Content-Type: multipart/mixed; boundary="-------------0003020209801" This is a multi-part message in MIME format. ---------------0003020209801 Content-Type: text/plain; name="00-96.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-96.keywords" Schroedinger operators, singular perturbation theory, Stark effect, magnetic fields. ---------------0003020209801 Content-Type: application/postscript; name="asif.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="asif.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.85 Copyright 1999 Radical Eye Software %%Title: asif.dvi %%Pages: 36 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips asif %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.03.02:0723 %%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/IE 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 IE 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/IE 256 array N 2 string 0 1 255{IE 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 TeXDict begin 40258431 52099146 1000 600 600 (asif.dvi) @start %DVIPSBitmapFont: Fa cmbx6 6 2 /Fa 2 100 df<3803FFC0000F13F8003F7FEB80FF80EC3F80A2EA0E00C7FC90B5FC1207 381FF03FEA7F80EAFE005AA36C137F007F13FF393FFFDFFC6C139F3803FC071E167E9522 >97 D99 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb cmsy6 6 1 /Fb 1 1 df0 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc cmex8 8 1 /Fc 1 81 df80 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmr6 6 6 /Fd 6 62 df<1438B2B712FEA3C70038C7FCB227277C9F2F>43 D<13FF000313C0380781 E0380F00F0001E137848133CA248131EA400F8131FAD0078131EA2007C133E003C133CA2 6C13786C13F0380781E03803FFC0C6130018227DA01E>48 D<13E01201120712FF12F912 01B3A7487EB512C0A212217AA01E>II<13FF 000313C0380F03E0381C00F014F8003E13FC147CA2001E13FC120CC712F8A2EB01F0EB03 E0EB0FC03801FF00A2380003E0EB00F01478147C143E143F1230127812FCA2143E48137E 0060137C003813F8381E03F0380FFFC00001130018227DA01E>I61 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe msbm8 8 1 /Fe 1 83 df82 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmbx8 8 7 /Ff 7 122 df48 D82 D97 D99 D117 D120 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmmi6 6 5 /Fg 5 109 df<13F8D803FE1318487E48EB803048EBC060EA3C03397000E0C000601360 48EB71801431C7FCEC3300141B141EA2141CA31418A31438A214301470A45CA35CA21D21 7E9520>13 D 34 D<140C141C143C1438A21478147014F014E0130114C0A21303148013071400A25B13 0E131E131CA2133C13381378137013F05BA212015B12035BA2120790C7FC5A120EA2121E 121C123C123812781270A212F05AA216317CA420>61 D<1418143C147CA214381400A7EB 0780EB1FE01338EB60F013C0A2EA0180A2380001E0A4EB03C0A4EB0780A4EB0F00A4131E A21238EA783CEAF8381378EA70F0EA7FC0001FC7FC162D81A119>106 D108 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh cmex10 12 5 /Fh 5 91 df<1B3FF3FFC0973803E0F0973807C03897380F801C081F137E97383F01FEF3 03FF1A7E1AFE1AFC1901A2963903F801FEF300781C004F5AA2190FA262191FA34F5AA44F 5AA319FF97C8FCA360A261A21803A3611807A461180FA44E5AA4183FA261A2187FA361A2 18FFA44D5BA55F96C9FCA35FA360A2170FEE01FF160F047F7F93B67E0303011F13E0DB07 F880DB1FC0EBFDFCDB3F80EBFCFEDB7E00EBF83F4B013F6D7E4A486F7E4B15074A486F7E 0207834B90387FF0014A486F7EA24AC7157CA2021E4A48133CA2023E173EA2023C49151E 60A55E023E4B133EA2021E173CA2021F4990C7127CA2DA0F805EA2DA07C04B5A6F484813 0302035FDA01F04B5A03F8150FDA00FC4B5A037E4949C8FCDB3F9F14FEDB1FDF495ADB07 FFEB0FF06FEC7FE0030090B512807091C9FC7013F818C005F8CAFC5FA2167FA35FA316FF 5FA54B5BA494CBFCA25DA35EA21507A25EA44B5AA45E151FA45E153FA35EA2157FA25EA3 15FF93CCFCA34A5AA44A5AA35D14075DA2140F5D121E397F801FC0EAFFC05D143F92CDFC 5C147E6C485AEA7E00383801F86C485A380F07C03803FF80D800FCCEFC58DD7B7F37>73 D80 D<17FF040313C093380F81F093381E0078043E137C93387C 01FC9338F803FEA2150116F01503EF01FC9338E0007003071400A3150FA45E151FA7153F A74B5AA715FFA85C93C8FCA95C5DA85DA74A5AA75DA75D140FA45DA3001C5C007F131FEA FF8092C9FC5C143EA26C485A007C5B003C5B381F03E03807FF80D801FECAFC376F7B7F2F >82 D88 D<1B3FF3FFC0973803E0F0973807C03897 380F801C081F137E97383F01FEF303FF1A7E1AFE1AFC1901A2963903F801FEF300781C00 4F5AA2190FA262191FA34F5AA44F5AA319FF97C8FCA360A261A21803A3611807A461180F A44E5AA4183FA261A2187FA361A218FFA44D5BA55F96C9FCA35FA360A2170FA460171FA4 60173FA54D5AA54D5AA45E60A55E60A44C90CAFCA54C5AA55F161FA45F163FA45FA2167F A35FA316FF5FA54B5BA494CBFCA25DA35EA21507A25EA44B5AA45E151FA45E153FA35EA2 157FA25EA315FF93CCFCA34A5AA44A5AA35D14075DA2140F5D121E397F801FC0EAFFC05D 143F92CDFC5C147E6C485AEA7E00383801F86C485A380F07C03803FF80D800FCCEFC58DD 7B7F37>90 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fi cmsy8 8 11 /Fi 11 113 df0 D<123C127E12FFA4127E123C08087A9414>I< 130C131EA50060EB01800078130739FC0C0FC0007FEB3F80393F8C7F003807CCF83801FF E038007F80011EC7FCEB7F803801FFE03807CCF8383F8C7F397F0C3F8000FCEB0FC03978 1E078000601301000090C7FCA5130C1A1D7C9E23>3 D20 D<12E012F812FEEA3F80EA0F E0EA03F8EA00FEEB3F80EB0FE0EB03F8EB00FC143FEC0FC0EC07F0EC01FCEC007FED1FC0 ED07F0ED01FCED007FEE1FC01607161FEE7F00ED01FCED07F0ED1FC0037FC7FCEC01FCEC 07F0EC0FC0023FC8FC14FCEB03F8EB0FE0EB3F8001FEC9FCEA03F8EA0FE0EA3F80007ECA FC12F812E0CBFCAD007FB71280B812C0A22A3B7AAB37>I<170EA3170F83841703841701 84717E1878187C84180FF007C0BA12F819FC19F8CBEA07C0F00F00183E601878604D5A60 170360170795C7FC5F170EA33E237CA147>33 D<137813FE1201A3120313FCA3EA07F8A3 13F0A2EA0FE0A313C0121F1380A3EA3F00A3123E127E127CA35AA35A0F227EA413>48 DI<91B512C01307131FD97F80C7FC01FCC8FCEA01F0EA03 C0485A48C9FC120E121E5A123812781270A212F05AA3B712C0A300E0C9FCA37E1270A212 781238123C7E120E120F6C7E6C7EEA01F0EA00FCEB7F80011FB512C013071300222B7AA5 2F>I<12E0B3B3B3AD034378B114>106 D<18031807180F180E181E181C183C1838187818 7018F018E01701EF03C01880170718005F170E171E171C173C17381778177017F05F1601 5F16035F160701C092C7FC486C5C0007151E486C141C003F153CD873F8143800E31578D8 01FC147016F06C6C5C1501017F5C1503D93F805B1507D91FC090C8FC5D90380FE00E151E 903807F01C153C903803F83815786D6C5A5DEB00FF5D147F5D143F92C9FC80141E140E38 427C823B>112 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fj cmsy10 12 31 /Fj 31 115 df<007FB912E0BA12F0A26C18E03C04789A4D>0 D<121FEA3F80EA7FC0EA FFE0A5EA7FC0EA3F80EA1F000B0B789E1C>I<16C04B7EB3AC007FBA1280BB12C0A26C19 80C8D801E0C9FCB3A9007FBA1280BB12C0A26C198042427BC14D>6 D8 D<007FBA1280BB12C0A26C1980 CEFCB0007FBA1280BB12C0A26C1980CEFCB0007FBA1280BB12C0A26C1980422C7BAE4D> 17 D<19E0F003F0180FF03FE0F0FF80943803FE00EF0FF8EF3FE0EFFF80DC03FEC7FCEE 0FF8EE3FE0EEFF80DB03FEC8FCED1FF8ED7FE0913801FF80DA07FEC9FCEC1FF0EC7FC049 48CAFCEB07FCEB1FF0EB7FC04848CBFCEA07FCEA1FF0EA7FC048CCFCA2EA7FC0EA1FF0EA 07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FC913801FF809138 007FE0ED1FF8ED07FE923800FF80EE3FE0EE0FF8EE03FE933800FF80EF3FE0EF0FF8EF03 FE943800FF80F03FE0F00FF01803F000E01900B0007FB912E0BA12F0A26C18E03C4E78BE 4D>20 D<127012FCB4FCEA7FC0EA1FF0EA07FCEA01FF38007FC0EB1FF0EB07FCEB01FF90 38007FC0EC1FF0EC07FC913801FF809138007FE0ED1FF8ED07FE923800FF80EE3FE0EE0F F8EE03FE933800FF80EF3FE0EF0FF8EF03FE943800FF80F03FE0F00FF0A2F03FE0F0FF80 943803FE00EF0FF8EF3FE0EFFF80DC03FEC7FCEE0FF8EE3FE0EEFF80DB03FEC8FCED1FF8 ED7FE0913801FF80DA07FEC9FCEC1FF0EC7FC04948CAFCEB07FCEB1FF0EB7FC04848CBFC EA07FCEA1FF0EA7FC048CCFC12FC1270CDFCB0007FB912E0BA12F0A26C18E03C4E78BE4D >I24 D<037FB612E00207B712F0143F91B812E00103 01C0C9FCD907FCCAFCEB0FE0EB3F8049CBFC13FC485A485A485A5B485A121F90CCFC123E A2123C127CA2127812F8A25AA87EA21278127CA2123C123EA27E7F120F6C7E7F6C7E6C7E 6C7E137E6D7EEB1FE0EB07FC6DB47E010090B712E0023F16F01407020016E03C3A78B54D >26 D<007FB612F0B712FEEEFFC06C16F0C9EA1FFCEE03FE9338007F80EF1FC0EF07E071 7E717E717E187E183E841980180FF007C0A2180319E0A2180119F0A21800A81801A219E0 1803A219C01807A2F00F80181F1900183E187E604D5A4D5AEF0FE04D5A057FC7FCEE03FE EE3FFC007FB712F0B812C04CC8FC6C15E03C3A78B54D>I<1AF0A3861A78A21A7C1A3CA2 1A3E1A1E1A1F747EA2747E747E87747E747E1B7E87757EF30FE0F303F8007FBC12FEBE12 80A26CF3FE00CEEA03F8F30FE0F31F8051C7FC1B7E63505A505A63505A505AA250C8FC1A 1E1A3E1A3CA21A7C1A78A21AF862A359347BB264>33 D39 D<49B4EF3FC0010F01E0923803FFF8013F01FC030F13FE4901FF92383FE01F48 B66C91397E0007C02603F80301E0D901F8EB01E02807E0007FF049486D7E01806D6CD907 C0147048C76C6C494880001EDA07FE49C87E001C6E6C013E150C486E6D48150E71481506 486E01E0160793387FF1F0006092263FF3E08193381FFBC000E004FF1780486F49150170 90C9FC82707F8482717E844D7E6C4B6D1503006004EF1700933803E7FE0070922607C7FF 5DDC0F837F003004816D140E00384BC6FC0018033E6D6C5C001C4B6D6C143C6C4BD91FFC 5C6C4A486D6C5C6DD907E06D6C13036C6C49486D9038E00FE0D801F0013FC890B55A2700 7C03FE6F91C7FC90263FFFF8031F5B010F01E0030313F8D901FECAEA7FC0592D7BAB64> 49 D<92B6FC02071580143F91B7120001030180C8FCD907FCC9FCEB1FE0EB3F80017ECA FC5B485A485A485A5B485A121F90CBFC123EA2123C127CA2127812F8A25AA2B9FC1880A2 180000F0CBFCA27EA21278127CA2123C123EA27E7F120F6C7E7F6C7E6C7E6C7E137E6D7E EB1FE0EB07FC6DB47E010090B6FC023F1580140702001500313A78B542>I<1706170F17 1FA2173EA2177CA217F8A2EE01F0A2EE03E0A2EE07C0A2EE0F80A2EE1F00A2163EA25EA2 5EA24B5AA24B5AA24B5AA24B5AA24BC7FCA2153EA25DA25DA24A5AA24A5AA24A5AA24A5A A24AC8FCA2143EA25CA25CA2495AA2495AA2495AA2495AA249C9FCA2133EA25BA25BA248 5AA2485AA2485AA2485AA248CAFCA2123EA25AA25AA25A1260305C72C600>54 D66 D I<031FB512C00203B7FC021F16E091B812F8010317FE010F717E90283FE07FC03F80D9FE 00020080D801F8041F7FD803E04A01077F48481601000F716C7E4848717E003F02FF151F 007F180F90C7707E00FE92C8FC488400F01A80008084C75AA24B81A414035DA21B00A24A 5AA24F5AA24A5A621903624A5A4F5AA24B4B5A023F5F191F4B5E027F4CC7FC197E92C912 7C4A5E4E5A4A4B5A01014C5AF01F804A033EC8FC01035E4A4A5AEF07E00107ED1FC04A02 FFC9FC010FEC07FC4AEBFFF091B612C0017F4ACAFC90B612F04815804802F8CBFC4891CC FC49447EC34D>I72 D77 DII<0060170C00F0171EB3B3A66C173EA20078 173C007C177C007E17FC003E17F86CEE01F06D15036C6CED07E06C6CED0FC0D803F8ED3F 80D801FEEDFF0026007FC0EB07FCD93FFCEB7FF8010FB612E001031580D9007F01FCC7FC 020713C0373D7BBA42>91 D<913807FFC0027F13FC0103B67E010F15E0903A3FFC007FF8 D97FC0EB07FCD801FEC8B4FCD803F8ED3F80D807E0ED0FC04848ED07E04848ED03F090C9 1201003EEE00F8007E17FC007C177C0078173C00F8173EA248171EB3B3A60060170C373D 7BBA42>I<1538157CA215FEA24A7EA215EF02037FA2913807C7C0A291380F83E0A29138 1F01F0A2EC1E00023E7FA24A137CA24A7FA249487FA24A7F010381A249486D7EA249486D 7EA249C76C7EA2011E1400013E81A249157CA24981A2484881A24848ED0F80A249150700 0717C0A24848ED03E0A248C9EA01F0A2003EEE00F8A2003C1778007C177CA248173EA248 171E0060170C373D7BBA42>94 D102 D<12FEEAFFE0EA07F8EA00FEEB7F806D7E6D7E130F6D7EA26D7E B3AD6D7EA26D7E806E7E6E7EEC0FE0EC03FC913800FFE0A2913803FC00EC0FE0EC3FC04A 5A4AC7FC5C495AA2495AB3AD495AA2495A131F495A495A01FEC8FCEA07F8EAFFE048C9FC 236479CA32>I<126012F0B3B3B3B3B3A81260046474CA1C>106 D<0070130700F01480B3 B3B3B3B3A800701400196474CA32>I<1B0C1B1E1B3EA21B7CA21BF8A2F201F0A2F203E0 A2F207C0A2F20F80A2F21F00A21A3EA262A262A24F5AA2621903A24F5AA24F5AA24FC7FC A2193EA261A261A24E5AA24E5AA24E5AA24E5AA2010C4CC8FC133C017C163EEA01FE0003 5F487E001E5F00387FD8707F4B5A00E07FD8003F4B5A80011F4B5AA26E4A5A130F6E4AC9 FC13076E143E13036E5C13016E5C7F6F5B027F1301A26F485A143F6F485A141F6F485A14 0F6F48CAFC1407EDFC3E14035E15FE02015B15FF6E5BA26F5AA26F5AA26F5AA26FCBFC15 0E4F647A8353>112 D114 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fk cmbx12 12 41 /Fk 41 122 df12 D46 D48 D III<163FA25E5E5D5DA25D5D5D5D A25D92B5FCEC01F7EC03E7140715C7EC0F87EC1F07143E147E147C14F8EB01F0EB03E013 0714C0EB0F80EB1F00133E5BA25B485A485A485A120F5B48C7FC123E5A12FCB91280A5C8 000F90C7FCAC027FB61280A531417DC038>I<0007150301E0143F01FFEB07FF91B6FC5E 5E5E5E5E16804BC7FC5D15E092C8FC01C0C9FCAAEC3FF001C1B5FC01C714C001DF14F090 39FFE03FFC9138000FFE01FC6D7E01F06D13804915C0497F6C4815E0C8FC6F13F0A317F8 A4EA0F80EA3FE0487E12FF7FA317F05B5D6C4815E05B007EC74813C0123E003F4A1380D8 1FC0491300D80FF0495AD807FEEBFFFC6CB612F0C65D013F1480010F01FCC7FC010113C0 2D427BC038>I<4AB47E021F13F0027F13FC49B6FC01079038807F8090390FFC001FD93F F014C04948137F4948EBFFE048495A5A1400485A120FA248486D13C0EE7F80EE1E00003F 92C7FCA25B127FA2EC07FC91381FFF8000FF017F13E091B512F89039F9F01FFC9039FBC0 07FE9039FF8003FF17804A6C13C05B6F13E0A24915F0A317F85BA4127FA5123FA217F07F 121FA2000F4A13E0A26C6C15C06D4913806C018014006C6D485A6C9038E01FFC6DB55A01 1F5C010714C0010191C7FC9038003FF02D427BC038>I<121E121F13FC90B712FEA45A17 FC17F817F017E017C0A2481680007EC8EA3F00007C157E5E00785D15014B5A00F84A5A48 4A5A5E151FC848C7FC157E5DA24A5A14035D14074A5AA2141F5D143FA2147F5D14FFA25B A35B92C8FCA35BA55BAA6D5A6D5A6D5A2F447AC238>I II65 D I68 D76 D78 D80 D82 D<003FBA12E0A59026FE000FEB8003D87FE09338003FF049171F90C71607A2007E180300 7C1801A300781800A400F819F8481978A5C81700B3B3A20107B8FCA545437CC24E>84 D<903801FFE0011F13FE017F6D7E48B612E03A03FE007FF84848EB1FFC6D6D7E486C6D7E A26F7FA36F7F6C5A6C5AEA00F090C7FCA40203B5FC91B6FC1307013F13F19038FFFC0100 0313E0000F1380381FFE00485A5B127F5B12FF5BA35DA26D5B6C6C5B4B13F0D83FFE013E EBFFC03A1FFF80FC7F0007EBFFF86CECE01FC66CEB8007D90FFCC9FC322F7DAD36>97 DIIII103 DI< 137C48B4FC4813804813C0A24813E0A56C13C0A26C13806C1300EA007C90C7FCAAEB7FC0 EA7FFFA512037EB3AFB6FCA518467CC520>I107 DI<90277F8007FEEC0FFCB590263FFFC090387FFF8092B5D8F001B512E002 816E4880913D87F01FFC0FE03FF8913D8FC00FFE1F801FFC0003D99F009026FF3E007F6C 019E6D013C130F02BC5D02F86D496D7EA24A5D4A5DA34A5DB3A7B60081B60003B512FEA5 572D7CAC5E>I<90397F8007FEB590383FFF8092B512E0028114F8913987F03FFC91388F 801F000390399F000FFE6C139E14BC02F86D7E5CA25CA35CB3A7B60083B512FEA5372D7C AC3E>II<90397FC00FF8B5 90B57E02C314E002CF14F89139DFC03FFC9139FF001FFE000301FCEB07FF6C496D13804A 15C04A6D13E05C7013F0A2EF7FF8A4EF3FFCACEF7FF8A318F017FFA24C13E06E15C06E5B 6E4913806E4913006E495A9139DFC07FFC02CFB512F002C314C002C091C7FCED1FF092C9 FCADB67EA536407DAC3E>I<90387F807FB53881FFE0028313F0028F13F8ED8FFC91389F 1FFE000313BE6C13BC14F8A214F0ED0FFC9138E007F8ED01E092C7FCA35CB3A5B612E0A5 272D7DAC2E>114 D<90391FFC038090B51287000314FF120F381FF003383FC00049133F 48C7121F127E00FE140FA215077EA27F01E090C7FC13FE387FFFF014FF6C14C015F06C14 FC6C800003806C15806C7E010F14C0EB003F020313E0140000F0143FA26C141F150FA27E A26C15C06C141FA26DEB3F8001E0EB7F009038F803FE90B55A00FC5CD8F03F13E026E007 FEC7FC232F7CAD2C>III119 DII E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fl cmti12 12 67 /Fl 67 128 df<4CB414FC040F9039C003FF80933B3F81F00783C0933B7C00781F01E04C 9038F83F03923C01F001FC3E07F003030103EB7E0F922607E007EB7C1F19FCDB0FC001F8 14E0943A03F0F80FC0DD01E1EB0780031FD9000190C7FC5E180361153F93C7FCA2180761 5D157EA2180F6115FE91B912F0A3DA00FCC7D81F80C7FC1401A25D183F96C8FCA214035D A260187E14075DA218FE60140F5DA2170160141F5DA2170360143F92C7FCA21707605C14 7EA2170F6014FE5CA24D5AA2495A95C9FC5F5C0103153E177E001CEBE038007F02FE137C 26FF07E114FC02C15C4C5AEB0F8100FE903901FC03E0D8F81F9038F007C03B701E00E00F 80D8783CD9F83ECAFCD81FF0EB3FF8D807C0EB0FE04C5A83C53C>11 DI I<167016E0ED01C0ED0380ED0700150E153C5D15F85D4A5A4A5A4A5A140F4AC7FC141E14 3E5C147814F8495A5C1303495AA2495AA249C8FCA25B133E137E137CA25BA212015BA212 035BA212075BA2120FA25BA2121FA290C9FCA25AA2123EA3127EA2127CA65AAB1278A612 7C123CA47EA2120E120FA27E6C7EA26C7EA26C7E1360246472CA28>40 D<1560A2157081A281151E150E150FA2811680A3ED03C0A516E0A21501A71503A91507A2 16C0A4150FA21680A2151FA21600A25DA2153EA2157EA2157C15FCA25D1401A25D14035D A214075D140F5DA24AC7FCA2143EA25C147814F8495AA2495A5C1307495A91C8FC131E13 3E5B13785B485A485A485A48C9FC121E5A5A12E05A23647FCA28>I<13F0EA03FC1207A2 EA0FFEA4EA07FCEA03CCEA000C131C1318A2133813301370136013E0EA01C013801203EA 0700120E5A5A5A5A5A0F1D7A891E>44 D<007FB5FCB6FCA214FEA21805789723>I<120F EA3FC0127FA212FFA31380EA7F00123C0A0A76891E>I<16C01501A215031507ED0F8015 1F153F157F913801FF005C140F147F903807FCFEEB0FF0EB0700EB00015DA314035DA314 075DA3140F5DA3141F5DA3143F5DA3147F92C7FCA35C5CA313015CA313035CA313075CA2 130FA2131F133FB612FCA25D224276C132>49 DIII<026014300278EB01F091397F801FE091B612 C01780170016FC4914F016C0DACFFEC7FC02C0C8FC13035CA3130791C9FCA35B130EA313 1E90381C07F8EC3FFE9138F80F8090393DC007C0D93F807F90383E0003013C80496D7E13 70A290C7FC82A41503A415075E120E123F486C130F00FF5DA390C7485A5A00F84A5A12E0 4B5A93C7FC15FE14016C5C0070495A0078495A6CEB1FC0003E495A261F80FEC8FC6CB45A 000313E0C66CC9FC2C4476C132>II<9026380FC0131C90 38787FE0902671FFF0133C01F3157801EF15F090B5FC4801E0EB01E09139007003C04848 EB380701F8EC1F804848EB1C3F4990381FFF004848EB07DF49EB001E48C85A121E5E4815 F800385D0078140100705D00F01403485D1507C8485AA24BC7FCA25D153E157E157C15FC 5D1401A24A5AA214075D140F5D141FA25D143FA24AC8FCA314FEA21301A25C1303A25C13 07A35C130FA35C5C6D5A2E4472C132>III<130FEB1FC0133FEB7FE013FFA214C0EB7F80140013 1E90C7FCB3A5120FEA3FC0127FA212FFA35B6CC7FC123C132B76AA1E>I<14F0EB01FC13 03EB07FE130FA214FCEB07F814F0EB01E090C7FCB3A513F0EA03F8487EA2120FA46C5AEA 03D8EA001813381330A21370136013E05B12015B120348C7FC1206120E5A5A5A5A5A173E 7AAA1E>I65 D<91B712FCF0FF8019E00201903980001FF06E90C7EA07F84A6F7E72 7E4B81841A800203167F5DA314075D19FFA2020F17004B5C611803021F5E4B4A5A180F4E 5A023F4B5A4BEC7F804EC7FCEF03FC027FEC0FF84BEBFFC092B6C8FC18E0913AFF800007 F892C7EA01FC717E187F49834A6F7EA30103835CA313075CA3010F5F4A157FA24E5A131F 4A4A90C7FC601703013F4B5A4A4A5A4D5A017F4B5A4D5A4A4948C8FC01FFEC0FFEB812F8 17C04CC9FC41447AC345>II<91B712F818FF19C00201903980003FF06E90C7EA0FF84AED03FCF000FE 4B157FA2F13F800203EE1FC05DF10FE0A214074B16F01907A2140F5D1AF8A2141F5DA219 0F143F5D1AF0A2147F4B151FA302FF17E092C9123FA34918C04A167F1A80A2010317FF4A 1700A24E5A13074A4B5A611807010F5F4A4B5A181F61011F4C5A4A4BC7FC18FE4D5A013F 4B5A4A4A5A4D5A017FED3FC005FFC8FC4AEB03FE01FFEC1FF8B812E094C9FC16F845447A C34A>I<91B912C0A30201902680000313806E90C8127F4A163F191F4B150FA30203EE07 005DA314074B5D190EA2140F4B1307A25F021F020E90C7FC5DA2171E023F141C4B133C17 7C17FC027FEB03F892B5FCA39139FF8003F0ED00011600A2495D5CA2160101034B13705C 19F061010791C8FC4A1501611803010F5F4A150796C7FC60131F4A151E183E183C013F16 7C4A15FC4D5A017F1503EF0FF04A143F01FF913803FFE0B9FCA26042447AC342>I<91B9 1280A30201902680000713006E90C8FC4A163FA24B81A30203160E5DA314074B151E191C A2140F5D17075F021F020E90C7FC5DA2171E023F141C4B133CA2177C027F5CED800392B5 FCA291B65AED00071601A2496E5A5CA2160101035D5CA2160301075D4A90CAFCA3130F5C A3131F5CA3133F5CA2137FA313FFB612E0A341447AC340>I<91B6D8803FB512E0A30201 0180C7387FE0006E90C86C5A4A167FA24B5EA219FF14034B93C7FCA26014074B5DA21803 140F4B5DA21807141F4B5DA2180F143F4B5DA2181F147F92B75AA3DAFF80C7123F92C85B A2187F5B4A5EA218FF13034A93C8FCA25F13074A5DA21703130F4A5DA21707131F4A5DA2 170F133F4A5DA2017F151FA24A5D496C4A7EB6D8803FB512E0A34B447AC348>72 D<027FB512E091B6FCA20200EBE000ED7F8015FFA293C7FCA35C5DA314035DA314075DA3 140F5DA3141F5DA3143F5DA3147F5DA314FF92C8FCA35B5CA313035CA313075CA3130F5C A3131F5CA2133FA25CEBFFE0B612E0A25D2B447BC326>I<031FB512F05DA29239000FFC 005FA35FA2161FA25FA2163FA25FA2167FA25FA216FFA294C7FCA25DA25EA21503A25EA2 1507A25EA2150FA25EA2151FA25EA2153FA25EA2157FA25EEA0F80D83FE013FF93C8FC12 7FA24A5AEAFFC04A5A1300007C495A0070495A4A5A6C5C003C495A6C01FEC9FC380F81F8 3803FFE0C690CAFC344679C333>I<91B612F0A25F020101C0C7FC6E5B4A90C8FCA25DA3 14035DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA314FF92C9FCA35B5CA3010316 104A1538A21878010716705C18F018E0010F15015C18C01703011F15074A1580170FA201 3FED1F004A5C5F017F15FE16034A130F01FFEC7FFCB8FCA25F35447AC33D>76 D<91B56C93387FFFC08298B5FC02014DEBC0006E614A5FA203DF4C6CC7FC1A0E63912603 CFE05D038F5F1A381A711407030FEEE1FCA2F101C3020FEE0383020E60F107036F6C1507 021E160E021C60191CF1380F143C023804705BA2F1E01F0278ED01C091267003F85EF003 801A3F02F0ED070002E0030E5CA24E137F130102C04B91C8FC606201036D6C5B02805F4D 5A943803800113070200DA07005BA2050E1303495D010E606F6C5A1907011E5D011C4B5C A27048130F133C01384B5C017892C7FC191F01F85C486C027E5DD807FE027C4A7EB500F0 0178013FB512C0A216705A447AC357>I<91B56C49B512E0A28202009239000FFC00F107 F0706E5A4A5F15DF705D1907EC03CFDB8FF892C7FCA203875D02077F0303150EA270141E EC0F01020E161C826F153C141E021C6E1338167F1978023C800238013F1470A27113F002 78131F02705E83040F130102F014F84A5E1607EFFC0313014A01035C17FE180701031401 4A02FF90C8FCA2705B0107168F91C8138E177F18DE5B010EED3FDC18FCA2011E151F011C 5EA2170F133C01386F5A1378A201F81503486C5EEA07FEB500F01401A2604B447AC348> II<91B712F018FEF0FF800201903980007FE06E90C7EA1FF04AED07F818034B15FC F001FE1403A24B15FFA21407A25DA2140FF003FE5DA2021F16FC18074B15F8180F023F16 F0F01FE04B15C0F03F80027FED7F0018FE4BEB03FCEF0FF002FFEC7FC092B6C7FC17F892 CAFC5BA25CA21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA25CA2137FA25C49 7EB67EA340447AC342>I<91B77E18F818FE020190398001FF806E90C7EA3FC04AED1FE0 F00FF04BEC07F8180319FC14034B15FEA314075DA3020FED07FC5DA2F00FF8141F4B15F0 F01FE0F03FC0023F16804BEC7F0018FEEF03F8027F4A5A4BEB1FC04CB4C7FC92B512F891 B612E092380003F8EE00FE177F496F7E4A6E7EA28413034A140FA2171F13075CA2173F13 0F5CA24D5A131F5CA3013F170E5CA2017FEE801E191C4A163C496C1638B66C90383FC070 051F13F094380FE1E0CA3803FF80943800FE003F467AC347>82 DI<48B912F85AA2913B0007FC001FF0D807F84A130701 E0010F140349160148485C90C71500A2001E021F15E05E121C123C0038143F4C13010078 18C0127000F0147F485DA3C800FF91C7FC93C9FCA35C5DA314035DA314075DA3140F5DA3 141F5DA3143F5DA3147F5DA314FF92CAFCA35B5CA21303A21307497E007FB612C0A25E3D 446FC346>I86 DI91 D93 D97 DII< EE07F0ED03FFA39238000FE01607160FA217C0A2161FA21780A2163FA21700A25EA2167E A216FEA25EEC1F80EC7FE1903801F071903907C039F890380F801D90381F001F013E130F 017E5C5B48481307A248485C120749130F120F5E485A151F123F495CA2153F127F90C790 C7FCA25DA200FE147EA29238FE0380160703FC1300A2007E13015E913803F80E1407003E 010F131E161C6C131C02385B3A0F80F078783A07C3E07C703A01FF801FE03A007E000780 2C4676C432>II< EE0F80EE3FE0EEF870923801F038923803E0F8923807E1FC16C3ED0FC7A2EE87F892381F 83F0EE81E0EE8000153F93C7FCA45D157EA415FE5DA349B512FEA390260001F8C7FCA314 035DA414075DA4140F5DA4141F5DA4143F92C8FCA55C147EA314FE5CA413015CA4495AA3 5C1307121C007F5B12FF495AA291C9FC485AEAF81E485AEA7878EA1FF0EA07C02E5A83C5 1E>I<15FCEC03FF91390F83838091393E01CFC091387C00EF4A13FF4948137F01031580 4948133F495A131F4A1400133F91C75A5B167E13FE16FE1201495CA215011203495CA215 03A2495CA21507A25EA2150F151F5E0001143F157F6C6C13FF913801DF8090387C039F90 383E0F3FEB0FFCD903F090C7FC90C7FC5DA2157EA215FEA25DA2001C495A127F48495A14 074A5A485C023FC8FC00F8137E387C01F8381FFFE0000390C9FC2A407BAB2D>I<14FE13 7FA3EB01FC13001301A25CA21303A25CA21307A25CA2130FA25CA2131FA25C157F90393F 83FFC091388F81F091381E00F802387F4948137C5C4A137EA2495A91C7FCA25B484814FE 5E5BA2000314015E5BA2000714035E5B1507000F5DA249130F5E001F1678031F13704914 80A2003F023F13F0EE00E090C7FC160148023E13C01603007E1680EE070000FEEC1E0FED 1F1E48EC0FF80038EC03E02D467AC432>I<143C147E14FE1301A3EB00FC14701400AE13 7C48B4FC3803C780380703C0000F13E0120E121C13071238A21278EA700F14C0131F00F0 138012E0EA003F1400A25B137EA213FE5B12015BA212035B141E0007131C13E0A2000F13 3CEBC038A21478EB807014F014E0EB81C0EA0783EBC7803803FE00EA00F8174378C11E> I<16F0ED03F8A21507A316F0ED01C092C7FCAEEC01F0EC07FCEC1E1EEC380F0270138014 E0130114C0EB03800107131F1400A2130E153F131E011C140090C7FC5DA2157EA215FEA2 5DA21401A25DA21403A25DA21407A25DA2140FA25DA2141FA25DA2143FA292C7FCA25C14 7EA214FE001C5B127F48485A495AA248485A495AD8F81FC8FCEA707EEA3FF8EA0FC02556 83C11E>I<14FE137FA3EB01FC13001301A25CA21303A25CA21307A25CA2130FA25CA213 1FA25C167E013F49B4FC92380783C09138000E07ED3C1F491370ED603F017E13E0EC01C0 9026FE03801380913907000E00D9FC0E90C7FC5C00015B5C495AEBF9C03803FB8001FFC9 FCA214F03807F3FCEBF07F9038E01FC06E7E000F130781EBC003A2001F150FA20180140E A2003F151E161C010013E0A2485DA2007E1578167000FE01015B15F1489038007F800038 021FC7FC2A467AC42D>IIIIII<91381F800C91387FE01C903901F070 3C903907C0387890390F801CF890381F001D013E130F017E14F05B48481307A2484814E0 12075B000F140F16C0485AA2003F141F491480A3007F143F90C71300A35D00FE147EA315 FE5DA2007E1301A24A5A1407003E130FA26C495A143B380F80F33807C3E73901FF87E038 007E071300140F5DA3141F5DA3143F92C7FCA25CA25C017F13FEA25D263F76AB2D>II< EC0FE0EC7FF8903801F01E903803C00F90390780078090380F0003011E14C0150749131F A2017CEB3F801378137CED0E0092C7FC137E137F14F014FF6D13C06D13F06D7F6D7F1300 EC0FFE14011400157F81120E003F141E487EA2153E48C7123CA200FC5C12705D0078495A 6C495A6CEB0F80260F803EC7FC3803FFF838007FC0222D7AAB28>I<1470EB01F8A31303 5CA313075CA3130F5CA3131F5CA2007FB512E0B6FC15C0D8003FC7FCA25B137EA313FE5B A312015BA312035BA312075BA3120F5BA2EC0780001F140013805C140E003F131EEB001C 143C14385C6C13F0495A6C485AEB8780D807FEC7FCEA01F81B3F78BD20>I<137C48B414 072603C780EB1F80380703C0000F7F000E153F121C0107150012385E1278D8700F147E5C 011F14FE00F05B00E05DEA003FEC0001A2495C137E150313FE495CA215071201495CA203 0F13380003167849ECC070A3031F13F0EE80E0153F00011581037F13C06DEBEF83000001 01148090397C03C787903A3E0F07C70090391FFE01FE903903F000782D2D78AB34>I<01 7C143848B414FC3A03C78001FE380703C0000F13E0120E001C14000107147E1238163E12 78D8700F141E5C131F00F049131C12E0EA003F91C7123C16385B137E167801FE14705BA2 16F0000115E05B150116C0A24848EB0380A2ED0700A2150E12015D6D5B000014786D5B90 387C01E090383F0780D90FFFC7FCEB03F8272D78AB2D>I<017CEE038048B4020EEB0FC0 2603C780013FEB1FE0380703C0000E7F5E001C037E130F01071607123804FE130300785D EA700F4A1501011F130100F001804914C012E0EA003FDA000314034C14805B137E030714 0701FE1700495CA2030F5C0001170E495CA260A24848495A60A2601201033F5C7F4B6C48 5A000002F713036D9039E7E0078090267E01C349C7FC903A1F0781F81E903A0FFF007FF8 D901FCEB0FE03B2D78AB41>I<02F8133FD907FEEBFFE0903A0F0F83C0F0903A1C07C780 F890393803CF03017013EE01E0EBFC07120101C013F8000316F00180EC01C000074AC7FC 13001407485C120EC7FC140F5DA3141F5DA3143F92C8FCA34AEB03C01780147EA202FEEB 0700121E003F5D267F81FC130E6E5BD8FF83143CD903BE5B26FE079E5B3A7C0F1F01E03A 3C1E0F83C0271FF803FFC7FC3907E000FC2D2D7CAB2D>I<137C48B414072603C780EB1F 80380703C0000F7F000E153F001C1600130712385E0078157EEA700F5C011F14FE00F049 5B12E0EA003FEC00015E5B137E150301FE5C5BA2150700015D5BA2150F00035D5BA2151F 5EA2153F12014BC7FC6D5B00005BEB7C0390383E0F7EEB1FFEEB03F090C712FE5DA21401 5D121F397F8003F0A24A5A4848485A5D48131F00F049C8FC0070137E007813F8383801F0 381E07C06CB4C9FCEA01FC294078AB2F>I<027C130749B4130F49EB800E010F141E49EB C03CEDE03890393F03F07890397C00FDF00178EB3FE00170EB03C001F0148049130790C7 EA0F00151E5D5D5D4A5A4A5A4A5A4AC7FC141E5C5C5C495A495A495A49C8FC011E14F049 14E05B491301485A4848EB03C0D807B0130701FEEB0F80390FCF801F3A1F07E07F00393E 03FFFED83C015B486C5B00705C00F0EB7FC048011FC7FC282D7BAB28>I<000FEB078039 3F801FC0397FC03FE000FF137FA4018013C0397F003F80003CEB1E001B0A66C232>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fm cmmi8 8 38 /Fm 38 123 df11 DI<131FD9 FFC013304801F0137000076D13604815E0D9807C13C0391E003C0148011E13800038EB0E 03480107130000605CEC030612E0C7138EEC018C159C159815B815B0A215F05DA35DA25D A21403A44AC7FCA4140EA45CA31418242C7F9D24>II< 3907C007E0390FE03FF8391CF8783E393879E01E39307B801F38707F00126013FEEAE0FC 12C05B0081143F120149133EA20003147EA249137CA2000714FCA24913F8A2000F1301A2 018013F0A2001F1303A2010013E0120EC71207A215C0A2140FA21580A2141FA21500A214 0E202C7E9D23>17 D<13FC13FFEB1FC0130F6D7EA36D7EA2130180A26D7EA3147EA280A3 6E7EA2140F81A24A7E143F147FECF3F0EB01E3EB03C190380781F8130F49C67E133E5B49 137E485A48487F1207485A4848EB1F8048C7FC127E48EC0FC048EC07E000701403232F7D AD29>21 D<131C013EEB0380ED07C0017E130F1680137CA201FC131F16005BA200015C15 3E5BA20003147E157C5BA20007ECFC08EDF8185BA2000F0101133816309038E003F00207 1370001F90380EF8609039F83C78E090397FF03FC090391FC00F0048C9FCA2123EA2127E A2127CA212FCA25AA21270252C7E9D2A>II<0103B512 F0131F137F90B612E03A01FC1F80003903F00FC03807C00748486C7E121F1300123EA25A A2140700FC5C5AA2140F5D141F92C7FC143E0078133C147C007C5B383C01E0381F07C0D8 07FFC8FCEA01F8241E7D9C28>27 D<1560A315E0A25DA21401A25DA21403A292C7FCA25C EC3FE0903803FFF890380FC63E90393E0E0F80017CEB07C03A01F00C03E0D803E0EB01F0 3807C01CD80F801300001F011813F81300003E1338A2481330A2EC700100FC15F0481360 150302E013E01507007801C013C0007CEC0F800101EB1F00003C143E003E495A001F5C39 0F8383E03903E39F802600FFFEC7FCEB1FF00107C8FCA21306A2130EA2130CA2131CA213 18A3253C7DAD2A>30 D34 D<123C127E12FFA4127E123C08087A8714>58 D<123C127EB4FCA21380A2127F123D1201 A312031300A25A1206120E5A5A5A126009157A8714>II<15C0140114031580A214071500A25C140EA2141E141CA2143C 143814781470A214F05CA213015CA213035C130791C7FCA25B130EA2131E131CA2133C13 38A21378137013F05BA212015BA212035BA2120790C8FC5A120EA2121E121CA2123C1238 A212781270A212F05AA21A437CB123>I<12E012F812FEEA3F80EA0FE0EA03F8EA00FEEB 3F80EB0FE0EB03F8EB00FC143FEC0FC0EC07F0EC01FCEC007FED1FC0ED07F0ED01FCED00 7FEE1FC01607161FEE7F00ED01FCED07F0ED1FC0037FC7FCEC01FCEC07F0EC0FC0023FC8 FC14FCEB03F8EB0FE0EB3F8001FEC9FCEA03F8EA0FE0EA3F8000FECAFC12F812E02A2B7A A537>I<1670A216F01501A24B7EA21507150DA2151915391531ED61FC156015C0EC0180 A2EC03005C14064A7F167E5C5CA25C14E05C4948137F91B6FC5B0106C7123FA25B131C13 18491580161F5B5B120112031207000FED3FC0D8FFF8903807FFFEA22F2F7DAE35>65 D<013FB512FEEEFFC0903A00FE0007F0EE01F84AEB007E8301018118804A140F18C00103 150718E05CA21307A25CA2130FA24A140FA2131F18C04A141FA2013F1680173F91C81300 A249157EA2017E5D5F01FE14014C5A494A5A4C5A00014BC7FC163E4914FCED03F00003EC 1FC0B7C8FC15F8332D7CAC3A>68 D<90263FFFFC90381FFF80A2D900FEC73803F80018E0 4AEC07804DC7FC0101151C5F4A14E04C5A01034A5A040EC8FC4A5B5E010714E04B5A9138 E00780030EC9FC010F131F157F4A487E14C190391FC71FC014CEEC9C0F02F07F90383FE0 0702C07FEC0003825B6F7E137E6F7E13FE167F5B707E1201161F4981831203B539E001FF FEA2392D7CAC3C>75 D78 D86 D97 D99 D<151FEC03FFA2EC003FA2153EA2 157EA2157CA215FCA215F8A21401EB07E190381FF9F0EB7C1DEBF80FEA01F03903E007E0 EA07C0120FEA1F8015C0EA3F00140F5A007E1480A2141F12FE481400A2EC3F021506143E 5AEC7E0E007CEBFE0C14FC0101131C393E07BE18391F0E1E38390FFC0FF03903F003C020 2F7DAD24>I<157C4AB4FC913807C380EC0F87150FEC1F1FA391383E0E0092C7FCA3147E 147CA414FC90383FFFF8A2D900F8C7FCA313015CA413035CA413075CA5130F5CA4131F91 C8FCA4133EA3EA383C12FC5BA25B12F0EAE1E0EA7FC0001FC9FC213D7CAE22>102 D<1307EB0F80EB1FC0A2EB0F80EB070090C7FCA9EA01E0EA07F8EA0E3CEA1C3E12381230 1270EA607EEAE07C12C013FC485A120012015B12035BA21207EBC04014C0120F13801381 381F01801303EB0700EA0F06131EEA07F8EA01F0122E7EAC18>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>II<90 387C01F89038FE07FE3901CF8E0F3A03879C0780D907B813C0000713F000069038E003E0 EB0FC0000E1380120CA2D8081F130712001400A249130F16C0133EA2017EEB1F80A2017C 14005D01FC133E5D15FC6D485A3901FF03E09038FB87C0D9F1FFC7FCEBF0FC000390C8FC A25BA21207A25BA2120FA2EAFFFCA2232B829D24>I<3807C01F390FF07FC0391CF8E0E0 383879C138307B8738707F07EA607E13FC00E0EB03804848C7FCA2128112015BA21203A2 5BA21207A25BA2120FA25BA2121FA290C8FC120E1B1F7E9D20>114 DI<130E131FA25BA2133EA2137E A2137CA213FCA2B512F8A23801F800A25BA21203A25BA21207A25BA2120FA25BA2001F13 10143013001470146014E0381E01C0EB0380381F0700EA0F0EEA07FCEA01F0152B7EA919 >II<011E1330EB3F809038FFC07048EBE0E0ECF1C03803C0FF9038803F8090 3800070048130EC75A5C5C5C495A495A49C7FC131E13385B491340484813C0485A380700 01000EEB0380380FE007391FF81F0038387FFF486C5A38601FFC38E00FF038C003C01C1F 7D9D21>122 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fn msbm10 12 2 /Fn 2 115 df<007FB712C0B812FCEFFF806C17E02800F807F00F13F8DBC00113FE0178 90398000FCFF94387C3F8094383E0FC0727E94381E03F0EF1F011800717FA21978A519F8 A24D5B1801EF1E034E5A94383E0FC094387E3F80DDFDFFC7FC933807FFFE92B612F818E0 95C8FC17F0ED87C1EEC0F8923883E0FC177C923881F03EA2923880F81F84EE7C0F717E16 3E93383F03E0041F7FEE0F81EF80F8EE07C0187C933803E07E183E706C7E85706C6C7E18 0794387C03E0057E7F94383E01F8716C7E197C01F86D6D6C7EF13F80007FB66C6CB512E0 B700C015F0836C4B6C14E044447EC33D>82 D<126012F87E127E7EEA1FC06C7EEA03F0EA 01FC6C7E133FEB1FC06D7EEB03F0EB01FC6D7E143FEC1FC06E7EEC03F0EC01FC6E7E153F ED1FC06F7EED03F0ED01FC6F7E163FEE1FC0707EEE03F0EE01FC707E173FEF1FC0717EEF 03F0EF01FC717E183FF01F80F00FC01807F001803A2D77AA4D>114 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fo cmr8 8 16 /Fo 16 111 df0 D<013FB5FCA29038007F806EC8FCA6903801FF E0011F13FE90397F3F3F80D801F8EB07E0D807E0EB01F8D80FC06D7ED81F80147ED83F00 80481680A2007E151F00FE16C0A5007E1680007F153FA26C1600D81F80147ED80FC05CD8 07E0495AD801F8EB07E0D8007FEB3F8090261FFFFEC7FC010113E0D9003FC8FCA64A7E01 3FB5FCA22A2D7CAC33>8 D<13031307130E131C1338137013F0EA01E013C01203EA0780 A2EA0F00A2121EA35AA45AA512F8A25AAB7EA21278A57EA47EA37EA2EA0780A2EA03C012 0113E0EA00F013701338131C130E1307130310437AB11B>40 D<12C07E12707E7E7E120F EA0780120313C0EA01E0A2EA00F0A21378A3133CA4131EA5131FA2130FAB131FA2131EA5 133CA41378A313F0A2EA01E0A2EA03C013801207EA0F00120E5A5A5A5A5A10437CB11B> I43 D48 D<130C133C137CEA03FC12FFEAFC7C1200B3B113FE387FFFFEA2172C7AAB23>III<140EA2141E143EA2147E14FEA2EB01BE130314 3E1306130E130C131813381330136013E013C0EA0180120313001206120E120C5A123812 305A12E0B612FCA2C7EA3E00A9147F90381FFFFCA21E2D7EAC23>I<000CEB0180380FC0 1F90B512005C5C14F014C0D80C7EC7FC90C8FCA8EB1FC0EB7FF8380DE07C380F801F0100 1380000E130F000CEB07C0C713E0A2140315F0A4127812FCA448EB07E012E0006014C000 70130F6C14806CEB1F006C133E380780F83801FFE038007F801C2D7DAB23>I57 D<123C127E12FFA4127E123C1200AD123C127E12FE12FFA312 7F123F1203A312071206A2120E120C121C1218123812701260082A7A9C14>59 D61 D108 D<3807C0FE39FFC3FF809038C703E0390FDE01F0EA07F849 6C7EA25BA25BB2486C487E3AFFFE1FFFC0A2221E7E9D27>110 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fp cmmi12 12 62 /Fp 62 123 df11 DI I<1578913807FFE0021F13FC91383C7FFEEC7007EC6003ECE0004A13381600A280A380A2 80147CA2147E143E143F816E7EA26E7E81140781EC3FFC14FF903803E1FEEB07C190381F 00FF133E49EB7F805B0001143F485A484814C049131F120F485AA248C7FC150F5A127EA3 00FEEC1F805AA316005A5DA2153E157E157CA26C5C127C4A5A6C495AA26C495A6C6C485A 6C6C48C7FC3803E07C3800FFF0EB1FC027487CC62B>I<01F8EB03FCD803FEEB1FFFD807 1F90387C0FC03B0E0F80E007E0001C9038C3C003271807C70013F002CE1301003801DC14 F8003013D8EB0FF800705B00605BA200E0491303D8C01F15F05C12001607133F91C713E0 A2160F5B017E15C0A2161F13FE491580A2163F1201491500A25E120349147EA216FE1207 495CA21501120F495CEA0380C81203A25EA21507A25EA2150FA25EA2151FA25EA2153FA2 93C7FC150E2D417DAB30>17 D21 D<147002F8140E0101153FA301035DA24A147EA201 0715FEA24A5CA2010F1401A24A5CA2011F1403A24A5CA2013F1407A291C75BA249140FA2 017E5DA201FE021F1318183849ED8030A20001033F13701860EE7F005E486C16E0DB01DF 13C09238039F016DD9071F1380489039801E0F83903BF7C078078700903AE1FFE003FE90 3AE07F8000F8000F90CAFCA25BA2121FA25BA2123FA290CBFCA25AA2127EA212FEA25A12 3835417DAB3B>II<010FB712E0013F 16F05B48B812E04817C02807E0060030C7FCEB800EEA0F00001E010C13705A0038011C13 605A0060011813E000E013381240C7FC5C4B5AA214F014E01301150314C01303A3EB0780 82130FA2EB1F00A34980133E137EA24980A2000114015BA26C48EB00E0342C7EAA37>25 DI<0203B612E0021F15 F091B7FC4916E0010716C090270FF80FF8C7FC90381FC00349486C7E017EC7FC49147E48 5A4848143E0007153F5B485AA2485AA2123F90C8FC5E48157E127EA216FE00FE5D5A1501 5EA24B5A007C5D15074B5A5E6C4AC8FC153E6C5C5D390F8003F03907C007C02601F03FC9 FC38007FFCEB1FE0342C7DAA37>I<161CA21618A21638A21630A21670A21660A216E0A2 5EA21501A25EA21503A293C8FCA25DED7FE0913807FFFE91391FC63F809139FE0E07C0D9 01F8EB03F0903A07E00C00F8D91FC08090263F001C137E017E814913184848ED1F800003 1438485A4848013014C0A248481370A248481360A248C712E0A24B133F481780481301A2 4B137F180014034816FE92C7FC4C5A6C49495AA2007E0106495A4C5A6C010E495A4C5A26 1F800C49C7FC000F15FC3A07C01C01F8D803E0EB07E03A01F8181F80D8007E01FEC8FC90 381FFFF801011380D90030C9FCA21470A21460A214E0A25CA21301A25CA21303A291CAFC A332597BC43A>30 D<137E48B46C150626078FE0150E260607F0151C260E03F81538000C 6D1570D81C0116E000006D15C0010015016EEC03806EEC0700170E6E6C5B5F5F6E6C1360 17E04C5A6E6C485A4CC7FC0207130E6F5A5E1630913803F8705EEDF9C06EB45A93C8FC5D 6E5A81A2157E15FF5C5C9138073F80140E141C9138181FC014381470ECE00FD901C07FEB 038049486C7E130E130C011C6D7E5B5B496D7E485A48488048C8FC000681000E6F137048 EE806048033F13E04892381FC0C048ED0FE348923803FF00CA12FC37407DAB3D>I<1730 A317701760A317E05FA316015FA3160394C8FCA35E1606A3160E160C013E1607D9FF80ED 1F802603C3C0011CEB3FC0260703E01318260601F0157F000E173F001C1538D818030230 131F0038170F0030170700701570D86007026013035CA2D8E00F02E0148000C049491301 EA001F4A150303011500013F5C1400604901031406017E91C7FC180E180C01FE49141C49 01061418183860030E1460030C14E04D5A4D5A031C49C7FC0318130E017E5D5F6D01385B 90261F80305BD90FC0EB03C0D907F0010FC8FC903901FE707C9039003FFFF002031380DA 0060C9FC15E05DA314015DA3140392CAFCA35C1406A3140E140C3A597DC43F>I34 D<121EEA7F80A2EAFFC0A4EA 7F80A2EA1E000A0A78891B>58 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413 E013C0A312011380120313005A1206120E5A5A5A12600B1D78891B>II<1618163C167CA2167816F8A216F01501A216E01503A216C01507A216 80150FA2ED1F00A2151E153EA2153C157CA2157815F8A25D1401A24A5AA25D1407A25D14 0FA292C7FC5CA2141E143EA2143C147CA25CA25C1301A25C1303A25C1307A25C130FA291 C8FC5BA2133EA2133C137CA2137813F8A25B1201A25B1203A2485AA25B120FA290C9FC5A A2121E123EA2123C127CA2127812F8A25A126026647BCA31>I<127012FCB4FCEA7FC0EA 1FF0EA07FCEA01FF38007FC0EB1FF0EB07FE903801FF809038007FE0EC1FF8EC03FE9138 00FF80ED3FE0ED0FF8ED03FF030013C0EE3FF0EE0FFCEE01FF9338007FC0EF1FF0EF07FC EF01FF9438007FC0F01FE0A2F07FC0943801FF00EF07FCEF1FF0EF7FC04C48C7FCEE0FFC EE3FF0EEFFC0030390C8FCED0FF8ED3FE0EDFF80DA03FEC9FCEC1FF8EC7FE0903801FF80 D907FECAFCEB1FF0EB7FC04848CBFCEA07FCEA1FF0EA7FC048CCFC12FC12703B3878B44C >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<4CB46C1318043F01F013384B B512FC0307D9007E1378DB1FF090380F80F0DB7F80EB03C1DA01FEC7EA01C34A48EC00E7 DA0FF0ED7FE04A48153F4A5A02FFC9121F494817C04948160F495A130F4A178049481607 495A137F4948170091CAFC5A485A1906485AA2485A96C7FC121F5BA2123F5BA3127F5BA4 485A4CB612805EA293C7EBE000725AA3007F60A218FF96C7FCA26C7E5F606C7EA2000F16 036D5E6C6C15070003160F6C6C151F6C6CED3DF8D97F8014786D6CEB01E0D91FF0903807 C078D907FE90387F00700101B500FC1330D9003F01F090C8FC020790CAFC45487CC54D> 71 D<91B6D8E003B61280A3020001E0C70003EB8000DB7F806E48C7FC03FF1503A293C8 5BA219075C4B5EA2190F14034B5EA2191F14074B5EA2193F140F4B5EA2197F141F4B5EA2 19FF143F92B8C8FCA3DA7FC0C712014B5DA2180314FF92C85BA218075B4A5EA2180F1303 4A5EA2181F13074A5EA2183F130F4A5EA2187F131F4A5EA2013F16FFA24A93C9FCD9FFE0 02037FB6D8E003B67EA351447CC351>I<027FB512F8A217F09139007FF000ED3FC0157F A25EA315FF93C7FCA35C5DA314035DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA3 14FF92C8FCA35B5CA313035CA313075CA3130F5CA2131FA25CEB7FF0007FB512F0B6FCA2 2D447DC32B>I<91B600E049B512C0A3020001E0C8383FF800DB7F80ED1FE003FF94C7FC 1A3E93C9127862F101C04A4C5A4B4BC8FC191C6102035E4B5DF003804EC9FC0207150E4B 14386060020F4A5A4B0107CAFC170E5F021F14784B13F84C7E1603023F130F4B487E163B EEE1FF91387FC1C1DB83807FED8700159CDAFFB86D7E5D03C06D7E5D4990C7FC4A6E7EA2 717E13034A811707A201076F7E5C717EA2130F4A6E7FA2727E131F5C727E133F854A82D9 FFE04B7EB600E0010FB512E05FA252447CC353>75 D<91B612F8A3020001E0C8FC6F5A4B 5AA293C9FCA35C5DA314035DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA314FF92 CAFCA35B4A16C0A21801010317804A15031900A201075E4A1506180E181E010F161C4A15 3C18381878011F16F84A4A5A1703013F150F4D5A4A14FF01FF02075BB9FCA2603A447CC3 42>I<91B500C0933803FFFE63630200F1FE00DB6FE0EE1BF803EF171F1B3703CFEF67F0 A21BCF0201EF018F038F60DB87F0ED030F1B1F020317060307040C5BA2F2183F02071730 0206616F6C15601B7F020E17C0020CDC018090C7FCA24F485A021C16060218606F6C5C1A 0102381618023004305BA2F16003027016C00260606F6CEB01801A0702E0ED03004A0306 5CA24E130F01015E4A60047F5B1A1F01035E91C74A5CA24D48133F494BC7FC010661EE3F 861A7F010E158C010C039892C8FCA205B05C011C15E001186001386E5A190101785D01FC 92C75BD803FFEF07FEB500F8011E0107B512FE161C160C5F447BC35E>I<91B500C0020F B5128082A2DA007F9239007FE00070ED1F8074C7FCDBEFF8150E15CF03C7160C70151C14 01DB83FE1518A2DB81FF1538140303001630831A704A6D7E02061760163F7114E0140E02 0C6D6C5CA2706C1301141C021801075D83190302386D7E023094C8FC1601715B14700260 6DEB8006A294387FC00E14E04A023F130C18E0191C0101ED1FF04A1618170FF0F8381303 91C83807FC30A2943803FE705B01060301136018FF19E0010E81010C5F187FA2131C0118 705A1338181F137801FC70C9FCEA03FFB512F884180651447CC34E>I<91B712FEF0FFE0 19F802009039C0000FFE6F48EB01FF03FF9138007F80F13FC093C8EA1FE0A24AEE0FF0A2 5D1AF81403A25DA21407F11FF05DA2020FEE3FE0A24B16C0197F021F1780F1FF004B4A5A 4E5A023F4B5A4E5A4BEC3FC006FFC7FC027FEC07FC92B612F018800380CAFC14FFA292CB FCA25BA25CA21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA25CEBFFE0B612E0 A345447CC33F>80 DI<91B712F018FF19E002 009039C0003FF86F48EB07FC03FFEC01FEF0007F93C8EA3F801AC0F11FE05C5D1AF0A214 035DA30207EE3FE05DA2F17FC0020F17804B15FF1A004E5A021F4B5A4B4A5AF00FE04E5A 023F037FC7FC4BEB03FCEF1FF092B612804A4AC8FC923980007F80EF0FC0EF07F002FF6E 7E92C77F1701845B4A1400A2170113035CA2170313075CA24D5A130F5CA3011F18185CA2 013F4C13381A304A6F1370D9FFE0020314E0B600E0ED01C00501EB0380943900FE0F00CB EA3FFEF007F045467CC34A>I<9339FF8001800307EBF003033F13FC9239FF007E07DA01 F8EB0F0FDA07E09038079F004A486DB4FC4AC77E023E804A5D187E5C495A183C495AA213 074A1538A3130F183080A295C7FC806D7E8014FF6D13E015FC6DEBFFC06D14FC6E13FF6E 14C0020F80020314F8EC003F03077F9238007FFE160F1603707E8283A283A21206A4000E 163EA2120C177E001E167CA25F5F003F15014C5A6D4A5A4C5A486C4AC8FC6D143ED87CF8 5CD8787E495A3AF01FC00FE0D8E007B51280010149C9FC39C0003FF039487BC53C>I<48 BA12C05AA291C7D980001380D807F092C7121F4949150F0180170748C75B1903120E4802 0316005E12181238003014074C5C00701806126000E0140F485DA3C8001F92C7FC5EA315 3F5EA3157F5EA315FF93CAFCA35C5DA314035DA314075DA3140F5DA3141F5DA3143F5DA2 147FA214FF01037F001FB612FCA25E42447EC339>I<003FB500F80103B512E0A326003F F8C8381FF800D91FE0ED07E0013F705A615C96C7FC60017F16065CA2180E01FF160C91C9 FCA2181C4817185BA21838000317305BA21870000717605BA218E0120F495EA21701121F 495EA21703123F4993C8FCA25F127F491506A2170E00FF160C90C9FC171CA21718173817 304816705F6C5E6C15014C5A4CC9FC6C150E6D141E001F5D6D5C6C6CEB01E06C6C495A6C 6CEB1F80C6B401FECAFC90387FFFF8011F13E0010190CBFC43467AC342>I<007FB56C91 381FFFF8B65DA2000101E0C8000313006C0180ED01FCF000F0614E5AA2017F4C5A96C7FC 1806A2606E5DA2013F5E1870186060A24D5A6E4AC8FCA2011F1506170E170C5FA26E5C5F A2010F5D16015F4CC9FCA26E13065EA201075C5EA25E16E06E5B4B5A13034BCAFC1506A2 5D151CECFE185D13015D5DA26E5AA292CBFC5C13005C5CA25CA25C45467BC339>I I97 D99 DIII<157E913803FF8091390FC1E0E091391F0073F0027E13334A13 3F4948131F010315E04948130F495AA2494814C0133F4A131F137F91C713805B163F5A49 1500A25E120349147EA216FEA2495CA21501A25EA21503150700015D150F0000141F6D13 3F017CEB77E090383E01E790381F078F903807FE0FD901F85B90C7FC151FA25EA2153FA2 93C7FCA2001C147E007F14FE485C4A5A140348495AEC0FC000F8495A007C01FEC8FC381F FFF8000313C02C407EAB2F>I<14FE137FA3EB01FC13001301A25CA21303A25CA21307A2 5CA2130FA25CA2131FA25CED3FC090393F81FFF0913887C0FC91380E007E023C133ED97F 70133F4A7F4A14805C13FF91C7FC5BA24848143F17005BA200035D167E5BA2000715FE5E 5B1501000F5DA24913035E001F1607030713064914E0150F003FEDC00E170C90C7141CEE 80184816381730007E167017E000FE91380781C0EEC38048913801FF000038EC007C3046 7BC438>I<141E143F5C5CA3147E143891C7FCAE133EEBFF803801C3C0380781E0380601 F0120E121CEA180312381230A2EA700700605BA2EAE00F00C05BEA001F5CA2133F91C7FC A25B137E13FE5BA212015BEC03800003140013F01207495A1406140E140CEBC01C141814 385C00035BEBE1C0C6B45A013EC7FC19437DC121>I<163C16FEA21501A316FCED007016 00AE15FCEC03FF91380F0780021C13C091383803E0147014E014C01301EC800713031400 5B0106130F130E010C14C090C7FC151FA21680A2153FA21600A25DA2157EA215FEA25DA2 1401A25DA21403A25DA21407A25DA2140FA25DA2141F5DA2143F001C91C7FC127F48137E 5CA248485AEB03E038F807C038781F80D83FFEC8FCEA07F0275681C128>I<14FE137FA3 EB01FC13001301A25CA21303A25CA21307A25CA2130FA25CA2131FA25C163F013FECFFC0 923803C0E09138000703ED1E0F491338ED701F017E13E0EC01C001FE018013C00203EB07 004948C8FC140E00015B5C495A5C3803FBC001FFC9FC8014F83807F1FE9038F03F809038 E00FE06E7E000F130381EBC001A2001FED01C017801380A2003F15031700010013F05E48 1506160E007E150C161C00FE01005BED787048EC3FE00038EC0F802B467BC433>II<01F8D903FCEC7F80D803FED91FFF903803FFE0D8 071F903B7C0FC00F81F83E0E0F80E007E01C00FC001C9026C3C0030178137C271807C700 D9F0E0137E02CE902601F1C0133E003801DCDAFB80133F003001D892C7FCD90FF814FF00 70495C0060495CA200E04949485CD8C01F187E4A5C1200040715FE013F6091C75BA2040F 14014960017E5D1903041F5D13FE494B130762043F160E0001060F130C4992C713C0191F 4CED801C00031A1849027E1638F2003004FE167000071A60494A16E0F201C0030192380F 0380000FF18700494AEC03FED80380D90070EC00F84F2D7DAB55>I<01F8EB03FCD803FE EB1FFFD8071F90387C0FC03B0E0F80E007E03A0C07C3C003001CD9C7007F001801CE1301 003801DC80003013D8EB0FF800705B00605BA200E0491303D8C01F5D5C12001607013F5D 91C7FCA2160F495D137E161F5F13FE49143F94C7FC187000014B136049147E16FE4C13E0 000317C049150104F81380170300071700495D170EEE781C000FED7C3849EC1FF0D80380 EC07C0342D7DAB3A>III< 01F8EB0FC0D803FEEB7FF0D8070FEBF038000E903883C07C3A0C07C701FC001C13CE0018 EBDC03003813D8003013F8D90FF013F800709038E000E0006015005C12E0EAC01F5C1200 A2133F91C8FCA35B137EA313FE5BA312015BA312035BA312075BA3120F5BEA0380262D7D AB2C>114 DI<141C147EA314FE 5CA313015CA313035CA313075CA2007FB512FCB6FC15F839000FC000A2131F5CA3133F91 C7FCA35B137EA313FE5BA312015BA312035BA21570000714605B15E015C0000F130101C0 13801403EC070000071306140E5C6C6C5A000113F03800FFC0013FC7FC1E3F7EBD23>I< 133ED9FF8014E02603C3C0EB03F0380703E0380601F0000E1507121CD818035D12380030 150FA2D870075D00605B161FEAE00F00C0495CEA001F4A133FA2013F92C7FC91C7FC5E5B 017E147EA216FE13FE495CA20301EB01801703484802F81300A25F0303130616F0000014 07030F130E6D010D130C017C011D131C033913186D9038F0F838903A1F03C07870903A07 FF803FE0903A01FC000F80312D7DAB38>I<02FCEB07E0903A03FF801FFC903A0F07C078 1E903A1C03E0E01F903A3801F1C07FD9700013804901FB13FF4848EBFF00495B000316FE 90C71438484A130012061401000E5C120CC7FC14035DA314075DA3140F5DA3021F143817 305D1770023F1460121E003F16E0267F807FEB01C0026F148000FF01EF1303D901CFEB07 0000FE903887C00E267C03835B3A3C0F01E0783A1FFC00FFE0D803F0EB3F80302D7EAB37 >120 D<027CEB018049B413034901801300010F6D5A49EBE00E6F5A90393F03F8389039 78007EF80170EB1FF00160EB01E001E05C49495A90C748C7FC150E5D5D5D5D4A5A4A5A4A C8FC140E5C5C5C5CEB03C049C9FC130E49141C4914185B49143848481430491470D80390 14F048B4495A3A0FEFC007C0391E03F01FD81C01B55A486C91C7FC485C00606D5A00E0EB 3FF048EB0FC0292D7CAB2D>122 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fq cmr12 12 86 /Fq 86 128 df0 D<1618163CA2167EA216FFA24B7FA24B6C7EA29238063FE0A24B6C7EA24B6C7EA2923838 07FC153092387003FE15609238E001FF15C002016D7F5D02036E7E92C7FC4A6E7E140602 0E6E7E140C021C6E7E141802386E7E143002706E7E146002E06E7E5C01016F7F5C010370 7E91C9FC183F010683181F4983180F49831807498318034983A249707EA24848701380A2 48CBEA7FC0A20006F03FE0A248F01FF0A2001FBA12F8A24819FCA24819FEA2BCFC48477C C651>I<0103B612FCA390C701F0C8FC6F5A6F5AA8913801FFF0023FEBFF80903A01FF3F DFF0D907F0EBC1FCD91FC0EBC07FD93F00EC1F8001FEED0FE048486F7E48486F7E48486F 7E48486F7E001F834982003F1880007F18C0A249163F00FF18E0A8007F18C06D167FA200 3F1880001F18006D5E000F5F6C6C4B5A6C6C4B5A6C6C4B5A6C6C4B5A013FED1F80D91FC0 027FC7FCD907F0EBC1FCD901FFEBDFF0D9003FB51280020101F0C8FC9138003FC0A84B7E 4B7E0103B612FCA33B447BC346>8 D<9239FFC001FC020F9038F80FFF913B3F803E3F03 C0913BFC00077E07E0D903F890390FFC0FF0494890383FF81F4948EB7FF0495A494814E0 49C7FCF00FE04991393FC0038049021F90C7FCAFB912F0A3C648C7D81FC0C7FCB3B2486C EC3FF0007FD9FC0FB512E0A33C467EC539>11 D<4AB4FC020F13E091387F80F8903901FC 001C49487FD907E0130F4948137F011FECFF80495A49C7FCA25B49EC7F00163E93C7FCAC EE3F80B8FCA3C648C7FC167F163FB3B0486CEC7FC0007FD9FC1FB5FCA330467EC536>I< 913801FFC0020FEBFB8091387F803F903801FC00494813FFEB07E0EB1FC0A2495A49C7FC 167F49143F5BAFB8FCA3C648C7123FB3B2486CEC7FC0007FD9FC1FB5FCA330467EC536> II<121EEA7F80EAFFC0A9EA7F80ACEA3F00AB121EAC 120CA5C7FCAA121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A4778C61B>33 D<001EEB03C0397F800FF000FF131F01C013F8A201E013FCA3007F130F391E6003CC0000 EB000CA401E0131C491318A3000114384913300003147090C712604814E0000614C0000E 130148EB038048EB070048130E0060130C1E1D7DC431>I<121EEA7F8012FF13C0A213E0 A3127FEA1E601200A413E013C0A312011380120313005A1206120E5A5A5A12600B1D78C4 1B>39 D<140C141C1438147014E0EB01C01303EB0780EB0F00A2131E5BA25B13F85B1201 5B1203A2485AA3485AA348C7FCA35AA2123EA2127EA4127CA312FCB3A2127CA3127EA412 3EA2123FA27EA36C7EA36C7EA36C7EA212017F12007F13787FA27F7FA2EB0780EB03C013 01EB00E014701438141C140C166476CA26>I<12C07E12707E7E7E120F6C7E6C7EA26C7E 6C7EA21378137C133C133E131E131FA2EB0F80A3EB07C0A3EB03E0A314F0A21301A214F8 A41300A314FCB3A214F8A31301A414F0A21303A214E0A3EB07C0A3EB0F80A3EB1F00A213 1E133E133C137C13785BA2485A485AA2485A48C7FC120E5A5A5A5A5A16647BCA26>I<16 C04B7EB3AB007FBAFCBB1280A26C1900C8D801E0C9FCB3AB6F5A41407BB84C>43 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013C0A312011380120313005A 1206120E5A5A5A12600B1D78891B>II<121EEA7F80A2EAFFC0A4 EA7F80A2EA1E000A0A78891B>I<14FF010713E090381F81F890383E007C01FC133F4848 EB1F8049130F4848EB07C04848EB03E0A2000F15F0491301001F15F8A2003F15FCA390C8 FC4815FEA54815FFB3A46C15FEA56D1301003F15FCA3001F15F8A26C6CEB03F0A36C6CEB 07E0000315C06D130F6C6CEB1F806C6CEB3F00013E137C90381F81F8903807FFE0010090 C7FC28447CC131>48 D<143014F013011303131F13FFB5FC13E713071200B3B3B0497E49 7E007FB6FCA3204278C131>II<49B4 FC010F13E0013F13FC9038FE01FE3A01F0007F80D803C0EB3FC048C7EA1FE0120EED0FF0 EA0FE0486C14F8A215077F5BA26C48130FEA03C0C813F0A3ED1FE0A2ED3FC01680ED7F00 15FE4A5AEC03F0EC1FC0D90FFFC7FC15F090380001FCEC007FED3F80ED1FC0ED0FE016F0 ED07F816FC150316FEA2150116FFA3121EEA7F80487EA416FE491303A2007EC713FC0070 1407003015F80038140F6C15F06CEC1FE06C6CEB3FC0D803E0EB7F803A01FE01FE003900 7FFFF8010F13E0010190C7FC28447CC131>II<000615C0D807C0130701FCEB 7F8090B612005D5D5D15E0158026063FFCC7FC90C9FCAE14FF010713C090381F01F09038 3800FC01F0137ED807C07F49EB1F8016C090C7120F000615E0C8EA07F0A316F81503A216 FCA5123E127F487EA416F890C712075A006015F0A20070140F003015E00038EC1FC07E00 1EEC3F806CEC7F006C6C13FE6C6C485A3901F807F039007FFFE0011F90C7FCEB07F82644 7BC131>II<121CA2EA1F8090B712C0A3481680A217 005E0038C8120C0030151C00705D0060153016705E5E4814014B5A4BC7FCC81206150E5D 151815385D156015E04A5AA24A5A140792C8FC5CA25C141E143EA2147E147CA214FCA213 01A3495AA41307A6130FAA6D5AEB01C02A457BC231>I<14FF010713E0011F13F890387F 00FE01FC133FD801F0EB1F804848EB0FC049EB07E00007EC03F048481301A290C713F848 1400A47FA26D130116F07F6C6CEB03E013FC6C6CEB07C09039FF800F806C9038C01F006C EBF03EECF87839007FFEF090383FFFC07F01077F6D13F8497F90381E7FFFD97C1F138049 6C13C02601E00313E048486C13F000079038007FF84848EB3FFC48C7120F003EEC07FE15 0148140016FF167F48153FA2161FA56C151E007C153EA2007E153C003E157C6C15F86DEB 01F06C6CEB03E06C6CEB07C0D803F8EB1F80C6B4EBFF0090383FFFFC010F13F001011380 28447CC131>I<14FF010713E0011F13F890387F80FC9038FC007E48487F4848EB1F8048 48EB0FC0000FEC07E0485AED03F0485A16F8007F140190C713FCA25AA216FE1500A516FF A46C5CA36C7E5D121F7F000F5C6C6C130E150C6C6C131C6C6C5BD8007C5B90383F01E090 390FFF80FE903801FE0090C8FC150116FCA4ED03F8A216F0D80F801307486C14E0486C13 0F16C0ED1F80A249EB3F0049137E001EC75A001C495A000F495A3907E01FE06CB51280C6 49C7FCEB1FF028447CC131>I<121EEA7F80A2EAFFC0A4EA7F80A2EA1E00C7FCB3A5121E EA7F80A2EAFFC0A4EA7F80A2EA1E000A2B78AA1B>I<121EEA7F80A2EAFFC0A4EA7F80A2 EA1E00C7FCB3A5121E127FEAFF80A213C0A4127F121E1200A512011380A3120313005A12 06120E120C121C5A5A12600A3E78AA1B>I<007FBAFCBB1280A26C1900CEFCB0007FBAFC BB1280A26C190041187BA44C>61 DI<16C04B7EA34B7EA34B7EA34B7E A3ED19FEA3ED30FFA203707FED607FA203E07FEDC03FA2020180ED801FA2DA03007F160F A20206801607A24A6D7EA34A6D7EA34A6D7EA20270810260147FA202E08191B7FCA24982 0280C7121FA249C87F170FA20106821707A2496F7EA3496F7EA3496F7EA201788313F848 6C83D80FFF03037FB500E0027FEBFFC0A342477DC649>65 DIIIIIIII<010FB512FEA3D9000313806E13 0080B3B3AB123F487E487EA44A5A13801300006C495A00705C6C13076C5C6C495A6CEB1F 802603E07FC7FC3800FFFCEB1FE027467BC332>IIIIIII82 D<49B41303010FEBE007013F13F89039FE 00FE0FD801F8131FD807E0EB079F49EB03DF48486DB4FC48C8FC4881003E81127E82127C 00FC81A282A37E82A27EA26C6C91C7FC7F7FEA3FF813FE381FFFE06C13FE6CEBFFE06C14 FC6C14FF6C15C0013F14F0010F80010180D9001F7F14019138001FFF03031380816F13C0 167F163F161F17E000C0150FA31607A37EA36C16C0160F7E17806C151F6C16006C5D6D14 7ED8FBC05CD8F9F0495AD8F07C495A90393FC00FE0D8E00FB51280010149C7FC39C0003F F02B487BC536>I<003FB912F8A3903BF0001FF8001F01806D481303003EC7150048187C 0078183CA20070181CA30060180CA5481806A5C81600B3B3A54B7EED7FFE49B77EA33F44 7DC346>III I<003FB500E0011FB5FCA3C691C7000713E0D93FFC020190C7FC6D4815FC010F6F5A6D6C 15E0A26D6C4A5A6D6C5D4DC8FC6D6D5B6E6C13065F6E6C131C6E6C13185F6E6C13706E6C 13605F913803FE01DA01FF5B4CC9FC6E1387ED7FC616CCED3FFC6F5A5E6F7E6F7EA26F7E 82A203067F150E92380C7FC04B6C7E15389238301FF04B6C7E15E04B6C7E4A486C7E1403 4B6C7E02066D7F140E020C6E7E4A6E7E143802306E7E4A6E7E14E04A6E7E49486E7E1303 49C86C7E496F7F5B496C8201FF83000701E0020313F8B500F8021FEBFFF0A344447EC349 >II<001FB81280A39126800001130001FCC7FC01F04A5A01C04A5A5B90C848 5A121E4C5A484B5AA200384B5A4C5AA24B90C7FC00304A5AA24B5AA24B5AC8485AA24B5A 4B5AA24B5A5C93C8FC4A5AA24A5A4A5AA24A5A4A5AA24A5A14FF5D4990C9FCEF0180495A 495AA2495A494814031800495AA2495A495A5F4890C8FC485A5F485A48485D5F48485D17 FE484814034848140F16FFB8FCA331447BC33C>II<01C01318000114384848137048C712E0000EEB01C0000C1480001C 13030018140000385B003013060070130E0060130CA300E0131C481318A400CFEB19E039 FFC01FF801E013FCA3007F130FA2003F130701C013F8390F0001E01E1D71C431>II97 DII<167FED3FFFA3 15018182B3EC7F80903803FFF090380FC07C90383F000E017E1307496D5AD803F87F4848 7F5B000F81485AA2485AA2127FA290C8FC5AAB7E7FA2123FA26C7EA2000F5D7F6C6C5B00 035C6C6C9038077F806C6C010E13C0013F011C13FE90380FC0F8903803FFE09026007F00 13002F467DC436>IIIIII<143C14FFA2 491380A46D1300A2143C91C7FCADEC7F80EB3FFFA31300147F143FB3B3AA123E127F39FF 807F00A2147EA25C6C485A383C01F06C485A3807FF80D801FEC7FC195785C21E>I III<3901FC01FE00FF90 3807FFC091381E07F091383801F8000701707F0003EBE0002601FDC07F5C01FF147F91C7 FCA25BA35BB3A8486CECFF80B5D8F83F13FEA32F2C7DAB36>II<3901FC03FC00FF90380FFF8091383C07E091387001F83A07FDE000 FE00030180137FD801FFEC3F8091C7EA1FC04915E049140F17F0160717F8160317FCA3EE 01FEABEE03FCA3EE07F8A217F0160F6D15E0EE1FC06D143F17806EEB7E00D9FDC05B9039 FCF003F891383C0FE091381FFF80DA03FCC7FC91C9FCAE487EB512F8A32F3F7DAB36>I< 91387F8003903903FFE00790380FE07890393F801C0F90387E000E496D5AD803F8EB039F 0007EC01BF4914FF48487F121F5B003F81A2485AA348C8FCAB6C7EA3123F7F121F6D5C12 0F6D5B12076C6C5B6C6C497E6C6C130E013F131C90380FC0F8903803FFE09038007F0091 C7FCAEEEFF80033F13FEA32F3F7DAB33>I<3903F803F000FFEB1FFCEC3C3EEC707F0007 EBE0FF3803F9C000015B13FBEC007E153C01FF13005BA45BB3A748B4FCB512FEA3202C7D AB26>I<90383FE0183901FFFC383907E01F78390F0003F8001E1301481300007C147812 7800F81438A21518A27EA27E6C6C13006C7E13FC383FFFE06C13FC6C13FF6C14C06C14E0 C614F0011F13F81300EC0FFC140300C0EB01FE1400157E7E153EA27EA36C143C6C147C15 786C14F86CEB01F039F38003E039F1F00F8039E07FFE0038C00FF01F2E7DAC26>I<1306 A5130EA4131EA3133E137EA213FE12011207001FB512F0B6FCA2C648C7FCB3A4150CAA01 7E131C017F1318A26D133890381F8030ECC070903807E0E0903801FFC09038007F001E3E 7EBC26>IIIIII<003FB612E0 A29038C0003F90C713C0003CEC7F800038ECFF00A20030495A0070495AA24A5A0060495A A24A5A4A5AA2C7485A4AC7FC5B5C495A13075C495A131F4A1360495A495AA249C712C048 5AA2485A485A1501485A48481303A24848EB07804848131F00FF14FF90B6FCA2232B7DAA 2B>I<01F81302D803FE13073907FF800E48EBE01C391F1FF8F8393807FFF0D8700113E0 39E0007FC00040EB1F00200978C131>126 D<001EEB0780007FEB0FE039FF801FF0EBC0 3FA4EB801F397F000FE0001EEB07801C0A76C231>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmbx12 17.28 37 /Fr 37 122 df45 DI<16F04B7E 1507151F153FEC01FF1407147F010FB5FCB7FCA41487EBF007C7FCB3B3B3B3007FB91280 A6395E74DD51>49 D<913801FFF8021FEBFFC091B612F8010315FF010F16C0013F829026 7FFC0114F89027FFE0003F7F4890C7000F7F48486E7FD807F86E148048486E14C048486E 14E048486F13F001FC17F8486C816D17FC6E80B56C16FE8380A219FFA283A36C5BA26C5B 6C90C8FCD807FC5DEA01F0CA14FEA34D13FCA219F85F19F04D13E0A294B512C019804C14 004C5B604C5B4C5B604C13804C90C7FC4C5A4C5A4B13F05F4B13804B90C8FC4B5AED1FF8 4B5A4B5A4B48143F4A5B4A48C8FC4A5A4A48157E4A5A4A5AEC7F8092C9FC02FE16FE495A 495A4948ED01FCD90FC0150749B8FC5B5B90B9FC5A4818F85A5A5A5A5ABAFCA219F0A440 5E78DD51>I<92B5FC020F14F8023F14FF49B712C04916F0010FD9C01F13FC90271FFC00 077FD93FE001017F49486D8049C86C7F484883486C6F7F14C0486D826E806E82487FA480 5CA36C5E4A5E6C5B6C5B6C495E011FC85A90C95CA294B55A614C91C7FC604C5B4C5B4C5B 4C5B047F138092260FFFFEC8FC020FB512F817E094C9FC17F817FF91C7003F13E0040713 F8040113FE707F717F7113E085717FA2717F85A285831A80A31AC0EA03FCEA0FFF487F48 7F487FA2B57EA31A80A34D14005C7E4A5E5F6C495E49C8485BD81FF85F000F5ED807FE92 B55A6C6C6C4914806C01F0010791C7FC6C9026FF803F5B6D90B65A011F16F0010716C001 014BC8FCD9001F14F0020149C9FC426079DD51>II<01C0EE01C0D801F8160F01FF167F02F0EC07FFDA FF8090B5FC92B7128019006060606060606095C7FC17FC5F17E0178004FCC8FC16E09026 FC3FFCC9FC91CBFCADED3FFE0203B512F0020F14FE023F6E7E91B712E001FDD9E00F7F90 27FFFE00037F02F801007F02E06EB4FC02806E138091C8FC496F13C04917E07113F0EA00 F090C914F8A219FC83A219FEA419FFA3EA03F0EA0FFC487E487E487FA2B57EA319FEA35C 4D13FC6C90C8FC5B4917F8EA3FF001804B13F06D17E0001F5E6C6C17C06D4B1380D807FC 92B512006C6C4A5B6C6C6C01075B6C01E0011F5BD97FFE90B55A6DB712C0010F93C7FC6D 15FC010115F0D9003F1480020301F0C8FC406078DD51>II58 D65 D<4DB5ED03C0057F02F014070407B600FE140F047FDBFFC0131F4BB800F0133F03 0F05FC137F033F9127F8007FFE13FF92B6C73807FF814A02F0020113C3020702C0913800 7FE74A91C9001FB5FC023F01FC16074A01F08291B54882490280824991CB7E4949844949 8449498449865D49498490B5FC484A84A2484A84A24891CD127FA25A4A1A3F5AA348491A 1FA44899C8FCA25CA3B5FCB07E071FB812F880A37EA296C70001ECC000A26C7FA37E807E A26C80A26C80A26C807F6D7F816D7F7F6D7F6D6D5F6D14C06D6E5E6E7F6E01FC5E020F01 FF5E6E02C0ED7FEF020102F8EDFFC76E02FF02071383033F02FC013F1301030F91B638FC 007F03014D131F6F6C04E01307040704801301DC007F02F8CAFC050191CBFC6D6677E37F >71 D73 D77 D80 D82 DI<913803FFFE027FEBFFF00103B612FE010F6F7E4916E090273FFE001F7FD97FE001 077FD9FFF801017F486D6D7F717E486D6E7F85717FA2717FA36C496E7FA26C5B6D5AEB1F C090C9FCA74BB6FC157F0207B7FC147F49B61207010F14C0013FEBFE004913F048B512C0 4891C7FC485B4813F85A5C485B5A5CA2B55AA45FA25F806C5E806C047D7F6EEB01F96C6D D903F1EBFF806C01FED90FE114FF6C9027FFC07FC01580000191B5487E6C6C4B7E011F02 FC130F010302F001011400D9001F90CBFC49437CC14E>97 D<903807FF80B6FCA6C6FC7F 7FB3A8EFFFF8040FEBFF80047F14F00381B612FC038715FF038F010014C0DBBFF0011F7F DBFFC001077F93C76C7F4B02007F03F8824B6F7E4B6F13804B17C0851BE0A27313F0A21B F8A37313FCA41BFEAE1BFCA44F13F8A31BF0A24F13E0A24F13C06F17804F1300816F4B5A 6F4A5B4AB402075B4A6C6C495B9126F83FE0013F13C09127F00FFC03B55A4A6CB648C7FC DAC00115F84A6C15E091C7001F91C8FC90C8000313E04F657BE35A>I<92380FFFF04AB6 7E020F15F0023F15FC91B77E01039039FE001FFF4901F8010113804901E0010713C04901 804913E0017F90C7FC49484A13F0A2485B485B5A5C5A7113E0485B7113C0487013809438 00FE0095C7FC485BA4B5FCAE7EA280A27EA2806C18FCA26C6D150119F87E6C6D15036EED 07F06C18E06C6D150F6D6DEC1FC06D01E0EC7F806D6DECFF00010701FCEB03FE6D9039FF C03FFC010091B512F0023F5D020F1580020102FCC7FCDA000F13C03E437BC148>II<92380F FFC04AB512FC020FECFF80023F15E091B712F80103D9FE037F499039F0007FFF011F01C0 011F7F49496D7F4990C76C7F49486E7F48498048844A804884485B727E5A5C48717EA35A 5C721380A2B5FCA391B9FCA41A0002C0CBFCA67EA380A27EA27E6E160FF11F806C183F6C 7FF17F006C7F6C6D16FE6C17016D6C4B5A6D6D4A5A6D01E04A5A6D6DEC3FE0010301FC49 B45A6D9026FFC01F90C7FC6D6C90B55A021F15F8020715E0020092C8FC030713F041437C C14A>III<903807FF80B6FCA6C6FC7F7FB3A8EF1FFF94B512F004 0714FC041F14FF4C8193267FE07F7F922781FE001F7FDB83F86D7FDB87F07FDB8FC0814C 7F039FC78015BE03BC8003FC825DA25DA25DA45DB3B2B7D8F007B71280A651647BE35A> II<903807FF 80B6FCA6C6FC7F7FB3B3B3B3ADB712E0A623647BE32C>108 D<902607FF80D91FFFEEFF F8B691B500F00207EBFF80040702FC023F14E0041F02FF91B612F84C6F488193267FE07F 6D4801037F922781FE001F9027E00FF0007FC6DA83F86D9026F01FC06D7F6DD987F06D4A 487F6DD98FC0DBF87EC7804C6D027C80039FC76E488203BEEEFDF003BC6E4A8003FC04FF 834B5FA24B5FA24B94C8FCA44B5EB3B2B7D8F007B7D8803FB612FCA67E417BC087>I<90 2607FF80EB1FFFB691B512F0040714FC041F14FF4C8193267FE07F7F922781FE001F7FC6 DA83F86D7F6DD987F07F6DD98FC0814C7F039FC78015BE03BC8003FC825DA25DA25DA45D B3B2B7D8F007B71280A651417BC05A>I<923807FFE092B6FC020715E0021F15F8027F15 FE494848C66C6C7E010701F0010F13E04901C001037F49496D7F4990C87F49486F7E4948 6F7E48496F13804819C04A814819E048496F13F0A24819F8A348496F13FCA34819FEA4B5 18FFAD6C19FEA46C6D4B13FCA36C19F8A26C6D4B13F0A26C19E06C6D4B13C0A26C6D4B13 806C6D4B13006D6C4B5A6D6D495B6D6D495B010701F0010F13E06D01FE017F5B010090B7 C7FC023F15FC020715E0020092C8FC030713E048437CC151>I<902607FF80EBFFF8B601 0FEBFF80047F14F00381B612FC038715FF038F010114C09227BFF0003F7FC6DAFFC0010F 7F6D91C76C7F6D496E7F03F86E7F4B6E7F4B17804B6F13C0A27313E0A27313F0A21BF885 A21BFCA3851BFEAE4F13FCA41BF861A21BF0611BE0611BC06F92B512801B006F5C6F4A5B 6F4A5B03FF4A5B70495B04E0017F13C09226CFFC03B55A03C7B648C7FC03C115F803C015 E0041F91C8FC040313E093CBFCB3A3B712F0A64F5D7BC05A>I114 D<913A3FFF8007800107B5EAF81F011FECFE7F017F91B5FC48B8FC48 EBE0014890C7121FD80FFC1407D81FF0801600485A007F167F49153FA212FF171FA27F7F 7F6D92C7FC13FF14E014FF6C14F8EDFFC06C15FC16FF6C16C06C16F06C826C826C826C82 013F1680010F16C01303D9007F15E0020315F0EC001F1500041F13F81607007C150100FC 81177F6C163FA2171F7EA26D16F0A27F173F6D16E06D157F6D16C001FEEDFF806D020313 0002C0EB0FFE02FCEB7FFC01DFB65A010F5DD8FE0315C026F8007F49C7FC48010F13E035 437BC140>I I<902607FFC0ED3FFEB60207B5FCA6C6EE00076D826D82B3B3A260A360A2607F60183E6D 6D147E4E7F6D6D4948806D6DD907F0ECFF806D01FFEB3FE06D91B55A6E1500021F5C0203 14F8DA003F018002F0C7FC51427BC05A>II<007FB600C0017FB512F8A6D8 001F01F8C70007EBF0006D040190C7FC6D6D5D6D6D4A5A6D6D4A5A70495A6D4C5A6E7F6E 6D495A6E6D495A7049C8FC6E4A5A6E6D485A6E6D485A6E13FFEF8FF06EEC9FE06FEBFFC0 6F5C6F91C9FC5F6F5B816F7F6F7F8481707F8493B57E4B805D4B80DB0FF37FDB1FE17F04 C080153F4B486C7F4B486C7F4A486D7F4A486D7F4A5A4B6D7F020F6E7F4A486D7F4A486D 804A5A4AC86C7F49486F7F4A6F7F0107707FEB3FFFB600F049B7FCA650407EBF55>120 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fs cmr10 10.95 39 /Fs 39 128 df12 D14 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013C0A312011380120313005A 120E5A1218123812300B1C798919>44 DI<121EEA7F80A2EAFF C0A4EA7F80A2EA1E000A0A798919>I<15074B7EA34B7EA34B7EA34B7EA34B7E15E7A291 3801C7FC15C3A291380381FEA34AC67EA3020E6D7EA34A6D7EA34A6D7EA34A6D7EA34A6D 7EA349486D7E91B6FCA249819138800001A249C87EA24982010E157FA2011E82011C153F A2013C820138151FA2017882170F13FC00034C7ED80FFF4B7EB500F0010FB512F8A33D41 7DC044>65 D73 D77 D81 DII<003FB91280A3903AF0007FE001018090393FC0003F48C7ED1FC0007E1707 127C00781703A300701701A548EF00E0A5C81600B3B14B7E4B7E0107B612FEA33B3D7DBC 42>II97 DI<49B4FC010F 13E090383F00F8017C131E4848131F4848137F0007ECFF80485A5B121FA24848EB7F0015 1C007F91C7FCA290C9FC5AAB6C7EA3003FEC01C07F001F140316806C6C13076C6C140000 03140E6C6C131E6C6C137890383F01F090380FFFC0D901FEC7FC222A7DA828>IIII<167C903903F801FF903A1FFF078F8090397E0FDE1F9038F803F83803 F001A23B07E000FC0600000F6EC7FC49137E001F147FA8000F147E6D13FE00075C6C6C48 5AA23901F803E03903FE0FC026071FFFC8FCEB03F80006CAFC120EA3120FA27F7F6CB512 E015FE6C6E7E6C15E06C810003813A0FC0001FFC48C7EA01FE003E140048157E825A82A4 6C5D007C153E007E157E6C5D6C6C495A6C6C495AD803F0EB0FC0D800FE017FC7FC90383F FFFC010313C0293D7EA82D>III<1478EB01FEA2EB03FFA4EB01 FEA2EB00781400AC147FEB7FFFA313017F147FB3B3A5123E127F38FF807E14FEA214FCEB 81F8EA7F01387C03F0381E07C0380FFF803801FC00185185BD1C>III<2701F801FE14FF00FF902707FFC00313E0913B 1E07E00F03F0913B7803F03C01F80007903BE001F87000FC2603F9C06D487F000101805C 01FBD900FF147F91C75B13FF4992C7FCA2495CB3A6486C496CECFF80B5D8F87FD9FC3F13 FEA347287DA74C>I<3901F801FE00FF903807FFC091381E07E091387803F000079038E0 01F82603F9C07F0001138001FB6D7E91C7FC13FF5BA25BB3A6486C497EB5D8F87F13FCA3 2E287DA733>I<14FF010713E090381F81F890387E007E01F8131F4848EB0F804848EB07 C04848EB03E0000F15F04848EB01F8A2003F15FCA248C812FEA44815FFA96C15FEA36C6C EB01FCA3001F15F86C6CEB03F0A26C6CEB07E06C6CEB0FC06C6CEB1F80D8007EEB7E0090 383F81FC90380FFFF0010090C7FC282A7EA82D>I<3901FC03FC00FF90381FFF8091387C 0FE09039FDE003F03A07FFC001FC6C496C7E6C90C7127F49EC3F805BEE1FC017E0A2EE0F F0A3EE07F8AAEE0FF0A4EE1FE0A2EE3FC06D1580EE7F007F6E13FE9138C001F89039FDE0 07F09039FC780FC0DA3FFFC7FCEC07F891C9FCAD487EB512F8A32D3A7EA733>I<02FF13 1C0107EBC03C90381F80F090397F00387C01FC131CD803F8130E4848EB0FFC1507484813 03121F485A1501485AA448C7FCAA6C7EA36C7EA2001F14036C7E15076C6C130F6C7E6C6C 133DD8007E137990383F81F190380FFFC1903801FE0190C7FCAD4B7E92B512F8A32D3A7D A730>I<3901F807E000FFEB1FF8EC787CECE1FE3807F9C100031381EA01FB1401EC00FC 01FF1330491300A35BB3A5487EB512FEA31F287EA724>I<90383FC0603901FFF8E03807 C03F381F000F003E1307003C1303127C0078130112F81400A27E7E7E6D1300EA7FF8EBFF C06C13F86C13FE6C7F6C1480000114C0D8003F13E0010313F0EB001FEC0FF800E01303A2 14017E1400A27E15F07E14016C14E06CEB03C0903880078039F3E01F0038E0FFFC38C01F E01D2A7DA824>I<131CA6133CA4137CA213FCA2120112031207001FB512C0B6FCA2D801 FCC7FCB3A215E0A912009038FE01C0A2EB7F03013F138090381F8700EB07FEEB01F81B39 7EB723>IIIIII<001C130E007FEB3F8039 FF807FC0A5397F003F80001CEB0E001A0977BD2D>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ft cmbx10 10.95 7 /Ft 7 117 df<16FCA24B7EA24B7EA34B7FA24B7FA34B7FA24B7FA34B7F157C03FC7FED F87FA2020180EDF03F0203804B7E02078115C082020F814B7E021F811500824A81023E7F 027E81027C7FA202FC814A147F49B77EA34982A2D907E0C7001F7F4A80010F835C83011F 8391C87E4983133E83017E83017C81B500FC91B612FCA5463F7CBE4F>65 D<903807FFC0013F13F848B6FC48812607FE037F260FF8007F6DEB3FF0486C806F7EA36F 7EA26C5A6C5AEA01E0C8FC153F91B5FC130F137F3901FFFE0F4813E0000F1380381FFE00 485A5B485A12FF5BA4151F7F007F143F6D90387BFF806C6C01FB13FE391FFF07F36CEBFF E100031480C6EC003FD91FF890C7FC2F2B7DA933>97 D<13FFB5FCA512077EAFEDFFE002 0713FC021FEBFF80027F80DAFF8113F09139FC003FF802F06D7E4A6D7E4A13074A807013 80A218C082A318E0AA18C0A25E1880A218005E6E5C6E495A6E495A02FCEB7FF0903AFCFF 01FFE0496CB55AD9F01F91C7FCD9E00713FCC7000113C033407DBE3A>II<3901FE01FE00FF903807FF804A13E04A13F0EC3F1F91387C3F F8000713F8000313F0EBFFE0A29138C01FF0ED0FE091388007C092C7FCA391C8FCB3A2B6 FCA525297DA82B>114 D<90383FFC1E48B512BE000714FE5A381FF00F383F800148C7FC 007E147EA200FE143EA27E7F6D90C7FC13F8EBFFE06C13FF15C06C14F06C806C806C806C 80C61580131F1300020713C014000078147F00F8143F151F7EA27E16806C143F6D140001 E013FF9038F803FE90B55A15F0D8F87F13C026E00FFEC7FC222B7DA929>II E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fu cmr12 14.4 33 /Fu 33 122 df<120FEA3FC0EA7FE012FF13F0A213F8A3127F123FEA0F381200A5137813 70A313F013E0A2120113C0120313801207EA0F00121EA25A5A12300D23768B21>44 D<120FEA3FC0EA7FE0EAFFF0A6EA7FE0EA3FC0EA0F000C0C768B21>46 D48 D<14075C5C147F5C1307133F 000FB5FCB6FC13F913C1EAF0011200B3B3B3A7497F010F13E0B712FEA4274F75CE3B>I< EC7FE0903803FFFE010F6D7E013F14E0D9FF0013F8D801F8EB1FFCD803E06D7E4848EB03 FF48486D138090C813C0001E16E0001C157F003CED3FF012380078ED1FF81270A2B4ED0F FC13C07FA66C5A6C5A000EC8FCC9EA1FF8A317F0163FA2EE7FE017C016FF17804B1300A2 4B5A4B5A5E4B5A4B5A4B5A5E4BC7FC15FE4A5A4A5A4A5A4A5A5D4A5A4AC8FC147E5C4948 141CEB03E0495A4948143891C8FC131E5B5B491578485A48481570484815F048B7FCA25A 5A5AB812E0A42E4F7ACE3B>I66 D68 D71 D77 DI< EE7FF80307B57E033F14F09239FFC00FFC913A03FE0001FFDA0FF89038007FC0DA3FE0EC 1FF0DA7F80EC07F84AC86C7E49486F7E49486F7E4948707E4948707E4948707E4948707E 017F844948707E91CA120348854848717EA24848711380A2000F1AC049187F001F1AE0A3 4848F03FF0A3007F1AF8A249181FA300FF1AFCAE6C6CF03FF8A5003F1AF06D187FA2001F 1AE0A26D18FF000F1AC0A26C6C4D1380A200031A006D5F6C616C6D4C5A6E160F017F606D 6C4C5A6D6C4C5AA26D6C4C5A6D6C4C5A6D6C4B90C7FCD900FFED03FCDA7FC0EC0FF86E6C 4A5ADA0FF8EC7FC0DA03FE4948C8FC913A00FFC00FFC033FB512F0030714809226007FF8 C9FC4E567AD35B>II82 D<003FBB12C0A449C79038F0000701F06E48130001C0183F48 C8EE0FE0007E1907007C1903A200781901A400701900A500F01AF0481A70A6C91700B3B3 AC4C7E030313FC027FB712E0A44C517CD055>84 DI97 D99 D<17FF4BB5FCA4ED0007160182B3A6EC0FF8EC7FFF49B512E0903907FC03F090391FE000 7C49487F49C7120F01FE80484880485A000781484880A2485AA2485AA2127FA35B12FFAB 127FA27FA2123FA27F121FA26C6C5C00075D7F6C6C5C6C6C5C6C6C021E7F6D6C017C13E0 D91FC049EBFF8090390FF807E00103B512800100495ADA1FF091C7FC39547CD241>II<157F913803FFE0020F13F091383FC0F891387F01FC90 3901FE03FE903803FC0714F81307EB0FF0A290391FE003FCED01F892C7FC495AB3B612FE A426003FC0C7FCB3B3A580EBFFF0007FEBFFF8A427547DD324>I104 D<1378EA01FE487E487FA66C90C7FC6C5AEA007890C8FCB0EB7F 80B5FCA41203C6FC137FB3B3A43801FFE0B61280A419507CCF21>I<01FFD907FEEC03FF B590261FFFC0010F13E0037F01F0013F13F8912701F80FFC9038FC07FE913D03C003FE01 E001FF000390260700019038038000C6010E6D6C48C76C7E6D48DA7F8E6E7E4A159CA24A DA3FF86E7E02605D14E04A5DA34A5DB3AD2601FFE0DAFFF0EC7FF8B6D8C07F9026FFE03F B512F0A45C347CB363>109 D<01FFEB07FCB590383FFF8092B512E0913901F00FF89139 03C007FC000349C66C7EC6010E13016D486D7E5C143002706E7E146014E05CA35CB3AD26 01FFE0903801FFE0B600C0B612C0A43A347CB341>II<90 397F8007FCB590387FFF800281B512E0913987F00FF891398F8003FC000390399E0001FF C601BC6D7FD97FF86E7E4A6E7E4A6E7E4A140F844A6E7EA2717EA3717EA4711380AB4D13 00A44D5AA24D5AA2606E140F4D5A6E5D6E4A5A6E4A5A02BC4AC7FC029E495A028FEB07FC 913987E01FF00281B512C0DA807F90C8FCED0FF892CAFCB13801FFE0B612C0A4394B7DB3 41>I<01FFEB1F80B5EB7FF0913801FFF8913803E1FC91380783FE0003EB0F07C6131EEB 7F1C1438143091387003FC91386000F0160014E05CA45CB3AA8048487EB612F0A427347D B32E>114 DIIII<007FB5D8800FB51280A4C69026FC0003EBF0 00D93FF86D1380011F4BC7FC010F15F801075D6D6C5C6E495A6D6C5C6D14076E6C48C8FC EDC01E6E6C5A021F133891380FF0786F5A913807FDE002035BEC01FF5E80157F6F7E824B 7E15FFEDE7F802017F913803C3FEEC07814AC67E020E80021E6D7E4A133F4A6D7E4A8070 7E4948130749486D7E010781010F6E7E013F8201FF8200076D010713F0B500F8011FEBFF E0A43B337FB23E>120 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fv cmr17 20.74 25 /Fv 25 122 df58 D65 D77 D<943801FFF0053FEBFF804CB612F0040F9038001FFEDC3FF0903801FF80DCFF80903800 3FE0DB03FEC8EA0FF8DB0FF8ED03FEDB1FE0ED00FFDB7FC0EE7FC04B48707E4A48CAEA0F F0DA07FCEF07FC4A48717E4A48717E4A48717F4A48727E02FF864949727E92CC121F4948 737E0107874948737E4A1903011F874948737FA24948737FA24948747E4889A24849747E A34890CE6C7EA24889491B0FA2001F89A34848751380A5007F1EC0A24987A400FF1EE0B1 6C6C5113C0A6003F1E806D63A3001F1E00A26D636C65A36C6D505AA36C6D505AA26C6D50 5AA26C656E61017F646D6C4F5BA26D6C4F90C7FCA26D6C4F5A6D6C4F5A6D636F183F6D6D 4E5A6D636E6C4E5A6E6C4D5B6E6C4D90C8FC6E6C4D5A6E6C4D5A6E6C4D5A02006DEE3FE0 DB7FE0EEFFC06F6C4B5BDB0FFCDB07FEC9FCDB03FEED0FF8922601FFC0EC7FF09226003F F0903801FF80DC0FFFD91FFECAFC040390B512F8DC003F1480050101F0CBFC6B7C78F87C >79 D<943801FFF0053FEBFF804CB612F0040F9038001FFEDC3FF0903801FF80DCFF8090 38003FE0DB03FEC8EA0FF8DB0FF8ED03FE4B486F7EDB7FC09238007FC04B48707E4A90CA 6C7EDA07FEEF0FFC4A48717E4A48717E4A48717F4A48717F4A48727E4949727EA24990CC 6C7E4948737E010F874A19074948737E013F884A85017F884A8501FF8848894A1A7F4889 4A1A3FA2488991CE121F4889A34848757EA3003F1E80A24987A3007F1EC0A44987A200FF 1EE0B1007F1EC0A26D63A4003F1E80A36D63001F1E00A36C6C515AA36C656E1A3FA26C65 6E1A7F6C65A26C6D505AA26D6CDB3FC04A5B013FDCFFF05E6E020301FC5C011F922607C0 3E93C7FC6E91270F800F805B010F92261F00035D6D6C021E6D6C495A6D6C4A6D6C495A05 3814606D6D0470495A6D6D01786E495A6E6C01704C5A6E6CDC18015B6E6CDC1C0390C8FC 6E6C4D5A6E6C70485A6E6C4D5A0200D98078ED3FE092267FC0384B5ADB3FF06F485A9226 0FF83C4BC9FC922603FE1EEC0FF8922601FF9FEC3FF0923C003FFF8001FF80040FD9C01F 1730040390B5EAFBC0DC003F14830501EBF00394C8FC871F701A01871FF07515E0A27514 011E03747E1E0775EC0FC076131F76133FE17FF0137F9939FC03FF8099B6FC751500A375 5C6687755C755C755C7514800A3F90C7FCF407F86C9A78F87C>81 D<913803FF80021F13F891B512FE903A03FC01FF80903A07E0003FE0D91F80EB0FF8013E C76C7E496E7E01F06E7E48486E7F717E4848153F4982D807A06F7E13FC487E6D6F7E80A2 717EA46C90C8FC6C5A6C5ACAFCA6EE07FF0303B5FC157F913903FFFE07021F138091387F F800903801FFC0010790C7FCEB1FFCEB3FF0EBFFE0485B485B4890C8FC5B485A485AA248 5A1A0E485AA312FF5B170FA4171FA26D153F007F163B177B6DDBF1FE131C003F16E16C6C 14016C6C912603C0FF13386C6CEC0F806C6C6C903A1F007F80706C6D017CECE1E028007F F803F8EB3FFF011FB500E06D1380010391C7000713009026003FF8EC01FC474D79CB4F> 97 D<14F8EA03FFB5FCA5C6FC133F131FA2130FB3B04CB47E041F13F8047F13FE923A01 FC01FF80923A07E0003FE0031FC7EA0FF0033EEC07FC0378EC01FE4B6E7EDAF9E06F7EDA FBC06F7EDAFF808292C96C7E737E5C4A707E864A160386851B80A37313C0A31BE0A31A7F A21BF0AE1BE0A21AFFA31BC0A2611B80A21B0061626E1607626E160F626E4C5A02F75FDA E7804B5ADAE3C0157FDAC1E04B5ADAC0F04A48C7FC03784A5A4A6CEC0FF8031F4A5A4A6C 6CEB7FC0922703F803FFC8FC0300B512FC010E023F13E090C8D807FEC9FC4C797BF758> II<19 1FF07FFF051FB5FCA5EF001F180784A284B3B0ED07FE92387FFFC00203B512F091390FFC 01FC91393FE0001FDAFF80EB07814990C7EA03E1D903FCEC01F14948EC0079D91FF0153D 4948151D4A151F49488101FF824890C9FC48835B0007835B120F5B121FA2123F5BA2127F A35BA212FFAE127FA27FA3123FA36C7EA36C7EA200075F7F00035F6C7E606C6D5D6D6C15 3D013F16396D6C03797F6D6C15F16D6CDA03E17FD903FEDA078113F0D900FFDA1F01EBFF F0DA7FC0137E91391FF803F80207B512E0020114809127001FF800EC80004C797AF758> IIII<14F8EA03FFB5FC A5C6FC133F131FA2130FB3B0933803FF80041F13F8047F13FE923A01FC03FF80923A03E0 007FE0DB0F80EB1FF0031EC76C7E5D4B6E7E4B6E7E5D14F9DAFBC06E7E5D14FF92C9FC86 5CA35CA45CB3B3A8496C4B7FD97FFF030713F0B7D8800FB612F8A54D787AF758>I<131E EB7F80497E487F487FA66C5B6C5B6D5A011EC7FC90C8FCB3A7EB01F0EA07FFB5FCA51201 EA007F133FA2131FB3B3B3A3497EEBFFFEB612FCA51E727AF12A>I108 DIII<02F849B47ED803FF021F13F8B502 7F13FE923A01FC01FF80923A07E0003FE0031FC76C7E033EEC0FFCC60278EC03FE013F49 6E7E90261FF9E06E7FDAFBC0826DB4486F7E92C96C7E737E5C4A707E864A160786851B80 A2851BC0A2851BE0A5F27FF0AEF2FFE0A54F13C0A34F1380A21B0061626E160F626E161F 626E4C5A4F5A6F5EDAFBC015FFDAF9E04A5BDAF8F04A48C7FC03784A5A6F4A5A031FEC3F F06F6CEBFFC0922603F80790C8FC0300B512FC043F13E0DC07FEC9FC93CBFCB3A7497EEB 7FFFB77EA54C6C7BCA58>I114 DI<1407A8 5CA65CA35CA35CA25CA25BA25B5B5B5B5B5B48B712FE120FB8FCA3D8000190C9FCB3B3A2 EF01C0B0EF03806D7FA3027FEC0700815F6E6C130E021F141E6F131C6E6C5B6E6C13F891 3901FF01F09139007FFFC0031F5BDB03FCC7FC326B7EE93D>I<02F8EE0F80D803FFEE3F FFB5030FB5FCA5C6EE000F013F1603011F82A2010F82B3B3A660A460A3601307606E150E 0103161E606E4B7F010116706D6C03F07F6FD903E013F86E6C4948EBFFF8DA1FE0EB1F00 DA0FFE13FE0203B512F8DA007F13E0030790C7EBC0004D4C7ACA58>II121 D E %EndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%EndSetup %%Page: 1 1 1 0 bop 536 976 a Fv(On)52 b(asymptotic)g(p)t(erturbation)h(theory)g (for)532 1183 y(Quan)l(tum)d(Mec)l(hanics:)70 b(Almost)53 b(in)l(v)-9 b(arian)l(t)498 1391 y(subspaces)52 b(and)g(gauge)f(in)l(v) -9 b(arian)l(t)54 b(magnetic)1194 1598 y(p)t(erturbation)f(theory)1617 1894 y Fu(G.)39 b(Nenciu)717 2043 y(Dept.)51 b(Theor.)g(Ph)m(ys.,)37 b(Univ)m(ersit)m(y)f(of)j(Buc)m(harest,)1210 2193 y(PO)g(Bo)m(x)f(MG)h (11,)f(Romania)1463 2427 y(2nd)h(Marc)m(h)f(2000)1676 2794 y Ft(Abstract)638 2962 y Fs(Singular)19 b(p)s(erturbation)i (theory)h(for)h(Quan)m(tum)e(Mec)m(hanics)i(is)e(considered)501 3075 y(in)39 b(a)h(framew)m(ork)g(generalising)f(the)h(sp)s(ectral)f (concen)m(tration)i(theory)-8 b(.)70 b(Un-)501 3188 y(der)32 b(v)m(ery)g(general)g(conditions,)f(asymptotic)i(estimations)e(on)h (the)g(Ra)m(yleigh-)501 3301 y(Sc)m(hr\177)-45 b(odinger)32 b(expansions)g(of)h(the)g(p)s(erturb)s(ed)e(sp)s(ectral)h(pro)5 b(jections)33 b(are)g(ob-)501 3414 y(tained.)59 b(As)37 b(a)g(consequence)g(almost)g(in)m(v)-5 b(arian)m(t)36 b(subspaces)f(of)i(exp)s(onen)m(tial)501 3527 y(order)31 b(are)h(constructed.)44 b(The)30 b(results)h(co)m(v)m(ers)h (practically)f(all)f(singular)f(p)s(er-)501 3640 y(turbations)38 b(considered)f(in)g(nonrelativistic)f(quan)m(tum)i(mec)m(hanics.)65 b(In)38 b(the)501 3753 y(magnetic)33 b(\014eld)e(case,)j(under)d(the)h (condition)f(that)i(the)f(magnetic)h(\014eld)e(do)s(es)501 3866 y(not)e(increase)g(at)g(in\014nit)m(y)-8 b(,)28 b(a)h(gauge)h(in)m(v)-5 b(arian)m(t)28 b(p)s(erturbation)f(theory)i (leading)501 3978 y(to)i(con)m(v)m(ergen)m(t)i(series)d(with)f(\014eld) g(dep)s(enden)m(t)g(co)s(e\016cien)m(ts)i(is)f(dev)m(elop)s(ed.)257 4311 y Fr(1)161 b(In)l(tro)t(duction)257 4530 y Fq(While)29 b(the)i(regular)e(p)s(erturbation)h(theory)h(ac)m(hiv)m(ed)g(its)e (\014nal)h(form)f(in)g(the)i(pap)s(ers)g(of)257 4651 y(Sz-Nagy)-8 b(,)31 b(Rellic)m(h)d(and)i(Kato)e(\(see)j([14])f(,)g ([22])f(and)h(references)i(therein\),)e(in)f(the)h(case)257 4771 y(singular)k(p)s(erturbations)i(\(Stark)g(and)f(Zeeman)g (e\013ects)i(in)e(quan)m(um)h(mec)m(hanics)g(are)257 4891 y(in)d(this)h(class\))f(the)h(progress)h(has)f(b)s(een)g(m)m(uc)m (h)h(slo)m(w)m(er,)f(at)g(least)f(for)g(the)h(degenerate)257 5012 y(eigen)m(v)-5 b(alues)35 b(case.)51 b(In)35 b(the)g(general)f (framew)m(ork)h(laid)e(do)m(wn)i(in)f(Kato's)h(b)s(o)s(ok)f(a)g(sys-) 257 5132 y(tematic)j(theory)h(of)f(asymptotic)g(Ra)m(yleigh-Sc)m (hr\177)-49 b(odinger)37 b(p)s(erturbation)f(theory)j(for)257 5253 y(degenerate)e(eigen)m(v)-5 b(alues)36 b(has)h(b)s(een)f(dev)m (elop)s(ed,)i(fairly)c(recen)m(tly)j(b)m(y)g(Hunzik)m(er)g(and)257 5373 y(co)m(w)m(ork)m(ers)j([27],)e([13])f(,)i([12)o(])f(\()f(see)h (the)g(b)s(eautiful)e(review)i(pap)s(er)f(b)m(y)h(Hunzik)m(er)g([12]) 1852 5637 y(1)p eop %%Page: 2 2 2 1 bop 257 573 a Fq(and)30 b(also)f(the)h(recen)m(t)h(b)s(o)s(ok)f(b)m (y)g(Hislop)f(and)g(Sigal)f([10])h(for)h(detailed)e(exp)s(osition)h (and)257 693 y(extensiv)m(e)42 b(bibliograph)m(y\).)63 b(The)41 b(k)m(ey)g(condition)d(whic)m(h)i(selects)h(the)f(allo)m(w)m (ed)f(class)257 814 y(of)31 b(p)s(erturbations)f(is)g(the)h(so)g (called)e(\\stabilit)m(y)g(condition";)h(in)g(the)h(cases)h(where)g (the)257 934 y(considered)j(eigen)m(v)-5 b(alues)34 b(are)g(not)g (stable)g(one)g(considers,)h(in)e(the)i(spirit)d(of)i(Aguilar-)257 1054 y(Balslev-Com)m(b)s(es)25 b(theory)i([1],)g([3])f(,)h(a)e(related) h(\(e.g.)41 b(dilated\))24 b(hamiltonian)e(for)k(whic)m(h)257 1175 y(the)36 b(stabilit)m(y)d(condition)g(holds)i(true.)50 b(The)36 b(main)d(p)s(oin)m(t)h(of)g(the)h(theory)h(is)e(that)h(the)257 1295 y(stabilit)m(y)21 b(condition,)i(together)g(with)f(the)g(exp)s (onen)m(tial)g(deca)m(y)i(of)e(unp)s(erturb)s(ed)h(eigen-)257 1416 y(v)m(ectors)30 b(giv)m(en)f(b)m(y)h(Com)m(b)s(es-Thomas)e(theory) h([4],)g([12])g(,)g(allo)m(ws)e(to)i(obtain)e(p)s(o)m(w)m(erfull)257 1536 y(asymptotic)33 b(estimates.)47 b(The)34 b(price)g(to)f(b)s(e)h (paid)f(are)g(strong)h(conditions)f(\(e.g.)47 b(dila-)257 1656 y(tion)31 b(analyticit)m(y\))g(on)h(the)g(p)s(erturbation.)42 b(Also,)32 b(at)g(the)g(tec)m(hnical)g(lev)m(el)f(the)i(theory)257 1777 y(is)i(far)g(from)f(b)s(eing)h(elemen)m(tary)g(since)h(one)f(has)h (to)f(deal)g(with)f(non-self-adjoin)m(t)g(op-)257 1897 y(erators,)41 b(analytic)d(con)m(tin)m(uation)g(etc.)63 b(In)39 b(addition,)g(as)g(far)f(as)h(w)m(e)h(understand,)i(it)257 2017 y(do)s(es)33 b(not)g(seem)g(easy)g(to)g(obtain)e(informations)f (on)j(the)g(p)s(erturb)s(ed)g(eigen)m(v)m(ectors.)404 2138 y(In)j(this)g(pap)s(er)g(whic)m(h)g(,)h(with)f(the)g(exception)h (of)e(the)h(last)g(section,)h(is)e(a)h(con)m(tin-)257 2258 y(uation)e(of)f([16])i(w)m(e)g(follo)m(w)d(a)j(di\013eren)m(t)f (route,)h(closer)f(in)g(its)g(spirit)f(with)h(the)h(theory)257 2379 y(of)f(sp)s(ectral)g(concen)m(tration,)h(in)f(whic)m(h)h(the)g (main)d(ob)5 b(jects)36 b(are)e(the)h(p)s(erturb)s(ed)g(sub-)257 2499 y(spaces.)54 b(The)36 b(aim)d(is)i(t)m(w)m(ofold:)48 b(\014rst,)37 b(using)e(the)g(results)h(in)f([17)o(])h(w)m(e)g(push)g (the)g(esti-)257 2619 y(mations)h(in)g([16],)i([20)o(])f(up)h(to)e(exp) s(onen)m(tial)g(order;)k(second,)g(and)d(more)f(imp)s(ortan)m(t,)257 2740 y(w)m(e)d(extend)h(the)f(analysis)e(to)h(cases)h(where)g(the)g (tec)m(hniques)h(in)d([16],)h([17])g(,)g(based)h(on)257 2860 y(adiabatic)d(expansions)j(do)e(not)h(apply)-8 b(.)404 2980 y(The)33 b(con)m(ten)m(t)g(of)f(the)g(pap)s(er)g(is)g(as)g(follo)m (ws.)42 b(In)32 b(Section)g(2)g(w)m(e)h(set)g(up)f(the)h(frame-)257 3101 y(w)m(ork.)79 b(Almost)43 b(in)m(v)-5 b(arian)m(t)42 b(subspaces)47 b(of)c(families)e(of)j(self-adjoin)m(t)e(op)s(erators)i (are)257 3221 y(de\014ned)33 b(and)e(it)e(is)i(sho)m(wn)h(that)e(their) h(existence)h(giv)m(es)f(\(up)g(to)g(some)f(errors\))h(similar)257 3342 y(sp)s(ectral)36 b(and)g(ev)m(olution)f(informations)e(as)j(the)g (existence)h(of)e(in)m(v)-5 b(arian)m(t)35 b(subspaces.)257 3462 y(In)e(this)f(con)m(text)h(w)m(e)h(giv)m(e)e(a)g(v)m(ery)i (general)d(form)g(of)h(the)h(Hylleraas-Wigner)d(\\2n+1")257 3582 y(theorem:)49 b(\\the)36 b(kno)m(wledge)h(of)e(eigenfunction)f(up) i(to)f(errors)h(of)f(order)h Fp(\016)t Fq(,)g(giv)m(es)g(the)257 3703 y(kno)m(vledge)e(of)d(the)i(corresp)s(onding)f(eigen)m(v)-5 b(alue)32 b(up)g(to)g(errors)h(of)e(order)i Fp(\016)3028 3667 y Fo(2)3067 3703 y Fq(".)43 b(Section)257 3823 y(3,)31 b(again)e(of)h(general)g(nature,)i(is)e(dev)m(oted)i(to)e(the)h (construction)g(of)f(almost)f(in)m(v)-5 b(arian)m(t)257 3944 y(subspaces)44 b(out)c(of)g(asymptotic)h(Ra)m(yleigh-)e(Sc)m (hr\177)-49 b(odinger)41 b(series)g(of)f(the)i(p)s(erturb)s(ed)257 4064 y(pro)5 b(jection.)86 b(Section)47 b(4)f(con)m(tains)h(one)g(of)f (the)h(main)e(results)i(of)g(the)g(pap)s(er:)72 b(for)257 4184 y(the)40 b(Sc)m(hr\177)-49 b(odinger)39 b(op)s(erator)f(in)g Fn(R)1557 4148 y Fo(3)1641 4184 y Fq(\(actually)f(all)g(the)i(argumen)m (ts)g(w)m(ork)h(in)e(an)h(ap-)257 4305 y(propriate)32 b(abstract)h(setting\),)g(under)g(v)m(ery)h(general)e(conditions)g(on)h (the)g(p)s(erturbing)257 4425 y(p)s(oten)m(tial,)k(w)m(e)h(are)f(able)g (to)g(construct)h(formal)c(Ra)m(yleigh-Sc)m(hr\177)-49 b(odinger)36 b(expansions)257 4545 y(for)j(the)g(p)s(erturb)s(ed)g(pro) 5 b(jection)39 b(\(see)h(also)e([11)o(]\))h(and)g(to)f(obtain)g(go)s(o) s(d)g(asymptotic)257 4666 y(estimates)i(on)g(the)h(co)s(e\016cien)m(ts) g(of)f(this)f(expansions.)67 b(The)41 b(results)g(\(or)f(their)f(easy) 257 4786 y(extensions)g(to)d Fn(R)921 4750 y Fm(n)974 4786 y Fq(\))h(co)m(v)m(er)h(the)g(p)s(erturbations)e(b)m(y)i(p)s (olynomially)33 b(b)s(ounded)38 b(p)s(oten-)257 4907 y(tials)33 b(of)h(b)s(ound)h(states)h(in)d(atomic,)h(molecular)e(and)j (solid)e(state)i(ph)m(ysics)h(.)50 b(Also)34 b(for)257 5027 y(p)s(erturbing)g(p)s(oten)m(tials)f(gro)m(wing)g(up)i(at)f (in\014nit)m(y)f(at)h(most)g(linearly)e(\(as)i(in)g(the)g(case)257 5147 y(of)k(the)h(Stark)f(e\013ect\))h(one)f(can)h(treat)e(the)i(p)s (erturbations)f(of)f(the)i(energy)g(bands)g(in)257 5268 y(solids)g(\(the)i(Stark-W)-8 b(annier)39 b(problem\).)65 b(In)40 b(all)e(this)i(cases)h(one)g(obtains)e(that)h(the)257 5388 y(p)s(erturb)s(ed)47 b(subspaces)i(are)d(almost)e(in)m(v)-5 b(arian)m(t)45 b(up)h(to)g(exp)s(onen)m(tially)f(long)g(times,)1852 5637 y(2)p eop %%Page: 3 3 3 2 bop 257 573 a Fq(whic)m(h)34 b(in)e(the)h(presen)m(t)i(setting)d (is)g(the)i(anlog)d(of)h(the)i(fact)e(that)h(the)g(imaginary)d(part)257 693 y(of)g(the)g(corresp)s(onding)f(resonances)j(\()d(in)g(the)h(cases) h(when)g(they)f(can)g(b)s(e)g(de\014ned\))h(are)257 814 y(exp)s(onen)m(tially)h(small.)404 934 y(Section)k(5,)g(the)h(last)e (one,)i(concernes)h(with)e(p)s(erturbations)f(giv)m(en)i(b)m(y)f (magnetic)257 1054 y(\014elds)26 b(whic)m(h)g(do)g(not)f(gro)m(w)h(at)g (in\014nit)m(y)f(\(the)h(most)f(imp)s(ortan)m(t)e(particular)h(case)j (b)s(eing)257 1175 y(the)f(Zeeman)f(e\013ect)h(i.e.)41 b(the)26 b(case)g(of)f(constan)m(t)h(magnetic)e(\014eld\))h(and)h(can)f (b)s(e)h(read)f(in-)257 1295 y(dep)s(enden)m(tly)g(of)e(the)h(rest)g (of)f(the)h(pap)s(er.)40 b(The)25 b(main)c(result)j(is)f(that)g(this)g (case)h(is)f(closer)257 1416 y(to)31 b(regular)g(p)s(erturbation)f (theory)-8 b(.)44 b(More)32 b(exactly)f(one)h(\014nds)g(that)f Fl(after)j(factorising)257 1536 y Fq(the)f(so)f(called)f(\\nonin)m (tegrable)g(gauge)h(phase)h(factor")e(on)h(can)g(write)g(do)m(wn)h(a)e (gauge)257 1656 y(in)m(v)-5 b(arian)m(t)42 b Fl(c)-5 b(onver)g(gent)43 b Fq(p)s(erturbation)f(series)i(for)f(the)g(in)m (tegral)f(k)m(ernel)i(of)f(the)h(p)s(er-)257 1777 y(turb)s(ed)31 b(resolv)m(en)m(t,)i(with)d(co)s(e\016cien)m(ts)h(dep)s(ending)g (themselvs)g(up)s(on)g(the)g(p)s(erturbing)257 1897 y(magnetic)h (\014eld.)44 b(This)33 b(dep)s(endence)i(comes)e(from)f Fl(phase)i(factors)e Fq(con)m(taining)g(\015uxes)257 2017 y(through)d(some)f(p)s(olygonal)f(con)m(tours)i(and)g(the)g (asymptotic)f(nature)h(of)f(the)i(usual)e(ex-)257 2138 y(pansions)j(comes)g Fl(solely)g Fq(from)e(expanding)i(these)i(phase)e (factors.)43 b(In)31 b(particular)f(this)257 2258 y(leads)d(\()f(for)g (the)h(\014rst)f(time)f(to)i(our)f(b)s(est)h(kno)m(wledge\))g(to)f(a)g (con)m(v)m(ergen)m(t)j(expansion)e(for)257 2379 y(the)g(usual)f(Zeeman) f(e\013ect.)43 b(The)27 b(fact)f(that)g(the)g(factorisation)e(of)i(the) h(non-in)m(tegrable)257 2499 y(gauge)32 b(phase)g(factor)f(mak)m(es)h (the)g(p)s(erturbation)f(giv)m(en)h(b)m(y)g(magnetic)e(\014elds)i(less) g(sin-)257 2619 y(gular)g(has)h(b)s(een)g(used)h(man)m(y)f(times)f(in)g (the)h(ph)m(ysical)g(literature)e(\(see)j(e.g.)44 b([15],)32 b([18])257 2740 y(and)i(references)i(therein)e(in)f(connection)h(with)f (the)h(P)m(eierls-Onsager)g(problem\))e(and)257 2860 y(has)40 b(b)s(een)g(fully)e(used)i(recen)m(tly)h([6])e(in)f (connection)i(with)f(deca)m(y)i(of)d(eigenfunctions)257 2980 y(\(see)30 b(also)d([5])h(for)f(the)i(use)g(of)f(the)g(same)g (ideea)g(to)g(con)m(trol)f(the)i(thermo)s(dynamic)e(limit)257 3101 y(for)32 b(quan)m(tum)h(gases)h(in)d(constan)m(t)j(magnetic)d (\014eld\).)257 3428 y Fr(2)161 b(Almost)65 b(in)l(v)-9 b(arian)l(t)67 b(subspaces.)105 b(General)65 b(the-)499 3610 y(ory)-13 b(.)257 3830 y Fq(W)-8 b(e)33 b(start)g(with)f(the)h (de\014nition)f(of)g(almost)f(in)m(v)-5 b(arian)m(t)31 b(subspaces)k(\(a.s.i.\).)257 3999 y Fk(De\014nition)h(2.1.)50 b Fl(L)-5 b(et)44 b Fp(H)1262 4014 y Fm(")1299 3999 y Fp(;)17 b(P)1406 4014 y Fm(")1442 3999 y Fp(;)107 b(")45 b Fj(2)h Fq([0)p Fp(;)17 b(")1945 4014 y Fo(0)1983 3999 y Fq(])p Fp(;)g(")2100 4014 y Fo(0)2184 3999 y Fp(>)45 b Fq(0)f Fl(b)-5 b(e)44 b(families)f(of)h(self-adjoint)257 4120 y(op)-5 b(er)g(ators)30 b(and)g(ortho)-5 b(gonal)29 b(pr)-5 b(oje)g(ctions)29 b(r)-5 b(esp)g(e)g(ctively,)31 b(in)f(a)f(sep)-5 b(ar)g(able)30 b(Hilb)-5 b(ert)30 b(sp)-5 b(ac)g(e)257 4240 y(,)35 b Fj(H)q Fl(,)g(satisfying)f(the)h(c)-5 b(onditions)404 4360 y(i.)1466 4541 y Fq(lim)1464 4601 y Fm(")p Fi(!)p Fo(0)1630 4541 y Fj(k)28 b Fp(P)1771 4556 y Fm(")1830 4541 y Fj(\000)22 b Fp(P)1992 4556 y Fo(0)2059 4541 y Fj(k)p Fq(=)28 b(0)p Fp(:)1006 b Fq(\(2.1\))404 4754 y Fl(ii.)44 b Fp(P)601 4769 y Fm(")638 4754 y Fj(D)s Fq(\()p Fp(H)837 4769 y Fm(")873 4754 y Fq(\))28 b Fj(\032)g(D)s Fq(\()p Fp(H)1243 4769 y Fm(")1279 4754 y Fq(\))35 b Fl(wher)-5 b(e)34 b Fj(D)s Fq(\()p Fp(H)1826 4769 y Fm(")1862 4754 y Fq(\))h Fl(is)f(the)h(domain)f(of)h Fp(H)2746 4769 y Fm(")2817 4754 y Fl(and)f(on)h Fj(D)s Fq(\()p Fp(H)3345 4769 y Fm(")3381 4754 y Fq(\))p Fl(:)827 5055 y Fj(k)28 b Fq([)p Fp(H)1013 5070 y Fm(")1049 5055 y Fp(P)1112 5070 y Fm(")1149 5055 y Fq(])g Fj(k\024)g Fp(\016)t Fq(\()p Fp(")p Fq(\))p Fp(;)116 b Fq(0)27 b Fj(\024)h Fp(\016)t Fq(\()p Fp(")p Fq(\))f Fp(<)h Fj(1)p Fp(;)117 b Fq(lim)2395 5115 y Fm(")p Fi(!)p Fo(0)2550 5055 y Fp(\016)t Fq(\()p Fp(")p Fq(\))27 b(=)g(0)p Fp(:)370 b Fq(\(2.2\))404 5268 y Fl(Then)40 b(we)h(say)g(that)h Fp(P)1262 5283 y Fm(")1299 5268 y Fj(H)g Fl(ar)-5 b(e)41 b(almost)g(invariant)f(subsp) -5 b(ac)g(es)41 b(of)g(or)-5 b(der)41 b Fp(\016)t Fq(\()p Fp(")p Fq(\))f Fl(for)257 5388 y Fp(H)338 5403 y Fm(")375 5388 y Fl(.)1852 5637 y Fq(3)p eop %%Page: 4 4 4 3 bop 404 573 a Fq(The)34 b(de\014nition)e(implies)f(that)i Fp(P)1646 588 y Fo(0)1685 573 y Fj(H)i Fq(is)d(an)i(in)m(v)-5 b(arian)m(t)31 b(subspace)36 b(of)c Fp(H)3049 588 y Fo(0)3089 573 y Fq(.)45 b(In)34 b(what)257 693 y(follo)m(ws)d(w)m(e)h(shall)e (restrict)h(ourselv)m(es)i(to)e(the)h(case)h(when)f Fp(P)2500 708 y Fo(0)2571 693 y Fq(is)f(the)h(sp)s(ectral)f(pro)5 b(jec-)257 814 y(tion)32 b(of)g Fp(H)650 829 y Fo(0)722 814 y Fq(corresp)s(onding)g(to)g(a)h(b)s(ounded)g(isolated)e(part,)i Fp(\033)2599 829 y Fo(0)2639 814 y Fq(,)f(of)g Fp(\033)t Fq(\()p Fp(H)2987 829 y Fo(0)3027 814 y Fq(\))g(i.e.)404 934 y(iii.)1354 1167 y Fp(P)1417 1182 y Fo(0)1484 1167 y Fq(=)1635 1100 y Fp(i)p 1598 1144 108 4 v 1598 1235 a Fq(2)p Fp(\031)1732 1032 y Fh(I)1788 1257 y Fo(\000)1836 1167 y Fq(\()p Fp(H)1955 1182 y Fo(0)2016 1167 y Fj(\000)23 b Fp(z)t Fq(\))2203 1126 y Fi(\000)p Fo(1)2298 1167 y Fp(dz)901 b Fq(\(2.3\))257 1424 y(where)40 b(\000)e(is)g(a)f(con)m (tour)i(enclosing)f Fp(\033)1674 1439 y Fo(0)1713 1424 y Fq(.)61 b(In)38 b(particular,)g(w)m(e)h(shall)e(not)h(consider)h(the) 257 1544 y(p)s(erturbation)32 b(of)g(em)m(b)s(edded)i(eigen)m(v)-5 b(alues.)404 1665 y Fl(R)g(emarks)404 1785 y Fq(2.1)43 b(The)i(condition)e(\(2.1\))g(implies)f(that)i(for)f(su\016cien)m(tly)i (small)d Fp(")h Fq(\(for)h(whic)m(h)257 1905 y Fj(k)28 b Fp(P)398 1920 y Fm(")456 1905 y Fj(\000)22 b Fp(P)618 1920 y Fo(0)685 1905 y Fj(k\024)28 b Fq(1\))k Fp(P)1022 1920 y Fm(")1091 1905 y Fq(and)g Fp(P)1343 1920 y Fo(0)1414 1905 y Fq(are)h(unitarily)d(equiv)-5 b(alen)m(t;)32 b(in)f(particular)g (dim)15 b Fp(P)3270 1920 y Fm(")3307 1905 y Fj(H)28 b Fq(=)257 2026 y Fp(dimP)489 2041 y Fo(0)529 2026 y Fj(H)q Fq(.)43 b(Notice)31 b(that)g(\(2.1\))f(cannot)h(b)s(e)h(replaced)f(b)m (y)h Fp(s)19 b Fj(\000)h Fq(lim)2694 2041 y Fm(")p Fi(!)p Fo(0)2853 2026 y Fp(P)2916 2041 y Fm(")2981 2026 y Fq(=)27 b Fp(P)3147 2041 y Fo(0)3218 2026 y Fq(ev)m(en)32 b(if)257 2146 y(dim)16 b Fp(P)500 2161 y Fo(0)539 2146 y Fj(H)29 b Fp(<)e Fj(1)32 b Fq(and)h Fp(H)1158 2161 y Fm(")1222 2146 y Fj(!)28 b Fp(H)1431 2161 y Fo(0)1502 2146 y Fq(in)k(the)h (strong)g(resolv)m(en)m(t)g(sense)i(\(see)e(e.g.)44 b([12]\))404 2267 y(2.2)30 b(The)h(condition)e(ii.)41 b(is)30 b(equiv)-5 b(alen)m(t)31 b(with)f(the)h(follo)m(wing)c(one:)43 b Fp(P)2939 2282 y Fm(")2976 2267 y Fj(D)3053 2282 y Fm(")3117 2267 y Fj(\032)28 b(D)s Fq(\()p Fp(H)3421 2282 y Fm(")3457 2267 y Fq(\))257 2387 y(where)34 b Fj(D)616 2402 y Fm(")685 2387 y Fq(is)e(a)h(core)g(of)f Fp(H)1263 2402 y Fm(")1332 2387 y Fq(and)g(2.2)h(holds)f(true)h(on)f Fj(D)2352 2402 y Fm(")2389 2387 y Fq(.)404 2507 y(If)43 b(w)m(e)i(write)e Fp(H)1008 2522 y Fm(")1088 2507 y Fq(in)g(a)g(matrix)f(form)h (according)g(to)g(the)h(decomp)s(osition)e Fj(H)47 b Fq(=)257 2628 y Fp(P)320 2643 y Fm(")357 2628 y Fj(H)23 b(\010)g Fq(\(1)f Fj(\000)g Fp(P)835 2643 y Fm(")872 2628 y Fq(\))p Fj(H)33 b Fq(then)h(the)f(o\013-diagonal)c(part,)k Fp(B)2259 2643 y Fm(")2295 2628 y Fq(:)727 2958 y Fp(B)801 2973 y Fm(")865 2958 y Fq(=)28 b Fp(P)1032 2973 y Fm(")1069 2958 y Fp(H)8 b Fq(\(1)21 b Fj(\000)i Fp(P)1429 2973 y Fm(")1465 2958 y Fq(\))f(+)g(\(1)g Fj(\000)h Fp(P)1895 2973 y Fm(")1931 2958 y Fq(\))p Fp(H)8 b(P)2121 2973 y Fm(")2185 2958 y Fq(=)28 b(\(1)22 b Fj(\000)g Fq(2)p Fp(P)2609 2973 y Fm(")2646 2958 y Fq(\)[)p Fp(H)2792 2973 y Fm(")2828 2958 y Fp(;)17 b(P)2935 2973 y Fm(")2971 2958 y Fq(])p Fp(:)270 b Fq(\(2.4\))257 3167 y(is)32 b(b)s(ounded)i(on)e Fj(D)s Fq(\()p Fp(H)1088 3182 y Fm(")1124 3167 y Fq(\))1593 3497 y Fj(k)c Fp(B)1745 3512 y Fm(")1809 3497 y Fj(k\024)g Fp(\016)t Fq(\()p Fp(")p Fq(\))p Fp(:)1135 b Fq(\(2.5\))404 3707 y(It)39 b(follo)m(ws)e(that)i(b)s(oth)g Fp(P)1360 3722 y Fm(")1397 3707 y Fp(H)1478 3722 y Fm(")1514 3707 y Fp(P)1577 3722 y Fm(")1653 3707 y Fq(and)g(\(1)26 b Fj(\000)h Fp(P)2129 3722 y Fm(")2166 3707 y Fq(\))p Fp(H)2285 3722 y Fm(")2321 3707 y Fq(\(1)f Fj(\000)h Fp(P)2601 3722 y Fm(")2638 3707 y Fq(\))39 b(are)g(self-adjoin)m(t)e (on)257 3827 y Fp(P)320 3842 y Fm(")357 3827 y Fj(D)s Fq(\()p Fp(H)556 3842 y Fm(")592 3827 y Fq(\))h(and)f(\(1)25 b Fj(\000)i Fp(P)1141 3842 y Fm(")1177 3827 y Fq(\))p Fj(D)s Fq(\()p Fp(H)1414 3842 y Fm(")1450 3827 y Fq(\))38 b(resp)s(ectiv)m(ely)-8 b(.)59 b(Notice)37 b(that)h Fp(P)2693 3842 y Fm(")2729 3827 y Fp(H)2810 3842 y Fm(")2847 3827 y Fp(P)2910 3842 y Fm(")2984 3827 y Fq(need)h(not)e(b)s(e)257 3948 y(b)s(ounded.)404 4068 y(In)c(what)f(follo)m(ws)g(w)m(e)h(shall)f (use)h(the)g(notations:)826 4398 y Fp(H)907 4413 y Fm(";D)1050 4398 y Fj(\021)28 b Fp(H)1236 4413 y Fm(")1295 4398 y Fj(\000)23 b Fp(B)1469 4413 y Fm(")1533 4398 y Fq(=)28 b Fp(P)1700 4413 y Fm(")1736 4398 y Fp(H)1817 4413 y Fm(")1854 4398 y Fp(P)1917 4413 y Fm(")1976 4398 y Fq(+)22 b(\(1)g Fj(\000)g Fp(P)2345 4413 y Fm(")2382 4398 y Fq(\))p Fp(H)2501 4413 y Fm(")2537 4398 y Fq(\(1)g Fj(\000)h Fp(P)2809 4413 y Fm(")2845 4398 y Fq(\))p Fp(;)385 b Fq(\(2.6\))1194 4697 y Fp(h)1250 4712 y Fm(")1315 4697 y Fj(\021)28 b Fp(P)1483 4712 y Fm(")1519 4697 y Fp(H)1600 4712 y Fm(")1637 4697 y Fp(P)1700 4712 y Fm(")1764 4697 y Fq(:)g Fp(P)1882 4712 y Fm(")1919 4697 y Fj(D)s Fq(\()p Fp(H)2118 4712 y Fm(")2154 4697 y Fq(\))f Fj(!)h Fp(P)2410 4712 y Fm(")2446 4697 y Fj(H)q Fp(:)737 b Fq(\(2.7\))404 4907 y(T)-8 b(aking)34 b(in)m(to)g(accoun)m(t)i(\(2.4-2.7\))d(the)i(ab) s(o)m(v)m(e)h(de\014nition)e(of)g(a.s.i.)49 b(coincide)35 b(with)257 5027 y(the)e(de\014nition)f(of)g(asymptotic)g(in)m(v)-5 b(arian)m(t)31 b(subspaces)k(in)d([16].)404 5147 y(One)38 b(exp)s(ects)h(that)e(the)i(existence)g(of)e(a.s.i.)58 b(to)37 b(giv)m(e)h(\(up)g(to)f(some)g(errors)h(con-)257 5268 y(trolled)h(b)m(y)h Fp(\016)t Fq(\()p Fp(")p Fq(\)\))g(similar)c (sp)s(ectral)k(informations)d(on)j Fp(H)2462 5283 y Fm(")2538 5268 y Fq(as)g(the)h(existence)g(of)e(in-)257 5388 y(v)-5 b(arian)m(t)32 b(subspaces.)1852 5637 y(4)p eop %%Page: 5 5 5 4 bop 257 573 a Fk(Prop)s(osition)36 b(2.2.)49 b Fl(If)34 b Fp(P)1251 588 y Fm(")1288 573 y Fj(H)i Fl(ar)-5 b(e)35 b(a.i.s.)44 b(of)34 b(or)-5 b(der)35 b Fp(\016)t Fq(\()p Fp(")p Fq(\))f Fl(for)h Fp(H)2639 588 y Fm(")2710 573 y Fl(then:)1108 913 y Fj(k)28 b Fq(\(1)21 b Fj(\000)i Fp(P)1457 928 y Fm(")1494 913 y Fq(\))p Fp(exp)p Fq(\()p Fj(\000)p Fp(itH)1945 928 y Fm(")1983 913 y Fq(\))p Fp(P)2084 928 y Fm(")2148 913 y Fj(k\024j)28 b Fp(t)g Fj(j)f Fp(\016)t Fq(\()p Fp(")p Fq(\))p Fp(:)650 b Fq(\(2.8\))404 1133 y Fl(If)34 b Fp(h)562 1148 y Fm(")634 1133 y Fl(has)g(an)h(eigenvalue)e Fp(E)1490 1148 y Fm(")1562 1133 y Fl(and)1610 1353 y Fp(h)1666 1368 y Fm(")1703 1353 y Fp( )1766 1368 y Fm(")1830 1353 y Fq(=)28 b Fp(E)2006 1368 y Fm(")2043 1353 y Fp( )2106 1368 y Fm(")3295 1353 y Fq(\(2.9\))257 1573 y Fl(then)1061 1793 y Fp(exp)p Fq(\()p Fj(\000)p Fp(itH)1474 1808 y Fm(")1512 1793 y Fq(\))p Fp( )1613 1808 y Fm(")1678 1793 y Fq(=)g Fp(exp)p Fq(\()p Fj(\000)p Fp(itE)2186 1808 y Fm(")2224 1793 y Fq(\))p Fp( )2325 1808 y Fm(")2384 1793 y Fq(+)22 b Fp(\037)2543 1808 y Fm(")2580 1793 y Fq(\()p Fp(t)p Fq(\))555 b(\(2.10\))257 2013 y Fl(with)1471 2233 y Fj(k)28 b Fp(\037)1610 2248 y Fm(")1646 2233 y Fq(\()p Fp(t)p Fq(\))g Fj(k\024j)g Fp(t)g Fj(j)f Fp(\016)t Fq(\()p Fp(")p Fq(\))p Fp(:)964 b Fq(\(2.11\))257 2453 y Fl(In)35 b(p)-5 b(articular)1078 2673 y Fj(j)p Fp(<)27 b( )1272 2688 y Fm(")1309 2673 y Fp(;)17 b(exp)p Fq(\()p Fj(\000)p Fp(itH)1766 2688 y Fm(")1804 2673 y Fq(\))p Fp( )1905 2688 y Fm(")1970 2673 y Fp(>)p Fj(j\025)28 b Fq(1)p Fj(\000)g(j)f Fp(t)h Fj(j)f Fp(\016)t Fq(\()p Fp(")p Fq(\))571 b(\(2.12\))257 2893 y Fl(i.e.)45 b(in)34 b(the)h(physicists)g(language)f Fp( )1621 2908 y Fm(")1693 2893 y Fl(describ)-5 b(es)33 b(a)i(metastable)f(state.)404 3097 y(Pr)-5 b(o)g(of.)43 b Fq(Direct)31 b(consequence)36 b(of)c(the)h(de\014nition)f(and)g(of)g(the)h(form)m(ula:)332 3374 y Fp(exp)p Fq(\()p Fj(\000)p Fp(itH)745 3389 y Fm(")783 3374 y Fq(\))28 b(=)g Fp(exp)p Fq(\()p Fj(\000)p Fp(itH)1366 3389 y Fm(";D)1483 3374 y Fq(\))22 b Fj(\000)h Fp(i)1693 3239 y Fh(Z)1792 3265 y Fm(t)1748 3464 y Fo(0)1839 3374 y Fp(exp)p Fq(\()p Fj(\000)p Fp(itH)2252 3389 y Fm(";D)2369 3374 y Fq(\()p Fp(t)f Fj(\000)h Fp(s)p Fq(\)\))p Fp(B)2760 3389 y Fm(")2797 3374 y Fp(exp)p Fq(\()p Fj(\000)p Fp(isH)3221 3389 y Fm(")3259 3374 y Fq(\))p Fp(ds:)3246 3557 y Fq(\(2.13\))404 3770 y Fl(R)-5 b(emarks)404 3890 y Fq(2.3.)43 b(Iterating)30 b(once)j(\(2.13\))e(and)h(taking)f(in)m(to)f(accoun)m(t)j(that)f Fp(B)2834 3905 y Fm(")2902 3890 y Fq(is)f(o\013-diagonal)257 4011 y(one)i(can)g(impro)m(v)m(e)f(\(2.12\))g(to)984 4231 y Fj(j)p Fp(<)27 b( )1178 4246 y Fm(")1215 4231 y Fp(;)17 b(exp)p Fq(\()p Fj(\000)p Fp(itH)1672 4246 y Fm(")1710 4231 y Fq(\))p Fp( )t(")27 b(>)p Fj(j\025)h Fq(1)p Fj(\000)g(j)f Fp(t)h Fj(j)2397 4190 y Fo(2)2464 4231 y Fp(\016)t Fq(\()p Fp(")p Fq(\))2633 4190 y Fo(2)2672 4231 y Fp(=)p Fq(2)476 b(\(2.14\))404 4451 y(Consider)35 b(no)m(w)g(the)g(case)g(when)h Fp(H)1734 4466 y Fm(")1802 4451 y Fj(!)31 b Fp(H)2014 4466 y Fo(0)2087 4451 y Fq(in)j(the)h (strong)g(resolv)m(en)m(t)h(sense)g(and)257 4571 y Fp(\033)312 4586 y Fo(0)380 4571 y Fq(=)27 b Fj(f)p Fp(\025)590 4586 y Fo(0)630 4571 y Fj(g)32 b Fq(with)g Fp(\025)991 4586 y Fo(0)1063 4571 y Fq(an)g(isolated,)g(m-degenerate)g(eigen)m(v)-5 b(alue)32 b(so)h(that)1461 4791 y Fp(dimP)1693 4806 y Fo(0)1732 4791 y Fj(H)c Fq(=)f Fp(m)g(<)f Fj(1)p Fp(:)954 b Fq(\(2.15\))257 5011 y(F)-8 b(rom)31 b(\(1.1\))h(it)g(follo)m(ws)f (that)i(for)f(su\016cien)m(tly)h(small)d Fp(")1462 5231 y(dimP)1694 5246 y Fm(")1731 5231 y Fj(H)f Fq(=)e Fp(m)h(<)g Fj(1)p Fp(:)955 b Fq(\(2.16\))1852 5637 y(5)p eop %%Page: 6 6 6 5 bop 257 573 a Fq(In)41 b(this)e(case)i(the)f(existence)h(of)f (a.i.s.)64 b(of)40 b(order)g Fp(\016)t Fq(\()p Fp(")p Fq(\))f(is)g(equiv)-5 b(alen)m(t)40 b([16)o(])g(with)g(the)257 693 y(sp)s(ectral)33 b(concen)m(tration)f(as)h(de\014ned)h(in)e([23)o (],)h([14].)43 b(Indeed)34 b(supp)s(ose)g Fp(P)2983 708 y Fm(")3020 693 y Fj(H)f Fq(are)g(a.i.s.)257 814 y(of)f(order)h Fp(\016)t Fq(\()p Fp(")p Fq(\))f(for)g Fp(H)1054 829 y Fm(")1091 814 y Fq(.)43 b(Since)33 b Fp(P)1479 829 y Fm(")1516 814 y Fj(H)g Fq(are)g(\014nite)f(dimensional)e(there)j (exist)558 999 y Fp(\036)616 1014 y Fm(";j)732 999 y Fj(2)28 b Fp(P)889 1014 y Fm(")926 999 y Fj(H)q Fp(;)114 b(<)28 b(\036)1314 1014 y Fm(";j)1402 999 y Fp(;)17 b(\036)1504 1014 y Fm(";k)1626 999 y Fp(>)p Fq(=)28 b Fp(\016)1849 1014 y Fm(j;k)1940 999 y Fp(;)2085 972 y Fq(~)2081 999 y Fp(\025)2138 1014 y Fm(";j)2254 999 y Fj(2)g Fk(R)p Fq(;)114 b Fp(j;)17 b(k)31 b Fq(=)d(1)p Fp(;)17 b Fq(2)p Fp(;)g(:::m)257 1184 y Fq(suc)m(h)34 b(that)1545 1369 y Fp(h)1601 1384 y Fm(")1638 1369 y Fp(\036)1696 1384 y Fm(";j)1812 1369 y Fq(=)1920 1342 y(~)1916 1369 y Fp(\025)1973 1384 y Fm(";j)2061 1369 y Fp(\036)2119 1384 y Fm(";j)3246 1369 y Fq(\(2.17\))257 1554 y(F)-8 b(rom)31 b(\(2.1\),)i(\(2.2\))f(and) g(\(2.6\))g(it)g(follo)m(ws)f(that)1393 1739 y(lim)1392 1798 y Fm(")p Fi(!)p Fo(0)1558 1739 y Fj(k)d Fq(\(1)21 b Fj(\000)i Fp(P)1907 1754 y Fo(0)1946 1739 y Fq(\))p Fp(\036)2042 1754 y Fm(";j)2158 1739 y Fj(k)p Fq(=)28 b(0)885 b(\(2.18\))1403 2038 y Fj(k)27 b Fq(\()p Fp(H)1599 2053 y Fm(")1658 2038 y Fj(\000)1761 2012 y Fq(~)1757 2038 y Fp(\025)1814 2053 y Fm(";j)1903 2038 y Fq(\))p Fp(\036)1999 2053 y Fm(";j)2115 2038 y Fj(k\024)h Fp(\016)2313 2053 y Fm(")3246 2038 y Fq(\(2.19\))257 2240 y(i.e.)69 b Fp(\036)509 2255 y Fm(";j)598 2240 y Fp(;)645 2213 y Fq(~)642 2240 y Fp(\025)699 2255 y Fm(";j)787 2240 y Fq(\))41 b(are)g(pseudo)h(eigen)m(v)m(ectors)h(and)e(pseudo)i(eigen)m (v)-5 b(alues)41 b(of)f(all)g(orders)257 2360 y Fp(p)32 b Fj(2)g Fk(N)i Fq(for)g(whic)m(h)i(lim)1126 2375 y Fm(")p Fi(!)p Fo(0)1285 2360 y Fp(")1331 2324 y Fi(\000)p Fm(p)1426 2360 y Fp(\016)t Fq(\()p Fp(")p Fq(\))31 b(=)g(0)j(and)h(this)g(is)f (equiv)-5 b(alen)m(t)35 b(with)f(the)h(sp)s(ectral)257 2481 y(concen)m(tration)f(according)e(to)h(the)g(Riddel's)g(theorem)g ([23)o(].)45 b(Con)m(v)m(ersely)35 b(if)3163 2454 y(~)3159 2481 y Fp(\025)3216 2496 y Fm(";j)3338 2481 y Fq(and)257 2601 y Fp(\036)315 2616 y Fm(";j)436 2601 y Fq(satisfying)d(\(2.18\))g (and)g(\(2.19\))g(exist)h(then)g Fp(P)2140 2616 y Fm(")2177 2601 y Fj(H)h Fq(with)1376 2865 y Fp(P)1439 2880 y Fm(")1503 2865 y Fq(=)1647 2740 y Fm(m)1606 2770 y Fh(X)1617 2980 y Fm(j)t Fo(=1)1778 2865 y Fp(<)28 b(\036)1940 2880 y Fm(";j)2028 2865 y Fp(;)17 b Fj(\001)27 b Fp(>)h(\036)2289 2880 y Fm(";j)3246 2865 y Fq(\(2.20\))257 3151 y(are)33 b(a.i.s.)43 b(of)32 b(order)h Fp(\016)t Fq(\()p Fp(")p Fq(\))f(for)g Fp(H)1456 3166 y Fm(")1492 3151 y Fq(.)404 3271 y(Finnaly)g(let)h(us)h(consider)g(the)g(case)h(when,)g(in)e (addition,)f Fp(\025)2658 3286 y Fo(0)2731 3271 y Fq(is)h(stable)g ([12])h(\()f(see)257 3391 y(also)d([14],)h([10]\).)43 b(F)-8 b(rom)29 b(the)i(de\014nition)f(of)h(stabilit)m(y)e(there)i (exist)g Fp(d)d(>)f Fq(0)p Fp(;)17 b(")3058 3406 y Fo(1)3125 3391 y Fp(>)27 b Fq(0)k(suc)m(h)257 3512 y(that)i(for)f Fp(")27 b Fj(2)h Fq([0)p Fp(;)17 b(")951 3527 y Fo(1)990 3512 y Fq(],)33 b(\000)27 b(=)h Fj(f)p Fp(z)k Fj(jj)27 b Fp(z)g Fj(\000)c Fp(\025)1708 3527 y Fo(0)1775 3512 y Fj(j)p Fq(=)k Fp(d)p Fj(g)g(\032)h Fp(\032)p Fq(\()p Fp(H)2308 3527 y Fm(")2345 3512 y Fq(\))k(and)1361 3697 y(inf)1357 3759 y Fm(z)s Fi(2)p Fo(\000)1500 3697 y Fp(dist)p Fq(\()p Fp(z)t(;)17 b(\033)t Fq(\()p Fp(H)1974 3712 y Fm(")2011 3697 y Fq(\)\))28 b Fj(\025)g Fp(d=)p Fq(2)p Fp(:)850 b Fq(\(2.21\))257 3915 y(In)33 b(addition,)e(if)1422 4143 y Fp(Q)1499 4158 y Fm(")1564 4143 y Fq(=)1667 4007 y Fh(I)1723 4233 y Fo(\000)1771 4143 y Fq(\()p Fp(H)1890 4158 y Fm(")1948 4143 y Fj(\000)23 b Fp(z)t Fq(\))2135 4102 y Fi(\000)p Fo(1)2230 4143 y Fp(dz)920 b Fq(\(2.22\))257 4375 y(then)1458 4560 y(lim)1457 4620 y Fm(")p Fi(!)p Fo(0)1623 4560 y Fj(k)28 b Fp(Q)1778 4575 y Fm(")1837 4560 y Fj(\000)22 b Fp(P)1999 4575 y Fo(0)2066 4560 y Fj(k)p Fq(=)28 b(0)p Fp(:)950 b Fq(\(2.23\))404 4777 y(F)-8 b(rom)29 b(\(2.1\),)i(\(2.2\))f(and)h(\(2.23\))e(there)j(exist)f Fp(")2138 4792 y Fo(2)2205 4777 y Fp(>)c Fq(0)k(suc)m(h)h(that)e(for)g (all)f Fp(")e Fj(2)h Fq([0)p Fp(;)17 b(")3429 4792 y Fo(2)3468 4777 y Fq(])257 4898 y(one)33 b(has:)803 5083 y Fj(k)28 b Fp(Q)958 5098 y Fm(")1017 5083 y Fj(\000)22 b Fp(P)1179 5098 y Fo(0)1246 5083 y Fj(k)p Fp(<)28 b Fq(1)p Fp(=)p Fq(6)p Fp(;)71 b Fj(k)28 b Fp(P)1786 5098 y Fm(")1844 5083 y Fj(\000)23 b Fp(P)2007 5098 y Fo(0)2074 5083 y Fj(k)p Fp(<)k Fq(1)p Fp(=)p Fq(6)p Fp(;)72 b(\016)t Fq(\()p Fp(")p Fq(\))27 b Fj(\024)h Fp(d=)p Fq(8)p Fp(:)296 b Fq(\(2.24\))404 5268 y(As)36 b(exp)s(ected)h Fp(Q)1037 5283 y Fm(")1110 5268 y Fq(and)f(the)g(sp)s(ectrum)g(of)f Fp(Q)2094 5283 y Fm(")2131 5268 y Fp(H)2212 5283 y Fm(")2249 5268 y Fp(Q)2326 5283 y Fm(")2398 5268 y Fq(\(as)h(an)f(op)s(erator)g (in)g(the)h(m-)257 5388 y(dimensional)30 b(space)k Fp(Q)1132 5403 y Fm(")1169 5388 y Fj(H)q Fq(\))f(are)f(close)h(to)f Fp(P)1903 5403 y Fm(")1972 5388 y Fq(and)h Fp(\033)t Fq(\()p Fp(h)2315 5403 y Fm(")2352 5388 y Fq(\))f(resp)s(ectiv)m(ely)-8 b(.)1852 5637 y(6)p eop %%Page: 7 7 7 6 bop 257 573 a Fk(Prop)s(osition)36 b(2.3.)49 b Fl(F)-7 b(or)34 b Fp(")27 b Fj(2)h Fq([0)p Fp(;)17 b(min)p Fj(f)p Fp(")1827 588 y Fo(1)1867 573 y Fp(;)g(")1957 588 y Fo(2)1995 573 y Fj(g)p Fq(])1390 780 y Fj(k)28 b Fp(P)1531 795 y Fm(")1590 780 y Fj(\000)22 b Fp(Q)1766 795 y Fm(")1831 780 y Fj(k\024)28 b Fp(c)2028 795 y Fo(1)2067 780 y Fq(\()p Fp(d)p Fq(\))p Fp(\016)t Fq(\()p Fp(")p Fq(\))883 b(\(2.25\))1124 1073 y Fp(dist)p Fq(\()p Fp(\033)t Fq(\()p Fp(Q)1501 1088 y Fm(")1538 1073 y Fp(H)1619 1088 y Fm(")1656 1073 y Fp(Q)1733 1088 y Fm(")1770 1073 y Fq(\))p Fp(;)17 b(\033)t Fq(\()p Fp(h)2005 1088 y Fm(")2042 1073 y Fq(\))27 b Fj(\024)h Fp(c)2254 1088 y Fo(2)2294 1073 y Fq(\()p Fp(d)p Fq(\))p Fp(\016)t Fq(\()p Fp(")p Fq(\))2590 1032 y Fo(2)3246 1073 y Fq(\(2.26\))404 1281 y Fl(R)-5 b(emarks)404 1401 y Fq(2.4.)46 b(The)34 b(pro)s(of)f(giv)m(es)g(the)h(follo)m(wing)d (\(nonoptimal\))g(v)-5 b(alues)33 b(for)g Fp(c)2955 1416 y Fm(i)2983 1401 y Fq(\()p Fp(d)p Fq(\):)45 b Fp(c)3224 1416 y Fo(1)3264 1401 y Fq(\()p Fp(d)p Fq(\))28 b(=)257 1521 y(16)p Fp(=)p Fq(3)p Fp(d)p Fq(,)k Fp(c)605 1536 y Fo(2)644 1521 y Fq(\()p Fp(d)p Fq(\))27 b(=)912 1482 y Fo(512)p 912 1498 106 4 v 929 1556 a(9)p Fm(d)1028 1521 y Fq(.)404 1642 y(2.5.)58 b(The)38 b(in)m(teresting)f(part)h(of)f (Prop)s(osition)e(2.3)j(is)f(\(2.26\))f(whic)m(h)i(is)f(a)h(general)257 1762 y(form)43 b(of)h(the)h(so)f(called)f(Hylleraas-Wigner)g("2)p Fp(n)30 b Fq(+)g(1")43 b(theorem)h(in)g(p)s(erturbation)257 1882 y(theory)49 b(\()f(see)g([8])g(and)g(references)i(therein\):)74 b(the)48 b(kno)m(wledge)h(of)e(the)i(p)s(erturb)s(ed)257 2003 y(w)m(a)m(v)m(e)e(function)e(to)f(order)h Fp(\016)t Fq(\()p Fp(")p Fq(\))g(giv)m(es)g(the)h(p)s(erturb)s(ed)f(eigen)m(v)-5 b(alue)45 b(with)f(errors)i(of)257 2123 y(order)37 b Fp(\016)t Fq(\()p Fp(")p Fq(\))685 2087 y Fo(2)724 2123 y Fq(.)57 b(T)-8 b(o)37 b(see)h(the)g(connection,)g(consider)f(the)h (elemen)m(tary)f(case)h(of)e(regular)257 2244 y(p)s(erturbations)i(of)f (nondegenerate)i(isolated)d(eigen)m(v)-5 b(alues.)59 b(Let)38 b Fp(H)2821 2259 y Fm(")2858 2244 y Fp( )2921 2259 y Fm(")2994 2244 y Fq(=)f Fp(\025)3164 2259 y Fm(")3200 2244 y Fp( )3263 2259 y Fm(")3338 2244 y Fq(and)257 2364 y(supp)s(ose)50 b(that)f(one)g(kno)m(ws)i(an)d(appro)m(ximate)g Fl(normalise)-5 b(d)47 b Fq(w)m(a)m(v)m(e)k(function)3285 2338 y(~)3265 2364 y Fp( )3328 2379 y Fm(")3366 2364 y Fq(,)h Fj(k)277 2458 y Fq(~)257 2484 y Fp( )320 2499 y Fm(")396 2484 y Fj(k)p Fq(=)39 b(1)g(with)g Fj(k)986 2458 y Fq(~)967 2484 y Fp( )1030 2499 y Fm(")1093 2484 y Fj(\000)27 b Fp( )1260 2499 y Fm(")1337 2484 y Fj(k\024)39 b Fp(\016)t Fq(\()p Fp(")p Fq(\),)h Fj(k)f Fp(H)1909 2499 y Fm(")1945 2484 y Fq(\()2003 2458 y(~)1983 2484 y Fp( )2046 2499 y Fm(")2110 2484 y Fj(\000)27 b Fp( )2277 2499 y Fm(")2314 2484 y Fq(\))39 b Fj(k\024)h Fp(M)10 b(\016)t Fq(\()p Fp(")p Fq(\))39 b(;)j Fp(M)50 b Fq(uniformly)257 2605 y(b)s(ounded)34 b(as)e Fp(")c Fj(!)f Fq(0.)43 b(Then)34 b(if)1521 2812 y Fp(P)1584 2827 y Fm(")1649 2812 y Fq(=)p Fp(<)1848 2785 y Fq(~)1828 2812 y Fp( )1891 2827 y Fm(")1928 2812 y Fp(;)17 b Fj(\001)27 b Fp(>)2151 2785 y Fq(~)2131 2812 y Fp( )2194 2827 y Fm(")257 3019 y Fq(one)33 b(has)1236 3139 y Fj(k)27 b Fq([)p Fp(P)1403 3154 y Fm(")1440 3139 y Fp(;)17 b(H)1565 3154 y Fm(")1601 3139 y Fq(])28 b Fj(k\024)g Fq(2\()p Fp(M)10 b Fq(+)28 b Fj(j)f Fp(\025)2218 3154 y Fm(")2283 3139 y Fj(j)p Fq(\))p Fp(\016)t Fq(\()p Fp(")p Fq(\))257 3308 y(whic)m(h)33 b(according)f(to)h(Prop)s(osition)e (2.3)h(giv)m(es)1156 3515 y Fj(j)27 b Fp(\025)1268 3530 y Fm(")1305 3515 y Fj(\000)h Fp(<)1534 3489 y Fq(~)1514 3515 y Fp( )1577 3530 y Fm(")1614 3515 y Fp(;)17 b(H)1739 3530 y Fm(")1795 3489 y Fq(~)1775 3515 y Fp( )1838 3530 y Fm(")1903 3515 y Fp(>)p Fj(j\024)28 b Fq(const.)q Fp(\016)t Fq(\()p Fp(")p Fq(\))2531 3474 y Fo(2)2569 3515 y Fp(:)650 b Fq(\(2.27\))257 3722 y(Actually)32 b(in)g(this)g(simple)f(case)i(one) g(can)g(obtain)f(\(2.27\))f(at)i(once)g(from:)918 3929 y Fj(j)27 b Fp(\025)1030 3944 y Fm(")1067 3929 y Fj(\000)h Fp(<)1295 3903 y Fq(~)1275 3929 y Fp( )1338 3944 y Fm(")1375 3929 y Fp(;)17 b(H)1500 3944 y Fm(")1557 3903 y Fq(~)1537 3929 y Fp( )1600 3944 y Fm(")1664 3929 y Fp(>)p Fj(j)p Fq(=)p Fj(j)p Fp(<)1995 3903 y Fq(~)1975 3929 y Fp( )2038 3944 y Fm(")2075 3929 y Fp(;)g Fq(\()p Fp(H)2238 3944 y Fm(")2297 3929 y Fj(\000)22 b Fp(\025)2453 3944 y Fm(")2490 3929 y Fq(\))2548 3903 y(~)2528 3929 y Fp( )2591 3944 y Fm(")2656 3929 y Fp(>)p Fj(j)p Fq(=)514 4136 y Fj(j)p Fp(<)665 4110 y Fq(~)646 4136 y Fp( )709 4151 y Fm(")746 4136 y Fp(;)17 b Fq(\()p Fp(H)909 4151 y Fm(")967 4136 y Fj(\000)23 b Fp(\025)1124 4151 y Fm(")1160 4136 y Fq(\)\()1256 4110 y(~)1236 4136 y Fp( )1299 4151 y Fm(")1358 4136 y Fj(\000)g Fp( )1521 4151 y Fm(")1558 4136 y Fq(\))28 b Fp(>)p Fj(j)p Fq(=)p Fj(j)p Fp(<)e Fq(\()p Fp(H)2053 4151 y Fm(")2112 4136 y Fj(\000)d Fp(\025)2269 4151 y Fm(")2305 4136 y Fq(\)\()2401 4110 y(~)2381 4136 y Fp( )2444 4151 y Fm(")2504 4136 y Fj(\000)f Fp( )2666 4151 y Fm(")2703 4136 y Fq(\))p Fp(;)2805 4110 y Fq(~)2785 4136 y Fp( )2848 4151 y Fm(")2907 4136 y Fj(\000)h Fp( )3070 4151 y Fm(")3135 4136 y Fp(>)p Fj(j)404 4305 y Fq(Notice)h(ho)m(w)m(ev)m(er)k(that)d (the)h(ab)s(o)m(v)m(e)g(argumen)m(t)e(cannot)i(b)s(e)f(generalised)g (directly)g(to)257 4425 y(the)e(degenerate)h(case)g(and)f(actually)e(w) m(e)j(do)e(not)h(kno)m(w)h(a)e(reference)i(for)e(the)i(Hylleraas-)257 4545 y(Wigner)k(theorem)g(in)f(the)h(degenerate)h(case)g(in)f(spite)g (of)f(the)i(fact)e(that)h(it)f(is)h(probably)257 4666 y(kno)m(wn)39 b(to)e(exp)s(erts.)59 b(The)38 b(reason)g(for)e(the)i (quadratic)f(estimation)e(in)i(\(2.26\))f(is)h(the)257 4786 y(fact)g(that)g Fp(B)744 4801 y Fm(")817 4786 y Fq(is)g(an)g(o\013-diagonal)c(p)s(erturbation)j(of)h Fp(H)2365 4801 y Fm(";D)2481 4786 y Fq(.)56 b(Let)37 b(us)h(stress)g(that)f(the)257 4907 y(condition)k Fp(P)758 4922 y Fm(")838 4907 y Fq(=)i Fp(P)1034 4870 y Fo(2)1020 4931 y Fm(")1115 4907 y Fq(is)e(essen)m(tial)h(for)g(\(2.26\))f(to)g (hold)g(true;)47 b(actually)41 b(it)g(replaces)257 5027 y(the)h(condition)e(that)1111 5001 y(~)1091 5027 y Fp( )1154 5042 y Fm(")1192 5027 y Fq(,)j(in)e(the)h(ab)s(o)m(v)m(e)g(pro)s(of)f (of)g(Hylleraas-Wigner)e(theorem,)44 b(is)257 5147 y(normalised.)404 5268 y(2.6.)82 b(Prop)s(osition)44 b(2.3)h(can)h(b)s(e)g(easily)f (extended)j(to)d(the)h(case)h(when)g Fp(\033)3250 5283 y Fo(0)3335 5268 y Fq(is)e(a)257 5388 y(stable,)f(b)s(ounded)e (isolated)e(part)h(of)g(the)g(sp)s(ectrum)h(oh)f Fp(H)2528 5403 y Fo(0)2609 5388 y Fq(but)h(the)f(constan)m(ts)i(in)1852 5637 y(7)p eop %%Page: 8 8 8 7 bop 257 573 a Fq(\(2.25\),)29 b(\(2.26\))f(b)s(ecomes)h(more)f (complicated)f(and)h(dep)s(end)i(also)e(up)s(on)g(the)h(diameter)257 693 y(of)j Fp(\033)423 708 y Fo(0)463 693 y Fq(.)404 814 y Fl(Pr)-5 b(o)g(of)42 b(of)g(Pr)-5 b(op)g(osition)41 b(2.3.)67 b Fq(W)-8 b(e)41 b(apply)g(regular)e(p)s(erturbation)h (theory)h(to)g(the)257 934 y(pair)32 b Fp(H)539 949 y Fm(";D)655 934 y Fp(;)17 b(H)780 949 y Fm(")844 934 y Fq(=)27 b Fp(H)1028 949 y Fm(";D)1167 934 y Fq(+)22 b Fp(B)1339 949 y Fm(")1375 934 y Fq(.)44 b(F)-8 b(rom)31 b(\(2.5\),\(2.6\))h(and)g(\(2.24\))g(,)1359 1143 y Fp(dist)p Fq(\(\000)p Fp(;)17 b(\033)t Fq(\()p Fp(H)1845 1158 y Fm(";D)1961 1143 y Fq(\)\))27 b Fj(\025)h Fq(3)p Fp(d=)p Fq(8)p Fp(:)852 b Fq(\(2.28\))257 1352 y(If)1320 1585 y Fp(T)1377 1600 y Fm(")1442 1585 y Fq(=)1593 1517 y Fp(i)p 1555 1562 108 4 v 1555 1653 a Fq(2)p Fp(\031)1690 1449 y Fh(I)1745 1675 y Fo(\000)1793 1585 y Fq(\()p Fp(H)1912 1600 y Fm(";D)2050 1585 y Fj(\000)23 b Fp(z)t Fq(\))2237 1544 y Fi(\000)p Fo(1)2332 1585 y Fp(dz)818 b Fq(\(2.29\))257 1841 y(then)879 2074 y Fp(Q)956 2089 y Fm(")1015 2074 y Fj(\000)22 b Fp(T)1171 2089 y Fm(")1236 2074 y Fq(=)27 b Fj(\000)1464 2006 y Fp(i)p 1426 2051 V 1426 2142 a Fq(2)p Fp(\031)1561 1938 y Fh(I)1617 2163 y Fo(\000)1665 2074 y Fq(\()p Fp(H)1784 2089 y Fm(";D)1922 2074 y Fj(\000)22 b Fp(z)t Fq(\))2108 2032 y Fi(\000)p Fo(1)2203 2074 y Fp(B)2277 2089 y Fm(")2314 2074 y Fq(\()p Fp(H)2433 2089 y Fm(")2492 2074 y Fj(\000)h Fp(z)t Fq(\))2679 2032 y Fi(\000)p Fo(1)2774 2074 y Fp(dz)376 b Fq(\(2.30\))257 2330 y(whic)m(h)33 b(giv)m(es)1235 2539 y Fj(k)28 b Fp(Q)1390 2554 y Fm(")1449 2539 y Fj(\000)23 b Fp(T)1606 2554 y Fm(")1670 2539 y Fj(k\024)28 b Fq(16)p Fp(\016)t Fq(\()p Fp(")p Fq(\))p Fp(=)p Fq(3)p Fp(d)e(<)i Fq(2)p Fp(=)p Fq(3)728 b(\(2.31\))404 2748 y(F)-8 b(rom)31 b(\(2.24\))h(and)g (\(2.31\))g(one)h(has)1551 2957 y Fj(k)28 b Fp(P)1692 2972 y Fm(")1751 2957 y Fj(\000)22 b Fp(T)1907 2972 y Fm(")1972 2957 y Fj(k)p Fp(<)27 b Fq(1)p Fp(:)1045 b Fq(\(2.32\))257 3166 y(No)m(w)34 b(\(2.32\))d(implies)f(that)1714 3375 y Fp(P)1777 3390 y Fm(")1841 3375 y Fq(=)e Fp(T)2002 3390 y Fm(")3246 3375 y Fq(\(2.33\))257 3584 y(whic)m(h)34 b(together)g(with)f(\(2.31\))f(giv)m(es)i(\(2.25\).)45 b(Indeed)35 b(since)f Fp(P)2611 3599 y Fm(")2648 3584 y Fp(;)74 b(T)2806 3599 y Fm(")2877 3584 y Fq(are)33 b(orthogonal)257 3705 y(pro)5 b(jections)49 b(comm)m(uting)d(with)i Fp(H)1616 3720 y Fm(";D)1732 3705 y Fq(,)k(in)47 b(a)h Fp(L)2103 3668 y Fo(2)2142 3705 y Fq(\()p Fp(M)5 b(;)17 b(d\026)p Fq(\))48 b(represen)m(tation)h(of)e Fp(H)3379 3720 y Fm(";D)257 3825 y Fq(they)34 b(are)f(m)m(ultiplications)c(with)j (some)h(c)m(haracteristic)g(functions)g(so)g Fp(P)2976 3840 y Fm(")3035 3825 y Fj(\000)23 b Fp(T)3192 3840 y Fm(")3261 3825 y Fq(is)33 b(the)257 3945 y(m)m(ultiplication)21 b(with)26 b(a)f(function)g(taking)g(only)g(the)i(v)-5 b(alues)25 b(0)h(and)f Fj(\006)p Fq(1.)42 b(As)26 b(the)g(v)-5 b(alues)257 4066 y Fj(\006)p Fq(1)33 b(are)g(ruled)f(out)g(b)m(y)i (\(2.32\))e(the)h(equalit)m(y)f(\(2.33\))g(follo)m(ws.)404 4186 y(F)-8 b(or)28 b(the)h(pro)s(of)f(of)g(\(2.26\))g(consider)h (consider)g(the)g(Sz-Nagy)g(transformation)e(ma-)257 4307 y(trix)32 b([14])h(corresp)s(onding)f(to)g(the)h(pair)f Fp(P)1806 4322 y Fm(")1842 4307 y Fp(;)45 b(Q)1991 4322 y Fm(")767 4516 y Fp(U)833 4531 y Fm(")898 4516 y Fq(=)28 b(\(1)22 b Fj(\000)g Fq(\()p Fp(Q)1325 4531 y Fm(")1384 4516 y Fj(\000)h Fp(P)1547 4531 y Fm(")1583 4516 y Fq(\))1621 4475 y Fo(2)1661 4516 y Fq(\))1699 4475 y Fi(\000)p Fo(1)p Fm(=)p Fo(2)1864 4516 y Fq([)p Fp(Q)1968 4531 y Fm(")2005 4516 y Fp(P)2068 4531 y Fm(")2127 4516 y Fq(+)f(\(1)f Fj(\000)i Fp(Q)2510 4531 y Fm(")2547 4516 y Fq(\)\(1)f Fj(\000)g Fp(P)2856 4531 y Fm(")2893 4516 y Fq(\)])p Fp(:)261 b Fq(\(2.34\))257 4725 y(As)33 b(w)m(ell)f(kno)m(wn)i Fp(U)974 4740 y Fm(")1044 4725 y Fq(is)e(unitary)g(and)h(in)m(tert)m (wines)g Fp(Q)2252 4740 y Fm(")2321 4725 y Fq(and)g Fp(P)2574 4740 y Fm(")1581 4934 y Fp(Q)1658 4949 y Fm(")1722 4934 y Fq(=)28 b Fp(U)1892 4949 y Fm(")1929 4934 y Fp(P)1992 4949 y Fm(")2029 4934 y Fp(U)2105 4893 y Fi(\003)2095 4958 y Fm(")2145 4934 y Fp(:)1074 b Fq(\(2.35\))404 5143 y(As)33 b(a)f(consequence)1260 5352 y Fp(U)1336 5311 y Fi(\003)1326 5377 y Fm(")1377 5352 y Fp(Q)1454 5367 y Fm(")1491 5352 y Fp(H)1572 5367 y Fm(")1608 5352 y Fp(Q)1685 5367 y Fm(")1722 5352 y Fp(U)1788 5367 y Fm(")1853 5352 y Fq(=)27 b Fp(P)2019 5367 y Fm(")2056 5352 y Fp(U)2132 5311 y Fi(\003)2122 5377 y Fm(")2172 5352 y Fp(H)2253 5367 y Fm(")2290 5352 y Fp(U)2356 5367 y Fm(")2393 5352 y Fp(P)2456 5367 y Fm(")3246 5352 y Fq(\(2.36\))1852 5637 y(8)p eop %%Page: 9 9 9 8 bop 257 573 a Fq(i.e.)43 b Fp(Q)502 588 y Fm(")539 573 y Fp(H)620 588 y Fm(")657 573 y Fp(Q)734 588 y Fm(")803 573 y Fq(is)33 b(unitarily)d(equiv)-5 b(alen)m(t)32 b(with)g(the)h ("e\013ectiv)m(e")h(hamiltonian)257 693 y Fp(P)320 708 y Fm(")357 693 y Fp(U)433 657 y Fi(\003)423 718 y Fm(")473 693 y Fp(H)554 708 y Fm(")591 693 y Fp(U)657 708 y Fm(")694 693 y Fp(P)757 708 y Fm(")793 693 y Fp(:)c Fq(On)f(the)h(other)f(hand)h (the)g(analytic)e(p)s(erturbation)g(theory)i(applied)f(to)257 814 y(the)k(pair)f Fp(H)707 829 y Fm(")o(;D)823 814 y Fp(;)44 b(H)975 829 y Fm(")1040 814 y Fq(=)27 b Fp(H)1224 829 y Fm(")o(;D)1362 814 y Fq(+)22 b Fp(B)1534 829 y Fm(")1604 814 y Fq(giv)m(es)33 b(for)f Fp(P)2055 829 y Fm(")2091 814 y Fp(U)2167 777 y Fi(\003)2157 838 y Fm(")2207 814 y Fp(H)2288 829 y Fm(")2325 814 y Fp(U)2391 829 y Fm(")2428 814 y Fp(P)2491 829 y Fm(")2560 814 y Fq([14]:)1083 1030 y Fp(P)1146 1045 y Fm(")1182 1030 y Fp(U)1258 988 y Fi(\003)1248 1054 y Fm(")1299 1030 y Fp(H)1380 1045 y Fm(")1416 1030 y Fp(U)1482 1045 y Fm(")1519 1030 y Fp(P)1582 1045 y Fm(")1646 1030 y Fq(=)c Fp(h)1806 1045 y Fm(")1865 1030 y Fq(+)22 b Fp(P)2026 1045 y Fm(")2063 1030 y Fp(B)2137 1045 y Fm(")2173 1030 y Fp(P)2236 1045 y Fm(")2295 1030 y Fq(+)g Fj(O)s Fq(\()p Fp(B)2592 988 y Fo(2)2587 1054 y Fm(")2632 1030 y Fq(\))576 b(\(2.37\))257 1246 y(so)33 b(that)f(\(since)h(the)g(second)h(term)e (in)g(the)h(r.h.s.)44 b(v)-5 b(anishes\))1233 1462 y Fj(k)27 b Fp(P)1373 1477 y Fm(")1410 1462 y Fp(U)1486 1420 y Fi(\003)1476 1486 y Fm(")1526 1462 y Fp(H)1607 1477 y Fm(")1643 1462 y Fp(U)1709 1477 y Fm(")1746 1462 y Fp(P)1809 1477 y Fm(")1868 1462 y Fj(\000)c Fp(h)2024 1477 y Fm(")2088 1462 y Fj(k\024)28 b(O)s Fq(\()p Fp(B)2442 1420 y Fo(2)2437 1486 y Fm(")2482 1462 y Fq(\))726 b(\(2.38\))257 1677 y(whic)m(h)41 b(in)e(particular)g(implies)e(\(2.26\).)66 b(The)41 b(computation)d(b)s(elo)m(w)i(giv)m(es)h(a)e(precise)257 1798 y(form)34 b(of)h(the)g(error)g(in)f(terms)h(of)g Fp(\016)t Fq(\()p Fp(")o Fq(\))g(and)h Fp(d)p Fq(.)50 b(Without)34 b(restricting)g(the)i(generalit)m(y)257 1918 y(on)d(can)g(tak)m(e)g Fp(\025)840 1933 y Fo(0)907 1918 y Fq(=)28 b(0.)43 b(Using)1201 2177 y Fp(Q)1278 2192 y Fm(")1315 2177 y Fp(H)1396 2192 y Fm(")1433 2177 y Fp(Q)1510 2192 y Fm(")1574 2177 y Fq(=)1725 2110 y Fp(i)p 1688 2154 108 4 v 1688 2245 a Fq(2)p Fp(\031)1822 2041 y Fh(I)1877 2267 y Fo(\000)1942 2177 y Fp(z)t Fq(\()p Fp(H)2110 2192 y Fm(")2170 2177 y Fj(\000)22 b Fp(z)t Fq(\))2356 2136 y Fi(\000)p Fo(1)2451 2177 y Fp(dz)699 b Fq(\(2.39\))257 2440 y(and)531 2656 y(\()p Fp(H)650 2671 y Fm(")709 2656 y Fj(\000)22 b Fp(z)t Fq(\))895 2615 y Fi(\000)p Fo(1)1018 2656 y Fq(=\()p Fp(H)1213 2671 y Fm(")o(;D)1351 2656 y Fj(\000)g Fp(z)t Fq(\))1537 2615 y Fi(\000)p Fo(1)1655 2656 y Fj(\000)g Fq(\()p Fp(H)1873 2671 y Fm(";D)2011 2656 y Fj(\000)h Fp(z)t Fq(\))2198 2615 y Fi(\000)p Fo(1)2293 2656 y Fp(B)2367 2671 y Fm(")2404 2656 y Fq(\()p Fp(H)2523 2671 y Fm(")o(;D)2661 2656 y Fj(\000)g Fp(z)t Fq(\))2848 2615 y Fi(\000)p Fo(1)1116 2801 y Fq(+)f(\()p Fp(H)1333 2816 y Fm(")1392 2801 y Fj(\000)g Fp(z)t Fq(\))1578 2760 y Fi(\000)p Fo(1)1673 2801 y Fp(B)1747 2816 y Fm(")1784 2801 y Fq(\()p Fp(H)1903 2816 y Fm(")o(;D)2041 2801 y Fj(\000)h Fp(z)t Fq(\))2228 2760 y Fi(\000)p Fo(1)2323 2801 y Fp(B)2397 2816 y Fm(")2434 2801 y Fq(\()p Fp(H)2553 2816 y Fm(")o(;D)2691 2801 y Fj(\000)f Fp(z)t Fq(\))2877 2760 y Fi(\000)p Fo(1)3246 2801 y Fq(\(2.40\))257 3017 y(one)33 b(has)590 3257 y Fp(P)653 3272 y Fm(")690 3257 y Fp(Q)767 3272 y Fm(")804 3257 y Fp(H)885 3272 y Fm(")921 3257 y Fp(Q)998 3272 y Fm(")1035 3257 y Fp(P)1098 3272 y Fm(")1163 3257 y Fq(=)p Fp(P)1302 3272 y Fm(")1385 3189 y Fp(i)p 1348 3234 V 1348 3325 a Fq(2)p Fp(\031)1482 3121 y Fh(I)1538 3347 y Fo(\000)1603 3257 y Fp(z)t Fq(\()p Fp(H)1771 3272 y Fm(";D)1909 3257 y Fj(\000)23 b Fp(z)t Fq(\))2096 3216 y Fi(\000)p Fo(1)2191 3257 y Fp(dz)t(P)2354 3272 y Fm(")1261 3516 y Fj(\000)f Fp(P)1423 3531 y Fm(")1507 3448 y Fp(i)p 1470 3493 V 1470 3584 a Fq(2)p Fp(\031)1604 3380 y Fh(I)1659 3605 y Fo(\000)1724 3516 y Fp(z)t Fq(\()p Fp(H)1892 3531 y Fm(";D)2031 3516 y Fj(\000)h Fp(z)t Fq(\))2218 3474 y Fi(\000)p Fo(1)2313 3516 y Fp(B)2387 3531 y Fm(")2424 3516 y Fq(\()p Fp(H)2543 3531 y Fm(";D)2681 3516 y Fj(\000)f Fp(z)t Fq(\))2867 3474 y Fi(\000)p Fo(1)2962 3516 y Fp(dz)t(P)3125 3531 y Fm(")1261 3774 y Fq(+)g Fp(P)1422 3789 y Fm(")1505 3707 y Fp(i)p 1468 3751 V 1468 3843 a Fq(2)p Fp(\031)1602 3639 y Fh(I)1658 3864 y Fo(\000)1723 3774 y Fp(z)t Fq(\()p Fp(H)1891 3789 y Fm(")1950 3774 y Fj(\000)h Fp(z)t Fq(\))2137 3733 y Fi(\000)p Fo(1)2232 3774 y Fp(B)2306 3789 y Fm(")2343 3774 y Fq(\()p Fp(H)2462 3789 y Fm(";D)2600 3774 y Fj(\000)f Fp(z)t Fq(\))2786 3733 y Fi(\000)p Fo(1)2881 3774 y Fp(B)2955 3789 y Fm(")1239 3982 y Fq(\()p Fp(H)1358 3997 y Fm(";D)1496 3982 y Fj(\000)g Fp(z)t Fq(\))1682 3941 y Fi(\000)p Fo(1)1777 3982 y Fp(dz)t(P)1940 3997 y Fm(")3246 3982 y Fq(\(2.41\))257 4198 y(No)m(w)35 b(from)d(\(2.39\))h(written)h(for)f Fp(H)1569 4213 y Fm(";D)1685 4198 y Fq(,)h(the)g(\014rst)g(term)f(in)g (the)h(r.h.s.)48 b(of)33 b(\(2.41\))g(giv)m(es)257 4318 y Fp(h)313 4333 y Fm(")387 4318 y Fq(while)j(the)h(second)h(v)-5 b(anishes)37 b(due)g(to)g(the)g(fact)f(that)h Fp(B)2494 4333 y Fm(")2567 4318 y Fq(is)f(o\013-diagonal)e(and)j Fp(P)3459 4333 y Fm(")257 4438 y Fq(comm)m(utes)g(with)e Fp(H)1026 4453 y Fm(";D)1142 4438 y Fq(.)55 b(Then)37 b(estimating)d(brutally)h(the)i(third)e(term)h(in)g(the)g(r.h.s.)257 4559 y(of)c(\(2.41\))g(one)h(obtains)1115 4775 y Fj(k)28 b Fp(P)1256 4790 y Fm(")1292 4775 y Fp(Q)1369 4790 y Fm(")1406 4775 y Fp(H)1487 4790 y Fm(")1524 4775 y Fp(Q)1601 4790 y Fm(")1638 4775 y Fp(P)1701 4790 y Fm(")1760 4775 y Fj(\000)22 b Fp(h)1915 4790 y Fm(")1980 4775 y Fj(k\024)28 b Fq(128)p Fp(\016)t Fq(\()p Fp(")p Fq(\))2451 4734 y Fo(2)2489 4775 y Fp(=)p Fq(9)p Fp(d)608 b Fq(\(2.42\))404 4991 y(On)32 b(the)h(other)g(hand)509 5207 y Fp(P)572 5222 y Fm(")609 5207 y Fp(U)685 5166 y Fi(\003)675 5231 y Fm(")725 5207 y Fp(H)806 5222 y Fm(")842 5207 y Fp(U)908 5222 y Fm(")945 5207 y Fp(P)1008 5222 y Fm(")1067 5207 y Fj(\000)23 b Fp(P)1230 5222 y Fm(")1266 5207 y Fp(Q)1343 5222 y Fm(")1380 5207 y Fp(H)1461 5222 y Fm(")1498 5207 y Fp(Q)1575 5222 y Fm(")1612 5207 y Fp(P)1675 5222 y Fm(")1739 5207 y Fq(=\()p Fp(P)1916 5222 y Fm(")1953 5207 y Fp(U)2029 5166 y Fi(\003)2019 5231 y Fm(")2091 5207 y Fj(\000)f Fp(P)2253 5222 y Fm(")2290 5207 y Fp(Q)2367 5222 y Fm(")2404 5207 y Fq(\))p Fp(Q)2519 5222 y Fm(")2556 5207 y Fp(H)2637 5222 y Fm(")2674 5207 y Fp(Q)2751 5222 y Fm(")2788 5207 y Fp(U)2854 5222 y Fm(")2891 5207 y Fp(P)2954 5222 y Fm(")1837 5352 y Fq(+)g Fp(P)1998 5367 y Fm(")2035 5352 y Fp(Q)2112 5367 y Fm(")2149 5352 y Fp(H)2230 5367 y Fm(")2266 5352 y Fp(Q)2343 5367 y Fm(")2380 5352 y Fq(\()p Fp(U)2484 5367 y Fm(")2521 5352 y Fp(P)2584 5367 y Fm(")2643 5352 y Fj(\000)h Fp(Q)2820 5367 y Fm(")2857 5352 y Fp(P)2920 5367 y Fm(")2956 5352 y Fq(\))252 b(\(2.43\))1852 5637 y(9)p eop %%Page: 10 10 10 9 bop 257 573 a Fq(and)33 b(then)308 793 y Fj(k)28 b Fp(U)462 752 y Fi(\003)452 817 y Fm(")502 793 y Fp(Q)579 808 y Fm(")616 793 y Fp(H)697 808 y Fm(")734 793 y Fp(Q)811 808 y Fm(")848 793 y Fp(U)914 808 y Fm(")973 793 y Fj(\000)22 b Fp(P)1135 808 y Fm(")1172 793 y Fp(Q)1249 808 y Fm(")1286 793 y Fp(H)1367 808 y Fm(")1404 793 y Fp(Q)1481 808 y Fm(")1518 793 y Fp(P)1581 808 y Fm(")1645 793 y Fj(k\024)28 b Fq(2)f Fj(k)h Fp(U)2020 808 y Fm(")2057 793 y Fp(Q)2134 808 y Fm(")2193 793 y Fj(\000)22 b Fp(Q)2369 808 y Fm(")2406 793 y Fp(P)2469 808 y Fm(")2534 793 y Fj(k)27 b(\001)h(k)f Fp(Q)2821 808 y Fm(")2858 793 y Fp(H)2939 808 y Fm(")2976 793 y Fp(Q)3053 808 y Fm(")3117 793 y Fj(k)79 b Fq(\(2.44\))257 1013 y(No)m(w)34 b(from)873 1233 y Fp(U)939 1248 y Fm(")976 1233 y Fp(P)1039 1248 y Fm(")1097 1233 y Fj(\000)23 b Fp(Q)1274 1248 y Fm(")1311 1233 y Fp(P)1374 1248 y Fm(")1438 1233 y Fq(=)28 b([\(1)22 b Fj(\000)g Fq(\()p Fp(Q)1892 1248 y Fm(")1951 1233 y Fj(\000)h Fp(P)2114 1248 y Fm(")2151 1233 y Fq(\))2189 1192 y Fo(2)2228 1233 y Fq(\))2266 1192 y Fi(\000)p Fo(1)p Fm(=)p Fo(2)2431 1233 y Fq(\))f Fj(\000)h Fq(1])p Fp(Q)2744 1248 y Fm(")2780 1233 y Fp(P)2843 1248 y Fm(")3246 1233 y Fq(\(2.45\))257 1453 y(one)33 b(obtains)f(\(see)i(\(2.31\))e(and)g(\(2.33\)\))815 1673 y Fj(k)c Fp(U)959 1688 y Fm(")996 1673 y Fp(P)1059 1688 y Fm(")1118 1673 y Fj(\000)22 b Fp(Q)1294 1688 y Fm(")1331 1673 y Fp(P)1394 1688 y Fm(")1458 1673 y Fj(k)p Fq(=)p Fj(k)28 b Fp(P)1725 1688 y Fm(")1761 1673 y Fp(U)1837 1632 y Fi(\003)1827 1698 y Fm(")1899 1673 y Fj(\000)23 b Fp(P)2062 1688 y Fm(")2099 1673 y Fp(Q)2176 1688 y Fm(")2240 1673 y Fj(k\024)28 b Fq(128)p Fp(\016)t Fq(\()p Fp(")p Fq(\))2711 1632 y Fo(2)2750 1673 y Fp(=)p Fq(3)p Fp(d)2899 1632 y Fo(2)3246 1673 y Fq(\(2.46\))257 1893 y(Since)33 b(\(remem)m(b)s(er)f(that)h(w)m(e)g(to)s(ok)f Fp(\025)1639 1908 y Fo(0)1706 1893 y Fq(=)c(0\))1477 2113 y Fj(k)g Fp(Q)1632 2128 y Fm(")1669 2113 y Fp(H)1750 2128 y Fm(")1787 2113 y Fp(Q)1864 2128 y Fm(")1928 2113 y Fj(k\024)g Fp(d=)p Fq(2)p Fp(;)987 b Fq(\(2.47\))257 2333 y(puting)32 b(all)f(together)i(one)f(obtains)1184 2592 y Fj(k)27 b Fp(P)1324 2607 y Fm(")1361 2592 y Fp(U)1437 2551 y Fi(\003)1427 2616 y Fm(")1477 2592 y Fp(H)1558 2607 y Fm(")1595 2592 y Fp(U)1661 2607 y Fm(")1698 2592 y Fp(P)1761 2607 y Fm(")1819 2592 y Fj(\000)c Fp(h)1975 2607 y Fm(")2040 2592 y Fj(k\024)2205 2524 y Fq(512)p 2205 2569 147 4 v 2228 2660 a(9)p Fp(d)2361 2592 y(\016)t Fq(\()p Fp(")o Fq(\))2529 2551 y Fo(2)3246 2592 y Fq(\(2.48\))257 2843 y(whic)m(h)33 b(giv)m(es)g(in)f(particular)f(\(2.26\)with)h(the)h (v)-5 b(alue)32 b(of)g Fp(c)2384 2858 y Fo(2)2456 2843 y Fq(giv)m(en)g(in)g(Remark)g(2.5.)257 3176 y Fr(3)161 b(P)l(erturbativ)l(e)45 b(construction)f(of)i(almost)g(in)l(v)-9 b(ari-)499 3358 y(an)l(t)54 b(subspaces)257 3577 y Fq(W)-8 b(e)39 b(turn)f(no)m(w)g(to)g(the)g(problem)f(of)h(constructing)g (a.i.s.)59 b(for)37 b(families,)g Fp(H)3105 3592 y Fm(")3141 3577 y Fq(,)j(of)d(self-)257 3698 y(adjoin)m(t)29 b(op)s(erators.)42 b(The)30 b(setting)f(w)m(e)h(shall)e(consider,)i(whic)m(h)g(is)f (su\016cien)m(tly)h(general)257 3818 y(to)45 b(co)m(v)m(er)h (practically)d(all)g(ph)m(ysical)i(situations)f(in)g(nonrelativistic)f (quan)m(tum)i(me-)257 3938 y(c)m(hanics,)d(is)d(as)g(follo)m(ws:)55 b Fp(")39 b(>)g Fq(0)f(is)h(a)g(small)e(parameter)h Fp(")h Fj(2)g Fq([0)p Fp(;)17 b(")2804 3953 y Fo(0)2843 3938 y Fq(];)43 b Fp(H)3021 3953 y Fo(0)3099 3938 y Fq(and)c Fp(V)22 b Fq(\()p Fp(")p Fq(\))257 4059 y(are)39 b(self-adjoin)m(t)f (op)s(erators)h(suc)m(h)h(that)f(there)h(exists)g(a)f(dense)h (subspace,)j Fj(D)s Fq(,)d Fj(D)i(\032)257 4179 y(D)s Fq(\()p Fp(H)456 4194 y Fo(0)495 4179 y Fq(\))24 b Fj(\\)g(D)s Fq(\()p Fp(V)d Fq(\()p Fp(")p Fq(\)\))35 b(on)g(whic)m(h)g Fp(H)1538 4194 y Fo(0)1601 4179 y Fq(+)24 b Fp("V)d Fq(\()p Fp(")p Fq(\))35 b(has)g(self-adjoin)m(t)e(extensions)k(\(in)d(most)257 4300 y(examples)e Fp(H)758 4315 y Fo(0)816 4300 y Fq(+)19 b Fp("V)j Fq(\()p Fp(")p Fq(\))30 b(is)h(essen)m(tially)g(self-adjoin)m (t)e(on)i Fj(D)s Fq(\);)h(w)m(e)g(shall)e(denote)i(b)m(y)g Fp(H)3459 4315 y Fm(")257 4420 y Fq(an)m(y)i(of)e(its)g(self-adjoin)m (t)f(extensions)i(with)g(the)g(prop)s(ert)m(y)g Fj(D)s Fq(\()p Fp(H)2648 4435 y Fm(")2684 4420 y Fq(\))28 b Fj(\033)g(D)s Fq(\()p Fp(H)3054 4435 y Fo(0)3093 4420 y Fq(\))22 b Fj(\\)g(D)s Fq(\()p Fp(V)3416 4435 y Fm(")3452 4420 y Fq(\))404 4540 y(W)-8 b(e)36 b(shall)f(consider)i(explicitely)d (only)i(the)h(particular)d(case)j(when)g Fp(V)22 b Fq(\()p Fp(")p Fq(\))33 b(=)h Fp(V)58 b Fq(is)257 4661 y(indep)s(enden)m(t)40 b(on)e Fp(")p Fq(,)i(and)e(in)g(this)f(case)j Fj(D)g Fq(=)d Fj(D)s Fq(\()p Fp(H)2222 4676 y Fo(0)2261 4661 y Fq(\))26 b Fj(\\)g(D)s Fq(\()p Fp(V)21 b Fq(\),)40 b(but)f(all)d(the)j(results)257 4781 y(generalise)33 b(as)g(far)g(as)g Fp(V)21 b Fq(\()p Fp(")p Fq(\))33 b(ob)s(eys)h(the)g (needed)g(tec)m(hnical)f(conditions)f(uniformly)f(in)257 4901 y Fp(")p Fq(.)43 b(Indeed)34 b(in)e(this)h(case)g(one)g(obtains)f (results)h(\(uniformly)d(in)i Fp(\021)f Fj(2)d Fq([0)p Fp(;)17 b(")2967 4916 y Fo(0)3006 4901 y Fq(]\))33 b(for)1477 5121 y Fp(H)1558 5136 y Fm(";\021)1680 5121 y Fq(=)27 b Fp(H)1864 5136 y Fo(0)1926 5121 y Fq(+)22 b Fp("V)f Fq(\()p Fp(\021)t Fq(\))257 5341 y(and)33 b(at)f(the)h(end)h(one)e (sets)i Fp(\021)e Fq(=)27 b Fp(")p Fq(.)1828 5637 y(10)p eop %%Page: 11 11 11 10 bop 404 573 a Fq(A)m(t)30 b(the)g(heuristic)g(lev)m(el)f(the)h (problem)f(of)g(\014nding)h(a.i.s)f(for)g Fp(H)2718 588 y Fm(")2785 573 y Fq(can)h(b)s(e)g(wiev)m(ed)h(as)257 693 y(the)i(searc)m(h)h(for)e(solutions)g(of)g(the)h(equations)1169 897 y([)p Fp(H)1277 912 y Fm(")1313 897 y Fp(;)17 b(P)1420 912 y Fm(")1457 897 y Fq(])28 b(=)f([)p Fp(H)1723 912 y Fo(0)1762 897 y Fp(;)17 b(P)1869 912 y Fm(")1906 897 y Fq(])22 b(+)g Fp(")p Fq([)p Fp(V)5 b(;)17 b(P)2295 912 y Fm(")2331 897 y Fq(])28 b Fj(')g Fq(0)p Fp(;)728 b Fq(\(3.1\))1603 1184 y Fp(P)1680 1143 y Fo(2)1666 1209 y Fm(")1741 1184 y Fj(\000)23 b Fp(P)1904 1199 y Fm(")1968 1184 y Fj(')28 b Fq(0)p Fp(:)1146 b Fq(\(3.2\))257 1388 y(Making)33 b(the)g(\\p)s(erturbational")d(ansatz)1584 1603 y Fp(P)1647 1618 y Fm(")1712 1603 y Fj(')1817 1509 y Fh(X)1873 1719 y Fm(j)1977 1603 y Fp(E)2049 1618 y Fm(j)2086 1603 y Fp(")2132 1562 y Fm(j)3295 1603 y Fq(\(3.3\))257 1902 y(one)d(is)f(led)g(to)g(the)h(problem)e(of)g(solving)h(the)g (follo)m(wing)e(equations)j(for)e Fp(E)2960 1917 y Fm(j)2997 1902 y Fq(,)j Fp(j)33 b Fq(=)28 b(1)p Fp(;)17 b Fq(2)p Fp(;)g(:::)1682 2106 y(P)1745 2121 y Fo(0)1812 2106 y Fq(=)27 b Fp(E)1987 2121 y Fo(0)2027 2106 y Fp(;)1241 b Fq(\(3.4\))1407 2393 y([)p Fp(H)1515 2408 y Fo(0)1554 2393 y Fp(;)17 b(E)1670 2408 y Fm(j)1707 2393 y Fq(])28 b(=)f Fj(\000)p Fq([)p Fp(V)5 b(;)17 b(E)2147 2408 y Fm(j)t Fi(\000)p Fo(1)2275 2393 y Fq(])p Fp(;)966 b Fq(\(3.5\))1576 2658 y Fm(j)1521 2692 y Fh(X)1537 2904 y Fm(l)q Fo(=0)1681 2786 y Fp(E)1753 2801 y Fm(l)1779 2786 y Fp(E)1851 2801 y Fm(j)t Fi(\000)p Fm(l)1993 2786 y Fq(=)27 b Fp(E)2168 2801 y Fm(j)2205 2786 y Fp(:)1063 b Fq(\(3.6\))257 3080 y(The)40 b(equation)f(\(3.4)f(\))h(is)g(nothing)f(but)h(\(2.1)f(\))h (for)f Fp(")h Fq(=)f(0)h(while)f(\(3.5)g(\))h(and)g(\(3.6)f(\))257 3201 y(follo)m(w)31 b(from)h(pluging)e(\(3.3)i(\))h(in)m(to)f(\(3.1\))g (and)g(\(3.2)h(\))f(resp)s(ectiv)m(ely)-8 b(.)404 3321 y(In)32 b(the)g(rest)g(of)f(this)g(section)h(w)m(e)h(shall)d(outline)g (ho)m(w)j(the)f(existence)h(of)e(solutions,)257 3441 y Fp(E)329 3456 y Fo(0)369 3441 y Fp(;)83 b(E)551 3456 y Fo(1)590 3441 y Fp(;)17 b(:::;)g(E)831 3456 y Fm(N)898 3441 y Fq(;)83 b Fp(N)43 b Fj(\024)34 b(1)h Fq(of)g(\(3.4-3.6)f(\))i (together)g(with)f(estimates)g(on)h(the)g(norm)257 3562 y(of)j Fp(E)447 3577 y Fm(j)522 3562 y Fq(and)g([)p Fp(V)5 b(;)17 b(E)923 3577 y Fm(j)960 3562 y Fq(])39 b(giv)m(es)g(the)g (existence)h(of)f(a.i.s.)61 b(in)38 b(the)h(sense)i(of)d(\(2.1-2.2\))g (with)257 3682 y(con)m(trol)27 b(on)h(the)g(constan)m(t)h Fp(\016)t Fq(\()p Fp(")p Fq(\).)41 b(Of)27 b(course)i(the)f(tec)m (hnical)g(core)g(of)f(the)h(theory)g(is)g(the)257 3803 y(pro)s(of)37 b(of)g(existence)h(of)f(solutions)g(of)f(\(3.4-3.6\))g (with)h(b)s(ounds)h(on)f(their)g(norms)g(and)257 3923 y(this)c(question)g(will)d(b)s(e)j(addressed)h(in)e(the)h(next)g (section.)404 4043 y(Supp)s(ose)f Fp(E)857 4058 y Fm(j)894 4043 y Fp(L)960 4007 y Fo(2)1000 4043 y Fq(\()p Fn(R)1104 4007 y Fo(3)1149 4043 y Fq(\))c Fj(\032)g(D)s Fq(\()p Fp(H)1519 4058 y Fo(0)1558 4043 y Fq(\))20 b Fj(\\)h(D)s Fq(\()p Fp(V)g Fq(\),)32 b Fp(j)h Fq(=)28 b(0)p Fp(;)17 b Fq(1)p Fp(;)g(:::N)41 b Fq(and)32 b(there)g(satisfy)g(\(3.4-)257 4164 y(3.6\).)43 b(In)33 b(addition:)1629 4488 y Fj(k)28 b Fp(E)1779 4503 y Fm(j)1843 4488 y Fj(k\024)g Fp(e)2043 4503 y Fm(j)2080 4488 y Fp(;)1188 b Fq(\(3.7\))1233 4775 y Fj(k)27 b Fq([)p Fp(V)5 b(;)17 b(E)1515 4790 y Fm(j)1552 4775 y Fq(])28 b Fj(k\024)g Fp(f)1810 4790 y Fm(j)1944 4775 y Fp(j)34 b Fq(=)27 b(1)p Fp(;)17 b Fq(2)p Fp(;)g(:::;)g(N)785 b Fq(\(3.8\))404 4978 y(Consider)1498 5250 y Fp(T)1555 5265 y Fm(N)1623 5250 y Fq(\()p Fp(")p Fq(\))27 b(=)1916 5125 y Fm(N)1876 5155 y Fh(X)1886 5365 y Fm(j)t Fo(=0)2036 5250 y Fp(E)2108 5265 y Fm(j)2145 5250 y Fp(")2191 5209 y Fm(j)2227 5250 y Fp(:)1041 b Fq(\(3.9\))1828 5637 y(11)p eop %%Page: 12 12 12 11 bop 257 573 a Fq(where)39 b Fp(N)48 b Fq(can)38 b(dep)s(end)h(up)s(on)f Fp(")p Fq(.)59 b(Unfortunately)37 b Fp(T)2275 588 y Fm(N)2342 573 y Fq(\()p Fp(")p Fq(\))h(is)f(not)g(a)h (pro)5 b(jection)37 b(\()h(in)257 693 y(particular)31 b Fp(T)764 708 y Fm(N)832 693 y Fq(\()p Fp(")p Fq(\))954 657 y Fo(2)1021 693 y Fj(6)p Fq(=)c Fp(T)1181 708 y Fm(N)1249 693 y Fq(\()p Fp(")p Fq(\))32 b(and)h(ev)m(en)h(if)e Fp(T)1962 708 y Fm(N)2029 693 y Fq(\()p Fp(")p Fq(\))g(is)g (self-adjoin)m(t)f Fp(i)p Fq([)2870 654 y Fm(d)p 2853 670 69 4 v 2853 728 a(d")2933 693 y Fp(T)2990 708 y Fm(N)3057 693 y Fq(\()p Fp(")p Fq(\))p Fp(;)17 b(T)3280 708 y Fm(N)3347 693 y Fq(\()p Fp(")p Fq(\)])257 814 y(is)33 b(not)g(self-adjoin)m(t;)e (this)i(p)s(oin)m(t)f(has)h(b)s(een)h(o)m(v)m(erlo)s(ok)m(ed)g(in)e ([11])h(but)g(without)g(conse-)257 934 y(quences)j(on)d(the)g(result)g (there\).)46 b(So)33 b(the)g(\014rst)h(step)g(is)e(to)h(construct)h (out)f(of)g Fp(T)3225 949 y Fm(N)3292 934 y Fq(\()p Fp(")p Fq(\))f(a)257 1054 y(pro)5 b(jection)35 b(op)s(erator,)p Fp(P)1174 1069 y Fm(N)1240 1054 y Fq(\()p Fp(")p Fq(\))f(whose)i (expansion)f(in)f Fp(")h Fq(should)f(coincide)g(with)g Fp(T)3306 1069 y Fm(N)3374 1054 y Fq(\()p Fp(")p Fq(\))257 1175 y(up)45 b(to)f(order)h Fp(N)10 b Fq(.)79 b(Of)44 b(course)i(suc)m(h)g(a)e(\\completion")e(is)i(far)g(from)f(b)s(eing)h (unique;)257 1295 y(w)m(e)c(shall)e(use)i(the)f(metho)s(d)g(emplo)m(y)m (ed)g(in)f([17])h(since)g(it)f(giv)m(es)h(an)g(easy)h(con)m(trol)e(on) 257 1416 y Fj(k)28 b Fq([)p Fp(H)443 1431 y Fm(")480 1416 y Fp(;)17 b(P)587 1431 y Fm(N)653 1416 y Fq(\()p Fp(")p Fq(\)])28 b Fj(k)404 1536 y Fq(W)-8 b(e)33 b(need)g(a)g(few)g (elemen)m(tary)g(facts)g(ab)s(out)f(pro)5 b(jections.)44 b(Here)33 b(w)m(e)h(consider)f(the)257 1656 y(self-adjoin)m(t)41 b(case;)47 b(for)42 b(the)g(non)g(self-adjoin)m(t)e(case)j(see)g(App)s (endix)g(where)g(a)f(more)257 1777 y(elab)s(orate)35 b(v)m(ersion)g(of)g(\(elemen)m(tary\))g(Prop)s(osition)f(3)h(in)g([17)o (])h(is)f(giv)m(en.)51 b(Let)36 b Fp(T)49 b Fq(b)s(e)35 b(a)257 1897 y(b)s(ounded)f(self-adjoin)m(t)d(op)s(erator)g(satisfying) 1419 2099 y Fj(k)d Fp(T)1568 2057 y Fo(2)1629 2099 y Fj(\000)23 b Fp(T)41 b Fj(k\024)28 b Fp(\016)k(<)27 b Fq(1)p Fp(=)p Fq(4)p Fp(:)912 b Fq(\(3.10\))257 2300 y(Then)34 b(b)m(y)g(the)f(sp)s(ectral)f(theorem)460 2565 y Fp(\032)p Fq(\()p Fp(T)14 b Fq(\))28 b Fj(\032)g Fq(\()838 2498 y(1)22 b Fj(\000)1008 2417 y(p)p 1091 2417 265 4 v 81 x Fq(1)g(+)g(4)p Fp(\016)p 838 2542 519 4 v 1073 2634 a Fq(2)1366 2565 y Fp(;)1420 2498 y Fq(1)g Fj(\000)1590 2417 y(p)p 1673 2417 267 4 v 81 x Fq(1)g Fj(\000)h Fq(4)p Fp(\016)p 1420 2542 520 4 v 1655 2634 a Fq(2)1950 2565 y(\))f Fj([)g Fq(\()2146 2498 y(1)g(+)2315 2417 y Fj(p)p 2398 2417 267 4 v 81 x Fq(1)g Fj(\000)h Fq(4)p Fp(\016)p 2146 2542 519 4 v 2381 2634 a Fq(2)2674 2565 y Fp(;)2728 2498 y Fq(1)f(+)2897 2417 y Fj(p)p 2980 2417 265 4 v 81 x Fq(1)g(+)g(4)p Fp(\016)p 2728 2542 517 4 v 2962 2634 a Fq(2)3254 2565 y(\))404 2823 y(In)33 b(particular)d(1)p Fp(=)p Fq(2)d Fj(2)h Fp(\032)p Fq(\()p Fp(T)14 b Fq(\))p Fp(;)72 b Fj(k)28 b Fp(T)41 b Fj(k\024)1880 2782 y Fo(1+)1970 2729 y Fi(p)p 2029 2729 160 3 v 53 x Fo(1+4)p Fm(\016)p 1880 2800 309 4 v 2017 2858 a Fo(2)2231 2823 y Fq(and)1268 3079 y(sup)1179 3165 y Fi(j)p Fm(z)s Fi(\000)p Fo(1)p Fi(j)p Fo(=1)p Fm(=)p Fo(2)1532 3079 y Fj(k)28 b Fq(\()p Fp(T)35 b Fj(\000)23 b Fp(z)t Fq(\))1927 3038 y Fi(\000)p Fo(1)2050 3079 y Fj(k\024)2365 3011 y Fq(2)p 2215 3056 350 4 v 2215 3076 a Fj(p)p 2298 3076 267 4 v 81 x Fq(1)f Fj(\000)g Fq(4)p Fp(\016)3246 3079 y Fq(\(3.11\))404 3350 y(No)m(w,)31 b(if)d Fp(P)44 b Fq(is)29 b(the)h(sp)s(ectral)g(pro)5 b(jection)29 b(of)h Fp(T)43 b Fq(corresp)s(onding)30 b(to)g(the)g(part)f(of)h(the)257 3470 y(sp)s(ectrum)j(around)g(1)f(:) 1251 3715 y Fp(P)41 b Fq(=)1506 3647 y Fp(i)p 1469 3692 108 4 v 1469 3783 a Fq(2)p Fp(\031)1603 3579 y Fh(I)1659 3804 y Fi(j)p Fm(z)s Fi(\000)p Fo(1)p Fi(j)p Fo(=1)p Fm(=)p Fo(2)1988 3715 y Fq(\()p Fp(T)36 b Fj(\000)23 b Fp(z)t Fq(\))2306 3673 y Fi(\000)p Fo(1)2401 3715 y Fp(dz)749 b Fq(\(3.12\))257 3980 y(then)34 b(again)d(b)m(y)i(the)g(sp)s (ectral)g(theorem)994 4245 y Fj(k)27 b Fp(T)36 b Fj(\000)23 b Fp(P)41 b Fj(k\024)1533 4177 y Fq(1)22 b Fj(\000)1703 4096 y(p)p 1786 4096 267 4 v 81 x Fq(1)g Fj(\000)h Fq(4)p Fp(\016)p 1533 4222 520 4 v 1768 4313 a Fq(2)2090 4245 y(=)2415 4177 y(2)p Fp(\016)p 2204 4222 519 4 v 2204 4322 a Fq(1)f(+)2373 4242 y Fj(p)p 2456 4242 267 4 v 80 x Fq(1)g Fj(\000)g Fq(4)p Fp(\016)2732 4245 y(:)487 b Fq(\(3.13\))404 4508 y(Consider)29 b(no)m(w)g Fp(T)1063 4523 y Fm(N)1130 4508 y Fq(\()p Fp(")p Fq(\).Due)f(to)g(\(3.6\))g(and)h (\(3.7\))f Fj(k)f Fp(T)2372 4523 y Fm(N)2440 4508 y Fq(\()p Fp(")p Fq(\))2562 4472 y Fo(2)2614 4508 y Fj(\000)14 b Fp(T)2762 4523 y Fm(N)2830 4508 y Fq(\()p Fp(")p Fq(\))27 b Fj(k\030)h(O)s Fq(\()p Fp(")3300 4472 y Fm(N)7 b Fo(+1)3457 4508 y Fq(\))257 4629 y(i.e.)43 b(for)32 b(su\016cien)m(tly)i(small)c Fp(")i Fq(,)1347 4830 y Fj(k)c Fp(T)1482 4845 y Fm(N)1549 4830 y Fq(\()p Fp(")p Fq(\))1671 4789 y Fo(2)1732 4830 y Fj(\000)23 b Fp(T)1889 4845 y Fm(N)1956 4830 y Fq(\()p Fp(")p Fq(\))28 b Fj(k)p Fp(<)f Fq(1)p Fp(=)p Fq(4)257 5032 y(and)33 b(one)g(can)g(de\014ne)h Fp(P)1150 5047 y Fm(N)1217 5032 y Fq(\()p Fp(")p Fq(\))e(b)m(y)1063 5282 y Fp(P)1126 5297 y Fm(N)1193 5282 y Fq(\()p Fp(")p Fq(\))27 b(=)1493 5214 y Fp(i)p 1455 5259 108 4 v 1455 5350 a Fq(2)p Fp(\031)1590 5146 y Fh(I)1645 5372 y Fi(j)p Fm(z)s Fi(\000)p Fo(1)p Fi(j)p Fo(=1)p Fm(=)p Fo(2)1975 5282 y Fq(\()p Fp(T)2070 5297 y Fm(N)2137 5282 y Fq(\()p Fp(")p Fq(\))22 b Fj(\000)g Fp(z)t Fq(\))2467 5240 y Fi(\000)p Fo(1)2563 5282 y Fp(dz)t(:)556 b Fq(\(3.14\))1828 5637 y(12)p eop %%Page: 13 13 13 12 bop 257 573 a Fp(P)320 588 y Fm(N)387 573 y Fq(\()p Fp(")p Fq(\))33 b(is)f(a)g(pro)5 b(jection)32 b(and)h(according)f(to)g (\(3.13\))1256 787 y Fj(k)c Fp(P)1397 802 y Fm(N)1464 787 y Fq(\()p Fp(")p Fq(\))21 b Fj(\000)i Fp(T)1764 802 y Fm(N)1831 787 y Fq(\()p Fp(")p Fq(\))28 b Fj(k\030)g(O)s Fq(\()p Fp(")2302 745 y Fm(N)7 b Fo(+1)2459 787 y Fq(\))749 b(\(3.15\))404 1000 y(W)-8 b(e)33 b(compute)f(no)m(w)i([)p Fp(H)1279 1015 y Fm(")1315 1000 y Fp(;)17 b(P)1422 1015 y Fm(N)1489 1000 y Fq(\()p Fp(")p Fq(\)].)43 b(Notice)32 b(\014rst)h(that)g(due)g(to)f(the)h(iden)m(tit)m(y)1168 1214 y(\()p Fp(T)i Fj(\000)23 b Fp(z)t Fq(\))1485 1173 y Fi(\000)p Fo(1)1608 1214 y Fq(=)k Fp(z)1760 1173 y Fi(\000)p Fo(1)1855 1214 y Fq(\()p Fp(T)14 b Fq(\()p Fp(T)36 b Fj(\000)22 b Fp(z)t Fq(\))2281 1173 y Fi(\000)p Fo(1)2399 1214 y Fj(\000)g Fq(1\))257 1428 y([)p Fp(H)365 1443 y Fm(")402 1428 y Fp(;)17 b(P)509 1443 y Fm(N)576 1428 y Fq(\()p Fp(")p Fq(\)])24 b(is)g(w)m(ell)f(de\014ned)j(on)e(a)g (dense)i(subspace)h(and)d(the)h(follo)m(wing)d(computation)257 1548 y(can)33 b(b)s(e)g(done.)44 b(F)-8 b(rom)31 b(\(3.5\))1329 1762 y([)p Fp(H)1437 1777 y Fm(")1473 1762 y Fp(;)17 b(T)1574 1777 y Fm(N)1641 1762 y Fq(\()p Fp(")p Fq(\)])28 b(=)f Fp(")1967 1721 y Fm(N)7 b Fo(+1)2124 1762 y Fq([)p Fp(V)e(;)17 b(E)2329 1777 y Fm(N)2397 1762 y Fq(])257 1976 y(and)33 b(then)333 2213 y([)p Fp(H)441 2228 y Fm(")477 2213 y Fp(;)17 b(P)584 2228 y Fm(N)651 2213 y Fq(\()p Fp(")p Fq(\)])27 b(=)22 b Fj(\000)1072 2146 y Fp(i)p 1035 2190 108 4 v 1035 2281 a Fq(2)p Fp(\031)1169 2078 y Fh(I)1225 2303 y Fi(j)p Fm(z)s Fi(\000)p Fo(1)p Fi(j)p Fo(=1)p Fm(=)p Fo(2)1555 2213 y Fq(\()p Fp(T)1650 2228 y Fm(N)1717 2213 y Fq(\()p Fp(")p Fq(\))g Fj(\000)g Fp(z)t Fq(\))2047 2172 y Fi(\000)p Fo(1)2142 2213 y Fq([)p Fp(H)2250 2228 y Fm(")2287 2213 y Fp(;)17 b(T)2388 2228 y Fm(N)2455 2213 y Fq(\()p Fp(")p Fq(\)]\()p Fp(T)2699 2228 y Fm(N)2766 2213 y Fq(\()p Fp(")p Fq(\))22 b Fj(\000)g Fp(z)t Fq(\))3096 2172 y Fi(\000)p Fo(1)3191 2213 y Fp(dz)931 2488 y Fq(=)28 b Fj(\000)p Fp(")1158 2447 y Fm(N)7 b Fo(+1)1362 2421 y Fp(i)p 1325 2466 V 1325 2557 a Fq(2)p Fp(\031)1459 2353 y Fh(I)1515 2578 y Fi(j)p Fm(z)s Fi(\000)p Fo(1)p Fi(j)p Fo(=1)p Fm(=)p Fo(2)1845 2488 y Fq(\()p Fp(T)1940 2503 y Fm(N)2007 2488 y Fq(\()p Fp(")p Fq(\))22 b Fj(\000)g Fp(z)t Fq(\))2337 2447 y Fi(\000)p Fo(1)2432 2488 y Fq([)p Fp(V)5 b(;)17 b(E)2637 2503 y Fm(N)2705 2488 y Fq(]\()p Fp(T)2827 2503 y Fm(N)2894 2488 y Fq(\()p Fp(")p Fq(\))22 b Fj(\000)g Fp(z)t Fq(\))3224 2447 y Fi(\000)p Fo(1)3319 2488 y Fp(dz)3246 2688 y Fq(\(3.16\))404 2895 y(W)-8 b(e)33 b(summarise)e(the)i(ab)s(o)m(v)m(e)g(discussion)g(in)f(the)h (follo)m(wing)d(lemma:)257 3093 y Fk(Lemma)37 b(3.1.)50 b Fl(Supp)-5 b(ose)38 b Fp(E)1325 3108 y Fm(j)1401 3093 y Fl(satisfy)h(\(3.4-3.8\),)g Fp(T)2210 3108 y Fm(N)2277 3093 y Fq(\()p Fp(")p Fq(\))g Fl(b)-5 b(e)39 b(given)f(by)h(\(3.9\))g (and)f(let)257 3213 y Fp(")303 3228 y Fm(N)405 3213 y Fl(b)-5 b(e)35 b(the)g(p)-5 b(ositive)34 b(solution)g(of)h(the)g(e)-5 b(quation:)1606 3389 y Fo(2)p Fm(N)1583 3419 y Fh(X)1540 3631 y Fm(l)q Fo(=)p Fm(N)7 b Fo(+1)1786 3514 y Fp(")1832 3473 y Fm(l)1858 3514 y Fp(a)1909 3529 y Fm(l)1963 3514 y Fq(=)27 b(1)p Fp(=)p Fq(8)1033 b(\(3.17\))257 3819 y Fl(wher)-5 b(e)1385 3940 y Fp(a)1436 3955 y Fm(l)1490 3940 y Fq(=)1804 3845 y Fh(X)1593 4057 y Fm(j)t Fo(+)p Fm(m)p Fo(=)p Fm(l)q(;)p Fo(0)p Fi(\024)p Fm(j;m)p Fi(\024)p Fm(N)2174 3940 y Fp(e)2219 3955 y Fm(j)2256 3940 y Fp(e)2301 3955 y Fm(m)404 4209 y Fl(Then)34 b(for)g Fq(0)28 b Fj(\024)g Fp(")f Fj(\024)i Fp(")1220 4224 y Fm(N)1286 4209 y Fl(:)900 4510 y Fj(k)e Fp(T)1034 4525 y Fm(N)1102 4510 y Fq(\()p Fp(")p Fq(\))1224 4469 y Fo(2)1285 4510 y Fj(\000)c Fp(T)1442 4525 y Fm(N)1509 4510 y Fq(\()p Fp(")p Fq(\))k Fj(k\024)1879 4385 y Fo(2)p Fm(N)1856 4415 y Fh(X)1813 4627 y Fm(l)q Fo(=)p Fm(N)7 b Fo(+1)2060 4510 y Fp(")2106 4469 y Fm(l)2131 4510 y Fp(a)2182 4525 y Fm(l)2236 4510 y Fj(\021)28 b Fp(\016)2384 4525 y Fm(N)2452 4510 y Fq(\()p Fp(")p Fq(\))f Fj(\024)h Fq(1)p Fp(=)p Fq(8)393 b(\(3.18\))257 4821 y Fl(and)35 b(if)f Fp(P)604 4836 y Fm(N)671 4821 y Fq(\()p Fp(")p Fq(\))h Fl(is)f(given)g(by)h(\()g(3.14\))1274 5034 y Fj(k)28 b Fp(P)1415 5049 y Fm(N)1482 5034 y Fq(\()p Fp(")p Fq(\))22 b Fj(\000)g Fp(T)1782 5049 y Fm(N)1850 5034 y Fq(\()p Fp(")p Fq(\))27 b Fj(k\024)h Fq(2)p Fp(\016)2246 5049 y Fm(N)2313 5034 y Fq(\()p Fp(")p Fq(\))p Fp(;)784 b Fq(\(3.19\))945 5352 y Fj(k)27 b Fq([)p Fp(H)1130 5367 y Fm(")1167 5352 y Fp(;)17 b(P)1274 5367 y Fm(N)1341 5352 y Fq(\()p Fp(")p Fq(\)])27 b Fj(k\024)h Fp(")1718 5311 y Fm(N)7 b Fo(+1)1903 5352 y Fj(k)27 b Fq([)p Fp(V)5 b(;)17 b(E)2185 5367 y Fm(N)2253 5352 y Fq(])27 b Fj(k\021)h Fp(")2508 5311 y Fm(N)7 b Fo(+1)2665 5352 y Fp(f)2713 5367 y Fm(N)2781 5352 y Fp(:)438 b Fq(\(3.20\))1828 5637 y(13)p eop %%Page: 14 14 14 13 bop 404 573 a Fl(Pr)-5 b(o)g(of.)404 693 y Fq(F)d(or)31 b(\(3.18\))h(notice)g(that)h(from)e(\(3.6\))859 1003 y Fp(T)916 1018 y Fm(N)984 1003 y Fq(\()p Fp(")p Fq(\))1106 962 y Fo(2)1167 1003 y Fj(\000)22 b Fp(T)1323 1018 y Fm(N)1391 1003 y Fq(\()p Fp(")p Fq(\))27 b(=)1709 878 y Fo(2)p Fm(N)1687 908 y Fh(X)1643 1120 y Fm(l)q Fo(=)p Fm(N)7 b Fo(+1)1890 1003 y Fp(")1936 962 y Fm(l)1962 1003 y Fq(\()2210 908 y Fh(X)2000 1120 y Fm(j)t Fo(+)p Fm(m)p Fo(=)p Fm(l)q(;)p Fo(0)p Fi(\024)p Fm(j;m)p Fi(\024)p Fm(N)2581 1003 y Fp(E)2653 1018 y Fm(j)2690 1003 y Fp(E)2762 1018 y Fm(m)2828 1003 y Fq(\))p Fp(:)257 1323 y Fq(Then)40 b(\(3.19\))d(follo)m(ws)g(from)g(\(3.18\))g(and)h(\(3.14\))g(while)f (\(3.20\))g(follo)m(ws)g(from)g(\(3.16\))257 1443 y(and)c(\(3.19\).)404 1563 y Fl(R)-5 b(emarks)404 1684 y Fq(3.1)32 b(The)h(main)e(p)s(oin)m (t)h(of)g(the)h(lemma)d(is)i(the)h(estimate)f(\(3.20\).)404 1804 y(3.2)e(Notice)g(that)g(in)g(eac)m(h)h(particular)e(case)i(one)g (has)g(still)d(to)i(pro)m(v)m(e)i(that)e Fp(P)3240 1819 y Fm(")3276 1804 y Fj(D)3353 1819 y Fm(")3418 1804 y Fj(\032)257 1924 y(D)s Fq(\()p Fp(H)456 1939 y Fm(")492 1924 y Fq(\))j(\(see)h(Remark)e(2.2)g(and)h(remind)e(that)i Fj(D)2092 1939 y Fm(")2161 1924 y Fq(is)f(a)g(core)h(of)f Fp(H)2738 1939 y Fm(")2775 1924 y Fq(\).)404 2045 y(A)m(t)41 b(a)f(more)g(elab)s(orate)f(lev)m(el)h(of)h(p)s(erturbation)e(theory)j (the)f(task)g(is)f(to)g(obtain,)257 2165 y(under)31 b(sp)s(eci\014c)g (conditions)f(on)g Fp(H)1547 2180 y Fo(0)1586 2165 y Fq(,)h Fp(V)52 b Fq(and)30 b Fp(P)2003 2180 y Fo(0)2073 2165 y Fq(the)h(dep)s(endence)h(on)e Fp(j)37 b Fq(of)30 b Fp(e)3123 2180 y Fm(j)3159 2165 y Fq(,)h Fp(f)3265 2180 y Fm(j)3302 2165 y Fq(;)g(the)257 2286 y(rate)j(of)g(increase)h (of)f(these)h(constan)m(ts)h(\\measures")e(the)h(singularit)m(y)d(of)i (the)g(p)s(ertur-)257 2406 y(bation.)58 b(The)39 b(simplest)d(case)j (is)e(when)i Fp(e)1825 2421 y Fm(j)1898 2406 y Fj(\024)e Fp(e)2057 2370 y Fm(j)2131 2406 y Fq(for)g(some)h(\014nite)f(constan)m (t)h Fp(e)g Fq(;)j(this)257 2526 y(corresp)s(onds)30 b(to)e(the)h(w)m(ell)f(kno)m(wn)i(regular)e(p)s(erturbations)g(and)g (there)i(is)e(no)g(need)i(for)257 2647 y(further)e(discussion.)42 b(The)29 b(most)d(in)m(teresting)h(case)i(is)e(when)h Fp(N)39 b Fq(=)27 b Fj(1)g Fq(and)h(there)g(exist)257 2767 y(constan)m(ts)34 b Fp(g)d(<)d Fj(1)p Fp(;)71 b(\013)29 b(>)e Fq(0)33 b(suc)m(h)h(that)873 2984 y Fp(e)918 2999 y Fm(j)983 2984 y Fj(\024)28 b Fp(g)1139 2943 y Fm(j)1175 2984 y Fq(\()p Fp(j)6 b Fq(!\))1324 2943 y Fm(\013)1373 2984 y Fp(;)72 b(f)1520 2999 y Fm(j)1584 2984 y Fj(\024)29 b Fp(g)1741 2943 y Fm(j)t Fo(+1)1867 2984 y Fq(\(\()p Fp(j)f Fq(+)22 b(1\)!\))2261 2943 y Fm(\013)2310 2984 y Fp(;)72 b(j)33 b Fq(=)28 b(1)p Fp(;)17 b Fq(2)p Fp(;)g(::::)366 b Fq(\(3.21\))257 3201 y(In)32 b(this)e(case)i(the)g(series)g(in)e (\(3.3\))g(is)h(not)g(con)m(v)m(ergen)m(t)i(and)e(the)h(idea)e(here)i (is)f(to)f(use)i(a)257 3321 y(b)s(est)24 b(remainder)e(estimate)h(a)f (la)g(Nec)m(harosev)k(i.e.)39 b(to)23 b(tak)m(e)h(N)f(dep)s(ending)h (on)f Fp(")f Fq(in)h(\(3.9\))257 3442 y(in)37 b(order)g(to)g(ha)m(v)m (e)h(the)g(r.h.s.)58 b(of)36 b(\(3.16\))h(as)g(small)e(as)i(p)s (ossible.)57 b(This)37 b(requiremen)m(t)257 3562 y(leads)c(to)f Fp(N)699 3577 y Fm(")768 3562 y Fq(giv)m(en)h(b)m(y)1486 3818 y Fp(N)1564 3833 y Fm(")1623 3818 y Fq(+)22 b(1)27 b(=)h([\()1999 3750 y(1)p 1976 3795 96 4 v 1976 3886 a Fp(g)t(")2082 3818 y Fq(\))2120 3776 y Fo(1)p Fm(=\013)2240 3818 y Fq(])979 b(\(3.22\))257 4085 y(where)34 b([)p Fp(:)17 b(:)g(:)f Fq(])33 b(means)g(the)g(in)m(teger)f(part.)43 b(Pluging)31 b(\(3.22\))h(in)m(to)g(\(3.9\))g(one)h(obtains:)257 4285 y Fk(Lemma)k(3.2.)50 b Fl(L)-5 b(et)36 b Fp(N)1125 4300 y Fm(")1197 4285 y Fl(b)-5 b(e)36 b(given)f(by)h(\(3.22\).)47 b(Then)35 b(ther)-5 b(e)35 b(exists)h Fp(")2849 4300 y Fo(0)2918 4285 y Fp(>)30 b Fq(0)35 b Fl(such)h(that)257 4406 y(for)f Fp(")27 b Fj(2)h Fq([0)p Fp(;)17 b(")746 4421 y Fo(0)785 4406 y Fq(])p Fl(,)35 b(\(3.18\))f(holds)g(true)h(and)g Fp(P)1878 4421 y Fm(")1942 4406 y Fj(\021)28 b Fp(P)2110 4421 y Fm(N)2166 4429 y Fg(")2238 4406 y Fl(satis\014es)35 b(\(2.2\))f(with)1122 4667 y Fp(\016)t Fq(\()p Fp(")o Fq(\))28 b(=)f(\(2)p Fp(\031)t Fq(\))1605 4626 y Fm(\013)1789 4600 y Fq(1)p 1664 4644 299 4 v 1664 4735 a(\()p Fp(g)20 b(")p Fq(\))1853 4707 y Fo(1)p Fm(=)p Fo(2)1973 4667 y Fp(exp)p Fq(\()p Fj(\000)2370 4600 y Fp(\013)p 2247 4644 309 4 v 2247 4735 a Fq(\()p Fp(g)g(")p Fq(\))2436 4707 y Fo(1)p Fm(=\013)2566 4667 y Fq(\))p Fp(:)615 b Fq(\(3.23\))404 4940 y Fl(Pr)-5 b(o)g(of.)43 b Fq(F)-8 b(rom)31 b(\(3.21\))1223 5250 y Fp(a)1274 5265 y Fm(l)1328 5250 y Fj(\024)1494 5125 y Fm(l)1433 5155 y Fh(X)1444 5365 y Fm(j)t Fo(=0)1593 5250 y Fp(e)1638 5265 y Fm(j)1675 5250 y Fp(e)1720 5265 y Fm(l)q Fi(\000)p Fm(j)1861 5250 y Fj(\024)e Fp(g)2018 5209 y Fm(l)2043 5250 y Fq(\()p Fp(l)24 b Fq(+)e(1\)\()p Fp(l)r Fq(!\))2453 5209 y Fm(\013)2503 5250 y Fp(:)716 b Fq(\(3.24\))1828 5637 y(14)p eop %%Page: 15 15 15 14 bop 257 573 a Fq(F)-8 b(rom)31 b(the)i(Stirling)d(form)m(ula)h (\()p Fp(K)40 b Fq(is)32 b(an)g(absolute)g(constan)m(t\))1538 774 y Fp(j)6 b Fq(!)28 b Fj(\024)g Fp(K)7 b(j)1880 733 y Fm(j)t Fo(+1)p Fm(=)p Fo(2)2078 774 y Fp(e)2123 733 y Fi(\000)p Fm(j)3246 774 y Fq(\(3.25\))257 976 y(and)33 b(the)g(fact)f(that)h(for)f(all)e Fp(N)1381 991 y Fm(")1440 976 y Fq(+)22 b(1)28 b Fj(\024)g Fp(k)j Fj(\024)d Fq(2)p Fp(N)2034 991 y Fm(")2070 976 y Fq(:)1149 1178 y Fp(\013)17 b Fq(ln)f Fp(k)31 b Fj(\024)d Fp(\013)17 b Fq(ln)f(2)p Fp(N)1817 1193 y Fm(")1881 1178 y Fj(\024)28 b Fp(\013)18 b Fq(ln)e(2)22 b Fj(\000)g Fq(ln)o(\()p Fp(g)e(")p Fq(\))642 b(\(3.26\))257 1379 y(one)33 b(obtains)1180 1581 y Fp(")1226 1540 y Fm(j)1279 1581 y Fp(a)1330 1596 y Fm(j)1394 1581 y Fj(\024)28 b Fp(K)1589 1540 y Fm(\013)1639 1581 y Fq(\()p Fp(j)g Fq(+)22 b(1\))1930 1540 y Fo(1+)p Fm(\013=)p Fo(2)2140 1581 y Fp(e)2185 1540 y Fi(\000)p Fm(j)t(\013)p Fo(\(1)p Fi(\000)p Fo(ln)12 b(2\))3246 1581 y Fq(\(3.27\))257 1782 y(so)1005 2035 y Fp(\016)1048 2050 y Fm(N)1115 2035 y Fq(\()p Fp(")p Fq(\))27 b Fj(\024)i Fp(K)1460 1994 y Fm(\013)1624 1910 y Fi(1)1587 1940 y Fh(X)1526 2152 y Fm(j)t Fo(=)p Fm(N)1670 2160 y Fg(")1703 2152 y Fo(+1)1793 2035 y Fq(\()p Fp(j)f Fq(+)22 b(1\))2084 1994 y Fm(\013)p Fo(+1)p Fm(=)p Fo(2)2294 2035 y Fp(e)2339 1994 y Fi(\000)p Fm(\013)p Fo(\(1)p Fi(\000)p Fm(l)q(n)p Fo(2\))p Fm(j)2721 2035 y Fp(:)498 b Fq(\(3.28\))257 2339 y(Since)35 b Fp(N)592 2354 y Fm(")661 2339 y Fj(!)c(1)j Fq(as)h Fp(")c Fj(!)g Fq(0)k(the)g(\014rst)g(part)g(of)f(the)h(lemma)e(is)h(pro)m(v)m(ed.)52 b(Then)36 b(\(3.23\))257 2459 y(follo)m(ws)c(from)f(\(3.20\),\(3.22\))g (and)i(the)g(Stirling)d(form)m(ula.)404 2579 y Fl(R)-5 b(emark)38 b Fq(3.3)h(If)g Fp(g)19 b(")39 b(<)f Fq(1,)j(in)d(the)h (limit)d Fp(\013)j Fj(!)g Fq(0)f(one)i(has)f Fp(\016)t Fq(\()p Fp(")o Fq(\))g Fj(!)f Fq(0)h(whic)m(h)h(\014ts)257 2700 y(with)33 b(the)g(fact)f(that)g(for)g Fp(\013)d Fq(=)e(0)33 b(the)g(series)g(in)f(\(3.3\))g(is)g(con)m(v)m(ergen)m(t)i (for)e Fp(")c(<)f(g)3214 2664 y Fi(\000)p Fo(1)3308 2700 y Fq(.)257 3030 y Fr(4)161 b(Asymptotic)53 b(estimations.)257 3249 y Fq(In)c(this)f(section)h(w)m(e)g(consider)g(the)g(problem)e(of)h (\014nding)g(solutions)f Fp(E)3040 3264 y Fm(j)3125 3249 y Fq(of)h(\()g(3.4-)257 3369 y(3.6\))42 b(satisfying)f(estimations)g (lik)m(e)g(\()h(3.7-3.8\).)71 b(In)42 b(spite)h(of)e(the)i(fact)f(that) g(all)e(the)257 3490 y(argumen)m(ts)29 b(b)s(elo)m(w)g(w)m(ork)h(in)e (an)h(appropriate)f(abstract)h(setting,)h(for)e(de\014niteness)j(w)m(e) 257 3610 y(shall)h(discuss)h(a)g(simple)e(Sc)m(hro)s(dinger)h(case)i (in)d Fn(R)2120 3574 y Fo(3)991 3932 y Fp(H)1072 3947 y Fm(")1136 3932 y Fq(=)d Fj(\000)p Fq(\001)23 b(+)f Fp(V)1576 3947 y Fo(0)1615 3932 y Fq(\()p Fk(x)p Fq(\))g(+)g Fp("V)f Fq(\()p Fk(x)p Fq(\))28 b Fj(\021)g Fp(H)2343 3947 y Fo(0)2405 3932 y Fq(+)22 b Fp("V)f Fq(\()p Fk(x)p Fq(\))533 b(\(4.1\))257 4134 y(with)916 4359 y Fp(V)973 4374 y Fo(0)1040 4359 y Fj(2)28 b Fp(L)1200 4318 y Fo(2)1200 4383 y Fm(ul)q(oc)1387 4359 y Fp(i:e:)i Fq(sup)1536 4446 y Ff(x)p Fi(2)p Fe(R)1674 4427 y Fd(3)1725 4223 y Fh(Z)1781 4449 y Fi(j)p Ff(x)p Fi(\000)p Ff(y)q Fi(j\024)p Fo(1)2084 4359 y Fj(j)d Fp(V)2196 4374 y Fo(0)2236 4359 y Fq(\()p Fk(y)r Fq(\))g Fj(j)2428 4318 y Fo(2)2495 4359 y Fp(d)p Fk(y)i Fp(<)e Fj(1)458 b Fq(\(4.2\))257 4624 y(and)862 4825 y Fj(j)27 b Fp(V)21 b Fq(\()p Fk(x)p Fq(\))28 b Fj(j\024)56 b Fp(<)27 b Fk(x)h Fp(>)1585 4784 y Fm(\014)1632 4825 y Fp(;)44 b(<)28 b Fk(x)g Fp(>)p Fq(=)f(\(1)22 b(+)g Fk(x)2339 4784 y Fo(2)2379 4825 y Fq(\))2417 4784 y Fo(1)p Fm(=)p Fo(2)2527 4825 y Fp(;)44 b(\014)33 b Fj(2)28 b Fk(R)p Fp(:)404 b Fq(\(4.3\))257 5027 y(In)24 b(particular)e Fp(H)892 5042 y Fo(0)955 5027 y Fq(is)h(essen)m(tially)g(self-adjoin)m (t)f(on)h Fj(C)2179 4991 y Fi(1)2173 5052 y Fo(0)2254 5027 y Fq(\()p Fn(R)2358 4991 y Fo(3)2403 5027 y Fq(\))h([21)o(].)41 b(I)24 b(what)f(follo)m(ws)g Fp(H)3370 5042 y Fm(")3430 5027 y Fq(is)257 5147 y Fl(any)28 b Fq(self-adjoin)m(t)f(extension)i (of)e Fp(H)1549 5162 y Fo(0)1602 5147 y Fq(+)13 b Fp("V)1794 5163 y Fo(\()1825 5147 y Fk(x)p Fq(\))28 b(de\014ned)i(on)e Fj(D)s Fq(\()p Fp(H)2612 5162 y Fo(0)2650 5147 y Fq(\))13 b Fj(\\)g(D)s Fq(\()p Fp(V)22 b Fq(\))28 b Fj(\033)g(C)3206 5111 y Fi(1)3200 5172 y Fo(0)3281 5147 y Fq(\()p Fn(R)3385 5111 y Fo(3)3430 5147 y Fq(\))p Fp(:)257 5268 y Fq(W)-8 b(e)23 b(remind)d(the)j(follo)m(wing)c(elemen)m(tary)j(result)f(from)g (Com)m(b)s(es-Thomas)h(theory)g(\([4]\),)257 5388 y(\([12]\),)33 b(\([18)o(]\):)1828 5637 y(15)p eop %%Page: 16 16 16 15 bop 257 573 a Fk(Prop)s(osition)36 b(4.1.)49 b Fl(L)-5 b(et)41 b Fp(K)k Fj(\032)39 b Fp(\032)p Fq(\()p Fp(H)1673 588 y Fo(0)1713 573 y Fq(\))h Fl(b)-5 b(e)41 b(c)-5 b(omp)g(act,)41 b Fp(\025)2387 588 y Fo(0)2467 573 y Fl(an)f(isolate)-5 b(d)40 b(\014nite)g(de)-5 b(gen-)257 693 y(er)g(ate)38 b(eigenvalue)f(of)g Fp(H)1175 708 y Fo(0)1215 693 y Fl(.)53 b(Then)37 b(ther)-5 b(e)38 b(exist)f Fp(\015)2081 708 y Fo(0)2153 693 y Fp(>)c Fq(0)p Fp(;)83 b(M)44 b(<)33 b Fj(1)k Fl(such)h(that)g(for)g(al)5 b(l)257 814 y Fp(\015)33 b Fj(2)28 b Fq([0)p Fp(;)17 b(\015)606 829 y Fo(0)645 814 y Fq(])p Fl(:)1424 1134 y Fj(k)27 b Fp(e)1546 1093 y Fm(\015)t(<)p Fi(\001)p Fm(>)1720 1134 y Fp(P)1783 1149 y Fo(0)1822 1134 y Fp(e)1867 1093 y Fm(\015)t(<)p Fi(\001)p Fm(>)2069 1134 y Fj(k\024)h Fp(M)977 b Fq(\(4.4\))1457 1535 y(sup)1457 1617 y Fm(z)s Fi(2)p Fm(K)1632 1535 y Fj(k)27 b Fp(R)1783 1550 y Fo(0)p Fm(;\015)1883 1535 y Fq(\()p Fp(z)t Fq(\))h Fj(k\024)g Fp(M)1010 b Fq(\(4.5\))257 1788 y Fl(wher)-5 b(e)1184 1989 y Fp(R)1258 2004 y Fo(0)p Fm(;\015)1357 1989 y Fq(\()p Fp(z)t Fq(\))29 b(=)e Fp(e)1659 1947 y Fm(\015)t(<)p Fi(\001)p Fm(>)1833 1989 y Fq(\()p Fp(H)1952 2004 y Fo(0)2013 1989 y Fj(\000)c Fp(z)t Fq(\))2200 1947 y Fi(\000)p Fo(1)2295 1989 y Fp(e)2340 1947 y Fi(\000)p Fm(\015)t(<)p Fi(\001)p Fm(>)3295 1989 y Fq(\(4.6\))257 2189 y Fl(and)35 b Fp(P)510 2204 y Fo(0)584 2189 y Fl(is)f(the)h(sp)-5 b(e)g(ctr)g(al)35 b(pr)-5 b(oje)g(ction)34 b(of)g Fp(H)1845 2204 y Fo(0)1920 2189 y Fl(c)-5 b(orr)g(esp)g(onding)33 b(to)i Fp(\025)2709 2204 y Fo(0)2748 2189 y Fl(.)404 2376 y Fq(If)h Fp(\014)k Fj(2)35 b Fq(\()p Fj(\0001)p Fp(;)17 b Fq(0])36 b(then)h Fp(V)22 b(R)1451 2391 y Fo(0)1490 2376 y Fq(\()p Fp(z)t Fq(\))37 b(\()g Fp(R)1801 2391 y Fo(0)1840 2376 y Fq(\()p Fp(z)t Fq(\))e(=)g(\()p Fp(H)2230 2391 y Fo(0)2294 2376 y Fj(\000)25 b Fp(z)t Fq(\))2483 2339 y Fi(\000)p Fo(1)2614 2376 y Fq(\))37 b(is)f(b)s(ounded)h(and)f Fp(E)3458 2391 y Fm(j)257 2496 y Fq(giv)m(en)d(b)m(y)h(the)f(w)m(ell)e(kno)m(wn)j(p)s (erturbation)e(form)m(ula)1122 2739 y Fp(E)1194 2754 y Fm(j)1258 2739 y Fq(=)c(\()p Fj(\000)p Fq(1\))1564 2698 y Fm(j)1648 2672 y Fp(i)p 1610 2716 108 4 v 1610 2808 a Fq(2)p Fp(\031)1745 2604 y Fh(I)1861 2739 y Fp(R)1935 2754 y Fo(0)1975 2739 y Fq(\()p Fp(z)t Fq(\)\()p Fp(V)22 b(R)2291 2754 y Fo(0)2330 2739 y Fq(\()p Fp(z)t Fq(\)\))2493 2698 y Fm(j)2530 2739 y Fp(dz)669 b Fq(\(4.7\))257 2988 y(satisfy)37 b(\()f(3.4-3.6\))f(and)i(\()f(3.7-3.8\))f(with)h Fp(\013)f Fq(=)g(0.)55 b(No)m(w,)38 b(if)d Fp(\014)40 b(>)34 b Fq(0)p Fp(;)85 b(V)22 b(R)3051 3003 y Fo(0)3090 2988 y Fq(\()p Fp(z)t Fq(\))37 b(is)f(not)257 3108 y(b)s(ounded)e(and)f (one)g(cannot)h(decide)f(at)g(once)g(from)f(\(4.7\))g(whether)j Fp(E)2897 3123 y Fm(j)2966 3108 y Fq(are)e(b)s(ounded)257 3229 y(or)j(not)f(and)h(the)g(main)e(idea)h(in)f(what)i(follo)m(ws)e (is)i(to)f(\014nd)h(alternativ)m(e)f(form)m(ulae)f(for)257 3349 y Fp(E)329 3364 y Fm(j)399 3349 y Fq(whic)m(h)f(mak)m(e)f(sense)j (for)d Fp(\014)h(>)28 b Fq(0.)257 3536 y Fk(Lemma)37 b(4.2.)50 b Fl(F)-7 b(or)37 b(al)5 b(l)38 b Fp(\014)i Fj(2)35 b Fn(R)5 b Fl(,)46 b Fp(E)1616 3551 y Fm(j)1691 3536 y Fl(given)38 b(by)g(\(4.7\))g(ar)-5 b(e)38 b(wel)5 b(l)38 b(de\014ne)-5 b(d)38 b(on)g Fj(C)3233 3500 y Fi(1)3227 3560 y Fo(0)3308 3536 y Fq(\()p Fn(R)3412 3500 y Fo(3)3457 3536 y Fq(\))257 3656 y Fl(and)d(satisfy)g(\(3.4-3.6\).)404 3843 y(Pr)-5 b(o)g(of.)40 b Fq(The)26 b(k)m(ey)g(observ)-5 b(ation)24 b(\([12)o(]\))h(is)f(that)g(due)i(to)e(\(4.3\))g(and)h (\(4.5-4.5\))e(expres-)257 3963 y(sions)36 b(lik)m(e)f Fp(R)756 3978 y Fo(0)796 3963 y Fq(\()p Fp(z)t Fq(\)\()p Fp(V)22 b(R)1112 3978 y Fo(0)1152 3963 y Fq(\()p Fp(z)t Fq(\)\))1315 3927 y Fm(l)1341 3963 y Fp(;)83 b(V)22 b(R)1604 3978 y Fo(0)1643 3963 y Fq(\()p Fp(z)t Fq(\)\()p Fp(V)h(R)1960 3978 y Fo(0)1999 3963 y Fq(\()p Fp(z)t Fq(\)\))2162 3927 y Fm(l)2189 3963 y Fp(V)57 b Fq(are)36 b(w)m(ell)f(de\014ned)i(on)f Fj(C)3206 3927 y Fi(1)3200 3988 y Fo(0)3281 3963 y Fq(\()p Fn(R)3385 3927 y Fo(3)3430 3963 y Fq(\).)257 4084 y(In)d(particular)e (from)g(\(4.7\),)i(for)f Fp(u)27 b Fj(2)h(C)1704 4047 y Fi(1)1698 4108 y Fo(0)1779 4084 y Fq(\()p Fn(R)1883 4047 y Fo(3)1928 4084 y Fq(\),)33 b Fp(E)2098 4099 y Fm(j)2134 4084 y Fp(u)f Fq(is)h(w)m(ell)e(de\014ned:)553 4335 y Fp(E)625 4350 y Fm(j)662 4335 y Fp(u)c Fq(=)h(\()p Fj(\000)p Fq(1\))1051 4294 y Fm(j)1135 4268 y Fp(i)p 1097 4312 V 1097 4404 a Fq(2)p Fp(\031)1232 4200 y Fh(I)1348 4335 y Fp(R)1422 4350 y Fo(0)1462 4335 y Fq(\()p Fp(z)t Fq(\)\()p Fp(V)22 b(e)1749 4289 y Fi(\000)1814 4258 y Fg(\015)p 1814 4274 36 3 v 1818 4315 a(j)1864 4335 y Fq(\))p Fp(R)1976 4353 y Fo(0)p Fm(;)2041 4322 y Fg(\015)p 2041 4338 V 2045 4379 a(j)2091 4335 y Fq(\()p Fp(z)t Fq(\))p Fp(:::R)2371 4350 y Fo(0)p Fm(;\015)2471 4335 y Fq(\()p Fp(z)t Fq(\))p Fp(e)2641 4294 y Fm(\015)t(<)p Fi(\001)p Fm(>)2816 4335 y Fp(udz)t(:)296 b Fq(\(4.8\))404 4588 y(F)-8 b(or)31 b(all)g Fp(z)h Fj(6)p Fq(=)c Fp(z)944 4552 y Fi(0)996 4588 y Fj(2)g Fp(\032)p Fq(\()p Fp(H)1259 4603 y Fo(0)1298 4588 y Fq(\))p Fp(;)72 b(j)34 b Fq(=)27 b(1)p Fp(;)17 b(:::)33 b Fq(and)f Fp(u)c Fj(2)g(C)2244 4552 y Fi(1)2238 4613 y Fo(0)2319 4588 y Fq(\()p Fk(R)2441 4552 y Fo(3)2480 4588 y Fq(\))992 4766 y(1)p 894 4810 245 4 v 894 4902 a Fp(z)f Fj(\000)22 b Fp(z)1114 4873 y Fi(0)1148 4833 y Fq([)p Fp(R)1249 4848 y Fo(0)1289 4833 y Fq(\()p Fp(z)t Fq(\)\()p Fp(V)g(R)1605 4848 y Fo(0)1645 4833 y Fq(\()p Fp(z)t Fq(\)\))1808 4792 y Fm(j)1867 4833 y Fj(\000)h Fp(R)2041 4848 y Fo(0)2080 4833 y Fq(\()p Fp(z)2167 4792 y Fi(0)2191 4833 y Fq(\)\()p Fp(V)f(R)2420 4848 y Fo(0)2459 4833 y Fq(\()p Fp(z)2546 4792 y Fi(0)2570 4833 y Fq(\)\))2646 4792 y Fm(j)2683 4833 y Fq(])p Fp(u)27 b Fq(=)931 5261 y(=)1090 5132 y Fm(j)1035 5166 y Fh(X)1051 5378 y Fm(l)q Fo(=0)1179 5261 y Fq([)p Fp(R)1280 5276 y Fo(0)1319 5261 y Fq(\()p Fp(z)t Fq(\)\()p Fp(V)22 b(R)1635 5276 y Fo(0)1675 5261 y Fq(\()p Fp(z)t Fq(\)\))1838 5220 y Fm(l)1865 5261 y Fq(][)p Fp(R)1993 5276 y Fo(0)2033 5261 y Fq(\()p Fp(z)2120 5220 y Fi(0)2144 5261 y Fq(\)\()p Fp(V)f(R)2372 5276 y Fo(0)2412 5261 y Fq(\()p Fp(z)2499 5220 y Fi(0)2523 5261 y Fq(\)\))2599 5220 y Fm(j)t Fi(\000)p Fm(l)2712 5261 y Fq(])p Fp(u:)473 b Fq(\(4.9\))1828 5637 y(16)p eop %%Page: 17 17 17 16 bop 257 573 a Fq(Indeed)34 b(from)e(the)h(resolv)m(en)m(t)g(form) m(ula)398 852 y(\()p Fp(z)27 b Fj(\000)22 b Fp(z)656 811 y Fi(0)681 852 y Fq(\))791 724 y Fm(j)735 758 y Fh(X)751 970 y Fm(l)q Fo(=0)879 852 y Fq([)p Fp(R)980 867 y Fo(0)1020 852 y Fq(\()p Fp(z)t Fq(\)\()p Fp(V)g(R)1336 867 y Fo(0)1376 852 y Fq(\()p Fp(z)t Fq(\)\))1539 811 y Fm(l)1565 852 y Fq(][)p Fp(R)1693 867 y Fo(0)1733 852 y Fq(\()p Fp(z)1820 811 y Fi(0)1844 852 y Fq(\)\()p Fp(V)f(R)2072 867 y Fo(0)2112 852 y Fq(\()p Fp(z)2199 811 y Fi(0)2223 852 y Fq(\)\))2299 811 y Fm(j)t Fi(\000)p Fm(l)2412 852 y Fq(])p Fp(u)27 b Fq(=)426 1097 y(=)g(\()p Fp(R)641 1112 y Fo(0)681 1097 y Fq(\()p Fp(z)t Fq(\))c Fj(\000)f Fp(R)1002 1112 y Fo(0)1042 1097 y Fq(\()p Fp(z)1129 1056 y Fi(0)1153 1097 y Fq(\)\))p Fp(V)f(R)1381 1112 y Fo(0)1421 1097 y Fq(\()p Fp(z)1508 1056 y Fi(0)1532 1097 y Fq(\))p Fp(:::V)h(R)1804 1112 y Fo(0)1843 1097 y Fq(\()p Fp(z)1930 1056 y Fi(0)1954 1097 y Fq(\))p Fp(u)g Fq(+)g Fp(R)2242 1112 y Fo(0)2282 1097 y Fq(\()p Fp(z)t Fq(\)\()p Fp(R)2519 1112 y Fo(0)2559 1097 y Fq(\()p Fp(z)t Fq(\))420 1242 y Fj(\000)h Fp(R)594 1257 y Fo(0)633 1242 y Fq(\()p Fp(z)720 1201 y Fi(0)744 1242 y Fq(\)\))p Fp(V)f(R)973 1257 y Fo(0)1013 1242 y Fq(\()p Fp(z)1100 1201 y Fi(0)1124 1242 y Fq(\))p Fp(:::V)f(R)1395 1257 y Fo(0)1435 1242 y Fq(\()p Fp(z)1522 1201 y Fi(0)1546 1242 y Fq(\))p Fp(u)h Fq(+)g Fp(:::)g Fq(+)g Fp(R)2035 1257 y Fo(0)2075 1242 y Fq(\()p Fp(z)t Fq(\))p Fp(V)g(R)2353 1257 y Fo(0)2393 1242 y Fq(\()p Fp(z)t Fq(\))p Fp(:::)p Fq(\()p Fp(R)2711 1257 y Fo(0)2751 1242 y Fq(\()p Fp(z)t Fq(\))h Fj(\000)g Fp(R)3073 1257 y Fo(0)3112 1242 y Fq(\()p Fp(z)3199 1201 y Fi(0)3223 1242 y Fq(\)\))p Fp(u)426 1397 y Fq(=)k Fp(R)603 1412 y Fo(0)643 1397 y Fq(\()p Fp(z)t Fq(\)\()p Fp(V)22 b(R)959 1412 y Fo(0)999 1397 y Fq(\()p Fp(z)t Fq(\)\))1162 1356 y Fm(j)1221 1397 y Fj(\000)h Fp(R)1395 1412 y Fo(0)1434 1397 y Fq(\()p Fp(z)1521 1356 y Fi(0)1545 1397 y Fq(\)\()p Fp(V)f(R)1774 1412 y Fo(0)1813 1397 y Fq(\()p Fp(z)1900 1356 y Fi(0)1924 1397 y Fq(\)\))2000 1356 y Fm(j)2037 1397 y Fp(u:)257 1606 y Fq(In)m(tegrating)37 b(\(4.9\))g(on)g(t)m(w)m(o)h(con)m(tours)g (\000)p Fp(;)88 b Fq(\000)1968 1570 y Fi(0)2027 1606 y Fj(\032)36 b Fp(\032)p Fq(\()p Fp(H)2309 1621 y Fo(0)2349 1606 y Fq(\))p Fp(;)88 b Fq(\000)2563 1570 y Fi(0)2622 1606 y Fj(\032)36 b Fp(I)8 b(nt)p Fq(\000)38 b(enclosing)e Fp(\033)3455 1621 y Fo(0)257 1726 y Fq(one)d(obtains)1465 2005 y Fp(E)1537 2020 y Fm(j)1574 2005 y Fp(u)27 b Fq(=)1816 1877 y Fm(j)1761 1911 y Fh(X)1777 2123 y Fm(l)q Fo(=0)1921 2005 y Fp(E)1993 2020 y Fm(l)2019 2005 y Fp(E)2091 2020 y Fm(j)t Fi(\000)p Fm(l)2205 2005 y Fp(u:)958 b Fq(\(4.10\))257 2309 y(F)-8 b(rom)31 b Fp(H)594 2324 y Fo(0)634 2309 y Fp(R)708 2324 y Fo(0)747 2309 y Fq(\()p Fp(z)t Fq(\))e(=)e(1)22 b(+)g Fp(z)t(R)1296 2324 y Fo(0)1337 2309 y Fq(\()p Fp(z)t Fq(\),)33 b(again)e(on)h Fp(u)c Fj(2)g(C)2153 2273 y Fi(1)2147 2333 y Fo(0)2228 2309 y Fq(\()p Fn(R)2332 2273 y Fo(3)2377 2309 y Fq(\))652 2518 y Fp(H)733 2533 y Fo(0)772 2518 y Fp(R)846 2533 y Fo(0)886 2518 y Fq(\()p Fp(z)t Fq(\)\()p Fp(V)22 b(R)1202 2533 y Fo(0)1242 2518 y Fq(\()p Fp(z)t Fq(\)\))1405 2477 y Fm(j)1442 2518 y Fp(V)g(R)1595 2533 y Fo(0)1634 2518 y Fq(\()p Fp(z)t Fq(\))p Fp(u)h Fj(\000)f Fp(R)2011 2533 y Fo(0)2051 2518 y Fq(\()p Fp(z)t Fq(\)\()p Fp(V)g(R)2367 2533 y Fo(0)2407 2518 y Fq(\()p Fp(z)t Fq(\)\))2570 2477 y Fm(j)2607 2518 y Fp(V)f(R)2759 2533 y Fo(0)2799 2518 y Fq(\()p Fp(z)t Fq(\))p Fp(H)3005 2533 y Fo(0)3045 2518 y Fp(u)680 2672 y Fq(=)27 b(\()p Fp(V)22 b(R)974 2687 y Fo(0)1014 2672 y Fq(\()p Fp(z)t Fq(\)\))1177 2631 y Fm(j)1214 2672 y Fp(V)f(R)1366 2687 y Fo(0)1406 2672 y Fq(\()p Fp(z)t Fq(\))p Fp(u)h Fj(\000)h Fp(R)1783 2687 y Fo(0)1823 2672 y Fq(\()p Fp(z)t Fq(\)\()p Fp(V)f(R)2139 2687 y Fo(0)2178 2672 y Fq(\()p Fp(z)t Fq(\)\))2341 2631 y Fm(j)2378 2672 y Fp(V)g(u)257 2881 y Fq(whic)m(h)33 b(after)g(the)g(in)m(tegration)e(on)h(\000)h(giv)m(es) 1373 3090 y([)p Fp(H)1481 3105 y Fo(0)1521 3090 y Fp(;)17 b(E)1637 3105 y Fm(j)t Fo(+1)1763 3090 y Fq(])p Fp(u)28 b Fq(=)f Fj(\000)p Fq([)p Fp(V)5 b(;)17 b(E)2259 3105 y Fm(j)2297 3090 y Fq(])p Fp(u)257 3299 y Fq(and)33 b(the)g(pro)s(of)f (is)g(\014nished.)404 3419 y(Consider)k(no)m(w)h(the)g(case)g(when)h Fp(\033)1717 3434 y Fo(0)1790 3419 y Fq(=)c Fj(f)p Fp(\025)2007 3434 y Fo(0)2046 3419 y Fj(g)i Fq(with)g Fp(\025)2415 3434 y Fo(0)2491 3419 y Fq(a)g(\014nite)g(degenerate,)i(iso-)257 3540 y(lated)32 b(eigen)m(v)-5 b(alue)32 b(of)g Fp(H)1159 3555 y Fo(0)1199 3540 y Fq(:)257 3749 y Fp(dist)p Fq(\()p Fp(\025)517 3764 y Fo(0)557 3749 y Fp(;)17 b(\033)t Fq(\()p Fp(H)779 3764 y Fo(0)818 3749 y Fq(\))22 b Fn(r)g Fp(\025)1034 3764 y Fo(0)1074 3749 y Fq(\))27 b(=)h Fp(d)1294 3764 y Fo(0)1361 3749 y Fp(>)f Fq(0)p Fp(;)72 b(dim)28 b(P)1872 3764 y Fo(0)1939 3749 y Fq(=)g Fp(m)g(<)f Fj(1)p Fp(;)44 b Fq(\000)28 b(=)f Fj(f)p Fp(z)33 b Fj(jj)27 b Fp(\025)2890 3764 y Fo(0)2951 3749 y Fj(\000)c Fp(z)32 b Fj(j)p Fq(=)27 b Fp(d)3310 3764 y Fo(0)3349 3749 y Fp(=)p Fq(2)p Fj(g)p Fp(:)3246 3869 y Fq(\(4.11\))257 4077 y(Then)34 b(b)m(y)g(the)f (residue)g(theorem)f([14])581 4303 y Fp(E)653 4318 y Fm(j)689 4303 y Fp(u)27 b Fq(=)h(\()p Fj(\000)p Fq(1\))1078 4262 y Fm(j)t Fi(\000)p Fo(1)1595 4208 y Fh(X)1221 4440 y Fm(\027)1256 4449 y Fd(1)1291 4440 y Fm(;\027)1346 4449 y Fd(2)1380 4440 y Fm(:::\027)1475 4450 y Fg(j)s Fd(+1)1583 4440 y Fo(;)p Fm(\027)1638 4452 y Fg(l)1661 4440 y Fi(\025)p Fo(0;)1771 4390 y Fc(P)1845 4407 y Fg(j)s Fd(+1)1845 4464 y Fg(l)p Fd(=1)1967 4440 y Fm(\027)2002 4452 y Fg(l)2026 4440 y Fo(=)p Fm(j)2129 4303 y Fp(S)2195 4262 y Fm(\027)2230 4271 y Fd(1)2269 4303 y Fp(V)21 b(S)2413 4262 y Fm(\027)2448 4271 y Fd(2)2486 4303 y Fp(V)5 b(:::V)23 b(S)2775 4262 y Fm(\027)2810 4272 y Fg(j)s Fd(+1)3246 4303 y Fq(\(4.12\))257 4640 y(where)34 b Fp(S)k Fq(is)33 b(the)g(reduced)h(resolv)m(en)m(t)f(at)g Fp(\025)1853 4655 y Fo(0)1892 4640 y Fq(:)1257 4887 y Fp(S)g Fq(=)1501 4820 y Fp(i)p 1464 4864 108 4 v 1464 4955 a Fq(2)p Fp(\031)1598 4751 y Fh(I)1653 4977 y Fo(\000)1718 4887 y Fp(R)1792 4902 y Fo(0)1832 4887 y Fq(\()p Fp(z)t Fq(\)\()p Fp(\025)2052 4902 y Fo(0)2114 4887 y Fj(\000)22 b Fp(z)t Fq(\))2300 4846 y Fi(\000)p Fo(1)2395 4887 y Fp(dz)755 b Fq(\(4.13\))257 5143 y(and)33 b(b)m(y)h(con)m(v)m(en)m(tion)1655 5352 y Fp(S)1721 5311 y Fo(0)1788 5352 y Fq(=)27 b Fj(\000)p Fp(P)2031 5367 y Fo(0)2071 5352 y Fp(:)1148 b Fq(\(4.14\))1828 5637 y(17)p eop %%Page: 18 18 18 17 bop 257 573 a Fq(Notice)33 b(that)f(from)f(\(4.5\))h(and)h (\(4.13\))1026 793 y Fj(k)27 b Fp(e)1148 752 y Fm(\015)t(<)p Fi(\001)p Fm(>)1322 793 y Fp(S)6 b(e)1433 752 y Fi(\000)p Fm(\015)t(<)p Fi(\001)p Fm(>)1689 793 y Fj(k\024)28 b Fp(M)5 b(;)115 b Fq(all)95 b Fp(\015)33 b Fj(2)28 b Fq([0)p Fp(;)17 b(\015)2634 808 y Fo(0)2673 793 y Fq(])p Fp(:)519 b Fq(\(4.15\))257 1013 y(The)41 b(main)c(p)s(oin)m(t)h(ab)s(out)h (\(4.12\))g(is)f(that)h(all)e(terms)j(in)e(its)h(r.h.s.)64 b(con)m(tain)39 b(at)f Fl(le)-5 b(ast)257 1133 y Fq(one)41 b Fp(P)507 1148 y Fo(0)546 1133 y Fq(;)j(this)39 b(together)i(with)e (the)i(deca)m(y)g(prop)s(ery)g(\(4.4\))e(of)h Fp(P)2683 1148 y Fo(0)2762 1133 y Fq(and)g(\(4.15\))g(whic)m(h)257 1254 y(allo)m(ws)35 b(to)g(propagate)g(this)f(deca)m(y)j(in)e(order)g (to)g(con)m(trol)g(the)g(un)m(b)s(oundedness)k(o)s(d)c Fp(V)257 1374 y Fq(implies)26 b(that)h Fp(E)862 1389 y Fm(j)927 1374 y Fq(de\014ned)i(initially)24 b(on)k Fj(C)1803 1338 y Fi(1)1797 1399 y Fo(0)1878 1374 y Fq(\()p Fn(R)1982 1338 y Fo(3)2027 1374 y Fq(\))g(are)g(b)s(ounded)h(and)f (their)f(extensions)257 1494 y(b)m(y)37 b(con)m(tin)m(uit)m(y)g(are)f (giv)m(en)g(b)m(y)h(\(4.12\))f(and)g(satisfy)g(\(3.5\)-\(3.6\).)53 b(More)37 b(precisely)f(w)m(e)257 1615 y(ha)m(v)m(e)257 1818 y Fk(Theorem)h(4.3.)50 b Fl(L)-5 b(et)41 b Fp(\014)6 b(;)56 b(\015)1325 1833 y Fo(0)1364 1818 y Fp(;)g(d)1498 1833 y Fo(0)1578 1818 y Fl(as)41 b(given)g(by)g(\(4.3)g(-4.5\))f(and)h (\(4.11\).)63 b(Then)40 b(the)257 1939 y(op)-5 b(er)g(ators)35 b Fp(E)753 1954 y Fm(j)790 1939 y Fp(;)72 b(j)33 b Fq(=)28 b(0)p Fp(;)17 b Fq(1)p Fp(;)g Fq(2)p Fp(;)g(:::)33 b Fl(de\014ne)-5 b(d)34 b(by)h(\(4.12\))f(have)g(the)h(fol)5 b(lowing)34 b(pr)-5 b(op)g(erties:)1682 2159 y Fp(E)1754 2174 y Fo(0)1821 2159 y Fq(=)28 b Fp(P)1988 2174 y Fo(0)2027 2159 y Fp(;)1192 b Fq(\(4.16\))1277 2478 y Fp(E)1349 2493 y Fm(j)1386 2478 y Fp(L)1452 2437 y Fo(2)1491 2478 y Fq(\()p Fn(R)1595 2437 y Fo(3)1641 2478 y Fq(\))27 b Fj(\032)i(D)s Fq(\()p Fp(H)2011 2493 y Fo(0)2049 2478 y Fq(\))22 b Fj(\\)h(D)s Fq(\()p Fp(V)e Fq(\))p Fp(;)787 b Fq(\(4.17\))1512 2904 y Fp(E)1584 2919 y Fm(j)1649 2904 y Fq(=)1808 2776 y Fm(j)1752 2810 y Fh(X)1768 3022 y Fm(l)q Fo(=0)1913 2904 y Fp(E)1985 2919 y Fm(l)2011 2904 y Fp(E)2083 2919 y Fm(j)t Fi(\000)p Fm(l)2197 2904 y Fp(;)1022 b Fq(\(4.18\))864 3324 y([)p Fp(H)972 3339 y Fo(0)1011 3324 y Fp(;)17 b(E)1127 3339 y Fm(j)1164 3324 y Fq(])p Fp(u)27 b Fq(=)g Fj(\000)p Fq([)p Fp(V)5 b(;)17 b(E)1659 3339 y Fm(j)1697 3324 y Fq(])p Fp(u;)71 b(al)r(l)58 b(u)28 b Fj(2)g(D)s Fq(\()p Fp(H)2424 3339 y Fo(0)2462 3324 y Fq(\))23 b Fj(\\)f(D)s Fq(\()p Fp(V)f Fq(\))p Fp(;)374 b Fq(\(4.19\))1564 3655 y Fj(k)27 b Fp(E)1713 3670 y Fm(j)1778 3655 y Fj(k\024)h Fp(g)1984 3613 y Fm(j)2020 3655 y Fp(j)2066 3613 y Fm(\014)s(j)2145 3655 y Fp(;)1074 b Fq(\(4.20\))1072 3989 y Fj(k)27 b Fq([)p Fp(V)5 b(;)17 b(E)1354 4004 y Fm(j)1391 3989 y Fq(])28 b Fj(k\024)g Fp(d)1652 4004 y Fo(0)1691 3989 y Fp(g)1742 3948 y Fm(j)t Fo(+1)1868 3989 y Fp(j)1914 3948 y Fm(\014)s Fo(\()p Fm(j)t Fo(+1\))2138 3989 y Fp(;)72 b(j)34 b Fq(=)28 b(1)p Fp(;)17 b Fq(2)p Fp(;)g(:::)564 b Fq(\(4.21\))257 4209 y Fl(with)1496 4429 y Fp(g)31 b Fq(=)c(3)p Fp(f)11 b(M)f Fq(\()p Fp(\014)c(=e\015)2133 4444 y Fo(0)2172 4429 y Fq(\))2210 4388 y Fm(\014)3246 4429 y Fq(\(4.22\))257 4649 y Fl(wher)-5 b(e)35 b Fp(f)45 b Fl(is)35 b(the)g(\(absolute\))f(c)-5 b(onstant)35 b(app)-5 b(e)g(aring)33 b(in)i(the)g(ine)-5 b(quality:)1472 4836 y Fm(j)t Fi(\000)p Fo(1)1461 4870 y Fh(X)1477 5082 y Fm(l)q Fo(=1)1800 4897 y Fq(1)p 1631 4942 385 4 v 1631 5033 a Fp(l)1662 5004 y Fo(2)1702 5033 y Fq(\()p Fp(j)28 b Fj(\000)23 b Fp(l)r Fq(\))1977 5004 y Fo(2)2054 4965 y Fj(\024)2183 4897 y Fp(f)p 2169 4942 86 4 v 2169 5033 a(j)2215 5004 y Fo(2)2265 4965 y Fp(:)954 b Fq(\(4.23\))1828 5637 y(18)p eop %%Page: 19 19 19 18 bop 404 573 a Fl(Pr)-5 b(o)g(of)p Fq(.)85 b(\(3.4,3.5,3.6\))46 b(can)h(b)s(e)g(read)g(directly)f(from)f(\(4.12\),)50 b(Prop)s(osition)44 b(3.1,)257 693 y(and)38 b(Lemma3.1.)56 b(Consider)37 b(no)m(w)h(the)g(estimations)e(\(4.20\),)i(\(4.21\))e(.) 58 b(One)37 b(can)h(see)257 814 y(directly)32 b(that)h(eac)m(h)g(term)f (in)g(\(4.12\))g(is)g(b)s(ounded)h(b)m(y)1459 1034 y Fp(M)1563 992 y Fm(j)t Fo(+1)1719 1034 y Fj(k)27 b Fp(V)1875 992 y Fm(j)1911 1034 y Fp(e)1956 992 y Fi(\000)p Fm(\015)2047 1001 y Fd(0)2082 992 y Fm(<)p Fi(\001)p Fm(>)2243 1034 y Fj(k)257 1254 y Fq(so)33 b(that)1156 1474 y Fj(k)27 b Fp(E)1305 1489 y Fm(j)1370 1474 y Fj(k\024)h Fp(N)10 b Fq(\()p Fp(j)c Fq(\))p Fp(M)1839 1433 y Fm(j)t Fo(+1)1994 1474 y Fj(k)28 b Fp(V)2150 1433 y Fm(j)2187 1474 y Fp(e)2232 1433 y Fi(\000)p Fm(\015)2323 1442 y Fd(0)2358 1433 y Fm(<)p Fi(\001)p Fm(>)2519 1474 y Fj(k)677 b Fq(\(4.24\))257 1694 y(where)29 b Fp(N)10 b Fq(\()p Fp(j)c Fq(\))28 b(is)e(the)i(n)m (um)m(b)s(er)g(of)f(terms)g(in)f(the)i(r.h.s.)43 b(of)26 b(\(4.12\).)41 b(Unfortunately)-8 b(,)28 b(due)257 1814 y(to)f(the)g(fact)g(that)f Fp(N)10 b Fq(\()p Fp(j)c Fq(\))27 b(is)g(at)f(least)g(of)h(order)g Fp(j)6 b Fq(!)26 b(\(4.24\))g(is)h(m)m (uc)m(h)g(w)m(eak)m(er)i(than)e(the)g(ex-)257 1934 y(p)s(ected)j (estimation;)e(for)h(example)f(if)g Fp(\014)33 b Fq(=)28 b(1)g(one)h(exp)s(ects)i Fj(k)d Fp(E)2615 1949 y Fm(j)2679 1934 y Fj(k\024)g Fp(const:e)3134 1898 y Fm(j)3171 1934 y Fp(j)6 b Fq(!)29 b(while)257 2055 y(\(4.24\))39 b(giv)m(es)g(at)g(b)s (est)g Fp(const:e)1432 2019 y Fm(j)1469 2055 y Fq(\()p Fp(j)6 b Fq(!\))1618 2019 y Fo(2)1658 2055 y Fq(.)62 b(W)-8 b(e)40 b(shall)d(pro)m(v)m(e)k(\(4.20,4.21\))c(b)m(y)j (recurrence)257 2175 y(using)33 b(the)g(iden)m(tit)m(y)f(\(4.18\))g(in) g(the)h(form:)1203 2490 y Fp(E)1275 2505 y Fm(j)1339 2490 y Fq(=)28 b Fp(P)1506 2505 y Fo(0)1545 2490 y Fp(E)1617 2505 y Fm(j)1676 2490 y Fq(+)22 b Fp(E)1846 2505 y Fm(j)1883 2490 y Fp(P)1946 2505 y Fo(0)2007 2490 y Fq(+)2116 2362 y Fm(j)t Fi(\000)p Fo(1)2105 2396 y Fh(X)2121 2608 y Fm(l)q Fo(=1)2266 2490 y Fp(E)2338 2505 y Fm(l)2364 2490 y Fp(E)2436 2505 y Fm(j)t Fi(\000)p Fm(l)3246 2490 y Fq(\(4.25\))257 2795 y(and)33 b(taking)g(the)g(adv)-5 b(an)m(tage)33 b(of)g(the)g(extra)h Fp(P)1974 2810 y Fo(0)2046 2795 y Fq(in)e Fp(P)2223 2810 y Fo(0)2262 2795 y Fp(E)2334 2810 y Fm(j)2404 2795 y Fq(and)h Fp(E)2666 2810 y Fm(j)2703 2795 y Fp(P)2766 2810 y Fo(0)2838 2795 y Fq(whic)m(h)h(allo)m(ws)e(to)257 2915 y(use)i(\(4.7\))e(instead)g(of) h(\(4.12\).)404 3036 y(F)-8 b(rom)1010 3256 y Fj(k)27 b Fp(V)1166 3215 y Fm(j)1202 3256 y Fp(e)1247 3215 y Fi(\000)p Fm(\015)t(<)p Fi(\001)p Fm(>)1504 3256 y Fj(k\024)h Fq(sup)1668 3335 y Fm(a>)p Fo(0)1822 3256 y Fp(a)1873 3215 y Fm(\014)s(j)1953 3256 y Fp(e)1998 3215 y Fi(\000)p Fm(\015)2089 3224 y Fd(0)2124 3215 y Fm(a)2193 3256 y Fq(=)g(\()p Fp(\014)6 b(j)g(=e\015)2587 3271 y Fo(0)2625 3256 y Fq(\))2663 3215 y Fm(\014)s(j)3246 3256 y Fq(\(4.26\))257 3526 y(4.4,)33 b(4.5)f(and)g(4.7)h(one)f(obtains)1309 3746 y Fj(k)c Fp(P)1450 3761 y Fo(0)1489 3746 y Fp(E)1561 3761 y Fm(j)1626 3746 y Fj(k)p Fq(=)p Fj(k)f Fp(E)1901 3761 y Fm(j)1938 3746 y Fp(P)2001 3761 y Fo(0)2068 3746 y Fj(k\024)g Fp(\026)2281 3705 y Fm(j)2318 3746 y Fp(j)2364 3705 y Fm(\014)s(j)3246 3746 y Fq(\(4.27\))257 3966 y(where)1532 4186 y Fp(\026)g Fq(=)g Fp(M)10 b Fq(\()p Fp(\014)c(=e\015)2069 4201 y Fo(0)2109 4186 y Fq(\))2147 4145 y Fm(\014)2194 4186 y Fp(:)1025 b Fq(\(4.28\))404 4406 y(In)33 b(particular)d(since)j Fp(E)1286 4421 y Fo(1)1354 4406 y Fq(=)27 b Fp(P)1520 4421 y Fo(0)1560 4406 y Fp(E)1632 4421 y Fo(1)1693 4406 y Fq(+)22 b Fp(E)1863 4421 y Fo(1)1903 4406 y Fp(P)1966 4421 y Fo(0)2038 4406 y Fq(one)33 b(has)1623 4626 y Fj(k)28 b Fp(E)1773 4641 y Fo(1)1840 4626 y Fj(k\024)g Fq(2)p Fp(\026:)1116 b Fq(\(4.29\))257 4846 y(Let)33 b(no)m(w)g Fp(f)44 b Fq(b)s(e)32 b(the)h(constan)m(t)h(app)s(earing)d(in)h(the)h (inequalit)m(y)1214 5027 y Fm(j)t Fi(\000)p Fo(1)1203 5061 y Fh(X)1219 5274 y Fm(l)q Fo(=1)1542 5089 y Fq(1)p 1374 5133 385 4 v 1374 5224 a Fp(l)1405 5196 y Fo(2)1444 5224 y Fq(\()p Fp(j)28 b Fj(\000)23 b Fp(l)r Fq(\))1719 5196 y Fo(2)1796 5156 y Fj(\024)1925 5089 y Fp(f)p 1911 5133 86 4 v 1911 5224 a(j)1957 5196 y Fo(2)2007 5156 y Fq(;)72 b Fp(j)34 b Fq(=)27 b(2)p Fp(;)17 b Fq(3)p Fp(;)g(:::)696 b Fq(\(4.30\))1828 5637 y(19)p eop %%Page: 20 20 20 19 bop 404 573 a Fq(Notice)32 b(that)g Fp(f)39 b Fj(\025)28 b Fq(4.)43 b(W)-8 b(e)33 b(no)m(w)g(pro)m(v)m(e)h(b)m(y)g(induction)d (that)h(for)g Fp(j)i Fq(=)28 b(1)p Fp(;)17 b Fq(2)p Fp(;)g(:::)1429 832 y Fj(k)27 b Fp(E)1578 847 y Fm(j)1643 832 y Fj(k\024)1826 764 y Fq(2)p 1808 809 86 4 v 1808 900 a Fp(j)1854 871 y Fo(2)1903 832 y Fp(A)1976 791 y Fm(j)t Fi(\000)p Fo(1)2103 832 y Fp(\026)2162 791 y Fm(j)2198 832 y Fp(j)2244 791 y Fm(\014)s(j)3246 832 y Fq(\(4.31\))257 1097 y(and)1204 1341 y Fj(k)h Fq([)p Fp(V)5 b(;)17 b(E)1487 1356 y Fm(j)1524 1341 y Fq(])28 b Fj(k\024)1743 1274 y Fq(2)p Fp(d)1843 1289 y Fo(0)p 1743 1318 139 4 v 1770 1409 a Fp(j)1816 1381 y Fo(2)1892 1341 y Fp(A)1965 1300 y Fm(j)t Fi(\000)p Fo(1)2092 1341 y Fp(\026)2151 1300 y Fm(j)t Fo(+1)2278 1341 y Fp(j)2324 1300 y Fm(\014)s Fo(\()p Fm(j)t Fo(+1\))3246 1341 y Fq(\(4.32\))257 1606 y(with)1710 1826 y Fp(A)f Fq(=)h(3)p Fp(f)5 b(:)1203 b Fq(\(4.33\))257 2046 y(Consider)45 b(\014rst)h(\(4.31\).)79 b(F)-8 b(rom)43 b(\(4.25-)g(4.27\))h(and)h (taking)f(in)m(to)g(accoun)m(t)h(that)g(for)257 2185 y Fp(\014)34 b Fj(\025)28 b Fq(0)k(,)h(\()640 2137 y Fm(l)662 2114 y Fg(l)686 2137 y Fo(\()p Fm(j)t Fi(\000)p Fm(l)q Fo(\))850 2114 y Fg(j)s Fb(\000)p Fg(l)p 640 2162 311 4 v 762 2219 a Fm(j)795 2200 y Fg(j)960 2185 y Fq(\))998 2149 y Fm(\014)1073 2185 y Fj(\024)28 b Fq(1)k(one)h(gets)1065 2476 y Fj(k)27 b Fp(E)1214 2491 y Fm(j)1279 2476 y Fj(k\024)h Fq(2)o Fp(\026)1541 2435 y Fm(j)1577 2476 y Fp(j)1623 2435 y Fm(\014)s(j)1725 2476 y Fq(+)1833 2409 y(4)p Fp(f)p 1833 2454 108 4 v 1844 2545 a(j)1890 2516 y Fo(2)1951 2476 y Fp(A)2024 2435 y Fm(j)t Fi(\000)p Fo(2)2150 2476 y Fp(\026)2209 2435 y Fm(j)2246 2476 y Fp(j)2292 2435 y Fm(\014)s(j)1510 2744 y Fq(=)1642 2676 y(2)p 1623 2721 86 4 v 1623 2812 a Fp(j)1669 2783 y Fo(2)1719 2744 y Fp(A)1792 2703 y Fm(j)t Fi(\000)p Fo(1)1919 2744 y Fp(\026)1978 2703 y Fm(j)2014 2744 y Fp(j)2060 2703 y Fm(\014)s(j)2139 2744 y Fq([)2234 2676 y Fp(j)2280 2640 y Fo(2)p 2176 2721 200 4 v 2176 2812 a Fp(A)2249 2783 y Fm(j)t Fi(\000)p Fo(1)2408 2744 y Fq(+)2516 2676 y(2)p Fp(f)p 2516 2721 108 4 v 2533 2812 a(A)2634 2744 y Fq(])p Fp(:)558 b Fq(\(4.34\))404 3014 y(F)-8 b(or)34 b Fp(A)f Fq(=)g(3)p Fp(f)46 b Fq(the)36 b(square)h(brak)m(et)f(in)f(the)h(r.h.s.)54 b(of)35 b(\(4.34\))g(is)g (smaller)e(than)j(one)257 3134 y(for)c(all)f Fp(j)j Fq(=)27 b(2)p Fp(;)17 b Fq(3)p Fp(;)g(:::)32 b Fq(and)h(the)g(pro)s(of)e(of)h (4.31)g(is)g(\014nished.)45 b(Consider)33 b(no)m(w)g(\(4.32\):)433 3545 y([)p Fp(V)5 b(;)17 b(E)638 3560 y Fm(j)675 3545 y Fq(])27 b(=)h([)p Fp(V)5 b(;)17 b(E)1038 3560 y Fm(j)1075 3545 y Fp(P)1138 3560 y Fo(0)1177 3545 y Fq(])22 b(+)g([)p Fp(V)5 b(;)17 b(P)1520 3560 y Fo(0)1560 3545 y Fp(E)1632 3560 y Fm(j)1668 3545 y Fq(])23 b(+)1826 3417 y Fm(j)t Fi(\000)p Fo(1)1816 3451 y Fh(X)1832 3663 y Fm(l)q Fo(=1)1960 3545 y Fq(\([)p Fp(V)5 b(;)17 b(E)2203 3560 y Fm(l)2229 3545 y Fq(])p Fp(E)2328 3560 y Fm(j)t Fi(\000)p Fm(l)2464 3545 y Fq(+)22 b Fp(E)2634 3560 y Fm(l)2660 3545 y Fq([)p Fp(V)5 b(;)17 b(E)2865 3560 y Fm(j)t Fi(\000)p Fm(l)2978 3545 y Fq(]\))p Fp(:)176 b Fq(\(4.35\))257 3850 y(As)33 b(b)s(efore)1165 4070 y Fj(k)28 b Fq([)p Fp(V)5 b(;)17 b(E)1448 4085 y Fm(j)1484 4070 y Fp(P)1547 4085 y Fo(0)1587 4070 y Fq(])28 b Fj(k\024)g Fp(d)1848 4085 y Fo(0)1887 4070 y Fp(\026)1946 4029 y Fm(j)t Fo(+1)2072 4070 y Fq(\()p Fp(j)g Fq(+)22 b(1\))2363 4029 y Fm(\014)s Fo(\()p Fm(j)t Fo(+1\))3246 4070 y Fq(\(4.36\))1426 4510 y Fj(k)28 b Fq([)p Fp(V)5 b(;)17 b(E)1709 4525 y Fo(1)1748 4510 y Fq(])28 b Fj(k\024)g Fq(2)p Fp(d)2058 4525 y Fo(0)2097 4510 y Fp(\026)2156 4469 y Fo(2)2195 4510 y Fq(2)2244 4469 y Fo(2)p Fm(\014)3246 4510 y Fq(\(4.37\))404 4730 y(Indeed,)34 b(for)e(example)458 4993 y Fj(k)c Fp(V)21 b(E)686 5008 y Fm(j)723 4993 y Fp(P)786 5008 y Fo(0)853 4993 y Fj(k)27 b(\024)1075 4925 y Fq(1)p 1046 4970 V 1046 5061 a(2)p Fp(\031)1180 4857 y Fh(Z)1235 5083 y Fo(\000)1311 4993 y Fj(k)h Fp(V)21 b(e)1512 4946 y Fi(\000)1596 4910 y Fg(\015)1628 4925 y Fd(0)p 1577 4931 106 3 v 1577 4972 a Fg(j)s Fd(+1)1693 4946 y Fm(<)p Fi(\001)p Fm(>)1826 4993 y Fp(R)1900 5016 y Fo(0)p Fm(;)1984 4980 y Fg(\015)2016 4995 y Fd(0)p 1965 5001 V 1965 5042 a Fg(j)s Fd(+1)2085 4993 y Fq(\()p Fp(z)t Fq(\))p Fp(:::R)2365 5008 y Fo(0)p Fm(;\015)t Fo(0)2500 4993 y Fq(\()p Fp(z)t Fq(\))p Fp(e)2670 4952 y Fm(\015)2706 4961 y Fd(0)2742 4952 y Fm(<)p Fi(\001)p Fm(>)2876 4993 y Fp(P)2939 5008 y Fo(0)3006 4993 y Fj(kj)27 b Fp(dz)32 b Fj(j)930 5252 y(\024)1046 5185 y Fp(d)1097 5200 y Fo(0)p 1046 5230 91 4 v 1066 5321 a Fq(2)1146 5252 y Fp(M)1250 5211 y Fm(j)t Fo(+1)1405 5252 y Fj(k)27 b Fp(V)1561 5211 y Fm(j)t Fo(+1)1688 5252 y Fp(e)1733 5211 y Fi(\000)p Fm(\015)1824 5220 y Fd(0)1859 5211 y Fm(<)p Fi(\001)p Fm(>)2020 5252 y Fj(k)g Fp(:)1828 5637 y Fq(20)p eop %%Page: 21 21 21 20 bop 257 573 a Fq(Inserting)28 b(in)f(\(4.35\))f(the)i (inequalities)e(\(4.36\),)i(\(4.37\))f(and)g(\(4.32\))g(for)g Fp(j)33 b Fq(=)28 b(2)p Fp(;)17 b(:::;)g(j)g Fj(\000)12 b Fq(1)257 693 y(one)33 b(obtains)681 896 y Fj(k)28 b Fq([)p Fp(V)5 b(;)17 b(E)964 911 y Fm(j)1001 896 y Fq(])27 b Fj(k)h(\024)g Fq(2)p Fp(d)1338 911 y Fo(0)1377 896 y Fp(\026)1436 855 y Fm(j)t Fo(+1)1562 896 y Fq(\()p Fp(j)g Fq(+)22 b(1\))1853 855 y Fm(\014)s Fo(\()p Fm(j)t Fo(+1\))2100 896 y Fq(+)2208 829 y(8)p Fp(d)2308 844 y Fo(0)2347 829 y Fp(f)p 2208 873 198 4 v 2264 965 a(j)2310 936 y Fo(2)2415 896 y Fp(A)2488 855 y Fm(j)t Fi(\000)p Fo(2)2615 896 y Fp(\026)2674 855 y Fm(j)t Fo(+1)2801 896 y Fp(j)2847 855 y Fm(\014)s Fo(\()p Fm(j)t Fo(+1\))1133 1163 y Fj(\024)1248 1096 y Fq(2)p Fp(d)1348 1111 y Fo(0)p 1248 1141 139 4 v 1275 1232 a Fp(j)1321 1203 y Fo(2)1397 1163 y Fp(A)1470 1122 y Fm(j)t Fi(\000)p Fo(1)1597 1163 y Fp(\026)1656 1122 y Fm(j)t Fo(+1)1782 1163 y Fp(j)1828 1122 y Fm(\014)s Fo(\()p Fm(j)t Fo(+1\))2053 1163 y Fq([)2147 1096 y Fp(j)2193 1060 y Fo(2)p 2090 1141 200 4 v 2090 1232 a Fp(A)2163 1203 y Fm(j)t Fi(\000)p Fo(1)2322 1163 y Fq(+)2430 1096 y(2)p Fp(f)p 2430 1141 108 4 v 2447 1232 a(A)2547 1163 y Fq(])257 1417 y(and)33 b(the)g(pro)s(of)f(is)g (\014nished)h(since)g(for)f(all)f Fp(j)i Fq(=)28 b(2)p Fp(;)17 b Fq(3)p Fp(;)g(:::)32 b Fq([)2411 1373 y Fm(j)2444 1350 y Fd(2)p 2363 1394 164 4 v 2363 1451 a Fm(A)2416 1432 y Fg(j)s Fb(\000)p Fd(1)2558 1417 y Fq(+)2666 1373 y Fo(2)p Fm(f)p 2666 1394 77 4 v 2678 1451 a(A)2753 1417 y Fq(])c Fj(\024)g Fq(1)p Fp(:)404 1537 y Fl(R)-5 b(emarks)404 1658 y Fq(4.1.)41 b(By)27 b(Stirling's)e(form)m(ula)g Fp(j)1549 1621 y Fm(\014)s(j)1657 1658 y Fj(\024)j Fp(e)1807 1621 y Fm(\014)s(j)1887 1658 y Fq(\()p Fp(j)6 b Fq(!\))2036 1621 y Fm(\014)2110 1658 y Fq(so)27 b(\(4.20\),)g(\(4.21\))g(can)g(b)s (e)g(rewritten)257 1778 y(as:)1529 1956 y Fj(k)g Fp(E)1678 1971 y Fm(j)1742 1956 y Fj(k\024)32 b Fq(~)-53 b Fp(g)1948 1915 y Fm(j)1984 1956 y Fq(\()p Fp(j)6 b Fq(!\))2133 1915 y Fm(\014)2180 1956 y Fp(;)1039 b Fq(\(4.38\))1258 2204 y Fj(k)27 b Fq([)p Fp(V)5 b(;)17 b(E)1540 2219 y Fm(j)1577 2204 y Fq(])28 b Fj(k\024)g Fp(d)1838 2219 y Fo(0)1880 2204 y Fq(~)-52 b Fp(g)1928 2163 y Fm(j)t Fo(+1)2054 2204 y Fq(\(\()p Fp(j)28 b Fq(+)22 b(1\)!\))2448 2163 y Fm(\014)3246 2204 y Fq(\(4.39\))257 2383 y(with)1508 2561 y(~)-52 b Fp(g)31 b Fq(=)c(3)p Fp(f)11 b(M)f Fq(\()p Fp(\014)c(=\015)2097 2576 y Fo(0)2136 2561 y Fq(\))2174 2520 y Fm(\014)2221 2561 y Fp(:)404 2739 y Fq(4.2.)71 b(As)42 b(already)g(said)f(the)h(result)g(in)f(Theorem)h(4.3.)71 b(can)42 b(b)s(e)g(extended)i(in)d(a)257 2860 y(v)-5 b(ariet)m(y)33 b(of)f(situations)f(as)i(for)f(example)g(for)g(Zeeman)g (e\013ect)1186 3038 y Fp(H)1267 3053 y Fm(")1332 3038 y Fq(=)27 b(\()p Fk(P)22 b Fj(\000)h Fp(")p Fk(a)p Fq(\))1810 2997 y Fo(2)1871 3038 y Fq(+)f Fp(V)2026 3053 y Fo(0)2065 3038 y Fq(;)73 b Fk(P)27 b Fq(=)g Fj(\000)p Fp(i)p Fj(r)257 3217 y Fq(with)33 b(\(4.3\))f(replaced)g(b)m(y)1359 3395 y Fj(j)27 b Fk(a)h Fj(j)g Fq(+)f Fj(j)g(r)c(\001)e Fk(a)29 b Fj(j\024)p Fp(<)e Fk(x)h Fp(>)2319 3354 y Fm(\014)257 3574 y Fq(Also)f(if)g Fp(V)614 3589 y Fo(0)653 3574 y Fq(\()p Fk(x)p Fq(\))h Fj(!)f(1)g Fq(as)g Fj(j)h Fk(x)g Fj(j!)f(1)g Fq(one)g(can)h(replace)f(the)h(deca)m(y)h(factor)e Fp(e)3029 3537 y Fi(\000)p Fm(\015)3120 3546 y Fd(0)3155 3537 y Fm(<)p Fi(\001)p Fm(>)3316 3574 y Fq(b)m(y)h(a)257 3694 y(b)s(etter)34 b(one;)h(e.g.)47 b(in)33 b(the)h("harmonic)e (oscillator")f(case)k Fp(V)2444 3709 y Fo(0)2483 3694 y Fq(\()p Fk(x)p Fq(\))30 b Fj(\030j)f Fk(x)h Fj(j)2899 3658 y Fo(2)2972 3694 y Fq(as)k Fj(j)29 b Fk(x)h Fj(j!)f(1)257 3814 y Fq(one)46 b(can)g(tak)m(e)g(\()g(see)g([12])f(\))h Fp(e)1445 3778 y Fi(\000)p Fm(\015)1536 3787 y Fd(0)1571 3778 y Fm(<)p Fi(\001)p Fm(>)1701 3755 y Fd(2)1784 3814 y Fq(instead)g(of)f Fp(e)2303 3778 y Fi(\000)p Fm(\015)2394 3787 y Fd(0)2429 3778 y Fm(<)p Fi(\001)p Fm(>)2608 3814 y Fq(and)g(in)g(this)g(case)i(the)257 3935 y(asymptotic)30 b(estimations)f(are)h(in)f(terms)i(of)e Fj(k)f Fp(V)2076 3899 y Fm(j)2113 3935 y Fp(e)2158 3899 y Fi(\000)p Fm(\015)2249 3908 y Fd(0)2284 3899 y Fm(<)p Fi(\001)p Fm(>)2414 3875 y Fd(2)2480 3935 y Fj(k)p Fq(.)42 b(Notice)30 b(also)f(that)i(ev)m(en) 257 4055 y(in)h(the)i(case)g(\(4.3\))e(one)h(can)g(allo)m(w)e Fj(j)d Fp(V)22 b Fq(\()p Fk(x)p Fq(\))28 b Fj(j\024)g Fp(e)2066 4019 y Fm(\021)r(<)p Fi(\001)p Fm(>)2271 4055 y Fq(with)k Fp(\021)37 b Fq(su\016cien)m(tly)c(small,)e(but)257 4175 y(in)h(this)g(case)i(only)e(a)g(\014nite)h(n)m(um)m(b)s(er)g(of)f Fp(E)1851 4190 y Fm(j)1920 4175 y Fq(can)h(b)s(e)f(con)m(trolled.)404 4296 y(4.3.There)e(is)f(another)h(w)m(a)m(y)g(of)f(pro)m(ving)g(that)h Fp(E)2216 4311 y Fm(j)2282 4296 y Fq(giv)m(en)f(b)m(y)i(\(4.12\))d (satisfy)i(\(4.18\))257 4416 y(and)52 b(\(4.19\))f(namely)g(starting)f (with)i(a)f(cut-o\013)g(p)s(erturbation)g(,)56 b Fp(V)2905 4431 y Fm(\025)2951 4416 y Fq(\()p Fk(x)p Fq(\))k(=)g Fp(V)21 b Fq(\()p Fk(x)p Fq(\))257 4537 y(if)49 b Fj(j)55 b Fp(V)22 b Fq(\()p Fk(x)p Fq(\))56 b Fj(j\024)h Fp(\025)49 b Fq(and)g(zero)h(orhervise,)k(for)49 b(whic)m(h)h(\(4.18\))e(and)i (\(4.19\))e(are)i(true)257 4657 y(\(since)30 b(the)g(series)f Fp(P)1016 4672 y Fm(";\025)1141 4657 y Fq(=)1245 4582 y Fh(P)1350 4609 y Fi(1)1350 4686 y Fm(j)t Fo(=0)1493 4657 y Fp(")1539 4621 y Fm(j)1576 4657 y Fp(E)1648 4672 y Fm(j;\025)1770 4657 y Fq(is)g(con)m(v)m(ergen)m(t)i(and)f (satis\014es)f Fj(\000)17 b Fp(")p Fq([)p Fp(P)3121 4672 y Fm(";\025)3219 4657 y Fp(;)g(V)3320 4672 y Fm(\025)3365 4657 y Fq(])27 b(=)257 4786 y Fj(\000)p Fq([)p Fp(P)424 4801 y Fm(";\025)522 4786 y Fp(;)17 b(H)647 4801 y Fo(0)686 4786 y Fq(])p Fp(;)100 b(P)903 4801 y Fm(";\025)1041 4786 y Fq(=)41 b Fp(P)1235 4750 y Fo(2)1221 4812 y Fm(";\025)1319 4786 y Fq(\))f(and)h(then)g(taking)e(the)i(limit)c Fp(\025)42 b Fj(!)e(1)p Fq(.)67 b(Y)-8 b(et)41 b(another)257 4907 y(pro)s(of)32 b(can)h(b)s(e)g(found)f(in)g([11].)404 5027 y(The)43 b(pro)s(of)e(of)g(Theorem)i(4.1)e(rests)j(hea)m(vily)d (on)h(exp)s(onen)m(tial)g(deca)m(y)h(of)f(eigen-)257 5147 y(functions)34 b(\(see)g(\(4.4\)\))f(that)g(is)g(on)g(the)h(fact)f (that)g Fp(dim)d(P)2430 5162 y Fo(0)2498 5147 y Fp(<)f Fj(1)p Fq(.)45 b(W)-8 b(e)34 b(consider)g(no)m(w)257 5268 y(the)j(case)h Fp(dim)d(P)907 5283 y Fo(0)980 5268 y Fq(=)g Fj(1)h Fq(when)i(\(4.4\))e(do)s(es)h(not)f(hold)g(true.)56 b(Still,)35 b(as)i(far)f(as)g Fp(\014)k Fj(\024)35 b Fq(1)257 5388 y(\(see)f(\(4.3\)\))e(on)g(can)h(\014nd)g(solutions)f(of) g(\(3.4\)-\(3.6\).)42 b(More)33 b(exactly:)1828 5637 y(21)p eop %%Page: 22 22 22 21 bop 257 573 a Fk(Theorem)37 b(4.4.)50 b Fl(Supp)-5 b(ose)32 b Fp(d)1376 588 y Fo(0)1443 573 y Fq(=)c Fp(dist)p Fq(\()p Fp(\033)1805 588 y Fo(0)1845 573 y Fp(;)17 b(\033)t Fq(\()p Fp(H)2067 588 y Fo(0)2106 573 y Fq(\))i Fn(r)h Fp(\033)2315 588 y Fo(0)2355 573 y Fq(\))27 b Fp(>)h Fq(0)p Fl(,)34 b Fp(D)2718 588 y Fo(0)2785 573 y Fq(=)27 b(sup)3035 596 y Fm(\025;\026)p Fi(2)p Fm(\033)3225 605 y Fd(0)3292 573 y Fp(<)h Fj(1)257 693 y Fl(and)1495 911 y Fq(sup)1482 998 y Ff(x)p Fi(2)p Fe(R)1620 980 y Fd(3)1682 911 y Fj(j)g(r)p Fp(V)21 b Fq(\()p Fk(x)p Fq(\))28 b Fj(j\024)g Fq(1)p Fp(:)975 b Fq(\(4.40\))404 1188 y Fl(i.)44 b Fp(E)580 1203 y Fm(j)652 1188 y Fl(given)34 b(r)-5 b(e)g(cur)g(ently)35 b(by)1682 1406 y Fp(E)1754 1421 y Fo(0)1821 1406 y Fq(=)28 b Fp(P)1988 1421 y Fo(0)2027 1406 y Fp(;)705 1782 y(E)777 1797 y Fm(j)t Fo(+1)931 1782 y Fq(=)1054 1714 y Fp(i)p 1017 1759 108 4 v 1017 1850 a Fq(2)p Fp(\031)1151 1646 y Fh(I)1207 1872 y Fo(\000)1272 1782 y Fp(R)1346 1797 y Fo(0)1385 1782 y Fq(\()p Fp(z)t Fq(\))p Fj(f)p Fp(P)1623 1797 y Fo(0)1663 1782 y Fq([)p Fp(V)5 b(;)17 b(E)1868 1797 y Fm(j)1905 1782 y Fq(])p Fp(Q)2009 1797 y Fo(0)2071 1782 y Fj(\000)22 b Fp(Q)2247 1797 y Fo(0)2287 1782 y Fq([)p Fp(V)5 b(;)17 b(E)2492 1797 y Fm(j)2529 1782 y Fq(])p Fp(P)2619 1797 y Fo(0)2658 1782 y Fj(g)p Fp(R)2782 1797 y Fo(0)2822 1782 y Fq(\()p Fp(z)t Fq(\))p Fp(dz)1029 1989 y Fq(+)22 b Fp(S)1187 2004 y Fm(j)1246 1989 y Fj(\000)h Fq(2)p Fp(P)1458 2004 y Fo(0)1497 1989 y Fp(S)1557 2004 y Fm(j)1593 1989 y Fp(P)1656 2004 y Fo(0)3246 1989 y Fq(\(4.41\))257 2208 y Fl(wher)-5 b(e)1594 2426 y Fp(Q)1671 2441 y Fo(0)1738 2426 y Fq(=)28 b(1)22 b Fj(\000)h Fp(P)2076 2441 y Fo(0)2115 2426 y Fp(;)1104 b Fq(\(4.42\))1479 2851 y Fp(S)1539 2866 y Fm(j)1603 2851 y Fq(=)1715 2722 y Fm(k)r Fo(=)p Fm(j)1706 2756 y Fh(X)1714 2968 y Fm(k)r Fo(=1)1867 2851 y Fp(E)1939 2866 y Fm(k)1982 2851 y Fp(E)2054 2866 y Fm(j)t Fo(+1)p Fi(\000)p Fm(k)3246 2851 y Fq(\(4.43\))257 3153 y Fl(have)35 b(the)f(pr)-5 b(op)g(erties:)1306 3372 y Fp(E)1378 3387 y Fm(j)1415 3372 y Fj(D)s Fq(\()p Fp(V)21 b Fq(\))28 b Fj(\032)g(D)s Fq(\()p Fp(H)1981 3387 y Fo(0)2020 3372 y Fq(\))22 b Fj(\\)g(D)s Fq(\()p Fp(V)g Fq(\))p Fp(;)816 b Fq(\(4.44\))1590 3697 y Fj(k)28 b Fp(E)1740 3712 y Fm(j)1804 3697 y Fj(k\024)g Fp(g)2010 3656 y Fm(j)2046 3697 y Fp(j)6 b Fq(!)p Fp(;)1100 b Fq(\(4.45\))1487 4022 y Fj(k)27 b Fq([)p Fp(V)5 b(;)17 b(E)1769 4037 y Fm(j)t Fi(\000)p Fo(1)1896 4022 y Fq(])28 b Fj(k\024)g Fp(g)2157 3981 y Fm(j)2193 4022 y Fp(j)6 b Fq(!)980 b(\(4.46\))257 4240 y Fl(for)35 b(some)f Fp(g)d(<)d Fj(1)34 b Fl(and)g(satisfy)h (\(3.4\)-\(3.6\).)404 4361 y(ii.)73 b(L)-5 b(et)44 b Fp(P)807 4376 y Fm(")888 4361 y Fl(as)g(given)g(by)g(\(3.14\),)i (\(3.9\)and)d(\(3.22\))g(with)h Fp(\013)i Fq(=)f(1)p Fl(.)73 b(L)-5 b(et)45 b Fp(H)3330 4376 y Fm(")3411 4361 y Fl(b)-5 b(e)257 4481 y(the)35 b(\(unique)f(by)h(F)-7 b(aris-L)i(avine)32 b(the)-5 b(or)g(em)34 b([21]\))g(self-adjoint)f (extension)h(of)g Fp(H)3197 4496 y Fo(0)3257 4481 y Fq(+)21 b Fp(")c(V)257 4601 y Fl(de\014ne)-5 b(d)34 b(on)h Fj(D)s Fq(\()p Fp(H)925 4616 y Fo(0)963 4601 y Fq(\))23 b Fj(\\)f(D)s Fq(\()p Fp(V)f Fq(\))p Fl(.)45 b(Then)1140 4820 y Fp(P)1203 4835 y Fm(")1239 4820 y Fj(D)s Fq(\()p Fp(H)1438 4835 y Fo(0)1477 4820 y Fq(\))22 b Fj(\\)h(D)s Fq(\()p Fp(V)e Fq(\))27 b Fj(\032)i(D)s Fq(\()p Fp(H)2192 4835 y Fo(0)2230 4820 y Fq(\))22 b Fj(\\)h(D)s Fq(\()p Fp(V)e Fq(\))633 b(\(4.47\))257 5038 y Fl(and)35 b(on)f Fj(D)s Fq(\()p Fp(H)785 5053 y Fo(0)824 5038 y Fq(\))22 b Fj(\\)h(D)s Fq(\()p Fp(V)e Fq(\))1297 5300 y Fj(k)27 b Fq([)p Fp(H)1482 5315 y Fm(")1519 5300 y Fp(;)17 b(P)1626 5315 y Fm(")1662 5300 y Fq(])28 b Fj(k\024)1884 5233 y Fq(2)p Fp(\031)p 1882 5278 113 4 v 1882 5369 a(g)20 b(")2021 5300 y Fq(exp)q(\()p Fj(\000)2327 5233 y Fq(1)p 2295 5278 V 2295 5369 a Fp(g)g(")2418 5300 y Fq(\))790 b(\(4.48\))1828 5637 y(22)p eop %%Page: 23 23 23 22 bop 404 573 a Fl(R)-5 b(emark)404 693 y Fq(4.4.)51 b(In)36 b(order)f(to)g(see)i(wh)m(y)g(b)s(ounded)f Fp(E)1954 708 y Fm(j)2026 693 y Fq(should)f(ho)m(w)m(ev)m(er)j(exist)d(consider)h Fp(E)3403 708 y Fm(j)3440 693 y Fp(u)257 814 y Fq(with)d Fp(u)27 b Fj(2)h(C)715 777 y Fi(1)709 838 y Fo(0)790 814 y Fq(\()p Fn(R)894 777 y Fo(3)939 814 y Fq(\))33 b(as)f(giv)m(en)h(b)m(y)h(\(4.7\).)43 b(Then)532 1038 y Fp(E)604 1053 y Fm(j)641 1038 y Fp(u)27 b Fq(=)h Fj(\000)952 971 y Fp(i)p 915 1015 108 4 v 915 1106 a Fq(2)p Fp(\031)1033 1038 y Fq(\()1071 902 y Fh(I)1126 1128 y Fo(\000)1191 1038 y Fp(R)1265 1053 y Fo(0)1305 1038 y Fq(\()p Fp(z)t Fq(\))p Fp(V)22 b(R)1583 1053 y Fo(0)1623 1038 y Fq(\()p Fp(z)t Fq(\))p Fp(dz)t Fq(\))p Fp(u)724 1297 y Fq(=)28 b Fj(\000)952 1229 y Fp(i)p 915 1274 V 915 1365 a Fq(2)p Fp(\031)1033 1297 y Fq(\()1071 1161 y Fh(I)1126 1387 y Fo(\000)1191 1297 y Fp(R)1266 1256 y Fo(2)1265 1321 y(0)1306 1297 y Fq(\()p Fp(z)t Fq(\))p Fp(V)22 b(dz)t Fq(\))p Fp(u)g Fj(\000)1873 1229 y Fp(i)p 1835 1274 V 1835 1365 a Fq(2)p Fp(\031)1953 1297 y Fq(\()1991 1161 y Fh(I)2046 1387 y Fo(\000)2111 1297 y Fp(R)2185 1312 y Fo(0)2225 1297 y Fq(\()p Fp(z)t Fq(\)[)p Fp(V)5 b(;)17 b(R)2557 1312 y Fo(0)2597 1297 y Fq(\()p Fp(z)t Fq(\)])p Fp(dz)t Fq(\))p Fp(u:)276 b Fq(\(4.49\))257 1529 y(No)m(w)34 b(for)e Fp(u)27 b Fj(2)h(C)864 1493 y Fi(1)858 1554 y Fo(0)939 1529 y Fq(\()p Fn(R)1043 1493 y Fo(3)1088 1529 y Fq(\))891 1705 y([)p Fp(V)5 b(;)17 b(R)1098 1720 y Fo(0)1138 1705 y Fq(\()p Fp(z)t Fq(\)])p Fp(u)28 b Fq(=)g Fp(iR)1585 1720 y Fo(0)1625 1705 y Fq(\()p Fp(z)t Fq(\)\()p Fk(P)22 b Fj(\001)g(r)p Fp(V)43 b Fq(+)22 b Fj(r)p Fp(V)44 b Fj(\001)22 b Fk(P)p Fq(\))p Fp(R)2641 1720 y Fo(0)2680 1705 y Fq(\()p Fp(z)t Fq(\))p Fp(u)385 b Fq(\(4.50\))257 1881 y(so)48 b(the)f(op)s(erator)g(in)f(the)i(last)e(term)h(in)f(the)i (r.h.s.)88 b(of)47 b(\(4.49\))f(is)h(b)s(ounded.)88 b(On)257 2002 y(the)54 b(other)f(hand)g(,)58 b(b)m(y)c(residue)g(theorem,)k(the) 53 b(\014rst)h(term)e(in)g(the)i(r.h.)105 b(s.)g(of)257 2122 y(\(4.49\))35 b(v)-5 b(anishes.)54 b(It)36 b(follo)m(ws)e(that)i Fp(E)1692 2137 y Fo(1)1767 2122 y Fq(as)g(giv)m(en)g(b)m(y)h(\(4.49\))e (is)g(b)s(ounded)h(on)g Fj(C)3271 2086 y Fi(1)3265 2147 y Fo(0)3346 2122 y Fq(\()p Fn(R)3450 2086 y Fo(3)257 2242 y Fq(and)j(then)h(it)e(has)i(an)e(b)s(ounded)i(extension)g(b)m(y)g (con)m(tin)m(uit)m(y)-8 b(.)63 b(In)39 b(addition)e(it)h(follo)m(vs)257 2363 y(that)43 b Fp(E)551 2378 y Fo(1)591 2363 y Fp(L)657 2327 y Fo(2)697 2363 y Fq(\()p Fn(R)800 2327 y Fo(3)891 2363 y Fj(\032)j(D)s Fq(\()p Fp(H)1213 2378 y Fo(0)1252 2363 y Fq(\))d(and)g(one)g(can)g(argue)g(that)g(\(see)h(the)f(pro)s(of) f(of)h(Lemma)257 2483 y(4.2\))k(that)h Fp(E)766 2498 y Fo(1)805 2483 y Fj(D)s Fq(\()p Fp(V)21 b Fq(\))53 b Fj(\032)h(D)s Fq(\()p Fp(V)21 b Fq(\).)88 b(Actually)46 b(one)i(can)g(pro)m(v)m(e)h(the)e(existence)j(of)c Fp(E)3458 2498 y Fm(j)257 2604 y Fq("rewriting")32 b(\(4.7\))h(in)f(a)i(form)e (in)g(whic)m(h)i Fp(V)55 b Fq(en)m(ters)35 b(only)e(in)f(m)m(ultiple)f (comm)m(utators)257 2724 y([)p Fp(V)5 b(;)17 b Fq([)p Fp(V)5 b(:::)p Fq([)p Fp(V)g(;)17 b(R)767 2739 y Fo(0)808 2724 y Fq(\()p Fp(z)t Fq(\)])p Fp(:::)p Fq(].)77 b(Instead)44 b(of)f(doing)f(that)h(w)m(e)h(shall)e(use)j(the)e(results)h(in)e([17]) 257 2844 y(ab)s(out)32 b(adiabatic)f(expansions.)404 2965 y Fl(Pr)-5 b(o)g(of)34 b(of)h(The)-5 b(or)g(em)34 b(4.4)404 3085 y Fq(As)43 b(already)f(said)g(Theorem)h(4.4)f(follo)m (ws)f(from)g(the)i(treatemen)m(t)g(of)f(the)h(Stark)257 3206 y(e\013ect)h(in)d([16],)k([20])d(com)m(bined)h(with)f(the)h (adiabatic)d(exp)s(onen)m(tial)i(estimations)f(in)257 3326 y([17],)33 b(so)f(w)m(e)i(shall)d(only)h(outline)g(the)h(main)d(p) s(oin)m(ts.)404 3446 y(Consider)j(the)g(family)d(of)i(self-adjoin)m(t)f (op)s(erators)1004 3622 y Fp(H)8 b Fq(\()p Fp(s)p Fq(\))27 b(=)h Fp(e)1391 3581 y Fm(isV)1508 3622 y Fp(H)1589 3637 y Fo(0)1629 3622 y Fp(e)1674 3581 y Fi(\000)p Fm(isV)1874 3622 y Fq(=)g(\()p Fk(P)21 b Fj(\000)i Fp(is)p Fj(r)p Fp(V)f Fq(\))2493 3581 y Fo(2)2554 3622 y Fq(+)g Fp(V)2709 3637 y Fo(0)3246 3622 y Fq(\(4.51\))257 3798 y(Let)33 b(\000)27 b Fj(\032)h Fp(\032)p Fq(\()p Fp(H)794 3813 y Fo(0)834 3798 y Fq(\))k(b)s(e)g(a)g(con)m(tour)h(of)e(\014nite)h (length)g(enclosing)f Fp(\033)2601 3813 y Fo(0)2641 3798 y Fq(.)44 b(By)32 b(regular)f(p)s(ertur-)257 3919 y(bation)h(theory)-8 b(,)34 b(\()p Fp(H)8 b Fq(\()p Fp(s)p Fq(\))22 b Fj(\000)h Fp(z)t Fq(\))1345 3882 y Fi(\000)p Fo(1)1473 3919 y Fq(is,)33 b(for)g(all)e Fp(z)i Fj(2)c Fq(\000,)34 b(analytic)d(in)i Fp(s)p Fq(,)g(in)g(a)f(strip)h(around)257 4039 y(real)f(axis)h(and)g (uniformly)e(b)s(ounded)i(there)h(so)f(w)m(e)h(can)f(apply)f(the)i (metho)s(ds)e(in)g([17].)257 4159 y(In)26 b(the)g(Lemma)e(b)s(elo)m(w)h (w)m(e)i(collect)d(some)i(results)f(from)g([17)o(])h(in)e(an)i (appropriate)e(form.)257 4325 y Fk(Lemma)37 b(4.5.)50 b Fl([17)o(])404 4445 y(i.)44 b(L)-5 b(et)398 4621 y Fp(P)461 4636 y Fo(0)500 4621 y Fq(\()p Fp(s)p Fq(\))27 b(=)h Fp(e)798 4580 y Fm(isV)916 4621 y Fp(P)979 4636 y Fo(0)1018 4621 y Fp(e)1063 4580 y Fi(\000)p Fm(isV)1236 4621 y Fp(;)116 b(R)q Fq(\()p Fp(s)p Fq(;)17 b Fp(z)t Fq(\))28 b Fj(\021)g Fq(\()p Fp(H)8 b Fq(\()p Fp(s)p Fq(\))22 b Fj(\000)g Fp(z)t Fq(\))2259 4580 y Fi(\000)p Fo(1)2382 4621 y Fq(=)27 b Fp(e)2530 4580 y Fm(isV)2648 4621 y Fq(\()p Fp(H)2767 4636 y Fo(0)2829 4621 y Fj(\000)22 b Fp(z)t Fq(\))3015 4580 y Fi(\000)p Fo(1)3110 4621 y Fp(e)3155 4580 y Fi(\000)p Fm(isV)3328 4621 y Fp(:)3246 4742 y Fq(\(4.52\))257 4916 y Fl(Then)34 b Fp(E)583 4931 y Fm(j)620 4916 y Fq(\()p Fp(s)p Fq(\))h Fl(given)f(by)h(the)g(r)-5 b(e)g(c)g(cur)g(enc)g(e)34 b(formula)271 5145 y Fp(E)343 5160 y Fm(j)t Fo(+1)470 5145 y Fq(\()p Fp(s)p Fq(\))27 b(=)742 5077 y Fp(i)p 705 5122 V 705 5213 a Fq(2)p Fp(\031)839 5009 y Fh(I)895 5235 y Fo(\000)960 5145 y Fp(R)q Fq(\()p Fp(s)p Fq(;)17 b Fp(z)t Fq(\))p Fj(f)p Fp(P)1363 5160 y Fo(0)1402 5145 y Fq(\()p Fp(s)p Fq(\))1557 5077 y Fp(d)p 1534 5122 97 4 v 1534 5213 a(ds)1640 5145 y(E)1712 5160 y Fm(j)1749 5145 y Fq(\()p Fp(s)p Fq(\))p Fp(Q)1948 5160 y Fo(0)1988 5145 y Fq(\()p Fp(s)p Fq(\))22 b Fj(\000)g Fp(Q)2308 5160 y Fo(0)2348 5145 y Fq(\()p Fp(s)p Fq(\))2503 5077 y Fp(d)p 2480 5122 V 2480 5213 a(ds)2586 5145 y(E)2658 5160 y Fm(j)2695 5145 y Fq(\()p Fp(s)p Fq(\))p Fp(P)2880 5160 y Fo(0)2919 5145 y Fq(\()p Fp(s)p Fq(\))p Fj(g)p Fp(R)q Fq(\()p Fp(s)p Fq(;)17 b Fp(z)t Fq(\))p Fp(dz)717 5352 y Fq(+)22 b Fp(S)875 5367 y Fm(j)912 5352 y Fq(\()p Fp(s)p Fq(\))g Fj(\000)g Fq(2)p Fp(P)1267 5367 y Fo(0)1307 5352 y Fq(\()p Fp(s)p Fq(\))p Fp(S)1489 5367 y Fm(j)1525 5352 y Fq(\()p Fp(s)p Fq(\))p Fp(P)1710 5367 y Fo(0)1749 5352 y Fq(\()p Fp(s)p Fq(\))1375 b(\(4.53\))1828 5637 y(23)p eop %%Page: 24 24 24 23 bop 257 573 a Fl(wher)-5 b(e)1472 786 y Fp(Q)1549 801 y Fo(0)1589 786 y Fq(\()p Fp(s)p Fq(\))27 b(=)h(1)22 b Fj(\000)h Fp(P)2076 801 y Fo(0)2115 786 y Fq(\()p Fp(s)p Fq(\))p Fp(;)982 b Fq(\(4.54\))1291 1201 y Fp(S)1351 1216 y Fm(j)1388 1201 y Fq(\()p Fp(s)p Fq(\))27 b(=)1641 1072 y Fm(k)r Fo(=)p Fm(j)t Fi(\000)p Fo(1)1677 1106 y Fh(X)1685 1318 y Fm(k)r Fo(=1)1873 1201 y Fp(E)1945 1216 y Fm(k)1988 1201 y Fq(\()p Fp(s)p Fq(\))p Fp(E)2182 1216 y Fm(j)t Fi(\000)p Fm(k)2312 1201 y Fq(\()p Fp(s)p Fq(\))p Fp(:)785 b Fq(\(4.55\))257 1498 y Fl(ar)-5 b(e)35 b(the)g(unique)g(solution)f(of)1252 1712 y Fp(E)1324 1727 y Fo(0)1364 1712 y Fq(\()p Fp(s)p Fq(\))27 b(=)h Fp(e)1662 1671 y Fm(isV)1780 1712 y Fp(P)1843 1727 y Fo(0)1882 1712 y Fp(e)1927 1671 y Fi(\000)p Fm(isV)2127 1712 y Fj(\021)h Fp(P)2296 1727 y Fo(0)2335 1712 y Fq(\()p Fp(s)p Fq(\))p Fp(;)762 b Fq(\(4.56\))1284 2078 y Fp(i)1350 2011 y(d)p 1327 2055 97 4 v 1327 2147 a(ds)1434 2078 y(E)1506 2093 y Fm(j)1543 2078 y Fq(\()p Fp(s)p Fq(\))27 b(=)h([)p Fp(H)8 b Fq(\()p Fp(s)p Fq(\))p Fp(;)17 b(E)2150 2093 y Fm(j)t Fo(+1)2276 2078 y Fq(\()p Fp(s)p Fq(\)])p Fp(;)794 b Fq(\(4.57\))1313 2525 y Fp(E)1385 2540 y Fm(j)1422 2525 y Fq(\()p Fp(s)p Fq(\))27 b(=)1684 2397 y Fm(k)r Fo(=)p Fm(j)1675 2431 y Fh(X)1682 2643 y Fm(k)r Fo(=0)1835 2525 y Fp(E)1907 2540 y Fm(k)1950 2525 y Fq(\()p Fp(s)p Fq(\))p Fp(E)2144 2540 y Fm(j)t Fi(\000)p Fm(k)2274 2525 y Fq(\()p Fp(s)p Fq(\))p Fp(;)823 b Fq(\(4.58\))257 2823 y Fl(ar)-5 b(e)35 b(norm)f(di\013er)-5 b(entiable,)34 b(and)g(satisfy)1365 3036 y Fp(E)1437 3051 y Fm(j)1474 3036 y Fq(\()p Fp(s)p Fq(\))p Fp(L)1662 2995 y Fo(2)1702 3036 y Fq(\()p Fn(R)1805 2995 y Fo(3)1851 3036 y Fq(\))27 b Fj(\032)i(D)s Fq(\()p Fp(H)8 b Fq(\()p Fp(s)p Fq(\)\))857 b(\(4.59\))906 3403 y Fj(k)28 b Fp(E)1056 3418 y Fm(j)1093 3403 y Fq(\()p Fp(s)p Fq(\))f Fj(k\024)h Fp(g)1448 3362 y Fm(j)1484 3403 y Fp(j)6 b Fq(!)p Fp(;)116 b Fj(k)1811 3335 y Fp(d)p 1788 3380 V 1788 3471 a(ds)1894 3403 y(E)1966 3418 y Fm(j)2003 3403 y Fq(\()p Fp(s)p Fq(\))28 b Fj(k\024)g Fp(g)2359 3362 y Fm(j)t Fo(+1)2485 3403 y Fq(\()p Fp(j)g Fq(+)22 b(1\)!)p Fp(;)416 b Fq(\(4.60\))257 3641 y Fl(for)35 b(some)f Fp(g)d(<)d Fj(1)404 3762 y Fl(ii.)45 b(L)-5 b(et)36 b Fp(P)771 3777 y Fm(")807 3762 y Fq(\()p Fp(s)p Fq(\))f Fl(as)g(given)f(by)i(\(3.14\),)e(\(3.9\)and)g(\(3.22\))g(with)h Fp(\013)29 b Fq(=)f(1)p Fl(.)45 b(Then)35 b Fp(P)3337 3777 y Fm(")3373 3762 y Fq(\()p Fp(s)p Fq(\))257 3882 y Fl(is)g(norm)f(di\013er)-5 b(entiable)1370 4095 y Fp(P)1433 4110 y Fm(")1469 4095 y Fq(\()p Fp(s)p Fq(\))p Fp(L)1657 4054 y Fo(2)1697 4095 y Fq(\()p Fn(R)1801 4054 y Fo(3)1846 4095 y Fq(\))28 b Fj(\032)g(D)s Fq(\()p Fp(H)8 b Fq(\()p Fp(s)p Fq(\)\))862 b(\(4.61\))257 4308 y Fl(and)35 b(ther)-5 b(e)35 b(exists)f Fp(")1002 4323 y Fo(0)1069 4308 y Fp(>)27 b Fq(0)35 b Fl(such)g(that)g(for)g Fp(")27 b Fj(2)h Fq([0)p Fp(;)17 b(")2165 4323 y Fo(0)2205 4308 y Fq(])p Fl(:)915 4571 y Fj(k)27 b Fp(")1088 4504 y(d)p 1065 4548 V 1065 4639 a(ds)1171 4571 y(P)1234 4586 y Fm(")1271 4571 y Fq(\()p Fp(s)p Fq(\))22 b Fj(\000)g Fq([)p Fp(H)8 b Fq(\()p Fp(s)p Fq(\))p Fp(;)17 b(P)1859 4586 y Fm(")1895 4571 y Fq(\()p Fp(s)p Fq(\)])28 b Fj(k\024)2269 4504 y Fq(1)p 2237 4548 113 4 v 2237 4639 a Fp(g)20 b(")2376 4571 y Fq(exp)q(\()p Fj(\000)2682 4504 y Fq(1)p 2650 4548 V 2650 4639 a Fp(g)g(")2773 4571 y Fq(\))p Fp(:)408 b Fq(\(4.62\))404 4841 y(Let)48 b Fp(A)h Fq(b)s(e)f(a)g(b)s(ounded)h(op)s(erator,)j(suc)m (h)e(that)e Fp(A)p Fq(\()p Fp(s)p Fq(\))54 b(=)g Fp(e)2702 4805 y Fm(isV)2820 4841 y Fp(Ae)2938 4805 y Fi(\000)p Fm(isV)3159 4841 y Fq(is)48 b(norm)257 4962 y(di\013eren)m(tiable.)42 b(Then)31 b(from)d(Stone's)j(theorem)f(for)f Fp(f)38 b Fj(2)28 b(D)s Fq(\()p Fp(V)22 b Fq(\))29 b(one)h(has)h Fp(Af)38 b Fj(2)28 b(D)s Fq(\()p Fp(V)21 b Fq(\))257 5082 y(and)1254 5320 y Fp(i)1320 5252 y(d)p 1297 5297 97 4 v 1297 5388 a(ds)1404 5320 y(A)p Fq(\()p Fp(s)p Fq(\))p Fp(f)38 b Fq(=)27 b Fp(e)1833 5279 y Fm(isV)1951 5320 y Fq([)p Fp(A;)17 b(V)22 b Fq(])p Fp(e)2246 5279 y Fi(\000)p Fm(isV)2418 5320 y Fp(f)5 b(:)748 b Fq(\(4.63\))1828 5637 y(24)p eop %%Page: 25 25 25 24 bop 257 573 a Fq(Notice)42 b(that)g([)p Fp(A;)17 b(V)22 b Fq(])42 b(is)g(b)s(ounded)h(on)f Fj(D)s Fq(\()p Fp(V)21 b Fq(\).)73 b(Since)42 b Fp(E)2417 588 y Fo(0)2457 573 y Fq(\()p Fp(s)p Fq(\))i(=)g Fp(e)2788 537 y Fm(isV)2906 573 y Fp(P)2969 588 y Fo(0)3008 573 y Fp(e)3053 537 y Fi(\000)p Fm(isV)3226 573 y Fq(,)g(from)257 693 y(\(4.53\))32 b(and)h(\(4.63\))f(it)f(follo)m(ws)h(that)g Fp(E)1712 708 y Fm(j)1749 693 y Fq(\()p Fp(s)p Fq(\))g(are)h(of)f(the)h(form)1451 913 y Fp(E)1523 928 y Fm(j)1560 913 y Fq(\()p Fp(s)p Fq(\))27 b(=)h Fp(e)1858 872 y Fm(isV)1975 913 y Fp(E)2047 928 y Fm(j)2084 913 y Fp(e)2129 872 y Fi(\000)p Fm(isV)3246 913 y Fq(\(4.64\))257 1132 y(where)34 b Fp(E)611 1147 y Fm(j)680 1132 y Fq(ha)m(v)m(e)g(all)d(the)i(prop)s(erties)f(stated)h (in)f(Theorem)h(4.4.i.)404 1253 y(By)g(the)g(same)f(argumen)m(t)g (\(see)i(\(3.14\)\))1017 1472 y Fp(P)1080 1487 y Fm(")1117 1472 y Fq(\()p Fp(s)p Fq(\))27 b(=)h Fp(e)1415 1431 y Fm(isV)1533 1472 y Fp(P)1596 1487 y Fm(")1632 1472 y Fp(e)1677 1431 y Fi(\000)p Fm(isV)1850 1472 y Fp(;)114 b(P)2054 1487 y Fm(")2091 1472 y Fj(D)s Fq(\()p Fp(V)21 b Fq(\))28 b Fj(\032)g(D)s Fq(\()p Fp(V)21 b Fq(\))p Fp(;)527 b Fq(\(4.65\))257 1692 y(and)1226 1936 y Fp(i)17 b(")1371 1868 y(d)p 1348 1913 97 4 v 1348 2004 a(ds)1455 1936 y(P)1518 1951 y Fm(")1555 1936 y Fq(\()p Fp(s)p Fq(\))27 b(=)h Fp(")16 b(e)1915 1895 y Fm(isV)2033 1936 y Fq([)p Fp(P)2123 1951 y Fm(")2159 1936 y Fp(;)h(V)22 b Fq(])p Fp(e)2354 1895 y Fi(\000)p Fm(isV)3246 1936 y Fq(\(4.66\))257 2191 y(whic)m(h)49 b(together)g(with)f Fp(P)1254 2206 y Fm(")1290 2191 y Fp(L)1356 2154 y Fo(2)1396 2191 y Fq(\()p Fn(R)1500 2154 y Fo(3)1545 2191 y Fq(\))55 b Fj(\032)f(D)s Fq(\()p Fp(H)1968 2206 y Fo(0)2007 2191 y Fq(\))48 b(and)h(\(4.23\))e(\014nishes)i(the)g(pro)s(of)f(ot)257 2311 y(Theorem)33 b(4.4.)257 2644 y Fr(5)161 b(Magnetic)44 b(Sc)l(hro)t(dinger)d(op)t(erators:)65 b(gauge)43 b(in-)499 2826 y(v)-9 b(arian)l(t)55 b(p)t(erturbation)e(theory)-13 b(.)257 3045 y Fq(The)36 b(aim)d(of)h(this)g(section)h(\(whic)m(h)g (can)f(b)s(e)h(read)g(indep)s(enden)m(tly)g(of)f(the)h(rest)h(of)e(the) 257 3166 y(pap)s(er)g(\))f(is)g(to)g(p)s(oin)m(t)g(out)g(ho)m(w)h(the)g (gauge)f(co)m(v)-5 b(ariance)33 b(can)h(b)s(e)f(used)i(to)e(con)m(trol) g(the)257 3286 y(singularitiy)k(of)i(p)s(erturbations)g(giv)m(en)g(b)m (y)i(magnetic)d(\014elds)h(whic)m(h)h(do)g(not)f(w)m(anish)257 3406 y(\(or)32 b(deca)m(y)i(v)m(ery)g(slo)m(wly)f(\))f(as)h Fj(j)27 b Fk(x)h Fj(j!)f(1)p Fq(.)43 b(In)33 b(order)f(not)h(to)f (obscure)i(the)f(main)d(ideas)257 3527 y(w)m(e)46 b(shall)e(consider)h (here)h(a)e(particular)g(case:)69 b(a)44 b(particle)g(in)g Fn(R)2745 3491 y Fo(3)2791 3527 y Fq(,)k(sub)5 b(jected)47 b(to)d(a)257 3647 y(magnetic)29 b(\014eld)h Fk(b)p Fq(\()p Fk(x)p Fq(\),)h(and)f(an)h(electric)e(p)s(oten)m(tial)g Fp(V)2257 3662 y Fo(0)2296 3647 y Fq(\()p Fk(x)p Fq(\).)43 b(Also)29 b(the)i(pro)s(ofs)f(of)f(some)257 3768 y(tec)m(hnical)k (Lemmae)e(b)s(elo)m(w)h(will)f(b)s(e)i(giv)m(en)f(at)g(the)h(end)h(of)e (the)h(section.)404 3888 y(Concerning)g(the)g(magnetic)e(\014eld,)h Fk(b)p Fq(\()p Fk(x)p Fq(\),)h(w)m(e)h(supp)s(ose)g(that:)1015 4296 y Fj(j)27 b Fk(b)p Fq(\()p Fk(x)p Fq(\))h Fj(j)f Fq(+)1496 4172 y Fm(d)1443 4202 y Fh(X)1453 4412 y Fm(j)t Fo(=1)1614 4296 y Fj(j)h Fp(@)1721 4311 y Fm(j)1758 4296 y Fk(b)p Fq(\()p Fk(x)p Fq(\))g Fj(j\024)p Fp(<)f Fj(\001)h Fp(>)2323 4255 y Fm(\014)2370 4296 y Fp(;)72 b(\014)33 b Fj(\024)28 b Fq(0)p Fp(:)557 b Fq(\(5.1\))257 4621 y(and)33 b(for)f Fp(V)653 4636 y Fo(0)692 4621 y Fq(\()p Fk(x)p Fq(\))h(as)g(b)s(efore,)f(w)m(e)i(supp)s(ose)g(to)e(b)s(e)h (uniformly)d Fp(L)2572 4585 y Fo(2)2572 4647 y Fm(l)q(oc)2696 4621 y Fq(i.e.)1354 4984 y(sup)1342 5071 y Ff(x)p Fi(2)p Fe(R)1480 5053 y Fd(3)1531 4849 y Fh(Z)1658 4984 y Fj(j)d Fp(V)1770 4999 y Fo(0)1810 4984 y Fq(\()p Fk(y)r Fq(\))g Fj(j)2002 4943 y Fo(2)2069 4984 y Fp(d)p Fk(y)i Fp(<)e Fj(1)884 b Fq(\(5.2\))404 5268 y(As)29 b(w)m(ell)f(kno)m(wn)h([7],)h (for)e(a)g(large)f(class)i(of)f(v)m(ector)i(p)s(oten)m(tials,)e Fk(a)p Fq(\()p Fk(x)p Fq(\),)i(corresp)s(ond-)257 5388 y(ing)i(to)g Fk(b)p Fq(\()p Fk(x)p Fq(\))h(and)g Fp(")27 b Fj(2)h Fq([0)p Fp(;)17 b Fj(1)p Fq(\))32 b(the)h(hamiltonian)1828 5637 y(25)p eop %%Page: 26 26 26 25 bop 1416 790 a Fp(H)1497 805 y Fm(";)p Ff(a)1620 790 y Fq(=)28 b(\()p Fk(P)22 b Fj(\000)g Fp(")p Fk(a)p Fq(\))2098 748 y Fo(2)2160 790 y Fq(+)g Fp(V)980 b Fq(\(5.3\))257 1006 y(is)45 b(essen)m(tially)h(self-adjoin)m(t)d(on)j Fj(C)1570 970 y Fi(1)1564 1031 y Fo(0)1645 1006 y Fq(\()p Fn(R)1749 970 y Fo(3)1794 1006 y Fq(\))g([7])f(and)h(w)m(e)g(shall)f (denote)h(b)m(y)g(the)g(same)257 1127 y(sym)m(b)s(ol)32 b(its)h(self-adjoin)m(t)e(closure.)44 b(As)33 b(b)s(efore,)g(the)h(unp) s(erturb)s(ed)f(hamiltonian)d(\(i.e.)257 1247 y Fp(")e Fq(=)f(0\))32 b(is)h(denoted)g(b)m(y)h Fp(H)1237 1262 y Fo(0)1588 1584 y Fp(H)1669 1599 y Fo(0)1736 1584 y Fq(=)27 b Fk(P)1916 1543 y Fo(2)1977 1584 y Fq(+)22 b Fp(V)5 b(:)1131 b Fq(\(5.4\))404 1801 y(The)34 b(results)g(b)s(elo)m(w) f(can)h(b)s(e)f(extended)j(in)c(man)m(y)i(directions:)44 b(arbitrary)33 b(dimen-)257 1921 y(sions;)46 b(more)40 b(\(lo)s(cally\))e(singular)i Fk(b)p Fq(\()p Fk(x)p Fq(\))h(and)g Fp(V)22 b Fq(\()p Fk(x)p Fq(\))41 b(;)k(man)m(y)c(particle)f(systems;) 47 b(the)257 2042 y(presence)c(of)e(a)g(\014xed)h(magnetic)d(\014eld)i (whic)m(h)h(amoun)m(ts)f(to)f(replace)h(in)f(\(4.4\))h Fk(P)f Fq(b)m(y)257 2162 y(\()p Fk(P)25 b Fj(\000)h Fk(A)585 2177 y Fo(0)624 2162 y Fq(\))38 b(where)h(where)f Fk(A)1358 2177 y Fo(0)1397 2162 y Fq(\()p Fk(x)p Fq(\))g(corresp)s(onds)h(to)e(a) g Fl(\014xe)-5 b(d)37 b Fq(magnetic)f(\014eld,)j Fk(B)3294 2177 y Fo(0)3333 2162 y Fq(\()p Fk(x)p Fq(\);)257 2282 y(Dirac)32 b(op)s(erators)g(etc.)404 2403 y(Supp)s(ose)j(that)f Fp(H)1082 2418 y Fo(0)1156 2403 y Fq(has)h(an)f(isolated)f(and)i(b)s (ounded)g(part,)g Fp(\033)2717 2418 y Fo(0)2757 2403 y Fq(,)g(of)f(the)h(sp)s(ectrum)257 2523 y(i.e.)1065 2740 y Fp(\033)t Fq(\()p Fp(H)1243 2755 y Fo(0)1282 2740 y Fq(\))28 b(=)g Fp(\033)1507 2755 y Fo(0)1569 2740 y Fj([)22 b Fp(\033)1712 2755 y Fo(1)1752 2740 y Fq(;)72 b Fp(dist)p Fq(\()p Fp(\033)2109 2755 y Fo(0)2149 2740 y Fp(;)17 b(\033)2248 2755 y Fo(1)2288 2740 y Fq(\))27 b(=)h Fp(d)f(>)h Fq(0)607 b(\(5.5\))257 2956 y(and)27 b(let)e Fp(P)638 2971 y Fo(0)703 2956 y Fq(b)s(e)i(the)f(sp)s(ectral)g (pro)5 b(jection)26 b(of)f Fp(H)1993 2971 y Fo(0)2059 2956 y Fq(corresp)s(onding)h(to)f Fp(\033)2846 2971 y Fo(0)2886 2956 y Fq(.)41 b(The)27 b(problem)257 3077 y(is)d(to)f(study)j(the)e("deformation")e(of)h Fp(\033)1668 3092 y Fo(0)1732 3077 y Fq(and)h Fp(P)1976 3092 y Fo(0)2039 3077 y Fq(when)h Fp(")i Fj(6)p Fq(=)h(0)p Fp(;)72 b(")27 b Fj(!)g Fq(0.)41 b(If)24 b Fp(\014)29 b Fq(in)23 b(\(5.1\))h(is)257 3197 y(less)j(than)f(0)f(then)i(there)g(exist)f(magnetic)e(p)s(oten)m (tials)h(whic)m(h)i(are)f(uniformly)d(b)s(ounded)257 3318 y(\(e.g.)56 b(the)36 b(transv)m(ersal)i(gauge,)f(see)g(b)s(elo)m (w\))g(there)g(is)f(nothing)f(new)j(to)e(sa)m(y)h(since)g(in)257 3438 y(that)c(case)754 3655 y Fp(H)835 3670 y Fm(";)p Ff(a)958 3655 y Fq(=)28 b Fp(H)1143 3670 y Fo(0)1204 3655 y Fj(\000)23 b Fp(")p Fq(\()p Fk(P)e Fj(\001)h Fk(a)g Fj(\000)h Fk(a)g Fj(\001)e Fk(P)h Fq(+)g Fp(")2081 3613 y Fo(2)2148 3655 y Fj(j)27 b Fk(a)h Fj(j)2313 3613 y Fo(2)2352 3655 y Fq(\))g Fj(\021)g Fp(H)2604 3670 y Fo(0)2666 3655 y Fj(\000)22 b Fp("W)2903 3670 y Fm(";)p Ff(a)3295 3655 y Fq(\(5.6\))257 3871 y(where)45 b Fp(W)642 3886 y Fm(";)p Ff(a)782 3871 y Fq(is)e(a)g(regular)g(p)s(erturbation)g(of)g Fp(H)2111 3886 y Fo(0)2194 3871 y Fq(and)h(the)g(analytic)e(p)s (erturbation)257 3992 y(theory)34 b(w)m(orks.)404 4112 y(On)40 b(the)i(con)m(trary)-8 b(,)43 b(if)d(in)g(\(5.1\),)i Fp(\014)47 b Fj(\025)42 b Fq(0)f(it)e(is)h(not)h(p)s(ossible)f(in)g (general)g(to)h(\014nd)257 4232 y Fk(a)f Fq(whic)m(h)h(is)e(uniformly)e (b)s(ounded)k(and)e(this)g(implies)f(that)h Fp(W)2642 4247 y Fm(";)p Ff(a)2738 4232 y Fq(\()p Fp(H)2857 4247 y Fo(0)2923 4232 y Fj(\000)28 b Fp(z)t Fq(\))3115 4196 y Fi(\000)p Fo(1)3249 4232 y Fq(is)39 b(not)257 4353 y(b)s(ounded)34 b(so)e(one)h(cannot)g(obtain)f(an)g(expansion)h(in)f Fp(")g Fq(of)g(\()p Fp(H)2588 4368 y Fm(";)p Ff(a)2706 4353 y Fj(\000)23 b Fp(z)t Fq(\))2893 4317 y Fi(\000)p Fo(1)3020 4353 y Fq(from)935 4570 y(\()p Fp(H)1054 4585 y Fm(";)p Ff(a)1171 4570 y Fj(\000)g Fp(z)t Fq(\)\()p Fp(H)1477 4585 y Fo(0)1539 4570 y Fj(\000)g Fp(z)t Fq(\))1726 4528 y Fi(\000)p Fo(1)1848 4570 y Fq(=)28 b(1)22 b Fj(\000)g Fp("W)2260 4585 y Fm(";)p Ff(a)2356 4570 y Fq(\()p Fp(H)2475 4585 y Fo(0)2536 4570 y Fj(\000)h Fp(z)t Fq(\))2723 4528 y Fi(\000)p Fo(1)3295 4570 y Fq(\(5.7\))404 4786 y(If)36 b Fp(\033)560 4801 y Fo(0)633 4786 y Fj(\032)e Fp(\033)799 4801 y Fm(discr)961 4786 y Fq(\()p Fp(H)1080 4801 y Fo(0)1119 4786 y Fq(\))i(the)g(asymptotic)f(p)s(erturbation)h(theory)g(dev)m (elop)s(ed)h(in)e(pre-)257 4907 y(vious)g(sections)g(w)m(orks)h(for)f (all)d Fp(\014)37 b Fj(2)32 b Fn(R)46 b Fq(and)34 b(giv)m(es)i (asymptotic)e(expansions)h(for)g(p)s(er-)257 5027 y(turb)s(ed)43 b(eigen)m(v)-5 b(alues)43 b(and)g(corresp)s(onding)f(eigen)m(v)m (ectors.)75 b(The)44 b(situation)d(is)h(more)257 5147 y(in)m(v)m(olv)m(ed)d(when)g Fp(\033)960 5162 y Fo(0)1038 5147 y Fq(is)f(not)g(con)m(tained)g(in)g(the)g(discrete)h(sp)s(ectrum)g (of)e Fp(H)3054 5162 y Fo(0)3094 5147 y Fq(.)60 b(On)38 b(one)257 5268 y(hand)d(for)f Fp(\014)j Fj(\024)32 b Fq(0)i(i.e.)49 b(for)34 b(magnetic)f(\014elds)i(whic)m(h)g(are)g (uniformly)d(b)s(ounded,)k(it)e(has)257 5388 y(b)s(een)e(pro)m(v)m(ed)h (b)m(y)e(Avron)h(and)f(Simon)e([2])i(in)f(a)h(particular)e(case)j(and)f (b)m(y)h(Nenciu)f([19])1828 5637 y(26)p eop %%Page: 27 27 27 26 bop 257 573 a Fq(and)43 b(Hel\013er)g(and)g(Sjostrand)g([9])g(in) f(the)h(general)f(case)i(that)e(the)i(spactrum)f(\(sa)f(a)257 693 y(set\))30 b(is)f(stable)g(and)g(then)h(for)f(su\016cien)m(tly)h (small)d Fp(")h Fq(one)i(can)f(still)e(de\014ne)k(the)e(sp)s(ectral)257 814 y(pro)5 b(jection)36 b Fp(P)787 829 y Fm(";)p Ff(a)918 814 y Fq(of)f Fp(H)1113 829 y Fm(";)p Ff(a)1244 814 y Fq(corresp)s(onding)g(to)h(the)g(part)f(of)g Fp(\033)t Fq(\()p Fp(H)2673 829 y Fm(";)p Ff(a)2769 814 y Fq(\))g(coinciding)f (with)257 934 y Fp(\033)312 949 y Fo(0)394 934 y Fq(in)41 b(the)h(limit)d Fp(")k Fj(!)g Fq(0.)71 b(No)m(w)43 b(the)f(singularit)m (y)e(of)h(the)i(p)s(erturbation)e(manifests)257 1054 y(itself)36 b(in)g(the)i("non)f(smo)s(oth")f(b)s(eha)m(viour)h(of)g Fp(P)2066 1069 y Fm(";)p Ff(a)2198 1054 y Fq(as)h Fp(")d Fj(!)g Fq(0;)k(There)f(are)f(examples)257 1175 y(in)32 b(solid)f(state)i(theory)h(\(i.e.)42 b Fp(V)55 b Fq(p)s(erio)s(dic)30 b(and)j Fk(b)g Fq(constan)m(t)g(where)h([18]\):)1450 1390 y(lim)1448 1450 y Fm(")p Fi(!)p Fo(0)1614 1390 y Fj(k)28 b Fp(P)1755 1405 y Fm(";)p Ff(a)1873 1390 y Fj(\000)22 b Fp(P)2035 1405 y Fo(0)2102 1390 y Fj(k)p Fq(=)28 b(1)990 b(\(5.8\))257 1639 y(\(of)29 b(course)h(one)f(still)e(has)i Fp(s)15 b Fj(\000)g Fq(lim)1519 1654 y Fm(")p Fi(!)p Fo(0)1678 1639 y Fp(P)1741 1654 y Fm(";)p Ff(a)1864 1639 y Fq(=)28 b Fp(P)2031 1654 y Fo(0)2070 1639 y Fq(\))h(so)g(there)h(is)f (no)g(ro)s(om)e(for)h(a)h(\(norm\))257 1759 y(asymptotic)37 b(expansion)g(of)g Fp(P)1400 1774 y Fm(";)p Ff(a)1496 1759 y Fq(.)56 b(Since,)39 b(in)d(con)m(trast)i(with)e Fp(\033)t Fq(\()p Fp(H)2771 1774 y Fm(";)p Ff(a)2867 1759 y Fq(\),)i Fp(P)3033 1774 y Fm(";)p Ff(a)3166 1759 y Fq(is)f(not)f(a)257 1879 y(gauge)e(in)m(v)-5 b(arian)m(t)32 b(quan)m(tit)m(y)-8 b(,)34 b(the)g(ab)s(o)m(v)m(e)g(discussion)g (suggests)h(that,)f(in)e(general,)i(the)257 2000 y(gauge)39 b(in)m(v)-5 b(arian)m(t)38 b(quan)m(tities)g(b)s(eha)m(v)m(e)j(m)m(uc)m (h)e(more)g(smo)s(othly)e(in)h(the)i(limit)35 b Fp(")j Fj(!)g Fq(0)257 2120 y(than)g(the)f(non)h(gauge)f(in)m(v)-5 b(arian)m(t)36 b(ones.)59 b(In)37 b(other)h(w)m(ords)g(if)e(one)i(w)m (an)m(ts)g(to)f(dev)m(elop)257 2240 y(a)h(p)s(erturbation)f(theory)-8 b(,)40 b(one)e(has)h(to)e(consider)h(gauge)g(in)m(v)-5 b(arian)m(t)37 b(quan)m(tities.)59 b(The)257 2361 y(problem)35 b(is)h(that,)h(in)e(almost)g(all)f(p)s(erturbation)h(sc)m(hemes,)k(the) e(basic)f(ob)5 b(ject)37 b(is)f(the)257 2481 y(resolv)m(en)m(t)e(whic)m (h)f(is)f(not)h(a)f(gauge)g(in)m(v)-5 b(arian)m(t)32 b(quan)m(tit)m(y)-8 b(.)404 2602 y(F)g(ortunately)g(,)48 b(one)e(can)f("factor")g(out)h(the)g(non)f(gauge)h(in)m(v)-5 b(arian)m(t)44 b(part)m(y)-8 b(.)83 b(Let)257 2722 y Fp(G)334 2737 y Fm(";)p Ff(a)430 2722 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fp(z)t Fq(\),)33 b Fp(z)f Fj(2)c Fp(\032)p Fq(\()p Fp(H)1163 2737 y Fm(";)p Ff(a)1259 2722 y Fq(\))k(b)s(e)h(the)f(in)m(tegral)f(k)m(ernel)i(\(w)m(e)g(shall) e(pro)m(v)m(e)j(b)s(elo)m(w)e(that)g(it)257 2842 y(exists\))e(of)e(\()p Fp(H)787 2857 y Fm(";)p Ff(a)896 2842 y Fj(\000)14 b Fp(z)t Fq(\))1074 2806 y Fi(\000)p Fo(1)1198 2842 y Fq(and)29 b Fp(\036)1442 2857 y Ff(a)1485 2842 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(\))28 b(b)s(e)h(the)g(famous)f(nonin)m(tegrable)f (phase)j(factor)257 2963 y(corresp)s(onding)j(to)f Fk(a)h Fq(i.e.)1370 3225 y Fp(\036)1428 3240 y Ff(a)1471 3225 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(\))27 b(=)1842 3089 y Fh(Z)1942 3115 y Ff(x)1897 3315 y(y)2005 3225 y Fk(a)p Fq(\()p Fk(u)p Fq(\))c Fj(\001)f Fp(d)p Fk(u)912 b Fq(\(5.9\))257 3500 y(where)41 b(the)f(in)m(tegral)e(ia)g(along)h(the)h(straigh)m (tline)d(joining)h Fk(x)h Fq(and)h Fk(y)r Fq(.)64 b(Alternativ)m(ely) 257 3621 y(one)33 b(can)g(write)f(\(5.9\))g(as)941 3901 y Fp(\036)999 3916 y Ff(a)1043 3901 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(\))27 b(=)g(\()1451 3766 y Fh(Z)1551 3792 y Fo(1)1507 3991 y(0)1607 3901 y Fk(a)p Fq(\()p Fk(y)d Fq(+)e Fp(t)p Fq(\()p Fk(x)h Fj(\000)f Fk(y)r Fq(\)\))p Fp(dt)p Fq(\))g Fj(\001)g Fq(\()p Fk(x)g Fj(\000)h Fk(y)r Fq(\))p Fp(:)434 b Fq(\(5.10\))257 4164 y(Then)34 b(as)f(one)g(can)g(easily)e(c)m(hec)m(k)1130 4380 y Fp(K)1213 4395 y Fm(")1250 4380 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fp(z)t Fq(\))28 b(=)f Fp(e)1759 4339 y Fi(\000)p Fm(i"\036)1913 4347 y Fa(a)1951 4339 y Fo(\()p Ff(x)p Fm(;)p Ff(y)q Fo(\))2116 4380 y Fp(G)2193 4395 y Fm(";)p Ff(a)2289 4380 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fp(z)t Fq(\))624 b(\(5.11\))257 4596 y(is)32 b(a)h(gauge)f(in)m(v)-5 b(arian)m(t)31 b(quan)m(tit)m(y)-8 b(.)44 b(In)33 b(other)g(w)m(ords)h (w)m(e)f(ha)m(v)m(e)h(to)e(mak)m(e)h(the)g(ansatz)1174 4811 y Fp(G)1251 4826 y Fm(";)p Ff(a)1347 4811 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fp(z)t Fq(\))28 b(=)f Fp(e)1856 4770 y Fm(i\036)1922 4778 y Fa(a)1960 4770 y Fo(\()p Ff(x)p Fm(;)p Ff(y)q Fo(\))2126 4811 y Fp(K)2209 4826 y Fm(")2245 4811 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fp(z)t Fq(\))668 b(\(5.12\))257 5027 y(and)46 b(seek)h(a)e(p)s(erturbation)g(expansion)h(for)f Fp(K)2072 5042 y Fm(")2109 5027 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fp(z)t Fq(\).)82 b(The)46 b(form)m(ula)e(\(5.12\))h(is)257 5147 y(crucial)c(for)g(all)f(what)i(follo)m(ws;)j(it)c(sa)m(ys)i(that)e (the)i(whole)e(gauge)h(nonin)m(v)-5 b(ariance)41 b(of)257 5268 y(the)c(in)m(tegral)e(k)m(ernel)i(of)e(the)i(resolv)m(en)m(t)g(\() f(and)h(then)g(for)e(other)i(related)e(ob)5 b(jects)38 b(lik)m(e)257 5388 y(as)g(sp)s(ectral)f(pro)5 b(jections,)38 b(ev)m(olution)e(etc\))i(is)f(con)m(tained)g(in)f(an)h(exp)s(onen)m (tial)g(factor)1828 5637 y(27)p eop %%Page: 28 28 28 27 bop 257 573 a Fp(e)302 537 y Fm(i\036)368 545 y Fa(a)406 537 y Fo(\()p Ff(x)p Fm(;)p Ff(y)q Fo(\))572 573 y Fq(.)43 b(Coming)31 b(bac)m(k)j(to)d(p)s(erturbation)h(theory)-8 b(,)33 b(\(5.12\))e(suggests)j(to)e(tak)m(e)h(as)g(the)257 693 y("zero'th")k(appro)m(ximation)c(for)j(\()p Fp(H)1609 708 y Fm(";)p Ff(a)1729 693 y Fj(\000)25 b Fp(z)t Fq(\))1918 657 y Fi(\000)p Fo(1)2049 693 y Fq(the)37 b(op)s(erator)e Fp(S)2677 708 y Fm(";)p Ff(a)2773 693 y Fq(\()p Fp(z)t Fq(\),)j(giv)m(en)e(b)m(y)h(the)257 814 y(follo)m(wing)30 b(in)m(tegral)h(k)m(ernel)i(\()g("t)m(wisted")g(unp)s(erturb)s(ed)g (resolv)m(en)m(t\))914 1065 y(\()p Fp(S)1012 1080 y Fm(";)p Ff(a)1108 1065 y Fq(\()p Fp(z)t Fq(\))p Fp(f)11 b Fq(\)\()p Fk(x)p Fq(\))28 b(=)1596 929 y Fh(Z)1652 1155 y Ff(R)1713 1136 y Fd(3)1768 1065 y Fp(e)1813 1024 y Fm(i\036)1879 1032 y Fa(a)1917 1024 y Fo(\()p Ff(x)p Fm(;)p Ff(y)q Fo(\))2083 1065 y Fp(G)2160 1080 y Fo(0)2199 1065 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fp(z)t Fq(\))p Fp(f)11 b Fq(\()p Fk(y)r Fq(\))p Fp(d)p Fk(y)408 b Fq(\(5.13\))257 1324 y(where)44 b Fp(G)626 1339 y Fo(0)665 1324 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fp(z)t Fq(\))43 b(is)f(the)g(in)m (tegral)f(k)m(ernel)i(of)f(\()p Fp(H)2228 1339 y Fo(0)2296 1324 y Fj(\000)30 b Fp(z)t Fq(\))2490 1288 y Fi(\000)p Fo(1)2585 1324 y Fq(.)73 b(Let's)43 b(consider)g(no)m(w)257 1445 y(\(5.7\))32 b(with)h(\()p Fp(H)832 1460 y Fo(0)893 1445 y Fj(\000)23 b Fp(z)t Fq(\))1080 1409 y Fi(\000)p Fo(1)1207 1445 y Fq(replaced)33 b(b)m(y)g Fp(S)1787 1460 y Fm(";)p Ff(a)1883 1445 y Fq(\()p Fp(z)t Fq(\))g(and)g(de\014ne)h Fp(T)2570 1460 y Fm(";)p Ff(a)2666 1445 y Fq(\()p Fp(z)t Fq(\))f(b)m(y)1202 1648 y(\()p Fp(H)1321 1663 y Fm(";)p Ff(a)1439 1648 y Fj(\000)23 b Fp(z)t Fq(\))p Fp(S)1686 1663 y Fm(";)p Ff(a)1782 1648 y Fq(\()p Fp(z)t Fq(\))28 b(=)g(1)22 b Fj(\000)h Fp(")16 b(T)2329 1663 y Fm(";)p Ff(a)2425 1648 y Fq(\()p Fp(z)t Fq(\))696 b(\(5.14\))257 1850 y(Before)33 b(computing)f Fp(T)1102 1865 y Fm(";)p Ff(a)1198 1850 y Fq(\()p Fp(z)t Fq(\))h(let)f(us)h(giv)m(e)f(the)h (follo)m(wing)d(gauge)j(co)m(v)-5 b(ariance)32 b(form)m(ula)257 2039 y Fk(Lemma)37 b(5.1.)50 b Fl(L)-5 b(et)52 b Fk(a)p Fq(\()p Fk(x)p Fq(\))g Fl(b)-5 b(e)52 b(an)f(arbitr)-5 b(ary)52 b(c)-5 b(ontinuous)52 b(ve)-5 b(ctor)51 b(p)-5 b(otential)52 b(c)-5 b(orr)g(e-)257 2159 y(sp)g(onding)30 b(to)h Fk(b)p Fq(\()p Fk(x)p Fq(\))p Fl(,)g Fk(c)d Fj(2)g Fk(R)1284 2123 y Fo(3)1354 2159 y Fl(and)i Fp(\036)1597 2174 y Ff(a)1640 2159 y Fq(\()p Fk(x)p Fp(;)17 b Fk(c)p Fq(\))31 b Fl(the)g(c)-5 b(orr)g(esp)g(onding)29 b(noninte)-5 b(gr)g(able)29 b(phase)257 2280 y(factor.)45 b(Then)867 2483 y Fp(e)912 2442 y Fi(\000)p Fm(i"\036)1066 2450 y Fa(a)1104 2442 y Fo(\()p Ff(x)p Fm(;)p Ff(c)p Fo(\))1262 2483 y Fq(\()p Fk(P)1377 2498 y Ff(x)1445 2483 y Fj(\000)23 b Fp(")p Fk(a)p Fq(\()p Fk(x)p Fq(\)\))p Fp(e)1863 2442 y Fm(i"\036)1962 2450 y Fa(a)2000 2442 y Fo(\()p Ff(x)p Fm(;)p Ff(c)p Fo(\))2185 2483 y Fq(=)k(\()p Fk(P)2403 2498 y Ff(x)2472 2483 y Fj(\000)c Fp(")p Fk(a)2672 2498 y Ff(c)2712 2483 y Fq(\()p Fk(x)p Fq(\)\))361 b(\(5.15\))257 2685 y Fl(wher)-5 b(e)36 b Fk(P)611 2700 y Ff(x)689 2685 y Fq(=)30 b Fj(\000)p Fp(i)p Fj(r)988 2700 y Ff(x)1073 2685 y Fl(and)35 b Fk(a)1317 2700 y Ff(c)1358 2685 y Fq(\()p Fk(x)p Fq(\))i Fl(is)f(the)h(ve)-5 b(ctor)36 b(p)-5 b(otential)36 b(given)g(by)g(the)h(tr)-5 b(ansversal)257 2806 y(\(Poinc)g(ar)g(e\))34 b(gauge)h(r)-5 b(elative)34 b(to)h(the)g(p)-5 b(oint)35 b Fk(c)p Fl(:)986 3074 y Fk(a)1040 3089 y Ff(c)1081 3074 y Fq(\()p Fk(x)p Fq(\))27 b(=)h(\()1385 2938 y Fh(Z)1484 2964 y Fo(1)1440 3164 y(0)1540 3074 y Fp(t)p Fk(b)p Fq(\()p Fk(c)23 b Fq(+)f Fp(t)p Fq(\()p Fk(x)g Fj(\000)h Fk(c)p Fq(\)\))p Fp(dt)p Fq(\))f Fj(^)g Fq(\()p Fk(x)g Fj(\000)h Fk(c)p Fq(\))479 b(\(5.16\))404 3329 y(Notice)32 b(that)g Fk(a)975 3344 y Ff(0)1021 3329 y Fq(\()p Fk(x)p Fq(\))g(is)g(the)h(usual)f(transv)m (ersal)i(gauge)e([26]:)1316 3597 y Fk(a)1370 3612 y Ff(0)1415 3597 y Fq(\()p Fk(x)p Fq(\))c(=)g(\()1720 3461 y Fh(Z)1819 3487 y Fo(1)1775 3687 y(0)1875 3597 y Fp(t)p Fk(b)p Fq(\()p Fp(t)p Fk(x)p Fq(\))p Fp(dt)p Fq(\))23 b Fj(^)f Fk(x)810 b Fq(\(5.17\))257 3852 y(The)38 b(most)e(in)m(teresting)h(prop)s (erties)g(of)f Fk(a)1822 3867 y Ff(c)1863 3852 y Fq(\()p Fk(x)p Fq(\))g(and)h Fp(\036)2286 3867 y Ff(a)2326 3875 y Fa(c)2365 3852 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(\))36 b(are)h(giv)m(en)g(in)f(the)h(fol-)257 3973 y(lo)m(wing:)257 4161 y Fk(Lemma)g(5.2.)50 b Fl(i.)421 4524 y Fp(\036)479 4539 y Ff(a)519 4547 y Fa(c)557 4524 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(\))27 b(=)h Fj(\000)p Fq(\()1043 4389 y Fh(Z)1143 4415 y Fo(1)1098 4614 y(0)1199 4524 y Fp(dt)1302 4389 y Fh(Z)1401 4415 y Fo(1)1357 4614 y(0)1457 4524 y Fp(dss)p Fk(b)p Fq(\()p Fp(st)p Fq(\()p Fk(x)22 b Fj(\000)h Fk(y)q Fq(\))f(+)g Fp(s)p Fq(\()p Fk(y)i Fj(\000)f Fk(c)p Fq(\)\))f Fj(\001)f Fq(\()p Fk(x)i Fj(\000)f Fk(c)g Fj(^)h Fk(y)h Fj(\000)e Fk(c)p Fq(\))3246 4707 y(\(5.18\))257 4908 y Fl(i.e.)45 b Fp(\036)495 4923 y Ff(a)535 4931 y Fa(c)573 4908 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(\))34 b Fl(is)h(the)g(\015ux)g(thr)-5 b(ough)35 b(the)g(triangle)f Fk(x)28 b(c)g(y)q Fl(.)404 5029 y(ii.)1434 5352 y Fj(j)f Fk(a)1543 5367 y Ff(c)1584 5352 y Fq(\()p Fk(x)p Fq(\))h Fj(j\024j)f Fk(x)c Fj(\000)f Fk(c)28 b Fj(j)f Fp(=)p Fq(2)927 b(\(5.19\))1828 5637 y(28)p eop %%Page: 29 29 29 28 bop 1389 573 a Fj(j)28 b(r)p Fk(a)1582 588 y Ff(c)1622 573 y Fq(\()p Fk(x)p Fq(\))g Fj(j\024)g Fq(2)f Fj(j)h Fk(x)22 b Fj(\000)h Fk(c)k Fj(j)910 b Fq(\(5.20\))257 793 y Fl(iii.)404 913 y(If)34 b Fk(b)p Fq(\()p Fk(x)p Fq(\))28 b(=)g Fk(b)897 928 y Fo(0)964 913 y Fq(=)f Fp(const)36 b Fl(then)1366 1133 y Fk(a)1420 1148 y Ff(c)1460 1133 y Fq(\()p Fk(x)p Fq(\))28 b(=)g Fk(b)1789 1148 y Fo(0)1851 1133 y Fj(^)22 b Fq(\()p Fk(x)g Fj(\000)h Fk(c)p Fq(\))p Fp(=)p Fq(2)p Fp(;)875 b Fq(\(5.21\))1087 1453 y Fp(\036)1145 1468 y Ff(a)1185 1476 y Fa(c)1223 1453 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(\))27 b(=)h Fk(b)1656 1468 y Fo(0)1717 1453 y Fj(\001)22 b Fq(\(\()p Fk(x)g Fj(\000)h Fk(c)p Fq(\))f Fj(^)g Fq(\()p Fk(y)i Fj(\000)f Fk(c)p Fq(\)\))p Fp(=)p Fq(2)579 b(\(5.22\))404 1673 y(Computing)35 b(the)h(r.h.s.)55 b(of)36 b(\(5.14\))f(applied)g(to)g Fp(f)45 b Fj(2)34 b(C)2472 1637 y Fi(1)2466 1698 y Fo(0)2547 1673 y Fq(\()p Fk(R)2669 1637 y Fo(3)2708 1673 y Fq(\))i(b)m(y)h(using)f(Lemma)257 1793 y(4.1.)43 b(one)33 b(\014nds:)892 2037 y(\()p Fp(T)987 2052 y Fm(";)p Ff(a)1083 2037 y Fq(\()p Fp(z)t Fq(\))p Fp(f)11 b Fq(\)\()p Fk(x)p Fq(\))28 b(=)1571 1901 y Fh(Z)1627 2127 y Ff(R)1688 2108 y Fd(3)1743 2037 y Fp(e)1788 1996 y Fm(i"\036)1887 2004 y Fa(a)1924 1996 y Fo(\()p Ff(x)p Fm(;)p Ff(y)q Fo(\))2090 2037 y Fp(M)2184 2052 y Fm(")2221 2037 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fp(z)t Fq(\))p Fp(f)11 b Fq(\()p Fk(y)r Fq(\))p Fp(d)p Fk(y)386 b Fq(\(5.23\))257 2304 y(where)426 2644 y Fp(M)520 2659 y Fm(")557 2644 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fp(z)t Fq(\))27 b(=)h(\()p Fj(\000)p Fq(2)p Fp(i)p Fk(a)1272 2659 y Ff(y)1321 2644 y Fq(\()p Fk(x)p Fq(\))22 b Fj(\001)g(r)1611 2659 y Ff(x)1680 2644 y Fj(\000)h Fp(i)p Fq(\()p Fj(r)1934 2659 y Ff(x)1981 2644 y Fk(a)2035 2659 y Ff(y)2084 2644 y Fq(\()p Fk(x)p Fq(\)\))f Fj(\000)h Fp(")k Fj(j)g Fk(a)2561 2659 y Ff(y)2610 2644 y Fq(\()p Fk(x)p Fq(\))h Fj(j)2801 2603 y Fo(2)2840 2644 y Fq(\))p Fp(G)2955 2659 y Fo(0)2994 2644 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fp(z)t Fq(\))3246 2765 y(\(5.24\))404 2978 y(Notice)34 b(that)h Fp(M)1020 2993 y Fm(")1092 2978 y Fq(dep)s(ends)h(only)e(up)s(on)h Fk(b)g Fq(i.e.)50 b(is)34 b(gauge)h(in)m(v)-5 b(arian)m(t.)49 b(Up)35 b(to)g(no)m(w)257 3098 y(the)d(discussion)g(w)m(as)h(at)e(the)h (heuristic)f(formal)e(lev)m(el.)43 b(The)33 b(main)d(tec)m(hnical)h(p)s (oin)m(t)g(of)257 3219 y(this)i(section)f(is)g(con)m(tained)h(in)257 3422 y Fk(Theorem)k(5.3.)50 b Fl(Supp)-5 b(ose)24 b(\(5.1\))g(and)g (\(5.2\))g(hold)h(true,)i Fp(z)32 b Fj(2)c Fp(\032)p Fq(\()p Fp(H)2731 3437 y Fo(0)2771 3422 y Fq(\))p Fp(;)72 b(")27 b Fj(2)h Fq([0)p Fp(;)17 b(")3241 3437 y Fo(0)3280 3422 y Fq(])p Fp(;)72 b(")3452 3437 y Fo(0)3519 3422 y Fp(<)257 3542 y Fj(1)41 b Fl(,)h Fp(K)k Fj(\032)39 b Fp(\032)p Fq(\()p Fp(H)884 3557 y Fo(0)923 3542 y Fq(\))i Fl(c)-5 b(omp)g(act.Then)39 b(\(5.23\),)i(\(5.24\))f(de\014ne)g(a)g(b) -5 b(ounde)g(d)40 b(op)-5 b(er)g(ator)41 b(in)257 3663 y Fp(L)323 3627 y Fo(2)363 3663 y Fq(\()p Fn(R)467 3627 y Fo(3)512 3663 y Fq(\))31 b Fl(and)g(ther)-5 b(e)31 b(exist)g Fp(t)p Fq(\()p Fp(z)t Fq(\))d Fp(<)g Fj(1)i Fl(\()h(dep)-5 b(endind)30 b(up)-5 b(on)30 b Fp(V)5 b(;)45 b Fk(b)p Fp(;)31 b Fl(and)g Fp(")2895 3678 y Fo(0)2965 3663 y Fl(but)g(not)g(up)-5 b(on)257 3783 y Fk(a)p Fl(\))36 b(such)e(that)h(uniformly)g(for)g Fp(")27 b Fj(2)h Fq([0)p Fp(;)17 b(")1741 3798 y Fo(0)1780 3783 y Fq(])35 b Fl(and)f Fp(z)f Fj(2)28 b Fp(K)34 b Fq(:)1513 4003 y Fj(k)27 b Fp(T)1647 4018 y Fm(";)p Ff(a)1743 4003 y Fq(\()p Fp(z)t Fq(\))i Fj(k\024)f Fp(t)p Fq(\()p Fp(z)t Fq(\))p Fp(:)1007 b Fq(\(5.25\))404 4223 y(As)29 b(a)g(direct)g(consequence)k(w)m(e)d (obtain)e(the)i(follo)m(wing)c(p)s(erturbation)i(form)m(ula)g(for)257 4344 y(\()p Fp(H)376 4359 y Fm(";)p Ff(a)494 4344 y Fj(\000)23 b Fp(z)t Fq(\))681 4307 y Fi(\000)p Fo(1)776 4344 y Fq(:)257 4547 y Fk(Lemma)37 b(5.4.)50 b Fl(If)34 b Fp(")27 b Fj(2)h Fq([0)p Fp(;)17 b(t)p Fq(\()p Fp(z)t Fq(\))1427 4511 y Fi(\000)p Fo(1)1522 4547 y Fq(])35 b Fl(then)g Fp(z)d Fj(2)c Fp(\032)p Fq(\()p Fp(H)2141 4562 y Fm(";)p Ff(a)2237 4547 y Fq(\))35 b Fl(and)1123 4846 y Fq(\()p Fp(H)1242 4861 y Fm(";)p Ff(a)1360 4846 y Fj(\000)23 b Fp(z)t Fq(\))1547 4861 y Fi(\000)p Fo(1)1670 4846 y Fq(=)k Fp(S)1833 4861 y Fm(";)p Ff(a)1929 4846 y Fq(\()p Fp(z)t Fq(\))2108 4721 y Fi(1)2071 4751 y Fh(X)2082 4961 y Fm(j)t Fo(=0)2232 4846 y Fp(")2278 4805 y Fm(j)2314 4846 y Fp(T)2371 4861 y Fm(";)p Ff(a)2467 4846 y Fq(\()p Fp(z)t Fq(\))2592 4805 y Fm(j)3246 4846 y Fq(\(5.26\))257 5157 y Fl(as)35 b(a)g(norm)f(c)-5 b(onver)g(gent)34 b(series.)1828 5637 y Fq(29)p eop %%Page: 30 30 30 29 bop 404 573 a Fq(Com)m(bining)38 b(\(5.13\),)k(\(5.23\))e(and)g (\(5.26\))g(one)h(can)f(write)h(do)m(wn)g(the)g(expansion)257 693 y(for)32 b Fp(K)489 708 y Fm(")526 693 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fp(z)t Fq(\):)918 979 y Fp(K)1001 994 y Fm(")1037 979 y Fq(\()p Fk(x)p Fp(;)g Fk(y)r Fq(;)g Fp(z)t Fq(\))28 b(=)g Fp(G)1579 994 y Fo(0)1618 979 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fp(z)t Fq(\))22 b(+)2108 855 y Fi(1)2071 885 y Fh(X)2082 1095 y Fm(j)t Fo(=1)2232 979 y Fp(")2277 938 y Fm(j)2330 979 y Fp(K)2413 994 y Fm(";j)2502 979 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fp(z)t Fq(\))411 b(\(5.27\))257 1285 y(where)761 1519 y Fp(K)844 1534 y Fm(";j)933 1519 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fp(z)t Fq(\))27 b(=)1386 1383 y Fh(Z)1502 1519 y Fj(\001)22 b(\001)f(\001)1646 1383 y Fh(Z)1762 1519 y Fp(d)p Fk(u)1875 1534 y Fo(1)1914 1519 y Fp(d)p Fk(u)2027 1534 y Fo(2)2067 1519 y Fp(:::d)p Fk(u)2261 1534 y Fm(j)2298 1519 y Fp(e)2343 1478 y Fm(i")p Fo(\010\()p Ff(x)p Fm(;)p Ff(u)2586 1487 y Fd(1)2620 1478 y Fm(;)p Ff(u)2685 1487 y Fd(2)2719 1478 y Fm(:::)p Ff(u)2824 1488 y Fg(j)2856 1478 y Fm(;)p Ff(y)q Fo(\))1369 1722 y Fp(G)1446 1737 y Fo(0)1485 1722 y Fq(\()p Fk(x)p Fp(;)c Fk(u)1688 1737 y Fo(1)1728 1722 y Fq(;)g Fp(z)t Fq(\))p Fp(M)1953 1737 y Fm(")1990 1722 y Fq(\()p Fk(u)2090 1737 y Fo(1)2130 1722 y Fp(;)g Fk(u)2236 1737 y Fo(2)2275 1722 y Fq(;)g Fp(z)t Fq(\))p Fp(:::M)2581 1737 y Fm(")2619 1722 y Fq(\()p Fk(u)2719 1737 y Fm(j)2756 1722 y Fp(;)g Fk(y)r Fq(;)g Fp(z)t Fq(\))254 b(\(5.28\))257 1932 y(and)320 2143 y(\010\()p Fk(x)p Fp(;)17 b Fk(u)593 2158 y Fo(1)633 2143 y Fp(;)g Fk(u)739 2158 y Fo(2)778 2143 y Fp(:::)p Fk(u)921 2158 y Fm(j)958 2143 y Fp(;)g Fk(y)r Fq(\))27 b(=)h Fp(\036)1290 2158 y Ff(a)1333 2143 y Fq(\()p Fk(y)r Fp(;)17 b Fk(u)1538 2158 y Fm(j)1574 2143 y Fq(\))22 b(+)g Fp(\036)1790 2158 y Ff(a)1834 2143 y Fq(\()p Fk(u)1934 2158 y Fm(j)1971 2143 y Fp(;)17 b Fk(u)2077 2158 y Fm(j)t Fi(\000)p Fo(1)2203 2143 y Fq(\))22 b(+)g Fp(:::)h Fq(+)f Fp(\036)2621 2158 y Ff(a)2664 2143 y Fq(\()p Fk(u)2764 2158 y Fo(1)2804 2143 y Fp(;)17 b Fk(x)p Fq(\))22 b(+)g Fp(\036)3123 2158 y Ff(a)3166 2143 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(\))p Fp(:)3246 2263 y Fq(\(5.29\))257 2466 y(Notice)23 b(that)h(\010)g(is)f(gauge)g(in)m(v)-5 b(arian)m(t)22 b(and)i(is)f(the)h(\015ux)g(through)g(the)f(p)s(olygonal)f(con)m(tour) 257 2587 y Fk(yu)379 2610 y Fm(j)415 2587 y Fk(u)477 2602 y Fm(j)t Fi(\000)p Fo(1)604 2587 y Fp(:::)p Fk(xy)r Fp(:)33 b Fq(In)g(terms)f(of)h(in)m(tegral)e(k)m(ernels,)i(\(5.26\))f (rewrites)h(as:)538 2885 y Fp(G)615 2900 y Fm(";)p Ff(a)711 2885 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fp(z)t Fq(\))27 b(=)h Fp(e)1220 2843 y Fm(i"\036)1319 2851 y Fa(a)1356 2843 y Fo(\()p Ff(x)p Fm(;)p Ff(y)q Fo(\))1522 2885 y Fj(f)p Fp(G)1649 2900 y Fo(0)1688 2885 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fp(z)t Fq(\))22 b(+)2178 2760 y Fi(1)2141 2790 y Fh(X)2152 3000 y Fm(j)t Fo(=1)2302 2885 y Fp(")2348 2843 y Fm(j)2384 2885 y Fp(K)2467 2900 y Fm(";j)2556 2885 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fp(z)t Fq(\))p Fj(g)p Fp(:)280 b Fq(\(5.30\))257 3196 y(The)34 b(form)m(ulae)d(\(5.28\),)h(\(5.30\))g(are)g(the)h(main)e (result)i(of)f(this)g(section.)404 3316 y(Starting)f(from)h(\(5.30\),)g (as)h(in)f(the)i(analytic)d(p)s(erturbations)i(case)g(,)g(on)g(can)g (com-)257 3436 y(pute)46 b(all)c(ph)m(ysical)j(quan)m(tities)f(as)h (con)m(v)m(ergen)m(t)i(series)e(in)f Fp(")g Fq(but)h(with)f(co)s (e\016cien)m(ts)257 3557 y(whic)m(h)32 b(dep)s(end)g(themselvs)g(up)s (on)f Fp(")p Fq(.)42 b(F)-8 b(or)31 b(example,)g(if)e Fp(\033)2403 3572 y Fo(0)2471 3557 y Fq(=)e Fj(f)p Fp(\025)2681 3572 y Fo(0)2721 3557 y Fj(g)j Fq(where)j Fp(\025)3139 3572 y Fo(0)3209 3557 y Fq(is)e(non-)257 3677 y(degenerate,)38 b(then)f(for)e(su\016cien)m(tly)h(small)e Fp(")h Fq(the)h(p)s(erturb)s (ed)h(eigen)m(v)-5 b(alue,)36 b Fp(\025)p Fq(\()p Fp(")p Fq(\),)g(has)257 3798 y(an)d(expansion:)844 4078 y Fp(\025)p Fq(\()p Fp(")p Fq(\))27 b(=)1190 3954 y Fi(1)1153 3984 y Fh(X)1164 4194 y Fm(j)t Fo(=0)1314 4078 y Fp(")1360 4037 y Fm(j)1396 4078 y Fp(\025)1453 4093 y Fm(";j)1542 4078 y Fp(;)72 b Fj(j)27 b Fp(\025)1753 4093 y Fm(";j)1869 4078 y Fj(j\024)h Fp(k)2056 4037 y Fm(j)2148 4078 y Fq(for)k(some)88 b Fp(k)31 b(<)c Fj(1)p Fp(:)337 b Fq(\(5.31\))257 4384 y(The)34 b(expressions)g(for)e Fp(\025)1172 4399 y Fm(";j)1293 4384 y Fq(can)h(b)s(e)g(obtained)f(from:)866 4645 y Fp(\025)p Fq(\()p Fp(")p Fq(\))27 b(=)h Fp(T)14 b(r)s Fq(\()p Fp(H)1413 4660 y Fm(";)p Ff(a)1508 4645 y Fp(P)1571 4660 y Fm(";)p Ff(a)1667 4645 y Fq(\))27 b(=)h Fp(T)14 b(r)2000 4578 y(i)p 1964 4623 108 4 v 1964 4714 a Fq(2)p Fp(\031)2098 4510 y Fh(I)2153 4735 y Fo(\000)2218 4645 y Fp(z)t Fq(\()p Fp(H)2386 4660 y Fm(";)p Ff(a)2504 4645 y Fj(\000)23 b Fp(z)t Fq(\))2691 4604 y Fi(\000)p Fo(1)2786 4645 y Fp(dz)257 4908 y Fq(where)33 b(\000)e(is)g(a)g(con)m(tour)h(enclosing)f Fp(\025)1635 4923 y Fo(0)1674 4908 y Fp(:)h Fq(Notice)f(that)g(\()p Fp( )2349 4923 y Fo(0)2389 4908 y Fq(\()p Fk(x)p Fq(\))g(is)g(the)h (eigenfunction)f(of)257 5029 y Fp(H)338 5044 y Fo(0)410 5029 y Fq(corresp)s(onding)i(to)f Fp(\025)1213 5044 y Fo(0)1252 5029 y Fq(\))1049 5298 y Fp(\025)1106 5313 y Fm(")o(;)p Fo(0)1225 5298 y Fq(=)1376 5231 y Fp(i)p 1339 5275 V 1339 5367 a Fq(2)p Fp(\031)1473 5163 y Fh(Z)1528 5388 y Ff(R)1589 5369 y Fd(3)1644 5298 y Fp(d)p Fk(x)1771 5163 y Fh(I)1826 5388 y Fo(\000)1891 5298 y Fp(z)1950 5231 y Fj(j)c Fp( )2069 5246 y Fo(0)2109 5231 y Fq(\()p Fk(x)p Fq(\))g Fj(j)2300 5195 y Fo(2)p 1951 5275 389 4 v 2011 5367 a Fp(\025)2068 5382 y Fo(0)2129 5367 y Fj(\000)23 b Fp(z)2349 5298 y(dz)32 b Fq(=)27 b Fp(\025)2637 5313 y Fo(0)2677 5298 y Fp(:)542 b Fq(\(5.32\))1828 5637 y(30)p eop %%Page: 31 31 31 30 bop 257 573 a Fq(F)-8 b(or)32 b Fp(j)i Fj(\025)28 b Fq(1,)k(due)i(to)e(the)h(fact)f(that)h Fp(K)1678 588 y Fm(j;")1762 573 y Fq(\()p Fk(x)p Fp(;)17 b Fk(x)p Fq(\))33 b(is)f(\014nite)g(one)h(can)g(write:)1141 815 y Fp(\025)1198 830 y Fm(";j)1315 815 y Fq(=)1465 747 y Fp(i)p 1428 792 108 4 v 1428 883 a Fq(2)p Fp(\031)1562 679 y Fh(Z)1618 905 y Ff(R)1679 886 y Fd(3)1734 815 y Fp(d)p Fk(x)1861 679 y Fh(I)1916 905 y Fo(\000)1981 815 y Fp(K)2064 830 y Fm(";j)2152 815 y Fq(\()p Fk(x)p Fp(;)17 b Fk(x)p Fq(;)g Fp(z)t Fq(\))p Fp(dz)t(:)636 b Fq(\(5.33\))257 1052 y(F)-8 b(urther)33 b(details)e(and)i(applications)e(of)h(Theorem)h(5.3)f(will) e(b)s(e)j(giv)m(en)f(elsewhere.)404 1172 y(The)k(rest)f(of)g(this)f (section)h(is)g(dev)m(oted)h(to)f(the)g(pro)s(ofs)g(of)f(Lemmae)g (5.1.-5.2)g(and)257 1293 y(Theorem)f(5.3.)404 1413 y Fl(Pr)-5 b(o)g(of)34 b(of)h(L)-5 b(emma)34 b(5.1.)43 b Fq(By)33 b(the)g(usual)f(computation)477 1603 y Fp(e)522 1562 y Fi(\000)p Fm(i"\036)676 1570 y Fa(a)713 1562 y Fo(\()p Ff(x)p Fm(;)p Ff(c)p Fo(\))871 1603 y Fq(\()p Fk(P)986 1618 y Ff(x)1054 1603 y Fj(\000)23 b Fp(")p Fk(a)p Fq(\()p Fk(x)p Fq(\)\))p Fp(e)1472 1562 y Fm(i"\036)1571 1570 y Fa(a)1609 1562 y Fo(\()p Ff(x)p Fm(;)p Ff(c)p Fo(\))1794 1603 y Fq(=)28 b(\()p Fk(P)2013 1618 y Ff(x)2081 1603 y Fj(\000)23 b Fp(")p Fq(\()p Fk(a)p Fq(\()p Fk(x)p Fq(\))f Fj(\000)h(r)p Fp(\036)2717 1618 y Ff(a)2760 1603 y Fq(\()p Fk(x)p Fp(;)17 b Fk(c)p Fq(\)\))219 b(\(5.34\))257 1794 y(T)-8 b(o)33 b(compute)g Fj(r)880 1809 y Ff(x)927 1794 y Fp(\036)985 1809 y Ff(a)1061 1794 y Fq(one)g(uses)h(\(5.10\))d (and)i(the)g(v)m(ector)h(analysis)e(form)m(ula)e([25])617 1984 y Fj(r)p Fq(\()p Fk(A)p Fq(\()p Fk(x)p Fq(\))22 b Fj(\001)g Fk(B)p Fq(\()p Fk(x)p Fq(\)\))27 b(=)p Fk(B)p Fq(\()p Fk(x)p Fq(\))22 b Fj(^)g Fq(\()p Fj(r)g(^)h Fk(A)p Fq(\()p Fk(x)p Fq(\)\))f(+)g Fk(A)p Fq(\()p Fk(x)p Fq(\))g Fj(^)g Fq(\()p Fj(r)g(^)h Fk(B)p Fq(\()p Fk(x)p Fq(\)\))1408 2129 y(+)f(\()p Fk(B)p Fq(\()p Fk(x)p Fq(\))g Fj(\001)g(r)p Fq(\))p Fk(A)p Fq(\()p Fk(x)p Fq(\))f(+)h(\()p Fk(B)p Fq(\()p Fk(x)p Fq(\))g Fj(\001)g(r)p Fq(\))p Fk(A)p Fq(\()p Fk(x)p Fq(\))289 b(\(5.35\))257 2320 y(with)1259 2550 y Fk(A)p Fq(\()p Fk(x)p Fq(\))27 b(=)1610 2415 y Fh(Z)1709 2441 y Fo(1)1665 2640 y(0)1766 2550 y Fk(a)p Fq(\()p Fk(c)22 b Fq(+)g Fp(t)p Fq(\()p Fk(x)g Fj(\000)h Fk(c)p Fq(\)\))p Fp(dt)1550 2873 y Fk(B)p Fq(\()p Fk(x)p Fq(\))k(=)h(\()p Fk(x)22 b Fj(\000)h Fk(c)p Fq(\))257 3063 y(One)34 b(obtains)f(four)g (terms;)h(the)g(\014rst)g(one)g(giv)m(es)f Fj(\000)p Fk(a)2232 3078 y Ff(c)2274 3063 y Fq(\()p Fk(x)p Fq(\),)g(the)h(second) h(v)-5 b(anishes)34 b(and)257 3193 y(the)f(fourth)g(giv)m(es)960 3113 y Fh(R)1026 3139 y Fo(1)1007 3227 y(0)1082 3193 y Fk(a)p Fq(\()p Fk(c)23 b Fq(+)f Fp(t)p Fq(\()p Fk(x)g Fj(\000)h Fk(c)p Fq(\)\))p Fp(dt)p Fq(.)43 b(Observing)33 b(that)806 3442 y Fp(t)869 3375 y(d)p 851 3419 86 4 v 851 3511 a(dt)947 3442 y Fk(a)p Fq(\()p Fk(c)22 b Fq(+)g Fp(t)p Fq(\()p Fk(x)h Fj(\000)f Fk(c)p Fq(\)\))28 b(=)f(\(\()p Fk(x)22 b Fj(\000)h Fk(c)p Fq(\))f Fj(\001)g(r)2220 3457 y Ff(x)2267 3442 y Fq(\))p Fk(a)p Fq(\()p Fk(c)g Fq(+)g Fp(t)p Fq(\()p Fk(x)h Fj(\000)f Fk(c)p Fq(\)\))257 3686 y(the)41 b(third)e(term)g(giv)m(es)1170 3605 y Fh(R)1236 3632 y Fo(1)1217 3720 y(0)1292 3686 y Fp(t)1350 3646 y Fm(d)p 1337 3663 62 4 v 1337 3720 a(dt)1409 3686 y Fk(a)p Fq(\()p Fk(c)28 b Fq(+)e Fp(t)p Fq(\()p Fk(x)i Fj(\000)f Fk(c)p Fq(\)\))p Fp(dt)40 b Fq(and)f(then,)k(after)c(in)m (tegration)f(b)m(y)257 3825 y(parts)k Fk(a)p Fq(\()p Fk(x)p Fq(\))28 b Fj(\000)838 3745 y Fh(R)905 3771 y Fo(1)885 3860 y(0)961 3825 y Fk(a)p Fq(\()p Fk(c)g Fq(+)g Fp(t)p Fq(\()p Fk(x)g Fj(\000)g Fk(c)p Fq(\)\))p Fp(dt)p Fq(.)68 b(Summing)39 b(up)i(all)e(the)j(terms)f(one)g(obtains)257 3946 y(\(5.34\).)404 4066 y Fl(Pr)-5 b(o)g(of)36 b(of)g(L)-5 b(emma)36 b(5.2.)49 b Fq(The)35 b(only)f(thing)g(to)g(b)s(e)h(pro)m(v)m (ed)h(is)e(\(5.18\))f(since)i(\(5.19\)-)257 4186 y(\(5.22\))e(can)g(b)s (e)g(read)g(directly)f(from)g(\(5.10\))g(and)h(\(5.1\).)44 b(No)m(w)33 b(\(5.18\))g(follo)m(ws,)e(either)257 4307 y(directly)g(b)m(y)h(direct)e(computation)g(from)g(\(5.10\))g(and)h (\(5.16\))f(or)h(from)e(Stok)m(es's)k(theo-)257 4427 y(rem)f(b)m(y)i(observing)f(that)1139 4528 y Fh(Z)1239 4554 y Ff(x)1194 4754 y(c)1302 4664 y Fk(a)1356 4679 y Ff(c)1397 4664 y Fq(\()p Fk(u)p Fq(\))22 b Fj(\001)g Fp(d)p Fk(u)28 b Fq(=)1851 4528 y Fh(Z)1951 4554 y Ff(y)1907 4754 y(c)2016 4664 y Fk(a)2070 4679 y Ff(c)2111 4664 y Fq(\()p Fk(u)p Fq(\))22 b Fj(\001)g Fp(d)p Fk(u)27 b Fq(=)h(0)404 4907 y Fl(Pr)-5 b(o)g(of)43 b(of)g(The)-5 b(or)g(em)42 b(5.3.)70 b Fq(W)-8 b(e)42 b(ha)m(v)m(e)h(to)e(estimate)g Fp(M)2473 4922 y Fm(")2510 4907 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fp(z)t Fq(\).)71 b(The)42 b(k)m(ey)h(fact)257 5027 y(is)j(that)f(\(see)i(\(5.19\))e(and)h(\(5.20\)\))f Fk(a)1687 5042 y Ff(y)1736 5027 y Fq(\()p Fk(x)p Fq(\))h(and)f Fj(r)2202 5042 y Ff(x)2281 5027 y Fj(\001)31 b Fk(a)2394 5042 y Ff(y)2442 5027 y Fq(\()p Fk(x)p Fq(\))46 b(div)m(erge)g(only)g (when)h Fj(j)257 5147 y Fk(x)23 b Fj(\000)g Fk(y)31 b Fj(j!)d(1)33 b Fq(and)h(this)f(div)m(ergence)h(is)f(canceled)h(b)m(y)g (the)g(deca)m(y)h(of)e Fp(G)2933 5162 y Fo(0)2972 5147 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fp(z)t Fq(\))33 b(and)257 5268 y Fj(r)340 5283 y Ff(x)418 5268 y Fj(\001)e Fp(G)554 5283 y Fo(0)593 5268 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fp(z)t Fq(\))45 b(as)g(giv)m(en)h(b)m(y)g(the)f(prop)s (osition)e(b)s(elo)m(w)i(whic)m(h)h(is)f(the)g(p)s(oin)m(t)m(wise)257 5388 y(analog)31 b(of)h(Prop)s(osition)f(4.1.)1828 5637 y(31)p eop %%Page: 32 32 32 31 bop 257 573 a Fk(Prop)s(osition)36 b(5.5.)49 b Fl(i.)43 b(F)-7 b(or)30 b(al)5 b(l)30 b Fp(z)i Fj(2)c Fp(\032)p Fq(\))p Fp(H)1842 588 y Fo(0)1882 573 y Fq(\))p Fp(;)72 b Fq(\()p Fp(H)2138 588 y Fo(0)2189 573 y Fj(\000)12 b Fp(z)t Fq(\))2365 537 y Fi(\000)p Fo(1)2491 573 y Fl(is)30 b(a)h(Carleman)e(op)-5 b(er)g(ator;)257 693 y(in)35 b(p)-5 b(articular)35 b(it)g(has)f(an)h(inte)-5 b(gr)g(al)34 b(kernel)g Fp(G)1948 708 y Fo(0)1988 693 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fp(z)t Fq(\))p Fl(.)404 814 y(ii.)44 b(L)-5 b(et)34 b Fp(K)h Fj(\032)28 b Fp(\032)p Fq(\()p Fp(H)1097 829 y Fo(0)1136 814 y Fq(\))34 b Fl(b)-5 b(e)33 b(c)-5 b(omp)g(act.)44 b(Then)33 b(ther)-5 b(e)33 b(exist)h Fp(C)g(<)28 b Fj(1)p Fp(;)44 b(\015)32 b(>)c Fq(0)33 b Fl(such)h(that)257 934 y(uniformly)h(in)g Fp(z)d Fj(2)c Fp(K)7 b Fl(:)1195 1153 y Fj(j)27 b Fp(G)1327 1168 y Fo(0)1367 1153 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fp(z)t Fq(\))27 b Fj(j\024)2008 1085 y Fp(C)p 1870 1130 353 4 v 1870 1221 a Fj(j)g Fk(x)c Fj(\000)f Fk(y)30 b Fj(j)2232 1153 y Fp(e)2277 1111 y Fi(\000)p Fm(\015)t Fi(j)p Ff(x)p Fi(\000)p Ff(y)q Fi(j)3246 1153 y Fq(\(5.36\))1096 1501 y Fj(j)e(r)1235 1516 y Ff(x)1282 1501 y Fp(G)1359 1516 y Fo(0)1398 1501 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fp(z)t Fq(\))28 b Fj(j\024)2059 1433 y Fp(C)p 1902 1478 392 4 v 1902 1569 a Fj(j)f Fk(x)22 b Fj(\000)h Fk(y)29 b Fj(j)2254 1540 y Fo(2)2303 1501 y Fp(e)2348 1459 y Fi(\000)p Fm(\015)t Fi(j)p Ff(x)p Fi(\000)p Ff(y)q Fi(j)2629 1501 y Fp(:)590 b Fq(\(5.37\))404 1726 y(F)-8 b(rom)943 1902 y(sup)955 1981 y Fm(r)r(>)p Fo(0)1107 1902 y Fp(r)1154 1860 y Fm(m)1220 1902 y Fp(e)1265 1860 y Fi(\000)p Fm(\027)t(r)1425 1902 y Fq(=)27 b Fp(M)10 b Fq(\()p Fp(m;)17 b(\027)6 b Fq(\))29 b Fp(<)f Fj(1)p Fp(;)113 b(m)28 b Fj(\025)h Fq(0)p Fp(;)17 b(\027)33 b(>)28 b Fq(0)436 b(\(5.38\))257 2133 y(\(5.19\),)45 b(\(5.20\),\(5.36\),)f(\(5.37\))e(and)h(\(5.24\))f(one)h(obtains)g (that)f(uniformly)f(for)h Fp(")j Fj(2)257 2254 y Fq([0)p Fp(;)17 b(")423 2269 y Fo(0)462 2254 y Fq(],)33 b Fp(z)f Fj(2)c Fp(K)1041 2474 y Fj(j)f Fp(M)1190 2489 y Fm(")1227 2474 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fp(z)t Fq(\))28 b Fj(j\024)g Fq(const)2105 2407 y(1)p 1953 2451 353 4 v 1953 2542 a Fj(j)g Fk(x)22 b Fj(\000)h Fk(y)29 b Fj(j)2316 2474 y Fp(e)2361 2433 y Fi(\000)p Fm(\015)t Fi(j)p Ff(x)p Fi(\000)p Ff(y)q Fi(j)p Fm(=)p Fo(2)3246 2474 y Fq(\(5.39\))257 2700 y(and)k(then)g(b)m(y)h(Y)-8 b(oung)32 b(inequalit)m(y)1093 2937 y Fj(k)c Fp(T)1228 2952 y Fm(")1265 2937 y Fp(f)38 b Fj(k\024)28 b Fq(const)q(\()1767 2801 y Fh(Z)1822 3027 y Fe(R)1870 3008 y Fd(3)1935 2869 y Fp(e)1980 2833 y Fi(\000)p Fm(\015)t Fi(j)p Ff(x)p Fi(j)p Fm(=)p Fo(2)p 1935 2914 298 4 v 1999 3005 a Fj(j)f Fk(x)h Fj(j)2243 2937 y Fp(d)p Fk(x)p Fq(\))f Fj(k)g Fp(f)39 b Fj(k)614 b Fq(\(5.40\))257 3163 y(and)33 b(the)g(pro)s(of)f (is)g(\014nished.)404 3283 y Fl(Pr)-5 b(o)g(of)31 b(of)g(Pr)-5 b(op)g(osition)30 b(5.5.)42 b Fq(The)29 b(prop)s(osition)e(follo)m(ws)g (from)g(the)j(results)f(in)e([24],)257 3404 y(so)37 b(w)m(e)g(shall)d (only)i(outline)e(the)j(pro)s(of.)53 b(The)37 b(fact)f(that)f(\()p Fp(H)2513 3419 y Fo(0)2577 3404 y Fj(\000)25 b Fp(z)t Fq(\))2766 3368 y Fi(\000)p Fo(1)2897 3404 y Fq(is)36 b(a)f(Carleman)257 3524 y(op)s(erator)28 b(follo)m(ws)g(from)f(the)j (fact)e(that)h Fp(V)1807 3539 y Fo(0)1875 3524 y Fq(is)f Fj(\000)p Fq(\001)h(b)s(ounded)h(and)f(then)g(\()p Fp(H)3074 3539 y Fo(0)3127 3524 y Fj(\000)14 b Fp(z)t Fq(\))3305 3488 y Fi(\000)p Fo(1)3429 3524 y Fj(2)257 3645 y(B)s Fq(\()p Fp(L)429 3608 y Fo(2)470 3645 y Fq(\()p Fn(R)573 3608 y Fo(3)619 3645 y Fq(\))p Fp(;)j(L)767 3608 y Fi(1)842 3645 y Fq(\()p Fn(R)946 3608 y Fo(3)991 3645 y Fq(\)\))22 b(\()f(see)i(Corollary)d(A.1.2)i(in)f([Si]\).)39 b(The)23 b(inequalit)m(y)d(\(5.36\))h(follo)m(ws)257 3765 y(from)31 b(Theorem)i(B)f(7.2.)43 b(in)32 b([24)o(])h(\(actually)e(Theorem)h(B)h (7.2.)43 b(in)31 b([24])h(is)g(stated)h(for)e(a)257 3885 y(\014xed)j Fp(z)j Fq(but)c(the)g(pro)s(of)f(co)m(v)m(ers)i(the)f(giv)m (en)g(case\).)404 4006 y(F)-8 b(or)31 b Fp(z)i Fq(=)27 b Fj(\000)p Fp(a;)45 b(a)28 b(>)g Fq(0)k(and)h(su\016cien)m(tly)g (large)403 4220 y Fp(G)480 4235 y Fo(0)520 4220 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fj(\000)p Fp(a)p Fq(\))27 b(=)1265 4153 y(1)p 1045 4198 488 4 v 1045 4289 a(4)p Fp(\031)32 b Fj(j)27 b Fk(x)22 b Fj(\000)h Fk(y)29 b Fj(j)1543 4220 y Fp(e)1588 4179 y Fi(\000)p Fm(a)1680 4156 y Fd(1)p Fg(=)p Fd(2)1777 4179 y Fi(j)p Ff(x)p Fi(\000)p Ff(y)q Fi(j)1057 4483 y Fj(\000)1157 4347 y Fh(Z)1212 4572 y Fe(R)1260 4554 y Fd(3)1546 4415 y Fq(1)p 1326 4460 490 4 v 1326 4551 a(4)p Fp(\031)i Fj(j)c Fk(x)c Fj(\000)f Fk(u)28 b Fj(j)1825 4483 y Fp(e)1870 4441 y Fi(\000)p Fm(a)1962 4418 y Fd(1)p Fg(=)p Fd(2)2059 4441 y Fi(j)p Ff(x)p Fi(\000)p Ff(u)p Fi(j)2246 4483 y Fp(V)21 b Fq(\()p Fp(u)p Fq(\))p Fp(G)2533 4498 y Fo(0)2572 4483 y Fq(\()p Fk(u)p Fp(;)c Fk(y)r Fq(;)g Fj(\000)p Fp(a)p Fq(\))p Fp(d)p Fk(u)146 b Fq(\(5.41\))257 4709 y(and)33 b(then)360 4904 y Fj(r)443 4919 y Ff(x)490 4904 y Fp(G)567 4919 y Fo(0)607 4904 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fj(\000)p Fp(a)p Fq(\))27 b(=)p Fj(r)1205 4919 y Ff(x)1252 4904 y Fq(\()1519 4837 y(1)p 1300 4881 488 4 v 1300 4972 a(4)p Fp(\031)k Fj(j)d Fk(x)22 b Fj(\000)h Fk(y)29 b Fj(j)1798 4904 y Fp(e)1843 4863 y Fi(\000)p Fm(a)1935 4840 y Fd(1)p Fg(=)p Fd(2)2032 4863 y Fi(j)p Ff(x)p Fi(\000)p Ff(y)q Fi(j)2217 4904 y Fq(\))1144 5166 y Fj(\000)1244 5031 y Fh(Z)1299 5256 y Fe(R)1347 5237 y Fd(3)1402 5166 y Fj(r)1485 5181 y Ff(x)1532 5166 y Fq(\()1801 5099 y(1)p 1580 5143 490 4 v 1580 5235 a(4)p Fp(\031)j Fj(j)27 b Fk(x)22 b Fj(\000)h Fk(u)28 b Fj(j)2080 5166 y Fp(e)2125 5125 y Fi(\000)p Fm(a)2217 5102 y Fd(1)p Fg(=)p Fd(2)2314 5125 y Fi(j)p Ff(x)p Fi(\000)p Ff(u)p Fi(j)2500 5166 y Fq(\))p Fp(V)22 b Fq(\()p Fp(u)p Fq(\))p Fp(G)2826 5181 y Fo(0)2865 5166 y Fq(\()p Fk(u)p Fp(;)17 b Fk(y)r Fq(;)g Fj(\000)p Fp(a)p Fq(\))p Fp(d)p Fk(u)3246 5352 y Fq(\(5.42\))1828 5637 y(32)p eop %%Page: 33 33 33 32 bop 257 573 a Fq(whic)m(h)34 b(giv)m(es)f(\(5.37\))f(for)g Fp(z)g Fq(=)c Fj(\000)p Fp(a)p Fq(.)44 b(Finally)30 b(\(5.37\))i(is)g (\\carried")g(from)g Fj(\000)p Fp(a)h Fq(to)f Fp(z)h Fj(2)28 b Fp(K)257 693 y Fq(via)k(the)h(resolv)m(en)m(t)h(equation)488 913 y Fj(r)571 928 y Ff(x)618 913 y Fp(G)695 928 y Fo(0)734 913 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fp(z)t Fq(\))28 b(=)p Fj(r)1254 928 y Ff(x)1301 913 y Fp(G)1378 928 y Fo(0)1417 913 y Fq(\()p Fk(x)p Fp(;)17 b Fk(y)r Fq(;)g Fj(\000)p Fp(a)p Fq(\))1417 b(\(5.43\))1193 1118 y(+)22 b(\()p Fp(z)27 b Fq(+)22 b Fp(a)p Fq(\))1605 983 y Fh(Z)1660 1208 y Fe(R)1708 1189 y Fd(3)1763 1118 y Fj(r)1846 1133 y Ff(x)1893 1118 y Fp(G)1970 1133 y Fo(0)2010 1118 y Fq(\()p Fk(x)p Fp(;)17 b Fk(u)p Fq(;)g Fj(\000)p Fp(a)p Fq(\))p Fp(G)2500 1133 y Fo(0)2539 1118 y Fq(\()p Fk(u)p Fp(;)g Fk(y)r Fq(;)g Fp(z)t Fq(\))p Fp(d)p Fk(u)p Fp(:)231 b Fq(\(5.44\))257 1466 y Fr(6)161 b(App)t(endix)257 1685 y Fq(Although)28 b(the)i(Prop)s(osition)d(3)h(in)g([17])h(pro)m(vides)g (the)g(nedeed)i(construction)e(w)m(e)h(shall)257 1806 y(giv)m(e)j(b)s(elo)m(w)f(a)h(more)e(aestetical)h(form)f(of)i(it.)404 1926 y(Let)42 b Fp(P)55 b Fq(b)s(e)41 b(a)h(b)s(ounded)g(idemp)s(oten)m (t.)70 b(Then)43 b(\(see)g(e.g)f([14)o(])g(Ch.)g(I)s(I)s(I.)g(4)g(\))f Fj(H)j Fq(=)257 2046 y Fj(M)23 b Fq(+)g Fj(N)47 b Fq(suc)m(h)36 b(that)d(if)f Fp(f)40 b Fq(=)30 b Fp(m)23 b Fq(+)g Fp(n;)114 b(m)30 b Fj(2)g(M)p Fp(;)113 b(n)30 b Fj(2)g(N)48 b Fq(then)34 b Fp(P)14 b(f)40 b Fq(=)29 b Fp(m)34 b Fq(i.e.)46 b Fp(P)h Fq(is)257 2167 y(a)33 b(pro)5 b(jection.)43 b(By)33 b(direct)f (computation)1229 2387 y(\()p Fp(P)k Fj(\000)23 b Fp(z)t Fq(\)\()p Fp(P)36 b Fq(+)22 b Fp(z)27 b Fj(\000)22 b Fq(1\))28 b(=)f Fp(z)t Fq(\()p Fp(z)h Fj(\000)22 b Fq(1\))257 2607 y(whic)m(h)32 b(in)e(particular)f(implies)f(that)j Fp(\033)t Fq(\()p Fp(P)14 b Fq(\))27 b(=)h Fj(f)p Fq(0)p Fp(;)17 b Fq(1)p Fj(g)p Fq(.)41 b(Moreo)m(v)m(er)33 b(it)d(giv)m(es)h (the)g(follo)m(w-)257 2727 y(ing)h(form)m(ula)f(for)h(the)h(resolv)m (en)m(t)g(of)f Fp(P)1345 2997 y Fq(\()p Fp(P)k Fj(\000)22 b Fp(z)t Fq(\))1668 2955 y Fi(\000)p Fo(1)1791 2997 y Fq(=)1905 2929 y(\()p Fp(P)35 b Fq(+)22 b Fp(z)27 b Fj(\000)c Fq(1\))p 1905 2974 493 4 v 1978 3065 a Fp(z)t Fq(\()p Fp(z)28 b Fj(\000)22 b Fq(1\))404 3272 y(Let)27 b(no)m(w)h Fp(T)41 b Fq(b)s(e)28 b(b)s(ounded)g(and)f(\\almost")e(idemp)s(oten)m (t\(but)i(not)g(necessarily)h(self-)257 3393 y(adjoin)m(t\).)43 b(The)33 b(problem)f(is)g(whether)i Fp(T)46 b Fq(is)32 b(close)h(to)f(an)g(idemp)s(oten)m(t.)257 3596 y Fk(Prop)s(osition)k (6.1.)49 b Fl(L)-5 b(et)47 b Fp(T)60 b Fl(b)-5 b(e)47 b(a)f(b)-5 b(ounde)g(d)46 b(op)-5 b(er)g(ator,)49 b Fq(\001)h(=)f Fp(T)2749 3560 y Fo(2)2819 3596 y Fj(\000)32 b Fp(T)14 b Fl(.)80 b(Then)45 b(for)257 3717 y Fj(k)28 b Fq(\001)g Fj(k\021)g Fp(\016)j Fj(\024)e Fq(1)p Fp(=)p Fq(4)p Fl(,)34 b Fj(fj)27 b Fp(z)g Fj(\000)22 b Fq(1)28 b Fj(j)p Fq(=)f(1)p Fp(=)p Fq(2)p Fj(g)g(\032)h Fp(\032)p Fq(\()p Fp(T)14 b Fq(\))35 b Fl(and)789 4087 y Fj(k)28 b Fp(T)36 b Fj(\000)1115 4020 y Fq(1)p 1069 4064 141 4 v 1069 4155 a(2)p Fp(\031)t(i)1237 3951 y Fh(I)1292 4177 y Fi(j)p Fm(z)s Fi(\000)p Fo(1)p Fi(j)p Fo(=1)p Fm(=)p Fo(2)1622 4087 y Fq(\()p Fp(T)g Fj(\000)22 b Fp(z)t Fq(\))1939 4046 y Fi(\000)p Fo(1)2034 4087 y Fp(dz)32 b Fj(k\024)2327 4020 y Fq(4)p Fp(\016)t Fq(\(2)27 b Fj(k)h Fp(T)41 b Fj(k)28 b Fq(+1\))p 2327 4064 627 4 v 2507 4155 a(1)22 b Fj(\000)h Fq(4)p Fp(\016)3295 4087 y Fq(\(6.1\))404 4371 y Fl(Pr)-5 b(o)g(of)404 4491 y Fq(By)33 b(direct)f(computation,)886 4711 y(\()p Fp(T)j Fj(\000)23 b Fp(z)t Fq(\)\()p Fp(T)37 b Fq(+)22 b Fp(z)k Fj(\000)d Fq(1\))k(=)h Fp(z)t Fq(\()p Fp(z)g Fj(\000)22 b Fq(1\)[1)g(+)g(\001)p Fp(=z)t Fq(\()p Fp(z)28 b Fj(\000)22 b Fq(1\)])257 4931 y(whereof)34 b(for)e Fj(k)27 b Fq(\001)h Fj(k)g Fp(=)f Fj(j)h Fp(z)t Fq(\()p Fp(z)f Fj(\000)c Fq(1\))k Fj(j)p Fp(<)g Fq(1)996 5199 y(\()p Fp(T)35 b Fj(\000)23 b Fp(z)t Fq(\))1313 5158 y Fi(\000)p Fo(1)1436 5199 y Fq(=)1549 5132 y Fp(T)36 b Fq(+)22 b Fp(z)27 b Fj(\000)c Fq(1)p 1549 5177 412 4 v 1582 5268 a Fp(z)t Fq(\(1)f Fj(\000)h Fp(z)t Fq(\))1970 5199 y([1)f(+)2309 5132 y(\001)p 2176 5177 346 4 v 2176 5268 a Fp(z)t Fq(\(1)h Fj(\000)f Fp(z)t Fq(\))2532 5199 y(])2559 5158 y Fi(\000)p Fo(1)2681 5199 y Fq(=)1828 5637 y(33)p eop %%Page: 34 34 34 33 bop 912 609 a Fq(\()1097 541 y Fp(T)p 960 586 346 4 v 960 677 a(z)t Fq(\(1)22 b Fj(\000)h Fp(z)t Fq(\))1338 609 y Fj(\000)1448 541 y Fq(1)p 1447 586 50 4 v 1447 677 a Fp(z)1507 609 y Fq(\)[1)f Fj(\000)1885 541 y Fq(\001)p 1753 586 346 4 v 1753 677 a Fp(z)t Fq(\(1)g Fj(\000)h Fp(z)t Fq(\))2109 609 y(\(1)e(+)2458 541 y(\001)p 2325 586 V 2325 677 a Fp(z)t Fq(\(1)i Fj(\000)g Fp(z)t Fq(\))2681 609 y(\))2719 568 y Fi(\000)p Fo(1)2814 609 y Fq(])454 b(\(6.2\))257 885 y(whereof)34 b(for)e Fp(\016)f(<)d Fq(1)p Fp(=)p Fq(4)316 1153 y Fp(P)36 b Fj(\000)22 b Fp(T)42 b Fq(=)27 b Fj(\000)p Fq(\001)931 1086 y(1)p 884 1130 141 4 v 884 1221 a(2)p Fp(\031)t(i)1052 1017 y Fh(I)1108 1243 y Fi(j)p Fm(z)s Fi(\000)p Fo(1)p Fi(j)p Fo(=1)p Fm(=)p Fo(2)1613 1086 y Fq(1)p 1464 1130 346 4 v 1464 1221 a Fp(z)t Fq(\(1)c Fj(\000)f Fp(z)t Fq(\))1820 1153 y(\()2006 1086 y Fp(T)p 1868 1130 V 1868 1221 a(z)t Fq(\(1)h Fj(\000)f Fp(z)t Fq(\))2246 1153 y(+)g(1\)\(1)g(+)2780 1086 y(\001)p 2648 1130 V 2648 1221 a Fp(z)t Fq(\(1)g Fj(\000)h Fp(z)t Fq(\))3003 1153 y(\))3041 1112 y Fi(\000)p Fo(1)3136 1153 y Fp(dz)63 b Fq(\(6.3\))257 1442 y(whic)m(h)33 b(giv)m(es)g(\()g(6.1\))404 1563 y Fl(R)-5 b(emarks)404 1683 y Fq(6.1.)43 b(If)32 b Fp(T)47 b Fq(is)32 b(self-adjoin)m(t)e (then)k Fp(P)46 b Fq(is)32 b(also)f(self-adjoin)m(t.)404 1803 y(6.2.)56 b(By)38 b(p)s(erforming)d(the)i(in)m(tegral)f(in)g(\()h (6.2\))f(using)h(the)h(residue)f(theorem)g(one)257 1924 y(can)c(sho)m(w)h(that)551 2144 y Fp(P)h Fj(\000)23 b Fp(T)41 b Fq(=)28 b(\()p Fp(T)36 b Fj(\000)22 b Fq(1)p Fp(=)p Fq(2\)[\(1)g(+)g(4\001\))1768 2103 y Fi(\000)p Fo(1)p Fm(=)p Fo(2)1955 2144 y Fj(\000)g Fq(1])847 2304 y(=)28 b(4\001\()p Fp(T)36 b Fj(\000)22 b Fq(1)p Fp(=)p Fq(2\)\(1)g(+)g(4\001\))1871 2263 y Fi(\000)p Fo(1)p Fm(=)p Fo(2)2036 2304 y Fq(\)[\(1)f(+)h(4\001\))2475 2263 y Fi(\000)p Fo(1)p Fm(=)p Fo(2)2662 2304 y Fq(+)g(1])2836 2263 y Fi(\000)p Fo(1)p Fm(=)p Fo(2)3295 2304 y Fq(\(6.4\))257 2524 y(whic)m(h)33 b(giv)m(es)g(the)g(estimation)1289 2793 y Fj(k)27 b Fp(P)36 b Fj(\000)22 b Fp(T)42 b Fj(k\024)1828 2726 y Fq(2)p Fp(\016)t Fq(\(2)27 b Fj(k)g Fp(T)42 b Fj(k)27 b Fq(+1\))p 1828 2771 627 4 v 2008 2862 a(1)22 b Fj(\000)g Fq(4)p Fp(\016)3295 2793 y Fq(\(6.5\))257 3053 y(whic)m(h)33 b(is)g(b)s(etter)g(than)f(\(6.1\))g(b)m(y)i(a)e (factor)g(of)g(2.)404 3409 y Fk(Ac)m(kno)m(wledgemen)m(ts)404 3530 y Fq(The)38 b(\014rst)g(draft)f(of)g(this)g(pap)s(er)g(w)m(as)i (written)e(during)g(a)g(one)g(mon)m(th)g(participa-)257 3650 y(tion)g(to)g(the)h(program)e(\\Magnetic)h(Fields)g(and)g(Sc)m (hr\177)-49 b(odinger)38 b(Equation")f(at)g(E.S.I.)257 3771 y(Vienna.)42 b(It)27 b(is)g(a)f(pleasure)i(to)f(thank)g(T.)h (Ho\013mann-)e(Ostenhof)h(for)g(making)f(the)h(visit)257 3891 y(p)s(ossible.)41 b(Also)26 b(the)h(\014nancial)e(supp)s(ort)i (from)e(E.S.I.)j(and)e(from)f(Romanian)g(Ministry)257 4011 y(of)32 b(Education)h(\(gran)m(t)f(CNCSU)i(13C\))f(is)f(ac)m(kno)m (wledged.)1828 5637 y(34)p eop %%Page: 35 35 35 34 bop 257 573 a Fr(References)306 792 y Fq([1])49 b(J.)28 b(Aguilar,)g(J.-M.)g(Com)m(b)s(es,)i Fl(A)i(class)e(of)h (analytic)f(p)-5 b(erturb)g(ations)31 b(for)g(one-b)-5 b(o)g(dy)458 912 y(Schr\177)-50 b(odinger)33 b(hamiltonians)p Fq(,)e(Comm)m(un.)i(Math.)f(Ph)m(ys.)j Fk(22)d Fq(\(1971\),)g(259-279.) 306 1116 y([2])49 b(J.)22 b(Avron,)j(B.)e(Simon,)g Fl(Stability)i(of)g (gaps)g(for)g(p)-5 b(eriop)g(dic)25 b(p)-5 b(otentials)25 b(under)g(varia-)458 1236 y(tions)k(of)f(the)i(magnetic)e(\014eld)p Fq(,)f(J.)f(Ph)m(ys.)i(A:)e(Math-Gen)57 b Fk(18)26 b Fq(\(1985\),)g(2199-3009.)306 1439 y([3])49 b(E.)30 b(Balslev,)g(J.-M.) g(Com)m(b)s(es,)h Fl(Sp)-5 b(e)g(ctr)g(al)32 b(pr)-5 b(op)g(erties)31 b(of)h(many)g(b)-5 b(o)g(dy)33 b(Schr\177)-50 b(odinger)458 1560 y(op)-5 b(er)g(ators)35 b(with)f(dilation)h (analytic)g(inter)-5 b(actions)p Fq(,)32 b(Comm)m(un.)g(Math.)h(Ph)m (ys.)i Fk(22)458 1680 y Fq(\(1971\),)c(280-294.)306 1884 y([4])49 b(J.-M.)34 b(Com)m(b)s(es,)h(L.)f(Thomas,)g Fl(Asymptotic)i(b)-5 b(ehaviour)36 b(of)f(Eigenfunctions)g(for)458 2004 y(multip)-5 b(article)39 b(Schr\177)-50 b(odinger)38 b(op)-5 b(er)g(ators)p Fq(,)38 b(Comm)m(un.)f(Math.)h(Ph)m(ys.)i Fk(34)d Fq(\(1973\),)458 2124 y(251-276.)306 2328 y([5])49 b(H.D.)24 b(Cornean,)j Fl(On)g(the)g(magnetization)f(of)h(a)g(char)-5 b(ge)g(d)27 b(Bose)f(gas)h(in)g(the)g(c)-5 b(anon-)458 2448 y(ic)g(al)34 b(ensemble)p Fq(,)d(Comm)m(un.)i(Math.)f(Ph)m(ys.)j (to)d(app)s(ear.)306 2652 y([6])49 b(H.D.)i(Cornean,)56 b(G.)50 b(Nenciu,)56 b Fl(On)c(the)f(eigenfunction)g(de)-5 b(c)g(ay)52 b(for)f(magnetic)458 2772 y(Schr)-5 b(o)g(dinger)33 b(op)-5 b(er)g(ators)41 b Fq(Comm)m(un.)32 b(Math.)h(Ph)m(ys.)71 b Fk(192)p Fq(,)33 b(\(1998\),)f(671-685.)306 2975 y([7])49 b(H.)32 b(L.)f(Cycon,)j(R.)d(G.)g(F)-8 b(ro)s(ese,)32 b(W.)g(Kirc)m(h,)g(B.)g(Simon,)e Fl(Schr\177)-50 b(odinger)33 b(op)-5 b(er)g(ators,)458 3096 y(with)36 b(applic)-5 b(ations)35 b(to)h(quantum)h(me)-5 b(chanics)35 b(and)g(glob)-5 b(al)36 b(ge)-5 b(ometry)36 b Fq(,)f(Springer)458 3216 y(V)-8 b(erlag,)31 b(New-Y)-8 b(ork,)34 b(1986.)306 3420 y([8])49 b(X.)30 b(Gonze,)h Fl(Perturb)-5 b(ation)33 b(exp)-5 b(ansions)30 b(of)j(variational)e(principles)g(at)i(arbitr)-5 b(ary)458 3540 y(or)g(der)p Fq(,)32 b(Ph)m(ys.)j(Rev.)e Fk(A52)f Fq(\(1995\),)g(1086-1095.)306 3743 y([9])49 b(B.)25 b(Hel\013er,)h(J)e(Sjostrand,)j Fl(Equation)g(de)g(Schr\177)-50 b(odinger)27 b(ave)-5 b(c)26 b(champ)h(magnetique)458 3864 y(et)39 b(e)-5 b(quation)38 b(de)g(Harp)-5 b(er)p Fq(,)38 b(Lectures)g(Notes)f(in)f(Ph)m(ysics)i Fk(345)e Fq(\(1989\),)h(118-197.)458 3984 y(Springer)32 b(Berlin)f(1989.)257 4187 y([10])49 b(P)-8 b(.)36 b(D.)e(Hislop,)h(I.)h(M.)g(Sigal,)e Fl(Intr)-5 b(o)g(duction)36 b(to)i(sp)-5 b(e)g(ctr)g(al)37 b(the)-5 b(ory.)38 b(With)f(applic)-5 b(a-)458 4308 y(tions)34 b(to)i(Schr\177)-50 b(odinger)33 b(op)-5 b(er)g(ators)p Fq(,)32 b(Springer)g(V)-8 b(erlag,)32 b(New-Y)-8 b(ork,)33 b(1996.)257 4511 y([11])49 b(J.)29 b(Ho)m(wland,)h Fl(A)i(note)f(on)g (sp)-5 b(e)g(ctr)g(al)32 b(c)-5 b(onc)g(entr)g(ation)30 b(for)i(non-isolarwd)e(eigenval-)458 4632 y(ues)p Fq(,)i(J.)h(Math.)g (Anal.)f(Appl.)g Fk(158)h Fq(\(1991\),)e(285-292.)257 4835 y([12])49 b(W.)42 b(Hunzik)m(er,)i Fl(Notes)f(on)g(asymptotic)g(p) -5 b(erturb)g(ation)43 b(the)-5 b(ory)43 b(for)g(Sch\177)-50 b(odinger)458 4955 y(eigenvalue)34 b(pr)-5 b(oblem)p Fq(,)31 b(Helv.)i(Ph)m(ys.)h(Acta)f Fk(61)g Fq(\(1988\),)e(219-233.)257 5159 y([13])49 b(W.)40 b(Hunzik)m(er,)i(C.)e(A.)g(Pillet,)f Fl(De)-5 b(gener)g(ate)40 b(asymptotic)h(p)-5 b(erturb)g(ation)42 b(the)-5 b(ory)p Fq(,)458 5279 y(Comm)m(un.)32 b(Math.)h(Ph)m(ys.)h Fk(90)f Fq(\(1983\),)f(219-233.)1828 5637 y(35)p eop %%Page: 36 36 36 35 bop 257 573 a Fq([14])49 b(T.)60 b(Kato,)66 b Fl(Perturb)-5 b(ation)60 b(The)-5 b(ory)59 b(for)h(line)-5 b(ar)59 b(op)-5 b(er)g(ators)p Fq(,)66 b(second)60 b(edition,)458 693 y(Springer)32 b(V)-8 b(erlag,)32 b(Berlin,)f(1980.)257 897 y([15])49 b(J.)28 b(M>)g(Luttinger,)g Fl(The)i(e\013e)-5 b(ct)30 b(of)g(a)g(magnetic)f(\014eld)h(on)g(ele)-5 b(ctr)g(ons)30 b(in)g(a)g(p)-5 b(erio)g(dic)458 1017 y(p)g(otentiasl)p Fq(,)32 b(Ph)m(ys.)i(Rev.,)g Fk(15)e Fq(\(1951\),)g(814-817.)257 1220 y([16])49 b(G.)56 b(Nenciu,)62 b Fl(A)-5 b(diab)g(atic)57 b(the)-5 b(or)g(em)56 b(and)g(sp)-5 b(e)g(ctr)g(al)56 b(c)-5 b(onc)g(entr)g(ation)p Fq(,)62 b(Comm)m(un.)458 1341 y(Math.)33 b(Ph)m(ys.)h Fk(82)f Fq(\(1981\),)e(121-135.)257 1544 y([17])49 b(G.)d(Nenciu,)k Fl(Line)-5 b(ar)47 b(adiab)-5 b(atic)47 b(the)-5 b(ory.)48 b(Exp)-5 b(onential)46 b(estimates)p Fq(,)k(Comm)m(un.)458 1665 y(Math.)33 b(Ph)m(ys.)h(textbf152)f (\(1993\),)e(479-496.)257 1868 y([18])49 b(G.)23 b(Nenciu,)k Fl(Dynamics)e(of)i(Band)f(Ele)-5 b(ctr)g(ons)27 b(in)f(Ele)-5 b(ctric)27 b(and)g(Magnetic)f(Fields:)458 1988 y(R)n(igor)-5 b(ous)36 b(Justi\014c)-5 b(ation)38 b(of)f(E\013e)-5 b(ctive)37 b(Hamiltonians)79 b Fq(Rev.)36 b(Mo)s(d.)g(Ph)m(ys.)77 b Fk(63)458 2109 y Fq(\(1991\),)31 b(91-127.)257 2312 y([19])49 b(G.)24 b(Nenciu,)j Fl(Stability)h(of)g(ener)-5 b(gy)27 b(gapsunder)g(variations)g(of)h(the)g(magnetic)f(\014eld)p Fq(,)458 2433 y(Lett.)33 b(Math.)g(Ph)m(ys.)h Fk(11)f Fq(\(1986\),)e(127-132.)257 2636 y([20])49 b(A.)30 b(Nenciu,)h(G.)e (Nenciu,)i Fl(Dynamics)g(of)h(Blo)-5 b(ch)32 b(ele)-5 b(ctr)g(ons)32 b(in)g(external)g(\014elds)f(II.)458 2756 y(The)f(existenc)-5 b(e)30 b(of)g(Stark-Wannier)g(ladder)g(r)-5 b(esonanc)g(es)p Fq(,)27 b(J.)i(Ph)m(ys.)p Fk(A14)g Fq(\(1981\),)458 2877 y(2817-2827.)257 3080 y([21])49 b(M.)28 b(Reed,)h(B.)f(Simon,)g Fl(Metho)-5 b(ds)30 b(of)g(mo)-5 b(dern)29 b(mathematic)-5 b(al)30 b(physics,)g(II.)g Fq(,)e(Aca-)458 3200 y(demic)k(Press,)i (New-Y)-8 b(ork,)33 b(1978.)257 3404 y([22])49 b(M.)26 b(Reed,)i(B.)e(Simon,)g Fl(Metho)-5 b(ds)29 b(of)f(mo)-5 b(dern)28 b(mathematic)-5 b(al)28 b(physics,)h(IV.)g(A)n(nal-)458 3524 y(ysis)35 b(of)f(op)-5 b(er)g(ators)p Fq(,)32 b(Academic)g(Press,) j(New-Y)-8 b(ork,)33 b(1978.)257 3728 y([23])49 b(R.)39 b(C.)h(Riddell,)f Fl(Sp)-5 b(e)g(ctr)g(al)41 b(c)-5 b(onc)g(entr)g (ation)40 b(for)h(self-adjoint)f(op)-5 b(er)g(ators)48 b Fq(P)m(ac.)40 b(J.)458 3848 y(Math.)33 b Fk(23)f Fq(\(1967\),)g (371-401.)257 4051 y([24])49 b(B.)29 b(Simon,)g Fl(Schr\177)-50 b(odinger)30 b(semigr)-5 b(oups)p Fq(,)29 b(Bull.)f(Am.)g(Math.)i(So)s (c.)f Fk(7)g Fq(\(1982\),)g(447-)458 4172 y(526.)257 4375 y([25])49 b(B.)33 b(Spain,)f Fl(V)-7 b(e)i(ctor)34 b(analysis)p Fq(,)e(V)-8 b(an)33 b(Nostrand,)g(London,)g(1965.)257 4579 y([26])49 b(B.)33 b(Thaller,)e Fl(The)j(Dir)-5 b(ac)34 b(e)-5 b(quation)p Fq(,)33 b(Springer)f(V)-8 b(erlag,)31 b(Berlin,)h(1992.)257 4782 y([27])49 b(E.)h(V)-8 b(o)s(c)m(k,)54 b(W.)c(Hunzik)m(er,)k Fl(Stability)d(of)f(Schr\177)-50 b(odinger)49 b(eigenvalue)g(pr)-5 b(oblems)p Fq(,)458 4902 y(Comm)m(un.)32 b(Math.)h(Ph)m(ys.)h Fk(83)f Fq(\(1982\),)f (281-302.)1828 5637 y(36)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0003020209801--