Content-Type: multipart/mixed; boundary="-------------0306271102194" This is a multi-part message in MIME format. ---------------0306271102194 Content-Type: text/plain; name="03-309.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-309.keywords" Periodic operator, singular perturbation, scattering theory, absolutely continuous spectrum ---------------0306271102194 Content-Type: application/postscript; name="diplomaweb.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="diplomaweb.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: diplomaweb.dvi %%CreationDate: Fri Jun 27 01:17:40 2003 %%Pages: 65 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: D:\PROGRA~1\MIKTEX-2.1\MIKTEX\BIN\DVIPS.EXE %+ diplomaweb.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.06.27:0116 %%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/IEn 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 IEn 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/IEn 256 array N 2 string 0 1 255{IEn 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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Fr(T)p Fw(\).)47 b(In)31 b(this)g(w)m(a)m(y)h(w)m(e)g(reduce)g(the)f(problem)f (of)h(\(p)s(ossibly)120 648 y(em)m(b)s(edded\))41 b(eigen)m(v)-5 b(alues)40 b(to)g(the)g(study)i(of)d(op)s(erators)h(with)g(compact)g (resolv)m(en)m(t.)67 b(The)120 768 y(Analytic)40 b(F)-8 b(redholm)40 b(Alternativ)m(e)g(and)i(a)f(con)m(tin)m(uit)m(y)g (principle)f(yield)g(information)e(on)120 888 y(the)28 b(lo)s(cation)e(of)i(eigen)m(v)-5 b(alues.)41 b(Under)29 b(the)g(condition)d(\(0.0.3\))h(w)m(e)i(can)g(pro)m(v)m(e)g(the)f (absence)120 1009 y(of)c(p)s(ositiv)m(e)h(eigen)m(v)-5 b(alues.)41 b(This)25 b(completes)f(the)h(study)i(of)d(the)h(sp)s (ectrum)g(of)g(the)g(op)s(erators)120 1129 y Fv(H)209 1093 y Fu(\()p Ft(\033)r Fu(\))310 1129 y Fw(\()p Fv(k)s Fw(\))33 b(in)f(the)h(case)g Fv(\033)f Fs(\025)c Fw(0.)120 1249 y(In)34 b(Section)g(3.4)f(w)m(e)i(study)f(the)h(scattering)e (matrix)f(in)h(more)g(detail)f(and)i(deriv)m(e)g(a)g(repre-)120 1370 y(sen)m(tation)f(in)e(terms)i(of)f(generalized)g(eigenfunctions.) 120 1490 y(Finally)-8 b(,)41 b(in)g(Section)g(4.2)g(w)m(e)i(deduce)g (our)e(main)f(result)h(\(Theorem)h(4.2.1\))f(ab)s(out)g Fv(H)3439 1454 y Fu(\()p Ft(\033)r Fu(\))120 1611 y Fw(under)g(the)f (condition)e(\(0.0.3\))h(from)g(the)h(results)g(ab)s(out)g(the)g(\014b) s(er)g(op)s(erators)g Fv(H)3282 1574 y Fu(\()p Ft(\033)r Fu(\))3383 1611 y Fw(\()p Fv(k)s Fw(\).)120 1731 y(W)-8 b(e)42 b(supplemen)m(t)h(this)f(theorem)f(with)h(information)d(on)j (the)g(case)h(of)f(general)f Fv(\033)t Fw(,)k(when)120 1851 y Fv(H)209 1815 y Fu(\()p Ft(\033)r Fu(\))343 1851 y Fw(ma)m(y)32 b(ha)m(v)m(e)i(negativ)m(e)f(sp)s(ectrum)g(with)f(band)h (structure.)120 2111 y Fp(5.)98 b Fw(In)33 b(the)g(case)h(when)g (condition)d(\(0.0.3\))h(is)h(violated)e(there)j(remain)d(some)i(op)s (en)g(ques-)120 2232 y(tions.)48 b(The)35 b(eigen)m(v)-5 b(alues)34 b(of)g(the)h(\014b)s(er)g(op)s(erators)f Fv(H)2142 2195 y Fu(\()p Ft(\033)r Fu(\))2243 2232 y Fw(\()p Fv(k)s Fw(\))g(ma)m(y)g(generate)h("new")g(zones)120 2352 y(in)43 b(the)h(sp)s(ectrum)h(of)e(the)h(op)s(erator)g Fv(H)1657 2316 y Fu(\()p Ft(\033)r Fu(\))1758 2352 y Fw(,)j(whic)m(h)d(pro)s (duce)h(w)m(a)m(v)m(eguide)g(e\013ects.)79 b(This)120 2472 y(phenomenon)33 b(is)f(w)m(orth)h(a)g(more)e(detailed)h(study)-8 b(.)120 2805 y Fq(Ac)l(kno)l(wledgemen)l(ts)120 3024 y Fw(The)42 b(results)g(of)e(the)i(presen)m(t)h(text)f(w)m(ere)g (obtained)f(under)h(the)f(sup)s(ervision)g(of)g(M.)h(Sh.)120 3145 y(Birman)28 b(\(St.)43 b(P)m(etersburg\))31 b(at)e(the)i (Mittag-Le\017er)e(Institute)h(\(Sto)s(c)m(kholm\))e(in)h(fall)f(2002.) 120 3265 y(The)42 b(author)f(is)f(deeply)i(grateful)e(to)g(Prof.)69 b(Birman)39 b(for)i(the)g(setting)g(of)g(the)g(problem,)120 3385 y(useful)33 b(discussions)g(and)g(constan)m(t)g(atten)m(tion)f(to) g(the)h(w)m(ork.)120 3506 y(It)41 b(is)f(also)g(a)g(pleasure)h(to)f (thank)h(Prof.)68 b(T.)41 b(A.)g(Suslina)e(\(St.)68 b(P)m(etersburg\)) 42 b(and)f(R.)f(G.)120 3626 y(Sh)m(teren)m(b)s(erg)26 b(\(St.)40 b(P)m(etersburg\))26 b(for)e(b)s(oth)g(linguistic)d(help)j (and)g(n)m(umerous)h(mathematical)120 3746 y(discussions.)120 3867 y(The)35 b(sta)m(y)h(at)e(Sto)s(c)m(kholm)f(w)m(ould)i(not)f(ha)m (v)m(e)i(b)s(een)f(p)s(ossible)f(without)g(the)h(generous)g(help)120 3987 y(of)j(Prof.)60 b(H.)38 b(Sieden)m(top)h(\(LMU)g(Munic)m(h\))g (and)f(Prof.)60 b(A.)38 b(Laptev)h(\(KTH)g(Sto)s(c)m(kholm\).)120 4108 y(The)f(author)g(appreciates)f(the)h(hospitalit)m(y)e(of)h(the)h (KTH)g(Sto)s(c)m(kholm)e(and)i(the)g(Mittag-)120 4228 y(Le\017er)f(Institute.)53 b(The)37 b(sta)m(y)f(in)g(Sw)m(eden)h(w)m (as)g(supp)s(orted)g(b)m(y)f(the)h(Studienstiftung)e(des)120 4348 y(deutsc)m(hen)g(V)-8 b(olk)m(es.)120 4469 y(Last)31 b(but)g(not)g(least)f(m)m(y)h(gratitudes)g(go)f(to)h(Marie)f(W)-8 b(einzierl)29 b(for)i(her)g(constan)m(t)h(patience)120 4589 y(and)h(encouragemen)m(t.)266 4809 y(The)c(presen)m(t)h(text)f(is) f(submitted)f(as)i(the)f(author's)h(master)f(thesis)g(\(Diplomarb)s (eit\))c(at)120 4930 y(the)33 b(LMU)g(Munic)m(h.)1789 5712 y(iii)p eop %%Page: 4 5 4 4 bop 120 1071 a Fk(Con)-6 b(ten)g(ts)120 1621 y Fp(1)90 b(Preliminary)35 b(material)2191 b(1)266 1741 y Fw(1.1)100 b(Notation)45 b(.)50 b(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g (.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.) g(.)166 b(1)266 1862 y(1.2)100 b(Sob)s(olev)32 b(spaces)103 b(.)50 b(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.) g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)166 b(2)266 1982 y(1.3)100 b(The)33 b(Gelfand)f(transformation)68 b(.)50 b(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) f(.)h(.)g(.)g(.)166 b(7)266 2103 y(1.4)100 b(Scattering)32 b(theory)69 b(.)49 b(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.) g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)117 b(10)120 2320 y Fp(2)90 b(De\014nition)36 b(of)i(the)f(op)s(erators.)51 b(Direct)36 b(in)m(tegral)f(decomp)s(osition)315 b(12)266 2441 y Fw(2.1)100 b(De\014niton)31 b(of)i(the)g(op)s(erators)f Fv(H)1717 2405 y Fu(\()p Ft(\033)r Fu(\))1851 2441 y Fw(and)g Fv(H)2129 2405 y Fu(\()p Ft(\033)r Fu(\))2231 2441 y Fw(\()p Fv(k)s Fw(\))91 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h (.)g(.)g(.)117 b(12)266 2561 y(2.2)100 b(The)33 b(unp)s(erturb)s(ed)h (op)s(erators)e Fv(H)1775 2525 y Fu(\(0\))1902 2561 y Fw(and)g Fv(H)2180 2525 y Fu(\(0\))2274 2561 y Fw(\()p Fv(k)s Fw(\))48 b(.)i(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)117 b(16)120 2779 y Fp(3)90 b(Sp)s(ectral)41 b(analysis)g(and)h(scattering) e(theory)i(of)f(the)g(op)s(erators)h Fv(H)3158 2743 y Fu(\()p Ft(\033)r Fu(\))3259 2779 y Fw(\()p Fv(k)s Fw(\))266 2900 y Fp(on)c(the)f(halfstrip)2402 b(20)266 3020 y Fw(3.1)100 b(The)33 b(absolutely)f(con)m(tin)m(uous)h(sp)s(ectrum)g(of)f(the)h(op) s(erators)g Fv(H)2869 2984 y Fu(\()p Ft(\033)r Fu(\))2970 3020 y Fw(\()p Fv(k)s Fw(\))44 b(.)50 b(.)g(.)117 b(20)266 3140 y(3.2)100 b(The)24 b(absence)g(of)f(singular)e(con)m(tin)m(uous)j (sp)s(ectrum)g(of)e(the)i(op)s(erators)f Fv(H)3158 3104 y Fu(\()p Ft(\033)r Fu(\))3259 3140 y Fw(\()p Fv(k)s Fw(\))53 b(25)266 3261 y(3.3)100 b(The)33 b(p)s(oin)m(t)f(sp)s(ectrum)h (of)f(the)h(op)s(erators)f Fv(H)2171 3225 y Fu(\()p Ft(\033)r Fu(\))2273 3261 y Fw(\()p Fv(k)s Fw(\))49 b(.)h(.)g(.)g(.)g(.)g(.)g(.)g (.)f(.)h(.)g(.)g(.)117 b(30)266 3381 y(3.4)100 b(Scattering)32 b(theory)h(for)f(the)h(op)s(erators)f Fv(H)2100 3345 y Fu(\()p Ft(\033)r Fu(\))2202 3381 y Fw(\()p Fv(k)s Fw(\))43 b(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)117 b(43)120 3599 y Fp(4)90 b(Sp)s(ectral)38 b(analysis)g(and)g(scattering) f(theory)h(of)g(the)g(op)s(erators)g Fv(H)3131 3563 y Fu(\()p Ft(\033)r Fu(\))3270 3599 y Fp(on)266 3719 y(the)g(halfplane) 2524 b(54)266 3840 y Fw(4.1)100 b(Scattering)32 b(theory)h(for)f(the)h (op)s(erators)f Fv(H)2100 3804 y Fu(\()p Ft(\033)r Fu(\))2298 3840 y Fw(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)117 b(54)266 3960 y(4.2)100 b(Sp)s(ectral)32 b(analysis)g(of)g(the)h(op)s (erators)f Fv(H)2039 3924 y Fu(\()p Ft(\033)r Fu(\))2221 3960 y Fw(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.) 117 b(56)1791 5712 y(iv)p eop %%Page: 1 6 1 5 bop 120 1071 a Fk(Chapter)78 b(1)120 1486 y(Preliminary)d(material) 120 2001 y Fq(1.1)161 b(Notation)120 2220 y Fw(In)39 b(this)g(section)g(w)m(e)h(in)m(tro)s(duce)e(the)i(notation)d(w)m(e)j (use)g(for)e(some)h(geometric)f(ob)5 b(jects)40 b(in)120 2340 y Fr(R)186 2304 y Fu(2)186 2365 y(+)284 2340 y Fw(and)32 b(for)g(spaces)i(of)e(smo)s(oth)g(functions.)44 b(General)31 b(con)m(v)m(en)m(tions)k(are)d(made.)120 2600 y Fp(1.)47 b(Sets.)98 b Fw(W)-8 b(e)24 b(shall)f(w)m(ork)i(in)e(t)m(w)m (o-dimensional)f(Euclidean)h(space)i Fr(R)2771 2564 y Fu(2)2841 2600 y Fw(with)e(co)s(ordinates)120 2720 y(denoted)34 b(b)m(y)f Fv(x)28 b Fw(=)g(\()p Fv(x)904 2735 y Fu(1)943 2720 y Fv(;)17 b(x)1042 2735 y Fu(2)1082 2720 y Fw(\).)43 b(In)33 b(particular,)e(w)m(e)j(consider)f(the)g Fj(halfplane)1047 2940 y Fr(R)1113 2899 y Fu(2)1113 2965 y(+)1206 2940 y Fw(:=)28 b Fs(f)p Fv(x)g Fs(2)g Fr(R)1630 2899 y Fu(2)1630 2965 y(+)1722 2940 y Fw(:)56 b Fv(x)1860 2955 y Fu(2)1927 2940 y Fv(>)28 b Fw(0)p Fs(g)f Fw(=)h Fr(R)33 b Fs(\002)22 b Fr(R)2520 2955 y Fu(+)2585 2940 y Fv(;)120 3160 y Fw(and)33 b(the)g Fj(halfstrip)643 3380 y Fw(\005)28 b(:=)f Fs(f)p Fv(x)h Fs(2)g Fr(R)1167 3339 y Fu(2)1167 3404 y(+)1260 3380 y Fw(:)55 b Fs(\000)p Fv(\031)33 b(<)27 b(x)1665 3395 y Fu(1)1733 3380 y Fv(<)g(\031)t(;)44 b(x)2021 3395 y Fu(2)2089 3380 y Fv(>)28 b Fw(0)p Fs(g)f Fw(=)g(\()p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(\))22 b Fs(\002)h Fr(R)2925 3395 y Fu(+)2990 3380 y Fv(;)120 3599 y Fw(where)34 b Fr(R)468 3614 y Fu(+)562 3599 y Fw(:=)28 b(\(0)p Fv(;)17 b Fw(+)p Fs(1)p Fw(\).)44 b(W)-8 b(e)33 b(\014nd)g(it)f(con)m(v)m (enien)m(t)j(to)e(denote)h(the)f(co)s(ordinates)g(on)f Fr(R)44 b Fw(\(or)120 3720 y(\()p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(\)\))30 b(and)f Fr(R)755 3735 y Fu(+)850 3720 y Fw(b)m(y)i Fv(x)1038 3735 y Fu(1)1108 3720 y Fw(and)e Fv(x)1349 3735 y Fu(2)1389 3720 y Fw(,)i(resp)s(ectiv)m(ely)-8 b(,)31 b(ev)m(en)h(if)c(w)m(orking)i(in)f(a)h(one-dimensional)120 3840 y(con)m(text.)120 3961 y(Moreo)m(v)m(er,)42 b(w)m(e)d(need)h(the)e (lattice)f(2)p Fv(\031)t Fr(Z)p Fw(.)59 b(Unless)39 b(stated)g (otherwise,)h(p)s(erio)s(dicit)m(y)d(condi-)120 4081 y(tions)e(are)h(understo)s(o)s(d)h(with)e(rep)s(ect)i(to)e(this)h (lattice.)52 b(W)-8 b(e)36 b(think)g(of)g(the)g(corresp)s(onding)120 4201 y(torus)d Fr(T)e Fw(:=)d Fr(R)5 b Fv(=)p Fw(2)p Fv(\031)t Fr(Z)35 b Fw(as)e(the)g(in)m(terv)-5 b(al)31 b([)p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(])33 b(with)f(endp)s(oin)m (ts)h(iden)m(ti\014ed.)120 4461 y Fp(2.)67 b(Spaces)44 b(of)f(di\013eren)m(tiable)e(functions.)98 b Fw(As)38 b(usual,)g(partial)d(di\013eren)m(tiation)h(\(in)120 4581 y(the)41 b(classical)e(and)h(in)g(the)h(distributional)c(sense\)) 43 b(with)d(resp)s(ect)h(to)g(the)g(co)s(ordinate)e Fv(x)3503 4596 y Ft(j)120 4702 y Fw(is)c(denoted)h(b)m(y)777 4663 y Ft(@)p 741 4679 114 4 v 741 4736 a(@)t(x)822 4746 y Fi(j)864 4702 y Fw(,)g(and)g Fv(D)1201 4717 y Ft(j)1270 4702 y Fw(:=)1416 4663 y Fu(1)p 1416 4679 36 4 v 1422 4736 a Ft(i)1507 4663 y(@)p 1471 4679 114 4 v 1471 4736 a(@)t(x)1552 4746 y Fi(j)1594 4702 y Fw(.)52 b(If)36 b Fv(x)1829 4717 y Ft(j)1901 4702 y Fw(is)f(the)h(co)s(ordinate)f(in)f (a)i(one-dimensional)120 4850 y(set,)43 b(w)m(e)e(write)762 4810 y Ft(d)p 726 4827 109 4 v 726 4884 a(dx)802 4894 y Fi(j)884 4850 y Fw(instead)f(of)1393 4810 y Ft(@)p 1357 4827 114 4 v 1357 4884 a(@)t(x)1438 4894 y Fi(j)1480 4850 y Fw(.)66 b(F)-8 b(or)39 b(higher)g(order)h(deriv)-5 b(ativ)m(es)40 b(w)m(e)h(use)g(the)f(usual)120 4981 y(m)m(ulti-index)30 b(notation.)120 5102 y(Let)40 b Fv(d)h Fs(2)g Fr(N)55 b Fw(\(w)m(e)41 b(are)g(particularly)d(in)m(terested)j(in)f(the)g (cases)i Fv(d)e Fw(=)h(1)p Fv(;)33 b Fw(2\).)66 b(F)-8 b(or)40 b(an)g(op)s(en)120 5222 y(set)i(\012)i Fs(\032)g Fr(R)582 5186 y Ft(d)670 5222 y Fw(w)m(e)f(denote)f(b)m(y)h Fv(C)1368 5186 y Fl(1)1443 5222 y Fw(\(\012\))f(the)g(space)h(of)e (complex-v)-5 b(alued)40 b(functions)i(on)g(\012)120 5342 y(whic)m(h)30 b(ha)m(v)m(e)h(con)m(tin)m(uous)g(deriv)-5 b(ativ)m(es)29 b(in)g(\012)h(up)g(to)g(an)m(y)g(order.)43 b Fv(C)2602 5306 y Fl(1)2595 5367 y Fu(0)2677 5342 y Fw(\(\012\))30 b(consists)g(of)f(those)120 5463 y(functions)k(in)e Fv(C)731 5427 y Fl(1)806 5463 y Fw(\(\012\))i(whose)h(supp)s(ort)f(is)f (a)g(compact)g(subset)i(of)e(\012.)1806 5712 y(1)p eop %%Page: 2 7 2 6 bop 120 407 a Fw(W)-8 b(e)42 b(need)i(some)d(more)h(notation)f (when)i(p)s(erio)s(dicit)m(y)d(is)i(in)m(v)m(olv)m(ed.)72 b(In)42 b(the)h(case)g(of)f(the)120 527 y(torus)g(w)m(e)h(denote)f(b)m (y)h Fv(C)1077 491 y Fl(1)1152 527 y Fw(\()p Fr(T)p Fw(\))h(the)f (space)f(of)g(functions)f(in)g Fv(C)2532 491 y Fl(1)2607 527 y Fw(\()p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(\))42 b(whic)m(h)g(can)g(b)s(e)120 648 y(extended)d(\(2)p Fv(\031)t Fw(-\)p)s(erio)s(dically)33 b(to)k(a)g(function)f(in)h Fv(C)2080 611 y Fl(1)2154 648 y Fw(\()p Fr(R)5 b Fw(\).)63 b(In)38 b(the)f(case)h(of)f(the)g(halfstrip)120 768 y(w)m(e)42 b(denote)g(b)m(y)762 743 y(~)740 768 y Fv(C)817 732 y Fl(1)891 768 y Fw(\(\005\))f(the)h(space)h(of)d(functions)i(in)e Fv(C)2276 732 y Fl(1)2351 768 y Fw(\(\005\))h(whic)m(h)h(can)f(b)s(e)h (extended)120 888 y(\(2)p Fv(\031)t Fw(-\)p)s(erio)s(dically)28 b(with)33 b(resp)s(ect)h(to)e Fv(x)1595 903 y Fu(1)1667 888 y Fw(to)g(a)h(function)f(in)g Fv(C)2441 852 y Fl(1)2515 888 y Fw(\()p Fr(R)2619 852 y Fu(2)2619 913 y(+)2684 888 y Fw(\).)120 1009 y(W)-8 b(e)34 b(also)f(need)h(the)g(notion)f(of)g (the)h(Sc)m(h)m(w)m(artz)i(class)e(on)f(an)h(arbitrary)e(op)s(en)i(set) h(\012)30 b Fs(\032)f Fr(R)3466 973 y Ft(d)3513 1009 y Fw(.)120 1129 y(Namely)-8 b(,)32 b(w)m(e)h(sa)m(y)h(that)e Fv(u)27 b Fs(2)h(S)7 b Fw(\(\012\))34 b(i\013)e Fv(u)27 b Fs(2)h Fv(C)1815 1093 y Fl(1)1890 1129 y Fw(\(\012\))k(and)770 1320 y(sup)774 1401 y Ft(x)p Fl(2)p Fu(\012)917 1320 y Fw(\(1)21 b(+)h Fs(j)p Fv(x)p Fs(j)1234 1279 y Ft(N)1301 1320 y Fw(\))p Fs(j)p Fv(D)1451 1279 y Ft(\013)1500 1320 y Fv(u)p Fw(\()p Fv(x)p Fw(\))p Fs(j)27 b Fv(<)h Fs(1)p Fv(;)211 b(N)39 b Fs(2)28 b Fr(N)2461 1335 y Fu(0)2506 1320 y Fv(;)33 b(\013)c Fs(2)f Fr(N)2817 1279 y Ft(d)2817 1344 y Fu(0)2863 1320 y Fv(:)120 1628 y Fp(3.)71 b(Con)m(v)m(en)m (tions.)97 b Fw(All)37 b(v)m(ector)j(spaces)g(are)f(considered)g(o)m(v) m(er)h(the)f(\014eld)f Fr(C)20 b Fw(.)68 b(All)37 b(sub-)120 1748 y(spaces)d(of)e(Hilb)s(ert)f(spaces)j(are)f(assumed)g(to)f(b)s(e)h (closed.)120 1869 y(Unless)f(stated)g(otherwise,)h(w)m(e)f(c)m(ho)s (ose)h(the)f(branc)m(h)g(of)f(the)i(square)f(ro)s(ot)f(on)g Fr(C)46 b Fs(n)20 b Fw(\()p Fs(\0001)p Fv(;)d Fw(0])120 1989 y(that)32 b(satis\014es)i(Re)822 1934 y Fs(p)p 905 1934 28 4 v 55 x Fv(:)28 b(>)g Fw(0.)120 2109 y(Statemen)m(ts)k(and)f (form)m(ulae)f(whic)m(h)i(con)m(tain)f(the)h(double)f(index)g(")p Fs(\006)p Fw(")h(are)f(understo)s(o)s(d)h(as)120 2230 y(t)m(w)m(o)h(indep)s(enden)m(t)h(assertions.)120 2558 y Fq(1.2)161 b(Sob)t(olev)53 b(spaces)120 2777 y Fw(In)38 b(this)f(section)h(w)m(e)h(presen)m(t)g(some)f(material)c(on)k(Sob)s (olev)f(spaces.)60 b(W)-8 b(e)38 b(do)g(not)g(aim)d(at)120 2898 y(completeness,)e(but)g(rather)g(try)g(to)f(standardize)h (notation.)120 3018 y(W)-8 b(e)22 b(are)g(particularly)e(in)m(terested) j(in)e(Sob)s(olev)h(spaces)h(with)f(p)s(erio)s(dicit)m(y)e(conditions.) 39 b(After)120 3138 y(ha)m(ving)31 b(recalled)f(in)g(Subsection)i(1)f (the)g(F)-8 b(ourier)30 b(expansion)h(in)g Fv(L)2600 3153 y Fu(2)2639 3138 y Fw(-spaces,)i(w)m(e)f(in)m(tro)s(duce)120 3259 y(p)s(erio)s(dic)25 b(Sob)s(olev)h(spaces)i(in)e(Subsection)h(2.) 41 b(Finally)-8 b(,)25 b(w)m(e)j(presen)m(t)g(some)f(auxiliary)d (state-)120 3379 y(men)m(ts)38 b(related)f(to)h(Sob)s(olev)f(em)m(b)s 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4725 y Ft(L)1168 4734 y Fm(2)1201 4725 y Fu(\(\012\))1339 4701 y Fw(:=)1470 4565 y Fo(Z)1525 4791 y Fu(\012)1597 4701 y Fs(j)p Fv(u)p Fw(\()p Fv(x)p Fw(\))p Fs(j)1840 4660 y Fu(2)1895 4701 y Fv(dx;)212 b(u)27 b Fs(2)h Fv(L)2483 4716 y Fu(2)2523 4701 y Fw(\(\012\))p Fv(:)120 4944 y Fw(W)-8 b(e)33 b(usually)f(omit)e(the)j(index)g Fv(L)1346 4959 y Fu(2)1386 4944 y Fw(\(\012\).)120 5064 y(The)39 b(space)f Fv(L)657 5079 y Fu(2)697 5064 y Fw(\()p Fr(T)p Fw(\))j(ma)m(y)c(b)s(e)h(formally)d(iden)m(ti\014ed)i(with)g Fv(L)2344 5079 y Fu(2)2384 5064 y Fw(\()p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(\).)59 b(It)37 b(has)h(a)g(complete)120 5185 y(orthonormal)30 b(system)k Fs(f)p Fv(e)1097 5200 y Ft(n)1171 5185 y Fw(:)56 b Fv(n)28 b Fs(2)g Fr(Z)p Fs(g)i Fw(consisting)h(of)i (the)g(trigonometric)d(functions)1144 5420 y Fv(e)1189 5435 y Ft(n)1237 5420 y Fw(\()p Fv(x)1330 5435 y Fu(1)1369 5420 y Fw(\))e(:=)1646 5353 y(1)p 1576 5397 191 4 v 1576 5417 a Fs(p)p 1659 5417 108 4 v 82 x Fw(2)p Fv(\031)1776 5420 y(e)1821 5379 y Ft(inx)1928 5388 y Fm(1)1967 5420 y Fv(;)212 b(x)2261 5435 y Fu(1)2328 5420 y Fs(2)28 b Fr(T)p Fv(:)1806 5712 y Fw(2)p eop %%Page: 3 8 3 7 bop 120 407 a Fw(Ev)m(ery)34 b(function)e Fv(f)39 b Fs(2)28 b Fv(L)1029 422 y Fu(2)1068 407 y Fw(\()p Fr(T)p Fw(\))36 b(can)d(b)s(e)f(expanded)i(in)e(its)g(\()p Fv(L)2344 422 y Fu(2)2384 407 y Fw(-con)m(v)m(ergen)m(t\))i Fj(F)-7 b(ourier)33 b(series)1561 629 y Fv(f)39 b Fw(=)1751 534 y Fo(X)1756 746 y Ft(n)p Fl(2)p Fn(Z)1933 603 y Fw(^)1912 629 y Fv(f)1960 644 y Ft(n)2007 629 y Fv(e)2052 644 y Ft(n)3264 629 y Fw(\(1.2.1\))120 922 y(with)32 b Fj(F)-7 b(ourier)34 b(c)-5 b(o)g(e\016cients)751 1146 y Fw(^)730 1172 y Fv(f)778 1187 y Ft(n)853 1172 y Fw(:=)27 b(\()p Fv(f)5 b(;)17 b(e)1163 1187 y Ft(n)1210 1172 y Fw(\))28 b(=)1461 1105 y(1)p 1390 1149 191 4 v 1390 1169 a Fs(p)p 1473 1169 108 4 v 83 x Fw(2)p Fv(\031)1607 1037 y Fo(Z)1706 1063 y Ft(\031)1662 1262 y Fl(\000)p Ft(\031)1781 1172 y Fv(f)11 b Fw(\()p Fv(x)1933 1187 y Fu(1)1972 1172 y Fw(\))p Fv(e)2055 1131 y Fl(\000)p Ft(inx)2217 1140 y Fm(1)2272 1172 y Fv(dx)2378 1187 y Fu(1)2418 1172 y Fv(;)212 b(n)27 b Fs(2)i Fr(Z)p Fv(:)331 b Fw(\(1.2.2\))120 1431 y(P)m(arsev)-5 b(al's)33 b(equalit)m(y)f(states)i(that)1477 1648 y Fs(k)p Fv(f)11 b Fs(k)1636 1607 y Fu(2)1702 1648 y Fw(=)1806 1553 y Fo(X)1811 1765 y Ft(n)p Fl(2)p Fn(Z)1966 1648 y Fs(j)2015 1621 y Fw(^)1994 1648 y Fv(f)2042 1663 y Ft(n)2089 1648 y Fs(j)2117 1607 y Fu(2)2156 1648 y Fv(:)1081 b Fw(\(1.2.3\))120 1946 y(Let)25 b(us)h(pass)g(to)e(the)h (case)h(of)f(the)g(halfstrip.)40 b(Considering)24 b Fv(L)2363 1961 y Fu(2)2403 1946 y Fw(\(\005\))h(as)g Fv(L)2755 1961 y Fu(2)2795 1946 y Fw(\()p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(\))7 b Fs(\012)g Fv(L)3267 1961 y Fu(2)3306 1946 y Fw(\()p Fr(R)3410 1961 y Fu(+)3475 1946 y Fw(\),)120 2067 y(the)33 b(expansion)g(\(1.2.1\))f(induces)h(an)g(\()p Fv(L)1637 2082 y Fu(2)1677 2067 y Fw(\(\005\)-con)m(v)m(ergen)m(t\))h (expansion)1498 2289 y Fv(u)27 b Fw(=)1685 2194 y Fo(X)1690 2406 y Ft(n)p Fl(2)p Fn(Z)1845 2289 y Fv(e)1890 2304 y Ft(n)1960 2289 y Fs(\012)h Fw(^)-55 b Fv(u)2115 2304 y Ft(n)3264 2289 y Fw(\(1.2.4\))120 2588 y(for)32 b(an)m(y)h Fv(u)27 b Fs(2)i Fv(L)697 2603 y Fu(2)736 2588 y Fw(\(\005\),)k(where) 403 2843 y(^)-55 b Fv(u)453 2858 y Ft(n)499 2843 y Fw(\()p Fv(x)592 2858 y Fu(2)632 2843 y Fw(\))28 b(:=)f(\()p Fv(u)p Fw(\()p Fv(:;)17 b(x)1086 2858 y Fu(2)1125 2843 y Fw(\))p Fv(;)g(e)1252 2858 y Ft(n)1299 2843 y Fw(\))1337 2859 y Ft(L)1385 2868 y Fm(2)1420 2859 y Fu(\()p Fn(T)p Fu(\))1551 2843 y Fw(=)1735 2776 y(1)p 1664 2820 191 4 v 1664 2840 a Fs(p)p 1747 2840 108 4 v 83 x Fw(2)p Fv(\031)1881 2708 y Fo(Z)1981 2734 y Ft(\031)1937 2933 y Fl(\000)p Ft(\031)2055 2843 y Fv(u)p Fw(\()p Fv(x)p Fw(\))p Fv(e)2287 2802 y Fl(\000)p Ft(inx)2449 2811 y Fm(1)2504 2843 y Fv(dx)2610 2858 y Fu(1)2650 2843 y Fv(;)212 b(x)2944 2858 y Fu(2)3011 2843 y Fs(2)28 b Fr(R)3171 2858 y Fu(+)3236 2843 y Fv(;)120 3108 y Fw(for)k(all)f Fv(n)c Fs(2)i Fr(Z)p Fw(.)40 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3855 y Fu(2)2377 3840 y Fw(\()p Fr(R)2481 3855 y Fu(+)2546 3840 y Fw(\))p Fv(:)653 b Fw(\(1.2.6\))120 4199 y Fp(2.)65 b(P)m(erio)s(dic)41 b(Sob)s(olev)h(spaces.)99 b Fw(F)-8 b(or)37 b(an)g(op)s(en)g(set)h (\012)d Fs(\032)h Fr(R)2578 4163 y Ft(d)2661 4199 y Fw(w)m(e)j(denote)e (b)m(y)h Fv(H)3357 4163 y Ft(s)3394 4199 y Fw(\(\012\))120 4319 y(the)c(Sob)s(olev)g(space)h(of)f(order)g Fv(s)c Fs(2)h Fr(R)45 b Fw(\(with)33 b(in)m(tegrabilit)m(y)f(index)i(2\).)48 b(F)-8 b(or)33 b(de\014nitions)h(as)120 4439 y(w)m(ell)e(as)h(the)g (Sob)s(olev)f(em)m(b)s(edding)g(theorems)h(w)m(e)g(refer)g(to)f([A)q (].)120 4560 y(F)-8 b(or)32 b Fv(s)27 b Fw(=)h Fv(m)g Fs(2)g Fr(N)745 4575 y Fu(0)823 4560 y Fw(w)m(e)34 b(use)f(the)g (standard)g(scalar)f(pro)s(duct)543 4808 y(\()p Fv(u;)17 b(v)t Fw(\))770 4823 y Ft(H)833 4804 y Fi(m)890 4823 y Fu(\(\012\))1028 4808 y Fw(:=)1187 4713 y Fo(X)1158 4929 y Fl(j)p Ft(\013)p Fl(j\024)p Ft(m)1377 4672 y Fo(Z)1432 4897 y Fu(\012)1504 4808 y Fv(D)1588 4766 y Ft(\013)1637 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a(2)1247 1342 y Fv(;)1300 1303 y Fu(1)p 1300 1319 V 1300 1377 a(2)1345 1342 y Fw(])p Fj(,)35 b(then)714 1528 y Fw(1)p 714 1573 49 4 v 714 1664 a(2)773 1596 y Fs(k)p Fv(u)p Fs(k)929 1555 y Fu(2)929 1622 y Ft(H)992 1603 y Fm(1)1026 1622 y Fu(\(\012\))1164 1596 y Fs(\024)28 b(k)p Fv(u)p Fs(k)1425 1555 y Fu(2)1425 1622 y Ft(H)1488 1603 y Fm(1)1522 1622 y Fu(\(\012\))p Ft(;k)1717 1596 y Fs(\024)g Fw(2)p Fs(k)p Fv(u)p Fs(k)2027 1555 y Fu(2)2027 1622 y Ft(H)2090 1603 y Fm(1)2124 1622 y Fu(\(\012\))2234 1596 y Fv(;)216 b(u)27 b Fs(2)h Fv(H)2743 1555 y Fu(1)2782 1596 y Fw(\(\012\))p Fv(:)266 1821 y Fw(Let)23 b(us)f(pass)h(to)f(the)h (de\014nition)e(of)g(p)s(erio)s(dic)g(Sob)s(olev)g(spaces.)42 b(W)-8 b(e)22 b(b)s(egin)g(with)g(the)g(case)120 1941 y(of)33 b(the)g(torus.)46 b(F)-8 b(or)32 b Fv(s)d Fs(\025)g Fw(0)k(w)m(e)i(denote)f(b)m(y)g Fv(H)1813 1905 y Ft(s)1849 1941 y Fw(\()p Fr(T)p Fw(\))i(the)e(closure)f(of)g Fv(C)2708 1905 y Fl(1)2783 1941 y Fw(\()p Fr(T)p Fw(\))j(in)d Fv(H)3162 1905 y Ft(s)3198 1941 y Fw(\()p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(\).)120 2062 y(One)43 b(has)h Fv(H)610 2025 y Ft(s)646 2062 y Fw(\()p Fr(T)p Fw(\))49 b(=)c Fv(H)1044 2025 y Ft(s)1081 2062 y Fw(\()p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(\))43 b(for)f(0)j Fs(\024)h Fv(s)g(<)2038 2022 y Fu(1)p 2038 2039 36 4 v 2038 2096 a(2)2127 2062 y Fw(and)d Fv(H)2416 2025 y Ft(s)2452 2062 y Fw(\()p Fr(T)p Fw(\))49 b(=)c Fs(f)p Fv(f)56 b Fs(2)46 b Fv(H)3116 2025 y Ft(s)3152 2062 y Fw(\()p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(\))46 b(:)134 2162 y Ft(d)170 2139 y Fi(l)195 2162 y Ft(f)p 130 2183 111 4 v 130 2245 a(dx)206 2222 y Fi(l)206 2266 y Fm(1)251 2206 y Fw(\()p Fv(\031)t Fw(\))39 b(=)556 2162 y Ft(d)592 2139 y Fi(l)616 2162 y Ft(f)p 551 2183 V 551 2245 a(dx)627 2222 y Fi(l)627 2266 y Fm(1)672 2206 y Fw(\()p Fs(\000)p Fv(\031)t Fw(\))p Fv(;)57 b(l)43 b Fw(=)d(0)p Fv(;)17 b(:::;)g Fw([)p Fv(s)26 b Fs(\000)1588 2167 y Fu(1)p 1588 2183 36 4 v 1588 2240 a(2)1633 2206 y Fw(])p Fs(g)40 b Fw(for)f Fv(s)h(>)2118 2167 y Fu(1)p 2118 2183 V 2118 2240 a(2)2164 2206 y Fw(,)h Fv(s)g Fs(6)p Fw(=)f Fv(m)27 b Fw(+)2660 2167 y Fu(1)p 2660 2183 V 2660 2240 a(2)2705 2206 y Fv(;)34 b(m)40 b Fs(2)h Fr(N)3064 2221 y Fu(0)3109 2206 y Fw(.)66 b(Here)40 b([)p Fv(r)s Fw(])120 2343 y(denotes)35 b(the)g(in)m(teger)e(part)h(of)g Fv(r)i Fw(and)e(the)h(b)s(oundary)f(v)-5 b(alues)34 b(of)f Fv(f)45 b Fw(exist)34 b(b)m(y)h(the)f(Sob)s(olev)120 2464 y(em)m(b)s(edding)e(theorem.)43 b(Finally)-8 b(,)30 b(for)i Fv(s)c Fw(=)g Fv(m)22 b Fw(+)1923 2424 y Fu(1)p 1923 2441 V 1923 2498 a(2)1968 2464 y Fv(;)34 b(m)28 b Fs(2)g Fr(N)2302 2479 y Fu(0)2347 2464 y Fw(,)33 b(one)f(has)960 2686 y Fv(H)1049 2645 y Ft(m)p Fu(+)1176 2618 y Fm(1)p 1176 2630 31 3 v 1176 2671 a(2)1220 2686 y Fw(\()p Fr(T)p Fw(\))f(=)d Fs(f)p Fv(f)38 b Fs(2)28 b Fv(H)1813 2645 y Ft(m)1879 2686 y Fw(\()p Fr(T)p Fw(\))j(:)55 b Fv(I)8 b Fw([)2219 2642 y Ft(d)2255 2619 y Fi(m)2314 2642 y Ft(f)p 2219 2663 137 4 v 2220 2720 a(dx)2296 2697 y Fi(m)2296 2741 y Fm(1)2365 2686 y Fw(])28 b Fv(<)g Fs(1g)p Fv(;)120 2932 y Fw(where)34 b Fv(I)8 b Fw([)p Fv(g)t Fw(])27 b(:=)715 2852 y Fo(R)782 2878 y Ft(\031)763 2967 y Fl(\000)p Ft(\031)881 2852 y Fo(R)947 2878 y Ft(\031)928 2967 y Fl(\000)p Ft(\031)1057 2885 y Fl(j)p Ft(g)r Fu(\()p Ft(x)1180 2894 y Fm(1)1214 2885 y Fu(\))p Fl(\000)p Ft(g)r Fu(\()p Ft(y)1394 2894 y Fm(1)1429 2885 y Fu(\))p Fl(j)1476 2861 y Fm(2)p 1057 2909 455 4 v 1068 2981 a Fl(j)12 b Fu(sin)o(\()1223 2945 y Fi(x)1258 2960 y Fm(1)1293 2945 y Fh(\000)p Fi(y)1373 2960 y Fm(1)p 1224 2966 184 3 v 1300 3007 a(2)1417 2981 y Fu(\))p Fl(j)1464 2962 y Fm(2)1537 2932 y Fv(dx)1643 2947 y Fu(1)1683 2932 y Fv(dy)1782 2947 y Fu(1)1821 2932 y Fw(.)120 3090 y(Another)36 b(imp)s(ortan)m(t)d(c)m(haracterization)i (of)f Fv(H)1886 3054 y Ft(s)1923 3090 y Fw(\()p Fr(T)p Fw(\))k(is)d(p)s(ossible)g(in)f(terms)h(of)g(F)-8 b(ourier)34 b(co-)120 3210 y(e\016cien)m(ts,)760 3353 y Fv(H)849 3311 y Ft(s)885 3353 y Fw(\()p Fr(T)p Fw(\))d(=)1158 3242 y Fo(n)1225 3353 y Fv(f)38 b Fs(2)28 b Fv(L)1471 3368 y Fu(2)1511 3353 y Fw(\()p Fr(T)p Fw(\))j(:)1764 3278 y Fo(P)1869 3382 y Ft(n)p Fl(2)p Fn(Z)2007 3353 y Fw(\(1)22 b(+)g Fv(n)2272 3316 y Fu(2)2311 3353 y Fw(\))2349 3316 y Ft(s)2386 3353 y Fs(j)2435 3326 y Fw(^)2414 3353 y Fv(f)2462 3368 y Ft(n)2509 3353 y Fs(j)2537 3316 y Fu(2)2604 3353 y Fv(<)27 b Fs(1)2807 3242 y Fo(o)2873 3353 y Fv(:)120 3546 y Fw(Throughout)33 b(the)g(follo)m(wing)d(w)m(e)j (will)e(use)i(in)f Fv(H)1941 3509 y Ft(s)1977 3546 y Fw(\()p Fr(T)p Fw(\))k(the)d(scalar)f(pro)s(duct)787 3763 y(\()p Fv(f)5 b(;)17 b(g)t Fw(\))1011 3778 y Ft(H)1074 3759 y Fi(s)1107 3778 y Fu(\()p Fn(T)p Fu(\))1238 3763 y Fw(=)1342 3668 y Fo(X)1346 3880 y Ft(n)p Fl(2)p Fn(Z)1485 3763 y Fw(\(1)22 b(+)g Fv(n)1750 3722 y Fu(2)1790 3763 y Fw(\))1828 3722 y Ft(s)1886 3736 y Fw(^)1865 3763 y Fv(f)1913 3778 y Ft(n)p 1960 3682 94 4 v 1963 3763 a Fw(^)-52 b Fv(g)2007 3778 y Ft(n)2053 3763 y Fv(;)212 b(f)5 b(;)34 b(g)d Fs(2)d Fv(H)2667 3722 y Ft(s)2703 3763 y Fw(\()p Fr(T)p Fw(\))p Fv(;)120 4051 y Fw(and)33 b(the)g(corresp)s(onding)f(norm)872 4262 y Fs(k)p Fv(f)11 b Fs(k)1031 4221 y Fu(2)1031 4287 y Ft(H)1094 4268 y Fi(s)1127 4287 y Fu(\()p Fn(T)p Fu(\))1258 4262 y Fw(=)1361 4168 y Fo(X)1366 4379 y Ft(n)p Fl(2)p Fn(Z)1505 4262 y Fw(\(1)22 b(+)g Fv(n)1770 4221 y Fu(2)1810 4262 y Fw(\))1848 4221 y Ft(s)1884 4262 y Fs(j)1933 4236 y Fw(^)1912 4262 y Fv(f)1960 4277 y Ft(n)2007 4262 y Fs(j)2035 4221 y Fu(2)2074 4262 y Fv(;)212 b(f)38 b Fs(2)28 b Fv(H)2582 4221 y Ft(s)2619 4262 y Fw(\()p Fr(T)p Fw(\))p Fv(:)120 4572 y Fw(W)-8 b(e)40 b(pro)s(ceed)h(to)e(the)i(case)f(of)g(the)g (halfstrip.)64 b(F)-8 b(or)39 b Fv(m)h Fs(2)h Fr(N)54 b Fw(w)m(e)41 b(denote)g(b)m(y)3086 4547 y(~)3061 4572 y Fv(H)3150 4536 y Ft(m)3216 4572 y Fw(\(\005\))f(the)120 4693 y(closure)31 b(of)575 4668 y(~)553 4693 y Fv(C)630 4657 y Fl(1)704 4693 y Fw(\(\005\))18 b Fs(\\)g Fv(H)1044 4657 y Ft(m)1110 4693 y Fw(\(\005\))31 b(in)f Fv(H)1491 4657 y Ft(m)1557 4693 y Fw(\(\005\).)42 b(A)31 b(function)f Fv(u)d Fs(2)h Fv(H)2525 4657 y Ft(m)2591 4693 y Fw(\(\005\))j(b)s (elongs)f(to)3261 4668 y(~)3236 4693 y Fv(H)3325 4657 y Ft(m)3391 4693 y Fw(\(\005\))120 4813 y(i\013)g(it)h(is)g(p)s(erio)s (dic)f(with)h(resp)s(ect)h(to)f Fv(x)1528 4828 y Fu(1)1600 4813 y Fw(in)f(the)i(sense)h(of)e(the)h(Sob)s(olev)f(trace)h(theorem,)f (i.e.)1365 5002 y Fv(@)1421 4966 y Ft(l)1448 5002 y Fv(u)p 1358 5047 152 4 v 1358 5139 a(@)5 b(x)1469 5105 y Ft(l)1469 5164 y Fu(1)1520 5070 y Fw(\()p Fv(\031)t(;)17 b(:)p Fw(\))27 b(=)1873 5002 y Fv(@)1929 4966 y Ft(l)1956 5002 y Fv(u)p 1867 5047 V 1867 5139 a(@)5 b(x)1978 5105 y Ft(l)1978 5164 y Fu(1)2029 5070 y Fw(\()p Fs(\000)p Fv(\031)t(;)17 b(:)p Fw(\))120 5342 y(for)48 b(all)e(0)53 b Fs(\024)i Fv(l)h(<)e(m)p Fw(.)90 b(It)49 b(is)e(con)m(v)m(enien)m(t)j(to)e(c)m (haracterize)2535 5317 y(~)2509 5342 y Fv(H)2598 5306 y Ft(m)2665 5342 y Fw(\(\005\))g(in)f(terms)h(of)g(the)120 5463 y(decomp)s(osition)31 b(\(1.2.6\))o(.)1806 5712 y(4)p eop %%Page: 5 10 5 9 bop 120 407 a Fp(Lemma)37 b(1.2.2.)42 b Fj(L)-5 b(et)35 b Fv(m)28 b Fs(2)g Fr(N)50 b Fj(and)35 b Fv(u)27 b Fs(2)h Fv(L)1735 422 y Fu(2)1775 407 y Fw(\(\005\))p Fj(.)44 b(Then)34 b Fv(u)28 b Fs(2)2455 382 y Fw(~)2430 407 y Fv(H)2519 371 y Ft(m)2585 407 y Fw(\(\005\))35 b Fj(i\013)857 624 y Fw(^)-55 b Fv(u)907 639 y Ft(n)982 624 y Fs(2)28 b Fv(H)1165 583 y Ft(m)1231 624 y Fw(\()p Fr(R)1335 639 y Fu(+)1400 624 y Fw(\))p Fv(;)216 b(n)27 b Fs(2)i Fr(Z)p Fv(;)868 703 y Fo(X)873 915 y Ft(n)p Fl(2)p Fn(Z)1028 798 y Fv(n)1086 757 y Fu(2)p Ft(\013)1166 766 y Fm(1)1206 798 y Fs(k)p Fv(D)1340 757 y Ft(\013)1385 766 y Fm(2)1430 798 y Fw(^)-55 b Fv(u)1480 813 y Ft(n)1526 798 y Fs(k)1576 757 y Fu(2)1643 798 y Fv(<)28 b Fs(1)p Fv(;)215 b(\013)29 b Fs(2)f Fr(N)2340 757 y Fu(2)2340 822 y(0)2385 798 y Fv(;)33 b Fs(j)p Fv(\013)q Fs(j)27 b(\024)h Fv(m:)3264 760 y Fw(\(1.2.8\))120 1112 y Fj(In)34 b(this)h(c)-5 b(ase,)34 b(the)h(series)f Fv(u)27 b Fw(=)1297 1037 y Fo(P)1402 1141 y Ft(n)p Fl(2)p Fn(Z)1557 1112 y Fv(e)1602 1127 y Ft(n)1671 1112 y Fs(\012)i Fw(^)-55 b Fv(u)1827 1127 y Ft(n)1908 1112 y Fj(c)-5 b(onver)g(ges)34 b(in)h Fv(H)2552 1076 y Ft(m)2618 1112 y Fw(\(\005\))f Fj(and)795 1358 y Fv(D)879 1316 y Ft(\013)929 1358 y Fv(u)27 b Fw(=)1115 1263 y Fo(X)1120 1475 y Ft(n)p Fl(2)p Fn(Z)1276 1358 y Fv(e)1321 1373 y Ft(n)1390 1358 y Fs(\012)c Fv(n)1548 1316 y Ft(\013)1593 1325 y Fm(1)1632 1358 y Fv(D)1716 1316 y Ft(\013)1761 1325 y Fm(2)1806 1358 y Fw(^)-55 b Fv(u)1856 1373 y Ft(n)1902 1358 y Fv(;)216 b(\013)29 b Fs(2)f Fr(N)2396 1316 y Fu(2)2396 1382 y(0)2441 1358 y Fv(;)33 b Fs(j)p Fv(\013)q Fs(j)27 b(\024)h Fv(m:)266 1666 y Fw(The)f(pro)s(of)f(is)g(straigh)m(tforw)m(ard,)h(so)f(w)m(e)h (can)g(omit)d(it.)40 b(W)-8 b(e)27 b(remark)f(that)g(it)f(is)h (su\016cien)m(t)120 1786 y(that)37 b(the)g(second)h(condition)e(in)g (\(1.2.8\))g(holds)h(only)g(for)f(the)h(m)m(ulti-indices)d Fv(\013)i Fw(=)f(\()p Fv(m;)17 b Fw(0\))120 1907 y(and)33 b Fv(\013)28 b Fw(=)g(\(0)p Fv(;)17 b(m)p Fw(\).)120 2167 y Fp(3.)99 b(Multiplication)50 b(on)k(the)f(b)s(oundary)-9 b(.)99 b Fw(In)47 b(this)f(subsection)i(w)m(e)g(assume)f(that)120 2287 y Fv(\033)32 b Fw(:)27 b Fr(R)39 b Fs(!)27 b Fr(R)44 b Fw(is)32 b(a)g(real-v)-5 b(alued,)31 b(\(2)p Fv(\031)t Fw(-\)p)s(erio)s(dic)f(function)i(suc)m(h)i(that)1616 2507 y Fv(\033)e Fs(2)c Fv(L)1863 2522 y Ft(q)1901 2507 y Fw(\()p Fr(T)p Fw(\))120 2727 y(for)k(some)h(1)28 b Fv(<)h(q)j Fs(\024)d(1)p Fw(.)44 b(W)-8 b(e)33 b(denote)h(b)m(y)g Fv(M)1771 2691 y Fu(\()p Ft(\033)r Fu(\))1907 2727 y Fw(the)f(self-adjoin)m(t)e(op)s(erator)i(of)f(m)m(ultiplica-)120 2847 y(tion)g(b)m(y)h Fv(\033)k Fw(in)31 b Fv(L)727 2862 y Fu(2)767 2847 y Fw(\()p Fr(T)p Fw(\),)1068 3071 y Fs(D)s Fw(\()p Fv(M)1290 3030 y Fu(\()p Ft(\033)r Fu(\))1392 3071 y Fw(\))d(:=)g Fs(f)p Fv(f)38 b Fs(2)28 b Fv(L)1885 3086 y Fu(2)1925 3071 y Fw(\()p Fr(T)p Fw(\))j(:)55 b Fv(\033)t(f)39 b Fs(2)28 b Fv(L)2483 3086 y Fu(2)2522 3071 y Fw(\()p Fr(T)p Fw(\))p 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Ft(dx)1920 4177 y Fm(2)1920 4221 y(1)1992 4164 y Fw(in)g Fv(L)2166 4179 y Fu(2)2206 4164 y Fw(\()p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(\))27 b(with)g(p)s(erio)s(dic)f(b)s(oundary)120 4299 y(conditions,)j(but)g(w) m(e)h(shall)d(not)i(use)h(this)e(fact.)42 b(Our)29 b(notation)e Fv(B)34 b Fw(is)29 b(motiv)-5 b(ated)27 b(b)m(y)j(its)e(use)120 4419 y(in)k(Chapter)h(3.)120 4540 y(By)39 b(H\177)-49 b(older's)38 b(inequalit)m(y)f(and)i(the)g(Sob)s(olev)e(em)m(b)s (edding)h Fv(H)2451 4503 y Fu(1)p Ft(=)p Fu(2)2561 4540 y Fw(\()p Fr(T)p Fw(\))j Fs(\032)d Fv(L)2922 4555 y Fu(2)p Ft(q)2991 4536 y Fh(0)3018 4540 y Fw(\()p Fr(T)p Fw(\))j(w)m(e)f(ha)m (v)m(e)120 4660 y(for)32 b Fv(f)38 b Fs(2)29 b Fv(H)539 4624 y Fu(1)p Ft(=)p Fu(2)648 4660 y Fw(\()p Fr(T)p Fw(\))298 4795 y Fo(Z)398 4821 y Ft(\031)354 5021 y Fl(\000)p Ft(\031)472 4931 y Fs(j)p Fv(\033)t Fw(\()p Fv(x)652 4946 y Fu(1)692 4931 y Fw(\))p Fs(jj)p Fv(f)11 b Fw(\()p Fv(x)938 4946 y Fu(1)976 4931 y Fw(\))p Fs(j)1042 4890 y Fu(2)1098 4931 y Fv(dx)1204 4946 y Fu(1)1271 4931 y Fs(\024)28 b(k)p Fv(\033)t Fs(k)1535 4946 y Ft(L)1583 4954 y Fi(q)1617 4946 y Fu(\()p Fn(T)p Fu(\))1721 4931 y Fs(k)p Fv(f)11 b Fs(k)1880 4890 y Fu(2)1880 4955 y Ft(L)1928 4974 y Fm(2)p Fi(q)1988 4960 y Fh(0)2014 4955 y Fu(\()p Fn(T)p Fu(\))2145 4931 y Fs(\024)28 b Fv(c)p Fs(k)p Fv(\033)t Fs(k)2451 4946 y Ft(L)2499 4954 y Fi(q)2533 4946 y Fu(\()p Fn(T)p Fu(\))2637 4931 y Fs(k)p Fv(f)11 b Fs(k)2796 4890 y Fu(2)2796 4963 y Ft(H)2859 4943 y Fm(1)p Fi(=)p Fm(2)2955 4963 y Fu(\()p Fn(T)p Fu(\))3058 4931 y Fv(:)179 b Fw(\(1.2.9\))120 5221 y(Since)32 b Fv(B)453 5185 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)650 5221 y Fw(maps)f Fv(L)970 5236 y Fu(2)1010 5221 y Fw(\()p Fr(T)p Fw(\))k(unitarily)29 b(on)m(to)j Fv(H)1889 5185 y Fu(1)p Ft(=)p Fu(2)1999 5221 y Fw(\()p Fr(T)p Fw(\),)j(it)c(follo)m(ws)f(that)i Fs(j)p Fv(M)2959 5185 y Fu(\()p Ft(\033)r Fu(\))3061 5221 y Fs(j)3089 5185 y Fu(1)p Ft(=)p Fu(2)3199 5221 y Fv(B)3278 5185 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)3474 5221 y Fw(is)120 5342 y(a)g(b)s(ounded)i(op)s (erator)e(on)g Fv(L)1194 5357 y Fu(2)1234 5342 y Fw(\()p Fr(T)p Fw(\).)120 5462 y(Indeed,)i(this)e(result)h(can)g(b)s(e)f (considerably)h(strenghened.)1806 5712 y(5)p eop %%Page: 6 11 6 10 bop 120 407 a Fp(Lemma)37 b(1.2.3.)42 b Fj(Under)35 b(the)g(ab)-5 b(ove)34 b(hyp)-5 b(otheses,)34 b Fs(j)p Fv(M)2184 371 y Fu(\()p Ft(\033)r Fu(\))2286 407 y Fs(j)2314 371 y Fu(1)p Ft(=)p Fu(2)2424 407 y Fv(B)2503 371 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)2696 407 y Fs(2)28 b Fw(\006)2860 422 y Fu(2)2900 407 y Fw(\()p Fv(L)3004 422 y Fu(2)3043 407 y Fw(\()p Fr(T)p Fw(\)\))p Fj(.)266 566 y Fw(Recall)45 b(\(cf.)84 b(Section)46 b(11.6)f(in)g([BS)q(]\))h(that)g(\006)2054 581 y Ft(p)2094 566 y Fw(\()p Fg(H)p Fw(\))p Fv(;)33 b Fw(0)50 b Fv(<)g(p)h(<)f Fs(1)p Fv(;)c Fw(is)g(the)g(class)g(of)120 695 y(compact)35 b(op)s(erators)h(K)f(on)h(a)g(Hilb)s(ert)e(space)j Fg(H)e Fw(for)g(whic)m(h)h(sup)2574 719 y Ft(m)2641 695 y Fw(\()p Fv(m)2764 659 y Fu(1)p Ft(=p)2874 610 y Fo(p)p 2974 610 420 4 v 85 x Fv(\025)3031 710 y Ft(m)3098 695 y Fw(\()p Fv(K)3226 666 y Fl(\003)3265 695 y Fv(K)7 b Fw(\)\))33 b Fv(<)120 815 y Fs(1)p Fw(,)46 b(where)f Fv(\025)643 830 y Ft(m)710 815 y Fw(\()p Fv(K)838 779 y Fl(\003)877 815 y Fv(K)7 b Fw(\))44 b(denotes)h(the)g Fv(m)p Fw(-th)e(eigen)m(v)-5 b(alue)44 b(of)f Fv(K)2536 779 y Fl(\003)2576 815 y Fv(K)50 b Fw(in)44 b(decreasing)g(order)120 936 y(coun)m(ting)32 b(m)m(ultiplicities.)120 1056 y(It)h(is)f(easy)h (to)g(see)g(that)1059 1224 y(\006)1129 1239 y Ft(r)1167 1224 y Fw(\()p Fg(H)p Fw(\))28 b Fs(\032)g Fg(S)1531 1239 y Ft(p)1570 1224 y Fw(\()p Fg(H)p Fw(\))f Fs(\032)i Fw(\006)1921 1239 y Ft(p)1961 1224 y Fw(\()p Fg(H)p Fw(\))p Fv(;)211 b(r)30 b(<)e(p;)120 1392 y Fw(where)35 b Fg(S)486 1407 y Ft(p)526 1392 y Fw(\()p Fg(H)p Fw(\))e(is)h(the)g(usual)g(Sc)m (hatten)i(class)e(of)f(order)i Fv(p)f Fw(on)g Fg(H)p Fw(.)48 b(The)35 b(dep)s(endence)h(on)e Fg(H)120 1513 y Fw(is)e(usually)g(dropp)s(ed)h(in)f(the)h(notation)e(when)j(the)f (underlying)f(Hilb)s(ert)f(space)i(is)f(clear.)120 1633 y(W)-8 b(e)33 b(will)d(often)j(use)g(the)g(fact)g(that)f Fv(K)1542 1648 y Fu(1)1609 1633 y Fs(2)c Fw(\006)1773 1648 y Ft(p)1809 1657 y Fm(1)1848 1633 y Fv(;)33 b(K)1991 1648 y Fu(2)2058 1633 y Fs(2)c Fw(\006)2223 1648 y Ft(p)2259 1657 y Fm(2)2330 1633 y Fw(implies)1413 1801 y Fv(K)1496 1816 y Fu(1)1536 1801 y Fv(K)1619 1816 y Fu(2)1686 1801 y Fs(2)f Fw(\006)1850 1826 y Fu(\()1906 1799 y Fm(1)p 1888 1811 67 3 v 1888 1852 a Fi(p)1920 1867 y Fm(1)1964 1826 y Fu(+)2047 1799 y Fm(1)p 2029 1811 V 2029 1852 a Fi(p)2061 1867 y Fm(2)2106 1826 y Fu(\))2133 1807 y Fh(\000)p Fm(1)2220 1801 y Fv(:)968 b Fw(\(1.2.10\))120 2010 y Fj(Pr)-5 b(o)g(of.)48 b Fw(W)-8 b(rite)32 b Fs(j)p Fv(M)828 1974 y Fu(\()p Ft(\033)r Fu(\))931 2010 y Fs(j)959 1974 y Fu(1)p Ft(=)p Fu(2)1068 2010 y Fv(B)1147 1974 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1340 2010 y Fw(=)1443 1930 y Fo(\000)1489 2010 y Fs(j)p Fv(M)1621 1974 y Fu(\()p Ft(\033)r Fu(\))1723 2010 y Fs(j)1751 1974 y Fu(1)p Ft(=)p Fu(2)1861 2010 y Fv(B)1940 1974 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)p Ft(q)2139 1930 y Fo(\001)2201 2010 y Fv(B)2280 1974 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)p Ft(q)2474 1951 y Fh(0)2534 2010 y Fw(\(without)g(loss,)g Fv(q)g(<)c Fs(1)p Fw(\).)120 2131 y(Under)i(F)-8 b(ourier)27 b(transformation)g (on)i Fv(L)1604 2146 y Fu(2)1644 2131 y Fw(\()p Fr(T)p Fw(\))j(the)d(last)f(factor)g(corresp)s(onds)j(to)d(m)m(ultiplica-)120 2251 y(tion)k(b)m(y)h(the)g(sequence)1028 2170 y Fo(\000)1074 2251 y Fw(\(1)22 b(+)g Fv(n)1339 2215 y Fu(2)1378 2251 y Fw(\))1416 2215 y Fl(\000)p Fu(1)p Ft(=)p Fu(4)p Ft(q)1610 2192 y Fh(0)1638 2170 y Fo(\001)1683 2290 y Ft(n)p Fl(2)p Fn(Z)1854 2251 y Fw(and)33 b(is)f(therefore)h(in)f(the)h(class)f(\006) 3131 2266 y Fu(2)p Ft(q)3200 2247 y Fh(0)3228 2251 y Fw(.)120 2372 y(The)39 b(pro)s(duct)e(of)g(the)h(\014rst)g(t)m(w)m(o)h (op)s(erators)e(corresp)s(onds)i(to)e(m)m(ultiplication)c(b)m(y)39 b(the)f(se-)120 2492 y(quence)438 2411 y Fo(\000)484 2492 y Fw(\(1)22 b(+)g Fv(n)749 2456 y Fu(2)788 2492 y Fw(\))826 2456 y Fl(\000)p Fu(1)p Ft(=)p Fu(4)p Ft(q)1025 2411 y Fo(\001)1071 2531 y Ft(n)p Fl(2)p Fn(Z)1237 2492 y Fs(2)28 b Fv(l)1360 2507 y Fu(2)p Ft(q)r(;w)1505 2492 y Fw(\()p Fr(Z)p Fw(\))e(in)h(F)-8 b(ourier)27 b(space)i(and)g(b)m(y)g (the)f(function)3232 2407 y Fo(p)p 3332 2407 115 4 v 85 x Fs(j)p Fv(\033)t Fs(j)f(2)120 2629 y Fv(L)186 2644 y Fu(2)p Ft(q)260 2629 y Fw(\()p Fr(T)p Fw(\))41 b(in)c (con\014guration)g(space.)61 b(Here)38 b Fv(l)1734 2644 y Ft(p;w)1846 2629 y Fw(\()p Fr(Z)p Fw(\))c(=)j Fs(f)p Fw(\()p Fv(\013)2288 2644 y Ft(n)2334 2629 y Fw(\))2372 2644 y Ft(n)p Fl(2)p Fn(Z)2547 2629 y Fw(:)54 b(sup)2775 2653 y Ft(m)2841 2629 y Fw(\()p Fv(m)2964 2593 y Fu(1)p Ft(=p)3075 2629 y Fv(\013)3138 2593 y Fl(\003)3137 2654 y Ft(m)3203 2629 y Fw(\))37 b Fv(<)g Fs(1g)120 2750 y Fw(where)27 b(\()p Fv(\013)496 2714 y Fl(\003)495 2774 y Ft(m)561 2750 y Fw(\))599 2765 y Ft(m)p Fl(2)p Fn(N)786 2750 y Fw(denotes)g(the)f(non-increasing)f(rearrangemen)m(t)g(of)g(\()p Fs(j)p Fv(\013)2804 2765 y Ft(n)2851 2750 y Fs(j)p Fw(\))2917 2765 y Ft(n)p Fl(2)p Fn(Z)3055 2750 y Fw(.)41 b(In)26 b([BKaS])120 2870 y(a)k(Cwik)m(el-t)m(yp)s(e)i(estimate)d(w)m(as)j (obtained)e(for)g(op)s(erators)h(of)f(this)g(form.)42 b(This)31 b(implies)d(\(cf.)120 2991 y(Theorem)33 b(4.8)f(in)g ([BKaS]\))g Fs(j)p Fv(M)1307 2954 y Fu(\()p Ft(\033)r Fu(\))1410 2991 y Fs(j)1438 2954 y Fu(1)p Ft(=)p Fu(2)1547 2991 y Fv(B)1626 2954 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)p Ft(q)1853 2991 y Fs(2)c Fw(\006)2017 3006 y Fu(2)p Ft(q)2091 2991 y Fw(.)120 3111 y(By)33 b(use)h(of)39 b(\(1.2.10\))31 b(w)m(e)j(can)f(conclude)g Fs(j)p Fv(M)1773 3075 y Fu(\()p Ft(\033)r Fu(\))1875 3111 y Fs(j)1903 3075 y Fu(1)p Ft(=)p Fu(2)2013 3111 y Fv(B)2092 3075 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)2284 3111 y Fs(2)c Fw(\006)2449 3126 y Fu(2)2488 3111 y Fw(,)k(as)g(claimed.)p 3469 3111 4 66 v 3473 3048 59 4 v 3473 3111 V 3531 3111 4 66 v 266 3291 a(It)41 b(follo)m(ws)f(in)f(particular,)j(that)1537 3211 y Fo(R)1603 3237 y Ft(\031)1584 3326 y Fl(\000)p Ft(\031)1703 3291 y Fs(j)p Fv(\033)t Fw(\()p Fv(x)1883 3306 y Fu(1)1922 3291 y Fw(\))p Fs(jj)p Fv(f)11 b Fw(\()p Fv(x)2168 3306 y Fu(1)2207 3291 y Fw(\))p Fs(j)2273 3255 y Fu(2)2329 3291 y Fv(dx)2435 3306 y Fu(1)2474 3291 y Fw(,)43 b Fv(f)52 b Fs(2)42 b Fv(H)2841 3255 y Fu(1)p Ft(=)p Fu(2)2951 3291 y Fw(\()p Fr(T)p Fw(\),)k(de\014nes)c(a)120 3427 y(compact)32 b(form)g(on)g Fv(H)965 3391 y Fu(1)p Ft(=)p Fu(2)1075 3427 y Fw(\()p Fr(T)p Fw(\).)46 b(This)33 b(implies)120 3586 y Fp(Corollary)k(1.2.4.)42 b Fj(L)-5 b(et)36 b Fv(\017)28 b(>)f Fw(0)p Fj(,)35 b(then)f(ther)-5 b(e)35 b(exists)g(a)g(c)-5 b(onstant)34 b Fv(C)2644 3601 y Fu(1)2683 3586 y Fw(\()p Fv(\017;)17 b(\033)t Fw(\))28 b Fv(>)f Fw(0)35 b Fj(such)g(that)342 3669 y Fo(Z)442 3695 y Ft(\031)397 3895 y Fl(\000)p Ft(\031)516 3805 y Fs(j)p Fv(\033)t Fw(\()p Fv(x)696 3820 y Fu(1)736 3805 y Fw(\))p Fs(jj)p Fv(f)11 b Fw(\()p Fv(x)982 3820 y Fu(1)1020 3805 y Fw(\))p Fs(j)1086 3764 y Fu(2)1142 3805 y Fv(dx)1248 3820 y Fu(1)1315 3805 y Fs(\024)28 b Fv(\017)p Fs(k)p Fv(f)11 b Fs(k)1618 3764 y Fu(2)1618 3836 y Ft(H)1681 3817 y Fm(1)p Fi(=)p Fm(2)1778 3836 y Fu(\()p Fn(T)p Fu(\))1904 3805 y Fw(+)22 b Fv(C)2072 3820 y Fu(1)2111 3805 y Fw(\()p Fv(\017;)17 b(\033)t Fw(\))p Fs(k)p Fv(f)11 b Fs(k)2488 3764 y Fu(2)2527 3805 y Fv(;)215 b(f)39 b Fs(2)28 b Fv(H)3039 3764 y Fu(1)p Ft(=)p Fu(2)3148 3805 y Fw(\()p Fr(T)p Fw(\))p Fv(:)266 4027 y Fw(No)m(w)38 b(let)e(us)h(pass)h(to)f(the)g(situation)e(on)i (the)g(halfstrip)e(and)i(on)g(the)g(halfplane.)55 b(The)120 4147 y(trace)37 b(op)s(erator)f Fv(u)e Fs(7!)g Fv(u)p Fw(\()p Fv(:;)17 b Fw(0\))36 b(is)g(b)s(ounded)h(from)f Fv(H)2105 4111 y Fu(1)2144 4147 y Fw(\(\005\))g(to)h Fv(H)2542 4111 y Fu(1)p Ft(=)p Fu(2)2651 4147 y Fw(\()p Fr(T)p Fw(\).)59 b(Hence)38 b(the)f(form)120 4187 y Fo(R)186 4214 y Ft(\031)167 4302 y Fl(\000)p Ft(\031)286 4268 y Fs(j)p Fv(\033)t Fw(\()p Fv(x)466 4283 y Fu(1)505 4268 y Fw(\))p Fs(jj)p Fv(u)p Fw(\()p Fv(x)748 4283 y Fu(1)787 4268 y Fv(;)17 b Fw(0\))p Fs(j)946 4231 y Fu(2)1001 4268 y Fv(dx)1107 4283 y Fu(1)1146 4268 y Fw(,)33 b Fv(u)27 b Fs(2)h Fv(H)1472 4231 y Fu(1)1511 4268 y Fw(\(\005\),)33 b(is)f(compact)g(on)h Fv(H)2433 4231 y Fu(1)2472 4268 y Fw(\(\005\))f(and)h(w)m(e)g(obtain)120 4426 y Fp(Corollary)k(1.2.5.) 42 b Fj(L)-5 b(et)36 b Fv(\017)28 b(>)f Fw(0)p Fj(,)35 b(then)f(ther)-5 b(e)35 b(exists)g(a)g(c)-5 b(onstant)34 b Fv(C)2644 4441 y Fu(2)2683 4426 y Fw(\()p Fv(\017;)17 b(\033)t Fw(\))28 b Fv(>)f Fw(0)35 b Fj(such)g(that)361 4510 y Fo(Z)461 4536 y Ft(\031)416 4735 y Fl(\000)p Ft(\031)535 4645 y Fs(j)p Fv(\033)t Fw(\()p Fv(x)715 4660 y Fu(1)754 4645 y Fw(\))p Fs(jj)p Fv(u)p Fw(\()p Fv(x)997 4660 y Fu(1)1036 4645 y Fv(;)17 b Fw(0\))p Fs(j)1195 4604 y Fu(2)1250 4645 y Fv(dx)1356 4660 y Fu(1)1423 4645 y Fs(\024)29 b Fv(\017)p Fs(k)p Fv(u)p Fs(k)1724 4604 y Fu(2)1724 4671 y Ft(H)1787 4652 y Fm(1)1821 4671 y Fu(\(\005\))1955 4645 y Fw(+)22 b Fv(C)2123 4660 y Fu(2)2162 4645 y Fw(\()p Fv(\017;)17 b(\033)t Fw(\))p Fs(k)p Fv(u)p Fs(k)2536 4604 y Fu(2)2575 4645 y Fv(;)216 b(u)27 b Fs(2)h Fv(H)3084 4604 y Fu(1)3123 4645 y Fw(\(\005\))p Fv(:)266 4877 y Fw(No)m(w)g(let)f Fv(u)g Fs(2)h Fv(H)885 4841 y Fu(1)924 4877 y Fw(\()p Fr(R)1028 4841 y Fu(2)1028 4902 y(+)1093 4877 y Fw(\).)42 b(F)-8 b(or)27 b(eac)m(h)h Fv(n)g Fs(2)g Fr(Z)d Fw(w)m(e)k(apply)e(Corollary)e(1.2.5)i(to)h(the)f(function)120 4997 y(\005)h Fs(3)g Fv(x)g Fs(7!)f Fv(u)p Fw(\()p Fv(x)674 5012 y Fu(1)736 4997 y Fw(+)22 b(2)p Fv(\031)t(n;)17 b(x)1099 5012 y Fu(2)1138 4997 y Fw(\))33 b(and)f(then)i(sum)e(o)m(v)m (er)i(all)c Fv(n)e Fs(2)g Fr(Z)p Fw(.)41 b(This)33 b(yields)120 5156 y Fp(Corollary)k(1.2.6.)42 b Fj(L)-5 b(et)36 b Fv(\017)28 b(>)f Fw(0)p Fj(,)35 b(then)f(ther)-5 b(e)35 b(exists)g(a)g(c)-5 b(onstant)34 b Fv(C)2644 5171 y Fu(2)2683 5156 y Fw(\()p Fv(\017;)17 b(\033)t Fw(\))28 b Fv(>)f Fw(0)35 b Fj(such)g(that)334 5237 y Fo(Z)389 5463 y Fn(R)458 5373 y Fs(j)p Fv(\033)t Fw(\()p Fv(x)638 5388 y Fu(1)678 5373 y Fw(\))p Fs(jj)p Fv(u)p Fw(\()p Fv(x)921 5388 y Fu(1)959 5373 y Fv(;)17 b Fw(0\))p Fs(j)1118 5332 y Fu(2)1173 5373 y Fv(dx)1279 5388 y Fu(1)1347 5373 y Fs(\024)28 b Fv(\017)p Fs(k)p Fv(u)p Fs(k)1647 5332 y Fu(2)1647 5403 y Ft(H)1710 5384 y Fm(1)1744 5403 y Fu(\()p Fn(R)1819 5380 y Fm(2)1819 5424 y(+)1870 5403 y Fu(\))1924 5373 y Fw(+)22 b Fv(C)2092 5388 y Fu(2)2131 5373 y Fw(\()p Fv(\017;)17 b(\033)t Fw(\))p Fs(k)p Fv(u)p Fs(k)2505 5332 y Fu(2)2544 5373 y Fv(;)216 b(u)27 b Fs(2)h Fv(H)3053 5332 y Fu(1)3092 5373 y Fw(\()p Fr(R)3196 5332 y Fu(2)3196 5398 y(+)3261 5373 y Fw(\))p Fv(:)1806 5712 y Fw(6)p eop %%Page: 7 12 7 11 bop 120 407 a Fq(1.3)161 b(The)53 b(Gelfand)h(transformation)120 626 y Fw(In)44 b(this)g(section)g(w)m(e)h(in)m(tro)s(duce)f(the)g (Gelfand)f(transformation)f(on)i(the)g(halfplane)f(and)120 746 y(discuss)26 b(its)e(basic)h(prop)s(erties.)41 b(This)25 b(unitary)g(op)s(erator)f(is)g(the)i(main)d(to)s(ol)g(in)h(the)h(Flo)s (quet-)120 867 y(Blo)s(c)m(h)32 b(theory)h(of)f(p)s(erio)s(dic)f(op)s (erators.)120 987 y(Because)i(of)f(the)g(geometric)f(structure)i(of)e Fr(R)1807 951 y Fu(2)1807 1012 y(+)1904 987 y Fw(w)m(e)i(\014nd)g(it)d (con)m(v)m(enien)m(t)k(to)e(in)m(tro)s(duce)f(\014rst)120 1107 y(the)c(Gelfand)e(transformation)f(on)j Fr(R)37 b Fw(in)26 b(Subsection)h(1.)41 b(Our)26 b(exp)s(osition)g(relies)g (gratefully)120 1228 y(on)d([Su)q(].)40 b(See)25 b(also)d(Section)i(XI) s(I)s(I.16)f(in)g([RS4])g(for)g(a)h(related)f(construction.)40 b(In)24 b(Subsection)120 1348 y(2)33 b(w)m(e)i(will)d(extend)j(the)f (op)s(erator)f(trivially)e(to)i Fr(R)1970 1312 y Fu(2)1970 1373 y(+)2035 1348 y Fw(.)47 b(W)-8 b(e)34 b(pro)m(v)m(e)h(the)f (material)d(needed)k(for)120 1469 y(the)e(direct)f(in)m(tegral)f (decomp)s(osition)g(in)h(Section)g(2.2.)120 1725 y Fp(1.)50 b(The)38 b(Gelfand)g(transformation)e(on)i Fr(R)5 b Fp(.)104 b Fw(F)-8 b(or)31 b(a)i(function)f Fv(f)38 b Fs(2)28 b(S)7 b Fw(\()p Fr(R)f Fw(\))38 b(w)m(e)c(de\014ne)276 1941 y(\()p Fs(U)376 1956 y Fu(1)415 1941 y Fv(f)11 b Fw(\)\()p Fv(k)s(;)17 b(x)703 1956 y Fu(1)743 1941 y Fw(\))27 b(:=)939 1847 y Fo(X)944 2058 y Ft(n)p Fl(2)p Fn(Z)1099 1941 y Fv(e)1144 1900 y Fl(\000)p Ft(ik)r Fu(\()p Ft(x)1329 1909 y Fm(1)1364 1900 y Fu(+2)p Ft(\031)r(n)p Fu(\))1571 1941 y Fv(f)11 b Fw(\()p Fv(x)1723 1956 y Fu(1)1785 1941 y Fw(+)22 b(2)p Fv(\031)t(n)p Fw(\))p Fv(;)212 b(k)30 b Fs(2)e Fw([)p Fs(\000)2615 1902 y Fu(1)p 2615 1918 36 4 v 2615 1976 a(2)2661 1941 y Fv(;)2715 1902 y Fu(1)p 2715 1918 V 2715 1976 a(2)2760 1941 y Fw(])p Fv(;)33 b(x)2902 1956 y Fu(1)2970 1941 y Fs(2)28 b Fw([)p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(])p Fv(:)120 2234 y Fw(The)32 b(inclusion)e Fv(f)38 b Fs(2)29 b(S)7 b Fw(\()p Fr(R)e Fw(\))38 b(implies)28 b(that)k(the)f(con)m(v)m(ergence)j(is)d (uniform,)f(so)i(w)m(e)g(easily)f(see)120 2354 y(that)f Fs(U)391 2369 y Fu(1)431 2354 y Fv(f)38 b Fs(2)28 b Fv(C)688 2318 y Fl(1)763 2354 y Fw(\(\()p Fs(\000)926 2315 y Fu(1)p 926 2331 V 926 2389 a(2)971 2354 y Fv(;)1025 2315 y Fu(1)p 1025 2331 V 1025 2389 a(2)1070 2354 y Fw(\))17 b Fs(\002)g Fw(\()p Fs(\000)p Fv(\031)t(;)g(\031)t Fw(\)\))30 b(and)g(\()p Fs(U)1889 2369 y Fu(1)1929 2354 y Fv(f)11 b Fw(\)\()p Fv(k)s(;)17 b(:)p Fw(\))27 b Fs(2)h Fv(C)2425 2318 y Fl(1)2500 2354 y Fw(\()p Fr(T)p Fw(\))33 b(for)c(\014xed)i Fv(k)g Fs(2)d Fw([)p Fs(\000)3341 2315 y Fu(1)p 3341 2331 V 3341 2389 a(2)3387 2354 y Fv(;)3441 2315 y Fu(1)p 3441 2331 V 3441 2389 a(2)3486 2354 y Fw(].)120 2475 y(Moreo)m(v)m(er,)34 b(if)e(w)m(e)h(de\014ne)587 2673 y(\()p Fv(E)6 b(F)14 b Fw(\)\()p Fv(k)s(;)j(x)1009 2688 y Fu(1)1048 2673 y Fw(\))28 b(:=)f Fv(e)1289 2632 y Ft(ik)r(x)1392 2641 y Fm(1)1430 2673 y Fv(F)14 b Fw(\()p Fv(k)s(;)j(x)1698 2688 y Fu(1)1738 2673 y Fw(\))p Fv(;)211 b(k)31 b Fs(2)d Fw([)p Fs(\000)2304 2634 y Fu(1)p 2304 2650 V 2304 2708 a(2)2350 2673 y Fv(;)2404 2634 y Fu(1)p 2404 2650 V 2404 2708 a(2)2449 2673 y Fw(])p Fv(;)33 b(x)2591 2688 y Fu(1)2659 2673 y Fs(2)28 b Fw([)p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(])p Fv(;)120 2872 y Fw(for)45 b(an)g(arbitrary)g(function)g Fv(F)59 b Fw(on)45 b([)p Fs(\000)1637 2833 y Fu(1)p 1637 2849 V 1637 2906 a(2)1683 2872 y Fv(;)1736 2833 y Fu(1)p 1736 2849 V 1736 2906 a(2)1782 2872 y Fw(])31 b Fs(\002)g Fw([)p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(],)49 b(w)m(e)d(see)h(that) e(\()p Fv(E)6 b Fs(U)3046 2887 y Fu(1)3086 2872 y Fv(f)11 b Fw(\)\()p Fv(:;)17 b(x)3347 2887 y Fu(1)3386 2872 y Fw(\))50 b Fs(2)120 2992 y Fv(C)197 2956 y Fl(1)272 2992 y Fw(\()p Fr(T)373 2956 y Fl(0)399 2992 y Fw(\),)31 b(whic)m(h)g(is)f (b)m(y)h(de\014nition)f(the)h(space)g(of)f(functions)h(in)e Fv(C)2573 2956 y Fl(1)2648 2992 y Fw(\()p Fs(\000)2773 2953 y Fu(1)p 2773 2970 V 2773 3027 a(2)2819 2992 y Fv(;)2872 2953 y Fu(1)p 2872 2970 V 2872 3027 a(2)2918 2992 y Fw(\))h(whic)m(h)h (can)g(b)s(e)120 3113 y(extended)h(1-p)s(erio)s(dically)26 b(to)k(a)g(function)g(in)f Fv(C)1903 3077 y Fl(1)1978 3113 y Fw(\()p Fr(R)5 b Fw(\).)48 b(This)31 b(pro)m(v)m(es)h(one)e (inclusion)f(of)h(the)120 3233 y(follo)m(wing)120 3418 y Fp(Lemma)37 b(1.3.1.)132 3589 y Fs(U)194 3604 y Fu(1)233 3589 y Fw(\()p Fs(S)7 b Fw(\()p Fr(R)f Fw(\)\))34 b(=)27 b Fs(f)p Fv(F)41 b Fs(2)28 b Fv(C)981 3548 y Fl(1)1056 3589 y Fw(\(\()p Fs(\000)1219 3550 y Fu(1)p 1219 3566 V 1219 3624 a(2)1265 3589 y Fv(;)1318 3550 y Fu(1)p 1318 3566 V 1318 3624 a(2)1364 3589 y Fw(\))22 b Fs(\002)g Fw(\()p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(\)\))28 b(:)g Fv(F)14 b Fw(\()p Fv(k)s(;)j(:)p Fw(\))27 b Fs(2)h Fv(C)2435 3548 y Fl(1)2509 3589 y Fw(\()p Fr(T)p Fw(\))p Fv(;)76 b(k)30 b Fs(2)f Fw([)p Fs(\000)3041 3550 y Fu(1)p 3041 3566 V 3041 3624 a(2)3086 3589 y Fv(;)3140 3550 y Fu(1)p 3140 3566 V 3140 3624 a(2)3185 3589 y Fw(])p Fv(;)1959 3735 y Fw(\()p Fv(E)6 b(F)14 b Fw(\)\()p Fv(:;)j(x)2354 3750 y Fu(1)2393 3735 y Fw(\))27 b Fs(2)h Fv(C)2629 3693 y Fl(1)2704 3735 y Fw(\()p Fr(T)2805 3693 y Fl(0)2832 3735 y Fw(\))p Fv(;)72 b(x)3024 3750 y Fu(1)3091 3735 y Fs(2)28 b Fw([)p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(])p Fs(g)120 3926 y Fj(Pr)-5 b(o)g(of.)48 b Fw(W)-8 b(e)39 b(only)f(ha)m(v)m(e)i(to)e(pro)m(v)m(e)i(")p Fs(\033)p Fw(".)61 b(Let)39 b Fv(F)52 b Fw(b)s(e)39 b(giv)m(en)g(with)f(the)h(ab) s(o)m(v)m(e)g(prop)s(erties,)120 4047 y(then)33 b(w)m(e)h(ha)m(v)m(e)g (to)e(\014nd)h Fv(f)38 b Fs(2)29 b(S)7 b Fw(\()p Fr(R)e Fw(\))39 b(suc)m(h)34 b(that)369 4168 y Fo(X)374 4380 y Ft(n)p Fl(2)p Fn(Z)529 4263 y Fv(e)574 4222 y Fl(\000)p Ft(ik)r Fu(\()p Ft(x)759 4231 y Fm(1)794 4222 y Fu(+2)p Ft(\031)r(n)p Fu(\))1001 4263 y Fv(f)11 b Fw(\()p Fv(x)1153 4278 y Fu(1)1215 4263 y Fw(+)22 b(2)p Fv(\031)t(n)p Fw(\))28 b(=)f Fv(F)14 b Fw(\()p Fv(k)s(;)j(x)1916 4278 y Fu(1)1955 4263 y Fw(\))p Fv(;)212 b(k)31 b Fs(2)d Fw([)p Fs(\000)2522 4224 y Fu(1)p 2522 4240 V 2522 4297 a(2)2568 4263 y Fv(;)2621 4224 y Fu(1)p 2621 4240 V 2621 4297 a(2)2667 4263 y Fw(])p Fv(;)33 b(x)2809 4278 y Fu(1)2877 4263 y Fs(2)28 b Fw([)p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(])p Fv(:)120 4563 y Fw(Multiplying)40 b(b)s(oth)i(sides)h(b)m(y)g Fv(e)1335 4527 y Ft(ik)r(x)1438 4536 y Fm(1)1519 4563 y Fw(w)m(e)g(see)h(that)e Fv(f)11 b Fw(\()p Fv(x)2213 4578 y Fu(1)2281 4563 y Fw(+)29 b(2)p Fv(\031)t(n)p Fw(\))42 b(m)m(ust)g(b)s(e)h(the)g(\()p Fs(\000)p Fv(n)p Fw(\)-th)120 4684 y(F)-8 b(ourier)31 b(co)s(e\016cien)m(t)j(of)e(\()p Fv(E)6 b(F)14 b Fw(\)\()p Fv(:;)j(x)1419 4699 y Fu(1)1458 4684 y Fw(\),)32 b(whence)690 4954 y Fv(f)11 b Fw(\()p Fv(x)842 4969 y Fu(1)904 4954 y Fw(+)22 b(2)p Fv(\031)t(n)p Fw(\))27 b(=)1337 4819 y Fo(Z)1437 4845 y Fu(1)p Ft(=)p Fu(2)1392 5044 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1574 4954 y Fv(F)14 b Fw(\()p Fv(k)s(;)j(x)1842 4969 y Fu(1)1881 4954 y Fw(\))p Fv(e)1964 4913 y Ft(ik)r Fu(\()p Ft(x)2094 4922 y Fm(1)2129 4913 y Fu(+2)p Ft(\031)r(n)p Fu(\))2353 4954 y Fv(dk)s(;)211 b(n)28 b Fs(2)g Fr(Z)p Fv(;)292 b Fw(\(1.3.1\))120 5222 y(for)38 b(ev)m(ery)i(\014xed)f Fv(x)834 5237 y Fu(1)911 5222 y Fs(2)f Fw([)p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(].)61 b(This)38 b(equation)g(uniquely)g (de\014nes)i(a)e(function)g Fv(f)49 b Fw(on)38 b Fr(R)120 5342 y Fw(and)33 b(it)g(su\016ces)i(to)e(pro)m(v)m(e)i(that)e(it)f(b)s (elongs)h(to)g(the)h(Sc)m(h)m(w)m(artz)h(class.)46 b(This)34 b(follo)m(ws)e(easily)120 5463 y(from)f(the)i(prop)s(erties)g(of)f Fv(F)14 b Fw(.)p 3469 5463 4 66 v 3473 5400 59 4 v 3473 5463 V 3531 5463 4 66 v 1806 5712 a(7)p eop %%Page: 8 13 8 12 bop 266 407 a Fw(W)-8 b(e)42 b(ha)m(v)m(e)h(seen)g(in)e(the)h (preceeding)h(pro)s(of)d(that)i(for)f Fv(f)54 b Fs(2)43 b(S)7 b Fw(\()p Fr(R)f Fw(\))48 b(and)41 b Fv(x)3054 422 y Fu(1)3137 407 y Fs(2)j Fw([)p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(])120 527 y(\(1.3.1\))24 b(are)g(the)h(F)-8 b(ourier)23 b(co)s(e\016cien)m(ts)j(of)d(\()p Fv(E)6 b(F)14 b Fw(\)\()p Fv(:;)j(x)2047 542 y Fu(1)2087 527 y Fw(\))27 b(=)h(\()p Fv(E)22 b Fs(U)2450 542 y Fu(1)2490 527 y Fv(f)11 b Fw(\)\()p Fv(:;)17 b(x)2751 542 y Fu(1)2790 527 y Fw(\),)26 b(so)f(b)m(y)g(P)m(arsev)-5 b(al's)120 648 y(equalit)m(y)374 768 y Fo(Z)474 794 y Fu(1)p Ft(=)p Fu(2)429 994 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)611 904 y Fs(j)p Fw(\()p Fs(U)739 919 y Fu(1)778 904 y Fv(f)11 b Fw(\)\()p Fv(k)s(;)17 b(x)1066 919 y Fu(1)1105 904 y Fw(\))p Fs(j)1171 862 y Fu(2)1227 904 y Fv(dk)31 b Fw(=)1463 809 y Fo(X)1468 1021 y Ft(n)p Fl(2)p Fn(Z)1624 904 y Fs(j)p Fv(f)11 b Fw(\()p Fv(x)1804 919 y Fu(1)1865 904 y Fw(+)22 b(2)p Fv(\031)t(n)p Fw(\))p Fs(j)2195 862 y Fu(2)2234 904 y Fv(;)212 b(x)2528 919 y Fu(1)2595 904 y Fs(2)28 b Fw([)p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(])p Fv(:)255 b Fw(\(1.3.2\))120 1187 y(In)m(tegration)32 b(with)g(resp)s(ect)i(to)e Fv(x)1354 1202 y Fu(1)1422 1187 y Fs(2)c Fw([)p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(])33 b(yields)235 1313 y Fo(Z)334 1339 y Fu(1)p Ft(=)p Fu(2)290 1539 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)471 1313 y Fo(Z)571 1339 y Ft(\031)527 1539 y Fl(\000)p Ft(\031)645 1449 y Fs(j)p Fw(\()p Fs(U)773 1464 y Fu(1)813 1449 y Fv(f)11 b Fw(\)\()p Fv(k)s(;)17 b(x)1101 1464 y Fu(1)1140 1449 y Fw(\))p Fs(j)1206 1408 y Fu(2)1262 1449 y Fv(dx)1368 1464 y Fu(1)1424 1449 y Fv(dk)30 b Fw(=)1660 1354 y Fo(X)1665 1566 y Ft(n)p Fl(2)p Fn(Z)1820 1313 y Fo(Z)1920 1339 y Ft(\031)r Fu(+2)p Ft(n\031)1876 1539 y Fl(\000)p Ft(\031)r Fu(+2)p Ft(n\031)2170 1449 y Fs(j)p Fv(f)11 b Fw(\()p Fv(x)2350 1464 y Fu(1)2389 1449 y Fw(\))p Fs(j)2455 1408 y Fu(2)2511 1449 y Fv(dx)2617 1464 y Fu(1)2684 1449 y Fw(=)2788 1313 y Fo(Z)2843 1539 y Fn(R)2912 1449 y Fs(j)p Fv(f)g Fw(\()p Fv(x)3092 1464 y Fu(1)3131 1449 y Fw(\))p Fs(j)3197 1408 y Fu(2)3253 1449 y Fv(dx)3359 1464 y Fu(1)3398 1449 y Fv(:)3264 1663 y Fw(\(1.3.3\))120 1783 y(This)33 b(allo)m(ws)e(us)i(to)g(extend)h Fs(U)1257 1798 y Fu(1)1329 1783 y Fw(b)m(y)f(con)m(tin)m(uit)m(y)g(to)f(an)h(isometric)e(op)s (erator)1133 2039 y Fs(U)1195 2054 y Fu(1)1263 2039 y Fw(:)d Fv(L)1384 2054 y Fu(2)1423 2039 y Fw(\()p Fr(R)5 b Fw(\))34 b Fs(!)1726 1904 y Fo(Z)1826 1930 y Fu(1)p Ft(=)p Fu(2)1781 2129 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1963 1945 y Fo(M)2130 2039 y Fv(L)2196 2054 y Fu(2)2236 2039 y Fw(\()p Fr(T)p Fw(\))17 b Fv(dk)s(;)120 2293 y Fw(the)33 b Fj(Gelfand)h(tr)-5 b(ansformation)34 b(on)h Fr(R)5 b Fw(.)120 2413 y(Indeed,)37 b Fs(U)529 2428 y Fu(1)603 2413 y Fw(is)d(unitary:)47 b(Being)34 b(isometric,)g Fs(R)p Fw(\()p Fs(U)1998 2428 y Fu(1)2038 2413 y Fw(\))g(is)h(a)f (subspace)j(whic)m(h,)e(according)f(to)120 2550 y(Lemma)d(1.3.1,)h(con) m(tains)h(a)f(dense)i(subset.)45 b(Th)m(us)35 b Fs(R)p Fw(\()p Fs(U)2227 2565 y Fu(1)2267 2550 y Fw(\))27 b(=)2436 2470 y Fo(R)2502 2496 y Fu(1)p Ft(=)p Fu(2)2483 2585 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)2664 2475 y Fo(L)2792 2550 y Fv(L)2858 2565 y Fu(2)2897 2550 y Fw(\()p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(\))g Fv(dk)s Fw(.)120 2689 y(Let)33 b(us)g(no)m(w)h(in)m(v)m(estigate)f(the)g(Gelfand)e (transformation)g(on)i(the)g(Sob)s(olev)f(space)i Fv(H)3326 2652 y Fu(1)3365 2689 y Fw(\()p Fr(R)5 b Fw(\).)120 2809 y(F)-8 b(or)32 b Fv(f)38 b Fs(2)28 b(S)7 b Fw(\()p Fr(R)f Fw(\))38 b(w)m(e)c(ha)m(v)m(e)1068 2999 y(\()p Fs(U)1168 3014 y Fu(1)1208 2999 y Fv(D)1289 3014 y Fu(1)1328 2999 y Fv(f)11 b Fw(\)\()p Fv(k)s(;)17 b(:)p Fw(\))27 b(=)h(\()p Fv(D)1876 3014 y Fu(1)1938 2999 y Fw(+)22 b Fv(k)s Fw(\)\()p Fs(U)2228 3014 y Fu(1)2267 2999 y Fv(f)11 b Fw(\)\()p Fv(k)s(;)17 b(:)p Fw(\))p Fv(;)672 b Fw(\(1.3.4\))120 3188 y(and)33 b(hence)h(using)f(\(1.3.3\))934 3314 y Fo(Z)1034 3340 y Fu(1)p Ft(=)p Fu(2)990 3540 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1171 3450 y Fs(k)p Fw(\()p Fs(U)1321 3465 y Fu(1)1361 3450 y Fv(f)11 b Fw(\)\()p Fv(k)s(;)17 b(:)p Fw(\))p Fs(k)1709 3409 y Fu(2)1709 3475 y Ft(H)1772 3457 y Fm(1)1806 3475 y Fu(\()p Fl(\000)p Ft(\031)r(;\031)r Fu(\))p Ft(;k)2100 3450 y Fv(dk)30 b Fw(=)e Fs(k)p Fv(f)11 b Fs(k)2495 3409 y Fu(2)2495 3475 y Ft(H)2558 3457 y Fm(1)2592 3475 y Fu(\()p Fn(R)p Fu(\))2699 3450 y Fv(:)120 3708 y Fw(Recall)50 b(that)g(the)i(norms)f Fs(k)p Fv(:)p Fs(k)1288 3725 y Ft(H)1351 3706 y Fm(1)1385 3725 y Fu(\()p Fl(\000)p Ft(\031)r(;\031)r Fu(\))p Ft(;k)1714 3708 y Fw(w)m(ere)h(de\014ned)h(in)d(\(1.2.7\))g(and)i(are)f(m)m(utually)120 3854 y(equiv)-5 b(alen)m(t,)46 b(uniformly)41 b(in)i Fv(k)49 b Fs(2)e Fw([)p Fs(\000)1531 3815 y Fu(1)p 1531 3831 36 4 v 1531 3888 a(2)1577 3854 y Fv(;)1630 3815 y Fu(1)p 1630 3831 V 1630 3888 a(2)1676 3854 y Fw(].)76 b(Hence)2107 3774 y Fo(R)2173 3800 y Fu(1)p Ft(=)p Fu(2)2154 3888 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)2335 3854 y Fs(k)p Fv(F)14 b Fw(\()p Fv(k)s(;)j(:)p Fw(\))p Fs(k)2713 3818 y Fu(2)2713 3885 y Ft(H)2776 3866 y Fm(1)2810 3885 y Fu(\()p Fl(\000)p Ft(\031)r(;\031)r Fu(\))p Ft(;k)3104 3854 y Fv(dk)46 b Fw(de\014nes)120 4017 y(an)32 b(equiv)-5 b(alen)m(t,)33 b(non-standard)g(norm)e(in)h(the)h(space)2140 3937 y Fo(R)2206 3963 y Fu(1)p Ft(=)p Fu(2)2187 4052 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)2368 3942 y Fo(L)2496 4017 y Fv(H)2585 3981 y Fu(1)2624 4017 y Fw(\()p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(\))g Fv(dk)s Fw(.)120 4156 y(It)31 b(follo)m(w)e(from)h(Lemma)f(1.3.1)h(that)h(\()p Fs(U)1620 4171 y Fu(1)1660 4156 y Fv(f)11 b Fw(\)\()p Fv(k)s(;)17 b(:)p Fw(\))27 b Fs(2)h Fv(H)2168 4119 y Fu(1)2207 4156 y Fw(\()p Fr(T)p Fw(\),)35 b(whence)d Fs(U)2809 4171 y Fu(1)2849 4156 y Fv(f)42 b Fw(b)s(elongs)30 b(to)h(the)120 4276 y(subspace)1419 4327 y Fo(Z)1518 4353 y Fu(1)p Ft(=)p Fu(2)1474 4553 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1655 4368 y Fo(M)1823 4463 y Fv(H)1912 4422 y Fu(1)1951 4463 y Fw(\()p Fr(T)p Fw(\))17 b Fv(dk)s(;)1025 b Fw(\(1.3.5\))120 4697 y(By)47 b(con)m(tin)m(uit)m(y)-8 b(,)50 b Fs(U)846 4712 y Fu(1)933 4697 y Fw(maps)c(the)h(Hilb)s(ert)e (space)j Fv(H)2098 4661 y Fu(1)2137 4697 y Fw(\()p Fr(R)5 b Fw(\))52 b(isometrically)44 b(in)m(to)i(the)h(space)120 4818 y(\(1.3.5\))o(,)42 b(endo)m(w)m(ed)f(with)f(the)g(non-standard)f (norm.)64 b(Since)40 b Fs(U)2513 4833 y Fu(1)2552 4818 y Fw(\()p Fs(S)7 b Fw(\()p Fr(R)f Fw(\)\))45 b(is)39 b(dense)i(in)e(this)120 4938 y(space)34 b(with)e(resp)s(ect)i(to)e(the) h(non-standard)g(norm)e(\(see)j(Lemma)d(1.3.1\),)h(w)m(e)i(obtain)1140 5200 y Fs(U)1202 5215 y Fu(1)1242 5200 y Fw(\()p Fv(H)1369 5159 y Fu(1)1408 5200 y Fw(\()p Fr(R)t Fw(\)\))g(=)1725 5064 y Fo(Z)1824 5091 y Fu(1)p Ft(=)p Fu(2)1780 5290 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1961 5105 y Fo(M)2129 5200 y Fv(H)2218 5159 y Fu(1)2256 5200 y Fw(\()p Fr(T)p Fw(\))17 b Fv(dk)120 5463 y Fw(and)33 b(\(1.3.4\))f(holds)g(for)g(an)m (y)h Fv(f)39 b Fs(2)28 b Fv(H)1477 5427 y Fu(1)1516 5463 y Fw(\()p Fr(R)t Fw(\).)1806 5712 y(8)p eop %%Page: 9 14 9 13 bop 120 407 a Fp(2.)114 b(The)60 b(Gelfand)f(transformation)f(on)h Fr(R)2043 371 y Fu(2)2043 431 y(+)2108 407 y Fp(.)98 b Fw(W)-8 b(e)52 b(de\014ne)g(the)g Fj(Gelfand)f(tr)-5 b(ans-)120 527 y(formation)53 b(on)g(the)g(halfplane)59 b Fw(as)53 b(op)s(erator)f(from)g Fv(L)2238 542 y Fu(2)2278 527 y Fw(\()p Fr(R)2382 491 y Fu(2)2382 552 y(+)2447 527 y Fw(\))62 b(=)g Fv(L)2751 542 y Fu(2)2791 527 y Fw(\()p Fr(R)5 b Fw(\))41 b Fs(\012)c Fv(L)3154 542 y Fu(2)3194 527 y Fw(\()p Fr(R)3298 542 y Fu(+)3363 527 y Fw(\))52 b(to)120 591 y Fo(R)186 617 y Fu(1)p Ft(=)p Fu(2)167 705 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)349 596 y Fo(L)476 671 y Fv(L)542 686 y Fu(2)582 671 y Fw(\(\005\))17 b Fv(dk)30 b Fw(=)983 591 y Fo(R)1050 617 y Fu(1)p Ft(=)p Fu(2)1030 705 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1212 596 y Fo(L)1339 671 y Fv(L)1405 686 y Fu(2)1445 671 y Fw(\()p Fr(T)p Fw(\))17 b Fv(dk)28 b Fs(\012)22 b Fv(L)1896 686 y Fu(2)1936 671 y Fw(\()p Fr(R)2040 686 y Fu(+)2105 671 y Fw(\))33 b(b)m(y)1564 891 y Fs(U)38 b Fw(:=)28 b Fs(U)1857 906 y Fu(1)1918 891 y Fs(\012)23 b Fv(I)8 b(:)120 1111 y Fw(F)-8 b(or)32 b Fv(u)27 b Fs(2)h(S)7 b Fw(\()p Fr(R)644 1075 y Fu(2)644 1135 y(+)709 1111 y Fw(\))32 b(w)m(e)i(can)f(rewrite)f (this)g(de\014nition)g(as)387 1355 y(\()p Fs(U)10 b Fv(u)p Fw(\)\()p Fv(k)s(;)17 b(x)p Fw(\))28 b(=)951 1260 y Fo(X)956 1472 y Ft(n)p Fl(2)p Fn(Z)1112 1355 y Fv(e)1157 1314 y Fl(\000)p Ft(ik)r Fu(\()p Ft(x)1342 1323 y Fm(1)1376 1314 y Fu(+2)p Ft(\031)r(n)p Fu(\))1584 1355 y Fv(u)p Fw(\()p Fv(x)1733 1370 y Fu(1)1794 1355 y Fw(+)22 b(2)p Fv(\031)t(n;)17 b(x)2157 1370 y Fu(2)2197 1355 y Fw(\))p Fv(;)211 b(k)31 b Fs(2)d Fw([)p Fs(\000)2763 1315 y Fu(1)p 2763 1332 36 4 v 2763 1389 a(2)2809 1355 y Fv(;)2863 1315 y Fu(1)p 2863 1332 V 2863 1389 a(2)2908 1355 y Fw(])p Fv(;)33 b(x)28 b Fs(2)g Fw(\005)p Fv(:)120 1698 y Fw(It)33 b(is)f(clear)g(that)g Fs(U)43 b Fw(is)32 b(a)g(unitary)g(op)s(erator)g (from)g Fv(L)2086 1713 y Fu(2)2125 1698 y Fw(\()p Fr(R)2229 1662 y Fu(2)2229 1722 y(+)2294 1698 y Fw(\))h(to)2484 1617 y Fo(R)2550 1644 y Fu(1)p Ft(=)p Fu(2)2531 1732 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)2713 1623 y Fo(L)2840 1698 y Fv(L)2906 1713 y Fu(2)2946 1698 y Fw(\(\005\))17 b Fv(dk)s Fw(.)120 1836 y(Let)33 b(us)g(in)m(v)m(estigate)g(the)g (Gelfand)e(transformation)f(on)j(the)g(Sob)s(olev)f(space)i Fv(H)3121 1800 y Fu(1)3160 1836 y Fw(\()p Fr(R)3264 1800 y Fu(2)3264 1861 y(+)3329 1836 y Fw(\).)120 2063 y Fp(Prop)s(osition)59 b(1.3.2.)52 b Fj(One)h(has)g Fs(U)10 b Fw(\()p Fv(H)1682 2027 y Fu(1)1721 2063 y Fw(\()p Fr(R)1825 2027 y Fu(2)1825 2087 y(+)1890 2063 y Fw(\)\))62 b(=)2166 1982 y Fo(R)2232 2009 y Fu(1)p Ft(=)p Fu(2)2213 2097 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)2395 1988 y Fo(L)2547 2038 y Fw(~)2522 2063 y Fv(H)2611 2027 y Fu(1)2650 2063 y Fw(\(\005\))17 b Fv(dk)56 b Fj(and)d(for)g Fv(u)62 b Fs(2)120 2201 y Fv(H)209 2165 y Fu(1)248 2201 y Fw(\()p Fr(R)352 2165 y Fu(2)352 2226 y(+)417 2201 y Fw(\))273 2421 y(\()p Fs(U)10 b Fv(D)464 2436 y Fu(1)503 2421 y Fv(u)p Fw(\)\()p Fv(k)s(;)17 b(:)p Fw(\))27 b(=)h(\()p Fv(D)1048 2436 y Fu(1)1109 2421 y Fw(+)22 b Fv(k)s Fw(\)\()p Fs(U)10 b Fv(u)p Fw(\)\()p Fv(k)s(;)17 b(:)p Fw(\))199 b(\()p Fs(U)10 b Fv(D)2094 2436 y Fu(2)2134 2421 y Fv(u)p Fw(\)\()p Fv(k)s(;)17 b(:)p Fw(\))26 b(=)i Fv(D)2640 2436 y Fu(2)2679 2421 y Fw(\()p Fs(U)10 b Fv(u)p Fw(\)\()p Fv(k)s(;)17 b(:)p Fw(\))p Fv(:)153 b Fw(\(1.3.6\))120 2641 y Fj(Pr)-5 b(o)g(of.)48 b Fw(F)-8 b(or)38 b(functions)h Fv(u)f Fs(2)h Fv(H)1321 2605 y Fu(1)1359 2641 y Fw(\()p Fr(R)1463 2605 y Fu(2)1463 2666 y(+)1528 2641 y Fw(\))g(of)f(the)i(form)d Fv(u)h Fw(=)g Fv(f)f Fs(\012)27 b Fv(g)42 b Fw(with)c Fv(f)49 b Fs(2)39 b Fv(H)3138 2605 y Fu(1)3177 2641 y Fw(\()p Fr(R)5 b Fw(\))p Fv(;)39 b(g)j Fs(2)120 2772 y Fv(H)209 2736 y Fu(1)248 2772 y Fw(\()p Fr(R)352 2787 y Fu(+)417 2772 y Fw(\))37 b(w)m(e)h(ha)m(v)m(e)h(\()p Fs(U)10 b Fv(u)p Fw(\)\()p Fv(k)s(;)17 b(:)p Fw(\))35 b Fs(2)1438 2747 y Fw(~)1413 2772 y Fv(H)1502 2736 y Fu(1)1540 2772 y Fw(\(\005\))j(and)f(\(1.3.6\))g(holds.)57 b(Moreo)m(v)m(er,)41 b(for)36 b(suc)m(h)j Fv(u)e Fw(it)120 2893 y(follo)m(ws)31 b(from)h(\(1.3.3\))g(that)981 3049 y Fo(Z)1080 3075 y Fu(1)p Ft(=)p Fu(2)1036 3274 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1217 3184 y Fs(k)p Fw(\()p Fs(U)10 b Fv(u)p Fw(\)\()p Fv(k)s(;)17 b(:)p Fw(\))p Fs(k)1722 3143 y Fu(2)1722 3210 y Ft(H)1785 3191 y Fm(1)1819 3210 y Fu(\(\005\))p Ft(;k)2006 3184 y Fv(dk)30 b Fw(=)e Fs(k)p Fv(u)p Fs(k)2398 3143 y Fu(2)2398 3214 y Ft(H)2461 3195 y Fm(1)2495 3214 y Fu(\()p Fn(R)2570 3191 y Fm(2)2570 3235 y(+)2621 3214 y Fu(\))2652 3184 y Fv(:)120 3474 y Fw(Recall)33 b(that)g(the)i(norms)f Fs(k)p Fv(:)p Fs(k)1220 3490 y Ft(H)1283 3471 y Fm(1)1317 3490 y Fu(\(\005\))p Ft(;k)1521 3474 y Fw(w)m(ere)i(de\014ned)f(in)f (\(1.2.7\))f(and)h(are)g(m)m(utually)f(equiv-)120 3619 y(alen)m(t,)i(uniformly)d(in)i Fv(k)h Fs(2)c Fw([)p Fs(\000)1250 3580 y Fu(1)p 1250 3596 V 1250 3653 a(2)1296 3619 y Fv(;)1350 3580 y Fu(1)p 1350 3596 V 1350 3653 a(2)1395 3619 y Fw(].)50 b(Hence)1791 3539 y Fo(R)1857 3565 y Fu(1)p Ft(=)p Fu(2)1838 3654 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)2019 3619 y Fs(k)p Fv(F)14 b Fw(\()p Fv(k)s(;)j(:)p Fw(\))p Fs(k)2397 3583 y Fu(2)2397 3650 y Ft(H)2460 3631 y Fm(1)2494 3650 y Fu(\(\005\))p Ft(;k)2681 3619 y Fv(dk)37 b Fw(de\014nes)f(an)f(equiv-) 120 3782 y(alen)m(t,)d(non-standard)h(norm)f(in)f(the)i(space)1782 3702 y Fo(R)1849 3728 y Fu(1)p Ft(=)p Fu(2)1829 3817 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)2011 3708 y Fo(L)2138 3782 y Fv(H)2227 3746 y Fu(1)2266 3782 y Fw(\(\005\))17 b Fv(dk)s Fw(.)120 3921 y(Since)36 b(ev)m(ery)i(function)d Fv(u)e Fs(2)h Fv(H)1302 3885 y Fu(1)1341 3921 y Fw(\()p Fr(R)1445 3885 y Fu(2)1445 3945 y(+)1510 3921 y Fw(\))i(can)g(b)s(e)g (appro)m(ximated)f(in)g(the)i Fv(H)2901 3885 y Fu(1)2940 3921 y Fw(\()p Fr(R)3043 3885 y Fu(2)3043 3945 y(+)3109 3921 y Fw(\)-norm)d(b)m(y)120 4041 y(linear)i(com)m(binations)f(of)i (function)f(of)h(the)h(ab)s(o)m(v)m(e)g(form,)f(it)f(follo)m(ws)g(b)m (y)i(con)m(tin)m(uit)m(y)f(that)120 4161 y Fs(U)43 b Fw(maps)32 b(the)h(Hilb)s(ert)e(space)j Fv(H)1334 4125 y Fu(1)1373 4161 y Fw(\()p Fr(R)1477 4125 y Fu(2)1477 4186 y(+)1542 4161 y Fw(\))e(isometrically)d(in)m(to)j(the)h(space)1415 4324 y Fo(Z)1515 4351 y Fu(1)p Ft(=)p Fu(2)1471 4550 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1652 4365 y Fo(M)1845 4435 y Fw(~)1819 4460 y Fv(H)1908 4419 y Fu(1)1947 4460 y Fw(\(\005\))17 b Fv(dk)s(;)120 4749 y Fw(endo)m(w)m(ed)53 b(with)d(the)h(non-standard)g(norm,)j(and)d(that)g(\(1.3.6\))f(holds)g (for)g(ev)m(ery)j Fv(u)58 b Fs(2)120 4870 y Fv(H)209 4833 y Fu(1)248 4870 y Fw(\()p Fr(R)352 4833 y Fu(2)352 4894 y(+)417 4870 y Fw(\).)43 b(Finally)-8 b(,)28 b(w)m(e)j(conclude)g (with)f(Lemma)f(1.3.1)h(that)h Fs(R)p Fw(\()p Fs(U)10 b Fw(\))31 b(con)m(tains)f(a)g(set)h(whic)m(h)120 5013 y(is)42 b(dense)h(in)626 4933 y Fo(R)693 4959 y Fu(1)p Ft(=)p Fu(2)673 5048 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)855 4938 y Fo(L)1007 4988 y Fw(~)982 5013 y Fv(H)1071 4977 y Fu(1)1110 5013 y Fw(\(\005\))17 b Fv(dk)44 b Fw(with)e(resp)s(ect)h (to)f(the)g(non-standard)h(norm.)71 b(By)42 b(iso-)120 5147 y(metricit)m(y)31 b(the)i(assertion)g(ab)s(out)f Fs(R)p Fw(\()p Fs(U)10 b Fw(\))33 b(follo)m(ws,)e(and)i(the)g(Prop)s (osition)e(is)h(pro)m(v)m(ed.)p 3469 5147 4 66 v 3473 5085 59 4 v 3473 5147 V 3531 5147 4 66 v 266 5342 a(The)27 b(next)g(assertion)e(sho)m(ws)j(ho)m(w)e(the)g(Gelfand)f (transformation)e(b)s(eha)m(v)m(es)28 b(under)f(m)m(ul-)120 5463 y(tiplication)i(b)m(y)34 b(a)e(p)s(erio)s(dic)f(function)h(on)g (the)h(b)s(oundary)-8 b(.)1806 5712 y(9)p eop %%Page: 10 15 10 14 bop 120 407 a Fp(Lemma)37 b(1.3.3.)42 b Fj(L)-5 b(et)35 b Fv(\033)d Fs(2)c Fv(L)1235 422 y Ft(q)1274 407 y Fw(\()p Fr(T)p Fw(\))38 b Fj(for)c(some)g Fw(1)28 b Fv(<)f(q)32 b Fs(\024)c(1)p Fj(.)44 b(Then)35 b(for)f Fv(u)27 b Fs(2)h Fv(H)3065 371 y Fu(1)3104 407 y Fw(\()p Fr(R)3208 371 y Fu(2)3208 431 y(+)3273 407 y Fw(\))494 570 y Fo(Z)594 596 y Fu(1)p Ft(=)p Fu(2)550 795 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)731 570 y Fo(Z)831 596 y Ft(\031)787 795 y Fl(\000)p Ft(\031)905 705 y Fv(\033)t Fw(\()p Fv(x)1057 720 y Fu(1)1097 705 y Fw(\))p Fs(j)p Fw(\()p Fs(U)10 b Fv(u)p Fw(\)\()p Fv(k)s(;)17 b(x)1558 720 y Fu(1)1597 705 y Fv(;)g Fw(0\))p Fs(j)1756 664 y Fu(2)1811 705 y Fv(dx)1917 720 y Fu(1)1973 705 y Fv(dk)30 b Fw(=)2209 570 y Fo(Z)2264 795 y Fn(R)2333 705 y Fv(\033)t Fw(\()p Fv(x)2485 720 y Fu(1)2525 705 y Fw(\))p Fs(j)p Fv(u)p Fw(\()p 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Fv(x)175 1485 y Fu(1)243 1470 y Fs(2)28 b Fw([)p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(].)43 b(This)33 b(pro)m(v)m(es)h(the)f(Lemma.)p 3469 1470 4 66 v 3473 1408 59 4 v 3473 1470 V 3531 1470 4 66 v 120 1803 a Fq(1.4)161 b(Scattering)53 b(theory)120 2022 y Fw(In)27 b(this)f(section)g(w)m(e)i (summarize)d(the)i(de\014nitions)f(and)g(basic)g(result)h(on)f (scattering)g(theory)-8 b(.)120 2142 y(F)g(or)32 b(pro)s(ofs)g(w)m(e)i (refer)e(to)h([Y)-8 b(a1)o(].)120 2402 y Fp(1.)84 b(Existence)48 b(and)i(completeness)e(of)h(the)f(w)m(a)m(v)m(e)h(op)s(erators.)99 b Fw(Let)42 b Fv(H)3209 2417 y Fu(0)3248 2402 y Fv(;)34 b(H)50 b Fw(b)s(e)120 2522 y(self-adjoin)m(t)36 b(op)s(erators)i(in)f (a)h(Hilb)s(ert)e(space)j Fg(H)p Fw(.)60 b(The)39 b(pro)5 b(jection)37 b(on)m(to)h(the)h(absolutely)120 2643 y(con)m(tin)m(uous) 34 b(part)e(of)h(the)g(sp)s(ectrum)g(and)g(the)h(unitary)e(group)h(of)f Fv(H)2698 2658 y Fu(0)2770 2643 y Fw(are)h(denoted)h(b)m(y)g Fv(P)3501 2658 y Fu(0)120 2763 y Fw(and)e Fv(U)375 2778 y Fu(0)414 2763 y Fw(\()p Fv(t)p Fw(\))c(=)g(exp)q(\()p Fs(\000)p Fv(itH)1070 2778 y Fu(0)1110 2763 y Fw(\),)j(resp)s(ectiv)m (ely)-8 b(.)45 b(Moreo)m(v)m(er,)33 b Fg(H)2294 2712 y Fu(\()p Ft(ac)p Fu(\))2294 2788 y(0)2448 2763 y Fw(:=)28 b Fs(R)p Fw(\()p Fv(P)2764 2778 y Fu(0)2804 2763 y Fw(\).)43 b(F)-8 b(or)30 b(the)i(similar)120 2884 y(ob)5 b(jects)34 b(related)e(to)g(the)h(op)s(erator)f Fv(H)40 b Fw(w)m(e)34 b(omit)c(the)j(index)g("0".)120 3004 y(In)g(case)g(of)f(existence,)j (the)e(limit)1019 3224 y Fv(W)1111 3239 y Fl(\006)1170 3224 y Fw(\()p Fv(H)r(;)17 b(H)1416 3239 y Fu(0)1455 3224 y Fw(\))27 b(:=)h Fv(s)22 b Fs(\000)66 b Fw(lim)1819 3283 y Ft(t)p Fl(!\0061)2057 3224 y Fv(U)10 b Fw(\()p Fs(\000)p Fv(t)p Fw(\))p Fv(U)2387 3239 y Fu(0)2428 3224 y Fw(\()p Fv(t)p Fw(\))p Fv(P)2602 3239 y Fu(0)120 3477 y Fw(is)32 b(called)f(the)i Fj(wave)i(op)-5 b(er)g(ator)43 b Fw(for)32 b(the)h(pair)e Fv(H)r(;)i(H)2035 3492 y Fu(0)2107 3477 y Fw(and)g(the)g(sign)f Fs(\006)p Fw(.)120 3598 y(Th)m(us)i(the)f(elemen)m(ts)g Fv(u)27 b Fw(=)h Fv(W)1213 3613 y Fl(\006)1272 3598 y Fw(\()p Fv(H)r(;)17 b(H)1518 3613 y Fu(0)1557 3598 y Fw(\))p Fv(u)1651 3556 y Fl(\006)1651 3622 y Fu(0)1737 3598 y Fs(2)28 b(R)p Fw(\()p Fv(W)2045 3613 y Fl(\006)2105 3598 y Fw(\()p Fv(H)r(;)17 b(H)2351 3613 y Fu(0)2390 3598 y Fw(\)\),)32 b Fv(u)2581 3556 y Fl(\006)2581 3622 y Fu(0)2667 3598 y Fs(2)c Fg(H)p Fw(,)33 b(satisfy)1252 3817 y(lim)1209 3877 y Ft(t)p Fl(!\0061)1448 3817 y Fs(k)p Fv(U)10 b Fw(\()p Fv(t)p Fw(\))p Fv(u)22 b Fs(\000)g Fv(U)1928 3832 y Fu(0)1968 3817 y Fw(\()p Fv(t)p Fw(\))p Fv(u)2135 3776 y Fl(\006)2135 3842 y Fu(0)2194 3817 y Fs(k)27 b Fw(=)h(0)p Fv(:)813 b Fw(\(1.4.1\))120 4102 y(The)44 b(w)m(a)m(v)m(e)i(op)s(erators)d(are)g (partial)e(isometries)h(with)h(initial)d(space)k Fg(H)2857 4051 y Fu(\()p Ft(ac)p Fu(\))2857 4127 y(0)2984 4102 y Fw(.)76 b(One)44 b(easily)120 4223 y(establishes)33 b(the)g Fj(intertwining)h(pr)-5 b(op)g(erty)1156 4442 y Fv(W)1248 4457 y Fl(\006)1307 4442 y Fw(\()p Fv(H)r(;)17 b(H)1553 4457 y Fu(0)1592 4442 y Fw(\))g Fv(H)1728 4457 y Fu(0)1794 4442 y Fw(=)28 b Fv(H)c(W)2095 4457 y Fl(\006)2154 4442 y Fw(\()p Fv(H)r(;)17 b(H)2400 4457 y Fu(0)2439 4442 y Fw(\))p Fv(:)120 4662 y Fw(It)25 b(follo)m(ws)e(that)h(the)h (subspace)i Fs(R)p Fw(\()p Fv(W)1507 4677 y Fl(\006)1566 4662 y Fw(\()p Fv(H)r(;)17 b(H)1812 4677 y Fu(0)1851 4662 y Fw(\)\))25 b(reduces)h Fv(H)32 b Fw(and)25 b(that)f(the)h(w)m(a) m(v)m(e)i(op)s(erator)120 4783 y(pro)m(vides)37 b(a)f(unitary)g(equiv) -5 b(alence)36 b(b)s(et)m(w)m(een)j(the)d(part)g(of)g Fv(H)44 b Fw(on)36 b(this)g(subspace)i(and)f(the)120 4903 y(absolutely)32 b(con)m(tin)m(uous)h(part)g(of)f Fv(H)1476 4918 y Fu(0)1515 4903 y Fw(.)43 b(In)33 b(particular,)1333 5123 y Fs(R)p Fw(\()p Fv(W)1547 5138 y Fl(\006)1607 5123 y Fw(\()p Fv(H)r(;)17 b(H)1853 5138 y Fu(0)1892 5123 y Fw(\)\))27 b Fs(\032)i Fg(H)2173 5082 y Fu(\()p Ft(ac)p Fu(\))2299 5123 y Fv(:)938 b Fw(\(1.4.2\))120 5342 y(The)33 b(w)m(a)m(v)m(e)h(op)s(erator)d Fv(W)1042 5357 y Fl(\006)1101 5342 y Fw(\()p Fv(H)r(;)17 b(H)1347 5357 y Fu(0)1386 5342 y Fw(\))32 b(is)f(said)h(to)f(b)s(e)h Fj(c)-5 b(omplete)39 b Fw(if)30 b(equalit)m(y)i(holds)f(in)h(\(1.4.2\))o(.)120 5463 y(It)26 b(is)g(easy)h(to)f(see)h(that)f(the)h(completeness)g(of)e Fv(W)1926 5478 y Fl(\006)1985 5463 y Fw(\()p Fv(H)r(;)17 b(H)2231 5478 y Fu(0)2270 5463 y Fw(\))26 b(is)g(equiv)-5 b(alen)m(t)26 b(to)g(the)g(existence)1781 5712 y(10)p eop %%Page: 11 16 11 15 bop 120 407 a Fw(of)23 b Fv(W)314 422 y Fl(\006)373 407 y Fw(\()p Fv(H)492 422 y Fu(0)532 407 y Fv(;)17 b(H)8 b Fw(\).)39 b(Th)m(us,)27 b(if)c(the)h(w)m(a)m(v)m(e)i(op)s(erator)d Fv(W)1982 422 y Fl(\006)2041 407 y Fw(\()p Fv(H)r(;)17 b(H)2287 422 y Fu(0)2326 407 y Fw(\))24 b(exists)g(and)g(is)f (complete,)i(then)120 527 y(the)33 b(absolutely)f(con)m(tin)m(uous)h (parts)g(of)f Fv(H)1682 542 y Fu(0)1754 527 y Fw(and)h Fv(H)40 b Fw(are)32 b(unitarily)f(equiv)-5 b(alen)m(t.)120 648 y(Let)32 b(us)h(conclude)g(this)e(brief)h(o)m(v)m(erview)h(with)f (an)g(imp)s(ortan)m(t)e(criterion)h(for)h(the)g(existence)120 768 y(and)h(completeness)g(of)f(the)h(w)m(a)m(v)m(e)i(op)s(erators,)d (the)h(Birman-Kre)-11 b(\025)-38 b(\020n)30 b(theorem.)120 971 y Fp(Theorem)40 b(1.4.1.)k Fj(L)-5 b(et)37 b Fv(H)1154 986 y Fu(0)1194 971 y Fv(;)c(H)45 b Fj(b)-5 b(e)37 b(self-adjoint)f(op) -5 b(er)g(ators)37 b(on)g(a)g(Hilb)-5 b(ert)38 b(sp)-5 b(ac)g(e)36 b Fg(H)h Fj(such)120 1092 y(that)e Fw(\()p Fv(H)29 b Fs(\000)22 b Fv(z)t Fw(\))653 1056 y Fl(\000)p Fu(1)769 1092 y Fs(\000)g Fw(\()p Fv(H)987 1107 y Fu(0)1048 1092 y Fs(\000)f Fv(z)t Fw(\))1233 1056 y Fl(\000)p Fu(1)1363 1092 y Fj(is)34 b(tr)-5 b(ac)g(e)35 b(class)f(for)g(some)g Fv(z)e Fs(2)c Fv(\032)p Fw(\()p Fv(H)2683 1107 y Fu(0)2723 1092 y Fw(\))21 b Fs(\\)h Fv(\032)p Fw(\()p Fv(H)8 b Fw(\))p Fj(.)44 b(Then)34 b(the)120 1212 y(wave)g(op)-5 b(er)g(ators)35 b Fv(W)875 1227 y Fl(\006)934 1212 y Fw(\()p Fv(H)r(;)17 b(H)1180 1227 y Fu(0)1219 1212 y Fw(\))34 b Fj(exist)h(and)f(ar)-5 b(e)35 b(c)-5 b(omplete.)120 1472 y Fp(2.)48 b(The)34 b(scattering)e(op)s(erator.)98 b Fw(W)-8 b(e)29 b(use)g(the)g(same)g(notation)e(as)i(ab)s(o)m(v)m(e)h (and)e(assume)120 1592 y(that)36 b(b)s(oth)h(w)m(a)m(v)m(e)h(op)s (erators)f Fv(W)1340 1607 y Fl(\006)1399 1592 y Fw(\()p Fv(H)r(;)17 b(H)1645 1607 y Fu(0)1684 1592 y Fw(\))37 b(exist.)56 b(In)37 b(this)f(case)i(the)f Fj(sc)-5 b(attering)38 b(op)-5 b(er)g(ator)120 1713 y Fw(is)32 b(de\014ned)i(b)m(y)1041 1833 y Fv(S)6 b Fw(\()p Fv(H)r(;)17 b(H)1353 1848 y Fu(0)1392 1833 y Fw(\))27 b(:=)h Fv(W)1680 1848 y Fu(+)1739 1833 y Fw(\()p Fv(H)r(;)17 b(H)1985 1848 y Fu(0)2024 1833 y Fw(\))2062 1792 y Fl(\003)2118 1833 y Fv(W)2210 1848 y Fl(\000)2269 1833 y Fw(\()p Fv(H)r(;)g(H)2515 1848 y Fu(0)2554 1833 y Fw(\))p Fv(:)120 2007 y Fw(The)40 b(scattering)e(op)s(erator)g(connects)i(the)g(asymptotics)e(of)g Fv(U)10 b Fw(\()p Fv(t)p Fw(\))p Fv(u)39 b Fw(as)g Fv(t)f Fs(!)g(\0001)h Fw(and)g(as)120 2141 y Fv(t)32 b Fs(!)g Fw(+)p Fs(1)j Fw(in)f(terms)h(of)g(the)g(free)h(problem:)47 b(If)35 b Fv(u)1972 2100 y Fl(\006)1972 2166 y Fu(0)2066 2141 y Fw(satisfy)h(\(1.4.1\))f(and)g(b)s(elong)f(to)h Fg(H)3386 2090 y Fu(\()p Ft(ac)p Fu(\))3386 2166 y(0)3513 2141 y Fw(,)120 2262 y(then)e Fv(S)6 b Fw(\()p Fv(H)r(;)17 b(H)654 2277 y Fu(0)693 2262 y Fw(\))p Fv(u)787 2220 y Fl(\000)787 2286 y Fu(0)873 2262 y Fw(=)28 b Fv(u)1033 2220 y Fu(+)1033 2286 y(0)1091 2262 y Fw(.)120 2396 y(The)h(scattering) f(op)s(erator)f(is)g(a)h(unitary)g(op)s(erator)f(on)h Fg(H)2251 2345 y Fu(\()p Ft(ac)p Fu(\))2251 2420 y(0)2406 2396 y Fw(pro)m(vided)g Fs(R)p Fw(\()p Fv(W)3016 2411 y Fu(+)3076 2396 y Fw(\()p Fv(H)r(;)17 b(H)3322 2411 y Fu(0)3361 2396 y Fw(\)\))27 b(=)120 2516 y Fs(R)p Fw(\()p Fv(W)334 2531 y Fl(\000)394 2516 y Fw(\()p Fv(H)r(;)17 b(H)640 2531 y Fu(0)679 2516 y Fw(\)\).)120 2636 y(By)48 b(the)g(in)m(tert)m(wining)f(prop)s(ert)m(y)h(the)g(scattering)g(op)s (erator)f(comm)m(utes)h(with)f Fv(H)3296 2651 y Fu(0)3383 2636 y Fw(and)120 2757 y(hence)j(decomp)s(oses)f(in)m(to)f(a)g (one-parameter)g(family)e(of)i(op)s(erators)g Fv(S)6 b Fw(\()p Fv(\025)p Fw(;)17 b Fv(H)r(;)g(H)3225 2772 y Fu(0)3264 2757 y Fw(\))p Fv(;)33 b(\025)55 b Fs(2)120 2877 y Fv(\033)t Fw(\()p Fv(H)298 2892 y Fu(0)337 2877 y Fw(\),)42 b(in)d(a)g(represen)m(tation)i(where)g Fv(H)1668 2892 y Fu(0)1747 2877 y Fw(acts)f(diagonal.)62 b(The)41 b(op)s(erators)f Fv(S)6 b Fw(\()p Fv(\025)p Fw(;)17 b Fv(H)r(;)g(H)3464 2892 y Fu(0)3502 2877 y Fw(\))120 2998 y(are)33 b(called)e Fj(sc)-5 b(attering)35 b(matrix)p Fw(.)1781 5712 y(11)p eop %%Page: 12 17 12 16 bop 120 1071 a Fk(Chapter)78 b(2)120 1486 y(De\014nition)g(of)f (the)h(op)6 b(erators.)120 1735 y(Direct)78 b(in)-6 b(tegral)76 b(decomp)6 b(osition)120 2188 y Fw(Throughout)30 b(this)f(c)m(hapter,)i (w)m(e)g(assume)f(without)f(further)g(commen)m(t)g(that)h Fv(\033)h Fw(:)d Fr(R)39 b Fs(7!)27 b Fr(R)40 b Fw(is)120 2308 y(a)32 b(p)s(erio)s(dic)f(function)h(suc)m(h)i(that)f(for)f(some)g (1)c Fv(<)f(q)32 b Fs(\024)c(1)1603 2501 y Fv(\033)j Fs(2)d Fv(L)1849 2516 y Ft(q)1888 2501 y Fw(\()p Fr(T)p Fw(\))p Fv(:)120 2830 y Fq(2.1)161 b(De\014niton)52 b(of)i(the)f(op)t (erators)f Ff(H)2506 2777 y Fw(\()p Fv(\033)t Fw(\))2700 2830 y Fq(and)h Ff(H)3138 2777 y Fw(\()p Fv(\033)t Fw(\))3278 2830 y Fe(\()p Ff(k)5 b Fe(\))120 3058 y Fw(Our)43 b(goal)f(in)h(this)g (section)g(is)g(to)g(de\014ne)i Fv(H)1831 3022 y Fu(\()p Ft(\033)r Fu(\))1975 3058 y Fw(as)f(self-adjoin)m(t)e(op)s(erator)g(on) i(the)g(half-)120 3179 y(plane)29 b(and)g(to)g(decomp)s(ose)h(it)e(in)m (to)h(a)g(direct)g(in)m(tegral)e(of)i(self-adjoin)m(t)f(op)s(erators)h Fv(H)3282 3142 y Fu(\()p Ft(\033)r Fu(\))3383 3179 y Fw(\()p Fv(k)s Fw(\),)120 3299 y Fv(k)i Fs(2)296 3218 y Fo(\002)337 3299 y Fs(\000)424 3260 y Fu(1)p 424 3276 36 4 v 424 3333 a(2)470 3299 y Fv(;)524 3260 y Fu(1)p 524 3276 V 524 3333 a(2)569 3218 y Fo(\003)610 3299 y Fw(,)i(on)g(the)g(halfstrip.)120 3428 y(The)40 b(de\014nition)f(of)g Fv(H)975 3392 y Fu(\()p Ft(\033)r Fu(\))1116 3428 y Fw(and)g Fv(H)1401 3392 y Fu(\()p Ft(\033)r Fu(\))1502 3428 y Fw(\()p Fv(k)s Fw(\))h(in)e(Subsection)j(1)e(and)g(2,)i(resp)s(ectiv)m (ely)-8 b(,)43 b(will)37 b(b)s(e)120 3548 y(giv)m(en)26 b(via)f(quadratic)g(forms.)40 b(This)26 b(allo)m(ws)f(us)h(to)f (include)g(a)h(wide)f(class)h(of)f(functions)h Fv(\033)t Fw(.)41 b(If)120 3669 y Fv(\033)35 b Fw(is)c(smo)s(oth)f(the)i (functions)f(in)g(the)g(domain)f(of)h(the)g(op)s(erator)g Fv(H)2618 3633 y Fu(\()p Ft(\033)r Fu(\))2750 3669 y Fw(satisfy)h(a)f(b)s(oundary)120 3789 y(condition)g(of)h(third)g(t)m (yp)s(e.)120 3910 y(Finally)-8 b(,)27 b(in)h(Subsection)h(3)g(w)m(e)h (use)f(the)h(Blo)s(c)m(h-Flo)s(quet)d(theory)i(\(i.e.,)g(the)g(Gelfand) f(trans-)120 4030 y(formation)41 b(from)h(Section)h(1.3\))g(to)g (decomp)s(ose)h Fv(H)2098 3994 y Fu(\()p Ft(\033)r Fu(\))2242 4030 y Fw(in)m(to)f(the)h(direct)f(in)m(tegral)f(of)h(the)120 4150 y(op)s(erators)38 b Fv(H)646 4114 y Fu(\()p Ft(\033)r Fu(\))748 4150 y Fw(\()p Fv(k)s Fw(\),)i(indexed)f(b)m(y)g(a)g (parameter)f Fv(k)j Fw(\("quasimomen)m(tum"\))c(from)g(the)i(in-)120 4271 y(terv)-5 b(all)421 4190 y Fo(\002)462 4271 y Fs(\000)549 4232 y Fu(1)p 549 4248 V 549 4305 a(2)595 4271 y Fv(;)649 4232 y Fu(1)p 649 4248 V 649 4305 a(2)694 4190 y Fo(\003)768 4271 y Fw(\("Brillouin)29 b(zone"\).)120 4535 y Fp(1.)48 b(De\014nition)29 b(of)i(the)g(op)s(erators)g Fv(H)1662 4499 y Fu(\()p Ft(\033)r Fu(\))1794 4535 y Fp(on)g(the)g(halfplane.)98 b Fw(In)27 b(the)h(Hilb)s(ert)d(space)120 4656 y Fv(L)186 4671 y Fu(2)226 4656 y Fw(\()p Fr(R)330 4620 y Fu(2)330 4680 y(+)395 4656 y Fw(\))32 b(consider)h(the)g(quadratic)f(form)735 4859 y Fs(D)s Fw([)p Fv(h)898 4818 y Fu(\()p Ft(\033)r Fu(\))1000 4859 y Fw(])c(:=)f Fv(H)1274 4818 y Fu(1)1313 4859 y Fw(\()p Fr(R)1417 4818 y Fu(2)1417 4883 y(+)1482 4859 y Fw(\))p Fv(;)759 5064 y(h)815 5023 y Fu(\()p Ft(\033)r Fu(\))917 5064 y Fw([)p Fv(u)p Fw(])h(:=)1185 4928 y Fo(Z)1241 5154 y Fn(R)1289 5131 y Fm(2)1289 5175 y(+)1360 5064 y Fs(j)p Fv(D)s(u)p Fw(\()p Fv(x)p Fw(\))p Fs(j)1687 5023 y Fu(2)1742 5064 y Fv(dx)22 b Fw(+)1968 4928 y Fo(Z)2024 5154 y Fn(R)2092 5064 y Fv(\033)t Fw(\()p Fv(x)2244 5079 y Fu(1)2284 5064 y Fw(\))p Fs(j)p Fv(u)p Fw(\()p Fv(x)2499 5079 y Fu(1)2538 5064 y Fv(;)17 b Fw(0\))p Fs(j)2697 5023 y Fu(2)2752 5064 y Fv(dx)2858 5079 y Fu(1)2898 5064 y Fv(:)3264 5000 y Fw(\(2.1.1\))120 5342 y(According)27 b(to)f(Corollary)f(1.2.6)i(the)g(in)m(tegral)1855 5262 y Fo(R)1902 5377 y Fn(R)1971 5342 y Fv(\033)t Fw(\()p Fv(x)2123 5357 y Fu(1)2163 5342 y Fw(\))p Fs(j)p Fv(u)p Fw(\()p Fv(x)2378 5357 y Fu(1)2417 5342 y Fv(;)17 b Fw(0\))p Fs(j)2576 5306 y Fu(2)2631 5342 y Fv(dx)2737 5357 y Fu(1)2803 5342 y Fw(is)26 b(w)m(ell-de\014ned)h(for)120 5463 y Fv(u)g Fs(2)h Fv(H)386 5427 y Fu(1)425 5463 y Fw(\()p Fr(R)529 5427 y Fu(2)529 5487 y(+)594 5463 y Fw(\))h(and)h(strongly)f (sub)s(ordinate)g(to)g(the)h(closed)g(and)f(lo)m(w)m(er)h(semib)s (ounded)f(form)1781 5712 y(12)p eop %%Page: 13 18 13 17 bop 120 407 a Fv(h)176 371 y Fu(\(0\))270 407 y Fw(.)44 b(Hence)34 b(the)f(form)e Fv(h)1085 371 y Fu(\()p Ft(\033)r Fu(\))1219 407 y Fw(is)h(closed)h(and)f(lo)m(w)m(er)h(semib)s (ounded)f(and)h(there)g(is)f(a)g(unique)120 527 y(self-adjoin)m(t)f(op) s(erator)h Fv(H)1104 491 y Fu(\()p Ft(\033)r Fu(\))1238 527 y Fw(with)g(domain)f Fs(D)s Fw(\()p Fv(H)2014 491 y Fu(\()p Ft(\033)r Fu(\))2114 527 y Fw(\))i(suc)m(h)h(that)762 707 y Fs(D)s Fw(\()p Fv(H)969 666 y Fu(\()p Ft(\033)r Fu(\))1070 707 y Fw(\))27 b Fs(\032)h Fv(H)1329 666 y Fu(1)1368 707 y Fw(\()p Fr(R)1472 666 y Fu(2)1472 731 y(+)1537 707 y Fw(\))p Fv(;)691 867 y Fw(\()p Fv(H)818 826 y Fu(\()p Ft(\033)r Fu(\))920 867 y Fv(u;)17 b(v)t Fw(\))26 b(=)i Fv(h)1295 826 y Fu(\()p Ft(\033)r Fu(\))1397 867 y Fw([)p Fv(u;)17 b(v)t Fw(])p Fv(;)211 b(u)27 b Fs(2)h(D)s Fw(\()p Fv(H)2224 826 y Fu(\()p Ft(\033)r Fu(\))2325 867 y Fw(\))p Fv(;)44 b(v)31 b Fs(2)e Fv(H)2696 826 y Fu(1)2735 867 y Fw(\()p Fr(R)2838 826 y Fu(2)2838 892 y(+)2903 867 y Fw(\))p Fv(:)3264 785 y Fw(\(2.1.2\))120 1046 y(T)-8 b(aking)41 b Fv(v)46 b Fs(2)d Fv(C)733 1010 y Fl(1)726 1070 y Fu(0)808 1046 y Fw(\()p Fr(R)912 1010 y Fu(2)912 1070 y(+)977 1046 y Fw(\))e(in)g(\(2.1.2\))g(w)m(e)h(see)g (that)g(the)f(distributional)e(Laplacian)g(\001)p Fv(u)i Fw(of)120 1166 y Fv(u)27 b Fs(2)h(D)s Fw(\()p Fv(H)504 1130 y Fu(\()p Ft(\033)r Fu(\))605 1166 y Fw(\))33 b(b)s(elongs)f(to)g Fv(L)1211 1181 y Fu(2)1251 1166 y Fw(\()p Fr(R)1355 1130 y Fu(2)1355 1191 y(+)1420 1166 y Fw(\))g(and)h(that)1521 1352 y Fv(H)1610 1311 y Fu(\()p Ft(\033)r Fu(\))1711 1352 y Fv(u)27 b Fw(=)h Fs(\000)p Fw(\001)p Fv(u:)120 1528 y Fw(The)g(case)g Fv(\033)j Fw(=)d(0)e(corresp)s(onds)j(to)d(the)h Fj(Neumann)j(L)-5 b(aplacian)33 b Fw(on)27 b(the)g(halfplane,)f (whereas)120 1649 y(the)36 b(case)h Fv(\033)g Fs(6)p Fw(=)32 b(0)k(implemen)m(ts)e(a)h(\(generalized\))g Fj(b)-5 b(oundary)38 b(c)-5 b(ondition)37 b(of)g(the)h(thir)-5 b(d)38 b(typ)-5 b(e)p Fw(.)120 1769 y(More)33 b(precisely)-8 b(,)33 b(w)m(e)g(ha)m(v)m(e)120 1912 y Fj(R)-5 b(emark)45 b Fw(2.1.1)p Fj(.)c Fw(It)33 b(is)f(w)m(ell-kno)m(wn)g(that)626 2120 y Fs(D)s Fw(\()p Fv(H)833 2079 y Fu(\()p Ft(\033)r Fu(\))934 2120 y Fw(\))c(=)1103 1980 y Fo(\032)1178 2120 y Fv(u)f Fs(2)h Fv(H)1444 2079 y Fu(2)1483 2120 y Fw(\()p Fr(R)1587 2079 y Fu(2)1587 2145 y(+)1652 2120 y Fw(\))g(:)50 b Fs(\000)1902 2053 y Fv(@)5 b(u)p 1882 2097 152 4 v 1882 2189 a(@)g(x)1993 2204 y Fu(2)2066 2120 y Fw(+)22 b Fv(\033)t(u)27 b Fw(=)h(0)f(on)h Fr(R)33 b Fs(\002)23 b(f)p Fw(0)p Fs(g)2960 1980 y Fo(\033)120 2348 y Fw(pro)m(vided)39 b(that)f Fv(\033)k Fw(is)c(smo)s(oth.)60 b(W)-8 b(e)38 b(are)h(able)e(to)h(pro)m(v)m(e)i(this)e(under)h(the)f(condition)f Fv(\033)42 b Fs(2)120 2468 y Fv(W)226 2432 y Fu(1)212 2493 y Ft(q)265 2468 y Fw(\()p Fr(T)p Fw(\))48 b(for)c(some)h(1)j Fv(<)h(q)j Fs(\024)d(1)p Fw(,)e(where)f Fv(W)1887 2432 y Fu(1)1873 2493 y Ft(q)1926 2468 y Fw(\()p Fr(T)p Fw(\))j(denotes)d (the)f(closure)g(of)f Fv(C)3197 2432 y Fl(1)3272 2468 y Fw(\()p Fr(T)p Fw(\))k(in)120 2589 y Fv(W)226 2552 y Fu(1)212 2613 y Ft(q)265 2589 y Fw(\()p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(\).)120 2853 y Fp(2.)56 b(De\014nition)39 b(of)g(the)h(op)s(erators)g Fv(H)1706 2817 y Fu(\()p Ft(\033)r Fu(\))1807 2853 y Fw(\()p Fv(k)s Fw(\))f Fp(on)h(the)g (halfstrip.)96 b Fw(Let)35 b Fv(k)f Fs(2)d Fw([)p Fs(\000)3341 2814 y Fu(1)p 3341 2830 36 4 v 3341 2888 a(2)3387 2853 y Fv(;)3441 2814 y Fu(1)p 3441 2830 V 3441 2888 a(2)3486 2853 y Fw(].)120 2974 y(In)i(the)g(Hilb)s(ert)e(space)j Fv(L)1073 2989 y Fu(2)1112 2974 y Fw(\(\005\))f(consider)g(the)g (quadratic)f(form)210 3154 y Fs(D)s Fw([)p Fv(h)373 3112 y Fu(\()p Ft(\033)r Fu(\))474 3154 y Fw(\()p Fv(k)s Fw(\)])c(:=)815 3128 y(~)790 3154 y Fv(H)879 3112 y Fu(1)918 3154 y Fw(\(\005\))p Fv(;)234 3361 y(h)290 3320 y Fu(\()p Ft(\033)r Fu(\))392 3361 y Fw(\()p Fv(k)s Fw(\)[)p Fv(u)p Fw(])f(:=)790 3225 y Fo(Z)845 3451 y Fu(\005)919 3280 y Fo(\000)964 3361 y Fs(j)p Fw(\()p Fv(D)1111 3376 y Fu(1)1173 3361 y Fw(+)22 b Fv(k)s Fw(\))p Fv(u)p Fw(\()p Fv(x)p Fw(\))p Fs(j)1578 3320 y Fu(2)1639 3361 y Fw(+)g Fs(j)p Fv(D)1846 3376 y Fu(2)1885 3361 y Fv(u)p Fw(\()p Fv(x)p Fw(\))p Fs(j)2100 3320 y Fu(2)2139 3280 y Fo(\001)2218 3361 y Fv(dx)g Fw(+)2444 3225 y Fo(Z)2543 3251 y Ft(\031)2499 3451 y Fl(\000)p Ft(\031)2618 3361 y Fv(\033)t Fw(\()p Fv(x)2770 3376 y Fu(1)2810 3361 y Fw(\))p Fs(j)p Fv(u)p Fw(\()p Fv(x)3025 3376 y Fu(1)3063 3361 y Fv(;)17 b Fw(0\))p Fs(j)3222 3320 y Fu(2)3278 3361 y Fv(dx)3384 3376 y Fu(1)3423 3361 y Fv(:)3264 3550 y Fw(\(2.1.3\))120 3733 y(According)38 b(to)g(Corollary)f(1.2.5)h(the)h(in)m(tegral)1925 3652 y Fo(R)1992 3679 y Ft(\031)1972 3767 y Fl(\000)p Ft(\031)2091 3733 y Fv(\033)t Fw(\()p Fv(x)2243 3748 y Fu(1)2283 3733 y Fw(\))p Fs(j)p Fv(u)p Fw(\()p Fv(x)2498 3748 y Fu(1)2536 3733 y Fv(;)17 b Fw(0\))p Fs(j)2695 3696 y Fu(2)2751 3733 y Fv(dx)2857 3748 y Fu(1)2935 3733 y Fw(is)38 b(w)m(ell-de\014ned) 120 3874 y(for)i Fv(u)h Fs(2)507 3849 y Fw(~)482 3874 y Fv(H)571 3838 y Fu(1)610 3874 y Fw(\(\005\))f(and)h(strongly)f(sub)s (ordinate)g(to)h(the)g(closed)g(and)f(lo)m(w)m(er)h(semib)s(ounded)120 3994 y(form)33 b Fv(h)408 3958 y Fu(\(0\))502 3994 y Fw(\(0\).)48 b(Because)36 b(of)d(the)i(t)m(w)m(o-sided)g(estimate)e(in) g(Lemma)g(1.2.1)h(b)s(et)m(w)m(een)i Fv(h)3321 3958 y Fu(\(0\))3415 3994 y Fw(\(0\))120 4115 y(and)24 b Fv(h)357 4078 y Fu(\(0\))451 4115 y Fw(\()p Fv(k)s Fw(\))g(the)g(form)f Fv(h)1042 4078 y Fu(\()p Ft(\033)r Fu(\))1144 4115 y Fw(\()p Fv(k)s Fw(\))g(is)h(closed)g(and)g(lo)m(w)m(er)g(semib)s (ounded)f(and)h(there)h(is)e(a)g(unique)120 4235 y(self-adjoin)m(t)31 b(op)s(erator)h Fv(H)1104 4199 y Fu(\()p Ft(\033)r Fu(\))1205 4235 y Fw(\()p Fv(k)s Fw(\))g(with)h(domain)e Fs(D)s Fw(\()p Fv(H)2144 4199 y Fu(\()p Ft(\033)r Fu(\))2244 4235 y Fw(\()p Fv(k)s Fw(\)\))i(suc)m(h)h(that)458 4415 y Fs(D)s Fw(\()p Fv(H)665 4374 y Fu(\()p Ft(\033)r Fu(\))765 4415 y Fw(\()p Fv(k)s Fw(\)\))28 b Fs(\032)1091 4390 y Fw(~)1066 4415 y Fv(H)1155 4374 y Fu(1)1194 4415 y Fw(\(\005\))p Fv(;)387 4575 y Fw(\()p Fv(H)514 4534 y Fu(\()p Ft(\033)r Fu(\))616 4575 y Fw(\()p Fv(k)s Fw(\))p Fv(u;)17 b(v)t Fw(\))26 b(=)i Fv(h)1121 4534 y Fu(\()p Ft(\033)r Fu(\))1222 4575 y Fw(\()p Fv(k)s Fw(\)[)p Fv(u;)17 b(v)t Fw(])p Fv(;)211 b(u)28 b Fs(2)g(D)s Fw(\()p Fv(H)2180 4534 y Fu(\()p Ft(\033)r Fu(\))2280 4575 y Fw(\()p Fv(k)s Fw(\)\))p Fv(;)45 b(v)31 b Fs(2)2718 4550 y Fw(~)2692 4575 y Fv(H)2781 4534 y Fu(1)2820 4575 y Fw(\(\005\))p Fv(:)3264 4493 y Fw(\(2.1.4\))120 4750 y(T)-8 b(aking)45 b Fv(v)53 b Fs(2)d Fv(C)751 4714 y Fl(1)744 4775 y Fu(0)826 4750 y Fw(\(\005\))45 b(in)g(\(2.1.4\))f(w)m(e)j(see)f(that)g(the)f (distributional)e(Laplacian)g(\001)p Fv(u)i Fw(of)120 4870 y Fv(u)27 b Fs(2)h(D)s Fw(\()p Fv(H)504 4834 y Fu(\()p Ft(\033)r Fu(\))605 4870 y Fw(\()p Fv(k)s Fw(\)\))33 b(b)s(elongs)f(to)g Fv(L)1341 4885 y Fu(2)1381 4870 y Fw(\(\005\))g(and)h(that)739 5046 y Fv(H)828 5005 y Fu(\()p Ft(\033)r Fu(\))929 5046 y Fv(u)27 b Fw(=)h(\()p Fv(D)1235 5061 y Fu(1)1296 5046 y Fw(+)22 b Fv(k)s Fw(\))1486 5005 y Fu(2)1526 5046 y Fv(u)g Fw(+)g Fv(D)1786 5005 y Fu(2)1783 5071 y(2)1825 5046 y Fv(u)27 b Fw(=)g Fs(\000)p Fw(\001)p Fv(u)c Fw(+)f(2)p Fv(k)s(D)2530 5061 y Fu(1)2569 5046 y Fv(u)g Fw(+)g Fv(k)2799 5005 y Fu(2)2838 5046 y Fv(u:)120 5222 y Fw(In)j(addition)d(to)i(the)h(Neumann)g(\(if)e Fv(\033)32 b Fw(=)27 b(0\))d(or)g(third)g(t)m(yp)s(e)h(\(if)e Fv(\033)32 b Fs(6)p Fw(=)c(0\))c(b)s(oundary)g(condition)120 5342 y(on)42 b(\()p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(\))28 b Fs(\002)h(f)p Fw(0)p Fs(g)42 b Fw(w)m(e)h(ha)m(v)m(e)g(p)s (erio)s(dic)d(b)s(oundary)i(conditions)f(at)h Fs(f)p Fv(x)2833 5357 y Fu(1)2916 5342 y Fw(=)i Fs(\006)p Fv(\031)t Fs(g)p Fw(.)72 b(More)120 5463 y(precisely)-8 b(,)33 b(w)m(e)g(ha)m(v)m(e)1781 5712 y(13)p eop %%Page: 14 19 14 18 bop 120 407 a Fj(R)-5 b(emark)45 b Fw(2.1.2)p Fj(.)c Fw(As)33 b(in)f(the)h(case)g(of)f(the)h(halfplane)e(\(see)j(Remark)e (2.1.1\))g(one)h(has)447 659 y Fs(D)s Fw(\()p Fv(H)654 618 y Fu(\()p Ft(\033)r Fu(\))755 659 y Fw(\()p Fv(k)s Fw(\)\))27 b(=)1054 519 y Fo(\032)1129 659 y Fv(u)g Fs(2)1331 634 y Fw(~)1306 659 y Fv(H)1395 618 y Fu(2)1434 659 y Fw(\(\005\))g(:)50 b Fs(\000)1795 592 y Fv(@)5 b(u)p 1774 636 152 4 v 1774 727 a(@)g(x)1885 742 y Fu(2)1959 659 y Fw(+)22 b Fv(\033)t(u)27 b Fw(=)h(0)f(on)h(\()p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(\))22 b Fs(\002)g(f)p Fw(0)p Fs(g)3095 519 y Fo(\033)3186 659 y Fv(;)120 916 y Fw(pro)m(vided)33 b(that)f Fv(\033)37 b Fw(is)32 b(smo)s(oth,)g(e.g.) 43 b Fv(\033)32 b Fs(2)c Fv(W)1770 879 y Fu(1)1756 940 y Ft(q)1809 916 y Fw(\()p Fr(T)p Fw(\))36 b(for)c(some)h(1)27 b Fv(<)h(q)j Fs(\024)d(1)p Fw(.)120 1183 y Fp(3.)78 b(Direct)44 b(in)m(tegral)h(decomp)s(osition)g(of)i(the)g(op)s(erator)g Fv(H)2701 1147 y Fu(\()p Ft(\033)r Fu(\))2802 1183 y Fp(.)98 b Fw(In)41 b(the)g(Hilb)s(ert)120 1315 y(space)381 1234 y Fo(R)447 1261 y Fu(1)p Ft(=)p Fu(2)428 1349 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)609 1240 y Fo(L)737 1315 y Fv(L)803 1330 y Fu(2)842 1315 y Fw(\(\005\))17 b Fv(dk)35 b Fw(w)m(e)f(consider)f(the)g(self-adjoin)m(t)e(op)s(erator)1245 1583 y(~)1219 1608 y Fv(H)1308 1567 y Fu(\()p Ft(\033)r Fu(\))1437 1608 y Fw(:=)1568 1472 y Fo(Z)1668 1498 y Fu(1)p Ft(=)p Fu(2)1623 1698 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1805 1513 y Fo(M)1972 1608 y Fv(H)2061 1567 y Fu(\()p Ft(\033)r Fu(\))2162 1608 y Fw(\()p Fv(k)s Fw(\))17 b Fv(dk)s(;)120 1863 y Fw(i.e.,)469 2140 y Fs(D)s Fw(\()612 2115 y(~)587 2140 y Fv(H)676 2099 y Fu(\()p Ft(\033)r Fu(\))777 2140 y Fw(\))28 b(=)946 1970 y Fo(\()1026 2140 y Fv(F)42 b Fs(2)1225 2004 y Fo(Z)1324 2031 y Fu(1)p Ft(=)p Fu(2)1280 2230 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1462 2045 y Fo(M)1629 2140 y Fv(L)1695 2155 y Fu(2)1735 2140 y Fw(\(\005\))17 b Fv(dk)30 b Fw(:)55 b Fv(F)14 b Fw(\()p Fv(k)s Fw(\))27 b Fs(2)h(D)s Fw(\()p Fv(H)2650 2099 y Fu(\()p Ft(\033)r Fu(\))2751 2140 y Fw(\()p Fv(k)s Fw(\)\))g(for)k (a.e.)c Fv(k)s(;)1232 2336 y Fo(Z)1331 2363 y Fu(1)p Ft(=)p Fu(2)1287 2562 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1468 2387 y Fo(\015)1468 2447 y(\015)1524 2472 y Fv(H)1613 2431 y Fu(\()p Ft(\033)r Fu(\))1714 2472 y Fw(\()p Fv(k)s Fw(\))p Fv(F)14 b Fw(\()p Fv(k)s Fw(\))2051 2387 y Fo(\015)2051 2447 y(\015)2106 2407 y Fu(2)2179 2472 y Fv(dk)30 b(<)e Fs(1)2515 2302 y Fo(\))2611 2472 y Fv(;)282 2634 y Fo(\020)367 2719 y Fw(~)342 2744 y Fv(H)431 2703 y Fu(\()p Ft(\033)r Fu(\))532 2744 y Fv(F)609 2634 y Fo(\021)685 2744 y Fw(\()p Fv(k)s Fw(\))g(=)f Fv(H)1035 2703 y Fu(\()p Ft(\033)r Fu(\))1136 2744 y Fw(\()p Fv(k)s Fw(\))p Fv(F)14 b Fw(\()p Fv(k)s Fw(\))p Fv(;)212 b(k)30 b Fs(2)1888 2664 y Fo(\002)1929 2744 y Fs(\000)2016 2705 y Fu(1)p 2016 2721 36 4 v 2016 2779 a(2)2062 2744 y Fv(;)2115 2705 y Fu(1)p 2115 2721 V 2115 2779 a(2)2161 2664 y Fo(\003)2219 2744 y Fv(:)120 2978 y Fw(The)k(op)s(erators)e Fv(H)841 2942 y Fu(\()p Ft(\033)r Fu(\))942 2978 y Fw(\()p Fv(k)s Fw(\))h(are)f(measurable)g (with)g(resp)s(ect)i(to)e Fv(k)f Fs(2)d Fw([)p Fs(\000)2742 2939 y Fu(1)p 2742 2955 V 2742 3012 a(2)2788 2978 y Fv(;)2842 2939 y Fu(1)p 2842 2955 V 2842 3012 a(2)2887 2978 y Fw(].)120 3098 y(Recall)j(the)i(de\014nition)f(of)g(the)h(Gelfand)e (transformation)g Fs(U)42 b Fw(in)32 b(Section)h(1.3.)120 3280 y Fp(Prop)s(osition)j(2.1.3.)1124 3542 y Fs(U)27 b Fv(H)1302 3501 y Fu(\()p Ft(\033)r Fu(\))1420 3542 y Fs(U)1492 3501 y Fl(\003)1559 3542 y Fw(=)1663 3406 y Fo(Z)1763 3432 y Fu(1)p Ft(=)p Fu(2)1718 3632 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1900 3447 y Fo(M)2067 3542 y Fv(H)2156 3501 y Fu(\()p Ft(\033)r Fu(\))2257 3542 y Fw(\()p Fv(k)s Fw(\))17 b Fv(dk)s(:)266 3800 y Fw(W)-8 b(e)48 b(pro)m(v)m(e)h(this)e (Prop)s(osition)f(as)i(a)f(statemen)m(t)h(ab)s(out)f(quadratic)g (forms.)88 b(In)48 b(the)120 3932 y(Hilb)s(ert)31 b(space)717 3852 y Fo(R)783 3878 y Fu(1)p Ft(=)p Fu(2)764 3966 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)945 3857 y Fo(L)1072 3932 y Fv(L)1138 3947 y Fu(2)1178 3932 y Fw(\(\005\))17 b Fv(dk)35 b Fw(consider)e(the)g(quadratic)f(form)1253 4199 y(~)1252 4225 y Fv(h)1308 4184 y Fu(\()p Ft(\033)r Fu(\))1437 4225 y Fw(:=)1568 4089 y Fo(Z)1668 4116 y Fu(1)p Ft(=)p Fu(2)1623 4315 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1805 4130 y Fo(M)1972 4225 y Fv(h)2028 4184 y Fu(\()p Ft(\033)r Fu(\))2130 4225 y Fw(\()p Fv(k)s Fw(\))17 b Fv(dk)s(;)120 4481 y Fw(i.e.,)502 4757 y Fs(D)s Fw([)610 4731 y(~)609 4757 y Fv(h)665 4716 y Fu(\()p Ft(\033)r Fu(\))767 4757 y Fw(])28 b(=)925 4587 y Fo(\()1005 4757 y Fv(F)42 b Fs(2)1204 4622 y Fo(Z)1303 4648 y Fu(1)p Ft(=)p Fu(2)1259 4847 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1441 4663 y Fo(M)1608 4757 y Fv(L)1674 4772 y Fu(2)1714 4757 y Fw(\(\005\))17 b Fv(dk)30 b Fw(:)55 b Fv(F)14 b Fw(\()p Fv(k)s Fw(\))27 b Fs(2)2448 4732 y Fw(~)2422 4757 y Fv(H)2511 4716 y Fu(1)2550 4757 y Fw(\(\005\))h(for)k(a.e.)c Fv(k)s(;)1211 4954 y Fo(Z)1310 4980 y Fu(1)p Ft(=)p Fu(2)1266 5179 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1447 5005 y Fo(\014)1447 5064 y(\014)1481 5089 y Fv(h)1537 5048 y Fu(\()p Ft(\033)r Fu(\))1639 5089 y Fw(\()p Fv(k)s Fw(\)[)p Fv(F)14 b Fw(\()p Fv(k)s Fw(\)])2030 5005 y Fo(\014)2030 5064 y(\014)2096 5089 y Fv(dk)30 b(<)e Fs(1)2432 4919 y Fo(\))2528 5089 y Fv(;)506 5380 y Fw(~)505 5406 y Fv(h)561 5365 y Fu(\()p Ft(\033)r Fu(\))663 5406 y Fw([)p Fv(F)14 b Fw(])28 b(=)925 5271 y Fo(Z)1025 5297 y Fu(1)p Ft(=)p Fu(2)980 5496 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1162 5406 y Fv(h)1218 5365 y Fu(\()p Ft(\033)r Fu(\))1320 5406 y Fw(\()p Fv(k)s Fw(\)[)p Fv(F)14 b Fw(\()p Fv(k)s Fw(\)])j Fv(dk)s(:)1781 5712 y Fw(14)p eop %%Page: 15 20 15 19 bop 120 407 a Fw(W)-8 b(e)37 b(remark)e(that)h Fv(h)899 371 y Fu(\()p Ft(\033)r Fu(\))1001 407 y Fw(\()p Fv(k)s Fw(\)[)p Fv(u)p Fw(])g(is)g(a)f(measurable)h(\(indeed,)h (quadratic\))f(function)g(of)f Fv(k)k Fw(for)120 527 y(eac)m(h)34 b Fv(u)28 b Fs(2)544 502 y Fw(~)519 527 y Fv(H)608 491 y Fu(1)647 527 y Fw(\(\005\).)45 b(Noting)32 b(that)h(the)g(constan)m(ts)i(in)d(the)h(t)m(w)m(o)h(sided)f(estimate)g (of)f(Lemma)120 648 y(1.2.1)25 b(do)h(not)f(dep)s(end)i(on)f Fv(k)i Fw(w)m(e)f(see)g(that)1674 621 y(~)1673 648 y Fv(h)1729 611 y Fu(\()p Ft(\033)r Fu(\))1857 648 y Fw(is)e(a)g(closed)h (and)g(lo)m(w)m(er)f(semib)s(ounded)h(form.)120 790 y(Since)44 b Fs(D)s Fw(\()529 765 y(~)504 790 y Fv(H)593 754 y Fu(\()p Ft(\033)r Fu(\))694 790 y Fw(\))j Fs(\032)g(D)s Fw([)1011 764 y(~)1010 790 y Fv(h)1066 754 y Fu(\()p Ft(\033)r Fu(\))1167 790 y Fw(])d(and)1439 679 y Fo(\020)1524 765 y Fw(~)1499 790 y Fv(H)1588 754 y Fu(\()p Ft(\033)r Fu(\))1689 790 y Fv(F)s(;)17 b(G)1876 679 y Fo(\021)1982 790 y Fw(=)2105 764 y(~)2104 790 y Fv(h)2160 754 y Fu(\()p Ft(\033)r Fu(\))2262 790 y Fw([)p Fv(F)s(;)g(G)p Fw(])44 b(for)f Fv(F)61 b Fs(2)47 b(D)s Fw(\()3087 765 y(~)3062 790 y Fv(H)3151 754 y Fu(\()p Ft(\033)r Fu(\))3252 790 y Fw(\))p Fv(;)33 b(G)47 b Fs(2)120 954 y(D)s Fw(\()263 929 y(~)238 954 y Fv(H)327 918 y Fu(\()p Ft(\033)r Fu(\))428 954 y Fw(\),)32 b(the)h(corresp)s(ondig)g(self-adjoin)m(t)e(op)s(erator)g (coincides)i(with)2820 929 y(~)2795 954 y Fv(H)2884 918 y Fu(\()p Ft(\033)r Fu(\))2985 954 y Fw(.)120 1075 y(The)h(pro)s(of)d (of)h(Prop)s(osition)f(2.1.3)h(is)g(no)m(w)i(easy)-8 b(.)120 1270 y Fj(Pr)j(o)g(of.)48 b Fw(By)33 b(the)g(ab)s(o)m(v)m(e)h (it)d(su\016ces)k(to)d(pro)m(v)m(e)970 1489 y Fs(U)1058 1409 y Fo(\000)1104 1489 y Fs(D)s Fw([)p Fv(h)1267 1448 y Fu(\()p Ft(\033)r Fu(\))1369 1489 y Fw(])1396 1409 y Fo(\001)1469 1489 y Fw(=)c Fs(D)s Fw([)1681 1463 y(~)1680 1489 y Fv(h)1736 1448 y Fu(\()p Ft(\033)r Fu(\))1837 1489 y Fw(])p Fv(;)1174 1651 y(h)1230 1610 y Fu(\()p Ft(\033)r Fu(\))1332 1651 y Fw([)p Fv(u)p Fw(])f(=)1574 1625 y(~)1573 1651 y Fv(h)1629 1610 y Fu(\()p Ft(\033)r Fu(\))1731 1651 y Fw([)p Fs(U)10 b Fv(u)p Fw(])p Fv(;)211 b(u)27 b Fs(2)i Fv(H)2418 1610 y Fu(1)2457 1651 y Fw(\()p Fr(R)2560 1610 y Fu(2)2560 1676 y(+)2625 1651 y Fw(\))p Fv(:)120 1864 y Fw(This)k(follo)m(ws)e(immediately)f(from)h(Prop)s (osition)g(1.3.2)h(and)g(Lemma)g(1.3.3.)p 3469 1864 4 66 v 3473 1802 59 4 v 3473 1864 V 3531 1864 4 66 v 266 2059 a(Prop)s(osition)j(2.1.3)h(allo)m(ws)f(us)i(to)f(in)m(v)m (estigate)g(the)h(op)s(erator)f Fv(H)2725 2023 y Fu(\()p Ft(\033)r Fu(\))2863 2059 y Fw(b)m(y)h(studying)g(the)120 2180 y(\014b)s(ers)45 b Fv(H)484 2144 y Fu(\()p Ft(\033)r Fu(\))585 2180 y Fw(\()p Fv(k)s Fw(\).)78 b(F)-8 b(rom)43 b(the)h(general)g(theory)g(of)g(direct)g(in)m(tegral)e(decomp)s (ositions)h(\(cf.)120 2300 y([RS,)33 b(Thm.)43 b(XI)s(I)s(I.85]\))32 b(w)m(e)i(kno)m(w)120 2504 y Fp(Corollary)j(2.1.4.)42 b Fj(L)-5 b(et)36 b Fv(\025)27 b Fs(2)h Fr(R)5 b Fj(,)41 b(then)35 b Fv(\025)27 b Fs(2)h Fv(\033)1877 2423 y Fo(\000)1923 2504 y Fv(H)2012 2467 y Fu(\()p Ft(\033)r Fu(\))2113 2423 y Fo(\001)2193 2504 y Fj(i\013)35 b(for)g(every)f Fv(\017)29 b(>)e Fw(0)708 2653 y Fo(\014)708 2712 y(\014)741 2657 y(\010)799 2737 y Fv(k)k Fs(2)975 2657 y Fo(\002)1026 2698 y Fu(1)p 1026 2714 36 4 v 1026 2772 a(2)1071 2737 y Fv(;)1125 2698 y Fu(1)p 1125 2714 V 1125 2772 a(2)1170 2657 y Fo(\003)1240 2737 y Fw(:)55 b Fv(\033)1398 2657 y Fo(\000)1443 2737 y Fv(H)1532 2696 y Fu(\()p Ft(\033)r Fu(\))1634 2737 y Fw(\()p Fv(k)s Fw(\))1764 2657 y Fo(\001)1831 2737 y Fs(\\)23 b Fw(\()p Fv(\025)f Fs(\000)h Fv(\017;)17 b(\025)22 b Fw(+)g Fv(\017)p Fw(\))28 b Fs(6)p Fw(=)f(0)2654 2657 y Fo(\011)2712 2653 y(\014)2712 2712 y(\014)2773 2737 y Fv(>)h Fw(0)p Fv(;)120 2966 y Fj(and)34 b Fv(\025)28 b Fs(2)g Fv(\033)543 2981 y Ft(p)600 2885 y Fo(\000)645 2966 y Fv(H)734 2930 y Fu(\()p Ft(\033)r Fu(\))836 2885 y Fo(\001)916 2966 y Fj(i\013)1023 3115 y Fo(\014)1023 3175 y(\014)1056 3119 y(\010)1114 3200 y Fv(k)j Fs(2)1290 3119 y Fo(\002)1341 3161 y Fu(1)p 1341 3177 V 1341 3234 a(2)1386 3200 y Fv(;)1440 3161 y Fu(1)p 1440 3177 V 1440 3234 a(2)1485 3119 y Fo(\003)1555 3200 y Fw(:)55 b Fv(\025)28 b Fs(2)g Fv(\033)1871 3215 y Ft(p)1927 3119 y Fo(\000)1973 3200 y Fv(H)2062 3159 y Fu(\()p Ft(\033)r Fu(\))2163 3200 y Fw(\()p Fv(k)s Fw(\))2293 3119 y Fo(\001\011)2397 3115 y(\014)2397 3175 y(\014)2458 3200 y Fv(>)g Fw(0)p Fv(:)120 3429 y Fj(Mor)-5 b(e)g(over,)46 b(if)d Fv(H)782 3392 y Fu(\()p Ft(\033)r Fu(\))884 3429 y Fw(\()p Fv(k)s Fw(\))h Fj(is)f(absolutely)h(c)-5 b(ontinuous)44 b(for)f(a.e.)72 b Fv(k)47 b Fs(2)e Fw([)2763 3389 y Fu(1)p 2763 3406 V 2763 3463 a(2)2808 3429 y Fv(;)2862 3389 y Fu(1)p 2862 3406 V 2862 3463 a(2)2907 3429 y Fw(])p Fj(,)h(then)e Fv(H)3325 3392 y Fu(\()p Ft(\033)r Fu(\))3470 3429 y Fj(is)120 3549 y(absolutely)35 b(c)-5 b(ontinuous.)120 3809 y Fp(4.)74 b(Commen)m(ts)44 b(and)i(Remarks.)98 b(Scaling)44 b(b)s(eha)m(vior.)64 b Fw(In)40 b(the)g(de\014nition)f(of) g(the)120 3929 y(op)s(erator)j Fv(H)612 3893 y Fu(\()p Ft(\033)r Fu(\))756 3929 y Fw(in)h(Subsection)g(1)g(it)f(w)m(as)i(not)e (essen)m(tial)h(that)g(the)h(p)s(erio)s(d)d(of)i Fv(\033)k Fw(is)42 b(2)p Fv(\031)t Fw(.)120 4050 y(Indeed,)50 b(if)f(~)-54 b Fv(\033)53 b Fw(:)d Fr(R)60 b Fs(7!)49 b Fr(R)56 b Fw(is)44 b Fv(a)p Fw(-p)s(erio)s(dic)g(for)g(some)h Fv(a)50 b(>)f Fw(0)c(and)g(b)s(elongs)g(to)g Fv(L)3236 4065 y Ft(q)r(;)11 b(l)q(oc)3392 4050 y Fw(\()p Fr(R)5 b Fw(\))120 4170 y(for)43 b(some)h(1)i Fv(<)g(q)51 b Fs(\024)c(1)p Fw(,)f(then)e(w)m(e)h(can)f(de\014ne)h(a)e(self-adjoin)m(t)f(op)s (erator)h Fv(H)3113 4134 y Fu(\()t(~)-39 b Ft(\033)s Fu(\))3258 4170 y Fw(b)m(y)45 b(the)120 4290 y(same)c(quadratic)g(form) f(expression)j(\(2.1.1\))e(with)g Fv(\033)k Fw(replaced)c(b)m(y)48 b(~)-54 b Fv(\033)t Fw(.)69 b(By)42 b(a)f(c)m(hange)h(of)120 4425 y(co)s(ordinates)29 b(it)f(is)h(easy)h(to)f(see)h(that)f Fv(H)1602 4389 y Fu(\()t(~)-39 b Ft(\033)s Fu(\))1732 4425 y Fw(is)29 b(unitarily)e(equiv)-5 b(alen)m(t)29 b(to)2798 4344 y Fo(\000)2854 4386 y Fu(2)p Ft(\031)p 2854 4402 79 4 v 2874 4459 a(a)2942 4344 y Fo(\001)2987 4366 y Fu(2)3044 4425 y Fv(H)3133 4389 y Fu(\()p Ft(\033)r Fu(\))3234 4425 y Fw(,)h(where)1098 4682 y Fv(\033)t Fw(\()p Fv(x)1250 4697 y Fu(1)1290 4682 y Fw(\))d(:=)1524 4615 y Fv(a)p 1496 4659 108 4 v 1496 4750 a Fw(2)p Fv(\031)1635 4682 y Fw(~)-54 b Fv(\033)1706 4572 y Fo(\020)1775 4615 y Fv(ax)1881 4630 y Fu(1)p 1775 4659 147 4 v 1795 4750 a Fw(2)p Fv(\031)1931 4572 y Fo(\021)2008 4682 y Fv(;)211 b(x)2301 4697 y Fu(1)2369 4682 y Fs(2)28 b Fr(R)5 b Fv(;)120 4933 y Fw(is)32 b(2)p Fv(\031)t Fw(-p)s(erio)s(dic)e(and)i(b)s(elongs)g (to)g Fv(L)1456 4948 y Ft(q)1495 4933 y Fw(\()p Fr(T)p Fw(\).)46 b(It)33 b(is)f(therefore)h(no)f(loss)g(to)g(assume)h(that)f Fv(\033)37 b Fw(has)120 5053 y(p)s(erio)s(d)31 b(2)p Fv(\031)t Fw(.)1781 5712 y(15)p eop %%Page: 16 21 16 20 bop 120 434 a Fq(2.2)161 b(The)53 b(unp)t(erturb)t(ed)f(op)t (erators)g Ff(H)2606 382 y Fw(\(0\))2789 434 y Fq(and)i Ff(H)3228 382 y Fw(\(0\))3357 434 y Fe(\()p Ff(k)5 b Fe(\))120 653 y Fw(Our)46 b(goal)e(in)h(this)h(section)g(is)f(to)h (study)h(the)g(sp)s(ectrum)f(of)f(the)i("unp)s(erturb)s(ed")f(\(i.e.,) 120 773 y Fv(\033)33 b Fw(=)c(0\))k(op)s(erators)g Fv(H)954 737 y Fu(\(0\))1081 773 y Fw(and)g Fv(H)1360 737 y Fu(\(0\))1454 773 y Fw(\()p Fv(k)s Fw(\))g(on)g(the)h(halfplane)e(and)h(on)g(the)h (halfstrip,)e(as)i(w)m(ell)120 894 y(as)f(to)g(giv)m(e)h(a)f(rather)g (explicit)f(form)m(ula)f(for)i(the)h(k)m(ernel)g(function)f(of)f(the)i (resolv)m(en)m(t)h(op)s(er-)120 1024 y(ator)326 943 y Fo(\000)372 1024 y Fv(H)461 988 y Fu(\(0\))554 1024 y Fw(\()p Fv(k)s Fw(\))23 b Fs(\000)f Fv(z)855 943 y Fo(\001)902 964 y Fl(\000)p Fu(1)996 1024 y Fw(.)120 1153 y(The)29 b(op)s(erators)f Fv(H)832 1117 y Fu(\(0\))954 1153 y Fw(and)g Fv(H)1228 1117 y Fu(\(0\))1322 1153 y Fw(\()p Fv(k)s Fw(\))g(are)g(diagonalized)e(b)m(y)j(v)-5 b(arian)m(ts)27 b(of)h(the)g(F)-8 b(ourier)27 b(trans-)120 1273 y(formation)k(in)i (Subsections)i(1)e(and)h(2,)g(resp)s(ectiv)m(ely)-8 b(.)47 b(The)35 b(form)m(ula)d(for)h(the)h(k)m(ernel)g(func-)120 1404 y(tion)j(of)h(the)g(resolv)m(en)m(t)h(op)s(erator)1430 1323 y Fo(\000)1475 1404 y Fv(H)1564 1367 y Fu(\(0\))1658 1404 y Fw(\()p Fv(k)s Fw(\))22 b Fs(\000)h Fv(z)1959 1323 y Fo(\001)2005 1344 y Fl(\000)p Fu(1)2138 1404 y Fw(\("free)38 b(Green's)h(function"\))e(will)f(b)s(e)120 1524 y(deriv)m(ed)d(b)m(y)h(separation)e(of)g(v)-5 b(ariables)31 b(from)h(the)h(one-dimensional)d(case.)120 1784 y Fp(1.)82 b(The)49 b(unp)s(erturb)s(ed)g(op)s(erator)g Fv(H)1750 1748 y Fu(\(0\))1892 1784 y Fp(on)f(the)g(halfplane.)98 b Fw(Using)42 b(the)g(direct)120 1904 y(in)m(tegral)24 b(decomp)s(osition)g(of)i(Section)f(1)h(it)f(is)g(p)s(ossible)g(to)h (carry)g(out)g(the)g(sp)s(ectral)g(analysis)120 2025 y(of)34 b(the)h(op)s(erator)e Fv(H)886 1988 y Fu(\(0\))1015 2025 y Fw(in)g(terms)i(of)f(the)g(op)s(erators)h Fv(H)2209 1988 y Fu(\(0\))2302 2025 y Fw(\()p Fv(k)s Fw(\))p Fv(;)f(k)g Fs(2)d Fw([)p Fs(\000)2789 1985 y Fu(1)p 2789 2002 36 4 v 2789 2059 a(2)2835 2025 y Fv(;)2888 1985 y Fu(1)p 2888 2002 V 2888 2059 a(2)2933 2025 y Fw(].)49 b(Ho)m(w)m(ev)m(er,)38 b(in)120 2145 y(this)32 b(subsection)i(w)m(e)f(c)m(ho)s(ose)h(a)e (direct)h(approac)m(h.)120 2265 y(F)-8 b(or)32 b Fv(u)27 b Fs(2)h(S)7 b Fw(\()p Fr(R)644 2229 y Fu(2)644 2290 y(+)709 2265 y Fw(\),)32 b(the)h(Sc)m(h)m(w)m(artz)i(class)e(on)f Fr(R)1819 2229 y Fu(2)1819 2290 y(+)1917 2265 y Fw(\(see)h(Section)g (1.1\),)f(let)627 2540 y(\(\010)735 2499 y Fu(\(0\))830 2540 y Fv(u)p Fw(\)\()p Fv(p)p Fw(\))27 b(:=)1221 2473 y(1)p 1216 2517 59 4 v 1216 2609 a Fv(\031)1302 2405 y Fo(Z)1357 2630 y Fn(R)1405 2607 y Fm(2)1405 2651 y(+)1477 2540 y Fv(u)p Fw(\()p Fv(x)p Fw(\))p Fv(e)1709 2499 y Fl(\000)p Ft(ip)1824 2508 y Fm(1)1858 2499 y Ft(x)1898 2508 y Fm(1)1953 2540 y Fw(cos\()p Fv(p)2170 2555 y Fu(2)2210 2540 y Fv(x)2265 2555 y Fu(2)2305 2540 y Fw(\))17 b Fv(dx;)211 b(p)28 b Fs(2)g Fr(R)2941 2499 y Fu(2)2941 2565 y(+)3006 2540 y Fv(:)120 2852 y Fw(Viewing)42 b Fv(L)573 2867 y Fu(2)613 2852 y Fw(\()p Fr(R)717 2816 y Fu(2)717 2877 y(+)782 2852 y Fw(\))j(=)g Fv(L)1052 2867 y Fu(2)1092 2852 y Fw(\()p Fr(R)5 b Fw(\))35 b Fs(\012)30 b Fv(L)1442 2867 y Fu(2)1482 2852 y Fw(\()p Fr(R)1585 2867 y Fu(+)1650 2852 y Fw(\),)46 b(\010)1831 2816 y Fu(\(0\))1969 2852 y Fw(is)c(the)i(tensor)f(pro)s(duct)g(of)g(the)g(F)-8 b(ourier)120 2972 y(transform)39 b(on)h Fv(L)784 2987 y Fu(2)824 2972 y Fw(\()p Fr(R)t Fw(\))46 b(and)40 b(the)g(F)-8 b(ourier)39 b(cosine)h(transform)f(on)h Fv(L)2687 2987 y Fu(2)2727 2972 y Fw(\()p Fr(R)2831 2987 y Fu(+)2896 2972 y Fw(\).)65 b(Therefore)41 b(it)120 3093 y(can)33 b(b)s(e)g(extended)h(b)m(y)g(con)m(tin)m(uit)m(y)e(to)h(a)f(unitary)g (op)s(erator)1303 3313 y(\010)1373 3272 y Fu(\(0\))1495 3313 y Fw(:)c Fv(L)1616 3328 y Fu(2)1656 3313 y Fw(\()p Fr(R)1760 3272 y Fu(2)1760 3337 y(+)1825 3313 y Fw(\))f Fs(!)h Fv(L)2084 3328 y Fu(2)2123 3313 y Fw(\()p Fr(R)2227 3272 y Fu(2)2227 3337 y(+)2292 3313 y Fw(\))p Fv(:)120 3533 y Fw(W)-8 b(e)29 b(remark)g(that)g(\010)892 3497 y Fu(\(0\))987 3533 y Fv(u)f Fw(coincides)h(with)g(the)h(restriction)e (of)g(the)i(F)-8 b(ourier)28 b(transform)g(of)35 b(~)-56 b Fv(u)120 3653 y Fw(to)29 b(the)g(upp)s(er)g(halfplane.)41 b(Here)36 b(~)-55 b Fv(u)27 b Fs(2)h Fv(L)1602 3668 y Fu(2)1642 3653 y Fw(\()p Fr(R)1746 3617 y Fu(2)1791 3653 y Fw(\))h(denotes)h(the)f(extension)h(of)f Fv(u)f Fw(b)m(y)i (re\015ection,)120 3774 y(i.e.,)1081 3947 y(~)-55 b Fv(u)o Fw(\()p Fv(x)p Fw(\))28 b(:=)1420 3806 y Fo(\032)1536 3886 y Fv(u)p Fw(\()p Fv(x)p Fw(\))353 b(if)96 b Fv(x)2285 3901 y Fu(2)2353 3886 y Fs(\025)28 b Fw(0)p Fv(;)1536 4006 y(u)p Fw(\()p Fv(x)1685 4021 y Fu(1)1724 4006 y Fv(;)17 b Fs(\000)p Fv(x)1900 4021 y Fu(2)1940 4006 y Fw(\))98 b(if)e Fv(x)2285 4021 y Fu(2)2353 4006 y Fv(<)27 b Fw(0)p Fv(:)120 4190 y Fw(F)-8 b(or)27 b Fv(u)g Fs(2)i Fv(H)557 4153 y Fu(2)595 4190 y Fw(\()p Fr(R)699 4153 y Fu(2)699 4214 y(+)764 4190 y Fw(\))f(one)h(has)34 b(~)-55 b Fv(u)27 b Fs(2)h Fv(H)1440 4153 y Fu(2)1479 4190 y Fw(\()p Fr(R)1583 4153 y Fu(2)1629 4190 y Fw(\))g(i\013)1833 4150 y Ft(@)t(u)p 1816 4167 116 4 v 1816 4224 a(@)t(x)1897 4233 y Fm(2)1970 4190 y Fw(=)f(0)h(on)g Fr(R)18 b Fs(\002)13 b(f)p Fw(0)p Fs(g)p Fw(.)47 b(Com)m(bining)27 b(this)g(with)120 4320 y(w)m(ell-kno)m(wn)33 b(facts)g(ab)s(out)f(the)h(F)-8 b(ourier)31 b(transformation)f(on)j Fr(R)2508 4284 y Fu(2)2586 4320 y Fw(w)m(e)g(obtain)665 4573 y(\010)735 4532 y Fu(\(0\))847 4432 y Fo(\022\032)995 4573 y Fv(u)27 b Fs(2)h Fv(H)1261 4532 y Fu(2)1300 4573 y Fw(\()p Fr(R)1404 4532 y Fu(2)1404 4597 y(+)1469 4573 y Fw(\))f(:)1647 4505 y Fv(@)5 b(u)p 1627 4550 152 4 v 1627 4641 a(@)g(x)1738 4656 y Fu(2)1816 4573 y Fw(=)28 b(0)f(on)h Fr(R)33 b Fs(\002)23 b(f)p Fw(0)p Fs(g)2470 4432 y Fo(\033)o(\023)2645 4573 y Fw(=)888 4875 y(=)992 4705 y Fo(\()1072 4875 y Fv(v)31 b Fs(2)d Fv(L)1310 4890 y Fu(2)1350 4875 y Fw(\()p Fr(R)1454 4834 y Fu(2)1454 4900 y(+)1519 4875 y Fw(\))g(:)1667 4739 y Fo(Z)1722 4965 y Fn(R)1770 4942 y Fm(2)1770 4986 y(+)1825 4875 y Fw(\(1)22 b(+)g Fs(j)p Fv(p)p Fs(j)2137 4834 y Fu(2)2176 4875 y Fw(\))2214 4834 y Fu(2)2253 4875 y Fs(j)p Fv(v)t Fw(\()p Fv(p)p Fw(\))p Fs(j)2485 4834 y Fu(2)2540 4875 y Fv(dp)28 b(<)f Fs(1)2871 4705 y Fo(\))2968 4875 y Fv(;)665 5124 y Fw(\(\010)773 5083 y Fu(\(0\))868 5124 y Fv(H)957 5083 y Fu(\(0\))1051 5124 y Fv(u)p Fw(\)\()p Fv(p)p Fw(\))g(=)g Fs(j)p Fv(p)p Fs(j)1505 5083 y Fu(2)1544 5124 y Fw(\(\010)1652 5083 y Fu(\(0\))1747 5124 y Fv(u)p Fw(\)\()p Fv(p)p Fw(\))p Fv(;)211 b(p)28 b Fs(2)g Fr(R)2441 5083 y Fu(2)2441 5148 y(+)2506 5124 y Fv(:)120 5337 y Fw(This)33 b(pro)m(v)m(es)1781 5712 y(16)p eop %%Page: 17 22 17 21 bop 120 407 a Fp(Prop)s(osition)44 b(2.2.1.)i Fj(The)40 b(op)-5 b(er)g(ator)42 b Fv(H)1726 371 y Fu(\(0\))1861 407 y Fj(is)f(unitarily)h(e)-5 b(quivalent)41 b(to)h(multiplic)-5 b(ation)120 527 y(by)35 b Fs(j)p Fv(p)p Fs(j)352 491 y Fu(2)426 527 y Fj(on)f Fv(L)631 542 y Fu(2)671 527 y Fw(\()p Fr(R)775 542 y Fu(+)840 527 y Fw(\))p Fj(.)45 b(In)34 b(p)-5 b(articular,)34 b(it)i(is)e(absolutely)h(c)-5 b(ontinuous)35 b(with)f(sp)-5 b(e)g(ctrum)1423 747 y Fv(\033)1499 666 y Fo(\000)1544 747 y Fv(H)1633 706 y Fu(\(0\))1727 666 y Fo(\001)1800 747 y Fw(=)28 b([0)p Fv(;)17 b Fw(+)p Fs(1)p Fw(\))120 967 y Fj(of)35 b(in\014nite)f (multiplicity.)120 1227 y Fp(2.)60 b(The)41 b(unp)s(erturb)s(ed)h(op)s (erator)f Fv(H)1705 1191 y Fu(\(0\))1799 1227 y Fw(\()p Fv(k)s Fw(\))f Fp(on)h(the)g(halfstrip.)97 b Fw(Let)35 b Fv(k)h Fs(2)d Fw([)p Fs(\000)3341 1188 y Fu(1)p 3341 1204 36 4 v 3341 1261 a(2)3387 1227 y Fv(;)3441 1188 y Fu(1)p 3441 1204 V 3441 1261 a(2)3486 1227 y Fw(].)120 1347 y(Recall)e(the)i(decomp)s(osition)1062 1579 y Fv(L)1128 1594 y Fu(2)1168 1579 y Fw(\(\005\))28 b(=)1448 1485 y Fo(X)1453 1696 y Ft(n)p Fl(2)p Fn(Z)1609 1485 y Fo(M)1776 1579 y Fw(span)q Fs(f)p Fv(e)2067 1594 y Ft(n)2114 1579 y Fs(g)21 b(\012)i Fv(L)2351 1594 y Fu(2)2391 1579 y Fw(\()p Fr(R)2495 1594 y Fu(+)2560 1579 y Fw(\))666 b(\(2.2.1\))120 1893 y(from)31 b(Section)i(1.2.)43 b(T)-8 b(ak)m(e)34 b Fv(u)27 b Fs(2)h Fv(L)1366 1908 y Fu(2)1406 1893 y Fw(\(\005\))k(and)h(decomp)s(ose)g(it)e(according)h(to)h(\(2.2.1\))f (as)1485 2131 y Fv(u)27 b Fw(=)1671 2036 y Fo(X)1676 2248 y Ft(n)p Fl(2)p Fn(Z)1832 2131 y Fv(e)1877 2146 y Ft(n)1946 2131 y Fs(\012)i Fw(^)-55 b Fv(u)2102 2146 y Ft(n)2148 2131 y Fv(:)120 2484 y Fw(F)-8 b(rom)24 b(Lemma)g(1.2.2)g (w)m(e)j(kno)m(w)f(that)f Fv(u)i Fs(2)h(D)1797 2404 y Fo(\000)1842 2484 y Fv(H)1931 2448 y Fu(\(0\))2025 2484 y Fw(\()p Fv(k)s Fw(\))2155 2404 y Fo(\001)2228 2484 y Fw(=)2332 2374 y Fo(n)2398 2484 y Fv(u)g Fs(2)2601 2459 y Fw(~)2576 2484 y Fv(H)2665 2448 y Fu(2)2704 2484 y Fw(\(\005\))f(:)2989 2445 y Ft(@)t(u)p 2973 2462 116 4 v 2973 2519 a(@)t(x)3054 2528 y Fm(2)3098 2484 y Fw(\()p Fv(:;)17 b Fw(0\))27 b(=)h(0)3474 2374 y Fo(o)120 2627 y Fw(i\013)989 2720 y(^)-55 b Fv(u)1039 2735 y Ft(n)1113 2720 y Fs(2)28 b Fv(H)1296 2684 y Fu(2)1335 2720 y Fw(\()p Fr(R)1439 2735 y Fu(+)1504 2720 y Fw(\))p Fv(;)1791 2681 y Ft(d)5 b Fu(^)-40 b Ft(u)1868 2689 y Fi(n)p 1791 2697 120 4 v 1796 2755 a Ft(dx)1872 2764 y Fm(2)1921 2720 y Fw(\(0\))27 b(=)g(0)p Fv(;)983 2804 y Fo(P)1088 2908 y Ft(n)p Fl(2)p Fn(Z)1243 2768 y Fo(\020)1303 2879 y Fw(\(1)21 b(+)h Fv(n)1567 2842 y Fu(2)1607 2879 y Fw(\))1645 2842 y Fu(2)1685 2879 y Fs(k)6 b Fw(^)-55 b Fv(u)1791 2894 y Ft(n)1837 2879 y Fs(k)1887 2842 y Fu(2)1948 2879 y Fw(+)22 b Fs(k)2106 2839 y Ft(d)2142 2816 y Fm(2)2182 2839 y Fu(^)-40 b Ft(u)2218 2847 y Fi(n)p 2106 2856 155 4 v 2128 2915 a Ft(dx)2204 2892 y Fm(2)2204 2936 y(2)2270 2879 y Fs(k)2320 2842 y Fu(2)2359 2768 y Fo(\021)2447 2879 y Fv(<)27 b Fs(1)p Fv(;)3264 2814 y Fw(\(2.2.2\))120 3075 y(and)33 b(in)e(this)i(case)851 3329 y Fv(H)940 3288 y Fu(\(0\))1034 3329 y Fw(\()p Fv(k)s Fw(\))p Fv(u)27 b Fw(=)1351 3234 y Fo(X)1355 3446 y Ft(n)p Fl(2)p Fn(Z)1511 3329 y Fv(e)1556 3344 y Ft(n)1625 3329 y Fs(\012)1725 3189 y Fo(\022)1798 3329 y Fs(\000)1885 3262 y Fv(d)1936 3225 y Fu(2)1999 3262 y Fw(^)-55 b Fv(u)2049 3277 y Ft(n)p 1885 3306 210 4 v 1918 3397 a Fv(dx)2024 3363 y Fu(2)2024 3422 y(2)2127 3329 y Fw(+)22 b(\()p Fv(n)g Fw(+)g Fv(k)s Fw(\))2533 3288 y Fu(2)2596 3329 y Fw(^)-56 b Fv(u)2645 3344 y Ft(n)2692 3189 y Fo(\023)2782 3329 y Fv(:)455 b Fw(\(2.2.3\))120 3643 y(Consider)33 b(the)g(self-adjoin)m(t)e(op)s (erator)h Fv(H)1671 3658 y Fu(\(0\))1797 3643 y Fw(on)h Fv(L)1999 3658 y Fu(2)2039 3643 y Fw(\()p Fr(R)2143 3658 y Fu(+)2208 3643 y Fw(\),)1415 3867 y Fs(D)s Fw([)p Fv(H)1603 3883 y Fu(\(0\))1697 3867 y Fw(])28 b(:=)g Fv(H)1972 3826 y Fu(2)2011 3867 y Fw(\()p Fr(R)2115 3882 y Fu(+)2180 3867 y Fw(\))p Fv(;)1490 4083 y(H)1571 4098 y Fu(\(0\))1666 4083 y Fv(f)38 b Fw(:=)28 b Fs(\000)1970 4016 y Fv(d)2021 3979 y Fu(2)2060 4016 y Fv(f)p 1970 4060 149 4 v 1972 4151 a(dx)2078 4117 y Fu(2)2078 4176 y(2)2129 4083 y Fv(:)120 4358 y Fw(It)33 b(follo)m(ws)e(from)g(\(2.2.2\))h(and)h (\(2.2.3\))f(that)g(the)h(subspaces)i(span)q Fs(f)p Fv(e)2700 4373 y Ft(n)2747 4358 y Fs(g)22 b(\012)g Fv(L)2984 4373 y Fu(2)3024 4358 y Fw(\()p Fr(R)3128 4373 y Fu(+)3193 4358 y Fw(\))33 b(reduce)120 4479 y Fv(H)209 4443 y Fu(\(0\))303 4479 y Fw(\()p Fv(k)s Fw(\))f(and)h(that)931 4716 y Fv(H)1020 4675 y Fu(\(0\))1114 4716 y Fw(\()p Fv(k)s Fw(\))27 b(=)1375 4622 y Fo(X)1380 4833 y Ft(n)p Fl(2)p Fn(Z)1535 4622 y Fo(M)1703 4636 y(\000)1748 4716 y Fv(I)j Fs(\012)23 b Fw(\()p Fv(H)2040 4732 y Fu(\(0\))2156 4716 y Fw(+)f(\()p Fv(n)g Fw(+)g Fv(k)s Fw(\))2562 4675 y Fu(2)2602 4716 y Fw(\))2640 4636 y Fo(\001)2702 4716 y Fv(:)535 b Fw(\(2.2.4\))120 5034 y(The)33 b(F)-8 b(ourier)30 b(cosine)i(transformation)d(on)j Fv(L)1807 5049 y Fu(2)1847 5034 y Fw(\()p Fr(R)1951 4998 y Fu(2)1951 5059 y(+)2016 5034 y Fw(\))f(pro)m(vides)i(a)e(unitary)g (equiv)-5 b(alence)32 b(b)s(e-)120 5155 y(t)m(w)m(een)g Fv(H)475 5170 y Fu(\(0\))600 5155 y Fw(and)e(m)m(ultiplication)c(b)m(y) 31 b Fv(\030)1592 5119 y Fu(2)1662 5155 y Fw(on)f Fr(R)1861 5170 y Fu(+)1926 5155 y Fw(.)43 b(Th)m(us,)32 b(the)f(sp)s(ectrum)g(of) f Fv(H)3048 5170 y Fu(\(0\))3173 5155 y Fw(is)f(simple)120 5275 y(and)k(absolutely)f(con)m(tin)m(uous)h(and)g(coincides)f(with)g ([0)p Fv(;)17 b Fw(+)p Fs(1)p Fw(\).)43 b(F)-8 b(rom)31 b(\(2.2.4\))h(w)m(e)h(obtain)1781 5712 y(17)p eop %%Page: 18 23 18 22 bop 120 407 a Fp(Prop)s(osition)30 b(2.2.2.)39 b Fj(L)-5 b(et)31 b Fv(k)g Fs(2)d Fw([)p Fs(\000)1474 368 y Fu(1)p 1474 384 36 4 v 1474 441 a(2)1519 407 y Fv(;)1573 368 y Fu(1)p 1573 384 V 1573 441 a(2)1618 407 y Fw(])p Fj(,)k(then)e(the)h(op)-5 b(er)g(ator)30 b Fv(H)2545 371 y Fu(\(0\))2639 407 y Fw(\()p Fv(k)s Fw(\))h Fj(is)f(unitarily)h(e) -5 b(quiv-)120 527 y(alent)39 b(to)g(multiplic)-5 b(ation)39 b(by)g Fv(\030)1286 491 y Fu(2)1350 527 y Fw(+)25 b(\()p Fv(n)h Fw(+)f Fv(k)s Fw(\))1766 491 y Fu(2)1845 527 y Fj(in)1969 452 y Fo(P)2074 556 y Ft(n)p Fl(2)p Fn(Z)2229 452 y Fo(L)2356 527 y Fv(L)2422 542 y Fu(2)2462 527 y Fw(\()p Fr(R)2566 542 y Fu(+)2631 527 y Fw(\))p Fj(.)57 b(In)39 b(p)-5 b(articular,)40 b(it)f(is)120 648 y(absolutely)c(c)-5 b(ontinuous)34 b(with)h(sp)-5 b(e)g(ctrum)1322 819 y Fv(\033)1398 738 y Fo(\000)1443 819 y Fv(H)1532 778 y Fu(\(0\))1626 819 y Fw(\()p Fv(k)s Fw(\))1756 738 y Fo(\001)1830 819 y Fw(=)27 b([)p Fv(k)2014 778 y Fu(2)2054 819 y Fv(;)17 b Fw(+)p Fs(1)p Fw(\))p Fv(;)120 990 y Fj(and)34 b(the)h(sp)-5 b(e)g(ctr)g(al)35 b(multiplicity)g(of)f Fv(\025)28 b Fs(2)g Fw([)p Fv(k)1706 954 y Fu(2)1746 990 y Fv(;)17 b Fw(+)p Fs(1)p Fw(\))34 b Fj(is)1062 1161 y Fv(N)10 b Fw(\()p Fv(\025;)17 b(k)s Fw(\))28 b(:=)f Fv(])p Fs(f)p Fv(n)h Fs(2)g Fr(Z)d Fw(:)56 b(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))2292 1120 y Fu(2)2359 1161 y Fs(\024)28 b Fv(\025)p Fs(g)p Fv(:)266 1332 y Fw(W)-8 b(e)33 b(note)g(that)f(for)g Fv(u)c Fs(2)g(S)7 b Fw(\(\005\))518 1552 y(\()p Fs(F)638 1511 y Fu(\(0\))732 1552 y Fv(u)p Fw(\))826 1567 y Ft(n)873 1552 y Fw(\()p Fv(\030)e Fw(\))27 b(:=)1169 1484 y(1)p 1164 1529 59 4 v 1164 1620 a Fv(\031)1250 1416 y Fo(Z)1305 1642 y Fu(\005)1379 1552 y Fv(u)p Fw(\()p Fv(x)p Fw(\))p Fv(e)1611 1511 y Fl(\000)p Ft(inx)1773 1520 y Fm(1)1828 1552 y Fw(cos\()p Fv(\030)5 b(x)2099 1567 y Fu(2)2138 1552 y Fw(\))17 b Fv(dx;)212 b(\030)32 b Fs(2)c Fr(R)2773 1567 y Fu(+)2838 1552 y Fv(;)399 b Fw(\(2.2.5\))120 1770 y(is)32 b(the)i(F)-8 b(ourier)31 b(cosine)i(transform)f(of)39 b(^)-55 b Fv(u)1628 1785 y Ft(n)1674 1770 y Fw(.)45 b(By)33 b(con)m(tin)m(uit)m(y)g(w)m(e)h(obtain)e(the)h(unitary)g(op)s(er-)120 1890 y(ator)1176 2011 y Fs(F)1258 1970 y Fu(\(0\))1379 2011 y Fw(:)28 b Fv(L)1500 2026 y Fu(2)1540 2011 y Fw(\(\005\))g Fs(!)1844 1916 y Fo(X)1849 2128 y Ft(n)p Fl(2)p Fn(Z)2004 1916 y Fo(M)2172 2011 y Fv(L)2238 2026 y Fu(2)2277 2011 y Fw(\()p Fr(R)2381 2026 y Fu(+)2446 2011 y Fw(\))120 2269 y(whic)m(h)34 b(pro)m(vides)g(the)f(unitary)g(equiv)-5 b(alence)34 b(b)s(et)m(w)m(een)h Fv(H)2281 2233 y Fu(\(0\))2375 2269 y Fw(\()p Fv(k)s Fw(\))e(and)g(m)m(ultiplication)c Fv(\030)3403 2233 y Fu(2)3464 2269 y Fw(+)120 2390 y(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))428 2353 y Fu(2)500 2390 y Fw(in)614 2315 y Fo(P)719 2419 y Ft(n)p Fl(2)p Fn(Z)874 2315 y Fo(L)1001 2390 y Fv(L)1067 2405 y Fu(2)1107 2390 y Fw(\()p Fr(R)1211 2405 y Fu(+)1276 2390 y Fw(\).)120 2510 y(Finally)-8 b(,)30 b(w)m(e)k(shall)d(realize)g Fv(L)1212 2525 y Fu(2)1252 2510 y Fw(\(\005\))i(as)g(direct)f(in)m (tegral)f Fg(h)p Fw(\()p Fv(k)s Fw(\))h(where)i Fv(H)2770 2474 y Fu(\(0\))2864 2510 y Fw(\()p Fv(k)s Fw(\))f(acts)g(as)g(m)m(ul-) 120 2630 y(tiplication)j(b)m(y)41 b(the)f(indep)s(enden)m(t)h(v)-5 b(ariable.)63 b(This)40 b(will)d(b)s(e)j(needed)i(in)d(Section)g(3.4)g (for)120 2751 y(the)33 b(construction)g(of)f(the)h(scattering)f (matrix.)42 b(By)33 b(Prop)s(osition)e(2.2.2)h(w)m(e)i(tak)m(e)1259 2967 y Fg(h)p Fw(\()p Fv(k)s Fw(\))27 b(:=)1599 2832 y Fo(Z)1698 2858 y Fl(1)1654 3057 y Ft(k)1693 3038 y Fm(2)1790 2872 y Fo(M)1957 2967 y Fr(C)2023 2926 y Ft(N)7 b Fu(\()p Ft(\025;k)r Fu(\))2267 2967 y Fv(d\025:)120 3202 y Fw(W)-8 b(e)38 b(think)g(of)g Fr(C)739 3166 y Ft(N)7 b Fu(\()p Ft(\025;k)r Fu(\))1004 3202 y Fw(as)38 b(space)h(of)f Fv(N)10 b Fw(\()p Fv(\025;)17 b(k)s Fw(\)-dimensional)35 b(v)m(ectors)k Fv(a)f Fw(whose)h(en)m(tries)g Fv(a)3493 3217 y Ft(n)120 3322 y Fw(ha)m(v)m(e)g(indices)f Fv(n)g Fs(2)f Fr(Z)f Fw(suc)m(h)j(that)f(\()p Fv(n)26 b Fw(+)g Fv(k)s Fw(\))1738 3286 y Fu(2)1814 3322 y Fs(\024)38 b Fv(\025)p Fw(.)60 b(With)37 b(this)h(agreemen)m(t)h(w)m(e)g(de\014ne) g(an)120 3443 y(op)s(erator)32 b(\000)574 3407 y Fu(\(0\))668 3443 y Fw(\()p Fv(k)s Fw(\))c(:)g Fv(L)947 3458 y Fu(2)986 3443 y Fw(\(\005\))g Fs(!)f Fg(h)p Fw(\()p Fv(k)s Fw(\),)133 3658 y(\(\000)232 3617 y Fu(\(0\))327 3658 y Fw(\()p Fv(k)s Fw(\))p Fv(u)p Fw(\))551 3673 y Ft(n)597 3658 y Fw(\()p Fv(\025)p Fw(\))g(:=)1254 3591 y(1)p 898 3636 761 4 v 898 3658 a Fs(p)p 981 3658 49 4 v 83 x Fw(2)1058 3703 y Fm(4)1033 3655 y Fo(p)p 1132 3655 527 4 v 1132 3741 a Fv(\025)22 b Fs(\000)h Fw(\()p Fv(n)f Fw(+)g Fv(k)s Fw(\))1619 3712 y Fu(2)1669 3658 y Fw(\()p Fs(F)1789 3617 y Fu(\(0\))1882 3658 y Fv(u)p Fw(\))1976 3673 y Ft(n)2023 3658 y Fw(\()2061 3568 y Fo(p)p 2160 3568 V 2160 3658 a Fv(\025)g Fs(\000)h Fw(\()p Fv(n)f Fw(+)g Fv(k)s Fw(\))2647 3630 y Fu(2)2687 3658 y Fw(\))p Fv(;)212 b Fw(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))3272 3617 y Fu(2)3339 3658 y Fv(<)28 b(\025:)120 3909 y Fw(\(This)38 b(de\014nition)f(is)h (not)g(complete)f(if)g Fv(\025)g Fw(=)g(\()p Fv(n)26 b Fw(+)g Fv(k)s Fw(\))2143 3873 y Fu(2)2182 3909 y Fv(;)33 b(n)38 b Fs(2)f Fr(Z)p Fw(.)57 b(Ho)m(w)m(ev)m(er,)42 b(it)37 b(su\016ces)j(of)120 4029 y(course)30 b(to)f(de\014ne)h(elemen) m(ts)g(\000)1266 3993 y Fu(\(0\))1360 4029 y Fw(\()p Fv(k)s Fw(\))p Fv(u)d Fs(2)h Fg(h)p Fw(\()p Fv(k)s Fw(\))i(only)e (almost)g(ev)m(erywhere)k(with)d(resp)s(ect)h(to)120 4150 y Fv(\025)p Fw(.\))120 4270 y(Since)i(\000)435 4234 y Fu(\(0\))530 4270 y Fw(\()p Fv(k)s Fw(\))g(di\013ers)g(from)f Fs(F)1294 4234 y Fu(\(0\))1420 4270 y Fw(only)g(b)m(y)j(the)e(c)m (hange)h(of)f(v)-5 b(ariables)31 b Fv(\030)2821 4234 y Fu(2)2881 4270 y Fw(+)21 b(\()p Fv(n)h Fw(+)f Fv(k)s Fw(\))3285 4234 y Fu(2)3352 4270 y Fw(=)28 b Fv(\025)p Fw(,)120 4391 y(w)m(e)g(\014nd)f(that)g(\000)714 4354 y Fu(\(0\))808 4391 y Fw(\()p Fv(k)s Fw(\))g(is)f(unitary)h(and)g(\000) 1641 4354 y Fu(\(0\))1735 4391 y Fw(\()p Fv(k)s Fw(\))17 b Fv(H)1971 4354 y Fu(\(0\))2064 4391 y Fw(\()p Fv(k)s Fw(\))g(\000)2272 4354 y Fu(\(0\))2366 4391 y Fw(\()p Fv(k)s Fw(\))2496 4354 y Fl(\003)2562 4391 y Fw(is)27 b(m)m(ultiplication)22 b(b)m(y)28 b(the)120 4511 y(indep)s(enden)m(t)34 b(v)-5 b(ariable)31 b Fv(\025)h Fw(in)g Fg(h)p Fw(\()p Fv(k)s Fw(\).)120 4631 y(It)h(follo)m(ws)e(from)g(\(2.2.5\))h(that)h (for)f Fv(u)27 b Fs(2)h(S)7 b Fw(\(\005\))601 4851 y(\(\000)700 4810 y Fu(\(0\))794 4851 y Fw(\()p Fv(k)s Fw(\))p Fv(u)p Fw(\))1018 4866 y Ft(n)1064 4851 y Fw(\()p Fv(\025)p Fw(\))28 b(=)1328 4715 y Fo(Z)1383 4941 y Fu(\005)1457 4851 y Fv(u)p Fw(\()p Fv(x)p Fw(\))p 1644 4738 492 4 v Fv( )1711 4800 y Fu(\(0\))1707 4861 y Ft(n)1805 4851 y Fw(\()p Fv(x)p Fw(;)17 b Fv(\025;)g(k)s Fw(\))g Fv(dx;)211 b Fw(\()p Fv(n)23 b Fw(+)f Fv(k)s Fw(\))2805 4810 y Fu(2)2872 4851 y Fv(<)27 b(\025;)120 5069 y Fw(where)565 5240 y Fv( )632 5199 y Fu(\(0\))628 5264 y Ft(n)726 5240 y Fw(\()p Fv(x)p Fw(;)17 b Fv(\025;)g(k)s Fw(\))27 b(:=)1295 5172 y(1)p 1224 5217 191 4 v 1224 5237 a Fs(p)p 1307 5237 49 4 v 82 x Fw(2)p Fv(\031)1435 5171 y(e)1480 5134 y Ft(inx)1587 5143 y Fm(1)1642 5171 y Fw(cos\()1810 5085 y Fo(p)p 1910 5085 527 4 v 86 x Fv(\025)22 b Fs(\000)h Fw(\()p Fv(n)f Fw(+)g Fv(k)s Fw(\))2397 5142 y Fu(2)2436 5171 y Fv(x)2491 5186 y Fu(2)2531 5171 y Fw(\))p 1435 5217 1135 4 v 1715 5284 a Fm(4)1690 5237 y Fo(p)p 1790 5237 527 4 v 85 x Fv(\025)g Fs(\000)h Fw(\()p Fv(n)f Fw(+)g Fv(k)s Fw(\))2277 5293 y Fu(2)2579 5240 y Fv(;)212 b(x)28 b Fs(2)g Fw(\005)p Fv(:)120 5463 y Fw(These)34 b("generalized)e(eigenfunctions")g(will)e(b)s(e)j(further)g(discussed)h (in)e(Section)h(3.4.)1781 5712 y(18)p eop %%Page: 19 24 19 23 bop 120 407 a Fp(3.)61 b(The)42 b(free)f(resolv)m(en)m(t)g Fv(R)1279 371 y Fu(\(0\))1373 407 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))42 b Fp(on)f(the)g(halfstrip.)97 b Fw(Let)36 b(us)h(deriv)m(e)f(a)g(form)m(ula)120 527 y(for)42 b(the)h(free)g (resolv)m(en)m(t)h(k)m(ernel.)74 b(W)-8 b(e)42 b(start)h(with)f(the)h (op)s(erator)f Fv(H)2744 543 y Fu(\(0\))2881 527 y Fw(on)g(the)h (hal\015ine,)120 648 y(as)c(in)m(tro)s(duced)g(in)f(the)h(previous)g (subsection.)63 b(Let)39 b Fv(z)k Fs(2)c Fr(C)52 b Fs(n)26 b Fw([0)p Fv(;)17 b Fw(+)p Fs(1)p Fw(\),)40 b(then)f(it)f(is)g(w)m (ell-)120 768 y(kno)m(wn)c(\(and)f(easily)f(v)m(eri\014ed)i(b)m(y)f (calculating)e(the)i(in)m(v)m(erse)h(F)-8 b(ourier)32 b(cosine)h(transform)e(of)120 888 y(the)i(function)f Fv(\030)g Fs(7!)944 849 y Fu(1)p 882 865 159 4 v 882 923 a Ft(\030)916 904 y Fm(2)951 923 y Fl(\000)p Ft(z)1051 888 y Fw(\))h(that)f(\()p Fv(H)1452 904 y Fu(\(0\))1568 888 y Fs(\000)23 b Fv(z)t Fw(\))1755 852 y Fl(\000)p Fu(1)1882 888 y Fw(is)32 b(an)h(in)m(tegral)e(op)s(erator)h(with)g(k)m (ernel)179 1121 y Fv(r)223 1136 y Fu(\(0\))317 1121 y Fw(\()p Fv(x)410 1136 y Fu(2)450 1121 y Fv(;)17 b(y)542 1136 y Fu(2)581 1121 y Fw(;)g Fv(z)t Fw(\))28 b(:=)986 1053 y(1)p 880 1098 260 4 v 880 1193 a(2)929 1118 y Fs(p)p 1012 1118 128 4 v 75 x(\000)p Fv(z)1166 1010 y Fo(\020)1225 1121 y Fv(e)1270 1080 y Fl(\000)1325 1029 y(p)p 1384 1029 91 3 v 51 x(\000)p Ft(z)15 b Fu(\()p Ft(x)1554 1089 y Fm(2)1588 1080 y Fu(+)p Ft(y)1678 1089 y Fm(2)1713 1080 y Fu(\))1767 1121 y Fw(+)22 b Fv(e)1910 1080 y Fl(\000)1965 1029 y(p)p 2023 1029 V 2023 1080 a(\000)p Ft(z)15 b Fl(j)p Ft(x)2186 1089 y Fm(2)2220 1080 y Fl(\000)p Ft(y)2310 1089 y Fm(2)2344 1080 y Fl(j)2368 1010 y Fo(\021)2444 1121 y Fv(;)212 b(x)2738 1136 y Fu(2)2778 1121 y Fv(;)33 b(y)2886 1136 y Fu(2)2953 1121 y Fs(2)28 b Fr(R)3113 1136 y Fu(+)3178 1121 y Fv(:)59 b Fw(\(2.2.6\))120 1336 y(Here)46 b(and)f(in)g(the)h(follo)m(wing)c(w)m(e)47 b(use)f(the)g(canonical)d(branc)m(h)k(of)d(the)i(square)h(ro)s(ot)d(on) 120 1456 y Fr(C)k Fs(n)22 b Fw(\()p Fs(\0001)p Fv(;)17 b Fw(0])32 b(satisfying)g(Re)1220 1401 y Fs(p)p 1303 1401 28 4 v 55 x Fv(:)27 b(>)h Fw(0.)120 1577 y(Using)k(the)h(decomp)s (osition)e(\(2.2.4\))h(w)m(e)h(obtain)f(for)g(the)h(resolv)m(en)m(t)h (op)s(erators)1219 1762 y Fv(R)1294 1721 y Fu(\(0\))1389 1762 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))28 b(:=)1771 1682 y Fo(\000)1816 1762 y Fv(H)1905 1721 y Fu(\(0\))1999 1762 y Fw(\()p Fv(k)s Fw(\))22 b Fs(\000)h Fv(z)2300 1682 y Fo(\001)2346 1698 y Fl(\000)p Fu(1)120 1928 y Fw(on)32 b(the)h(halfstrip)f(the)h(follo)m(wing)120 2085 y Fp(Prop)s(osition)41 b(2.2.3.)j Fj(L)-5 b(et)40 b Fv(k)e Fs(2)e Fw([)p Fs(\000)1514 2045 y Fu(1)p 1514 2062 36 4 v 1514 2119 a(2)1560 2085 y Fv(;)1614 2045 y Fu(1)p 1614 2062 V 1614 2119 a(2)1659 2085 y Fw(])j Fj(and)g Fv(z)h Fs(2)c Fr(C)51 b Fs(n)25 b Fw([)p Fv(k)2359 2049 y Fu(2)2399 2085 y Fv(;)17 b Fw(+)p Fs(1)p Fw(\))p Fj(.)56 b(Then)39 b(the)g(r)-5 b(esolvent)120 2205 y Fv(R)195 2169 y Fu(\(0\))289 2205 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))36 b Fj(is)e(an)h(inte)-5 b(gr)g(al)34 b(op)-5 b(er)g(ator)35 b(with)f(kernel)139 2374 y Fv(r)186 2333 y Fu(\(0\))280 2374 y Fw(\()p Fv(x;)17 b(y)t Fw(;)g Fv(z)t(;)g(k)s Fw(\))27 b(:=)266 2575 y(=)409 2508 y(1)p 380 2552 108 4 v 380 2644 a(4)p Fv(\031)514 2481 y Fo(X)519 2692 y Ft(n)p Fl(2)p Fn(Z)970 2508 y Fw(1)p 684 2552 619 4 v 684 2572 a Fo(p)p 784 2572 520 4 v 85 x Fw(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))1092 2629 y Fu(2)1154 2657 y Fs(\000)h Fv(z)1330 2575 y(e)1375 2534 y Ft(in)p Fu(\()p Ft(x)1509 2543 y Fm(1)1544 2534 y Fl(\000)p Ft(y)1634 2543 y Fm(1)1668 2534 y Fu(\))1716 2465 y Fo(\020)1776 2575 y Fv(e)1821 2534 y Fl(\000)1876 2467 y Fs(p)p 1959 2467 317 4 v 67 x Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))2150 2515 y Fm(2)2185 2534 y Fl(\000)p Ft(z)14 b Fu(\()p Ft(x)2354 2543 y Fm(2)2389 2534 y Fu(+)p Ft(y)2479 2543 y Fm(2)2513 2534 y Fu(\))2567 2575 y Fw(+)22 b Fv(e)2710 2534 y Fl(\000)2765 2467 y Fs(p)p 2848 2467 V 67 x Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))3039 2515 y Fm(2)3074 2534 y Fl(\000)p Ft(z)14 b Fl(j)p Ft(x)3236 2543 y Fm(2)3270 2534 y Fl(\000)p Ft(y)3360 2543 y Fm(2)3394 2534 y Fl(j)3418 2465 y Fo(\021)3494 2575 y Fv(;)2530 2824 y(x;)34 b(y)d Fs(2)d Fw(\005)p Fv(;)33 b(x)3007 2839 y Fu(2)3075 2824 y Fs(6)p Fw(=)27 b Fv(y)3226 2839 y Fu(2)3265 2824 y Fv(:)3264 2953 y Fw(\(2.2.7\))266 3117 y(Note)j(that)g(the)h(RHS)f(of)36 b(\(2.2.7\))30 b(con)m(v)m(erges)i(absolutely)d(and)i(uniformly)c(on)j (compact)120 3237 y(subsets)35 b(of)d Fs(f)p Fw(\()p Fv(x;)17 b(y)t Fw(\))26 b Fs(2)i Fw(\005)23 b Fs(\002)f Fw(\005)28 b(:)56 b Fv(x)1400 3252 y Fu(2)1467 3237 y Fs(6)p Fw(=)28 b Fv(y)1619 3252 y Fu(2)1658 3237 y Fs(g)p Fw(.)120 3417 y Fj(Pr)-5 b(o)g(of.)48 b Fw(Let)33 b Fv(u)27 b Fs(2)h Fv(L)843 3432 y Fu(2)883 3417 y Fw(\(\005\),)33 b(then)g(\(2.2.4\))f(implies)760 3600 y Fv(R)835 3559 y Fu(\(0\))929 3600 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))p Fv(u)28 b Fw(=)1340 3505 y Fo(X)1344 3717 y Ft(n)p Fl(2)p Fn(Z)1500 3600 y Fv(e)1545 3615 y Ft(n)1614 3600 y Fs(\012)1714 3519 y Fo(\000)1760 3600 y Fv(H)1841 3615 y Fu(\(0\))1957 3600 y Fs(\000)23 b Fw(\()p Fv(z)j Fs(\000)d Fw(\()p Fv(n)f Fw(+)g Fv(k)s Fw(\))2574 3559 y Fu(2)2614 3519 y Fo(\001)2659 3541 y Fl(\000)p Fu(1)2776 3600 y Fw(^)-55 b Fv(u)2826 3615 y Ft(n)2873 3600 y Fv(:)120 3859 y Fw(Using)34 b(\(2.2.6\))d(and)i(the)g(de\014nition)f(of)38 b(^)-55 b Fv(u)1663 3874 y Ft(n)1742 3859 y Fw(w)m(e)33 b(obtain)281 4076 y(\()p Fv(R)394 4034 y Fu(\(0\))488 4076 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))p Fv(u)p Fw(\)\()p Fv(x)p Fw(\))28 b(=)1148 4008 y(1)p 1077 4053 191 4 v 1077 4073 a Fs(p)p 1160 4073 108 4 v 82 x Fw(2)p Fv(\031)1295 3981 y Fo(X)1300 4193 y Ft(n)p Fl(2)p Fn(Z)1455 4076 y Fv(e)1500 4034 y Ft(inx)1607 4043 y Fm(1)1662 3940 y Fo(Z)1762 3966 y Fl(1)1718 4166 y Fu(0)1870 4076 y Fv(dy)1969 4091 y Fu(2)1131 4385 y Fv(r)1175 4400 y Fu(\(0\))1270 4385 y Fw(\()p Fv(x)1363 4400 y Fu(2)1402 4385 y Fv(;)17 b(y)1494 4400 y Fu(2)1533 4385 y Fw(;)g Fv(z)27 b Fs(\000)22 b Fw(\()p Fv(n)h Fw(+)f Fv(k)s Fw(\))2057 4343 y Fu(2)2096 4385 y Fw(\))2151 4244 y Fo(\022)2305 4317 y Fw(1)p 2234 4362 191 4 v 2234 4382 a Fs(p)p 2317 4382 108 4 v 82 x Fw(2)p Fv(\031)2451 4249 y Fo(Z)2551 4275 y Ft(\031)2507 4475 y Fl(\000)p Ft(\031)2625 4385 y Fv(u)p Fw(\()p Fv(y)t Fw(\))p Fv(e)2854 4343 y Fl(\000)p Ft(iny)3011 4352 y Fm(1)3065 4385 y Fv(dy)3164 4400 y Fu(1)3202 4244 y Fo(\023)3303 4385 y Fw(=)964 4650 y(=)1107 4583 y(1)p 1077 4627 V 1077 4719 a(2)p Fv(\031)1212 4556 y Fo(X)1216 4767 y Ft(n)p Fl(2)p Fn(Z)1372 4515 y Fo(Z)1427 4740 y Fu(\005)1501 4650 y Fv(e)1546 4609 y Ft(in)p Fu(\()p Ft(x)1680 4618 y Fm(1)1715 4609 y Fl(\000)p Ft(y)1805 4618 y Fm(1)1839 4609 y Fu(\))1871 4650 y Fv(r)1915 4666 y Fu(\(0\))2009 4650 y Fw(\()p Fv(x)2102 4665 y Fu(2)2142 4650 y Fv(;)17 b(y)2234 4665 y Fu(2)2273 4650 y Fw(;)g Fv(z)26 b Fs(\000)d Fw(\()p Fv(n)f Fw(+)g Fv(k)s Fw(\))2796 4609 y Fu(2)2836 4650 y Fw(\))p Fv(u)p Fw(\()p Fv(y)t Fw(\))17 b Fv(dy)28 b Fw(=)964 4950 y(=)1067 4815 y Fo(Z)1123 5040 y Fu(\005)1196 4950 y Fv(r)1243 4909 y Fu(\(0\))1337 4950 y Fw(\()p Fv(x;)17 b(y)t Fw(;)g Fv(z)t(;)g(k)s Fw(\))p Fv(u)p Fw(\()p Fv(y)t Fw(\))g Fv(dy)t(:)120 5163 y Fw(The)24 b(in)m(terc)m(hange)f(of) g(summation)e(and)i(in)m(tegration)e(in)h(the)h(last)f(equalit)m(y)h (is)f(justi\014ed)i(since)120 5283 y(the)35 b(RHS)g(of)41 b(\(2.2.7\))34 b(con)m(v)m(erges)j(in)e Fv(L)1577 5298 y Fu(2)1616 5283 y Fw(\(\005\))g(for)f(\014xed)i Fv(x)p Fw(.)51 b(This)35 b(pro)m(v)m(es)h(the)g(Prop)s(osition.)p 3469 5404 4 66 v 3473 5341 59 4 v 3473 5404 V 3531 5404 4 66 v 1781 5712 a(19)p eop %%Page: 20 25 20 24 bop 120 1071 a Fk(Chapter)78 b(3)120 1486 y(Sp)6 b(ectral)78 b(analysis)e(and)h(scattering)120 1735 y(theory)e(of)g(the) g(op)6 b(erators)76 b Fd(H)2734 1660 y Fz(\()p Fc(\033)6 b Fz(\))2962 1735 y Fb(\()p Fd(k)h Fb(\))75 b Fk(on)120 1984 y(the)j(halfstrip)120 2437 y Fw(Throughout)30 b(this)f(c)m (hapter,)i(w)m(e)g(assume)f(without)f(further)g(commen)m(t)g(that)h Fv(\033)h Fw(:)d Fr(R)39 b Fs(7!)27 b Fr(R)40 b Fw(is)120 2557 y(a)32 b(p)s(erio)s(dic)f(function)h(suc)m(h)i(that)f(for)f(some)g (1)c Fv(<)f(q)32 b Fs(\024)c(1)1603 2736 y Fv(\033)j Fs(2)d Fv(L)1849 2751 y Ft(q)1888 2736 y Fw(\()p Fr(T)p Fw(\))p Fv(:)120 3063 y Fq(3.1)161 b(The)59 b(absolutely)g(con)l(tin)l (uous)e(sp)t(ectrum)g(of)j(the)488 3246 y(op)t(erators)52 b Ff(H)1334 3194 y Fw(\()p Fv(\033)t Fw(\))1474 3246 y Fe(\()p Ff(k)5 b Fe(\))120 3465 y Fw(Our)38 b(goal)e(in)h(this)g (section)h(is)f(to)h(pro)m(v)m(e)h(that)e(the)i(absolutely)e(con)m(tin) m(uous)h(sp)s(ectrum)g(of)120 3585 y Fv(H)209 3549 y Fu(\()p Ft(\033)r Fu(\))310 3585 y Fw(\()p Fv(k)s Fw(\))47 b(coincides)g(with)f Fv(\033)1205 3600 y Ft(ac)1294 3504 y Fo(\000)1340 3585 y Fv(H)1429 3549 y Fu(\(0\))1523 3585 y Fw(\()p Fv(k)s Fw(\))1653 3504 y Fo(\001)1750 3585 y Fw(=)52 b([)p Fv(k)1959 3549 y Fu(2)1998 3585 y Fv(;)17 b Fw(+)p Fs(1)p Fw(\).)85 b(W)-8 b(e)48 b(will)c(deriv)m(e)j (this)g(b)m(y)h(the)120 3714 y(Birman-Kre)-11 b(\025)-38 b(\020n)32 b(theorem)i(from)f(the)i(fact)g(that)f(the)h(di\013erence)g (of)g(resolv)m(en)m(ts)h(of)e Fv(H)3309 3678 y Fu(\()p Ft(\033)r Fu(\))3410 3714 y Fw(\()p Fv(k)s Fw(\))120 3835 y(and)f Fv(H)399 3799 y Fu(\(0\))493 3835 y Fw(\()p Fv(k)s Fw(\))f(is)g(of)g(trace)h(class.)120 3955 y(Let)38 b(us)f(brie\015y)h(sk)m(etc)m(h)i(the)d(problem)g(w)m(e)h(are)f(facing) g(when)h(w)m(e)g(consider)g(the)g(resolv)m(en)m(t)120 4075 y(di\013erence.)44 b(F)-8 b(or)32 b(t)m(w)m(o)h(self-adjoin)m(t)e (op)s(erators)h Fv(H)8 b Fw(,)33 b Fv(H)2116 4090 y Fu(0)2187 4075 y Fw(in)f(a)g(Hilb)s(ert)f(space)j Fg(H)e Fw(satisfying)1491 4255 y Fs(D)s Fw(\()p Fv(H)8 b Fw(\))27 b(=)g Fs(D)s Fw(\()p Fv(H)2065 4270 y Fu(0)2104 4255 y Fw(\))p Fv(;)1095 b Fw(\(3.1.1\))120 4434 y(the)33 b(follo)m(wing)d(resolv)m(en)m(t)k (iden)m(tit)m(y)e(is)g(w)m(ell-kno)m(wn:)418 4605 y(\()p Fv(H)d Fs(\000)23 b Fv(z)t Fw(\))753 4564 y Fl(\000)p Fu(1)870 4605 y Fs(\000)g Fw(\()p Fv(H)1089 4620 y Fu(0)1150 4605 y Fs(\000)g Fv(z)t Fw(\))1337 4564 y Fl(\000)p Fu(1)1459 4605 y Fw(=)28 b Fs(\000)p Fw(\()p Fv(H)1759 4620 y Fu(0)1821 4605 y Fs(\000)22 b Fv(z)t Fw(\))2007 4564 y Fl(\000)p Fu(1)2102 4605 y Fw(\()p Fv(H)30 b Fs(\000)22 b Fv(H)2431 4620 y Fu(0)2471 4605 y Fw(\)\()p Fv(H)29 b Fs(\000)23 b Fv(z)t Fw(\))2844 4564 y Fl(\000)p Fu(1)2939 4605 y Fv(;)613 4750 y(z)32 b Fs(2)c Fv(\032)p Fw(\()p Fv(H)953 4765 y Fu(0)993 4750 y Fw(\))22 b Fs(\\)g Fv(\032)p Fw(\()p Fv(H)8 b Fw(\))p Fv(:)3264 4679 y Fw(\(3.1.2\))120 4922 y(Ho)m(w)m(ev)m(er,)46 b(if)40 b(the)i(op)s(erators)f Fv(H)1347 4937 y Fu(0)1387 4922 y Fw(,)i Fv(H)49 b Fw(are)42 b(de\014ned)h(via)d(quadratic)h(forms,)i(the)f(equalit)m(y)120 5043 y(\(3.1.1\))30 b(ma)m(y)h(not)f(hold,)h(so)g(the)g(op)s(erator)f Fv(H)c Fs(\000)19 b Fv(H)2010 5058 y Fu(0)2080 5043 y Fw(in)30 b(\(3.1.1\))g(ma)m(y)h(not)g(b)s(e)g(w)m(ell-de\014ned.)120 5163 y(In)44 b(Subsection)h(1)e(w)m(e)i(will)c(presen)m(t)46 b(a)d(di\013eren)m(t)h(approac)m(h)g(on)g(the)g(resolv)m(en)m(t)h(iden) m(tit)m(y)120 5284 y(whic)m(h)33 b(is)f(v)-5 b(alid)31 b(under)i(the)g(w)m(eak)m(er)i(assumption)1326 5463 y Fs(D)s Fw(\()p Fs(j)p Fv(H)8 b Fs(j)1589 5422 y Fu(1)p Ft(=)p Fu(2)1697 5463 y Fw(\))28 b(=)f Fs(D)s Fw(\()p Fs(j)p Fv(H)2093 5478 y Fu(0)2132 5463 y Fs(j)2160 5422 y Fu(1)p Ft(=)p Fu(2)2269 5463 y Fw(\))p Fv(:)1781 5712 y Fw(20)p eop %%Page: 21 26 21 25 bop 120 407 a Fw(After)25 b(ha)m(ving)f(in)m(tro)s(duced)h(some)g (auxiliary)d(op)s(erators)j(in)e(Subsection)j(2,)g(w)m(e)g(will)c (deriv)m(e)j(a)120 527 y(symmetric)30 b(factorization)e(of)j(the)g (resolv)m(en)m(t)g(di\013erence)h(in)e(Subsection)h(3.)43 b(The)31 b(required)120 648 y(compactness)j(statemen)m(t)f(follo)m(ws)e (then)i(immediately)d(from)h(Lemma)g(1.2.3.)120 903 y Fp(1.)49 b(A)35 b(general)g(approac)m(h)h(on)g(the)f(in)m(v)m(ersion)f (of)h(a)h(p)s(erturb)s(ed)g(op)s(erator.)98 b Fw(This)120 1024 y(subsection)41 b(presen)m(ts)i(the)d(material)e(from)g(Section)j (1.9)e(in)h([Y)-8 b(a1)o(].)67 b(W)-8 b(e)40 b(do)g(not)h(include)120 1144 y(pro)s(ofs.)i(F)-8 b(or)32 b(a)g(similar)d(construction)k(see)h (also)e(Section)g(5)g(in)g([BSu)q(].)120 1264 y(W)-8 b(e)29 b(w)m(ork)h(in)e(a)g(general)h(setting:)41 b(Let)29 b Fg(H)f Fw(b)s(e)h(a)g(Hilb)s(ert)e(space)j(and)f Fv(H)8 b Fw(,)29 b Fv(H)2874 1279 y Fu(0)2942 1264 y Fw(b)s(e)g(self-adjoin)m (t)120 1385 y(op)s(erators)j(satisfying)1326 1505 y Fs(D)s Fw(\()p Fs(j)p Fv(H)8 b Fs(j)1589 1464 y Fu(1)p Ft(=)p Fu(2)1697 1505 y Fw(\))28 b(=)f Fs(D)s Fw(\()p Fs(j)p Fv(H)2093 1520 y Fu(0)2132 1505 y Fs(j)2160 1464 y Fu(1)p Ft(=)p Fu(2)2269 1505 y Fw(\))p Fv(:)930 b Fw(\(3.1.3\))120 1667 y(W)-8 b(e)48 b(note)f(that)g Fv(H)55 b Fw(and)47 b Fv(H)1181 1682 y Fu(0)1267 1667 y Fw(are)h(not)f(assumed)h(to)e(b)s (e)i(semib)s(ounded.)87 b(Denote)47 b(their)120 1788 y(resolv)m(en)m(ts)34 b(b)m(y)1080 1900 y Fv(R)1154 1915 y Fu(0)1194 1900 y Fw(\()p Fv(z)t Fw(\))28 b(:=)g(\()p Fv(H)1597 1915 y Fu(0)1658 1900 y Fs(\000)23 b Fv(z)t Fw(\))1845 1863 y Fl(\000)p Fu(1)2135 1900 y Fv(z)32 b Fs(2)c Fv(\032)p Fw(\()p Fv(H)2475 1915 y Fu(0)2515 1900 y Fw(\))p Fv(;)1080 2020 y(R)q Fw(\()p Fv(z)t Fw(\))g(:=)g(\()p Fv(H)i Fs(\000)22 b Fv(z)t Fw(\))1774 1984 y Fl(\000)p Fu(1)2135 2020 y Fv(z)32 b Fs(2)c Fv(\032)p Fw(\()p Fv(H)8 b Fw(\))p Fv(:)120 2175 y Fw(Supp)s(ose)34 b(that)e(there)h(is)f(an)h ("auxiliary")d(Hilb)s(ert)h(space)i Fg(G)g Fw(and)g(op)s(erators)828 2367 y Fv(G)905 2382 y Fu(0)972 2367 y Fw(:)28 b Fg(H)f Fs(\033)h(D)s Fw(\()p Fv(G)1426 2382 y Fu(0)1465 2367 y Fw(\))g Fs(!)f Fg(G)p Fv(;)212 b(G)28 b Fw(:)g Fg(H)f Fs(\033)h(D)s Fw(\()p Fv(G)p Fw(\))f Fs(!)h Fg(G)p Fv(;)120 2558 y Fw(suc)m(h)34 b(that)e(the)h(follo)m(wing)d(is)i(true.)117 2737 y(\(H1\))49 b(The)33 b(op)s(erators)g Fv(G)1073 2752 y Fu(0)1112 2737 y Fw(,)g Fv(G)f Fw(are)h Fs(j)p Fv(H)1553 2752 y Fu(0)1592 2737 y Fs(j)1620 2701 y Fu(1)p Ft(=)p Fu(2)1729 2737 y Fw(-b)s(ounded,)g(i.e.,)486 2928 y Fs(D)s Fw(\()p Fs(j)p Fv(H)713 2943 y Fu(0)752 2928 y Fs(j)780 2892 y Fu(1)p Ft(=)p Fu(2)889 2928 y Fw(\))28 b Fs(\032)g(D)s Fw(\()p Fv(G)1255 2943 y Fu(0)1294 2928 y Fw(\))p Fv(;)572 b Fs(D)s Fw(\()p Fs(j)p Fv(H)2158 2943 y Fu(0)2196 2928 y Fs(j)2224 2892 y Fu(1)p Ft(=)p Fu(2)2334 2928 y Fw(\))28 b Fs(\032)g(D)s Fw(\()p Fv(G)p Fw(\))p Fv(;)486 3050 y(G)563 3065 y Fu(0)603 3050 y Fw(\()p Fs(j)p Fv(H)750 3065 y Fu(0)788 3050 y Fs(j)816 3014 y Fu(1)p Ft(=)p Fu(2)948 3050 y Fw(+)22 b Fv(I)8 b Fw(\))1135 3014 y Fl(\000)p Fu(1)1257 3050 y Fs(2)28 b Fg(B)p Fw(\()p Fg(H)p Fv(;)17 b Fg(G)p Fw(\))p Fv(;)195 b(G)p Fw(\()p Fs(j)p Fv(H)2155 3065 y Fu(0)2194 3050 y Fs(j)2222 3014 y Fu(1)p Ft(=)p Fu(2)2354 3050 y Fw(+)22 b Fv(I)8 b Fw(\))2541 3014 y Fl(\000)p Fu(1)2662 3050 y Fs(2)29 b Fg(B)p Fw(\()p Fg(H)p Fv(;)17 b Fg(G)p Fw(\))p Fv(:)3264 2989 y Fw(\(3.1.4\))117 3278 y(\(H2\))49 b(The)33 b(op)s(erators)g Fv(G)1073 3293 y Fu(0)1112 3278 y Fw(,)g Fv(G)f Fw(satisfy)974 3469 y(\()p Fv(G)1089 3484 y Fu(0)1128 3469 y Fv(f)5 b(;)17 b(Gg)t Fw(\))27 b(=)g(\()p Fv(Gf)5 b(;)17 b(G)1810 3484 y Fu(0)1849 3469 y Fv(g)t Fw(\))p Fv(;)212 b(f)5 b(;)33 b(g)e Fs(2)d(D)s Fw(\()p Fs(j)p Fv(H)2689 3484 y Fu(0)2728 3469 y Fs(j)2756 3428 y Fu(1)p Ft(=)p Fu(2)2865 3469 y Fw(\))p Fv(:)334 b Fw(\(3.1.5\))117 3698 y(\(H3\))49 b(The)33 b(relation)e Fv(H)k Fw(=)28 b Fv(H)1223 3713 y Fu(0)1284 3698 y Fw(+)22 b Fv(G)1459 3662 y Fl(\003)1499 3698 y Fv(G)1576 3713 y Fu(0)1648 3698 y Fw(holds)32 b(in)g(the)h(sense)h(of)e(forms,)g(i.e.,)479 3890 y(\()p Fv(H)8 b(f)d(;)17 b(f)751 3905 y Fu(0)790 3890 y Fw(\))27 b(=)h(\()p Fv(f)5 b(;)17 b(H)1175 3905 y Fu(0)1214 3890 y Fv(f)1262 3905 y Fu(0)1301 3890 y Fw(\))22 b(+)g(\()p Fv(Gf)5 b(;)17 b(G)1748 3905 y Fu(0)1788 3890 y Fv(f)1836 3905 y Fu(0)1875 3890 y Fw(\))p Fv(;)212 b(f)2200 3905 y Fu(0)2267 3890 y Fs(2)28 b(D)s Fw(\()p Fv(H)2560 3905 y Fu(0)2599 3890 y Fw(\))p Fv(;)33 b(f)39 b Fs(2)28 b(D)s Fw(\()p Fv(H)8 b Fw(\))p Fv(:)114 b Fw(\(3.1.6\))120 4119 y(The)53 b(assumption)e(\(H1\))g(guaran)m(tees)i(that)e(the)i(op)s (erators)e Fv(GR)2648 4134 y Fu(0)2688 4119 y Fw(\()p Fv(z)t Fw(\))61 b(:)f Fg(H)g Fs(!)g Fg(G)52 b Fw(and)120 4239 y Fv(G)197 4254 y Fu(0)236 4239 y Fw(\()p Fv(GR)425 4254 y Fu(0)465 4239 y Fw(\()p 503 4184 50 4 v Fv(z)5 b Fw(\)\))629 4203 y Fl(\003)714 4239 y Fw(:)47 b Fg(G)g Fs(!)f Fg(G)e Fw(are)f(w)m(ell-de\014ned)h(and)g(b)s(ounded)g(for)f Fv(z)51 b Fs(2)c Fv(\032)p Fw(\()p Fv(H)3047 4254 y Fu(0)3087 4239 y Fw(\).)76 b(Indeed,)120 4360 y(with)32 b Fv(L)408 4375 y Fu(0)476 4360 y Fw(:=)27 b(\()p Fs(j)p Fv(H)753 4375 y Fu(0)792 4360 y Fs(j)820 4324 y Fu(1)p Ft(=)p Fu(2)952 4360 y Fw(+)22 b Fv(I)8 b Fw(\)\()p Fs(j)p Fv(H)1286 4375 y Fu(0)1325 4360 y Fs(j)21 b Fw(+)i Fv(I)8 b Fw(\))1562 4324 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1754 4360 y Fs(2)28 b Fg(B)p Fw(\()p Fg(H)p Fw(\))k(w)m(e)i(see)f(that)460 4551 y Fv(GR)611 4566 y Fu(0)651 4551 y Fw(\()p Fv(z)t Fw(\))28 b(=)908 4471 y Fo(\000)953 4551 y Fv(G)p Fw(\()p Fs(j)p Fv(H)1177 4566 y Fu(0)1216 4551 y Fs(j)1244 4510 y Fu(1)p Ft(=)p Fu(2)1376 4551 y Fw(+)22 b Fv(I)8 b Fw(\))1563 4510 y Fl(\000)p Fu(1)1657 4471 y Fo(\001)1719 4551 y Fv(L)1785 4566 y Fu(0)1842 4551 y Fw(\(\()p Fs(j)p Fv(H)2027 4566 y Fu(0)2065 4551 y Fs(j)22 b Fw(+)g Fv(I)8 b Fw(\))17 b Fv(R)2393 4566 y Fu(0)2432 4551 y Fw(\()p Fv(z)t Fw(\)\))h(\()p Fs(j)p Fv(H)2760 4566 y Fu(0)2798 4551 y Fs(j)k Fw(+)g Fv(I)8 b Fw(\))3035 4510 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)120 4743 y Fw(and)120 4934 y Fv(G)197 4949 y Fu(0)236 4934 y Fw(\()p Fv(GR)425 4949 y Fu(0)465 4934 y Fw(\()p 503 4880 V Fv(z)d Fw(\)\))629 4893 y Fl(\003)696 4934 y Fw(=)799 4854 y Fo(\000)845 4934 y Fv(G)922 4949 y Fu(0)961 4934 y Fw(\()p Fs(j)p Fv(H)1108 4949 y Fu(0)1147 4934 y Fs(j)1175 4893 y Fu(1)p Ft(=)p Fu(2)1307 4934 y Fw(+)22 b Fv(I)8 b Fw(\))1494 4893 y Fl(\000)p Fu(1)1588 4854 y Fo(\001)1650 4934 y Fv(L)1716 4949 y Fu(0)1773 4934 y Fw(\(\()p Fs(j)p Fv(H)1958 4949 y Fu(0)1997 4934 y Fs(j)21 b Fw(+)h Fv(I)8 b Fw(\))17 b Fv(R)2324 4949 y Fu(0)2363 4934 y Fw(\()p 2401 4880 V Fv(z)5 b Fw(\)\))2527 4886 y Fl(\003)2583 4934 y Fv(L)2649 4893 y Fl(\003)2649 4959 y Fu(0)2705 4854 y Fo(\000)2751 4934 y Fv(G)p Fw(\()p Fs(j)p Fv(H)2975 4949 y Fu(0)3014 4934 y Fs(j)3042 4893 y Fu(1)p Ft(=)p Fu(2)3174 4934 y Fw(+)22 b Fv(I)8 b Fw(\))3361 4893 y Fl(\000)p Fu(1)3455 4854 y Fo(\001)3501 4870 y Fl(\003)120 5126 y Fw(are)33 b(pro)s(ducts)g(of)f(b)s(ounded)h(op)s(erators.)120 5246 y(With)f(sligh)m(t)f(abuse)j(of)e(notation)f(w)m(e)j(put)635 5438 y Fv(R)709 5453 y Fu(0)749 5438 y Fw(\()p Fv(z)t Fw(\))p Fv(G)951 5397 y Fl(\003)1018 5438 y Fw(:=)28 b(\()p Fv(GR)1338 5453 y Fu(0)1378 5438 y Fw(\()p 1416 5383 V Fv(z)t Fw(\)\))1541 5397 y Fl(\003)1581 5438 y Fv(;)211 b(G)1896 5453 y Fu(0)1936 5438 y Fv(R)2010 5453 y Fu(0)2050 5438 y Fw(\()p Fv(z)t Fw(\))p Fv(G)2252 5397 y Fl(\003)2319 5438 y Fw(:=)28 b Fv(G)2527 5453 y Fu(0)2566 5438 y Fw(\()p Fv(GR)2755 5453 y Fu(0)2795 5438 y Fw(\()p 2833 5383 V Fv(z)5 b Fw(\)\))2959 5397 y Fl(\003)2998 5438 y Fv(:)1781 5712 y Fw(21)p eop %%Page: 22 27 22 26 bop 120 407 a Fp(Prop)s(osition)50 b(3.1.1.)e Fj(L)-5 b(et)46 b Fv(H)1309 422 y Fu(0)1349 407 y Fv(;)17 b(H)53 b Fj(b)-5 b(e)46 b(self-adjoint)f(op)-5 b(er)g(ators)46 b(satisfying)54 b Fw(\(3.1.3\))46 b Fj(and)120 527 y(assume)29 b(that)h(the)f(op)-5 b(er)g(ators)29 b Fv(G)1301 542 y Fu(0)1341 527 y Fv(;)17 b(G)29 b Fj(satisfy)g(\(H1\)-\(H3\).)43 b(L)-5 b(et)30 b Fv(z)i Fs(2)d Fv(\032)p Fw(\()p Fv(H)2805 542 y Fu(0)2844 527 y Fw(\))p Fj(,)h(then)g Fv(z)i Fs(2)c Fv(\032)p Fw(\()p Fv(H)8 b Fw(\))120 648 y Fj(i\013)35 b Fv(I)30 b Fw(+)22 b Fv(G)493 663 y Fu(0)532 648 y Fv(R)606 663 y Fu(0)646 648 y Fw(\()p Fv(z)t Fw(\))p Fv(G)848 611 y Fl(\003)922 648 y Fj(is)35 b(b)-5 b(ounde)g(d)5 b(ly)34 b(invertible,)g(and)h(in)f(this)h(c)-5 b(ase)514 863 y Fv(R)q Fw(\()p Fv(z)t Fw(\))23 b Fs(\000)g Fv(R)911 878 y Fu(0)950 863 y Fw(\()p Fv(z)t Fw(\))29 b(=)e Fs(\000)p Fv(R)1358 878 y Fu(0)1398 863 y Fw(\()p Fv(z)t Fw(\))p Fv(G)1600 822 y Fl(\003)1657 863 y Fw(\()p Fv(I)j Fw(+)22 b Fv(G)1943 878 y Fu(0)1982 863 y Fv(R)2056 878 y Fu(0)2096 863 y Fw(\()p Fv(z)t Fw(\))p Fv(G)2298 822 y Fl(\003)2338 863 y Fw(\))2376 814 y Fl(\000)p Fu(1)2487 863 y Fv(G)2564 878 y Fu(0)2603 863 y Fv(R)2677 878 y Fu(0)2717 863 y Fw(\()p Fv(z)t Fw(\))p Fv(:)395 b Fw(\(3.1.7\))266 1078 y(F)-8 b(or)32 b(the)h Fj(pr)-5 b(o)g(of)53 b Fw(w)m(e)34 b(refer)f(to)f(Section)g(1.9)g(in)g([Y)-8 b(a1].)120 1337 y Fp(2.)50 b(Some)36 b(auxiliary)f(op)s(erators.)98 b Fw(F)-8 b(or)31 b Fv(k)g Fs(2)d Fw([)p Fs(\000)2076 1298 y Fu(1)p 2076 1314 36 4 v 2076 1371 a(2)2122 1337 y Fv(;)2175 1298 y Fu(1)p 2175 1314 V 2175 1371 a(2)2220 1337 y Fw(],)k Fv(z)h Fs(2)28 b Fr(C)46 b Fs(n)20 b Fw([)p Fv(k)2721 1301 y Fu(2)2761 1337 y Fv(;)d Fw(+)p Fs(1)p Fw(\))31 b(w)m(e)i(consider)120 1458 y(in)f Fv(L)300 1473 y Fu(2)340 1458 y Fw(\()p Fr(T)p Fw(\))j(the)e(op)s(erator)1008 1670 y Fs(D)s Fw(\()p Fv(B)5 b Fw(\()p Fv(z)t(;)17 b(k)s Fw(\)\))28 b(:=)g Fv(H)1714 1629 y Fu(1)1753 1670 y Fw(\()p Fr(T)p Fw(\))p Fv(;)1105 1844 y(B)5 b Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))p Fv(f)39 b Fw(:=)1625 1749 y Fo(X)1630 1961 y Ft(n)p Fl(2)p Fn(Z)1785 1754 y Fo(p)p 1885 1754 520 4 v 90 x Fw(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))2193 1815 y Fu(2)2255 1844 y Fs(\000)g Fv(z)2442 1818 y Fw(^)2421 1844 y Fv(f)2469 1859 y Ft(n)2532 1844 y Fv(e)2577 1859 y Ft(n)2625 1844 y Fv(:)3264 1806 y Fw(\(3.1.8\))120 2147 y(Recall)30 b(that)i(w)m(e)h(ha)m(v)m(e)h(agreed)e(to)g(c)m(ho)s (ose)h(the)f(canonical)f(branc)m(h)i(of)e(the)i(square)g(ro)s(ot)e(on) 120 2268 y Fr(C)48 b Fs(n)22 b Fw(\()p Fs(\0001)p Fv(;)17 b Fw(0])32 b(satisfying)g(Re)1225 2213 y Fs(p)p 1308 2213 28 4 v 55 x Fv(:)c(>)f Fw(0.)120 2388 y(The)i(op)s(erator)f Fv(B)5 b Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))28 b(is)g(in)m(v)m(ertible)f (for)h Fv(z)k Fs(2)c Fr(C)33 b Fs(n)13 b Fw([)p Fv(k)2095 2352 y Fu(2)2140 2388 y Fv(;)k Fw(+)p Fs(1)p Fw(\))27 b(and)h(its)g(square)h(ro)s(ot)f(is)f(w)m(ell-)120 2509 y(de\014ned)36 b(\(it)d(ma)m(y)g(b)s(e)i(considered)g(as)f(the)h (square)g(ro)s(ot)e(of)h(an)g(m-accretiv)m(e)g(op)s(erator\).)47 b(If)120 2629 y Fv(M)224 2593 y Fu(\()p Ft(\033)r Fu(\))362 2629 y Fw(denotes)36 b(as)g(in)e(Section)h(1.2)g(the)h(op)s(erator)e (of)h(m)m(ultiplication)c(b)m(y)36 b Fv(\033)j Fw(in)c Fv(L)3184 2644 y Fu(2)3224 2629 y Fw(\()p Fr(T)p Fw(\),)k(w)m(e)120 2749 y(de\014ne)908 2858 y Fv(T)979 2807 y Fu(\()p Ft(\033)r Fu(\))965 2883 y(0)1080 2858 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))28 b(:=)g(\(sgn)p Fv(M)1745 2822 y Fu(\()p Ft(\033)r Fu(\))1848 2858 y Fw(\))p Fs(j)p Fv(M)2018 2822 y Fu(\()p Ft(\033)r Fu(\))2120 2858 y Fs(j)2148 2822 y Fu(1)p Ft(=)p Fu(2)2258 2858 y Fv(B)5 b Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))2560 2822 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)2725 2858 y Fv(;)908 2980 y(T)979 2944 y Fu(\()p Ft(\033)r Fu(\))1080 2980 y Fw(\()p Fv(z)t(;)g(k)s Fw(\))28 b(:=)g Fs(j)p Fv(M)1594 2944 y Fu(\()p Ft(\033)r Fu(\))1696 2980 y Fs(j)1724 2944 y Fu(1)p Ft(=)p Fu(2)1834 2980 y Fv(B)5 b Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))2136 2944 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)2301 2980 y Fv(:)120 3152 y Fw(These)33 b(are)f(compact)f (op)s(erators)g(of)g(class)g(\006)1789 3167 y Fu(2)1829 3152 y Fw(\()p Fv(L)1933 3167 y Fu(2)1973 3152 y Fw(\()p Fr(T)p Fw(\)\).)46 b(Indeed,)33 b(let)e Fv(B)37 b Fw(as)31 b(in)g(Section)g(1.2,)120 3272 y(then)i Fv(B)421 3236 y Fu(1)p Ft(=)p Fu(2)531 3272 y Fv(B)5 b Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))833 3236 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1026 3272 y Fs(2)29 b Fg(B)p Fw(\()p Fv(L)1313 3287 y Fu(2)1352 3272 y Fw(\()p Fr(T)p Fw(\)\))36 b(and)d(hence)715 3487 y Fv(T)786 3446 y Fu(\()p Ft(\033)r Fu(\))887 3487 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))28 b(=)1242 3406 y Fo(\000)1288 3487 y Fs(j)p Fv(M)1420 3446 y Fu(\()p Ft(\033)r Fu(\))1522 3487 y Fs(j)1550 3446 y Fu(1)p Ft(=)p Fu(2)1659 3487 y Fv(B)1738 3446 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1903 3406 y Fo(\001)17 b(\000)2011 3487 y Fv(B)2090 3446 y Fu(1)p Ft(=)p Fu(2)2200 3487 y Fv(B)5 b Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))2502 3446 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)2668 3406 y Fo(\001)2741 3487 y Fs(2)28 b Fw(\006)2905 3502 y Fu(2)120 3726 y Fw(and)33 b Fv(T)381 3675 y Fu(\()p Ft(\033)r Fu(\))367 3750 y(0)482 3726 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))28 b(=)g(\(sgn)p Fv(M)1120 3690 y Fu(\()p Ft(\033)r Fu(\))1223 3726 y Fw(\))p Fv(T)1332 3690 y Fu(\()p Ft(\033)r Fu(\))1433 3726 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))28 b Fs(2)g Fw(\006)1848 3741 y Fu(2)1888 3726 y Fw(.)120 3846 y(Finally)-8 b(,)25 b(for)g Fv(z)33 b Fs(2)28 b Fr(C)h Fs(n)9 b Fw([)p Fv(k)995 3810 y Fu(2)1040 3846 y Fv(;)17 b Fw(+)p Fs(1)p Fw(\))25 b(w)m(e)i(consider)g(the)f(b)s (ounded)h(op)s(erator)f Fv(Y)21 b Fw(\()p Fv(z)t(;)c(k)s Fw(\))28 b(:)g Fv(L)3224 3861 y Fu(2)3264 3846 y Fw(\(\005\))f Fs(!)120 3967 y Fv(L)186 3982 y Fu(2)226 3967 y Fw(\()p Fr(T)p Fw(\))p Fv(;)190 4239 y(Y)21 b Fw(\()p Fv(z)t(;)c(k)s Fw(\))p Fv(u)28 b Fw(:=)706 4145 y Fo(X)710 4356 y Ft(n)p Fl(2)p Fn(Z)1163 4172 y Fw(1)p 876 4216 622 4 v 904 4284 a Fm(4)879 4236 y Fo(p)p 978 4236 520 4 v 978 4321 a Fw(\()p Fv(n)23 b Fw(+)f Fv(k)s Fw(\))1287 4293 y Fu(2)1348 4321 y Fs(\000)h Fv(z)1524 4099 y Fo(\022)1679 4172 y Fw(1)p 1608 4216 191 4 v 1608 4236 a Fs(p)p 1691 4236 108 4 v 83 x Fw(2)p Fv(\031)1825 4104 y Fo(Z)1880 4329 y Fu(\005)1954 4239 y Fv(u)p Fw(\()p Fv(x)p Fw(\))p Fv(e)2186 4198 y Fl(\000)p Ft(inx)2348 4207 y Fm(1)2386 4239 y Fv(e)2431 4198 y Fl(\000)2486 4131 y Fs(p)p 2569 4131 317 4 v 67 x Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))2760 4179 y Fm(2)2795 4198 y Fl(\000)p Ft(z)14 b(x)2937 4207 y Fm(2)2998 4239 y Fv(dx)3104 4099 y Fo(\023)3194 4239 y Fv(e)3239 4254 y Ft(n)3314 4239 y Fw(=)575 4535 y(=)679 4441 y Fo(X)683 4652 y Ft(n)p Fl(2)p Fn(Z)1604 4468 y Fw(1)p 849 4512 1560 4 v 877 4616 a Fm(4)852 4569 y Fo(p)p 951 4569 520 4 v 951 4654 a Fw(\()p Fv(n)23 b Fw(+)f Fv(k)s Fw(\))1260 4625 y Fu(2)1321 4654 y Fs(\000)h Fv(z)1471 4532 y Fo(q)p 1570 4532 839 4 v 1587 4654 a Fw(2)17 b(Re)1790 4569 y Fo(p)p 1889 4569 520 4 v 1889 4654 a Fw(\()p Fv(n)22 b Fw(+)h Fv(k)s Fw(\))2198 4625 y Fu(2)2259 4654 y Fs(\000)g Fv(z)2435 4425 y Fo(\020)2495 4535 y Fv(u;)17 b(e)2640 4550 y Ft(n)2708 4535 y Fs(\012)23 b Fv( )2871 4509 y Fs(p)p 2954 4509 317 4 v 67 x Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))3145 4557 y Fm(2)3180 4576 y Fl(\000)p 3235 4538 36 3 v Ft(z)3275 4425 y Fo(\021)3351 4535 y Fv(e)3396 4550 y Ft(n)3443 4535 y Fv(;)120 4880 y Fw(where)34 b(w)m(e)f(ha)m(v)m (e)h(put)f(for)f(Re)22 b Fv(\020)35 b(>)28 b Fw(0)1016 5095 y Fv( )1079 5110 y Ft(\020)1119 5095 y Fw(\()p Fv(x)1212 5110 y Fu(2)1252 5095 y Fw(\))f(:=)1448 5005 y Fo(p)p 1547 5005 253 4 v 1547 5095 a Fw(2)17 b(Re)22 b Fv(\020)i(e)1862 5054 y Fl(\000)p Ft(\020)5 b(x)1993 5063 y Fm(2)2031 5095 y Fv(;)212 b(x)2325 5110 y Fu(2)2392 5095 y Fs(2)28 b Fr(R)2552 5110 y Fu(+)2617 5095 y Fv(:)120 5311 y Fw(Since)h(the)g (functions)g Fv(e)997 5326 y Ft(n)1058 5311 y Fs(\012)14 b Fv( )1212 5284 y Fs(p)p 1297 5284 317 4 v 1297 5351 a Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))1488 5332 y Fm(2)1522 5351 y Fl(\000)p 1577 5314 36 3 v Ft(z)1646 5311 y Fw(are)29 b(normalized)d(in)i Fv(L)2475 5326 y Fu(2)2515 5311 y Fw(\(\005\))h(it)f(is)g(easy)i(to)e(see)i(that)120 5463 y Fv(Y)21 b Fw(\()p Fv(z)t(;)c(k)s Fw(\))33 b(is)f(a)h(compact)f(op)s (erator)g(of)g(class)g(\006)1826 5478 y Fu(1)1866 5463 y Fw(\()p Fv(L)1970 5478 y Fu(2)2010 5463 y Fw(\(\005\))p Fv(;)17 b(L)2269 5478 y Fu(2)2309 5463 y Fw(\()p Fr(T)p Fw(\)\).)1781 5712 y(22)p eop %%Page: 23 28 23 27 bop 120 407 a Fp(3.)80 b(The)48 b(resolv)m(en)m(t)f (di\013erence.)81 b(The)47 b(absolutely)g(con)m(tin)m(uous)h(sp)s (ectrum)f(of)120 527 y(the)e(op)s(erators)h Fv(H)919 491 y Fu(\()p Ft(\033)r Fu(\))1020 527 y Fw(\()p Fv(k)s Fw(\))p Fp(.)97 b Fw(W)-8 b(e)40 b(are)f(no)m(w)h(ready)g(to)f(apply)g (the)h(general)f(results)g(from)120 648 y(Subsection)j(1)f(to)g(our)g (situation.)67 b(Denote)41 b(the)h(resolv)m(en)m(ts)h(of)e(the)g(op)s (erators)g Fv(H)3289 611 y Fu(\(0\))3383 648 y Fw(\()p Fv(k)s Fw(\),)120 768 y Fv(H)209 732 y Fu(\()p Ft(\033)r Fu(\))310 768 y Fw(\()p Fv(k)s Fw(\))33 b(b)m(y)428 982 y Fv(R)503 946 y Fu(\(0\))597 982 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))28 b(:=)g(\()p Fv(H)1106 946 y Fu(\(0\))1200 982 y Fw(\()p Fv(k)s Fw(\))22 b Fs(\000)h Fv(z)t Fw(\))1539 946 y Fl(\000)p Fu(1)1634 982 y Fv(;)210 b(z)32 b Fs(2)c Fr(C)48 b Fs(n)22 b Fw([)p Fv(k)2289 946 y Fu(2)2329 982 y Fv(;)17 b Fw(+)p Fs(1)p Fw(\))27 b(=)g Fv(\032)2784 901 y Fo(\000)2830 982 y Fv(H)2919 946 y Fu(\(0\))3013 982 y Fw(\()p Fv(k)s Fw(\))3143 901 y Fo(\001)3205 982 y Fv(;)428 1104 y(R)503 1068 y Fu(\()p Ft(\033)r Fu(\))605 1104 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))28 b(:=)g(\()p Fv(H)1114 1068 y Fu(\()p Ft(\033)r Fu(\))1215 1104 y Fw(\()p Fv(k)s Fw(\))22 b Fs(\000)g Fv(z)t Fw(\))1553 1068 y Fl(\000)p Fu(1)1649 1104 y Fv(;)195 b(z)32 b Fs(2)c Fv(\032)2109 1023 y Fo(\000)2155 1104 y Fv(H)2244 1068 y Fu(\()p Ft(\033)r Fu(\))2345 1104 y Fw(\()p Fv(k)s Fw(\))2475 1023 y Fo(\001)2537 1104 y Fv(:)120 1313 y Fw(The)34 b(follo)m(wing)29 b(statemen)m(t)k(is)g(of)f(crucial)f(imp)s (ortance.)120 1512 y Fp(Prop)s(osition)g(3.1.2.)40 b Fj(L)-5 b(et)31 b Fv(k)g Fs(2)d Fw([)p Fs(\000)1476 1473 y Fu(1)p 1476 1489 36 4 v 1476 1546 a(2)1522 1512 y Fv(;)1576 1473 y Fu(1)p 1576 1489 V 1576 1546 a(2)1621 1512 y Fw(])k Fj(and)e Fv(z)j Fs(2)28 b Fr(C)35 b Fs(n)15 b Fw([)p Fv(k)2264 1476 y Fu(2)2309 1512 y Fv(;)i Fw(+)p Fs(1)p Fw(\))p Fj(.)42 b(Then)31 b Fv(z)h Fs(2)d Fv(\032)3129 1431 y Fo(\000)3174 1512 y Fv(H)3263 1476 y Fu(\()p Ft(\033)r Fu(\))3364 1512 y Fw(\()p Fv(k)s Fw(\))3494 1431 y Fo(\001)120 1656 y Fj(i\013)35 b(the)f(op)-5 b(er)g(ator)35 b Fv(I)30 b Fw(+)22 b Fv(T)1032 1605 y Fu(\()p Ft(\033)r Fu(\))1018 1680 y(0)1133 1656 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))23 b Fv(T)1450 1620 y Fu(\()p Ft(\033)r Fu(\))1551 1656 y Fw(\()p 1589 1601 50 4 v Fv(z)5 b(;)17 b(k)s Fw(\))1775 1620 y Fl(\003)1849 1656 y Fj(is)35 b(b)-5 b(ounde)g(d)5 b(ly)34 b(invertible,)g(and)g(in)h(this)g(c)-5 b(ase)218 1875 y Fv(R)293 1834 y Fu(\()p Ft(\033)r Fu(\))395 1875 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))22 b Fs(\000)h Fv(R)815 1834 y Fu(\(0\))909 1875 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))28 b(=)445 2076 y(=)f Fs(\000)p Fv(Y)22 b Fw(\()p 742 2021 V Fv(z)5 b(;)17 b(k)s Fw(\))928 2034 y Fl(\003)989 2076 y Fv(T)1060 2034 y Fu(\()p Ft(\033)r Fu(\))1162 2076 y Fw(\()p 1200 2021 V Fv(z)t(;)g(k)s Fw(\))1385 2034 y Fl(\003)1441 1965 y Fo(\020)1501 2076 y Fv(I)30 b Fw(+)22 b Fv(T)1743 2025 y Fu(\()p Ft(\033)r Fu(\))1729 2100 y(0)1844 2076 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))23 b Fv(T)2161 2034 y Fu(\()p Ft(\033)r Fu(\))2262 2076 y Fw(\()p 2300 2021 V Fv(z)5 b(;)17 b(k)s Fw(\))2486 2034 y Fl(\003)2525 1965 y Fo(\021)2584 1987 y Fl(\000)p Fu(1)2695 2076 y Fv(T)2766 2025 y Fu(\()p Ft(\033)r Fu(\))2752 2100 y(0)2868 2076 y Fw(\()p Fv(z)t(;)g(k)s Fw(\))22 b Fv(Y)g Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))p Fv(:)3264 2233 y Fw(\(3.1.9\))120 2447 y Fj(Pr)-5 b(o)g(of.)48 b Fw(W)-8 b(e)45 b(w)m(an)m(t)h(to)e(apply)g(the)h(results)g(of)f (Subsection)h(1)f(to)g(the)h(case)h Fg(H)h Fw(=)h Fv(L)3324 2462 y Fu(2)3364 2447 y Fw(\(\005\),)120 2567 y Fg(G)28 b Fw(=)f Fv(L)395 2582 y Fu(2)435 2567 y Fw(\()p Fr(T)p Fw(\),)36 b Fv(H)718 2582 y Fu(0)785 2567 y Fw(=)28 b Fv(H)978 2531 y Fu(\(0\))1071 2567 y Fw(\()p Fv(k)s Fw(\),)33 b Fv(H)i Fw(=)28 b Fv(H)1570 2531 y Fu(\()p Ft(\033)r Fu(\))1671 2567 y Fw(\()p Fv(k)s Fw(\))33 b(and)819 2787 y Fs(D)s Fw(\()p Fv(G)1014 2802 y Fu(0)1053 2787 y Fw(\))28 b(:=)f Fs(D)s Fw(\()p Fv(G)p Fw(\))g(:=)1665 2762 y(~)1640 2787 y Fv(H)1729 2746 y Fu(1)1768 2787 y Fw(\(\005\))p Fv(;)672 2954 y Fw(\()p Fv(G)787 2969 y Fu(0)827 2954 y Fv(u)p Fw(\)\()p Fv(x)1014 2969 y Fu(1)1053 2954 y Fw(\))h(:=)f(\(sgn)c Fv(\033)t Fw(\()p Fv(x)1603 2969 y Fu(1)1643 2954 y Fw(\)\))1719 2864 y Fo(p)p 1818 2864 286 4 v 1818 2954 a Fs(j)p Fv(\033)t Fw(\()p Fv(x)1998 2969 y Fu(1)2038 2954 y Fw(\))p Fs(j)16 b Fv(u)p Fw(\()p Fv(x)2269 2969 y Fu(1)2308 2954 y Fv(;)h Fw(0\))p Fv(;)211 b(x)2732 2969 y Fu(1)2800 2954 y Fs(2)28 b Fr(T)p Fv(;)712 3121 y Fw(\()p Fv(Gu)p Fw(\)\()p Fv(x)1014 3136 y Fu(1)1053 3121 y Fw(\))g(:=)1249 3031 y Fo(p)p 1349 3031 V 90 x Fs(j)p Fv(\033)t Fw(\()p Fv(x)1529 3136 y Fu(1)1568 3121 y Fw(\))p Fs(j)17 b Fv(u)p Fw(\()p Fv(x)1800 3136 y Fu(1)1839 3121 y Fv(;)g Fw(0\))p Fv(;)211 b(x)2263 3136 y Fu(1)2331 3121 y Fs(2)28 b Fr(T)p Fv(:)120 3330 y Fw(According)33 b(to)g(Corollary)f(1.2.5,)h(the)h(op)s(erators)f Fv(G)2075 3345 y Fu(0)2115 3330 y Fw(,)g Fv(G)h Fw(are)f(w)m(ell-de\014ned)h(and) f(b)s(ounded)120 3450 y(from)384 3425 y(~)359 3450 y Fv(H)448 3414 y Fu(1)487 3450 y Fw(\(\005\))42 b(to)f Fv(L)872 3465 y Fu(2)911 3450 y Fw(\()p Fr(T)p Fw(\).)74 b(Since)41 b(\()p Fs(j)p Fv(H)1561 3465 y Fu(0)1600 3450 y Fs(j)1628 3414 y Fu(1)p Ft(=)p Fu(2)1766 3450 y Fw(+)28 b Fv(I)8 b Fw(\))1959 3414 y Fl(\000)p Fu(1)2095 3450 y Fw(is)41 b(a)g(b)s(ounded)h(op)s(erator)f(from)f Fg(H)h Fw(to)145 3546 y(~)120 3571 y Fv(H)209 3535 y Fu(1)248 3571 y Fw(\(\005\),)33 b(the)g(assumption)f(\(H1\))g(in)g(Subsection)i (1)e(holds.)44 b(The)33 b(remaining)f(\(3.1.3\))o(,)h(\(H2\))120 3691 y(and)38 b(\(H3\))f(are)g(ob)m(vious,)j(so)d(that)h(the)g (statemen)m(t)g(of)f(Prop)s(osition)f(3.1.1)h(holds.)58 b(Let)37 b(us)120 3812 y(determine)32 b(the)h(pro)s(ducts)h Fv(G)1225 3827 y Fu(0)1264 3812 y Fv(R)1338 3827 y Fu(0)1378 3812 y Fw(\()p Fv(z)t Fw(\),)f Fv(GR)1714 3827 y Fu(0)1753 3812 y Fw(\()p Fv(z)t Fw(\))h(and)e Fv(G)2178 3827 y Fu(0)2218 3812 y Fv(R)2292 3827 y Fu(0)2331 3812 y Fw(\()p Fv(z)t Fw(\))p Fv(G)2533 3775 y Fl(\003)2573 3812 y Fw(.)120 3932 y(The)i(explicit)d(form)g(\(2.2.7\))h(of)g(the)h(free)g(resolv)m (en)m(t)h(implies)c(that)i(for)g Fv(u)27 b Fs(2)2974 3907 y Fw(~)2948 3932 y Fv(H)3037 3896 y Fu(1)3076 3932 y Fw(\(\005\))201 4196 y(\()p Fv(R)313 4211 y Fu(0)353 4196 y Fw(\()p Fv(z)t Fw(\))p Fv(u)p Fw(\)\()p Fv(x)665 4211 y Fu(1)705 4196 y Fv(;)17 b Fw(0\))27 b(=)1006 4128 y(1)p 976 4173 108 4 v 976 4264 a(2)p Fv(\031)1111 4101 y Fo(X)1115 4313 y Ft(n)p Fl(2)p Fn(Z)1566 4128 y Fw(1)p 1281 4173 619 4 v 1281 4193 a Fo(p)p 1381 4193 520 4 v 85 x Fw(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))1689 4249 y Fu(2)1751 4278 y Fs(\000)g Fv(z)1927 4060 y Fo(Z)1982 4286 y Fu(\005)2055 4196 y Fv(u)p Fw(\()p Fv(y)t Fw(\))17 b Fv(e)2301 4155 y Ft(in)p Fu(\()p Ft(x)2435 4164 y Fm(1)2468 4155 y Fl(\000)p Ft(y)2558 4164 y Fm(1)2593 4155 y Fu(\))2641 4196 y Fv(e)2686 4155 y Fl(\000)2741 4088 y Fs(p)p 2824 4088 317 4 v 67 x Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))3015 4136 y Fm(2)3050 4155 y Fl(\000)p Ft(z)d(y)3187 4164 y Fm(2)3253 4196 y Fv(dy)31 b Fw(=)863 4451 y(=)966 4370 y Fo(\000)1012 4451 y Fv(B)5 b Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))1314 4410 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1496 4451 y Fv(Y)k Fw(\()p Fv(z)t(;)c(k)s Fw(\))p Fv(u)1853 4370 y Fo(\001)1916 4451 y Fw(\()p Fv(x)2009 4466 y Fu(1)2048 4451 y Fw(\))p Fv(;)212 b(x)2380 4466 y Fu(1)2448 4451 y Fs(2)28 b Fr(T)p Fv(;)120 4660 y Fw(and)33 b(hence)516 4875 y Fv(G)593 4890 y Fu(0)632 4875 y Fv(R)706 4890 y Fu(0)746 4875 y Fw(\()p Fv(z)t Fw(\))28 b(=)g Fv(T)1074 4824 y Fu(\()p Ft(\033)r Fu(\))1060 4899 y(0)1175 4875 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))g Fv(Y)k Fw(\()p Fv(z)t(;)c(k)s Fw(\))p Fv(;)213 b(GR)2107 4890 y Fu(0)2146 4875 y Fw(\()p Fv(z)t Fw(\))29 b(=)e Fv(T)2474 4834 y Fu(\()p Ft(\033)r Fu(\))2576 4875 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))g Fv(Y)k Fw(\()p Fv(z)t(;)c(k)s Fw(\))p Fv(:)120 5090 y Fw(Moreo)m(v)m(er,)34 b(for)e Fv(f)39 b Fs(2)28 b Fv(L)971 5105 y Fu(2)1011 5090 y Fw(\()p Fr(T)p Fw(\))35 b(w)m(e)f(ha)m(v)m(e)278 5381 y(\()p Fv(Y)21 b Fw(\()p 432 5327 50 4 v Fv(z)t(;)c(k)s Fw(\))617 5340 y Fl(\003)657 5381 y Fv(f)11 b Fw(\)\()p Fv(x)p Fw(\))27 b(=)1097 5314 y(1)p 1026 5358 191 4 v 1026 5378 a Fs(p)p 1109 5378 108 4 v 83 x Fw(2)p Fv(\031)1243 5287 y Fo(X)1248 5499 y Ft(n)p Fl(2)p Fn(Z)1698 5288 y Fw(^)1677 5314 y Fv(f)1725 5329 y Ft(n)p 1413 5358 622 4 v 1441 5426 a Fm(4)1416 5378 y Fo(p)p 1516 5378 520 4 v 86 x Fw(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))1824 5435 y Fu(2)1886 5464 y Fs(\000)g Fv(z)2062 5381 y(e)2107 5340 y Ft(inx)2214 5349 y Fm(1)2269 5381 y Fv(e)2314 5340 y Fl(\000)2369 5273 y Fs(p)p 2452 5273 317 4 v 67 x Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))2643 5321 y Fm(2)2678 5340 y Fl(\000)p Ft(z)g(x)2828 5349 y Fm(2)2866 5381 y Fv(;)212 b(x)28 b Fs(2)g Fw(\005)p Fv(;)1781 5712 y Fw(23)p eop %%Page: 24 29 24 28 bop 120 407 a Fw(so)37 b(that)f Fv(Y)22 b Fw(\()p 576 352 50 4 v Fv(z)t(;)17 b(k)s Fw(\))761 371 y Fl(\003)800 407 y Fv(f)46 b Fs(2)1020 382 y Fw(~)995 407 y Fv(H)1084 371 y Fu(1)1123 407 y Fw(\(\005\))36 b(and)h(\()p Fv(Y)21 b Fw(\()p 1656 352 V Fv(z)5 b(;)17 b(k)s Fw(\))1842 371 y Fl(\003)1881 407 y Fv(f)11 b Fw(\)\()p Fv(:;)17 b Fw(0\))34 b(=)g Fv(B)5 b Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))2620 371 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)2786 407 y Fv(f)11 b Fw(.)55 b(It)37 b(follo)m(ws)e(that)120 541 y Fv(G)197 556 y Fu(0)236 541 y Fv(Y)22 b Fw(\()p 353 486 V Fv(z)t(;)17 b(k)s Fw(\))538 505 y Fl(\003)605 541 y Fw(=)28 b Fv(T)780 490 y Fu(\()p Ft(\033)r Fu(\))766 565 y(0)881 541 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))33 b(and)740 729 y Fv(G)817 744 y Fu(0)857 729 y Fv(R)931 744 y Fu(0)970 729 y Fw(\()p Fv(z)t Fw(\))p Fv(G)1172 688 y Fl(\003)1240 729 y Fw(=)28 b Fv(G)1421 744 y Fu(0)1477 729 y Fw(\()o Fv(GR)1665 744 y Fu(0)1705 729 y Fw(\()p 1743 674 V Fv(z)5 b Fw(\)\))1869 680 y Fl(\003)1936 729 y Fw(=)27 b Fv(T)2110 678 y Fu(\()p Ft(\033)r Fu(\))2096 753 y(0)2212 729 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))22 b Fv(T)2528 688 y Fu(\()p Ft(\033)r Fu(\))2630 729 y Fw(\()p 2668 674 V Fv(z)5 b(;)17 b(k)s Fw(\))2854 688 y Fl(\003)2893 729 y Fv(:)120 904 y Fw(This)33 b(concludes)g(the)g (pro)s(of)f(of)g(the)h(Prop)s(osition.)p 3469 904 4 66 v 3473 841 59 4 v 3473 904 V 3531 904 4 66 v 266 1086 a(As)g(an)g(easy)h(consequence)h(of)k(\(3.1.9\))32 b(w)m(e)i(obtain)120 1250 y Fp(Theorem)j(3.1.3.)42 b Fj(L)-5 b(et)35 b Fv(k)c Fs(2)d Fw([)p Fs(\000)1356 1211 y Fu(1)p 1356 1227 36 4 v 1356 1285 a(2)1402 1250 y Fv(;)1456 1211 y Fu(1)p 1456 1227 V 1456 1285 a(2)1501 1250 y Fw(])p Fj(,)35 b(then)1043 1438 y Fv(R)1118 1397 y Fu(\()p Ft(\033)r Fu(\))1219 1438 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))23 b Fs(\000)g Fv(R)1640 1397 y Fu(\(0\))1734 1438 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))28 b Fs(2)g Fw(\006)2149 1453 y Fu(1)p Ft(=)p Fu(3)2260 1438 y Fw(\()p Fv(L)2364 1453 y Fu(2)2403 1438 y Fw(\(\005\)\))p Fv(:)120 1613 y Fj(In)34 b(p)-5 b(articular,)926 1733 y Fv(\033)981 1748 y Ft(ac)1070 1652 y Fo(\000)1115 1733 y Fv(H)1204 1692 y Fu(\()p Ft(\033)r Fu(\))1306 1733 y Fw(\()p Fv(k)s Fw(\))1436 1652 y Fo(\001)1509 1733 y Fw(=)28 b Fv(\033)1668 1748 y Ft(ess)1787 1652 y Fo(\000)1832 1733 y Fv(H)1921 1692 y Fu(\()p Ft(\033)r Fu(\))2023 1733 y Fw(\()p Fv(k)s Fw(\))2153 1652 y Fo(\001)2226 1733 y Fw(=)g([)p Fv(k)2411 1692 y Fu(2)2450 1733 y Fv(;)17 b Fw(+)p Fs(1)p Fw(\))p Fv(:)120 1921 y Fj(Pr)-5 b(o)g(of.)48 b Fw(As)38 b(w)m(e)g(ha)m(v)m(e)g(remark)m(ed)g(in)e(Subsection)i(2,)g Fv(T)2180 1870 y Fu(\()p Ft(\033)r Fu(\))2166 1945 y(0)2281 1921 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))p Fv(;)42 b(T)2644 1885 y Fu(\()p Ft(\033)r Fu(\))2746 1921 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))36 b Fs(2)f Fw(\006)3176 1936 y Fu(2)3216 1921 y Fw(\()p Fv(L)3320 1936 y Fu(2)3360 1921 y Fw(\()p Fr(T)p Fw(\)\))120 2041 y(and)23 b Fv(Y)e Fw(\()p Fv(z)t(;)c(k)s Fw(\))29 b Fs(2)f Fw(\006)794 2056 y Fu(1)834 2041 y Fw(\()p Fv(L)938 2056 y Fu(2)977 2041 y Fw(\(\005\))p Fv(;)17 b(L)1236 2056 y Fu(2)1276 2041 y Fw(\()p Fr(T)p Fw(\)\),)28 b(whic)m(h)c(implies)c(\(cf.)41 b(\(1.2.10\))o(\))23 b Fv(R)2739 2005 y Fu(\()p Ft(\033)r Fu(\))2841 2041 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))s Fs(\000)s Fv(R)3222 2005 y Fu(\(0\))3317 2041 y Fw(\()p Fv(z)t(;)g(k)s Fw(\))120 2162 y Fs(2)58 b Fw(\006)314 2177 y Fu(1)p Ft(=)p Fu(3)424 2162 y Fw(\()p Fv(L)528 2177 y Fu(2)568 2162 y Fw(\(\005\)\).)95 b(Since)50 b Fv(\033)1204 2177 y Ft(ac)1294 2081 y Fo(\000)1339 2162 y Fv(H)1428 2126 y Fu(\(0\))1522 2162 y Fw(\()p Fv(k)s Fw(\))1652 2081 y Fo(\001)1755 2162 y Fw(=)57 b Fv(\033)1943 2177 y Ft(ess)2062 2081 y Fo(\000)2108 2162 y Fv(H)2197 2126 y Fu(\(0\))2291 2162 y Fw(\()p Fv(k)s Fw(\))2421 2081 y Fo(\001)2524 2162 y Fw(=)g([)p Fv(k)2738 2126 y Fu(2)2777 2162 y Fv(;)17 b Fw(+)p Fs(1)p Fw(\),)54 b(the)c(state-)120 2282 y(men)m(ts)d(ab)s(out)e(the)i(essen)m (tial)f(and)g(the)h(absolutely)e(con)m(tin)m(uous)i(sp)s(ectrum)f(of)g (the)h(op-)120 2403 y(erators)c Fv(H)545 2366 y Fu(\()p Ft(\033)r Fu(\))646 2403 y Fw(\()p Fv(k)s Fw(\))f(follo)m(w)f(no)m(w)i (from)f(the)g(W)-8 b(eyl)43 b(and)g(the)g(Birman-Kre)-11 b(\025)-38 b(\020n)39 b(theorem,)45 b(re-)120 2523 y(sp)s(ectiv)m(ely) -8 b(.)p 3469 2523 4 66 v 3473 2460 59 4 v 3473 2523 V 3531 2523 4 66 v 120 2776 a Fp(4.)48 b(Commen)m(ts)32 b(and)h(Remarks.)97 b(Compactness)33 b(of)g(the)f(resolv)m(en)m(t)g (di\013erence.)120 2896 y Fw(In)h(Theorem)g(3.1.3)f(w)m(e)h(pro)m(v)m (ed)h(that)1257 3071 y Fv(R)1332 3030 y Fu(\()p Ft(\033)r Fu(\))1433 3071 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))23 b Fs(\000)g Fv(R)1854 3030 y Fu(\(0\))1948 3071 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))28 b Fs(2)g Fw(\006)2363 3086 y Ft(p)3215 3071 y Fw(\(3.1.10\))120 3246 y(for)33 b Fv(p)d Fw(=)465 3206 y Fu(1)p 465 3223 36 4 v 465 3280 a(3)510 3246 y Fw(.)47 b(Our)34 b(pro)s(of)f(used)i(Lemma)d(1.2.3)h (for)h(the)g(compactness)h(result)e(on)h Fv(T)3215 3209 y Fu(\()p Ft(\033)r Fu(\))3317 3246 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))120 3366 y(and)36 b(th)m(us)h(did)f(rely)g(hea)m(vily)f(on) h(the)h(explicit)d(kno)m(wledge)j(of)f(the)h(free)f(resolv)m(en)m(t)h (k)m(ernel.)120 3486 y(A)c(m)m(uc)m(h)g(more)f(general)g(approac)m(h,)i (yielding)d(only)h(the)h(sligh)m(tly)e(w)m(eak)m(er)k(index)e Fv(p)28 b Fw(=)3381 3447 y Fu(1)p 3381 3463 V 3381 3521 a(2)3459 3486 y Fw(in)120 3607 y(\(3.1.10\))o(,)33 b(can)g(b)s(e)f (based)i(on)e(the)h(metho)s(d)f(in)g([B2],)h(see)h(also)d([B1].)120 3727 y Fp(Birman-Sc)m(h)m(winger)38 b(analysis.)48 b Fw(Let)34 b(us)g(brie\015y)h(explain)e(the)h(connection)g(of)g(the)h (ap-)120 3848 y(proac)m(h)23 b(in)e(Subsection)i(1)f(and)g(what)h(is)e (kno)m(wn)j(as)e(Birman-Sc)m(h)m(winger)g(analysis.)39 b(Supp)s(ose)120 3968 y(that)32 b(the)h(op)s(erators)1272 4088 y Fv(G)1349 4103 y Fu(0)1388 4088 y Fv(R)1462 4103 y Fu(0)1502 4088 y Fw(\()p Fv(z)t Fw(\))p Fv(G;)212 b(z)33 b Fs(2)28 b Fv(\032)p Fw(\()p Fv(H)2284 4103 y Fu(0)2323 4088 y Fw(\))p Fv(;)827 b Fw(\(3.1.11\))120 4243 y(are)30 b(compact.)42 b(It)30 b(can)g(b)s(e)g(sho)m(wn)h(\(see)g(Theorem)g (1.10.7)e(in)g([Y)-8 b(a1)o(]\))30 b(that)g(then)g Fv(\033)3169 4258 y Ft(ess)3272 4243 y Fw(\()p Fv(H)8 b Fw(\))27 b(=)120 4364 y Fv(\033)175 4379 y Ft(ess)278 4364 y Fw(\()p Fv(H)397 4379 y Fu(0)436 4364 y Fw(\).)42 b(Hence)31 b(b)m(y)f(Prop)s(osition)d (3.1.1,)i(a)g(n)m(um)m(b)s(er)h Fv(z)i Fs(2)c Fv(\032)p Fw(\()p Fv(H)2510 4379 y Fu(0)2550 4364 y Fw(\))h(b)s(elongs)g(to)f Fv(\033)3134 4379 y Ft(disc)3263 4364 y Fw(\()p Fv(H)8 b Fw(\))28 b(i\013)120 4484 y(the)33 b(op)s(erator)f(\(3.1.11\))g(has)h (eigen)m(v)-5 b(alue)32 b Fs(\000)p Fw(1.)120 4604 y(This)d(tec)m (hnique)h(is)f(frequen)m(tly)h(used)g(when)g Fv(H)1877 4619 y Fu(0)1945 4604 y Fw(is)e(a)h(self-adjoin)m(t)e(realization)f(of) i Fs(\000)p Fw(\001)i(and)120 4725 y Fv(G)197 4740 y Fu(0)236 4725 y Fv(;)k(G)j Fw(are)g(m)m(ultiplcation)d(b)m(y)k(real-v) -5 b(alued)36 b(functions)2248 4640 y Fs(p)p 2331 4640 79 4 v 85 x Fv(V)22 b(;)2470 4640 y Fo(p)p 2570 4640 134 4 v 85 x Fs(j)p Fv(V)f Fs(j)p Fw(,)38 b(resp)s(ectivly)-8 b(.)58 b(\(Here)120 4772 y Fs(p)p 203 4772 79 4 v 85 x Fv(V)49 b Fw(:=)28 b(\(sgn)17 b Fv(V)k Fw(\))752 4771 y Fo(p)p 852 4771 134 4 v 86 x Fs(j)p Fv(V)g Fs(j)p Fw(.\))43 b(Then)34 b(the)f(op)s(erator)f(\(3.1.11\))g(ma)m(y)g(b)s(e)h(written)f (as)1382 4958 y Fs(p)p 1465 4958 79 4 v 89 x Fv(V)22 b Fw(\()p Fs(\000)p Fw(\001)h Fs(\000)f Fv(z)t Fw(\))1949 5006 y Fl(\000)p Fu(1)2044 4957 y Fo(p)p 2144 4957 134 4 v 90 x Fs(j)p Fv(V)f Fs(j)937 b Fw(\(3.1.12\))120 5222 y(and)33 b(is)f(usually)g(called)f(Birman-Sc)m(h)m(winger)h(op)s (erator.)120 5342 y(Our)39 b(op)s(erator)e Fv(T)796 5292 y Fu(\()p Ft(\033)r Fu(\))782 5367 y(0)898 5342 y Fw(\()p Fv(z)t(;)17 b Fw(0\))p Fv(T)1187 5306 y Fu(\()p Ft(\033)r Fu(\))1288 5342 y Fw(\()p 1326 5288 50 4 v Fv(z)5 b(;)17 b Fw(0\))1507 5306 y Fl(\003)1585 5342 y Fw(is)38 b(formally)d (obtained)j(b)m(y)i(setting)e Fv(V)60 b Fw(=)37 b Fv(\033)31 b Fs(\012)26 b Fv(\016)43 b Fw(in)120 5463 y(\(3.1.12\))o(.)1781 5712 y(24)p eop %%Page: 25 30 25 29 bop 120 407 a Fq(3.2)161 b(The)87 b(absence)f(of)h(singular)h (con)l(tin)l(uous)d(sp)t(ec-)488 589 y(trum)52 b(of)i(the)f(op)t (erators)g Ff(H)2205 537 y Fw(\()p Fv(\033)t Fw(\))2344 589 y Fe(\()p Ff(k)5 b Fe(\))120 808 y Fw(Our)32 b(goal)e(in)h(this)g (section)h(is)f(to)g(pro)m(v)m(e)i(the)f(absence)i(of)d(singular)f(con) m(tin)m(uous)i(sp)s(ectrum.)120 929 y(This)40 b(will)e(b)s(e)j(ac)m (hiev)m(ed)g(b)m(y)g(con)m(trolling)d(the)j(b)s(eha)m(vior)f(of)f(the)i (resolv)m(en)m(t)g Fv(R)3087 893 y Fu(\()p Ft(\033)r Fu(\))3189 929 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))41 b(as)120 1049 y(the)31 b(sp)s(ectral)g(parameter)f Fv(z)35 b Fw(tends)d(to)e (the)h(real)f(axis.)43 b(Indeed,)32 b(for)e Fv(u)g Fw(from)g(a)g(dense) i(set)g(in)120 1170 y Fv(L)186 1185 y Fu(2)226 1170 y Fw(\(\005\))38 b(w)m(e)g(will)e(pro)m(v)m(e)j(that)f(lim)1371 1185 y Ft(\017)p Fl(!)p Fu(0+)1581 1170 y Fv(R)1656 1133 y Fu(\()p Ft(\033)r Fu(\))1758 1170 y Fw(\()p Fv(\025)25 b Fs(\006)i Fv(i\017;)17 b(k)s Fw(\))38 b(exists)g(in)f(a)h(suitable)f (top)s(ology)f(if)120 1290 y Fv(\025)g Fw(a)m(v)m(oids)g(an)g (exceptional)g(set.)54 b(This)36 b(assertion)g(is)g(usually)f(called)g (limiting)d(absorption)120 1410 y(principle.)120 1531 y(W)-8 b(e)36 b(will)e(pro)m(v)m(e)j(it)e(\014rst)h(in)f(Subsection)i (1)f(for)f(the)h(case)h Fv(\033)g Fs(\021)d Fw(0)i(using)f(the)i (explicit)d(form)120 1651 y(of)c(the)h(resolv)m(en)m(t)h(k)m(ernel.)43 b(The)32 b(role)d(of)h(the)h(exceptional)g(set)g(in)f(this)g(case)h (will)e(b)s(e)h(pla)m(y)m(ed)120 1772 y(b)m(y)g(the)f(p)s(oin)m(ts)g Fv(z)j Fw(=)c(\()p Fv(n)15 b Fw(+)g Fv(k)s Fw(\))1181 1735 y Fu(2)1220 1772 y Fv(;)33 b(n)28 b Fs(2)g Fr(Z)p Fw(,)f(where)j(the)g(sp)s(ectral)e(m)m(ultiplicit)m(y)e(of)i(the)i(op)s (erator)120 1892 y Fv(H)209 1856 y Fu(\(0\))303 1892 y Fw(\()p Fv(k)s Fw(\))c(c)m(hanges.)43 b(Then)28 b(in)d(Subsection)j (2)e(w)m(e)h(will)e(carry)i(o)m(v)m(er)g(the)g(result)f(to)h(the)g (general)120 2012 y(case)39 b(using)e(the)i(resolv)m(en)m(t)g(iden)m (tit)m(y)e(from)g(Section)h(1.)59 b(In)38 b(addition)e(to)i(the)g(p)s (oin)m(ts)g Fv(z)j Fw(=)120 2133 y(\()p Fv(n)16 b Fw(+)g Fv(k)s Fw(\))416 2097 y Fu(2)455 2133 y Fv(;)33 b(n)28 b Fs(2)g Fr(Z)p Fv(;)f Fw(the)i(exceptional)g(set)h(will)d(no)m(w)j (also)f(include)f(the)i(set)g(of)f Fv(\025)p Fw('s,)i(for)d(whic)m(h) 120 2253 y(one)33 b(of)f(the)h(equations)986 2455 y Fv(f)g Fw(+)22 b Fv(T)1236 2404 y Fu(\()p Ft(\033)r Fu(\))1222 2480 y(0)1338 2455 y Fw(\()p Fv(\025)g Fs(\006)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))g Fv(T)1860 2414 y Fu(\()p Ft(\033)r Fu(\))1961 2455 y Fw(\()p Fv(\025)22 b Fs(\007)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))2396 2414 y Fl(\003)2435 2455 y Fv(f)38 b Fw(=)28 b(0)120 2657 y(has)39 b(a)g(non-trivial)d (solution.)61 b(The)40 b(Analytic)e(F)-8 b(redholm)38 b(Alternativ)m(e)g(guaran)m(tees)i(that)120 2777 y(this)f(set)i(is)e (discrete)h(in)f Fr(R)f Fs(n)27 b(f)p Fw(\()p Fv(n)f Fw(+)h Fv(k)s Fw(\))1611 2741 y Fu(2)1690 2777 y Fw(:)67 b Fv(n)40 b Fs(2)g Fr(Z)p Fs(g)d Fw(and)j(can)g(therefore)g(not)f(supp) s(ort)h(a)120 2898 y(singular)31 b(con)m(tin)m(uous)i(sp)s(ectrum.)120 3155 y Fp(1.)103 b(The)55 b(limiting)c(absorption)k(principle)e(for)i (the)g(unp)s(erturb)s(ed)i(op)s(erator)120 3275 y Fv(H)209 3239 y Fu(\(0\))303 3275 y Fw(\()p Fv(k)s Fw(\))p Fp(.)97 b Fw(W)-8 b(e)38 b(w)m(an)m(t)g(to)f(extend)i(the)e(free)h(resolv)m(en) m(t)g Fv(R)2276 3239 y Fu(\(0\))2371 3275 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))37 b(from)f Fr(C)51 b Fs(n)25 b Fw([)p Fv(k)3119 3239 y Fu(2)3159 3275 y Fv(;)17 b Fw(+)p Fs(1)p Fw(\))36 b(to)120 3396 y(the)46 b(cut.)82 b(Let)46 b(us)g(consider)f(the)h (explicit)e(form)m(ula)f(\(2.2.7\))i(for)g(the)h(resolv)m(en)m(t)g(k)m (ernel.)120 3516 y(Because)40 b(of)e(the)h(p)s(oles)f(at)h(the)g(p)s (oin)m(ts)f Fv(z)k Fw(=)c(\()p Fv(n)27 b Fw(+)f Fv(k)s Fw(\))2160 3480 y Fu(2)2199 3516 y Fv(;)33 b(n)39 b Fs(2)f Fr(Z)p Fv(;)e Fw(w)m(e)k(ma)m(y)e(not)g(exp)s(ect)i(to)120 3637 y(extend)34 b(it)e(to)g(the)h(whole)f(cut.)120 3757 y(There)c(is)f(another)g(phenomenon:)41 b(W)-8 b(rite)27 b Fv(z)32 b Fw(=)c Fv(\025)11 b Fs(\006)g Fv(i\017)p Fw(,)29 b(where)f Fv(\017)g(>)g Fw(0)f(and)g Fv(\025)h(>)f Fw(\()p Fv(n)11 b Fw(+)g Fv(k)s Fw(\))3357 3721 y Fu(2)3423 3757 y Fw(for)120 3877 y(some)33 b Fv(n)c Fs(2)g Fr(Z)p Fw(.)43 b(Then)942 3792 y Fo(p)p 1041 3792 797 4 v 1041 3877 a Fw(\()p Fv(n)23 b Fw(+)f Fv(k)s Fw(\))1350 3849 y Fu(2)1411 3877 y Fs(\000)h Fw(\()p Fv(\025)f Fs(\006)g Fv(i\017)p Fw(\))30 b Fs(!)e(\007)p Fv(i)2122 3792 y Fo(p)p 2222 3792 527 4 v 85 x Fv(\025)22 b Fs(\000)h Fw(\()p Fv(n)f Fw(+)g Fv(k)s Fw(\))2709 3849 y Fu(2)2782 3877 y Fw(as)33 b Fv(\017)d Fs(!)e Fw(0+,)33 b(so)h(the)120 4031 y(factor)396 3920 y Fo(\020)455 4031 y Fv(e)500 3991 y Fl(\000)555 3924 y Fs(p)p 638 3924 485 4 v 67 x Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))829 3972 y Fm(2)864 3991 y Fl(\000)p Fu(\()p Ft(\025)p Fl(\006)p Ft(i\017)p Fu(\))12 b(\()p Ft(x)1201 4000 y Fm(2)1236 3991 y Fu(+)p Ft(y)1326 4000 y Fm(2)1360 3991 y Fu(\))1414 4031 y Fw(+)22 b Fv(e)1557 3991 y Fl(\000)1612 3924 y Fs(p)p 1695 3924 V 67 x Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))1886 3972 y Fm(2)1921 3991 y Fl(\000)p Fu(\()p Ft(\025)p Fl(\006)p Ft(i\017)p Fu(\))12 b Fl(j)p Ft(x)2251 4000 y Fm(2)2285 3991 y Fl(\000)p Ft(y)2375 4000 y Fm(2)2409 3991 y Fl(j)2433 3920 y Fo(\021)2522 4031 y Fw(in)29 b(\(2.2.7\))o(,)h(as)g(function)e (of)120 4173 y(either)j Fv(x)450 4188 y Fu(2)520 4173 y Fw(or)g Fv(y)686 4188 y Fu(2)725 4173 y Fw(,)g(ceases)i(to)d(b)s(e)i (square)g(in)m(tegrable)d(in)i(the)g(limit)c Fv(\017)h Fw(=)g(0.)43 b(W)-8 b(e)31 b(remedy)g(this)120 4293 y(problem)c(b)m(y)j (m)m(ultiplying)c(with)i(a)g(deca)m(ying)h(function.)42 b(Namely)-8 b(,)29 b(for)f Fv(s)f(>)h Fw(0)g(consider)i(the)120 4414 y(b)s(ounded)j(op)s(erator)f(\003)979 4372 y Fl(\000)p Ft(s)979 4438 y Fu(2)1103 4414 y Fw(on)h Fv(L)1305 4429 y Fu(2)1344 4414 y Fw(\(\005\),)953 4660 y(\(\003)1059 4619 y Fl(\000)p Ft(s)1059 4685 y Fu(2)1151 4660 y Fv(u)p Fw(\)\()p Fv(x)p Fw(\))27 b(:=)1743 4593 y(1)p 1544 4637 448 4 v 1544 4728 a(\(1)21 b(+)h Fv(x)1805 4694 y Fu(2)1805 4753 y(2)1845 4728 y Fw(\))1883 4700 y Ft(s=)p Fu(2)2001 4660 y Fv(f)11 b Fw(\()p Fv(x)p Fw(\))p Fv(;)211 b(x)28 b Fs(2)h Fw(\005)p Fv(:)120 4912 y Fw(The)35 b(follo)m(wing)d(result)j (is)f(called)f(the)i(limiting)30 b(absorption)k(principle)f(for)h(the)h (free)g(op)s(er-)120 5033 y(ator.)120 5221 y Fp(Prop)s(osition)h (3.2.1.)42 b Fj(L)-5 b(et)35 b Fv(s)27 b(>)1384 5182 y Fu(1)p 1384 5198 36 4 v 1384 5255 a(2)1464 5221 y Fj(and)34 b Fv(k)d Fs(2)d Fw([)p Fs(\000)1943 5182 y Fu(1)p 1943 5198 V 1943 5255 a(2)1989 5221 y Fv(;)2043 5182 y Fu(1)p 2043 5198 V 2043 5255 a(2)2088 5221 y Fw(])p Fj(.)45 b(Then)34 b(the)h(limits)741 5436 y Fw(lim)714 5496 y Ft(\017)p Fl(!)p Fu(0+)920 5436 y Fw(\003)988 5395 y Fl(\000)p Ft(s)988 5461 y Fu(2)1079 5436 y Fv(R)1154 5395 y Fu(\(0\))1249 5436 y Fw(\()p Fv(\025)22 b Fs(\006)g Fv(i\017;)17 b(k)s Fw(\)\003)1741 5395 y Fl(\000)p Ft(s)1741 5461 y Fu(2)1833 5436 y Fv(;)216 b(\025)28 b Fs(6)p Fw(=)f(\()p Fv(n)c Fw(+)f Fv(k)s Fw(\))2573 5395 y Fu(2)2612 5436 y Fv(;)33 b(n)28 b Fs(2)g Fr(Z)p Fv(;)1781 5712 y Fw(25)p eop %%Page: 26 31 26 30 bop 120 407 a Fj(exist)31 b(in)g(the)g(op)-5 b(er)g(ator)31 b(norm)f(top)-5 b(olo)g(gy)31 b(and)f(ar)-5 b(e)31 b(uniform)g(for)g Fv(\025)g Fj(fr)-5 b(om)30 b(c)-5 b(omp)g(act)31 b(intervals)120 527 y(of)k Fr(R)e Fs(n)22 b(f)p Fw(\()p Fv(n)g Fw(+)g Fv(k)s Fw(\))759 491 y Fu(2)826 527 y Fw(:)50 b Fv(n)28 b Fs(2)g Fr(Z)p Fs(g)p Fj(.)120 725 y(Pr)-5 b(o)g(of.)48 b Fw(Fix)32 b Fv(\025)c Fs(2)g Fr(R)33 b Fs(n)22 b(f)p Fw(\()p Fv(n)g Fw(+)g Fv(k)s Fw(\))1303 689 y Fu(2)1370 725 y Fw(:)50 b Fv(n)28 b Fs(2)g Fr(Z)p Fs(g)p Fw(,)i(and)j(de\014ne) 534 939 y Fv(r)581 898 y Fu(\(0\))675 939 y Fw(\()p Fv(x;)17 b(y)t Fw(;)g Fv(\025)k Fs(\006)i Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))27 b(:=)55 b(lim)1462 999 y Ft(\017)p Fl(!)p Fu(0+)1667 939 y Fv(r)1714 898 y Fu(\(0\))1808 939 y Fw(\()p Fv(x;)17 b(y)t Fw(;)g Fv(\025)k Fs(\006)i Fv(i\017;)17 b(k)s Fw(\))p Fv(;)212 b(x)28 b Fs(6)p Fw(=)g Fv(y)j Fs(2)d Fw(\005)p Fv(;)120 1193 y Fw(as)33 b(p)s(oin)m(t)m(wise)f(limit) d(using)j(form)m(ula)f(\(2.2.7\).)43 b(W)-8 b(e)33 b(will)d(pro)m(v)m (e)k(that)288 1320 y Fo(Z)343 1546 y Fu(\005)417 1320 y Fo(Z)472 1546 y Fu(\005)546 1371 y Fo(\014)546 1431 y(\014)579 1456 y Fv(r)626 1415 y Fu(\(0\))720 1456 y Fw(\()p Fv(x;)17 b(y)t Fw(;)g Fv(\025)k Fs(\006)h Fv(i\017;)17 b(k)s Fw(\))23 b Fs(\000)g Fv(r)1508 1415 y Fu(\(0\))1602 1456 y Fw(\()p Fv(x;)17 b(y)t Fw(;)g Fv(\025)k Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))2230 1371 y Fo(\014)2230 1431 y(\014)2263 1391 y Fu(2)2464 1389 y Fv(dx)p 2329 1433 377 4 v 2329 1524 a Fw(\(1)22 b(+)g Fv(x)2591 1490 y Fu(2)2591 1549 y(2)2631 1524 y Fw(\))2669 1496 y Ft(s)2861 1389 y Fv(dy)p 2726 1433 373 4 v 2726 1524 a Fw(\(1)f(+)h Fv(y)2984 1490 y Fu(2)2980 1549 y(2)3023 1524 y Fw(\))3061 1496 y Ft(s)3135 1456 y Fs(\000)-16 b(!)27 b Fw(0)529 1667 y(\()p Fv(\017)h Fs(!)f Fw(0+\))p Fv(:)3264 1795 y Fw(\(3.2.1\))120 2019 y(This)39 b(means)f(that)h(\003)939 1978 y Fl(\000)p Ft(s)939 2044 y Fu(2)1030 2019 y Fv(R)1105 1983 y Fu(\(0\))1199 2019 y Fw(\()p Fv(\025)26 b Fs(\006)h Fv(i\017;)17 b(k)s Fw(\)\003)1700 1978 y Fl(\000)p Ft(s)1700 2044 y Fu(2)1830 2019 y Fw(con)m(v)m(erges)41 b(in)d(\(Hilb)s(ert-Sc)m (hmidt\))e(norm)h(to)120 2140 y(an)29 b(in)m(tegral)e(op)s(erator)i (with)f(k)m(ernel)1614 2100 y Fu(1)p 1505 2117 253 4 v 1505 2176 a(\(1+)p Ft(x)1662 2153 y Fm(2)1662 2197 y(2)1697 2176 y Fu(\))1724 2157 y Fi(s)1784 2140 y Fv(r)1831 2104 y Fu(\(0\))1925 2140 y Fw(\()p Fv(x;)17 b(y)t Fw(;)g Fv(\025)e Fs(\006)g Fv(i)p Fw(0)p Fv(;)i(k)s Fw(\))2673 2100 y Fu(1)p 2566 2117 251 4 v 2566 2176 a(\(1+)p Ft(y)2720 2153 y Fm(2)2718 2197 y(2)2755 2176 y Fu(\))2782 2157 y Fi(s)2826 2140 y Fw(,)30 b(so)f(that)g(the)g(\014rst)120 2274 y(part)j(of)g(the)h(prop)s(osition)e(follo)m(ws.)120 2395 y(F)-8 b(or)41 b(the)h(pro)s(of)f(of)49 b(\(3.2.1\))41 b(w)m(e)i(restrict)e(ourselv)m(es)j(to)d(the)h("+"-case,)i(the)f(other) f(b)s(eing)120 2515 y(similar.)e(F)-8 b(or)32 b(simplicit)m(y)e(of)i (notation,)f(w)m(e)j(put)777 2729 y Fv(c)819 2744 y Ft(n)866 2729 y Fw(\()p Fv(\017)p Fw(\))28 b(:=)1139 2639 y Fo(p)p 1239 2639 721 4 v 90 x Fw(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))1547 2700 y Fu(2)1609 2729 y Fs(\000)h Fv(\025)f Fs(\000)g Fv(i\017)q(;)212 b(c)2241 2744 y Ft(n)2315 2729 y Fw(:=)55 b(lim)2446 2789 y Ft(\017)p Fl(!)p Fu(0+)2652 2729 y Fv(c)2694 2744 y Ft(n)2741 2729 y Fw(\()p Fv(\017)p Fw(\))p Fv(:)120 2982 y Fw(Using)32 b(P)m(arsev)-5 b(al's)33 b(iden)m(tit)m(y)g(w)m(e)g(can)g(rewrite)g(the)g(LHS)g(of)39 b(\(3.2.1\))32 b(as)216 3110 y Fo(Z)271 3336 y Fu(\005)345 3110 y Fo(Z)400 3336 y Fu(\005)474 3161 y Fo(\014)474 3221 y(\014)507 3246 y Fv(r)554 3205 y Fu(\(0\))648 3246 y Fw(\()p Fv(x;)17 b(y)t Fw(;)g Fv(\025)k Fs(\006)i Fv(i\017;)17 b(k)s Fw(\))22 b Fs(\000)h Fv(r)1436 3205 y Fu(\(0\))1530 3246 y Fw(\()p Fv(x;)17 b(y)t Fw(;)g Fv(\025)k Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))2158 3161 y Fo(\014)2158 3221 y(\014)2191 3181 y Fu(2)2392 3178 y Fv(dx)p 2257 3223 377 4 v 2257 3314 a Fw(\(1)22 b(+)g Fv(x)2519 3280 y Fu(2)2519 3338 y(2)2559 3314 y Fw(\))2597 3285 y Ft(s)2789 3178 y Fv(dy)p 2654 3223 373 4 v 2654 3314 a Fw(\(1)f(+)h Fv(y)2912 3280 y Fu(2)2908 3338 y(2)2951 3314 y Fw(\))2989 3285 y Ft(s)3063 3246 y Fw(=)485 3547 y(=)588 3411 y Fo(Z)688 3437 y Ft(\031)644 3636 y Fl(\000)p Ft(\031)779 3547 y Fv(dx)885 3562 y Fu(1)941 3411 y Fo(Z)1041 3437 y Ft(\031)996 3636 y Fl(\000)p Ft(\031)1132 3547 y Fv(dy)1231 3562 y Fu(1)1286 3411 y Fo(Z)1385 3437 y Fl(1)1341 3636 y Fu(0)1602 3479 y Fv(dx)1708 3494 y Fu(2)p 1487 3524 377 4 v 1487 3615 a Fw(\(1)g(+)g Fv(x)1749 3581 y Fu(2)1749 3639 y(2)1788 3615 y Fw(\))1826 3586 y Ft(s)1890 3411 y Fo(Z)1989 3437 y Fl(1)1945 3636 y Fu(0)2208 3479 y Fv(dy)2307 3494 y Fu(2)p 2091 3524 373 4 v 2091 3615 a Fw(\(1)g(+)g Fv(y)2350 3581 y Fu(2)2346 3639 y(2)2388 3615 y Fw(\))2426 3586 y Ft(s)2489 3372 y Fo(\014)2489 3432 y(\014)2489 3492 y(\014)2489 3551 y(\014)2489 3611 y(\014)2604 3479 y Fw(1)p 2533 3524 191 4 v 2533 3544 a Fs(p)p 2616 3544 108 4 v 82 x Fw(2)p Fv(\031)2750 3452 y Fo(X)2755 3664 y Ft(n)p Fl(2)p Fn(Z)2910 3547 y Fv(e)2955 3505 y Ft(inx)3062 3514 y Fm(1)689 3814 y Fv(e)734 3777 y Fl(\000)p Ft(iny)891 3786 y Fm(1)p 689 3858 241 4 v 714 3878 a Fs(p)p 797 3878 108 4 v 82 x Fw(2)p Fv(\031)956 3740 y Fo(\022)1039 3814 y Fv(e)1084 3777 y Fl(\000)p Ft(c)1170 3785 y Fi(n)1212 3777 y Fu(\()p Ft(\017)p Fu(\)\()p Ft(x)1362 3786 y Fm(2)1398 3777 y Fu(+)p Ft(y)1488 3786 y Fm(2)1522 3777 y Fu(\))1576 3814 y Fw(+)g Fv(e)1719 3777 y Fl(\000)p Ft(c)1805 3785 y Fi(n)1847 3777 y Fu(\()p Ft(\017)p Fu(\))p Fl(j)p Ft(x)1990 3786 y Fm(2)2024 3777 y Fl(\000)p Ft(y)2114 3786 y Fm(2)2148 3777 y Fl(j)p 1039 3858 1134 4 v 1479 3949 a Fw(2)p Fv(c)1570 3964 y Ft(n)1617 3949 y Fw(\()p Fv(\017)p Fw(\))2204 3881 y Fs(\000)2314 3814 y Fv(e)2359 3777 y Fl(\000)p Ft(c)2445 3785 y Fi(n)2487 3777 y Fu(\()p Ft(x)2554 3786 y Fm(2)2589 3777 y Fu(+)p Ft(y)2679 3786 y Fm(2)2713 3777 y Fu(\))2767 3814 y Fw(+)g Fv(e)2910 3777 y Fl(\000)p Ft(c)2996 3785 y Fi(n)3038 3777 y Fl(j)p Ft(x)3098 3786 y Fm(2)3132 3777 y Fl(\000)p Ft(y)3222 3786 y Fm(2)3256 3777 y Fl(j)p 2314 3858 967 4 v 2728 3949 a Fw(2)p Fv(c)2819 3964 y Ft(n)3290 3740 y Fo(\023)3363 3736 y(\014)3363 3796 y(\014)3363 3856 y(\014)3363 3916 y(\014)3397 3754 y Fu(2)485 4147 y Fw(=)588 4052 y Fo(X)593 4264 y Ft(n)p Fl(2)p Fn(Z)749 4011 y Fo(Z)848 4037 y Fl(1)804 4237 y Fu(0)1065 4079 y Fv(dx)1171 4094 y Fu(2)p 950 4124 377 4 v 950 4215 a Fw(\(1)g(+)g Fv(x)1212 4181 y Fu(2)1212 4239 y(2)1252 4215 y Fw(\))1290 4186 y Ft(s)1353 4011 y Fo(Z)1452 4037 y Fl(1)1408 4237 y Fu(0)1671 4079 y Fv(dy)1770 4094 y Fu(2)p 1554 4124 373 4 v 1554 4215 a Fw(\(1)g(+)g Fv(y)1813 4181 y Fu(2)1809 4239 y(2)1851 4215 y Fw(\))1889 4186 y Ft(s)669 4337 y Fo(\014)669 4396 y(\014)669 4456 y(\014)669 4516 y(\014)712 4414 y Fv(e)757 4378 y Fl(\000)p Ft(c)843 4386 y Fi(n)885 4378 y Fu(\()p Ft(\017)p Fu(\)\()p Ft(x)1035 4387 y Fm(2)1070 4378 y Fu(+)p Ft(y)1160 4387 y Fm(2)1195 4378 y Fu(\))1248 4414 y Fw(+)g Fv(e)1391 4378 y Fl(\000)p Ft(c)1477 4386 y Fi(n)1520 4378 y Fu(\()p Ft(\017)p Fu(\))p Fl(j)p Ft(x)1663 4387 y Fm(2)1697 4378 y Fl(\000)p Ft(y)1787 4387 y Fm(2)1821 4378 y Fl(j)p 712 4458 1134 4 v 1152 4549 a Fw(2)p Fv(c)1243 4564 y Ft(n)1290 4549 y Fw(\()p Fv(\017)p Fw(\))1877 4481 y Fs(\000)1987 4414 y Fv(e)2032 4378 y Fl(\000)p Ft(c)2118 4386 y Fi(n)2160 4378 y Fu(\()p Ft(x)2227 4387 y Fm(2)2262 4378 y Fu(+)p Ft(y)2352 4387 y Fm(2)2386 4378 y Fu(\))2440 4414 y Fw(+)g Fv(e)2583 4378 y Fl(\000)p Ft(c)2669 4386 y Fi(n)2711 4378 y Fl(j)p Ft(x)2771 4387 y Fm(2)2805 4378 y Fl(\000)p Ft(y)2895 4387 y Fm(2)2929 4378 y Fl(j)p 1987 4458 967 4 v 2401 4549 a Fw(2)p Fv(c)2492 4564 y Ft(n)2963 4337 y Fo(\014)2963 4396 y(\014)2963 4456 y(\014)2963 4516 y(\014)2996 4354 y Fu(2)3063 4481 y Fs(\024)485 4747 y(\024)28 b Fw(2)688 4652 y Fo(X)693 4864 y Ft(n)p Fl(2)p Fn(Z)848 4611 y Fo(Z)948 4638 y Fl(1)904 4837 y Fu(0)1165 4679 y Fv(dx)1271 4694 y Fu(2)p 1049 4724 377 4 v 1049 4815 a Fw(\(1)22 b(+)g Fv(x)1311 4781 y Fu(2)1311 4840 y(2)1351 4815 y Fw(\))1389 4786 y Ft(s)1452 4611 y Fo(Z)1552 4638 y Fl(1)1508 4837 y Fu(0)1770 4679 y Fv(dy)1869 4694 y Fu(2)p 1653 4724 373 4 v 1653 4815 a Fw(\(1)g(+)g Fv(y)1912 4781 y Fu(2)1908 4840 y(2)1951 4815 y Fw(\))1989 4786 y Ft(s)669 4915 y Fo( )748 4941 y(\014)748 5001 y(\014)748 5061 y(\014)748 5121 y(\014)791 5018 y Fv(e)836 4982 y Fl(\000)p Ft(c)922 4990 y Fi(n)964 4982 y Fu(\()p Ft(\017)p Fu(\)\()p Ft(x)1114 4991 y Fm(2)1149 4982 y Fu(+)p Ft(y)1239 4991 y Fm(2)1274 4982 y Fu(\))p 791 5063 515 4 v 922 5154 a Fw(2)p Fv(c)1013 5169 y Ft(n)1059 5154 y Fw(\()p Fv(\017)p Fw(\))1337 5086 y Fs(\000)1447 5018 y Fv(e)1492 4982 y Fl(\000)p Ft(c)1578 4990 y Fi(n)1620 4982 y Fu(\()p Ft(x)1687 4991 y Fm(2)1722 4982 y Fu(+)p Ft(y)1812 4991 y Fm(2)1846 4982 y Fu(\))p 1447 5063 431 4 v 1593 5154 a Fw(2)p Fv(c)1684 5169 y Ft(n)1888 4941 y Fo(\014)1888 5001 y(\014)1888 5061 y(\014)1888 5121 y(\014)1921 4959 y Fu(2)1982 5086 y Fw(+)2080 4941 y Fo(\014)2080 5001 y(\014)2080 5061 y(\014)2080 5121 y(\014)2124 5018 y Fv(e)2169 4982 y Fl(\000)p Ft(c)2255 4990 y Fi(n)2297 4982 y Fu(\()p Ft(\017)p Fu(\))p Fl(j)p Ft(x)2440 4991 y Fm(2)2474 4982 y Fl(\000)p Ft(y)2564 4991 y Fm(2)2599 4982 y Fl(j)p 2124 5063 499 4 v 2246 5154 a Fw(2)p Fv(c)2337 5169 y Ft(n)2384 5154 y Fw(\()p Fv(\017)p Fw(\))2654 5086 y Fs(\000)2764 5018 y Fv(e)2809 4982 y Fl(\000)p Ft(c)2895 4990 y Fi(n)2937 4982 y Fl(j)p Ft(x)2997 4991 y Fm(2)3031 4982 y Fl(\000)p Ft(y)3121 4991 y Fm(2)3155 4982 y Fl(j)p 2764 5063 416 4 v 2903 5154 a Fw(2)p Fv(c)2994 5169 y Ft(n)3189 4941 y Fo(\014)3189 5001 y(\014)3189 5061 y(\014)3189 5121 y(\014)3222 4959 y Fu(2)3262 4915 y Fo(!)3368 5086 y Fw(=)485 5381 y(=)27 b(2)686 5287 y Fo(X)691 5499 y Ft(n)p Fl(2)p Fn(Z)847 5246 y Fo(Z)946 5272 y Fl(1)902 5471 y Fu(0)1038 5246 y Fo(Z)1137 5272 y Fl(1)1093 5471 y Fu(0)1229 5381 y Fw(\()p Fv(t)1302 5396 y Ft(n;\017)1397 5381 y Fw(\()p Fv(x)1490 5396 y Fu(2)1552 5381 y Fw(+)22 b Fv(y)1698 5396 y Fu(2)1737 5381 y Fw(\))g(+)g Fv(t)1930 5396 y Ft(n;\017)2025 5381 y Fw(\()p Fv(x)2118 5396 y Fu(2)2180 5381 y Fs(\000)h Fv(y)2328 5396 y Fu(2)2367 5381 y Fw(\)\))2585 5314 y Fv(dx)2691 5329 y Fu(2)p 2469 5358 377 4 v 2469 5450 a Fw(\(1)f(+)g Fv(x)2731 5415 y Fu(2)2731 5474 y(2)2771 5450 y Fw(\))2809 5421 y Ft(s)2983 5314 y Fv(dy)3082 5329 y Fu(2)p 2866 5358 373 4 v 2866 5450 a Fw(\(1)g(+)g Fv(y)3125 5415 y Fu(2)3121 5474 y(2)3163 5450 y Fw(\))3201 5421 y Ft(s)3248 5381 y Fv(;)1781 5712 y Fw(26)p eop %%Page: 27 32 27 31 bop 120 407 a Fw(where)929 585 y Fv(t)964 600 y Ft(n;\017)1059 585 y Fw(\()p Fv(a)p Fw(\))28 b(:=)1345 441 y Fo(\014)1345 500 y(\014)1345 560 y(\014)1345 620 y(\014)1388 518 y Fv(e)1433 481 y Fl(\000)p Ft(c)1519 489 y Fi(n)1561 481 y Fu(\()p Ft(\017)p Fu(\))p Fl(j)p Ft(a)p Fl(j)p 1388 562 338 4 v 1430 653 a Fw(2)p Fv(c)1521 668 y Ft(n)1568 653 y Fw(\()p Fv(\017)p Fw(\))1757 585 y Fs(\000)1867 518 y Fv(e)1912 481 y Fl(\000)p Ft(c)1998 489 y Fi(n)2040 481 y Fl(j)p Ft(a)p Fl(j)p 1867 562 254 4 v 1925 653 a Fw(2)p Fv(c)2016 668 y Ft(n)2131 441 y Fo(\014)2131 500 y(\014)2131 560 y(\014)2131 620 y(\014)2164 458 y Fu(2)2220 585 y Fv(;)212 b(a)28 b Fs(2)g Fr(R)5 b Fv(:)120 806 y Fw(If)34 b(\()p Fv(n)24 b Fw(+)f Fv(k)s Fw(\))530 770 y Fu(2)600 806 y Fv(<)31 b(\025)j Fw(and)h Fv(\017)c Fs(2)g Fw(\(0)p Fv(;)17 b Fw(1],)35 b(the)f(functions)h Fv(t)2053 821 y Ft(n;\017)2183 806 y Fw(are)f(b)s(ounded)h(uniformly)d (in)i Fv(\017)h Fw(and)120 926 y(con)m(v)m(erge)f(p)s(oin)m(t)m(wise)f (to)f(0)g(as)h Fv(\017)28 b Fs(!)f Fw(0+.)43 b(It)33 b(follo)m(ws)e(from)h(Leb)s(esgue's)i(theorem)f(that)403 1062 y Fo(X)314 1279 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))505 1260 y Fm(2)540 1279 y Ft(<\025)653 1021 y Fo(Z)752 1047 y Fl(1)708 1246 y Fu(0)844 1021 y Fo(Z)943 1047 y Fl(1)899 1246 y Fu(0)1035 1157 y Fw(\()p Fv(t)1108 1172 y Ft(n;\017)1203 1157 y Fw(\()p Fv(x)1296 1172 y Fu(2)1358 1157 y Fw(+)22 b Fv(y)1504 1172 y Fu(2)1543 1157 y Fw(\))g(+)g Fv(t)1736 1172 y Ft(n;\017)1831 1157 y Fw(\()p Fv(x)1924 1172 y Fu(2)1986 1157 y Fs(\000)h Fv(y)2134 1172 y Fu(2)2173 1157 y Fw(\)\))2391 1089 y Fv(dx)2497 1104 y Fu(2)p 2275 1134 377 4 v 2275 1225 a Fw(\(1)f(+)g Fv(x)2537 1191 y Fu(2)2537 1249 y(2)2577 1225 y Fw(\))2615 1196 y Ft(s)2789 1089 y Fv(dy)2888 1104 y Fu(2)p 2672 1134 373 4 v 2672 1225 a Fw(\(1)g(+)g Fv(y)2931 1191 y Fu(2)2927 1249 y(2)2969 1225 y Fw(\))3007 1196 y Ft(s)3082 1157 y Fs(\000)-17 b(!)28 b Fw(0)p Fv(;)636 1423 y Fw(\()p Fv(\017)g Fs(!)f Fw(0+\))p Fv(:)120 1611 y Fw(Supp)s(ose)36 b(no)m(w)f(that)g(\()p Fv(n)24 b Fw(+)g Fv(k)s Fw(\))1236 1575 y Fu(2)1307 1611 y Fv(>)31 b(\025)p Fw(.)50 b(T)-8 b(o)35 b(con)m(trol)f(the)i(con)m(v)m (ergence)h(of)d(the)i Fv(t)3055 1626 y Ft(n;\017)3185 1611 y Fw(in)e(terms)120 1731 y(of)e Fv(n)p Fw(,)h(w)m(e)h(need)f(the)g (follo)m(wing)d(elemen)m(tary)j(estimates)1213 1916 y Fs(j)p Fw(1)22 b Fs(\000)h Fv(e)1457 1875 y Fl(\000)p Ft(\020)1551 1916 y Fs(j)28 b(\024)g(j)p Fv(\020)8 b Fs(j)p Fv(;)210 b Fw(Re)17 b Fv(\020)35 b Fs(\025)28 b Fw(0)p Fv(;)615 2146 y Fs(j)p Fw(1)21 b Fs(\000)813 2062 y(p)p 896 2062 238 4 v 84 x Fw(1)h(+)g Fv(it)p Fs(j)28 b(\024)1304 2079 y Fw(1)p 1304 2123 49 4 v 1304 2214 a(2)1363 2146 y Fs(j)p Fv(t)p Fs(j)p Fv(;)211 b Fs(j)p Fw(1)22 b Fs(\000)2036 2079 y Fw(1)p 1900 2123 321 4 v 1900 2143 a Fs(p)p 1983 2143 238 4 v 79 x Fw(1)g(+)g Fv(it)2231 2146 y Fs(j)27 b(\024)2401 2079 y Fw(1)p 2401 2123 49 4 v 2401 2214 a(2)2460 2146 y Fs(j)p Fv(t)p Fs(j)p Fv(;)211 b(t)28 b Fs(2)g Fr(R)5 b Fv(:)120 2355 y Fw(The)34 b(last)d(t)m(w)m(o)j(estimates)e(imply)841 2579 y Fs(j)p Fv(c)911 2594 y Ft(n)980 2579 y Fs(\000)23 b Fv(c)1122 2594 y Ft(n)1169 2579 y Fw(\()p Fv(\017)p Fw(\))p Fs(j)k(\024)1531 2512 y Fv(\017)p 1454 2556 194 4 v 1454 2647 a Fw(2)p Fs(j)p Fv(c)1573 2662 y Ft(n)1620 2647 y Fs(j)1657 2579 y Fv(;)212 b Fs(j)1954 2512 y Fw(1)p 1934 2556 89 4 v 1934 2647 a Fv(c)1976 2662 y Ft(n)2055 2579 y Fs(\000)2242 2512 y Fw(1)p 2164 2556 205 4 v 2164 2647 a Fv(c)2206 2662 y Ft(n)2253 2647 y Fw(\()p Fv(\017)p Fw(\))2379 2579 y Fs(j)27 b(\024)2646 2512 y Fv(\017)p 2549 2556 233 4 v 2549 2647 a Fw(2)p Fs(j)p Fv(c)2668 2662 y Ft(n)2715 2647 y Fs(j)2743 2619 y Fu(3)2792 2579 y Fv(:)120 2814 y Fw(Th)m(us)461 3057 y Fv(t)496 3072 y Ft(n;\017)592 3057 y Fw(\()p Fv(a)p Fw(\))g(=)860 2990 y Fv(e)905 2954 y Fl(\000)p Fu(2)p Ft(c)1026 2962 y Fi(n)1068 2954 y Fl(j)p Ft(a)p Fl(j)p 860 3034 290 4 v 980 3126 a Fw(4)1192 2913 y Fo(\014)1192 2973 y(\014)1192 3032 y(\014)1192 3092 y(\014)1236 2990 y Fv(e)1281 2954 y Ft(c)1312 2962 y Fi(n)1354 2954 y Fl(\000)p Ft(c)1440 2962 y Fi(n)1482 2954 y Fu(\()p Ft(\017)p Fu(\)\))p Fl(j)p Ft(a)p Fl(j)p 1236 3034 438 4 v 1352 3126 a Fv(c)1394 3141 y Ft(n)1441 3126 y Fw(\()p Fv(\017)p Fw(\))1706 3057 y Fs(\000)1835 2990 y Fw(1)p 1815 3034 89 4 v 1815 3126 a Fv(c)1857 3141 y Ft(n)1914 2913 y Fo(\014)1914 2973 y(\014)1914 3032 y(\014)1914 3092 y(\014)1947 2930 y Fu(2)2014 3057 y Fs(\024)746 3359 y(\024)862 3292 y Fv(e)907 3256 y Fl(\000)p Fu(2)p Ft(c)1028 3264 y Fi(n)1070 3256 y Fl(j)p Ft(a)p Fl(j)p 862 3337 290 4 v 982 3428 a Fw(2)1177 3189 y Fo( )1391 3292 y Fw(1)p 1266 3337 299 4 v 1266 3428 a Fs(j)p Fv(c)1336 3443 y Ft(n)1383 3428 y Fw(\()p Fv(\017)p Fw(\))p Fs(j)1526 3399 y Fu(2)1592 3275 y Fo(\014)1592 3335 y(\014)1625 3359 y Fv(e)1670 3318 y Ft(c)1701 3326 y Fi(n)1743 3318 y Fl(\000)p Ft(c)1829 3326 y Fi(n)1871 3318 y Fu(\()p Ft(\017)p Fu(\)\))p Fl(j)p Ft(a)p Fl(j)2085 3359 y Fs(\000)c Fw(1)2234 3275 y Fo(\014)2234 3335 y(\014)2267 3295 y Fu(2)2328 3359 y Fw(+)2426 3215 y Fo(\014)2426 3275 y(\014)2426 3335 y(\014)2426 3394 y(\014)2547 3292 y Fw(1)p 2469 3337 205 4 v 2469 3428 a Fv(c)2511 3443 y Ft(n)2558 3428 y Fw(\()p Fv(\017)p Fw(\))2706 3359 y Fs(\000)2835 3292 y Fw(1)p 2815 3337 89 4 v 2815 3428 a Fv(c)2857 3443 y Ft(n)2914 3215 y Fo(\014)2914 3275 y(\014)2914 3335 y(\014)2914 3394 y(\014)2947 3241 y Fu(2)2987 3189 y Fo(!)3093 3359 y Fs(\024)746 3671 y(\024)862 3603 y Fv(e)907 3567 y Fl(\000)p Fu(2)p Ft(c)1028 3575 y Fi(n)1070 3567 y Fl(j)p Ft(a)p Fl(j)p 862 3648 290 4 v 982 3739 a Fw(2)1177 3530 y Fo(\022)1265 3603 y Fv(\017)1304 3567 y Fu(2)1343 3603 y Fs(j)p Fv(a)p Fs(j)1450 3567 y Fu(2)p 1261 3648 233 4 v 1261 3739 a Fw(4)p Fs(j)p Fv(c)1380 3754 y Ft(n)1426 3739 y Fs(j)1454 3710 y Fu(4)1525 3671 y Fw(+)1710 3603 y Fv(\017)1749 3567 y Fu(2)p 1633 3648 V 1633 3739 a Fw(4)p Fs(j)p Fv(c)1752 3754 y Ft(n)1799 3739 y Fs(j)1827 3710 y Fu(6)1876 3530 y Fo(\023)1977 3671 y Fs(\024)746 3935 y(\024)862 3868 y Fw(1)p 862 3912 49 4 v 862 4003 a(8)920 3935 y Fv(\017)959 3894 y Fu(2)1009 3868 y Fv(\014)28 b Fw(+)22 b(1)p 1009 3912 230 4 v 1032 4003 a Fs(j)p Fv(c)1102 4018 y Ft(n)1149 4003 y Fs(j)1177 3975 y Fu(6)1249 3935 y Fv(;)120 4170 y Fw(where)1339 4291 y Fv(\014)34 b Fw(:=)27 b(max)1586 4354 y Ft(b)p Fl(2)p Fn(R)1773 4291 y Fv(b)1814 4250 y Fu(2)1854 4291 y Fv(e)1899 4250 y Fl(\000)p Fu(2)p Fl(j)p Ft(b)p Fl(j)2090 4291 y Fv(<)h Fs(1)p Fv(:)120 4484 y Fw(It)33 b(follo)m(ws)e(that)497 4600 y Fo(X)408 4817 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))599 4799 y Fm(2)633 4817 y Ft(>\025)746 4559 y Fo(Z)846 4586 y Fl(1)801 4785 y Fu(0)937 4559 y Fo(Z)1037 4586 y Fl(1)992 4785 y Fu(0)1128 4695 y Fw(\()p Fv(t)1201 4710 y Ft(n;\017)1296 4695 y Fw(\()p Fv(x)1389 4710 y Fu(2)1451 4695 y Fw(+)22 b Fv(y)1597 4710 y Fu(2)1636 4695 y Fw(\))g(+)g Fv(t)1829 4710 y Ft(n;\017)1925 4695 y Fw(\()p Fv(x)2018 4710 y Fu(2)2080 4695 y Fs(\000)g Fv(y)2227 4710 y Fu(2)2266 4695 y Fw(\)\))2484 4628 y Fv(dx)2590 4643 y Fu(2)p 2369 4672 377 4 v 2369 4763 a Fw(\(1)f(+)h Fv(x)2630 4729 y Fu(2)2630 4788 y(2)2670 4763 y Fw(\))2708 4735 y Ft(s)2882 4628 y Fv(dy)2981 4643 y Fu(2)p 2765 4672 373 4 v 2765 4763 a Fw(\(1)g(+)g Fv(y)3024 4729 y Fu(2)3020 4788 y(2)3062 4763 y Fw(\))3100 4735 y Ft(s)3175 4695 y Fs(\024)757 5038 y(\024)28 b Fw(2)1017 4944 y Fo(X)928 5161 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))1119 5142 y Fm(2)1153 5161 y Ft(>\025)1266 4898 y Fo(\022)1339 5038 y Fw(sup)1347 5120 y Ft(a)p Fl(2)p Fn(R)1503 5038 y Fv(t)1538 5053 y Ft(n;\017)1633 5038 y Fw(\()p Fv(a)p Fw(\))1760 4898 y Fo(\023)17 b(\022)1924 4903 y(Z)2023 4929 y Fl(1)1979 5128 y Fu(0)2240 4971 y Fv(dx)2346 4986 y Fu(2)p 2124 5015 377 4 v 2124 5107 a Fw(\(1)22 b(+)g Fv(x)2386 5072 y Fu(2)2386 5131 y(2)2426 5107 y Fw(\))2464 5078 y Ft(s)2511 4898 y Fo(\023)2584 4920 y Fu(2)2651 5038 y Fs(\024)757 5381 y(\024)872 5314 y Fv(\017)911 5278 y Fu(2)p 872 5358 79 4 v 887 5450 a Fw(4)978 5241 y Fo(\022)1051 5246 y(Z)1151 5272 y Fl(1)1106 5471 y Fu(0)1367 5314 y Fv(dx)1473 5329 y Fu(2)p 1252 5358 377 4 v 1252 5450 a Fw(\(1)g(+)g Fv(x)1514 5415 y Fu(2)1514 5474 y(2)1554 5450 y Fw(\))1592 5421 y Ft(s)1638 5241 y Fo(\023)1712 5263 y Fu(2)1768 5381 y Fw(\()p Fv(\014)27 b Fw(+)22 b(1\))2090 5287 y Fo(X)2095 5499 y Ft(n)p Fl(2)p Fn(Z)2328 5314 y Fw(1)p 2260 5358 184 4 v 2260 5450 a Fs(j)p Fv(c)2330 5465 y Ft(n)2377 5450 y Fs(j)2405 5421 y Fu(6)2454 5381 y Fv(:)1781 5712 y Fw(27)p eop %%Page: 28 33 28 32 bop 120 407 a Fw(The)38 b(RHS)g(is)f(\014nite)g(and)g(con)m(v)m (erges)j(to)d(0)g(as)g Fv(\017)f Fs(!)g Fw(0+,)i(whic)m(h)g(completes)f (the)h(pro)s(of)e(of)120 527 y(\(3.2.1\))o(.)120 648 y(Finally)-8 b(,)29 b(w)m(e)j(remark)g(that)f(the)h(limit)c(in)i (\(3.2.1\))h(is)g(uniform)f(in)g Fv(\025)i Fw(for)f Fv(\025)g Fw(from)f(a)h(compact)120 768 y(in)m(terv)-5 b(all)31 b(not)h(con)m(taining)f(an)m(y)j(of)e(the)h(p)s(oin)m(ts)g(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))2213 732 y Fu(2)2253 768 y Fv(;)33 b(n)28 b Fs(2)g Fr(Z)p Fw(.)42 b(This)33 b(follo)m(ws)e(from) h(the)120 888 y(fact,)g(that)h Fv(c)593 903 y Ft(n)667 888 y Fw(=)28 b Fv(c)813 903 y Ft(n)860 888 y Fw(\()p Fv(\025)p Fw(\))k(dep)s(ends)i(con)m(tin)m(uously)f(on)g Fv(\025)p Fw(.)p 3469 888 4 66 v 3473 826 59 4 v 3473 888 V 3531 888 4 66 v 120 1147 a Fp(2.)46 b(The)28 b(limiting)22 b(absorption)27 b(principle)e(for)i(the)g(p)s(erturb)s(ed)h(op)s (erator)f Fv(H)3278 1111 y Fu(\()p Ft(\033)r Fu(\))3379 1147 y Fw(\()p Fv(k)s Fw(\))p Fp(.)120 1267 y Fw(Using)j(the)h(resolv)m (en)m(t)h(iden)m(tit)m(y)g(\(3.1.9\))d(w)m(e)j(w)m(an)m(t)f(to)g(deriv) m(e)g(a)f(limiting)c(absorption)k(prin-)120 1388 y(ciple)i(for)g(the)h (op)s(erator)f Fv(H)1147 1351 y Fu(\()p Ft(\033)r Fu(\))1248 1388 y Fw(\()p Fv(k)s Fw(\).)120 1522 y(W)-8 b(e)33 b(start)g(our)f (considerations)g(with)g(the)h(op)s(erators)g Fv(T)2223 1471 y Fu(\()p Ft(\033)r Fu(\))2209 1546 y(0)2324 1522 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))p Fv(;)34 b(T)2679 1485 y Fu(\()p Ft(\033)r Fu(\))2780 1522 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\).)120 1718 y Fp(Lemma)37 b(3.2.2.)42 b Fj(L)-5 b(et)35 b Fv(k)c Fs(2)d Fw([)p Fs(\000)1278 1679 y Fu(1)p 1278 1695 36 4 v 1278 1752 a(2)1324 1718 y Fv(;)1378 1679 y Fu(1)p 1378 1695 V 1378 1752 a(2)1423 1718 y Fw(])p Fj(,)35 b(then)f(the)h(op)-5 b(er)g(ator)35 b(families)949 1952 y Fv(T)1020 1902 y Fu(\()p Ft(\033)r Fu(\))1006 1977 y(0)1122 1952 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))p Fv(;)33 b(T)1476 1911 y Fu(\()p Ft(\033)r Fu(\))1578 1952 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))p Fv(;)216 b(z)33 b Fs(2)28 b Fr(C)48 b Fs(n)22 b Fw([)p Fv(k)2463 1911 y Fu(2)2502 1952 y Fv(;)17 b Fs(1)p Fw(\))p Fv(;)120 2164 y Fj(c)-5 b(an)43 b(b)-5 b(e)44 b(extende)-5 b(d)43 b(norm-c)-5 b(ontinuously)43 b(to)h(the)g(cut)h(fr)-5 b(om)43 b(ab)-5 b(ove)43 b(and)h(fr)-5 b(om)43 b(b)-5 b(elow)43 b(with)120 2284 y(the)36 b(exc)-5 b(eption)34 b(of)i(the)f(p)-5 b(oints)35 b Fv(z)f Fw(=)29 b(\()p Fv(n)23 b Fw(+)f Fv(k)s Fw(\))1767 2248 y Fu(2)1807 2284 y Fv(;)33 b(n)c Fs(2)h Fr(Z)p Fj(.)44 b(Denoting)35 b(the)g(\(upp)-5 b(er)36 b(and)f(lower\))120 2405 y(b)-5 b(oundary)35 b(values)f(by)699 2616 y Fv(T)770 2565 y Fu(\()p Ft(\033)r Fu(\))756 2641 y(0)871 2616 y Fw(\()p Fv(\025)22 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))p Fv(;)216 b(T)1620 2575 y Fu(\()p Ft(\033)r Fu(\))1721 2616 y Fw(\()p Fv(\025)22 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))p Fv(;)215 b(\025)28 b Fs(6)p Fw(=)f(\()p Fv(n)c Fw(+)f Fv(k)s Fw(\))2895 2575 y Fu(2)2934 2616 y Fv(;)120 2828 y Fj(the)35 b(sets)120 3040 y Fs(N)202 3055 y Fl(\006)261 3040 y Fw(\()p Fv(k)s Fw(\))27 b(:=)h Fs(f)p Fv(\025)f Fs(2)i Fr(R)6 b Fs(n)q(f)p Fw(\()p Fv(n)q Fw(+)r Fv(k)s Fw(\))1213 2999 y Fu(2)1286 3040 y Fw(:)29 b Fv(n)f Fs(2)g Fr(Z)p Fs(g)d Fw(:)k Fs(N)15 b Fw(\()p Fv(I)9 b Fw(+)q Fv(T)2057 2989 y Fu(\()p Ft(\033)r Fu(\))2043 3064 y(0)2158 3040 y Fw(\()p Fv(\025)q Fs(\006)q Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))g Fv(T)2638 2999 y Fu(\()p Ft(\033)r Fu(\))2740 3040 y Fw(\()p Fv(\025)q Fs(\007)q Fv(i)p Fw(0)p Fv(;)g(k)s Fw(\))3132 2999 y Fl(\003)3173 3040 y Fw(\))27 b Fs(6)p Fw(=)h Fs(f)p Fw(0)p Fs(gg)120 3251 y Fj(ar)-5 b(e)35 b(discr)-5 b(ete)34 b(in)h Fr(R)e Fs(n)22 b(f)p Fw(\()p Fv(n)g Fw(+)g Fv(k)s Fw(\))1287 3215 y Fu(2)1354 3251 y Fw(:)50 b Fv(n)28 b Fs(2)g Fr(Z)p Fs(g)p Fj(.)120 3447 y(R)-5 b(emark)53 b Fw(3.2.3)p Fj(.)46 b Fw(W)-8 b(e)41 b(will)f(pro)m(v)m(e)i(in)f (Lemma)f(3.3.17)h(b)s(elo)m(w)g(that)g(the)h(sets)h Fs(N)3153 3462 y Fu(+)3211 3447 y Fw(\()p Fv(k)s Fw(\))f(and)120 3568 y Fs(N)202 3583 y Fl(\000)261 3568 y Fw(\()p Fv(k)s Fw(\))32 b(coincide.)120 3761 y Fj(Pr)-5 b(o)g(of.)48 b Fw(Let)552 3947 y(~)528 3972 y Fv(B)6 b Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))p Fv(;)212 b(z)32 b Fs(2)c Fr(C)48 b Fs(n)22 b(f)p Fw(\()p Fv(n)g Fw(+)g Fv(k)s Fw(\))1765 3931 y Fu(2)1827 3972 y Fw(+)g Fv(iy)31 b Fw(:)50 b Fv(n)28 b Fs(2)g Fr(Z)p Fv(;)j(y)g Fs(\024)d Fw(0)p Fs(g)f Fw(=:)h Fv(D)2943 3987 y Fu(+)3002 3972 y Fw(\()p Fv(k)s Fw(\))120 4184 y(b)s(e)37 b(the)h(analytic)d(family)g(of)h(op)s(erators)h(giv)m (en)g(b)m(y)h(the)f(same)g(formal)d(expression)39 b(\(3.1.8\))120 4304 y(as)49 b(the)g(op)s(erators)g Fv(B)5 b Fw(\()p Fv(z)t(;)17 b(k)s Fw(\),)53 b(but)c(with)f(the)h(branc)m(h)h(of)e(the)h (square)h(ro)s(ot)e(de\014ned)i(on)120 4425 y Fr(C)h Fs(n)25 b(f)p Fv(iy)39 b Fw(:)61 b Fv(y)38 b Fs(\024)e Fw(0)p Fs(g)p Fw(.)57 b(\(Hence)38 b(the)f(square)i(ro)s(ot)1961 4339 y Fo(p)p 2060 4339 520 4 v 2060 4425 a Fw(\()p Fv(n)23 b Fw(+)f Fv(k)s Fw(\))2369 4396 y Fu(2)2430 4425 y Fs(\000)h Fv(z)42 b Fw(in)36 b(the)i Fv(n)p Fw(-th)f(summand)120 4545 y(of)45 b(\(3.1.8\))37 b(is)h(w)m(ell-de\014ned)g(for)g Fv(z)k Fs(2)37 b Fv(D)1627 4560 y Fu(+)1686 4545 y Fw(\()p Fv(k)s Fw(\).\))60 b(Both)38 b(branc)m(hes)i(coincide)e(on)g(the)g(upp) s(er)120 4665 y(halfplane,)31 b(so)i(that)1201 4760 y(~)1178 4786 y Fv(B)5 b Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))28 b(=)f Fv(B)5 b Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))p Fv(;)213 b(z)32 b Fs(2)c Fr(C)2390 4801 y Fu(+)2455 4786 y Fv(:)782 b Fw(\(3.2.2\))120 4956 y(Let)27 b Fv(B)32 b Fw(b)s(e)26 b(as)h(in)f(Section)g(1.2,)i(then)f Fv(B)1553 4920 y Fu(1)p Ft(=)p Fu(2)1663 4956 y Fv(B)5 b Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))1965 4920 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)2131 4956 y Fv(;)33 b(z)g Fs(2)28 b Fv(D)2444 4971 y Fu(+)2503 4956 y Fw(\()p Fv(k)s Fw(\),)f(is)g(an)f(analytic)f(family)120 5077 y(of)32 b(b)s(ounded)h(op)s(erators)g(whic)m(h)g(are)f(giv)m(en)h (b)m(y)882 5381 y Fv(B)961 5340 y Fu(1)p Ft(=)p Fu(2)1094 5356 y Fw(~)1071 5381 y Fv(B)5 b Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))1373 5340 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1538 5381 y Fv(f)39 b Fw(=)1728 5287 y Fo(X)1733 5499 y Ft(n)p Fl(2)p Fn(Z)1917 5340 y Fm(4)1892 5204 y Fo(s)p 1991 5204 540 4 v 2128 5314 a Fv(n)2186 5285 y Fu(2)2247 5314 y Fw(+)22 b(1)p 2001 5358 520 4 v 2001 5450 a(\()p Fv(n)h Fw(+)f Fv(k)s Fw(\))2310 5421 y Fu(2)2371 5450 y Fs(\000)h Fv(z)2568 5355 y Fw(^)2547 5381 y Fv(f)2595 5396 y Ft(n)2659 5381 y Fv(e)2704 5396 y Ft(n)2751 5381 y Fv(:)486 b Fw(\(3.2.3\))1781 5712 y(28)p eop %%Page: 29 34 29 33 bop 120 407 a Fw(It)33 b(follo)m(ws)e(from)g(Lemma)g(1.2.3)h (that)474 581 y(~)455 606 y Fv(T)526 565 y Fu(\()p Ft(\033)r Fu(\))627 606 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))28 b(:=)g Fs(j)p Fv(M)1141 565 y Fu(\()p Ft(\033)r Fu(\))1243 606 y Fs(j)1271 565 y Fu(1)p Ft(=)p Fu(2)1404 581 y Fw(~)1381 606 y Fv(B)5 b Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))1683 565 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1876 606 y Fw(=)878 790 y(=)982 709 y Fo(\000)1027 790 y Fs(j)p Fv(M)1159 749 y Fu(\()p Ft(\033)r Fu(\))1262 790 y Fs(j)1290 749 y Fu(1)p Ft(=)p Fu(2)1399 790 y Fv(B)1478 749 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1643 709 y Fo(\001)1706 679 y(\020)1765 790 y Fv(B)1844 749 y Fu(1)p Ft(=)p Fu(2)1978 765 y Fw(~)1954 790 y Fv(B)5 b Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))2256 749 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)2422 679 y Fo(\021)2498 790 y Fv(;)212 b(z)32 b Fs(2)c Fv(D)2989 805 y Fu(+)3048 790 y Fw(\()p Fv(k)s Fw(\))p Fv(;)120 1037 y Fw(is)39 b(an)g(analytic)g(family)e(of)i(compact)g(op)s(erators.)64 b(Because)41 b(of)46 b(\(3.2.2\))39 b(it)f(is)h(a)g(b)s(ounded)120 1158 y(analytic)h(\(and)h(hence)h(norm-con)m(tin)m(uous\))f(extension)h (of)e(the)i(family)c Fv(T)2918 1122 y Fu(\()p Ft(\033)r Fu(\))3020 1158 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))41 b(across)120 1278 y(the)33 b(cut)g(from)e(ab)s(o)m(v)m(e.)45 b(W)-8 b(e)33 b(put)692 1498 y Fv(T)763 1457 y Fu(\()p Ft(\033)r Fu(\))865 1498 y Fw(\()p Fv(\025)22 b Fw(+)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))27 b(:=)1475 1473 y(~)1456 1498 y Fv(T)1527 1457 y Fu(\()p Ft(\033)r Fu(\))1628 1498 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))p Fv(;)212 b(\025)27 b Fs(6)p Fw(=)h(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))2594 1457 y Fu(2)2634 1498 y Fv(;)33 b(n)28 b Fs(2)g Fr(Z)p Fv(:)120 1718 y Fw(The)37 b(construction)g(of)e(the)i(op)s(erators)f Fv(T)1679 1682 y Fu(\()p Ft(\033)r Fu(\))1781 1718 y Fw(\()p Fv(\025)24 b Fs(\000)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))36 b(is)g(similar,)e(replacing)h Fv(D)3212 1733 y Fu(+)3271 1718 y Fw(\()p Fv(k)s Fw(\))h(b)m(y)120 1839 y Fv(D)201 1854 y Fl(\000)260 1839 y Fw(\()p Fv(k)s Fw(\))28 b(:=)f Fr(C)48 b Fs(n)22 b(f)p Fw(\()p Fv(n)g Fw(+)g Fv(k)s Fw(\))1072 1802 y Fu(2)1134 1839 y Fw(+)g Fv(iy)31 b Fw(:)50 b Fv(n)28 b Fs(2)g Fr(Z)p Fv(;)j(y)g Fs(\025)d Fw(0)p Fs(g)p Fw(.)120 1972 y(The)41 b(statemen)m(t)f(ab)s (out)f(the)h(op)s(erators)g Fv(T)1759 1922 y Fu(\()p Ft(\033)r Fu(\))1745 1997 y(0)1860 1972 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))40 b(follo)m(ws)f(b)m(y)i(applying)d(the)i(b)s(ounded)120 2093 y(op)s(erator)32 b(sgn)17 b Fv(M)775 2057 y Fu(\()p Ft(\033)r Fu(\))910 2093 y Fw(to)32 b Fv(T)1100 2057 y Fu(\()p Ft(\033)r Fu(\))1202 2093 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\).)120 2213 y(Let)40 b(us)h(pro)m(v)m(e)g(the)g(statemen)m(t) g(ab)s(out)e(the)i(sets)g Fs(N)2084 2228 y Fl(\006)2143 2213 y Fw(\()p Fv(k)s Fw(\).)66 b(It)40 b(follo)m(ws)f(from)g (\(3.2.3\))g(that)120 2334 y Fs(k)189 2308 y Fw(~)170 2334 y Fv(T)241 2297 y Fu(\()p Ft(\033)r Fu(\))342 2334 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))p Fs(k)40 b Fv(<)f Fw(1)g(if)g Fs(j)p Fw(Im)16 b Fv(z)t Fs(j)39 b Fw(is)g(large.)64 b(No)m(w)40 b(the)f(Analytic)g(F)-8 b(redholm)38 b(Alternativ)m(e)h (\(cf.)120 2468 y(Theorem)34 b(VI)s(I.1.9)g(in)f([K]\))h(applied)f(to)g (the)i(op)s(erators)2266 2442 y(~)2247 2468 y Fv(T)2318 2417 y Fu(\()p Ft(\033)r Fu(\))2304 2492 y(0)2420 2468 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))2679 2442 y(~)2660 2468 y Fv(T)2731 2431 y Fu(\()p Ft(\033)r Fu(\))2832 2468 y Fw(\()p 2870 2413 50 4 v Fv(z)5 b(;)17 b(k)s Fw(\))3056 2431 y Fl(\003)3129 2468 y Fw(yields)34 b(the)120 2588 y(discreteness)h(of)d(the)h(sets)h Fs(N)1205 2603 y Fl(\006)1263 2588 y Fw(\()p Fv(k)s Fw(\).)44 b(This)33 b(concludes)g(the)g(pro)s (of.)p 3469 2588 4 66 v 3473 2525 59 4 v 3473 2588 V 3531 2588 4 66 v 120 2791 a Fp(Prop)s(osition)j(3.2.4.)42 b Fj(L)-5 b(et)35 b Fv(s)27 b(>)1384 2752 y Fu(1)p 1384 2768 36 4 v 1384 2826 a(2)1464 2791 y Fj(and)34 b Fv(k)d Fs(2)d Fw([)p Fs(\000)1943 2752 y Fu(1)p 1943 2768 V 1943 2826 a(2)1989 2791 y Fv(;)2043 2752 y Fu(1)p 2043 2768 V 2043 2826 a(2)2088 2791 y Fw(])p Fj(.)45 b(Then)34 b(the)h(limits)479 3025 y Fw(lim)453 3084 y Ft(\017)p Fl(!)p Fu(0+)658 3025 y Fw(\003)726 2983 y Fl(\000)p Ft(s)726 3049 y Fu(2)818 3025 y Fv(R)893 2983 y Fu(\()p Ft(\033)r Fu(\))995 3025 y Fw(\()p Fv(\025)22 b Fs(\006)g Fv(i\017;)17 b(k)s Fw(\)\003)1487 2983 y Fl(\000)p Ft(s)1487 3049 y Fu(2)1579 3025 y Fv(;)216 b(\025)39 b(=)-61 b Fs(2)29 b(f)p Fw(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))2359 2983 y Fu(2)2426 3025 y Fw(:)50 b Fv(n)28 b Fs(2)g Fr(Z)p Fs(g)19 b([)k(N)2992 3040 y Fl(\006)3050 3025 y Fw(\()p Fv(k)s Fw(\))p Fv(;)120 3278 y Fj(exist)31 b(in)g(the)g(op)-5 b(er)g(ator)31 b(norm)f(top)-5 b(olo)g(gy)31 b(and)f(ar)-5 b(e)31 b(uniform)g(for)g Fv(\025)g Fj(fr)-5 b(om)30 b(c)-5 b(omp)g(act)31 b(intervals)120 3399 y(of)k Fr(R)e Fs(n)22 b Fw(\()p Fs(f)p Fw(\()p Fv(n)g Fw(+)g Fv(k)s Fw(\))797 3363 y Fu(2)864 3399 y Fw(:)50 b Fv(n)28 b Fs(2)g Fr(Z)p Fs(g)19 b([)k(N)1430 3414 y Fl(\006)1488 3399 y Fw(\()p Fv(k)s Fw(\)\))p Fj(.)120 3602 y(Pr)-5 b(o)g(of.)48 b Fw(W)-8 b(e)40 b(consider)g(the)h(resolv)m(en)m(t)f(iden)m(tit)m(y)h (\(3.1.9\))o(.)65 b(Since)40 b(the)g(norm-limits)35 b(of)40 b(the)120 3722 y(op)s(erators)32 b(\003)619 3681 y Fl(\000)p Ft(s)619 3747 y Fu(2)711 3722 y Fv(R)786 3686 y Fu(\(0\))880 3722 y Fw(\()p Fv(\025)22 b Fs(\006)h Fv(i\017;)17 b(k)s Fw(\)\003)1373 3681 y Fl(\000)p Ft(s)1373 3747 y Fu(2)1497 3722 y Fw(and)365 3986 y Fv(T)436 3945 y Fu(\()p Ft(\033)r Fu(\))537 3986 y Fw(\()p Fv(\025)22 b Fs(\007)h Fv(i\017;)17 b(k)s Fw(\))962 3945 y Fl(\003)1040 3876 y Fo(\020)1100 3986 y Fv(I)30 b Fw(+)22 b Fv(T)1342 3935 y Fu(\()p Ft(\033)r Fu(\))1328 4011 y(0)1443 3986 y Fw(\()p Fv(\025)g Fs(\006)h Fv(i\017;)17 b(k)s Fw(\))22 b Fv(T)1961 3945 y Fu(\()p Ft(\033)r Fu(\))2063 3986 y Fw(\()p Fv(\025)g Fs(\007)h Fv(i\017;)17 b(k)s Fw(\))2488 3945 y Fl(\003)2527 3876 y Fo(\021)2587 3898 y Fl(\000)p Fu(1)2698 3986 y Fv(T)2769 3935 y Fu(\()p Ft(\033)r Fu(\))2755 4011 y(0)2870 3986 y Fw(\()p Fv(\025)22 b Fs(\006)h Fv(i\017;)17 b(k)s Fw(\))120 4238 y(exist)35 b(and)f(are)h(uniform)d(on)i(compact)g(in)m(terv)-5 b(als)34 b(of)g Fr(R)g Fs(n)23 b Fw(\()p Fs(f)p Fw(\()p Fv(n)f Fw(+)g Fv(k)s Fw(\))2675 4202 y Fu(2)2743 4238 y Fw(:)27 b Fv(n)h Fs(2)g Fr(Z)p Fs(g)20 b([)i(N)3286 4253 y Fl(\006)3345 4238 y Fw(\()p Fv(k)s Fw(\)\),)120 4358 y(it)32 b(su\016ces)i(to)e(pro)m(v)m(e)i(that)f(the)g(limits)868 4578 y(lim)841 4638 y Ft(\017)p Fl(!)p Fu(0+)1047 4578 y Fv(Y)22 b Fw(\()p Fv(\025)g Fs(\006)g Fv(i\017;)17 b(k)s Fw(\)\003)1618 4537 y Fl(\000)p Ft(s)1618 4603 y Fu(2)1710 4578 y Fv(;)212 b(\025)27 b Fs(6)p Fw(=)h(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))2445 4537 y Fu(2)2485 4578 y Fv(;)33 b(n)28 b Fs(2)g Fr(Z)p Fv(;)120 4838 y Fw(exist)h(in)e(the)i (norm)e(sense)j(and)f(are)f(uniform)e(for)i Fv(\025)g Fw(from)f(compact)h(in)m(terv)-5 b(als)28 b(of)f Fr(R)19 b Fs(n)13 b(f)p Fw(\()p Fv(n)g Fw(+)120 4958 y Fv(k)s Fw(\))212 4922 y Fu(2)279 4958 y Fw(:)28 b Fv(n)g Fs(2)g Fr(Z)p Fs(g)p Fw(.)120 5079 y(W)-8 b(e)33 b(note)g(that)f Fv(Y)21 b Fw(\()p Fv(z)t(;)c(k)s Fw(\))33 b(is)f(an)h(in)m(tegral)e(op) s(erator)h(with)g(k)m(ernel)120 5343 y Fv(k)s Fw(\()p Fv(x)267 5358 y Fu(1)307 5343 y Fv(;)17 b(y)t Fw(;)g Fv(z)t(;)g(k)s Fw(\))27 b(:=)829 5276 y(1)p 799 5320 108 4 v 799 5411 a(2)p Fv(\031)933 5248 y Fo(X)938 5460 y Ft(n)p Fl(2)p Fn(Z)1390 5276 y Fw(1)p 1104 5320 622 4 v 1132 5387 a Fm(4)1107 5340 y Fo(p)p 1206 5340 520 4 v 1206 5425 a Fw(\()p Fv(n)c Fw(+)f Fv(k)s Fw(\))1515 5396 y Fu(2)1576 5425 y Fs(\000)h Fv(z)1752 5343 y(e)1797 5302 y Ft(in)p Fu(\()p Ft(x)1931 5311 y Fm(1)1966 5302 y Fl(\000)p Ft(y)2056 5311 y Fm(1)2090 5302 y Fu(\))2139 5343 y Fv(e)2184 5302 y Fl(\000)2239 5235 y Fs(p)p 2322 5235 317 4 v 67 x Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))2513 5283 y Fm(2)2547 5302 y Fl(\000)p Ft(z)15 b(y)2685 5311 y Fm(2)2723 5343 y Fv(;)212 b(x)3017 5358 y Fu(1)3085 5343 y Fs(2)28 b Fr(T)p Fv(;)36 b(y)31 b Fs(2)d Fw(\005)p Fv(:)1781 5712 y Fw(29)p eop %%Page: 30 35 30 34 bop 120 407 a Fw(Fix)32 b Fv(\025)27 b Fs(2)h Fr(R)34 b Fs(n)21 b(f)p Fw(\()p Fv(n)i Fw(+)f Fv(k)s Fw(\))998 371 y Fu(2)1065 407 y Fw(:)50 b Fv(n)28 b Fs(2)g Fr(Z)p Fs(g)h Fw(and)k(de\014ne)523 579 y Fv(k)s Fw(\()p Fv(x)670 594 y Fu(1)710 579 y Fv(;)17 b(y)t Fw(;)g Fv(\025)j Fs(\006)j Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))27 b(:=)55 b(lim)1403 639 y Ft(\017)p Fl(!)p Fu(0+)1609 579 y Fv(k)s Fw(\()p Fv(x)1756 594 y Fu(1)1796 579 y Fv(;)17 b(y)t Fw(;)g Fv(\025)j Fs(\006)j Fv(i\017;)17 b(k)s Fw(\))p Fv(;)212 b(x)28 b Fs(2)g Fr(T)p Fv(;)37 b(y)31 b Fs(2)d Fw(\005)p Fv(;)120 786 y Fw(as)33 b(p)s(oin)m(t)m(wise)f(limit.)40 b(W)-8 b(e)33 b(claim)d(that)184 867 y Fo(Z)239 1092 y Fu(\005)313 867 y Fo(Z)368 1092 y Fn(T)433 1002 y Fs(j)p Fv(k)s Fw(\()p Fv(x)608 1017 y Fu(1)648 1002 y Fv(;)17 b(y)t Fw(;)g Fv(\025)k Fs(\006)h Fv(i\017;)17 b(k)s Fw(\))23 b Fs(\000)f Fv(k)s Fw(\()p Fv(x)1442 1017 y Fu(1)1482 1002 y Fv(;)17 b(y)t Fw(;)g Fv(\025)k Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))p Fs(j)2045 954 y Fu(2)2117 1002 y Fv(dx)2223 1017 y Fu(1)2408 935 y Fv(dy)p 2273 979 373 4 v 2273 1071 a Fw(\(1)22 b(+)g Fv(y)2532 1036 y Fu(2)2528 1095 y(2)2570 1071 y Fw(\))2608 1042 y Ft(s)2683 1002 y Fs(!)27 b Fw(0)195 b(\()p Fv(\017)28 b Fs(!)f Fw(0+\))p Fv(:)3264 1179 y Fw(\(3.2.4\))120 1299 y(This)35 b(means)g(that)f Fv(Y)21 b Fw(\()p Fv(\025)j Fs(\006)g Fv(i\017;)17 b(k)s Fw(\)\003)1433 1258 y Fl(\000)p Ft(s)1433 1323 y Fu(2)1559 1299 y Fw(con)m(v)m(erges)37 b(in)d(\(Hilb)s(ert-Sc)m (hmidt\))e(norm)i(to)h(an)f(in-)120 1419 y(tegral)29 b(op)s(erator)g(with)g(k)m(ernel)i Fv(k)s Fw(\()p Fv(x)1435 1434 y Fu(1)1475 1419 y Fv(;)17 b(y)t Fw(;)g Fv(\025)g Fs(\006)g Fv(i)p Fw(0)p Fv(;)g(k)s Fw(\))2116 1380 y Fu(1)p 2011 1396 251 4 v 2011 1455 a(\(1+)p Ft(y)2165 1433 y Fm(2)2163 1477 y(2)2198 1455 y Fu(\))2225 1436 y Fi(s)2268 1419 y Fw(,)31 b(whic)m(h)f(establishes)g(the)h(Prop)s(o-) 120 1551 y(sition.)120 1671 y(The)36 b(pro)s(of)e(of)41 b(\(3.2.4\))34 b(uses)j(the)e(same)g(tec)m(hniques)h(as)f(the)h(pro)s (of)e(of)41 b(\(3.2.1\))o(,)36 b(so)f(w)m(e)h(can)120 1792 y(omit)31 b(it)g(here.)p 3469 1792 4 66 v 3473 1729 59 4 v 3473 1792 V 3531 1792 4 66 v 266 1973 a(As)i(an)g(easy)h (consequence)h(of)d(Prop)s(osition)f(3.2.4)h(w)m(e)i(obtain)120 2136 y Fp(Theorem)j(3.2.5.)42 b Fj(L)-5 b(et)35 b Fv(k)c Fs(2)d Fw([)p Fs(\000)1356 2097 y Fu(1)p 1356 2113 36 4 v 1356 2170 a(2)1402 2136 y Fv(;)1456 2097 y Fu(1)p 1456 2113 V 1456 2170 a(2)1501 2136 y Fw(])p Fj(,)35 b(then)1450 2322 y Fv(\033)1505 2337 y Ft(sc)1590 2241 y Fo(\000)1636 2322 y Fv(H)1725 2281 y Fu(\()p Ft(\033)r Fu(\))1826 2322 y Fw(\()p Fv(k)s Fw(\))1956 2241 y Fo(\001)2029 2322 y Fw(=)28 b Fs(;)p Fv(:)120 2503 y Fj(Pr)-5 b(o)g(of.)48 b Fw(By)k(Theorem)f(3.1.3,)j Fv(\033)1364 2518 y Ft(sc)1449 2423 y Fo(\000)1495 2503 y Fv(H)1584 2467 y Fu(\()p Ft(\033)r Fu(\))1685 2503 y Fw(\()p Fv(k)s Fw(\))1815 2423 y Fo(\001)1919 2503 y Fs(\032)59 b Fv(\033)2110 2518 y Ft(ess)2229 2423 y Fo(\000)2275 2503 y Fv(H)2364 2467 y Fu(\()p Ft(\033)r Fu(\))2465 2503 y Fw(\()p Fv(k)s Fw(\))2595 2423 y Fo(\001)2699 2503 y Fw(=)g([)p Fv(k)2915 2467 y Fu(2)2954 2503 y Fv(;)17 b Fw(+)p Fs(1)p Fw(\),)55 b(so)c(let)120 2624 y([)p Fv(a;)17 b(b)p Fw(])46 b Fs(\032)f Fw([)p Fv(k)559 2588 y Fu(2)599 2624 y Fv(;)17 b Fw(+)p Fs(1)p Fw(\))28 b Fs(n)h Fw(\()p Fs(f)p Fw(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))1360 2588 y Fu(2)1427 2624 y Fw(:)50 b Fv(n)28 b Fs(2)g Fr(Z)p Fs(g)19 b([)k(N)1993 2639 y Fl(\006)2052 2624 y Fw(\()p Fv(k)s 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3358 y Ft(sc)3019 3263 y Fo(\000)3065 3343 y Fv(H)3154 3307 y Fu(\()p Ft(\033)r Fu(\))3255 3343 y Fw(\()p Fv(k)s Fw(\))3385 3263 y Fo(\001)3463 3343 y Fs(\032)120 3464 y(f)p Fw(\()p Fv(n)25 b Fw(+)h Fv(k)s Fw(\))485 3428 y Fu(2)560 3464 y Fw(:)61 b Fv(n)36 b Fs(2)g Fr(Z)p Fs(g)22 b([)k(N)1159 3479 y Fu(+)1218 3464 y Fw(\()p Fv(k)s Fw(\))f Fs([)h(N)1547 3479 y Fl(\000)1605 3464 y Fw(\()p Fv(k)s Fw(\).)58 b(Since)37 b(the)h(latter)e(set)i (accum)m(ulates)g(only)e(at)120 3584 y Fs(f)p Fw(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))478 3548 y Fu(2)545 3584 y Fw(:)50 b Fv(n)28 b Fs(2)g Fr(Z)p Fs(g)i Fw(and)j(+)p Fs(1)p Fw(,)f(w)m(e)h(conclude)g Fv(\033)1975 3599 y Ft(sc)2060 3503 y Fo(\000)2105 3584 y Fv(H)2194 3548 y Fu(\()p Ft(\033)r Fu(\))2296 3584 y Fw(\()p Fv(k)s Fw(\))2426 3503 y Fo(\001)2499 3584 y Fw(=)28 b Fs(;)p Fw(,)k(as)h(claimed.)p 3469 3584 4 66 v 3473 3522 59 4 v 3473 3584 V 3531 3584 4 66 v 120 3910 a Fq(3.3)161 b(The)47 b(p)t(oin)l(t)g(sp)t(ectrum)e(of)j(the)f(op)t(erators)f Ff(H)3221 3858 y Fw(\()p Fv(\033)t Fw(\))3361 3910 y Fe(\()p Ff(k)5 b Fe(\))120 4139 y Fw(Our)26 b(goal)f(in)g(this)h (section)g(is)g(to)g(analyze)g(the)g(p)s(oin)m(t)g(sp)s(ectrum)g(of)g (the)h(op)s(erators)f Fv(H)3282 4102 y Fu(\()p Ft(\033)r Fu(\))3383 4139 y Fw(\()p Fv(k)s Fw(\).)120 4259 y(W)-8 b(e)24 b(will)e(pro)m(v)m(e)j(that)f(the)g(eigen)m(v)-5 b(alues)24 b(ha)m(v)m(e)h(\014nite)f(m)m(ultiplicities)c(and)k(can)g (accum)m(ulate)f(at)120 4379 y(+)p Fs(1)30 b Fw(only)-8 b(.)42 b(Under)31 b(the)g(assumption)e Fv(\033)j Fs(\025)c Fw(0)i(w)m(e)i(will)c(see)j(that)f(there)h(are)g(no)f(eigen)m(v)-5 b(alues.)120 4500 y(Our)46 b(metho)s(d)g(is)g(more)f(direct)i(than)f (in)g(the)g(previous)h(sections)g(of)f(this)g(c)m(hapter.)86 b(In)120 4620 y(Subsection)39 b(1)g(w)m(e)g(will)e(in)m(tro)s(duce)h ("b)s(oundary)h(op)s(erators")f Fv(B)2540 4584 y Fu(\()p Ft(\033)r Fu(\))2642 4620 y Fw(\()p Fv(\025)26 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))38 b(on)h Fv(L)3331 4635 y Fu(2)3371 4620 y Fw(\()p Fr(T)p Fw(\).)120 4741 y(These)33 b(are)e(op)s(erators)h(of)e(compact)h(resolv)m(en)m(t,)i (and)f(w)m(e)g(will)d(sho)m(w)j(in)f(Subsection)h(2)f(that)120 4861 y(an)47 b(eigen)m(v)-5 b(alue)46 b(0)g(of)g Fv(B)1049 4825 y Fu(\()p Ft(\033)r Fu(\))1151 4861 y Fw(\()p Fv(\025)32 b Fs(\006)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))47 b(corresp)s(onds)h(to) e(an)h(eigen)m(v)-5 b(alue)46 b Fv(\025)g Fw(of)h Fv(H)3282 4825 y Fu(\()p Ft(\033)r Fu(\))3383 4861 y Fw(\()p Fv(k)s Fw(\).)120 4981 y(\(Again)41 b(the)h(situation)e(is)h(sligh)m(tly)g (more)g(di\016cult)g(at)g(the)h(p)s(oin)m(ts)g Fv(\025)h Fw(=)g(\()p Fv(n)29 b Fw(+)f Fv(k)s Fw(\))3272 4945 y Fu(2)3312 4981 y Fv(;)33 b(n)44 b Fs(2)120 5102 y Fr(Z)p Fw(.\))f(Using)33 b(this)g(c)m(haracterization)f(w)m(e)j(will)c(deriv)m (e)j(the)f(ab)s(o)m(v)m(e)i(stated)e(prop)s(erties)h(of)f(the)120 5222 y(eigen)m(v)-5 b(alues)27 b(from)g(the)g(Analytic)g(F)-8 b(redholm)25 b(Alternativ)m(e.)42 b(W)-8 b(e)28 b(supplemen)m(t)f(this) h(general)120 5342 y(result)i(in)g(Subsection)h(4)g(with)f(conditions)f (on)i Fv(\033)j Fw(for)c(the)h(absence,)i(lo)s(cation)28 b(or)i(existence)120 5463 y(of)i(eigen)m(v)-5 b(alues.)1781 5712 y(30)p eop %%Page: 31 36 31 35 bop 120 407 a Fp(1.)60 b(De\014nition)40 b(of)h(the)g(op)s (erators)g Fv(B)1705 371 y Fu(\()p Ft(\033)r Fu(\))1807 407 y Fw(\()p Fv(\025)24 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))p Fp(.)97 b Fw(Let)36 b Fv(k)g Fs(2)d Fw([)p Fs(\000)2852 368 y Fu(1)p 2852 384 36 4 v 2852 441 a(2)2898 407 y Fv(;)2951 368 y Fu(1)p 2951 384 V 2951 441 a(2)2997 407 y Fw(])i(and)h Fv(\025)d Fs(2)g Fr(R)5 b Fw(.)120 527 y(In)33 b(the)g(Hilb)s(ert)e(space)j Fv(L)1073 542 y Fu(2)1112 527 y Fw(\()p Fr(T)p Fw(\))i(w)m(e)e(consider)f(the)g (quadratic)f(forms)248 751 y Fs(D)s Fw([)p Fv(b)396 710 y Fu(\()p Ft(\033)r Fu(\))497 751 y Fw(\()p Fv(\025)22 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\)])28 b(:=)f Fv(H)1206 710 y Fu(1)p Ft(=)p Fu(2)1316 751 y Fw(\()p Fr(T)p Fw(\))p Fv(;)268 925 y(b)309 884 y Fu(\()p Ft(\033)r Fu(\))412 925 y Fw(\()p Fv(\025)22 b Fs(\006)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\)[)p Fv(f)11 b Fw(])28 b(:=)f Fs(\007)p Fv(i)1334 830 y Fo(X)1244 1047 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))1435 1028 y Fm(2)1470 1047 y Ft(<\025)1583 835 y Fo(p)p 1683 835 527 4 v 90 x Fv(\025)22 b Fs(\000)g Fw(\()p Fv(n)h Fw(+)f Fv(k)s Fw(\))2170 896 y Fu(2)2226 925 y Fs(j)2275 899 y Fw(^)2254 925 y Fv(f)2302 940 y Ft(n)2348 925 y Fs(j)2376 884 y Fu(2)2415 925 y Fw(+)1106 1245 y(+)1293 1150 y Fo(X)1204 1367 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))1395 1349 y Fm(2)1430 1367 y Ft(>\025)1543 1155 y Fo(p)p 1642 1155 V 1642 1245 a Fw(\()p Fv(n)h Fw(+)f Fv(k)s Fw(\))1951 1216 y Fu(2)2012 1245 y Fs(\000)h Fv(\025)16 b Fs(j)2234 1219 y Fw(^)2213 1245 y Fv(f)2261 1260 y Ft(n)2308 1245 y Fs(j)2336 1204 y Fu(2)2397 1245 y Fw(+)2495 1109 y Fo(Z)2595 1136 y Ft(\031)2551 1335 y Fl(\000)p Ft(\031)2669 1245 y Fv(\033)t Fw(\()p Fv(x)2821 1260 y Fu(1)2861 1245 y Fw(\))p Fs(j)p Fv(f)11 b Fw(\()p Fv(x)3079 1260 y Fu(1)3118 1245 y Fw(\))p Fs(j)3184 1204 y Fu(2)3240 1245 y Fv(dx)3346 1260 y Fu(1)3385 1245 y Fv(:)3264 1487 y Fw(\(3.3.1\))120 1717 y(Of)32 b(course,)i Fv(b)625 1680 y Fu(\(0\))719 1717 y Fw(\()p Fv(\025)22 b Fs(\006)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))32 b(are)g(closed)h (sectorial)e(forms.)43 b(According)32 b(to)g(Corollary)e(1.2.4)120 1837 y(the)43 b(in)m(tegral)663 1757 y Fo(R)729 1783 y Ft(\031)710 1872 y Fl(\000)p Ft(\031)828 1837 y Fv(\033)t Fw(\()p Fv(x)980 1852 y Fu(1)1020 1837 y Fw(\))p Fs(j)p Fv(f)11 b Fw(\()p Fv(x)1238 1852 y Fu(1)1277 1837 y Fw(\))p Fs(j)1343 1801 y Fu(2)1399 1837 y Fv(dx)1505 1852 y Fu(1)1587 1837 y Fw(is)42 b(w)m(ell-de\014ned)h(for)f Fv(f)55 b Fs(2)45 b Fv(H)2701 1801 y Fu(1)p Ft(=)p Fu(2)2811 1837 y Fw(\()p Fr(T)p Fw(\))h(and)c(strongly)120 1973 y(sub)s(ordinate)30 b(to)g Fv(b)810 1937 y Fu(\(0\))905 1973 y Fw(\()p Fv(\025)18 b Fs(\006)g Fv(i)p Fw(0)p Fv(;)f(k)s Fw(\).)42 b(Hence)32 b Fv(b)1729 1937 y Fu(\()p Ft(\033)r Fu(\))1831 1973 y Fw(\()p Fv(\025)18 b Fs(\006)g Fv(i)p Fw(0)p Fv(;)f(k)s Fw(\))30 b(are)g(closed)h(sectorial)e(forms)h(and)120 2093 y(there)35 b(are)e(unique)i(m-sectorial)c(op)s(erators)j Fv(B)1865 2057 y Fu(\()p Ft(\033)r Fu(\))1966 2093 y Fw(\()p Fv(\025)23 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))33 b(with)h(domains)e Fs(D)s Fw(\()p Fv(B)3243 2057 y Fu(\()p Ft(\033)r Fu(\))3345 2093 y Fw(\()p Fv(\025)23 b Fs(\006)120 2213 y Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\)\))32 b(suc)m(h)i(that)575 2433 y Fs(D)s Fw(\()p Fv(B)772 2392 y Fu(\()p Ft(\033)r Fu(\))874 2433 y Fw(\()p Fv(\025)22 b Fs(\006)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\)\))83 b Fs(\032)h Fv(H)1679 2392 y Fu(1)p Ft(=)p Fu(2)1788 2433 y Fw(\()p Fr(T)p Fw(\))p Fv(;)508 2579 y Fw(\()p Fv(B)625 2538 y Fu(\()p Ft(\033)r Fu(\))727 2579 y Fw(\()p Fv(\025)22 b Fs(\006)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))p Fv(f)5 b(;)17 b(g)t Fw(\))83 b(=)h Fv(b)1631 2538 y Fu(\()p Ft(\033)r Fu(\))1733 2579 y Fw(\()p Fv(\025)22 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\)[)p Fv(f)5 b(;)17 b(g)t Fw(])p Fv(;)867 b Fw(\(3.3.2\))1590 2724 y Fv(f)38 b Fs(2)28 b(D)s Fw(\()p Fv(B)1967 2683 y Fu(\()p Ft(\033)r Fu(\))2069 2724 y Fw(\()p Fv(\025)22 b Fs(\006)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\)\))p Fv(;)44 b(g)31 b Fs(2)d Fv(H)2873 2683 y Fu(1)p Ft(=)p Fu(2)2983 2724 y Fw(\()p Fr(T)p Fw(\))p Fv(:)120 2944 y Fw(The)33 b(op)s(erators)f(Re)21 b Fv(B)966 2908 y Fu(\()p Ft(\033)r Fu(\))1068 2944 y Fw(\()p Fv(\025)g Fw(+)g Fv(i)p Fw(0)p Fv(;)c(k)s Fw(\))32 b(and)g(Re)22 b Fv(B)1936 2908 y Fu(\()p Ft(\033)r Fu(\))2038 2944 y Fw(\()p Fv(\025)f Fs(\000)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\),)32 b(asso)s(ciated)g(with)f(the)i(real)120 3064 y(parts)e(of)f(the)g(quadratic)g(forms)g Fv(b)1384 3028 y Fu(\()p Ft(\033)r Fu(\))1486 3064 y Fw(\()p Fv(\025)18 b Fs(\006)g Fv(i)p Fw(0)p Fv(;)f(k)s Fw(\),)30 b(are)h(self-adjoin)m(t) d(and)j(coincide.)42 b(W)-8 b(e)31 b(put)714 3284 y Fv(B)793 3243 y Fu(\()p Ft(\033)r Fu(\))895 3284 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))27 b(:=)h(Re)22 b Fv(B)1500 3243 y Fu(\()p Ft(\033)r Fu(\))1602 3284 y Fw(\()p Fv(\025)g Fw(+)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))28 b(=)f(Re)22 b Fv(B)2382 3243 y Fu(\()p Ft(\033)r Fu(\))2484 3284 y Fw(\()p Fv(\025)g Fs(\000)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))p Fv(:)318 b Fw(\(3.3.3\))120 3504 y(W)-8 b(e)33 b(remark)f(that)h Fv(B)912 3468 y Fu(\(0\))1006 3504 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))27 b(=)h Fv(B)5 b Fw(\()p Fv(\025;)17 b(k)s Fw(\))p Fv(;)44 b(\025)27 b(<)h(k)1991 3468 y Fu(2)2031 3504 y Fv(;)k Fw(in)g(the)h(notation)e(of)h(Section)h(1.)120 3625 y(Of)39 b(course,)i(the)f(op)s(erators)e Fv(B)1289 3589 y Fu(\(0\))1384 3625 y Fw(\()p Fv(\025)26 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))1827 3589 y Fl(\000)p Fu(1)1921 3625 y Fv(;)56 b(\025)38 b Fs(6)p Fw(=)g(\()p Fv(n)27 b Fw(+)f Fv(k)s Fw(\))2530 3589 y Fu(2)2570 3625 y Fv(;)55 b(n)39 b Fs(2)g Fr(Z)p Fv(;)d Fw(are)j(compact.)120 3745 y(More)33 b(generally)-8 b(,)32 b(w)m(e)h(ha)m(v)m(e)120 3949 y Fp(Lemma)59 b(3.3.1.)51 b Fj(L)-5 b(et)53 b Fv(k)62 b Fs(2)e Fw([)p Fs(\000)1390 3909 y Fu(1)p 1390 3926 36 4 v 1390 3983 a(2)1436 3949 y Fv(;)1490 3909 y Fu(1)p 1490 3926 V 1490 3983 a(2)1535 3949 y Fw(])52 b Fj(and)f Fv(\025)60 b Fs(2)g Fr(R)5 b Fj(.)102 b(Then)51 b(the)h(op)-5 b(er)g(ators)52 b Fv(B)3231 3912 y Fu(\()p Ft(\033)r Fu(\))3333 3949 y Fw(\()p Fv(\025)35 b Fs(\006)120 4069 y Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))p Fv(;)44 b(B)488 4033 y Fu(\()p Ft(\033)r Fu(\))590 4069 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))34 b Fj(have)h(c)-5 b(omp)g(act)34 b(r)-5 b(esolvent.)120 4272 y(Pr)g(o)g(of.)48 b Fw(It)37 b(su\016ces)i(\(cf.) 57 b(Theorem)37 b(VI.3.3)g(in)f([K]\))g(to)h(pro)m(v)m(e)h(that)f Fv(B)2787 4236 y Fu(\()p Ft(\033)r Fu(\))2888 4272 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))37 b(has)g(com-)120 4393 y(pact)i(resolv)m(en)m(t.)65 b(Since)39 b Fv(b)1113 4357 y Fu(\()p Ft(\033)r Fu(\))1215 4393 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))39 b(di\013ers)g(from)f Fv(b)2060 4357 y Fu(\(0\))2155 4393 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))38 b(b)m(y)i(a)f(strongly)g(sub)s(ordinate)120 4513 y(p)s(erturbation,)e (the)g(assertion)g(follo)m(ws)e(b)m(y)j(the)f(min-max-principle)32 b(from)k(the)h(fact)f(that)120 4634 y Fv(B)199 4597 y Fu(\(0\))293 4634 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))33 b(has)f(compact)h(resolv)m(en)m(t.)p 3469 4634 4 66 v 3473 4571 59 4 v 3473 4634 V 3531 4634 4 66 v 120 4893 a Fp(2.)72 b(Characterization)44 b(of)h(eigen)m(v)-6 b(alues)45 b(of)g Fv(H)2090 4857 y Fu(\()p Ft(\033)r Fu(\))2191 4893 y Fw(\()p Fv(k)s Fw(\))p Fp(.)98 b Fw(W)-8 b(e)39 b(shall)f(c)m(haracterize)i(the)120 5014 y(eigen)m(v)-5 b(alues)36 b(of)f(the)i(op)s(erator)e Fv(H)1399 4978 y Fu(\()p Ft(\033)r Fu(\))1500 5014 y Fw(\()p Fv(k)s Fw(\))h(as)g(the)g(v)-5 b(alues)36 b Fv(\025)g Fw(for)f(whic)m(h)i(0)e (is)h(an)f(eigen)m(v)-5 b(alue)120 5134 y(of)32 b(the)h(op)s(erators)f Fv(B)909 5098 y Fu(\()p Ft(\033)r Fu(\))1011 5134 y Fw(\()p Fv(\025)22 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\).)43 b(More)33 b(precisely)-8 b(,)33 b(w)m(e)h(ha)m(v)m(e)120 5338 y Fp(Prop)s(osition)i(3.3.2.)42 b Fj(L)-5 b(et)35 b Fv(k)c Fs(2)d Fw([)p Fs(\000)1487 5298 y Fu(1)p 1487 5315 36 4 v 1487 5372 a(2)1532 5338 y Fv(;)1586 5298 y Fu(1)p 1586 5315 V 1586 5372 a(2)1631 5338 y Fw(])35 b Fj(and)g Fv(\025)27 b Fs(2)h Fr(R)5 b Fj(.)1781 5712 y Fw(31)p eop %%Page: 32 37 32 36 bop 235 407 a Fj(1.)49 b(Supp)-5 b(ose)34 b(that)h Fv(u)27 b Fs(2)i(D)s Fw(\()p Fv(H)1319 371 y Fu(\()p Ft(\033)r Fu(\))1419 407 y Fw(\()p Fv(k)s Fw(\)\))35 b Fj(satis\014es)f Fv(H)2077 371 y Fu(\()p Ft(\033)r Fu(\))2179 407 y Fw(\()p Fv(k)s Fw(\))p Fv(u)27 b Fw(=)g Fv(\025u)p Fj(.)44 b(De\014ne)1322 624 y Fv(f)11 b Fw(\()p Fv(x)1474 639 y Fu(1)1514 624 y Fw(\))27 b(:=)h Fv(u)p Fw(\()p Fv(x)1859 639 y Fu(1)1898 624 y Fv(;)17 b Fw(0\))p Fv(;)216 b(x)2327 639 y Fu(1)2394 624 y Fs(2)28 b Fr(T)p Fv(:)686 b Fw(\(3.3.4\))364 842 y Fj(Then)632 815 y Fw(^)610 842 y Fv(f)658 857 y Ft(n)733 842 y Fw(=)28 b(0)f Fj(if)g Fw(\()p Fv(n)5 b Fw(+)g Fv(k)s Fw(\))1274 806 y Fu(2)1342 842 y Fs(\024)28 b Fv(\025)p Fj(,)h Fv(f)38 b Fs(2)28 b(D)s Fw(\()p Fv(B)1940 806 y Fu(\()p Ft(\033)r Fu(\))2042 842 y Fw(\()p Fv(\025)5 b Fs(\006)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\)\))28 b Fj(and)f Fv(B)2769 806 y Fu(\()p Ft(\033)r Fu(\))2870 842 y Fw(\()p Fv(\025)5 b Fs(\006)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))p Fv(f)40 b Fw(=)27 b(0)p Fj(.)364 962 y(Mor)-5 b(e)g(over,)655 1218 y Fv(u)p Fw(\()p Fv(x)p Fw(\))27 b(=)1054 1151 y(1)p 983 1196 191 4 v 983 1215 a Fs(p)p 1066 1215 108 4 v 83 x Fw(2)p Fv(\031)1289 1124 y Fo(X)1200 1341 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))1391 1322 y Fm(2)1426 1341 y Ft(>\025)1562 1192 y Fw(^)1538 1218 y Fv(f)1586 1233 y Ft(n)1650 1218 y Fv(e)1695 1177 y Ft(inx)1802 1186 y Fm(1)1857 1218 y Fv(e)1902 1177 y Fl(\000)1957 1110 y Fs(p)p 2040 1110 322 4 v 67 x Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))2231 1158 y Fm(2)2266 1177 y Fl(\000)p Ft(\025)12 b(x)2414 1186 y Fm(2)2452 1218 y Fv(;)216 b(x)28 b Fs(2)g Fw(\005)p Fv(:)292 b Fw(\(3.3.5\))235 1609 y Fj(2.)49 b(Supp)-5 b(ose)27 b(that)i Fv(f)38 b Fs(2)28 b(D)s Fw(\()p Fv(B)1298 1573 y Fu(\()p Ft(\033)r Fu(\))1400 1609 y Fw(\()p Fv(\025)7 b Fs(\006)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\)\))29 b Fj(satis\014es)e Fv(B)2309 1573 y Fu(\()p Ft(\033)r Fu(\))2411 1609 y Fw(\()p Fv(\025)7 b Fs(\006)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))p Fv(f)39 b Fw(=)27 b(0)h Fj(and)3286 1583 y Fw(^)3265 1609 y Fv(f)3313 1624 y Ft(n)3388 1609 y Fw(=)f(0)364 1730 y Fj(if)35 b Fw(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))767 1694 y Fu(2)834 1730 y Fw(=)28 b Fv(\025)p Fj(.)44 b(De\014ne)34 b Fv(u)g Fj(by)44 b Fw(\(3.3.5\))o Fj(.)364 1850 y(Then)29 b Fv(u)f Fs(2)g(D)s Fw(\()p Fv(H)998 1814 y Fu(\()p Ft(\033)r Fu(\))1098 1850 y Fw(\()p Fv(k)s Fw(\)\))j Fj(and)e Fv(H)1570 1814 y Fu(\()p Ft(\033)r Fu(\))1672 1850 y Fw(\()p Fv(k)s Fw(\))p Fv(u)e Fw(=)g Fv(\025u)p Fj(.)43 b(Mor)-5 b(e)g(over,)2650 1824 y Fw(^)2629 1850 y Fv(f)2677 1865 y Ft(n)2752 1850 y Fw(=)27 b(0)j Fj(if)g Fw(\()p Fv(n)12 b Fw(+)g Fv(k)s Fw(\))3312 1814 y Fu(2)3380 1850 y Fv(<)27 b(\025)364 1971 y Fj(and)44 b Fw(\(3.3.4\))35 b Fj(holds.)266 2172 y Fw(The)f(pro)s(of)e(relies)f (on)i(the)g(follo)m(wing)d(elemen)m(tary)i(observ)-5 b(ation.)120 2373 y Fp(Lemma)37 b(3.3.3.)42 b Fj(L)-5 b(et)35 b Fv(k)c Fs(2)d Fw([)p Fs(\000)1278 2334 y Fu(1)p 1278 2350 36 4 v 1278 2407 a(2)1324 2373 y Fv(;)1378 2334 y Fu(1)p 1378 2350 V 1378 2407 a(2)1423 2373 y Fw(])35 b Fj(and)f Fv(\025)28 b Fs(2)g Fr(R)5 b Fj(.)235 2588 y(1.)49 b(L)-5 b(et)35 b Fv(u)27 b Fs(2)735 2563 y Fw(~)709 2588 y Fv(H)798 2552 y Fu(1)837 2588 y Fw(\(\005\))p Fj(,)35 b(then)g(the)g(fol)5 b(lowing)33 b(ar)-5 b(e)35 b(e)-5 b(quivalent:)420 2790 y(\(i\))48 b Fv(u)28 b Fs(2)g(D)s Fw(\()p Fv(H)963 2754 y Fu(\()p Ft(\033)r Fu(\))1064 2790 y Fw(\()p Fv(k)s Fw(\)\))34 b Fj(and)h Fv(H)1545 2754 y Fu(\()p Ft(\033)r Fu(\))1646 2790 y Fw(\()p Fv(k)s Fw(\))p Fv(u)27 b Fw(=)h Fv(\025u)p Fj(,)390 2952 y(\(ii\))789 3084 y Fo(Z)888 3111 y Fl(1)844 3310 y Fu(0)963 3220 y Fv(D)1044 3235 y Fu(2)1089 3220 y Fw(^)-55 b Fv(u)1139 3235 y Ft(n)p 1186 3140 185 4 v 1186 3220 a Fv(D)1267 3235 y Fu(2)1306 3220 y Fv(')17 b(dx)1493 3235 y Fu(2)1555 3220 y Fw(+)1733 3152 y(1)p 1662 3197 191 4 v 1662 3217 a Fs(p)p 1746 3217 108 4 v 1746 3299 a Fw(2)p Fv(\031)1880 3084 y Fo(Z)1979 3111 y Ft(\031)1935 3310 y Fl(\000)p Ft(\031)2054 3220 y Fv(\033)t Fw(\()p Fv(x)2206 3235 y Fu(1)2245 3220 y Fw(\))p Fv(u)p Fw(\()p Fv(x)2432 3235 y Fu(1)2472 3220 y Fv(;)g Fw(0\))p Fv(e)2648 3179 y Fl(\000)p Ft(inx)2810 3188 y Fm(1)2864 3220 y Fv(dx)2970 3235 y Fu(1)p 3026 3133 189 4 v 3026 3220 a Fv(')p Fw(\(0\))28 b(=)991 3488 y(=)f(\()p Fv(\025)22 b Fs(\000)h Fw(\()p Fv(n)f Fw(+)g Fv(k)s Fw(\))1619 3447 y Fu(2)1659 3488 y Fw(\))1714 3352 y Fo(Z)1813 3379 y Fl(1)1768 3578 y Fu(0)1910 3488 y Fw(^)-55 b Fv(u)1960 3503 y Ft(n)p 2007 3433 64 4 v 2007 3488 a Fv(')16 b(dx)2193 3503 y Fu(2)2233 3488 y Fv(;)216 b(n)27 b Fs(2)i Fr(Z)p Fv(;)h(')e Fs(2)g Fv(H)3057 3447 y Fu(1)3096 3488 y Fw(\()p Fr(R)3200 3503 y Fu(+)3265 3488 y Fw(\))p Fv(:)235 3789 y Fj(2.)49 b(L)-5 b(et)35 b Fv(f)j Fs(2)29 b Fv(H)802 3753 y Fu(1)p Ft(=)p Fu(2)911 3789 y Fw(\()p Fr(T)p Fw(\))p Fj(,)38 b(then)d(the)g(fol)5 b(lowing)33 b(ar)-5 b(e)35 b(e)-5 b(quivalent:)420 3992 y(\(i\))48 b Fv(f)39 b Fs(2)28 b(D)s Fw(\()p Fv(B)956 3956 y Fu(\()p Ft(\033)r Fu(\))1057 3992 y Fw(\()p Fv(\025)22 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\)\))35 b Fj(and)f Fv(B)1833 3956 y Fu(\()p Ft(\033)r Fu(\))1935 3992 y Fw(\()p Fv(\025)22 b Fs(\006)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))p Fv(f)39 b Fw(=)27 b(0)p Fj(,)390 4153 y(\(ii\))1100 4395 y Fw(^)1079 4422 y Fv(f)1127 4437 y Ft(n)1202 4422 y Fw(=)1386 4354 y(1)p 1315 4399 191 4 v 1315 4419 a Fs(p)p 1398 4419 108 4 v 82 x Fw(2)p Fv(\031)1533 4286 y Fo(Z)1632 4312 y Ft(\031)1588 4511 y Fl(\000)p Ft(\031)1706 4422 y Fv(\033)t Fw(\()p Fv(x)1858 4437 y Fu(1)1898 4422 y Fw(\))p Fv(f)11 b Fw(\()p Fv(x)2088 4437 y Fu(1)2128 4422 y Fw(\))p Fv(e)2211 4380 y Fl(\000)p Ft(inx)2373 4389 y Fm(1)2428 4422 y Fv(dx)2534 4437 y Fu(1)2601 4422 y Fw(=)28 b(0)99 b Fj(if)55 b Fw(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))3276 4380 y Fu(2)3343 4422 y Fv(<)28 b(\025;)447 4599 y Fo(p)p 547 4599 527 4 v 91 x Fw(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))855 4661 y Fu(2)917 4690 y Fs(\000)h Fv(\025)1111 4663 y Fw(^)1090 4690 y Fv(f)1138 4705 y Ft(n)1207 4690 y Fw(+)1386 4622 y(1)p 1315 4667 191 4 v 1315 4687 a Fs(p)p 1398 4687 108 4 v 82 x Fw(2)p Fv(\031)1533 4554 y Fo(Z)1632 4580 y Ft(\031)1588 4779 y Fl(\000)p Ft(\031)1706 4690 y Fv(\033)t Fw(\()p Fv(x)1858 4705 y Fu(1)1898 4690 y Fw(\))p Fv(f)11 b Fw(\()p Fv(x)2088 4705 y Fu(1)2128 4690 y Fw(\))p Fv(e)2211 4648 y Fl(\000)p Ft(inx)2373 4657 y Fm(1)2428 4690 y Fv(dx)2534 4705 y Fu(1)2601 4690 y Fw(=)28 b(0)99 b Fj(if)55 b Fw(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))3276 4648 y Fu(2)3343 4690 y Fs(\025)29 b Fv(\025:)266 4981 y Fw(W)-8 b(e)23 b(emphasize)g(that,)h (according)e(to)h(our)f(con)m(v)m(en)m(tion,)k(statemen)m(ts)d(whic)m (h)g(con)m(tain)g(the)120 5102 y(double)37 b(index)g(")p Fs(\006)p Fw(")h(are)f(understo)s(o)s(d)g(as)h(t)m(w)m(o)g(indep)s (enden)m(t)g(assertions.)58 b(In)38 b(particular,)120 5222 y(in)f(the)h(second)g(part)f(of)g(the)h(lemma)d(the)j(statemen)m (ts)h(\(i\))d(corresp)s(onding)h(to)h(the)f(upp)s(er)120 5342 y(and)i(to)f(the)i(lo)m(w)m(er)e(sign)h(are)f(equiv)-5 b(alen)m(t,)41 b(since)e(the)g(statemen)m(t)g(\(ii\))e(is)h(indep)s (enden)m(t)i(of)120 5463 y(the)33 b(sign.)1781 5712 y(32)p eop %%Page: 33 38 33 37 bop 120 407 a Fj(Pr)-5 b(o)g(of.)48 b Fw(1\))33 b(The)g(assertion)g(\(1.i\))e(is)h(equiv)-5 b(alen)m(t)33 b(to)276 542 y Fo(Z)332 767 y Fu(\005)405 567 y Fo(\020)465 678 y Fw(\()p Fv(D)584 693 y Fu(1)645 678 y Fw(+)22 b Fv(k)s Fw(\))p Fv(u)p 891 591 422 4 v Fw(\()p Fv(D)1010 693 y Fu(1)1071 678 y Fw(+)g Fv(k)s Fw(\))p Fv(v)k Fw(+)c Fv(D)1513 693 y Fu(2)1553 678 y Fv(u)p 1609 598 172 4 v(D)1690 693 y Fu(2)1729 678 y Fv(v)1779 567 y Fo(\021)1872 678 y Fv(dx)g Fw(+)2098 542 y Fo(Z)2198 568 y Ft(\031)2154 767 y Fl(\000)p Ft(\031)2272 678 y Fv(\033)t Fw(\()p Fv(x)2424 693 y Fu(1)2464 678 y Fw(\))p Fv(u)p Fw(\()p Fv(x)2651 693 y Fu(1)2690 678 y Fv(;)17 b Fw(0\))p 2821 591 314 4 v Fv(v)t Fw(\()p Fv(x)2965 693 y Fu(1)3004 678 y Fv(;)g Fw(0\))f Fv(dx)3257 693 y Fu(1)3324 678 y Fw(=)482 943 y(=)28 b Fv(\025)660 808 y Fo(Z)715 1033 y Fu(\005)788 943 y Fv(u)p 844 888 51 4 v(v)20 b(dx;)212 b(v)32 b Fs(2)1454 918 y Fw(~)1429 943 y Fv(H)1518 902 y Fu(1)1557 943 y Fw(\(\005\))p Fv(:)3264 1126 y Fw(\(3.3.6\))120 1349 y(T)-8 b(aking)39 b Fv(v)k Fw(=)c Fv(e)702 1364 y Ft(n)775 1349 y Fs(\012)28 b Fv(';)55 b(n)40 b Fs(2)f Fr(Z)p Fv(;)17 b(')36 b Fs(2)k Fv(H)1637 1313 y Fu(1)1676 1349 y Fw(\()p Fr(R)1780 1364 y Fu(+)1845 1349 y Fw(\))p Fv(;)f Fw(and)g(in)m(tegrating)f(b)m(y)i(parts)f(w)m(e)i(see)f(that)120 1469 y(\(1.i\))31 b(implies)f(\(1.ii\).)120 1590 y(Con)m(v)m(ersely)-8 b(,)47 b(since)c(the)g(functions)g(of)f(this)h(sp)s(ecial)e(form)h(are) g(fundamen)m(tal)g(in)3288 1564 y(~)3263 1590 y Fv(H)3352 1553 y Fu(1)3391 1590 y Fw(\(\005\))120 1710 y(\(i.e.,)28 b(their)f(linear)e(span)j(is)f(dense,)j(cf.)42 b(Lemma)26 b(1.2.2\),)h(\(3.3.6\))g(holds)g(for)g(an)m(y)h Fv(v)j Fs(2)3288 1685 y Fw(~)3263 1710 y Fv(H)3352 1674 y Fu(1)3391 1710 y Fw(\(\005\))120 1830 y(if)d(it)f(holds)i(for)f(all)e Fv(v)32 b Fw(=)27 b Fv(e)1054 1845 y Ft(n)1116 1830 y Fs(\012)14 b Fv(';)45 b(n)28 b Fs(2)g Fr(Z)p Fv(;)17 b(')24 b Fs(2)k Fv(H)1907 1794 y Fu(1)1946 1830 y Fw(\()p Fr(R)2050 1845 y Fu(+)2115 1830 y Fw(\).)42 b(This)29 b(in)f(turn)h(follo)m(ws)f(from)f(\(1.ii\).)120 1951 y(2\))32 b(The)i(assertion)e(\(2.i\))g(is)g(equiv)-5 b(alen)m(t)32 b(to)154 2221 y Fs(\007)23 b Fv(i)359 2127 y Fo(X)270 2344 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))461 2325 y Fm(2)496 2344 y Ft(<\025)592 2131 y Fo(p)p 692 2131 527 4 v 90 x Fv(\025)f Fs(\000)g Fw(\()p Fv(n)h Fw(+)f Fv(k)s Fw(\))1179 2193 y Fu(2)1256 2195 y Fw(^)1235 2221 y Fv(f)1283 2236 y Ft(n)p 1346 2140 94 4 v 1349 2221 a Fw(^)-52 b Fv(g)1393 2236 y Ft(n)1462 2221 y Fw(+)1616 2127 y Fo(X)1527 2344 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))1718 2325 y Fm(2)1753 2344 y Fl(\025)p Ft(\025)1849 2131 y Fo(p)p 1948 2131 527 4 v 1948 2221 a Fw(\()p Fv(n)23 b Fw(+)f Fv(k)s Fw(\))2257 2193 y Fu(2)2318 2221 y Fs(\000)h Fv(\025)2513 2195 y Fw(^)2491 2221 y Fv(f)2539 2236 y Ft(n)p 2603 2140 94 4 v 2606 2221 a Fw(^)-52 b Fv(g)2650 2236 y Ft(n)2719 2221 y Fw(+)2817 2086 y Fo(Z)2917 2112 y Ft(\031)2872 2311 y Fl(\000)p Ft(\031)2991 2221 y Fv(\033)t(f)p 3109 2167 51 4 v 11 w(g)19 b(dx)3281 2236 y Fu(1)3349 2221 y Fw(=)27 b(0)p Fv(;)327 2495 y(g)k Fs(2)d Fv(H)588 2453 y Fu(1)p Ft(=)p Fu(2)698 2495 y Fw(\()p Fr(T)p Fw(\))p Fv(:)3264 2623 y Fw(\(3.3.7\))120 2863 y(T)-8 b(aking)47 b Fv(g)55 b Fw(=)e Fv(f)58 b Fw(and)47 b(considering)f(the)i(imaginary) d(part)h(only)h(w)m(e)h(obtain)3133 2836 y(^)3111 2863 y Fv(f)3159 2878 y Ft(n)3259 2863 y Fw(=)k(0)47 b(if)120 2983 y(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))428 2947 y Fu(2)495 2983 y Fv(<)28 b(\025)p Fw(,)33 b(and)f(taking)g Fv(g)f Fw(=)c Fv(e)1432 2998 y Ft(n)1480 2983 y Fv(;)44 b(n)28 b Fs(2)g Fr(Z)p Fv(;)i Fw(w)m(e)k(see)f(that)g(\(2.i\))e (implies)f(\(2.ii\).)120 3103 y(Con)m(v)m(ersely)-8 b(,)42 b(since)c Fs(f)p Fv(e)979 3118 y Ft(n)1026 3103 y Fv(;)54 b(n)37 b Fs(2)g Fr(Z)p Fs(g)e Fw(is)j(fundamen)m(tal)f(in)g Fv(H)2340 3067 y Fu(1)p Ft(=)p Fu(2)2450 3103 y Fw(\()p Fr(T)p Fw(\),)43 b(\(3.3.7\))37 b(holds)h(for)f(an)m(y)120 3224 y Fv(g)31 b Fs(2)d Fv(H)381 3188 y Fu(1)p Ft(=)p Fu(2)491 3224 y Fw(\()p Fr(T)p Fw(\))33 b(if)c(it)h(holds)g(for)g(all)e Fv(g)j Fw(=)d Fv(e)1606 3239 y Ft(n)1653 3224 y Fv(;)44 b(n)28 b Fs(2)g Fr(Z)p Fw(.)40 b(This)31 b(in)f(turn)g(follo)m(ws)f (from)h(\(2.ii\).)p 3469 3224 4 66 v 3473 3161 59 4 v 3473 3224 V 3531 3224 4 66 v 120 3419 a Fj(Pr)-5 b(o)g(of)35 b(\(Pr)-5 b(op)g(osition)34 b(3.3.2\).)47 b Fw(1\))23 b(Let)g Fv(u)28 b Fs(2)g(D)s Fw(\()p Fv(H)1927 3383 y Fu(\()p Ft(\033)r Fu(\))2028 3419 y Fw(\()p Fv(k)s Fw(\)\))23 b(suc)m(h)h(that)f Fv(H)2720 3383 y Fu(\()p Ft(\033)r Fu(\))2821 3419 y Fw(\()p Fv(k)s Fw(\))p Fv(u)28 b Fw(=)f Fv(\025u)p Fw(.)40 b(Then)120 3539 y(\(1.i\))h(in)h(Lemma)g(3.3.3)g (holds.)74 b(T)-8 b(aking)42 b Fv(')j Fs(2)g Fv(C)2023 3503 y Fl(1)2016 3564 y Fu(0)2098 3539 y Fw(\()p Fr(R)2202 3554 y Fu(+)2267 3539 y Fw(\))d(w)m(e)i(see)g(that)e(the)i(generalized) 120 3660 y(second)32 b(deriv)-5 b(ativ)m(e)29 b(of)36 b(^)-55 b Fv(u)1042 3675 y Ft(n)1119 3660 y Fw(b)s(elongs)29 b(to)h Fv(L)1649 3675 y Fu(2)1689 3660 y Fw(\()p Fr(R)1793 3675 y Fu(+)1858 3660 y Fw(\))g(and)g(satis\014es)h Fv(D)2554 3624 y Fu(2)2551 3684 y(2)2600 3660 y Fw(^)-55 b Fv(u)2650 3675 y Ft(n)2724 3660 y Fw(=)27 b(\(\()p Fv(\025)17 b Fs(\000)h Fw(\()p Fv(n)g Fw(+)f Fv(k)s Fw(\))3371 3624 y Fu(2)3416 3660 y Fw(^)-55 b Fv(u)3466 3675 y Ft(n)3513 3660 y Fw(.)120 3780 y(It)33 b(follo)m(ws)e(that)738 4062 y(^)-55 b Fv(u)788 4077 y Ft(n)835 4062 y Fw(\()p Fv(x)928 4077 y Fu(2)967 4062 y Fw(\))28 b(=)1136 3892 y Fo(\()1914 3986 y Fw(0)194 b(if)97 b(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))2620 3950 y Fu(2)2687 3986 y Fs(\024)29 b Fv(\025;)1258 4137 y(\013)1320 4152 y Ft(n)1367 4137 y Fv(e)1412 4097 y Fl(\000)1467 4030 y Fs(p)p 1550 4030 322 4 v 67 x Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))1741 4078 y Fm(2)1776 4097 y Fl(\000)p Ft(\025)12 b(x)1924 4106 y Fm(2)2157 4137 y Fw(if)97 b(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))2620 4101 y Fu(2)2687 4137 y Fv(>)28 b(\025;)120 4370 y Fw(for)k(some)g(constan)m(ts)i Fv(\013)1007 4385 y Ft(n)1082 4370 y Fs(2)28 b Fr(C)20 b Fw(,)39 b(and)32 b(hence)i(with)f Fv(f)43 b Fw(de\014ned)34 b(b)m(y)h(\(3.3.4\))o(,)1143 4621 y(^)1121 4648 y Fv(f)1169 4663 y Ft(n)1244 4648 y Fw(=)1348 4507 y Fo(\032)1524 4587 y Fw(0)195 b(if)96 b(\()p Fv(n)23 b Fw(+)f Fv(k)s Fw(\))2231 4551 y Fu(2)2298 4587 y Fs(\024)28 b Fv(\025;)1464 4707 y(\013)1526 4722 y Ft(n)1768 4707 y Fw(if)96 b(\()p Fv(n)23 b Fw(+)f Fv(k)s Fw(\))2231 4671 y Fu(2)2298 4707 y Fv(>)27 b(\025:)120 4942 y Fw(It)33 b(follo)m(ws)e(from)g Fv(u)c Fs(2)979 4917 y Fw(~)954 4942 y Fv(H)1043 4906 y Fu(1)1082 4942 y Fw(\(\005\),)32 b(that)629 5143 y Fo(X)540 5361 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))731 5342 y Fm(2)766 5361 y Ft(>\025)879 5157 y Fo(\000)924 5238 y Fw(\()p Fv(n)23 b Fw(+)f Fv(k)s Fw(\))1233 5197 y Fu(2)1294 5238 y Fs(\000)h Fv(\025)1451 5157 y Fo(\001)1766 5171 y Fs(j)1815 5144 y Fw(^)1794 5171 y Fv(f)1842 5186 y Ft(n)1888 5171 y Fs(j)1916 5135 y Fu(2)p 1523 5215 675 4 v 1523 5320 a Fw(2)1572 5235 y Fo(p)p 1672 5235 527 4 v 85 x Fw(\()p Fv(n)f Fw(+)g Fv(k)s Fw(\))1980 5292 y Fu(2)2041 5320 y Fs(\000)h Fv(\025)2236 5238 y Fw(=)2339 5143 y Fo(X)2344 5355 y Ft(n)p Fl(2)p Fn(Z)2500 5238 y Fs(k)p Fv(D)2631 5253 y Fu(2)2676 5238 y Fw(^)-55 b Fv(u)2726 5253 y Ft(n)2772 5238 y Fs(k)2822 5197 y Fu(2)2889 5238 y Fv(<)28 b Fs(1)p Fv(;)1781 5712 y Fw(33)p eop %%Page: 34 39 34 38 bop 120 407 a Fw(so)33 b(that)f Fv(f)39 b Fs(2)28 b Fv(H)721 371 y Fu(1)p Ft(=)p Fu(2)830 407 y Fw(\()p Fr(T)p Fw(\).)47 b(Moreo)m(v)m(er,)34 b(if)d(w)m(e)j(plug)e(the)h(iden) m(tit)m(y)276 535 y Fo(Z)375 561 y Fl(1)331 760 y Fu(0)467 670 y Fv(D)548 685 y Fu(2)593 670 y Fw(^)-55 b Fv(u)643 685 y Ft(n)p 690 590 185 4 v 690 670 a Fv(D)771 685 y Fu(2)810 670 y Fv(')17 b(dx)997 685 y Fu(2)1064 670 y Fw(=)1167 590 y Fo(\000)1213 670 y Fv(\025)22 b Fs(\000)h Fw(\()p Fv(n)f Fw(+)g Fv(k)s Fw(\))1700 629 y Fu(2)1739 590 y Fo(\001)1802 535 y(Z)1901 561 y Fl(1)1857 760 y Fu(0)1999 670 y Fw(^)-55 b Fv(u)2049 685 y Ft(n)p 2095 615 64 4 v 2095 670 a Fv(')17 b(dx)2282 685 y Fu(2)2343 670 y Fw(+)2441 580 y Fo(p)p 2541 580 527 4 v 90 x Fw(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))2849 641 y Fu(2)2911 670 y Fs(\000)h Fv(\025)3105 644 y Fw(^)3084 670 y Fv(f)3132 685 y Ft(n)p 3196 584 189 4 v 3196 670 a Fv(')p Fw(\(0\))120 936 y(in)m(to)32 b(\(1.ii\))e(of)i(Lemma)g(3.3.3,)g(w)m(e)h(obtain)f (\(2.ii\))e(and)j(hence)794 1148 y Fv(f)39 b Fs(2)28 b(D)s Fw(\()p Fv(B)1172 1107 y Fu(\()p Ft(\033)r Fu(\))1273 1148 y Fw(\()p Fv(\025)22 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\)\))p Fv(;)211 b(B)2063 1107 y Fu(\()p Ft(\033)r Fu(\))2165 1148 y Fw(\()p Fv(\025)22 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))p Fv(f)38 b Fw(=)28 b(0)p Fv(;)120 1361 y Fw(whic)m(h)33 b(concludes)h(the)f(pro)s(of)e(of)h(the) h(\014rst)g(part.)120 1481 y(2\))45 b(Let)h Fv(f)61 b Fs(2)50 b(D)s Fw(\()p Fv(B)862 1445 y Fu(\()p Ft(\033)r Fu(\))963 1481 y Fw(\()p Fv(\025)31 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\)\))45 b(suc)m(h)i(that)f Fv(B)2036 1445 y Fu(\()p Ft(\033)r Fu(\))2138 1481 y Fw(\()p Fv(\025)30 b Fs(\006)i Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))p Fv(u)49 b Fw(=)h(0)45 b(and)3139 1455 y(^)3118 1481 y Fv(f)3166 1496 y Ft(n)3263 1481 y Fw(=)50 b(0)45 b(if)120 1602 y(\()p Fv(n)28 b Fw(+)g Fv(k)s Fw(\))440 1566 y Fu(2)521 1602 y Fw(=)42 b Fv(\025)p Fw(.)68 b(Then)42 b(\(2.i\))e(in)g(Lemma)f (3.3.3)i(holds.)68 b(With)40 b Fv(u)g Fw(de\014ned)j(b)m(y)g(\(3.3.5\)) d(it)120 1722 y(follo)m(ws)31 b(from)h Fv(f)38 b Fs(2)28 b Fv(H)940 1686 y Fu(1)p Ft(=)p Fu(2)1050 1722 y Fw(\()p Fr(T)p Fw(\))35 b(that)254 1899 y Fo(X)258 2111 y Ft(n)p Fl(2)p Fn(Z)397 1994 y Fw(\(\(1)22 b(+)g Fv(n)700 1953 y Fu(2)740 1994 y Fw(\))p Fs(k)6 b Fw(^)-55 b Fv(u)884 2009 y Ft(n)930 1994 y Fs(k)980 1953 y Fu(2)1041 1994 y Fw(+)23 b Fs(k)p Fv(D)1271 2009 y Fu(2)1316 1994 y Fw(^)-55 b Fv(u)1366 2009 y Ft(n)1412 1994 y Fs(k)1462 1953 y Fu(2)1501 1994 y Fw(\))28 b(=)1760 1899 y Fo(X)1671 2117 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))1862 2098 y Fm(2)1896 2117 y Ft(>\025)2019 1927 y Fw(1)22 b(+)g Fv(n)2246 1891 y Fu(2)2308 1927 y Fw(+)g(\()p Fv(n)g Fw(+)g Fv(k)s Fw(\))2714 1891 y Fu(2)2776 1927 y Fs(\000)g Fv(\025)p 2019 1971 914 4 v 2138 2076 a Fw(2)2187 1991 y Fo(p)p 2287 1991 527 4 v 85 x Fw(\()p Fv(n)g Fw(+)g Fv(k)s Fw(\))2595 2048 y Fu(2)2656 2076 y Fs(\000)h Fv(\025)2959 1994 y Fs(j)3008 1968 y Fw(^)2987 1994 y Fv(f)3035 2009 y Ft(n)3081 1994 y Fs(j)3109 1953 y Fu(2)3176 1994 y Fv(<)28 b Fs(1)p Fv(;)120 2339 y Fw(so)37 b(that)g Fv(u)e Fs(2)678 2314 y Fw(~)653 2339 y Fv(H)742 2303 y Fu(1)781 2339 y Fw(\(\005\).)57 b(Clearly)-8 b(,)37 b(\(3.3.4\))g(holds.)57 b(\(Here)37 b(w)m(e)i(use)f(the)f(h)m(yp)s(othesis)3271 2313 y(^)3249 2339 y Fv(f)3297 2354 y Ft(n)3380 2339 y Fw(=)e(0)120 2460 y(if)i(\()p Fv(n)25 b Fw(+)h Fv(k)s Fw(\))530 2423 y Fu(2)606 2460 y Fw(=)36 b Fv(\025)p Fw(.\))59 b(It)38 b(remains)f(to)g(v)m(erify)i(\(1.ii\))c(in)i(Lemma)g(3.3.3)g(for)g(all) f Fv(n)h Fs(2)g Fr(Z)p Fv(;)50 b(')37 b Fs(2)120 2580 y Fv(H)209 2544 y Fu(1)248 2580 y Fw(\()p Fr(R)352 2595 y Fu(+)417 2580 y Fw(\).)43 b(This)33 b(in)f(turn)h(follo)m(ws)e(from)g (\(2.ii\))g(b)m(y)i(partial)d(in)m(tegration.)p 3469 2580 4 66 v 3473 2517 59 4 v 3473 2580 V 3531 2580 4 66 v 266 2773 a(Let)j(us)g(summarize)f(the)h(essen)m(tial)f(result)h (of)f(Prop)s(osition)f(3.3.2.)120 2970 y Fp(Corollary)37 b(3.3.4.)42 b Fj(L)-5 b(et)36 b Fv(k)30 b Fs(2)e Fw([)p Fs(\000)1380 2931 y Fu(1)p 1380 2947 36 4 v 1380 3004 a(2)1426 2970 y Fv(;)1480 2931 y Fu(1)p 1480 2947 V 1480 3004 a(2)1525 2970 y Fw(])35 b Fj(and)f Fv(\025)28 b Fs(2)g Fr(R)5 b Fj(,)41 b(then)170 3196 y Fw(dim)15 b Fs(N)g Fw(\()p Fv(H)573 3155 y Fu(\()p Ft(\033)r Fu(\))674 3196 y Fw(\()p Fv(k)s Fw(\))i Fs(\000)h Fv(\025)p Fw(\))27 b Fs(\024)h Fw(dim)15 b Fs(N)g Fw(\()p Fv(B)1536 3155 y Fu(\()p Ft(\033)r Fu(\))1638 3196 y Fw(\()p Fv(\025)i Fw(+)g Fv(i)p Fw(0)p Fv(;)g(k)s Fw(\)\))27 b(=)h(dim)15 b Fs(N)g Fw(\()p Fv(B)2623 3155 y Fu(\()p Ft(\033)r Fu(\))2724 3196 y Fw(\()p Fv(\025)i Fs(\000)h Fv(i)p Fw(0)p Fv(;)f(k)s Fw(\)\))p Fv(:)50 b Fw(\(3.3.8\))120 3409 y Fj(If)34 b Fv(\025)28 b Fs(6)p Fw(=)f(\()p Fv(n)c Fw(+)f Fv(k)s Fw(\))719 3373 y Fu(2)758 3409 y Fv(;)45 b(n)27 b Fs(2)i Fr(Z)p Fv(;)j Fj(then)j(e)-5 b(quality)35 b(holds.)120 3529 y(In)f(p)-5 b(articular,)35 b(the)g(eigenvalues)e(of)i Fv(H)1599 3493 y Fu(\()p Ft(\033)r Fu(\))1700 3529 y Fw(\()p Fv(k)s Fw(\))g Fj(have)f(\014nite)h(multiplicities.)120 3726 y(Pr)-5 b(o)g(of.)48 b Fw(The)43 b(op)s(erator)f Fs(N)15 b Fw(\()p Fv(H)1262 3690 y Fu(\()p Ft(\033)r Fu(\))1362 3726 y Fw(\()p Fv(k)s Fw(\))29 b Fs(\000)g Fv(\025)p Fw(\))43 b Fs(3)h Fv(u)g Fs(7!)f Fv(f)54 b Fs(2)45 b(N)15 b Fw(\()p Fv(B)2545 3690 y Fu(\()p Ft(\033)r Fu(\))2646 3726 y Fw(\()p Fv(\025)28 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\)\))42 b(giv)m(en)g(b)m(y)120 3847 y(\(3.3.4\))37 b(is)g(injectiv)m(e)g(and,)i(if)e Fv(\025)f Fs(6)p Fw(=)g(\()p Fv(n)26 b Fw(+)f Fv(k)s Fw(\))1767 3811 y Fu(2)1807 3847 y Fv(;)52 b(n)37 b Fs(2)f Fr(Z)p Fv(;)g Fw(also)g(surjectiv)m(e.)60 b(F)-8 b(or)37 b(the)h(second)120 3967 y(statemen)m(t)g(note)f(that)g Fs(N)15 b Fw(\()p Fv(B)1229 3931 y Fu(\()p Ft(\033)r Fu(\))1330 3967 y Fw(\()p Fv(\025)25 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\)\))36 b(is)h(\014nite)g(dimensional)d(b)s (ecause)39 b(of)d(Lemma)120 4088 y(3.3.1.)p 3469 4088 4 66 v 3473 4025 59 4 v 3473 4088 V 3531 4088 4 66 v 120 4346 a Fp(3.)53 b(The)39 b(p)s(oin)m(t)f(sp)s(ectrum)g(of)g(the)h (op)s(erators)f Fv(H)2191 4310 y Fu(\()p Ft(\033)r Fu(\))2293 4346 y Fw(\()p Fv(k)s Fw(\))p Fp(.)97 b Fw(Using)33 b(the)h(c)m (haracteriza-)120 4467 y(tion)28 b(of)g(the)i(eigen)m(v)-5 b(alues)28 b Fv(\025)h Fw(of)f Fv(H)1371 4430 y Fu(\()p Ft(\033)r Fu(\))1473 4467 y Fw(\()p Fv(k)s Fw(\))h(in)f(terms)g(of)h Fv(B)2196 4430 y Fu(\()p Ft(\033)r Fu(\))2298 4467 y Fw(\()p Fv(\025)15 b Fs(\006)g Fv(i)p Fw(0)p Fv(;)i(k)s Fw(\))28 b(w)m(e)i(shall)d(pro)m(v)m(e)j(no)m(w)120 4587 y(our)i(main)f(result)i(on)f(the)h(p)s(oin)m(t)f(sp)s(ectrum)h(of)f (the)h(op)s(erators)f(on)h(the)g(halfstrip.)120 4784 y Fp(Theorem)47 b(3.3.5.)g Fj(L)-5 b(et)44 b Fv(k)i Fs(2)d Fw([)p Fs(\000)1410 4745 y Fu(1)p 1410 4761 36 4 v 1410 4819 a(2)1456 4784 y Fv(;)1510 4745 y Fu(1)p 1510 4761 V 1510 4819 a(2)1555 4784 y Fw(])p Fj(,)i(then)e Fv(\033)1937 4799 y Ft(p)1994 4703 y Fo(\000)2040 4784 y Fv(H)2129 4748 y Fu(\()p Ft(\033)r Fu(\))2230 4784 y Fw(\()p Fv(k)s Fw(\))2360 4703 y Fo(\001)2449 4784 y Fj(c)-5 b(onsists)42 b(of)h(eigenvalues)f(of)120 4905 y(\014nite)35 b(multiplicities)f (which)g(may)h(ac)-5 b(cumulate)34 b(at)h Fw(+)p Fs(1)g Fj(only.)266 5102 y Fw(W)-8 b(e)42 b(will)d(pro)m(v)m(e)k(this)e (Theorem)h(in)f(t)m(w)m(o)h(steps.)71 b(First)41 b(\(Lemma)f(3.3.6\))h (w)m(e)h(use)h(the)120 5222 y(Analytic)26 b(F)-8 b(redholm)25 b(Alternativ)m(e)h(to)g(sho)m(w)i(that)e(the)h(only)f(p)s(ossible)g (\014nite)h(accum)m(ulation)120 5342 y(p)s(oin)m(ts)36 b(of)h Fv(\033)588 5357 y Ft(p)644 5262 y Fo(\000)690 5342 y Fv(H)779 5306 y Fu(\()p Ft(\033)r Fu(\))880 5342 y Fw(\()p Fv(k)s Fw(\))1010 5262 y Fo(\001)1093 5342 y Fw(are)f(the)i(p)s(oin)m(ts)e(\()p Fv(n)25 b Fw(+)g Fv(k)s Fw(\))2043 5306 y Fu(2)2082 5342 y Fv(;)52 b(n)35 b Fs(2)g Fr(Z)p Fw(.)54 b(Then)37 b(\(Lemma)f(3.3.7\))g(w)m(e)120 5463 y(use)49 b(a)e(con)m(tin)m(uit)m(y)h(and)g(monotonicit)m(y)e (principle)g(for)h(eigen)m(v)-5 b(alues)48 b(to)f(sho)m(w)i(that)f(the) 1781 5712 y(34)p eop %%Page: 35 40 35 39 bop 120 407 a Fw(eigen)m(v)-5 b(alues)33 b(do)f(actually)f(not)i (accum)m(ulate)f(at)g(these)i(p)s(oin)m(ts.)120 527 y(W)-8 b(e)36 b(state)f(\(and)g(pro)m(v)m(e\))i(these)f(statemen)m(ts)g(in)f (terms)g(of)g(the)g(op)s(erators)g Fv(B)3000 491 y Fu(\()p Ft(\033)r Fu(\))3102 527 y Fw(\()p Fv(\025)24 b Fs(\006)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))120 648 y(and)33 b(their)f(real)f (parts)i Fv(B)1061 611 y Fu(\()p Ft(\033)r Fu(\))1163 648 y Fw(\()p Fv(\025;)17 b(k)s Fw(\).)43 b(Note)33 b(that)1064 863 y Fs(N)15 b Fw(\()p Fv(B)1278 822 y Fu(\()p Ft(\033)r Fu(\))1380 863 y Fw(\()p Fv(\025)22 b Fs(\006)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\)\))28 b Fs(\032)g(N)15 b Fw(\()p Fv(B)2199 822 y Fu(\()p Ft(\033)r Fu(\))2300 863 y Fw(\()p Fv(\025;)i(k)s Fw(\)\))p Fv(:)668 b Fw(\(3.3.9\))120 1079 y(Indeed,)36 b(if)e Fv(B)637 1042 y Fu(\()p Ft(\033)r Fu(\))739 1079 y Fw(\()p Fv(\025)23 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))p Fv(f)41 b Fw(=)31 b(0)j(w)m(e)h(see)h (from)d(Im)23 b Fv(b)2173 1042 y Fu(\()p Ft(\033)r Fu(\))2275 1079 y Fw(\()p Fv(\025)g Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\)[)p Fv(f)11 b Fw(])31 b(=)f(0)k(that)3280 1052 y(^)3259 1079 y Fv(f)3307 1094 y Ft(n)3385 1079 y Fw(=)c(0)120 1199 y(if)k(\()p Fv(n)25 b Fw(+)f Fv(k)s Fw(\))525 1163 y Fu(2)597 1199 y Fv(<)33 b(\025)p Fw(.)53 b(Hence)37 b(\(Re)24 b Fv(b)1354 1163 y Fu(\()p Ft(\033)r Fu(\))1456 1199 y Fw(\()p Fv(\025)g Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\)\)[)p Fv(f)5 b(;)17 b(g)t Fw(])32 b(=)h Fv(b)2317 1163 y Fu(\()p Ft(\033)r Fu(\))2419 1199 y Fw(\()p Fv(\025)24 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\)[)p Fv(f)5 b(;)17 b(g)t Fw(])32 b(=)h(0)i(for)g(all)120 1319 y Fv(g)c Fs(2)d Fv(H)381 1283 y Fu(1)p Ft(=)p Fu(2)491 1319 y Fw(\()p Fr(T)p Fw(\),)35 b(i.e.)43 b Fv(f)c Fs(2)28 b(N)15 b Fw(\()p Fv(B)1255 1283 y Fu(\()p Ft(\033)r Fu(\))1356 1319 y Fw(\()p Fv(\025;)i(k)s Fw(\)\).)120 1519 y Fp(Lemma)37 b(3.3.6.)42 b Fj(L)-5 b(et)35 b Fv(k)c Fs(2)d Fw([)p Fs(\000)1278 1480 y Fu(1)p 1278 1496 36 4 v 1278 1553 a(2)1324 1519 y Fv(;)1378 1480 y Fu(1)p 1378 1496 V 1378 1553 a(2)1423 1519 y Fw(])35 b Fj(and)f Fv(\033)e Fs(6\021)c Fw(0)p Fj(.)45 b(Then)34 b(the)g(sets)241 1748 y Fs(N)15 b Fw(\()p Fv(k)s Fw(\))27 b(:=)h Fs(f)p Fv(\025)f Fs(2)h Fr(R)33 b Fs(n)22 b(f)p Fw(\()p Fv(n)g Fw(+)g Fv(k)s Fw(\))1378 1706 y Fu(2)1446 1748 y Fw(:)55 b Fv(n)28 b Fs(2)g Fr(Z)p Fs(g)d Fw(:)55 b(0)44 b Fj(is)35 b(eigenvalue)f(of)44 b Fv(B)2806 1706 y Fu(\()p Ft(\033)r Fu(\))2908 1748 y Fw(\()p Fv(\025)22 b Fs(\006)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))p Fs(g)p Fv(;)273 1883 y Fw(~)241 1908 y Fs(N)e Fw(\()p Fv(k)s Fw(\))27 b(:=)h Fs(f)p Fv(\025)f Fs(2)h Fr(R)33 b Fs(n)22 b(f)p Fw(\()p Fv(n)g Fw(+)g Fv(k)s Fw(\))1378 1867 y Fu(2)1446 1908 y Fw(:)55 b Fv(n)28 b Fs(2)g Fr(Z)p Fs(g)d Fw(:)55 b(0)44 b Fj(is)35 b(eigenvalue)f(of)44 b Fv(B)2806 1867 y Fu(\()p Ft(\033)r Fu(\))2908 1908 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))p Fs(g)120 2127 y Fj(ar)-5 b(e)35 b(discr)-5 b(ete)34 b(in)h Fr(R)e Fs(n)22 b(f)p Fw(\()p Fv(n)g Fw(+)g Fv(k)s Fw(\))1287 2091 y Fu(2)1326 2127 y Fv(;)45 b(n)27 b Fs(2)i Fr(Z)p Fs(g)p Fj(.)266 2326 y Fw(Note)40 b(that)e(the)i(set)g Fs(N)15 b Fw(\()p Fv(k)s Fw(\))38 b(do)s(es)h(not)g(dep)s(end)h(on)f(the)h(c)m(hoice)f (of)g(the)g Fs(\006)p Fw(-sign.)63 b(This)120 2447 y(follo)m(ws)31 b(from)h(Prop)s(osition)e(3.3.2.)120 2567 y(If)38 b Fv(\033)k Fs(\021)c Fw(0,)h(then)g Fs(N)15 b Fw(\()p Fv(k)s Fw(\))37 b(=)h Fs(;)p Fw(,)h(but)1489 2542 y(~)1457 2567 y Fs(N)14 b Fw(\()p Fv(k)s Fw(\))38 b(=)f([)p Fv(k)1915 2531 y Fu(2)1955 2567 y Fv(;)17 b Fw(+)p Fs(1)p Fw(\))25 b Fs(n)h(f)p Fw(\()p Fv(n)g Fw(+)g Fv(k)s Fw(\))2680 2531 y Fu(2)2719 2567 y Fv(;)54 b(n)38 b Fs(2)g Fr(Z)p Fs(g)e Fw(\(consider)120 2687 y Fv(f)i Fw(=)28 b Fv(e)355 2702 y Fu(0)395 2687 y Fw(\).)120 2881 y Fj(Pr)-5 b(o)g(of.)48 b Fw(Because)28 b(of)33 b(\(3.3.9\))25 b(it)g(su\016ces)k(to)c(consider)i(the)f(set) 2448 2856 y(~)2416 2881 y Fs(N)15 b Fw(\()p Fv(k)s Fw(\).)41 b(Let)26 b(\003)g(b)s(e)g(a)g(maximal)120 3002 y(op)s(en)33 b(in)m(terv)-5 b(al)31 b(in)h Fr(R)h Fs(n)22 b(f)p Fw(\()p Fv(n)h Fw(+)f Fv(k)s Fw(\))1347 2965 y Fu(2)1386 3002 y Fv(;)45 b(n)28 b Fs(2)g Fr(Z)p Fs(g)i Fw(and)j Fv(l)d Fw(:=)e(sup)17 b(\003.)44 b(W)-8 b(e)33 b(pro)m(v)m(e)h(that)e(the)h(p) s(oin)m(ts)120 3122 y Fv(\025)f Fw(for)g(whic)m(h)i Fv(B)717 3086 y Fu(\()p Ft(\033)r Fu(\))818 3122 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))33 b(has)f(eigen)m(v)-5 b(alue)33 b(0)f(do)g(not)h(accum)m (ulate)f(inside)g(\003.)120 3242 y(F)-8 b(or)32 b Fv(z)g Fs(2)c Fv(D)j Fw(:=)c Fs(f)p Fv(\020)35 b Fs(2)28 b Fr(C)54 b Fw(:)27 b(Re)c Fv(\020)34 b Fs(2)29 b Fw(\003)p Fs(g)j Fw(w)m(e)h(consider)g(the)g(quadratic)f(form)247 3463 y Fs(D)s Fw([)350 3437 y(~)354 3463 y Fv(b)395 3422 y Fu(\()p Ft(\033)r Fu(\))497 3463 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\)])28 b(:=)g Fv(H)995 3422 y Fu(1)p Ft(=)p Fu(2)1104 3463 y Fw(\()p Fr(T)p Fw(\))p Fv(;)264 3644 y Fw(~)268 3670 y Fv(b)309 3629 y Fu(\()p Ft(\033)r Fu(\))411 3670 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\)[)p Fv(f)11 b Fw(])28 b(:=)985 3576 y Fo(X)906 3793 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))1097 3774 y Fm(2)1132 3793 y Fl(\025)p Ft(l)1225 3580 y Fo(p)p 1325 3580 520 4 v 90 x Fw(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))1633 3642 y Fu(2)1695 3670 y Fs(\000)g Fv(z)f Fs(j)1909 3644 y Fw(^)1888 3670 y Fv(f)1936 3685 y Ft(n)1983 3670 y Fs(j)2011 3629 y Fu(2)2072 3670 y Fw(+)2170 3535 y Fo(Z)2270 3561 y Ft(\031)2226 3760 y Fl(\000)p Ft(\031)2344 3670 y Fv(\033)t Fw(\()p Fv(x)2496 3685 y Fu(1)2536 3670 y Fw(\))p Fs(j)p Fv(f)11 b Fw(\()p Fv(x)2754 3685 y Fu(1)2793 3670 y Fw(\))p Fs(j)2859 3629 y Fu(2)2915 3670 y Fv(dx)3021 3685 y Fu(1)3061 3670 y Fv(:)3215 3620 y Fw(\(3.3.10\))120 3997 y(\(As)31 b(usual)e(w)m(e)j(c)m (ho)s(ose)f(the)f(canonical)f(branc)m(h)i(of)e(the)i(square)g(ro)s(ot)f (on)g Fr(C)43 b Fs(n)17 b Fw(\()p Fs(\0001)p Fv(;)g Fw(0])29 b(with)120 4118 y(Re)235 4063 y Fs(p)p 318 4063 28 4 v 55 x Fv(:)f(>)f Fw(0.\))120 4259 y(Then)45 b(for)d(an)m(y)j Fv(f)56 b Fs(2)47 b Fv(H)1047 4223 y Fu(1)p Ft(=)p Fu(2)1157 4259 y Fw(\()p Fr(T)p Fw(\),)1368 4233 y(~)1372 4259 y Fv(b)1413 4223 y Fu(\()p Ft(\033)r Fu(\))1515 4259 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\)[)p Fv(f)11 b Fw(])44 b(is)f(an)g(analytic)f(function)h(of)g Fv(z)51 b Fs(2)c Fv(D)f Fw(and)120 4379 y(coincides)32 b(with)h(Re)22 b Fv(b)933 4343 y Fu(\()p Ft(\033)r Fu(\))1035 4379 y Fw(\()p Fv(\025)g Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\)[)p Fv(f)11 b Fw(])32 b(if)f Fv(z)i Fw(=)27 b Fv(\025)h Fs(2)g Fw(\003.)120 4500 y(It)47 b(follo)m(ws)e(b)m(y)i(argumen)m(ts)g (as)g(in)e(Subsection)i(1)g(that)2292 4474 y(~)2296 4500 y Fv(b)2337 4464 y Fu(\()p Ft(\033)r Fu(\))2439 4500 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))47 b(are)g(closed)f(sectorial)120 4620 y(forms.)41 b(The)30 b(corresp)s(ondig)f(op)s(erators)1642 4595 y(~)1619 4620 y Fv(B)1698 4584 y Fu(\()p Ft(\033)r Fu(\))1799 4620 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))30 b(form)d(an)i(analytic)f(family)e(of)i(t)m(yp)s(e)i(\(B\))120 4741 y(and)42 b(coincide)f(with)h Fv(B)1013 4704 y Fu(\()p Ft(\033)r Fu(\))1115 4741 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))41 b(if)g Fv(z)49 b Fw(=)43 b Fv(\025)h Fs(2)g Fw(\003.)72 b(It)42 b(follo)m(ws)e(from)h(Lemma)g(3.3.1)g(that)143 4836 y(~)120 4861 y Fv(B)199 4825 y Fu(\()p Ft(\033)r Fu(\))301 4861 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))33 b(has)g(compact)f (resolv)m(en)m(t)i(for)e(an)m(y)h Fv(z)f Fs(2)c Fv(D)s Fw(.)120 4981 y(Moreo)m(v)m(er,)47 b(0)c(is)g(not)f(an)h(eigen)m(v)-5 b(alue)43 b(of)1740 4956 y(~)1716 4981 y Fv(B)1795 4945 y Fu(\()p Ft(\033)r Fu(\))1897 4981 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))44 b(if)e(Im)o Fv(z)50 b Fs(6)p Fw(=)c(0.)74 b(Indeed,)47 b(giv)m(en)c Fv(f)57 b Fs(2)120 5102 y(N)15 b Fw(\()278 5077 y(~)255 5102 y Fv(B)334 5066 y Fu(\()p Ft(\033)r Fu(\))435 5102 y Fw(\()p Fv(\025;)i(k)s Fw(\)\))36 b(w)m(e)i(see)f(from)e(Im)1421 5075 y(~)1425 5102 y Fv(b)1466 5066 y Fu(\()p Ft(\033)r Fu(\))1568 5102 y Fw(\()p Fv(\025)24 b Fs(\006)i Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\)[)p Fv(f)11 b Fw(])34 b(=)g(0)i(that)2586 5075 y(^)2565 5102 y Fv(f)2613 5117 y Ft(n)2694 5102 y Fw(=)e(0)i(if)g(\()p Fv(n)25 b Fw(+)f Fv(k)s Fw(\))3296 5066 y Fu(2)3370 5102 y Fs(\025)35 b Fv(l)r Fw(.)120 5222 y(Hence)414 5142 y Fo(R)481 5168 y Ft(\031)461 5257 y Fl(\000)p Ft(\031)580 5222 y Fv(\033)t(f)p 698 5167 51 4 v 11 w(g)19 b(dx)870 5237 y Fu(1)945 5222 y Fw(=)1052 5196 y(~)1056 5222 y Fv(b)1097 5186 y Fu(\()p Ft(\033)r Fu(\))1199 5222 y Fw(\()p Fv(z)t(;)e(k)s Fw(\)[)p Fv(f)5 b(;)17 b(g)t Fw(])35 b(=)f(0)j(for)f(all)f Fv(g)j Fs(2)d Fv(H)2423 5186 y Fu(1)p Ft(=)p Fu(2)2533 5222 y Fw(\()p Fr(T)p Fw(\),)41 b(i.e.)56 b Fv(\033)t(f)45 b Fs(\021)36 b Fw(0.)56 b(Since)120 5342 y Fv(\033)34 b Fs(6\021)c Fw(0)k(and)g Fv(f)44 b Fw(as)35 b(trigonometric)c(p)s (olynomial)f(has)k(only)g(a)f(\014nite)h(n)m(um)m(b)s(er)g(of)g(zero)s (es,)h(w)m(e)120 5463 y(conclude)e Fv(f)38 b Fs(\021)29 b Fw(0.)1781 5712 y(35)p eop %%Page: 36 41 36 40 bop 120 407 a Fw(Finally)-8 b(,)27 b(it)i(follo)m(ws)f(from)g (the)h(Analytic)g(F)-8 b(redholm)27 b(Alternativ)m(e)i(\(cf.)43 b(Theorem)29 b(VI)s(I.1.10)120 527 y(in)37 b([K]\))g(applied)f(to)h (the)h(op)s(erators)1547 502 y(~)1524 527 y Fv(B)1603 491 y Fu(\()p Ft(\033)r Fu(\))1705 527 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))37 b(that)h(the)f(set)i Fs(f)p Fv(\025)c Fs(2)i Fw(\003)e(:)i(0)e(is)d(eigen)m(v)-5 b(alue)120 648 y(of)27 b Fv(B)305 611 y Fu(\()p Ft(\033)r Fu(\))407 648 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))p Fs(g)32 b Fw(is)g(a)g(discrete) i(subset)g(of)e(\003.)p 3469 648 4 66 v 3473 585 59 4 v 3473 648 V 3531 648 4 66 v 120 816 a Fp(Lemma)43 b(3.3.7.)i Fj(L)-5 b(et)40 b Fv(k)f Fs(2)e Fw([)p Fs(\000)1309 777 y Fu(1)p 1309 793 36 4 v 1309 851 a(2)1355 816 y Fv(;)1408 777 y Fu(1)p 1408 793 V 1408 851 a(2)1454 816 y Fw(])i Fj(and)g Fv(\033)i Fs(6\021)c Fw(0)p Fj(.)58 b(Then)39 b(the)h(sets)f Fs(N)15 b Fw(\()p Fv(k)s Fw(\))39 b Fj(and)3175 791 y Fw(~)3143 816 y Fs(N)15 b Fw(\()p Fv(k)s Fw(\))39 b Fj(ar)-5 b(e)120 937 y(discr)g(ete)34 b(in)h Fr(R)5 b Fj(.)120 1105 y(Pr)-5 b(o)g(of.)48 b Fw(Because)38 b(of)k(\(3.3.9\))35 b(it)g(su\016ces)j(again)c(to)i(consider)g(the)g (set)2799 1080 y(~)2767 1105 y Fs(N)14 b Fw(\()p Fv(k)s Fw(\).)54 b(Let)36 b(\003)f(b)s(e)h(a)120 1226 y(maximal)29 b(op)s(en)k(in)m(terv)-5 b(al)31 b(in)h Fr(R)h Fs(n)22 b(f)p Fw(\()p Fv(n)g Fw(+)g Fv(k)s Fw(\))1744 1190 y Fu(2)1784 1226 y Fv(;)44 b(n)28 b Fs(2)g Fr(Z)p Fs(g)p Fw(,)i(i.e.)43 b(either)1522 1405 y(\003)27 b(=)h(\()p Fs(\0001)p Fv(;)17 b(k)2034 1364 y Fu(2)2073 1405 y Fw(\))p Fv(;)1077 b Fw(\(3.3.11\))120 1585 y(or)32 b(for)g(some)h Fv(n)691 1600 y Fu(1)730 1585 y Fv(;)45 b(n)860 1600 y Fu(2)927 1585 y Fs(2)28 b Fr(Z)560 1765 y Fw(\003)g(=)f(\(\()p Fv(n)893 1780 y Fu(1)955 1765 y Fw(+)22 b Fv(k)s Fw(\))1145 1724 y Fu(2)1185 1765 y Fv(;)17 b Fw(\()p Fv(n)1325 1780 y Fu(2)1386 1765 y Fw(+)22 b Fv(k)s Fw(\))1576 1724 y Fu(2)1615 1765 y Fw(\))p Fv(;)212 b Fw(\()p Fv(n)23 b Fw(+)f Fv(k)s Fw(\))2201 1724 y Fu(2)2279 1765 y Fv(=)-60 b Fs(2)28 b Fw(\003)p Fv(;)44 b(n)28 b Fs(2)g Fr(Z)p Fv(:)438 b Fw(\(3.3.12\))120 1944 y(W)-8 b(e)31 b(pro)m(v)m(e)h(that)e (there)h(are)g(only)f(\014nitely)f(man)m(y)i(p)s(oin)m(ts)f Fv(\025)d Fs(2)i Fw(\003)h(for)g(whic)m(h)h Fv(B)3036 1908 y Fu(\()p Ft(\033)r Fu(\))3138 1944 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))30 b(has)120 2065 y(eigen)m(v)-5 b(alue)32 b(0.)120 2185 y(Recall)i(that)h Fv(B)710 2149 y Fu(\()p Ft(\033)r Fu(\))812 2185 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))35 b(is)g(a)g(self-adjoin)m(t)f(op)s(erator)h(with)g(compact)g (resolv)m(en)m(t.)54 b(W)-8 b(e)36 b(de-)120 2305 y(note)j(b)m(y)g Fv(\026)543 2320 y Ft(n)590 2305 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))p Fv(;)54 b(\025)37 b Fs(2)p 1101 2225 68 4 v 39 w Fw(\003)o Fv(;)i Fw(its)f Fv(n)p Fw(-th)g(eigen)m(v)-5 b(alue,)40 b(coun)m(ting)e(m)m(ultiplicities.)57 b(It)38 b(follo)m(ws)120 2426 y(from)28 b(the)i(min-max)d(principle)h(that)i Fv(\026)1584 2441 y Ft(n)1630 2426 y Fw(\()p Fv(:;)17 b(k)s Fw(\))30 b(is)f(a)g(con)m(tin)m(uous)h(function)f(on)p 3031 2346 V 30 w(\003)o(,)i(and)e(from)120 2546 y(the)k(inequalit)m(y) 1073 2667 y Fv(B)1152 2625 y Fu(\()p Ft(\033)r Fu(\))1253 2667 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))28 b Fs(\024)g Fv(B)1696 2625 y Fu(\()p Ft(\033)r Fu(\))1798 2667 y Fw(\()p Fv(\025)1893 2625 y Fl(0)1916 2667 y Fv(;)17 b(k)s Fw(\))p Fv(;)211 b(\025)2347 2625 y Fl(0)2398 2667 y Fs(\024)28 b Fv(\025;)120 2824 y Fw(that)50 b(it)g(is)g (non-increasing.)96 b(Moreo)m(v)m(er,)56 b(there)51 b(is)f(at)g(most)g (one)h Fv(\025)58 b Fs(2)p 3008 2744 V 58 w Fw(\003)51 b(for)e(whic)m(h)120 2944 y Fv(\026)179 2959 y Ft(n)226 2944 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))33 b(=)h(0,)j(since)g(otherwise) g(this)f(w)m(ould)g(hold)f(for)h(a)g(whole)g(in)m(terv)-5 b(al)35 b(of)h Fv(\025)p Fw('s)h(in)e(con-)120 3064 y(tradiction)c(to)h (Lemma)f(3.3.6.)120 3185 y(W)-8 b(e)33 b(consider)g(no)m(w)g(the)g (cases)h(\(3.3.11\))o(,)f(\(3.3.12\))f(separately)-8 b(.)120 3305 y(If)33 b(\003)g(is)g(of)g(the)g(form)f(\(3.3.11\))o(,)i (w)m(e)g(note)g(that)f(the)g(op)s(erator)g Fv(B)2530 3269 y Fu(\()p Ft(\033)r Fu(\))2632 3305 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))32 b(is)h(p)s(ositiv)m(e)g(de\014-)120 3426 y(nite)g(pro)m(vided)g Fs(\000)p Fv(\025)h Fw(is)e(large,)h(hence)h (the)g(function)e Fv(\026)2131 3441 y Ft(n)2178 3426 y Fw(\()p Fv(:;)17 b(k)s Fw(\))33 b(is)f(p)s(ositiv)m(e)h(for)f Fs(\000)p Fv(\025)i Fw(large.)44 b(It)120 3546 y(follo)m(ws)31 b(that)643 3725 y Fv(])681 3740 y Ft(cm)778 3725 y Fs(f)p Fv(\025)c Fs(2)h Fw(\()p Fs(\0001)p Fv(;)17 b(k)1319 3684 y Fu(2)1359 3725 y Fw(\))27 b(:)56 b(0)27 b(is)32 b(eigen)m(v)-5 b(alue)32 b(of)c Fv(B)2333 3684 y Fu(\()p Ft(\033)r Fu(\))2435 3725 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))p Fs(g)27 b Fw(=)726 3871 y(=)83 b Fv(])p Fs(f)p Fv(n)28 b Fs(2)g Fr(N)42 b Fw(:)56 b Fv(\026)1394 3886 y Ft(n)1440 3871 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))28 b(=)f(0)h(for)k(some)27 b Fv(\025)h Fs(2)g Fw(\()p Fs(\0001)p Fv(;)17 b(k)2759 3830 y Fu(2)2798 3871 y Fw(\))p Fs(g)28 b Fw(=)726 4016 y(=)83 b Fv(])p Fs(f)p Fv(n)28 b Fs(2)g Fr(N)42 b Fw(:)56 b Fv(\026)1394 4031 y Ft(n)1440 4016 y Fw(\()p Fv(k)1532 3975 y Fu(2)1572 4016 y Fv(;)17 b(k)s Fw(\))27 b Fv(<)h Fw(0)p Fs(g)f Fw(=)726 4161 y(=)83 b Fv(])923 4176 y Ft(cm)1020 4161 y Fs(f)p Fv(\026)27 b(<)h Fw(0)f(:)56 b Fv(\026)27 b Fw(is)32 b(eigen)m(v)-5 b(alue)32 b(of)27 b Fv(B)2254 4120 y Fu(\()p Ft(\033)r Fu(\))2356 4161 y Fw(\()p Fv(k)2448 4120 y Fu(2)2488 4161 y Fv(;)17 b(k)s Fw(\))p Fs(g)p Fv(;)514 b Fw(\(3.3.13\))120 4341 y(where)29 b Fv(])435 4356 y Ft(cm)532 4341 y Fs(f)p Fv(:::)p Fs(g)f Fw(means)f(that)h(the)g(cardinalit)m(y)e(of)h Fs(f)p Fv(:::)p Fs(g)g Fw(is)h(determined)f(according)g(to)h(m)m(ul-) 120 4461 y(tiplicities.)44 b(The)34 b(RHS)g(of)41 b(\(3.3.13\))32 b(is)i(\014nite)f(since)h Fv(B)2166 4425 y Fu(\()p Ft(\033)r Fu(\))2268 4461 y Fw(\()p Fv(k)2360 4425 y Fu(2)2400 4461 y Fv(;)17 b(k)s Fw(\))33 b(has)h(compact)g(resolv)m(en)m(t.)120 4582 y(This)f(pro)m(v)m(es)h(the)f(assertion)g(in)e(the)i(case)h (\(3.3.11\))o(.)120 4702 y(If)f(\003)f(is)g(of)g(the)h(form)e (\(3.3.12\))o(,)i(w)m(e)h(ha)m(v)m(e)391 4882 y Fv(])429 4897 y Ft(cm)526 4882 y Fs(f)p Fv(\025)28 b Fs(2)g Fw(\(\()p Fv(n)889 4897 y Fu(1)950 4882 y Fw(+)22 b Fv(k)s Fw(\))1140 4841 y Fu(2)1180 4882 y Fv(;)17 b Fw(\()p Fv(n)1320 4897 y Fu(2)1381 4882 y Fw(+)22 b Fv(k)s Fw(\))1571 4841 y Fu(2)1611 4882 y Fw(\))27 b(:)56 b(0)27 b(is)32 b(eigen)m(v)-5 b(alue)32 b(of)c Fv(B)2585 4841 y Fu(\()p Ft(\033)r Fu(\))2686 4882 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))p Fs(g)27 b Fw(=)474 5027 y(=)83 b Fv(])p Fs(f)p Fv(n)28 b Fs(2)g Fr(N)42 b Fw(:)56 b Fv(\026)1142 5042 y Ft(n)1188 5027 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))28 b(=)f(0)h(for)k(some)27 b Fv(\025)h Fs(2)g Fw(\(\()p Fv(n)2328 5042 y Fu(1)2390 5027 y Fw(+)22 b Fv(k)s Fw(\))2580 4986 y Fu(2)2619 5027 y Fv(;)17 b Fw(\()p Fv(n)2759 5042 y Fu(2)2821 5027 y Fw(+)22 b Fv(k)s Fw(\))3011 4986 y Fu(2)3050 5027 y Fw(\))p Fs(g)27 b Fw(=)474 5172 y(=)83 b Fv(])p Fs(f)p Fv(n)28 b Fs(2)g Fr(N)42 b Fw(:)56 b Fv(\026)1142 5187 y Ft(n)1188 5172 y Fw(\(\()p Fv(n)1322 5187 y Fu(2)1384 5172 y Fw(+)22 b Fv(k)s Fw(\))1574 5131 y Fu(2)1614 5172 y Fv(;)17 b(k)s Fw(\))27 b Fv(<)h Fw(0)f(and)h Fv(\026)2201 5187 y Ft(n)2247 5172 y Fw(\(\()p Fv(n)2381 5187 y Fu(1)2443 5172 y Fw(+)22 b Fv(k)s Fw(\))2633 5131 y Fu(2)2673 5172 y Fv(;)17 b(k)s Fw(\))27 b Fv(>)h Fw(0)p Fs(g)f Fw(=)474 5318 y(=)83 b Fv(])671 5333 y Ft(cm)768 5318 y Fs(f)p Fv(\026)27 b(<)h Fw(0)f(:)56 b Fv(\026)27 b Fw(is)32 b(eigen)m(v)-5 b(alue)32 b(of)27 b Fv(B)2002 5276 y Fu(\()p Ft(\033)r Fu(\))2104 5318 y Fw(\(\()p Fv(n)2238 5333 y Fu(2)2300 5318 y Fw(+)22 b Fv(k)s Fw(\))2490 5276 y Fu(2)2529 5318 y Fv(;)17 b(k)s Fw(\))p Fs(g)22 b(\000)633 5463 y(\000)p Fv(])748 5478 y Ft(cm)846 5463 y Fs(f)p Fv(\026)27 b Fs(\024)h Fw(0)f(:)56 b Fv(\026)27 b Fw(is)32 b(eigen)m(v)-5 b(alue)32 b(of)c Fv(B)2082 5422 y Fu(\()p Ft(\033)r Fu(\))2183 5463 y Fw(\(\()p Fv(n)2317 5478 y Fu(1)2379 5463 y Fw(+)22 b Fv(k)s Fw(\))2569 5422 y Fu(2)2609 5463 y Fv(;)17 b(k)s Fw(\))p Fs(g)p Fv(;)1781 5712 y Fw(36)p eop %%Page: 37 42 37 41 bop 120 407 a Fw(and)22 b(again)e(the)i(RHS)g(is)f(\014nite)h (since)g Fv(B)1559 371 y Fu(\()p Ft(\033)r Fu(\))1661 407 y Fw(\(\()p Fv(n)1795 422 y Fu(1)1835 407 y Fw(+)p Fv(k)s Fw(\))2003 371 y Fu(2)2042 407 y Fv(;)17 b(k)s Fw(\))p Fv(;)44 b(B)2328 371 y Fu(\()p Ft(\033)r Fu(\))2430 407 y Fw(\(\()p Fv(n)2564 422 y Fu(2)2603 407 y Fw(+)p Fv(k)s Fw(\))2771 371 y Fu(2)2811 407 y Fv(;)17 b(k)s Fw(\))k(ha)m(v)m(e)i(compact)120 527 y(resolv)m(en)m(t.)45 b(This)32 b(completes)h(the)g(pro)s(of)e(of)i(the)g(Lemma.)p 3469 527 4 66 v 3473 465 59 4 v 3473 527 V 3531 527 4 66 v 266 722 a(Theorem)g(3.3.5)f(follo)m(ws)g(no)m(w)h(immediately)-8 b(.)120 917 y Fj(Pr)j(o)g(of)35 b(\(The)-5 b(or)g(em)33 b(3.3.5\).)48 b Fw(Com)m(bine)32 b(Lemma)f(3.3.7)h(with)h(Corollary)d (3.3.4.)p 3469 917 V 3473 855 59 4 v 3473 917 V 3531 917 4 66 v 120 1177 a Fp(4.)63 b(F)-9 b(urther)42 b(results)f(on)h(the) g(eigen)m(v)-6 b(alues)41 b(of)h(the)g(op)s(erators)g Fv(H)2917 1141 y Fu(\()p Ft(\033)r Fu(\))3019 1177 y Fw(\()p Fv(k)s Fw(\))p Fp(.)97 b Fw(Under)120 1298 y(stronger)34 b(assumptions)g(on)g Fv(\033)k Fw(w)m(e)d(obtain)d(b)s(etter)j(results) f(on)g(the)g(lo)s(cation)e(\(or)h(absence\))120 1418 y(of)42 b(eigen)m(v)-5 b(alues)41 b(of)h(the)g(op)s(erators)g(on)g(the) h(halfstrip.)70 b(Our)42 b(main)e(to)s(ol)g(is)i(Prop)s(osition)120 1538 y(3.3.2.)120 1659 y(W)-8 b(e)28 b(start)g(with)g(a)f(simple)g (example)g(whic)m(h)i(sho)m(ws)g(that)f(the)g(case)h(of)e(an)h (in\014nite)f(sequence)120 1779 y(of)32 b(eigen)m(v)-5 b(alues)33 b(accum)m(ulating)d(at)j(+)p Fs(1)f Fw(can)h(actually)e (happ)s(en.)120 1982 y Fp(Example)36 b(3.3.8.)43 b Fw(Assume)33 b(that)f Fv(\033)g Fs(2)c Fr(R)44 b Fw(is)32 b(constan)m(t)h(and)g(let) f Fv(k)e Fs(2)f Fw([)p Fs(\000)2856 1943 y Fu(1)p 2856 1960 36 4 v 2856 2017 a(2)2901 1982 y Fv(;)2955 1943 y Fu(1)p 2955 1960 V 2955 2017 a(2)3000 1982 y Fw(].)44 b(Then)609 2270 y Fv(\033)664 2285 y Ft(p)721 2189 y Fo(\000)766 2270 y Fv(H)855 2229 y Fu(\()p Ft(\033)r Fu(\))957 2270 y Fw(\()p Fv(k)s Fw(\))1087 2189 y Fo(\001)1160 2270 y Fw(=)1264 2130 y Fo(\032)1380 2209 y Fs(;)1147 b Fw(if)97 b Fv(\033)31 b Fs(\025)e Fw(0)p Fv(;)1380 2329 y Fs(f\000)p Fv(\033)1566 2293 y Fu(2)1628 2329 y Fw(+)22 b(\()p Fv(n)g Fw(+)g Fv(k)s Fw(\))2034 2293 y Fu(2)2101 2329 y Fw(:)56 b Fv(n)27 b Fs(2)i Fr(Z)p Fs(g)94 b Fw(if)j Fv(\033)31 b(<)d Fw(0)p Fv(:)120 2543 y Fw(In)33 b(the)g(case)g Fv(\033)f(<)c Fw(0)k(the)h(eigenfunctions)f (are)h(giv)m(en)g(for)f Fv(n)27 b Fs(2)i Fr(Z)h Fw(b)m(y)1080 2800 y Fv(u)1136 2815 y Ft(n)1183 2800 y Fw(\()p Fv(x)p Fw(\))e(:=)1472 2678 y Fo(q)p 1572 2678 102 4 v 1582 2752 a Fl(j)p Ft(\033)r Fl(j)p 1582 2777 82 4 v 1601 2834 a Ft(\031)1690 2800 y Fv(e)1735 2758 y Ft(inx)1842 2767 y Fm(1)1897 2800 y Fv(e)1942 2758 y Ft(\033)r(x)2024 2767 y Fm(2)2064 2800 y Fv(;)212 b(x)28 b Fs(2)g Fw(\005)p Fv(:)120 3036 y Fw(By)39 b(separation)f(of)g(v)-5 b(ariables)37 b(\(similarly)e(as)k(in)e(Section)i(2.1\))f(this)g(is)g(deriv)m(ed)h (from)f(the)120 3156 y(w)m(ell-kno)m(wn)33 b(prop)s(erties)g(of)g(the)g (op)s(erator)g Fs(\000)1867 3117 y Ft(d)1903 3094 y Fm(2)p 1847 3133 111 4 v 1847 3192 a Ft(dx)1923 3169 y Fm(2)1923 3214 y(2)2001 3156 y Fw(on)g Fv(L)2203 3171 y Fu(2)2242 3156 y Fw(\()p Fr(R)2346 3171 y Fu(+)2411 3156 y Fw(\))g(with)g(b)s (oundary)g(condition)120 3297 y Fv(f)179 3260 y Fl(0)202 3297 y Fw(\(0\))27 b(=)h Fv(\033)t(f)11 b Fw(\(0\).)120 3500 y Fp(Prop)s(osition)42 b(3.3.9.)i Fj(Assume)c(that)g Fv(\033)k Fj(is)c(b)-5 b(ounde)g(d)39 b(fr)-5 b(om)39 b(b)-5 b(elow)39 b(and)g(let)h Fv(k)g Fs(2)d Fw([)p Fs(\000)3338 3461 y Fu(1)p 3338 3477 36 4 v 3338 3534 a(2)3384 3500 y Fv(;)3438 3461 y Fu(1)p 3438 3477 V 3438 3534 a(2)3483 3500 y Fw(])p Fj(.)120 3620 y(Then)587 3715 y Fv(\033)642 3730 y Ft(p)699 3634 y Fo(\000)745 3715 y Fv(H)834 3678 y Fu(\()p Ft(\033)r Fu(\))935 3715 y Fw(\()p Fv(k)s Fw(\))1065 3634 y Fo(\001)1138 3715 y Fw(=)28 b Fs(;)1373 b Fj(if)44 b Fv(\033)2824 3730 y Fu(0)2891 3715 y Fs(\025)29 b Fw(0)p Fv(;)587 3837 y(\033)642 3852 y Ft(p)699 3756 y Fo(\000)745 3837 y Fv(H)834 3801 y Fu(\()p Ft(\033)r Fu(\))935 3837 y Fw(\()p Fv(k)s Fw(\))1065 3756 y Fo(\001)1138 3837 y Fs(\032)1244 3762 y Fo(S)1327 3866 y Ft(n)p Fl(2)p Fn(Z)1465 3837 y Fw([)p Fs(\000)p Fv(\033)1628 3801 y Fu(2)1624 3861 y(0)1690 3837 y Fw(+)22 b(\()p Fv(n)g Fw(+)g Fv(k)s Fw(\))2096 3801 y Fu(2)2136 3837 y Fv(;)17 b Fw(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))2488 3801 y Fu(2)2527 3837 y Fw(\))100 b Fj(if)44 b Fv(\033)2824 3852 y Fu(0)2891 3837 y Fv(<)28 b Fw(0)p Fv(;)120 4021 y Fj(wher)-5 b(e)36 b Fv(\033)452 4036 y Fu(0)523 4021 y Fw(:=)31 b(essinf)895 4036 y Ft(x)935 4045 y Fm(1)970 4036 y Fl(2)p Fn(T)1082 4021 y Fv(\033)t Fw(\()p Fv(x)1234 4036 y Fu(1)1274 4021 y Fw(\))p Fj(.)50 b(Mor)-5 b(e)g(over,)37 b Fs(\000)p Fv(\033)1989 3985 y Fu(2)1985 4046 y(0)2052 4021 y Fw(+)23 b Fv(k)2205 3985 y Fu(2)2276 4021 y Fs(2)31 b Fv(\033)2428 4036 y Ft(p)2485 3940 y Fo(\000)2531 4021 y Fv(H)2620 3985 y Fu(\()p Ft(\033)r Fu(\))2721 4021 y Fw(\()p Fv(k)s Fw(\))2851 3940 y Fo(\001)2933 4021 y Fj(i\013)37 b Fv(\033)e Fs(\021)c Fv(\033)3313 4036 y Fu(0)3384 4021 y Fv(<)g Fw(0)120 4141 y Fj(is)k(\(a.e.\))44 b(a)34 b(ne)-5 b(gative)34 b(c)-5 b(onstant.)120 4345 y(Pr)g(o)g(of.)48 b Fw(Let)d Fv(\025)58 b(=)-60 b Fs(2)829 4270 y Fo(S)912 4374 y Ft(n)p Fl(2)p Fn(Z)1051 4345 y Fw([)p Fs(\000)17 b Fw(min)o Fs(f)p Fv(\033)1440 4360 y Fu(0)1479 4345 y Fv(;)g Fw(0)p Fs(g)1622 4309 y Fu(2)1691 4345 y Fw(+)30 b(\()p Fv(n)g Fw(+)g Fv(k)s Fw(\))2121 4309 y Fu(2)2160 4345 y Fv(;)17 b Fw(\()p Fv(n)30 b Fw(+)g Fv(k)s Fw(\))2528 4309 y Fu(2)2567 4345 y Fw(\))44 b(\(i.e.,)j Fv(\025)g Fs(2)h Fr(R)54 b Fw(if)43 b Fv(\033)3375 4360 y Fu(0)3463 4345 y Fs(\025)120 4465 y Fw(0\).)69 b(T)-8 b(o)41 b(arriv)m(e)g(at)g(a)f(con)m (tradiction)g(w)m(e)i(assume)g(that)f Fv(\025)g Fw(is)g(an)g(eigen)m(v) -5 b(alue)40 b(of)h Fv(H)3309 4429 y Fu(\()p Ft(\033)r Fu(\))3410 4465 y Fw(\()p Fv(k)s Fw(\))120 4586 y(with)k(corresp)s (ondig)f(eigenfunction)h Fv(u)j Fs(6)p Fw(=)h(0.)80 b(By)46 b(Prop)s(osition)d(3.3.2,)k Fv(f)60 b Fw(:=)49 b Fv(u)p Fw(\()p Fv(:;)17 b Fw(0\))48 b Fs(2)120 4706 y(N)15 b Fw(\()p Fv(B)334 4670 y Fu(\()p Ft(\033)r Fu(\))435 4706 y Fw(\()p Fv(\025)22 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\)\))22 b Fs(n)g(f)p Fw(0)p Fs(g)32 b Fw(and)1292 4900 y(^)1270 4926 y Fv(f)1318 4941 y Ft(n)1393 4926 y Fw(=)c(0)194 b(if)27 b(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))2133 4885 y Fu(2)2200 4926 y Fs(\024)29 b Fv(\025:)120 5146 y Fw(Hence)34 b Fv(b)451 5110 y Fu(\(0\))546 5146 y Fw(\()p Fv(\025)22 b Fs(\006)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\)[)p Fv(f)11 b Fw(])28 b Fs(\025)g Fv(\015)5 b Fs(k)p Fv(f)11 b Fs(k)1441 5110 y Fu(2)1512 5146 y Fw(where)914 5377 y Fv(\015)32 b Fw(:=)c(min)o Fs(f)1341 5287 y Fo(p)p 1440 5287 527 4 v 1440 5377 a Fw(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))1748 5348 y Fu(2)1810 5377 y Fs(\000)h Fv(\025)k Fw(:)56 b Fv(n)27 b Fs(2)i Fr(Z)p Fv(;)2416 5351 y Fw(^)2394 5377 y Fv(f)2442 5392 y Ft(n)2517 5377 y Fs(6)p Fw(=)f(0)p Fs(g)p Fv(;)1781 5712 y Fw(37)p eop %%Page: 38 43 38 42 bop 120 407 a Fw(and)1132 527 y Fv(b)1173 486 y Fu(\()p Ft(\033)r Fu(\))1276 527 y Fw(\()p Fv(\025)22 b Fs(\006)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\)[)p Fv(f)11 b Fw(])28 b Fs(\025)g Fw(\()p Fv(\015)f Fw(+)22 b Fv(\033)2225 542 y Fu(0)2265 527 y Fw(\))p Fs(k)p Fv(f)11 b Fs(k)2462 486 y Fu(2)2500 527 y Fv(:)120 693 y Fw(Since)34 b Fv(b)417 657 y Fu(\()p Ft(\033)r Fu(\))519 693 y Fw(\()p Fv(\025)23 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\)[)p Fv(f)11 b Fw(])29 b(=)h(0)j(and)h Fv(f)41 b Fs(6)p Fw(=)30 b(0)j(it)g(follo)m (ws)g(that)g Fv(\015)28 b Fw(+)23 b Fv(\033)2620 708 y Fu(0)2690 693 y Fs(\024)30 b Fw(0.)48 b(Ho)m(w)m(ev)m(er,)36 b(from)120 814 y(the)k(c)m(hoice)g(of)f Fv(\025)g Fw(it)f(follo)m(ws)h (that)g Fv(\015)32 b Fw(+)26 b Fv(\033)1696 829 y Fu(0)1776 814 y Fv(>)39 b Fw(0,)i(so)e(w)m(e)i(arriv)m(e)e(at)g(a)h(con)m (tradiction)e(and)120 934 y(ha)m(v)m(e)c(pro)m(v)m(en)g(the)f(\014rst)g (part)f(of)g(the)h(Prop)s(osition.)120 1055 y(Let)43 b(us)g(pro)m(v)m(e)h(the)f(second)g(one.)74 b(F)-8 b(rom)41 b(Example)h(3.3.8)g(w)m(e)i(kno)m(w)f(that)g Fs(\000)p Fv(\033)3162 1018 y Fu(2)3158 1079 y(0)3231 1055 y Fw(+)28 b Fv(k)3389 1018 y Fu(2)3474 1055 y Fs(2)120 1175 y Fv(\033)175 1190 y Ft(p)215 1175 y Fw(\()p Fv(H)342 1139 y Fu(\()p Ft(\033)r Fu(\))443 1175 y Fw(\()p Fv(k)s Fw(\)\))36 b(if)f Fv(\033)k Fw(is)d(a)f(negativ)m(e)h(constan)m(t.)54 b(T)-8 b(o)36 b(pro)m(v)m(e)h(the)f(opp)s(osite)f(implication)d(let)j Fv(\033)120 1295 y Fw(b)s(e)h(an)g(arbitrary)f(function)g(b)s(ounded)i (from)d(b)s(elo)m(w)i(b)m(y)g Fv(\033)2306 1310 y Fu(0)2382 1295 y Fw(suc)m(h)h(that)f Fv(\025)d Fw(:=)g Fs(\000)p Fv(\033)3182 1259 y Fu(2)3178 1320 y(0)3247 1295 y Fw(+)24 b Fv(k)3401 1259 y Fu(2)3474 1295 y Fs(2)120 1416 y Fv(\033)175 1431 y Ft(p)215 1416 y Fw(\()p Fv(H)342 1380 y Fu(\()p Ft(\033)r Fu(\))443 1416 y Fw(\()p Fv(k)s Fw(\)\))33 b(with)f(corresp)s(ondig)g(eigenfunction)g Fv(u)p Fw(.)43 b(Then)953 1617 y Fv(\025)p Fs(k)p Fv(u)p Fs(k)1166 1576 y Fu(2)1232 1617 y Fw(=)28 b Fv(h)1392 1576 y Fu(\()p Ft(\033)r Fu(\))1493 1617 y Fw(\()p Fv(k)s Fw(\)[)p Fv(u)p Fw(])g Fs(\025)g Fv(h)1922 1576 y Fu(\()p Ft(\033)1989 1585 y Fm(0)2024 1576 y Fu(\))2056 1617 y Fw(\()p Fv(k)s Fw(\)[)p Fv(u)p Fw(])f Fs(\025)i Fv(\025)p Fs(k)p Fv(u)p Fs(k)2642 1576 y Fu(2)2680 1617 y Fv(;)120 1818 y Fw(so)k(that)869 1829 y Fo(Z)969 1855 y Ft(\031)924 2054 y Fl(\000)p Ft(\031)1043 1964 y Fv(\033)t Fw(\()p Fv(x)1195 1979 y Fu(1)1235 1964 y Fw(\))p Fs(j)p Fv(u)p Fw(\()p Fv(x)1450 1979 y Fu(1)1489 1964 y Fv(;)17 b Fw(0\))p Fs(j)1648 1923 y Fu(2)1686 1964 y Fv(dx)1792 1979 y Fu(1)1860 1964 y Fw(=)1963 1829 y Fo(Z)2063 1855 y Ft(\031)2018 2054 y Fl(\000)p Ft(\031)2137 1964 y Fv(\033)2192 1979 y Fu(0)2232 1964 y Fs(j)p Fv(u)p Fw(\()p Fv(x)2409 1979 y Fu(1)2448 1964 y Fv(;)g Fw(0\))p Fs(j)2607 1923 y Fu(2)2645 1964 y Fv(dx)2751 1979 y Fu(1)3215 1964 y Fw(\(3.3.14\))120 2184 y(and)980 2346 y Fv(h)1036 2305 y Fu(\()p Ft(\033)1103 2314 y Fm(0)1139 2305 y Fu(\))1170 2346 y Fw(\()p Fv(k)s Fw(\)[)p Fv(u)p Fw(])28 b(=)f(\()131 b(inf)1579 2424 y Fu(0)p Fl(6)p Fu(=)p Ft(v)r Fl(2)1771 2407 y Fu(~)1752 2424 y Ft(H)1815 2405 y Fm(1)1851 2424 y Fu(\(\005\))1985 2279 y Fv(h)2041 2242 y Fu(\()p Ft(\033)2108 2251 y Fm(0)2144 2242 y Fu(\))2175 2279 y Fw(\()p Fv(k)s Fw(\)[)p Fv(v)t Fw(])p 1985 2323 425 4 v 2103 2414 a Fs(k)p Fv(v)t Fs(k)2254 2386 y Fu(2)2420 2346 y Fw(\))p Fs(k)p Fv(u)p Fs(k)2614 2305 y Fu(2)2653 2346 y Fv(:)120 2591 y Fw(It)46 b(follo)m(ws)e(from)g(general)h(principles)f(that)h Fv(u)50 b Fs(2)g(D)s Fw(\()p Fv(H)2269 2554 y Fu(\()p Ft(\033)2336 2563 y Fm(0)2370 2554 y Fu(\))2402 2591 y Fw(\()p Fv(k)s Fw(\)\))45 b(and)g Fv(H)2906 2554 y Fu(\()p Ft(\033)2973 2563 y Fm(0)3008 2554 y Fu(\))3040 2591 y Fw(\()p Fv(k)s Fw(\))p Fv(u)k Fw(=)g Fv(\025u)p Fw(.)120 2711 y(F)-8 b(rom)41 b(the)i(explicit)e(form)m(ulae)f(for)i (eigenfunctions)g(of)g Fv(H)2344 2675 y Fu(\()p Ft(\033)2411 2684 y Fm(0)2446 2675 y Fu(\))2478 2711 y Fw(\()p Fv(k)s Fw(\))g(w)m(e)h(kno)m(w)h(that)e Fv(u)p Fw(\()p Fv(:;)17 b Fw(0\))120 2831 y(can)28 b(ha)m(v)m(e)g(at)f(most)g(a)g(\014nite)g(n) m(um)m(b)s(er)h(of)f(zero)s(es,)i(so)f Fv(\033)k Fs(\021)c Fv(\033)2312 2846 y Fu(0)2379 2831 y Fw(b)m(y)i(\(3.3.14\))o(.)42 b(This)27 b(completes)120 2952 y(the)33 b(pro)s(of.)p 3469 2952 4 66 v 3473 2889 59 4 v 3473 2952 V 3531 2952 4 66 v 266 3141 a(Com)m(bining)e(this)h(result)h(with)f(Theorems)h (3.1.3,)g(3.2.5)f(w)m(e)h(obtain)f(the)h(imp)s(ortan)m(t)120 3329 y Fp(Theorem)38 b(3.3.10.)k Fj(Assume)36 b(that)g Fv(\033)j Fj(is)c(non-ne)-5 b(gative)34 b(and)h(let)h Fv(k)c Fs(2)d Fw([)p Fs(\000)2910 3289 y Fu(1)p 2910 3306 36 4 v 2910 3363 a(2)2956 3329 y Fv(;)3009 3289 y Fu(1)p 3009 3306 V 3009 3363 a(2)3055 3329 y Fw(])p Fj(.)46 b(Then)35 b(the)120 3449 y(op)-5 b(er)g(ator)35 b Fv(H)593 3413 y Fu(\()p Ft(\033)r Fu(\))694 3449 y Fw(\()p Fv(k)s Fw(\))g Fj(is)f(absolutely)h(c)-5 b(ontinuous.)120 3636 y Fp(Prop)s(osition)53 b(3.3.11.)d Fj(Assume)f(that)g Fv(\033)k Fj(is)c(a)g(trigonometric)f(p)-5 b(olynomial)48 b(of)h(de)-5 b(gr)g(e)g(e)120 3757 y Fv(N)38 b(>)28 b Fw(0)34 b Fj(and)h(let)g Fv(k)30 b Fs(2)f Fw([)p Fs(\000)1040 3717 y Fu(1)p 1040 3734 V 1040 3791 a(2)1085 3757 y Fv(;)1139 3717 y Fu(1)p 1139 3734 V 1139 3791 a(2)1184 3757 y Fw(])p Fj(.)45 b(Then)34 b(with)h Fv(\033)1807 3772 y Fu(0)1874 3757 y Fw(:=)28 b(essinf)2243 3772 y Ft(x)2283 3781 y Fm(1)2318 3772 y Fl(2)p Fn(T)2430 3757 y Fv(\033)t Fw(\()p Fv(x)2582 3772 y Fu(1)2622 3757 y Fw(\))1000 3971 y Fv(\033)1055 3986 y Ft(p)1112 3890 y Fo(\000)1158 3971 y Fv(H)1247 3930 y Fu(\()p Ft(\033)r Fu(\))1348 3971 y Fw(\()p Fv(k)s Fw(\))1478 3890 y Fo(\001)1551 3971 y Fs(\032)1656 3890 y Fo(\000)1702 3971 y Fs(\000)p Fv(\033)1838 3930 y Fu(2)1834 3996 y(0)1900 3971 y Fw(+)22 b Fv(k)2052 3930 y Fu(2)2092 3971 y Fv(;)17 b Fw(\()p Fv(N)32 b Fs(\000)23 b(j)p Fv(k)s Fs(j)p Fw(\))2532 3930 y Fu(2)2570 3890 y Fo(\001)2633 3971 y Fv(:)120 4172 y Fj(Pr)-5 b(o)g(of.)48 b Fw(The)31 b(lo)m(w)m(er)e(b)s(ound)h(follo)m(ws)e(from)g(Prop)s(osition)f(3.3.9.) 42 b(T)-8 b(o)29 b(arriv)m(e)h(at)f(a)g(con)m(tradic-)120 4292 y(tion)i(w)m(e)i(assume)f(that)f Fv(\025)d Fs(\025)g Fw(\()p Fv(N)j Fs(\000)21 b(j)p Fv(k)s Fs(j)p Fw(\))1592 4256 y Fu(2)1663 4292 y Fw(is)31 b(an)h(eigen)m(v)-5 b(alue)31 b(of)g Fv(H)2559 4256 y Fu(\()p Ft(\033)r Fu(\))2660 4292 y Fw(\()p Fv(k)s Fw(\))h(with)g(corresp)s(ond-)120 4413 y(ing)h(eigenfunction)g Fv(u)c Fs(6)p Fw(=)g(0.)46 b(Then)35 b(according)e(to)g(Prop)s(osition)f(3.3.2)h(and)h(Lemma)e (3.3.3,)120 4533 y Fv(f)38 b Fw(:=)28 b Fv(u)p Fw(\()p Fv(:;)17 b Fw(0\))27 b Fs(6)p Fw(=)g(0)33 b(and)1292 4627 y(^)1270 4654 y Fv(f)1318 4669 y Ft(n)1393 4654 y Fw(=)28 b(0)194 b(if)27 b(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))2133 4612 y Fu(2)2200 4654 y Fs(\024)29 b Fv(\025;)825 b Fw(\(3.3.15\))255 4780 y Fo(p)p 355 4780 527 4 v 91 x Fw(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))663 4842 y Fu(2)724 4871 y Fs(\000)h Fv(\025)919 4844 y Fw(^)898 4871 y Fv(f)946 4886 y Ft(n)1015 4871 y Fw(+)1194 4803 y(1)p 1123 4848 191 4 v 1123 4868 a Fs(p)p 1206 4868 108 4 v 82 x Fw(2)p Fv(\031)1340 4735 y Fo(Z)1440 4761 y Ft(\031)1395 4960 y Fl(\000)p Ft(\031)1514 4871 y Fv(\033)t Fw(\()p Fv(x)1666 4886 y Fu(1)1706 4871 y Fw(\))p Fv(f)11 b Fw(\()p Fv(x)1896 4886 y Fu(1)1935 4871 y Fw(\))p Fv(e)2018 4829 y Fl(\000)p Ft(inx)2180 4838 y Fm(1)2235 4871 y Fv(dx)2341 4886 y Fu(1)2408 4871 y Fw(=)28 b(0)195 b(if)26 b(\()p Fv(n)d Fw(+)f Fv(k)s Fw(\))3149 4829 y Fu(2)3216 4871 y Fs(\025)28 b Fv(\025:)120 5091 y Fw(W)-8 b(e)33 b(rewrite)g(the)g(second)g (equation)g(as)320 5289 y Fo(p)p 419 5289 527 4 v 419 5379 a Fw(\()p Fv(n)23 b Fw(+)f Fv(k)s Fw(\))728 5350 y Fu(2)789 5379 y Fs(\000)h Fv(\025)983 5353 y Fw(^)962 5379 y Fv(f)1010 5394 y Ft(n)1079 5379 y Fw(+)1258 5312 y(1)p 1187 5356 191 4 v 1187 5376 a Fs(p)p 1270 5376 108 4 v 82 x Fw(2)p Fv(\031)1491 5254 y Ft(N)1450 5284 y Fo(X)1405 5496 y Ft(m)p Fu(=)p Fl(\000)p Ft(N)1662 5379 y Fw(^)-54 b Fv(\033)1712 5394 y Ft(m)1800 5353 y Fw(^)1779 5379 y Fv(f)1827 5394 y Ft(n)p Fl(\000)p Ft(m)2019 5379 y Fw(=)27 b(0)195 b(if)27 b(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))2759 5338 y Fu(2)2826 5379 y Fs(\025)28 b Fv(\025:)200 b Fw(\(3.3.16\))1781 5712 y(38)p eop %%Page: 39 44 39 43 bop 120 407 a Fw(It)33 b(follo)m(ws)e(from)408 594 y Fv(])p Fs(f)p Fv(n)d Fs(2)g Fr(Z)d Fw(:)55 b(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))1160 552 y Fu(2)1228 594 y Fs(\024)28 b Fv(\025)p Fs(g)f(\025)h Fv(])p Fs(f)p Fv(n)g Fs(2)g Fr(Z)d Fw(:)56 b(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))2325 552 y Fu(2)2392 594 y Fs(\024)28 b Fw(\()p Fv(N)33 b Fs(\000)22 b(j)p Fv(k)s Fs(j)p Fw(\))2893 552 y Fu(2)2932 594 y Fs(g)28 b(\025)g Fw(2)p Fv(N)120 780 y Fw(and)38 b(\(3.3.15\))o(,)i(that)945 754 y(^)924 780 y Fv(f)972 795 y Ft(n)1056 780 y Fw(=)d(0)h(for)g(at)f(least)h(2)p Fv(N)49 b Fw(consecutiv)m(e)39 b Fv(n)p Fw(.)61 b(Using)43 b(^)-54 b Fv(\033)2946 795 y Ft(N)3050 780 y Fw(=)p 3163 699 178 4 v 42 w(^)g Fv(\033)3218 795 y Fl(\000)p Ft(N)3378 780 y Fs(6)p Fw(=)37 b(0)120 901 y(it)48 b(is)g(not)h(hard)f(to)h(see)h (from)d(\(3.3.16\))h(that)1960 875 y(^)1939 901 y Fv(f)1987 916 y Ft(n)2089 901 y Fw(=)55 b(0)48 b(for)h(all)d Fv(n)p Fw(.)93 b(Hence)50 b Fv(f)66 b Fw(=)55 b(0,)d(a)120 1021 y(con)m(tradiction.)p 3469 1021 4 66 v 3473 959 59 4 v 3473 1021 V 3531 1021 4 66 v 120 1207 a Fj(R)-5 b(emark)38 b Fw(3.3.12)p Fj(.)e Fw(The)26 b(same)g(idea)e(allo)m(ws)h(one)h(to)f (estimate)f(the)i(m)m(ultiplicit)m(y)c(of)j(an)h(eigen-)120 1327 y(v)-5 b(alue)44 b Fv(\025)49 b Fs(2)g Fw([\()p Fv(n)23 b Fs(\000)f Fw(1)g(+)g Fs(j)p Fv(k)s Fs(j)p Fw(\))1167 1291 y Fu(2)1206 1327 y Fv(;)17 b Fw(\()p Fv(n)22 b Fs(\000)h(j)p Fv(k)s Fs(j)p Fw(\))1616 1291 y Fu(2)1654 1327 y Fw(\))45 b(or)g Fv(\025)k Fs(2)g Fw([\()p Fv(n)22 b Fs(\000)h(j)p Fv(k)s Fs(j)p Fw(\))2483 1291 y Fu(2)2522 1327 y Fv(;)17 b Fw(\()p Fv(n)22 b Fw(+)g Fs(j)p Fv(k)s Fs(j)p Fw(\))2930 1291 y Fu(2)2968 1327 y Fw(\))45 b(in)g(terms)f(of)120 1448 y Fv(n)53 b Fs(2)g Fr(N)9 b Fw(.)94 b(It)47 b(is)g(necessary)i(to) e(study)i(the)e(n)m(um)m(b)s(er)h(of)f(linear)e(indep)s(enden)m(t)k (solutions)120 1568 y(\()179 1542 y(^)158 1568 y Fv(f)206 1583 y Ft(n)253 1568 y Fw(\))291 1583 y Ft(n)p Fl(2)p Fn(Z)463 1568 y Fs(2)34 b Fr(C)629 1532 y Fn(Z)720 1568 y Fw(\(without)i(requiring)f(summabilit)m(y\))e(of)43 b(\(3.3.16\))35 b(sub)5 b(jected)39 b(to)d(the)g(con-)120 1688 y(strain)m(t)c(\(3.3.15\))o(.)44 b(W)-8 b(e)33 b(do)f(not)h(go)f (in)m(to)g(details.)120 1863 y Fp(Example)k(3.3.13.)43 b Fw(Let)32 b Fv(\013)q(;)44 b(\014)6 b(;)44 b(\015)33 b Fs(2)28 b Fr(R)5 b Fw(,)39 b Fv(\014)1753 1827 y Fu(2)1814 1863 y Fw(+)22 b Fv(\015)1968 1827 y Fu(2)2035 1863 y Fs(6)p Fw(=)28 b(0)k(and)839 2050 y Fv(\033)t Fw(\()p Fv(x)991 2065 y Fu(1)1031 2050 y Fw(\))c(:=)f Fv(\013)c Fw(+)f Fv(\014)g Fw(sin\()p Fv(x)1700 2065 y Fu(1)1740 2050 y Fw(\))g(+)g Fv(\015)g Fw(cos\()p Fv(x)2194 2065 y Fu(1)2234 2050 y Fw(\))p Fv(;)212 b(x)2566 2065 y Fu(1)2633 2050 y Fs(2)28 b Fr(T)p Fv(:)120 2237 y Fw(Note)43 b(that)f(min)750 2252 y Ft(x)790 2261 y Fm(1)824 2252 y Fl(2)p Fn(T)953 2237 y Fv(\033)t Fw(\()p Fv(x)1105 2252 y Fu(1)1144 2237 y Fw(\))j(=)f Fv(\013)30 b Fs(\000)1545 2151 y Fo(p)p 1645 2151 316 4 v 86 x Fv(\014)1706 2208 y Fu(2)1767 2237 y Fw(+)22 b Fv(\015)1921 2208 y Fu(2)1961 2237 y Fw(.)73 b(Com)m(bining)41 b(Prop)s(ositions)g(3.3.9)h(and)120 2357 y(3.3.11)32 b(w)m(e)h(obtain)167 2526 y Fv(\033)222 2541 y Ft(p)279 2446 y Fo(\000)324 2526 y Fv(H)413 2490 y Fu(\()p Ft(\033)r Fu(\))515 2526 y Fw(\()p Fv(k)s Fw(\))645 2446 y Fo(\001)718 2526 y Fw(=)28 b Fs(;)1700 b Fw(if)43 b Fv(\013)29 b Fs(\025)2869 2440 y Fo(p)p 2969 2440 V 86 x Fv(\014)3030 2498 y Fu(2)3091 2526 y Fw(+)22 b Fv(\015)3245 2498 y Fu(2)3312 2526 y Fs(\025)28 b Fw(0)p Fv(;)167 2707 y(\033)222 2722 y Ft(p)279 2626 y Fo(\000)324 2707 y Fv(H)413 2671 y Fu(\()p Ft(\033)r Fu(\))515 2707 y Fw(\()p Fv(k)s Fw(\))645 2626 y Fo(\001)718 2707 y Fs(\032)823 2567 y Fo(\022)897 2707 y Fs(\000)991 2596 y Fo(\020)1050 2707 y Fv(\013)23 b Fs(\000)1235 2621 y Fo(p)p 1334 2621 V 1334 2707 a Fv(\014)1395 2678 y Fu(2)1456 2707 y Fw(+)f Fv(\015)1610 2678 y Fu(2)1650 2596 y Fo(\021)1709 2619 y Fu(2)1771 2707 y Fw(+)g Fv(k)1923 2671 y Fu(2)1962 2707 y Fv(;)17 b Fw(\(1)22 b Fs(\000)h(j)p Fv(k)s Fs(j)p Fw(\))2363 2671 y Fu(2)2401 2567 y Fo(\023)2572 2707 y Fw(if)43 b Fv(\013)29 b(<)2867 2621 y Fo(p)p 2967 2621 V 86 x Fv(\014)3028 2678 y Fu(2)3089 2707 y Fw(+)22 b Fv(\015)3243 2678 y Fu(2)3310 2707 y Fs(6)p Fw(=)28 b(0)p Fv(:)120 2962 y Fw(In)33 b(particular,)e(the)i(op)s(erators)f Fv(H)1407 2926 y Fu(\()p Ft(\033)r Fu(\))1508 2962 y Fw(\()p Fs(\006)1633 2923 y Fu(1)p 1633 2939 36 4 v 1633 2997 a(2)1679 2962 y Fw(\))h(do)f(not)g(ha)m(v)m(e)i(em)m(b)s(edded)g (eigen)m(v)-5 b(alues.)266 3137 y(In)40 b(Prop)s(osition)d(3.3.9)i(w)m (e)h(pro)m(v)m(ed)h(that)e Fv(H)1947 3101 y Fu(\()p Ft(\033)r Fu(\))2048 3137 y Fw(\()p Fv(k)s Fw(\))g(has)h(no)f(eigen)m(v)-5 b(alues)39 b(if)f Fv(\033)44 b Fw(is)38 b(non-)120 3258 y(negativ)m(e.)49 b(The)35 b(follo)m(wing)c(t)m(w)m(o)k(Prop)s (ositions)e(are)h(in)f(a)h(certain)g(sense)i(a)e(con)m(v)m(erse:)49 b(It)34 b(is)120 3378 y(sho)m(wn)24 b(that)e Fv(H)696 3342 y Fu(\()p Ft(\033)r Fu(\))797 3378 y Fw(\()p Fv(k)s Fw(\))g(has)h(an)f(eigen)m(v)-5 b(alue)21 b(if)g Fv(\033)27 b Fw(is)21 b("su\016cien)m(tly)i(negativ)m(e".)40 b(In)23 b(Prop)s(osition)120 3498 y(3.3.14)32 b(w)m(e)i(understand)h(this)e(in) f(a)h(global)d(sense)35 b(\(concerning)e(the)h(mean)e(v)-5 b(alue)33 b(of)g Fv(\033)t Fw(\),)g(in)120 3619 y(Prop)s(osition)26 b(3.3.15)h(w)m(e)h(understand)h(this)e(in)g(a)g(lo)s(cal)e(sense)30 b(\(concerning)d(negativ)m(e)h(p)s(eaks)120 3739 y(of)k Fv(\033)t Fw(\).)120 3914 y Fp(Prop)s(osition)k(3.3.14.)42 b Fj(Assume)34 b(that)1236 4146 y Fw(^)-55 b Fv(\033)1285 4161 y Fu(0)1353 4146 y Fw(=)1537 4079 y(1)p 1466 4123 191 4 v 1466 4143 a Fs(p)p 1550 4143 108 4 v 1550 4226 a Fw(2)p Fv(\031)1684 4011 y Fo(Z)1783 4037 y Ft(\031)1739 4236 y Fl(\000)p Ft(\031)1858 4146 y Fv(\033)t Fw(\()p Fv(x)2010 4161 y Fu(1)2049 4146 y Fw(\))17 b Fv(dx)2210 4161 y Fu(1)2277 4146 y Fv(<)28 b Fw(0)120 4404 y Fj(and)34 b(let)h Fv(k)c Fs(2)d Fw([)p Fs(\000)736 4365 y Fu(1)p 736 4381 36 4 v 736 4438 a(2)782 4404 y Fv(;)836 4365 y Fu(1)p 836 4381 V 836 4438 a(2)881 4404 y Fw(])p Fj(.)45 b(Then)34 b Fv(H)1326 4368 y Fu(\()p Ft(\033)r Fu(\))1427 4404 y Fw(\()p Fv(k)s Fw(\))h Fj(has)f(an)h(eigenvalue)e(smal)5 b(ler)34 b(or)h(e)-5 b(qual)35 b Fv(k)3148 4368 y Fu(2)3209 4404 y Fs(\000)3340 4365 y Fu(1)p 3319 4381 79 4 v 3319 4438 a(2)p Ft(\031)3412 4404 y Fw(^)-54 b Fv(\033)3466 4368 y Fu(2)3462 4428 y(0)3506 4404 y Fj(.)120 4592 y(Pr)-5 b(o)g(of.)48 b Fw(Consider)33 b(the)g(trial)e(function)h Fv(u)27 b Fs(2)1796 4567 y Fw(~)1771 4592 y Fv(H)1860 4556 y Fu(1)1899 4592 y Fw(\(\005\),)1214 4794 y Fv(u)p Fw(\()p Fv(x)p Fw(\))g(:=)h Fv(e)1608 4753 y Fu(^)-39 b Ft(\033)1644 4762 y Fm(0)1679 4753 y Ft(x)1719 4762 y Fm(2)1753 4753 y Ft(=)1788 4698 y Fl(p)p 1848 4698 79 3 v 1848 4753 a Fu(2)p Ft(\031)1930 4794 y Fv(;)212 b(x)28 b Fs(2)g Fw(\005)p Fv(:)120 4981 y Fw(It)33 b(is)f(easy)h(to)g (see)g(that)1196 5140 y Fv(h)1252 5099 y Fu(\()p Ft(\033)r Fu(\))1354 5140 y Fw(\()p Fv(k)s Fw(\)[)p Fv(u)p Fw(])27 b(=)h(\()p Fv(k)1817 5099 y Fu(2)1879 5140 y Fs(\000)2018 5073 y Fw(1)p 1988 5117 108 4 v 1988 5208 a(2)p Fv(\031)2111 5140 y Fw(^)-54 b Fv(\033)2165 5099 y Fu(2)2161 5165 y(0)2204 5140 y Fw(\))p Fs(k)p Fv(u)p Fs(k)2398 5099 y Fu(2)2437 5140 y Fv(:)120 5342 y Fw(Since)34 b(min)15 b Fv(\033)610 5357 y Ft(ess)729 5262 y Fo(\000)775 5342 y Fv(H)864 5306 y Fu(\()p Ft(\033)r Fu(\))965 5342 y Fw(\()p Fv(k)s Fw(\))1095 5262 y Fo(\001)1170 5342 y Fw(=)30 b Fv(k)1330 5306 y Fu(2)1370 5342 y Fw(,)k(the)g(Prop)s (osition)e(follo)m(ws)g(from)h(the)h(v)-5 b(ariational)30 b(prin-)120 5463 y(ciple.)p 3469 5463 4 66 v 3473 5400 59 4 v 3473 5463 V 3531 5463 4 66 v 1781 5712 a(39)p eop %%Page: 40 45 40 44 bop 120 407 a Fp(Prop)s(osition)30 b(3.3.15.)38 b Fj(L)-5 b(et)31 b Fv(k)g Fs(2)d Fw([)p Fs(\000)1529 368 y Fu(1)p 1529 384 36 4 v 1529 441 a(2)1575 407 y Fv(;)1629 368 y Fu(1)p 1629 384 V 1629 441 a(2)1674 407 y Fw(])i Fj(and)g(assume)g(that)h(ther)-5 b(e)30 b(is)h(a)f(non-empty)g (op)-5 b(en)120 527 y(c)g(onne)g(cte)g(d)34 b(subset)h Fw(\003)27 b Fs(\032)h Fr(T)39 b Fj(of)34 b(the)h(torus)g(and)g(a)f Fv(\016)e(>)c Fw(0)34 b Fj(such)h(that)959 794 y Fv(\033)t Fw(\()p Fv(x)1111 809 y Fu(1)1150 794 y Fw(\))28 b Fs(\024)g(\000)1398 640 y Fo(r)p 1498 640 649 4 v 43 x(\020)1590 754 y Ft(\031)p 1568 771 89 4 v 1568 828 a Fl(j)p Fu(\003)p Fl(j)1666 683 y Fo(\021)1725 705 y Fu(2)1787 794 y Fs(\000)22 b Fv(k)1940 765 y Fu(2)2002 794 y Fw(+)g Fv(\016)t(;)216 b(x)2445 809 y Fu(1)2513 794 y Fs(2)28 b Fw(\003)p Fv(:)120 1047 y Fj(Then)34 b Fv(H)463 1011 y Fu(\()p Ft(\033)r Fu(\))564 1047 y Fw(\()p Fv(k)s Fw(\))h Fj(has)f(an)h(eigenvalue)f (smal)5 b(ler)34 b(than)g Fv(k)2134 1011 y Fu(2)2196 1047 y Fs(\000)23 b Fv(\016)t Fj(.)120 1232 y(Pr)-5 b(o)g(of.)48 b Fw(Iden)m(tifying)34 b Fr(T)k Fw(with)c(\()p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(])34 b(and)g(shifting)f(the)i Fv(x)2351 1247 y Fu(1)2390 1232 y Fw(-co)s(ordinate)e(if)g(necessary)k (w)m(e)120 1352 y(ma)m(y)h(assume)g(that)g(\003)f(=)f(\()p Fs(\000)p Fv(\031)t(;)17 b(c)p Fw(\))38 b(for)g(some)f Fs(\000)p Fv(\031)42 b(<)36 b(c)h Fs(\024)h Fv(\031)t Fw(.)59 b(\(The)39 b(case)g(\003)d(=)h Fr(T)k Fw(can)e(b)s(e)120 1472 y(included)32 b(in)g(the)h(case)g Fv(c)28 b Fw(=)g Fv(\031)t Fw(.\))43 b(De\014ne)802 1790 y(~)-54 b Fv(\033)t Fw(\()p Fv(x)949 1805 y Fu(1)989 1790 y Fw(\))27 b(:=)1185 1586 y Fo(8)1185 1675 y(<)1185 1855 y(:)1315 1667 y Fv(\033)t Fw(\()p Fv(x)1467 1682 y Fu(1)1507 1667 y Fw(\))694 b(if)43 b Fv(x)2395 1682 y Fu(1)2462 1667 y Fs(2)28 b Fr(T)e Fs(n)c Fw(\003)p Fv(;)1315 1865 y Fs(\000)1392 1711 y Fo(r)p 1492 1711 649 4 v 43 x(\020)1584 1825 y Ft(\031)p 1562 1842 89 4 v 1562 1899 a Fl(j)p Fu(\003)p Fl(j)1660 1754 y Fo(\021)1719 1776 y Fu(2)1781 1865 y Fs(\000)g Fv(k)1934 1836 y Fu(2)1996 1865 y Fw(+)g Fv(\016)102 b Fw(if)43 b Fv(x)2395 1880 y Fu(1)2462 1865 y Fs(2)28 b Fw(\003)p Fv(;)120 2118 y Fw(and)f(denote)h(b)m(y)h Fv(g)795 2082 y Fu(\()t(~)-39 b Ft(\033)r Fu(\))896 2118 y Fw(\()p Fv(k)s Fw(\))27 b(the)h(closed)f(and)h(lo)m(w)m(er)f(semib)s (ounded)g(quadratic)g(form)f(obtained)120 2239 y(as)j(the)g (restriction)f(of)h Fv(h)1027 2203 y Fu(\()p Ft(\033)r Fu(\))1129 2239 y Fw(\()p Fv(k)s Fw(\))f(to)h(the)g(set)h Fs(f)p Fv(u)d Fs(2)h Fv(H)2032 2203 y Fu(1)2071 2239 y Fw(\(\005\))g(:)55 b Fv(u)p Fw(\()p Fs(\000)p Fv(\031)t(;)17 b(:)p Fw(\))27 b(=)h Fv(u)p Fw(\()p Fv(c;)17 b(:)p Fw(\))27 b(=)g Fv(u)p Fw(\()p Fv(\031)t(;)17 b(:)p Fw(\))27 b(=)120 2359 y(0)p Fs(g)p Fw(.)43 b(The)34 b(corresp)s(onding)e(self-adjoin)m (t)f(op)s(erator)h Fv(G)2088 2323 y Fu(\()t(~)-39 b Ft(\033)s Fu(\))2189 2359 y Fw(\()p Fv(k)s Fw(\))33 b(satis\014es)1436 2557 y Fv(H)1525 2516 y Fu(\()p Ft(\033)r Fu(\))1626 2557 y Fw(\()p Fv(k)s Fw(\))27 b Fs(\024)i Fv(G)1966 2516 y Fu(\()t(~)-39 b Ft(\033)s Fu(\))2067 2557 y Fw(\()p Fv(k)s Fw(\))p Fv(:)991 b Fw(\(3.3.17\))120 2755 y(Because)40 b(of)d(the)i(Diric)m(hlet)c(condition)i(at)h Fv(x)1810 2770 y Fu(1)1886 2755 y Fw(=)f Fv(c)h Fw(the)h(op)s(erator)e Fv(G)2728 2719 y Fu(\()t(~)-39 b Ft(\033)s Fu(\))2830 2755 y Fw(\()p Fv(k)s Fw(\))37 b(admits)g(a)h(de-)120 2889 y(comp)s(osition)44 b Fv(G)756 2853 y Fu(\()t(~)-39 b Ft(\033)s Fu(\))858 2889 y Fw(\()p Fv(k)s Fw(\))51 b(=)g Fv(G)1243 2838 y Fu(\()t(~)-39 b Ft(\033)s Fu(\))1243 2913 y(1)1345 2889 y Fw(\()p Fv(k)s Fw(\))31 b Fs(\010)h Fv(G)1692 2838 y Fu(\()t(~)-39 b Ft(\033)s Fu(\))1692 2913 y(2)1794 2889 y Fw(\()p Fv(k)s Fw(\))46 b(according)g(to)g Fv(L)2622 2904 y Fu(2)2662 2889 y Fw(\(\005\))51 b(=)g Fv(L)3055 2904 y Fu(2)3095 2889 y Fw(\(\()p Fs(\000)p Fv(\031)t(;)17 b(c)p Fw(\))32 b Fs(\002)120 3023 y Fr(R)186 3038 y Fu(+)251 3023 y Fw(\))g Fs(\010)h Fv(L)497 3038 y Fu(2)536 3023 y Fw(\(\()p Fv(c;)17 b(\031)t Fw(\))32 b Fs(\002)h Fr(R)1003 3038 y Fu(+)1068 3023 y Fw(\).)87 b(The)48 b(part)f Fv(G)1738 2972 y Fu(\()t(~)-39 b Ft(\033)s Fu(\))1738 3047 y(1)1840 3023 y Fw(\()p Fv(k)s Fw(\))47 b(on)g Fv(L)2233 3038 y Fu(2)2273 3023 y Fw(\(\()p Fs(\000)p Fv(\031)t(;)17 b(c)p Fw(\))32 b Fs(\002)g Fr(R)2816 3038 y Fu(+)2881 3023 y Fw(\))48 b(dep)s(ends)g(only)120 3217 y(on)58 b(~)-54 b Fv(\033)t Fs(j)363 3232 y Fu(\003)416 3217 y Fw(,)58 b(i.e.)104 b(on)53 b(the)h(constan)m(t)g Fs(\000)1566 3064 y Fo(r)p 1666 3064 649 4 v 43 x(\020)1758 3178 y Ft(\031)p 1735 3194 89 4 v 1735 3252 a Fl(j)p Fu(\003)p Fl(j)1833 3107 y Fo(\021)1893 3129 y Fu(2)1955 3217 y Fs(\000)22 b Fv(k)2108 3188 y Fu(2)2170 3217 y Fw(+)g Fv(\016)t Fw(.)105 b(Using)52 b(the)i(decomp)s(osition)120 3382 y Fv(L)186 3397 y Fu(2)226 3382 y Fw(\(\()p Fs(\000)p Fv(\031)t(;)17 b(c)p Fw(\))22 b Fs(\002)g Fr(R)749 3397 y Fu(+)814 3382 y Fw(\))28 b(=)984 3307 y Fo(P)1089 3411 y Ft(n)p Fl(2)p Fn(N)1247 3307 y Fo(L)1358 3382 y Fs(f)p Fv(b)1449 3397 y Ft(n)1496 3382 y Fw(\()p Fv(:;)17 b(k)s Fw(\))p Fs(g)22 b(\012)h Fv(L)1935 3397 y Fu(2)1974 3382 y Fw(\()p Fr(R)2078 3397 y Fu(+)2143 3382 y Fw(\))33 b(where)754 3620 y Fv(b)795 3635 y Ft(n)842 3620 y Fw(\()p Fv(x)935 3635 y Fu(1)975 3620 y Fv(;)17 b(k)s Fw(\))28 b(:=)1269 3509 y Fo(q)p 1369 3509 109 4 v 1405 3580 a Fu(2)p 1379 3597 89 4 v 1379 3654 a Fl(j)p Fu(\003)p Fl(j)1494 3620 y Fv(e)1539 3579 y Fl(\000)p Ft(ik)r(x)1697 3588 y Fm(1)1751 3620 y Fw(sin)1888 3509 y Fo(\020)1957 3572 y Ft(n\031)r Fu(\()p Ft(x)2110 3581 y Fm(1)2145 3572 y Fu(+)p Ft(\031)r Fu(\))p 1957 3597 313 4 v 2069 3654 a Fl(j)p Fu(\003)p Fl(j)2280 3509 y Fo(\021)2356 3620 y Fv(;)212 b(x)2650 3635 y Fu(1)2717 3620 y Fs(2)28 b Fw(\003)p Fv(;)120 3871 y Fw(and)40 b(pro)s(ceeding)f(as)h(in)f (Section)g(2.1)h(\(see)g(also)f(Example)g(3.3.8\))g(w)m(e)i(see)g(that) e Fv(G)3281 3820 y Fu(\()t(~)-39 b Ft(\033)s Fu(\))3281 3895 y(1)3383 3871 y Fw(\()p Fv(k)s Fw(\),)120 3991 y(and)48 b(th)m(us)h(also)e Fv(G)843 3955 y Fu(\()t(~)-39 b Ft(\033)s Fu(\))945 3991 y Fw(\()p Fv(k)s Fw(\),)52 b(has)c(an)g(eigen)m(v)-5 b(alue)47 b Fv(k)2029 3955 y Fu(2)2101 3991 y Fs(\000)34 b Fv(\016)t Fw(.)89 b(Since)48 b(min)15 b Fv(\033)2879 4006 y Ft(ess)2999 3911 y Fo(\000)3044 3991 y Fv(H)3133 3955 y Fu(\()p Ft(\033)r Fu(\))3235 3991 y Fw(\()p Fv(k)s Fw(\))3365 3911 y Fo(\001)3464 3991 y Fw(=)120 4121 y Fv(k)174 4084 y Fu(2)214 4121 y Fw(,)47 b(\(3.3.17\))c(together)i(with) f(the)h(v)-5 b(ariational)41 b(principle)i(imply)f(that)j Fv(H)2976 4084 y Fu(\()p Ft(\033)r Fu(\))3077 4121 y Fw(\()p Fv(k)s Fw(\))f(has)h(an)120 4241 y(eigen)m(v)-5 b(alue)32 b(smaller)f(or)h(equal)g Fv(k)1353 4205 y Fu(2)1415 4241 y Fs(\000)23 b Fv(\016)t Fw(.)120 4361 y(T)-8 b(o)43 b(complete)f(the)h(pro)s(of)f(w)m(e)i(w)m(an)m(t)f(to)g(arriv)m(e)f(at) h(a)f(con)m(tradiction)g(b)m(y)h(assuming)f(that)120 4482 y Fv(k)174 4445 y Fu(2)225 4482 y Fs(\000)11 b Fv(\016)31 b Fw(is)c(the)g(lo)m(w)m(est)h(eigen)m(v)-5 b(alue)27 b(of)g Fv(H)1586 4445 y Fu(\()p Ft(\033)r Fu(\))1687 4482 y Fw(\()p Fv(k)s Fw(\).)41 b(Consider)28 b(the)g(eigenfunction)e Fv(u)h Fw(of)f Fv(G)3308 4445 y Fu(\()t(~)-39 b Ft(\033)s Fu(\))3410 4482 y Fw(\()p Fv(k)s Fw(\))120 4602 y(corresp)s(onding)32 b(to)h(the)g(eigen)m(v)-5 b(alue)32 b Fv(k)1554 4566 y Fu(2)1615 4602 y Fs(\000)23 b Fv(\016)t Fw(,)752 4857 y Fv(u)p Fw(\()p Fv(x)p Fw(\))28 b(:=)1097 4716 y Fo(\032)1214 4715 y(p)p 1313 4715 156 4 v 1313 4800 a Fw(2)p Fs(j)p Fv(a)p Fs(j)16 b Fv(b)1526 4815 y Ft(n)1574 4800 y Fw(\()p Fv(x)1667 4815 y Fu(1)1706 4800 y Fv(;)h(k)s Fw(\))g Fv(e)1904 4764 y Fl(\000j)p Ft(a)p Fl(j)p Ft(x)2076 4773 y Fm(2)2114 4800 y Fv(;)195 b(x)2391 4815 y Fu(1)2459 4800 y Fs(2)28 b Fw([)p Fs(\000)p Fv(\031)t(;)17 b(c)p Fw(])p Fv(;)1214 4921 y Fw(0)1073 b Fv(x)2391 4936 y Fu(1)2459 4921 y Fs(2)28 b Fw(\()p Fv(c;)17 b(\031)t Fw(])p Fv(;)120 5204 y Fw(where)34 b Fv(a)28 b Fw(:=)f Fs(\000)688 5051 y Fo(r)p 788 5051 649 4 v 42 x(\020)881 5165 y Ft(\031)p 858 5181 89 4 v 858 5238 a Fl(j)p Fu(\003)p Fl(j)956 5093 y Fo(\021)1016 5116 y Fu(2)1077 5204 y Fs(\000)c Fv(k)1231 5175 y Fu(2)1292 5204 y Fw(+)f Fv(\016)t Fw(.)44 b(Then)33 b Fv(u)28 b Fs(2)1965 5179 y Fw(~)1940 5204 y Fv(H)2029 5168 y Fu(1)2068 5204 y Fw(\(\005\),)k Fs(k)p Fv(u)p Fs(k)27 b Fw(=)h(1)k(and)772 5463 y Fv(h)828 5422 y Fu(\()p Ft(\033)r Fu(\))930 5463 y Fw(\()p Fv(k)s Fw(\)[)p Fv(u)p Fw(])27 b(=)g(inf)7 b Fs(f)p Fv(h)1525 5422 y Fu(\()p Ft(\033)r Fu(\))1626 5463 y Fw(\()p Fv(k)s Fw(\)[)p Fv(v)t Fw(])28 b(:)55 b Fv(v)32 b Fs(2)2169 5438 y Fw(~)2144 5463 y Fv(H)2233 5422 y Fu(1)2272 5463 y Fw(\(\005\))p Fv(;)h Fs(k)p Fv(v)t Fs(k)27 b Fw(=)h(1)p Fs(g)p Fv(:)1781 5712 y Fw(40)p eop %%Page: 41 46 41 45 bop 120 407 a Fw(It)38 b(follo)m(ws)e(from)h(general)g (principles)f(that)i Fv(u)e Fs(2)h(D)s Fw(\()p Fv(H)2196 371 y Fu(\()p Ft(\033)r Fu(\))2296 407 y Fw(\()p Fv(k)s Fw(\)\))h(and)g Fv(H)2786 371 y Fu(\()p Ft(\033)r Fu(\))2887 407 y Fw(\()p Fv(k)s Fw(\))p Fv(u)e Fw(=)g Fv(\025u)p Fw(.)58 b(By)120 527 y(Elliptic)31 b(Regularit)m(y)h(w)m(e)i(m)m(ust)g (ha)m(v)m(e)h Fv(u)29 b Fs(2)g Fv(H)1825 491 y Fu(2)1817 553 y Ft(l)q(oc)1908 527 y Fw(\(\005\),)k(whic)m(h)i(is)e(ob)m(viously) g(not)g(true.)47 b(This)120 648 y(con)m(tradiction)31 b(completes)i(the)g(pro)s(of)e(of)i(the)g(Prop)s(osition.)p 3469 648 4 66 v 3473 585 59 4 v 3473 648 V 3531 648 4 66 v 120 836 a Fj(R)-5 b(emark)57 b Fw(3.3.16)p Fj(.)47 b Fw(Assume)f(that)g Fv(\033)j Fw(is)c(\014xed)i(with)j(^)-54 b Fv(\033)2184 851 y Fu(0)2273 836 y Fw(=)2460 797 y Fu(1)p 2409 813 137 4 v 2409 823 a Fl(p)p 2468 823 79 3 v 55 x Fu(2)p Ft(\031)2573 756 y Fo(R)2639 782 y Ft(\031)2620 870 y Fl(\000)p Ft(\031)2738 836 y Fv(\033)t Fw(\()p Fv(x)2890 851 y Fu(1)2930 836 y Fw(\))p Fv(dx)3074 851 y Fu(1)3163 836 y Fv(<)50 b Fw(0)45 b(and)120 981 y(consider)c(the)h (op)s(erators)f Fv(H)1214 944 y Fu(\()p Ft(\013\033)r Fu(\))1402 981 y Fw(with)g("coupling)e(constan)m(t")j Fv(\013)h(>)f Fw(0.)69 b(It)41 b(follo)m(ws)f(from)120 1101 y(Prop)s(osition)31 b(3.3.14)h(that)1108 1357 y Fv(\025)1165 1306 y Fu(\()p Ft(\033)r Fu(\))1165 1382 y(1)1267 1357 y Fw(\()p Fv(\013)q(;)17 b(k)s Fw(\))27 b(:=)160 b(inf)1662 1435 y Fu(0)p Fl(6)p Fu(=)p Ft(u)p Fl(2)1858 1419 y Fu(~)1840 1435 y Ft(H)1903 1416 y Fm(1)1937 1435 y Fu(\(\005\))2072 1290 y Fv(h)2128 1254 y Fu(\()p Ft(\013\033)r Fu(\))2275 1290 y Fw(\()p Fv(k)s Fw(\)[)p Fv(u)p Fw(])p 2072 1334 443 4 v 2196 1426 a Fs(k)p Fv(u)p Fs(k)2352 1397 y Fu(2)2525 1357 y Fv(:)120 1632 y Fw(is)32 b(an)h(eigen)m(v)-5 b(alue)32 b(of)g Fv(H)1020 1595 y Fu(\()p Ft(\013\033)r Fu(\))1166 1632 y Fw(\()p Fv(k)s Fw(\))h(and)f(satis\014es)1281 1872 y Fv(\025)1338 1821 y Fu(\()p Ft(\033)r Fu(\))1338 1896 y(1)1439 1872 y Fw(\()p Fv(\013)q(;)17 b(k)s Fw(\))27 b Fs(\024)i Fv(k)1863 1831 y Fu(2)1924 1872 y Fs(\000)2063 1804 y Fw(1)p 2034 1849 108 4 v 2034 1940 a(2)p Fv(\031)2157 1872 y Fw(^)-54 b Fv(\033)2211 1831 y Fu(2)2207 1896 y(0)2250 1872 y Fv(\013)2313 1831 y Fu(2)2352 1872 y Fv(:)120 2106 y Fw(This)38 b(result)h(can)f(b)s(e)h(strengthened:)57 b(F)-8 b(or)37 b(su\016cien)m(tly)i(small)d Fv(\013)i(>)g Fw(0)g(and)g Fv(k)j Fs(2)c Fw(\()p Fs(\000)3330 2067 y Fu(1)p 3330 2083 36 4 v 3330 2140 a(2)3376 2106 y Fv(;)3430 2067 y Fu(1)p 3430 2083 V 3430 2140 a(2)3475 2106 y Fw(\),)120 2249 y Fv(\025)177 2198 y Fu(\()p Ft(\033)r Fu(\))177 2274 y(1)279 2249 y Fw(\()p Fv(\013)q(;)17 b(k)s Fw(\))32 b(is)g(the)h(only)f(eigen)m(v)-5 b(alue)32 b(of)g Fv(H)1694 2213 y Fu(\()p Ft(\013\033)r Fu(\))1873 2249 y Fw(b)s(elo)m(w)g Fv(k)2203 2213 y Fu(2)2243 2249 y Fw(,)g(it)g(is)g(simple)f(and)i (satis\014es)413 2366 y Fo(q)p 513 2366 611 4 v 133 x Fv(k)567 2470 y Fu(2)628 2499 y Fs(\000)23 b Fv(\025)785 2448 y Fu(\()p Ft(\033)r Fu(\))785 2523 y(1)887 2499 y Fw(\()p Fv(\013)q(;)17 b(k)s Fw(\))27 b(=)g Fs(\000)1413 2432 y Fw(1)p 1341 2476 191 4 v 1341 2496 a Fs(p)p 1425 2496 108 4 v 1425 2578 a Fw(2)p Fv(\031)1547 2499 y Fw(^)-54 b Fv(\033)1597 2514 y Fu(0)1637 2499 y Fv(\013)23 b Fw(+)f Fv(C)7 b Fw(\()p Fv(\033)n(;)17 b(k)s Fw(\))p Fv(\013)2187 2458 y Fu(2)2248 2499 y Fw(+)22 b Fs(O)s Fw(\()p Fv(\013)2529 2458 y Fu(3)2569 2499 y Fw(\))195 b(\()p Fv(\013)28 b Fs(!)f Fw(0+\))p Fv(;)120 2750 y Fw(where)37 b Fv(C)7 b Fw(\()p Fv(\033)n(;)17 b(k)s Fw(\))34 b Fs(\025)g Fw(0)h(can)h(b)s(e) h(calculated)e(explicitly)-8 b(.)51 b(Moreo)m(v)m(er,)39 b(it)c(can)h(b)s(e)g(sho)m(wn)h(that)120 2884 y Fv(\025)177 2833 y Fu(\()p Ft(\033)r Fu(\))177 2908 y(1)279 2884 y Fw(\()p Fv(\013)q(;)17 b(k)s Fw(\))27 b(is)g(an)h(eigen)m(v)-5 b(alue)28 b(b)s(elo)m(w)g Fv(k)1555 2848 y Fu(2)1622 2884 y Fw(ev)m(en)i(if)i(^)-54 b Fv(\033)1980 2899 y Fu(0)2047 2884 y Fw(=)28 b(0,)h Fv(\033)i Fs(6\021)d Fw(0)g(\(indeed,)h(for)f(an)m(y)h Fv(\013)f Fs(2)g Fr(R)5 b Fw(\).)120 3140 y Fp(5.)53 b(Commen)m(ts)38 b(and)h(Remarks.)98 b(Diric)m(hlet-to-Neumann)36 b(op)s(erator.)46 b Fw(F)-8 b(or)33 b(suf-)120 3261 y(\014cien)m(tly)47 b(regular)f Fv(\033)t Fw(,)51 b(the)d(op)s(erator)f Fv(B)1655 3224 y Fu(\()p Ft(\033)r Fu(\))1757 3261 y Fw(\()p Fv(\025)32 b Fs(\006)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))47 b(is)g(closely)g (connected)i(with)e(the)120 3381 y(Diric)m(hlet-to-Neumann)42 b(op)s(erator)j(from)f(the)h(theory)h(of)f(Boundary)h(V)-8 b(alue)44 b(Problems.)120 3501 y(Indeed,)34 b(let)e Fv(\025)c Fs(2)g Fr(R)43 b Fw(and)33 b Fv(f)38 b Fs(2)28 b Fv(H)1347 3465 y Fu(3)p Ft(=)p Fu(2)1457 3501 y Fw(\()p Fr(T)p Fw(\))36 b(suc)m(h)e(that)1312 3671 y(^)1291 3697 y Fv(f)1339 3712 y Ft(n)1413 3697 y Fw(=)28 b(0)p Fv(;)212 b Fw(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))2113 3656 y Fu(2)2180 3697 y Fs(\024)28 b Fv(\025:)846 b Fw(\(3.3.18\))120 3893 y(Then)34 b(there)f(is)f(a)g(function)g Fv(u)c Fs(2)1388 3868 y Fw(~)1363 3893 y Fv(H)1452 3857 y Fu(2)1491 3893 y Fw(\(\005\))k(satisfying)1098 4006 y Fo(\000)1143 4086 y Fw(\()p Fv(D)1262 4101 y Fu(1)1324 4086 y Fw(+)22 b Fv(k)s Fw(\))1514 4045 y Fu(2)1575 4086 y Fw(+)g Fv(D)1757 4045 y Fu(2)1754 4111 y(2)1797 4006 y Fo(\001)1859 4086 y Fv(u)27 b Fw(=)h Fv(\025u;)211 b Fw(in)27 b(\005)1081 4232 y Fv(u)p Fw(\()p Fv(:;)17 b Fw(0\))27 b(=)g Fv(f)206 b Fw(in)27 b([)p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(])p Fv(:)120 4421 y Fw(This)46 b(can)g(b)s(e)h(seen)g(b)m(y)g(the)g(same)e (ideas)h(as)g(in)g(the)g(pro)s(of)f(of)h(Prop)s(osition)e(3.3.2.)84 b(In)120 4541 y(particular,)46 b(w)m(e)g(\014nd)g(that)f Fv(u)f Fw(is)h(giv)m(en)g(b)m(y)g(the)h(form)m(ula)d(\(3.3.5\))o(.)81 b(If)45 b Fv(\033)k Fw(is)44 b(su\016cien)m(tly)120 4661 y(regular)32 b(w)m(e)h(ha)m(v)m(e)1090 4826 y Fv(g)e Fw(:=)c Fs(\000)1406 4758 y Fv(@)5 b(u)p 1385 4803 152 4 v 1385 4894 a(@)g(x)1496 4909 y Fu(2)1548 4826 y Fw(\()p Fv(:;)17 b Fw(0\))k(+)h Fv(\033)t(u)p Fw(\()p Fv(:;)17 b Fw(0\))27 b Fs(2)h Fv(L)2361 4841 y Fu(2)2401 4826 y Fw(\()p Fr(T)p Fw(\))p Fv(:)120 5030 y Fw(The)41 b(op)s(erator)f Fv(f)52 b Fs(7!)40 b Fv(g)31 b Fs(\000)d Fv(\033)t(f)51 b Fw(is)40 b(usually)f(called)h(Diric)m(hlet-to-Neumann)d(op)s(erator.) 66 b(It)120 5150 y(follo)m(ws)31 b(from)h(\(3.3.5\))g(that)330 5364 y Fv(g)f Fw(=)601 5269 y Fo(X)512 5486 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))703 5467 y Fm(2)738 5486 y Ft(>\025)850 5274 y Fo(p)p 950 5274 527 4 v 90 x Fw(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))1258 5335 y Fu(2)1320 5364 y Fs(\000)g Fv(\025)1497 5337 y Fw(^)1476 5364 y Fv(f)1524 5379 y Ft(n)1571 5364 y Fv(e)1616 5379 y Ft(n)1686 5364 y Fw(+)g Fv(\033)t(f)38 b Fw(=)28 b Fv(B)2112 5323 y Fu(\()p Ft(\033)r Fu(\))2213 5364 y Fw(\()p Fv(\025)22 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))p Fv(f)38 b Fw(=)28 b Fv(B)2917 5323 y Fu(\()p Ft(\033)r Fu(\))3019 5364 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))p Fv(f)5 b(;)1781 5712 y Fw(41)p eop %%Page: 42 47 42 46 bop 120 407 a Fw(so)27 b(our)f(b)s(oundary)h(op)s(erators)f Fv(B)1338 371 y Fu(\()p Ft(\033)r Fu(\))1440 407 y Fw(\()p Fv(\025)10 b Fs(\006)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))p Fv(;)33 b(B)1989 371 y Fu(\()p Ft(\033)r Fu(\))2091 407 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))26 b(are)h(m-sectorial)d(extensions)j (of)120 527 y(the)33 b(Diric)m(hlet-to-Neumann)28 b(op)s(erator)k (di\013ering)e(only)i(on)g(the)h(orthogonal)d(complemen)m(t)120 648 y(of)i(functions)h Fv(f)38 b Fs(2)28 b Fv(L)898 663 y Fu(2)938 648 y Fw(\()p Fr(T)p Fw(\))36 b(satisfying)d(\(3.3.18\))o(.) 120 782 y Fp(Connection)53 b(b)s(et)m(w)m(een)g Fv(B)1260 745 y Fu(\()p Ft(\033)r Fu(\))1362 782 y Fw(\()p Fv(\025)31 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))53 b Fp(and)h Fv(T)2171 731 y Fu(\()p Ft(\033)r Fu(\))2157 806 y(0)2272 782 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))g Fv(T)2583 745 y Fu(\()p Ft(\033)r Fu(\))2685 782 y Fw(\()p 2723 727 50 4 v Fv(z)t(;)g(k)s Fw(\))2908 745 y Fl(\003)2948 782 y Fp(.)84 b Fw(Our)46 b(analy-)120 902 y(sis)c(in)f(Sections)i(1)e(and) h(2)g(w)m(as)h(based)g(on)f(the)g(resolv)m(en)m(t)i(iden)m(tit)m(y)f (\(3.1.9\))e(and)h(its)g(ex-)120 1022 y(tension)48 b(to)f(real)g(v)-5 b(alues)47 b(of)h Fv(z)t Fw(.)89 b(It)48 b(can)g(b)s(e)g(pro)m(v)m(ed)h (in)e(the)h(general)f(framew)m(ork)h(that)120 1143 y Fv(\025)28 b Fs(2)h Fw([)p Fv(k)381 1107 y Fu(2)421 1143 y Fv(;)17 b Fw(+)p Fs(1)p Fw(\))k Fs(n)i(f)p Fw(\()p Fv(n)f Fw(+)g Fv(k)s Fw(\))1131 1107 y Fu(2)1199 1143 y Fw(:)57 b Fv(n)29 b Fs(2)g Fr(Z)p Fs(g)h Fw(is)j(an)g(eigen)m(v)-5 b(alue)32 b(of)h Fv(H)2516 1107 y Fu(\()p Ft(\033)r Fu(\))2617 1143 y Fw(\()p Fv(k)s Fw(\))g(i\013)f Fs(\000)p Fw(1)h(is)g(an)g (eigen-)120 1277 y(v)-5 b(alue)23 b(of)h Fv(T)537 1226 y Fu(\()p Ft(\033)r Fu(\))523 1301 y(0)638 1277 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))g Fv(T)949 1240 y Fu(\()p Ft(\033)r Fu(\))1051 1277 y Fw(\()p 1089 1222 V Fv(z)5 b(;)17 b(k)s Fw(\))1275 1240 y Fl(\003)1314 1277 y Fw(,)26 b(see)f(Lemma)d(4.7.8)i (in)f([Y)-8 b(a1].)40 b(\(The)25 b(h)m(yp)s(othesis)g(of)f(strong)120 1397 y Fv(H)209 1361 y Fu(\(0\))303 1397 y Fw(\()p Fv(k)s Fw(\)-smo)s(othness)39 b(is)g(implicitly)c(v)m(eri\014ed)41 b(in)d(Lemma)g(3.4.4)h(b)s(elo)m(w.\))64 b(Ho)m(w)m(ev)m(er,)44 b(the)120 1531 y(op)s(erators)31 b Fv(T)621 1480 y Fu(\()p Ft(\033)r Fu(\))607 1555 y(0)723 1531 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))g Fv(T)1034 1495 y Fu(\()p Ft(\033)r Fu(\))1135 1531 y Fw(\()p 1173 1476 V Fv(z)5 b(;)17 b(k)s Fw(\))1359 1495 y Fl(\003)1429 1531 y Fw(ha)m(v)m(e)33 b(no)e(reasonable)g(limits) d(as)k Fv(\025)27 b Fs(!)h Fw(\()p Fv(n)19 b Fw(+)h Fv(k)s Fw(\))3167 1495 y Fu(2)3206 1531 y Fv(;)33 b(n)28 b Fs(2)g Fr(Z)p Fw(,)120 1651 y(so)g(it)f(seems)i(to)e(b)s(e)i(imp)s(ossible)c (to)j(decide)g(within)f(this)h(general)f(approac)m(h)h(whether)i(these) 120 1772 y(p)s(oin)m(ts)35 b(are)g(eigen)m(v)-5 b(alues)35 b(or)g(accum)m(ulation)f(p)s(oin)m(ts)h(of)f(eigen)m(v)-5 b(alues.)52 b(This)35 b(is)g(the)g(reason)120 1892 y(wh)m(y)24 b(w)m(e)g(had)g(to)e(c)m(hange)i(our)f(p)s(oin)m(t)f(of)h(view)g(and)g (in)m(tro)s(duce)g(the)h(op)s(erators)f Fv(B)3043 1856 y Fu(\()p Ft(\033)r Fu(\))3145 1892 y Fw(\()p Fv(\025)s Fs(\006)s Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))120 2012 y(in)32 b(this)g(section.)120 2133 y(Let)24 b(us)h(discuss)h(ho)m(w)e (they)h(are)g(connected.)42 b(Let)24 b Fv(\025)k Fs(6)p Fw(=)f(\()p Fv(n)5 b Fw(+)g Fv(k)s Fw(\))2400 2097 y Fu(2)2440 2133 y Fv(;)33 b(n)28 b Fs(2)g Fr(Z)p Fv(;)22 b Fw(and)i Fv(f)5 b(;)34 b(g)d Fs(2)d Fv(L)3331 2148 y Fu(2)3371 2133 y Fw(\()p Fr(T)p Fw(\).)120 2253 y(Then)34 b Fv(B)454 2217 y Fu(\(0\))548 2253 y Fw(\()p Fv(\025)22 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))983 2217 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1147 2253 y Fv(f)5 b(;)34 b(B)1340 2217 y Fu(\(0\))1435 2253 y Fw(\()p Fv(\025)21 b Fs(\007)i Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))1869 2217 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)2034 2253 y Fv(g)31 b Fs(2)d Fv(H)2295 2217 y Fu(1)p Ft(=)p Fu(2)2404 2253 y Fw(\()p Fr(T)p Fw(\))36 b(and)585 2477 y Fv(b)626 2436 y Fu(\()p Ft(\033)r Fu(\))728 2477 y Fw(\()p Fv(\025)22 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))1180 2396 y Fo(\002)1221 2477 y Fv(B)1300 2436 y Fu(\(0\))1394 2477 y Fw(\()p Fv(\025)22 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))1829 2436 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1993 2477 y Fv(f)5 b(;)34 b(B)2186 2436 y Fu(\(0\))2280 2477 y Fw(\()p Fv(\025)22 b Fs(\007)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))2715 2436 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)2880 2477 y Fv(g)2931 2396 y Fo(\003)2999 2477 y Fw(=)808 2661 y(=)27 b(\()p Fv(f)5 b(;)34 b(g)t Fw(\))21 b(+)1271 2550 y Fo(\020)1331 2661 y Fv(T)1402 2610 y Fu(\()p Ft(\033)r Fu(\))1388 2685 y(0)1503 2661 y Fw(\()p Fv(\025)h Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))p Fv(f)5 b(;)33 b(T)2122 2620 y Fu(\()p Ft(\033)r Fu(\))2224 2661 y Fw(\()p Fv(\025)22 b Fs(\007)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))p Fv(g)2709 2550 y Fo(\021)2784 2661 y Fv(:)120 2920 y Fw(Replacing)31 b Fv(f)43 b Fw(b)m(y)34 b Fv(T)869 2883 y Fu(\()p Ft(\033)r Fu(\))970 2920 y Fw(\()p Fv(\025)22 b Fs(\007)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))1405 2883 y Fl(\003)1444 2920 y Fv(f)43 b Fw(w)m(e)34 b(see)g(that)262 3143 y Fv(b)303 3102 y Fu(\()p Ft(\033)r Fu(\))405 3143 y Fw(\()p Fv(\025)22 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))857 3063 y Fo(\002)898 3143 y Fv(B)977 3102 y Fu(\(0\))1071 3143 y Fw(\()p Fv(\025)22 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))1506 3102 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1670 3143 y Fv(T)1741 3102 y Fu(\()p Ft(\033)r Fu(\))1843 3143 y Fw(\()p Fv(\025)22 b Fs(\007)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))2277 3102 y Fl(\003)2317 3143 y Fv(f)5 b(;)33 b(B)2509 3102 y Fu(\(0\))2604 3143 y Fw(\()p Fv(\025)22 b Fs(\007)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))3038 3102 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)3203 3143 y Fv(g)3254 3063 y Fo(\003)3322 3143 y Fw(=)485 3327 y(=)588 3217 y Fo(\020)q(\020)707 3327 y Fv(I)30 b Fw(+)22 b Fv(T)949 3276 y Fu(\()p Ft(\033)r Fu(\))935 3352 y(0)1051 3327 y Fw(\()p Fv(\025)g Fs(\006)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))g Fv(T)1573 3286 y Fu(\()p Ft(\033)r Fu(\))1674 3327 y Fw(\()p Fv(\025)22 b Fs(\007)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))2109 3286 y Fl(\003)2148 3217 y Fo(\021)2224 3327 y Fv(f)5 b(;)34 b(T)2409 3286 y Fu(\()p Ft(\033)r Fu(\))2510 3327 y Fw(\()p Fv(\025)22 b Fs(\007)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))p Fv(g)2996 3217 y Fo(\021)3071 3327 y Fv(:)120 3586 y Fw(Since)46 b Fv(B)467 3550 y Fu(\(0\))562 3586 y Fw(\()p Fv(\025)31 b Fs(\007)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))1014 3550 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1225 3586 y Fw(maps)45 b Fv(L)1559 3601 y Fu(2)1599 3586 y Fw(\()p Fr(T)p Fw(\))k(bijectiv)m(ely)d(on)m(to)g Fv(H)2589 3550 y Fu(1)p Ft(=)p Fu(2)2698 3586 y Fw(\()p Fr(T)p Fw(\),)53 b(it)45 b(follo)m(ws)f(that)120 3706 y Fv(B)199 3670 y Fu(\(0\))293 3706 y Fw(\()p Fv(\025)22 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))728 3670 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)893 3706 y Fv(T)964 3670 y Fu(\()p Ft(\033)r Fu(\))1065 3706 y Fw(\()p Fv(\025)22 b Fs(\007)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))1500 3670 y Fl(\003)1539 3706 y Fv(f)38 b Fs(2)28 b(D)s Fw(\()p Fv(B)1916 3670 y Fu(\()p Ft(\033)r Fu(\))2018 3706 y Fw(\()p Fv(\025)22 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\)\))32 b(and)g(that)148 3930 y Fv(B)227 3889 y Fu(\()p Ft(\033)r Fu(\))329 3930 y Fw(\()p Fv(\025)22 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))p Fv(B)843 3889 y Fu(\(0\))937 3930 y Fw(\()p Fv(\025)22 b Fs(\006)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))1371 3889 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1536 3930 y Fv(T)1607 3889 y Fu(\()p Ft(\033)r Fu(\))1709 3930 y Fw(\()p Fv(\025)22 b Fs(\007)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))2143 3889 y Fl(\003)2182 3930 y Fv(f)39 b Fw(=)371 4114 y(=)27 b Fv(B)553 4073 y Fu(\(0\))648 4114 y Fw(\()p Fv(\025)22 b Fs(\006)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))1082 4073 y Fu(1)p Ft(=)p Fu(2)1192 4114 y Fv(T)1263 4073 y Fu(\()p Ft(\033)r Fu(\))1365 4114 y Fw(\()p Fv(\025)22 b Fs(\007)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))1799 4073 y Fl(\003)1855 4003 y Fo(\020)1915 4114 y Fv(I)30 b Fw(+)22 b Fv(T)2157 4063 y Fu(\()p Ft(\033)r Fu(\))2143 4138 y(0)2258 4114 y Fw(\()p Fv(\025)g Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))g Fv(T)2781 4073 y Fu(\()p Ft(\033)r Fu(\))2882 4114 y Fw(\()p Fv(\025)22 b Fs(\007)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))3316 4073 y Fl(\003)3355 4003 y Fo(\021)3431 4114 y Fv(f)5 b(:)3215 4271 y Fw(\(3.3.19\))120 4484 y(In)33 b(particular,)e(w)m(e)i(can)g(pro)m(v)m(e)h(no)m(w)f(Remark)f (3.2.3.)120 4688 y Fp(Lemma)37 b(3.3.17.)42 b Fj(L)-5 b(et)35 b Fv(k)c Fs(2)d Fw([)p Fs(\000)1334 4649 y Fu(1)p 1334 4665 36 4 v 1334 4722 a(2)1380 4688 y Fv(;)1434 4649 y Fu(1)p 1434 4665 V 1434 4722 a(2)1479 4688 y Fw(])p Fj(,)35 b(then)362 4921 y Fs(N)444 4936 y Fu(+)503 4921 y Fw(\()p Fv(k)s Fw(\))28 b(=)f Fs(N)846 4936 y Fl(\000)905 4921 y Fw(\()p Fv(k)s Fw(\))h(=)f Fs(N)15 b Fw(\()p Fv(k)s Fw(\))27 b(=)h Fv(\033)1579 4936 y Ft(p)1636 4840 y Fo(\000)1681 4921 y Fv(H)1770 4880 y Fu(\()p Ft(\033)r Fu(\))1872 4921 y Fw(\()p Fv(k)s Fw(\))2002 4840 y Fo(\001)2069 4921 y Fs(n)22 b(f)p Fw(\()p Fv(n)g Fw(+)g Fv(k)s Fw(\))2499 4880 y Fu(2)2567 4921 y Fw(:)55 b Fv(n)28 b Fs(2)g Fr(Z)p Fs(g)p Fv(:)240 b Fw(\(3.3.20\))120 5141 y Fj(Pr)-5 b(o)g(of.)48 b Fw(The)43 b(last)d(equalit)m(y)i(in)e(\(3.3.20\))h(is)g(clear)g(from) f(Prop)s(osition)g(3.3.2.)69 b(Moreo)m(v)m(er,)120 5262 y(w)m(e)40 b(sa)m(w)f(in)f(the)h(pro)s(of)f(of)g(Theorem)h(3.2.5)f (that)h Fv(H)2099 5225 y Fu(\()p Ft(\033)r Fu(\))2200 5262 y Fw(\()p Fv(k)s Fw(\))g(is)f(absolutely)g(con)m(tin)m(uous)h(on) 120 5382 y([)p Fv(k)201 5346 y Fu(2)241 5382 y Fv(;)17 b Fw(+)p Fs(1)p Fw(\))32 b Fs(n)h Fw(\()p Fs(f)p Fw(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))1010 5346 y Fu(2)1077 5382 y Fw(:)55 b Fv(n)28 b Fs(2)g Fr(Z)p Fs(g)20 b([)i Fw(\()p Fs(N)1686 5397 y Fu(+)1745 5382 y Fw(\()p Fv(k)s Fw(\))g Fs(\\)g(N)2067 5397 y Fl(\000)2126 5382 y Fw(\()p Fv(k)s Fw(\)\)\).)91 b(Hence)50 b Fs(N)15 b Fw(\()p Fv(k)s Fw(\))55 b Fs(\032)g(N)3252 5397 y Fu(+)3311 5382 y Fw(\()p Fv(k)s Fw(\))33 b Fs(\\)1781 5712 y Fw(42)p eop %%Page: 43 48 43 47 bop 120 407 a Fs(N)202 422 y Fl(\000)261 407 y Fw(\()p Fv(k)s Fw(\).)120 527 y(Con)m(v)m(ersely)-8 b(,)35 b(if)c Fv(\025)d Fs(2)g(N)983 542 y Fl(\006)1042 527 y Fw(\()p Fv(k)s Fw(\))k(and)h Fv(f)38 b Fs(6)p Fw(=)28 b(0)k(suc)m(h)i(that)909 664 y Fo(\020)968 775 y Fv(I)c Fw(+)22 b Fv(T)1210 724 y Fu(\()p Ft(\033)r Fu(\))1196 799 y(0)1312 775 y Fw(\()p Fv(\025)g Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))g Fv(T)1835 733 y Fu(\()p Ft(\033)r Fu(\))1935 775 y Fw(\()p Fv(\025)22 b Fs(\007)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))2370 733 y Fl(\003)2409 664 y Fo(\021)2485 775 y Fv(f)39 b Fw(=)27 b(0)p Fv(;)464 b Fw(\(3.3.21\))120 1033 y(then)24 b(b)s(ecause)h(of)30 b(\(3.3.19\))23 b Fv(B)1222 997 y Fu(\()p Ft(\033)r Fu(\))1323 1033 y Fw(\()p Fv(\025)t Fs(\006)t Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))p Fv(g)31 b Fw(=)c(0)d(where)g Fv(g)31 b Fw(:=)d Fv(B)2535 997 y Fu(\(0\))2629 1033 y Fw(\()p Fv(\025)t Fs(\006)t Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))3027 997 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)3192 1033 y Fv(T)3263 997 y Fu(\()p Ft(\033)r Fu(\))3364 1033 y Fw(\()p Fv(\025)t Fs(\007)120 1154 y Fv(i)p Fw(0)p Fv(;)g(k)s Fw(\))338 1117 y Fl(\003)377 1154 y Fv(f)11 b Fw(.)43 b(No)m(w)35 b(\(3.3.21\))c(implies)f Fv(T)1489 1117 y Fu(\()p Ft(\033)r Fu(\))1591 1154 y Fw(\()p Fv(\025)22 b Fs(\007)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))2025 1117 y Fl(\003)2064 1154 y Fv(f)39 b Fs(6)p Fw(=)27 b(0,)32 b(so)h Fv(g)e Fs(6)p Fw(=)d(0)k(and)g Fv(\025)c Fs(2)g(N)15 b Fw(\()p Fv(k)s Fw(\).)p 3469 1154 4 66 v 3473 1091 59 4 v 3473 1154 V 3531 1154 4 66 v 266 1349 a(T)-8 b(ogether)43 b(with)g(Lemma)e (3.2.2,)j(this)e(is)g(another)h(pro)s(of)f(of)g(the)g(\014rst)h (statemen)m(t)g(of)120 1469 y(Lemma)35 b(3.3.6.)56 b(\(Ho)m(w)m(ev)m (er,)40 b(w)m(e)e(need)f(the)g(second)h(statemen)m(t)g(of)e(Lemma)f (3.3.6)h(for)h(the)120 1589 y(pro)s(of)32 b(of)g(Theorem)h(3.3.5)f(via) g(Lemma)f(3.3.7.\))120 1922 y Fq(3.4)161 b(Scattering)53 b(theory)f(for)i(the)f(op)t(erators)g Ff(H)3150 1870 y Fw(\()p Fv(\033)t Fw(\))3289 1922 y Fe(\()p Ff(k)5 b Fe(\))120 2141 y Fw(Our)46 b(goal)f(in)h(this)g(section)h(is)f(to)g (study)i(the)f(scattering)f(prop)s(erties)h(of)f(the)h(op)s(erator)120 2262 y Fv(H)209 2226 y Fu(\()p Ft(\033)r Fu(\))310 2262 y Fw(\()p Fv(k)s Fw(\))33 b(on)f(the)h(halfstrip.)42 b(The)34 b(comparison)d(op)s(erator)h(is)g Fv(H)2498 2226 y Fu(\(0\))2592 2262 y Fw(\()p Fv(k)s Fw(\).)120 2382 y(W)-8 b(e)31 b(ha)m(v)m(e)h(already)e(pro)m(v)m(ed)i(the)f (existence)h(and)f(completeness)h(of)e(the)h(w)m(a)m(v)m(e)h(op)s (erators)f(in)120 2502 y(Section)j(1.)46 b(Here)35 b(w)m(e)g(brie\015y) f(summarize)e(the)j(results)f(in)f(Subsection)h(1.)47 b(In)34 b(Subsection)120 2623 y(2)43 b(w)m(e)h(turn)g(our)f(atten)m (tion)g(to)g(the)h(scattering)f(matrix.)74 b(W)-8 b(e)44 b(deriv)m(e)g(a)f(represen)m(tation)120 2743 y(form)m(ula)32 b(from)g(the)i(general)f(Stationary)g(Scattering)g(Theory)-8 b(.)47 b(Finally)-8 b(,)31 b(in)i(Subsection)h(3)120 2864 y(w)m(e)f(touc)m(h)f(the)h(sub)5 b(ject)33 b(of)f(generalized)f (eigenfunctions.)43 b(In)32 b(particular,)f(w)m(e)h(express)j(the)120 2984 y(scattering)30 b(matrix)e(in)h(terms)h(of)g(the)g(asymptotics)g (of)f(these)j(solutions)d(of)g(the)i(stationary)120 3104 y(Sc)m(hr\177)-49 b(odinger)30 b(equation)f(\(with)g(quasimomen)m (tum\).)41 b(This)30 b(pro)m(vides)g(a)f(graphic)g(picture)g(of)120 3225 y(scattering)j(close)h(to)f(the)h(ph)m(ysical)g(in)m(tuition.)120 3485 y Fp(1.)78 b(Existence)46 b(and)i(completeness)e(of)h(the)g(w)m(a) m(v)m(e)g(op)s(erators.)98 b Fw(Recall)39 b(the)j(ab-)120 3605 y(stract)35 b(de\014nition)e(of)h(the)h(w)m(a)m(v)m(e)h(op)s (erators)f(from)e(Section)h(1.4.)49 b(In)35 b(case)g(of)f(existence)i (w)m(e)120 3725 y(will)30 b(use)k(the)f(notation)752 3945 y Fv(W)858 3894 y Fu(\()p Ft(\033)r Fu(\))844 3967 y Fl(\006)960 3945 y Fw(\()p Fv(k)s Fw(\))27 b(:=)h Fv(W)1340 3960 y Fl(\006)1399 3945 y Fw(\()p Fv(H)1526 3904 y Fu(\()p Ft(\033)r Fu(\))1627 3945 y Fw(\()p Fv(k)s Fw(\))p Fv(;)17 b(H)1890 3904 y Fu(\(0\))1984 3945 y Fw(\()p Fv(k)s Fw(\)\))p Fv(;)211 b(k)31 b Fs(2)2566 3865 y Fo(\002)2608 3945 y Fs(\000)2695 3906 y Fu(1)p 2695 3922 36 4 v 2695 3980 a(2)2740 3945 y Fv(;)2794 3906 y Fu(1)p 2794 3922 V 2794 3980 a(2)2839 3865 y Fo(\003)2881 3945 y Fv(:)120 4174 y Fw(Theorem)43 b(3.1.3)g(sho)m(ws)h(that)f(the)g(di\013erence)h(of)e (resolv)m(en)m(ts)j Fv(R)2578 4138 y Fu(\()p Ft(\033)r Fu(\))2680 4174 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))29 b Fs(\000)h Fv(R)3114 4138 y Fu(\(0\))3208 4174 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\))43 b(is)120 4294 y(trace)33 b(class.)44 b(Hence)33 b(the)g(Birman-Kre)-11 b(\025)-38 b(\020n)30 b(theorem)j(yields)120 4498 y Fp(Theorem)41 b(3.4.1.)i Fj(L)-5 b(et)39 b Fv(k)d Fs(2)d Fw([)p Fs(\000)1375 4459 y Fu(1)p 1375 4475 V 1375 4532 a(2)1421 4498 y Fv(;)1475 4459 y Fu(1)p 1475 4475 V 1475 4532 a(2)1520 4498 y Fw(])p Fj(,)38 b(then)g(the)g(wave)f(op)-5 b(er)g(ators)37 b Fv(W)2774 4447 y Fu(\()p Ft(\033)r Fu(\))2760 4520 y Fl(\006)2876 4498 y Fw(\()p Fv(k)s Fw(\))g Fj(on)h(the)g(half-)120 4618 y(strip)d(exist)g(and)f(ar)-5 b(e)34 b(c)-5 b(omplete.)266 4822 y Fw(Under)34 b(the)g(additional)c(assumption)i Fv(\033)g Fs(\025)d Fw(0)k(w)m(e)h(can)f(sa)m(y)h(more.)45 b(Indeed,)34 b(the)g(op)s(era-)120 4942 y(tors)f Fv(H)405 4906 y Fu(\()p Ft(\033)r Fu(\))506 4942 y Fw(\()p Fv(k)s Fw(\))f(are)h(absolutely)f(con)m(tin)m(uous)h(b)m(y)h(Theorem)f (3.3.10.)42 b(W)-8 b(e)33 b(obtain)120 5145 y Fp(Theorem)42 b(3.4.2.)i Fj(Assume)39 b(that)g Fv(\033)k Fj(is)c(non-ne)-5 b(gative)37 b(and)i(let)g Fv(k)f Fs(2)d Fw([)p Fs(\000)2896 5106 y Fu(1)p 2896 5123 V 2896 5180 a(2)2942 5145 y Fv(;)2996 5106 y Fu(1)p 2996 5123 V 2996 5180 a(2)3041 5145 y Fw(])p Fj(.)57 b(Then)38 b(the)120 5289 y(wave)c(op)-5 b(er)g(ators)35 b Fv(W)889 5238 y Fu(\()p Ft(\033)r Fu(\))875 5311 y Fl(\006)990 5289 y Fw(\()p Fv(k)s Fw(\))g Fj(on)f(the)h(halfstrip)f (exist)h(and)f(ar)-5 b(e)35 b(unitary.)1781 5712 y Fw(43)p eop %%Page: 44 49 44 48 bop 120 407 a Fp(2.)47 b(The)30 b(scattering)e(matrix.)97 b Fw(Let)25 b Fv(k)31 b Fs(2)d Fw([)p Fs(\000)1900 368 y Fu(1)p 1900 384 36 4 v 1900 441 a(2)1946 407 y Fv(;)2000 368 y Fu(1)p 2000 384 V 2000 441 a(2)2045 407 y Fw(].)41 b(Recall)24 b(from)g(Section)i(2.2)f(the)h(direct)120 527 y(in)m(tegral)g(decomp)s(osition)f Fg(h)p Fw(\()p Fv(k)s Fw(\))i(of)g Fv(L)1488 542 y Fu(2)1528 527 y Fw(\(\005\))g(with) g(\014b)s(er)h(spaces)g Fr(C)2500 491 y Ft(N)7 b Fu(\()q Ft(\025;k)r Fu(\))2755 527 y Fw(where)28 b Fv(H)3120 491 y Fu(\(0\))3214 527 y Fw(\()p Fv(k)s Fw(\))f(acts)120 648 y(diagonal.)120 768 y(The)48 b(completeness)g(of)f(the)h(w)m(a)m(v) m(e)h(op)s(erators)e Fv(W)2050 717 y Fu(\()p Ft(\033)r Fu(\))2036 790 y Fl(\006)2152 768 y Fw(\()p Fv(k)s Fw(\))g(implies)e (the)i(unitarit)m(y)g(of)g(the)120 888 y Fj(sc)-5 b(attering)35 b(op)-5 b(er)g(ator)1223 1009 y Fv(S)1289 968 y Fu(\()p Ft(\033)r Fu(\))1391 1009 y Fw(\()p Fv(k)s Fw(\))27 b(:=)h Fv(W)1785 958 y Fu(\()p Ft(\033)r Fu(\))1771 1030 y(+)1886 1009 y Fw(\()p Fv(k)s Fw(\))2016 968 y Fl(\003)2072 1009 y Fv(W)2178 958 y Fu(\()p Ft(\033)r Fu(\))2164 1030 y Fl(\000)2280 1009 y Fw(\()p Fv(k)s Fw(\))p Fv(:)120 1183 y Fw(Because)d(of)d(the)h(in)m(tert)m(wining)f(prop)s(ert)m(y)i(of)e (the)i(w)m(a)m(v)m(e)h(op)s(erators)d(the)i(scattering)f(op)s(erator) 120 1303 y(comm)m(utes)37 b(with)f Fv(H)898 1267 y Fu(\(0\))992 1303 y Fw(\()p Fv(k)s Fw(\))h(and)f(can)h(b)s(e)g(decomp)s(osed)h(in)e (the)h(space)h Fg(h)p Fw(\()p Fv(k)s Fw(\).)56 b(Namely)-8 b(,)36 b(for)120 1424 y(a.e.)41 b Fv(\025)28 b Fs(2)g Fw([)p Fv(k)567 1388 y Fu(2)607 1424 y Fv(;)17 b Fw(+)p Fs(1)p Fw(\))24 b(there)i(is)f(an)h(op)s(erator)f Fv(S)1803 1388 y Fu(\()p Ft(\033)r Fu(\))1904 1424 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))25 b(on)h Fr(C)2355 1388 y Ft(N)7 b Fu(\()p Ft(\025;k)r Fu(\))2582 1424 y Fw(,)27 b(the)f Fj(sc)-5 b(attering)28 b(matrix)p Fw(,)120 1544 y(suc)m(h)34 b(that)369 1764 y(\(\000)468 1723 y Fu(\(0\))562 1764 y Fw(\()p Fv(k)s Fw(\))p Fv(S)758 1723 y Fu(\()p Ft(\033)r Fu(\))860 1764 y Fw(\()p Fv(k)s Fw(\))p Fv(u)p Fw(\)\()p Fv(\025)p Fw(\))27 b(=)g Fv(S)1413 1723 y Fu(\()p Ft(\033)r Fu(\))1515 1764 y Fw(\()p Fv(\025;)17 b(k)s Fw(\)\(\000)1845 1723 y Fu(\(0\))1938 1764 y Fw(\()p Fv(k)s Fw(\))p Fv(u)p Fw(\)\()p Fv(\025)p Fw(\))p Fv(;)211 b Fw(a.e.)28 b Fv(\025)g Fs(2)g Fw([)p Fv(k)2967 1723 y Fu(2)3007 1764 y Fv(;)17 b Fw(+)p Fs(1)p Fw(\))p Fv(;)120 1984 y Fw(or)32 b(equiv)-5 b(alen)m(tly)295 2169 y Fo(\000)340 2249 y Fv(S)406 2208 y Fu(\()p Ft(\033)r Fu(\))508 2249 y Fw(\()p Fv(k)s Fw(\))p Fv(u;)17 b(v)789 2169 y Fo(\001)833 2288 y Ft(L)881 2297 y Fm(2)916 2288 y Fu(\(\005\))1056 2249 y Fw(=)1159 2114 y Fo(Z)1259 2140 y Fl(1)1214 2339 y Ft(k)1253 2320 y Fm(2)1350 2169 y Fo(\000)1396 2249 y Fv(S)1462 2208 y Fu(\()p Ft(\033)r Fu(\))1563 2249 y Fw(\()p Fv(\025;)g(k)s Fw(\)\(\000)1893 2208 y Fu(\(0\))1987 2249 y Fw(\()p Fv(k)s Fw(\))p Fv(u)p Fw(\)\()p Fv(\025)p Fw(\))p Fv(;)g Fw(\(\000)2487 2208 y Fu(\(0\))2580 2249 y Fw(\()p Fv(k)s Fw(\))p Fv(v)t Fw(\)\()p Fv(\025)p Fw(\))2932 2169 y Fo(\001)2977 2288 y Fn(C)3019 2269 y Fi(N)6 b Fm(\()p Fi(\025;k)q Fm(\))3258 2249 y Fv(d\025)120 2533 y Fw(for)35 b(all)e Fv(u;)g(v)j Fs(2)c Fv(L)773 2548 y Fu(2)813 2533 y Fw(\(\005\).)52 b(Since)35 b Fv(S)1364 2497 y Fu(\()p Ft(\033)r Fu(\))1465 2533 y Fw(\()p Fv(k)s Fw(\))h(is)e(unitary)-8 b(,)36 b(the)f(scattering)g(matrix)f Fv(S)3107 2497 y Fu(\()p Ft(\033)r Fu(\))3209 2533 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))34 b(is)120 2654 y(unitary)e(for)g(a.e.)44 b Fv(\025)p Fw(.)120 2774 y(In)f(this)f(de\014nition)g(the)h (scattering)g(matrix)e(is)h(determined)h(only)f(almost)f(ev)m (erywhere.)120 2894 y(All)33 b(the)j(prop)s(erties)f(of)f Fv(S)1093 2858 y Fu(\()p Ft(\033)r Fu(\))1195 2894 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))34 b(discussed)j(b)s(elo)m(w)e(are)g (also)f(only)h(satis\014es)g(for)g(a.e.)51 b Fv(\025)p Fw(,)120 3015 y(whic)m(h)33 b(is)f(not)h(men)m(tioned)f(eac)m(h)h (time.)120 3135 y(Let)j(us)g(in)m(tro)s(duce)f(for)g Fv(\025)d Fs(2)h Fw([)p Fv(k)1283 3099 y Fu(2)1323 3135 y Fv(;)17 b Fw(+)p Fs(1)p Fw(\))23 b Fs(n)h(f)p Fw(\()p Fv(n)g Fw(+)g Fv(k)s Fw(\))2040 3099 y Fu(2)2112 3135 y Fw(:)65 b Fv(n)32 b Fs(2)h Fr(Z)p Fs(g)g Fw(t)m(w)m(o)j(auxiliary)d (op)s(erators)120 3269 y Fv(Z)194 3218 y Fu(\()p Ft(\033)r Fu(\))187 3294 y(0)296 3269 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))p Fv(;)33 b(Z)661 3233 y Fu(\()p Ft(\033)r Fu(\))762 3269 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))27 b(:)h Fv(L)1141 3284 y Fu(2)1181 3269 y Fw(\()p Fr(T)p Fw(\))j Fs(!)c Fr(C)1544 3233 y Ft(N)7 b Fu(\()q Ft(\025;k)r Fu(\))1772 3269 y Fw(,)278 3534 y(\()p Fv(Z)390 3483 y Fu(\()p Ft(\033)r Fu(\))383 3558 y(0)492 3534 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))p Fv(f)11 b Fw(\))820 3549 y Ft(n)894 3534 y Fw(:=)1395 3466 y(1)p 1034 3511 771 4 v 1034 3544 a Fs(p)p 1117 3544 59 4 v 72 x Fv(\031)1204 3578 y Fm(4)1179 3531 y Fo(p)p 1279 3531 527 4 v 85 x Fv(\025)22 b Fs(\000)g Fw(\()p Fv(n)h Fw(+)f Fv(k)s Fw(\))1766 3587 y Fu(2)1832 3423 y Fo(\020)1891 3534 y Fw(\(sgn)17 b Fv(\033)t Fw(\))2184 3444 y Fo(p)p 2284 3444 115 4 v 90 x Fs(j)p Fv(\033)t Fs(j)o Fv(f)2457 3423 y Fo(\021)2512 3482 y(b)2516 3602 y Ft(n)2580 3534 y Fv(;)212 b Fw(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))3127 3492 y Fu(2)3194 3534 y Fv(<)28 b(\025;)278 3811 y Fw(\()p Fv(Z)390 3770 y Fu(\()p Ft(\033)r Fu(\))492 3811 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))p Fv(f)11 b Fw(\))820 3826 y Ft(n)894 3811 y Fw(:=)1395 3744 y(1)p 1034 3788 771 4 v 1034 3821 a Fs(p)p 1117 3821 59 4 v 72 x Fv(\031)1204 3855 y Fm(4)1179 3808 y Fo(p)p 1279 3808 527 4 v 85 x Fv(\025)22 b Fs(\000)g Fw(\()p Fv(n)h Fw(+)f Fv(k)s Fw(\))1766 3864 y Fu(2)1832 3700 y Fo(\020)1891 3721 y(p)p 1991 3721 115 4 v 90 x Fs(j)p Fv(\033)t Fs(j)o Fv(f)2164 3700 y Fo(\021)2219 3760 y(b)2223 3880 y Ft(n)2287 3811 y Fv(;)212 b Fw(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))2834 3770 y Fu(2)2901 3811 y Fv(<)28 b(\025:)120 4101 y Fw(The)34 b(adjoin)m(t)d(op)s(erators)h(are)h(giv)m(en)g(b)m(y)608 4338 y Fv(Z)682 4287 y Fu(\()p Ft(\033)r Fu(\))675 4363 y(0)784 4338 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))1015 4297 y Fl(\003)1054 4338 y Fv(a)28 b Fw(:=)f(\(sgn)17 b Fv(\033)t Fw(\))1556 4248 y Fo(p)p 1656 4248 V 90 x Fs(j)p Fv(\033)t Fs(j)1876 4244 y Fo(X)1787 4461 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))1978 4442 y Fm(2)2013 4461 y Ft(<\025)2472 4271 y Fv(a)2523 4286 y Ft(n)p 2135 4315 771 4 v 2135 4349 a Fs(p)p 2218 4349 59 4 v 71 x Fv(\031)2305 4383 y Fm(4)2280 4335 y Fo(p)p 2380 4335 527 4 v 85 x Fv(\025)22 b Fs(\000)g Fw(\()p Fv(n)h Fw(+)f Fv(k)s Fw(\))2867 4392 y Fu(2)2933 4338 y Fv(e)2978 4353 y Ft(n)3025 4338 y Fv(;)608 4631 y(Z)682 4590 y Fu(\()p Ft(\033)r Fu(\))784 4631 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))1015 4590 y Fl(\003)1054 4631 y Fv(a)28 b Fw(:=)1263 4541 y Fo(p)p 1363 4541 115 4 v 90 x Fs(j)p Fv(\033)t Fs(j)1583 4536 y Fo(X)1494 4753 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))1685 4734 y Fm(2)1720 4753 y Ft(<\025)2179 4563 y Fv(a)2230 4578 y Ft(n)p 1842 4608 771 4 v 1842 4641 a Fs(p)p 1925 4641 59 4 v 72 x Fv(\031)2012 4675 y Fm(4)1987 4628 y Fo(p)p 2087 4628 527 4 v 85 x Fv(\025)22 b Fs(\000)g Fw(\()p Fv(n)h Fw(+)f Fv(k)s Fw(\))2574 4684 y Fu(2)2640 4631 y Fv(e)2685 4646 y Ft(n)120 4974 y Fw(for)32 b Fv(a)c Fs(2)g Fr(C)508 4937 y Ft(N)7 b Fu(\()p Ft(\025)q(;k)r Fu(\))735 4974 y Fw(.)120 5094 y(W)-8 b(e)22 b(are)g(no)m(w)g(ready)h (to)e(state)i(the)f(main)e(result)i(of)f(this)h(section.)40 b(The)22 b(op)s(erators)g Fv(T)3188 5043 y Fu(\()p Ft(\033)r Fu(\))3174 5118 y(0)3289 5094 y Fw(\()p Fv(z)t(;)17 b(k)s Fw(\),)120 5214 y Fv(T)191 5178 y Fu(\()p Ft(\033)r Fu(\))292 5214 y Fw(\()p Fv(z)t(;)g(k)s Fw(\))31 b(and)f(their)g(b)s(oundary)g(v) -5 b(alues)31 b(for)e Fv(z)k Fw(=)27 b Fv(\025)17 b Fs(\006)h Fv(i)p Fw(0)30 b(ha)m(v)m(e)i(b)s(een)f(de\014ned)g(in)f(Sections)120 5335 y(1)i(and)h(2.)1781 5712 y(44)p eop %%Page: 45 50 45 49 bop 120 407 a Fp(Theorem)26 b(3.4.3.)35 b Fj(L)-5 b(et)27 b Fv(k)k Fs(2)d Fw([)p Fs(\000)1330 368 y Fu(1)p 1330 384 36 4 v 1330 441 a(2)1376 407 y Fv(;)1429 368 y Fu(1)p 1429 384 V 1429 441 a(2)1474 407 y Fw(])f Fj(and)f Fv(\025)h Fs(2)h Fw([)p Fv(k)1968 371 y Fu(2)2008 407 y Fv(;)17 b Fw(+)p Fs(1)p Fw(\))s Fs(n)s Fw(\()o Fs(f)p Fw(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))2717 371 y Fu(2)2784 407 y Fw(:)28 b Fv(n)g Fs(2)g Fr(Z)p Fs(g)19 b([)k(N)15 b Fw(\()p Fv(k)s Fw(\)\))o Fj(.)120 527 y(Then)126 750 y Fv(S)192 708 y Fu(\()p Ft(\033)r Fu(\))294 750 y Fw(\()p Fv(\025;)i(k)s Fw(\))27 b(=)h Fv(I)i Fs(\000)22 b Fw(2)p Fv(\031)t(iZ)1043 708 y Fu(\()p Ft(\033)r Fu(\))1145 750 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))1393 639 y Fo(\020)1451 750 y Fv(I)30 b Fw(+)22 b Fv(T)1693 699 y Fu(\()p Ft(\033)r Fu(\))1679 774 y(0)1795 750 y Fw(\()p Fv(\025)g Fw(+)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))g Fv(T)2316 708 y Fu(\()p Ft(\033)r Fu(\))2417 750 y Fw(\()p Fv(\025)22 b Fs(\000)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))2851 708 y Fl(\003)2891 639 y Fo(\021)2950 661 y Fl(\000)p Fu(1)3061 750 y Fv(Z)3135 699 y Fu(\()p Ft(\033)r Fu(\))3128 774 y(0)3237 750 y Fw(\()p Fv(\025;)g(k)s Fw(\))3468 708 y Fl(\003)3507 750 y Fv(:)3264 897 y Fw(\(3.4.1\))266 1087 y(W)-8 b(e)30 b(will)d(deriv)m(e)j(this)e(result)i(from)d(Theorem)j(5.7.1')f(in)f([Y) -8 b(a1].)42 b(W)-8 b(e)30 b(need)g(some)f(prepa-)120 1207 y(rations.)120 1327 y(Consider)k(a)f(closed)h(in)m(terv)-5 b(al)917 1531 y(\003)28 b(=)f([)p Fv(a;)17 b(b)p Fw(])29 b Fs(\032)f Fw([)p Fv(k)1521 1490 y Fu(2)1560 1531 y Fv(;)17 b Fw(+)p Fs(1)p Fw(\))22 b Fs(n)g(f)p Fw(\()p Fv(n)g Fw(+)g Fv(k)s Fw(\))2270 1490 y Fu(2)2337 1531 y Fw(:)55 b Fv(n)28 b Fs(2)g Fr(Z)p Fs(g)p Fv(;)519 b Fw(\(3.4.2\))120 1734 y(and)33 b(the)g(sp)s(ectral)f(pro)5 b(jection)32 b Fv(E)1385 1698 y Fu(\(0\))1480 1734 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))32 b(of)g Fv(H)1943 1698 y Fu(\(0\))2037 1734 y Fw(\()p Fv(k)s Fw(\))g(on)m(to)h(\003.)43 b(Put)704 1996 y Fv(\037)765 2011 y Ft(n)812 1996 y Fw(\()p Fv(\030)5 b Fw(\))27 b(:=)1094 1855 y Fo(\032)1210 1935 y Fw(1)195 b(if)27 b Fv(\030)1587 1898 y Fu(2)1653 1935 y Fs(2)h Fw([)p Fv(a)23 b Fs(\000)f Fw(\()p Fv(n)h Fw(+)f Fv(k)s Fw(\))2256 1898 y Fu(2)2295 1935 y Fv(;)17 b(b)23 b Fs(\000)f Fw(\()p Fv(n)g Fw(+)g Fv(k)s Fw(\))2810 1898 y Fu(2)2850 1935 y Fw(])p Fv(;)1210 2055 y Fw(0)195 b(otherwise)q Fv(:)120 2257 y Fw(\(The)28 b(dep)s(endence)h(on)d(\003)p Fv(;)33 b(k)d Fw(is)c(not)g(re\015ected)j(in)c(this)i(notation.\))40 b(Using)26 b(the)h(results)g(from)120 2377 y(Section)32 b(2.2)h(w)m(e)g(see)h(that)1078 2581 y Fv(E)1156 2540 y Fu(\(0\))1250 2581 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))p Fv(u)27 b Fw(=)1667 2486 y Fo(X)1672 2698 y Ft(n)p Fl(2)p Fn(Z)1828 2581 y Fv(e)1873 2596 y Ft(n)1942 2581 y Fs(\012)c(F)2124 2540 y Fl(\003)2114 2605 y Ft(c)2163 2581 y Fw(\()p Fv(\037)2262 2596 y Ft(n)2309 2581 y Fs(F)2381 2596 y Ft(c)2421 2581 y Fw(^)-55 b Fv(u)2471 2596 y Ft(n)2517 2581 y Fw(\))p Fv(;)682 b Fw(\(3.4.3\))120 2878 y(where)48 b Fs(F)488 2893 y Ft(c)569 2878 y Fw(denotes)f(the)g(F)-8 b(ourier)46 b(cosine)h(transformation)d(on)j Fv(L)2665 2893 y Fu(2)2704 2878 y Fw(\()p Fr(R)2808 2893 y Fu(+)2873 2878 y Fw(\).)86 b(The)47 b(sum)g(in)120 2999 y(\(3.4.3\))36 b(is)h(actually)e(a)i (\014nite)f(sum)h(since)g Fv(\037)1765 3014 y Ft(n)1847 2999 y Fs(\021)e Fw(0)i(if)f(\()p Fv(n)25 b Fw(+)g Fv(k)s Fw(\))2453 2962 y Fu(2)2527 2999 y Fv(>)35 b(b)p Fw(.)57 b(Moreo)m(v)m(er,)39 b Fv(\037)3284 3014 y Ft(n)3331 2999 y Fs(F)3403 3014 y Ft(c)3444 2999 y Fw(^)-55 b Fv(u)3494 3014 y Ft(n)120 3119 y Fw(has)32 b(b)s(ounded)f(supp)s(ort,)h(so)g Fs(F)1277 3083 y Fl(\003)1267 3144 y Ft(c)1316 3119 y Fw(\()p Fv(\037)1415 3134 y Ft(n)1462 3119 y Fs(F)1534 3134 y Ft(c)1574 3119 y Fw(^)-55 b Fv(u)1624 3134 y Ft(n)1671 3119 y Fw(\))31 b(is)f(a)h(smo)s(oth)f(function)h(on)p 2774 3038 131 4 v 31 w Fr(R)2840 3134 y Fu(+)2905 3119 y Fw(.)43 b(In)32 b(particular,)120 3239 y(w)m(e)i(ha)m(v)m(e)h Fv(E)568 3203 y Fu(\(0\))662 3239 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))p Fv(u)29 b Fs(2)1098 3214 y Fw(~)1073 3239 y Fv(H)1162 3203 y Fu(1)1201 3239 y Fw(\(\005\))k(for)g(an)m(y)h Fv(u)29 b Fs(2)g Fv(L)1964 3254 y Fu(2)2004 3239 y Fw(\(\005\).)46 b(It)33 b(is)g(easy)h(to)f(see)i(that)e Fv(E)3215 3203 y Fu(\(0\))3309 3239 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))120 3360 y(is)32 b(b)s(ounded)h(as)g(op)s(erator)f(from)f Fv(L)1425 3375 y Fu(2)1465 3360 y Fw(\(\005\))i(to)1791 3334 y(~)1766 3360 y Fv(H)1855 3324 y Fu(1)1894 3360 y Fw(\(\005\).)120 3480 y(No)m(w)26 b(consider)f(the)h(op)s(erators)f Fv(G)1369 3495 y Fu(0)1408 3480 y Fv(;)33 b(G)25 b Fw(from)f Fv(L)1859 3495 y Fu(2)1899 3480 y Fw(\(\005\))h(to)g Fv(L)2251 3495 y Fu(2)2291 3480 y Fw(\()p Fr(T)p Fw(\))j(as)e(in)m(tro) s(duced)f(in)f(the)i(pro)s(of)120 3600 y(of)32 b(Prop)s(osition)f (3.1.2,)819 3802 y Fs(D)s Fw(\()p Fv(G)1014 3817 y Fu(0)1053 3802 y Fw(\))d(:=)f Fs(D)s Fw(\()p Fv(G)p Fw(\))g(:=)1665 3777 y(~)1640 3802 y Fv(H)1729 3761 y Fu(1)1768 3802 y Fw(\(\005\))p Fv(;)672 3970 y Fw(\()p Fv(G)787 3985 y Fu(0)827 3970 y Fv(u)p Fw(\)\()p Fv(x)1014 3985 y Fu(1)1053 3970 y Fw(\))h(:=)f(\(sgn)c Fv(\033)t Fw(\()p Fv(x)1603 3985 y Fu(1)1643 3970 y Fw(\)\))1719 3880 y Fo(p)p 1818 3880 286 4 v 1818 3970 a Fs(j)p Fv(\033)t Fw(\()p Fv(x)1998 3985 y Fu(1)2038 3970 y Fw(\))p Fs(j)16 b Fv(u)p Fw(\()p Fv(x)2269 3985 y Fu(1)2308 3970 y Fv(;)h Fw(0\))p Fv(;)211 b(x)2732 3985 y Fu(1)2800 3970 y Fs(2)28 b Fr(T)p Fv(;)712 4137 y Fw(\()p Fv(Gu)p Fw(\)\()p Fv(x)1014 4152 y Fu(1)1053 4137 y Fw(\))g(:=)1249 4047 y Fo(p)p 1349 4047 V 90 x Fs(j)p Fv(\033)t Fw(\()p Fv(x)1529 4152 y Fu(1)1568 4137 y Fw(\))p Fs(j)17 b Fv(u)p Fw(\()p Fv(x)1800 4152 y Fu(1)1839 4137 y Fv(;)g Fw(0\))p Fv(;)211 b(x)2263 4152 y Fu(1)2331 4137 y Fs(2)28 b Fr(T)p Fv(:)120 4356 y Fw(W)-8 b(e)27 b(ha)m(v)m(e)g(remark)m(ed)g(there)h(that)e Fv(G)1448 4371 y Fu(0)1487 4356 y Fv(;)33 b(G)27 b Fw(are)f(b)s(ounded)h(op)s (erators)f(from)2874 4331 y(~)2848 4356 y Fv(H)2937 4320 y Fu(1)2976 4356 y Fw(\(\005\))h(to)f Fv(L)3331 4371 y Fu(2)3371 4356 y Fw(\()p Fr(T)p Fw(\).)120 4476 y(Hence)34 b Fv(G)487 4491 y Fu(0)526 4476 y Fv(E)604 4440 y Fu(\(0\))699 4476 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))p Fv(;)33 b(GE)1145 4440 y Fu(\(0\))1239 4476 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))32 b(are)h(b)s(ounded)g(op)s(erators)f(from)g Fv(L)2791 4491 y Fu(2)2830 4476 y Fw(\(\005\))h(to)f Fv(L)3197 4491 y Fu(2)3237 4476 y Fw(\()p Fr(T)p Fw(\).)120 4610 y(The)i(next)f(result)g(sho)m(ws)h(that)e(the)h(op)s(erators)g Fv(Z)1977 4559 y Fu(\()p Ft(\033)r Fu(\))1970 4635 y(0)2079 4610 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))p Fv(;)32 b(Z)2443 4574 y Fu(\()p Ft(\033)r Fu(\))2545 4610 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))32 b(coincide)g(with)h(the)120 4731 y(op)s(erators)42 b Fv(G)638 4746 y Fu(0)677 4731 y Fv(;)33 b(G)42 b Fw(in)f(the)h (direct)g(in)m(tegral)e(decomp)s(osition)g Fg(h)p Fw(\()p Fv(k)s Fw(\).)71 b(More)42 b(precisely)-8 b(,)45 b(w)m(e)120 4851 y(ha)m(v)m(e)120 5040 y Fp(Lemma)37 b(3.4.4.)42 b Fj(L)-5 b(et)35 b Fv(k)c Fs(2)d Fw([)p Fs(\000)1278 5001 y Fu(1)p 1278 5017 36 4 v 1278 5074 a(2)1324 5040 y Fv(;)1378 5001 y Fu(1)p 1378 5017 V 1378 5074 a(2)1423 5040 y Fw(])35 b Fj(and)f Fw(\003)h Fj(as)f(in)42 b Fw(\(3.4.2\))p Fj(.)i(Then)34 b(for)h Fv(f)k Fs(2)28 b Fv(L)3036 5055 y Fu(2)3075 5040 y Fw(\()p Fr(T)p Fw(\))680 5266 y Fv(Z)754 5216 y Fu(\()p Ft(\033)r Fu(\))747 5291 y(0)856 5266 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))p Fv(f)38 b Fw(=)27 b(\(\000)1375 5225 y Fu(\(0\))1469 5266 y Fw(\()p Fv(k)s Fw(\)\()p Fv(G)1714 5281 y Fu(0)1754 5266 y Fv(E)1832 5225 y Fu(\(0\))1926 5266 y Fw(\()p Fv(\025;)17 b(k)s Fw(\)\))2195 5225 y Fl(\003)2234 5266 y Fv(f)11 b Fw(\)\()p Fv(\025)p Fw(\))p Fv(;)215 b(\025)28 b Fs(2)g Fw(\003)p Fv(;)680 5427 y(Z)754 5386 y Fu(\()p Ft(\033)r Fu(\))856 5427 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))p Fv(f)38 b Fw(=)27 b(\(\000)1375 5386 y Fu(\(0\))1469 5427 y Fw(\()p Fv(k)s Fw(\)\()p Fv(GE)1792 5386 y Fu(\(0\))1887 5427 y Fw(\()p Fv(\025;)17 b(k)s Fw(\)\))2156 5386 y Fl(\003)2195 5427 y Fv(f)11 b Fw(\)\()p Fv(\025)p Fw(\))p Fv(;)215 b(\025)28 b Fs(2)g Fw(\003)p Fv(:)3264 5340 y Fw(\(3.4.4\))1781 5712 y(45)p eop %%Page: 46 51 46 50 bop 120 407 a Fj(Pr)-5 b(o)g(of.)48 b Fw(W)-8 b(e)37 b(pro)m(v)m(e)g(only)f(the)g(second)i(relation.)52 b(Let)36 b(us)h(\014rst)g(calculate)e(the)h(adjoin)m(t)g(op-)120 527 y(erator)c(\()p Fv(GE)600 491 y Fu(\(0\))694 527 y Fw(\()p Fv(\025;)17 b(k)s Fw(\)\))963 491 y Fl(\003)1002 527 y Fw(.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)h(for)e Fv(u)27 b Fs(2)h Fv(L)1858 542 y Fu(2)1898 527 y Fw(\(\005\))278 802 y(\()p Fv(e)361 817 y Ft(n)430 802 y Fs(\012)23 b(F)612 761 y Fl(\003)602 827 y Ft(c)651 802 y Fv(\037)712 817 y Ft(n)759 802 y Fv(;)17 b(u)p Fw(\))896 831 y Ft(L)944 840 y Fm(2)979 831 y Fu(\(\005\))1118 802 y Fw(=)1222 667 y Fo(Z)1321 693 y Fl(1)1277 892 y Fu(0)1413 802 y Fs(F)1495 761 y Fl(\003)1485 827 y Ft(c)1534 802 y Fv(\037)1595 817 y Ft(n)1642 802 y Fw(\()p Fv(x)1735 817 y Fu(2)1775 802 y Fw(\))p 1813 646 1321 4 v -140 x Fo(\022)1967 735 y Fw(1)p 1896 779 191 4 v 1896 799 a Fs(p)p 1979 799 108 4 v 83 x Fw(2)p Fv(\031)2113 667 y Fo(Z)2213 693 y Ft(\031)2168 892 y Fl(\000)p Ft(\031)2287 802 y Fv(u)p Fw(\()p Fv(x)2436 817 y Fu(1)2475 802 y Fv(;)g(x)2574 817 y Fu(2)2614 802 y Fw(\))p Fv(e)2697 774 y Fl(\000)p Ft(inx)2859 783 y Fm(1)2914 802 y Fv(dx)3020 817 y Fu(1)3059 662 y Fo(\023)3149 802 y Fv(dx)3255 817 y Fu(2)3323 802 y Fw(=)484 1070 y(=)588 935 y Fo(Z)687 961 y Fl(1)643 1160 y Fu(0)779 1070 y Fv(\037)840 1085 y Ft(n)887 1070 y Fw(\()p Fv(\030)5 b Fw(\))p 1011 984 408 4 v(\()p Fs(F)1121 1085 y Ft(c)1160 1070 y Fw(^)-55 b Fv(u)1210 1085 y Ft(n)1257 1070 y Fw(\)\()p Fv(\030)5 b Fw(\))16 b Fv(d\030)32 b Fw(=)1664 935 y Fo(Z)1764 961 y Fl(1)1719 1160 y Fu(0)p 1855 980 822 4 v 1855 1070 a Fs(F)1927 1085 y Ft(c)1961 1070 y Fw(\()p Fv(E)2077 1042 y Fu(\(0\))2172 1070 y Fw(\(\003)p Fv(;)17 b(k)s Fw(\))p Fv(u)p Fw(\))2504 1079 y Fo(b)2508 1095 y Ft(n)2553 1070 y Fw(\()p Fv(\030)5 b Fw(\))16 b Fv(d\030)32 b Fw(=)484 1347 y(=)588 1201 y Fo(r)p 687 1201 79 4 v 697 1280 a Fv(\031)p 697 1324 59 4 v 702 1416 a Fw(2)p 783 1257 717 4 v 783 1347 a(\()p Fv(E)899 1319 y Fu(\(0\))993 1347 y Fw(\(\003)p Fv(;)17 b(k)s Fw(\))p Fv(u)p Fw(\))1325 1356 y Fo(b)1329 1372 y Ft(n)1375 1347 y Fw(\(0\))p Fv(;)120 1597 y Fw(where)31 b(w)m(e)g(ha)m(v)m(e)h(used)f(\(3.4.3\))f(in)f(the)h(next)i(to)d(last)h (step)h(and)f(the)g(in)m(v)m(ersion)h(theorem)e(for)120 1718 y Fs(F)192 1733 y Ft(c)259 1718 y Fw(in)j(the)h(last)e(step.)120 1838 y(No)m(w)44 b(let)e Fv(f)56 b Fs(2)46 b Fv(L)786 1853 y Fu(2)826 1838 y Fw(\()p Fr(T)p Fw(\))g(b)s(e)d(b)s(ounded)h (\(so)f(that)1953 1753 y Fo(p)p 2053 1753 115 4 v 85 x Fs(j)p Fv(\033)t Fs(j)o Fv(f)56 b Fs(2)46 b Fv(L)2449 1853 y Fu(2)2489 1838 y Fw(\()p Fr(T)p Fw(\)\),)j(then)43 b(b)m(y)h(P)m(arsev)-5 b(al's)120 1958 y(equalit)m(y)137 2097 y Fo(\000)182 2178 y Fv(f)5 b(;)17 b(GE)434 2137 y Fu(\(0\))529 2178 y Fw(\()p Fv(\025;)g(k)s Fw(\))p Fv(u)816 2097 y Fo(\001)861 2217 y Ft(L)909 2226 y Fm(2)943 2217 y Fu(\()p Fn(T)p Fu(\))1074 2178 y Fw(=)1178 2068 y Fo(\020)1237 2088 y(p)p 1337 2088 V 90 x Fs(j)p Fv(\033)t Fs(j)o Fv(f)5 b(;)17 b(E)1626 2137 y Fu(\(0\))1721 2178 y Fw(\()p Fv(\025;)g(k)s Fw(\))p Fv(u)p Fw(\()p Fv(:;)g Fw(0\))2204 2068 y Fo(\021)2262 2247 y Ft(L)2310 2256 y Fm(2)2344 2247 y Fu(\()p Fn(T)p Fu(\))2475 2178 y Fw(=)343 2471 y(=)447 2377 y Fo(X)451 2588 y Ft(n)p Fl(2)p Fn(Z)590 2471 y Fw(\()628 2381 y Fo(p)p 728 2381 V 90 x Fs(j)p Fv(\033)t Fs(j)o Fv(f)11 b Fw(\))935 2467 y Fo(b)939 2496 y Ft(n)p 1003 2381 717 4 v 1003 2471 a Fw(\()p Fv(E)1119 2442 y Fu(\(0\))1213 2471 y Fw(\(\003)p Fv(;)17 b(k)s Fw(\))p Fv(u)p Fw(\))1545 2479 y Fo(b)1549 2496 y Ft(n)1595 2471 y Fw(\(0\))27 b(=)1851 2301 y Fo( )1929 2315 y(r)p 2029 2315 79 4 v 2044 2404 a Fw(2)p 2039 2448 59 4 v 2039 2540 a Fv(\031)2124 2377 y Fo(X)2129 2588 y Ft(n)p Fl(2)p Fn(Z)2268 2471 y Fw(\()2306 2381 y Fo(p)p 2406 2381 115 4 v 90 x Fs(j)p Fv(\033)t Fs(j)o Fv(f)11 b Fw(\))2613 2467 y Fo(b)2617 2496 y Ft(n)2681 2471 y Fv(e)2726 2486 y Ft(n)2795 2471 y Fs(\012)23 b(F)2977 2430 y Fl(\003)2967 2496 y Ft(c)3015 2471 y Fv(\037)3076 2486 y Ft(n)3124 2471 y Fv(;)17 b(u)3224 2301 y Fo(!)3302 2607 y Ft(L)3350 2616 y Fm(2)3384 2607 y Fu(\(\005\))3513 2471 y Fv(:)120 2784 y Fw(W)-8 b(e)33 b(remark)f(again)f(that)i(the)g(last)f(sum)g(is)g (a)g(\014nite)h(sum.)43 b(Th)m(us)34 b(for)e(an)m(y)i Fv(f)k Fs(2)28 b Fv(L)3152 2799 y Fu(2)3192 2784 y Fw(\()p Fr(T)p Fw(\))827 3059 y(\()p Fv(GE)1020 3018 y Fu(\(0\))1114 3059 y Fw(\()p Fv(\025;)17 b(k)s Fw(\)\))1383 3018 y Fl(\003)1422 3059 y Fv(f)38 b Fw(=)1612 2903 y Fo(r)p 1712 2903 79 4 v 1727 2992 a Fw(2)p 1722 3036 59 4 v 1722 3127 a Fv(\031)1807 2964 y Fo(X)1812 3176 y Ft(n)p Fl(2)p Fn(Z)1951 3059 y Fw(\()1989 2969 y Fo(p)p 2088 2969 115 4 v 2088 3059 a Fs(j)p Fv(\033)t Fs(j)p Fv(f)11 b Fw(\))2296 3055 y Fo(b)2300 3084 y Ft(n)2363 3059 y Fv(e)2408 3074 y Ft(n)2477 3059 y Fs(\012)23 b(F)2659 3018 y Fl(\003)2649 3084 y Ft(c)2698 3059 y Fv(\037)2759 3074 y Ft(n)2806 3059 y Fv(:)120 3351 y Fw(No)m(w)33 b(it)f(is)g(easy)h(to)g(pro)m(v)m(e)g(\(3.4.4\).)43 b(Indeed,)960 3626 y Fs(F)1032 3641 y Ft(c)1066 3626 y Fw(\(\()p Fv(GE)1297 3585 y Fu(\(0\))1391 3626 y Fw(\()p Fv(\025;)17 b(k)s Fw(\)\))1660 3585 y Fl(\003)1699 3626 y Fv(f)11 b Fw(\))1792 3622 y Fo(b)1796 3651 y Ft(n)1871 3626 y Fw(=)1974 3470 y Fo(r)p 2074 3470 79 4 v 2089 3559 a Fw(2)p 2084 3603 59 4 v 2084 3695 a Fv(\031)2153 3626 y Fw(\()2191 3536 y Fo(p)p 2290 3536 115 4 v 2290 3626 a Fs(j)p Fv(\033)t Fs(j)p Fv(f)g Fw(\))2498 3622 y Fo(b)2502 3651 y Ft(n)2565 3626 y Fv(\037)2626 3641 y Ft(n)2673 3626 y Fv(;)120 3870 y Fw(and)33 b(th)m(us)g(for)f Fv(\025)c Fs(2)g Fw(\003)k(and)h(\() p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))1450 3834 y Fu(2)1517 3870 y Fv(<)28 b(\025)432 4071 y Fw(\(\000)531 4030 y Fu(\(0\))625 4071 y Fw(\()p Fv(k)s Fw(\)\()p Fv(GE)948 4030 y Fu(\(0\))1043 4071 y Fw(\()p Fv(\025;)17 b(k)s Fw(\)\))1312 4030 y Fl(\003)1351 4071 y Fv(f)11 b Fw(\)\()p Fv(\025)p Fw(\))27 b(=)655 4272 y(=)1124 4205 y(1)p 769 4249 761 4 v 769 4272 a Fs(p)p 852 4272 49 4 v 83 x Fw(2)928 4317 y Fm(4)903 4269 y Fo(p)p 1003 4269 527 4 v 86 x Fv(\025)22 b Fs(\000)g Fw(\()p Fv(n)h Fw(+)f Fv(k)s Fw(\))1490 4326 y Fu(2)1539 4272 y Fs(F)1611 4287 y Ft(c)1645 4272 y Fw(\(\()p Fv(GE)1876 4231 y Fu(\(0\))1971 4272 y Fw(\()p Fv(\025;)17 b(k)s Fw(\)\))2240 4231 y Fl(\003)2279 4272 y Fv(f)11 b Fw(\))2372 4268 y Fo(b)2376 4297 y Ft(n)2422 4272 y Fw(\()2460 4182 y Fo(p)p 2560 4182 V 90 x Fv(\025)22 b Fs(\000)g Fw(\()p Fv(n)h Fw(+)f Fv(k)s Fw(\))3047 4244 y Fu(2)3086 4272 y Fw(\))28 b(=)655 4509 y(=)g Fv(Z)833 4468 y Fu(\()p Ft(\033)r Fu(\))934 4509 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))p Fv(f)120 4724 y Fw(since)33 b Fv(\037)420 4739 y Ft(n)467 4724 y Fw(\()505 4639 y Fo(p)p 605 4639 V 85 x Fv(\025)22 b Fs(\000)g Fw(\()p Fv(n)h Fw(+)f Fv(k)s Fw(\))1092 4695 y Fu(2)1131 4724 y Fw(\))28 b(=)f(1.)43 b(This)33 b(pro)m(v)m(es)i(the)e(second)g(relation)e(in)h(\(3.4.4\))o (.)p 3469 4724 4 66 v 3473 4662 59 4 v 3473 4724 V 3531 4724 4 66 v 266 4913 a(W)-8 b(e)33 b(can)g(rewrite)g(\(3.4.4\))f(as)467 5162 y(\()p Fv(G)582 5177 y Fu(0)621 5162 y Fv(u;)17 b(f)11 b Fw(\))818 5177 y Ft(L)866 5186 y Fm(2)899 5177 y Fu(\()p Fn(T)p Fu(\))1030 5162 y Fw(=)1134 5026 y Fo(Z)1233 5052 y Fl(1)1189 5252 y Ft(k)1228 5233 y Fm(2)1325 5051 y Fo(\020)1384 5162 y Fw(\(\000)1483 5121 y Fu(\(0\))1578 5162 y Fw(\()p Fv(k)s 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Fs(2)45 b Fv(L)1583 422 y Fu(2)1622 407 y Fw(\()p Fr(T)p Fw(\).)76 b(Indeed,)46 b(b)s(oth)c(sides)g(\(e.g.)72 b(of)42 b(the)h(second)120 527 y(equation\))32 b(are)h(equal)f(to)448 751 y(\()p Fv(u;)17 b Fw(\()p Fv(GE)779 710 y Fu(\(0\))872 751 y Fw(\(\003)p Fv(;)g(k)s Fw(\)\))1152 710 y Fl(\003)1191 751 y Fv(f)11 b Fw(\))1288 766 y Ft(L)1336 775 y Fm(2)1370 766 y Fu(\(\005\))1510 751 y Fw(=)670 958 y(=)774 823 y Fo(Z)874 849 y Fl(1)829 1048 y Ft(k)868 1029 y Fm(2)965 878 y Fo(\000)1011 958 y Fw(\(\000)1110 917 y Fu(\(0\))1204 958 y Fw(\()p Fv(k)s Fw(\))p Fv(u)p Fw(\)\()p Fv(\025)p Fw(\))p Fv(;)17 b Fw(\(\000)1704 917 y Fu(\(0\))1797 958 y Fw(\()p Fv(k)s Fw(\)\()p Fv(GE)2120 917 y Fu(\(0\))2214 958 y Fw(\()p Fv(\025;)g(k)s Fw(\)\))2483 917 y Fl(\003)2522 958 y Fv(f)11 b Fw(\)\()p Fv(\025)p Fw(\))2752 878 y Fo(\001)2797 997 y Fn(C)2839 978 y Fi(N)6 b Fm(\()p Fi(\025;k)q Fm(\))3078 958 y Fv(d\025:)120 1247 y Fw(Of)30 b(course,)i(\(3.4.5\))e (holds)g(for)g(all)e Fv(u)f Fs(2)h(R)p Fw(\()p Fv(E)1797 1211 y Fu(\(0\))1892 1247 y Fw(\()1939 1222 y(~)1930 1247 y(\003)p Fv(;)17 b(k)s Fw(\)\))30 b(where)2491 1222 y(~)2482 1247 y(\003)g(a)g(\014nite)g(union)g(of)g(sets)h(\003)120 1368 y(as)i(ab)s(o)m(v)m(e.)120 1488 y(No)m(w)g(the)g(pro)s(of)f(of)g (Theorem)h(3.4.3)f(is)g(immediate.)120 1683 y Fj(Pr)-5 b(o)g(of)35 b(\(The)-5 b(or)g(em)33 b(3.4.3\).)48 b Fw(W)-8 b(e)35 b(apply)g(Theorem)g(5.7.1')g(in)f([Y)-8 b(a1].)50 b(Assumptions)36 b(\(5.7.5\))120 1804 y(and)29 b(\(5.5.6\))e(in)h([Y)-8 b(a1])28 b(ha)m(v)m(e)i(already)e(b)s(een)h(c)m(hec)m(k)m(ed)j(in)27 b(Section)h(2.)42 b(Our)29 b(form)m(ulae)e(\(3.4.5\))120 1938 y(v)m(erify)i(the)g(de\014ning)g(relation)e(\(5.4.2\))h(in)g([Y)-8 b(a1])28 b(for)h(the)g(op)s(erators)f Fv(Z)2741 1887 y Fu(\()p Ft(\033)r Fu(\))2734 1962 y(0)2843 1938 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))p Fv(;)33 b(Z)3208 1901 y Fu(\()p Ft(\033)r Fu(\))3309 1938 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))120 2058 y(\()p Fv(Z)225 2073 y Fu(0)264 2058 y Fw(\()p Fv(\025;)g(G)480 2073 y Fu(0)519 2058 y Fw(\))p Fv(;)33 b(Z)684 2073 y Fu(0)724 2058 y Fw(\()p Fv(\025;)17 b(G)p Fw(\))40 b(in)h(Y)-8 b(afaev's)42 b(notation\).)68 b(The)43 b(represen)m(tation)f(\(5.7.6\))f(in)f([Y)-8 b(a1])120 2178 y(coincides)32 b(with)h(our)f(equation)g(\(3.4.1\).)p 3469 2178 4 66 v 3473 2116 59 4 v 3473 2178 V 3531 2178 4 66 v 120 2438 a Fp(3.)49 b(Generalized)32 b(eigenfunctions.)98 b Fw(Recall)27 b(from)h(Section)h(2.2)f(the)i(de\014nition)e(of)h(the) 120 2559 y(functions)39 b Fv( )614 2508 y Fu(\(0\))610 2569 y Ft(n)709 2559 y Fw(\()p Fv(:)p Fw(;)17 b Fv(\025;)g(k)s Fw(\).)63 b(These)41 b(are)e(generalized)g(eigenfunctions)g(in)f(the)i (unp)s(erturb)s(ed)120 2679 y(case)33 b Fv(\033)f Fw(=)c(0.)43 b(The)33 b(result)g(in)f(the)h(general)f(case)h(is)120 2882 y Fp(Prop)s(osition)24 b(3.4.5.)34 b Fj(L)-5 b(et)26 b Fv(k)31 b Fs(2)d Fw([)p Fs(\000)1458 2843 y Fu(1)p 1458 2859 36 4 v 1458 2917 a(2)1504 2882 y Fv(;)1558 2843 y Fu(1)p 1558 2859 V 1558 2917 a(2)1603 2882 y Fw(])p Fj(,)f Fv(\025)h Fs(2)g Fw([)p Fv(k)1947 2846 y Fu(2)1987 2882 y Fv(;)17 b Fw(+)p Fs(1)p Fw(\))r Fs(n)r Fw(\()n Fs(f)p Fw(\()p Fv(m)22 b Fw(+)g Fv(k)s Fw(\))2720 2846 y Fu(2)2787 2882 y Fw(:)28 b Fv(m)g Fs(2)g Fr(Z)p Fs(g)19 b([)k(N)15 b Fw(\()p Fv(k)s Fw(\)\))120 3016 y Fj(and)33 b Fv(n)28 b Fs(2)g Fr(Z)j Fj(such)i(that)h Fw(\()p Fv(n)19 b Fw(+)g Fv(k)s Fw(\))1308 2980 y Fu(2)1375 3016 y Fv(<)28 b(\025)p Fj(.)44 b(Then)32 b(ther)-5 b(e)34 b(ar)-5 b(e)33 b(functions)g Fv( )2757 2980 y Fl(\006)2753 3041 y Ft(n)2816 3016 y Fw(\()p Fv(:)p Fw(;)17 b Fv(\025;)g(k)s Fw(\))27 b Fs(2)3262 2991 y Fw(~)3239 3016 y Fv(C)3316 2980 y Fl(1)3391 3016 y Fw(\(\005\))120 3137 y Fj(satisfying)722 3176 y Fo(\000)767 3257 y Fw(\()p Fv(D)886 3272 y Fu(1)948 3257 y Fw(+)22 b Fv(k)s Fw(\))1138 3216 y Fu(2)1200 3257 y Fw(+)g Fv(D)1382 3216 y Fu(2)1379 3282 y(2)1421 3176 y Fo(\001)1483 3257 y Fv( )1550 3216 y Fl(\006)1546 3282 y Ft(n)1609 3257 y Fw(\()p Fv(:)p Fw(;)17 b Fv(\025;)g(k)s Fw(\))27 b(=)h Fv(\025 )2166 3216 y Fl(\006)2162 3282 y Ft(n)2224 3257 y Fw(\()p Fv(:)p Fw(;)17 b Fv(\025;)g(k)s Fw(\))199 b Fj(in)55 b Fw(\005)120 3431 y Fj(and)658 3583 y Fw(lim)631 3643 y Ft(\017)p Fl(!)p Fu(0+)847 3516 y Fv(@)5 b( )970 3480 y Fl(\006)966 3541 y Ft(n)1030 3516 y Fw(\()p Fv(:)p Fw(;)17 b Fv(\025;)g(k)s Fw(\))p 847 3560 485 4 v 1013 3652 a Fv(@)5 b(x)1124 3667 y Fu(2)1358 3583 y Fs(j)1385 3598 y Ft(x)1425 3607 y Fm(2)1460 3598 y Fu(=)p Ft(\017)1585 3583 y Fw(=)28 b Fv(\033)t( )1815 3542 y Fl(\006)1811 3608 y Ft(n)1874 3583 y Fw(\()p Fv(:)p Fw(;)17 b Fv(\025;)g(k)s Fw(\))g Fs(j)2219 3598 y Ft(x)2259 3607 y Fm(2)2294 3598 y Fu(=0)2614 3583 y Fj(in)55 b Fv(L)2820 3598 y Fu(1)2860 3583 y Fw(\()p Fr(T)p Fw(\))p Fv(:)120 3798 y Fj(Mor)-5 b(e)g(over,)34 b(let)419 4063 y Fv(a)470 4078 y Ft(mn)580 4063 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))27 b(:=)h Fv(i)1002 3987 y Fs(p)p 1085 3987 59 4 v 76 x Fv(\031)1161 3928 y Fo(Z)1260 3954 y Ft(\031)1216 4153 y Fl(\000)p Ft(\031)1335 4063 y Fv( )1402 4022 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4712 y Fm(1)1733 4739 y Fv(e)1778 4700 y Ft(i)1802 4633 y Fs(p)p 1885 4633 342 4 v 67 x Ft(\025)p Fl(\000)p Fu(\()p Ft(m)p Fu(+)p Ft(k)r Fu(\))2191 4681 y Fm(2)2238 4700 y Ft(x)2278 4709 y Fm(2)p 1522 4784 794 4 v 1619 4851 a(4)1594 4804 y Fo(p)p 1694 4804 554 4 v 85 x Fv(\025)22 b Fs(\000)h Fw(\()p Fv(m)f Fw(+)g Fv(k)s Fw(\))2208 4860 y Fu(2)2349 4807 y Fw(+)g Fv(o)p Fw(\()p Fv(e)2577 4765 y Fl(\000)p Ft(\013x)2717 4774 y Fm(2)2755 4807 y Fw(\))199 b(\()p Fv(x)3085 4822 y Fu(2)3153 4807 y Fs(!)27 b(1)p Fw(\))p Fv(;)3264 5048 y Fw(\(3.4.7\))120 5261 y Fj(uniformly)35 b(in)f Fv(x)739 5276 y Fu(1)779 5261 y Fj(.)1781 5712 y Fw(47)p eop %%Page: 48 53 48 52 bop 266 407 a Fw(Of)34 b(course,)i(the)e(functions)g Fv( )1392 371 y Fl(\006)1388 431 y Ft(n)1451 407 y Fw(\()p Fv(:)p Fw(;)17 b Fv(\025;)g(k)s Fw(\))34 b(dep)s(end)h(on)f Fv(\033)t Fw(.)48 b(This)35 b(is)e(not)h(re\015ected)i(in)e(the)120 541 y(notation.)42 b(F)-8 b(or)32 b Fv(\033)g Fw(=)27 b(0)32 b(w)m(e)i(obtain)e(again)f(the)i(functions)f Fv( )2357 490 y Fu(\(0\))2353 551 y Ft(n)2452 541 y Fw(\()p Fv(:)p Fw(;)17 b Fv(\025;)g(k)s Fw(\).)120 661 y(W)-8 b(e)45 b(emphasize)g(that)f(the)h(functions)g(ma)m(y)f(not)h(b)s(e)g(con)m (tin)m(uous)g(up)g(to)f(the)i(b)s(oundary)120 782 y([)p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(])22 b Fs(\002)h(f)p Fw(0)p Fs(g)p Fw(.)120 902 y(Let)43 b(us)h(construct)h Fv( )948 866 y Fl(\006)944 927 y Ft(n)1007 902 y Fw(\()p Fv(:)p Fw(;)17 b Fv(\025;)g(k)s Fw(\))42 b(as)i(solutions)e(of)h(an)g (equation)g(of)g(Lippman-Sc)m(h)m(winger)120 1022 y(t)m(yp)s(e)33 b(\(see)h(the)f(remarks)g(in)f(Subsection)h(4\).)120 1211 y Fp(Lemma)k(3.4.6.)42 b Fj(L)-5 b(et)35 b Fv(k)s(;)f(\025;)f(n)i Fj(as)g(ab)-5 b(ove.)43 b(Then)35 b(the)f(e)-5 b(quations)257 1412 y Fv( )324 1371 y Fl(\006)320 1437 y Ft(n)383 1412 y Fw(\()p Fv(x)p Fw(;)17 b Fv(\025;)g(k)s Fw(\))28 b(=)f Fv( )911 1371 y Fu(\(0\))907 1437 y Ft(n)1006 1412 y Fw(\()p Fv(x)p Fw(;)17 b Fv(\025;)g(k)s Fw(\))p Fs(\000)479 1619 y(\000)578 1484 y Fo(Z)678 1510 y Ft(\031)634 1709 y Fl(\000)p Ft(\031)752 1619 y Fv(r)799 1578 y Fu(\(0\))893 1619 y Fw(\()p Fv(x;)g(y)1078 1634 y Fu(1)1117 1619 y Fv(;)g Fw(0;)g Fv(\025)22 b Fs(\007)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))p Fv(\033)t Fw(\()p Fv(y)1795 1634 y Fu(1)1834 1619 y Fw(\))p Fv( )1939 1578 y Fl(\006)1935 1644 y Ft(n)1998 1619 y Fw(\()p Fv(y)2084 1634 y Fu(1)2123 1619 y Fv(;)g Fw(0;)g Fv(\025;)g(k)s Fw(\))g Fv(dy)2569 1634 y Fu(1)2606 1619 y Fv(;)216 b(x)28 b Fs(2)g Fw(\005)p Fv(;)3264 1544 y Fw(\(3.4.8\))120 1883 y Fj(have)34 b(unique)h(solutions)f Fv( )1133 1847 y Fl(\006)1129 1908 y Ft(n)1192 1883 y Fw(\()p Fv(:)p Fw(;)17 b Fv(\025;)g(k)s Fw(\))35 b Fj(satisfying)1256 2007 y Fo(p)p 1356 2007 115 4 v 90 x Fs(j)p Fv(\033)t Fs(j)16 b Fv( )1554 2056 y Fl(\006)1550 2122 y Ft(n)1613 2097 y Fw(\()p Fv(:;)h Fw(0;)g Fv(\025;)g(k)s Fw(\))26 b Fs(2)j Fv(L)2195 2112 y Fu(2)2234 2097 y Fw(\()p Fr(T)p Fw(\))p Fv(:)864 b Fw(\(3.4.9\))120 2300 y Fj(Pr)-5 b(o)g(of.)48 b Fw(With)f(sligh)m(t)e(abuse)j(of)e(notation,)k(w)m(e)d(put)2166 2228 y Fs(p)p 2250 2228 60 4 v 2250 2300 a Fv(\033)55 b Fw(:=)d(\(sgn)17 b Fv(\033)t Fw(\))2808 2215 y Fo(p)p 2908 2215 115 4 v 85 x Fs(j)p Fv(\033)t Fs(j)o Fw(.)87 b(Since)47 b(the)120 2439 y(function)502 2368 y Fs(p)p 585 2368 60 4 v 71 x Fv(\033)21 b( )728 2388 y Fu(\(0\))724 2450 y Ft(n)822 2439 y Fw(\()p Fv(:;)c Fw(0;)g Fv(\025;)g(k)s Fw(\))31 b(b)s(elongs)h(to)h Fv(L)1784 2454 y Fu(2)1824 2439 y Fw(\()p Fr(T)p Fw(\))i(w)m(e)f(can)f(de\014ne)h(for)e Fv(\025)g Fw(as)h(ab)s(o)m(v)m(e)295 2686 y Fv(')359 2645 y Fl(\006)359 2711 y Ft(n)418 2686 y Fw(\()p Fv(:)p Fw(;)17 b Fv(\025;)g(k)s Fw(\))27 b(:=)878 2576 y Fo(\020)937 2686 y Fv(I)j Fw(+)22 b Fv(T)1179 2635 y Fu(\()p Ft(\033)r Fu(\))1165 2711 y(0)1281 2686 y Fw(\()p Fv(\025)g Fs(\007)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))g Fv(T)1803 2645 y Fu(\()p Ft(\033)r Fu(\))1904 2686 y Fw(\()p Fv(\025)22 b Fs(\006)h Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))2339 2645 y 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3156 y Fw(\))p Fs(j)p Fv(;)212 b(x)2652 3171 y Fu(1)2692 3156 y Fv(;)33 b(y)2800 3171 y Fu(1)2867 3156 y Fs(2)28 b Fr(T)p Fv(;)120 3359 y Fw(w)m(e)34 b(see)f(that)155 3548 y Fv(')219 3507 y Fl(\006)219 3572 y Ft(n)278 3548 y Fw(\()p Fv(x)371 3563 y Fu(1)411 3548 y Fw(;)17 b Fv(\025;)g(k)s Fw(\))27 b(=)778 3458 y Fo(p)p 878 3458 230 4 v 90 x Fv(\033)t Fw(\()p Fv(x)1030 3563 y Fu(1)1070 3548 y Fw(\))16 b Fv( )1191 3507 y Fu(\(0\))1187 3572 y Ft(n)1286 3548 y Fw(\()p Fv(x)1379 3563 y Fu(1)1419 3548 y Fv(;)h Fw(0;)g Fv(\025;)g(k)s Fw(\))p Fs(\000)372 3755 y(\000)472 3619 y Fo(Z)572 3646 y Ft(\031)527 3845 y Fl(\000)p Ft(\031)646 3665 y Fo(p)p 746 3665 V 90 x Fv(\033)t Fw(\()p Fv(x)898 3770 y Fu(1)937 3755 y Fw(\))g Fv(r)1039 3714 y Fu(\(0\))1133 3755 y Fw(\()p Fv(x)1226 3770 y Fu(1)1266 3755 y Fv(;)g Fw(0)p Fv(;)g(y)1451 3770 y Fu(1)1489 3755 y Fv(;)g Fw(0;)g Fv(\025)k Fs(\007)i Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))2022 3665 y Fo(p)p 2121 3665 278 4 v 2121 3755 a Fs(j)p Fv(\033)t Fw(\()p 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Fu(1)2391 4530 y Fw(;)g Fv(\025;)g(k)s Fw(\))g Fv(dy)2744 4545 y Fu(1)2782 4530 y Fv(;)212 b(x)28 b Fs(2)g Fw(\005)p Fv(;)120 4792 y Fw(It)33 b(follo)m(ws)e(from)g(\(3.4.10\))h(that)777 5006 y Fv(')841 4965 y Fl(\006)841 5030 y Ft(n)900 5006 y Fw(\()p Fv(x)993 5021 y Fu(1)1033 5006 y Fw(;)17 b Fv(\025;)g(k)s Fw(\))27 b(=)1400 4916 y Fo(p)p 1500 4916 230 4 v 90 x Fv(\033)t Fw(\()p Fv(x)1652 5021 y Fu(1)1692 5006 y Fw(\))16 b Fv( )1813 4965 y Fl(\006)1809 5030 y Ft(n)1872 5006 y Fw(\()p Fv(x)1965 5021 y Fu(1)2005 5006 y Fv(;)h Fw(0;)g Fv(\025;)g(k)s Fw(\))p Fv(;)211 b(x)2628 5021 y Fu(1)2695 5006 y Fs(2)28 b Fr(T)p Fv(;)120 5209 y Fw(so)33 b Fv( )307 5172 y Fl(\006)303 5233 y Ft(n)366 5209 y Fw(\()p Fv(:)p Fw(;)17 b Fv(\025;)g(k)s Fw(\))32 b(satis\014es)h(\(3.4.8\))f(and)h(\(3.4.9\))o(.)120 5342 y(The)39 b(uniqueness)h(of)d(these)i(solutions)e(follo)m(ws)f (from)h(the)h(in)m(v)m(ertibilit)m(y)e(of)h Fv(I)d Fw(+)25 b Fv(T)3240 5292 y Fu(\()p Ft(\033)r Fu(\))3226 5367 y(0)3342 5342 y Fw(\()p Fv(\025)h Fs(\007)120 5463 y Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))g Fv(T)426 5427 y Fu(\()p Ft(\033)r Fu(\))527 5463 y Fw(\()p Fv(\025)22 b Fs(\006)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))961 5427 y Fl(\003)1001 5463 y Fw(.)p 3469 5463 4 66 v 3473 5400 59 4 v 3473 5463 V 3531 5463 4 66 v 1781 5712 a(48)p eop %%Page: 49 54 49 53 bop 266 407 a Fw(T)-8 b(o)23 b(deriv)m(e)h(the)f(prop)s(erties)f (of)h(the)g(functions)g Fv( )2022 371 y Fl(\006)2018 431 y Ft(n)2081 407 y Fw(\()p Fv(:)p Fw(;)17 b Fv(\025;)g(k)s Fw(\))22 b(w)m(e)i(shall)d(study)j(the)f(equation)120 527 y(\(3.4.8\))o(.)43 b(W)-8 b(e)31 b(need)h(the)f(follo)m(wing)d (result)j(on)g(the)g(free)g(resolv)m(en)m(t)h(k)m(ernel.)44 b(Recall)29 b(that)h(for)120 648 y Fv(\025)e Fs(6)p Fw(=)f(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))616 611 y Fu(2)656 648 y Fv(;)33 b(n)28 b Fs(2)g Fr(Z)p Fw(,)172 913 y Fv(r)219 872 y Fu(\(0\))313 913 y Fw(\()p Fv(x;)17 b(y)498 928 y Fu(1)537 913 y Fv(;)g Fw(0;)g Fv(\025)k Fs(\007)i Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))27 b(=)h Fs(\007)1325 846 y Fv(i)p 1288 890 108 4 v 1288 981 a Fw(2)p Fv(\031)1511 818 y Fo(X)1422 1036 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))1613 1017 y Fm(2)1648 1036 y Ft(<\025)2059 846 y Fw(1)p 1771 890 627 4 v 1771 910 a Fo(p)p 1870 910 527 4 v 1870 995 a Fv(\025)22 b Fs(\000)h Fw(\()p Fv(n)f Fw(+)g Fv(k)s Fw(\))2357 967 y Fu(2)2423 913 y Fv(e)2468 872 y Ft(in)p Fu(\()p Ft(x)2602 881 y Fm(1)2637 872 y Fl(\000)p Ft(y)2727 881 y Fm(1)2762 872 y Fu(\))2793 913 y Fv(e)2838 872 y Fl(\007)p Ft(i)2917 805 y Fs(p)p 3000 805 322 4 v 67 x Ft(\025)p Fl(\000)p Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))3287 853 y Fm(2)3334 872 y Ft(x)3374 881 y Fm(2)3412 913 y Fw(+)389 1227 y(+)527 1160 y(1)p 497 1204 108 4 v 497 1295 a(2)p Fv(\031)720 1132 y Fo(X)631 1349 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))822 1330 y Fm(2)857 1349 y Ft(>\025)1268 1160 y Fw(1)p 980 1204 627 4 v 980 1224 a Fo(p)p 1079 1224 527 4 v 1079 1309 a Fw(\()p Fv(n)h Fw(+)f Fv(k)s Fw(\))1388 1280 y Fu(2)1449 1309 y Fs(\000)h Fv(\025)1632 1227 y(e)1677 1186 y Ft(in)p Fu(\()p Ft(x)1811 1195 y Fm(1)1846 1186 y Fl(\000)p Ft(y)1936 1195 y Fm(1)1970 1186 y Fu(\))2002 1227 y Fv(e)2047 1186 y Fl(\000)2102 1119 y Fs(p)p 2185 1119 322 4 v 67 x Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))2376 1167 y Fm(2)2411 1186 y Fl(\000)p Ft(\025)12 b(x)2559 1195 y Fm(2)2597 1227 y Fv(;)212 b(x)28 b Fs(2)g Fw(\005)p Fv(;)34 b(y)3195 1242 y Fu(1)3261 1227 y Fs(2)28 b Fr(T)p Fv(:)3215 1468 y Fw(\(3.4.11\))120 1695 y Fp(Lemma)g(3.4.7.)35 b Fj(L)-5 b(et)28 b Fv(k)j Fs(2)d Fw([)p Fs(\000)1255 1655 y Fu(1)p 1255 1672 36 4 v 1255 1729 a(2)1301 1695 y Fv(;)1354 1655 y Fu(1)p 1354 1672 V 1354 1729 a(2)1400 1695 y Fw(])p Fj(,)g Fv(\025)g Fs(2)g Fw([)p Fv(k)1745 1658 y Fu(2)1785 1695 y Fv(;)17 b Fw(+)p Fs(1)p Fw(\))5 b Fs(n)g(f)p Fw(\()p Fv(n)g Fw(+)g Fv(k)s Fw(\))2427 1658 y Fu(2)2494 1695 y Fw(:)55 b Fv(n)28 b Fs(2)g Fr(Z)p Fs(g)d Fj(and)i Fv(f)38 b Fs(2)28 b Fv(L)3328 1710 y Fu(1)3368 1695 y Fw(\()p Fr(T)p Fw(\))p Fj(.)120 1815 y(L)-5 b(et)644 1961 y Fv(u)700 1920 y Fl(\006)759 1961 y Fw(\()p Fv(x)p Fw(\))28 b(:=)1048 1826 y Fo(Z)1148 1852 y Ft(\031)1104 2051 y Fl(\000)p Ft(\031)1222 1961 y Fv(r)1269 1920 y Fu(\(0\))1363 1961 y Fw(\()p Fv(x;)17 b(y)1548 1976 y Fu(1)1587 1961 y Fv(;)g Fw(0;)g Fv(\025)k Fs(\007)i Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))p Fv(f)11 b Fw(\()p Fv(y)2265 1976 y Fu(1)2303 1961 y Fw(\))17 b Fv(dy)2457 1976 y Fu(1)2496 1961 y Fv(;)216 b(x)28 b Fs(2)g Fw(\005)p Fv(:)120 2212 y Fj(Then)34 b Fv(u)430 2175 y Fl(\006)516 2212 y Fs(2)633 2186 y Fw(~)610 2212 y Fv(C)687 2175 y Fl(1)762 2212 y Fw(\(\005\))p Fj(,)1048 2351 y Fo(\000)1094 2432 y Fw(\()p Fv(D)1213 2447 y Fu(1)1275 2432 y Fw(+)22 b Fv(k)s Fw(\))1465 2390 y Fu(2)1526 2432 y Fw(+)g Fv(D)1708 2390 y Fu(2)1705 2456 y(2)1747 2351 y Fo(\001)1810 2432 y Fv(u)1866 2390 y Fl(\006)1952 2432 y Fw(=)27 b Fv(\025u)2168 2390 y Fl(\006)2426 2432 y Fj(in)g Fw(\005)120 2652 y Fj(and)1123 2804 y Fw(lim)1096 2863 y Ft(\017)p Fl(!)p Fu(0+)1312 2736 y Fv(@)5 b(u)1424 2700 y Fl(\006)p 1312 2781 172 4 v 1322 2872 a Fv(@)g(x)1433 2887 y Fu(2)1510 2804 y Fs(j)1538 2819 y Ft(x)1578 2828 y Fm(2)1612 2819 y Fu(=)p Ft(\017)1737 2804 y Fw(=)28 b Fs(\000)p Fv(f)210 b Fj(in)28 b Fv(L)2355 2819 y Fu(1)2394 2804 y Fw(\()p Fr(T)p Fw(\))p Fv(:)655 b Fw(\(3.4.12\))120 3018 y Fj(Mor)-5 b(e)g(over,)34 b(for)h(some)f Fv(\013)29 b(>)e Fw(0)188 3308 y Fv(u)244 3267 y Fl(\006)302 3308 y Fw(\()p Fv(x)p Fw(\))h(=)g Fs(\007)731 3241 y Fv(i)p 652 3286 191 4 v 652 3305 a Fs(p)p 735 3305 108 4 v 83 x Fw(2)p Fv(\031)959 3214 y Fo(X)870 3431 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))1061 3412 y Fm(2)1095 3431 y Ft(<\025)1505 3215 y Fw(^)1483 3241 y Fv(f)1531 3256 y Ft(n)p 1218 3286 627 4 v 1218 3305 a Fo(p)p 1318 3305 527 4 v 86 x Fv(\025)22 b Fs(\000)g Fw(\()p Fv(n)h Fw(+)f Fv(k)s Fw(\))1805 3362 y Fu(2)1871 3308 y Fv(e)1916 3267 y Ft(inx)2023 3276 y Fm(1)2061 3308 y Fv(e)2106 3267 y Fl(\007)p Ft(i)2185 3200 y Fs(p)p 2268 3200 322 4 v 67 x Ft(\025)p Fl(\000)p Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))2555 3248 y Fm(2)2602 3267 y Ft(x)2642 3276 y Fm(2)2702 3308 y Fw(+)g Fv(o)p Fw(\()p Fv(e)2930 3267 y Fl(\000)p Ft(\013x)3070 3276 y Fm(2)3109 3308 y Fw(\))387 3575 y(\()p Fv(x)480 3590 y Fu(2)548 3575 y Fs(!)27 b(1)p Fw(\))p Fv(;)3215 3404 y Fw(\(3.4.13\))120 3788 y Fj(uniformly)35 b(in)f Fv(x)739 3803 y Fu(1)779 3788 y Fj(.)266 3992 y Fw(The)g(main)d(tec)m(hnical)h (di\016cult)m(y)g(lies)g(in)f(the)i(v)m(eri\014cation)f(of)39 b(\(3.4.12\))o(.)44 b(It)32 b(consists)i(in)120 4133 y(making)h(the)h(formal)e(iden)m(tit)m(y)1322 4086 y Ft(@)t(r)1397 4062 y Fm(\(0\))1480 4086 y Fu(\()p Ft(x)1547 4095 y Fm(1)1582 4086 y Ft(;)p Fu(0)p Ft(;y)1692 4095 y Fm(1)1726 4086 y Ft(;)p Fu(0;)p Ft(\025)p Fl(\007)p Ft(i)p Fu(0)p Ft(;k)r Fu(\)\))p 1322 4110 747 4 v 1638 4167 a Ft(@)t(x)1719 4176 y Fm(2)2112 4133 y Fw(=)2253 4094 y Fu(1)p 2231 4110 79 4 v 2231 4167 a(2)p Ft(\031)2336 4058 y Fo(P)2441 4162 y Ft(n)p Fl(2)p Fn(Z)2596 4133 y Fv(e)2641 4097 y Ft(in)p Fu(\()p Ft(x)2775 4106 y Fm(1)2810 4097 y Fl(\000)p Ft(y)2900 4106 y Fm(1)2934 4097 y Fu(\))2999 4133 y Fw(=)g Fv(\016)t Fw(\()p Fv(x)3249 4148 y Fu(1)3313 4133 y Fs(\000)25 b Fv(y)3463 4148 y Fu(1)3502 4133 y Fw(\))120 4253 y(precise.)120 4449 y Fj(Pr)-5 b(o)g(of.)48 b Fw(The)35 b(series)g(\(3.4.11\))e(con)m(v)m(erges)k(uniformly)32 b(and)i(absolutely)f(on)h Fs(f)p Fw(\()p Fv(x;)17 b(y)3162 4464 y Fu(1)3201 4449 y Fw(\))30 b Fs(2)p 3366 4368 74 4 v 31 w Fw(\005)24 b Fs(\002)120 4569 y Fw([)p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(])45 b(:)89 b Fv(x)629 4584 y Fu(2)714 4569 y Fs(\025)45 b Fv(\017)p Fs(g)d Fw(for)g(ev)m(ery)j Fv(\017)g(>)f Fw(0.)73 b(Th)m(us)44 b(w)m(e)g(can)f(in)m(terc)m(hange)g (in)m(tegration)e(and)120 4689 y(summation,)31 b(and)h(obtain)322 4979 y Fv(u)378 4938 y Fl(\006)436 4979 y Fw(\()p Fv(x)p Fw(\))c(=)22 b Fs(\007)882 4912 y Fv(i)p 803 4957 191 4 v 803 4976 a Fs(p)p 886 4976 108 4 v 83 x Fw(2)p Fv(\031)1109 4885 y Fo(X)1020 5102 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))1211 5083 y Fm(2)1246 5102 y Ft(<\025)1655 4886 y Fw(^)1634 4912 y Fv(f)1682 4927 y Ft(n)p 1369 4957 627 4 v 1369 4976 a Fo(p)p 1468 4976 527 4 v 1468 5062 a Fv(\025)g Fs(\000)h Fw(\()p Fv(n)f Fw(+)g Fv(k)s Fw(\))1955 5033 y Fu(2)2021 4979 y Fv(e)2066 4938 y Ft(inx)2173 4947 y Fm(1)2212 4979 y Fv(e)2257 4938 y Fl(\007)p Ft(i)2336 4871 y Fs(p)p 2419 4871 322 4 v 67 x Ft(\025)p Fl(\000)p Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))2706 4919 y Fm(2)2752 4938 y Ft(x)2792 4947 y Fm(2)2831 4979 y Fw(+)693 5325 y(+)872 5257 y(1)p 801 5302 191 4 v 801 5322 a Fs(p)p 884 5322 108 4 v 82 x Fw(2)p Fv(\031)1108 5230 y Fo(X)1019 5447 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))1210 5428 y Fm(2)1244 5447 y Ft(>\025)1654 5231 y Fw(^)1632 5257 y Fv(f)1680 5272 y Ft(n)p 1367 5302 627 4 v 1367 5322 a Fo(p)p 1467 5322 527 4 v 85 x Fw(\()p Fv(n)g Fw(+)g Fv(k)s Fw(\))1775 5378 y Fu(2)1836 5407 y Fs(\000)h Fv(\025)2020 5325 y(e)2065 5283 y Ft(inx)2172 5292 y Fm(1)2210 5325 y Fv(e)2255 5283 y Fl(\000)2310 5216 y Fs(p)p 2393 5216 322 4 v 67 x Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))2584 5264 y Fm(2)2619 5283 y Fl(\000)p Ft(\025)12 b(x)2767 5292 y Fm(2)2805 5325 y Fv(;)212 b(x)3099 5340 y Fu(2)3167 5325 y Fs(\025)28 b Fv(\017:)1781 5712 y Fw(49)p eop %%Page: 50 55 50 54 bop 120 407 a Fw(It)29 b(is)g(easy)h(to)f(see)h(from)e(this)h (that)g Fv(u)1476 371 y Fl(\006)1564 407 y Fw(are)g(in\014nitely)f(smo) s(oth)g(on)h([)p Fs(\000)p Fv(\031)t(;)17 b(\031)t Fw(])e Fs(\002)g Fw([)p Fv(\017;)i Fw(+)p Fs(1)p Fw(\))30 b(and,)120 527 y(together)i(with)g(all)e(its)i(deriv)-5 b(ativ)m(es,)32 b(p)s(erio)s(dic)e(with)i(resp)s(ect)i(to)d Fv(x)2611 542 y Fu(1)2651 527 y Fw(.)43 b(Also,)32 b(the)h(equations)120 648 y(\(\()p Fv(D)277 663 y Fu(1)338 648 y Fw(+)22 b Fv(k)s Fw(\))528 611 y Fu(2)590 648 y Fw(+)g Fv(D)772 611 y Fu(2)769 672 y(2)811 648 y Fw(\))17 b Fv(u)922 611 y Fl(\006)1008 648 y Fw(=)28 b Fv(\025u)1225 611 y Fl(\006)1315 648 y Fw(and)33 b(the)g(asymptotics)f(\(3.4.13\))g(are)h (clear.)120 768 y(Let)c(us)h(pro)m(v)m(e)g(\(3.4.12\))o(.)42 b(W)-8 b(e)29 b(consider)h(the)f("-"-case)f(only)-8 b(,)29 b(the)h(other)f(b)s(eing)f(similar.)39 b(F)-8 b(or)120 888 y Fv(\017)28 b(>)g Fw(0)k(de\014ne)i(an)e(op)s(erator)g Fv(T)1239 903 y Ft(\017)1300 888 y Fw(:)27 b Fv(L)1420 903 y Fu(1)1460 888 y Fw(\()p Fr(T)p Fw(\))k Fs(!)c Fv(L)1823 903 y Fu(1)1863 888 y Fw(\()p Fr(T)p Fw(\))p Fv(;)561 1113 y(T)618 1128 y Ft(\017)651 1113 y Fv(g)k Fw(:=)948 1018 y Fo(X)859 1235 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))1050 1216 y Fm(2)1085 1235 y Ft(<\025)1201 1113 y Fw(^)-52 b Fv(g)1245 1128 y Ft(n)1291 1113 y Fv(e)1336 1072 y Ft(i\017)1389 1005 y Fs(p)p 1472 1005 322 4 v 67 x Ft(\025)p Fl(\000)p Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))1759 1053 y Fm(2)1798 1113 y Fv(e)1843 1128 y Ft(n)1912 1113 y Fw(+)2099 1018 y Fo(X)2010 1235 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))2201 1216 y Fm(2)2236 1235 y Ft(>\025)2352 1113 y Fw(^)g Fv(g)2396 1128 y Ft(n)2442 1113 y Fv(e)2487 1072 y Fl(\000)p Ft(\017)2571 1005 y Fs(p)p 2654 1005 V 67 x Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))2845 1053 y Fm(2)2880 1072 y Fl(\000)p Ft(\025)2980 1113 y Fv(e)3025 1128 y Ft(n)3072 1113 y Fv(;)120 1414 y Fw(so)33 b(that)1460 1499 y Fv(@)5 b(u)1572 1463 y Fl(\000)p 1460 1543 172 4 v 1470 1635 a Fv(@)g(x)1581 1650 y Fu(2)1641 1566 y Fs(j)1669 1581 y Ft(x)1709 1590 y Fm(2)1743 1581 y Fu(=)p Ft(\017)1859 1566 y Fw(=)27 b Fs(\000)p Fv(T)2096 1581 y Ft(\017)2130 1566 y Fv(f)5 b(:)120 1776 y Fw(Of)30 b(course,)i(w)m(e)g(ha)m(v)m(e)g Fv(T)1002 1791 y Ft(\017)1035 1776 y Fv(g)f Fs(!)c Fv(g)34 b Fw(in)c Fv(L)1499 1791 y Fu(1)1539 1776 y Fw(\()p Fr(T)p Fw(\))j(\()p Fv(\017)28 b Fs(!)g Fw(0\))i(for)g(ev)m(ery)i(trigonometric)d(p)s(olynomial)120 1896 y Fv(g)t Fw(,)j(so)h(to)f(conclude)h Fv(T)927 1911 y Ft(\017)987 1896 y Fs(!)28 b Fv(I)40 b Fw(strongly)32 b(\(i.e.,)h(\(3.4.12\))o(\))f(it)g(su\016ces)i(to)f(sho)m(w)1525 2092 y(sup)1492 2178 y Ft(\017)p Fl(2)p Fu(\(0)p Ft(;)p Fu(1])1721 2092 y Fs(k)p Fv(T)1828 2107 y Ft(\017)1861 2092 y Fs(k)27 b Fv(<)h Fs(1)p Fv(:)1046 b Fw(\(3.4.14\))120 2402 y(Let)33 b Fv( )358 2417 y Ft(\017)419 2402 y Fw(:=)549 2327 y Fo(P)654 2431 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))845 2412 y Fm(2)880 2431 y Ft(<\025)997 2402 y Fv(e)1042 2362 y Ft(i\017)1095 2296 y Fs(p)p 1178 2296 322 4 v 66 x Ft(\025)p Fl(\000)p Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))1465 2344 y Fm(2)1504 2402 y Fv(e)1549 2417 y Ft(n)1618 2402 y Fw(+)1716 2327 y Fo(P)1821 2431 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))2012 2412 y Fm(2)2047 2431 y Ft(>\025)2164 2402 y Fv(e)2209 2362 y Fl(\000)p Ft(\017)2293 2296 y Fs(p)p 2375 2296 V 2375 2362 a Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))2566 2344 y Fm(2)2601 2362 y Fl(\000)p Ft(\025)2701 2402 y Fv(e)2746 2417 y Ft(n)2793 2402 y Fw(,)g(then)868 2663 y(\()p Fv(T)963 2678 y Ft(\017)996 2663 y Fv(g)t Fw(\))1081 2658 y Fo(b)1085 2687 y Ft(n)1159 2663 y Fw(=)e(^)-53 b Fv(g)1309 2678 y Ft(n)1356 2663 y Fw(\()p Fv( )1457 2678 y Ft(\017)1490 2663 y Fw(\))1524 2658 y Fo(b)1528 2687 y Ft(n)1602 2663 y Fw(=)1787 2595 y(1)p 1716 2640 191 4 v 1716 2660 a Fs(p)p 1799 2660 108 4 v 82 x Fw(2)p Fv(\031)1917 2663 y Fw(\()p Fv(g)25 b Fs(\003)d Fv( )2162 2678 y Ft(\017)2195 2663 y Fw(\))2229 2658 y Fo(b)2233 2687 y Ft(n)2280 2663 y Fv(;)212 b(n)28 b Fs(2)g Fr(Z)p Fv(;)120 2921 y Fw(where)34 b(\()p Fv(g)25 b Fs(\003)d Fv( )647 2936 y Ft(\017)680 2921 y Fw(\)\()p Fv(x)811 2936 y Fu(1)851 2921 y Fw(\))28 b(:=)1047 2841 y Fo(R)1114 2867 y Ft(\031)1094 2956 y Fl(\000)p Ft(\031)1213 2921 y Fv(g)t Fw(\()p Fv(y)1350 2936 y Fu(1)1388 2921 y Fw(\))p Fv( )1489 2936 y Ft(\017)1522 2921 y Fw(\()p Fv(x)1615 2936 y Fu(1)1677 2921 y Fs(\000)23 b Fv(y)1825 2936 y Fu(1)1864 2921 y Fw(\))17 b Fv(dy)2018 2936 y Fu(1)2056 2921 y Fv(;)44 b(x)2182 2936 y Fu(1)2250 2921 y Fs(2)28 b Fr(T)p Fw(.)47 b(Hence)477 3178 y Fs(k)p Fv(T)584 3193 y Ft(\017)616 3178 y Fv(g)t Fs(k)717 3193 y Ft(L)765 3202 y Fm(1)799 3193 y Fu(\()p Fn(T)p Fu(\))930 3178 y Fw(=)1114 3110 y(1)p 1044 3155 191 4 v 1044 3175 a Fs(p)p 1127 3175 108 4 v 82 x Fw(2)p Fv(\031)1244 3178 y Fs(k)p Fv(g)25 b Fs(\003)d Fv( )1501 3193 y Ft(\017)1534 3178 y Fs(k)1584 3193 y Ft(L)1632 3202 y Fm(1)1667 3193 y Fu(\()p Fn(T)p Fu(\))1798 3178 y Fs(\024)1984 3110 y Fw(1)p 1913 3155 191 4 v 1913 3175 a Fs(p)p 1996 3175 108 4 v 82 x Fw(2)p Fv(\031)2114 3178 y Fs(k)p Fv(g)t Fs(k)2265 3193 y Ft(L)2313 3202 y Fm(1)2346 3193 y Fu(\()p Fn(T)p Fu(\))2449 3178 y Fs(k)p Fv( )2562 3193 y Ft(\017)2595 3178 y Fs(k)2645 3193 y Ft(L)2693 3202 y Fm(1)2728 3193 y Fu(\()p Fn(T)p Fu(\))2831 3178 y Fv(;)357 b Fw(\(3.4.15\))120 3429 y(and)33 b(so)f(for)g(\(3.4.14\))g(it)g(su\016ces)i(to)f(sho)m(w) 1428 3624 y(sup)1396 3710 y Ft(\017)p Fl(2)p Fu(\(0)p Ft(;)p Fu(1])1625 3624 y Fs(k)p Fv( )1738 3639 y Ft(\017)1771 3624 y Fs(k)1821 3640 y Ft(L)1869 3649 y Fm(1)1903 3640 y Fu(\()p Fn(T)p Fu(\))2034 3624 y Fv(<)28 b Fs(1)p Fv(:)950 b Fw(\(3.4.16\))120 3888 y(Let)33 b(us)g(write)820 4087 y Fv( )883 4102 y Ft(\017)943 4087 y Fw(=)1136 3993 y Fo(X)1047 4210 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))1238 4191 y Fm(2)1273 4210 y Ft(<\025)1385 3977 y Fo(\020)1445 4087 y Fv(e)1490 4046 y Ft(i\017)1543 3979 y Fs(p)p 1626 3979 322 4 v 67 x Ft(\025)p Fl(\000)p Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))1913 4027 y Fm(2)1974 4087 y Fs(\000)22 b Fv(e)2118 4046 y Fl(\000)p Ft(\017)p Fl(j)p Ft(n)p Fu(+)p Ft(k)r Fl(j)2381 3977 y Fo(\021)2457 4087 y Fv(e)2502 4102 y Ft(n)2550 4087 y Fw(+)1133 4385 y(+)1320 4291 y Fo(X)1231 4508 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))1422 4489 y Fm(2)1457 4508 y Ft(>\025)1569 4275 y Fo(\020)1629 4385 y Fv(e)1674 4344 y Fl(\000)p Ft(\017)1758 4277 y Fs(p)p 1840 4277 V 1840 4344 a Ft(\025)p Fl(\000)p Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))2127 4325 y Fm(2)2188 4385 y Fs(\000)h Fv(e)2333 4344 y Fl(\000)p Ft(\017)p Fl(j)p Ft(n)p Fu(+)p Ft(k)r Fl(j)2596 4275 y Fo(\021)2672 4385 y Fv(e)2717 4400 y Ft(n)2765 4385 y Fw(+)1133 4672 y(+)1231 4578 y Fo(X)1236 4789 y Ft(n)p Fl(2)p Fn(Z)1391 4672 y Fv(e)1436 4631 y Fl(\000)p Ft(\017)p Fl(j)p Ft(n)p Fu(+)p Ft(k)r Fl(j)1700 4672 y Fv(e)1745 4687 y Ft(n)943 4928 y Fw(=:)28 b Fv( )1141 4887 y Fu(\(1\))1137 4952 y Ft(\017)1257 4928 y Fw(+)22 b Fv( )1422 4887 y Fu(\(2\))1418 4952 y Ft(\017)1539 4928 y Fw(+)g Fv( )1704 4887 y Fu(\(3\))1700 4952 y Ft(\017)1798 4928 y Fv(:)120 5148 y Fw(Of)32 b(course,)i Fs(k)p Fv( )701 5097 y Fu(\(1\))697 5158 y Ft(\017)795 5148 y Fs(k)845 5163 y Ft(L)893 5172 y Fm(1)927 5163 y Fu(\()p Fn(T)p Fu(\))1063 5148 y Fw(is)e(b)s(ounded)i(uniformly)c(in) i Fv(\017)c Fs(2)g Fw(\(0)p Fv(;)17 b Fw(1].)43 b(Using)32 b(the)h(estimate)622 5286 y Fo(\014)622 5345 y(\014)622 5405 y(\014)655 5400 y Fv(e)700 5359 y Fl(\000)p Ft(\017)784 5292 y Fs(p)p 867 5292 V 67 x Ft(\025)p Fl(\000)p Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))1154 5340 y Fm(2)1215 5400 y Fs(\000)23 b Fv(e)1360 5359 y Fl(\000)p Ft(\017)p Fl(j)p Ft(n)p Fu(+)p Ft(k)r Fl(j)1623 5286 y Fo(\014)1623 5345 y(\014)1623 5405 y(\014)1684 5400 y Fs(\024)2088 5333 y Fv(\017\025)p 1799 5377 675 4 v 1799 5482 a Fw(2)1848 5397 y Fo(p)p 1947 5397 527 4 v 1947 5482 a Fw(\()p Fv(n)g Fw(+)f Fv(k)s Fw(\))2256 5453 y Fu(2)2317 5482 y Fs(\000)h Fv(\025)2500 5400 y(e)2545 5359 y Fl(\000)p Ft(\017)2629 5292 y Fs(p)p 2712 5292 322 4 v 67 x Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))2903 5340 y Fm(2)2938 5359 y Fl(\000)p Ft(\025)1781 5712 y Fw(50)p eop %%Page: 51 56 51 55 bop 120 408 a Fw(\(whic)m(h)35 b(is)g(established)f(as)h(in)g (the)g(pro)s(of)f(of)g(Prop)s(osition)f(2.2.1\),)i(w)m(e)h(see)g(that)e Fv( )3234 357 y Fu(\(2\))3230 418 y Ft(\017)3360 408 y Fs(!)d Fw(0)120 542 y(in)k Fv(L)303 557 y Fu(2)343 542 y Fw(\()p Fr(T)p Fw(\))k(as)d Fv(\017)d Fs(!)g Fw(0.)53 b(In)36 b(particular,)f Fs(k)p Fv( )1701 491 y Fu(\(2\))1697 552 y Ft(\017)1795 542 y Fs(k)1845 557 y Ft(L)1893 566 y Fm(1)1928 557 y Fu(\()p Fn(T)p Fu(\))2067 542 y Fw(is)g(b)s(ounded)i (uniformly)c(in)i Fv(\017)f Fs(2)g Fw(\(0)p Fv(;)17 b Fw(1].)120 662 y(Th)m(us,)34 b(for)e(\(3.4.16\))g(it)f(su\016ces)k(to)d (sho)m(w)1396 851 y(sup)1363 936 y Ft(\017)p Fl(2)p Fu(\(0)p Ft(;)p Fu(1])1592 851 y Fs(k)p Fv( )1709 809 y Fu(\(3\))1705 875 y Ft(\017)1803 851 y Fs(k)1853 866 y Ft(L)1901 875 y Fm(1)1936 866 y Fu(\()p Fn(T)p Fu(\))2067 851 y Fv(<)27 b Fs(1)p Fv(:)918 b Fw(\(3.4.17\))120 1161 y(It)23 b(is)f(con)m(v)m (enien)m(t)i(to)f(transfer)g(this)f(problem)g(from)f Fr(T)26 b Fw(to)d Fr(R)5 b Fw(.)46 b(Let)23 b Fv(')2574 1176 y Ft(\017)2606 1161 y Fw(\()p Fv(x)2699 1176 y Fu(1)2739 1161 y Fw(\))28 b(:=)2935 1047 y Fo(q)p 3035 1047 63 4 v 3049 1122 a Fu(2)p 3045 1138 43 4 v 3045 1196 a Ft(\031)3190 1122 y(\017)p 3108 1138 193 4 v 3108 1197 a(\017)3137 1178 y Fm(2)3171 1197 y Fu(+)p Ft(x)3266 1174 y Fm(2)3266 1218 y(1)3327 1161 y Fv(e)3372 1125 y Ft(ik)r(x)3475 1134 y Fm(1)3513 1161 y Fw(,)120 1300 y Fv(x)175 1315 y Fu(1)243 1300 y Fs(2)g Fr(R)5 b Fw(.)49 b(Then)1260 1420 y Fs(k)p Fv(')1374 1435 y Ft(\017)1406 1420 y Fs(k)1456 1436 y Ft(L)1504 1445 y Fm(1)1539 1436 y Fu(\()p Fn(R)p Fu(\))1673 1420 y Fw(=)28 b Fs(k)p Fv(')1891 1435 y Fu(1)1930 1420 y Fs(k)1980 1436 y Ft(L)2028 1445 y Fm(1)2062 1436 y Fu(\()p Fn(R)p Fu(\))2197 1420 y Fv(<)g Fs(1)814 b Fw(\(3.4.18\))120 1581 y(and)1275 1702 y(^)-73 b Fv(')1315 1717 y Ft(\017)1348 1702 y Fw(\()p Fv(\030)5 b Fw(\))27 b(=)h Fv(e)1648 1660 y Fl(\000)p Ft(\017)p Fl(j)p Ft(\030)s Fl(\000)p Ft(k)r Fl(j)1902 1702 y Fv(;)211 b(\030)32 b Fs(2)d Fr(R)t Fv(:)120 1862 y Fw(It)k(follo)m(ws)e(from)g(the)i(P)m (oisson)g(summation)e(form)m(ula)f(that)496 2070 y Fv( )563 2029 y Fu(\(3\))559 2095 y Ft(\017)657 2070 y Fw(\()p Fv(x)750 2085 y Fu(1)790 2070 y Fw(\))e(=)1040 2003 y(1)p 969 2047 191 4 v 969 2067 a Fs(p)p 1052 2067 108 4 v 83 x Fw(2)p Fv(\031)1187 1976 y Fo(X)1191 2187 y Ft(n)p Fl(2)p Fn(Z)1371 2070 y Fw(^)-73 b Fv(')1411 2085 y Ft(\017)1444 2070 y Fw(\()p Fv(n)p Fw(\))p Fv(e)1623 2029 y Ft(inx)1730 2038 y Fm(1)1796 2070 y Fw(=)1899 1976 y Fo(X)1904 2187 y Ft(n)p Fl(2)p Fn(Z)2060 2070 y Fv(')2124 2085 y Ft(\017)2157 2070 y Fw(\()p Fv(x)2250 2085 y Fu(1)2312 2070 y Fs(\000)22 b Fw(2)p Fv(\031)t(n)p Fw(\))p Fv(;)212 b(x)2909 2085 y Fu(1)2976 2070 y Fs(2)28 b Fr(T)p Fv(;)120 2352 y Fw(so)33 b(b)m(y)i(\(3.4.18\))c(w)m(e)j(conclude)f(that)711 2592 y Fs(k)p Fv( )828 2550 y Fu(\(3\))824 2616 y Ft(\017)922 2592 y Fs(k)972 2607 y Ft(L)1020 2616 y Fm(1)1055 2607 y Fu(\()p Fn(T)p Fu(\))1186 2592 y Fs(\024)1291 2497 y Fo(X)1296 2709 y Ft(n)p Fl(2)p Fn(Z)1451 2456 y Fo(Z)1551 2482 y Ft(\031)1507 2681 y Fl(\000)p Ft(\031)1625 2592 y Fs(j)p Fv(')1717 2607 y Ft(\017)1750 2592 y Fw(\()p Fv(x)1843 2607 y Fu(1)1905 2592 y Fs(\000)22 b Fw(2)p Fv(\031)t(n)p Fw(\))p Fs(j)17 b Fv(dx)2359 2607 y Fu(1)2426 2592 y Fw(=)27 b Fs(k)p Fv(')2643 2607 y Fu(1)2682 2592 y Fs(k)2732 2607 y Ft(L)2780 2616 y Fm(1)2815 2607 y Fu(\()p Fn(R)p Fu(\))2922 2592 y Fv(;)120 2874 y Fw(whic)m(h)33 b(pro)m(v)m(es)h(\(3.4.17\))p 3469 2874 4 66 v 3473 2811 59 4 v 3473 2874 V 3531 2874 4 66 v 120 3060 a Fj(R)-5 b(emark)46 b Fw(3.4.8)p Fj(.)41 b Fw(W)-8 b(e)34 b(note)g(that)f(if)f Fv(f)40 b Fs(2)29 b Fv(L)1703 3075 y Ft(p)1743 3060 y Fw(\()p Fr(T)p Fw(\))37 b(for)c(some)g(1)c Fs(\024)h Fv(p)f(<)g Fs(1)p Fw(,)k(then)h(the)g(con)m(v)m(er-)120 3180 y(gence)i(\(3.4.12\))e(holds)g(in)g Fv(L)1187 3195 y Ft(p)1227 3180 y Fw(\()p Fr(T)p Fw(\).)53 b(Indeed,)37 b(w)m(e)f(only)e(ha)m(v)m(e)i(to)e(consider)h Fv(T)2942 3195 y Ft(\017)3010 3180 y Fw(in)f(the)h(ab)s(o)m(v)m(e)120 3301 y(pro)s(of)d(as)g(op)s(erator)g(from)g Fv(L)1184 3316 y Ft(p)1224 3301 y Fw(\()p Fr(T)p Fw(\))k(to)c Fv(L)1584 3316 y Ft(p)1624 3301 y Fw(\()p Fr(T)p Fw(\))k(and)c(use)i(Y)-8 b(oung's)33 b(inequalit)m(y)e(in)h(\(3.4.15\))o(.)266 3449 y(No)m(w)h(Prop)s(osition)e(3.4.5)h(follo)m(ws)g(easily)-8 b(.)120 3635 y Fj(Pr)j(o)g(of)35 b(\(Pr)-5 b(op)g(osition)34 b(3.4.5\).)47 b Fw(Consider)h(the)g(functions)g Fv( )2375 3599 y Fl(\006)2371 3660 y Ft(n)2434 3635 y Fw(\()p Fv(:)p Fw(;)17 b Fv(\025;)g(k)s Fw(\))47 b(as)g(constructed)j(in)120 3755 y(Lemma)30 b(3.4.6)h(and)g(apply)g(Lemma)f(3.4.7)g(with)h Fv(f)39 b Fw(:=)27 b Fv(\033)t( )2296 3719 y Fl(\006)2292 3780 y Ft(n)2355 3755 y Fw(\()p Fv(:;)17 b Fw(0;)g Fv(\025;)g(k)s Fw(\))27 b Fs(2)h Fv(L)2937 3770 y Fu(1)2977 3755 y Fw(\()p Fr(T)p Fw(\).)46 b(W)-8 b(e)32 b(note)120 3876 y(that)g(in)g(the)h ("-"-case)239 4064 y(^)218 4090 y Fv(f)266 4105 y Ft(m)360 4090 y Fw(=)545 4023 y(1)p 474 4067 191 4 v 474 4087 a Fs(p)p 557 4087 108 4 v 82 x Fw(2)p Fv(\031)691 3954 y Fo(Z)791 3981 y Ft(\031)746 4180 y Fl(\000)p Ft(\031)865 4090 y Fv( )932 4049 y Fl(\000)928 4115 y Ft(n)991 4090 y Fw(\()p Fv(x)1084 4105 y Fu(1)1124 4090 y Fv(;)17 b Fw(0;)g Fv(\025;)g(k)s Fw(\))p Fv(\033)t Fw(\()p Fv(x)1606 4105 y Fu(1)1644 4090 y Fw(\))p Fv(e)1727 4049 y Fl(\000)p Ft(imx)1908 4058 y Fm(1)1964 4090 y Fv(dx)2070 4105 y Fu(1)2137 4090 y Fw(=)28 b Fs(\000)p Fv(i)2379 4046 y Fm(4)2354 4000 y Fo(p)p 2454 4000 554 4 v 90 x Fv(\025)22 b Fs(\000)g Fw(\()p Fv(m)h Fw(+)f Fv(k)s Fw(\))2968 4061 y Fu(2)3024 4090 y Fv(a)3075 4105 y Ft(mn)3185 4090 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))p Fv(:)3215 4270 y Fw(\(3.4.19\))120 4390 y(No)m(w)44 b(all)d(the)j(assertions)g(follo)m(w)e(immediately)e (from)i(the)i(Lippmann-Sc)m(h)m(winger)f(t)m(yp)s(e)120 4511 y(equation)32 b(\(3.4.8\).)p 3469 4511 4 66 v 3473 4448 59 4 v 3473 4511 V 3531 4511 4 66 v 266 4697 a(W)-8 b(e)45 b(can)f(think)f(of)h(the)g(n)m(um)m(b)s(ers)h Fv(a)1662 4712 y Ft(mn)1771 4697 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))44 b(as)g("scattering)f(amplitudes".)76 b(Their)120 4817 y(connection)33 b(with)f(the)h(scattering)f(matrix)f(is)h(giv)m (en)h(b)m(y)h(the)f(follo)m(wing)120 4993 y Fp(Prop)s(osition)28 b(3.4.9.)36 b Fj(L)-5 b(et)29 b Fv(k)i Fs(2)d Fw([)p Fs(\000)1467 4954 y Fu(1)p 1467 4970 36 4 v 1467 5028 a(2)1513 4993 y Fv(;)1566 4954 y Fu(1)p 1566 4970 V 1566 5028 a(2)1612 4993 y Fw(])p Fv(;)33 b(\025)28 b Fs(2)g Fw([)p Fv(k)1959 4957 y Fu(2)1998 4993 y Fv(;)17 b Fw(+)p Fs(1)p Fw(\))8 b Fs(n)g Fw(\()o Fs(f)p Fw(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))2717 4957 y Fu(2)2784 4993 y Fw(:)28 b Fv(n)g Fs(2)g Fr(Z)p Fs(g)19 b([)k(N)15 b Fw(\()p Fv(k)s Fw(\)\))o Fj(.)120 5114 y(Then)34 b(for)h Fv(b)28 b Fs(2)g Fr(C)759 5077 y Ft(N)7 b Fu(\()p Ft(\025)q(;k)r Fu(\))242 5233 y Fo(\000)288 5314 y Fv(S)354 5273 y Fu(\()p Ft(\033)r Fu(\))456 5314 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))g Fv(b)745 5233 y Fo(\001)790 5353 y Ft(m)884 5314 y Fw(=)28 b Fv(b)1029 5329 y Ft(m)1118 5314 y Fs(\000)23 b Fw(2)1267 5237 y Fs(p)p 1349 5237 59 4 v 1349 5314 a Fv(\031)1514 5219 y Fo(X)1425 5436 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))1616 5417 y Fm(2)1651 5436 y Ft(<\025)1763 5314 y Fv(a)1814 5329 y Ft(mn)1924 5314 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))g Fv(b)2213 5329 y Ft(n)2259 5314 y Fv(;)216 b Fw(\()p Fv(m)23 b Fw(+)f Fv(k)s Fw(\))2838 5273 y Fu(2)2905 5314 y Fv(<)27 b(\025:)123 b Fw(\(3.4.20\))1781 5712 y(51)p eop %%Page: 52 57 52 56 bop 120 407 a Fj(Pr)-5 b(o)g(of.)48 b Fw(W)-8 b(e)33 b(denote)h(b)m(y)f Fv(e)1088 371 y Ft(n)1168 407 y Fw(the)g Fv(n)p Fw(-th)f(canonical)f(unit)h(v)m(ector)i(in)e Fr(C)2658 371 y Ft(N)7 b Fu(\()p Ft(\025;k)r Fu(\))2885 407 y Fw(.)44 b(Then)539 666 y Fv(Z)613 615 y Fu(\()p Ft(\033)r Fu(\))606 690 y(0)714 666 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))945 625 y Fl(\003)984 666 y Fv(e)1029 625 y Ft(n)1104 666 y Fw(=)1208 589 y Fs(p)p 1291 589 60 4 v 77 x Fv(\033)1721 598 y Fw(1)p 1360 643 771 4 v 1360 676 a Fs(p)p 1443 676 59 4 v 72 x Fv(\031)1529 710 y Fm(4)1504 663 y Fo(p)p 1604 663 527 4 v 85 x Fv(\025)22 b Fs(\000)h Fw(\()p Fv(n)f Fw(+)g Fv(k)s Fw(\))2091 719 y Fu(2)2157 666 y Fv(e)2202 681 y Ft(n)2277 666 y Fw(=)2380 589 y Fs(p)p 2463 589 60 4 v 77 x Fv(\033)f( )2606 625 y Fu(\(0\))2602 690 y Ft(n)2700 666 y Fw(\()p Fv(:;)c Fw(0;)g Fv(\025;)g(k)s Fw(\))p Fv(:)120 955 y Fw(In)33 b(the)g(notation)e(of)h(Lemma)f(3.4.6)h (w)m(e)i(ha)m(v)m(e)435 1194 y Fv(')499 1153 y Fl(\000)499 1219 y Ft(n)558 1194 y Fw(\()p Fv(:)p Fw(;)17 b Fv(\025;)g(k)s Fw(\))27 b(=)991 1084 y Fo(\020)1050 1194 y Fv(I)j Fw(+)22 b Fv(T)1292 1143 y Fu(\()p Ft(\033)r Fu(\))1278 1219 y(0)1394 1194 y Fw(\()p Fv(\025)g Fw(+)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))g Fv(T)1915 1153 y Fu(\()p Ft(\033)r Fu(\))2016 1194 y Fw(\()p Fv(\025)22 b Fs(\000)g Fv(i)p Fw(0)p Fv(;)17 b(k)s Fw(\))2450 1153 y Fl(\003)2489 1084 y Fo(\021)2549 1106 y Fl(\000)p Fu(1)2660 1194 y Fv(Z)2734 1143 y Fu(\()p Ft(\033)r Fu(\))2727 1219 y(0)2835 1194 y Fw(\()p Fv(\025;)g(k)s Fw(\))3066 1153 y Fl(\003)3105 1194 y Fv(e)3150 1153 y Ft(n)3198 1194 y Fv(;)120 1442 y Fw(and)33 b(so)f(b)s(ecause)i(of)e(the)h(represen)m(tation)h(form)m (ula)c(\(3.4.1\))i(for)g(the)h(scattering)g(matrix)886 1706 y Fv(Z)960 1665 y Fu(\()p Ft(\033)r Fu(\))1061 1706 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))p Fv(')1356 1665 y Fl(\000)1356 1731 y Ft(n)1415 1706 y Fw(\()p Fv(:)p Fw(;)g Fv(\025;)g(k)s Fw(\))27 b(=)1904 1639 y(1)p 1857 1683 141 4 v 1857 1775 a(2)p Fv(\031)t(i)2008 1706 y Fw(\()p Fv(I)j Fs(\000)23 b Fv(S)2285 1665 y Fu(\()p Ft(\033)r Fu(\))2386 1706 y Fw(\()p Fv(\025;)17 b(k)s Fw(\)\))p Fv(e)2700 1665 y Ft(n)2747 1706 y Fv(:)120 1962 y Fw(Using)32 b Fv(')458 1926 y Fl(\000)458 1987 y Ft(n)517 1962 y Fw(\()p Fv(:)p Fw(;)17 b Fv(\025;)g(k)s Fw(\))27 b(=)950 1890 y Fs(p)p 1033 1890 60 4 v 72 x Fv(\033)t( )1159 1926 y Fl(\000)1155 1987 y Ft(n)1218 1962 y Fw(\()p Fv(:;)17 b Fw(0;)g Fv(\025;)g(k)s Fw(\))31 b(and)i(\(3.4.19\))e(w)m(e)j(can)f (rewrite)g(the)g(LHS)f(as)449 2146 y Fo(\000)495 2227 y Fv(Z)569 2186 y Fu(\()p Ft(\033)r Fu(\))670 2227 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))p Fv(')965 2186 y Fl(\000)965 2251 y Ft(n)1024 2227 y Fw(\()p Fv(:)p Fw(;)g Fv(\025;)g(k)s Fw(\))1326 2146 y Fo(\001)1371 2266 y Ft(m)1465 2227 y Fw(=)1953 2159 y(1)p 1578 2204 798 4 v 1578 2237 a Fs(p)p 1661 2237 59 4 v 72 x Fv(\031)1748 2271 y Fm(4)1723 2224 y Fo(p)p 1823 2224 554 4 v 85 x Fv(\025)22 b Fs(\000)g Fw(\()p Fv(m)h Fw(+)f Fv(k)s Fw(\))2337 2280 y Fu(2)2386 2227 y Fw(\()p Fv(\033)t( )2550 2186 y Fl(\000)2546 2251 y Ft(n)2609 2227 y Fw(\()p Fv(:;)17 b Fw(0;)g Fv(\025;)g(k)s Fw(\)\))3038 2223 y Fo(b)3042 2251 y Ft(m)3135 2227 y Fw(=)1465 2505 y(=)1594 2438 y Fs(\000)p Fv(i)p 1578 2482 142 4 v 1578 2502 a Fs(p)p 1661 2502 59 4 v 72 x Fv(\031)1747 2505 y(a)1798 2520 y Ft(mn)1908 2505 y Fw(\()p Fv(\025;)g(k)s Fw(\))p Fv(:)120 2780 y Fw(This)30 b(pro)m(v)m(es)i (\(3.4.20\))d(in)g(the)i(case)g Fv(b)d Fw(=)f Fv(e)1692 2744 y Ft(n)1740 2780 y Fw(.)42 b(The)31 b(general)f(case)h(follo)m(ws) d(b)m(y)j(linearit)m(y)-8 b(.)p 3469 2780 4 66 v 3473 2718 59 4 v 3473 2780 V 3531 2780 4 66 v 120 2984 a Fp(Example)41 b(3.4.10.)i Fw(Supp)s(ose)38 b(that)e Fv(\033)h Fs(2)e Fr(R)47 b Fw(is)35 b(constan)m(t.)55 b(Then)38 b(the)e(equations)h (\(3.4.8\))120 3104 y(of)32 b(Lippmann-Sc)m(h)m(winger-t)m(yp)s(e)h (can)f(b)s(e)h(solv)m(ed)g(explicitly)e(and)i(one)g(obtains)151 3363 y Fv( )218 3322 y Fl(\006)214 3388 y Ft(n)277 3363 y Fw(\()p Fv(x)p Fw(;)17 b Fv(\025;)g(k)s Fw(\))28 b(=)819 3296 y(1)p 748 3340 191 4 v 748 3360 a Fs(p)p 831 3360 108 4 v 83 x Fw(2)p Fv(\031)949 3363 y(e)994 3322 y Ft(inx)1101 3331 y Fm(1)1510 3296 y Fw(1)p 1149 3340 771 4 v 1149 3374 a Fs(p)p 1232 3374 59 4 v 71 x Fv(\031)1319 3408 y Fm(4)1294 3360 y Fo(p)p 1394 3360 527 4 v 85 x Fv(\025)22 b Fs(\000)g Fw(\()p Fv(n)h Fw(+)f Fv(k)s Fw(\))1881 3417 y Fu(2)1930 3363 y Fs(\002)271 3683 y(\002)371 3513 y Fo( )450 3683 y Fw(cos\()618 3593 y Fo(p)p 718 3593 V 90 x Fv(\025)g Fs(\000)g Fw(\()p Fv(n)h Fw(+)f Fv(k)s Fw(\))1205 3655 y Fu(2)1261 3683 y Fv(x)1316 3698 y Fu(2)1356 3683 y Fw(\))g Fs(\007)1916 3616 y Fv(\033)p 1525 3660 841 4 v 1525 3766 a(i)1558 3680 y Fo(p)p 1658 3680 527 4 v 86 x Fv(\025)g Fs(\000)h Fw(\()p Fv(n)f Fw(+)g Fv(k)s Fw(\))2145 3737 y Fu(2)2207 3766 y Fs(\006)g Fv(\033)2375 3683 y(e)2420 3642 y Fl(\007)p Ft(i)2499 3575 y Fs(p)p 2583 3575 322 4 v 2583 3642 a Ft(\025)p Fl(\000)p Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))2870 3623 y Fm(2)2916 3642 y Ft(x)2956 3651 y Fm(2)2994 3513 y Fo(!)3090 3683 y Fv(;)114 b(x)28 b Fs(2)g Fw(\005)p Fv(:)120 3985 y Fw(In)33 b(particular,)e(w)m(e)i(\014nd)926 4244 y Fv(a)977 4259 y Ft(mn)1087 4244 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))27 b(=)h Fs(\000)1583 4176 y Fw(1)p 1536 4221 142 4 v 1536 4241 a Fs(p)p 1619 4241 59 4 v 72 x Fv(\031)2105 4176 y(\033)p 1714 4221 841 4 v 1714 4326 a(i)1747 4241 y Fo(p)p 1847 4241 527 4 v 85 x Fv(\025)22 b Fs(\000)h Fw(\()p Fv(n)f Fw(+)g Fv(k)s Fw(\))2334 4297 y Fu(2)2396 4326 y Fs(\000)h Fv(\033)2581 4244 y(\016)2624 4259 y Ft(nm)120 4550 y Fw(and)33 b(for)f Fv(b)c Fs(2)g Fr(C)688 4514 y Ft(N)7 b Fu(\()p Ft(\025)q(;k)r Fu(\))391 4744 y Fo(\000)437 4825 y Fv(S)503 4783 y Fu(\()p Ft(\033)r Fu(\))604 4825 y Fw(\()p Fv(\025;)17 b(k)s Fw(\))g Fv(b)893 4744 y Fo(\001)939 4864 y Ft(m)1033 4825 y Fw(=)1225 4730 y Fo(X)1136 4947 y Fu(\()p Ft(n)p Fu(+)p Ft(k)r Fu(\))1327 4928 y Fm(2)1362 4947 y Ft(<\025)1486 4755 y Fv(i)1519 4670 y Fo(p)p 1619 4670 V 85 x Fv(\025)22 b Fs(\000)g Fw(\()p Fv(n)g Fw(+)g Fv(k)s Fw(\))2105 4727 y Fu(2)2167 4755 y Fw(+)g Fv(\033)p 1485 4802 841 4 v 1485 4907 a(i)1518 4822 y Fo(p)p 1618 4822 527 4 v 85 x Fv(\025)g Fs(\000)g Fw(\()p Fv(n)h Fw(+)f Fv(k)s Fw(\))2105 4878 y Fu(2)2166 4907 y Fs(\000)h Fv(\033)2351 4825 y(b)2392 4840 y Ft(n)2440 4825 y Fv(;)212 b Fw(\()p Fv(m)22 b Fw(+)g Fv(k)s Fw(\))3014 4783 y Fu(2)3081 4825 y Fv(<)28 b(\025:)1781 5712 y Fw(52)p eop %%Page: 53 58 53 57 bop 120 407 a Fp(4.)71 b(Commen)m(ts)44 b(and)h(Remarks.)98 b(Generalized)44 b(eigenfunctions.)61 b Fw(The)40 b(idea)e(of)120 527 y(connecting)30 b(scattering)h(theory)f(with)g(generalized)g (eigenfunctions)g(go)s(es)g(bac)m(k)i(to)e(A.)g(Y)-8 b(a.)120 648 y(P)m(o)m(vzner)28 b(and)e(T.)g(Ik)m(eb)s(e,)j(see)e (Subsection)g(1.3)e(in)g([Y)-8 b(a2])26 b(for)f(a)h(\(re\014ned\))h (presen)m(tation.)42 b(W)-8 b(e)120 768 y(remark)28 b(the)g(analogy)f (of)h(the)g(form)m(ulae)e(\(1.16\),)j(\(1.17\),)f(\(1.19\),)g(\(1.24\)) g(in)f([Y)-8 b(a2)o(])28 b(with)g(our)120 888 y(form)m(ulae)22 b(\(3.4.7\))o(,)k(\(3.4.6\))o(,)g(\(3.4.8\))o(,)g(\(3.4.20\))o(,)f (resp)s(ectiv)m(ely:)40 b(Up)24 b(to)f(n)m(umerical)g(constan)m(ts)120 1009 y(it)j(is)h(only)g(necessary)i(to)e(replace)g Fv(V)49 b Fw(b)m(y)28 b Fv(\033)15 b Fs(\012)c Fv(\016)32 b Fw(and)c(to)e (think)h(of)g(the)h(scattering)f(amplitude)120 1129 y(not)32 b(as)h(a)f(function)g(of)g(the)g(directions)g Fv(\022)s(;)i(!)d Fs(2)d Fr(S)1955 1093 y Fu(1)2021 1129 y Fw(of)k(the)h(incoming)d(and)i (outgoing)f(w)m(a)m(v)m(e,)120 1249 y(but)e(as)g(a)g(function)f(of)g (the)i(c)m(hannel)f(n)m(um)m(b)s(ers)h Fv(n;)j(m)28 b Fs(2)g Fr(Z)e Fw(of)j(the)g(incoming)e(and)h(outgoing)120 1370 y(w)m(a)m(v)m(e.)45 b(F)-8 b(or)29 b(\014xed)i(energy)g Fv(\025)f Fw(and)g(quasimomen)m(tum)f Fv(k)k Fw(only)d(the)g(op)s(en)g (c)m(hannels)h(\(i.e.)43 b(the)120 1490 y(n)m(um)m(b)s(ers)c Fv(n)e Fs(2)g Fr(Z)e Fw(suc)m(h)40 b(that)d(\()p Fv(n)26 b Fw(+)g Fv(k)s Fw(\))1579 1454 y Fu(2)1655 1490 y Fs(\024)37 b Fv(\025)p Fw(\))h(are)g(in)m(v)m(olv)m(ed)g(in)g(the)g(scattering)g (pro)s(cess.)120 1611 y(See)33 b(also)f(Chap.)44 b(XVI)s(I)s(I)33 b(in)f([LL])g(for)g(the)h(ph)m(ysical)g(in)m(terpretation.)120 1731 y(W)-8 b(e)47 b(ha)m(v)m(e)h(touc)m(hed)h(the)e(sub)5 b(ject)48 b(of)f(generalized)f(eigenfunctions)h(only)f(brie\015y)-8 b(.)86 b(It)47 b(is)120 1851 y(p)s(ossible)32 b(to)g(pro)m(v)m(e)i (that)e(the)h(de\014nition)622 2114 y(\(\000)721 2073 y Fl(\006)780 2114 y Fw(\()p Fv(k)s Fw(\))p Fv(u)p Fw(\))1004 2129 y Ft(n)1051 2114 y Fw(\()p Fv(\025)p Fw(\))27 b(:=)1342 1979 y Fo(Z)1397 2204 y Fu(\005)1471 2114 y Fv(u)p Fw(\()p Fv(x)p Fw(\))p 1658 2028 456 4 v Fv( )1725 2086 y Fl(\006)1721 2139 y Ft(n)1784 2114 y Fw(\()p Fv(x)p Fw(;)17 b Fv(\025;)g(k)s Fw(\))f Fv(dx;)212 b Fw(\()p Fv(n)22 b Fw(+)g Fv(k)s Fw(\))2783 2073 y Fu(2)2850 2114 y Fv(<)28 b(\025;)120 2388 y Fw(for)f Fv(u)g Fs(2)h(S)7 b Fw(\(\005\))29 b(can)e(b)s(e)h (extended)i(to)d(a)g(partially)e(isometric,)i(surjectiv)m(e)h(op)s (erator)f(\000)3296 2352 y Fl(\006)3355 2388 y Fw(\()p Fv(k)s Fw(\))h(:)120 2522 y Fv(L)186 2537 y Fu(2)226 2522 y Fw(\(\005\))k Fs(!)g Fg(h)p Fw(\()p Fv(k)s Fw(\))k(with)f (initial)c(space)37 b Fs(R)p Fw(\()p Fv(P)1729 2471 y Fu(\()p Ft(\033)r Fu(\))1715 2533 y Ft(ac)1830 2522 y Fw(\()p Fv(k)s Fw(\)\),)g(whic)m(h)e(diagonalizes)f Fv(H)2978 2486 y Fu(\()p Ft(\033)r Fu(\))3079 2522 y Fw(\()p Fv(k)s Fw(\).)52 b(More-)120 2643 y(o)m(v)m(er,)44 b(a)d(pro)s(of)f(for)h(the) g(existence)i(and)e(completeness)h(of)e(the)i(w)m(a)m(v)m(e)h(op)s (erators)d(can)i(b)s(e)120 2763 y(obtained)34 b(in)g(this)g(w)m(a)m(y)i (without)f(using)f(the)h(trace)g(class)g(prop)s(ert)m(y)g(of)f(the)i (resolv)m(en)m(t)f(dif-)120 2884 y(ference.)1781 5712 y(53)p eop %%Page: 54 59 54 58 bop 120 1071 a Fk(Chapter)78 b(4)120 1486 y(Sp)6 b(ectral)78 b(analysis)e(and)h(scattering)120 1735 y(theory)h(of)f(the) h(op)6 b(erators)79 b Fd(H)2745 1660 y Fz(\()p Fc(\033)6 b Fz(\))3050 1735 y Fk(on)120 1984 y(the)78 b(halfplane)120 2437 y Fw(Throughout)30 b(this)f(c)m(hapter,)i(w)m(e)g(assume)f (without)f(further)g(commen)m(t)g(that)h Fv(\033)h Fw(:)d Fr(R)39 b Fs(7!)27 b Fr(R)40 b Fw(is)120 2557 y(a)32 b(p)s(erio)s(dic)f(function)h(suc)m(h)i(that)f(for)f(some)g(1)c Fv(<)f(q)32 b Fs(\024)c(1)1603 2757 y Fv(\033)j Fs(2)d Fv(L)1849 2772 y Ft(q)1888 2757 y Fw(\()p Fr(T)p Fw(\))p Fv(:)120 3087 y Fq(4.1)161 b(Scattering)53 b(theory)f(for)i(the)f(op)t (erators)g Ff(H)3150 3035 y Fw(\()p Fv(\033)t Fw(\))120 3306 y(Our)27 b(goal)e(in)h(this)h(section)g(is)f(to)h(study)h(the)f (scattering)g(prop)s(erties)f(of)h(the)g(op)s(erators)g Fv(H)3439 3270 y Fu(\()p Ft(\033)r Fu(\))120 3427 y Fw(on)32 b(the)h(halfplane.)120 3547 y(In)42 b(Subsection)h(1)f(the)g (comparison)f(op)s(erator)g(is)h Fv(H)2141 3511 y Fu(\(0\))2277 3547 y Fw(and)g(w)m(e)h(will)d(deduce)j(existence)120 3667 y(of)35 b(the)g(w)m(a)m(v)m(e)i(op)s(erators)e(from)e(the)j (results)f(on)g(the)h(halfstrip)d(using)i(the)g(Gelfand)f(trans-)120 3788 y(formation.)53 b(In)37 b(Subsection)g(2)g(w)m(e)g(will)e(commen)m (t)h(brie\015y)g(on)h(other)g(natural)e(c)m(hoices)i(of)120 3908 y(comparison)31 b(op)s(erators.)120 4165 y Fp(1.)49 b(Existence)35 b(of)h(the)g(w)m(a)m(v)m(e)g(op)s(erators.)99 b Fw(W)-8 b(e)31 b(will)e(denote)j(the)g(w)m(a)m(v)m(e)h(op)s(erators)e (on)120 4285 y(the)i(halfplane)e(in)h(case)h(of)f(existence)j(b)m(y) 1312 4486 y Fv(W)1418 4435 y Fu(\()p Ft(\033)r Fu(\))1404 4507 y Fl(\006)1547 4486 y Fw(:=)27 b Fv(W)1769 4501 y Fl(\006)1829 4486 y Fw(\()p Fv(H)1956 4444 y Fu(\()p Ft(\033)r Fu(\))2057 4486 y Fv(;)17 b(H)2190 4444 y Fu(\(0\))2283 4486 y Fw(\))p Fv(:)120 4703 y Fw(Moreo)m(v)m(er,)41 b Fv(P)659 4652 y Fu(\()p Ft(\033)r Fu(\))645 4713 y Ft(ac)760 4703 y Fw(\()p Fv(k)s Fw(\))d(will)e(denote)j(the)g(pro)5 b(jection)38 b(in)f Fv(L)2266 4718 y Fu(2)2306 4703 y Fw(\(\005\))h(on)m(to)g(the)h(absolutely)e(con-)120 4823 y(tin)m(uous)32 b(subspace)i(of)d Fv(H)1070 4787 y Fu(\()p Ft(\033)r Fu(\))1171 4823 y Fw(\()p Fv(k)s Fw(\).)43 b(Recall)31 b(the)h(de\014nition)f(of)g(the)h(Gelfand)f(transformation) 120 4944 y Fs(U)43 b Fw(in)32 b(Section)g(1.3.)120 5130 y Fp(Theorem)37 b(4.1.1.)42 b Fj(The)34 b(wave)g(op)-5 b(er)g(ators)35 b Fv(W)1866 5079 y Fu(\()p Ft(\033)r Fu(\))1852 5152 y Fl(\006)2002 5130 y Fj(on)g(the)g(halfplane)e(exist)i (and)f(satisfy)1107 5406 y Fs(U)27 b Fv(W)1302 5355 y Fu(\()p Ft(\033)r Fu(\))1288 5428 y Fl(\006)1420 5406 y Fs(U)1492 5365 y Fl(\003)1559 5406 y Fw(=)1663 5271 y Fo(Z)1763 5297 y Fu(1)p Ft(=)p Fu(2)1718 5496 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1900 5311 y Fo(M)2067 5406 y Fv(W)2173 5355 y Fu(\()p Ft(\033)r Fu(\))2159 5428 y Fl(\006)2274 5406 y Fw(\()p Fv(k)s Fw(\))17 b Fv(dk)s(:)711 b Fw(\(4.1.1\))1781 5712 y(54)p eop %%Page: 55 60 55 59 bop 120 407 a Fj(In)34 b(p)-5 b(articular,)821 684 y Fs(R)p Fw(\()p Fv(W)1049 633 y Fu(\()p Ft(\033)r Fu(\))1035 706 y(+)1150 684 y Fw(\))28 b(=)g Fs(R)p Fw(\()p Fv(W)1548 633 y Fu(\()p Ft(\033)r Fu(\))1534 706 y Fl(\000)1649 684 y Fw(\))g(=)1818 549 y Fo(Z)1918 575 y Fu(1)p Ft(=)p Fu(2)1874 774 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)2055 590 y Fo(M)2222 684 y Fs(R)p Fw(\()p Fv(P)2421 643 y Fu(\()p Ft(\033)r Fu(\))2407 709 y Ft(ac)2523 684 y Fw(\()p Fv(k)s Fw(\)\))17 b Fv(dk)s(:)424 b Fw(\(4.1.2\))120 991 y Fj(Pr)-5 b(o)g(of.)48 b Fw(The)41 b(RHS)e(of)45 b(\(4.1.1\))39 b(mak)m(es)h(sense,)i(since)e(the)g(family)c Fv(W)2727 940 y Fu(\()p Ft(\033)r Fu(\))2713 1012 y Fl(\006)2829 991 y Fw(\()p Fv(k)s Fw(\))p Fv(;)43 b(k)f Fs(2)d Fw([)p Fs(\000)3341 951 y Fu(1)p 3341 968 36 4 v 3341 1025 a(2)3387 991 y Fv(;)3441 951 y Fu(1)p 3441 968 V 3441 1025 a(2)3486 991 y Fw(],)120 1111 y(is)46 b(measurable.)85 b(Indeed,)51 b(the)c(unitary)f(groups)h Fv(U)2128 1075 y Fu(\()p Ft(\033)r Fu(\))2231 1111 y Fw(\()p Fv(t;)17 b(k)s Fw(\))51 b(:=)g(exp)q(\()p Fs(\000)p Fv(itH)3066 1075 y Fu(\()p Ft(\033)r Fu(\))3168 1111 y Fw(\()p Fv(k)s Fw(\)\))c(and)120 1231 y Fv(U)196 1195 y Fu(\(0\))291 1231 y Fw(\()p Fv(t;)17 b(k)s Fw(\))42 b(:=)h(exp)q(\()p Fs(\000)p Fv(itH)1109 1195 y Fu(\(0\))1203 1231 y Fw(\()p Fv(k)s Fw(\)\))f(are)f(measurable)f(in)h Fv(k)j Fw(as)e(b)s(ounded)g(functions)f(of)g(the)120 1352 y(measurable)33 b Fv(H)720 1316 y Fu(\()p Ft(\033)r Fu(\))821 1352 y Fw(\()p Fv(k)s Fw(\))h(and)g Fv(H)1265 1316 y Fu(\(0\))1359 1352 y Fw(\()p Fv(k)s Fw(\),)g(and)f(so)h(the)g (strong)g(limit)c(of)k(their)f(pro)s(duct)h(is)f(also)120 1472 y(measurable.)120 1593 y(Put)42 b Fv(U)396 1556 y Fu(\()p Ft(\033)r Fu(\))498 1593 y Fw(\()p Fv(t)p Fw(\))h(:=)g(exp)q (\()p Fs(\000)p Fv(itH)1219 1556 y Fu(\()p Ft(\033)r Fu(\))1321 1593 y Fw(\))f(and)f Fv(U)1675 1556 y Fu(\(0\))1770 1593 y Fw(\()p Fv(t)p Fw(\))i(:=)g(exp)q(\()p Fs(\000)p Fv(itH)2491 1556 y Fu(\(0\))2586 1593 y Fw(\),)h(then)e(w)m(e)h(kno)m (w)f(from)120 1713 y(Theorem)33 b(XI)s(I)s(I.85)f(in)g([RS4])g(that)578 1996 y Fs(U)27 b Fv(U)743 1955 y Fu(\()p Ft(\033)r Fu(\))845 1996 y Fw(\()p Fv(t)p Fw(\))17 b Fs(U)1045 1955 y Fl(\003)1112 1996 y Fw(=)28 b(exp)q(\()p Fs(\000)p Fv(it)17 b Fs(U)27 b Fv(H)1743 1955 y Fu(\()p Ft(\033)r Fu(\))1861 1996 y Fs(U)1933 1955 y Fl(\003)1973 1996 y Fw(\))h(=)2142 1860 y Fo(Z)2242 1887 y Fu(1)p Ft(=)p Fu(2)2197 2086 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)2379 1901 y Fo(M)2546 1996 y Fv(U)2622 1955 y Fu(\()p Ft(\033)r Fu(\))2725 1996 y Fw(\()p Fv(t;)17 b(k)s Fw(\))g Fv(dk)s(;)586 2295 y Fs(U)26 b Fv(U)750 2254 y Fu(\(0\))845 2295 y Fw(\()p Fv(t)p Fw(\))17 b Fs(U)1045 2254 y Fl(\003)1112 2295 y Fw(=)28 b(exp)q(\()p Fs(\000)p Fv(it)17 b Fs(U)27 b Fv(H)1743 2254 y Fu(\(0\))1837 2295 y Fs(U)1909 2254 y Fl(\003)1949 2295 y Fw(\))h(=)2118 2159 y Fo(Z)2218 2185 y Fu(1)p Ft(=)p Fu(2)2173 2385 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)2355 2200 y Fo(M)2522 2295 y Fv(U)2598 2254 y Fu(\(0\))2693 2295 y Fw(\()p Fv(t;)17 b(k)s Fw(\))g Fv(dk)s(:)120 2579 y Fw(Hence)34 b(for)e(an)m(y)h Fv(u)27 b Fs(2)h Fv(L)986 2594 y Fu(2)1026 2579 y Fw(\()p Fr(R)1130 2543 y Fu(2)1130 2604 y(+)1195 2579 y Fw(\))246 2726 y Fo(\015)246 2786 y(\015)246 2846 y(\015)246 2906 y(\015)246 2965 y(\015)302 2901 y Fs(U)e Fv(U)466 2860 y Fu(\()p Ft(\033)r Fu(\))569 2901 y Fw(\()p Fs(\000)p Fv(t)p Fw(\))p Fv(U)833 2860 y Fu(\(0\))928 2901 y Fw(\()p Fv(t)p Fw(\))p Fv(u)c Fs(\000)1217 2730 y Fo( )1296 2765 y(Z)1395 2791 y Fu(1)p Ft(=)p Fu(2)1351 2991 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)1532 2806 y Fo(M)1700 2901 y Fv(W)1806 2850 y Fu(\()p Ft(\033)r Fu(\))1792 2922 y Fl(\006)1907 2901 y Fw(\()p Fv(k)s Fw(\))17 b Fv(dk)2159 2730 y Fo(!)2270 2901 y Fs(U)10 b Fv(u)2398 2726 y Fo(\015)2398 2786 y(\015)2398 2846 y(\015)2398 2906 y(\015)2398 2965 y(\015)2454 2753 y Fu(2)2521 2901 y Fw(=)452 3218 y(=)556 3082 y Fo(Z)656 3108 y Fu(1)p Ft(=)p Fu(2)611 3308 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)793 3103 y Fo(\015)793 3163 y(\015)793 3223 y(\015)848 3218 y Fv(U)924 3176 y Fu(\()p Ft(\033)r Fu(\))1027 3218 y Fw(\()p Fs(\000)p Fv(t;)17 b(k)s Fw(\))p Fv(U)1389 3176 y Fu(\(0\))1484 3218 y Fw(\()p Fv(t;)g(k)s Fw(\))g(\()p Fs(U)10 b Fv(u)p Fw(\)\()p Fv(k)s(;)17 b(:)p Fw(\))k Fs(\000)i Fv(W)2342 3167 y Fu(\()p Ft(\033)r Fu(\))2328 3239 y Fl(\006)2443 3218 y Fw(\()p Fv(k)s Fw(\))17 b(\()p Fs(U)10 b Fv(u)p Fw(\)\()p Fv(k)s(;)17 b(:)p Fw(\))2995 3103 y Fo(\015)2995 3163 y(\015)2995 3223 y(\015)3049 3129 y Fu(2)3122 3218 y Fv(dk)30 b Fs(!)e Fw(0)620 3442 y(\()p Fv(t)g Fs(!)f(\0061)p Fw(\))120 3663 y(b)m(y)33 b(Leb)s(esgue's)i(theorem.)43 b(This)33 b(means)f(that)h(the)g(strong)f (limits)e(of)i Fv(U)2818 3627 y Fu(\()p Ft(\033)r Fu(\))2920 3663 y Fw(\()p Fs(\000)p Fv(t)p Fw(\))p Fv(U)3184 3627 y Fu(\(0\))3280 3663 y Fw(\()p Fv(t)p Fw(\))g(for)120 3800 y Fv(t)45 b Fs(!)g(\0061)e Fw(exist)h(and)f(coincide)f(with)g Fs(U)1695 3764 y Fl(\003)1752 3800 y Fw(\()1790 3719 y Fo(R)1856 3746 y Fu(1)p Ft(=)p Fu(2)1837 3834 y Fl(\000)p Fu(1)p Ft(=)p Fu(2)2018 3725 y Fo(L)2145 3800 y Fv(W)2251 3749 y Fu(\()p Ft(\033)r Fu(\))2237 3821 y Fl(\006)2353 3800 y Fw(\()p Fv(k)s Fw(\))17 b Fv(dk)s Fw(\))g Fs(U)10 b Fw(,)44 b(whic)m(h)g(pro)m(v)m(es)h(the)120 3928 y(Theorem.)p 3469 3928 4 66 v 3473 3866 59 4 v 3473 3928 V 3531 3928 4 66 v 266 4121 a(In)34 b(the)f(case)h Fv(\033)e Fs(\025)d Fw(0)j(w)m(e)i(can)f(sa)m(y)h(more.)44 b(T)-8 b(o)33 b(a)m(v)m(oid)g(rep)s(etitions)f(w)m(e)i(p)s(ostp)s(one)f(this)f(to)120 4241 y(the)h(next)g(section.)120 4500 y Fp(2.)98 b(Scattering)52 b(with)g(other)h(comparison)g(op)s(erators.)98 b Fw(In)47 b(the)g(previous)g(sub-)120 4620 y(section)38 b(w)m(e)i(ha)m(v)m(e)f (considered)g(the)g(scattering)f(problem)f(for)h Fv(H)2573 4584 y Fu(\()p Ft(\033)r Fu(\))2712 4620 y Fw(with)g(resp)s(ect)i(to)e (the)120 4741 y(comparison)32 b(op)s(erator)g Fv(H)1118 4704 y Fu(\(0\))1212 4741 y Fw(,)g(the)i(Neumann)f(Laplacian)d(on)j (the)g(halfplane.)43 b(Similar)29 b(re-)120 4861 y(sults)39 b(hold)f(also)h(with)f(other)h(natural)f(c)m(hoices)i(of)f(comparison)f (op)s(erators,)i(namely)f(the)120 4981 y(Diric)m(hlet)30 b(Laplacian)h(on)i(the)g(halfplane)e(and)h(the)h(Laplacian)e(on)h(the)i (whole)e(plane.)120 5102 y(T)-8 b(o)31 b(see)g(this)g(it)e(su\016ces)k (to)e(consider)f(the)i(scattering)e(problem)f(b)s(et)m(w)m(een)k(these) f(op)s(erators)120 5222 y(and)38 b Fv(H)404 5186 y Fu(\(0\))498 5222 y Fw(.)59 b(This)38 b(can)g(studied)g(in)f(terms)h(of)f(the)h (asymptotic)f(b)s(eha)m(vior)h(of)f(the)h(unitary)120 5342 y(groups,)44 b(whic)m(h)e(are)f(giv)m(en)g(explicitly)e(after)i(F) -8 b(ourier)40 b(transformation.)68 b(W)-8 b(e)41 b(do)g(not)g(go)120 5463 y(in)m(to)32 b(details.)1781 5712 y(55)p eop %%Page: 56 61 56 60 bop 120 434 a Fq(4.2)161 b(Sp)t(ectral)54 b(analysis)h(of)f(the)f (op)t(erators)f Ff(H)3035 382 y Fw(\()p Fv(\033)t Fw(\))120 653 y(Our)34 b(goal)e(in)h(this)g(section)h(is)g(to)f(analyze)h(the)g (sp)s(ectrum)h(of)e(the)h(op)s(erators)g Fv(H)3133 617 y Fu(\()p Ft(\033)r Fu(\))3268 653 y Fw(on)g(the)120 773 y(halfplane.)120 894 y(In)c(Subsection)g(1)g(w)m(e)g(pro)m(v)m(e)h (a)e(complete)g(result)h(in)e(the)j(case)f(of)f(non-negativ)m(e)h Fv(\033)t Fw(,)g(namely)120 1014 y(unitary)25 b(equiv)-5 b(alence)25 b(of)g Fv(H)1157 978 y Fu(\()p Ft(\033)r Fu(\))1284 1014 y Fw(and)g Fv(H)1555 978 y Fu(\(0\))1649 1014 y Fw(.)41 b(In)26 b(the)f(case)i(of)d(general)h Fv(\033)t Fw(,)i(generically)d(another)120 1135 y(part)45 b(of)g Fv(H)557 1098 y Fu(\()p Ft(\033)r Fu(\))704 1135 y Fw(exists)i(and)e(is)g(resp)s(onsible)h(for)f(negativ)m(e)h(sp)s (ectrum)g(of)f Fv(H)3028 1098 y Fu(\()p Ft(\033)r Fu(\))3129 1135 y Fw(.)83 b(This)45 b(is)120 1255 y(discussed)37 b(in)e(Subsection)i(2,)f(where)h(w)m(e)g(establish)e(a)g(t)m(ypical)g (phenomenon)h(of)g(p)s(erio)s(dic)120 1375 y(op)s(erators,)c(a)h(gap)f (in)g(the)h(sp)s(ectrum.)120 1634 y Fp(1.)50 b(Main)38 b(result)e(in)h(the)g(case)h Fv(\033)32 b Fs(\025)c Fw(0)p Fp(.)97 b Fw(Our)33 b(main)e(result)h(is)120 1829 y Fp(Theorem)41 b(4.2.1.)j Fj(Assume)38 b(that)h Fv(\033)f Fs(\025)c Fw(0)p Fj(.)54 b(Then)38 b(the)g(wave)f(op)-5 b(er)g(ators)38 b Fv(W)3009 1778 y Fu(\()p Ft(\033)r Fu(\))2995 1851 y Fl(\006)3149 1829 y Fj(exist,)g(ar)-5 b(e)120 1949 y(unitary)35 b(and)g(satisfy)1339 2070 y Fv(H)1428 2028 y Fu(\()p Ft(\033)r Fu(\))1557 2070 y Fw(=)28 b Fv(W)1767 2019 y Fu(\()p Ft(\033)r Fu(\))1753 2091 y Fl(\006)1868 2070 y Fv(H)1957 2028 y Fu(\(0\))2051 2070 y Fv(W)2157 2019 y Fu(\()p Ft(\033)r Fu(\))p Fl(\003)2143 2091 y(\006)2294 2070 y Fv(:)943 b Fw(\(4.2.1\))120 2240 y Fj(In)34 b(p)-5 b(articular,)35 b Fv(H)812 2204 y Fu(\()p Ft(\033)r Fu(\))948 2240 y Fj(is)g(absolutely)g(c)-5 b(ontinuous)34 b(with)h(sp)-5 b(e)g(ctrum)35 b Fw([0)p Fv(;)17 b Fw(+)p Fs(1)p Fw(\))p Fj(.)120 2435 y(Pr)-5 b(o)g(of.)48 b Fw(By)26 b(Theorem)f(4.1.1,)h (together)f(with)f(Theorem)h(3.3.10,)h(the)f(w)m(a)m(v)m(e)i(op)s (erators)e(exist)120 2555 y(and)i(are)g(surjectiv)m(e,)j(so)d(they)h (are)f(unitary)-8 b(.)41 b(Th)m(us)29 b(\(4.2.1\))d(follo)m(ws)g(from)g (the)h(in)m(tert)m(wining)120 2676 y(prop)s(ert)m(y)-8 b(.)p 3469 2676 4 66 v 3473 2613 59 4 v 3473 2676 V 3531 2676 4 66 v 266 2868 a(It)30 b(is)g(clear)f(that)h(in)f(the)i(case)g (of)e(arbitrary)g Fv(\033)34 b Fw(there)d(is)e(the)i(part)f(of)f Fv(H)2908 2832 y Fu(\()p Ft(\033)r Fu(\))3039 2868 y Fw(on)h Fs(R)p Fw(\()p Fv(W)3400 2817 y Fu(\()p Ft(\033)r Fu(\))3386 2890 y Fl(\006)3502 2868 y Fw(\))120 2988 y(whic)m(h)j(is)f(unitarily)f(equiv)-5 b(alen)m(t)32 b(to)g Fv(H)1567 2952 y Fu(\(0\))1661 2988 y Fw(,)h(but)g(there)g(ma)m (y)f(b)s(e)h(also)f(a)g(non-trivial)e(part)i(of)120 3109 y Fv(H)209 3073 y Fu(\()p Ft(\033)r Fu(\))335 3109 y Fw(on)24 b Fs(R)p Fw(\()p Fv(W)690 3058 y Fu(\()p Ft(\033)r Fu(\))676 3131 y Fl(\006)792 3109 y Fw(\))830 3073 y Fl(?)889 3109 y Fw(,)i(whic)m(h)f(is)f(due)h(to)f(the)h(eigen)m(v)-5 b(alues)24 b(of)g(the)h(op)s(erators)g Fv(H)3022 3073 y Fu(\()p Ft(\033)r Fu(\))3123 3109 y Fw(\()p Fv(k)s Fw(\))f(on)h(the)120 3229 y(halfstrip.)53 b(This)36 b(part)g(can)h(b)s (e)f(b)s(ounded)h(\(e.g.,)h(if)d Fv(\033)40 b Fw(is)c(a)g (trigonometric)e(p)s(olynomial)e(of)120 3350 y(degree)24 b Fv(N)38 b(>)28 b Fw(1,)c(see)h(Corollary)c(3.3.11\))h(or)h(un)m(b)s (ounded)h(\(e.g.,)i(if)21 b Fv(\033)27 b Fw(is)c(a)g(negativ)m(e)g (constan)m(t,)120 3470 y(see)42 b(Example)e(3.3.8\).)66 b(If)41 b(it)e(has)i(non-empt)m(y)f(absolutely)g(con)m(tin)m(uous)h(sp) s(ectrum,)i(then)120 3604 y(the)30 b(w)m(a)m(v)m(e)i(op)s(erators)d (are)h(not)f(complete.)42 b(Usually)-8 b(,)30 b(the)g(part)f(of)g Fv(H)2668 3568 y Fu(\()p Ft(\033)r Fu(\))2799 3604 y Fw(on)h Fs(R)p Fw(\()p Fv(W)3160 3553 y Fu(\()p Ft(\033)r Fu(\))3146 3626 y Fl(\006)3262 3604 y Fw(\))3300 3568 y Fl(?)3388 3604 y Fw(will)120 3724 y(ha)m(v)m(e)k(a)e(negativ)m(e)h (sp)s(ectrum.)44 b(This)33 b(is)f(studied)h(in)e(the)i(next)h (subsection.)120 3983 y Fp(2.)50 b(The)38 b(negativ)m(e)f(sp)s(ectrum)g (of)g Fv(H)1638 3947 y Fu(\()p Ft(\033)r Fu(\))1740 3983 y Fp(.)97 b Fw(F)-8 b(or)32 b Fv(k)f Fs(2)d Fw([)p Fs(\000)2333 3943 y Fu(1)p 2333 3960 36 4 v 2333 4017 a(2)2379 3983 y Fv(;)2432 3943 y Fu(1)p 2432 3960 V 2432 4017 a(2)2478 3983 y Fw(])k(and)h Fv(n)28 b Fs(2)g Fr(N)47 b Fw(w)m(e)34 b(put)174 4200 y Fv(\025)231 4215 y Ft(n)278 4200 y Fw(\()p Fv(k)s Fw(\))28 b(:=)299 4377 y(=)234 b(sup)403 4463 y Ft(v)437 4472 y Fm(1)472 4463 y Ft(;:::;v)606 4472 y Fi(n)p Fh(\000)p Fm(1)725 4463 y Fl(2)p Ft(L)820 4472 y Fm(2)855 4463 y Fu(\(\005\))979 4377 y Fw(inf)6 b Fs(f)1157 4330 y Ft(h)1198 4306 y Fm(\()p Fi(\033)r Fm(\))1287 4330 y Fu(\()p Ft(k)r Fu(\)[)p Ft(u)p Fu(])p 1157 4354 304 4 v 1236 4412 a Fl(k)p Ft(u)p Fl(k)1347 4393 y Fm(2)1498 4377 y Fw(:)56 b Fv(u)27 b Fs(2)1783 4352 y Fw(~)1758 4377 y Fv(H)1847 4336 y Fu(1)1886 4377 y Fw(\(\005\))p Fv(;)33 b Fw(\()p Fv(u;)17 b(v)2280 4392 y Ft(j)2316 4377 y Fw(\))28 b(=)f(0)33 b(for)f(1)27 b Fs(\024)h Fv(j)34 b(<)27 b(n)p Fs(g)p Fv(:)3264 4328 y Fw(\(4.2.2\))120 4663 y(\(The)33 b(dep)s(endence)i(on)e Fv(\033)j Fw(is)d(not)f (re\015ected)i(in)e(this)g(notation.\))120 4783 y(The)39 b(v)-5 b(ariational)34 b(principle,)k(together)g(with)g(Theorem)g (3.3.5,)h(implies)d(that)i(for)f(eac)m(h)i Fv(k)120 4903 y Fw(only)23 b(\014nitely)h(man)m(y)g(of)f(the)h Fv(\025)1220 4918 y Ft(n)1267 4903 y Fw(\()p Fv(k)s Fw(\))p Fv(;)34 b(n)27 b Fs(2)i Fr(N)9 b Fw(,)31 b(are)24 b(di\013eren)m(t)g(form)f Fv(k)2568 4867 y Fu(2)2608 4903 y Fw(,)i(and)f(these)i(are)e(exactly) 120 5024 y(the)43 b(eigen)m(v)-5 b(alues)42 b(of)g Fv(H)1022 4988 y Fu(\()p Ft(\033)r Fu(\))1123 5024 y Fw(\()p Fv(k)s Fw(\))g(b)s(elo)m(w)g Fv(k)1635 4988 y Fu(2)1675 5024 y Fw(.)72 b(It)43 b(is)f(easy)h(to)f(see)h(that)f Fv(\025)2796 5039 y Ft(n)2885 5024 y Fw(is)g(a)g(con)m(tin)m(uous)120 5144 y(function)32 b(on)h([)p Fs(\000)752 5105 y Fu(1)p 752 5121 36 4 v 752 5178 a(2)797 5144 y Fv(;)851 5105 y Fu(1)p 851 5121 V 851 5178 a(2)896 5144 y Fw(])g(for)f(eac)m(h)h Fv(n)28 b Fs(2)g Fr(N)9 b Fw(.)50 b(Com)m(bining)31 b(this)h(with)g (Corollary)f(2.1.4)h(w)m(e)i(see)1040 5381 y Fv(\033)1116 5301 y Fo(\000)1162 5381 y Fv(H)1251 5340 y Fu(\()p Ft(\033)r Fu(\))1352 5301 y Fo(\001)1425 5381 y Fw(=)1542 5287 y Fo([)1529 5499 y Ft(n)p Fl(2)p Fn(N)1683 5381 y Fv(\025)1740 5396 y Ft(n)1787 5381 y Fw(\([)p Fs(\000)1939 5342 y Fu(1)p 1939 5358 V 1939 5416 a(2)1985 5381 y Fv(;)2039 5342 y Fu(1)p 2039 5358 V 2039 5416 a(2)2084 5381 y Fw(]\))22 b Fs([)h Fw([0)p Fv(;)17 b Fw(+)p Fs(1)p Fw(\))p Fv(;)643 b Fw(\(4.2.3\))1781 5712 y(56)p eop %%Page: 57 62 57 61 bop 120 407 a Fw(i.e.,)32 b(the)h(sp)s(ectrum)g(of)f Fv(H)1098 371 y Fu(\()p Ft(\033)r Fu(\))1232 407 y Fw(has)h(band)g (structure.)120 527 y(Moreo)m(v)m(er,)46 b(one)c(easily)f(sees)j(that)d (\(4.2.3\))h(coincides)f(with)h(the)g(essen)m(tial)g(sp)s(ectrum)g(of) 120 648 y Fv(H)209 611 y Fu(\()p Ft(\033)r Fu(\))310 648 y Fw(.)120 768 y(The)d(follo)m(wing)c(result)j(is)f(in)g(some)h (sense)h(a)f(con)m(v)m(erse)i(of)d(our)h(main)e(Theorem)i(4.2.1.)59 b(it)120 888 y(sa)m(ys)39 b(essen)m(tially)d(that)h Fv(H)1108 852 y Fu(\()p Ft(\033)r Fu(\))1247 888 y Fw(is)g(not)g(unitarily)e (equiv)-5 b(alen)m(t)37 b(to)f Fv(H)2611 852 y Fu(\(0\))2742 888 y Fw(if)h Fv(\033)k Fw(is)36 b("su\016cien)m(tly)120 1009 y(negativ)m(e".)120 1191 y Fp(Prop)s(osition)g(4.2.2.)42 b Fj(Assume)34 b(that)i(either)1457 1296 y Fo(Z)1557 1322 y Ft(\031)1512 1521 y Fl(\000)p Ft(\031)1631 1431 y Fv(\033)t Fw(\()p Fv(x)1783 1446 y Fu(1)1823 1431 y Fw(\))17 b Fv(dx)1984 1446 y Fu(1)2051 1431 y Fv(<)27 b Fw(0)120 1654 y Fj(or)1261 1775 y Fv(\033)t Fw(\()p Fv(x)1413 1790 y Fu(1)1453 1775 y Fw(\))h Fs(\024)g(\000)1743 1707 y Fv(\031)p 1711 1752 124 4 v 1711 1843 a Fs(j)p Fw(\003)p Fs(j)1844 1775 y Fv(;)216 b(x)2142 1790 y Fu(1)2210 1775 y Fs(2)28 b Fw(\003)p Fv(;)120 1989 y Fj(for)35 b(a)f(non-empty)h(c)-5 b(onne)g(cte)g(d)33 b(op)-5 b(en)35 b(subset)f Fw(\003)28 b Fs(\032)g Fr(T)38 b Fj(of)d(the)g(torus.)45 b(Then)1356 2184 y Fv(\033)1411 2199 y Ft(ess)1530 2103 y Fo(\000)1575 2184 y Fv(H)1664 2143 y Fu(\()p Ft(\033)r Fu(\))1766 2103 y Fo(\001)1839 2184 y Fr(\))28 b Fw([0)p Fv(;)17 b Fw(+)p Fs(1)p Fw(\))p Fv(:)120 2379 y Fj(Pr)-5 b(o)g(of.)48 b Fw(Because)35 b(of)40 b(\(4.2.3\))32 b(it)h(su\016ces)i (to)e(pro)m(v)m(e)h(that)f(there)h(exists)g(a)f Fv(k)f Fs(2)d Fw([)p Fs(\000)3147 2340 y Fu(1)p 3147 2357 36 4 v 3147 2414 a(2)3193 2379 y Fv(;)3247 2340 y Fu(1)p 3247 2357 V 3247 2414 a(2)3292 2379 y Fw(])k(suc)m(h)120 2500 y(that)k Fv(H)425 2464 y Fu(\()p Ft(\033)r Fu(\))526 2500 y Fw(\()p 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3298 36 4 v 582 3355 a(2)628 3321 y Fv(;)682 3282 y Fu(1)p 682 3298 V 682 3355 a(2)727 3321 y Fw(])i(w)m(e)i(in)m(tro)s(duce)e(self-adjoin)m(t)e(op)s(erators)i Fv(H)2374 3270 y Fu(\()p Ft(\033)r Fu(\))2366 3348 y Ft(D)2476 3321 y Fw(\()p Fv(k)s Fw(\))p Fv(;)j(H)2755 3270 y Fu(\()p Ft(\033)r Fu(\))2747 3348 y Ft(N)2856 3321 y Fw(\()p Fv(k)s Fw(\))d(in)g Fv(L)3194 3336 y Fu(2)3233 3321 y Fw(\(\005\))h(via)120 3441 y(the)i(quadratic)f(forms)210 3644 y Fs(D)s Fw([)p Fv(h)373 3594 y Fu(\()p Ft(\033)r Fu(\))373 3671 y Ft(D)474 3644 y Fw(\()p Fv(k)s Fw(\)])c(:=)g Fs(f)p Fv(u)f Fs(2)h Fv(H)1106 3603 y Fu(1)1145 3644 y Fw(\(\005\))f(:)56 b Fv(u)p Fw(\()p Fv(:;)17 b(\031)t Fw(\))27 b(=)g Fv(u)p Fw(\()p Fv(:;)17 b Fs(\000)p Fv(\031)t Fw(\))27 b(=)h(0)p Fs(g)p Fv(;)210 3814 y Fs(D)s Fw([)p Fv(h)373 3764 y Fu(\()p Ft(\033)r Fu(\))373 3842 y Ft(N)474 3814 y Fw(\()p Fv(k)s Fw(\)])g(:=)g Fv(H)879 3773 y Fu(1)918 3814 y Fw(\(\005\))p Fv(;)234 4022 y(h)290 3981 y Fu(\()p Ft(\033)r Fu(\))290 4046 y Ft(\027)392 4022 y Fw(\()p Fv(k)s Fw(\)[)p Fv(u)p Fw(])f(:=)790 3886 y Fo(Z)845 4112 y Fu(\005)919 3941 y Fo(\000)964 4022 y Fs(j)p Fw(\()p Fv(D)1111 4037 y Fu(1)1173 4022 y Fw(+)22 b Fv(k)s Fw(\))p Fv(u)p Fw(\()p Fv(x)p Fw(\))p Fs(j)1578 3981 y Fu(2)1639 4022 y Fw(+)g Fs(j)p Fv(D)1846 4037 y Fu(2)1885 4022 y Fv(u)p Fw(\()p Fv(x)p Fw(\))p Fs(j)2100 3981 y Fu(2)2139 3941 y Fo(\001)2218 4022 y Fv(dx)g Fw(+)2444 3886 y Fo(Z)2543 3912 y Ft(\031)2499 4112 y Fl(\000)p Ft(\031)2618 4022 y Fv(\033)t Fw(\()p Fv(x)2770 4037 y Fu(1)2810 4022 y Fw(\))p Fs(j)p Fv(u)p Fw(\()p Fv(x)3025 4037 y Fu(1)3063 4022 y Fv(;)17 b Fw(0\))p Fs(j)3222 3981 y Fu(2)3278 4022 y Fv(dx)3384 4037 y Fu(1)3423 4022 y Fv(;)827 4236 y(\027)34 b Fs(2)28 b(f)p Fv(D)s(;)17 b(N)10 b Fs(g)p Fv(:)120 4425 y Fw(Closedness)50 b(and)e(lo)m(w)m(er)h(semib)s (oundedness)g(is)f(established)g(as)h(in)e(Section)h(2.1.)90 b(The)120 4545 y(op)s(erators)26 b Fv(H)634 4494 y Fu(\()p Ft(\033)r Fu(\))626 4572 y Ft(D)736 4545 y Fw(\()p Fv(k)s Fw(\))p Fv(;)33 b(H)1015 4494 y Fu(\()p Ft(\033)r Fu(\))1007 4572 y Ft(N)1116 4545 y Fw(\()p Fv(k)s Fw(\))26 b(di\013er)g(from)f Fv(H)1831 4509 y Fu(\()p Ft(\033)r Fu(\))1933 4545 y Fw(\()p Fv(k)s Fw(\))h(b)m(y)h(Diric)m(hlet)e(and)h(natural)g(b)s (oundary)120 4665 y(conditions,)32 b(resp)s(ectiv)m(ely)-8 b(,)33 b(at)g Fv(x)1339 4680 y Fu(1)1406 4665 y Fw(=)28 b Fs(\006)p Fv(\031)t Fw(.)120 4786 y(De\014ne)42 b(the)g(n)m(um)m(b)s (ers)g Fv(\025)1067 4750 y Ft(\027)1067 4810 y(n)1114 4786 y Fv(;)33 b(n)43 b Fs(2)g Fr(N)9 b Fw(,)50 b(b)m(y)43 b(\(4.2.2\))e(with)g Fv(k)46 b Fw(=)d(0)e(and)2750 4761 y(~)2725 4786 y Fv(H)2814 4750 y Fu(1)2853 4786 y Fw(\(\005\))g (replaced)h(b)m(y)120 4920 y Fs(D)s Fw([)p Fv(h)283 4869 y Fu(\()p Ft(\033)r Fu(\))283 4930 y Ft(\027)385 4920 y Fw(\(0\)])e(for)g Fv(\027)48 b Fs(2)42 b(f)p Fv(D)s(;)17 b(N)10 b Fs(g)p Fw(.)68 b(A)41 b(su\016cien)m(t)g(condition)f(for)g (the)h(existence)h(of)f(a)f(gap)g(is)120 5040 y(giv)m(en)33 b(b)m(y)120 5222 y Fp(Prop)s(osition)j(4.2.4.)42 b Fj(L)-5 b(et)35 b Fv(n)28 b Fs(2)g Fr(N)50 b Fj(such)34 b(that)i Fv(\025)1962 5186 y Ft(D)1962 5247 y(n)2053 5222 y Fv(<)28 b(\025)2214 5186 y Ft(N)2214 5247 y(n)p Fu(+1)2351 5222 y Fj(.)44 b(Then)1275 5428 y Fv(\033)1350 5347 y Fo(\000)1396 5428 y Fv(H)1485 5387 y Fu(\()p Ft(\033)r Fu(\))1586 5347 y Fo(\001)1654 5428 y Fs(\\)23 b Fw(\()p Fv(\025)1838 5387 y Ft(D)1838 5453 y(n)1901 5428 y Fv(;)17 b(\025)2002 5387 y Ft(N)2002 5453 y(n)p Fu(+1)2139 5428 y Fw(\))28 b(=)f Fs(;)p Fv(:)1781 5712 y Fw(57)p eop %%Page: 58 63 58 62 bop 120 408 a Fj(Pr)-5 b(o)g(of.)48 b Fw(By)g(the)g(inclusions)e (of)h(the)g(form)f(domains)g(w)m(e)i(ha)m(v)m(e)h Fv(H)2676 357 y Fu(\()p Ft(\033)r Fu(\))2668 435 y Ft(N)2777 408 y Fw(\()p Fv(k)s Fw(\))k Fs(\024)g Fv(H)3179 372 y Fu(\()p Ft(\033)r Fu(\))3280 408 y Fw(\()p Fv(k)s Fw(\))g Fs(\024)120 544 y Fv(H)209 493 y Fu(\()p Ft(\033)r Fu(\))201 571 y Ft(D)310 544 y Fw(\()p Fv(k)s Fw(\),)33 b(so)g(b)m(y)g(the)g(v)-5 b(ariational)29 b(principle)1314 736 y Fv(\025)1371 695 y Ft(N)1371 760 y(n)1438 736 y Fw(\()p Fv(k)s Fw(\))f Fs(\024)g Fv(\025)1758 751 y Ft(n)1805 736 y Fw(\()p Fv(k)s Fw(\))g Fs(\024)g Fv(\025)2125 695 y Ft(D)2125 760 y(n)2189 736 y Fw(\()p Fv(k)s Fw(\))p Fv(;)918 b Fw(\(4.2.4\))120 941 y(where)43 b Fv(\025)468 905 y Ft(\027)468 966 y(n)515 941 y Fw(\()p Fv(k)s Fw(\))e(are)h(de\014ned)h(b)m(y)h (\(4.2.2\))d(with)1923 916 y(~)1898 941 y Fv(H)1987 905 y Fu(1)2026 941 y Fw(\(\005\))g(replaced)h(b)m(y)h Fs(D)s Fw([)p Fv(h)2918 890 y Fu(\()p Ft(\033)r Fu(\))2918 952 y Ft(\027)3019 941 y Fw(\()p Fv(k)s Fw(\)])f(for)f Fv(\027)50 b Fs(2)120 1062 y(f)p Fv(D)s(;)17 b(N)10 b Fs(g)p Fw(.)80 b(The)46 b(unitary)e(op)s(erator)g Fv(E)1589 1077 y Ft(k)1677 1062 y Fw(on)h Fv(L)1891 1077 y Fu(2)1931 1062 y Fw(\(\005\),)j(\()p Fv(E)2265 1077 y Ft(k)2307 1062 y Fv(u)p Fw(\)\()p Fv(x)p Fw(\))h(:=)f Fv(e)2777 1026 y Fl(\000)p Ft(ik)r(x)2935 1035 y Fm(1)2974 1062 y Fv(u)p Fw(\()p Fv(x)p Fw(\))p Fv(;)33 b(x)49 b Fs(2)g Fw(\005)p Fv(;)120 1196 y Fw(satis\014es)42 b Fv(H)577 1145 y Fu(\()p Ft(\033)r Fu(\))569 1206 y Ft(\027)679 1196 y Fw(\()p Fv(k)s Fw(\))g(=)h Fv(E)1042 1211 y Ft(k)1085 1196 y Fv(H)1174 1145 y Fu(\()p Ft(\033)r Fu(\))1166 1206 y Ft(\027)1275 1196 y Fw(\(0\))p Fv(E)1478 1159 y Fl(\003)1472 1222 y Ft(k)1518 1196 y Fv(;)i(\027)k Fs(2)43 b(f)p Fv(D)s(;)17 b(N)10 b Fs(g)p Fw(,)44 b(whence)f Fv(\025)2590 1159 y Ft(\027)2590 1220 y(n)2637 1196 y Fw(\()p Fv(k)s Fw(\))g(=)g Fv(\025)2986 1159 y Ft(\027)2986 1220 y(n)3074 1196 y Fw(for)e(all)f Fv(k)46 b Fs(2)120 1316 y Fw([)p Fs(\000)234 1277 y Fu(1)p 234 1293 36 4 v 234 1350 a(2)280 1316 y Fv(;)333 1277 y Fu(1)p 333 1293 V 333 1350 a(2)379 1316 y Fw(].)d(No)m(w)35 b(\(4.2.4\))d(implies) 924 1517 y(max)865 1593 y Ft(k)r Fl(2)p Fu([)p Fl(\000)1036 1565 y Fm(1)p 1035 1577 31 3 v 1035 1619 a(2)1075 1593 y Ft(;)1105 1565 y Fm(1)p 1105 1577 V 1105 1619 a(2)1145 1593 y Fu(])1182 1517 y Fv(\025)1239 1532 y Ft(n)1285 1517 y Fw(\()p Fv(k)s Fw(\))c Fs(\024)g Fv(\025)1605 1476 y Ft(D)1605 1541 y(n)1697 1517 y Fv(<)f(\025)1857 1476 y Ft(N)1857 1541 y(n)p Fu(+1)2022 1517 y Fs(\024)97 b Fw(min)2127 1593 y Ft(k)r Fl(2)p Fu([)p Fl(\000)2298 1565 y Fm(1)p 2297 1577 V 2297 1619 a(2)2338 1593 y Ft(;)2368 1565 y Fm(1)p 2367 1577 V 2367 1619 a(2)2408 1593 y Fu(])2444 1517 y Fv(\025)2501 1532 y Ft(n)p Fu(+1)2638 1517 y Fw(\()p Fv(k)s Fw(\))p Fv(;)120 1778 y Fw(whic)m(h)33 b(pro)m(v)m(es)h(the)f (Prop)s(osition.)p 3469 1778 4 66 v 3473 1715 59 4 v 3473 1778 V 3531 1778 4 66 v 266 1965 a(W)-8 b(e)33 b(remark)g(that)f (in)g(the)h(case)g(of)f(constan)m(t)i Fv(\033)d(<)d Fw(0)k(it)g(happ)s (ens)h(that)926 2157 y(max)867 2232 y Ft(k)r Fl(2)p Fu([)p Fl(\000)1038 2205 y Fm(1)p 1037 2217 31 3 v 1037 2258 a(2)1077 2232 y Ft(;)1107 2205 y Fm(1)p 1107 2217 V 1107 2258 a(2)1147 2232 y Fu(])1183 2157 y Fv(\025)1240 2172 y Ft(n)1287 2157 y Fw(\()p Fv(k)s Fw(\))28 b(=)f Fv(\025)1605 2115 y Ft(D)1605 2181 y(n)1697 2157 y Fw(=)g Fv(\025)1857 2115 y Ft(N)1857 2181 y(n)p Fu(+1)2022 2157 y Fw(=)96 b(min)2126 2232 y Ft(k)r Fl(2)p Fu([)p Fl(\000)2297 2205 y Fm(1)p 2296 2217 V 2296 2258 a(2)2336 2232 y Ft(;)2366 2205 y Fm(1)p 2366 2217 V 2366 2258 a(2)2406 2232 y Fu(])2442 2157 y Fv(\025)2499 2172 y Ft(n)p Fu(+1)2636 2157 y Fw(\()p Fv(k)s Fw(\))p Fv(:)120 2434 y Fw(Let)39 b(us)g(conclude)g(this)f (subsection)h(with)f(an)g(example)g(where)i(the)f(\014rst)g(gap)f(of)g Fv(H)3335 2398 y Fu(\()p Ft(\033)r Fu(\))3474 2434 y Fw(is)120 2555 y(op)s(en.)120 2734 y Fp(Example)e(4.2.5.)43 b Fw(Let)32 b Fv(a;)i(b)28 b Fs(2)g Fr(R)5 b Fw(,)39 b Fs(\000)p Fv(\031)32 b(<)27 b(c)h(<)f(\031)37 b Fw(and)1107 2978 y Fv(\033)t Fw(\()p Fv(x)1259 2993 y Fu(1)1299 2978 y Fw(\))28 b(:=)1495 2838 y Fo(\032)1612 2917 y Fv(a)195 b Fw(if)54 b Fv(x)2025 2932 y Fu(1)2093 2917 y Fs(2)28 b Fw([)p Fs(\000)p Fv(\031)t(;)17 b(c)p Fw(\))p Fv(;)1612 3038 y(b)205 b Fw(if)54 b Fv(x)2025 3053 y Fu(1)2093 3038 y Fs(2)28 b Fw([)p Fv(c;)17 b(\031)t Fw(\))p Fv(:)120 3223 y Fw(W)-8 b(e)33 b(claim)d(that)i(under)i(the)f(assumptions)1381 3415 y Fv(a)28 b(<)g Fs(\000)1694 3376 y Ft(\031)p 1651 3392 129 4 v 1651 3449 a(\031)r Fu(+)p Ft(c)1790 3415 y Fv(;)211 b(b)29 b Fs(\025)f Fw(0)p Fv(;)985 b Fw(\(4.2.5\))120 3607 y(one)33 b(has)1267 3727 y(max)1208 3803 y Ft(k)r Fl(2)p Fu([)p Fl(\000)1379 3776 y Fm(1)p 1378 3788 31 3 v 1378 3829 a(2)1418 3803 y Ft(;)1448 3776 y Fm(1)p 1448 3788 V 1448 3829 a(2)1488 3803 y Fu(])1525 3727 y Fv(\025)1582 3742 y Fu(1)1621 3727 y Fw(\()p Fv(k)s Fw(\))28 b Fv(<)96 b Fw(min)1882 3803 y Ft(k)r Fl(2)p Fu([)p Fl(\000)2053 3776 y Fm(1)p 2052 3788 V 2052 3829 a(2)2092 3803 y Ft(;)2122 3776 y Fm(1)p 2122 3788 V 2122 3829 a(2)2162 3803 y Fu(])2199 3727 y Fv(\025)2256 3742 y Fu(2)2295 3727 y Fw(\()p Fv(k)s Fw(\))p Fv(;)812 b Fw(\(4.2.6\))120 3964 y(with)32 b Fv(\025)399 3979 y Ft(n)446 3964 y Fw(\()p Fv(k)s Fw(\))h(de\014ned)h(b)m(y)g(\(4.2.2\).) 120 4084 y(Indeed,)39 b(applying)c(Prop)s(osition)f(3.3.15)i(with)g (\003)d(=)h(\()p Fs(\000)p Fv(\031)t(;)17 b(c)p Fw(\))36 b(and)h Fv(\016)g Fw(=)d Fv(k)2894 4048 y Fu(2)2958 4084 y Fw(+)25 b Fv(a)3110 4048 y Fu(2)3174 4084 y Fs(\000)g Fw(\()3367 4045 y Ft(\031)p 3324 4061 129 4 v 3324 4119 a(\031)r Fu(+)p Ft(c)3463 4084 y Fw(\))3501 4048 y Fu(2)120 4220 y Fw(\(whic)m(h)35 b(is)g(p)s(ositiv)m(e)f(b)m(y)j(\(4.2.5\))o(\)) e(w)m(e)h(\014nd)f(that)f Fv(\025)2004 4235 y Fu(1)2044 4220 y Fw(\()p Fv(k)s Fw(\))g(is)h(an)f(eigen)m(v)-5 b(alue)35 b(of)f Fv(H)3117 4184 y Fu(\()p Ft(\033)r Fu(\))3218 4220 y Fw(\()p Fv(k)s Fw(\))h(and)120 4340 y(satis\014es)1375 4461 y Fv(\025)1432 4476 y Fu(1)1472 4461 y Fw(\()p Fv(k)s Fw(\))27 b Fv(<)h Fs(\000)p Fv(a)1861 4419 y Fu(2)1923 4461 y Fw(+)22 b(\()2112 4421 y Ft(\031)p 2069 4438 V 2069 4495 a(\031)r Fu(+)p Ft(c)2207 4461 y Fw(\))2245 4419 y Fu(2)120 4634 y Fw(for)35 b(all)f Fv(k)j Fs(2)c Fw([)p Fs(\000)712 4595 y Fu(1)p 712 4611 36 4 v 712 4668 a(2)758 4634 y Fv(;)812 4595 y Fu(1)p 812 4611 V 812 4668 a(2)857 4634 y Fw(].)54 b(If)36 b(w)m(e)g(pro)s(ceed)h (similarly)32 b(as)37 b(in)e(the)h(pro)s(of)f(of)h(Prop)s(osition)e (3.3.15)120 4754 y(with)f(natural)f(b)s(oundary)h(conditions)g(at)f Fv(x)1766 4769 y Fu(1)1835 4754 y Fs(2)d(f\000)p Fv(\031)t(;)17 b(c;)g(\031)t Fs(g)32 b Fw(instead)i(of)e(Diric)m(hlet)f(b)s(ound-)120 4875 y(ary)24 b(conditions)f(\(i.e.,)i(considering)e(the)h(form)e Fv(g)1887 4839 y Fu(\()t(~)-39 b Ft(\033)s Fu(\))1988 4875 y Fw(\()p Fv(k)s Fw(\))24 b(on)f Fv(H)2357 4839 y Fu(1)2396 4875 y Fw(\(\()p Fs(\000)p Fv(\031)t(;)17 b(c)p Fw(\))t Fs(\002)t Fr(R)2884 4890 y Fu(+)2949 4875 y Fw(\))t Fs(\010)t Fv(H)3161 4839 y Fu(1)3200 4875 y Fw(\(\()p Fv(c;)g(\031)t Fw(\))t Fs(\002)120 4995 y Fr(R)186 5010 y Fu(+)251 4995 y Fw(\)\))32 b(and)h(use)h Fv(b)28 b Fs(\025)g Fw(0)k(w)m(e)i(\014nd)f(the)g(lo)m(w)m(er)g(b)s(ounds)g Fv(\025)2131 5010 y Fu(1)2170 4995 y Fw(\()p Fv(k)s Fw(\))28 b Fs(\025)g(\000)p Fv(a)2561 4959 y Fu(2)2634 4995 y Fw(and)1361 5187 y Fv(\025)1418 5202 y Fu(2)1457 5187 y Fw(\()p Fv(k)s Fw(\))g Fs(\025)g(\000)p Fv(a)1848 5146 y Fu(2)1910 5187 y Fw(+)22 b(\()2099 5148 y Ft(\031)p 2056 5164 129 4 v 2056 5221 a(\031)r Fu(+)p Ft(c)2195 5187 y Fw(\))2233 5146 y Fu(2)2272 5187 y Fv(:)120 5379 y Fw(\(Of)32 b(course,)i Fv(\025)679 5394 y Fu(2)718 5379 y Fw(\()p Fv(k)s Fw(\))f(need)g(not)g(b)s(e)f(an)h(eigen)m(v)-5 b(alue.\))43 b(This)33 b(pro)m(v)m(es)h(\(4.2.6\))o(.)1781 5712 y(58)p eop %%Page: 59 64 59 63 bop 120 1071 a Fk(Bibliograph)-6 b(y)120 1523 y Fw([A])49 b(R.)32 b(A.)h(Adams,)g Fj(Sob)-5 b(olev)33 b(Sp)-5 b(ac)g(es)p Fw(,)32 b(Academic)g(Press,)i(1975.)120 1727 y([B1])49 b(M.)33 b(Sh.)h(Birman,)e Fj(Perturb)-5 b(ation)36 b(of)f(the)h(sp)-5 b(e)g(ctrum)35 b(of)g(a)g(singular)g(el)5 b(liptic)35 b(op)-5 b(er)g(ator)296 1847 y(under)27 b(variation)e(of)i (the)f(b)-5 b(oundary)27 b(and)f(b)-5 b(oundary)26 b(c)-5 b(onditions)p Fw(.)22 b(So)m(viet.)i(Math.)g(Dokl.)296 1968 y Fp(2)33 b Fw(\(1961\),)e(326-328.)120 2171 y([B2])49 b(M.)37 b(Sh.)g(Birman,)f Fj(Perturb)-5 b(ations)38 b(of)h(the)f(c)-5 b(ontinuous)38 b(sp)-5 b(e)g(ctrum)39 b(of)f(a)g(singular)g(el-)296 2291 y(liptic)i(op)-5 b(er)g(ator)40 b(by)h(varying)e(the)i(b)-5 b(oundary)40 b(and)g(b)-5 b(oundary)40 b(c)-5 b(onditions)p Fw(.)37 b(\(Russian\))296 2412 y(V)-8 b(estnik)33 b(Leningrad.)f(Univ.) h Fp(1)p Fw(,)f(no.)h(1)f(\(1962\),)g(22-55.)120 2615 y([BKaS])49 b(M.)35 b(Sh.)f(Birman,)g(G.)g(E.)g(Karadzho)m(v,)i(M.)f (Z.)f(Solom)m(y)m(ak,)g Fj(Bounde)-5 b(dness)35 b(c)-5 b(ondi-)296 2736 y(tions)39 b(and)f(sp)-5 b(e)g(ctrum)39 b(estimates)f(for)h(the)g(op)-5 b(er)g(ators)39 b(b\(X\)a\(D\))f(and)g (their)h(analo)-5 b(gs)p Fw(,)296 2856 y(Estimates)44 b(and)g(asymptotics)g(for)g(discrete)g(sp)s(ectra)h(of)f(in)m(tegral)f (and)h(di\013eren)m(tial)296 2976 y(equations,)33 b(Adv.)g(So)m(viet.)g (Math.,)g(v)m(ol.)f(7,)h(Amer.)f(Math.)h(So)s(c.,)g(1991,)f(85-106.)120 3180 y([BSh)m(tSu])50 b(M.)39 b(Sh.)g(Birman,)f(R.G.)g(Sh)m(teren)m(b)s (erg,)j(T.)e(A.)g(Suslina,)g Fj(A)n(bsolute)h(c)-5 b(ontinuity)296 3300 y(of)50 b(the)f(sp)-5 b(e)g(ctrum)50 b(of)g(a)f(two-dimensional)f (Schr\177)-50 b(odinger)48 b(op)-5 b(er)g(ator)50 b(with)f(p)-5 b(otential)296 3421 y(supp)g(orte)g(d)37 b(on)g(a)g(p)-5 b(erio)g(dic)36 b(system)h(of)g(curves)p Fw(,)f(St.)f(P)m(etersburg)i (Math.)e(J.)g Fp(12)p Fw(,)h(no.)f(6)296 3541 y(\(2001\),)d(983-1012.) 120 3744 y([BKr])49 b(M.)38 b(Sh.)h(Birman,)f(M.)g(G.)g(Kre)-11 b(\025)-38 b(\020n,)39 b Fj(On)g(the)h(the)-5 b(ory)41 b(of)e(wave)g(op)-5 b(er)g(ators)40 b(and)f(sc)-5 b(at-)296 3865 y(tering)35 b(op)-5 b(er)g(ators)p Fw(,)32 b(So)m(viet.)h(Math.)g (Dokl.)e Fp(3)i Fw(\(1962\),)e(740-744.)120 4068 y([BS])49 b(M.)31 b(Sh.)g(Birman,)e(M.)i(Z.)f(Solom)m(y)m(ak,)h Fj(Sp)-5 b(e)g(ctr)g(al)32 b(the)-5 b(ory)33 b(of)g(self-adjoint)f(op) -5 b(er)g(ators)32 b(in)296 4188 y(Hilb)-5 b(ert)35 b(sp)-5 b(ac)g(e)p Fw(,)32 b(Reidel,)g(1987.)120 4392 y([BSu])49 b(M.)33 b(Sh.)g(Birman,)e(T.)i(A.)g(Suslina,)f Fj(A)n(bsolute)i(c)-5 b(ontinuity)36 b(of)e(the)h(two-dimensional)296 4512 y(p)-5 b(erio)g(dic)27 b(magnetic)g(Hamiltonian)g(with)g(disc)-5 b(ontinuous)28 b(ve)-5 b(ctor-value)g(d)27 b(p)-5 b(otential)p Fw(,)26 b(St.)296 4633 y(P)m(etersburg)34 b(Math.)f(J.)g Fp(10)p Fw(,)g(no.)f(4)h(\(1999\),)e(579-601.)120 4836 y([K])49 b(T.)33 b(Kato,)f Fj(Perturb)-5 b(ation)35 b(The)-5 b(ory)35 b(for)f(Line)-5 b(ar)35 b(Op)-5 b(er)g(ators)p Fw(,)32 b(Springer,)g(1976.)120 5039 y([LL])49 b(L.)32 b(D.)h(Landau,)f(E.)h(M.)g(Lifshitz,)f Fj(Quantum)j(Me)-5 b(chanics)p Fw(,)31 b(P)m(ergamon)i(Press,)h(1960.)120 5243 y([RS4])49 b(M.)g(Reed,)k(B.)c(Simon,)j Fj(Metho)-5 b(ds)49 b(of)h(Mo)-5 b(dern)49 b(Mathematic)-5 b(al)49 b(Physics)p Fw(,)k(v)m(ol.)48 b(4,)296 5363 y(Academic)32 b(Press,)i(1978.)1781 5712 y(59)p eop %%Page: 60 65 60 64 bop 120 407 a Fw([Su])49 b(T.)33 b(A.)g(Suslina,)f Fj(Perio)-5 b(dic)34 b(op)-5 b(er)g(ators)p Fw(,)32 b(Lecture)i(Notes,) f(Munic)m(h,)g(2002.)120 610 y([Y)-8 b(a1])49 b(D.)32 b(R.)g(Y)-8 b(afaev,)33 b Fj(Mathematic)-5 b(al)34 b(Sc)-5 b(attering)35 b(The)-5 b(ory)p Fw(,)32 b(Amer.)g(Math.)h(So)s(c.,)g (1992.)120 814 y([Y)-8 b(a2])49 b(D.)40 b(R.)g(Y)-8 b(afaev,)43 b Fj(Sc)-5 b(attering)42 b(The)-5 b(ory:)60 b(Some)41 b(Old)h(and)g(New)g(Pr)-5 b(oblems)p Fw(,)42 b(Lecture)296 934 y(Notes)33 b(in)f(Mathematics)g(1735,)g(Springer,)g(2000.)266 1262 y Fx(R)m(uper)-7 b(t)37 b(L.)h(Frank)266 1382 y Fj(Curr)-5 b(ent)41 b(addr)-5 b(ess:)61 b Fw(Mathematisc)m(hes)39 b(Institut,)i(LMU)e(M)s(\177)-51 b(unc)m(hen,)41 b(Theresienstra\031e) 120 1503 y(39,)32 b(80333)g(Munic)m(h,)h(German)m(y)266 1623 y Fj(E-mail)h(addr)-5 b(ess:)48 b Fa(rfrank@mathematik.uni-m)q (uenc)q(hen.)q(de)1781 5712 y Fw(60)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0306271102194--