Методы математической физики. Конспект лекций
TRANSCRIPT
Êîíñïåêò ëåêöèé ïî ìàòåìàòè÷åñêîé ôèçèêåÔÒÔ, îñåííèé ñåìåñòð
Ïëàí ñåìåñòðà1. Ñèíãóëÿðíàÿ çàäà÷à Øòóðìà-Ëèóâèëëÿ
2. Ïðåîáðàçîâàíèå Ëàïëàñà
3. Èíòåãðàëüíûå óðàâíåíèÿ
4. Âàðèàöèîííîå èñ÷èñëåíèå
Ðåêîìåíäóåìàÿ ëèòåðàòóðà1. Á.Ì. Ëåâèòàí, È.Ñ. Ñàðãñÿí. Çàäà÷à Øòóðìà-Ëèóâèëëÿ
2. Ý.Ò. Òèò÷ìàðø. Ðàçëîæåíèå ïî ñîáñòâåííûì ôóíêöèÿì, ñâÿçàííûì ñ äèôôåðåíöèàëüíûìèóðàâíåíèÿìè 2-îãî ïîðÿäêà (òîì 1)
3. Ì.À. Ëàâðåíòüåâ, Á.Â. Øàáàò. Ìåòîäû òåîðèè ôóíêöèé êîìïëåêñíîãî ïåðåìåííîãî
4. È.Ã. Ïåòðîâñêèé. Èíòåãðàëüíûå óðàâíåíèÿ
5. Â.È. Ñìèðíîâ. Êóðñ âûñøåé ìàòåìàòèêè, ò. IV
6. Â.Ñ. Áóñëàåâ. Âàðèàöèîííîå èñ÷èñëåíèå
7. Ýëüñãîëüö. Äèôôåðåíöèàëüíûå óðàâíåíèÿ è âàðèàöèîííîå èñ÷èñëåíèå
Äîïîëíèòåëüíî ïîòðåáóåòñÿ1. Èíòåãðàë Ñòèëòüåñà
2. Òåîðåìû Õåëëè (î ïðåäåëüíîì ïåðåõîäå â èíòåãðàëå Ñòèëòüåñà)
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Ãëàâà 1
Ñèíãóëÿðíàÿ çàäà÷àØòóðìà-Ëèóâèëëÿ
Ïóñòü f(x) óäîâëåòâîðÿåò óñëîâèÿì Äèðèõëå, òîãäà ñóùåñòâóåò èíòåãðàë :
f(ν) =
∞∫
−∞f(x)eiνxdx,
è èìååòñÿ ôîðìóëà îáðàùåíèÿ :
f(x) =12π
∞∫
−∞f(ν)e−iνxdν.
Ðàññìîòðèì óðàâíåíèåy′′ + λy = 0, −l < y < l
ñ ãðàíè÷íûìè óñëîâèÿìèy(−l) = y(l) è y′(−l) = y′(l).
Ðåøåíèå ýòîé ñèñòåìû :λ = 0 y0(x) = 1
λn =π2n2
l2y(1)
n = ei√
λnx è y(2)n = e−i
√λnx,
ñèñòåìà ýòèõ ôóíêöèé îáëàäàåò ñâîéñòâîì ïîëíîòû.Âûïîëíèì ïðåäåëüíûé ïåðåõîä l → ∞, ïðè ýòîì óðàâíåíèå ñîõðàíèòñÿ, âìåñòî ãðàíè÷íûõ
óñëîâèé ïîòðåáóåì |y(x)| < ∞, òîãäà ïîëó÷èì íàáîð ñîáñòâåííûõ ôóíêöèé
yλ(x) = ei√
λx
Îäíàêî òàêèå ôóíêöèè íå ÿâëÿþòñÿ áàçèñîì â ãèëüáåðòîâîì ïðîñòðàíñòâå, ïîýòîìó ïðîñòîå îáîá-ùåíèå çàäà÷è Øòóðìà-Ëèóâèëëÿ íà ñèíãóëÿðíûé ñëó÷àé íå ïîëó÷àåòñÿ.
Ìû óâèäèì, ÷òî ñèíãóëÿðíûå çàäà÷è Øòóðìà-Ëèóâèëëÿ ïðèâîäÿò ê èíòåãðàëüíûì ïðåîáðàçî-âàíèÿì.
1.1 Ðàâåíñòâî Ïàðñåâàëÿ íà ïîëóîñèÐàññìîòðèì äèôôåðåíöèàëüíîå óðàâíåíèå è ãðàíè÷íîå óñëîâèå 3-åãî ðîäà ïðè x = 0
y′′ + (λ− q(x))y = 0, 0 < x < ∞
y(0) cos α + y′(0) sin α = 0
Íàðÿäó ñ ýòîé çàäà÷åé ðàññìîòðèì ðåãóëÿðíóþ
y′′ + (λ− q(x))y = 0, 0 < x < b
2
y(0) cos α + y′(0) sin α = 0,y(b) cos β + y′(b) sin β = 0
Òàêàÿ çàäà÷à îáëàäàåò ñ÷åòíûì íàáîðîì ñîáñòâåííûõ ÷èñåë λn,b è ñîáñòâåííûõ ôóíêöèé yn,b.Âîçüìåì ôóíêöèþ f(x) ∈ L2(0, b), ïóñòü
fn =
b∫
0
f(x)yn,b(x)dx
/
b∫
0
y2n,b(x)dx
1/2
=
b∫
0
f(x)yn,b(x)dx
/Nn,b
Èìååò ìåñòî ðàâåíñòâî Ïàðñåâàëÿ
b∫
0
f(x)2dx =∞∑
n=1
f2n =
∞∑n=1
1N2
n,b
b∫
0
f(x)yn,b(x)dx
Ââåäåì ôóíêöèþ (ñì. ðèñ. 1.1)
ρb(λ) =
− ∑λ<λn,b<0
1N2
n,b, λ ≤ 0
∑λ>λn,b>0
1N2
n,b, λ > 0
r(l)b
l
Ðèñ. 1.1:
Ðàññìîòðèì Fb(λ) =b∫0
f(x)y(x, λ)dx, ãäå y(x, λ) ðåøåíèå íà÷àëüíîé çàäà÷è
y′′ + (λ− q(x))y = 0,y(0) = sin α,
y′(0) = − cosα
Òîãäàb∫
0
f2(x)dx =
∞∫
−∞F 2
b (λ)dρb(λ),
â ñàìîì äåëå, âû÷èñëÿÿ èíòåãðàë Ñòèëòüåñà∞∫
−∞F 2
b (λ)dρb(λ) =∑
n
F 2(λn)[ρ(λn + 0)− ρ(λn − 0)] =∑
n
(∫ b
0
f(x)yn,b(x)dx
)1
N2n,b
.
Áóäåì äîêàçûâàòü, ÷òî äëÿ âñÿêîé f(x) ∈ L2(0,∞) åñòü ðàâåíñòâî∞∫
0
f2(x)dx =
∞∫
−∞F 2(λ)dρ(λ),
3
2
2
b
p2
24
b
p2
29
b
p2
216
b
p2
225
b
p
llrp2)( =
)
)(lr)
l
Ðèñ. 1.2:
ãäå ôóíêöèè F è ρ ïîëó÷àþòñÿ èç Fb è ρb ïðåäåëüíûì ïåðåõîäîì b →∞ (ðèñ. 1.2).Ïðèìåð
y′′ + λy = 0, 0 < x < b
y′(0) = 0,y′(b) = 0
Ïîòðåáóåì äîïîëíèòåëüíî∫ b
0ydx = 0, òîãäà èìååì ïîñëåäîâàòåëüíîñòü ñîáñòâåííûõ ÷èñåë
λn =π2n2
b2,
yn = cosπnx
b,
êâàäðàò íîðìû N2n = b/2. Ëåãêî âèäåòü, ÷òî ρ(λ) = 2
π
√λ íå çàâèñèò îò b. Òîãäà ââåäåì
F (λ) =2π
∫ ∞
0
cos√
λf(x)dx,
íåïîñðåäñòâåííî âû÷èñëÿÿ èíòåãðàë∫ ∞
0
F 2(λ)d(2π
√λ) =
∫f2(x)dx.
êîíåö ïðèìåðàÄëÿ äîêàçàòåëüñòâà ðàññìîòðèì ïîñëåäîâàòåëüíîñòü fn(x) ∈ C2(O, b), fn(x) ≡ 0 äëÿ âñÿêîãî
x > n, n < b è ïóñòü fn(x) óäîâëåòâîðÿåò ãðàíè÷íîìó óñëîâèþ fn(0) cos α + f ′n(0) sin α = 0Äîïðåäåëüíàÿ ôîðìà ðàâåíñòâà Ïàðñåâàëÿ
∫ b
0
f2(x)dx =∫ ∞
−∞F 2
b (λ)dρ(λ), (A)
ïîäñòàâèâ f = fn(x) èìååì ∫ n
0
f2n(x)dx =
∫ ∞
−∞F 2
b,n(λ)dρ(λ),
è Fb,n(λ) =∫ n
0fn(x)y(x, λ)x.
Ïîñêîëüêó y′′ + (λ− q(x))y = 0
Fb,n(λ) = − 1λ
∫ n
0
fn(x)[y′′ − qy]dx = − 1λ
fn(x)y′|n0 +1λ
yf ′n|n0 −1λ
∫ n
0
(f ′′n − qfn)y(x, λ)dx =
− 1λ
∫ n
0
(f ′′n − qfn)y(x, λ)dx
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Îöåíèì õâîñòû èíòåãðàëîâ â (A)∫
|λ|>M
F 2b,ndρb(λ) =
∫
|λ|>M
1λ2
(∫ n
0
(f ′′n − qfn)y(x, λ)dx
)2
dρb(λ) ≤
1M2
∫
|λ|>M
(∫ n
0
(f ′′n − qfn)y(x, λ)dx
)2
dρb(λ) ≤ 1M2
∞∫
−∞
(∫ n
0
(f ′′n − qfn)y(x, λ)dx
)2
dρb(λ) =
(âîñïîëüçîâàâøèñü ðàâåíñòâîì Ïàðñåâàëÿ â îáðàòíóþ ñòîðîíó)
=1
M2
∫ n
0
(f ′′n − qfn)2dx,
òàêèì îáðàçîì ∫
|λ|>M
F 2b,ndρb(λ) ≤ 1
M2
∫ n
0
(f ′′n − qfn)2dx.
Òîãäà ∣∣∣∣∣∫ ∞
0
f2ndx−
∫ M
−M
F 2b,nλdρb(λ)
∣∣∣∣∣ ≤1
M2
∫ n
0
(f ′′n − qfn)2dx.
Äàëåå ñëåäóåò âûïîëíèòü ïðåäåëüíûé ïåðåõîä b → ∞ (îöåíêè äëÿ ïðåäåëüíîãî ïåðåõîäà ïîâòîðîé òåîðåìå Õåëëè îïóùåíû), ôóíêöèè ρb(λ) → ρ(λ) (ρ-ïðåäåëüíàÿ ôóíêöèÿ - ïîëó÷àåòñÿ ïîòåîðåìå Õåëëè). (Ïîäðîáíîñòè - ñì. À.Í. Êîëìîãîðîâ, Ñ.Â. Ôîìèí "Ýëåìåíòû òåîðèè ôóíêöèé èôóíêöèîíàëüíîãî àíàëèçà".) Òàêèì îáðàçîì èìååì
∣∣∣∣∣∫ ∞
0
f2ndx−
∫ M
−M
F 2b,nλdρ(λ)
∣∣∣∣∣ ≤1
M2
∫ n
0
(f ′′n − qfn)2dx.
âûïîëíèì ïðåäåëüíûé ïåðåõîä M →∞, ñëåäîâàòåëüíî
∫f2
ndx =
∞∫
−∞F 2
ndρ(λ). (A′)
Äëÿ çàâåðøåíèÿ äîêàçàòåëüñòâà âîñïîëüçóåìñÿ òåì, ÷òî ìíîæåñòâî ôóíêöèé fn âñþäó ïëîòíîâ L2(0,∞) :
∀f(x) ∈ L2(0,∞)∃fn :∫ ∞
0
(f(x)− fn(x))2dx → 0, n →∞,
ïðè÷åì ïîñëåäîâàòåëüíîñòü fn ñõîäèòñÿ â ñåáå, ò. å.∫∞0
(fn − fm)2dx →∞ ïðè n,m →∞.
Òîãäà ñîãëàñíî (A')∞∫−∞
(Fn − Fm)2dρ(λ) =∞∫0
(fn − fm)2dx → 0, ïðè n,m → ∞, ñëåäîâàòåëü-
íî ïîñëåäîâàòåëüíîñòü Fn(λ) ñõîäèòñÿ â ñåáå, à â ñèëó ïîëíîòû ïðîñòðàíñòâà L2(0,∞, ρ) ýòàïîñëåäîâàòåëüíîñòü ê ïðåäåëüíîé F (λ), çíà÷èò
∞∫
0
f2(x)dx =
∞∫
−∞F 2(λ)dρ(λ). (B)
Çàìå÷àíèå. Âòîðîå ãðàíè÷íîå óñëîâèå äëÿ çàäà÷è Øòóðìà-Ëèóâèëëÿ íå ïîëóîîñè íå ñòàâèòñÿ,ò.ê. óñëîâèå îãðàíè÷åííîñòè ñëèøêîì ñèëüíî îãðàíè÷èâàåò êëàññ ðåøåíèé, à óñëîâèå ïðèíàäëåæ-íîñòè ðåøåíèÿ ê L2(0,∞) ìîæåò è íå âûïîëíÿòüñÿ, ïîýòîìó çàäà÷à íà (0,∞) ðàññìàòðèâàåòñÿ êàêïðåäåë çàäà÷ íà (0, b).
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1.2 Îáîáùåííîå ðàâåíñòâî Ïàðñåâàëÿ. Òåîðåìà ðàçëîæåíèÿÐàññìîòðèì ïðîèçâîëüíûå ôóíêöèè f(x), g(x) ∈ L2(0,∞).Ñîãëàñíî ðåçóëüòàòàì ïàðàãðàôà ??, èìååì
∞∫
0
(f + g)2dx =
∞∫
−∞(F (λ) + G(λ))2dρ(λ)
∞∫
0
(f − g)2dx =
∞∫
−∞(F (λ)−G(λ))2dρ(λ)
Âû÷èòàÿ èç ïåðâîãî ðàâåíñòâà âòîðîå èìååì∞∫
0
fgdx =
∞∫
−∞F (λ)G(λ)dρ(λ)
Ïîëó÷åííîå ñîîòíîøåíèå íîñèò íàçâàíèå îáîáùåííîãî ðàâåíñòâà Ïàðñåâàëÿ.Ïðåæäå ÷åì ïðèñòóïèòü ê äîêàçàòåëüñòâó òåîðåìû ðàçëîæåíèÿ, äîêàæåì ñëåäóþùóþ âàæíóþ
ëåììó.
Ëåììà 1.2.1 (Ëàãðàíæà) Ïóñòü äëÿ ∀g(x) ∈ C(a, b),∫ b
af(x)g(x)dx = 0. Òîãäà f ≡ 0.
ÄîêàçàòåëüñòâîÏóñòü â íåêîòîðîé òî÷êå ξ ôóíêöèÿ f(ξ) > 0. Ñëåäîâàòåëüíî ∃ε > 0 : ∀x ∈ (ξ − ε, ξ + ε), f(x) > 0
âîçüìåì g(x) =
(x− ξ + ε)(x− ξ − ε), êîãäà |x− ξ| < ε0 èíà÷å
Î÷åâèäíî, ÷òî òàêàÿ g(x) ÿâëÿåòñÿ íåïðåðûâíîé ôóíêöèåé, è èíòåãðàë∫ b
af(x)g(x)dx > 0. Òàêèì
îáðàçîì ìû ïðèõîäèì ê ïðîòèâîðå÷èþ, ñëåäîâàòåëüíî ëåììà äîêàçàíà.
Ïåðåéäåì, íàêîíåö ê äîêàçàòåëüñòâó îñíîâíîé òåîðåìû äàííîãî ïàðàãðàôà.
Òåîðåìà 1.2.2 (ðàçëîæåíèÿ) Ðàññìîòðèì f(x) ∈ C(0,∞) è ïóñòü∞∫−∞
F (λ)y(x, λ)dρ(λ) - ñõî-
äèòñÿ ðàâíîìåðíî ïî x.Òîãäà f(x) =
∞∫−∞
F (λ)y(x, λ)dρ(λ)
ÄîêàçàòåëüñòâîÐàññìîòðèì ôóíêöèþ g(x) ∈ C(0, n) òàêóþ ÷òî g(x) = 0, x > n.
Òîãäà∫ n
0f(x)g(x)dx =
+∞∫−∞
F (λ)(∫ n
0g(x)y(x, λ)dx)dρ(λ) =
Ìåíÿÿ ïîðÿäîê èíòåãðèðîâàíèÿ, â ñèëó ðàâíîìåðíîé ñõîäèìîñòè, è ïðîäîëæàÿ ðàâåíñòâî èìå-åì =
∫ n
0dxg(x)
+∞∫−∞
F (λ)dρ(λ). Èç çà ïðîèçâîëüíîñòè âûáîðà g(x), ïî ëåììå Ëàãðàíæà ïîëó÷àåì
f(x) =∞∫−∞
F (λ)y(x, λ)dρ(λ), ÷òî è òðåáîâàëîñü
ÏðèìåðÂîçüìåì, êàê è â ïðåäûäóùåì ïðèìåðå, ρ(λ) = 2
π
√λ è y(x, λ) = cos
√λx . Òîãäà ðàçëîæåíèå ôóíê-
öèè áóäåò âûãëÿäåòü ñëåäóþùèì îáðàçîì f(x) = 2π
+∞∫−∞
F (λ) cos√
λxd√
λ, ãäå F (λ) =∞∫0
f(x) cos√
λxdx .
Ñäåëàåì çàìåíó s =√
λ , F (s2) = f(s) ,òîãäà
f(x) = 2π
+∞∫−∞
f(s) cos sxds
f(s) =∞∫0
f(x) cos sxdx
Òî åñòü ìû ïîëó÷èëè îáû÷íîå ðàçëîæåíèå â èíòåãðàë Ôóðüå.
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1.3 Êðóã è òî÷êà Âåéëÿ ïðåäûäóùèõ ïàðàãðàôàõ áûëî ïîêàçàíî, ÷òî ïðè èçâåñòíîé ñïåêòðàëüíîé ïëîòíîñòè, äëÿ ëþáîéäîñòàòî÷íî õîðîøåé ôóíêöèè ñóùåñòâóåò ðàâåíñòâî Ïàðñåâàëÿ è òåîðåìà ðàçëîæåíèÿ. Îäíàêîîñòàëñÿ íåðåøåííûì âîïðîñ î íàõîæäåíèè ñàìîé ñïåêòðàëüíîé ïëîòíîñòè äëÿ äàííîãî óðàâíåíèÿ.Ýòîìó âîïðîñó è áóäóò ïîñâÿùåíû òðè ñëåäóþùèõ ïàðàãðàôà.
Äëÿ íà÷àëà âûâåäåì îäíî ïîëåçíîå ñîîòíîøåíèå.Ðàññìîòðèì äèôôåðåíöèàëüíîå óðàâíåíèå y′′(x) + (λ − q(x))y(x) = 0, 0 < x < b (î ãðàíè÷íûõóñëîâèÿõ ðå÷è íå èäåò).Ïóñòü F (x) è G(x) - ðåøåíèÿ ýòîãî óðàâíåíèÿ ñ λ è λ′ ñîîòâåòñòâåííî. Òî åñòü,
F ′′ + (λ− q)F = 0G′′ + (λ′ − q)G = 0
Äîìíîæàÿ ïåðâîå èç ýòèõ óðàâíåíèé íà G, âòîðîå íà F , âû÷èòàÿ èç ïåðâîãî âòîðîå è èíòåãðèðóÿïîëó÷èâøååñÿ óðàâíåíèå ïî ïðîìåæóòêó (0, b), èìååì
∫ b
0(F ′′G−G′′F )dx + (λ− λ′)
∫ b
0FGdx = 0
Ïðîèçâîäÿ â ïåðâîì ñëàãàåìîì èíòåãðèðîâàíèå ïî ÷àñòÿì è çàìåòèâ, ÷òî âíåèíòåãðàëüíûé ÷ëåíñîâïàäàåò ñ âðîíñêèàíîì ôóíêöèé F è G ïîëó÷èì WF, G0 −WF, Gb = (λ′ − λ)
∫ b
0FGdx
Ðàññìîòðèì, â ÷àñòíîñòè, λ′ = λ . Òîãäà, â ñèëó ýðìèòîâîñòè èñõîäíîãî óðàâíåíèÿ, G = F .Îêîí÷àòåëüíî ïîëó÷àåì èñêîìîå ñîîòíîøåíèå
2Imλ
∫ b
0
|F |2dx = iWF, F0 − iWF, Fb (1.1)
Ïðèñîåäèíèì ê ðàññìàòðèâàåìîìó äèôôåðåíöèàëüíîìó óðàâíåíèþ êðàåâîå óñëîâèå y(0) cos α +y′(0) sin α = 0.Ðàññìîòðèì òàêæå ôóíêöèþ φ(x, λ), ÿâëÿþùóþñÿ ðåøåíèåì çàäà÷è Êîøè ñ íà÷àëüíûìè óñëîâèÿ-ìè φ(0, λ) = sin α; φ′(0, λ) = − cos α. Ýòà ôóíêöèÿ, î÷åâèäíî, óäîâëåòâîðÿåò ââåäåííîìó êðàåâî-ìó óñëîâèþ. Òàêàÿ ôóíêöèÿ φ(x, λ) íàçûâàåòñÿ ïåðâûì êàíîíè÷åñêèì ðåøåíèåì èñõîäíîé çàäà÷è.Ðàññìîòðèì íàðÿäó ñ ôóíêöèåé φ, ôóíêöèþ θ(x, λ) òàêóþ, ÷òî θ(0, λ) = cos α; θ′(0, λ) = sin α .Òàêàÿ ôóíêöèÿ íàçûâàåòñÿ âòîðûì êàíîíè÷åñêèì ðåøåíèåì. Íåòðóäíî âèäåòü, ÷òî Wφ, θ = 1 ,ñëåäîâàòåëüíî φ è θ ëèíåéíî íåçàâèñèìû, à çíà÷èò îáùèé èíòåãðàë óðàâíåíèÿ ìîæíî ïðåäñòàâèòü,ñ òî÷íîñòüþ äî ïîñòîÿííîãî ìíîæèòåëÿ, â âèäå θ(x, λ) + lφ(x, λ).Ðàññìîòðèì òåïåðü âòîðîå êðàåâîå óñëîâèå y(b) cos β + y′(b) sin β = 0. Ïîäñòàâëÿÿ âûðàæåíèå äëÿîáùåãî ðåøåíèÿ, ïîëó÷èì (θ(b) + lφ(b)) cos β + (θ′(b) + lφ′(b)) sin β = 0èëè,l = − θ(b)ctgβ+θ′(b)
φ(b)ctgβ+φ′(b) . Îáîçíà÷èâ ctgβ = z, ïîëó÷èì îêîí÷àòåëüíî l = − θ(b)z+θ′(b)φ(b)z+φ′(b) . Ïðè òàêîì
îòîáðàæåíèè z → l âåùåñòâåííàÿ îñü êîìïëåêñíîé ïëîñêîñòè z ïåðåéäåò â íåêîòîðóþ îêðóæíîñòü,ÿâëÿþùóþñÿ ãðàíèöåé íåêîòîðîãî êðóãà Cb íà ïëîñêîñòè l (ðèñ. 1.3). Ìîæíî ïîêàçàòü (ñì.[1]), ÷òîðàäèóñ ïîëó÷èâøåãîñÿ êðóãà rb = 1
2|Imλ| R b0 |φ|2dx
. Îêàçûâàåòñÿ òàêæå, ÷òî óñëîâèå ïðèíàäëåæíîñòèêðóãó çàïèñûâàåòñÿ â âèäå
iWbθ + lφ, θ + lφ > 0 êîãäà Imλ > 0iWbθ + lφ, θ + lφ < 0 êîãäà Imλ < 0
br
l(z)zÞ
Cb
Ðèñ. 1.3:
Âîñïîëüçóåìñÿ òåïåðü ïîëó÷åííûì â íà÷àëå ïàðàãðàôà ñîîòíîøåíèåì (1.1)äëÿ ôóíêöèè ψ =θ + lφ, ãäå l ïðèíàäëåæèò Cb. Èìååì: 2Imλ
∫ b
0|ψ(x, λ)|2dx = iWψ, ψ0 − iWψ, ψb
Òàê êàê Imλ è iWψ, ψ îäíîãî çíàêà ïîëó÷èì 2∫ b
0|ψ(xλ)|2dx < iWψ,ψ
Imλ . Äàëåå,
Wψ, ψ = Wθ, θ+ Wφ, φ+ lWφ, θ+ lWθ, φ =
 ñèëó äåéñòâèòåëüíûõ ãðàíè÷íûõ óñëîâèé íà ôóíêöèè θ è φ ïåðâûå äâà ñëàãàåìûõ ðàâíû íóëþ,òîãäà, ïðîäîëæàÿ ðàâåíñòâî,
= l − l = 2iImλ
7
Îêîí÷àòåëüíî ïîëó÷àåì ∫ b
0
|ψ(x, λ)|2dx < − Iml
Imλ(1.2)
Ðàññìîòðèì òåïåðü b′ > b è l ∈ Cb′ . Òîãäà∫ b
0|ψ(x, λ)|2dx <
∫ b′
0|ψ(x, λ)|2dx < − Iml
Imλ . Òî åñòül ∈ Cb, ñëåäîâàòåëüíî Cb′ ⊂ Cb. Óñòðåìèì òåïåðü b ê áåñêîíå÷íîñòè. Òîãäà âîçìîæíû âîçìîæíûäâà ñëó÷àÿ - êðóãè Cb áóäóò ñõîäèòñÿ ê íåêîòîðîìó êðóãó èëè â íåêîòóðóþ òî÷êó. Ïîëó÷èâøååñÿìíîæåñòâî íàçûâàåòñÿ ïðåäåëüíûì êðóãîì Âåéëÿ èëè ïðåäåëüíîé òî÷êîé Âåéëÿ ñîîòâåòñòâåííî.Ïóñòü äàëåå m - ïðåäåëüíàÿ òî÷êà èëè ëþáàÿ òî÷êà ïðåäåëüíîãî êðóãà. Ðàññìîòðèì ïðåäåëüíîåðåøåíèå ψ(x, λ) = θ(x, λ) + mφ(x, λ) . Òîãäà ïåðåõîäÿ â íåðàâåíñòâå (1.2) ê ïðåäåëó, ïîëó÷èì
∞∫
0
|ψ(x, λ)|2dx < −Imm
Imλ(1.3)
Òî åñòü ìû ïîëó÷èëè, ÷òî âíå äåéñòâèòåëüíîé îñè λ ñóùåñòâóåò êâàäðàòè÷íî ñóììèðóåìîå ðåøå-íèå. Ýòî ðåøåíèå íàçûâàåòñÿ ðåøåíèåì Âåéëÿ, à ôóíêöèÿ m(λ) - ôóíêöèåé Âåéëÿ-Òèò÷ìàðøà.Çàìåòèì, ÷òî â ñëó÷àå ïðåäåëüíîãî êðóãà, φ(x, λ) òîæå êâàäðàòè÷íî ñóììèðóåìàÿ ôóíêöèÿ (ýòîñëåäóåò èç âûðàæåíèÿ äëÿ ðàäèóñà êðóãà Cb). Ñëåäîâàòåëüíî â ñëó÷àå ïðåäåëüíîãî êðóãà ëþáîåðåøåíèå ñèíãóëÿðíîé çàäà÷è Øòóðìà-Ëèóâèëëÿ ÿâëÿåòñÿ êâàäðàòè÷íî ñóìèðóåìûì.Âîçíèêàåò âîïðîñ, çàâèñèò ëè àëüòåðíàòèâà ïðåäåëüíûé êðóã/ ïðåäåëüíàÿ òî÷êà îò λ. Îêàçûâà-åòñÿ, ñì. [1], ÷òî âûïîëíåíà ñëåäóþùàÿ òåîðåìàÒåîðåìà 1.3.1 Åñëè äëÿ íåêîòîðîãî λ âûïîëíåí ñëó÷àé ïðåäåëüíîãî êðóãà, òî îí âûïîëíåí è äëÿâñåõ îñòàëüíûõ λ.Î÷åâèäíî, òî æå âûïîëíåíî è äëÿ ñëó÷àÿ ïðåäåëüíîé òî÷êè.
1.4 Ôóíêöèÿ Ãðèíà êðàåâîé çàäà÷è íà ïîëóîñèÐàññìîòðèì çàäà÷ó
y′′ + (λ− q) = −δ(x− ξ), 0 < x < ∞y(0) cos α + y′(0) sin α = 0 (1.4)
Ïóñòü, êðîìå òîãî, Imλ 6= 0, ÷òî îáåñïå÷èò ñóùåñòâîâàíèå êâàäðàòè÷íî ñóììèðóåìîãî ðåøåíèÿψ(x, λ).  êà÷åñòâå âòîðîãî óñëîâèÿ íà y âîçüìåì êâàäðàòè÷íóþ ñóììèðóåìîñòü ðåøåíèÿ. Ðåøåíèåýòîé çàäà÷è - ôóíêöèÿ Ãðèíà G(x, ξ, λ), ïðè÷åì [G(x, ξ, λ]|x=ξ = 0 è [G′(x, ξ, λ]|x=ξ = −1 Ðàññìîò-ðèì ôóíêöèþ
C
φ(x, λ)ψ(ξ, λ), 0 6 x 6 ξ < +∞φ(ξ, λ)ψ(x, λ), 0 6 ξ 6 x < +∞
Î÷åâèäíî, ÷òî îíà íåïðåðûâíà â òî÷êå ξ è G(x, ξ, λ) ∈ L2. Íàéäåì C:[G′(x, ξ, λ]|x=ξ = C[φ(ξ, λ)ψ′(ξ, λ)− φ′(ξ, λ)ψ(ξ, λ)] = CWφ, ψ ⇒ C = −1
Wφ,ψ = −1 Èòàê,
G(x, ξ, λ) = −φ(x, λ)ψ(ξ, λ), 0 6 x 6 ξ < +∞−φ(ξ, λ)ψ(x, λ), 0 6 ξ 6 x < +∞ (1.5)
Íàðÿäó ñ çàäà÷åé (1.4) ðàññìîòðèì ðåãóëÿðíóþ çàäà÷ó
y′′ + (λ− q) = −δ(x− ξ), 0 < x < by(0) cos α + y′(0) sin α = 0y(b) cos β + y′(b) sin β = 0
(1.6)
Ðåøåíèå ýòîé çàäà÷è - Gb(x, ξ, λ)Èìååò ìåñòî áèëèíåéíàÿ ôîðìóëà Gb(x, ξ) def= Gb(x, ξ, 0) =
∑∞n=1
vn(x)vn(ξ)λn
(vn - îðòîíîðìèðîìàí-íûå ñ.ô. çàäà÷è Øòóðìà-Ëèóâèëëÿ, vn = φ(x,λn)
Nn,b). Êðîìå òîãî Gb(x, ξ, λ) =
∑∞n=1
vn(x)vn(ξ)λn−λ
Íàïèøåì äëÿ ïàðû Gb è Gb îáîáùåííîå ðàâåíñòâî Ïàðñåâàëÿ.∫ b
0
|Gb(x, ξ, λ|2dξ =∞∑
n=1
∣∣∣∣vn(x)λ− λn
∣∣∣∣2
=∞∑
n=1
∣∣∣∣φ(x, λn)λ− λn
∣∣∣∣2 1
N2n,b
=
+∞∫
−∞
∣∣∣∣φ(x, µ)λ− µ
∣∣∣∣2
dρb(µ)
8
Ïåðåõîäÿ ê ïðåäåëó b →∞, ïîëó÷èì∞∫
0
|G(x, ξ, λ|2dξ =
+∞∫
−∞
∣∣∣∣φ(x, µ)λ− µ
∣∣∣∣2
dρ(µ) (1.7)
1.5 Ñâÿçü ìåæäó ôóíêöèåé Âåéëÿ-Òèò÷ìàðøà è ñïåêòðàëü-íîé ïëîòíîñòüþ
Ñîãëàñíî ôîðìóëå (1.5), G(0, ξ, λ) = − sin αψ(ξ, λ) Òîãäà sin2 α∞∫0
|ψ(ξ, λ)|2dξ = sin2 α+∞∫−∞
dρ(µ)|λ−µ|2 Ñëå-
äîâàòåëüíî∞∫
0
|ψ(ξ, λ)|2dξ =
+∞∫
−∞
dρ(µ)|λ− µ|2 (1.8)
Ïðèìåì áåç äîêàçàòåëüñòâà ñëåäóþùóþ ëåììó:
Ëåììà 1.5.1 iWψ, ψb → 0, b →∞Òîãäà, ñîãëàñíî ôîðìóëå (1.1) èìååì 2Imλ
∫∞0|ψ(x, λ)|2dx = iWψ, ψ0 . Äåéñòâóÿ äàëåå èäåíòè÷íî
ïàðàãðàôó 1.3, ïîëó÷èì ÷òî∫∞0|ψ(x, λ)|2dx = − Imm(λ)
Imλ , èëè, ïîäñòàâëÿÿ â (1.8)
−Imm(λ) = Imλ
+∞∫
−∞
dρ(µ)|λ− µ|2
Ïóñòü λ = u + iv. Ïðîèíòåãðèðóåì òåïåðü îáå ÷àñòè ðàâåíñòâà ïî u.
−∫ u2
u1
Imm(λ)du = v
∫ u2
u1
du
+∞∫
−∞
dρ(µ)(µ− u)2 + v2
=
èíòåãðèðóÿ ïî ÷àñòÿì è ïðîäîëæàÿ ðàâåíñòâî ïîëó÷àåì
= v
∫ u2
u1
du
ρ(µ)(u− µ)2 + v2
∣∣∣∣+∞
−∞−
+∞∫
−∞ρ(µ)
∂
∂µ
1(u− µ)2 + v2
dµ
=
Ïðåäïîëàãàÿ, ÷òî ρ(µ) - ìåäëåííî ìåíÿþùàÿ ôóíêöèÿ, ïîëó÷èì äàëåå
= v
∫ u2
u1
du
+∞∫
−∞ρ(µ)
∂
∂u
1(u− µ)2 + v2
dµ = v
+∞∫
−∞dµρ(µ)
[1
u2 − µ)2 + v2− 1
(u1 − µ)2 + v2
]→
v→0π(ρ(u2)−ρ(u2))
Òîãäà ρ(u2)− ρ(u1) = − 1π lim
v→0
∫ u2
u1Imm(u + iv)du (1.9)
 òî÷êàõ àíàëèòè÷íîñòè Imm(u + i0), èìååì ρ′(λ) = − 1π Imm(u + i0) .
 ñëó÷àå íàëè÷èÿ ïîëþñà ïåðâîãî ïîðÿäêà íà âåùåñòâåííîé îñè, m(u+iv) = C−1(u+iv)−λ0
+ðåãóëÿðíàÿ ÷àñòüÄàëåå, − 1
π
∫ u2
u1Im lim
v→0
C−1(u+iv)−λ0
du = − 1π
∫ u2
u1Im
[v.p. 1
u−λ0− πiδ(u− λ0)
]du = C−1
Òîãäà òåîðåìà ðàçëîæåíèÿ ïðèíèìàåò âèä
f(x) =∑
k
C−1,kF (λk)y(x, λk) +∫
R\λkF (λ)y(x, λ)ρ′(λ)dλ (1.10)
Ïåðâàÿ ÷àñòü ýòîãî âûðàæåíèÿ îïèñûâàåò äèñêðåòíóþ, à âòîðàÿ - íåïðåðûâíóþ ÷àñòè ñïåêòðà.Èòàê, íà÷àâ ñ ïîèñêà êâàäðàòè÷íî ñóììèðóåìîãî ðåøåíèÿ ìû ïðèøëè ê ôîðìóëå äëÿ ñïåê-
òðàëüíîé ïëîòíîñòè, òî åñòü ïîñòðîåíà îáùàÿ òåîðèÿ íàõîæäåíèÿ ðàçëè÷íûõ ðàçëîæåíèé ôóíê-öèé.  ñëåäóþùèõ ïàðàãðàôàõ ìû ðàññìîòðèì êîíêðåòíûå ïðèìåðû.
9
1.6 Ïðåîáðàçîâàíèå Ôóðüå íà ïîëóîñèÐàññìîòðèì çàäà÷ó
y′′ + λy = 0, 0 < x < +∞y(0) cos α + y′(0) sin α = 0
Ïóñòü Imλ > 0.  òàêîì ñëó÷àå êâàäðàòè÷íî-ñóììèðóåìîå ðåøåíèå ðàâíî ei√
λx (Äåéñòâèòåëüíî,ei√
λx = eiRe√
λxe−Im√
λx. Ïóñòü λ = ρeiφ, 0 < φ < π ⇒ Im√
λ > 0.Çíà÷èò ei√
λx ∈ L2) Íàéäåìêàíîíè÷åñêèå ðåøåíèÿ: φ(x, λ) = sin α cos
√λx− cos α sin
√λx√
λ; θ(x, λ) = cos α cos
√λx + sin α sin
√λx√
λ.
Ðåøåíèå Âåéëÿ θ + m(λ)φ = (cos α + m(λ) sin α) cos√
λx + ( sin α√λ−m(λ) cos α√
λ) sin
√λx
Òàê êàê ðåøåíèå Âåéëÿ êâàäðàòè÷íî ñóììèðóåìî,
θ + m(λ)φ = constei√
λx = const(cos√
λx + i sin√
λx) ⇒ cos α + m(λ) sin αsin α√
λ−m(λ) cos α√
λ
=1i
Îòêóäà
m(λ) =sin α− i
√λ cos α
cos α + i√
λ sin α
 ñëó÷àå λ > 0, èìååì m(λ) = (sin α−i√
λ cos α)(cos α−i√
λ sin α)cos2 α+λ sin2 α
è Imm(λ) = −√
λcos2 α+λ sin2 α
, ñëåäîâà-òåëüíî
ρ′(λ) = 1π
√λ
cos2 α+λ sin2 α, λ > 0
Ïðè λ < 0 âîçìîæíû äâà ñëó÷àÿ:1)ctgα 6 0. Òîãäà m(λ) ∈ R. Ñëåäîâàòåëüíî ρ(λ) = const, ò.å. ñïåêòð íà îòðèöàòåëüíîé ïîëóîñèîòñóòñòâóåò. Ôîðìóëà ðàçëîæåíèÿ ïðèíèìàåò âèä:
f(x) = 1
π
∫∞0
F (λ)(sin α cos√
λx− cosα sin√
λx√λ
)√
λdλcos2 α+λ sin2 α
F (λ) =∫∞0
f(x)(sinα cos√
λx− cosα sin√
λx√λ
)dx(1.11)
2)ctgα > 0. Òîãäà ó m(λ) íà âåùåñòâåííîé îñè èìååòñÿ ïîëþñ â òî÷êå λ0 = −ctg2α (i√
λ0 = −ctgα).Èìååì, m(λ) = (sin α−i
√λ cos α)(cos α−i
√λ sin α)
cos2 α+λ sin2 α= 2 cos α
sin3 α(λ+ctg2α)+ ðåã. ÷àñòü . Äåéñòâóÿ ñîãëàñíî ôîð-
ìóëå (1.10), ïîëó÷èì C−1 = 2 cos αsin3 α
. Äàëåå, φ(x, λ0) = sin αe−ctg2αx. Îáîçíà÷èâ ctgα = h, ïîëó÷èì÷àñòü, îòâå÷àþùóþ çà äèñêðåòíûé ñïåêòð, â âèäå 2h
∫∞0
e−hxf(x)dx. È ôîðìóëà ðàçëîæåíèÿ áóäåòâûãëÿäåòü êàê
f(x) = 2h∞∫0
e−hxf(x)dx + 1π
∞∫0
F (λ)(sin α cos√
λx− cosα sin√
λx√λ
)√
λdλcos2 α+λ sin2 α
F (λ) =∞∫0
f(x)(sin α cos√
λx− cos α sin√
λx√λ
)dx(1.12)
ÏðèìåðÐàññìîòðèì α = 0.  ýòîì ñëó÷àå ôîðìóëà (1.11) äàåò
f(x) = − 1π
∞∫0
F (λ) sin√
λx√λ
√λdλ
F (λ) = −∞∫0
f(x) sin√
λx√λ
dx
Ñäåëàåì çàìåíû s =√
λ; F (λ) = − 1s F (s). Ïîëó÷èì îêîí÷àòåëüíî
f(x) = 2π
∞∫0
F (s) sin sxds
F (s) =∞∫0
f(x) sin sxdx
10
1.7 Ïðåîáðàçîâàíèå ÂåáåðàÄðóãîå ÷àñòî èñïîëüçóåìîå ïðåîáðàçîâàíèå ñâÿçàíî ñ óðàâíåíèåì Áåññåëÿ: (ρy′)′ + λρy = 0Ïðîèçâîäÿ ïðåîáðàçîâàíèå Ëèóâèëëÿ-Ãðèíà y = 1√
ρv(ρ), ïîëó÷èì
v′′ + (λ +1
4ρ2)v = 0, 0 < a < ρ < +∞ (1.13)
Îáùèé èíòåãðàë óðàâíåíèÿ (1.13) v(ρ) =√
ρ(AJ0(√
λρ) + BN0(√
λρ))Ïîëîæèì â íà÷àëüíûõ óñëîâèÿõ äëÿ êàíîíè÷åñêèõ ðåøåíèé ïàðàìåòð α ðàâíûì 0. Òîãäà
φ(a, λ) = 0φ′(a, λ) = −1θ(a, λ) = 1θ′(a, λ) = 0
Äàëåå, v′(ρ) = 12ρ−1/2(AJ0(
√λρ) + BN0(
√λρ)) − √
ρ√
λ(AJ1(√
λρ) + BN1(√
λρ)). Èç ãðàíè÷íûõóñëîâèé íà φ ïîëó÷àåì
AJ0(√
λa) + BN0(√
λa) = 0AJ1(
√λρ) + BN1(
√λρ) = 1√
λa
Îïðåäåëèòåëü ñèñòåìû íà (A,B) ∆ = −∣∣∣∣
J0 N0
J ′0 N ′0
∣∣∣∣ = − 2π√
λa
Îòêóäà
A = π√
a2 N0(
√λa)
B = −π√
a2 J0(
√λa)
Òîãäà φ(ρ, λ) =√
ρπ√
a
2(N0(
√λa)J0(
√λρ)− J0(
√λa)N0(
√λρ)) (1.14)
Èùåì θ(ρ, λ): √
a(AJ0(√
λa) + BN0(√
λa)) = 112a −
√λa(AJ1(
√λa) + BN1(
√λa)) = 0
Ïîëó÷èì A = −π
√λa
2 N1(√
λa) + π4√
aN0(
√λa)
B = π√
λa2 J1(
√λa)− π
4√
aJ0(√
λa)
Òàêèì îáðàçîì, θ(ρ, λ) =π√
λaρ
2(−N1(
√λa)J0(
√λρ) + J1(
√λa)N0(
√λρ)) +
φ(ρ, λ)2a
(1.15)
Âñïîìíèì àñèìïòîòèêè:
Jν(x) ∼ 2√πx
cos (x− πν + π4 )(1 + O( 1
x )), x →∞Nν(x) ∼ 2√
πxsin (x− πν + π
4 )(1 + O( 1x )), x →∞
H(1)ν (x) ∼ 2√
πxei(x−πν+ π
4 )(1 + O( 1x )), x →∞
H(2)ν (x) ∼ 2√
πxe−i(x−πν+ π
4 )(1 + O( 1x )), x →∞
Àíàëîãè÷íî ïàðàãðàôó 1.6, êâàäðàòè÷íî ñóììèðóìîå ðåøåíèå (ïðè Imλ > 0) äîëæíî áûòü ïðîïîð-öèîíàëüíî√ρH
(1)0 (
√λρ). Èìååì, ψ(ρ, λ) = θ(ρ, λ)+m(λ)φ(ρ, λ) = const
√ρH
(01)(
√λρ) = const
√ρ(J0(
√λρ)+
iN0(√
λρ)) . Îáîçíà÷èì m′ = m + 12a . Ïðèðàâíèâàÿ êîýôôèöèåíòû ïðè J0 è N0 ïîëó÷èì:
−√
λN1(√
λa) + m′N0(√
λa)√λJ1(
√λa)−m′J0(
√λa)
=1i
Òîãäà äëÿ m:
m =
√λ(J1(
√λa) + iN1(
√λa))
J0(√
λa) + iN0(√
λa)− 1
2a=
√λH
(1)1 (
√λa)
H(1)0 (
√λa)
− 12a
(1.16)
11
Ðàññìîòðèì äâà ñëó÷àÿ:1)λ > 0.  òàêîì ñëó÷àå,
Imm = Imm′ =
√λ(J1(
√λa) + iN1(
√λa))
J20 (√
λa) + N20 (√
λa)(J0(
√λa)− iN0(
√λa)) =
=
√λ(−J1(
√λa)N0(
√λa) + J0(
√λa)N1(
√λa))
J20 (√
λa) + N20 (√
λa)= − 2
πa
1J2
0 (√
λa) + N20 (√
λa)
2)λ < 0. Îáîçíà÷èì√
λ = is, s > 0. Òîãäà H(1)0 (isa) = 2
πiK0(sa) . Òîãäà
Imm = Imm′ =isH
(1)1 (isa)
H(1)0 (isa)
= −sK ′
0(sa)K0(sa)
∈ R⇒ Imm(λ) = 0
Îêîí÷àòåëüíî,
ρ′(x, λ) =
2
π2a1
J20 (√
λa)+N20 (√
λa), λ > 0
0, λ < 0
1.8 Ðàâåíñòâî Ïàðñåâàëÿ è òåîðåìà ðàçëîæåíèÿ íà âñåé îñè ïðåäûäóùèõ ïàðàãðàôàõ ìû ðàññìîòðåëè ñëó÷àé ïîëóáåñêîíå÷íîãî ïðîìåæóòêà. Çàéìåìñÿ òå-ïåðü ïîñòðîåíèåì òåîðèè äëÿ ñëó÷àÿ âñåé îñè. Ðàññìîòðèì óðàâíåíèå
y′′(x) + (λ− q(x))y(x) = 0, −∞ < x < +∞ (1.17)
Ââåäåì êàíîíè÷åñêèå ðåøåíèÿ φ(x, λ) è θ(x, λ), óäîâëåòâîðÿþùèå ãðàíè÷íûì óñëîâèÿì
φ(0, λ) = 0, φ′(0, λ) = −1θ(0, λ) = 1, θ′(0, λ) = 0 (1.18)
Òàê êàê Wφ, θ = 1, ýòè ðåøåíèÿ íåçàâèñèìû, è îáùåå ðåøåíèå (1.17) ïðåäñòàâèìî â âèäå y(x, λ) =C1φ(x, λ) + C2θ(x, λ).Ðàññìîòðèì, êàê è ðàíåå, ðåãóëÿðíóþ çàäà÷ó Øòóðìà-Ëèóâèëëÿ íà ïðîìåæóòêå (a, b)
y′′(x) + (λ− q(x))y(x) = 0, a < x < by(a) cos α + y′(a) sin α = 0y(b) cos β + y′(b) cos β = 0
(1.19)
Ïóñòü λk è yk(x) - ïîñëåäîâàòåëüíîñòè ñîáñòâåííûõ ÷èñåë è ñîîòâåòñòâóþùèõ èì îðòîíîðìè-ðîâàííûõ ñîáñòâåííûõ ôóíêöèé. Ïóñòü yk âûðàæàþòñÿ ÷åðåç êàíîíè÷åñêèå ðåøåíèÿ êàê yk(x) =αkφ(x, λk) + βkθ(x, λk) . Íàïèøåì â ýòîì ñëó÷àå äëÿ íåêîòîðîé ôóíêöèè f(x) ∈ L2(−∞, +∞)ðàâåíñòâî Ïàðñåâàëÿ
∫ b
a
f2(x)dx =∞∑
k=1
C2k =
∞∑
k=1
(∫ b
a
f(x)yk(x)dx)2 =∞∑
k=1
(∫ b
a
f(x)(αkφ(x, λk) + βkθ(x, λk))dx)2 =
=∞∑
k=1
[α2k(
∫ b
a
f(x)φ(x, λk)dx)2 +2αkβk
∫ b
a
f(x)φ(x, λk)dx
∫ b
a
f(x)θ(x, λk)dx+β2k(
∫ b
a
f(x)θ(x, λk)dx)2]
Ââîäÿ ñïåêòðàëüíûå ïëîòíîñòè êàê
ξa,b(λ) =
∑0<λk<λ
α2k, λ > 0
− ∑0>λk>λ
α2k, λ 6 0
ηa,b(λ) =
∑0<λk<λ
αkβk, λ > 0
− ∑0>λk>λ
αkβk, λ 6 0
12
ζa,b(λ) =
∑0<λk<λ
β2k, λ > 0
− ∑0>λk>λ
β2k, λ 6 0
è ïîëàãàÿF1,a,b(λ) =
∫ b
a
f(x)φ(x, λ)dx
F2,a,b(λ) =∫ b
a
f(x)θ(x, λ)dx
ïîëó÷èì ðàâåíñòâî Ïàðñåâàëÿ â âèäå
∫ b
a
f2(x)dx =
+∞∫
−∞F 2
1,a,b(λ)dξa,b(λ) + 2
+∞∫
−∞F1,a,b(λ)F2,a,b(λ)dηa,b(λ) +
+∞∫
−∞F 2
2,a,b(λ)dζa,b(λ)
Îïóñêàÿ òåõíèêó ïðåäåëüíîãî ïåðåõîäà (ñì. ??), ïîëó÷èì îêîí÷àòåëüíî+∞∫
−∞f2(x)dx =
+∞∫
−∞F 2
1 (λ)dξ(λ) + 2
+∞∫
−∞F1(λ)F2(λ)dη(λ) +
+∞∫
−∞F 2
2 (λ)dζ(λ) (1.20)
ãäå Fi(λ) = l · i ·ma,b→∞
Fi,a,b(λ)
Ðàññìîòðèì òåïåðü f(x), g(x) ∈ L2(−∞,+∞). Íàïèñàâ (1.20) äëÿ f + g è f − g, è âû÷èòàÿ èçïåðâîãî âòîðîå, ïîëó÷èì îáîáùåííîå ðàâåíñòâî Ïàðñåâàëÿ
+∞∫
−∞f(x)g(x)dx =
+∞∫
−∞F1(λ)G1(λ)dξ(λ) +
+∞∫
−∞F2(λ)G2(λ)dζ(λ) +
+∞∫
−∞(F1(λ)G2(λ) + F2(λ)G1(λ))dη(λ)
(1.21)
Òåîðåìà 1.8.1 (ðàçëîæåíèÿ) Ïóñòü f(x) ∈ L2(−∞, +∞) ∩ C2(−∞, +∞) è ïóñòü ñëåäóþùèåèíòåãðàëû ñõîäÿòñÿ àáñîëþòíî è ðàâíîìåðíî
+∞∫−∞
F1(λ)φ(x, λ)dξ(λ)+∞∫−∞
F2(λ)θ(x, λ)dζ(λ)
+∞∫−∞
F2(λ)φ(x, λ)dη(λ)+∞∫−∞
F1(λ)θ(x, λ)dη(λ)
òîãäà ñïðàâåäëèâî
f(x) =
+∞∫
−∞F1(λ)φ(x, λ)dξ(λ) +
+∞∫
−∞F2(λ)θ(x, λ)dζ(λ) +
+∞∫
−∞F2(λ)φ(x, λ)dη(λ) +
+∞∫
−∞F1(λ)θ(x, λ)dη(λ)
ÄîêàçàòåëüñòâîÏóñòü g(x) ∈ C(−∞, +∞) è g(x) = 0, ïðè |x| > N . Òîãäà G1(λ) =
∫ N
−Ng(x)φ(x, λ)dx è G2(λ) =∫ N
−Ng(x)θ(x, λ)dx. Ñîãëàñíî (1.21), èìååì
∫ N
−N
f(x)g(x)dx =
+∞∫
−∞F1(λ)
∫ N
−N
g(x)φ(x, λ)dxdξ(λ) +
+∞∫
−∞F2(λ)
∫ N
−N
g(x)θ(x, λ)dxdζ(λ)+
+
+∞∫
−∞F2(λ)
∫ N
−N
g(x)φ(x, λ)dxdη(λ) +
+∞∫
−∞F1(λ)
∫ N
−N
g(x)θ(x, λ)dxdη(λ)
Ìåíÿÿ ïîðÿäîê èíòåãðèðîâàíèÿ, ïîëó÷èì
∫ N
−N
f(x)g(x)dx =∫ N
−N
g(x)dx
+∞∫
−∞F1(λ)φ(x, λ)dξ(λ) +
+∞∫
−∞F2(λ)θ(x, λ)dζ(λ)+
13
+
+∞∫
−∞F2(λ)φ(x, λ)dη(λ) +
+∞∫
−∞F1(λ)θ(x, λ)dη(λ)
îòêóäà, ââèäó ïðîèçâîëüíîñòè g, ïîëó÷àåì òðåáóåìîå.Çàìå÷àíèå áîëåå àêêóðàòíîé òåîðåìå ðàçëîæåíèÿ, äëÿ ôóíêöèé f ∈ L2, â òî÷êàõ ðàçðûâà ïåðâîãî ðîäàèíòåãðàëû áóäóò ñõîäèñÿ ê ïîëóñóììå çíà÷åíèé ôóíêöèè ñëåâà è ñïðàâà.
1.9 Ôóíêöèÿ Ãðèíà êðàåâîé çàäà÷è äëÿ óðàâíåíèÿ Øòóðìà-Ëèóâèëëÿ íà âñåé îñè
Äëÿ íàõîæäåíèÿ ñïåêòðàëüíûõ ïëîòíîñòåé, íåîáõîäèìî, àíàëîãè÷íî ñëó÷àþ îäíîãî ñèíãóëÿðíîãîêîíöà, ïîñòðîèòü ôóíêöèþ Ãðèíà. Ðàññìîòðèì óðàâíåíèå
y′′(x) + (λ− q(x))y(x) = δ(x− ξ), −∞ < x < +∞y(x) ∈ L2(−∞,∞) (1.22)
Ïóñòü Imλ > 0. Ñîãëàñíî òåîðèè Âåéëÿ, ñóùåñòâóþò òàêèå m1(λ) è m2(λ), ÷òî ôóíêöèè ψ1(x, λ) =θ(x, λ)+m1(λ)φ(x, λ) è ψ2(x, λ) = θ(x, λ)+m2(λ)φ(x, λ) óäîâëåòâîðÿþò óñëîâèÿì ψ1(x, λ) ∈ L2(−∞, 0); ψ2(x, λ) ∈L2(0, +∞) .  òàêîì ñëó÷àå, ôóíêöèþ Ãðèíà ìîæíî çàïèñàòü â âèäå
G(x, ξ, λ) = const
ψ1(x, λ)ψ2(ξ, λ), x 6 ξψ1(ξ, λ)ψ2(x, λ), x > ξ
Îïðåäåëèì êîíñòàíòó:
∂G(x, ξ, λ)∂x
= const
ψ′1(x, λ)ψ2(ξ, λ), x 6 ξψ′2(x, λ)ψ1(ξ, λ), x > ξ
Èç óñëîâèÿ[
∂G(x,ξ,λ)∂x
]x=ξ
= 1, ïîëó÷èì
[∂G(x, ξ, λ)
∂x
]
x=ξ
= const(ψ′2(x, λ)ψ1(ξ, λ)− ψ′1(x, λ)ψ2(ξ, λ)) =
= constWψ1, ψ2 = const(m1Wφ, θ+ m2Wθ, φ) = const(m1 −m2) = 1
ÎòêóäàG(x, ξ, λ) =
1m1(λ)−m2(λ)
ψ1(x, λ)ψ2(ξ, λ), x 6 ξψ1(ξ, λ)ψ2(x, λ), x > ξ
(1.23)
Ïåðåïèøåì (1.22) â âèäå
G′′(x, ξ, λ) + (λ− q(x))G(x, ξ, λ) = 0, −∞ < x < ξ; ξ < x < ∞[G(x, ξ, λ)]x=ξ = 0[
∂G(x,ξ,λ)∂x
]x=ξ
= 1(1.24)
Âñïîìíèì óðàâíåíèå äëÿ ôóíêöèè φ(x, µ):
φ′′(x, µ) + (µ− q(x))φ(x, µ) = 0
Óìíîæàÿ ýòî óðàâíåíèå íà G(x, ξ, λ), âû÷èòàÿ èç (1.24), óìíîæåííîãî íà φ(x, µ), è èíòåãðèðóÿ ïîïðîìåæóòêó (−A, A) (ñì. ðèñ. 1.4) ïîëó÷èì
∫ A
−A
(G′′(x, ξ, λ)φ(x, µ)−G(x, ξ, λ)φ(′′x, µ))dx + (λ− µ)∫ A
−A
G(x, ξ, λ)φ(x, µ)dx = 0
Ñ äðóãîé ñòîðîíû,∫ A
−A
(G′′(x, ξ, λ)φ(x, µ)−G(x, ξ, λ)φ(′′x, µ))dx = (G′(x, ξ, λ)φ(x, µ)−G(x, ξ, λ)φ′(x, µ))|ξ−A +
14
-A Ax
Ðèñ. 1.4:
+ (G′(x, ξ, λ)φ(x, µ)−G(x, ξ, λ)φ′(x, µ))|Aξ −∫ A
−A
(G′(x, ξ, λ)φ′(x, µ)−G′(x, ξ, λ)φ′(x, µ))dx =
= Wφ,G|ξ−0 − Wφ,G|−A − Wφ,G|ξ+0 + Wφ,G|A =
Ñîãëàñíî ëåììå 1.5.1, Wφ,G|±A →A→∞
0 . Òîãäà, ïðîäîëæàÿ ðàâåíñòâî,
=1
m1(λ)−m2(λ)
ψ2(ξ, λ) Wφ, ψ1|ξ − ψ1(ξ, λ) Wφ, ψ2|ξ
=
=1
m1(λ)−m2(λ)
ψ2(ξ, λ)
∣∣∣∣φ(x, µ) ψ1(x, λ)φ′(x, µ) ψ′1(x, λ)
∣∣∣∣ξ
− ψ1(ξ, λ)∣∣∣∣
φ(x, µ) ψ2(x, λ)φ′(x, µ) ψ′2(x, λ)
∣∣∣∣ξ
=
=1
m1(λ)−m2(λ)φ(ξ, µ)(ψ′1(ξ, λ)ψ2(ξ, λ)− ψ1(ξ, λ)ψ′2(ξ, λ)) = −φ(ξ, µ)
Ñëåäîâàòåëüíî,
F1(λ) =
+∞∫
−∞G(x, ξ, λ)φ(x, µ)dx =
φ(ξ, µ)λ− µ
Òî÷íî òàêæå ìîæíî ïîëó÷èòü è
F2(λ) =
+∞∫
−∞G(x, ξ, λ)θ(x, µ)dx =
θ(ξ, µ)λ− µ
Îêîí÷àòåëüíî, èìååì:
G(x, ξ′, λ) =
+∞∫
−∞
φ(x, µ)φ(ξ′, µ)λ− µ
dξ(µ) +
+∞∫
−∞
φ(x, µ)θ(ξ′, µ)λ− µ
dη(µ) +
+
+∞∫
−∞
θ(x, µ)φ(ξ′, µ)λ− µ
dη(µ) +
+∞∫
−∞
θ(x, µ)θ(ξ′, µ)λ− µ
dζ(µ) (1.25)
1.10 Ñâÿçü ìåæäó ôóíêöèÿìè Âåéëÿ-Òèò÷ìàðøà è ýëåìåí-òàìè ìàòðèöû ñïåêòðàëüíîé ïëîòíîñòè
Èç ïðåäñòàâëåíèÿ (1.23) äëÿ ôóíêöèè Ãðèíà, èìååì G(0, 0, λ) = 1m1(λ)−m2(λ) . Ñ äðóãîé ñòîðîíû,
èç ôîðìóëû (1.25),
G(0, 0, λ) =
+∞∫
−∞
dζ(µ)λ− µ
⇒ 1m1(λ)−m2(λ)
=
+∞∫
−∞
dζ(µ)λ− µ
Âçÿâ ìíèìóþ ÷àñòü îò îáåèõ ñòîðîí ðàâåíñòâà, ïîëó÷èì
−Im1
m1(λ)−m2(λ)= Imλ
+∞∫
−∞
dζ(µ)|λ− µ|2
Îòêóäà, ïîëó÷èì (ñì.1.5)
ζ(λ)− ζ(0) = − 1π
∫ λ
0
Im1
m1(u + i0)−m2(u + i0)du (1.26)
15
Äàëåå,∂G(x, ξ, λ)
∂x
∣∣∣∣ ξ = 0x = +0
=1
m1(λ)−m2(λ)ψ1(0, λ)ψ′2(0, λ) = − m2(λ)
m1(λ)−m2(λ)
∂G(x, ξ, λ)∂x
∣∣∣∣ ξ = 0x = −0
=1
m1(λ)−m2(λ)ψ′1(0, λ)ψ2(0, λ) = − m2(λ)
m1(λ)−m2(λ)
Èç ïðåäñòàâëåíèÿ (1.25),∂G(x, ξ, λ)
∂x
∣∣∣∣ ξ = 0x = 0
= −+∞∫
−∞
dη(µ)λ− µ
Ñîãëàñíî çàìå÷àíèþ ê ïàðàãðàôó 1.8,+∞∫
−∞
dη(µ)λ− µ
=12
(m1(λ) + m2(λ)m1(λ)−m2(λ)
)= −1
2+
m1(λ)m1(λ)−m2(λ)
Ñëåäîâàòåëüíî,
η(λ)− η(0) = − 1π
∫ λ
0
Imm1(u + i0)
m1(u + i0)−m2(u + i0)du (1.27)
Íàêîíåö, äëÿ ïîäñ÷åòà ôóíêöèè ξ(λ) ðàññìîòðèì ñìåøàííóþ ïðîèçâîäíóþ
∂2G(x, ξ, λ)∂x∂ξ
∣∣∣∣x=ξ=0
=1
m1(λ)−m2(λ)ψ′1(0, λ)ψ′2(0, λ) = − m1(λ)m2(λ)
m1(λ)−m2(λ)
Ñ äðóãîé ñòîðîíû,∂2G(x, ξ, λ)
∂x∂ξ
∣∣∣∣x=ξ=0
=
+∞∫
−∞
dξ(µ)λ− µ
Îêîí÷àòåëüíî,
ξ(λ)− ξ(0) = − 1π
∫ λ
0
Imm1(u + i0)m2(u + i0)
m1(u + i0)−m2(u + i0)du (1.28)
Ðàçáåðåì äâà âàæíûõ ÷àñòíûõ ñëó÷àÿ1) m1(λ) ∈ R⇒ Imm = 0Ïóñòü, êðîìå òîãî, ∃ζ ′, η′, ξ′, òîãäà
ζ ′(λ) = − 1π
Im1
m1(λ)−m2(λ)
η′(λ) = − 1π
m1(λ)Im1
m1(λ)−m2(λ)= m1(λ)ζ ′(λ)
ξ′(λ) = − 1π
m1(λ)Imm2(λ)
m1(λ)−m2(λ)= − 1
πm1(λ)Im
m1(λ)m1(λ)−m2(λ)
= m21(λ)ζ ′(λ)
Òîãäà, ñîãëàñíî òåîðåìå 1.8.1, äëÿ ïðîèçâîëüíîé ôóíêöèè f(x), èìååì
f(x) =
+∞∫
−∞F1(λ)φ(x, λ)m2
1(λ)dζ(λ) +
+∞∫
−∞F2(λ)φ(x, λ)m1(λ)dζ(λ) +
+∞∫
−∞F1(λ)θ(x, λ)m1(λ)dζ(λ)+
+
+∞∫
−∞F2(λ)θ(x, λ)dζ(λ) =
+∞∫
−∞(F2(λ) + m1(λ)F1(λ))ψ1(x, λ)dζ(λ)
Îáîçíà÷èì F (λ) = F2(λ) + m1(λ)F1(λ). Òîãäà ïîëó÷èì
f(x) =+∞∫−∞
F (λ)ψ1(x, λ)dζ(λ)
F (λ) =+∞∫−∞
f(x)ψ1(x, λ)dx
(1.29)
16
2) Ïóñòü q(x) - ÷åòíàÿ ôóíêöèÿ.  ýòîì ñëó÷àå (ñì.(1.18)) φ(x, λ) - íå÷åòíàÿ, à θ(x, λ) - ÷åòíàÿ ôóíê-öèè. Òîãäà äëÿ x > 0, ψ2(x, λ) = θ(x, λ) + m2(λ)φ(x, λ) = θ(−x, λ)−m2(λ)φ(−x, λ). Ñëåäîâàòåëüíîôóíêöèÿ (θ(x, λ) − m2(λ)φ(x, λ)) ïðèíàäëåæèò êëàññó L2(−∞, 0). Òîãäà θ(x, λ) − m2(λ)φ(x, λ) =constψ1(x, λ) = const(θ(x, λ) + m1(λ)φ(x, λ)). Îòêóäà m1(λ) = −m2(λ). Ñïåêòðàëüíûå ïëîòíîñòè âòàêîì ñëó÷àå ïðèíèìàþò âèä
ζ(λ)− ζ(0) = − 12π
∫ λ
0
Im1
m1(u + i0)du
η(λ)− η(0) = − 1π
∫ λ
0
Imm1(u + i0)2m1(u + i0)
du = 0
ξ(λ)− ζ(0) = − 12π
∫ λ
0
Imm1(u + i0)du
Òîãäà òåîðåìà 1.8.1, äàåò
f(x) =
+∞∫
−∞F1(λ)φ(x, λ)dξ(λ) +
+∞∫
−∞F2(λ)θ(x, λ)dζ(λ) (1.30)
1.11 Ïðåîáðàçîâàíèå Ôóðüå íà âñåé îñèÏåðåéäåì òåïåðü íåïîñðåäñòâåííî ê ïðèìåðàì èíòåãðàëüíûõ ïðåîáðàçîâàíèé íà âñåé îñè. Ðàññìîò-ðèì çàäà÷ó
y′′ + λy = 0, −∞ < x < +∞ (1.31) ýòîì ñëó÷àå q(x) ≡ 0, òî åñòü ÿâëÿåòñÿ ÷åòíîé ôóíêöèåé. Ñëåäîâàòåëüíî m1(λ) = −m2(λ).Êàíîíè÷åñêèå ðåøåíèÿ òàêîé çàäà÷è èìåþò âèä φ(x, λ) = − sin
√λx√
λ, θ(x, λ) = cos
√λx . Àíàëîãè÷-
íî ïàðàãðàôó 1.6, êâàäðàòè÷íî ñóììèðóåìîå íà ëåâîé ïîëóîñè ðåøåíèå ψ2(x, λ) = constei√
λx =const(cos
√λx + i sin
√λx) . Ñëåäîâàòåëüíî, m2(λ) = −i
√λ, m1(λ) = i
√λ .
òîãäà ζ(λ)− ζ(0) = − 12π
∫ λ
0
Im1
m1(u + i0)du
⇒ ζ ′(λ) = 1
2π√
λ, λ > 0
0, λ < 0
ξ′(λ) = √
λ2π , λ > 00, λ < 0
Òîãäà
F1(λ) = −+∞∫
−∞f(x)
sin√
λx√λ
dx
F2(λ) =
+∞∫
−∞f(x) cos
√λxdx
Äàëåå, ñîãëàñíî (1.30),
f(x) =12π
+∞∫
0
F1(λ)
(− sin
√λx√
λ
)√λdλ +
12π
+∞∫
0
F2(λ)(cos
√λx
) 1√λ
dλ
Ñäåëàåì çàìåíû√
λ = ν, F1(ν) = −√
λF1(λ), F2(ν) = F2(λ) . Ïîëó÷èì îêîí÷àòåëüíî
f(x) = 1π
+∞∫0
F1(ν) sin νxdν + 1π
+∞∫0
F2(ν) cos νxdν
F1(ν) =+∞∫−∞
f(x) sin νxdx
F2(ν) =+∞∫−∞
f(x) cos νxdx
(1.32)
17
1.12 Ïðåîáðàçîâàíèå ÕàíêåëÿÐàññìîòðèì óðàâíåíèå (ρy′)′ + (λρ− ν2
ρ )y = 0, 0 < ρ < +∞Ïðîèçâîäÿ ïðåîáðàçîâàíèå Ëèóâèëëÿ-Ãðèíà y = 1√
ρv(ρ), ïîëó÷èì
v′′ + (λ +1/4− ν2
ρ2)v = 0, 0 < ρ < +∞ (1.33)
Îáùåå ðåøåíèå ýòîãî óðàâíåíèÿ v(ρ) =√
ρ(AJν(√
λρ) + BNν(√
λρ))Êàíîíè÷åñêèå ðåøåíèÿ (óñëîâèÿ íà íèõ ñòàâÿòñÿ â íåêîòîðîé òî÷êå a) èìåþò âèä (ñì.(1.14), (1.15))
φ(ρ, λ) =√
ρπ√
a
2(Nν(
√λa)Jν(
√λρ)− Jν(
√λa)Nν(
√λρ))
θ(ρ, λ) =π√
λaρ
2(N ′
ν(√
λa)Jν(√
λρ)− J ′ν(√
λa)Nν(√
λρ)) +φ(ρ, λ)
2a
Êâàäðàòè÷íî ñóììèðóåìîå ðåøåíèå ïðè X > a, (ñì.1.7) ψ2(ρ, λ) = const√
ρH(1)ν (
√λρ) . Â ýòîì
ñëó÷àå
m2(λ) = −√
λH(1)′ν (
√λa)
H(1)ν (
√λa)
− 12a
Äëÿ íàõîæäåíèÿ êâàäðàòè÷íî ñóììèðóåìîãî ðåøåíèÿ ïðè x < a, ðàññìîòðèì àñèìïòîòèêè â íóëå:
Jν(ρ) ∼ ρν
Nν(ρ) ∼ ρ−ν
Òîãäà (√
ρJν(ρ))2 ∼ ρ2ν+1, (√
ρNν(ρ))2 ∼ ρ−2ν+1. Òàêèì îáðàçîì ïðè ν > 1,√
ρJν(√
λρ) ∈ L2(0, a),à ïðè 0 < ν < 1, √ρJν(
√λρ) ∈ L2(0, a) è √ρNν(
√λρ) ∈ L2(0, a). Çíà÷èò ïðè 0 < ν < 1 ðåàëèçóåòñÿ
ñëó÷àé ïðåäåëüíîãî êðóãà. Ñîãëàñíî òåîðèè Âåéëÿ, ìû ìîæåì âûáðàòü ëþáîå ðåøåíèå. Âîçüìåìòîãäà, êàê è äëÿ ν > 1, ψ1(ρ, λ) = const
√ρJν(
√λρ) . Äëÿ òîãî, ÷òîáû îáíóëèòü êîýôôèöèåíò ïðè
Nν(√
λρ) â ψ1, ïîëó÷èì√
(λ)J ′ν(√
λa) +(m1(λ) + 1
2a
)Jν(
√λa) = 0, îòêóäà
m1(λ) = −√
λJ ′ν(√
λa)Jν(
√λa)
− 12a
Î÷åâèäíî, ÷òî ïðè λ > 0, Imm1(λ) = 0. Ðàññìîòðèì λ < 0. Âñïîìíèì: Jν(ix) ∼ iνIν(x). Òîãäàm1(λ) = −i
√|λ|J′ν(i
√|λ|)
Jν(i√|λ|) − 1
2a = −√|λ|I′ν(
√|λ|)
Iν(i√|λ|) − 1
2a ∈ R. Òàêèì îáðàçîì, âûïîëíÿåòñÿ ÷àñòíûéñëó÷àé 1 èç ïàðàãðàôà 1.10. Òîãäà òåîðåìà ðàçëîæåíèÿ ïðåäñòàâëÿåòñÿ ôîðìóëàìè (1.29). Èìååì:
ζ ′(λ) = − 1π
Im1
m1(λ)−m2(λ)=
1π
Im1
√λ
(J ′ν(
√λa)
Jν(√
λa)− H
(1)′ν (
√λa)
H(1)ν (
√λa)
) =1π
ImJν(
√λa)H(1)
ν (√
λa)√λWH(1)
ν , Jν√λa
=
=a
2ImiJν(
√λa)H(1)
ν (√
λa) =
a2J2
ν (√
λa), λ > 00 λ < 0
Èç óñëîâèÿ ψ1(a, λ) = 1, ïîëó÷èì ψ1(ρ, λ) =√
ρa
Jν(√
λρ)
Jν(√
λa). Òîãäà ôîðìóëû ðàçëîæåíèÿ çàïèøóòñÿ â
âèäå:
f(ρ) =+∞∫0
F (λ)√
ρa
Jν(√
λρ)
Jν(√
λa)a2J2
ν (√
λa)dλ
F (λ) =+∞∫0
f(ρ)√
ρa
Jν(√
λρ)
Jν(√
λa)dρ
Ïðèâåäåì èõ ê òðàäèöèîííîìó âèäó ñ ïîìîùüþ ïîäñòàíîâîê s =√
λ, f(ρ) =√
ρy(ρ), F (λ) =y(s)√
aJnu(√
λa)Òîãäà îêîí÷àòåëüíî ïîëó÷èì
y(ρ) =+∞∫0
sy(s)Jν(sρ)ds
y(s) =+∞∫0
ρy(ρ)Jν(sρ)dρ
(1.34)
18
1.13 Ëåììà Ðèìàíà-ËåáåãàÏóñòü ψ(x) íåïðåðûâíà íà èíòåðâàëå (a, b) è èíòåãðàë îò ψ(x) ïî (a, b) àáñîëþòíî ñõîäèòñÿ:
b∫
a
|ψ(x)|dx < ∞.
Òîãäà
b∫
a
ψ(x) cos νxdx → 0,
b∫
a
ψ(x) sin νxdx → 0 ïðè ν →∞ (1.35)
Äëÿ êðàòêîñòè èçëîæåíèÿ îãðàíè÷èìñÿ ñëó÷àåì a = −∞, b = ∞ è èíòåãðàëîì∫∞−∞ ψ(x) cos νxdx.
Âîçüìåì ε > 0.  ñèëó óñëîâèÿ àáñîëþòíîé ñõîäèìîñòè èíòåãðàë ñóùåñòâóåò äîñòàòî÷íî áîëüøîåïîëîæèòåëüíîå A òàêîå, ÷òî
−∞∫
−A
|ψ(x)|dx +∫ A
∞|ψ(x)|dx <
ε
3.
Îòðåçîê [−A,A] ðàçîáüåì òî÷êàìè äåëåíèÿ −A = x0 < x1 < ... < xn−1 < xn = A òàê, ÷òîáû|ψ(x′)− ψ(x′′)| < ε/6A, åñëè x′, x′′ ∈ [xi, xi+1]. Èìååì îöåíêó èíòåãðàë ïî îòðåçêó [−A,A]
∣∣∣∣∣∫ −A
A
ψ(x) cos νxdx
∣∣∣∣∣ =
∣∣∣∣∣n−1∑
i=0
∫ xi+1
xi
ψ(x) cos νxdx
∣∣∣∣∣ =
=
∣∣∣∣∣n−1∑
i=0
∫ xi+1
xi
[ψ(x)− ψ(xi)] cos νxdx + ψ(xi)∫ xi+1
xi
cos νxdx
∣∣∣∣∣ ≤
≤n−1∑
i=0
max
xi≤x≤xi+1|ψ(x)− ψ(xi)|(xi+1) + |ψ(xi)| | sin νxi+1 − sin νxi|
ν
≤
≤ ε
6A
n−1∑
i=0
(xi+1 − xi) +2ν
max|x|<A
|ψ(x)| = ε
3+
2ν
max|x|<A
|ψ(x)|.
Ïóñòü ν0 ñòîëü âåëèêî, ÷òî2ν
max|x|<A
|ψ(x)| < ε
3ïðè ν ≥ ν0.
Òîãäà ∣∣∣∣∫ ∞
−∞ψ(x) cos νxdx
∣∣∣∣ ≤∣∣∣∣∣∫ −A
−∞ψ(x) cos νxdx
∣∣∣∣∣ +
∣∣∣∣∣∫ A
−A
ψ(x) cos νxdx
∣∣∣∣∣ +
+∣∣∣∣∫ ∞
A
ψ(x) cos νxdx
∣∣∣∣ ≤ε
3+
∣∣∣∣∣∫ A
−A
ψ(x) cos νxdx
∣∣∣∣∣ ≤
≤ ε
3+
ε
3+
2ν
max|x|<A
|ψ(x)| ≤ ε
3+
ε
3+
ε
3= ε,
÷òî è äîêàçûâàåò ëåììó.Íà ÿçûêå òåîðèè îáîáùåííûõ ôóíêöèé ñîîòíîøåíèÿ (1.35) ìîæíî çàïèñàòü â âèäå
cos νx → 0, sin νx → 0 ïðè ν →∞,
ïîñêîëüêó ëåâûå ÷àñòè (1.35) ñîâïàäàþò ñ ñîîòâåòñòâóþùèìè ôóíêöèîíàëàìè (cos νx, ϕ(x)) è(sin νx, ϕ(x)).  çàäà÷àõ ñ íåïðåðûâíûì ñïåêòðîì ïîëåçíû ñëåäóþùèå âèäîèçìåíåíèÿ ôîðìóë(1.35): ∫ b
a
ψ(x) sin ν√
xdx → 0,
∫ b
a
ψ(x) cos ν√
xdx → 0, ν →∞
èëè sin ν√
xdx → 0, cos ν√
xdx → 0, ν →∞.Äëÿ äîêàçàòåëüñòâà äîñòàòî÷íî ñäåëàòü â èíòåãðàëàõ çàìåíó ïåðåìåííûõ √x = y è ïðèìåíèòü
ëåììó Ðèìàíà-Ëåáåãà.
19
1.14 ßäðî Äèðèõëå - δ-îáðàçíàÿ ïîñëåäîâàòåëüíîñòü. òåîðèè ðÿäîâ è èíòåãðàëîâ Ôóðüå âàæíóþ ðîëü èãðàåò ôóíêöèÿ
fν(x) =1π
sin ν(x− x1)x− x1
(0 < ν < ∞), (1.36)
òàê íàçûâàåìîå "ÿäðî Äèðèõëå". Â ïåðâîì ñåìåñòðå îíà ôèãóðèðîâàëà ó íàñ â ñïèñêå δ÷-îáðàçíûõïîñëåäîâàòåëüíîñòåé: íà ëþáîì èíòåðâàëå (a, b) ïðè ν →∞
fν → δ(x− x1), åñëè x1 ∈ (a, b),fν → 0, åñëè x1∈(a, b). (1.37)
Âòîðîå èç óòâåðæäåíèé (1.37) - ñëåäñòâèå ëåììû Ðèìàíà-Ëåáåãà, à ïåðâîå ýêâèâàëåíòíî ñëåäó-þùåé òåîðåìå.
Òåîðåìà 1.14.1 Ïóñòü ψ(x) ∈ C1(−∞,∞),∫ +∞−∞ |ψ(x)|dx < ∞.
Òîãäà
1π
∫ ∞
−∞
sin ν(x− x1)x− x1
ψ(x)dx → ψ(x1) ïðè ν →∞. (1.38)
Äîêàçàòåëüñòâî îñíîâàíî íà ëåììå Ðèìàíà-Ëåáåãà è èçâåñòíîì ðàâåíñòâå∫ ∞
0
sinx
xdx =
12
∫ ∞
−∞
sin x
xdx =
π
2. (1.39)
 ñèëó (1.39) ñîîòíîøåíèþ (1.38) ìîæíî ïðèäàòü âèä
1π
∫ ∞
−∞
sin ν(x− x1)x− x1
[ψ(x)− ψ(x1)]dx → 0, ν →∞. (1.40)
Åñëè A è B òàêèå ÷èñëà, ÷òî A < x1 < B, òî íà èíòåðâàëàõ (−∞, A) è (B.∞) ôóíêöèÿ ψ(x)/(x−x1) íåïðåðûâíà è àáñîëþòíî èíòåãðèðóåìà.  ñèëó ëåììû Ðèìàíà-Ëåáåãà
∫ A
−∞
ψ(x)x− x1
sin ν(x− x1)dx → 0,
∫ ∞
B
ψ(x)x− x1
sin ν(x− x1)dx → 0, ν →∞.
Ôóíêöèÿ (ψ(x)−ψ(x1))/(x−x1) íåïðåðûâíà íà [A,B], òàê êàê ψ(x), ïî ïðåäïîëîæåíèþ, íåïðå-ðûâíî äèôôåðåíöèðóåìà. Ñëåäîâàòåëüíî,
∫ B
A
ψ(x)− ψ(x1)x− x1
sin ν(x− x1)dx → 0, ν →∞
ïî ëåììå Ðèìàíà-Ëåáåãà. ñèëó ñõîäèìîñòè èíòåãðàëà (1.39) êàê íåñîáñòâåííîãî è íåðàâåíñòâ A − x1 < 0, B − x1 > 0
èìååì ∫ A
−∞
sin ν(x− x1)x− x1
ψ(x1)dx = ψ(x1)∫ ν(A−x1)
−∞
sin t
tdt → 0, ν →∞
∫ ∞
B
sin ν(x− x1)x− x1
ψ(x1)dx = ψ(x1)∫ ∞
ν(B−x1)
sin t
tdt → 0, ν →∞.
Ðàçáèâàÿ èíòåãðàë (1.40) íà ñóììó
1π
∫ ∞
−∞
sin ν(x− x1)x− x1
[ψ(x)− ψ(x1)]dx =1π
(∫ A
−∞+
∫ B
A
+∫ ∞
B
)=
=1π
(∫ A
−∞
ψ(x) sin ν(x− x1)x− x1
dx−∫ A
−∞ψ(x1)
sin ν(x− x1)x− x1
dx+
20
+∫ B
A
sin ν(x− x1)x− x1
[ψ(x)− ψ(x1)]dx +∫ ∞
B
ψ(x) sin ν(x− x1)x− x1
dx−
−∫ ∞
B
ψ(x1)sin ν(x− x1)
x− x1dx
)
è ïîëüçóÿñü ïîëó÷åííûìè ïðåäåëüíûìè ñîîòíîøåíèÿìè, óñòàíàâëèâàåì ñïðàâåäëèâîñòü (1.40)è, ñëåäîâàòåëüíî, (1.38).
Ñëåäñòâèå. Íà èíòåðâàëå (0,∞) èìååò ìåñòî ïðåäåëüíûé ïåðåõîä
fν = F (x)sin νx(
√x−√x1)√
x−√x1→ 2F (x1)π
√x1δ(x− x1), ν →∞, (1.41)
ãäå F (x) - íåïðåðûâíàÿ âìåñòå ñ F ′(x) ôóíêöèÿ, x1 - ïîëîæèòåëüíàÿ êîíñòàíòà.Äëÿ äîêàçàòåëüñòâà äîñòàòî÷íî â èíòåãðàëå
(fν , ϕ) =∫ ∞
0
fν(x)ϕ(x)dx
ñäåëàòü çàìåíó ïåðåìåííûõ è âîñïîëüçîâàòüñÿ ôîðìóëîé (1.38).
1.15 Îðòîãîíàëüíîñòü ñîáñòâåííûõ ôóíêöèé íåïðåðûâíîãî ñïåê-òðà.
Íåñìîòðÿ íà ñóùåñòâîâàíèå îòëè÷èÿ ñèíãóëÿðíîé çàäà÷è Øòóðìà-Ëèóâèëëÿ îò ðåãóëÿðíîé, íåêî-òîðûå ñâîéñòâà ðåøåíèé çàäà÷è ñ íåïðåðûâíûì ñïåêòðîì èìåþò ìíîãî îáùåãî ñî ñâîéñòâàìè ñîá-ñòâåííûõ ôóíêöèé ðåãóëÿðíîé çàäà÷è. Òàê, â ÷àñòíîñòè, ôóíêöèè íåïðåðûâíîãî ñïåêòðà ñîõðàíÿ-þò ñâîéñòâî îðòîãîíàëüíîñòè (â íåêîòîðîì íîâîì ñìûñëå).
Ðàññìîòðèì çàäà÷ó Øòóðìà-Ëèóâèëëÿ íà ïîëóîñè x > 0 äëÿ óðàâíåíèÿ y′′+λy = 0 ñ óñëîâèåìy(0) = 0. Îáîçíà÷èì
y1 = sin√
λ1x, y2 = sin√
λ2x, λi > 0, i = 1, 2.
Ôóíêöèè y1 è y2 óäîâëåòâîðÿþò òîæäåñòâàì
y′′1 = −λ1y1, y′′2 = λ2y2.
Áóäåì äåéñòâîâàòü òàê æå, êàê è â ðåãóëÿðíîé çàäà÷å: óìíîæàÿ ïåðâîå òîæäåñòâî íà y2, âòîðîå -íà y1 è âû÷èòàÿ ðåçóëüòàòû, ïîëó÷èì
d
dxW (y1, y2) =
d
dx(y′1y2 − y′2y1) = (λ2 − λ1)y1y2. (1.42)
Åñëè òî÷íî ñëåäîâàòü ìåòîäó, ïðèìåíèìîìó â çàäà÷å íà êîíå÷íîì ïðîìåæóòêå, òî ïîëàãàëîñüáû ïðîèíòåãðèðîâàòü îáå ÷àñòè ðàâåíñòâà (1.42) îò 0 äî ∞.  ðàññìàòðèâàåìîì ñëó÷àå, îäíàêî,èíòåãðàëû îò îáåèõ ÷àñòåé ðàâåíñòâà (1.42) ðàñõîäÿòñÿ íà áåñêîíå÷íîñòè. Íà ïîìîùü ïðèõîäèòòåîðèÿ îáîáùåííûõ ôóíêöèé.
Èíòåãðèðóÿ îáå ÷àñòè (1.42) ïî x îò 0 äî N è ó÷èòûâàÿ ãðàíè÷íûå óñëîâèÿ y1(0) = y2(0) = 0,ïîëó÷àåì ∫ N
0
y1y2dx =√
λ1 cos√
λ1N sin√
λ2N −√λ2 cos√
λ2N sin√
λ1N
λ2 − λ1=
=(√
λ1 −√
λ2) sin(√
λ1 +√
λ2)N + (√
λ1 +√
λ2) sin(√
λ2 −√
λ1)N2(λ2 − λ1)
=
=sin(
√λ1 +
√λ2)N
2(√
λ1 +√
λ2)+
sin(√
λ2 −√
λ1)N2(√
λ2 −√
λ1).
Ïåðåõîäÿ ê ïðåäåëó N → ∞ è èñïîëüçóÿ ëåììó Ðèìàíà-Ëåáåãà (äëÿ ïåðâîãî ñëàãàåìîãî) è δ-îáðàçíóþ ïîñëåäîâàòåëüíîñòü ((1.41) äëÿ âòîðîãî), ïîëó÷àåì
limN→∞
∫ N
0
y1y2dx = limN→∞
[sin(
√λ1 +
√λ2)N√
λ1 +√
λ2
+sin(
√λ2 −
√λ1)N
2(√
λ2 −√
λ1)
]=
21
= 0 +122π
√λ1δ(λ1 − λ2) = π
√λ1δ(λ1 − λ2),
èëè ∫ ∞
0
y1y2dx = π√
λ1δ(λ1 − λ2). (1.43)
Ïîñêîëüêó δ(λ1 − λ2) = 0 ïðè λ1 6= λ2, ìû ìîæåì òðàêòîâàòü ñîîòíîøåíèå (1.43) êàê "îðòî-ãîíàëüíîñòüñîáñòâåííûõ ôóíêöèé"y1(x) è y2(x). Êàê îòìå÷àåòñÿ â [], ðàâåíñòâî (1.43) ñîäåðæèò âñåáå íå òîëüêî îðòîãîíàëüíîñòü, íî è íîðìèðîâêó, êîòîðóþ õàðàêòåðèçóåò ìíîæèòåëü π
√λ1, íî,
áåçóñëîâíî, ýòî íå êâàäðàò íîðìû â L2(0,∞), "ñîáñòâåííûå"ôóíêöèè íå ÿâëÿþòñÿ êâàäðàòè÷íî-èíòåãðèðóåìûìè.
Îðòîãîíàëüíîñòü, âûðàæåííàÿ ôîðìóëîé (1.43), ïîçâîëÿåò íàõîäèòü ïðåîáðàçîâàíèå Ôóðüåôóíêöèè f(x) âïîëíå àíàëîãè÷íî íàõîæäåíèþ êîýôôèöèåíòîâ Ôóðüå â ðåãóëÿðíîé çàäà÷å.
Äåéñòâèòåëüíî, ïóñòü ôóíêöèÿ f(x) ïðåäñòàâèìà â âèäå ñóïåðïîçèöèè ôóíêöèé sin√
λx:
f(x) =∫ ∞
0
f(λ) sin√
λxdλ. (1.44)
Óìíîæàÿ îáå ÷àñòè ðàâåíñòâà (1.44) íà sin√
λx, èíòåãðèðóÿ ïî x îò 0 äî ∞ è ïîëüçóÿñü ôîðìóëîé(1.43), ïîëó÷àåì
f(x) =∫ ∞
0
f(λ) sin√
λ1xdx =∫ ∞
0
sin√
λ1xdx
∫ ∞
0
f(λ) sin√
λxdλ =
=∫ ∞
0
f(λ)dλ
∫ ∞
0
sin√
λ1x sin√
λdx = π
∫ ∞
0
f(λ)√
λδ(λ− λ1)dλ =
= πf(λ1)√
λ1,
èëèf(λ) =
1π
∫ ∞
0
f(x)sin√
λx√λ
dx. (1.45)
Îáîçíà÷àÿ√
λ = ν, π√
λf(λ) = f(λ) ïðèäàåì ôîðìóëàì (1.44) è (1.45) îáû÷íûé âèä îáðàòíîãîè ïðÿìîãî ñèíóñ-ïðåîáðàçîâàíèÿ Ôóðüå:
f(x) =2π
∫ ∞
0
f(ν) sin νxdν, f(ν) =∫ ∞
0
f(x) sin νxdx.
Ïîä÷åðêíåì, ÷òî ôîðìóëà (1.45) äîêàçàíà â ïðåäïîëîæåíèè, ÷òî âåðíà òåîðåìà ðàçëîæåíèÿ(1.44).
Çàìå÷àíèå  1.1 îïðåäåëåíà ôóíêöèÿ F (λ):
F (λ)∫ ∞
0
f(x)y(x, λ)dx.
Ñðàâíèâàÿ ñ (1.45), èìååì f(λ) = F (λ)/π√
λ. Ñ äðóãîé ñòîðîíû, ñîãëàñíî òåîðåìå ðàçëîæåíèÿ(1.22)
f(x) =∫ ∞
−∞F (λ)y(x, λ)dρ(λ).
Ñðàâíèâàÿ ñ (1.44), ïîëó÷àåì
ρ(λ) = 0, −∞ < λ < 0; ρ′(λ) =1
π√
λ, λ > 0,
îòêóäàρ(λ) =
2π
√λ, λ ≥ 0.
22
1.16 Àñèìïòîòèêà ðåøåíèé ñèíãóëÿðíîé çàäà÷èØòóðìà-Ëèóâèëëÿñ ñóììèðóåìûì ïîòåíöèàëîì.
Ïðîñìàòðèâàÿ âûâîä ñîîòíîøåíèÿ îðòîãîíàëüíîñòè (1.43), ìîæíî çàìåòèòü, ÷òî åãî ãëàâíàÿ ÷àñòüñîñòîèò â èñïîëüçîâàíèè çíà÷åíèé ôóíêöèé y1(x, λ) è y2(x, λ) è èõ ïðîèçâîäíûõ ïðè áîëüøèõçíà÷åíèÿõ x ñ äàëüíåéøèì ïðèìåíåíèåì δ-îáðàçíîé ïîñëåäîâàòåëüíîñòè (1.41). Ñóùåñòâóåò êëàñññèíãóëÿðíûõ çàäà÷ Øòóðìà-Ëèóâèëëÿ, ðåøåíèÿ êîòîðûõ îáëàäàþò ïîõîæåé "òðèãîíîìåòðè÷å-ñêîé"àñèìïòîòèêîé ïðè x → ∞. Ýòî òàê íàçûâàåìàÿ ñèíãóëÿðíàÿ çàäà÷à Øòóðìà-Ëèóâèëëÿ ññóììèðóåìûì ïîòåíöèàëîì. Ïîÿñíèì ýòî íàçâàíèå. Åñëè óðàâíåíèå Øòóðìà-Ëèóâèëëÿ ñ ïîìî-ùüþ ïðåîáðàçîâàíèÿ Ëèóâèëëÿ-Ãðèíà ïðèâåäåíî ê ôîðìå
y′′ + [λ− q(x)]y = 0, (1.46)
êîòîðóþ ìû èñïîëüçîâàëè â ïðåäûäóùèõ ïàðàãðàôàõ, òî îíî ñîâïàäàåò ïî ôîðìå ñ îäíîìåðíûìêâàíòîâî-ìåõàíè÷åñêèì óðàâíåíèåì Øðåäèíãåðà.  óðàâíåíèè Øðåäèíãåðà (1.46) ôóíêöèÿ q(x)èìååò ñìûñë ïîòåíöèàëà ñèëîâîãî ïîëÿ, â êîòîðîì íàõîäèòñÿ ÷àñòèöà, à y(x) - âîëíîâàÿ ôóíêöèÿýòîé ÷àñòèöû. Ïîòåíöèàë q(x) ïðåäïîëàãàåòñÿ äàëåå ñóììèðóåìûì:
q(x) ∈ L(0,∞), ò.å.∫ ∞
0
|q(x)|dx < ∞. (1.47)
Äëÿ âû÷èñëåíèÿ àñèìïòîòèêè ðåøåíèé óðàâíåíèÿ (1.46) ïðè x → ∞ íàì ïîíàäîáèòñÿ ëåììà,ÿâëÿþùàÿñÿ óïðîùåííûì âàðèàíòîì ëåììû 2.1 èç êíèãè [].
Ëåììà 1.16.1 Ïóñòü ôóíêöèè h(x) è g(x) íåîòðèöàòåëüíû íà ïðîìåæóòêå [0, X], h(x) - íåïðå-ðûâíà, à g(x) - èíòåãðèðóåìà íà ýòîì ïðîìåæóòêå.
Åñëè ïðè 0 ≤ x ≤ X
h(x) ≤ C +∫ x
0
h(ξ)g(ξ)dξ, (1.48)
ãäå C -ïîñòîÿííàÿ, òî ñïðàâåäëèâî íåðàâåíñòâî
h(x) ≤ C exp∫ x
0
g(ξ)dξ, 0 ≤ x ≤ X. (1.49)
Äîêàçàòåëüñòâî. Îáîçíà÷èìy(x) =
∫ x
0
h(ξ)g(ξ)dξ, (1.50)
òîãäày′(x) = h(x)g(x). (1.51)
Óìíîæèì íåðàâåíñòâî (1.49) íà g(x):
g(x)h(x) ≤ Cg(x) + g(x)∫ x
0
h(ξ)g(ξ)dξ,
òî åñòüy′(x) ≤ g(x)[C + y(x)].
Óìíîæèì ïîñëåäíåå íåðàâåíñòâî íà exp(− ∫ x
0g(ξ)dξ):
y′(x) exp(−
∫ x
0
g(ξ)dξ
)≤ g(x)[C + y(x)] exp
(−
∫ x
0
g(ξ)dξ
). (1.52)
Ðàññìîòðèì ïîëíóþ ïðîèçâîäíóþ
d
dx
y(x) exp
(−
∫ x
0
g(ξ)dξ
)= y′(x) exp
(−
∫ x
0
g(ξ)dξ
)− g(x)y(x) exp
(−
∫ x
0
g(ξ)dξ
). (1.53)
Ñ ó÷åòîì (1.53) íåðàâåíñòâî (1.52) ïåðåïèñûâàåòñÿ â âèäå
d
dx
y(x) exp
(−
∫ x
0
g(ξ)dξ
)≤ Cg(x) exp
(−
∫ x
0
g(ξ)dξ
).
23
Èíòåãðèðóÿ ýòî íåðàâåíñòâî, ïîëó÷àåì
y(x) exp(−
∫ x
0
g(ξ)dξ
)≤ C
[1− g(x) exp
(−
∫ x
0
g(ξ)dξ
)],
èëèy(x) ≤ Cg(x)
(exp
∫ x
0
g(ξ)dξ − 1)
. (1.54)
Ñ ó÷åòîì (1.48) è (1.50), òî åñòü íåðàâåíñòâà
h(x) ≥ C + y(x),
èç (1.54) ñëåäóåò íåðàâåíñòâî (1.49). Ëåììà äîêàçàíà.Äîêàæåì òåïåðü, ÷òî ïðèíàäëåæàùèå íåïðåðûâíîìó ñïåêòðó ðåøåíèÿ óðàâíåíèÿ (1.46) äîïóñ-
êàþò àñèìïòîòè÷åñêîå ïðåäñòàâëåíèå
ϕ(x, λ) = µ(λ) cos√
λx + ν(λ) sin√
λx + 0(1), x →∞, (1.55)
ãäå µ(λ) è ν(λ) - íåïðåðûâíî äèôôåðåíöèðóåìûå ôóíêöèè. Ïóñòü ðåøåíèÿ ϕ(x, λ) óäîâëåòâîðÿþòóñëîâèÿì
ϕ(0, λ) = sin α, ϕ′(0, λ) = − cos α.
Ïåðåïèñûâàÿ óðàâíåíèå (1.46) â âèäå
y′′ + λy = q(x)y,
ïîëàãàÿ λ = s2 è ïðèìåíÿÿ ìåòîä âàðèàöèè ïðîèçâîëüíûõ ïîñòîÿííûõ, ïðèõîäèì ê èíòåãðàëüíîìóóðàâíåíèþ Âîëüòåððà âèäà
ϕ(x, λ) = sin α cos sx− cosαsin sx
s+
1s
∫ x
0
sin s(x− ξ)q(ξ)ϕ(ξ, λ)dξ. (1.56)
Èç (1.56) ñëåäóåò îöåíêà
|ϕ(x, λ)| ≤ 1 +1|s| +
1|s|
∫ x
0
|q(ξ)||ϕ(ξ, λ)|dξ. (1.57)
Ïðèìåíÿÿ ê (1.57) ëåììó ïðè C = 1 + 1|s| , h(x) = |ϕ(x, λ)|, g(x) = |q(x)|/s, ïîëó÷àåì
|ϕ(x, λ)| ≤ 1 +1|s| exp
1|s|
∫ x
0
|q(ξ)|dξ, (1.58)
òî åñòü îãðàíè÷åííîñòü ôóíêöèè ϕ(x, λ) ïðè 0 < x < ∞, |s| ≥ p > 0. Òîãäà èç (1.56) ïîëó÷àåìèñêîìîå àñèìïòîòè÷åñêîå ñîîòíîøåíèå (1.57):
ϕ(x, λ) = sin α cos sx− cossin sx
s+
1s
∫ ∞
0
sin s(x− ξ)q(ξ)ϕ(ξ, λ)dξ−
−1s
∫ ∞
x
sin s(x− ξ)q(ξ)ϕ(ξ, λ)dξ = µ(λ) cos sx + ν(λ) sin sx + 0(1), x →∞,
ãäå îáîçíà÷åíîµ(λ) = sin α− 1
s
∞∫0
sin sξq(ξ)ϕ(ξ, λ)dξ,
ν(λ) = − cos αs + 1
s
∞∫0
cos sξq(ξ)ϕ(ξ, λ)dξ.
(1.59)
Ïîñêîëüêó èíòåãðàëû â ôîðìóëàõ (1.59) ðàâíîìåðíî ñõîäÿòñÿ ïðè s ≥ p > 0, òî µ(λ) è ν(λ)ÿâëÿþòñÿ íåïðåðûâíûìè ôóíêöèÿìè. Áîëåå ïîäðîáíûé àíàëèç ïîçâîëÿåò óáåäèòüñÿ â íåïðåðûâ-íîé äèôôåðåíöèðóåìîñòè ýòèõ ôóíêöèé (è äàæå àíàëèòè÷íîñòè à íåêîòîðîì ñïåêòðå ïëîñêîñòèλ, âêëþ÷àþùåì ïîëîæèòåëüíóþ ïîëóîñü, ïðè |λ| ≥ p2 > 0).
Äèôôåðåíöèðóÿ îáå ÷àñòè óðàâíåíèÿ (1.56) ïî x è ïîâòîðÿÿ îöåíêè (1.58) è (1.59), óáåæäàåìñÿ,÷òî àñèìïòîòè÷åñêîå ïðåäñòàâëåíèå (1.55) äîïóñêàåò äèôôåðåíöèðîâàíèå
ϕ′(x, λ) = s[−µ(λ) sin sx + ν(λ) cos sx] + 0(1), x →∞. (1.60)
24
1.17 Îðòîãîíàëüíîñòü ðåøåíèé ñèíãóëÿðíîé çàäà÷èØòóðìà-Ëèóâèëëÿ ñ ñóììèðóåìûì ïîòåíöèàëîì.
Ïóñòü y1(x) è y2(x) óäîâëåòâîðÿþò óðàâíåíèþ Øòóðìà-Ëèóâèëëÿ (1.46) ïðè çíà÷åíèÿõ ïàðàìåòðàλ, ðàâíûõ ñîîòâåòñòâåííî λ1 è λ2:
y′′1 + [λ1 − q(x)]y1 = 0,
y′′2 + [λ2 − q(x)]y2 = 0,
à òàêæå íà÷àëüíûì óñëîâèÿìy(0, λ) = sin α, y′(0, λ) = − cosα.
Òîãäà ñïðàâåäëèâî ñëåäóþùåå ñîîòíîøåíèå îðòîãîíàëüíîñòè:∫ ∞
0
y1(x)y2(x)dx = π√
λ1[µ2(λ1) + ν2(λ1)]δ(λ1 − λ2). (1.61)
Äëÿ äîêàçàòåëüñòâà îáû÷íûì îáðàçîì âû÷èñëÿåì èíòåãðàë∫ N
0
y1(x)y2(x)dx =1
λ1 − λ2Wy1, y2x=N
è ïîäñòàâëÿåì â ïðàâóþ ÷àñòü ðàâåíñòâà àñèìïòîòè÷åñêèå ïðåäñòàâëåíèÿ (1.55) è (1.60):∫ N
0
y1(x)y2(x)dx =1
λ1 − λ2[y′2(N)y1(N)− y′1(N)y2(N)] =
=1
λ1 − λ2
√λ2[−µ(λ2) sin
√λ2N + ν(λ2) cos
√λ2N ][µ(λ1) cos
√λ1N+
+ν(λ1) sin√
λ1N ]−√
λ1[−µ(λ1) sin√
λ1N + ν(λ1) cos√
λ1N ][µ(λ2) cos√
λ2N+
+ν(λ2) sin√
λ2N ] + 0(1)N→∞
=
=1
λ1 − λ2
sin
√λ1N sin
√λ2N [−
√λ2µ(λ2)ν(λ1) +
√λ1µ(λ1)ν(λ2)]+
+ cos√
λ1N sin√
λ2N [−√
λ2µ(λ2)ν(λ1)−√
λ1µ(λ1)ν(λ2)]+
+ sin√
λ1N cos√
λ2N [√
λ2ν(λ1)ν(λ2) +√
λ1µ(λ1)µ(λ2)]+
+cos√
λ1N cos√
λ2N [√
λ2µ(λ1)ν(λ2)−√
λ1ν(λ1)µ(λ2)] + 0(1)n→∞
=
=1
λ1 − λ2
(√
λ2 +√
λ1)[µ(λ1)ν(λ2)− ν(λ1)µ(λ2)] cos(√
λ1 −√
λ2)N+
+(√
λ2 −√
λ1)[µ(λ1)ν(λ2) + ν(λ1)µ(λ2)] cos(√
λ1 +√
λ2)N+
+(√
λ2 −√
λ1)[ν(λ1)ν(λ2) + µ(λ1)µ(λ2)] sin(√
λ1 +√
λ2)N+
+(√
λ2 +√
λ1)[ν(λ1)ν(λ2) + µ(λ1)µ(λ2)] sin(√
λ1 −√
λ2)N + 0(1)N→∞
=
12
µ(λ1)ν(λ2)− ν(λ1)µ(λ2)√
λ1 −√
λ2
cos(√
λ1 −√
λ2)N−
−µ(λ1)ν(λ2) + ν(λ1)µ(λ2)√λ1 +
√λ2
cos(√
λ1 +√
λ2)N−
−ν(λ1)ν(λ2) + µ(λ1)µ(λ2)√λ1 +
√λ2
sin(√
λ1 +√
λ2)N+
+[ν(λ1)ν(λ2) + µ(λ1)µ(λ2)]sin(
√λ1 −
√λ2)N√
λ1 −√
λ2
+ 0(1)N→∞
.
Ôóíêöèè, ñòîÿùèå â ïðàâîé ÷àñòè, ìû ñ÷èòàåì ôóíêöèÿìè ïåðåìåííîãî λ1 ñ ïàðàìåòðàìè λ2 èN . Ôóíêöèè ν(λ1)ν(λ2)+µ(λ1)µ(λ2), ν(λ1)ν(λ2)+µ(λ1)µ(λ2)/(
√λ1+
√λ2), µ(λ1)ν(λ2)+ν(λ1)µ(λ2)/(
√λ1+√
λ2), µ(λ1)ν(λ2)−ν(λ1)µ(λ2)/(√
λ1−√
λ2) ÿâëÿþòñÿ íåïðåðûâíûìè (â ïîñëåäíåì ñëó÷àå íàäî ðàñ-êðûòü íåîïðåäåëåííîñòü ïðè λ1 = λ2 ïî ïðàâèëó Ëîïèòàëÿ). Ïåðåõîäÿ ê ïðåäåëó n →∞ è ïðèìå-íÿÿ ê ïåðâûì òðåì ñëàãàåìûì ëåììó Ðèìàíà-Ëåáåãà, à ê ïîñëåäíåìó - ôîðìóëû (1.44), ïîëó÷àåìèñêîìîå ñîîòíîøåíèå (1.61).
25
1.18 Èíòåãðàëüíûå ðàçëîæåíèÿ ïî ôóíêöèÿì íåïðåðûâíîãîñïåêòðà ñèíãóëÿðíîé çàäà÷è Øòóðìà-Ëèóâèëëÿ ñ ñóì-ìèðóåìûì ïîòåíöèàëîì.
Ïóñòü ñïðàâåäëèâà òåîðåìà ðàçëîæåíèÿ: ïðîèçâîäíàÿ ôóíêöèÿ f(x) èç íåêîòîðîãî ôóíêöèîíàëü-íîãî êëàññà ïðåäñòàâèìà â âèäå ñóïåðïîçèöèè ðåøåíèé óðàâíåíèÿ
y′′ + [λ− q(x)]y = 0, 0 < x < ∞,
ñ ñóììèðóåìûì ïîòåíöèàëîì q(x) ïðè óñëîâèÿõ
y(0) = sin α, y′(0) = − cos α,
òî åñòüf(x) =
∫ ∞
0
f(y)y(x, λ)dλ. (1.62)
Ïîâòîðèì ðàññóæäåíèÿ èç 1.15. Óìíîæèì îáå ÷àñòè (1.62) íà y(x, λ1), ïðîèíòåãðèðóåì ïî x îò 0äî ∞ è èñïîëüçóåì ñîîòíîøåíèå îðòîãîíàëüíîñòè (1.61). Èìååì
∫ ∞
0
f(x)y(x, λ1)dx =∫ ∞
0
y(x, λ)dx
∫ ∞
0
f(y)y(x, λ)dλ =
=∫ ∞
0
f(λ)dλ
∫ ∞
0
y(x, λ1)y(x, λ)dλ = π
∫ ∞
0
f(λ)√
λ·
·[µ2(λ) + ν2(λ)δ(λ− λ1)]dλ = πf(λ1)[µ2(λ1) + ν2(λ1)],
èëèf(λ) =
1π√
λ[µ2(λ) + ν2(λ)]
∫ ∞
0
f(x)y(x, λ)dx. (1.63)
Ôîðìóëû (1.63) è (1.62) îáðàçóþò ïàðó ôîðìóë ïðÿìîãî (1.63) è îáðàòíîãî (1.62) èíòåãðàëüíî-ãî ïðåîáðàçîâàíèÿ ïî ôóíêöèÿì íåïðåðûâíîãî ñïåêòðà ñèíãóëÿðíîé çàäà÷è Øòóðìà-Ëèóâèëëÿ.Ôîðìóëû ñèíóñ- è êîñèíóñ-ïðåîáðàçîâàíèÿ Ôóðüå ÿâëÿþòñÿ ÷àñòíûìè ñëó÷àÿìè (1.62), (1.63).
Çàìå÷àíèå. Èñïîëüçóÿ îïðåäåëåíèå ôóíêöèè F (λ) èç 1.1, èìååì
F (λ) =∫ ∞
0
f(x)y(x, λ)dx = π√
λ[µ2(λ) + ν2(λ)]f(λ),
ñ äðóãîé ñòîðîíûf(λ) = F (λ)ρ′(λ),
ñëåäîâàòåëüíî,ρ = 0, −∞ < λ < 0; ρ′(λ) =
1π√
λ[µ2(λ) + ν2(λ)], λ > 0.
Óðàâíåíèå Øòóðìà-Ëèóâèëëÿ â ôîðìå Øðåäèíãåðà (1.46) áûëî ïîëó÷åíî ñ ïîìîùüþ ïðåîáðà-çîâàíèÿ Ëèóâèëëÿ-Ãðèíà (ñì. 1.41 I ñåìåñòðà). Ñ òî÷êè çðåíèÿ ïðèëîæåíèé óäîáíî âåðíóòüñÿ êèñõîäíûì ïåðåìåííûì. Ïàðà ôîðìóë (1.62), (1.63) ïåðåõîäèò ïðè ýòîì (âûâîä ïðåäîñòàâèì ÷èòà-òåëþ) â ôîðìóëû âèäà
f(x) =∫ ∞
0
f(λ)y(x, λ)dλ, (1.64)
f(λ) =1
π√
λ[µ2(λ) + ν2(λ)]
∫ ∞
a
r(x)f(x)y(x, λ)dx. (1.65)
×àñòî áûâàåò óäîáíî ïåðåéòè îò ïàðàìåòðà λ ê ïàðàìåòðó s: λ = s2. Íàðÿäó ñ ýòèì ïðîèçâåäåìçàìåíó ôóíêöèè f(λ):
2√
λf(λ) = f(s).
Òîãäà ôîðìóëû (1.64), (1.65), ïðèìóò âèä
f(x) =∫ ∞
0
f(s)y(x, s2)ds, (1.66)
26
f(s) =2
π[µ2(s2) + ν2(s2)]
∫ ∞
a
r(x)f(x)y(x, s2)dx. (1.67)
Ïðèìåðû.1. Îáîáùåííîå ïðåîáðàçîâàíèå Ôóðüå íà ïîëóîñè. Ñèíãóëÿðíàÿ çàäà÷à Øòóðìà-Ëèóâèëëÿ:
y′′ + λy = 0, 0 < x < ∞y′(0)− hy(0) = 0, h ≥ 0
(1.68)
Ðåøåíèÿ çàäà÷è (1.68) èìåþò âèä
y(x, λ) = cos√
λx +h√λ
sin√
λx, λ > 0,
ñëåäîâàòåëüíî, µ(λ) = 1, ν(λ) = h/√
λ,∫ ∞
0
y1(x)y2(x)dx = π√
λ1(1 +h2
λ1)δ(λ1 − λ2),
÷òî ïðèâîäèò ê ôîðìóëàì ïðÿìîãî è îáðàòíîãî ïðåîáðàçîâàíèÿ, ýêâèâàëåíòíûì (1.11).2. Ïðåîáðàçîâàíèå Âåáåðà.Çàäà÷à Øòóðìà-Ëèóâèëëÿ:
(ρy′)′ + (λρ− ν2
ρ)y = 0, a < ρ < ∞; y(a) = 0.
Ïðåîáðàçîâàíèå Ëèóâèëëÿ-Ãðèíà y = ρ−1/2v(ρ) ïåðåâîäèò óðàâíåíèå ê âèäó óðàâíåíèÿ Øðåäèí-ãåðà
v′′ + (λ +1/4− ν2
ρ2)v = 0
ñ ñóììèðóåìûì ïîòåíöèàëîì 1/4− ν2ρ−2. Ðåøåíèÿ, óäîâëåòâîðÿþùèå óñëîâèþ y(a) = 0 âûáåðåìâ âèäå
y(ρ, λ) = Jν(√
λρ)Nν(√
λa)− Jν(√
λa)Nν(√
λρ),
ñîîòâåòñòâåííî, ôóíêöèÿ v(ρ, λ) èìååò âèä
v(ρ, λ) =√
ρ[Jν(√
λρ)Nν(√
λa)− Jν(√
λa)Nν(√
λρ)].
Èñïîëüçóÿ àñèìïòîòè÷åñêèå ôîðìóëû äëÿ öèëèíäðè÷åñêèõ ôóíêöèé, èìååì
v(ρ, λ) =
√2
π√
λ[Nν(
√λa) cos(
√λ− ν
π
2− π
4)−
−Jν(√
λa) sin(√
λρ− νπ
2− π
4)] + 0(ρ−1)ρ→∞ =
=
√2
π√
λ
Nν(
√λa)[cos
√λρ cos(ν +
12)π
4+ sin
√λρ sin(ν +
12)π
4]−
−Jν(√
λa)[sin√
λρ cos(ν +12)π
4− cos
√λρ sin(ν +
12)π
4]−
+ 0(ρ−1)ρ→∞ =
=
√2
π√
λ
cos
√λρ[Nν(
√λa) cos(ν +
12)π
4+ Jν(
√λa) sin(ν +
12)π
4]+
+ sin√
λρ[Nν(√
λa) sin(ν +12)π
4− Jν(
√λa) cos(ν +
12)π
4]
+ 0(ρ−1)ρ→∞,
ñëåäîâàòåëüíî,
µ(λ) =
√2
π√
λ[Nν(
√λa) cos(ν +
12)π
4+ Jν(
√λa) sin(ν +
12)π
4],
27
ν(λ) =
√2
π√
λ[Nν(
√λa) sin(ν +
12)π
4− Jν(
√λa) cos(ν +
12)π
4].
Ïîäñòàâëÿÿ â â ôîðìóëû (1.66), (1.67), èìååì
f(ρ) =∫ ∞
0
f(s)[Jν(sρ)Nν(sa)− Jν(sa)Nν(ρa)]ds,
f(s) =s
J2ν (sa) + N2
ν (sa)
∫ ∞
a
ρf(ρ)[Jν(sρ)Nν(sa)− Jν(sa)Nν(ρa)]dρ.
Ïðîèçâîäÿ çàìåíó√
J2ν (sa) + N2
ν (sa)f(s)/s = f(s), ïðèâåäåì ôîðìóëû ïðåîáðàçîâàíèÿ Âåáåðà êáîëåå ñèììåòðè÷íîìó âèäó
f(ρ) =∞∫0
Jν(sρ)Nν(sa)−Jν(sa)Nν(ρa)√J2
ν (sa)+N2ν (sa)
f(s)sds,
f(s) =∞∫a
Jν(sρ)Nν(sa)−Jν(sa)Nν(ρa)√J2
ν (sa)+N2ν (sa)
f(ρ)ρdρ.(1.69)
 ýòèõ ôîðìóëàõ ëåãêî ïðîâåñòè ïðåäåëüíûé ïåðåõîä a → 0 è ïîëó÷èòü ôîðìóëû ïðåîáðàçîâàíèÿÕàíêåëÿ:
f(ρ) =∞∫0
Jν(sρ)f(s)sds,
f(s) =∞∫0
Jν(sρ)f(ρ)ρdρ.(1.70)
28
Ãëàâà 2
Ïðåîáðàçîâàíèå Ëàïëàñà
2.1 Ïðåîáðàçîâàíèå Ëàïëàñà. Îïðåäåëåíèå. Àíàëèòè÷íîñòüÎäíîñòîðîííåå ïðåîáðàçîâàíèå Ëàïëàñà :
f(p) =∫ ∞
0
e−ptf(t)dt (f(t)÷ f(p)) (2.1)
f(t) - îðèíèíàë, f(p) - èçîáðàæåíèå ïî Ëàïëàñó (îáðàç) Óñëîâèÿ :1. f(t) ≡ 0, ïðè t ≤ 0. Íàïðèìåð sin ωt ÷ ω
p2+ω2 , âîîáùå ãîâîðÿ òàêàÿ çàïèñü íåâåðíà, íóæíîïèñàòü η(t) sin ωt÷ ω
p2+ω2 , ãäå η(t) ôóíêöèÿ Õåâèñàéäà.
2. f(t) äîëæíà áûòü íåïðåðûâíî äèôôåðåíöèðóåìîé ôóíêöèåé (èëè óäîâëåòâîðÿòü óñëîâèþÃåëüäåðà), çà èñêëþ÷åíèåì áûòü ìîæåò êîíå÷íîãî ÷èñëà òî÷åê ðàçðûâà ïåðâîãî ðîäà â ëþáîìêîíå÷íîì ïðîìåæóòêå.
3. Óñëîâèå êîíå÷íîãî ðîñòà :∀t |f(t)| < Meσ0t, òî÷íóþ íèæíþþ ãðàíü òàêèõ σ0 íàçûâàþòïîêàçàòåëåì ðîñòà ôóíêöèè.
Ïðè âûïîëíåíèè óñëîâèé 1-3 èíòåãðàë Ëàïëàñà ñõîäèòñÿ â ïðàâîé âîëóïëîñêîñòè ïàðàìåòðàp = σ + iτ ïðè σ > σ0 (ðèñ 2.1).
t
s
s0
p
Ðèñ. 2.1:
Èìååò ìåñòî ñëåäóþùàÿ îöåíêà çíà÷åíèÿ èíòåãðàëà∣∣∣∣∫ ∞
0
e−ptf(t)dt
∣∣∣∣ ≤∫ ∞
0
e−σteσ0tdt =1
σ − σ0(2.2)
Óòâåðæäåíèå 1 Åñëè σ ≥ σ∗ > σ0, òî f(p) àíàëèòè÷åñêàÿ ôóíêöèÿ.
Äîê-âî Âûïîëíèì ðàâíîìåðíóþ ïî σ îöåíêó∣∣∣∣∫ ∞
0
e−ptf(t)dt
∣∣∣∣ ≤=1
σ∗ − σ0, (ðàâíîìåðíî ïî σ),
29
ñëåäîâàòåëüíî f(p) àíàëèòè÷íà â óêàçàííîé îáàëñòè, ÷òî è òðåáîâàëîñü.
Óòâåðæäåíèå 2 Ïðè σ = Rp →∞ f(p) → 0
Äîê-âî Î÷åâèäíî
Óòâåðæäåíèå 3 Ïðè arg p ≤ π2 − α (α < 0) è p →∞ f(p) → 0 (ñì. ðèñ. 2.2)
Äîê-âî Î÷åâèäíî
a
Ðèñ. 2.2:
Ïðèìåðûη(t)÷
∫ ∞
0
1 · e−ptdt =1p
(2.3)
η(t)eαt ÷ 1p− α
(2.4)
(ýòè ôîðìóëû äîïóñêàþò àíàëèòè÷åñêîå ïðîäîëæåíèå íà âñþ ïëîñêîñòü ïàðàìåòðà p)
η(t)tν ÷∫ ∞
0
tνe−ptdt =1
pν+1
∫ ∞
0
zνe−zdz =Γ(ν + 1)
pν+1, (2.5)
ïðè ν > −1 è J p = 0, p > 0, (ïî ïðàâèëó àíàëèòè÷åñêîãî ïðîäîëæåíèÿ ìîæíî èçáàâèòüñÿ îòóñëîâèÿ íà p) Ïðè ν < −1 ôóíêöèÿ tν èìååò ðàçðûâ âòîðîãî ðîäà, ïðè ýòîì óñëîâèå ãåëüäåðîâîñòèçàìåíÿåòñÿ íà óñëîâèå èíòåãðèðóåìîñòè (| ∫∞
0ep0tf(t)dt| < ∞)
(êîíåö ïðèìåðîâ)
2.2 Îñíîâíûå ñâîéñòâà ïðåîáðàçîâàíèÿ Ëàïëàñà1. Ëèíåéíîñòü Ïóñòü f1(t)÷ f1(p),f2(t)÷ f2(p), òîãäà
c1f1(t) + c2f2(t)÷ c1f1(p) + c2f2(p) (2.6)
(îáëàñòü îïðåäåëåíèÿ ïðåîáðàçîâàíèÿ - ïåðåñå÷åíèå îáëàñòåé îïðåäåëåíèÿ äëÿ êàæäîé ôóíê-öèè) (Ïðèìåð cosωt = 1/2(eiωt + e−iωt) ÷ 1/2
(1
p−iω + 1p+iω
)= p
p2+ω2 - îáðàòèì âíèìàíèå íàäâà ïîëþñà â òî÷êàõ ±iω)
2. Òåîðåìà ïîäîáèÿ f(t)÷ f(p), âûïîëíèì çàìåíó ïåðåìåííîãî t → αt, α > 0, èíòåãðèðóÿ∫ ∞
0
f(αt)e−ptdt =1α
∫ ∞
0
f(s)e−p/αsds =1α
fp/α (2.7)
Äâîéñòâåííàÿ òåîðåìà f(βp)÷ 1β f(t/β)
30
3. Òåîðåìà ñìåùåíèÿ f(t)÷ f(p)e−αtf(t)÷ f(p + α), (2.8)
â ÷åì óáåæäàåìñÿ íåïîñðåäñòâåííûì èíòåãðèðîâàíèåì
4. Òåîðåìà çàïàçäûâàíèÿ (ðèñ. 2.3)
η(t− τ)f(t− τ)÷ e−pτ f(p) (2.9)
Çàäà÷à f(αt− β)η(αt− β)÷? f(αp + β)÷?
t
t
Ðèñ. 2.3:
5. Äèôôåðåíöèðîâàíèå îðèãèíàëà f(t)÷ f(p)∫ ∞
0
f ′(t)e−ptdt = f(t)e−pt∣∣∞0
+ p
∫ ∞
0
f(t)e−ptdt = pf − f(0) (2.10)
 îáùåì ñëó÷àåf (k)(t)÷ pkf(p)− pk−1f(0)− . . .− f (k−1)(0) (2.11)
(ýòî ñâîéñòâî ïîçâîëÿåò ðåøàòü çàäà÷è Êîøè)
6. Äèôôåðåíöèðîâàíèå èçîáðàæåíèÿ Äèôôåðåíöèðóÿ ïî ïàðàìåòðó èíòåãðàë Ëàïëàñà èìååì :
d
dp
∫ ∞
0
e−ptf(t)dt =∫ ∞
0
(−t)e−ptf(t)dt, (2.12)
òîãäàdkf
dpk÷ (−1)ktkf(t) (2.13)
Ïðèìåð Óðàâíåíèå Áåññåëÿ
(ty′)′ + ty = 0, èëèty′′ + y′ + ty = 0 (2.14)
Ïóñòü y = J0(t), y(0) = 1, y′(0) = 0 (íî òî÷êà t = 0 îñîáàÿ) y(t)÷ y(p), y′ ÷ py − 1, ty ÷−dyp ,
êðîìå ýòîãî y′′÷p2y−p, ty′′÷− ddp (p2y−p) = −2py−p2 dy
p +1 (çäåñü ìû ïîòåðÿëè èíôîðìàöèþî y'(0)) Îòñþäà óðàâíåíèå íà y :
(p2 + 1)dy
dp= −py, (2.15)
ðàçäåëÿÿ ïåðåìåííûå è èíòåãðèðóÿ èìååì
y =const√p2 + 1
. (2.16)
Îïðåäåëèì êîíñòàíòó : çàìåòèì, ÷òî ïðè p →∞ y → const+O( 1p ), çíà÷èò, y → const+O(t2)
ïðè t → 0, ïîýòîìó const = 1! êîíåö ïðèìåðà
31
7. Èçîáðàæåíèå èíòåãðàëà Äîêàæåì, ÷òî∫ ∞
0
f(τ)dτ ÷ f(p)p
. (2.17)
Âûïîëíèì îöåíêó | ∫∞0
f(τ)dτ‖ ≤ maxτ∈[0,t] |f(τ)|t, à ïîñêîëüêó f ïðèíàäëåæèò ê êëàññóèçîáðàæàåìûõ ïî Ëàïëàñó, çíà÷èò è èíòåãðàë îò f òîæå. Òîãäà èíòåãðèðóÿ ïî ÷àñòÿì
∫ ∞
0
e−pt
∫ t
0
f(τ)dτ = −1p
e−pt
∫ t
0
f(τ)dτ
∣∣∣∣ 0∞ +1p
∫ ∞
0
e−ptf(t)ft =f(p)
p, (2.18)
÷òî è òðåáîâàëîñü.
8. Èíòåãðèðîâàíèå èçîáðàæåíèÿ Ïðåäïîëîæèì, ÷òî f(t)/t ïðèíàäëåæèò ê êëàññó îðèãèíàëîâ,è ïóñòü
J(p) =∫ ∞
0
f(t)t
e−ptdp (2.19)
Äèôôåðåíöèðóÿ ïî p èìååì
dJ
dp= −
∫ ∞
0
f(t)e−ptdt = −f(p), (2.20)
â êà÷åñòâå í.ó. äëÿ ýòîãî óðàâíåíèÿ ïðèìåì J(p) = 0 ïðè arg p ≤ π/2 − α, çíà÷èò, J =∫∞p
f(q)dq, òîãäà ∫ ∞
0
f(q)dq ÷ f(t)t
(2.21)
2.3 Òåîðåìà î ñâåðòêå (Ý. Áîðåëÿ)Îïðåäåëåíèå 1 Ñâåðòêà ôóíêöèé f(t) ∗ g(t) =
∫∞−∞ f(τ)g(t− τ)dτ) (=
∫∞−∞ f(t− τ)g(τ)dτ)
Ïîñêîëüêó ìû ðàññìàòðèâàåì ôóíêöèè, òîæäåñòâåííî ðàâíûå íóëè ïðè àðãóìåíòå ìåíüøåìíóëÿ, òî f(t) ∗ g(t) =
∫ t
0f(τ)g(t− τ)dτ .
Òåîðåìà 2.3.1f(t) ∗ g(t)÷ f(p)g(p) (2.22)
Äîê-âî Ïðîâåðèì, ÷òî ñâåðòêà ïîïàäàåò â êëàññ îðèãèíàëîâ. Â ñàìîì äåëå,
|f(t)| < Meα1t, |g(t)| < Neα2t,
òîãäà | ∫ t
0f(τ)g(t − τ)dτ | ≤ MN
∫ t
0eα1τeα2(t−τ)dτ ≤ MNteαt, ãäå α = max(α1, α2).  ýòîì ñëó÷àå
âû÷èñëÿåì èçîáðàæåíèå∫ t
0
e−ptdt
∫ t
0
f(τ)g(t− τ)dτ =∫ ∞
0
f(τ)dτ
∫ ∞
τ
e−ptg(t− τ)dt =
∫ ∞
0
f(τ)e−pτdτ
∫ ∞
0
e−psg(s)ds = f(p)g(p),
÷òî è òðåáîâàëîñü.Ïðèìåíèì òåîðåìó î ñâåðòêå ê ðåøåíèþ óðàâíåíèÿ Âîëüòåððà II-îãî ðîäà
y(t) +∫ t
0
y(τ)K(t, τ)dτ = f(τ). (2.23)
Ïóñòü K(t, τ) = K(t− τ), òîãäà óðàâíåíèå II-îãî ðîäà
y(t) +∫ t
0
y(τ)K(t− τ)τ = f(τ), (2.24)
32
è óðàâíåíèå ïåðâîãî ðîäà ∫ t
0
y(τ)K(t− τ)τ = f(τ), (2.25)
Ïåðåéäåì ê ïðåîáðàçîâàíèÿì Ëàïëàñà
y + yK = f , II ðîä,
yK = f , I ðîä,
òîãäày =
f
1 + K, II ðîä, y =
f
K, I ðîä.
Îáðàùåíèå ýòèõ ôîðìóë áóäåò äàíî ïîçæå. Íåîáõîäèìûé ïðèçíàê íàëè÷èÿ îðèãèíàëà : òàê êàêf(p) → 0, p →∞ è K(p) → 0, p →∞ ïðè Rp > 0, âèäíî, ÷òî äëÿ óðàâíåíèÿ âòîðîãî ðîäà y → 0 ïðèp → ∞, Rp > 0. Äëÿ óðàâíåíèÿ ïåðâîãî ðîäà íè÷åãî íå èçâåñòíî, ïðè÷åì ÷åì "ëó÷øå"ñâîéñòâàÿäðà, òåì õóæå ñâîéñòâà ðåøåíèÿ.
2.4 Îáîáùåííàÿ òåîðåìà î ñâåðòêåÏóñòü f(t)÷ f(p), G(p), q(p) ïðîèçâîëüíûå àíàëèòè÷åñêèå ôóíêöèè ïðè Rp > 0. Ïóñòü, êðîìå òîãî,
G(p)e−τq(p) ÷ g(t; τ)
Òåîðåìà 2.4.1 (Îáîáùåííàÿ òåîðåìà î ñâåðòêå, À.Ì. Ýôðîñ, 1935)
f(q(p))G(p)÷∫ ∞
0
f(τ)g(t; τ)dτ (2.26)
Äîê-âî Ðàññìîòðèì èíòåãðàë∫ ∞
0
e−ptdt
∫ ∞
0
f(τ)g(t; τ)dτ =∫ ∞
0
f(τ)dτ
∫e−ptg(t; τ)dt =
∫ ∞
0
f(τ)G(p)e−τq(p)dτ = G(p)f(q(p)).
(ñõîäèìîñòü èíòåãðàëîâ àíàëîãè÷åíà ïðåäûäóùåìó ïàðàãðàôó), ÷òî è òðåáîâàëîñü äîêàçàòü.Ïðèìåðû1. q(p) = p, g(p)÷ g(t). ñëåäîâàòåëüíî,
f(p)g(p)÷∫ t
0
f(τ)g(t− τ)dτ
(òåîðåìà Áîðåëÿ)
2. q(p) = p, G(p) = 1/√
p, íèæå áóäåò ïîêàçàíî, ÷òî
e−τ√
p
√p
÷ 1√πt
e−τ24t
Òîãäàf(√
p)√p
÷ 1√πt
∫ ∞
)
f(τ)e−τ24t
3. Íàéòè îðèãèíàë1
p + h√
p?
f(√
p)√p
=1
p + h√
p⇒ f(p) =
1p + h
⇒ f(t) = e−ht,
òîãäà1
p + h√
p÷ 1√
πt
∫ ∞
0
e−hte−τ2/4tdτ = eh2t Erf (h√
t)
33
2.5 Ôîðìóëà Ðèìàíà-Ìåëëèíà. Îáðàùåíèå ïðåîáðàçîâàíèÿËàïëàñà
Òåîðåìà 2.5.1 Ïóñòü f(t)÷ f(p), òîãäà
f(t) =1
2πiv.p.
σ+i∞∫
σ−i∞
f(p)eptdp, (2.27)
ãäå |f(t)| < Meat, σ > a
Äîê-âî Ðàññìîòðèì ôóíêöèþ φ(t) = f(t)e−σt, ïîêàçàòåëü ðîñòà φ(t) îòðèöàòåëüíûé.  òàêîìñëó÷àå φ ïðåäñòàâëÿåòñÿ èíòåãðàëîì Ôóðüå. Ïðåîáðàçîâàíèå Ôóðüå ôóíêöèè φ
φ(ν) =∫ ∞
−∞eiνtφ(t)dt,
îáðàùåíèåφ(t) =
12π
∫ ∞
−∞e−iνtφ(ν)dν =
12π
∫ ∞
−∞e−iνtdν
∫ ∞
0
eiνηφ(η)dη =
12π
∫ ∞
−∞dν
∫ ∞
0
eiν(η−t)φ(eta)dη =12π
∫ ∞
−∞dνeiνt
∫ ∞
0
e−ση+iνηf(η)dη =12π
∫ ∞
−∞f(p)dν =
(â ïîñëåäíåì ðàâåíñòâå ìû ïîëîæèëè p = σ − iν)
−e−σt
2πiv.p.
σ−i∞∫
σ+i∞
f(p)eptdp
(çäåñü ìû åùå ðàç çàìåíèëè ïåðåìåííóþ èíòåãðèðîâàíèÿ). Ñîêðàùàÿ íà ýêñïîíåíòó è ìåíÿÿ çíàêèìååì ôîðìóëó Ðèìàíà-Ìåëëèíà. ×òî è òðåáîâàëîñü äîêàçàòü. (Çàìå÷àíèå : Ïðåîáðàçîâàíèå Ëà-ïëàñà, êàê ìû âèäèì, òåñíî ñâÿçàíî ñ ïðåîáðàçîâàíèåì Ôóðüå.)
a
a
s
s
s>
t
Ðèñ. 2.4:
Òåîðåìà 2.5.2 Ïóñòü ôóíêöèÿ f(p) ðåãóëÿðíà â ïðàâîé ïîëóïëîñêîñòè ïàðàìåòðà p. Ïóñòü,êðîìå òîãî, f(p) → 0, p →∞ ïðè Rp > 0, è èìååò ìåñòî àáñîëþòíàÿ ñõîäèìîñòü èíòåãðàëà
σ+i∞∫
σ−i∞
f(p)eptdp.
Òîãäà ôóíêöèÿ
f(t) =1
2πi
σ+i∞∫
σ−i∞
f(p)eptdp
ÿâëÿåòñÿ îðèíãèíàëîì ôóíêöèè f(p).Äîê-âî Ñì. Òèõîíîâ, Ñâåøíèêîâ; Ëàâðåíòüåâ è Øàáàò.
34
2.6 Òåîðåìû ðàçëîæåíèÿÒåîðåìà 2.6.1 Ïóñòü f(p) ïðàâèëüíàÿ íà áåñêîíå÷íîñòè ôóíêöèÿ, ò. å.
f(p) =∞∑
k=1
Ck
pk, |p| > R, (2.28)
òîãäà îðèãèíàë
f(t) =
0, t < 0∑∞
k=1Cktk−1
(k−1)! , t > 0(2.29)
Äîê-âî Ïîñêîëüêó tν ÷ Γ(ν+1)pν+1 èìååì tk−1 ÷ (k−1)!
pk . Îñòàåòñÿ äîêàçàòü ñõîäèìîñòü ðÿäà (2.29), äëÿýòîãî ââåäåì q = 1/p, òîãäà â ñîîòâåòñòâóþùåì êðóãå ñ öåíòðîì â íà÷àëå êîîðäèíàò ïëîñêîñòè q(rq = 1/Rp) èìååì ðàçëîæåíèå
f(p) = Φ(q) =∞∑
k=1
Ckqk,
ïðè÷åì èç ôîðìóëû Êîøè äëÿ ïðîèçâîäíûõ ôóíêöèè êîìïëåêñíîãî ïåðåìåííîãî ñëåäóåò îöåíêà|Φ(n)(q)| ≤ Mn!Rn èëè |Ck| ≤ MRk. Òîãäà
∣∣∣∣Cktk−1
(k − 1)!
∣∣∣∣ ≤MRk|t|k−1
(k − 1)!= MReR|t|,
òàêèì îáðàçîì ñõîäèìîñòü äîêàçàíà. ×òî è òðåáîâàëîñü.Ïðèìåð äëÿ ôóíêöèè Áåññåëÿ
J0(t) =∞∑
k=0
(−1)k(t/2)2k
k!⇒ J0(p) =
1p
+C−3
p3+ . . . (2.30)
Òåîðåìà 2.6.2 Ïóñòü f(p) ìåðîìîðôíàÿ ôóíêöèÿ è ïðè Rp > c f(p) - àíàëèòè÷åñêàÿ ôóíêöèÿ.Òîãäà
f(t) =∑
(pk)
res(pk)
f(p)ept (2.31)
(òî÷êè pk - ïîëþñû ôóíêöèè f).
Äîê-âî Èçâåñòíî, ÷òî ∫
LA
f eptdp +∫
CR
f eptdp = 2πi∑
(pk)
res(pk)
f(p)ept.
 ñèëó ëåììû Æîðäàíà è óñëîâèé òåîðåìû èíòåãðàë∫CR
→ 0 ïðè R → ∞, òîãäà âûïîëíèâïðåäåëüíûé ïåðåõîä A →∞ è R →∞, ïîëó÷àåì òðåáóåìîå (ñì. ðèñ. 2.5).
s
s
s
s
t
c
+iA
-iA
pk
LA
CR
Ðèñ. 2.5:
35
2.7 Ïðèìåðû âû÷èñëåíèÿ èíòåãðàëîâ îò íåîäíîçíà÷íûõ îá-ðàçîâ
• f(p) = 1√p2+1
, íà ðèñóíêå 2.6 óêàçàíî êàê âûáèðàåòñÿ êîíòóð äëÿ íàõîæäåíèÿ îðèãèíàëà.Äóãè CR âûáèðàþòñÿ äëÿ èñïîëüçîâàíèÿ Ëåììû Æîðäàíà. Èíòåãðàëû ïî áåðåãàì ðàçðåçàâçàèìíî óíè÷òîæàòüñÿ.
s
t
i
-i
p
Ðèñ. 2.6:
• Ïðîâåðèì ôîðìóëó îáðàùåíèÿ äëÿ
tν ÷ Γ(ν + 1)pν+1
.
Ïóñòü −π < arg p < π, íåîáõîäèìî âû÷èñëèòü èíòåãðàë
12πi
∫
L
Γ(ν + 1)pν+1
eptdp,
çàìåòèì äëÿ ýòîãî, ÷òî ñóììà ñëåäóþùèõ èíòåãðàëîâ ðàâíà íóëþ (â îáëàñòè èíòåãðèðîâàíèÿïîäûíòåãðàëüíàÿ ôóíêöèÿ ðåãóëÿðíà), ñì. ðèñ. 2.7 :
∫LA
+∫CR
+∫
I+
∫Cρ
+∫
II= 0, êðîìå òîãî
èìååì îöåíêó : ∣∣∣∣∣∫
Cρ
ept
pν+1dp
∣∣∣∣∣ ≤∫ −π
π
eρ cos φρ
ρν+1dφ ≤ const
ρν.
Ïîòðåáóåì, ÷òîáû −1 < ν < 0, ò.å.∫Cρ→ 0 ïðè ρ → 0. Èíòåãðèðóÿ ïî áåðåãàì ðàçðåçà èìååì:
(âåðõíèé áåðåã) p = ρeiπ,∫
I
= Γ(ν + 1)e−iπ(ν+1)
∫ ∞
0
e−ρtρ−ν−1dρ,
(íèæíèé áåðåã) ∫
II
= −Γ(ν + 1)eiπ(ν+1)
∫ ∞
0
e−ρtρ−ν−1dρ,
èòîãî, ∫
L= −Γ(ν + 1)
∫ ∞
0
e−ρtρ−ν−1[eiπ(ν+1) − e−iπ(ν+1)
]dρ.
Òîãäà Γ(ν+1)pν+1 ÷ Γ(ν+1)
π sinπ(ν + 1)tνΓ(−ν) = tν !
• Äîêàæåì, ÷òîe−τ
√p
√p
÷ 1√πt
e−τ24t (2.32)
36
s
tp
Cr
CR
arg(p)=p
arg(p)=-p
LA
Ðèñ. 2.7:
(Ðèñóíîê 2.7)
∫
I
= −∫ 0
∞dρ
e−iτ√
ρe−ρt
i√
ρ
è ∫
II
=∫ 0
∞dρ
eiτ√
ρe−ρt
−i√
ρ,
òîãäà
12πi
∫
L= − 1
2πi(∫
I
+∫
II
) =12π
∫ ∞
0
e−ρt
√ρ
[eiτ
√ρ + e−iτ
√ρ]dρ =
1π
∫ ∞
0
e−ρt cos τ√
ρ√ρ
dρ.
Äëÿ âû÷èñëåíèÿ ýòîãî èíòåãðàëà ïîëîæèì √ρ = u, ââåäåì ôóêíöèþ
f(t; τ) =2π
∫ ∞
0
e−u2t cosuτdu, (2.33)
ïîñêîëüêó∂f
∂τ= − τ
2tf
(â ÷åì óáåæäàåìñÿ ïðîäèôôåðåíöèðîâàâ f ñ îäíîé ñòîðîíû è âçÿâ èíòåãðàë (2.33) ïî ÷àñòÿì),òî f = conste−τ2/4t, ïîäñòàâëÿÿ τ = 0 íàõîäèì const = 1/πt, òàêèì îáðàçîì ôîðìóëà (2.32)äîêàçàíà.
37
Ãëàâà 3
Èíòåãðàëüíûå óðàâíåíèÿ
3.1 Êëàññèôèêàöèÿ ëèíåéíûõ èíòåãðàëüíûõ óðàâíåíèéÈç ïðîøëîãî ñåìåñòðà
3.2 Àíàëîãèÿ ìåæäó ëèíåéíûì èòåãðàëüíûì óðàâíåíèåì Ôðåä-ãîëüìà 2-ãî ðîäà è ñèñòåìîé ëèíåíåéíûõ àëãåáðàè÷åñêèõóðàâíåíèé. Ôîðìóëèðîâêà òåîðåì Ôðåäãîëüìà
Ðàññìîòðèì îäíîìåðíîå óðàâíåíèå Ôðåäãîëüìà 2-ãî ðîäà
φ(x) = λ
∫ b
a
K(x, y)φ(y)dy + f(x) (3.1)
K(x, y) è f(x) ïðåäïîëàãàþòñÿ íåïðåðûâíûìèÂûïîëíèì ðàçáèåíèå îòðåçêà (a, b) íà N îòðåçêîâ: xi = a + ih, (i = 0, 1..N), ïðè÷åì x0 = a, xn = b.
Òîãäà∫ b
aK(x, y)φ(y)dy ≈
N∑i=1
K(x, yi)φ(yi)h. Ââîäÿ îáîçíà÷åíèÿ φ(xj) = zj ; f(xj) = Fj ; K(xj , yi)h =
Aji, è ïîñ÷èòàâ (3.1) â òî÷êå xj , ïîëó÷èì
zj = λ
N∑
i=1
Ajizi + Fj (3.2)
Òî åñòü èñõîäíîìó èíòåãðàëüíîìó óðàâíåíèþ ìîæíî ñîïîñòàâèòü ñèñòåìó ëèíåéíûõ àëãåáðàè÷å-ñêèõ óðàâíåíèé. Âñïîìíèì òåîðåìû, èçâåñòíûå èç êóðñà ëèíåéíîé àëãåáðû
Òåîðåìà 3.2.1 (Àëüòåðíàòèâà) Ëèáî ñèñòåìà (3.2) îäíîçíà÷íî ðàçðåøèìà ïðè ∀Fj, ëèáî ñîîò-âåòñòâóþùàÿ îäíîðîäíàÿ ñèñòåìà èìååò íåòðèâèàëüíîå ðåøåíèå.
Òåîðåìà 3.2.2 Ðàññìîòðèì òðàíñïîíèðîâàííóþ ñèñòåìó
zj = λ
N∑
i=1
Aijzi + Gj (3.3)
Àëüòåðíàòèâà âûïîëíåíà ïðè òåõ æå λÒî æå âåðíî è äëÿ ñîïðÿæåííîé ñèñòåìû
zj = λ
N∑
i=1
Aijzi + G′j (3.4)
Òåîðåìà 3.2.3 Ðàíãè õàðàêòåðèñòè÷åñêèõ ÷èñåë ñèñòåì (3.2) è (3.3) ñîâïàäàþò.
38
Òåîðåìà 3.2.4 Ïóñòü λ0 - õàðàêòåðèñòè÷åñêîå ÷èñëî (3.2), è ïóñòü V(p)j - ñîáñòâåííûå âåêòîðà
ñèñòåìû (3.3) ñ λ = λ0.
Òîãäà ñèñòåìà (3.2) c ïðàâîé ÷àñòüþ Fj ðàçðåøèìà, åñëèN∑
j=1
FjV(p)j = 0, p = 1..rankλ0.
(×åðåç ñîáñòâåííûå âåêòîðû ñèñòåìû (3.5) V ′j , óñëîâèå ðàçðåøèìîñòè çàïèñûâàåòñÿ â âèäå
N∑j=1
FjV ′(p)
j = 0)
Îêàçûâàåòñÿ, âñå ýòè òåîðåìû ïåðåíîñÿòñÿ íà óðàâíåíèÿ Ôðåäãîëüìà. Ïåðåôîðìóëèðóåì èõ äëÿèíòåãðàëüíûõ óðàâíåíèé:
Òåîðåìà 3.2.5 (Àëüòåðíàòèâà Ôðåäãîëüìà) Ëèáî óðàâíåíèå
φ(x) = λ
∫
Ω
K(x,y)φ(y)dy + f(x) (3.5)
îäíîçíà÷íî ðàçðåøèìî ïðè ∀f(x), ëèáî ñîîòâåòñòâóþùåå îäíîðîäíîå óðàâíåíèå èìååò íåòðèâè-àëüíîå ðåøåíèå. Äðóãèìè ñëîâàìè, λ ÿâëÿåòñÿ õàðàêòåðèñòè÷åñêèì ÷èñëîì.
Òåîðåìà 3.2.6 Äëÿ ñîþçíîãî óðàâíåíèÿ
φ(x) = λ
∫
Ω
K(x,y)φ(y)dy + g(x) (3.6)
è ñîïðÿæåííîãî óðàâíåíèÿφ(x) = λ
∫
Ω
K(x,y)φ(y)dy + g′(x) (3.7)
àëüòåðíàòèâà âûïîëíåíà ïðè òåõ æå λ.
Òåîðåìà 3.2.7 Ðàíãè õàðàêòåðèñòè÷åñêèõ ÷èñåë óðàâíåíèé (3.5),(3.6) è(3.7) ñîâïàäàþò.
Òåîðåìà 3.2.8 Ïóñòü λ0 - õàðàêòåðèñòè÷åñêîå ÷èñëî (3.5). Óñëîâèåì ðàçðåøèìîñòè ÿâëÿåòñÿ∫
Ω
f(x)ψ(p)(x)dx = 0,
äëÿ âñÿêîé ψ(p)(x) - ñîáñòâåííîé ôóíêöèè (3.6).Ïðè ïîìîùè ñîáñòâåííûõ ôóíêöèé (3.6) χ(p)(x), óñëîâèå ðàçðåøèìîñòè çàïèñûâàåòñÿ â âèäå
∫
Ω
f(x)χ(p)(x)dx = 0,
Çàìå÷àíèåÌû áóäåì ñòðîèòü òåîðèþ äëÿ óðàâíåíèé ñ íåïðåðûâíûì ÿäðîì. Òó æå òåîðèþ ìîæíî ðàñïðî-ñòðàíèòü íà ÿäðà ñî ñëàáîé îñîáåííîñòüþ K(x,y) = L(x,y)
|x−y|q , ãäå L(x,y) - íåïðåðûâíàÿ ôóíêöèÿ, àq < dimΩ
2 .
3.3 Óðàâíåíèÿ Ôðåäãîëüìà 2-ãî ðîäà ñ ìàëûìè ÿäðàìèÐàññìîòðèì óðàâíåíèå
φ(x) = λ
∫
Ω
K(x,y)φ(y)dy + f(x) (3.8)
Ïóñòü |Ω| - ìåðà îáëàñòè Ω, K0 = supΩ×Ω
|K(x,y)|. Ïåðåïèøåì (3.8) â îïåðàòîðíîì âèäå: φ = λKφ+f ,
ãäå K - èíòåãðàëüíûé îïåðàòîð Kφ =∫Ω
K(x,y)φ(y)dy. Ïóñòü v(x) = Kφ. Ðàññìîòðèì φ(x) ∈ C(Ω),Φ = sup
Ω|φ(x)|, òîãäà
|v(x + h)− v(x)| = |∫
Ω
[K(x + h,y)−K(x,y)] φ(y)dy| 6 ε|Ω|Φ
39
(Èç íåïðåðûâíîñòè K(x,y), ∀ε > 0 ∃h : |K(x + h,y)−K(x,y)| < ε)Òàêèì îáðàçîì, îïåðàòîð K ÿâëÿåòñÿ íåïðåðûâíûì (â òîì ñìûñëå, ÷òî îí ïåðåâîäèò íåïðåðûâíûåôóíêöèè â íåïðåðûâíûå).Ðàññìîòðèì äàëåå îïåðàòîð Lφ =
∫Ω
L(x,y)φ(y)dy, òîãäà
Lv = LKφ =∫
Ω
L(x,y)v(y)dy =∫
Ω
dyL(x,y)∫
Ω
K(y, t)φ(t)dt =∫
Ω
φ(y)dy∫
Ω
L(x, t)K(t,y)dt
ñëåäîâàòåëüíî îïåðàòîðó LK ñîîòâåòñòâóåò ÿäðî LK =∫Ω
L(x, t)K(t,y)dt. ßäðî LK íàçûâàåòñÿêîìïîçèöèåé ÿäåð. Íåòðóäíî âèäåòü, ÷òî â îáùåì ñëó÷àå L K 6= K L.Ïðèìåíèì äàëåå ê íàøåìó óðàâíåíèþ ìåòîä ïðîñòîé èòåðàöèè:
φ0 = fφ1 = f + λKf............φn = f + λKφn−1 = f + λKf + ... + λnKnf
Î÷åâèäíî, ÷òî åñëè ðÿä φn ñõîäèòñÿ ê íåêîòîðîé ôóíêöèè φ, òî φ = f + λKφ. Ïóñòü |f(x)| 6 f0,òîãäà |λKf | = |λ ∫
ΩK(x,y)f(y)dy| 6 |λ|K0f0|Ω|. Äëÿ n-ãî ñëàãàåìîãî, |λnKnf | 6 |λ|nKn
0 |Ω|nf0.Òàêèì îáðàçîì, ðÿä äëÿ φ ìàæîðèðóåòñÿ ãåîìåòðè÷åñêîé ïðîãðåññèåé, ñõîäÿùåéñÿ ïðè |λ|K0|Ω| <1. Òî åñòü äîêàçàíà ñëåäóþùàÿ òåîðåìà
Òåîðåìà 3.3.1 Ó óðàâíåíèÿ (3.8) â êðóãå |λ| < 1K0|Ω| íåò õàðàêòåðèñòè÷åñêèõ ÷èñåë
Ðåøåíèå ìîæíî çàïèñàòü â âèäå φ = f + λRf , ãäå R =∞∑
i=1
λi−1Ki - ðåçîëüâåíòà óðàâíåíèÿ (3.8).
3.4 Óðàâíåíèÿ Ôðåäãîëüìà ñ âûðîæäåííûìè ÿäðàìèÎïðåäåëèì âûðîæäåííîå ÿäðî, êàê ÿäðî,ïðåäñòàâèìîå â âèäå A(x,y) =
m∑i=1
αi(x)βi(y). Ôóíêöèèαi(x) è βi(y) ïðåäïîëàãàþòñÿ ëèíåéíî íåçàâèñèìûìè (èíà÷å ìîæíî óìåíüøèòü êîëè÷åñòâî ñëàãà-åìûõ). Ðàññìîòðèì óðàâíåíèå ñ âûðîæäåííûì ÿäðîì:
φ(x) = λ
∫
Ω
A(x,y)φ(y)dy = λ
m∑
i=1
αi(x)∫
Ω
βi(y)dy + f(x) (3.9)
Îáîçíà÷èì xi =∫Ω
βi(y)φ(y)dy, òîãäà φ(x) = λxiαi(x)+f(x). Ïîäñòàâëÿÿ ýòî âûðàæåíèå äëÿ φ(x)â îïðåäåëåíèå xi, ïîëó÷èì
xi = λ
m∑
i=1
Aijxj + fi (3.10)
ãäå îáîçíà÷åíî fi =∫Ω
βi(y)f(y)dy, Aij =∫Ω
βi(y)α(y)dy. Î÷åâèäíî, ÷òî (3.9) è (3.10) - ýêâèâà-ëåíòíû.Èç ëèíåéíîé àëãåáðû èçâåñòíî, ÷òî
xi =
m∑j=1
A′ijfj
D(λ),
ãäå A′ij - àëãåáðàè÷åñêîå äîïîëíåíèå ýëåìåíòîâ ìàòðèöû E − λA, D(λ) = |E − λA|. Ñëåäîâàòåëüíî
φ(x) = f(x) +λ
m∑i=1
m∑j=1
A′ij(λ)αi(x)∫Ω
f(y)βi(y)dy
D(λ)
Ñ äðóãîé ñòîðîíû, φ(x) = f(x) + λ∫Ω
R(x,y, λ)f(y)dy, ñëåäîâàòåëüíî
R(x,y, λ) =1
D(λ)
m∑
i=1
m∑
j=1
A′ij(λ)αi(x)βi(y) (3.11)
40
Òàêèì îáðàçîì, ðåçîëüâåíòà ÿâëÿåòñÿ ìåðîìîðôíîé ôóíêöèåé âî âñåé ïëîñêîñòè λ. Î÷åâèäíî, ÷òîïîëþñà ðåçîëüâåíòû ñîîòâåòñòâóþò õàðàêòåðèñòè÷åñêèì ÷èñëàì. áîëåå îáùåì ñëó÷àå, αi = αi(x, λ); βi = βi(y, λ). Ïóñòü â ïëîñêîñòè λ ∈ C βi, αi - öåëûå ôóíêöèè.Òîãäà âñå ôîðìóëû, ïðèâåäåííûå âûøå áóäóò âåðíû. Ôóíêöèÿ D(λ) áóäåò öåëîé, ñëåäîâàòåëüíîíå áóäåò èìåòü òî÷åê íàêîïëåíèÿ íóëåé âî âñÿêîé êîíå÷íîé ÷àñòè C . Ðåçîëüâåíòà îñòàíåòñÿ ìå-ðîìîðôíîé è â ýòîì ñëó÷àå, ñëåäîâàòåëüíî äëÿ ëþáûõ λ, çà èñêëþ÷åíèåì ñ÷åòíîãî ÷èñëà òî÷åê,èñõîäíîå óðàâíåíèå áóäåò èìåòü ðåøåíèèå ïðè ëþáîé f(x) . Áîëåå òîãî, ìîæíî ïîêàçàòü, ÷òî è âýòîì ñëó÷àå ïîëþñà ðåçîëüâåíòû ÿâëÿþòñÿ õàðàêòåðèñòè÷åñêèìè ÷èñëàìè.Âñå òåîðåìû Ôðåäãîëüìà â ñëó÷àå âûðîæäåííûõ ÿäåð âûïîëíåíû àâòîìàòè÷åñêè, êàê ïðÿìûåñëåäñòâèÿ ñîîòâåòñòâóþùèõ òåîðåì ëèíåéíîé àëãåáðû.
3.5 Èíòåãðàëüíûå óðàâíåíèÿ Ôðåäãîëüìà 2-ãî ðîäà, áëèçêèåê âûðîæäåííûì
Ðàññìîòðèì óðàâíåíèå φ(x) = λ∫Ω
K(x,y)φ(y)dy + f(x). Ïóñòü K(x,y) = A(x,y) + K1(x,y), ãäåA - âûðîæäåííîå ÿäðî, à K1 - ìàëîå, òî åñòü |λ|max
x,y∈Ω|K1(x,y)||Ω| < 1.
Ëåììà 3.5.1 Ïóñòü A(x,y) - âûðîæäåííîå ÿäðî, K(x,y) - íåïðåðûâíîå ÿäðî.Òîãäà K A è A K - âûðîæäåííûå ÿäðà.
ÄîêàçàòåëüñòâîA K =
∫Ω
A(x, t)K(t,y)dt =m∑
i=1
αi(x)∫Ω
βi(t)K(t,y)dt =m∑
i=1
αi(x)κ(y)
K A =m∑
i=1
βi(y)∫Ω
αi(t)K(x, t)dt =m∑
i=1
βi(y)κ′(x)
Ëåììà 3.5.2 (K1 K2)Ò = KÒ2 KÒ
1 , çíà÷îê ()Ò îáîçíà÷àåò òðàíñïîíèðîâàíèå.
Äîêàçàòåëüñòâî ßñíî. Èòàê, ðàññìîòðèì óðàâíåíèå ñ ÿäðîì, áëèçêèì ê âûðîæäåííîìó:
φ = λAφ + λK1φ + f (3.12)
Ïåðåïèøåì åãî â âèäå φ = λK1φ+η, ãäå η = φ+λAφ. Òîãäà ðåøåíèå çàïèøåòñÿ â âèäå φ = η+λR1η,ãäå R1 - ðåçîëüâåíòà K1. Ïîäñòàâëÿÿ â âûðàæåíèå äëÿ η, ïîëó÷èì
η = λA(η + λR1η) + f = f + (λA + λ2AR1)η (3.13)
Ðåçîëüâåíòà R1 îïðåäåëåíà è ÿâëÿåòñÿ öåëîé â êðóãå |λ| < 1|Ω||K1| , ñëåäîâàòåëüíî ÿäðî (3.13)
öåëàÿ ôóíêöèÿ â ýòîé îáëàñòè. Òàêèì îáðàçîì, â êðóãå ìàëîñòè K1, ñóùåñòâóåò ðåçîëüâåíòà -ìåðîìîðôíàÿ ôóíêöèÿ, ñëåäîâàòåëüíî âûïîëíåíà àëüòåðíàòèâà Ôðåäãîëüìà.Ðàññìîòðèì äàëåå ñîþçíîå óðàâíåíèå
ψ = λAÒψ + λKÒ1 ψ + g (3.14)
òîãäà ψ = g + λAÒψ + λR1(g + λAÒψ), èëè
ψ = g + λR1g + (λAÒ + λ2RÒ1 AÒ)ψ (3.15)
Î÷åâèäíî, óðàâíåíèÿ (3.13) è (3.15) - ñîþçíû. Òàê êàê (3.13) ýêâèâàëåíòíî (3.12), èõ óñëîâèÿ ðàçðå-øèìîñòè ñîâïàäàþò. Òàêèì îáðàçîì, óñëîâèå ðàçðåøèìîñòè (3.12) ïðèíèìàåò âèä
∫Ω
f(x)ψ(x)dx =0, ãäå ψ - ñîáñòâåííàÿ ôóíêöèÿ (3.13).Èòàê â ñëó÷àå ÿäåð, áëèçêèõ ê âûðîæäåííûì, â êðóãå ìàëîñòè K1 âûïîëíåíû âñå òåîðåìû Ôðåä-ãîëüìà.
3.6 Èíòåãðàëüíûå óðàâíåíèÿ Ôðåäãîëüìà ñ íåïðåðûâíûìèÿäðàìè
Ðàññìîòðèì óðàâíåíèå φ(x) = λ∫Ω
K(x,y)φ(y)dy+ f(x), ãäå K(x,y) ∈ C(Ω×Ω). Ñîãëàñíî òåîðåìåÂåéåðøòðàññà, ∀ε > 0∃PN (x,y) : K(x,y) = K1(x,y)+PN (x,y), ãäå |K1(x,y)| < ε, x, y ∈ Ω. Òàê êàê
41
PN (x,y) - âûðîæäåííîå ÿäðî, â êðóãå |λ| < 1|Ω|ε ñóùåñòâóåò ðåçîëüâåíòà - ìåðîìîðôíàÿ ôóíêöèÿ, è
âûïîëíåíà âñå òåîðåìû Ôðåäãîëüìà. Âûïîëíÿÿ ïðåäåëüíûé ïåðåõîä ε →∞, èìååì ìåðîìîðôíóþðåçîëüâåíòó âî âñåì ïðîñòðàíñòâå. Ìîæíî ïîêàçàòü, ÷òî òî÷êà íàêîïëåíèÿ ïîëþñîâ ðåçîëüâåíòû- áåñêîíå÷íîñòü, òî åñòü èìååòñÿ ñ÷åòíîå êîëè÷åñòâî õàðàêòåðèñòè÷åñêèõ ÷èñåëÄîïîëíåíèåÐåçîëüâåíòó ìîæíî ïðåäñòàâèòü â âèäå R(x,y, λ) = D(x,y,λ)
D(λ) , ãäå D èD - öåëûå ôóíêöèè λ. ÔóíêöèÿD(λ) íàçûâàåòñÿ çíàìåíàòåëåì Ôðåäãîëüìà. Ñïðàâåäëèâû óòâåðæäåíèÿ:1) Ëþáîå õàðàêòåðèñòè÷åñêîå ÷èñëî ÿâëÿåòñÿ íóëåì D(λ).2) Ëþáîé íóëü D(λ) ÿâëÿåòñÿ õàðàêòåðèñòè÷åñêèì ÷èñëîì.Ïîêàæåì âòîðîå èç ýòèõ óòâåðæäåíèé. Ëþáîé íóëü D(λ) ÿâëÿåòñÿ ïîëþñîì ðåçîëüâåíòû. Äàëåå,èç ñîîòíîøåíèé
φ = f + λKφφ = f + λRf
ïîëó÷èì óðàâíåíèå íà ðåçîëüâåíòó: R = K + λKR.Ïóñòü λ0 - ïîëþñ ðåçîëüâåíòû, òîãäà R(x,y, λ) = a−r(x,y)
(λ−λ0)r + a−r+1(x,y)(λ−λ0)r−1 + ... +
m∑i=1
(λ − λ0)iai(x,y).Äîìíîæàÿ óðàâíåíèå äëÿ ðåçîëüâåíòû íà (λ − λ0)r, è ïîëàãàÿ λ = λ0, ïîëó÷èì a−r = λ0Ka−r.Çíà÷èò, λ0 ÿâëÿåòñÿ õàðàêòåðèñòè÷åñêèì ÷èñëîì, à a−r - ñîáñòâåííîé ôóíêöèåé.
3.7 Èíòåãðàëüíîå óðàâíåíèå Ôðåäãîëüìà ñ ýðìèòîâûì ÿäðîìÝðìèòîâûì íàçûâàåòñÿ ÿäðî, óäîâëåòâîðÿþùåå ñîîòíîøåíèþ K(x,y) = K(y,x) ( âåùåñòâåííîìñëó÷àå K(x,y) ∈ R, K(x,y) = K(y,x) - ñèììåòðè÷íîå ÿäðî).Îïðåäåëèì ñêàëÿðíîå ïðîèçâåäåíèå êàê (u, v) =
∫Ω
u(x)v(x)dx. Çàìåòèì, ÷òî (v, u) =∫Ω
v(x)u(x)dx =(u, v).Òîãäà äëÿ ýðìèòîâîãî îïåðàòîðà âûïîëíåíî òîæäåñòâî Ëàãðàíæà: (φ,Kψ) = (Kφ,ψ). Ïðîâåðèì:
(φ,Kψ) =∫
Ω
φ(x)∫
Ω
K(x,y)ψ(y)dydx =∫
Ω
φ(x)∫
Ω
K(y,x)ψ(y)dydx =
=∫
Ω
[∫
Ω
K(y,x)φ(x)dx]
ψ(y)dx = (Kφ,ψ)
 ÷àñòíîñòè, ïðè ψ = φ, èìååì (φ,Kφ) = (Kφ, φ) = (φ,Kφ), ñëåäîâàòåëüíî (φ,Kφ) ∈ R. Òåïåðüìîæíî ïîêàçàòü âåùåñòâåííîñòü õàðàêòåðèñòè÷åñêèõ ÷èñåë K. Äåéñòâèòåëüíî, ïóñòü λ õàðàêòå-ðèñòè÷åñêîå ÷èñëî, φ - ñîáñòâåííàÿ ôóíêöèÿ îïåðàòîðà K, òîãäà (φ, φ) = (λKφ, φ) = λ(Kφ, φ) ⇒λ ∈ R. Äàëåå, ïóñòü φk è φl - ñîáñòâåííûå ôóíêöèè, îòâå÷àþùèå ðàçëè÷íûì õàðàêòåðèñòè÷åñêèì÷èñëàì λk è λl ñîîòâåòñòâåííî. Âûïîëíåíû ðàâåíñòâà
φk = λkKφk
φl = λlKφl
Óìíîæàÿ ïåðâîå èç ýòèõ ðàâåíñòâ ñêàëÿðíî íà φl ñëåâà, âòîðîå íà φk ñïðàâà è âû÷èòàÿ îäíî èçäðóãîãî, ïîëó÷èì
(1λl− 1
λk
)(φk, φl) = 0, ñëåäîâàòåëüíî φk îðòîãîíàëüíî φl ((φk, φl) = 0). Ñîá-
ñòâåííûõ ÷èñåë, îòíîñÿùèõñÿ ê îäíîìó è òîìó æå õàðàêòåðèñòè÷åñêîìó ÷èñëó êîíå÷íîå ÷èñëî,ñëåäîâàòåëüíî îíè îðòîãîíàëèçóþòñÿ ïî ìåòîäó Øìèäòà. Èòàê, ìû ïîëó÷èëè, ÷òî ñîáñòâåííûåôóíêöèè ýðìèòîâîãî îïåðàòîðà ïðåäñòàâëÿþò ñîáîé îðãîãîíàëüíûé íàáîð. Íèæå ìû óñòàíîâèì, âíåêîòîðîì ñìûñëå, ïîëíîòó ýòîãî íàáîðà.
3.8 Áèëèíåéíàÿ ôîðìóëàÐàññìîòðèì óðàâíåíåèå íà ñîáñòâåííóþ ôóíêöèþ îïåðàòîðà ñ ñèììåòðè÷íûì ÿäðîì φk(x) =λk
∫Ω
K(x,y)φk(y)dy. Ýòî óðàâíåíèå ìîæíî ïåðåïèñàòü â âèäå φk(x)λk
= (K(x,y), φ(y)). Ôîìàëü-
íî ñîïîñòàâèì ÿäðó ðÿä K(x,y) ∼∞∑
k=1
Ck(x)φk(y). Ðàññìàòðèâàÿ Ck, êàê êîýôôèöèåíòû Ôóðüå, è
ïðåäïîëàãàÿ φk - îðòîíîðìèðîâàííûì íàáîðîì, ïîëó÷èì Ck = (K, φk), òî åñòü
K(x,y) ∼∞∑
k=1
φk(x)φk(y)λk
(3.16)
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Êàê èçâåñòíî, â òàêîì ñëó÷àå ñïðàâåäëèâî íåðàâåíñòâî Áåññåëÿ∞∑
k=1
|φk(x)|2λ2
k
6∫
Ω
|K(x,y)|2dy (3.17)
Êðîìå òîãî, â ïðîøëîì ñåìåñòðå áûëà äîêàçàíà ñëåäóþùàÿ òåîðåìà
Òåîðåìà 3.8.1 Åñëè ðÿä (3.16) ñõîäèòñÿ ðàâíîìåðíî, òî
K(x,y) =∞∑
k=1
φk(x)φk(y)λk
(3.18)
3.9 Òîåðåìà Ãèëüáåðòà-Øìèäòà. Ðåãóëÿðíàÿ ñõîäèìîñòü ðÿ-äà
Îïðåäåëèì f(x), êàê ôóíêöèþ, èñòîêîîáðàçíî ïðåäñòàâèìóþ ÷åðåç ýðìèòîâî ÿäðî K(x,y), åñëèf(x) =
∫Ω
K(x,y)h(y)dy .Îòíîñèòåëüíî K(x,y) ïðåäïîëîæèì:
1. K(x,y) ∈ L2(Ω)
2. ∃K0 : ∀x ∈ Ω∫Ω|K(x,y)|2dy < K0 (ðàâíîìåðíàÿ ñõîäèìîñòü)
3. È ïóñòü, êðîìå òîãî, h(x) ∈ L(Ω)
Ðàññìîòðèì ðÿä
f(x) ∼∞∑
k=1
fkφk(x)
ãäå fk =∫Ω
f(x)φk(x)dx(3.19)
Òåîðåìà 3.9.1 (Ãèëüáåðòà-Øìèäòà) Ïóñòü f(x) - èñòîêîîáðàçíî ïðåäñòàâèìàÿ ôóíêöèÿ, òî-ãäà ðÿä (3.19) ñõîäèòñÿ ê f(x) ðåãóëÿðíî.
Äîêàçàòåëüñòâî (Çäåñü ìû äîêàæåì ðåãóëÿðíóþ ñõîäèìîñòü ðÿäà)Çàìåòèì, ÷òî fk =
∫Ω
f(x)φk(x)dx =∫Ω
φk(x)dx∫Ω
K(x,y)h(y)dy =∫Ω
h(y)dy∫Ω
K(x,y)φk(x)dx =∫Ω
h(y)dy∫Ω
K(y,x)φk(x)dx = 1λk
∫Ω
h(y)φk(y)dy = hk
λk. Ñëåäîâàòåëüíî, ðÿä (3.19) ýêâèâàëåíòåí
ðÿäó∞∑
k=1
hk
λkφk(x).
Ðàññìîòðèì äàëåå,m+p∑
k=m
∣∣∣∣hkφk(x)
λk
∣∣∣∣ 6(
m+p∑
k=m
|hk|2) 1
2(
m+p∑
k=m
∣∣∣∣φk(x)
λk
∣∣∣∣) 1
2
Òàê êàê∞∑
k=1
|hk|2 6∫Ω|h(x)|2dx < +∞, à
∞∑k=1
∣∣∣φk(x)λk
∣∣∣ 6∫Ω|K(x,y)|2dy < K0 (ñì.(3.17), òî ñóùå-
ñòâóåò òàêîå m(ε), ÷òîm+p∑k=m
∣∣∣hkφk(x)λk
∣∣∣ < ε. Ñëåäîâàòåëüíî ðÿä∞∑
k=1
∣∣∣hk
λkφk(x)
∣∣∣ ñõîäèòñÿ ðàâíîìåðíî, àçíà÷èò ðÿä (3.19) ñõîäèòñÿ ðåãóëÿðíî.
3.10 Âïîëíå íåïðåðûâíûå îïåðàòîðûÐàññìîòðèì ïîñëåäîâàòåëüíîñòü ïðîèçâîëüíûõ ôóíêöèé ψn(x), x ∈ Ω. Ïóñòü ‖ψn(x)‖2 = (ψn(x), ψn(x)) <A. Îïðåäåëèì ïîñëåäîâàòåëüíîñòü vn(x) êàê vn(x) = Lψn(x), ãäå L - ëèíåéíûé îïåðàòîð. Òîãäàîïåðàòîð L íàçûâàþò âïîëíå íåïðåðûâíûì, åñëè èç ïîñëåäîâàòåëüíîñòè vn(x) ìîæíî èçâëå÷üïîäïîñëåäîâàòåëüíîñòü vnk
(x) → v(x) ∈ L2(Ω). (Åñëè v(x) ∈ C(Ω), òî L íàçûâàþò óñèëåííîâïîëíå íåïðåðûâíûì îïåðàòîðîì)Ðàñìîòðèì èíòåãðàëüíûé îïåðàòîð K, òàêîé ÷òî K(x,y) = K(y,x);
∫Ω|K(x,y)|2dy < K0. Êðîìå
òîãî, ìû áóäåì ïðåäïîëàãàòü, ÷òî K(x,y) ∈ C(Ω× Ω).Óòâåðæäåíèå: K - âïîëíå íåïðåðûâíûé îïåðàòîð.Ïîêàæåì, ÷òî ýòî óòâåðæäåíèå ñëåäóåò èç ñëåäóþùåé òåîðåìû:
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Òåîðåìà 3.10.1 (Àðöåëà) Ðàâíîñòåïåííî íåïðåðûâíîå è ðàâíîìåðíî îãðàíè÷åííîå ìíîæåñòâîôóíêöèé êîìïàêòíî.
(ðàâíîñòåïåííàÿ íåïðåðûâíîñòü ∀ε > 0∃δ(ε) : |x− x′| < δ ⇒ |φ(x)− φ(x′)| < ε)Äîêàçàòåëüñòâî ñì. Ñìèðíîâ ò. IVÄîêàæåì, ÷òî ìíîæåñòâî ôóíêöèé Kψn ðàâíîìåðíî îãðàíè÷åííî è ðàâíîñòåïåííî íåïðåíûâíî.1)ðàâíîìåðíàÿ îãðàíè÷åííîñòü:
|Kψn|2 =∣∣∣∣∫
Ω
K(x,y)ψn(y)dy∣∣∣∣2
6∫
Ω
|K(x,y)|2dy∫
Ω
|ψn(y)|2dy 6 K0‖ψn‖ 6 K0A
2)ðàâíîñòåïåííàÿ íåïðåðûâíîñòü.
|Kψn(x)−Kψn(x′)|2 =∣∣∣∣∫
Ω
[K(x,y)−K(x′,y)]ψn(y)dy∣∣∣∣2
6
6∫
Ω
|K(x,y)−K(x′,y)|2dy∫
Ω
|ψn(y)|2dy 6∫
Ω
|K(x,y)−K(x′,y)|2dyA
Òàê êàê ÿäðî K(x,y) - íåïðåðûâíî, òî îíî ðàâíîìåðíî íåïðåðûâíî, òî åñòü ∀ε > 0∃δ 6= δ(y) : |x−x′| < δ ⇒ |K(x,y)−K(x′,y)| < ε . Òîãäà |K(x,y)−K(x′,y)| < ε
A|Ω| , îòêóäà |Kψn(x)−Kψn(x′)| 6 ε.Î÷åâèäíî, ÷òî ôóíêöèè vk ïîëó÷èëèñü íåïðåðûâíûìè, à êîìïàêòíîñòü ýêâèâàëåíòíà âïîëíåíåïðåðûâíîñòè, ñëåäîâàòåëüíî ìû äîêàçàëè óñèëåííóþ âïîëíå íåïðåðûâíîñòü K.
3.11 Òåîðåìà ñóùåñòâîâàíèÿ õàðàêòåðèñòè÷åñêèõ ÷èñåëÐàññìîòðèì ýðìèòîâ îïåðàòîð K 6= Θ, ïðè÷åì áóäåì ñ÷èòàòü, ÷òî ÿäðî K(x,y) ∈ C(Ω× Ω).
Òåîðåìà 3.11.1 Ñóùåñòâóåò õîòÿ áû îäíî λk: φk = λkKφk
Äîêàçàòåëüñòâî 1) Ëåãêî âèäåòü, ÷òî ñóùåñòâîâàíèå õàðàêòåðèñòè÷åñêîãî ÷èñëà ó îïåðàòîðàK2 âëå÷åò çà ñîáîé ñóùåñòâîâàíèå õàðàêòåðèñòè÷åñêîãî ÷èñëà ó îïåðàòîðà K. Äåéñòâèòåëüíî,φ = λK2φ ⇒ φ +
√λKφ −
√λKφ − λK2φ = 0. Îáîçíà÷èì ψ = φ −
√λφ, òîãäà ψ +
√λψ = 0.
Ñëåäîâàòåëüíî, ëèáî ψ ≡ 0, è√
λ - õàðàêòåðèñòè÷åñêîå ÷èñëî, à φ - ñîáñòâåííàÿ ôóíêöèÿ K, ëèáî−√
λ - õàðàêòåðèñòè÷åñêîå ÷èñëî, à ψ - ñîáñòâåííàÿ ôóíêöèÿ.2) ðàññìîòðèì âåëè÷èíû (φ,K2φ)
(φ,φ) = (Kφ,Kφ)(φ,φ) . Çàìåòèì, ÷òî
‖Kφ‖2 =∫
Ω
|Kφ|2dx 6 |Ω|max|Kφ|2 = |Ω|max
∣∣∣∣∫
Ω
K(x,y)φ(y)dy∣∣∣∣2
6
6 |Ω|max
∫Ω|K(x,y)|2dy
∫
Ω
|φ(y)|2dy 6 K20 |Ω|2‖φ‖
Òîãäà 0 6 (Kφ,Kφ)(φ,φ) 6 K2
0 |Ω|2 . Ñëåäîâàòåëüíî, ñóùåñòâóåò ïîñëåäîâàòåëüíîñòü φn: (Kφn,Kφn)
(φn,φn)→
κ1 , ãäå κ1 - òî÷íàÿ âåðõíÿÿ ãðàíü (Kφ,Kφ)(φ,φ) . Äàëåå èç (Kφ,Kφ)
(φ,φ) 6 κ1, äîìíîæàÿ ýòî íåðàâåíñòâîíà (K2φ,K2φ)
(Kφ,Kφ) 6 κ1, ïîëó÷èì (K2φ,K2φ)(φ,φ) 6 κ2
1 Ðàññìîòðèì ïîñëåäîâàòåëüíîñòü ôóíêöèé φn, òà-êóþ, ÷òî φn = φn
‖φn‖ . Î÷åâèäíî, ÷òî (φn, φn) = 1, à (Kφn,Kφn) → κ1 . Äîêàæåì òåïåðü ÷òîïîñëåäîâàòåëüíîñòü ψn, ãäå ψn = κ1φn −K2φn, ñõîäèòñÿ ê íóëþ â ñðåäíåì, òî åñòü ‖ψn‖2 → 0.Äåéñòâèòåëüíî,
‖ψn‖2 = (κ1φn −K2φn,κ1φn −K2φn) = κ21 − κ1(K2φn, φn)− κ1(φn, K2φn) + (K2φn,K2φn) 6
6 2κ21 − 2κ1(Kφn,Kφn) = 2κ1(κ1 − (Kφn,Kφn)) → 0
Ñîãëàñíî óñèëåííîé âïîëíå íåïðåðûâíîñòè K2, ìû ìîæåì âûáðàòü ïîäïîñëåäîâàòåëüíîñòü φnk,
òàêóþ ÷òî K2φnk→ v, ãäå v - íåïðåðûâíàÿ ôóíêöèÿ. Òàêèì îáðàçîì, ó φnk
åñòü ïðåäåë φ0, òàêîé÷òî κ1φ0 −K2φ0 ∼ 0 (ðàâíî ñ òî÷íîñòüþ äî ìíîæåñòâà ìåðû íîëü). Âîçüìåì φ0 = K2φ0
κ1= vκ1
, φ0
- íåïðåðûâíàÿ ôóíêöèÿ. Òîãäà φ0 ∼ φ0. Òàê êàê K2-èíòåãðàëüíûé îïåðàòîð, ìíîæåñòâî ìåðû
44
íîëü íå âëèÿåò íà åãî äåéñòâèå, ñëåäîâàòåëüíî K2φ0 = K2φ0. Çíà÷èò κ1φ0 − K2φ0 ∼ 0, à èçíåïðåðûâíîñòè φ0 ïîëó÷èì îêîí÷àòåëüíî
κ1φ0 −K2φ0 = 0
Òàêèì îáðàçîì ó K2 åñòü õàðàêòåðèñòè÷åñêîå ÷èñëî 1κ1
è ñîáñòâåííàÿ ôóíêöèÿ φ0. Ñëåäîâàòåëüíî,ïî äîêàçàííîìó â ï.1, è ó K åñòü õàðàêòåðèñòè÷åñêîå ÷èñëî.
3.12 Òåîðåìà Ãèëüáåðòà-Øìèäòà. Ïîñëåäîâàòåëüíîñòü õàðàê-òåðèñòè÷åñêèõ ÷èñåë.
 ïðåäûäóùåì ïàðàãðàôå áûëî óñòàíîâëåíî, ÷òî ó ýðìèòîâîãî èíòåãðàëüíîãî îïåðàòîðà K åñòüõàðàêòåðèñòè÷åñêîå ÷èñëî λ2
1 = 1κ1
, è íîðìèðîâàííàÿ ñîáñòâåííàÿ ôóíêöèÿ φ1. Áóäåì ïðîäîëæàòüèññëåäîâàòü ñïåêòð ýòîãî îïåðàòîðà.Ââåäåì îïåðàòîð K1 ñ ÿäðîì K1(x,y) = K(x,y)− φ1(x)φ(y)
λ1. Î÷åâèäíî, K1 îáëàäàåò âñåìè ôóíê-
öèÿìè K, êðîìå φ1.Ñëó÷àé 1. K1 ≡ Θ.  ýòîì ñëó÷àå, K(x,y) = φ1(x)φ(y)
λ1- âûðîæäåííîå ÿäðî.
Ñëó÷àé 2. K1 - íå îïåðàòîð àííóëèðîâàíèÿ. Î÷åâèäíî, îïåðàòîð K1 - ýðìèòîâ óñèëåííî âïîëíåíåïðåðûâíûé. Òîãäà, ñîãëàñíî ðåçóëüòàòàì ïàðàãðàôà 3.11, ó íåãî åñòü õàðàêòåðèñòè÷åñêîå ÷èñëîλ2, ïðè÷åì 0 < (K1φ,K1φ) 6 1
λ22(äëÿ (φ, φ) = 1). Òàê êàê ‖K1φ‖ = ‖Kφ‖ − 1
λ21(φ, φ1)2 6 1
λ21, à 1
λ22-
òî÷íàÿ âåðõíÿÿ ãðàíü, ïîëó÷èì 1λ2
26 1
λ21. Ñëåäîâàòåëüíî |λ2| > |λ1|.
Áóäåì ïðîäîëæàòü ýòîò ïðîöåññ äàëåå. Íà n-ì øàãå èìååì:
Kn+1(x,y) = K(x,y)−n∑
k=1
φk(x)φk(y)λk
Îïÿòü âîçìîæíû äâà ñëó÷àÿÑëó÷àé 1. Kn+1 ≡ Θ (ñëó÷àé âûðîæäåííîãî ÿäðà).  ýòîì ñëó÷àå, äëÿ ïðîèçâîëüíîé ôóíêöèè ω,èìååì
|λ1| 6 |λ2| 6 ... 6 |λn|n∑
k=1
φk(x)(ω, φk)λk
Òî åñòü äëÿ ïðîèçâîëüíîé èñòîêîîáðàçíî ïðåäñòàâèìîé ôóíêöèè èìååì òåîðåìó Ãèëüáåðòà-Øìèäòà.Ñëó÷àé 2. Ïðîöåññ íå îáðûâàåòñÿ. Òîãäà ñóùåñòâóåò λn+1,ïðè ýòîì (Kn+1ω,Kn+1ω) 6 1
λn+1(ω, ω).
Ñîãëàñíî îáùåé òåîðèè Ôðåäãîëüìà, òî÷êîé íàêîïëåíèÿ õàðàêòåðèñòè÷åñêèõ ÷èñåë ìîæåò áûòüòîëüêî áåñêîíå÷íîñòü. Ñëåäîâàòåëüíî ‖Kn+1ω‖ → 0. Çíà÷èò, Kω ∼
∞∑k=1
φk(x)(ω,φk)λk
. Ðàññìîòðèìèñòîêîîáðàçíî ïðåäñòàâèìóþ ôóíêöèþ f = Kω. Âûáèðàÿ, f íåïðåðûâíîé, ïîëó÷èì
f =∞∑
k=1
φk(x)(ω, φk)λk
Äîêàçàòåëüñòâî òåîðåìû Ãèëüáåðòà-Øìèäòà çàâåðøåíî.Òàêèì îáðàçîì, ìû ïîëó÷èëè ïîëíîòó ñèñòåìû ñîáñòâåííûõ ôóíêöèé îòíîñèòåëüíî èñòîêîîáðàçíîïðåäñòàâèìûõ ôóíêöèé.
3.13 Ôîðìóëà ØìèäòàÐàññìîòðèì óðàâíåíèå ñ ýðìèòîâûì îïåðàòîðîì φ = λKφ + f . Òîãäà φ− f - èñòîêîîáðàçíî ïðåä-ñòàâèìàÿ ÷åðåç K, φ ôóíêöèÿ. Òîãäà, ïî òåîðåìå 3.9.1
φ = λ
∞∑
k=1
Ck
λkφk + f (3.20)
45
ãäå Ck = (φ, φk). Óìíîæàÿ (3.20) ñêàëÿðíî íà φl, ïîëó÷èì Cl = λλl
Cl + fl.Ñëó÷àé 1. Äëÿ âñÿêîãî l, λ 6= λl. Òîãäà Cl = flλl
λl−λ Îòêóäà,
φ = λ
∞∑
k=1
(f, φk)φk
λk − λ+ f (3.21)
Ñëó÷àé 2. Ïóñòü λ = λl, φ(p)l - ñîîòâåòñòâóþùàÿ ñîáñòâåííàÿ ôóíêöèÿ. Òîãäà èìååì (f, φ
(p)l ) = 0.
Åñëè ýòî óñëîâèå âûïîëíåíî, èìååì
φ = λ
∞∑
k=1k 6=l
(f, φk)φk
λk − λ+
rankλl∑p=1
Cpl φ
(p)l + f (3.22)
ãäå Cpl - ïðîèçâîëüíûå êîýôôèöèåíòû.
Âñïîìíèì: φ = f + λRf . Òîãäà, èç ðÿäà (3.21) ïîëó÷èì
R(x,y, λ) =∞∑
k=1
φk(x)φk(y)λk − λ
(3.23)
Òàê êàê 1λk−λ = 1
λk+ λ
λ(λk−λ) ,
R(x,y, λ) = K(x,y) + λ
∞∑
k=1
φk(x)φk(y)λk(λk − λ)
(3.24)
Òàêèì îáðàçîì, ðåçîëüâåíòà - ìåðîìîðôíàÿ ôóíêöèÿ ñ ïîëþñàìè â òî÷êàõ õàðàêòåðèñòè÷åñêèõ÷èñåë (êàê è áûëî îáåùàíî)Âñå òåîðèÿ ìîæåò áûòü ðàñïðîñòðàíåíà íà ïîëÿðíûå ÿäðà è ÿäðà è ôóíêöèè èç L2
46
Ãëàâà 4
Âàðèàöèîííîå èñ÷èñëåíèå
4.1 Îñíîâíûå ëåììû âàðèàöèîííîãî èñ÷èñëåíèÿËåììà 4.1.1 (Ëàãðàíæ) Ðàññìîòðèì f(x) ∈ C[x0, x1], ïóñòü äëÿ ëþáîé η(x) ∈ C0[x0, x1] (ò. å.η ∈ C[x0, x1] è η(x0) = η(x1) = 0) ∫ x1
x0
F (x)η(x)dx = 0, (4.1)
òîãäà f(x) ≡ 0.
Äîê-âî Ïóñòü â òî÷êå ξ ∈ (x0, x1) ôóíêöèÿ íå ðàâíà íóëþ, è ïóñòü, äëÿ îïðåäåëåííîñòè, f(x) > 0,çíà÷èò â ε−îêðåñòíîñòè òî÷êè ξ f(x) äåðæèò çíàê. Âûáåðåì ôóíêöèþ η ñîãëàñíî
η(x) =
0, x0 < x < ξ − ε0, ξ + ε < x < x1
(x− ξ − ε)(−x + ξ + ε), ξ − ε < x < ξ − ε,
â ýòîì ñëó÷àå∫ x1
x0f(x)η(x)dx > 0, òàêèì îáðàçîì èìååì ïðîòèâîðå÷èå ñ óñëîâèåì ëåììû, ÷òî è
òðåáîâàëîñü.
Ëåììà 4.1.2 (Ëàãðàíæ - ìíîãîìåðíûé ñëó÷àé) Ðàññìîòðèì f(x) ∈ C(Ω), ïóñòü äëÿ ëþáîéη(x) ∈ C0(Ω) ∫
Ω
f(x)η(x)dx = 0, (4.2)
òîãäà f(x) ≡ 0 â Ω.
(äîêàçàòåëüñòâî ïðîâîäèòñÿ àíàëîãè÷íî, îêðåñòíîñòü |x− ξ| < ε,
η(x) =
0, |x− ξ| > εε2 − |x− ξ|2, |x− ξ| < ε
Ëåììà 4.1.3 (Äþáóà-Ðåéìîí) Ïóñòü f(x) ∈ C[x0, x1] è äëÿ âñÿêîé η(x) ∈ C10 [x0, x1] (òðåáîâà-
íèé ê çíà÷åíèÿì ïðîèçâîäíîé íà êðàÿõ îòðåçêà íåò) èíòåãðàë∫ x1
x0
f(x)η′(x)dx = 0, (4.3)
çíà÷èò f(x) = const
Äîê-âî Ââåäåì ñðåäíåå çíà÷åíèå ôóíêöèè f :
f =1
x1 − x0
∫ x1
x0
f(x)dx,
ïîñêîëüêó∫ x1
x0[f(x)− f ]dx = 0 âûáåðåì η(x) =
∫ x
x0[f(x′)− f ]dx′ (ëåãêî âèäåòü, ÷òî η(x) ∈ C1
0 [x0, x1]).Ïî óñëîâèþ ëåììû ∫ x1
x0
f(x)[f(x)− f ]dx = 0 (4.4)
47
è êðîìå òîãî, ∫ x1
x0
f [f(x)− f ]dx = 0, (4.5)
ñëåäîâàòåëüíî, âû÷èòàÿ èç (4.4) (4.5) èìååì∫ x1
x0[f(x)− f ]2dx = 0, à â ñèëó íåïðåðûâíîñòè f f(x) =
f = const, ÷òî è òðåáîâàëîñü.
Ëåììà 4.1.4 Ïóñòü a(x), b(x) ∈ C[x0, x1] è äëÿ âñÿêîé η(x) ∈ C10 [x0, x1]
∫ x1
x0
[a(x)η(x) + b(x)η′(x)]dx = 0, (4.6)
ñëåäîâàòåëüíî b(x) íåïðåðûâíî-äèôôåðåíöèðóåìà è b′(x) = a(x).
Äîê-âî Ââåäåì ôóíêöèþ A(x) =∫ x
x0a(x′)dx′. Ïðîèíòåãðèðîâàâ â ïåðâîì ñëàãàåìîì (4.6) ïî ÷àñòÿì
èìååì ∫ x1
x0
(b(x)−A(x))η′(x)dx = 0,
îòñþäà ïî ëåììå Äþáóà-Ðåéìîíà ïîëó÷àåì b(x) = A(x)+const èëè b′(x) = a(x), ÷òî è òðåáîâàëîñü.
4.2 Íåîáõîäèìûå óñëîâèÿ ýêñòåìóìà â ïðîñòåéøåé çàäà÷å âà-ðèàöèîííîãî èñ÷èñëåíèÿ
Ïðîñòåéøàÿ çàäà÷à : ôóíêöèîíàë
J [y(x)] =∫ x1
x0
F (x, y, y′)dx. (4.7)
Êðîìå ýòîãî, çàäàíû êðàåâûå óñëîâèÿ
y(x0) = y0
y(x1) = y1
y ∈ C1[x0, x1]
Îïðåäåëåíèå 2 Ìíîæåñòâî ôóíêöèé y(x) òàêèõ, ÷òî |y(x)−y(x)| < ε íàçûâàþò ε-îêðåñòíîñòüþíóëåâîãî ïîðÿäêà ôóíêöèè y(x).
Ìíîæåñòâî ôóíêöèé y(x) òàêèõ, ÷òî |y(x)−y(x)|+|y′(x)−y′(x)| < ε íàçûâàþò ε-îêðåñòíîñòüþ1-îãî ïîðÿäêà ôóíêöèè y(x).
Îïðåäåëåíèå 3 Ãîâîðÿò, ÷òî ôóíêöèÿ y(x) ðåàëèçóåò îòíîñèòåëüíûé ìàêñèìóì (ìèíèìóì)ôóíêöèîíàëà J , åñëè äëÿ âñÿêîé y(x) èç ε îêðåñòíîñòè y, J [y] ≥ (≤)J [y]
(Ïðè îòñóòñòâèè îãðàíè÷åíèé íà ôóíêöèþ y ãîâîðÿò îá àáñîëþòíîì ýêñòðåìóìå ôóíêöèîíàëà).Ñäåëàåì íåñêîëüêî ïðåäïîëîæåíèé î âèäå ôóíêöèè F . À èìåííî, áóäåì ñ÷èòàòü, ÷òî â ε-
îêðåñòíîñòè 1-îãî ïîðÿäêà (ýêñòðåìóìà ôóíêöèîíàëà)
F,∂F
∂x,∂F
∂y,∂F
∂y′,∂2F
∂x2,∂2F
∂y2,∂2F
∂y′2
íåïðåðûâíûå ôóíêöèè.Íåîáõîäèìûå óñëîâèÿ ýêñòðåìóìàÏóñòü y(x) - ôóíêöèÿ, íà êîòîðîé ðåàëèçóåòñÿ ýêñòðåìóì. Ïîëîæèì
y(x) = y(x) + αη(x), (4.8)
(η(x) ∈ C0[x0, x1]). Äëÿ òàêîé y(x) ôóíêöèîíàë ñâîäèòñÿ ê ôóíêöèè ÷èñëîâîãî ïàðàìåòðà α :
J [y(x)] =∫ x1
x0
F (x, y + αη, y′ + αη′)dx = Φ(α).
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Èçâåñòíî, ÷òî α = 0 - ýêñòðåìóì ôóíêöèè Φ, ïîýòîìó ñ íåîáõîäèìîñòüþ ïðè α = 0
dΦdα
∣∣∣∣α=0
= 0,
çíà÷èòdΦdα
∣∣∣∣α=0
=∫ x1
x0
[∂F
∂y
∣∣∣∣x,y,y′
η(x) +∂F
∂y′
∣∣∣∣x,y,y′
η′(x)
]dx.
Ïî ëåììå Äþáóà-Ðåéìîíà è ëåììå 4.1.4 (η(x) - ïðîèçâîëüíàÿ ôóíêöèÿ, ïîïàäàíèå â ε−îêðåñòíîñòüãàðàíòèðóåòñÿ ïî âûáîðó α) èìååì
d
dx
∂F
∂y′
∣∣∣∣y=y
− ∂F
∂y
∣∣∣∣y=y
(4.9)
Îïðåäåëåíèå 4 Âàðèàöèîííàÿ ïðîèçâîäíàÿ
δF
δy=
d
dx
∂F
∂y′
∣∣∣∣y=y
=∂F
∂y
∣∣∣∣y=y
.
Óðàâíåíèå (4.9) íàçûâàåòñÿ óðàâíåíèåì Ýéëåðà.Îïðåäåëåíèå 5 αη(x) = δy - âàðèàöèÿ ôóíêöèè
Îïðåäåëåíèå 6 αΦ′(0) = δJ - âàðèàöèÿ ôóíêöèîíàëà
Òîãäà èìååì
δJ =∫ x1
x0
[∂F
∂y
∣∣∣∣y=y
δy +∂F
∂y′
∣∣∣∣y=y
δy′]
dx.
Ðàñêðûâàÿ ïîëíóþ ïðîèçâîäíóþ ïî x èìååì óðàâíåíèå
∂F
∂y− ∂2F
∂x∂y′− ∂2F
∂y∂y′y′ − ∂2F
∂y′2y′′ = 0
(Äëÿ îáîñíîâàíèÿ ýòîãî ïåðåõîäà íåîáõîäèìî ñóùåñòâîâàíèå âòîðîé ïðîèçâîäíîé ðåøåíèÿ. Ïî-êàæåì, ÷òî ïðè ∂2F
∂y′2 6= 0 è y = y, x ∈ [x0, x1] ýòî âûïîëíåíî. Òîãäà
d
dx
∂F
∂y′= lim
∆x→0
1∆x
(∂F
∂y′(x + ∆x, y(x + ∆x), y′(x + ∆x))− ∂F
∂y′(x, y, y′)
)=
∂2F ∗
∂x∂y′+
∂2F ∗
∂y∂y′+
∂2F ∗
∂y′2∆y′
∆x
- çäåñü ∗ îçíà÷àåò çíà÷åíèå â òî÷êå x < x∗ < x + ∆x, â ýòîì ñëó÷àå, ðàçäåëèâ íà
∂2F ∗
∂y′2
èìååì ñóùåñòâîâàíèå y′′ èd
dx
∂F
∂y′=
∂2F
∂x∂y′+
∂2F
∂y∂y′y′ +
∂2F
∂y′2
÷òî è òðåáîâàëîñü.)Èòàê, â êà÷åñòâå íåîáõîäèìîãî óñëîâèÿ ýêñòðåìóìà èìååì êâàçèëèíåéíóþ êðàåâóþ çàäà÷ó
∂F
∂y− ∂2F
∂x∂y′− ∂2F
∂y∂y′y′ − ∂2F
∂y′2y′′ = 0
y(x0) = y0
y(x1) = y1
Îïðåäåëåíèå 7 Ðåøåíèå ýòîé êðàåâîé çàäà÷è - ýêñòðåìàëü.Ïðèìåð Âîçüìåì Ëàãðàíæèàí â âèäå F = p(x)y′2 + q(x)y2 + 2f(x)y, óðàâíåíèå íà ýêñòðåìàëè
d
dxp(x)
dy
dx− q(x)y = f(x)
(âèäíî, ÷òî èç âàðèàöèîííîãî ïðèíöèïà ñëåäóåò ñàìîñîïðÿæåííîå óðàâíåíèå.)
49
4.3 Íåîáõîäèìûå óñëîâèÿ ýêñòðåìóìà ôóíêöèîíàëà, çàâèñÿ-ùåãî îò íåñêîëüêèõ ôóíêöèîíàëüíûõ àðãóìåíòîâ
Ðàññìîòðèì ôóíêöèîíàë
J [y1(x), . . . , yn(x)] =∫ x1
x0
F (x, y1, y′1, . . . , yn, y′n)dx (4.10)
yi(x0) = y0
yi(x1) = y1 - ãðàíè÷íûå óñëîâèÿi = 1, . . . , n
Îáîçíà÷èì ÷åðòîé ñâåðõó ôóíêöèè, íà êîòîðûõ äîñòèãàåòñÿ ýêñòðåìóì, ïóñòü ôóíêöèè ñðàâíåíèÿyi = yi + αiηi(x), ηi ∈ C1
0 [x0, x1]. Òîãäà
J [y(x)] =∫ x1
x0
F (x, y1 + α1η1, y′1 + α1η
′1, . . . , yn + αnηn, y′n + αnη′n)dx = Φ(α1, . . . , αn).
Íåîáõîäèìûå óñëîâèÿ ýêñòðåìóìà ôóíêöèè Φ â íóëå.
∂Φ∂αi
∣∣∣∣α1=...=αn=0
= 0, i = 1, . . . , n
èëè ∫ x1
x0
(∂F
∂yi
∣∣∣∣y=y
ηi +∂F
∂y′i
∣∣∣∣y=y
η′i
)dx = 0.
Äàëåå ïðåäïîëîæèì ñóùåñòâîâàíèå âñåõ òåõ ïðîèçâîäíûõ, êîòîðûå ïîòðåáóþòñÿ, è ïðîèíòåãðèðó-åì ïî ÷àñòÿì. Èìååì ∫ x1
x0
(∂F
∂yi− d
dx
∂F
∂y′i
)
y=y
ηidx +∂F
∂y′iη
∣∣∣∣x1
x0
= 0,
ïðè ýòîì ïîñëåäíåå ñëàãàåìîå óõîäèò â ñèëó òîãî, ÷òî η(x0, x1) = 0, è ïî ëåììå Ëàãðàíæà ïîëó÷àåìñèñòåìó óðàâíåíèé Ýéëåðà
∂F
∂yi− d
dx
∂F
∂y′i= 0, i = 1, . . . , n (4.11)
4.4 Íåîáõîäèìîå óñëîâèå ýêñòðåìóìà ôóíêöèîíàëà, çàâèñÿ-ùåãî îò âûñøèõ ïðîèçâîäíûõ
Ðàññìîòðèì ôóíêöèîíàëJ [y(x)] =
∫ x1
x0
F (x, y, y′, . . . , , y(n))dx, (4.12)
ãðàíè÷íûå óñëîâèÿ
y(x0) = y00 y(x1) = y10
y′(x0) = y01 y′(x1) = y11
. . . . . .y(n−1)(x0) = y0,n−1 y(n−1)(x1) = y1,n−1
Ïóñòü y = y + αη, ãäå η ∈ Cn0 [x0, x1]. Òîãäà (äåéñòâóÿ àíàëîãè÷íî ïðåäûäóùèì ïàðàãðàôàì) ïîëó-
÷àåì J [y] = Φ(α). íåîáõîäèìîå óñëîâèå ýêñòðåìóìà ∂Φ∂α
∣∣α=0
= 0, âû÷èñëÿÿ ïðîèçâîäíóþ∫ x1
x0
(∂F
∂y
∣∣∣∣y=y
η +∂F
∂y′
∣∣∣∣y=y
η′ + . . . +∂F
∂y(n)
∣∣∣∣y=y
η(n)
)dx = 0
(ìû íå áóäåì îáîáùàòü ëåììó Äþáóà-Ðåéìîíà íà ñëó÷àé ïðîèçâîäíûõ âûñøèõ ïîðÿäêîâ, à ïðåä-ïîëîæèì n−êðàòíóþ äèôôåðåíöèðóåìîñòü ∂F
∂α , òîãäà èíòåãðèðóÿ ïî ÷àñòÿì)
∂F
∂y′
∣∣∣∣y=y
η
∣∣∣∣∣
x1
x0
+∂F
∂y′′
∣∣∣∣y=y
η′∣∣∣∣∣
x1
x0
+ . . . +∂F
∂y(n)
∣∣∣∣y=y
η(n−1)
∣∣∣∣∣
x1
x0
+
50
∫ x1
x0
η
(∂F
∂y
∣∣∣∣y=y
− d
dx
∂F
∂y′
∣∣∣∣y=y
)− d
dx
∂F
∂y′′η′ − . . .− d
dx
∂F
∂y(n)η(n−1)
dx =
è òàê äàëåå
= . . . =∫ x1
x0
[∂F
∂y− d
dx
∂F
∂y′+
d2
dx2
∂F
∂y′′− . . . + (−1)n dn
dxn
∂F
∂y(n)
]ηdx = 0
â ñèëó ïðîèçâîëüíîñòè η èìååì óðàâíåíèå Ýéëåðà-Ëàãðàíæà
∂F
∂y− d
dx
∂F
∂y′+
d2
dx2
∂F
∂y′′− . . . + (−1)n dn
dxn
∂F
∂y(n)= 0 (4.13)
Ïðèìåð (óðàâíåíèå ïðîãèáà áàëêè)
J [y] =∫ x1
x0
[EIy′′2 + 2fy]dx
y(x0) = 0, y(x1) = 0
y′(x0) = 0, y′(x1) = 0
Óðàâíåíèå Ýéëåðà-ËàãðàíæàEIy(IV) = −f
- ÷åòâåðòîãî ïîðÿäêà.
4.5 Íåîáõîäèìûå óñëîâèÿ ýêñòðåìóìà äëÿ ôóíêöèîíàëà, çà-âèñÿùåãî îò ôóíêöèè íåñêîëüêèõ ïåðåìåííûõ
Ôóíêöèîíàë è ãðàíè÷íûå óñëîâèÿ :
J [u(x)] =∫
Ω
F (x, u,∂u
∂x)dx, u|∂Ω = g(x), x ∈ Ω ⊂ Rn (4.14)
Ôóíêöèÿ ñðàâíåíèÿ u = u + αη, η|∂Ω = 0, è
J [u(x)] =∫
Ω
F
(x, u + αη,
∂u
∂x1+ α
∂η
∂x1, . . . ,
∂u
∂xn+ α
∂η
∂xn
)dx = Φ(α).
Íåîáõîäèìûå óñëîâèÿ ýêñòðåìóìà Φ′(α) = 0, ñ äðóãîé ñòîðîíû, ðàñêðûâàÿ ïðîèçâîäíóþ èìååì
0 =∫
Ω
(∂F
∂uη +
∂F
∂p1
∂η
∂x1+ . . . +
∂F
∂pm
∂η
∂xm
)dx =
çäåñü pi = ∂u∂xi
, äàëåå èíòåãðèðóÿ ïî ÷àñòÿì ïðîäîëæàåì ðàâåíñòâî
=∫
∂Ω
η
(∂F
∂p1cosnx1 + . . .
)dS +
∫
Ω
(∂F
∂u− ∂
∂x1
∂F
∂p1− . . .
)ηdx,
ïåðâîå ñëàãàåìîå óõîäèò â 0 ïî îïðåäåëåíèþ η, äàëåå ïî ëåììå Ëàãðàíæà ïîëó÷àåì óðàâíåíèåÝéëåðà
∂F
∂u− ∂
∂x1
∂F
∂p1− . . .− ∂
∂xn
∂F
∂pn= 0. (4.15)
Ïðèìåðû
1. Ôóíêöèîíàë Äèðèõëå
J [u] =∫
Ω
[(∂u
∂x1
)2
+ . . . +(
∂u
∂xn
)2]
dx,
óðàâíåíèå Ýéëåðà ïðèíèìàåò âèä∆u = 0.
51
2. J [u] =∫Ω
[(∇u)2 + 2fu
]dx ⇐⇒ ∆u = f
3. Óðàâíåíèå êîëåáàíèé ñòðóíû
J [u] =12
∫ t2
t1
∫ x1
x0
[T
(∂u
∂x
)2
− ρ
(∂u
∂t
)2]
dxdt ⇐⇒
⇐⇒ ∂
∂xT
∂u
∂x=
∂
∂tρ∂u
∂t,
íî ãðàíè÷íûå óñëîâèÿ òàêîâû u|t=t1 = µ1(x)u|t=t2 = µ2(x)
(à íå ñòàíäàðòíàÿ çàäà÷à Êîøè). Ìû âèäèì, ÷òî íåò ïîëíîé ýêâèâàëåíòíîñòè âàðèàöèîííîéçàäà÷è è íà÷àëüíî-êðàåâîé çàäà÷è.
4. Ïîïðîáóåì ïîëó÷èòü âàðèàöèîííûé ïðèíöèï äëÿ óðàâíåíèÿ òåïëîïðîâîäíîñòè, âîçüìåì ôóíê-öèîíàë
J [u, v] =∫ t2
t1
∫ x1
x0
[∂u
∂x
∂v
∂x− ∂u
∂tv
]dxdt
óðàâíåíèÿ Ýéëåðà-Ëàãðàíæà
∂v∂t − ∂2v
∂x2 = 0 - ñãëàæèâàåò íåîäíîðîäíîñòè∂u∂t + ∂2u
∂x2 = 0 - ñîáèðàåò íåîäíîðîäíîñòè
5.J [u] =
∫
Ω
aij∂u
∂xi
∂u
∂xjdx
m∂
∂xiaij
∂u
∂xj= 0
(è ãðàíè÷íûå óñëîâèÿ ïåðâîãî ðîäà.)
4.6 Èçîïåðèìåòðè÷åñêàÿ çàäà÷àÐàññìîòðèì ïðîñòåéøèé ôóíêöèîíàë
J [y(x)] =∫ x1
x0
F (x, y, y′)dx,
ïóñòü J [y] > J [y] ïðè óñëîâèè, ÷òî ôóíêöèîíàë
J1[y(x)] =∫ x1
x0
G(x, y, y′)dx = l = const
è y(x0) = y0 è y(x1) = y1. Áóäåì èñêàòü íåîáõîäèìîå óñëîâèå òàêîãî ýêñòðåìóìà (ò. å. ðåøåíèåèçîïåðèìåòðè÷åñêîé çàäà÷è) ïðè óñëîâèè, ÷òî y íå ÿâëÿåòñÿ ýêñòðåìàëüþ ôóíêöèîíàëà J1, ò. å.(
∂G∂y − d
dx∂G∂y′
)y=y
6= 0 äëÿ âñÿêîãî x ∈ (x0, x1).
Òåîðåìà 4.6.1 (Ýéëåð) Ïóñòü H = F + λG, λ = const,
δH
δy= 0 ⇔
δFδy = 0
J1[y] = l
52
Äîê-âî Ïóñòü y = y + α1η1 + α2η2, ηi(x) ∈ C[x0, x1], òîãäà J [y] = Φ(α1, α2) è J1[y] = Ψ(α1, α2) = l.Ëåãêî âèäåòü, ÷òî
∂Ψ∂α1
=∫ x1
x0
(∂G
∂y− d
dx
∂G
∂y′
)η1dx
è∂Ψ∂α2
=∫ x1
x0
(∂G
∂y− d
dx
∂G
∂y′
)η2dx 6= 0, (ïî âûáîðó η2(x)).
Òîãäà ïî òåîðåìå î íåÿâíîé ôóíöèè(Ψ(0, 0) = l, ∂Ψ/∂α2|0,0 6= 0) ñóùåñòâóåò ôóíêöèÿ α2 = α2(α1),òàêàÿ ÷òî Ψ(α1, α2(α1)) = l, ïðè ýòîì
dα2
dα1= −
∂Ψ∂α1
∂Ψ∂α2
Òîãäà
dα2
dα1= −
∫ x1
x0
(∂G∂y − d
dx∂G∂y′
)y=y
η1dx
∫ x1
x0
(∂G∂y − d
dx∂G∂y′
)y=y
η2dx.
Òîãäà ïîäñòàâèâ α2 = α2(α1) â Φ ïîëó÷àåì íåîáõîäèìîå óñëîâèå ýêñòðåìóìà
0 =dΦdα1
∣∣∣∣α1=0
=(
∂Φ∂α1
+∂Φ∂α2
dα2
dα1
)=
ïîäñòàâëÿÿ ñþäà âûðàæåíèÿ äëÿ ïðîèçâîäíûõ îêîí÷àòåëüíî ïîëó÷àåì
0 =∫ x1
x0
(∂F
∂y− d
dx
∂F
∂y′
)η1dx− λ
∫ x1
x0
(∂G
∂y− d
dx
∂G
∂y′
)η1dx,
ãäå
λ =
∫ x1
x0
(∂F∂y − d
dx∂F∂y′
)y=y
η2dx
∫ x1
x0
(∂G∂y − d
dx∂G∂y′
)y=y
η2dx.
- êîíñòàíòà, îïðåäåëÿåìàÿ âûáîðîì ôóíêöèè η2. Ïðèìåíÿÿ ëåììó Ëàãðàíæà ïîëó÷àåì òðåáóåìîå.Çàìå÷àíèå Èìååòñÿ ñîîòíîøåíèå äâîéñòâåííîñòè ìåæäó çàäà÷àìè
J [y] = extr J1[y] = const
èJ [y] = const J1[y] = extr
ïîñêîëüêó ïåðâàÿ çàäà÷à ÿâëÿåòñÿ çàäà÷åé íà ýêñòðåìóì ôóíêöèîíàëà F + λG, à âòîðàÿ - äëÿµF + G.
Ïðèìåðû
1. Èçîïåðèìåòðè÷åñêàÿ çàäà÷à - íàéòè êðèâóþ çàäàííîé äëèíû, îõâàòûâàþùóþ íàèáîëüøóþïëîùàäü
2.∫ x1
x0(py′2 + qy2)dx = min, p > 0, q > 0, âòîðîé ôóíêöèîíàë
∫ x1
x0ry2dx = 1, y(x0) = y(x1) = 0.
Ôóíêöèÿ H = py′2 + qy2 − λry2 òîãäà
d
dxpdy
dx+ (λr − q)y = 0
- çàäà÷à Øòóðìà-Ëèóâèëëÿ. (Ìèíèìàëüíîñòü ôóíêöèîíàëà áóäåò äîêàçàíà îòäåëüíî).
3.∫Ω|∇u|2dx = min,
∫Ω
u2dx = 1
53
4.7 Óñëîâíûé ýêñòðåìóìÔóíêöèîíàë
J [y, z] =∫ x1
x0
F (x, y, y′, z, z′)dx = extr
ãðàíè÷íûå óñëîâíèÿy(x0) = y0 z(x0) = z0
y(x1) = y1 z(x1) = z1
Ðàññìîòðèì óñëîâèå, çàïèñàííîå â âèäåΦ(x, y, z) = 0,
ïóñòü íà ôóíêöèÿõ, ðåàëèçóþùèõ óñëîâíûé ýêñòðåìóì y(x), z(x) ïðîèçâîäíàÿ∂Φ∂z
∣∣∣∣y,z
6= 0
äëÿ âñÿêîãî x ∈ (x0, x1).Òåîðåìà 4.7.1 Ïóñòü H = F + λ(x)Φ, òîãäà ýêñòðåìóì
δHδy = 0δHδz = 0
(4.16)
ýêâèâàëåíòåí ýêñòðåìóìó J ïðè óñëîâèè Φ = 0.Äîê-âî Ïî òåîðåìå î íåÿâíîé ôóíêöèè ñóùåñòâóåò òàêàÿ ôóíêöèÿ z = z(x, y), ÷òî Φ(x, y, z(x, y)) =Ψ(x, y) = 0, êðîìå òîãî
∂z
∂y= −
∂Φ∂y
∂Φ∂z
,
è J [y, z(x, y)] =∫ x1
x0G(x, y, y′)dx. Òîãäà íåîáõîäèìîå óñëîâèå ýêñòðåìóìà ôóíêöèîíàëà çàïèñûâà-
åòñÿ â âèäå :∂G
∂y− d
dx
∂G
∂y′= 0,
âû÷èñëèì ïðîèçâîäíûå :∂G
∂y=
∂F
∂y+
∂F
∂z
∂z
∂y+
∂F
∂z′
(∂2z
∂y∂x+
∂2z
∂y2y′
)
∂G
∂y′=
∂F
∂y′+
∂F
∂z′∂z
∂y
d
dx
∂G
∂y′=
d
dx
∂F
∂y′+
∂z
∂y
d
dx
∂F
∂z′+
∂F
∂z′
(∂2z
∂y∂x+
∂2z
∂y2y′
),
(çäåñü ìû ïîëüçîâàëèñü òåì, ÷òî G = F (x, y, y′, z(x, y), ∂z/∂x + (∂z/∂y)y′).) Âû÷èòàÿ è ïðèâîäÿïîäîáíûå íàõîäèì : [
∂F
∂y− d
dx
∂F
∂y′+
∂z
∂y
(∂F
∂z− d
dx
∂F
∂z′
)]
y=y,z=z
= 0
Ïîäñòàâèâ ñþäà ∂z/∂y ïîëó÷àåì :[
∂F
∂y− d
dx
∂F
∂y′− ∂Φ
∂y
∂F∂z − d
dx∂F∂z′
∂Φ∂z
]
y=y,z=z
= 0.
Çàìåòèì, ÷òî ìíîæèòåëü ïðè ∂Φ∂y åñòü íåêîòîðàÿ ôóíêöèÿ x. Îáîçíà÷èì
λ(x) = −∂F∂z − d
dx∂F∂z′
∂Φ∂z
,
òîãäà∂F
∂y− d
dx
∂F
∂y′+ λ(x)
∂Φ∂y
= 0
∂F
∂z− d
dx
∂F
∂z′+ λ(x)
∂Φ∂z
= 0,
îáîçíà÷àÿ F + λΦ çà H ïîëó÷àåì òðåáóåìîå.
54
4.8 Åñòåñòâåííûå êðàåâûå óñëîâèÿ
J [y(x)] =∫ x1
x0
F (x, y, y′)dx
Ðàññìîòðèì êëàññ ôóíêöèé ñðàâíåíèÿ òàêèõ, ÷òî íà ëåâîì êîíöå y(x0) = y0, à íà ïðàâîì êîíöåy(x1) ïðîèçâîëüíî (è ïóñòü y íàõîäèòñÿ â ε-îêðåñòíîñòè y) : y = y+αη, η(x0) = 0, η(x1) ïðîèçâîëüíî.Òîãäà J [y] = Φ(α). ïîòðåáóåì, ÷òîáû Φ′(0) = 0. Ïîñêîëüêó
Φ′(0) =∂F
∂y′
∣∣∣∣y=y,x=x1
η(x1) +∫ x1
x0
(∂F
∂y− d
dx
∂F
∂y′
)
y=y
ηdx, (4.17)
òî
1. Ò. ê. η(x1) ïðîèçâîëüíî, âîçâðàùàÿñü ê óñëîâèþ η(x1) = 0 ïî ëåììå Ëàãðàíæà
δF
δy= 0
2. Òîãäà èíòåãðàë â (4.17) óõîäèò â íóëü è Φ′(0) = ∂F∂y′ η(x1) = 0, à â ñèëó ïðîèçâîëüíîñòè η(x1)
ïîëó÷àåì∂F
∂y′
∣∣∣∣y,x1
= 0
Îêîí÷àòåëüíî, èìååì óðàâíåíèå Ýéëåðà
∂F
∂y− d
dx
∂F
∂y′= 0,
ãëàâíîå êðàåâîå óñëîâèåy(x0) = y0
è åñòåñòâåííîå êðàåâîå óñëîâèå∂F
∂y′
∣∣∣∣y,x1
= 0.
Ïðèìåð F = p(x)y′2 + q(x)y2 + 2f(x)y, ∂F∂y′ = 2p(x)y′, ïóñòü p(x1) 6= 0, òîãäà åñòåñòâåííîå
ãðàíè÷íîå óñëîâèå çàïèñûâàåòñÿ y′(x1) = 0 (óñëîâèå âòîðîãî ðîäà).Ñëó÷àé ïðîèçâîäíûõ âûñøèõ ïîðÿäêîâ
J [y(x)] =∫ x1
x0
F (x, y, y′, y′′)dx
Ïóñòü y(x1) è y′(x1) íå çàäàíû. Ïðèðàâíÿåì âàðèàöèþ ôóíêöèîíàëà íóëþ :
0 = δJ =∫ x1
x0
(∂F
∂yδy +
∂F
∂y′δy′ +
∂F
∂y′′δy′′
)dx =
èíòåãðèðóÿ ïî ÷àñòÿì
∂F
∂y′δy
∣∣∣∣x1
x0
+∂F
∂y′′δy′
∣∣∣∣x1
x0
− d
dx
∂F
∂y′′δy
∣∣∣∣x1
x0
+∫ x1
x0
(∂F
∂y− d
dx
∂F
∂y′+
d2
dx2
∂F
∂y′′
)dx
Ïîòðåáóåì
1. âûïîëíåíèå óðàâíåíèÿ Ýéëåðà
2. ïðè x = x0 êîíåö êðèâûõ ñðàâíåíèÿ çàêðåïëåí
3. âàðèàöèè δy è δy′ ïðè x = x1 ïðîèçâîëüíû.
55
Òîãäà∂F
∂y− d
dx
∂F
∂y′+
d2
dx2
∂F
∂y′′
è ãðàíè÷íûå óñëîâèÿ
(∂F∂y′ − d
dx∂F∂y′′
)
∂F∂y′′
∣∣∣x1
= 0
Ïðèìåð (áàëêà) F = EIy′′2, åñòåñòâåííûå ãðàíè÷íûå óñëîâèÿ (óñëîâèÿ ñâîáîäíîãî çàêðåïëå-íèÿ) :
2EIy′′′(x1) = 0, 2EIy′′(x1) = 0
Ñëó÷àé ìíîãèõ ïåðåìåííûõ
J [u(x)] =∫
Ω
F (x, u,∂u
∂x)dx,
ïóñòü u|∂Ω íå çàäàíî. Âàðèàöèÿ
δJ =n∑
i=1
∫
∂Ω
∂F
∂picosnxiδudS +
∫
Ω
[∂F
∂u−
n∑
i=1
∂
∂xi
∂F
∂pi
]δudx = 0.
Òîãäà ïîìèìî óðàâíåíèÿ Ýéëåðà èìååì ãðàíè÷íûå óñëîâèÿn∑
i=1
(∂F
∂picosnxi
)
u,x∈∂Ω
= 0
Ïðèìåðû
1. Ôóíêöèîíàë Äèðèõëå F = (∇u)2, åñòåñòâåííûå ãðàíè÷íûå óñëîâèÿ
∂u
∂n|∂Ω = 0
2. Ïðîèçâîëüíûé ýëëèïòè÷åñêèé ôóíêöèîíàë F = aij∂u∂xi
∂u∂xj
, òîãäà(
aij∂u
∂xj
)
∂Ω
= 0
- ðàâåíñòâî íóëþ ïðîèçâîäíîé ïî êîíîðìàëè.
4.9 Åñòåñòâåííûå êðàåâûå óñëîâèÿ äëÿ ñëó÷àÿ ôóíêöèîíà-ëîâ, ñîäåðæàùèõ âíåèíòåãðàëüíûå ÷ëåíû (èëè èíòåãðà-ëû ïî ïîâåðõíîñòè)
ÐàññìîòðèìJ [y(x)] =
∫ x1
x0
F (x, y, y′)dx + Φ(y(x1)),
y(x0) = y0, y(x1) - íå çàäàíî. Ïóñòü y = y + αη, òîãäà (ñâåäÿ êàê îáû÷íî ôóíêöèîíàë ê ôóíêöèè)ïîëó÷èì
Ψ(α) =∫ x1
x0
F (x, y + αη, y′ + αη′)dx + Φ( y + αη|x1),
äèôôåðåíöèðóÿ ïî α è èíòåãðèðóÿ ïî ÷àñòÿì
Ψ′(0) =
(∂F
∂y′
∣∣∣∣y=y,x=x1
+dΦdy
∣∣∣∣y=y,x=x1
)η(x1) +
∫ x1
x0
(∂F
∂y− d
dx
∂F
∂y′
)
y=y
ηdx,
òîãäà óðàâíåíèå íà ýêñòðåìàëè∂F
∂y− d
dx
∂F
∂y′= 0
56
è ãðàíè÷íîå óñëîâèå (∂F
∂y′+
dΦdy
)
x=x1
= 0
Ïðèìåð F = py′2 + qy2, Φ = hy2 + 2Qy, òîãäà ãðàíè÷íîå óñëîâèå ïðèíèìàåò âèä p(x1)y′(x1) +hy(x1) + Q = 0.
 ñëó÷àå ôóíêöèè ìíîãèõ ïåðåìåííûõ J [u] =∫Ω
. . . dx+∫
∂ΩΦ(x, u)dS, ãðàíè÷íîå óñëîâèå
(∂F
∂picosnxi +
∂Φ∂n
)
∂Ω
= 0.
4.10 Ìåòîä Ðèòöà. ÑõåìàÌåòîä Ðèòöà ïîçâîëÿåò ñòðîèòü ïðèáëèæåííûå ðåøåíèÿ âàðèàöèîííûõ çàäà÷, íå ðåøàÿ óðàâíåíèéÝéëåðà-Ëàãðàíæà. Îäíîìåðíàÿ çàäà÷à
J [y(x)] =∫ x1
x0
F (x, y, y′)dx, y(x0) = y0 = y(x1) = y1 = 0 (4.18)
Ìíîãîìåðíàÿ çàäà÷àJ [u(x)] =
∫
Ω
F (x, u,∂u
∂x)dx, u|∂Ω = 0. (4.19)
(Ìåòîä Ðèòöà ïðèìåíèì äëÿ îäíîðîäíûõ êðàåâûõ óñëîâèé.)Ðàññìîòðèì íàáîð ôóíêöèé uk(x) ∈ C1
0 [x0, x1] äëÿ (4.18) è vk ∈ C10(Ω) (äëÿ (4.19)). Ïîòðå-
áóåì ïîëíîòó è ëèíåéíóþ íåçàâèñèìîñòü íàáîðà ôóíêöèé. (Íàïîìíèì, ÷òî ïîëíîòà â C10 îçíà-
÷àåò, ÷òî äëÿ âñÿêîé y(x) ∈ C10 è ëþáîãî ε > 0 íàéäóòñÿ òàêîé íîìåð m è êîíñòàíòû Ck, ÷òî
|y(x)−∑mk=1 Ckuk(x)|+ |y′(x)−∑m
k=1 Cku′k(x)| 6 ε. Ìíîãîìåðíûé ñëó÷àé àíàëîãè÷åí. Åñëè âìåñòîvk ∈ C1
0(Ω) ïîòðåáîâàòü ïðèíàäëåæíîñòü ýòèõ ôóíêöèé ñîáîëåâñêîìó ïðîñòðàíñòâó W 12 (Ω) óñëîâèå
ïîëíîòû çàïèøåòñÿ òàê∫Ω|y(x)−∑m
k=1 Ckuk(x)|2 + |y′(x)−∑mk=1 Cku′k(x)|2dx ≤ ε. Òàêèå íàáîðû
ôóíêöèé vk(x) íàçûâàþò êîîðäèíàòíûìè.)Ïðèìåð
1. Íàáîð xk, åãî ïîëíîòà íå èçìåíèòñÿ, åñëè åãî äîìíîæèòü íà íåïðåðûâíóþ ôóíêöèþ, íåîáðàùàþùóþñÿ âíóòðè îòðåçêà â 0, òàê íàáîð uk = (x − x0)(x − x1)xk ÿâëÿåòñÿ íàáîðîìêîîðäèíàòíûõ ôóíêöèé äëÿ çàäà÷è (4.18).
2. Íàáîð xk11 · . . . · xkn
n ω(x), ω|∂Ω = 0 - êîîðäèíàòíûå ôóíêöèè äëÿ çàäà÷è (4.19).
3. ×àñòî èñïîëüçóþòñÿ ñîáñòâåííûå ôóíêöèè çàäà÷è Øòóðìà-Ëèóâèëëÿ (íàïðèìåð, sin πk(x−x0)x1−x0
èëè ñîáñòâåííûå ôóíêöèè çàäà÷è ∆u + λu = 0 c îäíîðîäíûìè ãðàíè÷íûìè óñëîâèÿìè.)
Âîçüìåì ëèíåéíóþ êîìáèíàöèþ
y(k) =k∑
i=1
C(k)i ui(x),
ïîäñòàâèì â ôóíêöèîíàë, ïðîèíòåãðèðóåì,
J [y(k)] = Φ(C1, . . . , Ck),
òîãäà íåîáõîäèìûå óñëîâèÿ ýêñòðåìóìà ∂Φ∂Ci
= 0 ñîñòàâëÿþò ñèñòåìó (íåëèíåéíûõ) óðàâíåíèé íàêîíñòàíòû, îïðåäåëèâ êîòîðûå ìîæíî íàéòè ðåøåíèå âàðèàöèîííîé çàäà÷è.
Ïðèìåð Çàäà÷à Øòóðìà-Ëèóâèëëÿ y′′ + λy = 0, y(0) = 0, y(1) = 0, ïåðâîå ñîáñòâåííîå ÷èñëîλ1 = π2. Ïîïðîáóåì îïðåäåëèòü åãî ìåòîäîì Ðèòöà. Äëÿ ýòîãî ðàññìîòðèì âàðèàöèîííóþ çàäà÷óJ [y] =
∫ 1
0y′2dx → min ïðè óñëîâèè J1[y] =
∫ 1
0y2dx = 1. Âîçüìåì îäíó ôóíêöèþ u1(x) = x(1 − x),
òîãäà y(1) = Cx(1− x), òîãäà∫ 1
0
(y′2 − λy2)dx =13
(C2 − λ
C2
10
)= Φ(C),
ïðèðàâíèâÿ Φ′(C) íóëþ èìååì λ = 10.
57
Ïðèìåð íåñàìîñîïðÿæåííîé çàäà÷è (ìåòîä Áóáíîâà-Ãàëåðêèíà)
y′′ + by′ + cy = f, (Ly = f),
ïðèáëèæåííîå ðåøåíèå y(m) =∑
Ciui(x), ïîòðåáóåì, ÷òîáû Lu − f ðàâíÿëîñü íóëþ íà ïîäïðî-ñòðàíñòâå u1, . . . , um, ò. å.
(Ly(m) − f, ui) = 0, i = 1, . . . ,m.
Ïðèìåð (ìåòîä êîíå÷íîãî ýëåìåíòà)
J [y] =∫ l
0
[p(x)y′2 + q(x)y2 − 2f(x)y]dx, y(0) = y(l) = 0
Ðàçîáüåì ïðîìåæóòîê ñ øàãîì l/n = h, âîçüìåì ui(x) êàê ïîêàçàíî íà ðèñóíêå (áóäåì ïðåäñòàâëÿòüôóíêöèþ ëîìàíûìè). (Åñëè â çàäà÷å òðåáóåòñÿ ãëàäêîñòü èñïîëüçóþò ñïëàéíû, ò. å. ôóíêöèè, íàêàæäîì îòðåçêå óäîâëåòâîðÿþùèå, íàïðèìåð, óðàâíåíèþ y(3) = 0.) Âû÷èñëÿÿ
J [y(m)] = . . . +
ih∫
(i−1)h
p(x)y′(m)2 + q(x)y(m)2 − 2f(x)y(m)]dx +
(i+1)h∫
ih
. . . dx + . . . =
= . . .+
ih∫
(i−1)h
[p(x)
(Ci−1u
′i−1 + Ciu
′i
)2 + q (Ci−1ui−1 + Ciui)2 − 2f (Ci−1ui−1 + Ciui)
]dx+
(i+1)h∫
ih
. . . dx+. . . ,
ìû âèäèì, ÷òî êîíñòàíòà Ci âõîäèò â äâà èíòåãðàëà, ïîýòîìó ñèñòåìà Áóáíîâà-Ãàëåðêèíà ïðèíè-ìàåò âèä
aiCi−1 + biCi + diCi+1 + fi = 0, (i = 0, . . . , 1)
- âàðèàöèîííî-ðàçíîñòíàÿ ñõåìà.
4.11 Ìåòîä Ðèòöà äëÿ ïðîñòåéøåãî êâàäðàòè÷íîãî ôóíêöèî-íàëà
Ðàññìîòðèì
J [y] =∫ l
0
[p(x)y′2 + q(x)y2 − 2f(x)y]dx, y(0) = y(l) = 0
Ïóñòü p(x) > 0, q(x) > 0. Âîçüìåì η ∈ C10 [0, l] è ðàññìîòðèì ðàçíîñòü
J [y + η]− J [η] =∫ l
0
[2p(x)y′η′ + p(x)η′2 + 2q(x)yη + q(x)η2 − 2fη]dx =
ïðîèíòåãðèðîâàâ â ïåðâîì ñëàãàåìîì ïî ÷àñòÿì
= 2py′η|l0 +∫ l
0
[−2η
d
dxp(x)
dy
dx+ 2q(x)yη − 2fη
]dx +
∫ l
0
(pη′2 + qη2
)dx =
è â ñèëó íåîáõîäèìîãî óñëîâèÿ ýêñòðåìóìà ñëàãàåìîå â êâàäðàòíûõ ñêîáêàõ îáðàùàåòñÿ â íóëü,âíåèíòåãðàëüíûé ÷ëåí ïðîïàäàåò èç-çà óñëîâèé íà η(x)
=∫ l
0
(pη′2 + qη2
)dx > 0,
ñëåäîâàòåëüíî äîñòèãàåòñÿ àáñîëþòíûé ýêñòðåìóì.Ðàññìîòðèì ïîëíóþ ñèñòåìó uk(x), óäîâëåòâîðÿþùóþ êðàåâûì óñëîâèÿì. Ïðåäñòàâèì
y(m)) =m∑1
C(m)k uk(x),
58
ïîäñòâàëÿÿ â ôóíêöèîíàë è èíòåãðèðóÿ èìååì
J [y(m)] = Φ(C1, . . . , Cm) =m∑
i,j=1
αijC(m)i C
(m)j +
m∑
i=1
βiC(m)i .
ßñíî, ÷òî ìàòðèöó αij ìîæíî âûáðàòü ñèììåòðè÷íîé. Êðîìå òîãî, ýòà ìàòðèöà ïîëîæèòåëüíîîïðåäåëíà. Íåîáõîäèìûå óñëîâèÿ ýêñòðåìóìà Φ ïðèâîäÿò ê ñèñòåìå ëèíåéíûõ óðàâíåíèé
2m∑
j=1
αkjC(m)j + βk = 0, k = 1, . . . , m, (4.20)
â ñèëó ïîëîæèòåëüíîé îïðåäåëåííîñòè αkj ñèñòåìà èìååò åäèíñòâåííîå ðåøåíèå.Çàêëþ÷åíèÿ
1. Ïðè äðóãîì âûáîðå êîíñòàíò, íåæëè ðåøåíèÿ ñèñòåìû (4.20), çíà÷åíèå ôóíêöèîíàëà ðàçâå÷òî âîçðàñòàåò.
2. Ïðè óâåëè÷åíèè ÷èñëà áàçèñíûõ ôóíêöèé çíà÷åíèå ôóíêöèîíàëà ðàçâå ÷òî óáûâàåò.
Äîêàæåì ñõîäèìîñòü ðåøåíèÿ.Ñëàáàÿ ñõîäèìîñòü (J [y(m)] → J [y]). Â ñèëó ïîëíîòû ñèñòåìû áàçèñíûõ ôóíêöèé, äëÿ ∀δ > 0
|y(x)−n∑
i=1
Aiui(x)| < δ
|y′(x)−n∑
i=1
Aiu′i(x)| < δ
Ïóñòü z =∑
Aiui, y(m) - ïðèáëèæåíèå Ðèòöà. Ïîñêîëüêó J [z]−J [y] < ε(δ), òî è J [y(m)]−J [y] < ε(δ),à ïîñêîëüêó ïðè p > n J [y(p)] 6 J [y(n)], òî è J [y(p)] − J [y] < ε(δ) äëÿ âñåõ p > m. Òàêèì îáðàçîìñëàáàÿ ñõîäèìîñòü (ñõîäèìîñòü ïî ôóíêöèîíàëó) äîêàçàíà.
Ðàâíîìåðíàÿ ñõîäèìîñòü Ïóñòü η(n) = y − y(n), òîãäà
J [y(n)]− J [y] =∫ l
0
[p(x)(y′(n) − y′)2 + q(x)((y(n) − y)2
]dx ≤ ε
(áëàãîäàðÿ ñëàáîé ñõîäèìîñòè). Äàëåå∫ l
0
[p(x)(y′(n) − y′)2 + q(x)(y(n) − y)2
]dx >
∫ l
0
p(x)(y′(n) − y′)2dx >
(ïîñêîëüêó q(x) > 0, ïóñòü p0 = min p(x) > 0)
> p0
∫ l
0
(y′(n) − y′)2dx,
çíà÷èò ∫ l
0
(y′(n) − y′)2dx ≤ ε
p0
Òîãäà
|y(n)(x)− y(x)|2 ≤∣∣∣∣∫ x
0
(y′(n)(t)− y′(t))dt
∣∣∣∣2
6∫ x
0
(y′(n)(t)− y′(t))2dt
∫ x
0
12dx 6 l
∫ l
0
(y′(n)− y′)2dx 6 ε′.
Ðàâíîìåðíàÿ ñõîäèìîñòü äîêàçàíà.
59
4.12 Çàêîíû ñîõðàíåíèÿ
J [y(x)] =∫ x1
x0
F (x, y, y′)dx
y(x0) = y0
y(x1) = y1
Óðàâíåíèå Ýéëåðà ∂F∂y − d
dx∂F∂y = 0.
1. Çàêîí ñîõðàíåíèÿ èìïóëüñà. Ïóñòü ∂F∂y = 0, òîãäà èç óðàâíåíèÿ Ýéëåðà d
dx∂F∂y = 0 èëè
p =∂F
∂y= const
âäîëü òðàåêòîðèè äâèæåíèÿ - çàêîí ñîõðàíåíèÿ èìïóëüñà.Îòìåòèì, êñòàòè, ÷òî èç óñëîâèÿ ∂F
∂y = 0 ñëåäóåò, ÷òî çàäà÷à íå ìåíÿåòñÿ ïðè ïðåîáðàçî-âàíèÿõ y → y + const. (Ñì. òåîðåìó Íåòåð - â êëàññè÷åñêîé ìåõàíèêå äîêàçûâàåòñÿ, ÷òîèíâàðèàíòíîñòè ñèñòåìû îòíîñèòåëüíî ïðåîáðàçîâàíèé êîîðäèíàò è âðåìåíè ñîîòâåòñòâóþòçàêîíû ñîõðàíåíèÿ.)
2. Çàêîí ñîõðàíåíèÿ ýíåðãèè. Ïóñòü ñèñòåìà íå ìåíÿåòñÿ ïðè ïðåîáðàçîâàíèÿõ x → x + const,èíà÷å ãîâîðÿ ∂F
∂x = 0. Òîãäà ñîõðàíÿåòñÿ âåëè÷èíà
E = y′∂F
∂y′− F.
 ñàìîì äåëå, èç óðàâíåíèÿ Ýéëåðà
∂F
∂y− ∂2F
∂y∂y′y′ − ∂2F
∂y′2y′′ = 0,
à ñ äðóãîé ñòîðîíû
dE
dx= y′′
∂F
∂y′+ y′
(∂2F
∂y∂y′y′ +
∂2F
∂y′2y′′
)− ∂F
∂yy′ − ∂F
∂y′y′′ =
= y′(−∂F
∂y+
∂2F
∂y∂y′y′ +
∂2F
∂y′2y′′
)= 0
3. Ìíîãîìåðíûé ñëó÷àé (ñëó÷àé íåñêîëüêèõ ôóíêöèîíàëüíûõ àðãóìåíòîâ)
J [y(x)] =∫ x1
x0
F (x,y,y′)dx
Ñèñòåìà óðàâíåíèé Ýéëåðà
∂F
∂yi− d
dx
∂F
∂yi= 0, i = 1, . . . , n
• Åñëè äëÿ êàêîãî-ëèáî íîìåðà k ∂F∂yk
= 0, òî
pk =∂F
∂y′k= const
(çàêîí ñîõðàíåíèÿ èìïóëüñà)• Åñëè ∂F
∂x = 0, òî ñîõðàíÿåòñÿ âåëè÷èíà
E =n∑
k=1
y′k∂F
∂y′k− F.
60
 ñàìîì äåëå, óðàâíåíèÿ Ýéëåðà
∂F
∂yi−
n∑
j=1
∂2F
∂yj∂y′iy′j −
n∑
j=1
∂2F
∂y′jy′i
y′′j = 0,
à ñ äðóãîé ñòîðîíû
dE
dx=
n∑
k=1
y′′k∂F
∂y′k+
n∑
k=1
y′n
n∑
j=1
∂2F
∂yj∂y′iy′j +
n∑
k=1
y′n
n∑
j=1
∂2F
∂y′jy′i
y′′j −n∑
k=1
∂F
∂yky′k −
n∑
k=1
∂F
∂y′ky′′k =
= −n∑
k=1
y′n
∂F
∂yi−
n∑
j=1
∂2F
∂yj∂y′iy′j −
n∑
j=1
∂2F
∂y′jy′i
y′′j
= 0.
(â ñèëó óðàâíåíèé Ýéëåðà)
4.13 Äèôôåðåíöèàëüíûå çàêîíû ñîõðàíåíèÿÐàññìàòðèâàåì ñëó÷àé ôóíêöèîíàëà, çàâèñÿùåãî îò ôóíêöèè ìíîãèõ ïåðåìåííûõ (òàêèå ôóíêöè-îíàëû îïèñûâàþò, íàïðèìåð, ïîëÿ èëè ìàòåðèàëüíûå ñðåäû) :
J [u(x)] =∫
Ω
F (x, u,∂u
∂x)dx
Óðàâíåíèå Ýéëåðà∂F
∂u−
n∑
i=1
∂
∂xi
∂F
∂qi= 0, qi =
∂u
∂xi
1. ∂F∂u = 0, òîãäà
∑ ∂
∂xi
∂F
∂qi= 0,
èíà÷å ãîâîðÿ div φ = 0, ãäå φ =(
∂F∂q1
, . . . , ∂F∂qn
).
2. Ðàññìîòðèì ôóíêöèîíàë (îïèñûâàþùèé, íàïðèìåð, êîëåáàíèÿ ñïëîøíîé ñðåäû)
J [u(t,x)] =∫ t2
t1
dt
∫
Ω
dxF (t,x,∂u
∂t,∂u
∂x),
îïðåäåëèì ïëîòíîñòü ýíåðãèèE =
∂u
∂t
∂F
∂ ∂u∂t
− F
è ïëîòíîñòü ïîòîêà èìïóëüñàQi =
∂u
∂t
∂F
∂ ∂u∂xi
.
Òîãäà, åñëè ∂F∂t = 0, òî
∂E∂t
+ div Q = 0 (4.21)
Ïðèìåð
F =12
(ρ
(∂u
∂t
)2
− T
(∂u
∂x
)2)
òîãäà
E =12
(ρ
(∂u
∂t
)2
+ T
(∂u
∂x
)2)
61
3. Äîêàæåì çàêîí ñîõðàíåíèÿ (4.21). Íà ðåøåíèÿõ u = u(x) óðàâíåíèé äâèæåíèÿ ýíåðãèÿ èïîòîê èìïóëüñà ÿâëÿþòñÿ ôóíêöèÿìè êîîðäèíàò è âðåìåíè, çàïèøåì ýòî â âèäå
E =∂u
∂tΨ(x, t)− Φ(x, t)
Qi =∂u
∂tχi(x, t).
Òîãäà∂E∂t
=∂2u
∂t2Ψ +
∂u
∂t
∂Ψ∂t
− ∂Φ∂t
è àíàëîãè÷íî äëÿ Q. Íåïîñðåäñòâåííî âû÷èñëÿÿ ïðîèçâîäíûå è ïîëüçóÿñü óðàâíåíèåì Ýé-ëåðà ìîæíî óáåäèòüñÿ â èñòèííîñòè (4.21) (âûêëàäêè, èç-çà ãðîìîçäêîñòè è î÷åâèäíîñòè,îïóùåíû). Îäíàêî âûäåëåííîñòü âðåìåíè ÿâëÿåòñÿ ñëåäñòâèåì ìåõàíè÷åñêîé èíòåðïðåòàöèèçàäà÷è, ïîýòîìó ðàññìîòðèì
4. Îáùèé ñëó÷àé. Ïóñòü u = u(x1, . . . , xn), ââåäåì òåíçîð
Tik = uxi
∂F
∂uxk
− δikF
(íèæíèé èíäåêñ xi îçíà÷àåò äèôôåðåíöèðîâàíèå ïî ñîîòâåòñòâóþùåé ïåðåìåííîé.) Ïóñòü∂F∂xk
= 0, òîãäà äèôôåðåíöèàëüíûé çàêîí ñîõðàíåíèÿ çàïèøåòñÿ â âèäå∑
i
∂Tik
∂xk= 0.
Åñëè æåF = F
(x, u1, . . . , ul,
∂u1
∂x, . . . ,
∂ul
∂x
),
òî
Tik =l∑
j=1
uj,xi
∂F
∂uj,xk
− δikF
(ïîäðîáíîñòè ñì. Äóáðîâñêèé è äð.)
4.14 Ïàðàìåòðè÷åñêàÿ ôîðìà íåîáõîäèìûõ óñëîâèé ýêñòðå-ìóìà
Ïðîñòåéøàÿ çàäà÷à âàðèàöèîííîãî èñ÷èñëåíèÿ
J [y(x)] =∫ x2
x1
F (x, y, y′)dx
Îäíàêî, ðåøåíèåì òàêîé çàäà÷è ìîæåò áûòü íåîäíîçíà÷íàÿ ôóíêöèÿ (íàïðèìåð, ðàññìîòðèì èçî-ïåðèìåòðè÷åñêóþ çàäà÷ó ïðè J1 = l > π(x2−x1)
2 , ñì. ðèñ. 4.1). Ïîýòîìó áóäåì èñêàòü ðåøåíèÿ âïàðàìåòðè÷åñêîì âèäå (ââåäåì ïàðàìåòð t).
x = x(t) y = y(t),
J [y] =∫ t2
t1
F (x, y,y
x)xdt
(òî÷êîé îáîçíà÷åíî äèôôåðåíöèðîâàíèå ïî t. Ëåãêî âèäåòü, ÷òî ïðè çàìåíå ïàðàìåòðèçàöèè âèäíîâîãî ôóíêöèîíàëà íå ìåíÿåòñÿ.)
Êðîìå ýòîãî, çàìåòèì, ÷òî ïðè ïðåîáðàçîâàíèè x → kx, y → ky, ïîäûíòåãðàëüíàÿ ôóíêöèÿ çà-ìåíÿåòñÿ F (x, y, y
x )x → kF (x, y, yx )x, ò. å. ïîäûíòåãðàëüíàÿ ôóíêöèÿ îäíîðîäíà ïî ïàðå ïåðåìåííûõ
x, y. Ïîýòîìó áóäåì ðàññìàòðèâàòü ôóíêöèîíàë áîëåå îáùåãî âèäà
J [x(t), y(t)] =∫ t2
t1
F (x, y, x, y)dt,
62
Íåò îäíîçíà÷íîñòè
Ðèñ. 4.1: Èçîïåðèìåòðè÷åñêàÿ çàäà÷à.
ïðè÷åì F áóäåì ñ÷èòàòü îäíîðîäíîé ôóíêöèåé äâóõ ïîñëåäíûõ àðãóìåíòîâ.Íåîáõîäèìûå óñëîâèÿ ýêñòðåìóìà J çàïèñûâàþòñÿ â ñëåäóþùåì âèäå :
∂F
∂x− d
dt
∂F
∂x= 0
∂F
∂y− d
dt
∂F
∂y= 0.
Íåñìîòðÿ íå òî, ÷òî ìû ïîëó÷èëè ñèñòåìó èç äâóõ óðàâíåíèé âìåñòî îäíîãî, òðåáîâàíèå îäíîðîä-íîñòè F ñóùåñòâåííî óïðîùàåò çàäà÷ó, à èìåííî, ñåé÷àñ ìû ïîêàæåì, ÷òî ïîëó÷åííûå óðàâíåíèÿëèíåéíîçàâèñèìû. Ñäåëàåì ýòî â ìíîãîìåðíîì ñëó÷àå :
J [x1(t), . . . , xn(t)] =∫ t2
t1
F (x, x)dt,
F (x, kx) = kF (x, x),
ýêñòðåìàëè óäîâëåâòîðÿþò ñèñòåìå óðàâíåíèé
∂F
∂xi− d
dt
∂F
∂xi= 0, (i = 1, . . . , n),
ðàñêðûâàÿ â ÿâíîì âèäå ïðîèçâîäíóþ ïî âðåìåíè
∂F
∂xi−
n∑
k=1
(∂2F
∂xkxixk +
∂2F
∂xkxixk
)= 0, (i = 1, . . . , n). (4.22)
Ëåãêî âèäåòü, ÷òî∑n
i=1 xiδFδxi
≡ 0, â ñàìîì äåëå, ïî òåîðåìå Ýéëåðà îá îäíîðîäíûõ ôóíêöèÿõ,âñÿêóþ îäíîðîäíóþ ôóíêöèþ ìîæíî ïðåäñòàâèòü â âèäå
F =n∑
i=1
xi∂F
∂xi. (4.23)
Äèôôåðåíöèðóÿ ýòî ðàâåíñòâî ïî xk ïîëó÷àåì
∂F
∂xk=
n∑
i=1
xi∂2F
∂xk∂xi,
à ïðîäèôôåðåíöèðîâàâ (4.23) ïî xk
∂F
∂xk=
∂F
∂xk+
n∑
i=1
xi∂2F
∂xk∂xi,
èíà÷å ãîâîðÿn∑
i=1
xi∂2F
∂xk∂xi≡ 0.
63
Óìíîæèâ óðàâíåíèÿ Ýéëåðà (4.22) íà xi è ïðîñóììèðîâàâ ïî i ïîëó÷èìn∑
i=1
xi
[∂F
∂xi−
n∑
k=1
(∂2F
∂xkxixk +
∂2F
∂xkxixk
)]=
ìåíÿÿ ïîðÿäîê ñóììðîâàíèÿ â ïîñëåäíåì ñëàãàåìîì
=n∑
k=1
xk
n∑
i=1
xi∂2F
∂xk∂xi+
∑
i
xi∂F
∂xi−
∑
k
xk∂F
∂xk= 0,
÷òî è òðåáîâàëîñü äîêàçàòü.Òàêèì îáðàçîì, ïîñëå ïàðàìåòðèçàöèè çàäà÷è îäíî èç óðàâíåíèé Ýéëåðà îêàçàëîñü ëèøíèì -
åãî ìîæíî óáðàòü. Êðîìå ýòîãî, åñòü äîïîëíèòåëüíàÿ ñâîáîäà â âûáîðå ïàðàìåòðà, íàïðèìåð :
1.∑
k x2k = 1 - t - äëèíà äóãè
2. F (x, x) = 1 íà ðåøåíèÿõ ñèñòåìû - t - "ôèêñëåðîâà"äëèíà äóãè.
4.15 Îáùàÿ ôîðìà âàðèàöèè ôóíêöèîíàëàÐàññìîòðèì ïðîñòåéøèé ôóíêöèîíàë
J [y] =∫ x2
x1
F (x, y, y′)dx,
â êà÷åñòâå êðèâûõ ñðàâíåíèÿ ðàññìîòðèì ôóíêöèè y = y(x, α), ïðè÷åì y(x, 0) = y (ýêñòðåìàëü).Ïóñòü, êðîìå ýòîãî,
x1 = x1(α), x2 = x2(α)
x1(0) = x1, x2(0) = x2.
 òàêîì ñëó÷àå
J [y(x, α)] =∫ x2(α)
x1(α)
F (x, y(α), y′(α))dx = Φ(α),
âàðèàöèÿ ôóíêöèîíàëàδJ = Φ′(0)α,
âàðèàöèÿ ôóíêöèèδy =
∂y
∂αα,
è âàðèàöèè êîíöîâ ïðîìåæóòêà
δx1 =∂x1
∂αα, δx2 =
∂x2
∂αα,
êðîìå ýòîãî δy′ = ddxδy.
Íåîáõîäèìîå óñëîâèå ýêñòðåìóìà δJ = 0 (ïðè α = 0). Âû÷èñëÿÿ âàðèàöèþ ôóíêöèîíàëà íàõî-äèì
δJ = F (x, y, y′)|y=y,x=x2δx2−F (x, y, y′)|y=y,x=x1
δx1+∫ x2
x1
(∂F
∂y
∣∣∣∣y=y
δy +∂F
∂y′
∣∣∣∣y=y
δy′)
dx = (4.24)
F (x, y, y′)|y=y,x=x2δx2 − F (x, y, y′)|y=y,x=x1
δx1 +∫ x2
x1
(∂F
∂y− d
dx
∂F
∂y′
)
y=y
δydx+
∂F
∂y′
∣∣∣∣y=y,x=x2
(δy)|x2 −∂F
∂y′
∣∣∣∣y=y,x=x1
(δy)|x1
Çàìåòèì, ÷òî (δy)|x1,2 ñâÿçàíî ñëåäóþùèì îáðàçîì ñ âàðèàöèåé ôóíêöèè (ðèñ. 4.2):
δy2 =∂y2(x2(α), α)
∂α
∣∣∣∣α=0
α +∂y2
∂x
∣∣∣∣x=x2
dx2
dα
∣∣∣∣α=0
α = (δy)|x2 + δx2y′(x2)
64
2)(
xyd
2yd
Ðèñ. 4.2:
(è àíàëîãè÷íî ïðè x = x1.)Ïðîäîëæàÿ (4.25) ïîëó÷àåì
δJ =(
F (x, y, y′)− y′∂F
∂y′
)∣∣∣∣y=y,x=x2
δx2 −(
F (x, y, y′)− y′∂F
∂y′
)∣∣∣∣y=y,x=x1
δx1+
∂F
∂y′
∣∣∣∣y=y,x=x2
δy2 − ∂F
∂y′
∣∣∣∣y=y,x=x1
δy1 +∫ x2
x1
(∂F
∂y− d
dx
∂F
∂y′
)
y=y
δydx (4.25)
(Òàêîé âèä âàðèàöèè ôóíêöèîíàëà ìîæåò èñïîëüçîâàòüñÿ â çàäà÷àõ ñ íåîïðåäåëåííîé ãðàíèöåé,íàïðèìåð, ïðè èçó÷åíèè ôàçîâûõ ïåðåõîäîâ.)
4.16 Óñëîâèå òðàíñâåðñàëüíîñòèÏóñòü êîíöû êðèâûõ ñðàâíåíèÿ ñêîëüçÿò ïî êàêèì-ëèáî êðèâûì (λ) (ïîäîáíûå çàäà÷è âñòðå÷àþòñÿâ ãåîìåòðè÷åñêîé îïòèêå). Çàäàäèì êðèâóþ â âèäå y = y(x), òîãäà
δy
δx= y′(x).
Ïðè ýòîì âíåèíòåãðàëüíûå ñëàãàåìûå â îáùåì âèäå âàðèàöèè ôóíêöèîíàëà (ñì. ôîðìóëó (4.25)èç ïðåäûäóùåãî ïàðàãðàôà) çàïèñûâàþòñÿ êàê
(F (x, y, y′)− y′
∂F
∂y′
)∣∣∣∣y=y,x=x2
δx2 +∂F
∂y′y′(x)δx2 = 0
èëèF |y=y + (y′ − y′)
∂F
∂y′= 0, x = x2. (4.26)
Òàêîå ãðàíè÷íîå óñëîâèå íàçûâàþò óñëîâèåì òðàíñâåðñàëüíîñòè (èëè óñëîâèåì ïåðåñå÷åíèÿ).Åñëè êðèâàÿ λ çàäàíà íåÿâíî óðàâíåíèåì ϕ(x, y) = 0, óñëîâèå òðàíñâåðñàëüíîñòè ïðèíèìàåò
âèä : (F (x, y, y′)− y′ ∂F
∂y′
)∣∣∣y=y,x=x2
ϕx=
∂F∂y′
ϕy,
â ñàìîì äåëå, âåêòîð êàñàòåëüíîé ê êðèâîé λ èìååò êîìïîíåíòû (ϕy,−ϕx), çäåñü íèæíèì èíäåêñîìîáîçíà÷åíî äèôôåðåíöèðîâàíèå ïî ñîîòâåòñòâóþùåé êîîðäèíàòå.
 ìíîãîìåðíîì ñëó÷àå F = F (x,y,y′), îáùèé âèä âàðèàöèè ôóíêöèîíàëà
δJ =
(F −
∑
i
y′i∂F
∂y′i
)δx2 +
∑
i
∂F
∂y′iδy2,i + . . . ,
ãäå òî÷êàìè îáîçíà÷åíû îñòàëüíûå ñëàãàåìûå (âíåèíòåãðàëüíûå ÷ëåíû â òî÷êå x1 è èíòåãðàë).Ïóñòü
λ : ϕ(x, y1, . . . , yn) = 0,
òîãäà óñëîâèå òðàíñâåðñàëüíîñòè çàïèøåòñÿ êàê
F (x, y, y′)−∑i y′i
∂F∂y′i
ϕx=
∂F∂y′1
ϕy1
= . . . =∂F∂y′n
ϕyn
.
65
4.17 Ïðåîáðàçîâàíèå ËåæàíäðàÐàññìîòðèì y = f(x), ïîòðåáóåì f ′′(x) > 0 è ñîñòàâèì F (x, p) = px− f(x), çíà÷åíèå ýòîé ôóíêöèèåñòü ðàññòîÿíèå ïî âåðòèêàëè ìåæäó ãðàôèêàìè f(x) è px. Òî÷êà íàèáîëüøåãî âåðòèêàëüíîãîóäàëåíèÿ îïðåäåëÿåòñÿ èç óðàâíåíèÿ
∂F
∂x= p− f ′(x) = 0,
ýòî óðàâíåíèå îïðåäåëÿåò ôóíêöèþ x(p).
Îïðåäåëåíèå 8 Íàçîâåìg(p) = F (x(p), p) = px− f(x)
ïðåîáðàçîâàíèåì Ëåæàíäðà ôóíêöèè f(x).
Ïðèìåðû
1. f(x) = x2, òîãäà F (x, p) = px− x2, óðàâíåíèå íà x ïðèíèìàåò âèä p− 2x = 0 èëè g(p) = p2/4.
2. f(x) = mx2
2 , ñëåäîâàòåëüíî g(p) = p2
2m .
3. f(x) = xα
α , ïóñòü α > 1. Çàïèøåì F (x, p) = px− xα
α , óðàâíåíèå p− xα−1 = 0, òîãäà
g(p) =(
1− 1α
)p
αα−1 .
Çàïèøåì ýòî íåñêîëüêî èíà÷å. Ïóñòü1α
+1β
= 1,
òîãäàxα
α→ pβ
β.
Ïîñêîëüêó âûïîëíåíî px− f(x) 6 g(p) èëè
f(x) + g(p) > px
ïîëó÷àåì äëÿ ôóíêöèé èç òðåòüåãî ïðèìåðà íåðàâåíñòâî Þíãà
xα
α+
pβ
β> px
ïðè1α
+1β
= 1, α, β > 1.
Îïðåäåëåíèå 9 Ôóíêöèè f(x) è g(p) íàçûâàþò äâîéñòâåííûìè ïî Þíãó
(Çàìå÷àíèå Ýòî îïðåäåëåíèå ñîäåðæèò óòâåðæäåíèå èíâîëþòèâíîñòè ïðåîáðàçîâàíèÿ Ëåæàíäðà
f(x) → g(p) ⇒ g(p) → f(x).
Äîêàçàòåëüñòâî èíâîëþòèâíîñòè ìîæíî íàéòè â êíèãàõ Â. È. Àðíîëüäà "Ìàòåìàòè÷åñêèå ìåòîäûêëàññè÷åñêîé ìåõàíèêè"èëè "Îáûêíîâåííûå äèôôåðåíöèàëüíûå óðàâíåíèÿ.)
Ïðåîáðàçîâàíèå Ëåæàíäðà äëÿ ôóíêöèé äâóõ ïåðåìåííûõ îïðåäåëÿåòñÿ ñëåäóþùèì îáðàçîì.Ðàññìîòðèì f(x), ïóñòü ìàòðèöà âòîðûõ ïðîèçâîäíûõ
[∂2f
∂xi∂xk
]
ïîëîæèòåëüíî îïðåäåëåíà. Ïóñòü x = (x1, . . . , xn) è p = (p1, . . . , pn). Ðàññìîòðèì ôóíêöèþ
F (x,p) =n∑
i=1
pixi − f(x),
66
ïðèðàâíèâàÿ íóëþ åå ïðîèçâîäíûå ïî xk ïîëó÷àåì ñèñòåìó óðàâíåíèé
pk =∂f
∂xk, k = 1, . . . , n,
èç êîòîðîé îïðåäåëÿåòñÿ x = x(p).
Îïðåäåëåíèå 10 Ïðåîáðàçîâàíèåì Ëåæàíäðà ôóíêöèè f(x) íàçûâàåòñÿ ôóíêöèÿ
g(p) =n∑
i=1
pixi(p)− f(x(p1, . . . , pn).
4.18 Êàíîíè÷åñêèå ïåðåìåííûå. Óðàâíåíèÿ ÃàìèëüòîíàÐàññìîòðèì ôóíêöèîíàë âèäà
J [y1(x), . . . , yn(x)] =∫ x2
x1
F (x,y,y′)dx. (4.27)
Áóäåì äåëàòü ïðåîáðàçîâàíèå Ëåæàíäðà îòíîñèòåëüíî y′ (ñ÷èòàÿ ïðè ýòîì ÷òî êðèòåðèé ïðè-ìåíèìîñòè ïðåîáðàçîâàíèÿ Ëåæàíäðà âûïîëíåí). Óðàâíåíèÿ íà p :
∂F
∂y′k= pk, k = 1, . . . , n,
òîãäà y′k = y′k(p, x,y), îïðåäåëèì (ïî àíàëîãèè ñ ìåõàíèêîé) ýíåðãèþ ñèñòåìû
E =n∑
i=1
y′i∂F
∂pi− F (x,y,y′)
è ïðåîáðàçîâàíèå Ëåæàíäðà ôóíêöèè F ôóíêöèþ Ãàìèëüòîíà
H(x,y,y′) =n∑
i=1
piy′i(p, x,y)− F (x,y,y′).
Óðàâíåíèÿ Ýéëåðà äëÿ çàäà÷è ýêñòðåìóìà ôóíêöèîíàëà J ãëàñÿò
∂F
∂yi− d
dx
∂F
∂y′i= 0, i = 1, . . . , n,
ëåãêî âèäåòü, ÷òî ∂H∂x = −∂F
∂x è
∂H
∂yk=
∑
i
pi∂y′i∂yk
− ∂F
∂yk−
∑
i
∂F
∂y′i
∂y′i∂yk
= − ∂F
∂yk,
êðîìå òîãî ∂H∂pk
= y′k(p, x,y).Ïîäñòàâëÿÿ ýòî â óðàâíåíèÿ Ýéëåðà ïîëó÷àåì ñèñòåìó óðàâíåíèé Ãàìèëüòîíà :
dyi
dx = ∂H∂pi
dpi
dx = −∂H∂yi
i = 1, . . . , n
(4.28)
4.19 Ïîëå ýêñòðåìàëåé. Äåéñòâèå êàê ôóíêöèÿ êîîðäèíàò.Ïðèíöèï Ãþéãåíñà
Çàïèøåì îáùóþ ôîðìó âàðèàöèè ôóíêöèîíàëà J , âçÿòîãî èç ïðåäûäóùåãî ïàðàãðàôà.
δJ =
(F −
n∑
i=1
y′i∂F
∂y′i
)δx +
n∑
i=1
∂F
∂y′iδyi +
n∑
i=1
∫ x2
x1
(∂F
∂yi− d
dx
∂F
∂y′i
)δyidx, (4.29)
67
ïåðåõîäÿ ê îáîçíà÷åíèÿì ïðåäûäóùåãî ïàðàãðàôà ïîëó÷àåì
δJ = −Hδx +∑
i
piδyi +∑
i
∫ x2
x1
δF
δyiδyidx.
Çàôèêñèðóåì â ïðîñòðàíñòâå (x,y) òî÷êó M1 è áóäåì âûïóñêàòü èç íåå ýêñòðåìàëè ïðè ðàç-ëè÷íûõ çíà÷åíèÿõ pi èëè y′i.
Îïðåäåëåíèå 11 Òàêîå ñåìåéñòâî êðèâûõ íàçûâàþò öåíòðàëüíûì ïîëåì ýêñòðåìàëåé (ëó÷åé).
Ñîïîñòàâèì êàæäîé êðèâîé òàêîå çíà÷åíèå âåðõíåãî ïðåäåëà èíòåãðèðîâàíèÿ x2 â (4.27) òàê, ÷òîáûJ = ρ (äëÿ íåêîòîðîãî ôèêñèðîâàííîãî ÷èñëà ρ).
Îïðåäåëåíèå 12 Ïîëó÷åííàÿ êðèâàÿ x2(p) íàçûâàåòñÿ êâàçèñôåðîé (âîëíîâûì ôðîíòîì).
Íàéäåì óñëîâèÿ ïåðåñå÷åíèÿ ïîëÿ ýêñòðåìàëåé è êâàçèñôåðû. Ïîñêîëüêó âäîëü êâàçèñôåðû(
F −n∑
i=1
y′i∂F
∂y′i
)δx +
n∑
i=1
∂F
∂y′iδyi = 0
èëè−Hδx +
n∑
i=1
piδyi = 0,
ïåðåñå÷åíèå ýêñòðåìàëåé è êâàçèñôåðû òðàíñâåðñàëüíî.Òàê êàê ýêñòðåìàëè íå ïåðåñåêàþòñÿ, à íà ëþáîé êâàçèñôåðå ôóíêöèîíàë J îïðåäåëåí, òî
îïðåäåëèì äåéñòâèå êàê ôóíêöèþ êîîðäèíàò è âðåìåíè ñëåäóþùèì îáðàçîì :
S(x,y) = J íà ñîîòâåòñòâóþùåé òî÷êå êâàçèñôåðû
Ôóíêöèþ S(x,y) ìîæíî ñòðîèòü èíà÷å. Çàôèêñèðóåì íåêîòîðóþ ãèïåðïîâåðõíîñòü (x,y), ïî-ñòðîèì ïîëå ýêñòðåìàëåé ïî óñëîâèþ òðàíñâåðñàëüíîñòè, ïîñëå ÷åãî ïîñòîèì êâàçèñôåðû è îïðå-äåëèì S(x,y) êàê áûëî ñäåëàíî âûøå. Çàìåòèì, ÷òî è â ýòîì ñëó÷àå ïåðåñå÷åíèå ïîëÿ ýêñòðåìàëåéñ êâàçèñôåðîé ïðîèñõîäèò òðàíñâåðñàëüíî.
4.20 Óðàâíåíèå Ãàìèëüòîíà-ßêîáèÇàïèøåì èçìåíåíèå (âàðèàöèþ) äåéñòâèÿ êàê ôóíêöèè êîîðäèíàò :
δS = −Hδx +∑
i
piδyi,
ïîñêîëüêó S ÿâëÿåòñÿ ôóíêöèåé x è y ïîëó÷àåì äèôôåðåíöèàë
dS = −Hdx +∑
i
pidyi.
Èç ýòîé çàïèñè âèäíî, ÷òî ∂S∂x = −H è ∂S
∂yi= pi (i = 1, . . . , n). Ïîñêîëüêó ôóíêöèÿ Ãàìèëüòîíà
H ÿâëÿåòñÿ çàäàííîé ôóíêöèåé òðîéêè ïåðåìåííûõ (x,y,p) òî äåéñòâèå êàê ôóíêöèÿ êîîðäèíàòóäîâëåòâîðÿåò ñëåäóþùåìó óðàâíåíèþ :
∂S
∂x+ H
(x, y1, . . . , yn,
∂S
∂y1, . . . ,
∂S
∂yn
)= 0 (4.30)
Ýòî óðàâíåíèå íàçûâàþò óðàâíåíèåì Ãàìèëüòîíà-ßêîáè. Çàìåòèì, ÷òî óðàâíåíèÿ Ãàìèëüòîíà ÿâ-ëÿþòñÿ óðàâíåíèÿìè õàðàêòåðèñòèê äëÿ óðàâíåíèÿ Ãàìèëüòîíà-ßêîáè.
Âñïîìíèì, ÷òî ïðåîáðàçîâàíèå Ëåæàíäðà ïðèìåíèìî â ñëó÷àå, åñëè ìàòðèöà[
∂2F
∂yi∂yk
]
68
ïîëîæèòåëüíî îïðåäåëåíà.  ñëó÷àå îäíîðîäíîãî ôóíêöèîíàëà F (x,y, ky′) = kF (x,y,y′) ïî òåî-ðåìå Ýéëåðà ïîëó÷àåì, ÷òî
0 =n∑
i=1
y′i∂2F
∂yi∂yk, k = 1, .., n,
ò. å. ïðåîáðàçîâàíèå Ëåæàíäðà íåâîçìîæíî (ìàòðèöà âòîðûõ ïðîèçâîäíûõ âûðîæäåíà). Ïîýòîìóäåéñòâèå êàê ôóíêöèþ êîîðäèíàò ââåäåì ñëåäóþùèì îáðàçîì
dS =
(F −
∑
i
y′i∂F
∂y′i
)dx +
∑
i
∂F
∂y′idyi,
èíà÷å ãîâîðÿ,∂S
∂x= 0
è∂S
∂yi=
∂F
∂y′i, i = 1, . . . , n
Ïîñêîëüêó F îäíîðîäíà ïî y′ ñ ïîêàçàòåëåì îäíîðîäíîñòè 1, òî ∂F/∂y′i îäíîðîäíà ïî y′ ñïîêàçàòåëåì îäíîðîäíîñòè 0.  òàêîì ñëó÷àå ââåäåì
y′iy′1
= ϕi
(y,
∂S
∂y
),
ïîäñòàâëÿÿ ýòè ôóíêöèè â òåîðåìó Ýéëåðà F =∑n
i=1 y′i∂F∂yi
ïîëó÷àåì àíàëîã óðàâíåíèÿ Ãàìèëüòîíà-ßêîáè.
Ìîæíî ïîñòóïèòü èíà÷å, à èìåííî, çàìåíèòü F íà F 2, ýòà ôóíêöèÿ óæå íå áóäåò îäíîðîäíîé èïîñòðîèòü S(x,y) ïî íåé (ñì. Ðóíä).
Ïðèìåð Ôóíêöèîíàë ãåîìåòðè÷åñêîé îïòèêè
F = n(x1, . . . , xn)√
x′12 + . . . + x′n
2,
∂F
∂x′i= n(x)
x′i√x′1
2 + . . . + x′n2
=∂S
∂xi,
òîãäà óðàâíåíèå Ãàìèëüòîíà-ßêîáè äëÿ ãåîìåòðè÷åñêîé îïòèêè çàïèñûâàåòñÿ
n2(x) =(
∂S
∂x1
)2
+ . . . +(
∂S
∂xn
)2
.
Ñôîðìóëèðóåì (áåç äîêàçàòåëüñòâ) íåñêîëüêî çàìå÷àíèé
4.21 Êîðîòêîâîëíîâàÿ (êâàçèêëàññè÷åñêàÿ) àñèìïòîòèêàÐàññìîòðèì óðàâíåíèå Ãåëüìãîëüöà (ñòàöèîíàðíîå âîëíîâîå óðàâíåíèå)
∆u + k2n2(x)u = 0. (4.31)
Çäåñü n(x) ïîêàçàòåëü ïðåëîìëåíèÿ.Ðàññìîòðèì ñëó÷àé k >> 1 (êîðîòêîâîëíîâûé ñëó÷àé äëÿ âîëíîâîãî óðàâíåíèÿ, êâàçèêëàññè-
÷åñêàÿ àñèìïòîòèêà - äëÿ óðàâíåíèÿ Øðåäèíãåðà). Äåáàé ïðåäëîæèë ñëåäóþùèé àíçàö (ensatz)âèäà ðåøåíèÿ:
u = eikS(x)
(a0 +
1k
a1 +1k2
a2 + . . .
),
ãäå ai ÿâëÿþòñÿ ôóíêöèÿìè x. Âû÷èñëèì ïðîèçâîäíûå ðåøåíèÿ :
∂u
∂xi= eikS(x)ik
∂S
∂xi
(a0 +
1k
a1 +1k2
a2 + . . .
)+ eikS(x)
(∂a0
∂xi+
1k
∂a1
∂xi+ . . .
)
69
∂2u
∂x2i
=
[−k2
(∂S
∂xi
)2
+ ik∂2S
∂x2i
]eikS(x)
(a0 +
1k
a1 +1k2
a2 + . . .
)
+2eikS(x)ik∂S
∂xi
(∂a0
∂xi+
1k
∂a1
∂xi+ . . .
)+ eikS(x)
(∂2a0
∂x2i
+1k
∂2a1
∂x2i
+ . . .
).
Ïîäñòàâëÿÿ ýòè ïðîèçâîäíûå â (4.31) ïîëó÷àåì
−k2
[(∂S
∂x1
)2
+ . . . +(
∂S
∂xn
)2]
eikS(x)a0 + k2n2(x)eikS(x)a0
+ik
[∆S + 2
∂S
∂xi
∂a0
∂xi
]eikS(x) − (∇S)2ka1e
ikS(x) + kn2(x)eikS(x)a1 + O(1) = 0, k →∞
Òîãäà, ñîáèðàÿ ñëàãàåìûå ïðè ðàçëè÷íûõ ñòåïåíÿõ k ïîëó÷àåì
k2 :[− (∇S)2 + n2(x)
]eikS(x)a0 = 0,
ïîëîæèì, ÷òî a0 6= 0, òîãäà ïîëó÷àåì óðàâíåíèå
(∇S)2 = n2(x), (4.32)
èk :
(∆S + 2
∂S
∂xi
∂a0
∂xi
)+
[− (∇S)2 + n2(x)
]a0 = 0,
ïðè÷åì ñëàãàåìîå â [] îáðàùàåòñÿ â íóëü â ñèëó (4.32).Óðàâíåíèå (4.32) - óðàâíåíèå ýéêîíàëà (ñð. ñ óðàâíåíèåì Ãàìèëüòîíà-ßêîáè). Ôóíêöèþ S íà-
çûâàþò ýéêîíàëîì (îò iconal - èçîáðàæåíèå).  ñëåäóþùåì ïðèáëèæåíèè (êîýôôèöèåíò ïðè kïîëó÷àåì íåîäíîðîäíîå óðàâíåíèå â ÷àñòíûõ ïðîèçâîäíûõ íà a0(x) è ò. ä.  êà÷åñòâå ãðàíè÷íûõóñëîâèé èñïîëüçóþò ëèáî óñëîâèÿ íà ïëîñêîñòè, ëèáî óñëîâèå íàëè÷èÿ (îòñóòñòâèÿ) ôðîíòà íàáåñêîíå÷íîñòè.
4.22 Èíâàðèàíòíûé èíòåãðàë Ãèëüáåðòà. Ôóíêöèÿ Âåéðøòðàñ-ñà
Ïóñòü S(x,y) äåéñòâèå êàê ôóíêöèÿ êîîðäèíàò. Òîãäà
dS = −Hdx +n∑
i=1
pidyi =(
F −∑
y′i∂F
∂y′i
)dx +
∑ ∂F
∂y′idyi.
Ââåäåì ôóíêöèè íàêëîíà ïîëÿ ti(x,y) : y′i = ti(x,x), òîãäà F = F (x,y, t(x,y)). (Çàìåòèì,÷òî ôóíêöèè ti è pi çàäàþòñÿ ÷åðåç ïîëå ýêñòðåìàëåé.) Òîãäà äèôôåðåíöèàë äåéñòâèÿ ìîæíîïðåäñòàâèòü â âèäå
dS =(
F −∑
ti∂F
∂y′i
)dx +
∑ ∂F
∂y′idyi,
â ýòîì ñëó÷àå
S(x,y) =∫ x
x0
(F +
n∑
i=1
(y′i − ti)∂F
∂y′i
)dx, (4.33)
ïðè÷åì èíòåãðàë íå çàâèñèò îò âûáîðà ïóòè èíòåãðèðîâàíèÿ. Ýòîò èíòåãðàë íàçûâàþò èíâàðè-àíòíûì èíòåãðàëîì Ãèëüáåðòà.
Ïóñòü êðèâàÿ λ : y(x) ÿâëÿåòñÿ ýêñòðåìàëüþ, òîãäà
J [y(x)] =∫
λ
F (x, y, y′)dx,
ñ äðóãîé ñòîðîíû
J [y(x)] = S(x,y) =∫
l
(F +
n∑
i=1
(y′i − ti)∂F
∂y′i
)dx,
70
ãäå l - ïðîèçâîëüíàÿ êðèâàÿ, çàäàííàÿ ôóíêöèåé y(x).  òàêîì ñëó÷àå
J [y]− J [y] =∫
l
[F (x,y,y′)− F (x, y, t)−
n∑
i=1
(y′i − ti)∂F
∂y′i
]dx.
Ââåäåì ôóíêöèþ Âåéðøòðàññà
E(x,y, ξ, η) = F (x,y, η)− F (x,y, ξ)−∑
i
∂F
∂ηi(ξi − ηi),
ñëåäîâàòåëüíîJ [y]− J [y] =
∫E(x,y, t,y′)dx.
Îáðàòèì âíèìàíèå íà òî, ÷òî ïîñëå òîãî, êàê ïîñòðîåíî ïîëå ýêñòðåìàëåé E ñòàíîâèòñÿ èçâåñòíîéôóíêöèåé. Ñôîðìóëèðóåì äîñòàòî÷íîå óñëîâèå ýêñòðåìóìà
Òåîðåìà 4.22.1 Åñëè ýêñòðåìàëü λ îêðóæåíà ïîëåì ýêñòðåìàëåé è E(x,y, t,y′) ≥ 0, òî íà λäîñòèãàåòñÿ ìèíèìóì ôóíêöèîíàëà.
 îêðåñòíîñòè ýêñòðåìàëè
E(x,y, t,y′) = E(x,y, t, t) +n∑
i=1
(y′i − ti)∂E
∂y′i+ . . . =
∑
i,k
∂2F
∂y′i∂y′k(y∗i − ti)(y∗k − tk) + . . . ,
ïîýòîìó èìååò ìåñòî òåîðåìà
Òåîðåìà 4.22.2 Åñëè êâàäðàòè÷íàÿ ôîðìà
∂2F
∂y′i∂y′kηiηk
ïîëîæèòåëüíî îïðåäåëåíà, òî íà λ äîñòèãàåòñÿ ìèíèìóì ôóíêöèîíàëà.
71
Îãëàâëåíèå
1 Ñèíãóëÿðíàÿ çàäà÷à Øòóðìà-Ëèóâèëëÿ 21.1 Ðàâåíñòâî Ïàðñåâàëÿ íà ïîëóîñè . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Îáîáùåííîå ðàâåíñòâî Ïàðñåâàëÿ. Òåîðåìà ðàçëîæåíèÿ . . . . . . . . . . . . . . . . . 61.3 Êðóã è òî÷êà Âåéëÿ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Ôóíêöèÿ Ãðèíà êðàåâîé çàäà÷è íà ïîëóîñè . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Ñâÿçü ìåæäó ôóíêöèåé Âåéëÿ-Òèò÷ìàðøà è ñïåêòðàëüíîé ïëîòíîñòüþ . . . . . . . 91.6 Ïðåîáðàçîâàíèå Ôóðüå íà ïîëóîñè . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.7 Ïðåîáðàçîâàíèå Âåáåðà . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.8 Ðàâåíñòâî Ïàðñåâàëÿ è òåîðåìà ðàçëîæåíèÿ íà âñåé îñè . . . . . . . . . . . . . . . . 121.9 Ôóíêöèÿ Ãðèíà êðàåâîé çàäà÷è äëÿ óðàâíåíèÿ Øòóðìà-Ëèóâèëëÿ íà âñåé îñè . . . 141.10 Ñâÿçü ìåæäó ôóíêöèÿìè Âåéëÿ-Òèò÷ìàðøà è ýëåìåíòàìè ìàòðèöû ñïåêòðàëüíîé
ïëîòíîñòè . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.11 Ïðåîáðàçîâàíèå Ôóðüå íà âñåé îñè . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.12 Ïðåîáðàçîâàíèå Õàíêåëÿ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.13 Ëåììà Ðèìàíà-Ëåáåãà . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.14 ßäðî Äèðèõëå - δ-îáðàçíàÿ ïîñëåäîâàòåëüíîñòü. . . . . . . . . . . . . . . . . . . . . . 201.15 Îðòîãîíàëüíîñòü ñîáñòâåííûõ ôóíêöèé íåïðåðûâíîãî ñïåêòðà. . . . . . . . . . . . . 211.16 Àñèìïòîòèêà ðåøåíèé ñèíãóëÿðíîé çàäà÷è Øòóðìà-Ëèóâèëëÿ ñ ñóììèðóåìûì ïî-
òåíöèàëîì. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.17 Îðòîãîíàëüíîñòü ðåøåíèé ñèíãóëÿðíîé çàäà÷è Øòóðìà-Ëèóâèëëÿ ñ ñóììèðóåìûì
ïîòåíöèàëîì. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.18 Èíòåãðàëüíûå ðàçëîæåíèÿ ïî ôóíêöèÿì íåïðåðûâíîãî ñïåêòðà ñèíãóëÿðíîé çàäà÷è
Øòóðìà-Ëèóâèëëÿ ñ ñóììèðóåìûì ïîòåíöèàëîì. . . . . . . . . . . . . . . . . . . . . 26
2 Ïðåîáðàçîâàíèå Ëàïëàñà 292.1 Ïðåîáðàçîâàíèå Ëàïëàñà. Îïðåäåëåíèå. Àíàëèòè÷íîñòü . . . . . . . . . . . . . . . . . 292.2 Îñíîâíûå ñâîéñòâà ïðåîáðàçîâàíèÿ Ëàïëàñà . . . . . . . . . . . . . . . . . . . . . . . 302.3 Òåîðåìà î ñâåðòêå (Ý. Áîðåëÿ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.4 Îáîáùåííàÿ òåîðåìà î ñâåðòêå . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.5 Ôîðìóëà Ðèìàíà-Ìåëëèíà. Îáðàùåíèå ïðåîáðàçîâàíèÿ Ëàïëàñà . . . . . . . . . . . 342.6 Òåîðåìû ðàçëîæåíèÿ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.7 Ïðèìåðû âû÷èñëåíèÿ èíòåãðàëîâ îò íåîäíîçíà÷íûõ îáðàçîâ . . . . . . . . . . . . . . 36
3 Èíòåãðàëüíûå óðàâíåíèÿ 383.1 Êëàññèôèêàöèÿ ëèíåéíûõ èíòåãðàëüíûõ óðàâíåíèé . . . . . . . . . . . . . . . . . . . 383.2 Àíàëîãèÿ ìåæäó ëèíåéíûì èòåãðàëüíûì óðàâíåíèåì Ôðåäãîëüìà 2-ãî ðîäà è ñèñòå-
ìîé ëèíåíåéíûõ àëãåáðàè÷åñêèõ óðàâíåíèé. Ôîðìóëèðîâêà òåîðåì Ôðåäãîëüìà . . 383.3 Óðàâíåíèÿ Ôðåäãîëüìà 2-ãî ðîäà ñ ìàëûìè ÿäðàìè . . . . . . . . . . . . . . . . . . . 393.4 Óðàâíåíèÿ Ôðåäãîëüìà ñ âûðîæäåííûìè ÿäðàìè . . . . . . . . . . . . . . . . . . . . 403.5 Èíòåãðàëüíûå óðàâíåíèÿ Ôðåäãîëüìà 2-ãî ðîäà, áëèçêèå ê âûðîæäåííûì . . . . . . 413.6 Èíòåãðàëüíûå óðàâíåíèÿ Ôðåäãîëüìà ñ íåïðåðûâíûìè ÿäðàìè . . . . . . . . . . . . 413.7 Èíòåãðàëüíîå óðàâíåíèå Ôðåäãîëüìà ñ ýðìèòîâûì ÿäðîì . . . . . . . . . . . . . . . 423.8 Áèëèíåéíàÿ ôîðìóëà . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.9 Òîåðåìà Ãèëüáåðòà-Øìèäòà. Ðåãóëÿðíàÿ ñõîäèìîñòü ðÿäà . . . . . . . . . . . . . . . 433.10 Âïîëíå íåïðåðûâíûå îïåðàòîðû . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.11 Òåîðåìà ñóùåñòâîâàíèÿ õàðàêòåðèñòè÷åñêèõ ÷èñåë . . . . . . . . . . . . . . . . . . . . 443.12 Òåîðåìà Ãèëüáåðòà-Øìèäòà. Ïîñëåäîâàòåëüíîñòü õàðàêòåðèñòè÷åñêèõ ÷èñåë. . . . . 45
72
3.13 Ôîðìóëà Øìèäòà . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4 Âàðèàöèîííîå èñ÷èñëåíèå 474.1 Îñíîâíûå ëåììû âàðèàöèîííîãî èñ÷èñëåíèÿ . . . . . . . . . . . . . . . . . . . . . . . 474.2 Íåîáõîäèìûå óñëîâèÿ ýêñòåìóìà â ïðîñòåéøåé çàäà÷å âàðèàöèîííîãî èñ÷èñëåíèÿ . 484.3 Íåîáõîäèìûå óñëîâèÿ ýêñòðåìóìà ôóíêöèîíàëà, çàâèñÿùåãî îò íåñêîëüêèõ ôóíêöè-
îíàëüíûõ àðãóìåíòîâ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.4 Íåîáõîäèìîå óñëîâèå ýêñòðåìóìà ôóíêöèîíàëà, çàâèñÿùåãî îò âûñøèõ ïðîèçâîäíûõ 504.5 Íåîáõîäèìûå óñëîâèÿ ýêñòðåìóìà äëÿ ôóíêöèîíàëà, çàâèñÿùåãî îò ôóíêöèè íåñêîëü-
êèõ ïåðåìåííûõ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.6 Èçîïåðèìåòðè÷åñêàÿ çàäà÷à . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.7 Óñëîâíûé ýêñòðåìóì . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.8 Åñòåñòâåííûå êðàåâûå óñëîâèÿ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.9 Åñòåñòâåííûå êðàåâûå óñëîâèÿ äëÿ ñëó÷àÿ ôóíêöèîíàëîâ, ñîäåðæàùèõ âíåèíòåãðàëü-
íûå ÷ëåíû (èëè èíòåãðàëû ïî ïîâåðõíîñòè) . . . . . . . . . . . . . . . . . . . . . . . . 564.10 Ìåòîä Ðèòöà. Ñõåìà . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.11 Ìåòîä Ðèòöà äëÿ ïðîñòåéøåãî êâàäðàòè÷íîãî ôóíêöèîíàëà . . . . . . . . . . . . . . 584.12 Çàêîíû ñîõðàíåíèÿ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.13 Äèôôåðåíöèàëüíûå çàêîíû ñîõðàíåíèÿ . . . . . . . . . . . . . . . . . . . . . . . . . . 614.14 Ïàðàìåòðè÷åñêàÿ ôîðìà íåîáõîäèìûõ óñëîâèé ýêñòðåìóìà . . . . . . . . . . . . . . . 624.15 Îáùàÿ ôîðìà âàðèàöèè ôóíêöèîíàëà . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.16 Óñëîâèå òðàíñâåðñàëüíîñòè . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.17 Ïðåîáðàçîâàíèå Ëåæàíäðà . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.18 Êàíîíè÷åñêèå ïåðåìåííûå. Óðàâíåíèÿ Ãàìèëüòîíà . . . . . . . . . . . . . . . . . . . . 674.19 Ïîëå ýêñòðåìàëåé. Äåéñòâèå êàê ôóíêöèÿ êîîðäèíàò. Ïðèíöèï Ãþéãåíñà . . . . . . 674.20 Óðàâíåíèå Ãàìèëüòîíà-ßêîáè . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.21 Êîðîòêîâîëíîâàÿ (êâàçèêëàññè÷åñêàÿ) àñèìïòîòèêà . . . . . . . . . . . . . . . . . . . 694.22 Èíâàðèàíòíûé èíòåãðàë Ãèëüáåðòà. Ôóíêöèÿ Âåéðøòðàññà . . . . . . . . . . . . . . 70
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