Content-Type: multipart/mixed; boundary="-------------0308251001228" This is a multi-part message in MIME format. ---------------0308251001228 Content-Type: text/plain; name="03-383.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-383.keywords" absolutely continuous spectrum, Schr\"odinger operators, slowly decaying potentials, trace formulae ---------------0308251001228 Content-Type: application/postscript; name="schr2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="schr2.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: schr2.dvi %%Pages: 26 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Times-Bold Times-Roman Times-Italic Courier %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o schr2.ps schr2.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.08.25:1648 %%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: 8r.enc % @@psencodingfile@{ % author = "S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry", % version = "0.6", % date = "1 July 1998", % filename = "8r.enc", % email = "tex-fonts@@tug.org", % docstring = "Encoding for TrueType or Type 1 fonts % to be used with TeX." % @} % % Idea is to have all the characters normally included in Type 1 fonts % available for typesetting. This is effectively the characters in Adobe % Standard Encoding + ISO Latin 1 + extra characters from Lucida. % % Character code assignments were made as follows: % % (1) the Windows ANSI characters are almost all in their Windows ANSI % positions, because some Windows users cannot easily reencode the % fonts, and it makes no difference on other systems. The only Windows % ANSI characters not available are those that make no sense for % typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen % (173). quotesingle and grave are moved just because it's such an % irritation not having them in TeX positions. % % (2) Remaining characters are assigned arbitrarily to the lower part % of the range, avoiding 0, 10 and 13 in case we meet dumb software. % % (3) Y&Y Lucida Bright includes some extra text characters; in the % hopes that other PostScript fonts, perhaps created for public % consumption, will include them, they are included starting at 0x12. % % (4) Remaining positions left undefined are for use in (hopefully) % upward-compatible revisions, if someday more characters are generally % available. % % (5) hyphen appears twice for compatibility with both % ASCII and Windows. % /TeXBase1Encoding [ % 0x00 (encoded characters from Adobe Standard not in Windows 3.1) /.notdef /dotaccent /fi /fl /fraction /hungarumlaut /Lslash /lslash /ogonek /ring /.notdef /breve /minus /.notdef % These are the only two remaining unencoded characters, so may as % well include them. /Zcaron /zcaron % 0x10 /caron /dotlessi % (unusual TeX characters available in, e.g., Lucida Bright) /dotlessj /ff /ffi /ffl /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef % very contentious; it's so painful not having quoteleft and quoteright % at 96 and 145 that we move the things normally found there to here. /grave /quotesingle % 0x20 (ASCII begins) /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash % 0x30 /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question % 0x40 /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O % 0x50 /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore % 0x60 /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o % 0x70 /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /.notdef % rubout; ASCII ends % 0x80 /.notdef /.notdef /quotesinglbase /florin /quotedblbase /ellipsis /dagger /daggerdbl /circumflex /perthousand /Scaron /guilsinglleft /OE /.notdef /.notdef /.notdef % 0x90 /.notdef /.notdef /.notdef /quotedblleft /quotedblright /bullet /endash /emdash /tilde /trademark /scaron /guilsinglright /oe /.notdef /.notdef /Ydieresis % 0xA0 /.notdef % nobreakspace /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen % Y&Y (also at 45); Windows' softhyphen /registered /macron % 0xD0 /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown % 0xC0 /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis % 0xD0 /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls % 0xE0 /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis % 0xF0 /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis ] def %%EndProcSet %%BeginProcSet: texps.pro %! TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]/Metrics exch def dict begin Encoding{exch dup type/integertype ne{pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def} ifelse}forall Metrics/Metrics currentdict end def[2 index currentdict end definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{ dup sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[exch{dup CharStrings exch known not{pop/.notdef/Encoding true def} if}forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def} def end %%EndProcSet TeXDict begin 39158280 55380996 1000 600 600 (schr2.dvi) @start /Fa 137[50 50 50 50 50 1[50 50 50 50 50 50 2[50 1[50 50 50 1[50 50 32[50 17[50 1[50 44[{TeXBase1Encoding ReEncodeFont} 20 83.022 /Courier rf /Fb 198[42 42 42 42 42 42 42 42 42 42 48[{TeXBase1Encoding ReEncodeFont}10 83.022 /Times-Bold rf %DVIPSBitmapFont: Fc msam10 12 1 /Fc 1 4 df<007FBA1280BB12C0A300F0CB1203B3B3B3A6BBFCA36C198042447BC34D>3 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd eufm8 8 2 /Fd 2 84 df<4AB4FC020F13C0027F7FECF01F903903C007F0EB078090380F0003131E13 1C133C137CA213FCA27FA216FF017FEB01FCED00C06E90C7FC133FA26D7EA2130F801307 A21303A25C5C91C9FC010615805B1338D801F01401120FD81FFEEC0300D9FFC05B4801F8 5B26787FFF137FD8701FEBFFFEEAE00748C65C021F5B4801035BC8EA7FC029307DAD2E> 76 D83 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe eufm10 12 4 /Fe 4 85 df66 D76 D<1638ED03FE92261FFF80140892B500E0140C020314F8020F 02FE1418DA3F016D7E91267C007F01E01330D901F0010F01F81370D903C001039038FF01 E049486DECFFC049C87E013E031F1480013C0307EBFE00496F5B49030013F00001EF3FC0 48480407C7FC95C8FC484813784A5A000F495A49485A001F495A141F003F133F1300147F 48EF3FE06F903807FFFC6F90B6FCDBF01F15804891B812C06E17E0A26EDAF00713F06E90 38FC0001020390C8EA7FF86DCA123FA2191F190FA27F1907127F7F1AF0A27F123F6D18E0 A26C6CEF0FC07F000F19806D171F6C6D17006E163E6C7F6E5E6C01FC5E6C6D4B5A6D6C6C EC07C06D01E04A5A6D01F8027FC7FC6D9039FF8007FC010391B512F06D16C06D6C4AC8FC 020F14F0020049C9FC46487AC453>83 D<19011903DA1FFF150649B500FC140E0107DAFF E0131C011F03FF13F849EEFFF090B9FC2701FC003F15E0D803E0010115C0D80780D9003F 1480000EC8FC000C92397C3FFC00001C4BC8FC484A5A4B5A485D4B5A4BC9FC00F05C151E 6C143E5D7E15FC7E38FF8001D87FC07F1380EB0003003E80003C811218C76C7F826E7F15 7F826F7E151F82150F6F7E816F7FA281167FA270C8FCA3163E163C167C16785E020E495B 91263F81C0EB0380DA7FC3EC07009126FFC780130E4901EFC7123C4901FE14F8499039FF E003F04991B55AD91F805D90263E001F5C0178010791C7FC01E0010113FC48486D6C5A6C 48EC1FE090C8EA038040487CC543>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmr6 6 5 /Ff 5 53 df<13FF000313C0380781E0380F00F0001E137848133CA248131EA400F8131F AD0078131EA2007C133E003C133CA26C13786C13F0380781E03803FFC0C6130018227DA0 1E>48 D<13E01201120712FF12F91201B3A7487EB512C0A212217AA01E>II<13FF000313C0380F03E0381C00F014F8003E13FC147CA2 001E13FC120CC712F8A2EB01F0EB03E0EB0FC03801FF00A2380003E0EB00F01478147C14 3E143F1230127812FCA2143E48137E0060137C003813F8381E03F0380FFFC00001130018 227DA01E>I<14E01301A213031307A2130D131D13391331136113E113C1EA01811203EA 07011206120C121C12181230127012E0B6FCA2380001E0A6EB03F0EB3FFFA218227DA11E >I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmsy6 6 3 /Fg 3 50 df0 D<136013701360A20040132000E0137038F861 F0387E67E0381FFF803807FE00EA00F0EA07FE381FFF80387E67E038F861F038E0607000 40132000001300A21370136014157B9620>3 D<01FEEC0FE02603FFC0EB3FF8000F01F0 EBFE3E3B1F0FF801F0073C3C01FC07C003803B3000FE0F00010070D93F1EEB00C00060EB 1F9C00E0D90FF81460485C14076E7E6E7E81020315E00060D9073F14C091390F1F80016C 90261E0FE01380003890397C07F0073C1C01F003FE1F003B0F8FE001FFFE3B03FF80007F F8C648C7EA0FE033177C953D>49 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh msbm8 8 3 /Fh 3 84 df81 DI<90383FE00C3901FFFC1C3907F03FFC390EE007EC391DC0038C393B80 01CC0073C712EC0063147C153C12E300C3141C7F00C1140C7FEAC0E01378D8E01E90C7FC 38600780387001F03838007C6CEB0F806CEB03C06CEB00F0D803C01338D800F07F013E7F 90380F8007903801E003D900781380EC1C0100C0010E13C0EC0700EC0380EC01C014006C 14E015606C146116806C146300DCECE30000CE5C00C714CE39C38001FC39DFC003F039FF F00FE026E07FFFC7FC38C00FF822307EAE31>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fi cmmi6 6 6 /Fi 6 113 df18 D100 D<1418143C147CA214381400A7EB0780 EB1FE01338EB60F013C0A2EA0180A2380001E0A4EB03C0A4EB0780A4EB0F00A4131EA212 38EA783CEAF8381378EA70F0EA7FC0001FC7FC162D81A119>106 D<13F8EA0FF0A21200A2485AA4485AA43807801E147FEB81C3EB8387380F060F495A1318 EB700E4848C7FCA213FCEA1E7EEA3C0F80EB0781158039780F0300A21402EB070600F013 8CEB03F8386000F019247CA221>I<000F017E13FC3A1F81FF83FF3B31C383C707803A61 EE03CC039026EC01F813C0D8C1F813F013F001E013E00003903903C0078013C0A2EE0F00 3907800780A2EE1E041706270F000F00130C163C1718A2001E011EEB1C70EE1FE0000C01 0CEB07802F177D9536>109 D<3801E01F3903F07FC0390639C1E0390C3F80F0EB3E0000 1814F8013C137815F8C65AA49038F001F0A3EC03E0D801E013C0EBF00715809038F80F00 3803DC3CEBCFF8EBC7E001C0C7FC485AA448C8FCA2EA7FF012FF1D20809520>112 D E %EndDVIPSBitmapFont /Fj 166[54 2[54 54 46 42 50 1[42 54 54 66 46 54 1[25 2[42 46 54 50 50 54 7[37 37 37 37 37 37 37 37 37 37 3[19 44[{TeXBase1Encoding ReEncodeFont}30 74.7198 /Times-Roman rf /Fk 87[28 45[32 37 37 55 37 42 23 32 32 42 42 42 42 60 23 37 23 23 42 42 23 37 42 37 42 42 9[69 1[60 46 42 51 1[51 60 55 69 46 1[37 28 60 60 51 51 60 55 51 51 6[28 11[21 28 21 38[28 2[42 2[{TeXBase1Encoding ReEncodeFont}53 83.022 /Times-Italic rf %DVIPSBitmapFont: Fl cmsy8 8 12 /Fl 12 111 df0 D<123C127E12FFA4127E123C08087A9414>I< 130C131EA50060EB01800078130739FC0C0FC0007FEB3F80393F8C7F003807CCF83801FF E038007F80011EC7FCEB7F803801FFE03807CCF8383F8C7F397F0C3F8000FCEB0FC03978 1E078000601301000090C7FCA5130C1A1D7C9E23>3 D<140381B3A3B812FCA3C7D80380 C7FCB3B812FCA32E2F7CAD37>6 DI20 D<12E012F812FEEA3F80EA0FE0EA03F8EA00FEEB3F80EB 0FE0EB03F8EB00FC143FEC0FC0EC07F0EC01FCEC007FED1FC0ED07F0ED01FCED007FEE1F C01607161FEE7F00ED01FCED07F0ED1FC0037FC7FCEC01FCEC07F0EC0FC0023FC8FC14FC EB03F8EB0FE0EB3F8001FEC9FCEA03F8EA0FE0EA3F80007ECAFC12F812E0CBFCAD007FB7 1280B812C0A22A3B7AAB37>I<170EA3170F8384170384170184717E1878187C84180FF0 07C0BA12F819FC19F8CBEA07C0F00F00183E601878604D5A60170360170795C7FC5F170E A33E237CA147>33 D<137813FE1201A3120313FCA3EA07F8A313F0A2EA0FE0A313C0121F 1380A3EA3F00A3123E127E127CA35AA35A0F227EA413>48 DI<12E0B3B3B3AD034378B114>106 D<12E0A27E1270A212781238A2123C121CA2121E12 0EA2120F7E7F1203A27F1201A27F1200A27F137013781338A2133C131CA2131E130EA213 0F7FA2801303801301A2801300A2801470A214781438143C141CA2141E140EA2140F80A2 15801403A215C0140114001A437CB123>110 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fm cmex10 12 23 /Fm 23 113 df0 D<12E07E12787E7E7E7F6C7E6C7E7F12016C7E7F137C137E7FA26D7EA26D7EA26D7EA36D 7EA2801301A2801300A280A2147EA2147FA4801580A7EC1FC0B3A5EC3F80A715005CA414 7EA214FEA25CA213015CA213035CA2495AA3495AA2495AA249C7FCA2137E137C13FC5B48 5A12035B485A485A90C8FC121E5A5A5A5A1A777C832E>II I<12F0B3B3B3AA0440728121>12 D<00F0EB03C0B3B3B3AA1A40728137>I<16F01501ED 03E0ED07C0ED0F80ED1F005D157E5D5D14014A5A4A5A4A5AA24A5A143F92C7FC147EA25C 13015C13035C13075C130F5C131FA2495AA349C8FCA213FEA312015BA212035BA21207A2 5BA2120FA25BA2121FA45BA2123FA55B127FA990C9FC5AB3AA7E7FA9123F7FA5121FA27F A4120FA27FA21207A27FA21203A27F1201A27F1200A3137FA26D7EA36D7EA2130F801307 801303801301801300147EA28081141F6E7EA26E7E6E7E6E7E140081157E8181ED0F80ED 07C0ED03E0ED01F0150024B26E833B>16 D<12F07E127C7E7E6C7E7F6C7E6C7E12017F6C 7E137E7FA26D7E80130F6D7EA26D7E80130180130080147E147F8081A26E7EA36E7EA26E 7EA3811403A2811401A281A21400A281A281A21680A4153FA216C0A5151F16E0A9150F16 F0B3AA16E0151FA916C0153FA51680A2157FA41600A25DA25DA21401A25DA214035DA214 075DA34A5AA24A5AA34A5AA292C7FC5C147E14FE5C13015C13035C495AA2495A131F5C49 C8FCA2137E5B485A5B1203485A485A5B48C9FC123E5A5A5A24B27C833B>I<171E173E17 7C17F8EE01F0EE03E0EE07C0160FEE1F80EE3F00167E167C16FC4B5A4B5A15075E4B5A4B 5A153F93C7FC5D15FE5D14015D14034A5AA24A5AA24A5AA24A5AA24AC8FCA214FEA21301 5C13035C1307A25C130F5C131FA25C133FA3495AA349C9FCA35A5BA312035BA31207A25B A2120FA35BA3121FA35BA3123FA55BA2127FAB485AB3B06C7EAB123FA27FA5121FA37FA3 120FA37FA31207A27FA21203A37F1201A37F7EA36D7EA36D7EA3131F80A2130F80130780 A21303801301801300A2147FA26E7EA26E7EA26E7EA26E7EA26E7E140181140081157F81 82151F6F7E6F7E8215036F7E6F7E167C167E82EE1F80EE0FC01607EE03E0EE01F0EE00F8 177C173E171E2FEE6B8349>I<12F07E127C7E7E6C7E6C7E7F6C7E6C7E6C7E137C137E7F 6D7E80130F6D7E6D7E801301806D7E147E147F80816E7EA26E7EA26E7EA26E7EA26E7EA2 6E7EA2818182153F82A2151F82150F82A2150782A36F7EA36F7EA38281A31780167FA317 C0A2163FA217E0A3161FA317F0A3160FA317F8A51607A217FCABEE03FEB3B0EE07FCAB17 F8A2160FA517F0A3161FA317E0A3163FA317C0A2167FA21780A316FF1700A35D5EA34B5A A34B5AA35E150FA25E151F5E153FA25E157F93C7FC5D5DA24A5AA24A5AA24A5AA24A5AA2 4A5AA24A5A92C8FC5C147E14FE495A5C13035C495A495A131F5C49C9FC137E137C13FC48 5A485A485A5B485A48CAFC123E5A5A5A2FEE7C8349>III[51 298 114 131 80 40 D80 D<17FF040313C093380F81F093381E 0078043E137C93387C01FC9338F803FEA2150116F01503EF01FC9338E0007003071400A3 150FA45E151FA7153FA74B5AA715FFA85C93C8FCA95C5DA85DA74A5AA75DA75D140FA45D A3001C5C007F131FEAFF8092C9FC5C143EA26C485A007C5B003C5B381F03E03807FF80D8 01FECAFC376F7B7F2F>82 D88 DI<1B3FF3FFC0973803E0F0973807C03897380F801C08 1F137E97383F01FEF303FF1A7E1AFE1AFC1901A2963903F801FEF300781C004F5AA2190F A262191FA34F5AA44F5AA319FF97C8FCA360A261A21803A3611807A461180FA44E5AA418 3FA261A2187FA361A218FFA44D5BA55F96C9FCA35FA360A2170FA460171FA460173FA54D 5AA54D5AA45E60A55E60A44C90CAFCA54C5AA55F161FA45F163FA45FA2167FA35FA316FF 5FA54B5BA494CBFCA25DA35EA21507A25EA44B5AA45E151FA45E153FA35EA2157FA25EA3 15FF93CCFCA34A5AA44A5AA35D14075DA2140F5D121E397F801FC0EAFFC05D143F92CDFC 5C147E6C485AEA7E00383801F86C485A380F07C03803FF80D800FCCEFC58DD7B7F37>I< B512FCA500F8C7FCB3B3B3B3B3B3B3B3B3A6B512FCA516B26A832F>104 DI110 D<12F012FE6C7E13E0EA 3FF8EA0FFCEA03FFC67F6D7E6D7E6D7E6D7E6D7E6D7EA26D7E7FA281147FB3B3AF81143F A281141F81140F8114076E7E6E7E6E7E6F7E6F7EED1FF0ED07F8ED01FE923800FF80163F A216FF923801FE00ED07F8ED1FF0ED3FC04B5A4BC7FC4A5A4A5A4A5A140F5D141F5D143F 5DA2147F5DB3B3AF14FF92C8FCA25B495AA2495A495A495A495A495A495A000390C9FCEA 0FFCEA3FF8EAFFE0138048CAFC12F029B2748342>I<1DC0F401E01C03A2F407C0A2F40F 80A2F41F00A21C3EA264A264A2641B01A2515AA2515AA2515AA251C7FCA21B3EA263A263 A2505AA2505AA2505AA2505AA250C8FCA21A3EA21A3C1A7CA262A24F5AA24F5AA24F5AA2 4F5AA24FC9FCA20104173E130E011E5F137F495F5A486D4B5A120F261C7FC04B5A123826 F03FE04B5A124000004D5A6D7E96CAFC6D6C5DA26D6C153EA2606D7E606D7E4D5A6D7F4D 5AA26E6C495AA26E6C495AA26E6C49CBFCA26E6C133EA25F6E7E5F6E7E4C5AEC01FF4C5A A26EEB83C01687ED7FC7EECF80ED3FEF04FFCCFCA26F5AA26F5AA26F5AA25E15035E6F5A 5B78758364>I E %EndDVIPSBitmapFont /Fn 134[44 44 66 44 50 28 39 39 50 50 50 50 72 28 44 1[28 50 50 28 44 50 44 50 50 9[83 2[55 1[61 1[61 2[83 55 2[33 72 1[61 61 3[61 6[33 4[50 50 50 50 50 2[25 33 25 41[50 2[{TeXBase1Encoding ReEncodeFont}45 99.6264 /Times-Italic rf %DVIPSBitmapFont: Fo cmmi8 8 32 /Fo 32 123 df14 D<3907C007E0390FE03FF8391CF878 3E393879E01E39307B801F38707F00126013FEEAE0FC12C05B0081143F120149133EA200 03147EA249137CA2000714FCA24913F8A2000F1301A2018013F0A2001F1303A2010013E0 120EC71207A215C0A2140FA21580A2141FA21500A2140E202C7E9D23>17 D<147C49B4FC903803C78090380783C090381F03E0EB1E01133E017C13F013F8A2EA01F0 120313E01207A2EA0FC01403A2EA1F80A21407003F14E0130090B5FCA2397F000FC0127E A2141F1580127C00FC14005CA2147EA248137C14FC00785B495AA2387C03E0383C07C049 5A001C90C7FCEA1E3EEA0FF8EA03E01C307DAE21>I<13FC13FFEB1FC0130F6D7EA36D7E A2130180A26D7EA3147EA280A36E7EA2140F81A24A7E143F147FECF3F0EB01E3EB03C190 380781F8130F49C67E133E5B49137E485A48487F1207485A4848EB1F8048C7FC127E48EC 0FC048EC07E000701403232F7DAD29>21 D<14C0A5ECFFE04913F8130790381F9FE0017F C7FC13FE5B485A12035B1207A25BA312037FA23801FBFE38007FFFA2EBF7FED803C0C7FC 485A48C8FC121EA25A127C1278A212F85A7EA37EB4FCEA7FC0EA3FF8EA1FFE380FFFC000 0313F038007FFCEB1FFEEB03FF1300141F80A3EB701EEB3C1CEB1FF8EB03E01D3C7EAD1F >24 D<1506A3150E150CA3151C1518A315381530A31570D801E0EB6007D807F8EC1F80EA 0E3CD81C3E01E013C0003814C00030150F0070150726607E011480D8E07CEB800312C013 FC3880F803000002001300120113F04A5B00030106130601E0140E160C020E131C020C13 1801C0143801E05C021C5B91381801C0D801F0495A030FC7FC3900FC381C90383F30F890 380FFFE0010190C8FCEB00701460A314E05CA313015CA42A3C7EAD2E>32 D<160ED80380143FA20007168090C8FC000E151F001E150F001C16000018811238003013 0C141E007015061260143E023C130E00E0150C5A0238131C6C15184A1338147802F85BD8 F00114F0496C485A397C0FBE073A7FFF9FFFC0021F5B263FFC0F90C7FC391FF807FC3907 E001F0291F7F9D2C>II<123C127EB4FCA213 80A2127F123D1201A312031300A25A1206120E5A5A5A126009157A8714>59 DI<15C0140114031580A21407 1500A25C140EA2141E141CA2143C143814781470A214F05CA213015CA213035C130791C7 FCA25B130EA2131E131CA2133C1338A21378137013F05BA212015BA212035BA2120790C8 FC5A120EA2121E121CA2123C1238A212781270A212F05AA21A437CB123>I<147F903801 FFE090380780F890380E003C497F497F49148001781307017C14C001FC130316E0A21370 90C7FC16F0A314FE903807FF8390381F01C390397C00E7E049137748481337D807E0133F 49131F484814C0121F48C7FCA2481580127EA2ED3F0012FE48147EA2157C15FC5D4A5A00 7C495AA26C495A001E49C7FC6C133E3807C0F83803FFE038007F8024307DAE25>64 D<013FB512FEEEFFC0903A00FE0007F0EE01F84AEB007E8301018118804A140F18C00103 150718E05CA21307A25CA2130FA24A140FA2131F18C04A141FA2013F1680173F91C81300 A249157EA2017E5D5F01FE14014C5A494A5A4C5A00014BC7FC163E4914FCED03F00003EC 1FC0B7C8FC15F8332D7CAC3A>68 D<90273FFFFC0FB5FCA2D900FEC7EA3F80A24A1500A2 01015D177E5CA2010315FE5F5CA2010714015F5CA2010F14035F5C91B6FC5B9139C00007 E05CA2013F140F5F91C7FCA249141F5F137EA201FE143F94C7FC5BA200015D167E5BA200 0315FEB539E03FFFF8A2382D7CAC3A>72 D<90383FFFFEA2010090C8FC5C5CA21301A25C A21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA291C7EA0180A2491403170001 7E5C160601FE140EA2495C163C12015E49EB01F84B5A0003141FB7FC5E292D7DAC30>76 D<000FB8FCA23B1FC003F8003F0100151F001C4A130E123C003801071406123000704A13 0EA20060010F140C12E0485CA2141FC715005DA2143FA292C8FCA25CA2147EA214FEA25C A21301A25CA21303A25CA21307A25C130F131F001FB512F0A2302D7FAC29>84 D<3B7FFFF801FFFEA2D801FCC7EA0FC0178049EC070016060003150E160C5BA20007151C 16185BA2000F153816305BA2001F157016605BA2003F15E05E90C8FCA24814015E127EA2 150300FE92C7FC5A5D1506150E007C5C151815386C5C5D6CEB03C0260F800FC8FC3803E0 3C3801FFF038003FC02F2E7BAC30>II99 D<151FEC03FFA2EC003FA2 153EA2157EA2157CA215FCA215F8A21401EB07E190381FF9F0EB7C1DEBF80FEA01F03903 E007E0EA07C0120FEA1F8015C0EA3F00140F5A007E1480A2141F12FE481400A2EC3F0215 06143E5AEC7E0E007CEBFE0C14FC0101131C393E07BE18391F0E1E38390FFC0FF03903F0 03C0202F7DAD24>I<1307EB0F80EB1FC0A2EB0F80EB070090C7FCA9EA01E0EA07F8EA0E 3CEA1C3E123812301270EA607EEAE07C12C013FC485A120012015B12035BA21207EBC040 14C0120F13801381381F01801303EB0700EA0F06131EEA07F8EA01F0122E7EAC18>105 D<15E0EC01F01403A3EC01C091C7FCA9147CEB03FE9038078F80EB0E07131C013813C013 30EB700F0160138013E013C0EB801F13001500A25CA2143EA2147EA2147CA214FCA25CA2 1301A25CA21303A25CA2130700385BEAFC0F5C49C7FCEAF83EEAF0F8EA7FF0EA1F801C3B 81AC1D>I<131FEA03FFA2EA003FA2133EA2137EA2137CA213FCA25BA2120115F89038F0 03FCEC0F0E0003EB1C1EEC387EEBE07014E03807E1C09038E3803849C7FC13CEEA0FDC13 F8A2EBFF80381F9FE0EB83F0EB01F81300481404150C123EA2007E141C1518007CEBF038 ECF83000FC1470EC78E048EB3FC00070EB0F801F2F7DAD25>I<137CEA0FFCA21200A213 F8A21201A213F0A21203A213E0A21207A213C0A2120FA21380A2121FA21300A25AA2123E A2127EA2127CA2EAFC08131812F8A21338133012F01370EAF860EA78E0EA3FC0EA0F000E 2F7DAD15>I<27078007F0137E3C1FE01FFC03FF803C18F0781F0783E03B3878E00F1E01 263079C001B87F26707F8013B00060010013F001FE14E000E015C0485A4914800081021F 130300015F491400A200034A13076049133E170F0007027EEC8080188149017C131F1801 000F02FCEB3F03053E130049495C180E001F0101EC1E0C183C010049EB0FF0000E6D48EB 03E0391F7E9D3E>I<3907C007E0391FE03FF83918F8783E393879E01E39307B801F3870 7F00126013FEEAE0FC12C05B00815C0001143E5BA20003147E157C5B15FC0007ECF80816 18EBC00115F0000F1538913803E0300180147016E0001F010113C015E390C7EAFF00000E 143E251F7E9D2B>I<90387C01F89038FE07FE3901CF8E0F3A03879C0780D907B813C000 0713F000069038E003E0EB0FC0000E1380120CA2D8081F130712001400A249130F16C013 3EA2017EEB1F80A2017C14005D01FC133E5D15FC6D485A3901FF03E09038FB87C0D9F1FF C7FCEBF0FC000390C8FCA25BA21207A25BA2120FA2EAFFFCA2232B829D24>112 D<3807C01F390FF07FC0391CF8E0E0383879C138307B8738707F07EA607E13FC00E0EB03 804848C7FCA2128112015BA21203A25BA21207A25BA2120FA25BA2121FA290C8FC120E1B 1F7E9D20>114 DI<130E131FA2 5BA2133EA2137EA2137CA213FCA2B512F8A23801F800A25BA21203A25BA21207A25BA212 0FA25BA2001F1310143013001470146014E0381E01C0EB0380381F0700EA0F0EEA07FCEA 01F0152B7EA919>I<013F137C9038FFC1FF3A01C1E383803A0380F703C0390700F60F00 0E13FE4813FC12180038EC0700003049C7FCA2EA200100005BA313035CA301075B5D14C0 00385CD87C0F130600FC140E011F130C011B131C39F03BE038D8707113F0393FE0FFC026 0F803FC7FC221F7E9D28>120 D<011E1330EB3F809038FFC07048EBE0E0ECF1C03803C0 FF9038803F80903800070048130EC75A5C5C5C495A495A49C7FC131E13385B4913404848 13C0485A38070001000EEB0380380FE007391FF81F0038387FFF486C5A38601FFC38E00F F038C003C01C1F7D9D21>122 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fp msbm10 12 6 /Fp 6 91 df<922601FFE01330033F01FC13784AB6FC020FEDC0F8023F9038C07FF8913A FFFE000FFF4901F81303902607FBF07F90260FE7E0EB007F90261F87C0EC3F7849484814 1FD97E1FED0FF801FC90C8FC2601F83E1507D803F0160348485A01C01601380F807802F8 1500EA1F004A1678EA3E01A2003C5B007C183019001278130312F85C12F0AC12F8A20078 7FA2EA7C01A2123C003E7FA2EA1F00A26C6C7E19062607C07C160F01E0171F6C6C6C163F D801F8177E6C6C6C167C017E6D15FCD93F0FED01F890261F87C0EC07F090260FE7E0EC0F E0902607FBF8EC3FC0902601FFFE903801FF806D903AFFC00FFE00023F90B55A020F15F0 020115C0DA003F49C7FC030113F040487CC52E>67 D<007FB54AB512C0B66C4914E0816C 6E6D14C02707F003F09039000FF8002601F801ED03F000006D7E017C6D6E5A017E137E01 7F133E8102807FECC00F6E6C7E017B80903979F003F0ECF801903978FC00F8027C7F6E13 7E023F133E6E6C7E020F1480913907C00FC0EDE007913903F003E0020114F0913900F801 F8EDFC00037E137C033E137E6F133EEE801FDB0FC013810307EB0FC1923803E0079338F0 03E1DB01F813F1923900FC01F9EE7C0070137D043F137F93381F803F040F131F933807C0 0F17E0933803F00704011303933800F80117FC177E173E171F1881EF0FC11707EF03E118 F1EF01F91700187D01FC167F183FD803FF161F007F01F8150FB57E18076C491503CB1201 725A43467DC339>78 D<923807FFC092B512FE0207ECFFC0021F15F0027F010113FC903B 01FFF8003FFF4901E0010F7F010F496D13E090261FCF80903803E7F0D93F1FC73801F1F8 017EEEF0FCD9FC3E913800F87ED801F88348484892387C1F8001E0170F48484892383E07 C0000F19E04948ED1E03D81F00EF01F00101161F001E1800003E496F13F8A2003C197800 7C197C010317800078193C4A150700F8193EA200F0191EAC00F8193EA20078193C6E150F 007C197C01011700003C1978003E19F8A2001E6D4B13F0001F18010100161ED80F80EF03 E06D6CED3E07000719C02603E07C92387C0F8001F0171F6C6C6C9238F83F00D800FC177E D97E1F4A485A013FEEF1F890261FCF80903803E7F06DB46C49B45A01036D4913806D01F8 013F90C7FC903B007FFF01FFFC021F90B512F0020715C002004AC8FC6F5B92387C007C6F 7F173F6F6D7E706C7E92390FC007F00307EB03FC923B03F000FF80707090387FFFF8DB00 FC131F047F010713F093263FC00013E093260FFC0113C070B61200040114FC706C13F005 0790C7FC47597DC53C>81 D<007FB712C0B812FCEFFF806C17E02800F807F00F13F8DBC0 0113FE017890398000FCFF94387C3F8094383E0FC0727E94381E03F0EF1F011800717FA2 1978A519F8A24D5B1801EF1E034E5A94383E0FC094387E3F80DDFDFFC7FC933807FFFE92 B612F818E095C8FC17F0ED87C1EEC0F8923883E0FC177C923881F03EA2923880F81F84EE 7C0F717E163E93383F03E0041F7FEE0F81EF80F8EE07C0187C933803E07E183E706C7E85 706C6C7E180794387C03E0057E7F94383E01F8716C7E197C01F86D6D6C7EF13F80007FB6 6C6CB512E0B700C015F0836C4B6C14E044447EC33D>I I<0003B812FE4883A301879039000F803ED98FF0011F137ED9BFC0EC007C01FFC7003E5B 13FC48484A485A49ECFC034902F85B4B48485A5B494948485A0307131F04C090C7FC90C7 380F803EA24B485A00064A13FCC8003E5B4B485AA24B485A0201130703F05B4A48485AA2 4A4848C8FC5E91380F803E021F5B1500023E5B1501027C5B9138FC03E014F84948485AA2 4948485A0107011F156002C090C812F090380F803EA249484814014913FC013E5B494848 EC03E0A249484814070001130701F049140F48484848141FA2484848C8EA3FC0000F4915 7FD9803E15FF484848EC01FBEF07F3003E49EC0FE7D87E01ED7F87007C49903903FF0780 BAFCA36C18003C447DC345>90 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fq cmr8 8 16 /Fq 16 116 df<156015F0A24A7E4A7EA24A7E1406EC0E7F140C91381C3F8014184A6C7E 150F02607F150702C07F1503D901807F1501D903007F496D7E1306010E147F130C011C6E 7E131801386E7E1330496E7E160749811603484881160148C87F486F7E1206000E167F12 0C001CEE3F801218003FB812C0A24817E0A2B912F0342F7DAE3B>1 D10 D<13031307130E131C1338137013F0EA01E013C01203EA0780A2EA0F00A2121EA35AA45A A512F8A25AAB7EA21278A57EA47EA37EA2EA0780A2EA03C0120113E0EA00F01370133813 1C130E1307130310437AB11B>40 D<12C07E12707E7E7E120FEA0780120313C0EA01E0A2 EA00F0A21378A3133CA4131EA5131FA2130FAB131FA2131EA5133CA41378A313F0A2EA01 E0A2EA03C013801207EA0F00120E5A5A5A5A5A10437CB11B>I43 D48 D<130C133C137CEA03FC12FFEAFC7C1200B3B113FE387FFFFEA2172C7AAB23>III<140EA2141E143EA2147E14FEA2EB01BE1303143E13 06130E130C131813381330136013E013C0EA0180120313001206120E120C5A123812305A 12E0B612FCA2C7EA3E00A9147F90381FFFFCA21E2D7EAC23>I<000CEB0180380FC01F90 B512005C5C14F014C0D80C7EC7FC90C8FCA8EB1FC0EB7FF8380DE07C380F801F01001380 000E130F000CEB07C0C713E0A2140315F0A4127812FCA448EB07E012E0006014C0007013 0F6C14806CEB1F006C133E380780F83801FFE038007F801C2D7DAB23>I56 D61 D105 D<3807C0FE39FFC3FF809038C703E0390FDE01F0EA07F8496C7EA25BA25BB2486C487E3A FFFE1FFFC0A2221E7E9D27>110 D<3801FE183807FFB8381E01F8EA3C00481378481338 A21418A27E7EB41300EA7FF06CB4FC6C13C06C13F0000113F838001FFC130138C0007E14 3EA26C131EA27EA26C133CA26C137838FF01F038E3FFC000C0130017207E9E1C>115 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmmi12 12 76 /Fr 76 123 df11 DI I<1578913807FFE0021F13FC91383C7FFEEC7007EC6003ECE0004A13381600A280A380A2 80147CA2147E143E143F816E7EA26E7E81140781EC3FFC14FF903803E1FEEB07C190381F 00FF133E49EB7F805B0001143F485A484814C049131F120F485AA248C7FC150F5A127EA3 00FEEC1F805AA316005A5DA2153E157E157CA26C5C127C4A5A6C495AA26C495A6C6C485A 6C6C48C7FC3803E07C3800FFF0EB1FC027487CC62B>I<1530A7ED33FF033F13C0ED7C00 92B5FCDA03C31300DA0780C7FC4AC8FC141C14785C495A5C495A130749C9FC131E131C13 3C5B5BA2485AA2485AA2485AA248CAFCA25A121EA2123E123CA3127C1278A412F8A67EA2 127C127EA2127F6C7E7F6C7E13F8EA0FFE3807FFC06C13F86C13FF6C6C13E0011F7F0103 13FC9038007FFEEC1FFF14039138007F80153F151FA2150FA393C7FCA20102131E13076D 6C5A903801E0789038007FE0EC1F802A597CC42B>16 D<01F8EB03FCD803FEEB1FFFD807 1F90387C0FC03B0E0F80E007E0001C9038C3C003271807C70013F002CE1301003801DC14 F8003013D8EB0FF800705B00605BA200E0491303D8C01F15F05C12001607133F91C713E0 A2160F5B017E15C0A2161F13FE491580A2163F1201491500A25E120349147EA216FE1207 495CA21501120F495CEA0380C81203A25EA21507A25EA2150FA25EA2151FA25EA2153FA2 93C7FC150E2D417DAB30>I<157E913801FF80913807C3E091381F01F0EC3E004A13F814 FC4948137C495A5C0107147E495A131F5C133F49C7127FA213FEA212015B12034914FF12 07A25B000F15FE1501A2485AA21503003F15FC5B90B6FCA24815F89038800007A2150F00 FF15F090C7FCA2ED1FE0A25AED3FC0A21680157F16005A15FEA24A5AA25D14035D4A5A00 7C495AA24A5A007E49C7FC003E133E5C001E5B6C485A380783C06CB4C8FCEA00FC28477C C52D>I21 D<147002F8140E0101153FA301035DA24A147EA2010715FEA24A5CA2010F 1401A24A5CA2011F1403A24A5CA2013F1407A291C75BA249140FA2017E5DA201FE021F13 18183849ED8030A20001033F13701860EE7F005E486C16E0DB01DF13C09238039F016DD9 071F1380489039801E0F83903BF7C078078700903AE1FFE003FE903AE07F8000F8000F90 CAFCA25BA2121FA25BA2123FA290CBFCA25AA2127EA212FEA25A123835417DAB3B>I<15 60A7ED7FFF92B512C0913807F80191381FDFFF91397F87FE004AC8FCEB03FC495A130F5C 495A133F5C137F5C13FFA291C9FCA57F80A2133F6D7E90390FE3FF806DB512E0903901FC 006049B512E0D90F8F1380011EC9FC5B13F8485A5B485A485A48CAFCA2121E123E123C12 7C1278A312F8A47EA27E127F7FEA3FE013F86CB4FC6C13C0000313F86C13FE39007FFFC0 010F13F0010313FC9038007FFF021F7F02037F1400151F150F1507A401025C903803800F D901C090C7FC903800F83EEC3FF8EC07E02A597EC42B>24 D<010FB712E0013F16F05B48 B812E04817C02807E0060030C7FCEB800EEA0F00001E010C13705A0038011C13605A0060 011813E000E013381240C7FC5C4B5AA214F014E01301150314C01303A3EB078082130FA2 EB1F00A34980133E137EA24980A2000114015BA26C48EB00E0342C7EAA37>II<0203B612E0021F15F091B7FC49 16E0010716C090270FF80FF8C7FC90381FC00349486C7E017EC7FC49147E485A4848143E 0007153F5B485AA2485AA2123F90C8FC5E48157E127EA216FE00FE5D5A15015EA24B5A00 7C5D15074B5A5E6C4AC8FC153E6C5C5D390F8003F03907C007C02601F03FC9FC38007FFC EB1FE0342C7DAA37>I<010FB612FC013F15FE5B48B712FC4816F82707E001C0C7FC0180 5B380F0003121E121C5A4849C8FC126012E000405BC7FC140E141EA45CA3147CA2147814 F8A4495AA31303A25C1307A3130FA25C6D5A2F2C7EAA2A>I<161CA21618A21638A21630 A21670A21660A216E0A25EA21501A25EA21503A293C8FCA25DED7FE0913807FFFE91391F C63F809139FE0E07C0D901F8EB03F0903A07E00C00F8D91FC08090263F001C137E017E81 4913184848ED1F8000031438485A4848013014C0A248481370A248481360A248C712E0A2 4B133F481780481301A24B137F180014034816FE92C7FC4C5A6C49495AA2007E0106495A 4C5A6C010E495A4C5A261F800C49C7FC000F15FC3A07C01C01F8D803E0EB07E03A01F818 1F80D8007E01FEC8FC90381FFFF801011380D90030C9FCA21470A21460A214E0A25CA213 01A25CA21303A291CAFCA332597BC43A>30 D<137E48B46C150626078FE0150E260607F0 151C260E03F81538000C6D1570D81C0116E000006D15C0010015016EEC03806EEC070017 0E6E6C5B5F5F6E6C136017E04C5A6E6C485A4CC7FC0207130E6F5A5E1630913803F8705E EDF9C06EB45A93C8FC5D6E5A81A2157E15FF5C5C9138073F80140E141C9138181FC01438 1470ECE00FD901C07FEB038049486C7E130E130C011C6D7E5B5B496D7E485A48488048C8 FC000681000E6F137048EE806048033F13E04892381FC0C048ED0FE348923803FF00CA12 FC37407DAB3D>I<1730A317701760A317E05FA316015FA3160394C8FCA35E1606A3160E 160C013E1607D9FF80ED1F802603C3C0011CEB3FC0260703E01318260601F0157F000E17 3F001C1538D818030230131F0038170F0030170700701570D86007026013035CA2D8E00F 02E0148000C049491301EA001F4A150303011500013F5C1400604901031406017E91C7FC 180E180C01FE49141C4901061418183860030E1460030C14E04D5A4D5A031C49C7FC0318 130E017E5D5F6D01385B90261F80305BD90FC0EB03C0D907F0010FC8FC903901FE707C90 39003FFFF002031380DA0060C9FC15E05DA314015DA3140392CAFCA35C1406A3140E140C 3A597DC43F>I<0110160E0138163F0178EE7F80137001F016FF4848167F5B0003173F49 161F120790CA120FA2000E1707A248180015060018140FA20038021E14061230A2180E00 704A140C1260181CA203381418183800E05C6015F86C01015D170114030078D907BC495A DA0FBE1307007CD91F3E495A007ED97E3F013FC7FC3B7F83FE1FE0FF263FFFFCEBFFFE4A 6C5B6C01F05C6CD9C0075B6CD9000113C0D801FC6D6CC8FC392D7FAB3D>II<121EEA7F80A2EAFFC0A4EA7F80A2 EA1E000A0A78891B>58 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013C0 A312011380120313005A1206120E5A5A5A12600B1D78891B>II<1618163C167CA2167816F8A216F01501A216E01503A216C01507A21680150F A2ED1F00A2151E153EA2153C157CA2157815F8A25D1401A24A5AA25D1407A25D140FA292 C7FC5CA2141E143EA2143C147CA25CA25C1301A25C1303A25C1307A25C130FA291C8FC5B A2133EA2133C137CA2137813F8A25B1201A25B1203A2485AA25B120FA290C9FC5AA2121E 123EA2123C127CA2127812F8A25A126026647BCA31>I<127012FCB4FCEA7FC0EA1FF0EA 07FCEA01FF38007FC0EB1FF0EB07FE903801FF809038007FE0EC1FF8EC03FE913800FF80 ED3FE0ED0FF8ED03FF030013C0EE3FF0EE0FFCEE01FF9338007FC0EF1FF0EF07FCEF01FF 9438007FC0F01FE0A2F07FC0943801FF00EF07FCEF1FF0EF7FC04C48C7FCEE0FFCEE3FF0 EEFFC0030390C8FCED0FF8ED3FE0EDFF80DA03FEC9FCEC1FF8EC7FE0903801FF80D907FE CAFCEB1FF0EB7FC04848CBFCEA07FCEA1FF0EA7FC048CCFC12FC12703B3878B44C>I64 D<1830187018F0A217011703A24D7EA2170F171FA21737A2176717E717C793380187FCA2 EE0307EE07031606160CA216181638163004607FA216C0030113011680ED0300A2150615 0E150C5D845D03707F15605DA24A5A4AB7FCA25C0206C87F5C021C157F14185CA25C14E0 5C495A8549C9FC49163F1306130E5B133C137C01FE4C7ED807FFED01FF007F01F0027FEB FFC0B5FC5C42477DC649>I<91B87E19F019FC02009039C00003FF6F480100138003FFED 3FC01AE093C8121FF10FF0A24A17F84B1507A314035D190FA2020717F04B151F1AE0193F 020F17C04BED7F80F1FF004E5A021F4B5A4B4A5AF01FF0F03FC0023F4AB4C7FC4BEB1FFC 92B612F018FEDA7FC0C7EA7F804BEC1FC0F00FF0727E02FF6F7E92C8FC727EA249835CA3 13035CA301075F4A1503A24E5A130F4A4B5A4E5AA2011F4C5A4A4B5A4D485A013F4B48C7 FCEF0FFC4AEC3FF801FF913801FFE0B9128005FCC8FC17C045447CC34A>I<4CB46C1318 043F01F013384BB512FC0307D9007E1378DB1FF090380F80F0DB7F80EB03C1DA01FEC7EA 01C34A48EC00E7DA0FF0ED7FE04A48153F4A5A02FFC9121F494817C04948160F495A130F 4A178049481607495A137F4948170091CAFC5A485A1906485AA2485A96C7FC121F5BA212 3F5BA3127F5BA4485AA419C0A2180161127F180396C7FC6018066C6C160E601818001F17 386D5E000F5F6D4B5A6C6C4B5A00034CC8FC6C6C150E6C6C153C017F5DD93FC0EB01E0D9 1FF0EB0FC0D907FE017FC9FC0101B512FCD9003F13E0020790CAFC45487CC546>I<91B8 7E19F019FC02009039C00007FF6F489038007FC003FFED1FE0737E93C86C7E737E19014A 707E5D1A7FA20203EF3F805DA21BC014075DA3140F4B17E0A3141F4B17C0A3143F4B167F A3027F18804B16FFA302FF180092C95A62A24917034A5F19076201034D5A5C4F5A620107 173F4A5F4FC7FC19FE010F4C5A4A15034E5AF00FE0011F4C5A4A4B5A06FFC8FC013FED01 FCEF0FF84AEC3FE001FF913803FF80B848C9FC17F094CAFC4B447CC351>I<91B912FCA3 020001C0C7123F6F48EC03F803FF1501190093C91278A21A385C5DA3020317305DA31407 4B1460A218E0020F4B13005DA21701021F5D4B13031707170F023F027FC8FC92B6FCA391 397FC0007E4B131EA2170E02FF140C92C7FCA2171C49031813035C611906010392C7FC4A 160E190C191C010717184A163819301970130F4A5E180161011F16034A15074E5A013F16 3F4EC7FC4AEC03FF01FFED3FFEB9FCA26046447CC348>I<91B912F8A3020001C0C7123F 6F48EC07F003FF1503190193C9FCA21A705C5DA3020317605DA314075D18C01701020F4B 13005DA21703021F92C8FC4B5BA25F023F141E4B13FE92B5FCA24A5CED8000173CA202FF 141892C7FCA217384915305CA21770010315604A91C9FCA313075CA3130F5CA3131F5CA2 133FA313FFB612F8A345447CC33F>I<4CB46C1318043F01F013384BB512FC0307D9007E 1378DB1FF090380F80F0DB7F80EB03C1DA01FEC7EA01C34A48EC00E7DA0FF0ED7FE04A48 153F4A5A02FFC9121F494817C04948160F495A130F4A178049481607495A137F49481700 91CAFC5A485A1906485AA2485A96C7FC121F5BA2123F5BA3127F5BA4485A4CB612805EA2 93C7EBE000725AA3007F60A218FF96C7FCA26C7E5F606C7EA2000F16036D5E6C6C150700 03160F6C6C151F6C6CED3DF8D97F8014786D6CEB01E0D91FF0903807C078D907FE90387F 00700101B500FC1330D9003F01F090C8FC020790CAFC45487CC54D>I<91B6D8E003B612 80A3020001E0C70003EB8000DB7F806E48C7FC03FF1503A293C85BA219075C4B5EA2190F 14034B5EA2191F14074B5EA2193F140F4B5EA2197F141F4B5EA219FF143F92B8C8FCA3DA 7FC0C712014B5DA2180314FF92C85BA218075B4A5EA2180F13034A5EA2181F13074A5EA2 183F130F4A5EA2187F131F4A5EA2013F16FFA24A93C9FCD9FFE002037FB6D8E003B67EA3 51447CC351>I<027FB512F8A217F09139007FF000ED3FC0157FA25EA315FF93C7FCA35C 5DA314035DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA314FF92C8FCA35B5CA313 035CA313075CA3130F5CA2131FA25CEB7FF0007FB512F0B6FCA22D447DC32B>I<031FB5 12FC5D18F89239000FFE00705AA35FA2160FA25FA2161FA25FA2163FA25FA2167FA25FA2 16FFA294C7FCA25DA25EA21503A25EA21507A25EA2150FA25EA2151FA25EA2153FA25EEA 0F80D83FE0137F5E127FA24BC8FC485A4A5A1300006C495A0060495A0070495A0030495A 0038EB3F806C49C9FC380F81FC3803FFF038007F80364679C336>I<91B600E049B512C0 A3020001E0C8383FF800DB7F80ED1FE003FF94C7FC1A3E93C9127862F101C04A4C5A4B4B C8FC191C6102035E4B5DF003804EC9FC0207150E4B14386060020F4A5A4B0107CAFC170E 5F021F14784B13F84C7E1603023F130F4B487E163BEEE1FF91387FC1C1DB83807FED8700 159CDAFFB86D7E5D03C06D7E5D4990C7FC4A6E7EA2717E13034A811707A201076F7E5C71 7EA2130F4A6E7FA2727E131F5C727E133F854A82D9FFE04B7EB600E0010FB512E05FA252 447CC353>I<91B612F8A3020001E0C8FC6F5A4B5AA293C9FCA35C5DA314035DA314075D A3140F5DA3141F5DA3143F5DA3147F5DA314FF92CAFCA35B4A16C0A21801010317804A15 031900A201075E4A1506180E181E010F161C4A153C18381878011F16F84A4A5A1703013F 150F4D5A4A14FF01FF02075BB9FCA2603A447CC342>I<91B500C0933803FFFE63630200 F1FE00DB6FE0EE1BF803EF171F1B3703CFEF67F0A21BCF0201EF018F038F60DB87F0ED03 0F1B1F020317060307040C5BA2F2183F020717300206616F6C15601B7F020E17C0020CDC 018090C7FCA24F485A021C16060218606F6C5C1A0102381618023004305BA2F160030270 16C00260606F6CEB01801A0702E0ED03004A03065CA24E130F01015E4A60047F5B1A1F01 035E91C74A5CA24D48133F494BC7FC010661EE3F861A7F010E158C010C039892C8FCA205 B05C011C15E001186001386E5A190101785D01FC92C75BD803FFEF07FEB500F8011E0107 B512FE161C160C5F447BC35E>I<91B500C0020FB5128082A2DA007F9239007FE00070ED 1F8074C7FCDBEFF8150E15CF03C7160C70151C1401DB83FE1518A2DB81FF153814030300 1630831A704A6D7E02061760163F7114E0140E020C6D6C5CA2706C1301141C021801075D 83190302386D7E023094C8FC1601715B147002606DEB8006A294387FC00E14E04A023F13 0C18E0191C0101ED1FF04A1618170FF0F838130391C83807FC30A2943803FE705B010603 01136018FF19E0010E81010C5F187FA2131C0118705A1338181F137801FC70C9FCEA03FF B512F884180651447CC34E>II<91B712FEF0FFE019F802009039C0000F FE6F48EB01FF03FF9138007F80F13FC093C8EA1FE0A24AEE0FF0A25D1AF81403A25DA214 07F11FF05DA2020FEE3FE0A24B16C0197F021F1780F1FF004B4A5A4E5A023F4B5A4E5A4B EC3FC006FFC7FC027FEC07FC92B612F018800380CAFC14FFA292CBFCA25BA25CA21303A2 5CA21307A25CA2130FA25CA2131FA25CA2133FA25CEBFFE0B612E0A345447CC33F>II<91B712F018FF19E002009039C0003FF86F48EB 07FC03FFEC01FEF0007F93C8EA3F801AC0F11FE05C5D1AF0A214035DA30207EE3FE05DA2 F17FC0020F17804B15FF1A004E5A021F4B5A4B4A5AF00FE04E5A023F037FC7FC4BEB03FC EF1FF092B612804A4AC8FC923980007F80EF0FC0EF07F002FF6E7E92C77F1701845B4A14 00A2170113035CA2170313075CA24D5A130F5CA3011F18185CA2013F4C13381A304A6F13 70D9FFE0020314E0B600E0ED01C00501EB0380943900FE0F00CBEA3FFEF007F045467CC3 4A>I<9339FF8001800307EBF003033F13FC9239FF007E07DA01F8EB0F0FDA07E0903807 9F004A486DB4FC4AC77E023E804A5D187E5C495A183C495AA213074A1538A3130F183080 A295C7FC806D7E8014FF6D13E015FC6DEBFFC06D14FC6E13FF6E14C0020F80020314F8EC 003F03077F9238007FFE160F1603707E8283A283A21206A4000E163EA2120C177E001E16 7CA25F5F003F15014C5A6D4A5A4C5A486C4AC8FC6D143ED87CF85CD8787E495A3AF01FC0 0FE0D8E007B51280010149C9FC39C0003FF039487BC53C>I<48BA12C05AA291C7D98000 1380D807F092C7121F4949150F0180170748C75B1903120E48020316005E121812380030 14074C5C00701806126000E0140F485DA3C8001F92C7FC5EA3153F5EA3157F5EA315FF93 CAFCA35C5DA314035DA314075DA3140F5DA3141F5DA3143F5DA2147FA214FF01037F001F B612FCA25E42447EC339>I<003FB500F80103B512E0A326003FF8C8381FF800D91FE0ED 07E0013F705A615C96C7FC60017F16065CA2180E01FF160C91C9FCA2181C4817185BA218 38000317305BA21870000717605BA218E0120F495EA21701121F495EA21703123F4993C8 FCA25F127F491506A2170E00FF160C90C9FC171CA21718173817304816705F6C5E6C1501 4C5A4CC9FC6C150E6D141E001F5D6D5C6C6CEB01E06C6C495A6C6CEB1F80C6B401FECAFC 90387FFFF8011F13E0010190CBFC43467AC342>I<007FB56C91381FFFF8B65DA2000101 E0C8000313006C0180ED01FCF000F0614E5AA2017F4C5A96C7FC1806A2606E5DA2013F5E 1870186060A24D5A6E4AC8FCA2011F1506170E170C5FA26E5C5FA2010F5D16015F4CC9FC A26E13065EA201075C5EA25E16E06E5B4B5A13034BCAFC1506A25D151CECFE185D13015D 5DA26E5AA292CBFC5C13005C5CA25CA25C45467BC339>II<023FB500E0011FB5FC A39126007FFEC7000313E0DB3FF8913801FE006F486E5A1AF06F6C4A5A626F6C4A5A0706 C7FC190E6F6C5C616F6C5C6171485A6F5D4EC8FC93387FC00660706C5A6060706C5A17F1 93380FFB8005FFC9FC5F705AA2707EA3707E5E04067F5E93381C7FC0163816704C6C7EED 01C04B486C7E160015064B6D7E5D4B6D7E5D5D4A486D7E14034AC76C7E140E5C4A6E7F14 3002E06F7E495A0103707E495A131F496C4B7E2603FFE04A487E007F01FC021FEBFFF0B5 FCA250447EC351>II<020FB812C05C1A809326800001130003 F8C7FCDA3FE04A5A03804A5A92C8485A027E4B5A027C4B5A02784B5A4A4B5AA24A4A90C7 FC4A4A5A01014B5A4D5A4A4A5A01034B5A91C8485A4D5AA290C84890C8FC4C5A4C5A4C5A 4C5A4C5A4C5A4C5AA24B90C9FC4B5A4B5A4B5A4B5A4B5A4B5AA24B5A4A90CAFC4A5A4A48 14064A5A4A5A4A48140E4A48140CA24A48141C4990C8121849481538495A49485D495A49 4815F049485D1701494814034890C8485A4848150F4848151F48484B5A484815FF484814 03043F90C8FC48B8FCB9FC5F42447BC343>I96 DIII< EE01FC16FFA3EE03F816011603A217F0A21607A217E0A2160FA217C0A2161FA21780A216 3FA21700EC0FC091387FF07F903801F838903907E01C7E90380FC00E90393F0007FE4913 0301FE5C485A491301120348485C120F491303121F5E485A1507127F495CA2150F12FF90 C75BA2151FA2485DA2033F13301770EE0060A24B13E017C015FE007E130102031301003E D9073E1380003F010E13036C011C14006C6C486C5A3A07C0F00F0E3A01FFC007FC3A007F 0001F02E467CC433>III<157E913803FF8091390FC1E0E091391F0073F0027E13334A133F 4948131F010315E04948130F495AA2494814C0133F4A131F137F91C713805B163F5A4915 00A25E120349147EA216FEA2495CA21501A25EA21503150700015D150F0000141F6D133F 017CEB77E090383E01E790381F078F903807FE0FD901F85B90C7FC151FA25EA2153FA293 C7FCA2001C147E007F14FE485C4A5A140348495AEC0FC000F8495A007C01FEC8FC381FFF F8000313C02C407EAB2F>I<141E143F5C5CA3147E143891C7FCAE133EEBFF803801C3C0 380781E0380601F0120E121CEA180312381230A2EA700700605BA2EAE00F00C05BEA001F 5CA2133F91C7FCA25B137E13FE5BA212015BEC03800003140013F01207495A1406140E14 0CEBC01C141814385C00035BEBE1C0C6B45A013EC7FC19437DC121>105 D<163C16FEA21501A316FCED00701600AE15FCEC03FF91380F0780021C13C091383803E0 147014E014C01301EC8007130314005B0106130F130E010C14C090C7FC151FA21680A215 3FA21600A25DA2157EA215FEA25DA21401A25DA21403A25DA21407A25DA2140FA25DA214 1F5DA2143F001C91C7FC127F48137E5CA248485AEB03E038F807C038781F80D83FFEC8FC EA07F0275681C128>I<14FE137FA3EB01FC13001301A25CA21303A25CA21307A25CA213 0FA25CA2131FA25C163F013FECFFC0923803C0E09138000703ED1E0F491338ED701F017E 13E0EC01C001FE018013C00203EB07004948C8FC140E00015B5C495A5C3803FBC001FFC9 FC8014F83807F1FE9038F03F809038E00FE06E7E000F130381EBC001A2001FED01C01780 1380A2003F15031700010013F05E481506160E007E150C161C00FE01005BED787048EC3F E00038EC0F802B467BC433>II<01F8D903FC EC7F80D803FED91FFF903803FFE0D8071F903B7C0FC00F81F83E0E0F80E007E01C00FC00 1C9026C3C0030178137C271807C700D9F0E0137E02CE902601F1C0133E003801DCDAFB80 133F003001D892C7FCD90FF814FF0070495C0060495CA200E04949485CD8C01F187E4A5C 1200040715FE013F6091C75BA2040F14014960017E5D1903041F5D13FE494B130762043F 160E0001060F130C4992C713C0191F4CED801C00031A1849027E1638F2003004FE167000 071A60494A16E0F201C0030192380F0380000FF18700494AEC03FED80380D90070EC00F8 4F2D7DAB55>I<01F8EB03FCD803FEEB1FFFD8071F90387C0FC03B0E0F80E007E03A0C07 C3C003001CD9C7007F001801CE1301003801DC80003013D8EB0FF800705B00605BA200E0 491303D8C01F5D5C12001607013F5D91C7FCA2160F495D137E161F5F13FE49143F94C7FC 187000014B136049147E16FE4C13E0000317C049150104F81380170300071700495D170E EE781C000FED7C3849EC1FF0D80380EC07C0342D7DAB3A>III<91380FC00391383FF0079138F83C0F903903E00E1E 90390FC0063E90381F800790393F00037E4914FC01FE1301485AA2484814F812075B000F 140316F0485AA2003F14074914E0A3007F140F4914C0A3151F90C713805AA2153F6C1500 A2127E5D007F14FE6C1301A214036C6C485A000F131E3807C0383803E0F13901FFC1F838 003F01130014035DA314075DA3140F5DA2141FA2143F011FB51280A21600283F7DAB2B> I<01F8EB0FC0D803FEEB7FF0D8070FEBF038000E903883C07C3A0C07C701FC001C13CE00 18EBDC03003813D8003013F8D90FF013F800709038E000E0006015005C12E0EAC01F5C12 00A2133F91C8FCA35B137EA313FE5BA312015BA312035BA312075BA3120F5BEA0380262D 7DAB2C>II<141C147EA314FE5C A313015CA313035CA313075CA2007FB512FCB6FC15F839000FC000A2131F5CA3133F91C7 FCA35B137EA313FE5BA312015BA312035BA21570000714605B15E015C0000F130101C013 801403EC070000071306140E5C6C6C5A000113F03800FFC0013FC7FC1E3F7EBD23>I<13 3ED9FF8014E02603C3C0EB03F0380703E0380601F0000E1507121CD818035D1238003015 0FA2D870075D00605B161FEAE00F00C0495CEA001F4A133FA2013F92C7FC91C7FC5E5B01 7E147EA216FE13FE495CA20301EB01801703484802F81300A25F0303130616F000001407 030F130E6D010D130C017C011D131C033913186D9038F0F838903A1F03C07870903A07FF 803FE0903A01FC000F80312D7DAB38>I<013E140ED9FF80EB3F802603C3C0137F380703 E0380601F0120E121CD81803143F0038151F0030150FA2D87007140700605BA2D8E00F15 0000C0497FEA001F4A5B1606133F91C7FC160E49140C137EA2161C01FE14185B16381630 16704848146016E05E150100005D15036D49C7FC1506017C130E017E5B6D137890380F81 E06DB45AD900FEC8FC292D7DAB2F>I<02FCEB07E0903A03FF801FFC903A0F07C0781E90 3A1C03E0E01F903A3801F1C07FD9700013804901FB13FF4848EBFF00495B000316FE90C7 1438484A130012061401000E5C120CC7FC14035DA314075DA3140F5DA3021F143817305D 1770023F1460121E003F16E0267F807FEB01C0026F148000FF01EF1303D901CFEB070000 FE903887C00E267C03835B3A3C0F01E0783A1FFC00FFE0D803F0EB3F80302D7EAB37> 120 D<133ED9FF8014E02603C3C0EB03F0380703E0380601F0000E1507001C16E0EA1803 12380030150F007016C0EA60075C161FD8E00F158000C05BEA001F4A133F1700133F91C7 FC5E49147E137EA216FE01FE5C5BA215015E485AA215035EA200001407150F6D5C017C13 1F153F6D13FF90391F03CFC0903807FF8F903801FC0F90C7121F5EA2153F93C7FCD807C0 5BD81FE0137E5DA24848485A4A5A01805B39380007C00018495A001C49C8FC6C137C3807 81F83803FFE0C66CC9FC2C407DAB30>I<027CEB018049B413034901801300010F6D5A49 EBE00E6F5A90393F03F838903978007EF80170EB1FF00160EB01E001E05C49495A90C748 C7FC150E5D5D5D5D4A5A4A5A4AC8FC140E5C5C5C5CEB03C049C9FC130E49141C4914185B 49143848481430491470D8039014F048B4495A3A0FEFC007C0391E03F01FD81C01B55A48 6C91C7FC485C00606D5A00E0EB3FF048EB0FC0292D7CAB2D>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fs cmr12 12 39 /Fs 39 127 df0 D<1618163CA2167EA216FFA24B7FA24B6C7EA29238063FE0A24B6C7EA24B6C7EA2923838 07FC153092387003FE15609238E001FF15C002016D7F5D02036E7E92C7FC4A6E7E140602 0E6E7E140C021C6E7E141802386E7E143002706E7E146002E06E7E5C01016F7F5C010370 7E91C9FC183F010683181F4983180F49831807498318034983A249707EA24848701380A2 48CBEA7FC0A20006F03FE0A248F01FF0A2001FBA12F8A24819FCA24819FEA2BCFC48477C C651>I<003FB812FCA60038CA121C0030170C0070170EA200601706A5CCFCAA01601506 A5017FB612FEA60160C81206A590CBFCAB00C01703A66C1707A2006017060070170E007F B812FEA638447CC341>4 D<0103B612FCA390C701F0C8FC6F5A6F5AA8913801FFF0023F EBFF80903A01FF3FDFF0D907F0EBC1FCD91FC0EBC07FD93F00EC1F8001FEED0FE048486F 7E48486F7E48486F7E48486F7E001F834982003F1880007F18C0A249163F00FF18E0A800 7F18C06D167FA2003F1880001F18006D5E000F5F6C6C4B5A6C6C4B5A6C6C4B5A6C6C4B5A 013FED1F80D91FC0027FC7FCD907F0EBC1FCD901FFEBDFF0D9003FB51280020101F0C8FC 9138003FC0A84B7E4B7E0103B612FCA33B447BC346>8 D<027FB67EA39126001FFEC9FC 6F5A6F5AA8B46CEFFF8001E01603D81FF0933807FC006C6C4C5A0007606D161F000360A2 6D163F000160AC6C6C5F187FA4D97F804BC7FCA2013F5E02C01401131F02E04A5A010F5E D907F01407D903F85DD901FC4A5AD900FE4A5A027F027FC8FCDA1FC713FE0207B512F802 0114C09126001FFCC9FCED07F8A84B7E4B7E027FB67EA341447BC34C>II<140C141C1438147014E0EB 01C01303EB0780EB0F00A2131E5BA25B13F85B12015B1203A2485AA3485AA348C7FCA35A A2123EA2127EA4127CA312FCB3A2127CA3127EA4123EA2123FA27EA36C7EA36C7EA36C7E A212017F12007F13787FA27F7FA2EB0780EB03C01301EB00E014701438141C140C166476 CA26>40 D<12C07E12707E7E7E120F6C7E6C7EA26C7E6C7EA21378137C133C133E131E13 1FA2EB0F80A3EB07C0A3EB03E0A314F0A21301A214F8A41300A314FCB3A214F8A31301A4 14F0A21303A214E0A3EB07C0A3EB0F80A3EB1F00A2131E133E133C137C13785BA2485A48 5AA2485A48C7FC120E5A5A5A5A5A16647BCA26>I<16C04B7EB3AB007FBAFCBB1280A26C 1900C8D801E0C9FCB3AB6F5A41407BB84C>43 D<14FF010713E090381F81F890383E007C 01FC133F4848EB1F8049130F4848EB07C04848EB03E0A2000F15F0491301001F15F8A200 3F15FCA390C8FC4815FEA54815FFB3A46C15FEA56D1301003F15FCA3001F15F8A26C6CEB 03F0A36C6CEB07E0000315C06D130F6C6CEB1F806C6CEB3F00013E137C90381F81F89038 07FFE0010090C7FC28447CC131>48 D<143014F013011303131F13FFB5FC13E713071200 B3B3B0497E497E007FB6FCA3204278C131>II<49B4FC010F13E0013F13FC9038FE01FE3A01F0007F80D803C0EB3FC048C7EA1F E0120EED0FF0EA0FE0486C14F8A215077F5BA26C48130FEA03C0C813F0A3ED1FE0A2ED3F C01680ED7F0015FE4A5AEC03F0EC1FC0D90FFFC7FC15F090380001FCEC007FED3F80ED1F C0ED0FE016F0ED07F816FC150316FEA2150116FFA3121EEA7F80487EA416FE491303A200 7EC713FC00701407003015F80038140F6C15F06CEC1FE06C6CEB3FC0D803E0EB7F803A01 FE01FE0039007FFFF8010F13E0010190C7FC28447CC131>II<121EEA7F80A2 EAFFC0A4EA7F80A2EA1E00C7FCB3A5121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A2B78AA 1B>58 D<121EEA7F80A2EAFFC0A4EA7F80A2EA1E00C7FCB3A5121E127FEAFF80A213C0A4 127F121E1200A512011380A3120313005A1206120E120C121C5A5A12600A3E78AA1B>I< 007FBAFCBB1280A26C1900CEFCB0007FBAFCBB1280A26C190041187BA44C>61 D91 D93 D<130C131E133F497EEBF3C03801E1E03803C0F03807 807848487E001E7F487F0070EB038048EB01C00040EB00801A0E75C331>I97 D99 D<167FED3FFFA315018182B3EC7F80903803 FFF090380FC07C90383F000E017E1307496D5AD803F87F48487F5B000F81485AA2485AA2 127FA290C8FC5AAB7E7FA2123FA26C7EA2000F5D7F6C6C5B00035C6C6C9038077F806C6C 010E13C0013F011C13FE90380FC0F8903803FFE09026007F0013002F467DC436>IIII105 D107 DII<3901FC01FE00FF903807FFC091381E07F0913838 01F8000701707F0003EBE0002601FDC07F5C01FF147F91C7FCA25BA35BB3A8486CECFF80 B5D8F83F13FEA32F2C7DAB36>II<3901FC 03FC00FF90380FFF8091383C07E091387001F83A07FDE000FE00030180137FD801FFEC3F 8091C7EA1FC04915E049140F17F0160717F8160317FCA3EE01FEABEE03FCA3EE07F8A217 F0160F6D15E0EE1FC06D143F17806EEB7E00D9FDC05B9039FCF003F891383C0FE091381F FF80DA03FCC7FC91C9FCAE487EB512F8A32F3F7DAB36>I<3903F803F000FFEB1FFCEC3C 3EEC707F0007EBE0FF3803F9C000015B13FBEC007E153C01FF13005BA45BB3A748B4FCB5 12FEA3202C7DAB26>114 D<90383FE0183901FFFC383907E01F78390F0003F8001E1301 481300007C1478127800F81438A21518A27EA27E6C6C13006C7E13FC383FFFE06C13FC6C 13FF6C14C06C14E0C614F0011F13F81300EC0FFC140300C0EB01FE1400157E7E153EA27E A36C143C6C147C15786C14F86CEB01F039F38003E039F1F00F8039E07FFE0038C00FF01F 2E7DAC26>I<1306A5130EA4131EA3133E137EA213FE12011207001FB512F0B6FCA2C648 C7FCB3A4150CAA017E131C017F1318A26D133890381F8030ECC070903807E0E0903801FF C09038007F001E3E7EBC26>II120 D<01F81302D803FE13073907FF800E48EBE0 1C391F1FF8F8393807FFF0D8700113E039E0007FC00040EB1F00200978C131>126 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ft cmsy10 12 19 /Ft 19 115 df<007FB912E0BA12F0A26C18E03C04789A4D>0 D<121FEA3F80EA7FC0EA FFE0A5EA7FC0EA3F80EA1F000B0B789E1C>I<16C04B7EB3AC007FBA1280BB12C0A26C19 80C8D801E0C9FCB3A9007FBA1280BB12C0A26C198042427BC14D>6 D<19E0F003F0180FF03FE0F0FF80943803FE00EF0FF8EF3FE0EFFF80DC03FEC7FCEE0FF8 EE3FE0EEFF80DB03FEC8FCED1FF8ED7FE0913801FF80DA07FEC9FCEC1FF0EC7FC04948CA FCEB07FCEB1FF0EB7FC04848CBFCEA07FCEA1FF0EA7FC048CCFCA2EA7FC0EA1FF0EA07FC EA01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FC913801FF809138007F E0ED1FF8ED07FE923800FF80EE3FE0EE0FF8EE03FE933800FF80EF3FE0EF0FF8EF03FE94 3800FF80F03FE0F00FF01803F000E01900B0007FB912E0BA12F0A26C18E03C4E78BE4D> 20 D<127012FCB4FCEA7FC0EA1FF0EA07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038 007FC0EC1FF0EC07FC913801FF809138007FE0ED1FF8ED07FE923800FF80EE3FE0EE0FF8 EE03FE933800FF80EF3FE0EF0FF8EF03FE943800FF80F03FE0F00FF0A2F03FE0F0FF8094 3803FE00EF0FF8EF3FE0EFFF80DC03FEC7FCEE0FF8EE3FE0EEFF80DB03FEC8FCED1FF8ED 7FE0913801FF80DA07FEC9FCEC1FF0EC7FC04948CAFCEB07FCEB1FF0EB7FC04848CBFCEA 07FCEA1FF0EA7FC048CCFC12FC1270CDFCB0007FB912E0BA12F0A26C18E03C4E78BE4D> I<037FB612E00207B712F0143F91B812E0010301C0C9FCD907FCCAFCEB0FE0EB3F8049CB FC13FC485A485A485A5B485A121F90CCFC123EA2123C127CA2127812F8A25AA87EA21278 127CA2123C123EA27E7F120F6C7E7F6C7E6C7E6C7E137E6D7EEB1FE0EB07FC6DB47E0100 90B712E0023F16F01407020016E03C3A78B54D>26 D<1AF0A3861A78A21A7C1A3CA21A3E 1A1E1A1F747EA2747E747E87747E747E1B7E87757EF30FE0F303F8007FBC12FEBE1280A2 6CF3FE00CEEA03F8F30FE0F31F8051C7FC1B7E63505A505A63505A505AA250C8FC1A1E1A 3E1A3CA21A7C1A78A21AF862A359347BB264>33 D<49B4EF3FC0010F01E0923803FFF801 3F01FC030F13FE4901FF92383FE01F48B66C91397E0007C02603F80301E0D901F8EB01E0 2807E0007FF049486D7E01806D6CD907C0147048C76C6C494880001EDA07FE49C87E001C 6E6C013E150C486E6D48150E71481506486E01E0160793387FF1F0006092263FF3E08193 381FFBC000E004FF1780486F4915017090C9FC82707F8482717E844D7E6C4B6D15030060 04EF1700933803E7FE0070922607C7FF5DDC0F837F003004816D140E00384BC6FC001803 3E6D6C5C001C4B6D6C143C6C4BD91FFC5C6C4A486D6C5C6DD907E06D6C13036C6C49486D 9038E00FE0D801F0013FC890B55A27007C03FE6F91C7FC90263FFFF8031F5B010F01E003 0313F8D901FECAEA7FC0592D7BAB64>49 D<92B6FC02071580143F91B7120001030180C8 FCD907FCC9FCEB1FE0EB3F80017ECAFC5B485A485A485A5B485A121F90CBFC123EA2123C 127CA2127812F8A25AA2B9FC1880A2180000F0CBFCA27EA21278127CA2123C123EA27E7F 120F6C7E7F6C7E6C7E6C7E137E6D7EEB1FE0EB07FC6DB47E010090B6FC023F1580140702 001500313A78B542>I<1706170F171FA2173EA2177CA217F8A2EE01F0A2EE03E0A2EE07 C0A2EE0F80A2EE1F00A2163EA25EA25EA24B5AA24B5AA24B5AA24B5AA24BC7FCA2153EA2 5DA25DA24A5AA24A5AA24A5AA24A5AA24AC8FCA2143EA25CA25CA2495AA2495AA2495AA2 495AA249C9FCA2133EA25BA25BA2485AA2485AA2485AA2485AA248CAFCA2123EA25AA25A A25A1260305C72C600>54 D<0060171800F0173C6C177CA200781778007C17F8A2003C17 F0003E1601A26CEE03E0A26C17C06D1507A2000717806D150FA26C6CED1F00A20001161E 6D153EA20000163C90B712FCA26D5DA2013CC85A013E1401A2011E5D011F1403A26D5D6E 1307A26D6C495AA2010392C7FC6E5BA20101141E6E133EA26D6C5BA202781378027C13F8 A2023C5BEC3E01A26E485AA2020F5B1587A202075B15CFA26EB4C8FCA26E5AA36E5AA315 781530364780C437>56 D<0060170C00F0171EB3B3A66C173EA20078173C007C177C007E 17FC003E17F86CEE01F06D15036C6CED07E06C6CED0FC0D803F8ED3F80D801FEEDFF0026 007FC0EB07FCD93FFCEB7FF8010FB612E001031580D9007F01FCC7FC020713C0373D7BBA 42>91 D102 D<12FEEAFFE0EA07F8EA00FEEB7F806D7E6D7E130F6D7EA26D7EB3AD6D7EA26D7E806E7E 6E7EEC0FE0EC03FC913800FFE0A2913803FC00EC0FE0EC3FC04A5A4AC7FC5C495AA2495A B3AD495AA2495A131F495A495A01FEC8FCEA07F8EAFFE048C9FC236479CA32>I<126012 F0B3B3B3B3B3A81260046474CA1C>106 D<0070130700F01480B3B3B3B3B3A800701400 196474CA32>I<126012F07EA21278127CA2123C123EA2121E121FA26C7EA212077FA212 037FA212017FA26C7EA21378137CA2133C133EA2131E131FA26D7EA2130780A2130380A2 130180A26D7EA21478147CA2143C143EA280A28081A2140781A2140381A26E7EA2140081 A21578157CA2153C153EA281A2811680A2150716C0A2150316E0A2ED01F0A2150016F8A2 1678167CA2163C163EA2161E160C27647BCA32>110 D<1B0C1B1E1B3EA21B7CA21BF8A2 F201F0A2F203E0A2F207C0A2F20F80A2F21F00A21A3EA262A262A24F5AA2621903A24F5A A24F5AA24FC7FCA2193EA261A261A24E5AA24E5AA24E5AA24E5AA2010C4CC8FC133C017C 163EEA01FE00035F487E001E5F00387FD8707F4B5A00E07FD8003F4B5A80011F4B5AA26E 4A5A130F6E4AC9FC13076E143E13036E5C13016E5C7F6F5B027F1301A26F485A143F6F48 5A141F6F485A140F6F48CAFC1407EDFC3E14035E15FE02015B15FF6E5BA26F5AA26F5AA2 6F5AA26FCBFC150E4F647A8353>112 D114 D E %EndDVIPSBitmapFont /Fu 167[58 2[58 49 44 53 58 44 58 58 71 49 58 1[27 58 58 44 49 58 53 53 58 65[{TeXBase1Encoding ReEncodeFont}21 79.701 /Times-Roman rf /Fv 87[33 19[44 44 24[44 50 50 72 50 50 28 39 33 50 50 50 50 78 28 50 28 28 50 50 33 44 50 44 50 44 3[33 1[33 61 72 72 94 72 1[61 55 66 1[55 72 72 89 61 72 39 33 72 72 55 61 72 66 66 72 6[28 50 50 50 50 50 50 50 50 50 50 28 25 33 25 2[33 33 33 33[33 2[55 2[{TeXBase1Encoding ReEncodeFont}75 99.6264 /Times-Roman rf %DVIPSBitmapFont: Fw cmsy10 10 1 /Fw 1 50 df49 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fx cmmi10 10 1 /Fx 1 60 df<121C127FEAFF80A213C0A3127F121C1200A412011380A2120313005A1206 120E5A5A5A12600A19798817>59 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fy cmr10 10 3 /Fy 3 92 df<12C07E12707E7E7E120F6C7E6C7EA26C7E6C7EA21378A2137C133C133E13 1EA2131F7FA21480A3EB07C0A6EB03E0B2EB07C0A6EB0F80A31400A25B131EA2133E133C 137C1378A25BA2485A485AA2485A48C7FC120E5A5A5A5A5A13527CBD20>41 D48 D91 D E %EndDVIPSBitmapFont /Fz 171[41 37 44 14[44 44 48 65[{TeXBase1Encoding ReEncodeFont}6 66.4176 /Times-Roman rf /FA 87[28 17[42 27[37 42 42 60 42 42 23 32 28 42 42 42 42 65 23 42 23 23 42 42 28 37 42 37 42 37 3[28 1[28 51 60 60 78 60 60 51 46 55 1[46 60 60 74 51 60 32 28 60 60 46 51 60 55 55 60 5[23 23 42 42 42 42 42 42 42 42 42 42 1[21 28 21 47 1[28 28 28 22[28 10[28 5[{TeXBase1Encoding ReEncodeFont}76 83.022 /Times-Roman rf /FB 87[33 51[33 39 44 1[55 50 55 83 3[28 55 2[44 3[50 7[72 1[100 1[72 66 55 72 1[61 78 72 94 66 2[39 78 78 61 66 72 72 66 72 8[50 50 1[50 50 50 50 50 50 1[25 46[{TeXBase1Encoding ReEncodeFont}41 99.6264 /Times-Bold rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: A4 %%EndSetup %%Page: 1 1 1 0 bop 488 716 a FB(ABSOLUTEL)-9 b(Y)34 b(CONTINUOUS)e(SPECTR)m(UM)h (OF)f(SCHR)2958 693 y(\250)2936 716 y(ODINGER)491 832 y(OPERA)-9 b(T)n(ORS)32 b(WITH)g(SLO)-5 b(WL)c(Y)32 b(DECA)-10 b(YING)32 b(AND)g(OSCILLA)-9 b(TING)1631 948 y(PO)l(TENTIALS)1125 1197 y FA(A.)20 b(LAPTEV)-11 b(,)20 b(S.)h(N)m(ABOK)n(O)f(AND)h(O.)f (SAFR)m(ONO)l(V)756 1437 y(A)t Fz(B)t(S)t(T)t(R)t(A)q(C)t(T)o FA(.)44 b(The)23 b(aim)h(of)f(this)h(paper)e(is)i(to)g(e)o(xtend)e(a)i (class)g(of)f(potentials)g(for)754 1537 y(which)d(the)g(absolutely)e (continuous)g(spectrum)h(of)h(the)g(corresponding)c(multidi-)754 1637 y(mensional)21 b(Schr)7 b(\250)-35 b(odinger)19 b(operator)g(is)k(essentially)f(supported)d(by)i Fy([0)p Fx(;)14 b Fw(1)p Fy(\))p FA(.)29 b(Our)754 1736 y(main)c(theorem)e (states)i(that)g(this)g(property)d(is)j(preserv)o(ed)e(for)g(slo)n(wly) i(decaying)754 1836 y(potentials)h(pro)o(vided)d(that)i(there)g(are)h (some)f(oscillations)h(with)g(respect)f(to)h(one)754 1936 y(of)20 b(the)h(v)n(ariables.)1555 2403 y Fv(1.)52 b(I)t Fu(N)t(T)t(R)q(O)t(D)t(U)t(C)t(T)5 b(I)t(O)g(N)555 2578 y Fv(In)25 b(this)f(paper)h(we)g(pro)o(v)o(e)e(that)h(the)h (absolutely)e(continuous)g(spectrum)h(of)h(a)g(class)456 2694 y(of)d(Schr)8 b(\250)-41 b(odinger)22 b(operators)g Ft(\000)p Fs(\001)12 b(+)g Fr(V)46 b Fv(in)21 b Fr(L)1989 2658 y Fq(2)2029 2694 y Fs(\()p Fp(R)2133 2658 y Fo(d)2180 2694 y Fs(\))p Fv(,)h Fr(d)28 b Ft(\025)g Fs(3)22 b Fv(is)g (essentially)e(supported)456 2810 y(by)36 b Fs([0)p Fr(;)17 b Ft(1)p Fs(\))p Fv(.)67 b(This)36 b(means)g(that)h(the)g(spectral)f (projection)g(corresponding)g(to)h(an)o(y)456 2926 y(subset)24 b(of)g(positi)n(v)o(e)f(Lebesgue)i(measure)f(is)h(not)f(zero.)555 3043 y(In)30 b(particular)l(,)i(our)e(results)f(can)i(be)f(applied)f (to)h(the)g(follo)n(wing)e(classes)i(of)g(long)456 3159 y(range)25 b(potentials:)555 3346 y Fn(Example)g(I)p Fv(.)g(Let)f Fr(d)k Fs(=)f(3)e Fv(and)1350 3532 y Fr(V)c Fs(\()p Fr(x;)c(y)t(;)g(z)t Fs(\))28 b(=)f Fr(v)1926 3547 y Fq(1)1966 3532 y Fs(\()p Fr(x)p Fs(\))p Fr(v)2144 3547 y Fq(2)2184 3532 y Fs(\()p Fr(y)t Fs(\))p Fr(v)2359 3547 y Fq(3)2397 3532 y Fs(\()p Fr(z)t Fs(\))p Fr(;)456 3737 y Fv(where)34 b Fr(v)780 3752 y Fo(j)817 3737 y Fs(\()p Fr(t)p Fs(\))44 b(=)1093 3662 y Fm(P)1198 3766 y Fo(n)1262 3737 y Fr(c)1304 3686 y Fq(\()p Fo(j)t Fq(\))1304 3747 y Fo(n)1395 3737 y Fr(\022)1440 3752 y Fo(j)1477 3737 y Fs(\()p Fr(t)29 b Ft(\000)g Fr(x)1740 3752 y Fo(n)1788 3737 y Fs(\()p Fr(j)6 b Fs(\)\))p Fv(,)36 b Fr(\022)2054 3752 y Fo(j)2135 3737 y Ft(2)45 b Fr(C)2323 3701 y Fl(1)2398 3737 y Fs(\()p Fp(R)5 b Fs(\))40 b Fv(and)2758 3662 y Fm(P)2863 3766 y Fo(n)2927 3737 y Ft(j)p Fr(c)2997 3686 y Fq(\()p Fo(j)t Fq(\))2997 3747 y Fo(n)3088 3737 y Ft(j)3116 3701 y Fq(4)3199 3737 y Fr(<)45 b Ft(1)p Fv(,)456 3853 y Fr(j)h Fs(=)40 b(1)p Fr(;)17 b Fs(2)p Fr(;)g Fs(3)p Fv(.)49 b(Assume)31 b(that)g(functions)g Fr(\022)1949 3868 y Fq(2)2020 3853 y Fv(and)h Fr(\022)2241 3868 y Fq(3)2312 3853 y Fv(are)h(of)e(\002nite)h(support)e(and)i(the)456 3969 y(support)18 b(of)i(the)g(F)o(ourier)f(transform)g(of)h Fr(\022)1885 3984 y Fq(1)1945 3969 y Fv(is)f(\002nite)g(and)h(does)f (not)h(intersect)f(a)h(neigh-)456 4086 y(bourhood)g(of)i(zero.)31 b(No)n(w)-6 b(,)21 b(choose)h(the)f(sequences)h Fr(x)2346 4101 y Fo(n)2394 4086 y Fs(\()p Fr(j)6 b Fs(\))21 b Fv(so)h(that)f Fr(v)2866 4101 y Fo(j)2931 4086 y Ft(2)28 b Fr(L)3091 4050 y Fq(4)3131 4086 y Fs(\()p Fp(R)5 b Fs(\))28 b Fv(and)456 4202 y(the)23 b(potentials)g Fr(v)1060 4217 y Fo(j)1097 4202 y Fv(,)h Fr(j)33 b Fs(=)28 b(1)p Fr(;)17 b Fs(2)p Fr(;)g Fs(3)p Fr(;)23 b Fv(are)h(sparse)g(\(see)h(the)e(papers)i(of)f (A.Kisele)n(v)-6 b(,)22 b(Y)-13 b(.Last)456 4318 y(and)28 b(B.Simon[13)o(])h(and)f(also)f(D.Pearson)i([30]\).)41 b(Note)28 b(that)g(in)f(this)h(case)g(for)h(each)456 4434 y Fr(j)50 b Fs(=)44 b(1)p Fr(;)17 b Fs(2)p Fr(;)g Fs(3)33 b Fv(there)h(is)g(a)g(set)f(of)h(a)h(positi)n(v)o(e)c(Lebesgue) j(measure)g(containing)e(only)456 4551 y(singular)23 b(continuous)g(spectrum)h(of)g(the)h(corresponding)e(one-dimensional)g (opera-)456 4667 y(tors)30 b(with)g(the)h(potential)f Fr(v)1424 4682 y Fo(j)1460 4667 y Fv(.)49 b(Ho)n(we)n(v)o(er)l(,)31 b(it)g(follo)n(ws)e(from)i(our)g(Theorem)f(2.1)h(that)456 4783 y(the)g(absolutely)f(continuous)f(spectrum)i(of)g(the)h(operator)f Ft(\000)p Fs(\001)d(+)f Fr(V)53 b Fv(in)30 b Fr(L)3119 4747 y Fq(2)3159 4783 y Fs(\()p Fp(R)3263 4747 y Fq(3)3309 4783 y Fs(\))h Fv(is)456 4899 y(essentially)23 b(supported)h(by)g Fs([0)p Fr(;)17 b Ft(1)p Fs(\))p Fv(.)p 456 5025 499 4 v 555 5116 a FA(1991)25 b Fk(Mathematics)h(Subject)f (Classi\002cation.)40 b FA(Primary)26 b(35P15;)i(Secondary)d(35L15,)h (47A75,)456 5216 y(35J10.)1931 5315 y Fj(1)p eop %%Page: 2 2 2 1 bop 456 251 a Fj(2)845 b(LAPTEV)-10 b(,)17 b(N)m(ABOK)n(O)i(AND)f (SAFR)m(ONO)l(V)555 450 y Fn(Example)k(II)p Fv(.)h(The)f(same)h (statement)e(is)h(true)g(for)h(a)f(3-dimensional)f(operator)h(with)456 566 y(the)i(potential)1123 767 y Fr(V)d Fs(\()p Fr(x;)c(y)t(;)g(z)t Fs(\))28 b(=)f Fr(v)1699 782 y Fq(1)1739 767 y Fs(\()p Fr(x)p Fs(\))p Fr(v)t Fs(\()p Fr(y)t(;)17 b(z)t Fs(\))p Fr(;)116 b(v)31 b Ft(2)d Fr(L)2523 726 y Fq(4)2563 767 y Fs(\()p Fp(R)2667 726 y Fq(2)2712 767 y Fs(\))p Fr(:)456 968 y Fv(Here)d Fr(v)721 983 y Fq(1)786 968 y Fv(is)f(a)h(so-called)g (W)l(igner)n(-v)n(on)f(Neumann)g(potential)983 1246 y Fr(v)1030 1261 y Fq(1)1070 1246 y Fs(\()p Fr(x)p Fs(\))k(=)1373 1122 y Fo(m)1332 1151 y Fm(X)1343 1361 y Fo(j)t Fq(=1)1493 1246 y Fr(c)1535 1261 y Fo(j)1581 1179 y Fs(sin\()p Fr(!)1800 1194 y Fo(j)1836 1179 y Fr(x)p Fs(\))23 b(+)f Fr(o)p Fs(\(1\))p 1581 1223 641 4 v 1725 1314 a(1)g(+)g Ft(j)p Fr(x)p Ft(j)2005 1284 y Fo(p)2041 1294 y Fi(j)2231 1246 y Fr(;)266 b Ft(j)p Fr(x)p Ft(j)28 b(!)f(1)p Fr(;)456 1552 y Fv(where)35 b Fr(!)795 1567 y Fo(j)877 1552 y Fr(>)45 b Fs(0)p Fv(,)37 b Fr(p)1158 1567 y Fo(j)1240 1552 y Fr(>)45 b Fs(1)p Fr(=)p Fs(4)p Fv(,)37 b Fr(c)1612 1567 y Fo(j)1694 1552 y Ft(2)46 b Fp(R)5 b Fv(,)43 b Fr(m)j Ft(2)g Fp(N)9 b Fv(,)43 b(is)34 b(a)h(function)e(whose)i(F)o (ourier)456 1668 y(transform)c(v)n(anishes)f(on)i(a)g(small)f(interv)n (al)f(containing)h(zero.)52 b(F)o(or)32 b(e)o(xample,)h(one)456 1785 y(can)25 b(consider)1136 2042 y Fr(v)1183 2057 y Fq(1)1222 2042 y Fs(\()p Fr(x)p Fs(\))j(=)g Fv(Re)1678 1917 y Fo(m)1637 1947 y Fm(X)1648 2157 y Fo(j)t Fq(=1)1798 1906 y Fm(Z)1897 1933 y Fq(1+)p Fo(!)2031 1943 y Fi(j)1853 2132 y Fo(!)1897 2142 y Fi(j)2105 1974 y Fr(C)2175 1989 y Fo(j)2228 1974 y Fs(exp)q(\()p Fr(ik)s(x)p Fs(\))p 2094 2019 512 4 v 2094 2110 a(\()p Fr(k)e Ft(\000)c Fr(!)2369 2125 y Fo(j)2406 2110 y Fs(\))2444 2080 y Fq(1)p Fl(\000)p Fo(p)2570 2090 y Fi(j)2632 2042 y Fr(dk)s(;)456 2341 y Fv(with)i(appropriate)g(constants)g Fr(C)1602 2356 y Fo(j)1638 2341 y Fv(.)555 2542 y(Our)f(w)o(ork)g(dif)n(fers)f(from)g (the)h(results)f(obtained)g(in)g(the)h(scattering)f(theory)-6 b(,)22 b(where)456 2658 y(the)i(e)o(xistence)h(of)g(w)o(a)n(v)o(e)f (operators)h(is)g(pro)o(v)o(ed)e(either)i(for)h(the)e(potentials)g Ft(j)p Fr(V)d Fs(\()p Fr(x)p Fs(\))p Ft(j)28 b(\024)456 2775 y Fr(C)7 b Fs(\(1)24 b(+)i Ft(j)p Fr(x)p Ft(j)p Fs(\))895 2739 y Fl(\000)p Fq(1)p Fl(\000)p Fo(")1076 2775 y Fv(,)k Fr(")35 b(>)g Fs(0)29 b Fv(or)g Ft(j)p Fr(V)21 b Fs(\()p Fr(x)p Fs(\))p Ft(j)26 b Fs(+)f Ft(jr)p Fr(V)c Fs(\()p Fr(x)p Fs(\))p Ft(j)p Fs(\(1)k(+)g Ft(j)p Fr(x)p Ft(j)p Fs(\))35 b Ft(\024)h Fr(C)7 b Fs(\(1)25 b(+)g Ft(j)p Fr(x)p Ft(j)p Fs(\))3202 2739 y Fl(\000)p Fo(")3293 2775 y Fv(.)43 b(In)456 2891 y(this)26 b(case)i(the)f (corresponding)f(a.c.)38 b(property)27 b(of)g(the)g(spectrum)g(is)g(a)g (byproduct)f(of)456 3007 y(much)j(stronger)g(results)g(on)h(the)f (unitary)g(equi)n(v)n(alence)g(of)h(the)f(operators)h Ft(\000)p Fs(\001)h Fv(and)456 3123 y Ft(\000)p Fs(\001)23 b(+)f Fr(V)f Fv(.)555 3240 y(As)35 b(in)g(our)h(pre)n(vious)e(paper)i ([15)o(])g(the)f(multidimensional)d(case)k(is)f(reduced)h(to)456 3356 y(a)e(problem)e(for)i(a)g(one-dimensional)e(second)i(order)g (elliptic)e(inte)o(gro-dif)n(ferential)456 3472 y(operator)-5 b(.)29 b(The)22 b(\223potential\224)f(type)h(term)g(appears)g(to)f(be)i (a)f(dissipati)n(v)o(e)d(Fredholm)j(in-)456 3588 y(te)o(gral)d (operator)h(depending)f(on)h(the)g(spectral)f(parameter)-5 b(.)29 b(Such)21 b(an)f(operator)g(might)456 3705 y(ha)n(v)o(e)30 b(poles)f(appearing)i(in)f(an)g(operator)g(v)o(ersion)g(of)g(the)g (so-called)g(\002rst)h(Buslae)n(v-)456 3821 y(F)o(addee)n(v-Zakharo)o (v)j(\(BFZ\))j(trace)f(formula.)62 b(Their)36 b(contrib)n(ution)d (appears)j(with)456 3937 y(the)24 b(\224right\224)h(sign)f(and)h (therefore)g(can)g(be)g(ignored.)555 4053 y(There)k(are)g(tw)o(o)e(ne)n (w)h(elements)f(compared)h(with)f([15].)41 b(One)28 b(of)g(them)g (suggests)456 4169 y(ne)n(w)19 b(\224spectrally)g(local\224)g (Lieb-Thirring)f(inequalities)g(for)i(the)f Fs(3)p Fr(=)p Fs(2)f Fv(moments)g(of)i(the)456 4286 y(ne)o(gati)n(v)o(e)31 b(eigen)l(v)n(alues)i(of)h(Schr)8 b(\250)-41 b(odinger)34 b(operators)g(\(compare)h(with)e(O.Safrono)o(v)456 4402 y([20)o(]\).)j(Before)27 b(applying)e(this)g(result)h(we)h(need)f(an)g (ar)n(gument)g(from)g(A.Lapte)n(v)g(and)456 4518 y(T)-7 b(.W)f(eidl)32 b([16])h(lifting)f(the)h(corresponding)f(eigen)l(v)n (alue)h(estimates)f(for)h(their)g Fs(1)p Fr(=)p Fs(2)p Fv(-)456 4634 y(moments)22 b(to)h Fs(3)p Fr(=)p Fs(2)p Fv(-moments)f(by)h(using)g(an)h(induction)e(with)h(respect)h(to)f (dimension.)456 4751 y(This)32 b(ar)n(gument)g(forces)h(us)g(to)f (consider)g(the)h(problem)f(starting)g(from)g(dimension)456 4867 y Fr(d)f Ft(\025)h Fs(3)p Fv(.)37 b(The)27 b(second)g(ne)n(w)f (element)h(is)f(concerned)h(with)g(a)g(parallel)g(consideration)456 4983 y(of)e(a)g(couple)f(of)h(Schr)8 b(\250)-41 b(odinger)25 b(operators)g(with)f(potentials)f Fr(V)47 b Fv(and)25 b Ft(\000)p Fr(V)d Fv(.)31 b(This)24 b(leads)456 5099 y(to)30 b(the)h(cancellation)f(of)h(the)f(term)1709 5019 y Fm(R)1775 5045 y Fl(1)1756 5134 y Fq(0)1866 5019 y Fm(R)1913 5134 y Fh(S)1956 5115 y Fi(d)p Fg(\000)p Ff(1)2086 5099 y Fr(V)21 b(d\022)s(dr)33 b Fv(appearing)e(in)f(the)h(BFZ)g (\002rst)456 5216 y(trace)25 b(formula.)p eop %%Page: 3 3 3 2 bop 1246 251 a Fj(ABSOLUTEL)-7 b(Y)22 b(CONTINUOUS)h(SPECTR)m(UM) 751 b(3)555 450 y Fv(Note)54 b(that)f(the)g(\002rst)h(result)f(based)g (on)h(Buslae)n(v-F)o(addee)n(v-Zakharo)o(v)e(trace)456 566 y(formulae)46 b(for)g(the)g(study)f(of)h(the)g(a.c.)96 b(properties)45 b(of)i(the)f(spectrum)f(of)h(one-)456 683 y(dimensional)31 b(Schr)8 b(\250)-41 b(odinger)34 b(operators)f(w)o(as)h(suggested)f(in)g(the)g(paper)h(by)f(P)-11 b(.Deift)456 799 y(and)34 b(R.Killip)f([10].)59 b(Their)35 b(theorem)e(ga)n(v)o(e)h(a)g(natural)h(generalization)e(of)i(the)f(re-) 456 915 y(sults)28 b(obtained)i(by)f(by)h(M.Christ,)h(A.Kisele)n(v)e (and)h(C.Remling)f(in)h([7],)h(M.Christ,)456 1031 y(A.Kisele)n(v)42 b([8])h(and)g(C.Remling[19].)86 b(R.Killip)42 b([12])h(w)o(as)g (\002rst)h(in)e(pro)o(ving)g(a)456 1155 y(\223local\224)c (one-dimensional)e(result.)71 b(That)38 b(is)g(if)2236 1130 y Fs(^)2221 1155 y Fr(V)74 b Ft(2)53 b Fr(L)2537 1119 y Fq(2)2577 1155 y Fs(\(2)p Fr(a;)17 b Fs(2)p Fr(b)p Fs(\))p Fv(,)41 b Fr(a)53 b(>)f Fs(0)p Fv(,)42 b(and)456 1272 y Fr(V)85 b Ft(2)64 b Fr(L)794 1236 y Fq(3)834 1272 y Fs(\()p Fp(R)5 b Fs(\))p Fv(,)55 b(then)43 b(the)i(absolutely)d (continuous)h(spectrum)h(\002lls)f(the)i(interv)n(al)456 1388 y Fs(\()p Fr(a)545 1352 y Fq(2)584 1388 y Fr(;)17 b(b)669 1352 y Fq(2)709 1388 y Fs(\))p Fv(.)555 1504 y(Our)33 b(theorems)f(requires)h(only)g Fr(V)64 b Ft(2)43 b Fr(L)2002 1468 y Fq(4)2042 1504 y Fs(\()p Fp(R)5 b Fs(\))39 b Fv(rather)33 b(than)g(the)f(condition)g Fr(V)64 b Ft(2)456 1620 y Fr(L)522 1584 y Fq(3)561 1620 y Fs(\()p Fp(R)5 b Fs(\))p Fv(.)51 b(Note)30 b(that)f(the)g(second)g Fr(L)1705 1584 y Fq(2)1745 1620 y Fv(-condition)g(\(2.10\))g(on)g(the)h (F)o(ourier)f(transform)456 1737 y(of)i Fr(V)53 b Fv(with)31 b(respect)h(to)f(one)h(of)f(v)n(ariables)g(near)h(the)f(origin,)i (becomes)e(interesting)456 1853 y(if)j(there)g(are)h(cancellations)f (pro)o(vided)f(by)h(oscillations)e(of)i(the)g(potential)f Fr(V)56 b Fv(near)456 1969 y(in\002nity)-6 b(.)555 2085 y(There)37 b(is)f(e)o(xtensi)n(v)o(e)e(literature)i(concerning)g(the)g (properties)g(of)g(the)g(spectrum)456 2202 y(of)g(oscillating)e (potentials)h(starting)g(from)h(the)g(classical)g(W)l(igner)n(-v)n(on)g (Neumann)456 2318 y(construction)21 b([31)o(],)i(see)g(also)f (M.Skrigano)o(v)f([29)o(])i(and)f(H.Behnck)o(e)h([3],)g([4].)30 b(Some)456 2434 y(e)o(xamples)i(of)h(oscillating)e(potentials)g(with)i (respect)g(to)g(the)f(radial)h(v)n(ariable)g(were)456 2550 y(gi)n(v)o(en)23 b(in)h(M.Reed)h(and)g(B.Simon)g([21)o(],)g(v)n (ol.3)f(Ch)i(XI.)555 2666 y(Our)k(Theorems)g(2.1)f(and)h(2.2)g(are)g (applied)g(to)f(a)h(class)g(of)g(potentials)f(described)456 2783 y(in)23 b(terms)f(of)i(the)f(F)o(ourier)g(transform)g(with)g (respect)g(to)g(one)g(of)h(the)f(v)n(ariables.)29 b(Some)456 2899 y(related)41 b(results)g(for)h(a)g(class)f(of)g(Schr)8 b(\250)-41 b(odinger)42 b(operators)f(with)g(anisotropic)f(be-)456 3015 y(ha)n(viour)23 b(of)i(potentials)d(at)i(in\002nity)g(were)h (considered)e(in)h(the)g(paper)h(by)f(V)-13 b(.G.Deich,)456 3131 y(E.L.K)m(orotjae)n(v)22 b(and)j(D.R.)g(Y)-10 b(af)o(ae)n(v)25 b([9].)555 3248 y(This)i(article)g(is)g(a)g(natural)g(de)n(v)o (elopment)e(of)i(our)g(pre)n(vious)f(paper)i([15)o(].)39 b(F)o(or)27 b(the)456 3364 y(sak)o(e)35 b(of)f(completeness)g(we)h (recall)g(the)g(ar)n(guments)f(of)h(Section)g(3-4)f(and)h(8)g(from)456 3480 y([15)o(])25 b(which)g(become)g(in)f(this)g(te)o(xt)g(Sections)g (4-6)h(and)g(10)f(respecti)n(v)o(ely)-6 b(.)1470 3693 y(2.)51 b(T)t Fu(H)t(E)31 b(M)t(A)t(I)t(N)g(R)t(E)t(S)t(U)t(L)m(T)t(S) 555 3868 y Fv(Let)23 b(us)f(consider)g(a)h(Schr)8 b(\250)-41 b(odinger)23 b(operator)g Ft(\000)p Fs(\001)14 b(+)g Fr(V)45 b Fv(in)22 b Fr(L)2644 3831 y Fq(2)2684 3868 y Fs(\()p Fp(R)2788 3831 y Fo(d)2835 3868 y Fs(\))p Fv(,)h Fr(d)k Ft(\025)h Fs(3)p Fv(,)23 b(where)456 4034 y(\(2.1\))465 b Fr(V)49 b Ft(2)28 b Fr(L)1378 3993 y Fl(1)1453 4034 y Fs(\()p Fp(R)1557 3993 y Fo(d)1604 4034 y Fs(\))p Fr(;)116 b(V)21 b Fs(\()p Fr(x)p Fs(\))28 b Ft(!)g Fs(0)p Fr(;)49 b Fs(as)34 b Ft(j)p Fr(x)p Ft(j)27 b(!)h(1)p Fr(:)456 4208 y Fv(Let)628 4183 y Fs(^)613 4208 y Fr(V)47 b Fv(be)25 b(the)f(F)o(ourier)h(transform)f(of)h Fr(V)47 b Fv(with)24 b(respect)h(to)f(the)h(\002rst)g(v)n(ariable)456 4419 y(\(2.2\))983 4394 y Fs(^)969 4419 y Fr(V)c Fs(\()p Fr(\030)5 b(;)17 b(y)t Fs(\))26 b(=)1396 4284 y Fm(Z)1452 4509 y Fh(R)1521 4419 y Fr(e)1566 4378 y Fl(\000)p Fo(i\030)s(s)1715 4419 y Fr(V)c Fs(\()p Fr(s;)17 b(y)t Fs(\))g Fr(ds;)114 b(x)28 b Fs(=)f(\()p Fr(s;)17 b(y)t Fs(\))27 b Ft(2)h Fp(R)2858 4378 y Fo(d)2904 4419 y Fr(:)456 4805 y FB(Theor)n(em)d(2.1.) 39 b Fn(Let)24 b Fr(d)k Ft(\025)g Fs(3)c Fn(and)f(let)g Fr(V)46 b Fn(be)23 b(a)h(r)l(eal)g(valued)f(function)f(on)i Fp(R)3062 4769 y Fo(d)3132 4805 y Fn(obe)m(ying)456 4921 y Fv(\(2.1\))g Fn(and)h(let)f(for)g(some)h Fr(\016)32 b(>)27 b Fs(0)761 5016 y Fm(Z)816 5242 y Fh(R)864 5223 y Fi(d)921 5152 y Fr(V)1000 5111 y Fq(4)1039 5152 y Fs(\()p Fr(x)p Fs(\))17 b Fr(dx)28 b(<)g Ft(1)p Fr(;)1767 5016 y Fm(Z)1823 5242 y Fh(R)1871 5223 y Fi(d)p Fg(\000)p Ff(1)1989 5041 y Fm(\020)2049 5016 y(Z)2148 5043 y Fo(\016)2104 5242 y Fl(\000)p Fo(\016)2213 5152 y Ft(j)2256 5127 y Fs(^)2241 5152 y Fr(V)22 b Fs(\()p Fr(\030)5 b(;)17 b(y)t Fs(\))p Ft(j)2568 5111 y Fq(2)2621 5152 y Fr(d\030)2720 5041 y Fm(\021)2779 5152 y Fr(dy)31 b(<)c Ft(1)p Fr(:)p eop %%Page: 4 4 4 3 bop 456 251 a Fj(4)845 b(LAPTEV)-10 b(,)17 b(N)m(ABOK)n(O)i(AND)f (SAFR)m(ONO)l(V)456 450 y Fn(Then)24 b(the)g(absolutely)f(continuous)f (spectrum)i(of)f(the)h(oper)o(ator)f Ft(\000)p Fs(\001)d(+)f Fr(V)46 b Fn(is)23 b(essen-)456 566 y(tially)g(supported)h(by)h Fs([0)p Fr(;)17 b Ft(1)p Fs(\))p Fn(.)555 874 y Fv(The)38 b(latter)g(theorem)f(gi)n(v)o(es)f(some)h(qualitati)n(v)o(e)f (information)g(about)h(the)h(abso-)456 990 y(lutely)f(continuous)g (spectrum)h(of)h(Schr)8 b(\250)-41 b(odinger)38 b(operators.)72 b(The)38 b(ne)o(xt)g(result)g(is)456 1106 y(related)31 b(to)g(more)g(delicate)g(properties)f(of)i(the)f(a.c.)50 b(spectrum.)f(It)31 b(pro)o(vides)f(some)456 1223 y(quantitati)n(v)o(e) c(characteristics)i(of)h(the)g(spectral)f(measure)h(which)f(is)h(a)g (multidimen-)456 1339 y(sional)21 b(continuous)g(analog)h(of)h(the)f (well-kno)n(wn)f(Sze)o(g)8 b(\005)-41 b(o)23 b(condition)e(for)i (orthogonal)456 1455 y(polynomials)f(and)j(Jacobi)f(matrices)h (\(compare)g(with)f([15]\).)555 1571 y(Let)c Fs(\012)778 1586 y Fq(1)837 1571 y Fv(be)g(the)f(unit)g(ball)g(in)g Fp(R)1599 1535 y Fo(d)1646 1571 y Fv(,)h Fr(@)5 b Fs(\012)1817 1586 y Fq(1)1886 1571 y Fs(=)28 b Fp(S)2051 1535 y Fo(d)p Fl(\000)p Fq(1)2175 1571 y Fv(,)21 b(and)f Fr(V)41 b Fv(be)20 b(a)g(real)g(v)n(alued)e(function)456 1688 y(on)26 b Fp(R)648 1651 y Fo(d)718 1688 y Ft(n)d Fs(\012)861 1703 y Fq(1)901 1688 y Fv(.)36 b(W)-8 b(e)27 b(consider)g(the)f (operator)h Fr(H)34 b Fv(in)26 b Fr(L)2276 1651 y Fq(2)2316 1688 y Fs(\()p Fp(R)2420 1651 y Fo(d)2490 1688 y Ft(n)d Fs(\012)2633 1703 y Fq(1)2673 1688 y Fs(\))k Fv(with)f(the)g(Dirichlet) 456 1804 y(boundary)e(conditions)f(on)h Fp(S)1483 1768 y Fo(d)p Fl(\000)p Fq(1)456 1981 y Fv(\(2.3\))318 b Fr(H)8 b(u)27 b Fs(=)h Fr(H)1322 1996 y Fq(0)1361 1981 y Fr(u)22 b Fs(+)g Fr(V)f(u)27 b Fs(=)h Ft(\000)p Fs(\001)p Fr(u)22 b Fs(+)g Fr(V)g(u;)215 b(u)p Ft(j)2597 1996 y Fo(@)t Fq(\012)2689 2005 y Ff(1)2755 1981 y Fs(=)28 b(0)p Fr(:)456 2158 y Fv(Let)c(us)h(assume)f(for)h(the)g(sak)o(e)g(of)g(simplicity)d (that)i(there)h(is)g Fr(c)2597 2173 y Fq(1)2664 2158 y Fr(>)i Fs(1)e Fv(such)g(that)456 2359 y(\(2.4\))605 b Fr(V)44 b Fs(+)1485 2292 y Fr(\013)1547 2307 y Fo(d)p 1461 2336 151 4 v 1461 2427 a Ft(j)p Fr(x)p Ft(j)1572 2399 y Fq(2)1649 2359 y Fs(=)28 b(0)99 b(for)g(1)27 b Fr(<)h Ft(j)p Fr(x)p Ft(j)f Fr(<)h(c)2581 2374 y Fq(1)2620 2359 y Fr(;)456 2624 y Fv(where)j Fr(\013)792 2639 y Fo(d)871 2624 y Fs(=)996 2577 y Fq(\()p Fo(d)p Fl(\000)p Fq(1\))1176 2554 y Ff(2)p 996 2602 216 4 v 1086 2659 a Fq(4)1248 2624 y Ft(\000)1362 2585 y Fo(d)p Fl(\000)p Fq(1)p 1362 2602 127 4 v 1408 2659 a(2)1499 2624 y Fv(.)49 b(Let)30 b Fr(E)1808 2639 y Fo(H)1876 2624 y Fs(\()p Fr(!)t Fs(\))p Fv(,)h Fr(!)42 b Ft(\032)e Fp(R)5 b Fv(,)38 b(be)31 b(the)g(spectral)f(projection)456 2741 y(of)e(the)g(operator)h Fr(H)8 b Fv(.)41 b(W)-8 b(e)28 b(construct)g(a)h(measure)f Fr(\026)g Fv(on)g(the)h(real)f(line)g(such)g(that)g(for)456 2857 y(spherically)c(symmetric)f(functions)h Fr(f)456 3083 y Fv(\(2.5\))199 b Fs(\()p Fr(E)956 3098 y Fo(H)1023 3083 y Fs(\()p Fr(!)t Fs(\))p Fr(f)5 b(;)17 b(f)11 b Fs(\))26 b(=)1488 2948 y Fm(Z)1543 3173 y Fo(!)1610 3083 y Ft(j)p Fr(F)14 b Fs(\()p Fr(\025)p Fs(\))p Ft(j)1876 3042 y Fq(2)1915 3083 y Fr(d\026)p Fs(\()p Fr(\025)p Fs(\))p Fr(;)115 b(!)31 b Ft(\032)d Fp(R)2563 3098 y Fq(+)2656 3083 y Fs(=)f(\(0)p Fr(;)17 b Ft(1)p Fs(\))p Fr(;)456 3310 y Fv(where)456 3427 y(\(2.6\))456 3590 y Fr(F)d Fs(\()p Fr(\025)p Fs(\))27 b(=)809 3523 y(1)p 806 3568 55 4 v 806 3659 a Fr(k)887 3455 y Fm(Z)987 3481 y Fo(c)1018 3490 y Ff(1)942 3680 y Fq(0)1072 3590 y Fs(sin\()p Fr(k)s Fs(\()p Fr(r)s Ft(\000)p Fs(1\)\))p Fr(f)11 b Fs(\()p Fr(r)s Fs(\))18 b Fr(r)1818 3549 y Fq(\()p Fo(d)p Fl(\000)p Fq(1\))p Fo(=)p Fq(2)2074 3590 y Fr(dr)m(;)111 b Fs(supp)q Fr(f)39 b Ft(\032)28 b(f)p Fr(x)g Fs(:)44 b(1)28 b Fr(<)f Ft(j)p Fr(x)p Ft(j)h Fr(<)f(c)3365 3605 y Fq(1)3405 3590 y Ft(g)p Fr(:)456 3829 y Fv(and)d Fr(k)678 3792 y Fq(2)746 3829 y Fs(=)j Fr(\025)h(>)f Fs(0)p Fv(.)555 3945 y(Let)e(us)f(e)o(xtend)g Fr(V)47 b Fv(by)24 b(zero)i(into)e Fs(\012)1790 3960 y Fq(1)1854 3945 y Fv(and)h(then)g(de\002ne)2503 3920 y Fs(^)2488 3945 y Fr(V)47 b Fv(as)24 b(in)h(\(2.2\).)555 4061 y(The)g(follo)n(wing)e(theorem)h(is)h(the)f(main)g(result)h(of)g (the)f(paper)-5 b(.)456 4308 y FB(Theor)n(em)33 b(2.2.)44 b Fn(Let)32 b Fr(d)40 b Ft(\025)g Fs(3)31 b Fn(and)g(let)h Fr(V)53 b Fn(be)31 b(a)h(r)l(eal)f(valued)g(function)f(on)h Fp(R)3184 4272 y Fo(d)3258 4308 y Ft(n)c Fs(\012)3405 4323 y Fq(1)456 4424 y Fn(obe)m(ying)d Fv(\(2.1\))h Fn(and)i Fv(\(2.4\))p Fn(.)j(Let)456 4669 y Fv(\(2.7\))810 4534 y Fm(Z)865 4759 y Fh(R)913 4740 y Fi(d)949 4759 y Fl(n)p Fq(\012)1035 4768 y Ff(1)1091 4669 y Fr(V)1169 4628 y Fq(4)1209 4669 y Fs(\()p Fr(x)p Fs(\))17 b Fr(dx)27 b(<)h Ft(1)p Fr(;)1937 4534 y Fm(Z)1992 4759 y Fh(R)2040 4740 y Fi(d)p Fg(\000)p Ff(1)2159 4559 y Fm(\020)2218 4534 y(Z)2318 4560 y Fo(\016)2273 4759 y Fl(\000)p Fo(\016)2383 4669 y Ft(j)2426 4644 y Fs(^)2411 4669 y Fr(V)21 b Fs(\()p Fr(\030)5 b(;)17 b(y)t Fs(\))p Ft(j)2737 4628 y Fq(2)2791 4669 y Fr(d\030)2890 4559 y Fm(\021)2949 4669 y Fr(dy)30 b(<)e Ft(1)456 4914 y Fn(for)c(some)g Fr(\016)32 b(>)c Fs(0)p Fn(.)i(Then)456 5146 y Fv(\(2.8\))1418 5010 y Fm(Z)1518 5037 y Fl(1)1473 5236 y Fq(0)1619 5079 y Fs(log\(1)p Fr(=\026)1940 5042 y Fl(0)1962 5079 y Fs(\()p Fr(t)p Fs(\)\))17 b Fr(dt)p 1619 5123 596 4 v 1662 5224 a Fs(\(1)22 b(+)g Fr(t)1904 5195 y Fq(3)p Fo(=)p Fq(2)2014 5224 y Fs(\))2052 5143 y Ft(p)p 2135 5143 36 4 v 81 x Fr(t)2252 5146 y(<)27 b Ft(1)p Fr(;)p eop %%Page: 5 5 5 4 bop 1246 251 a Fj(ABSOLUTEL)-7 b(Y)22 b(CONTINUOUS)h(SPECTR)m(UM) 751 b(5)456 450 y Fn(wher)l(e)25 b Fr(\026)g Fn(is)f(de\002ned)h(in)f Fv(\(2.5\))p Fn(.)31 b(If)39 b Fv(\(2.4\))25 b Fn(is)f(satis\002ed)g (then)g Fv(\(2.8\))h Fn(is)g(equivalent)e(to)456 681 y Fv(\(2.9\))1205 545 y Fm(Z)1305 572 y Fl(1)1261 771 y Fq(0)1406 607 y Fs(log)1532 526 y Fm(\000)1609 568 y Fo(d)p 1588 584 78 4 v 1588 642 a(d\025)1675 607 y Fs(\()p Fr(E)1785 622 y Fo(H)1853 607 y Fs(\()p Fr(\025)p Fs(\))p Fr(f)5 b(;)17 b(f)11 b Fs(\))2180 526 y Fm(\001)2242 607 y Fr(d\025)p 1406 658 943 4 v 1602 763 a Fs(\(1)22 b(+)g Fr(\025)1866 734 y Fq(3)p Fo(=)p Fq(2)1976 763 y Fs(\))2014 678 y Ft(p)p 2097 678 57 4 v 85 x Fr(\025)2387 681 y(>)27 b Ft(\0001)p Fr(;)456 914 y Fn(for)f(any)i(bounded)e (spherically)h(symmetric)f(function)h Fr(f)43 b Ft(6)p Fs(=)32 b(0)27 b Fn(with)g Fs(supp)q Fr(f)44 b Ft(\032)33 b(f)p Fr(x)f Fs(:)456 1030 y(1)27 b Fr(<)h Ft(j)p Fr(x)p Ft(j)f Fr(<)h(c)920 1045 y Fq(1)959 1030 y Ft(g)p Fn(.)456 1248 y(Remark)33 b(1.)55 b Fv(The)33 b(inequality)f(\(2.8\))h (guarantees)g(that)g(the)g(a.c.)h(spectrum)e(of)h Fr(H)41 b Fv(is)456 1364 y(essentially)25 b(supported)h(by)h Fs([0)p Fr(;)17 b Ft(1)p Fs(\))p Fv(,)26 b(since)h Fr(\026)2050 1328 y Fl(0)2104 1364 y Fr(>)k Fs(0)c Fv(almost)f(e)n(v)o(erywhere)g (and)h(gi)n(v)o(es)456 1481 y(quantitati)n(v)o(e)22 b(information)h (about)i(the)f(measure)h Fr(\026)p Fv(.)456 1597 y Fn(Remark)30 b(2.)46 b Fv(If)31 b Fr(d)37 b Fs(=)h(1)p Fv(,)31 b(then)f(the)g (conditions)e(\(2.7\))i(do)g(not)g(pro)o(vide)f(e)o(xistence)g(of)456 1713 y(the)d(absolutely)f(continuous)g(spectrum)h(on)g Fr(R)2096 1728 y Fq(+)2155 1713 y Fv(.)36 b(This)26 b(is)g(con\002rmed) g(by)h(e)o(xamples)456 1829 y(of)37 b(sparse)g(potentials)f (constructed)g(in)h([13].)68 b(The)37 b(v)n(alidity)e(of)i(Theorem)g (2.2)g(in)456 1946 y(dimension)23 b Fr(d)k Fs(=)h(2)c Fv(remains)h(open.)456 2062 y Fn(Remark)d(3.)30 b Fv(The)22 b(equi)n(v)n(alence)g(of)g(\(2.8\))g(and)h(\(2.9\))f(follo)n(ws)f(from) h(the)h(f)o(act)f(that)g(if)h Fr(F)456 2178 y Fv(is)g(de\002ned)h(as)f (in)h(\(2.6\),)f(then)h(the)f(function)g Fs(\(1)17 b(+)h Fr(\025)2262 2142 y Fq(2)2301 2178 y Fs(\))2339 2142 y Fl(\000)p Fq(1)2450 2178 y Fs(log\()p Ft(j)p Fr(F)c Fs(\()p Fr(\025)p Fs(\))p Ft(j)p Fs(\))22 b Fv(is)i(in)f Fr(L)3198 2142 y Fq(1)3238 2178 y Fs(\()p Fp(R)3341 2193 y Fq(+)3406 2178 y Fs(\))456 2294 y Fv(see,)i(for)g(e)o(xample,)f(P)-11 b(.)25 b(K)m(oosis)e([14])i(\(section)f(IIIG2\).)456 2411 y Fn(Remark)31 b(4.)50 b Fv(When)31 b(pro)o(ving)f(Theorem)h(2.2)g (we)h(use)f(the)g(projection)g(operator)g Fr(P)3405 2426 y Fq(0)456 2527 y Fv(on)38 b(the)g(spherical)g(function)g Fr(Y)1580 2542 y Fq(0)1658 2527 y Fv(which)g(leads)g(us)g(to)h(a)f (scalar)h(one-dimensional)456 2643 y(problem)30 b(\(4.2\))h(with)f(an)i (operator)f(v)n(alued)f(potential)g Fr(Q)2489 2658 y Fo(z)2529 2643 y Fv(.)50 b(Had)31 b(we)g(used)g(instead)456 2759 y(of)37 b Fr(P)639 2774 y Fq(0)716 2759 y Fv(the)g(projection)1317 2685 y Fm(P)1422 2711 y Fo(n)1422 2788 y(j)t Fq(=1)1566 2759 y Fr(P)1629 2774 y Fo(j)1665 2759 y Fv(,)k(where)d Fr(P)2075 2774 y Fo(j)2149 2759 y Fv(are)g(projections)e(on)h(the)h (spherical)456 2878 y(functions)f Fr(Y)922 2893 y Fo(j)958 2878 y Fv(,)k(then)d(we)g(w)o(ould)f(ha)n(v)o(e)h(obtained)f(the)h (corresponding)f(system)g(of)456 2994 y(one-dimensional)31 b(equations)g(with)h(an)h(operator)g(v)n(alued)f(potential)g(which)g (could)456 3110 y(be)24 b(treated)g(similarly)-6 b(.)29 b(This)23 b(w)o(ould)g(imply)g(that)h(the)g(multiplicity)d(of)j(the)g (a.c.)31 b(spec-)456 3226 y(trum)24 b(is)g(not)g(smaller)h(than)f Fr(n)p Fv(.)31 b(Since)25 b Fr(n)h Fv(is)e(arbitrary)-6 b(,)24 b(we)h(obtain)f(the)h(a.c.)31 b(spectrum)456 3343 y(is)24 b(of)h(in\002nite)f(multiplicity)-6 b(.)555 3464 y(Denote)25 b(by)1007 3439 y Fs(~)992 3464 y Fr(V)47 b Fv(the)24 b(F)o(ourier)h(transform)f(of)h(the)g Fr(V)46 b Fv(with)24 b(respect)i(to)e(all)g(v)n(ariables)1404 3647 y Fs(~)1390 3672 y Fr(V)d Fs(\()p Fr(\021)t Fs(\))27 b(=)1727 3536 y Fm(Z)1782 3762 y Fh(R)1830 3743 y Fi(d)1887 3672 y Fr(V)22 b Fs(\()p Fr(x)p Fs(\))p Fr(e)2142 3631 y Fl(\000)p Fo(i)p Fq(\()p Fo(\021)r(;x)p Fq(\))2377 3672 y Fr(dx:)456 3998 y FB(Theor)n(em)33 b(2.3.)44 b Fn(Let)32 b Fr(d)40 b Ft(\025)g Fs(3)31 b Fn(and)g(let)h Fr(V)53 b Fn(be)31 b(a)h(r)l(eal)f(valued)g(function)f(on)h Fp(R)3184 3961 y Fo(d)3258 3998 y Ft(n)c Fs(\012)3405 4013 y Fq(1)456 4114 y Fn(obe)m(ying)d Fv(\(2.1\))h Fn(and)i Fv(\(2.4\))p Fn(.)j(Let)456 4344 y Fv(\(2.10\))958 4209 y Fm(Z)1013 4434 y Fh(R)1061 4415 y Fi(d)1097 4434 y Fl(n)p Fq(\012)1183 4443 y Ff(1)1239 4344 y Fr(V)1317 4303 y Fq(4)1357 4344 y Fs(\()p Fr(x)p Fs(\))17 b Fr(dx)27 b(<)h Ft(1)p Fr(;)2084 4209 y Fm(Z)2140 4434 y Fh(R)2188 4415 y Fi(d)2255 4277 y Ft(j)2298 4252 y Fs(~)2283 4277 y Fr(V)21 b Fs(\()p Fr(\021)t Fs(\))p Ft(j)2517 4241 y Fq(2)p 2255 4321 301 4 v 2267 4413 a Fs(1)h(+)g Ft(j)p Fr(\021)t Ft(j)2582 4344 y Fr(d\021)31 b(<)d Ft(1)p Fr(:)456 4570 y Fn(Then)d(the)f(statements)g Fv(\(2.8\))h Fn(and)i Fv(\(2.9\))e Fn(of)f(Theor)l(em)h(2.2)g(hold)f(true)o(.)456 4746 y(Remark)h(5.)31 b Fv(Theorems)25 b(2.2)g(and)g(2.3)f(do)h(not)g (allo)n(w)f(us)h(to)f(resist)h(the)g(temptation)e(of)456 4862 y(formulating)g(the)i(follo)n(wing)d(conjecture:)456 4983 y(Let)30 b Fr(V)61 b Ft(2)40 b Fr(L)909 4947 y Fq(4)948 4983 y Fs(\()p Fp(R)1052 4947 y Fo(d)1099 4983 y Fs(\))p Fv(,)32 b Fs(\012)40 b Ft(\032)f Fp(R)1486 4947 y Fo(d)1564 4983 y Fv(be)31 b(an)g(open)g(domain.)48 b(Assume)30 b(that)2959 4958 y Fs(~)2944 4983 y Fr(V)61 b Ft(2)39 b Fr(L)3233 4947 y Fq(2)3273 4983 y Fs(\(\012\))p Fv(.)456 5099 y(Then)22 b(the)h(set)g Ft(f)p Fr(\025)k Fs(:)45 b Fr(\025)27 b Fs(=)h Ft(j)p Fr(\021)t Ft(j)1464 5063 y Fq(2)1502 5099 y Fr(;)34 b Fs(2)p Fr(\021)d Ft(2)d Fs(\012)p Ft(g)23 b Fv(is)g(contained)f(in)h(the)f(essential)g(support) g(of)456 5216 y(the)i(absolutely)g(continuous)f(spectrum)h(of)h(the)g (Schr)8 b(\250)-41 b(odinger)24 b(operator)h Ft(\000)p Fs(\001)e(+)f Fr(V)g Fv(.)p eop %%Page: 6 6 6 5 bop 456 251 a Fj(6)845 b(LAPTEV)-10 b(,)17 b(N)m(ABOK)n(O)i(AND)f (SAFR)m(ONO)l(V)987 450 y Fv(3.)51 b(E)t Fu(S)t(T)t(I)t(M)t(A)-5 b(T)t(E)5 b(S)35 b(F)t(O)t(R)d(T)t(H)t(E)e(D)t(I)t(S)t(C)t(R)t(E)t(T)t (E)35 b(S)t(P)t(E)t(C)t(T)t(R)q(U)5 b(M)555 624 y Fv(Throughout)24 b(the)g(paper)l(,)h Fr(T)1521 639 y Fl(\006)1606 624 y Fv(denotes)f(the)h(positi)n(v)o(e)d(and)j(ne)o(gati)n(v)o(e)e(part)h (of)h(a)h(self)456 741 y(adjoint)f(operator)h Fr(T)14 b Fv(,)27 b(i.e.)36 b Fs(2)p Fr(T)1504 756 y Fl(\006)1594 741 y Fs(=)30 b Ft(j)p Fr(T)14 b Ft(j)23 b(\006)h Fr(T)14 b Fv(.)35 b(Denote)26 b(by)h Fe(S)2606 756 y Fo(p)2645 741 y Fv(,)g Fr(p)k(>)f Fs(0)d Fv(the)f(standard)456 857 y(Neumann-Schatten)e(classes)h(of)g(compact)f(operators)1317 1055 y Fe(S)1400 1070 y Fo(p)1467 1055 y Fs(=)j Ft(f)p Fr(T)58 b Fs(:)j Fv(tr)25 b Fs(\()p Fr(T)2018 1014 y Fl(\003)2057 1055 y Fr(T)14 b Fs(\))2166 1014 y Fo(p=)p Fq(2)2303 1055 y Fr(<)28 b Ft(1g)p Fr(:)456 1264 y Fv(Consider)37 b(a)g(one)g(dimensional)f(Schr)8 b(\250)-41 b(odinger)37 b(operator)g Fr(J)60 b Fs(=)50 b Ft(\000)2881 1225 y Fo(d)2917 1201 y Ff(2)p 2861 1241 111 4 v 2861 1298 a Fo(dx)2937 1279 y Ff(2)3013 1264 y Fs(+)31 b Fr(V)21 b Fs(\()p Fr(x)p Fs(\))38 b Fv(in)456 1386 y Fr(L)522 1349 y Fq(2)561 1386 y Fs(\()p Fp(R)5 b Fs(\))31 b Fv(with)24 b(a)h(real)h(v)n(alued)e(potential)f Fr(V)49 b Ft(2)28 b Fr(C)2118 1349 y Fl(1)2111 1410 y Fq(0)2193 1386 y Fs(\()p Fp(R)5 b Fs(\))p Fv(.)456 1581 y FB(Theor)n(em)26 b(3.1.)41 b Fn(Let)25 b Fr(V)49 b Ft(2)29 b Fr(C)1487 1545 y Fl(1)1480 1606 y Fq(0)1561 1581 y Fs(\()p Fp(R)5 b Fs(\))p Fn(.)37 b(Then)25 b(for)f(any)h Fr(\016)32 b(>)27 b Fs(0)456 1847 y Fv(\(3.1\))254 b(tr)987 1737 y Fm(\020)1047 1847 y Ft(\000)1162 1780 y Fr(d)1213 1744 y Fq(2)p 1134 1825 146 4 v 1134 1916 a Fr(dx)1240 1887 y Fq(2)1312 1847 y Fs(+)22 b Fr(V)f Fs(\()p Fr(x)p Fs(\))1619 1737 y Fm(\021)1679 1759 y Fq(3)p Fo(=)p Fq(2)1679 1916 y Fl(\000)1817 1847 y Ft(\024)28 b Fr(C)1999 1737 y Fm(\020)2058 1712 y(Z)2114 1937 y Fh(R)2183 1847 y Fr(V)2261 1806 y Fq(4)2301 1847 y Fr(dx)22 b Fs(+)2527 1712 y Fm(Z)2626 1738 y Fo(\016)2582 1937 y Fl(\000)p Fo(\016)2692 1847 y Ft(j)2735 1822 y Fs(^)2720 1847 y Fr(V)f Fs(\()p Fr(\030)5 b Fs(\))p Ft(j)2950 1806 y Fq(2)3005 1847 y Fr(d\030)3104 1737 y Fm(\021)3162 1847 y Fr(;)456 2125 y Fn(wher)l(e)25 b(the)g(constant)e Fr(C)35 b Fs(=)28 b Fr(C)7 b Fs(\()p Fr(\016)n(;)17 b Ft(jj)p Fr(V)k Ft(jj)1828 2140 y Fl(1)1902 2125 y Fs(\))k Fn(and)2154 2099 y Fs(^)2139 2125 y Fr(V)c Fs(\()p Fr(\030)5 b Fs(\))27 b(=)2472 2044 y Fm(R)2555 2125 y Fs(exp)18 b(\()p Ft(\000)p Fr(i\030)5 b(x)p Fs(\))p Fr(V)22 b Fs(\()p Fr(x)p Fs(\))17 b Fr(dx)p Fn(.)555 2320 y(Pr)l(oof)o(.)72 b Fv(F)o(or)39 b(each)h Fr(T)68 b Ft(2)55 b Fe(S)1602 2335 y Fq(1)1641 2320 y Fv(,)42 b(one)d(can)h(de\002ne)g(a)f(comple)o(x-v)n(alued)e(function)456 2436 y Fs(det\(1)22 b(+)g Fr(T)14 b Fs(\))p Fv(,)24 b(so)h(that)1371 2630 y Ft(j)p Fs(det\(1)d(+)g Fr(T)14 b Fs(\))p Ft(j)27 b(\024)h Fs(exp)r(\()p Ft(k)p Fr(T)14 b Ft(k)2369 2645 y Fd(S)2426 2654 y Ff(1)2464 2630 y Fs(\))p Fr(:)456 2824 y Fv(F)o(or)24 b Fr(T)42 b Ft(2)28 b Fe(S)893 2839 y Fq(4)957 2824 y Fv(one)d(de\002nes)456 3022 y(\(3.2\))427 b Fs(det)1209 3037 y Fq(4)1249 3022 y Fs(\(1)21 b(+)h Fr(T)14 b Fs(\))28 b(=)f(det)q(\(\(1)22 b(+)g Fr(T)14 b Fs(\))p Fr(e)2230 2981 y Fl(\000)p Fo(T)c Fq(+)p Fo(T)2442 2957 y Ff(2)2476 2981 y Fo(=)p Fq(2)p Fl(\000)p Fo(T)2652 2957 y Ff(3)2687 2981 y Fo(=)p Fq(3)2761 3022 y Fs(\))p Fr(:)456 3216 y Fv(It)21 b(is)h(pro)o(v)o(ed)e(in)h([27],)i(Section)e (9,)i(Theorem)e(9.2\(b\),)h(that)f(there)h(is)g(a)g(constant)e Fr(c)28 b(>)g Fs(0)456 3332 y Fv(such)c(that)456 3526 y(\(3.3\))501 b Ft(j)p Fs(det)1311 3541 y Fq(4)1351 3526 y Fs(\(1)21 b(+)h Fr(T)14 b Fs(\))p Ft(j)27 b(\024)i Fs(exp)q(\()p Fr(c)p Ft(k)p Fr(T)14 b Ft(k)2227 3485 y Fq(4)2227 3550 y Fd(S)2284 3559 y Ff(4)2322 3526 y Fs(\))p Fr(;)116 b(c)28 b(>)f Fs(0)p Fr(:)456 3740 y Fv(Note)d(that)h(if)f Fr(J)988 3755 y Fq(0)1053 3740 y Fv(is)g(the)h(operator)f Fr(J)1701 3755 y Fq(0)1768 3740 y Fs(=)k Ft(\000)1979 3701 y Fo(d)2015 3678 y Ff(2)p 1959 3717 111 4 v 1959 3775 a Fo(dx)2035 3756 y Ff(2)2105 3740 y Fv(in)c Fr(L)2273 3704 y Fq(2)2313 3740 y Fs(\()p Fp(R)5 b Fs(\))p Fv(,)31 b(then)1122 3975 y Fs(lim)1121 4035 y Fo(")p Fl(!)p Fq(0)1276 3975 y Ft(j)17 b Fs(det)1456 3865 y Fm(\020)1515 3975 y Fr(I)30 b Fs(+)22 b Fr(V)g Fs(\()p Fr(J)1857 3990 y Fq(0)1918 3975 y Ft(\000)h Fs(\()p Fr(\025)f Ft(\006)g Fr(i")p Fs(\)\))2389 3934 y Fl(\000)p Fq(1)2483 3865 y Fm(\021)2543 3975 y Ft(j)27 b(\025)h Fs(1)p Fr(:)456 4199 y Fv(In)c(order)i(to)e(pro)o(v)o(e)g(Theorem)g (3.1)h(we)g(need)g(the)g(follo)n(wing)d(auxiliary)i(statement:)456 4394 y FB(Lemma)31 b(3.1.)45 b Fn(Let)31 b Fr(V)22 b Fs(\()p Fr(x)p Fs(\))32 b Fn(be)f(a)g(smooth)f(r)l(eal)h(valued)g (function)f(of)h(\002nite)f(support.)456 4511 y(F)-10 b(or)23 b(e)o(very)j Fr(\016)31 b(>)d Fs(0)c Fn(ther)l(e)h(is)g(a)f (constant)g Fr(C)35 b Fs(=)27 b Fr(C)7 b Fs(\()p Fr(\016)n(;)17 b Ft(jj)p Fr(V)k Ft(jj)2463 4526 y Fl(1)2537 4511 y Fs(\))k Fn(suc)o(h)f(that)g(for)g(all)g Fr(z)t Fn(:)1404 4706 y Ft(j)p Fr(z)i Ft(\000)d(jj)p Fr(V)e Ft(jj)1793 4721 y Fl(1)1867 4706 y Ft(j)27 b Fs(=)h Ft(jj)p Fr(V)21 b Ft(jj)2216 4721 y Fl(1)2312 4706 y Fs(+)h Fr(\016)2457 4664 y Fq(2)456 4899 y Fn(it)i(holds)456 5122 y Fv(\(3.4\))360 b Ft(j)17 b Fs(log)f(det)1329 5137 y Fq(4)1369 5122 y Fs(\()p Fr(I)30 b Fs(+)22 b Fr(V)f Fs(\()p Fr(J)1748 5137 y Fq(0)1810 5122 y Ft(\000)h Fr(z)t Fs(\))1996 5081 y Fl(\000)p Fq(1)2091 5122 y Fs(\))p Ft(j)28 b(\024)2401 5055 y Fr(C)p 2300 5099 281 4 v 2300 5191 a Ft(j)p Fv(Im)c Fr(z)t Ft(j)2540 5162 y Fq(4)2590 5122 y Ft(jj)p Fr(V)d Ft(jj)2780 5081 y Fq(4)2780 5148 y Fo(L)2828 5129 y Ff(4)2866 5122 y Fr(:)p eop %%Page: 7 7 7 6 bop 1246 251 a Fj(ABSOLUTEL)-7 b(Y)22 b(CONTINUOUS)h(SPECTR)m(UM) 751 b(7)555 450 y Fn(Pr)l(oof)p Fv(.)67 b(Let)37 b Fr(z)56 b Fs(=)51 b Fr(\025)31 b Fs(+)h Fr(i\021)t Fv(,)40 b(where)e Fr(\025)f Fv(and)h Fr(\021)j Fv(are)d(real.)69 b(One)37 b(can)h(repeat)g(the)456 566 y(ar)n(guments)26 b(of)h(R.Killip)f(and)h (B.Simon,)h(Proposition)d(5.2)i([11],)h(in)e(order)i(to)f(sho)n(w)456 683 y(that)1262 911 y Ft(j)1326 843 y Fr(d)p 1300 888 103 4 v 1300 979 a(d\021)1429 911 y Fs(log)16 b(det)1707 926 y Fq(4)1746 911 y Fs(\()p Fr(I)30 b Fs(+)22 b Fr(V)g Fs(\()p Fr(J)2126 926 y Fq(0)2187 911 y Ft(\000)h Fr(z)t Fs(\))2374 870 y Fl(\000)p Fq(1)2469 911 y Fs(\))p Ft(j)k Fs(=)1270 1120 y Ft(j)p Fv(tr)e Fs(\()p Fr(i)p Fs([\()p Fr(J)1574 1135 y Fq(0)1636 1120 y Ft(\000)d Fr(z)t Fs(\))1822 1079 y Fl(\000)p Fq(1)1917 1120 y Fr(V)g Fs(])2023 1079 y Fq(4)2062 1120 y Fs(\()p Fr(J)32 b Ft(\000)22 b Fr(z)t Fs(\))2372 1079 y Fl(\000)p Fq(1)2467 1120 y Fs(\))p Ft(j)27 b(\024)1375 1280 y(jj)p Fr(V)21 b Fs(\()p Fr(J)1601 1295 y Fq(0)1662 1280 y Ft(\000)i Fr(z)t Fs(\))1849 1239 y Fl(\000)p Fq(1)1944 1280 y Ft(jj)2000 1239 y Fq(4)2000 1305 y Fd(S)2057 1314 y Ff(4)2095 1280 y Ft(jj)p Fs(\()p Fr(J)31 b Ft(\000)22 b Fr(z)t Fs(\))2460 1239 y Fl(\000)p Fq(1)2555 1280 y Ft(jj)p Fr(:)456 1071 y Fv(\(3.5\))456 1478 y(On)i(the)h(other)g(hand,)1303 1676 y Fs(lim)1281 1736 y Fo(\021)r Fl(!1)1477 1676 y Fs(det)1612 1691 y Fq(4)1651 1676 y Fs(\()p Fr(I)30 b Fs(+)22 b Fr(V)g Fs(\()p Fr(J)2031 1691 y Fq(0)2092 1676 y Ft(\000)h Fr(z)t Fs(\))2279 1635 y Fl(\000)p Fq(1)2374 1676 y Fs(\))28 b(=)f(1)p Fr(:)456 1917 y Fv(Therefore)i(the)g(estimate)f(\(3.4\))h(follo)n(ws)f (from)h(\(3.5\))g(by)f(the)h(Fundamental)f(Theo-)456 2034 y(rem)c(of)h(Calculus.)130 b Fc(\003)555 2232 y Fv(It)25 b(w)o(as)g(established)e(in)i([12])g(that)f(for)h Fr(z)33 b Fs(=)27 b Fr(k)2131 2196 y Fq(2)2171 2232 y Fr(;)33 b(k)e Ft(2)d Fp(R)5 b Fr(;)1221 2504 y Ft(\000)p Fv(Re)26 b(tr)f Fs(\()p Fr(V)c Fs(\()p Fr(J)1728 2519 y Fq(0)1790 2504 y Ft(\000)h Fr(z)t Fs(\))1976 2463 y Fl(\000)p Fq(1)2071 2504 y Fs(\))2109 2463 y Fq(2)2176 2504 y Fs(=)2290 2436 y Ft(j)2333 2411 y Fs(^)2318 2436 y Fr(V)f Fs(\(2)p Fr(k)s Fs(\))p Ft(j)2603 2400 y Fq(2)p 2290 2481 353 4 v 2395 2572 a Fs(2)p Fr(k)2498 2543 y Fq(2)2652 2504 y Fr(:)456 2742 y Fv(Therefore)k(for)g Fr(z)33 b Fs(=)27 b Fr(k)1249 2705 y Fq(2)1289 2742 y Fr(;)33 b(k)e Ft(2)d Fp(R)5 b Fv(,)951 3052 y Fs(0)27 b Ft(\024)h Fs(log)17 b Ft(j)p Fs(det\()p Fr(I)30 b Fs(+)22 b Fr(V)g Fs(\()p Fr(J)1818 3067 y Fq(0)1879 3052 y Ft(\000)h Fr(z)t Fs(\))2066 3011 y Fl(\000)p Fq(1)2161 3052 y Fs(\))p Ft(j)f Fs(+)g(log)16 b Ft(j)p Fs(det\()p Fr(I)30 b Ft(\000)23 b Fr(V)e Fs(\()p Fr(J)3033 3067 y Fq(0)3095 3052 y Ft(\000)h Fr(z)t Fs(\))3281 3011 y Fl(\000)p Fq(1)3376 3052 y Fs(\))p Ft(j)1157 3212 y Fs(=)28 b Ft(\000)p Fv(Re)e(tr)f Fs(\()p Fr(V)c Fs(\()p Fr(J)1768 3227 y Fq(0)1830 3212 y Ft(\000)h Fr(z)t Fs(\))2016 3171 y Fl(\000)p Fq(1)2111 3212 y Fs(\))2149 3171 y Fq(2)2211 3212 y Fs(+)g(log)16 b Ft(j)p Fs(det)2614 3227 y Fq(4)2654 3212 y Fs(\()p Fr(I)30 b Fs(+)22 b Fr(V)f Fs(\()p Fr(J)3033 3227 y Fq(0)3095 3212 y Ft(\000)h Fr(z)t Fs(\))3281 3171 y Fl(\000)p Fq(1)3376 3212 y Fs(\))p Ft(j)458 3450 y Fs(+)17 b(log)f Ft(j)p Fs(det)856 3465 y Fq(4)896 3450 y Fs(\()p Fr(I)30 b Ft(\000)22 b Fr(V)g Fs(\()p Fr(J)1277 3465 y Fq(0)1338 3450 y Ft(\000)h Fr(z)t Fs(\))1525 3409 y Fl(\000)p Fq(1)1620 3450 y Fs(\))p Ft(j)k Fs(=)1827 3383 y Ft(j)1870 3358 y Fs(^)1855 3383 y Fr(V)21 b Fs(\(2)p Fr(k)s Fs(\))p Ft(j)2140 3347 y Fq(2)p 1827 3427 V 1931 3519 a Fs(2)p Fr(k)2034 3490 y Fq(2)2211 3450 y Fs(+)h(log)16 b Ft(j)p Fs(det)2614 3465 y Fq(4)2654 3450 y Fs(\()p Fr(I)30 b Fs(+)22 b Fr(V)f Fs(\()p Fr(J)3033 3465 y Fq(0)3095 3450 y Ft(\000)h Fr(z)t Fs(\))3281 3409 y Fl(\000)p Fq(1)3376 3450 y Fs(\))p Ft(j)2188 3640 y Fs(+)17 b(log)f Ft(j)p Fs(det)2586 3655 y Fq(4)2625 3640 y Fs(\()p Fr(I)30 b Ft(\000)23 b Fr(V)e Fs(\()p Fr(J)3006 3655 y Fq(0)3068 3640 y Ft(\000)h Fr(z)t Fs(\))3254 3599 y Fl(\000)p Fq(1)3349 3640 y Fs(\))p Ft(j)p Fr(:)456 2920 y Fv(\(3.6\))456 3836 y(Let)i(no)n(w)606 4034 y Fr(\033)t Fs(\()p Fr(k)s Fs(\))k(=)g Fr(k)981 3993 y Fq(2)1020 4034 y Fs(\()p Fr(k)1112 3993 y Fq(2)1174 4034 y Ft(\000)22 b Fr(\016)1320 3993 y Fq(2)1360 4034 y Fs(\))1398 3993 y Fq(4)1437 4034 y Fr(;)216 b Fe(L)28 b Fs(=)g Ft(f)p Fr(k)i Fs(:)44 b Ft(j)p Fr(k)2162 3993 y Fq(2)2224 4034 y Ft(\000)22 b(jj)p Fr(V)f Ft(jj)2513 4049 y Fl(1)2587 4034 y Ft(j)28 b Fs(=)f Ft(jj)p Fr(V)21 b Ft(jj)2936 4049 y Fl(1)3032 4034 y Fs(+)h Fr(\016)3177 3993 y Fq(2)3217 4034 y Ft(g)p Fr(:)456 4233 y Fv(Then)i(applying)g (\(3.4\))h(we)g(obtain)456 4481 y(\(3.7\))883 4366 y Fm(\014)883 4426 y(\014)883 4486 y(\014)916 4345 y(Z)972 4571 y Fd(L)1039 4481 y Fs(log)16 b(det)1317 4496 y Fq(4)1356 4481 y Fs(\()p Fr(I)30 b Fs(+)22 b Fr(V)g Fs(\()p Fr(J)1736 4496 y Fq(0)1797 4481 y Ft(\000)h Fr(k)1951 4440 y Fq(2)1990 4481 y Fs(\))2028 4440 y Fl(\000)p Fq(1)2122 4481 y Fs(\))p Fr(\033)t Fs(\()p Fr(k)s Fs(\))17 b Fr(dk)2471 4366 y Fm(\014)2471 4426 y(\014)2471 4486 y(\014)2532 4481 y Ft(\024)28 b Fr(C)7 b Ft(jj)p Fr(V)21 b Ft(jj)2904 4440 y Fq(4)2904 4506 y Fo(L)2952 4488 y Ff(4)2990 4481 y Fr(:)456 4740 y Fv(No)n(w)h(let)h Fr(i\014)881 4755 y Fo(j)917 4740 y Fs(\()p Fr(V)f Fs(\))h Fv(be)g(the)g(zeros)g(of)g Fs(log)17 b(det)1974 4755 y Fq(4)2014 4740 y Fs(\()p Fr(I)23 b Fs(+)15 b Fr(V)22 b Fs(\()p Fr(J)2380 4755 y Fq(0)2435 4740 y Ft(\000)15 b Fr(k)2581 4704 y Fq(2)2621 4740 y Fs(\))2659 4704 y Fl(\000)p Fq(1)2753 4740 y Fs(\))23 b Fv(and)g(let)g Fe(B)p Fs(\()p Fr(k)s(;)17 b(V)k Fs(\))456 4856 y Fv(be)k(the)f(Blaschk)o(e)h(product)1391 5111 y Fe(B)p Fs(\()p Fr(k)s(;)17 b(V)22 b Fs(\))27 b(=)1863 5016 y Fm(Y)1910 5226 y Fo(j)2017 5043 y Fr(k)e Ft(\000)d Fr(i\014)2280 5058 y Fo(j)2317 5043 y Fs(\()p Fr(V)g Fs(\))p 2017 5088 456 4 v 2018 5179 a Fr(k)j Fs(+)d Fr(i\014)2280 5194 y Fo(j)2316 5179 y Fs(\()p Fr(V)g Fs(\))2482 5111 y Fr(:)p eop %%Page: 8 8 8 7 bop 456 251 a Fj(8)845 b(LAPTEV)-10 b(,)17 b(N)m(ABOK)n(O)i(AND)f (SAFR)m(ONO)l(V)456 450 y Fv(Then)1792 778 y(Re)1944 642 y Fm(Z)2044 668 y Fo(\016)2000 867 y Fl(\000)p Fo(\016)2109 778 y Fs(log)f(det)2387 793 y Fq(4)2426 778 y Fs(\()p Fr(I)30 b Fs(+)22 b Fr(V)g Fs(\()p Fr(J)2806 793 y Fq(0)2867 778 y Ft(\000)h Fr(z)t Fs(\))3054 736 y Fl(\000)p Fq(1)3149 778 y Fs(\))p Fr(\033)t Fs(\()p Fr(k)s Fs(\))17 b Fr(dk)456 1047 y Fs(=)27 b Fv(Re)711 912 y Fm(Z)767 1137 y Fd(L)833 1047 y Fs(log)17 b(det)1111 1062 y Fq(4)1151 1047 y Fs(\()p Fr(I)30 b Fs(+)22 b Fr(V)f Fs(\()p Fr(J)1530 1062 y Fq(0)1592 1047 y Ft(\000)h Fr(z)t Fs(\))1778 1006 y Fl(\000)p Fq(1)1873 1047 y Fs(\))p Fr(\033)t Fs(\()p Fr(k)s Fs(\))17 b Fr(dk)25 b Ft(\000)e Fv(Re)2496 912 y Fm(Z)2551 1137 y Fd(L)2618 1047 y Fs(log\()p Fe(B)p Fs(\()p Fr(k)s(;)17 b(V)k Fs(\)\))p Fr(\033)t Fs(\()p Fr(k)s Fs(\))c Fr(dk)s(:)456 575 y Fv(\(3.8\))456 1260 y(Thus,)k(combining)e(the)i(inequality)f(\(3.6\))i (with)e(the)h(estimate)g(\(3.7\))g(and)g(the)g(relation)456 1376 y(\(3.8\))o(,)k(we)g(obtain)456 1604 y(\(3.9\))696 1509 y Fm(X)752 1719 y Fo(j)857 1604 y Fr(f)11 b Fs(\()p Fr(\014)1009 1619 y Fo(j)1045 1604 y Fs(\()p Fr(V)22 b Fs(\)\))14 b(+)1342 1509 y Fm(X)1398 1719 y Fo(j)1503 1604 y Fr(f)d Fs(\()p Fr(\014)1655 1619 y Fo(j)1691 1604 y Fs(\()p Ft(\000)p Fr(V)22 b Fs(\)\))28 b Ft(\024)g Fr(C)2171 1493 y Fm(\020)2230 1468 y(Z)2285 1694 y Fh(R)2354 1604 y Fr(V)2433 1563 y Fq(4)2472 1604 y Fr(dx)14 b Fs(+)2682 1468 y Fm(Z)2783 1494 y Fo(\016)2739 1694 y Fl(\000)p Fo(\016)2848 1604 y Ft(j)2891 1579 y Fs(^)2876 1604 y Fr(V)21 b Fs(\(2)p Fr(\030)5 b Fs(\))p Ft(j)3155 1563 y Fq(2)3210 1604 y Fr(d\030)3309 1493 y Fm(\021)3367 1604 y Fr(;)456 1864 y Fv(where)1060 2016 y Fr(f)11 b Fs(\()p Fr(t)p Fs(\))27 b(=)h Fv(Re)1513 1881 y Fm(Z)1568 2106 y Fd(L)1635 2016 y Fs(log)1761 1906 y Fm(\020)1830 1949 y Fr(k)e Ft(\000)c Fr(it)p 1830 1993 245 4 v 1831 2085 a(k)j Fs(+)d Fr(it)2085 1906 y Fm(\021)2144 2016 y Fr(\033)t Fs(\()p Fr(k)s Fs(\))17 b Fr(dk)s(;)116 b(t)28 b(>)f Fs(0)p Fr(:)456 2208 y Fv(Inte)o(grating)c(by)i(parts)f(and)h (using)f(the)g(f)o(act)i(that)e Fr(\033)29 b Fv(is)24 b(e)n(v)o(en)g(we)h(obtain)456 2420 y Fr(f)11 b Fs(\()p Fr(t)p Fs(\))27 b(=)h Fv(Re)909 2285 y Fm(Z)964 2510 y Fd(L)1014 2310 y Fm(\020)1182 2353 y Fs(1)p 1084 2397 V 1084 2489 a Fr(k)d Ft(\000)e Fr(it)1338 2420 y Ft(\000)1522 2353 y Fs(1)p 1425 2397 243 4 v 1425 2489 a Fr(k)j Fs(+)c Fr(it)1678 2310 y Fm(\021)1738 2420 y Fs(\004\()p Fr(k)s Fs(\))17 b Fr(dk)30 b Fs(=)d Fv(Re)2337 2285 y Fm(Z)2393 2510 y Fl(j)p Fo(k)r Fl(j)p Fq(=2)p Fo(t)2715 2353 y Fs(1)p 2617 2397 245 4 v 2617 2489 a Fr(k)e Ft(\000)e Fr(it)2871 2420 y Fs(\004\()p Fr(k)s Fs(\))17 b Fr(dk)30 b Fs(=)e(2)p Fr(\031)t Fs(\004\()p Fr(it)p Fs(\))p Fr(;)456 2649 y Fv(where)1542 2820 y Fs(\004\()p Fr(k)s Fs(\))g(=)1868 2684 y Fm(Z)1968 2711 y Fo(k)1924 2910 y Fq(0)2027 2820 y Fr(\033)t Fs(\()p Fr(\034)11 b Fs(\))17 b Fr(d\034)6 b(:)456 3012 y Fv(This)24 b(implies)1641 3194 y Fr(f)11 b Fs(\()p Fr(t)p Fs(\))27 b Ft(\025)1953 3127 y Fs(2)p Fr(\031)t(\016)2108 3090 y Fq(8)p 1953 3171 195 4 v 2026 3262 a Fs(3)2157 3194 y Fr(t)2192 3153 y Fq(3)2232 3194 y Fr(:)456 3365 y Fv(The)d(proof)h(is)f(complete.)55 b Fc(\003)845 3578 y Fv(4.)c(T)t Fu(H)t(E)31 b(B)t(E)t(G)t(I)t(N)t(N)t (I)t(N)t(G)j(O)t(F)d(T)t(H)t(E)f(P)t(R)q(O)t(O)t(F)j(O)t(F)e Fv(T)t Fu(H)t(E)t(O)t(R)t(E)t(M)j Fv(2)t(.)t(2)555 3753 y(In)i(this)f(section)g(we)h(reduce)g(problem)f(\(2.3\))g(to)h(a)f (one-dimensional)f(problem)456 3869 y(with)28 b(an)h(operator)g(v)n (alued)f(potential.)41 b(Such)29 b(a)h(reduction)e(has)h(been)g (already)g(used)456 3985 y(in)24 b([15].)555 4101 y(Assume)j(that)g Fr(V)55 b Ft(2)33 b Fr(C)1369 4065 y Fl(1)1362 4126 y Fq(0)1472 4101 y Fv(and)27 b(introduce)g(polar)h(coordinates)f Fs(\()p Fr(r)m(;)17 b(\022)s Fs(\))p Fr(;)50 b(x)33 b Fs(=)g Fr(r)s(\022)j Ft(2)456 4217 y Fp(R)522 4181 y Fo(d)568 4217 y Fr(;)d(\022)51 b Ft(2)d Fp(S)899 4181 y Fo(d)p Fl(\000)p Fq(1)1024 4217 y Fv(.)62 b(Denote)36 b(by)f Ft(f)p Fr(Y)1677 4232 y Fo(j)1713 4217 y Ft(g)1763 4181 y Fl(1)1763 4242 y Fo(j)t Fq(=0)1925 4217 y Fv(the)g(orthonormal)f (in)h Fr(L)2789 4181 y Fq(2)2829 4217 y Fs(\()p Fp(S)2928 4181 y Fo(d)p Fl(\000)p Fq(1)3053 4217 y Fs(\))h Fv(basis)e(of)456 4334 y(\(real\))24 b(spherical)g(functions,)f(i.e.)30 b(eigenfunctions)23 b(of)h(the)g(Laplace-Beltrami)g(oper)n(-)456 4450 y(ator)g Ft(\000)p Fs(\001)793 4465 y Fo(\022)833 4450 y Fv(,)h(and)g(let)f Fr(P)1239 4465 y Fo(j)1301 4450 y Fv(be)h(the)f(orthogonal)g(projection)g(gi)n(v)o(en)f(by)1125 4668 y Fr(P)1188 4683 y Fo(j)1224 4668 y Fr(u)p Fs(\()p Fr(r)m(;)17 b(\022)s Fs(\))27 b(=)h Fr(Y)1677 4683 y Fo(j)1713 4668 y Fs(\()p Fr(\022)s Fs(\))1854 4532 y Fm(Z)1909 4758 y Fh(S)1952 4739 y Fi(d)p Fg(\000)p Ff(1)2081 4668 y Fr(Y)2138 4683 y Fo(j)2174 4668 y Fs(\()p Fr(\022)2260 4627 y Fl(0)2284 4668 y Fs(\))p Fr(u)p Fs(\()p Fr(r)m(;)17 b(\022)2549 4627 y Fl(0)2571 4668 y Fs(\))g Fr(d\022)2725 4627 y Fl(0)2748 4668 y Fr(:)456 4881 y Fv(Clearly)25 b Fr(P)837 4896 y Fq(0)876 4881 y Fr(u)g Fv(depends)f(only)g(on)h Fr(r)s Fv(.)30 b(Denote)1294 5043 y Fr(V)1351 5058 y Fq(1)1418 5043 y Fs(=)d Fr(P)1584 5058 y Fq(0)1624 5043 y Fr(V)21 b(P)1765 5058 y Fq(0)1805 5043 y Fr(;)116 b(H)2029 5058 y Fq(0)p Fo(;)p Fq(1)2151 5043 y Fs(=)27 b Fr(P)2317 5058 y Fq(0)2357 5043 y Fr(H)2438 5058 y Fq(0)2477 5043 y Fr(P)2540 5058 y Fq(0)2579 5043 y Fr(;)1241 5211 y(V)1298 5226 y Fq(1)p Fo(;)p Fq(2)1420 5211 y Fs(=)g Fr(P)1586 5226 y Fq(0)1626 5211 y Fr(V)21 b Fs(\()p Fr(I)30 b Ft(\000)23 b Fr(P)1978 5226 y Fq(0)2017 5211 y Fs(\))p Fr(;)116 b(V)2255 5226 y Fq(2)p Fo(;)p Fq(1)2377 5211 y Fs(=)28 b Fr(V)2559 5170 y Fl(\003)2538 5236 y Fq(1)p Fo(;)p Fq(2)2632 5211 y Fr(;)p eop %%Page: 9 9 9 8 bop 1246 251 a Fj(ABSOLUTEL)-7 b(Y)22 b(CONTINUOUS)h(SPECTR)m(UM) 751 b(9)797 450 y Fr(V)854 465 y Fq(2)921 450 y Fs(=)27 b(\()p Fr(I)j Ft(\000)23 b Fr(P)1298 465 y Fq(0)1337 450 y Fs(\))p Fr(V)f Fs(\()p Fr(I)30 b Ft(\000)22 b Fr(P)1727 465 y Fq(0)1767 450 y Fs(\))p Fr(;)116 b(H)2029 465 y Fq(0)p Fo(;)p Fq(2)2151 450 y Fs(=)27 b(\()p Fr(I)j Ft(\000)23 b Fr(P)2528 465 y Fq(0)2567 450 y Fs(\))p Fr(H)2686 465 y Fq(0)2725 450 y Fs(\()p Fr(I)30 b Ft(\000)23 b Fr(P)2999 465 y Fq(0)3038 450 y Fs(\))p Fr(:)456 587 y Fv(Then)h(the)h(operator)g Fr(H)k Ft(\000)23 b Fr(z)30 b Fv(can)25 b(be)g(represented)g(as)g(a)g (matrix:)1055 802 y Fr(H)k Ft(\000)23 b Fr(z)32 b Fs(=)1446 662 y Fm(\022)1519 743 y Fr(H)1600 758 y Fq(0)p Fo(;)p Fq(1)1716 743 y Fs(+)22 b Fr(V)1871 758 y Fq(1)1933 743 y Ft(\000)h Fr(z)293 b(V)2428 758 y Fq(1)p Fo(;)p Fq(2)1725 859 y Fr(V)1782 874 y Fq(2)p Fo(;)p Fq(1)2165 859 y Fr(H)2246 874 y Fq(0)p Fo(;)p Fq(2)2363 859 y Fs(+)22 b Fr(V)2518 874 y Fq(2)2579 859 y Ft(\000)h Fr(z)2728 662 y Fm(\023)2818 802 y Fr(;)456 1014 y Fv(and)h(the)h(equation)1343 1131 y Fs(\()p Fr(H)30 b Ft(\000)22 b Fr(z)t Fs(\))p Fr(u)28 b Fs(=)g Fr(P)1929 1146 y Fq(0)1968 1131 y Fr(f)5 b(;)117 b Fv(Im)24 b Fr(z)33 b Ft(6)p Fs(=)27 b(0)p Fr(;)456 1268 y Fv(is)d(equi)n(v)n(alent)f(to)456 1425 y(\(4.1\))49 b Fs(\()p Fr(H)815 1440 y Fq(0)p Fo(;)p Fq(1)918 1425 y Fs(+)9 b Fr(T)1060 1440 y Fo(z)1108 1425 y Ft(\000)g Fr(z)t Fs(\))p Fr(P)1344 1440 y Fq(0)1384 1425 y Fr(u)27 b Fs(=)g Fr(P)1633 1440 y Fq(0)1673 1425 y Fr(f)5 b(;)116 b Fs(\()p Fr(H)1988 1440 y Fq(0)p Fo(;)p Fq(2)2091 1425 y Fs(+)9 b Fr(V)2233 1440 y Fq(2)2280 1425 y Ft(\000)g Fr(z)t Fs(\))2453 1384 y Fl(\000)p Fq(1)2548 1425 y Fr(V)2605 1440 y Fq(2)p Fo(;)p Fq(1)2699 1425 y Fr(P)2762 1440 y Fq(0)2802 1425 y Fr(u)27 b Fs(=)g(\()p Fr(P)3089 1440 y Fq(0)3137 1425 y Ft(\000)9 b Fr(I)f Fs(\))p Fr(u:)456 1582 y Fv(Here)25 b(the)g(operator)g Fr(T)1235 1597 y Fo(z)1299 1582 y Fv(is)g(de\002ned)g(by)1209 1739 y Fr(T)1266 1754 y Fo(z)1334 1739 y Fs(=)i Fr(V)1494 1754 y Fq(1)1556 1739 y Ft(\000)22 b Fr(V)1712 1754 y Fq(1)p Fo(;)p Fq(2)1806 1739 y Fs(\()p Fr(H)1925 1754 y Fq(0)p Fo(;)p Fq(2)2042 1739 y Fs(+)g Fr(V)2197 1754 y Fq(2)2258 1739 y Ft(\000)h Fr(z)t Fs(\))2445 1698 y Fl(\000)p Fq(1)2540 1739 y Fr(V)2597 1754 y Fq(2)p Fo(;)p Fq(1)456 1899 y Fv(on)h Fr(L)646 1862 y Fq(2)686 1899 y Fs(\(\(1)p Fr(;)17 b Ft(1)p Fs(\))p Fr(;)g(r)1084 1862 y Fo(d)p Fl(\000)p Fq(1)1229 1899 y Fr(dr)s Fs(\))p Fv(.)555 2015 y(By)138 b(using)d(the)i(unitary)f (operator)h(from)g Fr(L)2705 1979 y Fq(2)2745 2015 y Fs(\(\(1)p Fr(;)17 b Ft(1)p Fs(\))p Fr(;)g(dr)s Fs(\))135 b Fv(to)456 2131 y Fr(L)522 2095 y Fq(2)561 2131 y Fs(\(\(1)p Fr(;)17 b Ft(1)p Fs(\))p Fr(;)g(r)959 2095 y Fo(d)p Fl(\000)p Fq(1)1105 2131 y Fr(dr)s Fs(\))p Fv(,)1537 2255 y Fr(U)10 b(u)p Fs(\()p Fr(r)s Fs(\))27 b(=)h Fr(r)1970 2214 y Fl(\000)p Fq(\()p Fo(d)p Fl(\000)p Fq(1\))p Fo(=)p Fq(2)2280 2255 y Fr(u;)456 2392 y Fv(we)66 b(reduce)h(\(4.1\))f(to)g(the)h (problem)e(for)i(the)f(follo)n(wing)e(one-dimensional)456 2509 y(Schr)8 b(\250)-41 b(odinger)24 b(operator)h(in)g Fr(L)1493 2472 y Fq(2)1533 2509 y Fs(\(1)p Fr(;)17 b Ft(1)p Fs(\))456 2729 y Fv(\(4.2\))252 b Fr(L)965 2744 y Fo(z)1005 2729 y Fr(u)p Fs(\()p Fr(r)s Fs(\))27 b(=)h Ft(\000)1402 2662 y Fr(d)1453 2626 y Fq(2)1492 2662 y Fr(u)p 1402 2706 146 4 v 1406 2797 a(dr)1504 2769 y Fq(2)1580 2729 y Fs(+)22 b Fr(Q)1755 2744 y Fo(z)1795 2729 y Fr(u;)116 b(u)27 b Ft(2)h Fr(L)2237 2688 y Fq(2)2277 2729 y Fs(\(1)p Fr(;)17 b Ft(1)p Fs(\))p Fr(;)40 b(u)p Fs(\(1\))27 b(=)h(0)p Fr(;)456 2915 y Fv(where)456 3112 y Fr(Q)533 3127 y Fo(z)600 3112 y Fs(=)g Fr(V)761 3127 y Fq(1)815 3112 y Fs(+)916 3044 y Fr(\013)978 3059 y Fo(d)p 916 3089 103 4 v 924 3180 a Fr(r)971 3151 y Fq(2)1043 3112 y Ft(\000)15 b Fr(V)1192 3127 y Fq(1)p Fo(;)p Fq(2)1287 3112 y Fs(\()p Fr(U)1401 3071 y Fl(\003)1441 3112 y Fr(H)1522 3127 y Fq(0)p Fo(;)p Fq(2)1616 3112 y Fr(U)26 b Fs(+)15 b Fr(V)1856 3127 y Fq(2)1910 3112 y Ft(\000)g Fr(z)t Fs(\))2089 3071 y Fl(\000)p Fq(1)2184 3112 y Fr(V)2241 3127 y Fq(2)p Fo(;)p Fq(1)2335 3112 y Fr(;)117 b(\013)2541 3127 y Fo(d)2609 3112 y Fs(=)2722 3044 y(\()p Fr(d)22 b Ft(\000)h Fs(1\))3020 3008 y Fq(2)p 2722 3089 337 4 v 2866 3180 a Fs(4)3084 3112 y Ft(\000)3186 3044 y Fr(d)f Ft(\000)h Fs(1)p 3186 3089 222 4 v 3272 3180 a(2)3417 3112 y Fr(:)456 3297 y Fv(W)-8 b(e)36 b(are)g(going)f(to)g(approximate)g(the)g(problem)g(by) g(the)h(corresponding)e(problem)456 3414 y(with)25 b(a)h(smooth)f (compactly)g(supported)g(potential)g Fr(V)48 b Fv(and)26 b(the)f(term)h Fr(\013)2977 3429 y Fo(d)3018 3414 y Fr(=r)3114 3377 y Fq(2)3179 3414 y Fv(substi-)456 3530 y(tuted)33 b(by)g Fr(\020)865 3545 y Fo(")901 3530 y Fs(\()p Fr(r)s Fs(\))p Fr(\013)1086 3545 y Fo(d)1127 3530 y Fr(=r)1223 3494 y Fq(2)1261 3530 y Fv(,)j(where)f Fr(\020)1643 3545 y Fo(")1679 3530 y Fr(=r)1775 3494 y Fq(2)1858 3530 y Ft(!)44 b Fs(1)p Fr(=r)2147 3494 y Fq(2)2185 3530 y Fv(,)36 b(as)e Fr(")44 b Ft(!)g Fs(0)p Fv(,)36 b(in)d(the)g(both)g(spaces)456 3646 y Fr(L)522 3610 y Fq(1)561 3646 y Fs(\(1)p Fr(;)17 b Ft(1)p Fs(\))30 b Fv(and)h Fr(L)1101 3610 y Fq(2)1141 3646 y Fs(\(1)p Fr(;)17 b Ft(1)p Fs(\))30 b Fv(and)h Fr(\020)1658 3661 y Fo(")1733 3646 y Ft(2)39 b Fr(C)1915 3610 y Fl(1)1908 3671 y Fq(0)1990 3646 y Fs(\(1)p Fr(;)17 b Fs(+)p Ft(1)p Fs(\))p Fv(.)48 b(The)30 b(same)h(should)f(be)h(done) 456 3762 y(with)24 b(the)g(term)h Fs(\001)1093 3777 y Fo(\022)1132 3762 y Fr(u=r)1284 3726 y Fq(2)1323 3762 y Fv(,)f(i.e.)31 b(it)24 b(should)g(be)h(substituted)e(by)h Fr(\020)2639 3777 y Fo(")2675 3762 y Fs(\()p Fr(r)s Fs(\)\001)2879 3777 y Fo(\022)2918 3762 y Fr(u=r)3070 3726 y Fq(2)3109 3762 y Fv(.)555 3878 y(So)h(when)g(approximating)e(the)i(problem)f(we)h (al)o(w)o(ays)f(assume)h(that)456 4060 y(\(4.3\))355 b Fr(Q)1079 4075 y Fo(z)1147 4060 y Fs(=)28 b Fr(V)1308 4075 y Fq(1)1369 4060 y Fs(+)22 b Fr(\020)1510 4075 y Fo(")1546 4060 y Fs(\()p Fr(r)s Fs(\))1679 3992 y Fr(\013)1741 4007 y Fo(d)p 1679 4037 103 4 v 1687 4128 a Fr(r)1734 4099 y Fq(2)1814 4060 y Ft(\000)g Fr(V)1970 4075 y Fq(1)p Fo(;)p Fq(2)2064 4060 y Fs(\()p Fr(S)2162 4075 y Fo(")2221 4060 y Fs(+)g Fr(V)2376 4075 y Fq(2)2438 4060 y Ft(\000)g Fr(z)t Fs(\))2624 4018 y Fl(\000)p Fq(1)2719 4060 y Fr(V)2776 4075 y Fq(2)p Fo(;)p Fq(1)2870 4060 y Fr(;)456 4245 y Fv(where)456 4442 y(\(4.4\))469 b Fr(S)1176 4457 y Fo(")1213 4442 y Fr(u)27 b Fs(=)g Ft(\000)1486 4375 y Fr(d)1537 4339 y Fq(2)1577 4375 y Fr(u)p 1486 4419 146 4 v 1490 4510 a(dr)1588 4482 y Fq(2)1664 4442 y Ft(\000)c Fr(\020)1807 4457 y Fo(")1843 4442 y Fs(\()p Fr(r)s Fs(\))1976 4375 y(\001)2057 4390 y Fo(\022)2096 4375 y Fr(u)p 1976 4419 176 4 v 2021 4510 a(r)2068 4482 y Fq(2)2162 4442 y Fr(;)116 b(u)p Fs(\(1)p Fr(;)17 b(\022)s Fs(\))27 b(=)g(0)p Fr(:)555 4628 y Fv(According)e(to)f(\(4.1\))h(we)g(obtain)456 4785 y(\(4.5\))586 b Fr(P)1296 4800 y Fq(0)1335 4785 y Fs(\()p Fr(H)30 b Ft(\000)23 b Fr(z)t Fs(\))1671 4744 y Fl(\000)p Fq(1)1766 4785 y Fr(P)1829 4800 y Fq(0)1896 4785 y Fs(=)k Fr(U)10 b Fs(\()p Fr(L)2179 4800 y Fo(z)2242 4785 y Ft(\000)23 b Fr(z)t Fs(\))2429 4744 y Fl(\000)p Fq(1)2524 4785 y Fr(U)2600 4744 y Fl(\003)2640 4785 y Fr(:)456 4942 y Fv(W)-8 b(e)21 b(see)h(also)e(that)h(if)g(supp)16 b Fr(V)30 b Ft([)8 b Fv(supp)16 b Fr(\020)1804 4957 y Fo(")1841 4942 y Fs(\()p Ft(j)8 b(\001)g(j)p Fs(\))27 b Ft(\032)h(f)p Fr(x)g Ft(2)g Fp(R)2442 4906 y Fo(d)2516 4942 y Fs(:)70 b Fr(c)2655 4957 y Fq(1)2722 4942 y Fr(<)28 b Ft(j)p Fr(x)p Ft(j)f Fr(<)h(c)3110 4957 y Fq(2)3149 4942 y Ft(g)p Fr(;)33 b(c)3301 4957 y Fq(1)3369 4942 y Fr(>)456 5058 y Fs(1)p Fr(;)24 b Fv(then)h(for)g(the)g(operator)f (\(4.3\))h(we)g(ha)n(v)o(e)1569 5216 y Fr(Q)1646 5231 y Fo(z)1713 5216 y Fs(=)j Fr(Q)1894 5231 y Fo(z)1934 5216 y Fr(\037)g Fs(=)f Fr(\037Q)2264 5231 y Fo(z)2304 5216 y Fr(;)p eop %%Page: 10 10 10 9 bop 456 251 a Fj(10)808 b(LAPTEV)-10 b(,)17 b(N)m(ABOK)n(O)i(AND)f (SAFR)m(ONO)l(V)456 450 y Fv(where)k Fr(\037)g Fv(is)f(an)g(operator)h (of)g(multiplication)d(by)i(the)g(characteristic)h(function)f(of)h(the) 456 566 y(interv)n(al)e Fs(\()p Fr(c)859 581 y Fq(1)898 566 y Fr(;)d(c)984 581 y Fq(2)1023 566 y Fs(\))p Fr(;)33 b(c)1163 581 y Fq(1)1231 566 y Fr(>)27 b Fs(0)p Fv(.)i(It)22 b(is)e(important)g(for)i(us)f(that)f Fr(Q)2511 581 y Fo(z)2573 566 y Fv(is)g(an)i(analytic)e(operator)456 683 y(v)n(alued)26 b(function)g(of)h Fr(z)32 b Fv(with)26 b(a)h(ne)o(gati)n(v)o(e)e(imaginary)h(part)h(in)g(the)g(upper)g(half)g (plane)456 799 y(and)d(which)h(has)f(a)i(positi)n(v)o(e)c(imaginary)i (part)h(in)f(the)h(lo)n(wer)f(half)h(plane.)1443 1073 y(5.)52 b(G)t Fu(R)t(E)t(E)t(N)t Fv(')t Fu(S)35 b(F)t(U)t(N)t(C)t(T)t (I)t(O)t(N)t Fv(.)555 1247 y(In)21 b(sections)f(5-7)h(we)g(assume)f (that)g Fr(V)43 b Fv(is)20 b(not)h(a)g(potential)e(b)n(ut)i(the)f (operator)h Fr(P)14 b(V)21 b(P)14 b Fv(,)456 1363 y Fr(P)48 b Fs(=)677 1289 y Fm(P)783 1315 y Fo(n)783 1392 y(j)t Fq(=0)926 1363 y Fr(P)989 1378 y Fo(j)1025 1363 y Fv(,)30 b(which)e(approximates)g Fr(V)50 b Fv(for)29 b(lar)n(ge)g Fr(n)p Fv(.)43 b(It)28 b(can)h(be)g(interpreted)f(as)456 1482 y(an)f(operator)h(of)f(multiplication)e(by)i(a)h(matrix)e(v)n (alued)h(function)f(of)i Fr(r)s Fv(.)38 b(In)27 b(this)g(case)456 1598 y(the)22 b(function)f Fr(V)1011 1613 y Fq(1)1073 1598 y Fv(remains)h(the)g(same)h(as)f(before.)31 b(Since)23 b Fr(P)2505 1613 y Fo(j)2563 1598 y Fv(are)h(projections)d(on)h(real) 456 1714 y(spherical)f(functions,)g(this)g(matrix)g(is)h(real.)30 b(Recall)22 b(that)g(the)f(f)o(actor)i Fs(1)p Fr(=r)2986 1678 y Fq(2)3046 1714 y Fv(in)f(front)f(of)456 1830 y Ft(\000)p Fs(\001)614 1845 y Fo(\022)681 1830 y Fv(and)28 b Fr(\013)915 1845 y Fo(d)984 1830 y Fv(is)f(also)h(substituted)e(by)i (a)g(smooth)e(compactly)h(supported)g(function)456 1947 y Fr(\020)499 1962 y Fo(")535 1947 y Fr(=r)631 1910 y Fq(2)670 1947 y Fv(.)555 2063 y(Let)e(us)f(consider)h(the)f(equation) 456 2306 y(\(5.1\))299 b Ft(\000)1057 2239 y Fr(d)1108 2203 y Fq(2)p 1033 2284 137 4 v 1033 2375 a Fr(dr)1131 2346 y Fq(2)1180 2306 y Fr( )t Fs(\()p Fr(r)s Fs(\))22 b(+)g(\()p Fr(Q)1605 2321 y Fo(z)1645 2306 y Fr( )t Fs(\)\()p Fr(r)s Fs(\))27 b(=)h Fr(z)t( )t Fs(\()p Fr(r)s Fs(\))p Fr(;)116 b(r)31 b Ft(\025)d Fs(1)p Fr(;)41 b(z)33 b Ft(2)28 b Fp(C)20 b Fr(;)456 2525 y Fv(with)30 b Fr(Q)741 2540 y Fo(z)812 2525 y Fv(gi)n(v)o(en)g(by)h(\(4.3\))g(and)h(let)f Fr( )1784 2540 y Fo(k)1827 2525 y Fs(\()p Fr(r)s Fs(\))g Fv(be)g(the)g(solution)f(of)h(the)g(equation)g(\(5.1\))456 2641 y(satisfying)949 2825 y Fr( )1012 2840 y Fo(k)1055 2825 y Fs(\()p Fr(r)s Fs(\))c(=)h(exp)18 b(\()p Fr(ik)s(r)s Fs(\))o Fr(;)117 b(k)1882 2784 y Fq(2)1949 2825 y Fs(=)28 b Fr(z)t(;)42 b Fv(Im)24 b Fr(k)31 b(>)d Fs(0)p Fr(;)41 b Ft(8)p Fr(r)31 b(>)c(c)2884 2840 y Fq(2)2924 2825 y Fr(:)555 3009 y Fv(Then)e(this)f(solution)f(also)h(satis\002es)h(the)f (follo)n(wing)f(\223inte)o(gral\224)h(equation)456 3244 y(\(5.2\))290 b Fr( )1000 3259 y Fo(k)1044 3244 y Fs(\()p Fr(r)s Fs(\))27 b(=)g Fr(e)1342 3203 y Fo(ik)r(r)1465 3244 y Ft(\000)c Fr(k)1619 3203 y Fl(\000)p Fq(1)1730 3108 y Fm(Z)1829 3135 y Fl(1)1785 3334 y Fo(r)1921 3244 y Fs(sin)16 b Fr(k)s Fs(\()p Fr(r)25 b Ft(\000)e Fr(s)p Fs(\)\()p Fr(Q)2517 3259 y Fo(z)2556 3244 y Fr( )2619 3259 y Fo(k)2662 3244 y Fs(\)\()p Fr(s)p Fs(\))17 b Fr(ds:)555 3484 y Fv(In)23 b(order)f(to)g(describe)h(the)f(properties)g(of)h Fr( )2082 3499 y Fo(k)2125 3484 y Fs(\()p Fr(r)s Fs(\))f Fv(we)g(systematically)f(use)h(the)h(fol-)456 3601 y(lo)n(wing)e (analytic)i(Fredholm)f(theorem)h(\(see,)h(for)f(e)o(xample,)f(M.Reed)i (and)f(B.Simon)456 3717 y([21)o(],)i(Theorem)g(VI.14)f(or)h(D.Y)-10 b(af)o(ae)n(v)25 b(Ch.I,)g(Section)g(8\):)456 3906 y FB(Theor)n(em)h(5.1.)41 b Fn(Let)25 b Fr(D)30 b Ft(\032)e Fp(C)51 b Fn(be)25 b(an)g(open)f(connected)h(set)f(and)g(let)h Fe(T)p Fs(\()p Fr(k)s Fs(\))f Fn(be)h(an)g(an-)456 4023 y(alytic)g(oper)o(ator)f(valued)i(function)e(on)i Fr(D)i Fn(suc)o(h)d(that)g Fe(T)p Fs(\()p Fr(k)s Fs(\))h Fn(is)f(a)h(compact)f (oper)o(ator)456 4139 y(in)f(a)h(Hilbert)e(space)i(for)f(eac)o(h)h Fr(k)31 b Ft(2)d Fr(D)s Fn(.)i(Then)414 4255 y Fv(\(1\))42 b Fn(either)24 b Fs(\()p Fr(I)30 b Ft(\000)23 b Fe(T)p Fs(\()p Fr(k)s Fs(\)\))1275 4219 y Fl(\000)p Fq(1)1393 4255 y Fn(e)n(xists)i(for)f(no)g Fr(k)31 b Ft(2)d Fr(D)s Fn(,)414 4371 y Fv(\(2\))42 b Fn(or)24 b Fs(\()p Fr(I)29 b Ft(\000)22 b Fe(T)p Fs(\()p Fr(k)s Fs(\)\))1129 4335 y Fl(\000)p Fq(1)1247 4371 y Fn(e)n(xists)i(for)g(all)g Fr(k)31 b Ft(2)d Fr(D)c Ft(n)d Fr(D)2195 4386 y Fq(0)2234 4371 y Fn(,)k(wher)l(e)g Fr(D)2629 4386 y Fq(0)2693 4371 y Fn(is)f(a)h(discr)l(ete)f(subset)456 4488 y(of)30 b Fr(D)s Fn(.)47 b(In)30 b(this)g(case)g Fs(\()p Fr(I)k Ft(\000)27 b Fe(T)p Fs(\()p Fr(k)s Fs(\)\))1669 4451 y Fl(\000)p Fq(1)1793 4488 y Fn(is)j(mer)l(omorphic)f(in)h Fr(D)j Fn(with)d(possible)f(poles)456 4604 y(belonging)23 b(to)h Fr(D)1062 4619 y Fq(0)1102 4604 y Fn(.)555 4793 y Fv(W)-8 b(e)24 b(\002rst)g(apply)g(this)e(theorem)i(in)f(order)h(to)f (pro)o(v)o(e)g(the)h(statement)e(which)i(is)f(quite)456 4910 y(standard)h(in)g(the)h(resonance)g(theory)-6 b(.)456 5099 y FB(Lemma)39 b(5.1.)49 b Fn(The)39 b(oper)o(ator)e Fr(Q)1671 5114 y Fo(z)1750 5099 y Fn(has)h(a)g(mer)l(omorphic)f (continuation)g(into)h(the)456 5216 y(second)24 b(sheet)h(of)f(the)h (comple)n(x)g(plane)o(.)p eop %%Page: 11 11 11 10 bop 1246 251 a Fj(ABSOLUTEL)-7 b(Y)22 b(CONTINUOUS)h(SPECTR)m(UM) 714 b(11)555 458 y Fn(Pr)l(oof)o(.)28 b Fv(Let)23 b Fr(S)1047 473 y Fo(")1107 458 y Fv(be)g(the)f(same)h(operator)g(as)g(in)f (\(4.4\))h(and)g(let)2677 433 y Fs(~)2660 458 y Fr(S)34 b Fs(=)27 b Ft(\000)p Fr(d)2985 422 y Fq(2)3025 458 y Fr(=dr)3172 422 y Fq(2)3233 458 y Fv(be)c(an)456 575 y(operator)33 b(in)h Fr(L)999 538 y Fq(2)1038 575 y Fs(\(\(1)p Fr(;)17 b Ft(1)p Fs(\))p Fr(;)g(P)d(L)1532 538 y Fq(2)1570 575 y Fs(\()p Fp(S)1669 538 y Fo(d)p Fl(\000)p Fq(1)1794 575 y Fs(\)\))34 b Fv(with)f(the)g(Dirichlet)g(boundary)g(condition)456 691 y(at)d Fs(1)p Fv(.)47 b(Let)31 b Fr(\036)38 b Ft(2)g Fr(C)1120 655 y Fl(1)1113 715 y Fq(0)1195 691 y Fs(\()p Fp(R)1299 706 y Fq(+)1364 691 y Fs(\))30 b Fv(be)h(a)f(function)g (which)g(is)g(identically)f(equal)h(to)g(one)h(on)456 807 y(the)24 b(support)g(of)h(the)g(matrix-function)e Fr(V)46 b Fv(and)25 b Fr(\020)2146 822 y Fo(")2182 807 y Fv(.)31 b(Then)522 1018 y Fr(\036)p Fs(\()p Fr(S)678 1033 y Fo(")736 1018 y Fs(+)22 b Fr(V)891 1033 y Fq(2)953 1018 y Ft(\000)g Fr(z)t Fs(\))1139 977 y Fl(\000)p Fq(1)1234 1018 y Fr(\036)28 b Fs(=)1423 907 y Fm(\020)1483 1018 y Fr(I)i Fs(+)22 b Fr(\036)p Fs(\()1766 993 y(~)1750 1018 y Fr(S)28 b Ft(\000)22 b Fr(z)t Fs(\))2024 977 y Fl(\000)p Fq(1)2119 907 y Fm(\020)2179 1018 y Fr(V)2236 1033 y Fq(2)2297 1018 y Fs(+)g Fr(\020)2438 1033 y Fo(")2484 950 y Fs(\001)2565 965 y Fo(\022)p 2484 995 121 4 v 2502 1086 a Fr(r)2549 1057 y Fq(2)2615 907 y Fm(\021\021)2734 930 y Fl(\000)p Fq(1)2828 1018 y Fr(\036)p Fs(\()2940 993 y(~)2924 1018 y Fr(S)28 b Ft(\000)22 b Fr(z)t Fs(\))3198 977 y Fl(\000)p Fq(1)3293 1018 y Fr(\036:)456 1231 y Fv(Ob)o(viously)29 b(both)j(operators)g Fr(\036)p Fs(\()1626 1206 y(~)1610 1231 y Fr(S)g Ft(\000)c Fr(z)t Fs(\))1894 1195 y Fl(\000)p Fq(1)1989 1231 y Fs(\()p Fr(V)2084 1246 y Fq(2)2151 1231 y Fs(+)g Fr(\020)2298 1246 y Fo(")2344 1189 y Fq(\001)2403 1201 y Fi(\022)p 2344 1208 94 4 v 2357 1265 a Fo(r)2391 1247 y Ff(2)2448 1231 y Fs(\))k Fv(and)g Fr(\036)p Fs(\()2806 1206 y(~)2790 1231 y Fr(S)h Ft(\000)28 b Fr(z)t Fs(\))3075 1195 y Fl(\000)p Fq(1)3170 1231 y Fr(\036)k Fv(ha)n(v)o(e)456 1347 y(an)22 b(analytic)h (continuation)e(into)g(the)i(second)g(sheet)f(of)h(the)f(comple)o(x)g (plane)g(through)456 1464 y(the)i(positi)n(v)o(e)f(semi-axis.)29 b(By)d(using)d(Theorem)i(5.1)f(we)h(obtain)f(that)h(the)f(operator)1248 1564 y Fm(\020)1307 1674 y Fr(I)30 b Fs(+)22 b Fr(\036)p Fs(\()1591 1649 y(~)1574 1674 y Fr(S)28 b Ft(\000)22 b Fr(z)t Fs(\))1848 1633 y Fl(\000)p Fq(1)1943 1564 y Fm(\020)2003 1674 y Fr(V)2060 1689 y Fq(2)2121 1674 y Fs(+)g Fr(\020)2262 1689 y Fo(")2309 1607 y Fs(\001)2390 1622 y Fo(\022)p 2309 1651 121 4 v 2326 1743 a Fr(r)2373 1714 y Fq(2)2439 1564 y Fm(\021\021)2558 1586 y Fl(\000)p Fq(1)456 1864 y Fv(and)28 b(thus)g(the)h(operator)f Fr(Q)1411 1879 y Fo(z)1480 1864 y Fv(de\002ned)h(in)g(\(4.3\))f(ha)n(v)o(e)h (meromorphic)f(continuations)456 1981 y(into)c(the)g(second)h(sheet)f (of)h(the)g(comple)o(x)e(plane.)31 b Fc(\003)555 2142 y Fv(Let)25 b(us)f(no)n(w)g(apply)h(Theorem)f(5.1)h(to)f(the)h (operator)987 2355 y Fe(T)p Fs(\()p Fr(k)s Fs(\))p Fr( )t Fs(\()p Fr(r)s Fs(\))i(=)g Ft(\000)p Fr(k)1635 2314 y Fl(\000)p Fq(1)1747 2220 y Fm(Z)1846 2246 y Fl(1)1802 2445 y Fo(r)1938 2355 y Fs(sin)16 b Fr(k)s Fs(\()p Fr(r)25 b Ft(\000)d Fr(s)p Fs(\)\()p Fr(Q)2533 2370 y Fo(z)2573 2355 y Fr( )t Fs(\)\()p Fr(s)p Fs(\))17 b Fr(ds)456 2578 y Fv(in)31 b Fr(L)631 2542 y Fq(2)671 2578 y Fs(\(1)p Fr(;)17 b(c)844 2593 y Fq(2)883 2578 y Fs(\))p Fv(.)53 b(W)-8 b(e)32 b(conclude)g(that)g(the)g(equation)f(\(5.2\))h(is)g (uniquely)e(solv)n(able)h(for)456 2694 y(all)f Fr(k)35 b Fv(e)o(xcept)c(perhaps)g(a)g(discrete)g(sequence)h(of)f(points)f(and) h(that)g(its)f(solution)f Fr( )3401 2709 y Fo(k)456 2811 y Fv(is)h(a)i(meromorphic)e(with)h(respect)g(to)g Fr(k)k Fv(function)30 b(with)g(v)n(alues)h(in)g Fr(L)2944 2774 y Fq(2)2984 2811 y Fs(\(1)p Fr(;)17 b(c)3157 2826 y Fq(2)3195 2811 y Fs(\))p Fv(,)33 b(in)e(a)456 2927 y(neighbourhood)23 b(of)i(e)n(v)o(ery)f(Im)g Fr(k)31 b Ft(\025)d Fs(0)p Fv(,)d Fr(k)31 b Ft(6)p Fs(=)c(0)p Fv(.)k(Clearly)456 3089 y(\(5.3\))394 b Fr( )1104 3104 y Fo(k)1147 3089 y Fs(\()p Fr(x)p Fs(\))28 b(=)g Fr(a)p Fs(\()p Fr(k)s Fs(\))p Fr(e)1636 3048 y Fo(ik)r(x)1765 3089 y Fs(+)22 b Fr(b)p Fs(\()p Fr(k)s Fs(\))p Fr(e)2079 3048 y Fl(\000)p Fo(ik)r(x)2240 3089 y Fr(;)117 b Fs(1)27 b Fr(<)h(x)g(<)f(c)2792 3104 y Fq(1)2832 3089 y Fr(;)456 3251 y Fv(and)e(therefore)g(both)g Fr(a)p Fs(\()p Fr(k)s Fs(\))g Fv(and)g Fr(b)p Fs(\()p Fr(k)s Fs(\))h Fv(are)g(meromorphic)e(functions)g(\(e)n(v)o(en)g(in)h (some)456 3367 y(neighborhoods)e(of)i(points)e Fr(k)31 b Ft(6)p Fs(=)c(0)e Fv(of)g(the)g(real)g(axis\).)555 3483 y(Consider)g(the)f(resolv)o(ent)g(operator)h Fr(R)q Fs(\()p Fr(z)t Fs(\))j(=)g(\()p Fr(L)2268 3498 y Fo(z)2330 3483 y Ft(\000)22 b Fr(z)t Fs(\))2516 3447 y Fl(\000)p Fq(1)2611 3483 y Fv(,)j(where)g Fr(L)2995 3498 y Fo(z)3060 3483 y Fv(is)f(de\002ned)456 3600 y(in)19 b(\(4.2\).)29 b(If)21 b Fr(\037)946 3615 y Fo(c)977 3624 y Ff(1)1036 3600 y Fv(is)e(the)h(operator)h(of)f(multiplication)d(by)j(the)g (characteristic)g(function)456 3716 y(of)k Fs(\(1)p Fr(;)17 b(c)736 3731 y Fq(1)775 3716 y Fs(\))p Fv(.)31 b(Then)25 b Fr(R)q Fs(\()p Fr(z)t Fs(\))p Fr(\037)1360 3731 y Fo(c)1391 3740 y Ff(1)1455 3716 y Fv(is)f(an)h(inte)o(gral)f(operator)h(with)f (the)g(k)o(ernel:)456 3969 y(\(5.4\))225 b Fr(G)949 3984 y Fo(z)989 3969 y Fs(\()p Fr(r)m(;)17 b(s)p Fs(\))27 b(=)1327 3798 y Fm(\()1418 3845 y Fo( )1464 3857 y Fi(k)1502 3845 y Fq(\()p Fo(s)p Fq(\))p 1417 3870 175 4 v 1417 3927 a Fo( )1463 3939 y Fi(k)1501 3927 y Fq(\(1\))1628 3845 y(sin)o(\()p Fo(k)r Fq(\()p Fo(r)r Fl(\000)p Fq(1\)\))p 1628 3870 359 4 v 1788 3927 a Fo(k)1997 3893 y Fr(;)116 b Fs(for)24 b Fr(r)31 b(<)c(s)h(<)f(c)2678 3908 y Fq(1)2718 3893 y Fr(;)1418 3994 y Fo( )1464 4006 y Fi(k)1502 3994 y Fq(\()p Fo(r)r Fq(\))p 1417 4019 175 4 v 1417 4076 a Fo( )1463 4088 y Fi(k)1501 4076 y Fq(\(1\))1628 3994 y(sin)o(\()p Fo(k)r Fq(\()p Fo(s)p Fl(\000)p Fq(1\)\))p 1628 4019 358 4 v 1787 4076 a Fo(k)1995 4041 y Fr(;)117 b Fs(for)24 b Fr(s)k(<)f Fs(min)o Ft(f)p Fr(c)2712 4056 y Fq(1)2751 4041 y Fr(;)17 b(r)s Ft(g)p Fr(:)456 4222 y Fv(Indeed,)26 b(assuming)e(that)h(supp)p Fs(\()p Fr(f)11 b Fs(\))28 b Ft(\032)i Fs(\(1)p Fr(;)17 b(c)1987 4237 y Fq(1)2026 4222 y Fs(\))25 b Fv(we)h(can)g(easily)f(check)h(that)g (the)f(func-)456 4338 y(tion)999 4494 y Fr(u)p Fs(\()p Fr(r)s Fs(\))i(=)1409 4427 y(1)p 1318 4471 231 4 v 1318 4563 a Fr( )1381 4578 y Fo(k)1424 4563 y Fs(\(1\))1559 4384 y Fm(n)1625 4359 y(Z)1725 4385 y Fl(1)1681 4584 y Fo(r)1826 4427 y Fs(sin\()p Fr(k)s Fs(\()p Fr(r)e Ft(\000)e Fs(1\)\))p 1826 4471 543 4 v 2071 4563 a Fr(k)2379 4494 y( )2442 4509 y Fo(k)2485 4494 y Fs(\()p Fr(s)p Fs(\))p Fr(f)11 b Fs(\()p Fr(s)p Fs(\))17 b Fr(ds)1243 4768 y Fs(+)1336 4632 y Fm(Z)1435 4658 y Fo(r)1391 4858 y Fq(1)1490 4768 y Fr( )1553 4783 y Fo(k)1596 4768 y Fs(\()p Fr(r)s Fs(\))1729 4700 y(sin)o(\()p Fr(k)s Fs(\()p Fr(s)22 b Ft(\000)h Fs(1\)\))p 1729 4745 V 1972 4836 a Fr(k)2280 4768 y(f)11 b Fs(\()p Fr(s)p Fs(\))17 b Fr(ds)2592 4657 y Fm(o)456 4959 y Fv(satis\002es)24 b(the)h(equation)456 5180 y(\(5.5\))260 b Ft(\000)1018 5113 y Fr(d)1069 5077 y Fq(2)p 994 5158 137 4 v 994 5249 a Fr(dr)1092 5220 y Fq(2)1142 5180 y Fr(u)p Fs(\()p Fr(r)s Fs(\))21 b(+)h(\()p Fr(Q)1555 5195 y Fo(z)1595 5180 y Fr(u)p Fs(\)\()p Fr(r)s Fs(\))f Ft(\000)h Fr(z)t(u)p Fs(\()p Fr(r)s Fs(\))28 b(=)g Fr(f)11 b Fs(\()p Fr(r)s Fs(\))p Fr(;)115 b(r)30 b Ft(\025)f Fs(1)p Fr(;)41 b(z)32 b Ft(2)d Fp(C)19 b Fr(;)p eop %%Page: 12 12 12 11 bop 456 251 a Fj(12)808 b(LAPTEV)-10 b(,)17 b(N)m(ABOK)n(O)i(AND) f(SAFR)m(ONO)l(V)456 450 y Fv(and)24 b(moreo)o(v)o(er)g Fr(u)p Fs(\(1\))j(=)g(0)p Fv(.)555 566 y(Here)j(we)g(should)f(also)g (mention)f(that)h(since)g Fr( )2242 581 y Fo(k)2285 566 y Fs(\(1\))h Fv(is)f(meromorphic)f(in)h Fr(k)k Fv(in)c(a)456 683 y(neighborhood)23 b(of)j(an)o(y)f Fr(k)32 b Ft(6)p Fs(=)c(0)p Fv(,)d(we)h(conclude)f(that)g Fr( )2362 698 y Fo(k)2405 683 y Fs(\(1\))j(=)h(0)c Fv(only)g(on)g(a)h(discrete)456 799 y(subset)21 b(of)i(the)g(closed)f(upper)g(half)h(plane,)g(ha)n (ving)f(no)g(accumulation)g(points)f(e)o(xcept)456 915 y(perhaps)j(zero.)755 1134 y(6.)51 b(W)t Fu(R)q(O)t(N)t(S)t(K)t(I)t(A)t (N)35 b(A)t(N)t(D)30 b(P)t(R)q(O)t(P)t(E)t(RT)t(I)5 b(E)t(S)36 b(O)t(F)31 b(T)t(H)t(E)d Fr(M)12 b Fv(-)t Fu(F)t(U)t(N)t(C)t(T)t(I)5 b(O)t(N)g Fv(.)555 1309 y(Let)25 b(as)g(in)f(\(4.3\))1225 1473 y Fr(Q)1302 1488 y Fo(z)1369 1473 y Fs(=)k Fr(V)1530 1488 y Fq(1)1591 1473 y Ft(\000)23 b Fr(V)1748 1488 y Fq(1)p Fo(;)p Fq(2)1842 1473 y Fs(\()p Fr(S)1940 1488 y Fo(")1999 1473 y Fs(+)f Fr(V)2154 1488 y Fq(2)2215 1473 y Ft(\000)h Fr(z)t Fs(\))2402 1432 y Fl(\000)p Fq(1)2497 1473 y Fr(V)2554 1488 y Fq(2)p Fo(;)p Fq(1)2648 1473 y Fr(:)456 1637 y Fv(The)h(function)456 1837 y(\(6.1\))674 b Fr(M)10 b Fs(\()p Fr(k)s Fs(\))28 b(=)1697 1770 y Fr( )1764 1734 y Fl(0)1760 1796 y Fo(k)1803 1770 y Fs(\(1\))p 1697 1814 231 4 v 1697 1906 a Fr( )1760 1921 y Fo(k)1803 1906 y Fs(\(1\))1938 1837 y Fr(;)216 b Fv(Im)24 b Fr(k)31 b Ft(\025)d Fs(0)p Fr(;)456 2055 y Fv(is)h(no)n(w)f(well)i(de\002ned)g (and)f(called)h(the)f(W)-8 b(e)o(yl)29 b Fr(M)10 b Fv(-function)30 b(of)g(the)f(operator)h(\(5.1\))o(.)456 2171 y(Let)24 b(us)h(consider)f(the)h(Wronskian)456 2336 y(\(6.2\))437 b Fr(W)14 b Fs([)p 1217 2255 107 4 v Fr( )1280 2351 y Fo(k)1322 2336 y Fr(;)j( )1429 2351 y Fo(k)1472 2336 y Fs(]\()p Fr(r)s Fs(\))28 b(=)p 1753 2253 V 27 w Fr( )1820 2302 y Fl(0)1816 2364 y Fo(k)1859 2336 y Fs(\()p Fr(r)s Fs(\))p Fr( )2045 2351 y Fo(k)2088 2336 y Fs(\()p Fr(r)s Fs(\))21 b Ft(\000)p 2332 2255 V 23 w Fr( )2395 2351 y Fo(k)2438 2336 y Fs(\()p Fr(r)s Fs(\))p Fr( )2628 2295 y Fl(0)2624 2360 y Fo(k)2667 2336 y Fs(\()p Fr(r)s Fs(\))p Fr(:)456 2509 y Fv(Note)k(that)p 849 2428 V 24 w Fr( )912 2524 y Fo(k)981 2509 y Fv(satis\002es)g(the)g(equation)f(\(5.1\))h (with)g Fr(Q)p 2333 2487 40 3 v 15 x Fo(z)2398 2509 y Fv(and)p 2567 2454 50 4 v 25 w Fr(z)30 b Fv(instead)25 b(of)g Fr(Q)3135 2524 y Fo(z)3201 2509 y Fv(and)g Fr(z)t Fv(.)456 2625 y(Since)g Fr( )765 2640 y Fo(k)833 2625 y Fv(is)f(a)h(solution)e(of)i(the)g(equation)f(\(5.1\))h(we)g(\002nd) 489 2773 y Fr(d)p 466 2817 98 4 v 466 2908 a(dr)573 2840 y(W)14 b Fs([)p 706 2759 107 4 v Fr( )769 2855 y Fo(k)812 2840 y Fr(;)j( )919 2855 y Fo(k)961 2840 y Fs(]\()p Fr(r)s Fs(\))28 b(=)f(\()p Fr(z)6 b Ft(\000)p 1410 2785 50 4 v 2 w Fr(z)g Fs(\))p 1499 2759 107 4 v Fr( )1562 2855 y Fo(k)1605 2840 y Fs(\()p Fr(r)s Fs(\))p Fr( )1791 2855 y Fo(k)1834 2840 y Fs(\()p Fr(r)s Fs(\))r(+)r(\()p Fr(Q)p 2152 2818 40 3 v 15 x Fo(z)p 2191 2759 107 4 v 2191 2840 a Fr( )2254 2855 y Fo(k)2297 2840 y Fs(\)\()p Fr(r)s Fs(\))p Fr( )2521 2855 y Fo(k)2563 2840 y Fs(\()p Fr(r)s Fs(\))r Ft(\000)p 2767 2759 V 2 w Fr( )2830 2855 y Fo(k)2873 2840 y Fs(\()p Fr(r)s Fs(\)\()p Fr(Q)3111 2855 y Fo(z)3151 2840 y Fr( )3214 2855 y Fo(k)3257 2840 y Fs(\)\()p Fr(r)s Fs(\))p Fr(:)456 3033 y Fv(So)25 b(we)g(obtain)456 3197 y(\(6.3\))75 b Ft(\006)p Fv(Im)25 b Ft(f)p Fr(W)14 b Fs([)p 1118 3116 V Fr( )1181 3212 y Fo(k)1224 3197 y Fr(;)j( )1331 3212 y Fo(k)1374 3197 y Fs(]\()p Fr(c)1481 3212 y Fq(2)1520 3197 y Fs(\))22 b Ft(\000)h Fr(W)14 b Fs([)p 1813 3116 V Fr( )1876 3212 y Fo(k)1919 3197 y Fr(;)j( )2026 3212 y Fo(k)2068 3197 y Fs(]\()p Fr(c)2175 3212 y Fq(1)2215 3197 y Fs(\))p Ft(g)27 b(\025)h Fs(0)p Fr(;)116 b Fs(for)55 b Ft(\006)23 b Fv(Im)i Fr(z)32 b Ft(\025)c Fs(0+)p Fr(;)456 3362 y Fv(which)c(means)g(that)h(for)g(all)f (real)i Fr(k)i Fv(we)d(ha)n(v)o(e)f(the)h(follo)n(wing)e(inequality) 1680 3509 y Fr(k)p 1522 3554 371 4 v 1522 3645 a Fv(Im)i Fr(M)10 b Fs(\()p Fr(k)s Fs(\))1930 3577 y Ft(\024)28 b(j)p Fr( )2126 3592 y Fo(k)2169 3577 y Fs(\(1\))p Ft(j)2322 3535 y Fq(2)2361 3577 y Fr(:)456 3796 y Fv(Moreo)o(v)o(er)l(,)23 b(if)i(we)g(represent)g(the)g(solution)e Fr( )2064 3811 y Fo(k)2132 3796 y Fv(for)i(real)g Fr(k)j Fv(in)d(the)f(form)1131 3961 y Fr( )1194 3976 y Fo(k)1237 3961 y Fs(\()p Fr(x)p Fs(\))k(=)g Fr(a)p Fs(\()p Fr(k)s Fs(\))p Fr(e)1726 3920 y Fo(ik)r(x)1855 3961 y Fs(+)22 b Fr(b)p Fs(\()p Fr(k)s Fs(\))p Fr(e)2169 3920 y Fl(\000)p Fo(ik)r(x)2330 3961 y Fr(;)117 b(x)28 b(<)f(c)2702 3976 y Fq(1)2742 3961 y Fr(;)456 4125 y Fv(then)d(it)g(follo)n(ws)g(from)g(\(6.3\))h(that) 1402 4290 y Ft(j)p Fr(a)p Ft(j)1509 4249 y Fq(2)1571 4290 y Ft(\000)d(j)p Fr(b)p Ft(j)1767 4249 y Fq(2)1834 4290 y Ft(\025)28 b Fs(1)p Fr(;)216 b(k)31 b Fs(=)p 2416 4209 55 4 v 27 w Fr(k)s(:)456 4454 y Fv(Clearly)855 4619 y Fr(M)10 b Fs(\()p Fr(k)s Fs(\))29 b(=)e Fr( )1288 4578 y Fl(0)1284 4643 y Fo(k)1327 4619 y Fs(\(1\)\()p Fr( )1553 4634 y Fo(k)1596 4619 y Fs(\(1\)\))1759 4578 y Fl(\000)p Fq(1)1881 4619 y Fs(=)g Fr(ik)s Fs(\(1)22 b Ft(\000)h Fr(\032)p Fs(\()p Fr(k)s Fs(\)\)\(1)f(+)g Fr(\032)p Fs(\()p Fr(k)s Fs(\)\))2923 4578 y Fl(\000)p Fq(1)3018 4619 y Fr(;)456 4783 y Fv(where)1443 4907 y Fr(\032)p Fs(\()p Fr(k)s Fs(\))28 b(:=)f Fr(e)1826 4866 y Fl(\000)p Fq(2)p Fo(ik)1983 4907 y Fr(b)p Fs(\()p Fr(k)s Fs(\))p Fr(a)p Fs(\()p Fr(k)s Fs(\))2335 4866 y Fl(\000)p Fq(1)2430 4907 y Fr(:)456 5051 y Fv(The)d(latter)h(implies)1215 5216 y Fr(\032)p Fs(\()p Fr(k)s Fs(\))j(=)f(\()p Fr(ik)f Ft(\000)c Fr(M)10 b Fs(\()p Fr(k)s Fs(\)\)\()p Fr(ik)26 b Fs(+)c Fr(M)10 b Fs(\()p Fr(k)s Fs(\)\))2563 5174 y Fl(\000)p Fq(1)2658 5216 y Fr(:)p eop %%Page: 13 13 13 12 bop 1246 251 a Fj(ABSOLUTEL)-7 b(Y)22 b(CONTINUOUS)h(SPECTR)m(UM) 714 b(13)456 450 y Fv(Since)25 b Ft(j)p Fr(a)p Ft(j)809 414 y Fq(2)870 450 y Ft(\000)e(j)p Fr(b)p Ft(j)1067 414 y Fq(2)1134 450 y Ft(\025)28 b Fs(1)c Fv(we)h(obtain)f(that)h(for)g (real)g Fr(k)1138 701 y Ft(j)p Fr(a)p Fs(\()p Fr(k)s Fs(\))p Ft(j)1375 660 y Fl(\000)p Fq(2)1496 701 y Ft(\024)j Fs(1)22 b Ft(\000)h(j)p Fr(\032)p Fs(\()p Fr(k)s Fs(\))p Ft(j)2008 660 y Fq(2)2075 701 y Fs(=)2285 634 y(4)p Fr(k)s Fv(Im)i Fr(M)p 2188 678 538 4 v 2188 769 a Ft(j)p Fr(ik)g Fs(+)d Fr(M)10 b Fs(\()p Fr(k)s Fs(\))p Ft(j)2685 741 y Fq(2)2735 701 y Fr(:)456 953 y Fv(Note)24 b(that)h(since)f(Im)h Fr(M)38 b Ft(\025)28 b Fs(0)p Fv(,)d(then)f(for)h(an)o(y)g Fr(k)31 b(>)c Fs(0)e Fv(we)g(ha)n(v)o(e)1064 1150 y Ft(j)p Fr(ik)h Fs(+)c Fr(M)10 b Fs(\()p Fr(k)s Fs(\))p Ft(j)1562 1109 y Fq(2)1629 1150 y Fs(=)28 b Fr(k)1787 1109 y Fq(2)1848 1150 y Fs(+)22 b Ft(j)p Fr(M)10 b Ft(j)2106 1109 y Fq(2)2168 1150 y Fs(+)22 b(2)p Fr(k)s Fv(Im)i Fr(M)39 b Ft(\025)28 b Fr(k)2796 1109 y Fq(2)456 1348 y Fv(and)c(therefore)456 1545 y(\(6.4\))604 b Ft(j)p Fr(a)p Fs(\()p Fr(k)s Fs(\))p Ft(j)1488 1504 y Fl(\000)p Fq(2)1610 1545 y Ft(\024)28 b Fs(4)p Fr(k)1818 1504 y Fl(\000)p Fq(1)1912 1464 y Fm(\000)1958 1545 y Fv(Im)d Fr(M)2198 1464 y Fm(\001)2244 1545 y Fr(;)116 b(k)31 b(>)d Fs(0)p Fr(:)456 1743 y Fv(From)c(\(6.1\))h (and)g(\(6.2\))g(we)g(obtain)456 1940 y(\(6.5\))716 b(Im)24 b Fr(M)10 b Fs(\()p Fr(k)s Fs(\))29 b Fr(>)e Fs(0)99 b(if)31 b Fv(Im)25 b Fr(k)2291 1899 y Fq(2)2358 1940 y Fr(>)j Fs(0)p Fr(:)456 2137 y Fv(Thus,)j(there)h(are)f(constants)61 b Fr(C)1598 2152 y Fq(0)1677 2137 y Ft(2)39 b Fp(R)j Fv(and)31 b Fr(C)2130 2152 y Fq(1)2208 2137 y Ft(\025)40 b Fs(0)31 b Fv(and)g(a)g(positi)n(v)o(e)e(measure)i Fr(\026)p Fv(,)456 2254 y(such)24 b(that)1588 2310 y Fm(Z)1688 2336 y Fl(1)1644 2536 y(\0001)1812 2378 y Fr(d\026)p Fs(\()p Fr(t)p Fs(\))p 1800 2423 244 4 v 1800 2514 a(1)e(+)g Fr(t)2004 2485 y Fq(2)2081 2446 y Fr(<)28 b Ft(1)p Fr(;)456 2679 y Fv(where)456 2906 y(\(6.6\))201 b Fr(M)10 b Fs(\()p Fr(k)s Fs(\))28 b(=)g Fr(C)1284 2921 y Fq(0)1345 2906 y Fs(+)22 b Fr(C)1513 2921 y Fq(1)1553 2906 y Fr(z)k Fs(+)1722 2770 y Fm(Z)1778 2996 y Fh(R)1847 2795 y Fm(\020)1995 2839 y Fs(1)p 1916 2883 207 4 v 1916 2974 a Fr(t)c Ft(\000)h Fr(z)2155 2906 y Ft(\000)2369 2839 y Fr(t)p 2264 2883 244 4 v 2264 2974 a Fs(1)f(+)g Fr(t)2468 2946 y Fq(2)2518 2795 y Fm(\021)2577 2906 y Fr(d\026)p Fs(\()p Fr(t)p Fs(\))p Fr(;)116 b(k)2995 2865 y Fq(2)3062 2906 y Fs(=)28 b Fr(z)t(:)456 3164 y Fv(Finally)-6 b(,)39 b(note)e(that)f Fr(R)q Fs(\()p Fr(z)t Fs(\))52 b(=)e Fr(P)1633 3179 y Fq(0)1672 3164 y Fs(\()p Fr(U)1786 3128 y Fl(\003)1826 3164 y Fr(H)1907 3179 y Fq(0)1947 3164 y Fr(U)42 b Fs(+)31 b Fr(V)53 b Ft(\000)31 b Fr(z)t Fs(\))2467 3128 y Fl(\000)p Fq(1)2562 3164 y Fr(P)2625 3179 y Fq(0)2702 3164 y Fv(and)37 b(hence)h(we)f(can)456 3281 y(formally)24 b(write)g(that)653 3538 y Fr(M)10 b Fs(\()p Fr(k)s Fs(\))28 b(=)1084 3471 y Fr(@)1140 3434 y Fq(2)p 1029 3515 207 4 v 1029 3606 a Fr(@)5 b(r)s(@)g(s)1245 3538 y(G)1322 3553 y Fo(z)1362 3538 y Fs(\()p Fr(r)m(;)17 b(s)p Fs(\))p Ft(j)1597 3553 y Fq(\(1)p Fo(;)p Fq(1\))1773 3538 y Fs(=)28 b(\()p Fr(P)1978 3553 y Fq(0)2017 3538 y Fs(\()p Fr(U)2131 3497 y Fl(\003)2171 3538 y Fr(H)2252 3553 y Fq(0)2292 3538 y Fr(U)k Fs(+)22 b Fr(V)44 b Ft(\000)23 b Fr(z)t Fs(\))2776 3497 y Fl(\000)p Fq(1)2871 3538 y Fr(P)2934 3553 y Fq(0)2973 3538 y Fr(\016)3020 3497 y Fl(0)3016 3563 y Fq(1)3056 3538 y Fr(;)17 b(\016)3147 3497 y Fl(0)3143 3563 y Fq(1)3182 3538 y Fs(\))p Fr(;)456 3770 y Fv(where)23 b Fr(\016)769 3734 y Fl(0)765 3795 y Fq(1)827 3770 y Fv(is)f(the)g(deri)n(v)n(ati)n(v)o(e)f(of)h(the)h (delta)f(function)g Fr(\016)t Fs(\()p Fr(r)17 b Ft(\000)d Fs(1\))p Fv(.)29 b(Let)23 b Fr(\037)2889 3785 y Fo(c)2920 3794 y Ff(1)2981 3770 y Fv(be)f(the)h(char)n(-)456 3887 y(acteristic)d(function)g(of)i Fs(\(1)p Fr(;)17 b(c)1467 3902 y Fq(1)1505 3887 y Fs(\))p Fv(.)30 b(The)21 b(representation)f (\(5.4\))h(for)h(the)e(resolv)o(ent)g(oper)n(-)456 4003 y(ator)28 b(gi)n(v)o(es)e(us)i(the)g(representation)g(for)g(the)g (operator)h Fr(\037)2449 4018 y Fo(c)2480 4027 y Ff(1)2518 4003 y Fr(P)2581 4018 y Fq(0)2620 4003 y Fr(E)2692 4018 y Fo(U)2747 3999 y Fg(\003)2783 4018 y Fo(H)2841 4027 y Ff(0)2876 4018 y Fo(U)7 b Fq(+)p Fo(V)3047 4003 y Fs(\()p Fr(!)t Fs(\))p Fr(P)3251 4018 y Fq(0)3289 4003 y Fr(\037)3350 4018 y Fo(c)3381 4027 y Ff(1)3419 4003 y Fv(,)456 4119 y(where)25 b Fr(E)796 4134 y Fo(U)851 4115 y Fg(\003)887 4134 y Fo(H)945 4143 y Ff(0)979 4134 y Fo(U)7 b Fq(+)p Fo(V)1150 4119 y Fs(\()p Fr(!)t Fs(\))24 b Fv(is)h(the)f(spectral)h (measure)g(of)g Fr(U)2429 4083 y Fl(\003)2469 4119 y Fr(H)2550 4134 y Fq(0)2589 4119 y Fr(U)33 b Fs(+)22 b Fr(V)f Fv(:)456 4369 y(\(6.7\))402 b Fs(\()p Fr(P)1150 4384 y Fq(0)1189 4369 y Fr(E)1261 4384 y Fo(U)1316 4365 y Fg(\003)1352 4384 y Fo(H)1410 4393 y Ff(0)1445 4384 y Fo(U)7 b Fq(+)p Fo(V)1615 4369 y Fs(\()p Fr(!)t Fs(\))p Fr(P)1819 4384 y Fq(0)1858 4369 y Fr(f)e(;)17 b(f)11 b Fs(\))27 b(=)2183 4233 y Fm(Z)2238 4459 y Fo(!)2305 4369 y Ft(j)p Fr(F)14 b Fs(\()p Fr(\025)p Fs(\))p Ft(j)2571 4328 y Fq(2)2609 4369 y Fr(d\026)p Fs(\()p Fr(\025)p Fs(\))456 4616 y Fv(and)24 b(where)615 4845 y Fr(F)14 b Fs(\()p Fr(\025)p Fs(\))27 b(=)968 4778 y(1)p 965 4822 55 4 v 965 4914 a Fr(k)1046 4710 y Fm(Z)1146 4736 y Fo(c)1177 4745 y Ff(1)1101 4935 y Fq(0)1232 4845 y Fs(sin)o(\()p Fr(k)s Fs(\()p Fr(r)e Ft(\000)e Fs(1\)\))p Fr(f)11 b Fs(\()p Fr(r)s Fs(\))24 b Fr(dr)m(;)141 b Fs(supp)q Fr(f)38 b Ft(\032)28 b Fs(\(1)p Fr(;)17 b(c)2806 4860 y Fq(1)2845 4845 y Fs(\))p Fr(;)67 b(k)3031 4804 y Fq(2)3098 4845 y Fs(=)27 b Fr(\025:)456 5099 y Fv(Since)d Fr(F)38 b Fv(is)23 b(a)i(boundary)e(v)n(alue)g(of)i(an)f(analytic)f(function,)g (we)i(obtain)e(that)g Fr(F)14 b Fs(\()p Fr(\025)p Fs(\))28 b Ft(6)p Fs(=)456 5216 y(0)c Fv(for)h(a.e.)32 b Fr(\025)p Fv(.)e(This)24 b(means)h(that)f Fr(E)1680 5231 y Fo(H)1747 5216 y Fs(\()p Fr(!)t Fs(\))j Ft(6)p Fs(=)h(0)d Fv(if)f Fr(\026)2237 5179 y Fl(0)2288 5216 y Fr(>)k Fs(0)c Fv(a.e.)31 b(on)25 b Fr(!)t Fv(.)p eop %%Page: 14 14 14 13 bop 456 251 a Fj(14)808 b(LAPTEV)-10 b(,)17 b(N)m(ABOK)n(O)i(AND) f(SAFR)m(ONO)l(V)1426 450 y Fv(7.)51 b(T)t Fu(R)t(A)q(C)t(E)33 b(I)t(N)t(E)t(Q)s(U)q(A)t(L)t(I)t(T)t(I)t(E)t(S)555 624 y Fv(Recall)58 b(that)e(we)h(assume)f(that)g Fr(V)79 b Fv(is)56 b(not)g(a)h(potential)f(b)n(ut)g(the)g(operator)456 666 y Fm(P)561 692 y Fo(n)561 770 y(j)t Fq(=0)704 741 y Fr(P)767 756 y Fo(j)803 741 y Fr(V)899 666 y Fm(P)1004 692 y Fo(n)1004 770 y(j)t Fq(=0)1147 741 y Fr(P)1210 756 y Fo(j)1246 741 y Fv(,)26 b(which)f(approximates)g Fr(V)47 b Fv(for)26 b(lar)n(ge)g Fr(n)p Fv(.)33 b(As)26 b(before)g(we)g(sub-)456 870 y(stitute)h(the)i(term)g Ft(\000)p Fs(\001)1249 885 y Fo(\022)1288 870 y Fr(=r)1384 834 y Fq(2)1452 870 y Fv(and)g Fr(\013)1687 885 y Fo(d)1728 870 y Fr(=r)1824 834 y Fq(2)1891 870 y Fv(on)g Fs(\(1)p Fr(;)17 b Ft(1)p Fs(\))28 b Fv(by)h(a)g(\224compactly)f(supported\224) 456 986 y(approximations)h Ft(\000)p Fr(\020)1227 1001 y Fo(")1264 986 y Fs(\()p Fr(r)s Fs(\)\001)1468 1001 y Fo(\022)1507 986 y Fr(=r)1603 950 y Fq(2)1674 986 y Fv(and)i Fr(\020)1892 1001 y Fo(")1929 986 y Fs(\()p Fr(r)s Fs(\))p Fr(\013)2114 1001 y Fo(d)2154 986 y Fr(=r)2250 950 y Fq(2)2289 986 y Fv(,)i(where)f Fr(\020)2665 1001 y Fo(")2742 986 y Ft(2)41 b Fr(C)2926 950 y Fl(1)2919 1011 y Fq(0)3001 986 y Fs(\(1)p Fr(;)17 b Ft(1)p Fs(\))31 b Fv(and)456 1102 y Fr(\020)499 1117 y Fo(")535 1102 y Fs(\()p Fr(r)s Fs(\))p Fr(=r)754 1066 y Fq(2)821 1102 y Ft(!)e Fs(1)p Fr(=r)1095 1066 y Fq(2)1159 1102 y Fv(in)c Fr(L)1328 1066 y Fq(1)1368 1102 y Fs(\(1)p Fr(;)17 b Ft(1)p Fs(\))25 b Fv(and)g Fr(L)1897 1066 y Fq(2)1937 1102 y Fs(\(1)p Fr(;)17 b Ft(1)p Fs(\))24 b Fv(as)i Fr(")j Ft(!)f Fs(0)p Fv(.)33 b(Then)25 b(the)g(coef)n(\002cient)456 1219 y Fr(a)p Fs(\()p Fr(k)s Fs(\))j Fv(introduced)g(in)g(\(5.3\))h (will)e(depend)i(on)f Fr(")g Fv(and)h(we)f(shall)g(write)h Fr(a)2967 1234 y Fo(")3004 1219 y Fs(\()p Fr(k)s Fs(\))f Fv(instead)456 1335 y(of)c Fr(a)p Fs(\()p Fr(k)s Fs(\))p Fv(.)31 b(From)25 b(\(5.2\))g(and)g(\(4.3\))g(we)g(\002nd)g(that)456 1571 y Fs(exp)q(\()p Ft(\000)p Fr(ik)s(r)s Fs(\))p Fr( )955 1586 y Fo(k)998 1571 y Fs(\()p Fr(r)s Fs(\))i(=)h(1)p Ft(\000)1432 1503 y Fs(1)p 1388 1548 137 4 v 1388 1639 a(2)p Fr(ik)1551 1435 y Fm(Z)1650 1462 y Fl(1)1606 1661 y Fo(r)1725 1571 y Fs(\(1)p Ft(\000)p Fr(e)1934 1530 y Fq(2)p Fo(ik)r Fq(\()p Fo(s)p Fl(\000)p Fo(r)r Fq(\))2213 1571 y Fs(\)\()p Fr(\020)2332 1586 y Fo(")2368 1571 y Fs(\()p Fr(s)p Fs(\))p Fr(\013)2552 1586 y Fo(d)2592 1571 y Fr(=s)2687 1530 y Fq(2)2726 1571 y Fs(+)p Fr(V)2859 1586 y Fq(1)2899 1571 y Fs(\()p Fr(s)p Fs(\)\))17 b Fr(ds)p Fs(+)p Fr(o)p Fs(\(1)p Fr(=k)s Fs(\))456 1802 y Fv(and)24 b(thus)456 1918 y(\(7.1\))456 2080 y Fr(a)507 2095 y Fo(")544 2080 y Fs(\()p Fr(k)s Fs(\))j(=)75 b(lim)805 2140 y Fo(r)r Fl(!\0001)1051 2080 y Fs(exp)q(\()p Ft(\000)p Fr(ik)s(r)s Fs(\))p Fr( )1550 2095 y Fo(k)1594 2080 y Fs(\()p Fr(r)s Fs(\))27 b(=)g(1)s Ft(\000)2034 2012 y Fs(1)p 1989 2057 V 1989 2148 a(2)p Fr(ik)2153 1944 y Fm(Z)2253 2080 y Fs(\()p Fr(\020)2334 2095 y Fo(")2370 2080 y Fs(\()p Fr(r)s Fs(\))p Fr(\013)2555 2095 y Fo(d)2595 2080 y Fr(=r)2691 2039 y Fq(2)2733 2080 y Fs(+)s Fr(V)2869 2095 y Fq(1)2909 2080 y Fs(\))17 b Fr(dr)6 b Fs(+)s Fr(o)p Fs(\(1)p Fr(=k)s Fs(\))p Fr(;)456 2313 y Fv(as)32 b Fr(k)46 b Ft(!)c(1)p Fv(.)54 b(No)n(w)32 b(let)g Fr(i\014)1433 2328 y Fo(m)1533 2313 y Fv(and)h Fr(\015)1761 2328 y Fo(j)1830 2313 y Fv(be)g(zeros)g(and)f(poles)g(of)h Fr(a)2786 2328 y Fo(")2823 2313 y Fs(\()p Fr(k)s Fs(\))g Fv(in)f(the)h(open)456 2430 y(upper)19 b(half)g(plane.)29 b(Note)19 b(that)g Ft(\000)p 1605 2375 88 4 v Fr(\015)1656 2445 y Fo(j)1712 2430 y Fv(are)h(also)f(poles)g(of)g Fr(a)2416 2445 y Fo(")2453 2430 y Fs(\()p Fr(k)s Fs(\))h Fv(\(this)e(will)h(follo)n(w)f (from)456 2546 y(\(7.5\))o(\).)33 b(W)-8 b(e)26 b(shall)f(sho)n(w)f(in) h(Proposition)f(7.1)h(that)g Ft(f\000)p Fr(\014)2441 2510 y Fq(2)2435 2571 y Fo(m)2502 2546 y Ft(g)h Fv(are)g(the)f(eigen)l (v)n(alues)f(of)456 2662 y(a)k(certain)g(self-adjoint)f(operator)g(of)h (a)h(Schr)8 b(\250)-41 b(odinger)27 b(type.)40 b(Therefore)29 b(we)f(choose)456 2778 y Fr(\014)511 2793 y Fo(m)605 2778 y Fr(>)f Fs(0)p Fv(.)k(Let)25 b Fe(B)g Fv(be)g(the)f (corresponding)g(Blaschk)o(e)h(product)1217 3015 y Fe(B)p Fs(\()p Fr(k)s Fs(\))j(=)1566 2921 y Fm(Y)1599 3130 y Fo(m)1720 2948 y Fs(\()p Fr(k)d Ft(\000)e Fr(i\014)2022 2963 y Fo(m)2089 2948 y Fs(\))p 1720 2992 407 4 v 1721 3084 a(\()p Fr(k)i Fs(+)d Fr(i\014)2021 3099 y Fo(m)2088 3084 y Fs(\))2153 2921 y Fm(Y)2201 3131 y Fo(j)2307 2948 y Fs(\()p Fr(k)j Ft(\000)p 2521 2893 88 4 v 23 w Fr(\015)2572 2963 y Fo(j)2608 2948 y Fs(\))p 2307 2992 339 4 v 2307 3084 a(\()p Fr(k)g Ft(\000)e Fr(\015)2572 3099 y Fo(j)2608 3084 y Fs(\))2656 3015 y Fr(:)456 3321 y Fv(Clearly)i Ft(j)p Fe(B)p Fs(\()p Fr(k)s Fs(\))o Ft(j)j Fs(=)f(1)p Fr(;)p 1287 3235 219 4 v 33 w Fe(B)p Fs(\()p Fr(k)s Fs(\))h(=)g Fe(B)p Fs(\()p Ft(\000)p Fr(k)s Fs(\))p Fr(;)34 b(k)c Ft(2)e Fp(R)5 b Fv(,)31 b(and)25 b(we)g(obtain)456 3569 y(\(7.2\))1423 3433 y Fm(Z)1523 3460 y Fq(+)p Fl(1)1478 3659 y(\0001)1669 3569 y Fs(log\()p Fr(a)1884 3584 y Fo(")1921 3569 y Fs(\()p Fr(k)s Fs(\))p Fr(=)p Fe(B)p Fs(\()p Fr(k)s Fs(\)\))17 b Fr(dk)736 3899 y Fs(=)850 3831 y Fr(\031)p 850 3876 59 4 v 855 3967 a Fs(2)935 3763 y Fm(Z)1035 3899 y Fs(\()p Fr(\020)1116 3914 y Fo(")1152 3899 y Fs(\()p Fr(r)s Fs(\))p Fr(\013)1337 3914 y Fo(d)1377 3899 y Fr(=r)1473 3858 y Fq(2)1534 3899 y Fs(+)22 b Fr(V)1689 3914 y Fq(1)1728 3899 y Fs(\()p Fr(r)s Fs(\)\))17 b Fr(dr)24 b Fs(+)e(2)p Fr(\031)2231 3788 y Fm(\020)2290 3804 y(X)2451 3899 y Fr(\014)2506 3914 y Fo(m)2594 3899 y Ft(\000)2694 3804 y Fm(X)2855 3899 y Fv(Im)i Fr(\015)3041 3914 y Fo(j)3077 3788 y Fm(\021)3137 3899 y Fr(;)456 4112 y Fv(pro)o(vided)g(that)h(for) h(some)f(inte)o(ger)g Fr(l)31 b Ft(\025)f Fs(0)25 b Fv(the)g(coef)n (\002cient)h Fr(a)2571 4127 y Fo(")2608 4112 y Fs(\()p Fr(k)s Fs(\))g Fv(has)f(an)h(e)o(xpansion)456 4228 y Fr(a)507 4243 y Fo(")544 4228 y Fs(\()p Fr(k)s Fs(\))41 b(=)832 4154 y Fm(P)937 4257 y Fo(j)t Fl(\025\000)p Fo(l)1122 4228 y Fr(c)1164 4243 y Fo(j)1201 4228 y Fr(k)1255 4192 y Fo(j)1324 4228 y Fv(at)32 b(zero.)53 b(The)33 b(e)o(xistence)e(of)h (such)g(an)h(e)o(xpansion)e(as)h(well)456 4358 y(as)g(the)h(condition)e Ft(j)p Fr(a)1214 4373 y Fo(")1250 4358 y Fs(\()p Fr(k)s Fs(\))p Ft(j)d(\000)g Fs(1)42 b(=)g Fr(O)s Fs(\(1)p Fr(=)p Ft(j)p Fr(k)s Ft(j)2074 4321 y Fq(2)2112 4358 y Fs(\))33 b Fv(as)f Fr(k)45 b Ft(!)d(\0061)33 b Fv(will)f(be)h(pro)o(v)o(en)e(in) 456 4474 y(Appendix.)555 4598 y(Let)d Fr(P)47 b Fs(=)935 4523 y Fm(P)1040 4550 y Fo(n)1040 4627 y(j)t Fq(=0)1183 4598 y Fr(P)1246 4613 y Fo(j)1311 4598 y Fv(and)27 b(let)1635 4573 y Fs(^)1610 4598 y Fr(H)1691 4613 y Fo(")1755 4598 y Fv(be)h(the)g(operator)g(in)f Fr(L)2558 4562 y Fq(2)2598 4598 y Fs(\()p Fp(R)5 b Fr(;)17 b(P)d(L)2889 4562 y Fq(2)2934 4598 y Fs(\()p Fp(S)3033 4562 y Fo(d)p Fl(\000)p Fq(1)3158 4598 y Fs(\)\))28 b Fv(such)456 4716 y(that)456 4833 y(\(7.3\))481 4980 y Fs(^)456 5005 y Fr(H)537 5020 y Fo(")573 5005 y Fr(u)f Fs(=)h Ft(\000)847 4937 y Fr(d)898 4901 y Fq(2)937 4937 y Fr(u)p 847 4982 146 4 v 851 5073 a(dr)949 5044 y Fq(2)1021 5005 y Ft(\000)18 b Fr(\020)1159 5020 y Fo(")1205 4937 y Fs(\001)1286 4952 y Fo(\022)1325 4937 y Fr(u)p 1205 4982 176 4 v 1250 5073 a(r)1297 5044 y Fq(2)1391 5005 y Fr(;)116 b Fs(\()p Fr(I)25 b Ft(\000)18 b Fr(P)1798 5020 y Fq(0)1838 5005 y Fs(\))p Fr(u)p Fs(\(1)p Fr(;)f Ft(\001)p Fs(\))26 b(=)h(0)p Fr(;)117 b(u)p Fs(\()p Fr(r)m(;)17 b Ft(\001)p Fs(\))26 b Ft(2)i Fr(P)14 b(L)2959 4964 y Fq(2)2999 5005 y Fs(\()p Fp(S)3098 4964 y Fo(d)p Fl(\000)p Fq(1)3222 5005 y Fs(\))p Fr(;)34 b Ft(8)p Fr(r)m(;)456 5216 y Fv(where)25 b Fr(\020)767 5231 y Fo(")828 5216 y Fv(is)g(the)f(same)h(as)g(abo)o(v)o(e.)p eop %%Page: 15 15 15 14 bop 1246 251 a Fj(ABSOLUTEL)-7 b(Y)22 b(CONTINUOUS)h(SPECTR)m(UM) 714 b(15)456 450 y FB(Pr)n(oposition)23 b(7.1.)40 b Fn(Eac)o(h)23 b Ft(\000)p Fr(\014)1525 414 y Fq(2)1519 475 y Fo(m)1611 450 y Fn(is)g(one)h(of)g(the)f(eig)o(en)l(values)h Ft(\000)p Fr(\014)2744 414 y Fq(2)2738 475 y Fo(m)2805 450 y Fs(\()p Fr(V)d Fs(\))j Fn(of)g(the)g(oper)n(-)456 578 y(ator)672 552 y Fs(^)647 578 y Fr(H)728 593 y Fo(")786 578 y Fs(+)e Fr(V)g Fn(.)31 b(Mor)l(eo)o(ver)-11 b(,)456 848 y Fv(\(7.4\))843 712 y Fm(Z)943 739 y Fq(+)p Fl(1)899 938 y(\0001)1089 848 y Fs(log)17 b Ft(j)p Fr(a)1311 863 y Fo(")1347 848 y Fs(\()p Fr(k)s Fs(\))p Ft(j)g Fr(dk)30 b Ft(\024)e Fs(2)p Fr(\031)1867 737 y Fm(\020)1926 753 y(X)2087 848 y Fr(\014)2142 863 y Fo(m)2209 848 y Fs(\()p Fr(V)21 b Fs(\))h(+)2483 753 y Fm(X)2644 848 y Fr(\014)2699 863 y Fo(m)2765 848 y Fs(\()p Ft(\000)p Fr(V)g Fs(\))2997 737 y Fm(\021)1549 1250 y Fs(+)p Fr(\031)1700 1114 y Fm(Z)1800 1140 y Fl(1)1756 1340 y Fq(0)1901 1182 y Fr(\020)1944 1197 y Fo(")1980 1182 y Fs(\()p Fr(r)s Fs(\))p Fr(\013)2165 1197 y Fo(d)p 1901 1227 305 4 v 2010 1318 a Fr(r)2057 1289 y Fq(2)2232 1250 y Fr(dr)m(:)555 1507 y Fn(Pr)l(oof)p Fv(.)52 b(Ob)o(viously)-6 b(,)31 b(if)h Fr(s)41 b(<)g(c)1661 1522 y Fq(1)1742 1507 y Fr(<)g(c)1901 1522 y Fq(2)1982 1507 y Fr(<)g(r)s Fv(,)34 b(then)e(the)g(k)o(ernel)g(of)g(the)g (operator)456 1631 y Fr(P)519 1646 y Fq(0)558 1631 y Fs(\()621 1606 y(^)596 1631 y Fr(H)677 1646 y Fo(")736 1631 y Fs(+)22 b Fr(V)43 b Ft(\000)23 b Fr(z)t Fs(\))1121 1595 y Fl(\000)p Fq(1)1216 1631 y Fr(P)1279 1646 y Fq(0)1343 1631 y Fv(equals)456 1904 y(\(7.5\))726 b Fr(g)t Fs(\()p Fr(r)m(;)17 b(s;)g(k)s Fs(\))27 b(=)g Ft(\000)1946 1837 y Fs(exp)18 b Fr(ik)s Fs(\()p Fr(r)25 b Ft(\000)e Fr(s)p Fs(\))p 1946 1881 544 4 v 2041 1972 a(2)p Fr(ik)s(a)2228 1987 y Fo(")2265 1972 y Fs(\()p Fr(k)s Fs(\))2500 1904 y Fr(:)456 2171 y Fv(The)h(proof)h(of)g(the)f(latter)g(relation)h(is)f (a)h(counterpart)f(of)h(the)f(proof)h(of)g(\(5.4\))o(.)31 b(On)25 b(the)456 2287 y(other)33 b(hand)f(we)i(can)f(consider)g(the)g (e)o(xpansion)f(of)h Fr(g)j Fv(near)e(the)f(eigen)l(v)n(alue)f Ft(\000)p Fr(\014)3358 2251 y Fq(2)3352 2312 y Fo(m)3419 2287 y Fv(.)456 2404 y(Denote)e(by)g Fr(\036)962 2419 y Fo(m;j)1080 2404 y Fs(\()p Fr(r)m(;)17 b(\022)s Fs(\))p Fv(,)32 b Fr(j)44 b Fs(=)38 b(1)p Fr(;)17 b Fs(2)g Fr(:)g(:)g(:)e(n)31 b Fv(the)f(orthonormal)g(system)f(of)h(eigenfunc-)456 2540 y(tions)23 b(corresponding)h(to)h Ft(\000)p Fr(\014)1510 2504 y Fq(2)1504 2565 y Fo(m)1571 2540 y Fv(.)30 b(If)c Fr(\036)1776 2490 y Fq(\(0\))1776 2566 y Fo(m;j)1922 2540 y Fs(=)2025 2460 y Fm(R)2072 2575 y Fh(S)2115 2556 y Fi(d)p Fg(\000)p Ff(1)2244 2540 y Fr(\036)2302 2555 y Fo(m;j)2421 2540 y Fs(\()p Fr(r)m(;)17 b(\022)s Fs(\))g Fr(d\022)s Fv(,)24 b(then)456 2657 y(\(7.6\))562 2884 y Fr(g)t Fs(\()p Fr(r)m(;)17 b(s;)g(k)s Fs(\))27 b(=)1058 2729 y Fm(P)1163 2755 y Fo(n)1163 2832 y(j)t Fq(=1)1306 2803 y Fr(\036)1364 2752 y Fq(\(0\))1364 2829 y Fo(m;j)1483 2803 y Fs(\()p Fr(r)s Fs(\))p 1606 2691 299 4 v Fr(\036)1664 2752 y Fq(\(0\))1664 2829 y Fo(m;j)1782 2803 y Fs(\()p Fr(s)p Fs(\))p 1058 2861 846 4 v 1313 2952 a Fr(k)1367 2924 y Fq(2)1429 2952 y Fs(+)22 b Fr(\014)1588 2924 y Fq(2)1582 2977 y Fo(m)1936 2884 y Fs(+)g Fr(g)2081 2899 y Fq(0)2120 2884 y Fs(\()p Fr(r)m(;)17 b(s;)g(k)s Fs(\))p Fr(;)215 b(s)28 b(<)f(c)2886 2899 y Fq(1)2953 2884 y Fr(<)h(c)3099 2899 y Fq(2)3166 2884 y Fr(<)g(r)m(;)456 3158 y Fv(where)j Fr(g)777 3173 y Fq(0)816 3158 y Fs(\()p Fr(r)m(;)17 b(s;)g(k)s Fs(\))38 b(=)g Fr(O)s Fs(\(1\))p Fv(,)31 b(as)g Fr(k)41 b Ft(!)d Fr(i\014)1964 3173 y Fo(m)2031 3158 y Fv(.)48 b(This)30 b(pro)o(v)o(es)g(that)g Fr(a)2837 3173 y Fo(")2874 3158 y Fs(\()p Fr(k)s Fs(\))g Fv(is)h(a)f(mero-)456 3274 y(morphic)35 b(function)h(in)g(the)g(upper)g (half)h(plane)f(and)g(its)g(zeros)h(correspond)f(to)g(the)456 3398 y(eigen)l(v)n(alues)31 b Ft(\000)p Fr(\014)1090 3362 y Fq(2)1084 3423 y Fo(m)1184 3398 y Fv(of)h(the)g(operator)1843 3373 y Fs(^)1818 3398 y Fr(H)1899 3413 y Fo(")1963 3398 y Fs(+)c Fr(V)21 b Fv(.)54 b(Comparing)32 b(\(7.5\))g(and)h(\(7.6\))f (we)456 3514 y(\002nd)26 b(that)g(the)h(multiplicities)c(of)k(these)g (zeros)f(are)i(equal)e(to)g(one.)36 b(F)o(or)27 b(the)f(latter)h(ar)n (-)456 3631 y(guments)f(see)h([16].)39 b(T)-8 b(aking)26 b(into)h(account)g(the)g(estimate)g Ft(j)p Fr(a)2614 3646 y Fo(")2650 3631 y Fs(\()p Fr(k)s Fs(\))p Ft(j)32 b(\025)h Fs(1)p Fv(,)28 b(we)f(obtain)456 3747 y(the)32 b(statement)g(of)h(the)g(proposition)e(if)i(we)g(add)g(to)f(\(7.2\))h (the)g(same)g(identity)e(with)456 3863 y Ft(\000)p Fr(V)47 b Fv(instead)24 b(of)h Fr(V)c Fv(.)31 b Fc(\003)555 4077 y Fv(Observ)o(e)k(that)f(when)h Fr(")46 b Ft(!)g Fs(0)35 b Fv(the)f(eigen)l(v)n(alues)g(of)2476 4052 y Fs(^)2451 4077 y Fr(H)2532 4092 y Fo(")2598 4077 y Fs(+)c Fr(V)56 b Fv(con)l(v)o(er)n(ge)35 b(to)g(the)456 4201 y(eigen)l(v)n(alues)30 b(of)i(the)g(operator)1609 4176 y Fs(^)1583 4201 y Fr(H)j Fs(+)27 b Fr(V)22 b Fv(,)33 b(where)2240 4176 y Fs(^)2214 4201 y Fr(H)40 b Fv(is)31 b(the)g(follo)n(wing)f(operator)i(in)456 4317 y Fr(L)522 4281 y Fq(2)561 4317 y Fs(\()p Fp(R)5 b Fr(;)17 b(L)775 4281 y Fq(2)821 4317 y Fs(\()p Fp(S)920 4281 y Fo(d)p Fl(\000)p Fq(1)1045 4317 y Fs(\)\))851 4569 y(^)825 4595 y Fr(H)36 b Fs(=)27 b Ft(\000)1132 4527 y Fr(d)1183 4491 y Fq(2)1223 4527 y Fr(u)p 1132 4572 146 4 v 1136 4663 a(dr)1234 4634 y Fq(2)1310 4595 y Fs(+)1437 4527 y(1)p 1418 4572 87 4 v 1418 4663 a Fr(r)1465 4634 y Fq(2)1514 4595 y Fs(\()p Ft(\000)p Fs(\001)1710 4610 y Fo(\022)1750 4595 y Fr(u)22 b Fs(+)g Fr(\013)1988 4610 y Fo(d)2028 4595 y Fr(u)p Fs(\))p Fr(;)116 b Fs(\()p Fr(I)30 b Ft(\000)23 b Fr(P)2539 4610 y Fq(0)2578 4595 y Fs(\))p Fr(u)p Fs(\(1)p Fr(;)17 b Ft(\001)p Fs(\))26 b(=)i(0)p Fr(:)555 4870 y Fv(Denote)40 b(the)f(eigen)l(v)n(alues)f(of) 1695 4845 y Fs(^)1670 4870 y Fr(H)i Fs(+)33 b Fr(V)61 b Fv(by)40 b Ft(\000)2235 4789 y Fm(\000)2281 4870 y Fr(\014)2342 4819 y Fq(\(0\))2336 4880 y Fo(m)2436 4789 y Fm(\001)2481 4812 y Fq(2)2521 4870 y Fv(,)j(where)d Fr(\014)2933 4819 y Fq(\(0\))2927 4880 y Fo(m)3081 4870 y Fr(>)55 b Fs(0)40 b Fv(and)456 5008 y(let)601 4982 y Fs(^)586 5008 y Fr(V)52 b Fv(be)31 b(the)g(F)o(ourier)f(transform)g (of)h Fr(V)52 b Fv(with)30 b(respect)h(to)g(the)f(\002rst)h(v)n (ariable)f(as)h(in)456 5124 y(Theorem)24 b(2.2.)p eop %%Page: 16 16 16 15 bop 456 251 a Fj(16)808 b(LAPTEV)-10 b(,)17 b(N)m(ABOK)n(O)i(AND) f(SAFR)m(ONO)l(V)456 450 y FB(Pr)n(oposition)28 b(7.2.)42 b Fn(F)-10 b(or)28 b(any)g Fr(\016)37 b(>)d Fs(0)28 b Fn(ther)l(e)g(is)f(a)h(constant)f Fr(C)41 b Fs(=)33 b Fr(C)7 b Fs(\()p Fr(\016)n(;)17 b Ft(k)p Fr(V)22 b Ft(k)3140 465 y Fl(1)3214 450 y Fs(\))34 b Fr(>)g Fs(0)456 566 y Fn(suc)o(h)24 b(that)900 680 y Fm(X)1060 774 y Fr(\014)1121 733 y Fq(\(0\))1115 799 y Fo(m)1243 774 y Ft(\024)k Fr(C)1425 664 y Fm(\020)1484 639 y(Z)1540 864 y Fh(R)1588 845 y Fi(d)1645 774 y Fr(V)1723 733 y Fq(4)1763 774 y Fr(dx)22 b Fs(+)1989 639 y Fm(Z)2044 864 y Fh(R)2092 845 y Fi(d)p Fg(\000)p Ff(1)2228 639 y Fm(Z)2327 665 y Fo(\016)2283 864 y Fl(\000)p Fo(\016)2392 774 y Ft(j)2435 749 y Fs(^)2420 774 y Fr(V)f Fs(\()p Fr(\030)5 b(;)17 b(y)t Fs(\))p Ft(j)2746 733 y Fq(2)2800 774 y Fr(d\030)5 b(dy)2418 1023 y Fs(+)p Ft(k)p Fr(V)22 b Fs(\()p Fr(x)p Fs(\))p Ft(k)2804 982 y Fq(1)p Fo(=)p Fq(2)2804 1048 y Fl(1)2914 912 y Fm(\021)2973 1023 y Fr(:)456 878 y Fv(\(7.7\))555 1230 y Fn(Pr)l(oof)o(.)72 b Fv(F)o(or)39 b(an)o(y)f(self-adjoint)g(operator)h Fr(T)53 b Fv(and)39 b Fr(t)54 b(>)g Fs(0)39 b Fv(denote)g Fr(N)10 b Fs(\()p Fr(t;)17 b(T)d Fs(\))54 b(=)456 1346 y(rank)17 b Fr(E)737 1361 y Fo(T)792 1346 y Fs(\()p Ft(\0001)p Fr(;)g Ft(\000)p Fr(t)p Fs(\))p Fv(.)31 b(Then)1155 1487 y Fm(X)1316 1581 y Fr(\014)1377 1540 y Fq(\(0\))1371 1606 y Fo(m)1498 1581 y Fs(=)1602 1446 y Fm(Z)1701 1472 y Fl(jj)p Fo(V)15 b Fl(jj)1837 1480 y Fg(1)1657 1671 y Fq(0)1922 1581 y Fr(N)10 b Fs(\()p Fr(t;)2153 1556 y Fs(^)2127 1581 y Fr(H)30 b Fs(+)22 b Fr(V)g Fs(\))2503 1514 y Fr(dt)p 2463 1558 168 4 v 2463 1659 a Fs(2)2512 1578 y Ft(p)p 2595 1578 36 4 v 81 x Fr(t)2667 1581 y Ft(\024)692 1751 y Fm(Z)792 1777 y Fl(jj)p Fo(V)15 b Fl(jj)928 1785 y Fg(1)748 1976 y Fq(0)996 1886 y Fs(\(1)22 b(+)g Fr(N)10 b Fs(\()p Fr(t;)1434 1861 y Fs(^)1408 1886 y Fr(H)1489 1901 y Fo(D)1575 1886 y Fs(+)22 b Fr(V)g Fs(\)\))1878 1819 y Fr(dt)p 1838 1863 168 4 v 1838 1964 a Fs(2)1887 1883 y Ft(p)p 1970 1883 36 4 v 81 x Fr(t)2043 1886 y Fs(=)27 b Fv(tr)e Fs(\()2295 1861 y(^)2270 1886 y Fr(H)2351 1901 y Fo(D)2437 1886 y Fs(+)d Fr(V)f Fs(\))2651 1835 y Fq(1)p Fo(=)p Fq(2)2651 1908 y Fl(\000)2783 1886 y Fs(+)h Ft(jj)p Fr(V)f Ft(jj)3071 1845 y Fq(1)p Fo(=)p Fq(2)3071 1911 y Fl(1)3181 1886 y Fr(;)456 2077 y Fv(where)1107 2210 y Fs(^)1082 2235 y Fr(H)1163 2250 y Fo(D)1254 2235 y Fs(=)28 b Ft(\000)1445 2168 y Fr(d)1496 2132 y Fq(2)1535 2168 y Fr(u)p 1445 2213 146 4 v 1449 2304 a(dr)1547 2275 y Fq(2)1623 2235 y Ft(\000)1732 2168 y Fs(\001)1813 2183 y Fo(\022)1853 2168 y Fr(u)p 1732 2213 176 4 v 1777 2304 a(r)1824 2275 y Fq(2)1940 2235 y Fs(+)2048 2168 y Fr(\013)2110 2183 y Fo(d)p 2048 2213 103 4 v 2056 2304 a Fr(r)2103 2275 y Fq(2)2161 2235 y Fr(u;)116 b(u)p Fs(\(1)p Fr(;)17 b Ft(\001)p Fs(\))26 b(=)h(0)p Fr(:)456 2417 y Fv(Let)g Ft(\000)p Fs(\001)e(+)f Fr(V)49 b Fv(be)28 b(the)f(operator)h(in)f Fr(L)1807 2381 y Fq(2)1847 2417 y Fs(\()p Fp(R)1951 2381 y Fo(d)1997 2417 y Fs(\))p Fv(.)40 b(Then)27 b(the)g(mini-max)f (principle)h(tells)456 2533 y(us)d(that)456 2692 y(\(7.8\))1194 2597 y Fm(X)1354 2692 y Fr(\014)1415 2651 y Fq(\(0\))1409 2717 y Fo(m)1537 2692 y Ft(\024)k Fv(tr)d Fs(\()p Ft(\000)p Fs(\001)d(+)g Fr(V)g Fs(\))2161 2641 y Fq(1)p Fo(=)p Fq(2)2161 2714 y Fl(\000)2293 2692 y Fs(+)g Ft(k)p Fr(V)f Ft(k)2569 2651 y Fq(1)p Fo(=)p Fq(2)2569 2717 y Fl(1)2679 2692 y Fr(:)456 2866 y Fv(Applying)34 b(the)h(Lieb-Thirring)g (inequality)f(for)i(operator)f(v)n(alued)g(potentials)f(\(see)456 2982 y([16)o(]\))26 b(and)e(Theorem)h(3.1)f(we)h(obtain)913 3197 y(tr)f Fs(\()p Ft(\000)p Fs(\001)f(+)f Fr(V)g Fs(\))1432 3146 y Fq(1)p Fo(=)p Fq(2)1432 3219 y Fl(\000)1569 3197 y Ft(\024)29 b Fr(C)1768 3061 y Fm(Z)1823 3287 y Fh(R)1871 3268 y Fi(d)p Fg(\000)p Ff(1)1990 3086 y Fm(\020)2050 3197 y Ft(\000)2160 3130 y Fr(d)2211 3093 y Fq(2)p 2137 3174 137 4 v 2137 3265 a Fr(ds)2234 3236 y Fq(2)2305 3197 y Fs(+)22 b Fr(V)g Fs(\()p Fr(s;)17 b(y)t Fs(\))2700 3086 y Fm(\021)2758 3109 y Fo(d=)p Fq(2)2758 3266 y Fl(\000)2886 3197 y Fr(dy)1225 3486 y Ft(\024)29 b Fr(C)1401 3501 y Fq(0)1456 3351 y Fm(Z)1512 3576 y Fh(R)1560 3557 y Fi(d)p Fg(\000)p Ff(1)1678 3376 y Fm(\020)1738 3486 y Ft(\000)1848 3419 y Fr(d)1899 3383 y Fq(2)p 1825 3463 V 1825 3555 a Fr(ds)1922 3526 y Fq(2)1993 3486 y Fs(+)23 b Fr(V)e Fs(\()p Fr(s;)c(y)t Fs(\))2388 3376 y Fm(\021)2446 3398 y Fq(3)p Fo(=)p Fq(2)2446 3555 y Fl(\000)2573 3486 y Fr(dy)1012 3764 y Ft(\024)28 b Fr(C)1187 3779 y Fq(1)1226 3654 y Fm(\020)1286 3629 y(Z)1341 3854 y Fh(R)1389 3835 y Fi(d)1446 3764 y Fr(V)1525 3723 y Fq(4)1564 3764 y Fr(dx)22 b Fs(+)1790 3629 y Fm(Z)1846 3854 y Fh(R)1894 3835 y Fi(d)p Fg(\000)p Ff(1)2029 3629 y Fm(Z)2128 3655 y Fo(\016)2084 3854 y Fl(\000)p Fo(\016)2194 3764 y Ft(j)2237 3739 y Fs(^)2222 3764 y Fr(V)f Fs(\()p Fr(\030)5 b(;)17 b(y)t Fs(\))p Ft(j)2548 3723 y Fq(2)2602 3764 y Fr(d\030)5 b(dy)2804 3654 y Fm(\021)2861 3764 y Fr(;)456 3967 y Fv(where)34 b Fr(C)803 3982 y Fq(0)887 3967 y Fs(=)44 b Fr(C)7 b Fs(\()p Ft(k)p Fr(V)22 b Ft(k)1301 3982 y Fl(1)1375 3967 y Fs(\))34 b Fv(and)g Fr(C)1695 3982 y Fq(1)1779 3967 y Fs(=)44 b Fr(C)7 b Fs(\()p Fr(\016)n(;)17 b Ft(k)p Fr(V)22 b Ft(k)2278 3982 y Fl(1)2352 3967 y Fs(\))p Fv(.)58 b(The)34 b(latter)g(inequality)f(to-)456 4083 y(gether)24 b(with)h(\(7.8\))f(implies)g(\(7.7\))o(.)131 b Fc(\003)555 4242 y Fv(No)n(w)24 b(the)h(trace)g(formula)g(\(7.4\))g (and)f(the)h(inequality)e(\(7.7\))i(lead)g(us)g(to)2157 4464 y Fs(lim)17 b(sup)2238 4543 y Fo(")p Fl(!)p Fq(0)2473 4328 y Fm(Z)2573 4354 y Fq(+)p Fl(1)2528 4554 y(\0001)2719 4464 y Fs(log)f Ft(j)p Fr(a)2940 4479 y Fo(")2977 4464 y Fs(\()p Fr(k)s Fs(\))p Ft(j)h Fr(dk)835 4753 y Ft(\024)28 b Fr(C)1017 4643 y Fm(\020)1076 4618 y(Z)1132 4843 y Fh(R)1180 4824 y Fi(d)1237 4753 y Fr(V)1315 4712 y Fq(4)1355 4753 y Fr(dx)22 b Fs(+)1581 4618 y Fm(Z)1636 4843 y Fh(R)1684 4824 y Fi(d)p Fg(\000)p Ff(1)1820 4618 y Fm(Z)1919 4644 y Fo(\016)1875 4843 y Fl(\000)p Fo(\016)1984 4753 y Ft(j)2027 4728 y Fs(^)2012 4753 y Fr(V)f Fs(\()p Fr(\030)5 b(;)17 b(y)t Fs(\))p Ft(j)2338 4712 y Fq(2)2392 4753 y Fr(d\030)5 b(dy)24 b Fs(+)e Ft(k)p Fr(V)g Ft(k)2891 4712 y Fq(1)p Fo(=)p Fq(2)2891 4778 y Fl(1)3023 4753 y Fs(+)g(1)3170 4643 y Fm(\021)3229 4753 y Fr(:)456 4607 y Fv(\(7.9\))555 4969 y(F)o(or)29 b(a)h(perturbation)e Fr(V)51 b Fv(satisfying)28 b(the)h(conditions)f(of)h(Theorem)g(2.2)g(the)g(W)-8 b(e)o(yl)456 5094 y(function)27 b Fr(M)39 b Fv(can)29 b(also)e(be)i(de\002ned)f(as)h Fr(M)10 b Fs(\()p Fr(k)s Fs(\))34 b(=)2285 5055 y Fo(@)2326 5031 y Ff(2)p 2248 5071 149 4 v 2248 5128 a Fo(@)t(r)r(@)t(s)2407 5094 y Fr(G)2484 5109 y Fo(z)2524 5094 y Fs(\()p Fr(r)m(;)17 b(s)p Fs(\))p Ft(j)2759 5109 y Fq(\(1)p Fo(;)p Fq(1\))2907 5094 y Fv(,)29 b(where)g Fr(G)3310 5109 y Fo(z)3378 5094 y Fv(is)456 5216 y(the)24 b(inte)o(gral)g(k)o(ernel)h(of)g(the)f (operator)h Fr(P)1877 5231 y Fq(0)1917 5216 y Fs(\()p Fr(U)2031 5179 y Fl(\003)2071 5216 y Fr(H)8 b(U)32 b Ft(\000)23 b Fr(z)t Fs(\))2445 5179 y Fl(\000)p Fq(1)2540 5216 y Fr(P)2603 5231 y Fq(0)2642 5216 y Fv(.)p eop %%Page: 17 17 17 16 bop 1246 251 a Fj(ABSOLUTEL)-7 b(Y)22 b(CONTINUOUS)h(SPECTR)m(UM) 714 b(17)555 450 y Fv(F)o(or)27 b(an)o(y)e(pair)i(of)f(\002nite)h (numbers)e Fr(r)1823 465 y Fq(2)1893 450 y Fr(>)31 b(r)2044 465 y Fq(1)2114 450 y Ft(\025)g Fs(0)26 b Fv(and)g(for)h Fr(V)52 b Ft(2)31 b Fr(C)2893 414 y Fl(1)2886 475 y Fq(0)2968 450 y Fs(\()p Fp(R)3072 414 y Fo(d)3142 450 y Ft(n)23 b Fs(\012)3285 465 y Fq(1)3325 450 y Fs(\))j Fv(it)456 566 y(follo)n(ws)d(from)h(Corollary)h(5.3)g([11)o(])h(that)456 786 y(\(7.10\))963 719 y Fs(1)p 963 763 49 4 v 963 855 a(2)1039 651 y Fm(Z)1138 677 y Fo(r)1170 686 y Ff(2)1094 876 y Fo(r)1126 885 y Ff(1)1226 786 y Fs(log)1561 719 y Fr(k)p 1378 763 420 4 v 1378 855 a Fs(4)p Fv(Im)f Fr(M)10 b Fs(\()p Fr(k)s Fs(\))1824 786 y Fr(dk)30 b Ft(\024)e Fs(lim)17 b(sup)2142 865 y Fo(")p Fl(!)p Fq(0)2377 651 y Fm(Z)2477 677 y Fq(+)p Fl(1)2432 876 y(\0001)2623 786 y Fs(log)f Ft(j)p Fr(a)2844 801 y Fo(")2881 786 y Fs(\()p Fr(k)s Fs(\))p Ft(j)h Fr(dk)s(:)555 1002 y Fv(Therefore)26 b(\(7.9\))f(and)f(\(7.10\))h(imply)456 1176 y FB(Pr)n(oposition)j(7.3.) 43 b Fn(F)-10 b(or)28 b(any)g(pair)g(of)g(\002nite)f(number)o(s)h Fr(r)2451 1191 y Fq(2)2525 1176 y Fr(>)34 b(r)2679 1191 y Fq(1)2753 1176 y Ft(\025)h Fs(0)28 b Fn(and)g(for)g Fr(V)56 b Ft(2)456 1292 y Fr(C)533 1256 y Fl(1)526 1317 y Fq(0)607 1292 y Fs(\()p Fp(R)711 1256 y Fo(d)780 1292 y Ft(n)22 b Fs(\012)922 1307 y Fq(1)962 1292 y Fs(\))1519 1437 y(1)p 1519 1481 49 4 v 1519 1572 a(2)1594 1369 y Fm(Z)1694 1395 y Fo(r)1726 1404 y Ff(2)1650 1594 y Fo(r)1682 1603 y Ff(1)1781 1504 y Fs(log)2116 1437 y Fr(k)p 1934 1481 420 4 v 1934 1572 a Fs(4)p Fv(Im)i Fr(M)10 b Fs(\()p Fr(k)s Fs(\))2379 1504 y Fr(dk)860 1796 y Ft(\024)28 b Fr(C)1042 1686 y Fm(\020)1101 1661 y(Z)1157 1886 y Fh(R)1205 1867 y Fi(d)1262 1796 y Fr(V)1340 1755 y Fq(4)1380 1796 y Fr(dx)22 b Fs(+)1606 1661 y Fm(Z)1661 1886 y Fh(R)1709 1867 y Fi(d)p Fg(\000)p Ff(1)1844 1661 y Fm(Z)1944 1687 y Fo(\016)1900 1886 y Fl(\000)p Fo(\016)2009 1796 y Ft(j)2052 1771 y Fs(^)2037 1796 y Fr(V)f Fs(\()p Fr(\030)5 b(;)17 b(y)t Fs(\))p Ft(j)2363 1755 y Fq(2)2417 1796 y Fr(d\030)5 b(dy)24 b Fs(+)e Ft(k)p Fr(V)g Ft(k)2916 1755 y Fq(1)p Fo(=)p Fq(2)2916 1821 y Fl(1)3048 1796 y Fs(+)g(1)3195 1686 y Fm(\021)3254 1796 y Fr(;)456 1654 y Fv(\(7.11\))456 2017 y Fn(wher)l(e)j Fr(C)35 b Fs(=)27 b Fr(C)7 b Fs(\()p Fr(\016)n(;)17 b Ft(k)p Fr(V)22 b Ft(k)1307 2032 y Fl(1)1381 2017 y Fs(\))p Fn(.)456 2272 y Fv(Ob)o(viously)980 2362 y Fm(Z)1035 2588 y Fh(R)1083 2569 y Fi(d)p Fg(\000)p Ff(1)1218 2362 y Fm(Z)1318 2388 y Fo(\016)1274 2588 y Fl(\000)p Fo(\016)1383 2498 y Ft(j)1426 2473 y Fs(^)1411 2498 y Fr(V)f Fs(\()p Fr(\030)5 b(;)17 b(y)t Fs(\))p Ft(j)1737 2457 y Fq(2)1791 2498 y Fr(d\030)5 b(dy)30 b Fs(=)2122 2362 y Fm(Z)2178 2588 y Fl(j)p Fq(\()p Fo(\021)r(;\022)r Fq(\))p Fl(j)p Fo(<\016)2473 2498 y Ft(j)2516 2473 y Fs(~)2501 2498 y Fr(V)22 b Fs(\()p Fr(\021)t Fs(\))p Ft(j)2736 2457 y Fq(2)2791 2498 y Fr(d\021)t(;)456 2745 y Fv(where)31 b Fr(\022)44 b Fs(=)39 b(\(1)p Fr(;)17 b Fs(0)p Fr(;)g(:)g(:)g(:)e(;)i Fs(0\))31 b Fv(and)1641 2719 y Fs(~)1626 2745 y Fr(V)53 b Fv(is)31 b(the)g(F)o(ourier)g (transform)g(of)h Fr(V)53 b Fv(with)30 b(respect)456 2861 y(to)g(all)g(v)n(ariables.)47 b(Note)30 b(that)g(the)g(left)g (hand)h(side)f(of)g(\(7.11\))g(is)g(independent)g(of)g Fr(\022)s Fv(.)456 2977 y(Inte)o(grating)d(both)h(sides)h(of)g(this)f (inequality)f(o)o(v)o(er)h Fp(S)2332 2941 y Fo(d)p Fl(\000)p Fq(1)2485 2977 y Fv(we)i(obtain)e(the)g(follo)n(wing)456 3093 y(inequality:)456 3267 y FB(Pr)n(oposition)g(7.4.)43 b Fn(F)-10 b(or)28 b(any)g(pair)g(of)g(\002nite)f(number)o(s)h Fr(r)2451 3282 y Fq(2)2525 3267 y Fr(>)34 b(r)2679 3282 y Fq(1)2753 3267 y Ft(\025)h Fs(0)28 b Fn(and)g(for)g Fr(V)56 b Ft(2)456 3383 y Fr(C)533 3347 y Fl(1)526 3408 y Fq(0)607 3383 y Fs(\()p Fp(R)711 3347 y Fo(d)780 3383 y Ft(n)22 b Fs(\012)922 3398 y Fq(1)962 3383 y Fs(\))1183 3459 y Fm(Z)1282 3486 y Fo(r)1314 3495 y Ff(2)1238 3685 y Fo(r)1270 3694 y Ff(1)1380 3528 y Fs(1)p 1380 3572 49 4 v 1380 3663 a(2)1455 3595 y(log)1790 3528 y Fr(k)p 1608 3572 420 4 v 1608 3663 a Fs(4)p Fv(Im)i Fr(M)10 b Fs(\()p Fr(k)s Fs(\))2053 3595 y Fr(dk)945 3890 y Ft(\024)28 b Fr(C)1127 3780 y Fm(\020)1187 3755 y(Z)1242 3980 y Fh(R)1290 3961 y Fi(d)1347 3890 y Fr(V)1425 3849 y Fq(4)1465 3890 y Fr(dx)22 b Fs(+)1691 3755 y Fm(Z)1746 3980 y Fh(R)1794 3961 y Fi(d)1861 3823 y Ft(j)1904 3798 y Fs(~)1889 3823 y Fr(V)g Fs(\()p Fr(\021)t Fs(\))p Ft(j)2124 3787 y Fq(2)p 1861 3867 301 4 v 1874 3959 a Fs(1)g(+)g Ft(j)p Fr(\021)t Ft(j)2189 3890 y Fr(d\021)j Fs(+)d Ft(k)p Fr(V)g Ft(k)2590 3849 y Fq(1)p Fo(=)p Fq(2)2590 3915 y Fl(1)2722 3890 y Fs(+)g(1)2869 3780 y Fm(\021)2928 3890 y Fr(;)456 3745 y Fv(\(7.12\))456 4107 y Fn(wher)l(e)j Fr(C)35 b Fs(=)27 b Fr(C)7 b Fs(\()p Ft(k)p Fr(V)22 b Ft(k)1222 4122 y Fl(1)1296 4107 y Fs(\))p Fn(.)786 4444 y Fv(8.)52 b(T)t Fu(H)t(E)30 b(E)t(N)t(D)h(O)t(F)f(T)t(H)t(E)h(P)t(R)q(O)t(O)t(F)i(O)t (F)e Fv(T)t Fu(H)t(E)t(O)t(R)t(E)t(M)t(S)k Fv(2)t(.)t(2)c Fu(A)t(N)t(D)f Fv(2)t(.)t(3)555 4699 y(Let)c Fp(Q)48 b Fs(=)31 b([0)p Fr(;)17 b Fs(1\))1136 4663 y Fo(d)1175 4699 y Fv(.)36 b(The)26 b(the)g(cubes)g Fp(Q)1890 4714 y Fo(m)1993 4699 y Fs(=)k Fp(Q)41 b Fs(+)23 b Fr(m;)34 b(m)d Ft(2)f Fp(Z)2726 4663 y Fo(d)2764 4699 y Fv(,)d(form)f(a)h (partition)456 4816 y(of)22 b Fp(R)627 4779 y Fo(d)695 4816 y Fv(to)g(which)f(we)h(associate)g(classes)g(of)g(functions)f Fr(u)g Fv(such)h(that)f(the)h(sequence)g(of)456 4932 y(\(quasi-\))i(norms)g Ft(jj)p Fr(u)p Ft(jj)1232 4947 y Fo(L)1280 4928 y Fi(p)1314 4947 y Fq(\()p Fh(Q)1388 4955 y Fi(m)1452 4947 y Fq(\))1484 4932 y Fv(,)g Fr(q)32 b(>)c Fs(0)p Fv(,)c(belongs)g(to)g Fr(`)2288 4896 y Fl(1)2363 4932 y Fv(.)30 b(These)25 b(classes)g(are)g(denoted)456 5054 y(by)f Fr(`)621 5018 y Fl(1)696 5054 y Fs(\()p Fp(Z)803 5018 y Fo(d)841 5054 y Fs(;)17 b Fr(L)951 5018 y Fo(p)990 5054 y Fs(\()p Fp(Q)12 b Fs(\)\))p Fv(.)36 b(It)25 b(is)g(clear)g(that) f(\(2.1\))h(implies)456 5216 y(\(8.1\))701 b Fr(V)50 b Ft(2)28 b Fr(`)1590 5174 y Fl(1)1664 5216 y Fs(\()p Fp(Z)1771 5174 y Fo(d)1809 5216 y Fs(;)17 b Fr(L)1919 5174 y Fo(p)1959 5216 y Fs(\()p Fp(Q)11 b Fs(\)\))p Fr(;)123 b(p)27 b(>)h(d;)p eop %%Page: 18 18 18 17 bop 456 251 a Fj(18)808 b(LAPTEV)-10 b(,)17 b(N)m(ABOK)n(O)i(AND) f(SAFR)m(ONO)l(V)456 450 y Fv(and)61 b(therefore)h(by)f([6])h(it)f (guarantees)g(the)h(boundedness)e(of)h(the)g(operator)456 488 y Fm(p)p 555 488 134 4 v 555 573 a Ft(j)p Fr(V)21 b Ft(j)p Fs(\()p Ft(\000)p Fs(\001)i(+)f(1\))1093 537 y Fl(\000)p Fq(1)p Fo(=)p Fq(2)1258 573 y Fv(.)555 689 y(The)e(ne)o(xt)g(proposition)e(allo)n(ws)h(us)h(to)f(approximate)h Fr(V)41 b Fv(by)20 b(compactly)f(supported)456 806 y(smooth)k (functions)h Fr(V)1227 821 y Fo(n)1273 806 y Fv(.)456 986 y FB(Pr)n(oposition)30 b(8.1.)45 b Fn(Let)31 b Fr(V)53 b Fn(satisfy)29 b(the)i(conditions)e(of)i(Theor)l(em)g(2.2.)49 b(Then)31 b(ther)l(e)456 1102 y(e)n(xists)e(a)h(sequence)g Fr(V)1237 1117 y Fo(n)1314 1102 y Fn(of)g(compactly)g(supported)e (smooth)h(functions)g(con)l(ver)l(ging)456 1218 y(to)24 b Fr(V)456 1413 y Fv(\(8.2\))1103 1278 y Fm(Z)1219 1413 y Ft(j)p Fr(V)1304 1428 y Fo(n)1350 1413 y Ft(j)1378 1372 y Fq(4)1434 1413 y Fr(dx)k(<)f(C)7 b Fs(\()p Fr(V)22 b Fs(\))p Fr(;)216 b Ft(jj)p Fr(V)2259 1428 y Fo(n)2305 1413 y Ft(jj)2361 1428 y Fl(1)2463 1413 y Fr(<)27 b(C)7 b Fs(\()p Fr(V)22 b Fs(\))456 1626 y Fn(and)456 1845 y Fv(\(8.3\))1215 1710 y Fm(Z)1270 1935 y Fh(R)1318 1916 y Fi(d)p Fg(\000)p Ff(1)1454 1710 y Fm(Z)1553 1736 y Fo(\016)r(=)p Fq(2)1509 1935 y Fl(\000)p Fo(\016)r(=)p Fq(2)1689 1845 y Ft(j)1732 1820 y Fs(^)1717 1845 y Fr(V)1774 1860 y Fo(n)1821 1845 y Fs(\()p Fr(\030)5 b(;)17 b(y)t Fs(\))p Ft(j)2069 1804 y Fq(2)2122 1845 y Fr(d\030)5 b(dy)30 b(<)e(C)7 b Fs(\()p Fr(V)21 b Fs(\))456 2079 y Fn(suc)o(h)32 b(that)h(the)g(W)-9 b(e)m(yl)33 b(functions)f Fr(M)1727 2094 y Fo(n)1808 2079 y Fn(corr)l(esponding)g(to)h Fr(V)2587 2094 y Fo(n)2667 2079 y Fn(con)l(ver)l(g)o(e)h(uniformly)456 2195 y(to)e Fr(M)10 b Fs(\()p Fr(k)s Fs(\))33 b Fn(when)g Fr(k)1130 2159 y Fq(2)1202 2195 y Fn(belongs)e(to)i(any)f(compact)g (subset)g(of)g(the)g(upper)g(half)g(plane)o(.)456 2311 y(Ther)l(efor)l(e)37 b(the)g(sequence)g(of)g(measur)l(es)f Fr(\026)2022 2326 y Fo(n)2106 2311 y Fn(con)l(ver)l(g)o(es)h(weakly)h (to)e(the)h(spectr)o(al)456 2428 y(measur)l(e)24 b Fr(\026)p Fn(.)555 2607 y(Pr)l(oof)o(.)29 b Fv(Let)24 b Fr(W)1081 2622 y Fl(\006)1167 2607 y Fs(=)1271 2534 y Ft(p)p 1354 2534 116 4 v 73 x Fr(V)1411 2622 y Fl(\006)1470 2607 y Fv(.)30 b(Since)25 b(the)f(class)g Fr(C)2212 2571 y Fl(1)2205 2632 y Fq(0)2310 2607 y Fv(is)g(dense)g(in)g Fr(L)2820 2571 y Fo(p)2884 2607 y Fv(for)g(an)o(y)g Fr(p)j(>)h Fs(0)p Fv(,)456 2724 y(we)d(can)g(\002nd)g(a)g(pair)g(of)g(sequences)f Fr(W)1831 2688 y Fl(\000)1817 2748 y Fo(n)1918 2724 y Ft(2)k Fr(C)2089 2688 y Fl(1)2082 2748 y Fq(0)2188 2724 y Fv(and)d Fr(W)2463 2688 y Fq(+)2449 2748 y Fo(n)2550 2724 y Ft(2)j Fr(C)2721 2688 y Fl(1)2714 2748 y Fq(0)2820 2724 y Fv(satisfying)456 2894 y(\(8.4\))110 b Fr(W)863 2853 y Fl(\006)849 2919 y Fo(n)950 2894 y Ft(!)27 b Fr(W)1169 2909 y Fl(\006)1278 2894 y Fs(in)16 b Fr(L)1442 2853 y Fq(8)1481 2894 y Fs(\()p Fp(R)1585 2853 y Fo(d)1632 2894 y Fs(\);)50 b Fr(W)1853 2853 y Fl(\006)1839 2919 y Fo(n)1939 2894 y Ft(!)27 b Fr(W)2158 2909 y Fl(\006)2251 2894 y Fs(in)16 b Fr(`)2390 2853 y Fl(1)2464 2894 y Fs(\()p Fp(Z)2571 2853 y Fo(d)2609 2894 y Fs(;)h Fr(L)2719 2853 y Fo(p)2759 2894 y Fs(\()p Fp(Q)11 b Fs(\)\))p Fr(;)56 b Fs(2)p Fr(p)27 b(>)h(d:)456 3061 y Fv(Introduce)c(a)h(sequence)g(of)g (functions)f Ft(f)p Fr(V)1932 3076 y Fo(n)1979 3061 y Ft(g)2029 3025 y Fl(1)2029 3086 y Fo(n)p Fq(=1)1478 3229 y Fr(V)1535 3244 y Fo(n)1610 3229 y Fs(=)j(\()p Fr(W)1857 3188 y Fq(+)1843 3253 y Fo(n)1916 3229 y Fs(\))1954 3188 y Fq(2)2015 3229 y Ft(\000)c Fs(\()p Fr(W)2259 3188 y Fl(\000)2245 3253 y Fo(n)2317 3229 y Fs(\))2355 3188 y Fq(2)2395 3229 y Fr(:)456 3395 y Fv(The)h(sequences)h Fr(W)1170 3359 y Fl(\006)1156 3420 y Fo(n)1254 3395 y Fv(can)g(be)g(chosen)g(so)f(that)1321 3502 y Fm(Z)1420 3529 y Fo(\016)r(=)p Fq(2)1376 3728 y Fl(\000)p Fo(\016)r(=)p Fq(2)1556 3638 y Ft(j)1599 3613 y Fs(^)1584 3638 y Fr(V)1641 3653 y Fo(n)1688 3638 y Fs(\()p Fr(\030)5 b(;)17 b(y)t Fs(\))p Ft(j)1936 3597 y Fq(2)1990 3638 y Fr(d\030)5 b(dy)29 b(<)f(C)7 b Fs(\()p Fr(V)21 b Fs(\))p Fr(:)456 3871 y Fv(Then)j Fr(V)742 3886 y Fo(n)817 3871 y Ft(2)k Fr(C)988 3835 y Fl(1)981 3896 y Fq(0)1087 3871 y Fv(and)d(the)g (relations)f(\(8.2\),)h(\(8.4\))f(hold)g(true.)31 b(Let)456 4101 y(\(8.5\))582 b Fr(S)1289 4116 y Fq(0)1328 4101 y Fr(u)28 b Fs(=)f Ft(\000)1602 4034 y Fr(d)1653 3997 y Fq(2)1693 4034 y Fr(u)p 1602 4078 146 4 v 1606 4169 a(dr)1704 4141 y Fq(2)1780 4101 y Ft(\000)1890 4034 y Fs(\001)1971 4049 y Fo(\022)2010 4034 y Fr(u)p 1890 4078 176 4 v 1935 4169 a(r)1982 4141 y Fq(2)2076 4101 y Fr(;)116 b(u)p Fs(\(1)p Fr(;)17 b(\022)s Fs(\))27 b(=)g(0)456 4343 y Fv(acting)36 b(in)g Fr(L)916 4307 y Fq(2)955 4232 y Fm(\020)1015 4343 y Fs(\(1)p Fr(;)17 b Ft(1)p Fs(\);)g Fr(L)1394 4307 y Fq(2)1432 4343 y Fs(\()p Fp(S)1532 4307 y Fo(d)p Fl(\000)p Fq(1)1656 4343 y Fs(\))1694 4232 y Fm(\021)1754 4343 y Fv(.)65 b(Suppose)36 b(no)n(w)g(that)g Fs(\000)2671 4358 y Fq(0)2710 4343 y Fs(\()p Fr(z)t Fs(\))h Fv(and)g Fs(\000)3114 4358 y Fo(n)3161 4343 y Fs(\()p Fr(z)t Fs(\))g Fv(are)456 4489 y(the)c(resolv)o(ent)f(operators)h(of)g Fr(S)1588 4504 y Fq(0)1660 4489 y Fv(and)h Fr(S)1898 4504 y Fq(0)1965 4489 y Fs(+)29 b Fr(V)2127 4504 y Fo(n)2207 4489 y Fv(respecti)n(v)o(ely)-6 b(.)53 b(Recall)34 b(that)f(by)g Fr(\016)3409 4453 y Fl(0)3405 4513 y Fq(1)456 4605 y Fv(we)k(denote)g(the)g(deri)n(v)n(ati)n(v)o(e)d(of)k(the)f(delta)f (function)h Fr(\016)t Fs(\()p Fr(r)d Ft(\000)d Fs(1\))p Fv(.)68 b(The)37 b(e)o(xpression)456 4721 y Fs(\000)517 4736 y Fq(0)556 4721 y Fs(\()p Fr(z)t Fs(\))p Fr(\016)728 4685 y Fl(0)724 4746 y Fq(1)764 4721 y Fr(;)46 b Fv(Im)25 b Fr(z)41 b Ft(6)p Fs(=)36 b(0)p Fv(,)30 b(can)g(be)g(understood)e(as)h (an)h(e)o(xponentially)d(decaying)i(func-)456 4838 y(tion)k(\(Hank)o (el')-5 b(s)33 b(function\))g(which)h(coincides)f(with)g(the)h (corresponding)f(solution)456 4954 y(of)24 b(the)h(equation)456 5180 y(\(8.6\))587 b Ft(\000)1321 5113 y Fr(d)1372 5077 y Fq(2)1412 5113 y Fr( )p 1321 5158 158 4 v 1331 5249 a(dr)1429 5220 y Fq(2)1511 5180 y Fs(+)1619 5113 y Fr(\013)1681 5128 y Fo(d)p 1619 5158 103 4 v 1627 5249 a Fr(r)1674 5220 y Fq(2)1731 5180 y Fr( )32 b Fs(=)c Fr(z)t( )t(;)117 b( )t Fs(\(1\))27 b(=)g Ft(\000)p Fs(1)p Fr(:)p eop %%Page: 19 19 19 18 bop 1246 251 a Fj(ABSOLUTEL)-7 b(Y)22 b(CONTINUOUS)h(SPECTR)m(UM) 714 b(19)456 450 y Fv(According)24 b(to)g(assumptions)f(\(8.4\))i(we)g (ha)n(v)o(e)f(that)1392 610 y Fr(W)1498 569 y Fl(\006)1484 635 y Fo(n)1557 610 y Fs(\000)1618 625 y Fq(0)1657 610 y Fs(\()p Fr(z)t Fs(\))p Fr(\016)1829 569 y Fl(0)1825 635 y Fq(1)1893 610 y Ft(!)k Fr(W)2113 625 y Fl(\006)2172 610 y Fs(\000)2233 625 y Fq(0)2272 610 y Fs(\()p Fr(z)t Fs(\))p Fr(\016)2444 569 y Fl(0)2440 635 y Fq(1)2481 610 y Fr(;)456 770 y Fv(in)c Fr(L)624 734 y Fq(2)664 770 y Fs(\()p Fp(R)768 734 y Fo(d)814 770 y Fs(\))p Fv(.)31 b(Thus)24 b(in)g(order)i(to)e(pro)o(v)o(e)g(that)g(the)h(W)-8 b(e)o(yl)24 b(functions)1029 993 y Fr(M)1123 1008 y Fo(n)1171 993 y Fs(\()p Fr(k)s Fs(\))j(=)1497 926 y Fr(@)1553 890 y Fq(2)p 1442 970 207 4 v 1442 1061 a Fr(@)5 b(r)s(@)g(s)1658 993 y(G)1735 1008 y Fo(n;z)1837 993 y Fs(\()p Fr(r)m(;)17 b(s)p Fs(\))p Ft(j)2072 1009 y Fq(\(1)p Fo(;)p Fq(1\))2249 993 y Fs(=)27 b(\(\000)2451 1008 y Fo(n)2498 993 y Fs(\()p Fr(z)t Fs(\))p Fr(\016)2670 952 y Fl(0)2666 1018 y Fq(1)2706 993 y Fr(;)17 b(\016)2797 952 y Fl(0)2793 1018 y Fq(1)2833 993 y Fs(\))684 1198 y(=)27 b(\(\000)886 1213 y Fq(0)925 1198 y Fs(\()p Fr(z)t Fs(\))p Fr(\016)1097 1157 y Fl(0)1093 1222 y Fq(1)1134 1198 y Fr(;)17 b(\016)1225 1157 y Fl(0)1221 1222 y Fq(1)1260 1198 y Fs(\))22 b Ft(\000)h Fs(\(\()p Fr(W)1602 1157 y Fq(+)1588 1222 y Fo(n)1682 1198 y Ft(\000)g Fr(W)1888 1157 y Fl(\000)1874 1222 y Fo(n)1947 1198 y Fs(\)\000)2046 1213 y Fq(0)2085 1198 y Fs(\()p Fr(z)t Fs(\))p Fr(\016)2257 1157 y Fl(0)2253 1222 y Fq(1)2293 1198 y Fr(;)17 b Fs(\()p Fr(W)2481 1157 y Fq(+)2467 1222 y Fo(n)2562 1198 y Fs(+)22 b Fr(W)2766 1157 y Fl(\000)2752 1222 y Fo(n)2824 1198 y Fs(\)\000)2923 1213 y Fo(n)2970 1198 y Fs(\()p 3008 1143 50 4 v Fr(z)5 b Fs(\))p Fr(\016)3143 1157 y Fl(0)3139 1222 y Fq(1)3179 1198 y Fs(\))456 1337 y Fv(con)l(v)o(er)n(ge,)24 b(it)h(is)f(suf)n(\002cient)g(to)h(sho)n(w)e (that)456 1497 y(\(8.7\))194 b Fs(\()p Fr(W)985 1456 y Fq(+)971 1522 y Fo(n)1066 1497 y Fs(+)22 b Fr(W)1270 1456 y Fl(\000)1256 1522 y Fo(n)1329 1497 y Fs(\)\000)1428 1512 y Fo(n)1475 1497 y Fs(\()p 1513 1442 V Fr(z)t Fs(\))p Fr(\016)1647 1456 y Fl(0)1643 1522 y Fq(1)1711 1497 y Ft(!)27 b Fs(\()p Fr(W)1968 1512 y Fq(+)2049 1497 y Fs(+)22 b Fr(W)2239 1512 y Fl(\000)2298 1497 y Fs(\)\()p Fr(S)2434 1512 y Fq(0)2496 1497 y Fs(+)g Fr(V)43 b Ft(\000)p 2794 1442 V 23 w Fr(z)5 b Fs(\))2882 1456 y Fl(\000)p Fq(1)2976 1497 y Fr(\016)3023 1456 y Fl(0)3019 1522 y Fq(1)456 1657 y Fv(in)24 b Fr(L)624 1621 y Fq(2)664 1657 y Fs(\()p Fp(R)768 1621 y Fo(d)814 1657 y Fs(\))p Fv(.)555 1782 y(Let)32 b(us)g(denote)g Fr(W)1231 1797 y Fo(n)1320 1782 y Fs(=)41 b Fr(W)1543 1746 y Fq(+)1529 1806 y Fo(n)1629 1782 y Fs(+)28 b Fr(W)1839 1746 y Fl(\000)1825 1806 y Fo(n)1929 1782 y Fv(and)33 b Fr(W)2198 1797 y Fo(n)2245 1740 y Fq(\(0\))2380 1782 y Fs(=)42 b Fr(W)2604 1746 y Fq(+)2590 1806 y Fo(n)2690 1782 y Ft(\000)28 b Fr(W)2901 1746 y Fl(\000)2887 1806 y Fo(n)2960 1782 y Fv(.)53 b(Clearly)-6 b(,)34 b(if)456 1898 y Fr(W)562 1862 y Fl(\006)548 1923 y Fo(n)648 1898 y Ft(!)27 b Fr(W)867 1913 y Fl(\006)951 1898 y Fv(in)e(the)f(class)h(\(8.1\))g(with)f Fs(2)p Fr(p)j(>)h(d)p Fv(,)c(as)h Fr(n)j Ft(!)f(1)p Fv(,)e(then)456 2066 y(\(8.8\))331 b Fr(W)1070 2081 y Fo(n)1117 2066 y Fs(\000)1178 2081 y Fq(0)1217 2066 y Fs(\()p 1255 2011 V Fr(z)5 b Fs(\))p Fr(W)1449 2024 y Fq(\(0\))1435 2090 y Fo(n)1571 2066 y Ft(!)27 b Fs(\()p Fr(W)1828 2081 y Fq(+)1909 2066 y Fs(+)22 b Fr(W)2099 2081 y Fl(\000)2158 2066 y Fs(\)\000)2257 2081 y Fq(0)2297 2066 y Fs(\()p 2335 2011 V Fr(z)t Fs(\)\()p Fr(W)2552 2081 y Fq(+)2633 2066 y Ft(\000)h Fr(W)2825 2081 y Fl(\000)2884 2066 y Fs(\))456 2225 y Fv(in)h(the)h(operator)g(norm)f(topology)-6 b(.)555 2342 y(Then)25 b(\(8.7\))g(follo)n(ws)e(from)i(the)f(identity) 966 2506 y Fr(W)1058 2521 y Fo(n)1105 2506 y Fs(\000)1166 2521 y Fo(n)1213 2506 y Fs(\()p 1251 2451 V Fr(z)5 b Fs(\))p Fr(\016)1386 2465 y Fl(0)1382 2531 y Fq(1)1449 2506 y Fs(=)28 b(\()p Fr(I)i Fs(+)22 b Fr(W)1854 2521 y Fo(n)1901 2506 y Fs(\000)1962 2521 y Fq(0)2001 2506 y Fs(\()p 2039 2451 V Fr(z)5 b Fs(\))p Fr(W)2233 2465 y Fq(\(0\))2219 2531 y Fo(n)2327 2506 y Fs(\))2365 2465 y Fl(\000)p Fq(1)2459 2506 y Fr(W)2551 2521 y Fo(n)2598 2506 y Fs(\000)2659 2521 y Fq(0)2698 2506 y Fs(\()p 2736 2451 V Fr(z)g Fs(\))p Fr(\016)2871 2465 y Fl(0)2867 2531 y Fq(1)2907 2506 y Fr(:)456 2666 y Fc(\003)555 2826 y Fv(Similarly)22 b(we)i(can)f(pro)o(v)o(e)g(the)g(follo)n(wing)e(result) i(which)f(allo)n(ws)g(us)h(to)g(pass)g(from)456 2876 y Fm(P)561 2902 y Fo(l)561 2979 y(j)t Fq(=0)704 2950 y Fr(P)767 2965 y Fo(j)803 2950 y Fr(V)899 2876 y Fm(P)1004 2902 y Fo(l)1004 2979 y(j)t Fq(=0)1147 2950 y Fr(P)1210 2965 y Fo(j)1271 2950 y Fv(to)i Fr(V)c Fv(.)456 3129 y FB(Pr)n(oposition)31 b(8.2.)45 b Fn(Let)32 b Fr(V)54 b Fn(be)32 b(a)f(compactly)h(supported)e(smooth)h(function.)50 b(Then)456 3252 y(the)33 b(W)-9 b(e)m(yl)34 b(functions)f Fr(M)1326 3267 y Fo(l)1386 3252 y Fn(corr)l(esponding)g(to)g(the)h (potential)2654 3178 y Fm(P)2759 3204 y Fo(l)2759 3282 y(j)t Fq(=0)2902 3252 y Fr(P)2965 3267 y Fo(j)3001 3252 y Fr(V)3097 3178 y Fm(P)3202 3204 y Fo(l)3202 3282 y(j)t Fq(=0)3345 3252 y Fr(P)3408 3267 y Fo(j)456 3382 y Fn(con)l(ver)l(g)o (e)40 b(uniformly)d(to)i Fr(M)51 b Fn(when)39 b Fr(k)1843 3346 y Fq(2)1922 3382 y Fn(belongs)g(to)g(any)g(compact)g(subset)f Fr(K)47 b Fn(of)456 3498 y(the)30 b(upper)g(half)f(plane)h(and)f(ther)l (efor)l(e)i(the)f(sequence)g(of)g(measur)l(es)g Fr(\026)2993 3513 y Fo(l)3049 3498 y Fn(con)l(ver)l(g)o(es)456 3614 y(weakly)25 b(to)g(the)f(spectr)o(al)g(measur)l(e)g Fr(\026)h Fn(constructed)f(for)g(the)h(potential)e Fr(V)e Fn(.)456 3797 y(Pr)l(oof)o(.)51 b Fv(Let)33 b(us)f(denote)g Fr(V)1397 3812 y Fo(l)1465 3797 y Fs(=)1582 3723 y Fm(P)1688 3749 y Fo(l)1688 3826 y(j)t Fq(=0)1831 3797 y Fr(P)1894 3812 y Fo(j)1930 3797 y Fr(V)2025 3723 y Fm(P)2131 3749 y Fo(l)2131 3826 y(j)t Fq(=0)2274 3797 y Fr(P)2337 3812 y Fo(j)2406 3797 y Fv(let)g Fs(\000)2599 3812 y Fq(0)2638 3797 y Fs(\()p Fr(z)t Fs(\))i Fv(and)e(let)g Fs(\000)3166 3812 y Fo(l)3192 3797 y Fs(\()p Fr(z)t Fs(\))h Fv(be)456 3916 y(the)28 b(resolv)o(ent)g(operators)g(of)h Fr(S)1570 3931 y Fq(0)1638 3916 y Fv(de\002ned)g(in)f(\(8.5\))h(and)g Fr(S)2519 3931 y Fq(0)2583 3916 y Fs(+)c Fr(V)2741 3931 y Fo(l)2795 3916 y Fv(respecti)n(v)o(ely)-6 b(.)41 b(As)456 4032 y(in)29 b(Proposition)g(7.1)h(the)g(e)o(xpression)f Fs(\000)1876 4047 y Fq(0)1915 4032 y Fs(\()p Fr(z)t Fs(\))p Fr(\016)2087 3996 y Fl(0)2083 4057 y Fq(1)2124 4032 y Fr(;)47 b Fv(Im)24 b Fr(z)43 b Ft(6)p Fs(=)37 b(0)p Fv(,)32 b(is)d(understood)g(as)i(the)456 4148 y(e)o(xponentially)k(decaying)j (solution)e(of)i(the)f(equation)g(\(8.6\).)70 b(According)37 b(to)h(our)456 4264 y(assumptions)1094 4519 y Fr(V)1151 4534 y Fo(l)1177 4519 y Fs(\000)1238 4534 y Fq(0)1277 4519 y Fs(\()p Fr(z)t Fs(\))p Fr(\016)1449 4477 y Fl(0)1445 4543 y Fq(1)1513 4519 y Fs(=)1678 4394 y Fo(l)1617 4424 y Fm(X)1627 4634 y Fo(j)t Fq(=0)1777 4519 y Fr(P)1840 4534 y Fo(j)1877 4519 y Fr(V)21 b Fs(\000)2016 4534 y Fq(0)2056 4519 y Fs(\()p Fr(z)t Fs(\))p Fr(\016)2228 4477 y Fl(0)2224 4543 y Fq(1)2292 4519 y Ft(!)27 b Fr(V)21 b Fs(\000)2558 4534 y Fq(0)2598 4519 y Fs(\()p Fr(z)t Fs(\))p Fr(\016)2770 4477 y Fl(0)2766 4543 y Fq(1)456 4791 y Fv(in)j Fr(L)624 4755 y Fq(2)664 4791 y Fs(\()p Fp(R)768 4755 y Fo(d)814 4791 y Fs(\))p Fv(.)31 b(Thus)24 b(in)g(order)i(to)e(pro)o(v)o(e)g(that)g(the)h(W)-8 b(e)o(yl)24 b(functions)1050 5014 y Fr(M)1144 5029 y Fo(l)1171 5014 y Fs(\()p Fr(k)s Fs(\))j(=)1497 4947 y Fr(@)1553 4911 y Fq(2)p 1442 4991 207 4 v 1442 5083 a Fr(@)5 b(r)s(@)g(s)1658 5014 y(G)1735 5029 y Fo(n;z)1837 5014 y Fs(\()p Fr(r)m(;)17 b(s)p Fs(\))p Ft(j)2072 5030 y Fq(\(1)p Fo(;)p Fq(1\))2249 5014 y Fs(=)27 b(\(\000)2451 5029 y Fo(l)2477 5014 y Fs(\()p Fr(z)t Fs(\))p Fr(\016)2649 4973 y Fl(0)2645 5039 y Fq(1)2685 5014 y Fr(;)17 b(\016)2776 4973 y Fl(0)2772 5039 y Fq(1)2812 5014 y Fs(\))1179 5216 y(=)27 b(\(\000)1381 5231 y Fq(0)1421 5216 y Fs(\()p Fr(z)t Fs(\))p Fr(\016)1593 5174 y Fl(0)1589 5240 y Fq(1)1629 5216 y Fr(;)17 b(\016)1720 5174 y Fl(0)1716 5240 y Fq(1)1755 5216 y Fs(\))22 b Ft(\000)h Fs(\()p Fr(V)2010 5231 y Fo(l)2036 5216 y Fs(\000)2097 5231 y Fq(0)2136 5216 y Fs(\()p Fr(z)t Fs(\))p Fr(\016)2308 5174 y Fl(0)2304 5240 y Fq(1)2344 5216 y Fr(;)17 b Fs(\000)2449 5231 y Fo(l)2475 5216 y Fs(\()p 2513 5161 50 4 v Fr(z)5 b Fs(\))p Fr(\016)2648 5174 y Fl(0)2644 5240 y Fq(1)2683 5216 y Fs(\))p eop %%Page: 20 20 20 19 bop 456 251 a Fj(20)808 b(LAPTEV)-10 b(,)17 b(N)m(ABOK)n(O)i(AND) f(SAFR)m(ONO)l(V)456 450 y Fv(con)l(v)o(er)n(ge,)k(it)g(is)g(suf)n (\002cient)g(to)g(sho)n(w)f(that)h Fs(\000)1982 465 y Fo(l)2008 450 y Fs(\()p 2046 395 50 4 v Fr(z)5 b Fs(\))p Fr(\016)2181 414 y Fl(0)2177 475 y Fq(1)2239 450 y Fv(con)l(v)o(er)n (ges)22 b(to)g Fs(\()p Fr(S)2856 465 y Fq(0)2908 450 y Fs(+)13 b Fr(V)35 b Ft(\000)p 3179 395 V 13 w Fr(z)5 b Fs(\))3267 414 y Fl(\000)p Fq(1)3362 450 y Fr(\016)3409 414 y Fl(0)3405 475 y Fq(1)456 566 y Fv(in)20 b Fr(L)620 530 y Fq(2)660 566 y Fs(\()p Fp(R)763 530 y Fo(d)810 566 y Fs(\))h Fv(uniformly)e(on)h(compact)g(subsets)g Fr(K)28 b Fv(of)20 b(the)h(comple)o(x)e(plane.)29 b(The)21 b(latter)456 683 y(follo)n(ws)i(from)h(the)h(identity)616 844 y Fs(\000)677 859 y Fo(l)703 844 y Fs(\()p 741 790 V Fr(z)t Fs(\))p Fr(\016)875 803 y Fl(0)871 869 y Fq(1)939 844 y Fs(=)i(\()p Fr(S)1140 859 y Fq(0)1202 844 y Fs(+)22 b Fr(V)43 b Ft(\000)p 1500 790 V 23 w Fr(z)5 b Fs(\))1588 803 y Fl(\000)p Fq(1)1682 844 y Fr(\016)1729 803 y Fl(0)1725 869 y Fq(1)1787 844 y Ft(\000)23 b Fs(\000)1948 859 y Fo(l)1973 844 y Fs(\()p 2011 790 V Fr(z)5 b Fs(\)\()p Fr(V)2194 859 y Fo(l)2242 844 y Ft(\000)23 b Fr(V)e Fs(\)\()p Fr(S)2556 859 y Fq(0)2618 844 y Fs(+)h Fr(V)43 b Ft(\000)p 2916 790 V 23 w Fr(z)5 b Fs(\))3004 803 y Fl(\000)p Fq(1)3098 844 y Fr(\016)3145 803 y Fl(0)3141 869 y Fq(1)3208 844 y Fs(=)678 1109 y(=)28 b(\()p Fr(S)880 1124 y Fq(0)941 1109 y Fs(+)22 b Fr(V)44 b Ft(\000)p 1239 1054 V 22 w Fr(z)5 b Fs(\))1327 1068 y Fl(\000)p Fq(1)1421 1109 y Fr(\016)1468 1068 y Fl(0)1464 1134 y Fq(1)1526 1109 y Fs(+)22 b(\000)1685 1124 y Fo(l)1711 1109 y Fs(\()p 1749 1054 V Fr(z)5 b Fs(\)\()p Fr(I)30 b Ft(\000)2108 984 y Fo(l)2047 1014 y Fm(X)2058 1224 y Fo(j)t Fq(=0)2208 1109 y Fr(P)2271 1124 y Fo(j)2307 1109 y Fs(\))p Fr(V)22 b Fs(\()p Fr(S)2522 1124 y Fq(0)2583 1109 y Fs(+)g Fr(V)44 b Ft(\000)p 2882 1054 V 23 w Fr(z)t Fs(\))2969 1068 y Fl(\000)p Fq(1)3064 1109 y Fr(\016)3111 1068 y Fl(0)3107 1134 y Fq(1)3146 1109 y Fs(+)1017 1466 y(+\000)1154 1481 y Fo(l)1180 1466 y Fs(\()p 1218 1411 V Fr(z)t Fs(\))1383 1342 y Fo(l)1322 1372 y Fm(X)1337 1582 y Fo(i)p Fq(=0)1483 1466 y Fr(P)1546 1481 y Fo(i)1574 1466 y Fr(V)21 b Fs(\()p Fr(I)30 b Ft(\000)1924 1342 y Fo(l)1863 1372 y Fm(X)1873 1582 y Fo(j)t Fq(=0)2023 1466 y Fr(P)2086 1481 y Fo(j)2123 1466 y Fs(\)\()p Fr(S)2259 1481 y Fq(0)2320 1466 y Fs(+)22 b Fr(V)44 b Ft(\000)p 2618 1411 V 22 w Fr(z)5 b Fs(\))2706 1425 y Fl(\000)p Fq(1)2800 1466 y Fr(\016)2847 1425 y Fl(0)2843 1491 y Fq(1)456 1706 y Fv(and)24 b(from)h(the)g(bound)1304 1893 y Ft(jj)p Fs(\000)1421 1908 y Fo(l)1446 1893 y Fs(\()p 1484 1838 V Fr(z)5 b Fs(\))p Ft(jj)27 b(\024)1838 1825 y Fs(1)p 1770 1870 186 4 v 1770 1961 a Fv(Im)d Fr(z)1993 1893 y Ft(\024)k Fr(C)r(;)116 b(z)32 b Ft(2)c Fr(K)r(:)456 2082 y Fc(\003)555 2199 y Fv(Finally)34 b(according)h(to)f(inequality)f (\(7.11\))h(and)h(Propositions)d(8.1)j(and)f(8.2)g(we)456 2315 y(observ)o(e)22 b(that)g(there)h(e)o(xists)e(a)j(sequence)f(of)f (measures)h Fr(\026)2443 2330 y Fo(l)2492 2315 y Fv(weakly)f(con)l(v)o (er)n(gent)g(to)h Fr(\026)p Fv(,)456 2431 y(such)h(that)g(for)i(an)o(y) e(\002x)o(ed)g Fr(c)k(>)g Fs(0)1256 2513 y Fm(Z)1356 2539 y Fo(c)1311 2739 y Fq(0)1417 2581 y Fs(log\(1)p Fr(=\026)1738 2545 y Fl(0)1738 2607 y Fo(l)1763 2581 y Fs(\()p Fr(t)p Fs(\)\))17 b Fr(dt)p 1417 2626 598 4 v 1462 2727 a Fs(\(1)k(+)h Fr(t)1703 2698 y Fq(3)p Fo(=)p Fq(2)1814 2727 y Fs(\))1852 2646 y Ft(p)p 1935 2646 36 4 v 81 x Fr(t)2052 2649 y(<)28 b(C)7 b Fs(\()p Fr(V)21 b Fs(\))p Fr(;)116 b Ft(8)p Fr(l)r(;)456 2880 y Fv(where)37 b Fr(C)7 b Fs(\()p Fr(V)21 b Fs(\))37 b Fv(is)f(independent)g(of)h Fr(c)p Fv(.)66 b(Therefore)38 b(due)f(to)f(the)h(statement)e(on)i(the) 456 2996 y(upper)24 b(semi-continuity)f(of)i(an)g(entrop)o(y)f(\(see)h ([11]\))g(we)g(obtain)1418 3078 y Fm(Z)1518 3105 y Fl(1)1473 3304 y Fq(0)1619 3147 y Fs(log\(1)p Fr(=\026)1940 3111 y Fl(0)1962 3147 y Fs(\()p Fr(t)p Fs(\)\))17 b Fr(dt)p 1619 3191 596 4 v 1662 3292 a Fs(\(1)22 b(+)g Fr(t)1904 3263 y Fq(3)p Fo(=)p Fq(2)2014 3292 y Fs(\))2052 3211 y Ft(p)p 2135 3211 36 4 v 81 x Fr(t)2252 3214 y(<)27 b Ft(1)p Fr(:)456 3439 y Fv(The)d(proof)h(of)g(Theorem)g(2.2)f(is)g (complete.)456 3646 y(The)j(proof)g(of)h(Theorem)f(2.3)g(is)g(e)o (xactly)f(the)h(same)h(as)f(the)g(proof)g(of)h(Theorem)f(2.2)456 3763 y(b)n(ut)d(instead)g(of)h(Proposition)e(7.3)i(we)g(apply)f (Proposition)f(7.4.)1364 3974 y(9.)52 b(P)t Fu(R)q(O)t(O)t(F)33 b(O)t(F)e Fv(T)t Fu(H)t(E)t(O)t(R)t(E)t(M)i Fv(2)t(.)t(1)555 4149 y(The)c(proof)f(is)g(reduced)h(to)f(the)g(references)i(on)e([5],)i ([2])f(and)f(Theorem)g(2.2.)42 b(Let)456 4265 y Ft(\000)p Fs(\001)27 b Fv(be)g(the)f(Laplace)h(operator)f(in)g Fr(L)1780 4229 y Fq(2)1820 4265 y Fs(\()p Fp(R)1924 4229 y Fo(d)1970 4265 y Fs(\))p Fv(.)36 b(According)26 b(to)g([5],)h(if)g Fr(V)48 b Fv(satis\002es)26 b(the)456 4381 y(conditions)d(of)i(Theorem) f(2.2,)g(then)1096 4543 y Fs(\()p Ft(\000)p Fs(\001)e(+)h Fr(V)43 b Ft(\000)23 b Fr(z)t Fs(\))1700 4502 y Fl(\000)p Fo(n)1825 4543 y Ft(\000)f Fs(\()p Fr(H)30 b Fs(+)22 b Fr(V)43 b Ft(\000)23 b Fr(z)t Fs(\))2458 4502 y Fl(\000)p Fo(n)2588 4543 y Ft(2)28 b Fe(S)2765 4558 y Fq(1)456 4705 y Fv(for)35 b(some)g Fr(z)40 b Fv(and)35 b(suf)n(\002ciently)g (lar)n(ge)g Fr(n)48 b(>)f Fs(0)p Fv(.)62 b(The)35 b(latter)g(relation)g (implies)f(that)456 4821 y Ft(\000)p Fs(\001)25 b(+)f Fr(V)49 b Fv(and)27 b Fr(H)32 b Fs(+)24 b Fr(V)49 b Fv(ha)n(v)o(e)28 b(the)f(same)h(a.c.)39 b(spectrum.)g(No)n(w)26 b(by)i(Theorem)f(2.11) 456 4937 y(and)d(Corollary)g(2.13)f(of)i([2],)f(the)g(a.c.)31 b(spectrum)24 b(does)f(not)h(change)g(if)h(we)f(add)g(to)g Fr(V)456 5054 y Fv(an)o(y)g(real)h Fr(L)863 5018 y Fl(1)938 5054 y Fv(-function)f Fr(V)1385 5069 y Fq(0)1450 5054 y Fv(with)g(a)h(\002nite)f(support.)30 b(Indeed,)25 b(in)f(this)g(case) 952 5216 y Fs(\()p Ft(\000)p Fs(\001)f(+)f Fr(V)44 b Ft(\000)22 b Fr(z)t Fs(\))1556 5174 y Fl(\000)p Fo(n)1681 5216 y Ft(\000)h Fs(\()p Ft(\000)p Fs(\001)g(+)f Fr(V)43 b Fs(+)22 b Fr(V)2353 5231 y Fq(0)2415 5216 y Ft(\000)g Fr(z)t Fs(\))2601 5174 y Fl(\000)p Fo(n)2732 5216 y Ft(2)28 b Fe(S)2909 5231 y Fq(1)p eop %%Page: 21 21 21 20 bop 1246 251 a Fj(ABSOLUTEL)-7 b(Y)22 b(CONTINUOUS)h(SPECTR)m(UM) 714 b(21)456 450 y Fv(for)25 b(some)f Fr(z)30 b Fv(and)24 b(suf)n(\002ciently)g(lar)n(ge)h Fr(n)j(>)g Fs(0)p Fv(.)i(This)24 b(pro)o(v)o(es)g(Theorem)g(2.1.)1639 668 y(10.)51 b(A)t Fu(P)t(P)t(E)t(N)t(D)t(I)t(X)456 842 y Fv(Here)26 b(we)h(sho)n(w)e (that)h Fr(a)1279 857 y Fo(")1316 842 y Fs(\()p Fr(k)s Fs(\))g Fv(appearing)g(in)g(\(7.1\))g(is)g(a)g(meromorphic)f(function)g (in)h(a)456 958 y(neighborhood)e(of)j(zero)f(and)h Ft(j)p Fr(a)1592 973 y Fo(")1628 958 y Fs(\()p Fr(k)s Fs(\))p Ft(j)j Fs(=)g(1)23 b(+)h Fr(O)s Fs(\(1)p Fr(=)p Ft(j)p Fr(k)s Ft(j)2418 922 y Fq(2)2455 958 y Fs(\))p Fv(,)j(as)f Fr(k)33 b Ft(!)d(\0061)d Fv(which,)f(in)456 1074 y(particular)l(,)e (means)h(that)f Fs(log)17 b Ft(j)p Fr(a)1559 1089 y Fo(")1595 1074 y Fs(\()p Fr(k)s Fs(\))p Ft(j)28 b(2)g Fr(L)1941 1038 y Fq(1)1980 1074 y Fs(\()p Fp(R)5 b Fs(\))p Fv(.)555 1191 y FB(1.)54 b Fv(Let)32 b Fr(P)55 b Fs(=)1084 1116 y Fm(P)1190 1142 y Fo(n)1190 1220 y(j)t Fq(=0)1333 1191 y Fr(P)1396 1206 y Fo(j)1432 1191 y Fv(,)35 b Fr(V)63 b Fs(=)41 b Fr(P)14 b(V)21 b(P)14 b Fv(.)53 b(Introduce)32 b(matrices)g Fr(A)p Fs(\()p Fr(k)s Fs(\))h Fv(and)f Fr(B)5 b Fs(\()p Fr(k)s Fs(\))456 1323 y Fv(de\002ned)28 b(in)f(the)g(space)h Fr(P)14 b(L)1423 1287 y Fq(2)1463 1323 y Fs(\()p Fp(S)1562 1287 y Fo(d)p Fl(\000)p Fq(1)1687 1323 y Fs(\))p Fv(,)28 b(such)f(that)h(the)f(solution)f(of)i(the)f(equation)g(\(for)456 1439 y(the)d(matrix)g(v)n(alued)g(function)g Fs(\010)p Fv(\))456 1659 y(\(10.1\))389 b Ft(\000)1173 1592 y Fr(d)1224 1556 y Fq(2)1263 1592 y Fs(\010)p 1173 1636 161 4 v 1185 1728 a Fr(dr)1283 1699 y Fq(2)1366 1659 y Fs(+)1477 1592 y Fr(\020)1520 1607 y Fo(")p 1474 1636 87 4 v 1474 1728 a Fr(r)1521 1699 y Fq(2)1570 1549 y Fm(\020)1629 1659 y Ft(\000)p Fs(\001)1787 1674 y Fo(\022)1827 1659 y Fs(\010)23 b(+)f Fr(\013)2080 1674 y Fo(d)2120 1659 y Fs(\010)2190 1549 y Fm(\021)2272 1659 y Fs(+)g Fr(V)g Fs(\010)28 b(=)f Fr(k)2704 1618 y Fq(2)2744 1659 y Fs(\010)456 1852 y Fv(satis\002es)d(the)h(follo)n(wing)e(conditions)1306 2015 y Fs(\010)28 b(=)f(exp)q(\()p Fr(ik)s(r)s Fs(\))p Fr(P)s(;)133 b Fs(for)99 b Fr(r)30 b(>)e(c)2528 2030 y Fq(2)2567 2015 y Fr(;)456 2179 y Fv(and)985 2303 y Fs(exp)q(\()p Fr(ik)s(r)s Fs(\))p Fr(A)p Fs(\()p Fr(k)s Fs(\))22 b(+)g(exp)q(\()p Ft(\000)p Fr(ik)s(r)s Fs(\))p Fr(B)5 b Fs(\()p Fr(k)s Fs(\))100 b(for)g Fr(r)30 b(<)d(c)2848 2318 y Fq(1)2888 2303 y Fr(:)456 2446 y Fv(W)-8 b(e)32 b(shall)g(see)g(that)g Fr(A)p Fs(\()p Fr(k)s Fs(\))h Fv(and)f Fr(B)5 b Fs(\()p Fr(k)s Fs(\))32 b Fv(both)g(ha)n(v)o(e)g(at)g (most)f(a)i(simple)e(pole)h(at)g(zero)456 2562 y(and)24 b(therefore)i(by)e(\(10.2\))h Fr(a)1450 2577 y Fo(")1487 2562 y Fs(\()p Fr(k)s Fs(\))g Fv(could)f(also)g(ha)n(v)o(e)h(a)g(pole)g (at)f(zero.)456 2741 y FB(Pr)n(oposition)g(10.1.)41 b Fn(The)25 b(following)f(r)l(elation)g(holds)f(true:)456 2949 y Fv(\(10.2\))1058 2881 y Fs(1)p 973 2926 218 4 v 973 3017 a Fr(a)1024 3032 y Fo(")1061 3017 y Fs(\()p Fr(k)s Fs(\))1201 2949 y Fr(P)1264 2964 y Fq(0)1331 2949 y Fs(=)k Fr(P)1497 2964 y Fq(0)1537 2868 y Fm(\000)1582 2949 y Fr(A)p Fs(\()p Fr(k)s Fs(\))c(+)f(\()p Fr(I)30 b Ft(\000)22 b Fr(P)2179 2964 y Fq(0)2218 2949 y Fs(\))p Fr(e)2301 2908 y Fl(\000)p Fq(2)p Fo(ik)2458 2949 y Fr(B)5 b Fs(\()p Fr(k)s Fs(\))2667 2868 y Fm(\001)2713 2890 y Fl(\000)p Fq(1)2808 2949 y Fr(P)2871 2964 y Fq(0)2910 2949 y Fr(:)555 3207 y Fn(Pr)l(oof)p Fv(.)47 b(Let)31 b Fr(G)p Fs(\()p Fr(r)m(;)17 b(s;)g(k)s Fs(\))31 b Fv(be)f(the)h(k)o (ernel)g(of)g(the)g(operator)f Fs(\()2677 3182 y(^)2651 3207 y Fr(H)2732 3222 y Fo(")2795 3207 y Fs(+)d Fr(V)48 b Ft(\000)27 b Fr(z)t Fs(\))3194 3171 y Fl(\000)p Fq(1)3289 3207 y Fr(\037)3350 3222 y Fo(c)3381 3231 y Ff(1)3419 3207 y Fv(,)456 3323 y(where)f Fr(\037)786 3338 y Fo(c)817 3347 y Ff(1)881 3323 y Fv(is)g(the)f(operator)h(of)g(multiplication)d (by)j(the)f(characteristic)h(function)f(of)456 3439 y Fs(\(1)p Fr(;)17 b(c)629 3454 y Fq(1)667 3439 y Fs(\))p Fv(.)31 b(Then)807 3694 y Fr(G)p Fs(\()p Fr(r)m(;)17 b(s;)g(k)s Fs(\))27 b(=)1320 3524 y Fm(\()1400 3628 y Fs(\011\()p Fr(r)m(;)17 b(k)s Fs(\))p Fr(Z)1758 3643 y Fq(1)1797 3628 y Fs(\()p Fr(s;)g(k)s Fs(\))p Fr(;)116 b Fs(as)100 b Fr(r)30 b(<)e(s)f(<)h(c)2744 3643 y Fq(1)1400 3767 y Ft(\000)p Fs(\010\()p Fr(r)m(;)17 b(k)s Fs(\))p Fr(Z)1829 3782 y Fq(2)1869 3767 y Fs(\()p Fr(s;)g(k)s Fs(\))p Fr(;)116 b Fs(as)100 b Fr(s)28 b(<)f(c)2638 3782 y Fq(1)2678 3767 y Fr(;)33 b(s)27 b(<)h(r)m(:)555 3957 y Fv(Here)35 b Fs(\011\()p Fr(r)m(;)17 b(k)s Fs(\))45 b(=)f Fr(e)1284 3921 y Fl(\000)p Fo(ik)r(r)1440 3957 y Fr(P)1503 3972 y Fq(0)1571 3957 y Fs(+)29 b Fr(k)1730 3921 y Fl(\000)p Fq(1)1841 3957 y Fs(sin\()p Fr(k)s Fs(\()p Fr(r)j Ft(\000)d Fs(1\)\)\()p Fr(P)42 b Ft(\000)30 b Fr(P)2711 3972 y Fq(0)2750 3957 y Fs(\))k Fv(for)h Fr(r)47 b(<)e(c)3227 3972 y Fq(1)3301 3957 y Fv(and)456 4073 y Fs(\010\()p Fr(r)m(;)17 b(k)s Fs(\))31 b(=)g Fr(e)924 4037 y Fo(ik)r(r)1025 4073 y Fr(P)40 b Fv(for)27 b Fr(r)33 b(>)e(c)1497 4088 y Fq(2)1537 4073 y Fv(.)36 b(The)27 b(matrices)f Fr(Z)2211 4088 y Fq(1)2250 4073 y Fs(\()p Fr(s;)17 b(k)s Fs(\))26 b Fv(and)h Fr(Z)2734 4088 y Fq(2)2773 4073 y Fs(\()p Fr(s;)17 b(k)s Fs(\))26 b Fv(are)i(chosen)456 4190 y(such)c(that)g Fr(G)p Fs(\()p Fr(r)m(;)17 b(s;)g(k)s Fs(\))25 b Fv(is)f(continuous)f(at)i(the)g(diagonal)f(and)1161 4354 y Fs(lim)1115 4413 y Fo(r)r Fl(!)p Fo(s)p Fl(\000)p Fq(0)1359 4354 y Fr(G)1436 4312 y Fl(0)1436 4378 y Fo(r)1473 4354 y Fs(\()p Fr(r)m(;)17 b(s;)g(k)s Fs(\))28 b(=)73 b(lim)1909 4413 y Fo(r)r Fl(!)p Fo(s)p Fq(+0)2153 4354 y Fr(G)2230 4312 y Fl(0)2230 4378 y Fo(r)2268 4354 y Fs(\()p Fr(r)m(;)17 b(s;)g(k)s Fs(\))22 b(+)g Fr(P)s(:)456 4554 y Fv(The)i(tw)o(o)h(latter)f(equations)g(are)i(equi)n(v)n(alent)d (to)1084 4719 y Fs([)p Fr(e)1156 4678 y Fl(\000)p Fo(ik)r(r)1311 4719 y Fr(P)1374 4734 y Fq(0)1436 4719 y Fs(+)f Fr(k)1588 4678 y Fl(\000)p Fq(1)1699 4719 y Fs(sin\()p Fr(k)s Fs(\()p Fr(r)i Ft(\000)f Fs(1\)\)\()p Fr(P)35 b Ft(\000)23 b Fr(P)2541 4734 y Fq(0)2580 4719 y Fs(\)])p Fr(Z)2712 4734 y Fq(1)2751 4719 y Fs(+)1581 4882 y([)p Fr(e)1653 4841 y Fl(\000)p Fo(ik)r(r)1809 4882 y Fr(B)5 b Fs(\()p Fr(k)s Fs(\))22 b(+)g Fr(e)2183 4841 y Fo(ik)r(r)2284 4882 y Fr(A)p Fs(\()p Fr(k)s Fs(\)])p Fr(Z)2581 4897 y Fq(2)2648 4882 y Fs(=)27 b(0;)1073 5046 y([)p Ft(\000)p Fr(ik)s(e)1309 5005 y Fl(\000)p Fo(ik)r(r)1465 5046 y Fr(P)1528 5061 y Fq(0)1590 5046 y Fs(+)22 b(cos)q(\()p Fr(k)s Fs(\()p Fr(r)i Ft(\000)f Fs(1\)\)\()p Fr(P)35 b Ft(\000)23 b Fr(P)2541 5061 y Fq(0)2580 5046 y Fs(\)])p Fr(Z)2712 5061 y Fq(1)2751 5046 y Fs(+)1328 5210 y([)p Ft(\000)p Fr(ik)s(e)1564 5169 y Fl(\000)p Fo(ik)r(r)1721 5210 y Fr(B)5 b Fs(\()p Fr(k)s Fs(\))22 b(+)g Fr(ik)s(e)2182 5169 y Fo(ik)r(r)2283 5210 y Fr(A)p Fs(\()p Fr(k)s Fs(\)])p Fr(Z)2580 5225 y Fq(2)2647 5210 y Fs(=)28 b Fr(P)456 4965 y Fv(\(10.3\))p eop %%Page: 22 22 22 21 bop 456 251 a Fj(22)808 b(LAPTEV)-10 b(,)17 b(N)m(ABOK)n(O)i(AND) f(SAFR)m(ONO)l(V)456 461 y Fv(and)26 b(are)h(uniquely)e(solv)n(able)g (if)h(and)g(only)g(if)g Fr(k)2107 425 y Fq(2)2173 461 y Fv(is)g(not)g(an)g(eigen)l(v)n(alue)g(of)3126 436 y Fs(^)3101 461 y Fr(H)3182 476 y Fo(")3242 461 y Fs(+)d Fr(V)e Fv(.)456 578 y(The)j(\002rst)h(equation)f(of)h(the)g(system)f (\(10.3\))g(gi)n(v)o(es)550 793 y Fr(Z)617 808 y Fq(1)684 793 y Fs(=)j Ft(\000)864 712 y Fm(\002)906 793 y Fr(e)951 752 y Fo(ik)r(r)1052 793 y Fr(P)1115 808 y Fq(0)1177 793 y Fs(+)1529 726 y Fr(k)p 1284 770 543 4 v 1284 861 a Fs(sin\()p Fr(k)s Fs(\()p Fr(r)e Ft(\000)e Fs(1\)\))1837 793 y(\()p Fr(P)36 b Ft(\000)22 b Fr(P)2136 808 y Fq(0)2176 793 y Fs(\))2214 712 y Fm(\003)o(\002)2297 793 y Fr(e)2342 752 y Fl(\000)p Fo(ik)r(r)2497 793 y Fr(B)5 b Fs(\()p Fr(k)s Fs(\))22 b(+)g Fr(e)2871 752 y Fo(ik)r(r)2972 793 y Fr(A)p Fs(\()p Fr(k)s Fs(\))3175 712 y Fm(\003)3217 793 y Fr(Z)3284 808 y Fq(2)3323 793 y Fr(:)456 1012 y Fv(Therefore)j(we)g(obtain)f(from)h(the)g(second)f(equation)g(of)h (\(10.3\))g(that)830 1097 y Fm(\002)872 1178 y Fr(ik)s(P)1022 1193 y Fq(0)1083 1178 y Ft(\000)e Fr(k)d Fs(ctg)e(\()p Fr(k)s Fs(\()p Fr(r)25 b Ft(\000)d Fs(1\)\)\()p Fr(P)35 b Ft(\000)23 b Fr(P)2124 1193 y Fq(0)2163 1178 y Fs(\))2201 1097 y Fm(\003\002)2284 1178 y Fr(e)2329 1137 y Fl(\000)p Fo(ik)r(r)2485 1178 y Fr(B)5 b Fs(\()p Fr(k)s Fs(\))22 b(+)g Fr(e)2859 1137 y Fo(ik)r(r)2960 1178 y Fr(A)p Fs(\()p Fr(k)s Fs(\))3163 1097 y Fm(\003)3204 1178 y Fr(Z)3271 1193 y Fq(2)1720 1341 y Fs(+[)p Ft(\000)p Fr(ik)s(e)2032 1300 y Fl(\000)p Fo(ik)r(r)2188 1341 y Fr(B)5 b Fs(\()p Fr(k)s Fs(\))22 b(+)g Fr(ik)s(e)2649 1300 y Fo(ik)r(r)2750 1341 y Fr(A)p Fs(\()p Fr(k)s Fs(\)])p Fr(Z)3047 1356 y Fq(2)3114 1341 y Fs(=)28 b Fr(P)s(;)456 1260 y Fv(\(10.4\))456 1506 y(or)c(equi)n(v)n(alently)456 1699 y Fs(\()p Fr(P)14 b Ft(\000)p Fr(P)711 1714 y Fq(0)750 1699 y Fs(\))788 1588 y Fm(h)835 1618 y(\000)880 1699 y Ft(\000)p Fr(k)21 b Fs(ctg)q(\()p Fr(k)s Fs(\()p Fr(r)s Ft(\000)p Fs(1\)\))p Ft(\000)p Fr(ik)1703 1618 y Fm(\001)1749 1699 y Fr(e)1794 1658 y Fl(\000)p Fo(ik)r(r)1950 1699 y Fr(B)5 b Fs(\()p Fr(k)s Fs(\)+)2235 1618 y Fm(\000)2280 1699 y Ft(\000)p Fr(k)21 b Fs(ctg)q(\()p Fr(k)s Fs(\()p Fr(r)s Ft(\000)p Fs(1\)\)+)p Fr(ik)3102 1618 y Fm(\001)3148 1699 y Fr(e)3193 1658 y Fo(ik)r(r)3293 1699 y Fr(A)p Fs(\()p Fr(k)s Fs(\))3496 1588 y Fm(i)3544 1699 y Fr(Z)3611 1714 y Fq(2)1453 1917 y Fs(+2)p Fr(ik)s(P)1728 1932 y Fq(0)1768 1917 y Fr(e)1813 1875 y Fo(ik)r(r)1913 1917 y Fr(A)p Fs(\()p Fr(k)s Fs(\))p Fr(Z)2183 1932 y Fq(2)2250 1917 y Fs(=)28 b Fr(P)s(:)456 2061 y Fv(Ob)o(viously)1124 2253 y Ft(\000)p Fr(k)21 b Fs(ctg)q(\()p Fr(k)s Fs(\()p Fr(r)k Ft(\000)d Fs(1\)\))g Ft(\006)h Fr(ik)31 b Fs(=)c Ft(\000)2297 2186 y Fr(k)s(e)2396 2149 y Fl(\007)p Fo(ik)r Fq(\()p Fo(r)r Fl(\000)p Fq(1\))p 2254 2230 484 4 v 2254 2321 a Fs(sin)17 b Fr(k)s Fs(\()p Fr(r)25 b Ft(\000)e Fs(1\))2748 2253 y Fr(:)456 2451 y Fv(This)h(implies)456 2666 y Fs(\()p Fr(P)h Ft(\000)12 b Fr(P)734 2681 y Fq(0)774 2666 y Fs(\))812 2556 y Fm(h)1045 2599 y Ft(\000)p Fr(k)p 869 2644 V 869 2735 a Fs(sin)17 b Fr(k)s Fs(\()p Fr(r)25 b Ft(\000)d Fs(1\))1363 2556 y Fm(\020)1422 2666 y Fr(e)1467 2625 y Fl(\000)p Fo(ik)1589 2666 y Fr(B)5 b Fs(\()p Fr(k)s Fs(\))12 b(+)g Fr(e)1943 2625 y Fo(ik)2011 2666 y Fr(A)p Fs(\()p Fr(k)s Fs(\))2214 2556 y Fm(\021i)2320 2666 y Fr(Z)2387 2681 y Fq(2)2439 2666 y Fs(+)g(2)p Fr(ik)s(P)2726 2681 y Fq(0)2766 2666 y Fr(e)2811 2625 y Fo(ik)r(r)2911 2666 y Fr(A)p Fs(\()p Fr(k)s Fs(\))p Fr(Z)3181 2681 y Fq(2)3248 2666 y Fs(=)28 b Fr(P)s(:)456 2885 y Fv(Multiplying)22 b(both)i(sides)g(of)h(this)f (identity)f(by)1168 3046 y Ft(\000)17 b Fs(sin)g Fr(k)s Fs(\()p Fr(r)24 b Ft(\000)f Fs(1\))p 1168 3091 578 4 v 1430 3182 a Fr(k)1756 3113 y(e)1801 3072 y Fl(\000)p Fo(ik)1922 3113 y Fs(\()p Fr(P)36 b Ft(\000)23 b Fr(P)2222 3128 y Fq(0)2261 3113 y Fs(\))f(+)2429 3046 y Fr(e)2474 3010 y Fl(\000)p Fo(ik)r(r)p 2429 3091 201 4 v 2461 3182 a Fs(2)p Fr(ik)2640 3113 y(P)2703 3128 y Fq(0)456 3307 y Fv(we)j(deri)n(v)o(e)604 3472 y Fr(P)667 3487 y Fq(0)706 3472 y Fr(Z)773 3487 y Fq(2)813 3472 y Fs(\()p Fr(r)m(;)17 b(k)s Fs(\))p Fr(P)1091 3487 y Fq(0)1157 3472 y Fs(=)28 b(\(2)p Fr(ik)s Fs(\))1473 3431 y Fl(\000)p Fq(1)1567 3472 y Fr(e)1612 3431 y Fl(\000)p Fo(ik)r(r)1768 3472 y Fr(P)1831 3487 y Fq(0)1870 3392 y Fm(\000)1916 3472 y Fr(A)p Fs(\()p Fr(k)s Fs(\))22 b(+)g Fr(e)2284 3431 y Fl(\000)p Fq(2)p Fo(ik)2441 3472 y Fs(\()p Fr(P)36 b Ft(\000)22 b Fr(P)2740 3487 y Fq(0)2780 3472 y Fs(\))p Fr(B)5 b Fs(\()p Fr(k)s Fs(\))3027 3392 y Fm(\001)3072 3414 y Fl(\000)p Fq(1)3167 3472 y Fr(P)3230 3487 y Fq(0)3269 3472 y Fr(:)456 3637 y Fv(Finally)-6 b(,)23 b(since)1273 3763 y Fr(P)1336 3778 y Fq(0)1375 3763 y Fr(Z)1442 3778 y Fq(2)1481 3763 y Fs(\()p Fr(r)m(;)17 b(k)s Fs(\))p Fr(P)1759 3778 y Fq(0)1826 3763 y Fs(=)28 b(\(2)p Fr(ik)s(a)2155 3778 y Fo(")2192 3763 y Fs(\))2230 3722 y Fl(\000)p Fq(1)2324 3763 y Fr(e)2369 3722 y Fl(\000)p Fo(ik)r(r)2525 3763 y Fr(P)2588 3778 y Fq(0)456 3908 y Fv(we)d(obtain)f(\(10.2\))o(.)31 b Fc(\003)456 4073 y FB(2.)f Fv(In)23 b(this)f(subsection)g(we)h(adapt) g(the)g(ar)n(gument)g(from)f([16].)31 b(The)23 b(solution)e Fs(\010\()p Fr(r)m(;)c(k)s Fs(\))456 4189 y Fv(of)24 b(\(10.1\))h(satis\002es)g(the)f(inte)o(gral)g(equation)456 4406 y(\(10.5\))251 b Fs(\010\()p Fr(r)m(;)17 b(k)s Fs(\))28 b(=)f Fr(e)1409 4365 y Fo(ik)r(r)1510 4406 y Fr(P)35 b Ft(\000)1708 4270 y Fm(Z)1808 4297 y Fl(1)1763 4496 y Fo(r)1899 4406 y Fr(k)1953 4365 y Fl(\000)p Fq(1)2064 4406 y Fs(sin)16 b Fr(k)s Fs(\()p Fr(r)25 b Ft(\000)e Fr(s)p Fs(\))p Fr(V)2602 4421 y Fl(\003)2641 4406 y Fs(\()p Fr(s)p Fs(\)\010\()p Fr(s;)17 b(k)s Fs(\))g Fr(ds;)456 4632 y Fv(where)25 b Fr(V)781 4647 y Fl(\003)848 4632 y Fs(=)i Fr(V)44 b Ft(\000)23 b Fr(r)1199 4596 y Fl(\000)p Fq(2)1293 4632 y Fr(\020)1336 4647 y Fo(")1389 4632 y Fr(P)30 b Fs(\001)1563 4647 y Fo(\022)1602 4632 y Fv(.)h(Denote)1369 4797 y Fr(X)8 b Fs(\()p Fr(r)m(;)17 b(k)s Fs(\))27 b(=)h Fr(e)1849 4756 y Fl(\000)p Fo(ik)r(r)2004 4797 y Fs(\010\()p Fr(r)m(;)17 b(k)s Fs(\))23 b Ft(\000)f Fr(P)30 b(:)456 4963 y Fv(Then)456 5159 y(\(10.6\))229 b Fr(X)8 b Fs(\()p Fr(r)m(;)17 b(k)s Fs(\))28 b(=)1361 5023 y Fm(Z)1461 5050 y Fl(1)1416 5249 y Fo(r)1552 5159 y Fr(K)7 b Fs(\()p Fr(r)m(;)17 b(s;)g(k)s Fs(\))g Fr(ds)k Fs(+)2180 5023 y Fm(Z)2280 5050 y Fl(1)2235 5249 y Fo(r)2371 5159 y Fr(K)7 b Fs(\()p Fr(r)m(;)17 b(s;)g(k)s Fs(\))p Fr(X)8 b Fs(\()p Fr(s;)17 b(k)s Fs(\))g Fr(ds;)p eop %%Page: 23 23 23 22 bop 1246 251 a Fj(ABSOLUTEL)-7 b(Y)22 b(CONTINUOUS)h(SPECTR)m(UM) 714 b(23)456 450 y Fv(where)456 722 y(\(10.7\))602 b Fr(K)7 b Fs(\()p Fr(r)m(;)17 b(s;)g(k)s Fs(\))28 b(=)1835 655 y Fr(e)1880 619 y Fq(2)p Fo(ik)r Fq(\()p Fo(s)p Fl(\000)p Fo(r)r Fq(\))2181 655 y Ft(\000)22 b Fs(1)p 1835 700 494 4 v 2014 791 a(2)p Fr(ik)2339 722 y(V)2396 737 y Fl(\003)2435 722 y Fs(\()p Fr(s)p Fs(\))17 b Fr(:)456 976 y Fv(Note)24 b(that)456 1202 y(\(10.8\))546 b Ft(k)p Fr(K)7 b Fs(\()p Fr(r)m(;)17 b(s;)g(k)s Fs(\))p Ft(k)27 b(\024)h Fr(C)1940 1217 y Fq(1)1980 1202 y Fs(\()p Fr(V)2075 1217 y Fl(\003)2114 1202 y Fr(;)17 b(n)p Fs(\))p Fr(=)p Fs(\(1)k(+)h Ft(j)p Fr(k)s Ft(j)p Fs(\))456 1428 y Fv(for)31 b(all)h Fr(k)i Fv(with)d(Im)25 b Fr(k)43 b Ft(\025)e Fs(0)31 b Fv(and)g(all)h Fr(k)i Fv(with)d Fs(1)40 b Fr(<)g(r)j Ft(\024)e Fr(s)p Fv(.)51 b(Here)32 b(and)f(belo)n(w)g Ft(k)c(\001)g(k)456 1544 y Fv(denotes)d(the)h(norm)f(of)h(an)g (operator)f(in)h Fr(P)14 b(L)1996 1508 y Fq(2)2035 1544 y Fs(\()p Fp(S)2134 1508 y Fo(d)p Fl(\000)p Fq(1)2259 1544 y Fs(\))p Fv(.)555 1660 y(Solving)27 b(the)g(V)-13 b(olterra)28 b(equation)e(\(10.6\))h(we)h(obtain)e(the)i(follo)n(wing)d (con)l(v)o(er)n(gent)456 1776 y(series)839 2058 y Fr(X)8 b Fs(\()p Fr(r)m(;)17 b(k)s Fs(\))28 b(=)1315 1934 y Fl(1)1279 1963 y Fm(X)1274 2173 y Fo(m)p Fq(=1)1476 1923 y Fm(Z)1592 2058 y Ft(\001)17 b(\001)g(\001)1725 1923 y Fm(Z)1443 2205 y Fo(r)r Fl(\024)p Fo(r)1564 2214 y Ff(1)1599 2205 y Fl(\024\001\001\001)n(\024)p Fo(r)1799 2213 y Fi(m)1907 1934 y Fo(m)1874 1963 y Fm(Y)1882 2176 y Fo(l)q Fq(=1)2018 2058 y Fr(K)7 b Fs(\()p Fr(r)2190 2073 y Fo(l)q Fl(\000)p Fq(1)2306 2058 y Fr(;)17 b(r)2394 2073 y Fo(l)2420 2058 y Fr(;)g(k)s Fs(\))g Fr(dr)2668 2073 y Fq(1)2723 2058 y Ft(\001)g(\001)g(\001)e Fr(dr)2951 2073 y Fo(m)3034 2058 y Fr(:)456 2418 y Fv(From)38 b(\(10.8\))h(we)g (see)g(that)f Ft(j)p Fr(X)8 b Fs(\()p Fr(r)m(;)17 b(k)s Fs(\))p Ft(j)52 b(\024)i Fr(C)2112 2433 y Fq(2)2151 2418 y Fs(\()p Fr(V)2246 2433 y Fl(\003)2285 2418 y Fs(\))39 b Fv(for)g(all)f Fs(1)54 b Fr(<)f(r)s Fv(.)72 b(Ob)o(viously)456 2535 y Fr(X)8 b Fs(\()p Fr(r)m(;)17 b(k)s Fs(\))29 b Fv(is)g(an)h(entire)f(function)g(in)g Fr(k)s Fv(.)45 b(Inserting)28 b(this)h(estimate)g(back)g(into)g(\(10.6\))o(,)456 2651 y(we)c(conclude)f(that)h(the)f(inequality)456 2876 y(\(10.9\))569 b Ft(k)p Fr(X)8 b Fs(\()p Fr(r)m(;)17 b(k)s Fs(\))p Ft(k)27 b(\024)h Fr(C)1872 2891 y Fq(3)1911 2876 y Fs(\()p Fr(V)2006 2891 y Fl(\003)2046 2876 y Fr(;)17 b(n)p Fs(\)\(1)k(+)h Ft(j)p Fr(k)s Ft(j)p Fs(\))2540 2835 y Fl(\000)p Fq(1)456 3102 y Fv(holds)h(for)i(all)g Fr(r)j Fv(with)c Fs(1)j Fr(<)h(r)f Fv(and)e(all)f Fr(k)k Fv(with)d(Im)f Fr(k)31 b Ft(\025)d Fs(0)p Fv(.)555 3218 y(If)d(we)h(re)n(write)e(\(10.5\))h(as)g(follo)n(ws)575 3640 y Fs(\010\()p Fr(r)m(;)17 b(k)s Fs(\))28 b(=)g Fr(e)1037 3599 y Fo(ik)r(r)1154 3499 y Fm(\024)1206 3640 y Fr(P)36 b Ft(\000)1458 3572 y Fs(1)p 1415 3617 137 4 v 1415 3708 a(2)p Fr(ik)1577 3504 y Fm(Z)1677 3530 y Fl(1)1633 3730 y Fo(r)1768 3640 y Fr(V)1825 3655 y Fl(\003)1865 3640 y Fs(\()p Fr(s)p Fs(\))17 b Fr(ds)k Ft(\000)2275 3572 y Fs(1)p 2231 3617 V 2231 3708 a(2)p Fr(ik)2394 3504 y Fm(Z)2494 3530 y Fl(1)2449 3730 y Fo(r)2585 3640 y Fr(V)2642 3655 y Fl(\003)2681 3640 y Fs(\()p Fr(s)p Fs(\))p Fr(X)8 b Fs(\()p Fr(s;)17 b(k)s Fs(\))g Fr(ds)3226 3499 y Fm(\025)456 3422 y Fv(\(10.10\))982 3921 y Fs(+)1090 3854 y Fr(e)1135 3818 y Fl(\000)p Fo(ik)r(r)p 1090 3898 201 4 v 1122 3990 a Fs(2)p Fr(ik)1317 3781 y Fm(\024)1370 3786 y(Z)1469 3812 y Fl(1)1425 4011 y Fo(r)1561 3921 y Fr(e)1606 3880 y Fq(2)p Fo(ik)r(s)1741 3921 y Fr(V)1798 3936 y Fl(\003)1837 3921 y Fs(\()p Fr(s)p Fs(\))g Fr(ds)k Fs(+)2192 3786 y Fm(Z)2292 3812 y Fl(1)2247 4011 y Fo(r)2383 3921 y Fr(e)2428 3880 y Fq(2)p Fo(ik)r(s)2563 3921 y Fr(V)2620 3936 y Fl(\003)2659 3921 y Fs(\()p Fr(s)p Fs(\))p Fr(X)8 b Fs(\()p Fr(s;)17 b(k)s Fs(\))g Fr(dx)3213 3781 y Fm(\025)3298 3921 y Fr(;)456 4202 y Fv(then)37 b(the)h(e)o (xpressions)f(in)h(the)f(brack)o(ets)i(in)e(the)h(r)-5 b(.h.s.)37 b(do)h(not)g(depend)g(on)f Fr(r)k Fv(for)456 4318 y Fr(r)30 b Ft(\024)e Fs(1)p Fv(.)j(From)24 b(\(10.10\))h(it)f (follo)n(ws)f(that)620 4719 y Fr(A)p Fs(\()p Fr(k)s Fs(\))28 b(=)f Fr(P)36 b Ft(\000)1206 4651 y Fs(1)p 1162 4696 137 4 v 1162 4787 a(2)p Fr(ik)1325 4583 y Fm(Z)1425 4610 y Fq(+)p Fl(1)1381 4809 y(\0001)1571 4719 y Fr(V)1628 4734 y Fl(\003)1667 4719 y Fs(\()p Fr(s)p Fs(\))17 b Fr(ds)22 b Ft(\000)2078 4651 y Fs(1)p 2034 4696 V 2034 4787 a(2)p Fr(ik)2197 4583 y Fm(Z)2296 4610 y Fq(+)p Fl(1)2252 4809 y(\0001)2443 4719 y Fr(V)2500 4734 y Fl(\003)2539 4719 y Fs(\()p Fr(s)p Fs(\))p Fr(X)8 b Fs(\()p Fr(s;)17 b(k)s Fs(\))g Fr(ds)g(;)456 4522 y Fv(\(10.11\))614 5119 y Fr(B)5 b Fs(\()p Fr(k)s Fs(\))28 b(=)1008 5051 y(1)p 964 5096 V 964 5187 a(2)p Fr(ik)1127 4983 y Fm(Z)1227 5009 y Fq(+)p Fl(1)1182 5209 y(\0001)1373 5119 y Fr(e)1418 5078 y Fq(2)p Fo(ik)r(s)1553 5119 y Fr(V)1610 5134 y Fl(\003)1649 5119 y Fs(\()p Fr(s)p Fs(\))17 b Fr(ds)k Fs(+)2058 5051 y(1)p 2014 5096 V 2014 5187 a(2)p Fr(ik)2177 4983 y Fm(Z)2276 5009 y Fq(+)p Fl(1)2232 5209 y(\0001)2423 5119 y Fr(e)2468 5078 y Fq(2)p Fo(ik)r(s)2603 5119 y Fr(V)2660 5134 y Fl(\003)2699 5119 y Fs(\()p Fr(s)p Fs(\))p Fr(X)8 b Fs(\()p Fr(s;)17 b(k)s Fs(\))g Fr(ds)g(:)456 4922 y Fv(\(10.12\))p eop %%Page: 24 24 24 23 bop 456 251 a Fj(24)808 b(LAPTEV)-10 b(,)17 b(N)m(ABOK)n(O)i(AND) f(SAFR)m(ONO)l(V)555 458 y Fv(Recall)28 b(that)e(supp)1233 433 y Fs(~)1218 458 y Fr(V)53 b Ft(\032)32 b Fs(\(1)p Fr(;)17 b Ft(1)p Fs(\))p Fv(.)36 b(Thus)26 b(for)h(suf)n(\002ciently)f (lar)n(ge)h Ft(j)p Fr(k)s Ft(j)f Fv(the)h(smooth-)456 575 y(ness)d(of)h Fr(V)46 b Fv(and)25 b(\(10.9\))g(imply)620 787 y Fm(\015)620 847 y(\015)620 907 y(\015)620 966 y(\015)675 931 y Fr(A)p Fs(\()p Fr(k)s Fs(\))d Ft(\000)h Fr(P)36 b Fs(+)1250 864 y(1)p 1207 909 137 4 v 1207 1000 a(2)p Fr(ik)1369 796 y Fm(Z)1469 822 y Fq(+)p Fl(1)1425 1021 y(\0001)1615 931 y Fr(V)1672 946 y Fl(\003)1711 931 y Fs(\()p Fr(s)p Fs(\))p Fr(ds)1930 787 y Fm(\015)1930 847 y(\015)1930 907 y(\015)1930 966 y(\015)2013 931 y Ft(\024)28 b Fr(C)2188 946 y Fq(4)2227 931 y Fs(\()p Fr(V)2322 946 y Fl(\003)2362 931 y Fr(;)17 b(n)p Fs(\))p Ft(j)p Fr(k)s Ft(j)2612 890 y Fl(\000)p Fq(2)2705 931 y Fr(;)117 b Fv(Im)24 b Fr(k)31 b Ft(\025)d Fs(0)17 b Fr(;)456 735 y Fv(\(10.13\))1463 1068 y Fm(\015)1463 1128 y(\015)1519 1153 y Fr(e)1564 1112 y Fl(\000)p Fq(2)p Fo(ik)1721 1153 y Fr(B)5 b Fs(\()p Fr(k)s Fs(\))1930 1068 y Fm(\015)1930 1128 y(\015)2013 1153 y Ft(\024)28 b Fr(C)2188 1168 y Fq(5)2227 1153 y Fs(\()p Fr(V)2322 1168 y Fl(\003)2362 1153 y Fr(;)17 b(n)p Fs(\))p Ft(j)p Fr(k)s Ft(j)2612 1112 y Fl(\000)p Fq(2)2722 1153 y Fr(;)116 b Fv(Im)25 b Fr(k)30 b Ft(\025)f Fs(0)17 b Fr(:)-2825 b Fv(\(10.14\))555 1336 y(Note)27 b(that)f(from)h(\(10.2\))o(,)h (\(10.13\))e(and)h(\(10.14\))f(we)h(no)n(w)f(obtain)g(that)g Fr(a)3113 1351 y Fo(")3150 1336 y Fs(\()p Fr(k)s Fs(\))h Fv(is)f(a)456 1452 y(meromorphic)i(function)g(in)h(a)h(neighborhood)e (of)h(zero)h(and)f Ft(j)p Fr(a)2712 1467 y Fo(")2749 1452 y Fs(\()p Fr(k)s Fs(\))p Ft(j)g Fv(tends)f(to)h(1)g(as)456 1569 y Fr(O)s Fs(\(1)p Fr(=)p Ft(j)p Fr(k)s Ft(j)780 1533 y Fq(2)817 1569 y Fs(\))c Fv(when)g Fr(k)31 b Ft(!)c(\0061)p Fv(.)456 1746 y Fn(Ac)n(knowledgments.)44 b Fv(A.L)29 b(and)g(O.S.)h(are)g(grateful)g(for)f(the)h(partial)f(support)f(of)i (the)456 1862 y(ESF)39 b(European)f(programme)f(SPECT)-7 b(.)39 b(S.N.)f(w)o(ould)g(lik)o(e)f(to)h(thank)g(Gustafsson)456 1979 y(foundation)19 b(which)h(has)g(allo)n(wed)g(him)g(to)g(spend)g (one)g(month)f(at)i(the)f(Ro)o(yal)h(Institute)456 2095 y(of)f(T)-7 b(echnology)19 b(in)h(Stockholm.)28 b(This)19 b(research)j(w)o(as)e(also)g(partly)f(supported)h(by)g(the)456 2211 y(KBN)25 b(grant)f(5,)h(PO3A/026/21.)k(g1925l.)1671 2468 y(R)t Fu(E)t(F)t(E)t(R)t(E)t(N)t(C)5 b(E)g(S)497 2625 y FA([1])40 b(Z.)27 b(S.)g(Agrano)o(vich)c(and)j(V)-11 b(.)27 b(A.)f(Marchenk)o(o,)g Fk(Re-establishment)e(of)j(the)f (potential)f(fr)l(om)i(the)635 2725 y(scattering)22 b(matrix)f(for)h(a) g(system)g(of)g(dif)o(fer)m(ential)e(equations)p FA(,)g(\(Russian\))h (Dokl.)g(Akad.)g(Nauk)635 2825 y(SSSR)h(\(N.S.\))e Fb(113)f FA(\(1957\),)f(951-954.)497 2924 y([2])40 b(J.)19 b(A)-6 b(vron,)16 b(I.)i(Herbst)g(and)f(B.)i(Simon,)e Fk(Sc)o(hr)1907 2925 y(\250)1900 2924 y(oding)o(er)f(oper)o(ator)o(s)h(with)h(ma)o (gnetic)f(\002elds.)h(I.)f(Gen-)635 3024 y(er)o(al)k(inter)o(actions)p FA(,)e(Duk)o(e)g(Math.)h(J.)h Fb(45)f FA(\(1978\),)e(847-883.)497 3123 y([3])40 b(H.)24 b(Behnck)o(e,)g Fk(Absolute)e(continuity)g(of)i (Hamiltonians)e(with)i(von)f(Neumann-W)-5 b(igner)22 b(poten-)635 3223 y(tials.)f FA(Proc.)f(Amer)-5 b(.)20 b(Math.)f(Soc.)h Fb(111)g FA(\(1991\),)e(373-384.)497 3323 y([4])40 b(H.)24 b(Behnck)o(e,)g Fk(Absolute)e(continuity)g(of)i (Hamiltonians)e(with)i(von)f(Neumann-W)-5 b(igner)22 b(poten-)635 3422 y(tials.)f(II.)f FA(Manuscripta)f(Math.)g Fb(71)h FA(\(1991\),)e(163-181.)497 3522 y([5])40 b(M.Sh.)23 b(Birman,)f Fk(P)-7 b(erturbations)22 b(of)h(the)f(continuous)f (spectrum)h(of)h(a)g(singular)e(elliptic)i(oper)n(-)635 3622 y(ator)i(by)f(varying)g(the)h(boundary)d(and)h(the)i(boundary)d (conditions)p FA(,)i(\(Russian.)h(English)e(sum-)635 3721 y(mary\))c(V)-9 b(estnik)20 b(Leningrad.)e(Uni)n(v)-5 b(.,)19 b Fb(17)h FA(\(1962\),)d(22-55.)497 3821 y([6])40 b(M.Sh.)23 b(Birman,)g Fk(Discr)m(ete)g(Spectrum)g(in)g(the)f(gaps)h (of)g(a)g(continuous)e(one)h(for)h(perturbations)635 3920 y(with)e(lar)m(g)o(e)f(coupling)f(constants)p FA(,)g(Adv)n(ances)g (in)h(So)o(viet)g(Mathematics,)f Fb(7)i FA(\(1991\),)c(57-73.)497 4020 y([7])40 b(M.)26 b(Christ,)g(A.)f(Kisele)n(v)g(and)g(C.)g (Remling,)h Fk(The)f(absolutely)f(continuous)f(spectrum)i(of)g(one-)635 4120 y(dimensional)37 b(Sc)o(hr)1239 4121 y(\250)1232 4120 y(oding)o(er)e(oper)o(ator)o(s)j(with)g(decaying)e(potentials)p FA(.)h(Math.)g(Res.)i(Lett.)f Fb(4)635 4219 y FA(\(1997\),)18 b(719-723.)497 4319 y([8])40 b(M.)h(Christ)g(and)f(A.)h(Kisele)n(v)-5 b(,)45 b Fk(Absolutely)39 b(continuous)g(spectrum)h(for)h (one-dimensional)635 4419 y(Sc)o(hr)794 4420 y(\250)787 4419 y(oding)o(er)18 b(oper)o(ator)o(s)h(with)h(slowly)h(decaying)d (potentials:)24 b(some)19 b(optimal)g(r)m(esults)p FA(.)i(J.)f(Am.)635 4518 y(Math.)g(Soc.)g Fb(11)g FA(\(1998\),)e(771-797.)497 4618 y([9])40 b(V)-11 b(.G.)22 b(Deich,)e(E.L.)h(K)m(orotjae)n(v)f(and) g(D.R.)h(Y)-8 b(af)o(ae)n(v)j(,)20 b Fk(P)-7 b(otential)20 b(scattering)h(with)h(allowance)e(for)635 4717 y(spatial)g(anisotr)l (opy)p FA(,)f(\(Russian\))h(Dokl.)g(Akad.)f(Nauk)h(SSSR)h Fb(235)f FA(\(1977\),)d(749-752.)456 4817 y([10])39 b(P)-9 b(.)28 b(Deift)g(and)f(R.)h(Killip,)i Fk(On)d(the)h(absolutely)e (continuous)g(spectrum)h(of)h(one)f(-dimensional)635 4917 y(Sc)o(hr)794 4918 y(\250)787 4917 y(oding)o(er)20 b(oper)o(ator)o(s)g(with)i(squar)m(e)f(summable)f(potentials)p FA(,)h(Commun.)e(Math.)i(Phys.)g Fb(203)635 5016 y FA(\(1999\),)d (341-347.)456 5116 y([11])39 b(R.)19 b(Killip)g(and)f(B.)h(Simon,)e Fk(Sum)h(rules)h(for)g(J)m(acobi)e(matrices)i(and)e(their)h (applications)f(to)h(spec-)635 5216 y(tr)o(al)j(theory)p FA(,)f(Annals)f(of)h(Math.,)g(to)g(appear)-5 b(.)p eop %%Page: 25 25 25 24 bop 1246 251 a Fj(ABSOLUTEL)-7 b(Y)22 b(CONTINUOUS)h(SPECTR)m(UM) 714 b(25)456 450 y FA([12])39 b(R.)33 b(Killip,)i Fk(P)-7 b(erturbations)31 b(of)h(one-dimensional)d(Sc)o(hr)2323 451 y(\250)2316 450 y(oding)o(er)h(oper)o(ator)o(s)h(pr)m(eserving)h (the)635 550 y(absolutely)19 b(continuous)g(spectrum)p FA(,)g(Int.)h(Math.)g(Res.)h(Not.)f(\(2002\),)e(2029\2262061.)456 649 y([13])39 b(A.)31 b(Kisele)n(v)-5 b(,)32 b(Y)-11 b(.)31 b(Last)g(and)e(B.)i(Simon)f Fk(Modi\002ed)f(Pr)2243 650 y(\250)2236 649 y(ufer)h(and)f(EFGP)h(tr)o(ansforms)h(and)e(the)635 749 y(spectr)o(al)17 b(analysis)f(of)g(one-dimensional)e(Sc)o(hr)2019 750 y(\250)2012 749 y(oding)o(er)g(oper)o(ator)o(s)p FA(,)i(Comm.)g(Math.)g(Phys.)g Fb(194)635 849 y FA(\(1998\),)i(1-45.) 456 948 y([14])39 b(P)-9 b(.)21 b(K)m(oosis,)f Fk(The)g(lo)o(garithmic) g(inte)m(gr)o(al)f(I)p FA(,)h(Cambridge)f(uni)n(v)o(ersity)f(press)j (\(1988\).)456 1048 y([15])39 b(A.)33 b(Lapte)n(v)-5 b(,)34 b(S.)e(Nabok)o(o)f(and)h(O.Safrono)o(v)-5 b(,)32 b Fk(A)g(Sze)m(g)2251 1049 y(\005)2244 1048 y(o)f(condition)g(for)i(a)f (multidimensional)635 1147 y(Sc)o(hr)794 1148 y(\250)787 1147 y(oding)o(er)19 b(oper)o(ator)p FA(,)f(Preprint.)456 1247 y([16])39 b(A.)26 b(Lapte)n(v)f(and)g(T)-6 b(.)26 b(W)-7 b(eidl,)28 b Fk(Sharp)c(Lieb-Thirring)h(inequalities)g(in)h (high)f(dimensions)p FA(,)h(Acta)635 1347 y(Mathematica)20 b Fb(184)f FA(\(2000\),)f(87-111.)456 1446 y([17])39 b(E.H.)29 b(Lieb)g(and)g(W)-8 b(.)31 b(Thirring,)e Fk(Inequalities)f (for)i(the)f(moments)g(of)h(the)f(eig)o(en)m(values)e(of)j(the)635 1546 y(Sc)o(hr)794 1547 y(\250)787 1546 y(oding)o(er)18 b(Hamiltonian)g(and)g(their)i(r)m(elation)f(to)g(Sobole)o(v)f (inequalities)p FA(,)g(Studies)i(in)f(Math.)635 1646 y(Phys.,)h(Essays)g(in)h(Honor)e(of)h(V)-9 b(alentine)19 b(Bar)o(gmann,)f(Princeton,)h(\(1976\))f(269-303.)456 1745 y([18])39 b(V)-11 b(.)35 b(Maz'ya,)h Fk(Sobole)o(v)c(Spaces)p FA(,)k(Springer)n(-V)-9 b(erlag,)34 b(Berlin)g(Heidelber)o(g)e(Ne)n(w)i (Y)-9 b(ork)33 b(T)-7 b(okio)635 1845 y(\(1985\).)456 1944 y([19])39 b(C.)33 b(Remling,)i Fk(The)d(absolutely)f(continuous)f (spectrum)i(of)h(one-dimensional)c(Sc)o(hr)3195 1945 y(\250)3188 1944 y(oding)o(er)635 2044 y(oper)o(ator)o(s)20 b(with)h(decaying)d(potentials)p FA(.)h(Comm.)h(Math.)g(Phys.)f Fb(193)g FA(\(1998\),)f(151-170.)456 2144 y([20])39 b(O.)24 b(Safrono)o(v)-5 b(,)22 b Fk(The)h(spectr)o(al)g(measur)m(e)h(of)f(a)h (J)m(acobi)e(matrix)i(in)g(terms)h(of)e(the)h(F)-9 b(ourier)23 b(tr)o(ans-)635 2243 y(form)e(of)f(the)h(perturbation)p FA(,)d(Preprint.)456 2343 y([21])39 b(M.)16 b(Reed)f(and)g(B.)h(Simon,) g Fk(Methods)f(of)g(modern)g(mathematical)f(physics)p FA(,)i Fb(3)p FA(,)g(Academic)e(Press,)635 2443 y(San)21 b(Francisco,)e(London)f(\(1978\).)456 2542 y([22])39 b(G.)32 b(Sze)o(g)7 b(\005)-35 b(o,)34 b Fk(Beitr)1179 2543 y(\250)1172 2542 y(ag)o(e)d(zue)h(Theorie)f(der)h(T)-8 b(oeplitzsc)o(hen)31 b(F)-9 b(ormen,)34 b(II)p FA(,)d(Math.)g(Z.)h Fb(9)f FA(\(1921\),)635 2642 y(167-190.)456 2742 y([23])39 b(G.)c(Sze)o(g)7 b(\005)-35 b(o,)36 b Fk(Ortho)o(gonal)d(P)-7 b(olynomials)p FA(,)36 b(4th)e(edition.)f(American)g(Mathematical)g (Society)-5 b(,)635 2841 y(Colloquium)24 b(Publications,)h(V)-11 b(ol.)25 b(XXIII.)f(American)g(Mathematical)g(Society)-5 b(,)25 b(Pro)o(vidence,)635 2941 y(R.I.,)20 b(1975.)456 3040 y([24])39 b(B.)26 b(Simon,)g(D.)f(Damanik)f(and)h(D.)g (Hundertmark,)e Fk(Bound)h(states)h(and)g(the)g(Sze)m(g)3072 3041 y(\005)3065 3040 y(o)f(condition)635 3140 y(for)d(J)m(acobi)e (matrices)i(and)e(Sc)o(hr)1603 3141 y(\250)1596 3140 y(oding)o(er)f(oper)o(ator)o(s)p FA(,)i(J.)g(Funct.)g(Anal.,)g(to)g (appear)-5 b(.)456 3240 y([25])39 b(B.)23 b(Simon,)e Fk(Sc)o(hr)1153 3241 y(\250)1146 3240 y(oding)o(er)f(oper)o(ator)o(s)h (in)h(the)g(twentieth)g(century)p FA(,)g(J.)g(Math.)g(Phys.)f Fb(41)g FA(\(2000\),)635 3339 y(3523-3555.)456 3439 y([26])39 b(B.)23 b(Simon,)e Fk(Some)g(Sc)o(hr)1355 3440 y(\250)1348 3439 y(oding)o(er)e(oper)o(ator)o(s)i(with)h(dense)f(point)g(spectrum)p FA(,)h(Proc.)f(Am.)g(Math.)635 3539 y(Soc.)f Fb(125)g FA(\(1997\),)e(203-208.)456 3638 y([27])39 b(B.)31 b(Simon,)g Fk(T)-5 b(r)o(ace)30 b(ideals)g(and)f(their)h(applications)p FA(,)g(London.)d(Math.)i(Soc.,)j(Lecture)d(Note)635 3738 y(Series)21 b Fb(35)f FA(\(1979\).)456 3837 y([28])39 b(B.)19 b(Simon)f(and)g(A.)h(Zlatos,)f Fk(Sum)g(rules)h(and)f(the)g (Sze)m(g)2220 3838 y(\005)2213 3837 y(o)f(condition)g(for)i(ortho)o (gonal)d(polynomi-)635 3937 y(als)21 b(on)f(the)g(r)m(eal)g(line)p FA(,)h(Comm.)e(Math.)h(Phys.,)f(to)i(appear)-5 b(.)456 4037 y([29])39 b(M.)26 b(Skrigano)o(v)-5 b(,)24 b Fk(The)i(eig)o(en)m (values)e(of)i(the)g(Sc)o(hr)2082 4038 y(\250)2075 4037 y(oding)o(er)e(oper)o(ator)g(that)i(ar)m(e)g(located)f(on)g(the)635 4136 y(continuous)g(spectrum)p FA(,)k(\(Russian\))d(Boundary)f(v)n (alue)i(problems)e(of)i(mathematical)f(physics)635 4236 y(and)32 b(related)f(questions)g(in)h(the)g(theory)f(of)g(functions,)j (7.)d(Zap.)h(Nau)5 b(\020)-33 b(cn.)31 b(Sem.)g(Leningrad.)635 4336 y(Otdel.)20 b(Mat.)h(Inst.)f(Steklo)o(v)-5 b(.)18 b(\(LOMI\))h Fb(38)h FA(\(1973\),)e(149-152.)456 4435 y([30])39 b(D.B.)e(Pearson,)i Fk(Singular)34 b(continuous)g(measur)m (es)i(in)g(scattering)f(theory)p FA(,)k(Comm.)c(Math.)635 4535 y(Phys.)20 b Fb(60)g FA(\(1978\),)e(13-36.)456 4634 y([31])39 b(J.)18 b(v)n(on)f(Neumann)f(and)h(E.P)-9 b(.)17 b(W)m(igner)m(,)1776 4617 y Fk(\250)1755 4634 y(Uber)g(merkw)2171 4635 y(\250)2164 4634 y(ur)m(dig)o(e)h(diskr)m(ete)f(Eig)o(enwerte)p FA(,)h(Z.Phys.)e Fb(30)635 4734 y FA(\(1929\),)i(465-467.)456 4834 y([32])39 b(D.)27 b(Y)-8 b(af)o(ae)n(v)j(,)26 b Fk(Mathematical)f(scattering)h(theory)-5 b(.)26 b(Gener)o(al)g(theory)p FA(.)f(T)m(ranslations)h(of)g(Mathe-)635 4933 y(matical)k(Monographs,)f Fb(105)p FA(.)f(American)g(Mathematical)h(Society)-5 b(,)30 b(Pro)o(vidence,)g(RI,)g(1992.)635 5033 y(x+341)19 b(pp.)p eop %%Page: 26 26 26 25 bop 456 251 a Fj(26)808 b(LAPTEV)-10 b(,)17 b(N)m(ABOK)n(O)i(AND) f(SAFR)m(ONO)l(V)555 450 y Fk(E-mail)135 b(addr)m(ess)p FA(:)256 b Fa(laptev@math.kth.se,)46 b(naboko@math.uab.edu,)456 550 y(safronov@math.kth.se)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0308251001228--