Диссертация (1136178), страница 14
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Çàäà÷à (1.202) íà ïîäïðîñòðàíñòâå, îòâå÷àþùåì îòðèöà-107òåëüíîìó ñïåêòðóâE < 0 , ýêâèâàëåíòíà ñïåêòðàëüíîé çàäà÷å (1.204)L2− (R3 ).Îòìåòèì, ÷òî â îòñóòñòâèå ìàãíèòíîãî ïîëÿ ñïåêòð îïåðàòî-{~N |N = 1, 2, 3, ...}.  ÷àñòíîñòè, S0 èìååòñîáñòâåííîå çíà÷åíèå ~n = 1. Ïðè µ > 0 áóäåì ðàññìàòðèâàòü âåòâè ñîáñòâåííûõ çíà÷åíèé îïåðàòîðà S0 + µS1 , ðàâíûå 1 â ïðåäåëåïðè µ = 0. Îíè èìåþò âèä 1 + µλk,m,n (µ), ãäå λk,m,n (µ) - íåêîòîðûåãëàäêèå ôóíêöèè â îêðåñòíîñòè µ = 0, k - íîìåð âåòâè. Ñîîòâåòñòâóþùèå ñîáñòâåííûå ôóíêöèè çàäà÷è (1.204) îáîçíà÷èì ψ̃k,m,n (q, µ).( Íèæå äëÿ êðàòêîñòè îáîçíà÷åíèé íîìåðà m è n â èíäåêñàõ áóäóòîïóñêàòüñÿ.) Îïðåäåëèâ èç (1.204) ôóíêöèè λk (µ), ìîæíî çàòåì èçðàS0öåëî÷èñëåííûéñèñòåìû óðàâíåíèéµ = ε2 n6 ν 41 + µλk (µ) = ν,íàéòèν = νk (ε)è(1.205)µ = µk (ε).3.3.Ïðåäïîëîæèì, ÷òîÊâàíòîâîå óñðåäíåíèåε2 n7 .
1èν1.ïîðÿäêàÒîãäàµ1, è êçàäà÷å (1.204) ìîæíî ïðèìåíèòü êâàíòîâóþ âåðñèþ ìåòîäà óñðåäíåíèÿ[33; 38;îïåðàòîðU163].Ñîãëàñíî ýòîìó ìåòîäó íàéäåì òàêîé îáðàòèìûéè òàêîé îïåðàòîðS1 + µS2 ,÷òîáûU−1 (S0 + µS1 )U = S0 + µS1 + µ2 S2 + O(µ3 ),[S1 + µS2 , S0 ] = [S1 + µS2 , M3 ] = 0.Íîâûé âîçìóùàþùèé îïåðàòîðøåé ÷àñòüþS0è ñ îïåðàòîðîìS1 + µS2M3 .(1.206)êîììóòèðóåò ñî ñòàð-Ïîýòîìó ðåøåíèå ñïåêòðàëü-íîé çàäà÷è (1.204) ñâîäèòñÿ ê ðåøåíèþ òàêîé çàäà÷è äëÿ îïåðàòîðàS1 + µS2íàL[m, n] ⊂ L2− (R3 ) ñîâìåñòíîì ïîäïðîñòðàíñòâå ôóíê-öèé, îäíîâðåìåííî ÿâëÿþùèìèñÿ ñîáñòâåííûìè ôóíêöèÿìè îïåðà-108òîðîâS0èM3 .Òàê êàêν = 1 + µλ(µ),ψ̃(q, µ) = Uϕ(q, µ),(1.207)òî óñðåäíåííàÿ çàäà÷à ïðèíèìàåò âèäS0 ϕ = ϕ,M3 ϕ = ~mϕ,(1.208)(S1 + µS2 + O(µ2 ))ϕ = λϕ.(1.209)Îòìåòèì, ÷òî äëÿ ðåàëèçàöèè ýòîé èäåè äîëæíî âûïîëíÿòüñÿ óñëîâèåexp {ãäåI2πiS0 } = I,~(1.210)- åäèíè÷íûé îïåðàòîð.
Ïîñêîëüêó â ðàññìàòðèâàåìîé çàäà÷åñïåêòð îïåðàòîðàÎïåðàòîðU1S]1 =2πS0öåëî÷èñëåííûé, òî óñëîâèå (1.210) âûïîëíåíî.âû÷èñëÿåòñÿ ÿâíî [38]. ÎïðåäåëèìZ2πiτiτ(π − τ ) exp {− S0 } S1 exp { S0 }dτ.~~0ÒîãäàU = exp {−iµnS]1 } + O(ε2 n6 ).(1.211)Ïðåîáðàçîâàíèå (1.211) íàçûâàåòñÿ äåóñðåäíÿþùèì. Îíî ïîçâîëÿåòïåðåéòè ê ïðèáëèæåííûì ñîáñòâåííûì ôóíêöèÿì èñõîäíîé (íåóñðåäíåííîé) çàäà÷è.Ðåøåíèå çàäà÷è (1.208), (1.209) ìîæíî ïðåäñòàâèòü â âèäåϕ=ϕk + O(ε2 n6 ), ãäå {ϕk (q)} - îðòîíîðìèðîâàííûé áàçèñ ñîáñòâåííûõôóíêöèé S1 , îòâå÷àþùèõ íåêîòîðûì ñîáñòâåííûì çíà÷åíèÿì ξk :S0 ϕk = ϕk ,M3 ϕk = ~mϕk ,S1 ϕk = ξk ϕk .(1.212)109λ = λk (µ) ïîëíîé çàäà÷è (1.208), (1.209) èìåλk (µ) = ξk + µηk + O(µ2 ), ãäåÑîáñòâåííûå çíà÷åíèÿþò âèäηk = hS2 ϕk , ϕk i− .(1.213)Äàëåå, ïðèáëèæåííî ðåøàÿ ñèñòåìó (1.205), íàõîäèì:νk (ε) = 1 + ε2 n6 ξk + ε4 n12 (ηk + 4ξk2 ) + O(ε6 n18 ),µk (ε) = ε2 n6 + 4ε4 n12 ξk + O(ε6 n18 ).Ïîäñòàâëÿÿ ýòè âûðàæåíèÿ â (1.203), (1.207), (1.211), ïîëó÷àåì, ÷òîàñèìïòîòèêà ðåøåíèÿ èñõîäíîé çàäà÷è (1.202) çàäàåòñÿ ôîðìóëàìè[38]Ek = −1 2 41 4 101+εnξ+ε n (2ηk + 5ξk2 ) + O(ε6 n16 ),k24n24(1.214)1exp {−iε2 n7 S]1 } ϕk + O(ε2 n6 ),(1.215)2n2ãäå ôóíêöèè â ïðàâîé ÷àñòè (1.215) áåðóòñÿ â òî÷êå x/(n νk (ε)).Ðàññìîòðèì àëãåáðó Fquant Êàðàñåâà-Íîâèêîâîé, ñîñòîÿùóþ èçïåðâûõ èíòåãðàëîâ ïàðû S0 , M3 [38; 137].
Åå îáðàçóþùèå B0 , B1 ,B2 , B3 ïîä÷èíåíû ñëåäóþùèì êâàäðàòè÷íûì êîììóòàöèîííûì ñîψk (x) =îòíîøåíèÿì[B1 , B2 ] = i~B0 B3 ,[B0 , B1 ] = 2i~B2 ,i~(B0 B1 + B1 B0 ),2i~[B3 , B1 ] = − (B0 B2 + B2 B0 ),2[B2 , B3 ] = −[B0 , B2 ] = −2i~B1 ,[B0 , B3 ] = 0.S1 , S2 âûðàæàþòñÿ ÷åB = (B0 , B1 , B2 , B3 ) àëãåáðû Fquant . Ñïðàâåäëèâà ñèëó (1.206) óñðåäíåííûå ãàìèëüòîíèàíûðåç îáðàçóþùèå[38](1.216)110Ëåììà 1.31.Èìåþò ìåñòî ðàâåíñòâàS1 = S0 g0 (B),S2 = S0 g00 (B),ãäå îïåðàòîðû B0 , B1 , B2 , B3 ñèììåòðèçîâàíû ïî Âåéëþ, à ñèìâîëûg0 (b) = 12b3 − 8b2 − 4M32 + 4~2 ,(1.217)1g00 (b) = − (4(108b22 +239b23 −308b2 b3 )+4(−66S02 +100M32 −234~2 )b2 +3+4(65S02 −130M32 +249~2 )b3 −127S02 M32 +72M34 +127~2 S02 −277~2 M32 ++205~4 ).3.4.Ïóñòüm, n(1.218)Êîãåðåíòíîå ïðåîáðàçîâàíèå öåëûå,n > |m| ≥ 0.Îáîçíà÷èì ÷åðåçïðîñòðàíñòâî àíòèãîëîìîðôíûõ ìíîãî÷ëåíîâ íàäøåCP[m, n]ñòåïåíè íå âû-n−|m|−1, ñêàëÿðíîå ïðîèçâåäåíèå â êîòîðîì çàäàåòñÿ ôîðìóëîéZΦ1 (z)Φ2 (z)dµm,n (z).(Φ1 , Φ2 )P[m,n] =(1.219)CÇäåñüdµm,n (z) = %(|z|2 )dzdz ,%(r) =ôóíêöèÿn(n − |m|)(|m| + 1)n |m|r F (n + 1, n + |m| + 1; 2n + 2; 1 − r),2π(n + 1)n+1(1.220)ãäåF(l)M îïðåäåëÿåòñÿ≡ l(l + 1)...(l + M − 1), (l)0 ≡ 1.
Îòìåòèì, ÷òî ãèïåðãåîìåòðè÷åñêèé ðÿä [5], à îïåðàöèÿ(l)M%(r) ÿâëÿåòñÿïî ôîðìóëåôóíêöèÿåäèíñòâåííûì ðåøåíèåì ñëåäóþùåé çàäà÷èr(1 − r)%00 + ((1 − |m|) − (2n − |m| + 3)r)%0 − (n + 1)(n − |m| + 1)% = 0,(1.221)Z∞%(r)dr =01,2π%(r) > 0.(1.222)111×òîáû ðåøèòü çàäà÷ó (1.212), âîñïîëüçóåìñÿ êîãåðåíòíûì ïðåîáðàçîâàíèåì [38; 60;137]ZH(Φ) =Φ(z)Hz dµm,n (z).(1.223)CÇäåñü ãèïåðãåîìåòðè÷åñêèå êîãåðåíòíûå ñîñòîÿíèÿ èìåþò âèän−|m|−1Hz =XPj (z)χj ,j=0ãäåPj (z) =p jkj z ,kj =(n − j)j (n − |m| − j)j,j!(1 + |m|)jj = 0, ..., n − |m| − 1(1.224) îðòîíîðìèðîâàííûé áàçèñ âP[m, n],χj (q) = cj (q1 + ià ôóíêöèè|m|sgn(m)q2 )×|q| |m| |q| + q3|q|−q3|m|× exp −LjLn−|m|−1−j.2~2~2~ ôîðìóëå (1.225)êîíñòàíòûcjLMN (y) ïîëèíîìû Ëàãåððà(1.225)[6], à íîðìèðîâî÷íûåèìåþò âèäj|m|cj = (−1) / 2|m|+1π~q(n − |m| − j)|m| (1 + j)|m| .Êîãåðåíòíîå ïðåîáðàçîâàíèå (1.223) îòîáðàæàåò ãèëüáåðòîâîïðîñòðàíñòâîðàçîâàíèåP[m, n]HP[m, n]L[m, n].
ÏðåîáH−1 : L[m, n] →íà ãèëüáåðòîâî ïðîñòðàíñòâîóíèòàðíîå, îáðàòíîå ïðåîáðàçîâàíèåâû÷èñëÿåòñÿ ïî ôîðìóëåH−1 (χ) = hχ, Hz iL[m,n] .Çàìåòèì, ÷òî èíòåãðàëüíîå ïðåäñòàâëåíèåϕk = H(Φk )(1.226)112äàåò òî÷íûå ðåøåíèÿ ïåðâûõ äâóõ óðàâíåíèé (1.212) ïðè ëþáûõ àì-Φk . Ïîñêîëüêó â ðåçóëüòàòå êîãåðåíòíîãî ïðåîáðàçîâàíèÿB0 , B1 , B2 , B3 ïðåîáðàçóþòñÿ â äèôôåðåíöèàëüíûå îïåðà-ïëèòóäàõ(1.223)òîðû 1-ãî è 2-ãî ïîðÿäêà 0di~2d22z(z −1) 2 −((2n−|m|−3)z 2 +B 0 = ~ 2z −n+|m|+1 , B 1 =dz2dz0 0d~2d2+ (n − 1)(n − |m| − 1)z , B 2 =z(z 2 + 1) 2 −dz2dzd−((2n − |m| − 3)z 2 − |m| − 1) + (n − 1)(n − |m| − 1)z ,dz0d2d1B 3 = −~2 z 2 2 − (n − |m| − 2)z − (n − 1)(|m| + 1) , (1.227)dz 2dzòî â ñèëó (1.217), (1.212) ïîëó÷àåì ñëåäóþùåå óðàâíåíèå äëÿ Φk :+|m| + 1)00(12B 3 − 8B 2 − 4~2 m2 + 4~2 )Φk = ξk Φk .(1.228)Ñîáñòâåííûìè ÷èñëàìè óðàâíåíèÿ (1.228) íàçîâåì òàêèå çíà÷åíèÿïàðàìåòðàξk ,ïðè êîòîðûõ ýòî óðàâíåíèå èìååò ïîëèíîìèàëüíûåðåøåíèÿ â ïðîñòðàíñòâåP[m, n].Ïîòðåáóåì, ÷òîáûkΦk kP[m,n] = 1.Ïîêàæåì, êàê ñâÿçàíà òî÷êàg0 (B).Ïîëîæèìc = |m|/nz = 1(1.229)ñî ñïåêòðîì îïåðàòîðàè ïóñòün → ∞,|m| → ∞,5−1/2 < c < 1.(1.230)(1.231)Ðàññìîòðèì ñóæåíèå ôóíêöèè (1.217) íà ñèìïëåêòè÷åñêèé ëèñòΩàëãåáðû (1.216), çàäàâàåìûé óðàâíåíèÿìèb23 − b21 − b22 = c2 /4,b20 + 4b3 = c2 + 1.(1.232)113Ââåäåì íàΩêýëåðîâó ñòðóêòóðó ñ ïîìîùüþ êîìïëåêñíîé êîîðäè-íàòûz=Ïóñòü2(b3 + ib1 ).1 − 2b3 − b0(1.233)z êîìïëåêñíî ñîïðÿæåííàÿ ê (1.233) ôóíêöèÿ.
Òîãäà â ñèëó(1.232), (1.233) [38]p(c − 1)(−2|z|2 − c(|z|2 − 1) + κ(|z|2 ))pb0 =,2|z|2 − c(|z|2 − 1) + κ(|z|2 )(1.234)c2 + 1 − b20c2(c − 1)2 |z|2pb3 == +,42 c(|z|2 − 1)2 + 4|z|2 + (|z|2 + 1) κ(|z|2 )(1.235)−1p22 (z + z)(1 − c)(c(|z| + 1) + κ(|z|2 ))(z + z)p(1−2b3 −b0 ) =,b2 =4(c(|z|2 − 1)2 + 4|z|2 + (|z|2 + 1) κ(|z|2 ))(1.236)p2(z − z)(1 − c)(c(|z| + 1) + κ(|z| ))(z − z)b2pb1 ==.i(z + z)2i(c(|z|2 − 1)2 + 4|z|2 + (|z|2 + 1) κ(|z|2 ))2(1.237)Çäåñüκ(r) = c2 (r − 1)2 + 4r.Äàëåå ïåðåéäåì âíàòàìz, z[38;137].g0îò êîîðäèíàòb0 , b1 , b2 , b3(1.238)ê íîâûì êîîðäè- ðåçóëüòàòå, ñóæåíèå ôóíêöèèg0íàΩïðèìåòâèäg0,Ω (z, z) = 6c − 4c2 + 4~2 +p4(c − 1)[6(c − 1)|z|2 + (z + z)(c(|z|2 + 1) + κ(|z|2 ))]p,+c(|z|2 − 1)2 + 4|z|2 + (|z|2 + 1) κ(|z|2 )ãäåκ(r)èìååò âèä (1.238).Òåîðåìà 1.5.íåíèè óñëîâèéÃëîáàëüíûé ìèíèìóì ôóíêöèè g0,Ω (z, z) ïðè âûïîë(1.230), (1.231)äîñòèãàåòñÿ â òî÷êå z = z = 1.
Îíðàâåí g0,Ω (1, 1) = 1 + c2 + O(~2 ).114Äîêàçàòåëüñòâî. Ïóñòüg0,Ωz = x + iy , z = x − iy .Òîãäà ôóíêöèþìîæíî ïðåäñòàâèòü â âèäåg0,Ω (x, y) = g1 (x2 + y 2 ) + 2xg2 (x2 + y 2 ),ãäå224(c − 1)2 r2g1 (r) = 6c − 4c + 4~ +c(r −1)2p,+ 4r + (r + 1) κ(r)pκ(r)]pg2 (r) =.c(r − 1)2 + 4r + (r + 1) κ(r)4(c − 1)[c(r + 1) +Ñòàöèîíàðíûå òî÷êèg0,Ωóäîâëåòâîðÿþò ñëåäóþùåé ñèñòåìå∂g0,Ω (x, y)= 2[xg10 (x2 +y 2 )+2x2 g20 (x2 +y 2 )+g2 (x2 +y 2 )] = 0,∂x(1.239)∂g0,Ω (x, y)= 2y[g10 (x2 + y 2 ) + 2xg20 (x2 + y 2 )] = 0.∂y(1.240)Åñëèy 6= 0, x 6= ∞, òî âûðàçèìxèç óðàâíåíèÿ (1.240) èïîäñòàâèì â (1.239).
 ðåçóëüòàòå, ïðèõîäèì ê óðàâíåíèþy 2 ) = 0,g2 (x2 +êîòîðîå íå èìååò ðåøåíèé.Åñëè æåy = 0,òîxóäîâëåòâîðÿåò óðàâíåíèþxg10 (x2 ) + 2x2 g20 (x2 ) + g2 (x2 ) = 0.(1.241)Ïîñêîëüêóp22 224(c−1)(r−1)[c(r+1)+2r+c(r+1)κ(r)]pp,g10 (r) = −κ(r)[c(r − 1)2 + 4r + (r + 1) κ(r)]2g2 (r) + 2rg20 (r) = −8(c − 1)(r − 1)×pc(r + 1)[c2 (r2 + 1) − 2(c − 2)r] + [c2 (r + 1)2 − 2(c − 1)r] κ(r)pp×,κ(r)[c(r − 1)2 + 4r + (r + 1) κ(r)]2115òî (1.241) ïðèíèìàåò âèäg3 (x)(x2 − 1) = 0,(1.242)ãäåg3 (x) = 3(c − 1)x[c2 (x4 + 1) + 2x2 + c(x2 + 1)pκ(x2 )]+p+c(x2 + 1)[c2 (x4 + 1) − 2(c − 2)x2 ] + [c2 (x2 + 1)2 − 2(c − 1)x2 ] κ(x2 ).(1.243)Êîðíÿìè (1.242) ÿâëÿþòñÿx = ±1.
Ïîêàæåì, ÷òî ïðè âûïîë-íåíèè óñëîâèÿ (1.231) ýòî óðàâíåíèå äðóãèõ êîðíåé íå èìååò. Äåéñòâèòåëüíî, ïðèx ≤ 0 ôóíêöèÿ g3 (x) ïîëîæèòåëüíà, òàê êàê ïåðâîåñëàãàåìîå â (1.243) íåîòðèöàòèëüíî, à îñòàëüíûå ïîëîæèòåëüíû. ñëó÷àåx>0ïîñëå çàìåíûu = x + 1/xóðàâíåíèåg3 (x) = 0(1.244)ïðèíèìàåò âèäp[c2 u2 + 3c(c − 1)u − 2(c − 1)] c2 (u2 − 4) + 4 == −c3 u3 − 3c2 (c − 1)u2 + c(2c2 + 2c − 4)u + 6(c2 − 1)(c − 1).(1.245)Ïîñëå âîçâåäåíèÿ ïðàâîé è ëåâîé ÷àñòåé ðàâåíñòâà (1.245) â êâàäðàò,u2 +6u−5c−2 +9 = 0, êîðíè êîòîðîãî u± = −3±√ −15c ìåíüøå 2.
Ïîýòîìó óðàâíåíèå x+1/x = u± , à, ñëåäîâàòåëüíî,è óðàâíåíèå (1.244) ïðè x > 0 íå èìåþò ðåøåíèé.Ïðîâåðèì, ÷òî â òî÷êå z = z = 1 âûïîëíåíû äîñòàòî÷íûå óñëîïîëó÷àåì óðàâíåíèåâèÿ ëîêàëüíîãî ìèíèìóìà. Òàê êàê∂ 2 g0,Ω (1, 0)= 0,∂x∂y116òî äîñòàòî÷íî ïîêàçàòü, ÷òî∂ 2 g0,Ω (1, 0)= 2g10 (1) + 12g20 (1) + 4g100 (1) + 8g200 (1) > 0,2∂x∂ 2 g0,Ω (1, 0)= 2g10 (1) + 4g20 (1) > 0.2∂yÏîñêîëüêóg20 (1)g10 (1) = 0,1 − c23(c2 − 1)(5 − c2 )002 200=, g1 (1) = − (1 − c ) , g2 (1) =,288òî â ñèëó (1.231)∂ 2 g0,Ω (1, 0) (1 − c2 )(5c2 − 1)=> 0,∂x22à, çíà÷èò,z = z = 1∂ 2 g0,Ω (1, 0)= 2(1 − c2 ) > 0,2∂y òî÷êà ëîêàëüíîãî ìèíèìóìàg0,Ω .Òàê êàêçíà÷åíèÿ ôóíêöèèg0,Ω |z=z=1 = c2 + 1 + 4~2 ,g0,Ω |z=z=−1 = −3c2 + 5 + 4~2 ,g0,Ω |z=z=∞ = 6c − 4c2 + 4~2c2 +1 < 6c−4c2 < −3c2 +5, òî ýòà òî÷êàÿâëÿåòñÿ òî÷êîé ãëîáàëüíîãî ìèíèìóìà g0,Ω .
Òåîðåìà äîêàçàíà.×èñëî g0,Ω (1, 1) îïðåäåëÿåò íèæíþþ ãðàíèöó ñïåêòðàëüíîãîóäîâëåòâîðÿþò íåðàâåíñòâàìêëàñòåðà. Äàëåå áóäåò âû÷èñëåíà ïîïðàâêà ê ýòîìó ÷èñëó ( ñì. ôîðìóëó (1.292)).Íàðÿäó ñ çàäà÷åé î íàõîæäåíèè ñïåêòðà ðàññìîòðèì çàäà÷ó âû÷èñëåíèÿ ñðåäíèõ çíà÷åíèé äèôôåðåíöèàëüíûõ îïåðàòîðîâ íà ðåøåíèÿõ (1.202) âáëèçè ãðàíèö ñïåêòðàëüíûõ êëàñòåðîâ. Çàìåòèì,÷òî ïðèξk ∼ 1 + c2(1.246)g0,Ω (z, z)ϕk = ξk ϕk , ê êîòîðîìó ñâîäèòñÿ (1.204) íà ñîáñòâåííîì ïîäïðîñòðàíñòâå L[m, n], ñîáñòâåííûå ôóíêöèè ϕk áóäóòëîêàëèçîâàíû â ìàëîé îêðåñòíîñòè òî÷êè z = z = 1. Ïîýòîìó ôîðó óðàâíåíèÿ117ìóëà äëÿ êâàíòîâûõ ñðåäíèõ ïðèíèìàåò âèä(F (b0 , b1 , b2 , b3 )ϕk , ϕk ) ∼ F (0, 0, (1−c2 )/4, (1+c2 )/4)(ϕk , ϕk ).(1.247)b0 , b1 , b2 , b3 áûëè ïðèáëèæåííî çàìåíåíû èõ çíà÷åíèÿìè â òî÷êå z = z = 1, âû÷èñëåííûìè ïî ôîðìóëàì (1.234) (1.237).22Êðîìå òîãî ïðåäïîëàãàåòñÿ, ÷òî F (0, 0, (1 − c )/4, (1 + c )/4) 6= 0.Çäåñü ôóíêöèèÏðèâåäåííûå âûøå ôîðìàëüíûå ðàññóæäåíèÿ áóäóò äàëåå ñòðîãîîáîñíîâàíû ( ñì.
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