Content-Type: multipart/mixed; boundary="-------------0106220917660" This is a multi-part message in MIME format. ---------------0106220917660 Content-Type: text/plain; name="01-224.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-224.keywords" quasi-periodic Schr dinger equation; adiabatic limit, complex WKB method, monodromy matrix, absolutely continuous spectrum, Bloch-Floquet solutions ---------------0106220917660 Content-Type: application/postscript; name="adiab-bmm.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="adiab-bmm.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: adiab-bmm.dvi %%Pages: 28 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips adiab-bmm.dvi -o adiab-bmm.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.06.22:1606 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N /vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N /rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N /@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{ /hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B /@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{ /urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known {userdict/md get type/dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{ itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack} if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{ noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{ Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale }if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState save N userdict maxlength dict begin/magscale true def normalscale currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts /psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR/showpage{}N/erasepage{}N/copypage{}N/p 3 def @MacSetUp}N/doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N/endTexFig{end psf$SavedState restore}N/@beginspecial{SDict begin/SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count/ocount X/dcount countdictstack N}N/@setspecial{ CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR }{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if/showpage{}N/erasepage{}N/copypage{}N newpath}N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{end} repeat grestore SpecialSave restore end}N/@defspecial{SDict begin}N /@fedspecial{end}B/li{lineto}B/rl{rlineto}B/rc{rcurveto}B/np{/SaveX currentpoint/SaveY X N 1 setlinecap newpath}N/st{stroke SaveX SaveY moveto}N/fil{fill SaveX SaveY moveto}N/ellipse{/endangle X/startangle X /yrad X/xrad X/savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet TeXDict begin 39158280 55380996 1000 600 600 (adiab-bmm.dvi) @start %DVIPSBitmapFont: Fa cmtt9 9 24 /Fa 24 122 df<007FB512F8B612FCA46C14F81E067C9927>45 D<121EEA7F80A2EAFFC0 A4EA7F80A2EA1E000A0A728927>I<130E131FA25B5BA25B5A5A127FB5FCA213BFEA7E3F 1200B3AA003FB512805A15C01580A21A2F79AE27>49 D51 D64 D<3803FFC0000F13F04813FC4813FF811380EC1FC0381F000F000480 C71207A2EB0FFF137F0003B5FC120F5A383FFC07EA7FC0130012FE5AA46C130F007F131F EBC0FF6CB612806C15C07E000313F1C69038807F8022207C9F27>97 DI100 DII104 D<130F497E497EA46D5A6D C7FC90C8FCA7383FFF80487FA37EEA000FB3A4007FB512F0B6FC15F815F07E1D2F7BAE27 >I107 D<387FFF80B57EA37EEA000FB3B2007FB512F8B612FCA36C14F81E2E7CAD27>I<397F07 C01F3AFF9FF07FC09039FFF9FFE091B57E7E3A0FFC7FF1F89038F03FC001E0138001C013 00A3EB803EB03A7FF0FFC3FF486C01E3138001F913E701F813E36C4801C313002920819F 27>I<387FE07F39FFF1FFC001F713F090B5FC6C80000313C1EC01FCEBFE005B5BA25BB0 3A7FFF83FFE0B500C713F0A36C018313E024207F9F27>II<387FE0FFD8FFF313C090B512F0816C800003EB81FE49C6 7E49EB3F8049131F16C049130FA216E01507A6150F16C07F151F6DEB3F80157F6DEBFF00 9038FF83FEECFFFC5D5D01F313C0D9F0FEC7FC91C8FCAC387FFF80B57EA36C5B23317F9F 27>I<397FFC03FC39FFFE0FFF023F13804A13C0007F90B5FC39007FFE1F14F89138F00F 809138E002004AC7FC5CA291C8FCA2137EAD007FB57EB67EA36C5C22207E9F27>114 D<9038FFF3800007EBFFC0121F5A5AEB803F38FC000F5AA2EC07806C90C7FCEA7F8013FC 383FFFF06C13FC000713FF00011480D8000F13C09038003FE014070078EB03F000FC1301 A27E14036CEB07E0EBE01F90B512C01580150000FB13FC38707FF01C207B9F27>I<133C 137EA8007FB512F0B612F8A36C14F0D8007EC7FCAE1518157EA415FE6D13FC1483ECFFF8 6D13F06D13E0010313C0010013001F297EA827>I<397FE01FF8486C487EA3007F131F00 031300B21401A21403EBFC0F6CB612E016F07EEB3FFE90390FF87FE024207F9F27>I<3A 7FFC0FFF80486C4813C0A36C486C13803A07C000F800EBE00100035CA2EBF00300015CA2 EBF80700005CA390387C0F80A36D48C7FCA3EB3F3FEB1F3EA214FE6D5AA36D5AA26D5A22 207E9F27>I<3A7FFC0FFF80486C4813C0A36C486C13803A07E000F800000313015D13F0 0001130301F85B1200A26D485A137CA290387E0F80133EA2011F90C7FC5CA2130F149E14 BE130714FC1303A25C1301A25CA213035CA213075C1208EA3E0F007F5B131FD87E7FC8FC EA7FFE6C5A5B6C5AEA07C022317E9F27>121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb cmex8 8 1 /Fb 1 83 df82 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc cmsy6 6 5 /Fc 5 50 df0 D<136013701360A20040132000E0137038F861 F0387E67E0381FFF803807FE00EA00F0EA07FE381FFF80387E67E038F861F038E0607000 40132000001300A21370136014157B9620>3 D<14301438B1B712FCA3C70038C7FCAFB7 12FCA326277CA430>6 D48 D<01FEEC0FE02603FFC0EB3FF800 0F01F0EBFE3E3B1F0FF801F0073C3C01FC07C003803B3000FE0F00010070D93F1EEB00C0 0060EB1F9C00E0D90FF81460485C14076E7E6E7E81020315E00060D9073F14C091390F1F 80016C90261E0FE01380003890397C07F0073C1C01F003FE1F003B0F8FE001FFFE3B03FF 80007FF8C648C7EA0FE033177C953D>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmmi6 6 14 /Fd 14 111 df14 D<13031306A3EB07FC14FE 130FEB3CFCEB70005B485A485A48C7FC12065A121C121812385AA35AA77EA27E127E6C7E EA1FF86CB4FC6C1380C613C0EB0FE01301A21300A214C0EA01C13800FF80EB3E00172E7C A21C>16 D<0003B512FC120F5A4814F8397818180000E01338EAC03000001330A2137013 60EBE070A21478EA01C0A21203EB807C0007133C143E380F003C000613181E167D9424> 25 D<90381FFFFC90B5FC5A4814F83907C07C00380F003C001E131C48131E12381278A2 485BA35C1470007013F0495A6C485AD81C0FC7FCEA0FFEEA03F01E167E9424>27 D34 D77 D83 D100 DI< 1338137CA2137813701300A7EA0780EA1FC0EA38E01230EA60F0EAC1E0A3EA03C0A3EA07 80A2EA0F0013041306EA1E0CA21318121CEA1E70EA0FE0EA07800F237DA116>105 D<1418143C147CA214381400A7EB0780EB1FE01338EB60F013C0A2EA0180A2380001E0A4 EB03C0A4EB0780A4EB0F00A4131EA21238EA783CEAF8381378EA70F0EA7FC0001FC7FC16 2D81A119>I108 D<000F017E13FC3A1F81FF 83FF3B31C383C707803A61EE03CC039026EC01F813C0D8C1F813F013F001E013E0000390 3903C0078013C0A2EE0F003907800780A2EE1E041706270F000F00130C163C1718A2001E 011EEB1C70EE1FE0000C010CEB07802F177D9536>I<000F13FC381FC3FF3931C7078038 61EC0301F813C0EAC1F0A213E03903C00780A3EC0F00EA0780A2EC1E041506D80F00130C 143C15181538001EEB1C70EC1FE0000CEB07801F177D9526>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe cmti8 8 3 /Fe 3 112 df99 D<137CEA0FFCA21200A213F8A2 1201A213F0A21203A213E0A21207A213C0A2120FA21380A2121FA21300A25AA2123EA212 7EA2127CA2EAFC30137012F8A213F013E012F012F113C012FBEA7F80EA1E000E2F7AAD12 >108 D111 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmex10 10.95 17 /Ff 17 114 df<140E141E143C147814F01301EB03E0EB07C0A2EB0F80EB1F00A2133E13 7E137C13FC5B1201A2485AA3485AA2120F5BA2121FA25BA2123FA390C7FCA25AA6127E12 FEB3A4127E127FA67EA27FA3121FA27FA2120FA27F1207A26C7EA36C7EA212007F137C13 7E133E7FA2EB0F80EB07C0A2EB03E0EB01F013001478143C141E140E176C72832A>0 D<12E07E12787E7E121F6C7E6C7EA26C7E6C7EA26C7E7F137C137E133E133FA2EB1F80A3 EB0FC0A214E01307A214F0A21303A214F8A31301A214FCA6130014FEB3A414FC1301A614 F8A21303A314F0A21307A214E0A2130F14C0A2EB1F80A3EB3F00A2133E137E137C13FC5B 485AA2485A485AA2485A48C7FC121E5A5A5A5A176C7C832A>I<12F0B3B3B3A5043B7381 1E>12 D16 D<12F07E127C7E7E6C7E6C7E7F6C7E6C7E12007F137E7FA26D7E6D7EA26D 7EA26D7E6D7EA26D7EA280147E147F80A26E7EA281140FA281140781A21403A281A21401 81A3140081A4157E157FA5811680A9ED1FC0B3A9ED3F80A916005DA5157E15FEA45D1401 A35D1403A25DA21407A25D140F5DA2141F5DA24AC7FCA25C147E14FE5CA2495AA2495A49 5AA2495AA2495A49C8FCA2137E5B5B1201485A485A5B485A48C9FC123E5A5A5A22A37D83 36>I<17F01601EE03E0EE07C0EE0F80EE1F00163E5E16FC4B5A4B5A4B5A5E150F4B5A4B C7FCA2157E5D14015D4A5AA24A5A140F5D141F5D143F4AC8FCA214FEA2495AA2495AA249 5AA3495AA2495AA3495AA349C9FCA25B5BA312015BA21203A25BA21207A25BA2120FA35B A2121FA45BA2123FA65B127FAA48CAFCB3AE6C7EAA123F7FA6121FA27FA4120FA27FA312 07A27FA21203A27FA21201A27F1200A37F7FA26D7EA36D7EA36D7EA26D7EA36D7EA26D7E A26D7EA2147FA26E7E141F81140F8114076E7EA26E7E811400157E81A26F7E6F7E150782 6F7E6F7E6F7E167C8282EE0F80EE07C0EE03E0EE01F016002CDA6D8343>I<12F07E127C 7E7E6C7E6C7E6C7E7F6C7E6C7E137E133E133F6D7E6D7EA26D7E6D7E8013016D7EA2147E 147F8081141F816E7EA26E7EA26E7EA26E7EA26E7EA3157FA26F7EA36F7EA36F7EA28215 07A3821503A282A21501A282A21500A282A382A21780A4163FA217C0A6161F17E0AAEE0F F0B3AEEE1FE0AA17C0163FA61780A2167FA41700A25EA35EA21501A25EA21503A25EA215 075EA3150F5EA24B5AA34B5AA34BC7FCA215FEA34A5AA24A5AA24A5AA24A5AA24A5A5D14 3F92C8FC5C147E5CA2495A13035C495A495AA2495A49C9FC133E137E5B485A485A5B485A 485A48CAFC123E5A5A5A2CDA7D8343>I<1778EE01F81607161FEE7FE0EEFF8003031300 ED07FC4B5A4B5A4B5A4B5A4B5A93C7FC4A5A14035D14075D140F5DA34A5AB3B3B3A9143F 5DA2147F5DA24AC8FCA2495A13035C495A495A495A495A495A49C9FC485AEA07F8485AEA 3FC0B4CAFC12FCA2B4FCEA3FC0EA0FF06C7EEA01FE6C7E6D7E6D7E6D7E6D7E6D7E6D7E80 13016D7EA26E7EA281143FA281141FB3B3B3A96E7EA38114078114038114016E7E826F7E 6F7E6F7E6F7E6F7E6FB4FC03001380EE7FE0EE1FF816071601EE00782DDA758344>26 D<12F012FCB47E7FEA3FF0EA0FFCEA03FE6C7E6C13C0EB3FE06D7E6D7E6D7E1303806D7E 7F81147F81143FA381141FB3B3B3A881140FA36E7EA26E7EA26E7EA26E7E6F7E82153F6F 7E6F7E6F7EED01FE6F7EEE7F80EE1FE0EE07F81601A21607EE1FE0EE7F80EEFF004B5AED 07F84B5A4B5A4B5A157F5E4BC7FC4A5AA24A5AA24A5AA24A5AA3141F5DB3B3B3A8143F5D A3147F5D14FF92C8FC5B495A5C1307495A495A495AEBFFC04890C9FC485AEA0FFCEA3FF0 EAFFC05B00FCCAFC12F02DDA758344>I[<173E177E17FCEE01F8160317F0EE07E0EE0FC0 EE1F80163F1700167E16FE4B5A5E15034B5A5E150F4B5AA24B5A4BC7FCA215FEA24A5AA2 4A5AA24A5A140F5D141F5DA2143F5D147F92C8FC5CA2495AA25C1303A2495AA3495AA349 5AA3133F5CA3495AA313FF91C9FCA35A5BA31203A25BA31207A25BA3120FA35BA3121FA5 5BA2123FA75B127FAD5B12FFB3B3A4127F7FAD123F7FA7121FA27FA5120FA37FA31207A3 7FA21203A37FA21201A37F7EA380137FA36D7EA380131FA36D7EA36D7EA36D7EA2130180 A26D7EA28081143F81141FA281140F8114076E7EA26E7EA26E7EA2157FA26F7E6F7EA26F 7E1507826F7E1501826F7E167E821780161FEE0FC0EE07E0EE03F017F81601EE00FC177E 173E>47 272 107 131 72 32 D[<12F87E127E7E7F121F6C7E6C7E6C7E7F12016C7E7F 137F7F806D7E130F806D7EA26D7E6D7EA26D7EA2147FA26E7EA26E7E81140F811407A281 140381140181A26E7EA28182A26F7EA36F7EA36F7EA3821507A36F7EA3821501A38281A3 1780A2167FA317C0A2163FA317E0A3161FA317F0A5160FA217F8A7160717FCAD160317FE B3B3A417FC1607AD17F8160FA717F0A2161FA517E0A3163FA317C0A3167FA21780A316FF A21700A35D5EA315035EA34B5AA3150F5EA34B5AA34B5AA34B5AA293C7FC5DA24A5AA25D 14035D14075DA2140F5D141F5D4A5AA24AC8FCA214FEA2495AA2495A495AA2495A5C131F 495A91C9FC5B13FE5B485A12035B485A485A485A123F90CAFC127E5A5A>47 272 125 131 72 I82 D88 D90 D<003C1B0F007EF31F80B4F33FC0B3B3B3B3AF6D1A7F007F1C80A46D1AFF003F1C006D61 A26C6C4F5AA26C6C4F5AA26C6C4F5AA26C6C4F5A6D193F6C6D4E5A6E18FF6C6D4D5B6D6C 4D5B6D6C4D90C7FC6D6C4D5A6DB4EF3FFC6D6D4C5A6D01E04B485A6D01F803075B6D01FE 031F5B91267FFFC091B55A6E01FE011F91C8FC020F90B712FC6E5F020117E06E6C168003 1F4BC9FC030315F0DB007F1480040301F0CAFC5A7F7B7F65>I<1C601CF01B01A2F303E0 A2F307C0A2F30F80A2F31F00A21B3EA263A263A2631A01A2505AA2505AA2505AA250C7FC A21A3EA262A262A24F5AA24F5AA24F5AA24F5AA24FC8FCA2193EA261A261A24E5AA26118 03130C011C4C5A133E01FE4C5A487E484DC9FC380F7F80001C173E123826F03FC05D1240 C66C6C5DA26D6C4A5AA24D5A6D7E4D5A6D7E4D5A6D7E4DCAFCA26D6C143EA26E6C5BA26E 6C5BA24C5AEC1FE04C5AEC0FF05F913807F807A24C5AEC03FC4CCBFCEC01FE163EEC00FF 5EA26F5AA26F5AA26F5AA25E150F5E6FCCFC546D77835B>112 D<1C601CF0A21B01A21C E01B03A21CC01B07A21C801B0FA21C0063A21B1E1B3EA21B3C1B7CA21B78A21BF8A2631A 01A2631A03A2631A07A2631A0FA298C7FC62A21A1E1A3EA21A3C1A7CA21A78A21AF8A262 1901A2621903A2621907A262190FA297C8FC61A2191E193EA2193CA2197CA2197819F8A2 611801A26118031304010C5F011C1607131E013E5F017E160F13FE486C94C9FC485F5A26 067F80151E120C0018173E1238486C6C153C00E0177C1240C66C6C157818F8A2606D6C14 01A2606D6C1403A26017076D7E60170F6D7E95CAFCA25F6D7E171E173E6E7E173C177CA2 6E6C137817F8A26E6C5B1601A25F91380FF003A25F913807F807A25F160FEC03FC94CBFC A26E6C5AA2161E163EEC00FF163C167CED7FFC5EA36F5AA36F5AA35E150FA25E150793CC FC54A477835B>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmr6 6 5 /Fg 5 127 df<1438B2B712FEA3C70038C7FCB227277C9F2F>43 D<13FF000313C0380781E0380F00F0001E137848133CA248131EA400F8131FAD0078131E A2007C133E003C133CA26C13786C13F0380781E03803FFC0C6130018227DA01E>48 D<13E01201120712FF12F91201B3A7487EB512C0A212217AA01E>II<000F1380381F81C0383FF780387BFF00EAE07EEA403C12067AA1 1E>126 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh msbm8 8 3 /Fh 3 91 df78 D82 D<001FB612FCA23A183E00701801F0EB6038D81BC0EBE030001FC7EA C070003E010113E0003C90380380C01501003801071380EC0603003090380E0700EC1C06 EC180EC7EA380CEC301CEC7038ECE030ECC07001015B4A5AEB03819038070180EB0603D9 0E07C7FCEB0C06EB1C0EEB380CEB301CD970381303EBE030EBC070000101601307EB80E0 260381C0130626070180130ED80603141E000E90C7FCD80C07143ED81C0E1476D8380C14 E6D8301CEB01CED87018EB078CD86038EB3E0CB712FCA2282E7EAD38>90 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fi cmti10 10.95 59 /Fi 59 123 df11 D<933807FF80043F13E09338FE00F8DB01F0133EDB07E0130E4B48131F4C137F031F14FF 4BC7FCA218FE157E1878180015FE5DA31401A25DA414030103B712F0A218E0903A0003F0 00070207140F4B14C0A3171F020F15805DA2173F1800141F5D5F177EA2143F92C712FE5F A34A1301027EECF81CA3160302FEECF03C4A1538A21878187013014A010113F018E09338 00F1C0EF7F804948EC1F0094C7FCA35C1307A2001E5B127F130F00FF5BA249CAFC12FEEA F81EEA703CEA7878EA1FF0EA07C0385383BF33>I14 D34 D39 DI<14031580A2 EC01C0EC00E0A21570A215781538153CA3151EA4151FA2150FA7151FA9153FA2153EA315 7EA2157CA215FCA215F8A21401A215F0A2140315E0A2140715C0A2EC0F80A2141F15005C 143EA25CA25CA2495A5C1303495A5C130F49C7FC131E5B137C5B5B485A485A485A48C8FC 121E5A12705A5A205A7FC325>I 44 D<387FFFFCA3B5FCA21605799521>I<120FEA3FC0127FA212FFA31380EA7F00123C0A 0A77891C>I<15031507150F151F151E153E157EEC01FEEC03FC1407141FEB01FF90380F FBF8EB1FC3EB0E07130015F0A2140FA215E0A2141FA215C0A2143FA21580A2147FA21500 A25CA25CA21301A25CA21303A25CA21307A25CA2130FA25CA2131FA25CEB7FE0B612F0A2 15E0203D77BC2E>49 D<15FE913803FFC091380F01F091383C00F84A137C4A7F4948133F 49487F4A148049C7FC5BEB0E0C011E15C0EB1C0EEB3C06133813781370020E133FD9F00C 148013E0141C0218137F00011600EBC0384A13FEEC600102E05B3A00E3C003F89039FF00 07F0013C495A90C7485A5E037FC7FC15FC4A5A4A5AEC0FC04AC8FC147E14F8EB03E0495A 011FC9FC133E49141801F0143C48481438485A1678485A48C85A120E001E4A5AD83FE013 0301FF495A397C3FF01FD8780FB55AD8700391C7FCD8F0015B486C6C5A6E5AEC07C02A3F 79BC2E>II 54 DI<131EEB3F80137FEBFFC05AA214806C13005B133C90C7FCB3120FEA3FC012 7FA212FFA35B6CC7FC123C122777A61C>58 DI<171C173C177C A217FCA216011603A21607A24C7EA2161DA216391679167116E1A2ED01C1A2ED03811507 1601150EA2031C7FA24B7EA25D15F05D4A5AA24A5AA24AC7FC5C140E5C021FB6FC4A81A2 0270C7127FA25C13015C495AA249C8FCA2130E131E131C133C5B01F882487ED807FEEC01 FFB500E0017FEBFF80A25C39417BC044>65 D<49B712C018F818FE903B0003FC0001FF94 38007F804BEC3FC0A2F01FE014074B15F0180FA2140F5D181FA2021F16E05D183F19C002 3FED7F804B14FF19004D5A027F4A5A92C7EA07F0EF1FE0EF7F804AD903FEC7FC92B512F0 17FE4AC7EA3F800101ED1FE04A6E7E17078401036F7E5CA30107825CA3010F5E4A1407A2 60011F150F5C4D5A60013F153F4A4A5A4D5A017F4A90C7FC4C5A91C7EA0FF849EC3FF0B8 12C094C8FC16F83C3E7BBD40>I<9339FF8001C0030F13E0033F9038F803809239FF807E 07913A03FC001F0FDA0FF0EB071FDA1FC0ECBF00DA7F806DB4FC4AC77E495AD903F86E5A 495A130F4948157E4948157C495A13FF91C9FC4848167812035B1207491670120FA2485A 95C7FC485AA3127F5BA312FF5BA490CCFCA2170FA2170EA2171E171C173C173817786C16 706D15F04C5A003F5E6D1403001F4B5A6D4AC8FC000F151E6C6C5C6C6C14F86C6C495A6C 6CEB07C090397FC03F8090261FFFFEC9FC010713F0010013803A4272BF41>I<49B712C0 18F818FE903B0003FE0003FF9438007F804BEC1FC0F00FE0F007F014074BEC03F8F001FC A2140F4BEC00FEA3141F4B15FFA3143F5DA3027F5D5DA219FE14FF92C81203A34917FC4A 1507A219F813034A150F19F0A20107EE1FE05CF03FC0A2010FEE7F804A16006060011F4B 5A4A4A5A4D5AA2013F4B5A4AEC3FC04DC7FC017F15FEEE03FC4AEB0FF001FFEC7FE0B812 8004FCC8FC16E0403E7BBD45>I<49B812F0A390260003FEC7123F180F4B1403A2F001E0 14075DA3140F5D19C0A2141F5D1770EFF003023F02E013804B91C7FCA21601027F5CED80 03A2160702FFEB1F8092B5FCA349D9003FC8FC4A7F82A20103140E5CA2161E0107141C5C A293C9FC130F5CA3131F5CA3133F5CA2137FA25C497EB612E0A33C3E7BBD3B>70 D<49B648B6FC495DA2D9000390C7000313004B5D4B5DA2180714074B5DA2180F140F4B5D A2181F141F4B5DA2183F143F4B5DA2187F147F4B5DA218FF91B8FC96C7FCA292C712015B 4A5DA2170313034A5DA2170713074A5DA2170F130F4A5DA2171F131F4A5DA2173F133F4A 5DA2017F157FA24A5D496C4A7EB66CB67EA3483E7BBD44>72 D<49B6FC5BA2D900031300 5D5DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA314FF92C7FCA35B5CA313035CA3 13075CA3130F5CA3131F5CA3133F5CA2137FA25C497EB67EA3283E7BBD23>I<49B612C0 A25FD9000390C8FC5D5DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA314FF92C9FC A35B5CA313035C18C0EF01E0010716C05C17031880130F4A140718005F131F4A141EA217 3E013F5D4A14FC1601017F4A5A16074A131F01FFECFFF0B8FCA25F333E7BBD39>76 D<49B5933807FFFC496062D90003F0FC00505ADBBF805E1A771AEF1407033F923801CFE0 A2F1039F020FEE071F020E606F6C140E1A3F021E161C021C04385BA2F1707F143C023804 E090C7FCF001C0629126780FE0495A02705FF00700F00E0114F002E0031C5BA2F0380301 0116704A6C6C5D18E019070103ED01C00280DA03805BA2943807000F13070200020E5C5F DB03F8141F495D010E4B5CA24D133F131E011CDAF9C05CEEFB80197F013C6DB4C7FC0138 95C8FC5E01784A5C13F8486C4A5CD807FE4C7EB500F04948B512FE16E01500563E7BBD52 >I79 D<49B77E18F018FC903B0003FE0003FEEF00FF4BEC7F80F03FC00207151F19 E05DA2020F16F0A25DA2141FF03FE05DA2023F16C0187F4B1580A2027FEDFF00604B495A 4D5A02FF4A5A4D5A92C7EA3FC04CB4C7FC4990B512FC17E04ACAFCA21303A25CA21307A2 5CA2130FA25CA2131FA25CA2133FA25CA2137FA25C497EB67EA33C3E7BBD3E>I<49B612 FCEFFF8018F0903B0003FE000FF8EF03FE4BEB00FF8419800207ED3FC05DA219E0140F5D A3021FED7FC05DA2F0FF80143F4B15004D5A60027F4A5A4B495A4D5AEF3F8002FF02FEC7 FC92380007F892B512E01780499038000FE04A6D7E707E707E0103814A130083A213075C A25E130F5C5F1603131F5CA3013F020714404A16E05F017F160119C04A01031303496C16 80B6D8800113079438FE0F009338007E1ECAEA3FFCEF07F03B407BBD42>82 D<92391FE00380ED7FFC913A01FFFE0700913907F01F8F91390FC007DF4AC66CB4FC023E 6D5A4A130014FC495A4948147CA2495AA2010F15785CA3011F1570A46E91C7FCA2808014 FE90380FFFE015FC6DEBFF8016E06D806D806D6C7F141F02037FEC003FED07FF1501A281 A282A212075A167E120EA2001E15FE5EA25E003E14015E003F14034B5A486C5C150F6D49 5A6D49C8FCD8F9F0137C39F8FE01F839F03FFFF0D8E00F13C026C001FEC9FC314279BF33 >I<48B9FCA25A903AFE001FF00101F89138E0007FD807E0163E49013F141E5B48C75BA2 001E147FA2001C4B131C123C003814FFA2007892C7FC12704A153C00F01738485CC71600 1403A25DA21407A25DA2140FA25DA2141FA25DA2143FA25DA2147FA25DA214FFA292C9FC A25BA25CA21303A25CEB0FFE003FB67E5AA2383D71BC41>I<277FFFFE01B500FC90B512 E0B5FCA20003902680000790C7380FFC006C90C701FCEC07F049725A04035EA26350C7FC A20407150EA2040F5D1A3C041F153862163B6216734F5A6D14E303014B5A6C15C303034B C8FC1683DB0703140E191E030E151C61031C7F61ED380161157003F04A5A15E002014B5A 15C0DA03804AC9FC60DA0700140E60140E605C029C5D14B8D97FF85D5C715A5C4A5DA24A 92CAFC5F91C7FC705A137E5F137C5F137801705D53406EBD5B>87 D<010C1306011C130E0178133C01E01370484813E04913C0000313013907000380000EEB 0700000C1306001C130E0018130C0038131C003013180070133800601330A200E0137000 CFEB678039FFC07FE0A6018013C0397F003F80003CEB1E001F1C69BE2F>92 D<147E49B47E903907C1C38090391F80EFC090383F00FF017E137F4914804848133F485A A248481400120F5B001F5C157E485AA215FE007F5C90C7FCA21401485C5AA21403EDF038 5AA21407EDE078020F1370127C021F13F0007E013F13E0003E137FECF3E1261F01E313C0 3A0F8781E3803A03FF00FF00D800FC133E252977A72E>97 DIIII<167C4BB4FC923807C7 8092380F83C0ED1F87161FED3F3FA2157EA21780EE0E004BC7FCA414015DA414035DA301 03B512F8A390260007E0C7FCA3140F5DA5141F5DA4143F92C8FCA45C147EA414FE5CA413 015CA4495AA4495AA4495A121E127F5C12FF49C9FCA2EAFE1EEAF83C1270EA7878EA3FE0 EA0F802A5383BF1C>III<1478EB01FCA21303A314F8EB00E01400AD137C48B4FC38038F80EA0707000E13 C0121E121CEA3C0F1238A2EA781F00701380A2EAF03F140012005B137E13FE5BA212015B A212035B1438120713E0000F1378EBC070A214F0EB80E0A2EB81C01383148038078700EA 03FEEA00F8163E79BC1C>I<1507ED1FC0A2153FA31680ED0E0092C7FCADEC07C0EC3FF0 EC78F8ECE07CEB01C01303EC807EEB0700A2010E13FE5D131E131CEB3C01A201005BA214 03A25DA21407A25DA2140FA25DA2141FA25DA2143FA292C7FCA25CA2147EA214FEA25CA2 13015CA2121C387F03F012FF495A5C495A4848C8FCEAF83EEA707CEA3FF0EA0FC0225083 BC1C>I IIIII<903903E001F890390FF807FE903A1E7C1E0F80903A1C3E3C07 C0013C137801389038E003E0EB783F017001C013F0ED80019038F07F0001E015F8147E16 03000113FEA2C75AA20101140717F05CA20103140F17E05CA20107EC1FC0A24A1480163F 010F15005E167E5E131F4B5A6E485A4B5A90393FB80F80DA9C1FC7FCEC0FFCEC03E049C9 FCA2137EA213FEA25BA21201A25BA21203A2387FFFE0B5FCA22D3A80A72E>I<027E1360 903901FF81E0903807C1C390391F80E7C090383F00F7017E137F5B4848EB3F80485AA248 5A000F15005B121F5D4848137EA3007F14FE90C75AA3481301485CA31403485CA314074A 5A127C141F007E133F003E495A14FF381F01EF380F879F3903FF1F80EA00FC1300143F92 C7FCA35C147EA314FE5CA21301130390B512F05AA2233A77A72A>IIII<137C48B4141C26038F80137EEA0707000E7F001E15FE121CD83C 0F5C12381501EA781F007001805BA2D8F03F1303140000005D5B017E1307A201FE5C5B15 0F1201495CA2151F0003EDC1C0491481A2153F1683EE0380A2ED7F07000102FF13005C01 F8EBDF0F00009038079F0E90397C0F0F1C90391FFC07F8903907F001F02A2979A731>I< 017CEB01C048B4EB07F038038F80EA0707000E01C013F8121E001C1403EA3C0F0038EC01 F0A2D8781F130000705BA2EAF03F91C712E012005B017E130116C013FE5B150300011580 5BA2ED07001203495B150EA25DA25D1578000114706D5B0000495A6D485AD97E0FC7FCEB 1FFEEB03F0252979A72A>I<017C167048B491387001FC3A038F8001F8EA0707000E01C0 15FE001E1403001CEDF000EA3C0F0038177C1507D8781F4A133C00701380A2D8F03F130F 020049133812005B017E011F14784C137013FE5B033F14F0000192C712E05BA217010003 4A14C049137E17031880A2EF070015FE170E00010101141E01F86D131C0000D9039F5BD9 FC076D5A903A3E0F07C1E0903A1FFC03FFC0902703F0007FC7FC372979A73C>I<903903 F001F890390FFC07FE90393C1E0E0F9026780F1C138001F0EBB83FD801E013F89039C007 F07FEA0380000714E0D9000F140048151C000E4AC7FCA2001E131FA2C75BA2143F92C8FC A35C147EA314FE4A131CA30101143C001E1538003F491378D87F811470018314F000FF5D 9039077801C039FE0F7C033A7C0E3C078027783C1E1EC7FC391FF80FFC3907E003F02929 7CA72A>I<137C48B4143826038F8013FCEA0707000E7F001E1401001C15F8EA3C0F1238 1503D8781F14F000701380A2D8F03F1307020013E012005B017E130F16C013FE5B151F12 01491480A2153F000315005BA25D157EA315FE5D00011301EBF8030000130790387C1FF8 EB3FF9EB07E1EB00035DA21407000E5CEA3F80007F495AA24A5AD8FF0090C7FC143E007C 137E00705B387801F0383803E0381E0FC06CB4C8FCEA03F8263B79A72C>II E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fj cmti9 9 48 /Fj 48 123 df11 D<923803FF80031F13F092383F 00F803F8133C4A48133E4A48137E17FE4A5A17FC17384A481300A3141F92C8FCA55C143E 011FB612E0A217C09039007E0007147C160F1780A214FC4A131F1700A301015C4A133EA3 167E0103147C5C1718EEFC1CEEF83C010715385C1778177016F0010F15F04AEBF8E01679 EE3FC0011FEC0F0093C7FC91C9FCA3133EA21238EA7E3C137CEAFE7812FC485AEA79E0EA 3FC0000FCAFC2F4582B42B>I<130E131F133F137E13FCEA01F8EA03F0EA07E0EA0F80EA 1F00123E5A5A5A100E67B327>19 D<0060133000F813F8387C03F0383E0FC0381F1F8038 0FFE006C5AEA03F0EA01C015096CAE27>I39 D45 D<121C127F12FFA412FE12380808778718>I<161C163CA2167C16FCA21501 821503A2ED077E150F150E151CA21538A2157015F015E0EC01C0A2913803807F82EC0700 A2140E141E141C5CA25CA25C49B6FCA25B913880003F49C7EA1F80A2130E131E131C133C 13385B13F05B12011203D80FF0EC3FC0D8FFFE903807FFFEA32F367BB539>65 D67 D<0107B712F05B18E0903A003F80 001F1707170392C7FC17015C18C0147EA214FEA24A130EA20101EC1E03041C13804A91C7 FC163C13035E9138F001F891B5FC5B5EECE0011500130F5E5C1707011F01015BEEC00E02 80141E92C7121C133F173C91C812381778495DA2017E14014C5A01FE14074C5A49141F00 014AB45A007FB7FCB8FC94C7FC34337CB234>69 D<0107B712E05B18C0903A003F80003F 170F170792C7FC17035C1880147EA214FEA25C161C0101EC3C07043813004A91C7FCA201 03147816704A13F0150349B5FCA25EECE003130F6F5A14C0A2011F13035E1480A2013F90 C9FCA291CAFCA25BA2137EA213FEA25B1201387FFFFCB5FCA233337CB232>I<010FB512 80A216009038003FC05DA292C7FCA25CA2147EA214FEA25CA21301A25CA21303A25CA213 07A25CA2130FA25CA2131FA25CA2133FA291C8FCA25BA2137EA213FEA25B1201B512F8A2 5C21337BB21E>73 D<91381FFFFE5C16FC9138003F80A31600A25D157EA315FE5DA31401 5DA314035DA314075DA3140F5DA3141F5DA3143F92C7FCA2121C007E5B00FE137EA214FE 485BEAF80100E05B495A387007E038780FC06C48C8FCEA1FFCEA07F0273579B228>I<01 07B590380FFFF05B19E09026003FC0903803FE004B14F818E092C7485A4DC7FC4A141E17 7C027E5CEE01E002FE495A4C5A4A011FC8FC163E010114785E4A485AED07C00103495A4B 7EECF03F157F903907F1FFE0ECF3E79138E787F0ECEF0790380FFE0302FC7FECF80114E0 D91FC07F15004A7FA2013F147E167F91C77E835B161F017E81160F13FE835B000182267F FFF090B57EB55B95C7FC3C337BB23B>I<0107B512C05BA29026003FC0C7FC5DA292C8FC A25CA2147EA214FEA25CA21301A25CA21303A25CA21307A25CA2130FA25C17E0011F1401 17C05C1603013F1580160791C7FCEE0F005B5E017E143EA201FE5CED01FC4913030001EC 1FF8007FB6FCB7FC5E2B337CB230>I<902607FFC0ED7FFC4917FF81D9003F4B13006118 03023BED077CA2027BED0EFC610273151C1838DAF1F01439F071F014E118E10101ED01C3 6102C1EC0383EF070301031607050E5BEC80F8171C0107ED380F6102001470A249EDE01F DC01C090C7FC130EEE0380011E017C5C933807003E011C140EA2013C4A137E187C01385C 5E017816FC6F485B1370ED3FC001F0EC80016000011500D807F81503277FFF803E90B512 C0B5EB3C01151C46337BB245>I79 D<0107B612C04915F883903A003F8001FEEE003FEF1F8092C713C0170F5C18E0147EA214 FEEF1FC05CA201011680173F4A1500177E010315FE5F4AEB03F8EE07E00107EC3FC091B6 C7FC16F802E0C9FC130FA25CA2131FA25CA2133FA291CAFCA25BA2137EA213FEA25B1201 387FFFF0B5FCA233337CB234>I<0107B512FE49ECFFC017F0903A003F8007F8EE01FCEE 007E92C7127F835C1880147EA214FEEF7F005CA2010115FE5F4A13015F01034A5AEE0FC0 4A495A04FEC7FC49B512F016C09138E003E0ED01F8010F6D7E167C4A137EA2131FA25CA2 013F14FEA291C7FCA24913015E137EEF01C001FE150318805B00011607277FFFF0001400 B5ECFE0EEE7E1CC9EA1FF8EE07E032357BB238>82 D<913901FC018091380FFF03023F13 C791387E07EF903A01F801FF0049487E4A7F495A4948133E131F91C7FC5B013E143CA313 7E1638A293C7FC137FA26D7E14E014FE90381FFFC06D13F86D7F01017F6D6C7E020F7F14 00153F6F7E150FA4120EA2001E5D121CA2151F003C92C7FCA2003E143E5D127E007F5C6D 485A9038C007E039F3F80FC000F0B5C8FC38E03FFC38C00FF029377AB42B>I<0003B812 C05A1880903AF800FC003F260FC001141F0180150F01005B001EEE07001403121C003C4A 5BA200380107140E127800705CA2020F141E00F0161CC74990C7FCA2141FA25DA2143FA2 92C9FCA25CA2147EA214FEA25CA21301A25CA21303A25CA21307A25C497E001FB512F05A A2323374B237>I<3B3FFFF801FFFE485CA2D801FEC7EA1FC049EC0F80170049140EA216 1E120349141CA2163C1207491438A21678120F491470A216F0121F495CA21501123F90C7 5BA215035A007E5DA2150712FE4892C7FCA25D150E48141E151C153C153815786C5C5D00 7C1301007E495A003EEB0F806C011EC8FC380FC0FC6CB45A000113E06C6CC9FC2F3570B2 39>I97 D<137EEA0FFE121F5B1200A35BA21201A25BA21203A25BA21207A2EB C3E0EBCFF8380FDC3EEBF81F497E01E01380EA1FC0138015C013005AA2123EA2007E131F 1580127CA2143F00FC14005AA2147EA25CA2387801F85C495A6C485A495A6C48C7FCEA0F FCEA03F01A3578B323>I<14FCEB07FF90381F078090383E03C0EBFC013801F8033803F0 073807E00F13C0120F391F80070091C7FC48C8FCA35A127EA312FE5AA4007C14C0EC01E0 A2EC03C06CEB0F80EC1F006C137C380F81F03803FFC0C648C7FC1B2278A023>III<151FED7FC0EDF0E0020113F0EC03 E3A2EC07C316E0EDC1C091380FC0005DA4141F92C7FCA45C143E90381FFFFEA3D9007EC7 FC147CA414FC5CA513015CA413035CA413075CA3130FA25CA3131F91C8FCA35B133E1238 EA7E3CA2EAFE7812FC485AEA78E0EA3FC0000FC9FC244582B418>I<143FECFF80903803 E1E6903807C0FF90380F807FEB1F00133E017E133F49133EA24848137EA24848137CA215 FC12074913F8A21401A2D80FC013F0A21403120715E01407140F141F3903E03FC0000113 7FEBF0FF38007FCF90381F0F801300141FA21500A25C143E1238007E137E5C00FE5B4848 5A387803E0387C0F80D81FFFC7FCEA07F820317CA023>III<1538157C15FCA315701500AB143EECFF80903801E3C090380383E0EB0701 130FEB0E03131C133C133814071378013013C01300140FA21580A2141FA21500A25CA214 3EA2147EA2147CA214FCA25CA21301A25CA213035C1238387E07C0A238FE0F804848C7FC EAF83EEA787CEA3FF0EA0F801E4283B118>II<133FEA07FF5A13FEEA007EA3137CA213FCA213F8A212 01A213F0A21203A213E0A21207A213C0A2120FA21380A2121FA21300A25AA2123EA2127E A2127C1318EAFC1C133CEAF838A21378137012F013F0EAF8E01279EA3FC0EA0F00103579 B314>I<2703C003F8137F3C0FF00FFE01FFC03C1E783C1F07C1E03C1C7CF00F8F01F03B 3C3DE0079E0026383FC001FC7FD97F805B007001005B5E137ED8F0FC90380FC00100E05F D860F8148012000001021F130360491400A200034A13076049013E130FF081800007027E EC83C0051F138049017C1403A2000F02FC1407053E130049495CEF1E0E001F01015D183C 010049EB0FF0000E6D48EB03E03A227AA03F>I<3903C007F0390FF01FFC391E787C1E39 1C7CF01F393C3DE00F26383FC01380EB7F8000781300EA707EA2D8F0FC131F00E01500EA 60F8120000015C153E5BA20003147E157C4913FCEDF8180007153C0201133801C013F0A2 000F1578EDE070018014F016E0001FECE1C015E390C7EAFF00000E143E26227AA02B>I< 14FCEB07FF90381F07C090383E03E09038FC01F0EA01F83903F000F8485A5B120F484813 FCA248C7FCA214014814F8127EA2140300FE14F05AA2EC07E0A2007CEB0FC01580141FEC 3F006C137E5C381F01F0380F83E03803FF80D800FCC7FC1E2278A027>I<011E137C9038 7F81FF9039F3C387C09039E3EF03E03901E1FE01D9C1FC13F0EBC3F8000313F0018314F8 14E0EA07871307000313C01200010F130316F01480A2011F130716E01400A249EB0FC0A2 013EEB1F80A2017EEB3F00017F133E5D5D9038FF81F09038FDC3E09038F8FF80027EC7FC 000190C8FCA25BA21203A25BA21207A25BB5FCA325307FA027>I<903803F01890380FF8 3890383E1C7890387C0EF89038F807F0EA01F0EA03E000071303D80FC013E0A2EA1F8014 07D83F0013C0A348130F007E1480A300FE131F481400A35C143E147E127C14FE495AEA3C 03EA3E07EA1F0E3807FCF8EA01F0C7FC13015CA313035CA21307A25C48B5FCA25C1D3078 A023>I<3903C00FC0390FF03FF0391E78F078391C7DE03C393C3FC0FC00381380EB7F00 007814F8D8707E13701500EAF0FC12E0EA60F812001201A25BA21203A25BA21207A25BA2 120FA25BA2121FA290C8FC120E1E227AA020>II<1303EB0F80A3131FA21400A25BA2133EA2137EA2137C387FFFF8A2 B5FC3800F800A21201A25BA21203A25BA21207A25BA2120FA25B1460001F13F014E01300 130114C01303001E1380EB07005BEA0F1EEA07F8EA01E015307AAE19>II<01F01338D803FC13FCEA0F1E120E121C123C0038147CEA783E0070143CA2137E D8F07C1338EA60FCC65A1578000114705BA215F0000314E05BA2EC01C0A2EBC003158014 071500EBE00EA26C6C5A3800F878EB7FE0EB1F801E227AA023>II<011F137C90387FC1FF3A01E1E787803A 03C0F703C0903880FE0FEA07004813FC000E1580001E9038F80700001C91C7FC1301003C 5B1218120013035CA31307A25C1506010F130F150E14800038141ED87C1F131C00FC143C 1538013F5B39F07FC0E03970F3C3C0393FE1FF80260F807EC7FC22227CA023>I<13F0D8 03FC1307D80F1E130F000E141F121C123C0038143FD8783E133E1270A2017E137ED8F07C 137CEA60FCC65A15FC000114F85BA21401000314F013E0A2140315E0EA07C0A200031307 15C0EBE00F141F0001133F9038F07F8038007FEFEB1F8FEB001F1500A25C003E133E007E 137E147C5C007C5BEA7001495A38380780D83C1FC7FCEA0FFCEA07F020317AA025>I<90 3807801C90381FE03C90383FF038017F13789038FFF8F03901F07CE0EBE01F3903C003C0 9038800780EC0F00C7121E141C143C5C5C495AEB07C0495A011EC7FC5B5B4913704913F0 000114E0485A38078001390FC003C0381FF80790383E0F80393C1FFF00127838700FFE38 F007F838E001E01E227CA01F>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fk cmsy8 8 14 /Fk 14 111 df0 D<123C127E12FFA4127E123C08087A9414>I< 130C131EA50060EB01800078130739FC0C0FC0007FEB3F80393F8C7F003807CCF83801FF E038007F80011EC7FCEB7F803801FFE03807CCF8383F8C7F397F0C3F8000FCEB0FC03978 1E078000601301000090C7FCA5130C1A1D7C9E23>3 D<140381B3A3B812FCA3C7D80380 C7FCB3B812FCA32E2F7CAD37>6 DI20 D<12E012F812FEEA3F80EA0FE0EA03F8EA00FEEB3F80EB 0FE0EB03F8EB00FC143FEC0FC0EC07F0EC01FCEC007FED1FC0ED07F0ED01FCED007FEE1F C01607161FEE7F00ED01FCED07F0ED1FC0037FC7FCEC01FCEC07F0EC0FC0023FC8FC14FC EB03F8EB0FE0EB3F8001FEC9FCEA03F8EA0FE0EA3F80007ECAFC12F812E0CBFCAD007FB7 1280B812C0A22A3B7AAB37>I<137813FE1201A3120313FCA3EA07F8A313F0A2EA0FE0A3 13C0121F1380A3EA3F00A3123E127E127CA35AA35A0F227EA413>48 DI<91B512C01307131FD97F80C7FC01FCC8FCEA01F0EA03 C0485A48C9FC120E121E5A123812781270A212F05AA3B712C0A300E0C9FCA37E1270A212 781238123C7E120E120F6C7E6C7EEA01F0EA00FCEB7F80011FB512C013071300222B7AA5 2F>I54 D66 D<12E0B3B3B3AD034378B114>106 D<12E0A27E1270A212781238A2123C121CA2121E120EA2120F7E7F1203A27F1201A27F12 00A27F137013781338A2133C131CA2131E130EA2130F7FA2801303801301A2801300A280 1470A214781438143C141CA2141E140EA2140F80A215801403A215C0140114001A437CB1 23>110 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fl msbm10 10.95 4 /Fl 4 91 df67 D78 D<007FB612FCB812C06C16F83B03E007C07FFE0000903A0F001F7F80020E9038078FC093 380383E0EFC0F0040113788484EFE00E1600180F84A760180E0401131EEFC01C183C0403 5BEF81F093380787E093381F7FC04BB5C7FC020FB512FC17C004F7C8FC91390E1C078092 381E03C0ED0E01030F7FED078003037FEEC078923801E0380300133C707EEE780EEE380F 93383C0780EE1E03040E7F93380F01E093380780F004031370EFC078706C7E04007F717E 943878078094383803C00003D90F8090383C01E0007FB500FE90381FFFFCB6806C823E3E 7EBD39>82 D<0003B812F05AA2903B0FFC001C01E0D93FC0013C13C0D80F7EC7EA7803D8 0EF802701380D80FE0ECF00749903901E00F0049ECC00E90C70003131E4C5A001E020713 3892380F0078001C020E1370031E13F04B485AC800385BED780303705BEDF0074A4848C7 FCEDC00E0203131E4A485AED00384A1378020E1370021E13F04A485A02385BEC78039138 70078002F090C8FC49485AECC00E0103131E49485AEC0038491378010E01701406011E01 F0140E49485A01385BD97803151E49485A01E090C8FC000149153CEBC00E0003011E157C 48484815FCEB00384801781401000E49EC03DC001E49EC0F9CD83C01ED1F3C003849EC7E 38D87803EC01F8484848EB1FF0B912F8A3373E7DBD41>90 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fm cmr8 8 27 /Fm 27 127 df<156015F0A24A7E4A7EA24A7E1406EC0E7F140C91381C3F8014184A6C7E 150F02607F150702C07F1503D901807F1501D903007F496D7E1306010E147F130C011C6E 7E131801386E7E1330496E7E160749811603484881160148C87F486F7E1206000E167F12 0C001CEE3F801218003FB812C0A24817E0A2B912F0342F7DAE3B>1 D<013FB5FCA29038007F806EC8FCA6903801FFE0011F13FE90397F3F3F80D801F8EB07E0 D807E0EB01F8D80FC06D7ED81F80147ED83F0080481680A2007E151F00FE16C0A5007E16 80007F153FA26C1600D81F80147ED80FC05CD807E0495AD801F8EB07E0D8007FEB3F8090 261FFFFEC7FC010113E0D9003FC8FCA64A7E013FB5FCA22A2D7CAC33>8 D10 D<13031307130E131C1338137013F0EA01E013C01203EA0780A2EA0F00A2121EA35AA45A A512F8A25AAB7EA21278A57EA47EA37EA2EA0780A2EA03C0120113E0EA00F01370133813 1C130E1307130310437AB11B>40 D<12C07E12707E7E7E120FEA0780120313C0EA01E0A2 EA00F0A21378A3133CA4131EA5131FA2130FAB131FA2131EA5133CA41378A313F0A2EA01 E0A2EA03C013801207EA0F00120E5A5A5A5A5A10437CB11B>I43 D48 D<130C133C137CEA03FC12FFEAFC7C1200B3B113FE387FFFFEA2172C7AAB23>III<140EA2141E143EA2147E14FEA2EB01BE1303143E13 06130E130C131813381330136013E013C0EA0180120313001206120E120C5A123812305A 12E0B612FCA2C7EA3E00A9147F90381FFFFCA21E2D7EAC23>I<000CEB0180380FC01F90 B512005C5C14F014C0D80C7EC7FC90C8FCA8EB1FC0EB7FF8380DE07C380F801F01001380 000E130F000CEB07C0C713E0A2140315F0A4127812FCA448EB07E012E0006014C0007013 0F6C14806CEB1F006C133E380780F83801FFE038007F801C2D7DAB23>II<1230123C003FB512F8A215F05A15E039700001C00060 1480140348EB0700140E140CC7121C5C143014705C495AA2495AA249C7FCA25B130E131E A2133EA3133C137CA413FCA913781D2E7CAC23>III61 D73 D<13C0487E487E487EEA0F3CEA1E1E487E3870038038E001 C0EAC000120A78AD23>94 D<13FF000713C0380F01F0381C00F8003F137C80A2143F001E 7FC7FCA4EB07FF137F3801FE1FEA07F0EA1FC0EA3F80EA7F00127E00FE14065AA3143F7E 007E137F007FEBEF8C391F83C7FC390FFF03F83901FC01E01F207D9E23>97 D99 D<013F13F89038FFC3FE3903E1FF1E3807807C000F140C391F 003E00A2003E7FA76C133EA26C6C5A00071378380FE1F0380CFFC0D81C3FC7FC90C8FCA3 121E121F380FFFF814FF6C14C04814F0391E0007F848130048147C12F848143CA46C147C 007C14F86CEB01F06CEB03E03907E01F803901FFFE0038003FF01F2D7E9D23>103 D108 D<2607C07FEB07F03BFFC3FFC03FFC903AC783F0783F3C0FCE01F8E01F803B07DC00F9C0 0F01F8D9FF8013C04990387F000749137EA249137CB2486C01FEEB0FE03CFFFE0FFFE0FF FEA2371E7E9D3C>I<3807C0FE39FFC3FF809038C703E0390FDE01F0EA07F8496C7EA25B A25BB2486C487E3AFFFE1FFFC0A2221E7E9D27>II< 38078008380FE01C381FF838383FFFF038707FE038E01FC03840078016077AAC23>126 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fn cmsy10 10.95 33 /Fn 33 111 df<007FB812F8B912FCA26C17F83604789847>0 D<121EEA7F80A2EAFFC0 A4EA7F80A2EA1E000A0A799B19>I<0060166000F816F06C1501007E15036CED07E06C6C EC0FC06C6CEC1F806C6CEC3F006C6C147E6C6C5C6C6C495A017E495A6D495A6D6C485A6D 6C485A6D6C48C7FC903803F07E6D6C5A903800FDF8EC7FF06E5A6E5AA24A7E4A7EECFDF8 903801F8FC903803F07E49487E49486C7E49486C7E49486C7E017E6D7E496D7E48486D7E 4848147E4848804848EC1F804848EC0FC048C8EA07E0007EED03F0481501481500006016 602C2C73AC47>I<1506150FB3A9007FB912E0BA12F0A26C18E0C8000FC9FCB3A6007FB9 12E0BA12F0A26C18E03C3C7BBC47>6 D<007FB912E0BA12F0A26C18E0C8000FC9FCB3A6 007FB912E0BA12F0A26C18E0C8000FC9FCB3A915063C3C7BAC47>I15 D<007FB912E0BA12F0A26C18E0CDFC AE007FB912E0BA12F0A26C18E0CDFCAE007FB912E0BA12F0A26C18E03C287BAA47>17 D<1818187CEF01FCEF07F8EF1FF0EF7FC0933801FF00EE07FCEE1FF0EE7FC04B48C7FCED 07FCED1FF0ED7FC04A48C8FCEC07FCEC1FF0EC7FC04948C9FCEB07FCEB1FF0EB7FC04848 CAFCEA07FCEA1FF0EA7FC048CBFC5AEA7F80EA3FE0EA0FF8EA03FEC66C7EEB3FE0EB0FF8 EB03FE903800FF80EC3FE0EC0FF8EC03FE913800FF80ED3FE0ED0FF8ED03FE923800FF80 EE3FE0EE0FF8EE03FE933800FF80EF3FE0EF0FF8EF03FC170018381800AE007FB812F8B9 12FCA26C17F8364878B947>20 D<126012F812FEEA7F80EA3FE0EA0FF8EA03FEC66C7EEB 3FE0EB0FF8EB03FE903800FF80EC3FE0EC0FF8EC03FE913800FF80ED3FE0ED0FF8ED03FE 923800FF80EE3FE0EE0FF8EE03FE933800FF80EF3FE0EF0FF8EF03FC1701EF07F8EF1FF0 EF7FC0933801FF00EE07FCEE1FF0EE7FC04B48C7FCED07FCED1FF0ED7FC04A48C8FCEC07 FCEC1FF0EC7FC04948C9FCEB07FCEB1FF0EB7FC04848CAFCEA07FCEA1FF0EA7FC048CBFC 12FC1270CCFCAE007FB812F8B912FCA26C17F8364878B947>I24 D<0207B612F8023F15FC49B7FC4916F8D90FFCC9FCEB1FE0017FCAFC13FEEA01F8485A48 5A5B485A121F90CBFC123EA25AA21278A212F8A25AA87EA21278A2127CA27EA27E7F120F 6C7E7F6C7E6C7EEA00FE137FEB1FE0EB0FFC0103B712F86D16FCEB003F020715F8363678 B147>26 D<19301978A2197C193CA2193E191EA2191F737EA2737E737EA2737E737E1A7C 1A7EF21F80F20FC0F207F0007FBB12FCBDFCA26C1AFCCDEA07F0F20FC0F21F80F27E001A 7C624F5A4F5AA24F5A4F5AA24FC7FC191EA2193E193CA2197C1978A2193050307BAE5B> 33 D49 D<0207B512E0023F14F049B6FC4915E0D90FFCC8FCEB1FE001 7FC9FC13FEEA01F8485A485A5B485A121F90CAFC123EA25AA21278A212F8A25AA2B812E0 17F0A217E000F0CAFCA27EA21278A2127CA27EA27E7F120F6C7E7F6C7E6C7EEA00FE137F EB1FE0EB0FFC0103B612E06D15F0EB003F020714E02C3678B13D>I<176017F01601A2EE 03E0A2EE07C0A2EE0F80A2EE1F00A2163EA25EA25EA24B5AA24B5AA24B5AA24B5AA24BC7 FCA2153EA25DA25DA24A5AA24A5AA24A5AA24A5AA24AC8FCA2143EA25CA25CA2495AA249 5AA2495AA2495AA249C9FCA2133EA25BA25BA2485AA2485AA2485AA2485AA248CAFCA212 3EA25AA25AA25A12602C5473C000>54 D<126012F0AE12FC12FEA212FC12F0AE12600722 7BA700>I<0060EE018000F0EE03C06C1607A200781780007C160FA2003C1700003E5EA2 6C163EA26C163C6D157CA2000716786D15F8A26C6C4A5AA200015E6D140390B7FC6C5EA3 017CC7EA0F80A2013C92C7FC013E5CA2011E141E011F143EA26D6C5BA2010714786E13F8 A26D6C485AA201015CECF003A201005CECF807A291387C0F80A2023C90C8FCEC3E1FA2EC 1E1EEC1F3EA2EC0FFCA26E5AA36E5AA36E5A6E5A324180BE33>I<1518153CA2157CA290 3803FC7890380FFFF8EB3E0790387801F0EBF0004848487ED803C07FD807807FA2390F00 03EFA248ECCF80001EEB07C7003E15C01587A2140F007E15E0007C1403A2141FA2141E00 FC013E13F0A2143CA2147CA21478A214F8A214F01301A214E0A21303A214C0A21307A214 80D87C0F14E0A21400007E14075BA2D83E1E14C0A2133E001FEC0F80133CD80F7C1400A2 495B0007141E00035C00015C4913F83900F801E03901FE07C090B5C7FCEBE3FCD803E0C8 FCA25BA26C5A244D7CC52D>59 D<020EEC7FC0023E903803FFF802FE011F7F0103027F7F 010F49B6FC011F903803F81F013F90260FC0031380903A79FC1F00010101013E7F5D4B14 7F903803FDF002FF16005D5D187E4B14FE4990C85A604A4A5A4D5A4A4A5AEF1F80010F03 7EC7FC4A495AEE0FF04AEB7FC0DB03FFC8FC011F011F13E04A4813F84B13FE92B6FC4AC6 6C7F013F020F7F04037F4A1300717E173F49C86C7EA2170FA201FE1507A448485EA3495E 0003160F605B00074C5A4993C7FCD9E180143E260FE7C05CD9DFE05C48B46CEB03F0D9BF FCEB0FC09139FF80FF80D83F1FD9FFFEC8FC6D14F8D87E0714E0D8780191C9FC39E0003F F039427DBF3C>66 DI< 4AB512FC023FECFFE049B712FC0107EEFF80011F8390277FE1FC0114F02601FC01D9000F 7FD803F003017FD807C09238003FFE260F80036F7E001F1707D83F0070138084007E4A6E 13C012FE48187F00F019E000C00107163FC7FC5D191FA3140F5DA21AC0A24A5AA2F13F80 A24A5A1A0061197E4AC9FC61A2027E4B5A02FE5E18034A4B5A01015F4E5A4A4BC7FC0103 163E604A5D0107ED03F04AEC07C0EF1F80010F037EC8FC4A495A011FEC0FF04AEB7FC0DB 0FFFC9FC49B512FC90B612E04892CAFC4814F84891CBFC433E7EBD46>II< 033FB612F00207B7FC023F16E091B812800103EEFE0090280FFC0007C0C7FCD91F80130F 013EC7485A4992C8FC01FC5C48485C167E484814FE01C05C90C8FCC812015E1503A34B5A A35E150FA34B5AA44B5AA44BC9FCA415FEA35D1401A25D14035DA24A5A18704A48EB01F0 4D5A4A48130792C7485A023E5D4A023FC7FC0007B712FE001F16F8485E481680B700FCC8 FC3C3E83BD32>73 D<0438198004F81801030119030303190770180F1D1FF53F00A20307 61706064525A1C07A24B6C170F1C1F525A030E187F7017FD031E6DED01F91CFB031CEF03 F3983807E3F892263C3FC0ED0FC3F31F830338EF3F03F37E079238781FE009FC5B0370EE 01F8F203F09226F00FF0EC07E003E093380FC00FF21F800201173F4B6C6C03005B1A7E02 035F03806D495A04034A48131F02074C5A03004B5A706C131F020E4C5A4FC75B021E6E13 FE021C6D495AF083F84ADB87F0143F94387FCFE00278EDDFC002706EB45A96C8FC4A6E5A 6001016F5A49485D0030705A267C07805DD87F0F6F5AB5C890C9EBE1C094CA13E749F3FF 80491C00F41FFC491BF06C48F20FC0D81FE097C8FCEA078062457DBF6D>77 D83 D87 D<0060EE018000F0EE03C0B3B3A36C1607A200781780 007C160FA26CEE1F00003F5E6C6C157E6C6C5DD807F0EC03F8D803FCEC0FF06CB4EC3FE0 3B007FF003FF80011FB548C7FC010714F8010114E09026001FFEC8FC32397BB63D>91 DI<153FEC03FFEC0FE0EC3F80EC7E00495A5C495A A2495AB3AA130F5C131F495A91C7FC13FEEA03F8EA7FE048C8FCEA7FE0EA03F8EA00FE13 3F806D7E130F801307B3AA6D7EA26D7E80EB007EEC3F80EC0FE0EC03FFEC003F205B7AC3 2D>102 D<12FCEAFFC0EA07F0EA01FCEA007E6D7E131F6D7EA26D7EB3AA801303806D7E 1300147FEC1FC0EC07FEEC00FFEC07FEEC1FC0EC7F0014FC1301495A5C13075CB3AA495A A2495A133F017EC7FC485AEA07F0EAFFC000FCC8FC205B7AC32D>I<126012F0B3B3B3B3 B11260045B76C319>106 D<0060131800F0133CB3B3B3B3B000601318165A75C32D>I<12 6012F07EA21278127CA2123C123EA2121E121FA27E7FA212077FA212037FA212017FA212 007FA21378137CA27FA2131E131FA27F80A2130780A2130380A2130180A2130080A21478 147CA2143C143EA2141E141FA26E7EA2140781A2140381A2140181A2140081A21578157C A2153C153EA2151E151FA2811680A2150716C0A21503ED0180225B7BC32D>110 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fo cmmi8 8 47 /Fo 47 123 df12 D<131FD9FFC013304801F01370 00076D13604815E0D9807C13C0391E003C0148011E13800038EB0E03480107130000605C EC030612E0C7138EEC018C159C159815B815B0A215F05DA35DA25DA21403A44AC7FCA414 0EA45CA31418242C7F9D24>III<1460A5EC7FF0EC3FF814FF903803 CFE0903807800049C7FC131E5B5B5BA2485A485AA2485A48C8FCA25A121E123E123CA312 7C1278A412F8A4127CA2127E127F6C7E13E0EA1FFC6CB47E6C13F000017F38003FFCEB07 FE1300143F80A280141EA2EB601CEB7838EB1FF0EB07C01D3C7DAD1F>I<13E0486C133C 15FF00031303EC0F7F9038E01C7EEC381C000749C7FC5CEBC3C001C7C8FCEA0FDE13F8EB FF8014F8381F83FEEB803F496C7E140F48154016C0123EA2007E14811680007C1403ED83 0000FC1487EC078E48EB03FC0070EB00F0221F7D9D29>20 D<90B612F812035A4815F03A 1E0380C000003C130000701301130700E05CEAC00638000E03A3131CA2133C1407133813 78A201F07FA21201A2D803E07FA20007130313C0A26C486C5A251E7E9C29>25 D<0103B512F0131F137F90B612E03A01FC1F80003903F00FC03807C00748486C7E121F13 00123EA25AA2140700FC5C5AA2140F5D141F92C7FC143E0078133C147C007C5B383C01E0 381F07C0D807FFC8FCEA01F8241E7D9C28>27 D<1560A315E0A25DA21401A25DA21403A2 92C7FCA25CEC3FE0903803FFF890380FC63E90393E0E0F80017CEB07C03A01F00C03E0D8 03E0EB01F03807C01CD80F801300001F011813F81300003E1338A2481330A2EC700100FC 15F0481360150302E013E01507007801C013C0007CEC0F800101EB1F00003C143E003E49 5A001F5C390F8383E03903E39F802600FFFEC7FCEB1FF00107C8FCA21306A2130EA2130C A2131CA21318A3253C7DAD2A>30 D<1506A3150E150CA3151C1518A315381530A31570D8 01E0EB6007D807F8EC1F80EA0E3CD81C3E01E013C0003814C00030150F0070150726607E 011480D8E07CEB800312C013FC3880F803000002001300120113F04A5B00030106130601 E0140E160C020E131C020C131801C0143801E05C021C5B91381801C0D801F0495A030FC7 FC3900FC381C90383F30F890380FFFE0010190C8FCEB00701460A314E05CA313015CA42A 3C7EAD2E>32 D<160ED80380143FA20007168090C8FC000E151F001E150F001C16000018 8112380030130C141E007015061260143E023C130E00E0150C5A0238131C6C15184A1338 147802F85BD8F00114F0496C485A397C0FBE073A7FFF9FFFC0021F5B263FFC0F90C7FC39 1FF807FC3907E001F0291F7F9D2C>II39 D<123C127EB4FCA21380A2127F123D1201A312031300A25A1206120E5A5A5A126009157A 8714>59 D<15C0140114031580A214071500A25C140EA2141E141CA2143C143814781470 A214F05CA213015CA213035C130791C7FCA25B130EA2131E131CA2133C1338A213781370 13F05BA212015BA212035BA2120790C8FC5A120EA2121E121CA2123C1238A212781270A2 12F05AA21A437CB123>61 D<147F903801FFE090380780F890380E003C497F497F491480 01781307017C14C001FC130316E0A2137090C7FC16F0A314FE903807FF8390381F01C390 397C00E7E049137748481337D807E0133F49131F484814C0121F48C7FCA2481580127EA2 ED3F0012FE48147EA2157C15FC5D4A5A007C495AA26C495A001E49C7FC6C133E3807C0F8 3803FFE038007F8024307DAE25>64 D<1670A216F01501A24B7EA21507150DA215191539 1531ED61FC156015C0EC0180A2EC03005C14064A7F167E5C5CA25C14E05C4948137F91B6 FC5B0106C7123FA25B131C1318491580161F5B5B120112031207000FED3FC0D8FFF89038 07FFFEA22F2F7DAE35>I<92387FC003913903FFF80791391FC03E0F91397E00071FD901 F8EB03BF4948EB01FED90FC013004948147E49C8FC017E157C49153C485A120348481538 485AA2485A173048481500A2127F90CAFCA35A5AA5EE018016031700A2007E5D1606160E 6C5D5E6C6C5C000F5D6C6C495A6C6CEB0780D801F8011EC7FCD8007E13F890381FFFE001 0390C8FC302F7CAD32>67 D<013FB71280A2D900FEC7127F170F4A1407A2010115031800 5CA21303A25C16300107147094C7FC4A136016E0130F15019138C007C091B5FC5BECC007 4A6C5AA2133FA20200EB000CA249151C92C71218017E1538173001FE15705F5B4C5A0001 15034C5A49140F161F00034AB4C7FCB8FC5E312D7DAC34>69 D<90273FFFFC0FB5FCA2D9 00FEC7EA3F80A24A1500A201015D177E5CA2010315FE5F5CA2010714015F5CA2010F1403 5F5C91B6FC5B9139C00007E05CA2013F140F5F91C7FCA249141F5F137EA201FE143F94C7 FC5BA200015D167E5BA2000315FEB539E03FFFF8A2382D7CAC3A>72 D<90383FFFFCA2903800FE00A25CA21301A25CA21303A25CA21307A25CA2130FA25CA213 1FA25CA2133FA291C7FCA25BA2137EA213FEA25BA21201A25BA21203B512E0A21E2D7DAC 1F>I<91383FFFF8A29138007F00A2157EA215FE5DA314015DA314035DA314075DA3140F 5DA3141F5DA3143FA292C7FCA2003C5B127E00FE137E14FE5CEAFC0100F05B48485A3860 07E038781F80D81FFEC8FCEA07F0252E7BAC27>I<90263FFFFC90381FFF80A2D900FEC7 3803F80018E04AEC07804DC7FC0101151C5F4A14E04C5A01034A5A040EC8FC4A5B5E0107 14E04B5A9138E00780030EC9FC010F131F157F4A487E14C190391FC71FC014CEEC9C0F02 F07F90383FE00702C07FEC0003825B6F7E137E6F7E13FE167F5B707E1201161F49818312 03B539E001FFFEA2392D7CAC3C>I77 DI89 D97 D99 D<151FEC03FFA2EC003FA2153EA2157EA2 157CA215FCA215F8A21401EB07E190381FF9F0EB7C1DEBF80FEA01F03903E007E0EA07C0 120FEA1F8015C0EA3F00140F5A007E1480A2141F12FE481400A2EC3F021506143E5AEC7E 0E007CEBFE0C14FC0101131C393E07BE18391F0E1E38390FFC0FF03903F003C0202F7DAD 24>II<157C4AB4FC 913807C380EC0F87150FEC1F1FA391383E0E0092C7FCA3147E147CA414FC90383FFFF8A2 D900F8C7FCA313015CA413035CA413075CA5130F5CA4131F91C8FCA4133EA3EA383C12FC 5BA25B12F0EAE1E0EA7FC0001FC9FC213D7CAE22>I<131FEA03FFA2EA003FA2133EA213 7EA2137CA213FCA25BA21201143F9038F1FFC09038F3C1F03803FF0001FC7F5BA2485A5B A25B000F13015D1380A2001F13035D1300140748ECC04016C0003E130F1580007E148191 381F0180007C1403ED070000FCEB0F06151E48EB07F80070EB01E0222F7DAD29>104 D<1307EB0F80EB1FC0A2EB0F80EB070090C7FCA9EA01E0EA07F8EA0E3CEA1C3E12381230 1270EA607EEAE07C12C013FC485A120012015B12035BA21207EBC04014C0120F13801381 381F01801303EB0700EA0F06131EEA07F8EA01F0122E7EAC18>I<15E0EC01F01403A3EC 01C091C7FCA9147CEB03FE9038078F80EB0E07131C013813C01330EB700F0160138013E0 13C0EB801F13001500A25CA2143EA2147EA2147CA214FCA25CA21301A25CA21303A25CA2 130700385BEAFC0F5C49C7FCEAF83EEAF0F8EA7FF0EA1F801C3B81AC1D>I<131FEA03FF A2EA003FA2133EA2137EA2137CA213FCA25BA2120115F89038F003FCEC0F0E0003EB1C1E EC387EEBE07014E03807E1C09038E3803849C7FC13CEEA0FDC13F8A2EBFF80381F9FE0EB 83F0EB01F81300481404150C123EA2007E141C1518007CEBF038ECF83000FC1470EC78E0 48EB3FC00070EB0F801F2F7DAD25>I<137CEA0FFCA21200A213F8A21201A213F0A21203 A213E0A21207A213C0A2120FA21380A2121FA21300A25AA2123EA2127EA2127CA2EAFC08 131812F8A21338133012F01370EAF860EA78E0EA3FC0EA0F000E2F7DAD15>I<27078007 F0137E3C1FE01FFC03FF803C18F0781F0783E03B3878E00F1E01263079C001B87F26707F 8013B00060010013F001FE14E000E015C0485A4914800081021F130300015F491400A200 034A13076049133E170F0007027EEC8080188149017C131F1801000F02FCEB3F03053E13 0049495C180E001F0101EC1E0C183C010049EB0FF0000E6D48EB03E0391F7E9D3E>I<39 07C007E0391FE03FF83918F8783E393879E01E39307B801F38707F00126013FEEAE0FC12 C05B00815C0001143E5BA20003147E157C5B15FC0007ECF8081618EBC00115F0000F1538 913803E0300180147016E0001F010113C015E390C7EAFF00000E143E251F7E9D2B>II<90387C01F89038FE07FE39 01CF8E0F3A03879C0780D907B813C0000713F000069038E003E0EB0FC0000E1380120CA2 D8081F130712001400A249130F16C0133EA2017EEB1F80A2017C14005D01FC133E5D15FC 6D485A3901FF03E09038FB87C0D9F1FFC7FCEBF0FC000390C8FCA25BA21207A25BA2120F A2EAFFFCA2232B829D24>I<903807E03090381FF87090387C1CF0EBF80D3801F00F3903 E007E0EA07C0000F1303381F800715C0EA3F00A248130F007E1480A300FE131F481400A3 5C143E5A147E007C13FE5C1301EA3E07EA1F0E380FFCF8EA03F0C7FC13015CA313035CA2 1307A2EBFFFEA21C2B7D9D20>I<3807C01F390FF07FC0391CF8E0E0383879C138307B87 38707F07EA607E13FC00E0EB03804848C7FCA2128112015BA21203A25BA21207A25BA212 0FA25BA2121FA290C8FC120E1B1F7E9D20>I<130E131FA25BA2133EA2137EA2137CA213 FCA2B512F8A23801F800A25BA21203A25BA21207A25BA2120FA25BA2001F131014301300 1470146014E0381E01C0EB0380381F0700EA0F0EEA07FCEA01F0152B7EA919>116 DI<013F137C9038FFC1FF3A01C1E383803A0380F703C0390700F60F000E13FE 4813FC12180038EC0700003049C7FCA2EA200100005BA313035CA301075B5D14C000385C D87C0F130600FC140E011F130C011B131C39F03BE038D8707113F0393FE0FFC0260F803F C7FC221F7E9D28>120 DI<011E1330EB3F809038FFC07048EBE0E0ECF1C03803C0FF9038803F8090 3800070048130EC75A5C5C5C495A495A49C7FC131E13385B491340484813C0485A380700 01000EEB0380380FE007391FF81F0038387FFF486C5A38601FFC38E00FF038C003C01C1F 7D9D21>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fp cmmi10 10.95 75 /Fp 75 123 df11 DIIIIII<15FCEC03FF91380F87C091383E03E0EC7C 0102F813F01301903903F000F8495A010F14FC5C495A133F91C7FC4914FE13FEA212015B 12034913011207A25B000F15FC1503121F5BA21507003F15F890B6FCA33A7FC0000FF05B A2151F16E048C7FCA2ED3FC0A2481580157F1600A215FEA24A5AA24A5A007E5C14075D4A 5A003E5C141F4AC7FC6C137E5C380F81F03807C3E03801FF80D8007EC8FC27417DBF2B> I<131C013E141F017EEC7FC0ED01FFED07BF01FEEB1E3F033813804913709238E01E0000 014948C7FCEC0780D9F80EC8FC5C00035B14F0EBF3C001FFC9FC4813F0ECFF8001E013F0 EC07FC000FEB00FE157F496D7EA2001F141F17705BA2003F16F0033F13E090C71300A248 ED01C0A2007EEC1F03178000FE91380F0700168E48EC07FC0038EC01F02C297CA734>20 D<133F14E0EB07F0EB03FC13016D7EA3147FA26E7EA36E7EA36E7EA36E7EA36E7EA26E7E A36E7EA3157FA36F7E157F15FF4A7F5C913807CFE0EC0F8FEC1F0F91383E07F0147C14FC 49486C7EEB03F0EB07E049486C7EEB1F80EB3F00496D7E13FE4848147F485A485A4848EC 3F80485A123F4848EC1FC048C8FC4816E048150F48ED07F0007015032C407BBE35>II24 D<011FB612FE017F15FF48B8FC5A4816 FE3B0FC03801C000EA1F00003E1403003C01785B4813705AECF0075AC712E0010191C7FC A25DEB03C0A313071480A2010F5BA2EB1F0082A2133EA2137E825B150F0001815B120315 075BC648EB038030287DA634>I<020FB512FE027F14FF49B7FC1307011F15FE903A3FE0 3FE00090387F000F01FE6D7E4848130348488048481301485A5B121F5B123F90C7FC5A12 7EA2150300FE5D5AA24B5AA2150F5E4B5AA2007C4AC7FC157E157C6C5C001E495A001FEB 07E0390F800F802603E07EC8FC3800FFF8EB3FC030287DA634>27 D<16F0A25EA21501A25EA21503A25EA21507A293C7FCA25DA2150EA2151EA2151C4AB47E 020F13F091387F3CFC903901F8381FD907E0EB0F80903A0F807807C0D93F00EB03E0017E 90387001F04915F84848EBF000484815FC48485B4848157C1401EA1F805DEA3F00020314 FC5A007E5CA20207130100FE16F848140016034A14F01607020E14E0007CED0FC0141E00 7EED1F80003E011CEB3F00167E6C013C5B0180495A000F90383803E0D807E0EB0FC02701 F0783FC7FC3900FC79FC90381FFFE0D903FEC8FCEB00F0A25CA21301A25CA21303A25CA2 1307A291C9FCA25BA22E527BBE36>30 D<13FE2603FF80157026078FE015F0260F07F014 01000E6D15E00103ED03C0000C6DEC0780D80001ED0F006E141E01005D5F027F5C4C5A91 383F80035F4C5A6E6C48C7FC161E5E6E6C5A5EEDE1E0913807E3C015F75E6EB4C8FC5D5D 5D6E7EA2140314074A7EA2141EEC3C7F147814F049486C7EEB03C0EB078049486C7EA213 1E496D7E5B498048481307485A48486D7E48C7FC48EDFC03001E0201EB07804803FE1300 486E6C5A48ED7F1E0060ED1FFCC9EA03F0343B7EA739>II<0120ED01C00178ED07F001F8150F000117F85B485A 5B0007160749150348C9EA01F0A2121E1700121C003C023814E0003814FCA20078160114 0100704A14C0A217034B148000F0130317074B14005F5D0207141E6F133E6C010F5C4A7E 6C013F5C007E9038FFF8033B7F87FDFF0FF0D9FFF8EBFFE06C495C4A6C5B6C496C90C7FC 00079038001FFCD801F8EB03F035297EA739>II39 D<121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A 0A798919>58 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013C0A3120113 80120313005A120E5A1218123812300B1C798919>I<183818FC1703EF0FF8EF3FE0EFFF 80933803FE00EE0FF8EE3FE0EEFF80DB03FEC7FCED0FF8ED3FE0EDFF80DA03FEC8FCEC0F F8EC3FE0ECFF80D903FEC9FCEB0FF8EB3FE0EBFF80D803FECAFCEA0FF8EA3FE0EA7F8000 FECBFCA2EA7F80EA3FE0EA0FF8EA03FEC66C7EEB3FE0EB0FF8EB03FE903800FF80EC3FE0 EC0FF8EC03FE913800FF80ED3FE0ED0FF8ED03FE923800FF80EE3FE0EE0FF8EE03FE9338 00FF80EF3FE0EF0FF8EF03FC17001838363678B147>II<126012F8B4FCEA7FC0EA 1FF0EA07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FCEC01FF91 38007FC0ED1FF0ED07FCED01FF9238007FC0EE1FF0EE07FCEE01FF9338007FC0EF1FF0EF 07F8EF01FCA2EF07F8EF1FF0EF7FC0933801FF00EE07FCEE1FF0EE7FC04B48C7FCED07FC ED1FF0ED7FC04A48C8FCEC07FCEC1FF0EC7FC04948C9FCEB07FCEB1FF0EB7FC04848CAFC EA07FCEA1FF0EA7FC048CBFC12FC1270363678B147>I<15FF020713E091381F00F80278 133E4A7F4948EB0F804948EB07C04948EB03E091C7FC496CEB01F002E014F8131F160017 FCA25C0107C812FE90C9FCA7EC03FCEC3FFF9138FE03C1903903F000E149481371D91F80 133149C7123B017EEC1BFC5B0001151F4848140F484815F8A2485A121F17F0485A161F17 E0127F5BEE3FC0A200FF168090C8127F1700A216FEA2484A5A5E007E1403007F4A5A5E6C 4A5A6C6C495A4BC7FC6C6C13FE6C6C485A3903F80FF06CB512C06C6C90C8FCEB0FF82F43 7CC030>64 D<17075F84171FA2173F177FA217FFA25E5EA24C6C7EA2EE0E3F161E161C16 38A21670A216E0ED01C084ED0380171FED07005D150E5DA25D157815705D844A5A170F4A 5A4AC7FC92B6FC5CA2021CC7120F143C14384A81A24A140713015C495AA249C8FC5B130E 131E4982137C13FED807FFED1FFEB500F00107B512FCA219F83E417DC044>I<49B712F8 18FF19E090260001FEC7EA3FF0F007F84B6E7E727E850203815D1A80A20207167F4B15FF A3020F17004B5C611803021F5E4B4A5A180FF01FE0023F4B5A4B4A5ADD01FEC7FCEF07F8 027FEC7FE092B6C8FC18E092C7EA07F84AEC01FE4A6E7E727E727E13014A82181FA21303 4A82A301075F4A153FA261010F167F4A5E18FF4D90C7FC011F5E4A14034D5A013FED1FF0 4D5A4AECFFC0017F020790C8FCB812FC17F094C9FC413E7DBD45>II<49B712F818FF19C0D9000190C7EA3FF0F00FF84BEC 03FCF000FE197F0203EE3F805DF11FC0A20207EE0FE05D1AF0A2020F16075DA21AF8141F 5DA2190F143F5DA21AF0147F4B151FA302FF17E092C9123FA21AC049177F5C1A8019FF01 0318005C4E5A61010716034A5E4E5A180F010F4C5A4A5E4E5A4EC7FC011F16FE4A4A5AEF 07F8013FED0FE0EF3FC04A49B4C8FC017FEC0FFCB812F017C004FCC9FC453E7DBD4B>I< 49B912C0A3D9000190C71201F0003F4B151F190F1A80020316075DA314075D1A00A2140F 4B1307A24D5B021F020E130E4B92C7FC171EA2023F5C5D177CEE01FC4AB55AA3ED800302 FF6D5A92C7FCA3495D5C19380401147801034B13704A16F093C85AA2010716014A5E1803 61010F16074A4BC7FCA260011F163E4A157E60013F15014D5A4A140F017F15FFB95AA260 423E7DBD43>I<49B9FCA3D9000190C7120718004B157F193F191E14035DA314075D191C A2140F5D17074D133C021F020E13384B1500A2171E023F141C4B133C177C17FC027FEB03 F892B5FCA39139FF8003F0ED00011600A2495D5CA2160101035D5CA293C9FC13075CA313 0F5CA3131F5CA2133FA25C497EB612F8A3403E7DBD3A>I<49B6D8C03FB512F81BF01780 D900010180C7383FF00093C85B4B5EA2197F14034B5EA219FF14074B93C7FCA260140F4B 5DA21803141F4B5DA21807143F4B5DA2180F4AB7FC61A20380C7121F14FF92C85BA2183F 5B4A5EA2187F13034A5EA218FF13074A93C8FCA25F130F4A5DA21703131F4A5DA2013F15 07A24A5D496C4A7EB6D8E01FB512FCA2614D3E7DBD4C>72 D<49B612C05BA2D90001EB80 0093C7FC5DA314035DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA314FF92C8FCA3 5B5CA313035CA313075CA3130F5CA3131F5CA2133FA25CEBFFE0B612E0A32A3E7DBD28> I<92B612E0A39239003FF000161F5FA2163F5FA3167F5FA316FF94C7FCA35D5EA315035E A315075EA3150F5EA3151FA25EA2153FA25EA2157FA25EA2D80F8013FFEA3FC0486C91C8 FCA25CD8FFC05B140301805B49485A00FC5C0070495A0078495A0038495A001E017EC9FC 380F81FC3803FFE0C690CAFC33407ABD32>I<49B600C090387FFFF896B5FC5FD9000101 80C7000F130093C813F84B16E01A804FC7FC0203163C4B15F84E5AF003C002074B5A4B02 1FC8FC183E1878020F5D4BEB03E0EF07804DC9FC021F143E4B5B17F04C5A023F1307EDC0 0F4C7E163F027FEBFFF8ED81EFED83CF92388F87FC9138FF9F0792383C03FE15784B6C7E 4913E0158092C77F5C01036F7E5C717EA213074A6E7EA2717E130F4A6E7EA284011F1503 5C717E133F855C496C4A13E0B600E0017F13FFA34D3E7DBD4D>I<49B612F0A3D9000101 80C7FC93C8FC5DA314035DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA314FF92C9 FCA35B5C180C181E0103161C5C183C183813074A1578187018F0130F4AEC01E0A2170301 1FED07C04A140F171F013FED3F8017FF4A1303017F021F1300B9FCA25F373E7DBD3E>I< 49B56C93B512C050148062D90001F18000704B90C7FC03DF5F1A0E1A1D1403039FEE39FC 1A711A739126078FE015E3030F5FF101C3F10387140F020E93380707F0A2F10E0F021E16 1C91261C07F05E1938F1701F143C023804E05BA2953801C03F0278ED038091267003F85E F00700060E137F14F002E04B91C8FCA24E5B01015E4A6C6C5D60943801C00113030280DA 03805BA294380700030107150E91C700FE5D5F1907495D010E4B5CA24D130F011E6E5A01 1C60705A013C171F017C92C7FC01FE027E5DD803FF4D7EB500FC017C017FB512E0167804 385E5A3E7CBD58>I<49B56C49B512F81BF0A290C76D9039000FFE004AEE03F0705D735A 03DF150302037F038F5E82190791380787FC030793C7FC1503705C140F91260E01FF140E A26F151E021E80021C017F141C83193C023C6D7E02381638161F711378147802706D6C13 70A2040714F002F0804A01035C8318010101EC01FF4A5E82188313034A91387FC380A2EF 3FC7010716E791C8001F90C8FC18F718FF4981010E5E1707A2131E011C6F5AA2013C1501 137C01FE6F5AEA03FFB512FC187818704D3E7DBD49>II<49B712F018FF19C0D9000190C76C7EF00FF84BEC03FC18010203 82727E5DA214071A805DA2140F4E13005DA2021F5E18034B5D1807023F5E4E5A4B4A5A4E 5A027F4B5A06FEC7FC4BEB03FCEF3FF091B712C005FCC8FC92CBFCA25BA25CA21303A25C A21307A25CA2130FA25CA2131FA25CA2133FA25C497EB612E0A3413E7DBD3A>II<49B77E18F818FFD90001D900017F9438003FE04BEC0F F0727E727E14034B6E7EA30207825DA3020F4B5A5DA24E5A141F4B4A5A614E5A023F4B5A 4B4A5A06FEC7FCEF03FC027FEC0FF04BEBFF8092B500FCC8FC5F9139FF8001FE92C7EA7F 80EF1FC084496F7E4A1407A28413035CA2170F13075C60171F130F5CA3011F033F5B4AEE 038018E0013F17071A004A021F5B496C160EB600E090380FF01E05075B716C5ACBEAFFE0 F03F8041407DBD45>II<007FB500F090387FFFFE19FC 5D26007FE0C7000313804A913800FC004A5D187001FF16F0A291C95AA2481601605BA200 031603605BA20007160795C7FC5BA2000F5E170E5BA2001F161E171C5BA2003F163C1738 5BA2007F1678A2491570A200FF16F0A290C95AA216015F5A16035F16074CC8FC160E161E 5E007F5D5E6C4A5A6D495A6C6C495A6C6C011FC9FC6C6C137E3903FC03F8C6B512E0013F 1380D907FCCAFC3F407ABD3E>85 DII<027FB5D88007B512C091B6FCA2020101F8C7 EBF8009126007FE0EC7F804C92C7FC033F157C701478616F6C495A4E5A6F6C495A4EC8FC 180E6F6C5B606F6C5B6017016F6C485A4D5A6F018FC9FC179E17BCEE7FF85F705AA3707E A283163F167FEEF7FCED01E7EEC3FEED0383ED070392380E01FF151E4B6C7F5D5D4A486D 7E4A5A4A486D7E92C7FC140E4A6E7E5C4A6E7E14F0495A49486E7E1307D91F806E7ED97F C014072603FFE0EC1FFF007F01FC49B512FEB55CA24A3E7EBD4B>II<027FB712F0A3DAFFFCC7EA3FE0 03E0EC7FC092C8EAFF8049484A13004A4A5A5C4A4A5A49484A5A4A4A5A4D5A49484A5A4D 5A91C74890C7FC5B010E4A5A4C5A4C5A011E4A5A90C8485A4C5A4C5A4B90C8FCA24B5A4B 5A4B5A4B5A4B5A4B5A4B5AA24A90C9FC4A5A4A5A4A5A4A4814704A4814F04A485C14FF5D 4990C7120149485D49481403495A49485D49481407495A4DC7FC49485C4890C8FC48485D 4848157E484815FE484814034848EC0FFC16FF48B7FCB8FC5F3C3E7BBD3E>I<151EED7F 80913801F1C0EC03C1EC07C0ED80E0EC0F005C141E91383E01C0147CA214F81503D901F0 1380A21303ECE007010714005D90380FC00EA2151E90381F801C153C5D133F4A5A5D1401 49485A017E5B14074AC7FCEBFE1E13FC5C5C5C3801F9E0EBFBC0A2EBFF8091C8FC5B5B5B 5BA212031207120F121F123D127800F0140300E0EC0780C66CEB0F000178131E157C6D13 F04A5A90381E0F80D90FFEC7FCEB03F823417FBF26>96 DIIIII<163EEEFFC0923803E1E0923807C0F0ED0F811687ED1F8F160F153FA217E092387E 038093C7FCA45DA514015DA30103B512FCA390260003F0C7FCA314075DA4140F5DA5141F 5DA4143F92C8FCA45C147EA414FE5CA413015CA4495AA35CEA1E07127F5C12FF495AA200 FE90C9FCEAF81EEA703EEA7878EA1FF0EA07C02C537CBF2D>III<143C14FEA21301A314FCEB 00701400AD137E3801FF803803C7C0EA0703000F13E0120E121C13071238A2EA780F0070 13C0A2EAF01F14801200133F14005B137EA213FE5BA212015B0003130E13F0A20007131E EBE01CA2143CEBC0381478147014E013C13803E3C03801FF00EA007C173E7EBC1F>IIII<01F8D907F0EB07F8D803FED93FFEEB1FFE28078F80F81FEB781F3E0F0F81C00F81E0 0F803E0E07C78007C3C007C0001CD9CF00EBC78002FEDAEF007F003C4914FE0038495C49 485C12780070495CA200F0494948130F011F600000495CA2041F141F013F6091C75B193F 043F92C7FC5B017E92C75A197E5E01FE9438FE01C049027E14FCA204FE01011303000106 F81380495CF20700030115F00003190E494A151E1A1C03035E0007943800F8F0494AEC7F E0D801C0D900E0EC1F804A297EA750>I<01F8EB0FF0D803FEEB3FFC3A078F80F03E3A0F 0F83C01F3B0E07C7800F80001CEBCF0002FE80003C5B00385B495A127800705BA200F049 131F011F5D00005BA2163F013F92C7FC91C7FC5E167E5B017E14FE5EA201FE0101EB0380 4914F8A203031307000103F013005B170E16E000035E49153C17385F0007913801F1E049 6DB45AD801C0023FC7FC31297EA737>II I<91381F800C9138FFE01C903903F0707C90390FC0387890391F801CF890383F000F137E 4914F000011407485A485A16E0485A121F150F484814C0A3007F141F491480A300FF143F 90C71300A35D48147EA315FE007E495A1403A26C13074A5A381F801D000F13793807C1F3 3901FFC3F038007F03130014075DA3140F5DA3141F5DA2143F147F90381FFFFE5BA2263A 7DA729>I I<147014FC1301A25CA21303A25CA21307A25CA2130FA25CA2007FB512F0B6FC15E03900 1F8000133FA291C7FCA25BA2137EA213FEA25BA21201A25BA21203A25BA21207EC01C013 E01403000F1480A2EBC0071500140E141E5C000713385C3803E1E03801FF80D8003EC7FC 1C3A7EB821>116 D<017E147848B4EB01FC2603C7C013FED807031303000F13E0120E12 1C0107130100381400167ED8780F143E00705B161EEAF01F4A131C1200133F91C7123C16 385B137E167801FE14705B16F016E0120149EB01C0A2ED0380A2ED0700A20000140E5D6D 133C017C5B6D5B90381F03C0903807FF80D901FCC7FC27297EA72C>118 D<013EEE0380D9FF800107EB0FE02601C3E090381F801FD8038117F0380701F0000E153F 001E1600D81C03160F003C170700384BEB03E0D87807147E00705B1801D8F00F14FE4A49 14C01200131FDA800114034C1480133F140003031407494A1400137EA26001FE0107140E 495C60A360150F017C5E017E011F14F0705B6D0139495A6D903970F8038090280FC0E07C 0FC7FC903A03FFC01FFC903A007F0007F03C297EA741>II<137C48B4EC 03802603C7C0EB0FC0EA0703000F7F000E151F001C168013071238163FD8780F15000070 5BA2D8F01F5C4A137E1200133F91C712FE5E5B137E150113FE495CA2150300015D5BA215 075EA2150F151F00005D6D133F017C137F017E13FF90393F03DF8090380FFF1FEB01FC90 C7123F93C7FCA25DD80380137ED80FE013FE001F5C4A5AA24848485A4A5A6CC6485A001C 495A001E49C8FC000E137C380781F03803FFC0C648C9FC2A3B7EA72D>I<02F8130ED903 FE131ED90FFF131C49EB803C49EBC0784914F090397E07F1E09038F800FF49EB1FC049EB 07800001EC0F006C48131E90C75A5D5D4A5A4A5A4A5A4AC7FC143E14785C495A495A495A 49C8FC011E14E05B5B4913014848EB03C0485AD807F8EB078048B4131F3A1F87E07F0039 1E03FFFE486C5B00785CD870005B00F0EB7FC048011FC7FC27297DA72A>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fq cmcsc10 10.95 34 /Fq 34 128 df<121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A0A77891D>46 D<14E013011303130F137FEA07FFB5FC138FEAF80F1200B3B3ACEB3FF8B612FEA31F3D77 BC32>49 D65 D II70 D73 D75 D80 D82 DI<003FB912E0A390 3BF0003FF0007F01806D48130F48C7ED07F0007E1703007C170100781700A300701870A5 481838A5C81600B3B14B7E4B7E0103B7FCA33D3D7CBC47>I87 D97 D99 DIIIIII108 DIIIIII<90383FC00C9038FFF8 1C0003EBFE3C390FE03FFC381F8007EB0003003E1301481300157C5A153CA36C141CA27E 6C14006C7E13E013FE383FFFE06C13FE6CEBFF806C14E0000114F06C6C13F8010F13FC13 00EC07FE14011400157F153F12E0151FA37EA2151E6C143E6C143C6C147C6C14F89038C0 01F039FBF807E000F1B512C0D8E07F130038C007FC20317BAF2A>I<007FB712F8A39039 801FF0073A7E000FE00000781678A20070163800F0163CA348161CA5C71500B3A8EC3FF8 011FB512F0A32E2E7CAD36>II<3B7FFFF001FFFEA30003D9C00013 E0C649EB7F80017F027EC7FC167C6D6C13786D6C5B6D6C5B15016D6C485AD903FC5B1507 6D6C48C8FC903800FF1EEC7F9C15BCEC3FF86E5AA2140F6E7E14034A7E4A7EEC1EFF141C 91383C7F804A6C7E14709138F01FE049486C7E49486C7E148001076D7E49486C7E130E01 1E6D7E496E7E017C6E7E13FC000382D80FFEEC7FF8B549B512C0A3322F7DAE38>120 D I<003C130F007FEB3F804814C0EB807FA3EB003F6C1480003CEB0F001A0975BD32>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmr10 10.95 87 /Fr 87 128 df0 D<16E04B7EA24B7EA24B7EA24B7EA2 ED1DFFA203387FA29238787FC015709238F03FE015E002016D7E15C002036D7E15800207 6D7E15004A6D7E140E021E6D7E141C023C6D7F143802786E7E147002F06E7E5C01016F7E 5C01036F7E5C01076F7E91C8FC496F7E130E011E6F7E131C013C6F7F13380178707E1370 01F0707E5B0001717E5B0003717E5B0007717E90CAFC48717E120E001E717E001FBAFC48 1980A24819C0A2BB12E0A243417CC04C>II6 D<010FB612E0A3D900030180C7FCDA00FEC8FCA8913807FFC0027F13FC903A03FCFE7F80 D90FE0EB0FE0D93F80EB03F8D9FE00EB00FE4848157F4848ED3F804848ED1FC0000F17E0 4848ED0FF0003F17F8A24848ED07FCA200FF17FEA8007F17FCA26C6CED0FF8A2001F17F0 6C6CED1FE0000717C06C6CED3F806C6CED7F006C6C15FED93F80EB03F8D90FE0EB0FE0D9 03FCEB7F809027007FFFFCC7FC020713C0DA00FEC8FCA8913803FF80010FB612E0A3373E 7BBD42>8 D<49B612FCA390C7D87FF0C8FCED1FC0A8B4EF0FF001C0163FD81FE0EE7F80 6C6CEEFF006C6C4B5A00035FA26D150300015FAB12006D4B5AA4017F4B5AA26D5E028014 1FD91FC05D010F153F02E04AC7FCD907F0147ED903F85CD900FCEBC3F8027FEBC7E09139 1FDFFF80912607FFFEC8FC9138007FF0ED1FC0A8ED7FF049B612FCA33C3E7BBD47>I<91 3801FFC0021F13FC9139FF007F80D903F8EB0FE0D90FF0EB07F8D91FC0EB01FCD97F806D B4FC49C86C7E48486F7E00038348486F7E000F8349150F001F83491507003F83A348486F 7EAA6C6C4B5AA3001F5FA26C6C4B5AA200075F6D151F00035FA26C6C4B5A00005FA2017F 4BC7FC6D157EA26D6C5C010F5DA26D6C495A00E0EF0380010315E0D870019238C007006E 130301001580A36C0160EC000E003C017049131E263FFFF0ECFFFEA36C5FA339407CBF42 >I<4AB4EB0FE0021F9038E03FFC913A7F00F8FC1ED901FC90383FF03FD907F090397FE0 7F80494801FF13FF4948485BD93F805C137F0200ED7F00EF003E01FE6D91C7FC82ADB97E A3C648C76CC8FCB3AE486C4A7E007FD9FC3FEBFF80A339407FBF35>IIII22 D<001E130F397F803FC000FF137F01 C013E0A201E013F0A3007F133F391E600F3000001300A401E01370491360A3000114E049 13C00003130101001380481303000EEB070048130E0018130C0038131C003013181C1C7D BE2D>34 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013C0A31201138012 0313005A120E5A1218123812300B1C79BE19>39 D<1430147014E0EB01C0EB03801307EB 0F00131E133E133C5B13F85B12015B1203A2485AA2120F5BA2121F90C7FCA25AA3123E12 7EA6127C12FCB2127C127EA6123E123FA37EA27F120FA27F1207A26C7EA212017F12007F 13787F133E131E7FEB07801303EB01C0EB00E014701430145A77C323>I<12C07E12707E 7E121E7E6C7E7F12036C7E7F12007F1378137CA27FA2133F7FA21480130FA214C0A31307 14E0A6130314F0B214E01307A614C0130FA31480A2131F1400A25B133EA25BA2137813F8 5B12015B485A12075B48C7FC121E121C5A5A5A5A145A7BC323>I<1506150FB3A9007FB9 12E0BA12F0A26C18E0C8000FC9FCB3A915063C3C7BB447>43 D<121EEA7F8012FF13C0A2 13E0A3127FEA1E601200A413E013C0A312011380120313005A120E5A1218123812300B1C 798919>II<121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A0A7989 19>IIIIII<150E151E153EA2157EA215FE1401A21403EC07 7E1406140E141CA214381470A214E0EB01C0A2EB0380EB0700A2130E5BA25B5BA25B5B12 01485A90C7FC5A120E120C121C5AA25A5AB8FCA3C8EAFE00AC4A7E49B6FCA3283E7EBD2D >I<00061403D80780131F01F813FE90B5FC5D5D5D15C092C7FC14FCEB3FE090C9FCACEB 01FE90380FFF8090383E03E090387001F8496C7E49137E497F90C713800006141FC813C0 A216E0150FA316F0A3120C127F7F12FFA416E090C7121F12FC007015C012780038EC3F80 123C6CEC7F00001F14FE6C6C485A6C6C485A3903F80FE0C6B55A013F90C7FCEB07F8243F 7CBC2D>II<1238123C123F90B612 FCA316F85A16F016E00078C712010070EC03C0ED078016005D48141E151C153C5DC81270 15F04A5A5D14034A5A92C7FC5C141EA25CA2147C147814F8A213015C1303A31307A3130F 5CA2131FA6133FAA6D5A0107C8FC26407BBD2D>III<121E EA7F80A2EAFFC0A4EA7F80A2EA1E00C7FCB3121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A 2779A619>I<121EEA7F80A2EAFFC0A4EA7F80A2EA1E00C7FCB3121E127FEAFF80A213C0 A4127F121E1200A412011380A3120313005A1206120E120C121C5A1230A20A3979A619> I<007FB912E0BA12F0A26C18E0CDFCAE007FB912E0BA12F0A26C18E03C167BA147>61 D<15074B7EA34B7EA34B7EA34B7EA34B7E15E7A2913801C7FC15C3A291380381FEA34AC6 7EA3020E6D7EA34A6D7EA34A6D7EA34A6D7EA34A6D7EA349486D7E91B6FCA24981913880 0001A249C87EA24982010E157FA2011E82011C153FA2013C820138151FA2017882170F13 FC00034C7ED80FFF4B7EB500F0010FB512F8A33D417DC044>65 DIIIIIIII75 DIIIII82 DI<003FB912 80A3903AF0007FE001018090393FC0003F48C7ED1FC0007E1707127C00781703A3007017 01A548EF00E0A5C81600B3B14B7E4B7E0107B612FEA33B3D7DBC42>II II91 D<486C13C00003130101 001380481303000EEB070048130E0018130C0038131C003013180070133800601330A300 E01370481360A400CFEB678039FFC07FE001E013F0A3007F133FA2003F131F01C013E039 0F0007801C1C73BE2D>II<131813 3C137E13FF3801E7803803C3C0380781E0380F00F0001E137848133C48131E48130F0060 1306180D76BD2D>I97 DI<49B4FC010F13E090383F00F8017C131E4848131F 4848137F0007ECFF80485A5B121FA24848EB7F00151C007F91C7FCA290C9FC5AAB6C7EA3 003FEC01C07F001F140316806C6C13076C6C14000003140E6C6C131E6C6C137890383F01 F090380FFFC0D901FEC7FC222A7DA828>II II<167C903903F801 FF903A1FFF078F8090397E0FDE1F9038F803F83803F001A23B07E000FC0600000F6EC7FC 49137E001F147FA8000F147E6D13FE00075C6C6C485AA23901F803E03903FE0FC026071F FFC8FCEB03F80006CAFC120EA3120FA27F7F6CB512E015FE6C6E7E6C15E06C810003813A 0FC0001FFC48C7EA01FE003E140048157E825A82A46C5D007C153E007E157E6C5D6C6C49 5A6C6C495AD803F0EB0FC0D800FE017FC7FC90383FFFFC010313C0293D7EA82D>III<1478EB01FEA2EB03FFA4EB01FEA2EB00781400AC147FEB7FFFA313 017F147FB3B3A5123E127F38FF807E14FEA214FCEB81F8EA7F01387C03F0381E07C0380F FF803801FC00185185BD1C>II I<2701F801FE14FF00FF902707FFC00313E0913B1E07E00F03F0913B7803F03C01F80007 903BE001F87000FC2603F9C06D487F000101805C01FBD900FF147F91C75B13FF4992C7FC A2495CB3A6486C496CECFF80B5D8F87FD9FC3F13FEA347287DA74C>I<3901F801FE00FF 903807FFC091381E07E091387803F000079038E001F82603F9C07F0001138001FB6D7E91 C7FC13FF5BA25BB3A6486C497EB5D8F87F13FCA32E287DA733>I<14FF010713E090381F 81F890387E007E01F8131F4848EB0F804848EB07C04848EB03E0000F15F04848EB01F8A2 003F15FCA248C812FEA44815FFA96C15FEA36C6CEB01FCA3001F15F86C6CEB03F0A26C6C EB07E06C6CEB0FC06C6CEB1F80D8007EEB7E0090383F81FC90380FFFF0010090C7FC282A 7EA82D>I<3901FC03FC00FF90381FFF8091387C0FE09039FDE003F03A07FFC001FC6C49 6C7E6C90C7127F49EC3F805BEE1FC017E0A2EE0FF0A3EE07F8AAEE0FF0A4EE1FE0A2EE3F C06D1580EE7F007F6E13FE9138C001F89039FDE007F09039FC780FC0DA3FFFC7FCEC07F8 91C9FCAD487EB512F8A32D3A7EA733>I<02FF131C0107EBC03C90381F80F090397F0038 7C01FC131CD803F8130E4848EB0FFC150748481303121F485A1501485AA448C7FCAA6C7E A36C7EA2001F14036C7E15076C6C130F6C7E6C6C133DD8007E137990383F81F190380FFF C1903801FE0190C7FCAD4B7E92B512F8A32D3A7DA730>I<3901F807E000FFEB1FF8EC78 7CECE1FE3807F9C100031381EA01FB1401EC00FC01FF1330491300A35BB3A5487EB512FE A31F287EA724>I<90383FC0603901FFF8E03807C03F381F000F003E1307003C1303127C 0078130112F81400A27E7E7E6D1300EA7FF8EBFFC06C13F86C13FE6C7F6C1480000114C0 D8003F13E0010313F0EB001FEC0FF800E01303A214017E1400A27E15F07E14016C14E06C EB03C0903880078039F3E01F0038E0FFFC38C01FE01D2A7DA824>I<131CA6133CA4137C A213FCA2120112031207001FB512C0B6FCA2D801FCC7FCB3A215E0A912009038FE01C0A2 EB7F03013F138090381F8700EB07FEEB01F81B397EB723>IIIIII<001FB61280A2EBE0000180140049485A001E495A121C4A5A003C 495A141F00385C4A5A147F5D4AC7FCC6485AA2495A495A130F5C495A90393FC00380A2EB 7F80EBFF005A5B484813071207491400485A48485BA248485B4848137F00FF495A90B6FC A221277EA628>I<01F01308D803FC131C48B4133848EB8070391F3FF3E0393807FFC048 6C138048C613000040133C1E0979BC2D>126 D<001C130E007FEB3F8039FF807FC0A539 7F003F80001CEB0E001A0977BD2D>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fs cmcsc10 9 50 /Fs 50 122 df<1338137C13FC1201EA03F813F0EA07C0EA0F80EA1F00123E12785A5A5A 0E0E6FB329>19 D<123C127EB4FCA21380A2127F123D1201A4EA0300A35A1206120E120C 5A12385A12200917788718>44 DI<123C127E12FFA4127E123C 0808788718>I48 D<13075B5B13FF120FB5FC133F12F01200B3B3A2497EB612C0A31A32 78B129>I51 DI<00 0C1460390FC007E090B512C01580150014FC5C14C0000CC8FCAAEB0FE0EB3FF8EBF03E38 0DC00F390F800780010013C0000EEB03E0000C14F0A2C7EA01F8A315FCA41238127E12FE 7EA24814F8A200F81303006014F0A26CEB07E0003814C06CEB0F80000FEB1F003807C0FE 6CB45A6C13F038003F801E347BB129>I<1230123C003FB61280A3481500A25D0070C712 1C0060141815385D485C5D1401C7485A4AC7FC1406140E5C141814385CA25C1301A2495A A213075CA2130FA2131FA291C8FC5BA55BA9131C21347BB129>55 DII<156015F0 A34A7EA34A7EA24A7E1406A2EC0EFFEC0C7FA202187F153FA24A6C7EA202707FEC600FA2 02C07F1507A249486C7EA349486C7EA2498001061300A2010FB6FCA30118C7EA3F80A349 6E7EA20170810160140FA2498116071201707E487ED80FF0EC0FFCD8FFFE91B512F0A334 367CB53D>65 DIII70 DI73 D<013FB51280A39039001FF0006E5AB3B0121C127FEAFF80A44A5A1300007C495A12706C 49C7FC6C13FC380F81F83803FFE0C66CC8FC21357CB22A>I77 DI80 D82 DI85 DI<1418143CA3147EA214FFA39038019F80A201037F140F A201067F1407A2496C7EA2011C7FEB1801A2496C7EA201707FEB7FFFA29038C0003FA200 01158049131FA2000315C090C7120F486C14E0120F486CEB1FF0D8FFF090B5FCA228277E A62E>97 DI<02FF13100107EBE03090391FC0787090397E001CF001F8 13074848130348481301485A000F1400485A167048C8FCA2481530127EA200FE1500A812 7E1630127F7EA26C6C1460A26C7E000715C06C6CEB01806C6C13036C6CEB0700017E130E 90381FC078903807FFE00100138024287DA62C>IIII<02FF13100107EBE03090391FC0787090397E 001CF001F813074848130348481301485A000F1400485A167048C8FCA2481530127EA200 FE1500A64AB5FCA2007E90380007F8ED03F0127F7EA26C7EA26C7E12076C7E6C6C1307EA 00FC017E130C90391FC03870903907FFF0100100EB800028287DA630>III<48B5FCA2380007F8EB 03F0B3A8127812FCA3EB07E01270EB0FC0383C1F80380FFE00EA03F818277DA520>IIIII<49B4FC 010F13E090383F01F890387C007C4848133FD803E0EB0F80000715C04848EB07E0491303 001F15F048C7EA01F8A24815FCA2007E1400A200FE15FEA9007FEC01FCA36C15F86D1303 001F15F06D1307000F15E06C6CEB0FC06C6CEB1F806C6CEB3F006C6C137E90383F01F890 380FFFE0010190C7FC27287DA62F>II<49B4FC010F13E090383F01F890387C007C4848133F4848EB1F804848EB0F C04848EB07E0491303001F15F0003F15F890C712014815FCA2007E1400A200FE15FEA900 7E15FC007F1401A26C15F8A23A1F807C03F014FE3A0FC1C307E03A07E3818FC03A03F301 DF803A01FB00FF006CB46C5A90383FC1F890390FFFF006130190C71278160EED7C1EED7F FE16FC153F16F8151FED0FF0ED07C027327DA62F>III<007FB6 12F8A2397E00FC010078EC00780070153800601518A200E0151C48150CA5C71400B3A449 7E90387FFFF8A226267EA52C>III121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ft cmr9 9 77 /Ft 77 128 df<91393FE00FE0903A01FFF83FF8903A07E01EF83C903A1F800FF07E903A 3F001FE0FE017E133F4914C0485A1738484890381F8000ACB812C0A33B03F0001F8000B3 A7486C497EB50083B5FCA32F357FB42D>11 D15 DI<127812FCA27E7EEA7F80 121FEA0FC0EA07E01203EA00F01378133C13080E0E78B326>18 D<137813FCA212011203 EA07F813E0EA0FC0EA1F801300123C5A5A12400E0E71B326>I<00C01306A36C130E0060 130C0070131C00301318003813386C1370380F83E03807FFC06C13803800FE00170D79B3 26>21 D<123C127EB4FCA21380A2127F123D1201A412031300A25A1206120E120C121C5A 5A126009177AB315>39 D<14C01301EB0380EB0F00130E5B133C5B5BA2485A485AA21207 5B120F90C7FC5AA2121E123EA3123C127CA55AB0127CA5123C123EA3121E121FA27E7F12 077F1203A26C7E6C7EA213787F131C7F130FEB0380EB01C01300124A79B71E>I<12C07E 1270123C121C7E120F6C7E6C7EA26C7E6C7EA27F1378137C133C133EA2131E131FA37F14 80A5EB07C0B0EB0F80A514005BA3131E133EA2133C137C137813F85BA2485A485AA2485A 48C7FC120E5A123C12705A5A124A7CB71E>I<123C127EB4FCA21380A2127F123D1201A4 12031300A25A1206120E120C121C5A5A126009177A8715>44 D I<123C127E12FFA4127E123C08087A8715>I48 D<13075B5B137FEA07FFB5FC13BFEAF83F1200B3B3A2497E007FB5 1280A319327AB126>IIII<000C14C0380FC00F90B5128015005C5C14F014C0 D80C18C7FC90C8FCA9EB0FC0EB7FF8EBF07C380FC03F9038001F80EC0FC0120E000CEB07 E0A2C713F01403A215F8A41218127E12FEA315F0140712F8006014E01270EC0FC06C131F 003C14806CEB7F00380F80FE3807FFF8000113E038003F801D347CB126>I<14FE903807 FF80011F13E090383F00F0017C13703901F801F8EBF003EA03E01207EA0FC0EC01F04848 C7FCA248C8FCA35A127EEB07F0EB1FFC38FE381F9038700F809038E007C039FFC003E001 8013F0EC01F8130015FC1400A24814FEA5127EA4127F6C14FCA26C1301018013F8000F14 F0EBC0030007EB07E03903E00FC03901F81F806CB51200EB3FFCEB0FE01F347DB126>I< 1230123C003FB6FCA34814FEA215FC0070C7123800601430157015E04814C01401EC0380 C7EA07001406140E5C141814385CA25CA2495A1303A3495AA2130FA3131F91C7FCA25BA5 5BA9131C20347CB126>III<123C127E12FFA4127E123C1200B0123C127E12FFA4127E123C 08207A9F15>I<15E0A34A7EA24A7EA34A7EA3EC0DFE140CA2EC187FA34A6C7EA202707F EC601FA202E07FECC00FA2D901807F1507A249486C7EA301066D7EA2010E80010FB5FCA2 49800118C77EA24981163FA2496E7EA3496E7EA20001821607487ED81FF04A7ED8FFFE49 B512E0A333367DB53A>65 DIIIIIII I<017FB5FCA39038003FE0EC1FC0B3B1127EB4FCA4EC3F805A0060140000705B6C13FE6C 485A380F03F03803FFC0C690C7FC20357DB227>IIIIIIIII<90381FE00390387FFC 0748B5FC3907F01FCF390F8003FF48C7FC003E80814880A200788000F880A46C80A27E92 C7FC127F13C0EA3FF013FF6C13F06C13FF6C14C06C14F0C680013F7F01037F9038003FFF 140302001380157F153FED1FC0150F12C0A21507A37EA26CEC0F80A26C15006C5C6C143E 6C147E01C05B39F1FC03F800E0B512E0011F138026C003FEC7FC22377CB42B>I<007FB7 12FEA390398007F001D87C00EC003E0078161E0070160EA20060160600E01607A3481603 A6C71500B3AB4A7E011FB512FCA330337DB237>IIII<267FFFFC90B512C0A3000101E090381FF80026007F80EB0FC0013F6E5A6E91C7FC 6D6C130E010F140C6E5B6D6C133801035C6E13606D6C13E06D6C485A5EDA7F83C8FCEC3F C715C6EC1FECEC0FFC5D14076E7EA26E7E815C6F7E9138063FC0140E4A6C7E9138180FF0 EC380702707F91386003FCECC0010101804A6C7E49C77E4981010E6E7E010C6E7E131C49 6E7E01786E7E13FCD807FEEC1FFEB56C90B512F8A335337EB23A>I91 D93 D97 DII<153FEC0FFFA3EC007F81AEEB07F0EB3FFCEBFC0F3901F003BF39 07E001FF48487E48487F8148C7FCA25A127E12FEAA127E127FA27E6C6C5BA26C6C5B6C6C 4813803A03F007BFFC3900F81E3FEB3FFCD90FE0130026357DB32B>III<151F90391FC07F809039FFF8E3C03901F07FC73907E03F033A0FC01F8380 9039800F8000001F80EB00074880A66C5CEB800F000F5CEBC01F6C6C48C7FCEBF07C380E FFF8380C1FC0001CC9FCA3121EA2121F380FFFFEECFFC06C14F06C14FC4880381F000100 3EEB007F4880ED1F8048140FA56C141F007C15006C143E6C5C390FC001F83903F007E0C6 B51280D91FFCC7FC22337EA126>IIIIII<2703F01FE013FF00FF90267FF80313C0903BF1E07C0F03E0 903BF3803E1C01F02807F7003F387FD803FE1470496D486C7EA2495CA2495CB3486C496C 487EB53BC7FFFE3FFFF0A33C217EA041>I<3903F01FC000FFEB7FF09038F1E0FC9038F3 807C3907F7007EEA03FE497FA25BA25BB3486CEB7F80B538C7FFFCA326217EA02B>II<3903F03F8000FFEBFFE09038 F3C0F89038F7007ED807FE7F6C48EB1F804914C049130F16E0ED07F0A3ED03F8A9150716 F0A216E0150F16C06D131F6DEB3F80160001FF13FC9038F381F89038F1FFE0D9F07FC7FC 91C8FCAA487EB512C0A325307EA02B>I<903807F00390383FFC07EBFC0F3901F8038F38 07E001000F14DF48486CB4FC497F123F90C77E5AA25A5AA9127FA36C6C5B121F6D5B000F 5B3907E003BF3903F0073F3800F81EEB3FF8EB0FE090C7FCAAED7F8091380FFFFCA32630 7DA029>I<3803E07C38FFE1FF9038E38F809038E71FC0EA07EEEA03ECA29038FC0F8049 C7FCA35BB2487EB512E0A31A217FA01E>II<1330A51370A313F0A21201A212031207381FFFFEB5FCA23803F000AF1403 A814073801F806A23800FC0EEB7E1CEB1FF8EB07E0182F7FAD1E>IIIII<3A7FFF807FF8A33A07F8001FC00003EC0F800001EC070015066C6C5BA26D131C017E 1318A26D5BA2EC8070011F1360ECC0E0010F5BA2903807E180A214F3010390C7FC14FBEB 01FEA26D5AA31478A21430A25CA214E05CA2495A1278D8FC03C8FCA21306130EEA701CEA 7838EA1FF0EA0FC025307F9F29>I<003FB512F0A2EB000F003C14E00038EB1FC00030EB 3F800070137F1500006013FE495A13035CC6485A495AA2495A495A49C7FC153013FE485A 12035B48481370485A001F14604913E0485A387F000348130F90B5FCA21C207E9F22>I< B712F8A22502809426>I<001C1370387F01FC00FF13FEA4007F13FC381C0070170879B2 26>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fu cmbx10 10.95 61 /Fu 61 128 df12 D40 D<127012F8127C7EEA3F806C7E6C7E12076C7E7F6C7E6C7EA2137F8013 3F806D7EA280130FA280130780A36D7EA4807FA51580B01500A55B5CA4495AA35C130F5C A2131F5CA2495A5C137F91C7FC13FEA2485A485A5B485A120F485A485A003EC8FC5A5A12 70195A7AC329>I45 DI<140F143F5C495A130F48B5FCB6FCA313F7 EAFE071200B3B3A8007FB612F0A5243C78BB34>49 D<903803FF80013F13F890B512FE00 036E7E4881260FF80F7F261FC0037F4848C67F486C6D7E6D6D7E487E6D6D7EA26F1380A4 6C5A6C5A6C5A0007C7FCC8FC4B1300A25E153F5E4B5AA24B5A5E4A5B4A5B4A48C7FC5D4A 5AEC1FE04A5A4A5A9139FF000F80EB01FC495A4948EB1F00495AEB1F8049C7FC017E5C5B 48B7FC485D5A5A5A5A5AB7FC5EA4293C7BBB34>I<903801FFE0010F13FE013F6D7E90B6 12E04801817F3A03FC007FF8D807F06D7E82D80FFC131F6D80121F7FA56C5A5E6C48133F D801F05CC8FC4B5A5E4B5A4A5B020F5B902607FFFEC7FC15F815FEEDFFC0D9000113F06E 6C7E6F7E6F7E6F7E1780A26F13C0A217E0EA0FC0487E487E487E487EA317C0A25D491580 127F49491300D83FC0495A6C6C495A3A0FFE01FFF86CB65A6C5DC61580013F49C7FC0103 13E02B3D7CBB34>II<00 071538D80FE0EB01F801FE133F90B6FC5E5E5E5E93C7FC5D15F85D15C04AC8FC0180C9FC A9ECFFC0018713FC019F13FF90B67E020113E09039F8007FF0496D7E01C06D7E5B6CC77F C8120F82A31780A21207EA1FC0487E487E12FF7FA21700A25B4B5A6C5A01805C6CC7123F 6D495AD81FE0495A260FFC075B6CB65A6C92C7FCC614FC013F13F0010790C8FC293D7BBB 34>II<121F7F13F890B712F0 A45A17E017C0178017005E5E5A007EC7EA01F84B5A007C4A5A4B5A4B5A93C7FC485C157E 5DC7485A4A5AA24A5A140F5D141F143F5D147FA214FF92C8FC5BA25BA3495AA3130FA513 1FAA6D5A6D5A6D5A2C3F7ABD34>II58 D<16FCA24B7EA24B7EA34B7FA24B7F A34B7FA24B7FA34B7F157C03FC7FEDF87FA2020180EDF03F0203804B7E02078115C08202 0F814B7E021F811500824A81023E7F027E81027C7FA202FC814A147F49B77EA34982A2D9 07E0C7001F7F4A80010F835C83011F8391C87E4983133E83017E83017C81B500FC91B612 FCA5463F7CBE4F>65 DI<922607FFC0130E92B500FC131E020702FF133E023FEDC07E91B7EAE1FE 01039138803FFB499039F80003FF4901C01300013F90C8127F4948151FD9FFF8150F4849 1507485B4A1503481701485B18004890CAFC197E5A5B193E127FA349170012FFAC127F7F 193EA2123FA27F6C187E197C6C7F19FC6C6D16F86C6D150119F06C6D15036C6DED07E0D9 7FFEED0FC06D6CED3F80010F01C0ECFF006D01F8EB03FE6D9039FF801FFC010091B55A02 3F15E002071580020002FCC7FC030713C03F407ABE4C>IIII<922607FFC0130E92B500FC131E020702FF133E023FEDC07E91B7EAE1FE010391 38803FFB499039F80003FF4901C01300013F90C8127F4948151FD9FFF8150F4849150748 5B4A1503481701485B18004890CAFC197E5A5B193E127FA34994C7FC12FFAB0407B612FC 127F7FA3003F92C7383FFE00A27F7EA26C7FA26C7F6C7FA26C7F6C7FD97FFE157F6D6C7E 010F01E014FF6D01F813036D9038FF801F010091B512F3023F15C00207ED803E02009138 FE000E030701E090C7FC46407ABE52>III75 DIIII III<903A03FFC001C0011FEBF803017FEBFE 0748B6128F4815DF48010013FFD80FF8130F48481303497F4848EB007F127F49143F161F 12FF160FA27F1607A27F7F01FC91C7FCEBFF806C13F8ECFFC06C14FCEDFF806C15E016F8 6C816C816C816C16806C6C15C07F010715E0EB007F020714F0EC003F1503030013F8167F 163F127800F8151FA2160FA27EA217F07E161F6C16E06D143F01E015C001F8EC7F8001FE EB01FF9026FFE00713004890B55A486C14F8D8F81F5CD8F00314C027E0003FFEC7FC2D40 7ABE3A>I<003FB912FCA5903BFE003FFE003FD87FF0EE0FFE01C0160349160190C71500 197E127EA2007C183EA400FC183F48181FA5C81600B3AF010FB712F8A5403D7CBC49>I< B76C90B61280A526003FFEC9003EC7FCB3B3A4197E011F177C80A26D17FC616D6D14014E 5A6D6D4A5A6D6D140F6D01F8EC3FC0DA7FFEECFF8091273FFFC00F90C8FC020F90B512FC 02035D020015E0031F1480030101F8C9FC493F7DBD50>I87 D89 D<903807FFC0013F13F848B6FC48812607FE037F260FF8007F6DEB 3FF0486C806F7EA36F7EA26C5A6C5AEA01E0C8FC153F91B5FC130F137F3901FFFE0F4813 E0000F1380381FFE00485A5B485A12FF5BA4151F7F007F143F6D90387BFF806C6C01FB13 FE391FFF07F36CEBFFE100031480C6EC003FD91FF890C7FC2F2B7DA933>97 D<13FFB5FCA512077EAFEDFFE0020713FC021FEBFF80027F80DAFF8113F09139FC003FF8 02F06D7E4A6D7E4A13074A80701380A218C082A318E0AA18C0A25E1880A218005E6E5C6E 495A6E495A02FCEB7FF0903AFCFF01FFE0496CB55AD9F01F91C7FCD9E00713FCC7000113 C033407DBE3A>IIIII<903A03FF8007F0 013F9038F83FF8499038FCFFFC48B712FE48018313F93A07FC007FC34848EB3FE1001FED F1FC4990381FF0F81700003F81A7001F5DA26D133F000F5D6C6C495A3A03FF83FF8091B5 C7FC4814FC01BF5BD80F03138090CAFCA2487EA27F13F06CB6FC16F016FC6C15FF17806C 16C06C16E01207001F16F0393FE000034848EB003F49EC1FF800FF150F90C81207A56C6C EC0FF06D141F003F16E001F0147FD81FFC903801FFC02707FF800F13006C90B55AC615F8 013F14E0010101FCC7FC2F3D7DA834>I<13FFB5FCA512077EAFED1FF8EDFFFE02036D7E 4A80DA0FE07F91381F007F023C805C4A6D7E5CA25CA35CB3A4B5D8FE0FB512E0A5333F7C BE3A>III<13FFB5FCA512077EB092380FFFFEA5DB01FEC7FC4B5AED07F0ED1FE04B5A4B5A4B C8FCEC03FC4A5A4A5A141F4A7EECFFFCA2818102E77F02C37F148102007F826F7E6F7E15 1F6F7E826F7F6F7F816F7FB5D8FC07EBFFC0A5323F7DBE37>I<13FFB5FCA512077EB3B3 AFB512FCA5163F7CBE1D>I<01FFD91FF8ECFFC0B590B5010713F80203DAC01F13FE4A6E 487FDA0FE09026F07F077F91261F003FEBF8010007013EDAF9F0806C0178ECFBC04A6DB4 486C7FA24A92C7FC4A5CA34A5CB3A4B5D8FE07B5D8F03FEBFF80A551297CA858>I<01FF EB1FF8B5EBFFFE02036D7E4A80DA0FE07F91381F007F0007013C806C5B4A6D7E5CA25CA3 5CB3A4B5D8FE0FB512E0A533297CA83A>II<01FFEBFFE0B5000713FC021FEBFF80027F80DAFF8113F09139FC007FF8000701 F06D7E6C496D7E4A130F4A6D7E1880A27013C0A38218E0AA4C13C0A318805E18005E6E5C 6E495A6E495A02FCEBFFF0DAFF035B92B55A029F91C7FC028713FC028113C00280C9FCAC B512FEA5333B7DA83A>II<3901FE01 FE00FF903807FF804A13E04A13F0EC3F1F91387C3FF8000713F8000313F0EBFFE0A29138 C01FF0ED0FE091388007C092C7FCA391C8FCB3A2B6FCA525297DA82B>I<90383FFC1E48 B512BE000714FE5A381FF00F383F800148C7FC007E147EA200FE143EA27E7F6D90C7FC13 F8EBFFE06C13FF15C06C14F06C806C806C806C80C61580131F1300020713C01400007814 7F00F8143F151F7EA27E16806C143F6D140001E013FF9038F803FE90B55A15F0D8F87F13 C026E00FFEC7FC222B7DA929>IIII< B5D8FC03B51280A5C69026E0007FC7FC6E13FE6D6C5B6D6C485A6D6C485A010F13076D6C 485AED9FC06DEBFF806D91C8FC6D5B6E5AA2143F6E7E140F814A7F4A7F4A7F02FE7F9038 01FC7F49486C7E02F07F49486C7E49486C7E011F7F49486C7FD97F008001FE6D7FB5D8C0 07EBFFC0A532287EA737>120 DI127 D E %EndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: A4 %%EndSetup %%Page: 1 1 1 0 bop -51 506 a Fu(ON)45 b(THE)g(ABSOLUTEL)-9 b(Y)45 b(CONTINUOUS)f(SPECTR)m(UM)i(OF)e(ONE)h(DIMENSIONAL)-61 623 y(QUASI-PERIODIC)f(SCHR)1194 601 y(\177)1181 623 y(ODINGER)g(OPERA)-9 b(TORS)46 b(IN)e(THE)h(ADIABA)-9 b(TIC)45 b(LIMIT)951 877 y Ft(ALEXANDER)31 b(FEDOTO)n(V)i(AND)f(FR)2228 858 y(\023)2221 877 y(ED)2339 858 y(\023)2332 877 y(ERIC)h(KLOPP)63 1111 y Fs(Abstra)n(ct.)42 b Ft(In)23 b(this)h(pap)r(er)g(w)n(e)h(study) d(the)i(sp)r(ectral)h(prop)r(erties)f(of)h(families)g(of)g(quasi-p)r (erio)r(dic)g(Sc)n(hr\177)-38 b(odinger)23 b(op)r(erators)63 1202 y(on)k(the)f(real)i(line)f(in)g(the)g(adiabatic)h(limit)f(in)g (the)f(case)i(when)f(the)f(adiabatic)i(iso-energetic)h(curv)n(es)d(are) i(extended)e(along)63 1294 y(the)32 b(p)r(osition)i(direction.)57 b(W)-6 b(e)32 b(pro)n(v)n(e)h(that,)i(in)e(energy)f(in)n(terv)l(als)h (where)h(this)f(is)g(the)g(case,)j(most)c(of)i(the)e(sp)r(ectrum)g(is) 63 1385 y(purely)g(absolutely)h(con)n(tin)n(uous)f(in)h(the)f (adiabatic)i(limit,)h(and)d(that)g(the)h(asso)r(ciated)h(generalized)g (eigenfunctions)g(are)63 1476 y(Blo)r(c)n(h-Flo)r(quet)26 b(solutions.)63 1686 y Fs(R)123 1680 y(\023)123 1686 y(esum)301 1680 y(\023)301 1686 y(e.)32 b Ft(Cet)26 b(article)f(est)g (consacr)n(\023)-36 b(e)26 b(\022)-38 b(a)25 b(l')n(\023)-36 b(etude)25 b(du)f(sp)r(ectre)i(de)e(certaines)i(familles)g(d')n(\023) -36 b(equations)25 b(de)f(Sc)n(hr\177)-38 b(odinger)25 b(quasi-)63 1777 y(p)n(\023)-36 b(erio)r(diques)33 b(sur)g(l'axe)h(r)n (\023)-36 b(eel)34 b(lorsque)g(les)g(v)l(ari)n(\023)-36 b(et)n(\023)g(es)33 b(iso-)n(\023)-36 b(energetiques)34 b(adiabatiques)g(son)n(t)d(\023)-36 b(etendues)33 b(dans)g(la)h (direction)63 1868 y(des)e(p)r(ositions.)54 b(Nous)31 b(d)n(\023)-36 b(emon)n(trons)31 b(que,)i(dans)f(un)f(in)n(terv)l(alle) h(d')n(\023)-36 b(energie)33 b(o)r(\022)-41 b(u)33 b(ceci)f(est)g(le)g (cas,)j(le)d(sp)r(ectre)g(est)g(dans)g(sa)63 1960 y(ma)t(jeure)d (partie)h(puremen)n(t)e(absolumen)n(t)h(con)n(tin)n(u)g(et)h(que)e(les) j(fonctions)g(propres)e(g)n(\023)-36 b(en)n(\023)g(eralis)n(\023)g(ees) 32 b(corresp)r(ondan)n(tes)f(son)n(t)63 2051 y(des)25 b(fonctions)i(de)f(Blo)r(c)n(h-Flo)r(quet.)1520 2456 y Fr(0.)51 b Fq(Intr)n(oduction)-24 2618 y Fr(In)30 b(this)f(pap)s(er,) h(w)m(e)h(analyze)f(the)h(sp)s(ectrum)e(of)i(the)f(ergo)s(dic)g(family) f(of)i(Sc)m(hr\177)-45 b(odinger)29 b(equations)570 2825 y Fp(H)646 2839 y Fo(z)s(;")737 2825 y Fp( )g Fr(=)c Fn(\000)1028 2763 y Fp(d)1075 2730 y Fm(2)p 1002 2804 139 4 v 1002 2887 a Fp(dx)1101 2861 y Fm(2)1150 2825 y Fp( )s Fr(\()p Fp(x)p Fr(\))d(+)e(\()p Fp(V)g Fr(\()p Fp(x)h Fn(\000)f Fp(z)t Fr(\))h(+)e Fp(W)13 b Fr(\()p Fp("x)p Fr(\)\))p Fp( )s Fr(\()p Fp(x)p Fr(\))28 b(=)d Fp(E)5 b( )s Fr(\()p Fp(x)p Fr(\))p Fp(;)108 b(x)25 b Fn(2)g Fl(R)r Fp(;)-3424 b Fr(\(0.1\))-236 3007 y(where)27 b Fp(V)20 b Fr(\()p Fp(x)p Fr(\))28 b(and)f Fp(W)13 b Fr(\()p Fp(\030)t Fr(\))27 b(are)h(p)s(erio)s(dic,)d(and)i Fp(")h Fr(is)e(c)m(hosen)i(so)g(that)g(the)f(p)s(oten)m(tial)g Fp(V)20 b Fr(\()p Fn(\001)14 b(\000)g Fp(z)t Fr(\))g(+)g Fp(W)f Fr(\()p Fp(")p Fn(\001)p Fr(\))30 b(b)s(e)d(quasi-p)s(erio)s (dic.)-236 3115 y(Our)37 b(aim)i(is)f(to)h(study)g(of)g(the)g(sp)s (ectral)f(prop)s(erties)f(of)i Fp(H)1875 3129 y Fo(z)s(;")2006 3115 y Fr(in)e(the)j(limit)c(as)j Fp(")h Fn(!)g Fr(0.)67 b(In)38 b(the)h(pap)s(er)f([9)q(],)j(w)m(e)e(ha)m(v)m(e)-236 3223 y(studied)e(this)g(op)s(erator)i(near)f(the)h(b)s(ottom)g(of)f (the)h(sp)s(ectrum)e(when)h Fp(W)51 b Fr(is)37 b(the)i(cosine.)65 b(In)37 b(the)i(presen)m(t)g(pap)s(er,)g(w)m(e)-236 3331 y(consider)33 b(rather)g(general)h Fp(W)46 b Fr(but)33 b(in)g(a)h(di\013eren)m(t)f(energy)i(region.)50 b(W)-8 b(e)35 b(are)f(in)m(terested)g(in)f(the)h(sp)s(ectrum)f(situated)g(in) -236 3439 y(the)d(\\middle")f(of)i(a)g(sp)s(ectral)e(band)g(of)i(the)g (\\unp)s(erturb)s(ed")c(p)s(erio)s(dic)h(op)s(erator)1271 3590 y Fp(H)1347 3604 y Fm(0)1386 3590 y Fp( )s Fr(\()p Fp(x)p Fr(\))f(=)e Fn(\000)p Fp( )1826 3553 y Fk(00)1868 3590 y Fr(\()p Fp(x)p Fr(\))c(+)f Fp(V)g Fr(\()p Fp(x)p Fr(\))p Fp( )s Fr(\()p Fp(x)p Fr(\))p Fp(:)-2742 b Fr(\(0.2\))-236 3747 y(More)33 b(precisely)-8 b(,)32 b(w)m(e)g(assume)g(that)h(the)f (set)h Fp(E)27 b Fn(\000)21 b Fp(W)13 b Fr(\()p Fl(R)r Fr(\))39 b(lies)30 b(inside)g(a)j(sp)s(ectral)e(band)g(of)40 b(\(0.2\))r(,)32 b(see)h(Fig.)f(1,)i(left)e(part.)-236 3855 y(Suc)m(h)j(energy)i(regions)e(alw)m(a)m(ys)i(exist)f(as)g(the)h (length)e(of)h(the)h(bands)e(of)h(the)g(sp)s(ectrum)f(of)i Fp(H)3082 3869 y Fm(0)3157 3855 y Fr(increases)f(with)e(energy)-236 3963 y(\(see)f(e.g.)h([7)q(,)f(12)q(]\).)48 b(W)-8 b(e)34 b(roughly)e(pro)m(v)m(e)h(that,)h(for)f Fp(")g Fr(su\016cien)m(tly)e (small,)h(in)g(the)g(corresp)s(onding)f(energy)i(region,)g(most)-236 4071 y(of)d(the)f(sp)s(ectrum)g(of)h Fp(H)598 4085 y Fo(z)s(;")719 4071 y Fr(is)e(absolutely)h(con)m(tin)m(uous,)g(and)g (that)i(the)e(corresp)s(onding)f(generalized)h(eigenfunctions)f(are) -236 4179 y(Blo)s(c)m(h-Flo)s(quet)i(solutions)f(of)i(the)f(same)h (functional)e(structure)h(as)g(in)f([6)q(,)h(8)q(].)-24 4287 y(In)h([8],)h(Eliasson)e(has)g(considered)g(one-dimensional)f(Sc)m (hr\177)-45 b(odinger)30 b(op)s(erators)h(with)f(\(analytic\))h(almost) g(p)s(erio)s(dic)-236 4395 y(p)s(oten)m(tials.)46 b(He)33 b(has)f(pro)m(v)m(ed)h(that,)g(in)e(the)i(high)e(energy)i(region,)g (the)f(sp)s(ectrum)f(is)h(absolutely)f(con)m(tin)m(uous,)i(and)f(also) -236 4503 y(that,)i(if)e(the)h(almost)g(p)s(erio)s(dic)d(p)s(oten)m (tial)i(is)g(small,)g(all)g(the)h(sp)s(ectrum)f(is)g(absolutely)g(con)m (tin)m(uous.)47 b(The)33 b(no)m(v)m(elt)m(y)h(here)-236 4610 y(is)i(that)i(w)m(e)f(do)g(not)h(need)e(the)i(energy)f(to)h(b)s(e) e(large)i(or)f Fp(V)45 b Fr(+)24 b Fp(W)49 b Fr(to)38 b(b)s(e)f(small.)59 b(Our)36 b(result)g(holds)g(in)f(a)j(\\middle")e (of)-236 4718 y(an)m(y)j(sp)s(ectral)f(band)f(of)i Fp(H)717 4732 y Fm(0)795 4718 y Fr(as)f(so)s(on)h(as)g(the)f(amplitude)f(of)i Fp(W)51 b Fr(is)37 b(smaller)h(than)g(the)h(width)e(of)h(the)h(band)f (and)g Fp(")h Fr(is)-236 4826 y(su\016cien)m(tly)27 b(small.)38 b(In)28 b(our)f(case,)j Fp(V)49 b Fr(only)27 b(needs)h(to)g(b)s(e)g (square)g(in)m(tegrable.)40 b(T)-8 b(o)28 b(the)h(b)s(est)e(of)i(our)e (kno)m(wledge,)i(this)e(and)-236 4934 y(our)j(previous)g(pap)s(er)g ([9])h(con)m(tain)h(the)f(\014rst)f(results)g(on)g(the)i(nature)e(of)h (the)h(sp)s(ectrum)d(of)i(quasi-p)s(erio)s(dic)d(Sc)m(hr\177)-45 b(odinger)-236 5042 y(op)s(erators)30 b(with)f(p)s(oten)m(tials)h(of)h (lo)m(w)f(regularit)m(y)-8 b(.)-24 5150 y(F)g(or)28 b(small)d Fp(")p Fr(,)j(our)e(results)f(giv)m(e)j(a)f(qualitativ)m(e)f(criterion) f(for)i(the)g(existence)g(of)g(absolutely)e(con)m(tin)m(uous)i(sp)s (ectrum.)-236 5258 y(Our)g(assumption)g(on)h(the)g(relativ)m(e)h(p)s (osition)d(of)i(the)h(sp)s(ectral)f(bands)f(of)35 b(\(0.2\))30 b(and)e(of)g(the)h(in)m(terv)-5 b(al)27 b Fp(E)22 b Fn(\000)15 b Fp(W)e Fr(\()p Fl(R)s Fr(\))35 b(admits)-236 5366 y(a)i(geometric)h (in)m(terpretation.)60 b(It)37 b(is)e(kno)m(wn)i(\(see,)j(for)c (example,)h([9)q(,)g(2]\))h(that)f(for)g(adiabatic)f(problems)f(of)i (the)h(kind)-236 5474 y(studied)32 b(in)g(the)i(presen)m(t)g(pap)s(er,) g(the)g(\\e\013ectiv)m(e")i(Hamiltonian)c(obtained)i(b)m(y)f(taking)h Fn(E)8 b Fr(\()p Fp(\024)p Fr(\),)36 b(the)e(disp)s(ersion)c(relation) -236 5582 y(of)42 b(the)g(p)s(erio)s(dic)d(Sc)m(hr\177)-45 b(odinger)40 b(op)s(erator)i Fp(H)1370 5596 y Fm(0)1409 5582 y Fr(,)j(as)d(kinetic)f(energy)h(and)f(the)h(p)s(oten)m(tial)f Fp(W)13 b Fr(\()p Fp(\020)7 b Fr(\))42 b(as)f(p)s(oten)m(tial)h(energy) -8 b(,)p -236 5654 499 4 v -137 5745 a Ft(1991)29 b Fj(Mathematics)g (Subje)l(ct)g(Classi\014c)l(ation.)39 b Ft(34L40,)28 b(34E20,)f(81Q05,)h(81Q20.)-137 5837 y Fj(Key)d(wor)l(ds)g(and)f(phr)l (ases.)40 b Ft(quasi-p)r(erio)r(dic)23 b(Sc)n(hr\177)-38 b(odinger)22 b(equation,)g(absolutely)h(con)n(tin)n(uous)e(sp)r (ectrum,)h(Blo)r(c)n(h-Flo)r(quet)h(solutions,)h(com-)-236 5928 y(plex)h(WKB)h(metho)r(d,)e(mono)r(drom)n(y)g(matrix,)h(adiabatic) i(limit.)-137 6019 y(This)h(w)n(ork)g(w)n(as)g(done)f(while)i(A.F.)e (held)g(a)h(P)-6 b(AST)27 b(professorship)i(at)e(Univ)n(ersit)n(\023) -36 b(e)27 b(P)n(aris-Nord.)40 b(F.K.)28 b(gratefully)g(ac)n(kno)n (wledges)h(supp)r(ort)-236 6110 y(of)d(the)g(europ)r(ean)f(TMR)h(net)n (w)n(ork)g(ERBFMRX)n(CT960001.)1872 6210 y Fm(1)p eop %%Page: 2 2 2 1 bop 236 908 a @beginspecial 71 @llx 591 @lly 644 @urx 721 @ury 3968 @rwi @setspecial %%BeginDocument: DESSINS/figbmm1.eps %!PS-Adobe-2.0 EPSF-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: essai-fig.dvi %%BoundingBox: 71 591 644 721 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: /usr/bin/dvips -E -h pstricks.pro -h pst-dots.pro -o %+ essai-fig.eps essai-fig.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.06.12:2301 %%BeginProcSet: pstricks.pro %! % PostScript prologue for pstricks.tex. % Version 97 patch 3, 98/06/01 % For distribution, see pstricks.tex. % /tx@Dict 200 dict def tx@Dict begin /ADict 25 dict def /CM { matrix currentmatrix } bind def /SLW /setlinewidth load def /CLW /currentlinewidth load def /CP /currentpoint load def /ED { exch def } bind def /L /lineto load def /T /translate load def /TMatrix { } def /RAngle { 0 } def /Atan { /atan load stopped { pop pop 0 } if } def /Div { dup 0 eq { pop } { div } ifelse } def /NET { neg exch neg exch T } def /Pyth { dup mul exch dup mul add sqrt } def /PtoC { 2 copy cos mul 3 1 roll sin mul } def /PathLength@ { /z z y y1 sub x x1 sub Pyth add def /y1 y def /x1 x def } def /PathLength { flattenpath /z 0 def { /y1 ED /x1 ED /y2 y1 def /x2 x1 def } { /y ED /x ED PathLength@ } {} { /y y2 def /x x2 def PathLength@ } /pathforall load stopped { pop pop pop pop } if z } def /STP { .996264 dup scale } def /STV { SDict begin normalscale end STP } def /DashLine { dup 0 gt { /a .5 def PathLength exch div } { pop /a 1 def PathLength } ifelse /b ED /x ED /y ED /z y x add def b a .5 sub 2 mul y mul sub z Div round z mul a .5 sub 2 mul y mul add b exch Div dup y mul /y ED x mul /x ED x 0 gt y 0 gt and { [ y x ] 1 a sub y mul } { [ 1 0 ] 0 } ifelse setdash stroke } def /DotLine { /b PathLength def /a ED /z ED /y CLW def /z y z add def a 0 gt { /b b a div def } { a 0 eq { /b b y sub def } { a -3 eq { /b b y add def } if } ifelse } ifelse [ 0 b b z Div round Div dup 0 le { pop 1 } if ] a 0 gt { 0 } { y 2 div a -2 gt { neg } if } ifelse setdash 1 setlinecap stroke } def /LineFill { gsave abs CLW add /a ED a 0 dtransform round exch round exch 2 copy idtransform exch Atan rotate idtransform pop /a ED .25 .25 % DG/SR modification begin - Dec. 12, 1997 - Patch 2 %itransform translate pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a itransform pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a % DG/SR modification end Div cvi /x1 ED /y2 y2 y1 sub def clip newpath 2 setlinecap systemdict /setstrokeadjust known { true setstrokeadjust } if x2 x1 sub 1 add { x1 % DG/SR modification begin - Jun. 1, 1998 - Patch 3 (from Michael Vulis) % a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore } % def a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore pop pop } def % DG/SR modification end /BeginArrow { ADict begin /@mtrx CM def gsave 2 copy T 2 index sub neg exch 3 index sub exch Atan rotate newpath } def /EndArrow { @mtrx setmatrix CP grestore end } def /Arrow { CLW mul add dup 2 div /w ED mul dup /h ED mul /a ED { 0 h T 1 -1 scale } if w neg h moveto 0 0 L w h L w neg a neg rlineto gsave fill grestore } def /Tbar { CLW mul add /z ED z -2 div CLW 2 div moveto z 0 rlineto stroke 0 CLW moveto } def /Bracket { CLW mul add dup CLW sub 2 div /x ED mul CLW add /y ED /z CLW 2 div def x neg y moveto x neg CLW 2 div L x CLW 2 div L x y L stroke 0 CLW moveto } def /RoundBracket { CLW mul add dup 2 div /x ED mul /y ED /mtrx CM def 0 CLW 2 div T x y mul 0 ne { x y scale } if 1 1 moveto .85 .5 .35 0 0 0 curveto -.35 0 -.85 .5 -1 1 curveto mtrx setmatrix stroke 0 CLW moveto } def /SD { 0 360 arc fill } def /EndDot { { /z DS def } { /z 0 def } ifelse /b ED 0 z DS SD b { 0 z DS CLW sub SD } if 0 DS z add CLW 4 div sub moveto } def /Shadow { [ { /moveto load } { /lineto load } { /curveto load } { /closepath load } /pathforall load stopped { pop pop pop pop CP /moveto load } if ] cvx newpath 3 1 roll T exec } def /NArray { aload length 2 div dup dup cvi eq not { exch pop } if /n exch cvi def } def /NArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop } if f { ] aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def /Line { NArray n 0 eq not { n 1 eq { 0 0 /n 2 def } if ArrowA /n n 2 sub def n { Lineto } repeat CP 4 2 roll ArrowB L pop pop } if } def /Arcto { /a [ 6 -2 roll ] cvx def a r /arcto load stopped { 5 } { 4 } ifelse { pop } repeat a } def /CheckClosed { dup n 2 mul 1 sub index eq 2 index n 2 mul 1 add index eq and { pop pop /n n 1 sub def } if } def /Polygon { NArray n 2 eq { 0 0 /n 3 def } if n 3 lt { n { pop pop } repeat } { n 3 gt { CheckClosed } if n 2 mul -2 roll /y0 ED /x0 ED /y1 ED /x1 ED x1 y1 /x1 x0 x1 add 2 div def /y1 y0 y1 add 2 div def x1 y1 moveto /n n 2 sub def n { Lineto } repeat x1 y1 x0 y0 6 4 roll Lineto Lineto pop pop closepath } ifelse } def /Diamond { /mtrx CM def T rotate /h ED /w ED dup 0 eq { pop } { CLW mul neg /d ED /a w h Atan def /h d a sin Div h add def /w d a cos Div w add def } ifelse mark w 2 div h 2 div w 0 0 h neg w neg 0 0 h w 2 div h 2 div /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx setmatrix } def % DG modification begin - Jan. 15, 1997 %/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup 0 eq { %pop } { CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2 %div dup cos exch sin Div mul sub def } ifelse mark 0 d w neg d 0 h w d 0 %d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx %setmatrix } def /Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2 div dup cos exch sin Div mul sub def mark 0 d w neg d 0 h w d 0 d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx % DG/SR modification begin - Jun. 1, 1998 - Patch 3 (from Michael Vulis) % setmatrix } def setmatrix pop } def % DG/SR modification end /CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth def } def /CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth def } def /CC { /l0 l1 def /x1 x dx sub def /y1 y dy sub def /dx0 dx1 def /dy0 dy1 def CCA /dx dx0 l1 c exp mul dx1 l0 c exp mul add def /dy dy0 l1 c exp mul dy1 l0 c exp mul add def /m dx0 dy0 Atan dx1 dy1 Atan sub 2 div cos abs b exp a mul dx dy Pyth Div 2 div def /x2 x l0 dx mul m mul sub def /y2 y l0 dy mul m mul sub def /dx l1 dx mul m mul neg def /dy l1 dy mul m mul neg def } def /IC { /c c 1 add def c 0 lt { /c 0 def } { c 3 gt { /c 3 def } if } ifelse /a a 2 mul 3 div 45 cos b exp div def CCA /dx 0 def /dy 0 def } def /BOC { IC CC x2 y2 x1 y1 ArrowA CP 4 2 roll x y curveto } def /NC { CC x1 y1 x2 y2 x y curveto } def /EOC { x dx sub y dy sub 4 2 roll ArrowB 2 copy curveto } def /BAC { IC CC x y moveto CC x1 y1 CP ArrowA } def /NAC { x2 y2 x y curveto CC x1 y1 } def /EAC { x2 y2 x y ArrowB curveto pop pop } def /OpenCurve { NArray n 3 lt { n { pop pop } repeat } { BOC /n n 3 sub def n { NC } repeat EOC } ifelse } def /AltCurve { { false NArray n 2 mul 2 roll [ n 2 mul 3 sub 1 roll ] aload /Points ED n 2 mul -2 roll } { false NArray } ifelse n 4 lt { n { pop pop } repeat } { BAC /n n 4 sub def n { NAC } repeat EAC } ifelse } def /ClosedCurve { NArray n 3 lt { n { pop pop } repeat } { n 3 gt { CheckClosed } if 6 copy n 2 mul 6 add 6 roll IC CC x y moveto n { NC } repeat closepath pop pop } ifelse } def /SQ { /r ED r r moveto r r neg L r neg r neg L r neg r L fill } def /ST { /y ED /x ED x y moveto x neg y L 0 x L fill } def /SP { /r ED gsave 0 r moveto 4 { 72 rotate 0 r L } repeat fill grestore } def /FontDot { DS 2 mul dup matrix scale matrix concatmatrix exch matrix rotate matrix concatmatrix exch findfont exch makefont setfont } def /Rect { x1 y1 y2 add 2 div moveto x1 y2 lineto x2 y2 lineto x2 y1 lineto x1 y1 lineto closepath } def /OvalFrame { x1 x2 eq y1 y2 eq or { pop pop x1 y1 moveto x2 y2 L } { y1 y2 sub abs x1 x2 sub abs 2 copy gt { exch pop } { pop } ifelse 2 div exch { dup 3 1 roll mul exch } if 2 copy lt { pop } { exch pop } ifelse /b ED x1 y1 y2 add 2 div moveto x1 y2 x2 y2 b arcto x2 y2 x2 y1 b arcto x2 y1 x1 y1 b arcto x1 y1 x1 y2 b arcto 16 { pop } repeat closepath } ifelse } def /Frame { CLW mul /a ED 3 -1 roll 2 copy gt { exch } if a sub /y2 ED a add /y1 ED 2 copy gt { exch } if a sub /x2 ED a add /x1 ED 1 index 0 eq { pop pop Rect } { OvalFrame } ifelse } def /BezierNArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop } if n 1 sub neg 3 mod 3 add 3 mod { 0 0 /n n 1 add def } repeat f { ] aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def /OpenBezier { BezierNArray n 1 eq { pop pop } { ArrowA n 4 sub 3 idiv { 6 2 roll 4 2 roll curveto } repeat 6 2 roll 4 2 roll ArrowB curveto } ifelse } def /ClosedBezier { BezierNArray n 1 eq { pop pop } { moveto n 1 sub 3 idiv { 6 2 roll 4 2 roll curveto } repeat closepath } ifelse } def /BezierShowPoints { gsave Points aload length 2 div cvi /n ED moveto n 1 sub { lineto } repeat CLW 2 div SLW [ 4 4 ] 0 setdash stroke grestore } def /Parab { /y0 exch def /x0 exch def /y1 exch def /x1 exch def /dx x0 x1 sub 3 div def /dy y0 y1 sub 3 div def x0 dx sub y0 dy add x1 y1 ArrowA x0 dx add y0 dy add x0 2 mul x1 sub y1 ArrowB curveto /Points [ x1 y1 x0 y0 x0 2 mul x1 sub y1 ] def } def /Grid { newpath /a 4 string def /b ED /c ED /n ED cvi dup 1 lt { pop 1 } if /s ED s div dup 0 eq { pop 1 } if /dy ED s div dup 0 eq { pop 1 } if /dx ED dy div round dy mul /y0 ED dx div round dx mul /x0 ED dy div round cvi /y2 ED dx div round cvi /x2 ED dy div round cvi /y1 ED dx div round cvi /x1 ED /h y2 y1 sub 0 gt { 1 } { -1 } ifelse def /w x2 x1 sub 0 gt { 1 } { -1 } ifelse def b 0 gt { /z1 b 4 div CLW 2 div add def /Helvetica findfont b scalefont setfont /b b .95 mul CLW 2 div add def } if systemdict /setstrokeadjust known { true setstrokeadjust /t { } def } { /t { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } bind def } ifelse gsave n 0 gt { 1 setlinecap [ 0 dy n div ] dy n div 2 div setdash } { 2 setlinecap } ifelse /i x1 def /f y1 dy mul n 0 gt { dy n div 2 div h mul sub } if def /g y2 dy mul n 0 gt { dy n div 2 div h mul add } if def x2 x1 sub w mul 1 add dup 1000 gt { pop 1000 } if { i dx mul dup y0 moveto b 0 gt { gsave c i a cvs dup stringwidth pop /z2 ED w 0 gt {z1} {z1 z2 add neg} ifelse h 0 gt {b neg} {z1} ifelse rmoveto show grestore } if dup t f moveto g t L stroke /i i w add def } repeat grestore gsave n 0 gt % DG/SR modification begin - Nov. 7, 1997 - Patch 1 %{ 1 setlinecap [ 0 dx n div ] dy n div 2 div setdash } { 1 setlinecap [ 0 dx n div ] dx n div 2 div setdash } % DG/SR modification end { 2 setlinecap } ifelse /i y1 def /f x1 dx mul n 0 gt { dx n div 2 div w mul sub } if def /g x2 dx mul n 0 gt { dx n div 2 div w mul add } if def y2 y1 sub h mul 1 add dup 1000 gt { pop 1000 } if { newpath i dy mul dup x0 exch moveto b 0 gt { gsave c i a cvs dup stringwidth pop /z2 ED w 0 gt {z1 z2 add neg} {z1} ifelse h 0 gt {z1} {b neg} ifelse rmoveto show grestore } if dup f exch t moveto g exch t L stroke /i i h add def } repeat grestore } def /ArcArrow { /d ED /b ED /a ED gsave newpath 0 -1000 moveto clip newpath 0 1 0 0 b grestore c mul /e ED pop pop pop r a e d PtoC y add exch x add exch r a PtoC y add exch x add exch b pop pop pop pop a e d CLW 8 div c mul neg d } def /Ellipse { /mtrx CM def T scale 0 0 1 5 3 roll arc mtrx setmatrix } def /Rot { CP CP translate 3 -1 roll neg rotate NET } def /RotBegin { tx@Dict /TMatrix known not { /TMatrix { } def /RAngle { 0 } def } if /TMatrix [ TMatrix CM ] cvx def /a ED a Rot /RAngle [ RAngle dup a add ] cvx def } def /RotEnd { /TMatrix [ TMatrix setmatrix ] cvx def /RAngle [ RAngle pop ] cvx def } def /PutCoor { gsave CP T CM STV exch exec moveto setmatrix CP grestore } def /PutBegin { /TMatrix [ TMatrix CM ] cvx def CP 4 2 roll T moveto } def /PutEnd { CP /TMatrix [ TMatrix setmatrix ] cvx def moveto } def /Uput { /a ED add 2 div /h ED 2 div /w ED /s a sin def /c a cos def /b s abs c abs 2 copy gt dup /q ED { pop } { exch pop } ifelse def /w1 c b div w mul def /h1 s b div h mul def q { w1 abs w sub dup c mul abs } { h1 abs h sub dup s mul abs } ifelse } def /UUput { /z ED abs /y ED /x ED q { x s div c mul abs y gt } { x c div s mul abs y gt } ifelse { x x mul y y mul sub z z mul add sqrt z add } { q { x s div } { x c div } ifelse abs } ifelse a PtoC h1 add exch w1 add exch } def /BeginOL { dup (all) eq exch TheOL eq or { IfVisible not { Visible /IfVisible true def } if } { IfVisible { Invisible /IfVisible false def } if } ifelse } def /InitOL { /OLUnit [ 3000 3000 matrix defaultmatrix dtransform ] cvx def /Visible { CP OLUnit idtransform T moveto } def /Invisible { CP OLUnit neg exch neg exch idtransform T moveto } def /BOL { BeginOL } def /IfVisible true def } def end % END pstricks.pro %%EndProcSet %%BeginProcSet: pst-dots.pro %!PS-Adobe-2.0 %%Title: Dot Font for PSTricks 97 - Version 97, 93/05/07. %%Creator: Timothy Van Zandt %%Creation Date: May 7, 1993 10 dict dup begin /FontType 3 def /FontMatrix [ .001 0 0 .001 0 0 ] def /FontBBox [ 0 0 0 0 ] def /Encoding 256 array def 0 1 255 { Encoding exch /.notdef put } for Encoding dup (b) 0 get /Bullet put dup (c) 0 get /Circle put dup (C) 0 get /BoldCircle put dup (u) 0 get /SolidTriangle put dup (t) 0 get /Triangle put dup (T) 0 get /BoldTriangle put dup (r) 0 get /SolidSquare put dup (s) 0 get /Square put dup (S) 0 get /BoldSquare put dup (q) 0 get /SolidPentagon put dup (p) 0 get /Pentagon put (P) 0 get /BoldPentagon put /Metrics 13 dict def Metrics begin /Bullet 1000 def /Circle 1000 def /BoldCircle 1000 def /SolidTriangle 1344 def /Triangle 1344 def /BoldTriangle 1344 def /SolidSquare 886 def /Square 886 def /BoldSquare 886 def /SolidPentagon 1093.2 def /Pentagon 1093.2 def /BoldPentagon 1093.2 def /.notdef 0 def end /BBoxes 13 dict def BBoxes begin /Circle { -550 -550 550 550 } def /BoldCircle /Circle load def /Bullet /Circle load def /Triangle { -571.5 -330 571.5 660 } def /BoldTriangle /Triangle load def /SolidTriangle /Triangle load def /Square { -450 -450 450 450 } def /BoldSquare /Square load def /SolidSquare /Square load def /Pentagon { -546.6 -465 546.6 574.7 } def /BoldPentagon /Pentagon load def /SolidPentagon /Pentagon load def /.notdef { 0 0 0 0 } def end /CharProcs 20 dict def CharProcs begin /Adjust { 2 copy dtransform floor .5 add exch floor .5 add exch idtransform 3 -1 roll div 3 1 roll exch div exch scale } def /CirclePath { 0 0 500 0 360 arc closepath } def /Bullet { 500 500 Adjust CirclePath fill } def /Circle { 500 500 Adjust CirclePath .9 .9 scale CirclePath eofill } def /BoldCircle { 500 500 Adjust CirclePath .8 .8 scale CirclePath eofill } def /BoldCircle { 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 /.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: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N /vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N /rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N /@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{ /hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B /@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{ /urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known {userdict/md get type/dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{ itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack} if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{ noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{ Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale }if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState save N userdict maxlength dict begin/magscale true def normalscale currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts /psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR/showpage{}N/erasepage{}N/copypage{}N/p 3 def @MacSetUp}N/doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N/endTexFig{end psf$SavedState restore}N/@beginspecial{SDict begin/SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count/ocount X/dcount countdictstack N}N/@setspecial{ CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR }{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if/showpage{}N/erasepage{}N/copypage{}N newpath}N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{end} repeat grestore SpecialSave restore end}N/@defspecial{SDict begin}N /@fedspecial{end}B/li{lineto}B/rl{rlineto}B/rc{rcurveto}B/np{/SaveX currentpoint/SaveY X N 1 setlinecap newpath}N/st{stroke SaveX SaveY moveto}N/fil{fill SaveX SaveY moveto}N/ellipse{/endangle X/startangle X /yrad X/xrad X/savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet TeXDict begin 40258437 52099154 1000 600 600 (essai-fig.dvi) @start %DVIPSBitmapFont: Fa cmsy8 8 1 /Fa 1 1 df0 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb cmmi8 8 1 /Fb 1 111 df<3907C007E0391FE03FF83918F8783E393879E01E39307B801F38707F00 126013FEEAE0FC12C05B00815C0001143E5BA20003147E157C5B15FC0007ECF8081618EB C00115F0000F1538913803E0300180147016E0001F010113C015E390C7EAFF00000E143E 251F7E9D2B>110 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc cmr8 8 2 /Fc 2 51 df<130C133C137CEA03FC12FFEAFC7C1200B3B113FE387FFFFEA2172C7AAB23 >49 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd msbm10 12 1 /Fd 1 83 df<007FB712C0B812FCEFFF806C17E02800F807F00F13F8DBC00113FE017890 398000FCFF94387C3F8094383E0FC0727E94381E03F0EF1F011800717FA21978A519F8A2 4D5B1801EF1E034E5A94383E0FC094387E3F80DDFDFFC7FC933807FFFE92B612F818E095 C8FC17F0ED87C1EEC0F8923883E0FC177C923881F03EA2923880F81F84EE7C0F717E163E 93383F03E0041F7FEE0F81EF80F8EE07C0187C933803E07E183E706C7E85706C6C7E1807 94387C03E0057E7F94383E01F8716C7E197C01F86D6D6C7EF13F80007FB66C6CB512E0B7 00C015F0836C4B6C14E044447EC33D>82 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe cmr12 12 2 /Fe 2 42 df<140C141C1438147014E0EB01C01303EB0780EB0F00A2131E5BA25B13F85B 12015B1203A2485AA3485AA348C7FCA35AA2123EA2127EA4127CA312FCB3A2127CA3127E A4123EA2123FA27EA36C7EA36C7EA36C7EA212017F12007F13787FA27F7FA2EB0780EB03 C01301EB00E014701438141C140C166476CA26>40 D<12C07E12707E7E7E120F6C7E6C7E A26C7E6C7EA21378137C133C133E131E131FA2EB0F80A3EB07C0A3EB03E0A314F0A21301 A214F8A41300A314FCB3A214F8A31301A414F0A21303A214E0A3EB07C0A3EB0F80A3EB1F 00A2131E133E133C137C13785BA2485A485AA2485A48C7FC120E5A5A5A5A5A16647BCA26 >I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmsy10 12 1 /Ff 1 1 df<007FB912E0BA12F0A26C18E03C04789A4D>0 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmmi12 12 4 /Fg 4 88 df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ndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%EndSetup 1 0 bop 0 531 a tx@Dict begin gsave CM STV CP newpath moveto 3.0 neg 0 rmoveto clip setmatrix end 0 531 a 0 1063 4757 1063 v 4757 531 a currentpoint grestore moveto 4757 531 a -3442 w @beginspecial @setspecial tx@Dict begin STP newpath 0.5 SLW 0. setgray /ArrowA { moveto } def /ArrowB { } def [ 99.58455 -14.22636 0.0 -14.22636 /Lineto /lineto load def false Line gsave 0.5 SLW 0. setgray 1.0 1.0 0 0 add DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.5 SLW 0. setgray /ArrowA { moveto } def /ArrowB { } def [ 270.30092 56.90546 142.26364 56.90546 /Lineto /lineto load def false Line gsave 0.5 SLW 0. setgray 1.0 1.0 0 0 add DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.5 SLW 0. setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. scale false 0.4 1.4 1.5 2. Arrow EndArrow } def [ 270.30092 0.0 142.26364 0.0 /Lineto /lineto load def false Line gsave 0.5 SLW 0. setgray 1.0 1.0 -2 0 add DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.5 SLW 0. setgray /ArrowA { moveto } def /ArrowB { } def [ 270.30092 -56.90546 142.26364 -56.90546 /Lineto /lineto load def false Line gsave 0.5 SLW 0. setgray 1.0 1.0 0 0 add DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.5 SLW 0. setgray /ArrowA { BeginArrow 1. 1. scale false 0.4 1.4 1.5 2. Arrow EndArrow moveto } def /ArrowB { } def [ 149.37682 -64.01863 149.37682 64.01863 /Lineto /lineto load def false Line gsave 0.5 SLW 0. setgray 1.0 1.0 0 -1 add DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.5 SLW 0. setgray /ArrowA { moveto } def /ArrowB { } def [ 206.28227 -64.01863 206.28227 64.01863 /Lineto /lineto load def false Line gsave 0.5 SLW 0. setgray 1.0 1.0 0 0 add DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.5 SLW 0. setgray /ArrowA { moveto } def /ArrowB { } def [ 263.18773 -64.01863 263.18773 64.01863 /Lineto /lineto load def false Line gsave 0.5 SLW 0. setgray 1.0 1.0 0 0 add DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.5 SLW 0. setgray /ArrowA { BeginArrow 1. 1. scale 0.15 2.0 5. Bracket EndArrow moveto } def /ArrowB { } def [ 56.90546 14.22636 35.5659 14.22636 /Lineto /lineto load def false Line gsave 1.5 SLW 0. setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.5 SLW 0. setgray /ArrowA { BeginArrow 1. 1. scale 2.0 5. Tbar EndArrow moveto } def /ArrowB { BeginArrow 1. 1. scale 0.15 2.0 5. Bracket EndArrow } def [ 78.245 14.22636 56.90546 14.22636 /Lineto /lineto load def false Line gsave 1.5 SLW 0. setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.5 SLW 0. setgray /ArrowA { BeginArrow 1. 1. scale 0.15 2.0 5. Bracket EndArrow moveto } def /ArrowB { BeginArrow 1. 1. scale 0.15 2.0 5. Bracket EndArrow } def [ 99.58455 -14.22636 0.0 -14.22636 /Lineto /lineto load def false Line gsave 1.5 SLW 0. setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.5 SLW 0. setgray /ArrowA { moveto } def /ArrowB { } def [ 35.5659 -14.22636 35.5659 14.22636 /Lineto /lineto load def false Line gsave 0.5 SLW 0. setgray 1.0 1.0 0 0 add DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.5 SLW 0. setgray /ArrowA { moveto } def /ArrowB { } def [ 78.245 -14.22636 78.245 14.22636 /Lineto /lineto load def false Line gsave 0.5 SLW 0. setgray 1.0 1.0 0 0 add DashLine grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.0 SLW 0. setgray /ArrowA { moveto } def /ArrowB { } def [ 270.30092 46.94704 263.18773 42.6791 256.07455 38.41113 213.39546 46.94704 206.28227 42.6791 199.1691 38.41113 156.49 46.94704 149.37682 42.6791 142.26364 38.41113 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 1.0 SLW 0. setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.0 SLW 0. setgray /ArrowA { moveto } def /ArrowB { } def [ 270.30092 9.9584 263.18773 14.22636 256.07455 18.49431 213.39546 9.9584 206.28227 14.22636 199.1691 18.49431 156.49 9.9584 149.37682 14.22636 142.26364 18.49431 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 1.0 SLW 0. setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.0 SLW 0. setgray /ArrowA { moveto } def /ArrowB { } def [ 270.30092 -46.94704 263.18773 -42.6791 256.07455 -38.41113 213.39546 -46.94704 206.28227 -42.6791 199.1691 -38.41113 156.49 -46.94704 149.37682 -42.6791 142.26364 -38.41113 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 1.0 SLW 0. setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 1.0 SLW 0. setgray /ArrowA { moveto } def /ArrowB { } def [ 270.30092 -9.9584 263.18773 -14.22636 256.07455 -18.49431 213.39546 -9.9584 206.28227 -14.22636 199.1691 -18.49431 156.49 -9.9584 149.37682 -14.22636 142.26364 -18.49431 1. 0.1 0. /c ED /b ED /a ED false OpenCurve gsave 1.0 SLW 0. setgray 0 setlinecap stroke grestore end @endspecial @beginspecial @setspecial tx@Dict begin STP newpath 0.5 SLW 0. setgray /ArrowA { moveto } def /ArrowB { BeginArrow 1. 1. scale false 0.4 1.4 1.5 2. Arrow EndArrow } def [ 35.5659 21.33954 7.11317 36.98863 /Lineto /lineto load def false Line gsave 0.5 SLW 0. setgray 0 setlinecap stroke grestore end @endspecial 1788 413 a tx@Dict begin { 7.11317 9.4013 8.2 0.0 90. Uput UUput } PutCoor PutBegin end 1788 413 a 1749 447 a Fg(E)1788 413 y tx@Dict begin PutEnd end 1788 413 a 1197 177 a tx@Dict begin { 0.0 54.59827 9.0 3.0 0. Uput UUput } PutCoor PutBegin end 1197 177 a 971 202 a Fg(E)28 b Ff(\000)23 b Fg(W)14 b Fe(\()p Fd(R)t Fe(\))1197 177 y tx@Dict begin PutEnd end 1197 177 a 1197 768 a tx@Dict begin { 0.0 29.46663 8.2 2.63336 0. Uput UUput } PutCoor PutBegin end 1197 768 a 1075 791 a Fg(E)1147 806 y Fc(2)p Fb(n)p Fa(\000)p Fc(1)1197 768 y tx@Dict begin PutEnd end 1197 768 a 2024 768 a tx@Dict begin { 0.0 18.60536 8.2 1.79999 0. Uput UUput } PutCoor PutBegin end 2024 768 a 1947 794 a Fg(E)2019 809 y Fc(2)p Fb(n)2024 768 y tx@Dict begin PutEnd end 2024 768 a 3501 531 a tx@Dict begin { 7.11317 6.03816 8.33331 2.33331 135. Uput UUput } PutCoor PutBegin end 3501 531 a 3476 556 a Fg(\020)3501 531 y tx@Dict begin PutEnd end 3501 531 a 2556 59 a tx@Dict begin { 7.11317 6.76382 5.16667 0.0 225. Uput UUput } PutCoor PutBegin end 2556 59 a 2528 81 a Fg(\024)2556 59 y tx@Dict begin PutEnd end 2556 59 a eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF %%EndDocument @endspecial 1327 1116 a Fq(Figure)34 b(1.)46 b Fr(The)30 b(band)f(cen)m(ter)-236 1331 y Fp(H)7 b Fr(\()p Fp(\024;)15 b(\020)7 b Fr(\))35 b(=)f Fn(E)8 b Fr(\()p Fp(\024)p Fr(\))24 b(+)g Fp(W)13 b Fr(\()p Fp(\020)7 b Fr(\),)37 b(pla)m(ys)e(an)g(imp)s(ortan)m(t)g(part.)57 b(Semi-classical)34 b(\\wisdom")h(sa)m(ys)h(that)g(one)g(should)e(consider)-236 1439 y(the)g(iso-energy)f(curv)m(es)h Fn(E)8 b Fr(\()p Fp(\024)p Fr(\))24 b(+)e Fp(W)13 b Fr(\()p Fp(\020)7 b Fr(\))30 b(=)g Fp(E)5 b Fr(.)51 b(Our)32 b(assumption)g(means)h(that) i(these)f(curv)m(es)f(are)h(extended)g(along)f(the)-236 1547 y(p)s(osition)24 b(axis,)i(see)g(Fig.)g(1,)h(righ)m(t)e(part.)39 b(F)-8 b(rom)27 b(the)e(quan)m(tum)h(ph)m(ysicist's)e(p)s(oin)m(t)g(of) i(view,)g(existence)g(of)g(suc)m(h)g(iso-energy)-236 1655 y(curv)m(es)36 b(corresp)s(onds)f(to)i(extended)g(states:)53 b(a)37 b(semi-classical)e(particle)h(should)e(\\liv)m(e")j(near)f (these)h(curv)m(es,)h(hence)e(b)s(e)-236 1763 y(\\extended")31 b(in)e(the)i(p)s(osition)d(v)-5 b(ariable,)30 b(see)h([15)q(].)41 b(W)-8 b(e)31 b(justify)e(this)g(heuristics.)1554 1960 y(1.)51 b Fq(The)34 b(resul)-6 b(ts)-236 2121 y Fr(1.1.)53 b Fu(The)35 b(assumptions.)45 b Fr(W)-8 b(e)32 b(assume)e(that)-111 2251 y Fn(\017)42 b Fp(V)50 b Fr(and)30 b Fp(W)43 b Fr(are)30 b(p)s(erio)s(dic,)871 2401 y Fp(V)36 b Fr(\()p Fp(x)20 b Fr(+)g(1\))26 b(=)f Fp(V)35 b Fr(\()p Fp(x)p Fr(\))p Fp(;)107 b(W)28 b Fr(\()p Fp(x)21 b Fr(+)e(2)p Fp(\031)s Fr(\))27 b(=)e Fp(W)i Fr(\()p Fp(x)p Fr(\))p Fp(;)107 b(x)25 b Fn(2)g Fl(R)s Fr(;)-3123 b(\(1.1\))-111 2551 y Fn(\017)42 b Fp(")30 b Fr(is)g(\014xed)f(p)s(ositiv)m(e)h(n)m(um)m(b) s(er;)-111 2659 y Fn(\017)42 b Fp(V)50 b Fr(is)30 b(real)g(v)-5 b(alued)29 b(and)h(lo)s(cally)f(square)h(in)m(tegrable;)-111 2767 y Fn(\017)42 b Fp(W)g Fr(is)30 b(real)g(on)g Fl(R)39 b Fr(and)30 b(analytic)g(in)f(a)i(neigh)m(b)s(orho)s(o)s(d)d(of)i Fl(R)s Fr(;)-111 2875 y Fn(\017)42 b Fp(z)34 b Fr(is)c(a)g(real)g (parameter)h(indexing)d(the)j(equations)f(of)h(the)f(family)-8 b(.)-236 3046 y(1.2.)53 b Fu(P)m(erio)s(dic)30 b(Sc)m(hr\177)-52 b(odinger)30 b(op)s(erator.)46 b Fr(T)-8 b(o)25 b(form)m(ulate)g(our)f (results,)h(w)m(e)g(need)g(to)g(recall)g(some)g(information)e(ab)s(out) -236 3154 y(the)39 b(p)s(erio)s(dic)e(Sc)m(hr\177)-45 b(odinger)37 b(op)s(erator)j(\(0.2\))r(.)67 b(Its)39 b(sp)s(ectrum)f(on)h Fp(L)2194 3121 y Fm(2)2233 3154 y Fr(\()p Fl(R)s Fr(\))45 b(is)39 b(absolutely)f(con)m(tin)m(uous)g (and)h(consists)g(of)-236 3262 y Fi(sp)-5 b(e)g(ctr)g(al)35 b(b)-5 b(ands)32 b Fr(i.e.)40 b(in)m(terv)-5 b(als)30 b([)p Fp(E)950 3276 y Fm(1)989 3262 y Fp(;)h(E)1112 3276 y Fm(2)1151 3262 y Fr(],)g([)p Fp(E)1324 3276 y Fm(3)1364 3262 y Fp(;)g(E)1487 3276 y Fm(4)1526 3262 y Fr(],)g Fp(:)15 b(:)g(:)h Fr(,)31 b([)p Fp(E)1876 3276 y Fm(2)p Fo(n)p Fm(+1)2048 3262 y Fp(;)g(E)2171 3276 y Fm(2)p Fo(n)p Fm(+2)2344 3262 y Fr(],)f Fp(:)15 b(:)g(:)i Fr(,)30 b(of)h(the)f(real)g(axis)g(suc)m(h)g(that)842 3412 y Fp(E)909 3426 y Fm(1)974 3412 y Fp(<)25 b(E)1137 3426 y Fm(2)1202 3412 y Fn(\024)g Fp(E)1365 3426 y Fm(3)1430 3412 y Fp(<)g(E)1593 3426 y Fm(4)1647 3412 y Fp(:)15 b(:)g(:)h(E)1835 3426 y Fm(2)p Fo(n)1943 3412 y Fn(\024)25 b Fp(E)2106 3426 y Fm(2)p Fo(n)p Fm(+1)2304 3412 y Fp(<)g(E)2467 3426 y Fm(2)p Fo(n)p Fm(+2)2665 3412 y Fn(\024)g Fp(:)15 b(:)g(:)31 b(;)-3158 b Fr(\(1.2\))1424 3552 y Fp(E)1491 3566 y Fo(n)1563 3552 y Fn(!)26 b Fr(+)p Fn(1)p Fp(;)106 b(n)24 b Fn(!)i Fr(+)p Fn(1)p Fp(:)-2592 b Fr(\(1.3\))-236 3708 y(The)39 b(op)s(en)g(in)m(terv)-5 b(als)39 b(\()p Fp(E)665 3722 y Fm(2)705 3708 y Fp(;)31 b(E)828 3722 y Fm(3)867 3708 y Fr(\),)43 b(\()p Fp(E)1072 3722 y Fm(4)1112 3708 y Fp(;)30 b(E)1234 3722 y Fm(5)1274 3708 y Fr(\),)43 b Fp(:)15 b(:)g(:)h Fr(,)42 b(\()p Fp(E)1667 3722 y Fm(2)p Fo(n)1750 3708 y Fp(;)30 b(E)1872 3722 y Fm(2)p Fo(n)p Fm(+1)2045 3708 y Fr(\),)43 b Fp(:)15 b(:)g(:)h Fr(,)42 b(are)e(called)f(the)h Fi(sp)-5 b(e)g(ctr)g(al)44 b(gaps)p Fr(.)70 b(The)39 b(ends)h(of)-236 3816 y(the)f(bands)e(are)i(eigen)m(v) -5 b(alues)38 b(of)h(the)g(Sc)m(hr\177)-45 b(odinger)37 b(equation)h(\(0.2\))j(with)c(either)h(the)h(p)s(erio)s(dic)d(or)i(the) h(an)m(ti-p)s(erio)s(dic)-236 3924 y(b)s(oundary)f(conditions)h(at)h (the)h(ends)e(of)h(the)h(in)m(terv)-5 b(al)39 b(\(0)p Fp(;)15 b Fr(1\).)72 b(Some)40 b(gaps)h(can)f(b)s(e)g(closed)g(\(empt)m (y\).)71 b(In)39 b(this)g(case,)-236 4031 y(connected)31 b(comp)s(onen)m(ts)g(of)f(the)h(sp)s(ectrum)e(are)i(unions)d(of)j(sp)s (ectral)e(bands)g(with)g(common)i(ends.)-236 4203 y(1.3.)53 b Fu(The)44 b(sp)s(ectral)h(result.)g Fr(Let)39 b Fp(W)1176 4217 y Fm(+)1274 4203 y Fr(=)g(max)1401 4263 y Fo(x)p Fk(2)p Fh(R)1568 4203 y Fp(W)13 b Fr(\()p Fp(x)p Fr(\))39 b(and)f Fp(W)2099 4217 y Fk(\000)2197 4203 y Fr(=)g(min)2315 4263 y Fo(x)p Fk(2)p Fh(R)2473 4203 y Fp(W)13 b Fr(\()p Fp(x)p Fr(\).)65 b(F)-8 b(or)40 b(an)e(energy)h Fp(E)5 b Fr(,)41 b(consider)c(the)-236 4353 y(\\windo)m(w")74 b Fn(W)7 b Fr(\()p Fp(E)e Fr(\))38 b(=)f([)p Fp(E)31 b Fn(\000)24 b Fp(W)919 4367 y Fm(+)978 4353 y Fp(;)15 b(E)31 b Fn(\000)25 b Fp(W)1298 4367 y Fk(\000)1357 4353 y Fr(].)62 b(Let)38 b([)p Fp(E)1731 4367 y Fm(2)p Fo(n)1809 4376 y Fg(0)1844 4367 y Fk(\000)p Fm(1)1938 4353 y Fp(;)15 b(E)2045 4367 y Fm(2)p Fo(n)2123 4376 y Fg(0)2162 4353 y Fr(])38 b(b)s(e)f(one)h(of)f(the)h(sp)s(ectral)f(bands)f(of)i(the)f (p)s(erio)s(dic)-236 4461 y(op)s(erator)c Fp(H)209 4475 y Fm(0)248 4461 y Fr(.)47 b(W)-8 b(e)33 b(analyze)g(the)g(sp)s(ectrum)e (of)i(the)f(family)f(of)i(equations)f(\(0.1\))i(in)d(the)i(middle)d(of) j(this)e(band,)h(i.e.)47 b(w)m(e)-236 4569 y(pic)m(k)30 b Fp(J)k Fn(\032)25 b Fl(R)s Fr(,)36 b(a)31 b(compact)h(in)m(terv)-5 b(al,)29 b(suc)m(h)i(that)-124 4698 y Fu(\(A\):)41 b Fn(W)7 b Fr(\()p Fp(E)e Fr(\))26 b Fn(\032)p Fr(])p Fp(E)535 4712 y Fm(2)p Fo(n)613 4721 y Fg(0)648 4712 y Fk(\000)p Fm(1)743 4698 y Fp(;)15 b(E)850 4712 y Fm(2)p Fo(n)928 4721 y Fg(0)967 4698 y Fr([)30 b(for)h(all)e Fp(E)h Fn(2)25 b Fp(J)9 b Fr(.)-236 4827 y(Suc)m(h)32 b(an)g(in)m(terv)-5 b(al)32 b Fp(J)42 b Fr(exists)32 b(if)g Fp(W)962 4841 y Fm(+)1042 4827 y Fn(\000)22 b Fp(W)1221 4841 y Fk(\000)1280 4827 y Fr(,)33 b(the)g("size")g(of)g(the)g(adiabatic)f(p)s (erturbation,)g(is)g(smaller)f(than)h(the)h(size)g(of)-236 4935 y(the)d(sp)s(ectral)g(band)f([)p Fp(E)581 4949 y Fm(2)p Fo(n)659 4958 y Fg(0)694 4949 y Fk(\000)p Fm(1)789 4935 y Fp(;)15 b(E)896 4949 y Fm(2)p Fo(n)974 4958 y Fg(0)1013 4935 y Fr(].)-236 5043 y(Our)29 b(main)g(result)g(is)-236 5208 y Fu(Theorem)34 b(1.1.)46 b Fi(L)-5 b(et)37 b Fp(J)47 b Fi(b)-5 b(e)37 b(a)g(nonempty)i(close)-5 b(d)39 b(interval)e (satisfying)h(the)g(hyp)-5 b(othesis)40 b(\(A\).)d(Fix)g Fr(0)d Fp(<)f(\033)k(<)c Fr(1)p Fi(.)56 b(Then,)-236 5316 y(ther)-5 b(e)33 b(exists)g Fp(S)d(>)25 b Fr(0)33 b Fi(and)h Fp(D)28 b Fn(\032)d Fr(\(0)p Fp(;)15 b Fr(1\))p Fi(,)34 b(a)f(set)g(of)g(Diophantine)h(numb)-5 b(ers)33 b(such)g(that)-111 5445 y Fn(\017)42 b Fi(one)33 b(has)828 5566 y Fr(mes)15 b(\()p Fp(D)23 b Fn(\\)d Fr(\(0)p Fp(;)15 b(")p Fr(\)\))p 828 5606 616 4 v 1114 5690 a Fp(")1479 5627 y Fr(=)24 b(1)d(+)f Fp(o)15 b Fr(\()q Fp("\025)1921 5590 y Fo(\033)1968 5627 y Fr(\))48 b Fi(wher)-5 b(e)34 b Fp(\025)25 b Fr(=)g(exp)2636 5499 y Ff(\022)2703 5627 y Fn(\000)2784 5566 y Fp(S)p 2784 5606 61 4 v 2793 5690 a(")2854 5499 y Ff(\023)2937 5627 y Fp(:)-3198 b Fr(\(1.4\))-111 5827 y Fn(\017)42 b Fi(for)33 b(any)g Fp(")26 b Fn(2)f Fp(D)35 b Fi(su\016ciently)e(smal)5 b(l,)33 b(ther)-5 b(e)34 b(exists)f(a)g(Bor)-5 b(el)33 b(set)g Fp(B)d Fn(\032)25 b Fp(J)41 b Fi(of)33 b(smal)5 b(l)34 b(me)-5 b(asur)g(e)1497 5964 y Fr(mes)15 b(\()p Fp(B)5 b Fr(\))p 1497 6005 312 4 v 1504 6088 a(mes)15 b(\()p Fp(J)9 b Fr(\))1843 6025 y(=)25 b Fp(O)s Fr(\()p Fp(\025)2099 5988 y Fo(\033)r(=)p Fm(2)2217 6025 y Fr(\))p Fp(;)1872 6210 y Fm(2)p eop %%Page: 3 3 3 2 bop -24 241 a Fi(such)32 b(that)i Fp(J)c Fn(n)20 b Fp(B)37 b Fi(b)-5 b(elongs)34 b(to)f(the)g(absolutely)h(c)-5 b(ontinuous)33 b(sp)-5 b(e)g(ctrum)35 b(of)d(the)h(e)-5 b(quation)34 b(family)41 b Fr(\(0.1\))r Fi(;)-111 349 y Fn(\017)h Fi(for)j(al)5 b(l)46 b Fp(E)53 b Fn(2)48 b Fp(J)38 b Fn(n)30 b Fp(B)5 b Fi(,)47 b(ther)-5 b(e)46 b(exist)g(two)g(line)-5 b(arly)46 b(indep)-5 b(endent)47 b(Blo)-5 b(ch-Flo)g(quet)47 b(solutions)g Fp( )3351 363 y Fk(\006)3410 349 y Fr(\()p Fp(x;)15 b(E)5 b Fr(\))46 b Fi(of)65 b Fr(\(0.1\))-24 457 y Fi(satisfying)1203 616 y Fp( )1262 630 y Fk(\006)1321 616 y Fr(\()p Fp(x)p Fr(\))26 b(=)f Fp(e)1607 578 y Fk(\006)p Fo(ip)p Fm(\()p Fo(E)t Fm(\))p Fo(x)1891 616 y Fp(P)1949 630 y Fk(\006)2009 616 y Fr(\()p Fp(x)20 b Fn(\000)g Fp(z)t(;)15 b("x;)g(E)5 b Fr(\))p Fp(;)-2795 b Fr(\(1.5\))-24 775 y Fi(wher)-5 b(e)27 b Fp(p)p Fr(\()p Fp(E)5 b Fr(\))28 b Fi(is)e(a)h(monotonously)j (incr)-5 b(e)g(asing,)29 b(Lipschitz)e(c)-5 b(ontinuous)28 b(function)f(of)g Fp(E)5 b Fi(,)28 b(the)f(functions)g Fp(P)3634 789 y Fk(\006)3693 775 y Fr(\()p Fp(x;)15 b(\020)7 b(;)15 b(E)5 b Fr(\))-24 886 y Fi(di\013er)35 b(by)f(the)i(c)-5 b(omplex)36 b(c)-5 b(onjugation,)37 b Fp(P)1418 900 y Fk(\000)1506 886 y Fr(=)p 1606 813 118 4 v 29 w Fp(P)1664 900 y Fm(+)1724 886 y Fi(,)e(the)g(function)g Fp(P)2358 900 y Fm(+)2452 886 y Fi(is)g Fr(1)p Fi(-p)-5 b(erio)g(dic)37 b(in)d Fp(x)h Fi(and)h Fr(2)p Fp(\031)s Fi(-p)-5 b(erio)g(dic)37 b(in)d Fp(\020)7 b Fi(.)-24 994 y(This)37 b(function)f(b)-5 b(elongs)37 b(to)h Fp(H)1062 961 y Fm(2)1055 1022 y Fe(lo)l(c)1178 994 y Fi(in)f Fp(x)f Fi(and)i(is)e(analytic)i(in)e Fp(\020)43 b Fi(in)37 b(a)g(neighb)-5 b(orho)g(o)g(d)40 b(of)c(the)h(r)-5 b(e)g(al)38 b(line.)54 b(Mor)-5 b(e)g(over,)-24 1102 y Fp(P)34 1116 y Fm(+)126 1102 y Fi(is)32 b(a)h(Lipschitz)h(c)-5 b(ontinuous)33 b(function)g(of)g Fp(E)5 b Fi(.)-236 1268 y Fr(1.4.)53 b Fu(Commen)m(ts.)44 b Fr(The)e(co)s(e\016cien)m(t)h Fp(\025)f Fr(is)g(exp)s(onen)m(tially)e(small)h(in)g Fp(")h Fr(as)h Fp(")j Fn(!)f Fr(0.)76 b(Under)42 b(our)g(assumptions,)h (the)-236 1376 y(adiabatic)33 b(p)s(erturbation)e Fp(W)46 b Fr(b)s(eing)32 b(quite)h(general,)h(w)m(e)g(cannot)g(giv)m(e)g(its)f (optimal)f(v)-5 b(alue.)49 b(F)-8 b(or)35 b(more)e(details,)g(see)h (the)-236 1484 y(remark)c(follo)m(wing)f(Theorem)h(1.3.)-236 1639 y(The)e(Blo)s(c)m(h-Flo)s(quet)g(solutions)f Fp( )977 1653 y Fk(\006)1064 1639 y Fr(describ)s(ed)f(in)i(Theorem)g(1.1)h(ha)m (v)m(e)h(the)e(same)h(functional)d(structure)i(as)h(the)f(Blo)s(c)m(h-) -236 1747 y(Flo)s(quet)i(solutions)g(constructed)h(in)e([6)q(,)i(8])h (for)e(small)f(almost)i(p)s(erio)s(dic)e(p)s(oten)m(tials)h(or)h(high)e (energies.)42 b(The)30 b(regularit)m(y)-236 1861 y(of)e(the)g (solutions)f Fp(e)439 1828 y Fk(\006)p Fo(ip)p Fm(\()p Fo(E)t Fm(\))p Fo(x)723 1861 y Fp(P)781 1875 y Fk(\006)840 1861 y Fr(\()p Fp(x)16 b Fn(\000)f Fp(z)t(;)g("x;)g(E)5 b Fr(\))30 b(in)d(the)h(\\slo)m(w")h(v)-5 b(ariable)27 b Fp("x)h Fr(is)f(determined)g(b)m(y)h(the)g(function)f Fp(W)13 b Fr(,)28 b(and,)g(in)-236 1969 y(the)i(\\fast")i(v)-5 b(ariable)29 b Fp(x)20 b Fn(\000)g Fp(z)35 b Fr(b)m(y)30 b(the)h(function)e Fp(V)20 b Fr(.)-236 2124 y(In)j(a)h(previous)f(pap)s (er)f([9)q(],)k(w)m(e)e(ha)m(v)m(e)h(studied)d(equation)i(\(0.1\))i(in) d(the)h(case)h(when)d(the)j(p)s(oten)m(tial)e Fp(W)36 b Fr(is)23 b(the)h(cosine.)39 b(In)23 b(this)-236 2232 y(case)k(to)s(o,)g(w)m(e)f(ha)m(v)m(e)h(found)e(that)h(some)g(parts)g (of)g(the)f(sp)s(ectrum)g(are)h(absolutely)f(con)m(tin)m(uous;)i (though)e(it)h(w)m(as)g(not)g(stated)-236 2340 y(there,)g(the)g (results)e(dev)m(elop)s(ed)g(in)g(section)h(7)h(of)f(the)h(presen)m(t)f (pap)s(er)f(sho)m(w)h(that,)i(in)d([9],)j(the)e(generalized)g (eigenfunctions)-236 2448 y(asso)s(ciated)31 b(to)g(the)f(absolutely)g (con)m(tin)m(uous)g(sp)s(ectrum)f(ha)m(v)m(e)i(the)g(structure)f (describ)s(ed)e(in)h(Theorem)h(1.1.)-236 2603 y(Theorem)21 b(1.1)h(follo)m(ws)e(from)g(the)i(analysis)d(of)i(a)h(\014nite)e (di\013erence)g(equation,)j(the)e(mono)s(drom)m(y)g(equation.)37 b(The)21 b(reduction)-236 2711 y(to)35 b(the)g(mono)s(drom)m(y)f (equation)g(is)g(indep)s(enden)m(t)e(of)j Fp(")p Fr(.)54 b(In)33 b(the)i(adiabatic)f(limit,)g(the)h(mono)s(drom)m(y)f(equation)g (tak)m(es)i(a)-236 2819 y(simple)29 b(mo)s(del)h(form.)44 b(The)31 b(conditions)f(\\)p Fp(")e Fn(2)e Fp(D)s Fr(")32 b(and)f(\\)p Fp(")h Fr(small")e(only)h(app)s(ears)f(in)g(the)i (analysis)e(of)i(the)f(mono)s(drom)m(y)-236 2927 y(equation.)-236 3105 y(1.5.)53 b Fu(The)33 b(mono)s(drom)m(y)g(matrix.)45 b Fr(First,)28 b(follo)m(wing)g([9)q(],)h(w)m(e)h(recall)e(the)h (de\014nition)e(of)i Fi(the)j(mono)-5 b(dr)g(omy)35 b(matrix)30 b Fr(and)-236 3213 y(in)m(tro)s(duce)35 b Fi(the)k(mono)-5 b(dr)g(omy)42 b(e)-5 b(quation)p Fr(.)59 b(Then,)38 b(w)m(e)f(describ)s (e)d(the)j(asymptotics)g(of)f(a)h(mono)s(drom)m(y)f(matrix)g(for)g (\(0.1\))-236 3321 y(in)h(the)h(adiabatic)g(case,)k(and,)e(\014nally)-8 b(,)39 b(w)m(e)g(use)f(this)f(information)g(to)i(explain)e(the)i (heuristics)d(guiding)g(the)j(pro)s(of)f(of)-236 3429 y(Theorem)30 b(1.1.)-236 3607 y(1.5.1.)48 b Fi(De\014nition)d(of)h(the) f(mono)-5 b(dr)g(omy)49 b(matrix.)e Fr(Consider)42 b(a)i Fi(c)-5 b(onsistent)46 b(b)-5 b(asis)45 b Fr(\()p Fp( )2824 3621 y Fm(1)p Fo(;)p Fm(2)2919 3607 y Fr(\),)i(i.e.)81 b(a)45 b(basis)d(of)i(solutions)-236 3715 y(of)37 b(\(0.1\))c(whose)d (W)-8 b(ronskian)30 b(is)f(indep)s(enden)m(t)f(of)j Fp(z)j Fr(and)c(that)h(are)g(1-p)s(erio)s(dic)d(in)h Fp(z)1215 3868 y( )1274 3882 y Fm(1)p Fo(;)p Fm(2)1369 3868 y Fr(\()p Fp(x;)15 b(z)25 b Fr(+)20 b(1\))26 b(=)f Fp( )1915 3882 y Fm(1)p Fo(;)p Fm(2)2009 3868 y Fr(\()p Fp(x;)15 b(z)t Fr(\))p Fp(;)108 b Fn(8)p Fp(x;)15 b(z)t(:)-2800 b Fr(\(1.6\))-236 4027 y(The)30 b(functions)f Fp( )402 4041 y Fm(1)p Fo(;)p Fm(2)496 4027 y Fr(\()p Fp(x)21 b Fr(+)f(2)p Fp(\031)s(=";)15 b(z)26 b Fr(+)20 b(2)p Fp(\031)s(=")p Fr(\))32 b(b)s(eing)d(solutions)f (of)j(equation)f(\(0.1\))r(,)h(one)f(can)h(write)675 4240 y(\011\()p Fp(x)21 b Fr(+)f(2)p Fp(\031)s(=";)15 b(z)26 b Fr(+)20 b(2)p Fp(\031)s(=")p Fr(\))27 b(=)e Fp(M)10 b Fr(\()p Fp(E)5 b(;)15 b(z)t Fr(\)\011\()p Fp(x;)g(z)t Fr(\))p Fp(;)109 b Fr(\011)25 b(=)2607 4111 y Ff(\022)2674 4185 y Fp( )2733 4199 y Fm(1)2772 4185 y Fr(\()p Fp(x;)15 b(z)t Fr(\))2674 4292 y Fp( )2733 4306 y Fm(2)2772 4292 y Fr(\()p Fp(x;)g(z)t Fr(\))2982 4111 y Ff(\023)3064 4240 y Fp(;)-3325 b Fr(\(1.7\))-236 4448 y(where)33 b Fp(M)10 b Fr(\()p Fp(E)5 b(;)15 b(z)t Fr(\))36 b(is)d(a)i(2)23 b Fn(\002)f Fr(2)34 b(matrix)g(with)e(co)s(e\016cien)m(ts)j(indep)s (enden)m(t)d(of)i Fp(x)p Fr(.)52 b(This)32 b(matrix)h(is)g(called)h Fi(the)i(mono)-5 b(dr)g(omy)-236 4556 y(matrix)31 b Fr(corresp)s (onding)e(to)i(the)f(consisten)m(t)h(basis)e(\()p Fp( )1645 4570 y Fm(1)1685 4556 y Fp(;)15 b( )1784 4570 y Fm(2)1825 4556 y Fr(\).)-236 4711 y(Let)31 b(us)e(men)m(tion)h(t)m(w)m(o)i (elemen)m(tary)f(prop)s(erties)e(of)h(the)h(mono)s(drom)m(y)f(matrix) 913 4865 y(det)15 b Fp(M)10 b Fr(\()p Fp(E)5 b(;)15 b(z)t Fr(\))27 b Fn(\021)e Fr(1)p Fp(;)107 b(M)10 b Fr(\()p Fp(E)5 b(;)15 b(z)26 b Fr(+)20 b(1\))26 b(=)f Fp(M)10 b Fr(\()p Fp(E)5 b(;)15 b(z)t Fr(\))p Fp(;)108 b Fn(8)p Fp(z)t(:)-3103 b Fr(\(1.8\))-236 5048 y(1.5.2.)48 b Fi(The)34 b(mono)-5 b(dr)g(omy)37 b(e)-5 b(quation.)47 b Fr(Set)31 b Fp(h)c Fn(\021)g Fr(2)p Fp(\031)s(=")32 b Fr(mo)s(d)f(1.)44 b(Let)32 b Fp(M)41 b Fr(b)s(e)31 b(a)h(mono)s(drom)m(y)e(matrix)h (corresp)s(onding)e(to)j(a)-236 5156 y(consisten)m(t)f(basis)e(\()p Fp( )504 5170 y Fm(1)p Fo(;)p Fm(2)599 5156 y Fr(\).)41 b(Consider)28 b(the)j(equation)1065 5310 y Fp(F)13 b Fr(\()p Fp(n)20 b Fr(+)g(1\))26 b(=)f Fp(M)10 b Fr(\()p Fp(E)5 b(;)15 b(z)26 b Fr(+)20 b Fp(nh)p Fr(\))p Fp(F)13 b Fr(\()p Fp(n)p Fr(\))p Fp(;)107 b Fn(8)p Fp(n)24 b Fn(2)h Fl(Z)p Fp(:)-2955 b Fr(\(1.9\))-236 5476 y(where)30 b Fp(F)43 b Fr(is)29 b(a)i(function)e Fp(F)39 b Fr(:)25 b Fl(Z)c Fn(!)k Fl(C)1060 5443 y Fm(2)1105 5476 y Fr(.)41 b(Equation)30 b(\(1.9\))i(is)d(the)i Fi(mono)-5 b(dr)g(omy)36 b(e)-5 b(quation)p Fr(.)-236 5632 y(The)30 b(main)g(feature)h(of)g(the) g(mono)s(drom)m(y)f(equation)h(is)f(that)h(the)g(b)s(eha)m(vior)f(of)h (its)g(solutions)e(for)i Fp(n)25 b Fn(!)h(\0061)k Fr(mimics)f(the)-236 5739 y(b)s(eha)m(vior)g(of)i(solutions)e(of)h(the)h(input)d(equation)i (\(0.1\))i(for)f Fp(x)25 b Fn(!)g(\0071)p Fr(,)30 b(see)h(Theorem)f (7.1.)-236 5847 y(Moreo)m(v)m(er,)38 b(the)c(Blo)s(c)m(h-Flo)s(quet)g (solutions)f(to)i(the)g(mono)s(drom)m(y)f(equation)g(\(1.9\))i(are)f (related)f(to)h(the)g(Blo)s(c)m(h-Flo)s(quet)-236 5955 y(solutions)29 b(of)37 b(\(0.1\))r(,)31 b(see)g(Theorem)f(7.2.)41 b(Here,)32 b(w)m(e)e(state)i(only)e(one)g(of)h(the)f(results)g(giv)m (en)g(in)f(section)i(7.)-236 6110 y(Let)f(\002\()p Fp(E)5 b(;)15 b(z)t Fr(\))31 b(and)e Fp(\022)s Fr(\()p Fp(E)5 b(;)15 b(z)t Fr(\))30 b(b)s(e)f(the)h(Ly)m(apuno)m(v)f(exp)s(onen)m(ts) h(for)f(the)h(equations)f(\(0.1\))i(and)e(\(1.9\))j(resp)s(ectiv)m(ely) -8 b(.)40 b(One)29 b(has)1872 6210 y Fm(3)p eop %%Page: 4 4 4 3 bop -236 241 a Fu(Theorem)34 b(1.2)h Fr(\([9)q(]\))p Fu(.)46 b Fi(Assume)33 b(that)h Fp( )1202 255 y Fm(1)1241 241 y Fi(,)f Fp( )1361 255 y Fm(2)1400 241 y Fi(,)g Fp(d )1567 255 y Fm(1)1607 241 y Fp(=dx)g Fi(and)g Fp(d )2066 255 y Fm(2)2106 241 y Fp(=dx)h Fi(ar)-5 b(e)33 b(lo)-5 b(c)g(al)5 b(ly)35 b(b)-5 b(ounde)g(d)34 b(in)e Fp(x)h Fi(and)g Fp(z)t Fi(.)-236 349 y(Then,)g(the)g(Lyapunov)g(exp)-5 b(onents)35 b Fr(\002\()p Fp(E)5 b(;)15 b(z)t Fr(\))34 b Fi(and)f Fp(\022)s Fr(\()p Fp(E)5 b(;)15 b(z)t Fr(\))34 b Fi(satisfy)f(the)g(r)-5 b(elation)1461 520 y Fr(\002\()p Fp(E)5 b(;)15 b(z)t Fr(\))27 b(=)1922 459 y Fp(")p 1893 499 101 4 v 1893 582 a Fr(2)p Fp(\031)2018 520 y(\022)s Fr(\()p Fp(E)5 b(;)15 b(z)t Fr(\))p Fp(:)-2553 b Fr(\(1.10\))-236 709 y Fu(Remark)34 b(1.1.)46 b Fr(The)32 b(ab)s(o)m(v)m(e)h (de\014nition)d(of)i(the)h(mono)s(drom)m(y)e(matrix)h(applies)e(to)j (an)m(y)f(quasi-p)s(erio)s(dic)d(equation)j(with)-236 817 y(t)m(w)m(o)26 b(frequencies.)38 b(It)25 b(generalizes)f(the)h (de\014nitions)e(of)i(the)f(mono)s(drom)m(y)h(matrix)f(for)g(the)h (one-dimensional)e(p)s(erio)s(dic)f(dif-)-236 925 y(feren)m(tial)h (equations.)38 b(The)24 b(passage)g(to)h(the)f(mono)s(drom)m(y)f (equation)h(is)f(close)h(to)g(the)g(mono)s(dromization)f(idea)g(dev)m (elop)s(ed)-236 1033 y(in)29 b([5)q(])h(for)g(di\013erence)g(equations) g(with)f(p)s(erio)s(dic)f(co)s(e\016cien)m(ts.)41 b(F)-8 b(or)32 b(a)e(detailed)g(discussion,)e(w)m(e)j(refer)f(to)h([9].)-236 1196 y(1.6.)53 b Fu(Asymptotics)38 b(of)h(a)e(mono)s(drom)m(y)h (matrix.)44 b Fr(T)-8 b(o)34 b(describ)s(e)d(the)i(asymptotics)h(of)f (a)g(mono)s(drom)m(y)g(matrix,)g(w)m(e)-236 1304 y(return)f(to)i(the)g (p)s(erio)s(dic)c(Sc)m(hr\177)-45 b(odinger)32 b(op)s(erator)i(\(0.2\)) r(.)49 b(Consider)32 b(the)h(Blo)s(c)m(h)g(quasi-momen)m(tum)g(asso)s (ciated)h(to)g(this)-236 1412 y(op)s(erator,)d(see)g(section)g(2.)42 b(It)31 b(is)e(an)i(analytic)f(m)m(ulti-v)-5 b(alued)28 b(function)i(of)h(the)f(sp)s(ectral)g(parameter.)42 b(Its)31 b(branc)m(h)f(p)s(oin)m(ts)-236 1520 y(coincide)h(with)g(the)h(ends)f (of)h(the)g(connected)h(comp)s(onen)m(ts)f(of)h(the)f(sp)s(ectrum)f(of) h(the)g(p)s(erio)s(dic)d(op)s(erator)k(\(0.2\))r(.)45 b(There)-236 1628 y(exists)36 b(a)h(branc)m(h)g(of)g(the)g(Blo)s(c)m(h) f(quasi-momen)m(tum)g(that)i(conformally)d(maps)i(the)g(upp)s(er)d (half)i(complex)h(plane)f(on)m(to)-236 1735 y(the)j(\014rst)g(quadran)m (t)g(cut)g(along)h(\014nite)e(v)m(ertical)h(segmen)m(ts)h(b)s(eginning) d(at)j(the)g(p)s(oin)m(ts)e Fp(k)43 b Fr(=)d Fp(\031)s(n)p Fr(,)h Fp(n)f Fr(=)g(1)p Fp(;)15 b Fr(2)p Fp(;)g Fr(3)g Fp(:)g(:)g(:)j Fr(.)67 b(In)-236 1843 y(particular,)43 b(it)e(maps)g(\()p Fp(E)676 1857 y Fm(2)p Fo(n)p Fk(\000)p Fm(1)849 1843 y Fp(;)15 b(E)956 1857 y Fm(2)p Fo(n)1039 1843 y Fr(\),)45 b(the)d Fp(n)p Fr(-th)f(sp)s(ectral)g(band)g(of)g(the) h(p)s(erio)s(dic)d(Sc)m(hr\177)-45 b(odinger)41 b(op)s(erator,)k(on)m (to)d(the)-236 1951 y(in)m(terv)-5 b(al)29 b(\()p Fp(\031)s Fr(\()p Fp(n)21 b Fn(\000)f Fr(1\))p Fp(;)15 b(\031)s(n)p Fr(\).)42 b(W)-8 b(e)31 b(denote)g(this)e(branc)m(h)h(b)m(y)g Fp(k)1816 1965 y Fo(p)1856 1951 y Fr(.)-236 2107 y(No)m(w,)h(w)m(e)g (de\014ne)f Fi(the)j(phase)g(inte)-5 b(gr)g(al)32 b Fr(\010)e(b)m(y) 1438 2313 y(\010\()p Fp(E)5 b Fr(\))26 b(=)1768 2189 y Ff(Z)1859 2215 y Fm(2)p Fo(\031)1818 2395 y Fm(0)1956 2313 y Fp(\024)2008 2327 y Fo(n)2051 2336 y Fg(0)2090 2313 y Fr(\()p Fp(\020)7 b Fr(\))p Fp(d\020)g(;)-2562 b Fr(\(1.11\))-236 2506 y(where)30 b(the)g(function)f Fp(\024)591 2520 y Fo(n)634 2529 y Fg(0)704 2506 y Fr(is)g(giv)m(en)h (b)m(y)757 2709 y Fp(\024)809 2723 y Fo(n)852 2732 y Fg(0)891 2709 y Fr(\()p Fp(')p Fr(\))c(=)1142 2581 y Ff(\032)1252 2654 y Fp(k)1299 2668 y Fo(p)1339 2654 y Fr(\()p Fp(E)g Fn(\000)20 b Fp(W)13 b Fr(\()p Fp(')p Fr(\)\))21 b Fn(\000)f Fp(\031)s Fr(\()p Fp(n)2078 2668 y Fm(0)2138 2654 y Fn(\000)f Fr(1\))p Fp(;)115 b Fr(if)36 b Fp(n)2593 2668 y Fm(0)2663 2654 y Fr(is)29 b(o)s(dd)o(;)1252 2762 y Fp(\031)s(n)1362 2776 y Fm(0)1421 2762 y Fn(\000)20 b Fp(k)1559 2776 y Fo(p)1599 2762 y Fr(\()p Fp(E)26 b Fn(\000)20 b Fp(W)13 b Fr(\()p Fp(')p Fr(\)\))p Fp(;)342 b Fr(if)36 b Fp(n)2593 2776 y Fm(0)2663 2762 y Fr(is)29 b(ev)m(en)q Fp(:)-236 2912 y Fr(Under)38 b(the)h(condition)e(\(1.3\))r (,)42 b(the)d(function)e(\010)i(is)f(analytic)g(in)g(an)h(neigh)m(b)s (orho)s(o)s(d)d(of)j Fp(J)48 b Fr(and)39 b(it)f(is)g(real)h(for)f Fp(E)45 b Fn(2)39 b Fp(J)9 b Fr(.)-236 3019 y(Moreo)m(v)m(er,)33 b(in)c(section)h(8.3.2,)j(w)m(e)d(c)m(hec)m(k)i(that)f(the)g(deriv)-5 b(ativ)m(e)30 b(of)g(\010)g(do)s(es)g(not)h(v)-5 b(anish)29 b(on)h Fp(J)9 b Fr(.)-236 3127 y(As)30 b(the)h(p)s(oten)m(tial)f(in)f (\(0.1\))j(is)d(real,)h(it)g(is)g(p)s(ossible)e(to)j(construct)g(a)f (mono)s(drom)m(y)g(matrix)g(of)g(the)h(form)1230 3340 y Fp(M)10 b Fr(\()p Fp(z)t(;)15 b(E)5 b Fr(\))27 b(=)1679 3212 y Ff(\022)1787 3274 y Fp(a)p Fr(\()p Fp(z)t(;)15 b(E)5 b Fr(\))90 b Fp(b)p Fr(\()p Fp(z)t(;)15 b(E)5 b Fr(\))p 1792 3313 269 4 v 1792 3405 a Fp(b)p Fr(\()p 1866 3355 47 4 v Fp(z)g(;)p 1953 3332 73 4 v 15 w(E)g Fr(\))p 2148 3313 278 4 v 88 w Fp(a)p Fr(\()p 2231 3355 47 4 v Fp(z)g(;)p 2318 3332 73 4 v 15 w(E)h Fr(\))2468 3212 y Ff(\023)-236 3340 y Fr(\(1.12\))-236 3547 y(W)-8 b(e)31 b(pro)m(v)m(e)-236 3710 y Fu(Theorem)j(1.3.)46 b Fi(Assume)d(the)i(interval)f Fp(J)53 b Fi(satisfy)45 b(assumption)h(\(A\))d(and)i(\014x)f Fp(E)2738 3724 y Fm(0)2823 3710 y Fn(2)i Fp(J)9 b Fi(.)75 b(Fix)44 b Fp(y)49 b(>)c Fr(0)p Fi(.)76 b(Then,)47 b(for)-236 3818 y(su\016ciently)28 b(smal)5 b(l)30 b Fp(")p Fi(,)f(ther)-5 b(e)30 b(exist)e Fp(V)1038 3832 y Fm(0)1078 3818 y Fi(,)h(a)f(neighb)-5 b(orho)g(o)g(d)32 b(of)d Fp(E)1915 3832 y Fm(0)1983 3818 y Fi(indep)-5 b(endent)30 b(of)e Fp(")p Fi(,)i(and)f(a)g(c)-5 b(onsistent)29 b(b)-5 b(asis)30 b(of)e(solutions)-236 3926 y(of)33 b(the)f(family)i(of)f(e)-5 b(quations)41 b Fr(\(0.1\))34 b Fi(such)f(that)g(the)g(c)-5 b(orr)g(esp)g(onding)36 b(mono)-5 b(dr)g(omy)36 b(matrix)e(has)g(the)f(fol)5 b(lowing)33 b(pr)-5 b(op)g(erties)-111 4054 y Fn(\017)42 b Fi(it)32 b(is)h(analytic)h(in)e Fr(\()p Fp(E)5 b(;)15 b(z)t Fr(\))27 b Fn(2)e Fp(V)1013 4068 y Fm(0)1073 4054 y Fn(\002)20 b(fj)p Fr(Im)15 b Fp(z)t Fn(j)26 b(\024)f Fp(y)s Fn(g)p Fi(;)-111 4162 y Fn(\017)42 b Fi(it)32 b(is)h(of)g(the)g(form)40 b Fr(\(1.12\))s Fi(;)-111 4270 y Fn(\017)i Fi(if)32 b(we)h(r)-5 b(epr)g(esent)34 b(the)f(c)-5 b(o)g(e\016cient)33 b Fp(a)g Fi(of)g(the)g(mono)-5 b(dr)g(omy)36 b(matrix)e(as)1210 4476 y Fp(a)26 b Fr(=)f Fp(a)1428 4490 y Fm(0)1487 4476 y Fr(+)20 b Fp(a)1626 4490 y Fm(1)1666 4476 y Fr(\()p Fp(z)t Fr(\))p Fp(;)109 b(a)1964 4490 y Fm(0)2029 4476 y Fr(=)2125 4352 y Ff(Z)2216 4378 y Fm(1)2175 4558 y(0)2270 4476 y Fp(a)p Fr(\()p Fp(z)t Fr(\))p Fp(dz)t Fr(;)-24 4669 y Fi(then,)33 b(for)g Fp(")25 b Fn(!)h Fr(0)p Fi(,)32 b(one)h(has)300 4840 y Fp(a)348 4854 y Fm(0)413 4840 y Fr(=)25 b(exp\()p Fn(\000)p Fp(i)p Fr(\010\()p Fp(E)5 b Fr(\))p Fp(=")22 b Fr(+)e(\012\()p Fp(E)5 b Fr(\)\)\(1)22 b(+)e Fp(o)p Fr(\(1\)\))p Fp(;)142 b Fi(and)127 b Fp(a)2308 4854 y Fm(1)2373 4840 y Fr(=)25 b Fp(o)2528 4739 y Ff(\020)2582 4840 y Fp(a)2630 4854 y Fm(0)2685 4840 y Fp(e)2727 4802 y Fk(\000)2792 4775 y Fd(S)p 2792 4787 41 3 v 2798 4828 a(")2846 4739 y Ff(\021)2916 4840 y Fp(;)108 b Fn(j)p Fr(Im)15 b Fp(z)t Fn(j)26 b Fp(<)f(y)s(;)-3700 b Fr(\(1.13\))-24 5015 y Fi(wher)-5 b(e)33 b Fp(S)38 b Fi(is)32 b(a)h(p)-5 b(ositive)34 b(c)-5 b(onstant)34 b(indep)-5 b(endent)35 b(of)d Fp(")p Fi(,)h(and)h Fp(E)c Fn(7!)c Fr(\012\()p Fp(E)5 b Fr(\))33 b Fi(is)f(a)h(r)-5 b(e)g(al)34 b(analytic)g(function)f(in)g Fp(V)3754 5029 y Fm(0)3793 5015 y Fi(,)-111 5123 y Fn(\017)42 b Fi(the)33 b(c)-5 b(o)g(e\016cient)33 b Fp(b)f Fi(of)h(the)g(mono)-5 b(dr)g(omy)37 b(matrix)d(admits)g(the)f(estimate)1356 5293 y Fp(b)25 b Fr(=)g Fp(o)1575 5192 y Ff(\020)1629 5293 y Fp(e)1671 5256 y Fk(\000)1736 5228 y Fd(S)p 1737 5240 V 1743 5282 a(")1791 5192 y Ff(\021)1861 5293 y Fp(;)108 b Fn(j)p Fr(Im)15 b Fp(z)t Fn(j)26 b Fp(<)f(y)s(;)-2645 b Fr(\(1.14\))-24 5469 y Fi(with)33 b(the)g(same)g(c)-5 b(onstant)35 b Fp(S)i Fi(as)d(in)39 b Fr(\(1.13\))r Fi(;)-111 5576 y Fn(\017)j Fi(the)33 b(estimates)g(for)g Fp(a)g Fi(and)h Fp(b)e Fi(ar)-5 b(e)34 b(uniform)f(in)f Fr(\()p Fp(z)t(;)15 b(E)5 b Fr(\))27 b Fn(2)e Fp(V)2007 5590 y Fm(0)2067 5576 y Fn(\002)20 b(fj)p Fr(Im)15 b Fp(z)t Fn(j)26 b(\024)f Fp(y)s Fn(g)p Fi(.)-24 5739 y Fr(Theorem)20 b(1.3)i(is)d(pro)m(v)m(ed)i(using)d(the)j(complex)f(WKB)h(metho)s(d)e (for)h(adiabatic)g(p)s(erturbations)e(of)j(p)s(erio)s(dic)c(Sc)m (hr\177)-45 b(odin-)-236 5847 y(ger)31 b(equations)f(dev)m(elop)s(ed)f (in)g([10)r(].)-236 6003 y(Under)d(our)h(assumptions,)g(the)h (adiabatic)f(p)s(erturbation)e Fp(W)40 b Fr(b)s(eing)26 b(quite)h(general,)h(w)m(e)g(cannot)g(sa)m(y)g(more)g(ab)s(out)f Fp(a)3851 6017 y Fm(1)3890 6003 y Fr(.)40 b(If)-236 6110 y(one)31 b(mak)m(es)h(more)f(restrictiv)m(e)g(assumptions,)f(it)g(is)g (p)s(ossible)f(to)i(get)h(an)f(asymptotics)g(for)g Fp(a)3010 6124 y Fm(1)3050 6110 y Fr(.)42 b(The)30 b(leading)g(term)h Fp(a)3884 6124 y Fm(1)3955 6110 y Fr(is)1872 6210 y Fm(4)p eop %%Page: 5 5 5 4 bop -236 241 a Fr(giv)m(en)30 b(b)m(y)i(^)-47 b Fp(a)175 255 y Fm(1)215 241 y Fr(\()p Fn(\000)p Fr(1\))p Fp(e)443 208 y Fk(\000)p Fm(2)p Fo(i\031)r(z)661 241 y Fr(+)21 b(^)-46 b Fp(a)800 255 y Fm(1)840 241 y Fr(\(1\))p Fp(e)997 208 y Fm(2)p Fo(i\031)r(z)1140 241 y Fr(,)31 b(the)f(sum)g(of)g(the)h (\014rst)e(t)m(w)m(o)j(terms)e(of)h(the)f(F)-8 b(ourier)30 b(series)g(of)g Fp(a)3387 255 y Fm(1)3427 241 y Fr(.)41 b(The)29 b(mo)s(dulus)-236 349 y(of)44 b(the)h(F)-8 b(ourier)43 b(co)s(e\016cien)m(ts)i(are)g(exp)s(onen)m(tially)d(small)h(in)g Fp(")h Fr(and)g(can)h(b)s(e)e(in)m(terpreted)h(as)g(tunneling)e(co)s (e\016cien)m(ts)-236 457 y(measuring)26 b(the)i(complex)f(tunneling)e (b)s(et)m(w)m(een)k(distinct)d(branc)m(hes)h(of)h(the)g(iso-energy)f (curv)m(e)h(corresp)s(onding)e(to)i(Fig.)g(1.)-236 565 y(In)i(particular,)g(these)h(asymptotics)h(giv)m(es)f(the)g(optimal)f (v)-5 b(alue)31 b(of)g(the)g(co)s(e\016cien)m(t)h Fp(\025)f Fr(used)g(in)e(Theorem)i(1.1.)44 b(This)29 b(will)-236 672 y(b)s(e)h(analyzed)g(in)f(a)i(subsequen)m(t)e(pap)s(er.)-236 851 y(1.6.1.)48 b Fi(The)37 b(mono)-5 b(dr)g(omy)41 b(matrix)d (asymptotics)h(and)f(the)f(sp)-5 b(e)g(ctr)g(al)40 b(pr)-5 b(op)g(erties)39 b(of)56 b Fr(\(0.1\))r Fi(.)45 b Fr(Let)36 b(us)e(explain)f(ho)m(w)i(Theo-)-236 959 y(rem)30 b(1.3)h(is)f(used)f (to)i(deriv)m(e)f(Theorem)g(1.1.)42 b(By)31 b(Theorem)f(1.3,)i(the)e (mono)s(drom)m(y)g(matrix)g(is)f(the)i(form)1117 1163 y Fp(M)k Fr(=)1336 1035 y Ff(\022)1403 1108 y Fp(U)126 b Fr(0)1416 1216 y(0)97 b Fp(U)1630 1183 y Fk(\003)1670 1035 y Ff(\023)1757 1163 y Fr(+)20 b Fp(O)s Fr(\()p Fp(\025)p Fr(\))p Fp(;)107 b(U)2247 1126 y Fk(\003)2311 1163 y Fr(=)p 2407 1078 216 4 v 25 w Fp(U)10 b Fr(\()2536 1140 y(\026)2514 1163 y Fp(E)c Fr(\))p Fp(;)-236 1367 y Fr(where)30 b Fp(U)40 b Fr(is)29 b(indep)s(enden)m(t)g(of)h Fp(z)t Fr(,)1304 1538 y Fp(U)35 b Fr(=)25 b Fp(e)1539 1500 y Fk(\000)1608 1473 y Fd(i)p 1604 1485 30 3 v 1604 1526 a(")1643 1500 y Fm(\010+\012+)p Fo(o)p Fm(\(1\))1983 1538 y Fp(;)107 b(\025)25 b Fr(=)g Fp(e)2331 1500 y Fk(\000)2396 1473 y Fd(S)p 2396 1485 41 3 v 2402 1526 a(")2451 1538 y Fp(:)-236 1697 y Fr(As)32 b(w)m(e)h(shall)e(see)i(later,)h(the)f (relation)e(det)16 b Fp(M)39 b Fn(\021)29 b Fr(1)k(implies)c(that)34 b Fn(j)p Fp(U)10 b Fn(j)29 b Fr(=)g(1)22 b(+)f Fp(o)p Fr(\()p Fp(\025)p Fr(\))34 b(for)e Fp(E)i Fn(2)29 b Fp(J)9 b Fr(.)47 b(So,)34 b(up)d(to)j(error)e(terms)-236 1805 y(of)g(order)f Fp(O)s Fr(\()p Fp(\025)p Fr(\),)i Fp(M)42 b Fr(is)31 b(a)i(constan)m(t)g(diagonal)e(matrix)g(with)g(diagonal)g (elemen)m(ts)i(of)f(absolute)f(v)-5 b(alue)32 b(1.)45 b(No)m(w,)34 b(consider)-236 1913 y(the)f(mono)s(drom)m(y)f(equation)g (with)f(the)i(mono)s(drom)m(y)f(matrix)g Fp(M)10 b Fr(.)47 b(If)33 b(the)f(error)h(terms)f(could)g(b)s(e)g(omitted,)h(one)g(w)m (ould)-236 2021 y(immediately)e(obtain)i(that,)i(for)e(all)f Fp(E)k Fn(2)29 b Fp(J)9 b Fr(,)35 b(there)e(are)h(b)s(ounded)d (solutions)h(of)h(the)g(mono)s(drom)m(y)g(equation.)49 b(T)-8 b(o)34 b(tak)m(e)-236 2129 y(care)d(of)f(the)h(exp)s(onen)m (tially)d(small)h(error)g(terms,)i(w)m(e)g(apply)d(standard)i(ideas)f (of)i(the)f(sp)s(ectral)g(KAM)g(theory:)41 b(w)m(e)30 b(use)g(a)-236 2237 y(simple)d(v)m(ersion)j(\(prepared)f(in)f([9)q(]\)) i(and)f(construct)h(b)s(ounded)e(solutions)g(of)i(the)g(mono)s(drom)m (y)f(equation)h(for)f Fp(E)35 b Fr(outside)-236 2345 y(of)e(a)g(Borel)g(set)g Fp(B)5 b Fr(;)34 b(it)e(is)g(a)h(coun)m(table) g(union)e(of)i(in)m(terv)-5 b(als)32 b(of)g(small)g(total)h(measure.)48 b(These)33 b(in)m(terv)-5 b(als)31 b(con)m(tain)j(KAM)-236 2453 y(resonances)d(that)g(can)f(b)s(e)g(roughly)f(c)m(haracterized)j (b)m(y)e(the)g("quan)m(tization)h(condition")1324 2583 y(1)p 1298 2624 98 4 v 1298 2707 a Fp(\031)s(")1405 2645 y Fr(\010\()p Fp(E)5 b Fr(\))26 b(=)f Fp(k)f Fn(\001)c Fp(h)h Fr(+)f Fp(l)r(;)106 b(k)s(;)15 b(l)28 b Fn(2)d Fl(Z)p Fp(:)-236 2825 y Fr(Ha)m(ving)g(constructed)g(b)s(ounded)e (solutions)h(of)h(the)g(mono)s(drom)m(y)g(equation)g(outside)f(of)h (the)h(set)f Fp(B)5 b Fr(,)26 b(b)m(y)f(Corollary)f(1.2,)j(w)m(e)-236 2933 y(conclude)21 b(that)h(the)f(Ly)m(apuno)m(v)h(exp)s(onen)m(t)f(of) h(the)f(equation)h(family)f(\(0.1\))i(is)d(zero)j(on)e Fp(J)11 b Fn(n)r Fp(B)5 b Fr(.)38 b(By)22 b(the)g(Ishii-P)m (astur-Kotani)-236 3041 y(Theorem)33 b([13)r(],)i(this)e(implies)e (that)k(the)f(essen)m(tial)g(closure)f(of)h(the)g(set)h Fp(J)d Fn(n)23 b Fp(B)38 b Fr(b)s(elongs)33 b(to)i(the)f(absolutely)f (con)m(tin)m(uous)-236 3149 y(sp)s(ectrum)k(of)45 b(\(0.1\))r(.)65 b(Finally)-8 b(,)39 b(Theorem)f(7.2)h(allo)m(ws)f(us)f(to)i(analyze)g (the)g(functional)d(structure)i(of)h(the)f(generalized)-236 3257 y(eigenfunctions)29 b(on)h Fp(J)f Fn(n)21 b Fp(B)5 b Fr(.)-236 3412 y(In)28 b(Theorem)g(1.1,)i(w)m(e)f(ha)m(v)m(e)h(only)e (describ)s(ed)e(the)j(part)g(of)f(the)h(sp)s(ectrum)f(outside)g(of)g(a) h(small)e(set.)41 b(As)29 b(said)e(ab)s(o)m(v)m(e,)k(this)-236 3520 y(set)25 b(is)e(related)i(to)g(the)f(KAM)h(resonances)g(for)f(the) h(mono)s(drom)m(y)e(equation.)39 b(W)-8 b(e)26 b(b)s(eliev)m(e)d(that,) k(adapting)c(the)i(tec)m(hniques)-236 3628 y(dev)m(elop)s(ed)k(in)h ([8],)h(one)g(can)f(pro)m(v)m(e)i(that,)f(in)e(this)g(small)g(set,)i (the)g(sp)s(ectrum)e(is)g(purely)g(absolutely)g(con)m(tin)m(uous.)-236 3806 y(1.6.2.)48 b Fi(Outline)41 b(of)h(the)f(p)-5 b(ap)g(er.)48 b Fr(In)39 b(section)h(2,)j(w)m(e)d(recall)f(some)h(information)e(on)i (the)g(p)s(erio)s(dic)d(Sc)m(hr\177)-45 b(odinger)38 b(op)s(era-)-236 3914 y(tor)j(\(0.2\))q(.)71 b(Sections)40 b(3,)h(4)g(and)f(5)h(are)f(dev)m(oted)i(to)f(the)f(complex)h(WKB)f (metho)s(d)g(for)g(adiabatic)g(p)s(erturbations)f(of)-236 4022 y(p)s(erio)s(dic)31 b(Sc)m(hr\177)-45 b(odinger)32 b(op)s(erators.)50 b(In)33 b(section)h(6,)h(w)m(e)f(pro)m(v)m(e)g (Theorem)f(1.3.)52 b(In)33 b(section)g(7,)i(w)m(e)f(study)f(ho)m(w)h (the)f(solu-)-236 4130 y(tions)f(to)h(equation)f(\(0.1\))j(relate)d(to) i(those)f(of)f(the)h(mono)s(drom)m(y)f(equation)g(\(1.9\))r(.)47 b(Section)33 b(8)f(is)g(dev)m(oted)h(to)h(the)e(pro)s(of)-236 4238 y(of)e(Theorem)g(1.1.)1070 4446 y(2.)51 b Fq(Periodic)35 b(Schr)1847 4438 y(\177)1843 4446 y(odinger)e(opera)-6 b(tors)-24 4607 y Fr(In)44 b(this)e(section,)48 b(w)m(e)c(collect)h (kno)m(wn)e(information)f(\(see)j([14)q(,)f(7)q(,)g(11)q(,)g(12)q(]\))g (ab)s(out)g(the)g(p)s(erio)s(dic)d(Sc)m(hr\177)-45 b(odinger)-236 4715 y(op)s(erator)30 b(\(0.2\))r(.)41 b(W)-8 b(e)32 b(assume)e(that)h Fp(V)50 b Fr(is)30 b(a)g(real)g(v)-5 b(alued,)30 b(1-p)s(erio)s(dic,)f Fp(L)2321 4682 y Fm(2)2321 4743 y Fo(l)q(oc)2411 4715 y Fr(-function.)-236 4894 y(2.1.)53 b Fu(Blo)s(c)m(h)37 b(solutions.)46 b Fr(Let)31 b Fp( )j Fr(b)s(e)29 b(a)i(solution)e(of)i(the)f(equation)1124 5104 y Fn(\000)1231 5042 y Fp(d)1278 5009 y Fm(2)p 1205 5083 139 4 v 1205 5166 a Fp(dx)1304 5140 y Fm(2)1354 5104 y Fp( )18 b Fr(\()p Fp(x)p Fr(\))j(+)f Fp(V)35 b Fr(\()p Fp(x)p Fr(\))26 b(=)f Fp(E)5 b( )19 b Fr(\()p Fp(x)p Fr(\))p Fp(;)107 b(x)25 b Fn(2)g Fl(R)s Fp(:)-2885 b Fr(\(2.1\))-236 5289 y(satisfying)27 b(the)h(relation)f Fp( )19 b Fr(\()p Fp(x)d Fr(+)f(1\))27 b(=)e Fp(\026)15 b( )j Fr(\()p Fp(x)p Fr(\),)58 b Fn(8)p Fp(x)24 b Fn(2)h Fl(R)s Fr(,)35 b(with)26 b Fp(\026)i Fr(indep)s(enden)m(t)e(of)j Fp(x)p Fr(.)39 b(It)29 b(is)e(a)h Fi(Blo)-5 b(ch)30 b Fr(solution,)d(and)h Fp(\026)g Fr(is)-236 5396 y(the)f Fi(Flo)-5 b(quet)30 b(multiplier)d Fr(asso)s(ciated)g(to)g Fp( )s Fr(.)40 b(W)-8 b(rite)27 b(it)f(as)h Fp(\026)e Fr(=)g(exp\()p Fp(ik)s Fr(\);)k(then,)e Fp(k)j Fr(is)c(called)f(the)i Fi(Blo)-5 b(ch)30 b(quasi-momentum)p Fr(.)-236 5524 y(The)40 b(Blo)s(c)m(h)g(solution)f Fp( )44 b Fr(can)d(b)s(e)f(represen)m(ted)g (in)g(the)g(form)g Fp( )s Fr(\()p Fp(x)p Fr(\))k(=)d Fp(e)2378 5491 y(ik)s(x)2516 5524 y(p)p Fr(\()p Fp(x)p Fr(\))g(where)f Fp(x)i Fn(7!)g Fp(p)p Fr(\()p Fp(x)p Fr(\))f(is)f(a)h(1-p)s(erio)s(dic)-236 5632 y(function.)-236 5787 y(The)31 b(sp)s(ectrum)f(of)i(the)g(p)s(erio)s(dic)c(Sc)m(hr\177) -45 b(odinger)31 b(op)s(erator)g(\(0.2\))j(w)m(as)e(describ)s(ed)d(in)h (section)i(1.2.)45 b(Consider)30 b(t)m(w)m(o)j(copies)-236 5895 y(of)d(the)g(complex)g(plane)f Fp(E)i Fn(2)25 b Fl(C)53 b Fr(cut)31 b(along)f(the)g(sp)s(ectral)g(bands)f(of)h(the)g(p) s(erio)s(dic)e(Sc)m(hr\177)-45 b(odinger)28 b(op)s(erator.)41 b(P)m(aste)32 b(them)-236 6003 y(together)d(in)m(to)e(a)h(Riemann)e (surface)i(\000)f(with)f(square)h(ro)s(ot)h(branc)m(h)f(p)s(oin)m(ts.) 39 b(One)27 b(constructs)g(a)h(Blo)s(c)m(h)g(solution)e Fp( )s Fr(\()p Fp(x;)15 b(E)5 b Fr(\))-236 6110 y(of)28 b(equation)f(\(2.1\))i(meromorphic)e(on)g(this)f(Riemann)h(surface.)40 b(It)27 b(is)g(normalized)f(b)m(y)h(the)h(condition)e Fp( )s Fr(\(1)p Fp(;)15 b(E)5 b Fr(\))28 b Fn(\021)d Fr(1.)40 b(The)1872 6210 y Fm(5)p eop %%Page: 6 6 6 5 bop -236 241 a Fr(p)s(oles)33 b(of)h(this)e(solution)h(are)h(lo)s (cated)g(in)f(the)h(op)s(en)f(sp)s(ectral)h(gaps)g(\(eac)m(h)h(op)s(en) e(sp)s(ectral)h(gap)g(con)m(tains)g(precisely)e(one)-236 349 y(simple)c(p)s(ole\).)-236 504 y(Outside)38 b(the)i(edges)h(of)f (the)g(sp)s(ectrum,)h(the)f(t)m(w)m(o)h(branc)m(hes)f Fp( )2037 518 y Fk(\006)2136 504 y Fr(of)g(the)g(Blo)s(c)m(h)g (solution)e(are)i(linearly)e(indep)s(enden)m(t)-236 612 y(solutions)29 b(of)h(the)h(p)s(erio)s(dic)c(equation)k(\(2.1\))q(.)41 b(On)30 b(the)g(sp)s(ectral)g(bands,)f(they)i(di\013er)e(only)g(b)m(y)i (complex)f(conjugation.)-236 867 y(2.2.)53 b Fu(Blo)s(c)m(h)d (quasi-momen)m(tum.)43 b Fr(The)f(Flo)s(quet)g(m)m(ultiplier)d Fp(\026)15 b Fr(\()p Fp(E)5 b Fr(\))43 b(asso)s(ciated)g(to)g Fp( )s Fr(\()p Fp(x;)15 b(E)5 b Fr(\))44 b(is)d(also)h(analytic)g(on) -236 975 y(\000.)48 b(The)33 b(corresp)s(onding)e(Blo)s(c)m(h)i (quasi-momen)m(tum)f(is)g(an)h(analytic)g(m)m(ulti-v)-5 b(alued)31 b(function)g(of)j Fp(E)k Fr(and)32 b(has)h(the)g(same)-236 1082 y(branc)m(h)d(p)s(oin)m(ts)f(as)h Fp( )s Fr(\()p Fp(x;)15 b(E)5 b Fr(\).)-236 1238 y(Let)28 b Fp(D)i Fr(b)s(e)d(a)h (simply)d(connected)j(domain)e(con)m(taining)h(no)h(branc)m(h)f(p)s (oin)m(ts)f(of)i(the)f(Blo)s(c)m(h)h(quasi-momen)m(tum.)39 b(On)26 b Fp(D)s Fr(,)i(\014x)-236 1346 y Fp(k)-189 1360 y Fm(0)-149 1346 y Fr(,)k(an)g(analytic)f(single-v)-5 b(alued)30 b(branc)m(h)i(of)g Fp(k)s Fr(.)45 b(All)31 b(the)h(other)g(single-v)-5 b(alued)30 b(branc)m(hes)h(that)i(are)f (analytic)g(in)e Fp(E)j Fn(2)28 b Fp(D)s Fr(,)-236 1454 y(are)j(describ)s(ed)d(b)m(y)i(the)h(form)m(ulae)1235 1646 y Fp(k)1282 1661 y Fk(\006)p Fo(;l)1382 1646 y Fr(\()p Fp(E)5 b Fr(\))27 b(=)e Fn(\006)p Fp(k)1765 1660 y Fm(0)1804 1646 y Fr(\()p Fp(E)5 b Fr(\))21 b(+)f(2)p Fp(\031)s(l)r(;)107 b(l)27 b Fn(2)e Fl(Z)p Fp(:)-2785 b Fr(\(2.2\))-236 1843 y(Giv)m(en)31 b(a)h(branc)m(h)f Fp(k)s Fr(,)i(analytic)e(on)h Fp(D)s Fr(,)g(w)m(e)g(\014x)f Fp( )1449 1857 y Fk(\006)1508 1843 y Fr(\()p Fp(x;)15 b(E)5 b Fr(\),)34 b(t)m(w)m(o)f(branc)m(hes)e (of)h(the)g(Blo)s(c)m(h)f(solution)f Fp( )s Fr(\()p Fp(x;)15 b(E)5 b Fr(\),)35 b(analytic)c(on)-236 1951 y Fp(D)i Fr(so)d(that)h Fn(\006)p Fp(k)j Fr(b)s(e)29 b(their)h(quasi-momen)m (ta.)41 b(Then,)30 b(one)g(has)901 2078 y Ff(Z)992 2104 y Fm(1)951 2284 y(0)1046 2201 y Fp( )1105 2215 y Fm(+)1164 2201 y Fr(\()p Fp(t;)15 b(E)5 b Fr(\))p Fp( )1438 2215 y Fk(\000)1499 2201 y Fr(\()p Fp(t;)15 b(E)5 b Fr(\))p Fp(dt)27 b Fr(=)e Fn(\000)p Fp(ik)2069 2164 y Fk(0)2092 2201 y Fr(\()p Fp(E)5 b Fr(\))p Fp(w)r Fr(\()p Fp(E)g Fr(\))p Fp(;)109 b(E)31 b Fn(2)25 b Fp(D)s(;)-3100 b Fr(\(2.3\))-236 2501 y(where)30 b Fp(w)r Fr(\()p Fp(E)5 b Fr(\))32 b(is)d(the)i(W)-8 b(ronskian)29 b(of)i(the)f(solutions)f Fp( )1667 2515 y Fk(\006)1757 2501 y Fr(i.e.)40 b Fp(w)r Fr(\()p Fp(E)5 b Fr(\))27 b(=)2255 2439 y Fp(@)5 b( )2367 2453 y Fm(+)p 2255 2480 172 4 v 2288 2563 a Fp(@)g(x)2437 2501 y( )2496 2515 y Fk(\000)2575 2501 y Fn(\000)2676 2439 y Fp(@)g( )2788 2453 y Fk(\000)p 2676 2480 V 2709 2563 a Fp(@)g(x)2857 2501 y( )2916 2515 y Fm(+)2976 2501 y Fr(.)-236 2682 y(Consider)38 b Fl(C)213 2696 y Fm(+)278 2682 y Fr(,)k(the)e(upp)s(er)e(half)h(of)h(the)g(complex)g(plane.)68 b(There)40 b(exists)g Fp(k)2502 2696 y Fo(p)2542 2682 y Fr(,)i(an)e(analytic)f(branc)m(h)h(of)g(the)g(complex)-236 2790 y(momen)m(tum)27 b(that)g(conformally)f(maps)g Fl(C)1217 2804 y Fm(+)1309 2790 y Fr(on)m(to)i(the)f(quadran)m(t)g Fn(f)p Fr(Im)15 b Fp(k)28 b(>)d Fr(0)p Fp(;)47 b Fr(Re)15 b Fp(k)28 b(>)d Fr(0)p Fn(g)j Fr(cut)f(along)g(\014nite)f(v)m(ertical)h (slits)-236 2898 y(b)s(eginning)22 b(at)j(the)g(p)s(oin)m(ts)e Fp(\031)s(l)r Fr(,)j Fp(l)i Fr(=)c(1)p Fp(;)15 b Fr(2)p Fp(;)g Fr(3)g Fp(:)g(:)g(:)44 b Fr(The)24 b(branc)m(h)g Fp(k)1885 2912 y Fo(p)1949 2898 y Fr(is)g(con)m(tin)m(uous)g(on)h Fl(C)2664 2912 y Fm(+)2738 2898 y Fn([)9 b Fl(R)r Fr(.)45 b(It)25 b(is)e(real)i(and)f(monotonically)-236 3006 y(increasing)29 b(along)h(the)h(sp)s(ectrum;)e(it)h(maps)g(the)h(sp)s(ectral)e(band)h (\()p Fp(E)2167 3020 y Fm(2)p Fo(n)p Fk(\000)p Fm(1)2339 3006 y Fp(;)15 b(E)2446 3020 y Fm(2)p Fo(n)2529 3006 y Fr(\))31 b(on)m(to)g(the)g(in)m(terv)-5 b(al)29 b(\()p Fp(\031)s Fr(\()p Fp(n)21 b Fn(\000)f Fr(1\))p Fp(;)15 b(\031)s(n)p Fr(\).)-236 3261 y(2.3.)53 b Fu(Analytic)36 b(con)m(tin)m(uation)f(through)g(a)g(connected)h(comp)s(onen)m(t)e(of)h (the)g(sp)s(ectrum.)45 b Fr(W)-8 b(e)31 b(denote)g(b)m(y)g Fl(C)3928 3275 y Fo(n)3971 3284 y Fg(0)-236 3369 y Fr(the)k(complex)g (plane)f(cut)h(along)g(the)h(half-lines)c(\()p Fn(\0001)p Fp(;)15 b(E)1798 3383 y Fm(2)p Fo(n)1876 3392 y Fg(0)1912 3383 y Fk(\000)p Fm(1)2006 3369 y Fr(])35 b(and)g([)p Fp(E)2340 3383 y Fm(2)p Fo(n)2418 3392 y Fg(0)2457 3369 y Fp(;)15 b Fr(+)p Fn(1)p Fr(\),)37 b(where)d Fp(E)3090 3383 y Fm(2)p Fo(n)3168 3392 y Fg(0)3203 3383 y Fk(\000)p Fm(1)3332 3369 y Fr(and)h Fp(E)3581 3383 y Fm(2)p Fo(n)3659 3392 y Fg(0)3733 3369 y Fr(are)g(the)-236 3477 y(ends)29 b(of)i(the)f Fp(n)286 3491 y Fm(0)325 3477 y Fr(-th)h(sp)s(ectral)f (band)f([)p Fp(E)1133 3491 y Fm(2)p Fo(n)1211 3500 y Fg(0)1246 3491 y Fk(\000)p Fm(1)1340 3477 y Fp(;)15 b(E)1447 3491 y Fm(2)p Fo(n)1525 3500 y Fg(0)1565 3477 y Fr(].)-236 3632 y(The)30 b(function)g Fp(k)355 3646 y Fo(p)426 3632 y Fr(b)s(eing)f(con)m(tin)m(uous)i(and)f(real)h(on)g([)p Fp(E)1701 3646 y Fm(2)p Fo(n)1779 3655 y Fg(0)1814 3646 y Fk(\000)p Fm(1)1908 3632 y Fp(;)15 b(E)2015 3646 y Fm(2)p Fo(n)2093 3655 y Fg(0)2132 3632 y Fr(],)32 b(it)e(can)i(b)s(e)e (analytically)f(con)m(tin)m(ued)i(to)h(a)f(function)-236 3740 y(analytic)f(on)g Fl(C)296 3754 y Fo(n)339 3763 y Fg(0)414 3740 y Fr(b)m(y)g(the)g(relation)1394 3941 y Fp(k)1441 3955 y Fo(p)1481 3941 y Fr(\()p 1516 3867 47 4 v Fp(\020)7 b Fr(\))26 b(=)p 1720 3862 205 4 v 25 w Fp(k)1767 3955 y Fo(p)1807 3941 y Fr(\()p Fp(\020)7 b Fr(\))p Fp(;)106 b(\020)32 b Fn(2)25 b Fl(C)2272 3955 y Fo(n)2315 3964 y Fg(0)2360 3941 y Fp(:)-2621 b Fr(\(2.4\))-236 4134 y(F)-8 b(or)31 b(this)e(branc)m(h)h(de\014ned)f(on)h Fl(C)904 4148 y Fo(n)947 4157 y Fg(0)992 4134 y Fr(,)g(w)m(e)h(k)m(eep) g(the)g(\\old")f(notation)h Fp(k)2198 4148 y Fo(p)2238 4134 y Fr(.)-236 4289 y(The)f(branc)m(hes)g Fp( )387 4303 y Fk(\006)476 4289 y Fr(of)h(the)f(Blo)s(c)m(h)h(solution)e Fp( )s Fr(\()p Fp(x;)15 b(E)5 b Fr(\))32 b(are)f(analytic)f(on)g Fl(C)2348 4303 y Fo(n)2391 4312 y Fg(0)2466 4289 y Fr(and)g(satisfy) 1245 4491 y Fp( )1304 4505 y Fk(\000)1363 4491 y Fr(\()p Fp(x;)p 1490 4418 73 4 v 15 w(E)6 b Fr(\))26 b(=)p 1720 4411 354 4 v 25 w Fp( )1779 4505 y Fm(+)1838 4491 y Fr(\()p Fp(x;)15 b(E)5 b Fr(\))q Fp(;)107 b(\020)31 b Fn(2)25 b Fl(C)2422 4505 y Fo(n)2465 4514 y Fg(0)2509 4491 y Fp(:)-2770 b Fr(\(2.5\))-236 4688 y(They)23 b(are)h(linearly)d(indep)s (enden)m(t)h(solutions)g(of)30 b(\(2.1\))c(for)d Fp(E)31 b Fn(2)25 b Fl(C)2027 4702 y Fo(n)2070 4711 y Fg(0)2115 4688 y Fr(.)38 b(One)23 b(indexes)g Fp( )2737 4702 y Fk(\006)2820 4688 y Fr(so)g(that)i Fn(\006)p Fp(k)3233 4702 y Fo(p)3296 4688 y Fr(b)s(e)e(their)g(resp)s(ectiv)m(e)-236 4796 y(quasi-momen)m(ta)31 b(for)f Fp(E)g Fn(2)25 b Fl(C)778 4810 y Fo(n)821 4819 y Fg(0)865 4796 y Fr(.)701 5112 y(3.)51 b Fq(The)34 b(main)h(theorem)d(of)i(the)g(complex)e(WKB)i (method)-24 5274 y Fr(In)27 b(this)f(section,)j(follo)m(wing)d([10)q (],)i(w)m(e)g(brie\015y)e(describ)s(e)f(the)j(main)e(constructions)h (of)h(the)f(complex)g(WKB)h(metho)s(d)-236 5382 y(for)i(adiabatically)f (p)s(erturb)s(ed)f(p)s(erio)s(dic)f(Sc)m(hr\177)-45 b(odinger)29 b(equations)1004 5629 y Fn(\000)1111 5568 y Fp(d)1158 5535 y Fm(2)p 1085 5608 139 4 v 1085 5692 a Fp(dx)1184 5665 y Fm(2)1234 5629 y Fp( )s Fr(\()p Fp(x)p Fr(\))21 b(+)f(\()p Fp(V)g Fr(\()p Fp(x)p Fr(\))h(+)f Fp(W)13 b Fr(\()p Fp("x)p Fr(\)\))p Fp( )s Fr(\()p Fp(x)p Fr(\))27 b(=)e Fp(E)5 b( )s Fr(\()p Fp(x)p Fr(\))p Fp(;)-2994 b Fr(\(3.1\))-236 5847 y(where)31 b Fp(V)52 b Fr(is)31 b(a)h(real)g(v)-5 b(alued)31 b(1-p)s(erio)s(dic)f(function)g(of)i Fp(x)p Fr(,)h(and)e Fp(")h Fr(is)f(a)h(small)f(p)s(ositiv)m(e)g (parameter.)45 b(The)32 b(complex)f(WKB)-236 5955 y(metho)s(ds)e(allo)m (ws,)h(in)f(particular,)g(to)j(describ)s(e)c(the)j(exp)s(onen)m(tially) d(small)h(e\013ects)j(due)e(to)h(the)g(complex)f(tunneling.)-236 6110 y(In)f(this)h(section,)g(w)m(e)h(assume)f(that)h Fp(V)46 b Fn(2)25 b Fp(L)1275 6078 y Fm(2)1275 6138 y(lo)r(c)1367 6110 y Fr(,)31 b(and)e(that)i Fp(W)43 b Fr(is)29 b(analytic)h(in)g(a)g (neigh)m(b)s(orho)s(o)s(d)e Fn(D)s Fr(\()p Fp(W)13 b Fr(\))30 b(of)h(the)f(real)h(line.)1872 6210 y Fm(6)p eop %%Page: 7 7 7 6 bop -236 241 a Fr(3.1.)53 b Fu(Additional)34 b(complex)f (parameter.)44 b Fr(T)-8 b(o)29 b(decouple)f(the)h(\\slo)m(w)g(v)-5 b(ariable")28 b Fp(\030)h Fr(=)c Fp("x)30 b Fr(and)e(the)h(\\fast)g(v) -5 b(ariable")29 b Fp(x)p Fr(,)-236 349 y(one)h(in)m(tro)s(duces)g(an)g (additional)e(parameter)j Fp(\020)37 b Fr(so)30 b(that)h(\(3.1\))i(b)s (ecomes)753 597 y Fn(\000)860 535 y Fp(d)907 502 y Fm(2)p 834 576 139 4 v 834 659 a Fp(dx)933 633 y Fm(2)982 597 y Fp( )s Fr(\()p Fp(x)p Fr(\))22 b(+)e(\()p Fp(V)g Fr(\()p Fp(x)p Fr(\))h(+)f Fp(W)13 b Fr(\()p Fp("x)20 b Fr(+)g Fp(\020)7 b Fr(\)\))p Fp( )s Fr(\()p Fp(x)p Fr(\))27 b(=)e Fp(E)5 b( )s Fr(\()p Fp(x)p Fr(\))p Fp(;)108 b(x)25 b Fn(2)g Fl(R)r Fp(:)-3256 b Fr(\(3.2\))-236 815 y(Then,)31 b(one)h(studies)e(solutions)g(of)39 b(\(3.2\))34 b(on)d(the)h(complex)f (plane)g(of)h Fp(\020)38 b Fr(to)32 b(reco)m(v)m(er)i(information)c(on) h(their)g(b)s(eha)m(vior)g(on)-236 923 y Fl(R)r Fr(.)46 b(T)-8 b(o)26 b(con)m(trol)g(the)g(dep)s(endence)f(of)h(solutions)f(of) h(equation)g(\(3.2\))h(on)f Fp(\020)33 b Fr(w)m(e)26 b(assume)g(that)g(they)g(satisfy)g(the)g Fi(c)-5 b(onsistency)-236 1031 y(c)g(ondition)1296 1218 y Fp( )s Fr(\()p Fp(x)21 b Fr(+)e(1)p Fp(;)c(\020)7 b Fr(\))26 b(=)f Fp( )s Fr(\()p Fp(x;)15 b(\020)28 b Fr(+)20 b Fp(")p Fr(\))92 b Fn(8)p Fp(\020)7 b(:)-2721 b Fr(\(3.3\))-236 1409 y Fu(Remark)34 b(3.1.)46 b Fr(Condition)32 b(\(3.3\))k(pla)m(ys)e(a)h(crucial)e(role)g (for)h(the)h(asymptotic)f(analysis)f(of)42 b(\(3.2\))q(.)53 b(It)34 b(app)s(ears)f(that)i(it)-236 1517 y(leads)d(to)h(the)g (geometric)h(ob)5 b(jects)33 b(and)f(analytic)h(constructions)f (similar)e(to)j(those)h(t)m(ypical)e(for)g(the)h(classical)f(complex) -236 1625 y(WKB)f(metho)s(d.)-236 1780 y(Condition)21 b(\(1.6\))k(used)e(to)h(de\014ne)e(the)i(mono)s(drom)m(y)f(matrix)f(w)m (as)i(also)f(called)g(\\consistency)h(condition".)37 b(In)23 b(section)g(3,)h(4)-236 1888 y(and)33 b(5,)i(the)f(w)m(ords)f (\\consistency)h(condition")f(and)g(\\consisten)m(t")i(refer)e(to)i(ob) 5 b(jects)34 b(satisfying)g(\(3.3\))r(.)50 b(Though)33 b(b)s(eing)-236 1996 y(of)d(di\013eren)m(t)g(nature,)g(in)f(the)i (analysis)e(of)h(the)h(family)d(of)j(equations)f(\(0.1\))r(,)g (conditions)f(\(1.6\))j(and)e(\(3.3\))i(are)f(related)f(to)-236 2104 y(eac)m(h)h(other)g(b)m(y)f(a)h(c)m(hange)g(of)g(v)-5 b(ariables)29 b(as)i(w)m(e)f(shall)f(see)i(in)e(the)i(b)s(eginning)c (of)k(section)f(6.)-236 2290 y(3.2.)53 b Fu(Complex)c(momen)m(tum.)43 b Fr(The)g(cen)m(tral)g(analytic)g(ob)5 b(ject)44 b(of)f(the)g(complex) g(WKB)g(metho)s(d)g(is)f(the)h Fi(c)-5 b(omplex)-236 2398 y(momentum)31 b Fp(\024)p Fr(\()p Fp(\020)7 b Fr(\).)42 b(It)30 b(is)g(de\014ned)f(in)g(terms)h(of)h(the)f(Blo)s(c)m(h)h (quasi-momen)m(tum)e(of)38 b(\(0.2\))32 b(b)m(y)e(the)h(form)m(ula)1484 2585 y Fp(\024)p Fr(\()p Fp(\020)7 b Fr(\))26 b(=)f Fp(k)s Fr(\()p Fp(E)h Fn(\000)20 b Fp(W)13 b Fr(\()p Fp(\020)7 b Fr(\)\))-2531 b(\(3.4\))-236 2778 y(in)29 b Fn(D)s Fr(\()p Fp(W)13 b Fr(\))30 b(the)h(domain)e(of)h(analyticit)m(y)g(of)h (the)f(function)g Fp(W)13 b Fr(.)-236 2933 y(The)31 b(complex)g(momen)m (tum)g Fp(\024)h Fr(is)e(a)i(m)m(ulti-v)-5 b(alued)29 b(analytic)i(function.)42 b(Its)32 b(branc)m(h)e(p)s(oin)m(ts)h(are)g (related)h(to)g(the)f(branc)m(h)-236 3041 y(p)s(oin)m(ts)e(of)i(the)f (quasi-momen)m(tum)g(b)m(y)g(the)h(relations)1202 3228 y Fp(E)1269 3243 y Fo(l)1321 3228 y Fr(=)25 b Fp(E)g Fn(\000)20 b Fp(W)28 b Fr(\()p Fp(\020)7 b Fr(\))p Fp(;)106 b(l)28 b Fr(=)c(1)p Fp(;)31 b Fr(2)p Fp(;)g Fr(3)p Fp(;)g(:)15 b(:)g(:)i(;)-2798 b Fr(\(3.5\))-236 3421 y(where)30 b(\()p Fp(E)129 3436 y Fo(l)155 3421 y Fr(\))190 3436 y Fo(l)q Fk(\025)p Fm(1)337 3421 y Fr(are)h(the)f(ends)g(of)g(the)h(sp)s(ectral) f(gaps)g(of)h(the)f(op)s(erator)h Fp(H)2365 3435 y Fm(0)2404 3421 y Fr(.)-236 3529 y(W)-8 b(e)35 b(sa)m(y)f(that)h(a)f(set)g(is)f Fi(r)-5 b(e)g(gular)35 b Fr(if)e(it)g(is)g(in)f(the)i(domain)f(of)h (analyticit)m(y)f(of)h Fp(W)47 b Fr(and)33 b(con)m(tains)h(no)f(branc)m (h)h(p)s(oin)m(ts)e(of)i Fp(\024)p Fr(.)-236 3637 y(F)-8 b(or)32 b(regular)f(curv)m(es,)h(w)m(e)h(also)e(ask)h(that)g(they)g (are)g(simply)d(connected)k(and)e(piecewise)g Fp(C)2944 3604 y Fm(1)2983 3637 y Fr(.)44 b(F)-8 b(or)32 b(regular)f(domains,)g (w)m(e)-236 3745 y(ask)f(that)h(they)g(are)g(simply)d(connected.)-236 3900 y(Let)36 b Fp(D)i Fr(b)s(e)c(a)i(regular)e(domain.)54 b(Then,)36 b(in)e Fp(D)s Fr(,)j(one)e(can)h(\014x)e Fp(\024)1988 3914 y Fm(0)2028 3900 y Fr(,)j(an)e(analytic)g(branc)m(h)f(of)i Fp(\024)p Fr(.)55 b(By)37 b(\(3.7\))r(,)g(all)d(the)h(other)-236 4008 y(analytic)30 b(branc)m(hes)g(are)h(describ)s(ed)d(b)m(y)i(the)h (form)m(ulas)1522 4195 y Fp(\024)1574 4158 y Fk(\006)1574 4218 y Fo(m)1667 4195 y Fr(=)25 b Fn(\006)p Fp(\024)1886 4209 y Fm(0)1945 4195 y Fr(+)20 b(2)p Fp(\031)s(m;)-2477 b Fr(\(3.6\))-236 4383 y(where)30 b Fn(\006)g Fr(and)f Fp(m)i Fr(are)f(indexing)e(the)j(branc)m(hes.)-236 4629 y(3.3.)53 b Fu(Canonical)28 b(domains.)45 b Fr(The)23 b(canonical)h(domain)e(notion)h(is)g(the)h(main)f(geometric)i(notion)e (of)h(the)g(complex)f(WKB)-236 4736 y(metho)s(d.)40 b(Pro)s(ceed)30 b(to)i(the)e(de\014nitions.)-236 4892 y(A)f(regular)e(curv)m(e)i Fp(\015)34 b Fr(is)28 b(called)g Fi(vertic)-5 b(al)29 b Fr(if)f(it)g(in)m(tersects)h(the)g(lines)e Fn(f)p Fr(Im)15 b Fp(\020)31 b Fr(=)25 b(Const)p Fn(g)k Fr(at)h(non-zero)f(angles)f Fp(\022)s Fr(,)57 b(0)26 b Fp(<)f(\022)i(<)e(\031)s Fr(.)-236 5000 y(V)-8 b(ertical)30 b(lines)f(are)i(naturally)d(parameterized)j(b) m(y)f(Im)15 b Fp(\020)7 b Fr(.)-236 5155 y(Let)35 b Fp(\015)40 b Fr(b)s(e)34 b(a)h Fp(C)298 5122 y Fm(1)371 5155 y Fr(regular)f(v)m (ertical)h(curv)m(e.)54 b(On)33 b Fp(\015)5 b Fr(,)36 b(\014x)e(a)h(con)m(tin)m(uous)g(branc)m(h)f(of)h(the)f(momen)m(tum)h Fp(\024)p Fr(.)54 b(The)34 b(curv)m(e)h Fp(\015)40 b Fr(is)-236 5263 y Fi(c)-5 b(anonic)g(al)32 b Fr(if)-137 5410 y(1.)43 b(Im)115 5337 y Ff(R)175 5363 y Fo(\020)230 5410 y Fp(\024d\020)37 b Fr(is)30 b(strictly)f(increasing)g(when)h(Im) 14 b Fp(\020)37 b Fr(is)30 b(increasing)f(along)h Fp(\015)5 b Fr(,)-137 5540 y(2.)43 b(Im)115 5466 y Ff(R)175 5493 y Fo(\020)215 5540 y Fr(\()p Fp(\024)21 b Fn(\000)f Fp(\031)s Fr(\))p Fp(d\020)37 b Fr(is)30 b(strictly)f(decreasing)h(when)g(Im)14 b Fp(\020)37 b Fr(is)29 b(increasing)g(along)i Fp(\015)5 b Fr(.)-236 5691 y(Note)31 b(that)g(canonical)f(lines)f(are)i(stable)f (under)f(small)g Fn(C)1751 5658 y Fm(1)1790 5691 y Fr(-p)s (erturbations.)-236 5877 y Fu(De\014nition)35 b(3.1.)46 b Fr(Let)33 b Fp(K)39 b Fr(b)s(e)32 b(a)h(regular)e(domain.)46 b(On)32 b Fp(K)7 b Fr(,)33 b(\014x)e(a)i(con)m(tin)m(uous)f(branc)m(h)g (of)h(the)g(quasi-momen)m(tum,)f(sa)m(y)-236 5985 y Fp(\024)p Fr(.)41 b(The)29 b(domain)g Fp(K)37 b Fr(is)29 b(called)g Fi(c)-5 b(anonic)g(al)32 b Fr(if)e(it)f(is)g(the)i(union)d(of)i(curv)m (es)g(canonical)g(with)f(resp)s(ect)h(to)h Fp(\024)f Fr(and)g(connecting)-236 6093 y(t)m(w)m(o)h(p)s(oin)m(ts)f Fp(\020)249 6107 y Fm(1)318 6093 y Fr(and)g Fp(\020)535 6107 y Fm(2)605 6093 y Fr(lo)s(cated)g(on)g Fp(@)5 b(K)i Fr(.)1872 6210 y Fm(7)p eop %%Page: 8 8 8 7 bop -236 241 a Fr(3.4.)53 b Fu(Canonical)35 b(Blo)s(c)m(h)i (solutions.)46 b Fr(Consider)28 b(the)j(p)s(erio)s(dic)d(Sc)m(hr\177) -45 b(odinger)29 b(equation)709 478 y Fn(\000)815 416 y Fp(d)862 383 y Fm(2)p 790 457 139 4 v 790 540 a Fp(dx)889 514 y Fm(2)938 478 y Fp( )s Fr(\()p Fp(x)p Fr(\))21 b(+)f Fp(V)h Fr(\()p Fp(x)p Fr(\))p Fp( )s Fr(\()p Fp(x)p Fr(\))27 b(=)e Fn(E)8 b Fp( )s Fr(\()p Fp(x)p Fr(\))p Fp(;)107 b Fn(E)33 b Fr(=)25 b Fp(E)h Fn(\000)20 b Fp(W)13 b Fr(\()p Fp(\020)7 b Fr(\))p Fp(;)106 b(x)25 b Fn(2)g Fl(R)s Fp(:)-3301 b Fr(\(3.7\))-236 685 y(Here,)31 b Fp(\020)37 b Fr(pla)m(ys)30 b(the)g(role)g(of)h(a)f(parameter.)-236 840 y(The)23 b(function)f Fp( )s Fr(\()p Fp(x;)15 b(E)d Fn(\000)7 b Fp(W)13 b Fr(\()p Fp(\020)7 b Fr(\)\))24 b(is)e(a)i(solution)e(of)31 b(\(3.7\))26 b(meromorphic)c(in)g Fp(\020)7 b Fr(;)26 b(w)m(e)e(no)m(w)f(construct)h(solutions)e(of)31 b(\(3.7\))25 b(that)-236 948 y(are)e(analytic)f(in)f Fp(\020)7 b Fr(.)37 b(Let)23 b Fp(D)i Fr(b)s(e)d(a)h(regular)e(domain.)37 b(There)22 b(are)h(t)m(w)m(o)h(di\013eren)m(t)e(branc)m(hes)g(of)g(the) h(function)e Fp( )s Fr(\()p Fp(x;)15 b(E)10 b Fn(\000)t Fp(W)j Fr(\()p Fp(\020)7 b Fr(\)\))-236 1056 y(that)37 b(are)g(meromorphic)e(on)h Fp(D)s Fr(.)59 b(Denote)38 b(them)f(b)m(y)f Fp( )1724 1070 y Fk(\006)1783 1056 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\).)60 b(On)36 b Fp(D)s Fr(,)i(\014x)e Fp(\024)p Fr(,)i(an)f(analytic)f(branc)m(h)g(of)g(the)h (complex)-236 1164 y(momen)m(tum)30 b(so)h(that)g Fn(\006)p Fp(\024)p Fr(\()p Fp(\020)7 b Fr(\))30 b(b)s(e)g(the)h(Blo)s(c)m(h)f (quasi-momen)m(ta)h(of)f Fp( )2151 1178 y Fk(\006)2210 1164 y Fr(.)41 b(W)-8 b(e)32 b(can)e(represen)m(t)h Fp( )3051 1178 y Fk(\006)3140 1164 y Fr(in)e(the)i(form)1340 1356 y Fp( )1399 1370 y Fk(\006)1458 1356 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))26 b(=)f Fp(e)1831 1318 y Fk(\006)p Fo(i\024)p Fm(\()p Fo(\020)5 b Fm(\))p Fo(x)2085 1356 y Fp(p)2131 1370 y Fk(\006)2190 1356 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))p Fp(;)-2660 b Fr(\(3.8\))-236 1542 y(where)30 b Fp(p)73 1556 y Fk(\006)131 1542 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))31 b(are)g(1-p)s(erio)s(dic)e(functions)f(of) j Fp(x)p Fr(.)41 b(Let)1211 1816 y Fp(!)1268 1830 y Fk(\006)1327 1816 y Fr(\()p Fp(\020)7 b Fr(\))25 b(=)g Fn(\000)1646 1664 y Ff(R)1707 1690 y Fm(1)1689 1769 y(0)1761 1737 y Fp(p)1807 1751 y Fk(\007)1866 1737 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))2085 1693 y Fo(@)t(p)2162 1702 y Fc(\006)p 2085 1716 129 4 v 2112 1768 a Fo(@)t(\020)2224 1737 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))p Fp(dx)p 1646 1796 888 4 v 1668 1823 a Ff(R)1729 1849 y Fm(1)1711 1928 y(0)1783 1896 y Fp(p)1829 1910 y Fm(+)1888 1896 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))p Fp(p)2143 1910 y Fk(\000)2202 1896 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))p Fp(dx)2543 1816 y(:)-2804 b Fr(\(3.9\))-236 2080 y(On)29 b Fp(D)s Fr(,)i(\014x)e(a)i (con)m(tin)m(uous)f(branc)m(h)g(of)h(the)f(function)f Fp(k)1675 2047 y Fk(0)1699 2080 y Fr(\()p Fp(E)d Fn(\000)19 b Fp(W)13 b Fr(\()p Fp(\020)7 b Fr(\)\).)41 b(One)30 b(has)-236 2262 y Fu(Lemma)i(3.1.)47 b Fi(The)30 b(functions)h Fp(!)976 2276 y Fk(\006)1065 2262 y Fi(ar)-5 b(e)31 b(mer)-5 b(omorphic)33 b(on)e Fp(D)s Fi(.)41 b(The)30 b(set)h(of)f(p)-5 b(oles)32 b Fp(!)2745 2276 y Fm(+)2834 2262 y Fi(and)f Fp(!)3065 2276 y Fk(\000)3154 2262 y Fi(is)f(the)h(union)f(of)h(the)g (set)-236 2370 y(of)i(p)-5 b(oles)34 b(of)f Fp( )260 2384 y Fk(\006)351 2370 y Fi(and)h(the)f(set)g(of)f(p)-5 b(oints)35 b(wher)-5 b(e)33 b Fp(k)1501 2337 y Fk(0)1525 2370 y Fr(\()p Fp(E)26 b Fn(\000)20 b Fp(W)13 b Fr(\()p Fp(\020)7 b Fr(\)\))25 b(=)g(0)p Fi(.)-236 2552 y Fr(This)j(lemma)i (follo)m(ws)g(from)g(the)g(relation)g(\(2.3\))r(.)-236 2708 y(Fix)39 b Fp(\020)-24 2722 y Fm(0)56 2708 y Fn(2)h Fp(D)j Fr(so)d(that)g Fp(k)652 2675 y Fk(0)675 2708 y Fr(\()p Fp(E)33 b Fn(\000)26 b Fp(W)13 b Fr(\()p Fp(\020)1081 2722 y Fm(0)1120 2708 y Fr(\)\))41 b Fn(6)p Fr(=)g(0.)69 b(In)39 b(a)h(neigh)m(b)s(orho)s(o)s(d)d(of)j(this)f(p)s(oin)m(t,)i(c)m (ho)s(ose)g(an)e(analytic)g(branc)m(h)h(of)-236 2824 y Fp(q)s Fr(\()p Fp(\020)7 b Fr(\))25 b(=)46 2746 y Ff(p)p 137 2746 544 4 v 78 x Fp(k)187 2797 y Fk(0)210 2824 y Fr(\()p Fp(E)h Fn(\000)20 b Fp(W)13 b Fr(\()p Fp(\020)7 b Fr(\)\).)41 b(The)30 b(functions)1207 3050 y(\011)1278 3064 y Fk(\006)1336 3050 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))26 b(=)f Fp(q)s Fr(\()p Fp(\020)7 b Fr(\))p Fp(e)1870 2947 y Fb(R)1917 2968 y Fd(\020)1904 3025 y(\020)1932 3040 y Fg(0)1983 3001 y Fo(!)2027 3010 y Fc(\006)2079 3001 y Fm(\()p Fo(\020)e Fm(\))p Fo(d\020)2245 3050 y Fp( )2304 3064 y Fk(\006)2363 3050 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))-2808 b(\(3.10\))-236 3231 y(are)31 b(called)e(the)i Fi(c)-5 b(anonic)g(al)34 b(Blo)-5 b(ch)34 b(solutions)e Fr(normalized)d(at)i Fp(\020)1974 3245 y Fm(0)2013 3231 y Fr(.)41 b(They)30 b(are)g(analytic)g(in)f Fp(D)s Fr(.)-236 3386 y(The)h(W)-8 b(ronskian)29 b(of)i(the)g (canonical)f(Blo)s(c)m(h)g(solutions)f(is)g(giv)m(en)h(b)m(y)1031 3568 y Fp(w)r Fr(\(\011)1204 3582 y Fm(+)1264 3568 y Fp(;)15 b Fr(\011)1375 3582 y Fk(\000)1434 3568 y Fr(\))25 b(=)g Fp(q)1634 3530 y Fm(2)1674 3568 y Fr(\()p Fp(\020)1749 3582 y Fm(0)1788 3568 y Fr(\))p Fp(w)r Fr(\()p Fp( )1984 3582 y Fm(+)2045 3568 y Fr(\()p Fp(x;)15 b(\020)2212 3582 y Fm(0)2252 3568 y Fr(\))p Fp(;)g( )2386 3582 y Fk(\000)2446 3568 y Fr(\()p Fp(x;)g(\020)2613 3582 y Fm(0)2653 3568 y Fr(\)\))p Fp(:)-2984 b Fr(\(3.11\))-236 3762 y(As)30 b Fp(q)-58 3729 y Fm(2)-19 3762 y Fr(\()p Fp(\020)56 3776 y Fm(0)96 3762 y Fr(\))c(=)f Fp(k)303 3729 y Fk(0)326 3762 y Fr(\()p Fp(E)h Fn(\000)20 b Fp(W)13 b Fr(\()p Fp(\020)719 3776 y Fm(0)758 3762 y Fr(\)\))26 b Fn(6)p Fr(=)f(0,)31 b(the)f(canonical)g(Blo)s(c)m(h)h(solutions)d (are)j(linearly)d(indep)s(enden)m(t.)-236 3995 y(3.5.)53 b Fu(The)35 b(main)f(theorem)g(of)h(the)f(WKB)i(metho)s(d.)44 b Fr(One)30 b(has)-236 4178 y Fu(Theorem)k(3.1)h Fr(\([10)r(]\))p Fu(.)46 b Fi(Fix)37 b Fp(X)k(>)32 b Fr(0)38 b Fi(and)f Fp(E)i Fr(=)33 b Fp(E)1588 4192 y Fm(0)1660 4178 y Fn(2)g Fl(C)18 b Fi(.)60 b(L)-5 b(et)37 b Fp(K)44 b Fi(b)-5 b(e)37 b(a)g(b)-5 b(ounde)g(d)39 b(c)-5 b(anonic)g(al)38 b(domain)h(for)e(the)h(family)-236 4285 y(of)33 b(e)-5 b(quations)41 b Fr(\(3.2\))r Fi(,)32 b(and)i(let)f Fp(\024)g Fi(b)-5 b(e)32 b(the)i(br)-5 b(anch)34 b(of)f(the)g(c)-5 b(omplex)34 b(momentum)g(with)g(r)-5 b(esp)g(e)g(ct)34 b(to)f(which)h Fp(K)39 b Fi(is)33 b(c)-5 b(anonic)g(al.)-236 4393 y(F)e(or)31 b(su\016ciently)f(smal)5 b(l)31 b(p)-5 b(ositive)30 b Fp(")p Fi(,)h(ther)-5 b(e)31 b(exists)f(a)g(c)-5 b(onsistent)31 b(b)-5 b(asis)31 b Fr(\()p Fp(f)2313 4407 y Fk(\006)2372 4393 y Fr(\))f Fi(de\014ne)-5 b(d)31 b(for)f Fp(x)25 b Fn(2)g Fl(R)39 b Fi(and)30 b Fp(\020)i Fn(2)25 b Fp(K)36 b Fi(and)31 b(having)-236 4501 y(the)i(fol)5 b(lowing)33 b(pr)-5 b(op)g(erties:)-111 4646 y Fn(\017)42 b Fi(F)-7 b(or)33 b(any)h(\014xe)-5 b(d)33 b Fp(x)25 b Fn(2)g Fl(R)s Fi(,)38 b(the)33 b(functions)g Fp(f)1409 4660 y Fk(\006)1468 4646 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))33 b Fi(ar)-5 b(e)33 b(analytic)h(in)f Fp(\020)e Fn(2)25 b Fp(K)7 b Fi(.)-111 4754 y Fn(\017)42 b Fi(F)-7 b(or)33 b Fn(\000)p Fp(X)g Fn(\024)25 b Fp(x)g Fn(\024)g Fp(X)7 b Fi(,)33 b(and)g Fp(\020)f Fn(2)25 b Fp(K)7 b Fi(,)32 b(the)h(functions)g Fp(f)1805 4768 y Fk(\006)1863 4754 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))34 b Fi(have)f(the)g (asymptotic)h(r)-5 b(epr)g(esentations)909 5006 y Fp(f)954 5020 y Fk(\006)1012 5006 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))26 b(=)f Fp(e)1385 4951 y Fn(\006)1470 4916 y Fo(i)p 1466 4931 33 4 v 1466 4983 a(")1524 4878 y Ff(R)1584 4905 y Fo(\020)1567 4983 y(\020)1598 4992 y Fg(0)1651 4951 y Fp(\024d\020)1802 5006 y Fr(\(\011)1908 5020 y Fk(\006)1967 5006 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))21 b(+)f Fp(o)p Fr(\(1\)\))p Fp(;)109 b(")26 b Fn(!)f Fr(0)p Fp(:)-3106 b Fr(\(3.12\))-24 5188 y Fi(Her)-5 b(e,)45 b Fr(\011)305 5202 y Fk(\006)407 5188 y Fi(ar)-5 b(e)45 b(the)f(c)-5 b(anonic)g(al)45 b(Blo)-5 b(ch)45 b(solutions)f(c)-5 b(orr)g(esp)g(onding)47 b(to)d(the)g(domain)h Fp(K)7 b Fi(,)45 b(normalize)-5 b(d)46 b(at)e Fp(\020)3789 5202 y Fm(0)3872 5188 y Fi(and)-24 5295 y(indexe)-5 b(d)33 b(so)g(that)h Fp(\024)p Fr(\()p Fp(\020)7 b Fr(\))33 b Fi(b)-5 b(e)33 b(the)g(Blo)-5 b(ch)34 b(quasi-momentum)f(c)-5 b(orr)g(esp)g(onding)36 b(to)d(the)g(solution)h Fr(\011)3275 5309 y Fm(+)3334 5295 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))p Fi(.)-111 5403 y Fn(\017)42 b Fi(The)f(err)-5 b(or)42 b(estimates)g(in)e(\(3.12\))j(may)e(b)-5 b(e)41 b(di\013er)-5 b(entiate)g(d)42 b(onc)-5 b(e)42 b(in)e Fp(x)p Fi(.)66 b(Mor)-5 b(e)g(over,)44 b(they)d(ar)-5 b(e)42 b(uniform)g(in)e Fp(x)g Fn(2)-24 5511 y Fr([)p Fn(\000)p Fp(X)r(;)15 b(X)7 b Fr(])34 b Fi(and)f(lo)-5 b(c)g(al)5 b(ly)35 b(uniform)e(in)f Fp(\020)39 b Fi(in)33 b(the)g(interior)h(of)e Fp(K)7 b Fi(.)-236 5741 y Fr(One)30 b(can)g(easily)g(calculate)h(the)f(W)-8 b(ronskian)30 b(of)h(the)f(solutions)f Fp(f)2046 5755 y Fk(\006)2104 5741 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))1269 5922 y Fp(w)r Fr(\()p Fp(f)1416 5936 y Fm(+)1475 5922 y Fp(;)15 b(f)1560 5936 y Fk(\000)1619 5922 y Fr(\))26 b(=)f Fp(w)r Fr(\(\011)1949 5936 y Fm(+)2009 5922 y Fp(;)15 b Fr(\011)2120 5936 y Fk(\000)2179 5922 y Fr(\))20 b(+)g Fp(o)p Fr(\(1\))p Fp(:)-2745 b Fr(\(3.13\))-236 6110 y(By)32 b(\(3.11\))r(,)f(the)f(solutions)f Fp(f)777 6124 y Fk(\006)866 6110 y Fr(are)i(linearly)d(indep)s(enden)m(t)g(as)i Fp(k)2020 6078 y Fk(0)2044 6110 y Fr(\()p Fp(E)c Fn(\000)20 b Fp(W)13 b Fr(\()p Fp(\020)2437 6124 y Fm(0)2476 6110 y Fr(\)\))26 b Fn(6)p Fr(=)f(0.)1872 6210 y Fm(8)p eop %%Page: 9 9 9 8 bop -236 241 a Fr(3.5.1.)48 b Fi(Dep)-5 b(endenc)g(e)32 b(on)f(the)g(sp)-5 b(e)g(ctr)g(al)34 b(p)-5 b(ar)g(ameter)33 b(and)f(admissible)g(sub)-5 b(domains.)47 b Fr(T)-8 b(o)29 b(simplify)c(the)k(statemen)m(t)h(of)f(The-)-236 349 y(orem)k(3.1,)i(w)m(e)e(ha)m(v)m(e)h(not)f(considered)f(the)h(dep)s (endence)f(of)h(the)g(solutions)e(on)i(the)g(sp)s(ectral)f(parameter)i Fp(E)5 b Fr(.)48 b(Let)34 b Fp(K)39 b Fr(b)s(e)-236 457 y(a)29 b(canonical)f(domain.)39 b(W)-8 b(e)30 b(call)e Fp(K)35 b Fr(without)28 b(the)h Fp(\016)s Fr(-neigh)m(b)s(orho)s(o)s(d) e(of)i(its)f(b)s(oundary)f(the)h Fp(\016)s Fi(-admissible)33 b(sub-domain)d Fr(of)-236 565 y Fp(K)7 b Fr(.)40 b(One)30 b(has)-236 731 y Fu(Prop)s(osition)36 b(3.1)f Fr(\([9)q(]\))p Fu(.)47 b Fi(In)33 b(the)h(setting)f(of)h(The)-5 b(or)g(em)35 b(3.1,)f(the)g(solutions)g Fp(f)2549 745 y Fk(\006)2641 731 y Fi(ar)-5 b(e)34 b(analytic)h(in)e Fp(E)38 b Fi(in)c Fp(V)3524 745 y Fm(0)3563 731 y Fi(,)f(a)h(c)-5 b(omplex)-236 839 y(neighb)g(orho)g(o)g(d)40 b(of)c Fp(E)485 853 y Fm(0)561 839 y Fi(indep)-5 b(endent)38 b(of)f Fp(")p Fi(.)54 b(F)-7 b(or)37 b Fp(A)p Fi(,)h(an)f(admissible)g(sub-domain)h (of)e(the)h(c)-5 b(anonic)g(al)39 b(domain)f Fp(K)7 b Fi(,)37 b(ther)-5 b(e)-236 947 y(exists)29 b Fp(V)62 961 y Fo(A)144 947 y Fn(\032)c Fp(V)293 961 y Fm(0)333 947 y Fi(,)k(a)h(c)-5 b(omplex)30 b(neighb)-5 b(orho)g(o)g(d)33 b(of)c Fp(E)1513 961 y Fm(0)1582 947 y Fi(indep)-5 b(endent)30 b(of)g Fp(")f Fi(such)g(that)i(the)e(asymptotics)39 b Fr(\(3.12\))31 b Fi(ar)-5 b(e)30 b(uniform)-236 1055 y(in)i Fr(\()p Fp(\020)7 b(;)15 b(E)5 b(;)15 b(x)p Fr(\))27 b Fn(2)e Fp(A)20 b Fn(\002)g Fp(V)541 1069 y Fo(A)618 1055 y Fn(\002)g Fr([)p Fn(\000)p Fp(X)r(;)15 b(X)7 b Fr(])p Fi(.)43 b(It)33 b(is)g(onc)-5 b(e)33 b(di\013er)-5 b(entiable)33 b(in)g Fp(x)f Fi(without)i(lo)-5 b(osing)34 b(its)f(uniformity)g(pr)-5 b(op)g(erties.)-236 1221 y Fr(3.5.2.)48 b Fi(T)-7 b(erminolo)i(gy:)52 b(standar)-5 b(d)39 b(b)-5 b(ehavior.)47 b Fr(Fix)34 b Fp(E)j Fr(=)32 b Fp(E)1804 1235 y Fm(0)1876 1221 y Fn(2)f Fl(C)58 b Fr(and)34 b(let)h Fp(D)g Fn(\032)c Fl(C)58 b Fr(b)s(e)34 b(a)h(regular)f(domain.)52 b(Let)35 b Fp(\024)f Fr(b)s(e)g(a)-236 1329 y(branc)m(h)28 b(of)h(the)g(complex)g(momen)m(tum)g(con)m(tin)m (uous)f(in)g Fp(D)s Fr(,)h(and)g(let)f(\011)2217 1343 y Fk(\006)2305 1329 y Fr(b)s(e)g(the)h(canonical)g(Blo)s(c)m(h)g (solutions)e(asso)s(ciated)-236 1437 y(to)g Fp(D)i Fr(and)c(indexed)g (so)h(that)h Fp(\024)g Fr(b)s(e)e(the)i(quasi-momen)m(tum)e(for)h(\011) 2044 1451 y Fm(+)2103 1437 y Fr(.)39 b(W)-8 b(e)27 b(sa)m(y)g(that)g(a) g(consisten)m(t)f(solution)f Fp(f)35 b Fr(of)27 b(equation)-236 1600 y(\(3.2\))32 b(has)e Fi(standar)-5 b(d)36 b(b)-5 b(ehavior)31 b Fp(f)k Fn(\030)25 b Fp(e)1102 1523 y Fo(i)p 1098 1538 33 4 v 1098 1590 a(")1156 1485 y Ff(R)1216 1512 y Fo(\020)1271 1559 y Fp(\024)15 b(d\020)1452 1600 y Fr(\011)1523 1614 y Fm(+)1612 1600 y Fr(\(or)31 b Fp(f)j Fn(\030)25 b Fp(e)1976 1559 y Fk(\000)2046 1523 y Fo(i)p 2041 1538 V 2041 1590 a(")2099 1485 y Ff(R)2160 1512 y Fo(\020)2215 1559 y Fp(\024)15 b(d\020)2395 1600 y Fr(\011)2466 1614 y Fk(\000)2525 1600 y Fr(\))31 b(in)e Fp(D)k Fr(if)-111 1730 y Fn(\017)42 b Fr(There)36 b(exists)g Fp(V)555 1744 y Fm(0)594 1730 y Fr(,)j(a)e(neigh)m(b)s(orho)s(o)s(d)d (of)j Fp(E)1494 1744 y Fm(0)1570 1730 y Fr(and)f Fp(X)43 b(>)35 b Fr(0)i(suc)m(h)f(that)i Fp(f)45 b Fr(is)36 b(de\014ned)f(and)h (satis\014es)h(\(3.2\))h(for)e(an)m(y)-24 1838 y(\()p Fp(\020)7 b(;)15 b(E)5 b(;)15 b(x)p Fr(\))26 b Fn(2)f Fp(D)e Fn(\002)d Fp(V)651 1852 y Fm(0)711 1838 y Fn(\002)g Fr([)p Fn(\000)p Fp(X)r(;)15 b(X)7 b Fr(],)32 b(and)d Fp(f)40 b Fr(is)29 b(analytic)h(for)g Fp(\020)i Fn(2)25 b Fp(D)s Fr(.)-111 1946 y Fn(\017)42 b Fr(F)-8 b(or)31 b Fp(A)p Fr(,)g(an)f(admissible)d(sub-domain)i(of)i Fp(D)s Fr(,)f(there)h(exists)f Fp(V)2094 1960 y Fo(A)2176 1946 y Fn(\032)25 b Fp(V)2325 1960 y Fm(0)2365 1946 y Fr(,)30 b(a)h(neigh)m(b)s(orho)s(o)s(d)d(of)j Fp(E)3238 1960 y Fm(0)3308 1946 y Fr(suc)m(h)f(that,)h(for)f(an)m(y)-24 2106 y(\()p Fp(\020)7 b(;)15 b(E)5 b(;)15 b(x)p Fr(\))26 b Fn(2)f Fp(A)6 b Fn(\002)g Fp(V)613 2120 y Fo(A)677 2106 y Fn(\002)g Fr([)p Fn(\000)p Fp(X)r(;)15 b(X)7 b Fr(],)50 b Fp(f)35 b Fr(=)25 b Fp(e)1381 2030 y Fo(i)p 1377 2045 V 1377 2097 a(")1435 1992 y Ff(R)1495 2018 y Fo(\020)1550 2065 y Fp(\024)15 b(d\020)1731 2106 y Fr(\(\011)1837 2120 y Fm(+)1902 2106 y Fr(+)6 b Fp(o)15 b Fr(\(1\)\),)27 b(\(or)d Fp(f)34 b Fr(=)25 b Fp(e)2597 2065 y Fn(\000)2682 2030 y Fo(i)p 2678 2045 V 2678 2097 a(")2736 1992 y Ff(R)2797 2018 y Fo(\020)2852 2065 y Fp(\024)15 b(d\020)3032 2106 y Fr(\(\011)3138 2120 y Fk(\000)3204 2106 y Fr(+)6 b Fp(o)15 b Fr(\(1\)\))25 b(resp)s(ectiv)m(ely\))-24 2214 y(as)30 b Fp(")c Fn(!)f Fr(0.)-111 2322 y Fn(\017)42 b Fr(This)28 b(asymptotics)j(is)e(uniform) f(in)h(\()p Fp(\020)7 b(;)15 b(E)5 b(;)15 b(x)p Fr(\))27 b Fn(2)e Fp(A)20 b Fn(\002)g Fp(V)1891 2336 y Fo(A)1968 2322 y Fn(\002)g Fr([)p Fn(\000)p Fp(X)r(;)15 b(X)7 b Fr(].)-111 2430 y Fn(\017)42 b Fr(This)28 b(asymptotics)j(can)f(b)s(e)g (di\013eren)m(tiated)g(once)h(in)e Fp(x)h Fr(without)g(lo)s(osing)f (its)h(uniformit)m(y)e(prop)s(erties.)-236 2561 y(Giv)m(en)i(a)h (canonical)f(domain)f Fp(K)7 b Fr(,)30 b(Theorem)g(3.1)h(establishes)e (the)i(existence)g(of)f(t)m(w)m(o)i(consisten)m(t)f(solutions)d Fp(f)3611 2575 y Fk(\006)3700 2561 y Fr(analytic)-236 2669 y(in)h Fp(K)37 b Fr(and)30 b(ha)m(ving)g(standard)f(b)s(eha)m (vior)h(in)f Fp(K)7 b Fr(.)1391 2874 y(4.)50 b Fq(Canonical)34 b(domains)-24 3036 y Fr(Here,)41 b(follo)m(wing)36 b([9],)k(w)m(e)e (describ)s(e)e(a)i(simple)e(approac)m(h)h(to)i(\\constructing")f (canonical)f(domains.)61 b(Belo)m(w,)40 b(w)m(e)-236 3144 y(assume)c(that)g Fp(D)j Fr(is)c(a)i(regular)e(domain,)i(and)e (that)i Fp(\024)f Fr(is)f(a)i(branc)m(h)e(of)i(the)f(complex)g(momen)m (tum)g(analytic)f(in)g Fp(D)s Fr(.)58 b(A)-236 3252 y Fi(se)-5 b(gment)31 b Fr(of)f(a)h(curv)m(e)g(is)e(a)i(connected,)g (closed)g(subset)e(of)i(that)g(curv)m(e.)-236 3407 y Fu(1.)39 b Fr(Let)26 b Fp(\015)k Fn(\032)25 b Fp(D)k Fr(b)s(e)c(a)h(canonical)f(line.)38 b(Denote)27 b(its)e(ends)g(b)m(y)g Fp(\020)1902 3421 y Fm(1)1967 3407 y Fr(and)g Fp(\020)2179 3421 y Fm(2)2218 3407 y Fr(.)39 b(Let)27 b(a)f(domain)e Fp(K)32 b Fn(\032)25 b Fp(D)j Fr(b)s(e)d(a)i(canonical)e(domain)-236 3515 y(corresp)s(onding)i(to)i(the)g(triple)e Fp(\024)p Fr(,)j Fp(\020)999 3529 y Fm(1)1067 3515 y Fr(and)e Fp(\020)1282 3529 y Fm(2)1321 3515 y Fr(.)40 b(If)28 b Fp(\015)j Fn(2)25 b Fp(K)7 b Fr(,)29 b(then)f Fp(K)35 b Fr(is)28 b(called)g(a)h (canonical)g(domain)e Fi(enclosing)i Fp(\015)5 b Fr(.)41 b(As)28 b(an)m(y)-236 3623 y(line)g(close)j(enough)f(in)f(the)i Fp(C)801 3590 y Fm(1)840 3623 y Fr(-norm)f(to)h(a)g(canonical)f(line)e Fp(\015)36 b Fr(is)29 b(canonical,)h(one)h(has)-236 3789 y Fu(Lemma)h(4.1)k Fr(\([9)q(]\))p Fu(.)46 b Fi(One)32 b(c)-5 b(an)34 b(always)g(c)-5 b(onstruct)34 b(a)f(c)-5 b(anonic)g(al)34 b(domain)h(enclosing)d(any)i(given)e(c)-5 b(anonic)g(al)34 b(curve.)-236 3955 y Fr(Suc)m(h)29 b(canonical)h (domains)g(are)g(called)g Fi(lo)-5 b(c)g(al)p Fr(.)-236 4111 y Fu(2.)76 b Fr(Let)32 b Fp(\015)h Fn(\032)27 b Fp(D)34 b Fr(b)s(e)d(a)h(smo)s(oth)f(curv)m(e.)45 b(W)-8 b(e)33 b(sa)m(y)f(that)h Fp(\015)j Fr(is)31 b(a)h(line)e Fi(of)k(Stokes)g(typ)-5 b(e)33 b Fr(with)d(resp)s(ect)i(to)g Fp(\024)g Fr(if,)f(along)h Fp(\015)5 b Fr(,)32 b(one)-236 4219 y(has)602 4416 y(either)e(Im)999 4288 y Ff(\022)1066 4292 y(Z)1157 4318 y Fo(\020)1212 4416 y Fp(\024d\020)1358 4288 y Ff(\023)1450 4416 y Fr(=)25 b(Const)91 b(or)g(Im)2181 4288 y Ff(\022)2248 4292 y(Z)2338 4318 y Fo(\020)2378 4416 y Fr(\()p Fp(\024)21 b Fn(\000)f Fp(\031)s Fr(\))p Fp(d\020)2761 4288 y Ff(\023)2853 4416 y Fr(=)25 b(Const)p Fp(:)-236 4619 y Fr(Let)32 b Fp(\015)i Fn(\032)28 b Fp(D)35 b Fr(b)s(e)c(a)i(v)m(ertical)f(curv)m(e.)47 b(W)-8 b(e)33 b(call)e Fp(\015)38 b Fi(pr)-5 b(e-c)g(anonic)g(al)34 b Fr(if)e(it)f(consists)h(of)g(a)h(\014nite)e(union)f(of)j(b)s(ounded)d (segmen)m(ts)-236 4727 y(of)g(canonical)g(lines)f(and/or)h(lines)f(of)h (Stok)m(es)i(t)m(yp)s(e.)-236 4882 y Fu(3.)71 b Fr(W)-8 b(e)31 b(construct)g(\\global")g(canonical)f(domains)f(b)m(y)h(means)g (of)-236 5048 y Fu(Prop)s(osition)36 b(4.1)f Fr(\([9)q(]\))p Fu(.)47 b Fi(L)-5 b(et)46 b Fp(\015)51 b Fi(b)-5 b(e)45 b(a)i(c)-5 b(anonic)g(al)48 b(line)d(with)i(r)-5 b(esp)g(e)g(ct)48 b(to)e Fp(\024)p Fi(.)82 b(Assume)46 b(that)h Fp(K)56 b Fn(\032)49 b Fp(D)f Fi(is)e(a)h(simply)-236 5156 y(c)-5 b(onne)g(cte)g(d)32 b(domain)g(c)-5 b(ontaining)32 b Fp(\015)j Fi(\(without)d(its)f(ends\).)42 b(The)30 b(domain)j Fp(K)k Fi(is)30 b(a)h(c)-5 b(anonic)g(al)32 b(domain)h(enclosing)d Fp(\015)36 b Fi(if)30 b(it)h(is)-236 5264 y(the)f(union)f(of)h(pr)-5 b(e-c)g(anonic)g(al)32 b(lines)d(obtaine)-5 b(d)31 b(fr)-5 b(om)31 b Fp(\015)j Fi(by)c(r)-5 b(eplacing)30 b(some)g(of)g Fp(\015)5 b Fi('s)29 b(internal)i(se)-5 b(gments)30 b(by)f(pr)-5 b(e-c)g(anonic)g(al)-236 5372 y(lines.)1275 5578 y Fr(5.)51 b Fq(The)34 b(continua)-6 b(tion)33 b(tools)-24 5739 y Fr(Giv)m(en)h(a)g(canonical)f(line,)f(Lemma)i(4.1)h(giv)m(e)f(us)e(a) i(lo)s(cal)f(canonical)g(domain)f Fp(K)7 b Fr(.)50 b(So,)34 b(b)m(y)g(Theorem)f(3.1,)i(w)m(e)f(can)-236 5847 y(construct)d (solutions)d Fp(f)588 5861 y Fk(\006)677 5847 y Fr(ha)m(ving)i (standard)g(b)s(eha)m(vior)f(in)g(this)g(domain.)-236 6003 y(First)i(note)i(that)389 6017 y Fk(\006)480 6003 y Fr(can)f(b)s(e)f(analytically)g(con)m(tin)m(ued)g(outside)g(of)h Fp(K)7 b Fr(.)45 b(Indeed,)32 b(let)g Fp(S)5 b Fr(\()p Fp(Y)2859 6017 y Fm(1)2898 6003 y Fp(;)15 b(Y)2991 6017 y Fm(2)3031 6003 y Fr(\))28 b(=)g Fn(f)p Fp(Y)3291 6017 y Fm(1)3358 6003 y Fp(<)f Fr(Im)15 b Fp(\020)34 b(<)28 b(Y)3806 6017 y Fm(2)3845 6003 y Fn(g)k Fr(b)s(e)-236 6110 y(the)k(smallest)g(strip)e(con)m(taining)i(the)h(domain)e Fp(K)7 b Fr(.)58 b(Fix)36 b(0)f Fp(<)g(\016)k(<)c Fr(\()p Fp(Y)2235 6124 y Fm(2)2298 6110 y Fn(\000)24 b Fp(Y)2446 6124 y Fm(1)2486 6110 y Fr(\))p Fp(=)p Fr(2.)59 b(Consider)35 b(the)h(domain)f Fp(K)3649 6125 y Fo(\016)3722 6110 y Fr(=)g Fn(f)p Fp(\020)42 b Fn(2)1872 6210 y Fm(9)p eop %%Page: 10 10 10 9 bop -236 241 a Fp(K)7 b Fr(;)60 b Fp(Y)-14 255 y Fm(1)51 241 y Fr(+)25 b Fp(\016)42 b(<)c Fr(Im)15 b Fp(\020)44 b(<)38 b(Y)708 255 y Fm(2)773 241 y Fn(\000)25 b Fp(\016)s Fn(g)p Fr(,)41 b(and)d(its)f(horizon)m(tal)h(width,)h Fp(w)h Fr(=)140 b(sup)2328 325 y Fo(\020)5 b(;\020)2420 301 y Fc(0)2441 325 y Fk(2)p Fo(K)2548 337 y Fd(\016)2284 398 y Fm(Im)11 b Fo(\020)5 b Fm(=Im)11 b Fo(\020)2603 374 y Fc(0)2640 241 y Fn(j)p Fp(\020)32 b Fn(\000)25 b Fp(\020)2880 203 y Fk(0)2903 241 y Fn(j)p Fr(.)64 b(Clearly)-8 b(,)39 b Fp(w)i(>)d Fr(0.)64 b(Assume)-236 491 y(that)32 b Fp(")27 b(<)g(w)r Fr(.)44 b(Then,)31 b(the)g(functions)f Fp(f)1124 505 y Fk(\006)1183 491 y Fr(,)i(b)s(eing)e(de\014ned)g(for)h (all)f Fp(x)d Fn(2)g Fl(R)39 b Fr(and)31 b(analytic)g(in)f Fp(\020)k Fn(2)26 b Fp(K)7 b Fr(,)32 b(can)f(b)s(e)g(analytically)-236 599 y(con)m(tin)m(ued)f(in)f(the)i(whole)e(strip)g Fn(f)p Fp(Y)1009 613 y Fm(1)1069 599 y Fr(+)20 b Fp(\016)29 b(<)c Fr(Im)15 b Fp(\020)31 b(<)25 b(Y)1669 613 y Fm(2)1729 599 y Fn(\000)20 b Fp(\016)s Fn(g)31 b Fr(using)e(the)i(consistency)f (condition)f(\(3.3\))r(.)-236 754 y(T)-8 b(o)33 b(study)e(the)i (asymptotic)g(b)s(eha)m(vior)f(of)g Fp(f)1305 768 y Fk(\006)1396 754 y Fr(outside)g(of)h(the)f(canonical)g(domain)g Fp(K)7 b Fr(,)33 b(w)m(e)g(dev)m(elop)f(t)m(w)m(o)i(general)f(to)s(ols,)-236 862 y(the)d(Rectangle)i(Lemma)e(and)g(the)h(Adjacen)m(t)g(Canonical)e (Domain)h(Principle.)-236 1017 y(In)f(the)i(sequel,)f(a)h(set)g(is)e (called)h Fi(c)-5 b(onstant)32 b Fr(if)d(it)h(is)f(indep)s(enden)m(t)g (of)h Fp(")p Fr(.)-236 1198 y(5.1.)53 b Fu(Asymptotics)d(of)g (increasing)g(solutions.)d Fr(Here,)g(w)m(e)c(roughly)f(pro)m(v)m(e)i (that)g(the)f(standard)g(b)s(eha)m(vior)f(of)h(a)-236 1306 y(solution)30 b(sta)m(ys)j(v)-5 b(alid)30 b(along)i(a)h(horizon)m (tal)e(line)f(as)i(long)g(as)g(the)g(leading)f(term)h(of)g(the)g (asymptotics)g(is)f(gro)m(wing)h(along)-236 1414 y(that)f(line.)-236 1569 y(Fix)37 b Fp(\021)-21 1583 y Fo(m)84 1569 y Fp(<)h(\021)238 1583 y Fo(M)318 1569 y Fr(.)64 b(De\014ne)38 b Fp(S)43 b Fr(=)38 b Fn(f)p Fp(\020)45 b Fn(2)38 b Fl(C)62 b Fr(:)77 b Fp(\021)1384 1583 y Fo(m)1489 1569 y Fn(\024)38 b Fr(Im)15 b Fp(\020)44 b Fn(\024)38 b Fp(\021)1960 1583 y Fo(M)2039 1569 y Fn(g)p Fr(.)65 b(Let)39 b Fp(\015)2392 1583 y Fm(1)2469 1569 y Fr(and)f Fp(\015)2701 1583 y Fm(2)2778 1569 y Fr(b)s(e)g(t)m(w)m(o)i(v)m(ertical)e(lines)e(suc)m(h)i(that)-236 1677 y Fp(\015)-189 1691 y Fm(1)-128 1677 y Fn(\\)21 b Fp(\015)1 1691 y Fm(2)69 1677 y Fr(=)29 b Fn(;)p Fr(.)47 b(Assume)32 b(that)h(b)s(oth)f(lines)f(in)m(tersect)i(the)f(strip)f Fp(S)38 b Fr(at)33 b(the)f(lines)f(Im)15 b Fp(\020)35 b Fr(=)29 b Fp(\021)2912 1691 y Fo(m)3011 1677 y Fr(and)j(Im)14 b Fp(\020)35 b Fr(=)29 b Fp(\021)3533 1691 y Fo(M)3612 1677 y Fr(,)k(and)f(that)-236 1785 y Fp(\015)-189 1799 y Fm(1)-119 1785 y Fr(is)d(situated)h(to)h(the)g(left)f(of)g Fp(\015)898 1799 y Fm(2)938 1785 y Fr(.)-236 1893 y(Consider)e(the)j (compact)h(set)e Fp(R)h Fr(b)s(ounded)e(b)m(y)h Fp(\015)1450 1907 y Fm(1)1489 1893 y Fr(,)h Fp(\015)1592 1907 y Fm(2)1662 1893 y Fr(and)f(the)g(b)s(oundaries)e(of)j Fp(S)5 b Fr(.)40 b(Let)31 b Fp(D)s Fr(=)p Fp(R)21 b Fn(n)f Fr(\()p Fp(\015)3237 1907 y Fm(1)3297 1893 y Fn([)g Fp(\015)3425 1907 y Fm(2)3465 1893 y Fr(\).)-236 2048 y(One)30 b(has)-236 2215 y Fu(Lemma)i(5.1)k Fr(\(The)30 b(Rectangle)i(Lemma\))p Fu(.)46 b Fi(Fix)24 b(an)h Fp(E)30 b Fr(=)25 b Fp(E)1885 2229 y Fm(0)1925 2215 y Fi(.)38 b(Assume)24 b(that)i(the)e(\\r)-5 b(e)g(ctangle")25 b Fp(R)f Fi(is)g(in)g(c)-5 b(onstant)26 b(r)-5 b(e)g(gular)-236 2323 y(domain.)43 b(L)-5 b(et)33 b Fp(f)42 b Fi(b)-5 b(e)32 b(a)h(c)-5 b(onsistent)34 b(solution)g(of)52 b Fr(\(3.2\))r Fi(.)42 b(Then,)33 b(for)g(su\016ciently)g(smal)5 b(l)33 b Fp(")p Fi(,)g(one)g(has)-124 2495 y Fu(1:)42 b Fi(If)32 b Fr(Im)15 b Fp(\024)26 b(<)f Fr(0)32 b Fi(in)h Fp(D)i Fi(and)f(if,)e(in)g(a)h(neighb)-5 b(orho)g(o)g(d)36 b(of)d Fp(\015)1868 2509 y Fm(1)1907 2495 y Fi(,)g Fp(f)41 b Fi(has)34 b(standar)-5 b(d)35 b(b)-5 b(ehavior)34 b Fp(f)h Fn(\030)25 b Fp(e)3177 2418 y Fd(i)p 3174 2430 30 3 v 3174 2471 a(")3225 2392 y Fb(R)3272 2412 y Fd(\020)3258 2470 y(\020)3286 2485 y Fg(0)3337 2445 y Fo(\024d\020)3454 2495 y Fr(\011)3525 2509 y Fm(+)3584 2495 y Fi(,)32 b(then,)h(this)-24 2603 y(asymptotics)h(r)-5 b(emains)34 b(valid)g(in)e(a)h(c)-5 b(onstant)35 b(domain)f(c)-5 b(ontaining)34 b(the)f(\\r)-5 b(e)g(ctangle")33 b Fp(R)q Fi(.)-124 2745 y Fu(2:)42 b Fi(If)32 b Fr(Im)15 b Fp(\024)26 b(>)f Fr(0)32 b Fi(in)h Fp(D)i Fi(and)f(if,)e(in)g(a)h(neighb)-5 b(orho)g(o)g(d)36 b(of)d Fp(\015)1868 2759 y Fm(2)1907 2745 y Fi(,)g Fp(f)41 b Fi(has)34 b(standar)-5 b(d)35 b(b)-5 b(ehavior)34 b Fp(f)h Fn(\030)25 b Fp(e)3177 2669 y Fd(i)p 3174 2681 V 3174 2722 a(")3225 2643 y Fb(R)3272 2663 y Fd(\020)3258 2721 y(\020)3286 2736 y Fg(0)3337 2696 y Fo(\024d\020)3454 2745 y Fr(\011)3525 2759 y Fm(+)3584 2745 y Fi(,)32 b(then,)h(this)-24 2853 y(asymptotics)h(r)-5 b(emains)34 b(valid)g(in)e(a)h(c)-5 b(onstant)35 b(domain)f(c)-5 b(ontaining)34 b(the)f(\\r)-5 b(e)g(ctangle")33 b Fp(R)q Fi(.)-236 3020 y Fr(A)c(w)m(eak)m(er)h (result)e(can)h(b)s(e)f(found)f(in)h([9)q(].)40 b(Also,)29 b(a)g(similar)d(result)i(for)h(some)g(di\013erence)f(equations)h(w)m (as)g(obtained)f(in)g([3].)-236 3188 y Fu(Remark)34 b(5.1.)77 b(1.)42 b Fr(The)30 b(v)m(ertical)h(b)s(oundaries)d(of)j Fp(R)h Fr(can)f(b)s(e)f(lines)f(where)h(Im)15 b Fp(\024)26 b Fr(=)g(0.)42 b(So,)31 b(Lemma)g(5.1)h(sa)m(ys)f(that)g(the)-236 3296 y(asymptotic)20 b(of)h(a)g(solution)e(sta)m(ys)i(v)-5 b(alid)19 b(along)h(horizon)m(tal)g(lines)e(where)i(it)g(gro)m(ws)h (and,)h(actually)-8 b(,)23 b(ev)m(en)e(somewhat)f(b)s(ey)m(ond)-236 3404 y(the)31 b(p)s(oin)m(t)e(where)i(it)f(stops)g(gro)m(wing.)72 b Fu(2.)42 b Fr(If)30 b(a)h(solution)f(has)g(standard)g(b)s(eha)m(vior) g(in)f(a)i(domain)f Fp(D)j Fr(for)e Fp(E)g Fr(=)25 b Fp(E)3718 3418 y Fm(0)3758 3404 y Fr(,)31 b(then,)-236 3511 y(this)g(asymptotics)h(sta)m(ys)i(v)-5 b(alid)30 b(in)h(an)h(admissible)e(sub)s(domain)f(of)k Fp(D)i Fr(uniformly)29 b(for)j Fp(E)38 b Fr(in)31 b(a)h(constan)m(t)i(neigh)m(b)s(orho)s(o)s (d)-236 3619 y(of)f Fp(E)-63 3633 y Fm(0)43 3619 y Fu(3.)49 b Fr(The)33 b(pro)s(of)f(giv)m(en)h(b)s(elo)m(w)g(sho)m(ws)g(that)h (Lemma)f(5.1)h(sta)m(ys)g(v)-5 b(alid)32 b(if)g(the)h(curv)m(es)h Fp(\015)3072 3633 y Fm(1)p Fo(;)p Fm(2)3199 3619 y Fr(con)m(tin)m (uously)e(dep)s(end)-236 3727 y(on)e Fp(E)36 b Fr(in)29 b(a)i(neigh)m(b)s(orho)s(o)s(d)c(of)k Fp(E)916 3741 y Fm(0)956 3727 y Fr(.)-236 3942 y Fu(Pro)s(of.)41 b Fr(W)-8 b(e)32 b(pro)m(v)m(e)f(only)e(the)i(\014rst)f(statemen)m(t;)i(the)f (second)f(one)h(is)e(pro)m(v)m(ed)i(in)e(an)h(analogous)h(w)m(a)m(y)-8 b(.)-236 4050 y(As)31 b(a)g(starter)g(let)g(us)f(sk)m(etc)m(h)j(the)e (pro)s(of.)41 b(First,)31 b(w)m(e)g(pro)m(v)m(e)h(that)g(there)f (exists)f(a)i(constan)m(t)g(\014nite)e(set)h(of)g(constan)m(t)h(op)s (en)-236 4164 y(disks)26 b(\()p Fp(D)97 4178 y Fo(j)134 4164 y Fr(\))169 4178 y Fo(j)234 4164 y Fr(co)m(v)m(ering)j Fp(R)f Fr(suc)m(h)g(that,)h(in)d(eac)m(h)k(of)e(these)g(disks,)f(there) h(exists)f(a)i(basis)d(of)i(solutions)f Fp(f)3300 4120 y Fo(j)3290 4187 y Fk(\006)3376 4164 y Fr(ha)m(ving)h(standard)-236 4306 y(b)s(eha)m(vior,)h Fp(f)215 4262 y Fo(j)205 4329 y Fk(\006)289 4306 y Fn(\030)c Fp(e)427 4270 y Fk(\006)496 4243 y Fd(i)p 492 4255 V 492 4296 a(")543 4217 y Fb(R)590 4237 y Fd(\020)638 4270 y Fo(\024d\020)755 4306 y Fr(\011)826 4320 y Fk(\006)884 4306 y Fr(.)41 b(Next,)31 b(w)m(e)g(express)f Fp(f)40 b Fr(in)29 b(terms)h(of)h(these)f(basis)g(of)g(solutions)956 4480 y Fp(f)10 b Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))25 b(=)g Fp(a)1389 4494 y Fo(j)1426 4480 y Fr(\()p Fp(\020)7 b Fr(\))p Fp(f)1598 4436 y Fo(j)1588 4503 y Fm(+)1647 4480 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))20 b(+)g Fp(b)2006 4494 y Fo(j)2043 4480 y Fr(\()p Fp(\020)7 b Fr(\))p Fp(f)2215 4436 y Fo(j)2205 4503 y Fk(\000)2264 4480 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))p Fp(;)15 b(\020)32 b Fn(2)25 b Fp(D)2746 4494 y Fo(j)2783 4480 y Fp(;)-236 4635 y Fr(and)j(pro)m(v)m(e)h(that,)h (under)d(the)i(assumptions)e(of)i(Lemma)g(5.1,)h(the)f(co)s(e\016cien)m (ts)g Fp(b)2576 4649 y Fo(j)2641 4635 y Fr(are)g(small)e(enough.)40 b(T)-8 b(o)29 b(complete)g(the)-236 4743 y(pro)s(of)g(of)i(Lemma)f (5.1,)i(w)m(e)f(compute)g(the)f(asymptotics)h(of)f Fp(a)1915 4757 y Fo(j)1952 4743 y Fr(.)-236 4898 y Fi(First)j(step.)41 b Fr(The)30 b(\014rst)g(step)g(is)g(a)g(consequence)h(of)g(the)f(follo) m(wing)f(general)i(observ)-5 b(ation)-236 5065 y Fu(Lemma)32 b(5.2.)47 b Fi(L)-5 b(et)31 b Fp(\020)538 5079 y Fk(\003)608 5065 y Fi(b)-5 b(e)30 b(a)h(r)-5 b(e)g(gular)33 b(p)-5 b(oint.)42 b(Then,)32 b(ther)-5 b(e)31 b(exists)g(a)g(br)-5 b(anch)33 b(of)e(the)g(c)-5 b(omplex)32 b(momentum,)g(say)g Fp(\024)3773 5079 y Fk(\003)3813 5065 y Fi(,)f(and)-236 5173 y(a)i(domain)h Fp(K)245 5187 y Fk(\003)317 5173 y Fi(c)-5 b(anonic)g(al)35 b(with)e(r)-5 b(esp)g(e)g(ct)34 b(to)g Fp(\024)1372 5187 y Fk(\003)1444 5173 y Fi(that)g(c)-5 b(ontains)34 b Fp(\020)2030 5187 y Fk(\003)2069 5173 y Fi(.)-236 5388 y Fu(Pro)s(of.)63 b Fr(It)37 b(su\016ces)h(to)g(sho)m (w)f(that)i(there)e(exists)g(a)h(complex)g(n)m(um)m(b)s(er)e Fp(z)41 b Fn(2)c Fl(C)2609 5402 y Fm(+)2711 5388 y Fr(and)g(a)h(branc)m (h)f Fp(\024)3338 5402 y Fk(\003)3416 5388 y Fr(of)g(the)h(complex)-236 5496 y(momen)m(tum)30 b(suc)m(h)g(that)1006 5651 y(Im)15 b(\()p Fp(\024)1217 5665 y Fk(\003)1257 5651 y Fr(\()p Fp(\020)1332 5665 y Fk(\003)1372 5651 y Fr(\))p Fp(z)t Fr(\))26 b Fp(>)f Fr(0)30 b(and)g(Im)15 b(\(\()p Fp(\024)2108 5665 y Fk(\003)2148 5651 y Fr(\()p Fp(\020)2223 5665 y Fk(\003)2263 5651 y Fr(\))21 b Fn(\000)f Fp(\031)s Fr(\))p Fp(z)t Fr(\))26 b Fp(<)f Fr(0)p Fp(:)-3009 b Fr(\(5.1\))-236 5811 y(Let)31 b Fp(\024)g Fr(b)s(e)f(one)h(of)g(the)g (branc)m(hes)f(of)h(the)g(complex)g(momen)m(tum)f(analytic)h(in)e Fp(V)2540 5825 y Fk(\003)2579 5811 y Fr(,)i(a)g(neigh)m(b)s(orho)s(o)s (d)e(of)i Fp(\020)3427 5825 y Fk(\003)3466 5811 y Fr(.)42 b(By)32 b(\(3.6\))r(,)f(all)-236 5919 y(the)f(branc)m(hes)g(of)h(the)f (complex)h(momen)m(tum)f(analytic)g(in)f Fp(V)1896 5933 y Fk(\003)1966 5919 y Fr(are)h(describ)s(ed)f(b)m(y)1534 6074 y Fp(\024)1586 6036 y Fk(\006)1586 6096 y Fo(m)1678 6074 y Fr(=)c Fn(\006)p Fp(\024)c Fr(+)e(2)p Fp(im\031)s(:)-2480 b Fr(\(5.2\))1854 6210 y Fm(10)p eop %%Page: 11 11 11 10 bop -236 241 a Fr(W)-8 b(e)31 b(pic)m(k)f(a)h(branc)m(h,)f(sa)m (y)h Fp(\024)723 255 y Fk(\003)763 241 y Fr(,)f(suc)m(h)h(that)1524 425 y(0)25 b Fn(\024)g Fr(Re)16 b Fp(\024)1865 439 y Fk(\003)1905 425 y Fr(\()p Fp(\020)1980 439 y Fk(\003)2019 425 y Fr(\))26 b Fn(\024)f Fp(\031)s(:)-236 610 y Fr(There)30 b(are)g(three)h(p)s(ossible)d(cases:)-137 756 y(1.)43 b(if)29 b(0)d Fp(<)f Fr(Re)15 b Fp(\024)400 770 y Fk(\003)440 756 y Fr(\()p Fp(\020)515 770 y Fk(\003)554 756 y Fr(\))26 b Fp(<)f(\031)s Fr(,)31 b(then)f Fp(z)f Fr(=)c Fp(i)31 b Fr(and)f Fp(\024)1487 770 y Fk(\003)1557 756 y Fr(satisfy)h(\(5.1\))r (.)-137 864 y(2.)43 b(if)26 b(Re)16 b Fp(\024)231 878 y Fk(\003)270 864 y Fr(\()p Fp(\020)345 878 y Fk(\003)385 864 y Fr(\))26 b(=)f(0,)j(then)g(Im)p Fp(\024)1006 878 y Fk(\003)1045 864 y Fr(\()p Fp(\020)1120 878 y Fk(\003)1160 864 y Fr(\))d Fn(6)p Fr(=)g(0)j(\(as)g Fp(\020)1573 878 y Fk(\003)1640 864 y Fr(is)f(not)g(a)h(branc)m(h)f(p)s(oin)m(t\).)40 b(So,)28 b(c)m(hanging)f(the)h(branc)m(h)f(if)g(necessary)-8 b(,)-24 972 y(w)m(e)34 b(can)h(assume)e(that)i(Im)15 b Fp(\024)978 986 y Fk(\003)1018 972 y Fr(\()p Fp(\020)1093 986 y Fk(\003)1132 972 y Fr(\))32 b Fp(>)f Fr(0)k(and)e(that)i(Re)15 b Fp(\024)1936 986 y Fk(\003)1976 972 y Fr(\()p Fp(\020)2051 986 y Fk(\003)2091 972 y Fr(\))31 b(=)h(0.)52 b(In)33 b(this)g(case,)j(w)m(e)f(set)g Fp(z)g Fr(=)c Fp(ie)3438 939 y Fk(\000)p Fo(i\017)3551 972 y Fr(.)51 b(F)-8 b(or)35 b Fp(")d(>)f Fr(0)-24 1080 y(su\016cien)m(tly)e(small,)g Fp(z)34 b Fr(and)c Fp(\024)999 1094 y Fk(\003)1069 1080 y Fr(satisfy)h(\(5.1\))r(.)-137 1188 y(3.)43 b(if)30 b Fp(\024)112 1202 y Fk(\003)152 1188 y Fr(\()p Fp(\020)227 1202 y Fk(\003)267 1188 y Fr(\))e(=)f Fp(\031)s Fr(,)33 b(w)m(e)f(can)g(assume)g(that)g(Im)15 b Fp(\024)1534 1202 y Fk(\003)1574 1188 y Fr(\()p Fp(\020)1649 1202 y Fk(\003)1688 1188 y Fr(\))29 b Fp(>)e Fr(0.)46 b(In)31 b(this)f(case,)k(w)m(e)e(c)m(hose)h Fp(z)f Fr(=)c Fp(ie)3102 1155 y Fo(i\017)3159 1188 y Fr(,)k(and,)g(for)g Fp(\017)27 b(>)h Fr(0)k(b)s(eing)-24 1296 y(small)d(enough,)h(\(5.1\))i(is)d (satis\014ed.)-236 1442 y(This)f(completes)j(the)g(pro)s(of)e(of)i (Lemma)f(5.2.)p 3950 1442 4 62 v 3954 1384 55 4 v 3954 1442 V 4008 1442 4 62 v -236 1597 a(Using)f(Lemma)h(5.2,)h(w)m(e)f (construct)g(\014nitely)e(man)m(y)i(op)s(en)f(disks)f(\()p Fp(D)2152 1611 y Fo(j)2189 1597 y Fr(\))2224 1611 y Fo(j)2290 1597 y Fr(co)m(v)m(ering)j(the)f(rectangle)g Fp(R)h Fr(suc)m(h)e(that,) i(they)f(all)-236 1705 y(b)s(e)g(regular,)f(and,)h(on)h(eac)m(h)g (disk,)e(b)s(et)m(w)m(een)i Fp(\015)1367 1719 y Fm(1)1437 1705 y Fr(and)f Fp(\015)1661 1719 y Fm(2)1700 1705 y Fr(,)h(one)g(ha)m(v)m(e)g(Im)15 b Fp(\024)26 b(<)f Fr(0.)-236 1860 y Fi(The)39 b(se)-5 b(c)g(ond)40 b(step.)62 b Fr(Let)38 b Fp(\020)705 1874 y Fm(1)781 1860 y Fn(2)f Fp(R)25 b Fn(\\)g Fp(\015)1106 1874 y Fm(1)1145 1860 y Fr(.)62 b(Consider)35 b(the)i(compact)i(in)m(terv)-5 b(al)37 b Fp(I)43 b Fr(=)37 b([)p Fp(\020)2744 1874 y Fm(1)2783 1860 y Fp(;)15 b(\020)2863 1874 y Fm(2)2903 1860 y Fr(])37 b(=)g Fp(R)25 b Fn(\\)g(f)p Fr(Im)15 b Fp(\020)43 b Fr(=)37 b(Im)14 b Fp(\020)3776 1874 y Fm(1)3816 1860 y Fn(g)p Fr(.)62 b(It)-236 1968 y(su\016ces)30 b(to)h(pro)m(v)m(e)g(that)g Fp(f)40 b Fr(has)30 b(standard)f(b)s(eha)m(vior)h(in)f(a)i(constan)m(t) g(neigh)m(b)s(orho)s(o)s(d)d(of)j(this)e(in)m(terv)-5 b(al.)-236 2076 y(Assume)30 b(that)h Fp(I)37 b Fr(is)29 b(co)m(v)m(ered)j(b)m(y)f(constan)m(t)g(disks)e Fp(D)1586 2090 y Fo(j)1623 2076 y Fr(,)h Fp(j)h Fr(=)25 b(1)p Fp(;)15 b(:)g(:)g(:)i(;)e(J)40 b Fr(suc)m(h)30 b(that)-137 2222 y(1.)43 b Fp(\020)16 2236 y Fm(1)80 2222 y Fn(2)25 b Fp(D)241 2236 y Fm(0)311 2222 y Fr(and)30 b Fp(\020)528 2236 y Fm(2)592 2222 y Fn(2)25 b Fp(D)753 2236 y Fo(J)802 2222 y Fr(,)-137 2330 y(2.)43 b Fp(f)c Fr(has)30 b(standard)g(b)s(eha)m (vior)f(in)g Fp(D)1154 2344 y Fm(0)1194 2330 y Fr(,)-137 2438 y(3.)43 b(for)30 b Fp(j)h Fr(=)25 b(1)p Fp(;)15 b(:)g(:)g(:)i(;)e(J)9 b Fr(,)31 b Fp(D)716 2452 y Fo(j)773 2438 y Fn(\\)20 b Fp(D)929 2452 y Fo(j)t Fm(+1)1081 2438 y Fn(6)p Fr(=)25 b Fn(;)p Fr(,)-137 2546 y(4.)43 b(the)30 b(disks)f Fp(D)433 2560 y Fo(j)500 2546 y Fr(with)g(1)d Fp(<)f(j)31 b(<)25 b(J)39 b Fr(are)31 b(strictly)e(b)s(et)m(w)m(een)i Fp(\015)1988 2560 y Fm(1)2058 2546 y Fr(and)f Fp(\015)2282 2560 y Fm(2)2321 2546 y Fr(.)-236 2697 y(Let)i Fp(I)-32 2712 y Fo(\016)38 2697 y Fr(b)s(e)g(a)g(constan)m(t)i Fp(\016)s Fr(-neigh)m(b)s(orho)s(o)s(d)d(of)h(the)g(in)m(terv)-5 b(al)32 b Fp(I)39 b Fr(con)m(tained)32 b(in)f Fn([)2512 2664 y Fo(J)2512 2723 y(j)t Fm(=0)2638 2697 y Fp(D)2713 2711 y Fo(j)2750 2697 y Fr(.)46 b(Supp)s(ose)30 b(that)j(w)m(e)g(ha)m (v)m(e)g(already)-236 2849 y(pro)m(v)m(ed)g(that)g Fp(f)41 b Fr(has)33 b(standard)e(b)s(eha)m(vior)h(in)f Fp(I)1415 2864 y Fo(\016)1475 2849 y Fn(\\)1557 2748 y Ff(\020)1611 2849 y Fn([)1672 2808 y Fo(J)1711 2817 y Fg(0)1672 2876 y Fo(j)t Fm(=1)1798 2849 y Fp(D)1873 2863 y Fo(j)1910 2748 y Ff(\021)1964 2849 y Fr(,)j(1)29 b Fn(\024)f Fp(J)2246 2863 y Fm(0)2315 2849 y Fp(<)h(J)h Fn(\000)22 b Fr(1.)47 b(Let)33 b(us)f(sho)m(w)g(that)i Fp(f)41 b Fr(has)33 b(standard)-236 2979 y(b)s(eha)m(vior)c(in)g Fp(D)316 2993 y Fo(J)355 3002 y Fg(0)390 2993 y Fm(+1)504 2979 y Fn(\\)20 b Fp(I)625 2994 y Fo(\016)663 2979 y Fr(.)-236 3134 y(F)-8 b(or)38 b(the)g(sak)m(e)h(of)f(simplicit)m(y)-8 b(,)38 b(w)m(e)g(omit)f(the)h(index)f Fp(j)43 b Fr(and)37 b(write)g Fp(d)h Fr(=)f Fp(D)2405 3148 y Fo(J)2444 3157 y Fg(0)2483 3134 y Fr(,)j Fp(d)2595 3148 y Fm(1)2672 3134 y Fr(=)d Fp(D)2855 3148 y Fo(J)2894 3157 y Fg(0)2929 3148 y Fm(+1)3023 3134 y Fr(.)63 b(The)37 b(solution)f Fp(f)47 b Fr(can)38 b(b)s(e)-236 3242 y(represen)m(ted)30 b(b)m(y)g(a)h(linear)e(com)m(bination)h(of)g(the)h(solutions)d Fp(f)1904 3256 y Fk(\006)1993 3242 y Fr(corresp)s(onding)h(to)i Fp(d)2736 3256 y Fm(1)2806 3242 y Fr(i.e.)1384 3427 y Fp(f)j Fr(=)25 b Fp(a)15 b(f)1667 3441 y Fm(+)1746 3427 y Fr(+)20 b Fp(b)15 b(f)1936 3441 y Fk(\000)1995 3427 y Fp(;)106 b(\020)32 b Fn(2)25 b Fp(d)2331 3441 y Fm(1)2371 3427 y Fp(:)-2632 b Fr(\(5.3\))-236 3611 y(Here,)995 3843 y Fp(a)p Fr(\()p Fp(\020)7 b Fr(\))26 b(=)1319 3782 y Fp(w)r Fr(\()p Fp(f)5 b(;)15 b(f)1556 3796 y Fk(\000)1615 3782 y Fr(\))p 1292 3822 386 4 v 1292 3906 a Fp(w)r Fr(\()p Fp(f)1439 3920 y Fm(+)1498 3906 y Fp(;)g(f)1583 3920 y Fk(\000)1642 3906 y Fr(\))1808 3843 y(and)121 b Fp(b)p Fr(\()p Fp(\020)7 b Fr(\))26 b(=)2388 3782 y Fp(w)r Fr(\()p Fp(f)2535 3796 y Fm(+)2595 3782 y Fp(;)15 b(f)10 b Fr(\))p 2363 3822 V 2363 3906 a Fp(w)r Fr(\()p Fp(f)2510 3920 y Fm(+)2570 3906 y Fp(;)15 b(f)2655 3920 y Fk(\000)2714 3906 y Fr(\))2759 3843 y Fp(:)-3020 b Fr(\(5.4\))-236 4081 y(By)20 b(the)h(consistency)f(condition)f(\(3.3\))r(,)j(the)f(co)s (e\016cien)m(ts)g Fp(a)f Fr(and)g Fp(b)g Fr(are)h Fp(")p Fr(-p)s(erio)s(dic.)35 b(As)21 b(usual,)g Fp(w)r Fr(\()p Fp(f)3147 4095 y Fm(+)3206 4081 y Fp(;)15 b(f)3291 4095 y Fk(\000)3350 4081 y Fr(\))26 b(=)f Fp(w)r Fr(\(\011)3680 4095 y Fm(+)3739 4081 y Fp(;)15 b Fr(\011)3850 4095 y Fk(\000)3910 4081 y Fr(\)+)-236 4189 y Fp(o)p Fr(\(1\))29 b(uniformly)c(on)j(an)m(y)h(compact)g(in)e Fp(d)1170 4203 y Fm(1)1210 4189 y Fr(.)39 b(The)28 b(v)-5 b(alue)28 b(of)g(the)g(leading)f(term)h(dep)s(ends)f(on)h(the)g(c)m(hoice)h(of)f (normalization)-236 4297 y(p)s(oin)m(t)h(for)i Fp(f)186 4311 y Fk(\006)244 4297 y Fr(.)42 b(W)-8 b(e)32 b(c)m(ho)s(ose)g(the)e (normalization)g(p)s(oin)m(t)f Fp(\020)1767 4311 y Fm(0)1837 4297 y Fr(in)h Fp(d)20 b Fn(\\)g Fp(d)2139 4311 y Fm(1)2199 4297 y Fn(\\)g Fp(I)38 b Fr(so)31 b(that)g Fp(q)s Fr(\()p Fp(\020)2786 4311 y Fm(0)2826 4297 y Fr(\))26 b Fn(6)p Fr(=)f(0.)42 b(Then,)30 b(the)h(leading)f(term)-236 4405 y(of)g Fp(w)r Fr(\()p Fp(f)14 4419 y Fm(+)74 4405 y Fp(;)15 b(f)159 4419 y Fk(\000)218 4405 y Fr(\))30 b(is)g(b)s(ounded)e(a)m(w)m (a)m(y)k(from)e(0)h(uniformly)c(in)i Fp(\020)37 b Fr(and)30 b Fp(")p Fr(.)-236 4571 y(The)g(solution)f Fp(f)39 b Fr(is)29 b(normalized)g(at)i(a)g(p)s(oin)m(t)e(whic)m(h)h(can)g(b)s(e)g (di\013eren)m(t)g(of)g Fp(\020)2412 4585 y Fm(0)2451 4571 y Fr(.)41 b(Therefore,)30 b(w)m(e)h(represen)m(t)g Fp(f)39 b Fr(as)31 b Fp(f)j Fr(=)25 b Fp(f)3897 4585 y Fm(0)3956 4547 y Fr(^)3936 4571 y Fp(f)9 b Fr(,)-236 4684 y(where)23 b Fp(f)65 4698 y Fm(0)127 4684 y Fr(is)g(a)h(constan)m (t)h(factor)f(dep)s(ending)e(on)h(the)h(normalization)e(p)s(oin)m(ts,)i (and)2646 4660 y(^)2626 4684 y Fp(f)33 b Fr(is)23 b(a)h(solution)e(suc) m(h)h(that)h(the)g(leading)-236 4797 y(terms)30 b(of)h(the)f (asymptotics)h(of)904 4773 y(^)884 4797 y Fp(f)40 b Fr(and)30 b Fp(f)1191 4811 y Fm(+)1279 4797 y Fr(coincide)g(in)f Fp(d)21 b Fn(\\)e Fp(d)1929 4811 y Fm(1)1969 4797 y Fr(.)-236 4960 y(As)30 b(the)f(leading)g(terms)h(of)741 4936 y(^)722 4960 y Fp(f)38 b Fr(and)30 b Fp(f)1027 4974 y Fm(+)1115 4960 y Fr(coincide)f(in)f Fp(d)19 b Fn(\\)g Fp(d)1761 4974 y Fm(1)1801 4960 y Fr(,)30 b(and)f(as)h(the)g(leading)e(term)i(of) g(the)g(W)-8 b(ronskian)29 b Fp(w)r Fr(\()p Fp(f)3686 4974 y Fm(+)3746 4960 y Fp(;)15 b(f)3831 4974 y Fk(\000)3890 4960 y Fr(\))30 b(is)-236 5068 y(non-zero,)h(lo)s(cally)e(uniformly)e (in)i Fp(\020)j Fn(2)25 b Fp(d)20 b Fn(\\)g Fp(d)1319 5082 y Fm(1)1359 5068 y Fr(,)31 b(w)m(e)f(obtain)1498 5252 y Fp(a)p Fr(\()p Fp(\020)7 b Fr(\))26 b(=)f Fp(f)1830 5266 y Fm(0)1869 5252 y Fr(\(1)c(+)f Fp(o)p Fr(\(1\)\))p Fp(:)-2516 b Fr(\(5.5\))-236 5442 y(Due)30 b(to)i(the)e Fp(")p Fr(-p)s(erio)s(dicit)m(y)f(of)h Fp(a)p Fr(,)h(the)f(asymptotics) h(\(5.5\))h(sta)m(ys)f(v)-5 b(alid)29 b(and)h(uniform)e(in)h Fp(I)2946 5457 y Fo(\016)2984 5442 y Fr(.)-236 5652 y(As)f(for)h Fp(b)p Fr(,)g(one)g(has)g Fp(b)p Fr(\()p Fp(\020)7 b Fr(\))25 b(=)g Fp(f)775 5666 y Fm(0)831 5652 y Fn(\001)17 b Fp(o)932 5524 y Ff(\022)999 5652 y Fp(e)1051 5576 y Fg(2)p Fd(i)p 1052 5588 53 3 v 1063 5629 a(")1126 5549 y Fb(R)1173 5570 y Fd(\020)1159 5627 y(\020)1187 5642 y Fg(0)1238 5603 y Fo(\024d\020)1355 5524 y Ff(\023)1450 5652 y Fr(lo)s(cally)28 b(uniformly)d(in)j Fp(\020)k Fn(2)24 b Fp(d)18 b Fn(\\)e Fp(d)2600 5666 y Fm(1)2640 5652 y Fr(.)40 b(Due)29 b(to)g(the)g Fp(")p Fr(-p)s(erio)s(dicit)m(y)e (of)i Fp(b)p Fr(,)g(one)-236 5807 y(can)h(write)1191 6024 y Fp(b)p Fr(\()p Fp(\020)7 b Fr(\))25 b(=)g Fp(f)1513 6038 y Fm(0)1572 6024 y Fn(\001)c Fp(o)1677 5896 y Ff(\022)1744 6024 y Fp(e)1796 5948 y Fg(2)p Fd(i)p 1796 5960 V 1808 6001 a(")1871 5921 y Fb(R)1923 5929 y Fg(~)1918 5942 y Fd(\020)1904 5999 y(\020)1932 6014 y Fg(0)1983 5975 y Fo(\024d\020)2100 5896 y Ff(\023)2182 6024 y Fp(;)106 b(\020)32 b Fn(2)25 b Fp(I)2511 6039 y Fo(\016)2548 6024 y Fp(;)1854 6210 y Fm(11)p eop %%Page: 12 12 12 11 bop -236 242 a Fr(where)35 218 y(~)27 242 y Fp(\020)31 b Fn(2)25 b Fp(d)c Fn(\\)f Fp(d)380 256 y Fm(1)419 242 y Fr(,)31 b(Im)607 218 y(~)599 242 y Fp(\020)g Fr(=)25 b(Im)15 b Fp(\020)7 b Fr(,)30 b(and)g(Re)15 b Fp(\020)27 b Fn(\000)20 b Fr(Re)1579 218 y(~)1571 242 y Fp(\020)32 b Fr(=)25 b(0)31 b(mo)s(d)e Fp(")p Fr(.)41 b(This)29 b(estimate)i(is)f(uniform)e(in)h Fp(\020)7 b Fr(.)-236 350 y(Substituting)27 b(the)k(estimates)g(obtained)f(for)g Fp(a)g Fr(and)g Fp(b)g Fr(in)m(to)h(the)f(represen)m(tation)h(for)f Fp(f)10 b Fr(,)30 b(w)m(e)h(get)151 567 y Fp(f)j Fr(=)25 b Fp(a)p Fr(\()p Fp(\020)7 b Fr(\))p Fp(f)536 581 y Fm(+)615 567 y Fr(+)20 b Fp(b)p Fr(\()p Fp(\020)7 b Fr(\))p Fp(f)907 581 y Fk(\000)991 567 y Fr(=)25 b Fp(f)1132 581 y Fm(0)1171 567 y Fp(e)1226 490 y Fd(i)p 1223 502 30 3 v 1223 544 a(")1274 464 y Fb(R)1321 485 y Fd(\020)1307 542 y(\020)1335 557 y Fg(0)1386 518 y Fo(\024d\020)1518 439 y Ff(\022)1585 567 y Fr(\011)1656 581 y Fm(+)1715 567 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))21 b(+)f Fp(o)p Fr(\(1\))h(+)f Fp(o)2366 439 y Ff(\022)2433 567 y Fp(e)2475 512 y Fk(\000)2540 485 y Fg(2)p Fd(i)p 2540 497 53 3 v 2552 538 a(")2615 458 y Fb(R)2662 479 y Fd(\020)2653 534 y Fg(~)2648 547 y Fd(\020)2709 512 y Fo(\024d\020)2826 439 y Ff(\023)2913 567 y Fr(+)g Fp(o)p Fr(\(1\))3163 439 y Ff(\023)3246 567 y Fp(;)46 b Fn(8)p Fp(\020)31 b Fn(2)25 b Fp(d)3572 581 y Fm(1)3632 567 y Fn(\\)20 b Fp(I)3753 582 y Fo(\016)3791 567 y Fp(:)-4052 b Fr(\(5.6\))-236 864 y(Let)35 b Fp(\020)40 b Fn(2)33 b Fr(\()p Fp(I)180 879 y Fo(\016)242 864 y Fn(\\)23 b Fp(d)373 878 y Fm(1)412 864 y Fr(\))h Fn(n)g Fp(d)p Fr(.)55 b(As)35 b Fp(d)853 878 y Fm(1)928 864 y Fr(is)f(to)i(the)g(left)f(of)g Fp(\015)1621 878 y Fm(2)1660 864 y Fr(,)i(in)d Fp(d)1880 878 y Fm(1)1920 864 y Fr(,)i(one)g(has)e (Im)15 b Fp(\024)34 b(<)f Fr(0.)55 b(Hence,)38 b(Re)3197 735 y Ff(\022)3264 864 y Fp(i)3310 740 y Ff(Z)3402 766 y Fo(\020)3367 929 y Fm(~)3361 946 y Fo(\020)3457 864 y Fp(\024)3509 735 y Ff(\023)3609 864 y Fp(>)33 b Fr(0.)55 b(This)-236 1030 y(implies)27 b(that)k Fp(f)40 b Fr(has)30 b(standard)g(b)s(eha)m(vior)f(in)g Fp(I)1413 1045 y Fo(\016)1471 1030 y Fn(\\)20 b Fp(d)1599 1044 y Fm(1)1639 1030 y Fr(.)-236 1186 y(Clearly)-8 b(,)35 b(one)g(can)f(apply)g(the)g(ab)s(o)m(v)m(e)i (argumen)m(ts)f(and)f(get)i(\(5.6\))g(in)d(the)i(case)h(where)e Fp(d)f Fr(=)f Fp(D)3119 1200 y Fo(J)6 b Fk(\000)p Fm(1)3292 1186 y Fr(and)34 b Fp(d)3520 1200 y Fm(1)3592 1186 y Fr(=)e Fp(D)3770 1200 y Fo(J)3854 1186 y Fr(\(the)-236 1360 y(last)f(disk\).)41 b(No)m(w,)32 b(in)e(\(5.6\))r(,)h(one)h(has)e (Re)1250 1232 y Ff(\022)1317 1360 y Fp(i)1363 1236 y Ff(Z)1454 1263 y Fo(\020)1420 1425 y Fm(~)1414 1443 y Fo(\020)1509 1360 y Fp(\024)1561 1232 y Ff(\023)1655 1360 y Fp(>)c Fr(0)31 b(for)g(an)m(y)g Fp(\020)38 b Fr(b)s(eing)29 b(either)i(in)f(the)h(part)g(of)g Fp(d)3335 1374 y Fm(1)3395 1360 y Fn([)21 b Fp(I)3517 1375 y Fo(\016)3585 1360 y Fr(situated)31 b(to)-236 1532 y(the)h(left)g(of)g Fp(\015)235 1546 y Fm(2)274 1532 y Fr(,)h(or)e(in)g Fp(\015)599 1546 y Fm(2)660 1532 y Fn(\\)21 b Fp(I)782 1547 y Fo(\016)819 1532 y Fr(.)46 b(Moreo)m(v)m(er,)34 b(this)d(expression)g(sta)m(ys)i (non-negativ)m(e)g(in)d(a)i(small)f(enough)g(\(but)h(constan)m(t\))-236 1640 y(neigh)m(b)s(orho)s(o)s(d)27 b Fp(V)49 b Fr(of)30 b Fp(\015)586 1654 y Fm(2)643 1640 y Fn(\\)18 b Fp(I)762 1655 y Fo(\016)800 1640 y Fr(.)40 b(This)28 b(implies)e(that)k Fp(f)39 b Fr(has)29 b(standard)f(b)s(eha)m(vior)h(in)f Fp(I)2714 1655 y Fo(\016)2781 1640 y Fr(b)s(oth)h(to)h(the)f(left)g(of) h Fp(\015)3568 1654 y Fm(2)3637 1640 y Fr(and)e(in)g Fp(V)20 b Fr(.)-236 1748 y(This)28 b(completes)j(the)g(pro)s(of)e(of)i (Lemma)f(5.1.)p 3950 1748 4 62 v 3954 1690 55 4 v 3954 1748 V 4008 1748 4 62 v -236 1951 a(5.2.)53 b Fu(Estimates)42 b(of)h(decreasing)h(solutions.)j Fr(The)37 b(Rectangle)i(Lemma)e(allo)m (ws)g(us)g(to)h(\\con)m(tin)m(ue")h(standard)e(b)s(e-)-236 2059 y(ha)m(vior)c(as)g(long)g(as)g(its)g(leading)f(term)h(increases)g (along)g(a)h(horizon)m(tal)e(line.)48 b(If)33 b(the)g(leading)f(term)h (decreases,)i(then,)f(in)-236 2167 y(general,)c(w)m(e)h(can)g(only)e (estimate)j(the)e(solution,)f(but)h(not)h(get)g(an)f(asymptotic)h(b)s (eha)m(vior.)-236 2341 y Fu(Lemma)h(5.3.)47 b Fi(Fix)32 b Fp(E)f Fr(=)25 b Fp(E)765 2355 y Fm(0)805 2341 y Fi(.)41 b(L)-5 b(et)33 b Fp(\020)1071 2355 y Fm(1)1110 2341 y Fp(;)15 b(\020)1190 2355 y Fm(2)1230 2341 y Fp(;)g(\020)1310 2355 y Fm(0)1382 2341 y Fi(b)-5 b(e)33 b(\014xe)-5 b(d)33 b(p)-5 b(oints)34 b(such)f(that)-137 2478 y Fr(1.)43 b(Im)14 b Fp(\020)139 2492 y Fm(1)204 2478 y Fr(=)25 b(Im)15 b Fp(\020)464 2492 y Fm(2)503 2478 y Fi(;)-137 2586 y Fr(2.)43 b(Re)15 b Fp(\020)138 2600 y Fm(1)203 2586 y Fp(<)25 b Fr(Re)15 b Fp(\020)461 2600 y Fm(2)500 2586 y Fi(;)-137 2694 y Fr(3.)43 b Fi(the)33 b(se)-5 b(gment)33 b Fr([)p Fp(\020)538 2708 y Fm(1)577 2694 y Fp(;)15 b(\020)657 2708 y Fm(2)697 2694 y Fr(])33 b Fi(of)g(the)g(line)f Fr(Im)15 b Fp(\020)32 b Fr(=)25 b(Im)14 b Fp(\020)1644 2708 y Fk(\000)1736 2694 y Fi(is)32 b(r)-5 b(e)g(gular.)-236 2831 y(Fix)40 b(a)g(c)-5 b(ontinuous)41 b(br)-5 b(anch)42 b(of)e Fp(\024)g Fi(on)h Fr([)p Fp(\020)1195 2845 y Fm(1)1234 2831 y Fp(;)15 b(\020)1314 2845 y Fm(2)1354 2831 y Fr(])p Fi(.)64 b(Assume)39 b(that)j(Im)p Fr(\()p Fp(\024)p Fr(\()p Fp(\020)7 b Fr(\)\))40 b Fp(>)e Fr(0)i Fi(on)h(the)f(se)-5 b(gment)41 b Fr([)p Fp(\020)3308 2845 y Fm(1)3347 2831 y Fp(;)15 b(\020)3427 2845 y Fm(2)3467 2831 y Fr(])p Fi(.)64 b(L)-5 b(et)40 b Fp( )j Fi(b)-5 b(e)40 b(a)-236 2979 y(c)-5 b(onsistent)34 b(solution)f(having)h(in)e (a)h(neighb)-5 b(orho)g(o)g(d)36 b(of)d Fp(\020)1696 2993 y Fm(1)1768 2979 y Fi(standar)-5 b(d)35 b(b)-5 b(ehavior)34 b Fp( )29 b Fn(\030)c Fp(e)2736 2903 y Fd(i)p 2733 2915 30 3 v 2733 2956 a(")2784 2877 y Fb(R)2831 2897 y Fd(\020)2818 2954 y(\020)2846 2969 y Fg(1)2897 2930 y Fo(\024d\020)3013 2979 y Fr(\011)3084 2993 y Fm(+)3143 2979 y Fi(.)-236 3087 y(Then,)33 b(ther)-5 b(e)33 b(exists)g Fp(C)f(>)25 b Fr(0)33 b Fi(such)g(that,)g(for)g Fp(")g Fi(su\016ciently)g(smal)5 b(l,)34 b(one)f(has)804 3289 y Ff(\014)804 3343 y(\014)804 3398 y(\014)804 3452 y(\014)845 3359 y Fp(d )p 845 3400 110 4 v 850 3483 a(dx)964 3420 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))1173 3289 y Ff(\014)1173 3343 y(\014)1173 3398 y(\014)1173 3452 y(\014)1225 3420 y Fr(+)19 b Fn(j)p Fp( )s Fr(\()p Fp(x;)c(\020)7 b Fr(\))p Fn(j)27 b(\024)e Fp(C)7 b(e)1883 3254 y Fr(1)p 1883 3294 46 4 v 1885 3378 a Fp(")1954 3192 y Ff(Z)2044 3218 y Fo(\020)2004 3398 y(\020)2035 3407 y Fg(1)2099 3315 y Fn(j)p Fi(Im)q Fp(\024)p Fn(j)p Fp(d\020)2410 3420 y(;)108 b(\020)32 b Fn(2)24 b Fr([)p Fp(\020)2765 3434 y Fm(1)2805 3420 y Fp(;)15 b(\020)2885 3434 y Fm(2)2925 3420 y Fr(])p Fp(:)-3211 b Fr(\(5.7\))-236 3637 y Fi(uniformly)33 b(in)g Fp(E)38 b Fi(in)32 b(a)h(c)-5 b(onstant)35 b(neighb)-5 b(orho)g(o)g(d)36 b(of)d Fp(E)1662 3651 y Fm(0)1701 3637 y Fi(.)-236 3811 y Fu(Remark)h(5.2.)46 b Fr(Note)25 b(that)g(the)f(leading)e(term)i(of)g (the)g(asymptotics)g(of)g Fp(f)33 b Fr(is)22 b(de\014ned)h(and)g (decreases)i(along)f(the)g(segmen)m(t)-236 3919 y([)p Fp(\020)-171 3933 y Fm(1)-132 3919 y Fp(;)15 b(\020)-52 3933 y Fm(2)-12 3919 y Fr(])29 b(as)f Fp(\020)35 b Fr(increases)28 b(from)g Fp(\020)857 3933 y Fm(1)925 3919 y Fr(to)h Fp(\020)1074 3933 y Fm(2)1113 3919 y Fr(.)41 b(One)28 b(pro)m(v)m(es)h(the)f (\\symmetric")h(statemen)m(t)i(for)d(the)h(case)g(where)f(Im)15 b Fp(\024)25 b(<)g Fr(0)k(and)-236 4066 y Fp(f)39 b Fr(has)30 b(standard)g(b)s(eha)m(vior)g Fp(f)k Fn(\030)25 b Fp(e)992 3990 y Fd(i)p 989 4002 30 3 v 989 4043 a(")1040 3964 y Fb(R)1087 3984 y Fd(\020)1074 4042 y(\020)1102 4057 y Fg(2)1153 4017 y Fo(\024d\020)1269 4066 y Fr(\011)1340 4080 y Fm(+)1429 4066 y Fr(in)k(a)i(neigh)m(b)s(orho)s(o)s(d)d(of)j Fp(\020)2326 4080 y Fm(2)2365 4066 y Fr(.)-236 4306 y Fu(Pro)s(of.)60 b Fr(In)37 b(a)g(neigh)m(b)s(orho)s(o)s(d)d(of)j Fp(\020)1036 4320 y Fm(1)1075 4306 y Fr(,)i(using)c(Lemma)i(5.1,)j(w)m (e)d(construct)g(t)m(w)m(o)i(linearly)34 b(indep)s(enden)m(t)h (solutions)g Fp( )3921 4258 y Fm(\(1\))3918 4329 y Fk(\006)-236 4461 y Fr(of)i(\(3.2\))c(ha)m(ving)c(standard)h(b)s(eha)m(vior)f Fp( )1197 4413 y Fm(\(1\))1194 4485 y Fk(\006)1318 4461 y Fn(\030)c Fp(e)1469 4385 y Fd(i)p 1466 4397 V 1466 4438 a(")1517 4359 y Fb(R)1564 4379 y Fd(\020)1550 4437 y(\020)1578 4452 y Fg(1)1629 4412 y Fo(\024d\020)1746 4461 y Fr(\011)1817 4475 y Fk(\006)1876 4461 y Fr(.)40 b(W)-8 b(e)32 b(normalize)d(them)h(at)i(the)e(p)s(oin)m(t)g Fp(\020)3290 4475 y Fm(1)3329 4461 y Fr(.)-236 4645 y(By)g(Lemma)h (5.1,)h Fp( )465 4597 y Fm(\(1\))462 4668 y Fk(\000)590 4645 y Fr(has)e(the)g(standard)g(b)s(eha)m(vior)f(in)h(a)g(constan)m(t) i(neigh)m(b)s(orho)s(o)s(d)c(of)i(whole)g(in)m(terv)-5 b(al)30 b([)p Fp(\020)3534 4659 y Fm(1)3573 4645 y Fp(;)15 b(\020)3653 4659 y Fm(2)3693 4645 y Fr(].)-236 4828 y(In)35 b(the)i(same)g(w)m(a)m(y)-8 b(,)39 b(starting)e(in)e(a)h(neigh)m(b)s (orho)s(o)s(d)e(of)j Fp(\020)1755 4842 y Fm(2)1794 4828 y Fr(,)h(w)m(e)f(construct)g Fp( )2465 4781 y Fm(\(2\))2462 4852 y(+)2596 4828 y Fr(a)g(solution)e(of)43 b(\(3.2\))c(ha)m(ving)d (standard)-236 4984 y(b)s(eha)m(vior)29 b Fp( )197 4936 y Fm(\(2\))194 5007 y(+)317 4984 y Fn(\030)c Fp(e)468 4907 y Fd(i)p 466 4919 V 466 4961 a(")516 4881 y Fb(R)564 4902 y Fd(\020)550 4959 y(\020)578 4974 y Fg(1)629 4935 y Fo(\024d\020)746 4984 y Fr(\011)817 4998 y Fm(+)906 4984 y Fr(in)k(a)h(constan)m(t)i(neigh)m(b)s(orho)s(o)s(d)c(of)j([)p Fp(\020)2194 4998 y Fm(1)2233 4984 y Fp(;)15 b(\020)2313 4998 y Fm(2)2353 4984 y Fr(].)41 b(W)-8 b(e)31 b(normalize)e(this)h (solution)f(at)i Fp(\020)3687 4998 y Fm(1)3726 4984 y Fr(.)-236 5167 y(As,)j(in)d(the)i(pro)s(of)g(of)g(Lemma)g(5.1,)i(w)m(e) e(compute)h(the)f(W)-8 b(ronskian)32 b(of)h(\()p Fp( )2368 5119 y Fm(\(1\))2365 5191 y Fk(\000)2463 5167 y Fp(;)15 b( )2565 5119 y Fm(\(2\))2562 5191 y(+)2661 5167 y Fr(\))33 b(and)f(see)i(that)f(it)g(is)f(b)s(ounded)f(a)m(w)m(a)m(y)-236 5304 y(from)f(0)g(b)m(y)h(a)f(constan)m(t)i(indep)s(enden)m(t)c(of)j Fp(\020)37 b Fr(and)29 b Fp(")p Fr(.)41 b(Hence,)32 b Fp( )1961 5256 y Fm(\(1\))1958 5327 y Fk(\000)2086 5304 y Fr(and)e Fp( )2325 5256 y Fm(\(2\))2322 5327 y(+)2450 5304 y Fr(form)g(a)h(basis)e(of)h(solutions.)-236 5459 y(W)-8 b(e)31 b(decomp)s(ose)g Fp( )j Fr(as)1412 5641 y Fp( )29 b Fr(=)c Fp(a)p Fr(\()p Fp(\020)7 b Fr(\))p Fp( )1823 5593 y Fm(\(2\))1820 5665 y(+)1938 5641 y Fr(+)20 b Fp(b)p Fr(\()p Fp(\020)7 b Fr(\))p Fp( )2247 5593 y Fm(\(1\))2244 5665 y Fk(\000)2342 5641 y Fp(:)-2603 b Fr(\(5.8\))-236 5808 y(As)30 b(in)f(the)i(pro)s(of)e(of)i(the)f (Rectangle)i(Lemma,)f(w)m(e)g(get)1082 6024 y Fp(a)p Fr(\()p Fp(\020)7 b Fr(\))25 b(=)g(1)c(+)f Fp(o)p Fr(\(1\))p Fp(;)107 b(b)p Fr(\()p Fp(\020)7 b Fr(\))26 b(=)f Fp(o)2153 5896 y Ff(\022)2220 6024 y Fp(e)2272 5948 y Fg(2)p Fd(i)p 2272 5960 53 3 v 2284 6001 a(")2346 5921 y Fb(R)2393 5942 y Fd(\020)2380 5999 y(\020)2408 6014 y Fg(1)2459 5975 y Fo(\024d\020)2575 5896 y Ff(\023)2657 6024 y Fp(;)1854 6210 y Fm(12)p eop %%Page: 13 13 13 12 bop -236 241 a Fr(uniformly)34 b(in)i(a)i(constan)m(t)g(neigh)m (b)s(orho)s(o)s(d)d(of)j Fp(\020)1483 255 y Fm(1)1522 241 y Fr(.)62 b(Note)38 b(that)g(the)g(last)f(estimate)h(implies)d (that)i Fn(j)p Fp(b)p Fn(j)h(\024)e Fr(Const)h(in)f(the)-236 349 y Fp(")p Fr(-neigh)m(b)s(orho)s(o)s(d)28 b(of)j Fp(\020)551 363 y Fm(1)590 349 y Fr(.)41 b(As)30 b Fp(a)h Fr(and)e Fp(b)i Fr(are)f Fp(")p Fr(-p)s(erio)s(dic,)f(then)1222 498 y Fp(a)p Fr(\()p Fp(\020)7 b Fr(\))25 b(=)g(1)c(+)f Fp(o)p Fr(\(1\))p Fp(;)107 b Fn(j)p Fp(b)p Fr(\()p Fp(\020)7 b Fr(\))p Fn(j)26 b(\024)f Fr(Const)p Fp(;)-236 652 y Fr(uniformly)e(on)k([)p Fp(\020)362 666 y Fm(1)402 652 y Fp(;)15 b(\020)482 666 y Fm(2)521 652 y Fr(].)40 b(As)26 b(these)i(estimates)f(w)m(ere)g(obtained)f(using)f(standard)h(b)s(eha)m (vior,)h(they)g(are)g(uniform)d(for)j Fp(E)32 b Fr(in)25 b(a)-236 760 y(constan)m(t)f(neigh)m(b)s(orho)s(o)s(d)d(of)j Fp(E)851 774 y Fm(0)890 760 y Fr(.)38 b(Substituting)21 b(this)h(estimates)i(in)m(to)f(\(5.8\))r(,)i(and)e(taking)g(in)m(to)g (accoun)m(t)i(the)f(asymptotics)-236 889 y(of)30 b Fp( )-71 841 y Fm(\(2\))-74 912 y(+)54 889 y Fr(and)g Fp( )293 841 y Fm(\(1\))290 912 y Fk(\000)388 889 y Fr(,)h(w)m(e)f(get)629 1100 y Fn(j)p Fp( )s Fn(j)21 b Fr(+)853 969 y Ff(\014)853 1023 y(\014)853 1078 y(\014)853 1132 y(\014)894 1039 y Fp(d )p 894 1080 110 4 v 899 1163 a(dx)1013 969 y Ff(\014)1013 1023 y(\014)1013 1078 y(\014)1013 1132 y(\014)1069 1100 y Fn(\024)k Fr(Const)1413 972 y Ff(\022)1480 1100 y Fp(e)1522 1051 y Fk(\000)1587 1024 y Fg(1)p 1587 1036 31 3 v 1588 1077 a Fd(")1639 998 y Fb(R)1686 1018 y Fd(\020)1672 1076 y(\020)1700 1091 y Fg(1)1751 1051 y Fm(Im)11 b Fo(\024d\020)1984 1100 y Fr(+)20 b Fp(e)2127 1024 y Fg(1)p 2128 1036 V 2129 1077 a Fd(")2180 998 y Fb(R)2227 1018 y Fd(\020)2213 1076 y(\020)2241 1091 y Fg(1)2292 1051 y Fm(Im)11 b Fo(\024d\020)2505 972 y Ff(\023)2587 1100 y Fp(;)106 b(\020)32 b Fn(2)25 b Fr([)p Fp(\020)2941 1114 y Fm(1)2980 1100 y Fp(;)15 b(\020)3060 1114 y Fm(2)3100 1100 y Fr(])p Fp(:)-236 1304 y Fr(This)28 b(implies)g(\(5.7\))k(and)e(completes)h(the)f(pro)s (of)g(of)g(Lemma)h(5.3.)p 3950 1304 4 62 v 3954 1246 55 4 v 3954 1304 V 4008 1304 4 62 v -236 1473 a(5.3.)53 b Fu(Adjacen)m(t)37 b(Canonical)f(Domain)g(Principle.)46 b Fr(The)31 b(estimate)h(w)m(e)g(obtained)f(in)f(Lemma)i(5.3)g(can)g(b) s(e)f(v)m(ery)h(far)-236 1643 y(from)c(optimal:)39 b(the)29 b(estimates)g(only)e(sa)m(ys)j(that)f(the)g(solution)e Fp( )32 b Fr(can)d(not)g(increase)f(faster)h(than)f(exp)3350 1515 y Ff(\022)3427 1581 y Fr(1)p 3427 1622 46 4 v 3429 1705 a Fp(")3498 1519 y Ff(Z)3589 1545 y Fo(\020)3548 1725 y(\020)3579 1734 y Fg(1)3644 1643 y Fn(j)p Fr(Im)o Fp(\024)p Fn(j)p Fp(d\020)3948 1515 y Ff(\023)-236 1815 y Fr(whereas)e(it)f(can,)j(in)d(fact,)j(decrease)f(along)f([)p Fp(\020)1340 1829 y Fm(1)1380 1815 y Fp(;)15 b(\020)1460 1829 y Fm(2)1499 1815 y Fr(].)40 b(Under)25 b(additional)f(conditions,) i(the)g(follo)m(wing)f(construction)g(yields)-236 1923 y(a)31 b(b)s(etter)f(result.)-236 2078 y(Note)39 b(that)g(the)f(union)e (of)i(an)m(y)h(n)m(um)m(b)s(er)e(of)h(canonical)f(domains)g(enclosing)g Fp(\015)43 b Fr(is)37 b(a)i(canonical)e(domain)g(enclosing)g Fp(\015)5 b Fr(.)-236 2186 y(Consider)40 b(the)h(union)f(of)i(all)f (the)g(canonical)h(domains)e(enclosing)h Fp(\015)5 b Fr(.)74 b(It)42 b(is)f(a)h(canonical)f(domain.)74 b(W)-8 b(e)42 b(call)f(it)g(the)-236 2294 y Fi(maximal)34 b(c)-5 b(anonic)g(al)35 b(domain)d Fr(enclosing)d Fp(\015)5 b Fr(.)41 b(One)30 b(has)-236 2458 y Fu(Lemma)i(5.4)k Fr(\(The)30 b(Adjacen)m(t)i(Canonical)d(Domain)h(Principle\))p Fu(.)43 b Fi(Assume)d(that)i(a)f(c)-5 b(onsistent)41 b(solution)h Fp(f)50 b Fi(has)41 b(stan-)-236 2565 y(dar)-5 b(d)42 b(b)-5 b(ehavior)42 b(in)e(either)h(the)f(left)h(hand)g(side)g (or)g(the)f(right)h(hand)h(side)f(of)f(a)h(c)-5 b(onstant)42 b(neighb)-5 b(orho)g(o)g(d)44 b(of)c(a)h(vertic)-5 b(al)-236 2673 y(curve)43 b Fp(\015)5 b Fi(.)75 b(Assume)44 b(that)h Fp(\015)k Fi(is)44 b(c)-5 b(anonic)g(al)45 b(with)g(r)-5 b(esp)g(e)g(ct)45 b(to)g(some)f(br)-5 b(anch)45 b(of)f(the)h(c)-5 b(omplex)45 b(momentum,)j(and)c(that)-236 2781 y(the)37 b(c)-5 b(orr)g(esp)g(onding)39 b(maximal)f(enclosing)f(c)-5 b(anonic)g(al)39 b(domain)f(is)e(b)-5 b(ounde)g(d.)55 b(Then,)38 b Fp(f)45 b Fi(has)38 b(standar)-5 b(d)39 b(b)-5 b(ehavior)38 b(in)e(this)-236 2889 y(domain.)-236 3053 y Fu(Pro)s(of.)71 b Fr(Consider)39 b Fp(M)10 b Fr(,)43 b(the)e(maximal)e(canonical)h(domain)f(enclosing)g(the)i(canonical)f (curv)m(e)h Fp(\015)5 b Fr(.)71 b(Denote)41 b(b)m(y)g Fp(\024)f Fr(the)-236 3161 y(corresp)s(onding)28 b(branc)m(h)i(of)g (the)h(complex)f(momen)m(tum.)41 b(Let)31 b Fp(\020)1981 3175 y Fo(m)2077 3161 y Fr(and)f Fp(\020)2294 3175 y Fo(M)2403 3161 y Fr(b)s(e)g(the)g(lo)m(w)m(er)h(and)f(the)g(upp)s(er)f (ends)g(of)i Fp(\015)5 b Fr(.)-236 3316 y(By)43 b(Theorem)g(3.1,)48 b(in)42 b Fp(M)10 b Fr(,)47 b(w)m(e)d(construct)f(the)h(corresp)s (onding)d(consisten)m(t)j(basis)e(of)h(solutions)f Fp(f)3345 3330 y Fk(\006)3447 3316 y Fr(with)g(standard)-236 3462 y(asymptotics)31 b Fp(f)313 3476 y Fk(\006)372 3462 y Fr(\()p Fp(x;)15 b(\020)7 b(;)15 b(\020)626 3476 y Fm(0)666 3462 y Fr(\))27 b Fn(\030)f Fp(e)867 3413 y Fk(\006)936 3385 y Fd(i)p 933 3397 30 3 v 933 3439 a(")984 3359 y Fb(R)1031 3380 y Fd(\020)1017 3437 y(\020)1045 3452 y Fg(0)1096 3413 y Fo(\024d\020)1213 3462 y Fr(\011)1284 3476 y Fk(\006)1342 3462 y Fr(\()p Fp(x;)15 b(\020)7 b(;)15 b(\020)1596 3476 y Fm(0)1636 3462 y Fr(\).)44 b(Here,)32 b(w)m(e)g(ha)m(v)m(e)h(explicitly)c(indicated)g(the)j(dep)s (endence)e(on)h(the)-236 3570 y(normalization)c(p)s(oin)m(t)i Fp(\020)615 3584 y Fm(0)654 3570 y Fr(.)40 b(W)-8 b(e)30 b(c)m(ho)s(ose)g Fp(\020)1201 3584 y Fm(0)1266 3570 y Fn(2)25 b Fp(M)39 b Fr(so)29 b(that)h Fp(q)s Fr(\()p Fp(\020)1904 3584 y Fm(0)1943 3570 y Fr(\))c Fn(6)p Fr(=)f(0.)40 b(There)29 b(exists)g Fp(c)c(>)g Fr(0)30 b(suc)m(h)f(that,)h(for)f Fp(")g Fr(su\016cien)m(tly)-236 3678 y(small,)g(one)i(has)f Fn(j)p Fp(w)r Fr(\()p Fp(f)528 3692 y Fm(+)587 3678 y Fp(;)15 b(f)672 3692 y Fk(\000)731 3678 y Fr(\))p Fn(j)26 b Fp(>)f(c)31 b Fr(lo)s(cally)e(uniformly)e(in)i Fp(\020)j Fn(2)25 b Fp(M)10 b Fr(.)-236 3833 y(Inside)28 b(the)j(domain)e Fp(M)10 b Fr(,)31 b(the)f(function)g Fp(f)39 b Fr(admits)30 b(the)g(represen)m(tation)639 4030 y Fp(f)35 b Fr(=)25 b Fp(af)908 4044 y Fm(+)986 4030 y Fr(+)20 b Fp(bf)1161 4044 y Fk(\000)1220 4030 y Fp(;)106 b(a)p Fr(\()p Fp(\020)7 b Fr(\))26 b(=)1675 3968 y Fp(w)r Fr(\()p Fp(f)5 b(;)15 b(f)1912 3982 y Fk(\000)1971 3968 y Fr(\))p 1648 4009 386 4 v 1648 4092 a Fp(w)r Fr(\()p Fp(f)1795 4106 y Fm(+)1854 4092 y Fp(;)g(f)1939 4106 y Fk(\000)1998 4092 y Fr(\))2164 4030 y(and)121 b Fp(b)p Fr(\()p Fp(\020)7 b Fr(\))26 b(=)2744 3968 y Fp(w)r Fr(\()p Fp(f)2891 3982 y Fm(+)2951 3968 y Fp(;)15 b(f)10 b Fr(\))p 2720 4009 V 2720 4092 a Fp(w)r Fr(\()p Fp(f)2867 4106 y Fm(+)2926 4092 y Fp(;)15 b(f)3011 4106 y Fk(\000)3070 4092 y Fr(\))3115 4030 y Fp(:)-3376 b Fr(\(5.9\))-236 4227 y(The)30 b(functions)f Fp(a)h Fr(and)g Fp(b)g Fr(are)h Fp(")p Fr(-p)s(erio)s(dic)d(and)i (analytic)g(in)f(the)i(strip)d(Im)15 b Fp(\020)2402 4241 y Fo(m)2494 4227 y Fp(<)25 b Fr(Im)14 b Fp(\020)32 b(<)25 b(\020)2921 4241 y Fo(M)3000 4227 y Fr(.)-236 4383 y(T)-8 b(o)42 b(pro)m(v)m(e)h(Lemma)f(5.4,)47 b(w)m(e)42 b(need)g(asymptotic)h (estimates)f(of)h(the)f(co)s(e\016cien)m(ts)h Fp(a)f Fr(and)f Fp(b)p Fr(.)76 b(Therefore,)45 b(w)m(e)e(use)e(the)-236 4536 y(standard)30 b(b)s(eha)m(vior)f(of)i Fp(f)40 b Fr(in)29 b(a)i(neigh)m(b)s(orho)s(o)s(d)d(of)j(the)g(curv)m(e)g Fp(\015)5 b Fr(,)31 b Fp(f)k Fn(\030)25 b Fp(e)2300 4449 y Fd(i)p 2297 4461 30 3 v 2297 4502 a(")2348 4423 y Fb(R)2395 4443 y Fd(\020)2387 4499 y Fg(~)2381 4512 y Fd(\020)2409 4527 y Fg(0)2463 4476 y Fm(~)-38 b Fo(\024d\020)2590 4513 y Fr(~)2577 4536 y(\011)2648 4550 y Fm(+)2707 4536 y Fr(\()p Fp(x;)15 b(\020)7 b(;)2930 4512 y Fr(~)2921 4536 y Fp(\020)2961 4550 y Fm(0)3001 4536 y Fr(\).)41 b(Both)32 b(the)e(branc)m(h)k(~)-49 b Fp(\024)31 b Fr(and)-236 4652 y(the)f(normalization)f(p)s(oin)m(t)741 4628 y(~)733 4652 y Fp(\020)773 4666 y Fm(0)842 4652 y Fr(can)h(b)s(e)g(di\013eren)m (t)f(from)h(the)g(branc)m(h)f Fp(\024)i Fr(and)e(the)h(normalization)f (p)s(oin)m(t)g Fp(\020)3429 4666 y Fm(0)3468 4652 y Fr(;)i(the)f (indexing)-236 4762 y(of)g(the)h(canonical)f(Blo)s(c)m(h)g(solution) 1036 4739 y(~)1023 4762 y(\011)1094 4776 y Fm(+)1183 4762 y Fr(is)g(determined)f(b)m(y)k(~)-48 b Fp(\024)p Fr(.)41 b(W)-8 b(e)31 b(express)j(~)-49 b Fp(\024)31 b Fr(in)e(terms)h(of)h Fp(\024)1482 4911 y Fr(~)-48 b Fp(\024)p Fr(\()p Fp(\020)7 b Fr(\))25 b(=)g Fp(\033)s(\024)p Fr(\()p Fp(\020)7 b Fr(\))21 b(+)f(2)p Fp(\031)s(n;)-236 5065 y Fr(where)34 b Fp(\033)k Fr(tak)m(es)e(one)g(of)f(the)g(v)-5 b(alues)34 b Fn(f\000)p Fr(1)p Fp(;)15 b Fr(+1)p Fn(g)p Fr(,)38 b(and)c Fp(n)g Fr(is)g(an)h(en)m(tire)g(n)m(um)m(b)s(er.)53 b(Let)35 b(us)f(\014rst)g(discuss)f(the)i(case)h(where)-236 5173 y Fp(\033)28 b Fr(=)d(+1.)-236 5281 y(Let)31 b Fp(V)50 b Fr(b)s(e)30 b(the)h(part)f(of)h(the)g(neigh)m(b)s(orho)s(o)s(d)d(of)i Fp(\015)36 b Fr(where)30 b(all)f(the)i(solutions)e Fp(f)10 b Fr(,)30 b Fp(f)2607 5295 y Fm(+)2696 5281 y Fr(and)g Fp(f)2918 5295 y Fk(\000)3007 5281 y Fr(admit)g(standard)f(b)s(eha)m (vior.)-236 5389 y(W)-8 b(e)31 b(represen)m(t)g Fp(f)39 b Fr(in)29 b(the)i(form)1001 5544 y Fp(f)k Fr(=)25 b Fp(')p Fr(\()p Fp(\020)7 b Fr(\))1373 5520 y(~)1353 5544 y Fp(f)j(;)106 b(')p Fr(\()p Fp(\020)7 b Fr(\))26 b(=)f Fp(f)1882 5558 y Fm(0)1921 5544 y Fp(e)1963 5506 y Fm(2)p Fo(\031)r(n)p Fm(\()p Fo(\020)5 b Fk(\000)p Fo(\020)2233 5514 y Fd(m)2292 5506 y Fm(\))p Fo(=")2391 5544 y Fp(;)106 b(\020)32 b Fn(2)25 b Fp(V)5 b(;)-2999 b Fr(\(5.10\))-236 5714 y(where)34 b Fp(f)76 5728 y Fm(0)149 5714 y Fr(is)f(a)i(constan)m (t)h(factor)f(\(dep)s(ending)d(on)i(the)h(normalization)e(p)s(oin)m (ts\),)i(and)2845 5690 y(~)2825 5714 y Fp(f)44 b Fr(has)34 b(standard)g(b)s(eha)m(vior)f(with)-236 5822 y(the)d(same)h(leading)e (term)i(as)f Fp(f)835 5836 y Fm(+)924 5822 y Fr(in)f Fp(V)20 b Fr(.)41 b(This)29 b(and)g(the)i(standard)f(b)s(eha)m(vior)f (of)i Fp(f)2610 5836 y Fk(\006)2698 5822 y Fr(imply)994 6024 y Fp(a)26 b Fr(=)f Fp(')p Fr(\()p Fp(\020)7 b Fr(\)\(1)21 b(+)f Fp(o)p Fr(\(1\)\))p Fp(;)108 b(b)25 b Fr(=)g Fp(')p Fr(\()p Fp(\020)7 b Fr(\))p Fp(o)2254 5896 y Ff(\022)2322 6024 y Fp(e)2374 5948 y Fg(2)p Fd(i)p 2374 5960 53 3 v 2386 6001 a(")2449 5921 y Fb(R)2496 5942 y Fd(\020)2482 5999 y(\020)2510 6014 y Fg(0)2561 5975 y Fo(\024d\020)2678 5896 y Ff(\023)2760 6024 y Fp(:)-3021 b Fr(\(5.11\))1854 6210 y Fm(13)p eop %%Page: 14 14 14 13 bop -236 241 a Fr(T)-8 b(o)37 b(get)g(these)g(estimates,)i(w)m(e) e(ha)m(v)m(e)h(used)e Fn(j)p Fp(w)r Fr(\()p Fp(f)1475 255 y Fm(+)1534 241 y Fp(;)15 b(f)1619 255 y Fk(\000)1678 241 y Fr(\))p Fn(j)36 b Fp(>)g(c)p Fr(.)59 b(The)36 b(estimate)h (\(5.11\))i(are)e(uniform)d(for)i Fp(\020)42 b Fn(2)35 b Fp(V)3717 256 y Fo(\016)3780 241 y Fn(\\)23 b Fp(M)3952 256 y Fo(\016)3990 241 y Fr(,)-236 349 y(where)30 b Fp(V)80 364 y Fo(\016)148 349 y Fr(and)g Fp(M)413 364 y Fo(\016)481 349 y Fr(are)h(the)f Fp(\016)s Fr(-admissible)f(sub)s(domains)e(of)k Fp(V)50 b Fr(and)30 b Fp(M)41 b Fr(\()p Fp(\016)29 b(>)c Fr(0)31 b(\014xed\).)-236 504 y(The)36 b Fp(")p Fr(-p)s(erio)s(dicit)m (y)g(of)h Fp(a)g Fr(implies)d(that)k(its)f(asymptotics)g(remain)f(v)-5 b(alid)35 b(and)i(uniform)e(in)h(whole)g(strip)g(Im)14 b Fp(\020)3678 518 y Fo(m)3769 504 y Fr(+)25 b Fp(\016)40 b Fn(\024)-236 612 y Fr(Im)15 b Fp(\020)31 b Fn(\024)25 b Fp(\020)95 626 y Fo(M)194 612 y Fn(\000)20 b Fp(\016)s Fr(.)-236 767 y(The)42 b(analysis)f(of)h Fp(b)h Fr(is)e(the)i(main)e(p) s(oin)m(t)h(of)g(the)h(pro)s(of)f(of)g(Lemma)h(5.4.)78 b(Belo)m(w,)46 b Fp(C)j Fr(denotes)43 b(a)g(p)s(ositiv)m(e)e(constan)m (t)-236 875 y(indep)s(enden)m(t)28 b(of)i Fp(")p Fr(.)42 b(As)30 b Fp(b)g Fr(is)g Fp(")p Fr(-p)s(erio)s(dic,)e(the)j(estimate)g (for)f Fp(b)g Fr(from)g(\(5.11\))j(implies)28 b(that,)j(for)f (su\016cien)m(tly)f(small)g Fp(")p Fr(,)497 1081 y Fn(j)p Fp(b)p Fn(j)d(\024)e(j)p Fp(')p Fr(\()p Fp(\020)7 b Fr(\))p Fn(j)15 b Fp(C)7 b(e)1062 1044 y Fo(C)e(\016)r(=")1255 949 y Ff(\014)1255 1004 y(\014)1255 1059 y(\014)1255 1113 y(\014)1285 1081 y Fp(e)1337 1006 y Fg(2)p Fd(i)p 1337 1018 53 3 v 1349 1059 a(")1412 979 y Fb(R)1459 993 y Fd(\020)1487 1010 y(M)1445 1057 y(\020)1473 1072 y Fg(0)1570 1033 y Fo(\024d\020)1687 949 y Ff(\014)1687 1004 y(\014)1687 1059 y(\014)1687 1113 y(\014)1743 1081 y Fr(=)25 b Fp(C)7 b(e)1953 1044 y Fo(C)e(\016)r(=")2128 1081 y Fn(j)p Fp(A)p Fn(j)15 b(j)p Fp(B)5 b Fn(j)p Fp(;)107 b Fr(Im)15 b Fp(\020)32 b Fr(=)25 b(Im)15 b Fp(\020)2973 1095 y Fo(M)3072 1081 y Fn(\000)20 b Fp(\016)3203 1095 y Fm(1)3242 1081 y Fp(;)-3503 b Fr(\(5.12\))-236 1283 y(and)1014 1434 y Fn(j)p Fp(b)p Fn(j)26 b(\024)f(j)p Fp(')p Fr(\()p Fp(\020)7 b Fr(\))p Fn(j)15 b Fp(C)7 b(e)1580 1397 y Fo(C)e(\016)r(=")1757 1434 y Fn(j)p Fp(B)g Fn(j)p Fp(;)106 b Fr(Im)15 b Fp(\020)32 b Fr(=)25 b(Im)15 b Fp(\020)2468 1448 y Fo(m)2554 1434 y Fr(+)20 b Fp(\016)2685 1448 y Fm(1)2725 1434 y Fp(;)-2986 b Fr(\(5.13\))-236 1586 y(where)30 b Fp(\016)67 1600 y Fm(1)132 1586 y Fp(>)25 b(\016)34 b Fr(is)29 b(a)i(\014xed)f(n)m(um)m(b)s(er,)f(and)731 1799 y Fp(A)c Fr(=)g(exp)1074 1671 y Ff(\022)1151 1737 y Fr(2)p Fp(i)p 1151 1778 77 4 v 1168 1861 a(")1253 1675 y Ff(Z)1344 1701 y Fo(\020)1375 1712 y Fd(M)1303 1881 y Fo(\020)1334 1889 y Fd(m)1393 1881 y Fo(;)35 b Fm(along)i Fo(\015)1708 1799 y Fp(\024d\020)1854 1671 y Ff(\023)1937 1799 y Fp(;)106 b(B)30 b Fr(=)25 b(exp)2417 1671 y Ff(\022)2494 1737 y Fr(2)p Fp(i)p 2494 1778 V 2511 1861 a(")2596 1675 y Ff(Z)2686 1701 y Fo(\020)2717 1709 y Fd(m)2646 1881 y Fo(\020)2677 1890 y Fg(0)2795 1799 y Fp(\024d\020)2941 1671 y Ff(\023)3024 1799 y Fp(:)-236 2009 y Fr(As)30 b Fp(b)g Fr(is)g(analytic)g(in)f(the)h(strip)f(Im)15 b Fp(\020)1044 2023 y Fo(m)1136 2009 y Fp(<)25 b Fr(Im)14 b Fp(\020)32 b(<)25 b(\020)1563 2023 y Fo(M)1672 2009 y Fr(w)m(e)31 b(can)g(represen)m(t)f Fp(b)g Fr(in)f(the)i(form)1440 2176 y Fp(b)25 b Fr(=)g Fp(b)1639 2190 y Fk(\000)1718 2176 y Fr(+)20 b Fp(e)1861 2112 y Fg(2)p Fd(\031)r(i)p 1862 2124 91 3 v 1892 2165 a(")1962 2139 y Fm(\()p Fo(\020)5 b Fk(\000)p Fo(\020)2111 2147 y Fd(m)2170 2139 y Fm(\))2201 2176 y Fp(b)2240 2190 y Fm(+)2299 2176 y Fp(;)-236 2328 y Fr(where)31 b Fp(b)67 2342 y Fk(\006)158 2328 y Fr(are)h Fp(")p Fr(-p)s(erio)s(dic,)e Fp(b)798 2342 y Fk(\000)888 2328 y Fr(is)h(analytic)g(in)g(the)g(half-plane)f(Im)15 b Fp(\020)34 b(<)27 b Fr(Im)15 b Fp(\020)2476 2342 y Fo(M)2555 2328 y Fr(,)32 b(and)f Fp(b)2829 2342 y Fm(+)2919 2328 y Fr(is)g(analytic)g(in)g(the)g(half-plane)-236 2436 y(Im)15 b Fp(\020)38 b(>)31 b Fr(Im)15 b Fp(\020)232 2450 y Fo(m)298 2436 y Fr(.)53 b(One)34 b(can)g(obtain)g(estimates)h (of)f Fp(b)1579 2450 y Fk(\006)1673 2436 y Fr(b)m(y)g(estimating)g(the) g(F)-8 b(ourier)34 b(co)s(e\016cien)m(ts)h(of)g Fp(b)p Fr(.)52 b(Estimating)33 b(the)-236 2544 y(F)-8 b(ourier)29 b(co)s(e\016cien)m(ts)i(with)d(non-negativ)m(e)j(indexes)e(b)m(y)g (means)h(of)37 b(\(5.13\))32 b(and)d(the)h(F)-8 b(ourier)29 b(co)s(e\016cien)m(ts)i(with)d(negativ)m(e)-236 2652 y(indexes)h(b)m(y)h(means)h(of)37 b(\(5.12\))r(,)31 b(one)g(obtains)919 2814 y Fn(j)p Fp(b)983 2828 y Fk(\000)1043 2814 y Fn(j)25 b(\024)g(j)p Fp(')p Fr(\()p Fp(\020)7 b Fr(\))p Fn(j)p Fp(C)g(e)1529 2776 y Fo(C)e(\016)r(=")1706 2814 y Fn(j)p Fp(A)p Fn(j)15 b(j)p Fp(B)5 b Fn(j)p Fp(;)107 b Fr(Im)15 b Fp(\020)31 b Fn(\024)25 b Fr(Im)15 b Fp(\020)2550 2828 y Fo(M)2649 2814 y Fn(\000)20 b Fp(\016)2780 2828 y Fm(2)2820 2814 y Fp(;)985 2973 y Fn(j)p Fp(b)1049 2987 y Fm(+)1108 2973 y Fn(j)26 b(\024)f(j)p Fp(')p Fr(\()p Fp(\020)7 b Fr(\))p Fn(j)15 b Fp(C)7 b(e)1610 2935 y Fo(C)e(\016)r(=")1787 2973 y Fn(j)p Fp(B)g Fn(j)p Fp(;)106 b Fr(Im)15 b Fp(\020)31 b Fn(\025)25 b Fr(Im)15 b Fp(\020)2497 2987 y Fo(m)2584 2973 y Fr(+)20 b Fp(\016)2715 2987 y Fm(2)2754 2973 y Fp(;)-236 3124 y Fr(where)30 b Fp(\016)67 3138 y Fm(2)137 3124 y Fr(is)f(\014xed)h(constan)m(t)i(larger)e(than)g Fp(\016)1325 3138 y Fm(1)1365 3124 y Fr(.)40 b(Therefore,)388 3298 y Fn(j)p Fp(b)p Fn(j)26 b(\024)f Fp(C)7 b Fn(j)p Fp(')p Fr(\()p Fp(\020)g Fr(\))p Fn(j)15 b Fp(e)954 3261 y Fo(C)5 b(\016)r(=")1131 3298 y Fn(j)p Fp(B)g Fn(j)1285 3198 y Ff(\020)1340 3298 y Fn(j)p Fp(A)p Fn(j)20 b Fr(+)1569 3194 y Ff(\014)1569 3248 y(\014)1569 3303 y(\014)1600 3298 y Fp(e)1652 3234 y Fg(2)p Fd(\031)r(i)p 1652 3246 V 1683 3287 a(")1752 3261 y Fm(\()p Fo(\020)5 b Fk(\000)p Fo(\020)1901 3269 y Fd(m)1960 3261 y Fm(\))1992 3194 y Ff(\014)1992 3248 y(\014)1992 3303 y(\014)2022 3198 y(\021)2091 3298 y Fp(;)107 b Fr(Im)14 b Fp(\020)2386 3312 y Fo(m)2473 3298 y Fr(+)20 b Fp(\016)2604 3312 y Fm(2)2669 3298 y Fn(\024)25 b Fr(Im)15 b Fp(\020)31 b Fn(\024)25 b Fp(\020)3096 3312 y Fo(M)3195 3298 y Fn(\000)20 b Fp(\016)3326 3312 y Fm(2)3366 3298 y Fp(:)-3627 b Fr(\(5.14\))-236 3478 y(No)m(w,)35 b(the)f(represen)m(tation)g(\(5.9\))r(,)g(the)g (estimate)g(\(5.11\))i(for)e Fp(a)f Fr(and)g(\(5.14\))j(for)d Fp(b)p Fr(,)i(and)e(the)h(standard)f(b)s(eha)m(vior)f(of)i Fp(f)3957 3492 y Fk(\006)-236 3586 y Fr(imply)28 b(that,)j(uniformly)c (in)j(the)g Fp(\016)958 3600 y Fm(2)998 3586 y Fr(-admissible)e(sub)s (domain)f(of)k Fp(M)10 b Fr(,)161 3787 y Fp(f)35 b Fr(=)25 b Fp(')p Fr(\()p Fp(\020)7 b Fr(\))15 b Fp(e)583 3711 y Fd(i)p 581 3723 30 3 v 581 3764 a(")632 3684 y Fb(R)679 3705 y Fd(\020)665 3762 y(\020)693 3777 y Fg(0)744 3738 y Fo(\024d\020)876 3659 y Ff(\032)944 3787 y Fr(\011)1015 3801 y Fm(+)1074 3787 y Fr(\()p Fp(x;)g(\020)7 b(;)15 b(\020)1328 3801 y Fm(0)1368 3787 y Fr(\))20 b(+)g Fp(o)p Fr(\(1\))h(+)f Fp(O)1872 3659 y Ff(\022)1939 3787 y Fp(e)1981 3749 y Fo(C)5 b(\016)r(=")2157 3787 y Fp(e)2209 3717 y Fg(2)p Fd(i)p 2210 3729 53 3 v 2221 3770 a(")2284 3690 y Fb(R)2331 3704 y Fd(\020)2359 3721 y(M)2317 3767 y(\020)2443 3744 y Fo(\024d\020)2560 3659 y Ff(\023)2647 3787 y Fr(+)20 b Fp(O)2825 3686 y Ff(\020)2879 3787 y Fp(e)2921 3749 y Fo(C)5 b(\016)r(=")3097 3787 y Fp(e)3139 3743 y Fk(\000)3204 3716 y Fg(2)p Fd(i)p 3204 3728 V 3216 3769 a(")3279 3689 y Fb(R)3326 3710 y Fd(\020)3312 3767 y(\020)3340 3775 y(m)3403 3743 y Fm(\()p Fo(\024)p Fk(\000)p Fo(\031)r Fm(\))p Fo(d\020)3673 3686 y Ff(\021)3727 3659 y(\033)3810 3787 y Fp(;)-4071 b Fr(\(5.15\))-236 3988 y(where)32 b(w)m(e)i(in)m(tegrate)h(along)e(curv)m(es)g(in)f Fp(M)10 b Fr(.)49 b(The)33 b(last)g(t)m(w)m(o)i(terms)e(in)f(the)h(curly)f (brac)m(k)m(ets)i(are)g(small.)48 b(Indeed,)33 b(as)g(the)-236 4096 y(domain)g Fp(M)44 b Fr(is)33 b(canonical,)i(when)e(computing)g (these)h(t)m(w)m(o)i(terms,)f(w)m(e)f(can)h(in)m(tegrate)g(along)f(a)h (canonical)e(curv)m(e)i(going)-236 4204 y(from)28 b Fp(\020)17 4218 y Fo(m)112 4204 y Fr(to)i Fp(\020)262 4218 y Fo(M)369 4204 y Fr(via)e(the)h(p)s(oin)m(t)f Fp(\020)7 b Fr(.)40 b(It)29 b(follo)m(ws)f(from)g(the)h(de\014nition)d(of)j(canonical)g (curv)m(es)f(that)i(b)s(oth)e(the)h(exp)s(onen)m(tials)-236 4335 y Fp(E)-169 4349 y Fo(M)-55 4335 y Fr(=)36 b(exp)206 4234 y Ff(\020)270 4299 y Fm(2)p Fo(i)p 270 4314 60 4 v 283 4366 a(")354 4261 y Ff(R)415 4288 y Fo(\020)397 4367 y(\020)428 4378 y Fd(M)515 4335 y Fp(\024d\020)661 4234 y Ff(\021)752 4335 y Fr(and)g Fp(E)1002 4349 y Fo(m)1104 4335 y Fr(=)1210 4234 y Ff(\020)1265 4335 y Fn(\000)1346 4299 y Fm(2)p Fo(i)p 1346 4314 V 1359 4366 a(")1430 4261 y Ff(R)1490 4288 y Fo(\020)1473 4367 y(\020)1504 4375 y Fd(m)1566 4335 y Fr(\()p Fp(\024)21 b Fn(\000)f Fp(\031)s Fr(\))p Fp(d\020)1949 4234 y Ff(\021)2040 4335 y Fr(are)37 b(small)e(as)i Fp(")f Fn(!)f Fr(0.)60 b(W)-8 b(e)38 b(need)e(only)f(to) j(pro)m(v)m(e)f(that)-236 4485 y(they)g(are)h(small)e(enough)h(to)h (comp)s(ensate)g(the)f(factor)h Fp(e)1789 4452 y Fo(C)5 b(\016)r(=")1950 4485 y Fr(.)62 b(Fix)36 b Fp(\016)2246 4499 y Fm(3)2323 4485 y Fp(>)h Fr(0)g(su\016cien)m(tly)f(small,)i(but)e (indep)s(enden)m(t)g(of)-236 4615 y Fp(")p Fr(.)69 b(Consider)38 b(the)h(function)g Fp(i)851 4629 y Fo(M)930 4615 y Fr(\()p Fp(\020)7 b Fr(\))41 b(=)f(Im)1338 4514 y Ff(\020)1393 4542 y(R)1453 4568 y Fo(\020)1484 4579 y Fd(M)1436 4647 y Fo(\020)1571 4615 y Fp(\024d\020)1717 4514 y Ff(\021)1811 4615 y Fr(on)g(the)f(line)f(Im)15 b Fp(\020)47 b Fr(=)41 b(Im)15 b Fp(\020)2780 4629 y Fo(M)2885 4615 y Fn(\000)26 b Fp(\016)3022 4629 y Fm(3)3101 4615 y Fr(inside)38 b Fp(M)10 b Fr(.)68 b(Clearly)-8 b(,)41 b Fp(i)3936 4629 y Fo(M)-236 4746 y Fr(is)34 b(con)m(tin)m(uous)g(on)h(this)f(line.)52 b(As)35 b Fp(M)45 b Fr(is)34 b(canonical,)i(the)f(p)s(oin)m(ts)f Fp(\020)41 b Fr(and)34 b Fp(\020)2391 4760 y Fo(M)2505 4746 y Fr(can)h(b)s(e)f(connected)i(b)m(y)f(canonical)g(curv)m(es)-236 4854 y(inside)g Fp(M)10 b Fr(,)40 b(and,)g(therefore,)g(the)e(v)-5 b(alues)37 b(of)h Fp(i)1404 4868 y Fo(M)1521 4854 y Fr(are)g(p)s (ositiv)m(e.)62 b(So,)40 b(inside)c(a)i(\014xed)f(admissible)e(sub)s (domain,)i(on)g(the)-236 4962 y(line)j(Im)15 b Fp(\020)51 b Fr(=)44 b(Im)15 b Fp(\020)441 4976 y Fo(M)548 4962 y Fn(\000)27 b Fp(\016)686 4976 y Fm(3)726 4962 y Fr(,)45 b Fp(i)827 4976 y Fo(M)949 4962 y Fr(is)c(p)s(ositiv)m(e)g(and)g(b)s (ounded)f(a)m(w)m(a)m(y)k(from)d(zero)i(b)m(y)f(a)g(p)s(ositiv)m(e)f (constan)m(t)j(\(of)e(course,)-236 5111 y(indep)s(enden)m(t)28 b(of)i Fp(")p Fr(\).)42 b(As)30 b Fp(\016)700 5125 y Fm(2)740 5111 y Fr(,)61 b Fp(\016)866 5125 y Fm(1)936 5111 y Fr(and)30 b Fp(\016)k Fr(can)d(b)s(e)e(c)m(hosen)i(arbitrarily)d (small,)h(the)i(expression)e Fp(e)3106 5078 y Fo(C)5 b(\016)r(=")3282 5111 y Fp(e)3324 5060 y Fk(\000)3389 5033 y Fg(2)p Fd(i)p 3389 5045 53 3 v 3401 5086 a(")3464 5007 y Fb(R)3511 5027 y Fd(\020)3497 5085 y(\020)3525 5102 y(M)3609 5060 y Fo(\024d\020)3756 5111 y Fr(can)30 b(b)s(e)-236 5219 y(made)e(uniformly)d(small)h(for)h(all)g Fp(\020)34 b Fr(in)27 b(an)m(y)h(\014xed)f(admissible)e(sub)s(domain)g (of)j Fp(M)10 b Fr(.)40 b(Similarly)-8 b(,)25 b(one)j(studies)f(the)h (last)g(term)-236 5327 y(in)f(the)i(curly)f(brac)m(k)m(ets)i(in)e (\(5.15\))r(.)40 b(In)29 b(result,)f(one)h(sees)h(that,)g(uniformly)c (in)h(an)m(y)i(giv)m(en)g(admissible)d(sub)s(domain)g(of)j Fp(M)10 b Fr(,)1112 5519 y Fp(f)34 b Fr(=)25 b Fp(')p Fr(\()p Fp(\020)7 b Fr(\))15 b Fp(e)1533 5442 y Fd(i)p 1531 5454 30 3 v 1531 5495 a(")1582 5416 y Fb(R)1629 5436 y Fd(\020)1616 5494 y(\020)1644 5509 y Fg(0)1695 5469 y Fo(\024d\020)1826 5519 y Fr(\()q(\011)1933 5533 y Fm(+)1992 5519 y Fr(\()p Fp(x;)g(\020)7 b(;)15 b(\020)2246 5533 y Fm(0)2285 5519 y Fr(\))21 b(+)f Fp(o)p Fr(\(1\)\))d Fp(:)-236 5670 y Fr(This)29 b(pro)m(v)m(es)i(Lemma)g(5.4)h(in)e(the)h (case)h(of)f Fp(\033)e Fr(=)d(+1.)43 b(The)30 b(case)i(of)f Fp(\033)e Fr(=)d Fn(\000)p Fr(1)31 b(is)f(treated)i(similarly)-8 b(.)39 b(W)-8 b(e)32 b(note)g(only)e(that,)-236 5778 y(in)f(this)g(case,)j(one)e(starts)h(with)e(the)i(represen)m(tation) 1214 5935 y Fp(f)j Fr(=)40 b(^)-60 b Fp(')q Fr(\()p Fp(\020)7 b Fr(\))1586 5911 y(^)1566 5935 y Fp(f)i(;)121 b Fr(^)-59 b Fp(')25 b Fr(=)g Fp(f)1977 5949 y Fm(0)2016 5935 y Fp(e)2058 5897 y Fk(\000)p Fm(2)p Fo(\031)r(in)p Fm(\()p Fo(\020)5 b Fk(\000)p Fo(\020)2407 5905 y Fd(m)2466 5897 y Fm(\))p Fo(=")-236 6110 y Fr(where)26 b Fp(f)68 6124 y Fm(0)134 6110 y Fr(is)f(constan)m(t,)k(and,)e(in)f Fp(V)20 b Fr(,)28 b(the)e(solution)1550 6087 y(^)1531 6110 y Fp(f)36 b Fr(has)26 b(standard)g(b)s(eha)m(vior)g(with)f(the)i (same)g(leading)f(term)g(as)h Fp(f)3770 6124 y Fk(\000)3829 6110 y Fr(.)p 3950 6110 4 62 v 3954 6052 55 4 v 3954 6110 V 4008 6110 4 62 v 1854 6210 a Fm(14)p eop %%Page: 15 15 15 14 bop 919 241 a Fr(6.)51 b Fq(Asymptotics)32 b(of)i(the)f(monodr)n (omy)g(ma)-6 b(trix)-236 403 y Fr(This)33 b(section)j(is)e(dev)m(oted)i (to)g(the)f(pro)s(of)g(of)g(Theorem)g(1.3.)57 b(Therefore,)36 b(one)g(\014rst)e(p)s(erforms)g(the)h(c)m(hange)i(of)e(v)-5 b(ariable)-236 511 y Fp(x)25 b Fr(:=)g Fp(x)20 b Fn(\000)g Fp(z)35 b Fr(and)30 b Fp(\020)h Fr(:=)26 b Fp("z)35 b Fr(in)29 b(\(0.1\))q(.)41 b(One)30 b(sees)h(that)-111 640 y Fn(\017)42 b Fr(equation)30 b(\(0.1\))i(b)s(ecomes)f(\(3.2\))q(;) -111 748 y Fn(\017)42 b Fr(the)30 b(consistency)g(condition)f(\(1.6\))k (b)s(ecomes)d(\(3.3\))r(;)-111 856 y Fn(\017)42 b Fr(if)26 b Fp(f)101 870 y Fk(\006)187 856 y Fr(is)g(a)i(consisten)m(t)g(basis)e (of)h(solutions)f(of)35 b(\(0.1\))r(,)28 b(in)e(the)h(new)g(v)-5 b(ariables,)27 b(the)h(de\014nition)d(of)i(the)h(corresp)s(onding)-24 964 y(mono)s(drom)m(y)h(matrix)h(b)s(ecomes)1027 1164 y Fp(F)13 b Fr(\()p Fp(x;)i(\020)27 b Fr(+)20 b(2)p Fp(\031)s Fr(\))26 b(=)1674 1141 y(~)1640 1164 y Fp(M)10 b Fr(\()p Fp(\020)d Fr(\))p Fp(F)13 b Fr(\()p Fp(x;)i(\020)7 b Fr(\))p Fp(;)108 b(F)38 b Fr(=)2460 1036 y Ff(\022)2527 1109 y Fp(f)2572 1123 y Fm(+)2527 1217 y Fp(f)2572 1231 y Fk(\000)2630 1036 y Ff(\023)2712 1164 y Fr(;)-2973 b(\(6.1\))-111 1386 y Fn(\017)42 b Fr(the)30 b(mono)s(drom)m(y)g (matrix)965 1363 y(~)931 1386 y Fp(M)40 b Fr(is)29 b Fp(")p Fr(-p)s(erio)s(dic)g(in)g Fp(\020)7 b Fr(.)-236 1515 y(T)-8 b(o)28 b(the)g(\\new")g(equation)f(\(3.2\))r(,)i(w)m(e)f (apply)e(the)i(complex)f(WKB)h(metho)s(d)f(to)i(calculate)f(the)g(mono) s(drom)m(y)f(matrix)g(for)g(a)-236 1623 y(consisten)m(t)i(basis)f (constructed)h(b)m(y)g(means)g(of)g(Theorem)g(3.1.)41 b(Our)28 b(plan)f(is)h(the)h(follo)m(wing.)39 b(First,)29 b(w)m(e)g(\014nd)f(a)h(canonical)-236 1731 y(line.)56 b(Then,)36 b(in)f(a)h(lo)s(cal)f(canonical)h(domain)f(enclosing)g(this) g(line,)g(using)g(Theorem)h(3.1,)i(w)m(e)f(construct)f(a)g(consisten)m (t)-236 1839 y(basis.)51 b(Then,)35 b(w)m(e)g(apply)e(our)h(con)m(tin)m (uation)g(to)s(ols)g(and)g(\014nd)f(a)i(large)f(enough)g(domain)f Fp(D)s Fr(,)j(where)e(b)s(oth)f Fp(F)13 b Fr(\()p Fp(x;)i(\020)7 b Fr(\))35 b(and)-236 1947 y Fp(F)13 b Fr(\()p Fp(x;)i(\020)27 b Fr(+)21 b(2)p Fp(\031)s Fr(\))31 b(ha)m(v)m(e)h(standard)e(b)s(eha)m (vior.)41 b(A)m(t)31 b(last,)g(w)m(e)g(compute)g(the)g(mono)s(drom)m(y) f(matrix.)41 b(T)-8 b(o)32 b(carry)e(out)h(this)f(plan,)-236 2055 y(w)m(e)f(b)s(egin)e(with)h(discussing)e(prop)s(erties)h(of)i(the) g(complex)g(momen)m(tum)g(in)e(a)i(neigh)m(b)s(orho)s(o)s(d)e(of)i(the) g(real)f(line)f(under)g(the)-236 2163 y(condition)i(\(1.3\))r(.)-236 2318 y(In)g(the)i(sequel)f(w)m(e)h(\014x)e Fp(E)i Fr(=)25 b Fp(E)828 2332 y Fm(0)898 2318 y Fr(in)k(the)h(in)m(terv)-5 b(al)30 b Fp(J)39 b Fr(satisfying)29 b(assumption)g(\(A\).)-236 2490 y(6.1.)53 b Fu(The)32 b(complex)g(momen)m(tum)e(and)i(a)f (canonical)i(line.)46 b Fr(Recall)27 b(that)i(the)f(complex)f(momen)m (tum)h(is)f(related)h(to)-236 2598 y(the)h(Blo)s(c)m(h)h(quasi-momen)m (tum)e(of)i(the)g(p)s(erio)s(dic)c(Sc)m(hr\177)-45 b(odinger)28 b(op)s(erator)i(\(0.2\))h(b)m(y)f(the)f(form)m(ula)g(\(3.4\))r(.)40 b(F)-8 b(or)30 b(real)f(v)-5 b(alues)-236 2706 y(of)30 b Fp(E)5 b Fr(,)30 b(the)g(pre-image)g(of)g(the)g(sp)s(ectral)g(axis)f (with)f(resp)s(ect)i(to)h(the)f(mapping)e Fn(E)34 b Fr(:)40 b Fp(\020)32 b Fn(7!)25 b Fp(E)g Fn(\000)18 b Fp(W)13 b Fr(\()p Fp(\020)7 b Fr(\))30 b(is)f(the)h(set)g Fp(W)3760 2673 y Fk(\000)p Fm(1)3854 2706 y Fr(\()p Fl(R)s Fr(\).)-236 2814 y(This)e(set)j(pla)m(ys)f(an)g(imp)s(ortan)m(t)g(part)g(for)g(the) h(analytic)f(prop)s(erties)f(of)h(the)h(complex)f(momen)m(tum.)-236 2986 y(6.1.1.)48 b Fi(The)30 b(set)f Fp(W)442 2953 y Fk(\000)p Fm(1)536 2986 y Fr(\()p Fl(R)s Fr(\))p Fi(.)52 b Fr(As)26 b Fp(W)39 b Fr(is)26 b(2)p Fp(\031)s Fr(-p)s(erio)s(dic)f (and)h(real)g(analytic,)h(the)g(set)h Fp(W)2670 2953 y Fk(\000)p Fm(1)2763 2986 y Fr(\()p Fl(R)s Fr(\))33 b(is)26 b(a)h(2)p Fp(\031)s Fr(-p)s(erio)s(dic)e(analytic)h(set)-236 3093 y(symmetric)j(with)h(resp)s(ect)g(to)h(the)g(real)f(axis.)40 b(It)30 b(consists)g(of)-137 3223 y(1.)43 b(the)30 b(real)g(line.)-137 3331 y(2.)43 b(complex)30 b(branc)m(hes)g(b)s(eginning)d(at)k(an)m(y)g (extrem)m(um)g(of)f Fp(W)43 b Fr(along)30 b(the)h(real)f(line:)51 3439 y Fn(\017)42 b Fr(eac)m(h)31 b(of)g(these)g(branc)m(hes)f(is)f(an) h(analytic)g(curv)m(e)h(b)s(eginning)d(at)j(an)f(extrem)m(um;)51 3547 y Fn(\017)42 b Fr(at)31 b(the)f(real)g(extrem)m(um,)h(the)g(branc) m(hes)f(are)h(transv)m(ersal)f(to)h(the)f(real)g(line,)f(and)h(transv)m (ersal)g(to)h(eac)m(h)h(other.)-137 3655 y(3.)43 b(complex)30 b(branc)m(hes)g(of)g Fp(W)911 3622 y Fk(\000)p Fm(1)1005 3655 y Fr(\()p Fl(R)s Fr(\))37 b(separated)31 b(from)e(the)i(real)f (line.)-236 3827 y(6.1.2.)48 b Fi(The)39 b(c)-5 b(omplex)41 b(momentum.)47 b Fr(The)36 b(real)h(line)e(b)s(elongs)h(to)i(the)g (pre-image)f(of)g(the)h Fp(n)3005 3841 y Fm(0)3044 3827 y Fr(-th)f(sp)s(ectral)f(band)g(of)h(the)-236 3935 y(p)s(erio)s(dic)27 b(Sc)m(hr\177)-45 b(odinger)29 b(op)s(erator)i(\(0.2\))h(with)d(resp)s (ect)i(to)g(the)f(mapping)f Fn(E)k Fr(:)56 b Fp(\020)32 b Fn(!)25 b Fp(E)g Fn(\000)20 b Fp(W)13 b Fr(\()p Fp(\020)7 b Fr(\).)41 b(F)-8 b(or)31 b Fp(Y)45 b(>)25 b Fr(0,)31 b(de\014ne)e(the)-236 4042 y(strip)g Fn(S)32 4056 y Fo(Y)118 4042 y Fr(=)c Fn(f\000)p Fp(Y)45 b Fn(\024)25 b Fr(Im)15 b Fp(\020)31 b Fn(\024)25 b Fp(Y)20 b Fn(g)p Fr(.)42 b(Let)30 b Fp(Z)i Fr(=)25 b Fn(S)1407 4056 y Fo(Y)1488 4042 y Fn(\\)20 b Fp(W)1668 4009 y Fk(\000)p Fm(1)1762 4042 y Fr(\()p Fl(R)s Fr(\).)47 b(W)-8 b(e)31 b(c)m(ho)s(ose)h Fp(Y)50 b Fr(su\016cien)m(tly)29 b(small)g(so)i(that)-111 4172 y Fn(\017)42 b Fr(the)30 b(strip)f Fn(S)400 4186 y Fo(Y)491 4172 y Fr(is)h(con)m(tained)g(in)f(the)i(domain)e(of)i (analyticit)m(y)f(of)g Fp(W)13 b Fr(;)-111 4280 y Fn(\017)42 b Fr(the)32 b(set)g Fp(Z)38 b Fr(consists)31 b(of)h(the)g(real)g(line)e (and)h(of)h(complex)f(lines)f(b)s(eginning)f(at)k(the)f(extrema)g(of)g Fp(W)44 b Fr(situated)32 b(along)-24 4388 y(the)e(real)g(line,)-111 4496 y Fn(\017)42 b Fr(these)30 b(complex)h(lines)d(do)i(not)h(in)m (tersect)g(outside)f(of)g Fl(R)39 b Fr(and)30 b(are)h Fi(vertic)-5 b(al)p Fr(,)-111 4604 y Fn(\017)42 b Fr(the)34 b(image)h(of)f Fp(Z)41 b Fr(b)m(y)34 b Fn(E)41 b Fr(:)32 b Fp(\020)38 b Fn(!)32 b Fp(E)c Fn(\000)23 b Fp(W)13 b Fr(\()p Fp(\020)7 b Fr(\))34 b(is)f(con)m(tained)i(in)e(the)h Fp(n)2365 4618 y Fm(0)2404 4604 y Fr(-th)h(sp)s(ectral)e(band)h(and)g (the)g(distance)g(from)-24 4711 y(this)29 b(image)i(to)g(the)f(ends)g (of)g(the)h(band)e(is)h(p)s(ositiv)m(e.)-236 4888 y(Note)h(that)f(the)f (last)h(condition)e(implies)f(that)j(the)f(branc)m(h)g(p)s(oin)m(ts)g (of)g(the)h(complex)f(momen)m(tum)h(and)f(the)g(p)s(oles)g(of)g(the) -236 4996 y(functions)g Fp( )215 5010 y Fk(\006)274 4996 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))31 b(and)f Fp(!)748 5010 y Fk(\006)806 4996 y Fr(\()p Fp(\020)7 b Fr(\))31 b(sta)m(y)g(outside)f(of)g Fn(S)1618 5010 y Fo(Y)1679 4996 y Fr(.)-236 5151 y(Fix)g(a)g(branc)m(h)g(of)h(the)f(complex)g (momen)m(tum)h(analytic)f(in)f(the)h(strip)f Fn(S)2273 5165 y Fo(Y)2364 5151 y Fr(b)m(y)h(the)h(form)m(ula)1426 5302 y Fp(\024)1478 5316 y Fo(p)1518 5302 y Fr(\()p Fp(\020)7 b Fr(\))25 b(=)g Fp(k)1803 5316 y Fo(p)1843 5302 y Fr(\()p Fp(E)h Fn(\000)20 b Fp(W)13 b Fr(\()p Fp(\020)7 b Fr(\)\))p Fp(;)-2574 b Fr(\(6.2\))-236 5453 y(where)29 b Fp(k)73 5467 y Fo(p)142 5453 y Fr(is)g(the)g(branc)m(h)g(of)h(the)g(Blo)s(c)m (h)f(quasi-momen)m(tum)g(of)g(the)h(p)s(erio)s(dic)d(Sc)m(hr\177)-45 b(odinger)28 b(op)s(erator)i(describ)s(ed)d(in)h(the)-236 5561 y(sections)i(2.2)i(and)d(2.3.)-236 5725 y Fu(Lemma)j(6.1.)47 b Fi(The)32 b(br)-5 b(anch)34 b Fp(\024)874 5739 y Fo(p)947 5725 y Fi(has)g(the)f(fol)5 b(lowing)33 b(pr)-5 b(op)g(erties:)-137 5855 y Fr(1.)43 b Fp(\024)28 5869 y Fo(p)100 5855 y Fi(takes)33 b(r)-5 b(e)g(al)34 b(values)f(if)f(and)i(only)f(if)g Fp(\020)e Fn(2)25 b Fp(Z)7 b Fi(;)-137 5963 y Fr(2.)43 b Fp(\031)s Fr(\()p Fp(n)20 b Fn(\000)g Fr(1\))26 b Fp(<)f(\024)486 5977 y Fo(p)526 5963 y Fr(\()p Fp(\020)7 b Fr(\))25 b Fp(<)g(\031)s(n)32 b Fi(for)i(al)5 b(l)33 b Fp(\020)e Fn(2)25 b Fp(Z)7 b Fi(;)-137 6084 y Fr(3.)43 b Fp(\024)28 6098 y Fo(p)68 6084 y Fr(\()p 103 6010 47 4 v Fp(\020)7 b Fr(\))25 b(=)p 306 6005 210 4 v 25 w Fp(\024)358 6098 y Fo(p)398 6084 y Fr(\()p Fp(\020)7 b Fr(\))p Fi(.)1854 6210 y Fm(15)p eop %%Page: 16 16 16 15 bop -236 241 a Fu(Pro)s(of.)47 b Fr(P)m(oin)m(t)33 b(1)g(directly)e(follo)m(ws)g(from)h(the)h(de\014nitions)c(of)k Fp(\024)2025 255 y Fo(p)2097 241 y Fr(and)f(the)h(prop)s(erties)d(of)j Fp(k)3017 255 y Fo(p)3089 241 y Fr(\(see)h(section)e(2.3\).)48 b(P)m(oin)m(t)-236 349 y(2)31 b(follo)m(ws)e(from)h(the)g(last)h(prop)s (ert)m(y)e(of)i(the)g(strip)d Fn(S)1581 363 y Fo(Y)1642 349 y Fr(.)41 b(P)m(oin)m(t)31 b(3)f(holds)f(as)i Fp(\024)2429 363 y Fo(p)2499 349 y Fr(is)f(analytic)g(in)f Fn(S)3098 363 y Fo(Y)3189 349 y Fr(and)g(tak)m(es)j(real)e(v)-5 b(alues)-236 457 y(on)30 b Fl(R)s Fr(.)p 3950 457 4 62 v 3954 399 55 4 v 3954 457 V 4008 457 4 62 v -236 655 a(6.1.3.)48 b Fi(Canonic)-5 b(al)41 b(line.)46 b Fr(T)-8 b(o)38 b(\014nd)e(a)i(canonical)f(line,)h(let)g(us)f(discuss)e(the)j (function)e Fp(W)13 b Fr(.)62 b(Let)38 b Fp(\020)3196 669 y Fm(0)3273 655 y Fn(2)f Fl(R)46 b Fr(b)s(e)37 b(one)h(of)f(the) -236 763 y(minima)31 b(of)i Fp(W)13 b Fr(.)49 b(As)33 b Fp(W)45 b Fr(is)32 b(analytic)h(and)g(non-constan)m(t,)i(near)e Fp(\020)2082 777 y Fm(0)2121 763 y Fr(,)h(one)g(has)e Fp(W)13 b Fr(\()p Fp(\020)7 b Fr(\))22 b Fn(\000)g Fp(W)13 b Fr(\()p Fp(\020)3019 777 y Fm(0)3058 763 y Fr(\))30 b Fn(\030)g Fp(W)3310 777 y Fo(n)3357 763 y Fr(\()p Fp(\020)f Fn(\000)21 b Fp(\020)3593 777 y Fm(0)3632 763 y Fr(\))3667 730 y Fm(2)p Fo(n)3783 763 y Fr(where)-236 871 y Fp(W)-150 885 y Fo(n)-73 871 y Fr(is)30 b(p)s(ositiv)m(e)g(and)g Fp(n)h Fr(is)e(a)j(p)s(ositiv)m(e)d(in)m(teger.)43 b(This)29 b(implies)f(that)j(there)g(are)h(2)p Fp(n)20 b Fn(\000)g Fr(1)32 b(complex)e(branc)m(hes)h(of)g Fp(W)3786 838 y Fk(\000)p Fm(1)3879 871 y Fr(\()p Fl(R)s Fr(\))-236 979 y(b)s(eginning)d(at)33 b Fp(\020)334 993 y Fm(0)404 979 y Fr(and)e(going)g(up)m(w)m(ards.)43 b(One)31 b(of)h(these)g(lines) d(is)i(orthogonal)h(to)g Fl(R)40 b Fr(at)32 b Fp(\020)2873 993 y Fm(0)2912 979 y Fr(.)44 b(Denote)33 b(it)e(b)m(y)g Fp(C)3578 993 y Fm(0)3618 979 y Fr(.)43 b(The)31 b(line)p -236 1014 105 4 v -236 1087 a Fp(C)-171 1101 y Fm(0)-132 1087 y Fr(,)g(symmetric)e(of)i Fp(C)535 1101 y Fm(0)605 1087 y Fr(with)e(resp)s(ect)h(to)h Fl(R)s Fr(,)36 b(is)30 b(also)g(a)h(branc)m(h)f(of)g Fp(W)2208 1054 y Fk(\000)p Fm(1)2302 1087 y Fr(\()p Fl(R)s Fr(\).)47 b(De\014ne)1515 1254 y Fp(\014)30 b Fr(=)25 b Fp(C)1757 1268 y Fm(0)1817 1254 y Fn([)20 b(f)p Fr(0)p Fn(g)h([)p 2135 1181 V 20 w Fp(C)2200 1268 y Fm(0)2239 1254 y Fp(:)-2500 b Fr(\(6.3\))-236 1431 y(Clearly)-8 b(,)29 b Fp(\014)36 b Fr(is)29 b(a)i(v)m(ertical)g Fp(C)747 1398 y Fm(1)785 1431 y Fr(-curv)m(e.)-236 1586 y(Describ)s(e)g(the)i(branc)m(h)e(of)i(the)f(complex)g(momen)m(tum)g (with)f(resp)s(ect)h(to)h(whic)m(h)e Fp(\014)37 b Fr(is)32 b(canonical.)46 b(The)31 b(branc)m(hes)h(of)g(the)-236 1694 y(complex)e(momen)m(tum)g(analytic)g(in)f Fn(S)1104 1708 y Fo(Y)1195 1694 y Fr(is)h(describ)s(ed)e(b)m(y)j(\(3.6\))r(.)41 b(Therefore,)30 b(the)h(function)e Fp(\024)3071 1708 y Fo(n)3114 1717 y Fg(0)3153 1694 y Fr(\()p Fp(\020)7 b Fr(\))30 b(de\014ned)g(b)m(y)1061 1913 y Fp(\024)1113 1927 y Fo(n)1156 1936 y Fg(0)1220 1913 y Fr(=)1316 1785 y Ff(\032)1426 1858 y Fp(\024)1478 1872 y Fo(p)1538 1858 y Fn(\000)20 b Fp(\031)s Fr(\()p Fp(n)1774 1872 y Fm(0)1834 1858 y Fn(\000)f Fr(1\))p Fp(;)115 b Fr(if)36 b Fp(n)2289 1872 y Fm(0)2359 1858 y Fr(is)29 b(o)s(dd)o Fp(;)1426 1966 y(\031)s(n)1536 1980 y Fm(0)1595 1966 y Fn(\000)20 b Fp(\024)1738 1980 y Fo(p)1778 1966 y Fp(;)341 b Fr(if)36 b Fp(n)2289 1980 y Fm(0)2359 1966 y Fr(is)29 b(ev)m(en)q Fp(;)-236 1913 y Fr(\(6.4\))-236 2127 y(is)g(a)i(branc)m(h)f(of)g(the)h (complex)f(momen)m(tum)g(analytic)g(in)f Fn(S)1832 2141 y Fo(Y)1893 2127 y Fr(.)41 b(In)29 b(view)h(of)g(p)s(oin)m(t)g(2)h(in)e (Lemma)h(6.1,)i(one)e(has)1379 2291 y(0)25 b Fp(<)g(\024)1597 2305 y Fo(n)1640 2314 y Fg(0)1679 2291 y Fr(\()p Fp(\020)7 b Fr(\))26 b Fp(<)f(\031)s(;)106 b Fn(8)p Fp(\020)31 b Fn(2)25 b Fp(Z)q(:)-2636 b Fr(\(6.5\))-236 2460 y(As)30 b Fp(\014)36 b Fr(is)29 b(v)m(ertical,)i(equation)f(\(6.5\))i(implies)c (that,)j(along)f Fp(\014)5 b Fr(,)733 2629 y Fp(d)p 709 2669 96 4 v 709 2753 a(dy)830 2562 y Ff(\022)897 2690 y Fr(Im)1020 2567 y Ff(Z)1111 2593 y Fo(\020)1166 2690 y Fp(\024)1218 2704 y Fo(n)1261 2713 y Fg(0)1315 2690 y Fp(d\020)1409 2562 y Ff(\023)1501 2690 y Fp(>)25 b Fr(0)p Fp(;)46 b Fr(and)1924 2629 y Fp(d)p 1900 2669 V 1900 2753 a(dy)2020 2562 y Ff(\022)2087 2690 y Fr(Im)2211 2567 y Ff(Z)2302 2593 y Fo(\020)2342 2690 y Fr(\()p Fp(\024)2429 2704 y Fo(n)2472 2713 y Fg(0)2531 2690 y Fn(\000)20 b Fp(\031)s Fr(\))15 b Fp(d\020)2821 2562 y Ff(\023)2913 2690 y Fp(>)25 b Fr(0)p Fp(:)-236 2904 y Fr(So,)30 b Fp(\014)36 b Fr(is)29 b(canonical)h(with)f(resp)s(ect)i(to)g Fp(\024)1174 2918 y Fo(n)1217 2927 y Fg(0)1256 2904 y Fr(.)-236 3102 y(6.2.)53 b Fu(Solution)29 b Fp(f)10 b Fu(.)45 b Fr(W)-8 b(e)25 b(no)m(w)g(de\014ne)e(and)h(study)g(one)g(of)h (the)f(solutions)f(of)i(the)f(consisten)m(t)h(basis)e(for)h(whic)m(h)g (w)m(e)g(compute)-236 3210 y(the)30 b(mono)s(drom)m(y)g(matrix.)-236 3409 y(6.2.1.)48 b Fi(L)-5 b(o)g(c)g(al)38 b(c)-5 b(anonic)g(al)37 b(domain.)47 b Fr(The)33 b(curv)m(e)h Fp(\014)k Fr(b)s(eing)32 b(canonical,)i(b)m(y)g(Lemma)f(4.1,)j(there)d(exists)h(a)f(lo)s(cal)g (canonical)-236 3517 y(domain)d Fp(K)37 b Fr(enclosing)30 b Fp(\014)5 b Fr(.)43 b(Using)30 b(Theorem)g(3.1,)j(for)e Fp(\020)h Fn(2)26 b Fp(K)7 b Fr(,)31 b(w)m(e)g(construct)h Fp(f)10 b Fr(,)30 b(a)i(solution)d(of)i(equation)g(\(3.2\))r(,)g(ha)m (ving)-236 3652 y(standard)e(b)s(eha)m(vior)h Fp(f)k Fn(\030)25 b Fr(exp)844 3551 y Ff(\020)913 3617 y Fo(i)p 909 3632 33 4 v 909 3684 a(")966 3579 y Ff(R)1027 3605 y Fo(\020)1009 3684 y Fm(0)1082 3652 y Fp(\024d\020)1228 3551 y Ff(\021)1297 3652 y Fr(\011)1368 3666 y Fm(+)1427 3652 y Fr(.)-236 3830 y(Fix)32 b Fp(\016)i(>)c Fr(0.)49 b(Then,)33 b(for)g(su\016cien)m(tly)f(small)f Fp(")p Fr(,)k(the)e(solution)f Fp(f)42 b Fr(is)32 b(analytic)h(in)f(the)h (strip)f Fn(S)3006 3845 y Fo(Y)15 b Fk(\000)p Fo(\016)3155 3830 y Fr(,)34 b(see)g(the)f(b)s(eginning)e(of)-236 3938 y(the)36 b(section)f(5.)57 b(Belo)m(w,)38 b(to)e(simplify)d(the)i (notations,)j(w)m(e)e(write)f Fp(Y)55 b Fr(instead)35 b(of)h Fp(Y)44 b Fn(\000)23 b Fp(\016)s Fr(,)38 b Fn(S)2986 3952 y Fo(Y)3082 3938 y Fr(instead)d(of)h Fn(S)3565 3953 y Fo(Y)16 b Fk(\000)p Fo(\016)3750 3938 y Fr(and)35 b Fp(K)-236 4046 y Fr(instead)29 b(of)i Fp(K)258 4061 y Fo(\016)296 4046 y Fr(,)g(the)f Fp(\016)s Fr(-admissible)f(sub)s (domain)e(of)k Fp(K)7 b Fr(.)-236 4244 y(6.2.2.)48 b Fi(Asymptotics)38 b(of)e Fp(f)46 b Fi(outside)37 b(of)f Fp(K)7 b Fi(.)45 b Fr(Here,)37 b(w)m(e)d(study)g(the)g(asymptotics)h (of)f Fp(f)44 b Fr(in)33 b(the)h(strip)f Fn(S)3369 4258 y Fo(Y)3464 4244 y Fr(outside)h(of)g(the)-236 4352 y(domain)29 b Fp(K)7 b Fr(.)40 b(W)-8 b(e)32 b(pro)m(v)m(e)-236 4525 y Fu(Prop)s(osition)k(6.1.)46 b Fi(F)-7 b(or)38 b(any)f(\014xe)-5 b(d)38 b(p)-5 b(ositive)38 b Fp(l)c Fn(2)f Fl(N)6 b Fi(,)44 b(ther)-5 b(e)37 b(exists)g Fp(Y)2257 4540 y Fo(l)2316 4525 y Fp(>)32 b Fr(0)37 b Fi(such)g(that,)i(for)f Fp(")f Fi(su\016ciently)g(smal)5 b(l,)38 b(the)-236 4632 y(solution)c Fp(f)41 b Fi(has)34 b(the)f(standar)-5 b(d)36 b(b)-5 b(ehavior)34 b(in)e(the)h(domain)h Fp(R)1894 4647 y Fo(l)1953 4632 y Fi(b)-5 b(ounde)g(d)34 b(by)f(the)g(lines)f Fp(\014)5 b Fi(,)33 b Fp(\014)25 b Fr(+)20 b(2)p Fp(\031)s(l)35 b Fi(and)f Fr(Im)15 b Fp(\020)31 b Fr(=)25 b Fn(\006)p Fp(Y)3820 4647 y Fo(l)3846 4632 y Fi(.)-236 4805 y Fr(The)43 b(rest)g(of)g(this)g(subsection)f(is)g(dev)m(oted)j(to)f(the)f(pro)s (of)g(of)g(Prop)s(osition)f(6.1.)80 b(W)-8 b(e)45 b(start)e(with)f(a)i (more)g(detailed)-236 4913 y(discussion)28 b(of)i(the)h(set)g Fp(Z)7 b Fr(.)-236 5111 y(6.2.3.)48 b Fi(The)33 b(structur)-5 b(e)33 b(of)g(the)g(set)g Fp(Z)7 b Fi(.)45 b Fr(Recall)30 b(that)h Fp(\020)1644 5125 y Fm(0)1713 5111 y Fr(is)f(a)g(minim)m(um)e (of)i Fp(W)13 b Fr(.)40 b(Denote)32 b(b)m(y)e(\()p Fp(\020)3067 5125 y Fo(i)3096 5111 y Fr(\))3131 5125 y Fo(i)p Fk(2)p Fh(Z)3251 5111 y Fr(,)g(the)h(extrema)g(of)f Fp(W)-236 5219 y Fr(on)i Fl(R)s Fr(.)54 b(W)-8 b(e)34 b(order)e(them)h (increasingly)-8 b(.)46 b(Clearly)-8 b(,)32 b(the)h(set)g(of)g(the)g (extrema)h(is)d(2)p Fp(\031)s Fr(-p)s(erio)s(dic:)44 b(there)33 b(exists)f Fp(l)3584 5233 y Fm(0)3653 5219 y Fn(2)d Fl(N)45 b Fr(suc)m(h)-236 5327 y(that)31 b Fp(\020)1 5342 y Fo(l)q Fm(+)p Fo(l)99 5351 y Fg(0)162 5327 y Fr(=)25 b Fp(\020)298 5342 y Fo(l)344 5327 y Fr(+)20 b(2)p Fp(\031)34 b Fr(for)c(all)f Fp(l)f Fn(2)c Fl(Z)p Fr(.)-236 5435 y(Pic)m(k)31 b Fp(l)f Fn(2)d Fl(Z)p Fr(.)40 b(W)-8 b(e)33 b(ha)m(v)m(e,)g(sa)m(y)-8 b(,)34 b Fp(n)869 5450 y Fo(l)926 5435 y Fr(branc)m(hes)d(of)h(the)g(set)g Fp(Z)38 b Fr(starting)32 b(at)g Fp(\020)2303 5450 y Fo(l)2361 5435 y Fr(and)f(situated)g(in)f Fl(C)3056 5449 y Fm(+)3121 5435 y Fr(,)i(the)g(upp)s(er)e(half)g (plane.)-236 5543 y(These)i(branc)m(hes)f(are)i(v)m(ertical)f(and)g(ha) m(v)m(e)h(only)e(one)h(common)h(p)s(oin)m(t)e Fp(\020)2290 5558 y Fo(l)2316 5543 y Fr(.)46 b(Recall)31 b(that,)j(at)e Fp(\020)3039 5558 y Fo(l)3065 5543 y Fr(,)h(eac)m(h)g(of)f(these)h (branc)m(hes)-236 5651 y(is)i(transv)m(ersal)h(to)i(the)e(real)g(line)f (as)i(w)m(ell)e(as)i(to)g(all)f(the)g(other)h(branc)m(hes;)i(moreo)m(v) m(er,)h(our)c(c)m(hoice)h(of)g Fn(S)3496 5665 y Fo(Y)3593 5651 y Fr(guaran)m(tees)-236 5759 y(that)28 b(the)f(branc)m(hes)g(b)s (eginning)d(at)k(di\013eren)m(t)f(extrema)h(of)f Fp(W)40 b Fr(do)27 b(not)h(in)m(tersect)f(in)f(the)i(strip)d Fn(S)3102 5773 y Fo(Y)3163 5759 y Fr(.)40 b(Denote)29 b(the)e(complex)-236 5867 y(branc)m(hes)j(of)g Fp(Z)37 b Fr(b)s(eginning)28 b(at)j(the)g(extrema)g(of)f Fp(W)43 b Fr(and)30 b(going)g(up)m(w)m(ards)g(b)m(y)g(\()p Fp(C)2616 5881 y Fo(j)2653 5867 y Fr(\))2688 5881 y Fo(j)t Fk(2)p Fh(Z)2816 5867 y Fr(,)h(so)f(that)-111 6003 y Fn(\017)42 b Fp(C)41 6017 y Fm(0)110 6003 y Fr(is)30 b(the)g(branc)m(h)g(used)g (for)g(the)g(construction)g(of)h(the)f(canonical)g(line)f Fp(\014)5 b Fr(;)-111 6110 y Fn(\017)42 b Fr(in)29 b(the)h(strip)f Fn(f)p Fr(0)d Fp(<)f Fr(Im)15 b Fp(\020)32 b(<)25 b(Y)20 b Fn(g)p Fr(,)31 b(the)f(branc)m(h)g Fp(C)1651 6124 y Fo(j)t Fm(+1)1808 6110 y Fr(is)f(situated)h(to)h(the)g(righ)m(t)f(of)h Fp(C)2905 6124 y Fo(j)2971 6110 y Fr(for)f(all)g Fp(j)h Fn(2)24 b Fl(Z)p Fr(.)1854 6210 y Fm(16)p eop %%Page: 17 17 17 16 bop -236 241 a Fr(Of)31 b(course,)i(some)f(of)g(these)g(branc)m (hes)g(can)g(ha)m(v)m(e)h(common)f(b)s(eginning)d(p)s(oin)m(ts)i (\(situated)h(on)f Fl(R)s Fr(\).)51 b(The)32 b(set)g(of)g(lines)e Fp(C)3979 255 y Fo(j)-236 349 y Fr(b)s(eing)f(2)p Fp(\031)s Fr(-p)s(erio)s(dic,)g(there)h(exists)g Fp(j)1034 363 y Fm(0)1099 349 y Fn(2)25 b Fl(Z)h Fr(suc)m(h)k(that)h Fp(C)1743 363 y Fo(j)t Fm(+)p Fo(j)1860 372 y Fg(0)1923 349 y Fr(=)25 b Fp(C)2084 363 y Fo(j)2140 349 y Fr(+)20 b(2)p Fp(\031)34 b Fr(for)c(all)g Fp(j)h Fn(2)24 b Fl(Z)p Fr(.)-236 504 y(F)-8 b(or)27 b Fp(j)j Fn(2)25 b Fl(Z)p Fr(,)e(w)m(e)j(let)g Fp(D)520 518 y Fo(j)583 504 y Fr(b)s(e)f(the)i(op) s(en)e(\\rectangle")j(delimited)23 b(b)m(y)j(the)g(lines)f Fp(C)2483 518 y Fo(j)2519 504 y Fr(,)i(Im)15 b Fp(\020)32 b Fr(=)25 b Fp(Y)20 b Fr(,)27 b Fp(C)3053 518 y Fo(j)t Fm(+1)3205 504 y Fr(and)f Fl(R)r Fr(.)45 b(The)26 b(complex)-236 612 y(branc)m(hes)32 b(of)h Fp(Z)39 b Fr(situated)33 b(in)e Fl(C)870 626 y Fk(\000)968 612 y Fr(can)i(b)s(e)f(obtained)g (from)g(the)h(lines)e(\()p Fp(C)2325 626 y Fo(j)2362 612 y Fr(\))2397 626 y Fo(j)t Fk(2)p Fh(Z)2558 612 y Fr(b)m(y)i(symmetry)f(with)g(resp)s(ect)g(to)i Fl(R)s Fr(.)53 b(W)-8 b(e)-236 728 y(denote)31 b(them)f(b)m(y)p 415 655 102 4 v 30 w Fp(C)480 742 y Fo(j)517 728 y Fr(;)g(for)g Fp(j)h Fn(2)25 b Fl(Z)p Fr(,)h(the)31 b(symmetric)f(of)g Fp(D)1759 742 y Fo(j)1826 728 y Fr(with)f(resp)s(ect)i(to)g(the)f(real) g(line)f(is)h(denoted)g(b)m(y)p 3522 655 112 4 v 30 w Fp(D)3597 742 y Fo(j)3634 728 y Fr(.)-236 883 y(W)-8 b(e)31 b(\014nish)d(this)i(subsection)f(b)m(y)h(pro)m(ving)-236 1048 y Fu(Lemma)i(6.2.)47 b Fi(L)-5 b(et)32 b Fp(E)f Fr(=)25 b Fp(E)760 1062 y Fm(0)825 1048 y Fn(2)g Fp(J)9 b Fi(.)42 b(Then,)-137 1178 y Fr(1.)h Fi(for)33 b(al)5 b(l)33 b Fp(j)e Fn(2)25 b Fl(Z)p Fi(,)j(the)33 b(imaginary)h(p)-5 b(art)34 b(of)f Fp(\024)1456 1192 y Fo(n)1499 1201 y Fg(0)1570 1178 y Fi(do)-5 b(es)34 b(not)f(vanish)h(in)e(the)h(r)-5 b(e)g(ctangles)34 b Fp(D)2968 1192 y Fo(j)3038 1178 y Fi(and)p 3214 1105 V 33 w Fp(D)3289 1192 y Fo(j)3326 1178 y Fi(;)-137 1286 y Fr(2.)43 b Fi(for)33 b(al)5 b(l)33 b Fp(j)e Fn(2)25 b Fl(Z)p Fi(,)j(in)k(the)h(domains)i Fp(D)1226 1300 y Fo(j)1295 1286 y Fi(and)f Fp(D)1547 1300 y Fo(j)t Fm(+1)1673 1286 y Fi(,)f(the)g(imaginary)h(p)-5 b(arts)34 b(of)f Fp(\024)2701 1300 y Fo(n)2744 1309 y Fg(0)2816 1286 y Fi(ar)-5 b(e)33 b(of)g(opp)-5 b(osite)35 b(sign;)-137 1401 y Fr(3.)43 b Fi(for)33 b(al)5 b(l)33 b Fp(j)e Fn(2)25 b Fl(Z)p Fi(,)j(in)k(the)h(domains)i Fp(D)1226 1415 y Fo(j)1295 1401 y Fi(and)p 1472 1328 V 34 w Fp(D)1547 1415 y Fo(j)1583 1401 y Fi(,)e(the)g(imaginary)h(p)-5 b(arts)34 b(of)f Fp(\024)2611 1415 y Fo(n)2654 1424 y Fg(0)2726 1401 y Fi(ar)-5 b(e)33 b(of)g(opp)-5 b(osite)35 b(sign.)-236 1614 y Fu(Pro)s(of.)41 b Fr(P)m(oin)m(t)29 b(1)g(follo)m(ws)f(from)g(p)s(oin)m(t)g(1)h(of)g(Lemma)f(6.1.)42 b(P)m(oin)m(t)29 b(3)g(follo)m(ws)f(from)g(p)s(oin)m(t)g(1)h(and)f(p)s (oin)m(t)g(3)h(of)f(Lemma)h(6.1.)-236 1722 y(Chec)m(k)i(p)s(oin)m(t)e (2.)41 b(By)31 b(construction)f(of)g(the)h(strip)e Fn(S)1577 1736 y Fo(Y)1637 1722 y Fr(,)i(see)g(section)f(6.1.1,)j(one)e(has)-111 1852 y Fn(\017)42 b Fr(the)24 b(line)e Fp(C)356 1866 y Fo(j)t Fm(+1)506 1852 y Fr(separating)i Fp(D)1015 1866 y Fo(j)1075 1852 y Fr(from)g Fp(D)1359 1866 y Fo(j)t Fm(+1)1509 1852 y Fr(is)f(bijectiv)m(ely)g(mapp)s(ed)f(b)m(y)i(the)g (analytic)f(function)g Fn(E)32 b Fr(on)m(to)25 b(an)f(in)m(terv)-5 b(al)-24 1960 y Fn(I)36 b Fr(of)31 b Fl(R)r Fr(,)-111 2068 y Fn(\017)42 b Fr(the)30 b(in)m(terv)-5 b(al)30 b Fn(I)36 b Fr(is)30 b(con)m(tained)g(in)f(the)i Fp(n)1368 2082 y Fm(0)1407 2068 y Fr(-th)f(sp)s(ectral)g(band)f(of)i(the)f(p)s (erio)s(dic)e(op)s(erator)j(\(0.2\))r(,)-111 2176 y Fn(\017)42 b Fr(one)d(of)h(the)g(domains)e Fp(D)873 2190 y Fo(j)949 2176 y Fr(and)h Fp(D)1210 2190 y Fo(j)t Fm(+1)1377 2176 y Fr(is)f(mapp)s(ed)g(in)m(to)i Fl(C)2086 2190 y Fk(\000)2191 2176 y Fr(=)g Fn(f)p Fp(\020)48 b Fn(2)40 b Fl(C)18 b Fr(;)51 b(Im)15 b Fp(\020)47 b(<)40 b Fr(0)p Fn(g)h Fr(and)d(the)i (second)g(one)g(is)-24 2284 y(mapp)s(ed)29 b(in)m(to)h Fl(C)566 2298 y Fm(+)656 2284 y Fr(=)25 b Fn(f)p Fp(\020)32 b Fn(2)25 b Fl(C)18 b Fr(;)51 b(Im)15 b Fp(\020)32 b(>)25 b Fr(0)p Fn(g)p Fr(.)-236 2414 y(P)m(oin)m(t)31 b(2)g(holds)e(as)i(the) g(main)f(branc)m(h)g Fp(k)1167 2428 y Fo(p)1237 2414 y Fr(of)h(the)g(Blo)s(c)m(h)g(quasi-momen)m(tum)f(maps)g Fl(C)2766 2428 y Fm(+)2861 2414 y Fr(in)m(to)h Fl(C)3106 2428 y Fm(+)3201 2414 y Fr(and)f Fl(C)3438 2428 y Fk(\000)3534 2414 y Fr(in)m(to)g Fl(C)3778 2428 y Fk(\000)3843 2414 y Fr(,)h(see)-236 2522 y(sections)f(2.2)i(and)d(2.3.)p 3950 2522 4 62 v 3954 2464 55 4 v 3954 2522 V 4008 2522 4 62 v -236 2696 a(6.2.4.)48 b Fi(Pr)-5 b(o)g(of)40 b(of)f(Pr)-5 b(op)g(osition)42 b(6.1.)k Fr(Our)35 b(main)h(to)s(ols)h(are)g(the)g (Rectangle)h(Lemma,)h(Lemma)e(5.1)h(and)f(the)g(Adjacen)m(t)-236 2804 y(Canonical)g(Domain)g(Principle,)g(Lemma)h(5.4.)65 b(Recall)37 b(that)i(Im)14 b Fp(\024)2163 2818 y Fo(n)2206 2827 y Fg(0)2283 2804 y Fn(6)p Fr(=)38 b(0)g(in)e Fp(D)2663 2818 y Fm(0)2703 2804 y Fr(.)63 b(First,)40 b(w)m(e)e(treat)h(the)f (case)h(where)-236 2912 y(Im)15 b Fp(\024)-60 2926 y Fo(n)-17 2935 y Fg(0)47 2912 y Fp(<)25 b Fr(0)30 b(in)f Fp(D)399 2926 y Fm(0)439 2912 y Fr(.)41 b(The)30 b(pro)s(of)f(is)h (made)g(in)f(sev)m(eral)i(steps.)-236 3067 y Fu(1.)47 b Fr(Let)33 b(us)f(sho)m(w)g(that)h Fp(f)42 b Fr(has)32 b(standard)g(b)s(eha)m(vior)g(in)f(the)h(domain)g Fp(D)2276 3081 y Fm(0)2316 3067 y Fr(.)47 b(This)30 b(follo)m(ws)i(from)g(the)h (Rectangle)g(Lemma.)-236 3175 y(Indeed,)24 b(pic)m(k)e Fp(\016)29 b(>)c Fr(0)f(small)e(and)g(consider)g(the)h(compact)i Fp(R)1819 3190 y Fo(\016)1880 3175 y Fr(b)s(ounded)c(b)m(y)i(the)g (lines)e Fp(\014)5 b Fr(,)25 b Fp(C)2883 3189 y Fm(1)2923 3175 y Fr(,)f(Im)15 b Fp(\020)32 b Fr(=)25 b Fp(\016)i Fr(and)22 b(Im)15 b Fp(\020)32 b Fr(=)25 b Fp(Y)g Fn(\000)6 b Fp(\016)s Fr(.)-236 3283 y(In)30 b Fp(D)-47 3297 y Fm(0)-8 3283 y Fr(,)i(one)f(has)f(Im)15 b Fp(\024)26 b(<)g Fr(0,)31 b(so)g(the)g("rectangle")h Fp(R)1596 3298 y Fo(\016)1665 3283 y Fr(satis\014es)e(the)h(conditions)e(of)i(Lemma)g (5.1.)43 b(Th)m(us,)30 b(the)h(solution)e Fp(f)-236 3391 y Fr(ha)m(ving)e(standard)f(b)s(eha)m(vior)h(in)f(a)i(neigh)m(b)s(orho) s(o)s(d)d(of)j Fp(R)1713 3406 y Fo(\016)1750 3391 y Fr('s)g(left)f(b)s (oundary)-8 b(,)27 b(also)g(has)g(standard)g(b)s(eha)m(vior)f(in)h(a)g (constan)m(t)-236 3499 y(neigh)m(b)s(orho)s(o)s(d)j(of)i Fp(R)511 3514 y Fo(\016)549 3499 y Fr(.)46 b(As)32 b Fp(\016)g(>)c Fr(0)k(is)f(arbitrary)-8 b(,)32 b(this)f(implies)f(that)j Fp(f)41 b Fr(has)32 b(standard)f(b)s(eha)m(vior)g(in)g(the)h(whole)g (domain)-236 3607 y Fp(D)-161 3621 y Fm(0)-122 3607 y Fr(.)-236 3762 y Fu(2.)39 b Fr(No)m(w,)29 b(let)e(us)f(study)g(the)h (asymptotics)g(of)g Fp(f)36 b Fr(in)26 b(the)h(domain)f Fp(D)2087 3776 y Fm(1)2127 3762 y Fr(.)39 b(Note)28 b(that)g(as)f(Im)15 b Fp(\024)2885 3776 y Fo(n)2928 3785 y Fg(0)2992 3762 y Fp(<)25 b Fr(0)i(in)f Fp(D)3338 3776 y Fm(0)3378 3762 y Fr(,)h(b)m(y)g(Lemma)g(6.2,)-236 3870 y(Im)15 b Fp(\024)-60 3884 y Fo(n)-17 3893 y Fg(0)47 3870 y Fp(>)26 b Fr(0)31 b(in)e Fp(D)401 3884 y Fm(1)441 3870 y Fr(.)42 b(So,)30 b(w)m(e)i(can)f(not)g(apply)e(the)i(Rectangle)g(Lemma)g(to)h(\\con)m (tin)m(ue")g(the)f(asymptotics)f(of)h Fp(f)40 b Fr(in)m(to)31 b Fp(D)3951 3884 y Fm(1)3990 3870 y Fr(.)-236 3978 y(The)d(line)e Fp(C)183 3992 y Fm(1)251 3978 y Fr(is)i(canonical)g(with)f(resp)s(ect)h (to)h Fp(\024)1413 3992 y Fo(n)1456 4001 y Fg(0)1523 3978 y Fr(\(for)g(the)g(same)f(reason)h(as)g Fp(C)2533 3992 y Fm(0)2600 3978 y Fr(w)m(as,)h(see)f(\(6.5\))r(\).)40 b(So,)29 b(b)m(y)f(the)h(Adjacen)m(t)-236 4086 y(Canonical)k(Domain)h (Principle,)f(the)i(standard)f(b)s(eha)m(vior)f(for)i Fp(f)43 b Fr(remains)34 b(v)-5 b(alid)32 b(in)i(the)g(maximal)g (canonical)g(domain)-236 4194 y(enclosing)29 b Fp(C)218 4208 y Fm(1)258 4194 y Fr(.)-236 4350 y(Let)41 b(us)e(describ)s(e)g (the)i(part)820 4328 y(~)799 4350 y Fp(D)874 4364 y Fm(1)954 4350 y Fr(of)f(the)h(maximal)e(canonical)h(domain)f(enclosing)g Fp(C)2821 4364 y Fm(1)2901 4350 y Fr(situated)h(in)f(the)i(domain)e Fp(D)3951 4364 y Fm(1)3990 4350 y Fr(.)-236 4458 y(Denote)32 b(b)m(y)f Fp(\020)250 4425 y Fk(\003)319 4458 y Fr(the)g(upp)s(ermost)e (p)s(oin)m(t)h(of)g Fp(C)1335 4472 y Fm(1)1406 4458 y Fr(in)f Fn(S)1567 4472 y Fo(Y)1628 4458 y Fr(.)41 b(Let)31 b Fp(\033)j Fr(b)s(e)c(the)h(line)e(of)i(Stok)m(es)h(t)m(yp)s(e)e (starting)h(at)g Fp(\020)3486 4425 y Fk(\003)3556 4458 y Fr(and)f(de\014ned)-236 4577 y(b)m(y)g(Im)29 4504 y Ff(R)90 4531 y Fo(\020)72 4609 y(\020)108 4590 y Fc(\003)163 4577 y Fp(\024)215 4591 y Fo(n)258 4600 y Fg(0)297 4577 y Fp(d\020)i Fr(=)25 b(0.)41 b(One)30 b(pro)m(v)m(es)-236 4752 y Fu(Lemma)i(6.3.)47 b Fi(The)27 b(line)h Fp(\033)j Fi(enters)d(in)f Fp(D)1227 4766 y Fm(1)1295 4752 y Fi(at)h(the)g(p)-5 b(oint)29 b Fp(\020)1817 4719 y Fk(\003)1856 4752 y Fi(;)g(inside)f Fp(D)2248 4766 y Fm(1)2288 4752 y Fi(,)g(it)g(is)f(vertic)-5 b(al)29 b(and)f(go)-5 b(es)29 b(downwar)-5 b(ds.)43 b(It)28 b(le)-5 b(aves)-236 4862 y Fp(D)-161 4876 y Fm(1)-86 4862 y Fi(at)36 b(a)g(p)-5 b(oint)37 b Fp(\020)388 4829 y Fk(\003\003)493 4862 y Fn(2)31 b Fp(C)650 4876 y Fm(2)725 4862 y Fi(such)36 b(that)h Fr(0)31 b Fp(<)f Fr(Im)15 b Fp(\020)1472 4829 y Fk(\003\003)1577 4862 y Fp(<)31 b(Y)20 b Fi(.)51 b(The)36 b(domain)2369 4839 y Fr(~)2348 4862 y Fp(D)2423 4876 y Fm(1)2498 4862 y Fi(is)g(delimite)-5 b(d)37 b(by)e(the)i(lines)e Fp(\033)s Fi(,)i Fl(R)r Fi(,)43 b Fp(C)3797 4876 y Fm(1)3872 4862 y Fi(and)-236 4970 y Fp(C)-171 4984 y Fm(2)-132 4970 y Fi(.)-236 5182 y Fu(Pro)s(of.)f Fr(First,)30 b(w)m(e)h(c)m(hec)m(k)h(the)f(geometric)h (prop)s(erties)d(of)i Fp(\033)s Fr(.)41 b(W)-8 b(e)32 b(iden)m(tify)d Fl(C)55 b Fr(and)30 b Fl(R)2760 5149 y Fm(2)2836 5182 y Fr(in)f(the)i(usual)e(w)m(a)m(y)-8 b(.)43 b(Let)31 b Fp(\020)3760 5149 y Fm(0)3825 5182 y Fn(2)25 b Fp(C)3976 5196 y Fm(1)-236 5301 y Fr(and)j(0)e Fp(<)f Fr(Im)14 b Fp(\020)276 5268 y Fm(0)340 5301 y Fp(<)25 b(Y)20 b Fr(.)40 b(Consider)27 b Fp(\033)1004 5315 y Fm(0)1072 5301 y Fr(a)i(line)e(of)i(Stok)m(es)h(t)m(yp)s(e)f(Im) 2042 5228 y Ff(R)2102 5254 y Fo(\020)2084 5333 y(\020)2120 5314 y Fg(0)2174 5301 y Fp(\024)2226 5315 y Fo(n)2269 5324 y Fg(0)2308 5301 y Fp(d\020)j Fr(=)25 b(0)k(con)m(taining)f(the)h (p)s(oin)m(t)f Fp(\020)3475 5268 y Fm(0)3514 5301 y Fr(.)40 b(This)27 b(line)g(is)-236 5441 y(an)k(in)m(tegral)g(curv)m(e)h(of)f (the)h(v)m(ector)h(\014eld)d Fp(t)p Fr(\()p Fp(\020)7 b Fr(\))27 b(=)p 1476 5362 252 4 v 27 w Fp(\024)1528 5455 y Fo(n)1571 5464 y Fg(0)1610 5441 y Fr(\()p Fp(\020)7 b Fr(\).)43 b(As)32 b(along)f Fp(C)2239 5455 y Fm(1)2279 5441 y Fr(,)63 b Fp(\024)2419 5455 y Fo(n)2462 5464 y Fg(0)2532 5441 y Fr(is)30 b(real,)i(the)g(tangen)m(t)g(v)m(ector)h Fp(t)p Fr(\()p Fp(\020)3706 5408 y Fm(0)3746 5441 y Fr(\))e(to)h Fp(\033)3976 5455 y Fm(0)-236 5549 y Fr(at)h Fp(\020)-76 5516 y Fm(0)-5 5549 y Fr(is)e(horizon)m(tal,)i(Im)15 b Fp(t)p Fr(\()p Fp(\020)782 5516 y Fm(0)821 5549 y Fr(\))29 b(=)f(0.)46 b(So,)33 b Fp(\033)1306 5563 y Fm(0)1378 5549 y Fr(in)m(tersects)g Fp(C)1848 5563 y Fm(1)1919 5549 y Fr(transv)m(ersally)f(and)f(en)m(ters)i(in)e Fp(D)3085 5563 y Fm(1)3157 5549 y Fr(at)i Fp(\020)3317 5516 y Fm(0)3356 5549 y Fr(.)46 b(In)32 b Fp(D)3618 5563 y Fm(1)3657 5549 y Fr(,)h(one)g(has)-236 5657 y(Im)15 b Fp(\024)-60 5671 y Fo(n)-17 5680 y Fg(0)48 5657 y Fp(>)26 b Fr(0;)32 b(near)f Fp(C)515 5671 y Fm(1)554 5657 y Fr(,)h(Re)15 b Fp(\024)785 5671 y Fo(n)828 5680 y Fg(0)898 5657 y Fr(is)30 b(p)s(ositiv)m(e)g(as)i (it)e(is)g(p)s(ositiv)m(e)g(on)h Fp(C)2152 5671 y Fm(1)2192 5657 y Fr(.)43 b(Therefore,)31 b(the)g(line)e Fp(\033)3079 5671 y Fm(0)3150 5657 y Fr(is)h(v)m(ertical)h(in)f Fp(D)3748 5671 y Fm(1)3788 5657 y Fr(,)h(and,)-236 5765 y(inside)d Fp(D)97 5779 y Fm(1)137 5765 y Fr(,)i(it)g(go)s(es)h(do)m(wn)m(w)m (ards.)-236 5873 y(As)f Fp(\024)-50 5887 y Fo(n)-7 5896 y Fg(0)57 5873 y Fn(6)p Fr(=)25 b(0)31 b(in)e Fn(S)390 5887 y Fo(Y)481 5873 y Fr(\(see)j(\(6.5\))q(\),)f(the)g(lines)d(of)j (Stok)m(es)g(t)m(yp)s(e)g(\014brate)f Fn(S)2240 5887 y Fo(Y)2301 5873 y Fr(,)g(and)g(the)h(observ)-5 b(ations)30 b(on)g Fp(\033)3387 5887 y Fm(0)3457 5873 y Fr(imply)e(that)-111 6003 y Fn(\017)42 b Fr(either)29 b Fp(\033)34 b Fr(b)s(eginning)28 b(at)j Fp(\020)894 5970 y Fk(\003)963 6003 y Fr(con)m(tains)g(a)f (segmen)m(t)i(of)e(the)h(line)e(Im)14 b Fp(\020)32 b Fr(=)25 b(Im)15 b Fp(\020)2641 5970 y Fk(\003)2680 6003 y Fr(,)-111 6110 y Fn(\017)42 b Fr(or)30 b(it)g(en)m(ters)h(in)e Fp(D)625 6124 y Fm(1)695 6110 y Fr(at)i Fp(\020)853 6078 y Fk(\003)892 6110 y Fr(.)1854 6210 y Fm(17)p eop %%Page: 18 18 18 17 bop -236 241 a Fr(The)43 b(\014rst)g(alternativ)m(e)h(is)e(imp)s (ossible,)i(since)f(then)g Fp(\024)1730 255 y Fo(n)1773 264 y Fg(0)1855 241 y Fr(m)m(ust)h(b)s(e)f(real)g(on)g(that)h(segmen)m (t)h(whic)m(h)d(con)m(tradicts)i(the)-236 349 y(assumptions)28 b(on)i(the)g(strip)f Fn(S)830 363 y Fo(Y)891 349 y Fr(.)40 b(Therefore,)31 b Fp(\033)i Fr(en)m(ters)e Fp(D)1820 363 y Fm(1)1889 349 y Fr(at)g(the)f(p)s(oin)m(t)g Fp(\020)2441 316 y Fk(\003)2479 349 y Fr(,)h(and,)f(inside)d Fp(D)3069 363 y Fm(1)3109 349 y Fr(,)k(it)e(is)g(v)m(ertical)i(and)e(go)s(es)-236 457 y(do)m(wn)m(w)m(ards.)-236 565 y(Since)h Fp(\033)k Fr(go)s(es)e(do)m(wn)m(w)m(ards,)f(it)g(can)g(lea)m(v)m(e)i Fp(D)1343 579 y Fm(1)1414 565 y Fr(only)d(through)h Fp(C)2024 579 y Fm(2)2063 565 y Fr(,)g Fp(C)2184 579 y Fm(1)2255 565 y Fr(or)g Fl(R)s Fr(.)49 b(As)31 b Fl(R)40 b Fr(is)30 b(also)h(a)h(line)d(of)i(Stok)m(es)i(t)m(yp)s(e,)e(if)g Fp(\033)-236 684 y Fr(in)m(tersects)c Fl(R)s Fr(,)34 b(then)27 b(the)g(in)m(tersection)g(p)s(oin)m(t)f(is)g(a)h(critical)f (p)s(oin)m(t)h(for)f(Im)2327 610 y Ff(R)2388 637 y Fo(\020)2443 684 y Fp(\024)2495 698 y Fo(n)2538 707 y Fg(0)2577 684 y Fp(d\020)7 b Fr(.)39 b(A)m(t)28 b(this)e(p)s(oin)m(t)g Fp(\024)3323 698 y Fo(n)3366 707 y Fg(0)3432 684 y Fr(v)-5 b(anishes)26 b(whic)m(h)-236 792 y(is)f(imp)s(ossible.)36 b(So,)27 b(w)m(e)g(need)f(only)g(to)h(c)m(hec)m(k)g(that)g Fp(\033)j Fr(can)c(not)h(in)m(tersect)g Fp(C)2380 806 y Fm(1)2419 792 y Fr(.)39 b(Assume)26 b(it)g(in)m(tersects)h Fp(C)3367 806 y Fm(1)3432 792 y Fr(at,)i(sa)m(y)-8 b(,)28 b(a)f(p)s(oin)m(t)-236 899 y Fp(\020)-196 913 y Fo(e)-160 899 y Fr(.)41 b(Then,)30 b(one)g(has)930 1193 y(0)c(=)f(Im)1236 1070 y Ff(Z)1327 1096 y Fo(\020)1358 1104 y Fd(e)1286 1276 y Fo(\020)1322 1257 y Fc(\003)1358 1276 y Fo(;)e Fm(along)h Fo(\033)1651 1193 y Fp(\024)1703 1207 y Fo(n)1746 1216 y Fg(0)1785 1193 y Fp(d\020)32 b Fr(=)25 b(Im)2139 1070 y Ff(Z)2230 1096 y Fo(\020)2261 1104 y Fd(e)2189 1276 y Fo(\020)2225 1257 y Fc(\003)2261 1276 y Fo(;)e Fm(along)h Fo(C)2542 1285 y Fg(1)2596 1193 y Fp(\024)2648 1207 y Fo(n)2691 1216 y Fg(0)2730 1193 y Fp(d\020)7 b(:)-236 1485 y Fr(But,)33 b(the)f(last)f(in)m(tegral)h(is)f(non-zero)h(since)g Fp(\024)1376 1499 y Fo(n)1419 1508 y Fg(0)1486 1485 y Fp(>)27 b Fr(0)32 b(along)g Fp(C)1970 1499 y Fm(1)2042 1485 y Fr(and)f Fp(C)2285 1499 y Fm(1)2356 1485 y Fr(is)g(v)m(ertical.) 46 b(So,)32 b Fp(\033)j Fr(has)d(to)h(lea)m(v)m(e)g Fp(D)3631 1499 y Fm(1)3702 1485 y Fr(through)-236 1593 y Fp(C)-171 1607 y Fm(2)-91 1593 y Fr(at)42 b(a)f(p)s(oin)m(t)f Fp(\020)412 1560 y Fk(\003\003)527 1593 y Fr(with)g(a)h(p)s(ositiv)m(e)f(imaginary) g(part.)73 b(W)-8 b(e)42 b(ha)m(v)m(e)g(pro)m(v)m(ed)f(all)f(the)h (prop)s(erties)f(of)h Fp(\033)j Fr(describ)s(ed)39 b(in)-236 1700 y(Lemma)30 b(6.3.)-236 1856 y(No)m(w,)35 b(let)f(us)f(c)m(hec)m(k) j(that)725 1833 y(~)704 1856 y Fp(D)779 1870 y Fm(1)853 1856 y Fr(is)d(as)h(describ)s(ed)d(in)i(Lemma)h(6.3.)52 b(Therefore,)35 b(w)m(e)f(use)g(Prop)s(osition)d(4.1.)53 b(Pic)m(k)33 b(a)i(p)s(oin)m(t)-228 1942 y(~)-236 1966 y Fp(\020)d Fn(2)-57 1943 y Fr(~)-79 1966 y Fp(D)-4 1980 y Fm(1)36 1966 y Fr(.)40 b(Sho)m(w)27 b(that)h(there)f(exists)g(a)h (pre-canonical)f(curv)m(e)h(connecting)g(t)m(w)m(o)g(in)m(ternal)f(p)s (oin)m(ts)f(of)h Fp(C)3264 1980 y Fm(1)3331 1966 y Fr(and)g(con)m (taining)3952 1942 y(~)3944 1966 y Fp(\020)6 b Fr(.)-236 2122 y(As)31 b(ab)s(o)m(v)m(e,)i(consider)d(the)i(line)d(of)j(the)f (Stok)m(es)h(t)m(yp)s(e)g Fp(\033)1672 2136 y Fm(0)1742 2122 y Fr(starting)f(at)h Fp(\020)2241 2089 y Fm(0)2307 2122 y Fn(2)26 b Fp(C)2459 2136 y Fm(1)2499 2122 y Fr(.)43 b(If)31 b Fp(\020)2706 2089 y Fm(0)2772 2122 y Fr(=)26 b Fp(\020)2916 2089 y Fk(\003)2955 2122 y Fr(,)32 b(then)f Fp(\033)3272 2136 y Fm(0)3342 2122 y Fr(coincides)f(with)g Fp(\033)s Fr(.)-236 2241 y(If)g Fp(\020)-98 2208 y Fm(0)-34 2241 y Fn(2)25 b Fl(R)s Fr(,)36 b(then)30 b Fp(\033)432 2255 y Fm(0)502 2241 y Fr(coincides)f(with)h(the)g(real)g(line.)39 b(Since)30 b(the)h(lines)d(Im)2375 2167 y Ff(R)2436 2194 y Fo(\020)2491 2241 y Fp(\024)2543 2255 y Fo(n)2586 2264 y Fg(0)2624 2241 y Fp(d\020)33 b Fr(=)25 b(Const)30 b(\014brate)g Fn(S)3446 2255 y Fo(Y)3507 2241 y Fr(,)g(there)h(exists)-236 2361 y(a)f(p)s(oin)m(t)f Fp(\020)123 2328 y Fm(0)191 2361 y Fr(suc)m(h)g(that)i(the)f(line)e Fp(\033)971 2375 y Fm(0)1040 2361 y Fr(con)m(tains)1404 2337 y(~)1395 2361 y Fp(\020)7 b Fr(.)40 b(W)-8 b(e)31 b(denote)f(this)f(line)f(b)m (y)i Fp(\014)2480 2375 y Fo(u)2555 2361 y Fr(and)f(the)h(corresp)s (onding)d(starting)j(p)s(oin)m(t)-236 2472 y(b)m(y)g Fp(\020)-70 2486 y Fo(u)-25 2472 y Fr(.)40 b(Clearly)-8 b(,)30 b(the)h(segmen)m(t)g(of)g(line)d Fp(\014)1208 2486 y Fo(u)1284 2472 y Fr(b)s(et)m(w)m(een)j Fp(\020)1675 2486 y Fo(u)1750 2472 y Fr(and)1935 2448 y(~)1927 2472 y Fp(\020)37 b Fr(b)s(elongs)29 b(to)2462 2449 y(~)2441 2472 y Fp(D)2516 2486 y Fm(1)2556 2472 y Fr(,)h(is)g(v)m(ertical,)h (and)e(go)s(es)i(do)m(wn)m(w)m(ards.)-236 2638 y(Similarly)-8 b(,)25 b(one)j(pro)m(v)m(es)h(that)g(there)f(exists)f(a)i(v)m(ertical)f (segmen)m(t)h Fp(\014)2077 2653 y Fo(d)2146 2638 y Fr(of)f(the)g(line)f (of)h(Stok)m(es)h(t)m(yp)s(e)f(Im)3294 2565 y Ff(R)3354 2591 y Fo(\020)3343 2663 y Fm(~)3337 2681 y Fo(\020)3394 2638 y Fr(\()p Fp(\024)3481 2652 y Fo(n)3524 2661 y Fg(0)3579 2638 y Fn(\000)15 b Fp(\031)s Fr(\))p Fp(d\020)32 b Fr(=)25 b(0)-236 2787 y(connecting)224 2763 y(~)216 2787 y Fp(\020)34 b Fr(in)414 2764 y(~)393 2787 y Fp(D)468 2801 y Fm(1)535 2787 y Fr(with)27 b(a)h(p)s(oin)m(t)e Fp(\020)1087 2802 y Fo(d)1153 2787 y Fn(2)f Fp(C)1304 2801 y Fm(1)1371 2787 y Fr(suc)m(h)i(that)h(0)e Fp(<)f(\020)1974 2802 y Fo(d)2039 2787 y Fp(<)2143 2763 y Fr(~)2135 2787 y Fp(\020)7 b Fr(.)40 b(First,)27 b(w)m(e)i(pic)m(k)e(a)h(p)s(oin)m(t)e Fp(\020)3159 2802 y Fo(d)3227 2787 y Fr(inside)g Fp(C)3548 2801 y Fm(1)3587 2787 y Fr(.)40 b(Then,)27 b(w)m(e)-236 2906 y(consider)k(the)h(line)e(of)j(Stok)m(es)g(t)m(yp)s(e)f(Im)1187 2833 y Ff(R)1248 2859 y Fo(\020)1230 2938 y(\020)1261 2950 y Fd(d)1301 2906 y Fr(\()p Fp(\024)1388 2920 y Fo(n)1431 2929 y Fg(0)1492 2906 y Fn(\000)21 b Fp(\031)s Fr(\))p Fp(d\020)35 b Fr(=)28 b(0)k(b)s(eginning)e(at)j(this)e(p)s(oin)m(t.)44 b(This)31 b(lines)f(is)h(transv)m(ersal)h(to)-236 3044 y Fp(C)-171 3058 y Fm(1)-105 3044 y Fr(at)27 b Fp(\020)42 3059 y Fo(d)82 3044 y Fr(.)39 b(It)27 b(is)e(v)m(ertical)i(in)e Fp(D)825 3058 y Fm(1)891 3044 y Fr(and)h(go)s(es)h(up)m(w)m(ards)e (there.)40 b(If)26 b(Im)15 b Fp(\020)2133 3059 y Fo(d)2198 3044 y Fp(>)25 b Fr(Im)2426 3020 y(~)2418 3044 y Fp(\020)6 b Fr(,)28 b(then)2728 3020 y(~)2720 3044 y Fp(\020)33 b Fr(is)25 b(b)s(elo)m(w)h(this)f(line.)38 b(If)26 b(Im)14 b Fp(\020)3763 3059 y Fo(d)3829 3044 y Fn(!)25 b Fr(0,)-236 3152 y(then)g(this)g(line)g(approac)m(hes)h Fl(R)35 b Fr(\()p Fl(R)g Fr(is)25 b(a)h(line)f(of)h(Stok)m(es)h(t)m(yp)s(e)f(of)g (the)g(same)h(family\).)38 b(So,)27 b(there)f(exists)g Fp(\020)3378 3167 y Fo(d)3444 3152 y Fr(situated)f(b)s(elo)m(w)-228 3252 y(~)-236 3276 y Fp(\020)39 b Fr(and)33 b(ab)s(o)m(v)m(e)h Fl(R)42 b Fr(whic)m(h)32 b(is)g(connected)i(in)e Fp(D)1350 3290 y Fm(1)1423 3276 y Fr(to)1545 3252 y(~)1537 3276 y Fp(\020)39 b Fr(b)m(y)33 b(a)h(line)d(of)j(Stok)m(es)g(t)m(yp)s(e)f (Im)2737 3203 y Ff(R)2798 3229 y Fo(\020)2780 3308 y(\020)2811 3320 y Fd(d)2851 3276 y Fr(\()p Fp(\024)2938 3290 y Fo(n)2981 3299 y Fg(0)3043 3276 y Fn(\000)22 b Fp(\031)s Fr(\))p Fp(d\020)37 b Fr(=)29 b(0.)50 b(This)31 b(line)h(is)-236 3392 y(denoted)e(b)m(y)g Fp(\014)284 3407 y Fo(d)325 3392 y Fr(.)-236 3547 y(The)j(line)g Fp(\014)181 3562 y Fo(d)244 3547 y Fn([)23 b Fp(\014)379 3561 y Fo(u)458 3547 y Fr(is)33 b(not)i(pre-canonical:)47 b(it)34 b(is)f(not)i(v)m (ertical)f(at)h(its)e(ends)g(\(common)i(p)s(oin)m(ts)e(with)g Fp(C)3412 3561 y Fm(1)3451 3547 y Fr(\).)53 b(But,)35 b(as)f(the)-236 3655 y(canonical)d(lines)e(are)j(stable)f(under)f Fp(C)1120 3622 y Fm(1)1159 3655 y Fr(-p)s(erturbations,)g(w)m(e)i(can)g (deform)e(somewhat)i(the)g(canonical)f(line)e Fp(C)3615 3669 y Fm(1)3686 3655 y Fr(near)i(the)-236 3763 y(p)s(oin)m(ts)39 b Fp(\020)87 3778 y Fo(d)167 3763 y Fr(and)h Fp(\020)394 3777 y Fo(u)479 3763 y Fr(in)f(suc)m(h)h(a)h(w)m(a)m(y)g(that,)j(from)c Fp(\014)1601 3778 y Fo(d)1669 3763 y Fn([)26 b Fp(\014)1807 3777 y Fo(u)1893 3763 y Fr(and)39 b(the)i(deformed)f(segmen)m(ts)h(of)g Fp(C)3229 3777 y Fm(1)3268 3763 y Fr(,)i(w)m(e)e(construct)f(the)-236 3874 y(pre-canonical)35 b(line)e(con)m(taining)957 3850 y(~)948 3874 y Fp(\020)42 b Fr(and)35 b(connecting)h(in)1804 3851 y(~)1783 3874 y Fp(D)1858 3888 y Fm(1)1933 3874 y Fr(t)m(w)m(o)h(in)m(ternal)d(p)s(oin)m(ts)g(of)i Fp(C)2903 3888 y Fm(1)2942 3874 y Fr(.)56 b(Since)3274 3850 y(~)3266 3874 y Fp(\020)42 b Fr(w)m(as)36 b(an)f(arbitrary)-236 3983 y(p)s(oin)m(t)29 b(in)128 3960 y(~)107 3983 y Fp(D)182 3997 y Fm(1)222 3983 y Fr(,)h(then,)h(b)m(y)f(Prop)s(osition)e(4.1,) 1318 3960 y(~)1297 3983 y Fp(D)1372 3997 y Fm(1)1442 3983 y Fr(is)i(a)g(part)h(of)f(a)h(canonical)f(domain)f(enclosing)g Fp(C)3162 3997 y Fm(1)3202 3983 y Fr(.)p 3950 3983 4 62 v 3954 3925 55 4 v 3954 3983 V 4008 3983 4 62 v -236 4140 a Fu(3.)58 b Fr(The)36 b(righ)m(t)g(b)s(oundary)e(of)867 4117 y(~)846 4140 y Fp(D)921 4154 y Fm(1)997 4140 y Fr(is)h(the)i (segmen)m(t)g(of)f(the)h(line)d Fp(C)2128 4154 y Fm(2)2204 4140 y Fr(b)s(et)m(w)m(een)j(the)f(real)g(line)f(and)g(the)i(p)s(oin)m (t)e Fp(\020)3719 4107 y Fk(\003\003)3829 4140 y Fr(\(0)h Fp(<)-236 4255 y Fr(Im)15 b Fp(\020)-65 4222 y Fk(\003\003)34 4255 y Fp(<)25 b(Y)20 b Fr(\).)39 b(Denote)27 b(this)e(segmen)m(t)i(b)m (y)1271 4232 y(~)1250 4255 y Fp(C)1315 4269 y Fm(2)1355 4255 y Fr(.)39 b(As)25 b Fp(C)1613 4269 y Fm(1)1653 4255 y Fr(,)i(the)e(line)2044 4232 y(~)2023 4255 y Fp(C)2088 4269 y Fm(2)2153 4255 y Fr(is)g(canonical.)39 b(Consider)24 b(a)i(lo)s(cal)f(canonical)g(domain)-236 4370 y(\(see)31 b(Lemma)e(4.1\))j(enclosing)860 4347 y(~)839 4370 y Fp(C)904 4384 y Fm(2)944 4370 y Fr(.)40 b(By)30 b(the)g(Adjacen)m(t)h(Canonical) e(Domain)g(Principle,)e Fp(f)39 b Fr(has)30 b(standard)f(b)s(eha)m (vior)g(also)-236 4478 y(in)g(this)g(domain.)-236 4633 y Fu(4.)38 b Fr(T)-8 b(o)24 b(\\con)m(tin)m(ue")h(the)f(standard)f(b)s (eha)m(vior)f(of)i Fp(f)33 b Fr(to)24 b(the)g(righ)m(t)f(of)2105 4610 y(~)2084 4633 y Fp(C)2149 4647 y Fm(2)2189 4633 y Fr(,)i(one)f(essen)m(tially)e(only)h(has)g(to)i(rep)s(eat)e(p)s(erio) s(dically)-236 4741 y(the)33 b(steps)f(1.)48 b(-)33 b(3.)48 b(In)32 b(particular,)g(when)f(studying)g(the)i(asymptotic)g(b)s(eha)m (vior)f(of)h Fp(f)42 b Fr(in)31 b Fp(D)3008 4755 y Fm(2)3048 4741 y Fr(,)i(one)g(considers)f(the)g(op)s(en)-236 4850 y(\\rectangle")265 4828 y(~)244 4850 y Fp(D)319 4864 y Fm(2)390 4850 y Fr(b)s(ounded)c(b)m(y)908 4828 y(~)888 4850 y Fp(C)953 4864 y Fm(2)992 4850 y Fr(,)j Fl(R)s Fr(,)37 b Fp(C)1235 4864 y Fm(3)1305 4850 y Fr(and)30 b(the)h(line)e(Im)15 b Fp(\020)32 b Fr(=)26 b(Im)15 b Fp(\020)2275 4817 y Fk(\003\003)2349 4850 y Fr(.)42 b(As)2572 4828 y(~)2550 4850 y Fp(D)2625 4864 y Fm(2)2691 4850 y Fn(\032)25 b Fp(D)2862 4864 y Fm(2)2902 4850 y Fr(,)31 b(in)3086 4828 y(~)3065 4850 y Fp(D)3140 4864 y Fm(2)3179 4850 y Fr(,)g(one)g(has)g(Im)15 b Fp(\024)3741 4864 y Fo(n)3784 4873 y Fg(0)3848 4850 y Fp(<)26 b Fr(0.)-236 4958 y(T)-8 b(o)35 b(this)e(rectangle,)j(w)m(e)f(apply)e(the)h (Rectangle)i(Lemma)e(in)f(the)i(same)f(w)m(a)m(y)i(as)e(w)m(e)h(ha)m(v) m(e)g(applied)e(it)g(to)i(the)g(rectangle)-236 5068 y Fp(D)-161 5082 y Fm(0)-91 5068 y Fr(\(in)29 b(step)i(1.\))41 b(and)30 b(w)m(e)h(see)g(that)g Fp(f)39 b Fr(has)30 b(standard)g(b)s (eha)m(vior)f(in)2169 5045 y(~)2147 5068 y Fp(D)2222 5082 y Fm(2)2262 5068 y Fr(.)-236 5223 y(Con)m(tin)m(uing)h(the)j(pro)s (of)f(\\p)s(erio)s(dically",)e(one)j(sho)m(ws)f(that)h Fp(f)41 b Fr(has)32 b(standard)g(b)s(eha)m(vior)f(in)g(the)i(domain)e Fp(R)3542 5238 y Fm(+)p Fo(;l)3675 5223 y Fr(b)s(ounded)-236 5331 y(b)m(y)37 b(the)h(lines)e Fp(C)341 5345 y Fm(0)381 5331 y Fr(,)77 b Fl(R)r Fr(,)84 b Fp(C)716 5345 y Fm(0)780 5331 y Fr(+)25 b(2)p Fp(\031)s(l)40 b Fr(and)d(Im)15 b Fp(\020)44 b Fr(=)37 b Fp(Y)1596 5346 y Fm(+)p Fo(;l)1696 5331 y Fr(,)j(where)d Fp(l)j Fr(is)c(a)i(\014xed)f(arbitrary)g(p)s (ositiv)m(e)g(in)m(teger,)j(and)d Fp(Y)3817 5346 y Fm(+)p Fo(;l)3955 5331 y Fr(is)-236 5439 y(p)s(ositiv)m(e)29 b(and)h(indep)s(enden)m(t)e(of)j Fp(")p Fr(.)-236 5594 y Fu(5.)40 b Fr(Similarly)-8 b(,)28 b(one)i(studies)f(the)i(asymptotic) g(b)s(eha)m(vior)e(of)i Fp(f)39 b Fr(b)s(elo)m(w)30 b(the)g(real)g (line.)40 b(Outline)28 b(a)j(\\p)s(erio)s(d")e(of)h(the)h(pro)s(of.) -236 5749 y(In)e(the)i(domain)p 357 5676 115 4 v 29 w Fp(D)432 5763 y Fm(0)471 5749 y Fr(,)g(b)m(y)f(the)g(third)f(p)s(oin)m (t)g(of)i(Lemma)f(6.2,)h(Im)15 b Fp(\024)2049 5763 y Fo(p)2115 5749 y Fp(>)24 b Fr(0.)42 b(One)29 b(justi\014es)g(the)h (standard)g(b)s(eha)m(vior)f(of)i Fp(f)39 b Fr(in)-236 5886 y(a)29 b(sub)s(domain)333 5845 y(~)p 299 5813 V 299 5886 a Fp(D)374 5900 y Fm(0)442 5886 y Fr(of)p 543 5813 V 28 w Fp(D)618 5900 y Fm(0)687 5886 y Fr(b)m(y)f(the)h(Adjacen)m (t)g(Canonical)f(Domain)g(Principle.)37 b(The)28 b(domain)3074 5845 y(~)p 3040 5813 V 3040 5886 a Fp(D)3115 5900 y Fm(0)3183 5886 y Fr(is)f(b)s(ounded)g(b)m(y)p 3766 5813 105 4 v 28 w Fp(C)3831 5900 y Fm(0)3871 5886 y Fr(,)i Fl(R)r Fr(,)p -236 5921 V -236 5994 a Fp(C)-171 6008 y Fm(1)-100 5994 y Fr(and)i(the)h(line)f(of)h(Stok)m(es)h(t)m(yp)s(e)f(Im)1145 5921 y Ff(R)1220 5994 y Fp(\024)1272 6008 y Fo(n)1315 6017 y Fg(0)1354 5994 y Fp(d\020)j Fr(=)27 b(0)33 b(sta)m(ying)f(in)p 2075 5921 115 4 v 31 w Fp(D)2150 6008 y Fm(0)2221 5994 y Fr(and)f(connecting)i(the)f(lo)m(w)m(er)g(end)f(of)p 3531 5921 105 4 v 32 w Fp(C)3596 6008 y Fm(0)3667 5994 y Fr(with)g Fp(\020)3916 6008 y Fk(\003\003)3990 5994 y Fr(,)-236 6110 y(an)f(in)m(ternal)f(p)s(oin)m(t)h(of)p 567 6037 V 30 w Fp(C)632 6124 y Fm(1)671 6110 y Fr(.)1854 6210 y Fm(18)p eop %%Page: 19 19 19 18 bop -236 241 a Fr(The)25 b(line)p 113 168 105 4 v 24 w Fp(C)178 255 y Fm(1)243 241 y Fr(is)g(canonical.)39 b(By)26 b(the)g(Adjacen)m(t)g(Canonical)f(Domain)g(Principle,)f(the)i (standard)f(b)s(eha)m(vior)g(remains)f(v)-5 b(alid)-236 369 y(in)29 b(a)i(lo)s(cal)e(canonical)h(domain)f(enclosing)h(the)g (segmen)m(t)1809 327 y(~)p 1780 296 V 1780 369 a Fp(C)1845 383 y Fm(1)1915 369 y Fr(of)g(the)h(line)p 2346 296 V 28 w Fp(C)2411 383 y Fm(1)2481 369 y Fr(b)s(et)m(w)m(een)g Fl(R)39 b Fr(and)30 b Fp(\020)3145 383 y Fk(\003\003)3219 369 y Fr(.)-236 524 y(In)p -113 451 115 4 v 39 w Fp(D)-38 538 y Fm(1)1 524 y Fr(,)42 b(one)e(has)f(Im)15 b Fp(\024)591 538 y Fo(n)634 547 y Fg(0)713 524 y Fp(<)41 b Fr(0.)68 b(Applying)37 b(the)j(Rectangle)g(Lemma,)j(one)c(justi\014es)f(the)i (standard)f(b)s(eha)m(vior)f(in)g(the)-236 635 y(rectangle)31 b(b)s(ounded)d(b)m(y)p 650 562 105 4 v 30 w Fp(C)715 649 y Fm(1)755 635 y Fr(,)i Fl(R)s Fr(,)p 932 562 72 4 v 37 w Fp(C)1003 649 y Fm(2)1073 635 y Fr(and)g(the)g(line)f(Im)15 b Fp(\020)31 b Fr(=)25 b(Im)15 b Fp(\020)2033 649 y Fk(\003\003)2108 635 y Fr(.)-236 790 y(Con)m(tin)m(uing)30 b(the)j(pro)s(of)f(\\p)s (erio)s(dically",)e(one)j(sho)m(ws)f(that)h Fp(f)41 b Fr(has)32 b(standard)g(b)s(eha)m(vior)f(in)g(the)i(domain)e Fp(R)3542 805 y Fk(\000)p Fo(;l)3675 790 y Fr(b)s(ounded)-236 907 y(b)m(y)p -103 834 105 4 v 37 w Fp(C)-38 921 y Fm(0)1 907 y Fr(,)39 b Fl(R)s Fr(,)p 195 834 V 45 w Fp(C)260 921 y Fm(0)324 907 y Fr(+)25 b(2)p Fp(\031)s(l)40 b Fr(and)c(the)i (line)d(Im)15 b Fp(\020)43 b Fr(=)37 b Fn(\000)p Fp(Y)1551 922 y Fk(\000)p Fo(;l)1651 907 y Fr(,)i(where)e Fp(l)i Fr(is)d(an)h(\014xed)g(arbitrary)f(p)s(ositiv)m(e)g(in)m(teger,)k(and)c Fp(Y)3817 922 y Fk(\000)p Fo(;l)3955 907 y Fr(is)-236 1015 y(p)s(ositiv)m(e)29 b(and)h(indep)s(enden)m(t)e(of)j Fp(")p Fr(.)-236 1170 y Fu(6.)55 b Fr(Fix)35 b Fp(l)h Fn(2)d Fl(Z)p Fr(.)51 b(T)-8 b(o)36 b(sho)m(w)f(that)h Fp(f)45 b Fr(has)35 b(standard)f(b)s(eha)m(vior)h(in)f(a)i(domain)e Fp(R)2544 1185 y Fo(l)2605 1170 y Fr(as)i(describ)s(ed)d(in)h(Prop)s (osition)f(6.1,)38 b(w)m(e)-236 1278 y(need)27 b(only)f(to)i(c)m(hec)m (k)g(that)g(it)f(has)f(standard)h(b)s(eha)m(vior)f(in)f Fp(V)c Fr(,)28 b(a)f(constan)m(t)h(neigh)m(b)s(orho)s(o)s(d)d(of)i(the) h(in)m(terv)-5 b(al)26 b([)p Fp(\020)3579 1292 y Fm(0)3618 1278 y Fp(;)15 b(\020)3698 1292 y Fm(0)3751 1278 y Fr(+)f(2)p Fp(\031)s(l)r Fr(].)-236 1433 y(Let)31 b Fp(c)-34 1447 y Fm(0)37 1433 y Fr(b)s(e)f(the)h(suprem)m(um)e(of)i(the)g(v)-5 b(alues)30 b(of)h Fp(c)h Fr(suc)m(h)e(that)i(the)f(standard)f(b)s(eha)m (vior)g(is)g(v)-5 b(alid)29 b(in)g(a)j(constan)m(t)g(neigh)m(b)s(or-) -236 1541 y(ho)s(o)s(d)j(of)h(a)g(in)m(terv)-5 b(al)35 b(\()p Fp(\020)596 1555 y Fm(0)636 1541 y Fp(;)15 b(c)p Fr(\))37 b(in)e(the)h(half-plane)e Fn(f)p Fr(Re)16 b Fp(\020)41 b(<)35 b(c)p Fn(g)p Fr(.)58 b(The)35 b(standard)h(b)s(eha)m (vior)f(of)h Fp(f)45 b Fr(is)35 b(v)-5 b(alid)34 b(in)h(a)h(constan)m (t)-236 1649 y(neigh)m(b)s(orho)s(o)s(d)c(of)j(the)g(p)s(oin)m(t)f(of)h (in)m(tersection)f(of)h(the)g(canonical)f(line)g Fp(\014)39 b Fr(and)c Fl(R)43 b Fr(\(as)35 b(it)g(is)e(v)-5 b(alid)33 b(in)h(a)h(lo)s(cal)f(canonical)-236 1757 y(domain)29 b(enclosing)g Fp(\014)5 b Fr(\).)42 b(Therefore,)30 b Fp(c)1110 1771 y Fm(0)1175 1757 y Fp(>)25 b Fr(0.)-236 1912 y(Assume)30 b(that)h Fp(c)337 1926 y Fm(0)402 1912 y Fp(<)25 b(\020)538 1926 y Fm(0)597 1912 y Fr(+)20 b(2)p Fp(\031)s(l)r Fr(.)41 b(Note)32 b(that)-137 2039 y(1.)43 b Fp(f)c Fr(has)30 b(standard)g(b)s(eha)m(vior)f(in)g Fp(R)1148 2054 y Fo(l)1195 2039 y Fn(n)20 b Fr([)p Fp(c)1324 2053 y Fm(0)1364 2039 y Fp(;)15 b(\020)1444 2053 y Fm(0)1504 2039 y Fr(+)20 b(2)p Fp(\031)s(l)r Fr(],)-137 2147 y(2.)43 b(b)m(y)30 b(Lemma)g(5.2,)i(there)f(exists)f(a)g(canonical)g(domain)g (con)m(taining)g Fp(c)2361 2161 y Fm(0)2400 2147 y Fr(.)-236 2275 y(By)k(the)g(Adjacen)m(t)g(Canonical)f(Domain)g(Principle,)f (these)i(t)m(w)m(o)h(observ)-5 b(ations)33 b(imply)e(that)k Fp(f)42 b Fr(has)34 b(standard)f(b)s(eha)m(vior)-236 2383 y(in)40 b(a)h(constan)m(t)i(neigh)m(b)s(orho)s(o)s(d)38 b(of)k Fp(c)1080 2397 y Fm(0)1161 2383 y Fr(whic)m(h)e(is)g(imp)s (ossible.)70 b(So,)44 b Fp(c)2249 2397 y Fm(0)2332 2383 y Fn(\025)f Fp(\020)2486 2397 y Fm(0)2552 2383 y Fr(+)27 b(2)p Fp(\031)s(l)r Fr(.)74 b(This)39 b(completes)j(the)f(pro)s(of)g (of)-236 2491 y(Prop)s(osition)28 b(6.1)k(when)d(Im)15 b Fp(\024)813 2505 y Fo(p)878 2491 y Fp(<)25 b Fr(0)31 b(in)e Fp(D)1231 2505 y Fm(0)1271 2491 y Fr(.)40 b(When)30 b(Im)15 b Fp(\024)1777 2505 y Fo(n)1820 2514 y Fg(0)1884 2491 y Fp(>)25 b Fr(0,)31 b(the)g(analysis)e(is)g(similar;)f(w)m(e)j (omit)f(it.)p 3950 2491 4 62 v 3954 2433 55 4 v 3954 2491 V 4008 2491 4 62 v -236 2655 a(6.3.)53 b Fu(Consisten)m(t)e (basis.)46 b Fr(In)d(the)h(sequel,)j(for)c Fp(g)s Fr(,)48 b(a)c(function)f(of)h(complex)g(v)-5 b(ariables)42 b(\()p Fp(z)3079 2669 y Fm(1)3119 2655 y Fp(;)15 b(z)3201 2669 y Fm(2)3241 2655 y Fp(;)g(:)g(:)g(:)i(;)e(z)3485 2670 y Fo(k)3528 2655 y Fr(\),)48 b(w)m(e)c(de\014ne)-236 2772 y Fp(g)-190 2739 y Fk(\003)-150 2772 y Fr(\()p Fp(z)-73 2786 y Fm(1)-33 2772 y Fp(;)15 b(z)49 2786 y Fm(2)89 2772 y Fp(;)g(:)g(:)g(:)i(;)e(z)333 2787 y Fo(k)376 2772 y Fr(\))26 b(=)p 533 2693 609 4 v 25 w Fp(g)s Fr(\()p 614 2722 82 4 v Fp(z)656 2786 y Fm(1)696 2772 y Fp(;)p 736 2722 V 15 w(z)778 2786 y Fm(2)818 2772 y Fp(;)15 b(:)g(:)g(:)i(;)p 1020 2722 85 4 v 15 w(z)1062 2787 y Fo(k)1105 2772 y Fr(\))q(.)-236 2937 y(6.3.1.)48 b Fi(Solution)f Fp(f)448 2904 y Fk(\003)487 2937 y Fi(.)e Fr(Recall)f(that)h Fp(f)54 b Fr(is)43 b(analytic)h(in)f(a)i(constan)m(t)h(neigh)m(b)s (orho)s(o)s(d)c(of)i Fp(E)2983 2951 y Fm(0)3023 2937 y Fr(.)83 b(In)44 b(this)f(neigh)m(b)s(orho)s(o)s(d,)-236 3045 y(consider)28 b Fp(f)172 3012 y Fk(\003)211 3045 y Fr(\()p Fp(x;)15 b(\020)7 b(;)15 b(E)5 b Fr(\),)31 b(where)d(w)m(e)i(ha)m(v)m(e)g(indicated)e(explicitly)f(the)i(dep)s (endence)f(on)h Fp(E)5 b Fr(.)41 b(As)29 b Fp(V)38 b Fr(+)18 b Fp(W)41 b Fr(is)29 b(real)f(on)i Fl(R)r Fr(,)36 b Fp(f)3887 3012 y Fk(\003)3955 3045 y Fr(is)-236 3153 y(a)31 b(solution)d(of)38 b(\(3.2\))r(.)j(The)29 b(solution)g Fp(f)1132 3120 y Fk(\003)1201 3153 y Fr(also)i(satis\014es)f(the)g (consistency)g(condition,)g(and)f(it)h(is)g(analytic)g(in)f Fp(\020)37 b Fr(in)29 b Fp(S)3828 3167 y Fo(Y)3888 3153 y Fr(.)-236 3317 y(6.3.2.)48 b Fi(The)32 b(asymptotics)h(of)f Fp(f)861 3284 y Fk(\003)900 3317 y Fi(.)45 b Fr(Let)29 b(us)g(write)f(the)h(asymptotics)g(of)g Fp(f)38 b Fr(explicitly)-8 b(.)39 b(W)-8 b(e)30 b(assume)e(that)i Fp(f)38 b Fr(is)28 b(normalized)-236 3425 y(at)j Fp(\020)-85 3439 y Fm(0)-46 3425 y Fr(.)41 b(Then,)436 3607 y Fp(f)10 b Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))25 b(=)g Fp(e)876 3530 y Fd(i)p 874 3542 30 3 v 874 3584 a(")925 3504 y Fb(R)972 3525 y Fd(\020)958 3582 y(\020)986 3597 y Fg(0)1037 3558 y Fo(\024)1078 3566 y Fd(n)1116 3581 y Fg(0)1155 3558 y Fo(d\020)1231 3607 y Fr(\(\011)1337 3621 y Fm(+)1396 3607 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))21 b(+)e Fp(o)p Fr(\(1\)\))p Fp(;)108 b Fr(\011)2114 3621 y Fm(+)2173 3607 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))26 b(=)f Fp(q)s Fr(\()p Fp(\020)7 b Fr(\))p Fp(e)2707 3504 y Fb(R)2754 3525 y Fd(\020)2741 3582 y(\020)2769 3597 y Fg(0)2820 3558 y Fo(!)2864 3567 y Fg(+)2914 3558 y Fo(d\020)2990 3607 y Fp( )3049 3621 y Fm(+)3108 3607 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))p Fp(:)-3578 b Fr(\(6.6\))-236 3797 y(W)-8 b(e)25 b(c)m(ho)s(ose)g Fp(q)j Fr(=)361 3698 y Ff(q)p 451 3698 557 4 v 451 3797 a Fp(k)501 3770 y Fk(0)498 3819 y Fo(p)538 3797 y Fr(\()p Fp(E)e Fn(\000)20 b Fp(W)13 b Fr(\()p Fp(\020)7 b Fr(\)\).)39 b(Recall)23 b(that)i Fp(k)1580 3764 y Fk(0)1577 3819 y Fo(p)1617 3797 y Fr(\()p Fn(E)8 b Fr(\))24 b(is)f(p)s(ositiv)m(e)g (on)g(the)h Fp(n)2506 3811 y Fm(0)2545 3797 y Fr(-th)g(sp)s(ectral)f (band.)38 b(So,)25 b(w)m(e)f(\014x)f(a)h(branc)m(h)-236 3934 y(of)30 b Fp(q)j Fr(so)e(that)g Fp(q)s Fr(\()p Fp(\020)7 b Fr(\))25 b Fp(>)g Fr(0)31 b(on)f Fl(R)s Fr(.)47 b(The)29 b(asymptotics)i(\(6.6\))h(is)e(v)-5 b(alid)28 b(in)h(the)i(domain)e Fp(R)2739 3949 y Fo(l)2796 3934 y Fr(describ)s(ed)f(b)m(y)i(Prop)s (osition)e(6.1.)-236 4090 y(Note)j(that,)h(for)e(real)g Fp(E)36 b Fr(and)29 b Fp(\020)7 b Fr(,)p 465 4162 252 4 v 465 4241 a Fp(\024)517 4255 y Fo(n)560 4264 y Fg(0)599 4241 y Fr(\()p Fp(\020)g Fr(\))25 b(=)g Fp(\024)889 4255 y Fo(n)932 4264 y Fg(0)971 4241 y Fr(\()p Fp(\020)7 b Fr(\))p Fp(;)p 1220 4162 328 4 v 107 w( )1279 4255 y Fm(+)1338 4241 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))26 b(=)f Fp( )1728 4255 y Fk(\000)1787 4241 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))p Fp(;)p 2128 4162 233 4 v 107 w(!)2185 4255 y Fm(+)2244 4241 y Fr(\()p Fp(\020)g Fr(\))25 b(=)g Fp(!)2539 4255 y Fk(\000)2598 4241 y Fr(\()p Fp(\020)7 b Fr(\))p Fp(;)p 2846 4162 162 4 v 106 w(q)s Fr(\()p Fp(\020)g Fr(\))25 b(=)g Fp(q)s Fr(\()p Fp(\020)7 b Fr(\))p Fp(:)-3550 b Fr(\(6.7\))-236 4388 y(The)33 b(\014rst)h(relation)f (follo)m(ws)h(from)f(p)s(oin)m(t)h(3)g(of)h(Lemma)f(6.1.)53 b(The)34 b(second)g(one)h(holds)d(as,)k(on)e(the)h(sp)s(ectral)e(band,) h(the)-236 4496 y(branc)m(hes)25 b(of)h(the)h(Blo)s(c)m(h)e(solution)g Fp( )s Fr(\()p Fp(x;)15 b Fn(E)8 b Fr(\))28 b(di\013er)c(b)m(y)i (complex)g(conjugation.)39 b(The)25 b(second)h(relation)g(and)f(the)h (de\014nition)-236 4604 y(of)k(the)h(functions)e Fp(!)473 4618 y Fk(\006)562 4604 y Fr(imply)f(the)i(third)f(one.)41 b(The)30 b(last)g(relation)g(follo)m(ws)f(from)h(our)g(c)m(hoice)h(of)g Fp(q)s Fr(.)-236 4759 y(The)f(relations)f(\(6.7\))j(and)e(the)h (standard)e(b)s(eha)m(vior)h(of)g Fp(f)40 b Fr(imply)28 b(that)j Fp(f)2317 4726 y Fk(\003)2386 4759 y Fr(has)f(standard)f(b)s (eha)m(vior)332 4946 y Fp(f)387 4908 y Fk(\003)451 4946 y Fr(=)24 b Fp(e)588 4897 y Fk(\000)657 4869 y Fd(i)p 654 4881 30 3 v 654 4923 a(")705 4843 y Fb(R)752 4864 y Fd(\020)738 4921 y(\020)766 4936 y Fg(0)817 4897 y Fo(\024)858 4905 y Fd(n)896 4920 y Fg(0)935 4897 y Fo(d\020)1011 4946 y Fr(\(\011)1117 4960 y Fk(\000)1176 4946 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))21 b(+)e Fp(o)p Fr(\(1\)\))p Fp(;)108 b Fr(\011)1894 4960 y Fk(\000)1953 4946 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))26 b(=)f Fp(q)s Fr(\()p Fp(\020)7 b Fr(\))p Fp(e)2487 4843 y Fb(R)2534 4864 y Fd(\020)2521 4921 y(\020)2549 4936 y Fg(0)2600 4897 y Fo(!)2644 4906 y Fc(\000)2696 4897 y Fo(d\020)2772 4946 y Fp( )2831 4960 y Fk(\000)2890 4946 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))p Fp(;)46 b(\020)32 b Fn(2)25 b Fp(R)3397 4961 y Fo(l)3423 4946 y Fp(:)-236 5111 y Fr(6.3.3.)48 b Fi(The)30 b(Wr)-5 b(onskian)30 b(of)g Fp(f)38 b Fi(and)30 b Fp(f)1075 5078 y Fk(\003)1114 5111 y Fi(.)45 b Fr(Both)28 b(solutions)d Fp(f)36 b Fr(and)26 b Fp(f)2095 5078 y Fk(\003)2160 5111 y Fr(satisfy)g(the)h(consistency)f(condition.)38 b(T)-8 b(o)28 b(de\014ne)d(the)-236 5219 y(mono)s(drom)m(y)35 b(matrix)g(for)g(the)g(pair)g Fp(f)44 b Fr(and)35 b Fp(f)1397 5186 y Fk(\003)1436 5219 y Fr(,)i(w)m(e)f(ha)m(v)m(e)h(to)f(c)m(hec)m (k)h(that)f(these)g(solutions)e(are)i(linearly)d(indep)s(enden)m(t)-236 5327 y(and)d(that)h(their)e(W)-8 b(ronskian)30 b(is)f(indep)s(enden)m (t)f(of)j Fp(\020)7 b Fr(.)-236 5482 y(The)30 b(asymptotics)g(of)h Fp(f)39 b Fr(and)30 b Fp(f)874 5449 y Fk(\003)943 5482 y Fr(imply)e(that)1314 5629 y Fp(w)r Fr(\()p Fp(f)5 b(;)15 b(f)1561 5591 y Fk(\003)1600 5629 y Fr(\))25 b(=)g Fp(w)r Fr(\(\011)1929 5643 y Fm(+)1989 5629 y Fp(;)15 b Fr(\011)2100 5643 y Fk(\000)2159 5629 y Fr(\))21 b(+)f Fp(o)p Fr(\(1\))-2701 b(\(6.8\))-236 5783 y(lo)s(cally)24 b(uniformly)f(in)h Fp(\020)32 b Fn(2)25 b Fp(R)785 5798 y Fo(l)811 5783 y Fr(.)39 b(The)25 b(W)-8 b(ronskian)26 b(of)f(the)h(canonical)g(Blo)s (c)m(h)g(solutions)e(\011)2849 5797 y Fk(\006)2933 5783 y Fr(is)h(giv)m(en)h(b)m(y)h(\(3.11\))r(.)39 b(As)26 b Fp(k)3852 5750 y Fk(0)3849 5805 y Fo(p)3889 5783 y Fr(\()p Fn(E)8 b Fr(\))-236 5895 y(is)32 b(p)s(ositiv)m(e)f(on)i(the)g Fp(n)539 5909 y Fm(0)578 5895 y Fr(-th)g(sp)s(ectral)f(band,)g(and)g (as)h Fn(E)38 b Fr(=)28 b Fp(E)g Fn(\000)21 b Fp(W)13 b Fr(\()p Fp(\020)2165 5909 y Fm(0)2204 5895 y Fr(\))33 b(b)s(elongs)f(to)h(this)f(band)f(for)i Fp(E)i Fn(2)28 b Fp(J)9 b Fr(,)34 b(the)f(leading)-236 6003 y(term)d(in)f(\(6.8\))j (is)d(non-zero.)41 b(So,)31 b(for)f Fp(")g Fr(su\016cien)m(tly)f (small,)g Fp(f)40 b Fr(and)29 b Fp(f)2196 5970 y Fk(\003)2265 6003 y Fr(are)i(linearly)d(indep)s(enden)m(t)g(for)i(an)m(y)g Fp(\020)37 b Fr(in)29 b(a)h(\014xed)-236 6110 y(admissible)d(sub)s (domain)h(of)i Fp(R)839 6125 y Fo(l)865 6110 y Fr(.)1854 6210 y Fm(19)p eop %%Page: 20 20 20 19 bop -236 241 a Fr(The)29 b(leading)f(term)h(in)f(\(6.8\))j(is)d (indep)s(enden)m(t)f(of)j Fp(\020)7 b Fr(,)29 b(but,)g(the)g(error)g (term)h(in)d(\(6.8\))32 b(can)d(dep)s(end)f(on)h Fp(\020)7 b Fr(.)40 b(W)-8 b(e)30 b(mo)s(dify)d(the)-236 349 y(solutions)j Fp(f)40 b Fr(and)31 b Fp(f)463 316 y Fk(\003)533 349 y Fr(so)g(that)h(this)f(error)g(term)g(b)s(e)g(constan)m(t.)45 b(As)31 b(b)s(oth)g(solutions)e(satisfy)i(the)h(consistency)f (condition)-236 457 y(\(3.3\))r(,)25 b(this)d(error)i(term)g(is)e(2)p Fp(\031)s Fr(-p)s(erio)s(dic)g(in)h Fp(\020)7 b Fr(.)37 b(So,)26 b(w)m(e)e(can)g(rede\014ne)f(the)h(solution)e Fp(f)33 b Fr(b)m(y)23 b(m)m(ultiplying)e(it)i(b)m(y)g(a)h(2)p Fp(\031)s Fr(-p)s(erio)s(dic)-236 565 y(factor)31 b(of)g(the)f(form)g (\(1)21 b(+)f Fp(o)p Fr(\(1\)\))32 b(to)f(get)h(exactly)1437 719 y Fp(w)r Fr(\()p Fp(f)5 b(;)15 b(f)1684 681 y Fk(\003)1723 719 y Fr(\))26 b(=)e Fp(w)r Fr(\(\011)2052 733 y Fm(+)2112 719 y Fp(;)15 b Fr(\011)2223 733 y Fk(\000)2282 719 y Fr(\))p Fp(:)-2578 b Fr(\(6.9\))-236 874 y(The)33 b(\\new")g(solutions) f Fp(f)43 b Fr(and)32 b Fp(f)939 841 y Fk(\003)1011 874 y Fr(form)h(a)h(consisten)m(t)g(basis)e(and,)h(in)f Fp(R)2343 889 y Fo(l)2370 874 y Fr(,)i(they)f(admit)g(the)g(same)h(standard)f(b)s (eha)m(vior)-236 982 y(as)d(the)h(\\old")f(solutions)f Fp(f)40 b Fr(and)30 b Fp(f)971 949 y Fk(\003)1009 982 y Fr(.)-236 1161 y(6.4.)53 b Fu(The)26 b(mono)s(drom)m(y)g(matrix.)44 b Fr(The)23 b(mono)s(drom)m(y)f(matrix)g(asso)s(ciated)h(to)h(the)f (basis)e(\()p Fp(f)5 b(;)15 b(f)3150 1128 y Fk(\003)3189 1161 y Fr(\))23 b(has)g(the)g(form)f(\(1.12\))r(,)-236 1269 y(where)30 b(the)g(co)s(e\016cien)m(ts)h Fp(a)g Fr(and)e Fp(b)i Fr(are)g(giv)m(en)f(b)m(y)537 1471 y Fp(a)p Fr(\()p Fp(\020)7 b Fr(\))25 b(=)833 1409 y Fp(w)r Fr(\()p Fp(f)10 b Fr(\()p Fp(x;)15 b(\020)28 b Fn(\000)19 b Fr(2)p Fp(\031)s Fr(\))p Fp(;)c(f)1505 1376 y Fk(\003)1545 1409 y Fr(\()p Fp(x;)g(\020)7 b Fr(\)\))p 833 1450 958 4 v 939 1533 a Fp(w)r Fr(\()p Fp(f)j Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))p Fp(;)15 b(f)1400 1507 y Fk(\003)1440 1533 y Fr(\()p Fp(x;)g(\020)7 b Fr(\)\))1800 1471 y Fp(;)107 b(b)p Fr(\()p Fp(\020)7 b Fr(\))25 b(=)g Fn(\000)2290 1409 y Fp(w)r Fr(\()p Fp(f)10 b Fr(\()p Fp(x;)15 b(\020)27 b Fn(\000)20 b Fr(2)p Fp(\031)s Fr(\))p Fp(;)15 b(f)10 b Fr(\()p Fp(x;)15 b(\020)7 b Fr(\)\))p 2290 1450 918 4 v 2376 1533 a Fp(w)r Fr(\()p Fp(f)j Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))p Fp(;)15 b(f)2837 1507 y Fk(\003)2877 1533 y Fr(\()p Fp(x;)g(\020)7 b Fr(\)\))3218 1471 y Fp(:)-3479 b Fr(\(6.10\))-236 1679 y(As)30 b Fp(M)41 b Fr(is)29 b(unimo)s(dular)e(\(see)k(\(1.8\))r(\),)g(one)f(has)1377 1833 y Fp(a)p Fr(\()p Fp(\020)7 b Fr(\))p Fp(a)1590 1795 y Fk(\003)1630 1833 y Fr(\()p Fp(\020)g Fr(\))20 b Fn(\000)g Fp(b)p Fr(\()p Fp(\020)7 b Fr(\))p Fp(b)2053 1795 y Fk(\003)2093 1833 y Fr(\()p Fp(\020)g Fr(\))25 b(=)g(1)p Fp(:)-2637 b Fr(\(6.11\))-236 1988 y(T)-8 b(ogether)33 b(with)d(the)j(solutions)d Fp(f)41 b Fr(and)31 b Fp(f)1214 1955 y Fk(\003)1253 1988 y Fr(,)i(the)f(mono)s(drom)m(y)f(matrix)g(is)g(analytic)h(in)e Fp(\020)35 b Fn(2)27 b Fl(R)3041 2003 y Fo(l)3104 1988 y Fr(and)32 b(in)e Fp(E)37 b Fr(in)31 b(a)h(constan)m(t)-236 2096 y(neigh)m(b)s(orho)s(o)s(d)i(of)j Fp(E)518 2110 y Fm(0)558 2096 y Fr(.)60 b(Actually)-8 b(,)39 b(b)s(eing)d(p)s(erio)s (dic,)g(the)h(mono)s(drom)m(y)f(matrix)h(is)f(analytic)g(in)g Fp(\020)43 b Fr(in)36 b(the)h(whole)f(strip)-236 2204 y Fn(j)p Fr(Im)15 b Fp(\020)7 b Fn(j)25 b(\024)g Fp(Y)159 2219 y Fo(l)184 2204 y Fr(.)40 b(Belo)m(w,)29 b(w)m(e)g(compute)f(the)g (asymptotics)g(of)g(the)g(co)s(e\016cien)m(ts)g Fp(a)g Fr(and)f Fp(b)p Fr(.)40 b(This)26 b(computation)i(will)d(complete)-236 2312 y(the)30 b(pro)s(of)g(of)h(Theorem)f(1.3.)-236 2491 y(6.4.1.)48 b Fi(Co)-5 b(e\016cient)39 b Fp(a)p Fi(.)45 b Fr(W)-8 b(e)37 b(compute)f(the)g(W)-8 b(ronskian)36 b Fp(w)r Fr(\()p Fp(f)10 b Fr(\()p Fp(x;)15 b(\020)31 b Fn(\000)23 b Fr(2)p Fp(\031)s Fr(\))p Fp(;)15 b(f)2439 2458 y Fk(\003)2480 2491 y Fr(\()p Fp(x;)g(\020)7 b Fr(\)\).)58 b(Fix)35 b(an)h(in)m(teger)g Fp(l)g Fn(\025)f Fr(2.)57 b(Assume)-236 2599 y(that)30 b Fp(\020)36 b Fr(and)29 b Fp(\020)d Fn(\000)19 b Fr(2)p Fp(\031)33 b Fr(b)s(elong)c(to)h Fp(R)967 2614 y Fo(l)993 2599 y Fr(.)41 b(Then,)29 b(b)s(oth)g (functions)f Fp(f)10 b Fr(\()p Fp(x;)15 b(\020)26 b Fn(\000)18 b Fr(2)p Fp(\031)s Fr(\))31 b(and)e Fp(f)2660 2566 y Fk(\003)2699 2599 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))30 b(ha)m(v)m(e)h(standard)e(b)s(eha)m(vior.)39 b(In)-236 2707 y(particular,)458 2911 y Fp(f)10 b Fr(\()p Fp(x;)15 b(\020)27 b Fn(\000)20 b Fr(2)p Fp(\031)s Fr(\))26 b(=)f Fp(e)1110 2834 y Fd(i)p 1107 2846 30 3 v 1107 2888 a(")1158 2808 y Fb(R)1205 2829 y Fd(\020)s Fc(\000)p Fg(2)p Fd(\031)1192 2886 y(\020)1220 2901 y Fg(0)1369 2862 y Fo(\024)1410 2870 y Fd(n)1448 2885 y Fg(0)1499 2862 y Fo(d\020)1605 2782 y Ff(\022)1672 2911 y Fp(q)s Fr(\()p Fp(\020)i Fn(\000)20 b Fr(2)p Fp(\031)s Fr(\))p Fp(e)2086 2808 y Fb(R)2134 2829 y Fd(\020)s Fc(\000)p Fg(2)p Fd(\031)2120 2886 y(\020)2148 2901 y Fg(0)2298 2862 y Fo(!)2342 2871 y Fg(+)2404 2862 y Fo(d\020)2480 2911 y Fp( )2539 2925 y Fm(+)2598 2911 y Fr(\()p Fp(\020)27 b Fn(\000)20 b Fr(2)p Fp(\031)s Fr(\))h(+)f Fp(o)p Fr(\(1\))3197 2782 y Ff(\023)3281 2911 y Fp(;)-3542 b Fr(\(6.12\))-236 3115 y(where)31 b(w)m(e)h(in)m(tegrate)h(in)e Fp(R)723 3130 y Fo(l)749 3115 y Fr(.)45 b(Let)32 b(us)f(c)m(hec)m(k)i(that)g(the)f(functions)e Fp(\024)2149 3129 y Fo(n)2192 3138 y Fg(0)2231 3115 y Fr(,)i Fp(q)s Fr(,)g Fp(!)2446 3129 y Fm(+)2537 3115 y Fr(and)f Fp( )2774 3129 y Fm(+)2865 3115 y Fr(are)h(2)p Fp(\031)s Fr(-p)s(erio)s(dic)e(in)g Fn(S)3661 3129 y Fo(Y)3722 3115 y Fr(.)45 b(All)31 b(of)-236 3223 y(these)g(functions)e (b)s(eing)f(analytic,)j(it)f(su\016ces)g(to)h(c)m(hec)m(k)h(their)d(p)s (erio)s(dicit)m(y)f(on)i Fl(R)39 b Fr(and)30 b(for)g(real)g Fp(E)5 b Fr(.)41 b(In)30 b(this)f(case,)-111 3354 y Fn(\017)42 b Fp(k)23 3368 y Fo(n)97 3354 y Fr(and)26 b Fp( )329 3368 y Fk(\006)415 3354 y Fr(are)h(p)s(erio)s(dic)e(as)i Fn(E)8 b Fr(\()p Fp(\020)f Fr(\))25 b(=)g Fp(E)19 b Fn(\000)13 b Fp(W)g Fr(\()p Fp(\020)7 b Fr(\))26 b(is)g(2)p Fp(\031)s Fr(-p)s(erio)s(dic)f(and)i(tak)m(es)h(v)-5 b(alues)26 b(in)g(inside)e(the)j Fp(n)3522 3368 y Fm(0)3561 3354 y Fr(-th)g(sp)s(ectral)-24 3462 y(band;)-111 3570 y Fn(\017)42 b Fr(the)30 b(p)s(erio)s(dicit)m(y)e(of)j Fp(!)748 3584 y Fk(\006)836 3570 y Fr(follo)m(ws)f(from)g(their)f(de\014nition)f(and) i(the)h(previous)e(p)s(oin)m(t;)-111 3678 y Fn(\017)42 b Fr(the)30 b(function)f Fp(q)k Fr(is)d(p)s(erio)s(dic)e(as)i Fp(k)1164 3645 y Fk(0)1161 3700 y Fo(p)1201 3678 y Fr(\()p Fp(E)c Fn(\000)20 b Fp(W)13 b Fr(\()p Fp(\020)7 b Fr(\)\))30 b(is)g(p)s(erio)s(dic)d(and)j(p)s(ositiv)m(e)g(on)g Fl(R)s Fr(.)-236 3813 y(So,)g(one)h(rewrites)e(\(6.12\))k(in)c(the)i(form)750 4022 y Fp(f)10 b Fr(\()p Fp(x;)15 b(\020)27 b Fn(\000)20 b Fr(2)p Fp(\031)s Fr(\))26 b(=)f Fp(c)1386 4036 y Fm(0)1426 4022 y Fp(e)1481 3945 y Fd(i)p 1478 3957 V 1478 3999 a(")1529 3919 y Fb(R)1576 3940 y Fd(\020)1562 3997 y(\020)1590 4012 y Fg(0)1641 3973 y Fo(\024)1682 3981 y Fd(n)1720 3996 y Fg(0)1759 3973 y Fo(d\020)1850 3894 y Ff(\022)1917 4022 y Fp(q)s Fr(\()p Fp(\020)7 b Fr(\))p Fp(e)2120 3919 y Fb(R)2168 3940 y Fd(\020)2154 3997 y(\020)2182 4012 y Fg(0)2233 3973 y Fo(!)2277 3982 y Fg(+)2339 3973 y Fo(d\020)2415 4022 y Fp( )2474 4036 y Fm(+)2533 4022 y Fr(\()p Fp(\020)g Fr(\))21 b(+)f Fp(o)p Fr(\(1\))2921 3894 y Ff(\023)3004 4022 y Fp(:)-3265 b Fr(\(6.13\))-236 4226 y(where)30 b(w)m(e)g(in)m(tegrate)i(in)d Fp(R)717 4241 y Fo(l)773 4226 y Fr(and)h(ha)m(v)m(e)i(de\014ned)140 4442 y Fp(c)179 4456 y Fm(0)244 4442 y Fr(=)25 b Fp(e)382 4404 y Fk(\000)451 4377 y Fd(i)p 448 4389 V 448 4430 a(")487 4404 y Fm(\010+)p Fo(i)p Fm(\012)672 4442 y Fp(;)106 b Fr(\010)25 b(=)g Fn(\000)1076 4318 y Ff(Z)1167 4344 y Fo(\020)5 b Fk(\000)p Fm(2)p Fo(\031)1126 4524 y(\020)1355 4442 y Fp(\024)1407 4456 y Fo(n)1450 4465 y Fg(0)1489 4442 y Fp(d\020)32 b Fr(=)1704 4318 y Ff(Z)1794 4344 y Fm(2)p Fo(\031)1754 4524 y Fm(0)1892 4442 y Fp(\024)1944 4456 y Fo(n)1987 4465 y Fg(0)2026 4442 y Fp(d\020)7 b(;)106 b Fr(and)90 b(\012)25 b(=)g Fn(\000)p Fp(i)2792 4318 y Ff(Z)2883 4344 y Fo(\020)5 b Fk(\000)p Fm(2)p Fo(\031)2843 4524 y(\020)3071 4442 y Fp(!)3128 4456 y Fm(+)3187 4442 y Fp(d\020)32 b Fr(=)25 b Fp(i)3448 4318 y Ff(Z)3539 4344 y Fm(2)p Fo(\031)3499 4524 y Fm(0)3637 4442 y Fp(!)3694 4456 y Fm(+)3752 4442 y Fp(d\020)7 b(:)-4107 b Fr(\(6.14\))-236 4660 y(Represen)m(tation)31 b(\(6.13\))h(implies)c(that)228 4814 y Fp(w)17 b Fr(\()q Fp(f)10 b Fr(\()p Fp(x;)15 b(\020)26 b Fn(\000)20 b Fr(2)p Fp(\031)s Fr(\))p Fp(;)15 b(f)915 4777 y Fk(\003)955 4814 y Fr(\))26 b(=)f Fp(c)1151 4828 y Fm(0)1206 4814 y Fr(\()p Fp(w)r Fr(\(\011)1414 4828 y Fm(+)1474 4814 y Fp(;)15 b Fr(\011)1585 4828 y Fk(\000)1644 4814 y Fr(\))21 b(+)f Fp(o)p Fr(\(1\)\))26 b(=)f Fp(c)2146 4828 y Fm(0)2186 4814 y Fr(\()p Fp(w)r Fr(\()p Fp(f)5 b(;)15 b(f)2468 4777 y Fk(\003)2508 4814 y Fr(\))20 b(+)g Fp(o)p Fr(\(1\)\))27 b(=)e Fp(c)3010 4828 y Fm(0)3050 4814 y Fp(w)r Fr(\()p Fp(f)5 b(;)15 b(f)3297 4777 y Fk(\003)3336 4814 y Fr(\)\(1)21 b(+)f Fp(o)p Fr(\(1\)\))p Fp(:)-236 4974 y Fr(Here,)40 b(w)m(e)e(ha)m(v)m(e)h(used)d(relation)h(\(6.9\))i (and)e(the)h(fact)g(that)g Fp(w)r Fr(\(\011)2061 4988 y Fm(+)2121 4974 y Fp(;)15 b Fr(\011)2232 4988 y Fk(\000)2291 4974 y Fr(\))38 b(do)s(es)f(not)h(v)-5 b(anish)36 b(and)h(is)f(indep)s (enden)m(t)f(of)j Fp(")p Fr(.)-236 5082 y(No)m(w,)31 b(b)m(y)h(\(6.10\))r(,)f(w)m(e)f(obtain)g(the)h(form)m(ula)e Fp(a)p Fr(\()p Fp(\020)7 b Fr(\))26 b(=)f Fp(c)1644 5096 y Fm(0)1684 5082 y Fr(\(1)c(+)e Fp(o)p Fr(\(1\)\).)-236 5261 y(6.4.2.)48 b Fi(Pr)-5 b(o)g(of)34 b(of)e(the)g(statements)i(of)e (The)-5 b(or)g(em)34 b(1.3)f(c)-5 b(onc)g(erning)33 b(the)f(c)-5 b(o)g(e\016cient)33 b Fp(a)p Fi(.)45 b Fr(Fix)29 b Fp(\016)g(>)c Fr(0.)41 b(Being)30 b(a)g(consequence)-236 5368 y(of)i(the)g(standard)f (b)s(eha)m(vior)f(of)i Fp(f)41 b Fr(and)31 b Fp(f)1204 5335 y Fk(\003)1243 5368 y Fr(,)i(the)e(asymptotics)h(of)g Fp(a)g Fr(is)f(uniform)e(for)j Fp(\020)38 b Fr(in)31 b(the)g Fp(\016)s Fr(-admissible)f(sub)s(domain)-236 5476 y(of)g Fp(R)-64 5491 y Fo(l)-7 5476 y Fr(and)f(for)i Fp(E)k Fr(in)29 b(a)i(constan)m(t)h(neigh)m(b)s(orho)s(o)s(d)c Fp(V)1584 5491 y Fo(\016)1652 5476 y Fr(of)i Fp(E)1822 5490 y Fm(0)1862 5476 y Fr(.)-236 5632 y(As)g Fp(a)g Fr(is)g Fp(")p Fr(-p)s(erio)s(dic,)f(its)g(asymptotics)i(is)e(uniform)f (in)i(the)g(strip)f Fn(j)p Fr(Im)15 b Fp(\020)7 b Fn(j)25 b(\024)g Fp(Y)2448 5647 y Fo(l)2494 5632 y Fn(\000)20 b Fp(\016)s Fr(.)-236 5787 y(Clearly)-8 b(,)30 b(the)h(zeroth)g(F)-8 b(ourier)30 b(co)s(e\016cien)m(t)i(of)e Fp(a)h Fr(has)g(the)f (asymptotics)h Fp(a)2326 5801 y Fm(0)2391 5787 y Fr(=)26 b Fp(c)2527 5801 y Fm(0)2567 5787 y Fr(\(1)21 b(+)f Fp(o)p Fr(\(1\)\).)43 b(Consider)29 b(the)i(functions)e(\010)-236 5895 y(and)f(\012)g(determining)f(the)i(leading)e(term)i(for)g(the)g (asymptotics)g(of)g Fp(a)p Fr(.)40 b(As,)29 b(for)g Fp(E)h Fr(=)25 b Fp(E)2785 5909 y Fm(0)2825 5895 y Fr(,)k(the)g(branc)m(h)f(p) s(oin)m(ts)g(of)h Fp(\024)3759 5909 y Fo(n)3802 5918 y Fg(0)3869 5895 y Fr(and)-236 6003 y(the)34 b(p)s(oles)g(of)g Fp(!)323 6017 y Fk(\006)416 6003 y Fr(are)h(outside)e(of)i(the)f(strip) f Fn(S)1429 6017 y Fo(Y)1490 6003 y Fr(,)i(and)f(as)h(they)f(con)m(tin) m(uously)f(dep)s(end)g(on)h Fp(E)5 b Fr(,)36 b(the)e(de\014nitions)e (of)j Fp(\024)3934 6017 y Fo(n)3977 6026 y Fg(0)-236 6110 y Fr(and)28 b Fp(!)-4 6124 y Fk(\006)83 6110 y Fr(imply)f(that)i (\010)g(and)f(\012)h(are)g(analytic)f(in)g Fp(E)34 b Fr(in)27 b(a)j(constan)m(t)g(neigh)m(b)s(orho)s(o)s(d)c(of)j Fp(E)2879 6124 y Fm(0)2919 6110 y Fr(.)40 b(Note)30 b(that,)g(as)f Fp(\024)3585 6124 y Fo(n)3628 6133 y Fg(0)3667 6110 y Fr(\()p Fp(\020)7 b Fr(\))29 b(tak)m(es)1854 6210 y Fm(20)p eop %%Page: 21 21 21 20 bop -236 241 a Fr(real)29 b(v)-5 b(alues)30 b(on)g(the)g(real)f (line,)g(\010)h(is)f(real)g(on)h Fp(J)9 b Fr(.)41 b(Chec)m(k)30 b(that)h(\012)e(is)g(real)h(on)g Fp(J)9 b Fr(.)40 b(Using)31 b(\(6.7\))h(and)d(the)h(de\014nition)e(of)i Fp(!)3932 255 y Fk(\006)3990 241 y Fr(,)-236 349 y(w)m(e)h(get)425 562 y(\012)19 b Fn(\000)p 601 489 66 4 v 20 w Fr(\012)25 b(=)g Fp(i)834 438 y Ff(Z)926 464 y Fm(2)p Fo(\031)885 644 y Fm(0)1008 562 y Fr(\()p Fp(!)1100 576 y Fm(+)1179 562 y Fr(+)p 1270 512 116 4 v 20 w Fp(!)1327 576 y Fm(+)1386 562 y Fr(\))p Fp(d\020)32 b Fr(=)25 b Fp(i)1682 438 y Ff(Z)1773 464 y Fm(2)p Fo(\031)1733 644 y Fm(0)1856 562 y Fr(\()p Fp(!)1948 576 y Fm(+)2027 562 y Fr(+)20 b Fp(!)2175 576 y Fk(\000)2233 562 y Fr(\))p Fp(d\020)33 b Fr(=)25 b Fn(\000)p Fp(i)2601 438 y Ff(Z)2692 464 y Fm(2)p Fo(\031)2652 644 y Fm(0)2823 500 y Fp(@)p 2799 541 100 4 v 2799 624 a(@)5 b(\020)2924 562 y Fr(log)17 b Fp(g)s Fr(\()p Fp(\020)7 b Fr(\))p Fp(d\020)g(;)-236 762 y Fr(where)1284 957 y Fp(g)s Fr(\()p Fp(\020)g Fr(\))26 b(=)1569 834 y Ff(Z)1660 860 y Fm(1)1620 1040 y(0)1715 957 y Fp( )1774 971 y Fm(+)1833 957 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))p Fp( )2101 971 y Fk(\000)2161 957 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))p Fp(dx:)-236 1175 y Fr(In)29 b(view)g(of)37 b(\(6.7\))q(,)30 b Fp(g)k Fr(is)28 b(p)s(ositiv)m(e.)40 b(As)29 b(it)h(is)e(p)s(erio)s (dic,)g(w)m(e)i(get)h(\012)18 b Fn(\000)p 2122 1102 66 4 v 19 w Fr(\012)25 b(=)g(0.)41 b(W)-8 b(e)30 b(ha)m(v)m(e)h(c)m(hec)m (k)m(ed)h(all)c(the)i(prop)s(erties)e(of)i Fp(a)3976 1189 y Fm(0)-236 1283 y Fr(announced)f(in)g(Theorem)h(1.3.)-236 1438 y(The)39 b(asymptotics)h(of)g Fp(a)g Fr(implies)d(that)j Fp(a)1246 1452 y Fm(1)1325 1438 y Fr(satis\014es)g(the)g(uniform)d (estimate)k Fp(a)2608 1452 y Fm(1)2688 1438 y Fr(=)g Fp(o)p Fr(\()p Fp(a)2927 1452 y Fm(0)2967 1438 y Fr(\))f(for)g(\()p Fp(\020)7 b(;)15 b(E)5 b Fr(\))41 b Fn(2)g(S)3618 1453 y Fo(Y)3659 1465 y Fd(l)3683 1453 y Fk(\000)p Fo(\016)3802 1438 y Fn(\002)27 b Fp(V)3953 1453 y Fo(\016)3990 1438 y Fr(.)-236 1546 y(Therefore,)j(the)h(F)-8 b(ourier)30 b(co)s(e\016cien)m(ts)h(\()q(^)-46 b Fp(a)1216 1560 y Fm(1)1256 1546 y Fr(\()p Fp(n)p Fr(\)\))1416 1560 y Fo(n)1493 1546 y Fr(of)31 b Fp(a)1645 1560 y Fm(1)1715 1546 y Fr(are)g(exp)s (onen)m(tially)d(decreasing)i(i.e.)41 b(uniformly)27 b(in)i Fp(E)i Fn(2)25 b Fp(V)3781 1561 y Fo(\016)3819 1546 y Fr(,)937 1755 y Fn(j)q Fr(^)-46 b Fp(a)1010 1769 y Fm(1)1050 1755 y Fr(\()p Fp(n)p Fr(\))p Fn(j)25 b(\024)g Fp(C)7 b Fn(j)p Fp(a)1466 1769 y Fm(0)1506 1755 y Fn(j)15 b Fr(exp)1700 1627 y Ff(\022)1767 1755 y Fn(\000)1848 1694 y Fr(2)p Fp(\031)s Fn(j)p Fp(n)p Fn(j)p 1848 1735 206 4 v 1929 1818 a Fp(")2078 1755 y Fr(\()p Fp(Y)2166 1770 y Fo(l)2213 1755 y Fn(\000)20 b Fp(\016)s Fr(\))2382 1627 y Ff(\023)2465 1755 y Fp(;)106 b(n)25 b Fn(6)p Fr(=)g(0)p Fp(:)-3078 b Fr(\(6.15\))-236 1965 y(Come)36 b(bac)m(k)i(to)f(the)g(v) -5 b(ariable)35 b Fp(z)40 b Fr(=)35 b Fp(\020)7 b(=")37 b Fr(of)f(the)h(initial)d(equation)i(\(0.1\))r(.)59 b(Fix)36 b Fp(y)i(>)e Fr(0)h(indep)s(enden)m(t)d(of)i Fp(")p Fr(.)60 b(By)37 b(means)-236 2073 y(of)g(\(6.15\))s(,)30 b(for)g(su\016cien)m (tly)f(small)g Fp(")p Fr(,)i(w)m(e)g(get)960 2282 y Fn(j)p Fp(a)1033 2296 y Fm(1)1073 2282 y Fn(j)26 b(\024)f Fp(C)7 b Fn(j)p Fp(a)1365 2296 y Fm(0)1404 2282 y Fn(j)30 b Fr(exp)1613 2154 y Ff(\022)1680 2282 y Fn(\000)1761 2221 y Fr(2)p Fp(\031)p 1761 2261 101 4 v 1790 2345 a(")1887 2282 y Fr(\()p Fp(Y)1975 2297 y Fo(l)2021 2282 y Fn(\000)20 b Fp(\016)s Fr(\))2190 2154 y Ff(\023)2273 2282 y Fp(;)106 b Fn(j)p Fr(Im)15 b Fp(z)t Fn(j)26 b(\024)f Fp(y)s(:)-236 2487 y Fr(W)-8 b(e)31 b(ha)m(v)m(e)h(c)m(hec)m(k)m(ed)g(all)e(the)g (statemen)m(ts)i(of)f(Theorem)f(1.3)h(concerning)f(the)h(co)s (e\016cien)m(t)g Fp(a)p Fr(.)-236 2666 y(6.4.3.)48 b Fi(Co)-5 b(e\016cient)36 b Fp(b)p Fi(.)45 b Fr(Let)34 b(us)e(pro)m(v)m(e)i(the)f(estimate)h(\(1.14\))h(for)e(the)g(co)s (e\016cien)m(t)h Fp(b)p Fr(.)49 b(Fix)32 b(an)h(in)m(teger)g Fp(l)f Fn(\025)d Fr(2.)49 b(Using)32 b(the)-236 2774 y(standard)d(b)s(eha)m(vior)h(of)g Fp(f)40 b Fr(and)30 b(\(6.10\))r(,)h(w)m(e)f(get)635 2996 y Fp(b)25 b Fr(=)g Fp(o)854 2867 y Ff(\022)921 2996 y Fr(exp)1075 2867 y Ff(\022)1157 2934 y Fp(i)p 1152 2975 43 4 v 1152 3058 a(")1219 2872 y Ff(Z)1310 2898 y Fo(\020)5 b Fk(\000)p Fm(2)p Fo(\031)1270 3078 y(\020)1301 3087 y Fg(0)1498 2996 y Fp(\024)1550 3010 y Fo(n)1593 3019 y Fg(0)1632 2996 y Fp(d\020)27 b Fr(+)1853 2934 y Fp(i)p 1847 2975 V 1847 3058 a(")1915 2872 y Ff(Z)2006 2898 y Fo(\020)1965 3078 y(\020)1996 3087 y Fg(0)2060 2996 y Fp(\024)2112 3010 y Fo(n)2155 3019 y Fg(0)2194 2996 y Fp(d\020)2288 2867 y Ff(\023\023)2437 2996 y Fp(;)106 b(\020)7 b(;)15 b(\020)27 b Fn(\000)20 b Fr(2)p Fp(\031)29 b Fn(2)c Fp(R)3094 3011 y Fo(l)3120 2996 y Fp(:)-236 3239 y Fr(Consider)30 b(the)i(in)m(tegral)f Fp(I)k Fr(=)808 3166 y Ff(R)869 3192 y Fo(\020)5 b Fk(\000)p Fm(2)p Fo(\031)851 3271 y(\020)1057 3239 y Fp(\024d\020)i Fr(.)45 b(As)32 b Fp(\024)g Fr(is)e(2)p Fp(\031)s Fr(-p)s(erio)s(dic,)h(the)h(in)m(tegral)g(is)e (constan)m(t,)k(and)d(as)h Fp(\024)3412 3253 y Fo(n)3455 3262 y Fg(0)3526 3239 y Fr(is)f(real)g(on)h Fl(R)r Fr(,)-236 3355 y(the)e(in)m(tegral)g Fp(I)38 b Fr(is)29 b(real.)41 b(So,)904 3571 y Fp(b)25 b Fr(=)g Fp(o)1123 3443 y Ff(\022)1190 3571 y Fr(exp)1344 3443 y Ff(\022)1421 3510 y Fr(2)p Fp(i)p 1421 3550 77 4 v 1438 3634 a(")1523 3448 y Ff(Z)1614 3474 y Fo(\020)1573 3654 y(\020)1604 3663 y Fg(0)1639 3654 y Fm(+2)p Fo(\031)1791 3571 y Fp(\024)1843 3585 y Fo(n)1886 3594 y Fg(0)1925 3571 y Fp(d\020)2019 3443 y Ff(\023\023)2168 3571 y Fp(;)106 b(\020)7 b(;)15 b(\020)27 b Fn(\000)20 b Fr(2)p Fp(\031)29 b Fn(2)c Fp(R)2825 3586 y Fo(l)2851 3571 y Fp(:)-3112 b Fr(\(6.16\))-236 3790 y(Being)33 b(a)h(consequence)f(of)h(the)f(standard)g(b)s(eha)m(vior,)g (the)g(estimate)h(\(6.16\))i(is)c(uniform)f(in)h Fp(\020)39 b Fr(for)33 b Fp(\020)40 b Fr(and)33 b Fp(\020)28 b Fn(\000)22 b Fr(2)p Fp(\031)37 b Fr(in)31 b(the)-236 3898 y Fp(\016)s Fr(-admissible)c(sub)s(domain)g(of)j Fp(R)911 3913 y Fo(l)937 3898 y Fr(,)g(and)f(for)g Fp(E)35 b Fr(in)28 b Fp(V)20 b Fr(\()p Fp(\016)s Fr(\),)32 b(a)d(constan)m(t)i(neigh)m(b)s (orho)s(o)s(d)c(of)j Fp(E)2936 3912 y Fm(0)2976 3898 y Fr(.)40 b(If,)29 b(in)g(\(6.16\))r(,)h(w)m(e)g(c)m(ho)s(ose)g Fp(\020)-236 4006 y Fr(in)f(the)h Fp(")p Fr(-neigh)m(b)s(orho)s(o)s(d)f (of)i Fp(\014)25 b Fr(+)20 b(2)p Fp(\031)s Fr(,)31 b(w)m(e)g(get)570 4239 y Fp(b)p Fr(\()p Fp(\020)7 b Fr(\))25 b(=)g Fp(o)906 4084 y Ff( )978 4239 y Fr(exp)1132 4084 y Ff( )1214 4178 y Fr(2)p Fp(i)p 1214 4218 V 1231 4302 a(")1316 4116 y Ff(Z)1413 4124 y Fm(^)1407 4142 y Fo(\020)1367 4322 y(\020)1398 4331 y Fg(0)1432 4322 y Fm(+2)p Fo(\031)r(;)e Fm(along)j Fo(\014)1860 4239 y Fp(\024)1912 4253 y Fo(n)1955 4262 y Fg(0)1994 4239 y Fp(d\020)2088 4084 y Ff(!!)2247 4239 y Fp(;)2386 4215 y Fr(^)2378 4239 y Fp(\020)32 b Fn(2)25 b Fp(\014)5 b(;)106 b Fr(Im)2855 4215 y(^)2847 4239 y Fp(\020)32 b Fr(=)25 b(Im)14 b Fp(\020)7 b(:)-3446 b Fr(\(6.17\))-236 4476 y(As)30 b Fp(b)g Fr(is)g Fp(")p Fr(-p)s(erio)s(dic,)f(this)g(estimate)i(is)e(uniform)g(in)g(\()p Fp(\020)7 b(;)15 b(E)5 b Fr(\))26 b Fn(2)f(fj)p Fr(Im)15 b Fp(\020)7 b Fn(j)25 b(\024)g Fp(Y)2367 4491 y Fo(l)2413 4476 y Fn(\000)20 b Fp(\016)s Fn(g)h(\002)f Fp(V)h Fr(\()p Fp(\016)s Fr(\).)-236 4631 y(Sho)m(w)31 b(that)g(the)h(estimate)g (\(6.17\))h(implies)28 b(\(1.14\))s(.)43 b(Recall)31 b(that)h Fp(\014)26 b Fr(+)20 b(2)p Fp(\031)30 b Fn(\032)c Fp(Z)7 b Fr(.)43 b(So,)32 b(there)f(exists)g Fp(c)c(>)f Fr(0)32 b(suc)m(h)f(that,)h(for)-236 4739 y Fp(E)e Fr(=)25 b Fp(E)24 4753 y Fm(0)94 4739 y Fr(and)30 b Fp(\020)i Fn(2)25 b Fp(\014)g Fr(+)20 b(2)p Fp(\031)s Fr(,)31 b(one)g(has)f Fp(c)25 b(<)g(\024)1293 4753 y Fo(n)1336 4762 y Fg(0)1375 4739 y Fr(\()p Fp(\020)7 b Fr(\))26 b Fp(<)f(\031)e Fn(\000)d Fp(c)p Fr(.)41 b(Therefore,)30 b(for)g Fp(\020)i Fn(2)25 b Fp(\014)g Fr(+)20 b(2)p Fp(\031)34 b Fr(and)c Fp(\021)f Fr(=)24 b(Im)15 b Fp(\020)7 b Fr(,)847 4961 y Fp(c\021)29 b(<)c Fr(Im)1195 4837 y Ff(Z)1286 4863 y Fo(\020)1246 5043 y(\020)1277 5052 y Fg(0)1311 5043 y Fm(+2)p Fo(\031)r(;)e Fm(along)j Fo(\014)s Fm(+2)p Fo(\031)1872 4961 y Fp(\024)1924 4975 y Fo(n)1967 4984 y Fg(0)2006 4961 y Fp(d\020)32 b(<)25 b Fr(\()p Fp(\031)f Fn(\000)19 b Fp(c)p Fr(\))p Fp(\021)s(;)108 b(\021)28 b Fn(\025)d Fr(0)p Fp(;)862 5235 y Fr(\()p Fp(\031)f Fn(\000)c Fp(c)p Fr(\))p Fp(\021)29 b(<)c Fr(Im)1447 5111 y Ff(Z)1538 5137 y Fo(\020)1498 5317 y(\020)1529 5326 y Fg(0)1563 5317 y Fm(+2)p Fo(\031)r(;)e Fm(along)j Fo(\014)s Fm(+2)p Fo(\031)2124 5235 y Fp(\024)2176 5249 y Fo(n)2219 5258 y Fg(0)2258 5235 y Fp(d\020)32 b(<)25 b(c\021)s(;)107 b(\021)29 b Fn(\024)c Fr(0)p Fp(:)-236 5096 y Fr(\(6.18\))-236 5453 y(These)32 b(estimates)i(also)f(hold)e(in) h(a)h(constan)m(t)h(neigh)m(b)s(orho)s(o)s(d)d(of)i Fp(E)2124 5467 y Fm(0)2196 5453 y Fr(\(with)f(a)h(smaller)e(constan)m(t)k Fp(c)p Fr(\).)48 b(Equations)32 b(\(6.17\))-236 5561 y(and)e(\(6.18\))i(sho)m(w)e(that)956 5771 y Fp(b)p Fr(\()p Fp(\020)7 b Fr(\))26 b(=)f Fp(o)1293 5642 y Ff(\022)1360 5771 y Fr(exp\()p Fn(\000)1615 5709 y Fr(2)p Fp(c)p 1615 5750 85 4 v 1636 5833 a(")1710 5771 y Fr(\()p Fp(Y)1798 5786 y Fo(l)1844 5771 y Fn(\000)20 b Fp(\016)s Fr(\)\))2048 5642 y Ff(\023)2131 5771 y Fp(;)107 b Fr(Im)14 b Fp(\020)32 b Fr(=)25 b Fp(Y)2607 5786 y Fo(l)2653 5771 y Fn(\000)20 b Fp(\016)n(;)808 6024 y(b)p Fr(\()p Fp(\020)7 b Fr(\))25 b(=)g Fp(o)1144 5896 y Ff(\022)1211 6024 y Fr(exp\()1395 5963 y(2\()p Fp(\031)f Fn(\000)c Fp(c)p Fr(\))p 1395 6003 322 4 v 1535 6086 a Fp(")1727 6024 y Fr(\()p Fp(Y)1815 6039 y Fo(l)1861 6024 y Fn(\000)g Fp(\016)s Fr(\)\))2065 5896 y Ff(\023)2149 6024 y Fp(;)106 b Fr(Im)15 b Fp(\020)32 b Fr(=)25 b Fn(\000)p Fr(\()p Fp(Y)2731 6039 y Fo(l)2777 6024 y Fn(\000)20 b Fp(\016)s Fr(\))p Fp(:)1854 6210 y Fm(21)p eop %%Page: 22 22 22 21 bop -236 245 a Fr(Let)31 b(\()-41 221 y(^)-38 245 y Fp(b)1 259 y Fo(n)48 245 y Fr(\))83 259 y Fo(n)161 245 y Fr(b)s(e)e(the)i(F)-8 b(ourier)30 b(co)s(e\016cien)m(ts)h Fp(b)p Fr(.)41 b(The)29 b(last)i(estimates)g(imply)d(that)875 479 y Fn(j)897 455 y Fr(^)900 479 y Fp(b)939 493 y Fo(n)987 479 y Fn(j)d(\024)g Fp(C)d Fr(exp)1374 351 y Ff(\022)1441 479 y Fn(\000)1522 417 y Fr(2)p Fp(\031)s Fr(\()p Fp(n)e Fn(\000)g Fr(1\))h(+)f(2)p Fp(c)p 1522 458 579 4 v 1789 541 a(")2125 479 y Fr(\()p Fp(Y)2213 494 y Fo(l)2259 479 y Fn(\000)g Fp(\016)s Fr(\))2428 351 y Ff(\023)2511 479 y Fp(;)107 b(n)24 b(>)h Fr(0)p Fp(;)1067 732 y Fn(j)1089 708 y Fr(^)1092 732 y Fp(b)1131 746 y Fo(n)1179 732 y Fn(j)g(\024)g Fp(C)d Fr(exp)1566 604 y Ff(\022)1633 732 y Fn(\000)1714 671 y Fr(2)p Fp(\031)s Fn(j)p Fp(n)p Fn(j)e Fr(+)g(2)p Fp(c)p 1714 711 402 4 v 1893 795 a(")2140 732 y Fr(\()p Fp(Y)2228 747 y Fo(l)2274 732 y Fn(\000)g Fp(\016)s Fr(\))2443 604 y Ff(\023)2526 732 y Fp(;)107 b(n)25 b Fn(\024)f Fr(0)p Fp(;)-236 605 y Fr(\(6.19\))-236 966 y(uniformly)42 b(in)i Fp(E)50 b Fr(in)44 b(a)i(constan)m(t)g(neigh) m(b)s(orho)s(o)s(d)d(of)i Fp(E)1795 980 y Fm(0)1835 966 y Fr(.)85 b(Come)45 b(bac)m(k)h(to)g(the)f(v)-5 b(ariable)44 b Fp(z)55 b Fr(=)49 b Fp(\020)7 b(=")46 b Fr(of)f(the)g(initial)-236 1074 y(equation)33 b(\(0.1\))q(.)49 b(Fix)32 b Fp(y)h(>)c Fr(0)k(indep)s(enden)m(t)e(of)i Fp(")p Fr(.)49 b(Using)33 b(\(6.19\))s(,)h(for)e(su\016cien)m(tly)g(small)f Fp(")p Fr(,)k(uniformly)30 b(in)h Fp(z)37 b Fr(and)c(in)e Fp(E)-236 1182 y Fr(in)e(a)i(constan)m(t)g(neigh)m(b)s(orho)s(o)s(d)d(of)j Fp(E)1054 1196 y Fm(0)1093 1182 y Fr(,)g(w)m(e)g(get)957 1411 y Fn(j)p Fp(b)p Fr(\()p Fp(z)t Fr(\))p Fn(j)c Fr(=)e Fp(o)1344 1283 y Ff(\022)1411 1411 y Fr(exp)1565 1283 y Ff(\022)1632 1411 y Fn(\000)1713 1349 y Fr(2)p Fp(c)p 1713 1390 85 4 v 1734 1473 a(")1823 1411 y Fr(\()p Fp(Y)1911 1426 y Fo(l)1957 1411 y Fn(\000)20 b Fp(\016)s Fr(\))2126 1283 y Ff(\023)q(\023)2276 1411 y Fp(;)106 b Fn(j)p Fr(Im)15 b Fp(z)t Fn(j)26 b(\024)f Fp(y)s(:)-236 1639 y Fr(W)-8 b(e)39 b(ha)m(v)m(e)h(pro)m(v)m(ed)e(the)h(statemen)m(t)h(of)e(Theorem) g(1.3)i(concerning)d(the)i(co)s(e\016cien)m(t)g Fp(b)p Fr(.)64 b(This)37 b(completes)i(the)f(pro)s(of)g(of)-236 1747 y(Theorem)30 b(1.3.)1237 2026 y(7.)51 b Fq(The)34 b(monodr)n(omy)e(equa)-6 b(tion)-236 2188 y Fr(Let)31 b(\()p Fp( )21 2202 y Fm(1)p Fo(;)p Fm(2)116 2188 y Fr(\))f(b)s(e)g(a)h (consisten)m(t)f(basis)g(of)g(solutions)f(of)38 b(\(0.1\))q(,)31 b(and)f Fp(M)40 b Fr(b)s(e)30 b(the)h(corresp)s(onding)d(mono)s(drom)m (y)i(matrix.)-236 2416 y(7.1.)53 b Fu(Beha)m(vior)37 b(at)e(in\014nit)m(y.)45 b Fr(Consider)29 b(the)i(mono)s(drom)m(y)g (equation)f(\(1.9\))r(.)42 b(As)31 b(already)g(men)m(tioned,)g(the)g(b) s(eha)m(vior)-236 2524 y(of)f(its)g(solutions)f(for)h Fp(n)25 b Fn(!)g(\0061)30 b Fr(mimics)f(the)h(b)s(eha)m(vior)g(of)g (solutions)f(of)h(equation)h(\(0.1\))h(for)e Fp(x)25 b Fn(!)g(\0071)p Fr(.)41 b(One)29 b(has)-236 2705 y Fu(Theorem)34 b(7.1)h Fr(\([9)q(]\))p Fu(.)46 b Fi(Fix)39 b Fp(z)g Fn(2)c Fl(R)s Fi(.)64 b(Then,)39 b(for)g Fp(\037)p Fi(,)g(a)g(solution) g(of)f(e)-5 b(quation)46 b Fr(\(1.9\))r Fi(,)39 b(ther)-5 b(e)39 b(exists)g(a)f(unique)f(solution)-236 2813 y(of)52 b Fr(\(0.1\))r Fi(,)32 b(say)h Fp(f)10 b Fi(,)32 b(such)h(that)618 2918 y Ff(\022)696 2992 y Fp(f)10 b Fr(\()p Fp(x)20 b Fr(+)g(2)p Fp(\031)s(n=";)15 b(z)t Fr(\))685 3099 y Fp(f)740 3066 y Fk(0)762 3099 y Fr(\()p Fp(x)21 b Fr(+)f(2)p Fp(\031)s(n=";)15 b(z)t Fr(\))1325 2918 y Ff(\023)1418 3047 y Fr(=)25 b Fp(M)1602 3062 y Fo( )1654 3047 y Fr(\()p Fp(x;)15 b(z)26 b Fn(\000)20 b Fp(nh)p Fr(\))g Fn(\001)h Fp(\033)i Fn(\001)d Fp(\037)2325 3061 y Fk(\000)p Fo(n)2427 3047 y Fp(;)108 b Fn(8)p Fp(x)25 b Fn(2)g Fl(R)s Fp(;)51 b(n)25 b Fn(2)g Fl(Z)p Fp(:)-3402 b Fr(\(7.1\))-236 3275 y Fi(wher)-5 b(e)692 3491 y Fp(M)780 3506 y Fo( )858 3491 y Fr(=)954 3363 y Ff(\022)1082 3430 y Fp( )1141 3444 y Fm(1)1302 3430 y Fp( )1361 3444 y Fm(2)1072 3509 y Fo(d )1154 3518 y Fg(1)p 1072 3529 118 4 v 1093 3581 a Fo(dx)1292 3509 y(d )1374 3518 y Fg(2)p 1292 3529 V 1313 3581 a Fo(dx)1461 3363 y Ff(\023)1543 3491 y Fp(;)108 b(\033)29 b Fr(=)1853 3363 y Ff(\022)1961 3436 y Fr(0)84 b Fn(\000)p Fr(1)1961 3544 y(1)119 b(0)2247 3363 y Ff(\023)2329 3491 y Fp(;)109 b(h)25 b Fr(=)2646 3430 y(2)p Fp(\031)p 2646 3470 101 4 v 2675 3554 a(")2757 3491 y Fr(mo)s(d)14 b(\(1\))p Fp(:)-3322 b Fr(\(7.2\))-236 3730 y Fi(Mor)-5 b(e)g(over,)36 b(r)-5 b(e)g(cipr)g(o)g(c)g(al)5 b(ly,)38 b(for)d Fp(f)10 b Fi(,)34 b(a)g(solution)i(of)54 b Fr(\(0.1\))r Fi(,)34 b(ther)-5 b(e)36 b(exists)e(a)h(unique)f(ve)-5 b(ctor)35 b Fp(\037)p Fi(,)f(solution)i(of)54 b Fr(\(1.9\))r Fi(,)34 b(satisfy-)-236 3838 y(ing)e Fr(\(7.1\))r Fi(.)-24 4019 y Fr(As)40 b(\()p Fp( )214 4033 y Fm(1)p Fo(;)p Fm(2)309 4019 y Fr(\))f(is)f(a)i(consisten)m(t)g(basis,)g(det)p Fp(M)1473 4034 y Fo( )1566 4019 y Fr(is)e(a)i(non-v)-5 b(anishing)36 b(constan)m(t;)46 b(th)m(us,)41 b(Theorem)e(7.1)h (immediately)-236 4126 y(implies)27 b(Theorem)j(1.2.)-236 4355 y(7.2.)53 b Fu(Blo)s(c)m(h-Flo)s(quet)38 b(solutions.)46 b Fr(There)32 b(is)f(also)h(a)g(relation)f(b)s(et)m(w)m(een)i(Blo)s(c)m (h-Flo)s(quet)f(solutions)e(of)i(the)g(family)f(of)-236 4463 y(equation)f(\(0.1\))i(and)e(Blo)s(c)m(h-Flo)s(quet)g(solutions)f (of)i(the)f(mono)s(drom)m(y)g(equation.)41 b(Let)30 b(us)g(\014rst)g (de\014ne)f(these)i(solutions.)-236 4691 y(7.2.1.)48 b Fi(Blo)-5 b(ch-Flo)g(quet)35 b(solutions)f(for)f(di\013er)-5 b(enc)g(e)33 b(e)-5 b(quations.)46 b Fr(Consider)29 b(the)i(equation) 994 4870 y Fp(\037)p Fr(\()p Fp(x)20 b Fr(+)g Fp(h)p Fr(\))26 b(=)f Fp(M)10 b Fr(\()p Fp(x)p Fr(\))p Fp(\037)p Fr(\()p Fp(x)p Fr(\))p Fp(;)108 b(\037)p Fr(\()p Fp(x)p Fr(\))25 b Fn(2)g Fl(C)2340 4833 y Fm(2)2385 4870 y Fp(;)107 b(x)25 b Fn(2)g Fl(R)r Fp(;)-3000 b Fr(\(7.3\))-236 5054 y(where)27 b Fp(x)e Fn(7!)g Fp(M)10 b Fr(\()p Fp(x)p Fr(\))26 b Fn(2)f Fp(S)5 b(L)p Fr(\(2)p Fp(;)15 b Fl(C)k Fr(\))33 b(is)26 b(1-p)s(erio)s(dic)g(and)h Fp(h)g Fr(is)g(a)g(\014xed) g(p)s(ositiv)m(e)f(n)m(um)m(b)s(er.)39 b(The)26 b(set)i(of)g(solutions) d(of)35 b(\(7.3\))29 b(is)d(a)-236 5162 y(t)m(w)m(o)31 b(dimensional)c(mo)s(dule)h(o)m(v)m(er)j(the)f(ring)e(of)i Fp(h)p Fr(-p)s(erio)s(dic)e(functions;)g(hence,)j(it)e(is)f(natural)h (to)i(call)e Fp(\037)g Fr(a)h Fi(Blo)-5 b(ch)33 b(solution)-236 5270 y Fr(if)c(it)h(satis\014es)g(the)g(relation)1282 5449 y Fp(\037)p Fr(\()p Fp(x)21 b Fr(+)f(1\))26 b(=)f Fp(\026)p Fr(\()p Fp(x)p Fr(\))p Fp(\037)p Fr(\()p Fp(x)p Fr(\))p Fp(;)107 b(x)25 b Fn(2)g Fl(R)s Fp(;)-2712 b Fr(\(7.4\))-236 5633 y(where)30 b Fp(x)25 b Fn(7!)g Fp(\026)p Fr(\()p Fp(x)p Fr(\))31 b(is)e Fp(h)p Fr(-p)s(erio)s(dic,)g(see)i([5)q (].)40 b(If)30 b Fp(\026)g Fr(is)g(constan)m(t,)i(the)e(Blo)s(c)m(h)h (solution)e(can)h(b)s(e)g(represen)m(ted)g(in)f(the)i(form)1366 5819 y Fp(\037)p Fr(\()p Fp(x)p Fr(\))26 b(=)f Fp(e)1709 5781 y Fo(i\036x)1819 5819 y Fp(U)10 b Fr(\()p Fp(x)p Fr(\))p Fp(;)107 b(x)25 b Fn(2)g Fl(R)r Fp(;)-2628 b Fr(\(7.5\))-236 6003 y(where)31 b Fp(\036)g Fr(is)g(a)h(constan)m(t)h (and)d Fp(x)d Fn(7!)g Fp(U)10 b Fr(\()p Fp(x)p Fr(\))33 b(is)d(1-p)s(erio)s(dic.)42 b(In)31 b(this)f(case,)j(w)m(e)f(call)f Fp(\036)h Fr(the)f Fi(quasi-momentum)i Fr(of)f Fp(\037)f Fr(and)g Fp(U)-236 6110 y Fr(the)f Fi(p)-5 b(erio)g(dic)35 b(c)-5 b(omp)g(onent)33 b Fr(of)d Fp(\037)p Fr(.)41 b(The)29 b(solution)g Fp(\037)i Fr(is)e(called)h(a)g Fi(Blo)-5 b(ch-Flo)g(quet)32 b Fr(solution)d(of)38 b(\(7.3\))r(.)1854 6210 y Fm(22)p eop %%Page: 23 23 23 22 bop -236 241 a Fr(7.2.2.)48 b Fi(Blo)-5 b(ch-Flo)g(quet)38 b(solutions)g(for)f(di\013er)-5 b(ential)37 b(e)-5 b(quations.)46 b Fr(Fix)34 b Fp(a)e(>)f Fr(0)k(and)f Fp(b)d(>)h Fr(0)i(t)m(w)m(o)i (constan)m(ts,)h(and)c(\()p Fp(x;)15 b(y)s Fr(\))33 b Fn(7!)-236 349 y Fp(Q)p Fr(\()p Fp(x;)15 b(y)s Fr(\),)31 b(a)g(su\016cien)m(tly)e(regular)g(function)h(1-p)s(erio)s(dic)e(b)s (oth)i(in)f Fp(x)h Fr(and)g(in)f Fp(y)s Fr(.)40 b(Consider)29 b(the)h(di\013eren)m(tial)f(equation)1138 494 y Fn(\000)p Fp(f)1264 457 y Fk(0)o(0)1305 494 y Fr(\()p Fp(x)p Fr(\))21 b(+)f Fp(Q)p Fr(\()p Fp(ax;)15 b(bx)p Fr(\))p Fp(f)10 b Fr(\()p Fp(x)p Fr(\))26 b(=)f(0)p Fp(;)107 b(x)25 b Fn(2)g Fl(R)s Fp(:)-2872 b Fr(\(7.6\))-236 640 y(Assume)30 b(that)h(a)f(solution)f Fp(f)40 b Fr(can)31 b(b)s(e)e(written)h(as)1281 786 y Fp(f)10 b Fr(\()p Fp(x)p Fr(\))25 b(=)g Fp(e)1621 748 y Fo(ipx)1725 786 y Fp(P)13 b Fr(\()p Fp(ax;)i(bx)p Fr(\))p Fp(;)107 b(x)26 b Fn(2)e Fl(R)s Fp(;)-2713 b Fr(\(7.7\))-236 937 y(where)37 b Fp(p)g Fr(is)g(constan)m(t)h(and)f Fp(P)13 b Fr(\()p Fp(x;)i(y)s Fr(\))39 b(is)d(a)i(function)e(whic)m(h)h (is)f(1-p)s(erio)s(dic)g(in)g Fp(x)h Fr(and)g(in)g Fp(y)s Fr(.)61 b(W)-8 b(e)39 b(call)e(suc)m(h)g(a)h(solution)-236 1045 y(a)h Fi(Blo)-5 b(ch-Flo)g(quet)42 b Fr(solution)37 b(of)47 b(\(7.5\))r(,)42 b(and)c(w)m(e)i(call)f(the)g(constan)m(t)i Fp(p)e Fr(its)g Fi(quasi-momentum)p Fr(.)68 b(The)39 b(\014rst)g(example)g(of)-236 1152 y(Blo)s(c)m(h-Flo)s(quet)44 b(solutions)e(of)i(an)g(equation)g(of)g(the)g(form)f(\(7.5\))j(with)c (quasi-p)s(erio)s(dic)f(p)s(oten)m(tial)i(w)m(as)h(constructed)-236 1260 y(in)29 b([6)q(].)-236 1422 y(7.2.3.)48 b Fi(R)-5 b(elation)31 b(b)-5 b(etwe)g(en)30 b(Blo)-5 b(ch-Flo)g(quet)30 b(solutions)h(of)e(the)g(family)h(of)g(e)-5 b(quations)37 b Fr(\(0.1\))31 b Fi(and)f(of)f(the)g(mono)-5 b(dr)g(omy)33 b(e)-5 b(qua-)-236 1530 y(tion)45 b Fr(\(1.9\))r Fi(.)g Fr(No)m(w,)39 b(instead)c(of)h(the)g(mono)s(drom)m(y)f(equation)h (\(1.9\))i(on)e Fl(Z)p Fr(,)d(w)m(e)j(consider)f(its)g(con)m(tin)m (uous)h(analog)i(\(7.3\))q(.)-236 1638 y(The)26 b(matrix)h Fp(M)37 b Fr(is)26 b(the)h(mono)s(drom)m(y)g(matrix)f(for)h(a)g (consisten)m(t)h(basis)e(\()p Fp( )2340 1652 y Fm(1)p Fo(;)p Fm(2)2435 1638 y Fr(\),)i(and,)f(as)h(b)s(efore,)f Fp(h)f Fn(\021)f Fr(2)p Fp(\031)s(=")j Fr(mo)s(d)e(1.)40 b(Note)-236 1746 y(that,)31 b(if)e Fp(\037)i Fr(satis\014es)e(\(7.3\))r (,)i(then)f(the)g(sequence)h(\()p Fp(\037)1566 1760 y Fo(n)1613 1746 y Fr(\))1648 1760 y Fo(n)1726 1746 y Fr(de\014ned)e(b)m (y)h Fp(\037)2222 1760 y Fo(n)2295 1746 y Fr(=)24 b Fp(\037)p Fr(\()p Fp(x)d Fr(+)f Fp(nh)p Fr(\))30 b(satis\014es)g(\(1.9\))r(.)41 b(W)-8 b(e)31 b(pro)m(v)m(e)-236 1897 y Fu(Theorem)j(7.2.)46 b Fi(Assume)32 b(that)h(ther)-5 b(e)33 b(exists)g Fp(\037)p Fi(,)f(a)h(Blo)-5 b(ch-Flo)g(quet)34 b(solution)40 b Fr(\(7.5\))34 b Fi(of)e(the)h(\\c)-5 b(ontinuous")33 b(mono)-5 b(dr)g(omy)-236 2005 y(e)g(quation)40 b Fr(\(7.3\))r Fi(.)i(L)-5 b(et)1147 2150 y Fp(F)13 b Fr(\()p Fp(x;)i(z)t Fr(\))27 b(=)e Fp(M)1637 2165 y Fo( )1690 2150 y Fr(\()p Fp(x;)15 b(z)t Fr(\))p Fp(\033)s(U)10 b Fr(\()p Fp(z)t Fr(\))p Fp(;)111 b(x;)15 b(z)29 b Fn(2)c Fl(R)s Fp(;)-2847 b Fr(\(7.8\))-236 2301 y Fi(wher)-5 b(e)33 b Fp(U)42 b Fi(is)32 b(the)g(p)-5 b(erio)g(dic)33 b(c)-5 b(omp)g(onent)35 b(of)d Fp(\037)p Fi(,)g(and)h(the)f(matric)-5 b(es)33 b Fp(M)2165 2316 y Fo( )2250 2301 y Fi(and)g Fp(\033)i Fi(ar)-5 b(e)33 b(as)f(in)g(The)-5 b(or)g(em)34 b(7.1.)42 b(Then,)32 b Fp(f)10 b Fr(\()p Fp(x;)15 b(z)t Fr(\))p Fi(,)-236 2409 y(the)33 b(\014rst)g(c)-5 b(omp)g(onent)35 b(of)e(the)g(ve)-5 b(ctor)33 b Fp(F)13 b Fi(,)33 b(is)f(a)h(Blo)-5 b(ch-Flo)g(quet)35 b(solution)f(of)52 b Fr(\(0.1\))r Fi(;)32 b(it)h(c)-5 b(an)33 b(b)-5 b(e)32 b(written)i(as)1320 2565 y Fp(f)10 b Fr(\()p Fp(x;)15 b(z)t Fr(\))26 b(=)f Fp(e)1747 2528 y Fo(ip)p Fm(\()p Fo(E)t Fm(\))p Fo(x)1976 2565 y Fp(P)13 b Fr(\()p Fp(x;)i(x)21 b Fn(\000)f Fp(z)t Fr(\))p Fp(;)-2680 b Fr(\(7.9\))-236 2716 y Fi(wher)-5 b(e)31 b Fr(\()p Fp(x;)15 b(y)s Fr(\))26 b Fn(7!)f Fp(P)13 b Fr(\()p Fp(x;)i(y)s Fr(\))31 b Fi(is)f Fr(2)p Fp(\031)s(=")p Fi(-p)-5 b(erio)g(dic)33 b(in)d Fp(x)g Fi(and)h Fr(1)p Fi(-p)-5 b(erio)g(dic)32 b(in)e Fp(y)s Fi(.)40 b(The)31 b(quasi-momenta)g(of)g Fp(f)39 b Fi(and)31 b Fp(\037)f Fi(ar)-5 b(e)31 b(r)-5 b(elate)g(d)-236 2824 y(by)1520 3012 y Fp(p)p Fr(\()p Fp(E)5 b Fr(\))26 b(=)f Fn(\000)1914 2951 y Fp("h)p 1911 2991 101 4 v 1911 3075 a Fr(2)p Fp(\031)2037 3012 y(\036)p Fr(\()p Fp(E)5 b Fr(\))p Fp(:)-2494 b Fr(\(7.10\))-236 3189 y(Note)42 b(that,)j(b)s(et)m(w)m(een)c(the)h(Blo)s(c)m(h-Flo)s (quet)f(solutions)e(of)j(di\013erence)e(equations)h(related)g(b)m(y)g (the)g(mono)s(dromization)-236 3297 y(pro)s(cedure,)29 b(there)i(is)e(a)i(relation)f(similar)d(to)k(\(7.8\))r(,)g(see)g([4)q (].)-236 3452 y Fu(Pro)s(of.)81 b Fr(It)43 b(follo)m(ws)g(from)g(the)h (de\014nition)d(of)j(the)g(matrix)e Fp(M)2048 3467 y Fo( )2145 3452 y Fr(that)i Fp(f)53 b Fr(is)42 b(a)i(linear)e(com)m (bination)h(of)h Fp( )3619 3466 y Fm(1)3702 3452 y Fr(and)f Fp( )3951 3466 y Fm(2)3990 3452 y Fr(.)-236 3560 y(So,)h(it)d (satis\014es)f(\(0.1\))r(.)73 b(The)41 b(represen)m(tation)g(\(7.9\))j (and)c(form)m(ula)h(\(7.10\))i(follo)m(w)e(from)f(the)i(next)f(t)m(w)m (o)i(statemen)m(ts,)-236 3668 y(Prop)s(osition)28 b(7.1)k(and)d(Lemma)i (7.1)-236 3819 y Fu(Prop)s(osition)36 b(7.1.)46 b Fi(The)33 b(solution)h Fp(f)42 b Fi(satis\014es)33 b(the)g(r)-5 b(elations:)589 3966 y Fp(f)10 b Fr(\()p Fp(x)19 b Fr(+)h Fp(a;)15 b(z)25 b Fr(+)20 b Fp(a)p Fr(\))26 b(=)f Fp(e)1334 3928 y Fk(\000)p Fo(ih\036)1500 3966 y Fp(f)10 b Fr(\()p Fp(x;)15 b(z)t Fr(\))p Fp(;)109 b(f)10 b Fr(\()p Fp(x;)15 b(z)24 b Fr(+)c(1\))26 b(=)f Fp(f)10 b Fr(\()p Fp(x;)15 b(z)t Fr(\))p Fp(;)109 b(x;)15 b(z)30 b Fn(2)25 b Fl(R)s Fp(;)-3406 b Fr(\(7.11\))-236 4117 y Fi(wher)-5 b(e)33 b Fp(a)26 b Fr(=)f(2)p Fp(\031)s(=")34 b Fi(and)f Fp(\036)g Fi(is)f(the)i(quasi-momentum)f(of)g Fp(\037)p Fi(.)-236 4315 y Fu(Pro)s(of.)50 b Fr(As)34 b Fp(M)43 b Fr(is)33 b(the)h(mono)s(drom)m(y)e(matrix)h(corresp)s(onding)f(to)i(the)f (consisten)m(t)h(basis)f(\()p Fp( )3032 4329 y Fm(1)3072 4315 y Fp(;)15 b( )3171 4329 y Fm(2)3211 4315 y Fr(\))33 b(and)g(as)h(det)15 b Fp(M)41 b Fn(\021)30 b Fr(1,)-236 4423 y(w)m(e)h(get)373 4568 y Fp(F)13 b Fr(\()p Fp(x)21 b Fr(+)e Fp(a;)c(z)25 b Fr(+)20 b Fp(a)p Fr(\))26 b(=)f Fp(M)1181 4583 y Fo( )1234 4568 y Fr(\()p Fp(x;)15 b(z)t Fr(\))g Fp(M)1555 4531 y Fo(t)1586 4568 y Fr(\()p Fp(z)t Fr(\))g Fp(\033)k(U)10 b Fr(\()p Fp(z)25 b Fr(+)20 b Fp(a)p Fr(\))26 b(=)f Fp(M)2346 4583 y Fo( )2399 4568 y Fr(\()p Fp(x;)15 b(z)t Fr(\))g Fp(\033)20 b(M)2792 4531 y Fk(\000)p Fm(1)2886 4568 y Fr(\()p Fp(z)t Fr(\))15 b Fp(U)10 b Fr(\()p Fp(z)26 b Fr(+)20 b Fp(a)p Fr(\))p Fp(;)-3627 b Fr(\(7.12\))-236 4725 y(where)29 4692 y Fo(t)91 4725 y Fr(denotes)33 b(the)g(transp)s(osition.)45 b(As)33 b Fp(U)43 b Fr(is)31 b(1-p)s(erio)s(dic,)h(and)g Fp(h)d Fr(=)g Fp(a)15 b Fr(mo)s(d)g(1,)34 b(w)m(e)f(ha)m(v)m(e)h Fp(U)10 b Fr(\()p Fp(z)26 b Fr(+)21 b Fp(a)p Fr(\))30 b(=)e Fp(U)10 b Fr(\()p Fp(z)27 b Fr(+)21 b Fp(h)p Fr(\).)48 b(In)-236 4833 y(view)29 b(of)i(the)f(represen)m(tation)h(\(7.5\))h (and)e(equation)g(\(7.3\))r(,)g(one)h(has)345 4989 y Fp(M)443 4951 y Fk(\000)p Fm(1)537 4989 y Fr(\()p Fp(z)t Fr(\))15 b Fp(U)10 b Fr(\()p Fp(z)26 b Fr(+)20 b Fp(h)p Fr(\))26 b(=)f Fp(M)1241 4951 y Fk(\000)p Fm(1)1335 4989 y Fr(\()p Fp(z)t Fr(\))15 b Fp(\037)p Fr(\()p Fp(z)26 b Fr(+)20 b Fp(h)p Fr(\))15 b Fp(e)1861 4951 y Fk(\000)p Fo(i\036)p Fk(\001)p Fm(\()p Fo(z)s Fm(+)p Fo(h)p Fm(\))2218 4989 y Fr(=)25 b Fp(\037)p Fr(\()p Fp(z)t Fr(\))15 b Fp(e)2544 4951 y Fk(\000)p Fo(i\036)p Fk(\001)p Fm(\()p Fo(z)s Fm(+)p Fo(h)p Fm(\))2902 4989 y Fr(=)24 b Fp(U)10 b Fr(\()p Fp(z)t Fr(\))15 b Fp(e)3242 4951 y Fk(\000)p Fo(i\036h)3410 4989 y Fp(:)-236 5151 y Fr(Therefore,)22 b(the)f(righ)m(t)f(hand)g(side)g(of)28 b(\(7.12\))23 b(can)d(b)s(e)h(transformed)e(to)j(the)f(form)f Fp(M)2606 5166 y Fo( )2659 5151 y Fr(\()p Fp(x;)15 b(z)t Fr(\))g Fp(\033)20 b(U)10 b Fr(\()p Fp(z)t Fr(\))15 b Fp(e)3199 5118 y Fk(\000)p Fo(i\036h)3391 5151 y Fr(=)25 b Fp(F)13 b Fr(\()p Fp(x;)i(z)t Fr(\))g Fp(e)3823 5118 y Fk(\000)p Fo(i\036h)3990 5151 y Fr(.)-236 5259 y(This)30 b(implies)f(the)j (\014rst)f(of)h(the)g(relations)f(\(7.11\))s(.)45 b(Due)32 b(to)h(the)f(consistency)g(condition,)f Fp(M)3024 5274 y Fo( )3077 5259 y Fr(\()p Fp(x;)15 b(z)t Fr(\))33 b(is)e(1-p)s(erio)s (dic)f(in)g Fp(z)t Fr(.)-236 5366 y(Th)m(us,)f(the)i(second)f(equalit)m (y)g(follo)m(ws)g(from)g(the)g(1-p)s(erio)s(dicit)m(y)f(of)h Fp(U)10 b Fr(\()p Fp(z)t Fr(\).)p 3950 5366 4 62 v 3954 5308 55 4 v 3954 5366 V 4008 5366 4 62 v -236 5517 a Fu(Lemma)32 b(7.1.)47 b Fi(Assume)32 b(that)i(a)f(function)g Fp(g)j Fi(satis\014es)d(the)g(r)-5 b(elations)659 5663 y Fp(g)s Fr(\()p Fp(x)21 b Fr(+)f Fp(a;)15 b(z)25 b Fr(+)20 b Fp(a)p Fr(\))26 b(=)e Fp(e)1396 5625 y Fo(iq)1459 5663 y Fp(g)s Fr(\()p Fp(x;)15 b(z)t Fr(\))p Fp(;)110 b(g)s Fr(\()p Fp(x;)15 b(z)26 b Fr(+)20 b Fp(b)p Fr(\))26 b(=)e Fp(g)s Fr(\()p Fp(x;)15 b(z)t Fr(\))p Fp(;)111 b(x;)15 b(z)30 b Fn(2)25 b Fl(R)r Fp(;)-236 5809 y Fi(with)33 b Fp(a)26 b(>)f Fr(0)p Fi(,)32 b Fp(b)26 b(>)f Fr(0)33 b Fi(and)g Fp(q)28 b Fn(2)d Fl(C)18 b Fi(,)38 b(thr)-5 b(e)g(e)34 b(c)-5 b(onstants;)35 b(then)e(it)f(admits)i(the)f(r)-5 b(epr)g(esentation)1138 5960 y Fp(g)s Fr(\()p Fp(x;)15 b(z)t Fr(\))28 b(=)c Fp(e)1557 5922 y Fo(iq)r(x=a)1732 5960 y Fp(v)s Fr(\()p Fp(x;)15 b(x)22 b Fn(\000)d Fp(z)t Fr(\))p Fp(;)109 b(x;)15 b(z)30 b Fn(2)25 b Fl(R)s Fp(;)-236 6110 y Fi(with)33 b(a)g(function)g Fp(v)s Fr(\()p Fp(x;)15 b(y)s Fr(\))34 b Fi(which)f(is)g Fp(a)p Fi(-p)-5 b(erio)g(dic)34 b(in)e Fp(x)h Fi(and)g Fp(b)p Fi(-p)-5 b(erio)g(dic)34 b(in)f Fp(y)s Fi(.)1854 6210 y Fm(23)p eop %%Page: 24 24 24 23 bop -236 241 a Fu(Pro)s(of.)41 b Fr(One)30 b(just)g(de\014nes)f Fp(v)s Fr(\()p Fp(x;)15 b(y)s Fr(\))27 b(=)e Fp(e)1179 208 y Fk(\000)p Fo(iq)r(x=a)1409 241 y Fp(g)s Fr(\()p Fp(x;)15 b(x)21 b Fn(\000)f Fp(y)s Fr(\))30 b(and)g(c)m(hec)m(ks)i(the) f(p)s(erio)s(dicit)m(y)c(of)k Fp(v)s Fr(.)p 3950 241 4 62 v 3954 183 55 4 v 3954 241 V 4008 241 4 62 v 1333 482 a(8.)51 b Fq(The)34 b(spectral)e(resul)-6 b(ts)-236 644 y Fr(8.1.)53 b Fu(Lo)s(cal)39 b(v)m(ersion)f(of)g(Theorem)f(1.1.)46 b Fr(Pic)m(k)33 b Fp(E)1651 658 y Fm(0)1720 644 y Fn(2)d Fp(J)9 b Fr(,)33 b(and)g(let)g Fp(V)2295 658 y Fm(0)2367 644 y Fr(b)s(e)f(a)i(complex)e(neigh)m(b)s(orho)s(o)s(d)f(of)i Fp(E)3677 658 y Fm(0)3717 644 y Fr(,)g(and)g Fp(S)-236 752 y Fr(b)s(e)d(as)g(in)f(Theorem)h(1.3.)42 b(Fix)30 b Fp(I)7 b Fr(,)31 b(a)f(closed)h(real)f(in)m(terv)-5 b(al)29 b(in)g Fp(V)1945 766 y Fm(0)1985 752 y Fr(,)h(and)g(de\014ne)g Fp(\025)25 b Fr(=)g(exp\()p Fn(\000)p Fp(S=")p Fr(\).)42 b(W)-8 b(e)31 b(pro)m(v)m(e)-236 925 y Fu(Theorem)j(8.1.)46 b Fi(Fix)33 b Fp(\033)28 b Fn(2)d Fr(\(0)p Fp(;)15 b Fr(1\))p Fi(.)44 b(Ther)-5 b(e)33 b(exists)g Fp(D)28 b Fn(\032)d Fr(\(0)p Fp(;)15 b Fr(1\))p Fi(,)34 b(a)f(set)g(of)g (Diophantine)h(numb)-5 b(ers)33 b(such)g(that)1279 1083 y Fr(mes)15 b(\()p Fp(D)24 b Fn(\\)c Fr(\(0)p Fp(;)15 b(")p Fr(\)\))p 1279 1123 616 4 v 1566 1206 a Fp(")1930 1144 y Fr(=)25 b(1)c(+)f Fp(o)15 b Fr(\()p Fp("\025)2372 1106 y Fo(\033)2419 1144 y Fr(\))h Fp(;)-2731 b Fr(\(8.1\))-236 1336 y Fi(and)33 b(that,)h(for)f(any)h Fp(")25 b Fn(2)g Fp(D)s Fi(,)32 b(the)h(interval)h Fp(I)39 b Fi(c)-5 b(ontains)34 b(absolutely)g(c)-5 b(ontinuous)34 b(sp)-5 b(e)g(ctrum,)34 b(and)1110 1507 y Fr(mes)15 b(\()p Fp(I)28 b Fn(\\)19 b Fr(\006)1526 1521 y Fm(ac)1597 1507 y Fr(\))26 b(=)e(mes)16 b(\()p Fp(I)7 b Fr(\))21 b Fn(\001)f Fr(\(1)h(+)f Fp(O)s Fr(\()p Fp(\025)2456 1470 y Fo(\033)r(=)p Fm(2)2574 1507 y Fr(\)\))p Fp(:)-236 1678 y Fi(Her)-5 b(e,)31 b Fr(\006)74 1692 y Fo(ac)176 1678 y Fi(is)g(the)g(absolutely)h(c)-5 b(ontinuous)32 b(sp)-5 b(e)g(ctrum)32 b(for)f(the)g(family)h(of)f(e)-5 b(quations)39 b Fr(\(0.1\))r Fi(.)i(F)-7 b(or)32 b Fp(E)f Fn(2)24 b Fp(I)7 b Fi(,)31 b(e)-5 b(quation)39 b Fr(\(0.1\))-236 1786 y Fi(has)33 b(solutions)h Fp( )366 1800 y Fk(\006)458 1786 y Fi(as)f(describ)-5 b(e)g(d)34 b(in)f(The)-5 b(or)g(em)34 b(1.1.)-236 1960 y Fr(As)39 b Fp(J)48 b Fr(is)38 b(compact,)43 b(Theorem)38 b(1.1)i(immediately)e(follo)m(ws)g(from)g(Theorem)h(8.1.) 68 b(T)-8 b(o)39 b(pro)m(v)m(e)h(Theorem)f(8.1,)j(w)m(e)e(w)m(ork)-236 2067 y(with)30 b(\(7.3\))r(,)j(the)f(con)m(tin)m(uous)f(analog)h(of)g (the)h(mono)s(drom)m(y)e(equation,)h(for)g(the)g(mono)s(drom)m(y)f (matrix)g Fp(M)42 b Fr(describ)s(ed)30 b(b)m(y)-236 2175 y(Theorem)g(1.3.)43 b(First,)30 b(w)m(e)h(construct)g(solutions)e(of)38 b(\(7.3\))32 b(b)m(y)f(means)f(a)h(KAM)g(theory)g(construction)f (describ)s(ed)e(b)s(elo)m(w;)-236 2283 y(then,)k(w)m(e)g(use)f(Theorem) h(7.2)g(and)g(Corollary)e(1.2)j(to)f(obtain)f(Theorem)h(8.1.)45 b(Corollary)31 b(1.2)i(and)e(Ishii-P)m(astur-Kotani)-236 2391 y(Theorem)d(allo)m(w)f(us)g(to)i(con)m(trol)f(the)h(lo)s(cation)e (of)h(the)h(absolutely)e(con)m(tin)m(uous)g(sp)s(ectrum,)h(and)f (Theorem)h(7.2)h(allo)m(ws)e(us)-236 2499 y(to)k(describ)s(e)e(the)h (functional)f(structure)h(of)g(the)h(generalized)f(eigenfunctions.)-236 2702 y(8.2.)53 b Fu(P)m(o)s(or)35 b(man)e(KAM)g(theory.)46 b Fr(W)-8 b(e)31 b(recall)d(a)i(form)f(of)g(KAM)g(theory)h(suited)e (for)h(our)g(purp)s(ose;)f(it)h(w)m(as)g(dev)m(elop)s(ed)-236 2809 y(in)j([9)q(])i(using)e(standard)h(ideas)g(of)h(KAM)g(theory)g (\(see)h([6)q(,)f(1]\).)52 b(Let)34 b Fn(B)i Fr(b)s(e)d(a)i(Borel)f (subset)f(of)h Fl(R)r Fr(.)57 b(Let)35 b Fp(S)3475 2823 y Fo(r)3546 2809 y Fr(b)s(e)e(the)h(strip)-236 2917 y Fn(f)p Fp(z)c Fn(2)25 b Fl(C)17 b Fr(;)51 b Fn(j)p Fr(Im)15 b Fp(z)t Fn(j)26 b(\024)f Fp(r)s Fn(g)p Fr(.)41 b(F)-8 b(or)30 b(the)g(matrix)g(v)-5 b(alued)28 b(functions)h(of)h(\()p Fp(';)15 b(z)t Fr(\))31 b(that)g(are)f(Lipsc)m(hitz)f(in)f Fp(')e Fn(2)f(B)32 b Fr(and)d(analytic)h(in)e Fp(z)-236 3025 y Fr(in)h Fp(S)-74 3039 y Fo(r)-6 3025 y Fr(w)m(e)i(in)m(tro)s (duce)e(the)i(norm)515 3237 y Fn(k)p Fp(M)10 b Fn(k)703 3251 y Fo(r)n(;)p Fk(B)831 3237 y Fr(:=)39 b(sup)952 3319 y Fk(j)p Fo(y)r Fk(j\024)p Fo(r)964 3390 y(')p Fk(2B)1133 3237 y Fn(k)p Fp(M)10 b Fr(\()p Fp(z)t(;)15 b(')p Fr(\))p Fn(k)23 b Fr(+)66 b(sup)1681 3319 y Fk(j)p Fo(y)r Fk(j\024)p Fo(r)1650 3396 y(';')1762 3373 y Fc(0)1783 3396 y Fk(2B)1679 3469 y Fo(')p Fk(6)p Fm(=)p Fo(')1826 3446 y Fc(0)1904 3175 y Fn(k)p Fp(M)10 b Fr(\()p Fp(z)t(;)15 b(')p Fr(\))22 b Fn(\000)e Fp(M)10 b Fr(\()p Fp(z)t(;)15 b(')2653 3142 y Fk(0)2678 3175 y Fr(\))p Fn(k)p 1904 3216 856 4 v 2179 3299 a(j)p Fp(')21 b Fn(\000)f Fp(')2434 3273 y Fk(0)2458 3299 y Fn(j)2769 3237 y Fp(;)106 b(y)28 b Fr(=)d(Im)15 b Fp(z)t(:)-3500 b Fr(\(8.2\))-236 3633 y(Here,)28 b Fn(k)p Fp(:)p Fn(k)f Fr(is)f(the)g(matrix)g(norm)g(asso)s(ciated)h(to)g (the)g Fp(`)1636 3600 y Fm(1)1675 3633 y Fr(-norm)f(on)g Fl(C)2121 3600 y Fm(2)2166 3633 y Fr(.)40 b(If)26 b(the)g(matrix)g Fp(M)37 b Fr(is)25 b(indep)s(enden)m(t)f(of)j Fp(z)t Fr(,)h(w)m(e)e(write)-236 3741 y Fn(k)p Fp(M)10 b Fn(k)-48 3755 y Fk(B)35 3741 y Fr(instead)30 b(of)g Fn(k)p Fp(M)10 b Fn(k)640 3755 y Fo(r)n(;)p Fk(B)743 3741 y Fr(.)-236 3896 y(A)30 b(matrix)g(v)-5 b(alued)30 b(function)f(\()p Fp(z)t(;)15 b(')p Fr(\))28 b Fn(2)d Fp(S)1185 3910 y Fo(r)1243 3896 y Fn(\002)20 b(B)28 b(7!)e Fp(M)10 b Fr(\()p Fp(z)t(;)15 b(')p Fr(\))32 b(b)s(elongs)e(to)h(the)g(class)f Fn(M)h Fr(if)f(it)g(is)f(analytic)i(and)f(1-p)s(erio)s(dic)-236 4004 y(in)f Fp(S)-74 4018 y Fo(r)-6 4004 y Fr(and)h(is)f(of)i(the)f (form)1581 4216 y Fp(M)35 b Fr(=)1800 4088 y Ff(\022)1883 4161 y Fp(a)122 b(b)1867 4269 y(b)1906 4236 y Fk(\003)2029 4269 y Fp(a)2077 4236 y Fk(\003)2116 4088 y Ff(\023)-236 4432 y Fr(Let)31 b Fp(A)f Fr(and)g Fp(D)j Fr(b)s(e)d(t)m(w)m(o)h (matrices)g(in)e Fn(M)h Fr(suc)m(h)g(that)-111 4569 y Fn(\017)42 b Fr(the)30 b(matrix)g Fp(D)j Fr(is)c(diagonal,)1061 4786 y Fp(D)f Fr(=)d Fp(D)s Fr(\()p Fp(')p Fr(\))h(=)1589 4658 y Ff(\022)1656 4728 y Fp(d)84 b Fr(0)1657 4842 y(0)p 1787 4768 48 4 v 85 w Fp(d)1834 4658 y Ff(\023)1946 4786 y Fr(where)30 b Fp(d)c Fr(=)f(exp\()p Fp(i')p Fr(\);)-111 5009 y Fn(\017)42 b Fr(the)30 b(matrix)g Fp(A)p Fr(\()p Fp(z)t(;)15 b(')p Fr(\))32 b(satis\014es)e Fn(k)p Fp(A)p Fn(k)1236 5023 y Fo(r)n(;I)1352 5009 y Fn(\024)25 b Fr(1.)-236 5146 y(F)-8 b(or)31 b(\011\()p Fp(z)t Fr(\),)g(a)g(2)21 b Fn(\002)e Fr(2)31 b(matrix,)f(consider)f(the)i(equation)1180 5312 y(\011\()p Fp(z)24 b Fr(+)c Fp(h)p Fr(\))26 b(=)f(\()p Fp(D)e Fr(+)d Fp(\025A)p Fr(\)\011\()p Fp(z)t Fr(\))p Fp(;)108 b(z)29 b Fn(2)c Fl(R)s Fp(:)-2830 b Fr(\(8.3\))-236 5477 y(where)30 b Fp(\025)g Fr(is)f(a)i(p)s(ositiv)m(e)f(parameter,)h (and)e Fp(h)i Fr(is)e(\014xed,)h(0)c Fp(<)f(h)h(<)f Fr(1.)-236 5633 y(T)-8 b(o)44 b(solv)m(e)g(\(8.3\))h(using)e(a)h(KAM)f(t)m(yp)s(e) h(metho)s(d,)j(w)m(e)d(imp)s(ose)e(a)i(Diophan)m(tine)f(condition)f(on) i(the)g(n)m(um)m(b)s(er)e Fp(h)p Fr(.)81 b(Fix)-236 5741 y(0)25 b Fp(<)g(\033)k(<)c Fr(1)31 b(and)e(de\014ne)763 5907 y Fp(H)839 5921 y Fo(\033)911 5907 y Fr(:=)c Fn(f)p Fp(h)h Fn(2)f Fr(\(0)p Fp(;)15 b Fr(1\);)48 b(min)1531 5968 y Fo(l)q Fk(2)p Fh(N)1680 5907 y Fn(j)p Fp(h)21 b Fn(\000)f Fp(l)r(=k)s Fn(j)26 b(\025)f Fp(\025)2193 5869 y Fo(\033)2240 5907 y Fp(=k)2335 5869 y Fm(3)2375 5907 y Fp(;)46 b(k)28 b Fr(=)d(1)p Fp(;)15 b Fr(2)p Fp(;)g Fr(3)g Fp(:)g(:)g(:)k Fn(g)-3252 b Fr(\(8.4\))-236 6110 y(One)30 b(has)1854 6210 y Fm(24)p eop %%Page: 25 25 25 24 bop -236 241 a Fu(Prop)s(osition)36 b(8.1)f Fr(\([9)q(]\))p Fu(.)47 b Fi(L)-5 b(et)31 b Fp(A)h Fi(and)g Fp(D)i Fi(b)-5 b(e)31 b(chosen)h(as)g(ab)-5 b(ove;)32 b(assume)g(that)g Fr(det)q(\()p Fp(D)20 b Fr(+)d Fp(\025A)p Fr(\))26 b Fn(\021)f Fr(1)p Fi(.)41 b(Then,)32 b(ther)-5 b(e)32 b(exists)-236 349 y Fp(\025)-183 363 y Fm(0)-144 349 y Fr(\()p Fp(r)m(;)15 b(\033)n(;)g(I)7 b Fr(\))28 b Fp(>)d Fr(0)33 b Fi(such)f(that,)i(for)f(al)5 b(l)33 b Fn(j)p Fp(\025)p Fn(j)26 b Fp(<)f(\025)1316 363 y Fm(0)1356 349 y Fi(,)32 b(ther)-5 b(e)34 b(exists)f(a)g(Bor)-5 b(el)33 b(set)g Fr(\010)2420 363 y Fk(1)2494 349 y Fr(\()p Fp(r)m(;)15 b(\033)n(;)g(I)7 b(;)15 b(\025)p Fr(\))28 b Fn(\032)d Fp(I)40 b Fi(satisfying)1360 506 y Fr(mes)31 b(\010)1609 520 y Fk(1)1708 506 y Fn(\024)25 b Fp(K)1881 520 y Fm(0)1921 506 y Fr(\()p Fp(r)m(;)15 b(\033)n(;)g(I)7 b Fr(\))p Fp(\025)2259 469 y Fo(\033)r(=)p Fm(2)2379 506 y Fp(;)-2640 b Fr(\(8.5\))-236 659 y Fi(outside)33 b(of)g(which,)g(for)h Fp(')25 b Fn(2)g(B)841 673 y Fk(1)941 659 y Fr(=)g Fp(I)i Fn(n)21 b Fr(\010)1236 673 y Fk(1)1310 659 y Fi(,)33 b(e)-5 b(quation)40 b Fr(\(8.3\))34 b Fi(has)g(a)f (solution)h(of)f(the)g(form)1057 867 y Fr(\011\()p Fp(z)t(;)15 b(')p Fr(\))27 b(=)e Fp(U)10 b Fr(\()p Fp(z)t(;)15 b(')p Fr(\))1768 739 y Ff(\022)1837 812 y Fp(e)1879 779 y Fo(i')1949 787 y Fc(1)2014 779 y Fk(\001)p Fo(z)s(=h)2394 812 y Fr(0)1970 927 y(0)217 b Fp(e)2274 894 y Fk(\000)p Fo(i')2399 902 y Fc(1)2465 894 y Fk(\001)p Fo(z)s(=h)2600 739 y Ff(\023)2682 867 y Fp(;)-2943 b Fr(\(8.6\))-236 1070 y Fi(wher)-5 b(e)-111 1197 y Fn(\017)42 b Fp(U)35 b Fn(2)25 b(M)32 b Fi(and)i(det)p Fr(\()p Fp(U)10 b Fr(\))26 b(=)f(1)p Fi(;)-111 1305 y Fn(\017)42 b Fp(U)g Fi(is)33 b(de\014ne)-5 b(d)33 b(and)h(analytic)g(for)f Fn(j)p Fr(Im)15 b Fp(z)t Fn(j)26 b Fp(<)f(r)s(=)p Fr(2)p Fi(,)33 b(it)f(is)h Fr(1)p Fi(-p)-5 b(erio)g(dic)34 b(in)f Fp(z)t Fi(,)f(and)i Fn(k)p Fp(U)c Fn(\000)20 b Fr(1)p Fn(k)3006 1324 y Fo(r)r(=)p Fm(2)p Fo(;)13 b Fk(B)3189 1332 y Fc(1)3284 1305 y Fn(\024)25 b Fp(C)7 b(\025)3505 1272 y Fm(2)p Fk(\000)p Fo(\033)3641 1305 y Fi(;)-111 1417 y Fn(\017)42 b Fp(')35 1431 y Fk(1)142 1417 y Fi(is)33 b(a)g(r)-5 b(e)g(al)34 b(value)-5 b(d)33 b(Lipschitz)h(c)-5 b(ontinuous)34 b(function)e(of)h Fp(')g Fi(and)h(satis\014es)f Fn(k)p Fp(')2789 1431 y Fk(1)2865 1417 y Fr(\()p Fp(')p Fr(\))21 b Fn(\000)f Fp(')p Fn(k)3210 1431 y Fk(B)3256 1439 y Fc(1)3352 1417 y Fn(\024)25 b Fr(2)p Fp(\025)p Fi(.)-236 1544 y(Her)-5 b(e,)32 b Fp(\025)62 1558 y Fm(0)102 1544 y Fi(,)g Fr(\010)228 1558 y Fk(1)335 1544 y Fi(and)h Fp(K)588 1558 y Fm(0)661 1544 y Fi(only)g(dep)-5 b(end)34 b(on)f(the)g(ar)-5 b(guments)34 b(indic)-5 b(ate)g(d)34 b(explicitly)g(ab)-5 b(ove.)-236 1707 y Fr(The)30 b(set)h(\010)159 1721 y Fk(1)263 1707 y Fr(admits)f(the)g(follo)m(wing)f(description)651 1904 y(\010)717 1918 y Fk(1)816 1904 y Fr(=)c Fn([)973 1867 y Fk(1)973 1927 y Fo(j)t Fm(=0)1100 1904 y Fr(\010)1166 1918 y Fo(j)1202 1904 y Fp(;)106 b Fr(\010)1399 1918 y Fo(j)1461 1904 y Fr(=)1592 1818 y Ff([)1557 2015 y Fo(k)r(;l)q Fk(2)p Fh(Z)1743 1776 y Ff(\032)1811 1904 y Fp(')26 b Fn(2)f Fp(I)32 b Fr(:)56 b Fn(j)p Fp(\013)2218 1918 y Fo(j)2255 1904 y Fr(\()p Fp(')p Fr(\))21 b Fn(\000)f Fp(hk)k Fn(\000)c Fp(l)r Fn(j)25 b Fp(<)2895 1843 y(\025)2948 1810 y Fo(\033)p 2895 1883 100 4 v 2900 1967 a Fp(k)2950 1940 y Fm(2)3005 1776 y Ff(\033)3088 1904 y Fp(;)-3349 b Fr(\(8.7\))-236 2148 y(where)30 b(the)g(functions)f Fp(\013)633 2162 y Fo(j)700 2148 y Fr(are)i(Lipsc)m(hitz)e(con)m(tin)m (uous)h(and,)g(for)g(some)h Fp(C)h(>)25 b Fr(0,)31 b(satisfy)651 2342 y Fn(k)p Fp(\013)754 2356 y Fo(j)811 2342 y Fn(\000)20 b Fp(\013)960 2356 y Fo(j)t Fk(\000)p Fm(1)1087 2342 y Fn(k)1132 2356 y Fo(I)1198 2342 y Fn(\024)k Fp(\025)1346 2305 y Fo(\033)1393 2342 y Fr(\()p Fp(C)7 b(\025)1553 2305 y Fm(1)p Fk(\000)p Fo(\033)1690 2342 y Fr(\))1725 2305 y Fm(2)1760 2281 y Fd(j)1798 2342 y Fp(;)106 b Fn(k)p Fp(\013)2032 2356 y Fm(0)2072 2342 y Fr(\()p Fp(')p Fr(\))21 b Fn(\000)2328 2281 y Fr(1)p 2323 2321 56 4 v 2323 2405 a Fp(\031)2388 2342 y(')p Fn(k)2492 2356 y Fo(I)2558 2342 y Fn(\024)k Fr(2)p Fp(\025;)107 b(j)31 b Fn(2)25 b Fl(N)6 b Fp(:)-3358 b Fr(\(8.8\))-236 2533 y(8.3.)53 b Fu(Constructing)30 b(solutions)h(of)f(the)f(mono)s(drom)m(y)g (equation.)46 b Fr(First,)26 b(w)m(e)g(c)m(hec)m(k)i(that)e(the)g(mono) s(drom)m(y)f(equa-)-236 2641 y(tion)44 b(can)h(b)s(e)f(rewritten)g(in)f (the)i(form)f(required)f(b)m(y)i(Prop)s(osition)d(8.1;)53 b(then,)c(w)m(e)c(c)m(hec)m(k)h(that)f(Prop)s(osition)e(8.1)j(is)-236 2749 y(applicable)36 b(for)h(su\016cien)m(tly)g(small)g Fp(")h Fn(2)f Fp(D)k Fn(\032)d Fr(\(0)p Fp(;)15 b Fr(1\))40 b(where)d Fp(D)k Fr(is)c(a)h(set)h(with)d(the)i(prop)s(erties)f (describ)s(ed)f(ab)s(o)m(v)m(e;)43 b(at)-236 2857 y(last,)30 b(w)m(e)h(use)f(Prop)s(osition)f(8.1)i(to)h(construct)e(the)h (solutions)e(of)h(the)h(mono)s(drom)m(y)f(equation)g(outside)g(a)h(set) g Fp(E)3669 2871 y Fk(1)3769 2857 y Fn(\032)25 b Fp(I)38 b Fr(of)-236 2965 y(Leb)s(esgue)30 b(measure)g Fp(O)s Fr(\()p Fp(\025)671 2932 y Fo(\033)r(=)p Fm(2)789 2965 y Fr(\).)-236 3130 y(8.3.1.)48 b Fi(The)34 b(c)-5 b(o)g(e\016cient)34 b Fp(a)684 3144 y Fm(0)723 3130 y Fi(.)45 b Fr(Here,)32 b(w)m(e)g(c)m(hec)m(k)g(that)g(the)f(zeroth)g(F)-8 b(ourier)31 b(co)s(e\016cien)m(t)h(of)f(the)g(co)s(e\016cien)m(t)h Fp(a)p Fr(\()p Fp(z)t(;)15 b(E)5 b Fr(\))32 b(of)f(the)-236 3238 y(mono)s(drom)m(y)f(matrix)f(admits)h(the)g(represen)m(tation)803 3435 y Fp(a)851 3449 y Fm(0)916 3435 y Fr(=)25 b(exp)1166 3307 y Ff(\022)1233 3435 y Fn(\000)1319 3374 y Fp(i)p 1314 3414 43 4 v 1314 3498 a(")1366 3435 y Fr(\010\()p Fp(E)5 b Fr(\))21 b(+)f Fp(i)p Fr(\012\()p Fp(E)5 b Fr(\))21 b(+)f Fp(i\036)p Fr(\()p Fp(E)5 b Fr(\))2264 3307 y Ff(\023)2353 3435 y Fr(+)20 b Fp(O)s Fr(\()p Fp(e)2606 3371 y Fd(i)p 2603 3383 30 3 v 2603 3424 a(")2642 3398 y Fm(\010\()p Fo(E)t Fm(\))2808 3435 y Fp(\025)2861 3398 y Fm(2)2900 3435 y Fr(\))p Fp(;)-3196 b Fr(\(8.9\))-236 3637 y(where)34 b Fp(\036)p Fr(\()p Fp(E)5 b Fr(\))35 b(is)f(real)g(analytic)g(and)g Fp(\036)p Fr(\()p Fp(E)5 b Fr(\))33 b(=)f Fp(o)p Fr(\(1\))k(uniformly) 31 b(in)j Fp(E)j Fn(2)32 b Fp(V)2376 3651 y Fm(0)2415 3637 y Fr(,)k(the)f(constan)m(t)g(neigh)m(b)s(orho)s(o)s(d)e(of)h Fp(E)3757 3651 y Fm(0)3831 3637 y Fr(from)-236 3745 y(Theorem)c(1.3\).) -236 3856 y(Therefore,)24 b(w)m(e)f(represen)m(t)g Fp(a)754 3870 y Fm(0)816 3856 y Fr(in)e(the)i(form)g Fp(a)1319 3870 y Fm(0)1383 3856 y Fr(=)i(exp)1633 3782 y Ff(\000)1675 3856 y Fn(\000)1760 3820 y Fo(i)p 1756 3835 33 4 v 1756 3887 a(")1798 3856 y Fr(\010\()p Fp(E)5 b Fr(\))21 b(+)f Fp(i)p Fr(\012\()p Fp(E)5 b Fr(\))21 b(+)f Fp(ig)s Fr(\()p Fp(E)5 b Fr(\))2688 3782 y Ff(\001)2732 3856 y Fr(.)38 b(By)23 b(Theorem)f(1.3,)k(the)d(function)e Fp(g)-236 3995 y Fr(is)j(analytic)h(and)f Fp(g)s Fr(\()p Fp(E)5 b Fr(\))27 b(=)e Fp(o)p Fr(\(1\))h(uniformly)c(in)i Fp(E)31 b Fn(2)24 b Fp(V)1604 4009 y Fm(0)1644 3995 y Fr(.)39 b(Let)25 b Fp(\036)p Fr(\()p Fp(E)5 b Fr(\))27 b(=)e(\()p Fp(g)s Fr(\()p Fp(E)5 b Fr(\))10 b(+)p 2498 3903 190 4 v 10 w Fp(g)s Fr(\()p 2579 3922 73 4 v Fp(E)d Fr(\)\))p Fp(=)p Fr(2,)28 b(and)c(let)h Fp(\036)3217 4009 y Fm(1)3257 3995 y Fr(\()p Fp(E)5 b Fr(\))26 b(=)f Fp(g)s Fr(\()p Fp(E)5 b Fr(\))10 b Fn(\000)p 3800 3903 190 4 v 10 w Fp(g)s Fr(\()p 3881 3922 73 4 v Fp(E)d Fr(\).)-236 4102 y(The)38 b(function)g Fp(\036)h Fr(is)f(real)h(analytic,)i(and)e Fp(\036)p Fr(\()p Fp(E)5 b Fr(\))41 b(=)e Fp(o)p Fr(\(1\))h(uniformly)c (in)i Fp(E)45 b Fn(2)39 b Fp(V)2620 4116 y Fm(0)2660 4102 y Fr(.)67 b(Hence,)42 b(it)d(su\016ces)f(to)i(pro)m(v)m(e)g(that) -236 4210 y Fp(\036)-182 4224 y Fm(1)-117 4210 y Fr(=)25 b Fp(O)s Fr(\()p Fp(a)134 4224 y Fm(0)173 4210 y Fp(\025)226 4177 y Fm(2)266 4210 y Fr(\))30 b(uniformly)e(in)h Fp(E)h Fn(2)25 b Fp(V)1088 4224 y Fm(0)1128 4210 y Fr(.)40 b(F)-8 b(or)32 b(complex)e Fp(E)5 b Fr(,)31 b(the)f(relation)g Fp(M)10 b Fr(\()p Fp(z)t(;)15 b(E)5 b Fr(\))27 b Fn(\021)e Fr(1)31 b(implies)c(that)1204 4379 y Fp(a)p Fr(\()p Fp(z)t(;)15 b(E)5 b Fr(\))p 1480 4288 278 4 v Fp(a)p Fr(\()p 1563 4329 47 4 v Fp(z)i(;)p 1652 4306 73 4 v 15 w(E)f Fr(\))25 b(=)g(1)c(+)f Fp(b)p Fr(\()p Fp(z)t(;)15 b(E)5 b Fr(\))p 2305 4288 269 4 v Fp(b)p Fr(\()p 2379 4329 47 4 v Fp(z)h(;)p 2467 4306 73 4 v 15 w(E)g Fr(\))-236 4531 y(\(compare)34 b(with)d(\(6.11\))r(\).)49 b(No)m(w,)35 b(w)m(e)e(use)f(the)i(represen) m(tation)f Fp(a)p Fr(\()p Fp(z)t(;)15 b(E)5 b Fr(\))31 b(=)e Fp(a)2459 4545 y Fm(0)2498 4531 y Fr(\()p Fp(E)5 b Fr(\))23 b(+)f Fp(a)2804 4545 y Fm(1)2843 4531 y Fr(\()p Fp(z)t(;)15 b(E)5 b Fr(\),)36 b(the)d(fact)h(that)f(\010)g(and)f(\012) -236 4639 y(are)f(real)f(analytic,)g(and)g(the)g(estimate)h(\(1.14\))i (to)e(obtain)568 4808 y(exp\()p Fp(g)s Fr(\()p Fp(E)5 b Fr(\))22 b Fn(\000)p 1043 4717 190 4 v 20 w Fp(g)s Fr(\()p 1124 4735 73 4 v Fp(E)6 b Fr(\))q(\))20 b(+)g Fp(a)1427 4822 y Fm(0)1467 4808 y Fr(\()p Fp(E)5 b Fr(\))p 1609 4717 318 4 v Fp(a)1657 4822 y Fm(1)1697 4808 y Fr(\()p 1732 4758 47 4 v Fp(z)g(;)p 1819 4735 73 4 v 15 w(E)h Fr(\))20 b(+)g Fp(a)2086 4822 y Fm(1)2126 4808 y Fr(\()p Fp(z)t(;)15 b(E)5 b Fr(\))p 2354 4717 231 4 v Fp(a)2402 4822 y Fm(0)2443 4808 y Fr(\()p 2478 4735 73 4 v Fp(E)h Fr(\))25 b(=)g(1)c(+)f Fp(O)s Fr(\()p Fp(a)3019 4822 y Fm(0)3058 4808 y Fp(\025)3111 4771 y Fm(2)3151 4808 y Fr(\))p Fp(:)-236 4991 y Fr(In)m(tegrating)42 b(this)f(form)m(ula)g (in)f Fp(z)46 b Fr(from)41 b(0)h(to)h(1,)i(and)c(recalling)g(that)2307 4917 y Ff(R)2368 4944 y Fm(1)2350 5023 y(0)2422 4991 y Fp(a)2470 5005 y Fm(1)2510 4991 y Fp(dz)48 b Fr(=)c(0,)i(w)m(e)c(get) h(exp\()p Fp(g)s Fr(\()p Fp(E)5 b Fr(\))29 b Fn(\000)p 3676 4899 190 4 v 28 w Fp(g)s Fr(\()p 3757 4918 73 4 v Fp(E)6 b Fr(\)\))45 b(=)-236 5107 y(1)20 b(+)g Fp(O)s Fr(\()p Fp(a)75 5121 y Fm(0)115 5107 y Fp(\025)168 5074 y Fm(2)208 5107 y Fr(\).)41 b(This)28 b(implies)g Fp(\036)879 5121 y Fm(1)944 5107 y Fr(=)d Fp(O)s Fr(\()p Fp(a)1195 5121 y Fm(0)1234 5107 y Fp(\025)1287 5074 y Fm(2)1327 5107 y Fr(\),)31 b(hence,)f(\(8.9\))r(.)-236 5271 y(8.3.2.)48 b Fi(A)29 b(new)g(p)-5 b(ar)g(ameterization)33 b(of)d(the)f(mono)-5 b(dr)g(omy)33 b(matrix.)47 b Fr(W)-8 b(e)27 b(consider)f(the)h(mono)s (drom)m(y)e(matrix)h(as)h(a)g(function)-236 5379 y(of)32 b(the)h(parameter)g Fp(')p Fr(\()p Fp(E)5 b Fr(\))30 b(=)e Fn(\000)p Fr(\010\()p Fp(E)5 b Fr(\))p Fp(=")23 b Fr(+)e(\012\()p Fp(E)5 b Fr(\))22 b(+)g Fp(\036)p Fr(\()p Fp(E)5 b Fr(\))33 b(instead)f(of)g(the)h(parameter)g Fp(E)5 b Fr(.)46 b(Pic)m(k)33 b(\001)28 b Fp(>)g Fr(0)33 b(and)f(let)g Fp(I)3827 5393 y Fm(\001)3922 5379 y Fr(b)s(e)-236 5487 y(the)e(\001-neigh)m(b)s(orho)s(o)s(d)e(of)i(the)g(in)m(terv)-5 b(al)29 b Fp(I)38 b Fr(in)28 b Fl(C)18 b Fr(.)47 b(W)-8 b(e)31 b(\014x)e(\001)h(so)g(that)h Fp(I)2237 5501 y Fm(\001)2330 5487 y Fr(b)s(e)e(inside)f(a)j(constan)m(t)g(compact)g (subset)f(of)g Fp(V)3951 5501 y Fm(0)3990 5487 y Fr(.)-236 5595 y(W)-8 b(e)29 b(sho)m(w)f(that,)i(for)e(\001)g(su\016cien)m(tly)f (small)g(but)g(indep)s(enden)m(t)g(of)h Fp(")p Fr(,)h(and)f(su\016cien) m(tly)f(small)g Fp(")p Fr(,)i(the)g(mapping)d Fp(E)31 b Fn(7!)25 b Fp(')k Fr(is)-236 5703 y(an)h(analytic)g(isomorphism)e(of) i Fp(I)911 5717 y Fm(\001)1004 5703 y Fr(on)m(to)i(its)e(image.)-236 5858 y(First,)g(w)m(e)h(note)g(that)1142 5987 y Fp(d')p 1135 6027 120 4 v 1135 6110 a(dE)1265 6048 y Fr(\()p Fp(E)5 b Fr(\))26 b(=)f Fn(\000)1610 5987 y Fr(1)p 1610 6027 46 4 v 1612 6110 a Fp(")1665 6048 y Fr(\010)1731 6011 y Fk(0)1754 6048 y Fr(\()p Fp(E)5 b Fr(\))21 b(+)f Fp(O)s Fr(\(1\))p Fp(;)108 b(E)30 b Fn(2)25 b Fp(I)2551 6062 y Fm(\001)2614 6048 y Fp(;)-2875 b Fr(\(8.10\))1854 6210 y Fm(25)p eop %%Page: 26 26 26 25 bop -236 241 a Fr(where)30 b(the)g(estimate)h(on)g(the)f(error)g (term)h(follo)m(ws)e(from)h(the)h(Cauc)m(h)m(y)f(estimates)h(applied)d (to)k(\012)e(and)f Fp(\036)p Fr(.)-236 439 y(Then,)j(w)m(e)h(study)f (\010)484 406 y Fk(0)507 439 y Fr(.)47 b(By)34 b(\(6.4\))h(and)d (\(6.2\))q(,)1379 377 y Fp(@)5 b(\024)1484 391 y Fo(n)1527 400 y Fg(0)p 1379 418 188 4 v 1410 501 a Fp(@)g(E)1605 439 y Fr(=)28 b Fp(\033)s(k)1809 401 y Fk(0)1806 461 y Fo(p)1847 439 y Fr(\()p Fp(E)f Fn(\000)21 b Fp(W)13 b Fr(\()p Fp(\020)7 b Fr(\)\),)34 b(where)e Fp(\033)g Fn(2)c(f\000)p Fr(1)p Fp(;)15 b Fr(+1)p Fn(g)35 b Fr(dep)s(ends)30 b(only)i(on)g Fp(n)3951 453 y Fm(0)3990 439 y Fr(,)-236 572 y(and)e Fp(k)-12 586 y Fo(p)58 572 y Fr(is)f(the)i(main)e(branc)m (h)h(of)g(the)h(Blo)s(c)m(h)f(quasi-momen)m(tum.)40 b(Therefore,)1246 842 y(\010)1312 804 y Fk(0)1335 842 y Fr(\()p Fp(E)5 b Fr(\))26 b(=)f Fp(\033)1669 718 y Ff(Z)1760 744 y Fm(2)p Fo(\031)1720 924 y Fm(0)1858 842 y Fp(k)1908 804 y Fk(0)1905 864 y Fo(p)1945 842 y Fr(\()p Fp(E)h Fn(\000)20 b Fp(W)13 b Fr(\()p Fp(\020)7 b Fr(\)\))p Fp(d\020)g(:)-2770 b Fr(\(8.11\))-236 1096 y(Recall)26 b(that)i Fp(k)278 1063 y Fk(0)302 1096 y Fr(\()p Fn(E)8 b Fr(\))27 b(is)f(real)h(and)g(do)s (es)f(not)i(v)-5 b(anish)25 b(inside)g(the)i Fp(n)1991 1110 y Fm(0)2030 1096 y Fr(-th)g(sp)s(ectral)g(band,)f(and)h(that,)h (for)f Fp(E)k Fn(2)25 b Fp(J)36 b Fr(and)26 b(for)h(real)-236 1204 y Fp(\020)7 b Fr(,)56 b Fn(E)33 b Fr(=)25 b Fp(E)c Fn(\000)15 b Fp(W)e Fr(\()p Fp(\020)7 b Fr(\))27 b(is)g(inside)f(this)h (band.)38 b(Therefore,)29 b(\010)1769 1171 y Fk(0)1792 1204 y Fr(\()p Fp(E)5 b Fr(\))29 b(is)e(real)g(and)g(of)h(\014xed)g (sign)f(on)g Fp(J)9 b Fr(.)40 b(This)27 b(and)g(\(8.10\))j(imply)-236 1360 y(that)-24 1298 y Fp(d')p -31 1339 120 4 v -31 1422 a(dE)128 1360 y Fr(is)e(b)s(ounded)f(a)m(w)m(a)m(y)k(from)d(0)h (uniformly)d(on)j Fp(I)36 b Fr(for)29 b(su\016cien)m(tly)f(small)f Fp(")p Fr(.)41 b(Therefore,)29 b(for)g(su\016cien)m(tly)e(small)h (\001,)-236 1499 y(and)i(for)g(su\016cien)m(tly)f(small)g Fp(")p Fr(,)i(the)f(function)f Fp(')p Fr(\()p Fp(E)5 b Fr(\))32 b(is)d(analytic)h(isomorphism)e(of)i Fp(I)2728 1513 y Fm(\001)2822 1499 y Fr(on)m(to)h(its)f(image.)-236 1654 y(In)m(tro)s(duce)e(the)h(function)e Fp(E)5 b Fr(\()p Fp(')p Fr(\))30 b(in)m(v)m(erse)f(to)g Fp(')p Fr(\()p Fp(E)5 b Fr(\).)42 b(It)29 b(is)e(de\014ned)h(on)g Fp(')p Fr(\()p Fp(I)2347 1668 y Fm(\001)2411 1654 y Fr(\).)40 b(Note)30 b(that)g(that)f Fn(I)i Fr(=)25 b Fp(')p Fr(\()p Fp(I)7 b Fr(\))30 b(is)e(an)h(in)m(terv)-5 b(al)-236 1762 y(of)30 b(the)h(real)f(axis;)g(its)g(length)g(is)f(of)h(order)g Fp(O)s Fr(\(1)p Fp(=")p Fr(\).)-236 2036 y(8.3.3.)48 b Fi(The)34 b(mono)-5 b(dr)g(omy)38 b(e)-5 b(quation.)46 b Fr(Let)32 b(us)f(c)m(hec)m(k)j(that)e(one)g(can)g(apply)e(Prop)s (osition)g(8.1)j(to)f(equation)g(\(7.3\))q(.)45 b(First,)-236 2144 y(w)m(e)f(note)g(that)g(det)15 b Fp(M)58 b Fn(\021)47 b Fr(1,)g(and)c(that)h(the)g(mono)s(drom)m(y)f(matrix)f(already)h(has)h (the)f(form)g(\(1.12\))s(.)80 b(Consider)41 b(the)-236 2252 y(mono)s(drom)m(y)33 b(matrix)g(as)h(a)g(function)e(of)i Fp(')d Fn(2)f Fp(')p Fr(\()p Fp(I)1544 2266 y Fm(\001)1608 2252 y Fr(\).)51 b(It)34 b(can)g(b)s(e)f(represen)m(ted)g(in)g(the)h (form)f(\(8.3\))i(with)e(the)g(diagonal)-236 2415 y(matrix)c Fp(D)g Fr(=)258 2287 y Ff(\022)325 2360 y Fp(e)367 2327 y Fo(i')588 2360 y Fr(0)361 2469 y(0)119 b Fp(e)567 2436 y Fk(\000)p Fo(i')696 2287 y Ff(\023)794 2415 y Fr(and)29 b(the)i(matrix)f Fp(A)g Fr(de\014ned)f(b)m(y)401 2685 y Fp(\025A)522 2699 y Fm(11)597 2685 y Fr(\()p Fp(z)t(;)15 b(')p Fr(\))27 b(=)e Fp(a)983 2699 y Fm(1)1022 2685 y Fr(\()p Fp(z)t(;)15 b(E)5 b Fr(\))22 b(+)e(\()p Fp(a)1446 2699 y Fm(0)1486 2685 y Fr(\()p Fp(E)5 b Fr(\))21 b Fn(\000)f Fp(e)1782 2648 y Fo(i')1857 2685 y Fr(\))p Fp(;)106 b(\025A)2144 2699 y Fm(12)2219 2685 y Fr(\()p Fp(z)t(;)15 b(')p Fr(\))27 b(=)e Fp(b)p Fr(\()p Fp(z)t(;)15 b(E)5 b Fr(\))p Fp(;)108 b(E)31 b Fr(=)25 b Fp(E)5 b Fr(\()p Fp(')p Fr(\))p Fp(:)-236 2939 y Fr(Let)35 b(us)f(study)g(the)h(matrix)f Fp(A)p Fr(.)55 b(As)35 b(the)g(mono)s(drom)m(y)f(matrix,)h(it)g(is)f(analytic) g(in)g Fp(z)39 b Fr(in)33 b(the)i(strip)f Fn(j)p Fr(Im)15 b Fp(z)t Fn(j)33 b(\024)f Fp(y)s Fr(.)54 b(So,)36 b(w)m(e)-236 3047 y(need)30 b(only)f(to)j(c)m(hec)m(k)f(that)g(the)g(Lipsc)m(hitz)e (norm)h Fn(k)p Fp(A)p Fn(k)1669 3066 y Fo(C)q(;')p Fm(\()p Fo(I)5 b Fm(\))1912 3047 y Fr(is)29 b(b)s(ounded)f(b)m(y)j(a)f(constan) m(t)i(uniformly)27 b(in)i Fp(")p Fr(.)-236 3206 y(As)d Fp(A)g Fr(is)e(analytic)i(in)e(\()p Fp(z)t(;)15 b(')p Fr(\))27 b Fn(2)e(fj)p Fr(Im)16 b Fp(z)t Fn(j)25 b(\024)g Fp(y)s Fn(g)11 b(\002)g Fp(')p Fr(\()p Fp(I)1552 3220 y Fm(\001)1616 3206 y Fr(\),)28 b(w)m(e)e(ha)m(v)m(e)h(only)e(to)i(c)m (hec)m(k)g(that)f(the)g(elemen)m(ts)h(of)f Fp(A)f Fr(are)i(uniformly) -236 3314 y(b)s(ounded)e(in)h Fp(z)j Fn(2)c(fj)p Fr(Im)16 b Fp(z)t Fn(j)25 b Fp(<)g(y)s Fn(g)j Fr(and)e(for)h Fp(')h Fr(in)e(a)h(constan)m(t)i(neigh)m(b)s(orho)s(o)s(d)c(of)i Fp(')p Fr(\()p Fp(I)7 b Fr(\),)29 b(i.e.)40 b(that)28 b(its)e(elemen)m(ts,)j(considered)-236 3422 y(as)35 b(functions)f(of)i (\()p Fp(z)t(;)15 b(E)5 b Fr(\))37 b(are)f(uniformly)c(b)s(ounded)i(in) g Fn(fj)p Fr(Im)15 b Fp(z)t Fn(j)35 b(\024)e Fp(y)s Fn(g)24 b(\002)f Fr(\()p Fp(c")p Fr(-neigh)m(b)s(orho)s(o)s(d)34 b(of)i Fp(I)7 b Fr(\),)37 b(where)e Fp(c)h Fr(is)f(a)g(\014xed)-236 3530 y(p)s(ositiv)m(e)29 b(constan)m(t.)-236 3652 y(Since)h(\010\()p Fp(z)t(;)15 b(E)5 b Fr(\))32 b(is)e(real)h(analytic)f(in)g Fp(E)5 b Fr(,)31 b(in)f(suc)m(h)g(a)i(neigh)m(b)s(orho)s(o)s(d,)d Fn(j)p Fp(e)2245 3589 y Fd(i)p 2242 3601 30 3 v 2242 3642 a(")2281 3616 y Fm(\010)2337 3652 y Fn(j)i Fr(is)f(b)s(ounded)e(b) m(y)j(a)h(constan)m(t)g(indep)s(enden)m(t)d(of)-236 3760 y Fp(E)34 b Fr(and)28 b Fp(")p Fr(.)40 b(So,)30 b(b)m(y)g(\(8.9\))q(,)g Fp(a)713 3774 y Fm(0)752 3760 y Fr(\()p Fp(E)5 b Fr(\))18 b Fn(\000)f Fp(e)1042 3727 y Fo(i')1142 3760 y Fr(=)25 b Fp(O)s Fr(\()p Fp(\025)1398 3727 y Fm(2)1437 3760 y Fr(\),)30 b(and)e(b)m(y)h(Theorem)f(1.3,)59 b Fp(a)2456 3774 y Fm(1)2521 3760 y Fr(=)25 b Fp(o)p Fr(\()p Fp(\025)p Fr(\).)41 b(Therefore,)29 b Fn(j)p Fp(A)3379 3774 y Fm(11)3454 3760 y Fr(\()p Fp(z)t(;)15 b(E)5 b Fr(\))p Fn(j)27 b Fr(=)e Fp(o)p Fr(\(1\).)-236 3868 y(The)i(estimate)i Fn(j)p Fp(A)403 3882 y Fm(12)478 3868 y Fr(\()p Fp(z)t(;)15 b(E)5 b Fr(\))p Fn(j)27 b Fr(=)e Fp(o)p Fr(\(1\))k(follo)m(ws)e (directly)g(from)g(Theorem)g(1.3.)41 b(Both)29 b(estimates)f(are)h (uniform)c(in)i Fn(fj)p Fr(Im)15 b Fp(z)t Fn(j)26 b(\024)-236 3976 y Fp(y)s Fn(g)20 b(\002)g Fr(\()p Fp(c")p Fr(-neigh)m(b)s(orho)s (o)s(d)30 b(of)g Fp(I)7 b Fr(\).)41 b(As)31 b(w)m(e)f(ha)m(v)m(e)i (already)e(explained,)f(they)h(imply)e(that)j Fn(k)p Fp(A)p Fr(\()p Fp(z)t(;)15 b(')p Fr(\))p Fn(k)3198 3994 y Fo(y)r(;')p Fm(\()p Fo(I)5 b Fm(\))3424 3976 y Fr(=)25 b Fp(o)p Fr(\(1\).)-236 4254 y(8.4.)53 b Fu(Lo)s(cating)42 b(the)f(absolutely)g(con)m(tin)m(uous)i(sp)s(ectrum.)i Fr(The)35 b(length)g(of)h(the)g(in)m(terv)-5 b(al)35 b Fp(')p Fr(\()p Fp(I)7 b Fr(\))37 b(is)e(of)h(order)f(1)p Fp(=")p Fr(.)-236 4362 y(But,)i(Prop)s(osition)c(8.1)j(can)f(only)g(b)s (e)f(applied)f(to)j(the)g(mono)s(drom)m(y)e(equation)h(for)g Fp(')h Fr(in)d(an)j(in)m(terv)-5 b(al)34 b(of)h(\014nite)f(length)-236 4470 y(indep)s(enden)m(t)f(of)i Fp(")p Fr(.)56 b(So,)37 b(w)m(e)f(divide)d Fp(')p Fr(\()p Fp(I)7 b Fr(\))36 b(in)m(to)g Fp(L)1551 4484 y Fm(0)1590 4470 y Fr(\()p Fp(")p Fr(\))f(=)e Fp(O)s Fr(\(1)p Fp(=")p Fr(\))k(subin)m(terv)-5 b(als)33 b(of)j(unit)e(length)g(and)h(apply)f(Prop)s(osi-)-236 4577 y(tion)27 b(8.1)i(to)f(eac)m(h)h(of)f(them.)39 b(This)26 b(can)i(b)s(e)f(done)h(since)f(the)h(constan)m(ts)g(in)f(Prop)s (osition)e(8.1)k(dep)s(end)d(only)h(on)g(the)h(length)-236 4685 y(of)k(the)h(in)m(terv)-5 b(al)31 b(of)i Fp(')p Fr(,)g(on)f(the)h(size)f(of)g(the)h(band)e(of)i(analyticit)m(y)f(of)g Fp(A)g Fr(in)f Fp(z)37 b Fr(and)32 b(on)g(the)g(exp)s(onen)m(t)h Fp(\033)s Fr(.)46 b(In)32 b(result,)g(this)-236 4793 y(pro)m(v)m(es)f(that)g(if)-111 4948 y Fn(\017)42 b Fp(\033)33 b Fr(is)d(a)g(\014xed)g(n)m(um)m(b)s(er)f(in)g(\(0)p Fp(;)15 b Fr(1\),)-111 5056 y Fn(\017)42 b Fp(")30 b Fr(is)g(small)f(enough,)-111 5166 y Fn(\017)42 b Fp(')30 b Fr(is)g(outside)f(a)i(subset)835 5143 y(~)825 5166 y(\010)891 5180 y Fk(1)996 5166 y Fr(of)f Fp(')p Fr(\()p Fp(I)7 b Fr(\))32 b(of)e(measure)h Fp(O)s Fr(\()1882 5130 y Fm(1)p 1882 5145 36 4 v 1883 5197 a Fo(")1927 5166 y Fp(\025)1980 5133 y Fo(\033)r(=)p Fm(2)2098 5166 y Fr(\),)-111 5274 y Fn(\017)42 b Fp(h)30 b Fr(b)s(elongs)g(to)h(the)f (set)h Fp(H)870 5288 y Fo(\033)947 5274 y Fr(de\014ned)e(in)g(\(8.4\))r (,)-236 5429 y(then,)h(the)h(mono)s(drom)m(y)e(equation)i(\(7.3\))h (has)e(solutions)f(describ)s(ed)f(in)h(Prop)s(osition)f(8.1.)-236 5586 y Fi(The)k(sets)g Fp(D)j Fi(and)e Fp(H)489 5600 y Fo(\033)535 5586 y Fi(.)41 b Fr(De\014ne)29 b Fp(D)k Fr(as)d(the)g(set)g Fp(")c Fn(2)f Fr(\(0)p Fp(;)15 b Fr(1\))31 b(suc)m(h)f(that)g Fp(h)c Fr(=)2370 5550 y Fm(2)p Fo(\031)p 2370 5565 79 4 v 2392 5617 a(")2458 5586 y Fr(mo)s(d)14 b(1)30 b(b)s(elongs)f(to)h Fp(H)3234 5600 y Fo(\033)3281 5586 y Fr(.)40 b(Let)30 b(us)f(sho)m(w)h(that)-236 5693 y(mes)15 b(\()p Fp(D)s Fr(\))31 b(satis\014es)f(\(1.4\))q(.)41 b(One)30 b(has)978 5970 y(mes)15 b(\(\(0)p Fp(;)g(")p Fr(\))23 b Fn(n)e Fp(D)s Fr(\))k Fn(\024)1789 5856 y Fk(1)1758 5883 y Ff(X)1700 6085 y Fo(n)p Fm(=)p Fo(N)7 b Fm(\()p Fo(")p Fm(\))1963 5846 y Ff(Z)2014 6052 y Fm(\(0)p Fo(;)p Fm(1\))p Fk(n)p Fo(H)2251 6060 y Fd(\033)2437 5908 y Fr(2)p Fp(\031)p 2323 5949 329 4 v 2323 6032 a Fr(\()p Fp(h)21 b Fr(+)f Fp(n)p Fr(\))2612 6006 y Fm(2)2661 5970 y Fp(dh;)1854 6210 y Fm(26)p eop %%Page: 27 27 27 26 bop -236 241 a Fr(where)42 b Fp(N)10 b Fr(\()p Fp(")p Fr(\))43 b(is)e(equal)h(in)m(teger)h(part)f(of)h(2)p Fp(\031)s(=")p Fr(.)77 b(Recall)42 b(that)h Fp(\025)j Fr(=)e Fp(o)p Fr(\(exp)q(\()p Fn(\000)p Fp(S=")p Fr(\)\))g(where)d Fp(S)48 b Fr(is)41 b(p)s(ositiv)m(e)g(constan)m(t)-236 349 y(indep)s(enden)m(t)28 b(of)i Fp(")p Fr(.)42 b(Therefore,)344 603 y(mes)15 b Fn(f)p Fr(\(0)p Fp(;)g(")p Fr(\))23 b Fn(n)d Fp(D)s Fn(g)26 b(\024)f Fr(2)p Fp(\031)1290 489 y Fk(1)1260 516 y Ff(X)1202 718 y Fo(n)p Fm(=)p Fo(N)7 b Fm(\()p Fo(")p Fm(\))1499 541 y Fr(1)p 1475 582 95 4 v 1475 665 a Fp(n)1530 639 y Fm(2)1624 489 y Fk(1)1594 516 y Ff(X)1595 714 y Fo(k)r Fm(=1)1787 489 y Fo(k)1740 516 y Ff(X)1750 714 y Fo(l)q Fm(=0)1919 541 y Fr(1)p 1897 582 90 4 v 1897 665 a Fp(k)1947 639 y Fm(3)2012 603 y Fr(exp)2166 475 y Ff(\022)2233 603 y Fn(\000)2314 541 y Fp(\033)s(nS)p 2314 582 171 4 v 2348 665 a Fr(2)p Fp(\031)2494 475 y Ff(\023)990 930 y Fn(\024)25 b Fp(C)1261 816 y Fk(1)1231 844 y Ff(X)1172 1045 y Fo(n)p Fm(=)p Fo(N)7 b Fm(\()p Fo(")p Fm(\))1470 868 y Fr(1)p 1446 909 95 4 v 1446 992 a Fp(n)1501 966 y Fm(2)1565 930 y Fr(exp)1719 802 y Ff(\022)1786 930 y Fn(\000)1867 868 y Fp(\033)s(nS)p 1867 909 171 4 v 1902 992 a Fr(2)p Fp(\031)2047 802 y Ff(\023)2139 930 y Fn(\024)25 b Fp(C)2412 868 y Fr(1)p 2317 909 236 4 v 2317 992 a Fp(N)10 b Fr(\()p Fp(")p Fr(\))2512 966 y Fm(2)2577 930 y Fr(exp)2732 802 y Ff(\022)2798 930 y Fn(\000)2879 868 y Fp(\033)s(N)g Fr(\()p Fp(")p Fr(\))p Fp(S)p 2879 909 313 4 v 2985 992 a Fr(2)p Fp(\031)3201 802 y Ff(\023)3294 930 y Fn(\024)24 b Fp(C)7 b(")3503 892 y Fm(2)3543 930 y Fp(\025)3596 892 y Fo(\033)3643 930 y Fp(:)-236 1206 y Fr(This)28 b(pro)m(v)m(es)j(\(8.1\))r(.)-236 1368 y Fi(The)39 b(sets)150 1345 y Fr(~)140 1368 y(\010)206 1382 y Fk(1)319 1368 y Fi(and)g Fp(E)568 1382 y Fk(1)643 1368 y Fi(.)60 b Fr(No)m(w,)39 b(let)e Fp(E)1176 1382 y Fk(1)1286 1368 y Fr(=)f Fp(E)5 b Fr(\()1510 1345 y(~)1500 1368 y(\010)1566 1382 y Fk(1)1641 1368 y Fr(\))37 b(where)f(the)h(function)f Fp(E)5 b Fr(\()p Fp(')p Fr(\))38 b(is)e(the)h(in)m(v)m(erse)f(of)h Fp(')p Fr(\()p Fp(E)5 b Fr(\).)61 b(Clearly)-8 b(,)-236 1483 y(mes)15 b(\()p Fp(E)33 1497 y Fk(1)108 1483 y Fr(\))26 b(=)265 1410 y Ff(R)315 1498 y Fm(~)308 1515 y(\010)359 1523 y Fc(1)453 1447 y Fo(dE)p 453 1462 92 4 v 458 1514 a(d')555 1483 y Fp(d')p Fr(.)40 b(This,)26 b(the)h(estimate)g(for)g (mes)15 b(\()1817 1460 y(~)1807 1483 y(\010)1873 1497 y Fk(1)1948 1483 y Fr(\))27 b(and)f(the)h(asymptotics)g(\(8.10\))i (imply)c(that)i(mes)15 b(\()p Fp(E)3809 1497 y Fk(1)3884 1483 y Fr(\))26 b(=)-236 1618 y Fp(O)s Fr(\()p Fp(\025)-76 1585 y Fo(\033)r(=)p Fm(2)41 1618 y Fr(\))-236 1774 y Fi(The)39 b(absolutely)g(c)-5 b(ontinuous)40 b(sp)-5 b(e)g(ctrum.)61 b Fr(W)-8 b(e)39 b(ha)m(v)m(e)f(sho)m(wn)e(that,)j (under)c(the)j(assumptions)d(of)i(Theorem)f(8.1,)k(equa-)-236 1881 y(tion)31 b(\(7.3\))i(has)e(b)s(ounded)e(solutions)h(for)i Fp(E)g Fn(2)27 b Fp(I)h Fn(n)21 b Fp(E)1615 1895 y Fk(1)1690 1881 y Fr(.)44 b(Let)32 b Fp(\037)p Fr(\()p Fp(x)p Fr(\))g(b)s(e)e(a)i (b)s(ounded)e(v)m(ector)j(solution)d(of)h(equation)h(\(7.3\))q(,)-236 1989 y(then,)j Fp(\037)58 2004 y Fo(k)131 1989 y Fr(=)c Fp(\037)p Fr(\()p Fp(z)c Fr(+)c Fp(hk)s Fr(\))34 b(is)f(a)i(b)s(ounded) d(v)m(ector)j(solution)e(of)h(the)g(mono)s(drom)m(y)f(equation.)52 b(Therefore,)34 b(the)h(Ly)m(apuno)m(v)-236 2097 y(exp)s(onen)m(t)i (for)h(the)g(mono)s(drom)m(y)f(equation)g(is)g(zero)h(on)g Fp(I)32 b Fn(n)25 b Fp(E)2002 2111 y Fk(1)2077 2097 y Fr(.)62 b(Corollary)37 b(1.2)h(then)g(implies)d(that)j(the)g(Ly)m (apuno)m(v)-236 2205 y(exp)s(onen)m(t)27 b(for)h(the)g(family)e(of)h (equations)h(\(0.1\))h(is)e(zero)h(on)g Fp(I)22 b Fn(n)15 b Fp(E)2028 2219 y Fk(1)2103 2205 y Fr(.)39 b(Applying)26 b(the)h(Ishii-P)m(astur-Kotani)f(Theorem,)i(w)m(e)-236 2313 y(that)j(the)f(essen)m(tial)g(closure)g(of)h Fp(I)c Fn(n)21 b Fp(E)1087 2327 y Fk(1)1192 2313 y Fr(is)29 b(in)g(the)i(absolutely)e(con)m(tin)m(uous)h(sp)s(ectrum)f(of)i(the)g (equation)f(family)g(\(0.1\))r(.)-236 2518 y(8.5.)53 b Fu(Blo)s(c)m(h-Flo)s(quet)42 b(solutions)g(for)f Fr(\(0.1\))r Fu(.)k Fr(T)-8 b(o)36 b(complete)g(the)g(pro)s(of)f(of)h(Theorem)f (8.1,)j(w)m(e)e(ha)m(v)m(e)h(only)e(to)h(c)m(hec)m(k)-236 2626 y(the)30 b(statemen)m(ts)i(on)f(the)f(solutions)f Fp( )1100 2640 y Fk(\006)1159 2626 y Fr(.)41 b(W)-8 b(e)32 b(break)e(the)g(pro)s(of)g(in)m(to)g(a)h(few)f(steps.)-236 2781 y Fu(1.)82 b Fr(The)43 b(mono)s(drom)m(y)h(matrix)f Fp(M)10 b Fr(\()p Fp(x;)15 b(z)t Fr(\))45 b(describ)s(ed)d(in)h (Theorem)h(1.3)h(corresp)s(onds)e(to)h(\()p Fp( )3129 2795 y Fk(\006)3189 2781 y Fr(\),)k(a)c(consisten)m(t)h(basis)-236 2889 y(constructed)34 b(in)g(terms)g(of)g(the)h(functions)e Fp(f)10 b Fr(\()p Fp(x;)15 b(\020)7 b Fr(\))34 b(and)g Fp(f)1824 2856 y Fk(\003)1863 2889 y Fr(\()p Fp(x;)15 b(\020)7 b Fr(\),)36 b(solutions)c(of)j(equation)f(\(3.2\))r(.)53 b(More)35 b(precisely)-8 b(,)34 b Fp( )3956 2903 y Fm(+)-236 2997 y Fr(and)c Fp( )0 3011 y Fk(\000)89 2997 y Fr(are)h(related)f(to)h Fp(f)40 b Fr(and)30 b Fp(f)972 2964 y Fk(\003)1041 2997 y Fr(b)m(y)g(the)g(c)m(hange)i(of)f(v)-5 b(ariables)29 b(describ)s(ed)f(in)h(the)h(b)s(eginning)e(of)j(section)f(6)796 3164 y Fp( )855 3178 y Fm(+)914 3164 y Fr(\()p Fp(x;)15 b(z)t Fr(\))27 b(=)e Fp(f)10 b Fr(\()p Fp(x)20 b Fn(\000)g Fp(z)t(;)15 b(z)t(=")p Fr(\))32 b(and)e Fp( )2020 3178 y Fk(\000)2079 3164 y Fr(\()p Fp(x;)15 b(z)t Fr(\))27 b(=)e Fp(f)2465 3126 y Fk(\003)2503 3164 y Fr(\()p Fp(x)c Fn(\000)f Fp(z)t(;)15 b(z)t(=")p Fr(\))p Fp(:)-236 3332 y Fr(The)30 b(functions)f Fp(\020)i Fn(7!)25 b Fp(f)40 b Fr(and)30 b Fp(\020)i Fn(7!)25 b Fp(f)1035 3299 y Fk(\003)1104 3332 y Fr(are)31 b(analytic)e(in)h(a)g(constan)m(t)i(neigh)m(b)s(orho)s (o)s(d)c(of)i(the)h(real)f(axis.)-236 3487 y Fu(2.)78 b Fr(Consider)41 b(the)i(\\con)m(tin)m(uous")h(mono)s(drom)m(y)e (equation)h(\(7.3\))i(with)c(the)j(mono)s(drom)m(y)e(matrix)g Fp(M)53 b Fr(describ)s(ed)41 b(in)-236 3595 y(Theorem)36 b(1.3.)60 b(Let)37 b Fp(\037)580 3609 y Fk(\000)675 3595 y Fr(and)f Fp(\037)915 3609 y Fm(+)1011 3595 y Fr(b)s(e)f(the)i (\014rst)f(and)g(second)g(columns)f(of)i(the)g(matrix)e(solution)g(to)j (the)e(mono)s(drom)m(y)-236 3703 y(equation)f(\(7.3\))j(constructed)e (in)f(Prop)s(osition)e(8.1.)59 b(By)37 b(\(8.6\))r(,)g(they)f(are)g (Blo)s(c)m(h-Flo)s(quet)g(solutions)e(of)43 b(\(7.3\))r(.)57 b(Their)-236 3811 y(quasi-momen)m(tum)34 b(are)h(equal)f(to)i Fn(\006)p Fp(')1127 3825 y Fk(1)1202 3811 y Fr(\()p Fp(E)5 b Fr(\))34 b(=)e Fn(\006)p Fp(')1611 3825 y Fk(1)1686 3811 y Fr(\()p Fp(')p Fr(\()p Fp(E)5 b Fr(\)\).)56 b(Their)33 b(p)s(erio)s(dic)f(comp)s(onen)m(ts,)37 b Fp(U)3237 3825 y Fk(\006)3296 3811 y Fr(,)f(are)f(just)f(the)h(\014rst)-236 3919 y(and)i(second)g(columns)f(of)i(the)f(matrix)g Fp(U)10 b Fr(\()p Fp(z)t(;)15 b(E)5 b Fr(\))38 b(=)f Fp(U)10 b Fr(\()p Fp(z)t(;)15 b(')p Fr(\()p Fp(E)5 b Fr(\)\))40 b(in)c(\(8.6\))r(.)62 b(The)37 b(function)f Fp(E)42 b Fn(7!)37 b Fp(')p Fr(\()p Fp(E)5 b Fr(\))38 b(is)f(de\014ned)f(in)-236 4027 y(section)30 b(8.3.2.)43 b(Hence,)1333 4199 y Fp(\037)1390 4213 y Fk(\006)1449 4199 y Fr(\()p Fp(z)t Fr(\))27 b(=)e Fp(e)1730 4162 y Fk(\007)p Fo(i')1855 4170 y Fc(1)1920 4162 y Fm(\()p Fo(E)t Fm(\))p Fo(z)2070 4199 y Fp(U)2132 4213 y Fk(\006)2191 4199 y Fr(\()p Fp(z)t(;)15 b(E)5 b Fr(\))p Fp(:)-2680 b Fr(\(8.12\))-236 4371 y(The)25 b(functions)g(\()p Fp(z)t(;)15 b(E)5 b Fr(\))27 b Fn(7!)e Fp(U)767 4385 y Fk(\006)826 4371 y Fr(\()p Fp(z)t(;)15 b(E)5 b Fr(\))28 b(are)e(analytic)g(for)g Fp(z)k Fr(in)25 b(a)i(constan)m(t)g(neigh)m(b)s(orho)s(o)s(d)d(of)i(the)g(real)g(axis)f (and)h(Lipsc)m(hitz)-236 4479 y(con)m(tin)m(uous)k(for)g Fp(E)36 b Fr(in)29 b Fp(I)e Fn(n)21 b Fp(E)767 4493 y Fk(1)841 4479 y Fr(.)41 b(As)30 b Fp(U)36 b Fn(2)25 b(M)p Fr(,)30 b(one)h(has)1316 4701 y Fp(U)1378 4715 y Fm(+)1463 4701 y Fr(=)25 b Fp(\033)1611 4715 y Fm(1)1650 4701 y Fp(U)1722 4664 y Fk(\003)1712 4724 y(\000)1771 4701 y Fp(;)106 b(\033)1954 4715 y Fm(1)2019 4701 y Fr(=)2115 4573 y Ff(\022)2182 4646 y Fr(0)83 b(1)2182 4754 y(1)g(0)2356 4573 y Ff(\023)2438 4701 y Fp(:)-2699 b Fr(\(8.13\))-236 4918 y Fu(3.)40 b Fr(De\014ne)31 b(the)g(v)m(ectors)261 5123 y(\011)332 5137 y Fk(\006)391 5123 y Fr(\()p Fp(x;)15 b(z)t Fr(\))26 b(=)f Fp(iF)13 b Fr(\()p Fp(x)21 b Fn(\000)f Fp(z)t(;)15 b(z)t(=")p Fr(\))p Fp(\033)s(U)1393 5137 y Fk(\006)1455 5123 y Fr(\()p Fp(z)t Fr(\))31 b(where)f Fp(F)13 b Fr(\()p Fp(x;)i(z)t Fr(\))26 b(=)2266 4994 y Ff(\022)2354 5061 y Fp(f)127 b(f)2581 5028 y Fk(\003)2348 5140 y Fo(d)-12 b(f)p 2343 5161 77 4 v 2343 5213 a(dx)2522 5140 y(d)g(f)2587 5117 y Fc(\003)p 2522 5161 102 4 v 2535 5213 a Fo(dx)2634 4994 y Ff(\023)2746 5123 y Fr(and)30 b Fp(\033)f Fr(=)3100 4994 y Ff(\022)3167 5068 y Fr(0)83 b Fn(\000)p Fr(1)3167 5175 y(1)118 b(0)3411 4994 y Ff(\023)3493 5123 y Fp(:)-3754 b Fr(\(8.14\))-236 5350 y(By)29 b(Theorem)f(7.2,)j Fp( )517 5364 y Fm(+)605 5350 y Fr(\(resp.)40 b Fp( )927 5364 y Fk(\000)986 5350 y Fr(\),)29 b(the)g(\014rst)f(comp)s(onen)m(t)h (of)g(\011)2052 5364 y Fm(+)2140 5350 y Fr(\(resp.)40 b(\011)2474 5364 y Fm(+)2532 5350 y Fr(\))29 b(is)f(a)h(Blo)s(c)m (h-Flo)s(quet)g(solution)f(of)36 b(\(0.1\))q(.)-236 5458 y(Moreo)m(v)m(er,)d(b)m(y)d(the)g(same)h(theorem)g(and)f(\(8.12\))r(,)h (\011)1589 5472 y Fk(\006)1678 5458 y Fr(admit)e(the)i(represen)m (tations)1239 5635 y(\011)1310 5649 y Fk(\006)1369 5635 y Fr(\()p Fp(x;)15 b(z)t Fr(\))26 b(=)f Fp(e)1741 5598 y Fk(\006)p Fo(ip)p Fm(\()p Fo(E)t Fm(\))p Fo(x)2010 5635 y Fp(P)2068 5649 y Fk(\006)2128 5635 y Fr(\()p Fp(x)20 b Fn(\000)g Fp(z)t(;)15 b(x)p Fr(\))p Fp(;)-2760 b Fr(\(8.15\))-236 5864 y(where)35 b Fp(p)p Fr(\()p Fp(E)5 b Fr(\))36 b(=)375 5802 y Fp("h)p 372 5843 101 4 v 372 5926 a Fr(2)p Fp(\031)482 5864 y(')541 5878 y Fk(1)617 5864 y Fr(\()p Fp(E)5 b Fr(\))37 b(and)e(\()p Fp(x;)15 b(y)s Fr(\))36 b Fn(7!)f Fp(P)1408 5878 y Fk(\006)1467 5864 y Fr(\()p Fp(x;)15 b(y)s Fr(\))38 b(are)e(1-p)s(erio)s(dic)e(in)h Fp(x)h Fr(and)g(2)p Fp(\031)s(=")p Fr(-p)s(erio)s(dic)f Fp(y)s Fr(.)58 b(This)35 b(pro)m(v)m(es)h(the)-236 6003 y(represen)m(tation)30 b(\(1.5\))i(and)e(the)h(prop)s(erties)e(of)h Fp(p)p Fr(\()p Fp(E)5 b Fr(\))31 b(describ)s(ed)d(in)h(Theorem)i(1.1.)-236 6110 y(W)-8 b(e)31 b(also)g(see)g(that)g Fp(P)505 6124 y Fk(\006)594 6110 y Fr(has)f(the)h(announced)f(p)s(erio)s(dicit)m(y)d (prop)s(erties)i(and)h(is)f(Lipsc)m(hitz)g(con)m(tin)m(uous)h(in)g Fp(E)5 b Fr(.)1854 6210 y Fm(27)p eop %%Page: 28 28 28 27 bop -236 241 a Fr(Let)31 b(discuss)d(the)j(regularit)m(y)e(of)i Fp(P)964 255 y Fk(\006)1054 241 y Fr(in)e(\()p Fp(x;)15 b(y)s Fr(\).)41 b(By)32 b(\(8.14\))h(and)d(\(8.15\))r(,)g(one)h(has)950 397 y Fp(P)1008 411 y Fk(\006)1068 397 y Fr(\()p Fp(x;)15 b(y)s Fr(\))26 b(=)f Fp(ie)1473 359 y Fk(\007)p Fo(ip)p Fm(\()p Fo(E)t Fm(\))p Fo(x)1742 397 y Fp(F)13 b Fr(\()p Fp(x;)i Fr(\()p Fp(y)24 b Fn(\000)c Fp(x)p Fr(\))p Fp(=")p Fr(\))p Fp(\033)s(U)2461 411 y Fk(\006)2522 397 y Fr(\()p Fp(y)k Fn(\000)19 b Fp(x)p Fr(\))p Fp(:)-3064 b Fr(\(8.16\))-236 555 y(This)28 b(immediately)g(implies)g(that)j Fp(P)1052 569 y Fk(\006)1141 555 y Fr(are)f Fp(H)1375 522 y Fm(2)1368 583 y(lo)r(c)1460 555 y Fr(-functions)f(in)g Fp(x)h Fr(and)g(are)g (analytic)g(in)f Fp(y)j Fr(in)d(a)i(constan)m(t)g(neigh)m(b)s(orho)s(o) s(d)-236 663 y(of)f(the)h(real)f(line.)-236 818 y(Finally)-8 b(,)29 b(c)m(hec)m(k)j(that)p 533 745 118 4 v 31 w Fp(P)591 832 y Fm(+)675 818 y Fr(=)25 b Fp(P)829 832 y Fk(\000)919 818 y Fr(for)30 b Fp(z)35 b Fr(real.)40 b(By)32 b(\(8.13\))r(,)f (\(8.16\))i(and)c(the)i(de\014nition)d(of)i Fp(F)13 b Fr(,)31 b(one)g(has)p 1027 896 V 1027 969 a Fp(P)1085 983 y Fm(+)1169 969 y Fr(=)25 b Fn(\000)p Fp(ie)1409 931 y Fo(ipx)1513 969 y Fp(F)13 b(\033)1636 983 y Fm(1)1676 969 y Fp(\033)s(\033)1783 983 y Fm(1)1822 969 y Fp(U)1884 983 y Fk(\000)1969 969 y Fr(=)24 b Fp(ie)2137 931 y Fo(ipx)2242 969 y Fp(F)13 b(\033)s(U)2430 983 y Fk(\000)2514 969 y Fr(=)25 b Fp(P)2668 983 y Fk(\000)2728 969 y Fp(:)-236 1114 y Fr(This)j(completes)j(the)g(pro)s(of)e(of)i(Theorem)f(1.1.)p 3950 1114 4 62 v 3954 1056 55 4 v 3954 1114 V 4008 1114 4 62 v 1625 1298 a Fq(References)-198 1443 y Ft([1])40 b(J.)35 b(Bellissard,)40 b(R.)34 b(Lima,)j(and)e(D.)g(T)-6 b(estard.)35 b(Metal-insulator)i(transition)e(for)h(the)e(Almost)h (Mathieu)g(mo)r(del.)g Fj(Communic)l(ations)h(in)-78 1534 y(Mathematic)l(al)28 b(Physics)p Ft(,)f(88:207{234,)j(1983.)-198 1626 y([2])40 b(V.)20 b(Buslaev.)j(On)d(sp)r(ectral)i(prop)r(erties)f (of)h(adiabatically)h(p)r(erturb)r(ed)d(Sc)n(hr\177)-38 b(odinger)21 b(op)r(erators)h(with)g(p)r(erio)r(dic)g(p)r(oten)n (tials.)g(In)e Fj(S)n(\023)-37 b(eminair)l(es)-78 1717 y(d')n(\023)g(equations)28 b(aux)h(d)n(\023)-37 b(eriv)n(\023)g(ees)29 b(p)l(artiel)t(les)p Ft(,)e(v)n(olume)d(XVI)r(I)r(I,)h(P)n(alaiseau,)j (1991.)f(Ecole)g(P)n(olytec)n(hnique.)-198 1808 y([3])40 b(V.)20 b(Buslaev)i(and)e(A.)h(F)-6 b(edoto)n(v.)20 b(The)h(complex)f (WKB)h(metho)r(d)f(for)h(Harp)r(er's)h(equation.)f(Preprin)n(t,)h (Mittag-Le\017er)g(Institute,)f(Sto)r(c)n(holm,)-78 1900 y(1993.)-198 1991 y([4])40 b(V.)33 b(Buslaev)g(and)g(A.)g(F)-6 b(edoto)n(v.)33 b(Mono)r(dromization)i(and)d(Harp)r(er)h(equation.)h (In:)49 b Fj(Equations)35 b(aux)h(d)n(\023)-37 b(eriv)n(\023)g(ees)35 b(p)l(artiel)t(les)p Ft(,)i(1994,)f(Ecole)-78 2082 y(Polytec)n(hnique,) 25 b(Paris,)i(France.)-198 2174 y([5])40 b(V.)25 b(Buslaev)i(and)e(A.)g (F)-6 b(edoto)n(v.)26 b(Blo)r(c)n(h)g(solutions)h(of)g(di\013erence)e (equations.)i Fj(St)h(Petersbur)l(g)i(Math.)e(Journal)p Ft(,)f(7:561{594,)i(1996.)-198 2265 y([6])40 b(E.)22 b(I.)g(Dinaburg)g(and)g(Ja.)h(G.)f(Sina)-9 b(\025)-30 b(\020.)24 b(The)e(one-dimensional)g(Sc)n(hr\177)-38 b(odinger)22 b(equation)g(with)h(quasip)r(erio)r(dic)g(p)r(oten)n (tial.)g Fj(F)-6 b(unkcional.)24 b(A)n(nal.)-78 2356 y(i)j(Prilo)l(\024)-35 b(zen.)p Ft(,)26 b(9\(4\):8{21,)i(1975.)-198 2448 y([7])40 b(M.)26 b(Eastham.)g Fj(The)i(sp)l(e)l(ctr)l(al)h(the)l (ory)g(of)e(p)l(erio)l(dic)i(di\013er)l(ential)f(op)l(er)l(ators)p Ft(.)h(Scottish)c(Academic)g(Press,)i(Edin)n(burgh,)e(1973.)-198 2539 y([8])40 b(L.)31 b(H.)f(Eliasson.)j(Flo)r(quet)e(solutions)h(for)f (the)f(1-dimensional)h(quasi-p)r(erio)r(dic)g(Sc)n(hr\177)-38 b(odinger)31 b(equation.)g Fj(Communic)l(ations)i(in)e(Mathe-)-78 2630 y(matic)l(al)c(Physics)p Ft(,)g(146:447{482,)j(1992.)-198 2722 y([9])40 b(A.)22 b(F)-6 b(edoto)n(v)23 b(and)g(F.)g(Klopp.)g (Anderson)f(transitions)j(for)e(a)h(family)f(of)g(almost)h(p)r(erio)r (dic)g(Sc)n(hr\177)-38 b(odinger)23 b(equations)g(in)g(the)f(adiabatic) i(case.)-78 2813 y(Preprin)n(t,)h(Univ)n(ersit)n(\023)-36 b(e)26 b(P)n(aris-Nord,)g(2001.)-236 2904 y([10])40 b(A.)25 b(F)-6 b(edoto)n(v)25 b(and)h(F.)g(Klopp.)f(A)h(complex)e(WKB)i (analysis)h(for)f(adiabatic)h(problems.)f Fj(Asymptotic)j(A)n(nalysis)p Ft(,)d(2001.)i(to)d(app)r(ear.)-236 2996 y([11])40 b(V.)27 b(Marc)n(henk)n(o)h(and)f(I.)h(Ostro)n(vskii.)g(A)f(c)n (haracterization)i(of)g(the)e(sp)r(ectrum)f(of)j(Hill's)f(equation.)g Fj(Math.)i(USSR)f(Sb)l(ornik)p Ft(,)h(26:493{554,)-78 3087 y(1975.)-236 3178 y([12])40 b(H.)25 b(McKean)h(and)g(E.)g(T)-6 b(rub)r(o)n(witz.)26 b(The)g(sp)r(ectrum)f(of)h(Hill's)h(equation.)f Fj(Inventiones)j(Mathematic)l(ae)p Ft(,)e(30:217{274,)j(1975.)-236 3270 y([13])40 b(L.)26 b(P)n(astur)g(and)f(A.)g(Figotin.)i Fj(Sp)l(e)l(ctr)l(a)j(of)d(R)l(andom)h(and)g(A)n(lmost-Perio)l(dic)h (Op)l(er)l(ators)p Ft(.)f(Springer)e(V)-6 b(erlag,)26 b(Berlin,)h(1992.)-236 3361 y([14])40 b(E.C.)27 b(Titsc)n(hmarc)n(h.)f Fj(Eigenfunction)j(exp)l(ansions)h(asso)l(ciate)l(d)g(with)e(se)l(c)l (ond-or)l(der)k(di\013er)l(ential)d(e)l(quations.)g(Part)g(II)p Ft(.)c(Clarendon)i(Press,)-78 3452 y(Oxford,)e(1958.)-236 3544 y([15])40 b(M.)d(Wilkinson.)g(Critical)h(prop)r(erties)f(of)g (electron)g(eigenstates)h(in)f(incommensurate)e(systems.)h Fj(Pr)l(o)l(c.)i(R)l(oy.)f(So)l(c.)g(L)l(ondon)i(Ser.)e(A)p Ft(,)-78 3635 y(391\(1801\):305{350,)31 b(1984.)-24 3806 y(\(Alexander)19 b(F)-6 b(edoto)n(v\))19 b Fs(Dep)-5 b(ar)g(tement)20 b(of)j(Ma)-5 b(thema)g(tical)21 b(Physics,)i(St)e (Petersbur)n(g)i(St)-5 b(a)g(te)21 b(University,)h(1,)h(Uliano)n(vska-) -236 3898 y(ja,)k(198904)j(St)f(Petersbur)n(g-Petr)n(od)n(v)n(orets,)h (R)n(ussia)-24 3989 y Fj(E-mail)d(addr)l(ess)6 b Ft(:)37 b Fa(fedotov@mph.phys.spbu.ru)-24 4144 y Ft(\(F)-6 b(r)n(\023)-36 b(ed)n(\023)g(eric)21 b(Klopp\))g Fs(D)615 4138 y(\023)615 4144 y(ep)-5 b(ar)g(tement)21 b(de)i(Ma)-5 b(th)1392 4138 y(\023)1392 4144 y(ema)g(tique,)23 b(Institut)g(Galil)2323 4138 y(\023)2323 4144 y(ee,)g(U.M.R)g(7539)h(C.N.R.S,)e(Universit)3616 4138 y(\023)3616 4144 y(e)f(de)i(P)-7 b(aris-)-236 4235 y(Nord,)28 b(A)-9 b(venue)27 b(J.-B.)h(Cl)618 4229 y(\023)618 4235 y(ement,)f(F-93430)k(Villet)-5 b(aneuse,)27 b(France)-24 4327 y Fj(E-mail)g(addr)l(ess)6 b Ft(:)37 b Fa (klopp@math.univ-paris13.fr)1854 6210 y Fm(28)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0106220917660--