Content-Type: multipart/mixed; boundary="-------------0211282240242" This is a multi-part message in MIME format. ---------------0211282240242 Content-Type: text/plain; name="02-495.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-495.keywords" Aharonov-Bohm effect, Dirac-Weyl operator, boundary condition, spectral analysis ---------------0211282240242 Content-Type: application/postscript; name="preprint.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="preprint.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86e Copyright 2001 Radical Eye Software %%Title: preprint.dvi %%Pages: 14 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX12 CMR12 CMR10 CMMI10 CMBX10 CMMI8 CMSY8 CMR8 %%+ CMMI12 CMEX10 CMSY10 MSBM10 CMR6 CMTI12 CMTI10 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips.nonja preprint.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.11.29:1300 %%BeginProcSet: texc.pro %! 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Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 66 /B put dup 71 /G put dup 76 /L put dup 80 /P put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 121 /y put dup 122 /z put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 55 /seven put dup 56 /eight put dup 58 /colon put dup 66 /B put dup 67 /C put dup 73 /I put dup 80 /P put dup 82 /R put dup 84 /T put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-36 -251 1103 750}readonly def /UniqueID 5000829 def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 50 /two put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 78 /N put dup 82 /R put dup 90 /Z put dup 126 /planckover2pi1 put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6A66A4000A13D5F68BFF326D 1D432B0D064B56C598F4338C319309181D78E1629A31ECA5DD8536379B03C383 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%%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 2 /multiply put dup 6 /plusminus put dup 7 /minusplus put dup 8 /circleplus put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 27 /propersuperset put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 72 /H put dup 76 /L put dup 78 /N put dup 83 /S put dup 91 /union put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 107 /bardbl put dup 110 /backslash put dup 112 /radical put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 56 /bracelefttp put dup 57 /bracerighttp put dup 58 /braceleftbt put dup 59 /bracerightbt put dup 60 /braceleftmid put dup 61 /bracerightmid put dup 62 /braceex put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 66 /parenleftex put dup 67 /parenrightex put dup 80 /summationtext put dup 82 /integraltext put dup 91 /uniondisplay put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 112 /radicalbig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 18 /theta put dup 25 /pi put dup 27 /sigma put dup 28 /tau put dup 32 /psi put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 76 /L put dup 77 /M put dup 79 /O put dup 80 /P put dup 81 /Q put dup 84 /T put dup 85 /U put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 113 /q put dup 114 /r put dup 116 /t put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueID 5087386 def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 6 /plusminus put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 72 /H put dup 76 /L put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 18 /theta put dup 25 /pi put dup 27 /sigma put dup 28 /tau put dup 59 /comma put dup 61 /slash put dup 66 /B put dup 77 /M put dup 101 /e put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 114 /r put dup 116 /t put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 45 /hyphen put dup 46 /period put dup 51 /three put dup 52 /four put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 88 /X put dup 90 /Z put dup 99 /c put dup 100 /d put dup 101 /e put dup 103 /g put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 116 /t put dup 119 /w put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 14 /delta put dup 72 /H put dup 81 /Q put dup 114 /r put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E394A533A081C36D456A09920001A3D2199583EB9B84B4DEE08E3D12939E321 990CD249827D9648574955F61BAAA11263A91B6C3D47A5190165B0C25ABF6D3E 6EC187E4B05182126BB0D0323D943170B795255260F9FD25F2248D04F45DFBFB DEF7FF8B19BFEF637B210018AE02572B389B3F76282BEB29CC301905D388C721 59616893E774413F48DE0B408BC66DCE3FE17CB9F84D205839D58014D6A88823 D9320AE93AF96D97A02C4D5A2BB2B8C7925C4578003959C46E3CE1A2F0EAC4BF 8B9B325E46435BDE60BC54D72BC8ACB5C0A34413AC87045DC7B84646A324B808 6FD8E34217213E131C3B1510415CE45420688ED9C1D27890EC68BD7C1235FAF9 1DAB3A369DD2FC3BE5CF9655C7B7EDA7361D7E05E5831B6B8E2EEC542A7B38EE 03BE4BAC6079D038ACB3C7C916279764547C2D51976BABA94BA9866D79F13909 95AA39B0F03103A07CBDF441B8C5669F729020AF284B7FF52A29C6255FCAACF1 74109050FBA2602E72593FBCBFC26E726EE4AEF97B7632BC4F5F353B5C67FED2 3EA752A4A57B8F7FEFF1D7341D895F0A3A0BE1D8E3391970457A967EFF84F6D8 47750B1145B8CC5BD96EE7AA99DDC9E06939E383BDA41175233D58AD263EBF19 AFC0E2F840512D321166547B306C592B8A01E1FA2564B9A26DAC14256414E4C8 42616728D918C74D13C349F4186EC7B9708B86467425A6FDB3A396562F7EE4D8 40B43621744CF8A23A6E532649B66C2A0002DD04F8F39618E4F572819DD34837 B5A08E643FDCA1505AF6A1FA3DDFD1FA758013CAED8ACDDBBB334D664DFF5B53 9560176671A33FC55340CD04C2DE46FD812FC2711A85F27B177A8A2CD86CE84B D5D79C949049EEE98F15937C0B281E1BE7055482E990965C1D41D297814C29BA D54A202CC557F1C870A2258A95A1714C05A9B3CC3BC6415FB364975DC52F13AA 266EE88B76058F6206D0479063E1E109555C62BE0380E7CB35E4A434B576CA99 A45E351DF4BB0359C9602DF281E033999171F2BC9E3D55646BC3FE129CF8431B A0B038FC67DBEC3A2D24091A404604EDC5D5D7D8C0C0DB3811A2E98338570B86 5A909ACB993B61FBC5144088A505A2A1E76C51A88B4F9C5B3C384593493F20D5 BA3F1C3D9B470264ECDF5585FE4304CD499531D89D907CE7B9A53D90E4727A78 DAD01FBE1BF024BA7E8A2BA30D8E5CB56CC45642AFC623E97C81133FC34013E8 8E317E938DDC3E97B2E14B3B0DB0617EC88CF7AE9C8A2381C625DE34510935E6 EFAB991F9369A55CAE5EA8519EB72151D104EA1A48EE1CFF3A9A0687DFADA050 9A6F548BE35DC93A1F7EDE928BC2E05D53A3CA1888C0C72503B4779264CCB4DA 8826032919CF04562EAFA2C3620FAD1D9CCB3CBA9CB212E38045770073F88216 77C279CB2583DAE1F24916C786045E4EFEB4D22B1F47D20871CDB9992BA898C4 2F10FC294E3C127BE6F22CA83967D82C449AECDAB232FE238904D4B5D20FC09B 1A49CE1CC08E52F8CD6163D284EC0F89CACFC3D1CD26F132A750A755C2AB00B0 576AFC08A037C4A6F051987CB303C5A4734D17419D964CF411320275E19C3CC2 9910C2F58D514158B8173817A11BD224D99AC457C8C1003146939AF69321AA32 A92F213F1BE53C3D57FD72F0F2FB2791290BC64E2D5B0020E40ACA29D3DFB59A EA5427A7B52542C500EA1E7D2508588820C98AAAE89C231563EDBE246C6CDE50 8F0B81153F68F6F3AB5817CBAD2285041A10C47841080F2E3DE81167070D680D 9AE2C0D622539562B02116DC76A59E805854DCE20FF08DE5602C43FB2F117AD4 87F1936BDBA1C40CF53144A9CF3D4D8D08E0A51835F6DC322C2CD5910447F5C8 8D09921F65B94CC7BD028B3F7AD4016AA59252B2D4205E39DEEA8CC7310409F4 AE8845A65BF86600FF4AD3BFD5E4B5108D55242051E6CC0C0D567761C97FEF1B 5FBE9626489025BE93135D8E7C5CC8A5480EB506083D4A98EC2EB6202A45A583 81928B8C4EF612D7128AFD165ED856685C33298D0ACB2EB70425A2C7136076EA 44C009F1B45AAFE53145F333C7E9EB4A8C4D6A8A4BB4A13CED4C05147876B9BF 0B683FE1368DDCFD5A5435A8CC05465F34E4FD209E645352CD99092324E8E39E E40DCEDC3197853198184E32372D1B2943E5A1F0F1FF67AFADD109A3FC9218CF B8508B710338818B68122DF3B5F2FFE8BA4D002691996623461357F7FAF1D619 01CF1FAC29A49E2F50E8E39097A9D34DB80EDC9469938643F3187BE1F05B3B77 89FE0D2CFED6D490485D60051DA14C1BAFD9A1F50079BA3612BAC16824BD7B07 AD302798EFBBCA9FF1A200D28888A24ECF 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 9 /Psi put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 16 /dotlessi put dup 19 /acute put dup 20 /caron put dup 24 /cedilla put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 91 /bracketleft put dup 93 /bracketright put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put 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b(L)673 5292 y Fj(2)748 5329 y Fp(with)37 b Fi(D)s Fp(\()p Fi(Q)1173 5344 y Fl(m)1240 5329 y Fp(\))d(=)g Fi(C)1499 5292 y Fk(1)1492 5353 y Fj(0)1573 5329 y Fp(\(\(0)p Fi(;)17 b Fg(1)p Fp(\)\))24 b Fg(\010)h Fi(C)2121 5292 y Fk(1)2114 5353 y Fj(0)2196 5329 y Fp(\(\(0)p Fi(;)17 b Fg(1)p Fp(\)\).)54 b(W)-8 b(e)36 b(remark)i(that)e Fi(q)3393 5292 y Fk(\000)3389 5353 y Fl(m)3492 5329 y Fp(and)g Fi(q)3732 5287 y Fj(+)3728 5353 y Fl(m)p Fj(+1)0 5503 y Fp(are)d(formal)f(adjoin)m(t)g(as)h(op)s (erators)g(acting)f(on)h Fg(L)1848 5467 y Fj(2)1887 5503 y Fp(.)p eop %%Page: 4 4 4 3 bop 3837 -208 a Fp(4)125 83 y(W)-8 b(e)39 b(note)g(that)g(the)h (sequence)h(of)e(self-adjoin)m(t)g(extensions)i(of)e Fi(Q)2622 98 y Fl(m)2689 83 y Fp('s,)i Fg(f)p Fi(Q)2949 47 y Fl(e)2949 108 y(m)3016 83 y Fp(;)17 b Fi(m)38 b Fg(2)i Ff(Z)p Fg(g)p Fp(,)h(are)e(corre-)0 257 y(sp)s(onding)31 b(to)g(a)f(self-adjoin)m(t)h(extension)i Fi(Q)1617 221 y Fl(e)1685 257 y Fp(of)d Fi(Q)p Fp(:)43 b(Let)31 b Fi(P)2177 272 y Fl(m)2274 257 y Fp(b)s(e)g(the)g(pro)5 b(jection)31 b(op)s(erator)g(from)f Fg(H)3736 272 y Fj(total)0 432 y Fp(to)i Fg(H)203 447 y Fl(m)302 432 y Fp(giv)m(en)i(b)m(y)832 763 y Fi(P)895 778 y Fl(m)979 563 y Fh(0)979 742 y(@)1066 686 y Fi(f)11 b Fp(\()p Fi(r)m(;)17 b(\022)s Fp(\))1070 860 y Fi(g)t Fp(\()p Fi(r)m(;)g(\022)s Fp(\))1333 563 y Fh(1)1333 742 y(A)1448 763 y Fp(=)1552 563 y Fh(0)1552 742 y(@)1707 686 y Fp(\(2)p Fi(\031)t Fp(\))1891 650 y Fk(\000)p Fj(1)2002 606 y Fh(R)2068 632 y Fj(2)p Fl(\031)2049 721 y Fj(0)2167 686 y Fi(f)11 b Fp(\()p Fi(r)m(;)17 b(t)p Fp(\))g Fi(e)2484 650 y Fk(\000)p Fj(i)p Fl(m)p Fj(\()p Fl(t)p Fk(\000)p Fl(\022)r Fj(\))2812 686 y Fi(dt)1639 860 y Fp(\(2)p Fi(\031)t Fp(\))1823 824 y Fk(\000)p Fj(1)1933 780 y Fh(R)2000 806 y Fj(2)p Fl(\031)1981 895 y Fj(0)2099 860 y Fi(g)t Fp(\()p Fi(r)m(;)g(t)p Fp(\))g Fi(e)2408 824 y Fk(\000)p Fj(i\()p Fl(m)p Fj(+1\)\()p Fl(t)p Fk(\000)p Fl(\022)r Fj(\))2880 860 y Fi(dt)2966 563 y Fh(1)2966 742 y(A)0 1095 y Fp(and)58 b Fi(U)68 b Fp(b)s(e)57 b(the)h(natural)g (unitary)g(op)s(erator)f(from)g Fg(H)2192 1110 y Fl(m)2316 1095 y Fp(to)g Fg(H)q Fp(.)119 b(Then)59 b(w)m(e)f(ha)m(v)m(e)h(that)e Fi(Q)3702 1059 y Fl(e)3810 1095 y Fp(=)0 1269 y Fg(\010)77 1284 y Fl(m)p Fk(2)p Fe(Z)235 1269 y Fp(\()p Fi(U)10 b(P)412 1284 y Fl(m)479 1269 y Fp(\))517 1233 y Fk(\000)p Fj(1)612 1269 y Fi(Q)689 1233 y Fl(e)689 1294 y(m)756 1269 y Fi(U)g(P)895 1284 y Fl(m)962 1269 y Fp(,)32 b(where)i(\()p Fi(U)10 b(P)1480 1284 y Fl(m)1547 1269 y Fp(\))1585 1233 y Fk(\000)p Fj(1)1712 1269 y Fp(stands)33 b(for)f(the)h(lifting)g(from) f Fg(H)i Fp(to)e Fg(H)3169 1284 y Fl(m)3236 1269 y Fp(.)125 1443 y(In)45 b(the)g(rest)h(of)e(this)i(section,)j(w)m(e)d(examine)g (the)f(generalized)i(eigenfunctions)f(of)f Fi(Q)3433 1458 y Fl(m)3500 1443 y Fp(,)j(i.e.,)g(w)m(e)0 1618 y(\014nd)39 b(the)f(v)m(ectors)i(\011)d Fg(2)h Fi(D)s Fp(\()p Fi(Q)1127 1582 y Fk(\003)1127 1642 y Fl(m)1193 1618 y Fp(\))g(whic)m(h)i (satis\014es)g(that)e(there)h(exists)h(a)e(constan)m(t)h Fi(E)k Fg(2)38 b Ff(C)g Fp(suc)m(h)i(that)0 1792 y Fg(h)p Fi( )t(;)17 b(Q)227 1756 y Fk(\003)227 1817 y Fl(m)293 1792 y Fp(\011)p Fg(i)408 1807 y Fk(H)500 1792 y Fp(=)27 b Fi(E)6 b Fg(h)p Fi( )t(;)17 b Fp(\011)p Fg(i)946 1807 y Fk(H)1042 1792 y Fp(for)32 b(all)h Fi( )e Fg(2)e Fi(D)s Fp(\()p Fi(Q)1715 1807 y Fl(m)1781 1792 y Fp(\).)43 b(Let)33 b Fi(k)e Fp(=)d Fi(m)22 b Fp(+)g Fi(\013)q Fp(.)43 b(W)-8 b(e)33 b(ha)m(v)m(e)h(that)481 2129 y Fi(Q)558 2088 y Fj(2)558 2154 y Fl(m)653 2129 y Fp(=)756 1929 y Fh(0)756 2108 y(@)843 2052 y Fi(L)909 2067 y Fl(k)975 2052 y Fp(+)22 b Fi(B)230 b Fp(0)973 2226 y(0)180 b Fi(L)1268 2241 y Fl(k)r Fj(+1)1423 2226 y Fg(\000)23 b Fi(B)1602 1929 y Fh(1)1602 2108 y(A)1738 2129 y Fp(with)33 b Fi(L)2026 2144 y Fl(k)2097 2129 y Fp(=)27 b Fg(\000)2287 2062 y Fp(1)p 2287 2106 49 4 v 2288 2198 a Fi(r)2347 2129 y(@)2398 2144 y Fl(r)2436 2129 y Fi(r)s(@)2534 2144 y Fl(r)2594 2129 y Fp(+)2721 2062 y(1)p 2702 2106 87 4 v 2702 2198 a Fi(r)2749 2169 y Fj(2)2815 1989 y Fh(\022)2888 2129 y Fi(k)f Fp(+)3073 2062 y Fi(B)5 b(r)3199 2026 y Fj(2)p 3073 2106 166 4 v 3131 2198 a Fp(2)3248 1989 y Fh(\023)3321 2009 y Fj(2)3377 2129 y Fi(:)357 b Fp(\(2\))0 2461 y(Using)33 b(this,)f(w)m(e)h(obtain)f(the)h(t)m(w)m(o)f(linearly)h(indep)s(enden)m (t)i(generalized)e(eigenfunctions)h(of)d Fi(Q)3516 2476 y Fl(m)3615 2461 y Fp(with)i(a)0 2635 y(generalized)h(eigen)m(v)-5 b(alue)34 b Fi(E)g Fg(2)28 b Ff(C)k Fp(as)1217 2967 y(\011)1293 2926 y Fj(1)1332 2967 y Fp(\()p Fi(E)6 b Fp(;)17 b Fi(r)s Fp(\))27 b(=)1707 2766 y Fh(0)1707 2946 y(@)1795 2889 y Fp(2\()p Fi(k)d Fp(+)e(1\))17 b(\011)2235 2853 y Fj(1)2235 2914 y(+)2294 2889 y Fp(\()p Fi(E)6 b Fp(;)17 b Fi(r)s Fp(\))1916 3064 y(i)p Fi(E)23 b Fp(\011)2114 3028 y Fj(1)2114 3088 y Fk(\000)2172 3064 y Fp(\()p Fi(E)6 b Fp(;)17 b Fi(r)s Fp(\))2538 2766 y Fh(1)2538 2946 y(A)2642 2967 y Fi(;)1217 3359 y Fp(\011)1293 3317 y Fj(2)1332 3359 y Fp(\()p Fi(E)6 b Fp(;)17 b Fi(r)s 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Fi(r)1369 4177 y Fl(k)r Fj(+1)1502 4219 y Fi(F)1595 4078 y Fh(\022)1669 4219 y Fi(k)e Fp(+)d(1)g Fg(\000)2028 4151 y Fi(E)2106 4115 y Fj(2)p 2023 4196 128 4 v 2023 4287 a Fp(2)p Fi(B)2161 4219 y(;)17 b Fp(2)22 b(+)g Fi(k)s(;)2482 4151 y(B)5 b(r)2608 4115 y Fj(2)p 2482 4196 166 4 v 2540 4287 a Fp(2)2657 4078 y Fh(\023)2747 4219 y Fi(e)2792 4177 y Fk(\000)p Fl(B)s(r)2937 4154 y Fd(2)2972 4177 y Fl(=)p Fj(4)3046 4219 y Fi(;)812 4493 y Fp(\011)888 4452 y Fj(2)888 4517 y(+)947 4493 y Fp(\()p Fi(E)h Fp(;)17 b Fi(r)s Fp(\))27 b(=)g Fi(r)1369 4452 y Fk(\000)p Fl(k)1467 4493 y Fi(F)1560 4352 y Fh(\022)1633 4493 y Fp(1)22 b Fg(\000)1819 4425 y Fi(E)1897 4389 y Fj(2)p 1814 4470 128 4 v 1814 4561 a Fp(2)p Fi(B)1952 4493 y(;)17 b Fp(1)k Fg(\000)i Fi(k)s(;)2274 4425 y(B)5 b(r)2400 4389 y Fj(2)p 2274 4470 166 4 v 2332 4561 a Fp(2)2449 4352 y Fh(\023)2539 4493 y Fi(e)2584 4452 y Fk(\000)p Fl(B)s(r)2729 4428 y Fd(2)2764 4452 y Fl(=)p Fj(4)2838 4493 y Fi(;)812 4767 y Fp(\011)888 4726 y Fj(2)888 4792 y Fk(\000)947 4767 y Fp(\()p Fi(E)h Fp(;)17 b Fi(r)s Fp(\))27 b(=)g Fi(r)1369 4726 y Fk(\000)p Fl(k)r Fk(\000)p Fj(1)1557 4767 y Fi(F)1650 4626 y Fh(\022)1723 4767 y Fg(\000)1815 4700 y Fi(E)1893 4663 y Fj(2)p 1810 4744 128 4 v 1810 4835 a Fp(2)p Fi(B)1949 4767 y(;)17 b Fg(\000)p Fi(k)s(;)2178 4700 y(B)5 b(r)2304 4663 y Fj(2)p 2178 4744 166 4 v 2236 4835 a Fp(2)2353 4626 y Fh(\023)2443 4767 y Fi(e)2488 4726 y Fk(\000)p Fl(B)s(r)2633 4702 y Fd(2)2668 4726 y Fl(=)p Fj(4)2743 4767 y Fi(:)0 5044 y Fp(Here,)38 b Fi(F)14 b Fp(\()p Fi(\014)6 b(;)17 b(\015)5 b(;)17 b(z)t Fp(\))35 b(is)i(the)g(con\015uen)m(t)g(h)m(yp)s (ergeometric)h(function)f([4].)54 b(W)-8 b(e)37 b(kno)m(w)g(that)f(if)g Fi(\015)j Fg(62)34 b Ff(N)i Fp(then)0 5219 y Fi(F)14 b Fp(\()p Fi(\014)6 b(;)17 b(\015)5 b(;)17 b(z)t Fp(\))30 b(and)h Fi(z)674 5182 y Fj(1)p Fk(\000)p Fl(\015)809 5219 y Fi(F)14 b Fp(\()p Fi(\014)23 b Fg(\000)c Fi(\015)k Fp(+)18 b(1)p Fi(;)f Fp(2)h Fg(\000)g Fi(\015)5 b(;)17 b(z)t Fp(\))31 b(are)g(linearly)h(indep)s(enden)m(t.)45 b(Note)30 b(that)h Fi(k)g Fg(62)d Ff(N)p Fp(,)j(and)f(w)m(e)p eop %%Page: 5 5 5 4 bop 3837 -208 a Fp(5)0 83 y(can)33 b(rewrite)g(\011)585 47 y Fj(2)585 108 y Fk(\006)677 83 y Fp(as)31 390 y(\011)107 349 y Fj(2)107 415 y(+)194 390 y Fp(=)o(const.)q Fi(r)566 349 y Fl(k)625 250 y Fh(\022)709 323 y Fi(B)5 b(r)835 287 y Fj(2)p 709 368 166 4 v 767 459 a Fp(2)884 250 y Fh(\023)957 270 y Fj(1)p Fk(\000)p Fj(\(1+)p Fl(k)r Fj(\))1257 390 y Fg(\002)23 b Fi(F)1450 250 y Fh(\022)1523 390 y Fi(k)j Fp(+)c(1)g Fg(\000)1883 323 y Fi(E)1961 287 y Fj(2)p 1878 368 128 4 v 1878 459 a Fp(2)p Fi(B)2038 390 y Fg(\000)h Fp(\(1)f(+)g Fi(k)s Fp(\))g(+)g(1)p Fi(;)17 b Fp(2)k Fg(\000)i Fp(\(1)f(+)g Fi(k)s Fp(\))p Fi(;)3172 323 y(B)5 b(r)3298 287 y Fj(2)p 3172 368 166 4 v 3231 459 a Fp(2)3348 250 y Fh(\023)3437 390 y Fi(e)3482 349 y Fk(\000)p Fl(B)s(r)3627 326 y Fd(2)3662 349 y Fl(=)p Fj(4)3737 390 y Fi(;)31 688 y Fp(\011)107 647 y Fj(2)107 713 y Fk(\000)194 688 y Fp(=)o(const.)q Fi(r)566 647 y Fl(k)r Fj(+1)715 548 y Fh(\022)799 621 y Fi(B)g(r)925 585 y Fj(2)p 799 665 V 857 757 a Fp(2)974 548 y Fh(\023)1047 568 y Fj(1)p Fk(\000)p Fj(\(2+)p Fl(k)r Fj(\))1347 688 y Fg(\002)23 b Fi(F)1540 548 y Fh(\022)1614 688 y Fi(k)i Fp(+)d(1)g Fg(\000)1973 621 y Fi(E)2051 585 y Fj(2)p 1968 665 128 4 v 1968 757 a Fp(2)p Fi(B)2128 688 y Fg(\000)h Fp(\(2)f(+)g Fi(k)s Fp(\))g(+)g(1)p Fi(;)17 b Fp(2)k Fg(\000)i Fp(\(2)f(+)g Fi(k)s Fp(\))p Fi(;)3262 621 y(B)5 b(r)3388 585 y Fj(2)p 3262 665 166 4 v 3321 757 a Fp(2)3438 548 y Fh(\023)3528 688 y Fi(e)3573 647 y Fk(\000)p Fl(B)s(r)3718 624 y Fd(2)3753 647 y Fl(=)p Fj(4)3827 688 y Fi(:)0 980 y Fp(Since)31 b(these)h(\011)576 933 y Fl(j)576 1002 y Fk(\006)635 980 y Fp('s)f(and)f(\011)994 944 y Fl(j)1030 980 y Fp('s)h(are)g(dep)s(end)g(on)f Fi(m)p Fp(,)h(w)m(e)h(should)f (write)g(a)f(subscript)i Fi(m)f Fp(as)f(\011)3399 933 y Fl(j)3399 1002 y(m;)p Fk(\006)3540 980 y Fp(,)h(but)f(w)m(e)0 1154 y(omit)j(it.)0 1369 y Fc(R)-5 b(emark)44 b Fp(1)p Fc(.)c Fp(The)32 b(op)s(erator)e Fi(L)1146 1384 y Fl(k)1220 1369 y Fp(is)h(the)h(Sc)m(hr\177)-49 b(odinger)32 b(op)s(erator)f(on)f (the)i Fi(m)p Fp(-sector)g(treated)f(in)g(Ref.)g([8].)0 1544 y(Using)i(Eq.)h(\(2\),)e(w)m(e)i(ha)m(v)m(e)f(that)1364 1879 y Fi(Q)1441 1838 y Fj(2)1509 1879 y Fp(=)1612 1679 y Fh(0)1612 1858 y(@)1700 1802 y Fi(H)1781 1817 y Fj(0)1842 1802 y Fp(+)22 b Fi(B)191 b Fp(0)1835 1976 y(0)185 b Fi(H)2150 1991 y Fj(0)2211 1976 y Fg(\000)23 b Fi(B)2390 1679 y Fh(1)2390 1858 y(A)2494 1879 y Fi(;)0 2214 y Fp(where,)35 b Fi(H)391 2229 y Fj(0)464 2214 y Fp(is)f(the)f(formal)g(Sc)m(hr\177) -49 b(odinger)35 b(op)s(erator)e(on)g(the)h(whole)g(space)h(treated)f (in)f(Ref.)h([8].)46 b(Th)m(us,)0 2388 y Fi(Q)77 2352 y Fj(2)149 2388 y Fp(is)33 b(the)g(op)s(erator)f Fi(H)889 2404 y Fj(1)p Fl(=)p Fj(2)1032 2388 y Fp(giv)m(en)h(b)m(y)h(Eq.)f (\(1\).)0 2604 y Fc(R)-5 b(emark)55 b Fp(2)p Fc(.)47 b Fp(F)-8 b(or)43 b(later)h(use,)j(w)m(e)e(summarize)h(the)e(facts)g (on)f(the)h(con\015uen)m(t)i(h)m(yp)s(ergeometric)f(func-)0 2778 y(tion)33 b([4)o(].)123 3034 y(1.)50 b Fi(F)14 b Fp(\()p Fi(\014)6 b(;)17 b(\015)5 b(;)17 b(z)t Fp(\))32 b(is)h(an)f(en)m(tire)i(function)f(and)g Fi(F)14 b Fp(\()p Fi(\014)6 b(;)17 b(\015)5 b(;)17 b Fp(0\))26 b(=)i(1.)123 3290 y(2.)50 b(If)22 b Fi(\014)33 b Fg(2)28 b(\000)p Ff(N)667 3305 y Fj(0)729 3290 y Fp(then)22 b Fi(F)14 b Fp(\()p Fi(\014)6 b(;)17 b(\015)5 b(;)17 b(z)t Fp(\))22 b(is)g(a)g(p)s(olynomial)g(with)g(the)h(leading)f(term)g(\000\()p Fi(\015)5 b Fp(\))p Fi(=)p Fp(\000\()p Fi(\015)27 b Fg(\000)c Fi(\014)6 b Fp(\))o(\()p Fg(\000)p Fi(z)t Fp(\))3755 3254 y Fk(\000)p Fl(\014)3858 3290 y Fp(.)249 3465 y(Indeed,)34 b(it)f(holds)g(that)1433 3639 y Fi(F)14 b Fp(\()p Fg(\000)p Fi(n;)j Fp(1)22 b(+)g Fi(\033)n(;)17 b(z)t Fp(\))29 b(=)e(const)p Fi(:)q(L)2528 3598 y Fl(\033)2528 3664 y(n)2576 3639 y Fp(\()p Fi(z)t Fp(\))1060 b(\(3\))249 3867 y(for)32 b Fi(n)c Fg(2)g Ff(N)650 3882 y Fj(0)722 3867 y Fp(and)33 b Fi(L)978 3830 y Fl(\033)978 3891 y(n)1025 3867 y Fp(\()p Fi(z)t Fp(\))g(is)g(a)g(Laguerre)f(p)s(olynomial.)123 4123 y(3.)50 b(If)33 b Fi(\014)g Fg(62)28 b(\000)p Ff(N)678 4138 y Fj(0)718 4123 y Fp(,)k(w)m(e)i(ha)m(v)m(e)g(that)921 4396 y Fi(F)14 b Fp(\()p Fi(\014)6 b(;)17 b(\015)5 b(;)17 b(z)t Fp(\))27 b(=)1471 4328 y(\000\()p Fi(\015)5 b Fp(\))p 1468 4373 198 4 v 1468 4464 a(\000\()p Fi(\014)h Fp(\))1676 4396 y Fi(e)1721 4354 y Fl(z)1761 4396 y Fi(z)1810 4354 y Fl(\014)s Fk(\000)p Fl(\015)1953 4396 y Fp(\(1)22 b(+)g Fi(O)s Fp(\()p Fi(z)2325 4354 y Fk(\000)p Fj(1)2419 4396 y Fp(\)\))97 b(for)g Fi(z)33 b Fg(!)27 b Fp(+)p Fg(1)p Fi(:)547 b Fp(\(4\))123 4709 y(4.)50 b(Let)491 4981 y Fi(G)p Fp(\()p Fi(\014)6 b(;)17 b(\013)q(;)g(z)t Fp(\))26 b(=)1135 4914 y(\000\(1)c Fg(\000)h Fi(\015)5 b Fp(\))p 1045 4958 545 4 v 1045 5049 a(\000\()p Fi(\014)28 b Fg(\000)22 b Fi(\015)27 b Fp(+)22 b(1\))1599 4981 y Fi(F)14 b Fp(\()p Fi(\014)6 b(;)17 b(\015)5 b(;)17 b(z)t Fp(\))22 b(+)2136 4914 y(\000\()p Fi(\015)27 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b(and)h(\011)27 b Fg(2)h(H)2466 2131 y Fj(total)2616 2116 y Fi(:)1118 b Fp(\(8\))0 2344 y(W)-8 b(e)28 b(solv)m(e)h(this)f(problem)g(on)g(eac) m(h)g Fg(H)1385 2359 y Fl(m)1452 2344 y Fp(.)42 b(Put)28 b Ff(N)1779 2359 y Fl(k)1849 2344 y Fp(=)g Fg(f)p Fi(n)f Fg(2)h Ff(Z)p Fp(;)17 b Fi(n)29 b Fg(\025)f Fi(k)s Fg(g)p Fp(.)41 b(Then)29 b(w)m(e)g(obtain)e(the)h(follo)m(wing)0 2518 y(theorem.)0 2776 y Fb(Theorem)38 b(3.)k Fc(The)34 b(eigenvalue)g(pr)-5 b(oblem)34 b(\(7\))g(with)h(BC)f(\(8\))g(has)h (the)g(fol)5 b(lowing)33 b(solutions.)125 2950 y(\(1\))h(If)g Fi(m)28 b Fp(=)g Fg(\000)p Fp(1)p Fc(,)35 b(we)f(have)g(no)h (nontrivial)f(solution.)125 3124 y(\(2\))g(If)g Fi(m)28 b Fg(2)g(\000)p Ff(N)747 3139 y Fj(2)787 3124 y Fc(,)35 b(the)g(solutions)f(ar)-5 b(e)35 b Fp(\010)1658 3088 y Fj(2)1698 3124 y Fp(\()p Fi(E)6 b Fp(;)17 b Fi(r)s Fp(\))34 b Fc(with)h Fi(E)e Fp(=)28 b Fg(\006)2475 3040 y(p)p 2558 3040 187 4 v 84 x Fp(2)p Fi(B)5 b(n)35 b Fc(for)g(al)5 b(l)35 b Fi(n)28 b Fg(2)g Ff(N)3327 3139 y Fj(0)3366 3124 y Fc(.)125 3299 y(\(3\))e(If)h Fi(m)h Fg(2)g Ff(N)655 3314 y Fj(0)694 3299 y Fc(,)h(the)e(solutions)g(ar)-5 b(e)27 b Fp(\010)1536 3262 y Fj(1)1576 3299 y Fp(\()p Fi(E)6 b Fp(;)17 b Fi(r)s Fp(\))26 b Fc(with)h Fi(E)34 b Fp(=)28 b Fg(\006)2338 3213 y Fh(p)p 2438 3213 820 4 v 86 x Fp(2)p Fi(B)5 b Fp(\()p Fi(m)22 b Fp(+)g Fi(\013)h Fp(+)f Fi(n)g Fp(+)g(1\))27 b Fc(for)g(al)5 b(l)27 b Fi(n)h Fg(2)g Ff(N)3816 3314 y Fj(0)3856 3299 y Fc(.)0 3556 y(Pr)-5 b(o)g(of.)49 b Fp(P)m(art)33 b(\(1\):)43 b(Since)34 b Fg(\000)p Fp(1)28 b Fi(<)g(k)i(<)e Fp(0,)k(w)m(e)i(ha)m(v) m(e)g(that)782 3893 y(lim)752 3953 y Fl(r)r Fk(!)p Fj(0+)963 3893 y Fp(\011)1039 3852 y Fj(1)1079 3893 y Fp(\()p Fi(E)6 b Fp(;)17 b Fi(r)s Fp(\))27 b(=)1454 3693 y Fh(0)1454 3872 y(@)1541 3816 y Fg(1)1567 3990 y Fp(0)1641 3693 y Fh(1)1641 3872 y(A)1842 3893 y Fp(and)144 b(lim)2114 3953 y Fl(r)r Fk(!)p Fj(0+)2325 3893 y Fp(\011)2401 3852 y Fj(2)2440 3893 y Fp(\()p Fi(E)6 b Fp(;)17 b Fi(r)s Fp(\))27 b(=)2816 3693 y Fh(0)2816 3872 y(@)2928 3816 y Fp(0)2903 3990 y Fg(1)3002 3693 y Fh(1)3002 3872 y(A)3106 3893 y Fi(:)0 4234 y Fp(Therefore,)37 b(w)m(e)f(ha)m(v)m(e)g(that)e (lim)1194 4249 y Fl(r)r Fk(!)p Fj(0+)1409 4234 y Fp(\011\()p Fi(r)s Fp(\))d Fg(6)p Fp(=)h(0)i(for)g(all)h(nonzero)h(linear)f(com)m (binations)h(\011)f(of)f(\011)3654 4198 y Fj(1)3728 4234 y Fp(and)0 4409 y(\011)76 4373 y Fj(2)115 4409 y Fp(.)44 b(Th)m(us,)34 b(w)m(e)g(ha)m(v)m(e)g(no)e(non)m(trivial)h(solution.)125 4583 y(P)m(art)f(\(2\):)43 b(W)-8 b(e)33 b(ha)m(v)m(e)h Fi(k)26 b Fp(+)c(1)27 b Fi(<)h Fp(0.)43 b(In)33 b(the)g(case)g(when)h Fi(E)g Fg(6)p Fp(=)27 b(0,)903 4920 y(lim)873 4980 y Fl(r)r Fk(!)p Fj(0+)1084 4920 y Fp(\011)1160 4879 y Fj(1)1200 4920 y Fp(\()p Fi(E)6 b Fp(;)17 b Fi(r)s Fp(\))27 b(=)1575 4720 y Fh(0)1575 4899 y(@)1662 4843 y Fg(1)1662 5017 y(1)1762 4720 y Fh(1)1762 4899 y(A)1963 4920 y Fp(and)144 b(lim)2235 4980 y Fl(r)r Fk(!)p Fj(0+)2446 4920 y Fp(\011)2522 4879 y Fj(2)2561 4920 y Fp(\()p Fi(E)6 b Fp(;)17 b Fi(r)s Fp(\))27 b(=)g(0)p Fi(:)0 5261 y Fp(Th)m(us,)j(\011)346 5225 y Fj(2)412 5261 y Fp(is)d(a)f(candidate)i(of)e(the)h(solutions.)43 b(If)26 b(1)10 b Fg(\000)g Fi(E)2044 5225 y Fj(2)2085 5261 y Fi(=)p Fp(2)p Fi(B)32 b Fg(62)c(\000)p Ff(N)2532 5276 y Fj(0)2572 5261 y Fp(,)g(then)f(\011)2919 5225 y Fj(2)2919 5286 y(+)3005 5261 y Fp(b)s(eha)m(v)m(es)i(as)e(a)f(pro)s (duct)0 5436 y(of)39 b(a)g(p)s(olynomial)h(of)f Fi(r)j Fp(and)d(exp)q(\()p Fi(B)5 b(r)1430 5400 y Fj(2)1469 5436 y Fi(=)p Fp(4\))39 b(for)g Fi(r)j Fg(!)d Fp(+)p Fg(1)p Fp(,)h(and)g(th)m(us)g(\011)2762 5400 y Fj(2)2762 5460 y(+)2860 5436 y Fg(62)g(L)3035 5400 y Fj(2)3074 5436 y Fp(.)64 b(On)39 b(the)h(con)m(trary)-8 b(,)p eop %%Page: 7 7 7 6 bop 3837 -208 a Fp(7)0 83 y(if)36 b(1)24 b Fg(\000)g Fi(E)345 47 y Fj(2)385 83 y Fi(=)p Fp(2)p Fi(B)38 b Fg(2)33 b(\000)p Ff(N)843 98 y Fj(0)919 83 y Fp(\(and)j(th)m(us)g Fg(\000)p Fi(E)1522 47 y Fj(2)1562 83 y Fi(=)p Fp(2)p Fi(B)i Fg(2)33 b(\000)p Ff(N)2020 98 y Fj(0)2060 83 y Fp(\),)k(the)f Fi(F)49 b Fp(parts)36 b(in)g(\011)2891 47 y Fj(2)2891 108 y Fk(\006)2986 83 y Fp(are)g(p)s(olynomials)g(of)g Fi(r)s Fp(.)0 257 y(Since)e(\011)331 221 y Fj(2)331 282 y Fk(\006)417 257 y Fg(2)28 b(L)580 221 y Fj(2)620 257 y Fp(,)k(w)m(e)i(ha)m(v)m(e)g(\011)1124 221 y Fj(2)1191 257 y Fg(2)28 b(H)q Fp(.)43 b(In)33 b(the)g(case)h(when)f Fi(E)h Fp(=)28 b(0,)k(w)m(e)i(ha)m(v)m(e)g(that)1268 554 y(\011)1344 513 y Fj(2)1383 554 y Fp(\(0;)17 b Fi(r)s Fp(\))27 b(=)1729 354 y Fh(0)1729 533 y(@)2149 477 y Fp(0)1816 651 y Fg(\000)p Fp(2)p Fi(k)s(r)2043 615 y Fk(\000)p Fl(k)r Fk(\000)p Fj(1)2231 651 y Fi(e)2276 615 y Fk(\000)p Fl(B)s(r)2421 592 y Fd(2)2456 615 y Fl(=)p Fj(4)2531 354 y Fh(1)2531 533 y(A)0 862 y Fp(and)40 b(th)m(us)g(\011) 494 826 y Fj(2)533 862 y Fp(\(0;)17 b Fi(r)s Fp(\))39 b Fg(2)g(H)q Fp(.)64 b(Consequen)m(tly)-8 b(,)45 b(\011)1778 826 y Fj(2)1856 862 y Fg(2)40 b(H)g Fp(if)g(and)f(only)h(if)f Fi(E)46 b Fp(=)39 b Fg(\006)3006 778 y(p)p 3089 778 187 4 v 84 x Fp(2)p Fi(B)5 b(n)40 b Fp(with)g Fi(n)f Fg(2)h Ff(N)3819 877 y Fj(0)3858 862 y Fp(.)0 1036 y(Th)m(us,)34 b(w)m(e)g(can)f(obtain)f(the)h(desired)h(result.)125 1211 y(P)m(art)c(\(3\):)41 b(In)30 b(a)g(w)m(a)m(y)h(similar)f(as)g(in) g(the)h(pro)s(of)e(of)g(P)m(art)h(\(2\),)g(w)m(e)h(can)f(see)h(that)e (\011)3156 1175 y Fj(2)3225 1211 y Fp(do)s(es)i(not)e(satisfy)0 1385 y(the)34 b(standard)g(BC)g(and)g(that)f(\011)1229 1349 y Fj(1)1297 1385 y Fg(2)d(H)35 b Fp(if)e(and)h(only)g(if)f Fi(E)j Fp(=)29 b Fg(\006)2388 1300 y Fh(p)p 2488 1300 820 4 v 85 x Fp(2)p Fi(B)5 b Fp(\()p Fi(m)22 b Fp(+)g Fi(\013)h Fp(+)f(1)g(+)g Fi(n)p Fp(\))34 b(with)g Fi(n)29 b Fg(2)h Ff(N)3819 1400 y Fj(0)3858 1385 y Fp(.)0 1559 y(Th)m(us,)k(w)m(e)g(can)f(obtain)f(the)h(desired)h(result.)p 3815 1559 4 66 v 3819 1497 59 4 v 3819 1559 V 3877 1559 4 66 v 125 1795 a(The)44 b(part)f(\(1\))f(of)h(Theorem)i(3)e(sa)m(ys)h (that)f(the)h(eigen)m(v)-5 b(alue)45 b(problem)f(\(7\))f(with)g(BC)h (\(8\))f(has)g(no)0 1970 y(solution)30 b(in)f Fg(\000)p Fp(1-sector.)43 b(It)29 b(is)h(di\013er)f(from)g(the)h(spin-less)h (case)f(treated)f(in)h(Ref.)f([8]:)42 b(In)29 b(that)g(case,)i(the)0 2144 y(sp)s(ectral)i(problem)h(for)e(the)h(standard)g(AB)f(Hamiltonian) h(has)g(at)g(least)g(one)g(solution)g(in)f(all)h(sectors.)125 2318 y(W)-8 b(e)33 b(giv)m(e)g(an)g(imp)s(ortan)m(t)f(prop)s(osition.)0 2535 y Fb(Prop)s(osition)38 b(4.)j Fc(If)35 b Fi(m)28 b Fg(2)g(\000)p Ff(N)p Fc(,)35 b(the)g(set)1105 2763 y Fg(S)1165 2778 y Fj(2)1233 2763 y Fp(=)1336 2652 y Fh(n)1403 2763 y Fp(\011)1479 2721 y Fj(2)1518 2763 y Fp(\()p Fi(E)6 b Fp(;)17 b Fi(r)s Fp(\);)g Fi(E)33 b Fp(=)27 b Fg(\006)2092 2673 y(p)p 2176 2673 187 4 v 2176 2763 a Fp(2)p Fi(B)5 b(n;)33 b(n)28 b Fg(2)g Ff(N)2674 2778 y Fj(0)2714 2652 y Fh(o)0 2990 y Fc(is)35 b(c)-5 b(omplete)34 b(in)g Fg(H)q Fc(.)45 b(If)34 b Fi(m)28 b Fg(2)g Ff(N)1166 3005 y Fk(\000)p Fj(1)1261 2990 y Fc(,)34 b(the)h(set)780 3217 y Fg(S)840 3232 y Fj(3)908 3217 y Fp(=)1011 3106 y Fh(n)1078 3217 y Fp(\011)1154 3176 y Fj(1)1193 3217 y Fp(\()p Fi(E)6 b Fp(;)17 b Fi(r)s Fp(\);)g Fi(E)33 b Fp(=)27 b Fg(\006)1767 3127 y Fh(p)p 1868 3127 820 4 v 1868 3217 a Fp(2)p Fi(B)5 b Fp(\()p Fi(m)22 b Fp(+)g Fi(\013)h Fp(+)f(1)g(+)g Fi(n)p Fp(\))p Fi(;)33 b(n)28 b Fg(2)g Ff(N)2999 3232 y Fj(0)3039 3106 y Fh(o)0 3444 y Fc(is)35 b(c)-5 b(omplete)34 b(in)g Fg(H)q Fc(.)0 3661 y(Pr)-5 b(o)g(of.)49 b Fp(By)34 b(Eq.)f(\(3\),)f(w)m(e)i (ha)m(v)m(e)522 3951 y(\011)598 3910 y Fj(2)637 3951 y Fp(\()675 3862 y Fg(p)p 758 3862 187 4 v 89 x Fp(2)p Fi(B)5 b(n)p Fp(;)17 b Fi(r)s Fp(\))k Fg(\000)i Fp(\011)1270 3910 y Fj(2)1309 3951 y Fp(\()p Fg(\000)1424 3862 y(p)p 1508 3862 V 1508 3951 a Fp(2)p Fi(B)5 b(n)p Fp(;)17 b Fi(r)s Fp(\))27 b(=)1954 3751 y Fh(0)1954 3931 y(@)2041 3874 y Fp(const)p Fi(:)q(r)2338 3838 y Fk(\000)p Fl(k)2435 3874 y Fi(L)2501 3833 y Fk(\000)p Fl(k)2501 3899 y(n)p Fk(\000)p Fj(1)2638 3874 y Fp(\()p Fi(B)5 b(r)2802 3838 y Fj(2)2841 3874 y Fi(=)p Fp(2\))p Fi(e)3022 3838 y Fk(\000)p Fl(B)s(r)3167 3815 y Fd(2)3202 3838 y Fl(=)p Fj(4)2634 4049 y Fp(0)3277 3751 y Fh(1)3277 3931 y(A)0 4246 y Fp(for)35 b Fi(n)f Fg(2)f Ff(N)p Fp(.)53 b(Since)37 b(the)f(set)h Fg(f)p Fi(L)1196 4210 y Fl(\033)1196 4271 y(n)1243 4246 y Fp(\()p Fi(z)t Fp(\);)17 b Fi(n)28 b Fg(2)g Ff(N)1664 4261 y Fj(0)1704 4246 y Fg(g)35 b Fp(with)i Fi(\033)g(>)c Fg(\000)p Fp(1)j(is)g(complete)h(in)f Fi(L)3077 4210 y Fj(2)3117 4246 y Fp(\(\(0)p Fi(;)17 b Fg(1)p Fp(\);)g Fi(z)3517 4210 y Fl(\033)3563 4246 y Fi(e)3608 4210 y Fk(\000)p Fl(z)3720 4246 y Fi(dz)t Fp(\),)0 4420 y(w)m(e)34 b(can)e(see)i(that)f(the)g(set)1078 4537 y Fh(n)1145 4647 y Fp(\011)1221 4606 y Fj(2)1260 4647 y Fp(\()1298 4558 y Fg(p)p 1381 4558 V 89 x Fp(2)p Fi(B)5 b(n;)17 b(r)s Fp(\))k Fg(\000)i Fp(\011)1893 4606 y Fj(2)1932 4647 y Fp(\()p Fg(\000)2047 4558 y(p)p 2131 4558 V 2131 4647 a Fp(2)p Fi(B)5 b(n;)17 b(r)s Fp(\);)g Fi(n)27 b Fg(2)h Ff(N)2741 4537 y Fh(o)0 4875 y Fp(is)33 b(complete)h(in)f Fg(L)693 4838 y Fj(2)754 4875 y Fg(\010)23 b(f)p Fp(0)p Fg(g)p Fp(.)43 b(In)32 b(a)h(w)m(a)m(y)h(similar)f(as)g(ab)s(o)m(v)m (e,)g(w)m(e)h(can)f(see)g(that)g(the)g(set)1079 4991 y Fh(n)1145 5102 y Fp(\011)1221 5061 y Fj(2)1261 5102 y Fp(\()1299 5013 y Fg(p)p 1382 5013 V 89 x Fp(2)p Fi(B)5 b(n;)17 b(r)s Fp(\))k(+)h(\011)1892 5061 y Fj(2)1932 5102 y Fp(\()p Fg(\000)2047 5013 y(p)p 2130 5013 V 89 x Fp(2)p Fi(B)5 b(n;)17 b(r)s Fp(\);)g Fi(n)27 b Fg(2)h Ff(N)2740 4991 y Fh(o)0 5329 y Fp(is)37 b(complete)h(in)e Fg(f)p Fp(0)p Fg(g)24 b(\010)i(L)980 5293 y Fj(2)1019 5329 y Fp(.)55 b(Consequen)m(tly)-8 b(,)40 b Fg(S)1789 5344 y Fj(2)1865 5329 y Fp(is)d(complete)h(in)f Fg(H)e Fp(=)f Fg(L)2799 5293 y Fj(2)2863 5329 y Fg(\010)25 b(L)3034 5293 y Fj(2)3073 5329 y Fp(.)55 b(In)37 b(the)g(case)g(when)0 5503 y Fi(m)28 b Fg(2)g Ff(N)279 5518 y Fk(\000)p Fj(1)373 5503 y Fp(,)33 b(w)m(e)h(can)e(obtain)h(the)g(desired)h(result)g(as)e (in)h(a)f(w)m(a)m(y)i(similar)g(as)e(ab)s(o)m(v)m(e.)p 3815 5503 4 66 v 3819 5441 59 4 v 3819 5503 V 3877 5503 4 66 v eop %%Page: 8 8 8 7 bop 3837 -208 a Fp(8)1060 83 y Fm(IV.)104 b(SELF-ADJOINT)34 b(EXTENTIONS)125 375 y Fp(W)-8 b(e)33 b(calculate)g(the)g(de\014ciency) i(index)f(of)e Fi(Q)p Fp(.)44 b(Let)208 766 y(\011)284 725 y Fk(\006)p Fj(i)363 766 y Fp(\()p Fi(r)s Fp(\))27 b(=)616 566 y Fh(0)616 745 y(@)704 689 y Fp(\011)780 648 y Fk(\006)p Fj(i)780 711 y(+)858 689 y Fp(\()p Fi(r)s Fp(\))704 863 y(\011)780 822 y Fk(\006)p Fj(i)780 885 y Fk(\000)858 863 y Fp(\()p Fi(r)s Fp(\))981 566 y Fh(1)981 745 y(A)1096 766 y Fp(=)g Fg(\000)p Fp(2\()p Fi(k)f Fp(+)c(1\))1642 506 y Fh(0)1642 682 y(B)1642 741 y(B)1642 805 y(@)1774 647 y Fg(\006)p Fi(r)1898 611 y Fl(k)1957 647 y Fi(G)2051 506 y Fh(\022)2124 647 y Fi(k)j Fp(+)d(1)g(+)2517 579 y(1)p 2477 624 128 4 v 2477 715 a(2)p Fi(B)2615 647 y(;)17 b Fp(1)22 b(+)g Fi(k)s(;)2936 579 y(B)5 b(r)3062 543 y Fj(2)p 2936 624 166 4 v 2994 715 a Fp(2)3111 506 y Fh(\023)3201 647 y Fi(e)3246 611 y Fk(\000)p Fl(B)s(r)3391 587 y Fd(2)3426 611 y Fl(=)p Fj(4)1728 888 y Fi(B)g(r)1854 852 y Fl(k)r Fj(+1)2003 888 y Fi(G)2097 747 y Fh(\022)2170 888 y Fi(k)25 b Fp(+)d(1)g(+)2563 820 y(1)p 2523 865 128 4 v 2523 956 a(2)p Fi(B)2661 888 y(;)17 b Fp(2)22 b(+)g Fi(k)s(;)2981 820 y(B)5 b(r)3107 784 y Fj(2)p 2981 865 166 4 v 3040 956 a Fp(2)3157 747 y Fh(\023)3247 888 y Fi(e)3292 852 y Fk(\000)p Fl(B)s(r)3437 828 y Fd(2)3472 852 y Fl(=)p Fj(4)3546 506 y Fh(1)3546 682 y(C)3546 741 y(C)3546 805 y(A)3650 766 y Fi(;)0 1169 y Fp(where)34 b Fi(G)e Fp(is)h(giv)m(en)h(in)f(Remark)g(2.)43 b(Note)33 b(that)f(w)m(e)i(can)f(rewrite)g(\011)2520 1132 y Fk(\006)p Fj(i)2631 1169 y Fp(as)g(follo)m(ws:)501 1440 y(\011)577 1399 y Fk(\006)p Fj(i)656 1440 y Fp(\()p Fi(r)s Fp(\))27 b(=)g Fg(\007)1128 1373 y Fp(\000\()p Fg(\000)p Fi(k)s Fp(\))p 996 1417 532 4 v 996 1508 a(\000\(1)c(+)f(1)p Fi(=)p Fp(2)p Fi(B)5 b Fp(\))1538 1440 y(\011)1614 1399 y Fj(1)1653 1440 y Fp(\()p Fg(\006)p Fp(i;)17 b Fi(r)s Fp(\))22 b(+)2054 1373 y(2)2103 1336 y Fl(k)r Fj(+1)p 2054 1417 182 4 v 2084 1508 a Fi(B)2163 1480 y Fl(k)2429 1373 y Fp(\000\()p Fi(k)j Fp(+)d(1\))p 2256 1417 706 4 v 2256 1508 a(\000\()p Fi(k)j Fp(+)d(1)g(+)g(1)p Fi(=)p Fp(2)p Fi(B)5 b Fp(\))2971 1440 y(\011)3047 1399 y Fj(2)3086 1440 y Fp(\()p Fg(\006)p Fp(i;)17 b Fi(r)s Fp(\))p Fi(:)0 1712 y Fp(Th)m(us,)34 b(w)m(e)g(ha)m(v)m(e)g(that)e Fi(Q)931 1675 y Fk(\003)931 1736 y Fl(m)998 1712 y Fp(\011)1074 1675 y Fk(\006)p Fj(i)1180 1712 y Fp(=)c Fg(\006)p Fp(i)17 b(\011)1481 1675 y Fk(\006)p Fj(i)1559 1712 y Fi(:)0 1967 y Fb(Lemma)44 b(5.)g Fc(The)38 b(de\014ciency)f(index)h(of)g Fi(Q)1648 1982 y Fk(\000)p Fj(1)1782 1967 y Fc(is)g Fp(\(1)p Fi(;)17 b Fp(1\))38 b Fc(and)g(the)g(de\014ciency)g(sp)-5 b(ac)g(es)38 b Fg(N)3339 1982 y Fk(\006)3436 1967 y Fc(of)g Fi(Q)3631 1982 y Fk(\000)p Fj(1)3765 1967 y Fc(for)0 2141 y Fi(E)c Fp(=)27 b Fg(\006)p Fp(i)36 b Fc(ar)-5 b(e)1304 2316 y Fg(N)1386 2331 y Fk(\006)1472 2316 y Fp(=)28 b Fg(f)p Fi(c)17 b Fp(\011)1761 2274 y Fk(\006)p Fj(i)1839 2316 y Fp(;)g Fi(c)27 b Fg(2)h Ff(C)p Fi(;)17 b(m)28 b Fp(=)f Fg(\000)p Fp(1)p Fg(g)p Fi(:)0 2543 y Fc(If)34 b Fi(m)28 b Fg(6)p Fp(=)g Fg(\000)p Fp(1)p Fc(,)35 b(the)g(de\014ciency)f(index)g(of)g Fi(Q)1573 2558 y Fl(m)1675 2543 y Fc(is)g Fp(\(0)p Fi(;)17 b Fp(0\))p Fc(.)44 b(Thus,)35 b(the)g(de\014ciency)f(index)g(of)g Fi(Q)h Fc(is)g Fp(\(1)p Fi(;)17 b Fp(1\))p Fc(.)0 2798 y(Pr)-5 b(o)g(of.)49 b Fp(Let)33 b Fi(E)h Fp(=)27 b Fg(\006)p Fp(i.)45 b(By)33 b(Eq.)g(\(4\),)f(w)m(e)i(ha)m(v)m(e)g(that)e(for)g(b)s (oth)h Fi(j)g Fp(=)28 b(1)p Fi(;)17 b Fp(2)690 3070 y(\011)766 3022 y Fl(j)766 3091 y Fk(\006)825 3070 y Fp(\()p Fi(E)6 b Fp(;)17 b Fi(r)s Fp(\))27 b(=)h(`a)k(p)s(olynomial)h(of)f Fi(r)s Fp(')22 b Fg(\002)h Fp(exp)2285 2989 y Fh(\000)2331 3070 y Fi(B)5 b(r)2457 3028 y Fj(2)2496 3070 y Fi(=)p Fp(4)2594 2989 y Fh(\001)2656 3070 y Fp(\(1)22 b(+)g Fi(O)s Fp(\()p Fi(r)3026 3028 y Fk(\000)p Fj(2)3119 3070 y Fp(\)\))0 3341 y(for)32 b Fi(r)e Fg(!)e(1)p Fp(.)43 b(Th)m(us)34 b(\011)844 3305 y Fl(j)908 3341 y Fg(62)28 b(H)q Fp(.)44 b(Ho)m(w)m(ev)m(er,)35 b(b)m(y)e(Eq.)h(\(5\),)e(w)m(e)h (ha)m(v)m(e)h(that)703 3613 y(\011)779 3571 y Fk(\006)p Fj(i)779 3637 y Fk(\006)857 3613 y Fp(\()p Fi(r)s Fp(\))27 b(=)h(`a)k(p)s(olynomial)h(of)f Fi(r)s Fp(')22 b Fg(\002)h Fp(exp)2196 3532 y Fh(\000)2241 3613 y Fg(\000)p Fi(B)5 b(r)2444 3571 y Fj(2)2484 3613 y Fi(=)p Fp(4)2582 3532 y Fh(\001)2644 3613 y Fp(\(1)22 b(+)g Fi(O)s Fp(\()p Fi(r)3014 3571 y Fk(\000)p Fj(2)3107 3613 y Fp(\)\))0 3884 y(for)39 b Fi(r)j Fg(!)d(1)p Fp(.)64 b(Therefore)40 b(\011)1096 3843 y Fk(\006)p Fj(i)1096 3906 y Fk(\006)1175 3884 y Fp(\()p Fi(r)s Fp(\))f(are)g Fi(L)1572 3848 y Fj(2)1651 3884 y Fp(with)h(resp)s(ect)h(to)e(the)h(measure)h Fi(r)19 b(dr)42 b Fp(at)d(in\014nit)m(y)-8 b(.)65 b(In)40 b(the)0 4058 y(other)33 b(hand,)g(since)h Fi(F)14 b Fp(\()p Fi(\014)6 b(;)17 b(\015)5 b(;)17 b(z)t Fp(\))32 b(is)h(an)f(en)m(tire)i (function,)f(w)m(e)h(ha)m(v)m(e)f(that)1210 4311 y(\011)1286 4270 y Fk(\006)p Fj(i)1286 4336 y(+)1365 4311 y Fp(\()p Fi(r)s Fp(\))27 b(=)h Fi(c)1661 4326 y Fj(1)1700 4311 y Fi(r)1747 4270 y Fl(k)1812 4311 y Fp(+)22 b Fi(c)1952 4326 y Fj(2)1991 4311 y Fi(r)2038 4270 y Fk(\000)p Fl(k)2157 4311 y Fp(+)g Fi(O)s Fp(\()p Fi(r)2418 4270 y Fj(2)2457 4311 y Fp(\))p Fi(;)1210 4510 y Fp(\011)1286 4469 y Fk(\006)p Fj(i)1286 4535 y Fk(\000)1365 4510 y Fp(\()p Fi(r)s Fp(\))27 b(=)h Fi(c)1661 4525 y Fj(3)1700 4510 y Fi(r)1747 4469 y Fk(\000)p Fl(k)r Fk(\000)p Fj(1)1957 4510 y Fp(+)22 b Fi(c)2097 4525 y Fj(4)2136 4510 y Fi(r)2183 4469 y Fl(k)r Fj(+1)2338 4510 y Fp(+)g Fi(O)s Fp(\()p Fi(r)2599 4469 y Fj(2)2637 4510 y Fp(\))0 4754 y(with)33 b(constan)m(ts)g Fi(c)695 4769 y Fl(j)732 4754 y Fp('s)g(for)e Fi(r)g Fg(!)c Fp(0+.)43 b(Since)33 b Fi(k)e Fg(2)d Fp(\()p Fi(m;)17 b(m)22 b Fp(+)f(1\),)32 b(\011)2398 4712 y Fk(\006)p Fj(i)2398 4775 y Fk(\006)2477 4754 y Fp(\()p Fi(r)s Fp(\))f(are)i Fi(L)2860 4718 y Fj(2)2932 4754 y Fp(with)g(resp)s(ect)g(to)f Fi(r)19 b(dr)35 b Fp(at)0 4928 y(the)30 b(origin)f(if)h(and)g(only)g (if)f Fi(m)f Fp(=)g Fg(\000)p Fp(1.)43 b(Consequen)m(tly)-8 b(,)33 b(the)d(de\014ciency)i(index)f(of)e Fi(Q)3097 4943 y Fk(\000)p Fj(1)3221 4928 y Fp(is)i(\(1)p Fi(;)17 b Fp(1\))28 b(and)i(the)0 5102 y(one)j(of)f Fi(Q)367 5117 y Fl(m)466 5102 y Fp(with)h Fi(m)28 b Fg(6)p Fp(=)g(1)k(is)h(\(0)p Fi(;)17 b Fp(0\).)p 3815 5102 4 66 v 3819 5040 59 4 v 3819 5102 V 3877 5102 4 66 v 125 5351 a(W)-8 b(e)28 b(examine)i(the)e (b)s(oundary)h(conditions)g(of)f(all)g(self-adjoin)m(t)h(extensions)h (of)e Fi(Q)p Fp(.)42 b(In)29 b(the)f(case)h(when)0 5525 y Fi(m)f Fg(6)p Fp(=)g Fg(\000)p Fp(1,)h(b)m(y)g(Lemma)g(5,)g Fi(Q)1056 5540 y Fl(m)1151 5525 y Fp(is)f(essen)m(tially)j(self-adjoin) m(t.)42 b(W)-8 b(e)29 b(denote)g(the)f(self-adjoin)m(t)h(extension)h (of)p eop %%Page: 9 9 9 8 bop 3837 -208 a Fp(9)0 88 y Fi(Q)77 103 y Fl(m)168 88 y Fp(b)m(y)p 296 8 144 4 v 26 w Fi(Q)373 103 y Fl(m)440 88 y Fp(.)41 b(Consider)26 b Fi(Q)984 103 y Fk(\000)p Fj(1)1078 88 y Fp(.)41 b(W)-8 b(e)25 b(use)h(the)f(de\014ciency)i (space)f(tec)m(hnique)h(\(Theorem)f(X.2)f(in)g(Ref.)g([12)o(]\).)0 262 y(Since)39 b(the)f(de\014ciency)j(index)e(of)e(it)h(is)g(\(1)p Fi(;)17 b Fp(1\),)39 b(w)m(e)g(can)f(sp)s(ecify)h(self-adjoin)m(t)f (extensions)i(b)m(y)f(one)f(real)0 437 y(parameter)31 b Fi(\034)39 b Fg(2)28 b Fp([0)p Fi(;)17 b Fp(2)p Fi(\031)t Fp(\).)42 b(W)-8 b(e)30 b(in)m(tro)s(duce)h(a)f(quan)m(tit)m(y)h(whic)m (h)h(c)m(haracterizes)g(the)e(b)s(oundary)h(condition)0 611 y(of)h(self-adjoin)m(t)h(extensions)i Fi(Q)1160 575 y Fl(\034)1160 636 y Fk(\000)p Fj(1)1287 611 y Fp(of)d Fi(Q)1475 626 y Fk(\000)p Fj(1)1570 611 y Fp(.)43 b(W)-8 b(e)33 b(put)1304 939 y(\010)1374 954 y Fk(\000)p Fj(1)1469 939 y Fp(\(\011\))27 b(=)57 b(lim)1752 998 y Fl(r)r Fk(!)p Fj(0+)1963 738 y Fh(0)1963 918 y(@)2050 861 y Fi(r)2097 825 y Fj(1)p Fk(\000)p Fl(\013)2237 861 y Fp(\011)2313 876 y Fj(+)2372 861 y Fp(\()p Fi(r)s Fp(\))2095 1036 y Fi(r)2142 1000 y Fl(\013)2192 1036 y Fp(\011)2268 1051 y Fk(\000)2326 1036 y Fp(\()p Fi(r)s Fp(\))2494 738 y Fh(1)2494 918 y(A)0 1254 y Fp(up)33 b(to)f(a)g(nonzero)h(constan)m(t)g (m)m(ultiplication.)45 b(The)33 b(follo)m(wing)g(is)g(the)g(main)f (theorem)h(in)g(this)g(section.)0 1497 y Fb(Theorem)50 b(6.)e Fc(The)c(al)5 b(l)44 b(self-adjoint)f(extensions)h Fi(Q)2053 1461 y Fl(\034)2053 1522 y Fk(\000)p Fj(1)2192 1497 y Fc(ar)-5 b(e)45 b(sp)-5 b(e)g(ci\014e)g(d)43 b(by)i(the)g(b)-5 b(oundary)44 b(c)-5 b(ondition)0 1672 y(with)35 b(a)f(p)-5 b(ar)g(ameter)35 b Fi(\034)k Fg(2)28 b Fp([0)p Fi(;)17 b Fp(2)p Fi(\031)t Fp(\))p Fc(:)1175 1993 y Fp(\010)1245 2008 y Fk(\000)p Fj(1)1340 1993 y Fp(\(\011\))27 b(=)1623 1792 y Fh(0)1623 1972 y(@)1783 1915 y Fp(\()p Fi(e)1866 1879 y Fj(i)p Fl(\034)1951 1915 y Fg(\000)22 b Fp(1\)\000\(1)g Fg(\000)g Fi(\013)q Fp(\))1710 2090 y(\()p Fi(e)1793 2054 y Fj(i)p Fl(\034)1878 2090 y Fp(+)g(1\)\000\()p Fi(\013)q Fp(\))o(2)2311 2054 y Fl(\013)2361 2090 y Fi(B)2440 2054 y Fj(1)p Fk(\000)p Fl(\013)2579 1792 y Fh(1)2579 1972 y(A)2683 1993 y Fi(:)1051 b Fp(\(9\))0 2325 y Fc(Pr)-5 b(o)g(of.)49 b Fp(Since)28 b Fg(k)p Fp(\011)681 2283 y Fk(\006)p Fj(i)681 2346 y(+)759 2325 y Fg(k)f Fp(=)h Fg(k)p Fp(\011)1066 2283 y Fk(\006)p Fj(i)1066 2346 y Fk(\000)1144 2325 y Fg(k)p Fp(,)f(the)g(all)f(self-adjoin)m(t)g (extensions)j Fi(Q)2575 2288 y Fl(\034)2575 2349 y Fk(\000)p Fj(1)2695 2325 y Fp(with)e(a)e(parameter)i Fi(\034)39 b Fg(2)28 b Fp([0)p Fi(;)17 b Fp(2)p Fi(\031)t Fp(\))0 2499 y(is)33 b(giv)m(en)h(b)m(y)786 2732 y Fi(D)s Fp(\()p Fi(Q)985 2691 y Fl(\034)985 2757 y Fk(\000)p Fj(1)1079 2732 y Fp(\))27 b(=)h Fg(f)p Fp(\012)22 b(+)g Fi(c)17 b Fp(\(\011)1661 2691 y Fj(+i)1761 2732 y Fp(+)23 b Fi(e)1905 2691 y Fj(i)p Fl(\034)1967 2732 y Fp(\011)2043 2691 y Fk(\000)p Fj(i)2122 2732 y Fp(\);)33 b(\012)28 b Fg(2)g Fi(D)s Fp(\()p 2534 2652 172 4 v Fi(Q)2611 2747 y Fk(\000)p Fj(1)2706 2732 y Fp(\))p Fi(;)17 b(c)27 b Fg(2)h Ff(C)p Fg(g)p Fi(;)786 2931 y(Q)863 2890 y Fl(\034)863 2956 y Fk(\000)p Fj(1)957 2931 y Fp([\012)23 b(+)f Fi(c)p Fp(\(\011)1331 2890 y Fj(+i)1431 2931 y Fp(+)g Fi(e)1574 2890 y Fj(i)p Fl(\034)1637 2931 y Fp(\011)1713 2890 y Fk(\000)p Fj(i)1792 2931 y Fp(\)])28 b(=)p 1988 2851 V 27 w Fi(Q)2065 2946 y Fk(\000)p Fj(1)2160 2931 y Fp(\012)22 b(+)g(i)p Fi(c)p Fp(\(\011)2533 2890 y Fj(+i)2634 2931 y Fg(\000)h Fi(e)2779 2890 y Fj(i)p Fl(\034)2842 2931 y Fp(\011)2918 2890 y Fk(\000)p Fj(i)2996 2931 y Fp(\))p Fi(;)0 3163 y Fp(with)39 b(the)h(aid)e(of)h(the)g(de\014ciency)i(space) f(tec)m(hnique.)65 b(Here,)p 2344 3083 V 41 w Fi(Q)2421 3178 y Fk(\000)p Fj(1)2554 3163 y Fp(is)40 b(the)f(closure)h(of)e Fi(Q)3359 3178 y Fk(\000)p Fj(1)3454 3163 y Fp(.)62 b(W)-8 b(e)39 b(\014rst)0 3338 y(in)m(v)m(estigate)45 b(the)e(BC)h(to)e (satisfy)i(that)f(\012)j Fg(2)g Fi(D)s Fp(\()p 1873 3258 V Fi(Q)1950 3353 y Fk(\000)p Fj(1)2044 3338 y Fp(\).)74 b(W)-8 b(e)44 b(use)g(the)f(follo)m(wing)g(fact:)64 b(Let)43 b Fi(A)g Fp(b)s(e)h(a)0 3512 y(symmetric)d(op)s(erator)d(and)h Fg(N)1158 3527 y Fk(\006)1255 3512 y Fp(are)g(the)g(de\014ciency)i (spaces)f(of)e Fi(A)p Fp(.)62 b(F)-8 b(or)38 b Fi(f)48 b Fg(2)39 b Fi(D)s Fp(\()p Fi(A)3212 3476 y Fk(\003)3251 3512 y Fp(\),)h Fi(f)49 b Fg(2)38 b Fi(D)s Fp(\()p 3679 3432 74 4 v Fi(A)p Fp(\))h(if)0 3686 y(and)33 b(only)g(if)f Fg(h)p Fi(f)5 b(;)17 b(A)702 3650 y Fk(\003)741 3686 y Fi(g)t Fg(i)27 b Fp(=)h Fg(h)p Fi(A)1074 3650 y Fk(\003)1113 3686 y Fi(f)5 b(;)17 b(g)t Fg(i)32 b Fp(for)g(all)g Fi(g)f Fg(2)d(N)1870 3701 y Fj(+)1951 3686 y Fp(+)22 b Fg(N)2131 3701 y Fk(\000)2190 3686 y Fp(.)43 b(Let)1450 4007 y(\012)28 b(=)1652 3807 y Fh(0)1652 3986 y(@)1739 3930 y Fp(\012)1809 3945 y Fj(+)1739 4104 y Fp(\012)1809 4119 y Fk(\000)1868 3807 y Fh(1)1868 3986 y(A)1983 4007 y Fg(2)g Fi(D)s Fp(\()p Fi(Q)2276 3966 y Fk(\003)2276 4032 y(\000)p Fj(1)2370 4007 y Fp(\))p Fi(:)0 4341 y Fp(Then,)34 b(\012)28 b Fg(2)g Fi(D)s Fp(\()p 596 4261 172 4 v Fi(Q)673 4356 y Fk(\000)p Fj(1)767 4341 y Fp(\))33 b(if)f(and)h(only)g(if)1175 4465 y Fh(8)1175 4555 y(<)1175 4734 y(:)1280 4580 y Fg(h)p Fp(\011)1395 4539 y Fk(\006)p Fj(i)1395 4605 y Fk(\000)1474 4580 y Fi(;)g(q)1581 4539 y Fj(+)1577 4605 y(0)1640 4580 y Fp(\012)1710 4595 y Fj(+)1769 4580 y Fg(i)1808 4597 y Fk(L)1857 4578 y Fd(2)1923 4580 y Fp(=)28 b Fg(h)p Fi(q)2113 4539 y Fk(\000)2109 4605 y(\000)p Fj(1)2203 4580 y Fp(\011)2279 4539 y Fk(\006)p Fj(i)2279 4605 y Fk(\000)2357 4580 y Fi(;)34 b Fp(\012)2488 4595 y Fj(+)2547 4580 y Fg(i)2586 4597 y Fk(L)2635 4578 y Fd(2)2673 4580 y Fi(;)1280 4779 y Fg(h)p Fp(\011)1395 4738 y Fk(\006)p Fj(i)1395 4804 y(+)1474 4779 y Fi(;)f(q)1581 4738 y Fk(\000)1577 4804 y(\000)p Fj(1)1672 4779 y Fp(\012)1742 4794 y Fk(\000)1801 4779 y Fg(i)1840 4796 y Fk(L)1889 4777 y Fd(2)1955 4779 y Fp(=)27 b Fg(h)p Fi(q)2144 4738 y Fj(+)2140 4804 y(0)2203 4779 y Fp(\011)2279 4738 y Fk(\006)p Fj(i)2279 4804 y(+)2357 4779 y Fi(;)34 b Fp(\012)2488 4794 y Fk(\000)2547 4779 y Fg(i)2586 4796 y Fk(L)2635 4777 y Fd(2)2673 4779 y Fi(:)0 5005 y Fp(Here,)f Fi(q)304 4963 y Fk(\000)300 5029 y(\000)p Fj(1)427 5005 y Fp(and)g Fi(q)664 4963 y Fj(+)660 5029 y(0)756 5005 y Fp(are)f(de\014ned)i(in)f(Sec.)g(I)s(I.) 44 b(This)34 b(is)f(equiv)-5 b(alen)m(t)34 b(to)973 5134 y Fh(8)973 5223 y(>)973 5253 y(<)973 5433 y(>)973 5463 y(:)1124 5250 y Fp(lim)1095 5310 y Fl(r)r Fk(!)p Fj(0+)1306 5250 y Fp(\012)1376 5265 y Fj(+)1435 5250 y Fp(\()p Fi(r)s Fp(\)\011)1634 5209 y Fk(\006)p Fj(i)1634 5274 y Fk(\000)1712 5250 y Fp(\()p Fi(r)s Fp(\))17 b Fi(r)30 b Fp(=)57 b(lim)2029 5310 y Fl(r)r Fk(!)p Fj(0+)2241 5250 y Fp(\012)2311 5265 y Fj(+)2370 5250 y Fp(\()p Fi(r)s Fp(\))17 b Fi(r)2557 5209 y Fj(1)p Fk(\000)p Fl(\013)2723 5250 y Fp(=)28 b(0)p Fi(;)1124 5482 y Fp(lim)1095 5541 y Fl(r)r Fk(!)p Fj(0+)1306 5482 y Fp(\012)1376 5497 y Fk(\000)1435 5482 y Fp(\()p Fi(r)s Fp(\)\011)1634 5441 y Fk(\006)p Fj(i)1634 5506 y(+)1712 5482 y Fp(\()p Fi(r)s Fp(\))17 b Fi(r)30 b Fp(=)57 b(lim)2029 5541 y Fl(r)r Fk(!)p Fj(0+)2241 5482 y Fp(\012)2311 5497 y Fk(\000)2370 5482 y Fp(\()p Fi(r)s Fp(\))17 b Fi(r)2557 5441 y Fl(\013)2633 5482 y Fp(=)28 b(0)p Fi(:)p eop %%Page: 10 10 10 9 bop 3788 -208 a Fp(10)0 83 y(Th)m(us,)35 b(\010)345 98 y Fk(\000)p Fj(1)440 83 y Fp(\(\012\))29 b(=)f(0.)46 b(Let)33 b(\011)c(=)f(\012)23 b(+)g Fi(c)p Fp(\(\011)1573 47 y Fj(+i)1674 83 y Fp(+)f Fi(e)1817 47 y Fj(i)p Fl(\034)1880 83 y Fp(\011)1956 47 y Fk(\000)p Fj(i)2035 83 y Fp(\))28 b Fg(2)h Fi(D)s Fp(\()p Fi(Q)2395 47 y Fl(\034)2395 108 y Fk(\000)p Fj(1)2490 83 y Fp(\).)45 b(Then,)35 b(w)m(e)f(ha)m(v)m(e)g (that)f(\010)3534 98 y Fk(\000)p Fj(1)3629 83 y Fp(\(\011\))c(=)0 257 y(\010)70 272 y Fk(\000)p Fj(1)165 257 y Fp(\(\011)279 221 y Fj(+i)379 257 y Fp(+)22 b Fi(e)522 221 y Fj(i)p Fl(\034)585 257 y Fp(\011)661 221 y Fk(\000)p Fj(i)740 257 y Fp(\).)43 b(Th)m(us,)34 b(it)f(holds)g(that)508 632 y(\010)578 647 y Fk(\000)p Fj(1)673 632 y Fp(\(\011\))27 b(=)57 b(lim)956 692 y Fl(r)r Fk(!)p Fj(0+)1167 372 y Fh(0)1167 547 y(B)1167 607 y(B)1167 671 y(@)1338 512 y Fp(\()p Fi(e)1421 476 y Fj(i)p Fl(\034)1506 512 y Fg(\000)22 b Fp(1\))17 b(exp)1874 372 y Fh(\022)1948 512 y Fg(\000)2035 445 y Fi(B)5 b(r)2161 409 y Fj(2)p 2035 489 166 4 v 2093 581 a Fp(4)2210 372 y Fh(\023)2300 512 y Fi(G)2394 372 y Fh(\022)2467 512 y Fi(\013)23 b Fp(+)2700 445 y(1)p 2660 489 128 4 v 2660 581 a(2)p Fi(B)2798 512 y(;)17 b(\013)q(;)2958 445 y(B)5 b(r)3084 409 y Fj(2)p 2958 489 166 4 v 3016 581 a Fp(2)3133 372 y Fh(\023)1254 753 y Fp(\()p Fi(e)1337 717 y Fj(i)p Fl(\034)1422 753 y Fp(+)22 b(1\))17 b(exp)1789 613 y Fh(\022)1862 753 y Fg(\000)1949 686 y Fi(B)5 b(r)2075 650 y Fj(2)p 1949 730 V 2008 822 a Fp(4)2125 613 y Fh(\023)2215 753 y Fi(G)2309 613 y Fh(\022)2382 753 y Fi(\013)23 b Fp(+)2614 686 y(1)p 2575 730 128 4 v 2575 822 a(2)p Fi(B)2713 753 y(;)17 b Fp(1)k(+)h Fi(\013)q(;)3042 686 y(B)5 b(r)3168 650 y Fj(2)p 3042 730 166 4 v 3100 822 a Fp(2)3217 613 y Fh(\023)3290 372 y(1)3290 547 y(C)3290 607 y(C)3290 671 y(A)852 1085 y Fp(=)1207 1018 y(1)p 966 1062 532 4 v 966 1154 a(\000\(1)22 b(+)g(1)p Fi(=)p Fp(2)p Fi(B)5 b Fp(\))1524 885 y Fh(0)1524 1064 y(@)1743 1008 y Fp(\()p Fi(e)1826 972 y Fj(i)p Fl(\034)1911 1008 y Fg(\000)22 b Fp(1\)\000\(1)g Fg(\000)g Fi(\013)q Fp(\))1611 1182 y(\()p Fi(e)1694 1146 y Fj(i)p Fl(\034)1779 1182 y Fp(+)g(1\)\000\()p Fi(\013)q Fp(\))p Fi(B)5 b Fp(\()p Fi(B)g(=)p Fp(2\))2496 1146 y Fk(\000)p Fl(\013)2599 885 y Fh(1)2599 1064 y(A)2703 1085 y Fi(:)0 1393 y Fp(Therefore)34 b(w)m(e)f(obtain)g(the)g(desired)h(result.)p 3815 1393 4 66 v 3819 1330 59 4 v 3819 1393 V 3877 1393 4 66 v 0 1635 a Fc(R)-5 b(emark)44 b Fp(7)p Fc(.)c Fp(Lemma)32 b(5)g(and)f(Lemma)h(6)g(imply)g(that)f(the)h(self-adjoin)m(t)g (extension)h Fi(Q)3191 1599 y Fl(\034)3266 1635 y Fp(of)e Fi(Q)h Fp(is)g(param-)0 1809 y(eterized)i(b)m(y)g(one)e(real)h (parameter)g Fi(\034)39 b Fg(2)28 b Fp([0)p Fi(;)17 b Fp(2)p Fi(\031)t Fp(\))32 b(and)h(sp)s(eci\014ed)h(b)m(y)g(the)f(b)s (oundary)g(condition:)1244 2122 y(\010\(\011\))28 b(=)1598 1922 y Fh(0)1598 2101 y(@)1757 2045 y Fp(\()p Fi(e)1840 2009 y Fj(i)p Fl(\034)1925 2045 y Fg(\000)23 b Fp(1\)\000\(1)e Fg(\000)i Fi(\013)q Fp(\))1685 2219 y(\()p Fi(e)1768 2183 y Fj(i)p Fl(\034)1853 2219 y Fp(+)f(1\)\000\()p Fi(\013)q Fp(\))o(2)2286 2183 y Fl(\013)2335 2219 y Fi(B)2414 2183 y Fj(1)p Fk(\000)p Fl(\013)2554 1922 y Fh(1)2554 2101 y(A)0 2435 y Fp(with)33 b(\010\(\011\))28 b(=)g(\010)646 2450 y Fk(\000)p Fj(1)740 2435 y Fp(\()p Fi(U)10 b(P)917 2450 y Fk(\000)p Fj(1)1012 2435 y Fp(\011\).)43 b(This)34 b Fi(Q)1496 2398 y Fl(\034)1572 2435 y Fp(is)f(giv)m(en)h(b)m(y)830 2684 y Fi(Q)907 2643 y Fl(\034)978 2684 y Fp(=)28 b Fg(\010)1186 2695 y Fl(m)p Fk(2)p Fe(Z)1159 2755 y Fl(m)p Fk(6)p Fj(=)p Fk(\000)p Fj(1)1371 2684 y Fp(\()p Fi(U)10 b(P)1548 2699 y Fl(m)1615 2684 y Fp(\))1653 2643 y Fk(\000)p Fj(1)p 1747 2604 144 4 v 1747 2684 a Fi(Q)1824 2699 y Fl(m)1891 2684 y Fi(U)g(P)2030 2699 y Fl(m)2119 2684 y Fg(\010)23 b Fp(\()p Fi(U)10 b(P)2396 2699 y Fk(\000)p Fj(1)2491 2684 y Fp(\))2529 2643 y Fk(\000)p Fj(1)2623 2684 y Fi(Q)2700 2643 y Fl(\034)2700 2709 y Fk(\000)p Fj(1)2794 2684 y Fi(U)g(P)2933 2699 y Fk(\000)p Fj(1)3028 2684 y Fi(:)657 b Fp(\(10\))0 2934 y(Here)33 b(the)g(op)s(erators)g Fi(U)43 b Fp(and)33 b Fi(P)1192 2949 y Fl(m)1291 2934 y Fp(are)f(giv)m(en)i(in) f(Sec.)g(I)s(I.)679 3300 y Fm(V.)105 b(SEPCTR)m(UM)36 b(OF)e(SELF-ADJOINT)f(EXTENSIONS)125 3592 y Fp(In)g(this)h(section,)h (w)m(e)g(examine)f(the)g(sp)s(ectrum)h(of)e(the)h(self-adjoin)m(t)f (extensions)j Fi(Q)3246 3556 y Fl(\034)3290 3592 y Fp(.)45 b(W)-8 b(e)34 b(shall)g(see)0 3767 y(that)i(all)g(of)g Fi(Q)546 3731 y Fl(\034)626 3767 y Fp(exp)s(ect)h Fi(Q)1015 3731 y Fj(0)1091 3767 y Fp(has)g(same)g(sp)s(ectrum.)56 b(This)37 b(is)g(di\013er)g(from)f(the)h(spin-less)h(case)f(treated)0 3941 y(in)c(Ref.)f([8].)44 b(W)-8 b(e)33 b(denote)g(the)g(sp)s(ectrum)h (of)e(a)g(op)s(erator)g Fi(T)47 b Fp(b)m(y)33 b Fi(\033)t Fp(\()p Fi(T)14 b Fp(\).)0 4178 y Fb(Theorem)38 b(8.)k Fc(We)35 b(have)f(that)637 4427 y Fi(\033)t Fp(\()p Fi(Q)811 4386 y Fj(0)851 4427 y Fp(\))28 b(=)1020 4317 y Fh(n)1087 4427 y Fg(\006)1164 4338 y(p)p 1247 4338 187 4 v 89 x Fp(2)p Fi(B)5 b(n)p Fp(;)17 b Fi(n)28 b Fg(2)g Ff(N)1729 4442 y Fj(0)1768 4317 y Fh(o)1857 4427 y Fg([)1945 4317 y Fh(n)2012 4427 y Fg(\006)2089 4337 y Fh(p)p 2189 4337 614 4 v 90 x Fp(2)p Fi(B)5 b Fp(\()p Fi(\013)23 b Fp(+)f Fi(n)g Fp(+)g(1\);)17 b Fi(n)27 b Fg(2)h Ff(N)3098 4442 y Fj(0)3138 4317 y Fh(o)3221 4427 y Fi(:)0 4677 y Fc(The)77 b(multiplicity)g(of)h Fg(\006)1032 4593 y(p)p 1115 4593 187 4 v 84 x Fp(2)p Fi(B)5 b(n)78 b Fc(is)f(in\014nity)g(for)g(al)5 b(l)77 b Fi(n)107 b Fg(2)g Ff(N)2697 4692 y Fj(0)2814 4677 y Fc(and)77 b(the)g(multiplicity)h(of)0 4851 y Fg(\006)77 4766 y Fh(p)p 177 4766 614 4 v 85 x Fp(2)p Fi(B)5 b Fp(\()p Fi(\013)23 b Fp(+)f Fi(n)g Fp(+)g(1\))35 b Fc(is)f Fi(n)23 b Fp(+)f(1)34 b Fc(for)h(al)5 b(l)35 b Fi(n)27 b Fg(2)i Ff(N)1740 4866 y Fj(0)1779 4851 y Fc(.)45 b(If)34 b Fi(\034)39 b Fg(2)28 b Fp(\(0)p Fi(;)17 b Fp(2)p Fi(\031)t Fp(\))p Fc(,)34 b(we)g(have)h(that)720 5101 y Fi(\033)t Fp(\()p Fi(Q)894 5060 y Fl(\034)937 5101 y Fp(\))28 b(=)1107 4990 y Fh(n)1173 5101 y Fg(\006)1250 5012 y(p)p 1333 5012 187 4 v 89 x Fp(2)p Fi(B)5 b(n)q Fp(;)17 b Fi(n)27 b Fg(2)h Ff(N)1815 5116 y Fj(0)1855 4990 y Fh(o)1943 5101 y Fg([)2032 4990 y Fh(n)2098 5101 y Fg(\006)2175 5011 y Fh(p)p 2275 5011 445 4 v 90 x Fp(2)p Fi(B)5 b Fp(\()p Fi(\013)23 b Fp(+)f Fi(n)p Fp(\);)17 b Fi(n)28 b Fg(2)g Ff(N)3016 5116 y Fj(0)3055 4990 y Fh(o)3138 5101 y Fi(:)0 5351 y Fc(The)35 b(multiplicity)h(of)f Fg(\006)906 5266 y(p)p 989 5266 187 4 v 85 x Fp(2)p Fi(B)5 b(n)36 b Fc(is)f(in\014nity)h(for)f(al)5 b(l)35 b Fi(n)30 b Fg(2)f Ff(N)2207 5366 y Fj(0)2282 5351 y Fc(and)35 b(the)g(multiplicity)h(of)f Fg(\006)3340 5266 y Fh(p)p 3441 5266 445 4 v 3441 5351 a Fp(2)p Fi(B)5 b Fp(\()p Fi(\013)22 b Fp(+)g Fi(n)p Fp(\))0 5525 y Fc(is)35 b Fi(n)22 b Fp(+)g(1)35 b Fc(for)f(al)5 b(l)35 b Fi(n)28 b Fg(2)g Ff(N)914 5540 y Fj(0)953 5525 y Fc(.)p eop %%Page: 11 11 11 10 bop 3788 -208 a Fp(11)125 88 y(T)-8 b(o)32 b(pro)m(v)m(e)i(this)f (theorem,)h(w)m(e)f(m)m(ust)h(examine)g(the)f(sp)s(ectra)g(of)p 2513 8 144 4 v 32 w Fi(Q)2590 103 y Fl(m)2690 88 y Fp(and)f Fi(Q)2956 52 y Fl(\034)2956 113 y Fk(\000)p Fj(1)3051 88 y Fp(.)0 345 y Fb(Prop)s(osition)38 b(9.)j Fc(If)35 b Fi(m)28 b Fg(2)g(\000)p Ff(N)1193 360 y Fj(2)1232 345 y Fc(,)1309 619 y Fi(\033)t Fp(\()p 1406 539 V Fi(Q)1483 634 y Fl(m)1549 619 y Fp(\))g(=)1719 509 y Fh(n)1785 619 y Fg(\006)1862 530 y(p)p 1945 530 187 4 v 89 x Fp(2)p Fi(B)5 b(n)q Fp(;)17 b Fi(n)27 b Fg(2)h Ff(N)2427 634 y Fj(0)2467 509 y Fh(o)2550 619 y Fi(:)0 893 y Fc(If)34 b Fi(m)28 b Fg(2)g Ff(N)381 908 y Fj(0)421 893 y Fc(,)984 1068 y Fi(\033)t Fp(\()p 1081 988 144 4 v Fi(Q)1158 1083 y Fl(m)1225 1068 y Fp(\))f(=)1394 957 y Fh(n)1460 1068 y Fg(\006)1537 978 y Fh(p)p 1637 978 820 4 v 90 x Fp(2)p Fi(B)5 b Fp(\()p Fi(m)23 b Fp(+)f Fi(\013)g Fp(+)g Fi(n)h Fp(+)f(1\))o(;)17 b Fi(n)28 b Fg(2)g Ff(N)2752 1083 y Fj(0)2792 957 y Fh(o)2875 1068 y Fi(:)0 1296 y Fc(In)34 b(b)-5 b(oth)35 b(c)-5 b(ases,)34 b(the)h(multiplicity)g(of)f(these)h (eigenvalues)f(is)g(1.)0 1553 y(Pr)-5 b(o)g(of.)49 b Fp(In)31 b(the)g(case)h(when)f Fi(m)d Fg(2)g(\000)p Ff(N)1405 1568 y Fj(2)1445 1553 y Fp(,)j(w)m(e)h(can)f(easily)g(see)h(that)e Fg(S)2513 1568 y Fj(2)2581 1553 y Fg(\032)e Fi(D)s Fp(\()p 2808 1473 144 4 v Fi(Q)2885 1568 y Fl(m)2951 1553 y Fp(\))j(and)f(the)h (set)h Fg(S)3584 1568 y Fj(2)3654 1553 y Fp(is)f(the)0 1728 y(set)k(of)f(the)h(eigenfunctions)h(of)e Fi(Q)1266 1743 y Fl(m)1368 1728 y Fp(and)g(complete)i(in)f Fg(H)q Fp(.)49 b(Th)m(us,)37 b(w)m(e)e(obtain)g(the)g(desired)h(result.)50 b(In)0 1902 y(the)33 b(case)g(when)h Fi(m)28 b Fg(2)g Ff(N)908 1917 y Fj(0)948 1902 y Fp(,)k(w)m(e)i(can)f(obtain)f(the)h (desired)h(result)g(in)f(a)f(similar)h(w)m(a)m(y)h(as)f(ab)s(o)m(v)m (e.)p 3815 1902 4 66 v 3819 1840 59 4 v 3819 1902 V 3877 1902 4 66 v 0 2159 a Fb(Prop)s(osition)38 b(10.)j Fc(We)35 b(have)f(that)1295 2433 y Fi(\033)t Fp(\()p Fi(Q)1469 2392 y Fj(0)1469 2458 y Fk(\000)p Fj(1)1563 2433 y Fp(\))28 b(=)1732 2323 y Fh(n)1799 2433 y Fg(\006)1876 2344 y(p)p 1959 2344 187 4 v 89 x Fp(2)p Fi(B)5 b(n)p Fp(;)17 b Fi(n)28 b Fg(2)g Ff(N)2441 2448 y Fj(0)2481 2323 y Fh(o)2564 2433 y Fi(;)1121 b Fp(\(11\))1179 2721 y Fi(\033)t Fp(\()p Fi(Q)1353 2679 y Fl(\031)1353 2745 y Fk(\000)p Fj(1)1447 2721 y Fp(\))28 b(=)1617 2610 y Fh(n)1683 2721 y Fg(\006)1760 2631 y Fh(p)p 1860 2631 445 4 v 90 x Fp(2)p Fi(B)5 b Fp(\()p Fi(\013)23 b Fp(+)f Fi(n)p Fp(\);)17 b Fi(n)28 b Fg(2)g Ff(N)2601 2736 y Fj(0)2640 2610 y Fh(o)3712 2721 y Fp(\(12\))0 2949 y Fc(If)34 b Fi(\034)39 b Fg(2)29 b Fp(\(0)p Fi(;)17 b(\031)t Fp(\))k Fg([)i Fp(\()p Fi(\031)t(;)17 b Fp(2)p Fi(\031)t Fp(\))p Fc(,)33 b(we)i(have)f(that)831 3223 y Fi(\033)t Fp(\()p Fi(Q)1005 3182 y Fl(\034)1005 3248 y Fk(\000)p Fj(1)1100 3223 y Fp(\))27 b(=)1269 3112 y Fh(n)1335 3223 y Fg(\006)1412 3133 y Fh(p)p 1512 3133 431 4 v 90 x Fp(2)p Fi(B)5 b Fp(\()p Fi(n)23 b Fp(+)f(1\))o Fi(;)17 b Fg(\006)2064 3133 y Fh(p)p 2164 3133 445 4 v 90 x Fp(2)p Fi(B)5 b Fp(\()p Fi(\013)23 b Fp(+)f Fi(n)p Fp(\);)17 b Fi(n)28 b Fg(2)g Ff(N)2905 3238 y Fj(0)2944 3112 y Fh(o)3027 3223 y Fi(:)658 b Fp(\(13\))0 3497 y Fc(In)34 b(al)5 b(l)35 b(c)-5 b(ases,)34 b(the)h(multiplicity)g(of)f (these)h(eigenvalues)e(is)i(1.)0 3754 y(Pr)-5 b(o)g(of.)49 b Fp(W)-8 b(e)33 b(\014rst)g(examine)i(the)e(p)s(oin)m(t)f(sp)s(ectrum) i(of)e Fi(Q)2094 3718 y Fl(\034)2094 3779 y Fk(\000)p Fj(1)2189 3754 y Fp(.)43 b(Note)33 b(that)1142 4098 y(\010\(\011)1326 4057 y Fj(1)1366 4098 y Fp(\))28 b(=)1535 3898 y Fh(0)1535 4077 y(@)1622 4021 y Fp(1)1622 4195 y(0)1671 3898 y Fh(1)1671 4077 y(A)1872 4098 y Fp(and)98 b(\010\(\011)2311 4057 y Fj(2)2351 4098 y Fp(\))27 b(=)2520 3898 y Fh(0)2520 4077 y(@)2607 4021 y Fp(0)2607 4195 y(1)2656 3898 y Fh(1)2656 4077 y(A)3712 4098 y Fp(\(14\))0 4430 y(for)32 b(all)h Fi(E)g Fg(2)28 b Ff(C)p Fp(.)125 4646 y(In)34 b(the)g(case)h(when)g Fi(\034)41 b Fp(=)30 b(0,)j(the)i(functions)f(\011)c Fg(2)g Fi(D)s Fp(\()p Fi(Q)2171 4610 y Fj(0)2171 4671 y Fk(\000)p Fj(1)2266 4646 y Fp(\))j(m)m(ust)i(satisfy)g(BC,)f (\010\(\011\))c(=)3444 4446 y Fh(0)3444 4625 y(@)3531 4569 y Fp(0)3531 4743 y(1)3580 4446 y Fh(1)3580 4625 y(A)3667 4646 y Fp(,)k(and)0 4901 y(\011)76 4865 y Fj(2)150 4901 y Fg(2)i(H)i Fp(if)e(and)h(only)h(if)e Fi(E)41 b Fp(=)35 b Fg(\006)1274 4817 y(p)p 1358 4817 187 4 v 1358 4901 a Fp(2)p Fi(B)5 b(n)36 b Fp(with)i Fi(n)d Fg(2)h Ff(N)2074 4916 y Fj(0)2150 4901 y Fp(as)h(w)m(e)h(already)f(see)h(in)f (Sec.)h(I)s(I)s(I.)56 b(Therefore,)0 5075 y(the)33 b(set)g Fg(S)380 5090 y Fj(2)453 5075 y Fp(de\014ned)h(in)e(Prop)s(osition)h(4) g(is)g(a)f(subset)i(of)e Fi(D)s Fp(\()p Fi(Q)2297 5039 y Fj(0)2297 5100 y Fk(\000)p Fj(1)2392 5075 y Fp(\).)43 b(Moreo)m(v)m(er,)34 b(the)f(set)h Fg(S)3336 5090 y Fj(2)3408 5075 y Fp(is)f(complete)0 5250 y(in)i Fg(H)q Fp(.)49 b(Th)m(us,)36 b(w)m(e)g(obtain)e(Eq.)h(\(11\).)49 b(In)34 b(the)h(case)h(when)f Fi(\034)42 b Fp(=)31 b Fi(\031)t Fp(,)k(w)m(e)g(can)g(obtain)f(Eq.)i(\(12\))d(in)i(a)f(w)m(a)m(y)0 5424 y(similar)f(as)g(ab)s(o)m(v)m(e.)p eop %%Page: 12 12 12 11 bop 3788 -208 a Fp(12)125 83 y(W)-8 b(e)35 b(consider)h(the)f (case)h(when)f Fi(\034)43 b Fg(6)p Fp(=)31 b(0)k(and)g Fi(\034)43 b Fg(6)p Fp(=)31 b Fi(\031)t Fp(.)50 b(Since)35 b(the)h(de\014ciency)h(index)f(of)e Fi(Q)3439 47 y Fl(\034)3439 108 y Fk(\000)p Fj(1)3568 83 y Fp(is)i(\014nite)0 257 y(and)45 b(the)h(essen)m(tial)h(sp)s(ectrum)f(of)e Fi(Q)1422 221 y Fj(0)1422 282 y Fk(\000)p Fj(1)1562 257 y Fp(is)i(empt)m(y)-8 b(,)49 b(the)d(Krein's)g(form)m(ula)f([3])g(join)m(tly)g(with)h(W)-8 b(eyl's)0 432 y(theorem)30 b(\(Theorem)h(XI)s(I)s(I.14)e(in)g(Ref.)h ([13)o(]\))f(tells)i(us)f(that)f(the)g(essen)m(tial)j(sp)s(ectrum)e(of) f Fi(Q)3382 396 y Fl(\034)3382 456 y Fk(\000)p Fj(1)3506 432 y Fp(is)h(empt)m(y)-8 b(.)0 606 y(Th)m(us,)34 b(w)m(e)g(examine)g (the)f(p)s(oin)m(t)g(sp)s(ectrum)h(of)e(it.)43 b(Let)982 833 y Fi(\014)33 b Fp(=)28 b Fi(k)d Fp(+)d(1)g Fg(\000)1534 766 y Fi(E)1612 730 y Fj(2)p 1528 810 128 4 v 1528 902 a Fp(2)p Fi(B)1666 833 y(;)115 b(\015)1859 848 y Fj(+)1945 833 y Fp(=)27 b(1)22 b(+)g Fi(k)s(;)115 b(\015)2464 848 y Fk(\000)2550 833 y Fp(=)27 b(2)22 b(+)g Fi(k)s(;)0 1061 y Fp(and)419 1235 y(\011)495 1194 y Fj(3)534 1235 y Fp(\()p Fi(E)6 b Fp(;)17 b Fi(r)s Fp(\))27 b(=)h(i)p Fi(E)1032 1095 y Fh(\022)1115 1168 y Fi(B)p 1115 1212 80 4 v 1130 1303 a Fp(2)1204 1095 y Fh(\023)1277 1117 y Fl(\013)p Fk(\000)p Fj(1)1534 1168 y Fp(\000\(1)22 b Fg(\000)h Fi(\015)1855 1183 y Fj(+)1913 1168 y Fp(\))p 1444 1212 598 4 v 1444 1303 a(\000\()p Fi(\014)k Fg(\000)c Fi(\015)1776 1318 y Fj(+)1857 1303 y Fp(+)f(1\))2051 1235 y(\011)2127 1194 y Fj(1)2167 1235 y Fp(\()p Fi(E)6 b Fp(;)17 b Fi(r)s Fp(\))k(+)h(2)p Fi(\013)2653 1168 y Fp(\000\()p Fi(\015)2803 1183 y Fj(+)2883 1168 y Fg(\000)h Fp(1\))p 2653 1212 418 4 v 2762 1303 a(\000\()p Fi(\014)6 b Fp(\))3080 1235 y(\011)3156 1194 y Fj(2)3195 1235 y Fp(\()p Fi(E)g Fp(;)17 b Fi(r)s Fp(\))p Fi(:)0 1443 y Fp(Then,)34 b(w)m(e)f(ha)m(v)m(e)h(that)688 1728 y(\011)764 1687 y Fj(3)803 1728 y Fp(\()p Fi(E)6 b Fp(;)17 b Fi(r)s Fp(\))27 b(=)1178 1528 y Fh(0)1178 1707 y(@)1266 1651 y Fp(2)p Fi(\013)q Fp(i)p Fi(E)22 b Fp(\()p Fi(B)5 b(=)p Fp(2\))1751 1603 y Fl(\013)p Fk(\000)p Fj(1)1908 1651 y Fi(r)1955 1615 y Fl(\013)p Fk(\000)p Fj(1)2094 1651 y Fi(G)p Fp(\()p Fi(\014)h(;)17 b(\015)2365 1666 y Fj(+)2423 1651 y Fi(;)g(B)5 b(r)2593 1615 y Fj(2)2632 1651 y Fi(=)p Fp(2\))p Fi(e)2813 1615 y Fk(\000)p Fl(B)s(r)2958 1591 y Fd(2)2992 1615 y Fl(=)p Fj(4)1370 1825 y Fg(\000)p Fp(4)p Fi(\013)17 b Fp(\()p Fi(B)5 b(=)p Fp(2\))1828 1777 y Fl(\013)1894 1825 y Fi(r)1941 1789 y Fl(\013)1990 1825 y Fi(G)p Fp(\()p Fi(\014)h(;)17 b(\015)2261 1840 y Fk(\000)2319 1825 y Fi(;)g(B)5 b(r)2489 1789 y Fj(2)2528 1825 y Fi(=)p Fp(2\))p Fi(e)2709 1789 y Fk(\000)p Fl(B)s(r)2854 1766 y Fd(2)2888 1789 y Fl(=)p Fj(4)3067 1528 y Fh(1)3067 1707 y(A)3171 1728 y Fi(:)0 2023 y Fp(This)40 b(\011)305 1987 y Fj(3)383 2023 y Fp(b)s(eha)m(v)m(es)h(w)m(ell)f(at)e(in\014nit)m (y)-8 b(,)42 b(that)c(is,)j(\011)1871 1987 y Fj(3)1949 2023 y Fp(is)e Fi(L)2119 1987 y Fj(2)2198 2023 y Fp(with)g(resp)s(ect)h (to)e Fi(r)19 b(dr)42 b Fp(at)c(in\014nit)m(y)-8 b(,)42 b(for)c(an)m(y)0 2197 y Fi(E)c Fg(2)28 b Ff(C)p Fp(.)43 b(Ho)m(w)m(ev)m(er,)35 b(in)e(general,)g(it)g(do)s(es)g(not)f(satisfy)i (the)f(b)s(oundary)g(condition)g(\(9\))f(for)g Fi(Q)3453 2161 y Fl(\034)3453 2222 y Fk(\000)p Fj(1)3548 2197 y Fp(:)1119 2554 y(\010\(\011)1303 2513 y Fj(3)1343 2554 y Fp(\()p Fi(E)6 b Fp(;)17 b Fi(r)s Fp(\)\))27 b(=)1756 2294 y Fh(0)1756 2469 y(B)1756 2529 y(B)1756 2593 y(@)1876 2436 y Fp(i)p Fi(E)2107 2368 y Fp(\000\(1)22 b Fg(\000)g Fi(\013)q Fp(\))p 1991 2413 602 4 v 1991 2504 a(\000\(1)g Fg(\000)h Fi(E)2339 2475 y Fj(2)2378 2504 y Fi(=)p Fp(2)p Fi(B)5 b Fp(\))1843 2671 y Fg(\000)p Fi(B)2218 2604 y Fp(\000\()p Fi(\013)q Fp(\))p 2009 2648 616 4 v 2009 2739 a(\000\()p Fi(\013)24 b Fg(\000)e Fi(E)2371 2711 y Fj(2)2411 2739 y Fi(=)p Fp(2)p Fi(B)5 b Fp(\))2635 2294 y Fh(1)2635 2469 y(C)2635 2529 y(C)2635 2593 y(A)2739 2554 y Fi(:)0 2904 y Fp(In)34 b(the)f(other)h(hand,)f(since)i(Eq.)f (\(14\))f(holds,)h(c)m(ho)s(osing)g(go)s(o)s(d)e(co)s(e\016cien)m(ts,)k (the)d(linear)h(com)m(binations)0 3078 y(of)39 b(\011)194 3042 y Fj(1)233 3078 y Fp(,)i(\011)377 3042 y Fj(2)456 3078 y Fp(and)e(\011)728 3042 y Fj(3)806 3078 y Fp(satisfy)i(the)e(b)s (oundary)h(condition)g(\(9\))f(at)f(the)i(origin.)63 b(Ho)m(w)m(ev)m(er,)44 b(\011)3476 3042 y Fj(1)3554 3078 y Fp(b)s(eha)m(v)m(es)0 3253 y(w)m(ell)34 b(at)f(in\014nit)m(y)i(if)e (and)g(only)h(if)f Fi(\013)23 b Fg(\000)g Fi(E)1503 3217 y Fj(2)1543 3253 y Fi(=)p Fp(2)p Fi(B)33 b Fg(2)c(\000)p Ff(N)1992 3268 y Fj(0)2066 3253 y Fp(and)k(\011)2332 3217 y Fj(2)2404 3253 y Fp(b)s(eha)m(v)m(es)j(w)m(ell)e(at)f(in\014nit) m(y)i(if)e(and)g(only)0 3427 y(if)f(1)22 b Fg(\000)h Fi(E)338 3391 y Fj(2)377 3427 y Fi(=)p Fp(2)p Fi(B)33 b Fg(2)28 b(\000)p Ff(N)825 3442 y Fj(0)865 3427 y Fp(.)43 b(Therefore,)34 b(w)m(e)g(ha)m(v)m(e)g(that)830 3654 y Fi(\033)t Fp(\()p Fi(Q)1004 3613 y Fl(\034)1004 3679 y Fk(\000)p Fj(1)1099 3654 y Fp(\))28 b Fg(\033)1270 3544 y Fh(n)1336 3654 y Fg(\006)1413 3564 y Fh(p)p 1513 3564 431 4 v 90 x Fp(2)p Fi(B)5 b Fp(\()p Fi(n)22 b Fp(+)g(1\))p Fi(;)17 b Fg(\006)2065 3564 y Fh(p)p 2165 3564 445 4 v 90 x Fp(2)p Fi(B)5 b Fp(\()p Fi(\013)23 b Fp(+)f Fi(n)p Fp(\);)17 b Fi(n)27 b Fg(2)i Ff(N)2906 3669 y Fj(0)2945 3544 y Fh(o)3028 3654 y Fi(:)0 3882 y Fp(Since)36 b(all)g (eigenfunctions)h(is)f(giv)m(en)g(b)m(y)g(a)f(linear)h(com)m(bination)g (of)f(\011)2638 3846 y Fj(1)2677 3882 y Fp(,)h(\011)2816 3846 y Fj(2)2891 3882 y Fp(and)f(\011)3159 3846 y Fj(3)3199 3882 y Fp(,)h(w)m(e)g(can)g(see)g(the)0 4056 y(rev)m(erse)f(inclusion)f (relation.)p 3815 4056 4 66 v 3819 3994 59 4 v 3819 4056 V 3877 4056 4 66 v 125 4292 a(No)m(w,)f(w)m(e)g(are)g(in)g(the)g(p)s (osition)g(to)f(pro)m(v)m(e)i(Theorem)g(8.)0 4528 y Fc(Pr)-5 b(o)g(of)35 b(of)f(The)-5 b(or)g(em)34 b(8.)49 b Fp(Since)34 b(w)m(e)g(ha)m(v)m(e)g(that)1289 4755 y Fi(\033)t Fp(\()p Fi(Q)1463 4714 y Fl(\034)1507 4755 y Fp(\))27 b(=)p 1676 4638 920 4 v 1724 4660 a Fh([)1703 4872 y Fl(m)p Fk(2)p Fe(Z)1676 4931 y Fl(m)p Fk(6)p Fj(=)p Fk(\000)p Fj(1)1900 4755 y Fi(\033)t Fp(\()p 1997 4675 144 4 v Fi(Q)2074 4770 y Fl(m)2141 4755 y Fp(\))22 b Fg([)g Fi(\033)t Fp(\()p Fi(Q)2463 4721 y Fl(\034)2463 4779 y Fk(\000)p Fj(1)2558 4755 y Fp(\))0 5090 y(b)m(y)32 b(Eq.)f(\(10\),)g(w)m(e)h(can)f(easily)h (obtain)f(the)g(desired)h(result)g(using)f(Prop)s(osition)g(9)g(and)g (Prop)s(osition)g(10.)p 3815 5264 4 66 v 3819 5202 59 4 v 3819 5264 V 3877 5264 4 66 v 0 5500 a Fc(R)-5 b(emark)45 b Fp(11)p Fc(.)c Fp(W)-8 b(e)33 b(remark)g(that)g(\011)1330 5464 y Fk(\006)p Fj(i)1408 5500 y Fp(\()p Fi(r)s Fp(\))27 b(=)h(const)p Fi(:)23 b Fg(\002)f Fp(\011)2109 5464 y Fj(3)2149 5500 y Fp(\()p Fg(\006)p Fp(i;)17 b Fi(r)s Fp(\))p Fi(:)p eop %%Page: 13 13 13 12 bop 3788 -208 a Fp(13)733 83 y Fm(VI.)104 b(DISCUSSIOONS)34 b(AND)g(FUTURE)i(PR)m(OBLEMS)125 375 y Fp(In)d(our)h(case,)h(the)e (generalized)i(BC)f(of)f(a)h(Dirac-W)-8 b(eyl)33 b(op)s(erator)g Fi(Q)h Fp(is)g(giv)m(en)h(b)m(y)f(Remark)g(7.)46 b(This)0 550 y(BC)f(is)f(v)m(ery)i(simpler)f(than)f(the)h(generalized)g(BC)g(of) f(the)g(spin-less)i(Sc)m(hr\177)-49 b(odinger)45 b(op)s(erator)f Fi(H)3721 565 y Fj(0)3804 550 y Fp(in)0 724 y(Ref.)c([8].)66 b(Moreo)m(v)m(er,)43 b(w)m(e)e(can)g(obtain)f Fi(\033)t Fp(\()p Fi(Q)1695 688 y Fl(\034)1738 724 y Fp(\))g(explicitly)i(as)e (in)h(Theorem)g(8.)66 b(Ho)m(w)m(ev)m(er,)44 b(the)d(exact)0 899 y(v)-5 b(alues)39 b(of)f Fi(\033)t Fp(\()p Fi(H)591 914 y Fj(0)630 899 y Fp(\))g(can)h(not)f(b)s(e)h(calculated)g(in)g (general,)h(as)f(men)m(tioned)g(in)g(Ref.)f([8].)61 b(Th)m(us,)41 b(so)e(do)s(es)0 1073 y(the)h(sp)s(ectrum)g(of)f Fi(H)806 1088 y Fj(1)p Fl(=)p Fj(2)916 1073 y Fp(.)63 b(Ho)m(w)m(ev)m(er,)43 b(w)m(e)e(ha)m(v)m(e)f(naiv)m(ely)h(that)e Fi(H)2455 1088 y Fj(1)p Fl(=)p Fj(2)2604 1073 y Fp(=)g Fi(Q)2796 1037 y Fj(2)2835 1073 y Fp(,)i(th)m(us)g(for)d(all)h Fi(\034)51 b Fg(2)39 b Fp([0)p Fi(;)17 b Fp(2)p Fi(\031)t Fp(\))0 1247 y(there)38 b(exists)i(a)d(self-adjoin)m(t)h(extension)h Fi(H)1647 1211 y Fl(\034)1728 1247 y Fp(of)e Fi(H)1925 1263 y Fj(1)p Fl(=)p Fj(2)2072 1247 y Fp(suc)m(h)i(that)f Fi(H)2603 1211 y Fl(\034)2682 1247 y Fp(=)e(\()p Fi(Q)2909 1211 y Fl(\034)2952 1247 y Fp(\))2990 1211 y Fj(2)3030 1247 y Fp(.)58 b(Therefore,)40 b(w)m(e)f(can)0 1422 y(obtain)32 b(the)h(sp)s(ectrum)h(of)e(suc)m(h)i Fi(H)1317 1385 y Fl(\034)1392 1422 y Fp(explicitly)-8 b(.)46 b(These)34 b(situations)f(suggest)h(that)e(the)h(spin)g(structure)0 1596 y(strongly)41 b(restrict)g(the)g(self-adjoin)m(t)f(extensions)j (of)c(a)h(Dirac-W)-8 b(eyl)41 b(op)s(erator.)65 b(It)41 b(is)f(a)g(natural)g(and)0 1770 y(future)46 b(problem)h(to)f(iden)m (tify)h(the)f(relation)g(b)s(et)m(w)m(een)i(of)e Fi(Q)2341 1734 y Fl(\034)2430 1770 y Fp(w)m(e)h(obtained,)j(and)c(the)g (self-adjoin)m(t)0 1945 y(extensions)35 b(of)d Fi(H)662 1960 y Fj(1)p Fl(=)p Fj(2)772 1945 y Fp(,)g(whic)m(h)j(w)m(e)e(can)g (obtain)f(from)h(the)g(Exner)h(et.)f(al.'s)g(results.)1529 2314 y Fm(Ac)m(kno)m(wledgemen)m(t)125 2607 y Fp(This)h(w)m(ork)g(is)g (supp)s(orted)h(b)m(y)f(the)g(Gran)m(t-In-Aid)f(14740113)e(for)i (Encouragemen)m(t)i(of)e(Y)-8 b(oung)33 b(Sci-)0 2781 y(en)m(tists)h(from)f(Japan)f(So)s(ciet)m(y)i(for)e(the)h(Promotion)g (of)f(Science)i(\(JSPS\).)p 924 2999 2039 5 v 1178 3000 1529 7 v 1433 3001 1020 9 v 1688 3002 510 11 v 45 3335 a Fo([1])51 b(R.)34 b(Adami)h(and)e(A.)i(T)-8 b(eta.)53 b(On)33 b(the)h(Aharono)m(v-Bohm)h(Hamiltonian.)53 b Fa(L)-5 b(ett.)37 b(Math.)f(Phys.)p Fo(,)g(43:43{53,)191 3516 y(1998.)45 3696 y([2])51 b(Y.)28 b(Aharono)m(v)g(and)e(D.)i(Bohm.) 36 b(Signi\014cance)28 b(of)f(electromagnetic)k(p)s(oten)m(tials)d(in)f (the)h(quan)m(tum)e(theory)-8 b(.)191 3877 y Fa(Phys.)33 b(R)-5 b(ev.)33 b(A)f(\(3\))p Fo(,)f(115:485{491,)36 b(1959.)45 4057 y([3])51 b(N.)31 b(I.)f(Akhiezer)h(and)f(I.)h(M.)f (Glazman.)42 b Fa(The)-5 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3788 -208 a Fp(14)45 83 y Fo([7])51 b(L.)25 b(D)s(\030)-43 b(abro)m(wski)26 b(and)f(P)-8 b(.)1021 60 y(\024)1019 83 y(S)1065 66 y(\024)1070 83 y(to)m(v)e(\023)-35 b(\020)m(\024)-43 b(cek.)35 b(Aharono)m(v-Bohm)26 b(e\013ect)h(with)d Fn(\016)s Fo(-t)m(yp)s(e)j(in)m(teraction.)34 b Fa(J.)28 b(Math.)g(Phys.)p Fo(,)191 264 y(39\(1\):47{62,)35 b(1998.)43 b(DOI:10.1063/1.532307)q(.)45 444 y([8])51 b(P)-8 b(.)38 b(Exner,)g(P)-8 b(.)727 421 y(\024)725 444 y(St'o)m(v)e(\023)-35 b(\020)m(\024)-43 b(cek,)42 b(and)36 b(P)-8 b(.)38 b(Vyt)-5 b(\024)-41 b(ras.)62 b(Generalized)39 b(b)s(oundary)c(conditions)i(for) g(the)h(Aharono)m(v-)191 625 y(Bohm)31 b(e\013ect)i(com)m(bined)e(with) g(a)g(homogeneous)h(magnetic)g(\014eld.)43 b Fa(J.)32 b(Math.)i(Phys.)p Fo(,)e(43\(5\):2151{2168,)191 805 y(2002.)43 b(DOI:)30 b(10.1063/1.14637)q(12.)45 986 y([9])51 b(C.)32 b(R.)g(Hagen.)45 b(Aharono)m(v-Bohm)33 b(scattering)g(of)f(particles)h (with)e(spin.)44 b Fa(Phys.)34 b(R)-5 b(ev.)34 b(L)-5 b(ett.)p Fo(,)32 b(64\(5\):503{)191 1166 y(506,)g(1990.)0 1347 y([10])51 b(Masao)32 b(Hirok)-5 b(a)m(w)m(a)32 b(and)e(Osam)m(u)f (Ogurisu.)40 b(Ground)29 b(states)i(of)f(a)h(spin-1/2)g(c)m(harged)g (particle)g(in)f(a)h(t)m(w)m(o-)191 1528 y(dimensional)g(magnetic)h (\014eld.)40 b Fa(J.)32 b(Math.)h(Phys.)p Fo(,)e(42\(8\):3334{3343,)37 b(2001.)42 b(DOI:10.1063/1.13793)q(12)q(.)0 1708 y([11])51 b(M.)42 b(N.)f(Hounk)m(onnou)g(and)f(G.)h(Y.)h(H.)f(Av)m(ossev)m(ou.)74 b(Sp)s(ectral)41 b(and)f(resonance)i(prop)s(erties)e(of)h Fn(\016)k Fo(and)191 1889 y Fn(\016)234 1856 y Fk(0)258 1889 y Fo(-t)m(yp)s(e)35 b(in)m(teractions)h(in)e(relativistic)i(quan)m (tum)e(mec)m(hanics.)52 b Fa(J.)36 b(Math.)g(Phys.)p Fo(,)g(42\(1\):30{51,)k(1)34 b(2001.)191 2069 y(DOI:)d (10.1063/1.131606)q(0.)0 2250 y([12])51 b(Mic)m(hael)33 b(Reed)f(and)e(Barry)h(Simon.)42 b Fa(F)-7 b(unctional)35 b(A)n(nalysis)p Fo(,)c(v)m(olume)h(1)g(of)f Fa(Metho)-5 b(ds)35 b(of)e(Mo)-5 b(dern)34 b(Math-)191 2430 y(ematic)-5 b(al)34 b(Physics)p Fo(.)41 b(Academic,)32 b(New)f(Y)-8 b(ork,)31 b(1972.)0 2611 y([13])51 b(Mic)m(hael)46 b(Reed)f(and)f (Barry)g(Simon.)81 b Fa(A)n(nalysis)46 b(of)f(Op)-5 b(er)g(ators)p Fo(,)51 b(v)m(olume)45 b(4)g(of)f Fa(Metho)-5 b(ds)47 b(of)e(Mo)-5 b(dern)191 2792 y(Mathematic)g(al)35 b(Physics)p Fo(.)41 b(Academic,)32 b(New)e(Y)-8 b(ork,)32 b(1978.)0 2972 y([14])51 b(S.)26 b(N.)h(M.)g(Ruijsenaars.)34 b(The)25 b(Aharono)m(v-Bohm)j(e\013ect)g(and)d(scattering)j(theory)-8 b(.)35 b Fa(A)n(nn.)28 b(Physics)p Fo(,)g(146:1{)191 3153 y(34,)k(1983.)0 3333 y([15])51 b(Ic)m(hiro)29 b(Shigek)-5 b(a)m(w)m(a.)38 b(Sp)s(ectral)28 b(prop)s(erties)f(of)i(Sc)m(hr\177)-45 b(odinger)27 b(op)s(erators)h(with)g(magnetic)i(\014elds)d(for)h(a)g (spin)191 3514 y(1/2)k(particle.)42 b Fa(J.)32 b(F)-7 b(unct.)33 b(A)n(nal.)p Fo(,)d(101:255{285,)35 b(1991.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0211282240242--