Content-Type: multipart/mixed; boundary="-------------0811230701907" This is a multi-part message in MIME format. ---------------0811230701907 Content-Type: text/plain; name="08-221.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="08-221.keywords" quasi-periodic operators, singular spectrum, complex WKB method, Lyapunov exponent ---------------0811230701907 Content-Type: application/postscript; name="MarxNajar.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="MarxNajar.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.94b Copyright 2004 Radical Eye Software %%Title: MarxNajar.dvi %%CreationDate: Sun Nov 23 13:50:17 2008 %%Pages: 23 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: CMBX10 CMR9 CMCSC10 CMR10 CMMI10 CMMI8 CMSY10 CMR8 %%+ MSBM10 CMTI10 MSBM7 CMTI9 CMSY8 CMEX10 CMMI6 CMR6 TeX-cmex8 CMSY6 %%+ CMSL9 CMBX9 CMTT9 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips.exe MarxNajar.dvi 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gt {z1} {z1 z2 add neg} ifelse h 0 gt {b neg} {z1} ifelse rmoveto show grestore } if dup t f moveto g t L stroke /i i w add def } repeat grestore gsave n 0 gt % DG/SR modification begin - Nov. 7, 1997 - Patch 1 %{ 1 setlinecap [ 0 dx n div ] dy n div 2 div setdash } { 1 setlinecap [ 0 dx n div ] dx n div 2 div setdash } % DG/SR modification end { 2 setlinecap } ifelse /i y1 def /f x1 dx mul n 0 gt { dx n div 2 div w mul sub } if def /g x2 dx mul n 0 gt { dx n div 2 div w mul add } if def y2 y1 sub h mul 1 add dup 1000 gt { pop 1000 } if { newpath i dy mul dup x0 exch moveto b 0 gt { gsave c i a cvs dup stringwidth pop /z2 ED w 0 gt {z1 z2 add neg} {z1} ifelse h 0 gt {z1} {b neg} ifelse rmoveto show grestore } if dup f exch t moveto g exch t L stroke /i i h add def } repeat grestore } def /ArcArrow { /d ED /b ED /a ED gsave newpath 0 -1000 moveto clip newpath 0 1 0 0 b grestore c mul /e ED pop pop pop r a e d PtoC y add exch x add exch r a PtoC y add exch x add exch b pop pop pop pop a 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mul add sqrt z add } { q { x s div } { x c div } ifelse abs } ifelse a PtoC h1 add exch w1 add exch } def /BeginOL { dup (all) eq exch TheOL eq or { IfVisible not { Visible /IfVisible true def } if } { IfVisible { Invisible /IfVisible false def } if } ifelse } def /InitOL { /OLUnit [ 3000 3000 matrix defaultmatrix dtransform ] cvx def /Visible { CP OLUnit idtransform T moveto } def /Invisible { CP OLUnit neg exch neg exch idtransform T moveto } def /BOL { BeginOL } def /IfVisible true def } def end % END pstricks.pro %%EndProcSet %%BeginProcSet: pst-dots.pro 0 0 %!PS-Adobe-2.0 %%Title: Dot Font for PSTricks %%Creator: Timothy Van Zandt %%Creation Date: May 7, 1993 %% Version 97 patch 1, 99/12/16 %% Modified by Etienne Riga - Dec. 16, 1999 %% to add /Diamond, /SolidDiamond and /BoldDiamond 10 dict dup begin /FontType 3 def /FontMatrix [ .001 0 0 .001 0 0 ] def /FontBBox [ 0 0 0 0 ] def /Encoding 256 array def 0 1 255 { Encoding exch /.notdef put } for Encoding dup (b) 0 get /Bullet put 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0000000000000000000000000000000000000000000000000000000000000000 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 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: CMSL9 %!PS-AdobeFont-1.1: CMSL9 1.0 %%CreationDate: 1991 Aug 20 16:41:23 % 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 (CMSL9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -9.46 def /isFixedPitch false def end readonly def /FontName /CMSL9 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 39 /quoteright put dup 46 /period put dup 58 /colon put dup 65 /A put dup 66 /B put dup 72 /H put dup 73 /I put dup 75 /K put dup 77 /M put dup 79 /O put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 87 /W put dup 97 /a 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 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put dup 121 /y put dup 127 /dieresis put readonly def /FontBBox{-61 -250 1150 750}readonly def /UniqueID 5000797 def currentdict end currentfile eexec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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 69 /E put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: TeX-cmex8 %!PS-AdobeFont-1.0: TeX-cmex8 001.001 % Filtered by type1fix.pl 0.05 %%EndComments 13 dict dup begin /FontInfo 16 dict dup begin /Copyright (see\040copyright\040of\040original\040TeX\040font) def /FamilyName (TeX\040cmex8) def /FullName (TeX\040cmex8\040Regular) def /ItalicAngle 0 def /Notice (converted\040after\040April\0402001) def /UnderlinePosition -100 def /UnderlineThickness 50 def /Weight (Regular) def /isFixedPitch false def /version (001.001) def end readonly def /FontName /TeX-cmex8 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 80 /P put dup 82 /R put readonly def /FontBBox {-29 -2957 1554 772} readonly def /UniqueID 4314416 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA06DA87FC7163A5A2A756A598FAB07633 89DE8BAE4F093966CD2192CE95EB0F323A6BABFDACCFCF27D91F7869A0E46CA5 9AAF6905783E8AC1F3F9875A76F97187738432F8D14E61574CB292FFB9740871 66839799D8CAF6E0DFE00012EE6D46A2B3655F29705BE37FD5EDA1C765AA2CF5 C5AD37207ED1EE9DB82FF31A33307FFA16911406557336AF92F50B603C7BD336 73EC060F68318378A6F599DDADA5A21504CADBA1E1F4B1A22962BA1BB39ADC7B E8CC92F196549457877C9636A8A7EFAC1C3745644C0FD151C70B9FAD69B02C1F FE5ED071CA1CF3D4A70909B6A3986687D8FAD2D91A9ED24A74DD3F1FE9EA0B50 D076040D6B144A72A89636F74C32D8EADD4DFBB5CD4844DE4D6818A736438AA7 657B523D81D271CF0C94430F8268D4AAEA69261D7E8A3778A2C9BBE16D839984 8157032AE38230F9DE2484D0F0CC96DD3E4C2963466334F685D5F8F1468CFC23 915D2F3759DDE3825A635A29CE151E0AADC574423CC947A0D5F485B227CCE41C 9B7D99BEBB3D10A51E25DEC45074BE221A066734EEB07A150D715353E4AAD204 8CEC24861DDB44C7D4CF4AEC464D0E3D32E3D358608D6B4E5329E4511F9EEFBA 9CD16DAE6A75D65A7E8F58E92790E82C4B14C738ACC86A19E3AEC867FE1EA06A 75D3E1F3ED8744582EBF3E35F02E07F72C9FD51C6251265A4CE916D35D034F1A A5CDCE0ABD996147461504EA4DAE01E34624D1E58958E4928D006C9E1E2903A7 E5BA2F396CC231882119CB3BBAD5AD2B5895BF172B653B83324DF7DE04E9B135 E7B47A831DE5DDA3662991D22EDABA932D04141D6137DBB698141CDCA0538302 3BAEE93934748D65C75A34749C7D7A4D9B0268FEEE83C1EA9D80EE942A955DB3 F9889D0C76F78435A2040699D923FD1D7C91C47F55FD449E0EF716BB8D310EAB 82CA675D5ED19C29C5BB254E27BDB79D4115975666335B87713DCFEA85F78855 549320F2F5A6AB955118F5BF9141F5DDE027D9CD6D484606C64594520C5E9795 8D17B1046CA34CCB9DFA013E4F04E792A022564A431A47E1CA5AD16C4D5AB66A EC3DA6E6C788EA3D471012B48512701FF915D0DA63E58D96291FCD5496CB9851 F1B7CAEF9DAD5582A795DE8ABE5F903B1B1C81711662F04540DDA8E4A8D6BD60 9E809D57DC361134EE1263490A98D7BEE9E66BA8C2B7B8FDBCF14934A05C6993 4FE7D6D88E1FFE0ADB45F0CFFB5772E6A239C9A798780F0D7BED62F30DC56F6C B81AB478CA96AC988AAE9F6616EE758C15C6470A789DA66D7BBCC8F5DAE110CB E2A5AED90A49897FAD6BD3DAD55A21D78D0E754F337AB122A00C55026CCA6447 8C23634758C6F1DE4F51F7592943C795D0C11630C3E801070E86D9B30EF3AB2D FCED9F8C59C99F226D75431A3BFCBD5A3939A2B3AAE897C75D4A25A4F01ED949 AD223772BEE9DE675953C5B7071A922DBD9822D3503B65E837BEB5181F3FE1F3 772228D5243C104A2C7840EEE30C4F7771FFFF87C68C365E3CDFF07D9F7F68C8 B3BC4787502B679AF5929AFFC7F095936B78412D2C415D2B561F0D5E1EFD8628 2ADCEAA794A48FDF5F1E8AD0112483B341803B63A07F90C512FBD2B740169804 0A5D9F122AD022FB8112617F7A815CDD30AA76275A02FE6B3B1AC6A433F926D8 A13D14319DAC0A7DA9134A3140EC693FCBB3477E4F65E6C1DF64B39F4455DAD2 35A8B7116BE382CC2B 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two 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: 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 16 /zeta put dup 25 /pi put dup 34 /epsilon put dup 78 /N put dup 80 /P 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 readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE 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%%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 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 73 /contintegraldisplay put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 98 /hatwide put dup 101 /tildewide put dup 102 /tildewider put dup 112 /radicalbig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 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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 7 /minusplus put dup 20 /lessequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 55 /mapsto put dup 102 /braceleft put dup 103 /braceright put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 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 12 /fi put dup 19 /acute put dup 34 /quotedblright put dup 39 /quoteright put dup 45 /hyphen put dup 46 /period put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 71 /G put dup 72 /H put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M 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 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 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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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period 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 58 /colon put dup 59 /semicolon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F 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 80 /P 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 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 D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E3948FFB0B4E70F212EC976D65099D84E0D37A7A771C3101D6AD26A0513378F 21EC3643079EECE0C9AB54B4772E5DCA82D0D4ACC7F42FB493AA04A3BF4A1BD6 06ECE186315DBE9CFDCB1A0303E8D3E83027CD3AFA8F0BD466A8E8CA0E7164CF 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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 readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 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 10 /Omega put 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 dup 99 /c put dup 104 /h put dup 108 /l put dup 111 /o put dup 116 /t put dup 126 /tilde put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % 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 (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 7 /minusplus put dup 15 /bullet put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 59 /emptyset put dup 66 /B put dup 68 /D put dup 69 /E put dup 71 /G put dup 83 /S put dup 86 /V put dup 87 /W put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put dup 112 /radical put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec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ABF0EB8F7244058D2DF2485404E083164A83F5E3299C8820908D6C5EB0D176CD 3A6BBAC68057F1CCB1521EDAB7D5CC76C492C09BF254FBA2EA5F3C6F2D6E5141 6349AF4CFBF034C711FE66DF90087E0EFF2210 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 13 /gamma put dup 16 /zeta put dup 20 /kappa put dup 25 /pi put dup 27 /sigma 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 68 /D put dup 69 /E put dup 78 /N put dup 80 /P put dup 81 /Q put dup 83 /S put dup 84 /T put dup 89 /Y put dup 97 /a 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 121 /y 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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 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 16 /zeta put dup 18 /theta put dup 20 /kappa put dup 21 /lambda put dup 24 /xi put dup 25 /pi put dup 27 /sigma 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 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H 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 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y 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 readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E394A533A081C36D456A09920001A3D2199583EB9B84B4DEE08E3D12939E321 990CD249827D9648574955F61BAAA11263A91B6C3D47A5190165B0C25ABF6D3E 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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 19 /acute 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 55 /seven 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 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N 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 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 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put dup 127 /dieresis 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 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 2 /Theta put dup 6 /Sigma put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 34 /quotedblright 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 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 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 91 /bracketleft 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 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 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 18 /grave put dup 19 /acute put dup 34 /quotedblright put dup 39 /quoteright 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 59 /semicolon 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 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft 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 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 127 /dieresis put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueID 5000792 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 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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 14 /ffi put dup 40 /parenleft put dup 41 /parenright put dup 45 /hyphen put dup 46 /period 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 58 /colon 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 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U 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 127 /dieresis 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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H.S end -118 1331 a -118 1331 a SDict begin 20 H.A end -118 1331 a -118 1331 a SDict begin [ /View [/XYZ H.V] /Dest (section.1) cvn H.B /DEST pdfmark end -118 1331 a 1523 1497 a Fs(1.)46 b Fr(Intr)n(oduction)94 1717 y Fs(W)-8 b(e)32 b(consider)e(the)h(follo)m(wing)h(Sc)m(hr\177)-45 b(odinger)30 b(equation)-118 1922 y(\(1.1\))491 1922 y SDict begin H.S end 491 1922 a 491 1922 a SDict begin 20 H.A end 491 1922 a 491 1922 a SDict begin [ /View [/XYZ H.V] /Dest (equation.1.1) cvn H.B /DEST pdfmark end 491 1922 a Fs(\()p Fq(H)602 1936 y Fp(z)s(;")694 1922 y Fq( )s Fs(\)\()p Fq(x)p Fs(\))d(=)d Fo(\000)1142 1860 y Fq(d)1189 1827 y Fn(2)p 1116 1901 139 4 v 1116 1984 a Fq(dx)1215 1958 y Fn(2)1265 1922 y Fq( )s Fs(\()p Fq(x)p Fs(\))d(+)f([)p Fq(V)h Fs(\()p Fq(x)f Fo(\000)g Fq(z)t Fs(\))h(+)f Fq(W)13 b Fs(\()p Fq("x)p Fs(\)])p Fq( )s Fs(\()p Fq(x)p Fs(\))27 b(=)e Fq(E)5 b( )s Fs(\()p Fq(x)p Fs(\))p Fq(;)108 b(x)25 b Fo(2)g Fm(R)p Fq(;)-118 2127 y Fs(where)32 b Fq(x)c Fo(7!)g Fq(V)20 b Fs(\()p 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Fs(\))p Fq(:)p 0 TeXcolorgray 85 2010 a Fv(\(H2\):)p 0 TeXcolorgray 41 w Fq(V)51 b Fs(is)30 b(real-v)-5 b(alued)31 b(and)f(lo)s(cally)i(square-in)m (tegrable.)p 0 TeXcolorgray 85 2176 a Fv(\(H3\):)p 0 TeXcolorgray 41 w Fq(W)43 b Fs(is)30 b(real)h(analytic)h(in)e(the)h (strip)f Fq(S)1646 2190 y Fp(Y)1731 2176 y Fs(=)25 b Fo(f)p Fq(z)30 b Fo(2)25 b Fm(C)30 b Fs(;)46 b Fo(j)p Fs(Im)30 b Fq(z)t Fo(j)c Fq(<)f(Y)20 b Fo(g)p Fs(.)-118 2363 y(W)-8 b(e)31 b(de\014ne:)-118 2570 y(\(1.6\))1167 2570 y SDict begin H.S end 1167 2570 a 1167 2570 a SDict begin 20 H.A end 1167 2570 a 1167 2570 a SDict begin [ /View [/XYZ H.V] /Dest (equation.1.6) cvn H.B /DEST pdfmark end 1167 2570 a Fq(W)1253 2584 y Fi(\000)1337 2570 y Fs(=)f(inf)1433 2631 y Fp(t)p Fi(2)p Fk(R)1568 2570 y Fq(W)13 b Fs(\()p Fq(t)p Fs(\))p Fq(;)198 b(W)2079 2584 y Fn(+)2163 2570 y Fs(=)25 b(sup)2267 2648 y Fp(t)p Fi(2)p Fk(R)2411 2570 y Fq(W)13 b Fs(\()p Fq(t)p Fs(\))-118 2722 y SDict begin H.S end -118 2722 a -118 2722 a SDict begin 20 H.A end -118 2722 a -118 2722 a SDict begin [ /View [/XYZ H.V] /Dest (subsubsection.1.1.2) cvn H.B /DEST pdfmark end -118 2722 a 75 x Fs(1.1.2.)47 b Fl(Assumptions)33 b(on)f(the)g(ener)-5 b(gy)32 b(r)-5 b(e)g(gion.)46 b Fs(T)-8 b(o)30 b(describ)s(e)f(the)g (energy)h(regions)f(where)g(w)m(e)h(study)e(the)h(sp)s(ectral)-118 2963 y(prop)s(erties,)h(w)m(e)h(consider)f(the)h(p)s(erio)s(dic)e(Sc)m (hr\177)-45 b(odinger)30 b(op)s(erator)h Fq(H)2269 2977 y Fn(0)2338 2963 y Fs(acting)h(on)e Fq(L)2799 2930 y Fn(2)2839 2963 y Fs(\()p Fm(R)p Fs(\))g(and)g(de\014ned)f(b)m(y)h(\()p 0 0 1 TeXcolorrgb 3656 2963 a SDict begin H.S end 3656 2963 a 0 0 1 TeXcolorrgb Fs(1.2)p 0 0 1 TeXcolorrgb 3773 2904 a SDict begin H.R end 3773 2904 a 3773 2963 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.1.2) cvn H.B /ANN pdfmark end 3773 2963 a 0 TeXcolorgray Fs(\).)-118 3046 y SDict begin H.S end -118 3046 a -118 3046 a SDict begin 20 H.A end -118 3046 a -118 3046 a SDict begin [ /View [/XYZ H.V] /Dest (subsubsection.1.1.3) cvn H.B /DEST pdfmark end -118 3046 a 143 x Fs(1.1.3.)47 b Fl(The)33 b(p)-5 b(erio)g(dic)35 b(op)-5 b(er)g(ator.)48 b Fs(The)30 b(sp)s(ectrum)g(of)g(\()p 0 0 1 TeXcolorrgb 1787 3189 a SDict begin H.S end 1787 3189 a 0 0 1 TeXcolorrgb Fs(1.2)p 0 0 1 TeXcolorrgb 1904 3131 a SDict begin H.R end 1904 3131 a 1904 3189 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.1.2) cvn H.B /ANN pdfmark end 1904 3189 a 0 TeXcolorgray Fs(\))g(is)h(absolutely)g(con)m(tin)m (uous)g(and)f(consists)h(of)g(in)m(terv)-5 b(als)-118 3355 y(of)23 b(the)h(real)g(axis,)h(sa)m(y)f([)p Fq(E)744 3369 y Fn(2)p Fp(n)p Fn(+1)917 3355 y Fq(;)31 b(E)1040 3369 y Fn(2)p Fp(n)p Fn(+2)1212 3355 y Fs(])24 b(for)f Fq(n)i Fo(2)g Fm(N)e Fs(,)h(suc)m(h)g(that)f Fq(E)2152 3369 y Fn(1)2217 3355 y Fq(<)i(E)2380 3369 y Fn(2)2445 3355 y Fo(\024)g Fq(E)2608 3369 y Fn(3)2673 3355 y Fq(<)g(E)2836 3369 y Fn(4)2875 3355 y Fq(:::E)3017 3369 y Fn(2)p Fp(n)3126 3355 y Fo(\024)g Fq(E)3289 3369 y Fn(2)p Fp(n)p Fn(+1)3486 3355 y Fq(<)g(E)3649 3369 y Fn(2)p Fp(n)p Fn(+2)3822 3355 y Fq(:::)-118 3521 y Fs(and)g Fq(E)121 3535 y Fp(n)193 3521 y Fo(!)g Fs(+)p Fo(1)p Fq(;)15 b(n)25 b Fo(!)g Fs(+)p Fo(1)p Fs(.)39 b(The)25 b(p)s(oin)m(ts)h(\()p Fq(E)1486 3535 y Fp(j)1522 3521 y Fs(\))1557 3535 y Fp(j)t Fi(2)p Fk(N)1715 3521 y Fs(are)g(the)g(eigen)m(v)-5 b(alues)27 b(of)f(the)f(self-adjoin)m(t)i(op)s(erator)f(obtained)-118 3687 y(b)m(y)39 b(considering)g Fq(H)577 3701 y Fn(0)655 3687 y Fs(de\014ned)e(b)m(y)i(\()p 0 0 1 TeXcolorrgb 1146 3687 a SDict begin H.S end 1146 3687 a 0 0 1 TeXcolorrgb Fs(1.2)p 0 0 1 TeXcolorrgb 1263 3629 a SDict begin H.R end 1263 3629 a 1263 3687 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.1.2) cvn H.B /ANN pdfmark end 1263 3687 a 0 TeXcolorgray Fs(\))g(and)f(acting)j(in)d Fq(L)1980 3654 y Fn(2)2020 3687 y Fs(\([0)p Fq(;)15 b Fs(1]\))41 b(with)e(p)s(erio)s(dic)f(b)s(oundary)f(conditions)j(\(see)-118 3853 y([)p 0 1 0 TeXcolorrgb -93 3853 a SDict begin H.S end -93 3853 a 0 1 0 TeXcolorrgb Fs(6)p 0 1 0 TeXcolorrgb -47 3795 a SDict begin H.R end -47 3795 a -47 3853 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Eas:73) cvn H.B /ANN pdfmark end -47 3853 a 0 TeXcolorgray Fs(,)p 0 1 0 TeXcolorrgb 11 3853 a SDict begin H.S end 11 3853 a 0 1 0 TeXcolorrgb Fs(19)p 0 1 0 TeXcolorrgb 102 3795 a SDict begin H.R end 102 3795 a 102 3853 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.ReSi4) cvn H.B /ANN pdfmark end 102 3853 a 0 TeXcolorgray Fs(]\).)49 b(The)32 b(in)m(terv)-5 b(als)34 b([)p Fq(E)884 3867 y Fn(2)p Fp(n)p Fn(+1)1057 3853 y Fq(;)c(E)1179 3867 y Fn(2)p Fp(n)p Fn(+2)1352 3853 y Fs(],)k Fq(n)29 b Fo(2)g Fm(N)p Fs(,)34 b(are)f(the)g Fl(sp)-5 b(e)g(ctr)g(al)38 b(b)-5 b(ands)p Fs(,)35 b(and)d(the)h(in)m(terv)-5 b(als)34 b(\()p Fq(E)3459 3867 y Fn(2)p Fp(n)3542 3853 y Fq(;)c(E)3664 3867 y Fn(2)p Fp(n)p Fn(+1)3837 3853 y Fs(\),)-118 4020 y Fq(n)25 b Fo(2)h Fm(N)115 3987 y Fi(\003)154 4020 y Fs(,)31 b(the)g Fl(sp)-5 b(e)g(ctr)g(al)35 b(gaps)p Fs(.)43 b(When)30 b Fq(E)1262 4034 y Fn(2)p Fp(n)1371 4020 y Fq(<)25 b(E)1534 4034 y Fn(2)p Fp(n)p Fn(+1)1707 4020 y Fs(,)31 b(one)g(sa)m(ys)g(that)h(the)e Fq(n)p Fs(-th)h(gap)g(is)g Fl(op)-5 b(en)p Fs(;)32 b(when)e([)p Fq(E)3510 4034 y Fn(2)p Fp(n)p Fi(\000)p Fn(1)3683 4020 y Fq(;)15 b(E)3790 4034 y Fn(2)p Fp(n)3872 4020 y Fs(])-118 4186 y(is)36 b(separated)g(from)f(the)h(rest)g(of)g(the)g(sp)s(ectrum)e(b)m(y)i(op)s (en)f(gaps,)i(the)f Fq(n)p Fs(-th)g(band)e(is)i(said)g(to)g(b)s(e)f Fl(isolate)-5 b(d)p Fs(.)59 b(The)-118 4352 y(sp)s(ectral)31 b(bands)e(and)g(gaps)i(are)g(represen)m(ted)f(in)g(\014gure)p 0 0 1 TeXcolorrgb 1865 4352 a SDict begin H.S end 1865 4352 a 0 0 1 TeXcolorrgb Fs(1)p 0 0 1 TeXcolorrgb 1910 4293 a SDict begin H.R end 1910 4293 a 1910 4352 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (figure.1) cvn H.B /ANN pdfmark end 1910 4352 a 0 TeXcolorgray Fs(.)p 0 TeXcolorgray 0 TeXcolorgray -118 4599 a SDict begin H.S end -118 4599 a -118 4599 a SDict begin 20 H.A end -118 4599 a -118 4599 a SDict begin [ /View [/XYZ H.V] /Dest (subsubsection.1.1.4) cvn H.B /DEST pdfmark end -118 4599 a 166 x Fs(1.1.4.)47 b Fl(The)29 b(ge)-5 b(ometric)30 b(assumption.)47 b Fs(Let)27 b(us)f(describ)s(e)f (the)h(energy)h(region)g(where)e(w)m(e)i(study)e(\()p 0 0 1 TeXcolorrgb 3245 4765 a SDict begin H.S end 3245 4765 a 0 0 1 TeXcolorrgb Fs(1.1)p 0 0 1 TeXcolorrgb 3362 4706 a SDict begin H.R end 3362 4706 a 3362 4765 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.1.1) cvn H.B /ANN pdfmark end 3362 4765 a 0 TeXcolorgray Fs(\).)39 b(W)-8 b(e)28 b(assume)-118 4931 y(that)j Fq(J)39 b Fs(is)31 b(a)f(real)i(compact)f (in)m(terv)-5 b(al)32 b(suc)m(h)e(that:)p 0 TeXcolorgray 85 5118 a Fv(\(H4a\):)p 0 TeXcolorgray 41 w Fs(F)-8 b(or)31 b(all)g Fq(E)g Fo(2)25 b Fq(J)9 b Fs(,)30 b(the)h(connected)g(comp)s (onen)m(ts)g(of)f(\000)2251 5132 y Fk(R)2303 5118 y Fs(\()p Fq(E)5 b Fs(\))31 b(ha)m(v)m(e)h(a)f(non)f(empt)m(y)g(in)m(terior.)p 0 TeXcolorgray 85 5284 a Fv(\(H4b\):)p 0 TeXcolorgray 41 w Fs(The)e(iso-energy)h(curv)m(e)g(\000)1337 5298 y Fk(R)1389 5284 y Fs(\()p Fq(E)5 b Fs(\))29 b(con)m(tains)h(a)f(real)g (branc)m(h)e(that)i(is)g(an)f(un)m(b)s(ounded)e(v)m(ertical)k(curv)m (e.)p 0 TeXcolorgray 85 5450 a Fv(\(H4c\):)p 0 TeXcolorgray 41 w Fs(F)-8 b(or)35 b(all)g Fq(E)h Fo(2)31 b Fq(J)9 b Fs(,)35 b(the)f(set)g Fo(W)7 b Fs(\()p Fq(E)e Fs(\))33 b(=)e([)p Fq(E)d Fo(\000)22 b Fq(W)2006 5464 y Fn(+)2065 5450 y Fq(;)15 b(E)28 b Fo(\000)22 b Fq(W)2379 5464 y Fi(\000)2438 5450 y Fs(])34 b(con)m(tains)h(only)f(isolated)i(bands)c (of)i(the)185 5616 y(p)s(erio)s(dic)29 b(op)s(erator.)p 0 TeXcolorgray 1872 5715 a Fn(2)p 0 TeXcolorgray eop end %%Page: 3 3 TeXDict begin 3 2 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray -118 -406 a SDict begin H.S end -118 -406 a 0 TeXcolorgray 0 TeXcolorgray -118 -406 a SDict begin H.R end -118 -406 a -118 -406 a SDict begin [ /View [/XYZ H.V] /Dest (page.3) cvn H.B /DEST pdfmark end -118 -406 a 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray 981 619 a @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. scale false 0.4 1.4 1.5 2. Arrow EndArrow } def [ 131.40033 93.07523 76.65019 104.02525 21.90005 93.07523 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 1.09506 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial 1072 -63 a tx@Dict begin { 5.0 15.27527 8.2125 2.73749 180. Uput UUput } PutCoor PutBegin end 1072 -63 a 1009 -40 a Fs(\()p Fo(E)8 b Fs(\))1072 -63 y tx@Dict begin PutEnd end 1072 -63 a 2072 -63 a tx@Dict begin { 5.0 34.24263 8.2125 2.73749 180. Uput UUput } PutCoor PutBegin end 2072 -63 a 1929 -40 a Fs(\()p Fq(k)2011 -26 y Fn(0)2052 -40 y Fs(\()p Fo(E)g Fs(\)\))2072 -63 y tx@Dict begin PutEnd end 2072 -63 a 981 619 a @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ -43.80011 54.75014 -65.70016 54.75014 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 0.0 54.75014 -21.90005 54.75014 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 32.85008 54.75014 21.90005 54.75014 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray [ 21.90005 54.75014 0.0 54.75014 -21.90005 54.75014 -43.80011 54.75014 -65.70016 54.75014 false NArray /DS 2.0 2. CLW mul add 2 div def /PSTricksDotFont 0. [1.0 0.0 0.0 1.0 0.0 0.0] FontDot /Dot { moveto gsave 1. 1. scale (b) show grestore } bind def newpath n { transform floor .5 add exch floor .5 add exch itransform Dot } repeat end @endspecial 435 256 a tx@Dict begin { 5.0 12.8333 7.48248 1.64249 180. Uput UUput } PutCoor PutBegin end 435 256 a 382 280 a Fq(E)449 294 y Fn(1)435 256 y tx@Dict begin PutEnd end 435 256 a 617 256 a tx@Dict begin { 5.0 12.8333 7.48248 1.64249 180. Uput UUput } PutCoor PutBegin end 617 256 a 564 280 a Fq(E)631 294 y Fn(2)617 256 y tx@Dict begin PutEnd end 617 256 a 799 256 a tx@Dict begin { 5.0 12.8333 7.48248 1.64249 180. Uput UUput } PutCoor PutBegin end 799 256 a 746 280 a Fq(E)813 294 y Fn(3)799 256 y tx@Dict begin PutEnd end 799 256 a 981 256 a tx@Dict begin { 5.0 12.8333 7.48248 1.64249 180. Uput UUput } PutCoor PutBegin end 981 256 a 928 280 a Fq(E)995 294 y Fn(4)981 256 y tx@Dict begin PutEnd end 981 256 a 1162 256 a tx@Dict begin { 5.0 12.8333 7.48248 1.64249 180. Uput UUput } PutCoor PutBegin end 1162 256 a 1109 280 a Fq(E)1176 294 y Fn(5)1162 256 y tx@Dict begin PutEnd end 1162 256 a 981 619 a @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { BeginArrow 1. 1. scale true 0.4 1.4 1.5 2. Arrow EndArrow moveto } def /ArrowB { } def [ -70.08011 59.13008 -76.65019 59.13008 -87.60022 59.13008 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 3.0 0 -1 add DotLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def -65.70016 59.13008 /y ED /x ED /r 4.37994 def /c 57.2957 r Div def /angleA 0. 0.0 c mul 2 div add def /angleB 180. 0.0 c mul 2 div sub def x y r angleA angleB arc gsave 1.09506 SLW 0 setgray 3.0 0 0 add DotLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def -43.80011 59.13008 /y ED /x ED /r 4.37994 def /c 57.2957 r Div def /angleA 0. 0.0 c mul 2 div add def /angleB 180. 0.0 c mul 2 div sub def x y r angleA angleB arc gsave 1.09506 SLW 0 setgray 3.0 0 0 add DotLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def -21.90005 59.13008 /y ED /x ED /r 4.37994 def /c 57.2957 r Div def /angleA 0. 0.0 c mul 2 div add def /angleB 180. 0.0 c mul 2 div sub def x y r angleA angleB arc gsave 1.09506 SLW 0 setgray 3.0 0 0 add DotLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def 0.0 59.13008 /y ED /x ED /r 4.37994 def /c 57.2957 r Div def /angleA 0. 0.0 c mul 2 div add def /angleB 180. 0.0 c mul 2 div sub def x y r angleA angleB arc gsave 1.09506 SLW 0 setgray 3.0 0 0 add DotLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def 21.90005 59.13008 /y ED /x ED /r 4.37994 def /c 57.2957 r Div def /angleA 0. 0.0 c mul 2 div add def /angleB 180. 0.0 c mul 2 div sub def x y r angleA angleB arc gsave 1.09506 SLW 0 setgray 3.0 0 0 add DotLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ -48.18005 59.13008 -61.3202 59.13008 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 3.0 0 0 add DotLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ -26.28 59.13008 -39.42015 59.13008 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 3.0 0 0 add DotLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ -4.37994 59.13008 -17.5201 59.13008 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 3.0 0 0 add DotLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 17.5201 59.13008 4.37994 59.13008 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 3.0 0 0 add DotLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 32.85008 59.13008 26.28 59.13008 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 3.0 0 0 add DotLine grestore end @endspecial 290 28 a tx@Dict begin { 5.0 10.4192 4.71457 2.12917 180. Uput UUput } PutCoor PutBegin end 290 28 a 247 39 a Fq(\015)294 53 y Fn(1)290 28 y tx@Dict begin PutEnd end 290 28 a 981 619 a @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ -43.80011 -54.75014 -65.70016 -54.75014 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 0.0 -54.75014 -21.90005 -54.75014 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 32.85008 -54.75014 21.90005 -54.75014 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray [ 21.90005 -54.75014 0.0 -54.75014 -21.90005 -54.75014 -43.80011 -54.75014 -65.70016 -54.75014 false NArray /DS 2.0 2. CLW mul add 2 div def /PSTricksDotFont 0. [1.0 0.0 0.0 1.0 0.0 0.0] FontDot /Dot { moveto gsave 1. 1. scale (b) show grestore } bind def newpath n { transform floor .5 add exch floor .5 add exch itransform Dot } repeat end @endspecial 617 1210 a tx@Dict begin { 5.0 12.8333 7.48248 1.64249 180. Uput UUput } PutCoor PutBegin end 617 1210 a 564 1234 a Fq(E)631 1248 y Fn(2)617 1210 y tx@Dict begin PutEnd end 617 1210 a 799 1210 a tx@Dict begin { 5.0 12.8333 7.48248 1.64249 180. Uput UUput } PutCoor PutBegin end 799 1210 a 746 1234 a Fq(E)813 1248 y Fn(3)799 1210 y tx@Dict begin PutEnd end 799 1210 a 981 1210 a tx@Dict begin { 5.0 12.8333 7.48248 1.64249 180. Uput UUput } PutCoor PutBegin end 981 1210 a 928 1234 a Fq(E)995 1248 y Fn(4)981 1210 y tx@Dict begin PutEnd end 981 1210 a 1162 1210 a tx@Dict begin { 5.0 12.8333 7.48248 1.64249 180. Uput UUput } PutCoor PutBegin end 1162 1210 a 1109 1234 a Fq(E)1176 1248 y Fn(5)1162 1210 y tx@Dict begin PutEnd end 1162 1210 a 981 619 a @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ -65.70016 -59.13008 32.85008 -59.13008 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 5.0 3.0 0.0 0.0 2 DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def -65.70016 -54.75014 /y ED /x ED /r 4.37994 def /c 57.2957 r Div def /angleA 90. 0.0 c mul 2 div add def /angleB 270. 0.0 c mul 2 div sub def x y r angleA angleB arc gsave 1.09506 SLW 0 setgray 5.0 3.0 0.0 0.0 2 DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. scale false 0.4 1.4 1.5 2. Arrow EndArrow } def [ 32.85008 -50.37018 -65.70016 -50.37018 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 5.0 3.0 0.0 0.0 2 DashLine grestore end @endspecial 1208 965 a tx@Dict begin { 5.0 10.4192 4.71457 2.12917 180. Uput UUput } PutCoor PutBegin end 1208 965 a 1165 976 a Fq(\015)1212 990 y Fn(2)1208 965 y tx@Dict begin PutEnd end 1208 965 a 981 619 a @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 219.00055 76.65019 219.00055 -43.80011 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 317.5508 0.0 131.40033 0.0 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 186.15047 43.80011 186.15047 0.0 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 153.30038 16.42503 153.30038 0.0 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 251.85063 43.80011 251.85063 0.0 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 284.70071 16.42503 284.70071 0.0 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial 2253 674 a tx@Dict begin { 5.0 20.62636 7.05666 0.9125 180. Uput UUput } PutCoor PutBegin end 2253 674 a 2168 699 a Fo(\000)p Fs(2)p Fq(\031)2253 674 y tx@Dict begin PutEnd end 2253 674 a 3344 674 a tx@Dict begin { 5.0 12.10966 7.05666 0.0 180. Uput UUput } PutCoor PutBegin end 3344 674 a 3294 703 a Fs(2)p Fq(\031)3344 674 y tx@Dict begin PutEnd end 3344 674 a 2526 674 a tx@Dict begin { 5.0 15.15135 6.3875 0.9125 180. Uput UUput } PutCoor PutBegin end 2526 674 a 2463 697 a Fo(\000)p Fq(\031)2526 674 y tx@Dict begin PutEnd end 2526 674 a 3072 674 a tx@Dict begin { 5.0 6.63466 4.71457 0.0 180. Uput UUput } PutCoor PutBegin end 3072 674 a 3044 693 a Fq(\031)3072 674 y tx@Dict begin PutEnd end 3072 674 a 981 619 a @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 148.92043 4.37994 131.40033 4.37994 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 5.0 3.0 0.0 0.0 2 DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 148.92043 16.42503 148.92043 4.37994 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 5.0 3.0 0.0 0.0 2 DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def 153.30038 16.42503 /y ED /x ED /r 4.37994 def /c 57.2957 r Div def /angleA 0. 0.0 c mul 2 div add def /angleB 180. 0.0 c mul 2 div sub def x y r angleA angleB arc gsave 1.09506 SLW 0 setgray 5.0 3.0 0.0 0.0 2 DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 157.68033 4.37994 157.68033 16.42503 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 5.0 3.0 0.0 0.0 2 DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 181.77051 4.37994 157.68033 4.37994 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 5.0 3.0 0.0 0.0 2 DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 181.77051 43.80011 181.77051 4.37994 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 5.0 3.0 0.0 0.0 2 DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def 186.15047 43.80011 /y ED /x ED /r 4.37994 def /c 57.2957 r Div def /angleA 0. 0.0 c mul 2 div add def /angleB 180. 0.0 c mul 2 div sub def x y r angleA angleB arc gsave 1.09506 SLW 0 setgray 5.0 3.0 0.0 0.0 2 DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 190.53041 43.80011 190.53041 4.37994 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 5.0 3.0 0.0 0.0 2 DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. scale false 0.4 1.4 1.5 2. Arrow EndArrow } def [ 219.00055 4.37994 190.53041 4.37994 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 5.0 3.0 0.0 0.0 2 DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 247.47067 4.37994 219.00055 4.37994 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 5.0 3.0 0.0 0.0 2 DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 247.47067 43.80011 247.47067 4.37994 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 5.0 3.0 0.0 0.0 2 DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def 251.85063 43.80011 /y ED /x ED /r 4.37994 def /c 57.2957 r Div def /angleA 0. 0.0 c mul 2 div add def /angleB 180. 0.0 c mul 2 div sub def x y r angleA angleB arc gsave 1.09506 SLW 0 setgray 5.0 3.0 0.0 0.0 2 DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 256.23058 4.37994 256.23058 43.80011 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 5.0 3.0 0.0 0.0 2 DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 280.32076 4.37994 256.23058 4.37994 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 5.0 3.0 0.0 0.0 2 DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 280.32076 16.42503 280.32076 4.37994 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 5.0 3.0 0.0 0.0 2 DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def 284.70071 16.42503 /y ED /x ED /r 4.37994 def /c 57.2957 r Div def /angleA 0. 0.0 c mul 2 div add def /angleB 180. 0.0 c mul 2 div sub def x y r angleA angleB arc gsave 1.09506 SLW 0 setgray 5.0 3.0 0.0 0.0 2 DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 289.08066 16.42503 289.08066 4.37994 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 5.0 3.0 0.0 0.0 2 DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { BeginArrow 1. 1. scale true 0.4 1.4 1.5 2. Arrow EndArrow moveto } def /ArrowB { } def [ 317.5508 4.37994 289.08066 4.37994 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 5.0 3.0 0.0 0.0 2 DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 223.3805 54.75014 223.3805 76.65019 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 3.0 0 0 add DotLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { BeginArrow 1. 1. scale true 0.4 1.4 1.5 2. Arrow EndArrow moveto } def /ArrowB { } def [ 223.3805 4.37994 223.3805 54.75014 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 3.0 0 -1 add DotLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 247.47067 4.37994 223.3805 4.37994 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 3.0 0 0 add DotLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 247.47067 43.80011 247.47067 4.37994 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 3.0 0 0 add DotLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def 251.85063 43.80011 /y ED /x ED /r 4.37994 def /c 57.2957 r Div def /angleA 0. 0.0 c mul 2 div add def /angleB 180. 0.0 c mul 2 div sub def x y r angleA angleB arc gsave 1.09506 SLW 0 setgray 3.0 0 0 add DotLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 256.23058 4.37994 256.23058 43.80011 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 3.0 0 0 add DotLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 280.32076 4.37994 256.23058 4.37994 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 3.0 0 0 add DotLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 280.32076 16.42503 280.32076 4.37994 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 3.0 0 0 add DotLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def 284.70071 16.42503 /y ED /x ED /r 4.37994 def /c 57.2957 r Div def /angleA 0. 0.0 c mul 2 div add def /angleB 180. 0.0 c mul 2 div sub def x y r angleA angleB arc gsave 1.09506 SLW 0 setgray 3.0 0 0 add DotLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 289.08066 16.42503 289.08066 4.37994 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 3.0 0 0 add DotLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.09506 SLW 0 setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. scale false 0.4 1.4 1.5 2. Arrow EndArrow } def [ 317.5508 4.37994 289.08066 4.37994 /Lineto /lineto load def false Line gsave 1.09506 SLW 0 setgray 3.0 -2 0 add DotLine grestore end @endspecial 2635 528 a tx@Dict begin { 5.0 10.4192 4.71457 2.12917 180. Uput UUput } PutCoor PutBegin end 2635 528 a 2592 539 a Fq(\015)2639 553 y Fn(2)2635 528 y tx@Dict begin PutEnd end 2635 528 a 2908 165 a tx@Dict begin { 5.0 10.4192 4.71457 2.12917 180. Uput UUput } PutCoor PutBegin end 2908 165 a 2865 176 a Fq(\015)2912 190 y Fn(1)2908 165 y tx@Dict begin PutEnd end 2908 165 a 3526 528 a tx@Dict begin { 5.0 25.70505 4.71457 2.12917 180. Uput UUput } PutCoor PutBegin end 3526 528 a 3419 539 a Fq(\015)3466 553 y Fn(1)3506 539 y Fq(;)15 b(\015)3593 553 y Fn(2)3526 528 y tx@Dict begin PutEnd end 3526 528 a 851 1736 a Fr(Figure)34 b(1.)p 0 TeXcolorgray 1322 1736 a SDict begin H.S end 1322 1736 a 0 TeXcolorgray 0 TeXcolorgray 1322 1736 a SDict begin H.R end 1322 1736 a 1322 1736 a SDict begin [ /View [/XYZ H.V] /Dest (figure.1) cvn H.B /DEST pdfmark end 1322 1736 a 0 TeXcolorgray Fs(Bands,)c(gaps)h(and)f (the)g(quasi-momen)m(tum)h Fq(k)p 0 TeXcolorgray -118 2004 a(W)-32 2018 y Fn(+)57 2004 y Fs(and)f Fq(W)320 2018 y Fi(\000)409 2004 y Fs(are)h(de\014ned)e(in)h(\()p 0 0 1 TeXcolorrgb 1015 2004 a SDict begin H.S end 1015 2004 a 0 0 1 TeXcolorrgb Fs(1.6)p 0 0 1 TeXcolorrgb 1132 1945 a SDict begin H.R end 1132 1945 a 1132 2004 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.1.6) cvn H.B /ANN pdfmark end 1132 2004 a 0 TeXcolorgray Fs(\).)-118 2128 y SDict begin H.S end -118 2128 a -118 2128 a SDict begin 20 H.A end -118 2128 a -118 2128 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.1.2) cvn H.B /DEST pdfmark end -118 2128 a 144 x Fs(1.2.)47 b Fv(Geometric)35 b(description.)-118 2392 y SDict begin H.S end -118 2392 a -118 2392 a SDict begin 20 H.A end -118 2392 a -118 2392 a SDict begin [ /View [/XYZ H.V] /Dest (subsubsection.1.2.1) cvn H.B /DEST pdfmark end -118 2392 a 148 x Fs(1.2.1.)47 b Fl(The)36 b(set)g Fq(W)572 2507 y Fi(\000)p Fn(1)666 2540 y Fs(\()p Fm(R)p Fs(\))p Fl(.)46 b Fs(As)33 b Fq(E)j Fo(2)31 b Fm(R)p Fs(,)j(the)g(set)g(\()p Fq(E)28 b Fo(\000)22 b Fq(W)13 b Fs(\))1995 2507 y Fi(\000)p Fn(1)2090 2540 y Fs(\()p Fm(R)p Fs(\))34 b(coincide)g(with)g Fq(W)2921 2507 y Fi(\000)p Fn(1)3015 2540 y Fs(\()p Fm(R)p Fs(\).)51 b(It)33 b(is)h(2)p Fq(\031)s Fs(-p)s(erio)s(dic.)-118 2706 y(It)c(consists)h (of)g(the)f(real)i(line)e(and)g(of)h(complex)g(branc)m(hes)f(\(curv)m (es\))h(whic)m(h)f(are)h(symmetric)g(with)f(resp)s(ect)g(to)h(the)-118 2872 y(real)e(line.)40 b(There)28 b(are)g(complex)h(branc)m(hes)f(b)s (eginning)f(at)i(the)g(real)g(extrema)g(of)f Fq(W)41 b Fs(that)28 b(do)g(not)h(cross)f(again)h(the)-118 3038 y(real)i(line.)-118 3204 y(Consider)g(an)h(extrem)m(um)g(of)g Fq(W)45 b Fs(of)32 b(order)f Fq(n)1457 3218 y Fp(i)1517 3204 y Fs(on)h(the)g(real)h(line,)g(sa)m(y)f Fq(\020)2379 3218 y Fp(i)2407 3204 y Fs(.)46 b(Near)33 b Fq(\020)2740 3218 y Fp(i)2767 3204 y Fs(,)g(the)f(set)h Fq(W)3226 3171 y Fi(\000)p Fn(1)3320 3204 y Fs(\()p Fm(R)p Fs(\))f(consists)g(of) -118 3370 y(a)d(real)h(segmen)m(t,)h(and)e(of)g Fq(n)841 3384 y Fp(i)887 3370 y Fo(\000)17 b Fs(1)30 b(complex)g(curv)m(es)f (symmetric)g(with)g(resp)s(ect)g(to)h(the)g(real)g(axis)f(and)g(in)m (tersecting)-118 3536 y(the)h(real)h(axis)g(only)g(on)f Fq(\020)766 3550 y Fp(i)794 3536 y Fs(.)41 b(The)30 b(angle)h(b)s(et)m (w)m(een)g(t)m(w)m(o)h(neigh)m(b)s(oring)e(curv)m(es)g(is)h(equal)g(to) 3047 3475 y Fq(\031)p 3033 3515 83 4 v 3033 3599 a(n)3088 3613 y Fp(i)3125 3536 y Fs(.)-118 3702 y(Let)g Fq(Y)45 b(>)25 b Fs(0.)41 b(W)-8 b(e)32 b(set)f Fq(S)705 3716 y Fp(Y)790 3702 y Fs(=)25 b Fo(f\000)p Fq(Y)46 b Fo(\024)25 b Fs(Im)k Fq(\020)j Fo(\024)25 b Fq(Y)20 b Fo(g)p Fs(.)41 b(W)-8 b(e)32 b(assume)e(that)h Fq(Y)50 b Fs(is)31 b(so)f(small)h(that) p 0 TeXcolorgray 98 3899 a Fo(\017)p 0 TeXcolorgray 42 w Fq(S)241 3913 y Fp(Y)331 3899 y Fs(is)g(con)m(tained)g(in)g(the)f (domain)g(of)h(analyticit)m(y)i(of)d Fq(W)13 b Fs(;)p 0 TeXcolorgray 98 4065 a Fo(\017)p 0 TeXcolorgray 42 w Fs(the)27 b(set)g Fq(W)575 4032 y Fi(\000)p Fn(1)669 4065 y Fs(\()p Fm(R)p Fs(\))13 b Fo(\\)g Fq(S)948 4079 y Fp(Y)1035 4065 y Fs(consists)27 b(of)g(the)g(real)g(line)g(and)f(of)h (the)g(complex)g(lines)g(passing)g(through)f(the)h(real)185 4231 y(extrema)k(of)f Fq(W)13 b Fs(.)-118 4428 y(An)30 b(example)h(of)f(subset)g Fq(W)868 4395 y Fi(\000)p Fn(1)962 4428 y Fs(\()p Fm(R)p Fs(\))h(is)f(sho)m(wn)g(in)g(\014gure)p 0 0 1 TeXcolorrgb 1855 4428 a SDict begin H.S end 1855 4428 a 0 0 1 TeXcolorrgb Fs(2)p 0 0 1 TeXcolorrgb 1900 4370 a SDict begin H.R end 1900 4370 a 1900 4428 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (figure.2) cvn H.B /ANN pdfmark end 1900 4428 a 0 TeXcolorgray Fs(.)p 0 TeXcolorgray 0 TeXcolorgray -118 4727 a SDict begin H.S end -118 4727 a -118 4727 a SDict begin 20 H.A end -118 4727 a -118 4727 a SDict begin [ /View [/XYZ H.V] /Dest (subsubsection.1.2.2) cvn H.B /DEST pdfmark end -118 4727 a 166 x Fs(1.2.2.)47 b Fl(Notations)35 b(and)e(description)h(of)f Fs(\()p Fq(E)26 b Fo(\000)20 b Fq(W)13 b Fs(\))1660 4860 y Fi(\000)p Fn(1)1754 4893 y Fs(\()p Fq(\033)s Fs(\()p Fq(H)1955 4907 y Fn(0)1995 4893 y Fs(\)\))p Fl(.)46 b Fs(F)-8 b(or)31 b(all)g Fq(E)g Fo(2)25 b Fq(J)9 b Fs(,)31 b(w)m(e)f(write)737 5154 y(\()p Fq(E)c Fo(\000)20 b Fq(W)13 b Fs(\))1090 5117 y Fi(\000)p Fn(1)1184 5154 y Fs(\()p Fq(\033)s Fs(\()p Fq(H)1385 5168 y Fn(0)1425 5154 y Fs(\)\))21 b Fo(\\)e Fm(R)25 b Fs(=)1798 5068 y Fh([)1783 5266 y Fp(k)r Fi(2)p Fk(Z)1958 5041 y Fp(N)1939 5068 y Fh([)1928 5263 y Fp(j)t Fn(=1)2051 5154 y Fo(f)p Fs([)p Fq(')2180 5116 y Fi(\000)2180 5182 y Fp(j)2240 5154 y Fs(\()p Fq(E)5 b Fs(\))p Fq(;)15 b(')2481 5116 y Fn(+)2481 5182 y Fp(j)2542 5154 y Fs(\()p Fq(E)5 b Fs(\)])21 b(+)f(2)p Fq(k)s(\031)s Fo(g)p Fq(;)-118 5419 y Fs(with)30 b(the)h(follo)m(wing)g(prop)s(erties:)p 0 TeXcolorgray 98 5616 a Fo(\017)p 0 TeXcolorgray 42 w Fq(')244 5577 y Fi(\000)244 5642 y Fn(1)303 5616 y Fs(\()p Fq(E)5 b Fs(\))26 b Fq(<)f(')626 5577 y Fn(+)626 5642 y(1)686 5616 y Fs(\()p Fq(E)5 b Fs(\))26 b Fq(<)f(')1009 5577 y Fi(\000)1009 5642 y Fn(2)1069 5616 y Fs(\()p Fq(E)5 b Fs(\))26 b Fq(<)f Fo(\001)15 b(\001)g(\001)26 b Fq(<)f(')1619 5577 y Fi(\000)1619 5644 y Fp(N)1687 5616 y Fs(\()p Fq(E)5 b Fs(\))26 b Fq(<)f(')2010 5577 y Fn(+)2010 5644 y Fp(N)2078 5616 y Fs(\()p Fq(E)5 b Fs(\))p Fq(;)198 b Fs(0)26 b Fq(<)f(')2669 5577 y Fn(+)2669 5644 y Fp(N)2737 5616 y Fs(\()p Fq(E)5 b Fs(\))21 b Fo(\000)f Fq(')3050 5577 y Fi(\000)3050 5642 y 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Fq(dx)1118 2175 y Fn(2)1167 2139 y Fq( )s Fs(\()p Fq(x;)15 b Fo(E)8 b Fs(\))22 b(+)e Fq(V)g Fs(\()p Fq(x)p Fs(\))p Fq( )s Fs(\()p Fq(x;)15 b Fo(E)8 b Fs(\))28 b(=)d Fo(E)8 b Fq( )s Fs(\()p Fq(x;)15 b Fo(E)8 b Fs(\))p Fq(;)108 b(x)25 b Fo(2)g Fm(R)p Fq(:)-118 2380 y Fs(satisfying)31 b(the)g(relation)-118 2622 y(\(2.2\))1367 2622 y SDict begin H.S end 1367 2622 a 1367 2622 a SDict begin 20 H.A end 1367 2622 a 1367 2622 a SDict begin [ /View [/XYZ H.V] /Dest (equation.2.2) cvn H.B /DEST pdfmark end 1367 2622 a Fq( )s Fs(\()p Fq(x)21 b Fs(+)f(1)p Fq(;)15 b Fo(E)8 b Fs(\))26 b(=)f Fq(\025)p Fs(\()p Fo(E)8 b Fs(\))p Fq( )s Fs(\()p Fq(x;)15 b Fo(E)8 b Fs(\))p Fq(:)-118 2864 y Fs(for)31 b(all)h Fq(x)26 b Fo(2)g Fm(R)31 b Fs(and)f(some)i(non-v)-5 b(anishing)30 b(complex)i(n)m(um)m(b)s(er)e Fq(\025)p Fs(\()p Fo(E)8 b Fs(\))32 b(indep)s(enden)m(t)d(of)j Fq(x)p Fs(.)42 b(Suc)m(h)31 b(a)g(solution)h(exists)-118 3030 y(and)24 b(is)g(called)h(the)g Fl(Blo)-5 b(ch)28 b(solution)e Fs(and)d Fq(\025)p 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Fs(is)f(analytic)i(in)2116 2825 y Fo(p)p 2192 2825 277 4 v 69 x Fq(E)26 b Fo(\000)20 b Fq(E)2443 2909 y Fp(l)2469 2894 y Fs(,)30 b(and)-118 3116 y(\(2.4\))916 3116 y SDict begin H.S end 916 3116 a 916 3116 a SDict begin 20 H.A end 916 3116 a 916 3116 a SDict begin [ /View [/XYZ H.V] /Dest (equation.2.4) cvn H.B /DEST pdfmark end 916 3116 a Fq(k)963 3130 y Fp(p)1003 3116 y Fs(\()p Fo(E)8 b Fs(\))21 b Fo(\000)f Fq(k)1288 3130 y Fp(p)1328 3116 y Fs(\()p Fq(E)1430 3131 y Fp(l)1456 3116 y Fs(\))26 b(=)f Fq(c)1652 3131 y Fp(l)1678 3033 y Fh(p)p 1769 3033 261 4 v 83 x Fo(E)j(\000)20 b Fq(E)2003 3131 y Fp(l)2050 3116 y Fs(+)f Fq(o)p Fs(\()p Fo(E)29 b(\000)20 b Fq(E)2454 3131 y Fp(l)2480 3116 y Fs(\))91 b Fq(c)2645 3131 y Fp(l)2697 3116 y Fo(6)p Fs(=)25 b(0)p Fq(:)-118 3338 y Fs(Finally)-8 b(,)35 b(w)m(e)f(note)f(that)h(the)f (main)g(branc)m(h)f(can)h(b)s(e)g(con)m(tin)m(ued)g(analytically)j(to)e (the)f(complex)g(plane)g(cut)h(along)-118 3504 y(\()p Fo(\0001)p Fq(;)15 b(E)186 3518 y Fn(1)226 3504 y Fs(])30 b(and)g(the)h(sp)s(ectral)f(gaps)h(])p Fq(E)1256 3518 y Fn(2)p Fp(n)1339 3504 y Fq(;)15 b(E)1446 3518 y Fn(2)p Fp(n)p Fn(+1)1618 3504 y Fs([,)31 b Fq(n)25 b Fo(2)g Fm(N)1931 3471 y Fi(\003)1970 3504 y Fs(,)31 b(of)f(the)h(p)s(erio)s (dic)e(op)s(erator)i Fq(H)3077 3518 y Fn(0)3116 3504 y Fs(.)-118 3618 y SDict begin H.S end -118 3618 a -118 3618 a SDict begin 20 H.A end -118 3618 a -118 3618 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.2.3) cvn H.B /DEST pdfmark end -118 3618 a 143 x Fs(2.3.)47 b Fv(A)h(meromorphic)h (function.)d Fs(No)m(w)d(let)g(us)e(discuss)g(a)h(function)f(pla)m (ying)i(an)f(imp)s(ortan)m(t)f(role)i(in)e(the)-118 3927 y(adiabatic)32 b(constructions.)-118 4093 y(In)e([)p 0 1 0 TeXcolorrgb 21 4093 a SDict begin H.S end 21 4093 a 0 1 0 TeXcolorrgb Fs(12)p 0 1 0 TeXcolorrgb 112 4035 a SDict begin H.R end 112 4035 a 112 4093 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.FEKL2) cvn H.B /ANN pdfmark end 112 4093 a 0 TeXcolorgray Fs(],)g(it)h(is)g(sho)m(wn)f(that,)h(on)f Fo(S)7 b Fs(,)30 b(there)h(is)f(a)h(meromorphic)f(function)g Fq(!)j Fs(ha)m(ving)e(the)g(follo)m(wing)h(prop)s(erties:)p 0 TeXcolorgray 98 4287 a Fo(\017)p 0 TeXcolorgray 42 w Fs(the)e(di\013eren)m(tial)h(\012)25 b(=)g Fq(!)s(d)p Fo(E)38 b Fs(is)29 b(meromorphic;)h(its)g(p)s(oles)g(are)g(the)g(p)s (oin)m(ts)g(of)g Fq(P)i Fo([)18 b Fq(Q)p Fs(,)30 b(where)g Fq(P)42 b Fs(is)30 b(the)g(set)185 4453 y(of)g(p)s(oles)g(of)h Fo(E)i(7!)26 b Fq( )s Fs(\()p Fq(x;)15 b Fo(E)8 b Fs(\),)32 b(and)e Fq(Q)g Fs(is)g(the)h(set)g(of)f(zeros)h(of)g Fq(k)2311 4420 y Fi(0)2334 4453 y Fs(;)p 0 TeXcolorgray 98 4619 a Fo(\017)p 0 TeXcolorgray 42 w Fs(all)g(the)f(p)s(oles)h(of)f (\012)g(are)h(simple;)p 0 TeXcolorgray 98 4785 a Fo(\017)p 0 TeXcolorgray 42 w Fs(if)f(the)h(residue)e(of)i(\012)f(at)h(a)g(p)s (oin)m(t)f Fq(p)g Fs(is)h(denoted)f(b)m(y)g(res)2106 4799 y Fp(p)2146 4785 y Fs(\012,)g(one)h(has)1430 5008 y SDict begin H.S end 1430 5008 a 1430 5008 a SDict begin 20 H.A end 1430 5008 a 1430 5008 a SDict begin [ /View [/XYZ H.V] /Dest (equation.2.5) cvn H.B /DEST pdfmark end 1430 5008 a Fs(res)1542 5022 y Fp(p)1582 5008 y Fs(\012)25 b(=)g(1)p Fq(;)15 b Fo(8)p Fq(p)25 b Fo(2)g Fq(P)33 b Fo(n)21 b Fq(Q;)p 0 TeXcolorgray -2434 w Fs(\(2.5\))p 0 TeXcolorgray 1282 5201 a(res)1394 5215 y Fp(q)1432 5201 y Fs(\012)k(=)g Fo(\000)p Fs(1)p Fq(=)p Fs(2)p Fq(;)15 b Fo(8)p Fq(q)29 b Fo(2)c Fq(Q)20 b Fo(n)h Fq(P)s(;)p 0 TeXcolorgray -2434 w Fs(\(2.6\))p 0 TeXcolorgray 1343 5394 a(res)1455 5408 y Fp(r)1493 5394 y Fs(\012)k(=)g(1)p Fq(=)p Fs(2)p Fq(;)15 b Fo(8)p Fq(r)29 b Fo(2)c Fq(P)33 b Fo(\\)20 b Fq(Q:)p 0 TeXcolorgray -2449 w Fs(\(2.7\))p 0 TeXcolorgray 0 TeXcolorgray 98 5616 a Fo(\017)p 0 TeXcolorgray 42 w Fs(if)30 b Fo(E)j(2)25 b(S)37 b Fs(pro)5 b(jects)31 b(in)m(to)g(a)g(gap,)g(then)f Fq(!)s Fs(\()p Fo(E)8 b Fs(\))26 b Fo(2)f Fm(R)p Fs(.)p 0 TeXcolorgray 1872 5715 a Fn(6)p 0 TeXcolorgray eop end %%Page: 7 7 TeXDict begin 7 6 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray -118 -406 a SDict begin H.S end -118 -406 a 0 TeXcolorgray 0 TeXcolorgray -118 -406 a SDict begin H.R end -118 -406 a -118 -406 a SDict begin [ /View [/XYZ H.V] /Dest (page.7) cvn H.B /DEST pdfmark end -118 -406 a 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray 98 -201 a Fo(\017)p 0 TeXcolorgray 42 w Fs(if)30 b Fo(E)j(2)25 b(S)37 b Fs(pro)5 b(jects)31 b(inside)f(a)h(band,)e(then)p 1668 -280 187 4 v 31 w Fq(!)s Fs(\()p Fo(E)8 b Fs(\))25 b(=)g Fq(!)s Fs(\()2083 -224 y Fh(b)2070 -201 y Fo(E)9 b Fs(\).)-118 -114 y SDict begin H.S end -118 -114 a -118 -114 a SDict begin 20 H.A end -118 -114 a -118 -114 a SDict begin [ /View [/XYZ H.V] /Dest (subsubsection.2.3.1) cvn H.B /DEST pdfmark end -118 -114 a 143 x Fs(2.3.1.)47 b Fl(The)42 b(c)-5 b(omplex)43 b(momentum.)k Fs(It)40 b(is)h(the)f(main)g(analytic) i(ob)5 b(ject)41 b(of)f(the)g(complex)h(WKB)g(metho)s(d.)69 b(Let)-118 195 y Fq(\020)32 b Fo(2)24 b Fq(S)95 209 y Fp(Y)156 195 y Fs(.)41 b(W)-8 b(e)31 b(de\014ne)f Fq(\024)p Fs(,)h(in)f Fo(D)s Fs(\()p Fq(W)13 b Fs(\))30 b(the)h(domain)f(of)g (analyticit)m(y)j(of)e Fq(W)43 b Fs(b)m(y)30 b(b)m(y)g(:)-118 404 y(\(2.8\))1472 404 y SDict begin H.S end 1472 404 a 1472 404 a SDict begin 20 H.A end 1472 404 a 1472 404 a SDict begin [ /View [/XYZ H.V] /Dest (equation.2.8) cvn H.B /DEST pdfmark end 1472 404 a Fq(\024)p Fs(\()p Fq(\020)7 b Fs(\))26 b(=)f Fq(k)s Fs(\()p Fq(E)h Fo(\000)20 b Fq(W)13 b Fs(\()p Fq(\020)7 b Fs(\)\))p Fq(:)-118 614 y Fs(Here,)36 b Fq(k)h Fs(is)d(the)h(Blo)s(c)m(h)g(quasi-momen)m(tum)f(de\014ned)f (in)h(section)p 0 0 1 TeXcolorrgb 2178 614 a SDict begin H.S end 2178 614 a 0 0 1 TeXcolorrgb Fs(2.2)p 0 0 1 TeXcolorrgb 2294 555 a SDict begin H.R end 2294 555 a 2294 614 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (subsection.2.2) cvn H.B /ANN pdfmark end 2294 614 a 0 TeXcolorgray Fs(.)52 b(Though)33 b Fq(\024)i Fs(dep)s(ends)d(on)i Fq(E)5 b Fs(,)36 b(w)m(e)e(omit)h(the)-118 780 y Fq(E)5 b Fs(-dep)s(endence.)60 b(Relation)39 b(\()p 0 0 1 TeXcolorrgb 934 780 a SDict begin H.S end 934 780 a 0 0 1 TeXcolorrgb Fs(2.8)p 0 0 1 TeXcolorrgb 1050 721 a SDict begin H.R end 1050 721 a 1050 780 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.2.8) cvn H.B /ANN pdfmark end 1050 780 a 0 TeXcolorgray Fs(\))f(translates)g(the)f(prop)s(erties)g(of)g Fq(k)j Fs(in)m(to)e(prop)s(erties)f(of)g Fq(\024)p Fs(.)61 b(Hence,)40 b Fq(\020)j Fo(7!)36 b Fq(\024)p Fs(\()p Fq(\020)7 b Fs(\))-118 946 y(is)42 b(a)h(m)m(ulti-v)-5 b(alued)43 b(analytic)h(function,)h(and)c(its)i(branc)m(h)e(p)s(oin)m(ts)h(are)h (related)g(to)g(the)f(branc)m(h)g(p)s(oin)m(ts)g(of)g(the)-118 1112 y(quasi-momen)m(tum)31 b(b)m(y)f(the)g(relations)-118 1321 y(\(2.9\))1260 1321 y SDict begin H.S end 1260 1321 a 1260 1321 a SDict begin 20 H.A end 1260 1321 a 1260 1321 a SDict begin [ /View [/XYZ H.V] /Dest (equation.2.9) cvn H.B /DEST pdfmark end 1260 1321 a Fq(E)c Fo(\000)20 b Fq(W)13 b Fs(\()p Fq(\020)7 b Fs(\))25 b(=)g Fq(E)1848 1336 y Fp(l)1874 1321 y Fq(;)106 b(l)27 b Fs(=)e(1)p Fq(;)15 b Fs(2)p Fq(;)g Fs(3)p Fq(;)g Fo(\001)g(\001)g(\001)-118 1530 y Fs(Let)31 b Fq(\020)85 1544 y Fn(0)154 1530 y Fs(b)s(e)f(a)h(branc)m(h)f(p)s(oin)m(t)g(of)g Fq(\024)p Fs(.)41 b(If)30 b Fq(W)1303 1497 y Fi(0)1326 1530 y Fs(\()p Fq(\020)1401 1544 y Fn(0)1441 1530 y Fs(\))25 b Fo(6)p Fs(=)g(0,)31 b(then)f Fq(\020)1945 1544 y Fn(0)2015 1530 y Fs(is)g(a)h(branc)m(h)f(p)s(oin)m(t)g(of)h(square)f(ro)s(ot)h(t)m(yp) s(e.)-118 1696 y(If)d Fq(D)g Fo(\032)d Fq(D)s Fs(\()p Fq(W)13 b Fs(\))28 b(is)h(a)g(simply)f(connected)h(set)g(con)m(taining) h(no)e(branc)m(h)g(p)s(oin)m(ts)h(of)f Fq(\024)p Fs(,)i(w)m(e)f(call)g (it)g Fl(r)-5 b(e)g(gular)p Fs(.)42 b(Let)29 b Fq(\024)3736 1710 y Fp(p)3804 1696 y Fs(b)s(e)-118 1862 y(a)i(branc)m(h)f(of)h(the)f (complex)i(momen)m(tum)e(analytic)i(in)f(a)g(regular)f(domain)h Fq(D)s Fs(.)41 b(All)31 b(the)g(other)g(branc)m(hes)f(that)h(are)-118 2028 y(analytic)h(in)e Fq(D)j Fs(are)e(describ)s(ed)e(b)m(y)h(the)h (follo)m(wing)h(form)m(ulae:)-118 2237 y(\(2.10\))1530 2237 y SDict begin H.S end 1530 2237 a 1530 2237 a SDict begin 20 H.A end 1530 2237 a 1530 2237 a SDict begin [ /View [/XYZ H.V] /Dest (equation.2.10) cvn H.B /DEST pdfmark end 1530 2237 a Fq(\024)1582 2199 y Fi(\006)1582 2259 y Fp(m)1674 2237 y Fs(=)25 b Fo(\006)p Fq(\024)1893 2251 y Fp(p)1953 2237 y Fs(+)20 b(2)p Fq(\031)s(m:)-118 2446 y Fs(Here)31 b Fo(\006)f Fs(and)g Fq(m)25 b Fo(2)f Fm(Z)30 b Fs(are)h(indexing)f(the)h(branc)m(hes.)-118 2528 y SDict begin H.S end -118 2528 a -118 2528 a SDict begin 20 H.A end -118 2528 a -118 2528 a SDict begin [ /View [/XYZ H.V] /Dest (subsubsection.2.3.2) cvn H.B /DEST pdfmark end -118 2528 a 149 x Fs(2.3.2.)47 b Fl(Index)39 b(of)g(an)f(interval)i Fs([)p Fq(')1066 2638 y Fi(\000)1066 2704 y Fp(j)1125 2677 y Fs(\()p Fq(E)5 b Fs(\))p Fq(;)15 b(')1366 2638 y Fn(+)1366 2704 y Fp(j)1427 2677 y Fs(\()p Fq(E)5 b Fs(\)])p Fl(.)47 b Fs(Fix)37 b Fq(j)k Fo(2)35 b(f)p Fs(1)p Fq(;)15 b(:)g(:)g(:)j(;)d(N)10 b Fo(g)p Fs(.)59 b(Fix)37 b(a)g(con)m(tin)m(uous)h(branc)m(h)d Fq(\024)3588 2691 y Fp(j)3662 2677 y Fs(of)h(the)-118 2843 y(complex)31 b(momen)m(tum)f(on)g([)p Fq(')925 2804 y Fi(\000)925 2870 y Fp(j)985 2843 y Fs(\()p Fq(E)5 b Fs(\))p Fq(;)15 b(')1226 2804 y Fn(+)1226 2870 y Fp(j)1287 2843 y Fs(\()p Fq(E)5 b Fs(\)].)42 b(W)-8 b(e)32 b(de\014ne:)-118 3052 y(\(2.11\))939 3052 y SDict begin H.S end 939 3052 a 939 3052 a SDict begin 20 H.A end 939 3052 a 939 3052 a SDict begin [ /View [/XYZ H.V] /Dest (equation.2.11) cvn H.B /DEST pdfmark end 939 3052 a Fq(\024)991 3013 y Fn(+)991 3079 y Fp(j)1076 3052 y Fs(=)24 b Fq(\024)1223 3066 y Fp(j)1260 3052 y Fs(\()p Fq(')1354 3013 y Fn(+)1354 3079 y Fp(j)1414 3052 y Fs(\);)107 b Fq(\024)1633 3013 y Fi(\000)1633 3079 y Fp(j)1718 3052 y Fs(=)25 b Fq(\024)1866 3066 y Fp(j)1902 3052 y Fs(\()p Fq(')1996 3013 y Fi(\000)1996 3079 y Fp(j)2056 3052 y Fs(\);)107 b Fq(p)2269 3066 y Fp(j)2305 3052 y Fq(\031)29 b Fs(=)24 b Fq(\024)2533 3013 y Fn(+)2533 3079 y Fp(j)2613 3052 y Fo(\000)c Fq(\024)2756 3013 y Fi(\000)2756 3079 y Fp(j)2815 3052 y Fq(:)-118 3261 y(p)-72 3275 y Fp(j)-6 3261 y Fs(is)31 b(the)f Fl(index)h Fs(of)g([)p Fq(')670 3223 y Fi(\000)670 3288 y Fp(j)729 3261 y Fs(\()p Fq(E)5 b Fs(\))p Fq(;)15 b(')970 3223 y Fn(+)970 3288 y Fp(j)1031 3261 y Fs(\()p Fq(E)5 b Fs(\)])32 b(asso)s(ciated)g(to)f Fq(\024)1826 3275 y Fp(j)1863 3261 y Fs(.)-118 3427 y(Let)g(us)e(giv)m(e)j(some)f(prop)s(erties)f(of) g Fq(p)1153 3441 y Fp(j)1190 3427 y Fs(.)-118 3453 y SDict begin H.S end -118 3453 a -118 3453 a SDict begin 20 H.A end -118 3453 a -118 3453 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.1) cvn H.B /DEST pdfmark end -118 3453 a 0 TeXcolorgray 228 x Fv(Lemma)35 b(2.1.)p 0 TeXcolorgray 42 w Fl(Assume)d(that)i Fs(\()p Fq(H)7 b Fs(4\))34 b Fl(is)e(satis\014e)-5 b(d.)44 b(The)32 b(indexes)h Fq(p)2247 3695 y Fp(j)2316 3681 y Fl(have)g(the)g(fol)5 b(lowing)34 b(pr)-5 b(op)g(erties:)-118 3729 y SDict begin H.S end -118 3729 a -118 3729 a SDict begin 20 H.A end -118 3729 a -118 3729 a SDict begin [ /View [/XYZ H.V] /Dest (Item.1) cvn H.B /DEST pdfmark end -118 3729 a 0 TeXcolorgray 27 3868 a Fs(\(1\))p 0 TeXcolorgray 43 w Fl(F)e(or)33 b Fq(j)e Fo(2)25 b(f)p Fs(1)p Fq(;)15 b Fo(\001)g(\001)g(\001)33 b Fq(;)15 b(N)10 b Fo(g)p Fl(,)33 b Fq(p)1051 3882 y Fp(j)1112 3868 y Fo(2)25 b(f\000)p Fs(1)p Fq(;)15 b Fs(0)p Fq(;)g Fs(1)p Fo(g)p Fl(.)-118 3895 y SDict begin H.S end -118 3895 a -118 3895 a SDict begin 20 H.A end -118 3895 a -118 3895 a SDict begin [ /View [/XYZ H.V] /Dest (Item.2) cvn H.B /DEST pdfmark end -118 3895 a 0 TeXcolorgray 27 4034 a Fs(\(2\))p 0 TeXcolorgray 185 3966 a Fh(P)281 3992 y Fp(N)281 4061 y(j)t Fn(=1)422 4034 y Fo(j)p Fq(p)493 4048 y Fp(j)530 4034 y Fo(j)25 b(2)g Fs(2)p Fm(N)p Fq(:)94 4288 y Fv(Pro)s(of)k Fs(The)e(p)s(oin)m(ts)h(\()p Fq(E)21 b Fo(\000)15 b Fq(W)e Fs(\)\()p Fq(')1268 4250 y Fi(\000)1268 4315 y Fp(j)1327 4288 y Fs(\()p Fq(E)5 b Fs(\)\))29 b(and)f(\()p Fq(E)20 b Fo(\000)15 b Fq(W)e Fs(\)\()p Fq(')2144 4250 y Fn(+)2144 4315 y Fp(j)2204 4288 y Fs(\()p Fq(E)5 b Fs(\)\))29 b(are)f(the)g(ends)f(of)h(a)g(band)f(of)h Fq(\033)s Fs(\()p Fq(H)3584 4302 y Fn(0)3624 4288 y Fs(\):)39 b(they)-118 4454 y(are)c(distinct)g(or)g(coincide.)55 b(If)35 b(they)g(coincide,)i(i.e.)55 b(if)35 b(\()p Fq(E)29 b Fo(\000)23 b Fq(W)13 b Fs(\)\()p Fq(')2279 4416 y Fi(\000)2279 4481 y Fp(j)2338 4454 y Fs(\()p Fq(E)5 b Fs(\)\))35 b(=)d(\()p Fq(E)d Fo(\000)23 b Fq(W)13 b Fs(\)\()p Fq(')3106 4416 y Fn(+)3106 4481 y Fp(j)3165 4454 y Fs(\()p Fq(E)5 b Fs(\)\),)38 b(the)d(index)f Fq(p)3861 4468 y Fp(j)-118 4620 y Fs(satis\014es)g Fq(p)266 4634 y Fp(j)334 4620 y Fs(=)d(0.)52 b(Else,)35 b(w)m(e)f(consider)g Fq(k)1323 4634 y Fp(j)1394 4620 y Fs(the)g(branc)m(h)f(of)i(the)f(quasi-momen)m (tum)g(asso)s(ciated)h(to)g Fq(\024)3445 4634 y Fp(j)3482 4620 y Fs(.)51 b(W)-8 b(e)35 b(ha)m(v)m(e)-118 4786 y(that)869 4957 y Fo(j)p Fq(k)941 4971 y Fp(j)978 4957 y Fs(\(\()p Fq(E)26 b Fo(\000)20 b Fq(W)13 b Fs(\)\()p Fq(')1460 4919 y Fi(\000)1460 4984 y Fp(j)1520 4957 y Fs(\()p Fq(E)5 b Fs(\)\)\))22 b Fo(\000)e Fq(k)1892 4971 y Fp(j)1928 4957 y Fs(\(\()p Fq(E)27 b Fo(\000)20 b Fq(W)13 b Fs(\)\()p Fq(')2411 4919 y Fn(+)2411 4984 y Fp(j)2470 4957 y Fs(\()p Fq(E)5 b Fs(\)\)\))p Fo(j)28 b Fs(=)c Fq(\031)s(;)-118 5147 y Fs(see)31 b(\()p 0 0 1 TeXcolorrgb 64 5147 a SDict begin H.S end 64 5147 a 0 0 1 TeXcolorrgb Fs(2.2)p 0 0 1 TeXcolorrgb 180 5089 a SDict begin H.R end 180 5089 a 180 5147 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (subsection.2.2) cvn H.B /ANN pdfmark end 180 5147 a 0 TeXcolorgray Fs(\),)g(and)f Fo(j)p Fq(p)519 5161 y Fp(j)556 5147 y Fo(j)25 b Fs(=)g(1.)-118 5313 y(Let)31 b(us)e(pro)m(v)m(e)j(p)s(oin)m (t)e(\(2\).)42 b(W)-8 b(e)31 b(write:)1534 5404 y Fp(N)1499 5431 y Fh(X)1504 5627 y Fp(j)t Fn(=1)1646 5518 y Fo(j)p Fq(p)1717 5532 y Fp(j)1753 5518 y Fo(j)26 b(\021)1934 5404 y Fp(N)1900 5431 y Fh(X)1904 5627 y Fp(j)t Fn(=1)2046 5518 y Fq(p)2092 5532 y Fp(j)2159 5518 y Fs([2])p Fq(;)p 0 TeXcolorgray 1872 5715 a Fn(7)p 0 TeXcolorgray eop end %%Page: 8 8 TeXDict begin 8 7 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray -118 -406 a SDict begin H.S end -118 -406 a 0 TeXcolorgray 0 TeXcolorgray -118 -406 a SDict begin H.R end -118 -406 a -118 -406 a SDict begin [ /View [/XYZ H.V] /Dest (page.8) cvn H.B /DEST pdfmark end -118 -406 a 0 TeXcolorgray 0 TeXcolorgray 332 -235 a Fp(N)298 -208 y Fh(X)302 -13 y Fp(j)t Fn(=1)444 -122 y Fq(p)490 -108 y Fp(j)552 -122 y Fs(=)682 -235 y Fp(N)648 -208 y Fh(X)652 -13 y Fp(j)t Fn(=1)804 -197 y Fq(\024)856 -183 y Fp(j)893 -197 y Fs(\()p Fq(')987 -235 y Fn(+)987 -170 y Fp(j)1047 -197 y Fs(\))21 b Fo(\000)e Fq(\024)1245 -183 y Fp(j)1282 -197 y Fs(\()p Fq(')1376 -235 y Fi(\000)1376 -170 y Fp(j)1436 -197 y Fs(\))p 804 -143 668 4 v 1110 -59 a Fq(\031)1507 -122 y Fs(=)1613 -186 y Fq(\024)1665 -172 y Fp(N)1732 -186 y Fs(\()p Fq(')1826 -224 y Fn(+)1826 -157 y Fp(N)1894 -186 y Fs(\))i Fo(\000)f Fq(\024)2093 -172 y Fn(1)2133 -186 y Fs(\()p Fq(')2227 -224 y Fi(\000)2227 -160 y Fn(1)2287 -186 y Fs(\))p 1613 -143 710 4 v 1940 -59 a Fq(\031)2352 -122 y Fs(+)2443 -235 y Fp(N)7 b Fi(\000)p Fn(1)2454 -208 y Fh(X)2458 -13 y Fp(j)t Fn(=1)2621 -197 y Fq(\024)2673 -183 y Fp(j)t Fn(+1)2800 -197 y Fs(\()p Fq(')2894 -235 y Fi(\000)2894 -170 y Fp(j)t Fn(+1)3022 -197 y Fs(\))20 b Fo(\000)g Fq(\024)3220 -183 y Fp(j)3257 -197 y Fs(\()p Fq(')3351 -235 y Fn(+)3351 -170 y Fp(j)3411 -197 y Fs(\))p 2621 -143 826 4 v 3006 -59 a Fq(\031)3456 -122 y(:)-118 120 y Fs(F)-8 b(or)31 b Fq(j)g Fo(2)25 b(f)p Fs(1)p Fq(;)15 b Fo(\001)g(\001)g(\001)32 b Fq(;)15 b(N)31 b Fo(\000)20 b Fs(1)p Fo(g)p Fs(,)31 b(\()p Fq(E)26 b Fo(\000)20 b Fq(W)13 b Fs(\)\()p Fq(')1294 81 y Fi(\000)1294 147 y Fp(j)t Fn(+1)1422 120 y Fs(\()p Fq(E)5 b Fs(\)\))31 b(and)f(\()p Fq(E)c Fo(\000)20 b Fq(W)13 b Fs(\)\()p Fq(')2254 81 y Fn(+)2254 147 y Fp(j)2314 120 y Fs(\()p Fq(E)5 b Fs(\)\))31 b(are)g(the)g(ends)e(of)i(a)f(same)h(gap)g(and)823 328 y Fq(k)870 342 y Fp(j)t Fn(+1)997 328 y Fs(\(\()p Fq(E)26 b Fo(\000)20 b Fq(W)13 b Fs(\)\()p Fq(')1479 290 y Fi(\000)1479 355 y Fp(j)t Fn(+1)1606 328 y Fs(\()p Fq(E)5 b Fs(\)\)\))27 b Fo(\021)e Fq(k)1988 342 y Fp(j)2025 328 y Fs(\(\()p Fq(E)h Fo(\000)20 b Fq(W)13 b Fs(\)\()p Fq(')2507 290 y Fn(+)2507 355 y Fp(j)2567 328 y Fs(\()p Fq(E)5 b Fs(\)\)\)[2)p Fq(\031)s Fs(])p Fq(:)-118 537 y Fs(By)31 b(p)s(erio)s(dicit)m(y)-8 b(,)31 b(\()p Fq(E)26 b Fo(\000)20 b Fq(W)13 b Fs(\)\()p Fq(')944 498 y Fi(\000)944 563 y Fn(1)1004 537 y Fs(\()p Fq(E)5 b Fs(\)\))31 b(and)f(\()p Fq(E)c Fo(\000)20 b Fq(W)13 b Fs(\)\()p Fq(')1836 498 y Fn(+)1836 565 y Fp(N)1904 537 y Fs(\()p Fq(E)5 b Fs(\)\))31 b(are)g(the)g(ends)e(of)i(the)f(same)h(gap.)-118 703 y(This)e(ends)h(the)h(pro)s(of)e(of)i(Lemma)p 0 0 1 TeXcolorrgb 1119 703 a SDict begin H.S end 1119 703 a 0 0 1 TeXcolorrgb Fs(2.1)p 0 0 1 TeXcolorrgb 1235 644 a SDict begin H.R end 1235 644 a 1235 703 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.2.1) cvn H.B /ANN pdfmark end 1235 703 a 0 TeXcolorgray Fs(.)-118 869 y(If)d Fq(p)17 883 y Fp(j)78 869 y Fo(6)p Fs(=)d(0,)30 b(w)m(e)f(sa)m(y)g(that)g(w)m(e)g(cross)f(a)h (band.)39 b(In)28 b(this)g(case,)i(\()p Fq(E)22 b Fo(\000)16 b Fq(W)d Fs(\)\()p Fq(')2387 830 y Fi(\000)2387 896 y Fp(j)2447 869 y Fs(\))25 b Fo(6)p Fs(=)g(\()p Fq(E)d Fo(\000)16 b Fq(W)d Fs(\)\()p Fq(')3042 830 y Fn(+)3042 896 y Fp(j)3102 869 y Fs(\))29 b(and)f(the)g(asso)s(ciated)-118 1035 y(connected)j(comp)s(onen)m(t)g(of)f(the)h(iso-energy)g(curv)m(e)g (is)f(un)m(b)s(ounded)e(v)m(ertically)-8 b(.)-118 1201 y(W)g(e)31 b(notice)h(that)f Fq(p)549 1215 y Fn(1)614 1201 y Fo(6)p Fs(=)24 b(0,)31 b(and)f(th)m(us)1187 1133 y Fh(P)1283 1159 y Fp(N)1283 1228 y(j)t Fn(=1)1425 1201 y Fo(j)p Fq(p)1496 1215 y Fp(j)1533 1201 y Fo(j)25 b Fq(>)g Fs(0.)-118 1240 y SDict begin H.S end -118 1240 a -118 1240 a SDict begin 20 H.A end -118 1240 a -118 1240 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.2.2) cvn H.B /DEST pdfmark end -118 1240 a 0 TeXcolorgray 214 x Fv(Remark)35 b(2.2.)p 0 TeXcolorgray 42 w Fl(We)e(c)-5 b(an)33 b(shose)g(the)g (determination)i(of)e Fq(\024)g Fl(such)g(that)g Fq(p)2425 1468 y Fn(1)2490 1454 y Fs(=)25 b(1)p Fl(.)-118 1559 y SDict begin H.S end -118 1559 a -118 1559 a SDict begin 20 H.A end -118 1559 a -118 1559 a SDict begin [ /View [/XYZ H.V] /Dest (subsubsection.2.3.3) cvn H.B /DEST pdfmark end -118 1559 a 148 x Fs(2.3.3.)47 b Fl(T)-7 b(unneling)38 b(c)-5 b(o)g(e\016cients.)47 b Fs(F)-8 b(or)37 b Fq(j)j Fo(2)35 b(f)p Fs(1)p Fq(;)15 b(:)g(:)g(:)j(;)d(N)10 b Fo(g)p Fs(,)38 b(w)m(e)f(denote)g(b)m(y)f Fq(\015)2521 1721 y Fp(j)2594 1707 y Fs(a)g(smo)s(oth)g(closed)i(curv)m(e)e(that)h(go)s (es)-118 1873 y(once)23 b(around)e([)p Fq(')465 1835 y Fn(+)465 1900 y Fp(j)524 1873 y Fs(\()p Fq(E)5 b Fs(\))p Fq(;)15 b(')765 1835 y Fi(\000)765 1900 y Fp(j)t Fn(+1)894 1873 y Fs(\()p Fq(E)5 b Fs(\)].)39 b(Notice)24 b(that)f(this)f(curv)m (e)g(is)g(the)g(pro)5 b(jection)23 b(of)f(a)h(closed)f(curv)m(e)h(on)f (the)g(complex)-118 2039 y(Riemann)30 b(surface)g Fq(\024)p Fs(\()p Fq(\020)7 b Fs(\))26 b(=)f Fq(k)s Fs(\()p Fq(E)h Fo(\000)20 b Fq(W)13 b Fs(\()p Fq(\020)7 b Fs(\)\).)41 b(W)-8 b(e)31 b(consider)g(the)f(tunneling)g(actions)i Fq(S)2886 2053 y Fp(j)2952 2039 y Fs(giv)m(en)g(b)m(y:)-118 2248 y(\(2.12\))1101 2248 y SDict begin H.S end 1101 2248 a 1101 2248 a SDict begin 20 H.A end 1101 2248 a 1101 2248 a SDict begin [ /View [/XYZ H.V] /Dest (equation.2.12) cvn H.B /DEST pdfmark end 1101 2248 a Fq(S)1157 2262 y Fp(j)1193 2248 y Fs(\()p Fq(E)5 b Fs(\))27 b(=)e Fq(i)1504 2124 y Fh(I)1555 2330 y Fp(\015)1591 2340 y Fg(j)1643 2248 y Fq(\024)p Fs(\()p Fq(\020)7 b Fs(\))p Fq(d\020)g(;)137 b Fo(8)p Fq(j)30 b Fo(2)25 b(f)p Fs(1)p Fq(;)15 b(:)g(:)g(:)j(N)10 b Fo(g)p Fq(;)-118 2464 y Fs(It)29 b(is)g(straigh)m(tforw)m(ard)h(to)g (pro)m(v)m(e)f(that)h(for)f Fq(E)h Fo(2)25 b Fq(J)9 b Fs(,)30 b(eac)m(h)g(of)f(these)h(actions)g(is)f(real)g(and)g(non-zero)g (and)g(that)g Fq(S)3771 2478 y Fp(j)3837 2464 y Fs(is)-118 2630 y(analytic)g(in)d(a)i(complex)g(neigh)m(b)s(orho)s(o)s(d)d(of)j Fq(J)36 b Fs(\(for)27 b(analogous)h(statemen)m(ts,)i(w)m(e)e(refer)e (to)i([)p 0 1 0 TeXcolorrgb 3045 2630 a SDict begin H.S end 3045 2630 a 0 1 0 TeXcolorrgb Fs(8)p 0 1 0 TeXcolorrgb 3091 2571 a SDict begin H.R end 3091 2571 a 3091 2630 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.FK0) cvn H.B /ANN pdfmark end 3091 2630 a 0 TeXcolorgray Fs(,)p 0 1 0 TeXcolorrgb 3143 2630 a SDict begin H.S end 3143 2630 a 0 1 0 TeXcolorrgb Fs(17)p 0 1 0 TeXcolorrgb 3234 2571 a SDict begin H.R end 3234 2571 a 3234 2630 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Mg1) cvn H.B /ANN pdfmark end 3234 2630 a 0 TeXcolorgray Fs(]\).)40 b(By)28 b(de\014nition,)-118 2796 y(w)m(e)j(c)m(ho)s(ose)g(the)g(direction)g (of)f(the)h(in)m(tegration)h(so)f(that)g(all)g(the)g(tunneling)f (actions)h(b)s(e)f(p)s(ositiv)m(e.)42 b(W)-8 b(e)31 b(set)1654 3004 y Fq(t)1687 3018 y Fp(j)1748 3004 y Fs(=)25 b Fq(e)1886 2967 y Fi(\000)1966 2940 y Ff(1)p 1951 2952 60 3 v 1951 2993 a(2)p Fg(")2021 2967 y Fp(S)2064 2977 y Fg(j)2101 3004 y Fq(:)-118 3318 y Fs(\(2.13\))1411 3318 y SDict begin H.S end 1411 3318 a 1411 3318 a SDict begin 20 H.A end 1411 3318 a 1411 3318 a SDict begin [ /View [/XYZ H.V] /Dest (equation.2.13) cvn H.B /DEST pdfmark end 1411 3318 a Fq(T)13 b Fs(\()p Fq(E)5 b(;)15 b(")p Fs(\))27 b(=)e Fq(e)1866 3229 y Fi(\000)1959 3166 y Fg(N)1948 3180 y Fe(P)1933 3290 y Fg(j)s Ff(=1)2050 3229 y Fp(S)2093 3239 y Fg(j)2126 3229 y Fn(\()p Fp(E)t Fn(\))p Fp(=)p Fn(2)p Fp(")2343 3318 y Fq(:)-118 3526 y(T)43 b Fs(is)30 b(exp)s(onen)m (tially)i(small.)-118 3633 y SDict begin H.S end -118 3633 a -118 3633 a SDict begin 20 H.A end -118 3633 a -118 3633 a SDict begin [ /View [/XYZ H.V] /Dest (section.3) cvn H.B /DEST pdfmark end -118 3633 a 1209 3781 a Fs(3.)46 b Fr(The)35 b(Pr)n(oof)e(of)h(Theorem)p 0 0 1 TeXcolorrgb 2441 3781 a SDict begin H.S end 2441 3781 a 0 0 1 TeXcolorrgb Fr(1.2)p 0 0 1 TeXcolorrgb 2570 3723 a SDict begin H.R end 2570 3723 a 2570 3781 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.1.2) cvn H.B /ANN pdfmark end 2570 3781 a 0 TeXcolorgray -118 3835 a SDict begin H.S end -118 3835 a -118 3835 a SDict begin 20 H.A end -118 3835 a -118 3835 a SDict begin [ /View [/XYZ H.V] /Dest (subsubsection.3.0.4) cvn H.B /DEST pdfmark end -118 3835 a 166 x Fs(3.0.4.)47 b Fl(Sp)-5 b(e)g(ctr)g(al)40 b(r)-5 b(esults.)47 b Fs(One)35 b(of)g(the)h(main)f(ob)5 b(jects)36 b(of)g(the)g(sp)s(ectral)f(theory)h (of)g(quasi-p)s(erio)s(dic)f(op)s(erators)h(is)-118 4167 y(the)30 b(Ly)m(apuno)m(v)g(exp)s(onen)m(t)g(\(for)g(a)h(de\014nition)f (and)f(additional)i(information,)g(see,)f(for)g(example,)h([)p 0 1 0 TeXcolorrgb 3342 4167 a SDict begin H.S end 3342 4167 a 0 1 0 TeXcolorrgb Fs(4)p 0 1 0 TeXcolorrgb 3388 4109 a SDict begin H.R end 3388 4109 a 3388 4167 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.quatre) cvn H.B /ANN pdfmark end 3388 4167 a 0 TeXcolorgray Fs(]\).)41 b(The)30 b(main)-118 4333 y(result)g(of)h(this)f(section)i(is)-118 4333 y SDict begin H.S end -118 4333 a -118 4333 a SDict begin 20 H.A end -118 4333 a -118 4333 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.1) cvn H.B /DEST pdfmark end -118 4333 a 0 TeXcolorgray 253 x Fv(Theorem)i(3.1.)p 0 TeXcolorgray 41 w 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Fs(\))20 b Fo(\001)h Fq(T)13 b Fs(\()p Fq(E)5 b Fs(\))1829 2903 y Fi(\000)p Fn(1)1924 2941 y Fq(e)1966 2903 y Fn(2)p Fp(\031)r(P)10 b(y)2134 2912 y Ff(0)2168 2903 y Fp(=")2240 2941 y Fq(;)46 b Fs(Im)p Fq(z)29 b Fs(=)c Fo(\000)p Fq(y)2703 2955 y Fn(0)2742 2941 y Fq(=":)-118 3147 y Fs(Here,)g Fq(C)7 b Fs(\()p Fq(y)268 3161 y Fn(0)307 3147 y Fs(\))24 b(is)f(a)g(p)s(ositiv)m(e)h(constan)m(t)h(indep)s (enden)m(t)d(of)h Fq(")p Fs(,)i(Re)p Fq(z)t Fs(,)g(and)e Fq(E)5 b Fs(.)39 b(The)22 b(estimates)j(are)f(v)-5 b(alid)23 b(for)g(su\016cien)m(tly)-118 3313 y(small)30 b Fq(")p Fs(.)40 b(W)-8 b(e)31 b(recall)f(that)g Fq(M)39 b Fs(is)30 b(analytic)g(and)f(1-p)s(erio)s(dic)g(in)g Fq(z)t Fs(.)41 b(Equations)29 b(\()p 0 0 1 TeXcolorrgb 2659 3313 a SDict begin H.S end 2659 3313 a 0 0 1 TeXcolorrgb Fs(3.15)p 0 0 1 TeXcolorrgb 2821 3255 a SDict begin H.R end 2821 3255 a 2821 3313 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.3.15) cvn H.B /ANN pdfmark end 2821 3313 a 0 TeXcolorgray Fs(\),)h(\()p 0 0 1 TeXcolorrgb 2946 3313 a SDict begin H.S end 2946 3313 a 0 0 1 TeXcolorrgb Fs(3.16)p 0 0 1 TeXcolorrgb 3108 3255 a SDict begin H.R end 3108 3255 a 3108 3313 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.3.16) cvn H.B /ANN pdfmark end 3108 3313 a 0 TeXcolorgray Fs(\))g(and)e(the)i(maxim)m(um)-118 3479 y(principle)g(imply)g(that:) 926 3648 y Fo(j)p Fq(A)p Fo(j)p Fq(;)15 b Fo(j)p Fq(B)5 b Fo(j)27 b(\024)e Fs(2)p Fq(C)7 b Fs(\()p Fq(y)1528 3662 y Fn(0)1567 3648 y Fs(\))p Fq(T)13 b Fs(\()p Fq(E)5 b Fs(\))1810 3610 y Fi(\000)p Fn(1)1921 3648 y Fs(exp)o(\(2)p Fq(\031)s(P)13 b(y)2310 3662 y Fn(0)2350 3648 y Fq(=")p Fs(\))p Fq(;)108 b(z)29 b Fo(2)c Fm(R)p Fq(:)-118 3836 y Fs(This)k(leads)h(to)h(the)f(follo)m(wing)h(upp)s(er)d(b)s(ound)g (for)h(the)h(Ly)m(apuno)m(v)g(exp)s(onen)m(t)g(for)g(the)g(matrix)g(co) s(cycle)i(generated)-118 4002 y(b)m(y)e Fq(M)10 b Fs(\()p Fq(z)t(;)15 b(E)5 b Fs(\).)-118 4208 y(\(3.17\))1117 4208 y SDict begin H.S end 1117 4208 a 1117 4208 a SDict begin 20 H.A end 1117 4208 a 1117 4208 a SDict begin [ /View [/XYZ H.V] /Dest (equation.3.17) cvn H.B /DEST pdfmark end 1117 4208 a Fq(\022)s Fs(\()p Fq(E)g(;)15 b(")p Fs(\))26 b Fo(\024)f Fs(log)s(\()p Fq(T)13 b Fs(\()p Fq(E)5 b Fs(\))1870 4170 y Fi(\000)p Fn(1)1965 4208 y Fs(\))21 b(+)e Fq(C)27 b Fs(+)20 b(2)p Fq(\031)s(P)13 b(y)2510 4222 y Fn(0)2550 4208 y Fq(=")p Fs(;)-118 4414 y(where)30 b Fq(C)37 b Fs(is)30 b(a)h(constan)m(t)g(indep)s(enden)m(t)f(of)g Fq(E)36 b Fs(and)29 b Fq(")p Fs(.)42 b(Using)30 b(\()p 0 0 1 TeXcolorrgb 2077 4414 a SDict begin H.S end 2077 4414 a 0 0 1 TeXcolorrgb Fs(3.10)p 0 0 1 TeXcolorrgb 2239 4356 a SDict begin H.R end 2239 4356 a 2239 4414 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.3.10) cvn H.B /ANN pdfmark end 2239 4414 a 0 TeXcolorgray Fs(\))h(one)f(gets)-118 4621 y(\(3.18\))1059 4621 y SDict begin H.S end 1059 4621 a 1059 4621 a SDict begin 20 H.A end 1059 4621 a 1059 4621 a SDict begin [ /View [/XYZ H.V] /Dest (equation.3.18) cvn H.B /DEST pdfmark end 1059 4621 a Fs(\002\()p Fq(E)5 b(;)15 b(")p Fs(\))27 b Fo(\024)1516 4559 y Fq(")p 1487 4600 101 4 v 1487 4683 a Fs(2)p Fq(\031)1612 4621 y Fs(log)s(\()p Fq(T)13 b Fs(\()p Fq(E)5 b Fs(\))1973 4583 y Fi(\000)p Fn(1)2068 4621 y Fs(\))21 b(+)f Fq("C)27 b Fs(+)20 b(2)p Fq(\031)s(P)13 b(y)2656 4635 y Fn(0)2695 4621 y Fq(:)-118 4742 y SDict begin H.S end -118 4742 a -118 4742 a SDict begin 20 H.A end -118 4742 a -118 4742 a SDict begin [ /View [/XYZ H.V] /Dest (subsubsection.3.4.2) cvn H.B /DEST pdfmark end -118 4742 a 104 x Fs(3.4.2.)47 b Fl(The)30 b(lower)g(b)-5 b(ound.)47 b Fs(F)-8 b(or)27 b(\()p Fq(M)10 b Fs(\()p Fq(z)t(;)15 b(")p Fs(\))1355 4860 y Fn(0)p Fp(<"<)p Fn(1)1574 4846 y Fs(\))27 b(a)h(family)f(of)g Fq(S)5 b(L)p Fs(\(2)p Fq(;)15 b Fm(C)p Fs(\)-v)-5 b(alued,)29 b(1-p)s(erio)s(dic)d(functions) h(of)g Fq(z)i Fo(2)c Fm(C)-118 5012 y Fs(and)30 b Fq(h)g Fs(an)h(irrational)g(n)m(um)m(b)s(er,)e(w)m(e)i(recall)h(the)e(follo)m (wing)i(result)f(obtained)f(in)g([)p 0 1 0 TeXcolorrgb 2698 5012 a SDict begin H.S end 2698 5012 a 0 1 0 TeXcolorrgb Fs(10)p 0 1 0 TeXcolorrgb 2790 4953 a SDict begin H.R end 2790 4953 a 2790 5012 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.huit) cvn H.B /ANN pdfmark end 2790 5012 a 0 TeXcolorgray Fs(].)-118 5035 y SDict begin H.S end -118 5035 a -118 5035 a SDict begin 20 H.A end -118 5035 a -118 5035 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.6) cvn H.B /DEST pdfmark end -118 5035 a 0 TeXcolorgray 228 x Fv(Prop)s(osition)i(3.6.)p 0 TeXcolorgray 39 w Fl(Fix)d Fq(")844 5277 y Fn(0)909 5263 y Fq(>)c Fs(0)p Fl(.)41 b(Assume)29 b(that)h(ther)-5 b(e)30 b(exist)g Fq(y)2114 5277 y Fn(0)2182 5263 y Fl(and)g Fq(y)2400 5277 y Fn(1)2468 5263 y Fl(such)f(that)h Fs(0)c Fq(<)f(y)3065 5277 y Fn(0)3129 5263 y Fq(<)g(y)3270 5277 y Fn(1)3334 5263 y Fq(<)g Fo(1)k Fl(and)h(such)-118 5429 y(that,)k(for)f(any)g Fq(")26 b Fo(2)f Fs(\(0)p Fq(;)15 b(")731 5443 y Fn(0)772 5429 y Fs(\))32 b Fl(one)h(has)p 0 TeXcolorgray 98 5616 a Fo(\017)p 0 TeXcolorgray 42 w Fl(the)g(function)f Fq(z)e Fo(!)25 b Fq(M)10 b Fs(\()p Fq(z)t(;)15 b(")p Fs(\))35 b Fl(is)d(analytic)i(in)e(the)h(strip)h Fq(S)c Fs(=)25 b Fo(f)p Fq(z)30 b Fo(2)25 b Fm(C)p Fs(;)48 b(0)25 b Fo(\024)g Fl(Im)q Fq(z)k Fo(\024)3105 5575 y Fp(y)3140 5584 y Ff(1)p 3105 5595 70 4 v 3124 5647 a Fp(")3185 5616 y Fo(g)p Fl(;)p 0 TeXcolorgray 1854 5715 a Fn(11)p 0 TeXcolorgray eop end %%Page: 12 12 TeXDict begin 12 11 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray -118 -406 a SDict begin H.S end -118 -406 a 0 TeXcolorgray 0 TeXcolorgray -118 -406 a SDict begin H.R end -118 -406 a -118 -406 a SDict begin [ /View [/XYZ H.V] /Dest (page.12) cvn H.B /DEST pdfmark end -118 -406 a 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray 98 -207 a Fo(\017)p 0 TeXcolorgray 42 w Fl(in)32 b(the)h(strip)h Fq(S)716 -193 y Fn(1)780 -207 y Fs(=)25 b Fo(f)p Fq(z)30 b Fo(2)25 b Fm(C)p Fs(;)1228 -248 y Fp(y)1263 -239 y Ff(0)p 1228 -228 70 4 v 1246 -175 a Fp(")1332 -207 y Fo(\024)g Fl(Im)q Fq(z)k Fo(\024)1715 -248 y Fp(y)1750 -239 y Ff(1)p 1715 -228 V 1733 -175 a Fp(")1794 -207 y Fo(g)d(\032)f Fq(S;)15 b(M)10 b Fs(\()p Fq(z)t(;)15 b(")p Fs(\))35 b Fl(admits)f(the)f(r)-5 b(epr)g(esentation)-118 3 y Fs(\(3.19\))1040 3 y SDict begin H.S end 1040 3 a 1040 3 a SDict begin 20 H.A end 1040 3 a 1040 3 a SDict begin [ /View [/XYZ H.V] /Dest (equation.3.19) cvn H.B /DEST pdfmark end 1040 3 a Fq(M)10 b Fs(\()p Fq(z)t(;)15 b(")p Fs(\))27 b(=)e Fq(\025)p Fs(\()p Fq(")p Fs(\))p Fq(e)1666 -34 y Fp(i)p Fn(2)p Fp(\031)r(n)1811 -25 y Ff(0)1847 -34 y Fp(z)1907 3 y Fo(\001)1952 -70 y Fh(\000)1994 3 y Fq(M)2082 17 y Fn(0)2121 3 y Fs(\()p Fq(")p Fs(\))d(+)e Fq(M)2434 17 y Fn(1)2473 3 y Fs(\()p Fq(z)t(;)15 b(")p Fs(\))2671 -70 y Fh(\001)2715 3 y Fs(;)185 214 y Fl(for)33 b(some)g(c)-5 b(onstant)35 b Fq(\025)p Fs(\()p Fq(")p Fs(\))p Fl(,)e(some)g(inte)-5 b(ger)33 b Fq(n)1739 228 y Fn(0)1811 214 y Fl(and)g(a)g(matrix)h Fq(M)2446 228 y Fn(0)2486 214 y Fs(\()p Fq(")p Fs(\))p Fl(,)f(al)5 b(l)34 b(of)e(them)i(indep)-5 b(endent)34 b(of)f Fq(z)t Fl(;)p 0 TeXcolorgray 98 380 a Fo(\017)p 0 TeXcolorgray 42 w Fq(M)10 b Fs(\()p Fq(z)t(;)15 b(E)5 b Fs(\))27 b(=)634 251 y Fh(\022)804 325 y Fs(1)209 b(0)742 432 y Fq(\014)5 b Fs(\()p Fq(")p Fs(\))84 b Fq(\013)p Fs(\()p Fq(")p Fs(\))1208 251 y Fh(\023)1274 380 y Fl(;)p 0 TeXcolorgray 98 546 a Fo(\017)p 0 TeXcolorgray 42 w Fl(ther)-5 b(e)32 b(exist)f(c)-5 b(onstants)34 b Fq(\014)c(>)25 b Fs(0)32 b Fl(and)g Fq(\013)26 b Fo(2)f Fs(\(0)p Fq(;)15 b Fs(1\))33 b Fl(indep)-5 b(endent)33 b(of)e Fq(")h Fl(such)f(that)i Fo(j)26 b Fq(\013)p Fs(\()p Fq(")p Fs(\))g Fo(j\024)g Fq(\013)31 b Fl(and)i Fo(j)25 b Fq(\014)5 b Fs(\()p Fq(")p Fs(\))27 b Fo(j\024)185 712 y Fq(\014)5 b Fl(;)p 0 TeXcolorgray 98 878 a Fo(\017)p 0 TeXcolorgray 42 w Fq(m)p Fs(\()p Fq(")p Fs(\))26 b(=)f(sup)636 900 y Fp(z)s Fi(2)p Fp(S)762 909 y Ff(1)825 878 y Fo(k)h Fq(M)984 892 y Fn(1)1024 878 y Fs(\()p Fq(z)t(;)15 b(")p Fs(\))27 b Fo(k!)e Fs(0)33 b Fl(as)g Fq(")26 b Fo(!)f Fs(0)p Fl(.)-118 1066 y(Then,)34 b(ther)-5 b(e)34 b(exists)g Fq(C)f(>)27 b Fs(0)33 b Fl(and)i Fq(")1119 1080 y Fn(1)1185 1066 y Fq(>)27 b Fs(0)34 b Fl(\(b)-5 b(oth)35 b(dep)-5 b(ending)34 b(only)h(on)f Fq(y)2391 1080 y Fn(0)2430 1066 y Fq(;)48 b(y)2548 1080 y Fn(1)2587 1066 y Fq(;)h(\013;)g(\014)39 b Fl(and)34 b Fq(")27 b Fo(7!)g Fq(m)p Fs(\()p Fq(")p Fs(\)\))35 b Fl(such)e(that,)-118 1232 y(if)f Fs(0)26 b Fq(<)f(")g(<)g(")342 1246 y Fn(1)382 1232 y Fl(,)33 b(one)g(has)-118 1442 y Fs(\(3.20\))1224 1442 y SDict begin H.S end 1224 1442 a 1224 1442 a SDict begin 20 H.A end 1224 1442 a 1224 1442 a SDict begin [ /View [/XYZ H.V] /Dest (equation.3.20) cvn H.B /DEST pdfmark end 1224 1442 a Fq(\022)s Fs(\()p Fq(M)10 b Fs(\()p Fo(\001)p Fq(;)15 b(")p Fs(\))p Fq(;)g(h)p Fs(\))28 b Fq(>)d Fs(log)17 b Fo(j)p Fq(\025)p Fs(\()p Fq(")p Fs(\))p Fo(j)22 b(\000)e Fq(C)7 b(m)p Fs(\()p Fq(")p Fs(\))94 1697 y(Prop)s(osition)p 0 0 1 TeXcolorrgb 595 1697 a SDict begin H.S end 595 1697 a 0 0 1 TeXcolorrgb Fs(3.6)p 0 0 1 TeXcolorrgb 711 1638 a SDict begin H.R end 711 1638 a 711 1697 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.3.6) cvn H.B /ANN pdfmark end 711 1697 a 0 TeXcolorgray 42 w Fs(is)42 b(used)f(b)m(y)g(applying)h(the)g (argumen)m(ts)g(of)g([)p 0 1 0 TeXcolorrgb 2357 1697 a SDict begin H.S end 2357 1697 a 0 1 0 TeXcolorrgb Fs(9)p 0 1 0 TeXcolorrgb 2402 1638 a SDict begin H.R end 2402 1638 a 2402 1697 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.FEKL1) cvn H.B /ANN pdfmark end 2402 1697 a 0 TeXcolorgray Fs(,)p 0 1 0 TeXcolorrgb 2469 1697 a SDict begin H.S end 2469 1697 a 0 1 0 TeXcolorrgb Fs(12)p 0 1 0 TeXcolorrgb 2560 1638 a SDict begin H.R end 2560 1638 a 2560 1697 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.FEKL2) cvn H.B /ANN pdfmark end 2560 1697 a 0 TeXcolorgray Fs(])g(to)h(get)f(the)g (lo)m(w)m(er)h(b)s(ound)d(for)h(the)-118 1863 y(Ly)m(apuno)m(v)30 b(exp)s(onen)m(t.)-118 2029 y(First)c(for)g Fq(\033)i Fs(=)410 1901 y Fh(\022)518 1974 y Fs(0)84 b(1)518 2082 y(1)g(0)734 1901 y Fh(\023)827 2029 y Fs(w)m(e)26 b(pro)m(v)m(e)h(that) f(the)g(matrix)g Fq(\033)s(M)10 b Fs(\()p Fq(z)t(;)15 b(E)5 b Fs(\))p Fq(\033)31 b Fs(completes)c(the)f(assumption)g(of)g (Prop)s(osition)p 0 0 1 TeXcolorrgb -118 2195 a SDict begin H.S end -118 2195 a 0 0 1 TeXcolorrgb Fs(3.6)p 0 0 1 TeXcolorrgb -2 2136 a SDict begin H.R end -2 2136 a -2 2195 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.3.6) cvn H.B /ANN pdfmark end -2 2195 a 0 TeXcolorgray Fs(.)-118 2361 y(Let)40 b Fq(y)99 2375 y Fn(0)178 2361 y Fs(and)e Fq(y)408 2375 y Fn(1)487 2361 y Fs(\014xed)h(suc)m(h)g(that)h(0)h Fq(<)g(y)1379 2375 y Fn(0)1458 2361 y Fq(<)g(y)1615 2375 y Fn(1)1694 2361 y Fq(<)g(Y)20 b Fs(.)68 b(The)39 b(asymptotics)i(of)e (the)h(mono)s(drom)m(y)f(matrix)h(are)-118 2527 y(uniform)29 b(for)h Fq(z)35 b Fs(in)30 b Fq(S)g Fs(=)25 b Fo(f)p Fq(z)30 b Fo(2)25 b Fm(C)p Fs(;)45 b Fq(y)1111 2541 y Fn(0)1150 2527 y Fq(=")26 b Fo(\024)f Fs(Im)p Fq(z)k Fo(\024)c Fq(y)1680 2541 y Fn(1)1719 2527 y Fq(=")p Fo(g)32 b Fs(and)e Fq(E)g Fo(2)25 b Fq(V)2296 2541 y Fn(0)2336 2527 y Fs(.)-118 2693 y(Let)31 b(us)e(assume)i(that)g Fq(n)24 b Fs(=)h Fq(N)921 2707 y Fp(N)1019 2693 y Fs(in)30 b(\()p 0 0 1 TeXcolorrgb 1160 2693 a SDict begin H.S end 1160 2693 a 0 0 1 TeXcolorrgb Fs(5.12)p 0 0 1 TeXcolorrgb 1322 2634 a SDict begin H.R end 1322 2634 a 1322 2693 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.5.12) cvn H.B /ANN pdfmark end 1322 2693 a 0 TeXcolorgray Fs(\))h(is)f(ev)m(en)h (then)f(the)h(follo)m(wing)h(relations)f(holds)-118 2903 y(\(3.21\))1032 2903 y SDict begin H.S end 1032 2903 a 1032 2903 a SDict begin 20 H.A end 1032 2903 a 1032 2903 a SDict begin [ /View [/XYZ H.V] /Dest (equation.3.21) cvn H.B /DEST pdfmark end 1032 2903 a Fq(Q)1104 2917 y Fi(\000)1188 2903 y Fs(=)25 b Fq(Q)1356 2917 y Fn(+)1435 2903 y Fs(+)20 b(1)p Fq(;)46 b(Q)1714 2917 y Fi(\000)1799 2903 y Fs(=)25 b Fq(P)1953 2917 y Fi(\000)2032 2903 y Fs(+)20 b(1)31 b(and)f(Q)2447 2917 y Fi(\000)2531 2903 y Fs(=)25 b(P)2689 2917 y Fn(+)-118 3113 y Fs(F)-8 b(or)31 b Fq(E)g Fo(2)24 b Fq(V)281 3127 y Fn(0)341 3113 y Fo(\\)c Fm(R)30 b Fs(and)g Fq(z)f Fo(2)c Fq(S)5 b Fs(,)31 b(Theorem)p 0 0 1 TeXcolorrgb 1353 3113 a SDict begin H.S end 1353 3113 a 0 0 1 TeXcolorrgb Fs(3.4)p 0 0 1 TeXcolorrgb 1469 3055 a SDict begin H.R end 1469 3055 a 1469 3113 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.3.4) cvn H.B /ANN pdfmark end 1469 3113 a 0 TeXcolorgray 30 w Fs(implies)g(that)p 774 3244 298 4 v 774 3324 a Fq(A)p Fs(\()p 877 3274 47 4 v Fq(z)5 b(;)15 b(E)5 b Fs(\))26 b(=)p 1193 3244 354 4 v 25 w Fq(A)1261 3338 y Fi(\000)1320 3324 y Fs(\()p Fq(E)5 b(;)15 b(")p Fs(\))r Fq(e)1588 3286 y Fi(\000)p Fn(2)p Fp(\031)r(iQ)1801 3295 y Fd(\000)1853 3286 y Fp(z)1893 3324 y Fs(\(1)21 b(+)f Fq(o)p Fs(\(1\)\))p Fq(;)2391 3262 y(A)p Fs(\()p Fq(z)t(;)15 b(E)5 b Fs(\))p 2391 3303 298 4 v 2391 3321 V 2391 3400 a Fq(A)p Fs(\()p 2494 3350 47 4 v Fq(z)g(;)15 b(E)5 b Fs(\))2724 3324 y(=)25 b Fq(o)p Fs(\(1\))p Fq(;)p 1027 3489 304 4 v 1027 3568 a(B)5 b Fs(\()p 1136 3518 47 4 v Fq(z)t(;)15 b(E)5 b Fs(\))p 1027 3608 304 4 v 1030 3627 298 4 v 1030 3706 a Fq(A)p Fs(\()p 1133 3656 47 4 v Fq(z)g(;)15 b(E)5 b Fs(\))1366 3629 y(=)25 b Fq(o)p Fs(\(1\))p Fq(;)1733 3568 y(B)5 b Fs(\()p Fq(z)t(;)15 b(E)5 b Fs(\))p 1733 3608 304 4 v 1736 3627 298 4 v 1736 3706 a Fq(A)p Fs(\()p 1839 3656 47 4 v Fq(z)t(;)15 b(E)5 b Fs(\))2071 3629 y(=)25 b Fq(c)p Fs(\()p Fq(E)5 b Fs(\)\(1)23 b(+)d Fq(o)p Fs(\(1\)\))p Fq(;)-118 3830 y Fs(Where)31 b Fq(c)p Fs(\()p Fq(E)5 b Fs(\))31 b(is)g(indep)s(enden)m(t)e(of)h Fq(z)35 b Fs(and)30 b(b)s(ounded)e(b)m(y)i(a)h(constan)m(t)h(uniformly) d(in)h Fq(")h Fs(and)f Fq(E)5 b Fs(.)41 b(So,)30 b(w)m(e)h(ha)m(v)m(e) 688 4040 y Fq(\033)23 b Fo(\001)d Fq(M)10 b Fs(\()p Fq(z)t(;)15 b(E)5 b Fs(\))22 b Fo(\001)f Fq(\033)28 b Fs(=)p 1378 3961 354 4 v 25 w Fq(A)1446 4054 y Fi(\000)1506 4040 y Fs(\()p Fq(E)5 b(;)15 b(")p Fs(\))q Fq(e)1773 4003 y Fi(\000)p Fn(2)p Fp(\031)r(iQ)1986 4012 y Fd(\000)2038 4003 y Fp(z)2098 4040 y Fo(\001)2144 3939 y Fh(\020)2213 3912 y(\022)2390 3985 y Fs(1)152 b(0)2322 4093 y Fq(c)p Fs(\()p Fq(E)5 b Fs(\))84 b(0)2674 3912 y Fh(\023)2761 4040 y Fs(+)20 b Fq(o)p Fs(\(1\))3011 3939 y Fh(\021)3067 4040 y Fq(:)-118 4250 y Fs(This)29 b(giv)m(es)j(that)f(the)g(matrix-v) -5 b(alued)31 b(function)1499 4461 y Fq(z)e Fo(7!)c Fq(\033)f Fo(\001)c Fq(M)10 b Fs(\()p Fq(z)t(;)15 b(E)5 b Fs(\))22 b Fo(\001)f Fq(\033)s Fs(;)-118 4671 y(satis\014es)31 b(the)f(assumptions)g(of)h(Prop)s(osition)p 0 0 1 TeXcolorrgb 1483 4671 a SDict begin H.S end 1483 4671 a 0 0 1 TeXcolorrgb Fs(3.6)p 0 0 1 TeXcolorrgb 1599 4612 a SDict begin H.R end 1599 4612 a 1599 4671 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.3.6) cvn H.B /ANN pdfmark end 1599 4671 a 0 TeXcolorgray Fs(.)-118 4689 y SDict begin H.S end -118 4689 a -118 4689 a SDict begin 20 H.A end -118 4689 a -118 4689 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.3.7) cvn H.B /DEST pdfmark end -118 4689 a 0 TeXcolorgray 237 x Fv(Remark)k(3.7.)p 0 TeXcolorgray 42 w Fl(When)e Fq(n)25 b Fs(=)g Fq(N)995 4940 y Fp(N)1095 4926 y Fl(in)32 b(\()p 0 0 1 TeXcolorrgb 1243 4926 a SDict begin H.S end 1243 4926 a 0 0 1 TeXcolorrgb Fl(5.12)p 0 0 1 TeXcolorrgb 1411 4867 a SDict begin H.R end 1411 4867 a 1411 4926 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.5.12) cvn H.B /ANN pdfmark end 1411 4926 a 0 TeXcolorgray Fl(\))g(is)h(o)-5 b(dd)34 b(then)f(we)g(get)-118 5136 y Fs(\(3.22\))1010 5136 y SDict begin H.S end 1010 5136 a 1010 5136 a SDict begin 20 H.A end 1010 5136 a 1010 5136 a SDict begin [ /View [/XYZ H.V] /Dest (equation.3.22) cvn H.B /DEST pdfmark end 1010 5136 a Fq(Q)1082 5150 y Fn(+)1166 5136 y Fs(=)25 b Fq(Q)1334 5150 y Fi(\000)1413 5136 y Fs(+)20 b(1)p Fq(;)48 b(P)1680 5150 y Fi(\000)1765 5136 y Fs(=)25 b Fq(Q)1933 5150 y Fn(+)1992 5136 y Fq(;)80 b Fs(and)p Fq(P)2302 5150 y Fn(+)2381 5136 y Fs(+)20 b(1)26 b(=)f Fq(Q)2711 5150 y Fn(+)-118 5346 y Fl(F)-7 b(or)34 b Fq(E)c Fo(2)25 b Fq(V)288 5360 y Fn(0)348 5346 y Fo(\\)20 b Fm(R)32 b Fl(and)h Fq(z)d Fo(2)25 b Fq(S)5 b Fl(,)32 b(The)-5 b(or)g(em)p 0 0 1 TeXcolorrgb 1360 5364 a SDict begin H.S end 1360 5364 a 0 0 1 TeXcolorrgb -18 x Fl(3.4)p 0 0 1 TeXcolorrgb 1480 5287 a SDict begin H.R end 1480 5287 a 1480 5346 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.3.4) cvn H.B /ANN pdfmark end 1480 5346 a 0 TeXcolorgray 33 w Fl(implies)33 b(that)772 5564 y Fq(A)p Fs(\()p Fq(z)t(;)15 b(E)5 b Fs(\))27 b(=)e Fq(A)1259 5578 y Fn(+)1319 5564 y Fs(\()p Fq(E)5 b(;)15 b(")p Fs(\))p Fq(e)1585 5526 y Fi(\000)p Fn(2)p Fp(\031)r(iQ)1798 5535 y Ff(+)1850 5526 y Fp(z)1890 5564 y Fs(\(1)21 b(+)f Fq(o)p Fs(\(1\)\))p Fq(;)p 2393 5423 298 4 v 2393 5502 a(A)p Fs(\()p 2496 5452 47 4 v Fq(z)5 b(;)15 b(E)5 b Fs(\))p 2393 5543 298 4 v 2393 5626 a Fq(A)p Fs(\()p Fq(z)t(;)15 b(E)5 b Fs(\))2726 5564 y(=)25 b Fq(o)p Fs(\(1\))p Fq(;)p 0 TeXcolorgray 1854 5715 a Fn(12)p 0 TeXcolorgray eop end %%Page: 13 13 TeXDict begin 13 12 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray -118 -406 a SDict begin H.S end -118 -406 a 0 TeXcolorgray 0 TeXcolorgray -118 -406 a SDict begin H.R end -118 -406 a -118 -406 a SDict begin [ /View [/XYZ H.V] /Dest (page.13) cvn H.B /DEST pdfmark end -118 -406 a 0 TeXcolorgray 0 TeXcolorgray 1025 -283 304 4 v 1025 -203 a Fq(B)5 b Fs(\()p 1134 -253 47 4 v Fq(z)t(;)15 b(E)5 b Fs(\))p 1025 -163 304 4 v 1028 -80 a Fq(A)p Fs(\()p 1131 -130 47 4 v Fq(z)t(;)15 b(E)5 b Fs(\))1363 -142 y(=)25 b Fq(c)p Fs(\()p Fq(E)5 b Fs(\)\(1)23 b(+)d Fq(o)p Fs(\(1\)\))p Fq(;)2145 -203 y(B)5 b Fs(\()p Fq(z)t(;)15 b(E)5 b Fs(\))p 2145 -163 304 4 v 2148 -80 a Fq(A)p Fs(\()p Fq(z)t(;)15 b(E)5 b Fs(\))2483 -142 y(=)25 b Fq(o)p Fs(\(1\))p Fq(;)-118 43 y Fl(Wher)-5 b(e)33 b Fq(c)p Fs(\()p Fq(E)5 b Fs(\))34 b Fl(is)f(indep)-5 b(endent)34 b(of)f Fq(z)j Fl(and)e(b)-5 b(ounde)g(d)34 b(by)f(a)g(c)-5 b(onstant)34 b(uniformly)g(in)e Fq(")h Fl(and)h Fq(E)5 b Fl(.)42 b(So,)33 b(we)f(have)688 247 y Fq(\033)24 b Fo(\001)c Fq(M)10 b Fs(\()p Fq(z)t(;)15 b(E)5 b Fs(\))22 b Fo(\001)f Fq(\033)28 b Fs(=)d Fq(A)1447 261 y Fn(+)1506 247 y Fs(\()p Fq(E)5 b(;)15 b(")p Fs(\))p Fq(e)1772 209 y Fi(\000)p Fn(2)p Fp(\031)r(iQ)1985 218 y Ff(+)2038 209 y Fp(z)2098 247 y Fo(\001)2143 146 y Fh(\020)2213 119 y(\022)2321 192 y Fs(0)84 b Fq(c)p Fs(\()p Fq(E)5 b Fs(\))2321 300 y(0)152 b(1)2673 119 y Fh(\023)2761 247 y Fs(+)20 b Fq(o)p Fs(\(1\))3011 146 y Fh(\021)3066 247 y Fq(:)-118 451 y Fl(The)28 b(moste)i(imp)-5 b(ortant)31 b(pr)-5 b(op)g(erties)31 b(of)d(the)h(matrix)h Fq(M)10 b Fs(\()p Fq(z)t(;)15 b(E)5 b Fs(\))30 b Fl(in)e(Pr)-5 b(op)g(osition)p 0 0 1 TeXcolorrgb 2630 451 a SDict begin H.S end 2630 451 a 0 0 1 TeXcolorrgb Fl(3.6)p 0 0 1 TeXcolorrgb 2751 392 a SDict begin H.R end 2751 392 a 2751 451 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.3.6) cvn H.B /ANN pdfmark end 2751 451 a 0 TeXcolorgray Fl(,)29 b(is)f(that)i(the)e(bigger)g(eigenvalue)-118 617 y(is)k Fs(1)h([)p 0 1 0 TeXcolorrgb 82 617 a SDict begin H.S end 82 617 a 0 1 0 TeXcolorrgb Fs(10)p 0 1 0 TeXcolorrgb 174 558 a SDict begin H.R end 174 558 a 174 617 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.huit) cvn H.B /ANN pdfmark end 174 617 a 0 TeXcolorgray Fs(])p Fl(.)94 866 y Fs(Using)40 b(\()p 0 0 1 TeXcolorrgb 394 866 a SDict begin H.S end 394 866 a 0 0 1 TeXcolorrgb Fs(3.20)p 0 0 1 TeXcolorrgb 556 807 a SDict begin H.R end 556 807 a 556 866 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.3.20) cvn H.B /ANN pdfmark end 556 866 a 0 TeXcolorgray Fs(\),)i(w)m(e)d (get)i(that)e(the)h(Ly)m(apuno)m(v)f(exp)s(onen)m(t)g Fq(\022)s Fs(\()p Fq(E)5 b(;)15 b(")p Fs(\))40 b(of)g(the)f(matrix)g (co)s(cycle)i(asso)s(ciated)g(to)-118 1032 y(\()p Fq(M)10 b Fs(\()p Fo(\001)p Fq(;)15 b(E)5 b Fs(\))p Fq(;)15 b(h)p Fs(\),)34 b(satis\014es)d(the)f(estimates)-118 1235 y(\(3.23\))1390 1235 y SDict begin H.S end 1390 1235 a 1390 1235 a SDict begin 20 H.A end 1390 1235 a 1390 1235 a SDict begin [ /View [/XYZ H.V] /Dest (equation.3.23) cvn H.B /DEST pdfmark end 1390 1235 a Fq(\022)s Fs(\()p Fq(E)5 b(;)15 b(")p Fs(\))26 b Fo(\025)f Fs(log)18 b Fo(j)p Fq(A)2008 1249 y Fi(\000)2067 1235 y Fo(j)j Fs(+)f Fq(o)p Fs(\(1\))p Fq(:)-118 1439 y Fs(T)-8 b(aking)31 b(in)m(to)g(accoun)m(t)h(\()p 0 0 1 TeXcolorrgb 740 1439 a SDict begin H.S end 740 1439 a 0 0 1 TeXcolorrgb Fs(1.2)p 0 0 1 TeXcolorrgb 857 1381 a SDict begin H.R end 857 1381 a 857 1439 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.1.2) cvn H.B /ANN pdfmark end 857 1439 a 0 TeXcolorgray Fs(\))e(w)m(e)h(get)h(that)1311 1643 y(\002\()p Fq(E)5 b(;)15 b(")p Fs(\))27 b Fo(\025)1768 1582 y Fq(")p 1739 1622 101 4 v 1739 1705 a Fs(2)p Fq(\031)1865 1643 y Fs(log)17 b Fo(j)p Fq(A)2090 1657 y Fi(\000)2150 1643 y Fo(j)j Fs(+)g Fq(o)p Fs(\()p Fq(")p Fs(\))p Fq(:)-118 1847 y Fs(By)31 b(\()p 0 0 1 TeXcolorrgb 60 1847 a SDict begin H.S end 60 1847 a 0 0 1 TeXcolorrgb Fs(3.14)p 0 0 1 TeXcolorrgb 222 1788 a SDict begin H.R end 222 1788 a 222 1847 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.3.14) cvn H.B /ANN pdfmark end 222 1847 a 0 TeXcolorgray Fs(\))f(w)m(e)h(get)-118 2051 y(\(3.24\))1219 2051 y SDict begin H.S end 1219 2051 a 1219 2051 a SDict begin 20 H.A end 1219 2051 a 1219 2051 a SDict begin [ /View [/XYZ H.V] /Dest (equation.3.24) cvn H.B /DEST pdfmark end 1219 2051 a Fs(\002\()p Fq(E)5 b(;)15 b(")p Fs(\))27 b Fo(\025)1676 1989 y Fq(")p 1647 2030 101 4 v 1647 2113 a Fs(2)p Fq(\031)1773 2051 y Fs(log)r(\()p Fq(T)13 b Fs(\()p Fq(E)5 b Fs(\))2133 2013 y Fi(\000)p Fn(1)2229 2051 y Fs(\))20 b(+)g Fq(O)s Fs(\()p Fq(")p Fs(\))-118 2167 y SDict begin H.S end -118 2167 a -118 2167 a SDict begin 20 H.A end -118 2167 a -118 2167 a SDict begin [ /View [/XYZ H.V] /Dest (subsubsection.3.4.3) cvn H.B /DEST pdfmark end -118 2167 a 104 x Fs(3.4.3.)47 b Fl(Conclusion.)g Fs(No)m(w)31 b(w)m(e)g(obtain)g(\()p 0 0 1 TeXcolorrgb 1311 2271 a SDict begin H.S end 1311 2271 a 0 0 1 TeXcolorrgb Fs(3.1)p 0 0 1 TeXcolorrgb 1427 2212 a SDict begin H.R end 1427 2212 a 1427 2271 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.3.1) cvn H.B /ANN pdfmark end 1427 2271 a 0 TeXcolorgray Fs(\),)g(b)m(y)f(comparing)h(\()p 0 0 1 TeXcolorrgb 2124 2271 a SDict begin H.S end 2124 2271 a 0 0 1 TeXcolorrgb Fs(3.18)p 0 0 1 TeXcolorrgb 2286 2212 a SDict begin H.R end 2286 2212 a 2286 2271 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.3.18) cvn H.B /ANN pdfmark end 2286 2271 a 0 TeXcolorgray Fs(\))g(and)e(\()p 0 0 1 TeXcolorrgb 2563 2271 a SDict begin H.S end 2563 2271 a 0 0 1 TeXcolorrgb Fs(3.24)p 0 0 1 TeXcolorrgb 2725 2212 a SDict begin H.R end 2725 2212 a 2725 2271 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.3.24) cvn H.B /ANN pdfmark end 2725 2271 a 0 TeXcolorgray Fs(\).)41 b(Indeed)30 b(w)m(e)h(see)g(that)-118 2475 y(\(3.25\))1219 2475 y SDict begin H.S end 1219 2475 a 1219 2475 a SDict begin 20 H.A end 1219 2475 a 1219 2475 a SDict begin [ /View [/XYZ H.V] /Dest (equation.3.25) cvn H.B /DEST pdfmark end 1219 2475 a Fs(\002\()p Fq(E)5 b(;)15 b(")p Fs(\))27 b(=)1676 2413 y Fq(")p 1647 2454 101 4 v 1647 2537 a Fs(2)p Fq(\031)1773 2475 y Fs(log)r(\()p Fq(T)13 b Fs(\()p Fq(E)5 b Fs(\))2133 2437 y Fi(\000)p Fn(1)2228 2475 y Fs(\))21 b(+)f Fq(o)p Fs(\(1\))p Fq(:)-118 2678 y Fs(The)30 b(expression)g(of)g Fq(T)13 b Fs(\()p Fq(E)5 b Fs(\))32 b(giv)m(en)f(b)m(y)f(\()p 0 0 1 TeXcolorrgb 1248 2678 a SDict begin H.S end 1248 2678 a 0 0 1 TeXcolorrgb Fs(3.25)p 0 0 1 TeXcolorrgb 1410 2620 a SDict begin H.R end 1410 2620 a 1410 2678 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.3.25) cvn H.B /ANN pdfmark end 1410 2678 a 0 TeXcolorgray Fs(\))h(and)f(\()p 0 0 1 TeXcolorrgb 1688 2678 a SDict begin H.S end 1688 2678 a 0 0 1 TeXcolorrgb Fs(3.24)p 0 0 1 TeXcolorrgb 1850 2620 a SDict begin H.R end 1850 2620 a 1850 2678 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.3.24) cvn H.B /ANN pdfmark end 1850 2678 a 0 TeXcolorgray Fs(\))g(giv)m(e)i(\()p 0 0 1 TeXcolorrgb 2137 2678 a SDict begin H.S end 2137 2678 a 0 0 1 TeXcolorrgb Fs(3.1)p 0 0 1 TeXcolorrgb 2254 2620 a SDict begin H.R end 2254 2620 a 2254 2678 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.3.1) cvn H.B /ANN pdfmark end 2254 2678 a 0 TeXcolorgray Fs(\))e(for)g(an)m(y)h Fq(E)g Fo(2)25 b Fq(V)2867 2692 y Fn(0)2926 2678 y Fo(\\)20 b Fm(R)p Fs(.)-118 2844 y(Recall)33 b(that)e Fq(V)407 2858 y Fn(0)467 2844 y Fo(\\)21 b Fm(R)30 b Fs(is)h(an)g(op)s(en)g(in)m (terv)-5 b(al)32 b(con)m(taining)g Fq(E)1923 2858 y Fn(0)1989 2844 y Fo(2)26 b Fq(J)9 b Fs(.)43 b(The)31 b(ab)s(o)m(v)m(e)h (construction)f(can)h(b)s(e)e(carried)h(out)-118 3010 y(for)f(an)m(y)h Fq(E)260 3024 y Fn(0)325 3010 y Fo(2)24 b Fq(J)9 b Fs(.)41 b(The)30 b(end)g(of)g(the)h(pro)s(of)f(of)g(Theorem) p 0 0 1 TeXcolorrgb 1879 3010 a SDict begin H.S end 1879 3010 a 0 0 1 TeXcolorrgb Fs(1.2)p 0 0 1 TeXcolorrgb 1995 2952 a SDict begin H.R end 1995 2952 a 1995 3010 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.1.2) cvn H.B /ANN pdfmark end 1995 3010 a 0 TeXcolorgray 31 w Fs(follo)m(ws)h(from)f(the)h (compactness)g(of)f(the)h(in)m(terv)-5 b(al)31 b Fq(J)9 b Fs(.)-118 3104 y SDict begin H.S end -118 3104 a -118 3104 a SDict begin 20 H.A end -118 3104 a -118 3104 a SDict begin [ /View [/XYZ H.V] /Dest (section.4) cvn H.B /DEST pdfmark end -118 3104 a 647 3252 a Fs(4.)46 b Fr(The)34 b(complex)f(WKB)g(method)g(f)n(or)h(adiaba)-6 b(tic)33 b(pr)n(oblems)94 3472 y Fs(In)i(this)h(section,)i(follo)m(wing)f([)p 0 1 0 TeXcolorrgb 1145 3472 a SDict begin H.S end 1145 3472 a 0 1 0 TeXcolorrgb Fs(12)p 0 1 0 TeXcolorrgb 1236 3414 a SDict begin H.R end 1236 3414 a 1236 3472 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.FEKL2) cvn H.B /ANN pdfmark end 1236 3472 a 0 TeXcolorgray Fs(,)p 0 1 0 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3780 y Fq(d)937 3747 y Fn(2)p 864 3821 139 4 v 864 3904 a Fq(dx)963 3878 y Fn(2)1013 3842 y Fq( )s Fs(\()p Fq(x)p Fs(\))21 b(+)f([)p Fq(V)g Fs(\()p Fq(x)p Fs(\))h(+)f Fq(W)13 b Fs(\()p Fq("x)21 b Fs(+)e Fq(\020)7 b Fs(\)])p Fq( )s Fs(\()p Fq(x)p Fs(\))27 b(=)e Fq(E)5 b( )s Fs(\()p Fq(x)p Fs(\))p Fq(;)108 b(x)25 b Fo(2)g Fm(R)p Fq(:)-118 4046 y Fs(Here,)34 b Fq(V)53 b Fs(is)33 b(1-p)s(erio)s(dic)g(and)g(real)g(v)-5 b(alued,)34 b Fq(")g Fs(is)f(a)g(small)g(p)s(ositiv)m(e)h(parameter,)h(and)d(the)h (energy)g Fq(E)39 b Fs(is)33 b(complex;)-118 4212 y(one)e(assumes)f (that)h Fq(V)50 b Fs(is)30 b Fq(L)852 4179 y Fn(2)852 4239 y(lo)r(c)975 4212 y Fs(and)g(that)h Fq(W)43 b Fs(is)30 b(analytic)i(in)e(a)h(strip)f(in)g(the)g(neigh)m(b)s(orho)s(o)s(d)f Fq(S)3199 4226 y Fp(Y)3290 4212 y Fs(of)i(the)f(real)h(line.)-118 4378 y(The)26 b(parameter)i Fq(\020)33 b Fs(is)27 b(an)g(auxiliary)g (complex)h(parameter)f(used)f(to)i(decouple)f(the)g(slo)m(w)h(v)-5 b(ariable)27 b Fq(\020)32 b Fs(=)25 b Fq("x)i 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Fs(compute)30 b(the)g(mono)s(drom)m(y)f(matrix;)h(w)m(e)g(consider)g Fq(V)50 b Fs(and)29 b Fq(W)42 b Fs(as)30 b(\014xed)f(and)g(construct)h (the)g(WKB)g(ob)5 b(jects)30 b(and)-118 -41 y(solutions)h(in)f(an)g (uniform)f(w)m(a)m(y)j(for)e(energies)h(in)f(a)h(neigh)m(b)s(orho)s(o)s (d)e(of)h Fq(E)5 b Fs(.)-118 83 y SDict begin H.S end -118 83 a -118 83 a SDict begin 20 H.A end -118 83 a -118 83 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.1) cvn H.B /DEST pdfmark end -118 83 a 149 x Fs(4.1.)47 b Fv(Standard)e(b)s(eha) m(vior)g(of)h(consisten)m(t)g(solutions.)g Fs(W)-8 b(e)41 b(start)e(b)m(y)h(de\014ning)e(another)i(analytic)h(ob)5 b(ject)-118 398 y(cen)m(tral)32 b(to)g(the)g(complex)g(WKB)f(metho)s (d,)g(the)h Fl(c)-5 b(anonic)g(al)35 b(Blo)-5 b(ch)35 b(solutions)p Fs(.)44 b(Then,)31 b(w)m(e)h(describ)s(e)e(the)i Fl(standar)-5 b(d)-118 564 y(b)g(ehavior)32 b Fs(of)e(the)h(solutions.) -118 670 y SDict begin H.S end -118 670 a -118 670 a SDict begin 20 H.A end -118 670 a -118 670 a SDict begin [ /View [/XYZ H.V] /Dest (subsubsection.4.1.1) cvn H.B /DEST pdfmark end -118 670 a 166 x Fs(4.1.1.)47 b Fl(Canonic)-5 b(al)33 b(Blo)-5 b(ch)31 b(solutions.)47 b Fs(T)-8 b(o)29 b(describ)s(e)e(the)h (asymptotic)i(form)m(ulae)e(of)g(the)h(complex)f(WKB)h(metho)s(d,)-118 1002 y(one)i(needs)f(to)h(construct)f(Blo)s(c)m(h)i(solutions)f(to)g (the)f(equation)-118 1232 y(\(4.3\))571 1232 y SDict begin H.S end 571 1232 a 571 1232 a SDict begin 20 H.A end 571 1232 a 571 1232 a SDict begin [ /View [/XYZ H.V] /Dest (equation.4.3) cvn H.B /DEST pdfmark end 571 1232 a 20 w Fo(\000)718 1171 y Fq(d)765 1138 y Fn(2)p 692 1211 139 4 v 692 1295 a Fq(dx)791 1268 y Fn(2)841 1232 y Fq( )s Fs(\()p Fq(x)p Fs(\))21 b(+)f Fq(V)h Fs(\()p Fq(x)p Fs(\))p Fq( )s Fs(\()p Fq(x)p Fs(\))27 b(=)e Fo(E)8 b Fs(\()p Fq(\020)f Fs(\))p Fq( )s Fs(\()p Fq(x)p Fs(\))p Fq(;)107 b Fo(E)8 b Fs(\()p Fq(\020)f Fs(\))26 b(=)f Fq(E)g Fo(\000)20 b Fq(W)13 b Fs(\()p Fq(\020)7 b Fs(\))p Fq(;)106 b(x)25 b Fo(2)g Fm(R)p Fs(;)-118 1462 y(that)31 b(are)g(moreo)m(v)m(er)h(analytic)g(in) e Fq(\020)36 b Fs(on)31 b(a)f(giv)m(en)i(regular)e(domain.)-118 1628 y(Let)d Fq(\020)81 1642 y Fn(0)147 1628 y Fs(b)s(e)g(a)g(regular)g (p)s(oin)m(t)g(\(i.e.)41 b Fq(\020)1110 1642 y Fn(0)1176 1628 y Fs(is)27 b(not)g(a)h(branc)m(h)e(p)s(oin)m(t)h(of)g Fq(\024)p Fs(\).)40 b(Let)28 b Fo(E)2486 1642 y Fn(0)2550 1628 y Fs(=)d Fo(E)8 b Fs(\()p Fq(\020)2777 1642 y Fn(0)2817 1628 y Fs(\).)40 b(Assume)26 b(that)i Fo(E)3492 1642 y Fn(0)3566 1628 y Fq(=)-55 b Fo(2)25 b Fq(P)i Fo([)13 b Fq(Q)p Fs(.)-118 1794 y(Let)43 b Fq(U)119 1808 y Fn(0)201 1794 y Fs(b)s(e)f(a)h(su\016cien)m(tly)h(small)f(neigh)m(b)s(orho)s(o)s (d)e(of)i Fo(E)1890 1808 y Fn(0)1930 1794 y Fs(,)j(and)c(let)h Fq(V)2386 1808 y Fn(0)2468 1794 y Fs(b)s(e)f(a)i(neigh)m(b)s(orho)s(o)s (d)d(of)i Fq(\020)3432 1808 y Fn(0)3514 1794 y Fs(suc)m(h)f(that)-118 1960 y Fo(E)8 b Fs(\()p Fq(V)26 1974 y Fn(0)66 1960 y Fs(\))30 b Fo(\032)g Fq(U)294 1974 y Fn(0)333 1960 y Fs(.)49 b(In)33 b Fq(U)586 1974 y Fn(0)625 1960 y Fs(,)h(w)m(e)g(\014x) f(a)g(branc)m(h)g(of)g(the)h(function)1959 1883 y Fh(p)p 2050 1883 201 4 v 77 x Fq(k)2100 1934 y Fi(0)2124 1960 y Fs(\()p Fo(E)8 b Fs(\))34 b(and)e(consider)h Fq( )2879 1974 y Fi(\006)2939 1960 y Fs(\()p Fq(x;)15 b Fo(E)8 b Fs(\),)35 b(the)e(t)m(w)m(o)i(branc)m(hes)-118 2127 y(of)e(the)h(Blo)s(c)m(h)g(solution)g Fq( )s Fs(\()p Fq(x;)15 b Fo(E)8 b Fs(\))35 b(and)d(\012)1314 2141 y Fi(\006)1373 2127 y Fs(,)i(the)g(corresp)s(onding)e(branc)m(hes)h(of)g (\012)g(\(see)i(section)p 0 0 1 TeXcolorrgb 3256 2127 a SDict begin H.S end 3256 2127 a 0 0 1 TeXcolorrgb Fs(2.3)p 0 0 1 TeXcolorrgb 3372 2068 a SDict begin H.R end 3372 2068 a 3372 2127 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (subsection.2.3) cvn H.B /ANN pdfmark end 3372 2127 a 0 TeXcolorgray Fs(.)50 b(F)-8 b(or)34 b Fq(\020)i Fo(2)30 b Fq(V)3833 2141 y Fn(0)3872 2127 y Fs(,)-118 2293 y(w)m(e)h(set)-118 2523 y(\(4.4\))661 2523 y SDict begin H.S end 661 2523 a 661 2523 a SDict begin 20 H.A end 661 2523 a 661 2523 a SDict begin [ /View [/XYZ H.V] /Dest (equation.4.4) cvn H.B /DEST pdfmark end 661 2523 a Fs(\011)732 2537 y Fi(\006)791 2523 y Fs(\()p Fq(x;)15 b(\020)7 b Fs(\))26 b(=)f Fq(q)s Fs(\()p Fo(E)8 b Fs(\))p Fq(e)1334 2420 y Fe(R)1382 2440 y Fd(E)1368 2498 y(E)1401 2513 y Ff(0)1452 2473 y Fn(\012)1503 2482 y Fd(\006)1559 2523 y Fq( )1618 2537 y Fi(\006)1678 2523 y Fs(\()p Fq(x;)15 b Fo(E)8 b Fs(\))p Fq(;)107 b(q)s Fs(\()p Fo(E)8 b Fs(\))26 b(=)2320 2440 y Fh(p)p 2411 2440 201 4 v 83 x Fq(k)2461 2496 y Fi(0)2484 2523 y Fs(\()p Fo(E)8 b Fs(\))q Fq(;)106 b Fo(E)34 b Fs(=)25 b Fo(E)8 b Fs(\()p Fq(\020)f Fs(\))p Fq(:)-118 2753 y Fs(The)30 b(functions)g(\011)532 2767 y Fi(\006)621 2753 y Fs(are)g(called)i(the) f Fl(c)-5 b(anonic)g(al)34 b(Blo)-5 b(ch)34 b(solutions)g(normalize)-5 b(d)35 b(at)e(the)g(p)-5 b(oint)34 b Fq(\020)3208 2767 y Fn(0)3248 2753 y Fs(.)-118 2919 y(The)g(prop)s(erties)g(of)h(the)f (di\013eren)m(tial)i(\012)f(imply)f(that)h(the)g(solutions)g(\011)2409 2933 y Fi(\006)2502 2919 y Fs(can)g(b)s(e)f(analytically)j(con)m(tin)m (ued)e(from)-118 3085 y Fq(V)-65 3099 y Fn(0)5 3085 y Fs(to)c(an)m(y)f(regular)h(domain)f(con)m(taining)i Fq(V)1414 3099 y Fn(0)1454 3085 y Fs(.)-118 3251 y(The)e(W)-8 b(ronskian)31 b(of)f(\011)696 3265 y Fi(\006)785 3251 y Fs(satis\014es)h(\(see)g([)p 0 1 0 TeXcolorrgb 1327 3251 a SDict begin H.S end 1327 3251 a 0 1 0 TeXcolorrgb Fs(8)p 0 1 0 TeXcolorrgb 1373 3192 a SDict begin H.R end 1373 3192 a 1373 3251 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.FK0) cvn H.B /ANN pdfmark end 1373 3251 a 0 TeXcolorgray Fs(]\))-118 3481 y(\(4.5\))380 3481 y SDict begin H.S end 380 3481 a 380 3481 a SDict begin 20 H.A end 380 3481 a 380 3481 a SDict begin [ /View [/XYZ H.V] /Dest (equation.4.5) cvn H.B /DEST pdfmark end 380 3481 a Fq(w)r Fs(\(\011)553 3495 y Fn(+)613 3481 y Fs(\()p Fo(\001)p Fq(;)15 b(\020)7 b Fs(\))p Fq(;)15 b Fs(\011)906 3495 y Fi(\000)966 3481 y Fs(\()p Fo(\001)p Fq(;)g(\020)7 b Fs(\)\))26 b(=)f Fq(w)r Fs(\(\011)1478 3495 y Fn(+)1538 3481 y Fs(\()p Fo(\001)p Fq(;)15 b(\020)1678 3495 y Fn(0)1718 3481 y Fs(\))p Fq(;)g Fs(\011)1864 3495 y Fi(\000)1924 3481 y Fs(\()p Fo(\001)p Fq(;)g(\020)2064 3495 y Fn(0)2104 3481 y Fs(\)\))26 b(=)f Fq(k)2346 3443 y Fi(0)2369 3481 y Fs(\()p Fo(E)2452 3495 y Fn(0)2492 3481 y Fs(\))p Fq(w)r Fs(\()p Fq( )2688 3495 y Fn(+)2749 3481 y Fs(\()p Fo(\001)p Fq(;)15 b(\020)2889 3495 y Fn(0)2929 3481 y Fs(\))p Fq(;)g( )3063 3495 y Fi(\000)3123 3481 y Fs(\()p Fo(\001)p Fq(;)g(\020)3263 3495 y Fn(0)3303 3481 y Fs(\)\))p Fq(:)-118 3711 y Fs(F)-8 b(or)31 b Fo(E)93 3725 y Fn(0)168 3711 y Fq(=)-55 b Fo(2)25 b Fq(Q)20 b Fo([)f(f)p Fq(E)528 3726 y Fp(l)555 3711 y Fo(g)p Fs(,)31 b(the)g(W)-8 b(ronskian)30 b Fq(w)r Fs(\(\011)1438 3725 y Fn(+)1498 3711 y Fs(\()p Fo(\001)p Fq(;)15 b(\020)7 b Fs(\))p Fq(;)15 b Fs(\011)1791 3725 y Fi(\000)1851 3711 y Fs(\()p Fo(\001)p Fq(;)g(\020)7 b Fs(\)\))31 b(is)g(non-zero.)-118 3840 y SDict begin H.S end -118 3840 a -118 3840 a SDict begin 20 H.A end -118 3840 a -118 3840 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.2) cvn H.B /DEST pdfmark end -118 3840 a 143 x Fs(4.2.)47 b Fv(Solutions)35 b(ha)m(ving)g(standard)g(asymptotic)g(b)s (eha)m(vior.)46 b Fs(Fix)31 b Fq(E)f Fs(=)25 b Fq(E)2721 3997 y Fn(0)2761 3983 y Fs(.)40 b(Let)31 b Fq(D)i Fs(b)s(e)d(a)g (regular)h(domain)-118 4149 y(\()j(i.e)g Fq(D)g Fo(\032)c Fq(D)s Fs(\()p Fq(W)13 b Fs(\),)34 b(and)f(simply)g(connected)i(set)f (con)m(taining)h(no)f(branc)m(h)e(p)s(oin)m(ts)i(of)g Fq(\024)p Fs(.\))51 b(.)f(Fix)34 b Fq(\020)3344 4163 y Fn(0)3414 4149 y Fo(2)c Fq(D)36 b Fs(so)e(that)-118 4315 y Fo(E)8 b Fs(\()p Fq(\020)13 4329 y Fn(0)53 4315 y Fs(\))45 b Fq(=)-55 b Fo(2)34 b Fq(P)j Fo([)24 b Fq(Q)p Fs(.)58 b(Let)36 b Fq(\024)h Fs(b)s(e)e(a)i(con)m(tin)m(uous)f(branc)m (h)g(of)g(the)g(complex)h(momen)m(tum)f(in)g Fq(D)s Fs(,)i(and)d(let)i (\011)3547 4329 y Fi(\006)3642 4315 y Fs(b)s(e)e(the)-118 4481 y(canonical)24 b(Blo)s(c)m(h)g(solutions)g(normalized)f(at)h Fq(\020)1496 4495 y Fn(0)1558 4481 y Fs(de\014ned)e(on)g Fq(D)k Fs(and)c(indexed)h(so)g(that)h Fq(\024)f Fs(b)s(e)f(the)h (quasi-momen)m(tum)-118 4647 y(for)30 b(\011)92 4661 y Fn(+)151 4647 y Fs(.)-118 4668 y SDict begin H.S end -118 4668 a -118 4668 a SDict begin 20 H.A end -118 4668 a -118 4668 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.4.1) cvn H.B /DEST pdfmark end -118 4668 a 0 TeXcolorgray 252 x Fv(De\014nition)40 b(4.1.)p 0 TeXcolorgray 44 w Fl(L)-5 b(et)37 b Fq(\033)e Fo(2)c(f)p Fs(+)p Fq(;)15 b Fo(\000g)p Fl(.)54 b(We)35 b(say)i(that,)i(in)d Fq(D)s Fl(,)h(a)f(c)-5 b(onsistent)38 b(solution)f Fq(f)46 b Fl(has)37 b(standar)-5 b(d)39 b(b)-5 b(ehavior)-118 5086 y(\(or)33 b(standar)-5 b(d)36 b(asymptotics\))f(if)p 0 TeXcolorgray 98 5284 a Fo(\017)p 0 TeXcolorgray 42 w Fl(ther)-5 b(e)34 b(exists)g Fq(V)715 5298 y Fn(0)755 5284 y Fl(,)f(a)h(c)-5 b(omplex)35 b(neighb)-5 b(orho)g(o)g(d)38 b(of)33 b Fq(E)1957 5298 y Fn(0)1997 5284 y Fl(,)h(and)g Fq(X)h(>)26 b Fs(0)34 b Fl(such)g(that)h Fq(f)43 b Fl(is)33 b(de\014ne)-5 b(d)35 b(and)g(satis\014es)185 5450 y(\()p 0 0 1 TeXcolorrgb 222 5467 a SDict begin H.S end 222 5467 a 0 0 1 TeXcolorrgb -17 x Fl(4.1)p 0 0 1 TeXcolorrgb 343 5391 a SDict begin H.R end 343 5391 a 343 5450 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.4.1) cvn H.B /ANN pdfmark end 343 5450 a 0 TeXcolorgray Fl(\))d(and)i(\()p 0 0 1 TeXcolorrgb 626 5467 a SDict begin H.S end 626 5467 a 0 0 1 TeXcolorrgb -17 x Fl(4.2)p 0 0 1 TeXcolorrgb 747 5391 a SDict begin H.R end 747 5391 a 747 5450 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.4.2) cvn H.B /ANN pdfmark end 747 5450 a 0 TeXcolorgray Fl(\))f(for)g(any)g Fs(\()p Fq(x;)15 b(\020)7 b(;)15 b(E)5 b Fs(\))27 b Fo(2)d Fs(\()p Fo(\000)p Fq(X)r(;)15 b(X)7 b Fs(\))22 b Fo(\002)e Fq(D)j Fo(\002)d Fq(V)2264 5464 y Fn(0)2304 5450 y Fl(;)p 0 TeXcolorgray 98 5616 a Fo(\017)p 0 TeXcolorgray 42 w Fq(f)41 b Fl(is)33 b(analytic)h(in)e Fq(\020)g Fo(2)25 b Fq(D)35 b Fl(and)f(in)e Fq(E)f Fo(2)25 b Fq(V)1617 5630 y Fn(0)1656 5616 y Fl(;)p 0 TeXcolorgray 1854 5715 a Fn(14)p 0 TeXcolorgray eop end %%Page: 15 15 TeXDict begin 15 14 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray -118 -406 a SDict begin H.S end -118 -406 a 0 TeXcolorgray 0 TeXcolorgray -118 -406 a SDict begin H.R end -118 -406 a -118 -406 a SDict begin [ /View [/XYZ H.V] /Dest (page.15) cvn H.B /DEST pdfmark end -118 -406 a 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray 98 -207 a Fo(\017)p 0 TeXcolorgray 42 w Fl(for)47 b(any)g(c)-5 b(omp)g(act)49 b(set)d Fq(K)57 b Fo(\032)50 b Fq(D)s Fl(,)g(ther)-5 b(e)47 b(exists)g Fq(V)71 b Fo(\032)50 b Fq(V)2264 -193 y Fn(0)2303 -207 y Fl(,)g(a)d(neighb)-5 b(orho)g(o)g(d)50 b(of)d Fq(E)3216 -193 y Fn(0)3255 -207 y Fl(,)j(such)d(that,)k(for)185 -41 y Fs(\()p Fq(x;)15 b(\020)7 b(;)15 b(E)5 b Fs(\))26 b Fo(2)f Fs(\()p Fo(\000)p Fq(X)r(;)15 b(X)7 b Fs(\))22 b Fo(\002)e Fq(K)27 b Fo(\002)20 b Fq(V)g Fl(,)32 b Fq(f)42 b Fl(has)34 b(the)f(uniform)g(asymptotic) -118 172 y Fs(\(4.6\))1013 172 y SDict begin H.S end 1013 172 a 1013 172 a SDict begin 20 H.A end 1013 172 a 1013 172 a SDict begin [ /View [/XYZ H.V] /Dest (equation.4.6) cvn H.B /DEST pdfmark end 1013 172 a Fq(f)h Fs(=)25 b Fq(e)1230 123 y Fp(\033)1287 96 y Fg(i)p 1283 108 30 3 v 1283 149 a(")1334 70 y Fe(R)1381 90 y Fg(\020)1368 148 y(\020)1396 163 y Ff(0)1447 123 y Fp(\024)p Fn(\()p Fp(u)p Fn(\))p Fp(du)1664 172 y Fs(\(\011)1770 186 y Fp(\033)1837 172 y Fs(+)20 b Fq(o)p Fs(\(1\)\))34 b Fl(as)g Fq(")f Fl(tends)g(to)g Fs(0;)p 0 TeXcolorgray 98 385 a Fo(\017)p 0 TeXcolorgray 42 w Fl(this)g(asymptotic)i(c)-5 b(an)33 b(b)-5 b(e)32 b(di\013er)-5 b(entiate)g(d)35 b(onc)-5 b(e)33 b(in)f Fq(x)h Fl(without)h(lo)-5 b(osing)34 b(its)f(uniformity)g(pr)-5 b(op)g(erties.)-118 575 y(We)32 b(set)-118 788 y Fs(\(4.7\))1350 788 y SDict begin H.S end 1350 788 a 1350 788 a SDict begin 20 H.A end 1350 788 a 1350 788 a SDict begin [ /View [/XYZ H.V] /Dest (equation.4.7) cvn H.B /DEST pdfmark end 1350 788 a Fq(f)1405 750 y Fi(\003)1444 788 y Fs(\()p Fq(x;)15 b(\020)7 b(;)15 b(E)5 b(;)15 b(")p Fs(\))27 b(=)p 1970 695 460 4 v 25 w Fq(f)10 b Fs(\()p Fq(x;)p 2152 714 47 4 v 15 w(\020)d(;)p 2239 715 73 4 v 15 w(E)e(;)15 b(")p Fs(\))94 1045 y(W)-8 b(e)44 b(call)g Fq(\020)482 1059 y Fn(0)563 1045 y Fs(the)f(normalization)h(p)s(oin)m(t)e(for)g Fq(f)10 b Fs(.)76 b(T)-8 b(o)43 b(sa)m(y)g(that)g(a)f(consisten)m(t)i (solution)f Fq(f)51 b Fs(has)42 b(standard)-118 1211 y(b)s(eha)m(vior,)31 b(w)m(e)f(will)h(use)f(the)h(follo)m(wing)h (notation)1303 1427 y Fq(f)i Fo(\030)25 b Fs(exp)1632 1299 y Fh(\022)1699 1427 y Fq(\033)1770 1366 y(i)p 1764 1406 43 4 v 1764 1490 a(")1832 1304 y Fh(Z)1923 1330 y Fp(\020)1882 1510 y(\020)1913 1519 y Ff(0)1978 1427 y Fq(\024)p Fs(\()p Fq(u)p Fs(\))p Fq(du)2251 1299 y Fh(\023)2334 1427 y Fs(\011)2405 1441 y Fp(\033)2452 1427 y Fq(:)-118 1595 y SDict begin H.S end -118 1595 a -118 1595 a SDict begin 20 H.A end -118 1595 a -118 1595 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.3) cvn H.B /DEST pdfmark end -118 1595 a 71 x Fs(4.3.)47 b Fv(Some)35 b(results)g(on)g(the)g(con)m(tin)m(uation)g(of)g(asymptotics.)-118 1756 y SDict begin H.S end -118 1756 a -118 1756 a SDict begin 20 H.A end -118 1756 a -118 1756 a SDict begin [ /View [/XYZ H.V] /Dest (subsubsection.4.3.1) cvn H.B /DEST pdfmark end -118 1756 a 149 x Fs(4.3.1.)47 b Fl(Description)32 b(of)f(the)g(Stokes)h(lines)f(ne)-5 b(ar)32 b Fs([)p Fq(')1662 1866 y Fi(\000)1662 1931 y Fn(1)1722 1905 y Fs(\()p Fq(E)5 b Fs(\))p Fq(;)15 b(')1963 1866 y Fn(+)1963 1931 y(1)2024 1905 y Fs(\()p Fq(E)5 b Fs(\)])p Fl(.)46 b Fs(This)28 b(section)i(is)e(dev)m(oted)i(to)f(the)g(description)-118 2071 y(of)h(the)h(Stok)m(es)g(lines)g(under)e(assumption)h(\(H4\).)-118 2166 y SDict begin H.S end -118 2166 a -118 2166 a SDict begin 20 H.A end -118 2166 a -118 2166 a SDict begin [ /View [/XYZ H.V] /Dest (subsubsection.4.3.2) cvn H.B /DEST pdfmark end -118 2166 a 143 x Fs(4.3.2.)47 b Fl(De\014nition.)f Fs(The)d(de\014nition)g(of)h(the)g(Stok)m(es)h(lines)e(is)h(fairly)g (standard,)i([)p 0 1 0 TeXcolorrgb 2832 2309 a SDict begin H.S end 2832 2309 a 0 1 0 TeXcolorrgb Fs(13)p 0 1 0 TeXcolorrgb 2924 2250 a SDict begin H.R end 2924 2250 a 2924 2309 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Fe) cvn H.B /ANN pdfmark end 2924 2309 a 0 TeXcolorgray Fs(,)p 0 1 0 TeXcolorrgb 2992 2309 a SDict begin H.S end 2992 2309 a 0 1 0 TeXcolorrgb Fs(8)p 0 1 0 TeXcolorrgb 3038 2250 a SDict begin H.R end 3038 2250 a 3038 2309 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.FK0) cvn H.B /ANN pdfmark end 3038 2309 a 0 TeXcolorgray Fs(].)80 b(The)44 b(in)m(tegral)h Fq(\020)54 b Fo(7!)-118 2402 y Fh(R)-58 2428 y Fp(\020)-3 2475 y Fq(\024)p Fs(\()p Fq(u)p Fs(\))p Fq(du)35 b Fs(has)e(the)g(same)h(branc)m(h)e(p)s(oin)m (ts)h(as)g(the)h(complex)g(momen)m(tum.)48 b(Let)34 b Fq(\020)2798 2489 y Fn(0)2870 2475 y Fs(b)s(e)f(one)g(of)g(them.)49 b(Consider)-118 2641 y(the)30 b(curv)m(es)h(b)s(eginning)f(at)h Fq(\020)885 2655 y Fn(0)924 2641 y Fs(,)g(and)e(describ)s(ed)h(b)m(y)g (the)g(equation)-118 2858 y(\(4.8\))1358 2858 y SDict begin H.S end 1358 2858 a 1358 2858 a SDict begin 20 H.A end 1358 2858 a 1358 2858 a SDict begin [ /View [/XYZ H.V] /Dest (equation.4.8) cvn H.B /DEST pdfmark end 1358 2858 a Fs(Im)1482 2734 y Fh(Z)1573 2760 y Fp(\020)1532 2940 y(\020)1563 2949 y Ff(0)1612 2858 y Fs(\()p Fq(\024)p Fs(\()p Fq(\030)t Fs(\))22 b Fo(\000)e Fq(\024)p Fs(\()p Fq(\020)2053 2872 y Fn(0)2093 2858 y Fs(\)\))p Fq(d\030)30 b Fs(=)25 b(0)-118 3076 y(These)c(curv)m(es)g(are)g(the)g Fl(Stokes)k(lines)c Fs(b)s(eginning)f(at)i Fq(\020)1718 3090 y Fn(0)1757 3076 y Fs(.)37 b(According)22 b(to)g(equation)g(\()p 0 0 1 TeXcolorrgb 2741 3076 a SDict begin H.S end 2741 3076 a 0 0 1 TeXcolorrgb Fs(2.3)p 0 0 1 TeXcolorrgb 2857 3018 a SDict begin H.R end 2857 3018 a 2857 3076 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.2.3) cvn H.B /ANN pdfmark end 2857 3076 a 0 TeXcolorgray Fs(\),)i(the)d(Stok)m(es)h(line)f (de\014nition)-118 3242 y(is)30 b(indep)s(enden)m(t)f(of)i(the)g(c)m (hoice)h(of)e(the)h(branc)m(h)f(of)g Fq(\024)p Fs(.)-118 3408 y(Assume)j(that)h Fq(W)521 3375 y Fi(0)544 3408 y Fs(\()p Fq(\020)619 3422 y Fn(0)658 3408 y Fs(\))d Fo(6)p Fs(=)f(0.)50 b(Equation)34 b(\()p 0 0 1 TeXcolorrgb 1376 3408 a SDict begin H.S end 1376 3408 a 0 0 1 TeXcolorrgb Fs(2.4)p 0 0 1 TeXcolorrgb 1492 3350 a SDict begin H.R end 1492 3350 a 1492 3408 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.2.4) cvn H.B /ANN pdfmark end 1492 3408 a 0 TeXcolorgray Fs(\))g(implies)f(that)h(there)g(are)g(exactly)h (three)e(Stok)m(es)i(lines)e(b)s(eginning)-118 3575 y(at)e Fq(\020)33 3589 y Fn(0)72 3575 y Fs(.)41 b(The)30 b(angle)h(b)s(et)m(w) m(een)g(an)m(y)g(t)m(w)m(o)h(of)e(them)g(at)i(this)e(p)s(oin)m(t)g(is)h (equal)f(to)2571 3539 y Fn(2)p Fp(\031)p 2571 3554 79 4 v 2592 3606 a Fn(3)2659 3575 y Fs(.)40 b(Indeed)30 b(for)g Fq(\020)37 b Fs(near)30 b Fq(\020)3478 3589 y Fn(0)3517 3575 y Fs(,)h(w)m(e)g(ha)m(v)m(e)1198 3787 y Fq(\024)p Fs(\()p Fq(\020)7 b Fs(\))26 b(=)f Fq(\024)p Fs(\()p Fq(\020)1616 3801 y Fn(0)1655 3787 y Fs(\))c(+)f Fq(c)1841 3705 y Fh(p)p 1932 3705 237 4 v 82 x Fq(\020)27 b Fo(\000)20 b Fq(\020)2130 3801 y Fn(0)2169 3787 y Fs(\(1)h(+)f Fq(o)p Fs(\(1\)\))p Fq(:)-118 4000 y Fs(So)1005 4052 y Fh(Z)1096 4078 y Fp(\020)1056 4258 y(\020)1087 4267 y Ff(0)1136 4176 y Fs(\()p Fq(\024)p Fs(\()p Fq(\020)7 b Fs(\))21 b Fo(\000)f Fq(\024)p Fs(\()p Fq(\020)1579 4190 y Fn(0)1619 4176 y Fs(\)\))p Fq(d\020)33 b Fs(=)25 b Fq(c)p Fs(\()p Fq(\020)i Fo(\000)20 b Fq(\020)2177 4190 y Fn(0)2216 4176 y Fs(\))2251 4138 y Fn(3)p Fp(=)p Fn(2)2361 4176 y Fs(\(1)h(+)f Fq(o)p Fs(\(1\)\))p Fq(:)-118 4343 y SDict begin H.S end -118 4343 a -118 4343 a SDict begin 20 H.A end -118 4343 a -118 4343 a SDict begin [ /View [/XYZ H.V] /Dest (subsubsection.4.3.3) cvn H.B /DEST pdfmark end -118 4343 a 87 x Fs(4.3.3.)47 b Fl(Stokes)38 b(lines)f(for)g Fq(E)859 4444 y Fn(0)931 4430 y Fo(2)c Fq(J)9 b Fl(.)45 b Fs(W)-8 b(e)36 b(describ)s(e)e(the)h(Stok)m(es)h(lines)f(b)s (eginning)g(at)g Fq(')2931 4392 y Fi(\000)2931 4456 y Fn(1)2991 4430 y Fs(\()p Fq(E)5 b Fs(\))36 b(and)e Fq(')3409 4392 y Fn(+)3409 4456 y(1)3468 4430 y Fs(\()p Fq(E)5 b Fs(\).)56 b(Since)-118 4596 y Fq(W)43 b Fs(is)30 b(real)h(on)f Fm(R)p Fs(,)h(the)f(set)h(of)g(the)f(Stok)m(es)i(lines)e(is)g (symmetric)h(with)f(resp)s(ect)h(to)g(the)f(real)h(line.)-118 4762 y(First,)i Fq(\024)183 4776 y Fn(1)256 4762 y Fs(is)f(real)h(on)g (the)f(in)m(terv)-5 b(al)34 b([)p Fq(')1230 4724 y Fi(\000)1230 4788 y Fn(1)1290 4762 y Fs(\()p Fq(E)5 b Fs(\))p Fq(;)15 b(')1531 4724 y Fn(+)1531 4788 y(1)1592 4762 y Fs(\()p Fq(E)5 b Fs(\)];)35 b(therefore)d(this)h(set)g(is)f(a)h(Stok)m(es)h (line)e(starting)i(at)f Fq(')3670 4724 y Fi(\000)3670 4788 y Fn(1)3729 4762 y Fs(\()p Fq(E)5 b Fs(\).)-118 4928 y(The)24 b(t)m(w)m(o)j(other)e(Stok)m(es)h(lines)f(b)s(eginning)f (at)i Fq(')1518 4890 y Fi(\000)1518 4954 y Fn(1)1578 4928 y Fs(\()p Fq(E)5 b Fs(\))26 b(are)f(symmetric)g(with)g(resp)s(ect) g(to)h(the)f(real)g(axis.)40 b(W)-8 b(e)26 b(denote)-118 5094 y(b)m(y)f Fq(\033)58 5056 y Fi(\000)55 5120 y Fn(1)118 5094 y Fs(\()p Fq(E)5 b Fs(\))26 b(the)g(Stok)m(es)h(line)f(going)g(do) m(wn)m(w)m(ard)g(and)f(b)m(y)p 1842 5044 92 4 v 25 w Fq(\033)1894 5108 y Fn(1)1934 5094 y Fs(\()p Fq(E)5 b Fs(\))26 b(its)g(symmetric.)40 b(Similarly)-8 b(,)27 b(w)m(e)g(denote)f(b)m(y)f Fq(\033)3695 5056 y Fn(+)3692 5120 y(1)3755 5094 y Fs(\()p Fq(E)5 b Fs(\))-118 5260 y(and)p 59 5171 115 4 v 30 w Fq(\033)114 5222 y Fn(+)111 5286 y(1)173 5260 y Fs(\()p Fq(E)g Fs(\))32 b(the)f(t)m(w)m(o)h(other)f (Stok)m(es)g(lines)g(starting)h(at)f Fq(')1915 5222 y Fn(+)1915 5286 y(1)1974 5260 y Fs(\()p Fq(E)5 b Fs(\),)32 b Fq(\033)2228 5222 y Fn(+)2225 5286 y(1)2288 5260 y Fs(\()p Fq(E)5 b Fs(\))31 b(go)s(es)h(up)m(w)m(ard.)40 b(These)31 b(Stok)m(es)g(lines)g(are)-118 5426 y(represen)m(ted)f(in)g (\014gure)p 0 0 1 TeXcolorrgb 724 5426 a SDict begin H.S end 724 5426 a 0 0 1 TeXcolorrgb Fs(3)p 0 0 1 TeXcolorrgb 770 5368 a SDict begin H.R end 770 5368 a 770 5426 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (figure.3) cvn H.B /ANN pdfmark end 770 5426 a 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray -118 5639 a SDict begin H.S end -118 5639 a -118 5639 a SDict begin 20 H.A end -118 5639 a -118 5639 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.4.2) cvn H.B /DEST pdfmark end -118 5639 a 0 TeXcolorgray 1854 5715 a Fn(15)p 0 TeXcolorgray eop end %%Page: 16 16 TeXDict begin 16 15 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray -118 -406 a SDict begin H.S end -118 -406 a 0 TeXcolorgray 0 TeXcolorgray -118 -406 a SDict begin H.R end -118 -406 a -118 -406 a SDict begin [ /View [/XYZ H.V] /Dest (page.16) cvn H.B /DEST pdfmark end -118 -406 a 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray 1890 347 a @beginspecial @setspecial tx@Dict begin STP newpath 0.03836 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 76.64978 0.0 -76.64978 0.0 /Lineto /lineto load def false Line gsave 0.03836 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.38327 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 76.64978 61.31982 -76.64978 61.31982 /Lineto /lineto load def false Line gsave 0.38327 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.38327 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 76.64978 -61.31982 -76.64978 -61.31982 /Lineto /lineto load def false Line gsave 0.38327 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.76654 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ -49.82236 1.53296 -49.82236 -1.53296 /Lineto /lineto load def false Line gsave 0.76654 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial 1349 410 a tx@Dict begin { 5.0 31.50539 9.4384 3.13586 180. Uput UUput } PutCoor PutBegin end 1349 410 a 1218 436 a Fq(')1277 398 y Fi(\000)1277 462 y Fn(1)1337 436 y Fs(\()p Fq(E)5 b Fs(\))1349 410 y tx@Dict begin PutEnd end 1349 410 a 1890 347 a @beginspecial @setspecial tx@Dict begin STP newpath 0.76654 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 26.82742 1.53296 26.82742 -1.53296 /Lineto /lineto load def false Line gsave 0.76654 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial 2303 410 a tx@Dict begin { 5.0 31.50539 9.4384 3.13586 180. Uput UUput } PutCoor PutBegin end 2303 410 a 2173 436 a Fq(')2232 398 y Fn(+)2232 462 y(1)2291 436 y Fs(\()p Fq(E)g Fs(\))2303 410 y tx@Dict begin PutEnd end 2303 410 a 1890 347 a @beginspecial @setspecial tx@Dict begin STP newpath 0.76654 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 26.82742 0.0 -49.82236 0.0 /Lineto /lineto load def false Line gsave 0.76654 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.76654 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 44.4569 61.31982 41.39082 40.6244 26.82742 0.0 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.76654 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial 2361 60 a tx@Dict begin { 5.0 30.99211 9.4384 3.13586 180. Uput UUput } PutCoor PutBegin end 2361 60 a 2232 86 a Fq(\033)2287 48 y Fn(+)2284 112 y(1)2346 86 y Fs(\()p Fq(E)g Fs(\))2361 60 y tx@Dict begin PutEnd end 2361 60 a 1890 347 a @beginspecial @setspecial tx@Dict begin STP newpath 0.76654 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 44.4569 -61.31982 41.39082 -40.6244 26.82742 0.0 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.76654 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial 2361 633 a tx@Dict begin { 5.0 30.99211 11.62834 3.13586 180. Uput UUput } PutCoor PutBegin end 2361 633 a 2232 579 258 4 v 2232 668 a Fq(\033)2287 630 y Fn(+)2284 694 y(1)2346 668 y Fs(\()p Fq(E)g Fs(\))2361 633 y tx@Dict begin PutEnd end 2361 633 a 1890 347 a @beginspecial @setspecial tx@Dict begin STP newpath 0.76654 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ -59.78685 61.31982 -57.48734 30.65991 -49.82236 0.0 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.76654 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial 1285 124 a tx@Dict begin { 5.0 30.99211 11.62834 3.13586 180. Uput UUput } PutCoor PutBegin end 1285 124 a 1157 70 258 4 v 1157 159 a Fq(\033)1212 121 y Fi(\000)1209 185 y Fn(1)1271 159 y Fs(\()p Fq(E)g Fs(\))1285 124 y tx@Dict begin PutEnd end 1285 124 a 1890 347 a @beginspecial @setspecial tx@Dict begin STP newpath 0.76654 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ -59.78685 -61.31982 -57.48734 -30.65991 -49.82236 0.0 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.76654 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial 1285 569 a tx@Dict begin { 5.0 30.99211 9.4384 3.13586 180. Uput UUput } PutCoor PutBegin end 1285 569 a 1157 595 a Fq(\033)1212 557 y Fi(\000)1209 621 y Fn(1)1271 595 y Fs(\()p Fq(E)g Fs(\))1285 569 y tx@Dict begin PutEnd end 1285 569 a 1218 1190 a Fr(Figure)34 b(3.)p 0 TeXcolorgray 1689 1190 a SDict begin H.S end 1689 1190 a 0 TeXcolorgray 0 TeXcolorgray 1689 1190 a SDict begin H.R end 1689 1190 a 1689 1190 a SDict begin [ /View [/XYZ H.V] /Dest (figure.3) cvn H.B /DEST pdfmark end 1689 1190 a 0 TeXcolorgray Fs(Stok)m(es)d(lines)g (for)f Fq(E)g Fo(2)25 b Fq(J)p 0 TeXcolorgray 0 TeXcolorgray -118 1446 a Fv(Lemma)35 b(4.2.)p 0 TeXcolorgray 42 w Fl(The)e(Stokes)g(lines)f Fq(\033)1192 1408 y Fi(\000)1189 1472 y Fn(1)1252 1446 y Fs(\()p Fq(E)5 b Fs(\))33 b Fl(and)h Fq(\033)1659 1408 y Fn(+)1656 1472 y(1)1718 1446 y Fs(\()p Fq(E)5 b Fs(\))34 b Fl(satisfy)f(the)g(fol)5 b(lowing)34 b(pr)-5 b(op)g(erties:)p 0 TeXcolorgray 98 1644 a Fo(\017)p 0 TeXcolorgray 42 w Fl(The)33 b(Stokes)g(lines)f Fq(\033)920 1605 y Fi(\000)917 1670 y Fn(1)980 1644 y Fs(\()p Fq(E)5 b Fs(\))33 b Fl(and)h Fq(\033)1387 1605 y Fn(+)1384 1670 y(1)1446 1644 y Fs(\()p Fq(E)5 b Fs(\))34 b Fl(stay)f(vertic)-5 b(al.)p 0 TeXcolorgray 98 1810 a Fo(\017)p 0 TeXcolorgray 42 w Fq(\033)240 1771 y Fi(\000)237 1836 y Fn(1)299 1810 y Fs(\()p Fq(E)5 b Fs(\))33 b Fl(and)h Fq(\033)706 1771 y Fn(+)703 1836 y(1)765 1810 y Fs(\()p Fq(E)5 b Fs(\))34 b Fl(do)f(not)h(interse)-5 b(ct)33 b(one)g(another.)-118 2081 y Fs(The)d(pro)s(of)f(of)i(this)f(Lemma)h(is)f(similar)h(to)g(the) g(studies)f(done)g(in)g([)p 0 1 0 TeXcolorrgb 2222 2081 a SDict begin H.S end 2222 2081 a 0 1 0 TeXcolorrgb Fs(9)p 0 1 0 TeXcolorrgb 2267 2022 a SDict begin H.R end 2267 2022 a 2267 2081 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.FEKL1) cvn H.B /ANN pdfmark end 2267 2081 a 0 TeXcolorgray Fs(,)p 0 1 0 TeXcolorrgb 2323 2081 a SDict begin H.S end 2323 2081 a 0 1 0 TeXcolorrgb Fs(14)p 0 1 0 TeXcolorrgb 2414 2022 a SDict begin H.R end 2414 2022 a 2414 2081 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.KM1) cvn H.B /ANN pdfmark end 2414 2081 a 0 TeXcolorgray Fs(,)p 0 1 0 TeXcolorrgb 2470 2081 a SDict begin H.S end 2470 2081 a 0 1 0 TeXcolorrgb Fs(16)p 0 1 0 TeXcolorrgb 2560 2022 a SDict begin H.R end 2560 2022 a 2560 2081 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Mg) cvn H.B /ANN pdfmark end 2560 2081 a 0 TeXcolorgray Fs(,)p 0 1 0 TeXcolorrgb 2616 2081 a SDict begin H.S end 2616 2081 a 0 1 0 TeXcolorrgb Fs(17)p 0 1 0 TeXcolorrgb 2707 2022 a SDict begin H.R end 2707 2022 a 2707 2081 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Mg1) cvn H.B /ANN pdfmark end 2707 2081 a 0 TeXcolorgray Fs(].)41 b(W)-8 b(e)31 b(do)g(not)f(giv)m(e)i (the)f(details.)-118 2208 y SDict begin H.S end -118 2208 a -118 2208 a SDict begin 20 H.A end -118 2208 a -118 2208 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.4.4) cvn H.B /DEST pdfmark end -118 2208 a 143 x Fs(4.4.)47 b Fv(Construction)e(of) f(a)g(consisten)m(t)i(basis)f(near)f Fs([)p Fq(')2023 2313 y Fi(\000)2023 2377 y Fn(1)2083 2351 y Fs(\()p Fq(E)5 b Fs(\))p Fq(;)15 b(')2324 2313 y Fn(+)2324 2377 y(1)2385 2351 y Fs(\()p Fq(E)5 b Fs(\)])p Fv(.)46 b Fs(W)-8 b(e)40 b(recall)g(this)f(result,)i(pro)m(v)m(ed)e(in)-118 2517 y([)p 0 1 0 TeXcolorrgb -93 2517 a SDict begin H.S end -93 2517 a 0 1 0 TeXcolorrgb Fs(14)p 0 1 0 TeXcolorrgb -2 2459 a SDict begin H.R end -2 2459 a -2 2517 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.KM1) cvn H.B /ANN pdfmark end -2 2517 a 0 TeXcolorgray Fs(].)-118 2540 y SDict begin H.S end -118 2540 a -118 2540 a SDict begin 20 H.A end -118 2540 a -118 2540 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.4.3) cvn H.B /DEST pdfmark end -118 2540 a 0 TeXcolorgray 249 x Fv(Prop)s(osition)c(4.3.)p 0 TeXcolorgray 41 w Fs([)p 0 1 0 TeXcolorrgb 674 2789 a SDict begin H.S end 674 2789 a 0 1 0 TeXcolorrgb Fs(14)p 0 1 0 TeXcolorrgb 766 2730 a SDict begin 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1575 y Fs(\()p Fq(x;)g(\020)28 b Fo(\000)19 b Fs(2)p Fq(\031)s(;)c(E)5 b(;)15 b(")p Fs(\))2153 1683 y Fq(f)2208 1650 y Fi(\003)2198 1707 y Fn(1)2247 1683 y Fs(\()p Fq(x;)g(\020)27 b Fo(\000)20 b Fs(2)p Fq(\031)s(;)15 b(E)5 b(;)15 b(")p Fs(\))2905 1502 y Fh(\023)2987 1630 y Fq(:)-118 1855 y Fs(The)35 b(aim)h(of)g(this)g(section)g(is)g(the)g(computation)h(of)e Fq(M)10 b Fs(\()p Fq(\020)d(;)15 b(E)5 b(;)15 b(")p Fs(\).)59 b(The)35 b(de\014nition)g(of)h(the)g(mono)s(drom)m(y)f(matrix)-118 2021 y(implies)30 b(that)-118 2245 y(\(5.3\))257 2245 y SDict begin H.S end 257 2245 a 257 2245 a SDict begin 20 H.A end 257 2245 a 257 2245 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.3) cvn H.B /DEST pdfmark end 257 2245 a Fq(A)p Fs(\()p Fq(\020)7 b Fs(\))25 b(=)g Fq(M)651 2259 y Fn(1)p Fp(;)p Fn(1)771 2245 y Fs(=)877 2184 y Fq(w)r Fs(\()p Fq(f)1024 2198 y Fn(1)1064 2184 y Fs(\()p Fq(x)20 b Fs(+)g(2)p Fq(\031)s(;)15 b(\020)7 b Fs(\))p Fq(f)1539 2151 y Fi(\003)1529 2208 y Fn(1)1579 2184 y Fs(\()p Fq(x;)15 b(\020)7 b Fs(\)\))p 877 2225 947 4 v 963 2308 a Fq(w)r Fs(\()p Fq(f)1110 2322 y Fn(1)1149 2308 y Fs(\()p Fq(x;)15 b(\020)7 b Fs(\))p Fq(;)15 b(f)1453 2277 y Fi(\003)1443 2334 y Fn(1)1493 2308 y Fs(\()p Fq(x;)g(\020)7 b Fs(\)\))1834 2245 y Fq(;)45 b(B)5 b Fs(\()p Fq(\020)i Fs(\))25 b(=)g Fq(M)2304 2259 y Fn(12)2404 2245 y Fs(=)2510 2184 y Fq(w)r Fs(\()p Fq(f)2657 2198 y Fn(1)2697 2184 y Fs(\()p Fq(x;)15 b(\020)7 b Fs(\))p Fq(;)15 b(f)2991 2198 y Fn(1)3031 2184 y Fs(\()p Fq(x;)g(\020)27 b Fs(+)20 b(2)p Fq(\031)s Fs(\)\))p 2510 2225 978 4 v 2611 2308 a Fq(w)r Fs(\()p Fq(f)2758 2322 y Fn(1)2798 2308 y Fs(\()p Fq(x;)15 b(\020)7 b Fs(\))p Fq(;)15 b(f)3102 2277 y Fi(\003)3092 2334 y Fn(1)3142 2308 y Fs(\()p Fq(x;)g(\020)7 b Fs(\)\))3498 2245 y Fq(:)-118 2470 y Fs(This)45 b(giv)m(es)h(that)h(the)e(mono)s (drom)m(y)g(matrix)h(is)f(analytic)j(in)d Fq(\020)52 b Fs(in)45 b(the)g(strip)g Fq(S)2812 2484 y Fp(Y)2918 2470 y Fs(and)g(in)g Fq(E)51 b Fs(in)45 b(a)h(constan)m(t)-118 2636 y(neigh)m(b)s(orho)s(o)s(d)30 b(of)h Fq(E)625 2650 y Fn(0)665 2636 y Fs(.)43 b(By)31 b(the)h(de\014nition)f(of)g Fq(f)1588 2650 y Fn(1)1658 2636 y Fs(w)m(e)h(get)g(that)g Fq(A)f Fs(and)g Fq(B)36 b Fs(are)31 b Fq(")p Fs(-p)s(erio)s(dic)g(in)g Fq(\020)j Fo(2)26 b Fq(S)3423 2650 y Fp(Y)3483 2636 y Fs(.)43 b(This)31 b(is)g(a)-118 2802 y(an)f(immediate)i(consequence)f (of)f(the)h(prop)s(erties)f(of)g Fq(f)1802 2816 y Fn(1)1841 2802 y Fs(.)-118 2968 y(Therefore,)36 b(w)m(e)f(will)f(compute)h(the)g (F)-8 b(ourier)35 b(series)g(of)g Fq(A)g Fs(and)e Fq(B)5 b Fs(.The)34 b(strategy)i(of)f(the)g(computation)g(is)g(based)-118 3134 y(on)30 b(the)g(ideas)h(of)g([)p 0 1 0 TeXcolorrgb 521 3134 a SDict begin H.S end 521 3134 a 0 1 0 TeXcolorrgb Fs(9)p 0 1 0 TeXcolorrgb 566 3076 a SDict begin H.R end 566 3076 a 566 3134 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.FEKL1) cvn H.B /ANN pdfmark end 566 3134 a 0 TeXcolorgray Fs(])g(and)e(w)m(e)i(\014rst)f(recall)h(some)g (notions)f(presen)m(ted)h(there.)40 b(W)-8 b(e)32 b(refer)e(the)g (reader)g(to)h(this)g(pap)s(er)-118 3300 y(for)f(more)h(details.)-118 3467 y(Let)j Fq(h)g Fs(and)f Fq(g)k Fs(ha)m(ving)d(a)f(standard)g (asymptotic)i(b)s(eha)m(vior)e(in)h(regular)f(domains)g Fq(D)2854 3482 y Fp(h)2933 3467 y Fs(and)g Fq(D)3188 3481 y Fp(g)3262 3467 y Fs(and)g(solutions)g(of)-118 3633 y(\()p 0 0 1 TeXcolorrgb -83 3633 a SDict begin H.S end -83 3633 a 0 0 1 TeXcolorrgb Fs(4.1)p 0 0 1 TeXcolorrgb 33 3574 a SDict begin H.R end 33 3574 a 33 3633 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.4.1) cvn H.B /ANN pdfmark end 33 3633 a 0 TeXcolorgray Fs(\):)897 3820 y Fq(h)26 b Fo(\030)f Fq(e)1126 3741 y Fg(i)p 1123 3753 30 3 v 1123 3794 a(")1174 3714 y Fe(R)1221 3735 y Fg(\020)1207 3792 y(\020)1235 3810 y(h)1292 3768 y Fp(\024)1333 3780 y Fg(h)1373 3768 y Fp(d\020)1449 3820 y Fq( )1508 3835 y Fp(h)1553 3820 y Fs(\()p Fq(x;)15 b(\020)7 b Fs(\))p Fq(;)46 b Fs(and)30 b Fq(g)f Fo(\030)c Fq(e)2233 3740 y Fg(i)p 2230 3752 V 2230 3793 a(")2281 3714 y Fe(R)2328 3734 y Fg(\020)2315 3792 y(\020)2343 3800 y(g)2395 3767 y Fp(\024)2436 3775 y Fg(g)2472 3767 y Fp(d\020)2548 3820 y Fq( )2607 3834 y Fp(g)2647 3820 y Fs(\()p Fq(x;)15 b(\020)7 b Fs(\))p Fq(:)-118 4025 y Fs(Here,)31 b Fq(\024)174 4040 y Fp(h)219 4025 y Fs(,)g(\(resp.)41 b Fq(\024)591 4039 y Fp(g)631 4025 y Fs(\))31 b(is)g(an)f(analytic)i(branc)m(h)e(of)h (the)f(complex)h(momen)m(tum)g(in)f Fq(D)2837 4040 y Fp(h)2912 4025 y Fs(\(resp.)p Fq(D)3210 4039 y Fp(g)3251 4025 y Fs(\),)h(\011)3413 4040 y Fp(h)3488 4025 y Fs(\(resp.)40 b(\011)3822 4039 y Fp(g)3862 4025 y Fs(\))-118 4191 y(is)29 b(the)g(canonical)h(Blo)s(c)m(h)g(solution)g(de\014ned)d(on)i Fq(D)1633 4206 y Fp(h)1707 4191 y Fs(\(resp.)40 b Fq(D)2045 4205 y Fp(g)2085 4191 y Fs(\),)30 b(and)f Fq(\020)2391 4206 y Fp(h)2464 4191 y Fs(\(resp.)40 b Fq(\020)2767 4205 y Fp(g)2807 4191 y Fs(\))29 b(is)g(the)g(normalization)i(p)s(oin)m (t)-118 4358 y(for)f Fq(h)h Fs(\(resp.)40 b Fq(g)s Fs(\).)-118 4524 y(As)30 b(the)h(solutions)g Fq(h)f Fs(and)g Fq(g)k Fs(satisfy)d(the)f(consistency)i(condition)f(\()p 0 0 1 TeXcolorrgb 2242 4524 a SDict begin H.S end 2242 4524 a 0 0 1 TeXcolorrgb Fs(4.2)p 0 0 1 TeXcolorrgb 2358 4465 a SDict begin H.R end 2358 4465 a 2358 4524 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.4.2) cvn H.B /ANN pdfmark end 2358 4524 a 0 TeXcolorgray Fs(\),)g(their)f(W)-8 b(ronskian)31 b(is)g Fq(")p Fs(-p)s(erio)s(dic)f(in)g Fq(\020)7 b Fs(.)-118 4642 y SDict begin H.S end -118 4642 a -118 4642 a SDict begin 20 H.A end -118 4642 a -118 4642 a SDict begin [ /View [/XYZ H.V] /Dest (subsubsection.5.1.1) cvn H.B /DEST pdfmark end -118 4642 a 143 x Fs(5.1.1.)47 b Fl(A)n(r)-5 b(cs.)46 b Fs(W)-8 b(e)39 b(assume)f(that)h Fq(D)1157 4799 y Fp(g)1223 4785 y Fo(\\)25 b Fq(D)1384 4800 y Fp(h)1467 4785 y Fs(con)m(tains)39 b(a)g(simply)e(connected)i (domain)2998 4763 y(~)2977 4785 y Fq(D)r Fs(.)64 b(Let)39 b Fq(\015)k Fs(b)s(e)38 b(a)g(regular)-118 4952 y(curv)m(e)e(going)h (from)e Fq(\020)638 4966 y Fp(g)714 4952 y Fs(to)h Fq(\020)870 4967 y Fp(h)951 4952 y Fs(in)f(the)h(follo)m(wing)i(w)m(a)m(y:)52 b(sta)m(ying)37 b(in)f Fq(D)2350 4966 y Fp(g)2390 4952 y Fs(,)i(it)e(go)s(es)g(from)g Fq(\020)3015 4966 y Fp(g)3090 4952 y Fs(to)h(some)f(p)s(oin)m(t)g(in)3816 4929 y(~)3795 4952 y Fq(D)r Fs(,)-118 5118 y(then,)30 b(sta)m(ying)i(in)e Fq(D)609 5133 y Fp(h)654 5118 y Fs(,)h(it)f(go)s(es)h(to)h Fq(\020)1152 5133 y Fp(h)1196 5118 y Fs(.)41 b(W)-8 b(e)31 b(sa)m(y)g(that)g Fq(\015)36 b Fs(is)30 b(an)h(arc)f(asso)s(ciated)i (to)f(the)g(triple)f Fq(h;)15 b(g)35 b Fs(and)3535 5095 y(~)3514 5118 y Fq(D)s Fs(.)-118 5284 y(As)34 5261 y(~)13 5284 y Fq(D)30 b Fs(is)d(simply)g(connected,)i(all)f(the)f(arcs)g(asso) s(ciated)i(to)f(the)f(triple,)h Fq(h;)43 b(g)31 b Fs(and)2746 5261 y(~)2724 5284 y Fq(D)s Fs(.)40 b(As)3019 5261 y(~)2998 5284 y Fq(D)30 b Fs(is)d(simply)f(connected,)-118 5450 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Fs(and)d Fq(\024)754 -192 y Fp(g)824 -206 y Fs(analytically)k(along)e Fq(\015)5 b Fs(\()p Fq(h;)15 b(g)s(;)1843 -229 y Fs(~)1819 -206 y Fq(D)5 b Fs(\).)41 b(F)-8 b(rom)30 b(the)g(prop)s(erties)e(of)i Fq(\024)p Fs(,)g(w)m(e)g(deduce)f(that,)h(for)f Fq(V)-118 -40 y Fs(a)i(small)g(neigh)m(b)s(orho)s(o)s(d)d(of)j Fq(\015)5 b Fs(,)31 b(one)f(has)-118 170 y(\(5.4\))718 170 y SDict begin H.S end 718 170 a 718 170 a SDict begin 20 H.A end 718 170 a 718 170 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.4) cvn H.B /DEST pdfmark end 718 170 a Fq(\024)770 184 y Fp(g)811 170 y Fs(\()p Fq(\020)7 b Fs(\))25 b(=)g Fq(\033)s(\024)1156 185 y Fp(h)1222 170 y Fs(+)19 b(2)p Fq(\031)s(m;)77 b Fs(for)30 b(some)h(m)25 b Fo(2)g Fm(Z)p Fq(;)106 b Fs(and)90 b Fq(\033)28 b Fo(2)d(f)p Fs(1)p Fq(;)15 b Fo(\000)p Fs(1)p Fo(g)p Fq(:)-118 379 y(\033)28 b Fs(=)d Fq(\033)n(;)15 b(h;)g(g)s(;)350 356 y Fs(~)326 379 y Fq(D)36 b Fs(is)30 b(called)i(the)e(signature)h(of)g Fq(\015)5 b Fs(\()p Fq(h;)15 b(g)s(;)1729 356 y Fs(~)1706 379 y Fq(D)5 b Fs(\),)31 b(and)e Fq(m)c Fs(=)g Fq(m)p Fs(\()p Fq(h;)15 b(g)s(;)2571 356 y Fs(~)2547 379 y Fq(D)5 b Fs(\))31 b(the)f(index)g(of)h Fq(\015)5 b Fs(\()p Fq(h;)15 b(g)s(;)3486 356 y Fs(~)3463 379 y Fq(D)5 b Fs(\).)-118 468 y SDict begin H.S end -118 468 a -118 468 a SDict begin 20 H.A end -118 468 a -118 468 a SDict begin [ /View [/XYZ H.V] /Dest (subsubsection.5.1.2) cvn H.B /DEST pdfmark end -118 468 a 143 x Fs(5.1.2.)47 b Fl(The)35 b(me)-5 b(eting)35 b(domain.)47 b Fs(Let)1219 588 y(~)1198 611 y Fq(D)35 b Fs(b)s(e)d(as)h(ab)s(o)m(v)m(e.)49 b(W)-8 b(e)34 b(call)2197 588 y(~)2175 611 y Fq(D)i Fl(a)f(me)-5 b(eting)35 b(domain)p Fs(,)g(if,)e(in)3298 588 y(~)3277 611 y Fq(D)s Fs(,)g(the)g(function) -118 777 y(Im)d Fq(\024)73 792 y Fp(h)148 777 y Fs(and)g(Im)g Fq(\024)516 791 y Fp(g)587 777 y Fs(do)g(not)h(v)-5 b(anish)29 b(and)h(are)h(of)f(opp)s(osite)h(signs.)-118 943 y(Note)24 b(that,)i(for)d(small)g(v)-5 b(alues)24 b(of)f Fq(")p Fs(,)i(whether)d Fq(\020)32 b Fo(7!)25 b Fq(g)s Fs(\()p Fq(x;)15 b(\020)7 b Fs(\))24 b(and)f Fq(\020)32 b Fo(7!)25 b Fq(h)p Fs(\()p Fq(x;)15 b(\020)7 b Fs(\))24 b(increase)g(or)f (decrease)h(is)f(determined)-118 1109 y(b)m(y)35 b(the)g(exp)s(onen)m (tial)h(factor)g Fq(e)973 1004 y Fe(R)1021 1024 y Fg(\020)1007 1082 y(\020)1035 1090 y(g)1080 1109 y Fq(\024)1132 1123 y Fp(g)1172 1109 y Fq(d\020)42 b Fs(and)35 b Fq(e)1525 1004 y Fe(R)1572 1025 y Fg(\020)1558 1082 y(\020)1586 1100 y(h)1635 1109 y Fq(\024)1687 1124 y Fp(h)1733 1109 y Fq(d\020)7 b Fs(.)54 b(So,)37 b(roughly)-8 b(,)36 b(in)f(a)g(meeting) i(domain,)f(along)g(the)f(lines)-118 1275 y(Im)o Fq(\020)d Fs(=)25 b Fq(C)7 b(onst)p Fs(,)29 b(the)i(solutions)g Fq(h)f Fs(and)g Fq(g)k Fs(increase)d(in)f(opp)s(osite)h(directions.) -118 1359 y SDict begin H.S end -118 1359 a -118 1359 a SDict begin 20 H.A end -118 1359 a -118 1359 a SDict begin [ /View [/XYZ H.V] /Dest (subsubsection.5.1.3) cvn H.B /DEST pdfmark end -118 1359 a 148 x Fs(5.1.3.)47 b Fl(The)33 b(action)h(and)f(the) g(amplitude)i(of)d(an)i(ar)-5 b(c.)46 b Fs(.)40 b(W)-8 b(e)32 b(call)g(the)e(in)m(tegral)1358 1717 y Fq(S)5 b Fs(\()p Fq(h;)15 b(g)s(;)1656 1694 y Fs(~)1632 1717 y Fq(D)5 b Fs(\))26 b(=)1869 1593 y Fh(Z)1919 1799 y Fp(\015)t Fn(\()p Fp(h;g)r(;)2119 1783 y Fn(~)2103 1799 y Fp(D)r Fn(\))2210 1717 y Fq(\024)2262 1731 y Fp(g)2302 1717 y Fq(d\020)7 b(;)-118 1957 y Fs(the)30 b Fl(action)i Fs(of)e(the)h(arc)g Fq(\015)f Fs(=)25 b Fq(\015)5 b Fs(\()p Fq(h;)15 b(g)s(;)1184 1934 y Fh(e)1163 1957 y Fq(D)5 b Fs(\).)41 b(Clearly)-8 b(,)32 b(the)e(action)i(tak)m(es)g(the)e(same) h(v)-5 b(alue)31 b(for)f(equiv)-5 b(alen)m(t)32 b(arcs.)-118 2123 y(Assume)40 b(that)i Fo(E)8 b Fs(\()p Fq(\020)f Fs(\))53 b Fq(=)-55 b Fo(2)43 b Fq(P)d Fo([)27 b Fq(Q)41 b Fs(along)h Fq(\015)5 b Fs(\()p Fq(h;)15 b(f)5 b(;)1600 2100 y Fs(~)1578 2123 y Fq(D)s Fs(\).)73 b(Consider)40 b(the)i(function)e Fq(q)2754 2137 y Fp(g)2837 2123 y Fs(=)2951 2045 y Fh(p)p 3042 2045 201 4 v 78 x Fq(k)3092 2096 y Fi(0)3115 2123 y Fs(\()p Fo(E)8 b Fs(\))42 b(and)e(the)h(1-form) -118 2289 y(\012)-52 2303 y Fp(g)-12 2289 y Fs(\()p Fo(E)8 b Fs(\()p Fq(\020)f Fs(\)\))31 b(in)f(the)h(de\014nition)f(of)g(\011) 1103 2303 y Fp(g)1143 2289 y Fs(.)41 b(Con)m(tin)m(ue)31 b(them)f(analytically)j(along)e Fq(\015)5 b Fs(.)41 b(W)-8 b(e)32 b(set)-118 2498 y(\(5.5\))1342 2498 y SDict begin H.S end 1342 2498 a 1342 2498 a SDict begin 20 H.A end 1342 2498 a 1342 2498 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.5) cvn H.B /DEST pdfmark end 1342 2498 a Fq(A)p Fs(\()p Fq(h;)15 b(g)s(;)g(\015)5 b Fs(\))28 b(=)d(\()1881 2436 y Fq(q)1922 2450 y Fp(g)p 1879 2477 86 4 v 1879 2561 a Fq(q)1920 2576 y Fp(h)1974 2498 y Fs(\))p Fo(j)2034 2513 y Fp(\020)5 b Fn(=)p Fp(\020)2156 2525 y Fg(h)2201 2498 y Fq(e)2243 2401 y Fe(R)2277 2479 y Fg(\015)2328 2454 y Fp(!)2372 2462 y Fg(g)2413 2498 y Fq(:)-118 2708 y(A)39 b Fs(is)g(called)i(the)e Fl(the)i(amplitude)f Fs(of)g(the)f(arc)g Fq(\015)5 b Fs(.)67 b(The)39 b(prop)s(erties)f(of)i (\012)e(imply)h(that)h(the)f(amplitudes)g(of)g(t)m(w)m(o)-118 2874 y(equiv)-5 b(alen)m(t)32 b(arcs)e Fq(\015)5 b Fs(\()p Fq(h;)15 b(g)s(;)790 2851 y Fs(~)766 2874 y Fq(D)5 b Fs(\))31 b(coincide.)-118 2963 y SDict begin H.S end -118 2963 a -118 2963 a SDict begin 20 H.A end -118 2963 a -118 2963 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.2) cvn H.B /DEST pdfmark end -118 2963 a 143 x Fs(5.2.)47 b Fv(Results)35 b(on)g(the)g(F)-9 b(ourier)35 b(co)s(e\016cien)m(ts.)47 b Fs(W)-8 b(e)31 b(recall)h(the)f(follo)m(wing)h(result)e(from)g([)p 0 1 0 TeXcolorrgb 3137 3106 a SDict begin H.S end 3137 3106 a 0 1 0 TeXcolorrgb Fs(9)p 0 1 0 TeXcolorrgb 3182 3047 a SDict begin H.R end 3182 3047 a 3182 3106 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.FEKL1) cvn H.B /ANN pdfmark end 3182 3106 a 0 TeXcolorgray Fs(])-118 3129 y SDict begin H.S end -118 3129 a -118 3129 a SDict begin 20 H.A end -118 3129 a -118 3129 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.5.1) cvn H.B /DEST pdfmark end -118 3129 a 0 TeXcolorgray 231 x Fv(Prop)s(osition)50 b(5.1.)p 0 TeXcolorgray 49 w Fl(L)-5 b(et)44 b Fq(d)h Fs(=)g Fq(d)p Fs(\()p Fq(h;)15 b(g)s Fs(\))46 b Fl(b)-5 b(e)44 b(a)g(me)-5 b(eting)44 b(domain)h(for)f Fq(h)g Fl(and)h Fq(g)s Fl(,)i(and)d Fq(m)h Fs(=)g Fq(m)p Fs(\()p Fq(h;)15 b(g)s(;)g(d)p Fs(\))47 b Fl(b)-5 b(e)43 b(the)-118 3526 y(c)-5 b(orr)g(esp)g(onding)36 b(index.)42 b(Then)-118 3902 y Fs(\(5.6\))999 3902 y SDict begin H.S end 999 3902 a 999 3902 a SDict begin 20 H.A end 999 3902 a 999 3902 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.6) cvn H.B /DEST pdfmark end 999 3902 a Fq(w)r Fs(\()p Fq(h;)15 b(g)s Fs(\))28 b(=)d Fq(w)1463 3916 y Fp(m)1530 3902 y Fq(e)1582 3837 y Ff(2)p Fg(\031)r(im)p 1582 3849 146 3 v 1640 3891 a(")1737 3865 y Fn(\()p Fp(\020)5 b Fi(\000)p Fp(\020)1886 3877 y Fg(h)1926 3865 y Fn(\))1958 3902 y Fs(\(1)21 b(+)f Fq(o)p Fs(\(1\)\);)49 b Fq(\020)32 b Fo(2)25 b Fq(S)5 b Fs(\()p Fq(d)p Fs(\))p Fq(;)-118 4112 y Fl(wher)-5 b(e)34 b Fq(w)204 4126 y Fp(m)303 4112 y Fl(is)e(the)h(c)-5 b(onstant)35 b(given)d(by)-118 4322 y Fs(\(5.7\))965 4322 y SDict begin H.S end 965 4322 a 965 4322 a SDict begin 20 H.A end 965 4322 a 965 4322 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.7) cvn H.B /DEST pdfmark end 965 4322 a Fq(w)1030 4336 y Fp(m)1121 4322 y Fs(=)25 b Fq(A)p Fs(\()p Fq(h;)15 b(g)s(;)g(d)p Fs(\))p Fq(e)1635 4257 y Fg(i)p 1636 4269 30 3 v 1636 4310 a(")1675 4284 y Fp(S)t Fn(\()p Fp(h;g)r(;d)p Fn(\))1933 4322 y Fq(w)r Fs(\()p Fq( )2094 4336 y Fn(+)2154 4322 y Fs(\()p Fo(\001)p Fq(;)g(\020)2294 4337 y Fp(h)2339 4322 y Fs(\))p Fq(;)g( )2473 4336 y Fi(\000)2533 4322 y Fs(\()p Fo(\001)p Fq(;)g(\020)2673 4337 y Fp(h)2719 4322 y Fs(\)\))p Fq(;)-118 4531 y Fl(Her)-5 b(e)33 b Fq( )159 4545 y Fn(+)243 4531 y Fs(=)26 b Fq( )399 4546 y Fp(h)477 4531 y Fl(and)33 b Fq( )712 4545 y Fi(\000)804 4531 y Fl(is)g(c)-5 b(omplementary)36 b(to)d Fq( )1699 4545 y Fn(+)1758 4531 y Fl(.)43 b(The)33 b(asymptotic)i(\()p 0 0 1 TeXcolorrgb 2510 4531 a SDict begin H.S end 2510 4531 a 0 0 1 TeXcolorrgb Fl(5.6)p 0 0 1 TeXcolorrgb 2631 4473 a SDict begin H.R end 2631 4473 a 2631 4531 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.5.6) cvn H.B /ANN pdfmark end 2631 4531 a 0 TeXcolorgray Fl(\))e(is)f(uniform)i(in)f Fq(\020)39 b Fl(and)34 b Fq(E)k Fl(when)33 b Fq(\020)-118 4697 y Fl(stays)h(in)e(a)h(\014xe)-5 b(d)34 b(c)-5 b(omp)g(act)34 b(of)f Fq(S)5 b Fs(\()p Fq(d)p Fs(\))34 b Fl(and)f Fq(E)38 b Fl(in)33 b(a)g(smal)5 b(l)33 b(enough)g(neighb)-5 b(orho)g(o)g(d)36 b(of)d Fq(E)2915 4711 y Fn(0)2955 4697 y Fl(.)-118 4808 y SDict begin H.S end -118 4808 a -118 4808 a SDict begin 20 H.A end -118 4808 a -118 4808 a SDict begin [ /View [/XYZ H.V] /Dest (subsubsection.5.2.1) cvn H.B /DEST pdfmark end -118 4808 a 144 x Fs(5.2.1.)47 b Fl(The)36 b(index)g Fq(m)p Fl(.)45 b Fs(Let)35 b Fq(\020)933 4966 y Fn(0)1005 4952 y Fs(b)s(e)e(a)h (regular)g(p)s(oin)m(t.)51 b(consider)34 b(a)g(regular)f(curv)m(e)h Fq(\015)39 b Fs(going)c(from)e Fq(\020)3390 4966 y Fn(0)3463 4952 y Fs(to)h Fq(\020)3617 4966 y Fn(0)3679 4952 y Fs(+)22 b(2)p Fq(\031)s Fs(.)-118 5118 y(Let)37 b Fq(\024)g Fs(b)s(e)f(a)i (branc)m(h)e(of)h(the)g(complex)g(momen)m(tum)g(that)g(is)g(con)m(tin)m (uous)h(on)e Fq(\015)5 b Fs(.)60 b(W)-8 b(e)38 b(call)h(the)e(couple)g (\()p Fq(\015)5 b(;)15 b(\024)p Fs(\))38 b(a)-118 5284 y(p)s(erio)s(d.)h(Let)29 b(\()p Fq(\015)443 5298 y Fn(1)483 5284 y Fq(;)15 b(\024)575 5298 y Fn(1)615 5284 y Fs(\))30 b(and)e(\()p Fq(\015)937 5298 y Fn(2)977 5284 y Fq(;)15 b(\024)1069 5298 y Fn(2)1109 5284 y Fs(\))29 b(b)s(e)g(t)m(w)m(o)h(p)s (erio)s(ds.)39 b(Assume)28 b(that)i(one)f(can)g(con)m(tin)m(uously)h (deform)f Fq(\015)3560 5298 y Fn(1)3628 5284 y Fs(in)m(to)h Fq(\015)3858 5298 y Fn(2)-118 5450 y Fs(without)j(in)m(tersecting)i(an) m(y)f(branc)m(hing)f(p)s(oin)m(t.)50 b(By)33 b(this)g(w)m(e)h(de\014ne) f(an)g(analytic)i(con)m(tin)m(uation)g(of)f Fq(\024)3489 5464 y Fn(1)3562 5450 y Fs(to)g Fq(\015)3723 5464 y Fn(2)3762 5450 y Fs(.)50 b(If)-118 5616 y(the)30 b(analytic)i(con)m(tin)m(uation) h(coincide)e(with)f Fq(\024)1517 5630 y Fn(2)1557 5616 y Fs(,)h(w)m(e)f(sa)m(y)i(that)e(the)h(p)s(erio)s(ds)e(are)i(equiv)-5 b(alen)m(t.)p 0 TeXcolorgray 1854 5715 a Fn(18)p 0 TeXcolorgray eop end %%Page: 19 19 TeXDict begin 19 18 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray -118 -406 a SDict begin H.S end -118 -406 a 0 TeXcolorgray 0 TeXcolorgray -118 -406 a SDict begin H.R end -118 -406 a -118 -406 a SDict begin [ /View [/XYZ H.V] /Dest (page.19) cvn H.B /DEST pdfmark end -118 -406 a 0 TeXcolorgray 0 TeXcolorgray 199 x Fs(Consider)25 b(the)i(branc)m(h)f Fq(\024)g Fs(along)i(the)e(curv)m (e)h Fq(\015)k Fs(of)c(a)g(p)s(erio)s(d)e(\()p Fq(\015)5 b(;)15 b(\024)p Fs(\).)41 b(In)25 b(a)i(neigh)m(b)s(orho)s(o)s(d)e(of)h Fq(\020)3111 -193 y Fn(0)3151 -207 y Fs(,)h(the)f(starting)i(p)s(oin)m (t)-118 -41 y Fq(\015)5 b Fs(,)31 b(one)f(has)833 156 y Fq(\024)p Fs(\()p Fq(\020)e Fs(+)19 b(2)p Fq(\031)s Fs(\))27 b(=)e Fq(\033)s(\024)p Fs(\()p Fq(\020)7 b Fs(\))21 b(+)e(2)p Fq(\031)s(m;)198 b(\033)28 b Fo(2)d(f\007)p Fs(1)p Fo(g)p Fq(;)199 b(m)25 b Fo(2)g Fm(Z)p Fq(:)-118 372 y Fs(The)31 b(n)m(um)m(b)s(ers)e Fq(\033)h Fs(=)c Fq(\033)s Fs(\()p Fq(\015)5 b(;)15 b(\024)p Fs(\))33 b(and)e Fq(m)26 b Fs(=)h Fq(m)p Fs(\()p Fq(\015)5 b(;)15 b(\024)p Fs(\))32 b(are)g(called)g(resp)s(ectiv)m(ely)h(the)e (signature)g(and)g(the)g(index)g(of)g(the)-118 539 y(p)s(erio)s(d)e(\() p Fq(\015)5 b(;)15 b(\024)p Fs(\).)42 b(They)30 b(coincide)i(for)e (equiv)-5 b(alen)m(t)32 b(p)s(erio)s(ds.)94 705 y(Recall)g(that)1374 902 y Fo(G)f Fs(=)25 b Fo([)1616 917 y Fp(k)r Fi(2)p Fk(Z)1749 902 y Fo(f[)1855 864 y Fp(N)1855 924 y(j)t Fn(=1)1982 902 y Fo(G)2036 916 y Fp(j)2093 902 y Fs(+)20 b(2)p Fq(\031)s(k)s Fo(g)p Fs(;)-118 1118 y(is)30 b(the)h(pre-image)g (with)f(resp)s(ect)h(to)g Fo(E)38 b Fs(of)31 b(the)f(union)g(of)g(the)h (sp)s(ectral)g(gaps)f(of)h Fq(H)2767 1132 y Fn(0)2806 1118 y Fs(.)40 b(One)30 b(has)-118 1135 y SDict begin H.S end -118 1135 a -118 1135 a SDict begin 20 H.A end -118 1135 a -118 1135 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.5.2) cvn H.B /DEST pdfmark end -118 1135 a 0 TeXcolorgray 259 x Fv(Lemma)h(5.2.)p 0 TeXcolorgray 40 w Fs([)p 0 1 0 TeXcolorrgb 476 1394 a SDict begin H.S end 476 1394 a 0 1 0 TeXcolorrgb Fs(12)p 0 1 0 TeXcolorrgb 567 1336 a SDict begin H.R end 567 1336 a 567 1394 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.FEKL2) cvn H.B /ANN pdfmark end 567 1394 a 0 TeXcolorgray Fs(])f Fl(L)-5 b(et)31 b Fs(\()p Fq(\015)5 b(;)15 b(\024)p Fs(\))31 b Fl(b)-5 b(e)30 b(a)g(p)-5 b(erio)g(d)32 b(such)e(that)i Fq(\015)j Fl(starts)c(at)g(a)f(p)-5 b(oint)31 b Fq(\020)2644 1408 y Fn(0)2719 1394 y Fq(=)-55 b Fo(2)25 b(G)5 b Fl(.)41 b(Assume)30 b(that)h Fq(\015)k Fl(interse)-5 b(cts)-118 1560 y Fo(G)34 b Fl(exactly)c Fq(n)e Fl(times)i(\()p Fq(n)25 b Fo(2)f Fm(N)p Fl(\))29 b(and)h(that)g(at)g(al)5 b(l)29 b(interse)-5 b(ction)30 b(p)-5 b(oints,)31 b Fq(W)2395 1527 y Fi(0)2443 1560 y Fo(6)p Fs(=)25 b(0)p Fl(.)41 b(L)-5 b(et)30 b Fq(r)2848 1574 y Fn(1)2887 1560 y Fq(;)15 b Fo(\001)g(\001)g(\001)32 b Fq(;)15 b(r)3145 1574 y Fp(n)3221 1560 y Fl(b)-5 b(e)29 b(the)g(values)g(that)-118 1726 y(R)-5 b(e)33 b Fq(\024)g Fl(takes)g(c)-5 b(onse)g(cutively)33 b(at)g(these)g(interse)-5 b(ction)34 b(p)-5 b(oints)34 b(as)f Fq(\020)39 b Fl(moves)33 b(along)h Fq(\015)k Fl(fr)-5 b(om)34 b Fq(\020)3025 1740 y Fn(0)3096 1726 y Fl(to)g Fq(\020)3246 1740 y Fn(0)3305 1726 y Fs(+)20 b(2)p Fq(\031)s Fl(.)42 b(Then,)482 1961 y SDict begin H.S end 482 1961 a 482 1961 a SDict begin 20 H.A end 482 1961 a 482 1961 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.8) cvn H.B /DEST pdfmark end 482 1961 a Fq(\033)s Fs(\()p Fq(\015)5 b(;)15 b(\024)p Fs(\))27 b(=)e(\()p Fo(\000)p Fs(1\))1060 1924 y Fp(n)1107 1961 y Fq(;)48 b(m)p Fs(\()p Fq(\015)5 b(;)15 b(\024)p Fs(\))85 b(=)1727 1900 y(1)p 1723 1940 56 4 v 1723 2024 a Fq(\031)1788 1961 y Fs(\()p Fq(r)1864 1975 y Fp(n)1931 1961 y Fo(\000)20 b Fq(r)2063 1975 y Fp(n)p Fi(\000)p Fn(1)2220 1961 y Fs(+)g Fq(r)2352 1975 y Fp(n)p Fi(\000)p Fn(2)2510 1961 y Fo(\000)g(\001)15 b(\001)g(\001)21 b Fs(+)f(\()p Fo(\000)p Fs(1\))3004 1924 y Fp(n)p Fi(\000)p Fn(1)3142 1961 y Fq(r)3183 1975 y Fn(1)3222 1961 y Fs(\))p Fq(;)p 0 TeXcolorgray -3400 w Fs(\(5.8\))p 0 TeXcolorgray 1559 2196 a(=)1723 2135 y(\()p Fo(\000)p Fs(1\))1909 2102 y Fp(n)p Fi(\000)p Fn(1)p 1723 2175 325 4 v 1857 2258 a Fq(\031)2056 2196 y Fs(\()p Fq(r)2132 2210 y Fn(1)2193 2196 y Fo(\000)f Fq(r)2324 2210 y Fn(2)2384 2196 y Fs(+)h Fo(\001)15 b(\001)g(\001)21 b Fs(+)f(\()p Fo(\000)p Fs(1\))2878 2159 y Fp(n)p Fi(\000)p Fn(1)3016 2196 y Fq(r)3057 2210 y Fp(n)3104 2196 y Fs(\))p Fq(:)p 0 TeXcolorgray -3282 w Fs(\(5.9\))p 0 TeXcolorgray -118 2369 a SDict begin H.S end -118 2369 a -118 2369 a SDict begin 20 H.A end -118 2369 a -118 2369 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.3) cvn H.B /DEST pdfmark end -118 2369 a 104 x Fs(5.3.)47 b Fv(The)34 b(F)-9 b(ourier)36 b(co)s(e\016cien)m(ts.)-118 2589 y SDict begin H.S end -118 2589 a -118 2589 a SDict begin 20 H.A end -118 2589 a -118 2589 a SDict begin [ /View [/XYZ H.V] /Dest (subsubsection.5.3.1) cvn H.B /DEST pdfmark end -118 2589 a 166 x Fs(5.3.1.)47 b Fl(F)-7 b(or)36 b Fq(B)5 b Fl(.)45 b Fs(By)32 b(equation)i(\()p 0 0 1 TeXcolorrgb 1011 2755 a SDict begin H.S end 1011 2755 a 0 0 1 TeXcolorrgb Fs(5.3)p 0 0 1 TeXcolorrgb 1127 2696 a SDict begin H.R end 1127 2696 a 1127 2755 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.5.3) cvn H.B /ANN pdfmark end 1127 2755 a 0 TeXcolorgray Fs(\),)g(w)m(e)e(ha)m(v)m(e)i(to)f(compute)g Fq(w)r Fs(\()p Fq(f)10 b Fs(\()p Fo(\001)p Fq(;)15 b(\020)7 b Fs(\))p Fq(;)15 b(f)10 b Fs(\()p Fo(\001)p Fq(;)15 b(\020)29 b Fs(+)22 b(2)p Fq(\031)s Fs(\)\).)47 b(With)33 b(this)f(aim)h(in)f (view,)-118 2921 y(w)m(e)f(apply)f(the)g(construction)h(done)f(in)g (section)p 0 0 1 TeXcolorrgb 1570 2921 a SDict begin H.S end 1570 2921 a 0 0 1 TeXcolorrgb Fs(5.1)p 0 0 1 TeXcolorrgb 1686 2862 a SDict begin H.R end 1686 2862 a 1686 2921 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (subsection.5.1) cvn H.B /ANN pdfmark end 1686 2921 a 0 TeXcolorgray 30 w Fs(with:)447 3156 y Fq(h)p Fs(\()p Fq(x;)15 b(\020)7 b Fs(\))26 b(=)f Fq(f)10 b Fs(\()p Fq(x;)15 b(\020)7 b Fs(\))p Fq(;)45 b(g)s Fs(\()p Fq(x;)15 b(\020)7 b Fs(\))27 b(=)e(\()p Fq(T)13 b(f)d Fs(\)\()p Fq(x;)15 b(\020)7 b Fs(\))p Fq(;)76 b Fs(with)91 b(\()p Fq(T)13 b(f)d Fs(\)\()p Fq(x;)15 b(\020)7 b Fs(\))25 b(=)g Fq(f)10 b Fs(\()p Fq(x;)15 b(\020)27 b Fs(+)20 b(2)p Fq(\031)s Fs(\);)-118 3391 y(and)-118 3626 y(\(5.10\))965 3626 y SDict begin H.S end 965 3626 a 965 3626 a SDict begin 20 H.A end 965 3626 a 965 3626 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.10) cvn H.B /DEST pdfmark end 965 3626 a Fq(D)1040 3641 y Fp(h)1111 3626 y Fs(=)k Fo(D)s Fq(;)15 b(D)1394 3640 y Fp(g)1460 3626 y Fs(=)25 b Fo(D)e(\000)d Fs(2)p Fq(\031)s(;)46 b(\020)1951 3641 y Fp(h)2021 3626 y Fs(=)25 b Fq(\020)2157 3640 y Fn(0)2196 3626 y Fq(;)76 b(\020)2337 3640 y Fp(g)2402 3626 y Fs(=)25 b Fq(\020)2538 3640 y Fn(0)2598 3626 y Fo(\000)20 b Fs(2)p Fq(\031)s(;)-118 3929 y Fs(\(5.11\))806 3929 y SDict begin H.S end 806 3929 a 806 3929 a SDict begin 20 H.A end 806 3929 a 806 3929 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.11) cvn H.B /DEST pdfmark end 806 3929 a Fo(8)p Fq(\020)32 b Fo(2)24 b Fq(D)1089 3944 y Fp(h)1134 3929 y Fq(;)46 b(\024)1257 3944 y Fp(h)1302 3929 y Fs(\()p Fq(\020)7 b Fs(\))26 b(=)f Fq(\024)p Fs(\()p Fq(\020)7 b Fs(\))p Fq(;)76 b Fo(8)p Fq(\020)32 b Fo(2)24 b Fq(D)2094 3943 y Fp(g)2135 3929 y Fq(;)76 b(\024)2288 3943 y Fp(g)2328 3929 y Fs(\()p Fq(\020)7 b Fs(\))26 b(=)f Fq(\024)p Fs(\()p Fq(\020)i Fs(+)20 b(2)p Fq(\031)s Fs(\))p Fq(:)-118 4164 y Fs(W)-8 b(e)31 b(will)g(start)g(b)m(y)-118 4298 y SDict begin H.S end -118 4298 a -118 4298 a SDict begin 20 H.A end -118 4298 a -118 4298 a SDict begin [ /View [/XYZ H.V] /Dest (subsubsection.5.3.2) cvn H.B /DEST pdfmark end -118 4298 a 149 x Fs(5.3.2.)47 b Fl(A)n(b)-5 b(ove)27 b(the)g(r)-5 b(e)g(al)29 b(line.)45 b Fs(W)-8 b(e)26 b(tak)m(e)g(the)e(meeting)h(domain)g Fq(D)2151 4461 y Fn(0)2214 4447 y Fs(as)g(the)f(sub)s(domain)f(of)h (the)h(strip)f(0)h Fq(<)g Fs(Im)p Fq(\020)32 b(<)25 b(Y)-118 4613 y Fs(b)s(et)m(w)m(een)36 b(the)f(Stok)m(es)h(lines)f Fq(\033)954 4574 y Fn(+)951 4639 y(1)1037 4613 y Fo(\000)23 b Fs(2)p Fq(\031)38 b Fs(and)d Fq(\033)1503 4574 y Fn(+)1500 4639 y(1)1562 4613 y Fs(.\(In)g(this)g(domain)f(w)m(e)i(ha)m(v)m(e)g (Im)f Fq(\024)2799 4627 y Fp(g)2872 4613 y Fs(=)e Fo(\000)p Fs(Im)h Fq(\024)3242 4628 y Fp(h)3320 4613 y Fq(<)f Fs(0\).)56 b(The)34 b(arc)-118 4779 y Fq(\015)-71 4793 y Fn(0)-1 4779 y Fs(connects)d(the)f(p)s(oin)m(t)g Fq(\020)801 4793 y Fp(g)872 4779 y Fs(to)h Fq(\020)1023 4794 y Fp(h)1067 4779 y Fs(.)41 b(By)30 b(\()p 0 0 1 TeXcolorrgb 1310 4779 a SDict begin H.S end 1310 4779 a 0 0 1 TeXcolorrgb Fs(5.10)p 0 0 1 TeXcolorrgb 1472 4720 a SDict begin H.R end 1472 4720 a 1472 4779 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.5.10) cvn H.B /ANN pdfmark end 1472 4779 a 0 TeXcolorgray Fs(\),)h(this)f(de\014nes)g(the)g(p)s(erio)s(d)f (\()p Fq(\015)2560 4793 y Fn(0)2600 4779 y Fq(;)15 b(\024)2692 4793 y Fp(g)2733 4779 y Fs(\).)41 b(Using)30 b(\()p 0 0 1 TeXcolorrgb 3124 4779 a SDict begin H.S end 3124 4779 a 0 0 1 TeXcolorrgb Fs(5.11)p 0 0 1 TeXcolorrgb 3286 4720 a SDict begin H.R end 3286 4720 a 3286 4779 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.5.11) cvn H.B /ANN pdfmark end 3286 4779 a 0 TeXcolorgray Fs(\),)h(one)g(gets)g(that)-118 4945 y Fq(m)p Fs(\()p Fq(f)10 b Fs(\()p Fo(\001)p Fq(;)15 b(\020)7 b Fs(\))p Fq(;)15 b(f)10 b Fs(\()p Fo(\001)p Fq(;)15 b(\020)28 b Fs(+)19 b(2)p Fq(\031)s(;)c(D)802 4959 y Fn(0)843 4945 y Fs(\)\))26 b(=)f Fq(m)p Fs(\()p Fq(\015)1197 4959 y Fn(0)1257 4945 y Fs(+)20 b(2)p Fq(\031)s(;)15 b(\024)p Fs(\))32 b([)p 0 1 0 TeXcolorrgb 1632 4945 a SDict begin H.S end 1632 4945 a 0 1 0 TeXcolorrgb Fs(9)p 0 1 0 TeXcolorrgb 1678 4886 a SDict begin H.R end 1678 4886 a 1678 4945 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.FEKL1) cvn H.B /ANN pdfmark end 1678 4945 a 0 TeXcolorgray Fs(].)-118 5111 y(W)-8 b(e)30 b(use)f(Lemma)p 0 0 1 TeXcolorrgb 517 5111 a SDict begin H.S end 517 5111 a 0 0 1 TeXcolorrgb Fs(5.2)p 0 0 1 TeXcolorrgb 633 5052 a SDict begin H.R end 633 5052 a 633 5111 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.5.2) cvn H.B /ANN pdfmark end 633 5111 a 0 TeXcolorgray 29 w Fs(to)h(compute)g(the)f(index.)40 b(T)-8 b(o)30 b(do)f(this,)h(w)m(e)g(ha)m(v)m(e)g(to)g(compute)g(Re)f Fq(\024)h Fs(at)g(the)f(in)m(tersection)i(of)-118 5277 y Fq(\015)-71 5291 y Fn(0)-11 5277 y Fs(+)20 b(2)p Fq(\031)33 b Fs(and)d Fo(G)5 b Fs(.)-118 5443 y(As)31 b Fq(\020)j Fo(!)26 b Fs(Re)32 b Fq(\024)p Fs(\()p Fq(\020)7 b Fs(\))32 b(is)f(constan)m(t)i(on)e(an)m(y)h(connected)g(comp)s(onen)m(t)g Fo(G)2253 5457 y Fp(j)2321 5443 y Fs(of)f Fo(G)5 b Fs(.)44 b(Let)32 b(us)f(start)h(b)m(y)f(de\014ning)f(the)i(index)-118 5609 y Fq(\025)-65 5571 y Fn(+)-65 5636 y Fp(j)33 5609 y Fs(of)40 b Fo(G)200 5623 y Fp(j)237 5609 y Fs(,)i(the)d(result)h(of)g (alternation)h(\()p Fo(\001)15 b(\001)g(\001)31 b Fq(;)15 b Fs(+)p Fq(;)g Fo(\000)p Fq(;)g Fs(+)p Fq(;)g Fo(\000)p Fq(;)g Fo(\001)g(\001)g(\001)i Fs(\))40 b(due)f(to)h(the)g(crossing)g (of)g Fo(G)3159 5623 y Fp(j)3195 5609 y Fs(.)68 b(W)-8 b(e)41 b(notice)g(that)p 0 TeXcolorgray 1854 5715 a Fn(19)p 0 TeXcolorgray eop end %%Page: 20 20 TeXDict begin 20 19 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray -118 -406 a SDict begin H.S end -118 -406 a 0 TeXcolorgray 0 TeXcolorgray -118 -406 a SDict begin H.R end -118 -406 a -118 -406 a SDict begin [ /View [/XYZ H.V] /Dest (page.20) cvn H.B /DEST pdfmark end -118 -406 a 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray 981 165 a @beginspecial @setspecial tx@Dict begin STP newpath 0.05472 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 273.7503 32.85004 -54.75006 32.85004 /Lineto /lineto load def false Line gsave 0.05472 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.05472 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 273.7503 -32.85004 -54.75006 -32.85004 /Lineto /lineto load def false Line gsave 0.05472 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.05472 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 172.46269 32.85004 151.93141 16.42502 136.87515 0.0 121.81888 -16.42502 101.28761 -32.85004 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.05472 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.05472 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 139.61266 32.85004 136.87515 0.0 139.61266 -32.85004 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.05472 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.05472 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 101.28761 32.85004 121.81888 16.42502 136.87515 0.0 151.93141 -16.42502 172.46269 -32.85004 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.05472 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.05472 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 98.55011 32.85004 109.50012 0.0 98.55011 -32.85004 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.05472 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.05472 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 39.42001 32.85004 54.75006 0.0 68.43758 -16.42502 87.6001 -32.85004 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.05472 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.05472 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 87.6001 32.85004 68.43758 16.42502 54.75006 0.0 39.42001 -32.85004 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.05472 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray [ -52.01256 19.16252 false NArray /DS 2.0 2. CLW mul add 2 div def /Symbol 0. [1.8 0.0 0.0 1.8 -0.495 -0.4788] FontDot /Dot { moveto gsave 1. 1. scale show grestore } bind def newpath n { transform floor .5 add exch floor .5 add exch itransform Dot } repeat end @endspecial 481 97 a tx@Dict begin { 5.0 9.5407 7.60416 2.12917 180. Uput UUput } PutCoor PutBegin end 481 97 a 441 119 a Fq(\020)481 133 y Fn(0)481 97 y tx@Dict begin PutEnd end 481 97 a 981 165 a @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ -35.58754 1.09497 -35.58754 -1.09497 /Lineto /lineto load def false Line gsave 0.54753 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 19.16252 1.09497 19.16252 -1.09497 /Lineto /lineto load def false Line gsave 0.54753 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 19.16252 0.0 -35.58754 0.0 /Lineto /lineto load def false Line gsave 0.54753 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 31.75505 43.80005 29.56499 29.01755 19.16252 0.0 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.54753 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ -42.70506 -43.80005 -41.06255 -21.90002 -35.58754 0.0 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.54753 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 210.78773 1.09497 210.78773 -1.09497 /Lineto /lineto load def false Line gsave 0.54753 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 265.5378 1.09497 265.5378 -1.09497 /Lineto /lineto load def false Line gsave 0.54753 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 265.5378 0.0 210.78773 0.0 /Lineto /lineto load def false Line gsave 0.54753 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 278.13033 43.80005 275.94026 29.01755 265.5378 0.0 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.54753 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray [ 194.36272 19.16252 false NArray /DS 2.0 2. CLW mul add 2 div def /Symbol 0. [1.8 0.0 0.0 1.8 -0.495 -0.4788] FontDot /Dot { moveto gsave 1. 1. scale show grestore } bind def newpath n { transform floor .5 add exch floor .5 add exch itransform Dot } repeat end @endspecial 2526 97 a tx@Dict begin { 5.0 35.03363 7.60416 2.12917 180. Uput UUput } PutCoor PutBegin end 2526 97 a 2381 119 a Fq(\020)2421 133 y Fn(0)2480 119 y Fs(+)20 b(2)p Fq(\031)2526 97 y tx@Dict begin PutEnd end 2526 97 a 981 165 a @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray [ 194.36272 -19.16252 false NArray /DS 2.0 2. CLW mul add 2 div def /Symbol 0. [1.8 0.0 0.0 1.8 -0.495 -0.4788] FontDot /Dot { moveto gsave 1. 1. scale show grestore } bind def newpath n { transform floor .5 add exch floor .5 add exch itransform Dot } repeat end @endspecial 2526 233 a tx@Dict begin { 5.0 35.03363 7.60416 2.12917 180. Uput UUput } PutCoor PutBegin end 2526 233 a 2381 256 a Fq(\020)2421 270 y Fn(0)2480 256 y Fs(+)g(2)p Fq(\031)2526 233 y tx@Dict begin PutEnd end 2526 233 a 981 165 a @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray [ -52.01256 -19.16252 false NArray /DS 2.0 2. CLW mul add 2 div def /Symbol 0. [1.8 0.0 0.0 1.8 -0.495 -0.4788] FontDot /Dot { moveto gsave 1. 1. scale show grestore } bind def newpath n { transform floor .5 add exch floor .5 add exch itransform Dot } repeat end @endspecial 481 233 a tx@Dict begin { 5.0 9.5407 7.60416 2.12917 180. Uput UUput } PutCoor PutBegin end 481 233 a 441 256 a Fq(\020)481 270 y Fn(0)481 233 y tx@Dict begin PutEnd end 481 233 a 981 165 a @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 203.6702 -43.80005 205.31273 -21.90002 210.78773 0.0 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.54753 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 1 0 0 setrgbcolor /ArrowA { moveto } def /ArrowB { } def [ 194.36272 19.16252 112.23763 13.68752 35.58754 2.7375 24.63753 -10.95001 10.95001 0.0 -21.90002 13.68752 -30.11253 16.42502 -52.01256 19.16252 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.54753 SLW 1 0 0 setrgbcolor 0 setlinecap stroke grestore end @endspecial 617 -63 a tx@Dict begin { 5.0 10.4192 4.71457 2.12917 180. Uput UUput } PutCoor PutBegin end 617 -63 a 1 0 0 TeXcolorrgb 574 -52 a Fq(\015)621 -38 y Fn(1)p 0 TeXcolorgray 617 -63 a tx@Dict begin PutEnd end 617 -63 a 981 165 a @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 1 0 0 setrgbcolor /ArrowA { moveto } def /ArrowB { } def [ 194.36272 -19.16252 13.68752 -13.68752 0.0 0.0 -30.11253 10.95001 -46.53755 0.0 -52.01256 -19.16252 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.54753 SLW 1 0 0 setrgbcolor 0 setlinecap stroke grestore end @endspecial 981 301 a tx@Dict begin { 5.0 10.4192 4.71457 2.12917 180. Uput UUput } PutCoor PutBegin end 981 301 a 1 0 0 TeXcolorrgb 938 312 a Fq(\015)985 326 y Fn(0)p 0 TeXcolorgray 981 301 a tx@Dict begin PutEnd end 981 301 a 1030 827 a Fr(Figure)34 b(4.)p 0 TeXcolorgray 1501 827 a SDict begin H.S end 1501 827 a 0 TeXcolorgray 0 TeXcolorgray 1501 827 a SDict begin H.R end 1501 827 a 1501 827 a SDict begin [ /View [/XYZ H.V] /Dest (figure.4) cvn H.B /DEST pdfmark end 1501 827 a 0 TeXcolorgray Fs(P)m(erio)s(ds)c(equiv)-5 b(alen)m(t)32 b(to)f Fq(\015)2416 841 y Fn(0)2486 827 y Fs(and)f Fq(\015)2710 841 y Fn(1)p 0 TeXcolorgray -118 1070 a Fq(\025)-65 1032 y Fn(+)-65 1098 y Fp(j)19 1070 y Fo(2)25 b(f\000)p Fs(1)p Fq(;)15 b Fs(0)p Fq(;)g Fs(1)p Fo(g)p Fs(.)-118 1236 y(W)-8 b(e)31 b(set)-118 1464 y(\(5.12\))1368 1464 y SDict begin H.S end 1368 1464 a 1368 1464 a SDict begin 20 H.A end 1368 1464 a 1368 1464 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.12) cvn H.B /DEST pdfmark end 1368 1464 a Fq(N)1441 1478 y Fp(j)1503 1464 y Fs(=)25 b(1)c(+)1760 1346 y Fp(j)t Fi(\000)p Fn(1)1755 1378 y Fh(X)1764 1573 y Fp(i)p Fn(=1)1887 1464 y Fs(\()1953 1349 y Fp(n)1996 1359 y Fg(i)1922 1378 y Fh(X)1932 1576 y Fp(l)q Fn(=1)2053 1464 y Fs(\()p Fq(o)2132 1479 y Fp(l)2179 1464 y Fo(\000)f Fs(1\)\))p Fq(:)-118 1720 y Fs(Here)38 b Fq(n)159 1734 y Fp(j)233 1720 y Fs(is)g(the)g(n)m(um)m(b)s(er)e(of)i (extrem)m(um)g(in)f Fo(G)1549 1734 y Fp(j)1586 1720 y Fs(,)j(and)d Fq(o)1879 1735 y Fp(l)1943 1720 y Fs(is)g(the)h(order)g (of)g(the)f Fq(l)2754 1687 y Fn(th)2863 1720 y Fs(extrem)m(um.)63 b(The)37 b(follo)m(wing)-118 1886 y(relations)31 b(hold)-118 2101 y(\(5.13\))1400 2101 y SDict begin H.S end 1400 2101 a 1400 2101 a SDict begin 20 H.A end 1400 2101 a 1400 2101 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.13) cvn H.B /DEST pdfmark end 1400 2101 a Fq(\025)1453 2062 y Fn(+)1453 2127 y(1)1537 2101 y Fs(=)1643 2039 y(1)21 b(+)f(\()p Fo(\000)p Fs(1\))1986 1956 y Fe(P)2061 1968 y Fg(n)2099 1983 y Ff(1)2061 2029 y Fg(l)p Ff(=1)2162 2006 y Fn(\()p Fp(o)2223 2018 y Fg(l)2248 2006 y Fi(\000)p Fn(1\))p 1643 2080 727 4 v 1984 2163 a Fs(2)-118 2315 y(and)30 b(for)g(2)c Fo(\024)e Fq(j)31 b Fo(\024)25 b Fq(N)10 b Fs(,)-118 2554 y(\(5.14\))1158 2554 y SDict begin H.S end 1158 2554 a 1158 2554 a SDict begin 20 H.A end 1158 2554 a 1158 2554 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.14) cvn H.B /DEST pdfmark end 1158 2554 a Fq(\025)1211 2516 y Fn(+)1211 2581 y Fp(j)1295 2554 y Fs(=)25 b(\()p Fo(\000)p Fs(1\))1577 2516 y Fp(N)1633 2526 y Fg(j)1671 2453 y Fh(\020)1735 2492 y Fo(\000)p Fs(1)20 b(+)g(\()p Fo(\000)p Fs(1\))2148 2409 y Fe(P)2224 2410 y Fg(n)2262 2426 y(j)2224 2483 y(l)p Ff(=1)2325 2459 y Fn(\()p Fp(o)2386 2471 y Fg(l)2410 2459 y Fi(\000)p Fn(1\))p 1735 2533 798 4 v 2111 2616 a Fs(2)2542 2453 y Fh(\021)2596 2554 y Fq(:)-118 2768 y Fs(Without)31 b(loss)g(of)f(generalit)m(y)j(w)m(e)e (assume)f(that)h(Re)f Fq(\024)c Fs(=)f(0)31 b(on)f(\(0)p Fq(;)15 b(')2281 2730 y Fi(\000)2281 2794 y Fn(1)2342 2768 y Fs(\).)41 b(By)30 b(the)h(ab)s(o)m(v)m(e)g(notation,)h(w)m(e)f (get)h(that)-118 3022 y(\(5.15\))355 3022 y SDict begin H.S end 355 3022 a 355 3022 a SDict begin 20 H.A end 355 3022 a 355 3022 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.15) cvn H.B /DEST pdfmark end 355 3022 a Fq(m)p Fs(\()p Fq(\015)517 3036 y Fn(0)557 3022 y Fq(;)15 b(\024)p Fs(\))26 b(=)f(\()p Fo(\000)p Fs(1\))992 2985 y Fp(n)1040 2921 y Fh(\020)1095 3022 y Fq(p)1141 3036 y Fn(1)1180 3022 y Fq(\025)1233 2984 y Fn(+)1233 3048 y(1)1312 3022 y Fs(+)20 b(\()p Fq(p)1484 3036 y Fn(1)1544 3022 y Fs(+)g Fq(p)1681 3036 y Fn(2)1720 3022 y Fs(\))p Fq(\025)1808 2984 y Fn(+)1808 3048 y(2)1887 3022 y Fs(+)g Fo(\001)15 b(\001)g(\001)22 b Fs(+)d(\()2280 2904 y Fp(j)2230 2936 y Fh(X)2239 3131 y Fp(i)p Fn(=1)2377 3022 y Fq(p)2423 3036 y Fp(i)2451 3022 y Fs(\))p Fq(\025)2539 2984 y Fn(+)2539 3049 y Fp(j)2619 3022 y Fs(+)h Fo(\001)15 b(\001)g(\001)21 b Fs(+)f(\()2996 2908 y Fp(N)2962 2936 y Fh(X)2971 3131 y Fp(i)p Fn(=1)3109 3022 y Fq(p)3155 3036 y Fp(i)3183 3022 y Fs(\))p Fq(\025)3271 2984 y Fn(+)3271 3051 y Fp(N)3338 2921 y Fh(\021)3418 3022 y Fs(=)25 b Fq(P)3572 3036 y Fn(+)3631 3022 y Fq(:)-118 3260 y Fs(Here)31 b Fq(p)143 3274 y Fp(i)201 3260 y Fs(is)f(the)h(index)f(of)g Fo(B)857 3274 y Fp(i)916 3260 y Fs(and)f Fq(n)c Fs(=)g Fq(N)1341 3274 y Fp(N)1408 3260 y Fs(.)-118 3426 y(No)m(w)31 b(using)f(\()p 0 0 1 TeXcolorrgb 362 3426 a SDict begin H.S end 362 3426 a 0 0 1 TeXcolorrgb Fs(5.3)p 0 0 1 TeXcolorrgb 478 3367 a SDict begin H.R end 478 3367 a 478 3426 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.5.3) cvn H.B /ANN pdfmark end 478 3426 a 0 TeXcolorgray Fs(\))h(for)f Fq(B)35 b Fs(and)30 b(Prop)s(osition)p 0 0 1 TeXcolorrgb 1453 3426 a SDict begin H.S end 1453 3426 a 0 0 1 TeXcolorrgb Fs(5.1)p 0 0 1 TeXcolorrgb 1569 3367 a SDict begin H.R end 1569 3367 a 1569 3426 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.5.1) cvn H.B /ANN pdfmark end 1569 3426 a 0 TeXcolorgray 31 w Fs(w)m(e)g(get)-118 3640 y(\(5.16\))160 3640 y SDict begin H.S end 160 3640 a 160 3640 a SDict begin 20 H.A end 160 3640 a 160 3640 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.16) cvn H.B /DEST pdfmark end 160 3640 a Fq(B)f Fs(=)c Fq(A)p Fs(\()p Fq(f)5 b(;)15 b(T)e Fs(\()p Fq(f)d Fs(\))p Fq(;)800 3617 y Fs(~)778 3640 y Fq(D)853 3654 y Fn(0)893 3640 y Fs(\))p Fq(e)983 3575 y Fg(i)p 981 3587 30 3 v 981 3629 a(")1020 3603 y Fp(S)t Fn(\()p Fp(f)s(;T)g Fn(\()p Fp(f)d Fn(\))p Fp(;)1334 3586 y Fn(~)1317 3603 y Fp(D)1375 3612 y Ff(0)1410 3603 y Fn(\))1458 3640 y Fo(\001)16 b Fq(e)1551 3560 y Fd(\000)p Ff(2)p Fg(i\031)r(P)1728 3575 y Ff(+)1780 3560 y Fg(\020)1808 3575 y Ff(0)p 1552 3587 292 3 v 1683 3629 a Fg(")1882 3640 y Fs(=)25 b Fq(B)2047 3654 y Fn(+)2106 3640 y Fs(\()p Fq(E)5 b(;)15 b(")p Fs(\))j Fo(\001)e Fq(e)2431 3603 y Fi(\000)2534 3575 y Fg(i)p 2497 3587 98 3 v 2497 3629 a Ff(2)p Fg(\031)r(")2604 3603 y Fp(P)2649 3612 y Ff(+)2700 3603 y Fp(\020)2731 3612 y Ff(0)2770 3640 y Fq(;)43 b(T)13 b Fs(\()p Fq(f)d Fs(\)\()p Fq(x;)15 b(\020)7 b Fs(\))26 b(=)f Fq(f)10 b Fs(\()p Fq(x;)15 b(\020)22 b Fs(+)16 b(2)p Fq(\031)s Fs(\))p Fq(:)-118 3738 y SDict begin H.S end -118 3738 a -118 3738 a SDict begin 20 H.A end -118 3738 a -118 3738 a SDict begin [ /View [/XYZ H.V] /Dest (subsubsection.5.3.3) cvn H.B /DEST pdfmark end -118 3738 a 143 x Fs(5.3.3.)47 b Fl(Below)34 b(the)f(r)-5 b(e)g(al)34 b(line.)p 0 TeXcolorgray 0 TeXcolorgray 94 4261 a Fs(Belo)m(w)g(the)f(real)g(line,)g(w)m(e)g(tak)m(e)h(the)e (domain)g(of)h(the)f(strip)g Fo(\000)p Fq(Y)48 b(<)28 b Fs(Im)p Fq(\020)35 b(<)28 b Fs(0)k(lo)s(cated)i(b)s(et)m(w)m(een)f (the)f(stok)m(es)-118 4427 y(line)g Fq(\033)110 4389 y Fi(\000)107 4453 y Fn(1)190 4427 y Fo(\000)21 b Fs(2)p Fq(\031)35 b Fs(and)c Fq(\033)647 4389 y Fi(\000)644 4453 y Fn(1)738 4427 y Fs(as)h(a)g(regular)g(domain;)h(whic)m(h)e(w)m (e)i(denote)f(b)m(y)2442 4404 y(~)2407 4427 y Fq(D)2482 4441 y Fn(0)2522 4427 y Fs(.)45 b(W)-8 b(e)33 b(set)i(~)-48 b Fq(\015)2940 4441 y Fn(0)3007 4427 y Fs(=)27 b Fq(\015)5 b Fs(\()p Fq(f)g(;)15 b(T)e Fs(\()p Fq(f)d Fs(\))p Fq(;)3535 4404 y Fs(~)3513 4427 y Fq(D)3588 4441 y Fn(1)3628 4427 y Fs(\),)33 b(then)-118 4593 y(it)d(de\014nes)g(a)g(p)s(erio)s(d,)f (and)h(so)g Fq(m)p Fs(\()p Fq(f)5 b(;)15 b(T)e Fs(\()p Fq(f)d Fs(\))p Fq(;)1399 4570 y Fs(~)1378 4593 y Fq(D)1453 4607 y Fn(1)1493 4593 y Fs(\))25 b(=)g Fq(m)p Fs(\()s(~)-48 b Fq(\015)1811 4607 y Fn(0)1870 4593 y Fs(+)20 b(2)p Fq(\031)s(;)15 b(\024)p Fs(\).)42 b(The)30 b(curv)m(e)g(de\014ning)f(a) i(p)s(erio)s(d)d(equiv)-5 b(alen)m(t)32 b(to)-118 4759 y(\()s(~)-48 b Fq(\015)-36 4773 y Fn(0)24 4759 y Fs(+)20 b(2)p Fq(\031)s(;)15 b(\024)p Fs(\))32 b(is)e(represen)m(ted)h(in)f (\014gure)p 0 0 1 TeXcolorrgb 1307 4759 a SDict begin H.S end 1307 4759 a 0 0 1 TeXcolorrgb Fs(4)p 0 0 1 TeXcolorrgb 1353 4701 a SDict begin H.R end 1353 4701 a 1353 4759 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (figure.4) cvn H.B /ANN pdfmark end 1353 4759 a 0 TeXcolorgray Fs(.)-118 4925 y(The)g(computation)h(of)g(the)f(index)g Fq(\025)1157 4887 y Fi(\000)1157 4953 y Fp(j)1246 4925 y Fs(of)h Fo(G)1404 4939 y Fp(j)1471 4925 y Fs(giv)m(es)h(in)e(this)g (case)h(that)1351 5176 y Fq(\025)1404 5137 y Fi(\000)1404 5202 y Fn(1)1488 5176 y Fs(=)1594 5114 y Fo(\000)p Fs(1)20 b(+)g(\()p Fo(\000)p Fs(1\))2007 5031 y Fe(P)2083 5043 y Fg(n)2121 5058 y Ff(1)2083 5103 y Fg(i)p Ff(=1)2186 5081 y Fn(\()p Fp(o)2247 5093 y Fg(l)2272 5081 y Fi(\000)p Fn(1\))p 1594 5155 800 4 v 1971 5238 a Fs(2)2403 5176 y Fq(;)-118 5390 y Fs(and)1138 5587 y Fq(\025)1191 5548 y Fi(\000)1191 5614 y Fp(j)1275 5587 y Fs(=)25 b(\()p Fo(\000)p Fs(1\))1557 5549 y Fp(N)1613 5559 y Fg(j)1650 5486 y Fh(\020)1715 5525 y Fo(\000)p Fs(1)20 b(+)g(\()p Fo(\000)p Fs(1\))2128 5442 y Fe(P)2203 5443 y Fg(l)p Ff(=)p Fg(n)2308 5459 y(j)2203 5515 y(l)p Ff(=1)2345 5492 y Fn(\()p Fp(o)2406 5504 y Fg(l)2431 5492 y Fi(\000)p Fn(1\))p 1715 5566 838 4 v 2111 5649 a Fs(2)2562 5486 y Fh(\021)2617 5587 y Fq(:)p 0 TeXcolorgray 1854 5715 a Fn(20)p 0 TeXcolorgray eop end %%Page: 21 21 TeXDict begin 21 20 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray -118 -406 a SDict begin H.S end -118 -406 a 0 TeXcolorgray 0 TeXcolorgray -118 -406 a SDict begin H.R end -118 -406 a -118 -406 a SDict begin [ /View [/XYZ H.V] /Dest (page.21) cvn H.B /DEST pdfmark end -118 -406 a 0 TeXcolorgray 0 TeXcolorgray 199 x Fs(By)31 b(this)f(notation)h(w)m(e)g(get)h(that)-118 78 y(\(5.17\))355 78 y SDict begin H.S end 355 78 a 355 78 a SDict begin 20 H.A end 355 78 a 355 78 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.17) cvn H.B /DEST pdfmark end 355 78 a Fq(m)p Fs(\()p Fq(\015)517 92 y Fn(0)557 78 y Fq(;)15 b(\024)p Fs(\))26 b(=)f(\()p Fo(\000)p Fs(1\))992 41 y Fp(n)1040 -23 y Fh(\020)1095 78 y Fq(p)1141 92 y Fn(1)1180 78 y Fq(\025)1233 40 y Fi(\000)1233 104 y Fn(1)1312 78 y Fs(+)20 b(\()p Fq(p)1484 92 y Fn(1)1544 78 y Fs(+)g Fq(p)1681 92 y Fn(2)1720 78 y Fs(\))p Fq(\025)1808 40 y Fi(\000)1808 104 y Fn(2)1887 78 y Fs(+)g Fo(\001)15 b(\001)g(\001)22 b Fs(+)d(\()2280 -40 y Fp(j)2230 -8 y Fh(X)2239 187 y Fp(i)p Fn(=1)2377 78 y Fq(p)2423 92 y Fp(i)2451 78 y Fs(\))p Fq(\025)2539 40 y Fi(\000)2539 105 y Fp(j)2619 78 y Fs(+)h Fo(\001)15 b(\001)g(\001)21 b Fs(+)f(\()2996 -35 y Fp(N)2962 -8 y Fh(X)2971 187 y Fp(i)p Fn(=1)3109 78 y Fq(p)3155 92 y Fp(i)3183 78 y Fs(\))p Fq(\025)3271 40 y Fi(\000)3271 107 y Fp(N)3338 -23 y Fh(\021)3418 78 y Fs(=)25 b Fq(P)3572 92 y Fi(\000)3631 78 y Fq(:)-118 360 y Fs(Using)30 b(\()p 0 0 1 TeXcolorrgb 172 360 a SDict begin H.S end 172 360 a 0 0 1 TeXcolorrgb Fs(5.3)p 0 0 1 TeXcolorrgb 289 302 a SDict begin H.R end 289 302 a 289 360 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.5.3) cvn H.B /ANN pdfmark end 289 360 a 0 TeXcolorgray Fs(\))h(for)f Fq(B)35 b Fs(and)29 b(Prop)s(osition)p 0 0 1 TeXcolorrgb 1264 360 a SDict begin H.S end 1264 360 a 0 0 1 TeXcolorrgb Fs(5.1)p 0 0 1 TeXcolorrgb 1380 302 a SDict begin H.R end 1380 302 a 1380 360 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.5.1) cvn H.B /ANN pdfmark end 1380 360 a 0 TeXcolorgray 30 w Fs(w)m(e)i(get)-118 614 y(\(5.18\))171 614 y SDict begin H.S end 171 614 a 171 614 a SDict begin 20 H.A end 171 614 a 171 614 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.18) cvn H.B /DEST pdfmark end 171 614 a Fq(B)240 628 y Fp(P)285 637 y Fd(\000)366 614 y Fs(=)25 b Fq(A)p Fs(\()p Fq(f)5 b(;)15 b(T)e Fs(\()p Fq(f)d Fs(\))p Fq(;)907 591 y Fs(~)886 614 y Fq(D)961 628 y Fn(1)1001 614 y Fs(\))p Fq(e)1091 549 y Fg(i)p 1088 561 30 3 v 1088 603 a(")1128 577 y Fn(\()p Fp(S)t Fn(\()p Fp(f)s(;T)g(f)s(;)1410 560 y Fn(~)1394 577 y Fp(D)1452 586 y Ff(1)1486 577 y Fn(\))p Fi(\000)p Fn(2)p Fp(\031)r(P)1691 586 y Fd(\000)1744 577 y Fp(\020)1775 586 y Ff(0)1809 577 y Fn(\))1866 614 y Fs(=)25 b Fq(B)2031 628 y Fi(\000)2090 614 y Fs(\()p Fq(E)5 b(;)15 b(")p Fs(\))22 b Fo(\001)f Fq(e)2437 549 y Fg(i)p 2434 561 V 2434 603 a(")2473 577 y Fn(2)p Fp(\031)r(P)2596 586 y Fd(\000)2648 577 y Fp(\020)2679 586 y Ff(0)2718 614 y Fq(;)76 b Fs(\()p Fq(T)13 b(f)d Fs(\)\()p Fq(x;)15 b(\020)7 b Fs(\))26 b(=)e Fq(f)10 b Fs(\()p Fq(x;)15 b(\020)27 b Fs(+)20 b(2)p Fq(\031)s Fs(\))p Fq(:)-118 798 y SDict begin H.S end -118 798 a -118 798 a SDict begin 20 H.A end -118 798 a -118 798 a SDict begin [ /View [/XYZ H.V] /Dest (subsection.5.4) cvn H.B /DEST pdfmark end -118 798 a 136 x Fs(5.4.)47 b Fv(F)-9 b(or)24 b Fq(A)p Fv(.)46 b Fs(F)-8 b(or)22 b(the)f(computation)h(of)f Fq(A)h Fs(using)e (equation)i(\()p 0 0 1 TeXcolorrgb 2019 934 a SDict begin H.S end 2019 934 a 0 0 1 TeXcolorrgb Fs(5.3)p 0 0 1 TeXcolorrgb 2136 875 a SDict begin H.R end 2136 875 a 2136 934 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.5.3) cvn H.B /ANN pdfmark end 2136 934 a 0 TeXcolorgray Fs(\),)h(w)m(e)f(ha)m (v)m(e)g(to)g(compute)f Fq(w)r Fs(\()p Fq(f)3162 901 y Fi(\003)3202 934 y Fs(\()p Fo(\001)p Fq(;)15 b(\020)7 b Fs(\))p Fq(;)15 b(f)10 b Fs(\()p Fo(\001)p Fq(;)15 b(\020)8 b Fs(+)r(2)p Fq(\031)s Fs(\)\).)-118 1100 y(It)30 b(su\016ces)g(to)i(apply)e(the)g(metho)s(d)g(presen)m(ted)g(in)g (section)p 0 0 1 TeXcolorrgb 1955 1100 a SDict begin H.S end 1955 1100 a 0 0 1 TeXcolorrgb Fs(5.1)p 0 0 1 TeXcolorrgb 2071 1041 a SDict begin H.R end 2071 1041 a 2071 1100 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (subsection.5.1) cvn H.B /ANN pdfmark end 2071 1100 a 0 TeXcolorgray 30 w Fs(with:)427 1353 y Fq(h)p Fs(\()p Fq(x;)15 b(\020)7 b Fs(\))26 b(=)f Fq(f)865 1316 y Fi(\003)904 1353 y Fs(\()p Fq(x;)15 b(\020)7 b Fs(\))p Fq(;)46 b(g)s Fs(\()p Fq(x;)15 b(\020)7 b Fs(\))26 b(=)f(\()p Fq(T)13 b(f)d Fs(\)\()p Fq(x;)15 b(\020)7 b Fs(\))p Fq(;)77 b Fs(with)90 b(\()p Fq(T)13 b(f)d Fs(\)\()p Fq(x;)15 b(\020)7 b Fs(\))26 b(=)f Fq(f)10 b Fs(\()p Fq(x;)15 b(\020)27 b Fs(+)19 b(2)p Fq(\031)s Fs(\);)-118 1607 y(and)-118 1861 y(\(5.19\))945 1861 y SDict begin H.S end 945 1861 a 945 1861 a SDict begin 20 H.A end 945 1861 a 945 1861 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.19) cvn H.B /DEST pdfmark end 945 1861 a Fq(D)1020 1876 y Fp(h)1091 1861 y Fs(=)25 b Fo(D)1260 1823 y Fi(\003)1299 1861 y Fq(;)15 b(D)1414 1875 y Fp(g)1480 1861 y Fs(=)25 b Fo(D)e(\000)d Fs(2)p Fq(\031)s(;)46 b(\020)1971 1876 y Fp(h)2041 1861 y Fs(=)25 b Fq(\020)2177 1875 y Fn(0)2216 1861 y Fq(;)76 b(\020)2357 1875 y Fp(g)2422 1861 y Fs(=)25 b Fq(\020)2558 1875 y Fn(0)2617 1861 y Fo(\000)20 b Fs(2)p Fq(\031)s(;)-118 2202 y Fs(\(5.20\))771 2202 y SDict begin H.S end 771 2202 a 771 2202 a SDict begin 20 H.A end 771 2202 a 771 2202 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.20) cvn H.B /DEST pdfmark end 771 2202 a Fo(8)p Fq(\020)31 b Fo(2)25 b Fq(D)1054 2217 y Fp(h)1099 2202 y Fq(;)46 b(\024)1222 2217 y Fp(h)1267 2202 y Fs(\()p Fq(\020)7 b Fs(\))25 b(=)g Fo(\000)p 1576 2152 53 4 v Fq(\024)q Fs(\()p 1664 2128 47 4 v Fq(\020)6 b Fs(\))p Fq(;)77 b Fo(8)p Fq(\020)31 b Fo(2)25 b Fq(D)2130 2216 y Fp(g)2170 2202 y Fq(;)76 b(\024)2323 2216 y Fp(g)2364 2202 y Fs(\()p Fq(\020)7 b Fs(\))25 b(=)g Fq(\024)p Fs(\()p Fq(\020)j Fs(+)19 b(2)p Fq(\031)s Fs(\))p Fq(:)-118 2382 y SDict begin H.S end -118 2382 a -118 2382 a SDict begin 20 H.A end -118 2382 a -118 2382 a SDict begin [ /View [/XYZ H.V] /Dest (subsubsection.5.4.1) cvn H.B /DEST pdfmark end -118 2382 a 140 x Fs(5.4.1.)47 b Fl(A)n(b)-5 b(ove)35 b(the)h(r)-5 b(e)g(al)37 b(line.)45 b Fs(In)33 b(this)g(case,)1494 2499 y(~)1472 2522 y Fq(D)1547 2536 y Fn(0)1587 2522 y Fs(,)h(the)g(meeting)g(domain,)g(is)g(the)f(sub)s(domain)f(of)h(the)h (strip)f Fo(f)p Fs(0)e Fq(<)-118 2688 y Fs(Im)o Fq(\020)38 b(<)32 b(Y)19 b Fo(g)35 b Fs(lo)s(cated)g(b)s(et)m(w)m(een)g(the)f (lines)h Fq(\033)1424 2649 y Fn(+)1421 2714 y(1)1505 2688 y Fo(\000)23 b Fs(2)p Fq(\031)37 b Fs(and)p 1914 2598 115 4 v 34 w Fq(\033)1969 2649 y Fi(\000)1966 2714 y Fn(1)2028 2688 y Fs(,)e(the)g(symmetric)f(to)h Fq(\033)2865 2649 y Fi(\000)2862 2714 y Fn(1)2958 2688 y Fs(with)f(resp)s(ect)g(to)h Fm(R)p Fs(.)51 b(The)-118 2854 y(arc)31 b Fq(\015)5 b Fs(\()p Fq(f)176 2821 y Fi(\003)215 2854 y Fq(;)15 b(T)e Fs(\()p Fq(f)d Fs(\))p Fq(;)508 2831 y Fs(~)486 2854 y Fq(D)561 2868 y Fn(0)601 2854 y Fs(\))31 b(de\014nes)e(a)i(p)s(erio)s (d)e(\()s(~)-48 b Fq(\015)1406 2868 y Fn(0)1446 2854 y Fq(;)15 b(\024)1538 2868 y Fp(g)1579 2854 y Fs(\),)31 b(in)f(\014gure)p 0 0 1 TeXcolorrgb 2029 2854 a SDict begin H.S end 2029 2854 a 0 0 1 TeXcolorrgb Fs(5)p 0 0 1 TeXcolorrgb 2074 2795 a SDict begin H.R end 2074 2795 a 2074 2854 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (figure.5) cvn H.B /ANN pdfmark end 2074 2854 a 0 TeXcolorgray 30 w Fs(w)m(e)h(represen)m(t)g(the)f(curv)m(e)k(~)-48 b Fq(\015)3077 2868 y Fn(0)3137 2854 y Fs(+)19 b(2)p Fq(\031)s Fs(.)-118 3020 y(Similarly)31 b(to)g(the)f(computation)h(of)g Fq(B)5 b Fs(,)30 b(w)m(e)h(de\014ne)f(the)g(index)g Fq(\014)2153 2982 y Fn(+)2148 3047 y Fp(j)2243 3020 y Fs(of)g Fo(G)2400 3034 y Fp(j)2467 3020 y Fs(and)g(in)g(this)g(case,)i(w)m(e)f(ha)m(v)m (e:)1349 3309 y Fq(\014)1405 3271 y Fn(+)1400 3335 y(1)1490 3309 y Fs(=)1596 3248 y Fo(\000)p Fs(1)20 b(+)g(\()p Fo(\000)p Fs(1\))2009 3165 y Fe(P)2084 3177 y Fg(n)2122 3192 y Ff(1)2084 3237 y Fg(i)p Ff(=1)2188 3215 y Fn(\()p Fp(o)2249 3227 y Fg(l)2273 3215 y Fi(\000)p Fn(1\))p 1596 3289 800 4 v 1973 3372 a Fs(2)2405 3309 y Fq(;)-118 3563 y Fs(and)1136 3799 y Fq(\014)1192 3761 y Fn(+)1187 3826 y Fp(j)1277 3799 y Fs(=)k(\()p Fo(\000)p Fs(1\))1558 3762 y Fp(N)1614 3772 y Fg(j)1652 3698 y Fh(\020)1716 3738 y Fo(\000)p Fs(1)d(+)f(\()p Fo(\000)p Fs(1\))2130 3654 y Fe(P)2205 3656 y Fg(l)p Ff(=)p Fg(n)2310 3672 y(j)2205 3728 y(l)p Ff(=1)2347 3704 y Fn(\()p Fp(o)2408 3716 y Fg(l)2432 3704 y Fi(\000)p Fn(1\))p 1716 3778 838 4 v 2112 3862 a Fs(2)2564 3698 y Fh(\021)2618 3799 y Fq(:)-118 4034 y Fs(By)31 b(this)f(notation)h(w)m(e)g(get)h(that)-118 4319 y(\(5.21\))345 4319 y SDict begin H.S end 345 4319 a 345 4319 a SDict begin 20 H.A end 345 4319 a 345 4319 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.21) cvn H.B /DEST pdfmark end 345 4319 a Fq(m)p Fs(\()p Fq(\015)507 4333 y Fn(0)546 4319 y Fq(;)15 b(\024)p Fs(\))27 b(=)e(\()p Fo(\000)p Fs(1\))982 4281 y Fp(n)1030 4218 y Fh(\020)1084 4319 y Fq(p)1130 4333 y Fn(1)1169 4319 y Fq(\014)1225 4281 y Fn(+)1220 4345 y(1)1304 4319 y Fs(+)20 b(\()p Fq(p)1476 4333 y Fn(1)1536 4319 y Fs(+)g Fq(p)1673 4333 y Fn(2)1712 4319 y Fs(\))p Fq(\014)1803 4281 y Fn(+)1798 4345 y(2)1883 4319 y Fs(+)g Fo(\001)15 b(\001)g(\001)21 b Fs(+)f(\()2276 4200 y Fp(j)2226 4233 y Fh(X)2235 4428 y Fp(i)p Fn(=1)2373 4319 y Fq(p)2419 4333 y Fp(i)2447 4319 y Fs(\))p Fq(\014)2538 4281 y Fn(+)2533 4346 y Fp(j)2618 4319 y Fs(+)g Fo(\001)15 b(\001)g(\001)21 b Fs(+)f(\()2995 4205 y Fp(N)2961 4233 y Fh(X)2970 4428 y Fp(i)p Fn(=1)3108 4319 y Fq(p)3154 4333 y Fp(i)3181 4319 y Fs(\))p Fq(\014)3272 4281 y Fn(+)3267 4348 y Fp(N)3335 4218 y Fh(\021)3415 4319 y Fs(=)25 b Fq(Q)3583 4333 y Fn(+)3642 4319 y Fq(:)-118 4601 y Fs(Using)30 b(\()p 0 0 1 TeXcolorrgb 172 4601 a SDict begin H.S end 172 4601 a 0 0 1 TeXcolorrgb Fs(5.3)p 0 0 1 TeXcolorrgb 289 4543 a SDict begin H.R end 289 4543 a 289 4601 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.5.3) cvn H.B /ANN pdfmark end 289 4601 a 0 TeXcolorgray Fs(\))h(for)f Fq(A)g Fs(and)g(Prop)s(osition)p 0 0 1 TeXcolorrgb 1258 4601 a SDict begin H.S end 1258 4601 a 0 0 1 TeXcolorrgb Fs(5.1)p 0 0 1 TeXcolorrgb 1375 4543 a SDict begin H.R end 1375 4543 a 1375 4601 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.5.1) cvn H.B /ANN pdfmark end 1375 4601 a 0 TeXcolorgray 30 w Fs(w)m(e)h(get)g(that)1248 4855 y Fq(m)p Fs(\()p Fq(f)1418 4817 y Fi(\003)1457 4855 y Fq(;)15 b(T)e(f)5 b(;)1674 4832 y Fs(~)1653 4855 y Fq(D)1728 4869 y Fn(0)1768 4855 y Fs(\))25 b(=)g Fq(m)p Fs(\()s(~)-48 b Fq(\015)2086 4869 y Fn(0)2126 4855 y Fq(;)15 b(\024)p Fs(\))26 b(=)f Fq(Q)2447 4869 y Fn(+)2506 4855 y Fq(;)-118 5108 y Fs(and)-118 5362 y(\(5.22\))198 5362 y SDict begin H.S end 198 5362 a 198 5362 a SDict begin 20 H.A end 198 5362 a 198 5362 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.22) cvn H.B /DEST pdfmark end 198 5362 a Fq(A)h Fs(=)f Fq(A)p Fs(\()p Fq(f)546 5325 y Fi(\003)585 5362 y Fq(;)15 b(T)e(f)5 b(;)802 5339 y Fs(~)781 5362 y Fq(D)856 5376 y Fn(0)896 5362 y Fs(\))p Fq(e)986 5297 y Fg(i)p 983 5309 30 3 v 983 5351 a(")1022 5325 y Fn(\()p Fp(S)t Fn(\()p Fp(f)1164 5301 y Fd(\003)1201 5325 y Fp(;T)10 b(f)s(;)1345 5308 y Fn(~)1329 5325 y Fp(D)1387 5334 y Ff(0)1421 5325 y Fn(\))p Fi(\000)p Fn(2)p Fp(\031)r(Q)1637 5334 y Ff(+)1688 5325 y Fp(\020)1719 5334 y Ff(0)1753 5325 y Fn(\))1810 5362 y Fs(=)25 b Fq(A)1974 5376 y Fn(+)2033 5362 y Fs(\()p Fq(E)5 b(;)15 b(")p Fs(\))22 b Fo(\001)f Fq(e)2367 5325 y Fi(\000)2436 5297 y Fg(i)p 2432 5309 V 2432 5351 a(")2471 5325 y Fn(2)p Fp(\031)r(Q)2605 5334 y Ff(+)2660 5362 y Fq(;)106 b Fs(\()p Fq(T)13 b(f)d Fs(\)\()p Fq(x;)15 b(\020)7 b Fs(\))26 b(=)f Fq(f)10 b Fs(\()p Fq(x;)15 b(\020)26 b Fs(+)20 b(2)p Fq(\031)s Fs(\))p Fq(:)p 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray 1854 5715 a Fn(21)p 0 TeXcolorgray eop end %%Page: 22 22 TeXDict begin 22 21 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray -118 -406 a SDict begin H.S end -118 -406 a 0 TeXcolorgray 0 TeXcolorgray -118 -406 a SDict begin H.R end -118 -406 a -118 -406 a SDict begin [ /View [/XYZ H.V] /Dest (page.22) cvn H.B /DEST pdfmark end -118 -406 a 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray 981 165 a @beginspecial @setspecial tx@Dict begin STP newpath 0.05472 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 273.7503 32.85004 -54.75006 32.85004 /Lineto /lineto load def false Line gsave 0.05472 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.05472 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 273.7503 -32.85004 -54.75006 -32.85004 /Lineto /lineto load def false Line gsave 0.05472 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.05472 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 172.46269 32.85004 151.93141 16.42502 136.87515 0.0 121.81888 -16.42502 101.28761 -32.85004 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.05472 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.05472 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 139.61266 32.85004 136.87515 0.0 139.61266 -32.85004 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.05472 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.05472 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 101.28761 32.85004 121.81888 16.42502 136.87515 0.0 151.93141 -16.42502 172.46269 -32.85004 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.05472 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.05472 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 98.55011 32.85004 109.50012 0.0 98.55011 -32.85004 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.05472 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.05472 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 39.42001 32.85004 54.75006 0.0 68.43758 -16.42502 87.6001 -32.85004 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.05472 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.05472 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 87.6001 32.85004 68.43758 16.42502 54.75006 0.0 39.42001 -32.85004 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.05472 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray [ -52.01256 -27.37503 false NArray /DS 2.0 2. CLW mul add 2 div def /Symbol 0. [1.8 0.0 0.0 1.8 -0.495 -0.4788] FontDot /Dot { moveto gsave 1. 1. scale show grestore } bind def newpath n { transform floor .5 add exch floor .5 add exch itransform Dot } repeat end @endspecial 481 301 a tx@Dict begin { 5.0 9.5407 7.60416 2.12917 180. Uput UUput } PutCoor PutBegin end 481 301 a 441 324 a Fq(\020)481 338 y Fn(0)481 301 y tx@Dict begin PutEnd end 481 301 a 981 165 a @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ -35.58754 1.09497 -35.58754 -1.09497 /Lineto /lineto load def false Line gsave 0.54753 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 19.16252 1.09497 19.16252 -1.09497 /Lineto /lineto load def false Line gsave 0.54753 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 19.16252 0.0 -35.58754 0.0 /Lineto /lineto load def false Line gsave 0.54753 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 31.75505 -43.80005 29.56499 -29.01755 19.16252 0.0 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.54753 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ -42.70506 43.80005 -41.06255 21.90002 -35.58754 0.0 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.54753 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 210.78773 1.09497 210.78773 -1.09497 /Lineto /lineto load def false Line gsave 0.54753 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 265.5378 1.09497 265.5378 -1.09497 /Lineto /lineto load def false Line gsave 0.54753 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 265.5378 0.0 210.78773 0.0 /Lineto /lineto load def false Line gsave 0.54753 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 278.13033 43.80005 275.94026 29.01755 265.5378 0.0 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.54753 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray [ -52.01256 19.16252 false NArray /DS 2.0 2. CLW mul add 2 div def /Symbol 0. [1.8 0.0 0.0 1.8 -0.495 -0.4788] FontDot /Dot { moveto gsave 1. 1. scale show grestore } bind def newpath n { transform floor .5 add exch floor .5 add exch itransform Dot } repeat end @endspecial 481 97 a tx@Dict begin { 5.0 9.5407 7.60416 2.12917 180. Uput UUput } PutCoor PutBegin end 481 97 a 441 119 a Fq(\020)481 133 y Fn(0)481 97 y tx@Dict begin PutEnd end 481 97 a 981 165 a @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray [ 194.36272 -27.37503 false NArray /DS 2.0 2. CLW mul add 2 div def /Symbol 0. [1.8 0.0 0.0 1.8 -0.495 -0.4788] FontDot /Dot { moveto gsave 1. 1. scale show grestore } bind def newpath n { transform floor .5 add exch floor .5 add exch itransform Dot } repeat end @endspecial 2526 301 a tx@Dict begin { 5.0 35.03363 7.60416 2.12917 180. Uput UUput } PutCoor PutBegin end 2526 301 a 2381 324 a Fq(\020)2421 338 y Fn(0)2480 324 y Fs(+)20 b(2)p Fq(\031)2526 301 y tx@Dict begin PutEnd end 2526 301 a 981 165 a @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray [ 194.36272 19.16252 false NArray /DS 2.0 2. CLW mul add 2 div def /Symbol 0. [1.8 0.0 0.0 1.8 -0.495 -0.4788] FontDot /Dot { moveto gsave 1. 1. scale show grestore } bind def newpath n { transform floor .5 add exch floor .5 add exch itransform Dot } repeat end @endspecial 2526 97 a tx@Dict begin { 5.0 35.03363 7.60416 2.12917 180. Uput UUput } PutCoor PutBegin end 2526 97 a 2381 119 a Fq(\020)2421 133 y Fn(0)2480 119 y Fs(+)g(2)p Fq(\031)2526 97 y tx@Dict begin PutEnd end 2526 97 a 981 165 a @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 0 setgray /ArrowA { moveto } def /ArrowB { } def [ 203.6702 -43.80005 205.31273 -21.90002 210.78773 0.0 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.54753 SLW 0 setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 1 0 0 setrgbcolor /ArrowA { moveto } def /ArrowB { } def [ 194.36272 19.16252 112.23763 13.68752 -13.68752 10.95001 -21.90002 0.0 -30.11253 -10.95001 -52.01256 19.16252 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.54753 SLW 1 0 0 setrgbcolor 0 setlinecap stroke grestore end @endspecial 981 -63 a tx@Dict begin { 5.0 10.4192 7.90834 2.12917 180. Uput UUput } PutCoor PutBegin end 981 -63 a 1 0 0 TeXcolorrgb 939 -39 a Fh(e)-52 b Fq(\015)985 -25 y Fn(1)p 0 TeXcolorgray 981 -63 a tx@Dict begin PutEnd end 981 -63 a 228 x @beginspecial @setspecial tx@Dict begin STP newpath 0.54753 SLW 1 0 0 setrgbcolor /ArrowA { moveto } def /ArrowB { } def [ 194.36272 -27.37503 38.32504 -10.95001 13.68752 13.68752 0.0 0.0 -52.01256 -27.37503 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 0.54753 SLW 1 0 0 setrgbcolor 0 setlinecap stroke grestore end @endspecial 981 301 a tx@Dict begin { 5.0 10.4192 7.90834 2.12917 180. Uput UUput } PutCoor PutBegin end 981 301 a 1 0 0 TeXcolorrgb 939 325 a Fh(e)g Fq(\015)985 339 y Fn(0)p 0 TeXcolorgray 981 301 a tx@Dict begin PutEnd end 981 301 a 1030 827 a Fr(Figure)34 b(5.)p 0 TeXcolorgray 1501 827 a SDict begin H.S end 1501 827 a 0 TeXcolorgray 0 TeXcolorgray 1501 827 a SDict begin H.R end 1501 827 a 1501 827 a SDict begin [ /View [/XYZ H.V] /Dest (figure.5) cvn H.B /DEST pdfmark end 1501 827 a 0 TeXcolorgray Fs(P)m(erio)s(ds)c(equiv)-5 b(alen)m(t)32 b(to)g Fh(e)-52 b Fq(\015)2416 841 y Fn(0)2486 827 y Fs(and)31 b Fh(e)-52 b Fq(\015)2710 841 y Fn(1)p 0 TeXcolorgray -118 972 a SDict begin H.S end -118 972 a -118 972 a SDict begin 20 H.A end -118 972 a -118 972 a SDict begin [ /View [/XYZ H.V] /Dest (subsubsection.5.4.2) cvn H.B /DEST pdfmark end -118 972 a 84 x Fs(5.4.2.)47 b Fl(Below)30 b(the)f(r)-5 b(e)g(al)31 b(line.)45 b Fs(In)26 b(this)g(case,)1456 1033 y(~)1435 1056 y Fq(D)1510 1070 y Fn(1)1550 1056 y Fs(,)h(the)g(meeting)g(domain,)g(is)f(the)h(sub)s(domain)d(of)j(the)f (band)g Fo(f\000)p Fq(Y)45 b(<)-118 1222 y Fs(Im)o Fq(\020)39 b(<)32 b Fs(0)p Fo(g)k Fs(lo)s(cated)g(b)s(et)m(w)m(een)f(the)g(lines)p 1346 1133 115 4 v 35 w Fq(\033)1401 1184 y Fn(+)1398 1248 y(1)1460 1222 y Fs(,)h(symmetric)f(of)g Fq(\033)2131 1184 y Fn(+)2128 1248 y(1)2225 1222 y Fs(with)f(resp)s(ect)h(to)g Fm(R)p Fs(,)h(and)e Fq(\033)3230 1184 y Fi(\000)3227 1248 y Fn(1)3312 1222 y Fo(\000)23 b Fs(2)p Fq(\031)s Fs(.)54 b(The)34 b(arc)-118 1388 y Fq(\015)5 b Fs(\()p Fq(h;)15 b(g)s(;)170 1365 y Fs(~)147 1388 y Fq(D)222 1402 y Fn(1)264 1388 y Fs(\))31 b(de\014nes)e(a)i(p)s(erio)s(d)e(\()s (~)-48 b Fq(\015)1069 1402 y Fn(1)1109 1388 y Fq(;)15 b(\024)1201 1402 y Fp(g)1242 1388 y Fs(\);)31 b(the)f(curv)m(e)k(~)-48 b Fq(\015)1779 1402 y Fn(1)1838 1388 y Fs(+)20 b(2)p Fq(\031)34 b Fs(is)d(represen)m(ted)f(in)g(\014gure)p 0 0 1 TeXcolorrgb 2994 1388 a SDict begin H.S end 2994 1388 a 0 0 1 TeXcolorrgb Fs(5)p 0 0 1 TeXcolorrgb 3039 1329 a SDict begin H.R end 3039 1329 a 3039 1388 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (figure.5) cvn H.B /ANN pdfmark end 3039 1388 a 0 TeXcolorgray Fs(.)41 b(One)30 b(obtains)h(that) -118 1617 y(\(5.23\))1386 1617 y SDict begin H.S end 1386 1617 a 1386 1617 a SDict begin 20 H.A end 1386 1617 a 1386 1617 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.23) cvn H.B /DEST pdfmark end 1386 1617 a Fq(\014)1442 1578 y Fi(\000)1437 1643 y Fn(1)1526 1617 y Fs(=)1632 1555 y(1)21 b(+)f(\()p Fo(\000)p Fs(1\))1975 1472 y Fe(P)2050 1484 y Fg(n)2088 1499 y Ff(1)2050 1545 y Fg(l)p Ff(=1)2151 1522 y Fn(\()p Fp(o)2212 1534 y Fg(l)2237 1522 y Fi(\000)p Fn(1\))p 1632 1596 727 4 v 1973 1679 a Fs(2)2369 1617 y Fq(;)-118 1822 y Fs(and)30 b(for)g(2)c Fo(\024)e Fq(j)31 b Fo(\024)25 b Fq(N)10 b Fs(,)-118 2053 y(\(5.24\))1156 2053 y SDict begin H.S end 1156 2053 a 1156 2053 a SDict begin 20 H.A end 1156 2053 a 1156 2053 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.24) cvn H.B /DEST pdfmark end 1156 2053 a Fq(\014)1212 2014 y Fi(\000)1207 2080 y Fp(j)1297 2053 y Fs(=)25 b(\()p Fo(\000)p Fs(1\))1579 2015 y Fp(N)1635 2025 y Fg(j)1672 1952 y Fh(\020)1737 1991 y Fo(\000)p Fs(1)20 b(+)g(\()p Fo(\000)p Fs(1\))2150 1908 y Fe(P)2225 1909 y Fg(n)2263 1925 y(j)2225 1981 y(l)p Ff(=1)2326 1958 y Fn(\()p Fp(o)2387 1970 y Fg(l)2412 1958 y Fi(\000)p Fn(1\))p 1737 2032 798 4 v 2112 2115 a Fs(2)2544 1952 y Fh(\021)2598 2053 y Fq(:)-118 2258 y Fs(So)30 b(b)m(y)g(the)h(ab)s(o)m(v)m(e)h(notation)f(w)m(e)g(get)h (that)-118 2495 y(\(5.25\))357 2495 y SDict begin H.S end 357 2495 a 357 2495 a SDict begin 20 H.A end 357 2495 a 357 2495 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.25) cvn H.B /DEST pdfmark end 357 2495 a Fq(m)p Fs(\()p Fq(\015)519 2509 y Fn(0)559 2495 y Fq(;)15 b(\024)p Fs(\))26 b(=)f(\()p Fo(\000)p Fs(1\))994 2458 y Fp(n)1042 2394 y Fh(\020)1096 2495 y Fq(p)1142 2509 y Fn(1)1182 2495 y Fq(\014)1238 2457 y Fi(\000)1233 2521 y Fn(1)1317 2495 y Fs(+)20 b(\()p Fq(p)1489 2509 y Fn(1)1549 2495 y Fs(+)g Fq(p)1686 2509 y Fn(2)1725 2495 y Fs(\))p Fq(\014)1816 2457 y Fi(\000)1811 2521 y Fn(2)1896 2495 y Fs(+)g Fo(\001)15 b(\001)g(\001)21 b Fs(+)f(\()2289 2377 y Fp(j)2239 2409 y Fh(X)2248 2604 y Fp(i)p Fn(=1)2386 2495 y Fq(p)2432 2509 y Fp(i)2459 2495 y Fs(\))p Fq(\014)2550 2457 y Fi(\000)2545 2523 y Fp(j)2630 2495 y Fs(+)g Fo(\001)15 b(\001)g(\001)21 b Fs(+)f(\()3008 2382 y Fp(N)2973 2409 y Fh(X)2982 2604 y Fp(i)p Fn(=1)3120 2495 y Fq(p)3166 2509 y Fp(i)3194 2495 y Fs(\))p Fq(\014)3285 2457 y Fi(\000)3280 2524 y Fp(N)3348 2394 y Fh(\021)3428 2495 y Fs(=)25 b Fq(Q)3596 2509 y Fi(\000)-118 2745 y Fs(One)30 b(obtains)g(that)h Fq(m)p Fs(\()p Fq(f)759 2712 y Fi(\003)798 2745 y Fq(;)15 b(T)e Fs(\()p Fq(f)d Fs(\))p Fq(;)1091 2722 y Fs(~)1069 2745 y Fq(D)1144 2759 y Fn(1)1185 2745 y Fs(\))25 b(=)g Fq(m)p Fs(\()s(~)-48 b Fq(\015)1503 2759 y Fn(1)1563 2745 y Fs(+)20 b(2)p Fq(\031)s(;)15 b(\024)p Fs(\))27 b(=)e Fq(Q)2076 2759 y Fi(\000)2135 2745 y Fs(,)30 b(and)-118 2951 y(\(5.26\))192 2951 y SDict begin H.S end 192 2951 a 192 2951 a SDict begin 20 H.A end 192 2951 a 192 2951 a SDict begin [ /View [/XYZ H.V] /Dest (equation.5.26) cvn H.B /DEST pdfmark end 192 2951 a Fq(A)25 b Fs(=)g Fq(A)p Fs(\()p Fq(f)539 2913 y Fi(\003)578 2951 y Fq(;)15 b(T)e(f)5 b(;)795 2928 y Fs(~)774 2951 y Fq(D)849 2965 y Fn(1)889 2951 y Fs(\))p Fq(e)979 2886 y Fg(i)p 976 2898 30 3 v 976 2939 a(")1016 2913 y Fn(\()p Fp(S)t Fn(\()p Fp(f)1158 2890 y Fd(\003)1194 2913 y Fp(;T)10 b(f)s(;)1338 2896 y Fn(~)1322 2913 y Fp(D)1380 2922 y Ff(1)1414 2913 y Fn(\))p Fi(\000)p Fn(2)p Fp(\031)r(Q)1630 2922 y Fd(\000)1682 2913 y Fp(\020)1713 2922 y Ff(0)1748 2913 y Fn(\))1805 2951 y Fs(=)25 b Fq(A)1969 2965 y Fi(\000)2028 2951 y Fs(\()p Fq(E)5 b(;)15 b(")p Fs(\))22 b Fo(\001)f Fq(e)2375 2886 y Fg(i)p 2372 2898 V 2372 2939 a(")2411 2913 y Fn(2)p Fp(\031)r(Q)2545 2922 y Fd(\000)2597 2913 y Fp(\020)2628 2922 y Ff(0)2697 2951 y Fs(;)76 b(\()p Fq(T)13 b(f)d Fs(\)\()p Fq(x;)15 b(\020)7 b Fs(\))25 b(=)g Fq(f)10 b Fs(\()p Fq(x;)15 b(\020)27 b Fs(+)20 b(2)p Fq(\031)s Fs(\))p Fq(:)-118 3322 y Fv(Ac)m(kno)m(wledgemen)m(ts)p Fq(:)30 b Fl(The)g(authors)h(would)f(like)f(to)h(thank)g(F.)f(Klopp)i (for)f(pr)-5 b(op)g(osing)32 b(this)e(pr)-5 b(oblem)30 b(and)h(many)-118 3488 y(valuable)i(c)-5 b(omments)34 b(and)g(r)-5 b(emarks.)43 b(H.N)32 b(thanks)h(M.)f(Dimassi)h(for)h (suggesting)e(some)h(inter)-5 b(esting)33 b(r)-5 b(efer)g(enc)g(es)1625 3735 y Fr(References)-118 3789 y SDict begin H.S end -118 3789 a -118 3789 a SDict begin 20 H.A end -118 3789 a -118 3789 a SDict begin [ /View [/XYZ H.V] /Dest (section*.1) cvn H.B /DEST pdfmark end -118 3789 a -118 3789 a SDict begin H.S end -118 3789 a -118 3789 a SDict begin 11 H.A end -118 3789 a -118 3789 a SDict begin [ /View [/XYZ H.V] /Dest (cite.AvSi83) cvn H.B /DEST pdfmark end -118 3789 a 0 TeXcolorgray 67 3881 a Fu([1])p 0 TeXcolorgray 40 w(J.)39 b(Avron)e(and)h(B.)h(Simon.)f(Almost)h(p)r(erio)r(dic)g(Sc)n (hr\177)-38 b(odinger)38 b(op)r(erators,)43 b(I)r(I,)38 b(the)g(In)n(tegrated)g(densit)n(y)f(of)i(states)g Fj(Duke)187 3972 y(Mathematic)l(al)28 b(Journal)p Fu(,)f(50:369-391,)i(1983.)-118 3987 y SDict begin H.S end -118 3987 a -118 3987 a SDict begin 11 H.A end -118 3987 a -118 3987 a SDict begin [ /View [/XYZ H.V] /Dest (cite.deux) cvn H.B /DEST pdfmark end -118 3987 a 0 TeXcolorgray 67 4063 a Fu([2])p 0 TeXcolorgray 40 w(V.)c(S.)h(Buslaev)g(and)g(A.)f(F)-6 b(edoto)n(v:)34 b Fc(Blo)r(c)n(h)27 b(solutions)g(for)f(di\013erence)g(equations.)g Fu(Algebra)g(i)g(Analiz)g Fb(\(7\))h Fu(p74-122)g(\(1995\))-118 4082 y SDict begin H.S end -118 4082 a -118 4082 a SDict begin 11 H.A end -118 4082 a -118 4082 a SDict begin [ /View [/XYZ H.V] /Dest (cite.CaLa) cvn H.B /DEST pdfmark end -118 4082 a 0 TeXcolorgray 67 4155 a Fu([3])p 0 TeXcolorgray 40 w(R.)52 b(Carmona)h(and)e(J.)i(Lacroix:)88 b Fc(Sp)r(ectral)52 b(Theory)g(of)h(Random)e(Sc)n(hr\177)-38 b(odinger)52 b(Op)r(erators)h Fu(Birkh\177)-38 b(auser)52 b(Boston.)187 4246 y(Basel.Berlin.)-118 4246 y SDict begin H.S end -118 4246 a -118 4246 a SDict begin 11 H.A end -118 4246 a -118 4246 a SDict begin [ /View [/XYZ H.V] /Dest (cite.quatre) cvn H.B /DEST pdfmark end -118 4246 a 0 TeXcolorgray 67 4337 a Fu([4])p 0 TeXcolorgray 40 w(H.L)25 b(Cycon,)i(R.G.)f(F)-6 b(ro)r(ese,)26 b(W.)g(Kirsc)n(h)g(and)f(B.)i(Simon)e Fc(Sc)n(hr\177)-38 b(odinger)26 b(op)r(erators.)h Fu(Springer)f(V)-6 b(erlag,)26 b(Berlin,)h(1987.)-118 4356 y SDict begin H.S end -118 4356 a -118 4356 a SDict begin 11 H.A end -118 4356 a -118 4356 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Busl2) cvn H.B /DEST pdfmark end -118 4356 a 0 TeXcolorgray 67 4429 a Fu([5])p 0 TeXcolorgray 40 w(V.)j(S.)h(Buslaev)g(and)g(A.)f (F)-6 b(edoto)n(v:)44 b Fc(The)31 b(complex)g(WKB)g(metho)r(d)f(for)i (Harp)r(er's)f(equation)f Fu(St)h(P)n(etersburg)g(Math)f(Jour-)187 4520 y(nal,)p Fb(\(95\))d Fu(6:)35 b(p495-517)28 b(\(1995\).)-118 4539 y SDict begin H.S end -118 4539 a -118 4539 a SDict begin 11 H.A end -118 4539 a -118 4539 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Eas:73) cvn H.B /DEST pdfmark end -118 4539 a 0 TeXcolorgray 67 4611 a Fu([6])p 0 TeXcolorgray 40 w(R.)d(Eastham:)36 b Fj(The)28 b(sp)l(e)l(ctr)l(al)h(the)l(ory)g(of) e(p)l(erio)l(dic)i(di\013er)l(ential)f(op)l(er)l(ators)g Fu(Scottish)e(Academic)g(Press,)h(Edin)n(burgh)e(\(1973\))-118 4630 y SDict begin H.S end -118 4630 a -118 4630 a SDict begin 11 H.A end -118 4630 a -118 4630 a SDict begin [ /View [/XYZ H.V] /Dest (cite.troi) cvn H.B /DEST pdfmark end -118 4630 a 0 TeXcolorgray 67 4702 a Fu([7])p 0 TeXcolorgray 40 w(E.)g(Sorets)g(and)g(T.)g(Sp)r(encer:)34 b Fj(Positive)27 b(Lyapunov)i(exp)l(o)l(ents)g(for)d(Schr\177)-39 b(odinger)29 b(op)l(er)l(ators)g(with)e(quasi-p)l(erio)l(dic)i(p)l(otentials.)187 4794 y Fu(Comm.)e(Math.)f(Ph)n(ys.)f Fb(\(142\))i Fu(\(3\))f(p)f (543-566)j(\(1991\))-118 4812 y SDict begin H.S end -118 4812 a -118 4812 a SDict begin 11 H.A end -118 4812 a -118 4812 a SDict begin [ /View [/XYZ H.V] /Dest (cite.FK0) cvn H.B /DEST pdfmark end -118 4812 a 0 TeXcolorgray 67 4885 a Fu([8])p 0 TeXcolorgray 40 w(A.)d(F)-6 b(edoto)n(v)26 b(and)f(F.)h(Klopp:)34 b Fj(A)28 b(c)l(omplex)g(metho)l(d)h(for)e (adiab)l(atic)i(pr)l(oblems)d Fu(Asymp.)f(Anal.)h(\(2001\))h Fb(\(27\))g Fu(p)e(219-264.)-118 4904 y SDict begin H.S end -118 4904 a -118 4904 a SDict begin 11 H.A end -118 4904 a -118 4904 a SDict begin [ /View [/XYZ H.V] /Dest (cite.FEKL1) cvn H.B /DEST pdfmark end -118 4904 a 0 TeXcolorgray 67 4976 a Fu([9])p 0 TeXcolorgray 40 w(A.)30 b(F)-6 b(edoto)n(v)30 b(and)g(F.)h(Klopp:)87 b Fj(On)32 b(the)h(singular)f(Sp)l(e)l(ctrum)i (for)d(A)l(diab)l(atic)i(Quasi-p)l(erio)l(dic)g(Schr\177)-39 b(odinger)34 b(Op)l(er)l(ators)g(in)187 5068 y(the)28 b(r)l(e)l(al)g(Line.)e Fu(Ann.)e(henri)i(P)n(oincar)n(\023)-36 b(e)27 b(\(2005\))g Fb(5)e Fu(p1-50.)-118 5086 y SDict begin H.S end -118 5086 a -118 5086 a SDict begin 11 H.A end -118 5086 a -118 5086 a SDict begin [ /View [/XYZ H.V] /Dest (cite.huit) cvn H.B /DEST pdfmark end -118 5086 a 0 TeXcolorgray 29 5159 a Fu([10])p 0 TeXcolorgray 40 w(A.)j(F)-6 b(edoto)n(v)28 b(and)f(F.)i(Klopp:)79 b Fj(A)n(nderson)31 b(tr)l(ansitions)h(for)e(a)g(family)e(of)h(almost)i(p)l(erio)l(dic)f (Schr\177)-39 b(odinger)32 b(e)l(quations)f(in)f(the)187 5250 y(adiab)l(atic)e(c)l(ase)f Fu(Comm.)g(Math.)f(Ph)n(ys.)g(\(2002\)) h Fb(\(227\))g Fu(\(1\))e(p1-92.)-118 5269 y SDict begin H.S end -118 5269 a -118 5269 a SDict begin 11 H.A end -118 5269 a -118 5269 a SDict begin [ /View [/XYZ H.V] /Dest (cite.neuf) cvn H.B /DEST pdfmark end -118 5269 a 0 TeXcolorgray 29 5342 a Fu([11])p 0 TeXcolorgray 40 w(A.)g(F)-6 b(edoto)n(v)26 b(and)f(F.)h(Klopp:)34 b Fj(Ge)l(ometric)29 b(to)l(ols)g(of)e(the)h (adiab)l(atic)h(c)l(omplex)f(WKB)f(metho)l(d)g Fu(Asymp.)e(Anal.)h (\(2004\))-118 5360 y SDict begin H.S end -118 5360 a -118 5360 a SDict begin 11 H.A end -118 5360 a -118 5360 a SDict begin [ /View [/XYZ H.V] /Dest (cite.FEKL2) cvn H.B /DEST pdfmark end -118 5360 a 0 TeXcolorgray 29 5433 a Fu([12])p 0 TeXcolorgray 40 w(A.)18 b(F)-6 b(edoto)n(v)18 b(and)g(F.)g(Klopp:)63 b Fj(On)20 b(the)i(absolutely)g(c)l(ontinuous)h(sp)l(e)l(ctrum)f(of)f (one)g(dimensional)f(q)h(uasi-p)l(erio)l(dic)h(schr\177)-39 b(odinger)187 5524 y(op)l(er)l(ators)30 b(in)d(the)i(adiab)l(atic)f (limit.)c Fu(T)-6 b(rans.)27 b(Amer.)e(Math.)h(So)r(c)g(\(2005\).)-118 5543 y SDict begin H.S end -118 5543 a -118 5543 a SDict begin 11 H.A end -118 5543 a -118 5543 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Fe) cvn H.B /DEST pdfmark end -118 5543 a 0 TeXcolorgray 29 5616 a Fu([13])p 0 TeXcolorgray 40 w(M.)g(V.)g(F)-6 b(edoryuk:)68 b Fj(Asymptotic)29 b(Analysis")p Fu(,)e(Springer)f(V)-6 b(erlag,)26 b(Berlin;)h(\(1993\).) p 0 TeXcolorgray 1854 5715 a Fn(22)p 0 TeXcolorgray eop end %%Page: 23 23 TeXDict begin 23 22 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 0 TeXcolorgray 0 TeXcolorgray -118 -406 a SDict begin H.S end -118 -406 a 0 TeXcolorgray 0 TeXcolorgray -118 -406 a SDict begin H.R end -118 -406 a -118 -406 a SDict begin [ /View [/XYZ H.V] /Dest (page.23) cvn H.B /DEST pdfmark end -118 -406 a 0 TeXcolorgray 0 TeXcolorgray -118 -290 a SDict begin H.S end -118 -290 a -118 -290 a SDict begin 11 H.A end -118 -290 a -118 -290 a SDict begin [ /View [/XYZ H.V] /Dest (cite.KM1) cvn H.B /DEST pdfmark end -118 -290 a 0 TeXcolorgray 29 -207 a Fu([14])p 0 TeXcolorgray 40 w(F.)27 b(Klopp)g(and)g(M.)g(Marx,)57 b Fj(R)l(esonanc)l(es)31 b(for)e(slow)t(ly)g(varying)g(p)l(erturb)l(ations)j(of)d (one-dimensional)g(p)l(erio)l(dic)h(Schr\177)-39 b(odinger)187 -115 y(op)l(er)l(ators)27 b Fu(,)e(Exp)r(os)f(IV.)g(Actes)h(du)e 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