Content-Type: multipart/mixed; boundary="-------------0312091918723" This is a multi-part message in MIME format. ---------------0312091918723 Content-Type: text/plain; name="03-531.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-531.comments" University of Alabama at Birmingham, karpeshi@math.uab.edu ---------------0312091918723 Content-Type: text/plain; name="03-531.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-531.keywords" Periodic, Magnetic, Schroedinger Operator, pertubation theory ---------------0312091918723 Content-Type: application/postscript; name="Magnet9hb.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Magnet9hb.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.90a Copyright 2002 Radical Eye Software %%Title: Magnet9hb.dvi %%CreationDate: Thu Nov 20 21:43:55 2003 %%Pages: 39 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR6 CMBX10 CMR8 CMR9 CMMI9 CMSY9 CMR10 CMCSC10 CMMI10 %%+ CMEX10 CMR7 CMMI7 CMR5 CMSY10 CMSY7 CMMI5 CMTI10 CMSY5 CMSY8 CMMI8 %%+ CMMI6 CMSY6 CMTI8 CMBX8 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips.exe -P pdf -G0 Magnet9hb %DVIPSParameters: dpi=8000, compressed %DVIPSSource: TeX output 2003.11.20:2142 %%BeginProcSet: tex.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: alt-rule.pro %! % Patch by TVZ % Makes dvips files draw rules with stroke rather than fill. % Makes narrow rules more predictable at low resolutions % after distilling to PDF. % May have unknown consequences for very thick rules. % Tested only with dvips 5.85(k). 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CirclePath .8 .8 scale CirclePath eofill } def /TrianglePath { 0 660 moveto -571.5 -330 lineto 571.5 -330 lineto closepath } def /SolidTriangle { TrianglePath fill } def /Triangle { TrianglePath .85 .85 scale TrianglePath eofill } def /BoldTriangle { TrianglePath .7 .7 scale TrianglePath eofill } def /SquarePath { -450 450 moveto 450 450 lineto 450 -450 lineto -450 -450 lineto closepath } def /SolidSquare { SquarePath fill } def /Square { SquarePath .89 .89 scale SquarePath eofill } def /BoldSquare { SquarePath .78 .78 scale SquarePath eofill } def /PentagonPath { -337.8 -465 moveto 337.8 -465 lineto 546.6 177.6 lineto 0 574.7 lineto -546.6 177.6 lineto closepath } def /SolidPentagon { PentagonPath fill } def /Pentagon { PentagonPath .89 .89 scale PentagonPath eofill } def /BoldPentagon { PentagonPath .78 .78 scale PentagonPath eofill } def % DG/SR modification begin - Dec. 16, 1999 - From Etienne Riga /DiamondPath { 0 742.5 moveto -428.5 0 lineto 0 -742.5 lineto 428.5 0 lineto closepath } def /SolidDiamond { DiamondPath fill } def /Diamond { DiamondPath .85 .85 scale DiamondPath eofill } def /BoldDiamond { DiamondPath .7 .7 scale DiamondPath eofill } def % DG/SR modification end /.notdef { } def end /BuildGlyph { exch begin Metrics 1 index get exec 0 BBoxes 3 index get exec setcachedevice CharProcs begin load exec end end } def /BuildChar { 1 index /Encoding get exch get 1 index /BuildGlyph get exec } bind def end /PSTricksDotFont exch definefont pop %END pst-dots.pro %%EndProcSet %%BeginProcSet: texps.pro %! 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put readonly def /FontBBox{-59 -250 1235 750}readonly def /UniqueID 5000766 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTI8 %!PS-AdobeFont-1.1: CMTI8 1.0 %%CreationDate: 1991 Aug 18 21:07:42 % 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 (CMTI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI8 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 44 /comma put dup 45 /hyphen put dup 46 /period put dup 50 /two 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 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T 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 1190 750}readonly def /UniqueID 5000826 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /beta put dup 61 /slash put dup 100 /d put dup 110 /n put dup 113 /q put dup 114 /r put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 40 /braceleftBigg put dup 41 /bracerightBigg put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 112 /radicalbig put dup 113 /radicalBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 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 83 /S put dup 106 /bar put dup 112 /radical put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /asteriskmath put dup 48 /prime put dup 49 /infinity put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueID 5000815 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D5FC1B2109839E5B52DFBAF552B11EFFB6A16C F03FB920C15AE724EFDF0CCBF00A838D34440FF9FED532F44036AD22561184C5 283722DDFA7285E62754372D716D704AC0E00B2F6AB67154241C7449AA047833 94CEDB08E8C92907FE72A0B05AE36A7B9226ACD6E7890A0B528FDDE84A950FC6 801DE75CF2E739E9121149CCB8B1C87A106822648D84A3D3FBF295EE6C4BF403 BBE9A1C1F6DAEDD1E642ACC486E609703D7612BFFD10C324F5DC710811F7F614 3691B400E3773987424C0D2B0D8A736873C6371DDB2442F05E018A2B5CA9A4AA 17AABB95D09E5890CFFFED5AC01495D89A53D3CEB6E0F620D2A75CB26E31E1FD 557B4C4A08FAFA16D07D0880ADC1A4C17F227DCF6EF48D70FBA97C6215BC5CCA 595C934B200EFD814749D407013724C07EAC338D7373C176EC41EB79318E3357 A756656C6F73B5506D8C5CDC5E9E55D256EF3D88EFE1742746AB66818EEBA31A 8B130CA692D48A1A059C28051E62813247A5AE112D9866C2B302A827570BF2BA 43C6AA06C6095E23D6329D0047FB91404185A9CA5C66ADB337A7A3F698250D92 B4E3E90B773667EB62A670B1E9C832553CCC115250FA182864EB05C1E8244189 3FE6D542BC146EF58793A5D5161327B72E6CDB3E574C7E86014455280EBD588E FA33FF191700D51CFB25A1ABFBEE39EE6B493633D8C0FB1927DB60C92CC35009 695C2669DD13DAD901FE79C5EDEC73788665ED1222918EDD295EFD0C0CB49FF5 866F409B2F13039A76CDD6FB4B834698C002A7AAAA02D6C0E2CA1C6AFA82C024 6382D149C32D0B8B3D856CFF2D1DFD2115E47BE1546C67F00F42008E8B4807DA 80C4B869BC5CFE532B09B4B1D7CB54ED1DCF3FA1632C50995EF22A9CEF13A297 EA26B50F2B2F3337F3724C8239EAAF5CEB4EFD96098512712EC2993CCCE62BEA 5FC0DBBB009DB2BA32FE8510267989F666AB3BEF5789A67CFFF5E3FA2C735D8C AFF5641ABA5A9D49B370C734294316E72327A8A3CA299A15F7CCD785E3E83082 338565DC53A2A127E8148EADE6B37C60DB825E0E37306AF1D189CFC8BB3FE679 5F5325833989326A9CE3BF0F9F00A7F276FE59F2A3E68360D56FC5ABF30FB108 C7F403558E3106BA27DF9994BFAFE63C8D4C02B1F542392414C9652F10281B8F A04EFF956DC6F90CAE44120D40FA1D52846929DFF63FA3A6E51DDA3099535982 B84BC8C7D1245CEDFCAE441DC64A1F1FD9CCF79D3F15FF79D7322A 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 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 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 69 /E put dup 70 /F put dup 73 /I put dup 76 /L put dup 78 /N 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 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 14 /delta put dup 22 /mu put dup 23 /nu put dup 34 /epsilon put dup 105 /i put dup 106 /j put dup 109 /m put dup 113 /q put dup 114 /r put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueID 5087380 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA06DA87FC7163A5A2A756A598FAB07633 89DE8BB201D5DB4627484A80A431B6AFDBBBF23D4157D4AFE17E6B1C853DD417 25F84CD55402AB88AB7EEFDEDBF2C2C731BD25567C53B474CCF739188A930039 098A197F9C4BE7594D79442B2C8A67447DE44698321145D7689B91EF235EA80E B600AA8E238064F154284096C4C2554EFE8DDF13AFF8D3CE30E0999375C0FEE6 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cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % 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 (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 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 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 54 /negationslash 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{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D251491EBF65A98C9FE2B1CF8D725A70281949 8F4AFFE638BBA6B12386C7F32BA350D62EA218D5B24EE612C2C20F43CD3BFD0D F02B185B692D7B27BEC7290EEFDCF92F95DDEB507068DE0B0B0351E3ECB8E443 E611BE0A41A1F8C89C3BC16B352C3443AB6F665EAC5E0CC4229DECFC58E15765 424C919C273E7FA240BE7B2E951AB789D127625BBCB7033E005050EB2E12B1C8 E5F3AD1F44A71957AD2CC53D917BFD09235601155886EE36D0C3DD6E7AA2EF9C C402C77FF1549E609A711FC3C211E64E8F263D60A57E9F2B47E3480B978AAF63 868AEA25DA3D5413467B76D2F02F8097D2841D149F0E3AFBF0A45518B94CFBBC 27BDA50986D45A73EFC22205F50AA6BC09BB13D62F0FDCAEAACFB425B0D641F0 BBFEFE048045F365862F34057FA304A6235101F397C4C305889F871BA19266FE E58BC9647CA50B1E26D7E760995A3A10155A7A616A25389E33B7A390E1B7B957 4972DAB4DA20DAD457A32F0076CF590E4C47B5B2AAA8DAE1CDB8BD5CE43EAC94 2379E40B35A76A5700578ED5C2CA0D329189A194994198FA35EF6B0692613B17 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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 9 /circleminus put dup 17 /equivalence put dup 18 /reflexsubset put dup 20 /lessequal put dup 21 /greaterequal put dup 25 /approxequal put dup 26 /propersubset put dup 27 /propersuperset put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 57 /existential put dup 59 /emptyset put dup 60 /Rfractur put dup 61 /Ifractur put dup 63 /perpendicular put dup 75 /K put dup 83 /S 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 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 52 /four put dup 55 /seven put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueID 5000788 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 18 /theta put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 25 /pi put dup 26 /rho put dup 28 /tau put dup 31 /chi put dup 34 /epsilon put dup 39 /phi1 put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 65 /A put dup 66 /B put dup 68 /D 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 81 /Q put dup 82 /R put dup 83 /S put dup 87 /W put dup 90 /Z put dup 98 /b put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 120 /x put dup 126 /vector put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueID 5087382 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D77639DF1232A4D6233A9CAF69B151DFD33F C0962EAC6E3EBFB8AD256A3C654EAAF9A50C51BC6FA90B61B60401C235AFAB7B B078D20B4B8A6D7F0300CF694E6956FF9C29C84FCC5C9E8890AA56B1BC60E868 DA8488AC4435E6B5CE34EA88E904D5C978514D7E476BF8971D419363125D4811 4D886EDDDCDDA8A6B0FDA5CF0603EA9FA5D4393BEBB26E1AB11C2D74FFA6FEE3 FAFBC6F05B801C1C3276B11080F5023902B56593F3F6B1F37997038F36B9E3AB 76C2E97E1F492D27A8E99F3E947A47166D0D0D063E4E6A9B535DC9F1BED129C5 123775D5D68787A58C93009FD5DA55B19511B95168C83429BD2D878207C39770 012318EA7AA39900C97B9D3859E3D0B04750B8390BF1F1BC29DC22BCAD50ECC6 A3C633D0937A59E859E5185AF9F56704708D5F1C50F78F43DFAC43C4E7DC9413 44CEFE43279AFD3C167C942889A352F2FF806C2FF8B3EB4908D50778AA58CFFC 4D1B14597A06A994ED8414BBE8B26E74D49F6CF54176B7297CDA112A69518050 01337CBA5478EB984CDD22020DAED9CA8311C33FBCC84177F5CE870E709FC608 D28B3A7208EFF72988C136142CE79B4E9C7B3FE588E9824ABC6F04D141E589B3 914A73A42801305439862414F893D5B6C327A7EE2730DEDE6A1597B09C258F05 261969424F885C6B93B28E3223FDD3B040F5535D6AAE9201E5F49A143F3B65FA 75FDE4E5FB4FDEB79A695E89B66FB385A22222553A72131A7BEAC3F44DD0AC0B 0B566039AC5C1CB0A1304B882DD2497870AA5FB1FD17A704C4668F6F85F6E3CC 814D68758E24D9199B67A9395FD76257FE284913EF1B8897FFA602A54B39EB03 2A783B4A582A33F532481524A8BD8998A93DBFD4804FA77802FF52D5183117CF 80BC1398B9B1F9844EB62E162912873F37005CC29CEB7A2F0D0BD5BB237F61AF 6E8FF7D1D0F67E7FA88F3F548CE14C9FA71F15FA8587DE6EB5B2614D46668B8E C00C8BF6CC43A945145D6121AB8ECCD6A06B284410C472127379C23222B0365A 63F087FC4F467BE18B0B352D050C6196EA4D6A5C1175B280CE99D48B811E48FB 3FE1A80EC2C4AE791F6F7DF695C9578157EE90870DF9A83FBEF29D4918367E7B 2E5AB55E5B79215C5F084006C959F7BC01325277C987BBF168CA795534F5356D E1D4B4656DFFB4A4CFE05F42F9C7A969E23C07222932116B85E03526353EA390 C3B11782D2A8FCCF5968D6B66793F784526AB49FD76AC927B4ADF92B8E7F62AF C1487B3158044AECE521688BF8120148DC8F83BE7BE510448F15B0FD4AE03E97 76A8B2845B82B7CA49A1E6E81BBA13CD9BABD63026113155B02751BED6F3D22C E8CEC10A70B8F2165AEFCD29BEDF17003E8130A7DED29F64A73681942A1A27B2 2CC68E0CAACBE3FE732C3B59A3D98DE9CD4B59166969C10F005F57EC3E2C3B66 BE9273B3AB5DB8ABF6C3B1B5895468037F2DD4228DF35C9C0E0BF91617967085 4921F08ADBEBCEC9BE25C6BB481EDFA151A34B28B0FE7F3CAE539435A1154FB9 E9F7A0D7C0F3FA2739526623A7605E40CFF0B8D884F26A915E6F442713C79AC3 05265E4CA9158F514E21EE03FE556830C5E0686734650935169AB127B24D44ED 728FF48D36AAC7469ED994BFE29486C28E5ADF62552BD1C0109EB62120587B67 C7B729764E12D98009E7FFB95437EFB57885B92CD494546B63BA51065C374ED1 13F164877CBFF39FE3CE2ABDA64BAC616F5D8E923F8D1D09A55F4E36871A901E 0666C40C2A4B7B92C338B595DDE05AF9DFFB8FF11B784C3CC0393FBBAE207DA2 5B37476D121FD2C56F5C0E7A13EFD8165D992454E0D4D3B6F4EF212442A1D787 547ECAB6D9AB09B89F0EC18E3C6473C382FC6C857D0CE30B61C6191A8D1A853E 6D4184FA313ADA2EBFF0D330DA5932CA06C8E4C16E48B18665B51CC7DA82987C 600454D01CFA7F17FA8BE45471F67B410851D59A4EF436F8723F8F6BFEB3F69E A8A3BCA5DEC685F1A68EFF66233EE383046B638CAF89FF15A502DD0D0E39AF7D 36C0BEDF02A44962164CA9D5DAA1D010C35BD9CB8CFF74054F2869110D5B4262 16B3AAC54209178604A82FFF9B4D5DE6E75E1B772753B0EB0949CEB3E8210C2A 29015FD5173FC3C4CBC09E9377EBFB51D4AE1D6C0516524BA89DBC95B2A9EFE9 B18EE12A133832308B41A8CE5F00017CECC53D104E8A6E96325FBBE454003CAA A6E0B2A27D66186A3A1588400BA088AE01EF91FD2BE4AE8D08B04CC5A64AE1EE 12F5BD7E4267AECDC8CF5A9556C437E382B527968E259AA3E399712A1118B924 050B0A68134AF7B796432F5A3F717064F410C8C599F5BBCDE67BA49E4BE502A0 378C918048D50D0D13B3EB1E8DEBE66E44763480C562052854E16CF027BF08DF 817CDB68B1C3E739DFB044663016AA92578634566459349E7BAC3CB1068CEFAB AEACEF2F7AAE96A1CD9855DB98EEC05DF339446769B49C061FC260E52D63D082 B97C334A3A9F37D2D11104A4FF4169E40175F6287D84D9060863D46E4DDBBD3D DCC644FE9028982781DEB67E999438FC0B8C928B0B44AF12F1BDE2D683B33BDD 563F927A2A9509AE4ABDF571586B5AB0D849173CDFD63185154E3177C055C27B FCFEEE89019544D84BA9F0D8CA40887EA0C9E7F59D1C13C94027D9F6C719CCB5 D5864D6558F0B7688519E215C15014DA6462276FBACC6B29EE51103CD79C6395 AF1B5D0C30F067E6112EE5A2F5E3A5D0B3BFDEA38026C806EE7A0463FCDD681D B618EA1AA2CA47CCC346647AE8C2A8048B0B15E98A06A412B1E4202C4C499D47 AED2E262583BF93FE3E4F79209528D9526E12DB22EF72599176926F30D3C5088 67CC229C0AEAA0537B0B3E58F22B64075A7A4C3DA90112F5469A66E4CAC97AA8 8E3AB52721DC8700C2E40C5A216215A81DDA2A831462E4A79EC8D57099CAB8B3 E3C1E7B92139B08B5D4FBD044A346AAF45ABD0C922AC2181C4A23E60B660831C E1E04AFF3A8AEF38E1AEE5D4DD48B2D44537E48F9F0AEC40FAE566F8BE4BBDB8 807E1FAD4109E7CC01464C23F0C600F853E37E8781D0E0FC38BF4FFB261C0EFC 0E1A44EC04B3EDF8CE7A22464A2F535D095CD539F24032DEFA473DD79326D9C9 00CC5BFC37E58FE79E0807B1CC94CEFB9DB5EB2E233240F565E78D4E20B6B2D3 9F6ED3FFA16CADA6F5CF3AEFA441E1B12A31CA9590EE3DB10F9F8FE53EB7ADED 9662121C586942496CC432A0BDD29DC78711F061356E577979FB67EC0CEB7AE8 AA43EB4719E65B0D0DB65ED3ADCABEC0CD4B1187775E12C098AF34AA09F283F6 74D67ADBD21D1458B501F1FF881B7681E0676AA8EEFF7906F1FD4FB51D5AFA1E C0800084777C949C7E815B704CE273B3239179F093B2519BFD305E3981D44C13 85A367F81B56821F0F90DBBA06FAB97FE8C369EFAC59188F36F48390ECE28865 6F0BBCE7806F3143774E305847E7202CCDCD321D8B64DEB8FAF7E057DD8EEAAC 2304691E7163373E9355211C2F21CA93545928DDB025962AE3ECC2B3C6EF1648 B10BA00113BE3D61444B0D8BCD3E3DD178C3B7BA1028ADC8DAED538A80EABE89 AF1B2DFD6EAAFFB15E7FBCB716586CFCAF475D81FA3A4A6CBD1FB169D5A0E0BB B12B72FD1569CFA921E2DF26F1F122315A60A4B44336AA8AF94CA36F24E6F939 54197FF08A5EF0322DB6597AFFA26B6A42232383E154E16829D7A097D53E2E59 E1C31B1EA22E4D4236B701C04B0424DADE7180AB38381E2C382D223AA5AC5970 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cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 5 /Pi put dup 22 /macron 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 58 /colon put dup 61 /equal put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 89 /Y put dup 126 /tilde put dup 127 /dieresis put readonly def 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%!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 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 28 /tau put dup 31 /chi put dup 32 /psi put dup 34 /epsilon put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 85 /U put dup 86 /V put dup 87 /W put dup 90 /Z 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 120 /x put dup 126 /vector put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 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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 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 70 /F put dup 73 /I put dup 77 /M put dup 80 /P put dup 82 /R put dup 84 /T 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 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put readonly def /FontBBox{14 -250 1077 750}readonly def /UniqueID 5000772 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 3 /Lambda put dup 5 /Pi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 22 /macron put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 63 /question put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY9 %!PS-AdobeFont-1.1: CMSY9 1.0 %%CreationDate: 1991 Aug 15 07:22:27 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 106 /bar put dup 114 /nabla put readonly def /FontBBox{-30 -958 1146 777}readonly def /UniqueID 5000819 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI9 %!PS-AdobeFont-1.1: CMMI9 1.100 %%CreationDate: 1996 Jul 23 07:53:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 23 /nu put dup 58 /period put dup 59 /comma put dup 62 /greater put dup 72 /H put dup 97 /a put dup 105 /i put dup 107 /k put dup 112 /p put dup 120 /x put dup 126 /vector put readonly def /FontBBox{-29 -250 1075 750}readonly def /UniqueID 5087384 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 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 54 /six put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C 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 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 121 /y put 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4147BBC29FC27B34C649B0E0F1C9D200F20DD9711A4BE4A98C906B7889BFD86C C657CB6EDB44C168ECCD1B4A090F360CEC43DECABCF6C29A422BC002C0D5763A B79FAAF7BD7F918170A8B2167A9F6A4F09E51F892B554D5633C134D15F548BEE B8A6EFC9960689DE64A99E54A0C45AED633CC5CACB41D36312886BD7C7F69905 C03F845CD84BA193D7B6C57B7E9CB720DE1B313EFE60C9B24EB5DB 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 11 /ff put dup 12 /fi put dup 35 /numbersign 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 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 89 /Y put dup 91 /bracketleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 127 /dieresis put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 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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 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 87 /W put dup 89 /Y put dup 97 /a put dup 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 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w 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 FF524AAD7262B10351C50EBAD08FB4CD55D2E369F6E836C82C591606E1E5C73F DE3FA3CAD272C67C6CBF43B66FE4B8677DAFEEA19288428D07FEB1F4001BAA68 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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. 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32333 y Fs(+)246 b Fq(a)31215 32499 y Fo(2)31712 32333 y Fs(\()p Fq(x)p Fs(\))33205 30772 y Fp(\023)34020 31001 y Fo(2)34763 32333 y Fs(+)g Fq(\013)t(a)37167 32499 y Fo(0)37664 32333 y Fs(\()p Fq(x)p Fs(\))5354 b(\(1\))6075 34933 y(in)311 b Fq(L)8061 35099 y Fo(2)8558 34933 y Fs(\()p Fq(R)9838 34532 y Fo(2)10334 34933 y Fs(\),)324 b(where)311 b Fq(a)15123 35099 y Fo(0)15620 34933 y Fs(\()p Fq(x)p Fs(\),)324 b Fq(a)18329 35099 y Fo(1)18825 34933 y Fs(\()p Fq(x)p Fs(\),)g Fq(a)21534 35099 y Fo(2)22031 34933 y Fs(\()p Fq(x)p Fs(\))312 b(are)f(real)h(p)31 b(erio)g(dic)311 b(functions,)325 b Fq(\013)315 b Fs(is)c(a)h(real)f (parame-)6075 36262 y(ter.)471 b(The)307 b(op)31 b(erator)306 b(\(1\))i(describ)31 b(es)304 b(motion)309 b(of)d(an)h(electron)g(in)f (the)g(plane)h(\()p Fq(x)38706 36428 y Fo(1)39203 36262 y Fq(;)184 b(x)40327 36428 y Fo(2)40824 36262 y Fs(\))307 b(when)f(the)6075 37590 y(p)31 b(erio)g(dic)292 b(magnetic)h(\014eld)f Fq(B)56 b Fs(\()p Fq(x)p Fs(\))308 b(=)f Fq(a)21672 37756 y Fo(1)22169 37590 y Fs(\()p Fq(x)p Fs(\))23662 37756 y Fn(x)24164 37867 y Fm(2)24743 37590 y Fl(\000)91 b Fq(a)26280 37756 y Fo(2)26776 37590 y Fs(\()p Fq(x)p Fs(\))28269 37756 y Fn(x)28771 37867 y Fm(1)29551 37590 y Fs(acts)292 b(in)g(the)g(direction)h Fq(x)39969 37756 y Fo(3)40756 37590 y Fs(orthogonal)6075 38918 y(to)326 b(the)f(plane)h(and)g(the)f(p)31 b(erio)g(dic)326 b(electric)g(\014eld) f Fq(\013)t Fl(r)p Fq(a)28673 39084 y Fo(0)29171 38918 y Fs(\()p Fq(x)p Fs(\))h(acts)f(in)h(a)f(direction)i(in)e(the)h(plane.) 6075 40247 y(Note)344 b(that)g(not)g(ev)-31 b(ery)343 b(p)31 b(erio)g(dic)343 b(magnetic)i(\014eld)f(can)f(b)31 b(e)342 b(describ)31 b(ed)343 b(in)g(terms)g(of)h(a)f(p)31 b(erio)g(dic)6075 41575 y(magnetic)457 b(p)31 b(oten)-31 b(tial)389 b Fq(~)-484 b(a)p Fs(\()p Fq(x)p Fs(\))451 b(=)g(\()q Fq(a)20513 41741 y Fo(1)21009 41575 y Fs(\()p Fq(x)p Fs(\))p Fq(;)184 b(a)23578 41741 y Fo(2)24077 41575 y Fs(\()p Fq(x)p Fs(\)\))q(.)750 b(In)455 b(particular,)409 b Fq(~)-484 b(a)q Fs(\()p Fq(x)p Fs(\))455 b(is)g(a)h(linear)g (function)6075 42903 y(for)396 b(a)g(constan)-31 b(t)397 b(magnetic)h(\014eld.)573 b(The)396 b(case)g(of)g(a)g(p)31 b(erio)g(dic)396 b Fq(B)56 b Fs(\()p Fq(x)p Fs(\))396 b(with)h(non-p)31 b(erio)g(dic)327 b Fq(~)-483 b(a)p Fs(\()p Fq(x)p Fs(\))6075 44232 y(is)455 b(signi\014can)-31 b(tly)458 b(more)e(di\016cult)g(than)g(ours.)751 b(Most)455 b(of)h(the)g(results)e(in)i(this)g(situation)h(are)6075 45560 y(obtained)401 b(for)e(the)h(case)g(of)g(a)f(constan)-31 b(t)401 b(magnetic)h(\014eld)e(\(e.g)g([LL)q(,)f(DN,)h(Sj])g(and)g (literature)6075 46888 y(there\).)7403 48217 y(W)-92 b(e)350 b(assume)h(that)h Fq(a)16063 48383 y Fo(0)16559 48217 y Fs(\()p Fq(x)p Fs(\),)k Fq(a)19300 48383 y Fo(1)19796 48217 y Fs(\()p Fq(x)p Fs(\),)g Fq(a)22537 48383 y Fo(2)23034 48217 y Fs(\()p Fq(x)p Fs(\))351 b(are)f(trigonometric)k(p)31 b(olynomials)353 b(and)e(ha)-31 b(v)g(e)352 b(the)6075 49687 y(same)442 b(p)31 b(erio)g(ds)12754 49395 y Fq(~)12878 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Fs(=)307 b(0)p Fq(;)978 b Fi(when)396 b Fl(j)p Fq(r)31 b Fl(j)308 b Fq(>)g Fs(2)32231 48635 y Fn(n)32834 49092 y Fq(R)33675 49258 y Fo(0)34171 49092 y Fq(;)9479 b Fs(\(36\))6075 50912 y Fq(S)6818 50337 y Fo(\()p Fn(n)p Fo(\))6754 51042 y Fn(r)8112 50912 y Fi(,)325 b Fs(~)-568 b Fq(a)9346 50337 y Fo(\()p Fn(n)p Fo(\))9346 51042 y Fn(r)10640 50912 y Fi(,)310 b Fq(r)339 b Fl(2)307 b Fq(Z)14007 50511 y Fo(2)14792 50912 y Fi(b)-57 b(eing)289 b(F)-85 b(ourier)288 b(c)-57 b(o)g(e\016cients)289 b(of)f Fq(S)28851 50511 y Fo(\()p Fn(n)p Fo(\))30145 50912 y Fs(\()30519 50620 y Fq(~)30575 50912 y(k)36 b(;)184 b(x)p Fs(\))289 b Fi(and)303 b Fs(~)-568 b Fq(a)35657 50511 y Fo(\()p Fn(n)p Fo(\))36951 50912 y Fs(\()37325 50620 y Fq(~)37381 50912 y(k)36 b(;)184 b(x)p Fs(\))p Fi(,)311 b(r)-57 b(esp)g(e)g(ctively;)15557 52035 y Fp(\015)15557 52699 y(\015)15557 53363 y(\015)16172 53308 y Fq(S)16915 52851 y Fo(\()p Fn(n)p Fo(\))18209 53308 y Fs(\()18583 53016 y Fq(~)18639 53308 y(k)35 b(;)184 b(x)p Fs(\))20804 52035 y Fp(\015)20804 52699 y(\015)20804 53363 y(\015)21421 54083 y Fn(\027)22279 53308 y Fl(\024)307 b Fq(c)23926 53474 y Fn(n;\027)t(;b)26076 52035 y Fp(\015)26076 52699 y(\015)26076 53363 y(\015)26898 53028 y Fq(~)26691 53308 y(A)27521 52035 y Fp(\015)27521 52699 y(\015)27521 53363 y(\015)28136 52309 y Fn(n)p Fo(+1)28136 54083 y Fn(\027)52 b Fo(+)p Fn(n)30098 53308 y Fq(k)30709 52851 y Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(\(1+2)p Fn(n)p Fo(\))p Fn(\014)36136 53308 y Fq(;)7514 b Fs(\(37\))15265 55066 y Fp(\015)15265 55730 y(\015)15265 56394 y(\015)15880 56339 y Fl(<)p Fq(S)17422 55881 y Fo(\()p Fn(n)p Fo(\))18716 56339 y Fs(\()19090 56046 y Fq(~)19146 56339 y(k)36 b(;)184 b(x)p Fs(\))21312 55066 y Fp(\015)21312 55730 y(\015)21312 56394 y(\015)21928 57113 y Fn(\027)22786 56339 y Fl(\024)308 b Fq(c)24434 56505 y Fn(n;\027)t(;b)26584 55066 y Fp(\015)26584 55730 y(\015)26584 56394 y(\015)27406 56059 y Fq(~)27199 56339 y(A)28029 55066 y Fp(\015)28029 55730 y(\015)28029 56394 y(\015)28644 55339 y Fn(n)p Fo(+1)28644 57113 y Fn(\027)52 b Fo(+)p Fn(n)p Fo(+2)31727 56339 y Fq(k)32338 55881 y Fk(\000)p Fo(2)p Fn(\014)40 b Fo(\()p Fn(n)p Fo(+1\))36428 56339 y Fq(;)7222 b Fs(\(38\))11883 57864 y Fp(\015)11883 58528 y(\015)11883 59192 y(\015)12498 59137 y Fq(D)13445 58605 y Fn(\026)13372 59489 y(~)13414 59694 y(k)14039 59137 y Fq(S)14782 58680 y Fo(\()p Fn(n)p Fo(\))16076 59137 y Fs(\()16450 58845 y Fq(~)16506 59137 y(k)36 b(;)184 b(x)p Fs(\))18672 57864 y Fp(\015)18672 58528 y(\015)18672 59192 y(\015)19288 59912 y Fn(\027)20146 59137 y Fl(\024)308 b Fq(c)21794 59303 y Fn(n;\027)t(;b;\026)24745 57864 y Fp(\015)24745 58528 y(\015)24745 59192 y(\015)25567 58857 y Fq(~)25360 59137 y(A)26190 57864 y Fp(\015)26190 58528 y(\015)26190 59192 y(\015)26805 58137 y Fn(n)p Fo(+1)26805 59912 y Fn(\027)52 b Fo(+)p Fn(n)28767 59137 y Fq(k)29378 58680 y Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(\(1+2)p Fn(n)p Fo(\))p Fn(\014)40 b Fk(\000)p Fo(\(1)p Fn(=)p Fo(2+)p Fn(\014)g Fo(\))p Fk(j)p Fn(\026)p Fk(j)39809 59137 y Fq(;)3841 b Fs(\(39\))6075 61474 y Fq(D)7022 60943 y Fn(\026)6949 61827 y(~)6991 62032 y(k)8012 61474 y Fi(b)-57 b(eing)397 b(the)f(derivative)h(with)f(r)-57 b(esp)g(e)g(ct)396 b(to)25042 61182 y Fq(~)25098 61474 y(k)431 b Fi(of)396 b(or)-57 b(der)396 b Fq(\026)308 b Fs(=)f(\()p Fq(\026)33560 61640 y Fo(1)34057 61474 y Fq(;)184 b(\026)35215 61640 y Fo(2)35713 61474 y Fs(\))p Fi(,)397 b Fl(j)p Fq(\026)p Fl(j)308 b Fs(=)g Fq(\026)40304 61640 y Fo(1)41046 61474 y Fs(+)246 b Fq(\026)42820 61640 y Fo(2)43317 61474 y Fi(;)10665 62878 y Fp(\015)10665 63542 y(\015)10665 64206 y(\015)11280 62922 y(\020)11941 64151 y Fq(S)12684 63694 y Fo(\()p Fn(n)p Fo(\))13978 64151 y Fs(\()14352 63859 y Fq(~)14408 64151 y(k)35 b(;)184 b(x)p Fs(\))248 b Fl(\000)e Fq(S)18671 63694 y Fo(\()p Fn(n)p Fo(\))19965 64151 y Fs(\()20339 63859 y Fq(~)20395 64151 y(k)21006 63694 y Fk(0)21317 64151 y Fq(;)184 b(x)p Fs(\))22871 62922 y Fp(\021)23533 62878 y(\015)23533 63542 y(\015)23533 64206 y(\015)24148 64925 y Fn(\027)25006 64151 y Fl(\024)308 b Fq(c)26654 64317 y Fn(n;\027)t(;b)28804 62878 y Fp(\015)28804 63542 y(\015)28804 64206 y(\015)29626 63871 y Fq(~)29419 64151 y(A)30249 62878 y Fp(\015)30249 63542 y(\015)30249 64206 y(\015)30864 63151 y Fn(n)p Fo(+1)30864 64925 y Fn(\027)52 b Fo(+)p Fn(n)32826 64151 y Fl(j)33077 63859 y Fq(~)33133 64151 y(k)281 b Fl(\000)35041 63859 y Fq(~)35097 64151 y(k)35708 63694 y Fk(0)36019 64151 y Fl(j)p Fq(k)36937 63694 y Fk(\000)p Fo(2\()p Fn(n)p Fo(+1\))p Fn(\014)41028 64151 y Fq(;)2622 b Fs(\(40\))6075 66721 y Fi(if)397 b Fl(j)7401 66429 y Fq(~)7457 66721 y(k)281 b Fl(\000)9365 66429 y Fq(~)9421 66721 y(k)10032 66319 y Fk(0)10343 66721 y Fl(j)308 b Fq(<)f Fs(\(4)p Fq(b)13584 66887 y Fo(0)14082 66721 y Fs(\))14512 66319 y Fk(\000)p Fo(1)15701 66721 y Fq(k)16312 66319 y Fo(1)p Fn(=)p Fo(2+)p Fn(\014)40 b Fk(\000)p Fn(")20035 66721 y Fi(,)396 b Fq(b)21245 66887 y Fo(0)22050 66721 y Fs(=)307 b(max)q Fl(f)p Fs(2)p Fq(\031)40 b(=b)28083 66887 y Fo(2)28581 66721 y Fq(;)184 b Fs(1)p Fl(g)p Fi(;)16989 67844 y Fp(\015)16989 68508 y(\015)16989 69172 y(\015)17620 69117 y Fs(~)-569 b Fq(a)18189 68659 y Fo(\()p Fn(n)p Fo(\))19484 69117 y Fs(\()19858 68824 y Fq(~)19914 69117 y(k)35 b(;)184 b(x)p Fs(\))22079 67844 y Fp(\015)22079 68508 y(\015)22079 69172 y(\015)22696 69891 y Fn(\027)23554 69117 y Fl(\024)307 b Fq(c)25201 69283 y Fn(n;\027)t(;b)27351 67844 y Fp(\015)27351 68508 y(\015)27351 69172 y(\015)28173 68837 y Fq(~)27966 69117 y(A)28796 67844 y Fp(\015)28796 68508 y(\015)28796 69172 y(\015)29411 68117 y Fn(n)p Fo(+1)29411 69891 y Fn(\027)52 b Fo(+)p Fn(n)31373 69117 y Fq(k)31984 68659 y Fo(1)p Fk(\000)p Fo(2)p Fn(n\014)34703 69117 y Fq(;)8947 b Fs(\(41\))p eop end %%Page: 10 10 TeXDict begin 10 9 bop 6075 3345 a Fo(10)14514 b(YULIA)300 b(KARPESHINA)14211 5120 y Fp(\015)14211 5784 y(\015)14211 6448 y(\015)14826 6393 y Fi(Im)412 b Fs(~)-569 b Fq(a)17139 5936 y Fo(\()p Fn(n)p Fo(\))18434 6393 y Fs(\()18808 6101 y Fq(~)18864 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Fq(~)26635 9683 y(A)27465 8410 y Fp(\015)27465 9074 y(\015)27465 9738 y(\015)28080 8683 y Fn(n)p Fo(+1)28080 10457 y Fn(\027)52 b Fo(+)p Fn(n)30042 9683 y Fq(k)30653 9225 y Fo(1)p Fk(\000)p Fo(2)p Fn(n\014)40 b Fk(\000)p Fo(\(1)p Fn(=)p Fo(2+)p Fn(\014)g Fo(\))p Fk(j)p Fn(\026)p Fk(j)38377 9683 y Fq(;)5273 b Fs(\(43\))10492 11699 y Fp(\015)10492 12363 y(\015)10492 13027 y(\015)11123 12972 y Fs(~)-569 b Fq(a)11692 12515 y Fo(\()p Fn(n)p Fo(\))12987 12972 y Fs(\()13361 12680 y Fq(~)13417 12972 y(k)36 b(;)184 b(x)p Fs(\))247 b Fl(\000)261 b Fs(~)-568 b Fq(a)17522 12515 y Fo(\()p Fn(n)p Fo(\))19001 11743 y Fp(\020)19605 12680 y Fq(~)19662 12972 y(k)20273 12515 y Fk(0)20583 12972 y Fq(;)184 b(x)21707 11743 y Fp(\021)22369 11699 y(\015)22369 12363 y(\015)22369 13027 y(\015)22984 13747 y Fn(\027)23842 12972 y Fl(\024)308 b Fq(c)25490 13138 y Fn(n;\027)t(;b)27640 11699 y Fp(\015)27640 12363 y(\015)27640 13027 y(\015)28462 12692 y Fq(~)28255 12972 y(A)29085 11699 y Fp(\015)29085 12363 y(\015)29085 13027 y(\015)29700 11973 y Fn(n)p Fo(+1)29700 13747 y Fn(\027)52 b Fo(+)p Fn(n)31662 12972 y Fl(j)31913 12680 y Fq(~)31969 12972 y(k)282 b Fl(\000)33877 12680 y Fq(~)33934 12972 y(k)34545 12515 y Fk(0)34855 12972 y Fl(j)p Fq(k)35773 12515 y Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(\(2)p Fn(n)p Fo(+1\))p Fn(\014)41200 12972 y Fq(;)2450 b Fs(\(44\))6075 15555 y Fi(if)431 b Fl(j)7435 15263 y Fq(~)7491 15555 y(k)307 b Fl(\000)9449 15263 y Fq(~)9506 15555 y(k)10117 15154 y Fk(0)10427 15555 y Fl(j)371 b Fq(<)f Fs(\(4)p Fq(b)13794 15721 y Fo(0)14291 15555 y Fs(\))14721 15154 y Fk(\000)p Fo(1)15910 15555 y Fq(k)16521 15154 y Fo(1)p Fn(=)p Fo(2+)p Fn(\014)40 b Fk(\000)p Fn(")20244 15555 y Fi(.)613 b(The)431 b(estimates)f(\(37\))f({)h(\(42\))g(hold)f(for)i(any)g Fq(\027)440 b Fs(=)370 b(0)p Fq(;)184 b Fs(1)p Fq(;)g(:::)6075 17013 y Fi(and)462 b Fq(k)498 b Fi(lar)-57 b(ge)462 b(enough,)480 b Fq(k)463 b(>)428 b(k)19193 17179 y Fo(0)19690 17013 y Fs(\()20327 16734 y Fq(~)20120 17013 y(A)q(;)184 b(\013)t(;)g(N)59 b(;)184 b(\014)58 b(;)184 b(";)g(\027)70 b Fs(\))p Fi(,)486 b(namely)464 b(if)f Fq(k)g Fs(:)428 b Fq(k)463 b(>)428 b(\013)t(;)648 b(k)39545 16612 y Fn(")40446 17013 y Fq(>)428 b Fs(2)42288 16612 y Fn(N)83 b Fo(+2)44249 17013 y Fq(R)45090 17179 y Fo(0)45586 17013 y Fi(,)6075 18472 y Fq(k)6686 18070 y Fo(2)p Fn(\014)8032 18472 y Fq(>)307 b Fs(\(5)p Fq(b)10658 18638 y Fo(0)11156 18472 y Fs(\))11586 18070 y Fn(N)83 b Fo(+2+)p Fn(\027)14722 18472 y Fl(k)15482 18192 y Fq(~)15275 18472 y(A)p Fl(k)16658 18638 y Fn(\027)52 b Fo(+)p Fn(N)83 b Fo(+2)20040 18472 y Fs(+)246 b Fq(b)21622 18070 y Fo(2)21622 18746 y(0)22119 18472 y Fi(.)6075 20630 y Fx(Corollary)443 b(1.)18123 21246 y Fp(\015)18123 21910 y(\015)18123 22574 y(\015)18738 22519 y Fq(S)64 b Fs(\()19855 22227 y Fq(~)19911 22519 y(k)35 b(;)184 b(x)p Fs(\))22076 21246 y Fp(\015)22076 21910 y(\015)22076 22574 y(\015)22693 23294 y Fn(\027)23551 22519 y Fl(\024)307 b Fq(c)25198 22685 y Fn(N)35 b(;\027)t(;b)27538 21246 y Fp(\015)27538 21910 y(\015)27538 22574 y(\015)28360 22239 y Fq(~)28153 22519 y(A)28983 21246 y Fp(\015)28983 21910 y(\015)28983 22574 y(\015)29598 23294 y Fn(\027)30333 22519 y Fq(k)30944 22062 y Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fn(\014)33569 22519 y Fq(;)10081 b Fs(\(45\))17611 24218 y Fp(\015)17611 24882 y(\015)17611 25546 y(\015)18226 25491 y Fl(<)p Fq(S)64 b Fs(\()20142 25198 y Fq(~)20198 25491 y(k)35 b(;)184 b(x)p Fs(\))22363 24218 y Fp(\015)22363 24882 y(\015)22363 25546 y(\015)22979 26265 y Fn(\027)23838 25491 y Fl(\024)307 b Fq(c)25485 25657 y Fn(N)35 b(;\027)t(;b)27824 24218 y Fp(\015)27824 24882 y(\015)27824 25546 y(\015)28646 25211 y Fq(~)28439 25491 y(A)29270 24218 y Fp(\015)29270 24882 y(\015)29270 25546 y(\015)29885 26265 y Fn(\027)52 b Fo(+2)31741 25491 y Fq(k)32352 25033 y Fk(\000)p Fo(2)p Fn(\014)34082 25491 y Fq(;)9568 b Fs(\(46\))17776 27281 y Fp(\015)17776 27945 y(\015)17776 28610 y(\015)18391 27281 y(\014)18391 27945 y(\014)18391 28610 y(\014)18760 28554 y 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Fs(\(73\))6075 19605 y(It)369 b(is)g(easy)g(to)h(see)f(that)22352 21061 y Fq(\037)p Fs(\()p Fq(\025)p Fs(\))308 b(=)f Fl(K)16 b Fq(D)31 b Fs(\()p Fq(\025)p Fs(\))p Fq(;)14311 b Fs(\(74\))6075 22749 y(here)368 b Fl(K)324 b Fs(:)308 b Fq(R)11108 22347 y Fo(2)11911 22749 y Fl(!)g Fq(K)448 b Fs(is)369 b(the)h(map:)23337 24205 y Fl(K)25 b Fq(~)-562 b(p)24753 24371 y Fn(j)25219 24205 y Fs(\()p Fq(t)p Fs(\))309 b(=)e Fq(t;)6075 26075 y Fs(the)396 b(map)h(b)31 b(eing)396 b(correctly)h(de\014ned,)403 b(since)395 b(an)-31 b(y)27073 25783 y Fq(~)27129 26075 y(k)388 b Fl(2)352 b Fq(R)30033 25673 y Fo(2)30924 26075 y Fs(can)396 b(b)31 b(e)396 b(represen)-31 b(ted)395 b(in)h(the)h(form)6018 27192 y Fq(~)6075 27484 y(k)342 b Fs(=)317 b Fq(~)-562 b(p)8719 27650 y Fn(j)9185 27484 y Fs(\()p Fq(t)p Fs(\))368 b(in)g(a)f(unique)h(w)-31 b(a)g(y)-92 b(.)493 b(Ob)-31 b(viously)-92 b(,)370 b(to)e(obtain)h Fq(\037)p Fs(\()p Fq(\025)p Fs(\))e(w)-31 b(e)368 b(break)g Fq(D)31 b Fs(\()p Fq(\025)p Fs(\))368 b(b)-31 b(y)367 b(the)h(lattice)6075 28812 y(with)323 b(no)31 b(des)321 b(in)h Fl(f)9 b Fq(~)-562 b(p)13967 28978 y Fn(r)14457 28812 y Fs(\(0\))p Fl(g)16423 29009 y Fn(r)24 b Fk(2)p Fn(Z)18112 28788 y Fm(2)18923 28812 y Fs(in)-31 b(to)323 b(pieces)f(and)g(then)g(shift)g(them)g(in)g(a)g(parallel)i(manner)e(in) -31 b(to)6075 30141 y Fq(K)79 b Fs(.)493 b(Th)-31 b(us,)369 b Fq(\037)p Fs(\()p Fq(\025)p Fs(\))h(is)f(the)g(set)g Fq(D)31 b Fs(\()p Fq(\025)p Fs(\))370 b(pac)-31 b(k)g(ed)370 b(in)-31 b(to)371 b(the)e(bag)h Fq(K)79 b Fs(.)7403 31469 y(W)-92 b(e)369 b(call)h Fq(\037)p Fs(\()p Fq(\025)p Fs(\))f(a)h(mo)31 b(del)370 b(iso)31 b(energetic)370 b(set.)6075 33535 y Fx(Theorem)348 b(2.)493 b Fi(L)-57 b(et)334 b Fs(0)308 b Fq(<)g(")f(<)h Fs(1)p Fq(=)p Fs(100)p Fi(,)349 b Fs(1)p Fq(=)p Fs(3)309 b Fq(<)f(\014)366 b(<)307 b Fs(1)p Fq(=)p Fs(2)111 b Fl(\000)g Fs(6)p Fq(")p Fi(,)349 b Fs(0)308 b Fq(<)g(\016)349 b(<)308 b Fs(min)q Fl(f)p Fs(1)p Fq(=)p Fs(2)111 b Fl(\000)g Fq(\014)170 b Fl(\000)111 b Fs(6)p Fq(";)184 b(")p Fl(g)p Fi(,)6075 34993 y Fq(N)504 b Fl(\025)383 b Fs(31)p Fi(.)10794 34564 y Fo(2)11235 34993 y Fi(.)635 b(Then)438 b(for)g(any)h(su\016ciently)g(lar)-57 b(ge)438 b Fq(\025)p Fi(,)449 b Fq(\025)383 b(>)g(\025)31807 35159 y Fo(0)32303 34993 y Fs(\()32941 34713 y Fq(~)32733 34993 y(A)q Fs(\()p Fq(x)p Fs(\))p Fq(;)184 b(\014)58 b(;)184 b(";)g(\016)42 b Fs(\))p Fi(,)453 b(ther)-57 b(e)438 b(exists)g(a)6075 36321 y(subset)395 b Fq(\037)10049 36487 y Fk(\003)10558 36321 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\))397 b Fi(of)g Fq(\037)p Fs(\()p Fq(\025)p Fs(\))f Fi(with)g(the)f(fol)57 b(lowing)396 b(pr)-57 b(op)g(erties:)7403 37769 y(1\))437 b(F)-85 b(or)437 b(every)i Fq(t)383 b Fl(2)g Fq(\037)16534 37935 y Fk(\003)17043 37769 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\))439 b Fi(ther)-57 b(e)438 b(is)g(a)g(single)g Fq(j)446 b Fl(2)384 b Fq(J)106 b Fs(\()p Fq(\025)32020 37935 y Fo(0)32517 37769 y Fs(\))438 b Fi(such)g(that)e Fl(<)39094 37477 y Fs(^)39048 37769 y Fq(\025)39694 37935 y Fn(j)40161 37769 y Fs(\()p Fq(t)p Fs(\))384 b(=)f Fq(\025)p Fi(,)449 b(the)6075 39098 y(fol)57 b(lowing)396 b(ine)-57 b(quality)398 b(b)-57 b(eing)398 b(satis\014e)-57 b(d:)18285 41077 y Fs(min)16341 41842 y Fn(m)p Fk(2)p Fn(J)73 b Fo(\()p Fn(\025)19158 41953 y Fm(0)19589 41842 y Fo(\))p Fn(;m)p Fk(6)p Fo(=)p Fn(j)22258 41077 y Fl(j<)23410 40785 y Fs(^)23364 41077 y Fq(\025)24010 41243 y Fn(j)24662 41077 y Fs(\()p Fq(t)p Fs(\))247 b Fl(\000)f(<)28121 40785 y Fs(^)28075 41077 y Fq(\025)28721 41243 y Fn(m)29746 41077 y Fs(\()q Fq(t)p Fs(\))184 b Fl(j)308 b Fq(>)g Fs(2)p Fq(\025)34174 40620 y Fk(\000)p Fn(\016)35352 41077 y Fq(:)8298 b Fs(\(75\))6075 43911 y Fi(The)396 b(op)-57 b(er)g(ator)395 b Fq(H)90 b Fs(\()p Fq(t)p Fs(\))397 b Fi(has)e(a)h(simple)g(eigenvalue)i Fq(\025)27747 44077 y Fn(j)28213 43911 y Fs(\()p Fq(t)p Fs(\))f Fi(close)f(to)f Fl(<)34760 43619 y Fs(^)34714 43911 y Fq(\025)35360 44077 y Fn(j)35827 43911 y Fs(\()p Fq(t)p Fs(\))308 b(=)f Fq(\025)p Fi(,)19114 45992 y Fq(\025)19760 46158 y Fn(j)20226 45992 y Fs(\()p Fq(t)p Fs(\))h(=)g Fl(<)23808 45699 y Fs(^)23762 45992 y Fq(\025)24408 46158 y Fn(j)25059 45992 y Fs(\()p Fq(t)p Fs(\))246 b(+)g Fq(O)31 b Fs(\()p Fq(\025)29623 45534 y Fk(\000)p Fo(21)p Fn(=)p Fo(2)32148 45992 y Fs(\))p Fq(:)11072 b Fs(\(76\))6075 47912 y Fi(This)469 b(eigenvalue)j(is)d(unique:)659 b(ther)-57 b(e)469 b(ar)-57 b(e)470 b(no)g(other)f(eigenvalues)i(of)f Fq(H)90 b Fs(\()p Fq(t)p Fs(\))470 b Fi(in)g(the)g(interval)6075 49241 y Fq(\017)p Fs(\()p Fq(\025)p Fs(\))308 b(=)g(\()p Fq(\025)246 b Fl(\000)g Fq(\025)12582 48839 y Fk(\000)p Fn(\016)13759 49241 y Fq(;)184 b(\025)247 b Fs(+)f Fq(\025)16896 48839 y Fk(\000)p Fn(\016)18073 49241 y Fs(\))p Fi(.)7403 50569 y(2\))396 b(F)-85 b(ormula)395 b(\(76\))g(is)h(di\013er)-57 b(entiable)397 b(with)f(r)-57 b(esp)g(e)g(ct)397 b(to)e Fq(t)p Fi(:)17118 52610 y Fl(r)18040 52776 y Fn(t)18430 52610 y Fq(\025)19076 52776 y Fn(j)19543 52610 y Fs(\()p Fq(t)p Fs(\))308 b(=)f Fl(r)23201 52776 y Fn(t)23591 52610 y Fl(<)24436 52318 y Fs(^)24390 52610 y Fq(\025)25036 52776 y Fn(j)25503 52610 y Fs(\()p Fq(t)p Fs(\))247 b(+)f Fq(O)31 b Fs(\()p Fq(\025)30068 52153 y Fk(\000)p Fo(19)p Fn(=)p Fo(2+2)p Fn(\016)34144 52610 y Fs(\))p Fq(:)9076 b Fs(\(77\))7403 54531 y Fi(3\))317 b(A)-28 b(n)317 b(eigenfunction)j Fs(\011)18101 54697 y Fn(j)18567 54531 y Fs(\()p Fq(t;)184 b(x)p Fs(\))319 b Fi(c)-57 b(orr)g(esp)g(onding)318 b(to)f Fq(\025)30075 54697 y Fn(j)30541 54531 y Fs(\()p Fq(t)p Fs(\))h Fi(is)f(close)g(to)g(the)f(mo)-57 b(del)318 b(function)6228 55727 y Fs(^)6075 56006 y(\011)6936 56172 y Fn(j)7402 56006 y Fs(\()p Fq(t;)184 b(x)p Fs(\))p Fi(:)16081 56561 y Fp(\015)16081 57226 y(\015)16081 57890 y(\015)16696 57834 y Fs(\011)17557 58000 y Fn(j)18023 57834 y Fs(\()p Fq(t;)g(x)p Fs(\))248 b Fl(\000)21915 57555 y Fs(^)21762 57834 y(\011)22623 58000 y Fn(j)23089 57834 y Fs(\()p Fq(t;)184 b(x)p Fs(\))25473 56561 y Fp(\015)25473 57226 y(\015)25473 57890 y(\015)26089 58609 y Fn(L)26696 58720 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Fl(\000)g Fq(\025)p Fl(j)307 b Fq(<)6075 63072 y Fs(4)p Fq(\025)7274 62670 y Fk(\000)p Fo(2)p Fn(\016)8893 63072 y Fi(,)359 b(the)350 b(estimate)g(\(75\))e(b)-57 b(eing)352 b(satis\014e)-57 b(d.)494 b(The)350 b(op)-57 b(er)g(ator)349 b Fq(H)90 b Fs(\()p Fq(t)p Fs(\))351 b Fi(has)e(a)h(simple)g(eigenvalue)6075 64400 y Fq(\025)6721 64566 y Fn(j)7187 64400 y Fs(\()p Fq(t)p Fs(\))488 b Fi(ob)-57 b(eying)489 b(\(76\).)782 b(This)487 b(eigenvalue)i(is)e (unique:)693 b(ther)-57 b(e)488 b(ar)-57 b(e)487 b(no)g(other)g (eigenvalues)h(of)6075 65729 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))397 b Fi(in)f(the)g(interval)h Fq(\017)p Fs(\()p Fq(\025)p Fs(\))p Fi(,)g(the)f(estimates)g(\(77\))e(and)i(\(78\))f(b) -57 b(eing)398 b(true.)p 6075 67114 6642 45 v 7403 68091 a Fo(2)7844 68434 y Fw(This)247 b(is)g(not)g(the)g(optimal)h(set)g(of)e (parameters.)397 b(F)-78 b(or)247 b(a)g(example,)261 b(an)248 b(analogous)h(theorem)e(can)g(b)26 b(e)247 b(pro)-26 b(v)g(ed)6075 69541 y(for)313 b(smaller)i Ff(\014)46 b Fw(.)418 b(Ho)-26 b(w)g(ev)g(er,)313 b(it)h(w)-26 b(ould)313 b(require)h(longer)g(considerations.)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 6075 3345 a Fo(14)14514 b(YULIA)300 b(KARPESHINA)7403 6001 y Fi(5\))412 b(The)h(set)f Fq(\037)13549 6167 y Fk(\003)14057 6001 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\))414 b Fi(has)e(a)g(me)-57 b(asur)g(e)414 b(asymptotic)-57 b(al)57 b(ly)412 b(close)g(to)g(that)g(of)h(a)f(cir)-57 b(cle)414 b(of)f(the)6075 7421 y(r)-57 b(adius)9399 6477 y Fl(p)p 10321 6477 646 45 v 944 x Fq(\025)396 b Fi(in)h(the)f(fol)57 b(lowing)395 b(sense:)17373 9098 y Fq(l)207 b Fs(\()p Fq(\037)19033 9264 y Fk(\003)19542 9098 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\)\))p 17373 9593 5132 45 v 18543 10757 a(2)p Fq(\031)19767 9814 y Fl(p)p 20690 9814 646 45 v 20690 10757 a Fq(\025)22945 9847 y(>)308 b Fs(1)246 b Fl(\000)g Fq(c)26665 10023 y Fn(~)26499 10219 y(A)27220 9847 y Fq(\025)27866 9390 y Fk(\000)p Fn(\016)28 b(=)p Fo(4)29938 9847 y Fq(;)977 b(c)31867 10023 y Fn(~)31701 10219 y(A)32730 9847 y Fq(>)308 b Fs(0)p Fq(:)9198 b Fs(\(79\))6075 12452 y Fx(Corollary)443 b(4.)553 b Fi(Under)396 b(c)-57 b(onditions)397 b(of)f(The)-57 b(or)g(em)397 b(2)18377 14338 y Fl(r)19299 14504 y Fn(t)19689 14338 y Fq(\025)20335 14504 y Fn(j)20802 14338 y Fs(\()p Fq(t)p Fs(\))308 b(=)316 b Fq(~)-562 b(p)24095 14504 y Fn(j)24562 14338 y Fs(\()p Fq(t)p Fs(\))246 b(+)g Fq(O)31 b Fs(\()p Fq(\025)29126 13881 y Fo(\(1)p Fk(\000)p Fo(6)p Fn(\014)40 b Fo(\))p Fn(=)p Fo(4)32885 14338 y Fs(\))p Fq(:)10335 b Fs(\(80\))7403 16494 y(F)-92 b(orm)-31 b(ula)370 b(\(80\))h(follo)-31 b(ws)372 b(from)d(\(77\))i(and)f(\(53\),)h(where)e Fq(k)343 b Fs(=)307 b Fq(p)33324 16660 y Fn(j)33790 16494 y Fs(\()p Fq(t)p Fs(\))i Fl(\031)36527 15550 y(p)p 37450 15550 V 37450 16494 a Fq(\025)o Fs(.)7403 17822 y(An)445 b(iso)31 b(energetic)447 b(set)e Fq(I)87 b Fs(\()p Fq(\025)p Fs(\))445 b(of)h Fq(H)535 b Fs(is)445 b(de\014ned)g(as)g(the)h(set)f(of)h Fq(t)e Fs(for)i(whic)-31 b(h)446 b Fq(H)90 b Fs(\()p Fq(t)p Fs(\))446 b(has)f(an)6075 19151 y(eigen)-31 b(v)-61 b(alue)371 b(equal)f(to)g(a)f(giv)-31 b(en)371 b Fq(\025)p Fs(:)15538 21000 y Fq(I)87 b Fs(\()p Fq(\025)p Fs(\))308 b(=)g Fl(f)p Fq(t)f Fl(2)h Fq(K)387 b Fs(:)308 b Fl(9)p Fq(\025)24604 21166 y Fn(n)25206 21000 y Fs(\()p Fq(t)p Fs(\))p Fq(;)554 b(n)308 b Fl(2)g Fq(N)59 b(;)184 b(\025)31430 21166 y Fn(n)32034 21000 y Fs(\()p Fq(t)p Fs(\))308 b(=)f Fq(\025)p Fl(g)185 b Fq(:)6075 22892 y Fs(W)-92 b(e)453 b(denote)i(b)-31 b(y)455 b Fq(I)13828 23058 y Fk(\003)14336 22892 y Fs(\()p Fq(\025)p Fs(\))g(the)g(part)f(of)h Fq(I)87 b Fs(\()p Fq(\025)p Fs(\))454 b(whic)-31 b(h)455 b(is)g(in)f(the)h(\()p Fq(\025)33550 22490 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)37197 22892 y Fs(\)-neigh)-31 b(b)31 b(orho)g(o)g(d)456 b(of)6075 24220 y Fq(\037)6768 24386 y Fk(\003)7276 24220 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\).)6075 26245 y Fx(Corollary)443 b(5.)553 b Fi(F)-85 b(or)395 b(every)i Fq(t)308 b Fl(2)f Fq(I)20533 26411 y Fk(\003)21042 26245 y Fs(\()p Fq(\025)p Fs(\))396 b Fi(ther)-57 b(e)396 b(is)h(a)e(single)i Fq(j)371 b Fl(2)307 b Fq(J)106 b Fs(\()p Fq(\025)34632 26411 y Fo(0)35129 26245 y Fs(\))397 b Fi(such)f(that)f Fq(\025)41383 26411 y Fn(j)41849 26245 y Fs(\()p Fq(t)p Fs(\))308 b(=)f Fq(\025)p Fi(.)7403 28270 y Fs(Pro)31 b(of)370 b(of)g(the)f(corollary)-92 b(.)p 7403 28662 10633 45 v 556 w(If)554 b Fq(t)616 b Fl(2)f Fq(I)22738 28436 y Fk(\003)23247 28270 y Fs(\()p Fq(\025)p Fs(\),)601 b(then)555 b(there)f(is)g(an)g(eigen)-31 b(v)-61 b(alue)556 b Fq(\025)40525 28436 y Fn(n)41127 28270 y Fs(\()p Fq(t)p Fs(\))f(of)g(the)6075 29800 y(op)31 b(erator)522 b Fq(H)90 b Fs(\()p Fq(t)p Fs(\))523 b(suc)-31 b(h)522 b(that)h Fq(\025)19332 29966 y Fn(n)19934 29800 y Fs(\()p Fq(t)p Fs(\))563 b(=)g Fq(\025)p Fs(.)951 b(Since)522 b Fq(t)g Fs(is)g(in)g(the)g(\()p Fq(\025)34895 29398 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)38542 29800 y Fs(\)-neigh)-31 b(b)31 b(orho)g(o)g(d)6075 31128 y(of)394 b Fq(\037)8053 31294 y Fk(\003)8562 31128 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\),)402 b(b)-31 b(y)395 b(the)f(forth)h(part)f(of)h(Theorem)g(2,)401 b(there)394 b(is)g(a)g(unique)h Fq(j)412 b Fl(2)349 b Fq(J)106 b Fs(\()p Fq(\025)40052 31294 y Fo(0)40550 31128 y Fs(\))394 b(suc)-31 b(h)394 b(that)6075 32577 y Fq(\025)6721 32743 y Fn(j)7187 32577 y Fs(\()p Fq(t)p Fs(\))370 b(is)g(giv)-31 b(en)371 b(b)-31 b(y)370 b(\(76\))i(and)e Fl(j<)20003 32285 y Fs(^)19957 32577 y Fq(\025)20603 32743 y Fn(j)21070 32577 y Fs(\()p Fq(t)p Fs(\))248 b Fl(\000)e Fq(\025)p Fl(j)309 b Fq(<)g Fs(4)p Fq(\025)27316 32175 y Fk(\000)p Fo(2)p Fn(\016)28935 32577 y Fs(.)494 b(Since)370 b Fq(\025)33273 32743 y Fn(j)33740 32577 y Fs(\()p Fq(t)p Fs(\))g(is)g(a)g(unique)g (eigen)-31 b(v)-61 b(alue)6075 33905 y(of)369 b Fq(H)90 b Fs(\()p Fq(t)p Fs(\))370 b(in)g(the)f(in)-31 b(terv)-61 b(al)371 b Fq(")p Fs(\()p Fq(\025)p Fs(\),)f(w)-31 b(e)370 b(conclude)f(that)i Fq(\025)29098 34071 y Fn(n)29700 33905 y Fs(\()p Fq(t)p Fs(\))308 b(=)g Fq(\025)33083 34071 y Fn(j)33549 33905 y Fs(\()p Fq(t)p Fs(\).)493 b Fl(})7403 35233 y Fs(Since)396 b Fq(\037)11013 35399 y Fk(\003)11522 35233 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\))353 b Fl(\032)e Fq(\037)p Fs(\()p Fq(\025)p Fs(\))h(=)g Fl(K)16 b Fq(D)427 b Fs(\(see)396 b(\(74\)\),)405 b(there)395 b(is)h(a)g(subset)f Fq(D)36223 35399 y Fk(\003)36733 35233 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\))397 b(of)f Fq(D)31 b Fs(\()p Fq(\025)p Fs(\))396 b(suc)-31 b(h)6075 36562 y(that)428 b Fq(\037)9224 36728 y Fk(\003)10135 36562 y Fs(=)403 b Fl(K)16 b Fq(D)13174 36728 y Fk(\003)13684 36562 y Fs(.)665 b(Ob)-31 b(viously)-92 b(,)443 b Fq(D)21062 36728 y Fk(\003)21571 36562 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\))428 b(is)f(also)g(a)g(distorted)g(sphere)f(with)i(holes,)442 b(it)427 b(just)6075 37890 y(has)410 b(more)g(holes)g(than)h Fq(D)31 b Fs(\()p Fq(\025)p Fs(\).)616 b(Note)411 b(that)g Fq(D)25786 38056 y Fk(\003)26295 37890 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\))412 b(is)e(de\014ned)f(uniquely)-92 b(,)422 b(since,)f(b)-31 b(y)410 b(\(75\),)6075 39218 y Fq(\037)6768 39384 y Fk(\003)7276 39218 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\))371 b(do)31 b(es)369 b(not)h(ha)-31 b(v)g(e)370 b(sel\014n)-31 b(tersections.)493 b(Hence,)370 b Fq(l)22 b Fs(\()p Fq(D)30845 39384 y Fk(\003)31355 39218 y Fs(\))308 b(=)f Fq(l)22 b Fs(\()p Fq(\037)34736 39384 y Fk(\003)35245 39218 y Fs(\))370 b(and)f(b)-31 b(y)370 b(Theorem)f(2:)17288 40958 y Fq(l)207 b Fs(\()q Fq(D)19172 41124 y Fk(\003)19681 40958 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\)\))p 17288 41452 5356 45 v 18570 42617 a(2)p Fq(\031)19794 41673 y Fl(p)p 20717 41673 646 45 v 20717 42617 a Fq(\025)23084 41706 y(>)308 b Fs(1)246 b Fl(\000)g Fq(c)26804 41882 y Fn(~)26638 42078 y(A)27359 41706 y Fq(\025)28005 41249 y Fk(\000)p Fn(\016)28 b(=)p Fo(4)30077 41706 y Fq(;)923 b(c)31952 41882 y Fn(~)31786 42078 y(A)32815 41706 y Fq(>)307 b Fs(0)p Fq(:)9114 b Fs(\(81\))6075 44198 y(Let)369 b Fq(D)8974 44364 y Fk(\003)p Fo(1)9925 44198 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\))370 b(b)31 b(e)369 b(the)g(set)g(of)h(directions)g(corresp)31 b(onding)369 b(to)h Fq(D)33664 44364 y Fk(\003)34174 44198 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\):)14871 46048 y Fq(D)15787 46214 y Fk(\003)p Fo(1)16738 46048 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\))309 b(=)e Fl(f)-22 b Fq(~)-531 b(\027)379 b Fl(2)307 b Fq(D)24186 46214 y Fo(1)24683 46048 y Fs(\()p Fq(\025)p Fs(\))h(:)g Fq(k)35 b Fs(\()p Fq(\025;)162 b(~)-531 b(\027)72 b Fs(\))-22 b Fq(~)-531 b(\027)378 b Fl(2)308 b Fq(D)33226 46214 y Fk(\003)33735 46048 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\))p Fl(g)p Fq(;)6831 b Fs(\(82\))6075 47898 y Fq(D)6991 48064 y Fo(1)7488 47898 y Fs(\()p Fq(\025)p Fs(\))530 b(b)31 b(eing)530 b(de\014ned)f(b)-31 b(y)529 b(\(62\).)975 b(In)529 b(the)h(next)g(theorem)g(w)-31 b(e)530 b(sho)-31 b(w)530 b(that)h Fq(I)40054 48064 y Fk(\003)40562 47898 y Fs(\()p Fq(\025)p Fs(\))f(can)g(b)31 b(e)6075 49226 y(describ)g(ed)368 b(as)h(the)h(shift)f Fl(K)386 b Fs(of)370 b(a)f(distorted)h(sphere)e(whic)-31 b(h)370 b(is)f(close)h(to)g Fq(D)37483 49392 y Fk(\003)37992 49226 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\).)6075 51251 y Fx(Theorem)462 b(3.)573 b Fi(Under)426 b(c)-57 b(onditions)426 b(of)g(The)-57 b(or)g(em)427 b(2,)433 b(the)425 b(set)g Fq(I)33218 51417 y Fk(\003)33727 51251 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\))427 b Fi(c)-57 b(an)427 b(b)-57 b(e)426 b(describ)-57 b(e)g(d)427 b(by)6075 52657 y(the)395 b(formula)i Fq(I)12517 52823 y Fk(\003)13025 52657 y Fs(\()p Fq(\025)p Fs(\))308 b(=)g Fl(K)17125 52378 y Fs(~)16867 52657 y Fq(D)17783 52823 y Fk(\003)18292 52657 y Fs(\()p Fq(\025)p Fs(\))p Fi(,)398 b(wher)-57 b(e)23919 52378 y Fs(~)23661 52657 y Fq(D)24577 52823 y Fk(\003)25086 52657 y Fs(\()p Fq(\025)p Fs(\))397 b Fi(is)f(a)g(distorte)-57 b(d)395 b(spher)-57 b(e)396 b(with)f(holes:)6885 54372 y Fs(~)6627 54652 y Fq(D)7543 54818 y Fk(\003)8052 54652 y Fs(\()p Fq(\025)p Fs(\))308 b(=)g Fl(f)11532 54360 y Fq(~)11588 54652 y(k)343 b Fl(2)307 b Fq(R)14402 54195 y Fo(2)15205 54652 y Fs(:)15764 54360 y Fq(~)15820 54652 y(k)343 b Fs(=)409 b(~)-654 b Fq(\032)p Fs(\()p Fq(\025;)162 b(~)-531 b(\027)71 b Fs(\))-22 b Fq(~)-531 b(\027)71 b(;)162 b(~)-531 b(\027)379 b Fl(2)307 b Fq(D)25091 54818 y Fk(\003)p Fo(1)26042 54652 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\))p Fq(;)288 b Fs(~)-657 b Fq(\032)s Fs(\()p Fq(\025;)162 b(~)-531 b(\027)71 b Fs(\))308 b(=)33731 53655 y Fl(p)p 34653 53655 V 997 x Fq(\025)246 b Fs(+)g Fq(h)p Fs(\()p Fq(\025;)162 b(~)-531 b(\027)71 b Fs(\))247 b(+)41270 54360 y(~)41258 54652 y Fq(h)p Fs(\()p Fq(\025;)162 b(~)-531 b(\027)72 b Fs(\))p Fl(g)p Fq(;)43957 55981 y Fs(\(83\))6075 57399 y Fi(function)10443 57107 y Fs(~)10431 57399 y Fq(h)p Fs(\()p Fq(\025;)162 b(~)-531 b(\027)71 b Fs(\))397 b Fi(satisfying)g(the)f(estimates:)21487 59047 y Fs(~)21476 59340 y Fq(h)o Fs(\()p Fq(\025;)162 b(~)-531 b(\027)72 b Fs(\))308 b(=)f Fq(O)31 b Fs(\()p Fq(\025)28156 58883 y Fk(\000)p Fo(11)29786 59340 y Fs(\))p Fq(;)13434 b Fs(\(84\))19928 61336 y Fq(@)20589 61044 y Fs(~)20577 61336 y Fq(h)p Fs(\()p Fq(\025;)162 b(~)-531 b(\027)71 b Fs(\))p 19928 61831 3903 45 v 21280 62845 a Fq(@)61 b(\022)24271 62085 y Fs(=)308 b Fq(O)31 b Fs(\()p Fq(\025)27391 61628 y Fk(\000)p Fo(19)p Fn(=)p Fo(2+2)p Fn(\016)31467 62085 y Fs(\))p Fq(;)11753 b Fs(\(85\))6075 64263 y Fi(given)407 b Fq(~)-531 b(\027)436 b Fs(=)366 b(\(cos)185 b Fq(\022)31 b(;)184 b Fs(sin)h Fq(\022)31 b Fs(\))p Fi(.)605 b(The)429 b(set)22122 63983 y Fs(~)21863 64263 y Fq(D)22779 64429 y Fk(\003)23289 64263 y Fs(\()p Fq(\025)p Fs(\))f Fi(has)g(a)g(lengths) g(which)f(is)i(asymptotic)-57 b(al)57 b(ly)428 b(close)6075 65683 y(to)395 b(that)g(of)i(a)e(cir)-57 b(cle)398 b(of)e(r)-57 b(adius)19512 64739 y Fl(p)p 20434 64739 646 45 v 944 x Fq(\025)396 b Fi(in)h(the)f(fol)57 b(lowing)396 b(sense:)17544 67803 y Fq(l)18080 66574 y Fp(\020)19000 67523 y Fs(~)18741 67803 y Fq(D)19657 67969 y Fk(\003)20167 67803 y Fs(\()p Fq(\025)p Fs(\))21673 66574 y Fp(\021)p 17544 68700 4791 45 v 18543 69864 a Fs(2)p Fq(\031)19767 68921 y Fl(p)p 20690 68921 646 45 v 20690 69864 a Fq(\025)22775 68954 y(>)307 b Fs(1)247 b Fl(\000)f Fq(c)26495 69130 y Fn(~)26329 69326 y(A)27050 68954 y Fq(\025)27696 68497 y Fk(\000)p Fn(\016)28 b(=)p Fo(4)29768 68954 y Fq(;)977 b(c)31697 69130 y Fn(~)31531 69326 y(A)32560 68954 y Fq(>)307 b Fs(0)p Fq(:)9369 b Fs(\(86\))p eop end %%Page: 15 15 TeXDict begin 15 14 bop 7289 3345 a Fo(SPECTRAL)388 b(PR)-24 b(OPER)-71 b(TIES)386 b(OF)h(THE)h(PERIODIC)f(MA)-24 b(GNETIC)389 b(SCHR)35152 3149 y(\177)35033 3345 y(ODINGER)e(OPERA)-71 b(TOR)331 b(15)6075 6001 y Fi(The)425 b(p)-57 b(ar)g(al)57 b(lel)424 b(shift)h Fl(K)442 b Fi(establishes)424 b(one-to-one)h(c)-57 b(orr)g(esp)g(ondenc)g(e)426 b(b)-57 b(etwe)g(en)38980 5722 y Fs(~)38722 6001 y Fq(D)39638 6167 y Fk(\003)40572 6001 y Fi(and)425 b Fq(I)43238 6167 y Fk(\003)44171 6001 y Fi(and)6075 7330 y(b)-57 b(etwe)g(en)350 b(their)f Fs(\()p Fq(\025)13747 6928 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)17394 7330 y Fs(\))p Fi(-neighb)-57 b(orho)g(o)g(ds)351 b(in)f Fq(R)27365 6928 y Fo(2)28210 7330 y Fi(and)f Fq(K)79 b Fi(,)359 b(r)-57 b(esp)g(e)g(ctively.)497 b(The)349 b(length)g(of)h Fq(I)45417 7496 y Fk(\003)6075 8736 y Fi(is)396 b(e)-57 b(qual)396 b(to)g(the)g(length)g(of)18045 8456 y Fs(~)17786 8736 y Fq(D)18702 8902 y Fk(\003)19212 8736 y Fs(\()p Fq(\025)p Fs(\))p Fi(.)7403 10830 y Fs(Let)454 b Fq(~)-523 b(n)10165 10428 y Fk(\030)10165 11135 y Fn(D)10889 11246 y Fh(\003)11407 10830 y Fs(\()-22 b Fq(~)-531 b(\027)71 b Fs(\))484 b(and)454 b Fq(~)-523 b(n)16300 10996 y Fn(D)17024 11107 y Fh(\003)17542 10830 y Fs(\()-22 b Fq(~)-531 b(\027)71 b Fs(\))484 b(b)31 b(e)483 b(normal)i(v)-31 b(ectors)484 b(to)30595 10550 y(~)30337 10830 y Fq(D)31253 10996 y Fk(\003)32245 10830 y Fs(and)g Fq(D)35428 10996 y Fk(\003)35938 10830 y Fs(,)512 b(resp)31 b(ectiv)-31 b(ely)-92 b(,)514 b(corre-)6075 12158 y(sp)31 b(onding)319 b(to)h(the)f(same)297 b 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Fk(\003)19952 45946 y Fs(\()p Fq(\025)p Fs(\)\))p 17989 46440 3900 45 v 18543 47605 a(2)p Fq(\031)19767 46661 y Fl(p)p 20690 46661 646 45 v 20690 47605 a Fq(\025)22329 46694 y(>)308 b Fs(1)246 b Fl(\000)g Fq(c)26049 46870 y Fn(~)25883 47066 y(A)26604 46694 y Fq(\025)27250 46237 y Fk(\000)p Fn(\016)28 b(=)p Fo(4)29322 46694 y Fq(;)978 b(c)31252 46870 y Fn(~)31086 47066 y(A)32114 46694 y Fq(>)308 b Fs(0)p Fq(:)9814 b Fs(\(89\))6075 49352 y Fx(Theorem)475 b(4.)579 b Fs(\(The)413 b(v)-61 b(alidit)-31 b(y)415 b(of)d(the)h(Bethe-Sommerfeld)h(conjecture\))437 b Fi(The)e(op)-57 b(er)g(ator)435 b Fq(H)6075 50680 y Fi(has)395 b(only)h(a)g(\014nite)h(numb)-57 b(er)398 b(of)e(gaps)f(in)i(the)f(sp)-57 b(e)g(ctrum.)7403 52695 y Fs(The)562 b(v)-61 b(alidit)-31 b(y)563 b(of)f(the)g (Bethe-Sommerfeld)h(conjecture)f(is)f(a)h(simple)g(consequence)g(of) 6075 54023 y(Corollary)333 b(7.)480 b(Existence)331 b(of)h(an)f(iso)31 b(energetic)332 b(set)e(for)h(a)g(giv)-31 b(en)332 b Fq(\025)f Fs(implies)h(that)g 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b(mathematical)k(induction.)6075 23362 y(Using)383 b(\(22\))i(and)f(the)f(inequalit)-31 b(y)386 b(2)21340 22961 y Fn(N)83 b Fo(+2)23301 23362 y Fq(R)24142 23528 y Fo(0)24969 23362 y Fq(<)331 b(k)26772 22961 y Fn(")27245 23362 y Fs(,)387 b(giv)-31 b(en)385 b(in)e(the)h(statemen)-31 b(t)385 b(of)e(the)h(lemma,)6075 24691 y(w)-31 b(e)369 b(get)10732 25832 y Fp(\014)10732 26496 y(\014)10732 27161 y(\014)11101 27105 y Fl(j)11352 26813 y Fq(~)11408 27105 y(k)282 b Fs(+)254 b Fq(~)-561 b(p)13930 27271 y Fn(r)14420 27105 y Fs(\(0\))p Fl(j)16140 26648 y Fo(2)16884 27105 y Fl(\000)246 b(j)18242 26813 y Fq(~)18298 27105 y(k)36 b Fl(j)19217 26648 y Fo(2)19714 25832 y Fp(\014)19714 26496 y(\014)19714 27161 y(\014)20390 27105 y Fq(>)308 b(k)22170 26648 y Fo(1)p Fn(=)p Fo(2+)p Fn(\014)25521 27105 y Fs(for)369 b(all)h Fq(r)339 b Fl(2)308 b Fq(Z)31471 26648 y Fo(2)31967 27105 y Fq(;)184 b Fs(0)309 b Fq(<)e Fl(j)p Fq(r)31 b Fl(j)309 b(\024)e Fs(2)37662 26648 y Fn(N)83 b Fo(+2)39623 27105 y Fq(R)40464 27271 y Fo(0)40960 27105 y Fq(:)2690 b Fs(\(97\))6075 29734 y(Using)426 b(the)g(de\014nition)i(\(30\))f(of)g Fq(S)20637 29332 y Fo(\(0\))21825 29734 y Fs(\()p Fq(x)p Fs(\),)441 b(inequalit)-31 b(y)429 b(\(97\))e(and)f(the)g(estimate)i Fl(j)p Fq(\013)t Fl(j)403 b(\024)f Fq(k)35 b Fs(,)441 b(w)-31 b(e)6075 31062 y(easily)370 b(obtain:)19811 32809 y Fl(k)p Fq(S)21107 32352 y Fo(0)21604 32809 y Fs(\()p Fq(x)p Fs(\))p Fl(k)23650 32975 y Fn(\027)24509 32809 y Fq(<)308 b(c)p Fl(k)26917 32530 y Fq(~)26710 32809 y(A)q Fl(k)28094 32975 y Fn(\027)28645 32809 y Fq(k)29256 32352 y Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fn(\014)31881 32809 y Fq(;)11769 b Fs(\(98\))6075 34763 y(i.e.,)383 b(\(37\))d(for)f Fq(n)324 b Fs(=)g(0.)522 b(Next,)383 b(w)-31 b(e)379 b(get)h(\(41\))g(for)395 b(~)-569 b Fq(a)26938 34361 y Fo(\(1\))28127 34763 y Fs(\()p Fq(x)p Fs(\).)522 b(In)379 b(fact,)j(it)e(is)f(easy)g(to)h(see)e(that)i(for)6075 36233 y(an)-31 b(y)8319 35941 y Fq(~)8166 36233 y(f)119 b(;)137 b(~)-506 b(g)348 b Fs(:)308 b Fq(Q)f Fl(!)h Fq(R)14256 35831 y Fo(2)18378 36910 y Fp(\015)18378 37574 y(\015)18378 38238 y(\015)18993 36954 y(\020)19808 37891 y Fq(~)19655 38183 y(f)119 b Fs(\()p Fq(x)p Fs(\))p Fq(;)137 b(~)-506 b(g)41 b Fs(\()p Fq(x)p Fs(\))24362 36954 y Fp(\021)25024 36910 y(\015)25024 37574 y(\015)25024 38238 y(\015)25639 38958 y Fn(\027)26497 38183 y Fl(\024)307 b Fs(2)28218 37726 y Fn(\027)28770 38183 y Fl(k)29476 37891 y Fq(~)29323 38183 y(f)119 b Fl(k)30537 38349 y Fn(\027)31089 38183 y Fl(k)-47 b Fq(~)-506 b(g)40 b Fl(k)32763 38349 y Fn(\027)33314 38183 y Fq(:)10336 b Fs(\(99\))6075 40482 y(Applying)371 b(\(99\))f(in)g(\(33\))h(w)-31 b(e)370 b(obtain:)10569 42958 y Fl(k)16 b Fs(~)-569 b Fq(a)11707 42501 y Fo(1)12204 42958 y Fs(\()p Fq(x)p Fs(\))p Fl(k)14250 43124 y Fn(\027)15109 42958 y Fl(\024)308 b Fs(2)16831 42501 y Fn(\027)17567 41729 y Fp(\020)18228 42958 y Fs(2)p Fl(kr)p Fq(S)20999 42501 y Fo(0)21496 42958 y Fs(\()p Fq(x)p Fs(\))p Fl(k)23542 43124 y Fn(\027)24094 42958 y Fl(k)-69 b Fq(~)-484 b(a)q Fs(\()p Fq(x)p Fs(\))p Fl(k)27279 43124 y Fn(\027)28077 42958 y Fs(+)246 b Fl(kr)p Fq(S)31402 42501 y Fo(\(0\))32591 42958 y Fs(\()p Fq(x)p Fs(\))p Fl(k)34637 42501 y Fo(2)34637 43231 y Fn(\027)35435 42958 y Fs(+)g Fl(k)-69 b Fq(~)-484 b(a)p Fs(\()p Fq(x)p Fs(\))p Fl(k)39726 42501 y Fo(2)39726 43231 y Fn(\027)40278 41729 y Fp(\021)41124 42958 y Fq(:)6075 45434 y Fs(Using)370 b(\(98\))g(and)g(considering)g(that)18242 47675 y Fl(kr)p Fq(S)20460 47218 y Fo(\(0\))21649 47675 y Fs(\()p Fq(x)p Fs(\))p Fl(k)23695 47841 y Fn(\027)24554 47675 y Fl(\024)307 b Fs(2)p Fq(b)26750 47841 y Fo(0)27248 47675 y Fl(k)p Fq(S)28544 47218 y Fo(\(0\))29732 47675 y Fs(\()p Fq(x)p Fs(\))p Fl(k)31778 47841 y Fn(\027)52 b Fo(+1)33451 47675 y Fq(;)9646 b Fs(\(100\))6075 49819 y(w)-31 b(e)459 b(get)h(\(41\))g(for)475 b(~)-569 b Fq(a)14524 49417 y Fo(1)15021 49819 y Fs(\()p Fq(x)p Fs(\).)762 b(It)459 b(follo)-31 b(ws)461 b(from)e(\(34\))i(and)e(\(97\))i(that)f (for)f(all)h Fq(\027)527 b Fl(\025)457 b Fs(0)i(and)g Fq(n)e Fs(=)6075 51147 y(1)p Fq(;)184 b(::::;)g(N)371 b Fs(+)246 b(1,)18035 52862 y Fl(k)p Fq(S)19331 52405 y Fo(\()p Fn(n)p Fo(\))20625 52862 y Fs(\()p Fq(x)p Fs(\))p Fl(k)22671 53028 y Fn(\027)23531 52862 y Fl(\024)307 b(k)16 b Fs(~)-569 b Fq(a)25837 52405 y Fo(\()p Fn(n)p Fo(\))27132 52862 y Fs(\()p Fq(x)p Fs(\))p Fl(k)29178 53028 y Fn(\027)29730 52862 y Fq(k)30341 52405 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fn(\014)33658 52862 y Fq(:)9439 b Fs(\(101\))6075 54773 y(F)-92 b(orm)-31 b(ulae)370 b(\(35\))h(and)e(\(99\))i(yield:)22900 56549 y Fl(k)16 b Fs(~)-569 b Fq(a)24038 56092 y Fo(\()p Fn(n)p Fo(\))25333 56549 y Fs(\()p Fq(x)p Fs(\))p Fl(k)27379 56715 y Fn(\027)28239 56549 y Fl(\024)10115 59864 y Fs(2)10668 59407 y Fn(\027)52 b Fo(+1)12524 57971 y Fp( )13401 59864 y Fl(kr)p Fq(S)15619 59407 y Fo(\()p Fn(n)p Fk(\000)p Fo(1\))18047 59864 y Fs(\()p Fq(x)p Fs(\))p Fl(k)20093 60030 y Fn(\027)20645 59864 y Fl(k)-69 b Fq(~)-484 b(a)p Fs(\()p Fq(x)p Fs(\))p Fl(k)23829 60030 y Fn(\027)24627 59864 y Fs(+)246 b Fl(kr)p Fq(S)27952 59407 y Fo(\()p Fn(n)p Fk(\000)p Fo(1\))30380 59864 y Fs(\()p Fq(x)p Fs(\))p Fl(k)32426 60030 y Fn(\027)33163 58481 y(n)p Fk(\000)p Fo(1)33203 58813 y Fp(X)33233 61157 y Fn(s)p Fo(=0)35027 59864 y Fl(kr)p Fq(S)37245 59407 y Fo(\()p Fn(s)p Fo(\))38411 59864 y Fs(\()p Fq(x)p Fs(\))p Fl(k)40457 60030 y Fn(\027)41009 57971 y Fp(!)6075 62792 y Fs(for)369 b(an)-31 b(y)370 b Fq(n)308 b Fl(\025)f Fs(2.)493 b(Using)370 b(\(100\))h(and)f(\(101\),)i(w)-31 b(e)369 b(obtain)15111 65033 y Fl(k)16 b Fs(~)-569 b Fq(a)16249 64576 y Fo(\()p Fn(n)p Fo(\))17544 65033 y Fs(\()p Fq(x)p Fs(\))p Fl(k)19590 65199 y Fn(\027)20450 65033 y Fl(\024)307 b Fs(2)22171 64576 y Fn(\027)52 b Fo(+1)23843 65033 y Fq(b)24318 65199 y Fo(0)24815 65033 y Fq(k)25426 64576 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fn(\014)28743 65033 y Fl(k)16 b Fs(~)-569 b Fq(a)29881 64576 y Fo(\()p Fn(n)p Fk(\000)p Fo(1\))32309 65033 y Fs(\()p Fq(x)p Fs(\))p Fl(k)34355 65199 y Fn(\027)52 b Fo(+1)36028 65033 y Fl(\002)11030 66687 y Fp( )11906 68580 y Fl(k)-69 b Fq(~)-484 b(a)q Fs(\()p Fq(x)p Fs(\))p Fl(k)15091 68746 y Fn(\027)15889 68580 y Fs(+)246 b(2)p Fq(b)18024 68746 y Fo(0)18521 68580 y Fl(k)p Fq(S)19817 68123 y Fo(\(0\))21005 68580 y Fs(\()p Fq(x)p Fs(\))p Fl(k)23051 68746 y Fn(\027)52 b Fo(+1)24970 68580 y Fs(+)246 b(2)p Fq(b)27105 68746 y Fo(0)27602 68580 y Fq(k)28213 68123 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fn(\014)31715 67197 y(n)p Fk(\000)p Fo(1)31755 67529 y Fp(X)31786 69873 y Fn(s)p Fo(=1)33579 68580 y Fl(k)16 b Fs(~)-569 b Fq(a)34717 68123 y Fo(\()p Fn(s)p Fo(\))35883 68580 y Fs(\()p Fq(x)p Fs(\))p Fl(k)37929 68746 y Fn(\027)52 b Fo(+1)39602 66687 y Fp(!)40663 68580 y Fq(:)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 6075 3345 a Fo(18)14514 b(YULIA)300 b(KARPESHINA)6075 6001 y Fs(Supp)31 b(ose)482 b(\(41\))i(holds)f(for)f Fq(s)496 b Fs(=)g(1)p Fq(;)184 b(:::;)g(n)326 b Fl(\000)321 b Fs(1.)833 b(Using)483 b(\(37\))h(for)f Fq(n)496 b Fs(=)g(0)483 b(and)g(\(41\))h(for)e Fq(s)496 b Fs(=)6075 7330 y(1)p Fq(;)184 b(:::;)g(n)250 b Fl(\000)c Fs(1)369 b(on)h(the)f(righ)-31 b(t-hand)371 b(side)e(of)h(the)f(last)h(inequalit)-31 b(y)-92 b(,)373 b(w)-31 b(e)369 b(obtain)7807 10311 y Fl(k)-69 b Fq(~)-484 b(a)p Fl(k)9498 10477 y Fn(\027)10295 10311 y Fs(+)246 b(2)p Fq(b)12430 10477 y Fo(0)12928 10311 y Fl(k)p Fq(S)14224 9854 y Fo(\(0\))15412 10311 y Fs(\()p Fq(x)p Fs(\))p Fl(k)17458 10477 y Fn(\027)52 b Fo(+1)19377 10311 y Fs(+)246 b(2)p Fq(b)21512 10477 y Fo(0)22009 10311 y Fq(k)22620 9854 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fn(\014)26121 8927 y(n)p Fk(\000)p Fo(1)26162 9259 y Fp(X)26192 11604 y Fn(s)p Fo(=1)27986 10311 y Fl(k)16 b Fs(~)-569 b Fq(a)29124 9854 y Fo(\()p Fn(s)p Fo(\))30290 10311 y Fs(\()p Fq(x)p Fs(\))p Fl(k)32336 10477 y Fn(\027)52 b Fo(+1)34316 10311 y Fl(\024)307 b Fq(cb)36438 10477 y Fo(0)36935 10311 y Fq(k)37546 9854 y Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fn(\014)40171 10311 y Fl(k)40932 10031 y Fq(~)40724 10311 y(A)q Fl(k)42108 10477 y Fn(\027)52 b Fo(+)p Fn(n)43886 10311 y Fq(;)6075 13469 y Fs(the)435 b(term)h(2)p Fq(b)11789 13635 y Fo(0)12286 13469 y Fl(k)p Fq(S)13582 13068 y Fo(\(0\))14771 13469 y Fs(\()p Fq(x)p Fs(\))p Fl(k)16817 13635 y Fn(\027)52 b Fo(+1)18925 13469 y Fs(b)31 b(eing)435 b(the)h(biggest)h(on)e(the)h (left-hand)h(side)e(when)h Fq(k)470 b Fs(is)435 b(large)6075 14928 y(enough,)370 b(namely)h(when)e Fq(k)17540 14526 y Fo(2)p Fn(\014)18886 14928 y Fq(>)308 b Fs(\(5)p Fq(b)21513 15094 y Fo(0)22010 14928 y Fs(\))22440 14526 y Fn(N)83 b Fo(+)p Fn(\027)52 b Fo(+2)25576 14928 y Fl(k)26337 14648 y Fq(~)26129 14928 y(A)q Fl(k)27513 15094 y Fn(N)83 b Fo(+)p Fn(\027)52 b Fo(+2)30895 14928 y Fs(+)246 b Fq(b)32477 14526 y Fo(2)32477 15202 y(0)32973 14928 y Fs(.)7403 16386 y(Th)-31 b(us,)326 b Fl(k)16 b Fs(~)-569 b Fq(a)11609 15984 y Fo(\()p Fn(n)p Fo(\))12904 16386 y Fs(\()p Fq(x)p Fs(\))p Fl(k)14950 16552 y Fn(\027)15810 16386 y Fq(<)307 b(c)137 b Fl(\001)g Fs(2)18591 15984 y Fn(\027)19144 16386 y Fq(b)19619 15984 y Fo(2)19619 16660 y(0)20115 16386 y Fq(k)20726 15984 y Fk(\000)p Fo(2)p Fn(\014)22457 16386 y Fl(k)23217 16106 y Fq(~)23010 16386 y(A)p Fl(k)24393 16552 y Fn(\027)52 b Fo(+1)26065 16386 y Fl(k)16 b Fs(~)-569 b Fq(a)27203 15984 y Fo(\()p Fn(n)p Fk(\000)p Fo(1\))29632 16386 y Fs(\()p Fq(x)p Fs(\))p Fl(k)31678 16552 y Fn(\027)52 b Fo(+1)33350 16386 y Fs(.)475 b(Using)315 b(\(41\))i(for)330 b(~)-568 b Fq(a)41697 15984 y Fo(\()p Fn(n)p Fk(\000)p Fo(1\))44124 16386 y Fs(\()p Fq(x)p Fs(\),)6075 17756 y(w)-31 b(e)369 b(get)h(\(41\))h(for)385 b(~)-568 b Fq(a)14166 17354 y Fo(\()p Fn(n)p Fo(\))15460 17756 y Fs(\()p Fq(x)p Fs(\).)493 b(Estimate)372 b(\(37\))e(follo)-31 b(ws)372 b(from)e(\(41\))h(and)e (\(101\).)7403 19084 y(Let)394 b(us)g(pro)-31 b(v)g(e)395 b(the)f(pair)h(\(38\))h(and)e(\(42\).)569 b(By)395 b(elemen)-31 b(tary)396 b(calculation)h(w)-31 b(e)395 b(obtain)h(from)6075 20413 y(\(30\):)7364 23886 y Fl(<)p Fq(S)8906 23429 y Fo(\(0\))10095 23886 y Fs(\()p Fq(x)p Fs(\))308 b(=)f Fl(\000)15272 22834 y Fp(X)14109 25260 y Fn(r)24 b Fk(2)p Fn(Z)15798 25039 y Fm(2)16231 25260 y Fk(nf)p Fo(0)p Fk(g)22249 21506 y Fp(\020)22910 22735 y Fs(2\()23837 22443 y Fq(~)23893 22735 y(k)36 b(;)115 b(~)-484 b(a)25581 22901 y Fn(r)26073 22735 y Fs(\))246 b(+)g Fq(\013)t(a)29153 22901 y Fo(0)p Fn(r)30085 21506 y Fp(\021)30930 22735 y Fq(p)31487 22333 y Fo(2)31487 23008 y Fn(r)31984 22735 y Fs(\(0\))p 18350 23631 18948 45 v 18350 24847 a(\()p Fl(j)19031 24555 y Fq(~)19087 24847 y(k)282 b Fs(+)255 b Fq(~)-562 b(p)21609 25013 y Fn(r)22099 24847 y Fs(\(0\))p Fl(j)23819 24527 y Fo(2)24564 24847 y Fl(\000)246 b(j)25922 24555 y Fq(~)25978 24847 y(k)35 b Fl(j)26896 24527 y Fo(2)27393 24847 y Fs(\)\()p Fl(j)28504 24555 y Fq(~)28560 24847 y(k)283 b Fl(\000)255 b Fq(~)-562 b(p)31083 25013 y Fn(r)31573 24847 y Fs(\(0\))p Fl(j)33293 24527 y Fo(2)34037 24847 y Fl(\000)246 b(j)35395 24555 y Fq(~)35451 24847 y(k)36 b Fl(j)36370 24527 y Fo(2)36867 24847 y Fs(\))37430 23886 y Fq(e)37945 23429 y Fn(i)p Fo(\()13 b Fn(~)-467 b(p)39060 23540 y Fj(r)39507 23429 y Fo(\(0\))p Fn(;x)p Fo(\))41807 23886 y Fq(;)1290 b Fs(\(102\))6075 27167 y(The)425 b(estimate)h(\(38\))h(for)d Fq(n)401 b Fs(=)f(0)425 b(easily)g(follo)-31 b(ws)427 b(from)f(the)f(last)g(form)-31 b(ula)427 b(and)e(\(97\).)660 b(Using)6075 28711 y(this)349 b(estimate)i(and)f(the)f(form)-31 b(ula)351 b(\(33\))g(for)365 b(~)-569 b Fq(a)25264 28136 y Fo(\(1\))25264 29019 y Fn(j)26453 28711 y Fs(,)353 b(and)c(taking)j(in)-31 b(to)350 b(accoun)-31 b(t)351 b(\(37\))g(for)e Fq(n)308 b Fs(=)f(0,)6075 30351 y(it)393 b(is)f(not)i(di\016cult)g(to)f(sho)-31 b(w)393 b(that)h Fl(k=)16 b Fs(~)-569 b Fq(a)22916 29950 y Fo(\(1\))24106 30351 y Fl(k)24659 30517 y Fn(\027)25556 30351 y Fl(\024)347 b Fq(c)27243 30517 y Fn(n;\027)t(;b)29208 30351 y Fl(k)29968 30072 y Fq(~)29761 30351 y(A)q Fl(k)31145 29950 y Fo(2)31145 30626 y Fn(\027)52 b Fo(+3)32817 30351 y Fq(k)33428 29950 y Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(3)p Fn(\014)36494 30351 y Fs(,)399 b(i.e.,)h(\(42\))394 b(holds)f(for)6075 31680 y Fq(n)308 b Fs(=)f(1.)493 b(F)-92 b(orm)-31 b(ula)370 b(\(34\))h(for)e Fq(S)64 b Fs(\()p Fq(x)p Fs(\))369 b(yields)13986 33767 y Fl(k<)p Fq(S)16081 33310 y Fo(\()p Fn(n)p Fo(\))17376 33767 y Fl(k)17929 33933 y Fn(\027)18788 33767 y Fq(<)307 b Fl(k)16 b Fs(~)-569 b Fq(a)21094 33310 y Fo(\()p Fn(n)p Fo(\))22389 33767 y Fl(k)22942 33933 y Fn(\027)52 b Fo(+2)24614 33767 y Fq(k)25225 33310 y Fk(\000)p Fo(1)p Fk(\000)p Fo(2)p Fn(\014)28335 33767 y Fs(+)246 b Fl(k=)16 b Fs(~)-569 b Fq(a)31379 33310 y Fo(\()p Fn(n)p Fo(\))32674 33767 y Fl(k)33227 33933 y Fn(\027)33778 33767 y Fq(k)34389 33310 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fn(\014)37706 33767 y Fq(:)6075 35757 y Fs(Using)370 b(estimate)h(\(41\),)g(w)-31 b(e)370 b(obtain:)12364 37844 y Fl(k<)p Fq(S)14459 37387 y Fo(\()p Fn(n)p Fo(\))15754 37844 y Fl(k)16307 38010 y Fn(\027)17166 37844 y Fq(<)307 b(c)18813 38010 y Fn(n;\027)20174 37844 y Fl(k)p Fq(A)p Fl(k)22110 37371 y Fn(n)p Fo(+1)22110 38139 y Fn(\027)52 b Fo(+)p Fn(n)p Fo(+2)25010 37844 y Fq(k)25621 37387 y Fk(\000)p Fo(2)p Fn(\014)40 b Fo(\()p Fn(n)p Fo(+1\))29957 37844 y Fs(+)246 b Fl(k=)16 b Fs(~)-569 b Fq(a)33001 37387 y Fo(\()p Fn(n)p Fo(\))34296 37844 y Fl(k)34849 38010 y Fn(\027)35401 37844 y Fq(k)36012 37387 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fn(\014)39329 37844 y Fq(:)3768 b Fs(\(103\))6075 39834 y(F)-92 b(orm)-31 b(ulae)370 b(\(35\))h(and)e(\(99\))i(yield:)22501 41921 y Fl(k=)16 b Fs(~)-569 b Fq(a)24438 41464 y Fo(\()p Fn(n)p Fo(\))25733 41921 y Fs(\()p Fq(x)p Fs(\))p Fl(k)27779 42087 y Fn(\027)28638 41921 y Fl(\024)8650 45161 y Fs(2)9203 44704 y Fn(\027)52 b Fo(+1)11060 43268 y Fp( )11936 45161 y Fl(k;)923 b Fs(\()p Fq(I)25304 39570 y Fn(J)25923 39404 y Fs(\))26353 39570 y Fn(j)45 b(m)27913 39404 y Fs(=)307 b Fq(\016)29573 39570 y Fn(j)45 b(m)30825 39404 y Fq(;)554 b(j)r(;)184 b(m)308 b Fl(2)f Fq(J)106 b Fs(\()p Fq(\025)36756 39570 y Fo(0)37254 39404 y Fs(\))p Fq(:)6075 41413 y Fx(Lemma)954 b(6.)581 b Fi(L)-57 b(et)439 b Fs(0)384 b Fq(<)g(\014)443 b(<)384 b Fs(1)p Fq(=)p Fs(2)p Fi(,)451 b Fs(0)384 b Fq(<)g Fs(2)p Fq(")h(<)f Fs(min)q Fl(f)p Fs(1)p Fq(=)p Fs(2)278 b Fl(\000)f Fq(\014)58 b(;)184 b Fs(4)p Fq(\014)58 b Fl(g)p Fi(.)639 b(Then,)450 b Fq(G)385 b Fs(=)f Fq(I)42150 41579 y Fn(J)43044 41413 y Fs(+)277 b Fq(\016)42 b(G)p Fi(,)6075 42742 y(wher)-57 b(e)396 b Fq(\016)42 b(G)396 b Fi(ob)-57 b(eys)397 b(the)e(estimate:)16324 44574 y Fl(k)p Fq(\016)42 b(G)p Fl(k)309 b Fq(<)e(C)21102 44774 y Fk(k)p Fn(A)p Fk(k)22676 44885 y Fm(7)23163 44574 y Fq(\025)23809 44042 y Fk(\000)p Fn(\015)24966 44153 y Fm(1)23809 44869 y Fo(0)25453 44574 y Fq(;)1373 b(\015)27706 44740 y Fo(1)28510 44574 y Fs(=)307 b Fq(\014)e Fl(\000)246 b Fq("=)p Fs(2)308 b Fq(>)g Fs(0)p Fq(:)7729 b Fs(\(108\))7403 46584 y(Pro)31 b(of.)p 7403 46761 2971 45 v 370 w(Ob)-31 b(viously)-92 b(,)8674 48918 y Fq(<)9996 48638 y Fs(^)9843 48918 y(\011)10704 49084 y Fn(m)11544 48918 y Fq(;)12190 48638 y Fs(^)12035 48918 y(\011)12896 49084 y Fn(j)13671 48918 y Fq(>)p Fs(=)16302 48169 y(1)p 15833 48663 1491 45 v 15833 49677 a Fl(j)p Fq(Q)p Fl(j)17641 47411 y Fp(Z)18256 49926 y Fn(Q)19193 48918 y Fs(exp)21069 48021 y Fp(\000)21576 48918 y Fq(i)p Fs(\()9 b Fq(~)-562 b(p)22944 49084 y Fn(m)23786 48918 y Fs(\()p Fq(t)p Fs(\))246 b Fl(\000)255 b Fq(~)-562 b(p)26956 49084 y Fn(j)27423 48918 y Fs(\()p Fq(t)p Fs(\))p Fq(;)184 b(x)p Fs(\))247 b(+)f Fq(S)32270 49084 y Fn(m)33111 48918 y Fs(\()p Fq(x)p Fs(\))h(+)36144 48638 y(\026)35958 48918 y Fq(S)36637 49084 y Fn(j)37103 48918 y Fs(\()p Fq(x)p Fs(\))38596 48021 y Fp(\001)39288 48918 y Fq(dx:)2600 b Fs(\(109\))6075 51451 y(First)368 b(w)-31 b(e)370 b(consider)f Fq(m)308 b Fs(=)f Fq(j)63 b Fs(.)493 b(Clearly)-92 b(,)7638 53779 y Fq(<)8960 53499 y Fs(^)8806 53779 y(\011)9667 53945 y Fn(j)10133 53779 y Fq(;)10779 53499 y Fs(^)10624 53779 y(\011)11485 53945 y Fn(j)12260 53779 y Fq(>)p Fs(=)14891 53030 y(1)p 14422 53524 V 14422 54538 a Fl(j)p Fq(Q)p Fl(j)16230 52272 y Fp(Z)16845 54788 y Fn(Q)17782 53779 y Fs(exp)185 b(\(2)p Fl(<)p Fq(S)22119 53945 y Fn(j)22586 53779 y Fs(\()p Fq(x)p Fs(\)\))h Fq(dx)307 b Fs(=)h(1)246 b(+)g Fq(O)30346 52550 y Fp(\020)31007 53779 y Fq(\025)31653 53247 y Fk(\000)p Fn(\014)31653 54074 y Fo(0)32941 52550 y Fp(\021)33910 53779 y Fs(=)307 b(1)247 b(+)f Fq(O)38044 52882 y Fp(\000)38552 53779 y Fq(\025)39198 53247 y Fk(\000)p Fn(\015)40355 53358 y Fm(1)39198 54074 y Fo(0)40842 52882 y Fp(\001)41534 53779 y Fq(;)1563 b Fs(\(110\))19987 55645 y Fp(\014)19987 56309 y(\014)20356 56586 y Fq(O)21415 55689 y Fp(\000)21922 56586 y Fq(\025)22568 56054 y Fk(\000)p Fn(\015)23725 56165 y Fm(1)22568 56881 y Fo(0)24212 55689 y Fp(\001)24720 55645 y(\014)24720 56309 y(\014)25396 56586 y Fq(<)308 b(C)27356 56786 y Fk(k)p Fn(A)p Fk(k)28930 56897 y Fm(2)29416 56586 y Fq(\025)30062 56054 y Fk(\000)p Fn(\015)31219 56165 y Fm(1)30062 56881 y Fo(0)31706 56586 y Fq(;)6075 58384 y Fs(here)411 b(\(47\))j(and)f(the)f(relation)i(2)p Fq(p)20259 58550 y Fn(j)20726 58384 y Fs(\()p Fq(t)p Fs(\))380 b Fq(>)23607 57441 y Fl(p)p 24529 57441 646 45 v 943 x Fq(\025)25175 58559 y Fo(0)26083 58384 y Fs(ha)-31 b(v)g(e)414 b(b)31 b(een)412 b(used.)621 b(Supp)31 b(ose)412 b Fq(m)379 b Fl(6)p Fs(=)g Fq(j)63 b Fs(.)622 b(Let)413 b(us)6075 59720 y(consider)369 b Fq(S)11068 59886 y Fn(m)11908 59720 y Fs(\()p Fq(x)p Fs(\))247 b(+)14942 59440 y(\026)14755 59720 y Fq(S)15434 59886 y Fn(j)15900 59720 y Fs(\()p Fq(x)p Fs(\).)493 b(First,)370 b(w)-31 b(e)369 b(c)-31 b(hec)g(k)370 b(that)14188 60713 y Fp(\015)14188 61377 y(\015)14803 61654 y Fq(S)15482 61820 y Fn(m)16323 61654 y Fs(\()p Fq(x)p Fs(\))247 b(+)19356 61374 y(\026)19170 61654 y Fq(S)19849 61820 y Fn(j)20315 61654 y Fs(\()p Fq(x)p Fs(\))21808 60713 y Fp(\015)21808 61377 y(\015)22423 62097 y Fn(\027)23282 61654 y Fq(<)307 b(C)25241 61854 y Fk(k)p Fn(A)p Fk(k)26815 61965 y Fj(\027)39 b Fm(+2)28304 61654 y Fl(j)9 b Fq(~)-562 b(p)29168 61820 y Fn(m)30009 61654 y Fs(\()p Fq(t)p Fs(\))247 b Fl(\000)254 b Fq(~)-561 b(p)33180 61820 y Fn(j)33646 61654 y Fs(\()p Fq(t)p Fs(\))p Fl(j)p Fq(\025)35859 61122 y Fk(\000)p Fn(\015)37016 61233 y Fm(1)35859 61949 y Fo(0)37504 61654 y Fq(:)5593 b Fs(\(111\))6075 63533 y(Without)412 b(the)e(loss)h(of)g(generalit)-31 b(y)413 b(w)-31 b(e)411 b(ma)-31 b(y)411 b(assume)g Fq(p)29652 63699 y Fn(j)30118 63533 y Fs(\()p Fq(t)p Fs(\))377 b Fl(\025)f Fq(p)33549 63699 y Fn(m)34390 63533 y Fs(\()p Fq(t)p Fs(\).)617 b(If)410 b Fl(j)9 b Fq(~)-562 b(p)38586 63699 y Fn(m)39428 63533 y Fs(\()p Fq(t)p Fs(\))274 b Fl(\000)282 b Fq(~)-562 b(p)42653 63699 y Fn(j)43120 63533 y Fs(\()p Fq(t)p Fs(\))p Fl(j)377 b Fq(<)6075 64943 y Fs(\(4)p Fq(b)7533 65109 y Fo(0)8030 64943 y Fs(\))8460 64541 y Fk(\000)p Fo(1)9649 64943 y Fq(p)10206 65109 y Fn(j)10673 64943 y Fs(\()p Fq(t)p Fs(\))11933 64541 y Fo(1)p Fn(=)p Fo(2+)p Fn(\014)40 b Fk(\000)p Fn(")15656 64943 y Fs(,)500 b(then)474 b(\(111\))i(follo)-31 b(ws)475 b(from)g(\(46\))g(and)f(\(48\).)807 b(If)473 b Fl(j)9 b Fq(~)-562 b(p)38397 65109 y Fn(m)39239 64943 y Fs(\()p Fq(t)p Fs(\))316 b Fl(\000)324 b Fq(~)-562 b(p)42548 65109 y Fn(j)43015 64943 y Fs(\()p Fq(t)p Fs(\))p Fl(j)482 b(\025)6075 66353 y Fs(\(4)p Fq(b)7533 66519 y Fo(0)8030 66353 y Fs(\))8460 65952 y Fk(\000)p Fo(1)9649 66353 y Fq(p)10206 66519 y Fn(j)10673 66353 y Fs(\()p Fq(t)p Fs(\))11933 65952 y Fo(1)p Fn(=)p Fo(2+)p Fn(\014)40 b Fk(\000)p Fn(")15656 66353 y Fs(,)374 b(then)f(w)-31 b(e)374 b(estimate)g Fl(k)q Fq(S)26166 66519 y Fn(m)27007 66353 y Fs(\()p Fq(x)p Fs(\))p Fl(k)29054 66685 y Fn(\027)29977 66353 y Fs(and)32133 65412 y Fp(\015)32133 66077 y(\015)32935 66074 y Fs(\026)32748 66353 y Fq(S)33427 66519 y Fn(j)33894 66353 y Fs(\()p Fq(x)p Fs(\))35387 65412 y Fp(\015)35387 66077 y(\015)36002 66796 y Fn(\027)36926 66353 y Fs(using)f(\(45\).)505 b(Apply-)6075 67682 y(ing)370 b(the)f(triangle)i(inequalit)-31 b(y)-92 b(,)372 b(w)-31 b(e)370 b(obtain:)7809 68600 y Fp(\015)7809 69264 y(\015)8424 69541 y Fq(S)9103 69707 y Fn(m)9943 69541 y Fs(\()p Fq(x)p Fs(\))247 b(+)12977 69261 y(\026)12790 69541 y Fq(S)13469 69707 y Fn(j)13935 69541 y Fs(\()p Fq(x)p Fs(\))15428 68600 y Fp(\015)15428 69264 y(\015)16044 69984 y Fn(\027)16902 69541 y Fq(<)307 b(C)18861 69741 y Fk(k)p Fn(A)p Fk(k)20435 69852 y Fj(\027)39 b Fm(+2)21924 69541 y Fq(p)22481 69707 y Fn(j)22947 69541 y Fs(\()p Fq(t)p Fs(\))24207 69084 y Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fn(\014)27140 69541 y Fq(<)308 b(C)29100 69741 y Fk(k)p Fn(A)p Fk(k)30674 69852 y Fj(\027)39 b Fm(+2)32162 69541 y Fl(j)9 b Fq(~)-562 b(p)33026 69707 y Fn(m)33867 69541 y Fs(\()p Fq(t)p Fs(\))247 b Fl(\000)255 b Fq(~)-562 b(p)37038 69707 y Fn(j)37504 69541 y Fs(\()p Fq(t)p Fs(\))p Fl(j)p Fq(\025)39717 69009 y Fk(\000)p Fn(\015)40874 69120 y Fm(1)39717 69836 y Fo(0)41363 69541 y Fq(:)1734 b Fs(\(112\))p eop end %%Page: 20 20 TeXDict begin 20 19 bop 6075 3345 a Fo(20)14514 b(YULIA)300 b(KARPESHINA)6075 6001 y Fs(Th)-31 b(us,)401 b(\(111\))c(is)d(pro)-31 b(v)g(ed.)570 b(Second,)401 b(w)-31 b(e)395 b(represen)-31 b(t)394 b(in)-31 b(tegral)397 b(\(109\))f(in)f(the)g(form)40792 5110 y Fp(R)41315 6395 y Fn(Q)42252 6001 y Fq(v)40 b(D)31 b(udx)p Fs(,)6075 7487 y(where)247 b Fq(u)g Fs(is)h(just)f(the)h(in)-31 b(tegrand)249 b(itself,)274 b Fq(D)k Fs(is)248 b(the)g(deriv)-61 b(ativ)-31 b(e)249 b(in)e(the)h(direction)h(of)257 b Fq(~)-562 b(p)40673 7653 y Fn(m)41514 7487 y Fs(\()p Fq(t)p Fs(\))s Fl(\000)12 b Fq(~)-562 b(p)44198 7653 y Fn(j)44665 7487 y Fs(\()p Fq(t)p Fs(\))6075 8815 y(and)16634 10514 y Fq(v)347 b Fs(=)26596 9766 y(1)p 18819 10260 16106 45 v 18819 11334 a Fl(j)10 b Fq(~)-562 b(p)19684 11500 y Fn(m)20525 11334 y Fs(\()p Fq(t)p Fs(\))246 b Fl(\000)255 b Fq(~)-562 b(p)23695 11500 y Fn(j)24162 11334 y Fs(\()p Fq(t)p Fs(\))p Fl(j)247 b Fs(+)f Fq(D)28215 10437 y Fp(\000)28722 11334 y Fq(S)29401 11500 y Fn(m)30488 11334 y Fs(+)31781 11054 y(\026)31595 11334 y Fq(S)32274 11500 y Fn(j)32740 10437 y Fp(\001)33432 11334 y Fs(\()p Fq(x)p Fs(\))35058 10514 y Fq(:)6075 13021 y Fs(In)-31 b(tegrating)371 b(\(109\))h(b)-31 b(y)369 b(parts)g(and)h(considering)g (\(111\),)h(w)-31 b(e)370 b(arriv)-31 b(e)370 b(at)g(inequalit)-31 b(y:)15159 15049 y Fq(<)16482 14769 y Fs(^)16328 15049 y(\011)17189 15215 y Fn(m)18030 15049 y Fq(;)18675 14769 y Fs(^)18521 15049 y(\011)19382 15215 y Fn(j)20157 15049 y Fq(>)368 b(<)677 b(C)23715 15249 y Fk(k)p Fn(A)p Fk(k)25289 15360 y Fm(4)25775 15049 y Fq(\025)26421 14517 y Fk(\000)p Fn(\015)27578 14628 y Fm(1)26421 15344 y Fo(0)28065 15049 y Fl(j)9 b Fq(~)-562 b(p)28929 15215 y Fn(m)29770 15049 y Fs(\()p Fq(t)p Fs(\))247 b Fl(\000)255 b Fq(~)-562 b(p)32941 15215 y Fn(j)33408 15049 y Fs(\()p Fq(t)p Fs(\))p Fl(j)34975 14592 y Fk(\000)p Fo(1)36164 15049 y Fq(:)6933 b Fs(\(113\))6075 16969 y(Rep)31 b(eating)370 b(in)-31 b(tegration)372 b(b)-31 b(y)370 b(parts)f(four)g(times)h(yields:)15159 18935 y Fq(<)16482 18655 y Fs(^)16328 18935 y(\011)17189 19101 y Fn(m)18030 18935 y Fq(;)18675 18655 y Fs(^)18521 18935 y(\011)19382 19101 y Fn(j)20157 18935 y Fq(>)e(<)677 b(C)23715 19135 y Fk(k)p Fn(A)p Fk(k)25289 19246 y Fm(7)25775 18935 y Fq(\025)26421 18403 y Fk(\000)p Fn(\015)27578 18514 y Fm(1)26421 19230 y Fo(0)28065 18935 y Fl(j)9 b Fq(~)-562 b(p)28929 19101 y Fn(m)29770 18935 y Fs(\()p Fq(t)p Fs(\))247 b Fl(\000)255 b Fq(~)-562 b(p)32941 19101 y Fn(j)33408 18935 y Fs(\()p Fq(t)p Fs(\))p Fl(j)34975 18477 y Fk(\000)p Fo(4)36164 18935 y Fq(;)6933 b Fs(\(114\))6075 20854 y(Therefore,)17194 22469 y(sup)17822 23382 y Fn(j)19184 21418 y Fp(X)19045 23801 y Fn(m)p Fk(6)p Fo(=)p Fn(j)21106 22469 y Fl(j)308 b Fq(<)23043 22190 y Fs(^)22889 22469 y(\011)23750 22635 y Fn(m)24591 22469 y Fq(;)25237 22190 y Fs(^)25082 22469 y(\011)25943 22635 y Fn(j)26718 22469 y Fq(>)f Fl(j)h Fq(<)g(c)30149 22669 y Fk(k)p Fn(A)p Fk(k)31723 22780 y Fm(7)32209 22469 y Fq(\025)32855 21937 y Fk(\000)p Fn(\015)34012 22048 y Fm(1)32855 22765 y Fo(0)34499 22469 y Fq(:)8598 b Fs(\(115\))6075 25362 y(Estimate)371 b(\(108\))g(follo)-31 b(ws)372 b(from)e(\(110\))h(and)f (\(115\).)494 b Fl(})7403 26691 y Fs(Let)373 b Fq(P)10101 26857 y Fo(0)10971 26691 y Fs(b)31 b(e)372 b(the)i(pro)61 b(jection)375 b(on)e(the)h(linear)g(span)f(of)g(the)h(exp)31 b(onen)-31 b(ts)374 b(\011)38027 26289 y Fo(0)38027 26979 y Fn(j)38523 26691 y Fs(\()p Fq(x)p Fs(\),)h Fq(j)j Fl(2)314 b Fq(J)43304 26289 y Fk(0)43614 26691 y Fs(\()p Fq(\025)44690 26857 y Fo(0)45187 26691 y Fs(\),)6075 28141 y(here)323 b(\011)9292 27739 y Fo(0)9292 28429 y Fn(j)9788 28141 y Fs(\()p Fq(x)p Fs(\))h(are)g(de\014ned)f(b)-31 b(y)324 b(\(9\).)478 b(W)-92 b(e)323 b(recall)i(that)f Fq(P)477 b Fs(is)324 b(the)g(pro)61 b(jection)325 b(on)f(the)g(linear)g(span) 6075 29739 y(of)369 b(mo)31 b(del)371 b(functions)15552 29459 y(^)15398 29739 y(\011)16259 29905 y Fn(j)16725 29739 y Fs(,)f Fq(j)g Fl(2)308 b Fq(J)106 b Fs(\()p Fq(\025)21070 29905 y Fo(0)21567 29739 y Fs(\).)6075 31804 y Fx(Lemma)848 b(7.)554 b Fi(If)396 b Fs(0)308 b Fq(<)f(")h(<)f Fs(1)p Fq(=)p Fs(100)p Fi(,)399 b Fs(0)308 b Fq(<)g(\014)366 b(<)307 b Fs(1)p Fq(=)p Fs(2)248 b Fl(\000)e Fs(6)p Fq(")p Fi(,)397 b(then)17516 33756 y Fq(P)154 b(P)19092 33922 y Fo(0)19895 33756 y Fs(=)308 b Fq(P)21775 33922 y Fo(0)22517 33756 y Fs(+)246 b Fq(\016)42 b(P)31 b(;)977 b Fl(k)p Fq(\016)42 b(P)154 b Fl(k)307 b Fs(=)h Fq(O)31 b Fs(\()p Fq(\025)32116 33282 y Fk(\000)p Fo(10)32116 34051 y(0)33746 33756 y Fs(\))p Fq(:)8921 b Fs(\(116\))6075 35821 y Fx(Corollary)443 b(8.)16946 37276 y Fq(P)17657 37442 y Fo(0)18153 37276 y Fq(P)461 b Fs(=)308 b Fq(P)21205 37442 y Fo(0)21947 37276 y Fs(+)246 b Fq(\016)42 b(P)24453 36819 y Fk(\003)24961 37276 y Fq(;)977 b Fl(k)p Fq(\016)42 b(P)28197 36819 y Fk(\003)28706 37276 y Fl(k)308 b Fs(=)f Fq(O)31 b Fs(\()p Fq(\025)32686 36802 y Fk(\000)p Fo(10)32686 37571 y(0)34316 37276 y Fs(\))p Fq(:)8351 b Fs(\(117\))7403 39341 y(Pro)31 b(of.)p 7403 39518 2971 45 v 387 w(W)-92 b(e)386 b(pro)-31 b(v)g(e)386 b(that)i(eac)-31 b(h)386 b(\011)21469 38939 y Fo(0)21469 39629 y Fn(j)21966 39341 y Fs(\()p Fq(x)p Fs(\),)391 b Fq(j)399 b Fl(2)336 b Fq(J)26806 38939 y Fk(0)27116 39341 y Fs(,)391 b(can)386 b(b)31 b(e)386 b(represen)-31 b(ted)385 b(as)h(a)h(linear)g(com)-31 b(bi-)6075 40939 y(nation)371 b(of)10933 40659 y(^)10779 40939 y(\011)11640 41105 y Fn(m)12481 40939 y Fs(,)f Fq(m)307 b Fl(2)g Fq(J)476 b Fs(with)370 b(a)g(high)g(accuracy:)18274 43018 y(\011)19135 42561 y Fo(0)19135 43292 y Fn(j)19631 43018 y Fs(\()p Fq(x)p Fs(\))308 b(=)22774 41967 y Fp(X)22601 44341 y Fn(m)p Fk(2)p Fn(J)24730 43018 y Fq(c)25209 43184 y Fn(j)45 b(m)26615 42739 y Fs(^)26461 43018 y(\011)27322 43184 y Fn(m)28163 43018 y Fs(\()p Fq(x)p Fs(\))247 b(+)f Fq(\017)31459 43184 y Fn(j)31925 43018 y Fs(\()p Fq(x)p Fs(\))p Fq(;)9679 b Fs(\(118\))21843 46312 y Fl(k)p Fq(\017)22845 46478 y Fn(j)23312 46312 y Fl(k)23865 46478 y Fo(0)24670 46312 y Fs(=)307 b Fq(O)31 b Fs(\()p Fq(\025)27789 45839 y Fk(\000)p Fo(11)27789 46608 y(0)29419 46312 y Fs(\))p Fq(:)13248 b Fs(\(119\))6075 48000 y(Considering)414 b(that)f(the)g(set)f Fq(J)519 b Fs(con)-31 b(tains)414 b(no)f(more)f(than)h Fq(c\025)32041 48166 y Fo(0)32950 48000 y Fs(elemen)-31 b(ts,)425 b(w)-31 b(e)413 b(easily)g(deduce)6075 49328 y(\(116\))381 b(from)e(\(118\))i(and)e(\(119\).)524 b(It)379 b(is)f(enough)i(to)f(pro)-31 b(v)g(e)380 b(these)e(form)-31 b(ulae)381 b(for)e Fq(\014)j Fl(\025)323 b Fs(1)p Fq(=)p Fs(2)254 b Fl(\000)f Fs(7)p Fq(")6075 50657 y Fs(since)398 b(the)g(set)g Fq(J)106 b Fs(\()p Fq(\025)14304 50823 y Fo(0)14802 50657 y Fs(\))398 b(is)g(getting)j(larger)d(when)h Fq(\014)456 b Fs(is)398 b(decreasing.)581 b(Let)398 b(us)g(pro)-31 b(v)g(e)398 b(\(118\))j(and)6075 51985 y(\(119\).)495 b(W)-92 b(e)368 b(do)h(this)h(in)g(three)e(steps.)7403 53313 y Fi(Step)440 b(1.)635 b Fs(First)416 b(w)-31 b(e)418 b(pro)-31 b(v)g(e)417 b(that)h(eac)-31 b(h)417 b(\011)24751 52912 y Fo(0)24751 53601 y Fn(j)25248 53313 y Fs(\()p Fq(x)p Fs(\),)430 b Fq(j)450 b Fl(2)386 b Fq(J)30228 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Fn(~)-467 b(p)31846 50589 y Fj(r)32292 50478 y Fo(\()p Fn(t)p Fo(\))p Fn(;x)p Fo(\)+)p Fn(S)35653 50589 y Fj(m)36388 50478 y Fo(\()p Fn(x)p Fo(\))37637 50935 y Fq(dx:)6075 53608 y Fs(here)401 b Fl(k)p Fq(S)9741 53774 y Fn(m)10582 53608 y Fs(\()p Fq(x)p Fs(\))p Fl(k)12628 53774 y Fn(\027)13543 53608 y Fs(=)362 b Fq(O)31 b Fs(\()p Fl(j)p Fq(m)p Fl(j)17657 53206 y Fo(6)p Fn(")18573 53608 y Fs(\))403 b(for)f(all)h Fq(\027)70 b Fs(,)411 b(since)402 b Fl(j)p Fq(m)269 b Fl(\000)f Fq(j)63 b Fl(j)363 b Fq(<)f(\025)32518 53076 y Fn(\026)32518 53903 y Fo(0)33111 53608 y Fs(,)411 b Fq(\026)362 b(<)h Fs(1)268 b Fl(\000)g Fs(7)p Fq(")403 b Fs(and)f Fq(j)426 b Fl(2)362 b Fq(J)44391 53206 y Fk(0)45064 53608 y Fl(\032)6075 54936 y Fs(\005\(1)p Fq(=)p Fs(2)306 b Fl(\000)d Fs(6)p Fq(")p Fs(\).)754 b(In)-31 b(tegrating)458 b(b)-31 b(y)456 b(parts)g Fq(M)576 b Fs(times)457 b(in)f(the)g (direction)h(of)465 b Fq(~)-562 b(p)38054 55102 y Fn(m)38895 54936 y Fs(\()p Fq(t)p Fs(\))305 b Fl(\000)312 b Fq(~)-562 b(p)42181 55102 y Fn(r)42671 54936 y Fs(\()p Fq(t)p Fs(\))457 b(the)6075 56264 y(same)369 b(w)-31 b(a)g(y)371 b(as)e(w)-31 b(e)370 b(did)f(to)h(obtain)h(\(126\),)h(w)-31 b(e)370 b(arriv)-31 b(e)369 b(at)h(the)g(estimate:)13870 58366 y Fl(j)308 b Fq(<)15807 58086 y Fs(^)15653 58366 y(\011)16514 58532 y Fn(m)17355 58366 y Fq(;)184 b Fs(\011)18707 57909 y Fo(0)18707 58640 y Fn(r)19512 58366 y Fq(>)308 b Fl(j)g Fq(<)22464 57469 y Fp(\000)22972 58366 y Fq(c)23451 58566 y Fk(k)p Fn(A)p Fk(k)25025 58677 y Fm(0)25455 58566 y Fn(;";\014)40 b(;\026;M)28988 58366 y Fl(j)p Fq(m)p Fl(j)30574 57909 y Fo(6)p Fn(")31490 58366 y Fl(j)p Fq(m)246 b Fl(\000)g Fq(r)31 b Fl(j)34959 57909 y Fk(\000)p Fo(1)36148 57469 y Fp(\001)36655 57683 y Fn(M)37822 58366 y Fq(;)5275 b Fs(\(133\))6075 60196 y(for)359 b(an)-31 b(y)361 b Fq(r)338 b Fl(62)308 b Fq(W)12769 60362 y Fo(1)13266 60196 y Fs(.)489 b(Considering)361 b Fq(M)428 b(>)308 b Fs(2)359 b(and)h(summarizing)i(with)e(resp)31 b(ect)359 b(to)h Fq(r)31 b Fs(,)362 b(w)-31 b(e)360 b(obtain)18223 62180 y Fl(k)p Fs(\()p Fq(I)334 b Fl(\000)246 b Fq(P)21845 62346 y Fo(1)22341 62180 y Fs(\)\011)23632 62346 y Fn(m)24473 62180 y Fl(k)25026 62346 y Fo(0)25831 62180 y Fl(\024)307 b Fq(\025)27645 61605 y Fo(3)p Fn("M)83 b Fk(\000)p Fn(\026)p Fo(\()p Fn(M)g Fk(\000)p Fo(2\))27645 62475 y(0)33469 62180 y Fq(:)6075 64119 y Fs(Cho)31 b(osing)390 b Fq(M)460 b Fs(=)340 b(408,)396 b(w)-31 b(e)389 b(obtain)h Fl(k)p Fs(\()p Fq(I)347 b Fl(\000)259 b Fq(P)24785 64285 y Fo(1)25281 64119 y Fs(\)\011)26572 64285 y Fn(m)27413 64119 y Fl(k)27966 64285 y Fo(0)28803 64119 y Fs(=)340 b Fq(O)31 b Fs(\()p Fq(\025)31955 63587 y Fk(\000)p Fo(202)p Fn(\026)31955 64414 y Fo(0)34565 64119 y Fs(\).)551 b(Considering)390 b(that)g(the)6075 65556 y(sum)381 b(\(132\))j(con)-31 b(tains)383 b(no)f(more)g(than)g Fq(c\025)23740 65024 y Fo(2)p Fn(\026)23740 65851 y Fo(0)25156 65556 y Fs(terms)g(and)30417 64726 y Fp(P)31586 65888 y Fn(m)32611 65556 y Fl(j)p Fq(c)33397 65722 y Fn(j)45 b(m)34650 65556 y Fl(j)34957 65154 y Fo(2)35782 65556 y Fl(\031)328 b Fs(1)382 b(b)-31 b(y)382 b(Lemma)g(5,)k(w)-31 b(e)6075 66884 y(obtain)371 b(\(131\).)494 b(Th)-31 b(us,)370 b(\(129\))i(holds.)7403 68213 y Fi(Step)357 b(3.)479 b Fs(W)-92 b(e)326 b(consider)h Fq(\026)307 b Fs(=)h(1)p Fq(=)p Fs(10)328 b(It)f(is)g(easy)g(to)h(see)f (that)h Fl(\000)p Fq(\015)33498 68379 y Fo(2)34156 68213 y Fs(+)162 b Fq(\026)307 b(<)h Fs(0.)479 b(Let)327 b(us)f(consider)6075 69541 y(sets)272 b Fq(W)9188 69707 y Fn(n)9791 69541 y Fs(:)444 b Fq(W)11587 69707 y Fn(n)12497 69541 y Fs(:)308 b Fl(f)p Fq(m)g Fs(:)g Fl(j)p Fq(m)52 b Fl(\000)g Fq(j)63 b Fl(j)309 b Fq(<)e Fs(\()p Fq(n)52 b Fs(+)g(1\))p Fq(\025)23795 69009 y Fn(\026)23795 69836 y Fo(0)24391 69541 y Fl(g)272 b Fs(and)h(let)g Fq(P)29485 69707 y Fn(n)30359 69541 y Fs(b)31 b(e)272 b(the)g(corresp)31 b(onding)272 b(pro)61 b(jections)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 7289 3345 a Fo(SPECTRAL)388 b(PR)-24 b(OPER)-71 b(TIES)386 b(OF)h(THE)h(PERIODIC)f(MA)-24 b(GNETIC)389 b(SCHR)35152 3149 y(\177)35033 3345 y(ODINGER)e(OPERA)-71 b(TOR)331 b(23)6075 6051 y Fs(on)310 b(the)g(linear)h(spans)e(of)h (\011)17447 5649 y Fo(0)17447 6324 y Fn(m)18288 6051 y Fs(,)322 b Fq(m)308 b Fl(2)f Fq(W)22287 6217 y Fn(n)22890 6051 y Fs(.)473 b(Clearly)-92 b(,)323 b Fl([)28422 5559 y Fn(N)29123 5670 y Fm(1)28422 6346 y Fn(n)p Fo(=1)30146 6051 y Fq(W)31191 6217 y Fn(n)31794 6051 y Fs(,)f Fq(N)33312 6217 y Fo(1)34116 6051 y Fs(=)308 b([)p Fq(\025)36238 5476 y Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(6)p Fn(")p Fk(\000)p Fn(\026)36238 6346 y Fo(0)40411 6051 y Fs(])i(b)31 b(elongs)310 b(to)6075 7613 y(the)269 b(\()p Fq(\025)8957 7038 y Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(6)p Fn(")8957 7908 y Fo(0)11900 7613 y Fs(\)-neigh)-31 b(b)31 b(orho)g(o)g(d)271 b(of)f Fq(J)21433 7211 y Fk(0)21744 7613 y Fs(\()p Fq(\025)22820 7779 y Fo(0)23317 7613 y Fs(\),)290 b(whic)-31 b(h)271 b(is)e(up)g(to)h(tec)-31 b(hnical)272 b(m)-31 b(ultiplies)272 b(is)d(the)h(same)6075 9040 y(as)451 b Fq(J)8236 8638 y Fk(0)8547 9040 y Fs(\()p Fq(\025)9623 9206 y Fo(0)10120 9040 y Fs(\).)740 b(W)-92 b(e)451 b(can)g(apply)i(\(120\),)475 b(\(129\))454 b(to)e(eac)-31 b(h)452 b Fq(m)445 b Fl(2)f([)32443 8548 y Fn(N)33144 8659 y Fm(1)32443 9335 y Fn(n)p Fo(=1)34167 9040 y Fq(W)35212 9206 y Fn(n)35815 9040 y Fs(.)739 b(Applying)453 b(form)-31 b(ulae)6075 10368 y(\(120\),)372 b(\(129\))f(to)f(eac)-31 b(h)370 b Fq(m)307 b Fl(2)g Fq(W)19376 10534 y Fo(1)19874 10368 y Fs(,)369 b(w)-31 b(e)370 b(get)9732 12520 y Fq(P)10443 12686 y Fo(1)10939 12520 y Fq(")11455 12686 y Fn(j)12229 12520 y Fs(=)13921 11469 y Fp(X)13397 13843 y Fn(m)p Fk(2)p Fn(W)15613 13954 y Fm(1)16044 12520 y Fs(\()p Fq(\017)16923 12686 y Fn(j)17390 12520 y Fs(\))17820 12686 y Fn(m)18662 12520 y Fs(\011)19523 12063 y Fo(0)19523 12794 y Fn(m)20363 12520 y Fs(\()p Fq(x)p Fs(\))309 b(=)23682 11469 y Fp(X)23333 13843 y Fn(r)24 b Fk(2)p Fn(W)25198 13954 y Fm(2)25813 12520 y Fq(c)26292 12686 y Fn(j)45 b(r)27348 12240 y Fs(^)27194 12520 y(\011)28055 12686 y Fn(r)28545 12520 y Fs(\()p Fq(x)p Fs(\))247 b(+)31915 11469 y Fp(X)31391 13843 y Fn(m)p Fk(2)p Fn(W)33607 13954 y Fm(1)34038 12520 y Fs(\()p Fq(\017)34917 12686 y Fn(j)35384 12520 y Fs(\))35814 12686 y Fn(m)36656 12520 y Fq(\017)37105 12686 y Fn(m)37946 12520 y Fs(\()p Fq(x)p Fs(\))p Fq(;)3658 b Fs(\(134\))6075 15608 y(here)368 b Fq(c)8955 15774 y Fn(j)45 b(r)10164 15608 y Fs(=)11332 14777 y Fp(P)12501 15940 y Fn(m)13342 15608 y Fs(\()p Fq(\017)14221 15774 y Fn(j)14688 15608 y Fs(\))15118 15774 y Fn(m)15959 15608 y Fq(c)16438 15774 y Fn(mr)17714 15608 y Fs(.)493 b(Clearly)-92 b(,)12533 17815 y Fl(k)13795 16763 y Fp(X)13270 19138 y Fn(m)p Fk(2)p Fn(W)15486 19249 y Fm(1)15918 17815 y Fs(\()p Fq(\017)16797 17981 y Fn(j)17264 17815 y Fs(\))17694 17981 y Fn(m)18536 17815 y Fq(\017)18985 17981 y Fn(m)19826 17815 y Fs(\()p Fq(x)p Fs(\))p Fl(k)21872 17981 y Fo(0)22677 17815 y Fl(\024)307 b(k)p Fq(\017)24847 17981 y Fn(j)25315 17815 y Fl(k)25868 17981 y Fo(0)26549 17815 y Fs(sup)26990 18694 y Fn(m)28400 17815 y Fl(k)p Fq(\017)29402 17981 y Fn(m)30244 17815 y Fl(k)30797 17981 y Fo(0)31601 17815 y Fs(=)h Fq(O)31 b Fs(\()p Fq(\025)34721 17240 y Fk(\000)p Fo(2\()p Fn(\015)36665 17351 y Fm(2)37097 17240 y Fk(\000)p Fn(\026)p Fo(\))34721 18110 y(0)38729 17815 y Fs(\))p Fq(:)6075 21045 y Fs(By)386 b(\(131\),)393 b Fl(k)p Fs(\()p Fq(I)344 b Fl(\000)257 b Fq(P)14691 21211 y Fo(1)15188 21045 y Fs(\))p Fq(")16134 21211 y Fn(j)16601 21045 y Fl(k)336 b Fs(=)f Fq(O)31 b Fs(\()p Fq(\025)20637 20571 y Fk(\000)p Fo(20)20637 21340 y(0)22267 21045 y Fs(\).)544 b(Using)386 b(\(120\),)393 b(\(134\))388 b(and)e(the)h(last)f(t)-31 b(w)g(o)388 b(estimates,)6075 22373 y(w)-31 b(e)369 b(obtain:)12466 24304 y(\011)13327 23846 y Fo(0)13327 24577 y Fn(j)13824 24304 y Fs(\()p Fq(x)p Fs(\))308 b(=)17317 23252 y Fp(X)16793 25626 y Fn(m)p Fk(2)p Fn(W)19009 25737 y Fm(2)19624 24304 y Fq(c)20103 24470 y Fn(j)45 b(r)21159 24024 y Fs(^)21005 24304 y(\011)21866 24470 y Fn(r)22356 24304 y Fs(\()p Fq(x)p Fs(\))247 b(+)e Fq(\017)25651 24470 y Fn(j)26118 24304 y Fs(\()p Fq(x)p Fs(\))p Fq(;)924 b Fl(k)p Fq(\017)29844 24470 y Fn(j)30311 24304 y Fl(k)30864 24470 y Fo(0)31669 24304 y Fs(=)307 b Fq(O)31 b Fs(\()p Fq(\025)34788 23729 y Fk(\000)p Fo(2\()p Fn(\015)36732 23840 y Fm(2)37164 23729 y Fk(\000)p Fn(\026)p Fo(\))34788 24599 y(0)38796 24304 y Fs(\))p Fq(:)3871 b Fs(\(135\))6075 27391 y(Rep)31 b(eating)370 b(this)g(pro)31 b(cedure)368 b(12)p Fq(=)p Fs(\()p Fq(\015)21310 27557 y Fo(2)22055 27391 y Fl(\000)246 b Fq(\026)p Fs(\))369 b(times,)i(w)-31 b(e)369 b(get)h(\(118\).)495 b Fl(})6075 29658 y Fs(3.3.)555 b Fx(Mo)35 b(del)376 b(Op)35 b(erator.)555 b Fs(A)-31 b(t)326 b(the)g(end)f(of)h(Section)h (2.2)g(w)-31 b(e)326 b(in)-31 b(tro)31 b(duced)326 b(a)f(family)j(of)e (mo)31 b(del)6075 31032 y(op)g(erators)11404 30752 y(^)11114 31032 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\),)543 b Fq(t)537 b Fl(2)h Fq(K)79 b Fs(,)542 b(acting)509 b(in)e Fq(L)23956 31198 y Fo(2)24453 31032 y Fs(\()p Fq(Q)p Fs(\).)908 b(W)-92 b(e)506 b(recall)i(that)h(the)e(action)i(of)42269 30752 y(^)41979 31032 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))508 b(on)6075 32361 y Fq(P)154 b(L)7693 32527 y Fo(2)8189 32361 y Fs(\()p Fq(Q)p Fs(\))370 b(is)f(de\014ned)g(b)-31 b(y)370 b(the)f(form)-31 b(ula:)13668 34343 y(^)13378 34623 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))15802 34343 y(^)15648 34623 y(\011)16509 34789 y Fn(j)16976 34623 y Fs(\()p Fq(t;)184 b(x)p Fs(\))309 b(=)20838 33394 y Fp(\020)21499 34623 y Fl(<)22344 34331 y Fs(^)22298 34623 y Fq(\025)22944 34789 y Fn(j)23410 34623 y Fs(\()p Fq(t)p Fs(\))24670 33394 y Fp(\021)25670 34343 y Fs(^)25517 34623 y(\011)26378 34789 y Fn(j)26844 34623 y Fs(\()p Fq(t;)184 b(x)p Fs(\))p Fq(;)924 b Fs(when)739 b Fq(j)370 b Fl(2)308 b Fq(J)106 b Fs(\()p Fq(\025)37387 34789 y Fo(0)37884 34623 y Fs(\);)4783 b(\(136\))6075 36886 y(and)369 b(on)h(\()p Fq(I)333 b Fl(\000)245 b Fq(P)154 b Fs(\))p Fq(L)14169 37052 y Fo(2)14666 36886 y Fs(\()p Fq(Q)p Fs(\))370 b(it)g(is)f(de\014ned)g(as)g(follo)-31 b(ws:)18280 38644 y(^)17990 38924 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\)\()p Fq(I)334 b Fl(\000)246 b Fq(P)154 b Fs(\))307 b(=)h(\()p Fq(I)333 b Fl(\000)246 b Fq(P)154 b Fs(\))p Fq(H)90 b Fs(\()p Fq(I)333 b Fl(\000)245 b Fq(P)154 b Fs(\))p Fq(:)9395 b Fs(\(137\))6075 40962 y(The)442 b(op)31 b(erator)13250 40682 y(^)12961 40962 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))442 b(is)g(correctly)h(de\014ned)f(on) g Fq(L)27769 41128 y Fo(2)28266 40962 y Fs(\()p Fq(Q)p Fs(\),)462 b(since)442 b(the)g(mo)31 b(del)443 b(functions)g(ha)-31 b(v)g(e)6075 42290 y(basis-lik)g(e)370 b(prop)31 b(erties.)16462 41861 y Fo(3)6075 44361 y Fx(Lemma)1000 b(8.)593 b Fi(L)-57 b(et)457 b Fs(0)418 b Fq(<)f(")h(<)f Fs(1)p Fq(=)p Fs(4)p Fi(,)474 b Fs(1)p Fq(=)p Fs(3)419 b Fq(<)e(\014)476 b(<)417 b Fs(1)p Fq(=)p Fs(2)292 b Fl(\000)e Fs(2)p Fq(")457 b Fi(and)g Fq(N)538 b Fl(\025)418 b Fs(31)p Fi(.)692 b(Then,)473 b(for)457 b(any)6075 45690 y Fq(t)307 b Fl(2)g Fq(K)79 b Fi(,)19714 47201 y Fl(k)p Fq(H)90 b Fs(\()p Fq(t)p Fs(\))247 b Fl(\000)24181 46922 y Fs(^)23891 47201 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fl(k)309 b Fq(<)e(C)29148 47377 y Fn(~)28982 47573 y(A)29703 47201 y Fq(\025)30349 46728 y Fk(\000)p Fo(10)30349 47497 y(0)31979 47201 y Fq(:)11118 b Fs(\(138\))7403 49272 y(Pro)31 b(of.)p 7403 49449 2971 45 v 360 w(Considering)361 b(form)-31 b(ulae)360 b(\(136\))i(and)d(\(70\))i({)e(\(72\),)k(for)c Fq(\014)366 b(>)308 b Fs(1)p Fq(=)p Fs(3,)363 b Fq(N)428 b Fl(\025)308 b Fs(31,)362 b Fq(k)343 b Fl(\031)307 b Fq(\025)45429 49438 y Fo(0)6075 50601 y Fs(and)346 b(taking)h(in)-31 b(to)347 b(accoun)-31 b(t)347 b(that)g Fq(J)106 b Fs(\()p Fq(\025)22051 50767 y Fo(0)22548 50601 y Fs(\))346 b(con)-31 b(tains)347 b(no)f(more)g(than)h Fq(c\025)35588 50767 y Fo(0)36429 50601 y Fs(elemen)-31 b(ts,)352 b(w)-31 b(e)346 b(obtain)18467 51590 y Fp(\015)18467 52254 y(\015)18467 52919 y(\015)19082 51635 y(\020)19743 52863 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))247 b Fl(\000)23656 52583 y Fs(^)23366 52863 y Fq(H)91 b Fs(\()p Fq(t)p Fs(\))25637 51635 y Fp(\021)26483 52863 y Fq(P)27348 51590 y Fp(\015)27348 52254 y(\015)27348 52919 y(\015)28270 52863 y Fq(<)307 b(C)30395 53039 y Fn(~)30229 53235 y(A)30950 52863 y Fq(\025)31596 52390 y Fk(\000)p Fo(10)31596 53158 y(0)33226 52863 y Fq(;)9871 b Fs(\(139\))6075 55569 y(and,)501 b(hence,)12179 54296 y Fp(\015)12179 54960 y(\015)12179 55624 y(\015)12794 55569 y Fs(\()p Fq(I)333 b Fl(\000)246 b Fq(P)154 b Fs(\))16630 54340 y Fp(\020)17292 55569 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))247 b Fl(\000)21205 55289 y Fs(^)20915 55569 y Fq(H)91 b Fs(\()p Fq(t)p Fs(\))23186 54340 y Fp(\021)24032 55569 y Fq(P)24897 54296 y Fp(\015)24897 54960 y(\015)24897 55624 y(\015)25994 55569 y Fq(<)482 b(C)28294 55745 y Fn(~)28128 55941 y(A)28849 55569 y Fq(\025)29495 55095 y Fk(\000)p Fo(10)29495 55864 y(0)31125 55569 y Fs(.)807 b(F)-92 b(rom)474 b(the)g(de\014nition)i(of)43944 55289 y(^)43654 55569 y Fq(H)91 b Fs(\()p Fq(t)p 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b(y)392 b(a)e(sp)31 b(ecial)392 b(role:)535 b(eigen)-31 b(v)-61 b(alues)392 b(of)f(\()p Fq(I)347 b Fl(\000)6075 63386 y Fq(P)154 b Fs(\))p Fq(H)90 b Fs(\()p Fq(t)p Fs(\)\()p Fq(I)373 b Fl(\000)285 b Fq(P)154 b Fs(\))428 b(whic)-31 b(h)429 b(are)e(close)i(to)g Fq(\025)23699 63552 y Fo(0)24623 63386 y Fs(can)f(b)31 b(e)428 b(w)-31 b(ell)429 b(appro)-31 b(ximated)431 b(b)-31 b(y)428 b(eigen)-31 b(v)-61 b(alues)430 b(of)6075 64715 y(the)369 b(op)31 b(erators)370 b(of)f Fq(P)14854 64881 y Fn(q)15343 64715 y Fq(H)90 b(P)17064 64881 y Fn(q)17552 64715 y Fs(,)370 b Fq(q)347 b Fl(2)308 b Fs(\000\()p Fq(\025)21884 64313 y Fn(")21884 64990 y Fo(0)22380 64715 y Fs(\).)6075 66826 y Fx(Lemma)1016 b(9.)597 b Fi(L)-57 b(et)14792 66534 y Fs(^)14746 66826 y Fq(\025)15392 66992 y Fn(j)15858 66826 y Fs(\()p Fq(t)p Fs(\))464 b Fi(b)-57 b(e)464 b(an)f(eigenvalue)i (of)e Fs(\()p Fq(I)382 b Fl(\000)294 b Fq(P)154 b Fs(\))p Fq(H)90 b Fs(\()p Fq(t)p Fs(\)\()p Fq(I)383 b Fl(\000)294 b Fq(P)154 b Fs(\))463 b Fi(satisfying)i(the)d(in-)6075 68154 y(e)-57 b(quality:)21242 69541 y Fl(j)21595 69249 y Fs(^)21549 69541 y Fq(\025)22195 69707 y Fn(j)22662 69541 y Fs(\()p Fq(t)p Fs(\))246 b Fl(\000)g Fq(\025)25921 69707 y Fo(0)26418 69541 y Fl(j)308 b Fq(<)f(\025)28847 69707 y Fo(0)29343 69541 y Fq(=)p Fs(8)p Fq(:)12648 b Fs(\(145\))p eop end %%Page: 26 26 TeXDict begin 26 25 bop 6075 3345 a Fo(26)14514 b(YULIA)300 b(KARPESHINA)6075 6001 y Fi(Then,)471 b(ther)-57 b(e)456 b(is)g(an)g(eigenvalue)i Fq(\026)21158 5600 y Fn(q)21158 6275 y(n)21760 6001 y Fs(\()p Fq(t)p Fs(\))p Fi(,)472 b Fq(n)416 b Fl(2)g Fq(N)121 b Fi(,)472 b(of)456 b(an)g(op)-57 b(er)g(ator)455 b Fq(P)36018 6167 y Fn(q)36507 6001 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)39488 6167 y Fn(q)39977 6001 y Fi(,)471 b Fq(q)456 b Fl(2)416 b Fs(\000\()p Fq(\025)44659 5600 y Fn(")44659 6276 y Fo(0)45155 6001 y Fs(\))p Fi(,)6075 7330 y(such)395 b(that)20156 8662 y Fl(j)20509 8370 y Fs(^)20463 8662 y Fq(\025)21109 8828 y Fn(j)21575 8662 y Fs(\()p Fq(t)p Fs(\))247 b Fl(\000)f Fq(\026)24856 8205 y Fn(q)24856 8935 y(n)25458 8662 y Fs(\()p Fq(t)p Fs(\))p Fl(j)309 b Fq(<)f(c)29148 8838 y Fn(~)28982 9034 y(A)29703 8662 y Fq(\025)30349 8188 y Fk(\000)p Fo(9)30349 8957 y(0)31537 8662 y Fq(:)11560 b Fs(\(146\))6075 10871 y Fx(Corollary)357 b(9.)487 b Fi(If)328 b Fs(min)16085 11071 y Fn(j)45 b Fk(2)p Fn(Z)17750 10849 y Fm(2)18183 11071 y Fk(n)p Fn(J)73 b Fo(\()p Fn(\025)20071 11182 y Fm(0)20502 11071 y Fo(\))21088 10871 y Fl(j)21441 10579 y Fs(^)21395 10871 y Fq(\025)22041 11037 y Fn(j)22507 10871 y Fs(\()p Fq(t)p Fs(\))97 b Fl(\000)g Fq(\025)25468 11037 y Fo(0)25966 10871 y Fl(j)308 b Fq(<)f(\025)28395 10339 y Fk(\000)p Fn(\026)28395 11166 y Fo(0)29680 10871 y Fq(;)329 b(\026)307 b(<)g Fs(9)p Fi(,)343 b(then)327 b Fs(min)37930 11037 y Fn(n;q)39414 10871 y Fl(j)p Fq(\026)40388 10469 y Fn(q)40388 11145 y(n)40991 10871 y Fs(\()p Fq(t)p Fs(\))97 b Fl(\000)g Fq(\025)43952 11037 y Fo(0)44449 10871 y Fl(j)308 b Fq(<)6075 12387 y Fs(2)p Fq(\025)7274 11855 y Fk(\000)p Fn(\026)7274 12682 y Fo(0)8559 12387 y Fq(:)7403 14381 y Fs(Pro)31 b(of.)p 7403 14558 2971 45 v 370 w(The)370 b(pro)31 b(of)369 b(con)-31 b(tains)371 b(\014v)-31 b(e)369 b(steps.)7403 15762 y Fx(1.)782 b Fs(Let)11409 15482 y(^)11256 15762 y(\011)12117 15928 y Fn(j)12583 15762 y Fs(\()p Fq(t)p Fs(\))466 b(b)31 b(e)466 b(an)g(eigenfunction)i(of)f(\()p Fq(I)397 b Fl(\000)311 b Fq(P)154 b Fs(\))p Fq(H)90 b Fs(\()p Fq(t)p Fs(\)\()p Fq(I)398 b Fl(\000)310 b Fq(P)154 b Fs(\))466 b(corresp)31 b(onding)466 b(to)h(an)6075 17250 y(eigen)-31 b(v)-61 b(alue)11410 16958 y(^)11364 17250 y Fq(\025)12010 17416 y Fn(j)12476 17250 y Fs(\()p Fq(t)p Fs(\))369 b(whic)-31 b(h)371 b(satis\014es)e(\(145\).)495 b(It)369 b(follo)-31 b(ws)371 b(from)f(\(138\))i(that)18777 19207 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))21201 18927 y(^)21047 19207 y(\011)21908 19373 y Fn(j)22683 19207 y Fs(=)23897 18915 y(^)23851 19207 y Fq(\025)24497 19373 y Fn(j)24963 19207 y Fs(\()p Fq(t)p Fs(\))26377 18927 y(^)26223 19207 y(\011)27084 19373 y Fn(j)27797 19207 y Fs(+)246 b Fq(O)31 b Fs(\()p Fq(\025)30855 18734 y Fk(\000)p Fo(10)30855 19502 y(0)32485 19207 y Fs(\))p Fq(;)10182 b Fs(\(147\))6075 21004 y(here)368 b(and)i(b)31 b(elo)-31 b(w)370 b(in)f(this)h(lemma)h(w)-31 b(e)370 b(mean)g Fq(O)31 b Fs(\()p Fl(\001)p Fs(\))370 b(with)h(resp)31 b(ect)368 b(to)i Fq(L)36195 21170 y Fo(2)36692 21004 y Fs(\()p Fq(Q)p Fs(\)-norm.)7403 22440 y Fx(2.)489 b Fs(Let)11105 22160 y(^)10857 22440 y Fq(P)514 b Fs(b)31 b(e)360 b(the)g(pro)61 b(jection)363 b(in)e Fq(L)22766 22606 y Fo(2)23263 22440 y Fs(\()p Fq(Q)p Fs(\))g(with)h(a)f(diagonal)i(matrix)37045 22160 y(^)36797 22440 y Fq(P)37508 22606 y Fn(mr)39143 22440 y Fs(in)e(the)g(basis)g (of)6075 23768 y(exp)31 b(onen)-31 b(ts)369 b Fl(f)p Fs(\011)12691 23367 y Fo(0)12691 24042 y Fn(r)13188 23768 y Fs(\()p Fq(t;)184 b(x)p Fs(\))p Fl(g)16125 23965 y Fn(r)24 b Fk(2)p Fn(Z)17814 23744 y Fm(2)18303 23768 y Fs(:)19471 25950 y(^)19224 26229 y Fq(P)19935 26395 y Fn(r)g(r)21167 26229 y Fs(=)22335 24669 y Fp(\032)23719 25554 y Fs(1)p Fq(;)1108 b Fs(if)370 b Fq(r)338 b Fl(2)308 b Fq(M)121 b Fs(\()p Fq(\025)30856 25720 y Fo(0)31352 25554 y Fs(\);)23719 26883 y(0)p Fq(;)1108 b Fs(otherwise.)43404 26229 y(\(148\))6075 28909 y(W)-92 b(e)440 b(pro)-31 b(v)g(e)442 b(b)-31 b(y)441 b(standard)g(p)31 b(erturbation)442 b(metho)31 b(ds)442 b(that)g(the)f(function)37525 28629 y(^)37372 28909 y(\011)38233 29075 y Fn(j)39140 28909 y Fs(de\014ned)g(in)g(the)6075 30385 y(\014rst)368 b(step)h(essen)-31 b(tially)372 b(b)31 b(elongs)369 b(to)21577 30105 y(^)21329 30385 y Fq(P)154 b(L)22947 30551 y Fo(2)23444 30385 y Fs(\()p Fq(Q)p Fs(\):)20820 32009 y(^)20666 32289 y(\011)21527 32455 y Fn(j)22301 32289 y Fs(=)23718 32009 y(^)23470 32289 y Fq(P)24488 32009 y Fs(^)24334 32289 y(\011)25195 32455 y Fn(j)25908 32289 y Fs(+)246 b Fq(O)31 b Fs(\()p Fq(\025)28966 31816 y Fk(\000)p Fo(10)28966 32584 y(0)30596 32289 y Fs(\))p Fq(;)12071 b Fs(\(149\))18616 34237 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))21040 33958 y(^)20886 34237 y(\011)21747 34403 y Fn(j)22522 34237 y Fs(=)307 b Fq(H)90 b Fs(\()p Fq(t)p Fs(\))26209 33958 y(^)25960 34237 y Fq(P)26979 33958 y Fs(^)26826 34237 y(\011)27687 34403 y Fn(j)28399 34237 y Fs(+)246 b Fq(O)31 b Fs(\()p Fq(\025)31457 33764 y Fk(\000)p Fo(9)31457 34533 y(0)32646 34237 y Fs(\))p Fq(:)10021 b Fs(\(150\))6075 35949 y(Indeed,)369 b(let)h(us)e(consider)h Fq(<)18636 35670 y Fs(^)18482 35949 y(\011)19343 36115 y Fn(j)19809 35949 y Fq(;)184 b Fs(\011)21161 35548 y Fo(0)21161 36223 y Fn(r)21966 35949 y Fq(>)p Fs(,)370 b Fq(r)338 b Fs(:)308 b Fl(j)p Fq(p)25820 35548 y Fo(2)25820 36223 y Fn(r)26317 35949 y Fs(\(0\))248 b Fl(\000)e Fq(\025)29731 36115 y Fo(0)30227 35949 y Fl(j)308 b Fq(>)g(\025)32657 36115 y Fo(0)33153 35949 y Fq(=)p Fs(4.)493 b(Considering)371 b(that)7146 38450 y Fq(<)307 b(H)90 b Fs(\()p Fq(t)p Fs(\))10738 38171 y(^)10584 38450 y(\011)11445 38616 y Fn(j)11912 38450 y Fq(;)184 b Fs(\011)13264 37993 y Fo(0)13264 38724 y Fn(r)14070 38450 y Fq(>)p Fs(=)307 b Fq(p)16656 37993 y Fo(2)16656 38724 y Fn(r)17152 38450 y Fs(\()p Fq(t)p Fs(\))i Fq(<)20043 38171 y Fs(^)19889 38450 y(\011)20750 38616 y Fn(j)21217 38450 y Fq(;)184 b Fs(\011)22569 37993 y Fo(0)22569 38724 y Fn(r)23374 38450 y Fq(>)307 b Fs(+)p Fq(O)26463 36890 y Fp(\022)28512 38450 y Fs(max)27277 39216 y Fn(s)p Fo(:)p Fk(j)p Fn(s)p Fk(\000)p Fn(r)24 b Fk(j)p Fn()307 b Fl(j)41025 36890 y Fp(\023)42025 38450 y Fq(;)1072 b Fs(\(151\))6075 40868 y(and)369 b(using)h(elemen)-31 b(tary)371 b(p)31 b(erturbation)370 b(considerations)g(w)-31 b(e)370 b(obtain)h(that:) 13652 41889 y Fp(\014)13652 42553 y(\014)13652 43217 y(\014)14021 43162 y Fq(<)15344 42882 y Fs(^)15190 43162 y(\011)16051 43328 y Fn(j)16517 43162 y Fq(;)184 b Fs(\011)17869 42705 y Fo(0)17869 43435 y Fn(r)18674 43162 y Fq(>)19535 41889 y Fp(\014)19535 42553 y(\014)19535 43217 y(\014)20212 43162 y Fq(<)307 b(c)22025 43338 y Fn(~)21859 43534 y(A)22580 43162 y Fq(p)23137 43328 y Fn(r)23627 43162 y Fs(\(0\))25224 41889 y Fp(\014)25224 42553 y(\014)25224 43217 y(\014)25595 43162 y Fq(p)26152 43328 y Fn(r)26643 43162 y Fs(\(0\))28056 42705 y Fo(2)28800 43162 y Fl(\000)29953 42869 y Fs(^)29907 43162 y Fq(\025)30553 43328 y Fn(j)31019 43162 y Fs(\()p Fq(t)p Fs(\))32279 41889 y Fp(\014)32279 42553 y(\014)32279 43217 y(\014)32648 42162 y Fk(\000)p Fo(1)34144 43162 y Fq(<)36076 42358 y(c)36721 42534 y Fn(~)36555 42730 y(A)p 35446 42907 2462 45 v 35446 43921 a Fq(p)36003 44087 y Fn(r)36493 43921 y Fs(\(0\))38040 43162 y Fq(:)6075 45848 y Fs(T)-92 b(aking)352 b(in)-31 b(to)352 b(accoun)-31 b(t)353 b(that)18631 45568 y Fq(~)18424 45848 y(A)p Fs(\()p Fq(x)p Fs(\))e(is)g(a)g(trigonometric)i(p)31 b(olynomial)354 b(and)d(applying)i(the)e(last)6075 47176 y(estimate,)371 b(w)-31 b(e)370 b(obtain)h(that)f(for)f(an)-31 b(y)370 b Fq(r)339 b Fs(:)308 b Fl(j)p Fq(p)24390 46774 y Fo(2)24390 47450 y Fn(r)24887 47176 y Fs(\(0\))247 b Fl(\000)f Fq(\025)28300 47342 y Fo(0)28797 47176 y Fl(j)308 b Fq(>)f(\025)31226 47342 y Fo(0)31723 47176 y Fq(=)p Fs(4)246 b(+)g Fq(R)35023 47342 y Fo(0)35520 47176 y Fs(,)6922 48364 y Fp(\014)6922 49028 y(\014)6922 49692 y(\014)7291 49637 y Fq(<)8613 49357 y Fs(^)8459 49637 y(\011)9320 49803 y Fn(j)9787 49637 y Fq(;)184 b Fs(\011)11139 49180 y Fo(0)11139 49911 y Fn(r)11944 49637 y Fq(>)12805 48364 y Fp(\014)12805 49028 y(\014)12805 49692 y(\014)13481 49637 y Fq(<)308 b(c)15295 49813 y Fn(~)15129 50009 y(A)16034 48076 y Fp(\022)18084 49637 y Fs(max)16849 50403 y Fn(s)p Fo(:)p Fk(j)p Fn(s)p Fk(\000)p Fn(r)24 b Fk(j)p Fn()27446 48364 y Fp(\014)27446 49028 y(\014)27446 49692 y(\014)27815 48076 y(\023)28814 49637 y Fq(p)29371 49803 y Fn(r)29861 49637 y Fs(\(0\))31458 48364 y Fp(\014)31458 49028 y(\014)31458 49692 y(\014)31829 49637 y Fq(p)32386 49803 y Fn(r)32876 49637 y Fs(\(0\))34289 49180 y Fo(2)35033 49637 y Fl(\000)36187 49345 y Fs(^)36140 49637 y Fq(\025)36786 49803 y Fn(j)37253 49637 y Fs(\()p Fq(t)p Fs(\))38513 48364 y Fp(\014)38513 49028 y(\014)38513 49692 y(\014)38882 48638 y Fk(\000)p Fo(1)40378 49637 y Fq(<)42559 48834 y(c)43204 49010 y Fn(~)43038 49206 y(A)p 41680 49382 2959 45 v 41680 50396 a Fq(p)42237 50562 y Fn(r)42727 50396 y Fs(\(0\))44140 50077 y Fo(2)44771 49637 y Fq(;)6075 52187 y Fs(and)369 b(so)g(on.)493 b(If)369 b Fq(r)339 b Fs(:)308 b Fl(j)p Fq(p)14978 51786 y Fo(2)14978 52461 y Fn(r)15475 52187 y Fs(\(0\))247 b Fl(\000)f Fq(\025)18888 52353 y Fo(0)19385 52187 y Fl(j)307 b Fq(>)h(\025)21814 52353 y Fo(0)22310 52187 y Fq(=)p Fs(4)247 b(+)f Fq(nR)26275 52353 y Fo(0)26772 52187 y Fs(,)369 b(then)6309 53375 y Fp(\014)6309 54040 y(\014)6309 54704 y(\014)6678 54648 y Fq(<)8000 54369 y Fs(^)7846 54648 y(\011)8707 54814 y Fn(j)9173 54648 y Fq(;)184 b Fs(\011)10525 54191 y Fo(0)10525 54922 y Fn(r)11330 54648 y Fq(>)12191 53375 y Fp(\014)12191 54040 y(\014)12191 54704 y(\014)12868 54648 y Fq(<)307 b(c)14681 54824 y Fn(~)14515 55020 y(A)15421 53088 y Fp(\022)17470 54648 y Fs(max)16236 55414 y Fn(s)p Fo(:)p Fk(j)p Fn(s)p Fk(\000)p Fn(r)24 b Fk(j)p Fn()26833 53375 y Fp(\014)26833 54040 y(\014)26833 54704 y(\014)27202 53088 y(\023)28201 54648 y Fq(p)28758 54814 y Fn(r)29248 54648 y Fs(\(0\))30845 53375 y Fp(\014)30845 54040 y(\014)30845 54704 y(\014)31216 54648 y Fq(p)31773 54814 y Fn(r)32263 54648 y Fs(\(0\))33676 54191 y Fo(2)34420 54648 y Fl(\000)35573 54356 y Fs(^)35527 54648 y Fq(\025)36173 54814 y Fn(j)36639 54648 y Fs(\()p Fq(t)p Fs(\))37899 53375 y Fp(\014)37899 54040 y(\014)37899 54704 y(\014)38269 53649 y Fk(\000)p Fo(1)39765 54648 y Fq(<)42559 53845 y(c)43204 54021 y Fn(~)43038 54217 y(A)p 41066 54394 4185 45 v 41066 55408 a Fq(p)41623 55574 y Fn(r)42113 55408 y Fs(\(0\))43526 55088 y Fn(n)p Fo(+1)45384 54648 y Fq(:)6075 57066 y Fs(Cho)31 b(osing)370 b Fq(n)308 b Fs(=)g(20,)370 b(w)-31 b(e)370 b(obtain)20150 57868 y Fp(\014)20150 58532 y(\014)20150 59196 y(\014)20519 59141 y Fq(<)21841 58861 y Fs(^)21687 59141 y(\011)22548 59307 y Fn(j)23015 59141 y Fq(;)184 b Fs(\011)24367 58684 y Fo(0)24367 59414 y Fn(r)25172 59141 y Fq(>)26033 57868 y Fp(\014)26033 58532 y(\014)26033 59196 y(\014)26709 59141 y Fq(<)29110 58338 y(c)29755 58514 y Fn(~)29589 58710 y(A)p 28010 58886 3400 45 v 28010 59900 a Fq(p)28567 60066 y Fn(r)29058 59900 y Fs(\(0\))30471 59580 y Fo(21)31543 59141 y Fq(:)11554 b Fs(\(152\))6075 61657 y(This)476 b(estimate)i(ob)-31 b(viously)478 b(holds)e(for)g(all)i Fq(r)516 b Fs(:)486 b Fl(j)p Fq(p)27373 61255 y Fo(2)27373 61931 y Fn(r)27870 61657 y Fs(\(0\))319 b Fl(\000)e Fq(\025)31426 61823 y Fo(0)31922 61657 y Fl(j)486 b Fq(>)g(\025)34708 61823 y Fo(0)35204 61657 y Fq(=)p Fs(2.)814 b(Squaring)477 b(the)f(last)6075 63093 y(estimate)290 b(and)f(summarizing)i(it)f(o)-31 b(v)g(er)289 b(the)g Fq(r)31 b Fs(,)305 b(w)-31 b(e)290 b(obtain)g(\(149\).)468 b(Considering)290 b(that)g Fl(k)p Fq(H)43587 62813 y Fs(^)43338 63093 y Fq(P)154 b Fl(k)308 b Fq(<)6075 64421 y Fs(2)p Fq(\025)7274 64587 y Fo(0)7771 64421 y Fs(,)369 b(w)-31 b(e)370 b(get)g(\(150\).)7403 65750 y Fx(3.)492 b Fs(W)-92 b(e)369 b(pro)-31 b(v)g(e)369 b(the)h(relations:)20183 67313 y(^)19935 67593 y Fq(P)20953 67313 y Fs(^)20799 67593 y(\011)21660 67759 y Fn(j)22434 67593 y Fs(=)308 b Fq(P)24314 67759 y Fn(Q)25220 67313 y Fs(^)25066 67593 y(\011)25927 67759 y Fn(j)26639 67593 y Fs(+)246 b Fq(O)31 b Fs(\()p Fq(\025)29697 67119 y Fk(\000)p Fo(10)29697 67888 y(0)31327 67593 y Fs(\))p Fq(;)11340 b Fs(\(153\))18515 69541 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))21034 69261 y(^)20785 69541 y Fq(P)21804 69261 y Fs(^)21650 69541 y(\011)22511 69707 y Fn(j)23285 69541 y Fs(=)308 b Fq(H)90 b(P)26175 69707 y Fn(Q)27081 69261 y Fs(^)26927 69541 y(\011)27788 69707 y Fn(j)28500 69541 y Fs(+)246 b Fq(O)31 b Fs(\()p Fq(\025)31558 69067 y Fk(\000)p Fo(9)31558 69836 y(0)32747 69541 y Fs(\))p Fq(;)9920 b Fs(\(154\))p eop end %%Page: 27 27 TeXDict begin 27 26 bop 7289 3345 a Fo(SPECTRAL)388 b(PR)-24 b(OPER)-71 b(TIES)386 b(OF)h(THE)h(PERIODIC)f(MA)-24 b(GNETIC)389 b(SCHR)35152 3149 y(\177)35033 3345 y(ODINGER)e(OPERA)-71 b(TOR)331 b(27)6075 6001 y Fs(here)22072 7389 y Fq(P)22783 7555 y Fn(Q)23842 7389 y Fs(=)25824 6338 y Fp(X)25011 8764 y Fn(q)28 b Fk(2)p Fo(\000\()p Fn(\025)27460 8497 y Fj(")27460 9004 y Fm(0)27891 8764 y Fo(\))28421 7389 y Fq(P)29132 7555 y Fn(q)29621 7389 y Fq(:)6075 10566 y Fs(In)436 b(fact,)454 b(considering)437 b(\(142\),)456 b(w)-31 b(e)437 b(represen)-31 b(t)26029 10286 y(^)25781 10566 y Fq(P)590 b Fs(in)437 b(the)f(form:)33843 10286 y(^)33596 10566 y Fq(P)572 b Fs(=)420 b Fq(P)36871 10732 y Fo(0)37658 10566 y Fs(+)291 b Fq(P)39521 10732 y Fn(Q)40273 10566 y Fs(,)453 b(pro)61 b(jection)6075 11966 y Fq(P)6786 12132 y Fo(0)7650 11966 y Fs(is)367 b(de\014ned)h(b)31 b(efore)368 b(Lemma)g(7.)493 b(By)368 b(Corollary)i(8,)f Fq(P)29768 12132 y Fo(0)30264 11966 y Fs(\()p Fq(I)330 b Fl(\000)243 b Fq(P)154 b Fs(\))308 b(=)f Fq(O)31 b Fs(\()p Fq(\025)37337 11493 y Fk(\000)p Fo(10)37337 12261 y(0)38967 11966 y Fs(\).)493 b(Considering)6075 13421 y(also)370 b(that)10847 13141 y(^)10693 13421 y(\011)11554 13587 y Fn(j)12328 13421 y Fs(=)308 b(\()p Fq(I)332 b Fl(\000)246 b Fq(P)154 b Fs(\))17302 13141 y(^)17148 13421 y(\011)18009 13587 y Fn(j)18476 13421 y Fs(,)369 b(w)-31 b(e)370 b(obtain:)9881 15141 y(^)9634 15421 y Fq(P)10652 15141 y Fs(^)10498 15421 y(\011)11359 15587 y Fn(j)12133 15421 y Fs(=)13549 15141 y(^)13301 15421 y Fq(P)154 b Fs(\()p Fq(I)333 b Fl(\000)246 b Fq(P)154 b Fs(\))17972 15141 y(^)17818 15421 y(\011)18679 15587 y Fn(j)19453 15421 y Fs(=)307 b Fq(P)21332 15587 y Fn(Q)22084 15421 y Fs(\()p Fq(I)333 b Fl(\000)246 b Fq(P)154 b Fs(\))25890 15141 y(^)25736 15421 y(\011)26597 15587 y Fn(j)27309 15421 y Fs(+)246 b Fq(O)31 b Fs(\()p Fq(\025)30367 14947 y Fk(\000)p Fo(10)30367 15716 y(0)31997 15421 y Fs(\))308 b(=)g Fq(P)34615 15587 y Fn(Q)35521 15141 y Fs(^)35367 15421 y(\011)36228 15587 y Fn(j)36940 15421 y Fs(+)246 b Fq(O)31 b Fs(\()p Fq(\025)39998 14947 y Fk(\000)p Fo(10)39998 15716 y(0)41628 15421 y Fs(\))p Fq(:)6075 17421 y Fs(Th)-31 b(us,)394 b(\(153\))d(is)d(pro)-31 b(v)g(ed.)552 b(T)-92 b(aking)391 b(in)-31 b(to)390 b(accoun)-31 b(t)390 b(that)g Fl(k)p Fq(H)90 b Fs(\()p Fq(t)p Fs(\))32916 17141 y(^)32667 17421 y Fq(P)155 b Fl(k)340 b Fq(<)g Fs(2)p Fq(\025)36826 17587 y Fo(0)37711 17421 y Fs(and)389 b Fl(k)p Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)43417 17587 y Fn(Q)44171 17421 y Fl(k)340 b Fq(<)6075 18750 y Fs(2)p Fq(\025)7274 18916 y Fo(0)7771 18750 y Fs(,)369 b(w)-31 b(e)370 b(arriv)-31 b(e)369 b(at)h(\(154\).)7403 20186 y Fx(4.)845 b Fs(Com)-31 b(bining)490 b(\(147\),)519 b(\(149\),)g(\(150\),)g(\(153\))490 b(and)d(\(154\),)519 b(w)-31 b(e)487 b(obtain:)730 b Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)42480 20352 y Fn(Q)43387 19906 y Fs(^)43233 20186 y(\011)44094 20352 y Fn(j)45064 20186 y Fs(=)6121 21382 y(^)6075 21674 y Fq(\025)6721 21840 y Fn(j)7187 21674 y Fq(P)7898 21840 y Fn(Q)8804 21394 y Fs(^)8650 21674 y(\011)9511 21840 y Fn(j)10223 21674 y Fs(+)246 b Fq(O)31 b Fs(\()p Fq(\025)13281 21200 y Fk(\000)p Fo(9)13281 21969 y(0)14470 21674 y Fs(\),)370 b(and,)g(therefore,)17433 23686 y Fq(P)18144 23852 y Fn(Q)18896 23686 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)21877 23852 y Fn(Q)22784 23407 y Fs(^)22630 23686 y(\011)23491 23852 y Fn(j)24265 23686 y Fs(=)25480 23394 y(^)25434 23686 y Fq(\025)26080 23852 y Fn(j)26546 23686 y Fq(P)27257 23852 y Fn(Q)28163 23407 y Fs(^)28009 23686 y(\011)28870 23852 y Fn(j)29582 23686 y Fs(+)246 b Fq(O)31 b Fs(\()p Fq(\025)32640 23213 y Fk(\000)p Fo(9)32640 23982 y(0)33829 23686 y Fs(\))p Fq(:)8838 b Fs(\(155\))6075 25539 y(The)369 b(last)h(relation)h(means)f(that)7040 27392 y(there)f(is)g(an)g(eigen) -31 b(v)-61 b(alue)371 b Fq(\026)18478 26990 y Fn(Q)18478 27666 y(n)19599 27392 y Fs(of)f Fq(P)21571 27558 y Fn(Q)22323 27392 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)25304 27558 y Fn(Q)26426 27392 y Fs(suc)-31 b(h)369 b(that)h Fl(j)31674 27100 y Fs(^)31628 27392 y Fq(\025)32274 27558 y Fn(j)32741 27392 y Fs(\()p Fq(t)p Fs(\))247 b Fl(\000)f Fq(\026)36022 26935 y Fn(Q)36022 27666 y(n)36774 27392 y Fs(\()p Fq(t)p Fs(\))p Fl(j)309 b Fq(<)e(c\025)40943 26918 y Fk(\000)p Fo(9)40943 27687 y(0)42132 27392 y Fq(:)965 b Fs(\(156\))6075 29245 y Fx(5.)492 b Fs(W)-92 b(e)368 b(pro)-31 b(v)g(e)370 b(that)18469 30792 y Fq(P)19180 30958 y Fn(Q)19933 30792 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)22914 30958 y Fn(Q)23974 30792 y Fs(=)25956 29741 y Fp(X)25142 32167 y Fn(q)28 b Fk(2)p Fo(\000\()p Fn(\025)27591 31900 y Fj(")27591 32407 y Fm(0)28023 32167 y Fo(\))28553 30792 y Fq(P)29264 30958 y Fn(q)29753 30792 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)32734 30958 y Fn(q)33223 30792 y Fq(:)9874 b Fs(\(157\))6075 33689 y(It)369 b(is)g(enough)h(to)g(sho)-31 b(w)370 b(that)18675 35542 y Fq(P)19386 35708 y Fn(q)19875 35542 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)22856 35708 y Fn(q)23290 35487 y Fh(0)23952 35542 y Fs(=)307 b(0)p Fq(;)924 b Fs(when)738 b Fq(q)348 b Fl(6)p Fs(=)307 b Fq(q)32707 35085 y Fk(0)33017 35542 y Fq(:)10080 b Fs(\(158\))6075 37395 y(Indeed,)455 b(supp)31 b(ose)437 b(\(158\))j(is)e(not)h(true.) 699 b(Then,)456 b(there)437 b(is)h Fq(s)422 b Fl(2)g Fs(\012)33719 37561 y Fn(q)34646 37395 y Fs(and)439 b Fq(r)453 b Fl(2)422 b Fs(\012)39779 37561 y Fn(q)40213 37339 y Fh(0)40568 37395 y Fs(,)455 b(suc)-31 b(h)438 b(that)6075 38893 y Fq(H)6995 39059 y Fn(sr)8210 38893 y Fl(6)p Fs(=)308 b(0,)317 b(here)302 b Fq(H)13811 39059 y Fn(r)24 b(s)15023 38893 y Fs(is)303 b(a)g(matrix)i(elemen)-31 b(t)305 b(of)e Fq(H)90 b Fs(\()p Fq(t)p Fs(\))304 b(in)f(the)h(basis)f (of)h(exp)31 b(onen)-31 b(ts.)471 b(Since)43808 38613 y Fq(~)43601 38893 y(A)q Fs(\()p Fq(x)p Fs(\))6075 40221 y(is)360 b(a)g(trigonometric)j(p)31 b(olynomial,)366 b Fq(H)21917 40387 y Fn(r)24 b(s)23185 40221 y Fs(can)360 b(di\013er)g(from)g(zero)g(only)h(if)g Fl(j)p Fq(r)260 b Fl(\000)227 b Fq(s)p Fl(j)308 b Fq(<)g(R)41721 40387 y Fo(0)42217 40221 y Fs(.)489 b(In)360 b(the)6075 41765 y(same)332 b(time)g(from)g(the)g(de\014nition)h(of)f(\012)22646 41931 y Fn(q)23466 41765 y Fs(and)g(\012)26380 41931 y Fn(q)26814 41709 y Fh(0)27500 41765 y Fs(it)g(follo)-31 b(ws)334 b(that)f Fl(j)p Fq(r)202 b Fl(\000)170 b Fq(s)p Fl(j)308 b Fq(>)f(c\025)39993 41190 y Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(")39993 42060 y Fo(0)42936 41765 y Fs(,)339 b(since)6075 43165 y(the)d(angle)g(b)31 b(et)-31 b(w)g(een)346 b Fq(~)-562 b(p)15602 43331 y Fn(q)16091 43165 y Fs(\(0\))336 b(and)345 b Fq(~)-562 b(p)20516 43331 y Fn(q)20950 43110 y Fh(0)21304 43165 y Fs(\(0\))337 b(is)f(greater)g(than)g Fq(c\025)31532 42692 y Fk(\000)p Fo(2)p Fn(")31532 43460 y Fo(0)33138 43165 y Fs(.)482 b(The)335 b(con)-31 b(tradiction)339 b(pro)-31 b(v)g(es)6075 44493 y(\(158\),)394 b(and,)e(hence,)f(\(157\).)548 b(Recall)388 b(that)g(the)f(pro)61 b(jections)389 b Fq(P)32676 44659 y Fn(q)33164 44493 y Fs(,)j Fq(q)377 b Fl(2)337 b Fs(\000\()p Fq(\025)37577 44092 y Fn(")37577 44768 y Fo(0)38074 44493 y Fs(\))387 b(are)f(orthogonal)6075 45822 y(to)353 b(eac)-31 b(h)353 b(other.)487 b(Therefore,)357 b(the)c(sp)31 b(ectrum)352 b(of)h Fq(P)27181 45988 y Fn(Q)27933 45822 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)30914 45988 y Fn(Q)32019 45822 y Fs(is)353 b(the)f(union)i(of)f(sp)31 b(ectra)352 b(of)h(the)6075 47150 y(op)31 b(erators)314 b Fq(P)11632 47316 y Fn(q)12121 47150 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)15102 47316 y Fn(q)15591 47150 y Fs(,)326 b Fq(q)347 b Fl(2)308 b Fs(\000\()p Fq(\025)19879 46748 y Fn(")19879 47425 y Fo(0)20376 47150 y Fs(\).)474 b(Com)-31 b(bining)318 b(the)c(last)i(statemen)-31 b(t)316 b(with)g(\(156\),)328 b(w)-31 b(e)315 b(\014nish)6075 48478 y(the)369 b(pro)31 b(of.)493 b Fl(})7403 49807 y Fs(Next,)565 b(w)-31 b(e)526 b(in)-31 b(v)g(estigate)527 b(eigen)-31 b(v)-61 b(alues)527 b(of)e(the)g(op)31 b(erators)525 b Fq(P)33190 49973 y Fn(q)33679 49807 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)36660 49973 y Fn(q)37149 49807 y Fs(,)564 b Fq(q)607 b Fl(2)567 b Fs(\000\()p Fq(\025)42194 49405 y Fn(")42194 50082 y Fo(0)42691 49807 y Fs(\).)960 b(W)-92 b(e)6075 51135 y(en)-31 b(umerate)409 b(the)f(eigen)-31 b(v)-61 b(alues)409 b Fq(\026)19874 50733 y Fn(q)19874 51409 y(n)20476 51135 y Fs(\()p Fq(t)p Fs(\))g(of)f(an)g(op)31 b(erator)409 b Fq(P)30235 51301 y Fn(q)30723 51135 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)33704 51301 y Fn(q)34601 51135 y Fs(in)408 b(the)g(increasing)h(order,)6075 52535 y Fq(n)308 b Fs(=)f(1)p Fq(;)184 b(:::;)g(N)11560 52701 y Fn(q)12053 52535 y Fs(.)493 b(It)369 b(is)g(easy)g(to)h(see)f(from)h(the)f (de\014nition)i(of)f Fq(P)32158 52701 y Fn(q)33015 52535 y Fs(and)g(\012)35967 52701 y Fn(q)36825 52535 y Fs(that)g Fq(N)40112 52701 y Fn(q)40909 52535 y Fq(<)308 b(c\025)43203 52062 y Fo(1)p Fk(\000)p Fo(3)p Fn(")43203 52831 y Fo(0)45250 52535 y Fs(.)6075 54562 y Fx(Lemma)823 b(10.)544 b Fi(Eigenvalues)386 b Fq(\026)19855 54160 y Fn(q)19855 54836 y(n)20458 54562 y Fs(\()p Fq(t)p Fs(\))g Fi(of)h(an)f(op)-57 b(er)g(ator)385 b Fq(P)30027 54728 y Fn(q)30515 54562 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)33496 54728 y Fn(q)34372 54562 y Fi(have)385 b(the)h(fol)57 b(lowing)386 b(pr)-57 b(op-)6075 55890 y(erties)396 b(as)f(functions)i(of)g Fq(t)p Fi(:)7403 57219 y(1\))373 b(Eigenvalues)h Fq(\026)15480 56817 y Fn(q)15480 57492 y(n)16083 57219 y Fs(\()p Fq(t)p Fs(\))g Fi(ar)-57 b(e)374 b(di\013er)-57 b(entiable)375 b(with)e(r)-57 b(esp)g(e)g(ct)374 b(to)f Fq(t)g Fi(for)h(almost)e(al)57 b(l)373 b Fq(t)307 b Fl(2)h Fq(K)452 b Fi(and)21553 59287 y Fl(jr)22782 59453 y Fn(t)23173 59287 y Fq(\026)23840 58830 y Fn(q)23840 59560 y(n)24442 59287 y Fs(\()p Fq(t)p Fs(\))p Fl(j)309 b Fq(<)e(c)27965 59453 y Fn(b)28410 59287 y Fq(\025)29056 58712 y Fo(1)p Fn(=)p Fo(2)29056 59582 y(0)43404 59287 y Fs(\(159\))6075 61139 y Fi(at)395 b(p)-57 b(oints)397 b(of)f(di\013er)-57 b(entiability.)7403 62468 y(2\))371 b(Eigenvalues)h Fq(\026)15476 62066 y Fn(q)15476 62741 y(n)16078 62468 y Fs(\()p Fq(t)p Fs(\))g Fi(ar)-57 b(e)371 b(monotonous)g(functions)h(of)g Fq(t)f Fi(in)h(a)f(dir)-57 b(e)g(ction)358 b Fq(~)-539 b(u)40044 62634 y Fn(q)40904 62468 y Fi(ortho)-57 b(gonal)6075 63796 y(to)535 b Fq(~)-562 b(p)8092 63962 y Fn(q)8581 63796 y Fs(\(0\))528 b Fi(and)f(their)g(derivative)h Fs(\()p Fl(r)22140 63962 y Fn(t)22531 63796 y Fq(\026)23198 63394 y Fn(q)23198 64070 y(n)23800 63796 y Fs(\()p Fq(t)p Fs(\))p Fq(;)170 b(~)-539 b(u)26185 63962 y Fn(q)26675 63796 y Fs(\))528 b Fi(in)g(this)e(dir)-57 b(e)g(ction)529 b(satisfy)e(the)f(fol)57 b(lowing)6075 65124 y(estimate:)20623 66512 y Fl(j)q Fs(\()p Fl(r)22283 66678 y Fn(t)22673 66512 y Fq(\026)23340 66055 y Fn(q)23340 66786 y(n)23943 66512 y Fs(\()p Fq(t)p Fs(\))p Fq(;)170 b(~)-539 b(u)26328 66678 y Fn(q)26818 66512 y Fs(\))q Fl(j)308 b Fq(>)f(\025)29678 65937 y Fo(1)p Fn(=)p Fo(2)29678 66807 y(0)31069 66512 y Fq(:)12028 b Fs(\(160\))7403 68181 y Fi(3\))391 b(The)h(numb)-57 b(er)392 b(of)g(eigenvalues)h Fq(\026)22772 67779 y Fn(q)22772 68454 y(n)23374 68181 y Fs(\()p Fq(t)p Fs(\))g Fi(satisfying)f(the)g (ine)-57 b(quality)393 b Fl(j)p Fq(\026)37755 67779 y Fn(q)37755 68454 y(n)38358 68181 y Fs(\()p Fq(t)p Fs(\))237 b Fl(\000)f Fq(\025)p Fl(j)308 b Fq(<)f(c)43860 67320 y Fl(p)p 44783 67320 1143 45 v 44783 68181 a Fq(\025)45429 68347 y Fo(0)6075 69541 y Fi(for)396 b(a)g(given)h Fq(t)307 b Fl(2)h Fq(K)475 b Fi(and)396 b(a)g(p)-57 b(ositive)397 b(c)-57 b(onstant)396 b Fq(c)g Fi(do)-57 b(es)396 b(not)f(exc)-57 b(e)g(e)g(d)398 b Fq(C)35931 69717 y Fn(~)35765 69913 y(A)36486 69541 y Fq(\025)37132 69067 y Fo(1)p Fk(\000)p Fo(3)p Fn(")37132 69836 y Fo(0)39180 69541 y Fi(.)p eop end %%Page: 28 28 TeXDict begin 28 27 bop 6075 3345 a Fo(28)14514 b(YULIA)300 b(KARPESHINA)7403 6001 y Fs(Pro)31 b(of.)p 7403 6178 2971 45 v 337 w(1\))337 b(The)f(\014rst)f(statemen)-31 b(t)338 b(can)e(b)31 b(e)335 b(pro)-31 b(v)g(ed)337 b(b)-31 b(y)336 b(standard)g(considerations.)483 b(Indeed,)6075 7330 y(if)369 b Fq(\026)7756 6928 y Fn(q)7756 7603 y(n)8359 7330 y Fs(\()p Fq(t)p Fs(\))h(is)f(a)g(simple)h(eigen)-31 b(v)-61 b(alue)371 b(than)f(it)g(is)f(di\013eren)-31 b(tiable)371 b(with)f(resp)31 b(ect)369 b(to)h Fq(t)e Fs([Kato)r(])i(and)16857 9335 y Fl(r)17779 9501 y Fn(t)18169 9335 y Fq(\026)18836 8878 y Fn(q)18836 9609 y(n)19439 9335 y Fs(\()p Fq(t)p Fs(\))308 b(=)p Fq(<)f Fl(r)23958 9501 y Fn(t)24533 9335 y Fs(\()p Fq(P)25674 9501 y Fn(q)26163 9335 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)29144 9501 y Fn(q)29633 9335 y Fs(\))185 b(\011)31109 9501 y Fn(n)31712 9335 y Fq(;)f Fs(\011)33064 9501 y Fn(n)33975 9335 y Fq(>;)8261 b Fs(\(161\))6075 11340 y(\011)6936 11506 y Fn(n)7538 11340 y Fs(\()p Fq(x)p Fs(\))344 b(b)31 b(eing)344 b(a)g(normalized)h(eigenfunction)g(of)f Fq(P)27579 11506 y Fn(q)28068 11340 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)31049 11506 y Fn(q)31881 11340 y Fs(corresp)31 b(onding)344 b(to)g Fq(\026)40965 10938 y Fn(q)40965 11614 y(n)41568 11340 y Fs(\()p Fq(t)p Fs(\).)484 b(Note)6075 12838 y(that)304 b Fq(P)9118 13004 y Fn(q)9910 12838 y Fs(do)31 b(es)302 b(not)i(dep)31 b(end)303 b(on)g Fq(t)g Fs(and)g Fl(r)p Fq(H)90 b Fs(\()p Fq(t)p Fs(\))309 b(=)27123 12558 y Fq(~)26950 12838 y(D)31 b Fs(\()p Fq(t)p Fs(\))114 b(+)g(2)-69 b Fq(~)-484 b(a)q Fs(,)317 b(where)35317 12558 y Fq(~)35144 12838 y(D)31 b Fs(\()p Fq(t)p Fs(\))308 b(=)f Fl(r)p Fq(H)40669 13004 y Fo(0)41167 12838 y Fs(\()p Fq(t)p Fs(\))c(is)g(a)h(di-)6075 14336 y(agonal)339 b(v)-31 b(ector-matrix:)17061 14056 y Fq(~)16887 14336 y(D)32 b Fs(\()p Fq(t)p Fs(\))19095 14502 y Fn(mj)20655 14336 y Fs(=)307 b Fq(\016)22315 14502 y Fn(mj)23567 14336 y Fl(r)24489 14502 y Fn(t)24879 14336 y Fq(p)25436 13935 y Fo(2)25436 14624 y Fn(j)25933 14336 y Fs(\()p Fq(t)p Fs(\))338 b(and)268 b Fq(~)-484 b(a)338 b Fs(is)f(the)g(op)31 b(erator)338 b(of)g(m)-31 b(ultiplication)6075 15716 y(b)g(y)300 b Fq(~)-484 b(a)p Fs(\()p Fq(x)p Fs(\).)493 b(Hence,)17630 17386 y Fl(r)18552 17552 y Fn(t)18942 17386 y Fq(\026)19609 16929 y Fn(q)19609 17660 y(n)20212 17386 y Fs(\()p Fq(t)p Fs(\))308 b(=)p Fq(<)23982 17106 y(~)23809 17386 y(D)31 b Fs(\()p Fq(t)p Fs(\)\011)26877 17552 y Fn(n)27481 17386 y Fq(;)184 b Fs(\011)28833 17552 y Fn(n)29744 17386 y Fq(>)307 b Fs(+)p Fq(O)31 b Fs(\(1\))p Fq(:)6075 19288 y Fs(It)402 b(is)g(easy)h(to)g(sho)-31 b(w)403 b(that)g Fl(k)18241 19009 y Fq(~)18068 19288 y(D)32 b Fs(\()p Fq(t)p Fs(\)\011)21137 19454 y Fn(n)21740 19288 y Fl(k)22293 18887 y Fo(2)23153 19288 y Fs(=)363 b Fq(c)24856 19454 y Fn(b)25663 19288 y Fq(<)f(H)27806 19454 y Fo(0)28303 19288 y Fs(\011)29164 19454 y Fn(n)29767 19288 y Fq(;)184 b Fs(\011)31119 19454 y Fn(n)32085 19288 y Fq(>)p Fs(.)592 b(Elemen)-31 b(tary)404 b(p)31 b(erturbation)6075 20708 y(considerations)417 b(yield)g Fq(<)386 b(H)18221 20874 y Fo(0)18717 20708 y Fs(\011)19578 20874 y Fn(n)20181 20708 y Fq(;)184 b Fs(\011)21533 20874 y Fn(n)22522 20708 y Fq(>)416 b(<)801 b Fs(2)p Fq(\025)26660 20874 y Fo(0)27573 20708 y Fs(and,)428 b(therefore,)g Fl(jr)36335 20874 y Fn(t)36726 20708 y Fq(\026)37393 20306 y Fn(q)37393 20981 y(n)37996 20708 y Fs(\()p Fq(t)p Fs(\))p Fl(j)386 b Fq(<)g(c)41675 20874 y Fn(b)42119 19764 y Fl(p)p 43042 19764 646 45 v 43042 20708 a Fq(\025)43688 20883 y Fo(0)44600 20708 y Fs(for)6075 22036 y Fq(\025)6721 22202 y Fo(0)7586 22036 y Fs(large)370 b(enough.)7403 23365 y(2\))316 b(Note)h(that)f(a)g (set)f(\012)17010 23531 y Fn(q)17815 23365 y Fs(consists)h(of)g(t)-31 b(w)g(o)317 b(parts)e(\012)28666 22963 y Fo(+)28666 23638 y Fn(q)29717 23365 y Fs(and)h(\012)32615 22963 y Fk(\000)32615 23638 y Fn(q)33677 23365 y Fs(\(Fig.4\).)478 b(Corresp)31 b(ondingly)-92 b(,)6075 24781 y(w)-31 b(e)419 b(can)g(represen)-31 b(t)419 b Fq(P)15373 24947 y Fn(q)16280 24781 y Fs(as)g Fq(P)18400 24947 y Fn(q)19279 24781 y Fs(=)390 b Fq(P)21395 24379 y Fo(+)21241 25055 y Fn(q)22409 24781 y Fs(+)279 b Fq(P)24414 24379 y Fk(\000)24260 25055 y Fn(q)25161 24781 y Fs(,)432 b(here)418 b Fq(P)29216 24379 y Fo(+)29062 25055 y Fn(q)30369 24781 y Fs(and)h Fq(P)33436 24379 y Fk(\000)33282 25055 y Fn(q)34602 24781 y Fs(are)f(de\014ned)h (similarly)i(to)6075 26197 y Fq(P)6786 26363 y Fn(q)7676 26197 y Fs(up)401 b(to)i(the)e(replacemen)-31 b(t)404 b(of)e(\012)20940 26363 y Fn(q)21831 26197 y Fs(b)-31 b(y)401 b(\012)24199 25796 y Fo(+)24199 26471 y Fn(q)25336 26197 y Fs(and)h(\012)28320 25796 y Fk(\000)28320 26471 y Fn(q)29068 26197 y Fs(,)410 b(resp)31 b(ectiv)-31 b(ely)-92 b(.)591 b(Ob)-31 b(viously)403 b Fq(P)42357 25796 y Fo(+)42203 26471 y Fn(q)43091 26197 y Fq(P)43956 25796 y Fk(\000)43802 26471 y Fn(q)45064 26197 y Fs(=)6075 27614 y(0.)998 b(T)-92 b(aking)540 b(in)-31 b(to)539 b(accoun)-31 b(t)539 b(that)g(the)f (distance)g(b)31 b(et)-31 b(w)g(een)539 b(\012)32826 27212 y Fo(+)32826 27887 y Fn(q)34099 27614 y Fs(and)f(\012)37219 27212 y Fk(\000)37219 27887 y Fn(q)38504 27614 y Fs(is)f(greater)h (than)6075 29305 y Fq(c)6554 29471 y Fn(b)6998 29305 y Fq(\025)7644 28730 y Fo(1)p Fn(=)p Fo(2)7644 29600 y(0)9473 29305 y Fs(and)439 b(considering)g(that)20228 29025 y Fq(~)20021 29305 y(A)p Fs(\()p Fq(x)p Fs(\))g(is)f(a)h (trigonometric)h(p)31 b(olynomial,)459 b(w)-31 b(e)439 b(easily)g(obtain)6075 30633 y Fq(P)6940 30232 y Fo(+)6786 30907 y Fn(q)7674 30633 y Fq(H)90 b(P)9549 30232 y Fk(\000)9395 30907 y Fn(q)10786 30633 y Fs(=)490 b(0;)535 b(and,)507 b(therefore,)g Fq(P)21934 30799 y Fn(q)22422 30633 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)25403 30799 y Fn(q)26383 30633 y Fs(=)490 b Fq(P)28599 30232 y Fo(+)28445 30907 y Fn(q)29334 30633 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)32469 30232 y Fo(+)32315 30907 y Fn(q)33523 30633 y Fs(+)319 b Fq(P)35568 30232 y Fk(\000)35414 30907 y Fn(q)36315 30633 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)39450 30232 y Fk(\000)39296 30907 y Fn(q)40197 30633 y Fs(.)822 b(Since)479 b Fq(P)45191 30232 y Fo(+)45037 30907 y Fn(q)6075 32050 y Fs(and)457 b Fq(P)9180 31648 y Fk(\000)9026 32323 y Fn(q)10383 32050 y Fs(are)g(pro)61 b(jections)459 b(on)-31 b(to)458 b(orthogonal)i(subspaces,)478 b(the)457 b(sp)31 b(ectrum)457 b(of)g Fq(P)40765 32216 y Fn(q)41254 32050 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)44235 32216 y Fn(q)45181 32050 y Fs(is)6075 33466 y(the)387 b(union)h(of)g(the)f(sp) 31 b(ectra)387 b(of)h Fq(P)20310 33064 y Fo(+)20156 33740 y Fn(q)21044 33466 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)24179 33064 y Fo(+)24025 33740 y Fn(q)25301 33466 y Fs(and)388 b Fq(P)28337 33064 y Fk(\000)28183 33740 y Fn(q)29084 33466 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)32219 33064 y Fk(\000)32065 33740 y Fn(q)32966 33466 y Fs(.)547 b(Next,)393 b(w)-31 b(e)387 b(sho)-31 b(w)388 b(that)h(b)31 b(oth)6075 34794 y(of)484 b(these)f(op)31 b(erators)484 b(are)f(monotonous)j(with)e(resp)31 b(ect)483 b(to)h Fq(t)f Fs(in)h(the)g(direction)g(orthogonal)6075 36123 y(to)465 b Fq(~)-562 b(p)8071 36289 y Fn(q)8560 36123 y Fs(\(0\).)752 b(In)455 b(fact,)479 b(let)434 b Fq(~)-531 b(\027)17333 36289 y Fn(q)18277 36123 y Fs(b)31 b(e)455 b(the)h(unit)g(v)-31 b(ector)456 b(in)g(the)g(direction)g(of) 465 b Fq(~)-562 b(p)37672 36289 y Fn(q)38161 36123 y Fs(\(0\),)479 b(and)441 b Fq(~)-538 b(u)43233 36289 y Fn(q)43721 36123 y Fl(?)-22 b Fq(~)-531 b(\027)45129 36289 y Fn(q)45618 36123 y Fs(,)6075 37451 y Fl(j)-14 b Fq(~)-539 b(u)7016 37617 y Fn(q)7505 37451 y Fl(j)456 b Fs(=)g(1.)760 b(An)-31 b(y)468 b Fq(~)-562 b(p)14219 37617 y Fn(i)14587 37451 y Fs(\(0\),)483 b Fq(i)456 b Fl(2)g Fs(\012)19620 37617 y Fn(q)20567 37451 y Fs(admits)k(the)e (represen)-31 b(tation)468 b Fq(~)-561 b(p)34171 37617 y Fn(i)34539 37451 y Fs(\(0\))457 b(=)f Fq(\030)38210 37617 y Fn(i)38557 37451 y Fq(~)-531 b(\027)39126 37617 y Fn(q)39920 37451 y Fs(+)306 b Fq(\021)41637 37617 y Fn(i)41990 37451 y Fq(~)-538 b(u)42639 37617 y Fn(q)43127 37451 y Fq(;)459 b Fs(here)6075 39035 y Fq(\030)6559 39201 y Fn(i)7284 39035 y Fs(=)356 b(\()9 b Fq(~)-562 b(p)9488 39201 y Fn(i)9857 39035 y Fs(\(0\))p Fq(;)162 b(~)-531 b(\027)12308 39201 y Fn(q)12799 39035 y Fs(\),)406 b Fl(j)p Fq(\030)14733 39201 y Fn(i)15103 39035 y Fl(j)356 b Fq(<)h(c\025)18109 38460 y Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(3)p Fn(")18109 39330 y Fo(0)21051 39035 y Fs(,)406 b(since)398 b Fq(i)357 b Fl(2)f Fs(\012)27136 39201 y Fn(q)27625 39035 y Fs(,)406 b(and)399 b Fl(j)p Fq(\021)31377 39201 y Fn(i)31746 39035 y Fl(j)32053 38633 y Fo(2)32906 39035 y Fq(>)356 b(\025)34769 39201 y Fo(0)35266 39035 y Fq(=)p Fs(2)266 b(+)f Fq(o)p Fs(\()p Fq(\025)39377 39201 y Fo(0)39874 39035 y Fs(\).)580 b(Note)400 b(that)6075 40618 y(the)368 b(sign)h(of)g Fq(\021)12071 40784 y Fn(i)12808 40618 y Fs(is)f(di\013eren)-31 b(t)369 b(in)g(the)f(sets)g(\012)24448 40217 y Fo(+)24448 40892 y Fn(q)25552 40618 y Fs(and)h(\012)28503 40217 y Fk(\000)28503 40892 y Fn(q)29250 40618 y Fs(,)g(namely)-92 b(,)370 b Fq(\021)34534 40784 y Fn(i)35210 40618 y Fq(>)36875 40182 y Fo(1)p 36511 40363 1169 45 v 36511 40483 a Fk(p)p 37239 40483 442 38 v 37239 41125 a Fo(2)37813 40618 y Fq(\025)38459 40043 y Fo(1)p Fn(=)p Fo(2)38459 40914 y(0)40094 40618 y Fs(+)245 b Fq(o)p Fs(\()p Fq(\025)42813 40043 y Fo(1)p Fn(=)p Fo(2)42813 40914 y(0)44204 40618 y Fs(\))369 b(in)6075 42487 y(one)417 b(of)h(them)g(\(w)-31 b(e)419 b(denote)f(it)g(b)-31 b(y)417 b(\012)21601 42086 y Fo(+)21601 42761 y Fn(q)22337 42487 y Fs(\))g(and)h Fq(\021)25935 42653 y Fn(i)26691 42487 y Fq(<)388 b Fl(\000)29297 42052 y Fo(1)p 28934 42233 1169 45 v 28934 42352 a Fk(p)p 29661 42352 442 38 v 642 x Fo(2)30235 42487 y Fq(\025)30881 41912 y Fo(1)p Fn(=)p Fo(2)30881 42783 y(0)32550 42487 y Fs(+)278 b Fq(o)p Fs(\()p Fq(\025)35302 41912 y Fo(1)p Fn(=)p Fo(2)35302 42783 y(0)36693 42487 y Fs(\))418 b(in)g(\012)39680 42086 y Fk(\000)39680 42761 y Fn(q)40427 42487 y Fs(.)637 b(Similarly)-92 b(,)6075 43991 y(w)-31 b(e)436 b(write)g(a)f (quasimomen)-31 b(tum)439 b Fq(t)c Fs(in)g(the)h(form:)625 b Fq(t)418 b Fs(=)f Fq(\034)103 b(~)-531 b(\027)30628 44157 y Fn(q)31407 43991 y Fs(+)290 b Fq(\020)67 b(~)-538 b(u)33758 44157 y Fn(q)34247 43991 y Fq(;)435 b(\034)64 b(;)184 b(\020)501 b Fl(2)417 b Fq(R)9 b Fs(.)690 b(Let)436 b(us)e(sho)-31 b(w)6075 45319 y(that)382 b(the)e(op)31 b(erator)381 b Fq(P)15743 44918 y Fo(+)15589 45593 y Fn(q)16478 45319 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)19613 44918 y Fo(+)19459 45593 y Fn(q)20728 45319 y Fs(dep)31 b(ends)380 b(on)h Fq(\020)462 b Fs(in)381 b(a)g(monotonous)i(w)-31 b(a)g(y)-92 b(.)528 b(In)380 b(fact,)385 b(for)380 b(an)-31 b(y)6075 46736 y Fq(f)426 b Fl(2)308 b Fq(P)8954 46334 y Fo(+)8800 47010 y Fn(q)9688 46736 y Fq(L)10441 46902 y Fo(2)10938 46736 y Fs(\()p Fq(Q)p Fs(\),)12511 48716 y Fq(@)p 12228 49211 1216 45 v 12228 50225 a(@)61 b(\020)13884 49465 y(<)307 b(P)15917 49008 y Fo(+)15763 49739 y Fn(q)16651 49465 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)19786 49008 y Fo(+)19632 49739 y Fn(q)20522 49465 y Fq(f)58 b(;)184 b(f)427 b(>)p Fs(=)p Fq(<)307 b(P)26337 49008 y Fo(+)26183 49739 y Fn(q)27256 48237 y Fp(\020)28090 49185 y Fq(~)27917 49465 y(D)31 b Fs(\()p Fq(t)p Fs(\))247 b(+)177 b Fq(~)-484 b(a;)170 b(~)-539 b(u)33188 49631 y Fn(q)33678 48237 y Fp(\021)34523 49465 y Fq(P)35388 49008 y Fo(+)35234 49739 y Fn(q)36123 49465 y Fq(f)58 b(;)184 b(f)427 b(>)p Fs(=)17716 52712 y Fq(<)308 b(P)19750 52255 y Fo(+)19596 52986 y Fn(q)20669 51484 y Fp(\020)21503 52432 y Fq(~)21330 52712 y(D)31 b Fs(\()p Fq(t)p Fs(\))p Fq(;)170 b(~)-539 b(u)24662 52878 y Fn(q)25152 51484 y Fp(\021)25998 52712 y Fq(P)26863 52255 y Fo(+)26709 52986 y Fn(q)27597 52712 y Fq(f)58 b(;)184 b(f)427 b(>)308 b Fs(+)p Fq(O)31 b Fs(\(1\))p Fq(:)9122 b Fs(\(162\))6075 55058 y(Ob)-31 b(viously)-92 b(,)324 b Fq(P)12311 54656 y Fo(+)12157 55331 y Fn(q)13045 55058 y Fs(\()13648 54778 y Fq(~)13475 55058 y(D)32 b Fs(\()p Fq(t)p Fs(\))p Fq(;)170 b(~)-539 b(u)16808 55224 y Fn(q)17298 55058 y Fs(\))p Fq(P)18593 54656 y Fo(+)18439 55331 y Fn(q)19638 55058 y Fs(is)310 b(a)h(diagonal)i(op)31 b(erator)311 b(in)g(the)g(basis)f(of)h(exp)31 b(onen)-31 b(ts)311 b(\(9\))h(and)6075 56386 y(its)369 b(diagonal)j(elemen)-31 b(ts)370 b(are)f(giv)-31 b(en)371 b(b)-31 b(y)369 b(the)h(form)-31 b(ula:)9042 57433 y Fp(\020)9703 58662 y Fq(P)10568 58205 y Fo(+)10414 58935 y Fn(q)11487 57433 y Fp(\020)12321 58382 y Fq(~)12148 58662 y(D)31 b Fs(\()p Fq(t)p Fs(\))p Fq(;)170 b(~)-539 b(u)15480 58828 y Fn(q)15970 57433 y Fp(\021)16816 58662 y Fq(P)17681 58205 y Fo(+)17527 58935 y Fn(q)18415 57433 y Fp(\021)19076 59437 y Fn(ii)20065 58662 y Fs(=)308 b(2\()p Fq(\021)22767 58828 y Fn(i)23382 58662 y Fs(+)246 b Fq(\020)82 b Fs(\))308 b Fq(>)26961 57692 y Fl(p)p 27884 57692 554 45 v 27884 58662 a Fs(2)p Fq(\025)29083 58087 y Fo(1)p Fn(=)p Fo(2)29083 58957 y(0)30720 58662 y Fs(+)246 b Fq(o)p Fs(\()p Fq(\025)33440 58087 y Fo(1)p Fn(=)p Fo(2)33440 58957 y(0)34831 58662 y Fs(\))p Fq(;)924 b Fs(when)369 b Fq(i)308 b Fl(2)g Fs(\012)41916 58205 y Fo(+)41916 58935 y Fn(q)42651 58662 y Fq(:)6075 61054 y Fs(Substituting)371 b(the)f(last)g(estimate)h(in)-31 b(to)370 b(\(162\),)i(w)-31 b(e)370 b(obtain:)18401 62887 y Fq(@)p 18118 63382 1216 45 v 18118 64396 a(@)61 b(\020)19774 63636 y(<)307 b(P)21807 63179 y Fo(+)21653 63910 y Fn(q)22542 63636 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)25677 63179 y Fo(+)25523 63910 y Fn(q)26412 63636 y Fq(f)58 b(;)184 b(f)427 b(>)738 b(>)1045 b(\025)32623 63061 y Fo(1)p Fn(=)p Fo(2)32623 63931 y(0)43404 63636 y Fs(\(163\))6075 66280 y(for)369 b(an)-31 b(y)370 b Fq(f)427 b Fl(2)307 b Fq(P)12739 65878 y Fo(+)12585 66553 y Fn(q)13473 66280 y Fq(L)14226 66446 y Fo(2)14723 66280 y Fs(\()p Fq(Q)p Fs(\),)371 b Fl(k)p Fq(f)119 b Fl(k)18903 66480 y Fn(L)19510 66591 y Fm(2)19943 66480 y Fo(\()p Fn(Q)p Fo(\))21695 66280 y Fs(=)307 b(1.)493 b(Similarly)-92 b(,)17959 68260 y Fq(@)p 17676 68755 V 17676 69768 a(@)61 b(\020)19331 69009 y(<)308 b(P)21365 68552 y Fk(\000)21211 69283 y Fn(q)22111 69009 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)25246 68552 y Fk(\000)25092 69283 y Fn(q)25994 69009 y Fq(f)58 b(;)184 b(f)427 b(>)738 b(<)1045 b Fl(\000)p Fq(\025)33066 68434 y Fo(1)p Fn(=)p Fo(2)33066 69304 y(0)43404 69009 y Fs(\(164\))p eop end %%Page: 29 29 TeXDict begin 29 28 bop 7289 3345 a Fo(SPECTRAL)388 b(PR)-24 b(OPER)-71 b(TIES)386 b(OF)h(THE)h(PERIODIC)f(MA)-24 b(GNETIC)389 b(SCHR)35152 3149 y(\177)35033 3345 y(ODINGER)e(OPERA)-71 b(TOR)331 b(29)6075 6001 y Fs(for)501 b(an)-31 b(y)502 b Fq(f)646 b Fl(2)527 b Fq(P)13442 5600 y Fk(\000)13288 6275 y Fn(q)14189 6001 y Fq(L)14942 6167 y Fo(2)15439 6001 y Fs(\()p Fq(Q)p Fs(\),)535 b Fl(k)p Fq(f)119 b Fl(k)19783 6201 y Fn(L)20390 6312 y Fm(2)20824 6201 y Fo(\()p Fn(Q)p Fo(\))22795 6001 y Fs(=)527 b(1.)889 b(Let)501 b Fq(\026)28714 5600 y Fn(q)28 b Fo(+)28714 6275 y Fn(n)29883 6001 y Fs(\()p Fq(t)p Fs(\))501 b(and)g Fq(\026)34595 5600 y Fn(q)28 b Fk(\000)34595 6275 y Fn(n)35776 6001 y Fs(\()p Fq(t)p Fs(\))502 b(b)31 b(e)500 b(eigen)-31 b(v)-61 b(alues)503 b(of)6075 7418 y Fq(P)6940 7016 y Fo(+)6786 7691 y Fn(q)7674 7418 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)10809 7016 y Fo(+)10655 7691 y Fn(q)11928 7418 y Fs(and)384 b Fq(P)14960 7016 y Fk(\000)14806 7691 y Fn(q)15707 7418 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)18842 7016 y Fk(\000)18688 7691 y Fn(q)19589 7418 y 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Fs(\()p Fq(t)p Fs(\))39406 66275 y Fp(\014)39406 66940 y(\014)39406 67604 y(\014)40083 67548 y Fq(>)308 b Fs(4)p Fq(\025)42451 67091 y Fk(\000)p Fn(\016)43628 65988 y Fp(\033)44643 67548 y Fq(:)43404 69541 y Fs(\(166\))p eop end %%Page: 30 30 TeXDict begin 30 29 bop 6075 3345 a Fo(30)14514 b(YULIA)300 b(KARPESHINA)7403 6001 y Fs(T)-92 b(o)395 b(understand)f(prop)31 b(erties)394 b(of)g Fq(\037)22085 6167 y Fk(\003)22594 6001 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\))395 b(w)-31 b(e)395 b(represen)-31 b(t)393 b(it)i(as)f(a)h(di\013erence)e(of)i(t) -31 b(w)g(o)396 b(sets:)6075 7330 y Fq(\037)6768 7496 y Fk(\003)7276 7330 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\))309 b(=)f Fq(\037)11978 7496 y Fo(1)12474 7330 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\))23 b Fl(n)g Fq(\037)16297 7496 y Fo(2)16796 7330 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\).)457 b(The)258 b(de\014nition)h(of)f Fq(\037)28905 7496 y Fo(1)29402 7330 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\))259 b(includes)f(only)h(mo)31 b(del)258 b(n)-31 b(um)g(b)31 b(ers)6121 8486 y(^)6075 8778 y Fq(\025)6721 8944 y Fn(m)7561 8778 y Fs(\()p Fq(t)p Fs(\))370 b(corresp)31 b(onding)369 b(to)h Fq(m)308 b Fl(2)f Fq(J)106 b Fs(\()p Fq(\025)21780 8944 y Fo(0)22277 8778 y Fs(\):)6494 11504 y Fq(\037)7187 11670 y Fo(1)7991 11504 y Fs(=)9159 9943 y Fp(\032)9990 11504 y Fq(t)307 b Fl(2)g Fq(K)387 b Fs(:)308 b Fl(9)p Fq(j)371 b Fl(2)307 b Fq(J)106 b Fs(\()p Fq(\025)17967 11670 y Fo(0)18464 11504 y Fs(\))308 b(:)g Fl(<)20662 11212 y Fs(^)20616 11504 y Fq(\025)21262 11670 y Fn(j)21729 11504 y Fs(\()p Fq(t)p Fs(\))g(=)g Fq(\025;)2128 b Fs(min)25603 12269 y Fn(m)p Fk(2)p Fn(J)73 b Fo(\()p Fn(\025)28420 12380 y Fm(0)28851 12269 y Fo(\))p Fn(;m)p Fk(6)p Fo(=)p Fn(j)31521 10231 y Fp(\014)31521 10895 y(\014)31521 11559 y(\014)31890 11504 y Fl(<)32735 11212 y Fs(^)32689 11504 y Fq(\025)33335 11670 y Fn(j)33801 11504 y Fs(\()p Fq(t)p Fs(\))247 b Fl(\000)f(<)37260 11212 y Fs(^)37214 11504 y Fq(\025)37860 11670 y Fn(m)38701 11504 y Fs(\()p Fq(t)p 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Fp(\014)32224 17080 y(\014)32224 17744 y(\014)32901 17689 y Fl(\024)308 b Fs(4)p Fq(\025)35269 17232 y Fk(\000)p Fn(\016)36447 16128 y Fp(\033)37461 17689 y Fq(:)5636 b Fs(\(168\))6075 20664 y(Note)437 b(that)g(in)g(the)f(last)h(form)-31 b(ula)438 b Fl(<)21752 20371 y Fs(^)21706 20664 y Fq(\025)22352 20830 y Fn(m)23193 20664 y Fs(\()p Fq(t)p Fs(\))420 b(=)26200 20371 y(^)26154 20664 y Fq(\025)26800 20830 y Fn(m)27640 20664 y Fs(\()p Fq(t)p Fs(\),)454 b(since)32486 20371 y(^)32440 20664 y Fq(\025)33086 20830 y Fn(m)33927 20664 y Fs(\()p Fq(t)p Fs(\),)g Fq(m)419 b Fl(2)g Fq(Z)39331 20262 y Fo(2)40118 20664 y Fl(n)291 b Fq(J)106 b Fs(\()p Fq(\025)42758 20830 y Fo(0)43255 20664 y Fs(\),)454 b(are)6075 21992 y(eigen)-31 b(v)-61 b(alues)510 b(of)f(a)g(self-adjoin)-31 b(t)512 b(op)31 b(erator.)912 b(W)-92 b(e)508 b(start)h(with)h(in)-31 b(v)g(estigating)513 b(prop)31 b(erties)509 b(of)6075 23320 y Fq(\037)6768 23486 y Fo(1)7264 23320 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\).)475 b(Some)313 b(prop)31 b(erties)311 b(of)i Fq(\037)20520 23486 y Fk(\003)21028 23320 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\))314 b(will)f(follo)-31 b(w)315 b(immediately)h(from)c(those)g(of)h Fq(\037)42589 23486 y Fo(1)43085 23320 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\),)6075 24649 y(since)369 b Fq(\037)9480 24815 y Fk(\003)9988 24649 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\))309 b Fl(\032)f Fq(\037)14690 24815 y Fo(1)15186 24649 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\).)7403 25977 y(First,)431 b(w)-31 b(e)419 b(sho)-31 b(w)419 b(that)h(the)f(inequalities)i(de\014ning)e Fq(\037)30138 26143 y Fo(1)30634 25977 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\))420 b(are)e(preserv)-31 b(ed)418 b(in)h(its)g(small)6075 27305 y(neigh)-31 b(b)31 b(orho)g(o)g(d)497 b(\(Lemma)h(11\).)876 b(Second,)529 b(w)-31 b(e)497 b(describ)31 b(e)495 b(geometric)k (structure)d(of)h Fq(\037)42897 27471 y Fo(1)43393 27305 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\))6075 28634 y(\(Lemmas)451 b(12)h(and)e(13\).)738 b(Third,)471 b(w)-31 b(e)451 b(pro)-31 b(v)g(e)451 b(that)h Fq(\037)28845 28800 y Fo(1)29341 28634 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\))452 b(has)e(a)h(length)h (asymptotically)6075 30053 y(close)369 b(to)h(2)p Fq(\031)11302 29110 y Fl(p)p 12225 29110 646 45 v 12225 30053 a Fq(\025)f Fs(\(Lemma)h(14\).)6075 32241 y Fx(Lemma)833 b(11.)548 b Fi(Supp)-57 b(ose)390 b Fq(t)g Fi(b)-57 b(elongs)391 b(to)e(the)h Fs(\()p Fq(\025)26188 31839 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)29835 32241 y Fs(\))p Fi(-neighb)-57 b(orho)g(o)g(d)392 b(of)e Fq(\037)39221 32407 y Fo(1)39717 32241 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\))p Fi(.)509 b(Then,)6075 33569 y(ther)-57 b(e)396 b(is)g(a)g(unique)h Fq(j)371 b Fl(2)307 b Fq(J)106 b Fs(\()p Fq(\025)18215 33735 y Fo(0)18712 33569 y Fs(\))397 b Fi(such)f(that)20806 35751 y Fl(j<)21958 35459 y Fs(^)21912 35751 y Fq(\025)22558 35917 y Fn(j)23025 35751 y Fs(\()p Fq(t)p Fs(\))247 b Fl(\000)f Fq(\025)p Fl(j)308 b Fq(<)f Fs(4)p Fq(\025)29267 35294 y Fk(\000)p Fo(2)p Fn(\016)30886 35751 y Fq(;)12211 b Fs(\(169\))6075 37812 y Fi(and,)396 b(mor)-57 b(e)g(over,)19730 39467 y Fl(j<)20882 39175 y Fs(^)20836 39467 y Fq(\025)21482 39633 y Fn(m)22324 39467 y Fs(\()p Fq(t)p Fs(\))246 b Fl(\000)g(<)25782 39175 y Fs(^)25736 39467 y Fq(\025)26382 39633 y Fn(j)26849 39467 y Fs(\()p Fq(t)p Fs(\))p Fl(j)309 b Fq(>)e Fs(2)p Fq(\025)31092 39010 y Fk(\000)p Fn(\016)43404 39467 y Fs(\(170\))6075 41296 y Fi(for)396 b(al)57 b(l)395 b Fq(m)307 b Fl(2)h Fq(J)106 b Fs(\()p Fq(\025)13548 41462 y Fo(0)14045 41296 y Fs(\))p Fi(,)397 b Fq(m)307 b Fl(6)p Fs(=)h Fq(j)63 b Fi(.)7403 43484 y Fs(Pro)31 b(of.)p 7403 43661 2971 45 v 342 w(Let)341 b Fq(t)f Fs(b)31 b(elong)341 b(to)h(the)f(\()p Fq(\025)21195 43083 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)24842 43484 y Fs(\)-neigh)-31 b(b)31 b(orho)g(o)g(d)342 b(of)f Fq(\037)34490 43650 y Fo(1)34986 43484 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\).)485 b(Then,)347 b(there)340 b(is)h(a)6075 44855 y Fq(t)6475 45021 y Fo(0)7283 44855 y Fl(2)313 b Fq(\037)9027 45021 y Fo(1)9523 44855 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\))373 b(suc)-31 b(h)372 b(that)i Fl(j)p Fq(t)248 b Fl(\000)g Fq(t)19793 45021 y Fo(0)20289 44855 y Fl(j)313 b Fq(<)f(\025)22728 44453 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)26375 44855 y Fs(.)501 b(By)372 b(the)h(de\014nition)g(of)g Fq(\037)37712 45021 y Fo(1)38208 44855 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\),)375 b(there)d(is)g(a) 6075 46303 y(single)e Fq(j)g Fl(2)308 b Fq(J)106 b Fs(\()p Fq(\025)12824 46469 y Fo(0)13321 46303 y Fs(\))369 b(suc)-31 b(h)369 b(that)i Fl(<)19861 46011 y Fs(^)19815 46303 y Fq(\025)20461 46469 y Fn(j)20927 46303 y Fs(\()p Fq(t)21757 46469 y Fo(0)22254 46303 y Fs(\))308 b(=)g Fq(\025)368 b Fs(and,)i(moreo)-31 b(v)g(er,)19080 48524 y Fl(j<)20232 48232 y Fs(^)20186 48524 y Fq(\025)20832 48690 y Fn(m)21673 48524 y Fs(\()p Fq(t)22503 48690 y Fo(0)23000 48524 y Fs(\))247 b Fl(\000)f(<)25629 48232 y Fs(^)25583 48524 y Fq(\025)26229 48690 y Fn(j)26695 48524 y Fs(\()p Fq(t)27525 48690 y Fo(0)28022 48524 y Fs(\))p Fl(j)308 b Fq(>)g Fs(4)p Fq(\025)31435 48067 y Fk(\000)p Fn(\016)32613 48524 y Fq(;)10484 b Fs(\(171\))6075 50768 y(for)422 b(all)h Fq(m)395 b Fl(2)g Fq(J)106 b Fs(\()p Fq(\025)13708 50934 y Fo(0)14205 50768 y Fs(\),)435 b Fq(m)396 b Fl(6)p Fs(=)f Fq(j)63 b Fs(.)650 b(Using)422 b(\(51\))i(and)e(\(53\))h(for) 31316 50476 y Fq(~)31372 50768 y(k)431 b Fs(=)403 b Fq(~)-561 b(p)34192 50934 y Fn(j)34658 50768 y Fs(\()p Fq(t)p Fs(\))422 b(and)g(considering)h(the)6075 52468 y(notation)372 b(\(69\),)f(w)-31 b(e)370 b(obtain)18221 51195 y Fp(\014)18221 51859 y(\014)18221 52524 y(\014)18590 52468 y Fl(r)19558 52176 y Fs(^)19512 52468 y Fq(\025)20158 52634 y Fn(j)20624 51195 y Fp(\014)20624 51859 y(\014)20624 52524 y(\014)21301 52468 y Fq(<)307 b Fs(2)23022 51525 y Fl(p)p 23945 51525 646 45 v 23945 52468 a Fq(\025)369 b Fs(and,)h(hence,)19295 55093 y Fl(j<)20447 54800 y Fs(^)20401 55093 y Fq(\025)21047 55259 y Fn(j)21514 55093 y Fs(\()p Fq(t)p Fs(\))246 b Fl(\000)g(<)24972 54800 y Fs(^)24926 55093 y Fq(\025)25572 55259 y Fn(j)26039 55093 y Fs(\()p Fq(t)26869 55259 y Fo(0)27366 55093 y Fs(\))p Fl(j)308 b Fq(<)g Fs(4)p Fq(\025)30779 54635 y Fk(\000)p Fo(2)p Fn(\016)32398 55093 y Fq(:)10699 b Fs(\(172\))6075 57314 y(T)-92 b(aking)367 b(in)-31 b(to)368 b(accoun)-31 b(t)367 b(that)g Fl(<)19329 57022 y Fs(^)19283 57314 y Fq(\025)19929 57480 y Fn(j)20396 57314 y Fs(\()p Fq(t)21226 57480 y Fo(0)21722 57314 y Fs(\))308 b(=)g Fq(\025)p Fs(,)366 b(w)-31 b(e)367 b(get)f(\(169\).)494 b(Let)366 b(us)f(pro)-31 b(v)g(e)367 b(that)g(\(170\))h(holds)6075 58642 y(for)h(all)h Fq(m)308 b Fl(6)p Fs(=)f Fq(j)63 b Fs(.)493 b(Supp)31 b(ose)369 b Fq(m)307 b Fl(2)h Fq(J)106 b Fs(\()p Fq(\025)21536 58808 y Fo(0)22033 58642 y Fs(\),)370 b Fq(m)307 b Fl(6)p Fs(=)h Fq(j)63 b Fs(.)492 b(Using)370 b(again)h(\(51\))g(and\(53\))g(yields:)18920 60824 y Fl(j<)20072 60532 y Fs(^)20026 60824 y Fq(\025)20672 60990 y Fn(m)21514 60824 y Fs(\()p Fq(t)p Fs(\))246 b Fl(\000)g(<)24972 60532 y Fs(^)24926 60824 y Fq(\025)25572 60990 y Fn(m)26413 60824 y Fs(\()p Fq(t)27243 60990 y Fo(0)27740 60824 y Fs(\))p Fl(j)309 b Fq(<)e Fs(4)p Fq(\025)31153 60367 y Fk(\000)p Fo(2)p Fn(\016)32772 60824 y Fq(:)10325 b Fs(\(173\))6075 62885 y(Com)-31 b(bining)249 b(the)d(last)h (inequalit)-31 b(y)249 b(with)f(\(171\))g(and)f(\(172\),)273 b(w)-31 b(e)247 b(get)g(\(170\))h(for)e Fq(m)308 b Fl(2)f Fq(J)106 b Fs(\()p Fq(\025)42331 63051 y Fo(0)42828 62885 y Fs(\))p Fq(;)432 b(m)307 b Fl(6)p Fs(=)6075 64214 y Fq(j)63 b Fs(.)492 b Fl(})7403 65542 y Fs(Next,)356 b(w)-31 b(e)351 b(in)-31 b(v)g(estigate)353 b(geometric)f(prop)31 b(erties)351 b(of)g Fq(\037)29657 65708 y Fo(1)30153 65542 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\).)488 b(W)-92 b(e)349 b(start)i(with)h(considering)6075 66870 y(a)369 b(set)g Fq(D)9641 67036 y Fn(r)10132 66870 y Fs(\()p Fq(\025;)184 b(\014)58 b Fs(\),)372 b(whic)-31 b(h)370 b(is)f(the)g(set)g Fq(D)31 b Fs(\()p Fq(\025;)184 b(\014)58 b Fs(\))372 b(\(see)d(\(58\)\))i(shifted)f(in)g Fq(R)36139 66469 y Fo(2)37003 66870 y Fs(b)-31 b(y)378 b Fq(~)-561 b(p)39098 67036 y Fn(r)39588 66870 y Fs(\(0\),)371 b Fq(r)338 b Fl(6)p Fs(=)308 b(0:)16762 69264 y Fq(D)17678 69430 y Fn(r)18168 69264 y Fs(\()p Fq(\025;)184 b(\014)58 b Fs(\))310 b(=)22328 68035 y Fp(n)23009 68972 y Fq(~)23066 69264 y(k)342 b Fs(:)24543 68972 y Fq(~)24599 69264 y(k)281 b Fl(\000)255 b Fq(~)-562 b(p)27120 69430 y Fn(r)27610 69264 y Fs(\(0\))309 b Fl(2)f Fq(D)31 b Fs(\()p Fq(\025;)184 b(\014)58 b Fs(\))34006 68035 y Fp(o)34931 69264 y Fq(:)p eop end %%Page: 31 31 TeXDict begin 31 30 bop 7289 3345 a Fo(SPECTRAL)388 b(PR)-24 b(OPER)-71 b(TIES)386 b(OF)h(THE)h(PERIODIC)f(MA)-24 b(GNETIC)389 b(SCHR)35152 3149 y(\177)35033 3345 y(ODINGER)e(OPERA)-71 b(TOR)331 b(31)6075 6001 y Fs(Ob)-31 b(viously)-92 b(,)476 b(the)454 b(p)31 b(oin)-31 b(ts)455 b(of)f(in)-31 b(tersections)455 b(of)f Fq(D)31 b Fs(\()p Fq(\025;)184 b(\014)58 b Fs(\))456 b(and)e Fq(D)33393 6167 y Fn(r)33884 6001 y Fs(\()p Fq(\025;)184 b(\014)58 b Fs(\),)477 b Fq(r)j Fl(2)448 b Fq(Z)40349 5600 y Fo(2)41148 6001 y Fl(n)303 b(f)p Fs(0)p Fl(g)p Fs(,)476 b(are)6075 7330 y(describ)31 b(ed)368 b(b)-31 b(y)369 b(the)h(equations:)19804 6901 y Fo(4)8237 9394 y Fl(<)9082 9101 y Fs(^)9036 9394 y Fq(\025)q Fs(\()10057 9101 y Fq(~)10113 9394 y(k)35 b Fs(\))308 b(=)g Fq(\025;)922 b Fl(<)15351 9101 y Fs(^)15305 9394 y Fq(\025)16136 8165 y Fp(\020)16741 9101 y Fq(~)16797 9394 y(k)281 b Fl(\000)255 b Fq(~)-562 b(p)19318 9560 y Fn(r)19808 9394 y Fs(\(0\))21221 8165 y Fp(\021)22191 9394 y Fs(=)308 b Fq(\025;)25548 9101 y(~)25604 9394 y(k)36 b(;)27020 9101 y(~)27077 9394 y(k)281 b Fl(\000)254 b Fq(~)-561 b(p)29598 9560 y Fn(r)30088 9394 y Fs(\(0\))309 b Fl(2)e Fs(\005\()p Fq(\014)58 b Fs(\))248 b Fl(\\)e Fq(M)37535 9560 y Fo(1)38031 9394 y Fs(\()p Fq(\025)39107 9560 y Fo(0)39604 9394 y Fs(\))p Fq(;)555 b(r)338 b Fl(6)p Fs(=)308 b(0)p Fq(;)6075 11638 y(M)7149 11804 y Fo(1)7645 11638 y Fs(\()p Fq(\025)8721 11804 y Fo(0)9218 11638 y Fs(\))375 b(b)31 b(eing)374 b(the)h(cen)-31 b(tered)374 b(at)h(the)g(origin)h(ring)e(with)i (radiuses)34949 11202 y Fo(1)p 34949 11383 442 45 v 34949 12019 a(2)35523 10777 y Fl(p)p 36445 10777 1143 45 v 861 x Fq(\025)37091 11804 y Fo(0)37962 11638 y Fs(and)e(2)40672 10777 y Fl(p)p 41595 10777 V 41595 11638 a Fq(\025)42241 11804 y Fo(0)42738 11638 y Fs(.)507 b(If)375 b(w)-31 b(e)6075 12966 y(replace)369 b(the)h(second)f(equation)i(in)e(the)h(ab) 31 b(o)-31 b(v)g(e)370 b(form)-31 b(ula)371 b(b)-31 b(y)369 b(the)h(inequalit)-31 b(y)18863 13757 y Fp(\014)18863 14421 y(\014)18863 15085 y(\014)19232 15030 y Fl(<)20077 14738 y Fs(^)20031 15030 y Fq(\025)20861 13801 y Fp(\020)21466 14738 y Fq(~)21523 15030 y(k)281 b Fl(\000)254 b Fq(~)-561 b(p)24044 15196 y Fn(r)24534 15030 y Fs(\(0\))25947 13801 y Fp(\021)26855 15030 y Fl(\000)246 b Fq(\025)28608 13757 y Fp(\014)28608 14421 y(\014)28608 15085 y(\014)29284 15030 y Fq(<)308 b 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b Fs(\()p Fq(\025;)184 b(\014)58 b(;)184 b(\016)42 b Fs(\))310 b(=)7220 24045 y Fp(\()8055 25646 y Fq(~)8112 25938 y(k)342 b Fl(2)308 b Fq(D)31 b Fs(\()p Fq(\025;)184 b(\014)58 b Fs(\))310 b(:)5433 b(min)14629 26884 y Fn(r)24 b Fo(:)15273 26680 y Fn(~)15315 26884 y(k)17 b Fk(\000)c Fn(~)-467 b(p)16951 26995 y Fj(r)17398 26884 y Fo(\(0\))p Fk(2)p Fo(\005\()p Fn(\014)40 b Fo(\))p Fk(\\)p Fn(M)22456 26995 y Fm(1)22887 26884 y Fo(\()p Fn(\025)23758 26995 y Fm(0)24190 26884 y Fo(\))p Fn(;)369 b(r)24 b Fk(6)p Fo(=0)26908 24665 y Fp(\014)26908 25330 y(\014)26908 25994 y(\014)27277 25938 y Fl(<)28122 25646 y Fs(^)28076 25938 y Fq(\025)28907 24710 y Fp(\020)29511 25646 y Fq(~)29568 25938 y(k)281 b Fl(\000)255 b Fq(~)-562 b(p)32089 26104 y Fn(r)32579 25938 y Fs(\(0\))33992 24710 y Fp(\021)34900 25938 y Fl(\000)246 b Fq(\025)36653 24665 y Fp(\014)36653 25330 y(\014)36653 25994 y(\014)37330 25938 y Fl(\025)307 b Fs(4)p Fq(\025)39697 25481 y Fk(\000)p Fn(\016)40875 24045 y Fp(\))41951 25938 y Fq(:)1146 b Fs(\(174\))6075 28714 y(Next,)370 b(w)-31 b(e)370 b(pro)-31 b(v)g(e)370 b(that)g Fq(\037)16795 28880 y Fo(1)17291 28714 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\))371 b(is)e(just)23754 28434 y(^)23495 28714 y Fq(D)31 b Fs(\()p Fq(\025;)184 b(\014)58 b(;)184 b(\016)42 b Fs(\))372 b(pac)-31 b(k)g(ed)371 b(in)-31 b(to)370 b(the)g(bag)g Fq(K)79 b Fs(.)6075 30736 y Fx(Lemma)940 b(12.)578 b Fi(The)432 b(fol)57 b(lowing)433 b(r)-57 b(elation)433 b(holds:)582 b Fl(K)28910 30456 y Fs(^)28651 30736 y Fq(D)406 b Fs(=)373 b Fq(\037)31900 30902 y Fo(1)32397 30736 y Fq(:)433 b Fi(The)g(p)-57 b(ar)g(al)57 b(lel)432 b(shift)h Fl(K)449 b Fi(estab-)6075 32110 y(lishes)395 b(one-to-one)i(c)-57 b(orr)g(esp)g(ondenc)g(e)397 b(b)-57 b(etwe)g(en)26503 31830 y Fs(^)26245 32110 y Fq(D)427 b Fi(and)396 b Fq(\037)30431 32276 y Fo(1)30927 32110 y Fq(:)6075 34071 y Fx(Corollary)443 b(10.)554 b Fi(The)396 b(length)g(of)g Fq(\037)21446 34237 y Fo(1)22338 34071 y Fi(is)h(e)-57 b(qual)396 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Fs(\()p Fq(t)p Fs(\))465 b Fl(2)f Fq(D)31 b Fs(,)487 b Fq(D)494 b Fs(b)31 b(eing)464 b(de\014ned)f(b)-31 b(y)463 b(\(58\).)776 b(Let)464 b(us)6075 43165 y(pro)-31 b(v)g(e)401 b(that)411 b Fq(~)-561 b(p)12080 43331 y Fn(j)12546 43165 y Fs(\()p Fq(t)p Fs(\))361 b Fl(2)15524 42885 y Fs(^)15266 43165 y Fq(D)31 b Fs(.)588 b(Supp)31 b(ose)401 b(it)h(is)f(not)g(so.)589 b(Then)401 b(there)g(is)g Fq(r)391 b Fl(2)361 b Fq(Z)37406 42763 y Fo(2)38170 43165 y Fl(n)267 b(f)p Fs(0)p Fl(g)p Fs(,)411 b(suc)-31 b(h)401 b(that)6084 44865 y Fq(~)-562 b(p)6632 45031 y Fn(j)7098 44865 y Fs(\()p Fq(t)p Fs(\))294 b Fl(\000)302 b Fq(~)-562 b(p)10363 45031 y Fn(r)10853 44865 y Fs(\(0\))426 b Fl(2)f Fs(\005\()p Fq(\014)58 b Fs(\))295 b Fl(\\)e Fq(M)18629 45031 y Fo(1)19126 44865 y Fs(\()p Fq(\025)20202 45031 y Fo(0)20699 44865 y Fs(\))440 b(and)23792 43592 y Fp(\014)23792 44256 y(\014)23792 44920 y(\014)24161 44865 y Fl(<)25006 44573 y Fs(^)24960 44865 y Fq(\025)184 b Fs(\()10 b Fq(~)-562 b(p)26778 45031 y Fn(j)27244 44865 y Fs(\()p Fq(t)p Fs(\)\))247 b Fl(\000)f(<)31133 44573 y Fs(^)31087 44865 y Fq(\025)185 b Fs(\()9 b Fq(~)-562 b(p)32905 45031 y Fn(j)33372 44865 y Fs(\()p Fq(t)p Fs(\))246 b Fl(\000)255 b Fq(~)-562 b(p)36542 45031 y Fn(r)37033 44865 y Fs(\(0\)\))38877 43592 y Fp(\014)38877 44256 y(\014)38877 44920 y(\014)39672 44865 y Fq(<)425 b Fs(4)p Fq(k)42122 44463 y Fk(\000)p Fn(\016)43300 44865 y Fs(.)704 b(Let)6075 46968 y Fq(m)493 b Fs(=)g Fq(j)384 b Fl(\000)320 b Fq(r)31 b Fs(.)827 b(Then,)518 b Fq(~)-562 b(p)16473 47134 y Fn(j)16939 46968 y Fs(\()p Fq(t)p Fs(\))321 b Fl(\000)329 b Fq(~)-561 b(p)20259 47134 y Fn(r)20749 46968 y Fs(\(0\))494 b(=)502 b Fq(~)-561 b(p)24568 47134 y Fn(m)25408 46968 y Fs(\()p Fq(t)p Fs(\))481 b(and)29413 45695 y Fp(\014)29413 46360 y(\014)29413 47024 y(\014)29782 46968 y Fl(<)30627 46676 y Fs(^)30581 46968 y Fq(\025)185 b Fs(\()9 b Fq(~)-562 b(p)32399 47134 y Fn(j)32866 46968 y Fs(\()p Fq(t)p Fs(\)\))247 b Fl(\000)f(<)36755 46676 y Fs(^)36709 46968 y Fq(\025)185 b Fs(\()9 b Fq(~)-562 b(p)38527 47134 y Fn(m)39368 46968 y Fs(\()p Fq(t)p Fs(\)\))41059 45695 y Fp(\014)41059 46360 y(\014)41059 47024 y(\014)41921 46968 y Fq(<)494 b Fs(4)p Fq(k)44440 46567 y Fk(\000)p Fn(\016)45618 46968 y Fs(.)6075 48629 y(Since)411 b Fq(~)-562 b(p)9555 48795 y Fn(m)10396 48629 y Fs(\()p Fq(t)p Fs(\))362 b Fl(2)g Fs(\005\()p Fq(\014)58 b Fs(\))269 b Fl(\\)f Fq(M)17841 48795 y Fo(1)18338 48629 y Fs(\()p Fq(\025)19414 48795 y Fo(0)19910 48629 y Fs(\),)411 b(w)-31 b(e)403 b(ha)-31 b(v)g(e)402 b Fq(m)362 b Fl(2)g Fq(J)106 b Fs(\()p Fq(\025)29535 48795 y Fo(0)30032 48629 y Fs(\).)591 b(Recalling)404 b(again)f(notation)i(\(69\),)6075 50077 y(w)-31 b(e)374 b(get)g Fl(j<)10710 49785 y Fs(^)10664 50077 y Fq(\025)11310 50243 y Fn(j)11777 50077 y Fs(\()p Fq(t)p Fs(\))250 b Fl(\000)f(<)15242 49785 y Fs(^)15196 50077 y Fq(\025)15842 50243 y Fn(m)16683 50077 y Fs(\()p Fq(t)p Fs(\))p Fl(j)316 b Fq(<)f Fs(4)p Fq(k)20906 49675 y Fk(\000)p Fn(\016)22084 50077 y Fs(.)506 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Fq(m)395 b Fl(6)p Fs(=)g Fq(j)63 b Fs(.)651 b(Supp)31 b(ose)421 b(the)h(last)h(inequalit)-31 b(y)425 b(do)31 b(es)421 b(not)i(hold)f(for)g(some)g Fq(m)p Fs(.)651 b(Since)422 b Fq(m)395 b Fl(2)g Fq(J)106 b Fs(\()p Fq(\025)44690 57516 y Fo(0)45187 57350 y Fs(\),)6121 58507 y(^)6075 58799 y Fq(\025)6721 58965 y Fn(m)7561 58799 y Fs(\()p Fq(t)p Fs(\))312 b(=)10350 58507 y(^)10304 58799 y Fq(\025)p Fs(\()9 b Fq(~)-562 b(p)11937 58965 y Fn(m)12778 58799 y Fs(\()p Fq(t)p Fs(\)\),)382 b Fq(~)-562 b(p)15705 58965 y Fn(m)16546 58799 y Fs(\()p Fq(t)p Fs(\))312 b Fl(2)e Fs(\005\()p Fq(\014)58 b Fs(\))249 b Fl(\\)f Fq(M)23849 58965 y Fo(1)24345 58799 y Fs(\()p Fq(\025)25421 58965 y Fo(0)25918 58799 y Fs(\).)499 b(Th)-31 b(us,)372 b Fl(j<)31420 58507 y Fs(^)31374 58799 y Fq(\025)p Fs(\()9 b Fq(~)-562 b(p)33007 58965 y Fn(m)33849 58799 y Fs(\()p Fq(t)p Fs(\)\))248 b Fl(\000)f(<)37740 58507 y Fs(^)37694 58799 y Fq(\025)q Fs(\()9 b Fq(~)-562 b(p)39328 58965 y Fn(j)39795 58799 y Fs(\()p Fq(t)p Fs(\)\))p Fl(j)312 b(\024)f Fs(4)p Fq(k)44440 58397 y Fk(\000)p Fn(\016)45618 58799 y Fs(.)6075 60309 y(De\014ning)406 b Fq(r)436 b Fs(b)-31 b(y)406 b(the)g(relation)i Fq(m)368 b Fs(=)g Fq(j)334 b Fl(\000)270 b Fq(r)31 b Fs(,)415 b(w)-31 b(e)406 b(obtain)i Fl(j<)31116 60017 y Fs(^)31070 60309 y Fq(\025)p Fs(\()32090 60017 y Fq(~)32146 60309 y(k)306 b Fl(\000)279 b Fq(~)-561 b(p)34717 60475 y Fn(r)35207 60309 y Fs(\(0\)\))272 b Fl(\000)e(<)39298 60017 y Fs(^)39252 60309 y Fq(\025)p Fs(\()40272 60017 y Fq(~)40328 60309 y(k)36 b Fs(\))p Fl(j)369 b(\024)g Fs(4)p Fq(k)44440 59907 y Fk(\000)p Fn(\016)45618 60309 y Fs(.)6075 61780 y(This)g(con)-31 b(tradicts)371 b(to)f(the)f(assumption)i(that)25756 61487 y Fq(~)25812 61780 y(k)343 b Fl(2)28035 61500 y Fs(^)27777 61780 y Fq(D)31 b Fs(.)492 b(Therefore,)370 b Fl(K)35967 61500 y Fs(^)35708 61780 y Fq(D)339 b Fl(\032)308 b Fq(\037)38825 61946 y Fo(1)39321 61780 y Fs(.)7403 63154 y(Com)-31 b(bining)372 b(t)-31 b(w)g(o)371 b(parts)e(of)h(the)f(pro)31 b(of)370 b(w)-31 b(e)369 b(obtain)i Fl(K)30211 62874 y Fs(^)29952 63154 y Fq(D)340 b Fs(=)307 b Fq(\037)33069 63320 y Fo(1)33565 63154 y Fs(.)7403 64528 y(W)-92 b(e)524 b(establish)h(one-to-one)h(corresp)31 b(ondence)523 b(b)31 b(et)-31 b(w)g(een)32323 64249 y(^)32065 64528 y Fq(D)555 b Fs(and)525 b Fq(\037)36537 64694 y Fo(1)37557 64528 y Fs(b)-31 b(y)525 b(con)-31 b(tradiction.)6075 65903 y(Supp)31 b(ose)369 b(for)h(some)g Fq(t)308 b Fl(2)h Fq(\037)17334 66069 y Fo(1)18199 65903 y Fs(there)370 b(is)g(a)g(pair)g Fq(j)r(;)184 b(m)309 b Fs(:)317 b Fq(~)-561 b(p)28751 66069 y Fn(j)29217 65903 y Fs(\()p Fq(t)p Fs(\))p Fq(;)195 b(~)-564 b(p)31525 66069 y Fn(m)32367 65903 y Fs(\()p Fq(t)p Fs(\))310 b Fl(2)35241 65623 y Fs(^)34983 65903 y Fq(D)31 b Fs(,)370 b Fq(j)i Fl(6)p Fs(=)308 b Fq(m)p Fs(.)495 b(This)370 b(means)p 6075 66995 6642 45 v 7403 67997 a Fo(4)7844 68340 y Fw(Note)356 b(that)g(for)13899 68116 y Ff(~)13720 68340 y(A)330 b Fw(=)h(0,)367 b(the)355 b(equations)h(ha)-26 b(v)g(e)356 b(the)f(form)g(of)h(v)-26 b(on)355 b(Laue)g(di\013raction)h(conditions)h Fg(j)43579 68107 y Ff(~)43626 68340 y(k)24 b Fg(j)44401 68028 y Fy(2)45193 68340 y Fw(=)6075 69541 y Fg(j)6289 69307 y Ff(~)6336 69541 y(k)233 b Fg(\000)221 b Ff(~)-482 b(p)8474 69652 y Fd(r)8935 69541 y Fw(\(0\))p Fg(j)10398 69228 y Fy(2)11122 69541 y Fw(=)261 b Ff(\025)p Fw(.)p eop end %%Page: 32 32 TeXDict begin 32 31 bop 6075 3345 a Fo(32)14514 b(YULIA)300 b(KARPESHINA)6075 6001 y Fs(that)492 b Fl(<)9440 5709 y Fs(^)9394 6001 y Fq(\025)10040 6167 y Fn(j)10507 6001 y Fs(\()p Fq(t)p Fs(\))511 b(=)g Fl(<)14495 5709 y Fs(^)14449 6001 y Fq(\025)15095 6167 y Fn(m)15936 6001 y Fs(\()p Fq(t)p Fs(\))g(=)f Fq(\025)p Fs(.)858 b(This)492 b(is)f(in)g(con)-31 b(tradiction)494 b(with)e(the)g(inequalit)-31 b(y)494 b(in)d(the)6075 7330 y(de\014nition)371 b(of)e Fq(\037)12948 7496 y Fo(1)13444 7330 y Fs(.)493 b Fl(})7403 8704 y Fs(Next,)323 b(w)-31 b(e)309 b(establish)h(that)h(there)e(is)g (one-to-one)i(corresp)31 b(ondence)308 b(not)i(only)g(b)31 b(et)-31 b(w)g(een)311 b Fl(K)45236 8424 y Fs(^)44978 8704 y Fq(D)6075 10033 y Fs(and)369 b Fq(\037)8920 10199 y Fo(1)9785 10033 y Fs(but)h(also)g(b)31 b(et)-31 b(w)g(een)370 b(their)f(\()p Fq(\025)22032 9631 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)25679 10033 y Fs(\))p Fl(\000)p Fs(neigh)-31 b(b)31 b(orho)g(o)g(ds.)6075 12076 y Fx(Lemma)1023 b(13.)600 b Fi(The)467 b(p)-57 b(ar)g(al)57 b(lel)465 b(shift)i Fl(K)482 b Fi(establishes)466 b(one-to-one)g(c)-57 b(orr)g(esp)g(ondenc)g(e)468 b(b)-57 b(etwe)g(en)6075 13451 y(the)369 b Fs(\()p Fq(\025)8963 13049 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)12610 13451 y Fs(\))p Fl(\000)p Fi(neighb)-57 b(orho)g(o)g(d)372 b(of)21986 13171 y Fs(^)21728 13451 y Fq(D)401 b Fi(in)371 b Fq(R)25227 13049 y Fo(2)26092 13451 y Fi(and)f(the)g Fs(\()p Fq(\025)31105 13049 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)34752 13451 y Fs(\))p Fi(-neighb)-57 b(orho)g(o)g(d)371 b(of)f Fq(\037)44097 13617 y Fo(1)44963 13451 y Fi(in)6075 14779 y Fq(K)79 b Fi(.)6075 16822 y Fx(Corollary)533 b(11.)602 b Fi(The)469 b(ar)-57 b(e)g(a)469 b(of)g(the)f Fs(\()p Fq(\025)23234 16421 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)26881 16822 y Fs(\))p Fl(\000)p Fi(neighb)-57 b(orho)g(o)g(d)471 b(of)36455 16543 y Fs(^)36197 16822 y Fq(D)500 b Fi(in)469 b Fq(R)39893 16421 y Fo(2)40858 16822 y Fi(is)f(e)-57 b(qual)470 b(to)6075 18193 y(the)395 b(ar)-57 b(e)g(a)397 b(of)f(the)g Fs(\()p Fq(\025)14520 17791 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)18167 18193 y Fs(\))p Fi(-neighb)-57 b(orho)g(o)g(d)398 b(of)e Fq(\037)27565 18359 y Fo(1)28457 18193 y Fi(in)h Fq(K)79 b Fi(.)7403 20378 y Fs(Pro)31 b(of.)p 7403 20555 2971 45 v 434 w(Supp)g(ose)15155 20086 y Fq(~)15212 20378 y(k)468 b Fs(is)433 b(in)g(the)g(\()p Fq(\025)21834 19977 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)25481 20378 y Fs(\))p Fl(\000)p Fs(neigh)-31 b(b)31 b(orho)g(o)g(d)435 b(of)35371 20099 y(^)35113 20378 y Fq(D)464 b Fs(in)433 b Fq(R)38698 19977 y Fo(2)39194 20378 y Fs(.)684 b(Then,)449 b(there)6075 21849 y(is)474 b(a)8263 21557 y Fq(~)8320 21849 y(k)8896 22015 y Fo(0)9875 21849 y Fl(2)11353 21569 y Fs(^)11095 21849 y Fq(D)31 b Fs(,)500 b(suc)-31 b(h)474 b(that)h Fl(j)18205 21557 y Fq(~)18261 21849 y(k)351 b Fl(\000)20309 21557 y Fq(~)20365 21849 y(k)20941 22015 y Fo(0)21438 21849 y Fl(j)482 b Fq(<)g(\025)24216 21447 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)27863 21849 y Fs(.)806 b(This)475 b(means)f(that)h Fq(t)482 b Fs(=)g Fl(K)40633 21557 y Fq(~)40689 21849 y(k)509 b Fs(is)474 b(in)g(the)6075 23319 y(\()p Fq(\025)7151 22917 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)10798 23319 y Fs(\))p Fl(\000)p Fs(neigh)-31 b(b)31 b(orho)g(o)g(d)475 b(of)f Fq(t)20911 23485 y Fo(0)21889 23319 y Fs(=)481 b Fl(K)24034 23027 y Fq(~)24090 23319 y(k)24666 23485 y Fo(0)25637 23319 y Fs(on)474 b(the)g(torus)f Fq(K)79 b Fs(.)806 b(By)474 b(the)g(previous)f(lemma)6075 24689 y Fq(t)6475 24855 y Fo(0)7322 24689 y Fl(2)350 b Fq(\037)9103 24855 y Fo(1)9994 24689 y Fs(and,)402 b(therefore,)h Fq(t)394 b Fs(is)h(in)g(the)h(\()p Fq(\025)23735 24288 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)27382 24689 y Fs(\)-neigh)-31 b(b)31 b(orho)g(o)g(d)396 b(of)g Fq(\037)37139 24855 y Fo(1)38030 24689 y Fs(in)f Fq(K)79 b Fs(.)571 b(Th)-31 b(us,)402 b(the)6075 26125 y(parallel)371 b(shift)e(of)h(the)g(neigh)-31 b(b)31 b(orho)g(o)g(d)370 b(of)24149 25846 y(^)23891 26125 y Fq(D)400 b Fs(b)31 b(elongs)370 b(to)g(the)f(neigh)-31 b(b)31 b(orho)g(o)g(d)370 b(of)g Fq(\037)41345 26291 y Fo(1)41841 26125 y Fs(.)7403 27454 y(Supp)31 b(ose)338 b Fq(t)f Fs(is)i(in)f(the)g(\()p Fq(\025)17745 27052 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)21392 27454 y Fs(\))p Fl(\000)p Fs(neigh)-31 b(b)31 b(orho)g(o)g(d)340 b(of)f Fq(\037)31528 27620 y Fo(1)32362 27454 y Fs(in)f Fq(K)79 b Fs(.)483 b(Then,)345 b(there)338 b(is)g Fq(t)42887 27620 y Fo(0)43691 27454 y Fl(2)307 b Fq(\037)45429 27620 y Fo(1)6075 28824 y Fs(suc)-31 b(h)460 b(that)i Fl(j)p Fq(t)307 b Fl(\000)g Fq(t)13735 28990 y Fo(0)14232 28824 y Fl(j)460 b Fq(<)g(\025)16966 28422 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)20613 28824 y Fs(.)767 b(There)460 b(is)h(a)g(single)g Fq(j)524 b Fs(suc)-31 b(h)461 b(that)g Fq(\025)37074 28990 y Fn(j)37541 28824 y Fs(\()p Fq(t)38371 28990 y Fo(0)38867 28824 y Fs(\))g(=)f Fq(\025)p Fs(.)767 b(Let)461 b(us)6075 30334 y(consider)10365 30042 y Fq(~)10421 30334 y(k)10997 30500 y Fo(0)11855 30334 y Fs(=)371 b Fq(~)-562 b(p)13635 30500 y Fn(j)14101 30334 y Fs(\()p Fq(t)14931 30500 y Fo(0)15428 30334 y Fs(\))402 b(and)18388 30042 y Fq(~)18445 30334 y(k)396 b Fs(=)370 b Fq(~)-562 b(p)21196 30500 y Fn(j)21663 30334 y Fs(\()p Fq(t)p Fs(\).)590 b(Ob)-31 b(viously)-92 b(,)411 b Fl(j)29529 30042 y Fq(~)29585 30334 y(k)304 b Fl(\000)31537 30042 y Fq(~)31593 30334 y(k)32169 30500 y Fo(0)32666 30334 y Fl(j)362 b Fs(=)f Fl(j)p Fq(t)268 b Fl(\000)f Fq(t)37060 30500 y Fo(0)37557 30334 y Fl(j)362 b Fq(<)f(\025)40094 29933 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)44142 30334 y Fs(and)6018 31553 y Fq(~)6075 31845 y(k)6651 32011 y Fo(0)7617 31845 y Fl(2)9082 31565 y Fs(^)8824 31845 y Fq(D)31 b Fs(.)784 b(Therefore,)16252 31553 y Fq(~)16308 31845 y(k)502 b Fs(is)466 b(in)g(the)h(\()p Fq(\025)23064 31443 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)26711 31845 y Fs(\)-neigh)-31 b(b)31 b(orho)g(o)g(d)467 b(of)36175 31565 y(^)35917 31845 y Fq(D)497 b Fs(in)467 b Fq(R)39569 31443 y Fo(2)40064 31845 y Fs(.)784 b(Th)-31 b(us,)491 b(the)6075 33215 y(\()p Fq(\025)7151 32813 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)10798 33215 y Fs(\)-neigh)-31 b(b)31 b(orho)g(o)g(d)468 b(of)f Fq(\037)20698 33381 y Fo(1)21661 33215 y Fs(b)31 b(elongs)467 b(to)h(the)f(parallel)h (shift)g(of)f(the)g(corresp)31 b(onding)6075 34651 y(neigh)-31 b(b)31 b(orho)g(o)g(d)370 b(of)14546 34371 y(^)14288 34651 y Fq(D)31 b Fs(.)7403 35979 y(Let)403 b(us)f(establish)i(that)g (there)e(is)h(one-to-one)h(corresp)31 b(ondence)402 b(b)31 b(et)-31 b(w)g(een)403 b(the)g(neigh)-31 b(b)31 b(or-)6075 37308 y(ho)g(o)g(ds.)497 b(Supp)31 b(ose)370 b(for)h(a)g(certain)h Fq(t)e Fs(in)h(the)g(neigh)-31 b(b)31 b(orho)g(o)g(d)372 b(of)f Fq(\037)33252 37474 y Fo(1)34119 37308 y Fs(there)g(is)f(a)h (pair)g(of)h(v)-31 b(ectors)6084 38682 y Fq(~)-562 b(p)6632 38848 y Fn(j)7098 38682 y Fs(\()p Fq(t)p Fs(\))362 b(and)370 b Fq(~)-562 b(p)11421 38848 y Fn(m)12262 38682 y Fs(\()p Fq(t)p Fs(\),)363 b(b)31 b(oth)362 b(b)31 b(eing)362 b(in)f(the)g(neigh)-31 b(b)31 b(orho)g(o)g(d)362 b(of)31409 38402 y(^)31151 38682 y Fq(D)31 b Fs(.)490 b(Considering)362 b(\(53\),)j(w)-31 b(e)362 b(easily)6075 40171 y(obtain)370 b(that)f Fl(j<)13067 39878 y Fs(^)13021 40171 y Fq(\025)13667 40337 y Fn(j)14134 40171 y Fs(\()p Fq(t)p Fs(\))245 b Fl(\000)f Fq(\025)p Fl(j)308 b Fq(<)g(c\025)20299 39769 y Fk(\000)p Fo(2)p Fn(\016)22285 40171 y Fs(and)369 b Fl(j<)25589 39878 y Fs(^)25543 40171 y Fq(\025)26189 40337 y Fn(m)27030 40171 y Fs(\()p Fq(t)p Fs(\))245 b Fl(\000)f Fq(\025)p Fl(j)308 b Fq(<)g(c\025)33195 39769 y Fk(\000)p Fo(2)p Fn(\016)35181 40171 y Fs(By)369 b(Lemma)g(11,)h Fq(m)307 b Fs(=)h Fq(j)63 b Fs(.)6075 41499 y Fl(})6075 43542 y Fx(Lemma)949 b(14.)581 b Fi(Supp)-57 b(ose)436 b Fs(0)381 b Fq(<)g(")g(<)g Fs(1)p Fq(=)p Fs(8)p Fi(,)448 b Fs(1)p Fq(=)p Fs(4)383 b Fq(<)d(\014)440 b(<)380 b Fs(1)p Fq(=)p Fs(2)277 b Fl(\000)e Fs(2)p Fq(")p Fi(,)448 b Fs(0)381 b Fq(<)g(\016)423 b(<)381 b Fs(min)p Fl(f)p Fs(1)277 b Fl(\000)e Fs(2)p Fq(\014)334 b Fl(\000)6075 44978 y Fs(4)p Fq(";)184 b Fs(2)p Fq(\014)306 b Fl(\000)246 b Fs(1)p Fq(=)p Fs(2)p Fl(g)p Fi(.)512 b(Then,)397 b(the)f(length)g(of) 23166 44699 y Fs(^)22908 44978 y Fq(D)31 b Fs(\()p Fq(\025;)184 b(\014)58 b(;)184 b(\016)42 b Fs(\))399 b Fi(ob)-57 b(eys)397 b(the)f(estimate:)16446 47193 y Fq(l)16982 45964 y Fp(\020)17902 46913 y Fs(^)17644 47193 y Fq(D)31 b Fs(\()p Fq(k)k(;)184 b(\014)58 b(;)184 b(\016)42 b Fs(\))22262 45964 y Fp(\021)23234 47193 y Fs(=)307 b(2)p Fq(\031)25626 46196 y Fl(p)p 26549 46196 646 45 v 26549 47193 a Fq(\025)27379 45964 y Fp(\020)28040 47193 y Fs(1)247 b(+)f Fq(O)31 b Fs(\()p Fq(\025)31898 46736 y Fk(\000)p Fn(\016)d(=)p Fo(4)33971 47193 y Fs(\))34401 45964 y Fp(\021)35247 47193 y Fq(:)7850 b Fs(\(175\))6075 49569 y Fx(Corollary)443 b(12.)554 b Fi(The)396 b(length)g(of)g Fq(\037)21446 49735 y Fo(1)21942 49569 y Fs(\()p Fq(\025;)184 b(\014)58 b(;)184 b(\016)42 b Fs(\))400 b Fi(ob)-57 b(eys)396 b(the)g(estimate:)16556 51783 y Fq(l)206 b Fs(\()q Fq(\037)18216 51949 y Fo(1)18712 51783 y Fs(\()p Fq(k)35 b(;)184 b(\014)58 b(;)184 b(\016)42 b Fs(\)\))311 b(=)d(2)p Fq(\031)25517 50786 y Fl(p)p 26439 50786 V 997 x Fq(\025)27270 50555 y Fp(\020)27931 51783 y Fs(1)246 b(+)g Fq(O)31 b Fs(\()p Fq(\025)31788 51326 y Fk(\000)p Fn(\016)d(=)p Fo(4)33861 51783 y Fs(\))34291 50555 y Fp(\021)35137 51783 y Fq(:)7960 b Fs(\(176\))7403 54377 y(Pro)31 b(of.)p 7403 54554 2971 45 v 317 w(W)-92 b(e)314 b(see)h(from)h(form)-31 b(ula)317 b(\(174\))h(that)e(the)g(set)29815 54097 y(^)29556 54377 y Fq(D)31 b Fs(\()p Fq(\025;)184 b(\014)58 b(;)184 b(\016)42 b Fs(\))319 b(is)c(the)h(set)f Fq(D)31 b Fs(\()p Fq(\025;)184 b(\014)58 b Fs(\))317 b(min)-31 b(us)6075 55706 y(neigh)g(b)31 b(orho)g(o)g(ds)370 b(of)f(in)-31 b(tersections)371 b(of)f Fq(D)31 b Fs(\()p Fq(\025;)184 b(\014)58 b Fs(\))371 b(with)f(all)h Fq(D)31340 55872 y Fn(r)31830 55706 y Fs(\()p Fq(\025;)184 b(\014)58 b Fs(\),)372 b Fq(r)339 b Fl(2)307 b Fq(Z)37908 55304 y Fo(2)38651 55706 y Fl(n)246 b(f)p Fs(0)p Fl(g)p Fs(,)371 b(i.e.,)18469 57416 y(^)18211 57696 y Fq(D)31 b Fs(\()p Fq(\025;)184 b(\014)58 b(;)184 b(\016)42 b Fs(\))311 b(=)c Fq(D)31 b Fs(\()p Fq(\025;)184 b(\014)58 b Fs(\))249 b Fl(n)d Fq(L)p Fs(\()p Fq(\025;)184 b(\014)58 b(;)184 b(\016)42 b Fs(\))p Fq(;)6075 59578 y Fs(here)18043 60996 y Fq(L)p Fs(\()p Fq(\025;)184 b(\014)58 b(;)184 b(\016)42 b Fs(\))311 b(=)c Fl([)24719 61196 y Fn(r)24 b Fk(2)p Fn(Z)26408 60974 y Fm(2)26840 61196 y Fk(nf)p Fo(0)p Fk(g)28697 60996 y Fq(L)29450 61162 y Fn(r)29941 60996 y Fs(\()p Fq(\025;)184 b(\014)58 b(;)184 b(\016)42 b Fs(\))p Fq(;)22939 62741 y(L)23692 62907 y Fn(r)24183 62741 y Fs(\()p Fq(\025;)184 b(\014)58 b(;)184 b(\016)42 b Fs(\))311 b(=)7713 63584 y Fp(n)8394 64521 y Fq(~)8451 64813 y(k)342 b Fl(2)308 b Fq(D)31 b Fs(\()p Fq(\025;)184 b(\014)58 b Fs(\))310 b(:)14911 64521 y Fq(~)14968 64813 y(k)281 b Fl(\000)255 b Fq(~)-562 b(p)17489 64979 y Fn(r)17979 64813 y Fs(\(0\))309 b Fl(2)e Fs(\005\()p Fq(\014)58 b Fs(\))248 b Fl(\\)e Fq(M)25426 64979 y Fo(1)25922 64813 y Fs(\()p Fq(\025)26998 64979 y Fo(0)27495 64813 y Fs(\))p Fq(;)28416 63540 y Fp(\014)28416 64204 y(\014)28416 64868 y(\014)28833 64521 y Fs(^)28787 64813 y Fq(\025)o Fs(\()29806 64521 y Fq(~)29862 64813 y(k)36 b Fs(\))247 b Fl(\000)32304 64521 y Fs(^)32257 64813 y Fq(\025)33088 63584 y Fp(\020)33692 64521 y Fq(~)33749 64813 y(k)281 b Fl(\000)255 b Fq(~)-562 b(p)36270 64979 y Fn(r)36760 64813 y Fs(\(0\))38173 63584 y Fp(\021)38836 63540 y(\014)38836 64204 y(\014)38836 64868 y(\014)39512 64813 y Fq(<)307 b Fs(4)p Fq(\025)41879 64356 y Fk(\000)p Fn(\016)43057 63584 y Fp(o)43980 64813 y Fq(:)6075 66795 y Fs(T)-92 b(o)397 b(pro)-31 b(v)g(e)398 b(\(175\))h(it)f(is)f(enough)g(to)h(sho) -31 b(w)398 b(that)g(the)f(length)i(of)e Fq(L)p Fs(\()p Fq(\025;)184 b(\014)58 b(;)184 b(\016)42 b Fs(\))400 b(satis\014es)d(the)h(esti-)6075 68123 y(mate:)19911 69541 y Fq(l)207 b Fs(\()p Fq(L)p Fs(\()p Fq(\025;)184 b(\014)58 b(;)184 b(\030)51 b Fs(\)\))312 b Fq(<)307 b(c\025)28373 69084 y Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fn(\016)28 b(=)p Fo(4)31782 69541 y Fq(;)11315 b Fs(\(177\))p eop end %%Page: 33 33 TeXDict begin 33 32 bop 7289 3345 a Fo(SPECTRAL)388 b(PR)-24 b(OPER)-71 b(TIES)386 b(OF)h(THE)h(PERIODIC)f(MA)-24 b(GNETIC)389 b(SCHR)35152 3149 y(\177)35033 3345 y(ODINGER)e(OPERA)-71 b(TOR)331 b(33)6075 6001 y Fs(since)369 b Fq(D)31 b Fs(\()p Fq(\025;)184 b(\014)58 b Fs(\))371 b(satis\014es)f(\(66\).)494 b(Ob)-31 b(viously)-92 b(,)16857 8020 y Fq(l)207 b Fs(\()p Fq(L)p Fs(\()p Fq(\025;)184 b(\014)58 b(;)184 b(\016)42 b Fs(\)\))312 b Fl(\024)25355 6968 y Fp(X)24193 9394 y Fn(r)24 b Fk(2)p Fn(Z)25882 9173 y Fm(2)26314 9394 y Fk(nf)p Fo(0)p Fk(g)28300 8020 y Fq(l)207 b Fs(\()q Fq(L)30021 8186 y Fn(r)30511 8020 y Fs(\()p Fq(\025;)184 b(\014)58 b(;)184 b(\016)42 b Fs(\)\))188 b Fq(:)6075 11047 y Fs(W)-92 b(e)368 b(pro)-31 b(v)g(e)370 b(the)f(follo)-31 b(wing)373 b(estimates:)14748 13530 y Fq(l)207 b Fs(\()p Fq(L)16468 13696 y Fn(r)16959 13530 y Fs(\()p Fq(\025;)184 b(\014)58 b(;)184 b(\016)42 b Fs(\)\))311 b Fq(<)25594 12781 y(c\032)26645 12380 y Fk(\000)p Fo(2)p Fn(\016)28 b Fo(+1)p 22707 13276 9566 45 v 22707 14443 a Fq(p)23264 14609 y Fn(r)23754 14443 y Fs(\(0\))25167 13497 y Fp(p)p 26276 13497 5997 45 v 26276 14443 a Fq(\032)26848 14124 y Fo(2)27591 14443 y Fl(\000)246 b Fq(p)29255 14124 y Fo(2)29255 14717 y Fn(r)29751 14443 y Fs(\(0\))p Fq(=)p Fs(4)32405 13530 y Fq(;)923 b(\032)308 b Fs(=)35684 12533 y Fl(p)p 36606 12533 646 45 v 997 x Fq(\025)6152 b Fs(\(178\))6075 16343 y(if)369 b Fq(p)7646 16509 y Fn(r)8137 16343 y Fs(\(0\))p Fq(=)p Fs(2)309 b Fl(\024)f Fq(\032)246 b Fl(\000)g Fq(\032)14631 15942 y Fn(\016)28 b(=)p Fo(2)16012 16343 y Fs(,)370 b(and)20818 17730 y Fq(l)207 b Fs(\()p Fq(L)22538 17896 y Fn(r)23029 17730 y Fs(\()p Fq(\025;)184 b(\014)58 b(;)184 b(\030)51 b Fs(\)\))311 b Fq(<)c(c\032)29696 17273 y Fk(\000)p Fn(\016)30874 17730 y Fq(;)12223 b Fs(\(179\))6075 19336 y(if)369 b Fq(\032)247 b Fl(\000)f Fq(\032)9587 18935 y Fn(\016)28 b(=)p Fo(2)11275 19336 y Fq(<)308 b(p)13001 19502 y Fn(r)13491 19336 y Fs(\(0\))p Fq(=)p Fs(2)i Fq(<)d(\032)246 b Fs(+)g Fq(\032)19985 18935 y Fn(\016)28 b(=)p Fo(2)21366 19336 y Fs(,)370 b(and)21693 21133 y Fq(l)207 b Fs(\()p Fq(L)23413 21299 y Fn(r)23904 21133 y Fs(\()p Fq(\032;)184 b(\014)58 b(;)184 b(\016)42 b Fs(\)\))311 b(=)d(0)p Fq(;)13098 b Fs(\(180\))6075 22931 y(if)354 b Fq(p)7631 23097 y Fn(r)8121 22931 y Fs(\(0\))p Fq(=)p Fs(2)310 b Fl(\025)e Fq(\032)216 b Fs(+)g Fq(\032)14556 22529 y Fn(\016)15042 22931 y Fs(.)487 b(W)-92 b(e)353 b(estimate)j(the)f(sum)f(o)-31 b(v)g(er)354 b Fq(r)385 b Fs(of)355 b(the)f(righ)-31 b(t-hand)356 b(sides)d(of)i(\(178\))h(as)6075 24259 y(follo)-31 b(ws:)12463 25475 y Fp(X)9911 27926 y Fn(m)p Fo(:)p Fn(p)11404 28037 y Fj(r)11849 27926 y Fo(\(0\))p Fn(=)p Fo(2)p Fk(\024)p Fn(\032)p Fk(\000)p Fn(\032)16179 27705 y Fj(\016)16799 26527 y Fq(l)207 b Fs(\()p Fq(L)18519 26693 y Fn(r)19010 26527 y Fs(\()p Fq(\032;)184 b(\014)58 b(;)184 b(\016)42 b Fs(\)\))311 b Fl(\024)26929 25475 y Fp(X)24552 27926 y Fn(r)24 b Fo(:)p Fn(p)25694 28037 y Fj(r)26140 27926 y Fo(\(0\))p Fn(=)p Fo(2)p Fk(\024)p Fn(\032)p Fk(\000)p Fn(\032)30470 27705 y Fj(\016)34110 25778 y Fq(c\032)35161 25376 y Fk(\000)p Fo(2)p Fn(\016)k Fo(+1)p 31223 26272 9566 45 v 31223 27440 a Fq(p)31780 27606 y Fn(r)32270 27440 y Fs(\(0\))33683 26494 y Fp(p)p 34791 26494 5997 45 v 34791 27440 a Fq(\032)35363 27120 y Fo(2)36106 27440 y Fl(\000)246 b Fq(p)37770 27120 y Fo(2)37770 27714 y Fn(r)38267 27440 y Fs(\(0\))p Fq(=)p Fs(4)41228 26527 y Fl(\024)13312 30331 y Fq(c\032)14363 29874 y Fk(\000)p Fo(2)p Fn(\016)28 b Fo(+1)17287 28825 y Fp(Z)18107 31144 y Fo(~)17902 31340 y Fn(B)22191 29582 y Fq(dx)p 18982 30077 7628 45 v 18982 31245 a Fl(j)p Fq(x)p Fl(j)20229 30298 y Fp(p)p 21336 30298 5274 45 v 947 x Fq(\032)21908 30925 y Fo(2)22651 31245 y Fl(\000)246 b(j)p Fq(x)p Fl(j)25005 30925 y Fo(2)25502 31245 y Fq(=)p Fs(4)27050 30331 y Fl(\024)307 b Fq(c\032)29269 29874 y Fk(\000)p Fo(2)p Fn(\016)28 b Fo(+1)32317 30331 y Fs(=)307 b Fq(O)31 b Fs(\()p Fq(\025)35436 29874 y Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fn(\016)37950 30331 y Fs(\))p Fq(;)6075 32948 y Fs(here)8824 32668 y(~)8561 32948 y Fq(B)510 b Fs(is)454 b(the)g(disc)h(of)f(radius) h Fq(\032)303 b Fl(\000)22338 32512 y Fo(1)p 22338 32693 442 45 v 22338 33330 a(2)22912 32948 y Fq(\032)23484 32547 y Fn(\016)28 b(=)p Fo(2)25319 32948 y Fs(in)455 b Fq(R)27546 32547 y Fo(2)28042 32948 y Fs(.)747 b(Considering)456 b(that)g(the)e(sum)g(o)-31 b(v)g(er)455 b Fq(r)481 b Fs(:)6075 34424 y Fq(\032)246 b Fl(\000)g Fq(\032)8572 34022 y Fn(\016)28 b(=)p Fo(2)10260 34424 y Fq(<)308 b(p)11986 34590 y Fn(r)12476 34424 y Fs(\(0\))p Fq(=)p Fs(2)i Fq(<)d(\032)247 b Fs(+)f Fq(\032)18971 34022 y Fn(\016)28 b(=)p Fo(2)20720 34424 y Fs(con)-31 b(tains)371 b(no)f(more)f(than)h Fq(c\032)33004 34022 y Fo(1+)p Fn(\016)28 b(=)p Fo(2)35875 34424 y Fs(terms,)369 b(w)-31 b(e)370 b(obtain:)15626 35390 y Fp(X)11886 37841 y Fn(r)24 b Fo(:)p Fn(\032)p Fk(\000)p Fn(\032)14182 37620 y Fj(\016)14617 37841 y Fn()590 b(\031)40 b(=)p Fs(3.)6075 50507 y(F)-92 b(rom)482 b(\(196\))j(it)e(follo)-31 b(ws)485 b Fq(\032)184 b Fs(cos)h Fq(')322 b Fl(\000)f Fq(\032)184 b Fs(cos)i Fq(')24868 50673 y Fo(0)25861 50507 y Fs(=)496 b Fq(o)p Fs(\()p Fq(\032)p Fs(\),)512 b(and,)g(hence,)f Fq(')321 b Fl(\000)h Fq(')39084 50673 y Fo(0)40077 50507 y Fs(=)496 b Fq(o)p Fs(\(1\),)512 b(i.e.,)6075 51835 y Fq(')307 b(>)8408 51399 y Fn(\031)p 8408 51580 546 45 v 8460 52217 a Fo(3)9086 51835 y Fs(\(1)68 b(+)g Fq(o)p Fs(\(1\)\).)466 b(Considering)282 b(\(195\))g(and)f(taking)h(in)-31 b(to)282 b(accoun)-31 b(t)282 b(that)f Fq(\014)366 b(>)308 b Fs(1)p Fq(=)p Fs(4,)300 b(w)-31 b(e)280 b(obtain)6075 53310 y Fq(A)p Fs(\()p Fq(')p Fs(\))320 b(=)e Fq(\032)184 b Fs(sin)h Fq(')251 b Fs(+)f Fq(o)p Fs(\()p Fq(\032)p Fs(\))320 b(=)e Fq(\032)184 b Fs(sin)i Fq(')20867 53476 y Fo(0)21363 53310 y Fs(\(1)252 b(+)e Fq(o)p Fs(\(1\)\).)514 b(Second,)378 b(supp)31 b(ose)375 b Fq(\032)319 b Fl(\024)g Fq(p)37775 53476 y Fn(r)38265 53310 y Fs(\(0\))h Fq(<)e Fs(2)p Fq(\032)252 b Fl(\000)e Fq(\032)44237 52909 y Fn(\016)28 b(=)p Fo(2)45618 53310 y Fs(.)6075 54681 y(Then,)358 b(b)-31 b(y)356 b(\(190\))h(sin)185 b Fq(')15929 54847 y Fo(0)16733 54681 y Fq(>)308 b(\032)18474 54279 y Fk(\000)p Fo(1)p Fn(=)p Fo(2+)p Fn(\016)28 b(=)p Fo(4)22562 54681 y Fs(,)358 b(and,)h(b)-31 b(y)355 b(\(196\),)361 b Fq(\032)184 b Fs(cos)i Fq(')218 b Fl(\000)g Fq(\032)184 b Fs(cos)h Fq(')37978 54847 y Fo(0)38782 54681 y Fs(=)308 b Fq(o)p Fs(\(1\).)489 b(Hence,)6075 56033 y(cos)184 b Fq(')340 b Fl(\000)g Fs(cos)185 b Fq(')12397 56199 y Fo(0)13436 56033 y Fs(=)543 b Fq(o)p Fs(\()p Fq(\032)16379 55631 y Fk(\000)p Fo(1)17568 56033 y Fs(\),)j(sin)20210 55567 y Fo(2)21024 55539 y Fn(')p 21024 55778 583 45 v 21094 56415 a Fo(2)22281 56033 y Fs(=)d(sin)25044 55567 y Fo(2)25858 55539 y Fn(')26440 55650 y Fm(0)p 25858 55778 1015 45 v 26144 56415 a Fo(2)27345 56033 y Fs(+)340 b Fq(o)p Fs(\()p Fq(\032)30085 55631 y Fk(\000)p Fo(1)31274 56033 y Fs(\),)546 b(sin)33916 55567 y Fo(2)34730 55539 y Fn(')p 34730 55778 583 45 v 34800 56415 a Fo(2)35988 56033 y Fs(=)c(sin)38750 55567 y Fo(2)39564 55539 y Fn(')40146 55650 y Fm(0)p 39564 55778 1015 45 v 39851 56415 a Fo(2)40711 56033 y Fs(\(1)341 b(+)f Fq(o)p Fs(\(1\)\),)6075 57508 y(sin)184 b Fq(')506 b(>)f Fs(sin)184 b Fq(')12481 57674 y Fo(0)12978 57508 y Fs(\(1)326 b(+)f Fq(o)p Fs(\(1\)\))507 b Fq(>)e(\032)20298 57107 y Fk(\000)p Fo(1)p Fn(=)p Fo(2+)p Fn(\016)28 b(=)p Fo(4)24386 57508 y Fs(.)848 b(Using)488 b(\(195\),)521 b(w)-31 b(e)488 b(obtain)h Fq(A)p Fs(\()p Fq(')p Fs(\))507 b(=)e Fq(\032)184 b Fs(sin)h Fq(')325 b Fs(+)6075 58879 y Fq(O)31 b Fs(\()p Fq(\032)7952 58477 y Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(3)p Fn(\014)11019 58879 y Fs(\))308 b(=)f Fq(\032)184 b Fs(sin)i Fq(')15950 59045 y Fo(0)16446 58879 y Fs(\(1)234 b(+)g Fq(o)p Fs(\(1\)\).)491 b(Th)-31 b(us,)365 b(w)-31 b(e)363 b(ha)-31 b(v)g(e)364 b(pro)-31 b(v)g(ed)363 b(\(194\))i(and,)g(therefore,)f(\(178\).)7403 60249 y(Supp)31 b(ose)279 b(2)p Fq(\032)67 b Fl(\000)g Fq(\032)14347 59847 y Fn(\016)28 b(=)p Fo(2)16037 60249 y Fq(<)307 b(p)17762 60415 y Fn(r)18253 60249 y Fs(\(0\))i Fq(<)e Fs(2)p Fq(\032)67 b Fs(+)g Fq(\032)23835 59847 y Fn(\016)28 b(=)p Fo(2)25217 60249 y Fs(.)463 b(Using)280 b(\(186\))i(and)e(\(191\))i(in)e(\(193\),)301 b(w)-31 b(e)280 b(obtain:)6075 61577 y Fq(\032)p Fs(\(cos)185 b Fq(')j Fl(\000)g Fs(cos)d Fq(')13096 61743 y Fo(0)13593 61577 y Fs(\))308 b(=)f Fq(O)31 b Fs(\()p Fq(\032)17376 61176 y Fk(\000)p Fo(2)p Fn(\014)19107 61577 y Fs(\))189 b(+)f Fq(O)31 b Fs(\()p Fq(\032)22652 61176 y Fk(\000)p Fo(2)p Fn(\016)d Fk(\000)p Fo(1)25405 61577 y Fs(\))308 b(=)f Fq(O)31 b Fs(\()p Fq(\032)29188 61176 y Fk(\000)p Fo(2)p Fn(\014)30919 61577 y Fs(\).)484 b(Therefore,)346 b Fl(j)184 b Fs(cos)i Fq(')i Fl(\000)g Fs(cos)d Fq(')43953 61743 y Fo(0)44449 61577 y Fl(j)308 b Fs(=)6075 62948 y Fq(O)31 b Fs(\()p Fq(\032)7952 62546 y Fk(\000)p Fo(2)p Fn(\014)40 b Fk(\000)p Fo(1)10816 62948 y Fs(\))499 b(and,)532 b(hence,)f Fl(j)p Fq(')333 b Fl(\000)f Fq(')21192 63114 y Fo(0)21689 62948 y Fl(j)524 b Fq(<)f(c\032)24955 62546 y Fk(\000)p Fn(\014)40 b Fk(\000)p Fo(1)p Fn(=)p Fo(2)28273 62948 y Fs(.)881 b(Considering)500 b(the)f(last)g(relation)i(and)6075 64276 y(\(192\),)372 b(w)-31 b(e)369 b(obtain:)19889 65789 y Fq(R)9 b Fs(\()p Fq(')p Fs(\))247 b Fl(\000)f Fq(r)31 b Fs(\()p Fq(')25361 65955 y Fo(0)25858 65789 y Fs(\))308 b(=)f Fq(O)31 b Fs(\()p Fq(\032)29641 65332 y Fk(\000)p Fo(4)p Fn(\014)31373 65789 y Fs(\))p Fq(:)6075 67523 y Fs(T)-92 b(aking)371 b(in)-31 b(to)370 b(accoun)-31 b(t)371 b(that)f Fq(g)40 b Fs(\()p Fq(R)9 b Fs(\()p Fq(')p Fs(\))p Fq(;)184 b(')p Fs(\))23574 67121 y Fk(0)24194 67523 y Fs(=)308 b Fq(g)25891 67689 y Fn(R)26617 67523 y Fq(R)27467 67121 y Fk(0)28023 67523 y Fs(+)245 b Fq(g)29657 67689 y Fn(')30603 67523 y Fs(=)307 b Fq(O)31 b Fs(\()p Fq(\032)33648 67121 y Fo(3)p Fn(=)p Fo(2)p Fk(\000)p Fo(3)p Fn(\014)36715 67523 y Fs(\),)370 b(w)-31 b(e)370 b(get:)16616 69541 y Fq(g)40 b Fs(\()p Fq(R)9 b Fs(\()p Fq(')p Fs(\))p Fq(;)184 b(')p Fs(\))248 b Fl(\000)e Fq(g)40 b Fs(\()p Fq(R)9 b Fs(\()p Fq(')26050 69707 y Fo(0)26547 69541 y Fs(\))p Fq(;)184 b(')28192 69707 y Fo(0)28690 69541 y Fs(\))308 b(=)f Fq(O)31 b Fs(\()p Fq(\032)32473 69084 y Fo(1)p Fk(\000)p Fo(4)p Fn(\014)34646 69541 y Fs(\))p Fq(:)p eop end %%Page: 35 35 TeXDict begin 35 34 bop 7289 3345 a Fo(SPECTRAL)388 b(PR)-24 b(OPER)-71 b(TIES)386 b(OF)h(THE)h(PERIODIC)f(MA)-24 b(GNETIC)389 b(SCHR)35152 3149 y(\177)35033 3345 y(ODINGER)e(OPERA)-71 b(TOR)331 b(35)6075 6001 y Fs(Using)370 b(the)f(last)h(t)-31 b(w)g(o)371 b(estimates,)g(w)-31 b(e)370 b(can)f(reduce)g(\(193\))i(to) f(the)f(form:)16290 7901 y Fl(j)p Fq(R)9 b Fs(\()p Fq(')18601 8067 y Fo(0)19098 7901 y Fs(\))184 b(cos)i Fq(')246 b Fl(\000)g Fq(a)p Fl(j)308 b Fs(=)f Fq(O)31 b Fs(\()p Fq(\032)27702 7444 y Fk(\000)p Fo(2)p Fn(\016)d Fk(\000)p Fo(1)30455 7901 y Fs(\))247 b(+)f(\()p Fq(\032)33241 7444 y Fk(\000)p Fo(4)p Fn(\014)34972 7901 y Fs(\))p Fq(;)6075 9746 y Fs(here)502 b Fq(a)531 b Fs(=)g Fq(\032)184 b Fs(cos)h Fq(')14265 9912 y Fo(0)15097 9746 y Fl(\000)336 b Fq(g)40 b Fs(\()p Fq(R)9 b Fs(\()p Fq(')19296 9912 y Fo(0)19792 9746 y Fs(\))p Fq(;)184 b(')21437 9912 y Fo(0)21935 9746 y Fs(\))p Fq(=)p Fs(2)p Fq(p)24028 9912 y Fn(r)24520 9746 y Fs(\(0\))p Fq(:)505 b Fs(Considering)f(that)h(2)p Fq(\016)573 b(<)531 b Fs(4)p Fq(\014)394 b Fl(\000)335 b Fs(1,)538 b(w)-31 b(e)503 b(get)6075 11074 y Fl(j)p Fq(R)9 b Fs(\()p Fq(')8386 11240 y Fo(0)8882 11074 y Fs(\))184 b(cos)i Fq(')91 b Fl(\000)g Fq(a)p Fl(j)309 b Fs(=)f Fq(O)31 b Fs(\()p Fq(\032)17178 10672 y Fk(\000)p Fo(2)p Fn(\016)d Fk(\000)p Fo(1)19931 11074 y Fs(\))p Fq(:)292 b Fs(Hence,)308 b Fq(L)25249 11240 y Fn(r)25740 11074 y Fs(\()p Fq(\032;)184 b(\014)58 b(;)184 b(\016)42 b Fs(\))295 b(can)d(b)31 b(e)292 b(describ)31 b(ed)291 b(b)-31 b(y)292 b(the)g(inequalit)-31 b(y)6075 12402 y Fl(j)p Fq(')307 b Fl(\000)485 b Fs(~)-731 b Fq(')p Fl(j)461 b Fq(<)e(c\032)12444 12001 y Fk(\000)p Fn(\016)28 b Fk(\000)p Fo(1)15216 12402 y Fs(for)461 b(some)638 b(~)-730 b Fq(')p Fs(.)767 b(It)460 b(easily)i(follo)-31 b(ws)463 b(that)f Fq(l)22 b Fs(\()p Fq(L)33848 12568 y Fn(r)34339 12402 y Fs(\()p Fq(\032;)184 b(\014)58 b(;)184 b(\016)42 b Fs(\)\))464 b Fq(<)c(c\032)41237 12001 y Fk(\000)p Fn(\016)42415 12402 y Fq(:)g Fs(Th)-31 b(us,)6075 13731 y(estimate)456 b(\(179\))h(is)d(c)-31 b(hec)g(k)g(ed.)749 b(The)455 b(relation)h(\(180\))h(is)d(easy)h(to)g(see)f(considering)h (that)h(the)6075 15150 y(deviation)288 b(of)f Fq(D)318 b Fs(from)287 b(the)f(exact)i(sphere)d(of)i(radius)28222 14207 y Fl(p)p 29144 14207 646 45 v 943 x Fq(\025)f Fs(do)31 b(es)286 b(not)h(exceed)g Fq(c\025)38954 14748 y Fk(\000)p Fn(\014)40529 15150 y Fs(\(see)f(\(191\)\).)6075 16478 y Fl(})p Fs(.)7403 17807 y(The)394 b(prop)31 b(ert)-31 b(y)393 b(of)h Fq(\037)16231 17973 y Fo(1)17120 17807 y Fs(established)h(in)e(the)h(next)g(lemma)h(w)-31 b(e)394 b(will)h(use)e(later)h(to)g(estimate)6075 19135 y(the)369 b(area)h(of)f Fq(\037)12336 19301 y Fo(1)13079 19135 y Fl(n)246 b Fq(\037)14571 19301 y Fo(2)15067 19135 y Fs(.)6075 21157 y Fx(Lemma)839 b(15.)551 b Fi(L)-57 b(et)393 b Fq(")p Fi(,)h Fq(\014)451 b Fi(and)393 b Fq(\016)434 b Fi(satisfy)393 b(the)f(c)-57 b(onditions)394 b(in)f(L)-57 b(emma)394 b(14.)508 b(Supp)-57 b(ose)393 b Fq(S)371 b Fl(\022)307 b Fq(\037)45429 21323 y Fo(1)6075 22485 y Fi(and)441 b Fq(l)22 b Fs(\()p Fq(S)64 b Fs(\))391 b Fq(>)g(\025)12514 22084 y Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fn(\026)15135 22485 y Fi(,)454 b Fq(\026)390 b Fl(\025)h Fs(0)p Fi(.)647 b(Then)442 b(the)g(ar)-57 b(e)g(a)442 b(of)g(the)g Fs(\()p Fq(\025)31336 22084 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)34983 22485 y Fs(\))p Fi(-neighb)-57 b(orho)g(o)g(d)443 b(of)f Fq(S)505 b Fi(in)6075 23814 y Fq(K)475 b Fi(c)-57 b(an)397 b(b)-57 b(e)397 b(estimate)-57 b(d)396 b(fr)-57 b(om)397 b(b)-57 b(elow)396 b(by)g Fq(c\025)23933 23412 y Fk(\000)p Fn(\026)p Fk(\000)p Fo(2)p Fn(\016)26782 23814 y Fi(.)7403 25836 y Fs(Pro)31 b(of.)p 7403 26013 2971 45 v 473 w(First,)498 b(w)-31 b(e)473 b(pro)-31 b(v)g(e)472 b(the)h(lemma)g(for)g Fq(S)542 b Fs(=)479 b Fq(\037)29533 26002 y Fo(1)30029 25836 y Fs(,)498 b Fq(\026)479 b Fs(=)g(0)473 b(By)f(Lemmas)h(12)g(and) f(13,)6075 27216 y(the)i(parallel)h(shifts)f(establishes)h(one-to-one)g (corresp)31 b(ondence)473 b(b)31 b(et)-31 b(w)g(een)475 b Fq(\037)39187 27382 y Fo(1)40157 27216 y Fs(and)42672 26937 y(^)42414 27216 y Fq(D)504 b Fs(and,)6075 28591 y(moreo)-31 b(v)g(er,)418 b(b)31 b(et)-31 b(w)g(een)408 b(their)g(\()p Fq(\025)19276 28189 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)22923 28591 y Fs(\)-neigh)-31 b(b)31 b(orho)g(o)g(ds.)608 b(Since)34843 28311 y(^)34585 28591 y Fq(D)438 b Fs(is)407 b(just)h(a)g(sligh)-31 b(tly)409 b(dis-)6075 30010 y(torted)427 b(circle)g(with)h(holes)f (with)h(a)e(length)i(2)p Fq(\031)26039 29067 y Fl(p)p 26962 29067 646 45 v 26962 30010 a Fq(\025)p Fs(\(1)285 b(+)f Fq(O)31 b Fs(\()p Fq(\025)31972 29609 y Fk(\000)p Fn(\016)d(=)p Fo(4)34045 30010 y Fs(\)\),)442 b(its)427 b(neigh)-31 b(b)31 b(orho)g(o)g(d)428 b(can)6075 31381 y(b)31 b(e)331 b(estimated)i(from)f(b)31 b(elo)-31 b(w)333 b(b)-31 b(y)332 b(2)p Fq(\031)40 b(\025)21603 30979 y Fk(\000)p Fo(2)p Fn(\016)23221 31381 y Fs(\(1)171 b(+)g Fq(O)31 b Fs(\()p Fq(\025)27358 30979 y Fk(\000)p Fn(\016)d(=)p Fo(4)29431 31381 y Fs(\)\).)481 b(By)332 b(Corollary)h(11,)341 b(the)331 b(v)-31 b(olume)334 b(of)6075 32751 y(the)383 b(\()p Fq(\025)9071 32349 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)12718 32751 y Fs(\)-neigh)-31 b(b)31 b(orho)g(o)g(d)384 b(of)f Fq(\037)22450 32917 y Fo(1)23329 32751 y Fs(in)f Fq(K)462 b Fs(is)383 b(equal)g(to)h(the)f (v)-31 b(olume)384 b(of)f(the)g(\()p Fq(\025)41479 32349 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)45126 32751 y Fs(\)-)6075 34187 y(neigh)-31 b(b)31 b(orho)g(o)g(d)349 b(of)14505 33907 y(^)14247 34187 y Fq(D)379 b Fs(in)349 b Fq(R)17663 33785 y Fo(2)18159 34187 y Fq(:)g Fs(Th)-31 b(us,)353 b(the)c(lemma)h(is)f(pro)-31 b(v)g(ed)349 b(for)g Fq(S)371 b Fs(=)307 b Fq(\037)36600 34353 y Fo(1)37096 34187 y Fs(.)486 b(Supp)31 b(ose)348 b Fq(S)371 b Fl(\032)308 b Fq(\037)45122 34353 y Fo(1)45618 34187 y Fs(.)6075 35561 y(Then,)301 b(ob)-31 b(viously)-92 b(,)302 b(there)283 b(is)18205 35282 y(^)18018 35561 y Fq(S)371 b Fl(\032)20495 35282 y Fs(^)20237 35561 y Fq(D)314 b Fs(suc)-31 b(h)282 b(that)i Fl(K)27236 35282 y Fs(^)27048 35561 y Fq(S)372 b Fs(=)307 b Fq(S)347 b Fs(and)283 b(there)g(is)g(one-to)h(one)f(neigh) -31 b(b)31 b(or-)6075 36936 y(ho)g(o)g(ds)9392 36656 y(^)9206 36936 y Fq(S)359 b Fs(and)296 b Fq(S)359 b Fs(and,)312 b(moreo)-31 b(v)g(er,)312 b(b)31 b(et)-31 b(w)g(een)297 b(their)f(\()p Fq(\025)28635 36534 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)32282 36936 y Fs(\)-neigh)-31 b(b)31 b(orho)g(o)g(ds.)469 b(Ob)-31 b(viously)-92 b(,)6075 38372 y(the)450 b(area)h(of)g(the)g(neigh)-31 b(b)31 b(orho)g(o)g(d)451 b(of)22437 38092 y(^)22250 38372 y Fq(S)514 b Fs(exceeds)450 b Fq(c\025)28622 37970 y Fk(\000)p Fn(\026)p Fk(\000)p Fo(2)p Fn(\016)31921 38372 y Fs(and)g(the)h(same)g(is)f(true)g(for)h(the)6075 39700 y(neigh)-31 b(b)31 b(orho)g(o)g(d)370 b(of)g Fq(S)64 b Fs(.)492 b Fl(})7403 41028 y Fs(Let)374 b(us)g(consider)g(the)g(set)g Fq(\037)19473 41194 y Fo(2)19969 41028 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\).)509 b(In)373 b(the)h(follo)-31 b(wing)378 b(lemma)e(w)-31 b(e)374 b(obtain)i(the)e(estimate)6075 42399 y(for)369 b(the)g(area)h(of)g(its)f(\()p Fq(\025)15957 41997 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)19604 42399 y Fs(\)-neigh)-31 b(b)31 b(orho)g(o)g(d.)6075 44463 y Fx(Lemma)861 b(16.)557 b Fi(The)401 b(ar)-57 b(e)g(a)402 b(of)f(the)g Fs(\()p Fq(\025)22099 44061 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)25746 44463 y Fs(\))p Fi(-neighb)-57 b(orho)g(o)g(d)403 b(of)e Fq(\037)35154 44629 y Fo(2)35650 44463 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\))403 b Fi(do)-57 b(es)401 b(not)g(exc)-57 b(e)g(e)g(d)6075 45791 y Fq(c)6720 45967 y Fn(~)6554 46163 y(A)o(;b)7927 45791 y Fq(\025)8573 45389 y Fk(\000)p Fn(\016)28 b Fk(\000)p Fn(")10860 45791 y Fi(.)7403 47950 y Fs(Pro)j(of.)p 7403 48127 2971 45 v 370 w(It)370 b(follo)-31 b(ws)371 b(from)f(Corollary)h(9)f(that)g Fq(\037)27102 48116 y Fo(2)27906 47950 y Fl(\022)307 b Fq(\037)29767 47548 y Fk(0)29767 48225 y Fo(2)30263 47950 y Fs(,)370 b(where)8490 50125 y Fq(\037)9183 49668 y Fk(0)9183 50399 y Fo(2)9987 50125 y Fs(=)11155 48896 y Fp(n)11893 50125 y Fq(t)307 b Fl(2)h Fq(K)387 b Fs(:)307 b Fl(9)p Fq(q)348 b Fl(2)307 b Fs(\000\()p Fq(\025)19857 49668 y Fn(")20331 50125 y Fs(\))p Fq(;)923 b(n)308 b Fl(2)g(f)p Fs(1)p Fq(;)184 b(:::;)g(N)27907 50291 y Fn(q)28400 50125 y Fl(g)p Fq(;)924 b Fl(j)p Fq(\026)31158 49668 y Fo(\()p Fn(q)28 b Fo(\))31158 50399 y Fn(n)32339 50125 y Fs(\()p Fq(t)p Fs(\))247 b Fl(\000)f Fq(\025)p Fl(j)307 b Fq(<)h Fs(5)p Fq(\025)38581 49668 y Fk(\000)p Fn(\016)39759 48896 y Fp(o)40681 50125 y Fq(;)2416 b Fs(\(197\))6075 51772 y Fp(n)6813 53001 y Fq(\026)7480 52426 y Fo(\()p Fn(q)28 b Fo(\))7480 53130 y Fn(n)8660 53001 y Fs(\()p Fq(t)p Fs(\))9920 51772 y Fp(o)10659 52001 y Fn(N)11360 52112 y Fj(q)10659 53776 y Fo(1)12106 53001 y Fs(b)j(eing)250 b(eigen)-31 b(v)-61 b(alues)252 b(of)f(the)f(op)31 b(erator)251 b Fq(P)28562 53167 y Fn(q)29051 53001 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)32032 53167 y Fn(q)32771 53001 y Fs(n)-31 b(umerated)251 b(in)g(the)f(increasing)6075 54869 y(order.)823 b(Therefore,)508 b(the)480 b(\()p Fq(\025)18290 54467 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)21937 54869 y Fs(\)-neigh)-31 b(b)31 b(orho)g(o)g(d)481 b(of)f Fq(\037)31863 55035 y Fo(2)32839 54869 y Fs(b)31 b(elongs)480 b(to)h(the)f(\()p Fq(\025)41479 54467 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)45126 54869 y Fs(\)-)6075 56197 y(neigh)-31 b(b)31 b(orho)g(o)g(d)406 b(of)g Fq(\037)15053 55796 y Fk(0)15053 56472 y Fo(2)15549 56197 y Fs(.)601 b(W)-92 b(e)405 b(denote)h(the)g(latter)g(neigh)-31 b(b)31 b(orho)g(o)g(d)407 b(b)-31 b(y)405 b Fq(\037)36253 55796 y Fk(00)36253 56472 y Fo(2)36819 56197 y Fs(.)601 b(Since)405 b(the)h(area)g(of)6075 57566 y(the)467 b(\()p Fq(\025)9155 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Fk(00)30560 60958 y Fo(2)31468 60683 y Fs(can)e(b)31 b(e)343 b(describ)31 b(ed)342 b(b)-31 b(y)344 b(a)f(form)-31 b(ula)6075 62012 y(similar)370 b(to)g(\(197\):)9531 64189 y Fq(\037)10224 63731 y Fk(00)10224 64462 y Fo(2)11097 64189 y Fl(\032)12265 62960 y Fp(n)13003 64189 y Fq(t)307 b Fl(2)h Fq(K)387 b Fs(:)308 b Fl(9)p Fq(q)347 b Fl(2)307 b Fs(\000\()p Fq(\025)20967 63731 y Fn(")21441 64189 y Fs(\))p Fq(;)924 b(n)307 b Fl(2)h(f)p Fs(1)p Fq(;)184 b(:::;)g(N)29017 64355 y Fn(q)29511 64189 y Fl(g)p Fq(;)923 b Fl(j)p Fq(\026)32268 63731 y Fo(\()p Fn(q)28 b Fo(\))32268 64462 y Fn(n)33449 64189 y Fs(\()p Fq(t)p Fs(\))247 b Fl(\000)f Fq(\025)p Fl(j)308 b Fq(<)f(c)38971 64355 y Fn(b)39416 64189 y Fq(\025)40062 63731 y Fk(\000)p Fn(\016)41239 62960 y Fp(o)42162 64189 y Fq(:)6075 66368 y Fs(With)334 b(some)h(abuse)f(of)h(notations)i(since)d(no)-31 b(w)335 b(w)-31 b(e)335 b(denote)g(b)-31 b(y)334 b Fq(\037)33010 65966 y Fk(00)33010 66642 y Fo(2)33910 66368 y Fs(the)g(set)h(on)f(the) h(righ)-31 b(t-hand)6075 67696 y(side)369 b(of)g(the)h(last)g(form)-31 b(ula.)494 b(W)-92 b(e)369 b(represen)-31 b(t)368 b Fq(\037)25429 67294 y Fk(00)25429 67971 y Fo(2)26364 67696 y Fs(in)h(the)h(form:) 21371 69541 y Fq(\037)22064 69084 y Fk(00)22064 69815 y Fo(2)22937 69541 y Fs(=)308 b Fl([)24844 69707 y Fn(q)28 b(;n)26512 69541 y Fq(\037)27205 69084 y Fk(00)27205 69815 y Fo(2)27770 69541 y Fs(\()p Fq(q)40 b(;)184 b(n)p Fs(\))p Fq(;)p eop end %%Page: 36 36 TeXDict begin 36 35 bop 6075 3345 a Fo(36)14514 b(YULIA)300 b(KARPESHINA)6075 6001 y Fs(where)369 b Fq(\037)9969 5600 y Fk(00)9969 6276 y Fo(2)10534 6001 y Fs(\()p Fq(q)40 b(;)184 b(n)p Fs(\))371 b(is)e(the)h(set)f(corresp)31 b(onding)369 b(to)h(just)g(one)f(inequalit)-31 b(y:)15553 8340 y Fq(\037)16246 7883 y Fk(00)16246 8614 y Fo(2)16812 8340 y Fs(\()p Fq(q)40 b(;)184 b(n)p Fs(\))310 b(=)20839 7112 y Fp(n)21577 8340 y Fq(t)d Fl(2)h Fq(K)387 b Fs(:)308 b Fl(j)p Fq(\026)26246 7883 y Fo(\()p Fn(q)28 b Fo(\))26246 8614 y Fn(n)27427 8340 y Fs(\()p Fq(t)p Fs(\))246 b Fl(\000)g Fq(\025)p Fl(j)308 b Fq(<)g(c)32949 8506 y Fn(b)33393 8340 y Fq(\025)34039 7883 y Fk(\000)p Fn(\016)35217 7112 y Fp(o)36139 8340 y Fq(:)6075 10681 y Fs(First,)487 b(w)-31 b(e)464 b(estimate)h(the)f(n)-31 b(um)g(b)31 b(er)463 b(of)h(non-empt)-31 b(y)465 b(sets)e Fq(\037)31514 10279 y Fk(00)31514 10956 y Fo(2)32079 10681 y Fs(\()p Fq(n;)184 b(q)40 b Fs(\).)778 b(In)463 b(fact,)488 b(if)464 b Fq(\037)41601 10279 y Fk(00)41601 10956 y Fo(2)42167 10681 y Fs(\()p Fq(q)40 b(;)184 b(n)p Fs(\))465 b(is)6075 12225 y(not)443 b(empt)-31 b(y)-92 b(,)463 b(than)443 b Fl(j)p Fq(\026)15436 11650 y Fo(\()p Fn(q)28 b Fo(\))15436 12354 y Fn(n)16617 12225 y Fs(\()p Fq(t)p Fs(\))296 b Fl(\000)f Fq(\025)p Fl(j)430 b Fq(<)g(c)22482 12391 y Fn(b)22927 12225 y Fq(\025)23573 11823 y Fk(\000)p Fn(\016)24750 12225 y Fs(,)462 b(for)442 b(at)i(least)f(one)g Fq(t)g Fs(in)f Fq(K)79 b Fs(.)714 b(Considering)444 b(that)6075 13768 y Fl(jr)p Fq(\026)7971 13193 y Fo(\()p Fn(q)28 b Fo(\))7971 13898 y Fn(n)9152 13768 y Fs(\()p Fq(t)p Fs(\))p Fl(j)503 b Fq(<)f(c)13064 13934 y Fn(b)13509 12825 y Fl(p)p 14431 12825 646 45 v 943 x Fq(\025)485 b Fs(\(Lemma)i(10\),)517 b(w)-31 b(e)486 b(obtain:)728 b Fl(j)p Fq(\026)29248 13193 y Fo(\()p Fn(q)28 b Fo(\))29248 13898 y Fn(n)30429 13768 y Fs(\()p Fq(t)p Fs(\))324 b Fl(\000)g Fq(\025)p Fl(j)502 b Fq(<)526 b Fs(~)-578 b Fq(c)36494 13934 y Fn(b)36939 12825 y Fl(p)p 37861 12825 V 943 x Fq(\025)486 b Fs(for)f(all)i Fq(t)502 b Fl(2)f Fq(K)79 b Fs(.)6075 15097 y(By)349 b(Lemma)h(10)g(the)g(n)-31 b(um)g(b)31 b(er)349 b(of)h(eigen)-31 b(v)-61 b(alues)351 b(of)e Fq(P)27975 15263 y Fn(q)28464 15097 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fq(P)31445 15263 y Fn(q)32283 15097 y Fs(satisfying)352 b(the)d(last)h(inequalit)-31 b(y)6075 16425 y(do)31 b(es)300 b(not)h(exceed)f Fq(c)14514 16601 y Fn(~)14348 16797 y(A;b)15722 16425 y Fq(\025)16368 16023 y Fo(1)p Fk(\000)p Fo(3)p Fn(")18415 16425 y Fs(.)470 b(Considering)302 b(that)f(\000\()p Fq(\025)29318 16023 y Fn(")29792 16425 y Fs(\))g(con)-31 b(tains)302 b(no)e(more)h(than)g Fq(c)41959 16591 y Fn(b)42404 16425 y Fq(\025)43050 16023 y Fo(2)p Fn(")44265 16425 y Fs(ele-)6075 18214 y(men)-31 b(ts,)338 b(w)-31 b(e)329 b(obtain)i(that)f(the)f(n)-31 b(um)g(b)31 b(er)329 b(of)h(eigen)-31 b(v)-61 b(alues)330 b Fq(\026)30366 17639 y Fo(\()p Fn(q)28 b Fo(\))30366 18344 y Fn(n)31547 18214 y Fs(\()p Fq(t)p Fs(\),)338 b Fq(q)347 b Fl(2)308 b Fs(\000\()p Fq(\025)37107 17812 y Fn(")37580 18214 y Fs(\),)338 b Fq(n)308 b Fl(2)g(f)p Fs(1)p Fq(;)184 b(:::;)g(N)44571 18380 y Fn(q)45064 18214 y Fl(g)p Fs(,)6075 19798 y(satisfying)406 b(the)e(inequalit)-31 b(y)407 b Fl(j)p Fq(\026)19127 19223 y Fo(\()p Fn(q)28 b Fo(\))19127 19927 y Fn(n)20308 19798 y Fs(\()p Fq(t)p Fs(\))270 b Fl(\000)f Fq(\025)p Fl(j)365 b Fq(<)h(c)25992 19964 y Fn(b)26437 18854 y Fl(p)p 27359 18854 V 944 x Fq(\025)403 b Fs(for)h(all)i Fq(t)d Fs(do)31 b(es)404 b(not)g(exceed)g Fq(c)41263 19974 y Fn(~)41097 20170 y(A)41763 20169 y(;b)42471 19798 y Fq(\025)43117 19396 y Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fn(")45618 19798 y Fs(.)6075 21413 y(Therefore,)369 b(there)g(is)h(no)f(more)g(than)h Fq(c)22881 21589 y Fn(~)22715 21785 y(A;b)24089 21413 y Fq(\025)24735 21012 y Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fn(")27605 21413 y Fs(non-empt)-31 b(y)371 b(sets)d Fq(\037)35997 21012 y Fk(00)35997 21688 y Fo(2)36563 21413 y Fs(\()p Fq(q)40 b(;)184 b(n)p Fs(\).)7403 22879 y(Second,)557 b(w)-31 b(e)519 b(estimate)i(the)e(area)h(of)f(a)g(non-empt)-31 b(y)521 b Fq(\037)31480 22477 y Fk(00)31480 23153 y Fo(2)32045 22879 y Fs(\()p Fq(q)40 b(;)184 b(n)p Fs(\).)944 b(By)519 b(Lemma)h(10,)558 b(eac)-31 b(h)6075 24422 y Fq(\026)6742 23847 y Fo(\()p Fn(q)28 b Fo(\))6742 24552 y Fn(n)7922 24422 y Fs(\()p Fq(t)p Fs(\))476 b(is)e(a)h(monotonous)i(con)-31 b(tin)g(uous)477 b(function)f(of)f Fq(t)f Fs(in)h(a)g(direction)462 b Fq(~)-539 b(u)38335 24588 y Fn(q)39298 24422 y Fs(orthogonal)477 b(to)6084 26006 y Fq(~)-562 b(p)6632 26172 y Fn(q)7121 26006 y Fs(\(0\).)700 b(Since)438 b(functions)h Fq(\026)18008 25431 y Fo(\()p Fn(q)28 b Fo(\))18008 26135 y Fn(n)19189 26006 y Fs(\()p Fq(t)p Fs(\))439 b(can)f(b)31 b(e)437 b(exp)31 b(ended)438 b(p)31 b(erio)g(dically)440 b(from)f Fq(K)517 b Fs(to)439 b(the)f(whole)6075 27589 y(plane)422 b Fq(R)9929 27188 y Fo(2)10424 27589 y Fs(,)435 b(w)-31 b(e)422 b(can)g(de\014ne)f Fq(\026)18847 27014 y Fo(\()p Fn(q)28 b Fo(\))18847 27719 y Fn(n)20027 27589 y Fs(\()p Fq(t)p Fs(\))422 b(as)f(a)h(function)h(of)f(\()p Fq(\020)82 b(;)184 b(\034)125 b Fs(\))423 b(in)f Fq(R)34939 27188 y Fo(2)35434 27589 y Fs(,)435 b(here)421 b Fq(\020)476 b Fs(=)394 b(\()p Fq(t;)170 b(~)-539 b(u)42800 27755 y Fn(q)43290 27589 y Fs(\))422 b(and)6075 28918 y Fq(\034)607 b Fs(=)483 b(\()p Fq(t;)162 b(~)-531 b(\027)10378 29084 y Fn(q)10868 28918 y Fs(\),)479 b Fq(~)-531 b(\027)12653 29084 y Fn(q)13616 28918 y Fs(b)31 b(eing)475 b(a)f(unit)h(v)-31 b(ector)475 b(in)g(the)f(direction)i(of)483 b Fq(~)-562 b(p)33653 29084 y Fn(q)34142 28918 y Fs(\(0\).)809 b(W)-92 b(e)474 b(consider)g Fq(\020)556 b Fs(and)6075 30246 y Fq(\034)d Fs(c)-31 b(hanging)431 b(in)d(in)-31 b(terv)-61 b(als)430 b(big)f(enough)g(to)h(co)-31 b(v)g(er)429 b(the)f(cell)i Fq(K)79 b Fs(.)670 b(Because)429 b(of)g(monotonicit)-31 b(y)-92 b(,)6075 31728 y(an)447 b(equation)i Fq(\026)12955 31153 y Fo(\()p Fn(q)28 b Fo(\))12955 31858 y Fn(n)14135 31728 y Fs(\()p Fq(t)p Fs(\))438 b(=)f Fq(\025)446 b Fs(can)h(b)31 b(e)446 b(solv)-31 b(ed)448 b(with)g(resp)31 b(ect)446 b(to)i Fq(\020)519 b Fs(=)436 b(\()p Fq(t;)170 b(~)-539 b(u)37473 31894 y Fn(q)37963 31728 y Fs(\),)467 b(i.e.,)h(the)447 b(curv)-31 b(e)6075 33312 y Fq(\026)6742 32737 y Fo(\()p Fn(q)28 b Fo(\))6742 33441 y Fn(n)7922 33312 y Fs(\()p Fq(t)p Fs(\))334 b(=)f Fq(\025)384 b Fs(can)h(b)31 b(e)384 b(describ)31 b(ed)383 b(b)-31 b(y)385 b(the)g(equation)h Fq(\020)415 b Fs(=)333 b Fq(f)30865 33478 y Fn(n)31467 33312 y Fs(\()p Fq(\034)125 b Fs(\).)540 b(It)384 b(is)h(not)g(di\016cult)h(to)f(sho)-31 b(w)6075 34640 y(that)422 b Fq(f)9067 34806 y Fn(n)9670 34640 y Fs(\()p Fq(\034)125 b Fs(\))422 b(is)f(a)h(piece-wise)g(con)-31 b(tin)g(uous)423 b(function.)29440 34211 y Fo(5)30529 34640 y Fs(Again,)436 b(b)31 b(ecause)421 b(of)h(monotonicit)-31 b(y)6075 36184 y(and)488 b(estimate)j(\(160\),)521 b(the)488 b(set)h Fl(j)p Fq(\026)21094 35609 y Fo(\()p Fn(q)28 b Fo(\))21094 36313 y Fn(n)22275 36184 y Fs(\()p Fq(t)p Fs(\))326 b Fl(\000)g Fq(\025)p Fl(j)506 b Fq(<)h(c)28354 36350 y Fn(b)28798 36184 y Fq(\025)29444 35782 y Fk(\000)p Fn(\016)31110 36184 y Fs(is)488 b(included)h(in)g(a)g(neigh)-31 b(b)31 b(orho)g(o)g(d)6075 37554 y(of)434 b(the)h(curv)-31 b(e)433 b Fq(\020)498 b Fs(=)415 b Fq(f)15191 37720 y Fn(n)15794 37554 y Fs(\()p Fq(\034)125 b Fs(\),)452 b(namely)435 b(in)f(the)g(set)g Fl(f)p Fq(t)416 b Fs(=)g(\()p Fq(\020)82 b(;)184 b(\034)125 b Fs(\))417 b(:)f Fl(j)p Fq(\020)372 b Fl(\000)289 b Fq(f)36217 37720 y Fn(n)36820 37554 y Fs(\()p Fq(\034)125 b Fs(\))p Fl(j)417 b Fq(<)e(c)40768 37720 y Fn(b)41213 37554 y Fq(\025)41859 37152 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fn(\016)45064 37554 y Fl(g)p Fs(.)6075 38924 y(Ob)-31 b(viously)-92 b(,)460 b(the)440 b(area)h(of)g(this)g(set)f(is)g(less)g(than)h Fq(c)27823 39090 y Fn(b)28268 38924 y Fq(\025)28914 38522 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fn(\016)32119 38924 y Fs(.)707 b(Th)-31 b(us,)458 b(the)441 b(area)g(of)g Fq(\037)42809 38522 y Fk(00)42809 39199 y Fo(2)43374 38924 y Fs(\()p Fq(q)40 b(;)184 b(n)p Fs(\))6075 40294 y(do)31 b(es)279 b(not)h(exceed)f Fq(c)14285 40460 y Fn(b)14730 40294 y Fq(\025)15376 39893 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fn(\016)18581 40294 y Fs(.)463 b(Considering)280 b(that)h(the)e(n)-31 b(um)g(b)31 b(er)280 b(of)g(non-empt)-31 b(y)281 b(sets)e Fq(\037)42809 39893 y Fk(00)42809 40569 y Fo(2)43374 40294 y Fs(\()p Fq(q)40 b(;)184 b(n)p Fs(\))6075 41665 y(is)427 b(no)h(larger)f(than)h Fq(c)15328 41841 y Fn(~)15162 42037 y(A)15828 42036 y(;b)16536 41665 y Fq(\025)17182 41263 y Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fn(")19683 41665 y Fs(,)442 b(w)-31 b(e)428 b(obtain)h(that)f(the)g (area)g(of)f Fq(\037)34514 41263 y Fk(00)34514 41940 y Fo(2)35507 41665 y Fs(and,)443 b(therefore,)f(of)428 b(the)6075 43281 y(\()p Fq(\025)7151 42879 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)10798 43281 y Fs(\)-neigh)-31 b(b)31 b(orho)g(o)g(d)371 b(of)e Fq(\037)20503 43447 y Fo(2)21368 43281 y Fs(do)31 b(es)369 b(not)h(exceed)f Fq(c)30014 43457 y Fn(~)29848 43653 y(A)30514 43652 y(;b)31222 43281 y Fq(\025)31868 42879 y Fk(\000)p Fn(\016)28 b Fk(\000)p Fn(")34155 43281 y Fs(.)493 b Fl(})6075 45530 y Fx(Lemma)1043 b(17.)604 b Fi(L)-57 b(et)475 b Fs(0)449 b Fq(<)g(")f(<)h Fs(1)p Fq(=)p Fs(8)p Fi(,)495 b Fs(1)p Fq(=)p Fs(4)450 b Fq(<)f(\014)507 b(<)448 b Fs(1)p Fq(=)p Fs(2)304 b Fl(\000)e Fs(2)p Fq(")p Fi(,)495 b Fs(0)449 b Fq(<)f Fs(2)p Fq(\016)491 b(<)449 b Fs(min)q Fl(f)p Fs(1)p Fq(=)p Fs(2)304 b Fl(\000)e Fq(\014)361 b Fl(\000)6075 46858 y Fs(2)p Fq(";)184 b Fs(2)p Fq(\014)306 b Fl(\000)246 b Fs(1)p Fq(=)p Fs(2)p Fq(;)184 b("=)p Fs(2)p Fl(g)p Fi(.)514 b(Then)397 b(the)e(length)i(of)f Fs(\()p Fq(\037)25807 47024 y Fo(1)26549 46858 y Fl(\\)246 b Fq(\037)28226 47024 y Fo(2)28723 46858 y Fs(\))185 b(\()p Fq(\025;)f(\016)42 b Fs(\))397 b Fi(admits)f(the)g(estimate:)20426 48962 y Fq(l)206 b Fs(\()q Fq(\037)22086 49128 y Fo(1)22828 48962 y Fl(\\)246 b Fq(\037)24505 49128 y Fo(2)25001 48962 y Fs(\))308 b Fq(<)f(c)27552 49138 y Fn(~)27386 49334 y(A)28108 48962 y Fq(\025)28754 48505 y Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fn(\016)31267 48962 y Fq(:)11830 b Fs(\(198\))7403 51077 y(Pro)31 b(of.)p 7403 51254 2971 45 v 537 w(W)-92 b(e)534 b(pro)-31 b(v)g(e)536 b(\(198\))i(b)-31 b(y)536 b(con)-31 b(tradiction.)994 b(Supp)31 b(ose)535 b(\(198\))j(do)31 b(es)535 b(not)h(hold.)992 b(Let)6075 52447 y(us)515 b(consider)h(the)h(\()p Fq(\025)15233 52045 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)18880 52447 y Fs(\)-neigh)-31 b(b)31 b(orho)g(o)g(d)518 b(of)e Fq(\037)28879 52613 y Fo(1)29720 52447 y Fl(\\)344 b Fq(\037)31495 52613 y Fo(2)31991 52447 y Fs(.)933 b(On)516 b(one)h(hand,)554 b(b)-31 b(y)516 b(Lemma)6075 53775 y(15)467 b(\()p Fq(S)533 b Fs(=)470 b Fq(\037)11314 53941 y Fo(1)12121 53775 y Fl(\\)311 b Fq(\037)13863 53941 y Fo(2)14359 53775 y Fs(\))467 b(the)f(area)h(of)g(this)g(neigh)-31 b(b)31 b(orho)g(o)g(d)467 b(is)f(greater)h(than)g Fq(c)38645 53941 y Fn(b)39090 53775 y Fq(\025)39736 53374 y Fk(\000)p Fo(3)p Fn(\016)41355 53775 y Fs(.)784 b(On)466 b(the)6075 55146 y(other)408 b(hand,)418 b Fq(\037)12823 55312 y Fo(1)13592 55146 y Fl(\\)271 b Fq(\037)15294 55312 y Fo(2)16163 55146 y Fl(\032)372 b Fq(\037)18089 55312 y Fo(2)18993 55146 y Fs(and,)418 b(therefore,)h(the)408 b(\()p Fq(\025)29528 54744 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)33175 55146 y Fs(\)-neigh)-31 b(b)31 b(orho)g(o)g(d)410 b(of)e Fq(\037)42958 55312 y Fo(1)43726 55146 y Fl(\\)272 b Fq(\037)45429 55312 y Fo(2)6075 56516 y Fs(b)31 b(elongs)532 b(to)h(the)e(\()p Fq(\025)14870 56114 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)18517 56516 y Fs(\)-neigh)-31 b(b)31 b(orho)g(o)g(d)534 b(of)e Fq(\037)28548 56682 y Fo(2)29044 56516 y Fs(.)980 b(By)532 b(Lemma)h(16,)574 b(the)532 b(area)g(of)g(the)6075 57886 y(latter)456 b(neigh)-31 b(b)31 b(orho)g(o)g(d)457 b(is)e(less)g(than)i Fq(c)22692 58052 y Fn(b)23136 57886 y Fq(\025)23782 57485 y Fk(\000)p Fn(")p Fk(\000)p Fn(\016)26070 57886 y Fs(,)477 b(and,)h(hence)455 b(the)g(area)h(of)g(the)g(\()p Fq(\025)41479 57485 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)45126 57886 y Fs(\)-)6075 59215 y(neigh)-31 b(b)31 b(orho)g(o)g(d)480 b(of)f Fq(\037)15200 59381 y Fo(1)16015 59215 y Fl(\\)319 b Fq(\037)17765 59381 y Fo(2)18740 59215 y Fs(is)479 b(less)f(than)i Fq(c)25286 59381 y Fn(b)25730 59215 y Fq(\025)26376 58813 y Fk(\000)p Fn(")p Fk(\000)p Fn(\016)28664 59215 y Fs(.)821 b(Considering)480 b(that)g(2)p Fq(\016)533 b(<)490 b(")p Fs(,)506 b(w)-31 b(e)480 b(see)6075 60543 y(that)433 b(t)-31 b(w)g(o)434 b(estimates)f(for)f(the)h(area)f(of)g (the)h(\()p Fq(\025)26183 60141 y Fk(\000)p Fo(1)p Fn(=)p Fo(2)p Fk(\000)p Fo(2)p Fn(\016)29830 60543 y Fs(\)-neigh)-31 b(b)31 b(orho)g(o)g(d)433 b(of)g Fq(\037)39661 60709 y Fo(1)40445 60543 y Fl(\\)288 b Fq(\037)42164 60709 y Fo(2)43092 60543 y Fs(are)432 b(in)6075 61871 y(con)-31 b(tradiction)372 b(for)d Fq(\025)g Fs(large)h(enough.)493 b Fl(})7403 63200 y Fs(The)369 b(pro)31 b(of)370 b(of)g(Lemma)g(5)p 7403 63592 10935 45 v 574 w(Lemma)575 b(5)f(immediately)j(follo)-31 b(ws)576 b(from)e(Corollary)h(12)g(and)6075 64528 y(Lemma)370 b(17.)p 6075 65862 6642 45 v 7403 66965 a Fo(5)7844 67309 y Fw(T)-78 b(o)333 b(sho)-26 b(w)333 b(this)g(w)-26 b(e)333 b(notice)g(that)g(the)g(curv)-26 b(es)333 b Ff(\026)24813 66837 y Fy(\()p Fd(q)24 b Fy(\))24813 67449 y Fd(n)25911 67309 y Fw(\()p Ff(t)p Fw(\))293 b(=)g Ff(\025)p Fw(,)338 b Ff(n)294 b Fw(=)f(1)p Ff(;)158 b(:::;)g(N)34177 67420 y Fd(q)34635 67309 y Fw(,)339 b(form)332 b(a)i(family)f(of)g(algebraic) 6075 68415 y(curv)-26 b(es,)324 b(since)f(the)f(determinan)-26 b(t)323 b(of)f Ff(P)20164 68526 y Fd(q)20622 68415 y Ff(H)68 b(P)22060 68526 y Fd(q)22840 68415 y Fw(is)323 b(a)g(p)26 b(olynomial)324 b(with)e(resp)26 b(ect)322 b(to)h Ff(t)36338 68538 y Fy(1)36799 68415 y Fw(,)i Ff(t)37725 68538 y Fy(2)38186 68415 y Fw(.)445 b(These)322 b(curv)-26 b(es)322 b(can)6075 69541 y(ha)-26 b(v)g(e)313 b(only)h(a)g(\014nite)g (n)-26 b(um)g(b)26 b(er)312 b(of)i(in)-26 b(tersections,)315 b(except)e(the)h(case)g(when)f(they)g(are)h(iden)-26 b(tical.)p eop end %%Page: 37 37 TeXDict begin 37 36 bop 7289 3345 a Fo(SPECTRAL)388 b(PR)-24 b(OPER)-71 b(TIES)386 b(OF)h(THE)h(PERIODIC)f(MA)-24 b(GNETIC)389 b(SCHR)35152 3149 y(\177)35033 3345 y(ODINGER)e(OPERA)-71 b(TOR)331 b(37)6075 6001 y Fs(3.5.)555 b Fx(Pro)35 b(of)509 b(of)f(the)g(Main)f(Theorems.)555 b Fs(Pro)31 b(of)370 b(of)g(Theorem)f(2)p 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Fq(\025)20353 16225 y Fn(j)20820 16059 y Fs(\()p Fq(t)p Fs(\))p Fl(j)309 b Fq(<)e(C)24821 16235 y Fn(~)24655 16431 y(A)25376 16059 y Fq(\025)26022 15658 y Fk(\000)p Fo(21)p Fn(=)p Fo(2)28546 16059 y Fs(,)360 b(here)c Fl(<)32447 15767 y Fs(^)32401 16059 y Fq(\025)33047 16225 y Fn(j)33514 16059 y Fs(\()p Fq(t)p Fs(\))308 b(=)g Fq(\025)356 b Fs(when)h Fq(t)307 b Fl(2)h Fq(\037)42577 16225 y Fk(\003)43085 16059 y Fs(\()p Fq(\025;)184 b(\016)42 b Fs(\).)6075 17590 y(By)425 b(Lemma)g(8,)440 b Fl(k)p Fq(H)90 b Fs(\()p Fq(t)p Fs(\))284 b Fl(\000)17715 17310 y Fs(^)17425 17590 y Fq(H)90 b Fs(\()p Fq(t)p Fs(\))p Fl(k)401 b Fq(<)f(C)22867 17766 y Fn(~)22701 17962 y(A)23422 17590 y Fq(\025)24068 17188 y Fk(\000)p Fo(10)25698 17590 y Fs(.)659 b(Using)425 b(standard)g(p)31 b(erturbation)426 b(considera-)6075 18918 y(tions)341 b([Kato)r(])f(w)-31 b(e)341 b(obtain)h(that)f Fq(\025)20142 19084 y Fn(j)20608 18918 y Fs(\()p Fq(t)p Fs(\))g(is)f(a)h(unique)f(eigen)-31 b(v)-61 b(alue)342 b(of)f Fq(H)90 b Fs(\()p Fq(t)p Fs(\))341 b(in)f(the)h(in)-31 b(terv)-61 b(al)341 b Fq(\017)p Fs(\()p Fq(\025)p Fs(\))6075 20247 y(and)i(estimate)h(\(78\))h(holds.)484 b(The)343 b(pro)31 b(of)343 b(of)h(\(77\))g(is)f(somewhat)i(more)e(complicated.) 486 b(In)343 b(fact,)6075 21575 y(let)318 b(us)e(note)i(that)g(the)g (relations)g(\(70\))h(and)f(\(71\))h(hold)e(not)h(only)g(for)g(real)f Fq(t)g Fs(in)g Fq(K)397 b Fs(but)317 b(also)h(for)6075 23023 y(all)403 b Fq(t)362 b Fs(:)h Fl(<)p Fq(t)f Fl(2)h Fq(K)18 b(;)184 b Fl(j=)p Fq(t)p Fl(j)364 b Fq(<)e(p)17145 23189 y Fn(j)17611 23023 y Fs(\(0\))19024 22622 y Fo(1)p Fn(=)p Fo(2+)p Fn(\014)40 b Fk(\000)p Fo(2)p Fn(")23190 23023 y Fs(,)410 b(the)403 b(mo)31 b(del)403 b(v)-61 b(alues)32545 22731 y(^)32499 23023 y Fq(\025)33145 23189 y Fn(j)33611 23023 y Fs(\()p Fq(t)p Fs(\))403 b(and)f(co)31 b(e\016cien)-31 b(ts)403 b(of)g(the)6075 24499 y(function)10551 24220 y(^)10397 24499 y(\011)11258 24665 y Fn(j)11725 24499 y Fs(\()p Fq(t;)184 b(x)p Fs(\))357 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b(ontinuous.)407 b Fw(Algebra)h(i)g(Analiz)g Fa(9)g Fw(\(1997\),)433 b(1,)f(32)408 b({48;)456 b(translation)409 b(in)f(St.P)-26 b(etersburg)8479 47402 y(Math.)314 b(J,)g Fa(9)p Fw(,)h(\(1998\),)g(1,)g(21)f({32.)6075 48509 y([BiSu2])471 b(Birman,)357 b(M.Sh.;)365 b(Suslina,)358 b(T.A.)348 b Fb(A)-24 b(bsolute)371 b(c)-49 b(ontinuity)372 b(of)f(the)h(two-dimensional)e(p)-49 b(erio)g(dic)372 b(mag-)8479 49616 y(netic)380 b(Hamiltonian)e(with)g(disc)-49 b(ontinuous)379 b(ve)-49 b(ctor-value)g(d)380 b(p)-49 b(otential.)355 b Fw(Algebra)h(i)g(Analiz)h Fa(10)f Fw(\(1998\),)8479 50723 y(4,)315 b(translation)g(in)e(St.P)-26 b(etersburg)314 b(Math.)f(J,)h Fa(10)p Fw(,)h(\(1999\),)h(4,)e(1)h({)f(26.)6075 51830 y([BiSu3])471 b(Birman,)264 b(M.Sh.;)271 b(Suslina,)264 b(T.A.)250 b Fb(Perio)-49 b(dic)280 b(magnetic)h(Hamiltonian)g(with)f (a)g(variable)h(metric.)f(The)8479 52936 y(pr)-49 b(oblem)360 b(of)e(absolute)h(c)-49 b(ontinuity.)333 b Fw(Algebra)i(i)f(Analiz)h Fa(11)g Fw(\(1999\),)341 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b(bsolute)328 b(c)-49 b(ontinuity)327 b(of)g(the)g(p)-49 b(erio)g(dic)327 b(magnetic)i(Schr\177)-486 b(odinger)327 b(op)-49 b(er)g(ator.)300 b Fw(In)-26 b(v)g(en)g(tiones) 8479 54708 y(Mathematicae)316 b Fa(137)e Fw(\(1999\),)i(1,)f(85)f({)g (112.)6075 55815 y([Th])679 b(Thomas,)388 b(L.E.)372 b Fb(Time-dep)-49 b(endent)396 b(appr)-49 b(o)g(ach)394 b(to)g(sc)-49 b(attering)394 b(fr)-49 b(om)394 b(impuriries)f(in)h(a)g (crystal.)371 b Fw(Com-)8479 56922 y(m)-26 b(un.)314 b(Math.)g(Ph)-26 b(ys.)313 b Fa(33)h Fw(\(1973\),)i(pp)d(335)i({)f (343.)6075 58029 y([V1])707 b(V)-78 b(eliev,)291 b(O.A.)286 b Fb(Asymptotic)312 b(F)-73 b(ormulae)314 b(for)e(Eigenvalues)i(of)f(a) g(Perio)-49 b(dic)313 b(Schr\177)-486 b(odinger)314 b(Op)-49 b(er)g(ator)313 b(and)8479 59135 y(Bethe-Sommerfeld)336 b(Conje)-49 b(ctur)g(e.)309 b Fw(F)-78 b(unctional.)309 b(Anal.)g(i)g(Prilozhen.,)h Fa(21)f Fw(\(1987\),)i(no.)e(2,)h(1{15;)i (Engl.)8479 60242 y(transl.:)420 b(F)-78 b(unctional)314 b(Anal.)g(Appl.,)g Fa(21)g Fw(\(1987\),)i(pp.)e(87)g({99.)6075 61349 y([V2])707 b(V)-78 b(eliev)307 b(O.A.)p Fb(The)334 b(Perio)-49 b(dic)333 b(Multidimensional)f(Schr)-49 b(o)g(dinger)335 b(Op)-49 b(er)g(ator,)334 b(Part)f(2,)h(Asymptotic)f(F)-73 b(or-)8479 62456 y(mulae)341 b(for)e(Blo)-49 b(ch)341 b(F)-73 b(unctions)340 b(and)g(F)-73 b(ermi)340 b(Surfac)-49 b(es.)314 b Fw(preprin)-26 b(t,)313 b(mp-arc)g(01-463,)j(32)f(pp.)6075 63563 y([V3])707 b(V)-78 b(eliev,)645 b(O.A.)579 b Fb(Asymptotic)585 b(F)-73 b(ormulae)587 b(for)e(Blo)-49 b(ch)587 b(F)-73 b(unctions)586 b(of)g(Multidimensional)f(Perio)-49 b(dic)8479 64670 y(Schr\177)-486 b(odinger)392 b(op)-49 b(er)g(ator)392 b(and)f(Some)i(of)e(Their)g(Applic)-49 b(ations.)368 b Fw(Sp)26 b(ectral)369 b(Theory)g(of)g(Op)26 b(erators)369 b(and)8479 65777 y(its)314 b(Applications,)i Fa(9)p Fw(,)e(Baku,)g (1989,)i(pp.)d(59)i({76)g(\(in)f(Russian\).)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0312091918723--