13 b Ft(~)-55 b Fr(x)1384 5123 y Fo(2)1384 5188 y Fq(n)1432 5164 y Fr(e)1489 5054 y Fk(1)p 1487 5066 34 3 v 1487 5107 a Fl(k)1553 5012 y Fk(~)-34 b Fl(x)1584 5020 y(n)1558 5028 y Fd(R)1559 5145 y Fk(0)1632 5081 y Fm(j)p Fq(V)1708 5058 y Fk(\()p Fl(n)p Fk(\))1798 5081 y Fo(\()p Fq(y)r Fo(\))p Fm(j)13 b Fq(dy)r Fo(+)p Fq(C)5 b Fo(2)2140 5058 y Fl(n)2183 5081 y Fq(a)2220 5051 y Fe(\000)p Fk(2)2220 5102 y Fl(n)p Fk(+1)2340 5081 y Fq(g)2374 5089 y Fl(n)2428 5031 y Fd(P)2502 5102 y Fl(j)2547 5081 y Fq(f)2588 5046 y Fk(\()p Fl(n)p Fk(\))2581 5103 y Fl(j)2683 5164 y Fr(;)-2561 b Ft(\(3.10\))1134 5360 y Fr(P)1197 5375 y Fq(n)1271 5360 y Ft(=)28 b Fr(C)1445 5375 y Fo(1)1680 5292 y Ft(~)-55 b Fr(x)1729 5307 y Fq(n)p 1494 5337 463 4 v 1494 5449 a Fr(\014)6 b Ft(\()p Fr(k)1647 5398 y Fo(\()p Fq(n)p Fo(\))1644 5475 y Fq(j)1748 5449 y Ft(\))22 b Fp(\000)h Ft(1)1967 5360 y Fr(e)2012 5317 y Fm(\000)p Fq(C)5 b Fo(2)2157 5294 y Fl(n)2200 5317 y Fq(a)2237 5287 y Fe(\000)p Fk(2)2237 5338 y Fl(n)p Fk(+1)2357 5317 y Fq(g)2391 5325 y Fl(n)2445 5267 y Fd(P)2519 5338 y Fl(j)2563 5317 y Fq(f)2604 5282 y Fk(\()p Fl(n)p Fk(\))2597 5339 y Fl(j)2699 5360 y Fr(:)-2577 b Ft(\(3.11\))p eop %%Page: 20 20 20 19 bop 149 66 a Fn(20)1286 b(ALEXANDER)23 b(KISELEV)3659 266 y Ff(\003)249 436 y Ft(No)m(w)33 b(w)m(e)h(pro)m(v)m(e)g(the)f (principal)d(result)j(of)f(this)g(section.)149 624 y FA(Lemma)f(3.2)g Ft(\(Con)m(tin)m(uit)m(y)c(Lemma\))p FA(.)36 b Fi(Cho)-5 b(ose)32 b Ft(~)-53 b Fr(g)2031 639 y Fq(n)p Fo(+1)2198 624 y Fi(so)29 b(that)34 b Ft(~)-53 b Fr(g)2562 573 y Fo(1)p Fq(=)p Fo(4)2558 649 y Fq(n)p Fo(+1)2723 624 y Fp(\025)28 b Ft(2)2877 588 y Fq(n)2924 624 y Fr(D)3005 639 y Fq(n)3052 624 y Fr(P)3129 588 y Fm(\000)p Fo(1)3115 649 y Fq(n)3222 624 y Fr(:)i Fi(Then)f(ther)-5 b(e)149 740 y(exists)35 b Fr(x)471 704 y Fm(0)471 765 y Fq(n)553 740 y Fi(such)g(that)g(for)g(any)g Fr(x)1372 755 y Fq(n)p Fo(+1)1537 740 y Fr(>)27 b(x)1695 704 y Fm(0)1695 765 y Fq(n)1778 740 y Fi(the)35 b(fol)5 b(lowing)33 b(holds.)44 b(If)149 978 y Ft(\(3.12\))702 b Fp(j)p Fr(k)24 b Fp(\000)f Fr(k)1358 927 y Fo(\()p Fq(n)p Fo(\))1355 1004 y Fq(j)1460 978 y Fp(j)k(\024)i Ft(2)s(~)-52 b Fr(g)1721 927 y Fm(\000)p Fo(1)p Fq(=)p Fo(2)1717 1003 y Fq(n)p Fo(+1)1901 838 y Fj(\022)1987 911 y Fr(@)5 b(\022)p 1984 955 111 4 v 1984 1047 a(@)g(k)2105 978 y Ft(\()p Fr(x)2198 993 y Fq(n)p Fo(+1)2336 978 y Fr(;)17 b(k)2434 927 y Fo(\()p Fq(n)p Fo(\))2431 1004 y Fq(j)2536 978 y Ft(\))2574 838 y Fj(\023)2647 860 y Fm(\000)p Fo(1)2758 978 y Fr(;)149 1194 y Fi(then)149 1355 y Ft(\(3.13\))777 b Fr(R)q Ft(\()p Fr(x;)17 b(k)s Ft(\))1480 1313 y Fo(2)1547 1355 y Ft(=)28 b Fr(R)q Ft(\()p Fr(x;)17 b(k)1917 1304 y Fo(\()p Fq(n)p Fo(\))1914 1380 y Fq(j)2019 1355 y Ft(\))2057 1313 y Fo(2)2096 1355 y Ft(\(1)22 b(+)g Fr(O)s Ft(\()s(~)-52 b Fr(g)2470 1304 y Fm(\000)p Fo(1)p Fq(=)p Fo(4)2466 1379 y Fq(n)p Fo(+1)2634 1355 y Ft(\)\))149 1515 y Fi(and)149 1704 y Ft(\(3.14\))1173 1636 y Fr(@)5 b(\022)p 1170 1681 V 1170 1772 a(@)g(k)1291 1704 y Ft(\()p Fr(x;)17 b(k)s Ft(\))27 b(=)1664 1636 y Fr(@)5 b(\022)p 1661 1681 V 1661 1772 a(@)g(k)1782 1704 y Ft(\()p Fr(x;)17 b(k)1973 1653 y Fo(\()p Fq(n)p Fo(\))1970 1729 y Fq(j)2075 1704 y Ft(\)\(1)22 b(+)g Fr(O)s Ft(\()s(~)-52 b Fr(g)2487 1653 y Fm(\000)p Fo(1)p Fq(=)p Fo(4)2483 1728 y Fq(n)p Fo(+1)2650 1704 y Ft(\)\))149 1893 y Fi(for)35 b(al)5 b(l)35 b Fr(x)28 b Fp(\024)g Fr(x)688 1908 y Fq(n)p Fo(+1)825 1893 y Fr(:)149 2084 y Fi(Pr)-5 b(o)g(of.)42 b Ft(F)-8 b(rom)27 b(\(3.5\))i(it)f(follo)m(ws)g(that)1561 2045 y Fq(@)t(\022)p 1559 2061 80 4 v 1559 2119 a(@)t(k)1648 2084 y Ft(\()p Fr(x;)17 b(k)1839 2033 y Fo(\()p Fq(n)p Fo(\))1836 2110 y Fq(j)1942 2084 y Ft(\))27 b Fp(!)g(1)i Ft(as)h Fr(x)e Fp(!)f(1)p Fr(;)i Ft(and)g(th)m(us)h(w)m(e)h(can)e(c)m (ho)s(ose)149 2206 y Fr(x)204 2170 y Fm(0)204 2230 y Fq(n)281 2206 y Ft(so)h(that)g(for)f(an)m(y)h Fr(x)989 2221 y Fq(n)p Fo(+1)1154 2206 y Fp(\025)f Fr(x)1315 2170 y Fm(0)1315 2230 y Fq(n)1392 2206 y Ft(the)h(v)-5 b(alues)29 b(of)g Fr(k)k Ft(satisfying)d(\(3.12\))f(also)g(satisfy)i(\(3.6\))o(.) 43 b(Then)149 2322 y(Lemma)32 b(3.1)g(applies,)g(and)g(th)m(us)i(w)m(e) f(ha)m(v)m(e)h(for)e(an)m(y)i Fr(x)845 2472 y Fq(@)t(\022)p 843 2488 V 843 2546 a(@)t(k)933 2511 y Ft(\()p Fr(x;)17 b(k)s Ft(\))p 714 2561 568 4 v 726 2635 a Fq(@)t(\022)p 724 2651 80 4 v 724 2709 a(@)t(k)814 2674 y Ft(\()p Fr(x)907 2689 y Fq(n)p Fo(+1)1044 2674 y Fr(;)g(k)1142 2623 y Fo(\()p Fq(n)p Fo(\))1139 2700 y Fq(j)1244 2674 y Ft(\))1319 2584 y Fp(\024)28 b Fr(D)1505 2599 y Fq(n)1552 2584 y Fr(P)1629 2543 y Fm(\000)p Fo(1)1615 2609 y Fq(n)1740 2414 y Fj( )1819 2584 y Ft(1)22 b(+)1988 2444 y Fj(\022)2094 2517 y Fr(x)p 2071 2561 103 4 v 2077 2653 a Ft(~)-55 b Fr(x)2126 2668 y Fq(n)2183 2444 y Fj(\023)2257 2466 y Fq(\014)s Fo(\()p Fq(k)r Fo(\))2397 2414 y Fj(!)2493 2444 y(\022)2576 2517 y Fr(x)2631 2532 y Fq(n)p Fo(+1)p 2576 2561 193 4 v 2627 2653 a Ft(~)g Fr(x)2676 2668 y Fq(n)2778 2444 y Fj(\023)2852 2466 y Fm(\000)p Fq(\014)s Fo(\()p Fq(k)3016 2431 y Fk(\()p Fl(n)p Fk(\))3014 2488 y Fl(j)3106 2466 y Fo(\))3154 2584 y Fr(:)149 2834 y Ft(Hence,)34 b(b)m(y)g(de\014nition)d(of)36 b(~)-52 b Fr(g)1194 2849 y Fq(n)p Fo(+1)1330 2834 y Fr(;)33 b Ft(w)m(e)h(ha)m(v)m (e)149 3099 y(\(3.15\))273 b Fp(j)p Fr(\022)s Ft(\()p Fr(x;)17 b(k)s Ft(\))22 b Fp(\000)h Fr(\022)s Ft(\()p Fr(x;)17 b(k)1338 3048 y Fo(\()p Fq(n)p Fo(\))1335 3125 y Fq(j)1440 3099 y Ft(\))p Fp(j)27 b(\024)i Ft(2)1688 3058 y Fm(\000)p Fq(n)1793 3099 y Ft(~)-53 b Fr(g)1840 3048 y Fm(\000)p Fo(1)p Fq(=)p Fo(4)1836 3123 y Fq(n)p Fo(+1)2021 2929 y Fj( )2100 3099 y Ft(1)22 b(+)2269 2959 y Fj(\022)2375 3032 y Fr(x)p 2352 3076 103 4 v 2358 3167 a Ft(~)-55 b Fr(x)2407 3182 y Fq(n)2464 2959 y Fj(\023)2538 2981 y Fq(\014)s Fo(\()p Fq(k)r Fo(\))2678 2929 y Fj(!)2774 2959 y(\022)2857 3032 y Fr(x)2912 3047 y Fq(n)p Fo(+1)p 2857 3076 193 4 v 2908 3167 a Ft(~)g Fr(x)2957 3182 y Fq(n)3060 2959 y Fj(\023)3133 2981 y Fm(\000)p Fq(\014)s Fo(\()p Fq(k)3297 2946 y Fk(\()p Fl(n)p Fk(\))3295 3003 y Fl(j)3387 2981 y Fo(\))3435 3099 y Fr(:)149 3351 y Ft(Recall)31 b(from)h(\(2.2\))g(that)522 3625 y Fr(R)q Ft(\()p Fr(x;)17 b(k)s Ft(\))826 3584 y Fo(2)866 3625 y Fr(=R)q Ft(\()p Fr(x;)g(k)1181 3574 y Fo(\()p Fq(n)p Fo(\))1178 3651 y Fq(j)1283 3625 y Ft(\))1321 3584 y Fo(2)1388 3625 y Ft(=)27 b Fr(e)1548 3497 y Fk(1)p 1546 3509 34 3 v 1546 3550 a Fl(k)1615 3457 y(x)1602 3471 y Fd(R)1604 3588 y Fk(0)1661 3524 y Fq(V)1718 3501 y Fk(\()p Fl(n)p Fk(\))1808 3524 y Fo(\()p Fq(y)r Fo(\))13 b(sin)f(2)p Fq(\022)r Fo(\()p Fq(y)r(;k)r Fo(\))g Fq(dy)r Fm(\000)2428 3497 y Fk(1)p 2381 3509 125 3 v 2381 3579 a Fl(k)2415 3550 y Fk(\()p Fl(n)p Fk(\))2414 3605 y Fl(j)2540 3457 y(x)2527 3471 y Fd(R)2529 3588 y Fk(0)2586 3524 y Fq(V)2643 3501 y Fk(\()p Fl(n)p Fk(\))2733 3524 y Fo(\()p Fq(y)r Fo(\))h(sin)f(2)p Fq(\022)r Fo(\()p Fq(y)r(;k)3129 3489 y Fk(\()p Fl(n)p Fk(\))3127 3546 y Fl(j)3219 3524 y Fo(\))g Fq(dy)3336 3625 y Fr(:)149 3816 y Ft(First)35 b(of)h(all,)f(w)m(e)i(can)f(replace)g(1)p Fr(=k)i Ft(with)e(1)p Fr(=k)1900 3766 y Fo(\()p Fq(n)p Fo(\))1897 3842 y Fq(j)2037 3816 y Ft(in)f(the)h(\014rst)h(term)e(in)g(the)i(exp)s(onen)m(t)g(at)f (the)149 3933 y(exp)s(ense)f(of)d(creating)g(an)h(error)f(factor)g(of)g (size)h(at)f(most)1489 4199 y Fr(e)1534 4117 y Fq(C)5 b Fm(j)p Fq(k)r Fm(\000)p Fq(k)1742 4082 y Fk(\()p Fl(n)p Fk(\))1740 4139 y Fl(j)1831 4117 y Fm(j)1869 4034 y Fl(x)1904 4049 y(n)p Fk(+1)1916 4063 y Fd(R)1917 4180 y Fk(0)2028 4117 y Fm(j)p Fq(V)2105 4093 y Fk(\()p Fl(n)p Fk(\))2195 4117 y Fo(\()p Fq(y)r Fo(\))p Fm(j)13 b Fq(dy)149 4360 y Ft(for)38 b Fr(x)g Fp(\024)f Fr(x)566 4375 y Fq(n)p Fo(+1)704 4360 y Fr(:)h Ft(F)-8 b(rom)37 b(\(3.12\))g(and)h(a)g(b)s (ound)g(\(3.5\))g(of)g(Lemma)e(3.1)i(it)f(is)h(clear)g(that)f(w)m(e)j (can)149 4493 y(c)m(ho)s(ose)34 b Fr(x)514 4457 y Fm(0)514 4518 y Fq(n)594 4493 y Ft(so)f(that)f(this)g(factor)g(do)s(es)h(not)g (exceed)h(1)22 b(+)g Fr(O)s Ft(\()s(~)-52 b Fr(g)2435 4443 y Fm(\000)p Fo(1)p Fq(=)p Fo(4)2431 4518 y Fq(n)p Fo(+1)2599 4493 y Ft(\))p Fr(:)32 b Ft(Next,)i(w)m(e)f(need)h(to)e(b)s (ound)149 4771 y(\(3.16\))1213 4704 y(1)p 1159 4748 156 4 v 1159 4861 a Fr(k)1213 4810 y Fo(\()p Fq(n)p Fo(\))1210 4886 y Fq(j)1394 4616 y(x)1342 4636 y Fj(Z)1352 4917 y Fo(0)1458 4771 y Fr(V)1537 4730 y Fo(\()p Fq(n)p Fo(\))1639 4771 y Ft(\()p Fr(y)t Ft(\))p Fp(j)p Fr(\022)s Ft(\()p Fr(y)t(;)17 b(k)s Ft(\))i Fp(\000)k Fr(\022)s Ft(\()p Fr(y)t(;)17 b(k)2424 4720 y Fo(\()p Fq(n)p Fo(\))2421 4797 y Fq(j)2525 4771 y Ft(\))p Fp(j)g Fr(dy)t(:)149 5054 y Ft(Using)34 b(\(3.15\))o(,)f(w)m(e)h(can)e(estimate)g(the)h (expression)h(\(3.16\))e(for)g Fr(x)c Fp(\024)g Fr(x)2759 5069 y Fq(n)p Fo(+1)2929 5054 y Ft(b)m(y)149 5332 y(\(3.17\))467 b Fr(C)7 b Ft(2)992 5291 y Fm(\000)p Fq(n)1097 5332 y Ft(~)-53 b Fr(g)1144 5281 y Fm(\000)p Fo(1)p Fq(=)p Fo(4)1140 5357 y Fq(n)p Fo(+1)1308 5332 y Fr(x)1363 5266 y Fm(\000)p Fq(\014)s Fo(\()p Fq(k)1527 5231 y Fk(\()p Fl(n)p Fk(\))1525 5288 y Fl(j)1618 5266 y Fo(\))1363 5357 y Fq(n)p Fo(+1)1656 5332 y Ft(~)e Fr(x)1705 5266 y Fq(\014)s Fo(\()p Fq(k)1814 5231 y Fk(\()p Fl(n)p Fk(\))1812 5288 y Fl(j)1905 5266 y Fo(\))p Fm(\000)p Fq(\014)s Fo(\()p Fq(k)r Fo(\))1705 5342 y Fq(n)2166 5169 y(x)2206 5178 y Fl(n)p Fk(+1)2174 5197 y Fj(Z)2184 5478 y Fo(0)2320 5332 y Fr(y)2372 5291 y Fq(\014)s Fo(\()p Fq(k)r Fo(\))2512 5332 y Fp(j)p Fr(V)2618 5291 y Fo(\()p Fq(n)p Fo(\))2720 5332 y Ft(\()p Fr(y)t Ft(\))p Fp(j)17 b Fr(dy)t(:)p eop %%Page: 21 21 21 20 bop 1220 66 a Fn(SINGULAR)31 b(CONTINUOUS)g(SPECTR)n(UM)994 b(21)149 283 y Ft(The)27 b(con)m(tribution)e(to)g(\(3.17\))g(coming)f (from)g(the)i(in)m(tegral)e(o)m(v)m(er)j([0)p Fr(;)c Ft(~)-55 b Fr(x)2717 298 y Fq(n)2764 283 y Ft(])26 b(can)g(b)s(e)f (made)h Fr(O)s Ft(\()s(~)-52 b Fr(g)3535 232 y Fm(\000)p Fo(1)p Fq(=)p Fo(4)3531 308 y Fq(n)p Fo(+1)3698 283 y Ft(\))149 409 y(b)m(y)46 b(the)f(c)m(hoice)g(of)g Fr(x)958 373 y Fm(0)958 434 y Fq(n)1005 409 y Fr(:)g Ft(Using)f(an)h(inequalit)m (y)e Fp(j)p Fr(V)2082 373 y Fo(\()p Fq(n)p Fo(\))2184 409 y Ft(\()p Fr(y)t Ft(\))p Fp(j)k(\024)h Fr(C)7 b Ft(2)2638 373 y Fq(n)2685 409 y Fr(\014)f Ft(\()p Fr(k)s Ft(\))p Fr(y)2928 373 y Fm(\000)p Fo(1)3066 409 y Ft(for)44 b Fr(y)51 b(>)k Ft(~)-55 b Fr(x)3506 424 y Fq(n)3553 409 y Fr(;)45 b Ft(w)m(e)149 557 y(estimate)24 b(the)h(remaining)c(part)k (of)30 b(\(3.17\))24 b(b)m(y)h Fr(C)11 b Ft(~)-53 b Fr(g)1980 506 y Fm(\000)p Fo(1)p Fq(=)p Fo(4)1976 582 y Fq(n)p Fo(+1)2144 557 y Ft(\()p Fr(x)2237 572 y Fq(n)p Fo(+1)2375 557 y Fr(=)6 b Ft(~)-55 b Fr(x)2479 572 y Fq(n)2526 557 y Ft(\))2564 515 y Fq(\014)s Fo(\()p Fq(k)r Fo(\))p Fm(\000)p Fq(\014)s Fo(\()p Fq(k)2864 480 y Fk(\()p Fl(n)p Fk(\))2862 537 y Fl(j)2954 515 y Fo(\))2986 557 y Fr(:)24 b Ft(Notice)g(that)g (from)149 674 y(the)33 b(de\014nition)f(of)g Fr(\014)6 b Ft(\()p Fr(k)s Ft(\))p Fr(;)32 b Ft(w)m(e)i(ha)m(v)m(e)149 955 y(\(3.18\))284 b Fp(j)p Fr(\014)6 b Ft(\()p Fr(k)s Ft(\))21 b Fp(\000)i Fr(\014)6 b Ft(\()p Fr(k)1176 904 y Fo(\()p Fq(n)p Fo(\))1173 980 y Fq(j)1277 955 y Ft(\))p Fp(j)27 b(\024)h Fr(C)7 b(g)1599 970 y Fq(n)1646 955 y Fp(j)p Fr(k)25 b Fp(\000)e Fr(k)1904 904 y Fo(\()p Fq(n)p Fo(\))1901 980 y Fq(j)2006 955 y Fp(j)k(\024)h Fr(C)7 b(g)2290 970 y Fq(n)2340 955 y Ft(~)-52 b Fr(g)2388 904 y Fm(\000)p Fo(1)p Fq(=)p Fo(2)2384 979 y Fq(n)p Fo(+1)2568 814 y Fj(\022)2655 887 y Fr(@)5 b(\022)p 2652 932 111 4 v 2652 1023 a(@)g(k)2773 955 y Ft(\()p Fr(x)2866 970 y Fq(n)p Fo(+1)3003 955 y Fr(;)17 b(k)3101 904 y Fo(\()p Fq(n)p Fo(\))3098 980 y Fq(j)3203 955 y Ft(\))3241 814 y Fj(\023)3314 836 y Fm(\000)p Fo(1)3425 955 y Fr(:)149 1219 y Ft(By)36 b(\(3.5\))c(of)h(Lemma)f(3.1)h(and)g(the)h (de\014nition)e(of)k(~)-52 b Fr(g)2109 1234 y Fq(n)p Fo(+1)2245 1219 y Fr(;)34 b Ft(the)f(righ)m(t)g(hand)g(side)h(in)e (\(3.18\))g(do)s(es)149 1367 y(not)h(exceed)h Fr(C)7 b Ft(\()p Fr(x)804 1382 y Fq(n)p Fo(+1)942 1367 y Fr(=)f Ft(~)-55 b Fr(x)1046 1382 y Fq(n)1093 1367 y Ft(\))1131 1325 y Fm(\000)p Fq(\014)s Fo(\()p Fq(k)1295 1290 y Fk(\()p Fl(n)p Fk(\))1293 1347 y Fl(j)1385 1325 y Fo(\))1417 1367 y Fr(:)33 b Ft(Th)m(us,)149 1678 y(\(3.19\))884 1538 y Fj(\022)968 1611 y Fr(x)1023 1626 y Fq(n)p Fo(+1)p 968 1655 193 4 v 1019 1747 a Ft(~)-55 b Fr(x)1068 1762 y Fq(n)1170 1538 y Fj(\023)1243 1560 y Fq(\014)s Fo(\()p Fq(k)r Fo(\))p Fm(\000)p Fq(\014)s Fo(\()p Fq(k)1543 1525 y Fk(\()p Fl(n)p Fk(\))1541 1582 y Fl(j)1634 1560 y Fo(\))1693 1678 y Fp(\024)29 b Fr(e)1844 1637 y Fq(C)5 b Fo(\(log)q(\()p Fq(x)2084 1646 y Fl(n)p Fk(+1)2204 1637 y Fq(=)t Fo(~)-39 b Fq(x)2279 1645 y Fl(n)2322 1637 y Fo(\)\)\()p Fq(x)2443 1646 y Fl(n)p Fk(+1)2563 1637 y Fq(=)t Fo(~)g Fq(x)2638 1645 y Fl(n)2681 1637 y Fo(\))2708 1595 y Fe(\000)p Fl(\014)s Fk(\()p Fl(k)2852 1565 y Fk(\()p Fl(n)p Fk(\))2851 1620 y Fl(j)2942 1595 y Fk(\))2974 1678 y Fr(;)149 1937 y Ft(whic)m(h)30 b(is)f(b)s(ounded)h(b)m(y)g(some) f(\014xed)h(univ)m(ersal)f(constan)m(t.)44 b(Com)m(bining)27 b(the)j(estimates,)g(w)m(e)g(see)149 2053 y(that)39 b(if)f(w)m(e)i(c)m (ho)s(ose)g Fr(x)984 2017 y Fm(0)984 2078 y Fq(n)1071 2053 y Ft(to)f(satisfy)g(all)e(of)h(the)i(ab)s(o)m(v)m(e)g(requiremen)m (ts,)i(then)d(\(3.13\))g(holds)f(for)149 2170 y(an)m(y)c Fr(x)28 b Fp(\024)g Fr(x)577 2185 y Fq(n)p Fo(+1)715 2170 y Fr(;)k Ft(pro)m(vided)h(that)g Fr(x)1442 2185 y Fq(n)p Fo(+1)1607 2170 y Fp(\025)28 b Fr(x)1767 2133 y Fm(0)1767 2194 y Fq(n)1814 2170 y Fr(:)249 2286 y Ft(Notice)37 b(also)f(that)h(from)f(\(3.15\))h(and)g(\(3.19\))o(,)i(it)d(follo)m(ws) g(that)h(for)g Fr(x)f Fp(2)g Ft(\(0)p Fr(;)17 b(x)3183 2301 y Fq(n)p Fo(+1)3320 2286 y Ft(\))37 b(w)m(e)i(ha)m(v)m(e)149 2420 y Fp(j)p Fr(\022)s Ft(\()p Fr(x;)17 b(k)s Ft(\))24 b Fp(\000)g Fr(\022)s Ft(\()p Fr(x;)17 b(k)818 2369 y Fo(\()p Fq(n)p Fo(\))815 2445 y Fq(j)920 2420 y Ft(\))p Fp(j)32 b(\024)g Fr(C)11 b Ft(~)-53 b Fr(g)1255 2369 y Fm(\000)p Fo(1)p Fq(=)p Fo(4)1251 2444 y Fq(n)p Fo(+1)1419 2420 y Fr(:)35 b Ft(Th)m(us)h(\(3.14\))e(follo)m(ws)g(from)f(\(3.13\))h (and)h(\(3.8\),)g(whic)m(h)h(can)149 2536 y(also)c(b)s(e)h(written)f (as)149 2825 y(\(3.20\))667 2758 y Fr(@)5 b(\022)p 664 2802 111 4 v 664 2894 a(@)g(k)785 2825 y Ft(\()p Fr(x;)17 b(k)s Ft(\))28 b(=)f Fr(R)q Ft(\()p Fr(x;)17 b(k)s Ft(\))1449 2784 y Fm(\000)p Fo(2)1613 2670 y Fq(x)1561 2690 y Fj(Z)1571 2971 y Fo(0)1677 2825 y Fr(R)q Ft(\()p Fr(y)t(;)g(k)s Ft(\))1978 2784 y Fo(2)2033 2685 y Fj(\022)2106 2825 y Ft(1)22 b(+)2307 2758 y(1)p 2285 2802 94 4 v 2285 2894 a Fr(k)2339 2865 y Fo(2)2388 2825 y Fr(V)2467 2784 y Fo(\()p Fq(n)p Fo(\))2569 2825 y Ft(\()p Fr(y)t Ft(\)\(sin)15 b Fr(\022)s Ft(\))2956 2784 y Fo(2)2996 2685 y Fj(\023)3102 2825 y Fr(dy)t(:)149 3150 y Ft(W)-8 b(e)33 b(used)h(\(3.2\))e(in)g(the) h(last)f(estimate.)2033 b Ff(\003)1231 3480 y Ft(4.)49 b Fs(A)38 b(Brick)f(of)i(Constr)n(uction)249 3654 y Ft(Our)31 b(goal)f(is)g(to)h(organize)f(a)h(Can)m(tor-lik)m(e)f(pro)s(cess,)j (splitting)c(eac)m(h)j(eigen)m(v)-5 b(alue)30 b(w)m(e)j(ha)m(v)m(e)f (on)149 3770 y(the)40 b Fr(n)p Ft(th)f(step)h(in)m(to)e(t)m(w)m(o)h(on) g(the)h(\()p Fr(n)26 b Ft(+)g(1\)th)39 b(step.)64 b(T)-8 b(o)38 b(do)h(this)g(w)m(e)h(need)g(to)e(see)i(ho)m(w)g(norm)149 3887 y(splitting)32 b(w)m(orks)k(in)d(a)h(more)f(general)g(setting)h (than)g(Splitting)e(Lemma,)h(whic)m(h)h(starts)h(at)f(the)149 4003 y(origin.)59 b(The)39 b(needed)h(estimates)e(are)g(pro)m(vided)h (b)m(y)g(the)g(follo)m(wing)c(lemma,)j(whic)m(h)h(helps)f(to)149 4119 y(connect)c(di\013eren)m(t)f(steps.)249 4235 y(Assume)h(that)e (for)g(some)g Fr(j;)h Ft(w)m(e)h(ha)m(v)m(e)g(for)e Fr(k)k Ft(satisfying)1122 4511 y Fp(j)p Fr(k)25 b Fp(\000)e Fr(k)1380 4460 y Fo(\()p Fq(n)p Fo(\))1377 4536 y Fq(j)1482 4511 y Fp(j)k(\024)h Ft(2)s(~)-52 b Fr(g)1742 4460 y Fm(\000)p Fo(1)p Fq(=)p Fo(2)1738 4535 y Fq(n)p Fo(+1)1923 4370 y Fj(\022)2009 4443 y Fr(@)5 b(\022)p 2006 4488 111 4 v 2006 4579 a(@)g(k)2127 4511 y Ft(\()p Fr(x)2220 4526 y Fq(n)p Fo(+1)2358 4511 y Fr(;)17 b(k)2456 4460 y Fo(\()p Fq(n)p Fo(\))2453 4536 y Fq(j)2557 4511 y Ft(\))2595 4370 y Fj(\023)2669 4393 y Fm(\000)p Fo(1)149 4770 y Ft(the)33 b(b)s(ounds)h(of)e(Con)m(tin)m(uit)m(y)h(Lemma:)149 4985 y(\(4.1\))826 b Fr(R)q Ft(\()p Fr(x;)17 b(k)s Ft(\))1480 4944 y Fo(2)1547 4985 y Ft(=)28 b Fr(R)q Ft(\()p Fr(x;)17 b(k)1917 4934 y Fo(\()p Fq(n)p Fo(\))1914 5010 y Fq(j)2019 4985 y Ft(\))2057 4944 y Fo(2)2096 4985 y Ft(\(1)22 b(+)g Fr(O)s Ft(\()s(~)-52 b Fr(g)2470 4934 y Fm(\000)p Fo(1)p Fq(=)p Fo(4)2466 5009 y Fq(n)p Fo(+1)2634 4985 y Ft(\)\))149 5188 y(and)149 5419 y(\(4.2\))1173 5352 y Fr(@)5 b(\022)p 1170 5397 V 1170 5488 a(@)g(k)1291 5419 y Ft(\()p Fr(x;)17 b(k)s Ft(\))27 b(=)1664 5352 y Fr(@)5 b(\022)p 1661 5397 V 1661 5488 a(@)g(k)1782 5419 y Ft(\()p Fr(x;)17 b(k)1973 5368 y Fo(\()p Fq(n)p Fo(\))1970 5445 y Fq(j)2075 5419 y Ft(\)\(1)22 b(+)g Fr(O)s Ft(\()s(~)-52 b Fr(g)2487 5368 y Fm(\000)p Fo(1)p Fq(=)p Fo(4)2483 5444 y Fq(n)p Fo(+1)2650 5419 y Ft(\)\))p eop %%Page: 22 22 22 21 bop 149 66 a Fn(22)1286 b(ALEXANDER)23 b(KISELEV)149 266 y Ft(for)32 b(all)f Fr(x)d Fp(\024)g Fr(x)677 281 y Fq(n)p Fo(+1)815 266 y Fr(:)k Ft(Set)1191 526 y Fr(f)1250 475 y Fo(\()p Fq(n)p Fo(+1\))1239 551 y Fq(j)1469 526 y Ft(=)f(~)-52 b Fr(g)1624 475 y Fo(1)p Fq(=)p Fo(2)1620 550 y Fq(n)p Fo(+1)1773 385 y Fj(\022)1859 458 y Fr(@)5 b(\022)p 1856 503 111 4 v 1856 594 a(@)g(k)1977 526 y Ft(\()p Fr(x)2070 541 y Fq(n)p Fo(+1)2208 526 y Fr(;)17 b(k)2306 475 y Fo(\()p Fq(n)p Fo(\))2303 551 y Fq(j)2407 526 y Ft(\))2445 385 y Fj(\023)2519 407 y Fm(\000)p Fo(1)149 526 y Ft(\(4.3\))1090 819 y Fr(\016)t(k)1191 768 y Fo(\()p Fq(n)p Fo(+1\))1188 844 y Fq(j)1411 819 y Ft(=)31 b(~)-53 b Fr(g)1565 768 y Fm(\000)p Fo(1)p Fq(=)p Fo(2)1561 843 y Fq(n)p Fo(+1)1746 678 y Fj(\022)1832 751 y Fr(@)5 b(\022)p 1829 796 V 1829 887 a(@)g(k)1950 819 y Ft(\()p Fr(x)2043 834 y Fq(n)p Fo(+1)2181 819 y Fr(;)17 b(k)2279 768 y Fo(\()p Fq(n)p Fo(\))2276 844 y Fq(j)2380 819 y Ft(\))2418 678 y Fj(\023)2492 700 y Fm(\000)p Fo(1)2602 819 y Fr(;)-2480 b Ft(\(4.4\))149 1091 y(and)35 b(let)f Fr(k)538 1040 y Fo(\()p Fq(n)p Fo(+1\))535 1117 y(2)p Fq(j)730 1091 y Fr(;)h(k)846 1040 y Fo(\()p Fq(n)p Fo(+1\))843 1117 y(2)p Fq(j)t Fm(\000)p Fo(1)1072 1091 y Ft(b)s(e)g(the)g(ends)g(of)f (the)h(in)m(terv)-5 b(al)33 b(of)h(size)h Fr(\016)t(k)2639 1040 y Fo(\()p Fq(n)p Fo(+1\))2636 1117 y Fq(j)2866 1091 y Ft(cen)m(tered)h(at)e Fr(k)3430 1040 y Fo(\()p Fq(n)p Fo(\))3427 1117 y Fq(j)3532 1091 y Fr(:)h Ft(F)-8 b(or)149 1224 y Fr(x)45 b(>)e(x)423 1239 y Fq(n)p Fo(+1)561 1224 y Fr(;)f Ft(de\014ne)h(p)s(oten)m(tial)e Fr(V)1421 1188 y Fo(\()p Fq(n)p Fo(+1\))1613 1224 y Ft(\()p Fr(x)p Ft(\))h(to)g(b)s(e) g(giv)m(en)g(b)m(y)j(\(3.1\))c(with)h Fr(n)29 b Ft(+)f(1)42 b(replacing)f Fr(n:)149 1340 y Ft(W)-8 b(e)41 b(are)f(going)f(to)h (assume)g(also)g(that)g(the)g(conditions)g(\(3.3\))f(of)h(the)g (Splitting)e(Lemma)h(are)149 1474 y(satis\014ed)30 b(\(with)f Fr(n)h Ft(replaced)f(b)m(y)i Fr(n)16 b Ft(+)g(1)28 b(in)h(our)g(curren) m(t)h(situation\).)41 b(De\014ne)30 b(a)f(parameter)g Fr(A)3634 1423 y Fo(\()p Fq(n)p Fo(\))3634 1499 y Fq(j)149 1590 y Ft(b)m(y)149 1823 y(\(4.5\))835 b Fr(A)1258 1772 y Fo(\()p Fq(n)p Fo(\))1258 1849 y Fq(j)1388 1823 y Ft(=)1504 1756 y Fr(@)5 b(\022)p 1501 1800 V 1501 1892 a(@)g(k)1622 1823 y Ft(\()p Fr(x)1715 1838 y Fq(n)p Fo(+1)1853 1823 y Fr(;)17 b(k)1951 1772 y Fo(\()p Fq(n)p Fo(\))1948 1849 y Fq(j)2053 1823 y Ft(\))p Fr(R)q Ft(\()p Fr(x)2259 1838 y Fq(n)p Fo(+1)2396 1823 y Fr(;)g(k)2494 1772 y Fo(\()p Fq(n)p Fo(\))2491 1849 y Fq(j)2596 1823 y Ft(\))2634 1782 y Fo(2)2673 1823 y Fr(:)149 2038 y Ft(W)-8 b(e)33 b(ha)m(v)m(e)149 2228 y FA(Lemma)38 b(4.1)h Ft(\(Connection)33 b(Lemma\))p FA(.)41 b Fi(Under)35 b(the)h(ab)-5 b(ove)35 b(assumptions,)f Fr(k)3072 2177 y Fo(\()p Fq(n)p Fo(+1\))3069 2254 y(2)p Fq(j)t Fm(\000)p Fo(1)3300 2228 y Fi(and)h Fr(k)3544 2177 y Fo(\()p Fq(n)p Fo(+1\))3541 2254 y(2)p Fq(j)149 2344 y Fi(ar)-5 b(e)47 b(eigenvalues)f(of)h Fr(H)1058 2367 y Fq(V)1114 2347 y Fk(\()p Fl(n)p Fk(+1\))1286 2344 y Fr(;)g Fi(and)g(the)g(norms)f(of)h(the)g(c)-5 b(orr)g(esp)g(onding)46 b(eigenfunctions)f(on)149 2461 y Ft(\()p Fr(x)242 2476 y Fq(n)p Fo(+1)380 2461 y Fr(;)17 b Fp(1)p Ft(\))34 b Fi(satisfy)149 2663 y Ft(\(4.6\))484 b Fp(k)p Fr(R)q Ft(\()p Fr(x;)17 b(k)1150 2612 y Fo(\()p Fq(n)p Fo(+1\))1147 2689 y(2)p Fq(j;)p Fo(2)p Fq(j)t Fm(\000)p Fo(1)1392 2663 y Ft(\))p Fp(k)1480 2622 y Fo(2)1480 2689 y Fq(L)1528 2670 y Fk(2)1563 2689 y Fo(\()p Fq(x)1630 2698 y Fl(n)p Fk(+1)1749 2689 y Fq(;)p Fm(1)p Fo(\))1899 2663 y Ft(=)27 b Fr(A)2075 2612 y Fo(\()p Fq(n)p Fo(\))2075 2689 y Fq(j)2177 2663 y Ft(\(1)22 b(+)g Fr(O)s Ft(\()2517 2638 y(~)2500 2663 y Fp(E)2553 2678 y Fq(n)p Fo(+1)2689 2663 y Fr(;)e Ft(~)-52 b Fr(g)2784 2612 y Fm(\000)p Fo(1)p Fq(=)p Fo(4)2780 2688 y Fq(n)p Fo(+1)2948 2663 y Ft(\)\))p Fr(:)149 2857 y Fi(Pr)-5 b(o)g(of.)42 b Ft(T)-8 b(o)45 b(simplify)d(notation,)47 b(let)d(us)i(assume)f(without)g(loss)f(of)h (generalit)m(y)f(that)h Fr(j)54 b Ft(=)49 b(1)p Fr(:)149 2986 y Ft(According)35 b(to)f(our)g(assumptions,)h Fr(\016)t(\022)1590 2935 y Fo(\()p Fq(n)p Fo(+1\))1587 3010 y(1)1782 2986 y Ft(\()p Fr(x)1875 3001 y Fq(n)p Fo(+1)2013 2986 y Ft(\))c(=)j(~)-53 b Fr(g)2239 2935 y Fm(\000)p Fo(1)p Fq(=)p Fo(2)2235 3010 y Fq(n)p Fo(+1)2404 2986 y Ft(\(1)23 b(+)g Fr(O)s Ft(\()s(~)-52 b Fr(g)2780 2935 y Fm(\000)p Fo(1)p Fq(=)p Fo(4)2776 3010 y Fq(n)p Fo(+1)2944 2986 y Ft(\)\))p Fr(:)34 b Ft(If)g Fr(\016)t(\022)3275 2935 y Fo(\()p Fq(n)p Fo(+1\))3272 3010 y(1)3468 2986 y Ft(\()p Fr(x)3561 3001 y Fq(n)p Fo(+1)3698 2986 y Ft(\))149 3102 y(w)m(ere)k(zero,)g(w)m(e)f(w)m(ould)f (b)s(e)g(exactly)h(in)e(a)h(situation)f(of)h(the)g(Splitting)e(Lemma)h (construction,)149 3230 y(and)e(w)m(ould)g(ha)m(v)m(e)h(t)m(w)m(o)g (eigenfunctions)f(of)f(norm)g Fp(\030)d Ft(2\()p Fr(f)2290 3180 y Fo(\()p Fq(n)p Fo(+1\))2279 3256 y Fq(j)2481 3230 y Ft(\))2519 3194 y Fm(\000)p Fo(1)2613 3230 y Fr(R)q Ft(\()p Fr(x)2781 3245 y Fq(n)p Fo(+1)2919 3230 y Fr(;)17 b(k)3017 3180 y Fo(\()p Fq(n)p Fo(\))3014 3256 y Fq(j)3119 3230 y Ft(\))3157 3194 y Fo(2)3196 3230 y Fr(:)33 b Ft(W)-8 b(e)33 b(ha)m(v)m(e)h(to)149 3378 y(tak)m(e)e(in)m(to)f(accoun)m(t)h (the)f(fact)g(that)g(on)g(the)h(in)m(terv)-5 b(al)29 b(where)k Fr(\016)t(\022)2509 3327 y Fo(\()p Fq(n)p Fo(+1\))2506 3402 y(1)2701 3378 y Ft(\()p Fr(x)p Ft(\))28 b Fp(2)g Ft([0)p Fr(;)17 b(\016)t(\022)3169 3327 y Fo(\()p Fq(n)p Fo(+1\))3166 3402 y(1)3361 3378 y Ft(\()p Fr(x)3454 3393 y Fq(n)p Fo(+1)3591 3378 y Ft(\)])31 b(a)149 3494 y(signi\014can)m(t)36 b(deca)m(y)i(tak)m(es)g(place)e(in)g(the)g(Splitting)e(Lemma,)i(and)h (w)m(e)g(lose)g(this)f(deca)m(y)h(in)f(our)149 3610 y(situation.)249 3727 y(Consider)29 b(an)g(auxiliary)d(problem)i(for)p 1707 3647 76 4 v 28 w Fr(R)q(;)g(\016)p 1884 3646 49 4 v 4 w(\022)k Ft(satisfying)c(same)g(equations)h(\(2.10\))o(,)h (\(2.11\))e(as)149 3871 y Fr(R)q Ft(\()p Fr(x;)17 b(k)415 3820 y Fo(\()p Fq(n)p Fo(+1\))412 3896 y(1)608 3871 y Ft(\))p Fr(;)32 b(\016)t(\022)800 3820 y Fo(\()p Fq(n)p Fo(+1\))797 3896 y(1)1024 3871 y Ft(but)g(with)p 1424 3791 79 4 v 32 w Fr(V)1502 3810 y Fo(\()p Fq(n)p Fo(+1\))1727 3871 y Ft(giv)m(en)g(b)m(y)i(\(2.7\))e(with)f Fr(n)22 b Ft(+)f(1)31 b(instead)h(of)g Fr(n:)g Ft(In)h(other)149 3988 y(w)m(ords,)g(consider)f(the)g(case)h(of)e(\()p Fr(n)20 b Ft(+)g(1\)st)31 b(step)i(p)s(oten)m(tial)d(starting)g(righ)m (t)h(a)m(w)m(a)m(y)i(at)e(the)h(origin.)149 4104 y(The)i(Small)c(Angle) i(and)h(Splitting)d(Lemmas)h(imply)g(that)149 4301 y(\(4.7\))204 b Fp(k)p 604 4221 76 4 v Fr(R)q Ft(\()p Fr(x;)17 b(k)870 4250 y Fo(\()p Fq(n)p Fo(+1\))867 4325 y(1)1062 4301 y Ft(\))p Fp(k)1150 4260 y Fo(2)1150 4327 y Fq(L)1198 4308 y Fk(2)1232 4327 y Fo(\()p Fq(x)1299 4341 y Fl(\016)r(\022)1358 4356 y Fk(1)1394 4341 y(\()p Fl(x)1453 4356 y(n)p Fk(+1)1572 4341 y(\))1600 4327 y Fq(;)p Fm(1)p Fo(\))1749 4301 y Ft(=)28 b(\()p Fr(f)1950 4250 y Fo(\()p Fq(n)p Fo(+1\))1939 4325 y(1)2142 4301 y Ft(\))2180 4260 y Fm(\000)p Fo(1)2274 4301 y Ft(\(1)22 b(+)g Fr(O)s Ft(\()2613 4276 y(~)2597 4301 y Fp(E)2650 4316 y Fq(n)p Fo(+1)2786 4301 y Fr(;)e Ft(~)-52 b Fr(g)2881 4250 y Fm(\000)p Fo(1)p Fq(=)p Fo(2)2877 4325 y Fq(n)p Fo(+1)3045 4301 y Fr(g)3092 4316 y Fq(n)p Fo(+1)3228 4301 y Ft(\)\))p Fr(:)149 4541 y Ft(But)33 b(the)g(same)f(estimates)h(as)f(in)g(these)i(Lemmas)d(apply)i(to)f Fr(R)q Ft(\()p Fr(x;)17 b(k)2708 4490 y Fo(\()p Fq(n)p Fo(+1\))2705 4565 y(1)2900 4541 y Ft(\))p Fr(;)32 b(\016)t(\022)3089 4556 y Fo(1)3162 4541 y Ft(on)g(\()p Fr(x)3390 4556 y Fq(n)p Fo(+1)3528 4541 y Fr(;)17 b Fp(1)p Ft(\))p Fr(;)149 4657 y Ft(b)m(y)34 b(iden)m(tical)d(argumen)m(ts.)44 b(The)33 b(only)f(adjustmen)m(t)h(w)m(e)h(need)f(to)g(mak)m(e)f(is)g(b) m(y)i(a)e(factor)1141 4854 y Fr(R)q Ft(\()p Fr(x)1309 4869 y Fq(n)p Fo(+1)1446 4854 y Fr(;)17 b(k)1544 4804 y Fo(\()p Fq(n)p Fo(+1\))1541 4879 y(1)1736 4854 y Ft(\))1774 4813 y Fo(2)1814 4854 y Fr(=)p 1863 4774 V(R)p Ft(\()p Fr(x)2030 4870 y Fq(\016)r(\022)2096 4879 y Fk(1)2132 4870 y Fo(\()p Fq(x)2199 4879 y Fl(n)p Fk(+1)2319 4870 y Fo(\))2350 4854 y Fr(;)g(k)2448 4804 y Fo(\()p Fq(n)p Fo(+1\))2445 4879 y(1)2640 4854 y Ft(\))2678 4813 y Fo(2)2718 4854 y Fr(:)149 5039 y Ft(By)34 b(Lemma)d(2.6)h(\(Small)e(Angle)i (Lemma\),)149 5156 y(\(4.8\))p 214 5283 V 214 5363 a Fr(R)q Ft(\()p Fr(x)382 5379 y Fq(\016)r(\022)448 5388 y Fk(1)483 5379 y Fo(\()p Fq(x)550 5388 y Fl(n)p Fk(+1)670 5379 y Fo(\))701 5363 y Fr(;)17 b(k)799 5312 y Fo(\()p Fq(n)p Fo(+1\))796 5388 y(1)991 5363 y Ft(\))1029 5322 y Fo(2)1096 5363 y Ft(=)1200 5193 y Fj( )1752 5296 y Fr(\016)t(k)1853 5245 y Fo(\()p Fq(n)p Fo(+1\))1850 5320 y(1)p 1289 5340 1220 4 v 1289 5453 a Fr(\016)t(\022)1384 5402 y Fo(\()p Fq(n)p Fo(+1\))1381 5477 y(1)1576 5453 y Ft(\()p Fr(x)1669 5468 y Fq(n)p Fo(+1)1806 5453 y Ft(\))p Fr(f)1903 5402 y Fo(\()p Fq(n)p Fo(+1\))1892 5477 y(1)2117 5453 y Ft(+)22 b Fr(\016)t(k)2316 5402 y Fo(\()p Fq(n)p Fo(+1\))2313 5477 y(1)2518 5193 y Fj(!)2614 5283 y(\000)2659 5363 y Ft(1)g(+)g Fr(O)s Ft(\()p Fp(E)2997 5378 y Fq(n)p Fo(+1)3133 5363 y Fr(;)e Ft(~)-52 b Fr(g)3228 5322 y Fm(\000)p Fo(1)3224 5388 y Fq(n)p Fo(+1)3361 5363 y Fr(g)3408 5378 y Fq(n)p Fo(+1)3545 5363 y Ft(\))3583 5283 y Fj(\001)3645 5363 y Fr(:)p eop %%Page: 23 23 23 22 bop 1220 66 a Fn(SINGULAR)31 b(CONTINUOUS)g(SPECTR)n(UM)994 b(23)149 266 y Ft(According)33 b(to)f(\(4.7\),)g(\(4.8\))g(w)m(e)i(ha)m (v)m(e)149 382 y(\(4.9\))149 589 y Fp(k)p Fr(R)q Ft(\()p Fr(x;)17 b(k)465 539 y Fo(\()p Fq(n)p Fo(+1\))462 614 y(1)p Fq(;)p Fo(2)658 589 y Ft(\))p Fp(k)746 548 y Fo(2)746 615 y Fq(L)794 596 y Fk(2)828 615 y Fo(\()p Fq(x)895 624 y Fl(n)p Fk(+1)1015 615 y Fq(;)p Fm(1)p Fo(\))1164 589 y Ft(=)1277 522 y Fr(R)q Ft(\()p Fr(x)1445 537 y Fq(n)p Fo(+1)1583 522 y Fr(;)g(k)1681 471 y Fo(\()p Fq(n)p Fo(\))1678 546 y(1)1783 522 y Ft(\))1821 486 y Fo(2)p 1277 567 583 4 v 1443 679 a Fr(f)1502 628 y Fo(\()p Fq(n)p Fo(+1\))1491 704 y(1)1887 419 y Fj( )1975 522 y Fr(\016)t(\022)2070 471 y Fo(\()p Fq(n)p Fo(+1\))2067 546 y(1)2263 522 y Ft(\()p Fr(x)2356 537 y Fq(n)p Fo(+1)2493 522 y Ft(\))p Fr(f)2590 471 y Fo(\()p Fq(n)p Fo(+1\))2579 546 y(1)p 1975 567 807 4 v 2232 679 a Fr(\016)t(k)2333 628 y Fo(\()p Fq(n)p Fo(+1\))2330 704 y(1)2814 589 y Ft(+)22 b(1)2961 419 y Fj(!)3056 479 y(\020)3116 589 y Ft(1)g(+)g Fr(O)s Ft(\()3417 564 y(~)3401 589 y Fp(E)3454 604 y Fq(n)p Fo(+1)3590 589 y Fr(;)e Ft(~)-52 b Fr(g)3685 539 y Fm(\000)p Fo(1)p Fq(=)p Fo(4)3681 614 y Fq(n)p Fo(+1)3848 589 y Ft(\))3886 479 y Fj(\021)3962 589 y Fr(:)149 888 y Ft(W)-8 b(e)30 b(utilized)d(assumption)h(\(4.1\))g(in)g(the)h(error)g (estimate,)f(and)h(also)f(used)33 b(~)-52 b Fr(g)2969 847 y Fm(\000)p Fo(1)2965 913 y Fq(n)p Fo(+1)3102 888 y Fr(g)3149 903 y Fq(n)p Fo(+1)3313 888 y Fr(<)28 b(C)11 b Ft(~)-53 b Fr(g)3545 837 y Fm(\000)p Fo(1)p Fq(=)p Fo(4)3541 913 y Fq(n)p Fo(+1)3709 888 y Fr(:)149 1005 y Ft(Because)35 b(of)d(the)h(assumption)f(\(4.2\))o(,)h(w)m(e)g(ha)m(v) m(e)149 1271 y(\(4.10\))980 1204 y Fr(\016)t(\022)1075 1153 y Fo(\()p Fq(n)p Fo(+1\))1072 1228 y(1)1267 1204 y Ft(\()p Fr(x)1360 1219 y Fq(n)p Fo(+1)1497 1204 y Ft(\))p 980 1248 556 4 v 1111 1361 a Fr(\016)t(k)1212 1310 y Fo(\()p Fq(n)p Fo(+1\))1209 1385 y(1)1573 1271 y Ft(=)1689 1204 y Fr(@)5 b(\022)p 1686 1248 111 4 v 1686 1340 a(@)g(k)1807 1271 y Ft(\()p Fr(x)1900 1286 y Fq(n)p Fo(+1)2038 1271 y Fr(;)17 b(k)2136 1220 y Fo(\()p Fq(n)p Fo(\))2133 1296 y(1)2238 1271 y Ft(\)\(1)k(+)h Fr(O)s Ft(\()s(~)-52 b Fr(g)2649 1220 y Fm(\000)p Fo(1)p Fq(=)p Fo(4)2645 1296 y Fq(n)p Fo(+1)2813 1271 y Ft(\)\))p Fr(:)149 1560 y Ft(The)32 b(term)e(\()p Fr(f)676 1509 y Fo(\()p Fq(n)p Fo(+1\))665 1584 y(1)868 1560 y Ft(\))906 1524 y Fm(\000)p Fo(1)1000 1560 y Fr(R)q Ft(\()p Fr(x)1168 1575 y Fq(n)p Fo(+1)1306 1560 y Fr(;)17 b(k)1404 1509 y Fo(\()p Fq(n)p Fo(\))1401 1584 y(1)1506 1560 y Ft(\))1544 1524 y Fo(2)1611 1560 y Ft(=)27 b Fr(A)1787 1509 y Fo(\()p Fq(n)p Fo(\))1787 1585 y Fq(j)1893 1560 y Ft(~)-53 b Fr(g)1940 1509 y Fm(\000)p Fo(1)p Fq(=)p Fo(2)1936 1584 y Fq(n)p Fo(+1)2104 1560 y Fr(;)31 b Ft(and)g(hence)h(con)m(tributes)g(a)e(correction)g(of)149 1707 y(higher)j(order.)43 b(Th)m(us)34 b(b)m(y)g(de\014nition)d(of)h Fr(A)1740 1656 y Fo(\()p Fq(n)p Fo(\))1740 1733 y Fq(j)1875 1707 y Ft(and)h(\(4.10\))o(,)g(\(4.9\))f(leads)g(exactly)h(to)f (\(4.6\).)125 b Ff(\003)249 1909 y Ft(The)32 b(usefulness)g(of)e(Lemma) f(4.1)i(b)s(ecomes)g(clear)f(if)f(one)i(lo)s(oks)f(at)h(\(3.20\))o(,)g (whic)m(h)g(in)f(partic-)149 2025 y(ular)i(implies)833 2187 y Fr(@)5 b(\022)p 830 2232 V 830 2323 a(@)g(k)951 2254 y Ft(\()p Fr(x;)17 b(k)s Ft(\))27 b(=)h Fr(R)q Ft(\()p Fr(x;)17 b(k)s Ft(\))1615 2213 y Fm(\000)p Fo(2)1710 2254 y Fp(k)p Fr(R)q Ft(\()p Fr(y)t(;)g(k)s Ft(\))p Fp(k)2111 2213 y Fo(2)2111 2280 y Fq(L)2159 2261 y Fk(2)2192 2280 y Fo(\(0)p Fq(;x)p Fo(\))2345 2254 y Ft(\(1)22 b(+)g Fr(O)s Ft(\()p Fp(k)p Fr(V)f Fp(k)2846 2269 y Fq(L)2894 2250 y Fe(1)2963 2254 y Ft(\)\))p Fr(:)149 2466 y Ft(Th)m(us)35 b(in)c(the)i(asymptotic)f(regime)g(where)h Fr(V)54 b Ft(is)33 b(small,)d(one)j(can)g(exp)s(ect,)h(informally)-8 b(,)1179 2628 y Fr(@)5 b(\022)p 1176 2672 V 1176 2763 a(@)g(k)1297 2695 y Ft(\()p Fr(x;)17 b(k)s Ft(\))27 b Fp(\030)h Fr(R)q Ft(\()p Fr(x;)17 b(k)s Ft(\))1962 2654 y Fm(\000)p Fo(2)2057 2695 y Fp(k)p Fr(R)q Ft(\()p Fr(y)t(;)g(k)s Ft(\))p Fp(k)2458 2654 y Fo(2)2458 2721 y Fq(L)2506 2702 y Fk(2)2539 2721 y Fo(\(0)p Fq(;x)p Fo(\))2693 2695 y Fr(;)149 2916 y Ft(and)38 b(the)g(role)e(of)h Fr(A)g Ft(is)g(pla)m(y)m(ed)h(b)m(y)h Fp(k)p Fr(R)q Ft(\()p Fr(x;)17 b(k)s Ft(\))p Fp(k)1896 2880 y Fo(2)1896 2947 y Fq(L)1944 2928 y Fk(2)1978 2947 y Fo(\(0)p Fq(;x)p Fo(\))2132 2916 y Fr(:)37 b Ft(Lemma)f(4.1)h(allo)m(ws)f(to)h(get)h(t)m (w)m(o)g(eigen-)149 3051 y(v)-5 b(alues)36 b(with)g(the)g(norm)f(on)h (\()p Fr(x;)17 b Fp(1)p Ft(\))35 b(same)h(as)g Fr(A)e Ft(=)f Fp(k)p Fr(R)q Ft(\()p Fr(x;)17 b(k)s Ft(\))p Fp(k)2538 3015 y Fo(2)2538 3082 y Fq(L)2586 3063 y Fk(2)2620 3082 y Fo(\(0)p Fq(;x)p Fo(\))2774 3051 y Fr(:)36 b Ft(This)g(will)d(pro)m (vide)k(the)149 3176 y(mec)m(hanism)32 b(for)f(norm)h(doubling)e(in)i (the)g(asymptotic)g(regime)f(whic)m(h)h(is)g(crucial)f(for)h(the)g (con-)149 3292 y(struction.)1415 3560 y(5.)49 b Fs(The)37 b(Constr)n(uction)249 3734 y Ft(No)m(w)h(w)m(e)g(are)f(ready)h(to)e (pro)m(vide)i(a)f(precise)g(description)g(of)g(the)g(construction)g(of) g(our)g(p)s(o-)149 3850 y(ten)m(tial)h Fr(V)21 b Ft(\()p Fr(x)p Ft(\))p Fr(:)39 b Ft(In)g(the)g(b)s(eginning,)f(w)m(e)i(tak)m(e) f(a)f(p)s(oten)m(tial)f(giv)m(en)i(b)m(y)h(\(2.7\),)g(lik)m(e)e(in)f (Splitting)149 3984 y(Lemma)k(with)g Fr(n)i Ft(=)g(1.)70 b(Start)41 b(b)m(y)h(c)m(ho)s(osing)f Fr(\016)t(k)2012 3933 y Fo(\(1\))2009 4009 y(1)2107 3984 y Fr(;)g(k)2229 3933 y Fo(\(0\))2226 4009 y(1)2324 3984 y Fr(;)g(g)2439 3999 y Fo(1)2520 3984 y Ft(and)g Fr(f)2777 3933 y Fo(\(1\))2766 4009 y(1)2871 3984 y Fr(:)h Ft(The)g(precise)h(c)m(hoice)149 4101 y(of)38 b(the)h(parameters)g(on)f(the)h(\014rst)g(step)h(of)e (construction)g(is)g(not)h(so)f(imp)s(ortan)m(t,)h(but)f(w)m(e)i(are) 149 4229 y(going)30 b(to)g(k)m(eep)j(it)c(similar)f(to)i(the)i (subsequen)m(t)h(steps.)45 b(T)-8 b(o)31 b(satisfy)h(\(2.6\))e(w)m(e)i (c)m(ho)s(ose)f Fr(k)3404 4178 y Fo(\(0\))3401 4253 y(1)3530 4229 y Ft(su\016-)149 4363 y(cien)m(tly)c(large)e(and)i Fr(\016)t(k)975 4312 y Fo(\(1\))972 4387 y(1)1096 4363 y Ft(small)d(compared)i(to)h Fr(k)1951 4312 y Fo(\(0\))1948 4387 y(1)2045 4363 y Fr(:)g Ft(W)-8 b(e)26 b(tak)m(e)i Fr(g)2513 4378 y Fo(1)2578 4363 y Ft(large,)f(imp)s(ose)f(a)g (relationship)149 4496 y Fr(\016)t(k)250 4445 y Fo(\(1\))247 4521 y(1)374 4496 y Ft(=)33 b(~)-53 b Fr(g)530 4455 y Fm(\000)p Fo(1)526 4521 y(1)624 4496 y Fr(f)683 4445 y Fo(\(1\))672 4521 y(1)811 4496 y Ft(and)33 b(tak)m(e)i Fr(e)1259 4460 y Fq(g)1293 4469 y Fk(1)1331 4496 y Fr(f)1390 4445 y Fo(\(1\))1379 4521 y(1)1518 4496 y Ft(small.)44 b(The)35 b(smallness)d(of)h(the)h(latter)f(n)m(um)m(b)s(er)h(and)g Fr(\016)t(k)3642 4445 y Fo(\(1\))3639 4521 y(1)149 4630 y Ft(can)29 b(b)s(e)g(ac)m(hiev)m(ed)h(b)m(y)g(taking)d Fr(f)1329 4579 y Fo(\(1\))1318 4654 y(1)1452 4630 y Ft(su\016cien)m (tly)i(small)e(while)g(k)m(eeping)j Fr(g)2834 4645 y Fo(1)2901 4630 y Ft(\014xed.)44 b(In)29 b(particular,)149 4763 y(w)m(e)f(tak)m(e)f Fr(g)539 4778 y Fo(1)605 4763 y Ft(large)e(enough)i(and)f Fr(f)1409 4712 y Fo(\(1\))1398 4788 y(1)1530 4763 y Ft(small)e(enough)j(for)f(the)g(conditions)g (\(3.3\))g(of)g(the)g(Splitting)149 4880 y(to)31 b(b)s(e)h (satis\014ed.)44 b(The)32 b(only)f(additional)d(restriction)j(w)m(e)h (need)h(to)e(imp)s(ose)f(on)h(the)h(parameters)149 4996 y(comes)k(from)e(the)h(condition)f Fp(j)p Fr(V)21 b Ft(\()p Fr(x)p Ft(\))p Fp(j)33 b(\024)f Fr(h)p Ft(\()p Fr(x)p Ft(\))p Fr(=)p Ft(\(1)24 b(+)g Fr(x)p Ft(\))p Fr(:)35 b Ft(By)h(the)g(monotonicit)m(y)d(of)i Fr(h)g Ft(and)h(the)149 5112 y(de\014nition)c(of)g(p)s(oten)m(tial,)f(the)i(inequalit)m(y)149 5373 y(\(5.1\))1039 b(2)p Fr(f)1497 5322 y Fo(\(1\))1486 5398 y(1)1590 5373 y Fr(k)1644 5322 y Fo(\(0\))1641 5398 y(1)1766 5373 y Fp(\024)1900 5306 y Fr(h)p Ft(\(\()p Fr(f)2091 5255 y Fo(\(1\))2080 5330 y(1)2185 5306 y Ft(\))2223 5270 y Fm(\000)p Fo(1)2317 5306 y Fr(g)2364 5321 y Fo(1)2403 5306 y Ft(\))p 1882 5350 579 4 v 1882 5463 a(1)21 b(+)h(\()p Fr(f)2147 5412 y Fo(\(1\))2136 5487 y(1)2241 5463 y Ft(\))2279 5434 y Fm(\000)p Fo(1)2374 5463 y Fr(g)2421 5478 y 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Fr(f)197 3707 y Fq(n)p Fo(+1)371 3692 y Ft(for)37 b(ev)m(ery)h Fr(j:)g Ft(This)e(can)h(alw)m(a)m(ys)h(b)s(e)f(ac)m(hiev)m(ed)h(b)m(y)g (taking)d Fr(x)2650 3707 y Fq(n)p Fo(+1)2825 3692 y Ft(large)g(enough)j (b)s(ecause)149 3813 y(of)48 b(\(3.5\))o(.)69 b(Notice)41 b(that)f(suc)m(h)j(c)m(hoice)e(ensures)i(that)2221 3788 y(~)2204 3813 y Fp(E)2257 3828 y Fq(n)p Fo(+1)2436 3813 y Fp(\024)f Fr(C)7 b Ft(2)2681 3777 y Fm(\000)p Fq(n)p Fm(\000)p Fo(1)2873 3813 y Fr(:)41 b Ft(Also,)i(the)e(constan)m(t)149 3930 y Fr(C)g Ft(is)33 b(\(5.5\))h(is)f(c)m(hosen)i(so)f(that)g(the)g (\014rst,)h(third)e(and)h(fourth)g(conditions)f(of)40 b(\(3.3\))33 b(needed)i(for)149 4046 y(Splitting)30 b(and)j(Connection) g(Lemmas)f(on)g(the)h(\()p Fr(n)22 b Ft(+)g(1\)st)33 b(step)h(hold.)249 4162 y(The)43 b(second)h(condition)d(imp)s(osed)g (on)h Fr(x)1814 4177 y Fq(n)p Fo(+1)1994 4162 y Ft(is)f(that)h(Lemma)f (3.2)h(\(Con)m(tin)m(uit)m(y)g(Lemma\))149 4296 y(applies,)32 b(that)h(is,)f Fr(x)894 4311 y Fq(n)p Fo(+1)1059 4296 y Fr(>)c(x)1218 4260 y Fm(0)1218 4321 y Fq(n)1265 4296 y Fr(;)33 b(x)1380 4260 y Fm(0)1380 4321 y Fq(n)1460 4296 y Ft(is)f(de\014ned)i(in)d(Lemma)h(3.2,)g(and)h Fr(\016)t(k)2831 4245 y Fo(\()p Fq(n)p Fo(\))2828 4321 y Fq(j)2960 4296 y Fp(\024)3093 4257 y Fo(1)p 3075 4273 71 4 v 3075 4330 a(12)3156 4296 y Fr(a)3207 4311 y Fq(n)p Fo(+1)3377 4296 y Ft(for)f(all)e Fr(j:)249 4429 y Ft(The)43 b(third)e(condition)f(comes)i(from)e(the)i(constrain)m(t)g Fp(j)p Fr(V)2436 4392 y Fo(\()p Fq(n)p Fo(+1\))2628 4429 y Ft(\()p Fr(x)p Ft(\))p Fp(j)h(\024)h Fr(h)p Ft(\()p Fr(x)p Ft(\))p Fr(=)p Ft(\(1)28 b(+)g Fr(x)p Ft(\))p Fr(:)42 b Ft(The)149 4548 y(p)s(oten)m(tial)30 b Fr(V)638 4512 y Fo(\()p Fq(n)p Fo(+1\))861 4548 y Ft(on)h([)p Fr(x)1077 4563 y Fq(n)p Fo(+1)1215 4548 y Fr(;)17 b Fp(1)p Ft(\))30 b(is)h(going)f(to)g(b)s(e)i(de\014ned)g(b)m(y)i(\(3.1\))c (with)h Fr(n)19 b Ft(+)g(1)31 b(instead)g(of)g Fr(n;)149 4664 y Ft(so)i(the)g(latter)f(requiremen)m(t)h(is)f(going)f(to)h(b)s(e) h(satis\014ed)g(if)e(for)h(ev)m(ery)j Fr(j;)149 4928 y Ft(\(5.6\))732 b Fr(f)1141 4877 y Fo(\()p Fq(n)p Fo(+1\))1130 4954 y Fq(j)1361 4928 y Fp(\024)28 b Ft(2)1515 4887 y Fm(\000)p Fq(n)p Fm(\000)p Fo(1)1717 4851 y Fr(h)p Ft(\()p Fr(x)1866 4866 y Fq(n)p Fo(+1)2025 4851 y Ft(+)22 b(\()p Fr(f)2220 4800 y Fo(\()p Fq(n)p Fo(+1\))2209 4876 y Fq(j)2412 4851 y Ft(\))2450 4815 y Fm(\000)p Fo(1)2544 4851 y Fr(g)2591 4866 y Fq(n)p Fo(+1)2728 4851 y Ft(\))p 1717 4905 1050 4 v 1783 5018 a Fr(x)1838 5033 y Fq(n)p Fo(+1)1997 5018 y Ft(+)g(\()p Fr(f)2192 4967 y Fo(\()p Fq(n)p Fo(+1\))2181 5043 y Fq(j)2384 5018 y Ft(\))2422 4989 y Fm(\000)p Fo(1)2516 5018 y Fr(g)2563 5033 y Fq(n)p Fo(+1)2776 4928 y Fr(:)149 5198 y Ft(By)35 b(\(3.5\))d(w)m(e)i(ha)m(v)m(e)1418 5253 y Fj(\022)1505 5326 y Fr(@)5 b(\022)p 1502 5370 111 4 v 1502 5461 a(@)g(k)1623 5393 y Ft(\()p Fr(x;)17 b(k)1814 5342 y Fo(\()p Fq(n)p Fo(\))1811 5419 y Fq(j)1916 5393 y Ft(\))p Fr(=x)2058 5253 y Fj(\023)2159 5337 y Fq(x)p Fm(!1)2169 5393 y Fp(\000)-16 b(!)37 b(1)p eop %%Page: 26 26 26 25 bop 149 66 a Fn(26)1286 b(ALEXANDER)23 b(KISELEV)149 266 y Ft(\(since)45 b(w)m(e)g(assume)f(that)g Fr(g)1211 281 y Fq(n)1301 266 y Ft(is)g(large)e(enough\).)78 b(Th)m(us,)49 b(for)43 b(large)g(enough)h Fr(x)3207 281 y Fq(n)p Fo(+1)3389 266 y Ft(w)m(e)h(ha)m(v)m(e)149 400 y Fr(x)204 415 y Fq(n)p Fo(+1)384 400 y Fp(\024)e Ft(\()p Fr(f)601 349 y Fo(\()p Fq(n)p Fo(+1\))590 425 y Fq(j)793 400 y Ft(\))831 363 y Fm(\000)p Fo(1)925 400 y Fr(g)972 415 y Fq(n)p Fo(+1)1150 400 y Ft(\(recall)d(that)h Fr(f)1736 349 y Fo(\()p Fq(n)p Fo(+1\))1725 425 y Fq(j)1969 400 y Ft(is)f(de\014ned)j (b)m(y)h(\(5.4\))o(\).)69 b(Then)43 b(the)e(condition)149 521 y(\(5.6\))32 b(reduces)j(to)1153 672 y(2)1202 631 y Fq(n)p Fo(+2)1339 672 y Fr(g)1386 687 y Fq(n)p Fo(+1)1551 672 y Fp(\024)28 b Fr(h)p Ft(\()p Fr(x)1805 687 y Fq(n)p Fo(+1)1964 672 y Ft(+)23 b(\()p Fr(f)2160 622 y Fo(\()p Fq(n)p Fo(+1\))2149 698 y Fq(j)2351 672 y Ft(\))2389 631 y Fm(\000)p Fo(1)2483 672 y Fr(g)2530 687 y Fq(n)p Fo(+1)2667 672 y Ft(\))p Fr(:)149 826 y Ft(This)33 b(can)g(alw)m(a)m (ys)g(b)s(e)g(ac)m(hiev)m(ed)h(b)m(y)f(taking)f Fr(x)1880 841 y Fq(n)p Fo(+1)2050 826 y Ft(su\016cien)m(tly)i(large)d(while)h Fr(g)3079 841 y Fq(n)p Fo(+1)3248 826 y Ft(is)g(\014xed.)249 957 y(By)49 b(the)g(c)m(hoice)f(of)g Fr(\016)t(k)1135 906 y Fo(\()p Fq(n)p Fo(+1\))1132 983 y Fq(j)1327 957 y Fr(;)h(x)1458 972 y Fq(n)p Fo(+1)1643 957 y Ft(and)j(~)-52 b Fr(g)1896 972 y Fq(n)p Fo(+1)2033 957 y Fr(;)48 b Ft(the)h(result)f (of)g(Lemma)f(3.2)h(\(Con)m(tin)m(uit)m(y)149 1105 y(Lemma\))25 b(applies)g(to)h Fr(k)1015 1054 y Fo(\()p Fq(n)p Fo(+1\))1012 1130 y Fq(j)1207 1105 y Fr(;)g Ft(and)g(therefore)h(w)m(e)g(can)f(use)h (Lemma)d(4.1)i(\(Connection)g(Lemma\).)149 1252 y(T)-8 b(ogether,)32 b(Con)m(tin)m(uit)m(y)f(and)g(Connection)g(Lemmas)f(sho)m (w)i(that)f Fr(k)2660 1201 y Fo(\()p Fq(n)p Fo(+1\))2657 1278 y Fq(j)2883 1252 y Ft(are)g(eigen)m(v)-5 b(alues)30 b(with)149 1369 y(eigenfunctions)j(satisfying)285 1617 y Fp(k)p Fr(R)q Ft(\()p Fr(x;)17 b(k)601 1566 y Fo(\()p Fq(n)p Fo(+1\))598 1643 y Fq(j)793 1617 y Ft(\))p Fp(k)881 1576 y Fo(2)881 1643 y Fq(L)929 1624 y Fk(2)995 1617 y Fp(\024)28 b Fr(C)1170 1632 y Fo(1)1209 1617 y Ft(2)1258 1576 y Fq(n)1364 1493 y(n)1322 1523 y Fj(Y)1329 1735 y Fq(l)q Fo(=1)1449 1617 y Ft(\(1)22 b(+)g Fr(E)6 b(r)1778 1632 y Fq(l)1804 1617 y Ft(\))1859 1507 y Fj(\020)1918 1617 y Ft(\(1)22 b(+)g Fr(C)11 b Ft(~)-53 b Fr(g)2253 1566 y Fm(\000)p Fo(1)p Fq(=)p Fo(4)2249 1642 y Fq(n)p Fo(+1)2417 1617 y Ft(\))22 b(+)g(\(1)g(+)g Fr(C)7 b Ft(\()s(~)-52 b Fr(g)2948 1566 y Fm(\000)p Fo(1)p Fq(=)p Fo(4)2944 1642 y Fq(n)p Fo(+1)3134 1617 y Ft(+)3249 1592 y(~)3232 1617 y Fp(E)3285 1632 y Fq(n)p Fo(+1)3422 1617 y Ft(\)\))3498 1507 y Fj(\021)3574 1617 y Fr(:)149 1886 y Ft(Here)37 b(the)f(\014rst)g(summand)f(in)g(the)i(brac)m(k)m(ets)g(comes)f(from)f (\(0)p Fr(;)17 b(x)2591 1901 y Fq(n)p Fo(+1)2728 1886 y Ft(\))35 b(and)h(w)m(as)h(estimated)e(b)m(y)149 2002 y(the)i(induction)d(assumption)i(\(5.2\))f(and)h(Con)m(tin)m(uit)m(y)g (Lemma,)f(while)g(the)h(second)i(summand)149 2118 y(comes)45 b(from)f(\()p Fr(x)785 2133 y Fq(n)p Fo(+1)923 2118 y Fr(;)17 b Fp(1)p Ft(\))44 b(and)g(w)m(as)i(estimated)e(using)i (\(5.3\),)i(Con)m(tin)m(uit)m(y)d(and)g(Connection)149 2235 y(Lemmas.)e(Altogether,)32 b(w)m(e)i(obtain)149 2498 y(\(5.7\))753 b Fp(k)p Fr(R)q Ft(\()p Fr(x;)17 b(k)1419 2447 y Fo(\()p Fq(n)p Fo(+1\))1416 2523 y Fq(j)1611 2498 y Ft(\))p Fp(k)1699 2457 y Fo(2)1699 2524 y Fq(L)1747 2505 y Fk(2)1813 2498 y Fp(\024)28 b Fr(C)1988 2513 y Fo(1)2027 2498 y Ft(2)2076 2457 y Fq(n)p Fo(+1)2230 2373 y Fq(n)p Fo(+1)2233 2403 y Fj(Y)2240 2615 y Fq(l)q Fo(=1)2363 2498 y Ft(\(1)22 b(+)g Fr(E)6 b(r)2692 2513 y Fq(l)2718 2498 y Ft(\))p Fr(;)149 2771 y Ft(where)34 b Fr(E)6 b(r)553 2786 y Fq(n)p Fo(+1)718 2771 y Fp(\024)28 b Fr(C)893 2786 y Fo(2)932 2771 y Ft(2)981 2734 y Fm(\000)p Fq(n)p Fm(\000)p Fo(1)1206 2771 y Ft(and)k(th)m(us)i(satis\014es)f(the)g (required)g(b)s(ound.)44 b(Next,)162 2985 y Fr(@)5 b(\022)p 159 3030 111 4 v 159 3121 a(@)g(k)280 3053 y Ft(\()p Fr(x;)17 b(k)471 3002 y Fo(\()p Fq(n)p Fo(+1\))468 3078 y Fq(j)664 3053 y Ft(\))27 b(=)846 2985 y Fr(@)5 b(\022)p 843 3030 V 843 3121 a(@)g(k)964 3053 y Ft(\()p Fr(x)1057 3068 y Fq(n)p Fo(+1)1194 3053 y Fr(;)17 b(k)1292 3002 y Fo(\()p Fq(n)p Fo(+1\))1289 3078 y Fq(j)1484 3053 y Ft(\))1532 2976 y Fr(R)q Ft(\()p Fr(x)1700 2991 y Fq(n)p Fo(+1)1837 2976 y Fr(;)g(k)1935 2925 y Fo(\()p Fq(n)p Fo(+1\))1932 3001 y Fq(j)2127 2976 y Ft(\))2165 2939 y Fo(2)p 1532 3030 673 4 v 1600 3143 a Fr(R)q Ft(\()p Fr(x;)g(k)1866 3092 y Fo(\()p Fq(n)p Fo(+1\))1863 3168 y Fq(j)2059 3143 y Ft(\))2097 3114 y Fo(2)2215 3053 y Ft(+)2372 2897 y Fq(x)2320 2917 y Fj(Z)2269 3198 y Fq(x)2309 3207 y Fl(n)p Fk(+1)2478 2976 y Fr(R)q Ft(\()p Fr(y)t(;)g(k)2741 2925 y Fo(\()p Fq(n)p Fo(+1\))2738 3001 y Fq(j)2932 2976 y Ft(\))2970 2939 y Fo(2)p 2476 3030 536 4 v 2476 3143 a Fr(R)q Ft(\()p Fr(x;)g(k)2742 3092 y Fo(\()p Fq(n)p Fo(+1\))2739 3168 y Fq(j)2934 3143 y Ft(\))2972 3114 y Fo(2)3022 3053 y Ft(\(1+)p Fr(O)s Ft(\()p Fp(j)p Fr(V)j Ft(\()p Fr(y)t Ft(\))p Fp(j)p Ft(\)\))d Fr(dy)t(:)149 3360 y Ft(By)28 b(induction)f(assumption)g(and)g(Con)m(tin)m(uit)m(y)h (Lemma,)f(the)h(\014rst)g(summand)f(do)s(es)h(not)f(exceed)995 3609 y Fr(R)q Ft(\()p Fr(x;)17 b(k)1261 3558 y Fo(\()p Fq(n)p Fo(+1\))1258 3635 y Fq(j)1453 3609 y Ft(\))1491 3568 y Fm(\000)p Fo(2)1585 3609 y Fr(C)1655 3624 y Fo(1)1694 3609 y Ft(2)1743 3568 y Fq(n)1849 3484 y(n)1807 3514 y Fj(Y)1814 3727 y Fq(l)q Fo(=1)1934 3609 y Ft(\(1)22 b(+)g Fr(E)6 b(r)2263 3624 y Fq(l)2289 3609 y Ft(\)\(1)22 b(+)g Fr(C)11 b Ft(~)-53 b Fr(g)2662 3558 y Fm(\000)p Fo(1)p Fq(=)p Fo(4)2658 3633 y Fq(n)p Fo(+1)2826 3609 y Ft(\))p Fr(:)149 3872 y Ft(The)34 b(second)g(summand,)e(b)m(y)h (Connection)g(Lemma,)e(do)s(es)j(not)e(exceed)500 4121 y Fr(R)q Ft(\()p Fr(x;)17 b(k)766 4070 y Fo(\()p Fq(n)p Fo(+1\))763 4146 y Fq(j)959 4121 y Ft(\))997 4080 y Fm(\000)p Fo(2)1091 4121 y Fr(C)1161 4136 y Fo(1)1200 4121 y Ft(2)1249 4080 y Fq(n)1355 3996 y(n)1312 4026 y Fj(Y)1320 4238 y Fq(l)q Fo(=1)1440 4121 y Ft(\(1)22 b(+)g Fr(E)6 b(r)1769 4136 y Fq(l)1795 4121 y Ft(\)\(1)21 b(+)h Fr(C)7 b Ft(\()2172 4096 y(~)2154 4121 y Fp(E)2207 4136 y Fq(n)p Fo(+1)2366 4121 y Ft(+)26 b(~)-53 b Fr(g)2515 4070 y Fm(\000)p Fo(1)p Fq(=)p Fo(4)2511 4145 y Fq(n)p Fo(+1)2679 4121 y Ft(\)\)\(1)22 b(+)g Fr(C)7 b Ft(2)3088 4080 y Fq(n)3156 4095 y Ft(~)3135 4121 y Fr(f)3183 4136 y Fq(n)p Fo(+1)3320 4121 y Ft(\))p Fr(:)149 4416 y Ft(Therefore,)37 b(taking)c(in)m(to)h(accoun)m(t)i(the) f(ab)s(o)m(v)m(e)g(estimates)f(on)2520 4390 y(~)2503 4416 y Fp(E)2556 4431 y Fq(n)p Fo(+1)2692 4416 y Fr(;)39 b Ft(~)-53 b Fr(g)2805 4365 y Fm(\000)p Fo(1)p Fq(=)p Fo(4)2801 4440 y Fq(n)p Fo(+1)3004 4416 y Ft(and)34 b(the)h(c)m(hoice)g (of)171 4521 y(~)149 4548 y Fr(f)197 4563 y Fq(n)p Fo(+1)335 4548 y Fr(;)d Ft(w)m(e)i(get)e(that)886 4746 y Fr(@)5 b(\022)p 883 4790 111 4 v 883 4881 a(@)g(k)1004 4813 y Ft(\()p Fr(x;)17 b(k)1195 4762 y Fo(\()p Fq(n)p Fo(+1\))1192 4839 y Fq(j)1388 4813 y Ft(\))27 b(=)h Fr(R)q Ft(\()p Fr(x;)17 b(k)1823 4762 y Fo(\()p Fq(n)p Fo(+1\))1820 4839 y Fq(j)2015 4813 y Ft(\))2053 4772 y Fm(\000)p Fo(2)2147 4813 y Fr(C)2217 4828 y Fo(1)2257 4813 y Ft(2)2306 4772 y Fq(n)p Fo(+1)2459 4689 y Fq(n)p Fo(+1)2462 4718 y Fj(Y)2470 4931 y Fq(l)q Fo(=1)2592 4813 y Ft(\(1)22 b(+)g Fr(E)6 b(r)2921 4828 y Fq(l)2947 4813 y Ft(\))p Fr(;)149 5086 y Ft(with)22 b Fr(E)433 5101 y Fq(n)p Fo(+1)598 5086 y Fp(\024)28 b Fr(C)773 5101 y Fo(2)812 5086 y Ft(2)861 5050 y Fm(\000)p Fq(n)p Fm(\000)p Fo(1)1053 5086 y Fr(:)22 b Ft(The)h(third)e(comp)s(onen)m(t)h(of)g(the)g(induction)f(step)i(w)m (as)g(already)e(handled)149 5202 y(b)m(y)38 b(the)g(c)m(hoice)g(of)e Fr(x)927 5217 y Fq(n)p Fo(+1)1065 5202 y Fr(:)h Ft(The)h(fourth)f(comp) s(onen)m(t)g(is)g(easily)f(satis\014ed,)j(since)f(b)m(y)g(insp)s (ection)149 5333 y(of)30 b(the)g(argumen)m(t)g(it)e(is)i(clear)f(that)g Fr(\016)t(k)1586 5282 y Fo(\()p Fq(n)p Fo(\))1583 5359 y Fq(j)1718 5333 y Ft(deca)m(y)i(rapidly)e(in)g Fr(n:)h Ft(In)g(fact)g(w)m(e)h(can)f(assume)g(freely)149 5455 y(that)e Fr(g)403 5470 y Fq(n)478 5455 y Fr(>)f Ft(2)630 5418 y Fq(n)705 5455 y Ft(\(there)i(is)e(no)h(restriction)g(on)g(ho)m (w)g(large)f Fr(g)2285 5470 y Fq(n)2360 5455 y Ft(w)m(e)i(can)f(c)m(ho) s(ose)h(on)f(eac)m(h)h(step,)h(only)p eop %%Page: 27 27 27 26 bop 1220 66 a Fn(SINGULAR)31 b(CONTINUOUS)g(SPECTR)n(UM)994 b(27)149 283 y Ft(a)30 b(b)s(ound)g(from)f(b)s(elo)m(w\).)43 b(Since)30 b Fr(f)1417 232 y Fo(\()p Fq(n)p Fo(\))1406 309 y Fq(j)1546 283 y Fp(\024)e Fr(C)7 b Ft(2)1777 247 y Fm(\000)p Fo(3)p Fq(n)1914 283 y Fr(e)1959 247 y Fm(\000)p Fq(g)2048 255 y Fl(n)2124 283 y Ft(for)30 b(an)m(y)g Fr(j)36 b Ft(and)30 b Fr(\016)t(k)2816 232 y Fo(\()p Fq(n)p Fo(\))2813 309 y Fq(j)2948 283 y Ft(is)g(ev)m(en)h(smaller,)e(w) m(e)149 405 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Fr(V)22 b Ft(\()p Fr(x)p Ft(\))40 b(on)h(all)d(p)s(ositiv)m(e)i(semi-axis.)66 b(W)-8 b(e)41 b(pro)m(v)m(e)h(the)f(follo)m(wing)d(result,)k(whic)m(h)f (implies)149 5251 y(Theorem)33 b(1.1.)149 5435 y FA(Theorem)38 b(6.1.)j Fi(The)35 b(singular)f(c)-5 b(ontinuous)35 b(sp)-5 b(e)g(ctrum)34 b(of)h(the)g(op)-5 b(er)g(ator)34 b Fr(H)3053 5450 y Fq(V)3149 5435 y Fi(is)h(nonempty.)p eop %%Page: 28 28 28 27 bop 149 66 a Fn(28)1286 b(ALEXANDER)23 b(KISELEV)149 266 y Fi(Pr)-5 b(o)g(of.)42 b Ft(Let)j(us)h(denote)h(the)e(sp)s(ectral) h(measure)g(of)f Fr(H)2259 281 y Fq(V)2365 266 y Ft(b)m(y)h Fr(\026;)f Ft(and)h(sp)s(ectral)f(measures)h(of)149 382 y Fr(H)230 405 y Fq(V)287 385 y Fk(\()p Fl(n)p Fk(\))425 382 y Ft(b)m(y)e Fr(\026)630 397 y Fq(n)677 382 y Fr(:)f Ft(Notice)g(that)g(since)g Fr(V)68 b Ft(=)45 b Fr(V)1860 346 y Fo(\()p Fq(n)p Fo(\))2005 382 y Ft(for)e Fr(x)j(<)g(x)2443 397 y Fq(n)2533 382 y Ft(and)d Fr(x)2788 397 y Fq(n)2882 382 y Fp(!)i(1)p Fr(;)e Ft(the)h(op)s(erator)149 498 y Fr(H)230 521 y Fq(V)287 501 y Fk(\()p Fl(n)p Fk(\))413 498 y Ft(con)m(v)m(erges)33 b(to)e Fr(H)1047 513 y Fq(V)1139 498 y Ft(in)f(a)h(strong)g(resolv)m(en)m(t)i(sense.)45 b(The)32 b(strong)f(resolv)m(en)m(t)h(con)m(v)m(ergence)149 614 y(implies)44 b(w)m(eak)j(con)m(v)m(ergence)i(of)d Fr(\026)1488 629 y Fq(n)1581 614 y Ft(to)f Fr(\026)h Ft(\(see,)51 b(e.g.)84 b([17],)49 b(Theorem)e(VI)s(I.20\).)84 b(Consider)149 748 y(an)m(y)43 b(in)m(terv)-5 b(al)40 b Fr(I)755 697 y Fo(\()p Fq(n)p Fo(\))747 774 y Fq(j)899 748 y Ft(with)h Fr(n)h Ft(su\016cien)m(tly)h(large.)70 b(Then)43 b(b)m(y)f(Theorem)h(5.1)e(and)h(Lemma)e(2.1,)149 896 y(w)m(e)h(ha)m(v)m(e)f Fr(\026)590 911 y Fq(m)656 896 y Ft(\()p Fr(I)745 845 y Fo(\()p Fq(n)p Fo(\))737 921 y Fq(j)847 896 y Ft(\))e Fp(\025)h Fr(B)5 b Ft(2)1167 860 y Fm(\000)p Fq(n)1308 896 y Ft(for)39 b(eac)m(h)h Fr(m)f(>)f(n:)h Ft(This)h(implies)c(b)m(y)k(w)m(eak)g(con)m(v)m (ergence)i(that)149 1043 y Fr(\026)p Ft(\(2)p Fr(I)346 992 y Fo(\()p Fq(n)p Fo(\))338 1069 y Fq(j)448 1043 y Ft(\))27 b Fp(\025)h Fr(B)5 b Ft(2)746 1007 y Fm(\000)p Fq(n)876 1043 y Ft(for)28 b(ev)m(ery)j Fr(n)d Ft(and)h Fr(j)k Ft(=)28 b(1)p Fr(;)17 b(:)g(:)g(:)e(;)i Ft(2)2040 1007 y Fq(n)2087 1043 y Fr(:)28 b Ft(Assume)h(that)g Fr(\026)f Ft(do)s(es)g(not)h(ha)m(v)m(e)g(singular)149 1165 y(con)m(tin)m(uous)35 b(comp)s(onen)m(t,)g(so)f(that)g Fr(\026)c Ft(=)g Fr(\026)1755 1129 y Fo(ac)1849 1165 y Ft(+)23 b Fr(\026)2007 1129 y Fo(pp)2089 1165 y Fr(:)34 b Ft(T)-8 b(ak)m(e)36 b(a)e(small)d(n)m(um)m(b)s(er)k Fr(\017)c(<)f(B)5 b(=)p Ft(10)p Fr(;)33 b Ft(then)149 1281 y(there)d(exists)f(a)g Fr(\016)781 1296 y Fq(\017)842 1281 y Ft(suc)m(h)h(that)f(if)e Fp(j)p Fr(S)6 b Fp(j)27 b Fr(<)h(\016)1647 1296 y Fq(\017)1680 1281 y Fr(;)g(\026)1794 1245 y Fo(ac)1865 1281 y Ft(\()p Fr(S)6 b Ft(\))27 b Fr(<)h(\017)h Ft(\(w)m(e)g(denote)h(here)f(b)m(y)h Fp(j)p Fr(S)6 b Fp(j)28 b Ft(the)h(Leb)s(esgue)149 1397 y(measure)k(of)f(the)h (set)h Fr(S)6 b Ft(\).)43 b(Cho)s(ose)33 b Fr(n)1532 1412 y Fo(1)1604 1397 y Ft(so)g(that)f(for)h(an)m(y)g Fr(n)28 b(>)f(n)2516 1412 y Fo(1)2556 1397 y Fr(;)1361 1552 y Fo(2)1396 1529 y Fl(n)1328 1582 y Fj(X)1339 1792 y Fq(j)t Fo(=1)1489 1677 y Fp(j)p Ft(2)p Fr(I)1617 1626 y Fo(\()p Fq(n)p Fo(\))1609 1702 y Fq(j)1718 1677 y Fp(j)g(\024)h Fr(C)7 b Ft(2)2004 1635 y Fq(n)p Fo(+1)2141 1677 y Fr(e)2186 1635 y Fm(\000)p Fo(2)2276 1612 y Fl(n)2351 1677 y Fr(<)27 b(\016)2497 1692 y Fq(\017)2530 1677 y Fr(:)149 1959 y Ft(Then)149 2142 y(\(6.1\))990 b Fr(\026)1399 2101 y Fo(ac)1470 2142 y Ft(\()p Fp([)1574 2157 y Fq(j)t Fo(=1)1701 2142 y Ft(\(2)p Fr(I)1839 2091 y Fo(\()p Fq(n)p Fo(\))1831 2168 y Fq(j)1940 2142 y Ft(\)\))28 b Fp(\024)g Fr(\017)g(<)g(B)5 b(=)p Ft(10)149 2333 y(for)38 b(an)m(y)g Fr(n)e(>)h(n)758 2348 y Fo(1)797 2333 y Fr(:)h Ft(Let)g Fr(\026)1101 2296 y Fo(pp)1219 2333 y Ft(=)1331 2258 y Fj(P)1437 2284 y Fm(1)1437 2362 y Fq(l)q Fo(=1)1569 2333 y Fr(\032)1619 2348 y Fq(l)1646 2333 y Fr(\016)t Ft(\()p Fr(x)26 b Fp(\000)g Fr(x)1970 2348 y 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Ft(Then)i(w)m(e)g(ha)m(v)m(e)149 2894 y(\(6.2\))957 b Fr(\026)1366 2853 y Fo(pp)1448 2894 y Ft(\()p Fp([)1552 2909 y Fq(j)t Fm(2)p Fq(Q)1688 2917 y Fl(n)1734 2894 y Ft(\(2)p Fr(I)1872 2844 y Fo(\()p Fq(n)p Fo(\))1864 2920 y Fq(j)1974 2894 y Ft(\)\))27 b Fp(\024)h Fr(\017)g(<)g(B)5 b(=)p Ft(10)149 3078 y(for)32 b(an)m(y)i Fr(n)28 b(>)f(n)730 3093 y Fo(2)770 3078 y Fr(:)32 b Ft(A)m(t)h(the)g(same)g(time,)e(b)m(y) j(Theorem)e(5.1,)h(w)m(e)g(ha)m(v)m(e)1458 3273 y Fr(\026)p Ft(\()p Fp([)1621 3288 y Fq(j)t Fm(2)p Fq(Q)1757 3296 y Fl(n)1803 3273 y Ft(\(2)p Fr(I)1941 3222 y Fo(\()p Fq(n)p Fo(\))1933 3298 y Fq(j)2042 3273 y Ft(\)\))28 b Fp(\025)g Fr(B)5 b(=)p Ft(2)149 3461 y(if)28 b Fr(n)g(>)g(n)483 3476 y Fo(2)522 3461 y Fr(:)h Ft(F)-8 b(or)28 b Fr(n)g Fp(\025)g Ft(max\()p Fr(n)1218 3476 y Fo(1)1258 3461 y Fr(;)17 b(n)1360 3476 y Fo(2)1399 3461 y Ft(\))p Fr(;)29 b Ft(this)g(giv)m(es)g(us)g(a)g(con)m(tradiction)f(with)g(\(6.1\),)i (\(6.2\))e(and)h(our)149 3577 y(assumption)j(on)h(the)g(absence)h(of)e (singular)f(con)m(tin)m(uous)j(comp)s(onen)m(t.)870 b Ff(\003)249 3775 y Ft(Theorem)41 b(1.1)g(follo)m(ws)f(from)g(Theorem)h (6.1)g(immediately)d(since)j(from)f(Theorem)h(5.1)g(w)m(e)149 3891 y(kno)m(w)34 b(that)f Fp(j)p Fr(V)21 b Ft(\()p Fr(x)p Ft(\))p Fp(j)27 b(\024)i Fr(h)p Ft(\()p Fr(x)p Ft(\))p Fr(=)p Ft(\(1)22 b(+)g Fr(x)p Ft(\))p Fr(:)659 4164 y Ft(7.)49 b Fs(Asymptotic)37 b(Incompleteness)g(of)h(W)-12 b(a)j(ve)38 b(Opera)-7 b(tors)249 4338 y Ft(In)33 b([4],)f(it)g(w)m(as) h(pro)m(v)m(ed)h(that)f(the)g(M\177)-49 b(oller)31 b(w)m(a)m(v)m(e)j (op)s(erators)e(exist)h(if)f Fr(V)49 b Fp(2)28 b Fr(L)3077 4302 y Fq(p)3117 4338 y Fr(;)33 b(p)27 b(<)h Ft(2)p Fr(;)k Ft(and)h(in)149 4454 y(addition)28 b(lim)666 4469 y Fq(x)p Fm(!1)868 4374 y Fj(R)934 4400 y Fq(x)915 4489 y Fo(0)995 4454 y Fr(V)21 b Ft(\()p Fr(s)p Ft(\))c Fr(ds)28 b Ft(exists.)43 b(Therefore,)31 b(to)e(pro)m(v)m(e)i(Theorem)e(1.2,)g(w)m(e)i(only)d (need)i(to)149 4572 y(sho)m(w)j(that)e(the)h(p)s(oten)m(tial)e Fr(V)22 b Ft(\()p Fr(x)p Ft(\))p Fr(;)31 b Ft(constructed)j(in)c(the)i (previous)g(sections,)h(can)e(b)s(e)h(c)m(hosen)h(to)149 4688 y(b)s(e)g(conditionally)d(in)m(tegrable.)149 4898 y Fi(Pr)-5 b(o)g(of)35 b(of)g(The)-5 b(or)g(em)33 b(1.2.)41 b Ft(Consider)h(one)f(of)g(the)h(comp)s(onen)m(ts)f(of)g Fr(V)2775 4862 y Fo(\()p Fq(n)p Fo(\))2876 4898 y Ft(\()p Fr(x)p Ft(\))p Fr(;)h Ft(sa)m(y)g Fr(V)3332 4847 y Fo(\()p Fq(n)p Fo(\))3310 4924 y Fq(j)3433 4898 y Ft(\()p Fr(x)p Ft(\))p Fr(;)g Ft(on)149 5020 y(\()p Fr(x)242 5035 y Fq(n)290 5020 y Fr(;)17 b Fp(1)p Ft(\))p Fr(:)32 b Ft(Simple)e(in)m (tegration)h(b)m(y)j(parts)f(similar)c(to)j(Lemma)f(2.4)h(sho)m(ws)j (that)d(for)g(an)m(y)h Fr(x;)825 5120 y Fj(\014)825 5179 y(\014)825 5239 y(\014)825 5299 y(\014)825 5359 y(\014)825 5419 y(\014)927 5168 y Fq(x)875 5188 y Fj(Z)861 5469 y Fq(x)901 5477 y Fl(n)991 5324 y Fr(V)1070 5273 y Fo(\()p Fq(n)p Fo(\))1048 5349 y Fq(j)1172 5324 y Ft(\()p Fr(y)t Ft(\))17 b Fr(dy)1420 5120 y Fj(\014)1420 5179 y(\014)1420 5239 y(\014)1420 5299 y(\014)1420 5359 y(\014)1420 5419 y(\014)1478 5324 y Fp(\024)28 b Fr(C)1677 5124 y Fj(0)1677 5303 y(@)1764 5324 y Fr(f)1823 5273 y Fo(\()p Fq(n)p Fo(\))1812 5349 y Fq(j)1946 5324 y Ft(+)2081 5168 y Fm(1)2044 5188 y Fj(Z)2031 5469 y Fq(x)2071 5477 y Fl(n)2182 5298 y Ft(~)2161 5324 y Fr(f)2220 5273 y Fo(\()p Fq(n)p Fo(\))2209 5349 y Fq(j)2321 5324 y Ft(\()p Fr(y)t Ft(\))p Fp(j)p Fr(V)2555 5283 y Fo(\()p Fq(n)p Fo(\))2656 5324 y Ft(\()p Fr(y)t Ft(\))p Fp(j)17 b Fr(dy)t(;)2976 5124 y Fj(1)2976 5303 y(A)p eop %%Page: 29 29 29 28 bop 1220 66 a Fn(SINGULAR)31 b(CONTINUOUS)g(SPECTR)n(UM)994 b(29)149 283 y Ft(where)34 b(similarly)29 b(to)j(what)h(w)m(e)h(had)f (b)s(efore)f(in)g(\(2.34\))o(,)2263 257 y(~)2242 283 y Fr(f)2301 232 y Fo(\()p Fq(n)p Fo(\))2290 309 y Fq(j)2435 283 y Ft(is)g(giv)m(en)h(b)m(y)511 703 y(~)490 729 y Fr(f)549 679 y Fo(\()p Fq(n)p Fo(\))538 755 y Fq(j)650 729 y Ft(\()p Fr(y)t Ft(\))27 b(=)909 436 y Fj(8)909 525 y(>)909 555 y(>)909 585 y(>)909 615 y(<)909 794 y(>)909 824 y(>)909 854 y(>)909 884 y(:)1039 532 y Fp(\000)p Fr(f)1175 481 y Fo(\()p Fq(n)p Fo(\))1164 557 y Fq(j)1277 532 y Fr(;)731 b(x)2090 547 y Fq(n)2165 532 y Fp(\024)28 b Fr(x)g Fp(\024)g Fr(x)2513 481 y Fo(\()p Fq(n)p Fo(\))2513 586 y Fq(j;\031)r Fm(\000)p Fq(\013)2705 551 y Fk(\()p Fl(n)p Fk(\))2705 608 y Fl(j)1039 718 y Fr(f)1098 667 y Fo(\()p Fq(n)p Fo(\))1087 744 y Fq(j)1199 718 y Fr(;)809 b(x)2090 667 y Fo(\()p Fq(n)p Fo(\))2090 773 y Fq(j;\031)r Fm(\000)p Fq(\013)2282 737 y Fk(\()p Fl(n)p Fk(\))2282 795 y Fl(j)2404 718 y Fp(\024)28 b Fr(x)g Fp(\024)g Fr(x)2752 733 y Fq(n)2822 718 y Ft(+)22 b(\()p Fr(f)3017 667 y Fo(\()p Fq(n)p Fo(\))3006 744 y Fq(j)3118 718 y Ft(\))3156 682 y Fm(\000)p Fo(1)3250 718 y Fr(g)3297 733 y Fq(n)1039 940 y Fp(\000)p Ft(min)o(\()p Fr(\016)t(k)1418 889 y Fo(\()p Fq(n)p Fo(\))1415 965 y Fq(j)1520 940 y Fr(;)1573 896 y Fq(g)1607 904 y Fl(n)p 1573 917 77 4 v 1574 974 a Fo(2)p Fq(x)1659 940 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y(\014)1856 1459 y(\014)1856 1518 y(\014)1856 1578 y(\014)1856 1638 y(\014)1914 1543 y Fp(\024)29 b Fr(C)7 b Ft(2)2146 1502 y Fq(n)2192 1543 y Fr(g)2239 1558 y Fq(n)2303 1449 y Fj(X)2358 1659 y Fq(j)2463 1543 y Fr(f)2522 1492 y Fo(\()p Fq(n)p Fo(\))2511 1569 y Fq(j)149 1885 y Ft(for)38 b(an)m(y)g Fr(x)f(>)g(x)753 1900 y Fq(n)800 1885 y Fr(:)h Ft(According)g(to)f(our)h(c)m(hoice)g(of) g(parameters)g(in)f(the)h(construction,)h(in)f(par-)149 2005 y(ticular)478 1979 y(~)457 2005 y Fr(f)505 2020 y Fq(n)552 2005 y Fr(;)30 b Ft(the)i(b)s(ound)f(on)f(the)i(righ)m(t)e (hand)h(side)f(do)s(es)i(not)e(exceed)j Fr(C)7 b Ft(2)2870 1969 y Fm(\000)p Fq(n)2972 2005 y Fr(:)31 b Ft(Summing)d(up)k(the)149 2122 y(con)m(tributions)g(from)g(\()p Fr(x)1067 2137 y Fq(n)1114 2122 y Fr(;)17 b(x)1213 2137 y Fq(n)p Fo(+1)1350 2122 y Ft(\))33 b(for)f(di\013eren)m(t)h Fr(n;)f Ft(w)m(e)i(get)f(the)g (result.)846 b Ff(\003)1618 2491 y Ft(8.)49 b Fs(Discussion)249 2665 y Ft(In)43 b(this)f(pap)s(er,)j(w)m(e)e(did)f(not)g(try)g(to)g (study)i(\014ne)f(prop)s(erties)f(of)g(the)g(im)m(b)s(edded)g(singular) 149 2781 y(con)m(tin)m(uous)32 b(measure,)g(suc)m(h)g(as)f(Hausdor\013) g(dimension.)41 b(The)32 b(reasonable)f(conjecture)h(is)e(that)149 2897 y(for)37 b Fp(j)p Fr(V)21 b Ft(\()p Fr(x)p Ft(\))p Fp(j)35 b(\024)g Fr(C)7 b Ft(\(1)25 b(+)f Fr(x)p Ft(\))1097 2861 y Fm(\000)p Fq(\013)1202 2897 y Fr(;)37 b Ft(there)g(could)g(b)s (e)g(im)m(b)s(edded)f(singular)f(con)m(tin)m(uous)j(measures)f(of)149 3014 y(dimension)g(up)h(to)f(1)25 b Fp(\000)i Ft(2)p Fr(\013)38 b Ft(\(see)g([19,)g(20)o(]\).)59 b(One)38 b(migh)m(t)e(try)i(to)g(use)g(ideas)g(of)f(this)g(pap)s(er)h(to)149 3130 y(construct)k(suc)m(h)f(measures.)67 b(This,)42 b(ho)m(w)m(ev)m(er,)i(w)m(ould)c(ha)m(v)m(e)h(to)f(lead)g(to)f(a)h (di\013eren)m(t)g(plan)f(of)149 3246 y(attac)m(k)k(and)f(more)g(tec)m (hnical)g(e\013ort.)72 b(In)42 b(particular,)h(one)g(should)f(use)h (appro)m(ximations)d(of)149 3362 y(the)32 b(limiting)27 b(measure)32 b(b)m(y)g(absolutely)f(con)m(tin)m(uous,)i(rather)e(than)h (b)m(y)g(pure)g(p)s(oin)m(t,)f(measures,)149 3479 y(cutting)i Fr(V)565 3442 y Fo(\()p Fq(n)p Fo(\))666 3479 y Ft(\()p Fr(x)p Ft(\))h(o\013)f(at)f(an)i(appropriate)e(p)s(oin)m(t.)44 b(In)34 b(addition,)d(one)j(w)m(ould)f(ha)m(v)m(e)h(to)f(b)s(e)h(more) 149 3595 y(careful)j(with)f(the)i(separation)e(of)g(scales)i(on)f (di\013eren)m(t)g(steps)h(of)f(construction,)h(whic)m(h)f(is)g(ex-)149 3711 y(tensiv)m(ely)28 b(used)h(in)d(our)i(argumen)m(t)f(to)g(con)m (trol)f(the)i(errors.)42 b(Suc)m(h)28 b(an)g(approac)m(h)f(w)m(ould)h (require)149 3827 y(an)38 b(impro)m(v)m(emen)m(t)f(in)g(con)m(trol)g (of)g(man)m(y)h(estimates,)g(in)f(particular)f(in)h(con)m(tin)m(uit)m (y)g(estimates)149 3944 y(of)h(Section)f(3.)59 b(T)-8 b(o)37 b(k)m(eep)j(the)e(size)g(of)f(the)h(pap)s(er)g(in)e(c)m(hec)m (k,)42 b(w)m(e)c(did)g(not)f(try)h(to)f(pursue)i(this)149 4060 y(direction)32 b(here.)249 4176 y(Another)39 b(in)m(teresting)f (op)s(en)g(question)h(is)e(what)i(happ)s(ens)g(sharply)f(at)g(the)h (Coulom)m(b)e(rate)149 4292 y(of)h(deca)m(y)h(1)p Fr(=x:)f Ft(In)h([19)o(],)h(it)d(is)g(sho)m(wn)i(that)f(im)m(b)s(edded)g (singular)e(sp)s(ectrum)j(cannot)f(o)s(ccur)g(if)149 4408 y Fr(V)22 b Ft(\()p Fr(x)p Ft(\))41 b(=)g Fr(o)p Ft(\(1)p Fr(=x)p Ft(\))f(or)g(ab)s(o)m(v)m(e)h(a)f(certain)g(threshold) g(if)f Fp(j)p Fr(V)22 b Ft(\()p Fr(x)p Ft(\))p Fp(j)41 b(\024)g Fr(C)t(=)p Ft(\()p Fr(x)28 b Ft(+)f(1\))p Fr(:)40 b Ft(In)h([10)o(],)i(it)c(w)m(as)149 4525 y(sho)m(wn)27 b(that)d(an)m(y)h(im)m(b)s(edded)g(eigen)m(v)-5 b(alues)25 b Fr(\025)1808 4540 y Fq(n)1879 4525 y Ft(that)g(can)g(o)s(ccur)g(for)f Fp(j)p Fr(V)d Ft(\()p Fr(x)p Ft(\))p Fp(j)28 b(\024)g Fr(C)t(=)p Ft(\()p Fr(x)6 b Ft(+)g(1\))25 b(satisfy)149 4566 y Fj(P)255 4670 y Fq(n)318 4641 y Fr(\025)375 4656 y Fq(n)450 4641 y Fr(<)i Fp(1)p Fr(:)32 b Ft(This)f(sho)m(ws)i(that)f (a)f(construction)h(carried)f(out)g(in)g(this)g(pap)s(er)h(is)f(not)h (p)s(ossible)149 4757 y(for)e Fr(V)52 b Ft(deca)m(ying)30 b(at)g(a)g(Coulom)m(b)f(rate.)43 b(W)-8 b(e)31 b(conjecture)g(that)f (singular)f(con)m(tin)m(uous)i(sp)s(ectrum)149 4873 y(cannot)i(o)s (ccur)g(for)f(suc)m(h)i Fr(V)5 b(:)249 5106 y FA(Ac)m(kno)m(wledgmen)m (t.)109 b Ft(The)36 b(author)f(is)g(grateful)f(to)h(Mic)m(hael)f (Christ)i(and)f(Barry)g(Simon)149 5222 y(for)24 b(useful)g(and)h(stim)m (ulating)c(discussions.)42 b(The)25 b(author)f(also)f(wishes)i(to)f (express)j(his)d(gratitude)149 5338 y(to)36 b(F)-8 b(rancois)35 b(Germinet)f(and)i(Stephan)h(de)f(Bi)m(\022)-46 b(evre)36 b(at)f(Univ)m(ersit)m(\023)-46 b(e)36 b(de)h(Lille)c(I,)j(where)h(part) e(of)149 5455 y(this)e(w)m(ork)g(w)m(as)h(carried)e(out.)p eop %%Page: 30 30 30 29 bop 149 66 a Fn(30)1286 b(ALEXANDER)23 b(KISELEV)1652 266 y Fs(References)191 423 y Fz([1])41 b(R.)47 b(Carmona)d(and)j(J.)f (Lacroix,)j Fc(Sp)l(e)l(ctr)l(al)e(The)l(ory)h(of)g(R)l(andom)f (Schr\177)-42 b(odinger)49 b(Op)l(er)l(ators,)j Fz(Boston:)320 523 y(Birkhuser,)26 b(c1990.)191 623 y([2])41 b(M.)d(Christ)g(and)g(A.) g(Kiselev,)i Fc(A)n(bsolutely)f(c)l(ontinuous)g(sp)l(e)l(ctrum)f(for)i (one-dimensional)h(Schr\177)-42 b(odinger)320 722 y(op)l(er)l(ators)25 b(with)g(slow)t(ly)h(de)l(c)l(aying)g(p)l(otentials:)36 b(Some)25 b(optimal)g(r)l(esults,)g Fz(J.)d(Amer.)g(Math.)f(So)r(c.)h Fb(11)f Fz(\(1998\),)320 822 y(771{797)191 922 y([3])p 320 922 250 4 v 304 w(,)i Fc(WKB)i(and)f(sp)l(e)l(ctr)l(al)h(analysis)h (of)f(one-dimensional)h(Schr\177)-42 b(odinger)26 b(op)l(er)l(ators)g (with)e(slow)t(ly)i(vary-)320 1021 y(ing)k(p)l(otentials)p Fz(,)f(Comm.)e(Math.)h(Ph)n(ys.)f(218)f(\(2001\),)g(245-262.)191 1121 y([4])p 320 1121 V 304 w(,)j Fc(Sc)l(attering)i(and)h(wave)g(op)l (er)l(ators)g(for)f(one-dimensional)i(Schr\177)-42 b(odinger)33 b(op)l(er)l(ators)f(with)f(slow)t(ly)320 1220 y(de)l(c)l(aying)g (nonsmo)l(oth)f(p)l(otentials,)h Fz(preprin)n(t.)191 1320 y([5])41 b(E.A.)32 b(Co)r(ddington)f(and)g(N.)h(Levinson,)g 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y([9])k(R.)28 b(Killip)g(and)f(B.)h(Simon,)f Fc(in)j(pr)l(ep)l(ar) l(ation)149 2217 y Fz([10])41 b(A.)c(Kiselev,)g(Y.)g(Last,)h(and)e(B.)g (Simon,)j Fc(Mo)l(di\014e)l(d)g(Pr)q(\177)-43 b(ufer)38 b(and)g(EF)n(GP)h(tr)l(ansforms)f(and)g(the)g(sp)l(e)l(ctr)l(al)320 2316 y(analysis)31 b(of)g(one-dimensional)g(Schr\177)-42 b(odinger)32 b(op)l(er)l(ators,)f Fz(Comm)n(un.)c(Math.)h(Ph)n(ys.)f Fb(194)g Fz(\(1998\),)f(1{45)149 2416 y([11])41 b(T.)48 b(Kriec)n(herbauer)d(and)i(C.)g(Remling,)53 b Fc(Finite)48 b(gap)i(p)l(otentials)e(and)h(WKB)f(asymptotics)h(for)g(one-)320 2516 y(dimensional)32 b(Schr\177)-42 b(odinger)31 b(op)l(er)l(ators,)g Fz(to)d(app)r(ear)e(at)i(Comm)n(un.)f(Math.)h(Ph)n(ys.)149 2615 y([12])41 b(Y.)24 b(Last,)f Fc(Quantum)h(dynamics)j(and)f(de)l(c)l (omp)l(ositions)h(of)g(singular)f(c)l(ontinuous)e(sp)l(e)l(ctr)l(a,)j Fz(J.)c(F)-7 b(unct.)24 b(Anal.)320 2715 y Fb(142)j Fz(\(1996\),)f (406{445)149 2814 y([13])41 b(B.)28 b(Levitan,)f Fc(Inverse)j 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5398 y Fc(E-mail)g(addr)l(ess)7 b Fz(:)38 b Fa(kiselev@math.uch)o(ic)o(ag)o(o.e)o(du)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0111182052493--