Диссертация (1136178), страница 24
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Òàê êàê ýòî ðàçëîæåíèå ñîãëàñóåòñÿ ñ ÂÊÁ-ïðèáëèæåíèåì, òî èç (2.61) âûòåêàåò, ÷òî202ïîðÿäîê çíà÷åíèép(z)âáëèçè√√(1 + x20 )` ( 3 + 2)` expx0îïðåäåëÿåòñÿ ìíîæèòåëåì√(1 +2`(3x40 + 6x20 + 1)√√x20 )( 3(1 + x20 ) + 2)!.(2.71)Ñðàâíèâàÿ (2.71) ñ (2.67) (2.70), íàõîäèì, ÷òî äëÿ ýêñïîíåíöèàëü-N (z)íîé ìàëîñòèp(z)ïî ñðàâíåíèþ ñïðèp0 ≤| x0 |< 2e2 − 3(2.72)äîëæíî âûïîëíÿòüñÿ íåðàâåíñòâî√1 + 3x202 | x0 |pexp< exp1 + x203 + x20√(1 +2(3x40 + 6x20 + 1)√√x20 )( 3(1 + x20 ) + 2)√√×( 3 + 2),à ïðè| x0 |≥| x0 | exp√2e2 − 32x201 + x20!×(2.73) íåðàâåíñòâî√√< ( 3 + 2) exp√(1 +2(3x40 + 6x20 + 1)√√x20 )( 3(1 + x20 ) + 2)!.(2.74)Íåðàâåíñòâî (2.73) èìååò ìåñòî ïðè âñåõx0 , óäîâëåòâîðÿþùèõ (2.72),à íåðàâåíñòâî (2.74) âûïîëíÿåòñÿ ëèøü ïðè| x0 |< x∗ ,ãäåx∗ êîðåíü óðàâíåíèÿ2x2x exp1 + x2√√= ( 3 + 2) expïðèáëèæåííîå çíà÷åíèå êîòîðîãîäëÿN (z)(2.69).(2.75)√2(3x4 + 6x2 + 1)√√(1 + x2 )( 3(1 + x2 ) + 2)x∗ ≈ 4.94.!,Ýòî ñâÿçàíî ñ òåì, ÷òîâûøå áûëà èñïîëüçîâàíà äîâîëüíî ãðóáàÿ îöåíêà ñâåðõó203Äàëåå îãðàíè÷èìñÿ ðàññìîòðåíèåì çíà÷åíèéx0 , óäîâëåòâîðÿþ-ùèõ óñëîâèþ (2.75).
Äëÿ íàõîæäåíèÿ àñèìïòîòè÷åñêèõ ñîáñòâåííûõ| x0 |≥ x∗ôóíêöèé ïðèìåíåíèåìx0ìîæíî âîñïîëüçîâàòüñÿ ñâÿçüþ ìåæäó èç-è ñäâèãîì íà ïîëÿðíûé óãîë, êîòîðûé äîïóñêàþò ñîá-ñòâåííûå ôóíêöèè çàäà÷è (0.23), (0.24). Îïåðàòîð ñäâèãà íà óãîëϕ0 ,îïðåäåëåííûé íà ñîáñòâåííîì ïîäïðîñòðàíñòâå, ïîä äåéñòâèåìêîãåðåíòíîãî ïðåîáðàçîâàíèÿMϕ0 : P` → P` .I` (g)(1.24) ïðåîáðàçóåòñÿ â îïåðàòîðÑïðàâåäëèâàÎïåðàòîð Mϕ0 èìååò âèäËåììà 2.10.`Mϕ0 g(z) = (cos ϕ0 − z sin ϕ0 ) gz cos ϕ0 + sin ϕ0cos ϕ0 − z sin ϕ0.(2.76)Äîêàçàòåëüñòâî. Ïóñòüq1 = % cos ϕ,q1∗ = % cos(ϕ + ϕ0 ),q2 = % sin ϕ,Òîãäàq2∗ = % sin(ϕ + ϕ0 ).q1∗ + zq2∗q1 + z ∗ q 2√=p,1 + z21 + (z ∗ )2ãäå(2.77)z cos ϕ0 − sin ϕ0.z sin ϕ0 + cos ϕ0(2.78)z ∗ cos ϕ0 + sin ϕ0,cos ϕ0 − z ∗ sin ϕ0(2.79)z∗ =Ïîñêîëüêó â ñèëó (2.78)z=òî ñïðàâåäëèâû ñîîòíîøåíèÿ2 `/2(1 + z )(1 + (z ∗ )2 )`/2=,(cos ϕ0 − z ∗ sin ϕ0 )`dz ∗ dz ∗dzdz =×(cos ϕ0 − z ∗ sin ϕ0 )2×(cos ϕ0 − z ∗ sin ϕ0 )−2 ,1 + |z ∗ |21 + |z| =.(cos ϕ0 − z ∗ sin ϕ0 )(cos ϕ0 − z ∗ sin ϕ0 )2(2.80)(2.81)204Íàêîíåö, èç (1.24), (1.27), (2.77), (2.79) (2.81) âûòåêàåò ðàâåíñòâîq ∗ + zq ∗ (q ∗ )2 + (q2∗ )2 dzdzg(z)(1 + z 2 )`/2 H` √ 1√ 2 exp − 1=2~(1 + |z|2 )`+2~ 1 + z2CZz ∗ cos ϕ0 + sin ϕ0=)(cos ϕ0 − z ∗ sin ϕ0 )` (1 + (z ∗ )2 )`/2 ×g(∗cos ϕ0 − z sin ϕ0CZq 1 + z ∗ q2 q12 + q22 dz ∗ dz ∗exp −,×H` √ p2~(1 + |z ∗ |2 )`+2~ 1 + (z ∗ )2èç êîòîðîãî ñëåäóåò ôîðìóëà (2.76) äëÿ îïåðàòîðàMϕ0 .Ëåììà äî-êàçàíà.1.6.Àñèìïòîòèêà íîðìûÏóñòü ìíîãî÷ëåíΦ(z)Φ(z) çàäàí ôîðìóëîé (0.12), ãäå ôóíêöèÿ p(u) àñèìïòîòè÷åñêîå ðåøåíèå ìíîãîòî÷å÷íîé ñïåêòðàëüíîé çàäà÷è.Ïðè âûïîëíåíèè óñëîâèÿ (2.75) âû÷èñëèì àñèìïòîòèêó íîðìûâ ïðîñòðàíñòâåΦ(z)P` .Çàïèøåì ãëàâíûé ÷ëåí ÂÊÁ-ïðèáëèæåíèÿ (2.62) â âèäå (1.162),ãäå ôóíêöèÿp√−2(1 − x20 + 2x0 z) + 2(x0 z + 1) Λ(z)s(z) =+(1 + x20 )(1 + z 2 )p√+ ln( Λ(z) + 2(x0 z + 1)),à ìíîãî÷ëåíΛ(z)(2.82)çàäàí ôîðìóëîé (2.20).
Åñëè ïîäñòàâèòü (1.162) âôîðìóëó (1.25) äëÿ ñêàëÿðíîãî ïðîèçâåäåíèÿ, òî ïîëó÷èì èíòåãðàë(1.163), ãäå ôóíêöèÿΩ(z, z)èìååò âèä (1.164).Íàéäåì òî÷êó, ãäå äîñòèãàåòñÿ ãëîáàëüíûé ìàêñèìóìΩ(z, z).Òîãäà àñèìïòîòèêà èíòåãðàëà (1.163) áóäåò ðàâíà èíòåãðàëó ïî ìàëîé îêðåñòíîñòè ýòîé òî÷êè.Ïðåäâàðèòåëüíî äîêàæåì äâå ëåììû.Ëåììà 2.11.Ýêñòðåìóì ôóíêöèè Ω(z, z) ìîæåò äîñòèãàòüñÿëèøü â òî÷êàõ, ãäå z = z .205Äîêàçàòåëüñòâî. Èñïîëüçóÿ (2.82), íàõîäèì, ÷òî ñòàöèîíàðíûåòî÷êè óäîâëåòâîðÿþò ñëåäóþùåé ñèñòåìå óðàâíåíèép∂Ω −4x0 + (5 − 3x20 )z + 4x0 z 2 + (1 + x20 )z 3 − 2(z − x0 ) 2Λ(z)−=∂z(1 + x20 )(1 + z 2 )2−z= 0,1 + zz(2.83)p222 3−4x+(5−3x)z+4xz+(1+x)z−2(z−x)2Λ(z)∂Ω00000=−∂z(1 + x20 )(1 + z 2 )2−ÇäåñüΛ(z)z= 0.1 + zz(2.84)çàäàíà ôîðìóëîé (2.20).Ïðåîáðàçóåì óðàâíåíèå (2.83) ê âèäó−8x20 + 8x0 z + (1 + x20 )z 2p√=−4x0 + (5 − 3x20 )z + 4x0 z 2 + (1 + x20 )z 3 + 2 2(z − x0 ) Λ(z)=z,1 + zzèç êîòîðîãî âûòåêàåò ñîîòíîøåíèå−8x20 + 8x0 z + 4x0 z + (1 + x20 )z 2 − (5 + 5x20 )zz + 4x0 z 2 z =p√= 2 2z(z − x0 ) Λ(z).Äàëåå ïîñëå çàìåíûu = z − x0 , u = z − x0(2.85)âîçâåäåì ïðàâóþ è ëåâóþ÷àñòè (2.85) â êâàäðàò.
Èìååì:−u2 u4 (8+8x20 ) −u2 u3 (88x0 +24x30 )+ u2 u2 (1− 86x20 − 23x40 ) −uu4 (8x0 ++8x30 )−uu3 (10+100x20 +26x40 )−uu2 (100x0 +128x30 +28x50 )+u4 (1+2x20 ++x40 ) + u3 (10x0 + 12x30 + 2x50 ) + u2 (x20 + 2x40 + x60 ) + u2 (x20 + 2x40 + x60 )++u2 u(10x0 + 4x30 − 6x50 ) − uu(10x20 + 20x40 + 10x60 ) = 0.(2.86)206Óìíîæèì (2.86) íàρeiϕ , u = ρe−iϕ .uè ïåðåéäåì ê ïîëÿðíûì êîîðäèíàòàìu = ðåçóëüòàòå ïîëó÷àåì:−ρ7 eiϕ (8+8x20 )−ρ6 (88x0 +24x30 )+ρ5 e−iϕ (1−86x20 −23x40 )−ρ6 e2iϕ (8x0 ++8x30 )−ρ5 eiϕ (10+100x20 +26x40 )−ρ4 (100x0 +128x30 +28x50 )+ρ5 e3iϕ (1++2x20 +x40 )+ρ4 e2iϕ (10x0 +12x30 +2x50 )+ρ3 eiϕ (x20 +2x40 +x60 )+ρ3 e−3iϕ (x20 ++2x40 + x60 ) + ρ4 e−2iϕ (10x0 + 4x30 − 6x50 ) − ρ3 eiϕ (10x20 + 20x40 + 10x60 ) = 0.(2.87)Ïðèðàâíÿâ ê íóëþ ìíèìóþ è âåùåñòâåííóþ ÷àñòè (2.87), ïðèõîäèì ê óðàâíåíèÿì(1 + x20 )ρ3 sin ϕ{(1 + x20 )(ρ2 − x20 ) cos2 ϕ + 4(−ρ3 x0 + ρx30 ) cos ϕ−−[2ρ4 + ρ2 (3 + x20 ) + 2x20 (1 + x20 )]} = 0,(2.88)(1 + x20 )2 (ρ2 + x20 ) cos3 ϕ − 2x0 (1 + x20 )(2ρ3 + ρ(x20 − 5)) cos2 ϕ−−[2(1 + x20 )ρ4 + (3 + 48x20 + 13x40 )ρ2 + 3x20 (1 + x20 )2 ] cos ϕ−−2x0 [2ρ3 (5 + x20 ) + 3ρ(1 + x20 )(5 + x20 )] = 0.(2.89)sin ϕ ðàâåí íóëþ ëèøü ïðè z = z , òî ïîñëå äåëåíèÿ (2.88)(1 + x20 )2 ρ3 (ρ2 − x20 ) sin ϕ èìååì:Ïîñêîëüêóíà4ρx02ρ4 + ρ2 (3 + x20 ) + 2x20 (1 + x20 )cos ϕ −cos ϕ −= 0.1 + x20(1 + x20 )(ρ2 − x20 )2 ÷àñòíîñòè, ïðèx0 = 0óðàâíåíèå (2.90) ïðèíèìàåò âèäcos2 ϕ = 2ρ2 + 3.Òàê êàê (2.91) íå èìååò ðåøåíèé, òî óòâåðæäåíèå ëåììû ïðèäîêàçàíî.(2.90)(2.91)x0 = 0207x0 6= 0.Èç ðàâåíñòâ (2.89),2ρx0 (5 + x20 )cos ϕ = −,(1 + x20 )(ρ2 + x20 )(2.92)Ðàññìîòðèì äàëåå ñëó÷àé, êîãäà(2.90) âûòåêàåò ñîîòíîøåíèåà èç (2.89), (2.92) ñëåäóåò, ÷òî2ρx0 (5 + x20 )(2ρ2 + 3 + 3x20 ).cos ϕ = −(1 + x20 )(2ρ4 + (3 + 5x20 )ρ2 + 3x20 )(2.93)Óñëîâèåì ñîâìåñòíîñòè (2.92), (2.93) ÿâëÿåòñÿ ðàâåíñòâî12ρ2 + 3 + 3x20= 4,ρ2 + x202ρ + (3 + 5x20 )ρ2 + 3x20êîòîðîå ðàâíîñèëüíî óñëîâèþx0 = 0.Ñëåäîâàòåëüíî, ïðèx0 6= 0ñèñòåìà óðàâíåíèé (2.92), (2.93) íå ðàçðåøèìà, à, çíà÷èò, è â ýòîìñëó÷àå îòñóòñòâóþò ñòàöèîíàðíûå òî÷êè, äëÿ êîòîðûõz 6= z .
Ëåììàäîêàçàíà.Ëåììà 2.12.Ìàêñèìóì ôóíêöèè Ω(z, z) ïðè z = z äîñòèãàåòñÿ âòî÷êå z = z = x0 .z = z = x ∈ R ôóíêöèþâûòåêàåò, ÷òî ïðè x 6= −1/x0Äîêàçàòåëüñòâî. Ðàññìîòðèì ïðèΩ = Ω(x).Èç ôîðìóë (1.164), (2.82)Ω0 (x) = 2××p√−4x0 + (5 − 3x20 )x + 4x0 x2 + (1 + x20 )x3 − 2 2(x − x0 ) Λ(x)−(1 + x20 )(1 + x2 )2x−.(2.94)1 + x2ÓðàâíåíèåΩ0 (x) = 0, ãäå Ω0 (x) çàäàåòñÿ ðàâåíñòâîì (2.94), ïðåîáðà-çóåòñÿ ê âèäóp√4(x − x0 )(x0 x + 1) = 2 2(x − x0 ) Λ(x).(2.95)208x0 . Äðóãèõ âåùåñòâåííûõ êîðíåé ýòî óðàâíåíèå íå èìååò, ïîñêîëüêó ïîñëå äåëåíèÿ (2.95) íà x − x0 è âîçâåäåíèÿÊîðíåì (2.95) ÿâëÿåòñÿïðàâîé è ëåâîé ÷àñòåé (2.95) â êâàäðàò, ïðèõîäèì ê óðàâíåíèþ(1 + x20 )(1 + x2 ) = 0.x > 0 è Ω0 (x) > 0 ïðè x < 0.Ñëåäîâàòåëüíî, ìàêñèìóì ôóíêöèè Ω(x) äîñòèãàåòñÿ â òî÷êå x =p0.
 ñëó÷àå x0 > 0 â ñèëó âûáîðà âåòâåé Λ(x) ôóíêöèÿ Ω(x)00íåïðåðûâíà, Ω (x) > 0 ïðè x ∈ (−1/x0 , x0 ) è Ω (x) < 0 ïðè x ∈(−∞, −1/x0 ) ∪ (x0 , +∞). Òàê êàêÅñëèx0 = 0 ,òîΩ0 (x) < 0ïðèlim Ω(x) = lim Ω(x),x→−∞òî ìàêñèìóì ôóíêöèèx0 < 0Ω(x)x→+∞òàêæå äîñòèãàåòñÿ â òî÷êåx0 .Ñëó÷àéðàññìàòðèâàåòñÿ àíàëîãè÷íî. Ëåììà äîêàçàíà.Òåîðåìà 2.1.Ôóíêöèÿ Ω(z, z) äîñòèãàåò ìàêñèìàëüíîå çíà÷åíèåïðè z = z = x0 .Äîêàçàòåëüñòâî. Òàê êàê ñïðàâåäëèâû ëåììû 2.11., 2.12., òîîñòàåòñÿ ïðîâåðèòü, ÷òî â òî÷êåz = z = x0äëÿΩ(z, z)âûïîëíåíûäîñòàòî÷íûå óñëîâèÿ ñóùåñòâîâàíèÿ ëîêàëüíîãî ìàêñèìóìà ôóíêöèè äâóõ ïåðåìåííûõ.
Ýòè óñëîâèÿ èìåþò âèä (1.177), (1.178).Èç (2.83), (2.84) âûòåêàåò, ÷òî∂Ω(x0 , x0 ) = 0,∂zÄèôôåðåíöèðóÿ ôóíêöèè∂Ω(x0 , x0 ) = 0.∂z∂Ω/∂z, ∂Ω/∂zåùå ðàç, íàõîäèì, ÷òî√∂ 2Ω∂ 2Ω5 − 2 6 ∂ 2Ω1(x,x)=(x,x)=,(x,x)=−,000000∂z 2(1 + x20 )2 ∂z∂z(1 + x20 )2∂z 2è, ñëåäîâàòåëüíî,√ √√∂ 2 Ω 4 2( 2 − 3)< 0,=∂x2(1 + x20 )2√√4 6(5 − 2 6)D=> 0.(1 + x20 )4209Òåîðåìà äîêàçàíà. ñèëó òåîðåìû 2.1. îñíîâíîé âêëàä â íîðìó àñèìïòîòè÷åñêî-p(z) âíîñèò ìàëàÿz = x0 ôóíêöèÿ p(z)ãî ðåøåíèÿ ìíîãîòî÷å÷íîé ñïåêòðàëüíîé çàäà÷èîêðåñòíîñòü òî÷êèz = z = x0 .Òàê êàê âáëèçèçàäàåòñÿ ðàçëîæåíèåì (2.59), ïîäñòàâèì åãî â ôîðìóëó (1.25) äëÿñêàëÿðíîãî ïðîèçâåäåíèÿ è âû÷èñëèì àñèìïòîòèêó âîçíèêàþùåãîèíòåãðàëà.Îïðåäåëèìr !2 2θ(t, r) = exp (− 1 −t −3Ëåììà 2.13.kp(z)k2P`r!3− 1 r2 ).2Ñïðàâåäëèâî ðàâåíñòâîα12 µ2= k+2 `π2 R (x0 )r2Σ0 (k)(1 + O(~)),3~ → 0,(2.96)ãäå ôóíêöèÿ Σ0 (k) èìååò âèäZθ(t, r) | Hk (t + ir) |2 dt dr,Σ0 (k) =(2.97)R2à êîíñòàíòà µ çàäàíà ôîðìóëîé(2.58).Äîêàçàòåëüñòâî.
Ðàçëîæèì ôóíêöèþïî ñòåïåíÿìu, u,ãäåuèz(` + 1)/(2π(1+ | z |2 )`+2 )ñâÿçàíû ðàâåíñòâîì (2.47).  ðåçóëüòàòåïîëó÷àåì:√ r`+1aa 4 2√=exp−x0 (u + u)−2π(1+ | z |2 )`+2π~(R(x0 ))`+22 ~ 3rr√2 | u |2 −x20 (u2 + u2 )~x0 4 21 2 2 3√−{1 − √[u + u +[x0 (u + u3 )−48 32 a 38 6−3(u + u) | u |2 ]] + O(~(1+ | u |6 ))}.(2.98)210Ïîäñòàâëÿÿ çàòåì (2.59), (2.98) â ôîðìóëó (1.25), èìååì:√√√ 2 2(52−43−2x0 )(u + u2 )2√kp(z)kP`exp(−16 3C(r√2 | u |2 −x20 (u2 + u2 )u 2~x0 4 2√−) | Hk ( √ ) | − √[u + u+2 a 38 62r√1 2 2 3u~[x0 (u + u3 ) − 3(u + u) | u |2 ]] | Hk ( √ ) |2 + √ 3/4 ×+48 32 a62("#√√x0 (3(−5 + 2 6) + x20 )(u3 + u3 ) (2 − 6)B(x0 )(u + u)√√×+×12 22α12 µ2 a= k2 π~(R(x0 ))`+2Zu 2 √uux0 u2+ B(x0 )]Hk0 ( √ )Hk ( √ )+× | Hk ( √ ) | + 6[−2222√uux0 u2+ 6[−+ B(x0 )]Hk0 ( √ )Hk ( √ )dz dz (1 + O(~)).222ÇäåñüB(x0 )(2.99)îïðåäåëåíî ôîðìóëîé (2.50).Ââåäåì âåùåñòâåííûå ïåðåìåííûåtèrñîãëàñíî ôîðìóëå(1.184).
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