Content-Type: multipart/mixed; boundary="-------------0601031856556" This is a multi-part message in MIME format. ---------------0601031856556 Content-Type: text/plain; name="06-2.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-2.keywords" Limit Periodic Potential ---------------0601031856556 Content-Type: application/postscript; name="KarpLee.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="KarpLee.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.94a Copyright 2003 Radical Eye Software %%Title: KarpLee.dvi %%CreationDate: Tue Jan 03 18:36:20 2006 %%Pages: 69 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: CMBX10 CMR9 CMR6 CMTI10 CMCSC10 CMMI9 CMSY9 CMMI6 CMSY6 %%+ CMR10 CMMI10 CMSY10 CMMI8 CMSY8 CMEX10 CMR8 EUSM10 MSBM10 MSBM7 %%+ CMBX8 CMMI5 CMSY5 CMR5 CMR7 CMMI7 CMTI9 CMBX9 CMTT9 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips.exe -P pdf -t letter KarpLee %DVIPSParameters: dpi=8000 %DVIPSSource: TeX output 2006.01.03:1834 %%BeginProcSet: tex.pro 0 0 %! /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/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]{Ci}imagemask restore}B/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: alt-rule.pro 0 0 %! % Patch by TVZ % Makes dvips files draw rules with stroke rather than fill. % Makes narrow rules more predictable at low resolutions % after distilling to PDF. % May have unknown consequences for very thick rules. % Tested only with dvips 5.85(k). 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450 lineto 450 -450 lineto -450 -450 lineto closepath } def /SolidSquare { SquarePath fill } def /Square { SquarePath .89 .89 scale SquarePath eofill } def /BoldSquare { SquarePath .78 .78 scale SquarePath eofill } def /PentagonPath { -337.8 -465 moveto 337.8 -465 lineto 546.6 177.6 lineto 0 574.7 lineto -546.6 177.6 lineto closepath } def /SolidPentagon { PentagonPath fill } def /Pentagon { PentagonPath .89 .89 scale PentagonPath eofill } def /BoldPentagon { PentagonPath .78 .78 scale PentagonPath eofill } def % DG/SR modification begin - Dec. 16, 1999 - From Etienne Riga /DiamondPath { 0 742.5 moveto -428.5 0 lineto 0 -742.5 lineto 428.5 0 lineto closepath } def /SolidDiamond { DiamondPath fill } def /Diamond { DiamondPath .85 .85 scale DiamondPath eofill } def /BoldDiamond { DiamondPath .7 .7 scale DiamondPath eofill } def % DG/SR modification end /.notdef { } def end /BuildGlyph { exch begin Metrics 1 index get exec 0 BBoxes 3 index get exec setcachedevice CharProcs begin load exec 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Grant, and David Carlisle. %% All rights reserved. %% %% This file is part of the PSfrag package. %% userdict begin /PSfragLib 90 dict def /PSfragDict 6 dict def /PSfrag { PSfragLib begin load exec end } bind def end PSfragLib begin /RO /readonly load def /CP /currentpoint load def /CM /currentmatrix load def /B { bind RO def } bind def /X { exch def } B /MD { { X } forall } B /OE { end exec PSfragLib begin } B /S false def /tstr 8 string def /islev2 { languagelevel } stopped { false } { 2 ge } ifelse def [ /sM /tM /srcM /dstM /dM /idM /srcFM /dstFM ] { matrix def } forall sM currentmatrix RO pop dM defaultmatrix RO idM invertmatrix RO pop srcFM identmatrix pop /Hide { gsave { CP } stopped not newpath clip { moveto } if } B /Unhide { { CP } stopped not grestore { moveto } if } B /setrepl islev2 {{ /glob currentglobal def true setglobal array astore globaldict exch /PSfrags exch put glob setglobal }} {{ array astore /PSfrags X }} ifelse B /getrepl islev2 {{ globaldict /PSfrags get aload length }} {{ PSfrags aload length }} ifelse B /convert { /src X src length string /c 0 def src length { dup c src c get dup 32 lt { pop 32 } if put /c c 1 add def } repeat } B /Begin { /saver save def srcFM exch 3 exch put 0 ne /debugMode X 0 setrepl dup /S exch dict def { S 3 1 roll exch convert exch put } repeat srcM CM dup invertmatrix pop mark { currentdict { end } stopped { pop exit } if } loop PSfragDict counttomark { begin } repeat pop } B /End { mark { currentdict end dup PSfragDict eq { pop exit } if } loop counttomark { begin } repeat pop getrepl saver restore 7 idiv dup /S exch dict def { 6 array astore /mtrx X tstr cvs /K X S K [ S K known { S K get aload pop } if mtrx ] put } repeat } B /Place { tstr cvs /K X S K known { bind /proc X tM CM pop CP /cY X /cX X 0 0 transform idtransform neg /aY X neg /aX X S K get dup length /maxiter X /iter 1 def { iter maxiter ne { /saver save def } if tM setmatrix aX aY translate [ exch aload pop idtransform ] concat cX neg cY neg translate cX cY moveto /proc load OE iter maxiter ne { saver restore /iter iter 1 add def } if } forall /noXY { CP /cY X /cX X } stopped def tM setmatrix noXY { newpath } { cX cY moveto } ifelse } { Hide OE Unhide } ifelse } B /normalize { 2 index dup mul 2 index dup mul add sqrt div dup 4 -1 roll exch mul 3 1 roll mul } B /replace { aload pop MD CP /bY X /lX X gsave sM setmatrix str stringwidth abs exch abs add dup 0 eq { pop } { 360 exch div dup scale } ifelse lX neg bY neg translate newpath lX bY moveto str { /ch X ( ) dup 0 ch put false charpath ch Kproc } forall flattenpath pathbbox [ /uY /uX /lY /lX ] MD CP grestore moveto currentfont /FontMatrix get dstFM copy dup 0 get 0 lt { uX lX /uX X /lX X } if 3 get 0 lt { uY lY /uY X /lY X } if /cX uX lX add 0.5 mul def /cY uY lY add 0.5 mul def debugMode { gsave 0 setgray 1 setlinewidth lX lY moveto lX uY lineto uX uY lineto uX lY lineto closepath lX bY moveto uX bY lineto lX cY moveto uX cY lineto cX lY moveto cX uY lineto stroke grestore } if dstFM dup invertmatrix dstM CM srcM 2 { dstM concatmatrix } repeat pop getrepl /temp X S str convert get { aload pop [ /rot /scl /loc /K ] MD /aX cX def /aY cY def loc { dup 66 eq { /aY bY def } { % B dup 98 eq { /aY lY def } { % b dup 108 eq { /aX lX def } { % l dup 114 eq { /aX uX def } { % r dup 116 eq { /aY uY def } % t if } ifelse } ifelse } ifelse } ifelse pop } forall K srcFM rot tM rotate dstM 2 { tM concatmatrix } repeat aload pop pop pop 2 { scl normalize 4 2 roll } repeat aX aY transform /temp temp 7 add def } forall temp setrepl } B /Rif { S 3 index convert known { pop replace } { exch pop OE } ifelse } B /XA { bind [ /Kproc /str } B /XC { ] 2 array astore def } B /xs { pop } XA XC /xks { /kern load OE } XA /kern XC /xas { pop ax ay rmoveto } XA /ay /ax XC /xws { c eq { cx cy rmoveto } if } XA /c /cy /cx XC /xaws { ax ay rmoveto c eq { cx cy rmoveto } if } XA /ay /ax /c /cy /cx XC /raws { xaws { awidthshow } Rif } B /rws { xws { widthshow } Rif } B /rks { xks { kshow } Rif } B /ras { xas { ashow } Rif } B /rs { xs { show } Rif } B /rrs { getrepl dup 2 add -1 roll //restore exec setrepl } B PSfragDict begin islev2 not { /restore { /rrs PSfrag } B } if /show { /rs PSfrag } B /kshow { /rks PSfrag } B /ashow { /ras PSfrag } B /widthshow { /rws PSfrag } B /awidthshow { /raws PSfrag } B end PSfragDict RO pop end %%EndProcSet %%BeginProcSet: texps.pro 0 0 %! 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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cleartomark %%EndFont %%BeginFont: CMBX9 %!PS-AdobeFont-1.1: CMBX9 1.0 %%CreationDate: 1991 Aug 20 16:36:25 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put readonly def /FontBBox{-58 -250 1195 750}readonly def /UniqueID 5000767 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTI9 %!PS-AdobeFont-1.1: CMTI9 1.0 %%CreationDate: 1991 Aug 18 21:08:07 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 14 /ffi put dup 45 /hyphen put dup 46 /period put dup 58 /colon put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 127 /dieresis put readonly def /FontBBox{-35 -250 1148 750}readonly def /UniqueID 5000827 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: EUSM10 %!PS-AdobeFont-1.1: EUSM10 2.1 %%CreationDate: 1992 Nov 20 17:36:44 % Euler fonts were designed by Hermann Zapf. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUSM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUSM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 66 /B put dup 68 /D put dup 71 /G put dup 75 /K put dup 76 /L put dup 79 /O put readonly def /FontBBox{-8 -194 963 738}readonly def /UniqueID 5031988 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF4E9D2405B169DE483FC3C5DBA587E58D683 9E11948C1A3E8B5360ADE57410E9F910E561F679EBFBEB0730125606CD072EB1 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cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 25 /pi put dup 78 /N put dup 97 /a put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueID 5087382 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 50 /two put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueID 5000790 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 78 /N put dup 82 /R put dup 90 /Z put dup 123 /kappa put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueID 5000788 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueID 5000815 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 110 /n put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueID 5087380 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX8 %!PS-AdobeFont-1.1: CMBX8 1.0 %%CreationDate: 1991 Aug 20 16:36:07 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /one put dup 50 /two put dup 51 /three put dup 82 /R put readonly def /FontBBox{-59 -250 1235 750}readonly def /UniqueID 5000766 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 82 /R put dup 90 /Z put dup 123 /kappa put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 45 /hyphen put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 61 /equal put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 89 /Y put dup 91 /bracketleft put dup 93 /bracketright put dup 94 /circumflex put dup 100 /d put dup 109 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 28 /angbracketleftbigg put dup 29 /angbracketrightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 40 /braceleftBigg put dup 46 /slashBig put dup 68 /angbracketleftBig put dup 69 /angbracketrightBig put dup 72 /contintegraltext put dup 73 /contintegraldisplay put dup 80 /summationtext put dup 81 /producttext put dup 82 /integraltext put dup 84 /intersectiontext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 92 /intersectiondisplay put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 112 /radicalbig put dup 113 /radicalBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 6 /plusminus put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 110 /backslash put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 2 /Theta put dup 11 /alpha put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 17 /eta put dup 21 /lambda put dup 23 /nu put dup 25 /pi put dup 28 /tau put dup 39 /phi1 put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 67 /C put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 120 /x put dup 121 /y put dup 122 /z put dup 126 /vector put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 6 /plusminus put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 25 /approxequal put dup 26 /propersubset put dup 27 /propersuperset put dup 28 /lessmuch put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 57 /existential put dup 60 /Rfractur put dup 61 /Ifractur put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put dup 114 /nabla put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 2 /Theta put dup 8 /Phi put dup 11 /alpha put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 17 /eta put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 25 /pi put dup 28 /tau put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 126 /vector put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMCSC10 %!PS-AdobeFont-1.1: CMCSC10 1.0 %%CreationDate: 1991 Aug 18 17:46:49 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 76 /L put dup 77 /M put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{14 -250 1077 750}readonly def /UniqueID 5000772 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 19 /acute put dup 33 /exclam put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 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cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 6 /plusminus put dup 49 /infinity put dup 104 /angbracketleft put dup 105 /angbracketright put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 17 /eta put dup 59 /comma put dup 61 /slash put dup 76 /L put dup 77 /M put dup 81 /Q put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 113 /q put dup 114 /r put dup 115 /s put dup 120 /x put dup 126 /vector put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY9 %!PS-AdobeFont-1.1: CMSY9 1.0 %%CreationDate: 1991 Aug 15 07:22:27 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 20 /lessequal put dup 21 /greaterequal put dup 25 /approxequal put dup 50 /element put dup 106 /bar put dup 107 /bardbl put readonly def /FontBBox{-30 -958 1146 777}readonly def /UniqueID 5000819 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI9 %!PS-AdobeFont-1.1: CMMI9 1.100 %%CreationDate: 1996 Jul 23 07:53:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 21 /lambda put dup 31 /chi put dup 39 /phi1 put dup 59 /comma put dup 60 /less put dup 62 /greater put dup 68 /D put dup 72 /H put dup 75 /K put dup 81 /Q put dup 83 /S put dup 86 /V put dup 87 /W put dup 97 /a put dup 98 /b put dup 101 /e put dup 107 /k put dup 108 /l put dup 110 /n put dup 112 /p put dup 115 /s put dup 116 /t put dup 117 /u put dup 120 /x put dup 121 /y put dup 126 /vector put readonly def /FontBBox{-29 -250 1075 750}readonly def /UniqueID 5087384 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 34 /quotedblright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 67 /C put dup 68 /D put dup 70 /F put dup 71 /G put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 92 /quotedblleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 61 /equal put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /Delta put dup 9 /Psi put dup 12 /fi put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueID 5000792 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 87 /W put dup 89 /Y put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F00F963068B8B731A88D7740B0DDAED1B3F82 7DB9DFB4372D3935C286E39EE7AC9FB6A9B5CE4D2FAE1BC0E55AE02BFC464378 77B9F65C23E3BAB41EFAE344DDC9AB1B3CCBC0618290D83DC756F9D5BEFECB18 2DB0E39997F264D408BD076F65A50E7E94C9C88D849AB2E92005CFA316ACCD91 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68.5 114.9 arc gs col0 s gr % Arc n 154430.4 4229.5 3553.5 162.0 -158.5 arc gs col0 s gr % Arc n 153938.6 3799.8 2980.0 -156.8 -116.2 arc gs col0 s gr % Arc n 154462.5 4162.5 3606.6 117.9 154.8 arc gs col0 s gr % Polyline gs clippath 159015 4230 m 159015 4170 l 158830 4170 l 158980 4200 l 158830 4230 l cp eoclip n 150000 4200 m 159000 4200 l gs col0 s gr gr % arrowhead n 158830 4230 m 158980 4200 l 158830 4170 l 158860 4200 l 158830 4230 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 154530 60 m 154470 60 l 154470 245 l 154500 95 l 154530 245 l cp eoclip n 154500 8700 m 154500 75 l gs col0 s gr gr % arrowhead n 154530 245 m 154500 95 l 154470 245 l 154500 215 l 154530 245 l cp gs 0.00 setgray ef gr col0 s % Polyline n 151437 5608 m 151437 5609 l 151437 5614 l 151438 5626 l 151439 5646 l 151440 5672 l 151443 5701 l 151446 5730 l 151449 5757 l 151453 5782 l 151458 5803 l 151463 5820 l 151470 5835 l 151478 5847 l 151487 5858 l 151498 5867 l 151509 5875 l 151523 5883 l 151538 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2407 l 157412 2411 l 157400 2418 l 157389 2427 l 157378 2436 l 157368 2446 l 157357 2454 l 157343 2461 l 157328 2464 l 157310 2462 l 157292 2455 l 157277 2442 l 157266 2427 l 157258 2409 l 157253 2390 l 157250 2371 l 157249 2351 l 157248 2332 l 157248 2316 l 157248 2303 l 157249 2295 l 157249 2292 l 157249 2291 l gs col0 s gr $F2psEnd rs %%EndDocument @endspecial 0 TeXcolorgray 30313 23337 a Fv(Figure)474 b(2.)606 b Fw(Distorted)419 b(circle)d(with)p 0 TeXcolorgray 30313 24776 a(holes,)404 b Fo(D)34694 24958 y Fp(2)35220 24776 y Fw(\()p Fu(\025)p Fw(\))p 0 TeXcolorgray 475 28251 a(where)505 b Fo(B)5004 28433 y Fs(n)5629 28251 y Fw(\()p Fu(\025)p Fw(\))h(is)f(a)g(subset)h(of)f(the)h(unit)f(circle) f Fu(S)24375 28433 y Fp(1)24901 28251 y Fw(.)841 b(The)505 b(set)g Fo(B)31587 28433 y Fs(n)32213 28251 y Fw(\()p Fu(\025)p Fw(\))h(can)f(b)34 b(e)505 b(in)-34 b(terpreted)505 b(as)g(the)h(set)f(of)475 29690 y(p)34 b(ossible)497 b(directions)g(of)h(propagation)g(for)f(almost)h(plane)g(w)-34 b(a)g(v)g(es)498 b(\(1.11\).)818 b(It)498 b(has)f(an)h(asymptotically)f (full)475 31129 y(measure)404 b(on)g Fu(S)7625 31311 y Fp(1)8555 31129 y Fw(as)g Fu(\025)337 b Ft(!)g(1)p Fw(:)17060 32982 y Fu(L)17885 32000 y Fq(\000)18441 32982 y Fo(B)19364 33164 y Fs(n)19989 32982 y Fw(\()p Fu(\025)p Fw(\))21638 32000 y Fq(\001)23275 32982 y Fw(=)22531 33799 y Fs(\025)p Fr(!1)25299 32982 y Fw(2)p Fu(\031)313 b Fw(+)269 b Fu(O)29281 31637 y Fq(\020)30005 32982 y Fu(\025)30712 32482 y Fr(\000)p Fs(\015)31929 32605 y FD(3)32390 32482 y Fs(=)p Fp(2)p Fs(l)33677 31637 y Fq(\021)34603 32982 y Fu(:)13488 b Fw(\(1.14\))475 35359 y(The)412 b(set)f Fo(B)5796 35541 y Fs(n)6422 35359 y Fw(\()p Fu(\025)p Fw(\))h(has)g(only)f(a)h(\014nite)g(n)-34 b(um)g(b)34 b(er)412 b(of)g(holes,)h(ho)-34 b(w)g(ev)g(er)412 b(their)g(n)-34 b(um)g(b)34 b(er)412 b(is)f(gro)-34 b(wing)412 b(with)h Fu(n)p Fw(.)560 b(More)475 36877 y(and)431 b(more)e(holes)h(of)h(a)f (smaller)f(and)i(smaller)e(size)g(are)h(added)h(at)f(eac)-34 b(h)430 b(step.)617 b(The)431 b(v)-67 b(alue)429 b Fn({)44053 37059 y Fs(n)44680 36877 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))287 b Ft(\000)f Fu(\025)49767 36437 y Fp(1)p Fs(=)p Fp(2)p Fs(l)475 38394 y Fw(giv)-34 b(es)466 b(the)i(deviation)f(of)g Fo(D)13457 38576 y Fs(n)14083 38394 y Fw(\()p Fu(\025)p Fw(\))h(from)f(the)g(p)34 b(erfect)467 b(circle)e(of)i(the)g(radius)h Fu(\025)36626 37955 y Fp(1)p Fs(=)p Fp(2)p Fs(l)38851 38394 y Fw(in)f(the)g(direction)442 b Fu(~)-581 b(\027)77 b Fw(.)726 b(It)467 b(is)475 39834 y(pro)-34 b(v)g(en)405 b(that)g(the)f(deviation)h(is)e(asymptotically)h (small:)8324 42580 y Fn({)9132 42762 y Fs(n)9759 42580 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))337 b(=)f Fu(\025)14946 42079 y Fp(1)p Fs(=)p Fp(2)p Fs(l)16974 42580 y Fw(+)269 b Fu(O)19346 41598 y Fq(\000)19902 42580 y Fu(\025)20609 42079 y Fr(\000)p Fs(\015)21826 42202 y FD(4)22342 41598 y Fq(\001)23099 42580 y Fu(;)25387 41760 y(@)67 b Fn({)26906 41942 y Fs(n)27532 41760 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))p 25387 42301 5010 49 v 27140 43411 a Fu(@)67 b(')30866 42580 y Fw(=)337 b Fu(O)33306 41598 y Fq(\000)33862 42580 y Fu(\025)34569 42079 y Fr(\000)p Fs(\015)35786 42202 y FD(5)36302 41598 y Fq(\001)38271 42580 y Fu(\015)38899 42762 y Fp(4)39425 42580 y Fu(;)202 b(\015)40592 42762 y Fp(5)41454 42580 y Fu(>)336 b Fw(0)p Fu(;)4752 b Fw(\(1.15\))475 45337 y Fu(')404 b Fw(b)34 b(eing)404 b(an)g(angle)g(v)-67 b(ariable,)378 b Fu(~)-581 b(\027)413 b Fw(=)337 b(\(cos)202 b Fu(';)g Fw(sin)g Fu(')p Fw(\).)1803 46776 y(On)375 b(eac)-34 b(h)375 b(step)g(more)f(and)h(more)f(p)34 b(oin)-34 b(ts)376 b(are)e(excluded)g(from)h(the)g(non-resonan)-34 b(t)377 b(sets)d Fo(G)43145 46958 y Fs(n)43771 46776 y Fw(,)381 b(th)-34 b(us)376 b Ft(f)p Fo(G)48465 46958 y Fs(n)49091 46776 y Ft(g)49697 46336 y Fr(1)49697 47101 y Fs(n)p Fp(=1)475 48215 y Fw(is)403 b(a)h(decreasing)f(sequence)g(of)h (sets.)538 b(The)404 b(set)g Fo(G)22931 48397 y Fr(1)24330 48215 y 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b(no)f(suitable)g(form)-34 b(ulae)479 b(are)475 26545 y(kno)-34 b(wn.)527 b(Absence)369 b(of)f(form)-34 b(ulae)368 b(causes)g(di\016culties)g(in)g(estimating)h(the)f(size)f (of)i Fo(B)39082 26727 y Fs(n)p Fr(\000)p Fp(1)40910 26545 y Fw(\()p Fu(\025)p Fw(\))197 b Ft(n)g Fo(B)44482 26727 y Fs(n)45109 26545 y Fw(\()p Fu(\025)p Fw(\).)527 b(T)-101 b(o)368 b(deal)475 27984 y(with)485 b(this)f(problem)g(w)-34 b(e)485 b(start)f(with)h(in)-34 b(tro)34 b(ducing)485 b(an)f(angle)g(v)-67 b(ariable)483 b Fu(')470 b Ft(2)g Fw([0)p Fu(;)202 b Fw(2)p Fu(\031)43 b Fw(\),)479 b Fu(~)-582 b(\027)547 b Fw(=)470 b(\(cos)202 b Fu(';)g Fw(sin)g Fu(')p Fw(\))470 b Ft(2)475 29423 y Fu(S)1218 29605 y Fp(1)2344 29423 y Fw(and)602 b(consider)e(sets)h Fo(B)13308 29605 y Fs(n)13934 29423 y Fw(\()p Fu(\025)p Fw(\))g(in)g(terms)f(of)h (this)h(v)-67 b(ariable.)1126 b(Next,)650 b(w)-34 b(e)601 b(sho)-34 b(w)601 b(that)h(the)f(resonan)-34 b(t)602 b(set)475 30867 y Fo(B)1398 31049 y Fs(n)p Fr(\000)p Fp(1)3226 30867 y Fw(\()p Fu(\025)p Fw(\))195 b Ft(n)g Fo(B)6794 31049 y Fs(n)7421 30867 y Fw(\()p Fu(\025)p Fw(\))368 b(can)f(b)34 b(e)366 b(describ)34 b(ed)367 b(as)g(the)g(set)h(of)f(zeros)f(of)i(determinan)-34 b(ts)368 b(of)f(the)h(t)-34 b(yp)34 b(e)367 b Fu(D)34 b(et)45012 29885 y Fq(\000)45567 30867 y Fu(I)290 b Fw(+)195 b Fu(S)48271 31049 y Fs(n)48897 30867 y Fw(\()p Fu(')p Fw(\))50632 29885 y Fq(\001)51189 30867 y Fw(,)475 32311 y Fu(S)1218 32493 y Fs(n)1844 32311 y Fw(\()p Fu(')p Fw(\))405 b(b)34 b(eing)404 b(a)g(trace)g(t)-34 b(yp)34 b(e)404 b(op)34 b(erator,)7079 34999 y Fu(I)365 b Fw(+)269 b Fu(S)9932 35181 y Fs(n)10558 34999 y Fw(\()p Fu(')p Fw(\))338 b(=)13910 33654 y Fq(\020)14634 34999 y Fu(H)15740 34499 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))18300 34017 y Fq(\000)18888 34999 y Fu(~)-638 b Fn({)19664 35181 y Fs(n)p Fr(\000)p Fp(1)21493 34999 y Fw(\()p Fu(')p Fw(\))270 b(+)24574 34679 y Fu(~)24710 34999 y(b)25230 34017 y Fq(\001)26055 34999 y Ft(\000)f Fu(\025)g Ft(\000)h Fu(\017)29948 33654 y Fq(\021)o(\020)31395 34999 y Fu(H)32403 35181 y Fp(0)32929 34017 y Fq(\000)33517 34999 y Fu(~)-639 b Fn({)34292 35181 y Fs(n)p Fr(\000)p Fp(1)36121 34999 y Fw(\()p Fu(')p Fw(\))270 b(+)39202 34679 y Fu(~)39338 34999 y(b)39858 34017 y Fq(\001)40683 34999 y Fw(+)f Fu(\025)42602 33654 y Fq(\021)43326 33956 y Fr(\000)p Fp(1)44584 34999 y Fu(;)475 37468 y Fw(where)623 b Fu(~)-638 b Fn({)4975 37650 y Fs(n)p Fr(\000)p Fp(1)6803 37468 y Fw(\()p Fu(')p Fw(\))592 b(is)f(a)g(v)-34 b(ector-function)592 b(describing)f Fo(D)27235 37650 y Fs(n)p Fr(\000)p Fp(1)29063 37468 y Fw(\()p Fu(\025)p Fw(\):)945 b Fu(~)-638 b Fn({)32770 37650 y Fs(n)p Fr(\000)p Fp(1)34599 37468 y Fw(\()p Fu(')p Fw(\))649 b(=)f Fn({)39382 37650 y Fs(n)p Fr(\000)p Fp(1)41210 37468 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))-25 b Fu(~)-581 b(\027)78 b Fw(.)1099 b(T)-101 b(o)591 b(obtain)475 39067 y Fo(B)1398 39249 y Fs(n)p Fr(\000)p Fp(1)3226 39067 y Fw(\()p Fu(\025)p Fw(\))320 b Ft(n)f Fo(B)7043 39249 y Fs(n)7669 39067 y Fw(\()p Fu(\025)p Fw(\))480 b(w)-34 b(e)479 b(tak)-34 b(e)479 b(all)g(v)-67 b(alues)478 b(of)i Fu(\017)f Fw(in)g(a)g(small)f(in)-34 b(terv)-67 b(al)479 b(and)g(v)-34 b(ectors)38992 38747 y Fu(~)39128 39067 y(b)479 b Fw(in)g(a)g(\014nite)h(set,)48060 38747 y Fu(~)48196 39067 y(b)462 b Ft(6)p Fw(=)f(0.)475 40506 y(F)-101 b(urther,)347 b(w)-34 b(e)334 b(extend)f(our)g(considerations) h(to)f(a)g(complex)g(neigh)-34 b(b)34 b(orho)g(o)g(d)333 b Fu(\010)36024 40688 y Fp(0)36883 40506 y Fw(of)h([0)p Fu(;)202 b Fw(2)p Fu(\031)43 b Fw(\).)515 b(W)-101 b(e)332 b(sho)-34 b(w)335 b(that)f(the)475 41945 y(determinan)-34 b(ts)346 b(are)e(analytic)g(functions)j(of)e Fu(')f Fw(in)h Fu(\010)24037 42127 y Fp(0)24563 41945 y Fw(,)356 b(and,)h(b)-34 b(y)345 b(this,)357 b(reduce)344 b(the)i(problem)e(of)h(estimating)h (the)475 43384 y(size)324 b(of)i(the)g(resonance)g(set)f(to)h(a)g (problem)f(in)h(complex)f(analysis.)512 b(W)-101 b(e)325 b(use)g(theorems)h(for)g(analytic)f(functions)475 44823 y(to)515 b(coun)-34 b(t)515 b(zeros)f(of)h(the)g(determinan)-34 b(ts)516 b(and)f(to)g(in)-34 b(v)g(estigate)515 b(ho)-34 b(w)516 b(far)e(zeros)g(mo)-34 b(v)g(e)515 b(when)g Fu(")g Fw(c)-34 b(hanges.)870 b(It)475 46262 y(enables)343 b(us)g(to)h (estimate)f(the)g(size)f(of)i(the)f(zero)f(set)h(of)h(the)f(determinan) -34 b(ts,)356 b(and,)g(hence,)f(the)343 b(size)g(of)g(the)g(non-)475 47701 y(resonance)387 b(set)g Fu(\010)8673 47883 y Fs(n)9636 47701 y Ft(\032)336 b Fu(\010)11723 47883 y Fp(0)12249 47701 y Fw(,)390 b(whic)-34 b(h)388 b(is)e(de\014ned)i(as)f(a)g (non-zero)g(set)g(for)g(the)h(determinan)-34 b(ts.)534 b(Pro)-34 b(ving)387 b(that)h(the)475 49140 y(non-resonance)350 b(set)f Fu(\010)10954 49322 y Fs(n)11930 49140 y Fw(is)g(su\016cien)-34 b(tly)350 b(large,)359 b(w)-34 b(e)350 b(obtain)g(estimates)g(\(1.12\)) g(for)f Fo(G)39071 49322 y Fs(n)40046 49140 y Fw(and)i(\(1.14\))f(for)f Fo(B)48519 49322 y Fs(n)49145 49140 y Fw(,)360 b(the)475 50580 y(set)366 b Fo(B)3252 50762 y Fs(n)4244 50580 y Fw(b)34 b(eing)366 b(the)g(real)g(part)g(of)h Fu(\010)16586 50762 y Fs(n)17212 50580 y Fw(.)526 b(T)-101 b(o)366 b(obtain)h Fu(\010)24362 50762 y Fs(n)25355 50580 y Fw(w)-34 b(e)366 b(delete)g(from)g Fu(\010)34200 50762 y Fp(0)35092 50580 y Fw(more)g(and)h(more)f(holes)g(of)g(smaller)475 52019 y(and)491 b(smaller)e(radii)h(at)h(eac)-34 b(h)490 b(step.)798 b(Th)-34 b(us,)513 b(the)491 b(non-resonance)g(set)f Fu(\010)34234 52201 y Fs(n)35341 52019 y Ft(\032)480 b Fu(\010)37572 52201 y Fp(0)38588 52019 y Fw(has)491 b(a)f(structure)h(of)g(Swiss)475 53458 y(Cheese)472 b(\(Fig.)744 b(7,)489 b(8,)g(pages)473 b(26,)489 b(45\).)743 b(Deleting)472 b(resonance)g(set)h(from)g Fu(\010)35119 53640 y Fp(0)36117 53458 y Fw(at)f(eac)-34 b(h)473 b(step)g(of)g(the)f(recurren)-34 b(t)475 54897 y(pro)34 b(cedure)440 b(w)-34 b(e)442 b(call)e(a)h (\\Swiss)g(Cheese)g(Metho)34 b(d".)650 b(The)441 b(essen)-34 b(tial)441 b(di\013erence)f(of)i(our)f(metho)34 b(d)441 b(from)h(those)475 56336 y(applied)457 b(in)g(similar)g(situations)h(b) 34 b(efore)457 b(\(see)g(e.g.)697 b([17]{[20)o(]\))457 b(is)g(that)h(w)-34 b(e)458 b(construct)g(a)f(non-resonance)h(set)475 57935 y(not)410 b(only)f(in)g(the)g(whole)h(space)f(of)g(a)g(parameter) g(\()24107 57615 y Fu(~)24169 57935 y(k)384 b Ft(2)345 b Fn(R)27212 57495 y Fp(2)28147 57935 y Fw(here\),)410 b(but)g(also)f(on)h(iso)34 b(energetic)408 b(curv)-34 b(es)409 b Fo(D)49250 58117 y Fs(n)49875 57935 y Fw(\()p Fu(\025)p Fw(\))475 59374 y(in)337 b(the)g(space)g(of)g(the)h (parameter,)349 b(when)338 b Fu(\025)f Fw(is)g(su\016cien)-34 b(tly)337 b(large.)515 b(Estimates)338 b(for)f(the)g(size)f(of)h (non-resonance)475 60813 y(sets)430 b(on)g(a)g(curv)-34 b(e)429 b(require)f(more)i(subtle)g(tec)-34 b(hnical)430 b(considerations)g(than)h(those)g(su\016cien)-34 b(t)430 b(for)g(description)475 62252 y(of)404 b(a)g(non-resonan)-34 b(t)406 b(set)f(in)f(the)g(whole)h(space)f(of)g(the)h(parameter.)1803 63692 y(The)536 b(restriction)f Fu(l)579 b Ft(\025)556 b Fw(6)535 b(is)g(tec)-34 b(hnical,)568 b(it)536 b(is)f(needed)h(only)f (for)h(the)g(\014rst)g(t)-34 b(w)g(o)536 b(steps)h(of)e(the)h(recurren) -34 b(t)475 65131 y(pro)34 b(cedure.)853 b(The)510 b(requiremen)-34 b(t)509 b(for)g(sup)34 b(er)509 b(exp)34 b(onen)-34 b(tial)510 b(deca)-34 b(y)509 b(of)g Ft(k)p Fu(V)34549 65313 y Fs(r)35055 65131 y Ft(k)h Fw(as)f Fu(r)545 b Ft(!)512 b(1)e Fw(is)f(more)f(essen) -34 b(tial,)475 66570 y(since)460 b(it)h(is)f(needed)i(to)f(ensure)g (con)-34 b(v)g(ergence)460 b(of)h(the)g(recurren)-34 b(t)461 b(pro)34 b(cedure.)708 b(It)461 b(is)f(not)i(essen)-34 b(tial)461 b(that)h(the)475 68102 y(p)34 b(oten)-34 b(tial)436 b Fu(V)6331 68284 y Fs(r)7273 68102 y Fw(has)h(doubling)f(p)34 b(erio)g(ds,)443 b(the)437 b(p)34 b(erio)g(ds)435 b(of)h(the)g(t)-34 b(yp)34 b(e)436 b Fu(q)32388 67662 y Fs(r)26 b Fr(\000)p Fp(1)33961 67782 y Fu(~)34097 68102 y(b)34617 68284 y Fp(1)35143 68102 y Fu(;)638 b(q)36702 67662 y Fs(r)26 b Fr(\000)p Fp(1)38275 67782 y Fu(~)38411 68102 y(b)38931 68284 y Fp(2)39457 68102 y Fu(;)202 b(q)433 b Ft(2)389 b Fn(N)p Fu(;)436 b Fw(can)g(b)34 b(e)436 b(treated)475 69541 y(in)404 b(the)g(same)g(w)-34 b(a)g(y)-101 b(.)540 b(First)404 b(consideration)h(of)f(the)h(problem)f(w)-34 b(as)405 b(giv)-34 b(en)404 b(in)g([23].)p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 6 6 TeXDict begin 6 5 bop 0 TeXcolorgray 0 TeXcolorgray 475 3345 a Fp(6)14718 b(YULIA)313 b(KARPESHINA)f(AND)h(YOUNG-RAN)g(LEE)p 0 TeXcolorgray 1803 6001 a Fw(The)345 b(plan)g(of)g(the)g(pap)34 b(er)344 b(is)g(the)h(follo)-34 b(wing.)519 b(Sections)345 b(2)g({)f(6)g(describ)34 b(e)344 b(steps)h(of)g(the)g(recurren)-34 b(t)344 b(pro)34 b(cedure.)475 7440 y(Discussion)473 b(of)g(con)-34 b(v)g(ergence)472 b(of)h(the)h(pro)34 b(cedure)472 b(and)i(pro)34 b(ofs)473 b(of)g(the)g(results)g(1)g({)g (3,)489 b(listed)473 b(ab)34 b(o)-34 b(v)g(e,)490 b(are)472 b(in)475 8879 y(Section)404 b(7.)475 11893 y FF(Ac)-39 b(kno)g(wledgemen)g(t)394 b Fw(The)g(authors)h(are)e(v)-34 b(ery)393 b(grateful)h(to)h(Prof.)535 b(G.)393 b(Stolz)h(and)h(Prof.) 535 b(G.)394 b(Galla)-34 b(v)g(otti)394 b(for)475 13332 y(useful)404 b(discussions.)17436 16317 y(2.)606 b Fv(The)458 b(First)h(Appr)-27 b(o)g(xima)-81 b(tion)475 18476 y Fw(2.1.)605 b FF(Op)39 b(erator)505 b Fu(H)10141 18036 y Fp(\(1\))11398 18476 y FF(.)607 b Fw(W)-101 b(e)438 b(in)-34 b(tro)34 b(duce)439 b(the)g(\014rst)g(op)34 b(erator)439 b Fu(H)30627 18036 y Fp(\(1\))31885 18476 y Fw(,)447 b(whic)-34 b(h)439 b(corresp)34 b(onds)439 b(to)g(a)g(partial)g(sum)475 19915 y(in)404 b(the)g(series)g(\(1.2\):) 15937 22773 y Fu(H)17043 22273 y Fp(\(1\))18637 22773 y Fw(=)337 b(\()p Ft(\000)p Fw(\001\))22812 22273 y Fs(l)23428 22773 y Fw(+)269 b Fu(W)25785 22955 y Fp(1)26311 22773 y Fu(;)2626 b(W)30419 22955 y Fp(1)31281 22773 y Fw(=)32754 21243 y Fs(M)33658 21366 y FD(1)32561 21622 y Fq(X)32610 24213 y Fs(r)26 b Fp(=1)34513 22773 y Fu(V)35220 22955 y Fs(r)35727 22773 y Fu(;)12970 b Fw(\(2.1\))475 25930 y(where)456 b Fu(M)5208 26112 y Fp(1)6191 25930 y Fw(is)g(c)-34 b(hosen)457 b(in)g(suc)-34 b(h)457 b(a)g(w)-34 b(a)g(y)457 b(that)h(2)22504 25490 y Fs(M)23408 25613 y FD(1)24348 25930 y Ft(\031)424 b Fu(k)26384 25490 y Fs(s)26819 25613 y FD(1)27793 25460 y Fp(2)28720 25930 y Fw(for)456 b(a)h Fu(k)462 b(>)424 b Fw(1,)470 b Fu(s)36131 26112 y Fp(1)37081 25930 y Fw(=)424 b(\(2)p Fu(l)328 b Ft(\000)304 b Fw(11\))p Fu(=)p Fw(32.)697 b(Ob)-34 b(viously)-101 b(,)475 27389 y(the)453 b(p)34 b(erio)g(ds)452 b(of)g Fu(W)9485 27571 y Fp(1)10463 27389 y Fw(are)g(\()p Fu(a)13647 27571 y Fp(1)14173 27389 y Fu(;)202 b Fw(0\))418 b(=)f(2)18173 26949 y Fs(M)19077 27072 y FD(1)19538 26949 y Fr(\000)p Fp(1)20795 27389 y Fw(\()p Fu(b)21786 27571 y Fp(1)22313 27389 y Fu(;)202 b Fw(0\))452 b(and)i(\(0)p Fu(;)202 b(a)29044 27571 y Fp(2)29569 27389 y Fw(\))418 b(=)f(2)32424 26949 y Fs(M)33328 27072 y FD(1)33789 26949 y Fr(\000)p Fp(1)35047 27389 y Fw(\(0)p Fu(;)202 b(b)37183 27571 y Fp(2)37709 27389 y Fw(\),)464 b(and)454 b Fu(a)42028 27571 y Fp(1)42970 27389 y Ft(\031)418 b Fu(k)45000 26949 y Fs(s)45435 27072 y FD(1)45951 27389 y Fu(b)46471 27571 y Fp(1)46997 27389 y Fu(=)p Fw(2)p Fu(;)453 b(a)49640 27571 y Fp(2)50582 27389 y Ft(\031)475 28966 y Fu(k)1144 28526 y Fs(s)1579 28649 y FD(1)2096 28966 y Fu(b)2616 29148 y Fp(2)3142 28966 y Fu(=)p Fw(2.)538 b(Note)404 b(that)i Ft(k)p Fu(W)12536 29148 y Fp(1)13061 28966 y Ft(k)13667 29148 y Fr(1)15000 28966 y Ft(\024)16280 28057 y Fq(P)17559 28408 y Fs(M)18463 28531 y FD(1)17559 29324 y Fs(r)26 b Fp(=1)19470 28966 y Ft(k)p Fu(V)20783 29148 y Fs(r)21289 28966 y Ft(k)21895 29148 y Fr(1)23228 28966 y Fw(=)336 b Fu(O)34 b Fw(\(1\))405 b(as)f Fu(k)375 b Ft(!)336 b(1)p Fu(:)1803 30539 y Fw(It)e(is)g(w)-34 b(ell-kno)g(wn)335 b(\(see)f(e.g.)514 b([21]\))334 b(that)h(sp)34 b(ectral)333 b(analysis)h(of)h(a)e(p)34 b(erio)g(dic)333 b(op)34 b(erator)334 b Fu(H)42091 30099 y Fp(\(1\))43682 30539 y Fw(can)g(b)34 b(e)334 b(reduced)475 32057 y(to)432 b(analysis)g(of)h(a)f(family)g(of) h(op)34 b(erators)432 b Fu(H)20660 31617 y Fp(\(1\))21918 32057 y Fw(\()p Fu(t)p Fw(\),)439 b Fu(t)383 b Ft(2)h Fu(K)27116 32239 y Fp(1)27642 32057 y Fw(,)439 b(where)432 b Fu(K)32980 32239 y Fp(1)33939 32057 y Fw(is)g(the)h(elemen)-34 b(tary)431 b(cell)g(of)i(the)g(dual)475 33551 y(lattice,)651 b Fu(K)5792 33733 y Fp(1)6985 33551 y Fw(=)666 b([0)p Fu(;)202 b Fw(2)p Fu(\031)43 b(a)12057 33041 y Fr(\000)p Fp(1)12057 33898 y(1)13314 33551 y Fw(\))402 b Ft(\002)f Fw([0)p Fu(;)202 b Fw(2)p Fu(\031)43 b(a)18994 33041 y Fr(\000)p Fp(1)18994 33898 y(2)20251 33551 y Fw(\))p Fu(:)603 b Fw(The)f(v)-34 b(ector)602 b Fu(t)g Fw(is)f(called)h Fu(q)43 b(uasimomentum)p Fw(.)1132 b(An)602 b(op)34 b(erator)475 35113 y Fu(H)1581 34673 y Fp(\(1\))2838 35113 y Fw(\()p Fu(t)p Fw(\),)541 b Fu(t)519 b Ft(2)g Fu(K)8409 35295 y Fp(1)8935 35113 y Fw(,)540 b(acts)514 b(in)f Fu(L)14768 35295 y Fp(2)15294 35113 y Fw(\()p Fu(Q)16723 35295 y Fp(1)17249 35113 y Fw(\),)541 b Fu(Q)19556 35295 y Fp(1)20596 35113 y Fw(b)34 b(eing)513 b(the)h(elemen)-34 b(tary)513 b(cell)f(of)i(the)f(p)34 b(erio)g(ds)513 b(of)h(the)g(p)34 b(oten)-34 b(tial,)475 36631 y Fu(Q)1433 36813 y Fp(1)2423 36631 y Fw(=)463 b([0)p Fu(;)202 b(a)5952 36813 y Fp(1)6477 36631 y Fw(])320 b Ft(\002)g Fw([0)p Fu(;)202 b(a)10520 36813 y Fp(2)11045 36631 y Fw(])p Fu(:)480 b Fw(The)h(op)34 b(erator)480 b Fu(H)20838 36191 y Fp(\(1\))22096 36631 y Fw(\()p Fu(t)p Fw(\))h(is)f(describ)34 b(ed)480 b(b)-34 b(y)481 b(form)-34 b(ula)481 b(\(2.1\))g(and)h(the)e(quasip)34 b(erio)g(dic)475 38070 y(conditions)405 b(for)f(a)g(function)i(itself)e (and)g(its)h(deriv)-67 b(ativ)-34 b(es:)9639 40201 y Fu(u)p Fw(\()p Fu(a)11445 40383 y Fp(1)11971 40201 y Fu(;)202 b(x)13203 40383 y Fp(2)13728 40201 y Fw(\))338 b(=)e(exp\()p Fu(it)18995 40383 y Fp(1)19520 40201 y Fu(a)20161 40383 y Fp(1)20687 40201 y Fw(\))p Fu(u)p Fw(\(0)p Fu(;)202 b(x)24161 40383 y Fp(2)24687 40201 y Fw(\))p Fu(;)1010 b(u)p Fw(\()p Fu(x)28363 40383 y Fp(1)28889 40201 y Fu(;)202 b(a)30069 40383 y Fp(2)30594 40201 y Fw(\))337 b(=)g(exp\()p Fu(it)35861 40383 y Fp(2)36386 40201 y Fu(a)37027 40383 y Fp(2)37553 40201 y Fw(\))p Fu(u)p Fw(\()p Fu(x)39882 40383 y Fp(1)40408 40201 y Fu(;)202 b Fw(0\))p Fu(;)6673 b Fw(\(2.2\))7202 42660 y Fu(u)7896 42159 y Fp(\()p Fs(j)51 b Fp(\))7896 42959 y Fs(x)8426 43082 y FD(1)9115 42660 y Fw(\()p Fu(a)10227 42842 y Fp(1)10752 42660 y Fu(;)202 b(x)11984 42842 y Fp(2)12510 42660 y Fw(\))337 b(=)f(exp\()p Fu(it)17776 42842 y Fp(1)18302 42660 y Fu(a)18943 42842 y Fp(1)19468 42660 y Fw(\))p Fu(u)20633 42159 y Fp(\()p Fs(j)51 b Fp(\))20633 42959 y Fs(x)21163 43082 y FD(1)21852 42660 y Fw(\(0)p Fu(;)202 b(x)24161 42842 y Fp(2)24687 42660 y Fw(\))p Fu(;)1010 b(u)27199 42159 y Fp(\()p Fs(j)51 b Fp(\))27199 42959 y Fs(x)27729 43082 y FD(2)28418 42660 y Fw(\()p Fu(x)29582 42842 y Fp(1)30108 42660 y Fu(;)202 b(a)31288 42842 y Fp(2)31813 42660 y Fw(\))337 b(=)f(exp\()p Fu(it)37079 42842 y Fp(2)37605 42660 y Fu(a)38246 42842 y Fp(2)38771 42660 y Fw(\))p Fu(u)39936 42159 y Fp(\()p Fs(j)51 b Fp(\))39936 42959 y Fs(x)40466 43082 y FD(2)41155 42660 y Fw(\()p Fu(x)42319 42842 y Fp(1)42845 42660 y Fu(;)202 b Fw(0\))p Fu(;)475 44737 y Fw(0)337 b Fu(<)f(j)406 b(<)337 b Fw(2)p Fu(l)24 b Fw(.)538 b(Eac)-34 b(h)404 b(op)34 b(erator)404 b Fu(H)15757 44297 y Fp(\(1\))17015 44737 y Fw(\()p Fu(t)p Fw(\),)g Fu(t)336 b Ft(2)h Fu(K)22084 44919 y Fp(1)22610 44737 y Fw(,)404 b(has)h(a)f(discrete)f(b)34 b(ounded)406 b(b)34 b(elo)-34 b(w)404 b(sp)34 b(ectrum)405 b(\003)45500 44297 y Fp(\(1\))46757 44737 y Fw(\()p Fu(t)p Fw(\):)15346 47008 y(\003)16188 46507 y Fp(\(1\))17445 47008 y Fw(\()p Fu(t)p Fw(\))337 b(=)g Ft([)21250 46507 y Fr(1)21250 47307 y Fs(n)p Fp(=1)23079 47008 y Fu(\025)23786 46507 y Fp(\(1\))23786 47307 y Fs(n)25043 47008 y Fw(\()p Fu(t)p Fw(\))p Fu(;)606 b(\025)28073 46507 y Fp(\(1\))28073 47307 y Fs(n)29331 47008 y Fw(\()p Fu(t)p Fw(\))337 b Ft(!)32260 47190 y Fs(n)p Fr(!1)35105 47008 y Ft(1)p 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Fp(1)36141 34719 y Fw(\))36612 34218 y Fs(r)37118 34719 y Fu(dz)g(;)9725 b Fw(\(2.12\))9133 38726 y Fu(G)10086 38226 y Fp(\(1\))10086 39026 y Fs(r)11344 38726 y Fw(\()p Fu(k)38 b(;)202 b(t)p Fw(\))337 b(=)15681 37906 y(\()p Ft(\000)p Fw(1\))18172 37466 y Fs(r)26 b Fp(+1)p 15681 38447 4200 49 v 16902 39558 a Fw(2)p Fu(\031)43 b(i)20216 37076 y Fq(I)20889 39825 y Fs(C)21560 39948 y FD(1)22077 38726 y Fw(\(\()p Fu(H)24125 38087 y Fp(\(1\))24027 39073 y(0)25383 38726 y Fw(\()p Fu(t)p Fw(\))270 b Ft(\000)f Fu(z)53 b Fw(\))29333 38226 y Fr(\000)p Fp(1)30591 38726 y Fu(W)31736 38908 y Fp(1)32262 38726 y Fw(\))32733 38226 y Fs(r)33239 38726 y Fw(\()p Fu(H)34816 38087 y Fp(\(1\))34718 39073 y(0)36074 38726 y Fw(\()p Fu(t)p Fw(\))270 b Ft(\000)f Fu(z)53 b Fw(\))40024 38226 y Fr(\000)p Fp(1)41282 38726 y Fu(dz)g(:)5561 b Fw(\(2.13\))475 41784 y(T)-101 b(o)460 b(\014nd)i Fu(g)5417 41145 y Fp(\(1\))5374 41942 y Fs(r)6675 41784 y Fw(\()p Fu(k)38 b(;)202 b(t)p Fw(\))461 b(and)g Fu(G)13090 41145 y Fp(\(1\))13090 41942 y Fs(r)14347 41784 y Fw(\()p Fu(k)38 b(;)202 b(t)p Fw(\))461 b(it)f(is)g(necessary)g(to)g(compute)h(the)g (residues)f(of)h(a)f(rational)g(function)i(of)475 43223 y(a)431 b(simple)f(structure,)438 b(whose)431 b(n)-34 b(umerator)432 b(do)34 b(es)430 b(not)i(dep)34 b(end)431 b(on)h Fu(z)53 b Fw(,)437 b(while)430 b(the)h(denominator)h(is)f(a)f (pro)34 b(duct)475 44662 y(of)423 b(factors)h(of)f(the)h(t)-34 b(yp)34 b(e)423 b(\()p Fu(p)13176 44222 y Fp(2)p Fs(l)13176 45002 y(i)13993 44662 y Fw(\()p Fu(t)p Fw(\))283 b Ft(\000)e Fu(z)53 b Fw(\).)596 b(F)-101 b(or)423 b(all)g Fu(t)f Fw(in)h(the)h(nonresonance)g(set)f(within)h Fu(C)41269 44844 y Fp(1)42218 44662 y Fw(the)f(in)-34 b(tegrand)425 b(has)475 46167 y(a)489 b(single)f(p)34 b(ole)488 b(at)i(the)f(p)34 b(oin)-34 b(t)490 b Fu(z)530 b Fw(=)478 b Fu(k)17878 45727 y Fp(2)p Fs(l)19173 46167 y Fw(=)g Fu(p)21204 45727 y Fp(2)p Fs(l)21204 46507 y(j)22021 46167 y Fw(\()p Fu(t)p Fw(\))p Fu(:)489 b 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Fp(SPECTRAL)406 b(PR)-26 b(OPER)-78 b(TIES)406 b(OF)h(A)h(POL)-78 b(YHARMONIC)405 b(OPERA)-78 b(TOR)406 b(WITH)h(LIMIT-PERIODIC)e(POTENTIAL)1430 b(11)p 0 TeXcolorgray 475 6197 a Fu(E)1440 5559 y Fp(\(1\))1370 6559 y Fs(j)2697 6197 y Fw(\()p Fu(\013)t(;)202 b(t)p Fw(\))482 b(=)e Ft(\000)8822 5720 y Fp(1)p 8376 5918 1363 49 v 8376 6615 a(2)p Fs(\031)32 b(i)10074 5221 y Fq(H)10646 6623 y Fs(C)11317 6746 y FD(1)11833 6197 y Fw(\()p Fu(H)13410 5559 y Fp(\(1\))13312 6355 y Fs(\013)14668 6197 y Fw(\()p Fu(t)p Fw(\))328 b Ft(\000)e Fu(z)53 b Fw(\))18733 5758 y Fr(\000)p Fp(1)19991 6197 y Fu(dz)544 b Fw(and)491 b(in)-34 b(tegrating)491 b(term)-34 b(wise,)512 b(w)-34 b(e)491 b(arriv)-34 b(e)490 b(at)g(\(2.17\).)799 b(F)-101 b(orm)-34 b(ula)475 7766 y(\(2.16\))404 b(is)g(obtained)h(in)g(the)f(analogous)h(w)-34 b(a)g(y)-101 b(.)539 b(Estimates)405 b(\(2.18\))g(follo)-34 b(w)404 b(from)h(\(2.22\),)f(for)g(details)g(see)g([22].)p 50750 9371 665 831 v 1803 11479 a(Next,)436 b(w)-34 b(e)431 b(sho)-34 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Fu(;)1010 b Ft(j)p Fu(m)p Ft(j)336 b Fw(:=)h Fu(m)28849 18647 y Fp(1)29644 18465 y Fw(+)269 b Fu(m)31920 18647 y Fp(2)32446 18465 y Fu(;)1010 b(m)p Fw(!)337 b(:=)f Fu(m)38211 18647 y Fp(1)38737 18465 y Fw(!)p Fu(m)40138 18647 y Fp(2)40664 18465 y Fw(!)p Fu(;)605 b(T)168 b Fw(\(0\))p Fu(f)469 b Fw(:=)336 b Fu(f)63 b(:)p 0 TeXcolorgray 475 21851 a FF(Lemma)408 b(2.6.)p 0 TeXcolorgray 514 w FC(The)389 b(c)-62 b(o)g(e\016cients)386 b Fu(g)17234 21212 y Fp(\(1\))17191 22009 y Fs(r)18492 21851 y Fw(\()p Fu(k)38 b(;)202 b(t)p Fw(\))p Fu(;)390 b FC(and)f Fu(G)25070 21212 y Fp(\(1\))25070 22009 y Fs(r)26327 21851 y Fw(\()p Fu(k)38 b(;)202 b(t)p Fw(\))390 b FC(c)-62 b(an)389 b(b)-62 b(e)388 b(c)-62 b(ontinue)g(d)388 b(as)g(holomorphic)e(functions)475 23319 y(of)315 b(two)g(variables)e(fr)-62 b(om)315 b(the)f(r)-62 b(e)g(al)315 b Fw(\(2)p Fu(k)17419 22880 y Fr(\000)p Fp(1)p Fr(\000)p Fp(4)p Fs(s)20258 23003 y FD(1)20721 22880 y Fr(\000)p Fp(2)p Fs(\016)22428 23319 y Fw(\))p FC(-neighb)-62 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y Fs(r)33767 33320 y Fw(\()p Fu(k)38 b(;)202 b(t)p Fw(\))569 b(can)g(b)34 b(e)569 b(con)-34 b(tin)g(ued)570 b(from)f(the)475 34789 y(real)363 b(\(2)p Fu(k)4541 34349 y Fr(\000)p Fp(1)p Fr(\000)p Fp(4)p Fs(s)7380 34472 y FD(1)7843 34349 y Fr(\000)p Fp(2)p Fs(\016)9550 34789 y Fw(\)-neigh)-34 b(b)34 b(orho)g(o)g(d)365 b(of)g Fu(t)f Fw(to)h(the)g(complex)e(\(2)p Fu(k)30084 34349 y Fr(\000)p Fp(1)p Fr(\000)p Fp(4)p Fs(s)32923 34472 y FD(1)33386 34349 y Fr(\000)p Fp(2)p Fs(\016)35093 34789 y Fw(\)-neigh)-34 b(b)34 b(orho)g(o)g(d)365 b(as)g(holomorphic) 475 36228 y(functions)416 b(of)g(t)-34 b(w)g(o)416 b(v)-67 b(ariables,)417 b(and)f(inequalities)e(\(2.18\))i(are)f(hereb)-34 b(y)415 b(preserv)-34 b(ed.)571 b(Estimating)416 b(b)-34 b(y)415 b(means)h(of)475 37667 y(the)380 b(Cauc)-34 b(h)g(y)382 b(in)-34 b(tegral)380 b(form)-34 b(ula,)385 b(the)c(v)-67 b(alue)379 b(of)h(the)h(deriv)-67 b(ativ)-34 b(e)379 b(with)i(resp)34 b(ect)379 b(to)i Fu(t)e Fw(in)h(terms)h(of)f(the)h(v) -67 b(alue)379 b(of)475 39106 y(the)407 b(function)g(itself)g(on)f(the) h(b)34 b(oundary)407 b(of)g(the)g(\(2)p Fu(k)24707 38666 y Fr(\000)p Fp(1)p Fr(\000)p Fp(4)p Fs(s)27546 38789 y FD(1)28008 38666 y Fr(\000)p Fp(2)p Fs(\016)29715 39106 y Fw(\)-neigh)-34 b(b)34 b(orho)g(o)g(d)408 b(of)f Fu(t)f Fw(\(form)-34 b(ulas)407 b(\(2.18\)\),)h(w)-34 b(e)475 40545 y(obtain)405 b(\(2.24\))f(and)h(\(2.25\).)p 50750 40711 665 831 v 1803 42820 a(F)-101 b(rom)404 b(this)h(lemma)f(the)g (follo)-34 b(wing)405 b(theorem)f(easily)f(follo)-34 b(ws.)p 0 TeXcolorgray 475 45052 a FF(Theorem)726 b(2.7.)p 0 TeXcolorgray 679 w FC(The)643 b(series)f(\(2.16\),)f(\(2.17\))h(c)-62 b(an)643 b(b)-62 b(e)643 b(c)-62 b(ontinue)g(d)642 b(as)g(holomorphic)f (functions)h(of)h(two)475 46521 y(variables)600 b(fr)-62 b(om)603 b(the)f(r)-62 b(e)g(al)602 b Fw(\(2)p Fu(k)15119 46081 y Fr(\000)p Fp(1)p Fr(\000)p Fp(4)p Fs(s)17958 46204 y FD(1)18420 46081 y Fr(\000)p Fp(2)p Fs(\016)20128 46521 y Fw(\))p FC(-neighb)-62 b(orho)g(o)g(d)599 b(of)k(the)f(nonr)-62 b(esonanc)g(e)602 b(set)g Fu(\037)42742 46703 y Fp(1)43871 46521 y FC(to)h(its)f(c)-62 b(omplex)475 47989 y Fw(\(2)p Fu(k)2221 47550 y Fr(\000)p Fp(1)p Fr(\000)p Fp(4)p Fs(s)5060 47673 y FD(1)5522 47550 y Fr(\000)p Fp(2)p Fs(\016)7230 47989 y Fw(\))p FC(-neighb)g(orho)g(o)g(d)430 b(and)j(the)f(fol)62 b(lowing)432 b(estimates)f(hold)h(in)i(the)e(c)-62 b(omplex)433 b(neighb)-62 b(orho)g(o)g(d:)9063 50340 y Ft(j)p Fu(T)168 b Fw(\()p Fu(m)p Fw(\)\()p Fu(\025)13460 49701 y Fp(\(1\))13460 50702 y Fs(j)14719 50340 y Fw(\()p Fu(\013)t(;)202 b(t)p Fw(\))271 b Ft(\000)e Fu(p)19510 49839 y Fp(2)p Fs(l)19510 50639 y(j)20327 50340 y Fw(\()p Fu(t)p Fw(\)\))p Ft(j)337 b Fu(<)f Fw(2)p Fu(m)p Fw(!)p Fu(\013)26917 49839 y Fp(2)27444 50340 y Fu(k)28113 49839 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)31714 49962 y FD(1)32176 49839 y Fr(\000)p Fs(\016)33 b Fr(\000)p Fp(2)p Fs(\015)35044 49962 y FD(0)35506 49839 y Fp(+)p Fr(j)p Fs(m)p Fr(j)p Fp(\(1+4)p Fs(s)40065 49962 y FD(1)40528 49839 y Fp(+2)p Fs(\016)g Fp(\))42601 50340 y Fu(;)5490 b Fw(\(2.26\))12235 53051 y Ft(k)p Fu(T)168 b Fw(\()p Fu(m)p Fw(\)\()p Fu(E)17159 52412 y Fp(\(1\))17089 53413 y Fs(j)18418 53051 y Fw(\()p Fu(\013)t(;)202 b(t)p Fw(\))271 b Ft(\000)e Fu(E)23494 53233 y Fs(j)23980 53051 y Fw(\))p Ft(k)25057 53233 y Fp(1)25920 53051 y Fu(<)337 b Fw(2)p Fu(m)p Fw(!)p Fu(\013)t(k)30655 52550 y Fr(\000)p Fs(\015)31872 52673 y FD(0)32334 52550 y Fp(+)p Fr(j)p Fs(m)p Fr(j)p Fp(\(1+4)p Fs(s)36893 52673 y FD(1)37355 52550 y Fp(+2)p Fs(\016)33 b Fp(\))39429 53051 y Fu(:)8662 b Fw(\(2.27\))p 0 TeXcolorgray 475 55331 a FF(Corollary)483 b(2.8.)p 0 TeXcolorgray 12972 57201 a Ft(r)p Fu(\025)14689 56562 y Fp(\(1\))14689 57563 y Fs(j)15947 57201 y Fw(\()p Fu(\013)t(;)202 b(t)p Fw(\))338 b(=)e(2)p Fu(l)24 b(k)21923 56701 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(2)23881 56881 y Fu(~)23943 57201 y(k)307 b Fw(+)269 b Fu(o)26883 55856 y Fq(\020)27607 57201 y Fu(k)28276 56701 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1)30295 55856 y Fq(\021)31221 57201 y Fu(;)33867 56881 y(~)33928 57201 y(k)375 b Fw(=)346 b Fu(~)-615 b(p)36824 57383 y Fs(j)37310 57201 y Fw(\()p Fu(t)p Fw(\))p Fu(;)9401 b Fw(\(2.28\))15225 59914 y Fu(T)168 b Fw(\()p Fu(m)p Fw(\))p Fu(\025)18814 59275 y Fp(\(1\))18814 60276 y Fs(j)20073 59914 y Fw(\()p Fu(\013)t(;)202 b(t)p Fw(\))338 b Fu(<)e Fw(4)p Fu(l)25380 59414 y Fp(2)25906 59914 y Fu(k)26575 59414 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(2)28594 59914 y Fu(;)1503 b FC(if)1301 b Ft(j)p Fu(m)p Ft(j)336 b Fw(=)g(2)p Fu(:)11653 b Fw(\(2.29\))1803 62457 y(The)419 b(next)g(lemma)g(will)f(b)34 b(e)419 b(used)g(in)g(the)g(second)g(step) g(of)h(appro)-34 b(ximation)420 b(where)e(the)h(op)34 b(erator)419 b Fu(H)48887 62017 y Fp(\(1\))50145 62457 y Fw(\()p Fu(t)p Fw(\))475 63896 y(will)403 b(pla)-34 b(y)405 b(a)f(role)f(of)h(the)h(initial)f(\(unp)34 b(erturb)g(ed\))406 b(op)34 b(erator.)p 0 TeXcolorgray 475 66158 a FF(Lemma)626 b(2.9.)p 0 TeXcolorgray 631 w FC(F)-93 b(or)564 b(any)f Fu(z)616 b FC(on)564 b(the)e(cir)-62 b(cle)562 b Fu(C)22646 66340 y Fp(1)23736 66158 y FC(given)g(by)h(\(2.9\))f(and)h Fu(t)g FC(in)h(the)e(c)-62 b(omplex)562 b Fw(\(2)p Fu(k)45611 65718 y Fr(\000)p Fp(1)p Fr(\000)p Fp(4)p Fs(s)48450 65841 y FD(1)48913 65718 y Fr(\000)p Fp(2)p Fs(\016)50620 66158 y Fw(\))p FC(-)475 67597 y(neighb)-62 b(orho)g(o)g(d)430 b(of)j Fu(\037)9860 67779 y Fp(1)10386 67597 y FC(,)16743 69238 y Ft(k)p Fw(\()p Fu(H)18926 68737 y Fp(\(1\))20184 69238 y Fw(\()p Fu(t)p Fw(\))270 b Ft(\000)f Fu(z)53 b Fw(\))24134 68737 y Fr(\000)p Fp(1)25392 69238 y Ft(k)337 b(\024)f Fw(4)p Fu(k)28889 68737 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+2+4)p Fs(s)33222 68860 y FD(1)33684 68737 y Fp(+)p Fs(\016)34921 69238 y Fu(:)13170 b Fw(\(2.30\))p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 12 12 TeXDict begin 12 11 bop 0 TeXcolorgray 0 TeXcolorgray 475 3345 a Fp(12)14248 b(YULIA)313 b(KARPESHINA)f(AND)h(YOUNG-RAN)g (LEE)p 0 TeXcolorgray 0 TeXcolorgray 475 6001 a FC(Pr)-62 b(o)g(of.)p 0 TeXcolorgray 552 w Fw(Considering)405 b(the)f(follo)-34 b(wing)405 b(Hilb)34 b(ert)404 b(relation)6587 8163 y(\()p Fu(H)8164 7663 y Fp(\(1\))9422 8163 y Fw(\()p Fu(t)p Fw(\))270 b Ft(\000)f Fu(z)53 b Fw(\))13372 7663 y Fr(\000)p Fp(1)14967 8163 y Fw(=)336 b(\()p Fu(H)17823 7524 y Fp(\(1\))17725 8510 y(0)19081 8163 y Fw(\()p Fu(t)p Fw(\))270 b Ft(\000)f Fu(z)53 b Fw(\))23031 7663 y Fr(\000)p Fp(1)24559 8163 y Fw(+)269 b(\()p Fu(H)27348 7524 y Fp(\(1\))27250 8510 y(0)28606 8163 y Fw(\()p Fu(t)p Fw(\))h Ft(\000)f Fu(z)53 b Fw(\))32556 7663 y Fr(\000)p Fp(1)33814 8163 y Fw(\()p Ft(\000)p Fu(W)36373 8345 y Fp(1)36899 8163 y Fw(\)\()p Fu(H)38947 7663 y Fp(\(1\))40205 8163 y Fw(\()p Fu(t)p Fw(\))270 b Ft(\000)f Fu(z)53 b Fw(\))44155 7663 y Fr(\000)p Fp(1)475 10437 y Fw(and)405 b(\(2.10\),)f(i.e.,)f Ft(k)p Fw(\()p Fu(H)11143 9798 y Fp(\(1\))11045 10784 y(0)12401 10437 y Fw(\()p Fu(t)p Fw(\))270 b Ft(\000)f Fu(z)53 b Fw(\))16351 9997 y Fr(\000)p Fp(1)17609 10437 y Ft(k)336 b(\024)h Fw(2)p Fu(k)21106 9997 y 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y Fw(\()p Fu(\013)t(;)202 b Fn({)26114 21995 y Fp(1)26616 21813 y Fu(~)-581 b(\027)77 b Fw(\))337 b(=)f Fu(\025:)17980 b Fw(\(2.38\))475 23799 y FC(F)-93 b(urthermor)-62 b(e,)431 b Ft(j)p Fn({)8993 23981 y Fp(1)9788 23799 y Ft(\000)269 b Fu(k)38 b Ft(j)337 b(\024)g Fu(ck)14817 23359 y Fr(\000)p Fp(1)p Fr(\000)p Fp(4)p Fs(s)17656 23482 y FD(1)18117 23359 y Fr(\000)p Fp(2)p Fs(\015)19804 23482 y FD(0)20265 23359 y Fr(\000)p Fs(\016)21502 23799 y FC(.)p 0 TeXcolorgray 475 25965 a(Pr)-62 b(o)g(of.)p 0 TeXcolorgray 552 w Fw(1\))505 b(Let)f(us)h(sho)-34 b(w)506 b(that)g(there)f(exists)f(suc)-34 b(h)506 b(an)f Fn({)27637 26147 y Fp(1)28163 25965 y Fw(.)840 b(Giv)-34 b(en)480 b Fu(~)-582 b(\027)581 b Ft(2)505 b Fo(B)36366 26147 y Fp(1)36891 25965 y Fw(\()p Fu(\025)p Fw(\),)530 b(b)-34 b(y)505 b(Theorem)g(2.3,)529 b(there)475 27483 y(exists)355 b(an)g(eigen)-34 b(v)-67 b(alue)355 b Fu(\025)11857 27043 y Fp(\(1\))13115 27483 y Fw(\()p Fu(\013)t(;)202 b Fn({)24 b Fu(~)-581 b(\027)77 b Fw(\))356 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Fw(\()p Fu(t)p Fw(\))337 b(=)g Fn({)24 b Fu(~)-581 b(\027)77 b Fw(,)475 30727 y(where)454 b Fu(j)r(;)656 b(t)p Fw(,)466 b(and)455 b Fu(t)9610 30909 y Fp(0)10589 30727 y Fw(are)f(uniquely)g (determined)g(b)-34 b(y)455 b Fn({)24 b Fu(~)-581 b(\027)531 b Fw(and)454 b Fu(k)14 b(~)-582 b(\027)78 b Fw(.)688 b(Let)454 b Fo(L)35599 30287 y Fp(\(1\))36857 30727 y Fw(\()p Fu(\013)t(;)177 b(~)-581 b(\027)78 b Fw(\))420 b(:=)g Ft(f)p Fu(\025)g Fw(=)g Fu(\025)45717 30287 y Fp(\(1\))46974 30727 y Fw(\()p Fu(\013)t(;)202 b Fn({)24 b Fu(~)-581 b(\027)78 b Fw(\))421 b(:)475 32245 y Fn({)663 b Ft(2)614 b Fu(I)3901 32427 y Fp(1)4427 32245 y Ft(g)p Fu(:)570 b Fw(Using)h(the)g(de\014nition)h(of)f Fu(I)19402 32427 y Fp(1)20498 32245 y Fw(and)h(considering)e(that)i Fu(\025)33028 31805 y Fp(\(1\))34286 32245 y Fw(\(0)p Fu(;)202 b Fn({)24 b Fu(~)-581 b(\027)77 b Fw(\))615 b(=)f Fn({)40935 31805 y Fp(2)p Fs(l)41752 32245 y Fw(,)e(w)-34 b(e)571 b(easily)e(obtain)475 33762 y Fo(L)1342 33323 y Fp(\(1\))2600 33762 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b(tiat-)475 41194 y(ing)404 b(\(2.34\))h(with)g(resp)34 b(ect)403 b(to)i Fn({)453 b Fw(and)404 b(using)h(\(2.36\),)g(w)-34 b(e)404 b(get:)15154 43229 y Fu(@)67 b(\025)16572 42789 y Fp(\(1\))17830 43229 y Fw(\()p Fu(\013)t(;)202 b Fn({)24 b Fu(~)-581 b(\027)78 b Fw(\))p 15154 43770 6470 49 v 17605 44880 a Fu(@)67 b Fn({)22094 44049 y Fw(=)336 b(2)p Fu(l)24 b Fn({)25222 43548 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1)27510 44049 y Fw(+)270 b Fu(O)34 b Fw(\()p Fu(k)30822 43548 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1)p Fr(\000)p Fp(2)p Fs(\015)34473 43671 y FD(0)34934 43548 y Fp(+)p Fs(\016)36170 44049 y Fw(\))p Fu(:)11450 b Fw(\(2.39\))475 46350 y(Therefore,)17281 47813 y Fu(@)67 b(\025)18699 47373 y Fp(\(1\))19957 47813 y Fw(\()p Fu(\013)t(;)202 b Fn({)24 b Fu(~)-581 b(\027)77 b Fw(\))p 17281 48355 V 19732 49465 a Fu(@)67 b Fn({)24220 48633 y Ft(\025)337 b Fw(2)p Fu(l)24 b(k)27161 48133 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1)29180 47652 y Fq(\000)29736 48633 y Fw(1)269 b(+)g Fu(o)p Fw(\(1\))33959 47652 y Fq(\001)34515 48633 y Fu(;)13576 b Fw(\(2.40\))475 50945 y(and)504 b(this)g(implies)e(that)j Fu(\025)13040 50505 y Fp(\(1\))14297 50945 y Fw(\()p Fu(\013)t(;)202 b Fn({)24 b Fu(~)-581 b(\027)78 b Fw(\))504 b(is)f(monotone)h(with)g(resp)34 b(ect)503 b(to)g Fn({)552 b Fw(in)504 b Fu(I)37699 51127 y Fp(1)38224 50945 y Fw(.)836 b(Th)-34 b(us,)529 b(there)503 b(is)g(a)h(single)475 52384 y Fn({)1283 52566 y Fp(1)2145 52384 y Ft(2)337 b Fu(I)3823 52566 y Fp(1)4753 52384 y Fw(satisfying)404 b(\(2.38\).)1803 53823 y(3\))h(No)-34 b(w)405 b(let)e(us)i(estimate)f Ft(j)p Fn({)15352 54005 y Fp(1)16147 53823 y Ft(\000)269 b Fu(k)38 b Ft(j)p Fw(.)539 b(By)403 b(\(2.34\))i(and)f(\(2.35\))h(w)-34 b(e)405 b(ha)-34 b(v)g(e)11090 55918 y Fu(\025)11797 55418 y Fp(\(1\))13055 55918 y Fw(\()p Fu(\013)t(;)202 b(k)14 b(~)-582 b(\027)78 b Fw(\))337 b(=)g Fu(k)18947 55418 y Fp(2)p Fs(l)20033 55918 y Fw(+)269 b Fu(f)21838 56100 y Fp(1)22365 55918 y Fw(\()p Fu(\013)t(;)202 b(k)14 b(~)-582 b(\027)78 b Fw(\))337 b(=)f Fu(k)28256 55418 y Fp(2)p Fs(l)29343 55918 y Fw(+)269 b Fu(O)34 b Fw(\()p Fu(k)32654 55418 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)36255 55541 y FD(1)36717 55418 y Fr(\000)p Fp(2)p Fs(\015)38404 55541 y FD(0)38865 55418 y Fr(\000)p Fs(\016)40102 55918 y Fw(\))p Fu(:)7518 b Fw(\(2.41\))475 57874 y(Using)404 b(\(2.41\),)h(\(2.38\),)f(\(2.40\),)g(and)h(the)g(Mean)f(V)-101 b(alue)404 b(Theorem,)g(w)-34 b(e)405 b(obtain)13251 59970 y Fu(ck)14445 59469 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)18046 59592 y FD(1)18507 59469 y Fr(\000)p Fp(2)p Fs(\015)20194 59592 y FD(0)20655 59469 y Fr(\000)p Fs(\016)22229 59970 y Ft(\025)336 b(j)p Fu(\025)24552 59469 y Fp(\(1\))25809 59970 y Fw(\()p Fu(\013)t(;)202 b Fn({)28406 60152 y Fp(1)28909 59970 y Fu(~)-582 b(\027)77 b Fw(\))270 b Ft(\000)f Fu(\025)32269 59469 y Fp(\(1\))33527 59970 y Fw(\()p Fu(\013)t(;)202 b(k)14 b(~)-582 b(\027)78 b Fw(\))p Ft(j)337 b(\025)14275 63035 y Fw(min)14203 63842 y Fm({)40 b Fr(2)p 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66147 665 831 v 1803 68102 a(Here)269 b(and)h(b)34 b(elo)-34 b(w,)296 b(w)-34 b(e)270 b(use)g(the)g (notations)g Fn({)21981 68284 y Fp(1)22507 68102 y Fw(\()p Fu(')p Fw(\))338 b(:=)e Fn({)27004 68284 y Fp(1)27530 68102 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))p Fu(;)447 b(~)-582 b(\027)414 b Fw(=)336 b(\(cos)203 b Fu(';)f Fw(sin)g Fu(')p Fw(\),)328 b Fu(~)-638 b Fn({)41518 68284 y Fp(1)42044 68102 y Fw(\()p Fu(')p Fw(\))337 b(=)g Fn({)46204 68284 y Fp(1)46730 68102 y Fw(\()p Fu(')p Fw(\))-25 b Fu(~)-581 b(\027)77 b(;)471 b(h)50647 68284 y Fp(1)51173 68102 y Fw(\()p Fu(')p Fw(\))338 b(=)475 69541 y Fn({)1283 69723 y Fp(1)1809 69541 y Fw(\()p Fu(')p Fw(\))270 b Ft(\000)f Fu(k)38 b Fw(.)p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 14 14 TeXDict begin 14 13 bop 0 TeXcolorgray 0 TeXcolorgray 475 3345 a Fp(14)14248 b(YULIA)313 b(KARPESHINA)f(AND)h(YOUNG-RAN)g (LEE)p 0 TeXcolorgray 0 TeXcolorgray 475 6001 a FF(Lemma)463 b(2.11.)p 0 TeXcolorgray 0 TeXcolorgray 2489 w Fw(\(1\))p 0 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Fw(The)405 b(set)f Fu(\037)30060 20151 y Fr(\003)30060 20916 y Fp(1)30586 20591 y Fw(\()p Fu(\025;)202 b(\016)46 b Fw(\))p 0 TeXcolorgray 0 TeXcolorgray 2409 24189 a(\(4\))p 0 TeXcolorgray 554 w(Considering)434 b(that)f Fu(L)p Fw(\()p Fo(D)16174 24371 y Fp(1)16700 24189 y Fw(\()p Fu(\025)p Fw(\)\))385 b(=)20532 23213 y Fq(R)21105 24615 y Fs(\002)21819 24738 y FD(1)22280 24615 y Fp(\()p Fs(\025)p Fp(\))23818 23135 y Fq(p)p 25030 23135 8196 49 v 1054 x Fn({)25887 23772 y Fp(2)25838 24536 y(1)26413 24189 y Fw(\()p Fu(')p Fw(\))270 b(+)f Fn({)30487 23772 y Fr(0)30438 24536 y Fp(1)30964 24189 y Fw(\()p Fu(')p Fw(\))32699 23839 y Fp(2)33427 24189 y Fu(d')433 b Fw(and)g(taking)g(in)-34 b(to)433 b(accoun)-34 b(t)434 b(\(2.44\))4511 25856 y(and)405 b(\(2.47\),)g(w)-34 b(e)404 b(obtain)7939 28686 y Fu(L)p Fw(\()p Fo(D)10242 28868 y Fp(0)10768 28686 y Fw(\()p Fu(\025)p Fw(\)\))270 b Ft(\000)f Fu(L)p Fw(\()p Fo(D)16673 28868 y Fp(1)17199 28686 y Fw(\()p Fu(\025)p Fw(\)\))338 b(=)20937 27036 y Fq(Z)21610 29785 y Fp([0)p Fs(;)p Fp(2)p Fs(\031)32 b Fp(])p Fr(n)p Fs(\002)25089 29908 y FD(1)25553 29785 y Fp(\()p Fs(\025)p Fp(\))27091 28686 y Fu(k)240 b(d')269 b Fw(+)30867 27036 y Fq(Z)31541 29785 y Fs(\002)32255 29908 y FD(1)32771 27341 y Fq(\020)33495 28686 y Fu(k)308 b Ft(\000)35646 27210 y Fq(q)p 36858 27210 V 1476 x Fn({)37715 28269 y Fp(2)37666 29033 y(1)38241 28686 y Fw(\()p Fu(')p Fw(\))270 b(+)f Fn({)42315 28269 y Fr(0)42266 29033 y Fp(1)42792 28686 y Fw(\()p Fu(')p Fw(\))44527 28336 y Fp(2)45053 27341 y Fq(\021)45979 28686 y Fu(d')11425 32127 y Ft(\024)12705 30477 y Fq(Z)13378 33226 y Fp([0)p Fs(;)p Fp(2)p Fs(\031)32 b Fp(])p Fr(n)p Fs(\002)16857 33349 y FD(1)17321 33226 y Fp(\()p Fs(\025)p Fp(\))18859 32127 y Fu(k)240 b(d')269 b Fw(+)22635 30477 y Fq(Z)23309 33226 y Fs(\002)24023 33349 y FD(1)24484 33226 y Fp(\()p Fs(\025)p Fp(\))25820 31145 y Fq(\000)26376 32127 y Fu(h)27074 32309 y Fp(1)27600 32127 y Fw(\()p Fu(')p Fw(\))h(+)f Fu(h)31515 31627 y Fr(0)31515 32427 y Fp(1)32041 32127 y Fw(\()p Fu(')p Fw(\))33776 31145 y Fq(\001)11425 35083 y Fu(<)337 b(ck)307 b Ft(\001)269 b Fu(k)15443 34582 y Fr(\000)p Fs(\016)33 b(=)p Fp(2)17890 35083 y Fw(+)269 b Fu(O)34 b Fw(\()p Fu(k)21201 34582 y Fr(\000)p Fp(1)p Fr(\000)p Fp(4)p Fs(s)24040 34705 y FD(1)24502 34582 y Fr(\000)p Fp(2)p Fs(\015)26189 34705 y FD(0)26650 34582 y Fr(\000)p Fs(\016)27887 35083 y Fw(\))270 b(+)f Fu(O)34 b Fw(\()p Fu(k)31939 34582 y Fr(\000)p Fp(2)p Fs(\015)33626 34705 y FD(0)34087 34582 y Fp(+1)35344 35083 y Fw(\))337 b(=)g Fu(k)307 b Ft(\001)269 b Fu(O)34 b Fw(\()p Fu(k)41075 34582 y Fr(\000)p Fs(\016)f(=)p Fp(2)43253 35083 y Fw(\))p Fu(;)4511 37356 y Fw(here)423 b(and)h(b)34 b(elo)-34 b(w,)428 b Fo(D)14338 37538 y Fp(0)14863 37356 y Fw(\()p Fu(\025)p Fw(\),)g(corresp)34 b(onding)423 b(to)h(the)f(free)g(op)34 b(erator,)427 b(is)c(the)g(p)34 b(erfect)423 b(circle)e(of)j(radius) 4511 38874 y Fu(\025)5218 38434 y Fp(1)p Fs(=)p Fp(2)p Fs(l)6977 38874 y Fw(.)538 b(Hence,)403 b(\(2.45\))i(is)f(v)-67 b(alid.)p 50750 40479 665 831 v 1803 42763 a(Next,)471 b(w)-34 b(e)458 b(de\014ne)g(a)f(nonresonance)i(subset)f Fu(\037)23693 42323 y Fr(\003)23693 43088 y Fp(1)24219 42763 y Fw(\()p Fu(\025)p Fw(\))h(of)f(iso)34 b(energetic)456 b(set)i Fu(S)37079 42945 y Fp(1)37605 42763 y Fw(\()p Fu(\025)p Fw(\))g(as)g(the)g(parallel)e(shift)j(of)475 44202 y Fo(D)1482 44384 y Fp(1)2007 44202 y Fw(\()p Fu(\025)p Fw(\))405 b(in)-34 b(to)405 b Fu(K)7548 44384 y Fp(1)8478 44202 y Fw(\(Fig.)539 b(5\):)21035 45904 y Fu(\037)21793 45403 y Fr(\003)21793 46203 y Fp(1)22320 45904 y Fw(\()p Fu(\025)p Fw(\))337 b(:=)f Fo(K)26919 46086 y Fp(1)27446 45904 y Fo(D)28453 46086 y Fp(1)28978 45904 y Fw(\()p Fu(\025)p Fw(\))p Fu(:)17464 b Fw(\(2.48\))p 0 TeXcolorgray 475 48242 a FF(Lemma)571 b(2.12.)p 0 TeXcolorgray 604 w FC(The)519 b(set)g Fu(\037)14292 47802 y Fr(\003)14292 48567 y Fp(1)14818 48242 y Fw(\()p Fu(\025)p Fw(\))h FC(b)-62 b(elongs)517 b(to)i(the)f Fw(\()p Fu(ck)26529 47802 y 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Fu(W)12905 39712 y Fp(1)13431 39530 y Fw(.)586 b(Ho)-34 b(w)g(ev)g(er,)424 b(the)c(family)g(of)g(op)34 b(erators)420 b Ft(f)p Fu(H)33910 39091 y Fp(\(1\))35167 39530 y Fw(\()p Fu(t)p Fw(\))p Ft(g)37153 39718 y Fs(t)p Fr(2)p Fs(K)38913 39841 y FD(1)39851 39530 y Fw(do)34 b(es)420 b(c)-34 b(hange,)424 b(when)d(w)-34 b(e)475 41056 y(replace)301 b(the)h(p)34 b(erio)g(ds)302 b Fu(a)11261 41238 y Fp(1)11786 41056 y Fu(;)202 b(a)12966 41238 y Fp(2)13794 41056 y Fw(b)-34 b(y)302 b Fu(N)16349 41238 y Fp(1)16875 41056 y Fu(a)17516 41238 y Fp(1)18041 41056 y Fu(;)202 b(N)19554 41238 y Fp(1)20080 41056 y Fu(a)20721 41238 y Fp(2)21246 41056 y Fw(.)505 b(The)302 b(family)g(of)g(op)34 b(erators)302 b Ft(f)p Fu(H)36334 40616 y Fp(\(1\))37592 41056 y Fw(\()p Fu(t)p Fw(\))p Ft(g)39578 41244 y Fs(t)p Fr(2)p Fs(K)41338 41367 y FD(1)42158 41056 y Fw(has)h(to)f(b)34 b(e)302 b(replaced)475 42593 y(b)-34 b(y)459 b(a)h(family)e(of)i(op)34 b(erators)459 b Ft(f)14819 42287 y Fw(~)14501 42593 y Fu(H)15607 42153 y Fp(\(1\))16865 42593 y Fw(\()p Fu(\034)137 b Fw(\))p Ft(g)19080 42781 y Fs(\034)102 b Fr(2)p Fs(K)21020 42904 y FD(2)21997 42593 y Fw(acting)460 b(in)f Fu(L)27983 42775 y Fp(2)28509 42593 y Fw(\()p Fu(Q)29938 42775 y Fp(2)30464 42593 y Fw(\),)473 b(where)459 b Fu(Q)36263 42775 y Fp(2)37218 42593 y Fw(=)428 b([0)p Fu(;)202 b(N)41045 42775 y Fp(1)41571 42593 y Fu(a)42212 42775 y Fp(1)42737 42593 y Fw(])306 b Ft(\002)g Fw([0)p Fu(;)202 b(N)47085 42775 y Fp(1)47610 42593 y Fu(a)48251 42775 y Fp(2)48777 42593 y Fw(])458 b(and)475 44317 y Fu(K)1504 44499 y Fp(2)2480 44317 y Fw(=)449 b([0)p Fu(;)202 b Fw(2)p Fu(\031)43 b(=)-67 b(N)8207 44499 y Fp(1)8733 44317 y Fu(a)9374 44499 y Fp(1)9899 44317 y Fw(\))315 b Ft(\002)g Fw([0)p Fu(;)202 b Fw(2)p Fu(\031)43 b(=)-67 b(N)16278 44499 y Fp(1)16803 44317 y Fu(a)17444 44499 y Fp(2)17970 44317 y Fw(\).)742 b(W)-101 b(e)471 b(denote)i(eigen)-34 b(v)-67 b(alues)471 b(of)33752 44011 y(~)33435 44317 y Fu(H)34541 43878 y Fp(\(1\))35798 44317 y Fw(\()p Fu(\034)137 b Fw(\))473 b(b)-34 b(y)39682 43998 y(~)39632 44317 y Fu(\025)40339 43679 y Fp(\(1\))40339 44475 y Fs(n)41596 44317 y Fw(\()p Fu(\034)137 b Fw(\),)490 b Fu(n)449 b Ft(2)h Fn(N)472 b Fw(and)h(its)475 45847 y(sp)34 b(ectrum)549 b(b)-34 b(y)7863 45541 y(~)7745 45847 y(\003)8587 45407 y Fp(\(1\))9845 45847 y Fw(\()p Fu(\034)137 b Fw(\).)975 b(The)550 b(next)f(lemma)h (establishes)f(a)h(connection)g(b)34 b(et)-34 b(w)g(een)550 b(sp)34 b(ectra)549 b(of)h(op)34 b(erators)475 47377 y Fu(H)1581 46937 y Fp(\(1\))2838 47377 y Fw(\()p Fu(t)p Fw(\))405 b(and)7297 47070 y(~)6980 47377 y Fu(H)8086 46937 y Fp(\(1\))9344 47377 y Fw(\()p Fu(\034)137 b Fw(\).)539 b(It)404 b(easily)f(follo)-34 b(ws)405 b(from)f(Blo)34 b(c)-34 b(h)404 b(theory)g(\(see)g(e.g.)538 b([21)o(]\).)p 0 TeXcolorgray 475 49641 a FF(Lemma)463 b(3.1.)p 0 TeXcolorgray 554 w FC(F)-93 b(or)434 b(any)f Fu(\034)474 b Ft(2)337 b Fu(K)15903 49823 y Fp(2)16429 49641 y FC(,)20267 51139 y Fw(~)20149 51445 y(\003)20991 50945 y Fp(\(1\))22249 51445 y Fw(\()p Fu(\034)137 b Fw(\))337 b(=)25716 50294 y Fq([)25475 52919 y Fs(p)p Fr(2)p Fs(P)27505 51445 y Fw(\003)28347 50945 y Fp(\(1\))29604 51445 y Fw(\()p Fu(t)30513 51627 y Fs(p)31042 51445 y Fw(\))p Fu(;)17184 b Fw(\(3.3\))475 54602 y FC(wher)-62 b(e)10163 56226 y Fu(P)505 b Fw(=)337 b Ft(f)p Fu(p)f Fw(=)h(\()p Fu(p)16639 56408 y Fp(1)17165 56226 y Fu(;)202 b(p)18314 56408 y Fp(2)18839 56226 y Fw(\))337 b Ft(2)g Fn(Z)21600 55726 y Fp(2)22463 56226 y Fw(:)f(0)h Ft(\024)f Fu(p)25968 56408 y Fp(1)26831 56226 y Ft(\024)g Fu(N)29084 56408 y Fp(1)29879 56226 y Ft(\000)269 b Fw(1)p Fu(;)636 b Fw(0)336 b Ft(\024)h Fu(p)35502 56408 y Fp(2)36364 56226 y Ft(\024)g Fu(N)38618 56408 y Fp(1)39413 56226 y Ft(\000)269 b Fw(1)p Ft(g)7197 b Fw(\(3.4\))475 58102 y FC(and)433 b Fu(t)3267 58284 y Fs(p)4132 58102 y Fw(=)337 b(\()p Fu(t)6321 58284 y Fs(p;)p Fp(1)7582 58102 y Fu(;)202 b(t)8559 58284 y Fs(p;)p Fp(2)9819 58102 y Fw(\))337 b(=)f(\()p Fu(\034)12907 58284 y Fp(1)13703 58102 y Fw(+)269 b(2)p Fu(\031)43 b(p)16865 58284 y Fp(1)17391 58102 y Fu(=)-67 b(N)18904 58284 y Fp(1)19430 58102 y Fu(a)20071 58284 y Fp(1)20596 58102 y Fu(;)202 b(\034)21665 58284 y Fp(2)22460 58102 y Fw(+)269 b(2)p Fu(\031)43 b(p)25622 58284 y Fp(2)26148 58102 y Fu(=)-67 b(N)27661 58284 y Fp(1)28187 58102 y Fu(a)28828 58284 y Fp(2)29353 58102 y Fw(\))337 b Ft(2)g Fu(K)32335 58284 y Fp(1)32861 58102 y FC(,)434 b(se)-62 b(e)432 b(Fig.)557 b(6.)1803 60496 y Fw(W)-101 b(e)459 b(de\014ned)g(iso)34 b(energetic)458 b(set)h Fu(S)17496 60678 y Fp(1)18022 60496 y Fw(\()p Fu(\025)p Fw(\))429 b Ft(\032)f Fu(K)22500 60678 y Fp(1)23485 60496 y Fw(of)460 b Fu(H)26027 60056 y Fp(\(1\))27743 60496 y Fw(b)-34 b(y)460 b(form)-34 b(ula)460 b(\(2.31\).)703 b(Ob)-34 b(viously)-101 b(,)473 b(this)459 b(de\014nition)475 62014 y(is)541 b(directly)g(asso)34 b(ciated)541 b(with)h(the)g(family) g(of)g(op)34 b(erators)541 b Fu(H)29384 61574 y Fp(\(1\))30642 62014 y Fw(\()p Fu(t)p Fw(\))h(and,)576 b(therefore,)f(with)543 b(p)34 b(erio)g(ds)541 b Fu(a)48958 62196 y Fp(1)49483 62014 y Fu(;)202 b(a)50663 62196 y Fp(2)51189 62014 y Fw(,)475 63453 y(whic)-34 b(h)347 b(w)-34 b(e)347 b(assigned)g(to)g Fu(W)12918 63635 y Fp(1)13443 63453 y Fw(\()p Fu(x)p Fw(\).)520 b(No)-34 b(w,)359 b(assuming)347 b(that)h(the)f(p)34 b(erio)g(ds)345 b(are)h(equal)g(to)h Fu(N)40493 63635 y Fp(1)41019 63453 y Fu(a)41660 63635 y Fp(1)42185 63453 y Fu(;)202 b(N)43698 63635 y Fp(1)44224 63453 y Fu(a)44865 63635 y Fp(2)45390 63453 y Fw(,)358 b(w)-34 b(e)347 b(giv)-34 b(e)346 b(an)475 65170 y(analogous)416 b(de\014nition)g(of)g(the)g(iso) 34 b(energetic)414 b(set)23666 64863 y(~)23462 65170 y Fu(S)24205 65352 y Fp(1)24731 65170 y Fw(\()p Fu(\025)p Fw(\))i(in)f Fu(K)29250 65352 y Fp(2)29776 65170 y Fw(:)30879 64863 y(~)30674 65170 y Fu(S)31417 65352 y Fp(1)31943 65170 y Fw(\()p Fu(\025)p Fw(\))357 b(:=)e Ft(f)p Fu(\034)492 b Ft(2)356 b Fu(K)39405 65352 y Fp(2)40287 65170 y Fw(:)f Ft(9)p Fu(n)g Ft(2)h Fn(N)g Fw(:)46703 64850 y(~)46652 65170 y Fu(\025)47359 64531 y Fp(\(1\))47359 65327 y Fs(n)48617 65170 y Fw(\()p Fu(\034)137 b Fw(\))356 b(=)475 66699 y Fu(\025)p Ft(g)p Fu(:)404 b Fw(By)f(Lemma)h(3.1,)11252 66393 y(~)11047 66699 y Fu(S)11790 66881 y Fp(1)12316 66699 y Fw(\()p Fu(\025)p Fw(\))h(can)f(b)34 b(e)404 b(expressed)g(as)g(follo)-34 b(ws:)9939 68889 y(~)9734 69196 y Fu(S)10477 69378 y Fp(1)11003 69196 y Fw(\()p Fu(\025)p Fw(\))338 b(=)14269 67850 y Fq(n)15077 69196 y Fu(\034)474 b Ft(2)337 b Fu(K)18255 69378 y Fp(2)19118 69196 y Fw(:)f Ft(9)p Fu(n)h Ft(2)g Fn(N)p Fu(;)606 b(p)336 b Ft(2)h Fu(P)505 b Fw(:)741 b Fu(\025)29651 68695 y Fp(\(1\))29651 69495 y Fs(n)30909 68214 y Fq(\000)31464 69196 y Fu(\034)407 b Fw(+)269 b(2)p Fu(\031)43 b(p=)-67 b(N)37076 69378 y Fp(1)37602 69196 y Fu(a)38243 68214 y Fq(\001)39134 69196 y Fw(=)337 b Fu(\025)41121 67850 y Fq(o)41929 69196 y Fu(;)p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 17 17 TeXDict begin 17 16 bop 0 TeXcolorgray 0 TeXcolorgray 2848 3345 a Fp(SPECTRAL)406 b(PR)-26 b(OPER)-78 b(TIES)406 b(OF)h(A)h(POL)-78 b(YHARMONIC)405 b(OPERA)-78 b(TOR)406 b(WITH)h(LIMIT-PERIODIC)e(POTENTIAL)1430 b(17)p 0 TeXcolorgray 0 TeXcolorgray 17533 17824 a /PSfrag where{pop(2p/a1)[[0(Bl)1 0]](2p/a2)[[1(Bl)1 0]](2p/Na1)[[2(Bl)1 0]](2p/Na2)[[3(Bl)1 0]]4 0 -1/Begin PSfrag}{userdict /PSfrag{pop}put}ifelse 17533 17824 a @beginspecial 173 @llx 310 @lly 414 @urx 494 @ury 1163 @rhi @setspecial %%BeginDocument: fig1.eps %!PS-Adobe-3.0 EPSF-3.0 %%BoundingBox: 173 310 414 494 %%Title: fig1.eps %%Creator: Canvas %%CreationDate: Sunday, August 03, 2003 07:13:59 %%DocumentProcSets: Deneba_CanvasCustomColors %%+ Deneba_CanvasGraphics %%DocumentFonts: ArialMT %%DocumentSuppliedFonts: ArialMT %%DocumentProcessColors: Cyan Magenta Yellow Black %%Copyright: (C) 1996 Deneba Systems. All Rights Reserved %%EndComments %%BeginProlog %%BeginResource: procset Deneba_CanvasGraphics %%Title: Deneba standard graphics %%Version: 1.0 %%CreationDate: 10/1/94 %%Copyright: (C) 1994 Deneba Systems. All Rights Reserved userdict /Deneba_CanvasGraphics 300 dict put Deneba_CanvasGraphics begin /invalidcolortable? true def /level2 /languagelevel where {pop languagelevel 2 ge}{false} ifelse def /initialize {Deneba_CanvasGraphics {dup xcheck{bind}if pop pop}forall currentscreen /_sp exch def /_sa exch def /_sf exch def /DefaultScreen{_sf _sa /_sp load}def /DefaultScreenProc{ /_sp load}def /setstrokeadjust where{pop true}{false}ifelse { /sa /setstrokeadjust load def /li /lineto load def /mt /moveto load def /ct /curveto load def /lh { currentpoint exch pop lineto}bind def /lv { currentpoint pop exch lineto}bind def} { /sa { {/_xl {transform .25 sub round .25 add exch .25 sub round .25 add exch itransform }bind def /li{ _xl lineto}bind def /mt{ _xl moveto}bind def /ct{ _xl curveto}bind def /lh { dup _xl pop currentpoint exch pop lineto}bind def /lv { dup _xl pop currentpoint pop exch lineto}bind def /rp{_xl/cv_b xdf/cv_r xdf _xl/cv_t xdf/cv_l xdf cv_l cv_t moveto cv_r cv_t lineto cv_r cv_b lineto cv_l cv_b lineto closepath }def} {/li{lineto} def /mt{moveto} def /ct{curveto} def /lh{currentpoint exch pop lineto}def /lv{currentpoint pop exch lineto}def /rp {/cv_b xdf/cv_r xdf /cv_t xdf/cv_l xdf cv_l cv_t moveto cv_r cv_t lineto cv_r cv_b lineto cv_l cv_b lineto closepath}def}ifelse}def }ifelse }def /terminate { currentdict Deneba_CanvasGraphics eq{end}if}def /mxMax 8 def /mxStk [mxMax {matrix} repeat]def /mxIdx 0 def /stm{mxStk mxIdx get currentmatrix pop /mxIdx mxIdx 1 add store}def /rtm{/mxIdx mxIdx 1 sub store mxStk mxIdx get setmatrix}def /_cvmtx matrix def /_txtmtx matrix def /_initmtx matrix def /xdf{exch def}def /min{2 copy gt{exch}if pop}def /max{2 copy lt{exch}if pop}def /CVvec 256 array def /T true def /F false def /_id 0 def /_encodindex 0 def /_encodArr 10 array def /_fd 0 def /_textSave null def /_letterSpace 0 def /_spaceXtra 0 def /_PageSave null def /_penh 1.0 def /_penv 1.0 def /save0 null def /_w 0 def /_h 0 def /st0 null def /st1 null def /st2 null def /st3 null def /st4 null def /stb 256 string def /pw 0 def /wdif 0 def /temp 0 def /currot 0 def /savehandler null def /debug {dup /$eps exch not def{/savehandler errordict /handleerror get def errordict /handleerror{$error begin newerror{/newerror false def showpage 72 72 scale /Helvetica findfont .2 scalefont setfont .25 9.6 moveto (Error: )show errorname {dup type/stringtype ne {(converted error name will end up in this string)cvs}if show}exec .25 9.4 moveto (Offending Command: )show /command load {dup type/stringtype ne {(converted command name will end up in this string)cvs}if show}exec .25 9.2 moveto (Stack: )show ostack {.25 currentpoint .2 sub exch pop moveto dup type/stringtype ne{dup type/integertype eq 1 index type/realtype eq or 1 index type/nametype eq or not {type}if (converted stack will end up in this string)cvs}if show}forall showpage Deneba_CanvasGraphics /savehandler get exec}if end}put}if}def /lc{setlinecap}def /lj{setlinejoin}def /lm{setmiterlimit}def /lf{setflat}def /cp{closepath}def /np{newpath}def /$psz 6 array def /$pix 0 def /gs{gsave $psz dup $pix _penh put $pix 1 add _penv put /$pix $pix 2 add def}def /gr{grestore/$pix $pix 2 sub def $psz dup $pix get /_penh exch def $pix 1 add get /_penv exch def}def /li{lineto} def /lh{currentpoint exch pop lineto}def /lv{currentpoint pop exch lineto}def /mt{moveto} def /mh{currentpoint exch pop moveto}def /mv{currentpoint pop exch moveto}def /ct{curveto} def /rt{4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath}def /_hr { {/@err false def currentflat dup{gsave{eoclip}stopped{grestore 2 mul dup setflat dup 200 ge{(%% Canvas:Clip failed\r)print /@err true def pop exit}if}{grestore setflat exit}ifelse}loop @err not {eoclip}if setflat} {/@err false def currentflat dup{gsave{clip}stopped{grestore 2 mul dup setflat dup 200 ge{(%% Canvas:Clip failed\r)print /@err true def pop exit}if}{grestore setflat exit}ifelse}loop @err not {clip}if setflat} }def /_cs null def /fg{$eps{systemdict /setgray get exec}{setgray}ifelse}def /setcmykcolor where {pop true}{false}ifelse {/f4{ setcmykcolor}def}{/f4{1 sub 4 1 roll 3{3 index add neg dup 0 lt{pop 0}if 3 1 roll}repeat setrgbcolor pop}def}ifelse /setcolorspace where {pop true}{false}ifelse {/scs{ dup setcolorspace /_cs exch def}def/f3{ _cs setcolorspace setcolor}def }{/f3{ setrgbcolor}def/scs{ pop}def}ifelse /$cc 5 array def /findcmykcustomcolor where {pop true}{false}ifelse not {/findcmykcustomcolor{$cc astore}def}if /setcustomcolor where {pop true}{false}ifelse not{ /setcmykcolor where {pop true}{false}ifelse {/setcustomcolor{exch aload pop pop 4{4 index mul 4 1 roll}repeat setcmykcolor pop}def} {/setcustomcolor{exch aload pop pop 4{4 index mul 4 1 roll}repeat 1 sub 4 1 roll 3{3 index add neg dup 0 lt{pop 0}if 3 1 roll}repeat setrgbcolor pop pop}def}ifelse }if /fc{6 1 roll findcmykcustomcolor exch setcustomcolor}def /setseparationgray where {pop true}{false}ifelse not{/setseparationgray {systemdict /setgray get exec} def}if /sg{ setseparationgray}def /sp{/_penv xdf/_penh xdf}def /rp {/cv_b xdf/cv_r xdf /cv_t xdf/cv_l xdf cv_l cv_t moveto cv_r cv_t lineto cv_r cv_b lineto cv_l cv_b lineto closepath}def /rr{/yradius xdf /xradius xdf /cv_b xdf /cv_r xdf /cv_t xdf /cv_l xdf cv_l xradius add cv_t moveto cv_r xradius sub cv_t lineto _cvmtx currentmatrix pop currentpoint translate xradius yradius scale 1 0 1 1 1 arcto 4{pop}repeat _cvmtx setmatrix cv_r cv_b yradius sub lineto _cvmtx currentmatrix pop currentpoint translate xradius yradius scale 0 1 -1 1 1 arcto 4{pop}repeat _cvmtx setmatrix cv_l xradius add cv_b lineto _cvmtx currentmatrix pop currentpoint translate xradius yradius scale -1 0 -1 -1 1 arcto 4{pop}repeat _cvmtx setmatrix cv_l cv_t yradius add lineto _cvmtx currentmatrix pop currentpoint translate xradius yradius scale 0 -1 1 -1 1 arcto 4{pop}repeat _cvmtx setmatrix closepath}def /ep{_cvmtx currentmatrix pop translate scale 1 0 moveto 0 0 1 0 360 arc closepath _cvmtx setmatrix}def /wd{_cvmtx currentmatrix pop translate scale 0 0 moveto 0 0 1 5 -2 roll arcn closepath _cvmtx setmatrix}def /as{_cvmtx currentmatrix pop translate scale 0 0 1 5 -2 roll arcn _cvmtx setmatrix}def /ds {_penv _penh ne dup{stm _penh dup 0 le{pop 1}if _penv dup 0 le{pop 1}if scale}{_penh setlinewidth}ifelse stroke {rtm}if}def /df{fill}def /de{eofill}def /_gstk 5 array def /_gstkc 0 def /_gst{ 9 dict begin level2{/_ccs currentcolorspace def /_cc [currentcolor] def}{/_cc [currentrgbcolor]def}ifelse {currentpoint}stopped {/_cp null def}{ 2 array astore /_cp exch def}ifelse /_cf currentfont def /_clw currentlinewidth def /_clc currentlinecap def /_clj currentlinejoin def /_cml currentmiterlimit def currentdict end}def /hs{newpath gsave stm}def /hc{/_hr load exch get exec newpath}def /he{_gstk _gstkc _gst put /_gstkc _gstkc 1 add store grestore rtm /_gstkc _gstkc 1 sub store _gstk _gstkc get level2{dup /_ccs get setcolorspace dup /_cc get aload pop setcolor}{dup /_cc get aload pop setrgbcolor}ifelse dup /_cp get dup null eq not{aload pop moveto}{pop}ifelse dup /_cf get setfont dup/_clw get setlinewidth dup/_clc get setlinecap dup/_clj get setlinejoin dup/_cml get setmiterlimit pop}def /id{ /_id xdf}def /_imatrix{_w 0 0 _h _id 0 ne{neg 0 _h}{0 0}ifelse _cvmtx astore}def % Copyright (C) 1990, 1991 by Adobe Systems Incorporated. % All rights reserved. level2 { save /dontloadlevel1 xdf }if /startnoload{{/noload save def} if}def /endnoload{{noload restore}if }def /testsystemdict{where{systemdict eq{true}{false}ifelse}{false}ifelse}def /ncolors 1 def /colorimage where{pop true}{false}ifelse{/ncolors 0 statusdict begin /processcolors where{pop pop processcolors}{/deviceinfo where{pop deviceinfo /Colors known {pop {deviceinfo /Colors get}}if}if}ifelse end def ncolors 0 ne{/colorimage testsystemdict /setcolortransfer testsystemdict /currentcolortransfer testsystemdict /currentcmykcolor testsystemdict and and and not{/ncolors 0 def}if}if}if ncolors dup 1 ne exch dup 3 ne exch 4 ne and and {/ncolors 0 def}if ncolors 1 eq dup dup not startnoload {/doclutimage{[{255 mul round cvi bwclut exch get 255 div} /exec load dup currenttransfer exch] cvx bind settransfer _w _h 8 _imatrix {currentfile st0 readhexstring pop}image}def}if not endnoload ncolors dup 3 eq exch 4 eq or dup dup not startnoload {/nullproc{{}}def/concatutil{/exec load 7 -1 roll /exec load}def /defsubclut {1 add getinterval def}def /spconcattransfer {/Dclut exch def/Cclut exch def/Bclut exch def/Aclut exch def/ncompute exch ld currentcolortransfer [{Aclut ncompute}concatutil] cvx [{Bclut ncompute}concatutil] cvx [{Cclut ncompute}concatutil] cvx [{Dclut ncompute}concatutil] cvx setcolortransfer}def /setuprgbcluts {/bit3x rgbclut length 3 sub def /bit1x bit3x 3 idiv def/rclut rgbclut def /gclut rclut 1 bit3x defsubclut/bclut rclut 2 bit3x defsubclut}def}if not endnoload ncolors 3 eq dup dup not startnoload {/3compute{exch bit3x mul round cvi get 255 div}def /doclutimage{%/rgbclut xdf pop setuprgbcluts/3compute rclut gclut bclut dup spconcattransfer _w _h 8 _imatrix [{currentfile st0 readhexstring pop}/exec load /dup load dup]cvx nullproc nullproc true 3 colorimage}def}if not endnoload ncolors 4 eq dup dup not startnoload {/stuffclut{cmykindex 3 -1 roll put}def /ftoint{1 exch sub 255 mul round cvi}def /4compute { exch bit4x mul round cvi get 255 div } def /computecmykclut{setuprgbcluts/bit4x rgbclut length 3 idiv 4 mul 4 sub def /cmykclut bit4x 4 add string def /cclut cmykclut def /mclut cclut 1 bit4x defsubclut /yclut cclut 2 bit4x defsubclut /kclut cclut 3 bit4x defsubclut /cmykindex 0 def 0 1 bit1x {dup /cmykindex exch bit1x exch sub 4 mul def 3 mul dup rclut exch get 255 div exch dup gclut exch get 255 div exch bclut exch get 255 div setrgbcolor currentcmykcolor ftoint kclut stuffclut ftoint yclut stuffclut ftoint mclut stuffclut ftoint cclut stuffclut }for}def /doclutimage{invalidcolortable?{computecmykclut}if /4compute cclut mclut yclut kclut spconcattransfer _w _h 8 _imatrix[{currentfile st0 readhexstring pop}/exec load/dup load dup dup] cvx nullproc nullproc nullproc true 4 colorimage}def}if not endnoload ncolors 0 eq dup dup not startnoload {/colorexpand{mystringexp 0 rgbclut 3 copy 7 -1 roll {3 mul 3 getinterval putinterval exch 3 add exch 3 copy}forall pop pop pop pop pop}def /createexpandstr{/mystringexp exch st0 length mul string def}def /doclutimage{/rgbclut xdf pop 3 createexpandstr _w _h 8 _imatrix [{currentfile st0 readhexstring pop} /exec load /colorexpand load /exec load] cvx false 3 colorimage}def}if not endnoload level2 { dontloadlevel1 restore }if level2 not{ save /dontloadlevel2 xdf }if /rgbclut 0 def/doclutimage{/hival 255 def [/Indexed /DeviceRGB hival rgbclut] setcolorspace _imagedict dup begin /Width _w def /Height _h def /Decode [0 hival] def /ImageMatrix _imatrix def /DataSource{currentfile st0 readhexstring pop}def /BitsPerComponent 8 def /Interpolate false def end image}def level2 not{ dontloadlevel2 restore }if %%EndProcSet /_imagedict 12 dict begin /ImageType 1 def/BitsPerComponent 8 def/Interpolate false def currentdict end def /$rhs0 {currentfile st0 readhexstring pop}def /$rhs1 {currentfile st1 readhexstring pop}def /$rhs2 {currentfile st2 readhexstring pop}def /$rhs3 {currentfile st3 readhexstring pop}def /$rhs4 {currentfile st4 readhexstring pop}def /$rs0 {currentfile st0 readstring pop}def /$rs1 {currentfile st1 readstring pop}def /$rs2 {currentfile st2 readstring pop}def /$rs3 {currentfile st3 readstring pop}def /$rs4 {currentfile st4 readstring pop}def /hascolor /deviceinfo where {pop deviceinfo /Colors known {deviceinfo /Colors get exec 1 gt} {false} ifelse} {/statusdict where {pop statusdict /processcolors known {statusdict /processcolors get exec 1 gt} {false} ifelse}{false} ifelse}ifelse def /level2 systemdict /languagelevel known {languagelevel 2 ge}{false}ifelse def /foureq {4 index eq 8 1 roll 4 index eq 8 1 roll 4 index eq 8 1 roll 4 index eq 8 1 roll pop pop pop pop and and and} def /setupbands{gsave /_settransfer systemdict /settransfer get def /_currenttransfer systemdict /currenttransfer get def /_image systemdict /image get def /blank 0 _currenttransfer exec 1 _currenttransfer exec eq def /negative blank{0 _currenttransfer exec 0.5 lt} {0 _currenttransfer exec 1 _currenttransfer exec gt}ifelse def /inverted? negative def /_currentgray systemdict /currentgray get def hascolor{/$b 0 def} {/$b 5 def} ifelse /setcmykcolor where{pop 1 0 0 0 setcmykcolor _currentgray 1 exch sub 0 1 0 0 setcmykcolor _currentgray 1 exch sub 0 0 1 0 setcmykcolor _currentgray 1 exch sub 0 0 0 1 setcmykcolor _currentgray 1 exch sub 4 {4 copy} repeat 1 0 0 0 foureq{/$b 1 store}if 0 1 0 0 foureq{/$b 2 store}if 0 0 1 0 foureq{/$b 3 store}if 0 0 0 1 foureq{/$b 4 store}if 0 0 0 0 foureq{/$b 6 store}if}if blank{/$b 6 store}if $b 0 eq $b 5 eq or{/$dt 0 def}if/$dt -1 def grestore}def /doimage{exch 1 index 0 eq{dup dup/Width get exch /Decode get length 2 div mul cvi}{dup/Width get}ifelse /st0 exch string def begin /MultipleDataSources false def dup 0 eq{pop/DataSource /$rhs0 load def} {1 eq{/DataSource /$rs0 load def}{/DataSource currentfile /ASCII85Decode filter def}ifelse}ifelse currentdict end %level2{image} true {begin Width Height BitsPerComponent ImageMatrix Decode length 2 eq {/DataSource load end image}{ Decode length 6 eq {MultipleDataSources{DataSource 0 get DataSource 1 get DataSource 2 get}{/DataSource load}ifelse MultipleDataSources 3 end DSIColorImage}{ Decode length 8 eq{MultipleDataSources{DataSource 0 get DataSource 1 get DataSource 2 get DataSource 3 get}{/DataSource load}ifelse MultipleDataSources 4 end DSIColorImage}if}ifelse }ifelse }if}def /dt{/rgbclut xdf /bwclut xdf}def /cl{pop gsave $i setmatrix translate scale cvi/_h xdf cvi/_w xdf/st0 _w string def pop doclutimage grestore}def /c3{/imagesave save def /$src xdf $i setmatrix translate scale cvi/_h xdf cvi/_w xdf level2{_cs setcolorspace 0 0 0 setrgbcolor}if _imagedict begin /Width _w def /Height _h def /ImageMatrix _imatrix def /Decode [0 1 0 1 0 1]def currentdict end exch doimage imagesave restore}def /ca{/imagesave save def /$src xdf $i setmatrix translate scale cvi/_h xdf cvi/_w xdf level2{_cs setcolorspace}if _imagedict begin /Decode[0 100 -128 127 -128 127]def /Width _w def /Height _h def /ImageMatrix _imatrix def currentdict end exch doimage imagesave restore}def /c4{/imagesave save def /$src xdf $i setmatrix translate scale cvi/_h xdf cvi/_w xdf level2 {/DeviceCMYK setcolorspace 0 0 0 0 setcmykcolor}if _imagedict begin /Decode [0 1 0 1 0 1 0 1] def /Width _w def /Height _h def /ImageMatrix _imatrix def currentdict end exch doimage imagesave restore}def /im{/imagesave save def pop $i setmatrix translate scale cvi/_h xdf cvi/_w xdf level2{/DeviceGray setcolorspace 0 setgray}if _imagedict begin /Decode [0 1] def /Width _w def /Height _h def /ImageMatrix _imatrix def currentdict end exch doimage imagesave restore}def /colorimage where{pop/DSIColorImage {colorimage} def}{/DSIColorImage{ /$nc xdf pop /proc0 xdf 3 index 2 index mul 7 add 8 div cvi string /st1 xdf $nc 3 eq{{proc0 dup/st0 xdf 0 exch 3 exch length 3 sub {dup st0 exch get .3 mul 1 index 1 add st0 exch get .59 mul add 1 index 2 add st0 exch get .11 mul add cvi exch 3 div cvi exch st1 3 1 roll put}for st1}}{{proc0 dup/st0 xdf 0 exch 4 exch length 4 sub{dup st0 exch get .3 mul 1 index 1 add st0 exch get .59 mul add 1 index 2 add st0 exch get .11 mul add 1 index 3 add st0 exch get add dup 255.0 gt{pop 255.0}if 255.0 exch sub cvi exch 4 div cvi exch st1 3 1 roll put}for st1}}ifelse image}def}ifelse /OPIimage{image}def /$i matrix def /bi{scale translate $i currentmatrix pop translate scale setupbands}def /ei{$i setmatrix}def /ik{/imagesave save def $i setmatrix translate scale cvi/_h xdf cvi/_w xdf /st0 _w 7 add 8 div cvi string def _w _h false _imatrix 5 -1 roll 0 eq {{$rhs0}}{{$rs0}}ifelse imagemask imagesave restore}def /rs{2 copy translate 3 -1 roll neg rotate neg exch neg exch translate }def /cs{_cvmtx astore concat} def /fd{/_fd xdf}def /_fontMatrix{1 0 0 _fd 0 ne{-1 0 0}{1 0 0}ifelse _cvmtx astore}def /$cvfontdict 150 dict begin /Symbol {/Symbol /Symbol /Symbol -1} def currentdict end def /rfp{/_encodindex xdf FontDirectory 3 index known not{ dup FontDirectory exch known{exch pop}{pop}ifelse findfont dup length 1 add dict begin{1 index/FID ne{def}{pop pop}ifelse}forall dup /FontName exch def FontType 0 ne{/Encoding Encoding 256 array copy def _encodindex 0 ge{0 _encodArr _encodindex get{dup type/nametype eq{Encoding 2 index 2 index put pop 1 add} {exch pop}ifelse}forall pop}if}if dup currentdict end definefont pop}{pop pop}ifelse} def /rf{3 index 6 1 roll counttomark array astore cvx exch pop $cvfontdict 3 1 roll put}def /cvsf{$cvfontdict exch get exec rfp findfont ftSize 0 0 ftSize _fd 0 ne{-1 mul}if 0 0 _cvmtx astore makefont setfont}def /mf{3 -1 roll $cvfontdict exch get exec rfp findfont 3 1 roll exch ftSize mul exch ftSize mul 0 exch 0 exch _fd 0 ne{-1 mul}if 0 0 _cvmtx astore makefont setfont}def /wrk1 ( ) def /wdict 16 dict def /Work75 75 string def /Nmk{Work75 cvs dup}def /Npt{put cvn}def /dhOdh{Nmk 2 79 Npt}def /dhBdh{Nmk 2 111 Npt}def /dhSdh{Nmk 2 83 Npt}def /sfWidth{gsave 0 0 moveto 0 0 lineto 0 0 lineto 0 0 lineto closepath clip stringwidth grestore}def /fts{/ftSize exch def}def /mkFT{dup/FontType get 0 eq{dup maxlength dict begin {1 index/FID ne 2 index /UniqueID ne and{def}{pop pop}ifelse}forall currentdict end dup/FDepVector 2 copy get [exch 6 index exch 6 index exch{mkFT 3 1 roll}forall pop pop] put dup}{/tempFT 12 dict def tempFT begin /FontMatrix [1 0 0 1 0 0] def /FontType 3 def dup/Encoding get/Encoding exch def /proc2 2 index def dup/ft1 exch def /ft2 1 array def /FontBBox [0 0 1 1] def /BuildChar{Deneba_CanvasGraphics begin exch dup/ftdt xdf begin wdict begin /chr xdf/chrst wrk1 dup 0 chr put def ft2 0 get null eq{ft1 dup length 2 add dict begin {1 index/FID ne 2 index /UniqueID ne and{def}{pop pop}ifelse}forall /PaintType 2 def/StrokeWidth .024 0 FontMatrix idtransform pop def currentdict end /q exch definefont ft2 exch 0 exch put}if ftdt/proc2 get exec end end end}def end tempFT}ifelse 3 index exch definefont exch pop}def /mb{dup dhBdh FontDirectory 1 index known 1 index $cvfontdict exch known and {exch pop}{dup 2 index $cvfontdict exch get exec rfp findfont{boldproc}exch mkFT pop pop pop dup 3 -1 roll mark 3 1 roll dup -1 counttomark array astore cvx exch pop $cvfontdict exch 2 index exch put}ifelse}def /mo{dup dhOdh FontDirectory 1 index known 1 index $cvfontdict exch known and {exch pop}{dup 2 index $cvfontdict exch get exec rfp findfont{outproc} exch mkFT pop pop pop dup 3 -1 roll mark 3 1 roll dup -1 counttomark array astore cvx exch pop $cvfontdict exch 2 index exch put}ifelse}def /ms{ dup dhSdh FontDirectory 1 index known 1 index $cvfontdict exch known and {exch pop}{dup 2 index $cvfontdict exch get exec rfp findfont{ShadChar} exch mkFT pop pop pop dup 3 -1 roll mark 3 1 roll dup -1 counttomark array astore cvx exch pop $cvfontdict exch 2 index exch put}ifelse}def /outproc{ft1 setfont gsave chrst sfWidth grestore setcharwidth dblsh}def /boldproc{ft1 setfont gsave chrst sfWidth 1 index 0 ne{exch .05 add exch}if grestore setcharwidth ft1 /PaintType get 2 ne {0 0 moveto gsave ft2 0 get setfont chrst show grestore chrst show} {0 0 moveto gsave chrst show grestore .05 0 moveto chrst show}ifelse}def /dblsh{currentgray 1.0 setgray chrst 0 0 moveto show setgray ft2 0 get setfont chrst 0 0 moveto show}def /ShadChar{ ft1 setfont gsave chrst sfWidth 1 index 0 ne{exch .05 add exch}if grestore setcharwidth chrst .06 0 moveto show 0 .05 translate dblsh}def /ts{show}def /ta{/_letterSpace xdf/_spaceXtra xdf}def /tf{/_textWidth xdf /_numSpaces xdf /_stringLength xdf dup stringwidth pop _letterSpace _stringLength 1 sub mul add _spaceXtra _numSpaces mul add _textWidth exch sub /wdif xdf _spaceXtra 0 32 wdif _stringLength div _letterSpace add 0 6 -1 roll awidthshow}def /pb{/_PageSave save def/showpage{}def}def /pe{_PageSave restore}def /pm{_pm currentmatrix pop}def /_pm matrix def /ps{showpage _pm setmatrix}def level2{/ef {gsave 6 array astore concat dup execform grestore}def} {/ef{gsave 6 array astore concat newpath dup begin Matrix concat BBox dup dup 0 get exch 1 get moveto dup dup 0 get exch 3 get lineto dup dup 2 get exch 3 get lineto dup 2 get exch 1 get lineto closepath clip newpath PaintProc end grestore}def}ifelse /DSIBeginPaintProc{/pp mark{currentfile token not{}if dup/DSIEndPaintProc eq{pop counttomark array astore cvx bind /exec load]cvx def DSIEndPaintProc exit} {counttomark 300 gt{counttomark array astore cvx bind /exec load}if}ifelse}loop}def /bc{save 11 1 roll 10 dict begin/BBox 5 1 roll 4 array astore def/Matrix 7 1 roll 6 array astore def /FormType 1 def/PaintProc{/pp get exec}def}def /ec{pop restore}def /printvm{1 vmreclaim vmstatus pop exch pop ( )cvs print (\n)print flush}def /DSIEndPaintProc{currentdict end}def /setoverprint where {pop}{/setoverprint{pop}def}ifelse /initfonts{level2{$cvfontdict dup { pop 1 index exch undef}forall pop}if $cvfontdict begin /Symbol {/Symbol /Symbol /Symbol -1}def end}def initialize end %%EndResource Deneba_CanvasGraphics begin _encodArr 0 [ 39/quotesingle 96/grave 128/bullet/bullet/quotesinglbase/florin/quotedblbase/ellipsis/dagger/daggerdbl/circumflex/perthousand/Scaron/guilsinglleft/OE/bullet/bullet/bullet /bullet/quoteleft/quoteright/quotedblleft/quotedblright/bullet/endash/emdash/tilde/trademark /scaron /guilsinglright/oe/bullet/bullet/Ydieresis /bullet/exclamdown/cent/sterling/currency/yen/brokenbar/section/dieresis/copyright/ordfeminine/guillemotleft/logicalnot/hyphen/registered/macron /degree/plusminus/twosuperior/threesuperior/acute/mu/paragraph/periodcentered/cedilla/onesuperior/ordmasculine/guillemotright/onequarter/onehalf/threequarters/questiondown /Agrave/Aacute/Acircumflex/Atilde/Adieresis /Aring 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/infinity/plusminus/lessequal/greaterequal/yen/mu/partialdiff/summation /product/pi/integral/ordfeminine/ordmasculine/Omega/ae/oslash /questiondown/exclamdown/logicalnot/radical/florin/approxequal/Delta/guillemotleft /guillemotright/ellipsis/blank/Agrave/Atilde/Otilde/OE/oe /endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide/lozenge /ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright/fi/fl /daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand/Acircumflex/Ecircumflex/Aacute /Edieresis/Egrave/Iacute/Icircumflex/Idieresis/Igrave/Oacute/Ocircumflex /apple/Ograve/Uacute/Ucircumflex/Ugrave/dotlessi/circumflex/tilde /macron/breve/dotaccent/ring/cedilla/hungarumlaut/ogonek/caron ] put mark /___FN1 /ArialMT dup -1 rf F debug 0 id 1 fd 0 804 translate 1 -1 scale %%EndSetup /DeviceRGB scs 0.0000 fg 0.1250 0.1250 sp 1 lc T sa %%DSIBeginObject: #1 - Ref:21 Group %%DSIBeginObject: #2 - Ref:2 Rectangle 1.0000 fg np 226.7716 325.9842 368.5039 467.7164 rp 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Fu(e)7582 20763 y Fr(\000)8447 20401 y FD(1)p 8447 20562 406 40 v 8447 21110 a(4)8986 20763 y Fs(k)9500 20451 y Fy(\021)24 b(s)10340 20650 y FD(1)10911 21241 y Fw(.)538 b(W)-101 b(e)404 b(de\014ne)g Fu(\037)18133 21423 y Fp(2)18660 21241 y Fw(\()p Fu(\025)p Fw(\))h(b)-34 b(y)404 b(the)h(form)-34 b(ula:)18674 23209 y Fu(\037)19432 23391 y Fp(2)19959 23209 y Fw(\()p Fu(\025)p Fw(\))337 b(=)g Fo(K)24222 23391 y Fp(2)24748 22227 y Fq(\000)25303 23209 y Fu(\037)26061 22708 y Fr(\003)26061 23509 y Fp(1)26588 23209 y Fw(\()p Fu(\025)p Fw(\))28237 22227 y Fq(\001)29062 23209 y Ft(n)270 b Fw(\012)30813 23391 y Fp(1)31339 23209 y Fw(\()p Fu(\025)p Fw(\))p Fu(:)15709 b Fw(\(3.7\))475 25408 y(3.2.)605 b FF(P)-39 b(erturbation)686 b(F)-116 b(orm)-39 b(ulae.)606 b Fw(Before)595 b(pro)-34 b(ving)596 b(the)g(main)f(result,)643 b(w)-34 b(e)596 b(form)-34 b(ulate)596 b(the)g(Geometric)475 26847 y(Lemma:)p 0 TeXcolorgray 475 29017 a FF(Lemma)521 b(3.5)i Fw(\(Geometric)454 b(Lemma\))p FF(.)p 0 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52310 a(\(2\))p 0 TeXcolorgray 554 w FC(F)-93 b(or)414 b(any)f Fu(\034)551 b FC(in)413 b(the)g Fw(\()p Fu(\017)14561 52492 y Fp(1)15087 52310 y Fu(k)15756 51870 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fs(\016)19689 52310 y Fw(\))p FC(-neighb)-62 b(orho)g(o)g(d)410 b(in)j Fn(C)30117 51870 y Fp(2)31057 52310 y FC(of)g Fu(\037)33220 52492 y Fp(2)33746 52310 y FC(,)418 b(ther)-62 b(e)412 b(exists)g(a)h(unique) f Fu(p)336 b Ft(2)h Fu(P)582 b FC(such)4511 53778 y(that)433 b Fu(\034)406 b Fw(+)269 b(2)p Fu(\031)43 b(p=)-67 b(N)12601 53960 y Fp(1)13127 53778 y Fu(a)433 b FC(is)g(in)h(the)f Fw(\()p Fu(\017)19966 53960 y Fp(1)20492 53778 y Fu(k)21161 53339 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fs(\016)25094 53778 y Fw(\))p FC(-neighb)-62 b(orho)g(o)g(d)430 b(in)k Fn(C)35563 53339 y Fp(2)36522 53778 y FC(of)f Fu(\037)38705 53339 y Fr(\003)38705 54103 y Fp(1)39665 53778 y FC(and)16898 56041 y Ft(j)p Fu(\025)17942 55402 y Fp(\(1\))17942 56403 y Fs(j)19200 56041 y Fw(\()p Fu(\034)407 b Fw(+)269 b(2)p Fu(\031)43 b(p=)-67 b(N)25283 56223 y Fp(1)25808 56041 y Fu(a)p Fw(\))270 b Ft(\000)f Fu(k)29071 55540 y Fp(2)p Fs(l)29888 56041 y Ft(j)337 b Fu(<)f(\017)32333 56223 y Fp(1)32859 56041 y Fu(k)33528 55540 y Fr(\000)p Fs(\016)34765 56041 y Fu(;)13932 b Fw(\(3.9\))4511 58119 y Fu(j)479 b FC(b)-62 b(eing)408 b(uniquely)f(de\014ne)-62 b(d)409 b(by)f Fu(\034)353 b Fw(+)216 b(2)p Fu(\031)43 b(p=)-67 b(N)24464 58301 y Fp(1)24990 58119 y Fu(a)409 b FC(as)f(it)h(is)f(describ)-62 b(e)g(d)406 b(in)k(Part)e(2)h(of)f(Ge) -62 b(ometric)407 b(L)-62 b(emma)4511 59558 y(for)433 b(the)g(pr)-62 b(evious)431 b(step.)p 0 TeXcolorgray 2409 60997 a Fw(\(3\))p 0 TeXcolorgray 554 w FC(The)362 b(se)-62 b(c)g(ond)361 b(non-r)-62 b(esonanc)g(e)361 b(set)h Fu(\037)21201 61179 y Fp(2)22089 60997 y FC(has)g(an)g (asymptotic)-62 b(al)62 b(ly)358 b(ful)62 b(l)361 b(me)-62 b(asur)g(e)361 b(in)h Fu(\037)42570 60557 y Fr(\003)42570 61322 y Fp(1)43459 60997 y FC(in)h(the)e(fol)62 b(lowing)4511 62436 y(sense:)19273 63476 y Fu(L)p Fw(\()p Fo(K)21566 63658 y Fp(2)22093 63476 y Fu(\037)22851 63036 y Fr(\003)22851 63801 y Fp(1)23646 63476 y Ft(n)270 b Fu(\037)25280 63658 y Fp(2)25806 63476 y Fw(\)\))p 19273 64017 7476 49 v 21485 65127 a Fu(L)p Fw(\()p Fu(\037)23539 64710 y Fr(\003)23539 65475 y Fp(1)24065 65127 y Fw(\))27218 64296 y Fu(<)337 b(k)29167 63795 y Fr(\000)p Fp(2)p Fr(\000)p Fp(2)p Fs(s)32006 63918 y FD(1)32523 64296 y Fu(:)15568 b Fw(\(3.10\))1803 67158 y(The)405 b(pro)34 b(of)404 b(of)h(the)f(lemma)g(will)f(b)34 b(e)404 b(presen)-34 b(ted)405 b(in)f(section)g(3.3.)p 0 TeXcolorgray 475 68218 6642 45 v 1803 69155 a Fp(7)2274 69541 y FE(F)-85 b(rom)341 b(geometric)h(p)28 b(oin)-28 b(t)342 b(of)f(view)h(this)g(means)g(that)f FA(\037)23820 69652 y FD(2)24282 69541 y FE(\()p FA(\025)p FE(\))h(do)28 b(es)342 b(not)f(ha)-28 b(v)g(e)341 b(self-in)-28 b(tersections.)p 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 19 19 TeXDict begin 19 18 bop 0 TeXcolorgray 0 TeXcolorgray 2848 3345 a Fp(SPECTRAL)406 b(PR)-26 b(OPER)-78 b(TIES)406 b(OF)h(A)h(POL)-78 b(YHARMONIC)405 b(OPERA)-78 b(TOR)406 b(WITH)h(LIMIT-PERIODIC)e(POTENTIAL)1430 b(19)p 0 TeXcolorgray 0 TeXcolorgray 475 6001 a FF(Corollary)328 b(3.6.)p 0 TeXcolorgray 443 w FC(If)310 b Fu(\034)447 b FC(b)-62 b(elongs)308 b(to)h(the)g(c)-62 b(omplex)308 b Fw(\()p Fu(\017)23954 6183 y Fp(1)24480 6001 y Fu(k)25149 5562 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fs(\016)29082 6001 y Fw(\))p Ft(\000)p FC(neighb)-62 b(orho)g(o)g(d)306 b(of)j(the)g(se)-62 b(c)g(ond)308 b(non-r)-62 b(esonanc)g(e)475 7470 y(set)348 b Fu(\037)3038 7652 y Fp(2)3564 7470 y Fw(\()p Fu(\025;)202 b(\016)46 b Fw(\))p FC(,)366 b(then)348 b(for)h(any)f Fu(z)403 b FC(lying)348 b(on)h(the)f(cir)-62 b(cle)347 b Fu(C)25197 7652 y Fp(2)26060 7470 y Fw(=)337 b Ft(f)p Fu(z)389 b Fw(:)337 b Ft(j)p Fu(z)139 b Ft(\000)86 b Fu(k)32311 7030 y Fp(2)p Fs(l)33127 7470 y Ft(j)336 b Fw(=)h Fu(\017)35572 7652 y Fp(1)36097 7470 y Fu(=)p Fw(2)p Ft(g)p FC(,)366 b(the)348 b(fol)62 b(lowing)348 b(ine)-62 b(qualities)475 8909 y(hold:)13808 11287 y Ft(k)p Fw(\()15203 10981 y(~)14885 11287 y Fu(H)15991 10787 y Fp(\(1\))17250 11287 y Fw(\()p Fu(\034)137 b Fw(\))270 b Ft(\000)f Fu(z)53 b Fw(\))21429 10787 y Fr(\000)p Fp(1)22687 11287 y Ft(k)337 b(\024)25248 10467 y Fw(4)p 25042 11009 1018 49 v 25042 12119 a Fu(\017)25534 12301 y Fp(1)26193 11287 y Fu(;)21898 b Fw(\(3.11\))13808 14602 y Ft(k)p Fw(\()15203 14295 y(~)14885 14602 y Fu(H)15991 14101 y Fp(\(1\))17250 14602 y Fw(\()p Fu(\034)137 b Fw(\))270 b Ft(\000)f Fu(z)53 b Fw(\))21429 14101 y Fr(\000)p Fp(1)22687 14602 y Ft(k)23293 14784 y Fp(1)24155 14602 y Ft(\024)25568 13782 y Fu(c)26093 13964 y Fp(0)26618 13782 y Fu(k)27287 13342 y Fp(2+2)p Fs(s)29394 13465 y FD(2)p 25568 14323 4344 49 v 27231 15433 a Fu(\017)27723 15615 y Fp(1)30045 14602 y Fu(;)1502 b(c)32409 14784 y Fp(0)33271 14602 y Fw(=)337 b(32)p Fu(b)36283 14784 y Fp(1)36809 14602 y Fu(b)37329 14784 y Fp(2)37855 14602 y Fu(:)10236 b Fw(\(3.12\))1803 17380 y(Corollary)403 b(is)h(pro)-34 b(v)g(en)405 b(in)f(App)34 b(endix)405 b(A.1.)p 0 TeXcolorgray 475 19632 a FF(Remark)508 b(3.7.)p 0 TeXcolorgray 574 w Fw(Note)443 b(that)g(ev)-34 b(ery)441 b(p)34 b(oin)-34 b(t)443 b(2)p Fu(\031)43 b(m=)-67 b(N)24551 19814 y Fp(1)25078 19632 y Fu(a)442 b Fw(\()p Fu(m)400 b Ft(2)h Fn(Z)30113 19192 y Fp(2)30639 19632 y Fw(\))442 b(of)h(a)f(dual)h(lattice)f(corresp)34 b(onding)442 b(to)h(the)475 21071 y(larger)344 b(p)34 b(erio)g(ds)345 b Fu(N)9025 21253 y Fp(1)9551 21071 y Fu(a)10192 21253 y Fp(1)10717 21071 y Fu(;)202 b(N)12230 21253 y Fp(1)12756 21071 y Fu(a)13397 21253 y Fp(2)14268 21071 y Fw(can)345 b(b)34 b(e)345 b(uniquely)h(represen)-34 b(ted)345 b(in)h(the)g(form)f(2)p Fu(\031)43 b(m=)-67 b(N)39373 21253 y Fp(1)39900 21071 y Fu(a)336 b Fw(=)g(2)p Fu(\031)43 b(j)69 b(=a)152 b Fw(+)g(2)p Fu(\031)43 b(p=)-67 b(N)50021 21253 y Fp(1)50548 21071 y Fu(a)p Fw(,)475 22510 y(where)516 b Fu(m)524 b Fw(=)f Fu(N)8120 22692 y Fp(1)8646 22510 y Fu(j)413 b Fw(+)344 b Fu(p)516 b Fw(and)h(2)p Fu(\031)43 b(j)69 b(=a)517 b Fw(is)f(a)g(p)34 b(oin)-34 b(t)517 b(of)g(a)f(dual)h (lattice)f(for)g(p)34 b(erio)g(ds)516 b Fu(a)40052 22692 y Fp(1)40577 22510 y Fw(,)544 b Fu(a)42099 22692 y Fp(2)42625 22510 y Fw(,)g(while)516 b Fu(p)523 b Ft(2)h Fu(P)684 b Fw(is)475 23949 y(resp)34 b(onsible)404 b(for)g(re\014ning)g(the)h (lattice.)1803 25399 y(Let)654 b(us)i(consider)e(a)h(normalized)f (eigenfunction)i Fu( )27198 25581 y Fs(n)27823 25399 y Fw(\()p Fu(t;)202 b(x)p Fw(\))655 b(of)g Fu(H)33827 24959 y Fp(\(1\))35085 25399 y Fw(\()p Fu(t)p Fw(\))g(in)f Fu(L)39609 25581 y Fp(2)40135 25399 y Fw(\()p Fu(Q)41564 25581 y Fp(1)42091 25399 y Fw(\).)1290 b(W)-101 b(e)654 b(extend)h(it)475 26942 y(quasip)34 b(erio)g(dically)332 b(to)j Fu(Q)12078 27124 y Fp(2)12604 26942 y Fw(,)348 b(renormalize)333 b(in)i Fu(L)21928 27124 y Fp(2)22453 26942 y Fw(\()p Fu(Q)23882 27124 y Fp(2)24409 26942 y Fw(\))g(and)g(denote)g(the)g(new)g(function)h(b)-34 b(y)42318 26622 y(~)42069 26942 y Fu( )42859 27124 y Fs(n)43485 26942 y Fw(\()p Fu(\034)70 b(;)202 b(x)p Fw(\),)348 b Fu(\034)474 b Fw(=)337 b Fo(K)50225 27124 y Fp(2)50751 26942 y Fu(t)p Fw(.)475 28485 y(The)568 b(F)-101 b(ourier)567 b(represen)-34 b(tations)568 b(of)g Fu( )18393 28667 y Fs(n)19019 28485 y Fw(\()p Fu(t;)202 b(x)p Fw(\))567 b(in)g Fu(L)24600 28667 y Fp(2)25126 28485 y Fw(\()p Fu(Q)26555 28667 y Fp(1)27081 28485 y Fw(\))h(and)30889 28166 y(~)30640 28485 y Fu( )31430 28667 y Fs(n)32056 28485 y Fw(\()p Fu(\034)70 b(;)202 b(x)p Fw(\))568 b(in)f Fu(L)37800 28667 y Fp(2)38326 28485 y Fw(\()p Fu(Q)39755 28667 y Fp(2)40281 28485 y Fw(\))h(are)f(simply)g(related.)475 29925 y(If)536 b(w)-34 b(e)536 b(denote)g(F)-101 b(ourier)536 b(co)34 b(e\016cien)-34 b(ts)535 b(of)h Fu( )20671 30107 y Fs(n)21297 29925 y Fw(\()p Fu(t;)202 b(x)p Fw(\))536 b(with)h(resp)34 b(ect)535 b(to)h(the)g(basis)g(of)h(exp)34 b(onen)-34 b(tial)536 b(functions)475 31468 y Ft(j)p Fu(Q)1770 31650 y Fp(1)2296 31468 y Ft(j)2633 31028 y Fr(\000)p Fp(1)p Fs(=)p Fp(2)4831 31468 y Fu(e)5395 31028 y Fs(i)p Fr(h)p Fs(t)p Fp(+2)p Fs(\031)32 b(j)51 b(=a;x)p Fr(i)10810 31468 y Fw(,)360 b Fu(j)406 b Ft(2)337 b Fn(Z)14365 31028 y Fp(2)14891 31468 y Fu(;)349 b Fw(in)h Fu(L)17762 31650 y Fp(2)18288 31468 y Fw(\()p Fu(Q)19717 31650 y Fp(1)20244 31468 y Fw(\))g(b)-34 b(y)350 b Fu(C)23560 31650 y Fs(nj)24618 31468 y Fw(,)360 b(then,)i(the)350 b(F)-101 b(ourier)349 b(co)34 b(e\016cien)-34 b(ts)40902 31161 y(~)40628 31468 y Fu(C)41494 31650 y Fs(nm)43303 31468 y Fw(of)44877 31148 y(~)44629 31468 y Fu( )45419 31650 y Fs(n)46045 31468 y Fw(\()p Fu(\034)70 b(;)202 b(x)p Fw(\))349 b(with)475 33036 y(resp)34 b(ect)355 b(to)i(the)g(basis)g(of)f(exp)34 b(onen)-34 b(tial)357 b(functions)h Ft(j)p Fu(Q)25237 33218 y Fp(2)25762 33036 y Ft(j)26099 32597 y Fr(\000)p Fp(1)p Fs(=)p Fp(2)28298 33036 y Fu(e)28862 32597 y Fs(i)p Fr(h)p Fs(\034)102 b Fp(+2)p Fs(\031)32 b(m=)-52 b(N)33839 32720 y FD(1)34301 32597 y Fs(a;x)p Fr(i)36013 33036 y Fw(,)366 b Fu(m)337 b Ft(2)f Fn(Z)40069 32597 y Fp(2)40595 33036 y Fu(;)356 b Fw(in)h Fu(L)43480 33218 y Fp(2)44005 33036 y Fw(\()p Fu(Q)45434 33218 y Fp(2)45961 33036 y Fw(\))g(are)e(giv)-34 b(en)475 34475 y(b)g(y)404 b(the)h(form)-34 b(ula:)17869 36800 y(~)17594 37106 y Fu(C)18460 37288 y Fs(nm)20256 37106 y Fw(=)21535 35033 y Fq(\()22512 36285 y Fu(C)23378 36467 y Fs(nj)24436 36285 y Fu(;)1212 b Fw(if)404 b Fu(m)337 b Fw(=)f Fu(j)69 b(N)31318 36467 y Fp(1)32114 36285 y Fw(+)269 b Fu(p)p Fw(;)22512 38012 y(0)p Fu(;)2530 b Fw(otherwise)p Fu(;)475 40056 y(p)386 b Fw(b)34 b(eing)387 b(de\014ned)h(from)f(the)h(relation)e Fu(t)337 b Fw(=)f Fu(\034)372 b Fw(+)235 b(2)p Fu(\031)43 b(p=)-67 b(N)25824 40238 y Fp(1)26350 40056 y Fu(a;)588 b(p)336 b Ft(2)h Fu(P)168 b Fw(.)533 b(Hence,)390 b(matrices)c(of)i(the)f(pro)67 b(jections)388 b(on)475 41599 y Fu( )1265 41781 y Fs(n)1891 41599 y Fw(\()p Fu(t;)202 b(x)p Fw(\))404 b(and)7512 41279 y(~)7263 41599 y Fu( )8053 41781 y Fs(n)8679 41599 y Fw(\()p Fu(\034)70 b(;)202 b(x)p Fw(\))404 b(with)h(resp)34 b(ect)404 b(to)g(the)h(ab)34 b(o)-34 b(v)g(e)404 b(bases)h(are)e (simply)h(related:)12045 44800 y(\()12796 44494 y(~)12516 44800 y Fu(E)13411 44982 y Fs(n)14037 44800 y Fw(\))14508 45202 y Fs(j)14921 44998 y Fp(^)14940 45202 y Fs(j)15764 44800 y Fw(=)17043 42727 y Fq(\()18019 43979 y Fw(\()p Fu(E)19385 44161 y Fs(n)20012 43979 y Fw(\))20483 44176 y Fs(m)181 b Fp(^)-651 b Fs(m)22203 43979 y Fu(;)1212 b Fw(if)404 b Fu(m)337 b Fw(=)g Fu(j)69 b(N)29086 44161 y Fp(1)29881 43979 y Fw(+)269 b Fu(p;)835 b Fw(^)-835 b Fu(m)337 b Fw(=)35308 43702 y(^)35326 43979 y Fu(j)70 b(N)36869 44161 y Fp(1)37664 43979 y Fw(+)269 b Fu(p)p Fw(;)18019 45706 y(0)p Fu(;)4790 b Fw(otherwise)p Fu(;)755 47902 y Fw(~)475 48209 y Fu(E)1370 48391 y Fs(n)2400 48209 y Fw(and)405 b Fu(E)5652 48391 y Fs(n)6682 48209 y Fw(b)34 b(eing)404 b(pro)67 b(jections)405 b(in)f Fu(L)18426 48391 y 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51479 y Fw(\()p Fu(Q)30076 51661 y Fp(2)30603 51479 y Fw(\).)1803 54009 y(By)527 b(analogy)g(with)i(\(2.12\),)558 b(\(2.13\),)h(w)-34 b(e)528 b(de\014ne)g(functions)h Fu(g)30803 53370 y Fp(\(2\))30760 54166 y Fs(r)32061 54009 y Fw(\()p Fu(k)38 b(;)202 b(\034)137 b Fw(\))528 b(and)h(op)34 b(erator-v)-67 b(alued)527 b(functions)475 55725 y Fu(G)1428 55086 y Fp(\(2\))1428 55883 y Fs(r)2685 55725 y Fw(\()p Fu(k)38 b(;)202 b(\034)137 b Fw(\),)405 b Fu(r)370 b Fw(=)337 b(1)p Fu(;)202 b Fw(2)p Fu(;)g Ft(\001)g(\001)g(\001)e Fw(,)404 b(as)g(follo)-34 b(ws:)12731 58535 y Fu(g)13352 58034 y Fp(\(2\))13309 58834 y Fs(r)14610 58535 y Fw(\()p Fu(k)38 b(;)202 b(\034)137 b Fw(\))338 b(=)19177 57715 y(\()p Ft(\000)p Fw(1\))21668 57275 y Fs(r)p 19177 58256 2998 49 v 19506 59366 a Fw(2)p Fu(\031)43 b(ir)22307 58535 y Fw(T)-101 b(r)23759 56885 y Fq(I)24432 59634 y Fs(C)25103 59757 y FD(2)25619 57189 y Fq(\020)26343 57553 y(\000)27216 58228 y Fw(~)26899 58535 y Fu(H)28005 58034 y Fp(\(1\))29262 58535 y Fw(\()p Fu(\034)137 b Fw(\))271 b Ft(\000)e Fu(z)32971 57553 y Fq(\001)33526 57855 y Fr(\000)p Fp(1)34784 58535 y Fu(W)35929 58717 y Fp(2)36455 57189 y Fq(\021)37178 57491 y Fs(r)37684 58535 y Fu(dz)53 b(;)9159 b Fw(\(3.13\))8368 62528 y Fu(G)9321 62027 y Fp(\(2\))9321 62827 y Fs(r)10579 62528 y Fw(\()p Fu(k)38 b(;)202 b(\034)137 b Fw(\))337 b(=)15146 61708 y(\()p Ft(\000)p Fw(1\))17637 61268 y Fs(r)26 b Fp(+1)p 15146 62249 4200 49 v 16367 63359 a Fw(2)p Fu(\031)43 b(i)19680 60878 y Fq(I)20354 63627 y Fs(C)21025 63750 y FD(2)21541 61182 y Fq(\020)22265 61546 y(\000)23138 62221 y Fw(~)22821 62528 y Fu(H)23927 62027 y Fp(\(1\))25184 62528 y Fw(\()p Fu(\034)137 b Fw(\))270 b Ft(\000)g Fu(z)28893 61546 y Fq(\001)29448 61848 y Fr(\000)p Fp(1)30706 62528 y Fu(W)31851 62710 y Fp(2)32376 61182 y Fq(\021)33100 61484 y Fs(r)33606 61546 y Fq(\000)34479 62221 y Fw(~)34162 62528 y Fu(H)35268 62027 y Fp(\(1\))36526 62528 y Fw(\()p Fu(\034)137 b Fw(\))270 b Ft(\000)f Fu(z)40234 61546 y Fq(\001)40790 61848 y Fr(\000)p Fp(1)42047 62528 y Fu(dz)53 b(:)4796 b Fw(\(3.14\))475 65578 y(W)-101 b(e)380 b(consider)g(the)h(op)34 b(erators)381 b Fu(H)15753 64939 y Fp(\(2\))15655 65736 y Fs(\013)17347 65578 y Fw(=)337 b Fu(H)19733 65138 y Fp(\(1\))21213 65578 y Fw(+)222 b Fu(\013)t(W)24302 65760 y Fp(2)25209 65578 y Fw(and)381 b(the)g(family)f Fu(H)34392 64939 y Fp(\(2\))34294 65736 y Fs(\013)35650 65578 y Fw(\()p Fu(\034)137 b Fw(\),)386 b Fu(\034)474 b Ft(2)336 b Fu(K)41159 65760 y Fp(2)41686 65578 y Fw(,)385 b(acting)380 b(in)h Fu(L)48236 65760 y Fp(2)48762 65578 y Fw(\()p Fu(Q)50191 65760 y Fp(2)50717 65578 y Fw(\).)p 0 TeXcolorgray 475 67859 a FF(Theorem)630 b(3.8.)p 0 TeXcolorgray 632 w FC(Supp)-62 b(ose)565 b Fu(\034)703 b FC(b)-62 b(elongs)564 b(to)i(the)f Fw(\()p Fu(\017)24323 68041 y Fp(1)24849 67859 y Fu(k)25518 67419 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fs(\016)29451 67859 y Fw(\))p FC(-neighb)-62 b(orho)g(o)g(d)563 b(in)j Fu(K)40339 68041 y Fp(2)41432 67859 y FC(of)f(the)g(se)-62 b(c)g(ond)565 b(non-)475 69541 y(r)-62 b(esonanc)g(e)524 b(set)h Fu(\037)8898 69723 y Fp(2)9424 69541 y Fw(\()p Fu(\025;)202 b(\016)46 b Fw(\))p FC(,)549 b Fw(0)504 b Fu(<)f Fw(7)p Fu(\016)549 b(<)504 b Fw(2)p Fu(l)360 b Ft(\000)336 b Fw(11)g Ft(\000)g Fw(16)p Fu(s)26029 69723 y Fp(1)26555 69541 y FC(,)549 b Fu(\017)27968 69723 y Fp(1)28997 69541 y Fw(=)504 b Fu(e)31008 69063 y Fr(\000)31873 68701 y FD(1)p 31873 68862 406 40 v 31873 69410 a(4)32411 69063 y Fs(k)32925 68751 y Fy(\021)24 b(s)33765 68950 y FD(1)34336 69541 y FC(.)834 b(Then,)549 b(for)525 b(su\016ciently)e(lar)-62 b(ge)524 b Fu(\025)p FC(,)p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 20 20 TeXDict begin 20 19 bop 0 TeXcolorgray 0 TeXcolorgray 475 3345 a Fp(20)14248 b(YULIA)313 b(KARPESHINA)f(AND)h(YOUNG-RAN)g (LEE)p 0 TeXcolorgray 475 6197 a Fu(\025)337 b(>)f(\025)3505 6379 y Fp(0)4031 6197 y Fw(\()p Fu(V)67 b(;)202 b(\016)46 b Fw(\))418 b FC(and)e(for)h(al)62 b(l)416 b Fu(\013)t FC(,)422 b Fw(0)336 b Ft(\024)h Fu(\013)k Ft(\024)c Fw(1)p FC(,)420 b(ther)-62 b(e)416 b(exists)g(a)h(unique)e(eigenvalue)g(of)h (the)g(op)-62 b(er)g(ator)415 b Fu(H)47187 5559 y Fp(\(2\))47089 6355 y Fs(\013)48445 6197 y Fw(\()p Fu(\034)137 b Fw(\))418 b FC(in)475 7666 y(the)432 b(interval)g Fu(")7498 7848 y Fp(2)8024 7666 y Fw(\()p Fu(k)38 b Fw(\))338 b(:=)e(\()p Fu(k)12729 7226 y Fp(2)p Fs(l)13816 7666 y Ft(\000)269 b Fu(\017)15520 7848 y Fp(1)16046 7666 y Fu(=)p Fw(2)p Fu(;)202 b(k)18466 7226 y Fp(2)p Fs(l)19553 7666 y Fw(+)269 b Fu(\017)21257 7848 y Fp(1)21783 7666 y Fu(=)p Fw(2\))p FC(.)557 b(It)434 b(is)f(given)g(by)f(the)h(series:)8363 10941 y Fu(\025)9070 10302 y Fp(\(2\))9051 11312 y(~)9070 11516 y Fs(j)10328 10941 y Fw(\()p Fu(\013)t(;)202 b(\034)137 b Fw(\))338 b(=)f Fu(\025)15580 10302 y Fp(\(1\))15580 11303 y Fs(j)16837 9959 y Fq(\000)17393 10941 y Fu(\034)406 b Fw(+)269 b(2)p Fu(\031)43 b(p=)-67 b(N)23004 11123 y Fp(1)23530 10941 y Fu(a)24171 9959 y Fq(\001)24996 10941 y Fw(+)26613 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37779 y(~)13113 37983 y Fs(j)14440 37409 y Fw(\()p Fu(\013)t(;)202 b(\034)137 b Fw(\))271 b Ft(\000)19131 37102 y Fw(~)18850 37409 y Fu(E)19815 36770 y Fp(\(1\))19745 37771 y Fs(j)21073 37409 y Fw(\()p Fu(\034)407 b Fw(+)269 b(2)p Fu(\031)43 b(p=)-67 b(N)27156 37591 y Fp(1)27681 37409 y Fu(a)p Fw(\))28793 36063 y Fq(\021)29517 36015 y(\015)29517 36742 y(\015)29517 37469 y(\015)30191 38252 y Fp(1)31053 37409 y Fu(<)337 b(\013)t(C)33978 37591 y Fs(m)34867 37409 y Fu(\017)35359 36770 y Fp(3)p Fr(\000j)p Fs(m)p Fr(j)35359 37756 y Fp(1)37971 37409 y Fu(k)38640 36908 y Fr(j)p Fs(m)p Fr(j)p Fp(\(2)p Fs(l)11 b Fr(\000)p Fp(1+)p Fs(\016)33 b Fp(\))43928 37409 y Fu(;)4163 b Fw(\(3.25\))475 40037 y FC(her)-62 b(e)432 b(and)h(b)-62 b(elow)433 b Fu(C)9534 40219 y Fs(m)10759 40037 y Fw(=)336 b(48)p Fu(m)p Fw(!2)15257 39597 y Fr(j)p Fs(m)p Fr(j)16668 40037 y FC(.)p 0 TeXcolorgray 475 42234 a FF(Corollary)483 b(3.13.)p 0 TeXcolorgray 10576 44454 a Ft(r)p Fu(\025)12293 43815 y Fp(\(2\))12293 44816 y Fs(j)13551 44454 y Fw(\()p Fu(\013)t(;)202 b(\034)137 b Fw(\))338 b(=)e(2)p Fu(l)24 b(k)19756 43954 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(2)21714 44134 y Fu(~)21775 44454 y(k)308 b Fw(+)269 b Fu(o)p Fw(\()p Fu(k)25654 43954 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1)27674 44454 y Fw(\))p Fu(;)29923 44134 y(~)29985 44454 y(k)375 b Fw(=)346 b Fu(~)-616 b(p)32880 44636 y Fs(j)33367 44454 y Fw(\()p Fu(\034)407 b Fw(+)269 b(2)p Fu(\031)43 b(p=)-67 b(N)39450 44636 y Fp(1)39976 44454 y Fu(a)p Fw(\))p Fu(:)7003 b Fw(\(3.26\))14887 47040 y Fu(T)168 b Fw(\()p Fu(m)p Fw(\))p Fu(\025)18476 46401 y Fp(\(2\))18476 47401 y Fs(j)19735 47040 y Fw(\()p Fu(\013)t(;)202 b(\034)137 b Fw(\))338 b(=)f Fu(O)25239 46058 y Fq(\000)25793 47040 y Fu(k)26462 46539 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(2)28482 46058 y Fq(\001)30772 47040 y FC(if)1301 b Ft(j)p Fu(m)p Ft(j)336 b Fw(=)h(2)p Fu(:)11314 b Fw(\(3.27\))1803 49546 y(The)377 b(next)g(lemma)g(will)f(b)34 b(e)377 b(used)g(in)g(the)g(third)g(step)h(of)f(appro)-34 b(ximation)378 b(where)e(the)i(op)34 b(erator)376 b Fu(H)47378 49106 y Fp(\(2\))48636 49546 y Fw(\()p Fu(\034)137 b Fw(\))337 b(=)475 51263 y Fu(H)1581 50624 y Fp(\(2\))1483 51610 y(1)2838 51263 y Fw(\()p Fu(\034)137 b Fw(\))405 b(will)f(pla)-34 b(y)404 b(a)g(role)g(of)g(the)h(initial)e(\(unp)34 b(erturb)g(ed\))406 b(op)34 b(erator.)p 0 TeXcolorgray 475 53533 a FF(Lemma)508 b(3.14.)p 0 TeXcolorgray 575 w FC(F)-93 b(or)469 b(any)g Fu(z)523 b FC(on)469 b(the)f(cir)-62 b(cle)468 b Fu(C)22604 53715 y Fp(2)23599 53533 y FC(and)h Fu(\034)606 b FC(in)470 b(the)e(c)-62 b(omplex)468 b Fw(\()p Fu(\017)36285 53715 y Fp(1)36811 53533 y Fu(k)37480 53094 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fs(\016)41413 53533 y Fw(\))p Ft(\000)469 b FC(neighb)-62 b(orho)g(o)g(d)466 b(of)475 54972 y Fu(\037)1233 55154 y Fp(2)1759 54972 y FC(,)19640 56987 y Ft(k)p Fw(\()p Fu(H)21823 56486 y Fp(\(2\))23081 56987 y Fw(\()p Fu(\034)137 b Fw(\))270 b Ft(\000)f Fu(z)53 b Fw(\))27260 56486 y Fr(\000)p Fp(1)28518 56987 y Ft(k)337 b(\024)31079 56167 y Fw(8)p 30873 56708 1018 49 v 30873 57818 a Fu(\017)31365 58000 y Fp(1)32024 56987 y Fu(:)16067 b Fw(\(3.28\))p 0 TeXcolorgray 475 59710 a FC(Pr)-62 b(o)g(of.)p 0 TeXcolorgray 552 w Fw(Considering)405 b(the)f(Hilb)34 b(ert)404 b(relation)5960 61792 y(\()p Fu(H)7537 61292 y Fp(\(2\))8795 61792 y Fw(\()p Fu(\034)137 b Fw(\))270 b Ft(\000)f Fu(z)53 b Fw(\))12974 61292 y Fr(\000)p Fp(1)14569 61792 y Fw(=)336 b(\()16637 61486 y(~)16319 61792 y Fu(H)17425 61292 y Fp(\(1\))18684 61792 y Fw(\()p Fu(\034)137 b Fw(\))270 b Ft(\000)f Fu(z)53 b Fw(\))22863 61292 y Fr(\000)p Fp(1)24390 61792 y Fw(+)269 b(\()26391 61486 y(~)26073 61792 y Fu(H)27179 61292 y Fp(\(1\))28437 61792 y Fw(\()p Fu(\034)137 b Fw(\))271 b Ft(\000)e Fu(z)53 b Fw(\))32617 61292 y Fr(\000)p Fp(1)33875 61792 y Fw(\()p Ft(\000)p Fu(W)36434 61974 y Fp(2)36960 61792 y Fw(\)\()p Fu(H)39008 61292 y Fp(\(2\))40266 61792 y Fw(\()p Fu(\034)137 b Fw(\))270 b Ft(\000)f Fu(z)53 b Fw(\))44445 61292 y Fr(\000)p Fp(1)45703 61792 y Fu(;)475 63802 y Fw(and)405 b(the)f(estimate)g (\(3.11\),)h(together)f(with)h(the)g(estimate)f Ft(k)p Fu(W)29852 63984 y Fp(2)30378 63802 y Ft(k)336 b Fu(<)h(\017)33092 63362 y Fp(4)33092 64127 y(1)33618 63802 y Fw(,)403 b(w)-34 b(e)405 b(obtain:)4675 66744 y Ft(k)p Fw(\()p Fu(H)6858 66243 y Fp(\(2\))8116 66744 y Fw(\()p Fu(\034)137 b Fw(\))270 b Ft(\000)g Fu(z)53 b Fw(\))12296 66243 y Fr(\000)p Fp(1)13554 66744 y Ft(k)336 b(\024)17788 65924 y(k)p Fw(\()19182 65617 y(~)18865 65924 y Fu(H)19971 65484 y Fp(\(1\))21229 65924 y Fw(\()p Fu(\034)137 b Fw(\))270 b Ft(\000)f Fu(z)53 b Fw(\))25408 65484 y Fr(\000)p Fp(1)26666 65924 y Ft(k)p 15909 66465 13243 49 v 15909 67726 a Fw(1)269 b Ft(\000)g(k)p Fw(\()19391 67420 y(~)19073 67726 y Fu(H)20179 67374 y Fp(\(1\))21437 67726 y Fw(\()p Fu(\034)137 b Fw(\))271 b Ft(\000)e Fu(z)53 b Fw(\))25617 67376 y Fr(\000)p Fp(1)26875 67726 y Fu(W)28020 67908 y Fp(2)28545 67726 y Ft(k)29621 66744 y(\024)336 b Fw(2)p Ft(k)p Fw(\()32901 66437 y(~)32583 66744 y Fu(H)33689 66243 y Fp(\(1\))34948 66744 y Fw(\()p Fu(\034)137 b Fw(\))270 b Ft(\000)f Fu(z)53 b Fw(\))39127 66243 y Fr(\000)p Fp(1)40385 66744 y Ft(k)337 b(\024)42946 65924 y Fw(8)p 42740 66465 1018 49 v 42740 67575 a Fu(\017)43232 67757 y Fp(1)43891 66744 y Fu(:)4200 b Fw(\(3.29\))p 50750 69541 665 831 v 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 22 22 TeXDict begin 22 21 bop 0 TeXcolorgray 0 TeXcolorgray 475 3345 a Fp(22)14248 b(YULIA)313 b(KARPESHINA)f(AND)h(YOUNG-RAN)g (LEE)p 0 TeXcolorgray 475 6001 a Fw(3.3.)605 b FF(Pro)39 b(of)464 b(of)h(the)g(Geometric)f(Lemma.)475 8865 y Fw(3.3.1.)605 b FC(Pr)-62 b(o)g(of)511 b(of)h(Statement)f(1.)605 b Fw(A)-34 b(t)490 b(the)g(end)g(of)g(Section)g(3.1.1)f(\(page)h(18\))f (w)-34 b(e)490 b(de\014ned)h(the)f(second)f(non-)475 10304 y(resonance)555 b(set,)593 b Fu(\037)9334 10486 y Fp(2)9861 10304 y Fw(\()p Fu(\025)p Fw(\))c Ft(\032)g Fu(K)14660 10486 y Fp(2)15187 10304 y Fw(,)k(b)-34 b(y)555 b(the)h(form)-34 b(ula)557 b Fu(\037)25548 10486 y Fp(2)26074 10304 y Fw(\()p Fu(\025)p Fw(\))590 b(=)f Fo(K)30842 10486 y Fp(2)31368 10304 y Fw(\()p Fu(\037)32597 9864 y Fr(\003)32597 10629 y Fp(1)33123 10304 y Fw(\))371 b Ft(n)f Fw(\012)35816 10486 y Fp(1)36343 10304 y Fw(,)593 b(the)556 b(set)f(\012)42430 10486 y Fp(1)43512 10304 y Fw(b)34 b(eing)555 b(giv)-34 b(en)556 b(b)-34 b(y)475 11743 y(\(3.6\).)614 b(Supp)34 b(ose)431 b Fu(\034)515 b Ft(2)379 b Fu(\037)11687 11925 y Fp(2)12213 11743 y Fw(\()p Fu(\025)p Fw(\).)615 b(Since)429 b Fu(\037)18762 11925 y Fp(2)19288 11743 y Fw(\()p Fu(\025)p Fw(\))380 b Ft(\032)e Fo(K)23635 11925 y Fp(2)24161 11743 y Fw(\()p Fu(\037)25390 11303 y Fr(\003)25390 12068 y Fp(1)25917 11743 y Fw(\),)436 b(there)429 b(is)g(a)g Fu(p)378 b Ft(2)h Fu(P)598 b Fw(suc)-34 b(h)430 b(that)g Fu(\034)424 b Fw(+)286 b(2)p Fu(\031)43 b(p=)-67 b(N)47173 11925 y Fp(1)47698 11743 y Fu(a)379 b Ft(2)f Fu(\037)50662 11303 y Fr(\003)50662 12068 y Fp(1)51189 11743 y Fw(.)475 13482 y(By)469 b(Lemma)h(2.12,)487 b(there)470 b(is)h(an)g(eigen)-34 b(v)-67 b(alue)469 b Fu(\025)22582 12843 y Fp(\(1\))22582 13639 y Fs(n)23840 13482 y Fw(\()p Fu(\034)451 b Fw(+)313 b(2)p Fu(\031)43 b(p=)-67 b(N)30011 13664 y Fp(1)30537 13482 y Fu(a)p Fw(\))447 b(=)h Fu(\025)p Fw(,)486 b(whic)-34 b(h)472 b(is)e(giv)-34 b(en)470 b(b)-34 b(y)471 b(p)34 b(erturbation)475 15199 y(series)349 b(\(2.16\).)522 b(By)349 b(the)i(same)g(lemma)f Fu(\025)19051 14560 y Fp(\(1\))19051 15356 y Fs(n)20309 15199 y Fw(\()p Fu(\034)300 b Fw(+)162 b(2)p Fu(\031)43 b(p=)-67 b(N)26178 15381 y Fp(1)26704 15199 y Fu(a)p Fw(\))350 b(is)h(a)f(unique)h(eigen)-34 b(v)-67 b(alue)350 b(of)h Fu(H)42345 14759 y Fp(\(1\))43804 15199 y Fw(\()q Fu(\034)406 b Fw(+)269 b(2)p Fu(\031)43 b(p=)-67 b(N)49887 15381 y Fp(1)50413 15199 y Fu(a)p Fw(\))475 16667 y(in)404 b Fu(")2454 16849 y Fp(1)3317 16667 y Fw(=)336 b(\()p Fu(k)5736 16227 y Fp(2)p Fs(l)6823 16667 y Ft(\000)269 b Fu(k)8704 16227 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)12305 16350 y FD(1)12767 16227 y Fr(\000)p Fs(\016)14004 16667 y Fu(;)202 b(k)15212 16227 y Fp(2)p Fs(l)16298 16667 y Fw(+)270 b Fu(k)18180 16227 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)21781 16350 y FD(1)22242 16227 y Fr(\000)p Fs(\016)23479 16667 y Fw(\).)539 b(This)405 b(means)7215 17937 y Fq(\014)7215 18664 y(\014)7215 19392 y(\014)7619 19331 y Fu(\025)8326 18831 y Fp(\(1\))8326 19631 y Fs(n)9584 19331 y Fw(\()p Fu(\034)i Fw(+)269 b(2)p Fu(\031)43 b(p=)-67 b(N)15667 19513 y Fp(1)16193 19331 y Fu(a)p Fw(\))269 b Ft(\000)g Fu(\025)19493 18692 y Fp(\(1\))19543 19722 y(^)-520 b Fs(n)20751 19331 y Fw(\()p Fu(\034)407 b Fw(+)269 b(2)p Fu(\031)43 b(p=)-67 b(N)26834 19513 y Fp(1)27360 19331 y Fu(a)p Fw(\))28472 17937 y Fq(\014)28472 18664 y(\014)28472 19392 y(\014)29213 19331 y Ft(\025)336 b Fu(k)31161 18831 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)34762 18954 y FD(1)35224 18831 y Fr(\000)p Fs(\016)36461 19331 y Fu(;)1010 b Fw(when)869 b(^)-666 b Fu(n)336 b Ft(6)p Fw(=)g Fu(n:)3643 b Fw(\(3.30\))475 21928 y(Using)404 b(the)h(de\014nition)g(of)f(\012)13612 22110 y Fp(1)14139 21928 y Fw(,)f(w)-34 b(e)405 b(obtain:)9741 24433 y Ft(j)p Fu(\025)10785 23933 y Fp(\(1\))10785 24733 y Fs(n)12043 24433 y Fw(\()p Fu(\034)h Fw(+)270 b(2)p Fu(\031)43 b(p=)-67 b(N)18126 24615 y Fp(1)18651 24433 y Fu(a)p Fw(\))270 b Ft(\000)f Fu(\025)21952 23794 y Fp(\(1\))22002 24825 y(^)-520 b Fs(n)23210 24433 y Fw(\()p Fu(\034)406 b Fw(+)270 b(2)p Fu(\031)146 b Fw(^)-709 b Fu(p=)-67 b(N)29293 24615 y Fp(1)29818 24433 y Fu(a)p Fw(\))p Ft(j)337 b Fu(>)f(\017)33375 24615 y Fp(1)35113 24433 y Fw(when)1213 b Fu(p)337 b Ft(6)p Fw(=)439 b(^)-709 b Fu(p:)475 26729 y Fw(Com)-34 b(bining)529 b(the)e(last)h(t)-34 b(w)g(o)528 b(inequalities,)557 b(w)-34 b(e)528 b(obtain)g(Statemen)-34 b(t)529 b(1\(c\).)909 b(It)527 b(remains)g(to)h(pro)-34 b(v)g(e)527 b(that)h Fu(p)f Fw(is)475 28168 y(de\014ned)500 b(uniquely)f(b)-34 b(y)500 b(the)f(relation)g Fu(\034)470 b Fw(+)333 b(2)p Fu(\031)43 b(p=)-67 b(N)23999 28350 y Fp(1)24524 28168 y Fu(a)495 b Ft(2)h Fu(\037)27722 27728 y Fr(\003)27722 28492 y Fp(1)28248 28168 y Fw(.)824 b(Supp)34 b(ose)500 b(it)f(is)g(not)i(so,)522 b(and)500 b(there)g(is)601 b(^)-708 b Fu(p)495 b Ft(6)p Fw(=)g Fu(p)p Fw(:)475 29906 y Fu(\034)424 b Fw(+)286 b(2)p Fu(\031)146 b Fw(^)-709 b Fu(p)q(=)-67 b(N)6122 30088 y Fp(1)6647 29906 y Fu(a)380 b Ft(2)g Fu(\037)9614 29466 y Fr(\003)9614 30231 y Fp(1)10141 29906 y Fw(.)616 b(Applying)431 b(Lemma)f(2.12)g(to)h(b)34 b(oth)431 b(p)34 b(oin)-34 b(ts)431 b(in)f Fu(\037)33574 29466 y Fr(\003)33574 30231 y Fp(1)34100 29906 y Fw(,)437 b(w)-34 b(e)431 b(obtain)g Fu(\025)41189 29267 y Fp(\(1\))41189 30064 y Fs(n)42447 29906 y Fw(\()p Fu(\034)424 b Fw(+)286 b(2)p Fu(\031)43 b(p=)-67 b(N)48564 30088 y Fp(1)49090 29906 y Fu(a)p Fw(\))381 b(=)475 31645 y Fu(\025)1182 31006 y Fp(\(1\))1232 32036 y(^)-520 b Fs(n)2439 31645 y Fw(\()p Fu(\034)139 b Fw(+)r(2)p Fu(\031)146 b Fw(^)-709 b Fu(p)q(=)-67 b(N)7988 31827 y Fp(1)8514 31645 y Fu(a)p Fw(\))p Fu(;)270 b Fw(the)h(eigen)-34 b(v)-67 b(alues)270 b(b)34 b(eing)270 b(giv)-34 b(en)271 b(b)-34 b(y)271 b(p)34 b(erturbation)271 b(series.)493 b(The)271 b(last)g(equalit)-34 b(y)270 b(con)-34 b(tradicts)475 33084 y(to)404 b(\(3.30\).)475 35977 y(3.3.2.)605 b FC(Pr)-62 b(o)g(of)540 b(of)g(the)f(Statement)h (2.)605 b Fw(If)520 b Fu(\034)658 b Fw(is)520 b(in)g(the)h(complex)f (\()p Fu(\017)31769 36159 y Fp(1)32295 35977 y Fu(k)32964 35537 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fs(\016)36897 35977 y Fw(\)-neigh)-34 b(b)34 b(orho)g(o)g(d)521 b(of)g Fu(\037)47756 36159 y Fp(2)48282 35977 y Fw(,)549 b(then)475 37445 y(there)390 b(exists)h(a)f Fu(\034)8422 37627 y Fp(0)9338 37445 y Fw(in)h Fu(\037)11497 37627 y Fp(2)12414 37445 y Fw(suc)-34 b(h)391 b(that)h Ft(j)p Fu(\034)379 b Ft(\000)242 b Fu(\034)20708 37627 y Fp(0)21234 37445 y Ft(j)336 b Fu(<)h(\017)23679 37627 y Fp(1)24205 37445 y Fu(k)24874 37006 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fs(\016)29197 37445 y Fw(and)391 b Fu(\034)32070 37627 y Fp(0)32987 37445 y Fw(satis\014es)g(Statemen)-34 b(t)392 b(1)f(of)g(the)g(lemma.)475 38914 y(Ob)-34 b(viously)383 b Fu(\034)365 b Fw(+)228 b(2)p Fu(\031)43 b(p=)-67 b(N)11680 39096 y Fp(1)12206 38914 y Fu(a)383 b Fw(is)g(in)h(the)g(\()p Fu(k)19029 38474 y Fr(\000)p Fp(1)p Fr(\000)p Fp(4)p Fs(s)21868 38597 y FD(1)22330 38474 y Fr(\000)p Fp(2)p Fs(\016)24037 38914 y Fw(\)-neigh)-34 b(b)34 b(orho)g(o)g(d)385 b(of)f Fu(\037)34623 39096 y Fp(1)35149 38914 y Fw(,)j(where)d(Theorem) f(2.3)h(holds.)532 b(By)475 40353 y(Corollary)403 b(3.13,)475 40699 y Fq(\014)475 41427 y(\014)475 42154 y(\014)879 42093 y Fu(\025)1586 41455 y Fp(\(1\))1586 42455 y Fs(j)2844 42093 y Fw(\()p Fu(\034)j Fw(+)269 b(2)p Fu(\031)43 b(p=)-67 b(N)8926 42275 y Fp(1)9452 42093 y Fu(a)p Fw(\))270 b Ft(\000)f Fu(\025)12753 41455 y Fp(\(1\))12753 42455 y Fs(j)14011 42093 y Fw(\()p Fu(\034)15012 42275 y Fp(0)15807 42093 y Fw(+)g(2)p Fu(\031)43 b(p=)-67 b(N)20482 42275 y Fp(1)21008 42093 y Fu(a)p Fw(\))22120 40699 y Fq(\014)22120 41427 y(\014)22120 42154 y(\014)23265 42093 y Ft(\024)741 b Fw(sup)26825 42384 y Fp(~)-521 b Fs(\034)27551 40699 y Fq(\014)27551 41427 y(\014)27551 42154 y(\014)27955 42093 y Ft(r)28965 42275 y Fs(\034)29540 42093 y Fu(\025)30247 41455 y Fp(\(1\))30247 42455 y Fs(j)31505 42093 y Fw(\()64 b(~)-670 b Fu(\034)407 b Fw(+)269 b(2)p Fu(\031)43 b(p=)-67 b(N)37588 42275 y Fp(1)38114 42093 y Fu(a)p Fw(\))39226 40699 y Fq(\014)39226 41427 y(\014)39226 42154 y(\014)39832 42093 y Ft(j)p Fu(\034)568 b Ft(\000)431 b Fu(\034)43171 42275 y Fp(0)43697 42093 y Ft(j)646 b Fw(=)741 b Fu(O)34 b Fw(\()p Fu(\017)48286 42275 y Fp(1)48811 42093 y Fu(k)49480 41654 y Fr(\000)p Fs(\016)50717 42093 y Fw(\))p Fu(:)475 44295 y Fw(Considering)405 b(that)g Fu(\025)10485 43656 y Fp(\(1\))10485 44657 y Fs(j)11743 44295 y Fw(\()p Fu(\034)12744 44477 y Fp(0)13539 44295 y Fw(+)269 b(2)p Fu(\031)43 b(p=)-67 b(N)18214 44477 y Fp(1)18740 44295 y Fu(a)p Fw(\))337 b(=)f Fu(k)22137 43855 y Fp(2)p Fs(l)22954 44295 y Fw(,)404 b(w)-34 b(e)405 b(get)f(\(3.9\).)475 47206 y(3.3.3.)605 b FC(Determinants.)533 b(Interse)-62 b(ctions)362 b(and)i(Quasi-interse)-62 b(ctions.)529 b(Description)362 b(of)h(the)g(set)g Fw(\012)44904 47388 y Fp(1)45794 47206 y FC(in)i(terms)e(of)475 48669 y(determinants.)604 b Fw(W)-101 b(e)329 b(considered)g(self-in)-34 b(tersections)331 b(of)26770 48362 y(~)26566 48669 y Fu(S)27309 48851 y Fp(1)27835 48669 y Fw(\()p Fu(\025)p Fw(\))f(b)34 b(elonging)330 b(to)g Fo(K)37632 48851 y Fp(2)38158 48669 y Fu(\037)38916 48229 y Fr(\003)38916 48993 y Fp(1)39772 48669 y Fw(\(see)g (De\014nition)g(3.2)f(and)475 50108 y(Lemma)376 b(3.4,)382 b(page)377 b(17\).)530 b(Here)376 b(w)-34 b(e)377 b(describ)34 b(e)377 b(self-in)-34 b(tersections)377 b(as)g(zeros)f(of)h(determinan) -34 b(ts)379 b(of)e(op)34 b(erators)377 b(of)475 51625 y(the)362 b(t)-34 b(yp)34 b(e)362 b Fu(I)280 b Fw(+)185 b Fu(A)p Fw(,)369 b Fu(A)337 b Ft(2)g FF(S)11926 51807 y Fl(1)12886 51625 y Fw(\(see)362 b(e.g.)523 b([21]\).)i(In)362 b(fact,)370 b(let)362 b(us)g(represen)-34 b(t)362 b(the)g(op)34 b(erator)362 b(\()p Fu(H)41883 51185 y Fp(\(1\))43141 51625 y Fw(\()p Fu(t)p Fw(\))185 b Ft(\000)g Fu(\025)p Fw(\)\()p Fu(H)48491 51807 y Fp(0)49017 51625 y Fw(\()p Fu(t)p Fw(\))g(+)475 53064 y Fu(\025)p Fw(\))1653 52625 y Fr(\000)p Fp(1)3315 53064 y Fw(in)404 b(the)h(form)f Fu(I)364 b Fw(+)269 b Fu(A)12700 53246 y Fp(1)13226 53064 y Fw(:)4477 55499 y(\()p Fu(H)6054 54999 y Fp(\(1\))7312 55499 y Fw(\()p Fu(t)p Fw(\))h Ft(\000)f Fu(\025)p Fw(\)\()p Fu(H)12831 55681 y Fp(0)13357 55499 y Fw(\()p Fu(t)p Fw(\))h(+)f Fu(\025)p Fw(\))17397 54999 y Fr(\000)p Fp(1)18992 55499 y Fw(=)336 b Fu(I)364 b Fw(+)269 b Fu(A)23289 55681 y Fp(1)23815 55499 y Fw(\()p Fu(t)p Fw(\))p Fu(;)1415 b(A)27856 55681 y Fp(1)28382 55499 y Fw(\()p Fu(t)p Fw(\))337 b(=)f(\()p Fu(W)32994 55681 y Fp(1)33790 55499 y Ft(\000)269 b Fw(2)p Fu(\025)p Fw(\)\()p Fu(H)38265 55681 y Fp(0)38791 55499 y Fw(\()p Fu(t)p Fw(\))h(+)f Fu(\025)p Fw(\))42831 54999 y Fr(\000)p Fp(1)44089 55499 y Fu(:)4002 b Fw(\(3.31\))475 57794 y(Ob)-34 b(viously)-101 b(,)403 b Fu(A)7316 57976 y Fp(1)7842 57794 y Fw(\()p Fu(t)p Fw(\))337 b Ft(2)g FF(S)11478 57976 y Fl(1)12076 57794 y Fw(.)p 0 TeXcolorgray 475 60154 a FF(Remark)419 b(3.15.)p 0 TeXcolorgray 522 w Fw(F)-101 b(rom)365 b(prop)34 b(erties)365 b(of)h(determinan)-34 b(ts)367 b(and)f(the)g(de\014nition)g(of)g Fu(S)38346 60336 y Fp(1)38872 60154 y Fw(\()p Fu(\025)p Fw(\))g(it)g(easily)e (follo)-34 b(ws)366 b(that)475 61962 y(the)404 b(iso)34 b(energetic)403 b(set)h Fu(S)11773 62144 y Fp(1)12299 61962 y Fw(\()p Fu(\025)p Fw(\))h(is)f(the)h(zero)e(set)h(of)h(det) 25178 60617 y Fq(\020)25902 61962 y Fu(I)364 b Fw(+)270 b Fu(A)28921 62144 y Fp(1)29446 61962 y Fw(\()p Fu(t)p Fw(\))30826 60617 y Fq(\021)31955 61962 y Fw(in)404 b Fu(K)34398 62144 y Fp(1)34924 61962 y Fw(.)p 0 TeXcolorgray 475 64886 a FF(Remark)494 b(3.16.)p 0 TeXcolorgray 568 w Fw(Matrices)430 b Fu(H)15230 65068 y Fp(0)15756 64886 y Fw(\()p Fu(t)p Fw(\),)437 b Fu(H)19016 64446 y Fp(\(1\))20274 64886 y Fw(\()p Fu(t)p Fw(\))431 b(and)g Fu(A)25377 65068 y Fp(1)25903 64886 y Fw(\()p Fu(t)p Fw(\))g(can)f(b)34 b(e)430 b(extended)h(as)g(holomorphic)g(functions)h(of)475 66403 y(t)-34 b(w)g(o)412 b(v)-67 b(ariables)409 b(to)i(a)f(vicinit)-34 b(y)410 b(of)h Fn(R)16920 65964 y Fp(2)17446 66403 y Fw(.)558 b(If)400 b Fu(~)-596 b(y)391 b Fw(=)357 b Fu(~)-616 b(p)22444 66585 y Fs(j)22931 66403 y Fw(\(0\))274 b(+)g Fu(t)p Fw(,)411 b(then)g(matrices)f Fu(H)35796 66585 y Fp(0)36322 66403 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\),)412 b Fu(H)39757 65964 y Fp(\(1\))41015 66403 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\))411 b(and)g Fu(A)46278 66585 y Fp(1)46804 66403 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\))411 b(di\013er)475 67972 y(from)433 b Fu(H)4377 68154 y Fp(0)4902 67972 y Fw(\()p Fu(t)p Fw(\),)440 b Fu(H)8165 67532 y Fp(\(1\))9423 67972 y Fw(\()p Fu(t)p Fw(\))433 b(and)g Fu(A)14530 68154 y Fp(1)15056 67972 y Fw(\()p Fu(t)p Fw(\))g(only)g(b)-34 b(y)433 b(a)g(shift)g(of)g(indices:)596 b Fu(H)31997 68154 y Fp(0)32522 67972 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\))34102 68154 y Fs(mm)36207 67972 y Fw(=)384 b Fu(H)38542 68154 y Fp(0)39067 67972 y Fw(\()p Fu(t)p Fw(\))40447 68154 y Fs(m)p Fp(+)p Fs(j)o(;m)p Fp(+)p Fs(j)44703 67972 y Fw(,)440 b(etc.)624 b(Hence,)475 68511 y Fq(\015)475 69238 y(\015)1148 69541 y Fw(\()p Fu(H)2725 69101 y Fp(\(1\))3983 69541 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\))270 b Ft(\000)f Fu(z)53 b Fw(\))8133 69101 y Fr(\000)p Fp(1)9391 68511 y Fq(\015)9391 69238 y(\015)10401 69541 y Fw(=)11681 68511 y Fq(\015)11681 69238 y(\015)12354 69541 y Fw(\()p Fu(H)13931 69101 y Fp(\(1\))15189 69541 y Fw(\()p Fu(t)p Fw(\))270 b Ft(\000)f Fu(z)53 b Fw(\))19139 69101 y Fr(\000)p Fp(1)20397 68511 y Fq(\015)20397 69238 y(\015)21475 69541 y Fw(and)405 b(det)25515 68559 y Fq(\000)26071 69541 y Fu(I)364 b Fw(+)269 b Fu(A)29089 69723 y Fp(1)29615 69541 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\))31195 68559 y Fq(\001)32088 69541 y Fw(=)336 b(det)35051 68559 y Fq(\000)35606 69541 y Fu(I)365 b Fw(+)269 b Fu(A)38625 69723 y Fp(1)39151 69541 y Fw(\()p Fu(t)p Fw(\))40531 68559 y Fq(\001)41087 69541 y Fu(:)p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 23 23 TeXDict begin 23 22 bop 0 TeXcolorgray 0 TeXcolorgray 2848 3345 a Fp(SPECTRAL)406 b(PR)-26 b(OPER)-78 b(TIES)406 b(OF)h(A)h(POL)-78 b(YHARMONIC)405 b(OPERA)-78 b(TOR)406 b(WITH)h(LIMIT-PERIODIC)e(POTENTIAL)1430 b(23)p 0 TeXcolorgray 1803 6001 a Fw(Next,)599 b(w)-34 b(e)560 b(recall)f(that)i(the)f(set)h Fo(D)18711 6183 y Fp(1)19236 6001 y Fw(\()p Fu(\025)p Fw(\))g(can)f(b)34 b(e)560 b(describ)34 b(ed)559 b(in)h(terms)h(of)f(v) -34 b(ectors)592 b Fu(~)-638 b Fn({)49 b Fw(\()p Fu(')p Fw(\))597 b(=)f Fn({)49 b Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))-25 b Fu(~)-581 b(\027)78 b Fw(,)450 7440 y Fu(~)-581 b(\027)663 b Fw(=)587 b(\(cos)202 b Fu(';)g Fw(sin)g Fu(')p Fw(\),)591 b Fu(')c Ft(2)f Fw(\002)14494 7622 y Fp(1)15020 7440 y Fw(,)591 b(see)554 b(\(2.33\),)591 b(\(2.42\).)989 b(By)553 b(de\014nition,)593 b Fu(\037)35228 7001 y Fr(\003)35228 7765 y Fp(1)35754 7440 y Fw(\()p Fu(\025)p Fw(\))587 b(=)g Fo(K)40517 7622 y Fp(1)41043 7440 y Fo(D)42050 7622 y Fp(1)42575 7440 y Fw(\()p Fu(\025)p Fw(\).)989 b(By)554 b(Lemma)475 8879 y(2.12,)509 b(page)489 b(15,)510 b Fu(\037)9206 8440 y Fr(\003)9206 9204 y Fp(1)9733 8879 y Fw(\()p Fu(\025)p Fw(\))489 b(do)34 b(es)489 b(not)h(ha)-34 b(v)g(e)489 b(self-in)-34 b(tersections,)510 b(i.e.,)f(for)489 b(ev)-34 b(ery)488 b Fu(t)477 b Ft(2)h Fu(\037)39948 8440 y Fr(\003)39948 9204 y Fp(1)40475 8879 y Fw(\()p Fu(\025)p Fw(\),)510 b(there)489 b(is)g(a)g(single)507 10319 y Fu(~)-638 b Fn({)49 b Fw(\()p Fu(')p Fw(\))539 b Ft(2)h Fo(D)5961 10501 y Fp(1)6486 10319 y Fw(\()p Fu(\025)p Fw(\),)557 b(suc)-34 b(h)526 b(that)h Fu(t)539 b Fw(=)g Fo(K)18088 10501 y Fp(1)18646 10319 y Fu(~)-638 b Fn({)49 b Fw(\()p Fu(')p Fw(\).)903 b(Next,)556 b(if)526 b Fu(\034)676 b Ft(2)539 b Fo(K)30681 10501 y Fp(2)31207 10319 y Fu(\037)31965 9879 y Fr(\003)31965 10643 y Fp(1)32492 10319 y Fw(\()p Fu(\025)p Fw(\),)556 b(then)526 b(there)g(is)f Fu(p)539 b Ft(2)g Fu(P)695 b Fw(suc)-34 b(h)526 b(that)475 11758 y Fu(\034)453 b Fw(+)316 b(2)p Fu(\031)43 b(p=)-67 b(N)6180 11940 y Fp(1)6706 11758 y Fu(a)454 b Ft(2)h Fu(\037)9822 11318 y Fr(\003)9822 12082 y Fp(1)10348 11758 y Fw(\()p Fu(\025)p Fw(\).)751 b(Note)475 b(that)h Fu(p)e Fw(is)g(not)i(uniquely)e(de\014ned)i(b)-34 b(y)475 b Fu(\034)137 b Fw(,)492 b(since)474 b Fo(K)39900 11940 y Fp(2)40427 11758 y Fu(\037)41185 11318 y Fr(\003)41185 12082 y Fp(1)41711 11758 y Fw(\()p Fu(\025)p Fw(\))h(ma)-34 b(y)475 b(ha)-34 b(v)g(e)475 b(self-)475 13197 y(in)-34 b(tersections.)808 b(Hence,)516 b(ev)-34 b(ery)492 b Fu(\034)624 b Ft(2)486 b Fo(K)18990 13379 y Fp(2)19517 13197 y Fu(\037)20275 12757 y Fr(\003)20275 13521 y Fp(1)20801 13197 y Fw(\()p Fu(\025)p Fw(\))495 b(can)f(b)34 b(e)493 b(represen)-34 b(ted)494 b(as)g Fu(\034)624 b Fw(=)486 b Fo(K)38688 13379 y Fp(2)39247 13197 y Fu(~)-639 b Fn({)49 b Fw(\()p Fu(')p Fw(\),)517 b(here)526 b Fu(~)-639 b Fn({)49 b Fw(\()p Fu(')p Fw(\))495 b(is)e(not)475 14748 y(necessary)289 b(uniquely)h(de\014ned.)501 b(The)291 b(next)f(lemma)g(describ)34 b(es)289 b(self-in)-34 b(tersection)291 b(of)38972 14442 y(~)38767 14748 y Fu(S)39510 14930 y Fp(1)40326 14748 y Fw(b)34 b(elonging)290 b(to)h Fo(K)48065 14930 y Fp(2)48591 14748 y Fu(\037)49349 14308 y Fr(\003)49349 15073 y Fp(1)49875 14748 y Fw(\()p Fu(\025)p Fw(\))475 16187 y(as)404 b(zeros)f(of)i(a)f(group)h(of)f(determinan)-34 b(ts.)p 0 TeXcolorgray 475 18515 a FF(Lemma)528 b(3.17.)p 0 TeXcolorgray 584 w FC(If)485 b Fu(\034)622 b FC(is)485 b(a)f(p)-62 b(oint)484 b(of)g(self-interse)-62 b(ction)481 b(of)28941 18208 y Fw(~)28737 18515 y Fu(S)29480 18697 y Fp(1)30006 18515 y Fw(\()p Fu(\025)p Fw(\))486 b FC(b)-62 b(elonging)483 b(to)h Fo(K)39962 18697 y Fp(2)40488 18515 y Fu(\037)41246 18075 y Fr(\003)41246 18840 y Fp(1)41773 18515 y Fw(\()p Fu(\025)p Fw(\))p FC(,)498 b(then)484 b Fu(\034)622 b FC(c)-62 b(an)485 b(b)-62 b(e)475 19954 y(r)g(epr)g(esente)g(d)431 b(in)j(the)e(form)h Fu(\034)474 b Fw(=)337 b Fo(K)16502 20136 y Fp(2)17060 19954 y Fu(~)-638 b Fn({)49 b Fw(\()p Fu(')p Fw(\))p FC(,)434 b(wher)-62 b(e)433 b Fu(')336 b Ft(2)h Fw(\002)27066 20136 y Fp(1)28025 19954 y FC(and)434 b(satis\014es)d(the)i(e)-62 b(quation)10343 22521 y Fw(det)12026 21175 y Fq(\020)12750 22521 y Fu(I)364 b Fw(+)270 b Fu(A)15769 22703 y Fp(1)16294 21539 y Fq(\000)16840 22521 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))19223 21539 y Fq(\001)19779 21175 y(\021)20840 22521 y Fw(=)336 b(0)p Fu(;)1493 b(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))337 b(=)369 b Fu(~)-639 b Fn({)49 b Fw(\()p Fu(')p Fw(\))270 b(+)32492 22201 y Fu(~)32629 22521 y(b;)34419 22201 y(~)34555 22521 y(b)337 b Fw(=)f(2)p Fu(\031)43 b(p=)-67 b(N)40154 22703 y Fp(1)40680 22521 y Fu(a;)6770 b Fw(\(3.32\))475 25087 y FC(for)463 b(some)g Fu(p)391 b Ft(2)g Fu(P)460 b Ft(n)291 b(f)p Fw(0)p Ft(g)p FC(.)648 b(Conversely,)469 b(if)464 b(\(3.32\))d(is)i(satis\014e)-62 b(d)462 b(for)h(some)g Fu(p)391 b Ft(2)g Fu(P)460 b Ft(n)291 b(f)p Fw(0)p Ft(g)p FC(,)472 b(then)463 b Fu(\034)528 b Fw(=)391 b Fo(K)48406 25269 y Fp(2)48965 25087 y Fu(~)-638 b Fn({)49 b Fw(\()p Fu(')p Fw(\))475 26526 y FC(is)433 b(a)g(p)-62 b(oint)433 b(of)f(self-interse)-62 b(ction.)p 0 TeXcolorgray 475 28854 a(Pr)g(o)g(of.)p 0 TeXcolorgray 552 w Fw(Supp)34 b(ose,)452 b Fu(\034)579 b Fw(is)441 b(a)h(p)34 b(oin)-34 b(t)443 b(of)f(self-in)-34 b(tersection)442 b(of)27689 28548 y(~)27484 28854 y Fu(S)28227 29036 y Fp(1)28753 28854 y Fw(\()p Fu(\025)p Fw(\))h(b)34 b(elonging)442 b(to)g Fo(K)38887 29036 y Fp(2)39413 28854 y Fu(\037)40171 28414 y Fr(\003)40171 29179 y Fp(1)40697 28854 y Fw(\()p Fu(\025)p Fw(\).)652 b(By)441 b(Lemma)h(3.4,)475 30293 y(page)412 b(17,)h(there)f(is)514 b(~)-709 b Fu(p)350 b Ft(2)f Fu(P)580 b Fw(\(here)412 b(w)-34 b(e)412 b(use)515 b(~)-709 b Fu(p)412 b Fw(instead)g(of)g Fu(p)g Fw(in)g(Lemma)f(3.4\),)i (suc)-34 b(h)413 b(that)g Fu(\034)f Fw(+)274 b(2)p Fu(\031)146 b Fw(~)-709 b Fu(p=)-67 b(N)47231 30475 y Fp(1)47757 30293 y Fu(a)349 b Ft(2)g Fu(\037)50662 29853 y Fr(\003)50662 30618 y Fp(1)51189 30293 y Fw(,)475 31732 y(and)508 b(^)-709 b Fu(p)336 b Ft(2)h Fu(P)168 b Fw(,)507 b(^)-709 b Fu(p)337 b Ft(6)p Fw(=)439 b(~)-709 b Fu(p)p Fw(,)404 b(suc)-34 b(h)405 b(that)g Fu(\034)i Fw(+)269 b(2)p Fu(\031)146 b Fw(^)-709 b Fu(p=)-67 b(N)21159 31914 y Fp(1)21684 31732 y Fu(a)337 b Ft(2)g Fu(S)24550 31914 y Fp(1)25076 31732 y Fw(\()p Fu(\025)p Fw(\).)539 b(This)405 b(means)f(that)6755 34299 y(det)8439 32954 y Fq(\020)9162 34299 y Fu(I)365 b Fw(+)269 b Fu(A)12181 34481 y Fp(1)12707 33317 y Fq(\000)13262 34299 y Fu(\034)407 b Fw(+)269 b(2)p Fu(\031)146 b Fw(~)-709 b Fu(p=)-67 b(N)18874 34481 y Fp(1)19400 34299 y Fu(a)20041 33317 y Fq(\001)20596 32954 y(\021)21657 34299 y Fw(=)336 b(0)1212 b(and)1415 b(det)29805 32954 y Fq(\020)30529 34299 y Fu(I)364 b Fw(+)269 b Fu(A)33547 34481 y Fp(1)34073 33317 y Fq(\000)34629 34299 y Fu(\034)406 b Fw(+)269 b(2)p Fu(\031)146 b Fw(^)-709 b Fu(p=)-67 b(N)40240 34481 y Fp(1)40766 34299 y Fu(a)41407 33317 y Fq(\001)41962 32954 y(\021)43023 34299 y Fw(=)336 b(0)p Fu(:)3183 b Fw(\(3.33\))475 36866 y(By)336 b(Lemma)h(2.13,)350 b(page)338 b(15,)351 b(there)337 b(is)g(a)g(single)370 b Fu(~)-639 b Fn({)49 b Fw(\()p Fu(')p Fw(\))338 b Ft(2)e Fo(D)27591 37048 y Fp(1)28117 36866 y Fw(\()p Fu(\025)p Fw(\):)506 b Fo(K)31606 37048 y Fp(1)32164 36866 y Fu(~)-638 b Fn({)386 b Fw(=)336 b Fu(\034)273 b Fw(+)136 b(2)p Fu(\031)146 b Fw(~)-709 b Fu(p=)-67 b(N)39950 37048 y Fp(1)40476 36866 y Fu(a)337 b Fw(\(this)h(means)g Fo(K)48863 37048 y Fp(2)49421 36866 y Fu(~)-638 b Fn({)385 b Fw(=)475 38674 y Fu(\034)137 b Fw(\).)522 b(According)355 b(to)g(Remark)f(3.16)g (and)i(form)-34 b(ulae)355 b(\(3.33\),)364 b(det)29368 37328 y Fq(\020)30092 38674 y Fu(I)265 b Fw(+)170 b Fu(A)32912 38856 y Fp(1)33438 38674 y Fw(\()32 b Fu(~)-638 b Fn({)50 b Fw(\))35238 37328 y Fq(\021)36299 38674 y Fw(=)336 b(0,)364 b(det)40569 37328 y Fq(\020)41293 38674 y Fu(I)265 b Fw(+)170 b Fu(A)44113 38856 y Fp(1)44639 38674 y Fw(\()32 b Fu(~)-638 b Fn({)220 b Fw(+)47115 38354 y Fu(~)47251 38674 y(b)p Fw(\))48242 37328 y Fq(\021)49303 38674 y Fw(=)336 b(0,)475 40758 y(where)3844 40438 y Fu(~)3980 40758 y(b)h Fw(=)6116 39776 y Fq(\002)6621 40758 y Fw(2)p Fu(\031)43 b Fw(\()103 b(^)-709 b Fu(p)270 b Ft(\000)372 b Fw(~)-709 b Fu(p)p Fw(\))p Fu(=)-67 b(N)13118 40940 y Fp(1)13644 40758 y Fu(a)14285 39776 y Fq(\003)14790 41238 y Fp(mo)26 b(d)314 b Fs(K)17700 41361 y FD(1)18217 40758 y Fw(.)538 b(Th)-34 b(us,)406 b(w)-34 b(e)404 b(ha)-34 b(v)g(e)405 b(obtained)g(\(3.32\).)1803 42380 y(Supp)34 b(ose)479 b(no)-34 b(w)478 b(that)h(\(3.32\))f(holds)g(for)f(some)g Fu(p)459 b Ft(2)g Fu(P)168 b Fw(.)759 b(Let)477 b(us)g(sho)-34 b(w)479 b(that)f Fu(\034)596 b Fw(=)459 b Fo(K)41339 42562 y Fp(2)41897 42380 y Fu(~)-638 b Fn({)49 b Fw(\()p Fu(')p Fw(\))478 b(is)f(a)h(p)34 b(oin)-34 b(t)478 b(of)475 43819 y(self-in)-34 b(tersection.)814 b(Let)598 b(~)-709 b Fu(p)490 b Ft(2)f Fu(P)168 b Fw(:)723 b Fu(\034)467 b Fw(+)331 b(2)p Fu(\031)146 b Fw(~)-709 b Fu(p=)-67 b(N)22258 44001 y Fp(1)22783 43819 y Fu(a)490 b Fw(=)f Fo(K)26343 44001 y Fp(1)26902 43819 y Fu(~)-639 b Fn({)49 b Fw(;)542 b(suc)-34 b(h)599 b(~)-709 b Fu(p)p Fw(,)519 b(ob)-34 b(viously)-101 b(,)518 b(exists,)g(since)496 b Fo(K)46463 44001 y Fp(1)47021 43819 y Fu(~)-638 b Fn({)539 b Ft(2)489 b Fu(K)50662 44001 y Fp(1)51189 43819 y Fw(,)475 45258 y Fu(\034)474 b Ft(2)336 b Fu(K)3652 45440 y Fp(2)4179 45258 y Fw(.)538 b(Since)436 b Fu(~)-638 b Fn({)49 b Fw(\()p Fu(')p Fw(\))337 b Ft(2)g Fo(D)13300 45440 y Fp(1)13826 45258 y Fw(\()p Fu(\025)p Fw(\),)404 b(w)-34 b(e)405 b(ha)-34 b(v)g(e)404 b Fu(\034)j Fw(+)269 b(2)p Fu(\031)146 b Fw(~)-709 b Fu(p=)-67 b(N)26407 45440 y Fp(1)26933 45258 y Fu(a)336 b Ft(2)h Fu(\037)29813 44819 y Fr(\003)29813 45583 y Fp(1)30676 45258 y Ft(\032)f Fu(S)32698 45440 y Fp(1)33224 45258 y Fw(\()p Fu(\025)p Fw(\).)539 b(Therefore,)404 b(there)g(is)g Fu(n)336 b Ft(2)h Fn(N)p Fw(:)19827 47618 y Fu(\025)20534 47117 y Fp(\(1\))20534 47917 y Fs(n)21792 47618 y Fw(\()p Fu(\034)407 b Fw(+)269 b(2)p Fu(\031)146 b Fw(~)-709 b Fu(p=)-67 b(N)27875 47800 y Fp(1)28401 47618 y Fu(a)p Fw(\))337 b(=)f Fu(\025:)16255 b Fw(\(3.34\))475 49816 y(Next,)356 b(it)344 b(follo)-34 b(ws)345 b(from)f(\(3.32\))h(and)g(Remarks)f (3.15,)356 b(3.16)344 b(that)h Fo(K)31237 49998 y Fp(1)31754 49816 y Fu(~)-596 b(y)43 b Fw(\()p Fu(')p Fw(\))338 b Ft(2)f Fu(S)36362 49998 y Fp(1)36888 49816 y Fw(\()p Fu(\025)p Fw(\).)519 b(Hence,)356 b(there)344 b(is)408 b(^)-670 b Fu(\034)473 b Ft(2)337 b Fu(K)50662 49998 y Fp(2)51189 49816 y Fw(,)578 51255 y(^)-709 b Fu(p)336 b Ft(2)h Fu(P)573 b Fw(suc)-34 b(h)405 b(that)469 b(^)-670 b Fu(\034)406 b Fw(+)269 b(2)p Fu(\031)146 b Fw(^)-709 b Fu(p)q(=)-67 b(N)14889 51437 y Fp(1)15414 51255 y Fu(a)337 b Fw(=)f Fo(K)18668 51437 y Fp(1)19184 51255 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))338 b Ft(2)e Fu(S)23792 51437 y Fp(1)24318 51255 y Fw(\()p Fu(\025)p Fw(\).)540 b(This)404 b(means)h(that)g(there)f(is)g(an)465 b(^)-667 b Fu(n)337 b Ft(2)f Fn(N)405 b Fw(suc)-34 b(h)405 b(that)19827 53731 y Fu(\025)20534 53092 y Fp(\(1\))20584 54122 y(^)-520 b Fs(n)21792 53731 y Fw(\()64 b(^)-670 b Fu(\034)407 b Fw(+)269 b(2)p Fu(\031)146 b Fw(^)-709 b Fu(p=)-67 b(N)27875 53913 y Fp(1)28401 53731 y Fu(a)p Fw(\))337 b(=)f Fu(\025:)16255 b Fw(\(3.35\))475 56178 y(Note)434 b(that)h Fu(\034)427 b Ft(\000)353 b Fw(^)-670 b Fu(\034)524 b Fw(=)387 b Fo(K)11660 56360 y Fp(2)12176 56178 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))290 b Ft(\000)f Fo(K)17078 56360 y Fp(2)17637 56178 y Fu(~)-639 b Fn({)49 b Fw(\()p Fu(')p Fw(\))388 b(=)e Fo(K)22910 56360 y Fp(2)23301 55858 y Fu(~)23437 56178 y(b)h Fw(=)f(0,)442 b(i.e.,)e Fu(\034)524 b Fw(=)451 b(^)-670 b Fu(\034)137 b Fw(.)628 b(Considering)435 b(this)g(equalit)-34 b(y)434 b(together)475 57617 y(with)h(\(3.34\),)442 b(\(3.35\),)g(w)-34 b(e)435 b(see)f(that)i Fu(\034)571 b Fw(is)434 b(a)h(p)34 b(oin)-34 b(t)435 b(of)g(self-in)-34 b(tersection,)442 b(pro)-34 b(vided)434 b(that)i Fu(p)387 b Ft(6)p Fw(=)490 b(^)-709 b Fu(p)p Fw(.)629 b(It)434 b(remains)475 59056 y(to)i(sho)-34 b(w)437 b(that)f Fu(p)390 b Ft(6)p Fw(=)491 b(^)-708 b Fu(p)o Fw(.)633 b(In)436 b(fact,)444 b Fu(\034)427 b Fw(+)290 b(2)p Fu(\031)43 b(p=)-67 b(N)21558 59238 y Fp(1)22084 59056 y Fu(a)389 b Fw(=)g Fo(K)25443 59238 y Fp(1)26002 59056 y Fu(~)-639 b Fn({)49 b Fw(,)444 b Fu(\034)427 b Fw(+)290 b(2)p Fu(\031)146 b Fw(^)-709 b Fu(p=)-67 b(N)33260 59238 y Fp(1)33786 59056 y Fu(a)389 b Fw(=)g Fo(K)37145 59238 y Fp(1)37661 59056 y Fu(~)-596 b(y)44 b Fw(,)443 b(hence)436 b(2)p Fu(\031)43 b Fw(\()103 b(^)-709 b Fu(p)291 b Ft(\000)f Fu(p)p Fw(\))p Fu(=)-67 b(N)49027 59238 y Fp(1)49553 59056 y Fu(a)388 b Fw(=)475 60655 y Fo(K)1472 60837 y Fp(1)1998 59674 y Fq(\000)2544 60655 y Fu(~)-596 b(y)265 b Ft(\000)254 b Fu(~)-638 b Fn({)48 b Fw(\))338 b(=)e Fo(K)8519 60837 y Fp(1)8909 60336 y Fu(~)9045 60655 y(b)h Fw(=)11046 60336 y Fu(~)11182 60655 y(b)g Ft(6)p Fw(=)f(0,)385 b(the)c(last)f(equalit)-34 b(y)380 b(holding,)385 b(b)34 b(ecause)32523 60336 y Fu(~)32659 60655 y(b)337 b Ft(2)g Fu(K)35690 60837 y Fp(1)36216 60655 y Fw(.)531 b(By)379 b(Lemma)h(3.4,)385 b Fu(\034)517 b Fw(is)380 b(a)g(p)34 b(oin)-34 b(t)475 62306 y(of)404 b(self-in)-34 b(tersection)405 b(of)12077 62000 y(~)11872 62306 y Fu(S)12615 62488 y Fp(1)13545 62306 y Fw(b)34 b(elonging)404 b(to)h Fo(K)21512 62488 y Fp(2)22038 62306 y Fu(\037)22796 61866 y Fr(\003)22796 62631 y Fp(1)23322 62306 y Fw(\()p Fu(\025)p Fw(\).)p 50750 62472 665 831 v 0 TeXcolorgray 475 64662 a FF(De\014nition)466 b(3.18.)p 0 TeXcolorgray 554 w Fw(Let)403 b Fu(\010)13382 64844 y Fp(1)14312 64662 y Fw(b)34 b(e)404 b(the)h(complex)22797 63680 y Fq(\000)23353 64662 y Fu(k)24022 64222 y Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)26861 64345 y FD(1)27323 64222 y Fr(\000)p Fp(2)p Fs(\016)29030 63680 y Fq(\001)29586 64662 y Fw(-neigh)-34 b(b)34 b(orho)g(o)g(d)405 b(of)f(\002)39926 64844 y Fp(1)40452 64662 y Fw(.)1803 67456 y(By)322 b(Lemma)f(2.11,)371 b Fu(~)-638 b Fn({)48 b Fw(\()p Fu(')p Fw(\))323 b(is)f(an)h(analytic)f (function)i(in)e Fu(\010)27689 67638 y Fp(1)28215 67456 y Fw(,)338 b(and,)h(hence,)g(det)36840 66111 y Fq(\020)37564 67456 y Fu(I)201 b Fw(+)106 b Fu(A)40256 66475 y Fq(\000)40801 67456 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')42713 66475 y Fq(\001)43268 66111 y(\021)43992 67456 y Fw(,)329 b Fu(~)-596 b(y)380 b Fw(=)369 b Fu(~)-638 b Fn({)48 b Fw(\()p Fu(')p Fw(\))106 b(+)50532 67137 y Fu(~)50668 67456 y(b)q Fw(,)339 69221 y Fu(~)475 69541 y(b)337 b Ft(2)f Fu(K)3505 69723 y Fp(1)4032 69541 y Fw(,)403 b(is)h(analytic)g(to)34 b(o.)p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 24 24 TeXDict begin 24 23 bop 0 TeXcolorgray 0 TeXcolorgray 475 3345 a Fp(24)14248 b(YULIA)313 b(KARPESHINA)f(AND)h(YOUNG-RAN)g (LEE)p 0 TeXcolorgray 0 TeXcolorgray 475 6010 a FF(De\014nition)498 b(3.19.)p 0 TeXcolorgray 570 w Fw(W)-101 b(e)431 b(sa)-34 b(y)433 b(that)g Fu(')384 b Ft(2)g Fu(\010)20520 6192 y Fp(1)21478 6010 y Fw(is)432 b(a)g(quasi-in)-34 b(tersection)433 b(of)f Fo(K)35843 6192 y Fp(2)36370 6010 y Fu(\037)37128 5570 y Fr(\003)37128 6335 y Fp(1)38086 6010 y Fw(with)41080 5704 y(~)40875 6010 y Fu(S)41618 6192 y Fp(1)42144 6010 y Fw(\()p Fu(\025)p Fw(\))h(if)g(the)f(relation)475 7449 y(\(3.32\))404 b(holds)h(for)f(some)g Fu(p)337 b Ft(2)g Fu(P)438 b Ft(n)269 b(f)p Fw(0)p Ft(g)p Fu(:)1803 9431 y Fw(Th)-34 b(us,)447 b(real)437 b(in)-34 b(tersections)437 b(corresp)34 b(ond)438 b(to)f(real)g(zeros)f(of)i(the)g(determinan)-34 b(t,)446 b(while)438 b(quasi-in)-34 b(tersections)475 10870 y(ma)g(y)404 b(ha)-34 b(v)g(e)405 b(a)f(small)f(imaginary)h(part) h(\(quasi-in)-34 b(tersections)405 b(include)f(in)-34 b(tersections\).)1803 12309 y(Next)404 b(w)-34 b(e)405 b(describ)34 b(e)403 b(the)h(non-resonance)h(set)g(in)f(terms)g(of)g (determinan)-34 b(ts.)p 0 TeXcolorgray 475 14290 a FF(Lemma)463 b(3.20.)p 0 TeXcolorgray 554 w FC(If)434 b Fu(\034)474 b Ft(2)337 b Fw(\012)13133 14472 y Fp(1)13659 14290 y FC(,)433 b(then)g Fu(\034)474 b Fw(=)336 b Fo(K)20439 14472 y Fp(2)20998 14290 y Fu(~)-638 b Fn({)49 b Fw(\()p Fu(')p Fw(\))434 b FC(wher)-62 b(e)433 b Fu(')336 b Ft(2)h Fw(\002)30632 14472 y Fp(1)31591 14290 y FC(and)434 b(satis\014es)d (the)i(e)-62 b(quation)7993 17329 y Fw(det)9878 15256 y Fq( )10971 16432 y Fu(H)12077 15992 y Fp(\(1\))13334 15450 y Fq(\000)13880 16432 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))16263 15450 y Fq(\001)17088 16432 y Ft(\000)270 b Fu(k)18970 15992 y Fp(2)p Fs(l)20056 16432 y Ft(\000)f Fu(\017)p 10971 17050 10790 49 v 12373 18226 a(H)13381 18408 y Fp(0)13906 17244 y Fq(\000)14452 18226 y Fu(~)-596 b(y)43 b Fw(\()p Fu(')p Fw(\))16834 17244 y Fq(\001)17660 18226 y Fw(+)269 b Fu(k)19541 17876 y Fp(2)p Fs(l)21893 15256 y Fq(!)23189 17329 y Fw(=)337 b(0)p Fu(;)1493 b(~)-596 b(y)43 b Fw(\()p Fu(')p Fw(\))338 b(=)369 b Fu(~)-639 b Fn({)49 b Fw(\()p Fu(')p Fw(\))270 b(+)34842 17009 y Fu(~)34978 17329 y(b)q(;)36768 17009 y(~)36905 17329 y(b)337 b Fw(=)f(2)p Fu(\031)43 b(p=)-67 b(N)42504 17511 y Fp(1)43030 17329 y Fu(a;)4420 b Fw(\(3.36\))475 20367 y FC(for)482 b(some)g Fu(p)425 b Ft(2)h Fu(P)474 b Ft(n)305 b(f)p Fw(0)p Ft(g)483 b FC(and)f Ft(j)p Fu(\017)p Ft(j)425 b Fu(<)h(\017)18195 20549 y Fp(1)18721 20367 y FC(.)704 b(Conversely,)493 b(if)483 b(\(3.36\))d(is)i(satis\014e)-62 b(d)481 b(for)h(some)g Fu(p)425 b Ft(2)h Fu(P)474 b Ft(n)305 b(f)p Fw(0)p Ft(g)p FC(,)495 b(and)475 21806 y Ft(j)p Fu(\017)p Ft(j)336 b Fu(<)g(\017)3748 21988 y Fp(1)4274 21806 y FC(,)433 b(then)g Fu(\034)474 b Fw(=)337 b Fo(K)11055 21988 y Fp(2)11613 21806 y Fu(~)-638 b Fn({)49 b Fw(\()p Fu(')p Fw(\))434 b FC(b)-62 b(elongs)432 b(to)h Fw(\012)21089 21988 y Fp(1)21615 21806 y FC(.)p 0 TeXcolorgray 475 23788 a(Pr)-62 b(o)g(of.)p 0 TeXcolorgray 552 w Fw(The)410 b(pro)34 b(of)410 b(is)f(analogous)i(to)f(the)g(pro)34 b(of)410 b(of)f(the)h(previous)g(lemma)f(after)h(w)-34 b(e)410 b(notice)f(that)i(the)f(set)g(of)475 25227 y(p)34 b(oin)-34 b(ts)405 b Fu(\034)541 b Fw(satisfying)405 b(the)f(inequalit)-34 b(y)475 25573 y Fq(\014)475 26300 y(\014)475 27028 y(\014)879 26967 y Fu(\025)1586 26328 y Fp(\(1\))1586 27125 y Fs(n)2844 26967 y Fw(\()p Fu(\034)406 b Fw(+)269 b(2)p Fu(\031)43 b(p=)-67 b(N)8926 27149 y Fp(1)9452 26967 y Fu(a)p Fw(\))270 b Ft(\000)f Fu(\025)12753 26328 y Fp(\(1\))12803 27359 y(^)-520 b Fs(n)14011 26967 y Fw(\()p Fu(\034)406 b Fw(+)269 b(2)p Fu(\031)146 b Fw(^)-709 b Fu(p)q(=)-67 b(N)20094 27149 y Fp(1)20619 26967 y Fu(a)p Fw(\))21731 25573 y Fq(\014)21731 26300 y(\014)21731 27028 y(\014)22692 26967 y Fu(<)557 b(\017)24684 27149 y Fp(1)25745 26967 y Fw(can)537 b(b)34 b(e)535 b(describ)34 b(ed)536 b(as)g(the)h(set)f(of)h(p)34 b(oin)-34 b(ts)537 b(satisfying)475 29169 y Fu(\025)1182 28530 y Fp(\(1\))1182 29326 y Fs(n)2439 29169 y Fw(\()p Fu(\034)407 b Fw(+)269 b(2)p Fu(\031)43 b(p=)-67 b(N)8522 29351 y Fp(1)9048 29169 y Fu(a)p Fw(\))270 b Ft(\000)f Fu(\025)12349 28530 y Fp(\(1\))12399 29560 y(^)-520 b Fs(n)13606 29169 y Fw(\()p Fu(\034)407 b Fw(+)269 b(2)p Fu(\031)146 b Fw(^)-709 b Fu(p)q(=)-67 b(N)19690 29351 y Fp(1)20215 29169 y Fu(a)p Fw(\))337 b(=)f Fu(\017)404 b Fw(for)h(some)f Ft(j)p Fu(\017)p Ft(j)336 b Fu(<)g(\017)32005 29351 y Fp(1)32531 29169 y Fw(.)p 50750 29335 665 831 v 1803 31272 a(Let)457 b(us)h(discuss)f(separately)g(the)h(case)f Fu(W)21476 31454 y Fp(1)22427 31272 y Fw(=)425 b(0.)698 b(If)457 b Fu(')g Fw(is)g(a)h(p)34 b(oin)-34 b(t)458 b(of)f(in)-34 b(tersection,)471 b(then)458 b(\(3.32\))g(means)475 32711 y(that)17324 34150 y Ft(j)32 b Fu(~)-638 b Fn({)49 b Fw(\()p Fu(')p Fw(\))270 b(+)f(2)p Fu(\031)43 b(j)69 b(=a)270 b Fw(+)f(2)p Fu(\031)43 b(p=)-67 b(N)29835 34332 y Fp(1)30361 34150 y Fu(a)p Ft(j)31339 33592 y Fp(2)32201 34150 y Fw(=)337 b Fu(k)34150 33650 y Fp(2)48428 34150 y Fw(\(3.37\))475 35841 y(for)327 b(some)h Fu(j)406 b Ft(2)337 b Fn(Z)8072 35401 y Fp(2)8925 35841 y Fw(and)329 b Fu(p)336 b Ft(2)h Fu(P)168 b Fw(,)343 b(and)328 b(vise)f(v)-34 b(ersa.)513 b(Th)-34 b(us,)344 b(\(3.32\))328 b(means)g(that)361 b Fu(~)-638 b Fn({)49 b Fw(\()p Fu(')p Fw(\))328 b(is)f(a)h(p)34 b(oin)-34 b(t)328 b(of)g(in)-34 b(tersection)475 37384 y(of)483 b(the)g(circle)e(of)i(the)g(radius)g Fu(k)521 b Fw(cen)-34 b(tered)483 b(at)g(the)g(origin)f(\()p Ft(j)32 b Fu(~)-638 b Fn({)49 b Fw(\()p Fu(')p Fw(\))p Ft(j)31726 36825 y Fp(2)32719 37384 y Fw(=)467 b Fu(k)34798 36944 y Fp(2)35807 37384 y Fw(when)484 b Fu(W)40196 37566 y Fp(1)41189 37384 y Fw(=)467 b(0\))483 b(and)h(the)f(circle)475 38823 y(of)466 b(the)g(same)g(radius)g(cen)-34 b(tered)466 b(at)h(a)f(p)34 b(oin)-34 b(t)466 b(2)p Fu(\031)43 b(j)69 b(=a)312 b Fw(+)e(2)p Fu(\031)43 b(p=)-67 b(N)29778 39005 y Fp(1)30304 38823 y Fu(a)465 b Fw(of)i(a)f(dual)g(lattice)f(corresp)34 b(onding)466 b(to)h(the)475 40262 y(p)34 b(erio)g(ds)427 b Fu(N)5725 40444 y Fp(1)6251 40262 y Fu(a)6892 40444 y Fp(1)7417 40262 y Fw(,)434 b Fu(N)9162 40444 y Fp(1)9687 40262 y Fu(a)10328 40444 y Fp(2)10854 40262 y Fw(.)609 b(If)428 b(there)g(is)f(a)h(quasi-in)-34 b(tersection,)434 b(then)428 b(\(3.37\))h(holds)f(for)g(some)g Fu(')376 b Ft(2)g Fu(\010)46056 40444 y Fp(1)46582 40262 y Fw(.)610 b(W)-101 b(e)427 b(will)475 41701 y(refer)408 b(to)h(suc)-34 b(h)410 b Fu(')e Fw(as)h(a)g(p)34 b(oin)-34 b(t)410 b(of)f(quasi-in)-34 b(tersection)409 b(of)g(t)-34 b(w)g(o)410 b(circles.)551 b(Geometrically)-101 b(,)408 b(suc)-34 b(h)410 b(terminology)e(is)475 43140 y(justi\014ed,)431 b(since)424 b(v)-67 b(alidit)-34 b(y)424 b(of)h(\(3.37\))g(for)g(some)f Fu(')h Fw(with)g(a)g(small)f (imaginary)g(part)h(means,)430 b(that)c(t)-34 b(w)g(o)426 b(circles)475 44579 y(come)403 b(close)h(together,)g(ev)-34 b(en)404 b(if)g(they)g(do)h(not)g(in)-34 b(tersect.)1803 46018 y(W)-101 b(e)363 b(denote)i(b)-34 b(y)364 b Fu(!)10113 46200 y Fp(1)11003 46018 y Fw(the)g(set)g(of)h Fu(')336 b Ft(2)h Fw(\002)19460 46200 y Fp(1)20349 46018 y Fw(corresp)34 b(onding)364 b(to)g(\012)30416 46200 y Fp(1)30943 46018 y Fw(,)371 b(i.e.,)g Fu(!)34663 46200 y Fp(1)35526 46018 y Fw(=)336 b Ft(f)p Fu(')h Ft(2)f Fw(\002)40628 46200 y Fp(1)41491 46018 y Fw(:)g Fo(K)43161 46200 y Fp(2)43720 46018 y Fu(~)-639 b Fn({)44495 46200 y Fp(1)45021 46018 y Fw(\()p Fu(')p Fw(\))338 b Ft(2)f Fw(\012)49114 46200 y Fp(1)49640 46018 y Ft(g)f(\032)475 47457 y Fw([0)p Fu(;)202 b Fw(2)p Fu(\031)43 b Fw(\).)475 49616 y(3.3.4.)605 b FC(Complex)517 b(r)-62 b(esonant)517 b(set.)605 b Fw(F)-101 b(urther)496 b(w)-34 b(e)496 b(consider)f(a)g(complex)g(resonance)h (set)f Fu(!)41940 49176 y Fr(\003)41897 49941 y Fp(1)42467 49616 y Fw(\()p Fu(\025)p Fw(\),)519 b(whic)-34 b(h)496 b(is)g(the)475 51055 y(set)528 b(of)h(zeros)f(of)h(the)g(determinan)-34 b(ts)530 b(\(3.36\))f(in)g Fu(\010)24292 51237 y Fp(1)25346 51055 y Fw(\()p Fu(p)544 b Ft(2)g Fu(P)521 b Ft(n)353 b(f)p Fw(0)p Ft(g)p Fw(,)559 b Ft(j)p Fu(\017)p Ft(j)543 b Fu(<)h(\017)36983 51237 y Fp(1)37509 51055 y Fw(\).)912 b(By)528 b(Lemma)g(3.20,)559 b Fu(!)49512 51237 y Fp(1)50582 51055 y Fw(=)475 52494 y Fu(!)1272 52054 y Fr(\003)1229 52819 y Fp(1)2164 52494 y Ft(\\)365 b Fw(\002)4280 52676 y Fp(1)4806 52494 y Fw(.)970 b(W)-101 b(e)547 b(prefer)h(to)g(consider) g(quasi-in)-34 b(tersections,)584 b(instead)548 b(of)h(in)-34 b(tersections,)584 b(and)548 b(the)h(complex)475 53933 y(resonance)413 b(set,)i(instead)f(of)g(just)g(the)g(real)f(one,)i(for) e(the)h(follo)-34 b(wing)414 b(reason:)557 b(the)414 b(determinan)-34 b(ts)415 b(\(3.32\))e(and)475 55372 y(\(3.36\),)558 b(in)-34 b(v)g(olv)g(ed)527 b(in)g(the)h(de\014nitions) g(of)g(quasi-in)-34 b(tersections)527 b(and)h(the)f(complex)g (resonance)g(set)g Fu(!)48852 54932 y Fr(\003)48809 55697 y Fp(1)49906 55372 y Fw(are)475 56811 y(holomorphic)343 b(functions)h(of)f Fu(')g Fw(in)g Fu(\010)17133 56993 y Fp(1)17659 56811 y Fw(.)518 b(W)-101 b(e)342 b(can)h(apply)g(complex) f(analysis)h(theorem)g(to)g(these)g(determinan)-34 b(ts.)475 58250 y(Rouc)g(h)g(\023)-572 b(e's)335 b(theorem)g(is)g(particularly)f (imp)34 b(ortan)-34 b(t)336 b(here,)348 b(since)334 b(it)h(states)g (stabilit)-34 b(y)336 b(of)f(zeros)f(of)h(a)g(holomorphic)475 59689 y(function)420 b(with)g(resp)34 b(ect)418 b(to)h(small)g(p)34 b(erturbations)419 b(of)h(the)f(function.)584 b(W)-101 b(e)418 b(tak)-34 b(e)419 b(the)h(determinan)-34 b(t)420 b(\(3.32\))f(as)475 61128 y(a)470 b(holomorphic)h(function,)488 b(its)471 b(zeros)f(b)34 b(eing)470 b(quasi-in)-34 b(tersections:)672 b(the)471 b(initial)f(determinan)-34 b(t)472 b(corresp)34 b(onds)475 62567 y(to)424 b(the)g(case)g Fu(W)7814 62749 y Fp(1)8709 62567 y Fw(=)369 b(0,)429 b(the)424 b(p)34 b(erturbation)425 b(is)e(obtained)i(b)-34 b(y)425 b(\\switc)-34 b(hing)425 b(on")f(a)g(p)34 b(oten)-34 b(tial)424 b Fu(W)44183 62749 y Fp(1)44709 62567 y Fw(.)598 b(Since)423 b(there)475 64006 y(is)465 b(no)h(analog)f(of)h(Rouc)-34 b(h)g(\023)-572 b(e's)466 b(theorem)f(\(e.g.)722 b(see)465 b([24]\))g(for)h(real)e (functions)j(on)f(the)g(real)e(axis,)480 b(in)-34 b(tro)34 b(ducing)475 65445 y(the)498 b(region)f Fu(\010)7197 65627 y Fp(1)8221 65445 y Fw(and)h(analytic)g(extension)g(of)g(the)g (determinan)-34 b(ts)499 b(in)-34 b(to)499 b(this)f(region)g(is)f(in)h (the)g(core)f(of)h(our)475 66884 y(considerations.)539 b(W)-101 b(e)403 b(will)h(use)g(also)g(a)g(w)-34 b(ell-kno)g(wn)406 b(inequalit)-34 b(y)403 b(for)i(the)f(determinan)-34 b(ts)406 b(\(e.g.)538 b(see)404 b([21)o(]\):)4902 67802 y Fq(\014)4902 68529 y(\014)4902 69256 y(\014)5306 69196 y Fw(det)q(\()p Fu(I)364 b Fw(+)269 b Fu(A)p Fw(\))h Ft(\000)f Fw(det)q(\()p Fu(I)365 b Fw(+)269 b Fu(B)61 b Fw(\))18148 67802 y Fq(\014)18148 68529 y(\014)18148 69256 y(\014)18889 69196 y Ft(\024)336 b(k)p Fu(A)270 b Ft(\000)f Fu(B)61 b Ft(k)24751 69378 y Fp(1)25277 69196 y Fw(exp)27129 67850 y Fq(\020)27853 69196 y Ft(k)p Fu(A)p Ft(k)29974 69378 y Fp(1)30769 69196 y Fw(+)269 b Ft(k)p Fu(B)61 b Ft(k)34173 69378 y Fp(1)34969 69196 y Fw(+)269 b(1)36787 67850 y Fq(\021)37511 69196 y Fu(;)606 b(A;)202 b(B)398 b Ft(2)336 b FF(S)43137 69378 y Fp(1)43664 69196 y Fu(:)4427 b Fw(\(3.38\))p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 25 25 TeXDict begin 25 24 bop 0 TeXcolorgray 0 TeXcolorgray 2848 3345 a Fp(SPECTRAL)406 b(PR)-26 b(OPER)-78 b(TIES)406 b(OF)h(A)h(POL)-78 b(YHARMONIC)405 b(OPERA)-78 b(TOR)406 b(WITH)h(LIMIT-PERIODIC)e(POTENTIAL)1430 b(25)p 0 TeXcolorgray 475 6001 a Fw(Note)446 b(that)i Fu(!)6912 5562 y Fr(\003)6869 6326 y Fp(1)7846 6001 y Fw(=)408 b Ft([)10005 6248 y Fs(p)p Fr(2)p Fs(P)131 b Fr(nf)p Fp(0)p Fr(g)13770 6001 y Fu(!)14567 5562 y Fr(\003)14524 6326 y Fp(1)p Fs(;p)15785 6001 y Fw(,)457 b(where)447 b Fu(!)20924 5562 y Fr(\003)20881 6326 y Fp(1)p Fs(;p)22588 6001 y Fw(corresp)34 b(onds)447 b(to)g(a)f(\014xed)h Fu(p)f Fw(in)h(\(3.36\);)468 b(and)447 b(similarly)-101 b(,)456 b Fu(!)49649 6183 y Fp(1)50582 6001 y Fw(=)475 7461 y Ft([)1283 7708 y Fs(p)p Fr(2)p Fs(P)131 b Fr(nf)p Fp(0)p Fr(g)5048 7461 y Fu(!)5802 7643 y Fp(1)p Fs(;p)7063 7461 y Fw(.)549 b(W)-101 b(e)407 b(\014x)h Fu(p)342 b Ft(2)h Fu(P)576 b Fw(and)408 b(study)g(eac)-34 b(h)408 b Fu(!)24449 7021 y Fr(\003)24406 7786 y Fp(1)p Fs(;p)26075 7461 y Fw(separately)-101 b(.)548 b(W)-101 b(e)407 b(start)h(with)h(the)f(case)f Fu(W)45793 7643 y Fp(1)46661 7461 y Fw(=)342 b(0.)549 b(The)475 8978 y(corresp)34 b(onding)404 b(determinan)-34 b(t)405 b(\(3.32\))g(is)4161 11682 y(det)5845 10337 y Fq(\020)6568 11682 y Fu(I)365 b Fw(+)269 b Fu(A)9587 11864 y Fp(0)10113 10700 y Fq(\000)10658 11682 y Fu(~)-596 b(y)11262 11864 y Fp(0)11788 11682 y Fw(\()p Fu(')p Fw(\))13523 10700 y Fq(\001)14080 10337 y(\021)14804 11682 y Fu(;)605 b(I)365 b Fw(+)269 b Fu(A)18765 11864 y Fp(0)19291 10700 y Fq(\000)19836 11682 y Fu(~)-596 b(y)20440 11864 y Fp(0)20966 11682 y Fw(\()p Fu(')p Fw(\))22701 10700 y Fq(\001)23594 11682 y Fw(=)24874 10337 y Fq(\020)25598 11682 y Fu(H)26606 11864 y Fp(0)27131 10700 y Fq(\000)27677 11682 y Fu(~)g(y)28281 11864 y Fp(0)28807 11682 y Fw(\()p Fu(')p Fw(\))30542 10700 y Fq(\001)31367 11682 y Ft(\000)269 b Fu(\025)33286 10337 y Fq(\021\020)34734 11682 y Fu(H)35742 11864 y Fp(0)36268 10700 y Fq(\000)36813 11682 y Fu(~)-596 b(y)37417 11864 y Fp(0)37943 11682 y Fw(\()p Fu(')p Fw(\))39678 10700 y Fq(\001)40504 11682 y Fw(+)269 b Fu(\025)42423 10337 y Fq(\021)43147 10639 y Fr(\000)p Fp(1)44405 11682 y Fu(;)3686 b Fw(\(3.39\))465 14443 y Fu(~)-596 b(y)1069 14625 y Fp(0)1595 14443 y Fw(\()p Fu(')p Fw(\))337 b(=)g Fu(k)38 b Fw(\(cos)202 b Fu(';)g Fw(sin)g Fu(')p Fw(\))139 b(+)13283 14123 y Fu(~)13419 14443 y(b)q Fw(.)517 b(This)340 b(determinan)-34 b(t)340 b(can)f(b)34 b(e)339 b(in)-34 b(v)g(estigated)340 b(b)-34 b(y)340 b(elemen)-34 b(tary)338 b(means.)518 b(W)-101 b(e)338 b(easily)475 15882 y(c)-34 b(hec)g(k,)409 b(that)g(the)g(n)-34 b(um)g(b)34 b(er)409 b(of)f(zeros)g(of)h(this)f(determinan)-34 b(t)410 b(in)e Fu(\010)30473 16064 y Fp(1)31407 15882 y Fw(do)34 b(es)408 b(not)h(exceed)f Fu(c)40706 16064 y Fp(0)41231 15882 y Fu(k)41900 15442 y Fp(2+2)p Fs(s)44007 15565 y FD(1)44933 15882 y Fw(\(Prop)34 b(osition)475 17910 y(3.21,)428 b(Corollary)423 b(3.22\).)597 b(The)424 b(resolv)-34 b(en)g(t)19960 16928 y Fq(\000)20515 17910 y Fu(H)21523 18092 y Fp(0)22049 16928 y Fq(\000)22594 17910 y Fu(~)-596 b(y)23198 18092 y Fp(0)23724 17910 y Fw(\()p Fu(')p Fw(\))25459 16928 y Fq(\001)26298 17910 y Ft(\000)282 b Fu(k)28192 17470 y Fp(2)p Fs(l)29010 16564 y Fq(\021)29734 16866 y Fr(\000)p Fp(1)31415 17910 y Fw(has)424 b(p)34 b(oles)423 b(at)h(zeros)f(of)h(the)g(determinan)-34 b(t.)475 19650 y(The)394 b(resolv)-34 b(en)g(t)395 b(norm)f(at)h Fu(')337 b Ft(2)f Fu(\010)15753 19832 y Fp(1)16674 19650 y Fw(can)394 b(b)34 b(e)394 b(estimated)h(b)-34 b(y)394 b(the)h(distance,)h(whic)-34 b(h)395 b Fu(')g Fw(has)f(to)h(the)g(nearest)f(zero)475 21182 y(of)414 b(the)h(determinan)-34 b(t)415 b(\(Lemma)f(3.23\).)569 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Fw(it)g(has)h(the)f(same)h (prop)34 b(erties)440 b(as)i(those)f(inside)h Fu(\010)39434 67820 y Fp(1)39959 67638 y Fw(.)650 b(F)-101 b(urther)442 b(w)-34 b(e)441 b(consider)475 69238 y(\000\()1568 68918 y Fu(~)1704 69238 y(b)p Fw(\))337 b Ft(\032)f Fu(\010)5119 69420 y Fp(1)5645 69238 y Fw(,)404 b Fo(O)p Fw(\()7579 68918 y Fu(~)7715 69238 y(b)p Fw(\))337 b Ft(\032)g Fu(\010)11131 69420 y Fp(1)11657 69238 y Fw(.)p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 28 28 TeXDict begin 28 27 bop 0 TeXcolorgray 0 TeXcolorgray 475 3345 a Fp(28)14248 b(YULIA)313 b(KARPESHINA)f(AND)h(YOUNG-RAN)g (LEE)p 0 TeXcolorgray 0 TeXcolorgray 475 6080 a FF(Remark)538 b(3.28.)p 0 TeXcolorgray 589 w Fw(F)-101 b(or)468 b(an)-34 b(y)469 b Fu(')443 b Ft(2)h Fu(\010)17190 6262 y Fp(1)18028 6080 y Ft(n)313 b Fo(O)p Fw(\()20140 5760 y Fu(~)20276 6080 y(b)p Fw(\))469 b(and)g(an)-34 b(y)469 b Fu(')27304 5640 y Fr(\006)27304 6380 y Fs(m)28660 6080 y Fw(the)f(estimate)h Ft(j)p Fu(')312 b Ft(\000)g Fu(')39221 5640 y 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Fs(m)45383 10231 y Fw(\(0\))p Ft(j)47268 9791 y Fp(2)p Fs(l)47268 10531 y Fr(\003)48423 10231 y Fw(=)336 b Fu(k)50371 9791 y Fp(2)p Fs(l)51189 10231 y Fw(,)475 12185 y Fu(m)578 b Ft(2)h Fn(Z)4312 11745 y Fp(2)4838 12185 y Fw(,)585 b Fu(')578 b Ft(2)g Fu(\010)9325 12367 y Fp(1)9851 12185 y Fw(,)585 b(whic)-34 b(h)550 b(can)f(b)34 b(e)549 b(rewritten)g(as)g Ft(j)-10 b Fu(~)-596 b(y)26516 12367 y Fp(0)27042 12185 y Fw(\()p Fu(')p Fw(\))367 b(+)f Fu(h)31151 12367 y Fp(1)31677 12185 y Fw(\()p Fu(')p Fw(\))-25 b Fu(~)-581 b(\027)444 b Fw(+)375 b Fu(~)-616 b(p)36373 12367 y Fs(m)37261 12185 y Fw(\(0\))p Ft(j)39146 11745 y Fp(2)p Fs(l)39146 12484 y Fr(\003)40542 12185 y Fw(=)578 b Fu(k)42732 11745 y Fp(2)p Fs(l)43549 12185 y Fw(,)585 b(the)550 b(functions)507 13624 y Fu(~)-638 b Fn({)49 b Fw(\()p Fu(')p Fw(\))494 b Ft(\021)f Fn({)49 b Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))498 b(and)h Fu(h)12365 13806 y Fp(1)12891 13624 y Fw(\()p Fu(')p Fw(\))494 b Ft(\021)f Fu(h)17254 13806 y Fp(1)17780 13624 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))499 b(b)34 b(eing)497 b(de\014ned)i(in)g(Lemma)e(2.11.)820 b(It)498 b(is)g(easy)g(to)g(see)g(that)h(the)475 15063 y(v)-34 b(ectors)430 b Fu(~)-638 b Fn({)49 b Fw(\()p Fu(')p Fw(\))399 b(satisfying)f(this)h(equation)g(with)g(a)f(real)g Fu(')g Fw(are)g(just)h(in)-34 b(tersections)399 b(of)g Fo(D)41351 15245 y Fp(1)41876 15063 y Fw(\()p Fu(\025)p Fw(\))g(with)g(the)g(circle)475 16662 y(of)517 b(radius)g Fu(k)554 b Fw(cen)-34 b(tered)517 b(at)g Ft(\000)14244 16342 y Fu(~)14380 16662 y(b)344 b Ft(\000)354 b Fu(~)-616 b(p)17141 16844 y Fs(m)18028 16662 y Fw(\(0\).)877 b(Since)506 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))517 b(di\013er)f(from)507 b Fu(~)-596 b(y)33776 16844 y Fp(0)34302 16662 y Fw(\()p Fu(')p Fw(\))517 b(b)-34 b(y)517 b(a)g(small)f(v)-67 b(alue)516 b Fu(h)46699 16844 y Fp(1)47225 16662 y Fw(\()p Fu(')p Fw(\),)545 b(the)475 18131 y(estimates)350 b(\(3.44\),)360 b(\(3.45\))350 b(and)h(\(3.46\))f(\(3.47\))g(with)g Fu(\017)25390 18313 y Fp(0)26253 18131 y Fw(=)336 b Fu(k)28201 17691 y Fr(\000)p Fp(4)p Fr(\000)p Fp(6)p Fs(s)31040 17814 y FD(1)31502 17691 y Fr(\000)p Fp(3)p Fs(\016)33559 18131 y Fw(are)349 b(preserv)-34 b(ed)349 b(for)h Fu(')336 b Ft(2)h Fu(\010)45757 18313 y Fp(1)46443 18131 y Ft(n)160 b Fo(O)p Fw(,)360 b(when)475 19570 y(w)-34 b(e)404 b(replace)393 b Fu(~)-596 b(y)6964 19752 y Fp(0)7490 19570 y Fw(\()p Fu(')p Fw(\))405 b(b)-34 b(y)395 b Fu(~)-596 b(y)43 b Fw(\()p Fu(')p Fw(\).)540 b(W)-101 b(e)403 b(pro)-34 b(v)g(e)404 b(this)h(result)f(in)g(the)h(follo)-34 b(wing)405 b(lemma.)p 0 TeXcolorgray 475 21913 a FF(Lemma)463 b(3.29.)p 0 TeXcolorgray 554 w FC(If)434 b Fu(b)10629 22095 y Fp(0)11492 21913 y Fu(>)336 b(k)13440 21473 y Fp(7+8)p Fs(s)15547 21596 y FD(1)16009 21473 y Fr(\000)p Fp(2)p Fs(\015)17696 21596 y FD(0)18158 21473 y Fp(+6)p Fs(\016)19865 21913 y FC(,)433 b Fu(\015)21298 22095 y Fp(0)22160 21913 y Fw(=)337 b(2)p Fu(l)293 b Ft(\000)269 b Fw(2)g Ft(\000)g Fw(4)p Fu(s)29174 22095 y Fp(1)29970 21913 y Ft(\000)g Fw(2)p Fu(\016)46 b FC(,)433 b(then:)p 0 TeXcolorgray 2409 24041 a Fw(\(1\))p 0 TeXcolorgray 554 w FC(The)500 b(numb)-62 b(er)500 b(of)f(zer)-62 b(os)499 b(of)h Fw(det)19228 22695 y Fq(\020)19952 24041 y Fu(I)413 b Fw(+)317 b Fu(A)23067 24223 y Fp(0)23593 23059 y Fq(\000)24139 24041 y Fu(~)-596 b(y)43 b Fw(\()p Fu(')p Fw(\))26521 23059 y Fq(\001)27078 22695 y(\021)28302 24041 y FC(in)500 b(any)g Fw(\000\()33338 23721 y Fu(~)33474 24041 y(b)p Fw(\))h FC(is)f(the)f(same)g(as)g(that)g (of)h Fw(det)48913 22695 y Fq(\020)49637 24041 y Fu(I)413 b Fw(+)4511 26333 y Fu(A)5420 26515 y Fp(0)5946 25351 y Fq(\000)6492 26333 y Fu(~)-596 b(y)7096 26515 y Fp(0)7622 26333 y Fw(\()p Fu(')p Fw(\))9357 25351 y Fq(\001)9913 24988 y(\021)10637 26333 y FC(.)p 0 TeXcolorgray 2409 28418 a Fw(\(2\))p 0 TeXcolorgray 554 w FC(The)434 b(fol)62 b(lowing)432 b(estimates)f(hold)h(for)h Fu(')336 b Ft(2)h Fu(\010)25080 28600 y Fp(1)25875 28418 y Ft(n)270 b Fo(O)p Fw(\()27944 28098 y Fu(~)28080 28418 y(b)p Fw(\))p FC(:)14060 29827 y Fq(\015)14060 30554 y(\015)14060 31281 y(\015)14734 29875 y(\020)15458 31221 y Fu(H)16466 31403 y Fp(0)16991 30239 y Fq(\000)17537 31221 y Fu(~)-596 b(y)43 b Fw(\()p Fu(')p Fw(\))19919 30239 y Fq(\001)20745 31221 y Ft(\000)269 b Fu(k)22626 30720 y Fp(2)p Fs(l)23443 29875 y Fq(\021)24167 30177 y Fr(\000)p Fp(1)25425 29827 y Fq(\015)25425 30554 y(\015)25425 31281 y(\015)26435 31221 y Fu(<)337 b(cb)28760 30711 y Fr(\000)p Fp(1)28760 31568 y(0)30017 31221 y Fu(k)30686 30720 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+9+12)p Fs(s)35489 30843 y FD(1)35951 30720 y Fp(+6)p Fs(\016)37658 31221 y Fu(;)10433 b Fw(\(3.48\))13534 32699 y Fq(\015)13534 33426 y(\015)13534 34154 y(\015)14208 32748 y(\020)14932 34093 y Fu(H)15940 34275 y Fp(0)16465 33111 y Fq(\000)17011 34093 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))19394 33111 y Fq(\001)20219 34093 y Ft(\000)269 b Fu(k)22100 33593 y Fp(2)p Fs(l)22918 32748 y Fq(\021)23641 33050 y Fr(\000)p Fp(1)24899 32699 y Fq(\015)24899 33426 y(\015)24899 34154 y(\015)25573 34936 y Fp(1)26435 34093 y Fu(<)337 b(cb)28760 33583 y Fr(\000)p Fp(1)28760 34440 y(0)30017 34093 y Fu(k)30686 33593 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+11+14)p Fs(s)35959 33716 y FD(1)36422 33593 y Fp(+6)p Fs(\016)38129 34093 y Fu(:)9962 b Fw(\(3.49\))p 0 TeXcolorgray 475 37315 a FF(Corollary)483 b(3.30.)p 0 TeXcolorgray 554 w FC(Ther)-62 b(e)433 b(ar)-62 b(e)433 b(no)g(zer)-62 b(os)433 b(of)f Fw(det)23553 35970 y Fq(\020)24277 37315 y Fu(I)364 b Fw(+)269 b Fu(A)27295 37497 y Fp(0)27821 36334 y Fq(\000)28367 37315 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))30750 36334 y Fq(\001)31306 35970 y(\021)32463 37315 y FC(in)434 b Fu(\010)34758 37497 y Fp(1)35553 37315 y Ft(n)270 b Fo(O)p Fw(\()37622 36996 y Fu(~)37758 37315 y(b)p Fw(\))p FC(.)p 0 TeXcolorgray 475 40483 a(L)-62 b(emma)433 b(3.29.)p 0 TeXcolorgray 552 w Fw(Note)370 b(that)i(det)15093 39137 y Fq(\020)15817 40483 y Fu(I)297 b Fw(+)202 b Fu(A)18701 40665 y Fp(0)19226 39501 y Fq(\000)19772 40483 y Fu(~)-596 b(y)20376 40665 y Fp(0)20902 40483 y Fw(\()p Fu(')p Fw(\))22637 39501 y Fq(\001)23193 39137 y(\021)24254 40483 y Fw(=)336 b(0)371 b(if)f(and)h(only)f(if)g Ft(j)-10 b Fu(~)-596 b(y)34544 40665 y Fp(0)35070 40483 y Fw(\()p Fu(')p Fw(\))202 b(+)212 b Fu(~)-616 b(p)38762 40665 y Fs(m)39650 40483 y Fw(\(0\))p Ft(j)41535 40043 y Fp(2)p Fs(l)41535 40782 y Fr(\003)42689 40483 y Fw(=)337 b Fu(k)44638 40043 y Fp(2)p Fs(l)45825 40483 y Fw(for)371 b(at)f(least)475 42775 y(one)339 b Fu(m)e Ft(2)g Fn(Z)5986 42335 y Fp(2)6512 42775 y Fw(.)517 b(The)340 b(analogous)g(statemen)-34 b(t)341 b(holds)f(for)g(det)27470 41430 y Fq(\020)28194 42775 y Fu(I)235 b Fw(+)140 b Fu(A)30954 42957 y Fp(0)31480 41793 y Fq(\000)32025 42775 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))34408 41793 y Fq(\001)34964 41430 y(\021)35688 42775 y Fw(.)517 b(It)340 b(is)f(easy)g(to)h(sho)-34 b(w)340 b(\(App)34 b(endix)475 44583 y(A.2\))404 b(that)8653 45979 y Fq(\014)8653 46706 y(\014)8653 47433 y(\014)9057 47373 y Ft(j)-10 b Fu(~)-596 b(y)9988 47555 y Fp(0)10514 47373 y Fw(\()p Fu(')p Fw(\))270 b(+)279 b Fu(~)-616 b(p)14341 47555 y Fs(m)15229 47373 y Fw(\(0\))p Ft(j)17114 46872 y Fp(2)p Fs(l)17114 47672 y Fr(\003)18201 47373 y Ft(\000)269 b(j)-10 b Fu(~)-596 b(y)43 b Fw(\()p Fu(')p Fw(\))270 b(+)279 b Fu(~)-615 b(p)24215 47555 y Fs(m)25102 47373 y Fw(\(0\))p Ft(j)26987 46872 y Fp(2)p Fs(l)26987 47672 y Fr(\003)27805 45979 y Fq(\014)27805 46706 y(\014)27805 47433 y(\014)28545 47373 y Fu(<)29958 46553 y Fw(1)p 29958 47094 607 49 v 29958 48204 a(2)30697 45979 y Fq(\014)30697 46706 y(\014)30697 47433 y(\014)31101 47373 y Ft(j)-10 b Fu(~)-596 b(y)32032 47555 y Fp(0)32557 47373 y Fw(\()p Fu(')p Fw(\))271 b(+)278 b Fu(~)-615 b(p)36385 47555 y Fs(m)37272 47373 y Fw(\(0\))p Ft(j)39157 46872 y Fp(2)p Fs(l)39157 47672 y Fr(\003)40244 47373 y Ft(\000)269 b Fu(k)42125 46872 y Fp(2)p Fs(l)42943 45979 y Fq(\014)42943 46706 y(\014)42943 47433 y(\014)48428 47373 y Fw(\(3.50\))475 50276 y(for)443 b(an)-34 b(y)443 b Fu(')g Fw(in)g Fu(\010)8194 50458 y Fp(1)9016 50276 y Ft(n)295 b Fo(O)p Fw(\()11110 49956 y Fu(~)11246 50276 y(b)p Fw(\))444 b(and)f Fu(m)402 b Ft(2)g Fn(Z)18560 49836 y Fp(2)19086 50276 y Fw(.)655 b(By)442 b(Rouc)-34 b(h)g(\023)-572 b(e's)444 b(theorem)f([24],)452 b(the)443 b(n)-34 b(um)g(b)34 b(er)445 b(of)e(zeros)f(of)i Ft(j)-10 b Fu(~)-596 b(y)48026 50458 y Fp(0)48551 50276 y Fw(\()p Fu(')p Fw(\))296 b(+)484 51875 y Fu(~)-615 b(p)1085 52057 y Fs(m)1972 51875 y Fw(\(0\))p Ft(j)3857 51435 y Fp(2)p Fs(l)3857 52175 y Fr(\003)5019 51875 y Ft(\000)343 b Fu(k)6974 51435 y Fp(2)p Fs(l)8307 51875 y Fw(in)516 b(ev)-34 b(ery)514 b(\000\()14239 51555 y Fu(~)14375 51875 y(b)q Fw(\))h(is)h(the)g(same)g(as)f(that)i(of)f Ft(j)-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))344 b(+)353 b Fu(~)-615 b(p)33341 52057 y Fs(m)34228 51875 y Fw(\(0\))p Ft(j)36113 51435 y Fp(2)p Fs(l)36113 52175 y Fr(\003)37274 51875 y Ft(\000)344 b Fu(k)39230 51435 y Fp(2)p Fs(l)40047 51875 y Fw(.)873 b(Therefore,)544 b(the)516 b(\014rst)475 53314 y(statemen)-34 b(t)405 b(of)g(the)f(lemma)g(is)g(pro)-34 b(v)g(ed.)1803 54753 y(T)-101 b(o)405 b(pro)-34 b(v)g(e)404 b(the)h(second)f(statemen)-34 b(t,)405 b(w)-34 b(e)405 b(use)f(the)h(Hilb)34 b(ert)403 b(relation:)5860 56143 y Fq(\020)6584 57489 y Fu(H)7592 57671 y Fp(0)8117 56507 y Fq(\000)8663 57489 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))11046 56507 y Fq(\001)11871 57489 y Ft(\000)269 b Fu(k)13752 56988 y Fp(2)p Fs(l)14570 56143 y Fq(\021)15294 56445 y Fr(\000)p Fp(1)16888 57489 y Fw(=)18167 56143 y Fq(\020)18891 57489 y Fu(H)19899 57671 y Fp(0)20425 56507 y Fq(\000)20970 57489 y Fu(~)-596 b(y)21574 57671 y Fp(0)22100 57489 y Fw(\()p Fu(')p Fw(\))23835 56507 y Fq(\001)24661 57489 y Ft(\000)269 b Fu(k)26542 56988 y Fp(2)p Fs(l)27359 56143 y Fq(\021)28083 56445 y Fr(\000)p Fp(1)9279 60361 y Fw(+)10491 59016 y Fq(\020)11215 60361 y Fu(H)12223 60543 y Fp(0)12748 59379 y Fq(\000)13294 60361 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))15677 59379 y Fq(\001)16502 60361 y Ft(\000)269 b Fu(k)18383 59861 y Fp(2)p Fs(l)19201 59016 y Fq(\021)19925 59318 y Fr(\000)p Fp(1)21182 59016 y Fq(\020)21906 60361 y Fu(H)22914 60543 y Fp(0)23440 59379 y Fq(\000)23985 60361 y Fu(~)-596 b(y)24589 60543 y Fp(0)25115 60361 y Fw(\()p Fu(')p Fw(\))26850 59379 y Fq(\001)27676 60361 y Ft(\000)269 b Fu(H)29896 60543 y Fp(0)30421 59379 y Fq(\000)30967 60361 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))33350 59379 y Fq(\001)33906 59016 y(\021\020)35354 60361 y Fu(H)36362 60543 y Fp(0)36887 59379 y Fq(\000)37433 60361 y Fu(~)-596 b(y)38037 60543 y Fp(0)38563 60361 y Fw(\()p Fu(')p Fw(\))40298 59379 y Fq(\001)41123 60361 y Ft(\000)269 b Fu(k)43004 59861 y Fp(2)p Fs(l)43822 59016 y Fq(\021)44546 59318 y Fr(\000)p Fp(1)45803 60361 y Fu(:)475 63090 y Fw(By)403 b(Lemma)h(3.23)g(with)h Fu(\017)12516 63272 y Fp(0)13378 63090 y Fw(=)337 b Fu(k)15327 62650 y Fr(\000)p Fp(4)p Fr(\000)p Fp(6)p Fs(s)18166 62773 y FD(1)18628 62650 y Fr(\000)p Fp(3)p Fs(\016)20335 63090 y Fw(,)13819 64431 y Fq(\015)13819 65159 y(\015)13819 65886 y(\015)14493 64480 y(\020)15216 65825 y Fu(H)16224 66007 y Fp(0)16750 64843 y Fq(\000)17295 65825 y Fu(~)-596 b(y)17899 66007 y Fp(0)18425 65825 y Fw(\()p Fu(')p Fw(\))20160 64843 y Fq(\001)20986 65825 y Ft(\000)269 b Fu(k)22867 65325 y Fp(2)p Fs(l)23685 64480 y Fq(\021)24408 64782 y Fr(\000)p Fp(1)25666 64431 y Fq(\015)25666 65159 y(\015)25666 65886 y(\015)26676 65825 y Fu(<)337 b(cb)29001 65315 y Fr(\000)p Fp(1)29001 66172 y(0)30258 65825 y Fu(k)30927 65325 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+9+12)p Fs(s)35730 65448 y FD(1)36192 65325 y Fp(+6)p Fs(\016)37900 65825 y Fu(;)13293 67304 y Fq(\015)13293 68031 y(\015)13293 68758 y(\015)13967 67352 y(\020)14691 68698 y Fu(H)15699 68880 y Fp(0)16224 67716 y Fq(\000)16770 68698 y Fu(~)-596 b(y)17374 68880 y Fp(0)17900 68698 y Fw(\()p Fu(')p Fw(\))19635 67716 y Fq(\001)20460 68698 y Ft(\000)269 b Fu(k)22341 68197 y Fp(2)p Fs(l)23159 67352 y Fq(\021)23883 67654 y Fr(\000)p Fp(1)25140 67304 y Fq(\015)25140 68031 y(\015)25140 68758 y(\015)25814 69541 y Fp(1)26676 68698 y Fu(<)337 b(cb)29001 68188 y Fr(\000)p Fp(1)29001 69045 y(0)30258 68698 y Fu(k)30927 68197 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+11+14)p Fs(s)36200 68320 y FD(1)36663 68197 y Fp(+6)p Fs(\016)38370 68698 y Fu(:)p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 29 29 TeXDict begin 29 28 bop 0 TeXcolorgray 0 TeXcolorgray 2848 3345 a Fp(SPECTRAL)406 b(PR)-26 b(OPER)-78 b(TIES)406 b(OF)h(A)h(POL)-78 b(YHARMONIC)405 b(OPERA)-78 b(TOR)406 b(WITH)h(LIMIT-PERIODIC)e(POTENTIAL)1430 b(29)p 0 TeXcolorgray 475 6001 a Fw(W)-101 b(e)403 b(easily)g(obtain)i(from)g(\(3.50\))f (that)11131 7139 y Fq(\015)11131 7866 y(\015)11131 8593 y(\015)11804 7187 y(\020)12528 8533 y Fu(H)13536 8715 y Fp(0)14061 7551 y Fq(\000)14607 8533 y Fu(~)-596 b(y)15211 8715 y Fp(0)15737 8533 y Fw(\()p Fu(')p Fw(\))17472 7551 y Fq(\001)18297 8533 y Ft(\000)270 b Fu(H)20518 8715 y Fp(0)21043 7551 y Fq(\000)21589 8533 y Fu(~)-596 b(y)43 b Fw(\()p Fu(')p Fw(\))23971 7551 y Fq(\001)24528 7187 y(\021)o(\020)25975 8533 y Fu(H)26983 8715 y Fp(0)27509 7551 y Fq(\000)28054 8533 y Fu(~)-596 b(y)28658 8715 y Fp(0)29184 8533 y Fw(\()p Fu(')p Fw(\))30919 7551 y Fq(\001)31745 8533 y Ft(\000)269 b Fu(k)33626 8032 y Fp(2)p Fs(l)34443 7187 y Fq(\021)35167 7489 y Fr(\000)p Fp(1)36425 7139 y Fq(\015)36425 7866 y(\015)36425 8593 y(\015)37435 8533 y Fu(<)337 b Fw(1)p Fu(=)p Fw(2)p Fu(:)7558 b Fw(\(3.51\))475 10844 y(No)-34 b(w,)404 b(using)h(a)f(standard)i (argumen)-34 b(t)405 b(as)f(in)g(Lemma)g(2.9,)g(w)-34 b(e)404 b(arriv)-34 b(e)403 b(at)i(\(3.48\))f(and)h(\(3.49\).)p 50750 11010 665 831 v 0 TeXcolorgray 475 12979 a FF(Lemma)463 b(3.31.)p 0 TeXcolorgray 554 w FC(If)434 b Fu(b)10629 13161 y Fp(0)11492 12979 y Fu(>)336 b(k)13440 12539 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+9+12)p Fs(s)18243 12662 y FD(1)18705 12539 y Fp(+7)p Fs(\016)20413 12979 y FC(,)433 b(then)g(the)f(fol)62 b(lowing)432 b(estimates)f(hold)h(for)h(any)h Fu(')336 b Ft(2)h Fu(\010)46340 13161 y Fp(1)47135 12979 y Ft(n)270 b Fo(O)p Fw(\()49204 12659 y Fu(~)49340 12979 y(b)p Fw(\))p FC(:)13982 13896 y Fq(\015)13982 14624 y(\015)13982 15351 y(\015)14655 15290 y Fw(\()p Fu(H)16232 14790 y Fp(\(1\))17490 15290 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\)\))271 b Ft(\000)e Fu(k)22957 14790 y Fp(2)p Fs(l)23774 15290 y Fw(\))24245 14790 y Fr(\000)p Fp(1)25503 13896 y Fq(\015)25503 14624 y(\015)25503 15351 y(\015)26514 15290 y Fu(<)336 b(cb)28838 14780 y Fr(\000)p Fp(1)28838 15637 y(0)30095 15290 y Fu(k)30764 14790 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+9+12)p Fs(s)35567 14913 y FD(1)36030 14790 y Fp(+6)p Fs(\016)37737 15290 y Fu(;)10354 b Fw(\(3.52\))13456 16549 y Fq(\015)13456 17276 y(\015)13456 18003 y(\015)14129 17942 y Fw(\()p Fu(H)15706 17442 y Fp(\(1\))16965 17942 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\)\))270 b Ft(\000)f Fu(k)22431 17442 y Fp(2)p Fs(l)23249 17942 y Fw(\))23720 17442 y Fr(\000)p Fp(1)24978 16549 y Fq(\015)24978 17276 y(\015)24978 18003 y(\015)25651 18786 y Fp(1)26514 17942 y Fu(<)336 b(cb)28838 17433 y Fr(\000)p Fp(1)28838 18290 y(0)30095 17942 y Fu(k)30764 17442 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+11+14)p Fs(s)36037 17565 y FD(1)36500 17442 y Fp(+6)p Fs(\016)38207 17942 y Fu(:)9884 b Fw(\(3.53\))p 0 TeXcolorgray 475 20617 a FF(Corollary)483 b(3.32.)p 0 TeXcolorgray 554 w FC(The)433 b(determinant)f(of)h Fu(I)365 b Fw(+)269 b Fu(A)23866 20799 y Fp(1)24392 19636 y Fq(\000)24937 20617 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))27320 19636 y Fq(\001)28310 20617 y FC(has)432 b(no)i(zer)-62 b(os)432 b(in)i Fu(\010)37559 20799 y Fp(1)38354 20617 y Ft(n)270 b Fo(O)p Fw(\()40423 20297 y Fu(~)40559 20617 y(b)p Fw(\))p FC(.)p 0 TeXcolorgray 475 23301 a(Pr)-62 b(o)g(of.)p 0 TeXcolorgray 552 w Fw(W)-101 b(e)404 b(use)g(Hilb)34 b(ert)403 b(relation)17028 21956 y Fq(\020)17752 23301 y Fu(H)18858 22862 y Fp(\(1\))20116 22320 y Fq(\000)20661 23301 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))23044 22320 y Fq(\001)23870 23301 y Ft(\000)269 b Fu(k)25751 22862 y Fp(2)p Fs(l)26568 21956 y Fq(\021)27292 22258 y Fr(\000)p Fp(1)28886 23301 y Fw(=)6861 24972 y Fq(\020)7585 26318 y Fu(H)8593 26500 y Fp(0)9118 25336 y Fq(\000)9664 26318 y Fu(~)-596 b(y)43 b Fw(\()p Fu(')p Fw(\))12046 25336 y Fq(\001)12872 26318 y Ft(\000)269 b Fu(k)14753 25817 y Fp(2)p Fs(l)15571 24972 y Fq(\021)16294 25274 y Fr(\000)p Fp(1)17821 26318 y Fw(+)19034 24972 y Fq(\020)19757 26318 y Fu(H)20863 25817 y Fp(\(1\))22121 25336 y Fq(\000)22667 26318 y Fu(~)-596 b(y)43 b Fw(\()p Fu(')p Fw(\))25049 25336 y Fq(\001)25875 26318 y Ft(\000)269 b Fu(k)27756 25817 y Fp(2)p Fs(l)28574 24972 y Fq(\021)29297 25274 y Fr(\000)p Fp(1)30555 26318 y Fw(\()p Ft(\000)p Fu(W)33114 26500 y Fp(1)33640 26318 y Fw(\))34111 24972 y Fq(\020)34835 26318 y Fu(H)35843 26500 y Fp(0)36369 25336 y Fq(\000)36914 26318 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))39297 25336 y Fq(\001)40122 26318 y Ft(\000)270 b Fu(k)42004 25817 y Fp(2)p Fs(l)42821 24972 y Fq(\021)43545 25274 y Fr(\000)p Fp(1)44802 26318 y Fu(:)475 28775 y Fw(Since)421 b Fu(b)4177 28957 y Fp(0)5069 28775 y Fu(>)366 b(k)7047 28335 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+9+12)p Fs(s)11850 28458 y FD(1)12312 28335 y Fp(+7)p Fs(\016)14386 28775 y Fu(>)366 b(k)16364 28335 y Fp(7+8)p Fs(s)18471 28458 y FD(1)18933 28335 y Fr(\000)p Fp(2)p Fs(\015)20620 28458 y FD(0)21081 28335 y Fp(+6)p Fs(\016)23632 28775 y Fw(\()p Fu(\015)24731 28957 y Fp(0)25622 28775 y Fw(=)g(2)p Fu(l)305 b Ft(\000)281 b Fw(2)g Ft(\000)g Fw(4)p Fu(s)32713 28957 y Fp(1)33520 28775 y Ft(\000)g Fw(2)p Fu(\016)46 b Fw(\),)426 b(the)c(previous)f(lemma)h(can)f(b)34 b(e)475 30214 y(applied.)539 b(It)404 b(yields:)11115 31284 y Fq(\015)11115 32011 y(\015)11115 32739 y(\015)11788 31333 y(\020)12512 32678 y Fu(H)13520 32860 y Fp(0)14045 31696 y Fq(\000)14591 32678 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))16974 31696 y Fq(\001)17799 32678 y Ft(\000)269 b Fu(k)19680 32178 y Fp(2)p Fs(l)20498 31333 y Fq(\021)21222 31635 y Fr(\000)p Fp(1)22479 31284 y Fq(\015)22479 32011 y(\015)22479 32739 y(\015)23489 32678 y Fw(=)337 b Fu(O)25929 31333 y Fq(\020)26653 32678 y Fu(b)27173 32168 y Fr(\000)p Fp(1)27173 33025 y(0)28431 32678 y Fu(k)29100 32178 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+9+12)p Fs(s)33903 32301 y FD(1)34365 32178 y Fp(+6)p Fs(\016)36072 31333 y Fq(\021)37133 32678 y Fw(=)336 b Fu(o)p Fw(\(1\))p Fu(:)475 34989 y Fw(Using)404 b(a)g(standard)i(argumen)-34 b(t)405 b(as)f(in)g(Lemma)g (2.9,)g(w)-34 b(e)404 b(arriv)-34 b(e)403 b(at)i(\(3.52\))f(and)h (\(3.53\).)p 50750 35155 V 0 TeXcolorgray 475 37241 a FF(Remark)425 b(3.33.)p 0 TeXcolorgray 526 w Fw(Ob)-34 b(viously)370 b(\(3.52\))h(and)h(\(3.53\))f(hold)g(for)f(an)-34 b(y)371 b Fu(H)31891 36602 y Fp(\(1\))31793 37399 y Fs(\013)33485 37241 y Fw(=)337 b Fu(H)35773 37423 y Fp(0)36500 37241 y Fw(+)202 b Fu(\013)t(W)39569 37423 y Fp(1)40096 37241 y Fw(,)376 b(0)337 b Ft(\024)g Fu(\013)k Ft(\024)c Fw(1)370 b(uniformly)475 38680 y(in)404 b Fu(\013)t Fw(.)1803 40654 y(F)-101 b(urther,)404 b(w)-34 b(e)405 b(will)f(use)g(the)g (notation)19778 40156 y Fs(T)20326 40279 y FD(1)p 19778 40376 1010 49 v 19778 41072 a Fs(T)20326 41195 y FD(2)21256 40654 y Fw(=)337 b Fu(T)23244 40836 y Fp(1)23770 40654 y Fu(T)24646 40145 y Fr(\000)p Fp(1)24478 41002 y(2)26308 40654 y Fw(for)404 b(a)h(pair)f(of)g(op)34 b(erators)404 b Fu(T)39124 40836 y Fp(1)39650 40654 y Fw(,)g Fu(T)41099 40836 y Fp(2)41625 40654 y Fw(.)p 0 TeXcolorgray 475 43235 a FF(Lemma)566 b(3.34.)p 0 TeXcolorgray 602 w FC(If)516 b Fu(b)10862 43417 y Fp(0)11873 43235 y Fu(>)485 b(k)13970 42795 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+9+12)p Fs(s)18773 42918 y FD(1)19235 42795 y Fp(+7)p Fs(\016)20943 43235 y FC(,)535 b(then)515 b(the)g(numb)-62 b(er)515 b(of)f(zer)-62 b(os)515 b(of)f Fw(det)38956 41890 y Fq(\020)39680 43235 y Fu(I)423 b Fw(+)329 b Fu(A)42817 43417 y Fp(1)43343 42253 y Fq(\000)43888 43235 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))46271 42253 y Fq(\001)46827 41890 y(\021)48067 43235 y FC(in)516 b(any)475 45528 y Fw(\000\()1568 45208 y Fu(~)1704 45528 y(b)p Fw(\))434 b FC(is)f(the)f(same)h(as)g(that)f (of)h Fw(det)16676 44182 y Fq(\020)17400 45528 y Fu(I)365 b Fw(+)269 b Fu(A)20419 45710 y Fp(0)20945 44546 y Fq(\000)21490 45528 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))23873 44546 y Fq(\001)24429 44182 y(\021)25153 45528 y FC(.)p 0 TeXcolorgray 475 49083 a(Pr)-62 b(o)g(of.)p 0 TeXcolorgray 552 w Fw(Let)404 b(us)g(consider)g(det)14557 47010 y Fq( )15649 48186 y Fu(H)16657 48368 y Fp(0)17183 47204 y Fq(\000)17728 48186 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))20111 47204 y Fq(\001)20937 48186 y Fw(+)269 b Fu(\013)22924 48368 y Fp(1)23450 48186 y Fu(W)24595 48368 y Fp(1)25390 48186 y Ft(\000)g Fu(k)27271 47746 y Fp(2)p Fs(l)p 15649 48804 12440 49 v 15649 49979 a Fu(H)16657 50161 y Fp(0)17183 48998 y Fq(\000)17728 49979 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))20111 48998 y Fq(\001)20937 49979 y Fw(+)269 b Fu(\013)22924 50161 y Fp(2)23450 49979 y Fu(W)24595 50161 y Fp(1)25390 49979 y Ft(\000)g Fu(k)27271 49629 y Fp(2)p Fs(l)28221 47010 y Fq(!)29181 49083 y Fw(,)404 b(where)g(0)336 b Ft(\024)h Fu(\013)36424 49265 y Fp(1)37287 49083 y Ft(\024)f Fu(\013)39341 49265 y Fp(2)40204 49083 y Ft(\024)g Fw(1.)539 b(Ob)-34 b(viously)-101 b(,)7358 53333 y(det)9244 51261 y Fq( )10336 52436 y Fu(H)11344 52618 y Fp(0)11870 51454 y Fq(\000)12415 52436 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))14798 51454 y Fq(\001)15624 52436 y Fw(+)269 b Fu(\013)17611 52618 y Fp(1)18137 52436 y Fu(W)19282 52618 y Fp(1)20077 52436 y Ft(\000)g Fu(k)21958 51996 y Fp(2)p Fs(l)p 10336 53054 V 10336 54230 a Fu(H)11344 54412 y Fp(0)11870 53248 y Fq(\000)12415 54230 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))14798 53248 y Fq(\001)15624 54230 y Fw(+)269 b Fu(\013)17611 54412 y Fp(2)18137 54230 y Fu(W)19282 54412 y Fp(1)20077 54230 y Ft(\000)g Fu(k)21958 53880 y Fp(2)p Fs(l)22908 51261 y Fq(!)24205 53333 y Fw(=)336 b(det)27370 51261 y Fq( )28329 53333 y Fu(I)364 b Fw(+)33442 52513 y(\()p Fu(\013)34688 52695 y Fp(1)35484 52513 y Ft(\000)269 b Fu(\013)37471 52695 y Fp(2)37998 52513 y Fw(\))p Fu(W)39614 52695 y Fp(1)p 30571 53054 V 30571 54230 a Fu(H)31579 54412 y Fp(0)32105 53248 y Fq(\000)32650 54230 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))35033 53248 y Fq(\001)35859 54230 y Fw(+)269 b Fu(\013)37846 54412 y Fp(2)38372 54230 y Fu(W)39517 54412 y Fp(1)40312 54230 y Ft(\000)g Fu(k)42193 53880 y Fp(2)p Fs(l)43143 51261 y Fq(!)44305 53333 y Fu(:)475 56372 y Fw(Using)550 b(\(3.53\),)586 b(whic)-34 b(h)551 b(is)f(uniform)g(with)h(resp)34 b(ect)549 b(to)h Fu(\013)t Fw(,)587 b(w)-34 b(e)550 b(obtain,)587 b(b)-34 b(y)550 b(Rouc)-34 b(h)g(\023)-572 b(e's)551 b(theorem,)586 b(that)551 b(this)475 57811 y(determinan)-34 b(t)378 b(has)g(the)g(same)f(n)-34 b(um)g(b)34 b(er)379 b(of)e(p)34 b(oles)377 b(and)i(zeros)d(if)i Ft(j)p Fu(\013)30818 57993 y Fp(2)31559 57811 y Ft(\000)216 b Fu(\013)33493 57993 y Fp(1)34019 57811 y Ft(j)377 b Fw(is)g(su\016cien)-34 b(tly)378 b(small:)525 b Ft(j)p Fu(\013)46708 57993 y Fp(2)47449 57811 y Ft(\000)216 b Fu(\013)49383 57993 y Fp(1)49909 57811 y Ft(j)336 b Fu(<)475 59280 y(b)995 59462 y Fp(0)1521 59280 y Fu(k)2190 58840 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(11)p Fr(\000)p Fp(14)p Fs(s)6731 58963 y FD(1)7194 58840 y Fr(\000)p Fp(7)p Fs(\016)8901 59280 y Fw(.)1247 b(W)-101 b(e)640 b(mak)-34 b(e)641 b(a)f(\014nite)h(n)-34 b(um)g(b)34 b(er)642 b(of)e(steps)i(b)34 b(et)-34 b(w)g(een)641 b Fu(\013)736 b Fw(=)730 b(0)640 b(and)i Fu(\013)735 b Fw(=)c(1.)1247 b(Indeed,)475 61747 y(det)2360 59674 y Fq( )3453 60850 y Fu(H)4461 61032 y Fp(0)4986 59868 y Fq(\000)5532 60850 y Fu(~)-596 b(y)43 b Fw(\()p Fu(')p Fw(\))7914 59868 y Fq(\001)8740 60850 y Fw(+)269 b Fu(\013)10727 61032 y Fp(1)11253 60850 y Fu(W)12398 61032 y Fp(1)13193 60850 y Ft(\000)g Fu(k)15074 60410 y Fp(2)p Fs(l)p 3453 61468 V 3453 62644 a Fu(H)4461 62826 y Fp(0)4986 61662 y Fq(\000)5532 62644 y Fu(~)-596 b(y)43 b Fw(\()p Fu(')p Fw(\))7914 61662 y Fq(\001)8740 62644 y Fw(+)269 b Fu(\013)10727 62826 y Fp(2)11253 62644 y Fu(W)12398 62826 y Fp(1)13193 62644 y Ft(\000)g Fu(k)15074 62294 y Fp(2)p Fs(l)16025 59674 y Fq(!)17388 61747 y Fw(has)405 b(the)f(same)g(n)-34 b(um)g(b)34 b(er)406 b(of)e(zeros)f(and)i(p)34 b(oles)404 b(and)h(so)f(do)34 b(es)475 65494 y(det)2360 63421 y Fq( )3453 64597 y Fu(H)4461 64779 y Fp(0)4986 63615 y Fq(\000)5532 64597 y Fu(~)-596 b(y)43 b Fw(\()p Fu(')p Fw(\))7914 63615 y Fq(\001)8740 64597 y Fw(+)269 b Fu(\013)10727 64779 y Fp(2)11253 64597 y Fu(W)12398 64779 y Fp(1)13193 64597 y Ft(\000)g Fu(k)15074 64157 y Fp(2)p Fs(l)p 3453 65215 V 3453 66391 a Fu(H)4461 66573 y Fp(0)4986 65409 y Fq(\000)5532 66391 y Fu(~)-596 b(y)43 b Fw(\()p Fu(')p Fw(\))7914 65409 y Fq(\001)8740 66391 y Fw(+)269 b Fu(\013)10727 66573 y Fp(3)11253 66391 y Fu(W)12398 66573 y Fp(1)13193 66391 y Ft(\000)g Fu(k)15074 66041 y Fp(2)p Fs(l)16025 63421 y Fq(!)16984 65494 y Fw(.)668 b(Both)447 b(determinan)-34 b(ts)449 b(can)e(b)34 b(e)447 b(represen)-34 b(ted)448 b(as)f(fractions)h(of)f(holomor-)475 68849 y(phic)564 b(functions:)9412 68029 y Fu(f)10005 68211 y Fp(1)10531 68029 y Fw(\()p Fu(')p Fw(\))p 9412 68570 2855 49 v 9412 69681 a Fu(f)10005 69863 y Fp(2)10531 69681 y Fw(\()p Fu(')p Fw(\))12964 68849 y(and)15614 68029 y Fu(g)16192 68211 y Fp(1)16717 68029 y Fw(\()p Fu(')p Fw(\))p 15614 68570 2840 49 v 15614 69681 a Fu(g)16192 69863 y Fp(2)16717 69681 y Fw(\()p Fu(')p Fw(\))18586 68849 y(,)604 b(resp)34 b(ectiv)-34 b(ely)-101 b(.)1018 b(The)564 b(n)-34 b(um)g(b)34 b(er)566 b(of)e(zeros)g(b)34 b(eing)564 b(the)h(same)f(for)h(the)p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 30 30 TeXDict begin 30 29 bop 0 TeXcolorgray 0 TeXcolorgray 475 3345 a Fp(30)14248 b(YULIA)313 b(KARPESHINA)f(AND)h(YOUNG-RAN)g (LEE)p 0 TeXcolorgray 475 7016 a Fw(pairs)551 b Fu(f)4188 7198 y Fp(1)4714 7016 y Fw(,)588 b Fu(f)6232 7198 y Fp(2)7310 7016 y Fw(and)552 b Fu(g)10392 7198 y Fp(1)10918 7016 y Fw(,)587 b Fu(g)12420 7198 y Fp(2)12946 7016 y Fw(.)980 b(Note)552 b(that)g(det)21999 4943 y Fq( )23091 6119 y Fu(H)24099 6301 y Fp(0)24625 6119 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))27469 5137 y Fq(\001)28294 6119 y Fw(+)270 b Fu(\013)30282 6301 y Fp(1)30808 6119 y Fu(W)31953 6301 y Fp(1)32748 6119 y Ft(\000)f Fu(k)34629 5679 y Fp(2)p Fs(l)p 23091 6737 12355 49 v 23091 7913 a Fu(H)24099 8095 y Fp(0)24625 7913 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))27469 6931 y Fq(\001)28294 7913 y Fw(+)270 b Fu(\013)30282 8095 y Fp(3)30808 7913 y Fu(W)31953 8095 y Fp(1)32748 7913 y Ft(\000)f Fu(k)34629 7562 y Fp(2)p Fs(l)35579 4943 y Fq(!)37121 7016 y Fw(=)38778 6196 y Fu(f)39371 6378 y Fp(1)39898 6196 y Fu(g)40476 6378 y Fp(1)p 38778 6737 2224 49 v 38778 7847 a Fu(f)39371 8029 y Fp(2)39898 7847 y Fu(g)40476 8029 y Fp(2)41134 7016 y Fu(:)551 b Fw(Ob)-34 b(viously)-101 b(,)588 b(it)551 b(also)475 10763 y(has)587 b(the)f(same)h(n)-34 b(um)g(b)34 b(er)587 b(of)g(zeros)e(and)i(p)34 b(oles.)1085 b(Hence,)631 b(det)30370 8690 y Fq( )31462 9866 y Fu(H)32470 10048 y Fp(0)32996 8884 y Fq(\000)33541 9866 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))35924 8884 y Fq(\001)36749 9866 y Fw(+)270 b Fu(W)39107 10048 y Fp(1)39902 9866 y Ft(\000)f Fu(k)41783 9426 y Fp(2)p Fs(l)p 31462 10484 11138 49 v 33080 11660 a Fu(H)34088 11842 y Fp(0)34614 11660 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))37458 10678 y Fq(\001)38283 11660 y Ft(\000)270 b Fu(k)40165 11310 y Fp(2)p Fs(l)42733 8690 y Fq(!)44279 10763 y Fw(has)587 b(the)f(same)475 13574 y(n)-34 b(um)g(b)34 b(er)444 b(of)f(zeros)g(and) g(p)34 b(oles)443 b(inside)g(\000\()19505 13255 y Fu(~)19641 13574 y(b)p Fw(\).)656 b(Considering)443 b(that)i(the)e(last)g (determinan)-34 b(t)445 b(is)d(the)i(quotien)-34 b(t)444 b(of)475 15382 y(det)2158 14037 y Fq(\020)2882 15382 y Fu(I)173 b Fw(+)78 b Fu(A)5518 15564 y Fp(1)6044 14400 y Fq(\000)6590 15382 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))8973 14400 y Fq(\001)9529 14037 y(\021)10562 15382 y Fw(and)309 b(det)14506 14037 y Fq(\020)15230 15382 y Fu(I)173 b Fw(+)78 b Fu(A)17866 15564 y Fp(0)18393 14400 y Fq(\000)18938 15382 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))21321 14400 y Fq(\001)21877 14037 y(\021)22910 15382 y Fw(and)309 b(these)g(t)-34 b(w)g(o)310 b(determinan)-34 b(ts)310 b(are)e(holomorphic)h(functions,)475 17467 y(w)-34 b(e)404 b(obtain,)h(that)g(they)g(ha)-34 b(v)g(e)404 b(the)h(same)f(n)-34 b(um)g(b)34 b(er)405 b(of)f(zeros)g(inside)g (\000\()33009 17147 y Fu(~)33145 17467 y(b)p Fw(\).)p 50750 17633 665 831 v 0 TeXcolorgray 475 19802 a FF(Lemma)463 b(3.35.)p 0 TeXcolorgray 0 TeXcolorgray 2489 w Fw(\(1\))p 0 TeXcolorgray 554 w FC(If)434 b Fu(b)14666 19984 y Fp(0)15528 19802 y Fu(>)337 b(k)17477 19363 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+9+12)p Fs(s)22280 19486 y FD(1)22742 19363 y Fp(+7)p Fs(\016)24449 19802 y FC(,)434 b(then)f(the)f(numb)-62 b(er)433 b(of)g(zer)-62 b(os)432 b(of)4511 21610 y Fw(det)6195 20265 y Fq(\020)6919 21610 y Fu(I)364 b Fw(+)269 b Fu(A)9937 21792 y Fp(1)10463 20628 y Fq(\000)11009 21610 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))13392 20628 y Fq(\001)13948 20265 y(\021)15105 21610 y FC(in)434 b(any)f Fw(\000\()20008 21290 y Fu(~)20144 21610 y(b)q Fw(\))h FC(is)f(the)f(same)h(as)g(that)f (of)h Fw(det)35117 20265 y Fq(\020)35841 21610 y Fu(I)364 b Fw(+)269 b Fu(A)38859 21792 y Fp(0)39385 20628 y Fq(\000)39931 21610 y Fu(~)-596 b(y)40535 21792 y Fp(0)41061 21610 y Fw(\()p Fu(')p Fw(\))42796 20628 y Fq(\001)43352 20265 y(\021)44076 21610 y FC(.)p 0 TeXcolorgray 2409 23903 a Fw(\(2\))p 0 TeXcolorgray 554 w FC(The)392 b(distanc)-62 b(e)390 b(fr)-62 b(om)391 b(a)h(zer)-62 b(o)391 b(of)g Fw(det)20999 22557 y Fq(\020)21723 23903 y Fu(I)274 b Fw(+)179 b Fu(A)24561 24085 y Fp(1)25087 22921 y Fq(\000)25632 23903 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))28015 22921 y Fq(\001)28571 22557 y(\021)29687 23903 y FC(to)391 b(the)g(ne)-62 b(ar)g(est)391 b(zer)-62 b(o)391 b(of)g Fw(det)42747 22557 y Fq(\020)43471 23903 y Fu(I)274 b Fw(+)179 b Fu(A)46309 24085 y Fp(0)46834 22921 y Fq(\000)47380 23903 y Fu(~)-596 b(y)47984 24085 y Fp(0)48510 23903 y Fw(\()p Fu(')p Fw(\))50245 22921 y Fq(\001)50801 22557 y(\021)4511 25710 y FC(do)-62 b(es)432 b(not)i(exc)-62 b(e)g(e)g(d)432 b Fu(r)34 b(=)p Fw(2)p FC(.)p 0 TeXcolorgray 475 28385 a FF(Corollary)342 b(3.36.)p 0 TeXcolorgray 453 w FC(The)320 b(total)f(numb)-62 b(er)320 b(of)g(zer)-62 b(os)319 b(of)h Fw(det)26366 27040 y Fq(\020)27090 28385 y Fu(I)119 b Fw(+)24 b Fu(A)29618 28567 y Fp(1)30143 27403 y Fq(\000)30688 28385 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))33071 27403 y Fq(\001)33627 27040 y(\021)34672 28385 y FC(in)321 b Fo(O)p Fw(\()37239 28065 y Fu(~)37375 28385 y(b)p Fw(\))g FC(do)-62 b(es)319 b(not)i(exc)-62 b(e)g(e)g(d)319 b Fu(c)47335 28567 y Fp(0)47860 28385 y Fu(k)48529 27945 y Fp(2+2)p Fs(s)50636 28068 y FD(1)51154 28385 y FC(.)1803 31337 y Fw(The)333 b(Corollary)g(immediately)f(follo) -34 b(ws)333 b(from)h(the)f(\014rst)g(statemen)-34 b(t)335 b(of)e(the)g(lemma,)347 b(the)333 b(de\014nition)h(of)g Fo(O)p Fw(\()50398 31017 y Fu(~)50534 31337 y(b)p Fw(\))475 33145 y(as)404 b(a)g(collection)f(of)i(disks)f(surrounding)h(zeros)f (of)g(det)25603 31799 y Fq(\020)26327 33145 y Fu(I)364 b Fw(+)269 b Fu(A)29345 33327 y Fp(0)29871 32163 y Fq(\000)30416 33145 y Fu(~)-595 b(y)31021 33327 y Fp(0)31547 33145 y Fw(\()p Fu(')p Fw(\))33282 32163 y Fq(\001)33838 31799 y(\021)34966 33145 y Fw(and)405 b(Remark)e(3.25,)h(page)g(27.)p 0 TeXcolorgray 475 35921 a FF(Corollary)743 b(3.37.)p 0 TeXcolorgray 678 w FC(The)641 b(numb)-62 b(er)642 b(of)f (self-interse)-62 b(ctions)637 b(of)k Fo(K)31431 36103 y Fp(2)31958 35921 y Fu(\037)32716 35481 y Fr(\003)32716 36246 y Fp(1)33242 35921 y Fw(\()p Fu(\025)p Fw(\))h FC(do)-62 b(es)640 b(not)i(exc)-62 b(e)g(e)g(d)640 b Fu(c)45144 36103 y Fp(0)45669 35921 y Fu(k)46338 35481 y Fp(2+2)p Fs(s)48445 35604 y FD(1)49605 35921 y FC(and)475 37360 y Fu(L)202 b Fw(\()p Fo(K)2970 37542 y Fp(2)3496 37360 y Fu(\037)4254 36920 y Fr(\003)4254 37685 y Fp(1)4781 37360 y Fw(\()p Fu(\025)p Fw(\)\))338 b(=)e Fu(L)202 b Fw(\()p Fu(\037)10774 36920 y Fr(\003)10774 37685 y Fp(1)11301 37360 y Fw(\()p Fu(\025)p Fw(\)\))q FC(.)1803 39666 y Fw(The)405 b(latter)f(equalit)-34 b(y)404 b(holds,)g(since)g Fo(K)19686 39848 y Fp(2)20212 39666 y Fu(\037)20970 39227 y Fr(\003)20970 39991 y Fp(1)21496 39666 y Fw(\()p Fu(\025)p Fw(\))h(has)g(only)f(a)g(\014nite)g(n)-34 b(um)g(b)34 b(er)405 b(of)g(self-in)-34 b(tersections.)p 0 TeXcolorgray 475 42111 a FC(L)-62 b(emma)433 b(3.35.)p 0 TeXcolorgray 552 w Fw(The)320 b(\014rst)g(statemen)-34 b(t)321 b(immediately)e (follo)-34 b(ws)320 b(Lemmas)f(3.29)g(\(P)-34 b(art)320 b(1\))g(and)h(3.34.)510 b(Next,)336 b(note)475 43710 y(that)362 b(Lemmas)f(3.29)g({)g(3.34)g(hold)g(not)h(only)f(for)g Fo(O)p Fw(\()24203 43390 y Fu(~)24339 43710 y(b)q Fw(\),)369 b(but)363 b(also)e(for)g(a)g(set)35359 43381 y(~)35233 43710 y Fo(O)p Fw(\()36426 43390 y Fu(~)36562 43710 y(b)q Fw(\),)369 b(constructed)362 b(from)g(disks)f(of)475 45149 y(t)-34 b(wice)355 b(smaller)g(radius)g Fu(r)34 b(=)p Fw(2)355 b(with)h(the)g(same)f(cen)-34 b(ters,)365 b(since)355 b(all)g(estimates)h(in)f(the)h(lemmas)f(preserv)-34 b(ed)355 b(under)475 46957 y(suc)-34 b(h)406 b(c)-34 b(hange)406 b(up)g(to)g(some)f(insigni\014can)-34 b(t)407 b(constan)-34 b(ts.)545 b(This)406 b(pro)-34 b(v)g(es)405 b(that)i(all)e(zeros)f(of)i(det)43047 45612 y Fq(\020)43771 46957 y Fu(I)365 b Fw(+)270 b Fu(A)46791 47139 y Fp(1)47317 45975 y Fq(\000)47862 46957 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))50245 45975 y Fq(\001)50801 45612 y(\021)475 49041 y Fw(in)404 b Fo(O)p Fw(\()3082 48722 y Fu(~)3218 49041 y(b)p Fw(\))h(are,)e(in)h(fact,)g(in)g(the)h(smaller)e(set)20820 48712 y(~)20694 49041 y Fo(O)p Fw(\()21887 48722 y Fu(~)22023 49041 y(b)q Fw(\),)h(i.e.,)e(the)j(second)f(statemen)-34 b(t)406 b(of)f(the)f(lemma)g(holds.)p 50750 49207 V 1803 51486 a(Let)460 b(us)g(recall)e(that)j Fu(!)12412 51046 y Fr(\003)12369 51810 y Fp(1)p Fs(;p)14090 51486 y Fw(is)f(the)g(set)g (of)h(zeros)e(in)h Fu(\010)26267 51668 y Fp(1)27252 51486 y Fw(of)g(the)h(determinan)-34 b(t)461 b(\(3.36\))f(when)h Ft(j)p Fu(\017)p Ft(j)429 b Fu(<)g(\017)47963 51668 y Fp(1)48489 51486 y Fw(,)474 b(here)475 53002 y Fu(p)336 b Ft(2)h Fu(P)438 b Ft(n)269 b(f)p Fw(0)p Ft(g)405 b Fw(and)g(\014xed.)538 b(The)405 b(set)f Fu(!)17775 53184 y Fp(1)p Fs(;p)19440 53002 y Fw(is)g(the)h(real)e(part)i(of)f Fu(!)29914 52563 y Fr(\003)29871 53327 y Fp(1)p Fs(;p)31132 53002 y Fw(:)539 b Fu(!)32762 53184 y Fp(1)p Fs(;p)34360 53002 y Fw(=)336 b Fu(!)36436 52563 y Fr(\003)36393 53327 y Fp(1)p Fs(;p)37924 53002 y Ft(\\)269 b Fw(\002)39944 53184 y Fp(1)40470 53002 y Fu(:)p 0 TeXcolorgray 475 55532 a FF(Lemma)463 b(3.38.)p 0 TeXcolorgray 554 w FC(If)434 b Fu(b)10629 55714 y Fp(0)11492 55532 y Fu(>)336 b(k)13440 55092 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+9+12)p Fs(s)18243 55215 y FD(1)18705 55092 y Fp(+7)p Fs(\016)20413 55532 y FC(,)433 b Ft(j)p Fu(\017)p Ft(j)336 b Fu(<)h(b)24520 55714 y Fp(0)25046 55532 y Fu(k)25715 55092 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(11)p Fr(\000)p Fp(14)p Fs(s)30256 55215 y FD(1)30718 55092 y Fr(\000)p Fp(7)p Fs(\016)32426 55532 y FC(,)433 b(then)p 0 TeXcolorgray 2409 58251 a Fw(\(1\))p 0 TeXcolorgray 554 w FC(The)h(determinant)e(of)15811 57431 y Fu(H)16917 56991 y Fp(\(1\))18175 57431 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\)\))271 b Ft(\000)e Fu(k)23642 56991 y Fp(2)p Fs(l)24728 57431 y Ft(\000)g Fu(\017)p 15811 57972 10622 49 v 17213 59083 a(H)18221 59265 y Fp(0)18747 59083 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\)\))270 b(+)f Fu(k)24213 58733 y Fp(2)p Fs(l)26999 58251 y FC(has)432 b(no)i(zer)-62 b(os)432 b(in)i Fu(\010)36248 58433 y Fp(1)37043 58251 y Ft(n)270 b Fo(O)p Fw(\()39112 57931 y Fu(~)39248 58251 y(b)p Fw(\))p FC(.)p 0 TeXcolorgray 2409 61618 a Fw(\(2\))p 0 TeXcolorgray 554 w FC(The)434 b(numb)-62 b(er)433 b(of)g(zer)-62 b(os)432 b(of)h Fw(det)19097 59545 y Fq( )20190 60798 y Fu(H)21296 60358 y Fp(\(1\))22553 60798 y Fw(\()-10 b Fu(~)-596 b(y)45 b Fw(\()p Fu(')p Fw(\)\))270 b Ft(\000)f Fu(k)28020 60358 y Fp(2)p Fs(l)29107 60798 y Ft(\000)g Fu(\017)p 20190 61339 V 21592 62450 a(H)22600 62632 y Fp(0)23125 62450 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\)\))271 b(+)e Fu(k)28592 62099 y Fp(2)p Fs(l)30944 59545 y Fq(!)32337 61618 y FC(in)434 b Fw(\000\()34917 61298 y Fu(~)35053 61618 y(b)p Fw(\))g FC(is)f(the)g(same)f(as)h(that)f (of)4511 65365 y Fw(det)6397 63292 y Fq( )7489 64545 y Fu(H)8595 64105 y Fp(\(1\))9853 64545 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\)\))271 b Ft(\000)e Fu(k)15320 64105 y Fp(2)p Fs(l)p 7489 65086 8648 49 v 7905 66197 a Fu(H)8913 66379 y Fp(0)9438 66197 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\)\))271 b(+)e Fu(k)14905 65847 y Fp(2)p Fs(l)16270 63292 y Fq(!)17230 65365 y FC(.)p 0 TeXcolorgray 475 69044 a FF(Corollary)483 b(3.39.)p 0 TeXcolorgray 554 w FC(If)11251 68724 y Fu(~)11387 69044 y(b)337 b Fw(=)f(2)p Fu(\031)43 b(p=)-67 b(N)16986 69226 y Fp(1)17512 69044 y Fu(a)p FC(,)433 b Fu(!)19755 68604 y Fr(\003)19712 69369 y Fp(1)p Fs(;p)21310 69044 y Ft(\032)336 b Fo(O)p Fw(\()23782 68724 y Fu(~)23918 69044 y(b)q Fw(\))p Fu(;)1503 b(!)27504 69226 y Fp(1)p Fs(;p)29102 69044 y Ft(\032)336 b Fo(O)p Fw(\()31574 68724 y Fu(~)31710 69044 y(b)q Fw(\))269 b Ft(\\)h Fw(\002)34992 69226 y Fp(1)35517 69044 y Fu(:)p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 31 31 TeXDict begin 31 30 bop 0 TeXcolorgray 0 TeXcolorgray 2848 3345 a Fp(SPECTRAL)406 b(PR)-26 b(OPER)-78 b(TIES)406 b(OF)h(A)h(POL)-78 b(YHARMONIC)405 b(OPERA)-78 b(TOR)406 b(WITH)h(LIMIT-PERIODIC)e(POTENTIAL)1430 b(31)p 0 TeXcolorgray 0 TeXcolorgray 475 6001 a FC(Pr)-62 b(o)g(of.)p 0 TeXcolorgray 552 w Fw(T)-101 b(o)283 b(pro)-34 b(v)g(e)284 b(Lemmas)e(3.31)h(and)h (3.34,)306 b(w)-34 b(e)284 b(used)f(the)h(estimate)f Ft(k)p Fu(W)33949 6183 y Fp(1)34474 6001 y Ft(k)337 b Fw(=)f Fu(O)34 b Fw(\(1\).)499 b(Ob)-34 b(viously)-101 b(,)306 b Ft(k)p Fu(W)47625 6183 y Fp(1)48178 6001 y Ft(\000)27 b Fu(\017)p Ft(k)336 b Fw(=)475 7440 y Fu(O)34 b Fw(\(1\).)570 b(Hence,)417 b(the)f(analogous)f(lemmas)g(hold)g(when)h (w)-34 b(e)415 b(replace)f Fu(W)32884 7622 y Fp(1)33825 7440 y Fw(b)-34 b(y)415 b Fu(W)36664 7622 y Fp(1)37466 7440 y Ft(\000)276 b Fu(\017)p Fw(,)417 b(i.e.,)f(when)g(w)-34 b(e)415 b(consider)475 8958 y Fu(H)1581 8518 y Fp(\(1\))2838 8958 y Fw(\()-10 b Fu(~)-596 b(y)45 b Fw(\()p Fu(')p Fw(\)\))149 b Ft(\000)g Fu(k)8064 8518 y Fp(2)p Fs(l)9030 8958 y Ft(\000)g Fu(\017)344 b Fw(instead)g(of)g(just)h Fu(H)19842 8518 y Fp(\(1\))21100 8958 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\)\))149 b Ft(\000)g Fu(k)26325 8518 y Fp(2)p Fs(l)27143 8958 y Fw(.)518 b(Mo)34 b(di\014ed)345 b(Lemmas)e(3.31)h(and)g(3.34)g(immediately)475 10397 y(yield)403 b(the)i(statemen)-34 b(t)405 b(of)g(the)f(presen)-34 b(t)405 b(lemma.)p 50750 10563 665 831 v 475 12719 a(3.3.5.)605 b FC(Pr)-62 b(o)g(of)587 b(of)f(statement)g(3)h(of)g(Ge)-62 b(ometric)585 b(L)-62 b(emma.)606 b Fw(By)570 b(Corollary)h(3.39,)612 b Fu(!)39703 12901 y Fp(1)p Fs(;p)41580 12719 y Ft(\032)j Fo(O)p Fw(\()44331 12399 y Fu(~)44467 12719 y(b)p Fw(\))382 b Ft(\\)f Fw(\002)47972 12901 y Fp(1)48497 12719 y Fw(,)49311 12399 y Fu(~)49447 12719 y(b)615 b Fw(=)475 14369 y(2)p Fu(\031)43 b(p=)-67 b(N)3938 14551 y Fp(1)4464 14369 y Fu(a)p Fw(.)810 b(It)495 b(is)g(easy)g(to)g(see)g(that)h Fu(b)18584 14551 y Fp(0)19598 14369 y Ft(\025)488 b Fu(c)21554 14566 y Fs(b)22011 14369 y Fu(k)22680 13929 y Fr(\000)p Fs(s)23847 14052 y FD(2)24364 14369 y Fw(,)517 b Fu(c)25743 14566 y Fs(b)26689 14369 y Fw(=)28253 13892 y Fp(1)p 28253 14090 471 49 v 28253 14787 a(4)29058 14369 y Fw(min)p Ft(f)p Fu(b)32204 14551 y Fp(1)32730 14369 y Fu(;)202 b(b)33789 14551 y Fp(2)34315 14369 y Ft(g)p Fw(.)811 b(Suc)-34 b(h)38953 14049 y Fu(~)39089 14369 y(b)495 b Fw(satis\014es)h(conditions)g(of)475 16084 y(Lemmas)347 b(3.29)g({)h(3.38)f(for)h(an)-34 b(y)348 b Fu(p)336 b Ft(2)h Fu(P)325 b Ft(n)156 b(f)p Fw(0)p Ft(g)p Fw(.)520 b(T)-101 b(aking)348 b(in)-34 b(to)348 b(accoun)-34 b(t)349 b(that)g(that)f Fo(O)p Fw(\()38983 15764 y Fu(~)39119 16084 y(b)q Fw(\))g(is)f(formed)h(b)-34 b(y)348 b(no)g(more)475 17552 y(than)379 b Fu(c)3802 17734 y Fp(0)4327 17552 y Fu(k)4996 17112 y Fp(2+2)p Fs(s)7103 17235 y FD(1)7998 17552 y Fw(disks)f(of)g(the)g(radius)g Fu(r)370 b Fw(=)336 b Fu(k)20882 17112 y Fr(\000)p Fp(4)p Fr(\000)p Fp(6)p Fs(s)23721 17235 y FD(1)24184 17112 y Fr(\000)p Fp(3)p Fs(\016)25891 17552 y Fw(,)382 b(w)-34 b(e)378 b(obtain)h(that)g(the)f (total)g(length)g(of)g Fu(!)45428 17734 y Fp(1)p Fs(;p)47067 17552 y Fw(do)34 b(es)377 b(not)475 19072 y(exceed)453 b(2)p Fu(c)5528 19254 y Fp(0)6053 19072 y Fu(k)6722 18632 y Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)9561 18755 y FD(1)10023 18632 y Fr(\000)p Fp(3)p Fs(\016)11730 19072 y Fw(.)688 b(Since)454 b(the)h(set)f Fu(P)622 b Fw(con)-34 b(tains)455 b(no)g(more)e(than)i(4)p Fu(k)35224 18632 y Fp(2)p Fs(s)36129 18755 y FD(2)36591 18632 y Fr(\000)p Fp(2)p Fs(s)38228 18755 y FD(1)39199 19072 y Fw(elemen)-34 b(ts)454 b(and)h Fu(s)47180 19254 y Fp(2)48126 19072 y Fw(=)419 b(2)p Fu(s)50662 19254 y Fp(1)51189 19072 y Fw(,)475 20511 y(the)349 b(total)g(length)h(of)f Fu(!)11075 20693 y Fp(1)11950 20511 y Fw(do)34 b(es)349 b(not)g(exceed)f(8)p Fu(c)21676 20693 y Fp(0)22201 20511 y Fu(k)22870 20071 y Fr(\000)p Fp(2)p Fr(\000)p Fp(2)p Fs(s)25709 20194 y FD(1)26172 20071 y Fr(\000)p Fp(3)p Fs(\016)27879 20511 y Fw(.)520 b(Therefore,)359 b(the)349 b(length)h(of)f(\012)42404 20693 y Fp(1)43279 20511 y Fw(do)34 b(es)349 b(not)g(exceed)475 21950 y(20)p Fu(\031)43 b(c)2946 22132 y Fp(0)3472 21950 y Fu(k)4141 21510 y Fr(\000)p Fp(1)p Fr(\000)p Fp(2)p Fs(s)6980 21633 y FD(1)7442 21510 y Fr(\000)p Fp(3)p Fs(\016)9149 21950 y Fw(.)538 b(Considering)405 b(that)g Fu(\037)20085 21510 y Fr(\003)20085 22275 y Fp(1)20612 21950 y Fw(\()p Fu(\025)p Fw(\))g(has)f(a)g(length)h(2)p Fu(\031)43 b(k)31550 20968 y Fq(\000)32106 21950 y Fw(1)270 b(+)f Fu(o)p Fw(\(1\))36330 20968 y Fq(\001)36886 21950 y Fw(,)403 b(w)-34 b(e)405 b(obtain)g(Statemen)-34 b(t)406 b(3.)475 24577 y(3.4.)605 b FF(Non-resonan)-39 b(t)567 b(part)g(of)g(the)g(iso)39 b(energetic)566 b(set)g(of)h Fu(H)31939 23938 y Fp(\(2\))31841 24735 y Fs(\013)33197 24577 y FF(.)606 b Fw(Let)492 b Fu(S)37193 24759 y Fp(2)37719 24577 y Fw(\()p Fu(\025)p Fw(\))i(b)34 b(e)492 b(an)i(iso)34 b(energetic)491 b(set)475 26294 y(of)470 b(the)h(op)34 b(erator)469 b Fu(H)10135 25655 y Fp(\(2\))10037 26451 y Fs(\013)11393 26294 y Fw(:)670 b Fu(S)13143 26476 y Fp(2)13670 26294 y Fw(\()p Fu(\025)p Fw(\))447 b(=)f Ft(f)p Fu(\034)584 b Ft(2)447 b Fu(K)21159 26476 y Fp(2)22131 26294 y Fw(:)g Ft(9)p Fu(n)f Ft(2)h Fn(N)g Fw(:)916 b Fu(\025)29299 25655 y Fp(\(2\))29299 26451 y Fs(n)30557 26294 y Fw(\()p Fu(\013)t(;)202 b(\034)137 b Fw(\))448 b(=)e Fu(\025)p Ft(g)p Fw(,)487 b(here)469 b Ft(f)p Fu(\025)41466 25655 y Fp(\(2\))41466 26451 y Fs(n)42724 26294 y Fw(\()p Fu(\013)t(;)202 b(\034)137 b Fw(\))p Ft(g)46257 25854 y Fr(1)46257 26618 y Fs(n)p Fp(=1)48557 26294 y Fw(is)470 b(the)475 28032 y(sp)34 b(ectrum)460 b(of)h Fu(H)8370 27393 y Fp(\(2\))8272 28190 y Fs(\013)9628 28032 y Fw(\()p Fu(\034)137 b Fw(\).)708 b(No)-34 b(w)462 b(w)-34 b(e)460 b(construct)i(a)e(non-resonance)h(subset)h Fu(\037)35870 27592 y Fr(\003)35870 28357 y Fp(2)36396 28032 y Fw(\()p Fu(\025)p Fw(\))f(of)g Fu(S)40686 28214 y Fp(2)41212 28032 y Fw(\()p Fu(\025)p Fw(\).)709 b(It)460 b(corresp)34 b(onds)475 29771 y(to)524 b(non-resonance)h(eigen)-34 b(v)-67 b(alues)524 b Fu(\025)17182 29132 y Fp(\(2\))17163 30141 y(~)17182 30345 y Fs(j)18440 29771 y Fw(\()p Fu(\034)137 b Fw(\))525 b(giv)-34 b(en)524 b(b)-34 b(y)525 b(the)f(p)34 b(erturbation)526 b(series)d(\(3.15\).)899 b(W)-101 b(e)524 b(start)h(with)g(a)475 31537 y(de\014nition)486 b(of)f Fo(D)8411 31719 y Fp(1)8936 31537 y Fw(\()p Fu(\025)p Fw(\))10585 31719 y Fs(nonr)26 b(es)13560 31537 y Fw(.)781 b(Recall)484 b(that)i Fu(\037)21877 31719 y Fp(2)22874 31537 y Ft(\032)471 b Fo(K)25285 31719 y Fp(2)25811 31537 y Fu(\037)26569 31097 y Fr(\003)26569 31862 y Fp(1)27096 31537 y Fw(\()p Fu(\025)p Fw(\))485 b(\(see)g(the)g(Geometric)f (Lemma\))h(and)g Fu(\037)47935 31097 y Fr(\003)47935 31862 y Fp(1)48461 31537 y Fw(\()p Fu(\025)p Fw(\))472 b(=)475 32976 y Fo(K)1472 33158 y Fp(1)1998 32976 y Fo(D)3005 33158 y Fp(1)3531 32976 y Fw(\()p Fu(\025)p Fw(\),)404 b(see)g(\(2.48\).)539 b(Hence,)403 b Fu(\037)16551 33158 y Fp(2)17414 32976 y Ft(\032)337 b Fo(K)19691 33158 y Fp(2)20217 32976 y Fo(D)21224 33158 y Fp(1)21749 32976 y Fw(\()p Fu(\025)p Fw(\).)540 b(Let)403 b Fo(D)27453 33158 y Fp(1)27979 32976 y Fw(\()p Fu(\025)p Fw(\))29628 33158 y Fs(nonr)26 b(es)33007 32976 y Fw(b)34 b(e)404 b(the)g(preimage)g(of)g Fu(\037)44071 33158 y Fp(2)45002 32976 y Fw(in)g Fo(D)47423 33158 y Fp(1)47948 32976 y Fw(\()p Fu(\025)p Fw(\):)15501 34954 y Fo(D)16508 35136 y Fp(1)17033 34954 y Fw(\()p Fu(\025)p Fw(\))18682 35136 y Fs(nonr)26 b(es)21994 34954 y Fw(=)337 b Ft(f)32 b Fu(~)-638 b Fn({)385 b Ft(2)337 b Fo(D)27225 35136 y Fp(1)27751 34954 y Fw(\()p Fu(\025)p Fw(\))g(:)f Fo(K)31407 35136 y Fp(2)31966 34954 y Fu(~)-638 b Fn({)385 b Ft(2)337 b Fu(\037)35030 35136 y Fp(2)35556 34954 y Ft(g)p Fu(:)11929 b Fw(\(3.54\))p 0 TeXcolorgray 475 37132 a FF(Lemma)508 b(3.40.)p 0 TeXcolorgray 574 w FC(The)469 b(formula)e Fo(K)16954 37314 y Fp(2)17480 37132 y Fo(D)18487 37314 y Fp(1)19013 37132 y Fw(\()p Fu(\025)p Fw(\))20662 37314 y Fs(nonr)26 b(es)24038 37132 y Fw(=)400 b Fu(\037)26139 37314 y Fp(2)27134 37132 y FC(establishes)465 b(one-to-one)h(c)-62 b(orr)g(esp)g(ondenc)g(e)466 b(b)-62 b(etwe)g(en)475 38571 y Fo(D)1482 38753 y Fp(1)2007 38571 y Fw(\()p Fu(\025)p Fw(\))3656 38753 y Fs(nonr)26 b(es)7065 38571 y FC(and)433 b Fu(\037)10177 38753 y Fp(2)10703 38571 y FC(.)p 0 TeXcolorgray 475 40750 a(Pr)-62 b(o)g(of.)p 0 TeXcolorgray 552 w Fw(Supp)34 b(ose)475 b(there)e(is)g(a)g(pair)506 b Fu(~)-639 b Fn({)17953 40932 y Fp(1)18479 40750 y Fu(;)234 b(~)-638 b Fn({)19826 40932 y Fp(2)20804 40750 y Ft(2)452 b Fo(D)23071 40932 y Fp(1)23597 40750 y Fw(\()p Fu(\025)p Fw(\))25246 40932 y Fs(nonr)26 b(es)28694 40750 y Fw(suc)-34 b(h)474 b(that)h Fo(K)35190 40932 y Fp(2)35748 40750 y Fu(~)-638 b Fn({)36524 40932 y Fp(1)37502 40750 y Fw(=)452 b Fo(K)39894 40932 y Fp(2)40452 40750 y Fu(~)-638 b Fn({)41228 40932 y Fp(2)42206 40750 y Fw(=)452 b Fu(\034)137 b Fw(,)490 b Fu(\034)589 b Ft(2)452 b Fu(\037)48232 40932 y Fp(2)48759 40750 y Fw(.)746 b(W)-101 b(e)475 42189 y(in)-34 b(tro)34 b(duce)286 b(also)g Fu(t)8498 42371 y Fp(1)9360 42189 y Fw(=)337 b Fo(K)11637 42371 y Fp(1)12195 42189 y Fu(~)-638 b Fn({)12971 42371 y Fp(1)13783 42189 y Fw(and)287 b Fu(t)16460 42371 y Fp(2)17322 42189 y Fw(=)336 b Fo(K)19598 42371 y Fp(1)20157 42189 y Fu(~)-638 b Fn({)20933 42371 y Fp(2)21458 42189 y Fw(.)499 b(The)287 b(de\014nition)g(\(2.48\))g(of)f Fu(\037)35341 41749 y Fr(\003)35341 42513 y Fp(1)35867 42189 y Fw(\()p Fu(\025)p Fw(\))h(implies)e(that)i Fu(t)44745 42371 y Fp(1)45271 42189 y Fu(;)202 b(t)46248 42371 y Fp(2)47110 42189 y Ft(2)336 b Fu(\037)49012 41749 y Fr(\003)49012 42513 y Fp(1)49539 42189 y Fw(\()p Fu(\025)p Fw(\),)475 43628 y(since)481 b Fu(~)-638 b Fn({)4298 43810 y Fp(1)4824 43628 y Fu(;)234 b(~)-638 b Fn({)6171 43810 y Fp(2)7109 43628 y Ft(2)413 b Fo(D)9337 43810 y Fp(1)9863 43628 y Fw(\()p Fu(\025)p Fw(\))11512 43810 y Fs(nonr)26 b(es)14899 43628 y Ft(\032)413 b Fo(D)17262 43810 y Fp(1)17787 43628 y Fw(\()p Fu(\025)p Fw(\).)676 b(Clearly)-101 b(,)460 b Fo(K)25951 43810 y Fp(2)26477 43628 y Fu(t)26915 43810 y Fp(1)27854 43628 y Fw(=)412 b Fo(K)30206 43810 y Fp(2)30732 43628 y Fu(t)31170 43810 y Fp(2)32109 43628 y Fw(=)g Fu(\034)587 b Fw(and,)461 b(hence,)g Fu(t)41530 43810 y Fp(1)42468 43628 y Fw(=)413 b Fu(\034)437 b Fw(+)299 b(2)p Fu(\031)43 b(p)47983 43810 y Fp(1)48509 43628 y Fu(=)-67 b(N)50022 43810 y Fp(1)50548 43628 y Fu(a)p Fw(,)475 45067 y Fu(t)913 45249 y Fp(2)1775 45067 y Fw(=)336 b Fu(\034)385 b Fw(+)247 b(2)p Fu(\031)43 b(p)7109 45249 y Fp(2)7635 45067 y Fu(=)-67 b(N)9148 45249 y Fp(1)9674 45067 y Fu(a)393 b Fw(for)g(some)g Fu(p)16188 45249 y Fp(1)16714 45067 y Fu(;)202 b(p)17863 45249 y Fp(2)18725 45067 y Ft(2)336 b Fu(P)168 b Fw(.)536 b(No)-34 b(w,)396 b(b)-34 b(y)393 b(P)-34 b(art)394 b(1a)f(of)g(Geometric)g(Lemma)f(3.5,) j Fu(p)45226 45249 y Fp(1)46089 45067 y Fw(=)336 b Fu(p)47978 45249 y Fp(2)48504 45067 y Fw(,)395 b(and,)475 46506 y(therefore,)403 b Fu(t)6340 46688 y Fp(1)7203 46506 y Fw(=)336 b Fu(t)8920 46688 y Fp(2)9446 46506 y Fw(.)538 b(Next,)404 b(b)-34 b(y)404 b(Lemma)g(2.13,)436 b Fu(~)-639 b Fn({)23334 46688 y Fp(1)24197 46506 y Fw(=)369 b Fu(~)-639 b Fn({)26284 46688 y Fp(2)26810 46506 y Fw(.)p 50750 46672 665 831 v 1803 48655 a(W)-101 b(e)404 b(de\014ne)g Fo(B)8315 48837 y Fp(2)8841 48655 y Fw(\()p Fu(\025)p Fw(\))h(as)f(the)g(set)h(of)f(directions)g(corresp)34 b(onding)404 b(to)h Fo(D)33555 48837 y Fp(1)34080 48655 y Fw(\()p Fu(\025)p Fw(\))35729 48837 y Fs(nonr)26 b(es)38704 48655 y Fw(:)13479 50632 y Fo(B)14402 50814 y Fp(2)14928 50632 y Fw(\()p Fu(\025)p Fw(\))337 b(=)g Ft(f)-25 b Fu(~)-581 b(\027)413 b Ft(2)337 b Fo(B)21880 50814 y Fp(1)22406 50632 y Fw(\()p Fu(\025)p Fw(\))g(:)f Fn({)25873 50814 y Fp(1)26399 50632 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))-25 b Fu(~)-581 b(\027)414 b Ft(2)337 b Fo(D)32428 50814 y Fp(1)32954 50632 y Fw(\()p Fu(\025)p Fw(\))34603 50814 y Fs(nonr)26 b(es)37578 50632 y Ft(g)p Fu(;)475 52610 y Fw(where)520 b Fn({)4904 52792 y Fp(1)5430 52610 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))520 b(is)g(de\014ned)i(b)-34 b(y)520 b(Lemma)g(2.11,)549 b Fn({)24532 52792 y Fp(1)25058 52610 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))530 b Ft(\021)g Fn({)30733 52792 y Fp(1)31259 52610 y Fw(\()p Fu(')p Fw(\).)888 b(Note)520 b(that)h Fo(B)40929 52792 y Fp(2)41455 52610 y Fw(\()p Fu(\025)p Fw(\))g(is)f(a)g(unit)h(circle)475 54049 y(with)422 b(holes,)j(cen)-34 b(tered)422 b(at)g(the)f(origin,)k (and)e Fo(B)22181 54231 y Fp(2)22706 54049 y Fw(\()p Fu(\025)p Fw(\))366 b Ft(\032)f Fo(B)26952 54231 y Fp(1)27478 54049 y Fw(\()p Fu(\025)p Fw(\).)591 b(W)-101 b(e)421 b(denote)h(b)-34 b(y)421 b(\002)38726 54231 y Fp(2)39252 54049 y Fw(\()p Fu(\025)p Fw(\))h(the)g(set)g(of)f(angles)h Fu(')p Fw(,)475 55488 y(corresp)34 b(onding)404 b(to)g Fo(B)10670 55670 y Fp(2)11196 55488 y Fw(\()p Fu(\025)p Fw(\):)10432 57465 y(\002)11375 57647 y Fp(2)11901 57465 y Fw(\()p Fu(\025)p Fw(\))337 b(=)g Ft(f)p Fu(')f Ft(2)h Fw([0)p Fu(;)202 b Fw(2)p Fu(\031)43 b Fw(\))337 b(:)741 b(\(cos)202 b Fu(';)g Fw(sin)g Fu(')p Fw(\))337 b Ft(2)g Fo(B)31742 57647 y Fp(2)32267 57465 y Fw(\()p Fu(\025)p Fw(\))p Ft(g)p Fu(;)1819 b Fw(\002)37621 57647 y Fp(2)38483 57465 y Ft(\032)337 b Fw(\002)40706 57647 y Fp(1)41231 57465 y Fu(:)475 59443 y Fw(Let)503 b Fu(~)-639 b Fn({)498 b Ft(2)448 b Fo(D)6282 59625 y Fp(1)6808 59443 y Fw(\()p Fu(\025)p Fw(\))8457 59625 y Fs(nonr)26 b(es)11432 59443 y Fw(.)739 b(By)471 b(\(3.54\),)488 b Fu(\034)585 b Ft(\021)448 b Fo(K)21903 59625 y Fp(2)22462 59443 y Fu(~)-638 b Fn({)497 b Ft(2)448 b Fu(\037)25749 59625 y Fp(2)26275 59443 y Fw(\()p Fu(\025)p Fw(\).)741 b(According)471 b(to)g(Theorem)g(3.8,)488 b(for)471 b(su\016cien)-34 b(tly)475 61160 y(large)407 b Fu(\025)p Fw(,)h(there)g(exists)g(an)g(eigen)-34 b(v)-67 b(alue)407 b(of)h(the)h(op)34 b(erator)407 b Fu(H)28309 60521 y Fp(\(2\))28211 61317 y Fs(\013)29567 61160 y Fw(\()p Fu(\034)137 b Fw(\),)410 b(giv)-34 b(en)407 b(b)-34 b(y)409 b(\(3.15\).)550 b(It)408 b(is)g(con)-34 b(v)g(enien)g(t)409 b(here)475 62876 y(to)516 b(denote)g Fu(\025)6792 62238 y Fp(\(2\))6773 63247 y(~)6792 63451 y Fs(j)8049 62876 y Fw(\()p Fu(\013)t(;)202 b(\034)137 b Fw(\))517 b(b)-34 b(y)516 b Fu(\025)13995 62437 y Fp(\(2\))15253 62876 y Fw(\()p Fu(\013)t(;)235 b(~)-639 b Fn({)50 b Fw(\).)873 b(W)-101 b(e)515 b(can)g(do)h(this,)544 b(since,)e(b)-34 b(y)516 b(Lemma)f(3.40,)543 b(there)515 b(is)g(one-to-one)475 64821 y(corresp)34 b(ondence)439 b(b)34 b(et)-34 b(w)g(een)473 b Fu(~)-638 b Fn({)445 b Ft(2)397 b Fo(D)17156 65003 y Fp(1)17681 64821 y Fw(\()p Fu(\025)p Fw(\))19330 65003 y Fs(nonr)26 b(es)22745 64821 y Fw(and)441 b(the)f(pair)g(\()p Fu(\034)70 b(;)31383 64543 y Fw(~)31402 64821 y Fu(j)f Fw(\),)481 b Fu(~)-638 b Fn({)445 b Fw(=)396 b(2)p Fu(\031)37141 64543 y Fw(~)37159 64821 y Fu(j)70 b(=a)293 b Fw(+)g Fu(\034)137 b Fw(.)646 b(W)-101 b(e)439 b(rewrite)h(\(3.15\))475 66260 y(in)404 b(the)g(form:)10479 68423 y Fu(\025)11186 67923 y Fp(\(2\))12444 68423 y Fw(\()p Fu(\013)t(;)235 b(~)-639 b Fn({)50 b Fw(\))337 b(=)f Fu(\025)17885 67923 y Fp(\(1\))19143 68423 y Fw(\()32 b Fu(~)-638 b Fn({)49 b Fw(\))270 b(+)f Fu(f)23017 68605 y Fp(2)23544 68423 y Fw(\()p Fu(\013)t(;)235 b(~)-639 b Fn({)49 b Fw(\))p Fu(;)1415 b(f)29006 68605 y Fp(2)29532 68423 y Fw(\()p Fu(\013)t(;)235 b(~)-639 b Fn({)50 b Fw(\))337 b(=)34671 66908 y Fr(1)34266 67272 y Fq(X)34315 69862 y Fs(r)26 b Fp(=1)36219 68423 y Fu(\013)36998 67923 y Fs(r)37505 68423 y Fu(g)38126 67923 y Fp(\(2\))38083 68723 y Fs(r)39384 68423 y Fw(\()32 b Fu(~)-638 b Fn({)50 b Fw(\))p Fu(;)6907 b Fw(\(3.55\))p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 32 32 TeXDict begin 32 31 bop 0 TeXcolorgray 0 TeXcolorgray 475 3345 a Fp(32)14248 b(YULIA)313 b(KARPESHINA)f(AND)h(YOUNG-RAN)g (LEE)p 0 TeXcolorgray 475 6197 a Fw(here)403 b Fu(g)3725 5559 y Fp(\(2\))3682 6355 y Fs(r)4984 6197 y Fw(\()32 b Fu(~)-638 b Fn({)49 b Fw(\))404 b(is)g(giv)-34 b(en)404 b(b)-34 b(y)405 b(\(3.13\).)539 b(The)404 b(function)i Fu(f)25060 6379 y Fp(2)25586 6197 y Fw(\()p Fu(\013)t(;)235 b(~)-639 b Fn({)50 b Fw(\))404 b(satis\014es)h(the)g(estimates:)19633 8438 y Ft(j)p Fu(f)20563 8620 y Fp(2)21089 8438 y Fw(\()p Fu(\013)t(;)235 b(~)-639 b Fn({)50 b Fw(\))p Ft(j)337 b(\024)f Fw(12)p Fu(\013)t(\017)28643 7938 y Fp(4)28643 8738 y(1)29170 8438 y Fu(;)18921 b Fw(\(3.56\))18623 10511 y Ft(jr)p Fu(f)20563 10693 y Fp(2)21089 10511 y Fw(\()p Fu(\013)t(;)235 b(~)-639 b Fn({)50 b Fw(\))p Ft(j)337 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Fp(4)30880 31469 y(1)31407 31170 y Fu(k)32076 30669 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)34828 31170 y Fu(:)13263 b Fw(\(3.60\))p 0 TeXcolorgray 475 33472 a FC(Pr)-62 b(o)g(of.)p 0 TeXcolorgray 552 w Fw(Let)606 b(us)h(sho)-34 b(w)608 b(that)g(there)f(exists)f(a)h Fn({)23261 33654 y Fp(2)23787 33472 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))607 b(satisfying)g(\(3.59\).)1147 b(Since)606 b Fn({)41617 33654 y Fp(1)42118 33472 y Fu(~)-581 b(\027)751 b Ft(2)674 b Fo(D)45982 33654 y Fp(1)46508 33472 y Fw(\()p Fu(\025)p Fw(\))607 b(when)450 34990 y Fu(~)-581 b(\027)618 b Ft(2)542 b Fo(B)3965 35172 y Fp(2)4490 34990 y Fw(\()p Fu(\025)p Fw(\),)558 b(for)527 b(all)g Fn({)576 b Fw(in)527 b(the)g(in)-34 b(terv)-67 b(al)526 b Fu(I)20986 35172 y Fp(2)21512 34990 y Fw(,)557 b(there)527 b(is)g(an)g(eigen)-34 b(v)-67 b(alue)526 b Fu(\025)35399 34550 y Fp(\(2\))36657 34990 y Fw(\()p Fu(\013)t(;)202 b Fn({)24 b Fu(~)-581 b(\027)78 b Fw(\))527 b(giv)-34 b(en)527 b(b)-34 b(y)527 b(\(3.55\).)908 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44664 y Fp(1)38096 44482 y Fu(\017)38588 44664 y Fp(1)39114 44482 y Fu(k)39783 44042 y Fr(\000)p Fs(\016)41020 44482 y Fu(=)p Fw(2])p Fu(:)567 b Fw(Th)-34 b(us)570 b(for)e(ev)-34 b(ery)450 46022 y Fu(~)-581 b(\027)413 b Ft(2)337 b Fo(B)3555 46204 y Fp(2)4080 46022 y Fw(\()p Fu(\025)p Fw(\),)405 b(there)f(exists)g(a)g Fn({)14737 46204 y Fp(2)15263 46022 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))337 b Ft(2)g Fu(I)20142 46204 y Fp(2)21072 46022 y Fw(suc)-34 b(h)405 b(that)g Fu(\025)27139 45582 y Fp(\(2\))28397 46022 y Fw(\()p Fu(\013)t(;)202 b Fn({)30994 46204 y Fp(2)31496 46022 y Fu(~)-582 b(\027)77 b Fw(\))337 b(=)g Fu(\025)p Fw(.)1803 47461 y(Using)405 b(\(3.26\),)f(w)-34 b(e)404 b(obtain)17281 49231 y Fu(@)67 b(\025)18699 48791 y Fp(\(2\))19957 49231 y Fw(\()p Fu(\013)t(;)202 b Fn({)24 b Fu(~)-581 b(\027)77 b Fw(\))p 17281 49772 6470 49 v 19732 50882 a Fu(@)67 b Fn({)24220 50051 y Fw(=)337 b(2)p Fu(l)24 b(k)27161 49550 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1)29180 49069 y Fq(\000)29736 50051 y Fw(1)269 b(+)g Fu(o)p Fw(\(1\))33959 49069 y Fq(\001)34515 50051 y Fu(;)13576 b Fw(\(3.61\))475 52601 y(and)450 b(this)g(implies)g(that)g Fu(\025)12825 52161 y Fp(\(2\))14083 52601 y Fw(\()p Fu(\013)t(;)202 b Fn({)24 b Fu(~)-581 b(\027)78 b Fw(\))450 b(is)f(monotone)i(with)g(resp)34 b(ect)449 b(to)h Fn({)499 b Fw(in)449 b Fu(I)37056 52783 y Fp(2)37582 52601 y Fw(.)675 b(Hence,)460 b(there)450 b(is)g(only)f(one)475 54040 y Fn({)1283 54222 y Fp(2)1809 54040 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))404 b(satisfying)h(\(3.59\))g(in)f Fu(I)15927 54222 y Fp(2)16452 54040 y Fw(.)1803 55558 y(No)-34 b(w)595 b(let)g(us)f(estimate)h Ft(j)p Fn({)14632 55740 y Fp(2)15158 55558 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))270 b Ft(\000)f Fn({)20312 55740 y Fp(1)20838 55558 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))p Ft(j)p Fw(.)1109 b(By)594 b(\(3.55\),)642 b(w)-34 b(e)595 b(ha)-34 b(v)g(e)595 b Fu(\025)37321 55118 y Fp(\(2\))38579 55558 y Fw(\()p Fu(\013)t(;)202 b Fn({)41176 55740 y Fp(1)41678 55558 y Fu(~)-581 b(\027)77 b Fw(\))654 b(=)g Fu(\025)45808 55118 y Fp(\(1\))47065 54576 y Fq(\000)47621 55558 y Fn({)48429 55740 y Fp(1)48930 55558 y Fu(~)-581 b(\027)49631 54576 y Fq(\001)50582 55558 y Fw(+)475 57197 y Fu(f)1068 57379 y Fp(2)1594 56215 y Fq(\000)2150 57197 y Fu(\013)t(;)202 b Fn({)4276 57379 y Fp(1)4777 57197 y Fu(~)-581 b(\027)5478 56215 y Fq(\001)6370 57197 y Fw(=)337 b Fu(\025)p Fw(+)p Fu(f)9893 57379 y Fp(2)10419 56215 y Fq(\000)10974 57197 y Fu(\013)t(;)202 b Fn({)13100 57379 y Fp(1)13602 57197 y Fu(~)-581 b(\027)14303 56215 y Fq(\001)14858 57197 y Fw(.)494 b(Using)270 b(\(3.56\))g(and)g (\(3.59\),)296 b(w)-34 b(e)270 b(easily)f(obtain:)37312 56167 y Fq(\014)37312 56894 y(\014)37716 57197 y Fu(\025)38423 56757 y Fp(\(2\))39681 56215 y Fq(\000)40236 57197 y Fu(\013)t(;)202 b Fn({)42362 57379 y Fp(2)42864 57197 y Fu(~)-581 b(\027)43564 56215 y Fq(\001)44120 57197 y Ft(\000)p Fu(\025)45770 56757 y Fp(\(2\))47028 56215 y Fq(\000)47583 57197 y Fu(\013)t(;)202 b Fn({)49709 57379 y Fp(1)50211 57197 y Fu(~)-581 b(\027)50912 56215 y Fq(\001)51467 56167 y(\014)51467 56894 y(\014)52208 57197 y Fu(<)475 58742 y Fw(12)p Fu(\013)t(\017)2958 58303 y Fp(4)2958 59067 y(1)3485 58742 y Fw(.)744 b(Applying)473 b(the)g(Mean)g(V)-101 b(alue)472 b(Theorem)h(at)g(the)g(left-hand)h (side)e(and)i(using)f(\(3.61\),)490 b(w)-34 b(e)473 b(arriv)-34 b(e)472 b(at)475 60233 y(the)404 b(inequalit)-34 b(y:)539 b Fu(l)24 b(k)9712 59793 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1)11731 60233 y Ft(j)p Fn({)12876 60415 y Fp(2)13671 60233 y Ft(\000)269 b Fn({)15691 60415 y Fp(1)16217 60233 y Ft(j)336 b Fu(<)h Fw(12)p Fu(\013)t(\017)20653 59793 y Fp(4)20653 60558 y(1)21179 60233 y Fw(.)539 b(The)404 b(estimate)h(\(3.60\))f(immediately)g(follo)-34 b(ws.)p 50750 60399 665 831 v 1803 62669 a(Let)573 b(us)h(recall)e(that)i Fu(\010)12877 62851 y Fp(2)13976 62669 y Fw(is)g(the)f(second)h (complex)f(non-resonan)-34 b(t)575 b(set)f(giv)-34 b(en)573 b(b)-34 b(y)574 b(\(3.40\),)616 b(see)572 b(Fig.)1046 b(7.)475 64108 y(F)-101 b(urther,)404 b(w)-34 b(e)405 b(use)f(the)g(notations:)540 b Fn({)17784 64290 y Fp(2)18310 64108 y Fw(\()p Fu(')p Fw(\))338 b Ft(\021)e Fn({)22470 64290 y Fp(2)22996 64108 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\),)404 b Fu(h)27299 64290 y Fp(2)27825 64108 y Fw(\()p Fu(')p Fw(\))338 b(=)e Fn({)31985 64290 y Fp(2)32511 64108 y Fw(\()p Fu(')p Fw(\))270 b Ft(\000)f Fn({)36536 64290 y Fp(1)37062 64108 y Fw(\()p Fu(')p Fw(\),)437 b Fu(~)-638 b Fn({)40347 64290 y Fp(2)40873 64108 y Fw(\()p Fu(')p Fw(\))337 b(=)f Fn({)45032 64290 y Fp(2)45558 64108 y Fw(\()p Fu(')p Fw(\))-25 b Fu(~)-581 b(\027)78 b Fw(.)p 0 TeXcolorgray 475 66410 a FF(Lemma)463 b(3.42.)p 0 TeXcolorgray 0 TeXcolorgray 2489 w Fw(\(1\))p 0 TeXcolorgray 554 w FC(The)480 b(set)f Fo(D)18341 66592 y Fp(2)18866 66410 y Fw(\()p Fu(\025)p Fw(\))i FC(is)f(a)g(distorte)-62 b(d)477 b(cir)-62 b(cle)479 b(with)g(holes:)649 b(it)480 b(c)-62 b(an)479 b(b)-62 b(e)480 b(describ)-62 b(e)g(d)477 b(by)i(the)4511 67850 y(formula:)14180 69541 y Fo(D)15187 69723 y Fp(2)15712 69541 y Fw(\()p Fu(\025)p Fw(\))338 b(=)18978 68559 y Fq(\010)19717 69541 y Fu(~)-638 b Fn({)386 b Ft(2)337 b Fn(R)22899 69040 y Fp(2)23762 69541 y Fw(:)368 b Fu(~)-638 b Fn({)386 b Fw(=)368 b Fu(~)-638 b Fn({)27716 69723 y Fp(2)28242 69541 y Fw(\()p Fu(')p Fw(\))p Fu(;)1070 b(')336 b Ft(2)h Fw(\002)34601 69723 y Fp(2)35127 69541 y Fw(\()p Fu(\025)p Fw(\))36776 68559 y Fq(\011)37484 69541 y Fu(;)10607 b Fw(\(3.62\))p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 33 33 TeXDict begin 33 32 bop 0 TeXcolorgray 0 TeXcolorgray 2848 3345 a Fp(SPECTRAL)406 b(PR)-26 b(OPER)-78 b(TIES)406 b(OF)h(A)h(POL)-78 b(YHARMONIC)405 b(OPERA)-78 b(TOR)406 b(WITH)h(LIMIT-PERIODIC)e(POTENTIAL)1430 b(33)p 0 TeXcolorgray 4511 6001 a FC(wher)-62 b(e)554 b Fn({)8863 6183 y Fp(2)9388 6001 y Fw(\()p Fu(')p Fw(\))i(=)e Fn({)13984 6183 y Fp(1)14510 6001 y Fw(\()p Fu(')p Fw(\))357 b(+)f Fu(h)18599 6183 y Fp(2)19125 6001 y Fw(\()p Fu(')p Fw(\))p FC(,)585 b Fn({)22625 6183 y Fp(1)23151 6001 y Fw(\()p Fu(')p Fw(\))554 b FC(is)f(the)g(\\r)-62 b(adius")551 b(of)i Fo(D)36554 6183 y Fp(1)37080 6001 y Fw(\()p Fu(\025)p Fw(\))i FC(and)e Fu(h)42456 6183 y Fp(2)42982 6001 y Fw(\()p Fu(')p Fw(\))i FC(satis\014es)c(the)4511 7440 y(estimates)15540 9309 y Ft(j)p Fu(h)16575 9491 y Fp(2)17101 9309 y Ft(j)337 b(\024)f Fw(2)p Fu(\013)t(\017)20931 8808 y Fp(4)20931 9608 y(1)21458 9309 y Fu(k)22127 8808 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)24878 9309 y Fu(;)26718 7551 y Fq(\014)26718 8278 y(\014)26718 9006 y(\014)26718 9733 y(\014)27255 8489 y Fu(@)67 b(h)28664 8671 y Fp(2)p 27255 9030 1936 49 v 27470 10140 a Fu(@)g(')29323 7551 y Fq(\014)29323 8278 y(\014)29323 9006 y(\014)29323 9733 y(\014)30063 9309 y Ft(\024)337 b Fw(4)p Fu(\013)t(\017)33220 8808 y Fp(3)33220 9608 y(1)33747 9309 y Fu(k)34416 8808 y Fp(1+)p Fs(\016)36123 9309 y Fu(:)11968 b Fw(\(3.63\))p 0 TeXcolorgray 2409 11732 a(\(2\))p 0 TeXcolorgray 554 w FC(The)434 b(total)d(length)i(of)g Fo(B)15688 11914 y Fp(2)16213 11732 y Fw(\()p Fu(\025)p Fw(\))h FC(satis\014es)e(the)g (estimate:)19335 13720 y Fu(L)202 b Fw(\()p Fo(B)21756 13902 y Fp(1)22551 13720 y Ft(n)269 b Fo(B)24349 13902 y Fp(2)24875 13720 y Fw(\))337 b Fu(<)f Fw(4)p Fu(\031)43 b(k)28971 13220 y Fr(\000)p Fp(2)p Fr(\000)p Fp(2)p Fs(s)31810 13343 y FD(1)32328 13720 y Fu(:)15763 b Fw(\(3.64\))p 0 TeXcolorgray 2409 15663 a(\(3\))p 0 TeXcolorgray 554 w FC(The)578 b(function)f Fn({)12857 15845 y Fp(2)13382 15663 y Fw(\()p Fu(')p Fw(\))i FC(c)-62 b(an)578 b(b)-62 b(e)577 b(extende)-62 b(d)576 b(as)i(a)f(holomorphic)e(function)h(of)i Fu(')g FC(into)f(the)g(c)-62 b(omplex)4511 17102 y(non-r)g(esonanc)g(e) 433 b(set)f Fu(\010)15216 17284 y Fp(2)15742 17102 y FC(,)i(estimates)d(\(3.63\))g(b)-62 b(eing)432 b(pr)-62 b(eserve)g(d.)p 0 TeXcolorgray 2409 18541 a Fw(\(4\))p 0 TeXcolorgray 554 w FC(The)313 b(curve)f Fo(D)11024 18723 y Fp(2)11549 18541 y Fw(\()p Fu(\025)p Fw(\))i FC(has)e(a)h(length)e(which)h(is)h(asymptotic)-62 b(al)62 b(ly)308 b(close)k(to)g(that)g(of)g Fo(D)41069 18723 y Fp(1)41594 18541 y Fw(\()p Fu(\025)p Fw(\))j FC(in)e(the)f(fol)62 b(lowing)4511 19980 y(sense:)14009 21485 y Fu(L)14834 20139 y Fq(\020)15558 21485 y Fo(D)16565 21667 y Fp(2)17091 21485 y Fw(\()p Fu(\025)p Fw(\))18740 20139 y Fq(\021)20545 21485 y Fw(=)19801 22302 y Fs(\025)p Fr(!1)22569 21485 y Fu(L)23394 20139 y Fq(\020)24117 21485 y Fo(D)25124 21667 y Fp(1)25650 21485 y Fw(\()p Fu(\025)p Fw(\))27299 20139 y Fq(\021)q(\020)28748 21485 y Fw(1)269 b(+)g Fu(O)31794 20503 y Fq(\000)32349 21485 y Fu(k)33018 20984 y Fr(\000)p Fp(2)p Fr(\000)p Fp(2)p Fs(s)35857 21107 y FD(1)36374 20503 y Fq(\001)36930 20139 y(\021)37654 21485 y Fu(:)10437 b Fw(\(3.65\))p 0 TeXcolorgray 475 24284 a FC(Pr)-62 b(o)g(of.)p 0 TeXcolorgray 0 TeXcolorgray 2487 w Fw(\(1\))p 0 TeXcolorgray 554 w(By)405 b(Lemma)h(3.41,)h Ft(j)p Fu(h)18424 24466 y Fp(2)18950 24284 y Ft(j)340 b Fw(=)g Ft(j)p Fn({)22055 24466 y Fp(2)22852 24284 y Ft(\000)270 b Fn({)24873 24466 y Fp(1)25399 24284 y Ft(j)340 b(\024)h Fw(2)p Fu(\013)t(\017)29237 23844 y Fp(4)29237 24609 y(1)29763 24284 y Fu(k)30432 23844 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)33184 24284 y Fw(.)545 b(Let)406 b(us)h(pro)-34 b(v)g(e)407 b(the)f(second)h(estimate)4511 25824 y(in)533 b(\(3.63\).)924 b(Considering)534 b(that)f Fu(\025)551 b Fw(=)g Fu(\025)23431 25384 y Fp(\(2\))24890 25824 y Fw(\()q Fu(\013)t(;)234 b(~)-638 b Fn({)27488 26006 y Fp(2)28014 25824 y Fw(\()p Fu(')p Fw(\)\))q(,)565 b Fu(\025)550 b Fw(=)h Fu(\025)34581 25384 y Fp(\(1\))36041 25824 y Fw(\()32 b Fu(~)-638 b Fn({)37320 26006 y Fp(1)37846 25824 y Fw(\()p Fu(')p Fw(\)\))203 b Fu(;)532 b Fw(di\013eren)-34 b(tiating)533 b(b)34 b(oth)4511 27263 y(equations)405 b(with)g(resp)34 b(ect)404 b(to)g Fu(')g Fw(and)h(subtracting)g(the)g (results,)f(w)-34 b(e)404 b(obtain:)18621 29217 y Fu(@)67 b(\025)20039 28777 y Fp(\(2\))p 18621 29758 2676 49 v 19175 30868 a Fu(@)g Fn({)21632 30037 y Fw(\()p Fu(\013)t(;)235 b(~)-639 b Fn({)24229 30219 y Fp(2)24755 30037 y Fw(\))25562 29217 y Fu(@)67 b(h)26971 29399 y Fp(2)p 25562 29758 1936 49 v 25777 30868 a Fu(@)g(')27966 30037 y Fw(=)337 b Fu(F)168 b Fw(\()p Fu(\025;)202 b(')p Fw(\))p Fu(;)14917 b Fw(\(3.66\))23396 33131 y Fu(F)168 b Fw(\()p Fu(\025;)202 b(')p Fw(\))338 b(=)19823 b(\(3.67\))6695 34549 y Fq(\020)7418 35894 y Ft(r)p Fu(\025)9135 35394 y Fp(\(1\))10595 35894 y Fw(\()33 b Fu(~)-638 b Fn({)11875 36076 y Fp(1)12401 35894 y Fw(\))269 b Ft(\000)g(r)p Fu(\025)16070 35394 y Fp(\(2\))17530 35894 y Fw(\()q Fu(\013)t(;)234 b(~)-638 b Fn({)20128 36076 y Fp(2)20654 35894 y Fw(\))21126 34549 y Fq(\021)21982 35074 y Fu(@)67 b Fn({)23501 35256 y Fp(1)p 21982 35616 2045 49 v 22253 36726 a Fu(@)g(')24429 35894 y Fw(+)25641 34549 y Fq(D)26382 35894 y Fn({)27190 36076 y Fp(1)27716 35894 y Ft(r)p Fu(\025)29433 35394 y Fp(\(1\))30893 35894 y Fw(\()32 b Fu(~)-638 b Fn({)32172 36076 y Fp(1)32698 35894 y Fw(\))270 b Ft(\000)f Fn({)35459 36076 y Fp(2)35985 35894 y Ft(r)p Fu(\025)37702 35394 y Fp(\(2\))39162 35894 y Fw(\()p Fu(\013)t(;)235 b(~)-639 b Fn({)41759 36076 y Fp(2)42286 35894 y Fw(\))202 b Fu(;)i(~)-608 b(\026)44228 34549 y Fq(E)44969 35894 y Fu(;)4511 38233 y Fw(where)407 b Fu(~)-608 b(\026)337 b Fw(=)f(\()p Ft(\000)202 b Fw(sin)g Fu(';)g Fw(cos)g Fu(')p Fw(\).)539 b(It)404 b(is)g(easy)f(to)i(c)-34 b(hec)g(k)404 b(\(App)34 b(endix)405 b(A.3\))f(that)19816 39236 y Fq(\014)19816 39963 y(\014)20220 40266 y Fu(F)168 b Fw(\()p Fu(\025;)202 b(')p Fw(\))24148 39236 y Fq(\014)24148 39963 y(\014)24890 40266 y Fu(<)337 b Fw(27)p Fu(\013)t(k)28830 39765 y Fp(2)p Fs(l)11 b Fp(+)p Fs(\016)30829 40266 y Fu(\017)31321 39765 y Fp(3)31321 40566 y(1)31847 40266 y Fu(:)16244 b Fw(\(3.68\))4511 42214 y(Then,)587 b(the)550 b(second)g(estimate)f(in)h(\(3.63\))g (easily)f(follo)-34 b(ws)550 b(from)g(form)-34 b(ulae)550 b(\(3.66\))g(and)g(estimates)4511 43653 y(\(3.61\))405 b(and)g(\(3.68\).)p 0 TeXcolorgray 2409 45092 a(\(2\))p 0 TeXcolorgray 554 w(Considering)400 b(that)g Fu(\002)14729 45274 y Fp(1)15653 45092 y Fw(is)f(the)g(set)g(of)g(angles)g(corresp)34 b(onding)399 b(to)g Fo(D)36118 45274 y Fp(1)36643 45092 y Fw(\()p Fu(\025)p Fw(\))h(\(Lemma)f(2.11\))g(and)g Fu(\002)49786 45274 y Fp(2)50710 45092 y Fw(is)4511 46531 y(the)405 b(set)f(of)h(angles)f(corresp)34 b(onding)404 b(to)h Fo(D)23795 46713 y Fp(1)24320 46531 y Fw(\()p Fu(\025)p Fw(\))25969 46713 y Fs(nonr)26 b(es)28944 46531 y Fw(,)404 b(w)-34 b(e)404 b(obtain)11051 49098 y Fu(L)11876 47753 y Fq(\020)12600 49098 y Fo(D)13607 49280 y Fp(1)14132 49098 y Fw(\()p Fu(\025)p Fw(\))270 b Ft(n)g Fo(D)17934 49280 y Fp(1)18459 49098 y Fw(\()p Fu(\025)p Fw(\))20108 49280 y Fs(nonr)26 b(es)23083 47753 y Fq(\021)24144 49098 y Fw(=)25424 47448 y Fq(Z)26097 50197 y Fs(\002)26811 50320 y FD(1)27272 50197 y Fr(n)p Fs(\002)28456 50320 y FD(2)29175 47621 y Fq(q)p 30387 47621 8600 49 v 1477 x Fn({)31244 48681 y Fp(2)31195 49445 y(1)31770 49098 y Fw(\()p Fu(')p Fw(\))270 b(+)f Fn({)35844 48681 y Fr(0)35795 49445 y Fp(1)36321 49098 y Fw(\()p Fu(')p Fw(\))38461 48748 y Fp(2)39189 49098 y Fu(d':)7478 b Fw(\(3.69\))4511 51942 y(Using)405 b(\(2.43\),)f(w)-34 b(e)405 b(easily)e(sho)-34 b(w)405 b(that)g(the)g(in)-34 b(tegrand)405 b(is)f(close)f(to)i Fu(k)38 b Fw(.)539 b(Hence,)12535 54253 y Fu(L)13360 52908 y Fq(\020)14083 54253 y Fo(D)15090 54435 y Fp(1)15616 54253 y Fw(\()p Fu(\025)p Fw(\))270 b Ft(n)g Fo(D)19418 54435 y Fp(1)19943 54253 y Fw(\()p Fu(\025)p Fw(\))21592 54435 y Fs(nonr)26 b(es)24567 52908 y Fq(\021)25628 54253 y Fw(=)336 b Fu(L)202 b Fw(\()q Fu(\002)29331 54435 y Fp(1)30125 54253 y Ft(n)270 b Fu(\002)31926 54435 y Fp(2)32451 54253 y Fw(\))202 b Fu(k)33793 53271 y Fq(\000)34349 54253 y Fw(1)270 b(+)f Fu(o)p Fw(\(1\))38573 53271 y Fq(\001)39129 54253 y Fu(:)8962 b Fw(\(3.70\))4511 56497 y(Let)404 b(us)h(sho)-34 b(w)405 b(that)13295 58506 y Fu(L)14120 57160 y Fq(\020)14844 58506 y Fo(D)15851 58688 y Fp(1)16376 58506 y Fw(\()p Fu(\025)p Fw(\))270 b Ft(n)g Fo(D)20178 58688 y Fp(1)20703 58506 y Fw(\()p Fu(\025)p Fw(\))22352 58688 y Fs(nonr)26 b(es)25327 57160 y Fq(\021)26388 58506 y Fu(<)336 b Fw(2)p Fu(\031)43 b(k)29676 58005 y Fr(\000)p Fp(1)p Fr(\000)p Fp(2)p Fs(s)32515 58128 y FD(1)33033 57524 y Fq(\000)33589 58506 y Fw(1)269 b(+)g Fu(o)p Fw(\(1\))37812 57524 y Fq(\001)38368 58506 y Fu(:)9723 b Fw(\(3.71\))4511 60938 y(In)458 b(fact,)471 b(b)-34 b(y)458 b(Lemma)f(3.40,)470 b Fu(L)18778 59956 y Fq(\000)19333 60938 y Fo(D)20340 61120 y Fp(1)20866 60938 y Fw(\()p Fu(\025)p Fw(\))22515 61120 y Fs(nonr)26 b(es)25490 59956 y Fq(\001)26471 60938 y Fw(=)425 b Fu(L)p Fw(\()p Fu(\037)29893 61120 y Fp(2)30420 60938 y Fw(\).)699 b(By)456 b(Lemma)h(2.13,)470 b Fu(L)42049 59956 y Fq(\000)42605 60938 y Fo(D)43612 61120 y Fp(1)44137 60938 y Fw(\()p Fu(\025)p Fw(\))45786 59956 y Fq(\001)46768 60938 y Fw(=)426 b Fu(L)p Fw(\()p Fu(\037)50191 60498 y Fr(\003)50191 61263 y Fp(1)50717 60938 y Fw(\).)4511 62503 y(By)461 b(Corollary)f(3.37,)474 b Fu(L)15679 61522 y Fq(\000)16235 62503 y Fu(\037)16993 62064 y Fr(\003)16993 62828 y Fp(1)17519 61522 y Fq(\001)18506 62503 y Fw(=)431 b Fu(L)20705 61522 y Fq(\000)21260 62503 y Fo(K)22257 62685 y Fp(2)22784 62503 y Fu(\037)23542 62064 y Fr(\003)23542 62828 y Fp(1)24068 61522 y Fq(\001)24623 62503 y Fw(,)475 b(and,)h(therefore,)e Fu(L)34523 61522 y Fq(\000)35079 62503 y Fo(D)36086 62685 y Fp(1)36611 62503 y Fw(\()p Fu(\025)p Fw(\))38260 61522 y Fq(\001)39248 62503 y Fw(=)431 b Fu(L)p Fw(\()p Fo(K)42915 62685 y Fp(2)43442 62503 y Fu(\037)44200 62064 y Fr(\003)44200 62828 y Fp(1)44726 62503 y Fw(\).)709 b(It)461 b(follo)-34 b(ws:)4511 64069 y Fu(L)5336 63087 y Fq(\000)5892 64069 y Fo(D)6899 64251 y Fp(1)7424 64069 y Fw(\()p Fu(\025)p Fw(\))84 b Ft(n)g Fo(D)10854 64251 y Fp(1)11381 64069 y Fw(\()p Fu(\025)p Fw(\))13030 64251 y Fs(nonr)26 b(es)16005 63087 y Fq(\001)16898 64069 y Fw(=)336 b Fu(L)19002 63087 y Fq(\000)19558 64069 y Fo(D)20565 64251 y Fp(1)21090 64069 y Fw(\()p Fu(\025)p Fw(\))22739 63087 y Fq(\001)23380 64069 y Ft(\000)84 b Fu(L)25232 63087 y Fq(\000)25787 64069 y Fo(D)26794 64251 y Fp(1)27320 64069 y Fw(\()p Fu(\025)p Fw(\))28969 64251 y Fs(nonr)26 b(es)31944 63087 y Fq(\001)32836 64069 y Fw(=)337 b Fu(L)p Fw(\()p Fo(K)36409 64251 y Fp(2)36935 64069 y Fu(\037)37693 63629 y Fr(\003)37693 64393 y Fp(1)38219 64069 y Fw(\))84 b Ft(\000)g Fu(L)p Fw(\()p Fu(\037)41855 64251 y Fp(2)42383 64069 y Fw(\))337 b(=)f Fu(L)45295 63087 y Fq(\000)45851 64069 y Fo(K)46848 64251 y Fp(2)47374 64069 y Fu(\037)48132 63629 y Fr(\003)48132 64393 y Fp(1)48743 64069 y Ft(n)84 b Fu(\037)50191 64251 y Fp(2)50717 64069 y Fw(\),)4511 65513 y(the)477 b(last)f(equalit)-34 b(y)475 b(b)34 b(eing)476 b(v)-67 b(alid)475 b(since)h Fu(\037)23862 65695 y Fp(2)24864 65513 y Fw(is)g(a)g(subset)g(of)h Fo(K)33475 65695 y Fp(2)34001 65513 y Fu(\037)34759 65073 y Fr(\003)34759 65838 y Fp(1)35285 65513 y Fw(.)754 b(Using)476 b(the)h(estimate)f(\(3.10\))g(in)4511 66952 y(the)405 b(Geometric)e(Lemma,)h(w)-34 b(e)404 b(obtain)13905 69196 y Fu(L)14730 67850 y Fq(\020)15454 69196 y Fo(D)16461 69378 y Fp(1)16986 69196 y Fw(\()p Fu(\025)p Fw(\))271 b Ft(n)e Fo(D)20788 69378 y Fp(1)21314 69196 y Fw(\()p Fu(\025)p Fw(\))22963 69378 y Fs(nonr)26 b(es)25938 67850 y Fq(\021)26998 69196 y Fu(<)337 b(L)29103 67850 y Fq(\020)29826 69196 y Fo(D)30833 69378 y Fp(1)31359 69196 y Fw(\()p Fu(\025)p Fw(\))33008 67850 y Fq(\021)33733 69196 y Fu(k)34402 68695 y Fr(\000)p Fp(2)p Fr(\000)p Fp(2)p Fs(s)37241 68818 y FD(1)37758 69196 y Fu(:)10333 b Fw(\(3.72\))p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 34 34 TeXDict begin 34 33 bop 0 TeXcolorgray 0 TeXcolorgray 475 3345 a Fp(34)14248 b(YULIA)313 b(KARPESHINA)f(AND)h(YOUNG-RAN)g (LEE)p 0 TeXcolorgray 4511 6001 a Fw(No)-34 b(w)361 b(considering)f (that)i Fu(L)16932 5020 y Fq(\000)17487 6001 y Fo(D)18494 6183 y Fp(1)19020 6001 y Fw(\()p Fu(\025)p Fw(\))20669 5020 y Fq(\001)21562 6001 y Fw(=)336 b(2)p Fu(\031)43 b(k)24850 5020 y Fq(\000)25406 6001 y Fw(1)181 b(+)g Fu(o)p Fw(\(1\))29453 5020 y Fq(\001)30370 6001 y Fw(\(Lemma)360 b(2.11\),)369 b(w)-34 b(e)360 b(arriv)-34 b(e)359 b(at)i(the)f (estimate)4511 7445 y(\(3.71\).)540 b(Com)-34 b(bining)405 b(it)f(with)h(\(3.70\),)g(w)-34 b(e)404 b(get)g(\(3.64\).)p 0 TeXcolorgray 2409 8884 a(\(3\))p 0 TeXcolorgray 554 w(By)557 b(the)h(third)g(statemen)-34 b(t)559 b(of)f(Lemma)f(2.11,)596 b(the)558 b(function)g Fn({)34829 9066 y Fp(1)35355 8884 y Fw(\()p Fu(')p Fw(\))g(is)g(holomorphic)f(in)h Fu(\010)48489 9066 y Fp(1)49572 8884 y Fw(and)4511 10402 y Fu(\025)5218 9962 y Fp(\(1\))6476 9420 y Fq(\000)7032 10402 y Fn({)7840 10584 y Fp(1)8366 10402 y Fw(\()p Fu(')p Fw(\))10101 9420 y Fq(\001)11311 10402 y Fw(=)655 b Fu(\025)p Fw(.)1110 b(If)595 b Fu(')654 b Ft(2)h Fw(\002)20319 10584 y Fp(2)20844 10402 y Fw(,)643 b(then)595 b Fu(\025)25482 9962 y Fp(\(2\))26740 9420 y Fq(\000)27295 10402 y Fn({)28103 10584 y Fp(1)28629 10402 y Fw(\()p Fu(')p Fw(\))30364 9420 y Fq(\001)31515 10402 y Fw(is)g(de\014ned)h(b)-34 b(y)595 b(the)g(series)f(\(3.55\),) 643 b(since)4511 12041 y Fn({)5319 12223 y Fp(1)5845 12041 y Fw(\()p Fu(')p Fw(\))373 b Ft(2)f Fo(D)10140 12223 y Fp(1)10665 12041 y Fw(\()p Fu(\025)p Fw(\))12314 12223 y Fs(nonr)26 b(es)15714 12041 y Fw(for)426 b(an)-34 b(y)425 b Fu(')372 b Ft(2)g Fw(\002)23189 12223 y Fp(2)23715 12041 y Fw(.)601 b(No)-34 b(w)426 b(w)-34 b(e)426 b(sho)-34 b(w)426 b(that)h Fu(\025)35620 11601 y Fp(\(2\))36877 11059 y Fq(\000)37433 12041 y Fn({)38241 12223 y Fp(1)38767 12041 y Fw(\()p Fu(')p Fw(\))40502 11059 y Fq(\001)41483 12041 y Fw(can)e(b)34 b(e)425 b(analytically)4511 13485 y(extended)518 b(from)f(\002)13697 13667 y Fp(2)14740 13485 y Fw(to)g Fu(\010)17142 13667 y Fp(1)18013 13485 y Ft(n)344 b Fo(O)p Fw(.)877 b(It)518 b(is)e(enough)i(to)g(c)-34 b(hec)g(k)517 b(that)h(the)f(series)g(\(3.55\))g(con)-34 b(v)g(erges)517 b(in)4511 15202 y Fu(\010)5319 15384 y Fp(1)5899 15202 y Ft(n)54 b Fo(O)p Fw(.)502 b(W)-101 b(e)296 b(need)h(an)f(estimate)h(for)f Fu(g)21648 14563 y Fp(\(2\))21605 15360 y Fs(r)22907 14220 y Fq(\000)23462 15202 y Fn({)24270 15384 y Fp(1)24796 15202 y Fw(\()p Fu(')p Fw(\))26531 14220 y Fq(\001)27087 15202 y Fw(,)318 b(whic)-34 b(h)297 b(is)f(similar)f(to)i(\(3.17\).)503 b(Indeed,)318 b(considering)4511 17040 y(form)-34 b(ula)446 b(\(3.13\))g(for)g Fu(g)15061 16401 y Fp(\(2\))15018 17198 y Fs(r)16764 17040 y Fw(and)g(taking)f(in)-34 b(to)446 b(accoun)-34 b(t)447 b(that)f Ft(k)p Fu(W)34377 17222 y Fp(2)34903 17040 y Ft(k)405 b Fu(<)g(\017)37754 16600 y Fp(4)37754 17365 y(1)38280 17040 y Fw(,)455 b(w)-34 b(e)446 b(easily)e(conclude)h(that)4511 18479 y(it)405 b(su\016ces)f(to)g(pro)-34 b(v)g(e)405 b(the)f(estimate:)17122 21152 y(sup)16917 22195 y Fs(z)37 b Fr(2)p Fs(C)18690 22318 y FD(2)19151 19758 y Fq(\015)19151 20486 y(\015)19151 21213 y(\015)19825 19807 y(\020)20866 20846 y Fw(~)20549 21152 y Fu(H)21655 20652 y Fp(\(1\))22912 20170 y Fq(\000)23500 21152 y Fu(~)-638 b Fn({)24276 21334 y Fp(1)24802 21152 y Fw(\()p Fu(')p Fw(\))26537 20170 y Fq(\001)27362 21152 y Ft(\000)270 b Fu(z)29192 19807 y Fq(\021)29915 20109 y Fr(\000)p Fp(1)31173 19758 y Fq(\015)31173 20486 y(\015)31173 21213 y(\015)32183 21152 y Fu(<)33801 20332 y Fw(1)p 33595 20873 1018 49 v 33595 21984 a Fu(\017)34087 22166 y Fp(1)34746 21152 y Fu(:)13345 b Fw(\(3.73\))4511 24379 y(F)-101 b(rom)405 b(the)f(de\014nition)h(of)16869 24072 y(~)16552 24379 y Fu(H)17658 23939 y Fp(\(1\))18916 24379 y Fw(,)e(w)-34 b(e)405 b(see:)8145 25658 y Fq(\015)8145 26385 y(\015)8145 27112 y(\015)8818 25706 y(\020)9860 26745 y Fw(~)9542 27052 y Fu(H)10648 26551 y Fp(\(1\))11906 26070 y Fq(\000)12494 27052 y Fu(~)-638 b Fn({)13270 27234 y Fp(1)13795 27052 y Fw(\()p Fu(')p Fw(\))15530 26070 y Fq(\001)16356 27052 y Ft(\000)269 b Fu(z)18185 25706 y Fq(\021)18909 26008 y Fr(\000)p Fp(1)20167 25658 y Fq(\015)20167 26385 y(\015)20167 27112 y(\015)21177 27052 y Fw(=)336 b(max)22670 27859 y Fs(p)p Fr(2)p Fs(P)24712 25658 y Fq(\015)24712 26385 y(\015)24712 27112 y(\015)25386 25706 y(\020)26110 27052 y Fu(H)27216 26551 y Fp(\(1\))28473 26070 y Fq(\000)29061 27052 y Fu(~)-638 b Fn({)29837 27234 y Fp(1)30363 27052 y Fw(\()p Fu(')p Fw(\))270 b(+)f(2)p Fu(\031)43 b(p=)-67 b(N)37043 27234 y Fp(1)37569 27052 y Fu(a)38210 26070 y Fq(\001)39034 27052 y Ft(\000)269 b Fu(z)40863 25706 y Fq(\021)41587 26008 y Fr(\000)p Fp(1)42845 25658 y Fq(\015)42845 26385 y(\015)42845 27112 y(\015)43518 27052 y Fu(:)4511 29817 y Fw(T)-101 b(o)362 b(estimate)g(the)g(righ)-34 b(t-hand)363 b(side,)370 b(w)-34 b(e)362 b(start)g(with)g(the)g(case)f Fu(p)336 b Fw(=)h(0.)524 b(W)-101 b(e)361 b(obtain)h(an)g(upp)34 b(er)362 b(b)34 b(ound)4511 31488 y(on)398 b Ft(k)6996 30506 y Fq(\000)7551 31488 y Fu(H)8657 31048 y Fp(\(1\))9915 30506 y Fq(\000)10503 31488 y Fu(~)-638 b Fn({)11279 31670 y Fp(1)11804 31488 y Fw(\()p Fu(')p Fw(\))13539 30506 y Fq(\001)14365 31488 y Ft(\000)269 b Fu(z)16194 30506 y Fq(\001)16750 30734 y Fr(\000)p Fp(1)18007 30458 y Fq(\015)18007 31185 y(\015)19078 31488 y Fw(using)397 b(results)g(of)g(Section)g(2.)536 b(Indeed,)399 b Fo(K)38445 31670 y Fp(1)39003 31488 y Fu(~)-638 b Fn({)39779 31670 y Fp(1)40305 31488 y Fw(\()p Fu(')p Fw(\))397 b(b)34 b(elongs)397 b(to)g Fu(\037)49012 31048 y Fr(\003)49012 31813 y Fp(1)49539 31488 y Fw(\()p Fu(\025)p Fw(\),)4511 33078 y(whic)-34 b(h)444 b(is)e(in)g(the)g(complex)g(\()p Fu(k)18778 32638 y Fr(\000)p Fp(1)p Fr(\000)p Fp(4)p Fs(s)21617 32761 y FD(1)22080 32638 y Fr(\000)p Fp(2)p Fs(\016)23787 33078 y Fw(\)-neigh)-34 b(b)34 b(orho)g(o)g(d)443 b(of)g(the)f(\014rst)h(non-resonance)g(set)f Fu(\037)49012 33260 y Fp(1)49539 33078 y Fw(\()p Fu(\025)p Fw(\).)4511 34517 y(Hence,)410 b(w)-34 b(e)409 b(can)g(apply)g(Lemma)f(A.1)h(from)g (App)34 b(endix)409 b(A.4)g(with)g Fu(t)345 b Fw(=)f Fo(K)38926 34699 y Fp(1)39485 34517 y Fu(~)-639 b Fn({)40260 34699 y Fp(1)40786 34517 y Fw(\()p Fu(')p Fw(\).)553 b(By)409 b(this)g(lemma,)4511 35199 y Fq(\000)5067 36181 y Fu(H)6173 35741 y Fp(\(1\))7431 35199 y Fq(\000)8018 36181 y Fu(~)-638 b Fn({)8794 36363 y Fp(1)9320 36181 y Fw(\()p Fu(')p Fw(\))11055 35199 y Fq(\001)11950 36181 y Ft(\000)338 b Fu(z)13848 35199 y Fq(\001)14403 35501 y Fr(\000)p Fp(1)16168 36181 y Fw(has)508 b(a)g(single)f(p)34 b(ole)507 b Fu(z)26284 36363 y Fp(0)27316 36181 y Fw(inside)h(the)g (con)-34 b(tour)508 b Fu(C)38435 36363 y Fp(1)38961 36181 y Fw(.)849 b(It)507 b(is)g(easy)g(to)h(see)f(that)4511 37820 y Fu(z)5075 38002 y Fp(0)5938 37820 y Fw(=)336 b Fu(\025)p Fw(,)404 b(since)g Fu(\025)12342 37380 y Fp(\(1\))13599 36838 y Fq(\000)14187 37820 y Fu(~)-638 b Fn({)14963 38002 y Fp(1)15489 37820 y Fw(\()p Fu(')p Fw(\))17224 36838 y Fq(\001)18117 37820 y Fw(=)336 b Fu(\025)p Fw(.)539 b(Again,)404 b(b)-34 b(y)404 b(Lemma)g(A.1,)15873 39288 y Fq(\015)15873 40015 y(\015)15873 40742 y(\015)16547 40682 y Fw(\()p Fu(z)323 b Ft(\000)269 b Fu(k)19786 40181 y Fp(2)p Fs(l)20603 40682 y Fw(\))21276 39336 y Fq(\020)22000 40682 y Fu(H)23106 40181 y Fp(\(1\))24364 39700 y Fq(\000)24952 40682 y Fu(~)-638 b Fn({)25728 40864 y Fp(1)26254 40682 y Fw(\()p Fu(')p Fw(\))27989 39700 y Fq(\001)28814 40682 y Ft(\000)269 b Fu(z)30643 39336 y Fq(\021)31367 39638 y Fr(\000)p Fp(1)32625 39288 y Fq(\015)32625 40015 y(\015)32625 40742 y(\015)33635 40682 y Fu(<)336 b Fw(16)12302 b(\(3.74\))4511 43348 y(for)466 b(all)f Fu(z)518 b Fw(inside)465 b Fu(C)13623 43530 y Fp(1)14149 43348 y Fw(.)722 b(Considering)466 b Fu(z)492 b Ft(2)439 b Fu(C)25115 43530 y Fp(2)25641 43348 y Fw(,)480 b(that)467 b(is,)480 b Ft(j)p Fu(z)362 b Ft(\000)310 b Fu(k)33963 42908 y Fp(2)p Fs(l)34781 43348 y Ft(j)438 b Fw(=)h Fu(\017)37430 43530 y Fp(1)37955 43348 y Fu(=)p Fw(2,)481 b(w)-34 b(e)465 b(easily)g(obtain)h(from)4511 44787 y(\(3.74\))405 b(that)15316 45630 y Fq(\015)15316 46357 y(\015)15316 47084 y(\015)15990 45678 y(\020)16714 47024 y Fu(H)17820 46523 y Fp(\(1\))19077 46042 y Fq(\000)19665 47024 y Fu(~)-638 b Fn({)20441 47206 y Fp(1)20967 47024 y Fw(\()p Fu(')p Fw(\))22702 46042 y Fq(\001)23527 47024 y Ft(\000)270 b Fu(z)25357 45678 y Fq(\021)26080 45980 y Fr(\000)p Fp(1)27338 45630 y Fq(\015)27338 46357 y(\015)27338 47084 y(\015)28348 47024 y Fu(<)29761 46204 y Fw(32)p 29761 46745 1213 49 v 29858 47855 a Fu(\017)30350 48037 y Fp(1)31105 47024 y Fu(;)1414 b(z)390 b Ft(2)337 b Fu(C)35821 47206 y Fp(2)36347 47024 y Fu(:)11744 b Fw(\(3.75\))4511 49794 y(Next,)390 b(w)-34 b(e)386 b(consider)g(the)h(case)e Fu(p)337 b Ft(2)g Fu(P)401 b Ft(n)234 b(f)p Fw(0)p Ft(g)p Fw(.)532 b(W)-101 b(e)386 b(use)g(Lemma)g(3.31)g(with)39286 49474 y Fu(~)39423 49794 y(b)336 b Fw(=)h(2)p Fu(\031)43 b(p=)-67 b(N)45022 49976 y Fp(1)45548 49794 y Fu(a)p Fw(,)389 b Fu(p)336 b Ft(6)p Fw(=)h(0.)532 b(It)4511 51263 y(is)435 b(p)34 b(ossible)435 b(to)h(replace)e Fu(k)16675 50823 y Fp(2)p Fs(l)17927 51263 y Fw(in)h(\(3.52\))h(b)-34 b(y)435 b(an)-34 b(y)436 b Fu(z)441 b Ft(2)389 b Fu(C)30008 51445 y Fp(2)30534 51263 y Fw(,)442 b(since)435 b(the)h(di\013erence)e (b)34 b(et)-34 b(w)g(een)436 b Fu(z)488 b Fw(and)436 b Fu(k)50708 50823 y Fp(2)p Fs(l)4511 52731 y Fw(is)404 b(small:)538 b Ft(j)p Fu(z)322 b Ft(\000)270 b Fu(k)12478 52292 y Fp(2)p Fs(l)13295 52731 y Ft(j)336 b Fw(=)h Fu(\017)15740 52913 y Fp(1)16266 52731 y Fu(=)p Fw(2,)403 b Fu(\017)18710 52913 y Fp(1)19573 52731 y Fw(=)336 b Fu(o)21642 51750 y Fq(\000)22198 52731 y Fu(b)22718 52913 y Fp(0)23244 52731 y Fu(k)23913 52292 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(9)p Fr(\000)p Fp(12)p Fs(s)27984 52415 y FD(1)28446 52292 y Fr(\000)p Fp(6)p Fs(\016)30153 51750 y Fq(\001)30709 52731 y Fw(.)538 b(Th)-34 b(us,)15090 54199 y Fq(\015)15090 54926 y(\015)15090 55654 y(\015)15763 54248 y(\020)16487 55593 y Fu(H)17593 55093 y Fp(\(1\))18851 54611 y Fq(\000)19439 55593 y Fu(~)-638 b Fn({)20215 55775 y Fp(1)20741 55593 y Fw(\()p Fu(')p Fw(\))270 b(+)f(2)p Fu(\031)43 b(p=)-67 b(N)27421 55775 y Fp(1)27947 55593 y Fu(a)28588 54611 y Fq(\001)29412 55593 y Ft(\000)269 b Fu(z)31241 54248 y Fq(\021)31965 54550 y Fr(\000)p 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Fu(')p Fw(\),)450 b(namely)-101 b(,)450 b(they)441 b(hold)h(at)f(all)f(p)34 b(oin)-34 b(ts)4511 62643 y Fn({)24 b Fu(~)-581 b(\027)77 b Fw(:)507 b Ft(j)p Fn({)192 b Ft(\000)143 b Fn({)10119 62825 y Fp(1)10644 62643 y Fw(\()p Fu(')p Fw(\))p Ft(j)337 b Fu(<)g(\017)14825 62825 y Fp(1)15351 62643 y Fu(k)16020 62203 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fs(\016)19953 62643 y Fw(.)517 b(Hence,)353 b(the)341 b(p)34 b(erturbation)342 b(series)e(for)h(the)g(eigen)-34 b(v)-67 b(alue)340 b(con)-34 b(v)g(erges)4511 64082 y(for)405 b(suc)-34 b(h)405 b Fn({)453 b Fw(to)34 b(o,)404 b(the)g(follo)-34 b(wing)405 b(estimates)f(b)34 b(eing)404 b(v)-67 b(alid:)17274 65141 y Fq(\014)17274 65868 y(\014)17274 66595 y(\014)17678 66535 y Fu(\025)18385 66034 y Fp(\(2\))19643 66535 y Fw(\()p Fu(\013)t(;)202 b Fn({)24 b Fu(~)-581 b(\027)78 b Fw(\))270 b Ft(\000)f Fu(\025)25626 66034 y Fp(\(1\))26883 66535 y Fw(\()p Fn({)24 b Fu(~)-581 b(\027)78 b Fw(\))29359 65141 y Fq(\014)29359 65868 y(\014)29359 66595 y(\014)30100 66535 y Fu(<)336 b Fw(12)p Fu(\013)t(\017)33862 66034 y Fp(4)33862 66834 y(1)34389 66535 y Fu(;)13702 b Fw(\(3.77\))16614 69541 y Ft(r)p Fu(\025)18331 69040 y Fp(\(2\))19589 69541 y Fw(\()p Fu(\013)t(;)202 b Fn({)24 b Fu(~)-581 b(\027)78 b Fw(\))337 b(=)f(2)p Fu(l)24 b(k)26660 69040 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(2)28618 69221 y Fu(~)28680 69541 y(k)307 b Fw(+)269 b Fu(o)p Fw(\()p Fu(k)32558 69040 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1)34578 69541 y Fw(\))p Fu(:)13042 b Fw(\(3.78\))p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 35 35 TeXDict begin 35 34 bop 0 TeXcolorgray 0 TeXcolorgray 2848 3345 a Fp(SPECTRAL)406 b(PR)-26 b(OPER)-78 b(TIES)406 b(OF)h(A)h(POL)-78 b(YHARMONIC)405 b(OPERA)-78 b(TOR)406 b(WITH)h(LIMIT-PERIODIC)e(POTENTIAL)1430 b(35)p 0 TeXcolorgray 4511 6001 a Fw(Next,)403 b(w)-34 b(e)403 b(w)-34 b(an)g(t)404 b(to)f(sho)-34 b(w)404 b(that)f(the)g(equation)g Fu(\025)27401 5562 y Fp(\(2\))28659 6001 y Fw(\()p Fu(\013)t(;)202 b Fn({)24 b Fu(~)-581 b(\027)78 b Fw(\))337 b(=)f Fu(\025)403 b Fw(has)g(a)f(solution)h Fn({)43739 6183 y Fp(2)44265 6001 y Fw(\()p Fu(')p Fw(\))46000 5532 y Fp(9)46874 6001 y Fw(for)f(ev)-34 b(ery)4511 7440 y Fu(')337 b Ft(2)g Fu(\010)7594 7622 y Fp(1)8389 7440 y Ft(n)270 b Fo(O)p Fu(:)403 b Fw(It)h(easily)f(follo)-34 b(ws)405 b(from)f(\(2.34\))h({)f (\(2.36\))h(that)11243 9730 y Fu(@)67 b(\025)12661 9290 y Fp(\(1\))13919 9730 y Fw(\()p Fn({)24 b Fu(~)-581 b(\027)77 b Fw(\))p 11243 10271 5152 49 v 13035 11381 a Fu(@)67 b Fn({)16864 10550 y Fw(=)336 b(2)p Fu(l)24 b(k)19804 10049 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1)21824 10550 y Fw(\(1)269 b(+)h Fu(o)p Fw(\(1\)\))p Fu(;)28874 9730 y(@)29585 9290 y Fp(2)30111 9730 y Fu(\025)30818 9290 y Fp(\(1\))32075 9730 y Fw(\()p Fn({)24 b Fu(~)-581 b(\027)78 b Fw(\))p 28874 10271 5677 49 v 30666 11381 a Fu(@)67 b Fn({)32234 11031 y Fp(2)35020 10550 y Fw(=)337 b Fu(O)34 b Fw(\()p Fu(k)38399 10049 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(2)40418 10550 y Fw(\))7539 b(\(3.79\))4511 13318 y(for)345 b(an)-34 b(y)346 b Fu(')336 b Ft(2)h Fu(\010)11620 13500 y Fp(1)12491 13318 y Fw(and)345 b Fn({)386 b Fw(:)336 b Ft(j)p Fn({)200 b Ft(\000)151 b Fu(k)38 b Ft(j)336 b Fu(<)g(k)22384 12878 y Fr(\000)p Fp(1)p Fr(\000)p Fp(4)p Fs(s)25223 13001 y FD(1)25685 12878 y Fr(\000)p Fp(2)p Fs(\016)27392 13318 y Fw(.)519 b(Using)345 b(these)g(estimates)g(w)-34 b(e)345 b(readily)f(obtain)i(that)4511 14836 y Ft(j)p Fu(\025)5555 14396 y Fp(\(1\))6813 14836 y Fw(\()p Fn({)24 b Fu(~)-581 b(\027)77 b Fw(\))286 b Ft(\000)g Fu(\025)p Ft(j)377 b Fu(>)g(k)14213 14396 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1)16233 14836 y Fu(\017)16725 14396 y Fp(4)16725 15161 y(1)17679 14836 y Fw(for)429 b(all)f Fn({)478 b Fw(on)429 b(the)g(circle)e Ft(j)p Fn({)334 b Ft(\000)286 b Fn({)33082 15018 y Fp(1)33608 14836 y Fw(\()p Fu(')p Fw(\))p Ft(j)378 b Fw(=)f Fu(\017)37870 14396 y Fp(4)37870 15161 y(1)38396 14836 y Fw(.)612 b(Considering)429 b(\(3.77\))g(and)4511 16275 y(applying)328 b(Rouc)-34 b(h)g(\023)-572 b(e's)329 b(theorem,)342 b(w)-34 b(e)328 b(obtain)h(that)g(there)e(is)h(a)f (single)g(solution)i Fn({)41076 16457 y Fp(2)41602 16275 y Fw(\()p Fu(')p Fw(\))f(of)g(the)g(equation)4511 17793 y Fu(\025)5218 17353 y Fp(\(2\))6476 17793 y Fw(\()p Fu(\013)t(;)202 b Fn({)24 b Fu(~)-581 b(\027)78 b Fw(\))463 b(=)g Fu(\025)480 b Fw(suc)-34 b(h)481 b(that)g Ft(j)p Fn({)19983 17975 y Fp(2)20508 17793 y Fw(\()p Fu(')p Fw(\))321 b Ft(\000)f Fn({)24635 17975 y Fp(1)25160 17793 y Fw(\()p Fu(')p Fw(\))p Ft(j)464 b(\024)f Fu(\017)29594 17353 y Fp(4)29594 18118 y(1)30119 17793 y Fw(.)766 b(Applying)480 b(\(3.78\))h(and)g(Implicit)e(function)4511 19232 y(theorem,)514 b(w)-34 b(e)493 b(obtain)g(that)g Fn({)18929 19414 y Fp(2)19455 19232 y Fw(\()p Fu(')p Fw(\))f(is)g(a)g(holomorphic)h (function)g(of)g Fu(')e Fw(in)i Fu(\010)41014 19414 y Fp(1)41867 19232 y Ft(n)328 b Fo(O)492 b Fw(and)h(estimates)4511 20671 y(\(3.63\))405 b(hold.)5840 22110 y(Let)439 b(us)g(mak)-34 b(e)439 b(here)g(a)g(remark)f(for)h(the)g(future.)645 b(Con)-34 b(v)g(ergence)439 b(of)h(the)f(series)f(for)i(the)f(resolv) -34 b(en)g(t)4511 22799 y Fq(\000)5067 23781 y Fu(H)6173 23341 y Fp(\(2\))7633 23781 y Fw(\()32 b Fu(~)-638 b Fn({)8912 23963 y Fp(1)9438 23781 y Fw(\()p Fu(')p Fw(\)\))270 b Ft(\000)g Fu(z)13744 22799 y Fq(\001)14299 23027 y Fr(\000)p Fp(1)15557 23781 y Fw(,)366 b Fu(z)390 b Ft(2)337 b Fu(C)19225 23963 y Fp(2)19751 23781 y Fw(,)366 b(follo)-34 b(wing)358 b(from)g(\(3.73\),)367 b(means)358 b(that)h(the)f(resolv)-34 b(en)g(t)357 b(has)h(a)g(single)4511 25420 y(p)34 b(ole)466 b Fu(z)493 b Fw(=)439 b Fu(\025)10312 24980 y Fp(\(2\))11570 25420 y Fw(\()32 b Fu(~)-638 b Fn({)12849 25602 y Fp(1)13375 25420 y Fw(\()p Fu(')p Fw(\))-25 b Fu(~)-581 b(\027)78 b Fw(\))466 b(inside)g Fu(C)21093 25602 y Fp(2)21619 25420 y Fw(.)724 b(Similar)465 b(result)h(holds)g(when)h(w)-34 b(e)466 b(replace)497 b Fu(~)-638 b Fn({)43646 25602 y Fp(1)44172 25420 y Fw(\()p Fu(')p Fw(\))467 b(b)-34 b(y)498 b Fu(~)-638 b Fn({)48927 25602 y Fp(2)49453 25420 y Fw(\()p Fu(')p Fw(\),)4511 26937 y(since)384 b Fu(~)-638 b Fn({)8237 27119 y Fp(1)8763 26937 y Fw(\()p Fu(')p Fw(\))352 b(and)385 b Fu(~)-638 b Fn({)13963 27119 y Fp(2)14489 26937 y Fw(\()p Fu(')p Fw(\))352 b(are)g(close:)21895 25907 y Fq(\014)21895 26634 y(\014)22332 26937 y Fu(~)-639 b Fn({)23107 27119 y Fp(2)23633 26937 y Fw(\()p Fu(')p Fw(\))165 b Ft(\000)197 b Fu(~)-638 b Fn({)27449 27119 y Fp(1)27975 26937 y Fw(\()p Fu(')p Fw(\))p Ft(j)338 b Fw(=)e Fu(o)p Fw(\()p Fu(\017)33215 27119 y Fp(1)33741 26937 y Fw(\).)521 b(Considering)353 b(that)g Fu(\025)44976 26497 y Fp(\(2\))46233 26937 y Fw(\()32 b Fu(~)-638 b Fn({)47512 27119 y Fp(2)48039 26937 y Fw(\()p Fu(')p Fw(\)\))337 b(=)4511 28796 y Fu(\025)p Fw(,)388 b(w)-34 b(e)383 b(obtain)i(that)f(\()p Fu(z)282 b Ft(\000)227 b Fu(\025)p Fw(\))17929 27815 y Fq(\000)18485 28796 y Fu(H)19591 28357 y Fp(\(2\))21051 28796 y Fw(\()33 b Fu(~)-639 b Fn({)22330 28978 y Fp(2)22856 28796 y Fw(\()p Fu(')p Fw(\)\))271 b Ft(\000)e Fu(z)27162 27815 y Fq(\001)27717 28043 y Fr(\000)p Fp(1)29358 28796 y Fw(is)384 b(holomorphic)f(inside)h Fu(C)41762 28978 y Fp(2)42671 28796 y Fw(and)g(the)g(estimate)4511 30241 y(similar)404 b(to)g(\(3.74\))h(holds:)16095 31646 y Fq(\015)16095 32373 y(\015)16095 33100 y(\015)16768 33040 y Fw(\()p Fu(z)323 b Ft(\000)269 b Fu(\025)p Fw(\))20718 31694 y Fq(\020)21442 33040 y Fu(H)22548 32539 y Fp(\(2\))23806 32058 y Fq(\000)24394 33040 y Fu(~)-638 b Fn({)25170 33222 y Fp(2)25696 33040 y Fw(\()p Fu(')p Fw(\))27431 32058 y Fq(\001)28256 33040 y Ft(\000)269 b Fu(z)30085 31694 y Fq(\021)30809 31996 y Fr(\000)p Fp(1)32067 31646 y Fq(\015)32067 32373 y(\015)32067 33100 y(\015)33077 33040 y Fu(<)336 b Fw(32)p Fu(:)12523 b Fw(\(3.80\))p 0 TeXcolorgray 2409 35619 a(\(4\))p 0 TeXcolorgray 554 w(First,)404 b(w)-34 b(e)405 b(sho)-34 b(w)405 b(that)12943 38198 y Fu(L)13768 37216 y Fq(\000)14324 38198 y Fo(D)15331 38380 y Fp(2)15856 38198 y Fw(\()p Fu(\025)p 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5020 y Fq(\000)22336 6001 y Fw(2)p Fu(\013)t(\017)24213 5562 y Fp(4)24213 6326 y(1)24740 6001 y Fu(k)25409 5562 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)28161 5020 y Fq(\001)29138 6001 y Fw(neigh)-34 b(b)34 b(orho)g(o)g(d)423 b(of)f Fo(D)39174 6183 y Fp(1)39699 6001 y Fw(\()p Fu(\025)p Fw(\))41348 6183 y Fs(nonr)26 b(es)44324 6001 y Fw(.)592 b(Considering)475 7445 y(that)553 b Fu(\037)4007 7006 y Fr(\003)4007 7770 y Fp(2)4533 7445 y Fw(\()p Fu(\025)p Fw(\))584 b(=)f Fo(K)9289 7627 y Fp(2)9815 7445 y Fo(D)10822 7627 y Fp(2)11347 7445 y Fw(\()p Fu(\025)p Fw(\))553 b(and)g Fu(\037)16812 7627 y Fp(2)17338 7445 y Fw(\()p Fu(\025)p Fw(\))584 b(=)f Fo(K)22094 7627 y Fp(2)22620 7445 y Fo(D)23627 7627 y Fp(1)24152 7445 y Fw(\()p Fu(\025)p Fw(\))25801 7627 y Fs(nonr)26 b(es)29328 7445 y Fw(\(see)552 b(\(3.54\)\),)590 b(w)-34 b(e)552 b(immediately)f(obtain)i(that)475 8936 y Fu(\037)1233 8496 y Fr(\003)1233 9260 y Fp(2)1759 8936 y Fw(\()p Fu(\025)p Fw(\))467 b(is)f(in)g(the)h(\(2)p Fu(\013)t(\017)11130 8496 y Fp(4)11130 9260 y(1)11657 8936 y Fu(k)12326 8496 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)15078 8936 y Fw(\)-neigh)-34 b(b)34 b(orho)g(o)g(d)467 b(of)g Fu(\037)25829 9118 y Fp(2)26355 8936 y Fw(\()p Fu(\025)p Fw(\).)726 b(The)466 b(size)g(of)g(this)h(neigh)-34 b(b)34 b(orho)g(o)g(d)467 b(is)f(less)g(than)475 10426 y Fu(\017)967 10608 y Fp(1)1493 10426 y Fu(k)2162 9986 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fs(\016)6094 10426 y Fw(,)542 b(hence)514 b(Theorem)g(3.1)g(holds)h(in)g(it,)541 b(i.e.,)f(for)515 b(an)-34 b(y)515 b Fu(\034)657 b Ft(2)520 b Fu(\037)34303 9986 y Fr(\003)34303 10751 y Fp(2)34830 10426 y Fw(\()p Fu(\025)p Fw(\))515 b(there)f(is)g(a)g(single)g(eigen) -34 b(v)-67 b(alue)475 12165 y(of)498 b Fu(H)3055 11526 y Fp(\(2\))2957 12322 y Fs(\013)4312 12165 y Fw(\()p Fu(\034)137 b Fw(\))499 b(in)e(the)h(in)-34 b(terv)-67 b(al)497 b Fu(")15147 12347 y Fp(2)15674 12165 y Fw(\()p Fu(k)38 b(;)202 b(\016)46 b Fw(\).)819 b(Since)497 b Fu(\037)23581 11725 y Fr(\003)23581 12489 y Fp(2)24107 12165 y Fw(\()p Fu(\025)p Fw(\))d Ft(\032)e Fu(S)28428 12347 y Fp(2)28954 12165 y Fw(\()p Fu(\025)p Fw(\),)522 b(this)498 b(eigen)-34 b(v)-67 b(alue)496 b(is)i(equal)f(to)h Fu(\025)p Fw(.)819 b(By)497 b(the)475 13611 y(theorem,)537 b(the)511 b(eigen)-34 b(v)-67 b(alue)510 b(is)g(giv)-34 b(en)511 b(b)-34 b(y)511 b(the)g(series)e(\(3.15\),)538 b(where)511 b Fu(p)j Ft(2)g Fu(P)34 b(;)202 b(j)583 b Ft(2)515 b Fn(Z)40284 13171 y Fp(2)41320 13611 y Fw(are)510 b(describ)34 b(ed)510 b(as)h(in)475 15050 y(Geometric)403 b(Lemma)h(3.5,)f(part)i(2.)p 50750 15216 665 831 v 0 TeXcolorgray 475 17281 a FF(Lemma)463 b(3.44.)p 0 TeXcolorgray 554 w FC(F)-93 b(ormula)433 b(\(3.83\))e(establishes)f(one-to-one)h(c) -62 b(orr)g(esp)g(ondenc)g(e)430 b(b)-62 b(etwe)g(en)433 b Fu(\037)42802 16842 y Fr(\003)42802 17606 y Fp(2)43328 17281 y Fw(\()p Fu(\025)p Fw(\))i FC(and)e Fo(D)48773 17463 y Fp(2)49298 17281 y Fw(\()p Fu(\025)p Fw(\))p FC(.)p 0 TeXcolorgray 475 19513 a FF(Remark)309 b(3.45.)p 0 TeXcolorgray 443 w Fw(F)-101 b(rom)269 b(geometric)g(p)34 b(oin)-34 b(t)270 b(of)g(view)f(this)h(means)g(that)g Fu(\037)33357 19073 y Fr(\003)33357 19838 y Fp(2)33884 19513 y Fw(\()p Fu(\025)p Fw(\))g(do)34 b(es)269 b(not)h(ha)-34 b(v)g(e)270 b(self-in)-34 b(tersections.)p 0 TeXcolorgray 475 21745 a FC(Pr)-62 b(o)g(of.)p 0 TeXcolorgray 552 w Fw(Supp)34 b(ose)482 b(there)g(is)e(a)i(pair)513 b Fu(~)-638 b Fn({)49 b Fu(;)234 b(~)-638 b Fn({)19438 21305 y Fr(\003)20428 21745 y Ft(2)465 b Fo(D)22708 21927 y Fp(2)23234 21745 y Fw(\()p Fu(\025)p Fw(\))482 b(suc)-34 b(h)482 b(that)g Fo(K)31876 21927 y Fp(2)32435 21745 y Fu(~)-638 b Fn({)514 b Fw(=)465 b Fo(K)36130 21927 y Fp(2)36688 21745 y Fu(~)-638 b Fn({)37513 21305 y Fr(\003)38504 21745 y Fw(=)465 b Fu(\034)137 b Fw(,)500 b Fu(\034)602 b Ft(2)466 b Fu(\037)44580 21305 y Fr(\003)44580 22070 y Fp(2)45106 21745 y Fw(\()p Fu(\025)p Fw(\).)770 b(By)481 b(the)475 23484 y(de\014nition)419 b(\(3.58\))g(of)f Fo(D)11793 23666 y Fp(2)12319 23484 y Fw(\()p Fu(\025)p Fw(\),)k(w)-34 b(e)419 b(ha)-34 b(v)g(e)418 b Fu(\025)20041 23044 y Fp(\(2\))21299 23484 y Fw(\()p Fu(\013)t(;)235 b(~)-639 b Fn({)50 b Fw(\))361 b(=)e Fu(\025)26787 23044 y Fp(\(2\))28045 23484 y Fw(\()p Fu(\013)t(;)235 b(~)-639 b Fn({)30691 23044 y Fr(\003)31218 23484 y Fw(\))360 b(=)g Fu(\025)p Fw(,)422 b(i.e.,)e(the)f(eigen)-34 b(v)-67 b(alue)417 b Fu(\025)h Fw(of)h Fu(H)48658 22845 y Fp(\(2\))48560 23641 y Fs(\013)49915 23484 y Fw(\()p Fu(\034)137 b Fw(\))475 24923 y(is)404 b(not)g(simple.)539 b(This)404 b(con)-34 b(tradicts)405 b(to)g(the)f(previous)g(lemma.)p 50750 25089 V 475 27338 a(3.5.)605 b FF(Preparation)465 b(for)g(the)g(Next)f (Appro)-39 b(ximation.)475 29846 y Fw(3.5.1.)605 b FC(Contr)-62 b(acte)g(d)623 b(set)h Fo(O)13188 30028 y Fs(s)13679 29846 y Fw(\()14014 29526 y Fu(~)14150 29846 y(b)p Fw(\))p FC(.)606 b Fw(Let)612 b(us)h(recall)e(that)i Fo(O)p Fw(\()27734 29526 y Fu(~)27870 29846 y(b)q Fw(\))f(is)g(the)h(union)g(of)g(disks)f (of)h(the)g(radius)f Fu(r)717 b Fw(=)475 31314 y Fu(k)1144 30874 y Fr(\000)p Fp(4)p Fr(\000)p Fp(6)p Fs(s)3983 30997 y FD(1)4445 30874 y Fr(\000)p Fp(3)p Fs(\016)6545 31314 y Fw(cen)-34 b(tered)393 b(at)g(the)h(zeros)e(of)h(the)h(unp)34 b(erturb)g(ed)394 b(determinan)-34 b(t)394 b(det)36883 30332 y Fq(\000)37439 31314 y Fu(I)342 b Fw(+)247 b Fu(A)40413 31496 y Fp(0)40939 31314 y Fw(\()-10 b Fu(~)-596 b(y)42004 31496 y Fp(0)42530 31314 y Fw(\()p Fu(')p Fw(\)\))44736 30332 y Fq(\001)45293 31314 y Fw(,)395 b(see)d(De\014ni-)475 32880 y(tion)416 b(3.24.)575 b(W)-101 b(e)416 b(pro)-34 b(v)g(ed)417 b(that)g(the)g(\\p)34 b(erturb)g(ed")417 b(determinan)-34 b(t)417 b(det)32260 31898 y Fq(\000)32815 32880 y Fu(I)373 b Fw(+)277 b Fu(A)35850 33062 y Fp(1)36376 32880 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))39220 31898 y Fq(\001)40192 32880 y Fw(has)417 b(the)g(same)f(n)-34 b(um)g(b)34 b(er)475 34600 y(of)488 b(zeros)f(inside)h(eac)-34 b(h)488 b(\000\()12492 34281 y Fu(~)12628 34600 y(b)p 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38337 y Fq(\001)15236 39319 y Fw(in)329 b(the)g(set)g Fo(O)p Fw(\()21597 38999 y Fu(~)21733 39319 y(b)p Fw(\).)514 b(Ob)-34 b(viously)329 b(eac)-34 b(h)328 b(of)i(these)e(disks)h(is)g (in)f Fo(O)p Fw(\()42792 38999 y Fu(~)42928 39319 y(b)q Fw(\),)343 b(since)329 b Fu(r)48076 38879 y Fr(0)48723 39319 y Fu(<<)336 b(r)475 40758 y Fw(and)445 b(the)f(distance)h(b)34 b(et)-34 b(w)g(een)445 b(zeros)f(of)h(the)f(p)34 b(erturb)g(ed)445 b(and)g(unp)34 b(erturb)g(ed)445 b(determinan)-34 b(ts)446 b(is)e(smaller)f(than)475 42290 y Fu(r)34 b(=)p Fw(2)363 b(\(Lemma)g(3.35,)371 b(P)-34 b(art)363 b(2\).)526 b(Next,)371 b(w)-34 b(e)363 b(tak)-34 b(e)364 b(the)f(union)i(of)e(these)h(smaller) e(disks)h(and)i(denote)e(it)h(b)-34 b(y)363 b Fo(O)49235 42472 y Fs(s)49726 42290 y Fw(\()50061 41971 y Fu(~)50197 42290 y(b)p Fw(\),)475 43890 y(here)518 b(the)h(index)g Fu(s)g Fw(stands)h(for)f(\\small".)882 b(Clearly)-101 b(,)546 b Fo(O)26406 44072 y Fs(s)26896 43890 y Fw(\()27231 43570 y Fu(~)27367 43890 y(b)q Fw(\))528 b Ft(\032)g Fo(O)p Fw(\()31551 43570 y Fu(~)31687 43890 y(b)p Fw(\).)883 b(W)-101 b(e)518 b(denote)h(b)-34 b(y)519 b(\000)42676 44072 y Fs(s)43166 43890 y Fw(\()43501 43570 y Fu(~)43637 43890 y(b)q Fw(\))g(a)g(connected)475 45490 y(comp)34 b(onen)-34 b(t)390 b(of)f(these)g(new)h(disks)f(whic)-34 b(h)390 b(in)-34 b(tersects)389 b(with)h Fu(\010)28943 45672 y Fp(1)29858 45490 y Fw(and)f(b)-34 b(y)390 b Fu(\015)34496 45672 y Fs(s)34986 45490 y Fw(\()35321 45170 y Fu(~)35457 45490 y(b)p Fw(\))g(its)f(b)34 b(oundary)-101 b(,)392 b Fu(\015)44822 45672 y Fs(s)45649 45490 y Fw(=)336 b Fu(@)67 b Fw(\000)48397 45672 y Fs(s)48887 45490 y Fw(.)534 b(Let)475 47089 y(us)404 b(remind)g(that)i(in)e(all)f(our)i(previous)f (considerations)26608 46769 y Fu(~)26744 47089 y(b)g Fw(satis\014es)h(the)g(conditons)g Fu(b)40202 47271 y Fp(0)41065 47089 y Fu(>)336 b(k)43013 46649 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+9+12)p Fs(s)47816 46772 y FD(1)48278 46649 y Fp(+7)p Fs(\016)49985 47089 y Fw(.)50322 46619 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57763 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+11+14)p Fs(s)33790 57886 y FD(1)34252 57763 y Fp(+6)p Fs(\016)35960 58264 y Fw(\(2)p Fu(r)34 b(J)49 b(=r)39526 57763 y Fr(0)39836 58264 y Fw(\))40307 57763 y Fs(J)40960 58264 y Fu(:)7131 b Fw(\(3.85\))p 0 TeXcolorgray 475 61065 a FF(Corollary)483 b(3.47.)p 0 TeXcolorgray 554 w FC(If)434 b Fu(b)11907 61247 y Fp(0)12769 61065 y Fu(>)337 b(k)14718 60626 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+9+12)p Fs(s)19521 60749 y FD(1)19983 60626 y Fp(+7)p Fs(\016)21690 61065 y FC(,)18032 62264 y Fq(\015)18032 62991 y(\015)18032 63718 y(\015)18705 62312 y(\020)19429 63658 y Fu(H)20535 63157 y Fp(\(1\))21793 63658 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\)\))270 b Ft(\000)g Fu(k)27260 63157 y Fp(2)p Fs(l)28077 62312 y Fq(\021)28801 62614 y Fr(\000)p Fp(1)30058 62264 y Fq(\015)30058 62991 y(\015)30058 63718 y(\015)31069 63658 y Fu(<)32687 62838 y Fw(1)p 32481 63379 1018 49 v 32481 64489 a Fu(\017)32973 64671 y Fp(1)33631 63658 y Fu(;)14460 b Fw(\(3.86\))p 0 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Fq(\015)28132 6058 y(\015)28132 6786 y(\015)28806 7568 y Fp(1)29668 6725 y Fu(<)31081 5905 y(c)31606 6087 y Fp(0)32131 5905 y Fu(k)32800 5465 y Fp(2+2)p Fs(s)34907 5588 y FD(1)p 31081 6446 4344 49 v 32744 7556 a Fu(\017)33236 7738 y Fp(1)35557 6725 y Fu(:)12534 b Fw(\(3.87\))p 0 TeXcolorgray 475 9802 a FC(Cor)-62 b(ol)62 b(lary)432 b(3.47.)p 0 TeXcolorgray 551 w Fw(W)-101 b(e)386 b(use)h(the)f (condition)i(on)f Fu(\021)43 b Fw(:)530 b Fu(\021)380 b(>)336 b Fw(2)234 b(+)g(64)p Fu(=)p Fw(\(2)p Fu(l)258 b Ft(\000)234 b Fw(11\))337 b(=)g(2)234 b(+)g(2)p Fu(=s)39784 9984 y Fp(1)40696 9802 y Fw(and)387 b(the)g(form)-34 b(ulae)387 b(for)475 11484 y Fu(\017)967 11666 y Fp(1)1829 11484 y Fw(=)337 b Fu(e)3673 11006 y Fr(\000)4538 10644 y FD(1)p 4538 10805 406 40 v 4538 11353 a(4)5076 11006 y Fs(k)5590 10693 y Fy(\021)24 b(s)6430 10892 y FD(1)7002 11484 y Fw(,)392 b Fu(r)34 b(=r)9499 11044 y Fr(0)10146 11484 y Fw(=)336 b Fu(k)12094 11044 y Fp(2+4)p Fs(s)14201 11167 y FD(2)14663 11044 y Fp(+)p 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27016 y Fw(,)475 29822 y Fu(J)1264 29382 y Fr(\003)2126 29822 y Ft(\024)336 b Fu(J)117 b Fw(.)539 b(Ob)-34 b(viously)-101 b(,)403 b(\()p Fu(H)12579 29382 y Fp(\(1\))13837 29822 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\)\))271 b Ft(\000)e Fu(k)19304 29382 y Fp(2)p Fs(l)20121 29822 y Fw(\))20592 29382 y Fr(\000)p Fp(1)22052 28913 y Fq(Q)23197 29264 y Fs(J)23794 28951 y Fx(\003)23197 30180 y Fs(j)51 b Fp(=1)24886 29822 y Fw(\()p Fu(')270 b Ft(\000)f Fu(')28425 30004 y Fs(j)o(;)p Fp(0)29592 29822 y Fw(\))404 b(is)g(holomorphic)g (in)g(\000.)538 b(F)-101 b(or)404 b(an)-34 b(y)405 b Fu(')f Fw(on)h Fu(\015)67 b Fw(,)5850 32399 y Fq(\015)5850 33126 y(\015)5850 33853 y(\015)6523 33793 y Fw(\()p Fu(H)8100 33292 y Fp(\(1\))9358 33793 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\)\))271 b Ft(\000)e Fu(k)14825 33292 y Fp(2)p Fs(l)15642 33793 y Fw(\))16113 33292 y Fr(\000)p Fp(1)17852 32278 y Fs(J)18449 31965 y Fx(\003)17616 32641 y Fq(Y)17573 35247 y Fs(j)51 b Fp(=1)19207 33793 y 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b(WITH)h(LIMIT-PERIODIC)e(POTENTIAL)1430 b(39)p 0 TeXcolorgray 475 6080 a Fw(form)-34 b(ula)407 b(for)f(nonzero)g Fu(W)12444 6262 y Fp(1)12969 6080 y Fw(.)545 b(F)-101 b(or)406 b(a)g(detailed)g (pro)34 b(of)406 b(see)g(App)34 b(endix)406 b(A.6.)544 b(In)406 b(the)h(case)f(when)44248 5760 y Fu(~)44384 6080 y(b)h Fw(is)e(close)h(to)g(a)475 7519 y(v)-34 b(ertex)403 b(other)i(than)g(\(0)p Fu(;)202 b Fw(0\),)404 b(the)h(considerations)f (are)g(the)g(same)g(up)h(to)g(a)f(parallel)f(shift.)p 50750 7685 665 831 v 0 TeXcolorgray 475 10195 a FF(De\014nition)470 b(3.50.)p 0 TeXcolorgray 555 w Fw(Let)407 b(\000)13341 9755 y Fr(\006)13341 10495 y Fs(s)14128 10195 y Fw(\()14463 9875 y Fu(~)14599 10195 y(b)p Fw(\))h(b)34 b(e)407 b(op)34 b(en)407 b(disks)g(cen)-34 b(tered)407 b(at)h Fu(')30695 9685 y Fr(\006)30695 10542 y Fp(0)31824 10195 y Ft(2)33176 9888 y Fw(~)32974 10195 y Fu(\010)33782 10377 y Fp(1)34715 9725 y(11)36063 10195 y Fw(with)g(the)g(radius)f Fu(r)45148 9755 y Fr(0)45865 10195 y Fw(in)-34 b(tro)34 b(duced)475 11839 y(in)329 b(the)h(previous)f(subsection,)346 b Fu(r)15308 11399 y Fr(0)15954 11839 y Fw(=)337 b Fu(r)34 b(k)18484 11399 y Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)21323 11522 y FD(2)21784 11399 y Fr(\000)p Fs(\016)23021 11839 y Fw(;)354 b Fu(\015)24407 11399 y Fr(\006)24340 12138 y Fs(s)25194 11839 y Fw(\()25529 11519 y Fu(~)25665 11839 y(b)q Fw(\))330 b(b)34 b(e)329 b(their)g(b)34 b(oundary)330 b(circles)e(and)i Fo(O)43481 12021 y Fs(s)43972 11839 y Fw(\()44307 11519 y Fu(~)44443 11839 y(b)p Fw(\))337 b(=)g(\000)47809 11399 y Fp(+)47809 12138 y Fs(s)48716 11839 y Ft([)120 b Fw(\000)50402 11399 y Fr(\000)50402 12138 y Fs(s)51189 11839 y Fw(.)p 0 TeXcolorgray 475 14514 a FF(Lemma)463 b(3.51.)p 0 TeXcolorgray 554 w FC(F)-93 b(or)434 b(any)f Fu(')h FC(in)g Fu(\010)16944 14696 y Fp(1)17739 14514 y Ft(n)269 b Fo(O)19472 14696 y Fs(s)19962 14514 y Fw(\()20297 14194 y Fu(~)20433 14514 y(b)q Fw(\))p FC(,)17659 17220 y Ft(j)p Fu(\025)18703 16719 y Fp(\(1\))19960 17220 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\)\))271 b Ft(\000)e Fu(k)25427 16719 y Fp(2)p Fs(l)26244 17220 y Ft(j)337 b(\025)f Fu(b)28717 17402 y Fp(0)29243 17220 y Fu(k)29912 16719 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1)p Fr(\000)p Fs(\016)33113 17220 y Fu(r)33694 16719 y Fr(0)34004 17220 y Fu(:)14087 b Fw(\(3.95\))p 0 TeXcolorgray 475 19895 a FC(Pr)-62 b(o)g(of.)p 0 TeXcolorgray 552 w Fw(Supp)34 b(ose)323 b(\(3.95\))f(do)34 b(es)321 b(not)i(hold)f(for)f (some)h Fu(')f Fw(in)h Fu(\010)27627 20077 y Fp(1)28257 19895 y Ft(n)104 b Fo(O)29825 20077 y Fs(s)30316 19895 y Fw(\()30651 19575 y Fu(~)30787 19895 y(b)p Fw(\).)512 b(This)322 b(means)g(that)g Fu(')g Fw(satis\014es)g(equation)475 21364 y(\(3.92\))416 b(with)g(some)f Fu(")10373 21546 y Fp(0)10899 21364 y Fw(:)561 b Ft(j)p Fu(")12699 21546 y Fp(0)13225 21364 y Ft(j)355 b Fu(<)g(b)15735 21546 y Fp(0)16261 21364 y Fu(k)16930 20924 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1)18950 21364 y Fu(r)19531 20924 y Fr(0)20197 21364 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Fr(\000)p Fp(1)34942 60576 y Fq(\015)34942 61303 y(\015)34942 62030 y(\015)34942 62757 y(\015)35953 62333 y Ft(\024)336 b Fw(16)p Fu(:)475 65632 y Fw(Using)272 b(\(3.95\),)299 b(w)-34 b(e)272 b(easily)f(get)h(\(3.96\).)495 b(Using)272 b(\(3.96\))g(and)h(considering)f(\(3.89\),)298 b(\(3.90\),)h(w)-34 b(e)272 b(obtain)h(\(3.97\).)p 50750 65798 665 831 v 0 TeXcolorgray 475 68218 6642 45 v 1803 69155 a Fp(11)2744 69541 y FA(')3413 69105 y Fx(\006)3413 69794 y FD(0)4392 69541 y FE(=)285 b FA(')6142 69118 y Fx(\006)6142 69716 y Fy(\017)6490 69844 y Fh(0)6976 69541 y FE(,)342 b(when)f FA(\017)10691 69652 y FD(0)11437 69541 y FE(=)284 b(0.)p 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 40 40 TeXDict begin 40 39 bop 0 TeXcolorgray 0 TeXcolorgray 475 3345 a Fp(40)14248 b(YULIA)313 b(KARPESHINA)f(AND)h(YOUNG-RAN)g (LEE)p 0 TeXcolorgray 17277 6001 a Fw(4.)606 b Fv(The)458 b(Third)h(Appr)-27 b(o)g(xima)-81 b(tion)475 8160 y Fw(4.1.)605 b FF(The)433 b(Op)39 b(erator)433 b Fu(H)12885 7521 y Fp(\(3\))12787 8318 y Fs(\013)14143 8160 y FF(.)606 b Fw(Cho)34 b(osing)377 b Fu(s)20969 8342 y Fp(3)21832 8160 y Fw(=)337 b(2)p Fu(s)24286 8342 y Fp(2)24812 8160 y Fw(,)381 b(w)-34 b(e)377 b(de\014ne)g(the)g(third)g(op)34 b(erator)377 b Fu(H)41795 7521 y Fp(\(3\))41697 8318 y Fs(\013)43429 8160 y Fw(b)-34 b(y)377 b(the)g(form)-34 b(ula:)11028 11977 y Fu(H)12134 11477 y Fp(\(3\))12036 12277 y Fs(\013)13728 11977 y Fw(=)336 b Fu(H)16113 11477 y Fp(\(2\))17641 11977 y Fw(+)269 b Fu(\013)t(W)20777 12159 y Fp(3)21303 11977 y Fu(;)1414 b Fw(\(0)337 b Ft(\024)g Fu(\013)k Ft(\024)c Fw(1\))p Fu(;)2626 b(W)33329 12159 y Fp(3)34191 11977 y Fw(=)36663 10447 y Fs(M)37567 10570 y FD(3)36470 10826 y Fq(X)35471 13451 y Fs(r)26 b Fp(=)p Fs(M)37558 13574 y FD(2)38018 13451 y Fp(+1)39423 11977 y Fu(V)40130 12159 y Fs(r)40636 11977 y Fu(;)475 16066 y Fw(where)394 b Fu(M)5146 16248 y Fp(3)6065 16066 y Fw(is)g(c)-34 b(hosen)395 b(in)f(suc)-34 b(h)395 b(a)f(w)-34 b(a)g(y)395 b(that)g(2)21941 15626 y Fs(M)22845 15749 y FD(3)23698 16066 y Ft(\031)336 b Fu(k)25646 15626 y Fs(s)26081 15749 y FD(3)26598 16066 y Fw(.)535 b(Ob)-34 b(viously)-101 b(,)396 b(the)e(p)34 b(erio)g(ds)394 b(of)g Fu(W)42230 16248 y Fp(3)43150 16066 y Fw(are)f(2)45769 15626 y Fs(M)46673 15749 y FD(3)47134 15626 y Fr(\000)p Fp(1)48392 16066 y Fw(\()p Fu(b)49383 16248 y Fp(1)49909 16066 y Fu(;)202 b Fw(0\))475 17524 y(and)411 b(2)3444 17084 y Fs(M)4348 17207 y FD(3)4809 17084 y Fr(\000)p Fp(1)6067 17524 y Fw(\(0)p Fu(;)202 b(b)8203 17706 y Fp(2)8729 17524 y Fw(\).)558 b(W)-101 b(e)409 b(write)i(them)g(in)f (the)h(form:)551 b Fu(N)26237 17706 y Fp(2)26763 17524 y Fu(N)27737 17706 y Fp(1)28263 17524 y Fw(\()p Fu(a)29375 17706 y Fp(1)29901 17524 y Fu(;)202 b Fw(0\))410 b(and)i Fu(N)35265 17706 y Fp(2)35790 17524 y Fu(N)36764 17706 y Fp(1)37290 17524 y Fw(\(0)p Fu(;)202 b(a)39547 17706 y Fp(2)40073 17524 y Fw(\),)412 b(here)e Fu(N)44903 17706 y Fp(2)45776 17524 y Fw(=)347 b(2)47672 17084 y Fs(M)48576 17207 y FD(3)49037 17084 y Fr(\000)p Fs(M)50673 17207 y FD(2)51189 17524 y Fw(,)608 18509 y Fp(1)p 608 18707 471 49 v 608 19404 a(4)1211 18986 y Fu(k)1880 18546 y Fs(s)2315 18669 y FD(3)2776 18546 y Fr(\000)p Fs(s)3943 18669 y FD(2)4797 18986 y Fu(<)336 b(N)7050 19168 y Fp(2)7912 18986 y Fu(<)h Fw(4)p Fu(k)10467 18546 y Fs(s)10902 18669 y FD(3)11364 18546 y Fr(\000)p Fs(s)12531 18669 y FD(2)13047 18986 y Fw(.)538 b(Note)405 b(that)10198 22986 y Ft(k)p Fu(W)11949 23168 y Fp(3)12474 22986 y Ft(k)13080 23168 y Fr(1)14413 22986 y Ft(\024)16885 21455 y Fs(M)17789 21578 y FD(3)16692 21834 y Fq(X)15693 24460 y Fs(r)26 b Fp(=)p Fs(M)17780 24583 y FD(2)18240 24460 y Fp(+1)19644 22986 y Ft(k)p Fu(V)20957 23168 y Fs(r)21464 22986 y Ft(k)22070 23168 y Fr(1)23403 22986 y Ft(\024)25875 21455 y Fs(M)26779 21578 y FD(3)25682 21834 y Fq(X)24682 24460 y Fs(r)g Fp(=)p Fs(M)26769 24583 y FD(2)27230 24460 y Fp(+1)28634 22986 y Fw(exp\()p Ft(\000)p Fw(2)32506 22485 y Fs(\021)32 b(r)33516 22986 y Fw(\))337 b Fu(<)g Fw(exp)o(\()p Ft(\000)p Fu(k)39538 22485 y Fs(\021)32 b(s)40477 22608 y FD(2)40994 22986 y Fw(\))p Fu(:)475 28060 y Fw(4.2.)605 b FF(Multiple)333 b(P)-39 b(erio)39 b(ds)332 b(of)f Fu(W)16017 28242 y Fp(2)16542 28060 y Fw(\()p Fu(x)p Fw(\))p FF(.)607 b Fw(The)289 b(op)34 b(erator,)310 b Fu(H)27785 27620 y Fp(\(2\))29380 28060 y Fw(=)336 b Fu(H)31667 28242 y Fp(1)32230 28060 y Fw(+)37 b Fu(W)34355 28242 y Fp(2)34880 28060 y Fw(\()p Fu(x)p Fw(\),)312 b(has)288 b(the)h(p)34 b(erio)g(ds)287 b Fu(N)46291 28242 y Fp(1)46817 28060 y Fu(a)47458 28242 y Fp(1)47984 28060 y Fu(;)202 b(N)49497 28242 y Fp(1)50022 28060 y Fu(a)50663 28242 y Fp(2)51189 28060 y Fw(.)475 29577 y(The)463 b(corresp)34 b(onding)463 b(family)f(of)h(op)34 b(erators,)477 b Ft(f)p Fu(H)23565 29138 y Fp(\(2\))24823 29577 y Fw(\()p Fu(\034)137 b Fw(\))p Ft(g)27038 29765 y Fs(\034)102 b Fr(2)p Fs(K)28978 29888 y FD(2)29496 29577 y Fw(,)477 b(acts)463 b(in)f Fu(L)35164 29759 y Fp(2)35690 29577 y Fw(\()p Fu(Q)37119 29759 y Fp(2)37645 29577 y Fw(\),)478 b(where)462 b Fu(Q)43452 29759 y Fp(2)44413 29577 y Fw(=)434 b([0)p Fu(;)202 b(N)48246 29759 y Fp(1)48771 29577 y Fu(a)49412 29759 y Fp(1)49937 29577 y Fw(])308 b Ft(\002)475 31016 y Fw([0)p Fu(;)202 b(N)2931 31198 y Fp(1)3456 31016 y Fu(a)4097 31198 y Fp(2)4622 31016 y Fw(])575 b(and)h Fu(K)9091 31198 y Fp(2)10240 31016 y Fw(=)621 b([0)p Fu(;)202 b Fw(2)p Fu(\031)43 b(=)-67 b(N)16139 31198 y Fp(1)16665 31016 y Fu(a)17306 31198 y Fp(1)17831 31016 y Fw(\))384 b Ft(\002)f Fw([0)p Fu(;)202 b Fw(2)p Fu(\031)43 b(=)-67 b(N)24347 31198 y Fp(1)24873 31016 y Fu(a)25514 31198 y Fp(2)26039 31016 y Fw(\).)1052 b(Since)575 b(no)-34 b(w)577 b(on)e(w)-34 b(e)576 b(denote)g(quasimomen)-34 b(tum)577 b Fu(t)475 32534 y Fw(from)535 b(the)g(\014rst)g(step)g(b)-34 b(y)535 b Fu(t)13269 32094 y Fp(\(1\))14526 32534 y Fw(,)567 b(quasimomen)-34 b(tum)536 b Fu(\034)672 b Fw(from)535 b(the)g(second)g(step)g(b)-34 b(y)535 b Fu(t)40065 32094 y Fp(\(2\))41322 32534 y Fw(.)930 b(Corresp)34 b(ondingly)-101 b(,)475 34184 y(quasimomen)-34 b(tum)411 b(for)e Fu(H)12544 33545 y Fp(\(3\))12446 34341 y Fs(\013)14211 34184 y Fw(w)-34 b(e)409 b(denote)h(b)-34 b(y)410 b Fu(t)22038 33744 y Fp(\(3\))23295 34184 y Fw(.)554 b(Eigen)-34 b(v)-67 b(alues)409 b(of)g Fu(H)33238 33744 y Fp(\(2\))34698 33202 y Fq(\000)35254 34184 y Fu(t)35692 33744 y Fp(\(2\))36949 33202 y Fq(\001)37914 34184 y Fw(are)f(denoted)j(b)-34 b(y)409 b Fu(\025)46922 33545 y Fp(\(2\))46922 34341 y Fs(n)48382 33202 y Fq(\000)48938 34184 y Fu(t)49376 33744 y Fp(\(2\))50633 33202 y Fq(\001)51189 34184 y Fw(,)475 35823 y Fu(n)336 b Ft(2)h Fn(N)404 b Fw(and)h(its)f(sp)34 b(ectrum)405 b(b)-34 b(y)404 b(\003)15832 35383 y Fp(\(2\))17292 34841 y Fq(\000)17847 35823 y Fu(t)18285 35383 y Fp(\(2\))19543 34841 y Fq(\001)20098 35823 y Fw(.)1803 37267 y(Next,)450 b(let)440 b(us)i(consider)e Fu(W)14434 37449 y Fp(2)14960 37267 y Fw(\()p Fu(x)p Fw(\))h(as)g(a)g(p)34 b(erio)g(dic)439 b(function)k(with)e(the)h(p)34 b(erio)g(ds)440 b Fu(N)39264 37449 y Fp(2)39789 37267 y Fu(N)40763 37449 y Fp(1)41289 37267 y Fu(a)41930 37449 y Fp(1)42456 37267 y Fw(,)449 b Fu(N)44216 37449 y Fp(2)44742 37267 y Fu(N)45716 37449 y Fp(1)46242 37267 y Fu(a)46883 37449 y Fp(2)47408 37267 y Fw(.)649 b(When)475 38785 y(c)-34 b(hanging)535 b(the)g(p)34 b(erio)g(ds,)567 b(the)534 b(family)h(of)g(op)34 b(erators)25720 37803 y Fq(\010)26427 38785 y Fu(H)27533 38345 y Fp(\(2\))28993 37803 y Fq(\000)29548 38785 y Fu(t)29986 38345 y Fp(\(2\))31244 37803 y Fq(\001)o(\011)32506 39264 y Fs(t)32846 39003 y FD(\(2\))33947 39264 y Fr(2)p Fs(K)35367 39387 y FD(2)36419 38785 y Fw(is)534 b(replaced)g(b)-34 b(y)535 b(the)g(family)f(of)475 40614 y(op)34 b(erators)5804 39632 y Fq(\010)6828 40307 y Fw(~)6511 40614 y Fu(H)7617 40174 y Fp(\(2\))9077 39632 y Fq(\000)9632 40614 y Fu(t)10070 40174 y Fp(\(3\))11327 39632 y Fq(\001\011)12590 41093 y Fs(t)12930 40832 y FD(\(3\))14031 41093 y Fr(2)p Fs(K)15451 41216 y FD(3)15968 40614 y Fw(,)373 b(acting)367 b(in)f Fu(L)22478 40796 y Fp(2)23004 40614 y Fw(\()p Fu(Q)24433 40796 y Fp(3)24959 40614 y Fw(\),)374 b(where)366 b Fu(Q)30566 40796 y Fp(3)31428 40614 y Fw(=)337 b([0)p Fu(;)202 b(N)35164 40796 y Fp(2)35689 40614 y Fu(N)36663 40796 y Fp(1)37189 40614 y Fu(a)37830 40796 y Fp(1)38355 40614 y Fw(])193 b Ft(\002)g Fw([0)p Fu(;)202 b(N)42477 40796 y Fp(2)43003 40614 y Fu(N)43977 40796 y Fp(1)44502 40614 y Fu(a)45143 40796 y Fp(2)45669 40614 y Fw(])366 b(and)h Fu(K)49720 40796 y Fp(3)50582 40614 y Fw(=)475 42630 y([0)p Fu(;)202 b Fw(2)p Fu(\031)43 b(=)-67 b(N)4810 42812 y Fp(2)5335 42630 y Fu(N)6309 42812 y Fp(1)6835 42630 y Fu(a)7476 42812 y Fp(1)8001 42630 y Fw(\))274 b Ft(\002)g Fw([0)p Fu(;)202 b Fw(2)p Fu(\031)43 b(=)-67 b(N)14298 42812 y Fp(2)14823 42630 y Fu(N)15797 42812 y Fp(1)16323 42630 y Fu(a)16964 42812 y Fp(2)17489 42630 y Fw(\).)558 b(W)-101 b(e)410 b(denote)h(eigen)-34 b(v)-67 b(alues)410 b(of)32841 42324 y(~)32524 42630 y Fu(H)33630 42190 y Fp(\(2\))35089 41648 y Fq(\000)35645 42630 y Fu(t)36083 42190 y Fp(\(3\))37340 41648 y Fq(\001)38306 42630 y Fw(b)-34 b(y)40047 42310 y(~)39996 42630 y Fu(\025)40703 41991 y Fp(\(2\))40703 42788 y Fs(n)42163 41648 y Fq(\000)42718 42630 y Fu(t)43156 42190 y Fp(\(3\))44414 41648 y Fq(\001)44969 42630 y Fw(,)412 b Fu(n)347 b Ft(2)g Fn(N)p Fw(,)412 b(and)475 44281 y(its)442 b(sp)34 b(ectrum)442 b(b)-34 b(y)9376 43975 y(~)9258 44281 y(\003)10100 43841 y Fp(\(2\))11559 43299 y Fq(\000)12115 44281 y Fu(t)12553 43841 y Fp(\(3\))13810 43299 y Fq(\001)14366 44281 y Fw(.)651 b(W)-101 b(e)442 b(denote)g(no)-34 b(w)443 b(b)-34 b(y)442 b Fu(P)26653 43841 y Fp(\(1\))28353 44281 y Fw(the)g(set)g Fu(P)168 b Fw(,)452 b(in)-34 b(tro)34 b(duced)442 b(b)-34 b(y)443 b(\(3.4\),)451 b(its)442 b(elemen)-34 b(ts)475 45920 y(b)34 b(eing)404 b Fu(p)4351 45480 y Fp(\(1\))5608 45920 y Fw(.)539 b(By)403 b(Blo)34 b(c)-34 b(h)404 b(theory)g(\(see)g (e.g.[21)o(]\),)f(for)i(an)-34 b(y)404 b Fu(t)27541 45480 y Fp(\(3\))29135 45920 y Ft(2)337 b Fu(K)31309 46102 y Fp(3)31835 45920 y Fw(,)13046 48636 y(~)12928 48942 y(\003)13770 48442 y Fp(\(2\))15229 47597 y Fq(\020)15953 48942 y Fu(t)16391 48442 y Fp(\(3\))17649 47597 y Fq(\021)18709 48942 y Fw(=)21330 47790 y Fq([)19989 50570 y Fs(p)20463 50309 y FD(\(2\))21563 50570 y Fr(2)p Fs(P)22918 50309 y FD(\(2\))24220 48942 y Fw(\003)25062 48442 y Fp(\(2\))26522 47597 y Fq(\020)27246 48942 y Fu(t)27684 48442 y Fp(\(3\))29210 48942 y Fw(+)270 b(2)p Fu(\031)43 b(p)32373 48442 y Fp(\(2\))33631 48942 y Fu(=)-67 b(N)35144 49124 y Fp(2)35669 48942 y Fu(N)36643 49124 y Fp(1)37169 48942 y Fu(a)37810 47597 y Fq(\021)38735 48942 y Fu(;)9962 b Fw(\(4.1\))475 53583 y(where)470 b Fu(P)4992 53143 y Fp(\(2\))6697 53583 y Fw(=)8086 52237 y Fq(n)8895 53583 y Fu(p)9505 53143 y Fp(\(2\))11099 53583 y Fw(=)12378 52601 y Fq(\000)12934 53583 y Fu(p)13544 52944 y Fp(\(2\))13544 53930 y(1)14801 53583 y Fu(;)202 b(p)15950 52944 y Fp(\(2\))15950 53930 y(2)17207 52601 y Fq(\001)18100 53583 y Ft(2)336 b Fn(Z)20052 53143 y Fp(2)20915 53583 y Fw(:)g(0)h Ft(\024)g Fu(p)24421 52944 y Fp(\(2\))24421 53930 y(1)26015 53583 y Ft(\024)f Fu(N)28268 53765 y Fp(2)29063 53583 y Ft(\000)269 b Fw(1)p Fu(;)606 b Fw(0)337 b Ft(\024)g Fu(p)34657 52944 y Fp(\(2\))34657 53930 y(2)36251 53583 y Ft(\024)f Fu(N)38504 53765 y Fp(2)39299 53583 y Ft(\000)269 b Fw(1)41117 52237 y Fq(o)41926 53583 y Fw(,)486 b(2)p Fu(\031)43 b(p)44699 53143 y Fp(\(2\))45957 53583 y Fu(=)-67 b(N)47470 53765 y Fp(2)47995 53583 y Fu(N)48969 53765 y Fp(1)49495 53583 y Fu(a)446 b Fw(=)475 54530 y Fq(\020)1199 55875 y Fw(2)p Fu(\031)43 b(p)3149 55236 y Fp(\(2\))3149 56222 y(1)4407 55875 y Fu(=a)5654 56057 y Fp(1)6179 55875 y Fu(;)202 b Fw(2)p Fu(\031)43 b(p)8668 55236 y Fp(\(2\))8668 56222 y(2)9926 55875 y Fu(=a)11173 56057 y Fp(2)11699 54530 y Fq(\021)12624 55875 y Fu(=)-67 b(N)14137 56057 y Fp(2)14663 55875 y Fu(N)15637 56057 y Fp(1)16163 55875 y Fw(.)714 b(An)463 b(iso)34 b(energetic)461 b(set)28048 55569 y(~)27844 55875 y Fu(S)28587 56057 y Fp(2)29113 55875 y Fw(\()p Fu(\025)30291 56057 y Fp(0)30817 55875 y Fw(\))435 b Ft(\032)f Fu(K)34129 56057 y Fp(3)35118 55875 y Fw(of)463 b(the)g(op)34 b(erator)43968 55569 y(~)43650 55875 y Fu(H)44756 55435 y Fp(\(2\))46477 55875 y Fw(is)462 b(de\014ned)475 57616 y(b)-34 b(y)404 b(the)h(form)-34 b(ula:)6106 60264 y(~)5901 60570 y Fu(S)6644 60752 y Fp(2)7170 60570 y Fw(\()p Fu(\025)p Fw(\))338 b(=)10436 59225 y Fq(n)11244 60570 y Fu(t)11682 60070 y Fp(\(3\))13276 60570 y Ft(2)f Fu(K)15450 60752 y Fp(3)16313 60570 y Fw(:)g Ft(9)p Fu(n)f Ft(2)h Fn(N)g Fw(:)22209 60250 y(~)22159 60570 y Fu(\025)22866 60070 y Fp(\(2\))22866 60870 y Fs(n)24123 59588 y Fq(\000)24679 60570 y Fu(t)25117 60070 y Fp(\(3\))26374 59588 y Fq(\001)27266 60570 y Fw(=)g Fu(\025)29253 59225 y Fq(o)9157 63223 y Fw(=)10436 61877 y Fq(n)11244 63223 y Fu(t)11682 62722 y Fp(\(3\))13276 63223 y Ft(2)g Fu(K)15450 63405 y Fp(3)16313 63223 y Fw(:)g Ft(9)p Fu(n)f Ft(2)h Fn(N)p Fu(;)606 b(p)22297 62722 y Fp(\(2\))23891 63223 y Ft(2)337 b Fu(P)25982 62722 y Fp(\(2\))27577 63223 y Fw(:)f Fu(\025)28957 62722 y Fp(\(2\))28957 63522 y Fs(n)30417 61877 y Fq(\020)31141 63223 y Fu(t)31579 62722 y Fp(\(3\))33106 63223 y Fw(+)269 b(2)p Fu(\031)43 b(p)36268 62722 y Fp(\(2\))37526 63223 y Fu(=)-67 b(N)39039 63405 y Fp(2)39564 63223 y Fu(N)40538 63405 y Fp(1)41064 63223 y Fu(a)41705 61877 y Fq(\021)42765 63223 y Fw(=)337 b Fu(\025)44752 61877 y Fq(o)45762 63223 y Fu(:)475 66451 y Fw(Ob)-34 b(viously)-101 b(,)6504 66145 y(~)6300 66451 y Fu(S)7043 66633 y Fp(2)7905 66451 y Fw(=)337 b Fo(K)10182 66633 y Fp(3)10708 66451 y Fu(S)11451 66633 y Fp(2)11977 66451 y Fu(;)269 b Fw(where)h Fo(K)16951 66633 y Fp(3)17746 66451 y Fw(is)g(the)f(parallel)g(shift)h(in)-34 b(to)270 b Fu(K)30913 66633 y Fp(3)31440 66451 y Fw(,)296 b(that)270 b(is,)296 b Fo(K)37009 66633 y Fp(3)37872 66451 y Fw(:)337 b Fn(R)39421 66012 y Fp(2)40283 66451 y Ft(!)g Fu(K)42861 66633 y Fp(3)43388 66451 y Fw(,)296 b Fo(K)45018 66633 y 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b(N)29838 69723 y Fp(1)30364 69541 y Fu(:)p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 41 41 TeXDict begin 41 40 bop 0 TeXcolorgray 0 TeXcolorgray 2848 3345 a Fp(SPECTRAL)406 b(PR)-26 b(OPER)-78 b(TIES)406 b(OF)h(A)h(POL)-78 b(YHARMONIC)405 b(OPERA)-78 b(TOR)406 b(WITH)h(LIMIT-PERIODIC)e(POTENTIAL)1430 b(41)p 0 TeXcolorgray 475 6001 a Fw(4.3.)605 b FF(P)-39 b(erturbation)397 b(F)-116 b(orm)-39 b(ulae.)606 b Fw(The)344 b(results)g(of)g(this)g(section)f (are)g(analogous)i(to)f(those)g(in)f(the)h(previous)475 7440 y(step,)490 b(the)473 b(index)f(2)h(just)h(b)34 b(eing)472 b(replaced)h(b)-34 b(y)473 b(3.)744 b(Let)472 b(us)h(start)g(with)h(establishing)f(a)g(lo)-34 b(w)g(er)473 b(b)34 b(ound)474 b(for)e Fu(k)38 b Fw(.)475 8879 y(Since)404 b Fu(\021)43 b(s)4853 9061 y Fp(1)5716 8879 y Fu(>)336 b Fw(2)270 b(+)f(2)p Fu(s)10257 9061 y Fp(1)10783 8879 y Fw(,)404 b(there)g(is)f(a)i(n)-34 b(um)g(b)34 b(er)405 b Fu(k)21899 9061 y Fr(\003)22761 8879 y Fu(>)337 b(e)404 b Fw(suc)-34 b(h)405 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Fu(m)337 b Ft(2)g Fn(Z)50663 7079 y Fp(2)51189 7519 y Fw(,)475 9037 y Fu(p)1085 8597 y Fp(\(2\))2679 9037 y Ft(2)g Fu(P)4770 8597 y Fp(\(2\))6028 9037 y Fw(.)514 b(Note)332 b(that)h(2)p Fu(\031)43 b(m=)-67 b(N)16207 9219 y Fp(1)16733 9037 y Fu(a)331 b Fw(is)g(a)h(p)34 b(oin)-34 b(t)332 b(of)g(a)f(dual)h(lattice)g(for)f(p)34 b(erio)g(ds)331 b Fu(N)38315 9219 y Fp(1)38841 9037 y Fu(a)39482 9219 y Fp(1)40007 9037 y Fw(,)346 b Fu(N)41664 9219 y Fp(1)42190 9037 y Fu(a)42831 9219 y Fp(2)43688 9037 y Fw(and)332 b Fu(p)46582 8597 y Fp(\(2\))48176 9037 y Ft(2)337 b Fu(P)50267 8597 y Fp(\(2\))475 10476 y Fw(is)294 b(resp)34 b(onsible)295 b(for)f(re\014ning)h(the)g (lattice.)502 b(By)294 b(Remark)g(3.7,)316 b(2)p Fu(\031)43 b(q)g(=)-67 b(N)31742 10658 y Fp(2)32269 10476 y Fu(N)33243 10658 y Fp(1)33768 10476 y Fu(a)295 b Fw(also)f(can)h(b)34 b(e)294 b(uniquely)h(represen)-34 b(ted)475 11994 y(as)404 b(2)p Fu(\031)43 b(q)g(=)-67 b(N)5400 12176 y Fp(2)5926 11994 y Fu(N)6900 12176 y Fp(1)7426 11994 y Fu(a)337 b Fw(=)f(2)p Fu(\031)43 b(j)69 b(=a)270 b Fw(+)f(2)p Fu(\031)43 b(p)16270 11554 y Fp(\(1\))17528 11994 y Fu(=)-67 b(N)19041 12176 y Fp(1)19567 11994 y Fu(a)269 b Fw(+)g(2)p Fu(\031)43 b(p)23639 11554 y Fp(\(2\))24897 11994 y Fu(=)-67 b(N)26410 12176 y Fp(2)26935 11994 y Fu(N)27909 12176 y Fp(1)28435 11994 y Fu(a)p Fw(,)404 b(here)f Fu(j)j Ft(2)337 b Fn(Z)35304 11554 y Fp(2)35830 11994 y Fw(,)404 b Fu(p)37181 11554 y Fp(\(1\))38775 11994 y Ft(2)337 b Fu(P)40866 11554 y Fp(\(1\))42124 11994 y Fw(,)404 b Fu(p)43475 11554 y Fp(\(2\))45069 11994 y Ft(2)337 b Fu(P)47160 11554 y Fp(\(2\))48418 11994 y Fw(.)1803 13512 y(Let)437 b(us)h(consider)g(a)f(normalized)h(eigenfunction)g Fu( )25895 13694 y Fs(n)26723 12530 y Fq(\000)27279 13512 y Fu(t)27717 13072 y Fp(\(2\))28974 13512 y Fu(;)202 b(x)30206 12530 y Fq(\001)31198 13512 y Fw(of)438 b Fu(H)33718 13072 y Fp(\(2\))35178 12530 y Fq(\000)35734 13512 y Fu(t)36172 13072 y Fp(\(2\))37429 12530 y Fq(\001)38422 13512 y Fw(in)g Fu(L)40695 13694 y Fp(2)41221 13512 y Fw(\()p Fu(Q)42650 13694 y Fp(2)43176 13512 y Fw(\).)639 b(W)-101 b(e)437 b(extend)h(it)475 15176 y(quasip)34 b(erio)g(dically)277 b(to)i Fu(L)11834 15358 y Fp(2)12360 15176 y Fw(\()p Fu(Q)13789 15358 y Fp(3)14315 15176 y Fw(\),)304 b(renormalize)278 b(and)h(denote)h(the)f(new)g(function)i(b) -34 b(y)38610 14856 y(~)38362 15176 y Fu( )39152 15358 y Fs(n)39980 14194 y Fq(\000)40535 15176 y Fu(t)40973 14736 y Fp(\(3\))42231 15176 y Fu(;)202 b(x)43463 14194 y Fq(\001)44018 15176 y Fw(,)303 b Fu(t)45096 14736 y Fp(\(3\))46690 15176 y Fw(=)337 b Fo(K)48967 15358 y Fp(3)49493 15176 y Fu(t)49931 14736 y Fp(\(2\))51189 15176 y Fw(.)475 16840 y(The)358 b(F)-101 b(ourier)358 b(represen)-34 b(tations)358 b(of)h Fu( )17555 17022 y Fs(n)18383 15858 y Fq(\000)18938 16840 y Fu(t)19376 16400 y Fp(\(2\))20633 16840 y Fu(;)202 b(x)21865 15858 y Fq(\001)22778 16840 y Fw(in)358 b Fu(L)24971 17022 y Fp(2)25497 16840 y Fw(\()p Fu(Q)26926 17022 y Fp(2)27452 16840 y Fw(\))h(and)30840 16520 y(~)30592 16840 y Fu( )31382 17022 y Fs(n)32008 16840 y Fw(\()p Fu(t)32917 16400 y Fp(\(3\))34175 16840 y Fu(;)202 b(x)p Fw(\))358 b(in)f Fu(L)38428 17022 y Fp(2)38954 16840 y Fw(\()p Fu(Q)40383 17022 y Fp(3)40910 16840 y Fw(\))h(are)f(simply)h(related.)475 18718 y(If)352 b(w)-34 b(e)352 b(denote)h(F)-101 b(ourier)352 b(co)34 b(e\016cien)-34 b(ts)352 b(of)g Fu( )19569 18900 y Fs(n)20397 17736 y Fq(\000)20953 18718 y Fu(t)21391 18278 y Fp(\(2\))22648 18718 y Fu(;)202 b(x)23880 17736 y Fq(\001)24787 18718 y Fw(with)353 b(resp)34 b(ect)352 b(to)g(the)h(basis)f Ft(j)p Fu(Q)39280 18900 y Fp(2)39806 18718 y Ft(j)40143 18278 y Fr(\000)p Fp(1)p Fs(=)p Fp(2)42341 18718 y Fu(e)42905 18203 y Fs(i)43225 18285 y Fw(\()43698 18203 y Fp(2)p Fs(\031)32 b(m=)-52 b(N)46737 18326 y FD(1)47199 18203 y Fs(a)p Fp(+)p Fs(t)48771 17890 y FD(\(2\))49871 18203 y Fs(;x)50662 18285 y Fw(\))51189 18718 y(,)475 20382 y Fu(m)337 b Ft(2)f Fn(Z)3828 19942 y Fp(2)4354 20382 y Fu(;)305 b Fw(in)g 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y Fw(,)403 b(in)i Fu(L)28172 22442 y Fp(2)28697 22260 y Fw(\()p Fu(Q)30126 22442 y Fp(3)30653 22260 y Fw(\))f(are)g(giv)-34 b(en)404 b(b)-34 b(y)404 b(the)h(form)-34 b(ula)17222 25056 y(~)16947 25362 y Fu(C)17813 25544 y Fs(nq)19229 25362 y Fw(=)20508 23289 y Fq(\()21485 24541 y Fu(C)22351 24723 y Fs(nm)23809 24541 y Fu(;)1212 b Fw(if)404 b Fu(q)381 b Fw(=)336 b Fu(mN)30708 24723 y Fp(2)31503 24541 y Fw(+)270 b Fu(p)33326 24101 y Fp(\(2\))34583 24541 y Fw(;)21485 26268 y(0)p Fu(;)2930 b Fw(otherwise)p Fu(;)475 28660 y(p)1085 28220 y Fp(\(2\))2740 28660 y Fw(b)34 b(eing)399 b(de\014ned)g(from)g(the)f (relation)g Fu(t)19991 28220 y Fp(\(2\))21585 28660 y Fw(=)337 b Fu(t)23303 28220 y Fp(\(3\))24818 28660 y Fw(+)257 b(2)p Fu(\031)43 b(p)27968 28220 y Fp(\(2\))29226 28660 y Fu(=)-67 b(N)30739 28842 y Fp(1)31265 28660 y Fu(N)32239 28842 y Fp(2)32765 28660 y Fu(a)p Fw(.)536 b(Corresp)34 b(ondingly)-101 b(,)399 b(matrices)f(of)h(the)475 30203 y(pro)67 b(jections)405 b(on)f Fu( )9187 30385 y 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y(\020)22371 46293 y Fw(~)22054 46600 y Fu(H)23160 46099 y Fp(\(2\))24418 45618 y Fq(\000)24973 46600 y Fu(t)25411 46099 y Fp(\(3\))26669 45618 y Fq(\001)27493 46600 y Ft(\000)270 b Fu(z)29323 45254 y Fq(\021)30046 45556 y Fr(\000)p Fp(1)31304 46600 y Fu(W)32449 46782 y Fp(3)32975 44891 y Fq(\023)33867 45193 y Fs(r)34575 45254 y Fq(\020)35616 46293 y Fw(~)35299 46600 y Fu(H)36405 46099 y Fp(\(2\))37663 45618 y Fq(\000)38218 46600 y Fu(t)38656 46099 y Fp(\(3\))39914 45618 y Fq(\001)40738 46600 y Ft(\000)g Fu(z)42568 45254 y Fq(\021)43292 45556 y Fr(\000)p Fp(1)44549 46600 y Fu(dz)53 b(:)p 0 TeXcolorgray 475 49793 a FF(Theorem)622 b(4.4.)p 0 TeXcolorgray 629 w FC(Supp)-62 b(ose)557 b Fu(t)14528 49353 y Fp(\(3\))16345 49793 y FC(b)-62 b(elongs)558 b(to)h(the)g Fw(\()p Fu(\017)25307 49975 y Fp(2)25833 49793 y Fu(k)26502 49353 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fs(\016)30435 49793 y Fw(\))p FC(-neighb)-62 b(orho)g(o)g(d)556 b(in)k Fu(K)41310 49975 y Fp(3)42396 49793 y FC(of)f(the)g(thir)-62 b(d)558 b(non-)475 51261 y(r)-62 b(esonanc)g(e)548 b(set)h Fu(\037)8946 51443 y Fp(3)9473 51261 y Fw(\()p Fu(\025;)202 b(\016)46 b Fw(\))p FC(,)579 b Fw(0)547 b Fu(<)h Fw(7)p Fu(\016)593 b(<)547 b Fw(2)p Fu(l)378 b Ft(\000)353 b Fw(11)h Ft(\000)f Fw(16)p Fu(s)26353 51443 y Fp(1)26880 51261 y FC(.)906 b(Then,)579 b(for)549 b(su\016ciently)f(lar)-62 b(ge)548 b Fu(\025)p FC(,)579 b Fu(\025)548 b(>)f(k)48237 50821 y Fp(2)p Fs(l)48199 51561 y Fr(\003)49605 51261 y FC(and)475 52978 y(for)481 b(al)62 b(l)480 b Fu(\013)t FC(,)494 b Fw(0)424 b Ft(\024)f Fu(\013)428 b Ft(\024)423 b Fw(1)p FC(,)494 b(ther)-62 b(e)480 b(exists)g(a)h(unique)e(eigenvalue)g(of)i (the)f(op)-62 b(er)g(ator)479 b Fu(H)39170 52339 y Fp(\(3\))39072 53136 y Fs(\013)40630 51996 y Fq(\000)41185 52978 y Fu(t)41623 52538 y Fp(\(3\))42881 51996 y Fq(\001)43918 52978 y FC(in)j(the)e(interval)475 54781 y Fu(")1040 54963 y Fp(3)1566 54781 y Fw(\()p Fu(k)38 b Fw(\))337 b(:=)g(\()p Fu(k)6271 54341 y Fp(2)p 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(de\014nition)i(of)e Fu(!)28591 52287 y Fr(\003)28548 53052 y Fp(1)29118 52727 y Fw(\()p Fu(\025)p Fw(\))g(\(Section)h (3.3.4\),)i(w)-34 b(e)416 b(consider)g(a)g(complex)475 54166 y(resonance)404 b(set)g Fu(!)8696 53726 y Fr(\003)8653 54491 y Fp(2)9223 54166 y Fw(\()p Fu(\025)p Fw(\),)g(whic)-34 b(h)405 b(is)f(the)h(set)f(of)h Fu(')336 b Ft(2)h Fu(\010)24743 54348 y Fp(2)25673 54166 y Fw(satisfying)4126 57588 y(det)6012 55516 y Fq( )7104 56691 y Fu(H)8210 56251 y Fp(\(2\))9468 55709 y Fq(\000)10013 56691 y Fu(~)-596 b(y)10660 56251 y Fp(\(2\))11919 56691 y Fw(\()p Fu(')p Fw(\))13654 55709 y Fq(\001)14479 56691 y Ft(\000)269 b Fu(k)16360 56251 y Fp(2)p Fs(l)17447 56691 y Ft(\000)g Fu(\017)p 7104 57310 12048 49 v 8506 58485 a(H)9514 58667 y Fp(0)10039 57503 y Fq(\000)10585 58485 y Fu(~)-596 b(y)11232 58133 y Fp(\(2\))12490 58485 y Fw(\()p Fu(')p Fw(\))14225 57503 y Fq(\001)15051 58485 y Fw(+)269 b Fu(k)16932 58135 y Fp(2)p Fs(l)19284 55516 y Fq(!)20580 57588 y Fw(=)337 b(0)p Fu(;)1404 b(~)-596 b(y)24854 57088 y Fp(\(2\))26112 57588 y Fw(\()p Fu(')p Fw(\))338 b(=)368 b Fu(~)-638 b Fn({)30272 57770 y Fp(2)30798 57588 y Fw(\()p Fu(')p Fw(\))270 b(+)33879 57268 y Fu(~)34015 57588 y(b)34535 57088 y Fp(\(2\))35793 57588 y Fu(;)37004 57268 y(~)37140 57588 y(b)37660 57088 y Fp(\(2\))39254 57588 y Fw(=)40883 56768 y(2)p Fu(\031)43 b(p)42833 56328 y Fp(\(2\))p 40667 57310 3641 49 v 40667 58420 a Fu(N)41641 58602 y Fp(2)42166 58420 y Fu(N)43140 58602 y Fp(1)43666 58420 y Fu(a)44440 57588 y(;)3651 b Fw(\(4.14\))475 61184 y(for)436 b(some)h Fu(p)6042 60744 y Fp(\(2\))7690 61184 y Ft(2)390 b Fu(P)9834 60744 y Fp(\(2\))11383 61184 y Ft(n)291 b(f)p Fw(0)p Ft(g)437 b Fw(and)g Ft(j)p Fu(\017)p Ft(j)390 b Fu(<)g(\017)20305 61366 y Fp(2)20831 61184 y Fw(.)635 b(Let)436 b Fu(!)24761 61366 y Fp(2)25288 61184 y Fw(\()p Fu(\025)p Fw(\))h(b)34 b(e)436 b(the)h(set)f(of)h Fu(')391 b Ft(2)f Fw(\002)37838 61366 y Fp(2)38800 61184 y Fw(corresp)34 b(onding)437 b(to)f(\012)49012 61366 y Fp(2)49539 61184 y Fw(\()p Fu(\025)p Fw(\).)475 62623 y(Considering)405 b(as)f(in)g(the)g(pro)34 b(of)405 b(of)f(Lemma)g(3.20,)f(it)i(is)e(easy)h(to)h(sho)-34 b(w)405 b(that)g Fu(!)36875 62805 y Fp(2)37738 62623 y Fw(=)337 b Fu(!)39815 62183 y Fr(\003)39772 62947 y Fp(2)40611 62623 y Ft(\\)269 b Fw(\002)42631 62805 y Fp(2)43157 62623 y Fw(.)1803 65288 y(By)320 b(analogy)h(with)g (\(3.41\),)338 b(let)320 b(us)h(consider)g(an)g(arbitrary)28788 64968 y Fu(~)28924 65288 y(b)29444 64848 y Fp(\(2\))31039 65288 y Ft(2)337 b Fu(K)33213 65470 y Fp(2)34059 65288 y Fw(and)322 b(its)f(distance)g Fu(b)43097 64649 y Fp(\(2\))43097 65635 y(0)44675 65288 y Fw(to)g(the)g(closest)475 66727 y(v)-34 b(ertex)403 b(of)i Fu(K)6558 66909 y Fp(2)7084 66727 y Fw(:)14315 68775 y Fu(b)14835 68136 y Fp(\(2\))14835 69122 y(0)16429 68775 y Fw(=)4369 b(min)17709 69641 y Fs(m)p Fp(=\(0)p Fs(;)p Fp(0\))p Fs(;)p Fp(\(0)p Fs(;)p Fp(1\))p Fs(;)p Fp(\(1)p Fs(;)p Fp(0\))p Fs(;)p Fp(\(1)p Fs(;)p Fp(1\))27995 67381 y Fq(\014)27995 68108 y(\014)27995 68835 y(\014)28263 68455 y Fu(~)28399 68775 y(b)28919 68274 y Fp(\(2\))30447 68775 y Ft(\000)269 b Fw(2)p Fu(\031)43 b(m=)-67 b(N)35576 68957 y Fp(1)36102 68775 y Fu(a)36743 67381 y Fq(\014)36743 68108 y(\014)36743 68835 y(\014)37349 68775 y Fu(:)p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 44 44 TeXDict begin 44 43 bop 0 TeXcolorgray 0 TeXcolorgray 475 3345 a Fp(44)14248 b(YULIA)313 b(KARPESHINA)f(AND)h(YOUNG-RAN)g (LEE)p 0 TeXcolorgray 475 6288 a Fw(No)-34 b(w)582 b(w)-34 b(e)582 b(construct)g Fo(O)11740 5849 y Fp(\(2\))13200 4943 y Fq(\020)13787 5969 y Fu(~)13923 6288 y(b)14443 5849 y Fp(\(2\))15701 4943 y Fq(\021)17057 6288 y Ft(\032)632 b Fu(\010)19440 6470 y Fp(2)19966 6288 y Fw(,)626 b(assuming)582 b Fu(b)26892 5649 y Fp(\(2\))26892 6635 y(0)28782 6288 y Fu(>)632 b Fw(0.)1071 b(Indeed,)625 b(for)582 b(eac)-34 b(h)41669 5969 y Fu(~)41806 6288 y(b)42326 5849 y Fp(\(2\))43971 6288 y Fw(+)387 b(2)p Fu(\031)43 b(p)47251 5849 y Fp(\(1\))48509 6288 y Fu(=)-67 b(N)50022 6470 y Fp(1)50548 6288 y Fu(a)p Fw(,)475 8490 y Fu(p)1085 8050 y Fp(\(1\))2679 8490 y Ft(2)337 b Fu(P)4770 8050 y Fp(\(1\))6028 8490 y Fw(,)367 b(w)-34 b(e)358 b(tak)-34 b(e)358 b(the)h(set)f Fo(O)15796 7851 y Fp(\(1\))15796 8648 y Fs(s)17053 7508 y Fq(\000)17473 8170 y Fu(~)17609 8490 y(b)18129 8050 y Fp(\(2\))19564 8490 y Fw(+)177 b(2)p Fu(\031)43 b(p)22634 8050 y Fp(\(1\))23892 8490 y Fu(=)-67 b(N)25405 8672 y Fp(1)25930 8490 y Fu(a)26571 7508 y Fq(\001)27484 8490 y Fw(describ)34 b(ed)358 b(in)g(Section)g (3.5)g(and)h(consider)f(a)g(union:)13249 10923 y Fo(O)14107 10422 y Fp(\(2\))15566 9577 y Fq(\020)16154 10603 y Fu(~)16290 10923 y(b)16810 10422 y Fp(\(2\))18068 9577 y Fq(\021)19128 10923 y Fw(=)337 b Ft([)21216 11265 y Fs(p)21690 11003 y FD(\(1\))22790 11265 y Fr(2)p Fs(P)24145 11003 y FD(\(1\))25301 10923 y Fo(O)26159 10422 y Fp(\(1\))26159 11222 y Fs(s)27618 9577 y Fq(\020)28206 10603 y Fu(~)28342 10923 y(b)28862 10422 y Fp(\(2\))30389 10923 y Fw(+)270 b(2)p Fu(\031)43 b(p)33552 10422 y Fp(\(1\))34810 10923 y Fu(=)-67 b(N)36323 11105 y Fp(1)36848 10923 y Fu(a)37489 9577 y Fq(\021)38415 10923 y Fu(:)9676 b Fw(\(4.15\))475 13719 y(A)404 b(connected)g(comp)34 b(onen)-34 b(t)406 b(of)e Fo(O)15878 13279 y Fp(\(2\))17338 12373 y Fq(\020)17925 13399 y Fu(~)18061 13719 y(b)18581 13279 y Fp(\(2\))19839 12373 y Fq(\021)20967 13719 y Fw(w)-34 b(e)405 b(denote)g(b)-34 b(y)404 b(\000)29099 13279 y Fp(\(2\))30558 12373 y Fq(\020)31146 13399 y Fu(~)31282 13719 y(b)31802 13279 y Fp(\(2\))33060 12373 y Fq(\021)34188 13719 y Fw(and)h(its)f(b)34 b(oundary)405 b(b)-34 b(y)405 b Fu(\015)46072 13279 y Fp(\(2\))47531 12373 y Fq(\020)48119 13399 y Fu(~)48255 13719 y(b)48775 13279 y Fp(\(2\))50033 12373 y Fq(\021)50757 13719 y Fw(.)p 0 TeXcolorgray 475 16723 a FF(Remark)466 b(4.7.)p 0 TeXcolorgray 555 w Fw(The)406 b(set)g Fo(O)13743 16284 y Fp(\(2\))15203 15378 y Fq(\020)15790 16404 y Fu(~)15926 16723 y(b)16446 16284 y Fp(\(2\))17704 15378 y Fq(\021)18834 16723 y Fw(consists)h(of)f(disks)g(of)g(the)h(radius)f Fu(r)35396 16284 y Fr(0)36112 16723 y Fw(around)h(all)e(zeros)h(of)g (the)g(deter-)475 19016 y(minan)-34 b(t)405 b(det)6300 17671 y Fq(\020)7024 19016 y Fu(I)364 b Fw(+)9453 18710 y(~)9133 19016 y Fu(A)10042 19198 y Fp(1)10568 18034 y Fq(\000)11113 19016 y Fu(~)-596 b(y)11760 18576 y Fp(\(1\))13019 19016 y Fw(\()p Fu(')p Fw(\))14754 18034 y Fq(\001)15310 17671 y(\021)16034 19016 y Fw(,)404 b(here)f(and)i(b)34 b(elo)-34 b(w:)20034 21604 y Fu(~)-596 b(y)20681 21104 y Fp(\(1\))21939 21604 y Fw(\()p Fu(')p Fw(\))337 b(=)369 b Fu(~)-638 b Fn({)26099 21786 y Fp(1)26625 21604 y Fw(\()p Fu(')p Fw(\))270 b(+)29706 21284 y Fu(~)29842 21604 y(b)30362 21104 y Fp(\(2\))31620 21604 y Fu(;)16471 b Fw(\(4.16\))12528 24136 y Fu(I)364 b Fw(+)14957 23829 y(~)14637 24136 y Fu(A)15546 24318 y Fp(1)16072 23154 y Fq(\000)16618 24136 y Fu(~)-596 b(y)17265 23635 y Fp(\(1\))18523 23154 y Fq(\001)19415 24136 y Fw(=)20695 22790 y Fq(\020)21736 23829 y Fw(~)21419 24136 y Fu(H)22525 23635 y Fp(\(1\))23783 23154 y Fq(\000)24328 24136 y Fu(~)g(y)24975 23635 y Fp(\(1\))26233 23154 y Fq(\001)27058 24136 y Ft(\000)270 b Fu(\025)28978 22790 y Fq(\021)o(\020)30743 23829 y Fw(~)30425 24136 y Fu(H)31433 24318 y Fp(0)31959 23154 y Fq(\000)32504 24136 y Fu(~)-596 b(y)33151 23635 y Fp(\(1\))34410 23154 y Fq(\001)35235 24136 y Fw(+)269 b Fu(\025)37154 22790 y Fq(\021)37878 23092 y Fr(\000)p Fp(1)39135 24136 y Fu(:)1803 27049 y Fw(This)355 b(remark)e(easily)g(follo)-34 b(ws)355 b(from)f(the)h(fact)f(that)i(eac)-34 b(h)354 b Fo(O)29144 26411 y Fp(\(1\))29144 27207 y Fs(s)30401 26068 y Fq(\000)30821 26730 y Fu(~)30957 27049 y(b)31477 26610 y Fp(\(2\))32904 27049 y Fw(+)169 b(2)p Fu(\031)43 b(p)35966 26610 y Fp(\(1\))37224 27049 y Fu(=)-67 b(N)38737 27231 y Fp(1)39263 27049 y Fu(a)39904 26068 y Fq(\001)40813 27049 y Fw(is)354 b(built)h(around)g(zeros)475 28978 y(of)517 b(det)3651 27633 y Fq(\020)4375 28978 y Fu(I)440 b Fw(+)344 b Fu(A)7544 29160 y Fp(1)8272 27633 y Fq(\020)9028 28978 y Fu(~)-638 b Fn({)9804 29160 y Fp(1)10330 28978 y Fw(\()p Fu(')p Fw(\))270 b(+)13411 28658 y Fu(~)13547 28978 y(b)14067 28538 y Fp(\(2\))15594 28978 y Fw(+)f(2)p Fu(\031)43 b(p)18756 28538 y Fp(\(1\))20014 28978 y Fu(=)-67 b(N)21527 29160 y Fp(1)22053 28978 y Fu(a)22694 27633 y Fq(\021)o(\021)24141 28978 y Fw(,)545 b(and)27812 28672 y(~)27492 28978 y Fu(A)28401 29160 y Fp(1)28927 27997 y Fq(\000)29473 28978 y Fu(~)-596 b(y)30120 28538 y Fp(\(1\))31378 28978 y Fw(\()p Fu(')p Fw(\))33113 27997 y Fq(\001)34186 28978 y Fw(is)517 b(a)f(direct)h(sum)g(of)g(the)g(op)34 b(erators)475 31271 y Fu(A)1384 31453 y Fp(1)2112 29925 y Fq(\020)2868 31271 y Fu(~)-638 b Fn({)3693 30831 y Fp(\(1\))5219 31271 y Fw(+)6295 30951 y Fu(~)6432 31271 y(b)6952 30831 y Fp(\(2\))8479 31271 y Fw(+)269 b(2)p Fu(\031)43 b(p)11641 30831 y Fp(\(1\))12899 31271 y Fu(=)-67 b(N)14412 31453 y Fp(1)14937 31271 y Fu(a)15578 29925 y Fq(\021)16302 31271 y Fw(,)404 b Fu(p)17653 30831 y Fp(\(1\))19247 31271 y Ft(2)337 b Fu(P)21338 30831 y Fp(\(1\))22596 31271 y Fw(.)p 0 TeXcolorgray 475 34276 a FF(Remark)596 b(4.8.)p 0 TeXcolorgray 617 w Fw(Let)519 b(us)h(estimate)g(the)g(n)-34 b(um)g(b)34 b(er)520 b(of)g(disks)g(in)f Fo(O)31360 33836 y Fp(\(2\))32820 32930 y Fq(\020)33407 33956 y Fu(~)33543 34276 y(b)34063 33836 y Fp(\(2\))35321 32930 y Fq(\021)36565 34276 y Fw(and)h(the)g(size)e(of)i(a)g(connected) 475 36568 y(comp)34 b(onen)-34 b(t)458 b(\000)7481 36128 y Fp(\(2\))8940 35223 y Fq(\020)9528 36248 y Fu(~)9664 36568 y(b)10184 36128 y Fp(\(2\))11442 35223 y Fq(\021)12166 36568 y Fw(.)697 b(Indeed,)470 b(the)458 b(set)f Fo(O)22486 36128 y Fp(\(2\))23945 35223 y Fq(\020)24533 36248 y Fu(~)24669 36568 y(b)25189 36128 y Fp(\(2\))26447 35223 y Fq(\021)27628 36568 y Fw(con)-34 b(tains)458 b(no)g(more)e(than)j(4)p Fu(c)41271 36750 y Fp(0)41796 36568 y Fu(k)42465 36128 y Fp(2+2)p Fs(s)44572 36251 y FD(2)45547 36568 y Fw(disks,)470 b(since)475 38770 y(eac)-34 b(h)311 b Fo(O)3967 38131 y Fp(\(1\))3967 38927 y Fs(s)5535 38770 y Fw(consists)g(of)h(no)f(more) g(than)h Fu(c)18985 38952 y Fp(0)19510 38770 y Fu(k)20179 38330 y Fp(2+2)p Fs(s)22286 38453 y FD(1)23115 38770 y Fw(disks)e(\(Lemma)h(3.46,)329 b(De\014nition)312 b(3.50\),)329 b(and)312 b Fu(P)45606 38330 y Fp(\(1\))47175 38770 y Fw(con)-34 b(tains)475 40577 y(no)477 b(more)f(than)i(4)p Fu(k)9513 40138 y Fp(2\()p Fs(s)10784 40261 y FD(2)11246 40138 y Fr(\000)p Fs(s)12413 40261 y FD(1)12874 40138 y Fp(\))13772 40577 y Fw(elemen)-34 b(ts.)756 b(The)477 b(set)g Fo(O)24804 40138 y Fp(\(2\))26263 39232 y Fq(\020)26851 40258 y Fu(~)26987 40577 y(b)27507 40138 y Fp(\(2\))28765 39232 y Fq(\021)29965 40577 y Fw(is)g(formed)g(b)-34 b(y)477 b(disks)f(of)h(the)g(radius)g Fu(r)48158 40138 y Fr(0)48468 40577 y Fw(,)495 b(here)475 42779 y Fu(r)1056 42339 y Fr(0)1809 42779 y Fw(=)444 b Fu(r)34 b(k)4446 42339 y Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)7285 42462 y FD(2)7747 42339 y Fr(\000)p Fs(\016)8983 42779 y Fw(,)484 b Fu(r)502 b Fw(b)34 b(eing)468 b(the)h(radius)g(of)f(disks)h (constituting)g Fo(O)32227 42140 y Fp(\(1\))32227 42968 y Fr(\003)33485 42779 y Fw(,)484 b Fu(r)477 b Fw(=)443 b Fu(k)37385 42339 y Fr(\000)p Fp(4)p Fr(\000)p Fp(6)p Fs(s)40224 42462 y FD(1)40686 42339 y Fr(\000)p Fp(3)p Fs(\016)42394 42779 y Fw(.)731 b(Hence,)483 b(the)469 b(size)475 44519 y(of)497 b(eac)-34 b(h)497 b(connected)h(comp)34 b(onen)-34 b(t)498 b(of)f Fo(O)19137 44080 y Fp(\(2\))20597 43174 y Fq(\020)21185 44200 y Fu(~)21321 44519 y(b)21841 44080 y Fp(\(2\))23099 43174 y Fq(\021)24319 44519 y Fw(do)34 b(es)497 b(not)h(exceed)e(8)p Fu(c)34490 44701 y Fp(0)35016 44519 y Fu(r)35597 44080 y Fr(0)35907 44519 y Fu(k)36576 44080 y Fp(2+2)p Fs(s)38683 44203 y FD(2)39692 44519 y Fw(=)491 b(8)p Fu(c)42257 44701 y Fp(0)42783 44519 y Fu(r)34 b(k)44033 44080 y Fr(\000)p Fs(\016)f Fr(\000)p Fp(2)p Fs(s)46851 44203 y FD(2)47368 44519 y Fw(.)817 b(Th)-34 b(us,)475 46812 y(the)427 b(size)f(of)h(\000)7065 46372 y Fp(\(2\))8525 45466 y Fq(\020)9112 46492 y Fu(~)9249 46812 y(b)9769 46372 y Fp(\(2\))11026 45466 y Fq(\021)12177 46812 y Fw(is)g(m)-34 b(uc)g(h)428 b(smaller)e(than)i(the)f(radius)g (of)h(disks)e(constituting)j Fo(O)41570 46173 y Fp(\(1\))41570 47000 y Fr(\003)42827 46812 y Fw(.)607 b(Here,)432 b(ev)-34 b(en)426 b(if)h(a)475 49104 y(comp)34 b(onen)-34 b(t)407 b(\000)7430 48665 y Fp(\(2\))8890 47759 y Fq(\020)9477 48785 y Fu(~)9613 49104 y(b)10133 48665 y Fp(\(2\))11391 47759 y Fq(\021)12522 49104 y Fw(is)f(not)h(strictly)f(inside)g Fu(\010)24305 49286 y Fp(2)24831 49104 y Fw(,)g(it)h(has)g(the)g(same)f (prop)34 b(erties)406 b(as)g(a)h(comp)34 b(onen)-34 b(t)407 b(inside)475 51397 y Fu(\010)1283 51579 y Fp(2)1809 51397 y Fw(.)538 b(F)-101 b(urther,)404 b(w)-34 b(e)405 b(consider)f(\000) 14688 50957 y Fp(\(2\))16147 50052 y Fq(\020)16735 51077 y Fu(~)16871 51397 y(b)17391 50957 y Fp(\(2\))18649 50052 y Fq(\021)19709 51397 y Ft(\032)337 b Fu(\010)21797 51579 y Fp(2)22323 51397 y Fw(,)404 b Fo(O)23922 50957 y 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b Fw(+)22048 68447 y(~)21729 68753 y Fu(A)22638 68935 y Fp(1)23365 67771 y Fq(\000)23911 68753 y Fu(~)-596 b(y)24558 68313 y Fp(\(1\))25816 68753 y Fw(\()p Fu(')p Fw(\))27551 67771 y Fq(\001)28108 67408 y(\021)29265 68753 y FC(in)434 b Fu(\010)31560 68935 y Fp(1)32520 68753 y FC(do)-62 b(es)432 b(not)h(exc)-62 b(e)g(e)g(d)432 b Fw(4)p Fu(c)42112 68935 y Fp(0)42637 68753 y Fu(k)43306 68313 y Fp(2+2)p Fs(s)45413 68436 y FD(2)45931 68753 y FC(.)p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 46 46 TeXDict begin 46 45 bop 0 TeXcolorgray 0 TeXcolorgray 475 3345 a Fp(46)14248 b(YULIA)313 b(KARPESHINA)f(AND)h(YOUNG-RAN)g (LEE)p 0 TeXcolorgray 0 TeXcolorgray 475 6288 a FF(Corollary)483 b(4.11.)p 0 TeXcolorgray 554 w FC(Ther)-62 b(e)433 b(ar)-62 b(e)433 b(no)g(zer)-62 b(os)433 b(of)f Fw(det)23553 4943 y Fq(\020)24277 6288 y Fu(I)364 b Fw(+)26706 5982 y(~)26386 6288 y Fu(A)27295 6470 y Fp(1)28023 5307 y Fq(\000)28569 6288 y Fu(~)-596 b(y)29216 5849 y Fp(\(1\))30474 6288 y Fw(\()p Fu(')p Fw(\))32209 5307 y Fq(\001)32765 4943 y(\021)33923 6288 y FC(in)434 b Fu(\010)36218 6470 y Fp(1)37013 6288 y Ft(n)270 b Fo(O)38747 5849 y Fp(\(2\))40206 4943 y Fq(\020)40794 5969 y Fu(~)40930 6288 y(b)41450 5849 y Fp(\(2\))42708 4943 y Fq(\021)43432 6288 y FC(.)p 0 TeXcolorgray 475 9119 a(Pr)-62 b(o)g(of.)p 0 TeXcolorgray 0 TeXcolorgray 2487 w Fw(\(1\))p 0 TeXcolorgray 554 w(Considering)405 b(the)f(relation)g(b)34 b(et)-34 b(w)g(een)405 b Fu(H)27255 8679 y Fp(\(1\))28917 9119 y Fw(and)31591 8813 y(~)31274 9119 y Fu(H)32380 8679 y Fp(\(1\))33637 9119 y Fw(,)f(w)-34 b(e)405 b(easily)e(see)4929 10426 y Fq(\015)4929 11154 y(\015)4929 11881 y(\015)5602 10475 y(\020)6644 11514 y Fw(~)6326 11820 y Fu(H)7432 11320 y Fp(\(1\))8690 10839 y Fq(\000)9236 11820 y Fu(~)-596 b(y)9883 11320 y Fp(\(1\))11141 11820 y Fw(\()p Fu(')p Fw(\))12876 10839 y Fq(\001)13702 11820 y Ft(\000)269 b Fu(k)15583 11320 y Fp(2)p Fs(l)16400 10475 y Fq(\021)17124 10777 y Fr(\000)p Fp(1)18382 10426 y Fq(\015)18382 11154 y(\015)18382 11881 y(\015)19392 11820 y Fw(=)1223 b(max)20671 12782 y Fs(p)21145 12521 y FD(\(1\))22246 12782 y Fr(2)p Fs(P)23601 12521 y FD(\(1\))24701 10426 y Fq(\015)24701 11154 y(\015)24701 11881 y(\015)25374 10475 y(\020)26098 11820 y Fu(H)27204 11320 y Fp(\(1\))28664 10475 y Fq(\020)29378 11820 y Fu(~)-596 b(y)30025 11320 y Fp(\(1\))31283 11820 y Fw(\()p Fu(')p Fw(\))270 b(+)f(2)p Fu(\031)43 b(p)36450 11320 y Fp(\(1\))37708 11820 y Fu(=)-67 b(N)39221 12002 y Fp(1)39747 11820 y Fu(a)40388 10475 y Fq(\021)41381 11820 y Ft(\000)269 b Fu(k)43262 11320 y Fp(2)p Fs(l)44079 10475 y Fq(\021)44803 10777 y Fr(\000)p Fp(1)46061 10426 y Fq(\015)46061 11154 y(\015)46061 11881 y(\015)46734 11820 y Fu(:)4511 15253 y Fw(The)560 b(condition)f Fu(')595 b Ft(2)g Fu(\010)16231 15435 y Fp(1)17129 15253 y Ft(n)373 b Fo(O)18966 14813 y Fp(\(2\))20425 13908 y Fq(\020)21013 14933 y Fu(~)21149 15253 y(b)21669 14813 y Fp(\(2\))22927 13908 y Fq(\021)24209 15253 y Fw(yields)559 b Fu(')594 b Ft(2)h Fu(\010)31370 15435 y Fp(1)32268 15253 y Ft(n)373 b Fo(O)34105 14614 y Fp(\(1\))34105 15411 y Fs(s)35564 13908 y Fq(\020)36152 14933 y Fu(~)36288 15253 y(b)36808 14813 y Fp(\(2\))38335 15253 y Fw(+)269 b(2)p Fu(\031)43 b(p)41497 14813 y Fp(\(1\))42755 15253 y Fu(=)-67 b(N)44268 15435 y Fp(1)44794 15253 y Fu(a)45435 13908 y Fq(\021)46717 15253 y Fw(for)559 b(ev)-34 b(ery)4511 17338 y Fu(p)5121 16898 y Fp(\(1\))6866 17338 y Ft(2)487 b Fu(P)9107 16898 y Fp(\(1\))10365 17338 y Fw(.)809 b(If)12677 17018 y Fu(~)12813 17338 y(b)13333 16898 y Fp(\(2\))14921 17338 y Fw(+)329 b(2)p Fu(\031)43 b(p)18143 16898 y Fp(\(1\))19401 17338 y Fu(=)-67 b(N)20914 17520 y Fp(1)21439 17338 y Fu(a)494 b Fw(satis\014es)h Fu(b)27649 17520 y Fp(0)28662 17338 y Ft(\025)487 b Fu(k)30761 16898 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+9+12)p Fs(s)35564 17021 y FD(1)36026 16898 y Fp(+7)p Fs(\016)37733 17338 y Fw(,)517 b(where)494 b Fu(b)42702 17520 y Fp(0)43722 17338 y Fw(is)g(the)h(distance)4511 18937 y(from)7164 18617 y Fu(~)7301 18937 y(b)7821 18497 y Fp(\(2\))9197 18937 y Fw(+)118 b(2)p Fu(\031)43 b(p)12208 18497 y Fp(\(1\))13466 18937 y Fu(=)-67 b(N)14979 19119 y Fp(1)15505 18937 y Fu(a)328 b Fw(to)h(the)g(nearest)g(v)-34 b(ertex)328 b(of)g Fu(K)29933 19119 y Fp(1)30460 18937 y Fw(,)343 b(then)329 b(w)-34 b(e)329 b(use)g(Lemma)f(3.46)h(and)g (Corollary)4511 20376 y(3.47:)11915 20877 y Fq(\015)11915 21604 y(\015)11915 22331 y(\015)12589 20925 y(\020)13313 22271 y Fu(H)14419 21770 y Fp(\(1\))15878 20925 y Fq(\020)16592 22271 y Fu(~)-596 b(y)17239 21770 y Fp(\(1\))18498 22271 y Fw(\()p Fu(')p Fw(\))270 b(+)f(2)p Fu(\031)43 b(p)23665 21770 y Fp(\(1\))24923 22271 y Fu(=)-67 b(N)26436 22453 y Fp(1)26961 22271 y Fu(a)27602 20925 y Fq(\021)28595 22271 y Ft(\000)269 b Fu(k)30476 21770 y Fp(2)p Fs(l)31294 20925 y Fq(\021)32018 21227 y Fr(\000)p Fp(1)33275 20877 y Fq(\015)33275 21604 y(\015)33275 22331 y(\015)34285 22271 y Fu(<)35904 21451 y Fw(1)p 35698 21992 1018 49 v 35698 23102 a Fu(\017)36190 23284 y Fp(1)37185 22271 y Fu(<)38803 21451 y Fw(1)p 38597 21992 V 38597 23125 a Fu(\017)39089 22708 y Fp(2)39089 23473 y(1)39748 22271 y Fu(:)8343 b Fw(\(4.21\))4511 25191 y(Supp)34 b(ose)9319 24871 y Fu(~)9455 25191 y(b)9975 24751 y Fp(\(2\))11629 25191 y Fw(+)395 b(2)p Fu(\031)43 b(p)14917 24751 y Fp(\(1\))16175 25191 y Fu(=)-67 b(N)17688 25373 y Fp(1)18214 25191 y Fu(a)593 b Fw(satis\014es)h Fu(b)24622 25373 y Fp(0)25800 25191 y Fu(<)653 b(k)28065 24751 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+9+12)p Fs(s)32868 24874 y FD(1)33330 24751 y Fp(+7)p Fs(\016)35037 25191 y Fw(.)1107 b(It)594 b(is)f(easy)g(to)h(see)f(that) i Fu(b)49404 25373 y Fp(0)50582 25191 y Fu(>)4511 26907 y(\017)5003 27089 y Fp(1)5529 26907 y Fu(k)6198 26467 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fp(2)p Fs(\016)10602 26907 y Fw(,)513 b(since)492 b Fu(b)15030 26268 y Fp(\(2\))15030 27254 y(0)16771 26907 y Fu(>)484 b(\017)18690 27089 y Fp(1)19215 26907 y Fu(k)19884 26467 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fp(2)p Fs(\016)24780 26907 y Fw(b)-34 b(y)492 b(the)h(h)-34 b(yp)34 b(othesis)493 b(of)g(the)f(lemma.)803 b(Using)492 b(Corollary)4511 28346 y(3.53)418 b(to)f(obtain)h(an)g(estimate)f (similar)g(to)g(\(4.21\).)579 b(Th)-34 b(us,)421 b(\(4.19\))d(is)f(pro) -34 b(v)g(ed.)579 b(The)417 b(estimate)h(\(4.20\))4511 29785 y(can)436 b(b)34 b(e)434 b(c)-34 b(hec)g(k)g(ed)436 b(in)f(the)g(analogous)h(w)-34 b(a)g(y)436 b(from)f(\(3.87\))h(and)g (\(3.99\),)443 b(when)436 b(w)-34 b(e)435 b(tak)-34 b(e)435 b(in)-34 b(to)436 b(accoun)-34 b(t)4511 31303 y(that)406 b Fu(P)8084 30863 y Fp(\(1\))9746 31303 y Fw(con)-34 b(tains)405 b(less)f(than)h(4)p Fu(k)20839 30863 y Fp(2)p Fs(s)21744 30986 y FD(2)22206 30863 y Fr(\000)p Fp(2)p Fs(s)23843 30986 y FD(1)24764 31303 y Fw(elemen)-34 b(ts.)p 0 TeXcolorgray 2409 32742 a(\(2\))p 0 TeXcolorgray 554 w(The)527 b(determinan)-34 b(t)527 b(of)f Fu(I)445 b Fw(+)18178 32436 y(~)17858 32742 y Fu(A)18767 32924 y Fp(1)19495 31760 y Fq(\000)20041 32742 y Fu(~)-596 b(y)20688 32302 y Fp(\(1\))21946 32742 y Fw(\()p Fu(')p Fw(\))23681 31760 y Fq(\001)24763 32742 y Fw(is)525 b(equal)h(to)g(zero)f(if)g(and) i(only)e(if)h(at)g(least)g(one)g(of)g(the)4511 34671 y(determinan)-34 b(ts)356 b(det)13462 33326 y Fq(\020)14186 34671 y Fu(I)264 b Fw(+)170 b Fu(A)17005 34853 y Fp(1)17733 33689 y Fq(\000)18278 34671 y Fu(~)-596 b(y)18925 34231 y Fp(\(1\))20184 34671 y Fw(\()p Fu(')p Fw(\))270 b(+)f(2)p Fu(\031)43 b(p)25351 34231 y Fp(\(1\))26609 34671 y Fu(=)-67 b(N)28122 34853 y Fp(1)28647 34671 y Fu(a)29288 33689 y Fq(\001)29843 33326 y(\021)30567 34671 y Fw(,)364 b Fu(p)31878 34231 y Fp(\(1\))33472 34671 y Ft(2)337 b Fu(P)35563 34231 y Fp(\(1\))36821 34671 y Fw(,)364 b(is)354 b(zero.)521 b(Since)355 b(eac)-34 b(h)354 b(of)h(them)4511 36674 y(has)490 b(no)f(more)f(than)i Fu(c)15082 36856 y Fp(0)15608 36674 y Fu(k)16277 36234 y Fp(2+2)p Fs(s)18384 36357 y FD(1)19390 36674 y Fw(zeros)e(\(Corollary)g(3.36,)510 b(De\014nition)489 b(3.50\))g(and)h Fu(P)43660 36234 y Fp(\(1\))45407 36674 y Fw(con)-34 b(tains)490 b(no)4511 38481 y(more)471 b(than)i(4)p Fu(k)11783 38042 y Fp(2\()p Fs(s)13054 38165 y FD(2)13516 38042 y Fr(\000)p Fs(s)14683 38165 y FD(1)15144 38042 y Fp(\))16036 38481 y Fw(elemen)-34 b(ts,)488 b(the)472 b(total)g(n)-34 b(um)g(b)34 b(er)472 b(of)g(zeros)e(of)i(det)38698 37136 y Fq(\020)39422 38481 y Fu(I)409 b Fw(+)41941 38175 y(~)41621 38481 y Fu(A)42530 38663 y Fp(1)43258 37500 y Fq(\000)43804 38481 y Fu(~)-596 b(y)44451 38042 y Fp(\(1\))45709 38481 y Fw(\()p Fu(')p Fw(\))47444 37500 y Fq(\001)48000 37136 y(\021)49195 38481 y Fw(do)34 b(es)4511 40391 y(not)405 b(exceed)e(4)p Fu(c)11669 40573 y Fp(0)12195 40391 y Fu(k)12864 39951 y Fp(2+2)p Fs(s)14971 40074 y FD(2)15488 40391 y Fw(.)p 50750 41996 665 831 v 1803 44156 a(The)i(follo)-34 b(wing)405 b(\014v)-34 b(e)404 b(lemmas)g(are)f(the)i(analogs)f(of)h(Lemmas)e (3.29)h({)g(3.38.)p 0 TeXcolorgray 475 46622 a FF(Lemma)463 b(4.12.)p 0 TeXcolorgray 554 w FC(If)9973 46302 y Fu(~)10109 46622 y(b)10629 46182 y Fp(\(2\))12224 46622 y Ft(2)336 b Fu(K)14397 46804 y Fp(2)15357 46622 y FC(and)433 b Fu(b)18231 45983 y Fp(\(2\))18231 46969 y(0)19826 46622 y Fu(>)336 b(\017)21597 46804 y Fp(1)22123 46622 y Fu(k)22792 46182 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fp(2)p Fs(\016)27196 46622 y FC(,)433 b(then:)p 0 TeXcolorgray 2409 48811 a Fw(\(1\))p 0 TeXcolorgray 554 w FC(The)546 b(numb)-62 b(er)546 b(of)f(zer)-62 b(os)544 b(of)i Fw(det)19457 47465 y Fq(\020)20181 48811 y Fu(I)446 b Fw(+)22773 48504 y(~)22453 48811 y Fu(A)23362 48993 y Fp(1)23888 47829 y Fq(\000)24434 48811 y Fu(~)-596 b(y)25081 48371 y Fp(\(2\))26339 48811 y Fw(\()p Fu(')p Fw(\))28074 47829 y Fq(\001)28630 47465 y(\021)29900 48811 y FC(inside)545 b Fw(\000)34302 48371 y Fp(\(2\))35761 47465 y Fq(\020)36349 48491 y Fu(~)36485 48811 y(b)37005 48371 y Fp(\(2\))38263 47465 y Fq(\021)39533 48811 y FC(is)g(the)g(same)g(as)h(that)e(of)4511 51103 y Fw(det)6195 49758 y Fq(\020)6919 51103 y Fu(I)364 b Fw(+)9348 50797 y(~)9028 51103 y Fu(A)9937 51285 y Fp(1)10463 50121 y Fq(\000)11009 51103 y Fu(~)-596 b(y)11656 50663 y Fp(\(1\))12914 51103 y Fw(\()p Fu(')p Fw(\))14649 50121 y Fq(\001)15205 49758 y(\021)15929 51103 y Fu(:)p 0 TeXcolorgray 2409 53396 a Fw(\(2\))p 0 TeXcolorgray 554 w FC(F)-93 b(or)434 b(any)f Fu(')337 b Ft(2)g Fu(\010)12180 53578 y Fp(2)12975 53396 y Ft(n)269 b Fo(O)14708 52956 y Fp(\(2\))16168 52050 y Fq(\020)16755 53076 y Fu(~)16892 53396 y(b)17412 52956 y Fp(\(2\))18669 52050 y Fq(\021)19393 53396 y FC(,)434 b(the)e(fol)62 b(lowing)432 b(ine)-62 b(qualities)430 b(hold:)17049 55188 y Fq(\015)17049 55915 y(\015)17049 56642 y(\015)17723 55236 y(\020)18764 56275 y Fw(~)18447 56582 y Fu(H)19553 56081 y Fp(\(1\))21012 55236 y Fq(\020)21726 56582 y Fu(~)-596 b(y)22373 56081 y Fp(\(2\))23632 56582 y Fw(\()p Fu(')p Fw(\))25367 55236 y Fq(\021)26361 56582 y Ft(\000)269 b Fu(k)28242 56081 y Fp(2)p Fs(l)29059 55236 y Fq(\021)29783 55538 y Fr(\000)p Fp(1)31041 55188 y Fq(\015)31041 55915 y(\015)31041 56642 y(\015)32051 56582 y Fu(<)33669 55762 y Fw(2)p 33463 56303 1018 49 v 33463 57437 a Fu(\017)33955 57019 y Fp(2)33955 57784 y(1)34614 56582 y Fu(;)13477 b Fw(\(4.22\))15174 59174 y Fq(\015)15174 59901 y(\015)15174 60628 y(\015)15847 59222 y(\020)16889 60261 y Fw(~)16571 60568 y Fu(H)17677 60067 y Fp(\(1\))18935 60568 y Fw(\()-10 b Fu(~)-596 b(y)20043 60067 y Fp(\(2\))21302 60568 y Fw(\()p Fu(')p Fw(\)\))270 b Ft(\000)f Fu(k)25659 60067 y Fp(2)p Fs(l)26477 59222 y Fq(\021)27201 59524 y Fr(\000)p Fp(1)28458 59174 y Fq(\015)28458 59901 y(\015)28458 60628 y(\015)29132 61411 y Fp(1)29994 60568 y Fu(<)31406 59748 y Fw(8)p Fu(c)32537 59930 y Fp(0)33063 59748 y Fu(k)33732 59308 y Fp(2+2)p Fs(s)35839 59431 y FD(2)p 31406 60289 4951 49 v 33373 61423 a Fu(\017)33865 61005 y Fp(2)33865 61770 y(1)36489 60568 y Fu(:)11602 b Fw(\(4.23\))p 0 TeXcolorgray 475 64091 a FF(Corollary)483 b(4.13.)p 0 TeXcolorgray 554 w FC(Ther)-62 b(e)433 b(ar)-62 b(e)433 b(no)g(zer)-62 b(os)433 b(of)f Fw(det)23553 62745 y Fq(\020)24277 64091 y Fu(I)364 b Fw(+)26706 63784 y(~)26386 64091 y Fu(A)27295 64273 y Fp(1)28023 63109 y Fq(\000)28569 64091 y Fu(~)-596 b(y)29216 63651 y Fp(\(2\))30474 64091 y Fw(\()p Fu(')p Fw(\))32209 63109 y Fq(\001)32765 62745 y(\021)33923 64091 y FC(in)434 b Fu(\010)36218 64273 y Fp(2)37013 64091 y Ft(n)270 b Fo(O)38747 63651 y Fp(\(2\))40206 62745 y Fq(\020)40794 63771 y Fu(~)40930 64091 y(b)41450 63651 y Fp(\(2\))42708 62745 y Fq(\021)43432 64091 y FC(.)p 0 TeXcolorgray 475 66715 a(Pr)-62 b(o)g(of.)p 0 TeXcolorgray 0 TeXcolorgray 2487 w Fw(\(1\))p 0 TeXcolorgray 554 w(Direct)403 b(calculation)h(giv)-34 b(es:)11943 69196 y(det)13626 67850 y Fq(\020)14350 69196 y Fu(I)364 b Fw(+)16779 68889 y(~)16459 69196 y Fu(A)17368 69378 y Fp(1)17894 68214 y Fq(\000)18440 69196 y Fu(~)-596 b(y)19087 68695 y Fp(\(2\))20345 68214 y Fq(\001)20901 67850 y(\021.)22810 69196 y Fw(det)24493 67850 y Fq(\020)25217 69196 y Fu(I)365 b Fw(+)27647 68889 y(~)27327 69196 y Fu(A)28236 69378 y Fp(1)28762 68214 y Fq(\000)29307 69196 y Fu(~)-596 b(y)29954 68695 y Fp(\(1\))31213 68214 y Fq(\001)31768 67850 y(\021)32829 69196 y Fw(=)336 b(det)q(\()p Fu(I)364 b Fw(+)270 b Fu(T)168 b Fw(\))p Fu(:)p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 47 47 TeXDict begin 47 46 bop 0 TeXcolorgray 0 TeXcolorgray 2848 3345 a Fp(SPECTRAL)406 b(PR)-26 b(OPER)-78 b(TIES)406 b(OF)h(A)h(POL)-78 b(YHARMONIC)405 b(OPERA)-78 b(TOR)406 b(WITH)h(LIMIT-PERIODIC)e(POTENTIAL)1430 b(47)p 0 TeXcolorgray 5775 6509 a Fu(T)505 b Fw(=)8268 5163 y Fq(\020)8992 6509 y Fu(W)10137 6691 y Fp(1)10932 6509 y Ft(\000)269 b Fw(2)p Fu(\025)13457 5163 y Fq(\021\020)15222 6202 y Fw(~)14905 6509 y Fu(H)15913 6691 y Fp(0)16438 5527 y Fq(\000)16984 6509 y Fu(~)-596 b(y)17631 6008 y Fp(\(2\))18889 5527 y Fq(\001)19714 6509 y Fw(+)269 b Fu(\025)21633 5163 y Fq(\021)22357 5465 y Fr(\000)p Fp(1)23615 5163 y Fq(\020)24656 6202 y Fw(~)24339 6509 y Fu(H)25347 6691 y Fp(0)25872 5527 y Fq(\000)26418 6509 y Fu(~)-596 b(y)27065 6008 y Fp(\(1\))28323 5527 y Fq(\001)29148 6509 y Ft(\000)30677 6202 y Fw(~)30360 6509 y Fu(H)31368 6691 y Fp(0)31894 5527 y Fq(\000)32439 6509 y Fu(~)g(y)33086 6008 y Fp(\(2\))34345 5527 y Fq(\001)34900 5163 y(\021\020)36665 6202 y Fw(~)36348 6509 y Fu(H)37454 6008 y Fp(\(1\))38712 5527 y Fq(\000)39257 6509 y Fu(~)g(y)39904 6008 y Fp(\(1\))41163 5527 y Fq(\001)41987 6509 y Ft(\000)270 b Fu(\025)43907 5163 y Fq(\021)44631 5465 y Fr(\000)p Fp(1)45888 6509 y Fu(:)4511 8828 y Fw(The)429 b(estimate)e(\(3.63\))i(yields:)19356 7797 y Fq(\014)19356 8525 y(\014)19750 8828 y Fu(~)-596 b(y)20397 8388 y Fp(\(2\))21656 8828 y Fw(\()p Fu(')p Fw(\))286 b Ft(\000)274 b Fu(~)-596 b(y)25541 8388 y Fp(\(1\))26800 8828 y Fw(\()p Fu(')p Fw(\))28535 7797 y Fq(\014)28535 8525 y(\014)29315 8828 y Fu(<)376 b Fw(2)p Fu(\013)t(\017)32511 8388 y Fp(4)32511 9152 y(1)33038 8828 y Fu(k)33707 8388 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)36458 8828 y Fw(.)609 b(Using)428 b(the)g(last)g (estimate)g(w)-34 b(e)4511 10267 y(readily)404 b(sho)-34 b(w)405 b(that)11125 11402 y Fq(\015)11125 12129 y(\015)11125 12856 y(\015)11798 11450 y(\020)12840 12489 y Fw(~)12522 12796 y Fu(H)13530 12978 y Fp(0)14056 11814 y Fq(\000)14601 12796 y Fu(~)-596 b(y)15248 12295 y Fp(\(2\))16507 11814 y Fq(\001)17331 12796 y Fw(+)270 b Fu(\025)19251 11450 y Fq(\021)19975 11752 y Fr(\000)p Fp(1)21232 11450 y Fq(\020)22273 12489 y Fw(~)21956 12796 y Fu(H)22964 12978 y Fp(0)23490 11814 y Fq(\000)24035 12796 y Fu(~)-596 b(y)24682 12295 y Fp(\(1\))25941 11814 y Fq(\001)26765 12796 y Ft(\000)28295 12489 y Fw(~)27978 12796 y Fu(H)28986 12978 y Fp(0)29511 11814 y Fq(\000)30056 12796 y Fu(~)g(y)30703 12295 y Fp(\(2\))31962 11814 y Fq(\001)32517 11450 y(\021)33241 11402 y(\015)33241 12129 y(\015)33241 12856 y(\015)34251 12796 y Fu(<)337 b Fw(2)p Fu(l)24 b(\013)t(\017)37794 12295 y Fp(4)37794 13095 y(1)38320 12796 y Fu(k)38989 12295 y Fr(\000)p Fp(2)p Fs(l)40538 12796 y Fu(:)7553 b Fw(\(4.24\))4511 15172 y(Considering)405 b(\(4.20\),)g(w)-34 b(e)404 b(obtain)10389 17548 y Ft(k)p Fu(T)168 b Ft(k)12477 17730 y Fp(1)13340 17548 y Ft(\024)337 b Fw(4)p Fu(k)15895 17047 y Fp(2)p Fs(l)16914 16202 y Fq(\020)17638 17548 y Fw(4)p Fu(l)24 b(\017)19122 17047 y Fp(4)19122 17847 y(1)19647 17548 y Fu(k)20316 17047 y Fr(\000)p Fp(2)p Fs(l)21866 16202 y Fq(\021)22792 16566 y(\000)23347 17548 y Fw(4)p Fu(c)24478 17730 y Fp(0)25004 17548 y Fu(k)25673 17047 y Fp(2+2)p Fs(s)27780 17170 y FD(2)28297 17548 y Fu(=\017)29395 17047 y Fp(2)29395 17847 y(1)29921 16566 y Fq(\001)30813 17548 y Ft(\024)337 b Fw(64)p Fu(l)24 b(c)34216 17730 y Fp(0)34741 17548 y Fu(\017)35233 17047 y Fp(2)35233 17847 y(1)35759 17548 y Fu(k)36428 17047 y Fp(2+2)p Fs(s)38535 17170 y FD(2)39389 17548 y Fu(<)336 b Fw(1)p Fu(:)6817 b Fw(\(4.25\))4511 20408 y(Hence,)530 b Ft(j)202 b Fw(det\()p Fu(I)432 b Fw(+)336 b Fu(T)168 b Fw(\))338 b Ft(\000)e Fw(1)p Ft(j)505 b Fu(<)f Fw(1)h(for)g(an)-34 b(y)505 b Fu(')g Ft(2)g Fu(\015)28136 19969 y Fp(\(2\))29595 19063 y Fq(\020)30183 20089 y Fu(~)30319 20408 y(b)30839 19969 y Fp(\(2\))32097 19063 y Fq(\021)32821 20408 y Fw(.)841 b(Applying)505 b(Rouc)-34 b(h)g(\023)-572 b(e's)506 b(Theorem,)530 b(w)-34 b(e)4511 22149 y(complete)404 b(the)h(pro)34 b(of.)p 0 TeXcolorgray 2409 23588 a(\(2\))p 0 TeXcolorgray 554 w(Let)404 b(us)h(consider)f(the)g(Hilb)34 b(ert)404 b(relation:)5666 24839 y Fq(\020)6707 25878 y Fw(~)6390 26184 y Fu(H)7496 25684 y Fp(\(1\))8753 25202 y Fq(\000)9299 26184 y Fu(~)-596 b(y)9946 25684 y Fp(\(2\))11204 26184 y Fw(\()p Fu(')p Fw(\))12939 25202 y Fq(\001)13765 26184 y Ft(\000)269 b Fu(\025)15684 24839 y Fq(\021)16408 25141 y Fr(\000)p Fp(1)18002 26184 y Fw(=)19282 24839 y Fq(\020)20323 25878 y Fw(~)20006 26184 y Fu(H)21112 25684 y Fp(\(1\))22370 25202 y Fq(\000)22915 26184 y Fu(~)-596 b(y)23562 25684 y Fp(\(1\))24821 26184 y Fw(\()p Fu(')p Fw(\))26556 25202 y Fq(\001)27381 26184 y Ft(\000)269 b Fu(\025)29300 24839 y Fq(\021)30024 25141 y Fr(\000)p Fp(1)31551 26184 y Fw(+)g Fu(T)33471 26366 y Fr(\003)33998 24839 y Fq(\020)35039 25878 y Fw(~)34721 26184 y Fu(H)35827 25684 y Fp(\(1\))37085 25202 y Fq(\000)37631 26184 y Fu(~)-596 b(y)38278 25684 y Fp(\(2\))39536 26184 y Fw(\()p Fu(')p Fw(\))41271 25202 y Fq(\001)42097 26184 y Ft(\000)269 b Fu(\025)44016 24839 y Fq(\021)44740 25141 y Fr(\000)p Fp(1)45997 26184 y Fu(;)10666 29330 y(T)11374 29512 y Fr(\003)12237 29330 y Fw(=)13517 27984 y Fq(\020)14558 29023 y Fw(~)14240 29330 y Fu(H)15346 28829 y Fp(\(1\))16604 28348 y Fq(\000)17150 29330 y Fu(~)-596 b(y)17797 28829 y Fp(\(1\))19055 29330 y Fw(\()p Fu(')p Fw(\))20790 28348 y Fq(\001)21616 29330 y Ft(\000)269 b Fu(\025)23535 27984 y Fq(\021)24259 28286 y Fr(\000)p Fp(1)25517 27984 y Fq(\020)26558 29023 y Fw(~)26240 29330 y Fu(H)27248 29512 y Fp(0)27774 28348 y Fq(\000)28319 29330 y Fu(~)-596 b(y)28966 28829 y Fp(\(1\))30225 29330 y Fw(\()p Fu(')p Fw(\))31960 28348 y Fq(\001)32785 29330 y Ft(\000)34315 29023 y Fw(~)33998 29330 y Fu(H)35006 29512 y Fp(0)35531 28348 y Fq(\000)36076 29330 y Fu(~)g(y)36723 28829 y Fp(\(2\))37982 29330 y Fw(\()p Fu(')p Fw(\))39717 28348 y Fq(\001)40273 27984 y(\021)40997 29330 y Fu(:)4511 31454 y Fw(T)-101 b(o)405 b(estimate)f Ft(k)p Fu(T)12465 31636 y Fr(\003)12991 31454 y Ft(k)p Fw(,)g(w)-34 b(e)405 b(represen)-34 b(t)404 b Fu(T)22063 31636 y Fr(\003)22993 31454 y Fw(in)g(the)h(form)f Fu(T)30068 31636 y Fr(\003)30931 31454 y Fw(=)337 b(\()p Fu(I)364 b Fw(+)269 b Fu(T)35499 31636 y Fp(1)36025 31454 y Fw(\))p Fu(T)37204 31636 y Fp(2)37731 31454 y Fw(,)404 b(here)14979 34050 y Fu(T)15687 34232 y Fp(1)16549 34050 y Fw(=)17829 32705 y Fq(\020)18870 33744 y Fw(~)18553 34050 y Fu(H)19659 33550 y Fp(\(1\))20917 33068 y Fq(\000)21462 34050 y Fu(~)-596 b(y)22109 33550 y Fp(\(1\))23368 34050 y Fw(\()p Fu(')p Fw(\))25103 33068 y Fq(\001)25928 34050 y Ft(\000)269 b Fu(\025)27847 32705 y Fq(\021)28571 33007 y Fr(\000)p Fp(1)29829 32705 y Fq(\020)30553 34050 y Ft(\000)p Fu(W)32641 34232 y Fp(1)33435 34050 y Fw(+)h(2)p Fu(\025)35961 32705 y Fq(\021)36685 34050 y Fu(;)11081 37196 y(T)11789 37378 y Fp(2)12652 37196 y Fw(=)13932 35850 y Fq(\020)14973 36889 y Fw(~)14656 37196 y Fu(H)15664 37378 y Fp(0)16189 36214 y Fq(\000)16735 37196 y Fu(~)-596 b(y)17382 36695 y Fp(\(1\))18640 37196 y Fw(\()p Fu(')p Fw(\))20375 36214 y Fq(\001)21201 37196 y Fw(+)269 b Fu(\025)23120 35850 y Fq(\021)23844 36152 y Fr(\000)p Fp(1)25101 35850 y Fq(\020)26143 36889 y Fw(~)25825 37196 y Fu(H)26833 37378 y Fp(0)27359 36214 y Fq(\000)27904 37196 y Fu(~)-596 b(y)28551 36695 y Fp(\(1\))29810 37196 y Fw(\()p Fu(')p Fw(\))31545 36214 y Fq(\001)32370 37196 y Ft(\000)33900 36889 y Fw(~)33582 37196 y Fu(H)34590 37378 y Fp(0)35116 36214 y Fq(\000)35661 37196 y Fu(~)g(y)36308 36695 y Fp(\(2\))37567 37196 y Fw(\()p Fu(')p Fw(\))39302 36214 y Fq(\001)39858 35850 y(\021)40582 37196 y Fu(:)4511 39491 y Fw(Using)309 b(\(4.19\),)328 b(w)-34 b(e)309 b(get)f Ft(k)p Fu(T)16512 39673 y Fp(1)17038 39491 y Ft(k)337 b(\024)f Fw(4)p Fu(\025\017)21065 38982 y Fr(\000)p Fp(2)21065 39838 y(1)22323 39491 y Fw(.)507 b(The)309 b(norm)f(of)h Fu(T)30628 39673 y Fp(2)31462 39491 y Fw(can)g(b)34 b(e)308 b(b)34 b(ounded)309 b(as)g(\(4.24\).)507 b(Multiplying)4511 40960 y(the)563 b(estimates)f(for)g Ft(k)p Fu(T)15575 41142 y Fp(1)16101 40960 y Ft(k)p Fw(,)601 b Ft(k)p Fu(T)18959 41142 y Fp(2)19485 40960 y Ft(k)p Fw(,)g(w)-34 b(e)562 b(obtain:)855 b Ft(k)p Fu(T)28843 41142 y Fr(\003)29369 40960 y Ft(k)599 b Fu(<)h Fw(8)p Fu(\013)t(l)24 b(")34453 40520 y Fp(2)34453 41284 y(1)35579 40960 y Fu(<)599 b Fw(1)p Fu(=)p Fw(2.)1011 b(Substituting)565 b(the)d(last)4511 42399 y(estimate)599 b(in)-34 b(to)599 b(the)g(Hilb)34 b(ert)598 b(relation)g(and)h(considering)f(\(4.19\),)647 b(w)-34 b(e)599 b(arriv)-34 b(e)597 b(at)i(\(4.22\).)1122 b(Using)4511 43838 y(\(4.20\),)405 b(w)-34 b(e)404 b(pro)-34 b(v)g(e)405 b(\(4.23\))f(in)h(the)f(analogous)h(w)-34 b(a)g(y)-101 b(.)p 50750 45443 665 831 v 0 TeXcolorgray 475 47613 a FF(Lemma)587 b(4.14.)p 0 TeXcolorgray 613 w FC(If)10255 47293 y Fu(~)10391 47613 y(b)10911 47173 y Fp(\(2\))12685 47613 y Ft(2)517 b Fu(K)15039 47795 y Fp(2)16098 47613 y FC(and)532 b Fu(b)19071 46974 y Fp(\(2\))19071 47960 y(0)20846 47613 y Fu(>)516 b(\017)22797 47795 y Fp(1)23323 47613 y Fu(k)23992 47173 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fp(2)p Fs(\016)28395 47613 y FC(,)557 b(then)532 b(the)g(fol)62 b(lowing)531 b(estimates)f(hold)i(for)g(any)475 49465 y Fu(')336 b Ft(2)h Fu(\010)3557 49647 y Fp(2)4352 49465 y Ft(n)270 b Fo(O)6086 49025 y Fp(\(2\))7545 48119 y Fq(\020)8133 49145 y Fu(~)8269 49465 y(b)8789 49025 y Fp(\(2\))10047 48119 y Fq(\021)10771 49465 y FC(:)17708 50648 y Fq(\015)17708 51376 y(\015)17708 52103 y(\015)18382 50697 y(\020)19106 52042 y Fu(H)20212 51542 y Fp(\(2\))21469 51060 y Fq(\000)22015 52042 y Fu(~)-596 b(y)22662 51542 y Fp(\(2\))23920 52042 y Fw(\()p Fu(')p Fw(\))25655 51060 y Fq(\001)26481 52042 y Ft(\000)269 b Fu(\025)28400 50697 y Fq(\021)29124 50999 y Fr(\000)p Fp(1)30382 50648 y Fq(\015)30382 51376 y(\015)30382 52103 y(\015)31392 52042 y Fu(<)33010 51222 y Fw(4)p 32804 51763 1018 49 v 32804 52897 a Fu(\017)33296 52480 y Fp(2)33296 53244 y(1)33955 52042 y Fu(;)14136 b Fw(\(4.26\))15176 54173 y Fq(\015)15176 54900 y(\015)15176 55627 y(\015)15850 54221 y(\020)16574 55567 y Fu(H)17680 55066 y Fp(\(2\))18937 54585 y Fq(\000)19483 55567 y Fu(~)-596 b(y)20130 55066 y Fp(\(2\))21388 55567 y Fw(\()p Fu(')p Fw(\))23123 54585 y Fq(\001)23949 55567 y Ft(\000)269 b Fu(\025)25868 54221 y Fq(\021)26592 54523 y Fr(\000)p Fp(1)27850 54173 y Fq(\015)27850 54900 y(\015)27850 55627 y(\015)28523 56410 y Fp(1)29386 55567 y Fu(<)30798 54747 y Fw(16)p Fu(c)32535 54929 y Fp(0)33060 54747 y Fu(k)33729 54307 y Fp(2+2)p Fs(s)35836 54430 y FD(2)p 30798 55288 5557 49 v 33067 56422 a Fu(\017)33559 56004 y Fp(2)33559 56769 y(1)36487 55567 y Fu(:)11604 b Fw(\(4.27\))p 0 TeXcolorgray 475 59030 a FF(Corollary)483 b(4.15.)p 0 TeXcolorgray 554 w FC(Ther)-62 b(e)433 b(ar)-62 b(e)433 b(no)g(zer)-62 b(os)433 b(of)f Fw(det)23553 57684 y Fq(\020)24277 59030 y Fu(I)364 b Fw(+)269 b Fu(A)27295 59212 y Fp(2)28023 58048 y Fq(\000)28569 59030 y Fu(~)-596 b(y)29216 58590 y Fp(\(2\))30474 59030 y Fw(\()p Fu(')p Fw(\))32209 58048 y Fq(\001)32765 57684 y(\021)33923 59030 y FC(in)434 b Fu(\010)36218 59212 y Fp(2)37013 59030 y Ft(n)270 b Fo(O)38747 58590 y Fp(\(2\))40206 57684 y Fq(\020)40794 58710 y Fu(~)40930 59030 y(b)41450 58590 y Fp(\(2\))42708 57684 y Fq(\021)43432 59030 y FC(.)p 0 TeXcolorgray 475 61594 a(Pr)-62 b(o)g(of.)p 0 TeXcolorgray 552 w Fw(The)296 b(pro)34 b(of)295 b(is)g(analogous)h(to)f(that)i(of)e(Lemma)g(3.31;)331 b(w)-34 b(e)295 b(use)g(Hilb)34 b(ert)295 b(relation,)316 b(the)296 b(estimates)f(\(4.22\),)475 63033 y(\(4.23\))423 b(and)g(the)g(inequalit)-34 b(y)422 b Ft(k)p Fu(W)15868 63215 y Fp(2)16394 63033 y Ft(k)367 b Fu(<)g(\017)19169 62593 y Fp(4)19169 63358 y(1)19695 63033 y Fw(,)427 b(whic)-34 b(h)423 b(immediately)f(follo)-34 b(ws)423 b(from)g(\(3.2\))g(and)g (the)g(de\014nition)h(of)475 64472 y Fu(\017)967 64654 y Fp(1)1493 64472 y Fw(.)p 50750 64638 665 831 v 0 TeXcolorgray 475 66824 a FF(Lemma)309 b(4.16.)p 0 TeXcolorgray 443 w FC(If)9584 66504 y Fu(~)9719 66824 y(b)10239 66384 y Fp(\(2\))11834 66824 y Ft(2)337 b Fu(K)14008 67006 y Fp(2)14844 66824 y FC(and)309 b Fu(b)17594 66185 y Fp(\(2\))17594 67171 y(0)19189 66824 y Fu(>)336 b(\017)20960 67006 y Fp(1)21486 66824 y Fu(k)22155 66384 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fp(2)p Fs(\016)26558 66824 y FC(,)335 b(then)309 b(the)f(numb)-62 b(er)309 b(of)g(zer)-62 b(os)309 b(of)g Fw(det)43337 65842 y Fq(\000)43893 66824 y Fu(I)364 b Fw(+)269 b Fu(A)46911 67006 y Fp(2)47437 66824 y Fw(\()-10 b Fu(~)-596 b(y)48545 66384 y Fp(\(2\))49804 66824 y Fw(\()p Fu(')p Fw(\)\))52010 65842 y Fq(\001)475 68753 y FC(inside)432 b Fw(\000)4764 68313 y Fp(\(2\))6021 68753 y Fw(\()6356 68433 y Fu(~)6492 68753 y(b)7012 68313 y Fp(\(2\))8270 68753 y Fw(\))j FC(is)e(the)f(same)h(as)f(that)h(of)f Fw(det)22925 67408 y Fq(\020)23649 68753 y Fu(I)364 b Fw(+)26078 68447 y(~)25758 68753 y Fu(A)26667 68935 y Fp(1)27193 68753 y Fw(\()-10 b Fu(~)-596 b(y)28301 68313 y Fp(\(2\))29560 68753 y Fw(\()p Fu(')p Fw(\)\))31766 67408 y Fq(\021)32491 68753 y FC(.)p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 48 48 TeXDict begin 48 47 bop 0 TeXcolorgray 0 TeXcolorgray 475 3345 a Fp(48)14248 b(YULIA)313 b(KARPESHINA)f(AND)h(YOUNG-RAN)g (LEE)p 0 TeXcolorgray 0 TeXcolorgray 475 6001 a FC(Pr)-62 b(o)g(of.)p 0 TeXcolorgray 552 w Fw(Using)405 b(\(4.23\),)f(w)-34 b(e)404 b(obtain)h(that)h(for)e(an)-34 b(y)405 b Fu(')336 b Ft(2)h Fu(\015)26762 5562 y Fp(\(2\))28019 6001 y Fw(:)5089 7094 y Fq(\014)5089 7821 y(\014)5089 8548 y(\014)5089 9276 y(\014)5089 10003 y(\014)5493 9215 y Fw(det)7379 7142 y Fq( )8471 8318 y Fu(H)9577 7878 y Fp(\(2\))11037 7336 y Fq(\000)11582 8318 y Fu(~)-596 b(y)12229 7878 y Fp(\(2\))13488 8318 y Fw(\()p Fu(')p Fw(\))15223 7336 y Fq(\001)16048 8318 y Ft(\000)269 b Fu(k)17929 7878 y Fp(2)p Fs(l)p 8471 8936 10276 49 v 8788 9891 a Fw(~)8471 10198 y Fu(H)9577 9846 y Fp(\(1\))11037 9216 y Fq(\000)11582 10198 y Fu(~)-596 b(y)12229 9846 y Fp(\(2\))13488 10198 y Fw(\()p Fu(')p Fw(\))15223 9216 y Fq(\001)16048 10198 y Ft(\000)269 b Fu(k)17929 9847 y Fp(2)p Fs(l)18880 7142 y Fq(!)20109 9215 y Ft(\000)g Fw(det)202 b Fu(I)23834 7094 y Fq(\014)23834 7821 y(\014)23834 8548 y(\014)23834 9276 y(\014)23834 10003 y(\014)24575 9215 y Fw(=)25854 7094 y Fq(\014)25854 7821 y(\014)25854 8548 y(\014)25854 9276 y(\014)25854 10003 y(\014)26258 9215 y Fw(det)28144 7142 y Fq( )29104 9215 y Fu(I)364 b Fw(+)35648 8395 y Fu(W)36793 8577 y Fp(2)p 31346 8936 V 31663 9891 a Fw(~)31346 10198 y Fu(H)32452 9846 y Fp(\(1\))33912 9216 y Fq(\000)34457 10198 y Fu(~)-596 b(y)35104 9846 y Fp(\(2\))36362 10198 y Fw(\()p Fu(')p Fw(\))38097 9216 y Fq(\001)38923 10198 y Ft(\000)269 b Fu(k)40804 9847 y Fp(2)p Fs(l)41754 7142 y Fq(!)42983 9215 y Ft(\000)g Fw(det)203 b Fu(I)46709 7094 y Fq(\014)46709 7821 y(\014)46709 8548 y(\014)46709 9276 y(\014)46709 10003 y(\014)5224 13031 y Ft(\024)336 b Fw(2)p Ft(k)p Fu(W)8860 13213 y Fp(2)9386 13031 y Ft(k)9992 11637 y Fq(\015)9992 12364 y(\015)9992 13092 y(\015)10665 11686 y(\020)11707 12725 y Fw(~)11389 13031 y Fu(H)12495 12531 y Fp(\(1\))13753 12049 y Fq(\000)14298 13031 y Fu(~)-595 b(y)14946 12531 y Fp(\(2\))16204 13031 y Fw(\()p Fu(')p Fw(\))17939 12049 y Fq(\001)18765 13031 y Ft(\000)269 b Fu(k)20646 12531 y Fp(2)p Fs(l)21463 11686 y Fq(\021)22187 11988 y Fr(\000)p Fp(1)23445 11637 y Fq(\015)23445 12364 y(\015)23445 13092 y(\015)24118 13874 y Fp(1)24980 13031 y Ft(\024)337 b Fw(2)p Fu(\017)27358 12531 y Fp(4)27358 13331 y(1)28153 13031 y Ft(\001)28892 12211 y Fw(4)p Fu(c)30023 12393 y Fp(0)30548 12211 y Fu(k)31217 11771 y Fp(2+2)p Fs(s)33324 11894 y FD(2)p 28892 12752 4951 49 v 30858 13886 a Fu(\017)31350 13468 y Fp(2)31350 14233 y(1)34312 13031 y Fw(=)f(8)p Fu(c)36722 13213 y Fp(0)37248 13031 y Fu(\017)37740 12531 y Fp(2)37740 13331 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20262 y FC(in)434 b(any)g Fw(\000)9080 19822 y Fp(\(2\))10337 20262 y Fw(\()10672 19942 y Fu(~)10808 20262 y(b)11328 19822 y Fp(\(2\))12586 20262 y Fw(\))g FC(is)f(the)f(same)h(as)g(that)f (of)h Fw(det)27240 18917 y Fq(\020)27964 20262 y Fu(I)365 b Fw(+)30394 19956 y(~)30074 20262 y Fu(A)30983 20444 y Fp(1)31509 20262 y Fw(\()-10 b Fu(~)-596 b(y)32617 19822 y Fp(\(1\))33875 20262 y Fw(\()p Fu(')p Fw(\)\))36081 18917 y Fq(\021)36806 20262 y FC(.)p 0 TeXcolorgray 2409 22265 a Fw(\(2\))p 0 TeXcolorgray 554 w FC(The)434 b(distanc)-62 b(e)431 b(fr)-62 b(om)433 b(a)g(zer)-62 b(o)433 b(of)g Fw(det)21451 21283 y Fq(\000)22007 22265 y Fu(I)364 b Fw(+)269 b Fu(A)25025 22447 y Fp(2)25551 22265 y Fw(\()-10 b Fu(~)-596 b(y)26659 21825 y Fp(\(2\))27918 22265 y Fw(\()p Fu(')p Fw(\)\))30124 21283 y Fq(\001)31114 22265 y FC(to)433 b(the)g(ne)-62 b(ar)g(est)432 b(zer)-62 b(o)433 b(of)4511 24194 y Fw(det)6397 22848 y Fq(\020)7121 24194 y Fu(I)364 b Fw(+)9550 23887 y(~)9230 24194 y Fu(A)10139 24376 y Fp(1)10665 24194 y Fw(\()-10 b Fu(~)-596 b(y)11773 23754 y Fp(\(1\))13032 24194 y Fw(\()p Fu(')p Fw(\)\))15238 22848 y Fq(\021)16397 24194 y FC(do)-62 b(es)432 b(not)h(exc)-62 b(e)g(e)g(d)432 b Fu(r)25439 23754 y Fp(\(2\))26696 24194 y Fu(=)p Fw(2)p FC(.)p 0 TeXcolorgray 475 27016 a FF(Corollary)483 b(4.18.)p 0 TeXcolorgray 554 w FC(The)433 b(numb)-62 b(er)433 b(of)g(zer)-62 b(os)432 b(of)h Fw(det)24700 26034 y Fq(\000)25255 27016 y Fu(I)365 b Fw(+)269 b Fu(A)28274 27198 y Fp(2)28800 27016 y Fw(\()-10 b Fu(~)-596 b(y)29908 26576 y Fp(\(2\))31166 27016 y Fw(\()p Fu(')p Fw(\)\))33372 26034 y Fq(\001)34363 27016 y FC(do)-62 b(es)432 b(not)h(exc)-62 b(e)g(e)g(d)432 b Fw(4)p Fu(c)43955 27198 y Fp(0)44480 27016 y Fu(k)45149 26576 y Fp(2+2)p Fs(s)47256 26699 y FD(2)47774 27016 y FC(.)1803 29274 y Fw(The)365 b(corollary)f(is)g (the)h(com)-34 b(bination)366 b(of)g(the)f(\014rst)g(part)g(of)h(the)f (lemma)f(with)i(the)f(second)g(part)h(of)f(Lemma)475 30713 y(4.10.)p 0 TeXcolorgray 475 33047 a FC(L)-62 b(emma)433 b(4.16.)p 0 TeXcolorgray 552 w Fw(The)306 b(\014rst)h(statemen)-34 b(t)307 b(immediately)e(follo)-34 b(ws)306 b(from)h(Lemmas)e(4.12)h (\(P)-34 b(art)306 b(1\))g(and)h(4.16.)506 b(Next,)475 34647 y(note)328 b(that)h(Lemmas)e(4.10)h({)f(4.16)h(hold)g(not)h(only) e(for)h Fo(O)26185 34207 y Fp(\(2\))27442 34647 y Fw(\()27777 34327 y Fu(~)27913 34647 y(b)28433 34207 y Fp(\(2\))29692 34647 y Fw(\),)343 b(but)329 b(also)e(for)h(a)g(set)39997 34317 y(~)39872 34647 y Fo(O)40730 34207 y Fp(\(2\))41987 34647 y Fw(\()42322 34327 y Fu(~)42458 34647 y(b)42978 34207 y Fp(\(2\))44236 34647 y Fw(\),)343 b(constructed)475 36164 y(from)484 b(disks)f(of)h(t)-34 b(wice)484 b(smaller)f(radius)h Fu(r)19752 35725 y Fp(\(2\))21009 36164 y Fu(=)p Fw(2)g(with)h(the)f (same)f(cen)-34 b(ters,)504 b(since)483 b(all)g(estimates)h(in)g(the)g (lem-)475 37604 y(mas)443 b(preserv)-34 b(ed)443 b(under)i(suc)-34 b(h)444 b(c)-34 b(hange)444 b(up)g(to)g(some)f(insigni\014can)-34 b(t)445 b(constan)-34 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Fp(\(2\))18231 43321 y(0)19826 42974 y Ft(\025)336 b Fu(\017)21597 43156 y Fp(1)22123 42974 y Fu(k)22792 42534 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fp(2)p Fs(\016)27196 42974 y FC(,)433 b Ft(j)p Fu(\017)p Ft(j)336 b Fu(<)h(\017)31275 42534 y Fp(4)31275 43299 y(1)31800 42974 y FC(,)434 b(then)p 0 TeXcolorgray 2409 45695 a Fw(\(1\))p 0 TeXcolorgray 554 w FC(The)g(determinant)e(of)15811 44875 y Fu(H)16917 44435 y Fp(\(2\))18175 44875 y Fw(\()-10 b Fu(~)-596 b(y)19283 44435 y Fp(\(2\))20542 44875 y Fw(\()p Fu(')p Fw(\)\))270 b Ft(\000)f Fu(k)24899 44435 y Fp(2)p Fs(l)25986 44875 y Ft(\000)g Fu(\017)p 15811 45417 11879 49 v 17213 46578 a(H)18221 46760 y Fp(0)18747 46578 y Fw(\()-10 b Fu(~)-596 b(y)19855 46226 y Fp(\(2\))21113 46578 y Fw(\()p Fu(')p Fw(\)\))271 b(+)e Fu(k)25471 46228 y Fp(2)p Fs(l)28257 45695 y FC(has)432 b(no)i(zer)-62 b(os)432 b(in)i Fu(\010)37506 45877 y Fp(2)38301 45695 y Ft(n)269 b Fo(O)40034 45255 y Fp(\(2\))41494 44350 y Fq(\020)42081 45375 y Fu(~)42218 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y Fp(\(2\))13645 52040 y Fw(\()p Fu(')p Fw(\)\))270 b Ft(\000)f Fu(k)18002 51600 y Fp(2)p Fs(l)p 8914 52581 9906 49 v 9329 53743 a Fu(H)10337 53925 y Fp(0)10863 53743 y Fw(\()-10 b Fu(~)-596 b(y)11971 53391 y Fp(\(2\))13230 53743 y Fw(\()p Fu(')p Fw(\)\))270 b(+)f Fu(k)17587 53393 y Fp(2)p Fs(l)18953 50787 y Fq(!)19912 52860 y FC(.)p 0 TeXcolorgray 475 56608 a FF(Corollary)483 b(4.20.)p 0 TeXcolorgray 554 w FC(The)433 b(fol)62 b(lowing)432 b(r)-62 b(elation)432 b(hold:)556 b Fu(!)26074 56169 y Fr(\003)26031 56933 y Fp(2)26937 56608 y Ft(\032)337 b Fo(O)29075 55970 y Fp(\(2\))29075 56797 y Fr(\003)30332 56608 y FC(.)p 0 TeXcolorgray 475 58867 a(L)-62 b(emma)433 b(4.19)f(and)h(Cor)-62 b(ol)62 b(lary)432 b(4.20.)p 0 TeXcolorgray 552 w Fw(Pro)34 b(of)308 b(of)h(the)g(lemma)f(is)h (completely)f(analogous)h(to)g(that)h(of)f(the)g(previ-)475 60306 y(ous)c(t)-34 b(w)g(o)306 b(lemmas)f(up)g(to)g(replacemen)-34 b(t)305 b(of)g(the)g(estimate)g Ft(k)p Fu(W)28493 60488 y Fp(2)29019 60306 y Ft(k)336 b Fu(<)h(\017)31733 59866 y Fp(4)31733 60631 y(1)32563 60306 y Fw(b)-34 b(y)306 b Ft(k)p Fu(W)35899 60488 y Fp(2)36495 60306 y Fw(+)71 b Fu(\017)p Ft(k)336 b Fu(<)g Fw(2)p Fu(\017)41320 59866 y Fp(4)41320 60631 y(1)41846 60306 y Fw(.)506 b(Corollary)303 b(follo)-34 b(ws)475 61745 y(from)430 b(the)g(lemma)g(and)h (De\014nition)f(4.6.)616 b(In)430 b(fact,)437 b Fu(!)25059 61305 y Fr(\003)25016 62070 y Fp(2)25586 61745 y Fw(\()p Fu(\025)p Fw(\))380 b(=)g Ft([)29746 62087 y Fs(p)30220 61826 y FD(\(2\))31320 62087 y Fr(2)p Fs(P)32675 61826 y FD(\(2\))33776 62087 y Fr(nf)p Fp(0)p Fr(g)35713 61745 y Fu(!)36467 62087 y Fp(2)p Fs(;p)37672 61826 y FD(\(2\))38829 61745 y Fw(,)436 b(where)430 b Fu(!)43887 62087 y Fp(2)p Fs(;p)45092 61826 y FD(\(2\))46679 61745 y Fw(is)g(the)g(set)475 63813 y(of)393 b(zeros)g(of)h(the)f(determinan)-34 b(t)395 b(\(4.14\))f(for)f(a)g(\014xed)h Fu(p)25011 63373 y Fp(\(2\))26268 63813 y Fw(.)535 b(By)393 b(Lemma)g(4.19)g(\(P)-34 b(art)394 b(1\),)h Fu(!)41667 64155 y Fp(2)p Fs(;p)42872 63894 y FD(\(2\))44366 63813 y Ft(\032)337 b Fo(O)46504 63373 y Fp(\(2\))47963 62468 y Fq(\020)48551 63493 y Fu(~)48687 63813 y(b)49207 63373 y Fp(\(2\))50465 62468 y Fq(\021)51189 63813 y Fw(,)339 65750 y Fu(~)475 66070 y(b)995 65630 y Fp(\(2\))2861 66070 y Fw(=)4695 65522 y Fp(2)p Fs(\031)32 b(p)6211 65210 y FD(\(2\))p 4546 65792 2916 49 v 4546 66488 a Fs(N)5293 66611 y FD(2)5754 66488 y Fs(N)6501 66611 y FD(1)6962 66488 y Fs(a)7594 66070 y Fw(,)608 b(since)567 b Fu(\017)12164 66252 y Fp(2)13298 66070 y Fu(<)609 b(\017)15342 65630 y Fp(4)15342 66395 y(1)15868 66070 y Fw(.)1028 b(Note)567 b(that)i(suc)-34 b(h)25876 65750 y Fu(~)26012 66070 y(b)26532 65630 y Fp(\(2\))28357 66070 y Fw(satis\014es)568 b(the)g(condition)g(of)f(the)h(lemma,)608 b(since)475 68025 y Fu(N)1449 68207 y Fp(2)1974 68025 y Fu(N)2948 68207 y Fp(1)3474 68025 y Fu(a)4115 68207 y Fp(1)p Fs(;)p Fp(2)5709 68025 y Ft(\024)337 b Fw(2)p Fu(k)8264 67585 y Fs(s)8699 67708 y FD(3)9216 68025 y Fw(.)530 b(Using)381 b(the)f(de\014nition)i(\(4.17\))f(of)f Fo(O)26586 67386 y Fp(\(2\))26586 68214 y Fr(\003)28224 68025 y Fw(and)h(the)f(form)-34 b(ula)381 b Fu(!)37841 67585 y Fr(\003)37798 68350 y Fp(2)38368 68025 y Fw(\()p Fu(\025)p Fw(\))338 b(=)e Ft([)42442 68367 y Fs(p)42916 68106 y FD(\(2\))44017 68367 y Fr(2)p Fs(P)45372 68106 y FD(\(2\))46472 68367 y Fr(nf)p Fp(0)p Fr(g)48409 68025 y Fu(!)49163 68367 y Fp(2)p Fs(;p)50368 68106 y FD(\(2\))475 69541 y Fw(yields)403 b(the)i(corollary)-101 b(.)p 50750 69707 665 831 v 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 49 49 TeXDict begin 49 48 bop 0 TeXcolorgray 0 TeXcolorgray 2848 3345 a Fp(SPECTRAL)406 b(PR)-26 b(OPER)-78 b(TIES)406 b(OF)h(A)h(POL)-78 b(YHARMONIC)405 b(OPERA)-78 b(TOR)406 b(WITH)h(LIMIT-PERIODIC)e(POTENTIAL)1430 b(49)p 0 TeXcolorgray 1803 6197 a Fw(F)-101 b(rom)410 b(no)-34 b(w)411 b(on)f(w)-34 b(e)411 b(denote)f(the)h(radius)f Fu(r)443 b Fw(of)411 b(the)f(disks)g(in)g Fo(O)30426 5559 y Fp(\(1\))30426 6386 y Fr(\003)32093 6197 y Fw(b)-34 b(y)410 b Fu(r)34363 5758 y Fp(\(1\))36030 6197 y Fw(and)h(the)f(radius)h Fu(r)44720 5758 y Fr(0)45440 6197 y Fw(of)f(the)g(disks)475 7648 y(in)404 b Fo(O)2747 7208 y Fp(\(2\))4408 7648 y Fw(b)-34 b(y)405 b Fu(r)6673 7208 y Fp(\(2\))7930 7648 y Fw(.)p 0 TeXcolorgray 475 9946 a FF(Lemma)414 b(4.21.)p 0 TeXcolorgray 519 w FC(\(pr)-62 b(o)g(of)392 b(of)i(Statement)f(3\))g (Under)h(c)-62 b(onditions)392 b(of)i(Ge)-62 b(ometric)392 b(lemma)h(4.1)h(,)401 b(estimate)392 b(\(4.5\))475 11385 y(holds.)p 0 TeXcolorgray 475 13683 a(Pr)-62 b(o)g(of.)p 0 TeXcolorgray 552 w Fw(By)401 b(Corollary)g(4.20,)h Fu(!)15169 13243 y Fr(\003)15126 14008 y Fp(2)16032 13683 y Ft(\032)336 b Fo(O)18169 13044 y Fp(\(2\))18169 13872 y Fr(\003)19427 13683 y Fw(.)537 b(Hence,)402 b Fu(!)24993 13865 y Fp(2)25856 13683 y Ft(\032)336 b Fo(O)27993 13044 y Fp(\(2\))27993 13872 y Fr(\003)29515 13683 y Ft(\\)265 b Fw(\002)31531 13865 y Fp(2)32056 13683 y Fw(,)402 b(the)g(set)g Fu(!)37524 13865 y Fp(2)38452 13683 y Fw(b)34 b(eing)401 b(giv)-34 b(en)402 b(in)g(De\014nition)475 15422 y(4.6.)524 b(Considering)361 b(that)i(that)f Fo(O)15543 14783 y Fp(\(2\))15543 15610 y Fr(\003)17162 15422 y Fw(is)e(formed)i(b)-34 b(y)361 b(no)h(more)f(than)h(16)p Fu(c)33166 15604 y Fp(0)33692 15422 y Fu(k)34361 14982 y Fp(2+2)p Fs(s)36468 15105 y FD(3)37346 15422 y Fw(disks)f(of)h(the)g(radius)f Fu(r)47880 14982 y Fp(\(2\))49498 15422 y Fw(\(see)475 16940 y(Remark)442 b(4.9\),)452 b(w)-34 b(e)443 b(obtain)h(that)g(the)f (total)g(length)g(of)g Fu(!)27161 17122 y Fp(2)28130 16940 y Fw(do)34 b(es)443 b(not)g(exceed)f(32)p Fu(c)38744 17122 y Fp(0)39269 16940 y Fu(k)39938 16500 y Fp(2+2)p Fs(s)42045 16623 y FD(3)42563 16940 y Fu(r)43144 16500 y Fp(\(2\))44401 16940 y Fw(.)654 b(T)-101 b(aking)443 b(in)-34 b(to)475 18457 y(accoun)g(t)500 b(that)g Fu(L)p Fw(\(\012)9940 18639 y Fp(2)10467 18457 y Fw(\))495 b Ft(\031)g Fu(k)38 b(L)p Fw(\()p Fu(!)15590 18639 y Fp(2)16117 18457 y Fw(\))500 b(and)g Fu(r)20121 18018 y Fp(\(2\))21873 18457 y Fw(=)495 b Fu(r)23892 18018 y Fp(\(1\))25149 18457 y Fu(k)25818 18018 y Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)28657 18141 y FD(2)29119 18018 y Fr(\000)p Fs(\016)30851 18457 y Fw(=)g Fu(k)32958 18018 y Fr(\000)p Fp(6)p Fr(\000)p Fp(6)p Fs(s)35797 18141 y FD(1)36259 18018 y Fr(\000)p Fp(4)p Fs(s)37896 18141 y FD(2)38358 18018 y Fr(\000)p Fp(4)p Fs(\016)40065 18457 y Fw(,)523 b Fu(s)41493 18639 y Fp(3)42514 18457 y Fw(=)495 b(2)p Fu(s)45126 18639 y Fp(2)46147 18457 y Fw(=)g(4)p Fu(s)48759 18639 y Fp(1)49285 18457 y Fw(,)523 b(w)-34 b(e)475 19926 y(obtain)368 b(that)h(the)g(the)f(length)g(of)g(\012)16788 20108 y Fp(2)17682 19926 y Fw(do)34 b(es)368 b(not)g(exceed)f(64)p Fu(c)28071 20108 y Fp(0)28596 19926 y Fu(k)29265 19486 y Fr(\000)p Fp(3)p Fr(\000)p Fp(2)p Fs(s)32104 19609 y FD(1)32567 19486 y Fr(\000)p Fp(2)p Fs(s)34204 19609 y FD(2)34665 19486 y Fr(\000)p Fp(4)p Fs(\016)36372 19926 y Fw(.)527 b(Considering)368 b(that)h Fu(\037)47224 19486 y Fr(\003)47224 20251 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Fp(\(3\))4936 30834 y Fw(\))344 b(giv)-34 b(en)344 b(b)-34 b(y)344 b(the)h(p)34 b(erturbation)345 b(series)e(\(4.8\).)519 b(W)-101 b(e)343 b(start)h(with)h(a)f (de\014nition)h(of)f Fo(D)42271 31016 y Fp(2)42797 30834 y Fw(\()p Fu(\025)p Fw(\))44446 31016 y Fs(nonr)26 b(es)47421 30834 y Fw(.)518 b(Recall)475 32562 y(that)605 b Fu(\037)4059 32744 y Fp(3)5256 32562 y Ft(\032)670 b Fo(K)7866 32744 y Fp(3)8393 32562 y Fu(\037)9151 32122 y Fr(\003)9151 32887 y Fp(2)9677 32562 y Fw(\()p Fu(\025)p Fw(\))605 b(\(see)f(Geometric)g(Lemma)g(4.1\))g(and)i Fu(\037)31100 32122 y Fr(\003)31100 32887 y Fp(2)31626 32562 y Fw(\()p Fu(\025)p Fw(\))671 b(=)f Fo(K)36556 32744 y Fp(2)37083 32562 y Fo(D)38090 32744 y Fp(2)38615 32562 y Fw(\()p Fu(\025)p Fw(\),)655 b(see)604 b(\(3.83\).)1140 b(Hence,)475 34001 y Fu(\037)1233 34183 y Fp(3)2096 34001 y Ft(\032)336 b Fo(K)4372 34183 y Fp(3)4898 34001 y Fo(D)5905 34183 y Fp(2)6431 34001 y Fw(\()p Fu(\025)p Fw(\).)539 b(Let)404 b Fo(D)12135 34183 y Fp(2)12660 34001 y Fw(\()p Fu(\025)p Fw(\))14309 34183 y Fs(nonr)26 b(es)17689 34001 y Fw(b)34 b(e)403 b(the)i(preimage)e(of)i Fu(\037)28753 34183 y Fp(3)29683 34001 y Fw(in)f Fo(D)32104 34183 y Fp(2)32630 34001 y Fw(\()p Fu(\025)p Fw(\):)15501 36188 y Fo(D)16508 36370 y Fp(2)17033 36188 y Fw(\()p Fu(\025)p Fw(\))18682 36370 y Fs(nonr)26 b(es)21994 36188 y Fw(=)337 b Ft(f)32 b Fu(~)-638 b Fn({)385 b Ft(2)337 b Fo(D)27225 36370 y Fp(2)27751 36188 y Fw(\()p Fu(\025)p Fw(\))g(:)f Fo(K)31407 36370 y Fp(3)31966 36188 y Fu(~)-638 b Fn({)385 b Ft(2)337 b Fu(\037)35030 36370 y Fp(3)35556 36188 y Ft(g)p Fu(:)11929 b Fw(\(4.28\))475 38375 y(The)404 b(follo)-34 b(wing)405 b(lemma)f(is)g(analogous)h(to)f(Lemma)g(3.40)g(in)g(the)g(second)h (step.)p 0 TeXcolorgray 475 40673 a FF(Lemma)508 b(4.22.)p 0 TeXcolorgray 574 w FC(The)469 b(formula)e Fo(K)16954 40855 y Fp(3)17480 40673 y Fo(D)18487 40855 y Fp(2)19013 40673 y Fw(\()p Fu(\025)p Fw(\))20662 40855 y Fs(nonr)26 b(es)24038 40673 y Fw(=)400 b Fu(\037)26139 40855 y Fp(3)27134 40673 y FC(establishes)465 b(one-to-one)h(c)-62 b(orr)g(esp)g(ondenc)g (e)466 b(b)-62 b(etwe)g(en)475 42112 y Fo(D)1482 42294 y Fp(2)2007 42112 y Fw(\()p Fu(\025)p Fw(\))3656 42294 y Fs(nonr)26 b(es)7065 42112 y FC(and)433 b Fu(\037)10177 42294 y Fp(3)10703 42112 y FC(.)p 0 TeXcolorgray 475 44410 a(Pr)-62 b(o)g(of.)p 0 TeXcolorgray 552 w Fw(The)344 b(pro)34 b(of)344 b(is)f(analogous)h(to)g(that)h(of)e(Lemma)g(3.40)g (up)i(to)e(the)h(shift)h(of)e(indices)h(b)-34 b(y)343 b(1,)356 b(i.e.,)e Fu(\037)47493 44592 y Fp(2)48356 44410 y Ft(!)336 b Fu(\037)50662 44592 y Fp(3)51189 44410 y Fw(,)475 45861 y Fu(\037)1233 45421 y Fr(\003)1233 46186 y Fp(1)1759 45861 y Fw(\()p Fu(\025)p Fw(\))i Ft(!)e Fu(\037)6052 45421 y Fr(\003)6052 46186 y Fp(2)6579 45861 y Fw(\()p Fu(\025)p Fw(\),)386 b Fu(\034)474 b Fw(=)337 b Fu(t)11673 45421 y Fp(\(2\))13267 45861 y Ft(!)g Fu(t)15254 45421 y Fp(\(3\))16511 45861 y Fw(;)389 b(w)-34 b(e)382 b(use)h(form)-34 b(ula)382 b(\(3.83\))h(instead)g(of)f(\(2.48\),)387 b(P)-34 b(art)382 b(1a)g(of)g(the)h(Geometric)475 47300 y(Lemma)454 b(for)i(the)f(third)h(step)g(instead)g(of)f(P)-34 b(art)456 b(1a)f(of)g(the)h(Geometric)e(Lemma)h(for)g(the)h(second)f (step,)468 b(and)475 48739 y(Lemma)403 b(3.44)h(instead)h(of)g(2.13.)p 50750 48905 V 1803 51164 a(W)-101 b(e)404 b(de\014ne)g Fo(B)8315 51346 y Fp(3)8841 51164 y Fw(\()p Fu(\025)p Fw(\))h(as)f(the)g(set)h(of)f(directions)g(corresp)34 b(onding)404 b(to)h Fo(D)33555 51346 y Fp(2)34080 51164 y Fw(\()p Fu(\025)p Fw(\))35729 51346 y Fs(nonr)26 b(es)38704 51164 y Fw(:)13648 53351 y Fo(B)14571 53533 y Fp(3)15096 53351 y Fw(\()p Fu(\025)p Fw(\))338 b(=)e Ft(f)-25 b Fu(~)-581 b(\027)414 b Ft(2)337 b Fo(B)22049 53533 y Fp(2)22574 53351 y Fw(\()p Fu(\025)p Fw(\))g(:)g Fn({)26042 53533 y Fp(2)26568 53351 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))-25 b Fu(~)-581 b(\027)414 b Ft(2)337 b Fo(D)32597 53533 y Fp(2)33122 53351 y Fw(\()p Fu(\025)p Fw(\))34771 53533 y Fs(nonr)26 b(es)37746 53351 y Ft(g)475 55538 y Fw(where)520 b Fn({)4904 55720 y Fp(2)5430 55538 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))520 b(is)g(de\014ned)i(b)-34 b(y)520 b(Lemma)g(3.42,)549 b Fn({)24532 55720 y Fp(2)25058 55538 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))530 b Ft(\021)g Fn({)30733 55720 y Fp(2)31259 55538 y Fw(\()p Fu(')p Fw(\).)888 b(Note)520 b(that)h Fo(B)40929 55720 y Fp(3)41455 55538 y Fw(\()p Fu(\025)p Fw(\))g(is)f(a)g(unit)h(circle)475 56977 y(with)470 b(holes)f(cen)-34 b(tered)469 b(at)h(the)f(origin)g (and)h Fo(B)21833 57159 y Fp(3)22358 56977 y Fw(\()p Fu(\025)p Fw(\))446 b Ft(\032)f Fo(B)26764 57159 y Fp(2)27289 56977 y Fw(\()p Fu(\025)p Fw(\).)735 b(W)-101 b(e)468 b(denote)i(b)-34 b(y)469 b(\002)38824 57159 y Fp(3)39350 56977 y Fw(\()p Fu(\025)p Fw(\))h(the)f(set)h(of)f(angles)g Fu(')475 58416 y Fw(corresp)34 b(onding)404 b(to)g Fo(B)10670 58598 y Fp(3)11196 58416 y Fw(\()p Fu(\025)p Fw(\):)10432 60604 y(\002)11375 60786 y Fp(3)11901 60604 y Fw(\()p Fu(\025)p Fw(\))337 b(=)g Ft(f)p Fu(')f Ft(2)h Fw([0)p Fu(;)202 b Fw(2)p Fu(\031)43 b Fw(\))337 b(:)741 b(\(cos)202 b Fu(';)g Fw(sin)g Fu(')p Fw(\))337 b Ft(2)g Fo(B)31742 60786 y Fp(3)32267 60604 y Fw(\()p Fu(\025)p Fw(\))p Ft(g)p Fu(;)1819 b Fw(\002)37621 60786 y Fp(3)38483 60604 y Ft(\032)337 b Fw(\002)40706 60786 y Fp(2)41231 60604 y Fu(:)6860 b Fw(\(4.29\))475 62869 y(W)-101 b(e)403 b(de\014ne)i Fo(D)7071 63051 y Fp(3)7596 62869 y Fw(\()p Fu(\025)p Fw(\))g(as)g(a)f(lev)-34 b(el)403 b(set)h(for)g Fu(\025)19364 62429 y Fp(\(3\))20622 62869 y Fw(\()p Fu(\013)t(;)235 b(~)-639 b Fn({)50 b Fw(\))404 b(in)g(the)27646 61887 y Fq(\000)28201 62869 y Fu(\017)28693 63051 y Fp(2)29219 62869 y Fu(k)29888 62429 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fs(\016)33821 61887 y Fq(\001)34377 62869 y Fw(-neigh)-34 b(b)34 b(orho)g(o)g(d)404 b(of)h Fo(D)44781 63051 y Fp(2)45306 62869 y Fw(\()p Fu(\025)p Fw(\))46955 63051 y Fs(nonr)26 b(es)49930 62869 y Fw(:)5081 65546 y Fo(D)6088 65728 y Fp(3)6613 65546 y Fw(\()p Fu(\025)p Fw(\))338 b(:=)10216 64201 y Fq(n)11056 65546 y Fu(~)-638 b Fn({)386 b Fw(=)336 b Fn({)24 b Fu(~)-581 b(\027)414 b Fw(:)312 b Fu(~)-582 b(\027)414 b Ft(2)337 b Fo(B)19121 65728 y Fp(3)19646 65546 y Fw(\()p Fu(\025)p Fw(\))p Fu(;)22643 64516 y Fq(\014)22643 65243 y(\014)23047 65546 y Fn({)318 b Ft(\000)269 b Fn({)26193 65728 y Fp(2)26719 65546 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))29583 64516 y Fq(\014)29583 65243 y(\014)30324 65546 y Fu(<)337 b(\017)32096 65728 y Fp(2)32621 65546 y Fu(k)33290 65046 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fs(\016)37223 65546 y Fu(;)606 b(\025)38873 65046 y Fp(\(3\))40131 65546 y Fw(\()p Fu(\013)t(;)235 b(~)-639 b Fn({)50 b Fw(\))337 b(=)f Fu(\025)45572 64201 y Fq(o)46582 65546 y Fu(:)475 68102 y Fw(Next)362 b(t)-34 b(w)g(o)364 b(lemmas)f(are)f(to) g(pro)-34 b(v)g(e)363 b(that)h Fo(D)20228 68284 y Fp(3)20754 68102 y Fw(\()p Fu(\025)p Fw(\))f(is)f(a)h(distorted)g(circle)e(with)i (holes.)525 b(Their)362 b(form)-34 b(ulations)364 b(and)475 69541 y(pro)34 b(ofs)404 b(are)g(analogous)h(to)f(those)h(of)f(Lemmas)g (3.41)g(and)h(3.42.)p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 50 50 TeXDict begin 50 49 bop 0 TeXcolorgray 0 TeXcolorgray 475 3345 a Fp(50)14248 b(YULIA)313 b(KARPESHINA)f(AND)h(YOUNG-RAN)g (LEE)p 0 TeXcolorgray 0 TeXcolorgray 475 6001 a FF(Lemma)503 b(4.23.)p 0 TeXcolorgray 573 w FC(F)-93 b(or)466 b(every)439 b Fu(~)-581 b(\027)471 b Ft(2)395 b Fo(B)17624 6183 y Fp(3)18149 6001 y Fw(\()p Fu(\025)p Fw(\))466 b FC(and)f(every)f Fu(\013)t FC(,)475 b Fw(0)394 b Ft(\024)g Fu(\013)400 b Ft(\024)394 b Fw(1)p FC(,)473 b(ther)-62 b(e)465 b(is)f(a)i(unique)d Fn({)444 b Fw(=)394 b Fn({)46616 6183 y Fp(3)47142 6001 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))466 b FC(in)475 7470 y(the)432 b(interval)g Fu(I)7466 7652 y Fp(3)8329 7470 y Fw(:=)9945 6488 y Fq(\002)10450 7470 y Fn({)11258 7652 y Fp(2)11784 7470 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))270 b Ft(\000)f Fu(\017)16622 7652 y Fp(2)17147 7470 y Fu(k)17816 7030 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fs(\016)21749 7470 y Fu(;)202 b Fn({)23096 7652 y Fp(2)23622 7470 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))270 b(+)f Fu(\017)28460 7652 y Fp(2)28986 7470 y Fu(k)29655 7030 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fs(\016)33588 6488 y Fq(\003)34526 7470 y FC(such)432 b(that)21552 9673 y Fu(\025)22259 9173 y Fp(\(3\))23517 9673 y Fw(\()p Fu(\013)t(;)202 b Fn({)26114 9855 y Fp(3)26616 9673 y Fu(~)-581 b(\027)77 b Fw(\))337 b(=)f Fu(\025:)17980 b Fw(\(4.30\))475 11616 y FC(F)-93 b(urthermor)-62 b(e,)16836 13078 y Ft(j)p Fn({)17981 13260 y Fp(3)18506 13078 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))270 b Ft(\000)f Fn({)23660 13260 y Fp(2)24186 13078 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))p Ft(j)337 b(\024)f Fw(2)p Fu(\013)t(\017)30880 12577 y Fp(4)30880 13377 y(2)31407 13078 y Fu(k)32076 12577 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)34828 13078 y Fu(:)13263 b Fw(\(4.31\))1803 15210 y(Let)367 b(us)h(recall)e(that)i Fu(\010)12053 15392 y Fp(3)12947 15210 y Fw(is)f(the)h(third)g(complex)e(non-resonan) -34 b(t)370 b(set)d(giv)-34 b(en)368 b(b)-34 b(y)367 b(\(4.17\),)375 b(see)367 b(Fig.)526 b(8.)h(F)-101 b(urther)475 16649 y(w)-34 b(e)404 b(use)h(the)f(notations)i Fn({)12576 16831 y Fp(3)13102 16649 y Fw(\()p Fu(')p Fw(\))337 b Ft(\021)f Fn({)17261 16831 y Fp(3)17787 16649 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\),)405 b Fu(h)22091 16831 y Fp(3)22617 16649 y Fw(\()p Fu(')p Fw(\))337 b Ft(\021)g Fn({)26777 16831 y Fp(3)27302 16649 y Fw(\()p Fu(')p Fw(\))271 b Ft(\000)e Fn({)31328 16831 y Fp(2)31854 16649 y Fw(\()p Fu(')p Fw(\),)436 b Fu(~)-638 b Fn({)35138 16831 y Fp(3)35664 16649 y Fw(\()p Fu(')p Fw(\))337 b(=)g Fn({)39824 16831 y Fp(3)40350 16649 y Fw(\()p Fu(')p Fw(\))-25 b Fu(~)-581 b(\027)77 b Fw(.)p 0 TeXcolorgray 475 18780 a FF(Lemma)463 b(4.24.)p 0 TeXcolorgray 0 TeXcolorgray 2489 w Fw(\(1\))p 0 TeXcolorgray 554 w FC(The)480 b(set)f Fo(D)18341 18962 y Fp(3)18866 18780 y Fw(\()p Fu(\025)p Fw(\))i FC(is)f(a)g(distorte)-62 b(d)477 b(cir)-62 b(cle)479 b(with)g(holes:)649 b(it)480 b(c)-62 b(an)479 b(b)-62 b(e)480 b(describ)-62 b(e)g(d)477 b(by)i(the)4511 20219 y(formula:)14180 21659 y Fo(D)15187 21841 y Fp(3)15712 21659 y Fw(\()p Fu(\025)p Fw(\))338 b(=)18978 20677 y Fq(\010)19717 21659 y Fu(~)-638 b Fn({)386 b Ft(2)337 b Fn(R)22899 21158 y Fp(2)23762 21659 y Fw(:)368 b Fu(~)-638 b Fn({)386 b Fw(=)368 b Fu(~)-638 b Fn({)27716 21841 y Fp(3)28242 21659 y Fw(\()p Fu(')p Fw(\))p Fu(;)1070 b(')336 b Ft(2)h Fw(\002)34601 21841 y Fp(3)35127 21659 y Fw(\()p Fu(\025)p Fw(\))36776 20677 y Fq(\011)37484 21659 y Fu(;)10607 b Fw(\(4.32\))4511 23354 y FC(wher)-62 b(e)558 b Fn({)8867 23536 y Fp(3)9393 23354 y Fw(\()p Fu(')p Fw(\))k(=)g Fn({)14003 23536 y Fp(2)14529 23354 y Fw(\()p Fu(')p Fw(\))360 b(+)f Fu(h)18624 23536 y Fp(3)19150 23354 y Fw(\()p Fu(')p Fw(\))p Fu(;)558 b Fn({)22588 23536 y Fp(2)23114 23354 y Fw(\()p Fu(')p Fw(\))h FC(is)e(the)g(\\r)-62 b(adius")555 b(of)i Fo(D)36538 23536 y Fp(2)37064 23354 y Fw(\()p Fu(\025)p Fw(\))h FC(and)g Fu(h)42448 23536 y Fp(3)42974 23354 y Fw(\()p Fu(')p Fw(\))h FC(satis\014es)c(the)4511 24793 y(estimates)15540 26662 y Ft(j)p Fu(h)16575 26844 y Fp(3)17101 26662 y Ft(j)337 b Fu(<)f Fw(2)p Fu(\013)t(\017)20931 26161 y Fp(4)20931 26961 y(2)21458 26662 y Fu(k)22127 26161 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)24878 26662 y Fu(;)26718 24904 y Fq(\014)26718 25631 y(\014)26718 26359 y(\014)26718 27086 y(\014)27255 25842 y Fu(@)67 b(h)28664 26024 y Fp(3)p 27255 26383 1936 49 v 27470 27493 a Fu(@)g(')29323 24904 y Fq(\014)29323 25631 y(\014)29323 26359 y(\014)29323 27086 y(\014)30063 26662 y Ft(\024)337 b Fw(4)p Fu(\013)t(\017)33220 26161 y Fp(3)33220 26961 y(2)33747 26662 y Fu(k)34416 26161 y Fp(1+)p Fs(\016)36123 26662 y Fu(:)11968 b Fw(\(4.33\))p 0 TeXcolorgray 2409 29085 a(\(2\))p 0 TeXcolorgray 554 w FC(The)434 b(total)d(length)i(of)g Fo(B)15688 29267 y Fp(3)16213 29085 y Fw(\()p Fu(\025)p Fw(\))h FC(satis\014es)e(the)g(estimate:)18286 31073 y Fu(L)202 b Fw(\()p Fo(B)20707 31255 y Fp(2)21502 31073 y Ft(n)269 b Fo(B)23300 31255 y Fp(3)23825 31073 y Fw(\))338 b Fu(<)e Fw(4)p Fu(\031)43 b(k)27922 30573 y Fr(\000)p Fp(4)p Fr(\000)p Fp(2)p Fs(s)30761 30696 y FD(1)31224 30573 y Fr(\000)p Fp(2)p Fs(s)32861 30696 y FD(2)33378 31073 y Fu(:)14713 b Fw(\(4.34\))p 0 TeXcolorgray 2409 33016 a(\(3\))p 0 TeXcolorgray 554 w FC(The)449 b(function)e Fn({)12598 33198 y Fp(3)13124 33016 y Fw(\()p Fu(')p Fw(\))i FC(c)-62 b(an)448 b(b)-62 b(e)448 b(extende)-62 b(d)447 b(as)h(a)g(holomorphic)d(function)i(of)h Fu(')h FC(to)f(the)f(c)-62 b(omplex)447 b(non-)4511 34455 y(r)-62 b(esonc)g(e)433 b(set)f Fu(\010)11499 34637 y Fp(3)12025 34455 y FC(,)h(estimates)e(\(4.33\))h(b)-62 b(eing)432 b(pr)-62 b(eserve)g(d.)p 0 TeXcolorgray 2409 35894 a Fw(\(4\))p 0 TeXcolorgray 554 w FC(The)313 b(curve)f Fo(D)11024 36076 y Fp(3)11549 35894 y Fw(\()p Fu(\025)p Fw(\))i 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b(P)-34 b(art)447 b(2,)456 b(when)448 b(pro)-34 b(ving)475 43092 y(con)g(v)g(ergence)403 b(of)i(the)f(series)g(for)g(the)g(resolv)-34 b(en)g(t)22933 42110 y Fq(\000)23488 43092 y Fu(H)24594 42652 y Fp(\(3\))25852 43092 y Fw(\()32 b Fu(~)-638 b Fn({)27131 43274 y Fp(2)27657 43092 y Fw(\()p Fu(')p Fw(\)\))271 b Ft(\000)e Fu(z)31963 42110 y Fq(\001)32518 42338 y Fr(\000)p Fp(1)33776 43092 y Fw(,)404 b(w)-34 b(e)404 b(use)g(the)h(estimate)17109 45745 y(sup)16904 46788 y Fs(z)37 b Fr(2)p Fs(C)18677 46911 y FD(3)19138 44351 y Fq(\015)19138 45078 y(\015)19138 45805 y(\015)19812 44399 y(\020)20853 45438 y Fw(~)20536 45745 y Fu(H)21642 45244 y Fp(\(2\))22899 45745 y Fw(\()32 b Fu(~)-638 b Fn({)24178 45927 y Fp(2)24705 45745 y Fw(\()p Fu(')p Fw(\)\))270 b Ft(\000)f Fu(z)29010 44399 y Fq(\021)29734 44701 y Fr(\000)p Fp(1)30992 44351 y Fq(\015)30992 45078 y(\015)30992 45805 y(\015)32002 45745 y Fu(<)33414 44925 y Fw(64)p 33414 45466 1213 49 v 33511 46576 a Fu(\017)34003 46758 y Fp(2)34759 45745 y Fu(;)13332 b Fw(\(4.36\))475 48762 y(analogous)397 b(to)f(\(3.73\),)i(the)f(op)34 b(erator)18646 48456 y(~)18329 48762 y Fu(H)19435 48322 y Fp(\(2\))21089 48762 y Fw(acting)396 b(in)g Fu(L)26948 48944 y Fp(2)27474 48762 y Fw(\()p Fu(Q)28903 48944 y Fp(3)29430 48762 y Fw(\).)536 b(The)396 b(estimate)h(\(4.36\))f(follo) -34 b(ws)397 b(from)g(\(4.26\))475 50362 y(for)2254 50042 y Fu(~)2390 50362 y(b)437 b Fw(=)g(2)p Fu(\031)43 b(p)6677 49922 y Fp(\(2\))7935 50362 y Fu(=)-67 b(N)9448 50544 y Fp(2)9973 50362 y Fu(N)10947 50544 y Fp(1)11473 50362 y Fu(a)p Fw(,)479 b Fu(p)13540 49922 y Fp(\(2\))15234 50362 y Ft(6)p Fw(=)437 b(0,)479 b(and)465 b(estimate)f(\(3.80\),)480 b(whic)-34 b(h)465 b(tak)-34 b(es)464 b(care)g(ab)34 b(out)465 b(the)g(case)e Fu(p)47508 49922 y Fp(\(2\))49203 50362 y Fw(=)436 b(0.)475 51801 y(As)467 b(a)g(side)f(result)h(of)h (these)f(considerations,)483 b(w)-34 b(e)467 b(obtain)h(an)f(estimate)g (analogous)h(to)f(\(3.80\))g(for)g(the)h(new)475 53539 y(resolv)-34 b(en)g(t)5569 52557 y Fq(\000)6125 53539 y Fu(H)7231 53099 y Fp(\(3\))8488 53539 y Fw(\()32 b Fu(~)-638 b Fn({)9767 53721 y Fp(3)10294 53539 y Fw(\()p Fu(')p Fw(\)\))270 b Ft(\000)f Fu(z)14599 52557 y Fq(\001)15155 52785 y Fr(\000)p Fp(1)16412 53539 y Fw(.)p 50750 53705 665 831 v 1803 55647 a(W)-101 b(e)404 b(de\014ne)g(the)h(non-resonance) g(set,)f Fu(\037)20356 55207 y Fr(\003)20356 55972 y Fp(3)20882 55647 y Fw(\()p Fu(\025)p Fw(\))h(in)f Fu(S)25093 55829 y Fp(3)25619 55647 y Fw(\()p Fu(\025)p Fw(\))h(b)-34 b(y)404 b(the)h(form)-34 b(ula)405 b(analogous)g(to)f(\(3.83\):)21035 57590 y Fu(\037)21793 57089 y Fr(\003)21793 57889 y Fp(3)22320 57590 y Fw(\()p Fu(\025)p Fw(\))337 b(:=)f Fo(K)26919 57772 y Fp(3)27446 57590 y Fo(D)28453 57772 y Fp(3)28978 57590 y Fw(\()p Fu(\025)p Fw(\))p Fu(:)17464 b Fw(\(4.37\))475 59532 y(The)404 b(follo)-34 b(wing)405 b(lemmas)f(are)g(analogous)h(to) f(Lemmas)g(3.43)g(and)h(3.44.)p 0 TeXcolorgray 475 61675 a FF(Lemma)453 b(4.25.)p 0 TeXcolorgray 546 w FC(The)425 b(set)f Fu(\037)13927 61236 y Fr(\003)13927 62000 y Fp(3)14453 61675 y Fw(\()p Fu(\025)p Fw(\))i FC(b)-62 b(elongs)423 b(to)i(the)24123 60694 y Fq(\000)24678 61675 y Fw(2)p Fu(\013)t(\017)26555 61236 y Fp(4)26555 62000 y(2)27082 61675 y Fu(k)27751 61236 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)30502 60694 y Fq(\001)31058 61675 y FC(-neighb)-62 b(orho)g(o)g(d)421 b(of)k Fu(\037)40860 61857 y Fp(3)41386 61675 y Fw(\()p Fu(\025)p Fw(\))h FC(in)f Fu(K)45968 61857 y Fp(3)46494 61675 y FC(.)555 b(If)425 b Fu(t)49123 61236 y Fp(\(3\))50717 61675 y Ft(2)475 63513 y Fu(\037)1233 63074 y Fr(\003)1233 63838 y Fp(3)1759 63513 y Fw(\()p Fu(\025)p Fw(\))p FC(,)486 b(then)474 b(the)g(op)-62 b(er)g(ator)473 b Fu(H)14975 62875 y Fp(\(3\))14877 63671 y Fs(\013)16232 63513 y Fw(\()p Fu(t)17141 63074 y Fp(\(3\))18399 63513 y Fw(\))j FC(has)d(a)i(simple)e(eigenvalue)f(e)-62 b(qual)473 b(to)h Fu(\025)p FC(.)682 b(This)474 b(eigenvalue)e(is)i (given)475 64952 y(by)433 b(the)f(p)-62 b(erturb)g(ation)431 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475 6001 a Fw(4.6.)605 b FF(Preparation)465 b(for)g(the)g(Next)f(Appro) -39 b(ximation.)475 9753 y Fw(4.6.1.)605 b FC(Contr)-62 b(acte)g(d)439 b(set)h Fo(O)12820 9114 y Fp(\(2\))12820 9910 y Fs(s)14280 8407 y Fq(\020)14867 9433 y Fu(~)15004 9753 y(b)15524 9313 y Fp(\(2\))16781 8407 y Fq(\021)17505 9753 y FC(.)606 b Fw(F)-101 b(or)412 b(an)-34 b(y)22828 9433 y Fu(~)22964 9753 y(b)23484 9313 y Fp(\(2\))25154 9753 y Fw(suc)g(h)413 b(that)g Fu(b)31050 9114 y Fp(\(2\))31050 10100 y(0)32658 9753 y Fu(>)350 b(\017)34443 9935 y Fp(1)34969 9753 y Fu(k)35638 9313 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fp(2)p Fs(\016)40041 9753 y Fw(,)414 b(w)-34 b(e)413 b(ha)-34 b(v)g(e)412 b(constructed)475 12045 y(the)320 b(set)f Fo(O)5143 11605 y Fp(\(2\))6401 12045 y Fw(\()6736 11725 y Fu(~)6872 12045 y(b)7392 11605 y Fp(\(2\))8650 12045 y Fw(\))337 b Ft(\032)f Fu(\010)11545 12227 y Fp(2)12071 12045 y Fw(,)g(whic)-34 b(h)321 b(surrounds)f(zeros) f(of)h(the)g(determinan)-34 b(t)320 b(det)36344 10700 y Fq(\020)37068 12045 y Fu(I)195 b Fw(+)39159 11739 y(~)38839 12045 y Fu(A)39748 12227 y Fp(1)40476 11063 y Fq(\000)41021 12045 y Fu(~)-596 b(y)41668 11605 y Fp(\(1\))42927 12045 y Fw(\()p Fu(')p Fw(\))44662 11063 y Fq(\001)45218 10700 y(\021)45942 12045 y Fw(,)326 b Fu(~)-596 b(y)47252 11605 y Fp(\(1\))48510 12045 y Fw(\()p Fu(')p Fw(\))337 b(=)507 14130 y Fu(~)-638 b Fn({)1283 14312 y Fp(1)1809 14130 y Fw(\()p Fu(')p Fw(\))279 b(+)4909 13810 y Fu(~)5045 14130 y(b)5565 13690 y Fp(\(2\))6823 14130 y Fw(,)422 b(see)d(Remark)f(4.7.)582 b(The)419 b(set)g(consists)h(of)f(disks)f(of) i(the)f(radius)g Fu(r)37657 13690 y Fp(\(2\))39275 14130 y Fw(=)361 b Fu(r)41160 13690 y Fp(\(1\))42417 14130 y Fu(k)43086 13690 y Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)45925 13813 y FD(2)46387 13690 y Fr(\000)p Fs(\016)47624 14130 y Fw(,)422 b Fu(r)48964 13690 y Fp(\(1\))50582 14130 y Fw(=)475 15598 y Fu(k)1144 15158 y Fr(\000)p Fp(4)p Fr(\000)p Fp(6)p Fs(s)3983 15281 y FD(1)4445 15158 y Fr(\000)p Fp(3)p Fs(\016)6699 15598 y Fw(and)548 b(con)-34 b(tains)548 b(no)f(more)g(than)h(4)p Fu(c)23201 15780 y Fp(0)23727 15598 y Fu(k)24396 15158 y Fp(2+2)p Fs(s)26503 15281 y FD(2)27567 15598 y Fw(disks,)582 b(see)547 b(Remark)g(4.8.)966 b(W)-101 b(e)547 b(pro)-34 b(v)g(ed)548 b(that)g(the)475 17338 y(new)503 b(determinan)-34 b(t)504 b(det)11685 15993 y Fq(\020)12409 17338 y Fu(I)430 b Fw(+)335 b Fu(A)15559 17520 y Fp(2)16085 16357 y Fq(\000)16631 17338 y Fu(~)-596 b(y)17278 16899 y Fp(\(2\))18536 17338 y Fw(\()p Fu(')p Fw(\))20271 16357 y Fq(\001)20827 15993 y(\021)21551 17338 y Fw(,)517 b Fu(~)-596 b(y)23052 16899 y Fp(\(2\))24311 17338 y Fw(\()p Fu(')p Fw(\))501 b(=)533 b Fu(~)-638 b Fn({)28799 17520 y Fp(2)29325 17338 y Fw(\()p Fu(')p Fw(\))336 b(+)32538 17019 y Fu(~)32673 17338 y(b)33193 16899 y Fp(\(2\))34451 17338 y Fw(,)527 b(has)503 b(the)h(same)e(n)-34 b(um)g(b)34 b(er)504 b(of)f(zeros)475 19631 y(inside)553 b(eac)-34 b(h)554 b(\000)7700 19191 y Fp(\(2\))8957 19631 y Fw(\()9292 19311 y Fu(~)9428 19631 y(b)9948 19191 y Fp(\(2\))11206 19631 y Fw(\))g(as)f(det)15552 18286 y Fq(\020)16276 19631 y Fu(I)463 b Fw(+)18904 19325 y(~)18584 19631 y Fu(A)19493 19813 y Fp(1)20019 18649 y Fq(\000)20564 19631 y Fu(~)-596 b(y)21211 19191 y Fp(\(1\))22470 19631 y Fw(\()p Fu(')p Fw(\))24205 18649 y Fq(\001)24761 18286 y(\021)26038 19631 y Fw(\(Lemma)553 b(5.11,)591 b(P)-34 b(art)553 b(1\))h(and)g(b)34 b(oth)554 b(ha)-34 b(v)g(e)554 b(no)g(zeros)475 21715 y(in)543 b Ft(2)569 b Fu(\010)4213 21897 y Fp(2)5100 21715 y Ft(n)363 b Fo(O)6927 21276 y Fp(\(2\))8184 21715 y Fw(\()8519 21396 y Fu(~)8655 21715 y(b)9175 21276 y Fp(\(2\))10433 21715 y Fw(\))544 b(\(Corollaries)e(4.11,)578 b(4.15\).)956 b(T)-101 b(o)543 b(prepare)g(the)h(next)f(appro)-34 b(ximation,)579 b(w)-34 b(e)543 b(con)-34 b(tract)475 23523 y(the)476 b(set)f Fo(O)5455 22884 y Fp(\(2\))5455 23681 y Fs(s)6712 23523 y Fw(\()7047 23203 y Fu(~)7183 23523 y(b)7703 23083 y Fp(\(2\))8962 23523 y Fw(\))g(around)i(the)f(zeros)e(of)i(the)g(new)g (determinan)-34 b(t)476 b(det)34128 22178 y Fq(\020)34852 23523 y Fu(I)412 b Fw(+)316 b Fu(A)37965 23705 y Fp(2)38491 22541 y Fq(\000)39037 23523 y Fu(~)-596 b(y)39684 23083 y Fp(\(2\))40942 23523 y Fw(\()p Fu(')p Fw(\))42677 22541 y Fq(\001)43233 22178 y(\021)43957 23523 y Fw(.)753 b(W)-101 b(e)474 b(consider)475 25725 y(the)564 b(disks)g(with)g(the)h(radius)f Fu(r)15447 25285 y Fp(\(3\))17307 25725 y Fw(=)602 b Fu(r)19433 25285 y Fp(\(2\))20691 25725 y Fu(k)21360 25285 y Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)24199 25408 y FD(3)24661 25285 y Fr(\000)p Fs(\016)26461 25725 y Fw(cen)-34 b(tered)564 b(at)g(eac)-34 b(h)564 b(zero)f Fu(')39482 25086 y Fp(\(2\))39482 26087 y Fs(j)o(;)p Fp(0)41303 25725 y Fw(of)i(the)f(determinan)-34 b(t)475 27764 y(det)2158 26418 y Fq(\020)2882 27764 y Fu(I)393 b Fw(+)298 b Fu(A)5958 27946 y Fp(2)6484 26782 y Fq(\000)7029 27764 y Fu(~)-596 b(y)7676 27324 y Fp(\(2\))8935 27764 y Fw(\()p Fu(')p Fw(\))10670 26782 y Fq(\001)11226 26418 y(\021)11950 27764 y Fw(.)667 b(Ob)-34 b(viously)447 b(eac)-34 b(h)447 b(of)g(these)g(disks)g(is)g(in)g Fo(O)32664 27324 y Fp(\(2\))33921 26782 y Fq(\000)34341 27444 y Fu(~)34477 27764 y(b)34997 27324 y Fp(\(2\))36255 26782 y Fq(\001)36810 27764 y Fw(,)457 b(since)447 b Fu(r)41198 27324 y Fp(\(3\))42863 27764 y Fu(<<)408 b(r)45738 27324 y Fp(\(2\))47442 27764 y Fw(and)448 b(the)475 29766 y(distance)352 b(b)34 b(et)-34 b(w)g(een)354 b(zeros)d(of)i(the)g(old)f(and)h(new)g (determinan)-34 b(ts)353 b(is)f(smaller)g(than)h Fu(r)38760 29326 y Fp(\(2\))40017 29766 y Fu(=)p Fw(2)g(\(Lemma)f(5.11,)362 b(P)-34 b(art)475 31483 y(2\).)813 b(Next,)518 b(w)-34 b(e)496 b(tak)-34 b(e)495 b(the)h(union)g(of)g(these)g(smaller)f(disks) g(and)h(denote)h(it)e(b)-34 b(y)496 b Fo(O)38774 30844 y Fp(\(2\))38774 31641 y Fs(s)40031 30501 y Fq(\000)40451 31163 y Fu(~)40587 31483 y(b)41107 31043 y Fp(\(2\))42365 30501 y Fq(\001)42920 31483 y Fw(.)813 b(W)-101 b(e)495 b(denote)h(b)-34 b(y)475 33321 y(\000)1233 32682 y Fp(\(2\))1233 33479 y Fs(s)2490 32339 y Fq(\000)2909 33001 y Fu(~)3046 33321 y(b)3566 32339 y Fq(\001)4496 33321 y Fw(a)374 b(connected)h(comp)34 b(onen)-34 b(t)376 b(of)f(these)g(new)g(disks,) 380 b(whic)-34 b(h)376 b(in)-34 b(tersects)374 b(with)i Fu(\010)39799 33503 y Fp(2)40325 33321 y Fw(,)k(and)375 b(b)-34 b(y)375 b Fu(\015)45718 32682 y Fp(\(2\))45651 33479 y Fs(s)46976 32339 y Fq(\000)47395 33001 y Fu(~)47531 33321 y(b)48051 32881 y Fp(\(2\))49309 32339 y Fq(\001)50239 33321 y Fw(its)475 35159 y(b)34 b(oundary)-101 b(,)404 b Fu(\015)6863 34520 y Fp(\(2\))6796 35317 y Fs(s)8458 35159 y Fw(=)336 b Fu(@)67 b Fw(\000)11206 34520 y Fp(\(2\))11206 35317 y Fs(s)12463 35159 y Fw(.)p 0 TeXcolorgray 475 37897 a FF(Lemma)463 b(4.28.)p 0 TeXcolorgray 0 TeXcolorgray 2489 w Fw(\(1\))p 0 TeXcolorgray 554 w FC(The)f(set)g Fo(O)18157 37258 y Fp(\(2\))18157 38055 y Fs(s)19414 36915 y Fq(\000)19834 37577 y Fu(~)19970 37897 y(b)20490 37457 y Fp(\(2\))21748 36915 y Fq(\001)22766 37897 y FC(c)-62 b(ontains)461 b(the)h(same)f(numb)-62 b(er)463 b(of)f(disks)e(as)i Fo(O)44008 37457 y Fp(\(2\))45266 36915 y Fq(\000)45685 37577 y Fu(~)45821 37897 y(b)46341 37457 y Fp(\(2\))47599 36915 y Fq(\001)48155 37897 y FC(.)644 b(This)4511 39442 y(numb)-62 b(er)434 b(do)-62 b(es)432 b(not)h(exc)-62 b(e)g(e)g(d)432 b Fw(4)p Fu(c)18425 39624 y Fp(0)18950 39442 y Fu(k)19619 39003 y Fp(2+2)p Fs(s)21726 39126 y FD(2)22244 39442 y FC(.)p 0 TeXcolorgray 2409 41038 a Fw(\(2\))p 0 TeXcolorgray 554 w FC(F)-93 b(or)434 b(any)f Fu(')337 b Ft(2)g Fu(\010)12180 41220 y Fp(2)12975 41038 y Ft(n)269 b Fo(O)14708 40399 y Fp(\(2\))14708 41196 y Fs(s)15966 40056 y Fq(\000)16385 40718 y Fu(~)16521 41038 y(b)17041 40598 y Fp(\(2\))18299 40056 y Fq(\001)18855 41038 y FC(,)7676 43910 y Fq(\015)7676 44637 y(\015)7676 45364 y(\015)8350 43958 y(\020)9074 45304 y Fu(H)10180 44803 y Fp(\(2\))11437 44322 y Fq(\000)11983 45304 y Fu(~)-596 b(y)12630 44803 y Fp(\(2\))13888 45304 y Fw(\()p Fu(')p Fw(\))15623 44322 y Fq(\001)16449 45304 y Ft(\000)269 b Fu(k)18330 44803 y Fp(2)p Fs(l)19147 43958 y Fq(\021)19871 44260 y Fr(\000)p Fp(1)21129 43910 y Fq(\015)21129 44637 y(\015)21129 45364 y(\015)22139 45304 y Ft(\024)23757 44484 y Fw(4)p 23551 45025 1018 49 v 23551 46159 a Fu(\017)24043 45741 y Fp(2)24043 46506 y(1)24904 43231 y Fq( )25996 44484 y Fw(2)p Fu(r)27183 44044 y Fp(\(2\))28441 44484 y Fu(J)29230 44044 y Fp(\(2\))p 25996 45025 4491 49 v 27323 46186 a Fu(r)27904 45835 y Fp(\(3\))30620 43231 y Fq(!)31579 43533 y Fs(J)32176 43220 y FD(\(2\))33535 45304 y Fu(;)1502 b(J)36163 44803 y Fp(\(2\))37758 45304 y Fw(=)336 b(4)p Fu(c)40168 45486 y Fp(0)40693 45304 y Fu(k)41362 44803 y Fp(2+2)p Fs(s)43469 44926 y FD(2)43987 45304 y Fu(;)4104 b Fw(\(4.38\))11822 49912 y Fq(\015)11822 50639 y(\015)11822 51367 y(\015)12496 49960 y(\020)13220 51306 y Fu(H)14326 50805 y Fp(\(2\))15583 50324 y Fq(\000)16129 51306 y Fu(~)-596 b(y)16776 50805 y Fp(\(2\))18034 51306 y Fw(\()p Fu(')p Fw(\))19769 50324 y Fq(\001)20595 51306 y Ft(\000)269 b Fu(k)22476 50805 y Fp(2)p Fs(l)23293 49960 y Fq(\021)24017 50263 y Fr(\000)p Fp(1)25275 49912 y Fq(\015)25275 50639 y(\015)25275 51367 y(\015)25948 52149 y Fp(1)26811 51306 y Ft(\024)28223 50486 y Fw(4)p Fu(J)29618 50046 y Fp(\(2\))p 28223 51027 2653 49 v 29040 52161 a Fu(\017)29532 51743 y Fp(2)29532 52508 y(1)31210 49233 y Fq( )32303 50486 y Fw(2)p Fu(r)33490 50046 y Fp(\(2\))34747 50486 y Fu(J)35536 50046 y Fp(\(2\))p 32303 51027 4491 49 v 33629 52189 a Fu(r)34210 51837 y Fp(\(3\))36926 49233 y Fq(!)37886 49535 y Fs(J)38483 49223 y FD(\(2\))39841 51306 y Fu(:)8250 b Fw(\(4.39\))1803 54862 y(The)545 b(pro)34 b(of)544 b(is)f(analogous)i(to)g(that)g(of)f (Lemma)g(3.46,)578 b(Lemma)543 b(4.14)h(b)34 b(eing)544 b(used)h(instead)f(of)h(Lemma)475 56312 y(3.31.)577 b(Considering)418 b(the)f(expressions)g(for)h Fu(r)21117 55873 y Fp(\(2\))22374 56312 y Fu(;)202 b(r)23494 55873 y Fp(\(3\))24751 56312 y Fu(;)g(J)26079 55873 y Fp(\(2\))27336 56312 y Fu(;)g(\017)28367 56494 y Fp(2)28892 56312 y Fw(,)420 b(w)-34 b(e)418 b(easily)e(obtain)i (that)g(the)g(righ)-34 b(t-hand)419 b(part)475 57751 y(of)404 b(\(4.38\))h(do)34 b(es)404 b(not)h(exceed)e(1)p Fu(=\017)15822 57933 y Fp(2)16348 57751 y Fw(.)538 b(Hence,)404 b(the)g(follo)-34 b(wing)405 b(corollary)e(holds:)p 0 TeXcolorgray 475 60557 a FF(Corollary)483 b(4.29.)p 0 TeXcolorgray 554 w FC(If)434 b Fu(b)11907 59918 y Fp(\(2\))11907 60904 y(0)13501 60557 y Fu(>)337 b(\017)15273 60739 y Fp(1)15799 60557 y Fu(k)16468 60117 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fp(2)p Fs(\016)20871 60557 y FC(,)433 b(then)17319 62384 y Fq(\015)17319 63111 y(\015)17319 63839 y(\015)17992 62433 y(\020)18716 63778 y Fu(H)19822 63278 y Fp(\(2\))21080 62796 y Fq(\000)21625 63778 y Fu(~)-596 b(y)22272 63278 y Fp(\(2\))23531 63778 y Fw(\()p Fu(')p Fw(\))25266 62796 y Fq(\001)26091 63778 y Ft(\000)270 b Fu(k)27973 63278 y Fp(2)p Fs(l)28790 62433 y Fq(\021)29514 62735 y Fr(\000)p Fp(1)30771 62384 y Fq(\015)30771 63111 y(\015)30771 63839 y(\015)31782 63778 y Fu(<)33400 62958 y Fw(1)p 33194 63499 1018 49 v 33194 64610 a Fu(\017)33686 64792 y Fp(2)34344 63778 y Fu(;)13747 b Fw(\(4.40\))15090 67134 y Fq(\015)15090 67861 y(\015)15090 68588 y(\015)15763 67182 y(\020)16487 68528 y Fu(H)17593 68027 y Fp(\(2\))18851 67546 y Fq(\000)19396 68528 y Fu(~)-596 b(y)20043 68027 y Fp(\(2\))21302 68528 y Fw(\()p Fu(')p Fw(\))23037 67546 y Fq(\001)23862 68528 y Ft(\000)270 b Fu(k)25744 68027 y Fp(2)p Fs(l)26561 67182 y Fq(\021)27285 67484 y Fr(\000)p Fp(1)28542 67134 y Fq(\015)28542 67861 y(\015)28542 68588 y(\015)29216 69371 y Fp(1)30078 68528 y Fu(<)31491 67708 y Fw(4)p Fu(c)32622 67890 y Fp(0)33147 67708 y Fu(k)33816 67268 y Fp(2+2)p Fs(s)35923 67391 y FD(2)p 31491 68249 4951 49 v 33457 69359 a Fu(\017)33949 69541 y Fp(2)36574 68528 y Fu(:)11517 b Fw(\(4.41\))p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 52 52 TeXDict begin 52 51 bop 0 TeXcolorgray 0 TeXcolorgray 475 3345 a Fp(52)14248 b(YULIA)313 b(KARPESHINA)f(AND)h(YOUNG-RAN)g (LEE)p 0 TeXcolorgray 475 6288 a Fw(4.6.2.)605 b FC(The)309 b(set)g Fo(O)8887 5649 y Fp(\(2\))8887 6446 y Fs(s)10346 4943 y Fq(\020)10934 5969 y Fu(~)11070 6288 y(b)11590 5849 y Fp(\(2\))12848 4943 y Fq(\021)13882 6288 y FC(for)g(smal)62 b(l)18657 5969 y Fu(~)18791 6288 y(b)19311 5849 y Fp(\(2\))20569 6288 y FC(.)606 b Fw(Ev)-34 b(erything)270 b(w)-34 b(e)269 b(considered)h(so)f(far)h(is)f(v)-67 b(alid)269 b(if)g Fu(b)43562 5649 y Fp(\(2\))43562 6635 y(0)45157 6288 y Fu(>)336 b(\017)46928 6470 y Fp(1)47454 6288 y Fu(k)48123 5849 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fp(2)p Fs(\016)52526 6288 y Fw(.)475 8490 y(Ho)-34 b(w)g(ev)g(er,)474 b(in)461 b(the)g(next)f(c)-34 b(hapter)461 b(and)g(later,)474 b Fu(b)22740 7851 y Fp(\(2\))22740 8837 y(0)24458 8490 y Fw(is)460 b(tak)-34 b(en)461 b(smaller,)474 b(since)460 b(the)g(recipro)34 b(cal)459 b(lattice)h(is)g(get-)475 10134 y(ting)439 b(\014ner)f(with)h(eac)-34 b(h)439 b(step.)642 b(T)-101 b(o)438 b(prepare)g(for)h(this,)447 b(let)438 b(us)h(consider)33362 9814 y Fu(~)33498 10134 y(b)34018 9694 y Fp(\(2\))35714 10134 y Fw(b)34 b(eing)438 b(close)g(to)h(a)f(v) -34 b(ertex)438 b(of)g Fu(K)50662 10316 y Fp(2)51189 10134 y Fw(:)475 12094 y(0)462 b Fu(<)f(b)3467 11455 y Fp(\(2\))3467 12441 y(0)5187 12094 y Ft(\024)h Fu(\017)7084 12276 y Fp(1)7610 12094 y Fu(k)8279 11654 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fp(2)p Fs(\016)12682 12094 y Fw(.)764 b(W)-101 b(e)478 b(sho)-34 b(w)481 b(that)f(for)f(suc) -34 b(h)26328 11774 y Fu(~)26465 12094 y(b)26985 11654 y Fp(\(2\))28242 12094 y Fw(,)498 b(the)480 b(resolv)-34 b(en)g(t)36409 10749 y Fq(\020)37133 12094 y Fu(H)38239 11654 y Fp(\(2\))39497 11112 y Fq(\000)40042 12094 y Fu(~)-596 b(y)40689 11654 y Fp(\(2\))41948 12094 y Fw(\()p Fu(')p Fw(\))43683 11112 y Fq(\001)44558 12094 y Ft(\000)320 b Fu(k)46490 11654 y Fp(2)p Fs(l)47307 10749 y Fq(\021)48031 11051 y Fr(\000)p Fp(1)49768 12094 y Fw(has)475 13835 y(no)402 b(more)g(than)h(t)-34 b(w)g(o)403 b(p)34 b(oles)402 b(in)g Fu(\010)15590 14017 y Fp(2)16518 13835 y Fw(and)h(ev)-34 b(en)402 b(its)g(small)f(neigh)-34 b(b)34 b(orho)g(o)g(d.)539 b(W)-101 b(e)401 b(surround)i(these)g(p)34 b(oles)401 b(b)-34 b(y)403 b(t)-34 b(w)g(o)475 15795 y(con)g(tours)364 b Fu(\015)6021 15156 y Fr(\006)404 b Fp(\(2\))5954 15953 y Fs(s)8778 15795 y Fw(and)364 b(pro)-34 b(v)g(e)363 b(an)h(estimate)g(for)f(the)h(norm)f(of)29108 14450 y Fq(\020)29832 15795 y Fu(H)30938 15355 y Fp(\(2\))32398 14813 y Fq(\000)32943 15795 y Fu(~)-596 b(y)33590 15355 y Fp(\(2\))34849 15795 y Fw(\()p Fu(')p Fw(\))36584 14813 y Fq(\001)37327 15795 y Ft(\000)188 b Fu(k)39127 15355 y Fp(2)p Fs(l)39944 14450 y Fq(\021)40668 14752 y Fr(\000)p Fp(1)42289 15795 y Fw(when)364 b Fu(')f Fw(is)g(outside)475 17535 y(these)404 b(con)-34 b(tours.)1803 18974 y(Let)4108 18668 y(~)3906 18974 y Fu(\010)4714 19156 y Fp(2)5575 18974 y Fw(b)34 b(e)334 b(the)9307 18497 y Fs(r)9758 18185 y FD(\(1\))p 9307 18695 1552 49 v 9847 19392 a Fp(2)10991 18974 y Fw(-neigh)-34 b(b)34 b(orho)g(o)g(d)336 b(of)f Fu(\010)21058 19156 y Fp(2)21584 18974 y Fw(.)515 b(It)335 b(is)g(easy)g(to)g(see)f(that)33490 18668 y(~)33288 18974 y Fu(\010)34096 19156 y Fp(2)34958 18974 y Ft(\032)36440 18668 y Fw(~)36238 18974 y Fu(\010)37046 19156 y Fp(1)37572 18974 y Fw(.)515 b(Supp)34 b(ose)336 b Ft(j)43310 18655 y Fu(~)43446 18974 y(b)43966 18535 y Fp(\(2\))45224 18974 y Ft(j)g Fw(=)g Fu(b)47696 18336 y Fp(\(2\))47696 19322 y(0)48954 18974 y Fw(,)349 b(i.e.,)475 20689 y(the)356 b(closest)g(v)-34 b(ertex)355 b(of)i Fu(K)12305 20871 y Fp(2)13187 20689 y Fw(for)14858 20369 y Fu(~)14994 20689 y(b)15514 20249 y Fp(\(2\))17128 20689 y Fw(is)e(\(0)p Fu(;)202 b Fw(0\).)523 b(W)-101 b(e)356 b(consider)f(the)i(functions)g Fu(\025)36490 20249 y Fp(\(1\))37748 20689 y Fw(\()-10 b Fu(~)-596 b(y)38856 20249 y Fp(\(1\))40115 20689 y Fw(\()p Fu(')p Fw(\)\))357 b(and)g Fu(\025)45694 20249 y Fp(\(2\))46951 20689 y Fw(\()-10 b Fu(~)-596 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Fp(2)29812 32775 y Fw(.)777 b(W)-101 b(e)483 b(base)h(our)f(further)i (considerations)f(on)475 34307 y(these)398 b(p)34 b(erturbation)399 b(series)f(expansions.)537 b(F)-101 b(or)22532 33987 y Fu(~)22668 34307 y(b)23188 33867 y Fp(\(2\))24844 34307 y Fw(b)34 b(eing)398 b(close)f(to)i(a)f(v)-34 b(ertex)351 b Fu(~)-559 b(e)398 b Fw(other)h(than)g(\(0)p Fu(;)202 b Fw(0\),)399 b(w)-34 b(e)399 b(tak)-34 b(e)465 35907 y Fu(~)-596 b(y)1112 35467 y Fp(\(2\))2370 35907 y Fw(\()p Fu(')p Fw(\))338 b(=)368 b Fu(~)-638 b Fn({)6530 36089 y Fp(2)7056 35907 y Fw(\()p Fu(')p Fw(\))270 b(+)10137 35587 y Fu(~)10273 35907 y(b)10793 35467 y Fp(\(2\))12320 35907 y Ft(\000)223 b Fu(~)-560 b(e)q Fw(.)1803 37425 y(F)-101 b(rom)395 b(no)-34 b(w)396 b(on,)h(w)-34 b(e)395 b(denote)h(the)f(solutions)h Fu(')23106 36985 y Fr(\006)23106 37724 y Fs(\017)23486 37847 y FD(0)24397 37425 y Fw(of)f(the)h (equation)f Fu(\025)33494 36985 y Fp(\(1\))34752 36443 y Fq(\000)35297 37425 y Fu(~)-596 b(y)35944 36985 y 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Fp(\(1\))33638 69541 y Fw(\()p Fu(')p Fw(\)\))338 b(=)f Fu(k)38131 69101 y Fp(2)p Fs(l)39270 69541 y Fw(inside)321 b(this)i(circle.)509 b(Th)-34 b(us,)p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 53 53 TeXDict begin 53 52 bop 0 TeXcolorgray 0 TeXcolorgray 2848 3345 a Fp(SPECTRAL)406 b(PR)-26 b(OPER)-78 b(TIES)406 b(OF)h(A)h(POL)-78 b(YHARMONIC)405 b(OPERA)-78 b(TOR)406 b(WITH)h(LIMIT-PERIODIC)e(POTENTIAL)1430 b(53)p 0 TeXcolorgray 475 6197 a Fw(an)-34 b(y)283 b(solution)i(of)e(\(4.42\))h(is)f(in)g (the)h(circle)e(of)h(the)h(radius)g Fu(r)26488 5758 y Fp(\(3\))27745 6197 y Fu(=)p Fw(4)f(surrounding)i Fu(')36660 5559 y Fr(\006)p Fp(\(1\))36660 6545 y(0)38649 6197 y Fw(,)307 b(the)284 b(p)34 b(oin)-34 b(t)284 b Fu(')45097 5559 y Fr(\006)p Fp(\(1\))45097 6545 y(0)47370 6197 y Fw(b)34 b(eing)283 b(in)475 7771 y(the)404 b(\()p Fu(r)3614 7331 y Fp(\(3\))4872 7771 y Fu(=)p Fw(4\)-neigh)-34 b(b)34 b(orho)g(o)g(d)405 b(of)16154 7465 y(~)15952 7771 y Fu(\010)16760 7953 y Fp(2)17286 7771 y Fw(.)538 b(It)404 b(remains)g(to)g(c)-34 b(hec)g(k)404 b(that)h(\(4.42\))f(has)g(no)h(more)e(than)i(t)-34 b(w)g(o)405 b(solutions)475 9488 y(and)514 b(no)g(more)f(than)i(one)f (in)g(a)f(vicinit)-34 b(y)513 b(of)h(eac)-34 b(h)514 b Fu(')25363 8849 y Fr(\006)p Fp(\(1\))25363 9835 y(0)27353 9488 y Fw(.)867 b(W)-101 b(e)513 b(construct)h(the)g(disk)g(of)g(the)g (radius)g Fu(r)49056 9048 y Fp(\(3\))50313 9488 y Fu(=)p Fw(4)475 11249 y(cen)-34 b(tered)389 b(at)g Fu(')7537 10610 y Fr(\006)p Fp(\(1\))7537 11596 y(0)9527 11249 y Fw(,)i(describ)34 b(ed)389 b(ab)34 b(o)-34 b(v)g(e,)392 b(and)e(note)f(that)i Ft(j)p Fu(\025)28039 10809 y Fp(\(1\))29296 11249 y Fw(\()-10 b Fu(~)-596 b(y)30404 10809 y Fp(\(1\))31663 11249 y Fw(\()p Fu(')p Fw(\)\))240 b Ft(\000)f Fu(k)35960 10809 y Fp(2)p Fs(l)36777 11249 y Ft(j)337 b Fw(=)38863 10772 y Fp(1)p 38863 10970 471 49 v 38863 11667 a(2)39466 11249 y Fu(l)24 b(k)40521 10809 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1)42541 11249 y Fu(b)43061 10610 y Fp(\(2\))43061 11596 y(0)44318 11249 y Fu(r)44899 10809 y Fp(\(3\))46359 11249 y Fw(\(1)269 b(+)h Fu(o)p Fw(\(1\)\))475 12688 y(on)451 b(the)h(circle.)679 b(Using)451 b(\(4.44\))h(and)g(Rouc)-34 b(h)g(\023)-572 b(e's)452 b(Theorem,)463 b(w)-34 b(e)452 b(obtain)g(that)h(there)e(is)g(only)g(one)h(solution)g(of)475 14288 y(\(4.42\))438 b(in)g(the)g(disk.)639 b(If)11792 13968 y Fu(~)11928 14288 y(b)438 b Fw(is)f(close)g(to)h(a)g(v)-34 b(ertex)437 b(other)h(than)g(\(0)p Fu(;)202 b Fw(0\),)447 b(the)438 b(considerations)g(are)f(the)h(same)f(up)475 15727 y(to)404 b(a)g(parallel)f(shift.)p 50750 15893 665 831 v 0 TeXcolorgray 475 18371 a FF(Lemma)601 b(4.31.)p 0 TeXcolorgray 620 w FC(Supp)-62 b(ose)541 b Fw(0)c Fu(<)f(b)16907 17732 y Fp(\(2\))16907 18718 y(0)18701 18371 y Ft(\024)g Fu(\017)20672 18553 y Fp(1)21198 18371 y Fu(k)21867 17931 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fp(2)p Fs(\016)26815 18371 y FC(and)543 b Fu(')536 b Ft(2)32155 18065 y Fw(~)31953 18371 y Fu(\010)32761 18553 y Fp(2)33831 18371 y FC(ob)-62 b(eys)541 b(the)i(ine)-62 b(quality)541 b(analo)-62 b(gous)541 b(to)475 20223 y(\(4.43\):)4626 18829 y Fq(\014)4626 19556 y(\014)4626 20284 y(\014)5030 20223 y Fu(')269 b Ft(\000)g Fu(')8097 19584 y Fr(\006)p Fp(\(1\))8097 20570 y(0)10086 18829 y Fq(\014)10086 19556 y(\014)10086 20284 y(\014)10827 20223 y Fu(<)337 b(r)12688 19783 y Fp(\(3\))13945 20223 y FC(.)557 b(Then,)13952 22949 y Fu(@)p 13556 23490 1504 49 v 13556 24601 a(@)67 b(')15192 23769 y(\025)15899 23269 y Fp(\(2\))17359 22424 y Fq(\020)18073 23769 y Fu(~)-596 b(y)18720 23269 y Fp(\(2\))19978 23769 y Fw(\()p Fu(')p Fw(\))21713 22424 y Fq(\021)22775 23769 y Fw(=)23718 23966 y Fs(k)24 b Fr(!1)26505 23769 y Ft(\006)p Fw(2)p Fu(l)g(b)28960 23130 y Fp(\(2\))28960 24116 y(0)30217 23769 y Fu(k)30886 23269 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1)32906 22787 y Fq(\000)33461 23769 y Fw(1)270 b(+)f Fu(o)p Fw(\(1\))37685 22787 y Fq(\001)38241 23769 y Fu(:)9850 b Fw(\(4.45\))1803 26784 y(The)405 b(pro)34 b(of)404 b(is)g(completely)f(analogous)i(to)g(that)g(of)f (Lemma)g(3.49.)p 0 TeXcolorgray 475 29519 a FF(De\014nition)352 b(4.32.)p 0 TeXcolorgray 472 w Fw(Let)305 b(\000)13038 28880 y Fr(\006)p Fp(\(2\))13038 29677 y Fs(s)15229 28174 y Fq(\020)15817 29200 y Fu(~)15953 29519 y(b)16473 29080 y Fp(\(2\))17731 28174 y Fq(\021)18760 29519 y Fw(b)34 b(e)305 b(the)g(op)34 b(en)306 b(disks)f(cen)-34 b(tered)305 b(at)g Fu(')34935 28880 y Fr(\006)p Fp(\(2\))34935 29867 y(0)37230 29519 y Fw(with)h(radius)f Fu(r)44020 29080 y Fp(\(3\))45277 29519 y Fw(;)338 b Fu(\015)46647 28880 y Fr(\006)p Fp(\(2\))46580 29677 y Fs(s)48636 28538 y Fq(\000)49056 29200 y Fu(~)49192 29519 y(b)49712 29080 y Fp(\(2\))50970 28538 y Fq(\001)475 31721 y Fw(b)34 b(e)404 b(their)g(b)34 b(oundary)404 b(circles)f(and)i Fo(O)17343 31082 y Fp(\(2\))17343 31879 y Fs(s)18600 30739 y Fq(\000)19020 31401 y Fu(~)19156 31721 y(b)19676 31281 y Fp(\(2\))20934 30739 y Fq(\001)21826 31721 y Fw(=)336 b(\000)23863 31082 y Fp(+\(2\))23863 31879 y Fs(s)26122 31721 y Ft([)269 b Fw(\000)27957 31082 y Fr(\000)p Fp(\(2\))27957 31879 y Fs(s)29946 31721 y Fw(.)p 0 TeXcolorgray 475 34554 a FF(Lemma)463 b(4.33.)p 0 TeXcolorgray 554 w FC(F)-93 b(or)434 b(any)f Fu(')h FC(in)g Fu(\010)16944 34736 y Fp(2)17739 34554 y Ft(n)269 b Fo(O)19472 33915 y Fp(\(2\))19472 34712 y Fs(s)20730 34554 y Fw(\()21065 34234 y Fu(~)21201 34554 y(b)21721 34114 y Fp(\(2\))22979 34554 y Fw(\))p FC(,)16191 37257 y Ft(j)p Fu(\025)17235 36756 y Fp(\(2\))18492 37257 y Fw(\()-10 b Fu(~)-596 b(y)19600 36756 y Fp(\(2\))20859 37257 y Fw(\()p Fu(')p Fw(\)\))270 b Ft(\000)f Fu(k)25216 36756 y Fp(2)p Fs(l)26034 37257 y Ft(j)336 b(\025)h Fu(b)28507 36618 y Fp(\(2\))28507 37604 y(0)29764 37257 y Fu(k)30433 36756 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1)p Fr(\000)p Fs(\016)33634 37257 y Fu(r)34215 36756 y Fp(\(3\))35473 37257 y Fu(:)12618 b Fw(\(4.46\))1803 39691 y(The)405 b(pro)34 b(of)404 b(is)g(analogous)h(to)f(that)i(of)e (Lemma)g(3.51.)p 0 TeXcolorgray 475 42427 a FF(Lemma)463 b(4.34.)p 0 TeXcolorgray 554 w FC(F)-93 b(or)434 b(any)f Fu(')337 b Ft(2)f Fu(\010)16504 42609 y Fp(2)17300 42427 y Ft(n)269 b Fo(O)19033 41788 y Fp(\(2\))19033 42584 y Fs(s)20493 41081 y Fq(\020)21080 42107 y Fu(~)21216 42427 y(b)21736 41987 y Fp(\(2\))22994 41081 y Fq(\021)23718 42427 y FC(,)13815 44531 y Fq(\015)13815 45259 y(\015)13815 45986 y(\015)14489 44580 y(\020)15213 45925 y Fu(H)16319 45425 y Fp(\(2\))17778 44580 y Fq(\020)18492 45925 y Fu(~)-596 b(y)19139 45425 y Fp(\(2\))20398 45925 y Fw(\()p Fu(')p Fw(\))22133 44580 y Fq(\021)23127 45925 y Ft(\000)269 b Fu(k)25008 45425 y Fp(2)p Fs(l)25825 44580 y Fq(\021)26549 44882 y Fr(\000)p Fp(1)27807 44531 y Fq(\015)27807 45259 y(\015)27807 45986 y(\015)28817 45925 y Fu(<)33366 45105 y Fw(16)p 30229 45646 7487 49 v 30229 47095 a Fu(b)30749 46456 y Fp(\(2\))30749 47442 y(0)32007 47095 y Fu(r)32588 46743 y Fp(\(3\))33845 47095 y Fu(k)34514 46745 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1)p Fr(\000)p Fs(\016)37848 45925 y Fu(;)10243 b Fw(\(4.47\))13552 49458 y Fq(\015)13552 50185 y(\015)13552 50912 y(\015)14226 49506 y(\020)14950 50852 y Fu(H)16056 50351 y Fp(\(2\))17515 49506 y Fq(\020)18229 50852 y Fu(~)-596 b(y)18876 50351 y Fp(\(2\))20135 50852 y Fw(\()p Fu(')p Fw(\))21870 49506 y Fq(\021)22864 50852 y Ft(\000)269 b Fu(k)24745 50351 y Fp(2)p Fs(l)25562 49506 y Fq(\021)26286 49808 y Fr(\000)p Fp(1)27544 49458 y Fq(\015)27544 50185 y(\015)27544 50912 y(\015)28217 51695 y Fp(1)29080 50852 y Fu(<)31457 50032 y Fw(64)p Fu(c)33194 50214 y Fp(0)33719 50032 y Fu(k)34388 49592 y Fp(2+2)p Fs(s)36495 49715 y FD(2)p 30492 50573 V 30492 52021 a Fu(b)31012 51382 y Fp(\(2\))31012 52368 y(0)32270 52021 y Fu(r)32851 51669 y Fp(\(3\))34108 52021 y Fu(k)34777 51671 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1)p Fr(\000)p Fs(\016)38111 50852 y Fu(:)9980 b Fw(\(4.48\))p 0 TeXcolorgray 475 54868 a FF(Corollary)483 b(4.35.)p 0 TeXcolorgray 554 w FC(If)434 b Fu(\017)11879 55050 y Fp(2)12405 54868 y Fu(k)13074 54428 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fp(2)p Fs(\016)17814 54868 y Fu(<)336 b(b)19613 54229 y Fp(\(2\))19613 55215 y(0)21208 54868 y Ft(\024)g Fu(\017)22979 55050 y Fp(1)23505 54868 y Fu(k)24174 54428 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fp(2)p Fs(\016)28577 54868 y FC(,)434 b(and)f Fu(')336 b Ft(2)h Fu(\010)34819 55050 y Fp(2)35614 54868 y Ft(n)270 b Fo(O)37348 54229 y Fp(\(2\))37348 55026 y Fs(s)38807 53523 y Fq(\020)39395 54548 y Fu(~)39531 54868 y(b)40051 54428 y Fp(\(2\))41309 53523 y Fq(\021)42033 54868 y FC(,)433 b(then)17049 56973 y Fq(\015)17049 57700 y(\015)17049 58427 y(\015)17723 57021 y(\020)18447 58367 y Fu(H)19553 57866 y Fp(\(2\))21012 57021 y Fq(\020)21726 58367 y Fu(~)-596 b(y)22373 57866 y Fp(\(2\))23632 58367 y Fw(\()p Fu(')p Fw(\))25367 57021 y Fq(\021)26361 58367 y Ft(\000)269 b Fu(k)28242 57866 y Fp(2)p Fs(l)29059 57021 y Fq(\021)29783 57323 y Fr(\000)p Fp(1)31041 56973 y Fq(\015)31041 57700 y(\015)31041 58427 y(\015)32051 58367 y Fu(<)33669 57547 y Fw(1)p 33463 58088 1018 49 v 33463 59222 a Fu(\017)33955 58804 y Fp(2)33955 59569 y(2)34614 58367 y Fu(;)13477 b Fw(\(4.49\))14820 61584 y Fq(\015)14820 62312 y(\015)14820 63039 y(\015)15494 61633 y(\020)16218 62978 y Fu(H)17324 62478 y Fp(\(2\))18783 61633 y Fq(\020)19497 62978 y Fu(~)-596 b(y)20144 62478 y Fp(\(2\))21403 62978 y Fw(\()p Fu(')p Fw(\))23138 61633 y Fq(\021)24132 62978 y Ft(\000)269 b Fu(k)26013 62478 y Fp(2)p Fs(l)26830 61633 y Fq(\021)27554 61935 y Fr(\000)p Fp(1)28812 61584 y Fq(\015)28812 62312 y(\015)28812 63039 y(\015)29485 63822 y Fp(1)30348 62978 y Fu(<)31760 62158 y Fw(4)p Fu(c)32891 62340 y Fp(0)33416 62158 y Fu(k)34085 61718 y Fp(2+2)p Fs(s)36192 61841 y FD(2)p 31760 62700 4951 49 v 33726 63833 a Fu(\017)34218 63416 y Fp(2)34218 64180 y(2)36843 62978 y Fu(:)11248 b Fw(\(4.50\))1803 66589 y(The)625 b(corollary)e(follo)-34 b(ws)626 b(from)e(the)h (condition)h(on)f Fu(b)27318 65950 y Fp(\(2\))27318 66936 y(0)28576 66589 y Fw(,)679 b(for)625 b(the)g(form)-34 b(ula)625 b Fu(r)39225 66149 y Fp(\(3\))41186 66589 y Fw(=)704 b Fu(r)43414 66149 y Fp(\(2\))44671 66589 y Fu(k)45340 66149 y Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)48179 66272 y FD(3)48642 66149 y Fr(\000)p Fs(\016)50582 66589 y Fw(=)475 68102 y Fu(k)1144 67662 y Fr(\000)p Fp(8)p Fr(\000)p Fp(6)p Fs(s)3983 67785 y FD(1)4445 67662 y Fr(\000)p Fp(4)p Fs(s)6082 67785 y FD(2)6543 67662 y Fr(\000)p Fp(4)p Fs(s)8180 67785 y FD(3)8642 67662 y Fr(\000)p Fp(5)p Fs(\016)10812 68102 y Fw(and)463 b(estimate)g (\(4.2\).)714 b(The)462 b(pro)34 b(of)463 b(of)g(the)g(lemma)f(is)g (analogous)h(to)g(that)h(of)f(Lemma)475 69541 y(3.52.)p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 54 54 TeXDict begin 54 53 bop 0 TeXcolorgray 0 TeXcolorgray 475 3345 a Fp(54)14248 b(YULIA)313 b(KARPESHINA)f(AND)h(YOUNG-RAN)g (LEE)p 0 TeXcolorgray 7827 6001 a Fw(5.)605 b Fv(The)459 b Fu(n)p Fv(-th)f(Step)g(of)h(Appr)-27 b(o)g(xima)-81 b(tion.)602 b(Swiss)458 b(Cheese)g(Method.)475 8160 y Fw(5.1.)605 b FF(P)-39 b(erturbation)678 b(F)-116 b(orm)-39 b(ulae.)606 b Fw(On)588 b(the)h Fu(n)p Fw(-th)f(step,)634 b Fu(n)643 b Ft(\025)g Fw(4,)633 b(w)-34 b(e)589 b(c)-34 b(ho)34 b(ose)588 b Fu(s)39710 8342 y Fs(n)40979 8160 y Fw(=)643 b(2)p Fu(s)43739 8342 y Fs(n)p Fr(\000)p Fp(1)46156 8160 y Fw(de\014ne)589 b(the)475 9848 y(op)34 b(erator)404 b Fu(H)6470 9209 y Fp(\()p Fs(n)p Fp(\))6372 10006 y Fs(\013)8232 9848 y Fw(b)-34 b(y)404 b(the)h(form)-34 b(ula:)9650 13045 y Fu(H)10756 12545 y Fp(\()p Fs(n)p Fp(\))10658 13345 y Fs(\013)12451 13045 y Fw(=)337 b Fu(H)14837 12545 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))17666 13045 y Fw(+)269 b Fu(\013)t(W)20802 13227 y Fs(n)21429 13045 y Fu(;)1414 b Fw(\(0)337 b Ft(\024)g Fu(\013)k Ft(\024)c Fw(1\))p Fu(;)2626 b(W)33455 13227 y Fs(n)34418 13045 y Fw(=)37413 11527 y Fs(M)38317 11638 y Fy(n)37272 11894 y Fq(X)35697 14519 y Fs(r)26 b Fp(=)p Fs(M)37784 14642 y Fy(n)p Fx(\000)p FD(1)39396 14519 y Fp(+1)40800 13045 y Fu(V)41507 13227 y Fs(r)42013 13045 y Fu(;)475 16617 y Fw(where)317 b Fu(M)5069 16799 y Fs(n)6013 16617 y Fw(is)g(c)-34 b(ho)34 b(osen)318 b(in)g(suc)-34 b(h)318 b(a)g(w)-34 b(a)g(y)318 b(that)h(2)21993 16178 y Fs(M)22897 16289 y Fy(n)23855 16617 y Ft(\031)337 b Fu(k)25804 16178 y Fs(s)26239 16289 y Fy(n)26861 16617 y Fw(.)510 b(Ob)-34 b(viously)-101 b(,)334 b(the)318 b(p)34 b(erio)g(ds)317 b(of)h Fu(W)42177 16799 y Fs(n)43121 16617 y Fw(are)f(2)45664 16178 y Fs(M)46568 16289 y Fy(n)47134 16178 y Fr(\000)p Fp(1)48392 16617 y Fw(\()p Fu(b)49383 16799 y Fp(1)49909 16617 y Fu(;)202 b Fw(0\))475 18076 y(and)399 b(2)3432 17636 y Fs(M)4336 17747 y Fy(n)4903 17636 y Fr(\000)p Fp(1)6160 18076 y Fw(\(0)p Fu(;)202 b(b)8296 18258 y Fp(2)8822 18076 y Fw(\).)537 b(W)-101 b(e)399 b(write)f(them)i(in)f (the)g(form:)536 b Fu(N)26237 18258 y Fs(n)p Fr(\000)p Fp(1)28324 18076 y Ft(\001)259 b Fu(:::)f Ft(\001)g Fu(N)31758 18258 y Fp(1)32284 18076 y Fw(\()p Fu(a)33396 18258 y Fp(1)33922 18076 y Fu(;)202 b Fw(0\))399 b(and)g Fu(N)39262 18258 y Fs(n)p Fr(\000)p Fp(1)41350 18076 y Ft(\001)258 b Fu(:::)g Ft(\001)g Fu(N)44783 18258 y Fp(1)45309 18076 y Fw(\(0)p Fu(;)202 b(a)47566 18258 y Fp(2)48092 18076 y Fw(\),)400 b(here)475 19654 y Fu(N)1449 19836 y Fs(n)p Fr(\000)p Fp(1)3689 19654 y Fw(=)412 b(2)5650 19214 y Fs(M)6554 19325 y Fy(n)7120 19214 y Fr(\000)p Fs(M)8756 19337 y Fy(n)p Fx(\000)p FD(1)10423 19654 y Fw(,)11353 19176 y Fp(1)p 11353 19374 471 49 v 11353 20072 a(4)11956 19654 y Fu(k)12625 19214 y Fs(s)13060 19325 y Fy(n)13627 19214 y Fr(\000)p Fs(s)14794 19337 y Fy(n)p Fx(\000)p FD(1)16873 19654 y Fu(<)g(N)19202 19836 y Fs(n)p Fr(\000)p Fp(1)21443 19654 y Fu(<)f Fw(4)p Fu(k)24072 19214 y Fs(s)24507 19325 y Fy(n)25074 19214 y Fr(\000)p Fs(s)26241 19337 y Fy(n)p Fx(\000)p FD(1)27909 19654 y Fw(.)674 b(Note)449 b(that)h Ft(k)p Fu(W)36316 19836 y Fs(n)36943 19654 y Ft(k)37549 19836 y Fr(1)38957 19654 y Ft(\024)40312 18744 y Fq(P)41591 19095 y Fs(M)42495 19206 y Fy(n)41591 20012 y Fs(r)26 b Fp(=)p Fs(M)43678 20135 y Fy(n)p Fx(\000)p FD(1)45289 20012 y Fp(+1)46749 19654 y Ft(k)p Fu(V)48062 19836 y Fs(r)48568 19654 y Ft(k)49174 19836 y Fr(1)50582 19654 y Fu(<)475 21247 y Fw(exp\()p Ft(\000)p Fu(k)4410 20807 y Fs(\021)32 b(s)5349 20930 y Fy(n)p Fx(\000)p FD(1)7016 21247 y Fw(\))p Fu(:)1803 22686 y Fw(The)401 b Fu(n)p Fw(-th)g(step)h(is)e(analogous)i(to)f(the)g(third)g(step)h(up) f(to)g(replacemen)-34 b(t)401 b(the)g(indices)g(3)f(b)-34 b(y)401 b Fu(n)p Fw(,)g(2)g(b)-34 b(y)401 b Fu(n)262 b Ft(\000)g Fw(1,)475 24125 y(the)404 b(pro)34 b(duct)405 b Fu(N)8085 24307 y Fp(2)8611 24125 y Fu(N)9585 24307 y Fp(1)10515 24125 y Fw(b)-34 b(y)404 b Fu(N)13172 24307 y Fs(n)p Fr(\000)p Fp(1)15270 24125 y Ft(\001)269 b Fu(:::)f Ft(\001)h Fu(N)18735 24307 y Fp(1)19261 24125 y Fw(,)404 b(etc.)1803 25564 y(W)-101 b(e)404 b(note)g(that)i Fu(k)38 b Fw(,)403 b(satisfying)i(\(4.2\),)f(ob)34 b(eys)404 b(the)h(analogous)g(condition)g(for)f(the)g Fu(n)p Fw(-th)h(step:)17375 27426 y(^)17100 27732 y Fu(C)87 b Fw(\(1)270 b(+)f Fu(s)21180 27914 y Fs(n)p Fr(\000)p Fp(2)23009 27732 y Fw(\))p Fu(k)24149 27232 y Fp(2+2)p Fs(s)26256 27355 y Fy(n)p Fx(\000)p FD(2)28127 27732 y Fw(ln)202 b Fu(k)375 b(<)336 b(k)32293 27232 y Fs(\021)32 b(s)33232 27355 y Fy(n)p Fx(\000)p FD(2)49034 27732 y Fw(\(5.1\))475 29901 y(with)384 b(the)g(same)f (constan)-34 b(t)13441 29594 y(^)13166 29901 y Fu(C)87 b Fw(.)532 b(The)384 b(inequalit)-34 b(y)383 b(\(5.1\))h(can)g(b)34 b(e)383 b(obtained)h(from)g(\(4.2\))g(b)-34 b(y)384 b(induction.)532 b(This)384 b(is)475 31340 y(an)401 b(imp)34 b(ortan)-34 b(t)403 b(fact:)538 b(it)401 b(means)h(that)g(the)g(lo)-34 b(w)g(er)401 b(b)34 b(ound)403 b(for)f Fu(k)439 b Fw(do)34 b(es)401 b(not)h(gro)-34 b(w)402 b(with)g Fu(n)p Fw(,)g(i.e.,)e(all)h (steps)h(hold)475 32779 y(uniformly)i(in)g Fu(k)442 b Fw(for)405 b Fu(k)374 b(>)337 b(k)13259 32961 y Fr(\003)13785 32779 y Fw(,)404 b Fu(k)15157 32961 y Fr(\003)16086 32779 y Fw(b)34 b(eing)404 b(in)-34 b(tro)34 b(duced)405 b(b)-34 b(y)405 b(\(4.2\).)539 b(F)-101 b(urther)405 b(w)-34 b(e)404 b(assume)h Fu(k)375 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b(lemma)f(is)h(pro)-34 b(v)g(ed)472 b(b)-34 b(y)472 b(induction)g(pro)34 b(cedure.)741 b(F)-101 b(or)471 b Fu(m)450 b Fw(=)f(2)471 b(it)h(holds)g(b)-34 b(y)472 b(P)-34 b(art)472 b(1)f(of)h(Lemma)f(4.28)475 42684 y(and)391 b(b)-34 b(y)390 b(De\014nition)h(4.32.)533 b(F)-101 b(or)390 b Fu(m)337 b(>)g Fw(2)390 b(it)g(follo)-34 b(ws)390 b(from)g(Corollary)g(5.21)f(and)i(De\014nition)g(5.26.)533 b(Rigorously)475 44123 y(sp)34 b(eaking,)392 b(w)-34 b(e)390 b(assume)h(for)f(no)-34 b(w)391 b(that)g(the)f(lemma)g(holds)g (for)g Fu(m)337 b Ft(\024)f Fu(n)241 b Ft(\000)g Fw(2.)533 b(A)-34 b(t)391 b(the)f(end)h(of)f(this)g(section,)j(w)-34 b(e)475 45562 y(obtain)444 b(the)h(analog)f(of)g(the)g(lemma)g(for)g (the)g(next)g(step)g(\()p Fu(n)403 b Ft(!)g Fu(n)295 b Fw(+)h(1\),)454 b(this)444 b(will)f(complete)h(the)g(induction)475 47080 y(pro)34 b(cedure.)1000 b(Corollaries)558 b(5.2)g(and)h(5.3)f (are)f(based)i(on)g(the)g(fact)g(that)g Fu(P)36048 46640 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))39167 47080 y Fw(con)-34 b(tains)559 b(no)g(more)f(than)475 48530 y(4)p Fu(k)1750 48090 y Fp(2\()p Fs(s)3021 48201 y Fy(n)3588 48090 y Fr(\000)p Fs(s)4755 48213 y Fy(n)p Fx(\000)p FD(1)6367 48090 y Fp(\))7192 48530 y Fw(elemen)-34 b(ts)404 b(and)h(a)g(similar)e (estimate)h(holds)h(for)f Fu(P)30327 48090 y Fp(\()p Fs(n)p Fr(\000)p Fp(2\))32888 48530 y Fw(.)1803 49969 y(Ob)-34 b(viously)-101 b(,)350 b Fu(\010)8490 50151 y Fs(n)9452 49969 y Fw(has)337 b(a)g(structure)g(of)g(Swiss)g(c)-34 b(heese,)349 b(more)336 b(and)h(more)f(holes)h(of)f(smaller)g(and)h (smaller)f(radii)475 51408 y(app)34 b(ear)409 b(at)h(eac)-34 b(h)410 b(step)g(of)h(appro)-34 b(ximation)410 b(\(Fig.)556 b(8\).)f(Note)410 b(that)h(the)f(disks)g(are)f(more)g(and)h(more)g (precisely)475 53125 y(\\targeted")304 b(at)h(eac)-34 b(h)304 b(step)g(of)g(appro)-34 b(ximation.)507 b(A)-34 b(t)304 b(the)h(n-th)g(step)f(the)h(disks)f(of)g Fo(O)38313 52486 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))38313 53314 y Fr(\003)41177 53125 y Fw(are)f(cen)-34 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b(is)g(completely)g (analogous)h(to)f(that)i(in)e(the)h(third)f(step,)372 b(w)-34 b(e)364 b(only)f(ha)-34 b(v)g(e)363 b(to)h(pro)-34 b(vide)363 b(that)h(condition)475 69541 y(\(5.1\))457 b(is)g(su\016cien)-34 b(t)458 b(for)f(all)f(necessary)g(estimates.)697 b(W)-101 b(e)457 b(start)g(with)h(an)f(analog)g(of)h(Lemma)e(4.10:)644 b(w)-34 b(e)457 b(shift)p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 56 56 TeXDict begin 56 55 bop 0 TeXcolorgray 0 TeXcolorgray 475 3345 a Fp(56)14248 b(YULIA)313 b(KARPESHINA)f(AND)h(YOUNG-RAN)g (LEE)p 0 TeXcolorgray 475 6001 a Fw(indices)367 b(1)337 b Ft(!)f Fu(n)196 b Ft(\000)g Fw(2)p Fu(;)202 b Fw(2)336 b Ft(!)h Fu(n)196 b Ft(\000)g Fw(1)366 b(and)i(replace)e(the)i(co)34 b(e\016cien)-34 b(t)367 b(4)p Fu(c)30839 6183 y Fp(0)31731 6001 y Fw(in)h(the)g(estimate)f(\(4.20\))h(b)-34 b(y)368 b(4)45696 5562 y Fs(n)p Fr(\000)p Fp(2)47524 6001 y Fu(c)48049 6183 y Fp(0)48575 6001 y Fw(.)526 b(The)475 7531 y(op)34 b(erator)5755 7225 y(~)5438 7531 y Fu(H)6544 7091 y Fp(\()p Fs(n)p Fr(\000)p Fp(2\))9583 7531 y Fw(is)478 b Fu(H)11982 7091 y Fp(\()p Fs(n)p Fr(\000)p Fp(2\))15021 7531 y Fw(extended)h(from) f Fu(Q)24144 7713 y Fs(n)p Fr(\000)p Fp(2)26451 7531 y Fw(to)h Fu(Q)28965 7713 y Fs(n)p Fr(\000)p Fp(1)30794 7531 y Fw(.)761 b(By)478 b(analogy)h(with)g(\(4.16\),)487 b Fu(~)-596 b(y)45825 7091 y Fp(\()p Fs(n)p Fr(\000)p Fp(2\))48386 7531 y Fw(\()p Fu(')p Fw(\))461 b(=)507 9131 y Fu(~)-638 b Fn({)1283 9313 y Fs(n)p Fr(\000)p Fp(2)3111 9131 y Fw(\()p Fu(')p Fw(\))275 b(+)6201 8811 y Fu(~)6337 9131 y(b)6857 8691 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))9418 9131 y Fw(,)411 b(the)g(function)h(b)34 b(eing)410 b(de\014ned)i(in)e Fu(\010)26696 9313 y Fs(n)p Fr(\000)p Fp(2)28525 9131 y Fw(.)557 b(Th)-34 b(us,)413 b(w)-34 b(e)411 b(ha)-34 b(v)g(e)411 b(the)g(follo)-34 b(wing)411 b(lemma)f(and)h(a)475 10570 y(corollary:)p 0 TeXcolorgray 475 12980 a FF(Lemma)463 b(5.4.)p 0 TeXcolorgray 554 w FC(If)9276 12660 y Fu(~)9412 12980 y(b)9932 12540 y Fp(\()p Fs(n)p 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18209 y Fw(\()p Fu(')p Fw(\))26187 16864 y Fq(\021)27180 18209 y Ft(\000)270 b Fu(k)29062 17709 y Fp(2)p Fs(l)29879 16864 y Fq(\021)30603 17166 y Fr(\000)p Fp(1)31861 16815 y Fq(\015)31861 17543 y(\015)31861 18270 y(\015)32871 18209 y Fu(<)35140 17389 y Fw(1)p 34283 17931 2321 49 v 34283 19064 a Fu(\017)34775 18647 y Fp(2)34775 19411 y Fs(n)p Fr(\000)p Fp(2)49034 18209 y Fw(\(5.5\))12196 20700 y Fq(\015)12196 21427 y(\015)12196 22155 y(\015)12870 20749 y(\020)13911 21788 y Fw(~)13594 22094 y Fu(H)14700 21594 y Fp(\()p Fs(n)p Fr(\000)p Fp(2\))17462 20749 y Fq(\020)18176 22094 y Fu(~)-596 b(y)18823 21594 y Fp(\()p Fs(n)p Fr(\000)p Fp(2\))21384 22094 y Fw(\()p Fu(')p Fw(\))23119 20749 y Fq(\021)24113 22094 y Ft(\000)269 b Fu(k)25994 21594 y Fp(2)p Fs(l)26812 20749 y Fq(\021)27536 21051 y Fr(\000)p Fp(1)28793 20700 y Fq(\015)28793 21427 y(\015)28793 22155 y(\015)29467 22937 y Fp(1)30329 22094 y Fu(<)31741 21274 y Fw(4)32347 20834 y Fs(n)p Fr(\000)p Fp(2)34176 21274 y Fu(c)34701 21456 y Fp(0)35226 21274 y Fu(k)35895 20834 y Fp(2+2)p Fs(s)38002 20957 y Fy(n)p Fx(\000)p FD(1)p 31741 21815 7930 49 v 34546 22949 a Fu(\017)35038 22531 y Fp(2)35038 23296 y Fs(n)p Fr(\000)p Fp(2)49034 22094 y Fw(\(5.6\))p 0 TeXcolorgray 2409 25218 a(\(2\))p 0 TeXcolorgray 554 w FC(The)338 b(numb)-62 b(er)338 b(of)g(zer)-62 b(os)337 b(of)g Fw(det)18419 23872 y Fq(\020)19143 25218 y Fu(I)157 b Fw(+)21157 24911 y(~)20838 25218 y Fu(A)21747 25400 y Fs(n)p Fr(\000)p Fp(2)23575 24236 y Fq(\000)24121 25218 y Fu(~)-596 b(y)24768 24778 y Fp(\()p Fs(n)p Fr(\000)p Fp(2\))27329 25218 y Fw(\()p Fu(')p Fw(\))29064 24236 y Fq(\001)29620 23872 y(\021)30682 25218 y FC(in)339 b Fu(\010)32882 25400 y Fs(n)p Fr(\000)p Fp(2)35049 25218 y FC(do)-62 b(es)337 b(not)h(exc)-62 b(e)g(e)g(d)336 b Fw(4)43830 24778 y Fs(n)p Fr(\000)p Fp(2)45659 25218 y Fu(c)46184 25400 y Fp(0)46709 25218 y Fu(k)47378 24778 y Fp(2+2)p Fs(s)49485 24901 y Fy(n)p Fx(\000)p FD(1)51154 25218 y FC(.)p 0 TeXcolorgray 475 28271 a FF(Corollary)483 b(5.5.)p 0 TeXcolorgray 554 w FC(Ther)-62 b(e)433 b(ar)-62 b(e)433 b(no)g(zer)-62 b(os)433 b(of)g Fw(det)22856 26926 y Fq(\020)23580 28271 y Fu(I)364 b Fw(+)26009 27965 y(~)25689 28271 y Fu(A)26598 28453 y Fs(n)p Fr(\000)p Fp(2)28427 27289 y Fq(\000)28973 28271 y Fu(~)-596 b(y)29620 27831 y Fp(\()p Fs(n)p Fr(\000)p Fp(2\))32181 28271 y Fw(\()p Fu(')p Fw(\))33916 27289 y Fq(\001)34472 26926 y(\021)35630 28271 y FC(in)475 30564 y Fu(\010)1283 30746 y Fs(n)p Fr(\000)p Fp(2)3381 30564 y Ft(n)269 b Fo(O)5114 30124 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))7876 29218 y Fq(\020)8464 30244 y Fu(~)8600 30564 y(b)9120 30124 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))11681 29218 y Fq(\021)12405 30564 y FC(.)p 0 TeXcolorgray 475 33065 a(Pr)-62 b(o)g(of.)p 0 TeXcolorgray 552 w Fw(Pro)34 b(of)383 b(is)g(analogous)h(to)g(that)h (of)e(Lemma)g(4.10.)532 b(W)-101 b(e)382 b(use)i(Corollary)e(4.29)h (for)h Fu(n)336 b Fw(=)h(4)383 b(and)h(Corollary)475 34504 y(5.23)492 b(for)g Fu(n)483 b(>)h Fw(4,)513 b(as)493 b(w)-34 b(ell)492 b(as)g(Corollary)g(4.35)g(\()p Fu(n)483 b Fw(=)g(4\))493 b(and)g(Corollary)e(5.29)h(\()p Fu(n)484 b(>)f Fw(4\).)42695 34034 y Fp(13)44439 34504 y Fw(P)-34 b(art)493 b(2)f(of)h(the)475 36312 y(lemma)403 b(follo)-34 b(ws)405 b(from)g(the)f(de\014nition)h(of)g Fo(O)20939 35872 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))23701 34966 y Fq(\020)24289 35992 y Fu(~)24425 36312 y(b)24945 35872 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))27505 34966 y Fq(\021)28633 36312 y Fw(and)g(Corollary)f(5.2.)p 50750 36478 665 831 v 1803 38812 a(Next,)605 b(w)-34 b(e)566 b(rep)34 b(eat)565 b(Lemmas)f(4.12)h({)h(4.21)f(up)h(to)f(replacemen)-34 b(t)565 b(of)h(indices.)1021 b(Pro)34 b(ofs)565 b(are)g(completely)475 40251 y(analogous)466 b(to)g(those)g(for)g(Step)g(3.)723 b(In)466 b(Lemma)f(4.21)g(w)-34 b(e)466 b(sho)-34 b(w)467 b(that)f(the)g(total)g(length)h(of)f(\012)44685 40433 y Fs(n)p Fr(\000)p Fp(1)46979 40251 y Fw(do)34 b(es)465 b(not)475 41822 y(exceed)290 b Fu(k)4903 41382 y Fr(\000)p Fs(S)6207 41493 y Fy(n)6829 41822 y Fw(,)314 b(here)291 b(\012)10872 42004 y Fs(n)p Fr(\000)p Fp(1)12992 41822 y Fw(is)h(the)g(analog)f(of)h(\012)21942 42004 y Fp(1)22468 41822 y Fw(,)314 b(\012)23994 42004 y Fp(2)24520 41822 y Fw(,)g(see)291 b(\(3.6\),)314 b(\(4.13\).)502 b(Let)281 b Fu(~)-596 b(y)36792 41382 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))39353 41822 y Fw(\()p Fu(')p Fw(\))337 b(=)369 b Fu(~)-638 b Fn({)43513 42004 y Fs(n)p Fr(\000)p Fp(1)45341 41822 y Fw(\()p Fu(')p Fw(\))44 b(+)47971 41502 y Fu(~)48107 41822 y(b)48627 41382 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))51189 41822 y Fw(,)475 43261 y(the)404 b(function)i(b)34 b(eing)404 b(de\014ned)h(in)f Fu(\010)16973 43443 y Fs(n)p Fr(\000)p Fp(1)18802 43261 y Fw(.)p 0 TeXcolorgray 475 45710 a FF(Lemma)463 b(5.6.)p 0 TeXcolorgray 554 w FC(If)9276 45390 y Fu(~)9412 45710 y(b)9932 45270 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))12829 45710 y Ft(2)337 b Fu(K)15003 45892 y Fs(n)p Fr(\000)p Fp(1)17266 45710 y FC(and)433 b Fu(b)20140 45071 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))20140 46057 y(0)23037 45710 y Fu(>)337 b(\017)24809 45892 y Fs(n)p Fr(\000)p Fp(2)26637 45710 y Fu(k)27306 45270 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fp(2)p Fs(\016)31710 45710 y FC(,)433 b(then:)p 0 TeXcolorgray 2409 47850 a Fw(\(1\))p 0 TeXcolorgray 554 w FC(The)j(numb)-62 b(er)436 b(of)g(zer)-62 b(os)435 b(of)h Fw(det)18909 46505 y Fq(\020)19633 47850 y Fu(I)366 b Fw(+)22066 47544 y(~)21746 47850 y Fu(A)22655 48032 y Fs(n)p Fr(\000)p Fp(2)24484 46868 y Fq(\000)25030 47850 y Fu(~)-596 b(y)25677 47410 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))28238 47850 y Fw(\()p Fu(')p Fw(\))29973 46868 y Fq(\001)30529 46505 y(\021)31689 47850 y FC(in)437 b(any)f Fw(\000)36263 47410 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))39025 46505 y Fq(\020)39613 47530 y Fu(~)39749 47850 y(b)40269 47410 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))42829 46505 y Fq(\021)43990 47850 y FC(is)g(the)f(same)g(as)4511 50143 y(that)e(of)f Fw(det)10098 48797 y Fq(\020)10822 50143 y Fu(I)364 b Fw(+)13251 49836 y(~)12931 50143 y Fu(A)13840 50325 y Fs(n)p Fr(\000)p Fp(2)15669 49161 y Fq(\000)16214 50143 y Fu(~)-595 b(y)16862 49703 y Fp(\()p Fs(n)p Fr(\000)p Fp(2\))19423 50143 y Fw(\()p Fu(')p Fw(\))21158 49161 y Fq(\001)21714 48797 y(\021)22438 50143 y FC(.)p 0 TeXcolorgray 2409 52435 a Fw(\(2\))p 0 TeXcolorgray 554 w FC(F)-93 b(or)434 b(any)f Fu(')337 b Ft(2)g Fu(\010)12180 52617 y Fs(n)p Fr(\000)p Fp(1)14278 52435 y Ft(n)269 b Fo(O)16011 51995 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))18773 51090 y Fq(\020)19361 52115 y Fu(~)19497 52435 y(b)20017 51995 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))22578 51090 y Fq(\021)23302 52435 y FC(,)433 b(the)f(fol)62 b(lowing)433 b(ine)-62 b(qualities)430 b(hold:)15365 54130 y Fq(\015)15365 54857 y(\015)15365 55585 y(\015)16038 54179 y(\020)17079 55218 y Fw(~)16762 55524 y Fu(H)17868 55024 y Fp(\()p Fs(n)p Fr(\000)p Fp(2\))20428 54542 y Fq(\000)20974 55524 y Fu(~)-596 b(y)21621 55024 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))24182 55524 y Fw(\()p Fu(')p Fw(\))25917 54542 y Fq(\001)26743 55524 y Ft(\000)269 b Fu(k)28624 55024 y Fp(2)p Fs(l)29441 54179 y Fq(\021)30165 54481 y Fr(\000)p Fp(1)31423 54130 y Fq(\015)31423 54857 y(\015)31423 55585 y(\015)32433 55524 y Fu(<)34702 54704 y Fw(2)p 33845 55245 2321 49 v 33845 56379 a Fu(\017)34337 55962 y Fp(2)34337 56726 y Fs(n)p Fr(\000)p Fp(2)36299 55524 y Fu(;)12398 b Fw(\(5.7\))11557 58015 y Fq(\015)11557 58742 y(\015)11557 59469 y(\015)12230 58063 y(\020)13271 59102 y Fw(~)12954 59409 y Fu(H)14060 58908 y Fp(\()p Fs(n)p Fr(\000)p Fp(2\))16620 58427 y Fq(\000)17166 59409 y Fu(~)-596 b(y)17813 58908 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))20374 59409 y Fw(\()p Fu(')p Fw(\))22109 58427 y Fq(\001)22935 59409 y Ft(\000)269 b Fu(k)24816 58908 y Fp(2)p Fs(l)25633 58063 y Fq(\021)26357 58365 y Fr(\000)p Fp(1)27615 58015 y Fq(\015)27615 58742 y(\015)27615 59469 y(\015)28288 60252 y Fp(1)29151 59409 y Fu(<)30563 58589 y Fw(2)g Ft(\001)g Fw(4)32650 58149 y Fs(n)p Fr(\000)p Fp(2)34479 58589 y Fu(c)35004 58771 y Fp(0)35529 58589 y Fu(k)36198 58149 y Fp(2+2)p Fs(s)38305 58272 y Fy(n)p Fx(\000)p FD(1)p 30563 59130 9411 49 v 34108 60264 a Fu(\017)34600 59846 y Fp(2)34600 60611 y Fs(n)p Fr(\000)p Fp(2)40107 59409 y Fu(:)8590 b Fw(\(5.8\))p 0 TeXcolorgray 475 62969 a FF(Corollary)483 b(5.7.)p 0 TeXcolorgray 554 w FC(Ther)-62 b(e)433 b(ar)-62 b(e)433 b(no)g(zer)-62 b(os)433 b(of)g Fw(det)22856 61623 y Fq(\020)23580 62969 y Fu(I)364 b Fw(+)26009 62662 y(~)25689 62969 y Fu(A)26598 63151 y Fs(n)p Fr(\000)p Fp(2)28427 61987 y Fq(\000)28973 62969 y Fu(~)-596 b(y)29620 62529 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))32181 62969 y Fw(\()p Fu(')p Fw(\))33916 61987 y Fq(\001)34472 61623 y(\021)35630 62969 y FC(in)475 65261 y Fu(\010)1283 65443 y Fs(n)p Fr(\000)p Fp(1)3381 65261 y Ft(n)269 b Fo(O)5114 64822 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))7876 63916 y Fq(\020)8464 64942 y Fu(~)8600 65261 y(b)9120 64822 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))11681 63916 y Fq(\021)12405 65261 y FC(.)p 0 TeXcolorgray 475 67000 6642 45 v 1803 67937 a Fp(13)2744 68323 y FE(Rigorously)403 b(sp)28 b(eaking,)420 b(w)-28 b(e)403 b(ha)-28 b(v)g(e)403 b(to)g(assume)g(that)h(the)f (lemma)i(holds)e(for)f FA(n)p FE(-th)h(step.)642 b(F)-85 b(or)402 b(\()p FA(n)269 b FE(+)g(1\)-th)403 b(step)h(it)g(follo)-28 b(ws)475 69541 y(from)341 b(Corollaries)g(5.23)h(and)f(5.29.)p 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 57 57 TeXDict begin 57 56 bop 0 TeXcolorgray 0 TeXcolorgray 2848 3345 a Fp(SPECTRAL)406 b(PR)-26 b(OPER)-78 b(TIES)406 b(OF)h(A)h(POL)-78 b(YHARMONIC)405 b(OPERA)-78 b(TOR)406 b(WITH)h(LIMIT-PERIODIC)e(POTENTIAL)1430 b(57)p 0 TeXcolorgray 0 TeXcolorgray 475 6001 a FC(Pr)-62 b(o)g(of.)p 0 TeXcolorgray 552 w Fw(The)367 b(pro)34 b(of)366 b(is)g(analogous)h(to)f(that)h(of)g (Lemma)f(4.12.)525 b(In)366 b(P)-34 b(art)367 b(1)f(w)-34 b(e)366 b(use)h(the)f(estimate)45051 4971 y Fq(\014)45051 5698 y(\014)45445 6001 y Fu(~)-596 b(y)46092 5562 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))48654 6001 y Fw(\()p Fu(')p Fw(\))193 b Ft(\000)465 7640 y Fu(~)-596 b(y)1112 7201 y Fp(\()p Fs(n)p Fr(\000)p Fp(2\))3673 7640 y Fw(\()p Fu(')p Fw(\))5408 6610 y Fq(\014)5408 7337 y(\014)6176 7640 y Fu(<)363 b Fw(2)p Fu(\013)t(\017)9359 7201 y Fp(4)9359 7965 y Fs(n)p Fr(\000)p Fp(2)11189 7640 y Fu(k)11858 7201 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)14609 7640 y Fw(,)424 b(whic)-34 b(h)421 b(follo)-34 b(ws)421 b(from)f(assumed)h(prop)34 b(erties)420 b(of)h Fu(\037)38484 7201 y Fr(\003)38484 7965 y Fs(n)p Fr(\000)p Fp(1)40313 7640 y Fw(,)i(see)d(\(4.33\))h(with) g(3)363 b Ft(!)475 9084 y Fu(n)390 b Ft(\000)h Fw(1.)1084 b(W)-101 b(e)586 b(apply)g(\(5.6\))h(instead)g(of)f(\(4.20\))h(and)g (pro)34 b(duce)586 b(an)h(analog)f(of)h(\(4.25\),)632 b(the)586 b(estimate)h(16)391 b Ft(\001)475 10523 y Fw(4)1081 10084 y Fs(n)p Fr(\000)p Fp(2)2909 10523 y Fu(c)3434 10705 y Fp(0)3960 10523 y Fu(\017)4452 10084 y Fp(2)4452 10848 y Fs(n)p Fr(\000)p Fp(2)6280 10523 y Fu(k)6949 10084 y Fp(2+2)p Fs(s)9056 10207 y Fy(n)p Fx(\000)p FD(1)11302 10523 y Fu(<)578 b Fw(1)548 b(follo)-34 b(wing)550 b(from)e(\(5.1\).) 973 b(In)549 b(P)-34 b(art)549 b(2)f(w)-34 b(e)549 b(use)g(the)g(ob)-34 b(vious)550 b(estimate)e Fu(\017)48176 10084 y Fp(2)48176 10848 y Fs(n)p Fr(\000)p Fp(2)50582 10523 y Fu(<)475 11963 y Fw(1)p Fu(=)p Fw(4.)p 50750 12129 665 831 v 0 TeXcolorgray 475 14601 a FF(Lemma)432 b(5.8.)p 0 TeXcolorgray 532 w FC(If)9198 14282 y Fu(~)9334 14601 y(b)9854 14162 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))12751 14601 y Ft(2)337 b Fu(K)14925 14783 y Fs(n)p Fr(\000)p Fp(1)17163 14601 y FC(and)408 b Fu(b)20012 13963 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))20012 14949 y(0)22909 14601 y Fu(>)337 b(\017)24681 14783 y Fs(n)p Fr(\000)p Fp(2)26509 14601 y Fu(k)27178 14162 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fp(2)p Fs(\016)31582 14601 y FC(,)413 b(then)408 b(the)g(fol)62 b(lowing)407 b(estimates)f(hold)h(for)475 16453 y(any)433 b Fu(')337 b Ft(2)f Fu(\010)5880 16635 y Fs(n)p Fr(\000)p Fp(1)7978 16453 y Ft(n)270 b Fo(O)9712 16013 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))12474 15108 y Fq(\020)13062 16133 y Fu(~)13198 16453 y(b)13718 16013 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))16278 15108 y Fq(\021)17002 16453 y FC(:)15095 18430 y Fq(\015)15095 19158 y(\015)15095 19885 y(\015)15769 18479 y(\020)16493 19824 y Fu(H)17599 19324 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))20361 18479 y Fq(\020)21075 19824 y Fu(~)-596 b(y)21722 19324 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))24283 19824 y Fw(\()p Fu(')p Fw(\))26018 18479 y Fq(\021)27012 19824 y Ft(\000)269 b Fu(k)28893 19324 y Fp(2)p Fs(l)29711 18479 y Fq(\021)30435 18781 y Fr(\000)p Fp(1)31692 18430 y Fq(\015)31692 19158 y(\015)31692 19885 y(\015)32702 19824 y Fu(<)34972 19004 y Fw(4)p 34115 19545 2321 49 v 34115 20679 a Fu(\017)34607 20262 y Fp(2)34607 21026 y Fs(n)p Fr(\000)p Fp(2)36568 19824 y Fu(;)12129 b Fw(\(5.9\))12028 22879 y Fq(\015)12028 23606 y(\015)12028 24333 y(\015)12701 22927 y(\020)13425 24273 y Fu(H)14531 23772 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))17294 22927 y Fq(\020)18008 24273 y Fu(~)-596 b(y)18655 23772 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))21216 24273 y Fw(\()p Fu(')p Fw(\))22951 22927 y Fq(\021)23945 24273 y Ft(\000)269 b Fu(k)25826 23772 y Fp(2)p Fs(l)26643 22927 y Fq(\021)27367 23229 y Fr(\000)p Fp(1)28625 22879 y Fq(\015)28625 23606 y(\015)28625 24333 y(\015)29298 25116 y Fp(1)30161 24273 y Fu(<)31573 23453 y Fw(4)32179 23013 y Fs(n)p Fr(\000)p Fp(1)34008 23453 y Fu(c)34533 23635 y Fp(0)35058 23453 y Fu(k)35727 23013 y Fp(2+2)p Fs(s)37834 23136 y Fy(n)p Fx(\000)p FD(1)p 31573 23994 7930 49 v 34377 25127 a Fu(\017)34869 24710 y Fp(2)34869 25475 y Fs(n)p Fr(\000)p Fp(2)39635 24273 y Fu(:)8456 b Fw(\(5.10\))p 0 TeXcolorgray 475 27994 a FF(Corollary)483 b(5.9.)p 0 TeXcolorgray 554 w FC(Ther)-62 b(e)433 b(ar)-62 b(e)433 b(no)g(zer)-62 b(os)433 b(of)g Fw(det)22856 26648 y Fq(\020)23580 27994 y Fu(I)364 b Fw(+)269 b Fu(A)26598 28176 y Fs(n)p Fr(\000)p Fp(1)28629 27012 y Fq(\000)29175 27994 y Fu(~)-596 b(y)29822 27554 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))32383 27994 y Fw(\()p Fu(')p Fw(\))34118 27012 y Fq(\001)34674 26648 y(\021)35832 27994 y FC(in)475 30286 y Fu(\010)1283 30468 y Fs(n)p Fr(\000)p Fp(1)3381 30286 y Ft(n)269 b Fo(O)5114 29846 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))7876 28941 y Fq(\020)8464 29966 y Fu(~)8600 30286 y(b)9120 29846 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))11681 28941 y Fq(\021)12405 30286 y FC(.)p 0 TeXcolorgray 475 33501 a FF(Lemma)387 b(5.10.)p 0 TeXcolorgray 499 w FC(If)9780 33181 y Fu(~)9916 33501 y(b)10436 33061 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))13334 33501 y Ft(2)336 b Fu(K)15507 33683 y Fs(n)p Fr(\000)p Fp(1)17709 33501 y FC(and)372 b Fu(b)20522 32862 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))20522 33848 y(0)23419 33501 y Fu(>)337 b(\017)25191 33683 y Fs(n)p Fr(\000)p Fp(2)27019 33501 y Fu(k)27688 33061 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fp(2)p Fs(\016)32092 33501 y FC(,)384 b(then)372 b(the)f(numb)-62 b(er)372 b(of)f(zer)-62 b(os)371 b(of)h Fw(det)49094 32155 y Fq(\020)49818 33501 y Fu(I)231 b Fw(+)475 35793 y Fu(A)1384 35975 y Fs(n)p Fr(\000)p Fp(1)3212 34811 y Fq(\000)3758 35793 y Fu(~)-596 b(y)4405 35353 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))6966 35793 y Fw(\()p Fu(')p Fw(\))8701 34811 y Fq(\001)9257 34448 y(\021)10415 35793 y FC(in)434 b(any)f Fw(\000)14983 35353 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))17745 34448 y Fq(\020)18333 35473 y Fu(~)18469 35793 y(b)18989 35353 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))21549 34448 y Fq(\021)22707 35793 y FC(is)g(the)g(same)f(as)h(that)f(of)h Fw(det)36254 34448 y Fq(\020)36978 35793 y Fu(I)365 b Fw(+)39408 35487 y(~)39088 35793 y Fu(A)39997 35975 y Fs(n)p Fr(\000)p Fp(2)41825 34811 y Fq(\000)42371 35793 y Fu(~)-596 b(y)43018 35353 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))45579 35793 y Fw(\()p Fu(')p Fw(\))47314 34811 y Fq(\001)47870 34448 y(\021)48594 35793 y FC(.)p 0 TeXcolorgray 475 38917 a FF(Lemma)463 b(5.11.)p 0 TeXcolorgray 0 TeXcolorgray 2489 w Fw(\(1\))p 0 TeXcolorgray 554 w FC(If)14160 38597 y Fu(~)14296 38917 y(b)14816 38477 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))17987 38917 y Ft(2)609 b Fu(K)20433 39099 y Fs(n)p Fr(\000)p Fp(1)22847 38917 y FC(and)584 b Fu(b)25872 38278 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))25872 39264 y(0)29042 38917 y Fu(>)609 b(\017)31086 39099 y Fs(n)p Fr(\000)p Fp(2)32915 38917 y Fu(k)33584 38477 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fp(2)p Fs(\016)37987 38917 y FC(,)622 b(then)584 b(the)f(numb)-62 b(er)584 b(of)f(ze-)4511 40769 y(r)-62 b(os)535 b(of)f Fw(det)9821 39423 y Fq(\020)10545 40769 y Fu(I)437 b Fw(+)343 b Fu(A)13710 40951 y Fs(n)p Fr(\000)p Fp(1)15539 39787 y Fq(\000)16084 40769 y Fu(~)-596 b(y)16731 40329 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))19292 40769 y Fw(\()p Fu(')p Fw(\))21027 39787 y Fq(\001)21584 39423 y(\021)22843 40769 y FC(in)535 b(any)f Fw(\000)27613 40329 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))30173 40769 y Fw(\()30508 40449 y Fu(~)30644 40769 y(b)31164 40329 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))33725 40769 y Fw(\))i FC(is)e(the)g(same)g(as)g(that)g(of)g Fw(det)48888 39423 y Fq(\020)49612 40769 y Fu(I)437 b Fw(+)4831 42755 y(~)4511 43061 y Fu(A)5420 43243 y Fs(n)p Fr(\000)p Fp(2)7249 42080 y Fq(\000)7795 43061 y Fu(~)-596 b(y)8442 42622 y Fp(\()p Fs(n)p Fr(\000)p Fp(2\))11003 43061 y Fw(\()p Fu(')p Fw(\))12738 42080 y Fq(\001)13294 41716 y(\021)14018 43061 y FC(.)p 0 TeXcolorgray 2409 45354 a Fw(\(2\))p 0 TeXcolorgray 554 w FC(The)551 b(distanc)-62 b(e)550 b(fr)-62 b(om)551 b(a)g(zer)-62 b(o)550 b(of)h Fw(det)21956 44008 y Fq(\020)22680 45354 y Fu(I)450 b Fw(+)354 b Fu(A)25869 45536 y Fs(n)p Fr(\000)p Fp(1)27698 44372 y Fq(\000)28243 45354 y Fu(~)-596 b(y)28890 44914 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))31452 45354 y Fw(\()p Fu(')p Fw(\))33187 44372 y Fq(\001)33743 44008 y(\021)35018 45354 y FC(to)551 b(the)f(ne)-62 b(ar)g(est)550 b(zer)-62 b(o)551 b(of)f Fw(det)48876 44008 y Fq(\020)49600 45354 y Fu(I)449 b Fw(+)4831 47340 y(~)4511 47646 y Fu(A)5420 47828 y Fs(n)p Fr(\000)p Fp(2)7249 46665 y Fq(\000)7795 47646 y Fu(~)-596 b(y)8442 47207 y Fp(\()p Fs(n)p Fr(\000)p Fp(2\))11003 47646 y Fw(\()p Fu(')p Fw(\))12738 46665 y Fq(\001)13294 46301 y(\021)14452 47646 y FC(do)-62 b(es)432 b(not)h(exc)-62 b(e)g(e)g(d)432 b Fu(r)23494 47207 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))26054 47646 y Fu(=)p Fw(2)p FC(.)p 0 TeXcolorgray 475 50571 a FF(Corollary)328 b(5.12.)p 0 TeXcolorgray 443 w FC(The)310 b(numb)-62 b(er)309 b(of)g(zer)-62 b(os)308 b(of)h Fw(det)23815 49589 y Fq(\000)24370 50571 y Fu(I)365 b Fw(+)269 b Fu(A)27389 50753 y Fs(n)p Fr(\000)p Fp(1)29419 49589 y Fq(\000)29965 50571 y Fu(~)-596 b(y)30612 50131 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))33173 50571 y Fw(\()p Fu(')p Fw(\))34908 49589 y Fq(\001\001)36330 50571 y FC(do)-62 b(es)308 b(not)h(exc)-62 b(e)g(e)g(d)308 b Fw(4)45025 50131 y Fs(n)p Fr(\000)p Fp(2)46854 50571 y Fu(c)47379 50753 y Fp(0)47904 50571 y Fu(k)48573 50131 y Fp(2+2)p Fs(s)50680 50254 y Fy(n)p Fx(\000)p FD(1)52349 50571 y FC(.)1803 52932 y Fw(The)365 b(corollary)f(is)g(the)h(com)-34 b(bination)366 b(of)g(the)f(\014rst)g(part)g(of)h(the)f(lemma)f(with)i (the)f(second)g(part)h(of)f(Lemma)475 54372 y(5.4.)538 b(The)405 b(follo)-34 b(wing)404 b(is)g(an)h(analog)f(of)h (De\014nition)f(4.6.)p 0 TeXcolorgray 475 56733 a FF(De\014nition)466 b(5.13.)p 0 TeXcolorgray 554 w Fw(A)404 b(complex)f(resonance)h(set)h Fu(!)24685 56293 y Fr(\003)24642 57057 y Fs(n)p Fr(\000)p Fp(1)26471 56733 y Fw(\()p Fu(\025)p Fw(\))g(is)e(the)i(set)f(of)h Fu(')336 b Ft(2)h Fu(\010)38186 56915 y Fs(n)p Fr(\000)p Fp(1)40419 56733 y Fw(satisfying)6741 60240 y(det)8627 58167 y Fq( )9719 59343 y Fu(H)10825 58903 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))13386 58361 y Fq(\000)13931 59343 y Fu(~)-596 b(y)14578 58903 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))17139 59343 y Fw(\()p Fu(')p Fw(\))18874 58361 y Fq(\001)19700 59343 y Ft(\000)269 b Fu(k)21581 58903 y Fp(2)p Fs(l)22668 59343 y Ft(\000)g Fu(\017)p 9719 59961 14653 49 v 11773 61137 a(H)12781 61319 y Fp(0)13306 60155 y Fq(\000)13852 61137 y Fu(~)-596 b(y)14499 60785 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))17060 61137 y Fw(\()p Fu(')p Fw(\))18795 60155 y Fq(\001)19620 61137 y Fw(+)269 b Fu(k)21501 60787 y Fp(2)p Fs(l)24505 58167 y Fq(!)25801 60240 y Fw(=)337 b(0)p Fu(;)1404 b(~)-596 b(y)30075 59739 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))32636 60240 y Fw(\()p Fu(')p Fw(\))337 b(=)369 b Fu(~)-638 b Fn({)36796 60422 y Fs(n)p Fr(\000)p Fp(1)38624 60240 y Fw(\()p Fu(')p Fw(\))270 b(+)41705 59920 y Fu(~)41841 60240 y(b)42361 59739 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))44922 60240 y Fu(;)3169 b Fw(\(5.11\))475 63910 y(for)406 b(some)5236 63590 y Fu(~)5372 63910 y(b)5892 63470 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))8793 63910 y Fw(=)340 b(2)p Fu(\031)43 b(p)12026 63470 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))14587 63910 y Fu(=)15544 63603 y Fw(^)15193 63910 y Fu(N)16167 64092 y Fs(n)p Fr(\000)p Fp(1)17996 63910 y Fu(a)p Fw(,)406 b Fu(p)19990 63470 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))22891 63910 y Ft(2)341 b Fu(P)24986 63470 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))27818 63910 y Ft(n)271 b(f)p Fw(0)p Ft(g)p Fw(,)407 b(and)g Ft(j)p Fu(\017)p Ft(j)340 b Fu(<)g(\017)36897 64092 y Fs(n)p Fr(\000)p Fp(1)38726 63910 y Fw(.)545 b(W)-101 b(e)406 b(denote)h(b)-34 b(y)407 b Fu(!)48046 64092 y Fs(n)p Fr(\000)p Fp(1)49875 63910 y Fw(\()p Fu(\025)p Fw(\))475 65349 y(the)417 b(set)g(of)g Fu(')358 b Ft(2)g Fw(\002)9133 65531 y Fs(n)p Fr(\000)p Fp(1)11378 65349 y Fw(corresp)34 b(onding)417 b(to)g(\012)21551 65531 y Fs(n)p Fr(\000)p Fp(1)23380 65349 y Fw(\()p Fu(\025)p Fw(\).)577 b(Considering)418 b(as)f(in)f(the)i(pro)34 b(of)417 b(of)g(Lemma)f(3.20,)k(it)c(is)475 66788 y(easy)403 b(to)i(sho)-34 b(w)405 b(that)h Fu(!)11007 66970 y Fs(n)p Fr(\000)p Fp(1)13172 66788 y Fw(=)337 b Fu(!)15249 66348 y Fr(\003)15206 67113 y Fs(n)p Fr(\000)p Fp(1)17304 66788 y Ft(\\)270 b Fw(\002)19325 66970 y Fs(n)p Fr(\000)p Fp(1)21153 66788 y Fw(.)p 0 TeXcolorgray 475 69541 a FF(Lemma)463 b(5.14.)p 0 TeXcolorgray 554 w FC(If)9973 69221 y Fu(~)10109 69541 y(b)10629 69101 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))13526 69541 y Ft(2)337 b Fu(K)15700 69723 y Fs(n)p Fr(\000)p Fp(1)17963 69541 y FC(and)433 b Fu(b)20837 68902 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))20837 69888 y(0)23734 69541 y Ft(\025)337 b Fu(\017)25506 69723 y Fs(n)p Fr(\000)p Fp(2)27334 69541 y Fu(k)28003 69101 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fp(2)p Fs(\016)32407 69541 y FC(,)433 b Ft(j)p Fu(\017)p Ft(j)336 b Fu(<)g(\017)36485 69101 y Fp(4)36485 69866 y Fs(n)p Fr(\000)p Fp(2)38314 69541 y FC(,)433 b(then)p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 58 58 TeXDict begin 58 57 bop 0 TeXcolorgray 0 TeXcolorgray 475 3345 a Fp(58)14248 b(YULIA)313 b(KARPESHINA)f(AND)h(YOUNG-RAN)g (LEE)p 0 TeXcolorgray 0 TeXcolorgray 2409 6895 a Fw(\(1\))p 0 TeXcolorgray 554 w FC(The)434 b(determinant)e(of)15811 5998 y Fu(H)16917 5559 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))19478 5017 y Fq(\000)20023 5998 y Fu(~)-596 b(y)20670 5559 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))23231 5998 y Fw(\()p Fu(')p Fw(\))24966 5017 y Fq(\001)25792 5998 y Ft(\000)269 b Fu(k)27673 5559 y Fp(2)p Fs(l)28760 5998 y Ft(\000)g Fu(\017)p 15811 6617 14653 49 v 17864 7792 a(H)18872 7974 y Fp(0)19398 6811 y Fq(\000)19943 7792 y Fu(~)-596 b(y)20590 7441 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))23152 7792 y Fw(\()p Fu(')p Fw(\))24887 6811 y Fq(\001)25712 7792 y Fw(+)269 b Fu(k)27593 7442 y Fp(2)p Fs(l)31030 6895 y FC(has)433 b(no)g(zer)-62 b(os)433 b(in)4511 9721 y Fu(\010)5319 9903 y Fs(n)p Fr(\000)p Fp(1)7417 9721 y Ft(n)270 b Fo(O)9151 9281 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))11913 8376 y Fq(\020)12501 9401 y Fu(~)12637 9721 y(b)13157 9281 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))15717 8376 y Fq(\021)16441 9721 y FC(.)p 0 TeXcolorgray 2409 12741 a Fw(\(2\))p 0 TeXcolorgray 554 w FC(The)434 b(numb)-62 b(er)433 b(of)g(zer)-62 b(os)432 b(of)h Fw(det)19097 10668 y Fq( )20190 11844 y Fu(H)21296 11404 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))23856 10862 y Fq(\000)24402 11844 y Fu(~)-596 b(y)25049 11404 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))27610 11844 y Fw(\()p Fu(')p Fw(\))29345 10862 y Fq(\001)30170 11844 y Ft(\000)270 b Fu(k)32052 11404 y Fp(2)p Fs(l)33138 11844 y Ft(\000)f Fu(\017)p 20190 12462 V 22243 13638 a(H)23251 13820 y Fp(0)23776 12656 y Fq(\000)24322 13638 y Fu(~)-596 b(y)24969 13286 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))27530 13638 y Fw(\()p Fu(')p Fw(\))29265 12656 y Fq(\001)30091 13638 y Fw(+)269 b Fu(k)31972 13288 y Fp(2)p Fs(l)34975 10668 y Fq(!)36368 12741 y FC(in)434 b(any)4511 16488 y Fw(\000)5269 16048 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))8031 15143 y Fq(\020)8619 16168 y Fu(~)8755 16488 y(b)9275 16048 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))11836 15143 y Fq(\021)12994 16488 y FC(is)f(the)f(same)h(as)f(that)h(of)f Fw(det)26743 14415 y Fq( )27835 15591 y Fu(H)28941 15151 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))31502 14609 y Fq(\000)32047 15591 y Fu(~)-596 b(y)32694 15151 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))35256 15591 y Fw(\()p Fu(')p Fw(\))36991 14609 y Fq(\001)37816 15591 y Ft(\000)269 b Fu(k)39697 15151 y Fp(2)p Fs(l)p 27835 16209 12680 49 v 28902 17385 a Fu(H)29910 17567 y Fp(0)30435 16403 y Fq(\000)30981 17385 y Fu(~)-596 b(y)31628 17033 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))34189 17385 y Fw(\()p Fu(')p Fw(\))35924 16403 y Fq(\001)36750 17385 y Fw(+)269 b Fu(k)38631 17035 y Fp(2)p Fs(l)40647 14415 y Fq(!)41607 16488 y FC(.)p 0 TeXcolorgray 475 20315 a FF(Corollary)483 b(5.15.)p 0 TeXcolorgray 554 w FC(The)433 b(fol)62 b(lowing)432 b(r)-62 b(elation)432 b(holds:)555 b Fu(!)26569 19876 y Fr(\003)26526 20640 y Fs(n)p Fr(\000)p Fp(1)28692 20315 y Ft(\032)337 b Fo(O)30830 19677 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))30830 20504 y Fr(\003)33390 20315 y Fu(:)p 0 TeXcolorgray 475 22653 a FF(Lemma)463 b(5.16.)p 0 TeXcolorgray 554 w FC(The)433 b(estimate)f(\(5.2\))f(holds.)p 0 TeXcolorgray 475 25268 a(Pr)-62 b(o)g(of.)p 0 TeXcolorgray 552 w Fw(By)325 b(Corollary)g (5.15,)341 b Fu(!)14956 24828 y Fr(\003)14913 25593 y Fs(n)p Fr(\000)p Fp(1)17079 25268 y Ft(\032)336 b Fo(O)19216 24629 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))19216 25456 y Fr(\003)21776 25268 y Fw(.)513 b(Hence,)341 b Fu(!)27257 25450 y Fs(n)p Fr(\000)p Fp(1)29422 25268 y Ft(\032)c Fo(O)31560 24629 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))31560 25456 y Fr(\003)34233 25268 y Ft(\\)113 b Fw(\002)36097 25450 y Fs(n)p Fr(\000)p Fp(1)37925 25268 y Fw(.)512 b(By)325 b(Corollary)g(5.3,)341 b Fo(O)48965 24629 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))48965 25456 y Fr(\003)475 26900 y Fw(is)429 b(formed)i(b)-34 b(y)430 b(no)g(more)f(than)i(4)15760 26460 y Fs(n)p Fr(\000)p Fp(1)17589 26900 y Fu(c)18114 27082 y Fp(0)18639 26900 y Fu(k)19308 26460 y Fp(2+2)p Fs(s)21415 26571 y Fy(n)22468 26900 y Fw(disks)f(of)g(the)h(radius)f Fu(r)33277 26460 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))36216 26900 y Fw(=)380 b Fu(r)38120 26460 y Fp(\()p Fs(n)p Fr(\000)p Fp(2\))40680 26900 y Fu(k)41349 26460 y Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)44188 26583 y Fy(n)p Fx(\000)p FD(1)45800 26460 y Fr(\000)p Fs(\016)47037 26900 y Fw(.)616 b(Hence,)475 28350 y(the)424 b(total)h(length)g(of)g Fu(!)11377 28532 y Fs(n)p Fr(\000)p Fp(1)13630 28350 y Fw(do)34 b(es)424 b(not)h(exceed)f(2)282 b Ft(\001)h Fw(4)24566 27910 y Fs(n)p Fr(\000)p Fp(1)26395 28350 y Fu(c)26920 28532 y Fp(0)27445 28350 y Fu(r)28026 27910 y Fp(\()p Fs(n)p Fr(\000)p Fp(2\))30586 28350 y Fu(k)31255 27910 y Fr(\000)p Fs(\016)32492 28350 y Fw(.)599 b(Considering)425 b(the)f(recurren)-34 b(t)424 b(form)-34 b(ula)475 29839 y(for)456 b Fu(r)2963 29399 y Fp(\()p Fs(m)p Fp(+1\))5785 29839 y Fw(,)469 b Fu(r)7172 29399 y Fp(\()p Fs(m)p Fp(+1\))10417 29839 y Fw(=)424 b Fu(r)12365 29399 y Fp(\()p Fs(m)p Fp(\))13984 29839 y Fu(k)14653 29399 y Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)17492 29522 y Fy(m)p FD(+1)19302 29399 y Fr(\000)p Fs(\016)20538 29839 y Fw(,)469 b(w)-34 b(e)457 b(easily)f(get)g(that)i(4)31931 29399 y Fs(n)32557 29839 y Fu(c)33082 30021 y Fp(0)33608 29839 y Fu(r)34189 29399 y Fp(\()p Fs(n)p Fr(\000)p Fp(2\))37173 29839 y Fu(<)423 b(k)39208 29399 y Fr(\000)p Fs(S)40512 29510 y Fy(n)41135 29839 y Fw(,)469 b Fu(n)423 b(>)h Fw(1.)696 b(Therefore,)475 31278 y(the)362 b(length)g(of)f(\012)8393 31460 y Fs(n)p Fr(\000)p Fp(1)10584 31278 y Fw(do)34 b(es)361 b(not)h(exceed)19349 30801 y Fs(k)p 19349 30999 514 49 v 19371 31696 a Fp(2)19996 31278 y Fu(k)20665 30838 y Fr(\000)p Fs(S)21969 30949 y Fy(n)22591 31278 y Fw(.)524 b(Considering)362 b(that)g Fu(\037)33427 30838 y Fr(\003)33427 31603 y Fs(n)p Fr(\000)p Fp(1)35256 31278 y Fw(\()p Fu(\025)p Fw(\))h(has)e(a)h(length)g(2)p Fu(\031)43 b(k)46024 30296 y Fq(\000)46580 31278 y Fw(1)184 b(+)g Fu(o)p Fw(\(1\))50633 30296 y Fq(\001)51189 31278 y Fw(,)475 32722 y(w)-34 b(e)404 b(obtain)h(\(5.2\).)p 50750 32888 665 831 v 475 35784 a(5.3.)605 b FF(Nonresonan)-39 b(t)508 b(part)g(of)f(the)h(iso)39 b(energetic)507 b(set)g(of)g Fu(H)31059 35145 y Fp(\()p Fs(n)p Fp(\))30961 35941 y Fs(\013)32417 35784 y FF(.)606 b Fw(No)-34 b(w)443 b(w)-34 b(e)441 b(construct)h(a)f(nonresonance)475 37433 y(subset)531 b Fu(\037)5076 36993 y Fr(\003)5076 37733 y Fs(n)5702 37433 y Fw(\()p Fu(\025)p Fw(\))g(of)f(the)g(iso)34 b(energetic)528 b(surface)i Fu(S)23255 37615 y Fs(n)23882 37433 y Fw(\()p Fu(\025)p Fw(\))g(of)g Fu(H)28673 36794 y Fp(\()p Fs(n)p Fp(\))28575 37591 y Fs(\013)30561 37433 y Fw(in)g Fu(K)33130 37615 y Fs(n)33756 37433 y Fw(,)561 b Fu(S)35397 37615 y Fs(n)36024 37433 y Fw(\()p Fu(\025)p Fw(\))546 b Ft(\032)g Fu(K)40737 37615 y Fs(n)41364 37433 y Fu(:)529 b Fw(It)h(corresp)34 b(onds)530 b(to)475 38872 y(nonresonance)423 b(eigen)-34 b(v)-67 b(alues)422 b(giv)-34 b(en)423 b(b)-34 b(y)423 b(p)34 b(erturbation)424 b(series.)594 b(The)423 b(sets)h Fu(\037)35717 38432 y Fr(\003)35717 39197 y Fp(1)36243 38872 y Fw(\()p Fu(\025)p Fw(\),)k Fu(\037)39415 38432 y Fr(\003)39415 39197 y Fp(2)39941 38872 y Fw(\()p Fu(\025)p Fw(\),)g Fu(\037)43113 38432 y Fr(\003)43113 39197 y Fp(3)43639 38872 y Fw(\()p Fu(\025)p Fw(\))c(are)e(de\014ned)475 40311 y(in)488 b(the)h(previous)e(steps)i(as)g(w)-34 b(ell)487 b(as)i(the)f(non-resonance)h(sets)g Fu(\037)31225 40493 y Fp(1)31751 40311 y Fw(\()p Fu(\025)p Fw(\),)510 b Fu(\037)35005 40493 y Fp(2)35531 40311 y Fw(\()p Fu(\025)p Fw(\),)f Fu(\037)38784 40493 y Fp(3)39311 40311 y Fw(\()p Fu(\025)p Fw(\).)791 b(Let)488 b(us)g(recall)f(that)475 41750 y(w)-34 b(e)472 b(started)i(with)f(the)f(de\014nition)i(of)f Fu(\037)19161 41932 y Fp(1)19687 41750 y Fw(\()p Fu(\025)p Fw(\))g(and)g(then)g(use)g(it)f(to)h(de\014ne)g Fu(\037)36384 41310 y Fr(\003)36384 42075 y Fp(1)36910 41750 y Fw(\()p Fu(\025)p Fw(\).)744 b(Considering)473 b Fu(\037)47143 41310 y Fr(\003)47143 42075 y Fp(1)47669 41750 y Fw(\()p Fu(\025)p Fw(\),)490 b(w)-34 b(e)475 43189 y(constructed)492 b Fu(\037)7862 43371 y Fp(2)8388 43189 y Fw(\()p Fu(\025)p Fw(\))g(\(Step)g(2\).)800 b(Next,)512 b(w)-34 b(e)492 b(de\014ned)g Fu(\037)26362 42750 y Fr(\003)26362 43514 y Fp(2)26888 43189 y Fw(\()p Fu(\025)p Fw(\),)513 b(using)492 b Fu(\037)33404 43371 y Fp(2)33930 43189 y Fw(\()p Fu(\025)p Fw(\).)800 b(Using)491 b Fu(\037)40968 42750 y Fr(\003)40968 43514 y Fp(2)41495 43189 y Fw(\()p Fu(\025)p Fw(\),)513 b(w)-34 b(e)491 b(in)-34 b(tro)34 b(duced)475 44628 y Fu(\037)1233 44810 y Fp(3)1759 44628 y Fw(\()p Fu(\025)p Fw(\))511 b(and,)537 b(then)511 b Fu(\037)10370 44189 y Fr(\003)10370 44953 y Fp(3)10896 44628 y Fw(\()p Fu(\025)p Fw(\).)858 b(Th)-34 b(us,)538 b(the)510 b(pro)34 b(cess)510 b(lo)34 b(oks)509 b(lik)-34 b(e)510 b Fu(\037)30102 44810 y Fp(1)31141 44628 y Ft(!)k Fu(\037)33625 44189 y Fr(\003)33625 44953 y Fp(1)34665 44628 y Ft(!)f Fu(\037)37148 44810 y Fp(2)38188 44628 y Ft(!)g Fu(\037)40671 44189 y Fr(\003)40671 44953 y Fp(2)41711 44628 y Ft(!)h Fu(\037)44195 44810 y Fp(3)45234 44628 y Ft(!)g Fu(\037)47718 44189 y Fr(\003)47718 44953 y Fp(3)48244 44628 y Fw(.)857 b(The)475 46067 y(geometric)417 b(lemma)i(in)f(this)h(section)g(giv)-34 b(es)418 b(us)h Fu(\037)23115 46249 y Fp(4)24060 46067 y Fw(and)g(ev)-34 b(ery)418 b(next)h Fu(\037)33148 46249 y Fs(n)34193 46067 y Fw(if)f Fu(\037)36076 45628 y Fr(\003)36076 46392 y Fs(n)p Fr(\000)p Fp(1)38324 46067 y Fw(is)g(de\014ned.)583 b(T)-101 b(o)419 b(ensure)g(the)475 47507 y(reccuren)-34 b(t)404 b(pro)34 b(cedure)403 b(w)-34 b(e)405 b(sho)-34 b(w)405 b(no)-34 b(w)405 b(ho)-34 b(w)406 b(to)e(de\014ne)h Fu(\037)26977 47067 y Fr(\003)26977 47806 y Fs(n)27604 47507 y Fw(\()p Fu(\025)p Fw(\))f(using)h Fu(\037)33587 47689 y Fs(n)34214 47507 y Fw(\()p Fu(\025)p Fw(\).)1803 48946 y(As)411 b(in)f(previous)g(sections,)i(w)-34 b(e)410 b(start)h(with)g(the)g(de\014nition)h(of)e Fo(D)32072 49128 y Fs(n)p Fr(\000)p Fp(1)33900 48946 y Fw(\()p Fu(\025)p Fw(\))35549 49128 y Fs(nonr)26 b(es)38525 48946 y Fw(.)557 b(First)410 b(note)h(that)g Fu(\037)48471 48506 y Fr(\003)48471 49270 y Fs(n)p Fr(\000)p Fp(1)50710 48946 y Fw(is)475 50385 y(de\014ned)439 b(b)-34 b(y)438 b(the)h(form)-34 b(ula)439 b Fu(\037)13763 49945 y Fr(\003)13763 50709 y Fs(n)p Fr(\000)p Fp(1)15985 50385 y Fw(=)393 b Fo(K)18318 50567 y Fs(n)p Fr(\000)p Fp(1)20147 50385 y Fo(D)21154 50567 y Fs(n)p Fr(\000)p Fp(1)22983 50385 y Fw(,)446 b(where)438 b Fo(D)28312 50567 y Fs(n)p Fr(\000)p Fp(1)30578 50385 y Fw(is)g(a)g(distorted)h(circle)e(with)i(holes,)446 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Fu(\025)p Fw(\))18582 57139 y Fs(nonr)26 b(es)21894 56957 y Fw(=)336 b Ft(f)32 b Fu(~)-638 b Fn({)386 b Ft(2)337 b Fo(D)27125 57139 y Fs(n)p Fr(\000)p Fp(1)28953 56957 y Fw(\()p Fu(\025)p Fw(\))g(:)g Fo(K)32610 57139 y Fs(n)33269 56957 y Fu(~)-638 b Fn({)385 b Ft(2)337 b Fu(\037)36333 57139 y Fs(n)36960 56957 y Ft(g)p Fu(;)10525 b Fw(\(5.12\))475 59213 y Fo(D)1482 59395 y Fs(n)p Fr(\000)p Fp(1)3310 59213 y Fw(\()p Fu(\025)p Fw(\))4959 59395 y Fs(nonr)26 b(es)8272 59213 y Fw(b)34 b(eing)338 b(de\014ned)g(since)g Fu(\037)19242 59395 y Fs(n)20205 59213 y Ft(\032)f Fu(\037)22243 58773 y Fr(\003)22243 59538 y Fs(n)p Fr(\000)p Fp(1)24071 59213 y Fu(:)h Fw(W)-101 b(e)337 b(de\014ne)i Fo(B)31126 59395 y Fs(n)32089 59213 y Fw(as)f(the)g(set)g(of)g(directions)g(corresp)34 b(onding)475 60652 y(to)404 b Fo(D)2963 60834 y Fs(n)p Fr(\000)p Fp(1)4791 60652 y Fw(\()p Fu(\025)p Fw(\))6440 60834 y Fs(nonr)26 b(es)9416 60652 y Fw(:)13041 62404 y Fo(B)13964 62586 y Fs(n)14590 62404 y Fw(\()p Fu(\025)p 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Fy(n)39428 9117 y Fr(\000)p Fs(\016)41103 9557 y Fw(cen)-34 b(tered)440 b(at)f(zeros)g(of)475 10996 y(the)498 b(new)h(determinan)-34 b(t.)821 b(Ob)-34 b(viously)498 b(eac)-34 b(h)498 b(of)g(these)h(disks)f(is)f(in)h Fo(O)32900 10556 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))35461 10015 y Fq(\000)35880 10676 y Fu(~)36016 10996 y(b)36536 10556 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))39097 10015 y Fq(\001)39652 10996 y Fw(,)521 b(since)498 b Fu(r)44155 10556 y Fp(\()p Fs(n)p Fp(\))46006 10996 y Fu(<<)492 b(r)48965 10556 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))475 12635 y Fw(and)347 b(the)f(distance)g(b)34 b(et)-34 b(w)g(een)347 b(zeros)f(of)g(the)h (old)f(and)g(new)h(determinan)-34 b(ts)347 b(is)f(smaller)f(than)i Fu(r)43013 12195 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))45573 12635 y Fu(=)p Fw(2)f(\(Lemma)475 14352 y(5.11\).)527 b(Next,)376 b(w)-34 b(e)369 b(tak)-34 b(e)370 b(the)g(union)g(of)g (these)f(smaller)g(disks)g(and)h(denote)g(it)f(b)-34 b(y)370 b Fo(O)38381 13713 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))38381 14510 y Fs(s)40941 13370 y Fq(\000)41360 14032 y Fu(~)41497 14352 y(b)42017 13912 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))44577 13370 y Fq(\001)45133 14352 y Fw(.)527 b(Ob)-34 b(viously)-101 b(,)475 15791 y(the)404 b(follo)-34 b(wing)405 b(lemma)f(holds.)p 0 TeXcolorgray 475 18097 a FF(Lemma)e(5.20.)p 0 TeXcolorgray 511 w FC(The)384 b(set)g Fo(O)13860 17458 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))13860 18255 y Fs(s)16420 17115 y Fq(\000)16839 17777 y Fu(~)16975 18097 y(b)17495 17657 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))20056 17115 y Fq(\001)20996 18097 y FC(b)-62 b(elongs)383 b(to)h Fo(O)27363 17657 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))29924 17115 y Fq(\000)30343 17777 y Fu(~)30479 18097 y(b)30999 17657 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))33560 17115 y Fq(\001)34500 18097 y FC(and)g(c)-62 b(onsists)383 b(of)h(the)g(same)g (numb)-62 b(er)475 19818 y(of)433 b(disks)e(as)i Fo(O)7280 19378 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))9840 18836 y Fq(\000)10260 19498 y Fu(~)10396 19818 y(b)10916 19378 y Fp(\()p Fs(n)p 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b(us)g(form)-34 b(ulate)343 b(the)f(corresp)34 b(onding)342 b(results)g(for)g(the)h(next)f(step)475 15738 y(of)368 b(appro)-34 b(ximation)368 b(\()p Fu(n)196 b Ft(\000)g Fw(2)337 b Ft(!)g Fu(n)196 b Ft(\000)g Fw(1\).)526 b(The)368 b(form)-34 b(ulations)369 b(and)f(pro)34 b(ofs)368 b(of)g(the)g(follo)-34 b(wing)368 b(four)g(lemmas)f(and)475 17177 y(are)456 b(analogous)h(to)g(those)g(for)g(Lemmas)f(4.30)h({)f (4.34.)696 b(De\014nition)457 b(5.26)g(is)f(analogous)h(to)g (De\014nition)h(4.32,)475 18616 y(and)405 b(Corollary)e(5.29)h(is)g(an) g(analog)h(of)f(Corollary)f(4.35.)p 0 TeXcolorgray 475 20778 a FF(Lemma)463 b(5.24.)p 0 TeXcolorgray 554 w FC(If)434 b Fw(0)337 b Fu(<)f(b)12851 20139 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))12851 21125 y(0)15748 20778 y Ft(\024)h Fu(\017)17520 20960 y Fs(n)p Fr(\000)p Fp(2)19348 20778 y Fu(k)20017 20338 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fp(2)p Fs(\016)24854 20778 y FC(and)434 b Ft(j)p Fu(\017)28038 20960 y Fp(0)28563 20778 y Ft(j)337 b Fu(<)f(b)31036 20139 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))31036 21125 y(0)33597 20778 y Fu(k)34266 20338 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1)p Fr(\000)p Fs(\016)37467 20778 y Fu(r)38048 20338 y Fp(\()p Fs(n)p Fp(\))39405 20778 y FC(,)434 b(then)f(the)f(e)-62 b(quation)18274 23134 y Fu(\025)18981 22633 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))21743 21788 y Fq(\020)22457 23134 y Fu(~)-596 b(y)23104 22633 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))25665 23134 y Fw(\()p Fu(')p Fw(\))27400 21788 y Fq(\021)28461 23134 y Fw(=)337 b Fu(k)30410 22633 y Fp(2)p Fs(l)31497 23134 y Fw(+)269 b Fu(\017)33201 23316 y Fp(0)48428 23134 y Fw(\(5.23\))475 25839 y FC(has)469 b(no)j(mor)-62 b(e)470 b(than)g(two)h(solutions,)478 b Fu(')18926 25200 y Fr(\006)433 b Fp(\()p Fs(n)p Fr(\000)p Fp(1\))18926 25997 y Fs(\017)19306 26120 y FD(0)22652 25839 y FC(,)480 b(in)25230 25532 y Fw(~)25028 25839 y Fu(\010)25836 26021 y Fs(n)p Fr(\000)p Fp(1)27665 25839 y FC(.)669 b(F)-93 b(or)471 b(any)g Fu(')34160 25200 y Fr(\006)433 b Fp(\()p Fs(n)p Fr(\000)p Fp(1\))34160 25997 y Fs(\017)34540 26120 y FD(0)38357 25839 y FC(ther)-62 b(e)469 b(is)i Fu(')43546 25200 y Fr(\006)433 b Fp(\()p Fs(n)p Fr(\000)p Fp(2\))43546 26186 y(0)47676 25839 y Ft(2)49091 25532 y Fw(~)48889 25839 y Fu(\010)49697 26021 y Fs(n)p Fr(\000)p Fp(2)475 27278 y FC(such)f(that)18286 28717 y Ft(j)p Fu(')19416 28217 y Fr(\006)p Fp(\()p Fs(n)p Fr(\000)p Fp(1\))19416 29017 y Fs(\017)19796 29140 y FD(0)22977 28717 y Ft(\000)269 b Fu(')24982 28078 y Fr(\006)p Fp(\()p Fs(n)p Fr(\000)p Fp(2\))24982 29064 y(0)28274 28717 y Ft(j)336 b Fu(<)h(r)30808 28217 y Fp(\()p Fs(n)p Fp(\))32166 28717 y Fu(=)p Fw(4)p Fu(:)14713 b Fw(\(5.24\))p 0 TeXcolorgray 475 31066 a FF(Lemma)440 b(5.25.)p 0 TeXcolorgray 538 w FC(Supp)-62 b(ose)413 b Fw(0)336 b Fu(<)h(b)16136 30427 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))16136 31413 y(0)19033 31066 y Ft(\024)g Fu(\017)20805 31248 y Fs(n)p Fr(\000)p Fp(2)22633 31066 y Fu(k)23302 30626 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fp(2)p Fs(\016)28121 31066 y FC(and)415 b Fu(')336 b Ft(2)32933 30760 y Fw(~)32731 31066 y Fu(\010)33539 31248 y Fs(n)p Fr(\000)p Fp(1)35783 31066 y FC(ob)-62 b(eys)413 b(the)h(ine)-62 b(quality)412 b(analo)-62 b(gous)475 32827 y(to)433 b(\(5.24\):)554 b Ft(j)p Fu(')269 b Ft(\000)h Fu(')9486 32188 y Fr(\006)p Fp(\()p Fs(n)p Fr(\000)p Fp(2\))9486 33174 y(0)12778 32827 y Ft(j)336 b Fu(<)h(r)15312 32387 y Fp(\()p Fs(n)p Fp(\))16669 32827 y FC(.)557 b(Then,)11998 34630 y Fu(@)p 11601 35171 1504 49 v 11601 36281 a(@)67 b(')13238 35450 y(\025)13945 34950 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))16708 34105 y Fq(\020)17421 35450 y Fu(~)-596 b(y)18068 34950 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))20630 35450 y Fw(\()p Fu(')p Fw(\))22365 34105 y Fq(\021)23426 35450 y Fw(=)24369 35647 y Fs(k)24 b Fr(!1)27156 35450 y Ft(\006)p Fw(2)p Fu(l)g(b)29611 34811 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))29611 35797 y(0)32171 35450 y Fu(k)32840 34950 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1)34860 34468 y Fq(\000)35415 35450 y Fw(1)270 b(+)f Fu(o)p Fw(\(1\))39639 34468 y Fq(\001)40195 35450 y Fu(:)7896 b Fw(\(5.25\))p 0 TeXcolorgray 475 38443 a FF(De\014nition)571 b(5.26.)p 0 TeXcolorgray 603 w Fw(Let)495 b(\000)13578 37805 y Fr(\006)p Fp(\()p Fs(n)p Fr(\000)p Fp(1\))13578 38601 y Fs(s)16870 37462 y Fq(\000)17289 38124 y Fu(~)17426 38443 y(b)17946 38004 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))20506 37462 y Fq(\001)21557 38443 y Fw(b)34 b(e)495 b(the)h(op)34 b(en)495 b(disks)g(cen)-34 b(tered)495 b(at)h Fu(')38873 37805 y Fr(\006)p Fp(\()p Fs(n)p Fr(\000)p Fp(2\))38873 38791 y(0)42660 38443 y Fw(with)g(radius)g Fu(r)49831 38004 y Fp(\()p Fs(n)p Fp(\))51189 38443 y Fw(;)475 40281 y Fu(\015)1170 39643 y Fr(\006)p Fp(\()p Fs(n)p Fr(\000)p Fp(1\))1103 40439 y Fs(s)4462 39300 y Fq(\000)4881 39962 y Fu(~)5017 40281 y(b)5537 39842 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))8098 39300 y Fq(\001)9058 40281 y Fw(b)34 b(e)403 b(their)h(b)34 b(oundary)405 b(circles)e(and)i Fo(O)25926 39643 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))25926 40439 y Fs(s)28486 39300 y Fq(\000)28905 39962 y Fu(~)29041 40281 y(b)29561 39842 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))32122 39300 y Fq(\001)33014 40281 y Fw(=)337 b(\000)35052 39643 y Fp(+\()p Fs(n)p Fr(\000)p Fp(1\))35052 40439 y Fs(s)38613 40281 y Ft([)269 b Fw(\000)40448 39643 y Fr(\000)p Fp(\()p Fs(n)p Fr(\000)p Fp(1\))40448 40439 y Fs(s)43740 40281 y Fw(.)p 0 TeXcolorgray 475 42632 a FF(Lemma)463 b(5.27.)p 0 TeXcolorgray 554 w FC(F)-93 b(or)434 b(any)f Fu(')h FC(in)g Fu(\010)16944 42814 y Fs(n)p Fr(\000)p Fp(1)19042 42632 y Ft(n)269 b Fo(O)20775 41993 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))20775 42790 y Fs(s)23335 42632 y Fw(\()23670 42312 y Fu(~)23806 42632 y(b)24326 42192 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))26887 42632 y Fw(\))p FC(,)13765 43549 y Fq(\014)13765 44277 y(\014)13765 45004 y(\014)14169 44943 y Fu(\025)14876 44443 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))17639 43598 y Fq(\020)18353 44943 y Fu(~)-596 b(y)19000 44443 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))21561 44943 y Fw(\()p Fu(')p Fw(\))23296 43598 y Fq(\021)24290 44943 y Ft(\000)269 b Fu(k)26171 44443 y Fp(2)p Fs(l)26988 43549 y Fq(\014)26988 44277 y(\014)26988 45004 y(\014)27729 44943 y Ft(\025)337 b Fu(b)29529 44304 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))29529 45290 y(0)32089 44943 y Fu(k)32758 44443 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1)p Fr(\000)p Fs(\016)35959 44943 y Fu(r)36540 44443 y Fp(\()p Fs(n)p Fp(\))37898 44943 y Fu(:)10193 b Fw(\(5.26\))p 0 TeXcolorgray 475 47657 a FF(Lemma)463 b(5.28.)p 0 TeXcolorgray 554 w FC(F)-93 b(or)434 b(any)f Fu(')337 b Ft(2)f Fu(\010)16504 47839 y Fs(n)p Fr(\000)p Fp(1)18602 47657 y Ft(n)270 b Fo(O)20336 47019 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))20336 47815 y Fs(s)22896 46676 y Fq(\000)23315 47338 y Fu(~)23452 47657 y(b)23972 47218 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))26532 46676 y Fq(\001)27088 47657 y FC(,)11811 48916 y Fq(\015)11811 49644 y(\015)11811 50371 y(\015)12484 48965 y(\020)13208 50310 y Fu(H)14314 49810 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))17077 48965 y Fq(\020)17791 50310 y Fu(~)-596 b(y)18438 49810 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))20999 50310 y Fw(\()p Fu(')p Fw(\))22734 48965 y Fq(\021)23728 50310 y Ft(\000)269 b Fu(k)25609 49810 y Fp(2)p Fs(l)26426 48965 y Fq(\021)27150 49267 y Fr(\000)p Fp(1)28408 48916 y Fq(\015)28408 49644 y(\015)28408 50371 y(\015)29418 50310 y Fu(<)34669 49490 y Fw(16)p 30830 50031 8890 49 v 30830 51480 a Fu(b)31350 50841 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))31350 51827 y(0)33911 51480 y Fu(r)34492 51128 y Fp(\()p Fs(n)p Fp(\))35849 51480 y Fu(k)36518 51130 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1)p Fr(\000)p Fs(\016)39852 50310 y Fu(;)8239 b Fw(\(5.27\))10984 52878 y Fq(\015)10984 53605 y(\015)10984 54333 y(\015)11658 52927 y(\020)12382 54272 y Fu(H)13488 53772 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))16250 52927 y Fq(\020)16964 54272 y Fu(~)-596 b(y)17611 53772 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))20172 54272 y Fw(\()p Fu(')p Fw(\))21907 52927 y Fq(\021)22901 54272 y Ft(\000)269 b Fu(k)24782 53772 y Fp(2)p Fs(l)25600 52927 y Fq(\021)26323 53229 y Fr(\000)p Fp(1)27581 52878 y Fq(\015)27581 53605 y(\015)27581 54333 y(\015)28254 55115 y Fp(1)29117 54272 y Fu(<)30529 53452 y Fw(16)h Ft(\001)f Fw(4)33223 53012 y Fs(n)p Fr(\000)p Fp(2)35051 53452 y Fu(c)35576 53634 y Fp(0)36102 53452 y Fu(k)36771 53012 y Fp(2+2)p Fs(s)38878 53135 y Fy(n)p Fx(\000)p FD(1)p 30529 53993 10017 49 v 31093 55442 a Fu(b)31613 54803 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))31613 55789 y(0)34174 55442 y Fu(r)34755 55090 y Fp(\()p Fs(n)p Fp(\))36112 55442 y Fu(k)36781 55092 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1)p Fr(\000)p Fs(\016)40679 54272 y Fu(:)7412 b Fw(\(5.28\))p 0 TeXcolorgray 475 57806 a FF(Corollary)328 b(5.29.)p 0 TeXcolorgray 443 w FC(If)310 b Fu(\017)11489 57988 y Fs(n)p Fr(\000)p Fp(1)13318 57806 y Fu(k)13987 57366 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fp(2)p Fs(\016)18727 57806 y Fu(<)336 b(b)20526 57167 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))20526 58153 y(0)23424 57806 y Ft(\024)g Fu(\017)25195 57988 y Fs(n)p Fr(\000)p Fp(2)27024 57806 y Fu(k)27693 57366 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fp(2)p Fs(\016)32096 57806 y FC(,)334 b(and)310 b Fu(')f FC(b)-62 b(elongs)308 b(to)h Fu(\010)42302 57988 y Fs(n)p Fr(\000)p Fp(1)44131 57806 y Ft(n)p Fo(O)45595 57167 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))45595 57964 y Fs(s)48357 56461 y Fq(\020)48945 57486 y Fu(~)49081 57806 y(b)49601 57366 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))52161 56461 y Fq(\021)52885 57806 y FC(,)475 59546 y(then)15095 59877 y Fq(\015)15095 60604 y(\015)15095 61332 y(\015)15769 59926 y(\020)16493 61271 y Fu(H)17599 60771 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))20361 59926 y Fq(\020)21075 61271 y Fu(~)-596 b(y)21722 60771 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))24283 61271 y Fw(\()p Fu(')p Fw(\))26018 59926 y Fq(\021)27012 61271 y Ft(\000)269 b Fu(k)28893 60771 y Fp(2)p Fs(l)29711 59926 y Fq(\021)30435 60228 y Fr(\000)p Fp(1)31692 59877 y Fq(\015)31692 60604 y(\015)31692 61332 y(\015)32702 61271 y Fu(<)34972 60451 y Fw(1)p 34115 60992 2321 49 v 34115 62126 a Fu(\017)34607 61709 y Fp(2)34607 62473 y Fs(n)p Fr(\000)p Fp(1)36568 61271 y Fu(;)11523 b Fw(\(5.29\))12028 63365 y Fq(\015)12028 64092 y(\015)12028 64819 y(\015)12701 63413 y(\020)13425 64759 y Fu(H)14531 64258 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))17294 63413 y Fq(\020)18008 64759 y Fu(~)-596 b(y)18655 64258 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))21216 64759 y Fw(\()p Fu(')p Fw(\))22951 63413 y Fq(\021)23945 64759 y Ft(\000)269 b Fu(k)25826 64258 y Fp(2)p Fs(l)26643 63413 y Fq(\021)27367 63715 y Fr(\000)p Fp(1)28625 63365 y Fq(\015)28625 64092 y(\015)28625 64819 y(\015)29298 65602 y Fp(1)30161 64759 y Fu(<)31573 63939 y Fw(4)32179 63499 y Fs(n)p Fr(\000)p Fp(2)34008 63939 y Fu(c)34533 64121 y Fp(0)35058 63939 y Fu(k)35727 63499 y Fp(2+2)p Fs(s)37834 63622 y Fy(n)p Fx(\000)p FD(1)p 31573 64480 7930 49 v 34377 65613 a Fu(\017)34869 65196 y Fp(2)34869 65961 y Fs(n)p Fr(\000)p Fp(1)39635 64759 y Fu(:)8456 b Fw(\(5.30\))1803 67979 y(The)578 b(corollary)e(follo)-34 b(ws)579 b(from)e(the)i(condition)f (on)g Fu(b)26989 67340 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))26989 68326 y(0)29550 67979 y Fw(,)620 b(estimate)578 b(\(5.1\))g(and)h(the)f (form)-34 b(ula)578 b Fu(r)48599 67539 y Fp(\()p Fs(n)p Fp(\))50582 67979 y Fw(=)475 69541 y Fu(r)1056 69101 y Fp(\()p Fs(n)p Fr(\000)p Fp(1\))3616 69541 y Fu(k)4285 69101 y Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)7124 69212 y Fy(n)7691 69101 y Fr(\000)p Fs(\016)8928 69541 y Fw(.)p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 62 62 TeXDict begin 62 61 bop 0 TeXcolorgray 0 TeXcolorgray 475 3345 a Fp(62)14248 b(YULIA)313 b(KARPESHINA)f(AND)h(YOUNG-RAN)g (LEE)p 0 TeXcolorgray 11863 6001 a Fw(6.)606 b Fv(Limit-Isoener)-27 b(getic)461 b(Set)d(and)h(Eigenfunctions)475 8160 y Fw(6.1.)605 b FF(Limit-Iso)39 b(energetic)469 b(Set)g(and)h(Pro)39 b(of)469 b(of)g(the)h(Bethe-Sommerfeld)e(Conjecture.)606 b Fw(A)-34 b(t)409 b(ev)-34 b(ery)475 9599 y(step)303 b Fu(n)g Fw(w)-34 b(e)303 b(constructed)h(a)f(set)g Fo(B)15717 9781 y Fs(n)16343 9599 y Fw(\()p Fu(\025)p Fw(\),)323 b Fo(B)19575 9781 y Fs(n)20201 9599 y Fw(\()p Fu(\025)p Fw(\))338 b Ft(\032)e Fo(B)24390 9781 y Fs(n)p Fr(\000)p Fp(1)26218 9599 y Fw(\()p Fu(\025)p Fw(\))i Ft(\032)e Fu(S)30227 9781 y Fp(1)30753 9599 y Fw(\()p Fu(\025)p Fw(\),)324 b(and)304 b(a)f(function)h Fn({)41682 9781 y Fs(n)42308 9599 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\),)299 b Fu(~)-581 b(\027)413 b Ft(2)337 b Fo(B)48913 9781 y Fs(n)49539 9599 y Fw(\()p Fu(\025)p Fw(\),)475 11038 y(with)449 b(the)g(follo)-34 b(wing)449 b(prop)34 b(erties.)671 b(The)448 b(set)h Fo(D)22361 11220 y Fs(n)22987 11038 y Fw(\()p Fu(\025)p Fw(\))g(of)g(v)-34 b(ectors)480 b Fu(~)-638 b Fn({)459 b Fw(=)410 b Fn({)34100 11220 y Fs(n)34727 11038 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))-25 b Fu(~)-581 b(\027)77 b Fw(,)435 b Fu(~)-582 b(\027)488 b Ft(2)410 b Fo(B)42291 11220 y Fs(n)42917 11038 y Fw(\()p Fu(\025)p Fw(\),)460 b(is)448 b(a)g(sligh)-34 b(tly)475 12477 y(distorted)504 b(circle)d(with)j(holes,)528 b(see)502 b(Fig.1,)528 b(Fig.2,)f(form)-34 b(ula)504 b(\(1.13\))g(and)g(Lemmas)e(2.11,)528 b(3.42,)f(4.24,)h(5.17.)475 13995 y(F)-101 b(or)528 b(an)-34 b(y)560 b Fu(~)-638 b Fn({)5995 14177 y Fs(n)6622 13995 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))543 b Ft(2)g Fo(D)12387 14177 y Fs(n)13013 13995 y Fw(\()p Fu(\025)p Fw(\))529 b(there)f(is)g(a)g (single)f(eigen)-34 b(v)-67 b(alue)528 b(of)g Fu(H)32916 13555 y Fp(\()p Fs(n)p Fp(\))34274 13995 y Fw(\()32 b Fu(~)-638 b Fn({)35553 14177 y Fs(n)36180 13995 y Fw(\))528 b(equal)g(to)g Fu(\025)g Fw(and)h(giv)-34 b(en)528 b(b)-34 b(y)528 b(a)475 15449 y(p)34 b(erturbation)413 b(series.)11187 14979 y Fp(15)12690 15449 y Fw(Let)f Fo(B)15793 15631 y Fr(1)16788 15449 y Fw(\()p Fu(\025)p Fw(\))351 b(=)20080 14540 y Fq(T)21090 14890 y Fr(1)21090 15807 y Fs(n)p Fp(=1)23121 15449 y Fo(B)24044 15631 y Fs(n)24670 15449 y Fw(\()p Fu(\025)p Fw(\))p Fu(:)412 b Fw(Since)g Fo(B)31164 15631 y Fs(n)p Fp(+1)33342 15449 y Ft(\032)349 b Fo(B)35557 15631 y Fs(n)36595 15449 y Fw(for)412 b(ev)-34 b(ery)410 b Fu(n)p Fw(,)j Fo(B)44068 15631 y Fr(1)45064 15449 y Fw(\()p Fu(\025)p Fw(\))g(is)f(a)f(unit)475 16888 y(circle)452 b(with)i(in\014nite)h(n)-34 b(um)g(b)34 b(er)455 b(of)f(holes,)466 b(more)453 b(and)i(more)e(holes)h(of)g(smaller)f(and)h(smaller)f(size)g (app)34 b(earing)475 18327 y(at)404 b(eac)-34 b(h)405 b(step.)p 0 TeXcolorgray 475 20392 a FF(Lemma)463 b(6.1.)p 0 TeXcolorgray 554 w FC(The)433 b(length)g(of)g Fo(B)16528 20574 y Fr(1)17523 20392 y Fw(\()p Fu(\025)p Fw(\))i FC(satis\014es)c(estimate)h(\(1.9\))f(with)i Fu(\015)35289 20574 y Fp(3)36151 20392 y Fw(=)337 b Fu(\016)46 b(=)p Fw(2)p FC(.)p 0 TeXcolorgray 475 22457 a(Pr)-62 b(o)g(of.)p 0 TeXcolorgray 552 w Fw(Using)385 b(\(2.44\),)k(\(3.64\),)g(\(4.34\))d (and)g(\(5.16\))f(and)h(consideirng)f(that)h Fu(S)36566 22639 y Fs(n)37529 22457 y Ft(\031)336 b Fw(2)39414 22017 y Fs(n)40041 22457 y Fu(s)40609 22639 y Fp(1)41135 22457 y Fw(,)388 b(w)-34 b(e)385 b(easily)f(conclude)475 23975 y(that)486 b Fu(L)202 b Fw(\()p Fo(B)5603 24157 y Fs(n)6229 23975 y Fw(\))472 b(=)8586 22993 y Fq(\000)9141 23975 y Fw(1)270 b(+)f Fu(O)34 b Fw(\()p Fu(k)13328 23535 y Fr(\000)p Fs(\016)f(=)p Fp(2)15505 23975 y Fw(\))15976 22993 y Fq(\001)16532 23975 y Fw(,)505 b Fu(k)510 b Fw(=)471 b Fu(\025)20636 23535 y Fp(1)p Fs(=)p Fp(2)p Fs(l)22879 23975 y Fw(uniformly)485 b(in)g Fu(n)p Fw(.)780 b(Since)485 b Fo(B)35994 24157 y Fs(n)37104 23975 y Fw(is)g(a)g(decreasing)f (sequence)h(of)475 25419 y(sets,)404 b(\(1.9\))g(holds.)p 50750 25585 665 831 v 1803 27522 a(Let)g(us)g(consider)g Fn({)11062 27704 y Fr(1)12059 27522 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))337 b(=)f(lim)18223 27704 y Fs(n)p Fr(!1)20933 27522 y Fn({)21741 27704 y Fs(n)22367 27522 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))p Fu(;)1390 b(~)-581 b(\027)413 b Ft(2)337 b Fo(B)30063 27704 y Fr(1)31059 27522 y Fw(\()p Fu(\025)p Fw(\))p Fu(:)p 0 TeXcolorgray 475 29587 a FF(Lemma)382 b(6.2.)p 0 TeXcolorgray 496 w FC(The)369 b(limit)f Fn({)14038 29769 y Fr(1)15034 29587 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))369 b FC(exists)e(for)i(any)343 b Fu(~)-581 b(\027)414 b Ft(2)336 b Fo(B)28731 29769 y Fr(1)29727 29587 y Fw(\()p Fu(\025)p Fw(\))370 b FC(and)e(the)g(fol)62 b(lowing)367 b(estimates)f(hold)h(when)475 31027 y Fu(n)336 b Ft(\025)g Fw(1)p FC(:)7527 32903 y Ft(j)p Fn({)8672 33085 y Fr(1)9668 32903 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)78 b Fw(\))269 b Ft(\000)g Fn({)14822 33085 y Fs(n)15449 32903 y Fw(\()p Fu(\025;)177 b(~)-581 b(\027)77 b Fw(\))p Ft(j)337 b Fu(<)f Fw(4)p Fu(\017)21364 32403 y Fp(4)21364 33203 y Fs(n)21990 32903 y Fu(k)22659 32403 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)25411 32903 y Fu(;)1069 b(\017)27309 33085 y Fs(n)28272 32903 y Fw(=)336 b(exp\()p Ft(\000)32950 32083 y Fw(1)p 32950 32624 607 49 v 32950 33735 a(4)33689 32903 y Fu(k)34358 32403 y Fs(\021)32 b(s)35297 32514 y Fy(n)35919 32903 y Fw(\))p Fu(;)1070 b(s)38365 33085 y Fs(n)39328 32903 y Fw(=)336 b(2)41213 32403 y Fs(n)p 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Fp(1)29916 40324 y Ft(\000)g Fw(2)p Fu(\016)46 b Fw(.)1803 41839 y(Estimates)439 b(\(3.63\),)448 b(\(4.33\))439 b(and)h(\(5.15\))f(justify)h(con)-34 b(v)g(ergence)438 b(of)h(the)g(series)38141 40930 y Fq(P)39421 41280 y Fr(1)39421 42197 y Fs(m)p Fp(=1)41845 41352 y Fs(@)52 b(h)42935 41463 y Fy(n)p 41845 41560 1657 49 v 42093 42257 a Fs(@)g(')43635 41839 y Fu(;)438 b Fw(and)i(hence,)447 b(of)475 43641 y(the)404 b(sequence)7719 43155 y Fs(@)52 b Fm({)8894 43266 y Fy(n)p 7719 43362 1742 49 v 8009 44059 a Fs(@)g(')9593 43641 y Fu(:)404 b Fw(W)-101 b(e)404 b(denote)g(the)h(limit)e(of)i(this)g(sequence)e(b)-34 b(y)31895 43155 y Fs(@)52 b Fm({)33070 43266 y Fx(1)p 31895 43362 2043 49 v 32335 44059 a Fs(@)g(')34070 43641 y Fu(:)p 0 TeXcolorgray 475 45877 a FF(Lemma)415 b(6.4.)p 0 TeXcolorgray 519 w FC(The)394 b(estimate)e(\(1.17\))h(with)h Fu(\015)22127 46059 y Fp(5)22989 45877 y Fw(=)336 b(\(4)p Fu(l)209 b Ft(\000)185 b Fw(5)g Ft(\000)g Fw(8)p Fu(s)30137 46059 y Fp(1)30847 45877 y Ft(\000)g 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56218 y Fs(n)11381 56036 y Fw(\()p Fu(\025)p Fw(\).)958 b(Applying)545 b(p)34 b(erturbation)545 b(form)-34 b(ulae)544 b(for)g Fu(n)p Fw(-th)h(step,)579 b(w)-34 b(e)544 b(easily)f(obtain)i(that)475 57554 y(there)430 b(is)h(an)g(eigen)-34 b(v)-67 b(alue)430 b Fu(\025)13083 57114 y Fp(\()p Fs(n)p Fp(\))14441 57554 y Fw(\()32 b Fu(~)-638 b Fn({)49 b Fw(\))431 b(of)g Fu(H)19184 57114 y Fp(\()p Fs(n)p Fp(\))20543 57554 y Fw(\()32 b Fu(~)-638 b Fn({)49 b Fw(\))431 b(satisfying)g(the)g(estimate)g Fu(\025)35903 57114 y Fp(\()p Fs(n)p Fp(\))37261 57554 y Fw(\()32 b Fu(~)-638 b Fn({)50 b Fw(\))381 b(=)g Fu(\025)287 b Fw(+)g Fu(\016)43529 57736 y Fs(n)44155 57554 y Fw(,)437 b Fu(\016)45468 57736 y Fs(n)46475 57554 y Fw(=)381 b Fu(O)48959 56572 y Fq(\000)49515 57554 y Fu(\017)50007 57114 y Fp(4)50007 57853 y Fs(n)50633 56572 y Fq(\001)51189 57554 y Fw(,)475 59193 y(the)442 b(eigen)-34 b(v)-67 b(alue)442 b Fu(\025)9137 58753 y Fp(\()p Fs(n)p Fp(\))10495 59193 y Fw(\()32 b Fu(~)-638 b Fn({)49 b 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b(of)g(the)g(sp)34 b(ectrum)476 b(of)h Fu(H)33437 6183 y Fs(n)34538 6001 y Fw(in)g(the)f Fu(\016)38723 6183 y Fs(n)39349 6001 y Fw(-neigh)-34 b(b)34 b(orho)g(o)g(d)477 b(of)g Fu(\025)f Fw(for)475 7440 y(ev)-34 b(ery)315 b Fu(\025)337 b(>)f(k)6581 7001 y Fp(2)p Fs(l)6543 7740 y Fr(\003)7399 7440 y Fw(,)d Fu(k)8700 7622 y Fr(\003)9543 7440 y Fw(b)34 b(eing)316 b(in)-34 b(tro)34 b(duced)317 b(b)-34 b(y)317 b(\(4.2\).)510 b(Since)316 b Ft(k)p Fu(H)28322 7622 y Fs(n)29042 7440 y Ft(\000)94 b Fu(H)k Ft(k)337 b Fu(<)f(\017)33899 7001 y Fp(4)33899 7740 y Fs(n)34525 7440 y Fw(,)e(there)316 b(is)h(a)f(p)34 b(oin)-34 b(t)317 b(of)g(the)g(sp)34 b(ectrum)475 8879 y(of)411 b Fu(H)508 b Fw(in)411 b(the)g Fu(\016)7478 8440 y Fr(\003)7432 9179 y Fs(n)8058 8879 y Fw(-neigh)-34 b(b)34 b(orho)g(o)g(d)411 b(of)g Fu(\025)p Fw(,)h Fu(\016)19509 8440 y Fr(\003)19463 9179 y Fs(n)20437 8879 y Fw(=)347 b Fu(\016)22266 9061 y Fs(n)23166 8879 y Fw(+)273 b Fu(\017)24874 8440 y Fp(4)24874 9179 y Fs(n)25501 8879 y Fw(.)558 b(Since)410 b(it)h(is)f(true)h(for)g (ev)-34 b(ery)409 b Fu(n)h Fw(and)i(the)f(sp)34 b(ectrum)411 b(of)475 10319 y Fu(H)502 b Fw(is)404 b(closed,)f Fu(\025)h Fw(is)g(in)g(the)h(sp)34 b(ectrum)404 b(of)h Fu(H)98 b Fw(.)p 50750 10485 665 831 v 475 13048 a(6.2.)605 b FF(Generalized)544 b(Eigenfunctions)h(of)e Fu(H)98 b FF(.)607 b Fw(A)473 b(plane)g(w)-34 b(a)g(v)g(e)473 b(is)g(usually)g (denoted)h(b)-34 b(y)473 b Fu(e)44168 12608 y Fs(i)p Fr(h)44807 12375 y Fs(~)44854 12608 y(k)24 b(;x)p Fr(i)46581 13048 y Fw(,)47346 12728 y Fu(~)47407 13048 y(k)490 b Ft(2)451 b Fn(R)50662 12608 y Fp(2)51189 13048 y Fw(.)475 14581 y(Here)409 b(w)-34 b(e)411 b(use)443 b Fu(~)-638 b Fn({)459 b Fw(instead)411 b(of)14018 14261 y Fu(~)14080 14581 y(k)448 b Fw(to)411 b(comply)f(with)h(our)g(previous)f (notations.)559 b(W)-101 b(e)410 b(sho)-34 b(w)412 b(that)f(for)g(ev) -34 b(ery)441 b Fu(~)-638 b Fn({)459 b Fw(in)475 16020 y(a)404 b(set)17905 17636 y Fo(G)18638 17818 y Fr(1)19971 17636 y Fw(=)336 b 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Fw(.)475 7440 y(Hence,)646 b Ft(j)p Fo(G)5727 7622 y Fr(1)6992 7440 y Ft(\\)270 b FF(B)9062 7630 y Fl(R)9927 7440 y Ft(j)661 b Fw(=)f(lim)14212 7622 y Fs(n)p Fr(!1)16922 7440 y Ft(j)p Fo(G)17992 7622 y Fs(n)18887 7440 y Ft(\\)269 b FF(B)20956 7630 y Fl(R)21822 7440 y Ft(j)p Fw(.)1121 b(Summarizing)599 b(v)-34 b(olumes)599 b(of)g(the)g(regions)f Fu(U)45000 7622 y Fs(n)45626 7440 y Fw(,)647 b(w)-34 b(e)599 b(easily)475 8958 y(conclude)475 b Ft(j)p Fo(G)6599 9140 y Fs(n)7494 8958 y Ft(\\)269 b FF(B)9563 9148 y Fl(R)10429 8958 y Ft(j)454 b Fw(=)h Ft(j)p FF(B)13947 9148 y Fl(R)14812 8958 y Ft(j)15351 7976 y Fq(\000)15907 8958 y Fw(1)269 b(+)g Fu(O)34 b Fw(\()p Fu(R)20353 8518 y Fr(\000)p Fs(\016)f(=)p Fp(2)22531 8958 y Fw(\))23002 7976 y Fq(\001)24033 8958 y Fw(uniformly)475 b(in)h Fu(n)p Fw(.)750 b(Th)-34 b(us,)495 b(w)-34 b(e)475 b(ha)-34 b(v)g(e)476 b(obtained)g(\(1.7\))f(with)475 10402 y Fu(\015)1103 10584 y Fp(3)1965 10402 y Fw(=)336 b Fu(\016)46 b(=)p Fw(2.)p 50750 10568 665 831 v 18425 12859 a Fv(Appendix)458 b Fw(A.)606 b Fv(Appendices.)475 15018 y Fw(A.1.)f FF(Pro)39 b(of)310 b(of)f(Corollary)h(3.6.)606 b Fw(Let)269 b Fu(\034)19785 15200 y Fp(0)20647 15018 y Ft(2)337 b Fu(\037)22550 15200 y Fp(2)23076 15018 y Fw(.)494 b(T)-101 b(aking)270 b(in)-34 b(to)270 b(accoun)-34 b(t)270 b(the)g(relation)f Fu(\025)41452 14379 y Fp(\(1\))41452 15379 y Fs(j)42710 15018 y Fw(\()p Fu(\034)43711 15200 y Fp(0)44237 15018 y Fw(+2)p Fu(\031)43 b(p=)-67 b(N)48643 15200 y Fp(1)49169 15018 y Fu(a)p Fw(\))336 b(=)475 16965 y Fu(k)1144 16526 y Fp(2)p Fs(l)2544 16965 y Fw(and)584 b(the)f(de\014nition)h(of)f Fu(C)15337 17147 y Fp(2)15864 16965 y Fw(,)627 b(w)-34 b(e)583 b(see)g(that)h Ft(j)p Fu(\025)24779 16326 y Fp(\(1\))24779 17327 y Fs(j)26036 16965 y Fw(\()p Fu(\034)27037 17147 y Fp(0)27952 16965 y Fw(+)388 b(2)p Fu(\031)43 b(p=)-67 b(N)32746 17147 y Fp(1)33272 16965 y Fu(a)p Fw(\))389 b Ft(\000)f Fu(z)53 b Ft(j)634 b Fw(=)h Fu(\017)39762 17147 y Fp(1)40288 16965 y Fu(=)p Fw(2.)1075 b(Using)583 b(\(3.8\))g(and)475 18913 y(the)522 b(last)g(equalit)-34 b(y)-101 b(,)550 b(w)-34 b(e)522 b(easily)f(obtain:)775 b Ft(j)p Fu(\025)20971 18274 y Fp(\(1\))20971 19071 y Fs(n)22228 18913 y Fw(\()p Fu(\034)23229 19095 y Fp(0)24103 18913 y Fw(+)347 b(2)p Fu(\031)146 b Fw(^)-709 b Fu(p)q(=)-67 b(N)28857 19095 y Fp(1)29382 18913 y Fu(a)p Fw(\))348 b Ft(\000)g Fu(z)53 b Ft(j)532 b(\025)g Fu(\017)35586 19095 y Fp(1)36112 18913 y Fu(=)p Fw(2)522 b(for)g Fu(\025)40526 18274 y Fp(\(1\))40526 19071 y Fs(n)41784 18913 y Fw(\()p Fu(\034)42785 19095 y Fp(0)43658 18913 y Fw(+)348 b(2)p Fu(\031)146 b Fw(^)-709 b Fu(p=)-67 b(N)48412 19095 y Fp(1)48938 18913 y Fu(a)p Fw(\))532 b Ft(6)p Fw(=)475 20630 y Fu(\025)1182 19991 y Fp(\(1\))1182 20992 y Fs(j)2439 20630 y Fw(\()p Fu(\034)3440 20812 y Fp(0)4236 20630 y Fw(+)269 b(2)p Fu(\031)43 b(p=)-67 b(N)8911 20812 y Fp(1)9437 20630 y Fu(a)p Fw(\).)538 b(Therefore,)404 b(for)g(an)-34 b(y)405 b Fu(z)389 b Ft(2)337 b Fu(C)24365 20812 y Fp(2)24891 20630 y Fw(,)18972 22947 y Ft(k)p Fw(\()20367 22640 y(~)20049 22947 y Fu(H)21155 22446 y Fp(\(1\))22413 22947 y Fw(\()p Fu(\034)23414 23129 y Fp(0)23940 22947 y Fw(\))270 b Ft(\000)f Fu(z)53 b Fw(\))26981 22446 y Fr(\000)p Fp(1)28239 22947 y Ft(k)337 b(\024)f Fw(2)p Fu(=\017)32165 23129 y Fp(1)32691 22947 y Fu(;)15703 b Fw(\(A.1\))475 24893 y(i.e.,)402 b(\(3.11\))j(is)f(pro)-34 b(v)g(ed)405 b(for)f Fu(\034)13813 25075 y Fp(0)14675 24893 y Ft(2)337 b Fu(\037)16578 25075 y Fp(2)17104 24893 y Fw(.)539 b(No)-34 b(w)405 b(w)-34 b(e)404 b(consider)g Fu(\034)541 b Fw(in)404 b(the)h(complex)475 26362 y(\()p Fu(\017)1438 26544 y Fp(1)1964 26362 y Fu(k)2633 25922 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fs(\016)6566 26362 y Fw(\))p Ft(\000)p Fw(neigh)-34 b(b)34 b(orho)g(o)g(d)405 b(of)f Fu(\037)17731 26544 y Fp(2)18257 26362 y Fw(.)539 b(By)403 b(Hilb)34 b(ert)404 b(relation,)10877 28447 y(\()11666 28141 y(~)11348 28447 y Fu(H)12454 27947 y Fp(\(1\))13712 28447 y Fw(\()p Fu(\034)137 b Fw(\))271 b Ft(\000)e Fu(z)53 b Fw(\))17892 27947 y Fr(\000)p Fp(1)19486 28447 y Fw(=)337 b(\()21555 28141 y(~)21237 28447 y Fu(H)22343 27947 y Fp(\(1\))23601 28447 y Fw(\()p Fu(\034)24602 28629 y Fp(0)25128 28447 y Fw(\))270 b Ft(\000)f Fu(z)53 b Fw(\))28169 27947 y Fr(\000)p Fp(1)29696 28447 y Fw(+)269 b Fu(T)31616 28629 y Fp(1)32143 28447 y Fu(T)32851 28629 y Fp(2)33377 28447 y Fw(\()34165 28141 y(~)33848 28447 y Fu(H)34954 27947 y Fp(\(1\))36212 28447 y Fw(\()p Fu(\034)137 b Fw(\))270 b Ft(\000)f Fu(z)53 b Fw(\))40391 27947 y Fr(\000)p Fp(1)41649 28447 y Fu(;)5570 30521 y(T)6278 30703 y Fp(1)7140 30521 y Fw(=)337 b(\()9209 30214 y(~)8891 30521 y Fu(H)9899 30703 y Fp(0)10425 30521 y Fw(\()p Fu(\034)11426 30703 y Fp(0)11952 30521 y Fw(\))270 b Ft(\000)f Fu(z)53 b Fw(\))14993 30020 y Fr(\000)p Fp(1)16251 30521 y Fw(\()17039 30214 y(~)16722 30521 y Fu(H)17730 30703 y Fp(0)18255 30521 y Fw(\()p Fu(\034)19256 30703 y Fp(0)19783 30521 y Fw(\))269 b(+)g Fu(k)22404 30020 y Fp(2)p Fs(l)23222 30521 y Fw(\))p Fu(;)1414 b(T)26152 30703 y Fp(2)27015 30521 y Fw(=)336 b(\()29083 30214 y(~)28765 30521 y Fu(H)29773 30703 y Fp(0)30299 30521 y Fw(\()p Fu(\034)31300 30703 y Fp(0)31826 30521 y Fw(\))270 b(+)f Fu(k)34448 30020 y Fp(2)p Fs(l)35265 30521 y Fw(\))35736 30020 y Fr(\000)p Fp(1)36994 30521 y Fw(\()37783 30214 y(~)37465 30521 y Fu(H)38473 30703 y Fp(0)38999 30521 y Fw(\()p Fu(\034)40000 30703 y Fp(0)40526 30521 y Fw(\))h Ft(\000)42796 30214 y Fw(~)42479 30521 y Fu(H)43487 30703 y Fp(0)44012 30521 y Fw(\()p Fu(\034)137 b Fw(\)\))p Fu(:)475 32681 y Fw(Supp)34 b(ose)423 b(w)-34 b(e)422 b(ha)-34 b(v)g(e)422 b(c)-34 b(hec)g(k)g(ed)422 b(that)h Ft(k)p Fu(T)18281 32863 y Fp(1)18807 32681 y Fu(T)19515 32863 y Fp(2)20042 32681 y Ft(k)366 b Fu(<)g(k)22992 32241 y Fr(\000)p Fs(\016)24229 32681 y Fw(.)591 b(Then,)427 b(using)422 b(\(A.1\))h(and)f(a)g (standard)h(argumen)-34 b(t)423 b(as)f(in)475 34149 y(Lemma)469 b(2.9,)485 b(w)-34 b(e)470 b(easily)e(arriv)-34 b(e)469 b(at)h(\(3.11\).)735 b(The)470 b(estimate)f Ft(k)p Fu(T)30507 34331 y Fp(1)31034 34149 y Ft(k)445 b Fu(<)h Fw(4)p Fu(k)34749 33709 y Fp(2)p Fs(l)35566 34149 y Fu(=\017)36664 34331 y Fp(1)37190 34149 y Fw(,)486 b(easily)468 b(follo)-34 b(ws)470 b(from)g(\(A.1\).)475 35618 y(The)407 b(estimate)f Ft(k)p Fu(T)9140 35800 y Fp(2)9667 35618 y Ft(k)340 b Fu(<)h Fw(2)p Fu(l)24 b(\017)13381 35800 y Fp(1)13906 35618 y Fu(k)14575 35178 y Fr(\000)p Fp(2)p Fs(l)11 b Fr(\000)p Fs(\016)17712 35618 y Fw(easily)406 b(follo)-34 b(ws)407 b(from)g Ft(j)p Fu(\034)h Ft(\000)270 b Fu(\034)30923 35800 y Fp(0)31449 35618 y Ft(j)341 b Fu(<)f(\017)33902 35800 y Fp(1)34428 35618 y Fu(k)35097 35178 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)p Fr(\000)p Fs(\016)39030 35618 y Fw(.)546 b(Th)-34 b(us,)408 b Ft(k)p Fu(T)44637 35800 y Fp(1)45163 35618 y Fu(T)45871 35800 y Fp(2)46397 35618 y Ft(k)341 b Fu(<)g Fw(8)p Fu(l)24 b(k)50289 35178 y Fr(\000)p Fs(\016)475 37057 y Fw(and,)404 b(hence,)g(\(3.11\))h(is)f (pro)-34 b(v)g(ed.)539 b(Estimate)404 b(\(3.12\))h(follo)-34 b(ws)405 b(from)f(the)h(Hilb)34 b(ert)403 b(relation)3388 39143 y(\()4176 38836 y(~)3859 39143 y Fu(H)4965 38642 y Fp(\(1\))6223 39143 y Fw(\()p Fu(\034)137 b Fw(\))270 b Ft(\000)f Fu(z)53 b Fw(\))10402 38642 y Fr(\000)p Fp(1)11997 39143 y Fw(=)336 b(\()14065 38836 y(~)13747 39143 y Fu(H)14755 39325 y Fp(0)15281 39143 y Fw(\()p Fu(\034)137 b Fw(\))270 b(+)f Fu(k)19041 38642 y Fp(2)p Fs(l)19859 39143 y Fw(\))20330 38642 y Fr(\000)p Fp(1)21857 39143 y Fw(+)g(\()23858 38836 y(~)23540 39143 y Fu(H)24548 39325 y Fp(0)25074 39143 y Fw(\()p Fu(\034)137 b Fw(\))270 b(+)f Fu(k)28834 38642 y Fp(2)p Fs(l)29652 39143 y Fw(\))30123 38642 y Fr(\000)p Fp(1)31381 39143 y Fw(\()p Ft(\000)p Fu(W)33940 39325 y Fp(1)34735 39143 y Fw(+)g Fu(k)36616 38642 y Fp(2)p Fs(l)37703 39143 y Fw(+)g Fu(z)53 b Fw(\)\()40792 38836 y(~)40474 39143 y Fu(H)41580 38642 y Fp(\(1\))42838 39143 y Fw(\()p Fu(\034)137 b Fw(\))270 b Ft(\000)g Fu(z)53 b Fw(\))47018 38642 y Fr(\000)p Fp(1)48276 39143 y Fu(;)475 41179 y Fw(an)404 b(elemen)-34 b(tary)404 b(estimate)g Ft(k)p Fw(\()14607 40873 y(~)14289 41179 y Fu(H)15297 41361 y Fp(0)15823 41179 y Fw(\()p Fu(\034)137 b Fw(\))270 b(+)f Fu(k)19583 40740 y Fp(2)p Fs(l)20401 41179 y Fw(\))20872 40740 y Fr(\000)p Fp(1)22130 41179 y Ft(k)22736 41361 y Fp(1)23598 41179 y Fu(<)337 b(b)25398 41361 y Fp(1)25924 41179 y Fu(b)26444 41361 y Fp(2)26970 41179 y Fu(k)27639 40740 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+2+2)p Fs(s)31972 40863 y FD(2)32893 41179 y Fw(and)405 b(\(3.11\).)475 43345 y(A.2.)605 b FF(Pro)39 b(of)465 b(of)f(estimate)g(\(3.50\).)606 b Fw(Note)405 b(that)15874 44266 y Fq(\014)15874 44994 y(\014)15874 45721 y(\014)16278 45660 y Ft(j)-10 b Fu(~)-596 b(y)17209 45842 y Fp(0)17735 45660 y Fw(\()p Fu(')p Fw(\))270 b(+)279 b Fu(~)-616 b(p)21562 45842 y Fs(m)22449 45660 y Fw(\(0\))p Ft(j)24334 45160 y Fp(2)24334 45960 y Fr(\003)25130 45660 y Ft(\000)269 b(j)-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))270 b(+)278 b Fu(~)-615 b(p)31144 45842 y Fs(m)32031 45660 y Fw(\(0\))p Ft(j)33916 45160 y Fp(2)33916 45960 y Fr(\003)34443 44266 y Fq(\014)34443 44994 y(\014)34443 45721 y(\014)35183 45660 y Fw(=)3108 47070 y Fq(\014)3108 47797 y(\014)3108 48524 y(\014)3512 48464 y Ft(j)-10 b Fu(~)-596 b(y)4443 48646 y Fp(0)4968 48464 y Fw(\()p Fu(')p Fw(\))270 b(+)279 b Fu(~)-615 b(p)8796 48646 y Fs(m)9683 48464 y Fw(\(0\))p Ft(j)11568 47963 y Fp(2)11568 48763 y Fr(\003)12364 48464 y Ft(\000)269 b(j)-10 b Fu(~)-596 b(y)14507 48646 y Fp(0)15033 48464 y Fw(\()p Fu(')p Fw(\))270 b(+)18170 48144 y Fu(~)18250 48464 y(h)18948 48646 y Fp(1)19474 48464 y Fw(\()p Fu(')p Fw(\))g(+)279 b Fu(~)-616 b(p)23301 48646 y Fs(m)24189 48464 y Fw(\(0\))p Ft(j)26074 47963 y Fp(2)26074 48763 y Fr(\003)26600 47070 y Fq(\014)26600 47797 y(\014)26600 48524 y(\014)27341 48464 y Fw(=)336 b Fu(O)29579 47118 y Fq(\020)30302 48464 y Ft(j)p Fu(h)31337 48646 y Fp(1)31863 48464 y Ft(j)269 b(\001)g(j)-10 b Fu(~)-596 b(y)34006 48646 y Fp(0)34532 48464 y Fw(\()p Fu(')p Fw(\))270 b(+)279 b Fu(~)-616 b(p)38359 48646 y Fs(m)39247 48464 y Fw(\(0\))p Ft(j)270 b Fw(+)f Ft(j)p Fu(h)43649 48646 y Fp(1)44175 48464 y Ft(j)44512 47963 y Fp(2)45037 47118 y Fq(\021)45761 48464 y Fu(:)2633 b Fw(\(A.2\))475 50628 y(If)404 b Fu(p)2297 50810 y Fs(m)3184 50628 y Fw(\(0\))338 b Fu(<)e Fw(4)p Fu(k)38 b Fw(,)404 b(then,)h(ob)-34 b(viously)-101 b(,)404 b Ft(j)-10 b Fu(~)-596 b(y)17990 50810 y Fp(0)18516 50628 y Fw(\()p Fu(')p Fw(\))270 b(+)279 b Fu(~)-616 b(p)22343 50810 y Fs(m)23230 50628 y Fw(\(0\))p Ft(j)25115 50188 y Fp(2)25115 50928 y Fr(\003)25978 50628 y Fw(=)337 b Fu(O)34 b Fw(\()p Fu(k)29357 50188 y Fp(2)29882 50628 y Fw(\).)539 b(Considering)405 b(\(2.43\),)f(w)-34 b(e)405 b(arriv)-34 b(e)403 b(at)5748 51549 y Fq(\014)5748 52277 y(\014)5748 53004 y(\014)6152 52943 y Ft(j)-10 b Fu(~)-596 b(y)7083 53125 y Fp(0)7608 52943 y Fw(\()p Fu(')p Fw(\))270 b(+)279 b Fu(~)-615 b(p)11436 53125 y Fs(m)12323 52943 y Fw(\(0\))p Ft(j)14208 52443 y Fp(2)p Fs(l)14208 53243 y Fr(\003)15295 52943 y Ft(\000)269 b(j)-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))270 b(+)278 b Fu(~)-615 b(p)21309 53125 y Fs(m)22196 52943 y Fw(\(0\))p Ft(j)24081 52443 y Fp(2)p Fs(l)24081 53243 y Fr(\003)24899 51549 y Fq(\014)24899 52277 y(\014)24899 53004 y(\014)25640 52943 y Fw(=)336 b Fu(O)34 b Fw(\()p Ft(j)p Fu(h)29384 53125 y Fp(1)29910 52943 y Ft(j)269 b(\001)g Fu(k)31791 52443 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1)33810 52943 y Fw(\))337 b(=)g Fu(O)34 b Fw(\()p Fu(k)37997 52443 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)41598 52566 y FD(1)42060 52443 y Fr(\000)p Fp(2)p Fs(\015)43747 52566 y FD(0)44208 52443 y Fr(\000)p Fs(\016)45444 52943 y Fw(\))p Fu(:)475 55404 y Fw(Com)-34 b(bining)405 b(this)g(relation)f(with)h(\(3.44\))f(where)h Fu(\017)23708 55586 y Fp(0)24570 55404 y Fw(=)336 b Fu(k)26518 54964 y Fr(\000)p Fp(4)p Fr(\000)p Fp(6)p Fs(s)29357 55087 y FD(1)29820 54964 y Fr(\000)p Fp(3)p Fs(\016)31527 55404 y Fw(,)404 b(w)-34 b(e)404 b(obtain)1784 56802 y Fq(\014)1784 57529 y(\014)1784 58257 y(\014)1784 58984 y(\014)2321 57740 y Ft(j)-10 b Fu(~)-596 b(y)3252 57922 y Fp(0)3777 57740 y Fw(\()p Fu(')p Fw(\))270 b(+)279 b Fu(~)-615 b(p)7605 57922 y Fs(m)8492 57740 y Fw(\(0\))p Ft(j)10377 57300 y Fp(2)p Fs(l)10377 58039 y Fr(\003)11464 57740 y Ft(\000)269 b(j)-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))270 b(+)279 b Fu(~)-616 b(p)17478 57922 y Fs(m)18365 57740 y Fw(\(0\))p Ft(j)20250 57300 y Fp(2)p Fs(l)20250 58039 y Fr(\003)p 2321 58281 18748 49 v 5773 59391 a Ft(j)-10 b Fu(~)-596 b(y)6704 59573 y Fp(0)7230 59391 y Fw(\()p Fu(')p Fw(\))270 b(+)279 b Fu(~)-616 b(p)11057 59573 y Fs(m)11945 59391 y Fw(\(0\))p Ft(j)13830 59041 y Fp(2)p Fs(l)13830 59691 y Fr(\003)14917 59391 y Ft(\000)269 b Fu(k)16798 59041 y Fp(2)p Fs(l)21201 56802 y Fq(\014)21201 57529 y(\014)21201 58257 y(\014)21201 58984 y(\014)11831 62057 y Ft(\024)14666 61237 y Fu(c)p Ft(j)p Fu(h)16226 61419 y Fp(1)16751 61237 y Ft(j)p Fu(k)17757 60797 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1)p 13243 61778 7956 49 v 13243 62888 a Fu(b)13763 63070 y Fp(0)14289 62888 y Fu(k)14958 62538 y Fp(2)p Fs(l)g Fr(\000)p Fp(9)p Fr(\000)p Fp(12)p Fs(s)19029 62661 y FD(1)19492 62538 y Fr(\000)p Fp(6)p Fs(\016)21669 62057 y Fw(=)336 b Fu(O)34 b Fw(\()p Fu(b)24898 61547 y Fr(\000)p Fp(1)24898 62404 y(0)26156 62057 y Ft(j)p Fu(h)27191 62239 y Fp(1)27716 62057 y Ft(j)p Fu(k)28722 61556 y Fp(8+12)p Fs(s)31299 61679 y FD(1)31762 61556 y Fp(+6)p Fs(\016)33469 62057 y Fw(\))337 b(=)g Fu(O)34 b Fw(\()p Fu(b)37507 61547 y Fr(\000)p Fp(1)37507 62404 y(0)38764 62057 y Fu(k)39433 61556 y Fp(7+8)p Fs(s)41540 61679 y FD(1)42002 61556 y Fr(\000)p Fp(2)p Fs(\015)43689 61679 y FD(0)44150 61556 y Fp(+5)p Fs(\016)45858 62057 y Fw(\))337 b(=)f Fu(o)p Fw(\(1\))p Fu(;)475 64597 y Fw(the)359 b(last)h(estimate)f(follo)-34 b(wing)360 b(from)g(the)f(restriction)g(on)h Fu(b)27434 64779 y Fp(0)28319 64597 y Fw(in)f(the)h(statemen)-34 b(t)361 b(of)e(the)h(lemma.)523 b(Th)-34 b(us,)370 b(\(3.50\))475 66036 y(holds)404 b(for)h Fu(p)6112 66218 y Fs(m)6999 66036 y Fw(\(0\))338 b Fu(<)e Fw(4)p Fu(k)38 b Fw(.)539 b(Supp)34 b(ose)405 b Fu(p)17679 66218 y Fs(m)18567 66036 y Fw(\(0\))337 b Ft(\025)f Fw(4)p Fu(k)38 b(:)404 b Fw(Using)h (\(3.45\))g(and)f(\(A.2\),)h(w)-34 b(e)404 b(easily)f(obtain:)8021 67075 y Fq(\014)8021 67802 y(\014)8021 68529 y(\014)8021 69256 y(\014)8558 68012 y Ft(j)-10 b Fu(~)-596 b(y)9489 68194 y Fp(0)10015 68012 y Fw(\()p Fu(')p Fw(\))270 b(+)278 b Fu(~)-615 b(p)13842 68194 y Fs(m)14729 68012 y Fw(\(0\))p Ft(j)16614 67572 y Fp(2)p Fs(l)16614 68312 y Fr(\003)17701 68012 y Ft(\000)269 b(j)-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))270 b(+)279 b Fu(~)-616 b(p)23715 68194 y Fs(m)24603 68012 y Fw(\(0\))p Ft(j)26488 67572 y Fp(2)p Fs(l)26488 68312 y Fr(\003)p 8558 68553 18748 49 v 12011 69664 a Ft(j)-10 b Fu(~)-596 b(y)12942 69846 y Fp(0)13467 69664 y Fw(\()p Fu(')p Fw(\))270 b(+)279 b Fu(~)-616 b(p)17294 69846 y Fs(m)18182 69664 y Fw(\(0\))p Ft(j)20067 69314 y Fp(2)p Fs(l)20067 69964 y Fr(\003)21154 69664 y Ft(\000)269 b Fu(k)23035 69314 y Fp(2)p Fs(l)27438 67075 y Fq(\014)27438 67802 y(\014)27438 68529 y(\014)27438 69256 y(\014)28179 68832 y Ft(\024)30165 68012 y(j)p Fu(h)31200 68194 y Fp(1)31726 68012 y Ft(j)p 29591 68553 3047 49 v 29591 69664 a Fu(p)30201 69846 y Fs(m)31089 69664 y Fw(\(0\))33107 68832 y(=)336 b Fu(O)34 b Fw(\()p Fu(k)36485 68332 y Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)39324 68455 y FD(1)39786 68332 y Fr(\000)p Fp(2)p Fs(\015)41473 68455 y FD(0)41934 68332 y Fr(\000)p Fs(\016)43171 68832 y Fw(\))p Fu(:)p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 66 66 TeXDict begin 66 65 bop 0 TeXcolorgray 0 TeXcolorgray 475 3345 a Fp(66)14248 b(YULIA)313 b(KARPESHINA)f(AND)h(YOUNG-RAN)g (LEE)p 0 TeXcolorgray 475 6001 a Fw(This)404 b(means)h(\(3.50\))g (holds)f(in)g(this)h(case)f(to)34 b(o.)475 8442 y(A.3.)605 b FF(Pro)39 b(of)414 b(of)g(estimate)f(\(3.68\).)607 b Fw(W)-101 b(e)360 b(use)g(\(3.67\).)524 b(First,)369 b(w)-34 b(e)361 b(represen)-34 b(t)360 b Ft(r)p Fu(\025)38670 8002 y Fp(\(1\))39928 8442 y Fw(\()32 b Fu(~)-638 b Fn({)41207 8624 y Fp(1)41734 8442 y Fw(\))182 b Ft(\000)g(r)p Fu(\025)45229 8002 y Fp(\(2\))46486 8442 y Fw(\()p Fu(\013)t(;)235 b(~)-639 b Fn({)49083 8624 y Fp(2)49610 8442 y Fw(\))360 b(as)475 9881 y(the)404 b(sum)h(of)f Fu(A)7417 10063 y Fp(1)8347 9881 y Fw(and)h Fu(A)11613 10063 y Fp(2)12139 9881 y Fw(:)6591 11943 y Fu(A)7500 12125 y Fp(1)8363 11943 y Fw(=)336 b Ft(r)p Fu(\025)11359 11443 y Fp(\(1\))12617 11943 y Fw(\()p Fn({)13896 12125 y Fp(1)14398 11943 y Fu(~)-582 b(\027)77 b Fw(\))270 b Ft(\000)f(r)p Fu(\025)18768 11443 y Fp(\(2\))20026 11943 y Fw(\()p Fu(\013)t(;)202 b Fn({)22623 12125 y Fp(1)23125 11943 y Fu(~)-581 b(\027)77 b Fw(\))p Fu(;)1414 b(A)26957 12125 y Fp(2)27819 11943 y Fw(=)337 b Ft(r)p Fu(\025)30816 11443 y Fp(\(2\))32074 11943 y Fw(\()p Fu(\013)t(;)202 b Fn({)34671 12125 y Fp(1)35173 11943 y Fu(~)-581 b(\027)76 b Fw(\))270 b Ft(\000)f(r)p Fu(\025)39543 11443 y Fp(\(2\))40801 11943 y Fw(\()p Fu(\013)t(;)202 b Fn({)43398 12125 y Fp(2)43900 11943 y Fu(~)-581 b(\027)77 b Fw(\))p Fu(:)475 14201 y Fw(Considering)428 b(\(3.57\),)434 b(w)-34 b(e)428 b(obtain)h Ft(j)p Fu(A)17892 14383 y Fp(1)18418 14201 y Ft(j)375 b Fu(<)h Fw(24)p Fu(\013)t(\017)22932 13761 y Fp(3)22932 14526 y(1)23459 14201 y Fu(k)24128 13761 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1+)p Fs(\016)27329 14201 y Fw(.)610 b(Using)428 b(the)g(Mean)g(V)-101 b(alue)427 b(Theorem)h(and)h(\(3.27\),)475 15691 y(w)-34 b(e)387 b(get:)530 b Ft(j)p Fu(A)5971 15873 y Fp(2)6497 15691 y Ft(j)336 b Fu(<)h(ck)9644 15251 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(2)11663 15691 y Ft(j)p Fn({)12808 15873 y Fp(2)13569 15691 y Ft(\000)235 b Fn({)15555 15873 y Fp(1)16080 15691 y Ft(j)p Fu(:)387 b Fw(Applying)g(\(3.60\))h(yields:) 529 b Ft(j)p Fu(A)30977 15873 y Fp(2)31503 15691 y Ft(j)336 b Fu(<)h(c\013)t(\017)35252 15251 y Fp(4)35252 16016 y(1)35778 15691 y Fu(k)36447 15251 y Fr(\000)p Fp(1)37705 15691 y Fu(:)386 b Fw(Adding)i(the)g(estimates)f(for)475 17130 y Fu(A)1384 17312 y Fp(1)2314 17130 y Fw(and)405 b Fu(A)5580 17312 y Fp(2)6510 17130 y Fw(w)-34 b(e)404 b(arriv)-34 b(e)403 b(at:)13852 18162 y Fq(\014)13852 18889 y(\014)14256 19192 y Ft(r)p Fu(\025)15973 18692 y Fp(\(1\))17231 19192 y Fw(\()p Fn({)18510 19374 y Fp(1)19012 19192 y Fu(~)-582 b(\027)77 b Fw(\))270 b Ft(\000)f(r)p Fu(\025)23382 18692 y Fp(\(2\))24640 19192 y Fw(\()p Fu(\013)t(;)202 b Fn({)27237 19374 y Fp(2)27739 19192 y Fu(~)-581 b(\027)77 b Fw(\))28911 18162 y Fq(\014)28911 18889 y(\014)29652 19192 y Fu(<)336 b Fw(25)p Fu(\013)t(\017)33414 18692 y Fp(3)33414 19492 y(1)33941 19192 y Fu(k)34610 18692 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1+)p Fs(\016)37811 19192 y Fu(:)10583 b Fw(\(A.3\))475 21237 y(Similarly)-101 b(,)13119 21819 y Fq(\014)13119 22546 y(\014)13523 22849 y Fn({)14331 23031 y Fp(1)14857 22849 y Ft(r)p Fu(\025)16574 22349 y Fp(\(1\))17832 22849 y Fw(\()p Fn({)19111 23031 y Fp(1)19613 22849 y Fu(~)-582 b(\027)77 b Fw(\))270 b Ft(\000)f Fn({)23074 23031 y Fp(2)23600 22849 y Ft(r)p Fu(\025)25317 22349 y Fp(\(2\))26575 22849 y Fw(\()p Fu(\013)t(;)202 b Fn({)29172 23031 y Fp(2)29674 22849 y Fu(~)-581 b(\027)77 b Fw(\))30846 21819 y Fq(\014)30846 22546 y(\014)31587 22849 y Fu(<)336 b Fw(26)p Fu(\013)t(\017)35349 22349 y Fp(3)35349 23149 y(1)35876 22849 y Fu(k)36545 22349 y Fp(2)p Fs(l)11 b Fp(+)p Fs(\016)38544 22849 y Fu(:)9850 b Fw(\(A.4\))475 24851 y(By)443 b(Lemma)g(2.11,)9861 24352 y Fs(@)52 b Fm({)11036 24475 y FD(1)p 9861 24572 1637 49 v 10098 25269 a Fs(@)g(')12032 24851 y Fw(=)13510 24352 y Fs(@)g(h)14600 24475 y FD(1)p 13510 24572 1552 49 v 13706 25269 a Fs(@)g(')15597 24851 y Fw(=)402 b Fu(O)34 b Fw(\()p Fu(k)19041 24411 y Fr(\000)p Fp(2)p Fs(\015)20728 24534 y FD(0)21189 24411 y Fp(+1)22447 24851 y Fw(\).)657 b(Substituting)447 b(\(A.3\),)453 b(\(A.4\))444 b(and)h(the)f(last)g(estimate)f(in)-34 b(to)475 26461 y(the)404 b(form)-34 b(ula)405 b(for)f Fu(F)168 b Fw(\()p Fu(\025;)177 b(~)-581 b(\027)78 b Fw(\),)404 b(w)-34 b(e)405 b(obtain)g(\(3.68\).)475 28823 y(A.4.)605 b FF(Lemma)464 b(A.1.)p 0 TeXcolorgray 475 31039 a(Lemma)397 b(A.1.)p 0 TeXcolorgray 506 w FC(If)381 b Fu(t)f FC(is)g(in)h(the)e(c)-62 b(omplex)379 b Fw(\(2)p Fu(k)21344 30599 y Fr(\000)p Fp(1)p Fr(\000)p Fp(4)p Fs(s)24183 30722 y FD(1)24646 30599 y Fr(\000)p Fp(2)p Fs(\016)26353 31039 y Fw(\))p FC(-neighb)-62 b(orho)g(o)g(d)377 b(of)j(the)f(nonr)-62 b(esonanc)g(e)380 b(set)f Fu(\037)47854 31221 y Fp(1)48380 31039 y Fw(\()p Fu(\025;)202 b(\016)46 b Fw(\))p FC(,)475 32777 y(then)433 b(the)f(op)-62 b(er)g (ator-function)14724 31795 y Fq(\000)15279 32777 y Fu(H)16385 32337 y Fp(\(1\))17643 32777 y Fw(\()p Fu(t)p Fw(\))270 b Ft(\000)f Fu(z)21122 31795 y Fq(\001)21678 32024 y Fr(\000)p Fp(1)23369 32777 y FC(has)432 b(a)h(single)g(p)-62 b(ole)432 b Fu(z)33040 32959 y Fp(0)33999 32777 y FC(inside)g(the)h(c) -62 b(ontour)432 b Fu(C)44824 32959 y Fp(1)45784 32777 y FC(and)17304 33917 y Fq(\015)17304 34644 y(\015)17304 35372 y(\015)17304 36099 y(\015)17977 35675 y Fw(\()p Fu(z)323 b Ft(\000)269 b Fu(z)21111 35857 y Fp(0)21636 35675 y Fw(\))22309 34329 y Fq(\020)23034 35675 y Fu(H)24140 35174 y Fp(\(1\))25397 35675 y Fw(\()p Fu(t)p Fw(\))h Ft(\000)f Fu(z)28876 34329 y Fq(\021)29600 34631 y Fr(\000)p Fp(1)30858 33917 y Fq(\015)30858 34644 y(\015)30858 35372 y(\015)30858 36099 y(\015)31868 35675 y Fu(<)337 b Fw(16)p Fu(;)14034 b Fw(\(A.5\))475 38384 y FC(for)433 b(every)f Fu(z)486 b FC(inside)433 b Fu(C)11065 38566 y Fp(1)11591 38384 y FC(.)p 0 TeXcolorgray 475 40600 a(Pr)-62 b(o)g(of.)p 0 TeXcolorgray 552 w Fw(By)403 b(\(3.38\),)13359 41646 y Fq(\014)13359 42373 y(\014)13359 43100 y(\014)13763 43040 y Fw(det)15782 42220 y Fu(H)16888 41780 y Fp(\(1\))18145 42220 y Fw(\()p Fu(t)p Fw(\))270 b Ft(\000)f Fu(z)p 15782 42761 5843 49 v 16197 43871 a(H)17205 44053 y Fp(0)17730 43871 y Fw(\()p Fu(t)p Fw(\))h Ft(\000)f Fu(z)22026 43040 y Ft(\000)g Fw(1)23844 41646 y Fq(\014)23844 42373 y(\014)23844 43100 y(\014)24585 43040 y Fu(<)337 b Fw(2)p Ft(k)p Fu(W)28222 43222 y Fp(1)28747 43040 y Ft(k)29555 41646 y Fq(\015)29555 42373 y(\015)29555 43100 y(\015)30229 43040 y Fw(\()p Fu(H)31708 43222 y Fp(0)32234 43040 y Fw(\()p Fu(t)p Fw(\))270 b Ft(\000)f Fu(z)53 b Fw(\))36184 42481 y Fr(\000)p Fp(1)37442 41646 y Fq(\015)37442 42373 y(\015)37442 43100 y(\015)38115 43883 y Fp(1)48731 43040 y Fw(\(A.6\))475 45547 y(for)364 b(ev)-34 b(ery)364 b Fu(z)417 b Fw(on)365 b(the)g(con)-34 b(tour)366 b Fu(C)15367 45729 y Fp(1)15893 45547 y Fw(.)525 b(Using)365 b(the)g(estimate)f(\(2.10\))h(and)h (considering)e(that)i Fu(H)42661 45729 y Fp(0)43186 45547 y Fw(\()p Fu(t)p Fw(\))f(is)f(a)h(diagonal)475 47090 y(op)34 b(erator,)299 b(w)-34 b(e)274 b(easily)f(obtain)14101 46060 y Fq(\015)14101 46787 y(\015)14774 47090 y Fw(\()q Fu(H)16254 47272 y Fp(0)16779 47090 y Fw(\()p Fu(t)p Fw(\))d Ft(\000)f Fu(z)53 b Fw(\))20729 46532 y Fr(\000)p Fp(1)21987 46060 y Fq(\015)21987 46787 y(\015)22660 47570 y Fp(1)23523 47090 y Fu(<)336 b(k)25471 46650 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+4+6)p Fs(s)29804 46773 y FD(1)30266 46650 y Fp(+)p Fs(\016)31503 47090 y Fw(.)495 b(It)274 b(immediately)f(follo)-34 b(ws)274 b(that)h(the)g(righ)-34 b(t-)475 48590 y(hand)390 b(part)f(of)g(\(A.6\))h(is)e(less)h(then)h (1.)533 b(Applying)389 b(Rouc)-34 b(h)g(\023)-572 b(e's)390 b(theorem,)h(w)-34 b(e)390 b(conclude)f(that)h(the)f(determinan)-34 b(t)475 50133 y(has)447 b(the)h(same)f(n)-34 b(um)g(b)34 b(er)448 b(of)g(zeros)e(and)i(p)34 b(oles)447 b(inside)g Fu(C)26718 50315 y Fp(1)27244 50133 y Fw(.)668 b(Considering)448 b(that)g(\()p Fu(H)39117 50315 y Fp(0)39643 50133 y Fw(\()p Fu(t)p Fw(\))270 b Ft(\000)f Fu(z)53 b Fw(\))43593 49574 y Fr(\000)p Fp(1)45298 50133 y Fw(has)447 b(a)h(single)475 51871 y(p)34 b(ole,)612 b(w)-34 b(e)571 b(obtain)h(that)12295 50889 y Fq(\000)12850 51871 y Fu(H)13956 51431 y Fp(\(1\))15214 51871 y Fw(\()p Fu(t)p Fw(\))270 b Ft(\000)f Fu(z)18693 50889 y Fq(\001)19249 51117 y Fr(\000)p Fp(1)21077 51871 y Fw(has)571 b(a)g(single)g(p)34 b(ole)570 b(to)34 b(o.)1039 b(W)-101 b(e)571 b(denote)g(it)g(b)-34 b(y)571 b Fu(z)44095 52053 y Fp(0)44621 51871 y Fw(.)1039 b(Ob)-34 b(viously)-101 b(,)475 53730 y(\()p Fu(z)366 b Ft(\000)313 b Fu(z)3696 53912 y Fp(0)4221 53730 y Fw(\))4894 52748 y Fq(\000)5450 53730 y Fu(H)6556 53290 y Fp(\(1\))7814 53730 y Fw(\()p Fu(t)p Fw(\))270 b Ft(\000)f Fu(z)11293 52748 y Fq(\001)11848 52976 y Fr(\000)p Fp(1)13575 53730 y Fw(is)470 b(holomorphic)f(inside)h Fu(C)26237 53912 y Fp(1)26763 53730 y Fw(.)734 b(The)470 b(de\014nition)h(of)f Fu(C)38156 53912 y Fp(1)38682 53730 y Fw(,)485 b(giv)-34 b(en)470 b(in)f(Corollary)g(2.2,)475 55320 y(yields:)523 b Ft(j)p Fu(z)264 b Ft(\000)211 b Fu(z)7222 55502 y Fp(0)7747 55320 y Ft(j)337 b Fu(<)f Fw(2)p Fu(k)10975 54880 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)14576 55003 y FD(1)15038 54880 y Fr(\000)p Fs(\016)16275 55320 y Fw(.)529 b(Using)375 b(the)g(last)g(inequalit)-34 b(y)375 b(and)h(estimate)f(\(2.30\),)381 b(w)-34 b(e)376 b(obtain)g(\(A.5\))f(for)475 56759 y(an)-34 b(y)404 b Fu(z)390 b Ft(2)337 b Fu(C)5729 56941 y Fp(1)6255 56759 y Fw(.)538 b(By)404 b(the)g(Maxim)-34 b(um)405 b(principle)e(it)h(can)h(b)34 b(e)404 b(extended)g(inside)g Fu(C)36407 56941 y Fp(1)36933 56759 y Fw(.)p 50750 56925 665 831 v 475 59120 a(A.5.)605 b FF(Pro)39 b(of)433 b(of)h(Lemma)f (3.48.)606 b Fw(By)376 b(Lemma)h(2.11,)382 b(P)-34 b(art)377 b(2,)383 b(the)377 b(function)411 b Fu(~)-638 b Fn({)37712 59302 y Fp(1)38238 59120 y Fw(\()p Fu(')p Fw(\))377 b(is)g(holomorphic) h(in)f(the)475 60638 y(\()p Fu(k)1615 60198 y Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)4454 60321 y FD(1)4916 60198 y Fr(\000)p Fp(2)p Fs(\016)6623 60638 y Fw(\))p Ft(\000)p Fw(neigh)-34 b(b)34 b(orho)g(o)g(d)357 b(of)g Fu(\010)17743 60820 y Fp(1)18625 60638 y Fw(and)f Fu(\025)21640 60198 y Fp(\(1\))22898 60638 y Fw(\()32 b Fu(~)-638 b Fn({)24177 60820 y Fp(1)24703 60638 y Fw(\()p Fu(')p Fw(\)\))338 b(=)f Fu(k)29196 60198 y Fp(2)p Fs(l)30013 60638 y Fu(:)356 b Fw(Hence,)364 b(the)357 b(equation)f(\(3.92\))h(is)f(equiv)-67 b(alen)-34 b(t)475 62237 y(to)470 b Fu(\025)2729 61798 y Fp(\(1\))3986 62237 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\)\))447 b(=)e Fu(\025)9843 61798 y Fp(\(1\))11101 62237 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))314 b Ft(\000)15379 61918 y Fu(~)15514 62237 y(b)p Fw(\))g(+)e Fu(\017)18566 62419 y Fp(0)19092 62237 y Fw(.)734 b(W)-101 b(e)469 b(use)h(p)34 b(erturbation)470 b(form)-34 b(ula)470 b(\(2.34\):)670 b Ft(j)-10 b Fu(~)-596 b(y)43 b Fw(\()p Fu(')p Fw(\))p Ft(j)43315 61798 y Fp(2)p Fs(l)43315 62537 y Fr(\003)44446 62237 y Fw(+)313 b Fu(f)46295 62419 y Fp(1)46821 62237 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\)\))446 b(=)475 63837 y Ft(j)-10 b Fu(~)-596 b(y)43 b Fw(\()p Fu(')p Fw(\))270 b Ft(\000)4530 63517 y Fu(~)4666 63837 y(b)q Ft(j)5524 63397 y Fp(2)p Fs(l)5524 64137 y Fr(\003)6610 63837 y Fw(+)f Fu(f)8415 64019 y Fp(1)8941 63837 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))270 b Ft(\000)13131 63517 y Fu(~)13267 63837 y(b)q Fw(\))f(+)h Fu(\017)16233 64019 y Fp(0)16758 63837 y Fu(:)404 b Fw(This)h(equation)f(can)h(b)34 b(e)403 b(rewritten)i(as)1582 65264 y Fq(\020)2306 66610 y Fw(2)p Ft(h)-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))p Fu(;)6159 66290 y(~)6295 66610 y(b)p Ft(i)7286 66792 y Fr(\003)8082 66610 y Ft(\000)269 b(j)9495 66290 y Fu(~)9631 66610 y(b)p Ft(j)10488 66109 y Fp(2)11014 65264 y Fq(\021)o(\020)12461 66610 y Ft(j)-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))p Ft(j)15508 66109 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(2)15508 66909 y Fr(\003)17797 66610 y Fw(+)269 b Ft(\001)202 b(\001)g(\001)269 b Fw(+)g Ft(j)-10 b Fu(~)-596 b(y)43 b Fw(\()p Fu(')p Fw(\))270 b Ft(\000)25960 66290 y Fu(~)26096 66610 y(b)q Ft(j)26954 66109 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(2)26954 66909 y Fr(\003)28973 65264 y Fq(\021)28999 69054 y Fw(+)269 b Fu(f)30804 69236 y Fp(1)31331 69054 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\)\))270 b Ft(\000)f Fu(f)36721 69236 y Fp(1)37248 69054 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))270 b Ft(\000)41438 68734 y Fu(~)41574 69054 y(b)p Fw(\))g Ft(\000)f Fu(\017)44539 69236 y Fp(0)45401 69054 y Fw(=)337 b(0)p Fu(;)1107 b Fw(\(A.7\))p 0 TeXcolorgray 0 TeXcolorgray eop end %%Page: 67 67 TeXDict begin 67 66 bop 0 TeXcolorgray 0 TeXcolorgray 2848 3345 a Fp(SPECTRAL)406 b(PR)-26 b(OPER)-78 b(TIES)406 b(OF)h(A)h(POL)-78 b(YHARMONIC)405 b(OPERA)-78 b(TOR)406 b(WITH)h(LIMIT-PERIODIC)e(POTENTIAL)1430 b(67)p 0 TeXcolorgray 475 6080 a Fw(where)443 b Ft(h)-10 b Fu(~)-596 b(y)44 b(;)5531 5760 y(~)5667 6080 y(b)p Ft(i)6658 6262 y Fr(\003)7586 6080 y Fw(=)402 b Fu(y)9525 6262 y Fp(1)10051 6080 y Fu(b)10571 6262 y Fp(1)11393 6080 y Fw(+)295 b Fu(y)13225 6262 y Fp(2)13751 6080 y Fu(b)14271 6262 y Fp(2)14797 6080 y Fw(.)657 b(Using)443 b(the)h(notation)26116 5760 y Fu(~)26252 6080 y(b)402 b Fw(=)g Fu(b)29039 6262 y Fp(0)29565 6080 y Fw(\(cos)202 b Fu(')32654 6277 y Fs(b)33112 6080 y Fu(;)g Fw(sin)g Fu(')36134 6277 y Fs(b)36592 6080 y Fw(\),)453 b(dividing)443 b(b)34 b(oth)445 b(sides)e(of)h(the)475 7888 y(equation)340 b(\(A.7\))g(b)-34 b(y)341 b(2)p Fu(b)11240 8070 y Fp(0)11766 7888 y Fu(k)12435 6542 y Fq(\020)13159 7888 y Ft(j)-10 b Fu(~)-596 b(y)43 b Fw(\()p Fu(')p Fw(\))p Ft(j)16205 7448 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(2)16205 8188 y Fr(\003)18366 7888 y Fw(+)141 b Ft(\001)202 b(\001)g(\001)139 b Fw(+)i Ft(j)-10 b Fu(~)-596 b(y)43 b Fw(\()p Fu(')p Fw(\))141 b Ft(\000)25886 7568 y Fu(~)26022 7888 y(b)p Ft(j)26879 7448 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(2)26879 8188 y Fr(\003)28899 6542 y Fq(\021)29622 7888 y Fw(,)353 b(and)340 b(considering)g(that)331 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))337 b(=)369 b Fu(~)-638 b Fn({)46240 8070 y Fp(1)46766 7888 y Fw(\()p Fu(')p Fw(\))141 b(+)49590 7568 y Fu(~)49726 7888 y(b)336 b Fw(=)475 9972 y(\()p Fu(k)308 b Fw(+)269 b Fu(h)3795 10154 y Fp(1)4321 9972 y Fw(\))-25 b Fu(~)-581 b(\027)347 b Fw(+)6814 9653 y Fu(~)6950 9972 y(b)p Fw(,)404 b(w)-34 b(e)404 b(obtain:)16845 11599 y(cos\()p Fu(')270 b Ft(\000)f Fu(')22007 11796 y Fs(b)22465 11599 y Fw(\))g Ft(\000)g Fu(\017)24909 11781 y Fp(0)25435 11599 y Fu(g)26013 11781 y Fp(1)26539 11599 y Fw(\()p Fu(')p Fw(\))h(+)f Fu(g)30334 11781 y Fp(2)30860 11599 y Fw(\()p Fu(')p Fw(\))338 b(=)e(0)p Fu(;)13576 b Fw(\(A.8\))475 14067 y(where)404 b Fu(g)4558 14249 y Fp(1)5084 14067 y Fw(\()p Fu(')p Fw(\))337 b(=)8436 12721 y Fq(h)9008 14067 y Fw(2)p Fu(b)10134 14249 y Fp(0)10660 14067 y Fu(k)11329 12721 y Fq(\020)12053 14067 y Ft(j)-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))p Ft(j)15100 13627 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(2)15100 14367 y Fr(\003)17389 14067 y Fw(+)269 b Ft(\001)202 b(\001)g(\001)268 b Fw(+)i Ft(j)-10 b Fu(~)-596 b(y)43 b Fw(\()p Fu(')p Fw(\))270 b Ft(\000)25552 13747 y Fu(~)25688 14067 y(b)p Ft(j)26545 13627 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(2)26545 14367 y Fr(\003)28565 12721 y Fq(\021)o(i)29861 13023 y Fr(\000)p Fp(1)31523 14067 y Fw(and)5375 17663 y Fu(g)5953 17845 y Fp(2)6479 17663 y Fw(\()p Fu(')p Fw(\))338 b(=)9964 16843 y Ft(h)10354 16523 y Fu(~)10435 16843 y(h)11133 17025 y Fp(1)11659 16843 y Fw(\()p Fu(')p Fw(\))p Fu(;)13797 16523 y(~)13933 16843 y(b)q Ft(i)14925 17025 y Fr(\003)p 9964 17384 5488 49 v 11850 18494 a Fu(b)12370 18676 y Fp(0)12896 18494 y Fu(k)15853 17663 y Fw(+)17313 16843 y Fu(b)17833 17025 y Fp(0)p 17198 17384 1276 49 v 17198 18494 a Fw(2)p Fu(k)18876 17663 y Fw(+)20088 16317 y Fq(\020)20812 17663 y Fu(f)21405 17845 y Fp(1)21931 17663 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\)\))271 b Ft(\000)e Fu(f)27322 17845 y Fp(1)27848 17663 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))270 b Ft(\000)32038 17343 y Fu(~)32174 17663 y(b)q Fw(\))33166 16317 y Fq(\021)33890 17663 y Fu(g)34468 17845 y Fp(1)34994 17663 y Fw(\()p Fu(')p Fw(\))p Fu(;)37996 17343 y(~)38076 17663 y(h)38774 17845 y Fp(1)39300 17663 y Fw(\()p Fu(')p Fw(\))338 b(=)e Fu(h)43350 17845 y Fp(1)43876 17663 y Fw(\()p Fu(')p Fw(\))-25 b Fu(~)-581 b(\027)78 b(:)475 20559 y Fw(Ob)-34 b(viously)485 b Fu(g)6831 20741 y Fp(1)7830 20559 y Fw(=)472 b Fu(O)10406 19577 y Fq(\000)10961 20559 y Fu(b)11481 20049 y Fr(\000)p Fp(1)11481 20906 y(0)12739 20559 y Fu(k)13408 20119 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)16160 19577 y Fq(\001)16715 20559 y Fw(.)783 b(Let)486 b(us)g(estimate)g Fu(g)27241 20741 y Fp(2)27767 20559 y Fw(\()p Fu(')p Fw(\).)784 b(Using)486 b(the)g(inequalit)-34 b(y)485 b(\(2.43\))i(for)e Fu(h)48203 20741 y Fp(1)48730 20559 y Fw(,)505 b(and)475 22149 y(considering)404 b(that)h Fu(b)9961 22331 y Fp(0)10824 22149 y Fw(=)336 b Fu(o)p Fw(\()p Fu(k)13831 21709 y Fr(\000)p Fp(1)p Fr(\000)p Fp(4)p Fs(s)16670 21832 y FD(1)17132 21709 y Fr(\000)p Fp(2)p Fs(\016)18839 22149 y Fw(\),)405 b(w)-34 b(e)404 b(easily)f(obtain:)9201 23254 y Fq(\014)9201 23981 y(\014)9201 24708 y(\014)9201 25436 y(\014)9201 26163 y(\014)9738 24555 y Ft(h)10128 24235 y Fu(~)10209 24555 y(h)10907 24737 y Fp(1)11434 24555 y Fw(\()p Fu(')p Fw(\))p Fu(;)13572 24235 y(~)13708 24555 y(b)p Ft(i)14699 24737 y Fr(\003)p 9738 25096 5488 49 v 11624 26207 a Fu(b)12144 26389 y Fp(0)12670 26207 y Fu(k)15358 23254 y Fq(\014)15358 23981 y(\014)15358 24708 y(\014)15358 25436 y(\014)15358 26163 y(\014)16099 25375 y Ft(\024)17511 24555 y Fw(2)p Fu(h)18815 24737 y Fp(1)p 17511 25096 1831 49 v 18092 26207 a Fu(k)19811 25375 y Fw(=)337 b Fu(O)34 b Fw(\()p Fu(k)23190 24875 y Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)26029 24998 y FD(1)26490 24875 y Fr(\000)p Fp(2)p Fs(\015)28177 24998 y FD(0)28638 24875 y Fr(\000)p Fs(\016)29875 25375 y Fw(\))p Fu(;)32345 24555 y(b)32865 24737 y Fp(0)p 32230 25096 1276 49 v 32230 26207 a Fw(2)p Fu(k)33975 25375 y Fw(=)336 b Fu(o)p Fw(\()p Fu(k)36982 24875 y Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)39821 24998 y FD(1)40284 24875 y Fr(\000)p Fp(2)p Fs(\016)41991 25375 y Fw(\))p Fu(:)475 28602 y Fw(By)403 b(\(2.26\),)4347 29707 y Fq(\014)4347 30434 y(\014)4347 31161 y(\014)4751 31101 y Fu(f)5344 31283 y Fp(1)5870 31101 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\)\))271 b Ft(\000)e Fu(f)11261 31283 y Fp(1)11787 31101 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))270 b Ft(\000)15977 30781 y Fu(~)16113 31101 y(b)q Fw(\))17105 29707 y Fq(\014)17105 30434 y(\014)17105 31161 y(\014)17846 31101 y Ft(\024)336 b Fw(sup)203 b Ft(jr)p Fu(f)23092 31283 y Fp(1)23618 31101 y Ft(j)p Fu(b)24475 31283 y Fp(0)25338 31101 y Fw(=)336 b Fu(O)34 b Fw(\()p Fu(b)28567 31283 y Fp(0)29093 31101 y Fu(k)29762 30600 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(1)p Fr(\000)p Fp(2)p Fs(\015)33413 30723 y FD(0)33874 30600 y Fp(+)p Fs(\016)35111 31101 y Fw(\))337 b(=)f Fu(O)34 b Fw(\()p Fu(b)39148 31283 y Fp(0)39674 31101 y Fu(k)40343 30600 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+3+8)p Fs(s)44676 30723 y FD(1)45138 30600 y Fp(+5)p Fs(\016)46845 31101 y Fw(\))p Fu(;)475 34353 y Fw(and)633 b(therefore,)8773 32959 y Fq(\014)8773 33686 y(\014)8773 34413 y(\014)9177 33371 y(\000)9732 34353 y Fu(f)10325 34535 y Fp(1)10852 34353 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\)\))270 b Ft(\000)f Fu(f)16242 34535 y Fp(1)16769 34353 y Fw(\()-10 b Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))270 b Ft(\000)20959 34033 y Fu(~)21095 34353 y(b)p Fw(\))22086 33371 y Fq(\001)22642 34353 y Fu(g)23220 34535 y Fp(1)23746 34353 y Fw(\()p Fu(')p Fw(\))25481 32959 y Fq(\014)25481 33686 y(\014)25481 34413 y(\014)26603 34353 y Fw(=)717 b Fu(O)34 b Fw(\()p Fu(k)30362 33913 y Fr(\000)p Fp(4)p Fs(l)11 b Fp(+4+8)p Fs(s)34695 34036 y FD(1)35156 33913 y Fp(+5)p Fs(\016)36863 34353 y Fw(\))p Fu(:)633 b Fw(Since)f Fu(s)42265 34535 y Fp(1)43508 34353 y Fw(=)45301 33533 y(2)p Fu(l)293 b Ft(\000)269 b Fw(11)p 45301 34074 3686 49 v 46537 35184 a(32)49119 34353 y(,)689 b(w)-34 b(e)475 36375 y(ha)g(v)g(e)506 b Fu(g)3949 36557 y Fp(2)4475 36375 y Fw(\()p Fu(')p Fw(\))g(=)f Fu(o)p Fw(\()p Fu(k)9892 35935 y Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)12731 36058 y FD(1)13193 35935 y Fr(\000)p Fp(2)p Fs(\016)14900 36375 y Fw(\))p Fu(:)h Fw(Using)f Fu(g)20300 36557 y Fp(1)20826 36375 y Fw(\()p Fu(')p Fw(\))h(=)f Fu(O)25676 35394 y Fq(\000)26231 36375 y Fu(b)26751 35866 y Fr(\000)p Fp(1)26751 36722 y(0)28009 36375 y Fu(k)28678 35935 y Fr(\000)p Fp(2)p Fs(l)11 b Fp(+1)31430 35394 y Fq(\001)32490 36375 y Fw(and)507 b Fu(\017)35441 36557 y Fp(0)36472 36375 y Fu(<)e(b)38440 36557 y Fp(0)38966 36375 y Fu(k)39635 35935 y Fp(2)p Fs(l)11 b Fr(\000)p Fp(3)p Fr(\000)p Fp(4)p Fs(s)43236 36058 y FD(1)43698 35935 y Fr(\000)p Fp(3)p Fs(\016)45405 36375 y Fw(,)530 b(w)-34 b(e)506 b(obtain)475 37965 y Fu(\017)967 38147 y Fp(0)1493 37965 y Fu(g)2071 38147 y Fp(1)2597 37965 y Fw(\()p Fu(')p Fw(\))337 b(=)f Fu(o)p Fw(\()p Fu(k)7676 37525 y Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)10515 37648 y FD(1)10977 37525 y Fr(\000)p Fp(2)p Fs(\016)12685 37965 y Fw(\))p Fu(:)404 b Fw(Th)-34 b(us,)17330 40186 y Fu(g)17908 40368 y Fp(2)18434 40186 y Fw(\()p Fu(')p Fw(\))270 b Ft(\000)g Fu(\017)22144 40368 y Fp(0)22669 40186 y Fu(g)23247 40368 y Fp(1)23773 40186 y Fw(\()p Fu(')p Fw(\))338 b(=)e Fu(o)p Fw(\()p Fu(k)28853 39685 y Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)31692 39808 y FD(1)32154 39685 y Fr(\000)p Fp(2)p Fs(\016)33861 40186 y Fw(\))p Fu(:)14062 b Fw(\(A.9\))475 42633 y(Supp)34 b(ose)589 b(that)g Fu(')9016 42830 y Fs(b)9866 42633 y Ft(\006)11334 41813 y Fu(\031)p 11334 42354 735 49 v 11398 43465 a Fw(2)12789 42633 y(is)f(in)g(the)18061 41651 y Fq(\000)18750 42156 y Fp(1)p 18750 42354 471 49 v 18750 43051 a(8)19353 42633 y Fu(k)20022 42193 y Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)22861 42316 y FD(1)23323 42193 y Fr(\000)p Fp(2)p Fs(\016)25030 41651 y Fq(\001)25586 42633 y Fw(-neigh)-34 b(b)34 b(orho)g(o)g(d)589 b(of)35553 42327 y(~)35351 42633 y Fu(\010)36159 42815 y Fp(1)36685 42633 y Fw(.)1090 b(W)-101 b(e)588 b(dra)-34 b(w)589 b(t)-34 b(w)g(o)589 b(circles)e Fu(C)50738 42815 y Fr(\006)475 44917 y Fw(cen)-34 b(tered)387 b(at)g Fu(')7533 45114 y Fs(b)8226 44917 y Ft(\006)9537 44097 y Fu(\031)p 9537 44638 735 49 v 9601 45749 a Fw(2)10791 44917 y(with)h(the)f (radius)19367 44440 y Fp(1)p 19367 44638 471 49 v 19367 45335 a(8)19971 44917 y Fu(k)20640 44477 y Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)23479 44600 y FD(1)23941 44477 y Fr(\000)p Fp(2)p Fs(\016)25648 44917 y Fw(.)533 b(They)387 b(are)f(b)34 b(oth)388 b(inside)f(the)g(complex)44841 44440 y Fp(3)p 44841 44638 V 44841 45335 a(4)45444 44917 y Fu(k)46113 44477 y Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)48952 44600 y FD(1)49414 44477 y Fr(\000)p Fp(2)p Fs(\016)51121 44917 y Fw(-)475 46768 y(neigh)-34 b(b)34 b(orho)g(o)g(d)293 b(of)g Fu(\010)10053 46950 y Fp(1)10579 46768 y Fw(,)314 b(the)293 b(p)34 b(erturbation)293 b(series)f(con)-34 b(v)g(erging)293 b(and)g(the)g(estimate)f(\(A.9\))h(holds.)502 b(F)-101 b(or)293 b(an)-34 b(y)292 b Fu(')h Fw(on)475 48237 y Fu(C)1341 48419 y Fr(\006)2128 48237 y Fw(,)305 b Ft(j)p Fu(')22 b Ft(\000)g Fw(\()p Fu(')6151 48434 y Fs(b)6630 48237 y Ft(\006)g Fu(\031)43 b(=)p Fw(2\))p Ft(j)336 b Fw(=)12098 47760 y Fp(1)p 12098 47958 V 12098 48655 a(8)12701 48237 y Fu(k)13370 47797 y Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)16209 47920 y FD(1)16671 47797 y Fr(\000)p Fp(2)p Fs(\016)18658 48237 y Fw(and,)306 b(therefore,)e Ft(j)202 b Fw(cos\()p Fu(')22 b Ft(\000)g Fu(')31787 48434 y Fs(b)32244 48237 y Fw(\))p Ft(j)337 b Fu(>)35037 47760 y Fp(1)p 34801 47958 941 49 v 34801 48655 a(16)35875 48237 y Fu(k)36544 47797 y Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)39383 47920 y FD(1)39845 47797 y Fr(\000)p Fp(2)p Fs(\016)41889 48237 y Fu(>)g Ft(j)p Fu(g)44084 48419 y Fp(2)44609 48237 y Fw(\()p Fu(')p Fw(\))22 b Ft(\000)g Fu(\017)47823 48419 y Fp(0)48349 48237 y Fu(g)48927 48419 y Fp(1)49453 48237 y Fw(\()p Fu(')p Fw(\))p Ft(j)475 49676 y Fw(for)303 b(an)-34 b(y)303 b Fu(')336 b Ft(2)h Fu(C)7557 49858 y Fr(\006)8345 49676 y Fw(.)504 b(By)302 b(Rouc)-34 b(h)g(\023)-572 b(e's)304 b(Theorem,)323 b(there)302 b(is)h(only)f(one)h(solution)h(of)f(the)g (equation)g(\(A.8\))g(inside)g(eac)-34 b(h)475 51206 y Fu(C)1341 51388 y Fr(\006)2128 51206 y Fw(.)497 b(If)281 b Fu(')4844 51403 y Fs(b)5324 51206 y Fw(+)22 b Fu(\031)43 b(=)p Fw(2)280 b(is)g(not)i(in)e(the)h(\()15500 50728 y Fp(1)p 15500 50926 471 49 v 15500 51624 a(8)16103 51206 y Fu(k)16772 50766 y Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)19611 50889 y FD(1)20074 50766 y Fr(\000)p Fp(2)p Fs(\016)21781 51206 y Fw(\))p Ft(\000)p Fw(neigh)-34 b(b)34 b(orho)g(o)g(d)281 b(of)32143 50899 y(~)31941 51206 y Fu(\010)32749 51388 y Fp(1)33275 51206 y Fw(,)304 b(then)282 b Ft(j)202 b Fw(cos)o(\()p Fu(')22 b Ft(\000)g Fu(')41759 51403 y Fs(b)42217 51206 y Fw(\))p Ft(j)337 b Fu(>)45009 50728 y Fp(1)p 44774 50926 941 49 v 44774 51624 a(16)45848 51206 y Fu(k)46517 50766 y Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)49356 50889 y FD(1)49818 50766 y Fr(\000)p Fp(2)p Fs(\016)475 52850 y Fw(in)473 b Fu(\010)2766 53032 y Fp(1)3765 52850 y Fw(and,)491 b(hence,)f(equation)474 b(\(A.8\))g(has)g(no)f(solution.)747 b(Th)-34 b(us,)492 b(there)473 b(are)g(at)g(most)h(t)-34 b(w)g(o)475 b(solutions)f(in)50393 52544 y(~)50191 52850 y Fu(\010)50999 53032 y Fp(1)475 54319 y Fw(and)405 b Ft(j)p Fu(')3962 53879 y Fr(\006)3962 54618 y Fs(\017)4342 54741 y FD(0)5128 54319 y Ft(\000)269 b Fw(\()p Fu(')7604 54516 y Fs(b)8331 54319 y Ft(\006)g Fu(\031)43 b(=)p Fw(2\))p Ft(j)338 b Fu(<)14047 53842 y Fp(1)p 14047 54040 471 49 v 14047 54737 a(8)14650 54319 y Fu(k)15319 53879 y Fr(\000)p Fp(2)p Fr(\000)p Fp(4)p Fs(s)18158 54002 y FD(1)18620 53879 y Fr(\000)p Fp(2)p Fs(\016)20328 54319 y Fu(:)475 56857 y Fw(A.6.)605 b FF(Pro)39 b(of)465 b(of)f(Lemma)g (3.49.)606 b Fw(Using)405 b(the)f(p)34 b(erturbation)405 b(form)-34 b(ula)405 b(\(2.34\),)f(w)-34 b(e)405 b(obtain:)10166 58804 y Fu(@)p 9770 59345 1504 49 v 9770 60455 a(@)67 b(')11406 59624 y(\025)12113 59123 y Fp(\(1\))13371 58642 y Fq(\000)13916 59624 y Fu(~)-596 b(y)44 b Fw(\()p Fu(')p Fw(\))16299 58642 y Fq(\001)17192 59624 y Fw(=)19001 58804 y Fu(@)p 18604 59345 V 18604 60455 a(@)67 b(')20443 58278 y Fq(h)21015 59624 y 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