Диссертация (1136178), страница 29
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Åñëè òàêæå ó÷åñòü (2.168), (2.152), òî ïîëó÷àåì àñèìïòîòèêó(0.31). Çäåñük = 0, 1, 2, . . . ,à ÷èñëî`èìååò ïîðÿäîê~−1 .Ïîëó÷åí-íàÿ ñåðèÿ îïèñûâàåò ðàñùåïëåíèå ñïåêòðà îïåðàòîðà òèïà Õàðòðèè ñîäåðæèò ÷ëåíû äî 3 ïîðÿäêà ïîÇàìå÷àíèå~âêëþ÷èòåëüíî.2.1. Àíàëîãè÷íî ìîãóò áûòü íàéäåíû àñèìïòîòè÷åñêèåñîáñòâåííûå çíà÷åíèÿ âáëèçè íèæíèõ ãðàíèö ñïåêòðàëüíûõ êëàñòå-240ðîâ. Ïðèb>7îíè èìåþò âèäλ = λk,` = `~+~+[151]`2 ~4`~4 p(9−2b)+( (b − 6)(b − 7)(2k+1)−2b+9)+42+O(~4 ),Çäåñük = 0, 1, 2, . . .
,à ÷èñëî`~ → 0.èìååò ïîðÿäîê~−1 .241Ãëàâà 3Àñèìïòîòè÷åñêèå ðåøåíèÿ óðàâíåíèéÕàðòðè ñ ñèíãóëÿðíûìè ïîòåíöèàëàìèñàìîäåéñòâèÿ 1.Êâàçèêëàññè÷åñêàÿ àñèìïòîòèêà ñïåêòðàòðåõìåðíîãî îïåðàòîðà Õàðòðè âáëèçèâåðõíèõ ãðàíèö ñïåêòðàëüíûõ êëàñòåðîâ.Àñèìïòîòè÷åñêèå ðåøåíèÿ,ñîñðåäîòî÷åííûå âáëèçè îêðóæíîñòè1.1.Ââåäåíèå ê 1Ðàññìîòðèì çàäà÷ó íà ñîáñòâåííûå çíà÷åíèÿ (0.32), (0.33) äëÿíåëèíåéíîãî îïåðàòîðà Õàðòðè ñ êóëîíîâñêèì âçàèìîäåéñòâèåì âL2 (R3 ), ãäå ∆ îïåðàòîð Ëàïëàñà, ε > 0 ìàëûé ïàðàìåòð. Óðàâ-íåíèå (0.32) èãðàåò ôóíäàìåíòàëüíóþ ðîëü â êâàíòîâîé òåîðèè èíåëèíåéíîé îïòèêå ( ñì.
1 âòîðîé ãëàâû ).Ðàññìîòðèì ñëó÷àé, êîãäà êâàíòîâîå ÷èñëîn, çàäàþùåå íåâîç-ìóùåííîå ñîáñòâåííîå çíà÷åíèå, âåëèêî ( äëÿ îïðåäåëåííîñòè áóäåìñ÷èòàòü, ÷òîλèìååò ïîðÿäîêε)[64; 70]. Ïóñòüp = n − m − 1,ãäåm ìàãíèòíîå êâàíòîâîå ÷èñëî.  äàííîì ïàðàãðàôå äëÿ êàæäî-ãîp = 0, 1, 2, . . .÷åíèÿ (0.34), ãäåáóäóò íàéäåíû àñèìïòîòè÷åñêèå ñîáñòâåííûå çíà-i = 0, .
. . , Ip ,êîòîðûå ðàñïîëîæåíû âáëèçè âåðõ-íèõ ãðàíèö ñïåêòðàëüíûõ êëàñòåðîâ, îáðàçóþùèõñÿ âîêðóã óðîâíåéýíåðãèè íåâîçìóùåííîãî îïåðàòîðà (ïðèε=0). Îòìåòèì, ÷òî íàíèæíåé ãðàíèöå êëàñòåðàλn (ε) ∼ −εEmin1+,4n2n2n → ∞,242ãäå ÷èñëîEminEminÇäåñüK(κ)óäîâëåòâîðÿåò íåðàâåíñòâó1≤ 32ππZZ√πK0sin θ sin θ0sin((θ + θ0 )/2)0!dθ0 dθ.sin((θ + θ0 )/2) ïîëíûé ýëëèïòè÷åñêèé èíòåãðàë 1 ðîäà [4].Àíàëîãè÷íàÿ (0.32), (0.33) çàäà÷à íà ñîáñòâåííûå çíà÷åíèÿ âL2 (R2 )äëÿ âîçìóùåííîãî äâóìåðíîãî ðåçîíàíñíîãî îñöèëëÿòîðà,âîçáóæäàþùèé ïîòåíöèàë êîòîðîãî çàäàåòñÿ èíòåãðàëüíîé íåëèíåéíîñòüþ òèïà Õàðòðè ñ ãëàäêèì ïîòåíöèàëîì ñàìîäåéñòâèÿ, ðàññìàòðèâàëàñü ðàíåå â 1 âòîðîé ãëàâû.
Åñëè ñðàâíèòü íàéäåííóþ òàì ñåðèþ (0.25) ñ (0.34), òî îíà íå ñîäåðæèò ëîãàðèôìè÷åñêèõ ïîïðàâîê,à ðàñùåïëåíèå ñïåêòðà â (0.25) ïðîèñõîäèò â ñëåäóþùåì ïðèáëèæåíèè.1.2.Àñèìïòîòèêà ñîáñòâåííûõ ôóíêöèé íåâîçìóùåííîéçàäà÷èÏîëüçóÿñü ðàñòÿæåíèåìq = x/ε, ψ = ε3/2 v, λ = εE,ïðèâåäåìçàäà÷ó (0.32), (0.33) ê ñòàíäàðòíîìó äëÿ òåîðèè êâàçèêëàññè÷åñêèõïðèáëèæåíèé âèäó1+ε(−ε∆ −|x|ZR3| v(x0 ) |2 0dx )v(x) = Ev(x),| x − x0 |kvkL2 (R3 ) = 1.(3.1)(3.2)Ïðè ïîñòðîåíèè àñèìïòîòè÷åñêèõ ðåøåíèé (3.1), (3.2) íàì ïîòðåáóåòñÿ àñèìïòîòèêà ñîáñòâåííûõ ôóíêöèé íåâîçìóùåííîé çàäà÷è(−ε∆ −1)v(x) = Ev(x),|x|kvkL2 (R3 ) = 1.Äèñêðåòíûì ñîáñòâåííûì çíà÷åíèÿìEn = −1,4εn2n = 1, 2, .
. . ,243â ñôåðè÷åñêèõ êîîðäèíàòàõ0 ≤ ϕ ≤ 2π ,(r, θ, ϕ), ãäå0 ≤ r ≤ ∞, 0 ≤ θ ≤ π ,îòâå÷àþò ñîáñòâåííûå ôóíêöèè [92]vn,k,nr = Y`m (θ, ϕ)Rn` (r).`, nr îðáèòàëüíîå1 + nr , k = ` − m,ÇäåñüsY`m (θ, ϕ) =(3.3)è ðàäèàëüíîå êâàíòîâûå ÷èñëà,(2` + 1)(`− | m |)! |m|P` (cos θ)eimϕ ,4π(`+ | m |)!n = `+(3.4)2 r `r−r/(2εn) 2`+1pRn` (r) =.eLn−`−1εn(2εn)3/2 n(n − ` − 1)!(n + `)! εn(3.5)Ôóíêöèè (3.4), (3.5) ñîäåðæàò ïðèñîåäèíåííûé ïîëèíîì ËåæàíäðàP`m (x) =`+m(−1)`2 m/2 d(1−x)(1 − x2 )` ,``+m2 `!dxà òàêæå ïîëèíîì ËàãåððàLsn (x)dn −x n+s(e x ).=e xdxnx −sÏóñòüa = 2`2 ε.Áóäåì ñ÷èòàòü, ÷òî ïðèïîâåäåíèå ôóíêöèéa, θ = π/2}âR3 .ε→0vn,k,nr÷èñëî`(3.6)ε−1/2 . Èçó÷èìΓa = {(r, θ, ϕ) | r =èìååò ïîðÿäîêâáëèçè îêðóæíîñòèÂâåäåì íîâûå ïåðåìåííûåπ √τ = (θ − ) `,2√rs = ( − 1) `.aÑïðàâåäëèâàÒåîðåìà 3.1.Ïðè ` → ∞ è íåáîëüøèõ nr = 0, 1, 2, .
. . è k =0, 1, 2, . . . ôóíêöèè vn,k,nr ïî mod O(`−∞ ) ñîñðåäîòî÷åíû âáëèçè îê-244ðóæíîñòè Γa , ãäå ïðè s6 + τ 4 ` ñïðàâåäëèâà àñèìïòîòèêàvn,k,nr√(−1)p `22√ eimϕ e−(s +τ )/2 Hnr (s)Hk (τ ) 1+=√a3/2 2(p+1)/2 π nr ! k! 2| s |3 +1s +1 0√√ Hnr (s) ++OHnr (s) + O`` 4| τ |3 +1 0τ +1Hk (τ ) + OHk (τ ) .+O``(3.7)Çäåñü Hn ïîëèíîì Ýðìèòà.Äîêàçàòåëüñòâî. Íà÷íåì ñ àñèìïòîòèêè ïîëèíîìà Ëåæàíäðà.Ïîñêîëüêó ïîëèíîì Ýðìèòà èìååò âèä [6][k/2]X (−1)j k!Hk (τ ) =(2τ )k−2j ,j!(k − 2j)!j=0ãäå[α] öåëàÿ ÷àñòü ÷èñëàα,òî çàìåíÿÿ â ðàâåíñòâå[k/2]X (−1)j `!(2` − 2j)!d2`−k2 ``(1 − x ) = (−1)xk−2j2`−kdx(` − j)!j!(k − 2j)!j=0ôàêòîðèàëû ïî ôîðìóëå Ñòèðëèíãà, à òàêæå ðàçëàãàÿ ôóíêöèþx2 )(`−k)/2(1−ñ ïîìîùüþ ôîðìóëû Òåéëîðà, èìååì:(−1)k ``−k/2 2`−k+1/2 e−` −τ 2 /2 eHk (τ )+k! 4| τ |3 +1 0τ +1+OHk (τ ) + OHk (τ ) .``P``−k (cos θ) =Çäåñüτ 4 `.×òîáû íàéòè àñèìïòîòèêó ïîëèíîìà Ëàãåððà, âîñïîëüçóåìñÿèíòåãðàëüíûìè ïðåäñòàâëåíèÿìè [6]:n!Lsn (x) =2πie−xω/(1−ω)dω,s+1 ω n+1|ω|=ρ (1 − ω)I(3.8)245n!Hn (x) =2πiÇäåñüρ < 1,2e2xz−zdz.n+1|z|=R zI(3.9)êîíòóðû èíòåãðèðîâàíèÿ îðèåíòèðîâàíû ïðîòèâ ÷àñî-âîé ñòðåëêè.
Äåëàÿ â èíòåãðàëå (3.8) çàìåíó√ω = z/ `è ðàçëàãàÿäàëåå ôóíêöèè ïî ôîðìóëå Òåéëîðà, èìååì:√√r n!Ie−[2(`−nr −1)+2 `s+O(s/ `)+O(1/`)]ω/(1−ω)rL2`+1=dω =nrεn2πi |ω|=ρ(1 − ω)2`+2 ω nr +12(2nr + 4)z − 2sz 2 − 4z 3 /3e−2sz−z√[1 + O]dz =√nr +1z`|z|= `ρ 23s+1|s|+1√√ Hn0 r (s) .= `nr /2 (−1)nr Hnr (s) + OHnr (s) + O```nr /2 nr !=2πiÇäåñüIs6 `.×òîáû ïîëó÷èòü ôîðìóëó (3.7), îñòàåòñÿ ðàçëîæèòü ôóíêöèþr` e−r/(2εn)âáëèçè òî÷êèr = a è ïðèìåíèòü ê âõîäÿùèì â (3.4), (3.5)ôàêòîðèàëàì ôîðìóëó Ñòèðëèíãà. Òåîðåìà äîêàçàíà.1.3.Ïîñòðîåíèå àñèìïòîòè÷åñêîãî ðåøåíèÿÏåðåõîäÿ â ñôåðè÷åñêóþ ñèñòåìó êîîðäèíàò, à òàêæå äåëàÿïîäñòàíîâêó√v(x) = eimϕ g(r, θ)/ 2π ,ïðåîáðàçóåì çàäà÷ó (3.1), (3.2)ê âèäó [41]Zπ ∂22 ∂1 ∂2∂m2 1−ε+++ ctg θ −− +∂r2 r ∂r r2 ∂θ2∂θ sin2 θrZ∞W (r, r0 , θ, θ0 ) | g(r0 , θ0 ) |2 (r0 )2 sin θ0 dr0 dθ0 − E g(r, θ) = 0,+ε00Z0πZ(3.10)∞| g(r, θ) |2 r2 sin θdrdθ = 1,0ãäå ÿäðî2×W (r, r0 , θ, θ0 ) = pπ r2 + (r0 )2 − 2rr0 cos (θ + θ0 )(3.11)246×K√2 rr0 sin θ sin θ0!.pr2 + (r0 )2 − 2rr0 cos(θ + θ0 )Äëÿ êâàíòîâûõ ÷èñåë`, n, mïîðÿäêàε−1/2 ,(3.12)è ñëåäîâàòåëüíî,nr , k è p = nr +k íèæå áóäóò ïîñòðîåíû àñèìïòîòè÷åñêèå−1ðåøåíèÿ çàäà÷è (3.1), (3.2) ïî mod O(` ) ëîêàëèçîâàííûå âáëèçèîêðóæíîñòè Γa .
Àñèìïòîòè÷åñêèå ðåøåíèÿ çàäà÷è (3.10), (3.11) áóíåáîëüøèõäåì èñêàòü â âèäå√g2 (τ, s)g3 (τ, s)g=a+O,`g0 (τ, s) + g1 (τ, s) + √``1E0 ln ` E1ln `E=−++ 2 + O 5/2 .2a(1 + (nr + 1)/`)2`2``−3/2(3.13)(3.14)` → ∞, ôóíêöèè gj (τ, s), j = 0, 1, 2, 3, ýêñïîíåíöèàëüíî óáû22âàþò ïðè τ + s → ∞; E0 , E1 íåêîòîðûå êîíñòàíòû. (Äëÿ óïðîùåíèÿ îáîçíà÷åíèé èíäåêñû i è p ó g0 è E1 îïóùåíû.)ÇäåñüÐàçëîæèì âõîäÿùèå â (3.10), (3.11) ôóíêöèè ñ ïîìîùüþ ôîðìóëû Òåéëîðà ïî ñòåïåíÿìτès.Ïîñêîëüêó ôóíêöèÿëîãàðèôìè÷åñêóþ îñîáåííîñòü ïðèκ→1K(κ)[4]pK(κ) = ln(4/ 1 − κ2 ) + O((1 − κ2 )) ln(1 − κ2 )),W ôîðìóëàt = (τ − τ 0 )2 + (s − s0 )2 . Òîãäàèìååò(3.15)òî íåïîñðåäñòâåííî êÒåéëîðà íå ïðèìåíèìà.
Îáîçíà-÷èìèç (3.12), (3.15) âûòåêàåòËåììà 3.1.Ïðè ` → ∞, t ` èìååò ìåñòî àñèìïòîòèêà√18`ln √ + OW (r, r0 , θ, θ0 ) =πat√ !s+s`√ ln √ + Ot`√ !t`ln √ .`t0(3.16)Äàëåå ðàçëîæèì(1 + (nr + 1)/`)−2ïî ñòåïåíÿì`è ïîäñòàâèìàñèìïòîòèêè (3.13), (3.14) â óðàâíåíèÿ (3.10), (3.11).  ñèëó (3.6),(3.14), (3.16), (3.11) äëÿ îòñóòñòâèÿ â ëåâîé ÷àñòè (3.10) ñëàãàåìûõïîðÿäêà`−2 ln `äîñòàòî÷íî ïîëîæèòüE0 = 1/(4π).Ïðèðàâíèâàÿ ê247`−1 , `−3/2g0 , g1 è g2 :íóëþ ñëàãàåìûå ïîðÿäêà÷è äëÿ îïðåäåëåíèÿè`−2 , ïîëó÷àåì ñëåäóþùèå çàäà-Lg0 = 0,Z(3.17)| g0 |2 dτ ds = 1;R2Lg1 = F1 ,ãäå(3.18)∂ 2 g0∂g0− s 2 + (s3 − 2ks + sτ 2 )g0 ,F1 =∂s∂τZZ(g0 g 1 + g 0 g1 )dτ ds = −2s | g0 |2 dτ ds;R2R2Lg2 = F2,1 + F2,2 ,(3.19)ãäåF2,1∂g0 3s2 ∂ 2 g0 τ ∂g0 1 2 4= −s+−−τ − 2kτ 2 + k 2 − 6ks2 + 3s2 τ 2 +2∂s2 ∂τ2 ∂τ2 3 ∂g1∂ 2 g1+3s4 + 3(nr + 1)2 g0 +− s 2 + (s3 − 2ks + sτ 2 )g1 ,∂s∂τ1F2,2 =(4πE1 − 6 ln 2+4πZ0 20 20 0 20 0+ln((τ − τ ) + (s − s ) ) | g0 (τ , s ) | dτ ds g0 ,R2ZR2Z=−R2(g0 g 2 + g 0 g2 )dτ ds =| g1 |2 +2s(g0 g 1 + g 0 g1 ) + (s2 − τ 2 /2) | g0 |2 dτ ds.Çäåñü îïåðàòîðLçàäàí ôîðìóëîé (0.39).Ðåøåíèÿìè óðàâíåíèÿ (3.17) èçL2 (R2 )ÿâëÿþòñÿ ñîáñòâåííûåôóíêöèè äâóìåðíîãî îñöèëëÿòîðà, îòâå÷àþùèå ñîáñòâåííîìó çíà÷åíèþp + 1 (p = 0, 1, 2, .
. . ).Îíè îáðàçóþò ïîäïðîñòðàíñòâîHp ⊂L2 (R2 ), îðòîíîðìèðîâàííûé áàçèñ â êîòîðîì ñîñòîèò èç ôóíêöèé22βj,p−j (τ, s), j = 0, . . . , p. Çäåñü βj,i (τ, s) = θj,i e−(s +τ )/2 Hj (s)Hi (τ ),248ãäåθj,i(−1)j+i=√ √ √ .2(j+i)/2 π j! i!Ñëåäîâàòåëüíî,g0 =pXcj βj,p−j ,(3.20)j=0ãäåcj íåêîòîðûå êîíñòàíòû, óäîâëåòâîðÿþùèå óñëîâèþ íîðìè-ðîâêèpX| cj |2 = 1.j=0Îíè íàõîäÿòñÿ èç óñëîâèé ðàçðåøèìîñòè äëÿ ñëåäóþùèõ ïðèáëèæåíèé. Èñïîëüçóÿ (3.17), à òàêæå èçâåñòíûå ñâîéñòâà ïîëèíîìîâ Ýðìèòà[6]sHj (s) = Hj+1 (s)/2 + jHj−1 (s), Hj0 (s) = 2jHj−1 (s),ïðåîá-ðàçóåì ïðàâóþ ÷àñòü óðàâíåíèÿ (3.18).
Òàê êàê 2 ∂2∂2+ 2s(nr + 1) e−s /2 Hj (s) = e−s /2 s(s2 − 1)Hj (s)−s 2+∂s∂s−4s2 jHj−1 (s) + 4sj(j − 1)Hj−2 (s) − sHj (s) + 2jHj−1 (s)+13 332+2s(nr +1)Hj (s) = e−s /2 Hj+3 (s)+( j+ )Hj+1 (s)+ j 2 Hj−1 (s)+84 4211+j(j − 1)(j − 2)Hj−3 (s) + 2nr Hj+1 (s) + jHj−1 (s) − 4j Hj+1 (s)+241+(j − )Hj−1 (s) + (j − 1)(j − 2)Hj−3 (s) + 2jHj−1 (s) + 4j(j − 1)×211 j 32× Hj−1 (s) + (j − 2)Hj−3 (s) = e−s /2 Hj+3 (s) + − + + nr ×284 4 jHj+1 (s) + j − + 2nr + 2 Hj−1 (s) + j(j − 1)(j − 2)Hj−3 (s) ,2òî F1 ïðèíèìàåò âèäF1 = s=pXj=02cj θj,p−j e−(s∂ 2 g0 ∂g0++ 2s(nr + 1)g0 =∂s2∂s+τ 2 )/2Hp−j (τ ) Hj+3 (s)/8 + ((3 − j)/4 + nr )×249×Hj+1 (s)+j(−j/2+2nr +2)Hj−1 (s)+j(j −1)(j −2)Hj−3 (s) .Ïîñêîëüêó (3.21) íå ñîäåðæèò ôóíêöèé èçHp ,(3.21)òî óðàâíåíèå (3.18)ðàçðåøèìî. Åãî ðåøåíèå èìååò âèäp∂g0 X ∗1 ∂ 3 g0g1 = −− 2(nr + 1)+cj βj,p−j ,3 ∂s3∂sj=0ãäåc∗j íåêîòîðûå êîíñòàíòû.Àíàëîãè÷íî äîêàçûâàåòñÿ, ÷òîèçF2,1 òàêæå íå ñîäåðæèò ôóíêöèéHp .
Ïîýòîìó óñëîâèÿ ðàçðåøèìîñòè óðàâíåíèÿ (3.19) ïðèíèìàþòâèäZF2,2 βj,p−j dτ ds = 0,j = 0, . . . , p.(3.22)R2Ïðè âûïîëíåíèè (3.22) ôóíêöèÿg2 ∈ L2 (R2 )ìîæåò áûòü ïðåäñòàâ-ëåíà â âèäå ñóììû ñëåäóþùåãî ðÿäà [6; 81]g2 =∞Xj,i=0j+i6=p12(j + i − p)Z(F2,1 (τ 0 , s0 ) + aF2,2 (τ 0 , s0 ))βj,i (τ 0 , s0 )dτ 0 ds0 ×R2×βj,i (τ, s) +pXc∗∗j βj,p−j ,(3.23)j=0ãäåc∗∗j íåêîòîðûå êîíñòàíòû.Óñëîâèå (3.22) ïîçâîëÿþò íàéòè âõîäÿùèå â (3.20) êîýôôèöèåíòû1cj =6 ln 2 − 4πE1Zln((τ − τ 0 )2 + (s − s0 )2 ) | g0 (τ 0 , s0 ) |2 ×R4×g0 (τ, s)βj,p−j (τ, s)dτ 0 ds0 dτ ds,j = 0, . . . , p. ðåçóëüòàòå, ïðèõîäèì ê ñëåäóþùåé íå ñîäåðæàùåéìàëûõ ïàðàìåòðîâ çàäà÷å íà ñîáñòâåííûå çíà÷åíèÿZ(6 ln 2 − 4πE1 )g0 =0000ZΩ(τ, s, τ , s )R2R2ln((τ 0 − τ 00 )2 + (s0 − s00 )2 )×250× | g0 (τ 0 , s0 ) |2 dτ 0 ds0 g0 (τ 00 , s00 )dτ 00 ds00 ,Z| g0 (τ, s) |2 dτ ds = 1.(3.24)(3.25)R2Çäåñü ôóíêöèÿΩ(τ, s, τ 00 , s00 ) =2=e−(spXHj (s)Hj (s00 )Hp−j (τ )Hp−j (τ 00 )+τ 2 +(s00 )2 +(τ 00 )2 )/22p πj!(p − j)!j=0.Ïîñêîëüêó â ðåçóëüòàòå ïðåîáðàçîâàíèÿ Ãàóññà [6]1√πòî äëÿ ÿäðàΩZ2(2ix)n e(ix−y) dx = Hn (y),Rèìååò ìåñòî èíòåãðàëüíîå ïðåäñòàâëåíèå(−1)p 2p −(s2 +τ 2 +(s00 )2 +(τ 00 )2 )/2e×Ω(τ, s, τ , s ) =p!π 300Z×002(s̃s̃0 + τ̃ τ̃ 0 )p e(is̃−s)+(iτ̃ −τ )2 +(is̃0 −s0 )2 +(iτ̃ 0 −τ 0 )2dτ̃ ds̃dτ̃ 0 ds̃0 .R4Èç (3.24), (3.25) ñëåäóåò, ÷òî ÷èñëîE1ìîæåò áûòü çàïèñàíî â âèäåZ Z3 ln 21E1 =−Ω(τ, s, τ 00 , s00 )g0 (τ 00 , s00 )g 0 (τ, s)dτ ds×2π4π R2 R2Z×ln((τ 0 − τ 00 )2 + (s0 − s00 )2 ) | g0 (τ 0 , s0 ) |2 dτ 0 ds0 dτ 00 ds00 .
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