Диссертация (1136178), страница 7
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Ïîýòîìó ðåøåíèå ñïåêòðàëüíîé çàäà÷è äëÿ îïåðàòîðàH0 + ~2 V ñâîäèòñÿ ê ðåøåíèþ òàêîé çàäà÷è äëÿ îïåðàòîðà V0 íàñîáñòâåííîì ïîäïðîñòðàíñòâå îïåðàòîðà H0 . Îòìåòèì, ÷òî ðåàëèçî÷àñòüþâàòü ýòó èäåþ óäàåòñÿ ëèøü ïðè äîñòàòî÷íî æåñòêèõ îãðàíè÷åíèÿõíà îïåðàòîðH0 .À èìåííî, äîëæíî áûòü âûïîëíåíî óñëîâèåexp {ãäåI- åäèíè÷íûé îïåðàòîð.2πiH0 } = I,~43Îïðåäåëèì1V0 =2πZ2πexp {−01V] =2πiτiτH0 }V (q , q ) exp { H0 }dτ,~~(1.13)Z2πiτiτ(π − τ ) exp {− H0 }V (q , q ) exp { H0 }dτ,~~0U = exp {−i~V] }.(1.14)Ñïðàâåäëèâà [38]ÎïåðàòîðûËåììà 1.2.(1.13), (1.14)óäîâëåòâîðÿþò ðàâåíñòâàì(1.11), (1.12).Îòìåòèì, ÷òî â [38] óêàçàíû òàêæå ôîðìóëû è äëÿ ñëåäóþùåãî(ïîðÿäêà~4) ÷ëåíà ðàçëîæåíèÿ â ïðàâîé ÷àñòè (1.11).Âû÷èñëèìV0â ñëó÷àå äâóìåðíîãî îñöèëëÿòîðà (0.15).
Ïî-ñêîëüêóexp {−iτiτ−τH0 }V (q , q ) exp { H0 } = γHV,0~~òðàåêòîðèÿì ñèñòåìû Ãàìèëüòîíà H0ãäå−τγH0−τ ,òî ïîëó÷àåì îïåðàòîð- ñäâèã ïîV0 = V0 (q1 , q2 , −i~∂∂, −i~),∂q1∂q2íà âðåìÿ(1.15)ñèìâîë êîòîðîãî1V0 (q1 , q2 , p1 , p2 ) =2πZ2πV (q1 cos τ − p1 sin τ, q2 cos τ − p2 sin τ )dτ.0(1.16)ÎïåðàòîðûÂåéëþ.qj , pj = −i~∂/∂qj (j = 1, 2)â (1.15) óïîðÿäî÷åíû ïî44Òàê êàê âûïîëíåíî ðàâåíñòâî (1.12), òî îïåðàòîðV0ìîæåòáûòü ïðåäñòàâëåí â âèäåV0 = f (S1 , S2 , S3 ),ãäåS1 , S2 , S3(1.17) ñèììåòðèçèðîâàííûå ïî Âåéëþ øâèíãåðîâñêèå îá-ðàçóþùèå àëãåáðû âðàùåíèé [159], óäîâëåòâîðÿþùèå öèêëè÷åñêèìêîììóòàöèîííûì ñîîòíîøåíèÿì[S1 , S2 ] = i~S3 ,[S2 , S3 ] = i~S1 , ïðåäñòàâëåíèè Øâèíãåðàùèõ îïåðàòîðîâ âS1 , S2 , S3[S3 , S1 ] = i~S2 .(1.18)ðåàëèçóþòñÿ â âèäå ñëåäóþ-L2 (R2 )q1 q2 + p1 p2,S1 =2q1 p 2 − q2 p1S2 =,2q12 + p21 q22 + p22S3 =−.44(1.19)f (S1 , S2 , S3 ) , ðàçðåøèì ñèñòåqj , pj (j = 1, 2) è ïîäñòàâèì ïîëó÷èâøèåñÿ×òîáû íàéòè ñèìâîë îïåðàòîðàìó (1.19) îòíîñèòåëüíîâûðàæåíèÿ â (1.16).
 ðåçóëüòàòå, ïîëó÷àåì, ÷òîf (S1 , S2 , S3 ) =12πZ2πrq√V ( 2 S3 + S12 + S22 + S32 cos τ,0√rq2 −S3 + S12 + S22 + S32S1S2!pcos τ − p 2sin τ )dτ.S12 + S22S1 + S22V (q1 , q2 ) - ìíîãî÷ëåí 4 ñòåïåíè, f (S1 , S2 , S3 ) ÿâëÿåòñÿìíîãî÷ëåíîì 2 ñòåïåíè.  ÷àñòíîñòè, åñëè V (q1 , q2 ) èìååò âèä (0.17), ñëó÷àå, åñëèòî1f (S1 , S2 ) =2πZ2π{4 cos2 τ (S1 cos τ − S2 sin τ )2 +013+2B cos τ (S1 cos τ − S2 sin τ )}dτ = S12 + S22 + BS1 .22(1.20)45Äàëåå ðàññìîòðèì ãèëüáåðòîâî ïîäïðîñòðàíñòâîñîñòîÿùåå èç ñîáñòâåííûõ ôóíêöèéçíà÷åíèþ~(` + 1) , ` = 0, 1, 2, . . .H0 ,H` ⊂ L2 (R2 ),îòâå÷àþùèõ ñîáñòâåííîìó. Îðòîíîðìèðîâàííûé áàçèñ âH`îáðàçóþò ôóíêöèè1χk = pH`−kπ~2` k!(` − k)!k = 0, . .
. , `,ãäåHk (z)q12 + q22q2q1√√Hkexp {−},~~~( k=0,1,2, ... ) - ïîëèíîìû Ýðìèòà[6]. Âîñ-ïîëüçóåìñÿ ðàâåíñòâàìè (1.11), (1.17).  ðåçóëüòàòå, íà ãèëüáåðòîâîì ïîäïðîñòðàíñòâåH`ïîëó÷àåì ñïåêòðàëüíóþ çàäà÷óf (S1 , S2 , S3 )ϕ = ξϕ,(1.21)kϕkH` = 1,(1.22)ê êîòîðîé ñâîäèòñÿ èñõîäíàÿ çàäà÷à (0.13), (0.14). Ñîáñòâåííûå çíà÷åíèÿ (1.21), (1.22) îáîçíà÷èìξ = ξk,` (k = 0, 1, . . . ) è óïîðÿäî÷èì èõïî óáûâàíèþ. Ñïåêòð èñõîäíîé çàäà÷è (0.13), (0.14) èìååò òîãäà äëÿ` ïîðÿäêà ~−1ñëåäóþùóþ àñèìïòîòèêóλ = ~(` + 1) + ~2 ξk,` + O(~4 ).Àñèìïòîòèêà ñîîòâåòñòâóþùèõ ñîáñòâåííûõ ôóíêöèé äàåòñÿ ôîðìóëîéψ = Uϕk,` + O(~3 ),ãäåϕk,`(1.23) ñîáñòâåííàÿ ôóíêöèÿ çàäà÷è (1.21), (1.22), îòâå÷àþùàÿñîáñòâåííîìó çíà÷åíèþ2.3.ξk,` .ÎïåðàòîðUèìååò âèä (1.14).Êîãåðåíòíîå ïðåîáðàçîâàíèå×òîáû ðåøèòü çàäà÷ó (1.21), (1.22) âîñïîëüçóåìñÿ êîãåðåíòíûìïðåîáðàçîâàíèåì [60;137]`+1I` (g) =2πZg(z) | z >Cdz dz,(1+ | z |2 )`+2(1.24)46ãäå| z > êîãåðåíòíûå ñîñòîÿíèÿ äëÿ àëãåáðû âðàùåíèé.
Îíîîòîáðàæàåò ãèëüáåðòîâî ïðîñòðàíñòâîP`àíòèãîëîìîðôíûõ ìíîãî-` íà ãèëüáåðòîâî ïðîñòðàíñòâî H` . ÑêàëÿðP` çàäàåòñÿ ñëåäóþùåé ôîðìóëîé [137]÷ëåíîâ ñòåïåíè íå âûøåíîå ïðîèçâåäåíèå â(g1 , g2 )P`ÏðåîáðàçîâàíèåI``+1=2πZg1 (z)g2 (z)Cdz dz.(1+ | z |2 )`+2(1.25)I`−1 : H` →óíèòàðíîå, îáðàòíîå ïðåîáðàçîâàíèåI`−1 (ϕ) = (ϕ, | z >)H` .Êîãåðåíòíûå ñîñòîÿíèÿ | z > îïðåäåëÿþòñÿ ðàâåíñòâîì | z >=ez(S1 −iS2 )/~ | 0 >, ãäå | 0 > âàêóóìíûé âåêòîð â ãèëüáåðòîâîìïðîñòðàíñòâå, òàêîé, ÷òî S3 | 0 >= a | 0 >, (S1 + iS2 ) | 0 >= 0,k | 0 > k = 1. Äëÿ àëãåáðû âðàùåíèé ñîáñòâåííîå çíà÷åíèåP`âû÷èñëÿåòñÿ ïî ôîðìóëåa = `~/2,` = 0, 1, 2, .
. . .(1.26)Ñïðàâåäëèâà [63]Ëåììà 1.3. ïðåäñòàâëåíèè Øâèíãåðà êîãåðåíòíûå ñîñòîÿíèÿèìåþò âèäq1 + zq2 (1 + z 2 )`/2q 2 + q22H` √ √).| z >= √exp(− 12~~ 1 + z2π2` `!~Áóäåì èñêàòü ðåøåíèå çàäà÷è (1.21), (1.22) â âèäåI` (Φk,` (z)).îïåðàòîðû 1-ãî0ϕk,` =Ïîñêîëüêó â ðåçóëüòàòå êîãåðåíòíîãî ïðåîáðàçîâàíèÿS1 , S2 , S3 ïðåîáðàçóþòñÿïîðÿäêà [137](1.24) îïåðàòîðûS1 =(1.27)â äèôôåðåíöèàëüíûåd 0i~d 0`d~z` + (1 − z 2 ), S2 =z` − (1 + z 2 ), S3 = ~ − z,2dz2dz2dz(1.28)òî ïîëó÷àåì ñëåäóþùåå óðàâíåíèå äëÿ00Φk,` (z):0f (S 1 , S 2 , S 3 )Φk,` (z) = ξk,` Φk,` (z),k = 0, 1, 2, .
. ..(1.29)470Çäåñü îïåðàòîðû00S 1, S 2, S 3óïîðÿäî÷åíû ïî Âåéëþ. Ñîáñòâåííûìè÷èñëàìè óðàâíåíèÿ (1.29) íàçîâåì òàêèå çíà÷åíèÿ ïàðàìåòðàξk,` ,ïðè êîòîðûõ ýòî óðàâíåíèå èìååò ïîëèíîìèàëüíîå ðåøåíèå â ïðîñòðàíñòâåP`.  ñèëó óíèòàðíîñòè êîãåðåíòíîãî ïðåîáðàçîâàíèÿkΦk,` kP` = 1.(1.30) äàííîì ïàðàãðàôå àñèìïòîòè÷åñêîå ðåøåíèå â ïðîñòðàíñòâåP`áóäåò ïîñòðîåíî íà ïðèìåðå óðàâíåíèÿ (0.13) ñ âîçìóùàþùèìïîòåíöèàëîì (0.17) â ñëó÷àå, êîãäà ïàðàìåòðpB = b a(a + ~).Çäåñü ÷èñëîëî`a > 0ïîðÿäêàb>√(1.31)6, à a çàäàíî ôîðìóëîé (1.26).
Îòìåòèì, ÷òî ÷èñ-èìååò ïîðÿäîê 1, òàê êàê äàëåå áóäóò ðàññìàòðèâàòüñÿ~−1 .Äëÿ òàêîãî ïîòåíöèàëàf (S1 , S2 )Àíàëîãè÷íî ìîæåò áûòü èçó÷åí è ñëó÷àé, êîãäàèìååò âèä (1.20).V (q1 , q2 ) ïðî-èçâîëüíûé ìíîãî÷ëåí 4 ñòåïåíè. Îäíàêî â ýòîì ñëó÷àå òðåáóþòñÿáîëåå ãðîìîçäêèå âûêëàäêè.z = 1 ñî ñïåêòðîì îïåðàòîðà. Ðàñ(1.20) íà ñôåðó Ω` :Ïîêàæåì, êàê ñâÿçàíà òî÷êàñìîòðèì ñóæåíèå ôóíêöèèS12 + S22 + S32 = a(a + ~),(1.32)êîòîðàÿ ÿâëÿåòñÿ ñèìïëåêòè÷åñêèì ëèñòîì äëÿ àëãåáðû âðàùåíèé.Ââåäåì íàΩ`êýëåðîâó ñòðóêòóðó ñ ïîìîùüþ êîìïëåêñíîé êîîðäè-íàòûÏóñòüS1 + iS2.z=pa(a + ~) + S3z(1.33) êîìïëåêñíî ñîïðÿæåííàÿ ê (1.33) ôóíêöèÿ.
Òîãäà â ñèëó(1.32), (1.33) [137]S1 =pz+za(a + ~),1+ | z |2S2 = ipz−za(a + ~),1+ | z |2S3 =48=Äàëåå ïåðåéäåì âfp1− | z |2a(a + ~).1+ | z |2îò êîîðäèíàòS1 , S2(1.34)ê íîâûì êîîðäèíàòàì ðåçóëüòàòå, ñóæåíèå ôóíêöèè (1.20) íà ñôåðóΩ`z, z .ïðèìåò âèäz 2 + 4zz + z 2 b(z + z) +.fΩ` (z, z) = a(a + ~)(1 + zz)21 + zzÃëîáàëüíûé ìàêñèìóì ôóíêöèè fΩ` (z, z) ïðè b >√6äîñòèãàåòñÿ â òî÷êå z = z = 1.
Îí ðàâåí fΩ` (1, 1) = a(a + ~)(b +3/2).Ëåììà 1.4.Äîêàçàòåëüñòâî. Îáîçíà÷èìïîëÿðíûì êîîðäèíàòàìz = ρeiϕ .P = fΩ` /(a(a + ~))è ïåðåéäåì êÒîãäàP (ρ, ϕ) = 2[ρ2 (2 cos2 ϕ + 1) + bρ(1 + ρ2 ) cos ϕ](1 + ρ2 )−2 ,è, ñëåäîâàòåëüíî,∂P(ρ, ϕ) = −2[4ρ2 cos ϕ + bρ(1 + ρ2 )](1 + ρ2 )−2 sin ϕ = 0∂ϕëèáî ïðèsin ϕ = 0, ëèáî ïðècos ϕ = −b(ρ + 1/ρ)/4.rb1− (ρ + ) < −4ρÏîýòîìó òî÷êè ýêñòðåìóìà ôóíêöèèPÍî ïðèb>√63.2ìîãóò ëåæàòü ëèøü íà âå-ùåñòâåííîé îñè.Ïîëîæèìz = x + iy .Òîãäàbx3x2 + y 2P (x, y) = 2+(1 + x2 + y 2 )2 1 + x2 + y 2.(1.35)Äèôôåðåíöèðóÿ (1.35), íàõîäèì∂P(x, 0) = 2(1 − x2 )(bx2 + 6x + b)(1 + x2 )−3 ,∂x∂P(x, 0) = 0.∂y49√9 − b2 )/b( ïðè(0)(0)(0)x1 = 1, x2 = −1, x3,4 = (−3 ±√√6 < b < 3).
Òàê êàê ïðè b > 6Ñòàöèîíàðíûìè ÿâëÿþòñÿ òî÷êè∂ 2P(1, 0) = −b − 3 < 0,∂x2∂ 2P∂ 2P(1, 0) 2 (1, 0) = (b + 3)(b + 2) > 0,∂x2∂yz = 1 ÿâëÿåòñÿ òî÷êîé ìàêñèìóìà ôóíêöèè P . Ïîñêîëüêó(0)2çíà÷åíèÿ P (−1, 0) = 3/2 − b, P (x3,4 , 0) = −b /6, lim|z|→∞ P (x, y) = 0ìåíüøå P (1, 0) = 3/2 + b, òî â òî÷êå z = 1 äîñòèãàåòñÿ ãëîáàëüíûéìàêñèìóì fΩ` .
Ëåììà äîêàçàíà.Òàêèì îáðàçîì, ÷èñëî fΩ` (1, 1) îïðåäåëÿåò âåðõíþþ ãðàíèöóòî÷êàñïåêòðàëüíîãî êëàñòåðà. Äàëåå áóäåò âû÷èñëåíà ïîïðàâêà ê ýòîìó÷èñëó ( ñì. ôîðìóëó (1.193) ).Íàðÿäó ñ çàäà÷åé î íàõîæäåíèè ñïåêòðà ðàññìîòðèì çàäà÷ó âû÷èñëåíèÿ ñðåäíèõ çíà÷åíèé äèôôåðåíöèàëüíûõ îïåðàòîðîâ íà ðåøåíèÿõ (0.13), (0.14) âáëèçè ãðàíèö ñïåêòðàëüíûõ êëàñòåðîâ.
Êâàíòîâûå ñðåäíèå èãðàþò âàæíóþ ðîëü â ïðèëîæåíèÿõ è ìîãóò áûòüíàéäåíû ýêñïåðèìåíòàëüíî. Çàìåòèì, ÷òî ïðè3ξk,` ∼ a(a + ~)(b + )2(1.36)fΩ` (z, z)ϕk,` = ξk,` ϕk,` , ê êîòîðîìó ñâîäèòñÿ (0.13) íàñîáñòâåííîì ïîäïðîñòðàíñòâå H` , ñîáñòâåííûå ôóíêöèè ϕk,` áóäóòëîêàëèçîâàíû â ìàëîé îêðåñòíîñòè òî÷êè z = z = 1. Ïîýòîìó ôîðó óðàâíåíèÿìóëà äëÿ êâàíòîâûõ ñðåäíèõ ïðèíèìàåò âèä(F (S1 , S2 , S3 )ϕk,` , ϕk,` ) ∼ F (a, 0, 0)(ϕk,` , ϕk,` ).(1.37)S1 , S2 , S3 áûëè ïðèáëèæåííî çàìåíåíû èõ çíà÷åíèÿìè â òî÷êå z = z = 1, âû÷èñëåííûìè ïî ôîðìóëàì (1.34).
Êðîìåòîãî ïðåäïîëàãàåòñÿ, ÷òî F (a, 0, 0) 6= 0.Çäåñü ôóíêöèèÏðèâåäåííûå âûøå ôîðìàëüíûå ðàññóæäåíèÿ áóäóò äàëååñòðîãî îáîñíîâàíû ( ñì. òåîðåìó 1.3.). Àíàëîãè÷íî ðàññìàòðèâàþòñÿ è ïðîèçâîëüíûå ìíîãî÷ëåíû 4-îé ñòåïåíèV (q1 , q2 ),äëÿ êîòîðûõ50ñóæåíèå ñèìâîëàf (S1 , S2 , S3 )íà ñôåðóΩ`äîñòèãàåò ãëîáàëüíîãîìàêñèìóìà èëè ìèíèìóìà â åäèíñòâåííîé òî÷êå.2.4.Èíòåãðàëüíîå ïðåäñòàâëåíèå äëÿ àñèìïòîòè÷åñêèõñîáñòâåííûõ ôóíêöèé.Ðàññìîòðèì óðàâíåíèå (1.29), ãäå ôóíêöèÿfèìååò âèä (1.20).Ó÷èòûâàÿ ôîðìóëû (1.28), ïðèõîäèì ê óðàâíåíèþ dΦd2 Φ~ [z − 4z + 1] 2 + 2~ −(2a − ~)z 3 − Bz 2 + 2(2a − ~)z + B+dzdz(1.38)+2 a(2a − ~)z 2 + 2Baz − 2ξk,` + 2a~ Φ = 0.242( Äëÿ óïðîùåíèÿ îáîçíà÷åíèé èíäåêñûk, `ó ôóíêöèèΦ(z)áóäóòíèæå îïóùåíû ).
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