Convergence of expansions for eigenfunctions and asymptotics of the   spectral data of the Sturm-Liouville problem

A.A. Pahlevanyan

arXiv:1902.06594·math.SP·February 19, 2019

Convergence of expansions for eigenfunctions and asymptotics of the spectral data of the Sturm-Liouville problem

A.A. Pahlevanyan

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TL;DR

This paper proves uniform convergence of eigenfunction expansions for a Sturm-Liouville problem with summable potential and uses this to derive more accurate asymptotic formulas for eigenvalues and norming constants.

Contribution

It establishes uniform convergence of eigenfunction expansions for Sturm-Liouville problems with summable potentials, enabling refined asymptotic analysis of spectral data.

Findings

01

Proved uniform convergence for eigenfunction expansions.

02

Derived more precise asymptotic formulas for eigenvalues.

03

Enhanced understanding of spectral data asymptotics.

Abstract

Uniform convergence of the expansion of an absolutely continuous function for eigenfunctions of the Sturm-Liouville problem $- y " + q (x) y = μ y,$ $y (0) = 0,$ $y (π) cos β + y^{'} (π) sin β = 0,$ $β \in (0, π)$ with summable potential $q \in L_{R}^{1} [0, π]$ is proved. This result is used to obtain more precise asymptotic formulae for eigenvalues and norming constants of this problem.

Equations152

- y^{''} + q (x) y = μ y \equiv λ^{2} y, x \in (0, π), μ \in C,

- y^{''} + q (x) y = μ y \equiv λ^{2} y, x \in (0, π), μ \in C,

y (0) cos α + y^{'} (0) sin α = 0, α \in (0, π],

y (0) cos α + y^{'} (0) sin α = 0, α \in (0, π],

y (π) cos β + y^{'} (π) sin β = 0, β \in [0, π),

y (π) cos β + y^{'} (π) sin β = 0, β \in [0, π),

μ_{0} (q, α, β) < μ_{1} (q, α, β) < \dots < μ_{n} (q, α, β) < \dots .

μ_{0} (q, α, β) < μ_{1} (q, α, β) < \dots < μ_{n} (q, α, β) < \dots .

φ (0, μ, α) = sin α, φ^{'} (0, μ, α) = - cos α,

φ (0, μ, α) = sin α, φ^{'} (0, μ, α) = - cos α,

ψ (π, μ, β) = sin β, ψ^{'} (π, μ, β) = - cos β .

ψ (π, μ, β) = sin β, ψ^{'} (π, μ, β) = - cos β .

W_{α, β} (x, μ) := φ (x, μ, α) ψ^{'} (x, μ, β) - φ^{'} (x, μ, α) ψ (x, μ, β) .

W_{α, β} (x, μ) := φ (x, μ, α) ψ^{'} (x, μ, β) - φ^{'} (x, μ, α) ψ (x, μ, β) .

ψ_{n} (x) = β_{n} φ_{n} (x), β_{n} \neq = 0.

ψ_{n} (x) = β_{n} φ_{n} (x), β_{n} \neq = 0.

a_{n} = a_{n} (q, α, β) = 0 \int π φ_{n}^{2} (x) d x, b_{n} = b_{n} (q, α, β) = 0 \int π ψ_{n}^{2} (x) d x,

a_{n} = a_{n} (q, α, β) = 0 \int π φ_{n}^{2} (x) d x, b_{n} = b_{n} (q, α, β) = 0 \int π ψ_{n}^{2} (x) d x,

N \to \infty lim x \in [0, π] max f (x) - n = 0 \sum N c_{n} φ_{n} (x) = 0, c_{n} = \frac{1}{a _{n}} \int_{0}^{π} f (t) φ_{n} (t) d t,

N \to \infty lim x \in [0, π] max f (x) - n = 0 \sum N c_{n} φ_{n} (x) = 0, c_{n} = \frac{1}{a _{n}} \int_{0}^{π} f (t) φ_{n} (t) d t,

N \to \infty lim x \in [a, π] max f (x) - n = 0 \sum N c_{n} φ_{n} (x) = 0, c_{n} = \frac{1}{a _{n}} \int_{0}^{π} f (t) φ_{n} (t) d t,

N \to \infty lim x \in [a, π] max f (x) - n = 0 \sum N c_{n} φ_{n} (x) = 0, c_{n} = \frac{1}{a _{n}} \int_{0}^{π} f (t) φ_{n} (t) d t,

N \to \infty lim x \in [0, b] max f (x) - n = 0 \sum N c_{n} φ_{n} (x) = 0, c_{n} = \frac{1}{a _{n}} \int_{0}^{π} f (t) φ_{n} (t) d t,

N \to \infty lim x \in [0, b] max f (x) - n = 0 \sum N c_{n} φ_{n} (x) = 0, c_{n} = \frac{1}{a _{n}} \int_{0}^{π} f (t) φ_{n} (t) d t,

\frac{π}{2} = n = 0 \sum \infty \frac{sin ( n + \frac{1}{2} ) x}{n + \frac{1}{2}}, 0 < x < 2 π,

\frac{π}{2} = n = 0 \sum \infty \frac{sin ( n + \frac{1}{2} ) x}{n + \frac{1}{2}}, 0 < x < 2 π,

δ_{n} (α, β) := μ_{n} (0, α, β) - n = λ_{n} (0, α, β) - λ_{n} (0, \frac{π}{2}, \frac{π}{2}), n \geq 2,

δ_{n} (α, β) := μ_{n} (0, α, β) - n = λ_{n} (0, α, β) - λ_{n} (0, \frac{π}{2}, \frac{π}{2}), n \geq 2,

δ_{n} (α, β) = \frac{1}{π} arccos \frac{cos α}{( n + δ _{n} ( α , β ) ) ^{2} sin ^{2} α + cos ^{2} α} - - \frac{1}{π} arccos \frac{cos β}{( n + δ _{n} ( α , β ) ) ^{2} sin ^{2} β + cos ^{2} β} .

δ_{n} (α, β) = \frac{1}{π} arccos \frac{cos α}{( n + δ _{n} ( α , β ) ) ^{2} sin ^{2} α + cos ^{2} α} - - \frac{1}{π} arccos \frac{cos β}{( n + δ _{n} ( α , β ) ) ^{2} sin ^{2} β + cos ^{2} β} .

λ_{n} (q, α, β) = n + δ_{n} (α, β) + \frac{[ q ]}{2 ( n + δ _{n} ( α , β ) )} + l_{n} (q, α, β) + O (\frac{1}{n ^{2}}),

λ_{n} (q, α, β) = n + δ_{n} (α, β) + \frac{[ q ]}{2 ( n + δ _{n} ( α , β ) )} + l_{n} (q, α, β) + O (\frac{1}{n ^{2}}),

l_{n} (q, α, β) = \frac{1}{2 π ( n + δ _{n} ( α , β ) )} \int_{0}^{π} q (x) cos 2 (n + δ_{n} (α, β)) x d x, α \in (0, π),

l_{n} (q, α, β) = \frac{1}{2 π ( n + δ _{n} ( α , β ) )} \int_{0}^{π} q (x) cos 2 (n + δ_{n} (α, β)) x d x, α \in (0, π),

l_{n} = l_{n} (q, π, β) = - \frac{1}{2 π ( n + δ _{n} ( π , β ) )} \int_{0}^{π} q (x) cos 2 (n + δ_{n} (π, β)) x d x .

l_{n} = l_{n} (q, π, β) = - \frac{1}{2 π ( n + δ _{n} ( π , β ) )} \int_{0}^{π} q (x) cos 2 (n + δ_{n} (π, β)) x d x .

l (x) = n = 2 \sum \infty l_{n} (q, α, β) sin (n + δ_{n} (α, β)) x,

l (x) = n = 2 \sum \infty l_{n} (q, α, β) sin (n + δ_{n} (α, β)) x,

a_{n} (q, α, β) = \frac{π}{2} [1 + \frac{2 s _{n} ( q , α , β )}{π [ n + δ ( α , β ) ]} + r_{n}] sin^{2} α +

a_{n} (q, α, β) = \frac{π}{2} [1 + \frac{2 s _{n} ( q , α , β )}{π [ n + δ ( α , β ) ]} + r_{n}] sin^{2} α +

+ \frac{π}{2 [ n + δ _{n} ( α , β ) ] ^{2}} [1 + \frac{2 s _{n} ( q , α , β )}{π [ n + δ ( α , β ) ]} + \tilde{r}_{n}] cos^{2} α,

+ \frac{π}{2 [ n + δ _{n} ( α , β ) ] ^{2}} [1 + \frac{2 s _{n} ( q , α , β )}{π [ n + δ ( α , β ) ]} + \tilde{r}_{n}] cos^{2} α,

b_{n} (q, α, β) = \frac{π}{2} [1 + \frac{2 s _{n} ( q , α , β )}{π [ n + δ ( α , β ) ]} + p_{n}] sin^{2} β + + \frac{π}{2 [ n + δ _{n} ( α , β ) ] ^{2}} [1 + \frac{2 s _{n} ( q , α , β )}{π [ n + δ ( α , β ) ]} + \tilde{p}_{n}] cos^{2} β,

b_{n} (q, α, β) = \frac{π}{2} [1 + \frac{2 s _{n} ( q , α , β )}{π [ n + δ ( α , β ) ]} + p_{n}] sin^{2} β + + \frac{π}{2 [ n + δ _{n} ( α , β ) ] ^{2}} [1 + \frac{2 s _{n} ( q , α , β )}{π [ n + δ ( α , β ) ]} + \tilde{p}_{n}] cos^{2} β,

s_{n} = s_{n} (q, α, β) = - \frac{1}{2} \int_{0}^{π} (π - t) q (t) sin 2 [n + δ_{n} (α, β)] t d t,

s_{n} = s_{n} (q, α, β) = - \frac{1}{2} \int_{0}^{π} (π - t) q (t) sin 2 [n + δ_{n} (α, β)] t d t,

s (x) = n = 2 \sum \infty \frac{s _{n}}{n + δ _{n} ( α , β )} cos [n + δ_{n} (α, β)] x

s (x) = n = 2 \sum \infty \frac{s _{n}}{n + δ _{n} ( α , β )} cos [n + δ_{n} (α, β)] x

φ (x, μ, π) := φ_{π} (x, μ) \equiv φ_{π} (x, λ^{2}) = \frac{sin λ x}{λ} + O (\frac{e ^{∣ I mλ ∣ x}}{∣ λ ∣ ^{2}}),

φ (x, μ, π) := φ_{π} (x, μ) \equiv φ_{π} (x, λ^{2}) = \frac{sin λ x}{λ} + O (\frac{e ^{∣ I mλ ∣ x}}{∣ λ ∣ ^{2}}),

φ^{'} (x, μ, π) := φ_{π}^{'} (x, μ) \equiv φ_{π}^{'} (x, λ^{2}) = cos λ x + O (\frac{e ^{∣ I mλ ∣ x}}{∣ λ ∣}),

φ^{'} (x, μ, π) := φ_{π}^{'} (x, μ) \equiv φ_{π}^{'} (x, λ^{2}) = cos λ x + O (\frac{e ^{∣ I mλ ∣ x}}{∣ λ ∣}),

ψ (x, μ, β) := ψ_{β} (x, μ) \equiv ψ_{β} (x, λ^{2}) = cos λ (π - x) sin β + \frac{sin λ ( π - x )}{λ} cos β +

ψ (x, μ, β) := ψ_{β} (x, μ) \equiv ψ_{β} (x, λ^{2}) = cos λ (π - x) sin β + \frac{sin λ ( π - x )}{λ} cos β +

+ O (\frac{e ^{∣ I mλ ∣ (π - x)}}{∣ λ ∣}) sin β + O (\frac{e ^{∣ I mλ ∣ (π - x)}}{∣ λ ∣ ^{2}}) cos β,

+ O (\frac{e ^{∣ I mλ ∣ (π - x)}}{∣ λ ∣}) sin β + O (\frac{e ^{∣ I mλ ∣ (π - x)}}{∣ λ ∣ ^{2}}) cos β,

ψ^{'} (x, μ, β) := ψ_{β}^{'} (x, μ) \equiv ψ_{β}^{'} (x, λ^{2}) = (λ sin λ (π - x) + O (e^{∣ I mλ ∣ (π - x)})) sin β -

ψ^{'} (x, μ, β) := ψ_{β}^{'} (x, μ) \equiv ψ_{β}^{'} (x, λ^{2}) = (λ sin λ (π - x) + O (e^{∣ I mλ ∣ (π - x)})) sin β -

- (cos λ (π - x) + O (\frac{e ^{∣ I mλ ∣ (π - x)}}{∣ λ ∣})) cos β .

- (cos λ (π - x) + O (\frac{e ^{∣ I mλ ∣ (π - x)}}{∣ λ ∣})) cos β .

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TopicsSpectral Theory in Mathematical Physics · Differential Equations and Boundary Problems · Numerical methods in inverse problems

Full text

, , 52, . 6, 2017, . 77-90

-

Аннотация.

$-y^{\prime\prime}+q\left(x\right)y=\mu y,$ $y\left(0\right)=0,$ $y\left(\pi\right)\cos\beta+y^{\prime}\left(\pi\right)\sin\beta=0,$ $\beta\in\left(0,\pi\right)$ $q\in L_{\mathbb{R}}^{1}\left[0,\pi\right].$ , .

111 No. 16YR–1A017.

E-mail: [email protected]

MSC2010 number: 34B24, 34L05, 34L10, 34L20

: - ; ; .

1.

$L\left(q,\alpha,\beta\right)$ -

[TABLE]

$q$ , $\left[0,\pi\right]$ ( $q\in L_{\mathbb{R}}^{1}\left[0,\pi\right]$ ). $L\left(q,\alpha,\beta\right)$ , (1.1)–(1.3) $L^{2}\left[0,\pi\right]$ ( . [1, 2]). , $L\left(q,\alpha,\beta\right)$ , ( . [1]—[3]), ${\mu}_{n}\left(q,\alpha,\beta\right),$ $n=0,1,2,\dots,$ $q,$ $\alpha$ $\beta.$ , $\mu_{n}$ :

[TABLE]

$\varphi\left(x,\mu,\alpha\right)$ $\psi\left(x,\mu,\beta\right)$ (1.1),

[TABLE]

([1, 2, 4]), $x,$ $\varphi,$ $\varphi^{\prime},$ $\psi,$ $\psi^{\prime}$ $\mu.$ $W_{\alpha,\beta}\left(x,\mu\right)$ $\varphi\left(x,\mu,\alpha\right)$ $\psi\left(x,\mu,\beta\right):$

[TABLE]

( . , [5]) $W_{\alpha,\beta}\left(x,\mu\right)$ $x.$ ${\varphi_{n}}\left(x\right):=\varphi(x,\mu_{n},\alpha)$ ${\psi_{n}}\left(x\right):=\psi\left(x,\mu_{n},\beta\right),$ $n=0,1,2,\dots,$ , $\mu_{n}.$ , $\beta_{n}=\beta_{n}\left(q,\alpha,\beta\right),$ $n=0,1,2,\dots,$

[TABLE]

$L^{2}$ - :

[TABLE]

.

( . [1]) :

1.1.

([1, . 90]) .

, . - ( . , [6]), $\sin\alpha\neq 0,$ $\sin\beta\neq 0$ ( . . $\alpha,\beta\in\left(0,\pi\right)$ ) , ( . [5, 2, 7])

1.2.

([6]) $q\in L_{\mathbb{R}}^{2}\left[0,\pi\right],$ $\alpha,\beta\in\left(0,\pi\right)$ $f$ $\left[0,\pi\right].$

[TABLE]

$\varphi_{n}\left(x\right)\equiv\varphi\left(x,\mu_{n}\left(q,\alpha,\beta\right),\alpha\right).$ **

, $L\left(q,\pi,\beta\right),$ $\beta\in\left(0,\pi\right)$ $L\left(q,\alpha,0\right),$ $\alpha\in\left(0,\pi\right),$ $q\in L_{\mathbb{R}}^{1}\left[0,\pi\right]:$

1.3.

$q\in L_{\mathbb{R}}^{1}\left[{0,\pi}\right],$ * $\alpha=\pi,$ $\beta\in\left(0,\pi\right)$ $f$ $\left[0,\pi\right].$ $a\in\left(0,\pi\right)$ *

[TABLE]

$\varphi_{n}\left(x\right)\equiv\varphi\left(x,\mu_{n}\left(q,\pi,\beta\right),\pi\right)\equiv\varphi\left(x,\mu_{n},\pi\right).$ **

1.4.

$q\in L_{\mathbb{R}}^{1}\left[{0,\pi}\right],$ * $\alpha\in\left(0,\pi\right),$ $\beta=0$ $f$ $\left[0,\pi\right].$ $b\in\left(0,\pi\right)$ *

[TABLE]

$\varphi_{n}\left(x\right)\equiv\varphi\left(x,\mu_{n}\left(q,\alpha,0\right),\alpha\right).$ **

1.1.

, 1.3 1.4 ( (1.7) (1.8) $\left[0,\pi\right]$ ) . , $f\equiv\dfrac{\pi}{2},$ $x\in\left[0,2\pi\right],$ ( . , [8, 37 . 578])

[TABLE]

. . 1.3 , $\mathop{\operatorname{max}}\limits_{x\in\left[a,\pi\right]}\left|\dots\right|$ $\mathop{\operatorname{max}}\limits_{x\in\left[0,\pi\right]}\left|\dots\right|,$ . , 1.3 $f\left(0\right)=0,$ (1.7) $\left[0,\pi\right].$ 1.4, $f\left(\pi\right)=0.$

$L\left(q,\pi,\beta\right)$ $q\in L_{\mathbb{R}}^{1}\left[0,\pi\right]$ $\beta\in\left(0,\pi\right)$ ( . . $\sin\beta\neq 0$ ). , , [9] . . $\delta_{n}\left(\alpha,\beta\right),$

[TABLE]

$-1\leq\delta_{n}\left(\alpha,\beta\right)\leq 1$ $\delta_{n}\left(\alpha,\beta\right)$ :

[TABLE]

1.5.

$q\in L_{\mathbb{R}}^{1}\left[0,\pi\right]$ * $\lambda_{n}^{2}\left(q,\alpha,\beta\right)={\mu_{n}}\left(q,\alpha,\beta\right).$ *

(a)

$\left(n\to\infty\right)$ **

[TABLE]

$\left[q\right]=\dfrac{1}{\pi}\displaystyle\int_{0}^{\pi}q\left(t\right)dt,$ **

[TABLE]

$O\left(\dfrac{1}{n^{2}}\right)$ * (1.10) $\alpha,\beta\in[0,\pi]$ $q$ $L_{\mathbb{R}}^{1}\left[{0,\pi}\right]$ ( $q\in{BL}^{1}_{\mathbb{R}}\left[0,\pi\right]$ ).* 2. (b)

$l,$ **

[TABLE]

$\left[a,b\right]\subset\left(0,2\pi\right),$ * . . $l\in AC\left(0,2\pi\right).$ *

[10] (b) 1.5 $\alpha,\beta\in\left(0,\pi\right)$ $\alpha=\pi,$ $\beta=0.$ , $\alpha=\pi,$ $\beta\in\left(0,\pi\right).$ 3. $a_{n}$ $b_{n}$ ( . (1.6)).

1.6.

$a_{n}$ * $b_{n}$ *

(a)

$\left(n\to\infty\right):$ **

[TABLE]

$r_{n}=r_{n}\left(q,\alpha,\beta\right)=O\left(\dfrac{1}{n^{2}}\right)$ * $\tilde{r}_{n}=\tilde{r}_{n}\left(q,\alpha,\beta\right)=O\left(\dfrac{1}{n^{2}}\right)$ ( $p_{n}$ $\tilde{p}_{n}$ ), $n\to\infty,$ $\alpha,\beta\in\left[0,\pi\right]$ $q\in{BL}^{1}_{\mathbb{R}}\left[0,\pi\right].$ * 2. (b)

$s,$ **

[TABLE]

$\left[a,b\right]\subset\left(0,2\pi\right),$ * . . $s\in AC\left(0,2\pi\right).$ *

[11] (b) 1.6 $\alpha,\beta\in\left(0,\pi\right)$ $\alpha=\pi,$ $\beta=0.$ , 1.5 , $\alpha=\pi,$ $\beta\in\left(0,\pi\right).$

2. 1.3 1.4

1.3. 1.4 .

Доказательство.

$\left|\lambda\right|\to\infty,$ ([1, 4, 10])

[TABLE]

(1.4) (2.3) , $W_{\pi,\beta}\left(\mu\right)$

[TABLE]

${\mathbb{Z}}_{1/6}$ $\mathbb{C}$ :

[TABLE]

[12], [13].

2.1.

([12]) $\lambda\in{\mathbb{Z}}_{1/6}$ ,

[TABLE]

(2.5) (2.6) $\lambda^{*}>0,$ $C_{1}>0$

[TABLE]

, $y\left(x,\mu,f\right)$ (2.8)–(2.9) ( . , [1, 7])

[TABLE]

$\varphi,$ $\psi$ $W_{\pi,\beta}$ $\mu$ , $y\left(x,\mu,f\right)$ $\mu,$ $W_{\pi,\beta}$ , , $\mu_{n},n=0,1,2,\dots.$ $\dot{W}_{\pi,\beta}\left(\mu_{n}\right)\equiv\dfrac{d}{d\mu}W_{\pi,\beta}\left(\mu_{n}\right)=\beta_{n}a_{n}$ ( . [6, 1.1.1]), (1.5),

[TABLE]

(2.1), (2.3), (2.7) (2.10) , $C,$ $C_{2},$ $C_{3},$ $C_{4}$ $\lambda\in{\mathbb{Z}}_{1/6},$ $\left|\lambda\right|>\lambda^{*}$

[TABLE]

$f\in AC\left[0,\pi\right].$ , $\varphi_{\pi}\left(x,\mu\right)$ $\psi_{\beta}\left(x,\mu\right)$ (1.1), (2.10) $y\left(x,\mu,f\right)$ ( [6]):

[TABLE]

,

[TABLE]

, $f^{\prime}$ $\left[0,\pi\right].$ $f^{\prime\prime}\in L^{1}\left[{0,\pi}\right]$ (2.14)

[TABLE]

(2.1)–(2.4) (2.7) , $C>0,$

[TABLE]

(2.16) $f^{\prime}\in AC\left[0,\pi\right].$

$g:=f^{\prime}\in L^{1}\left[0,\pi\right].$ $\epsilon>0$ $g_{\epsilon},$

[TABLE]

, (2.1)–(2.4), (2.7) (2.14) $\lambda\in{\mathbb{Z}}_{1/6},\;\left|\lambda\right|>\lambda^{*},$

[TABLE]

, $\lambda_{\epsilon}^{*}=\dfrac{2C\left(\epsilon\right)}{\epsilon},$ $\lambda\in{\mathbb{Z}}_{1/6}$ $\left|\lambda\right|>\lambda_{\epsilon}^{*}$ $\mathop{\operatorname{max}}\limits_{x\in\left[0,\pi\right]}\left|Z_{1}\left(x,\mu,\pi,\beta\right)\right|\leq\epsilon.$ $\epsilon>0,$ (2.16). $Z_{2}\left(x,\mu,\pi,\beta\right)$ ( . (2.15)). $qf\in L^{1}\left[{0,\pi}\right],$ (2.12) $y\left(x,\mu,qf\right).$ (2.1), (2.7), (2.12) , $\sin\beta\neq 0$ ( $\lambda\in{\mathbb{Z}}_{1/6},\;\left|\lambda\right|>\lambda^{*}$ ):

[TABLE]

$C_{5}-C_{7}$ .

[TABLE]

$\Gamma_{N}=\left\{\mu:\left|\mu\right|=\left(N+\dfrac{3}{4}\right)^{2}\right\}$ ( ). , ( . [14]), (2.11)

[TABLE]

, (2.13), (2.16) (2.17) ,

[TABLE]

${\epsilon_{N}}\left(x\right),$ (2.16) (2.17), $0:$

[TABLE]

, $\mu=0$ $L\left(q,\pi,\beta\right).$ , , $c$ , $\mu_{n}+c\neq 0,$ $n=0,1,2,\dots,$ $L\left(q+c,\pi,\beta\right)$ $\varphi_{n}$ $a_{n},$ $L\left(q,\pi,\beta\right).$ $\dfrac{\psi_{\beta}\left(x,\mu\right)}{\mu W_{\pi,\beta}\left(\mu\right)}$ ,

[TABLE]

, $\phi_{N}\left(x\right)$ [math] ( $N\to\infty$ ) $\left[a,\pi\right],$ $a\in\left(0,\pi\right).$

$\varphi_{n}\left(x\right)=\dfrac{\sin\left(n+\frac{1}{2}\right)x}{n+\frac{1}{2}}+O\left(\dfrac{1}{n^{2}}\right)$ $\left[0,\pi\right]$ ( . (2.1)), $\mu_{n}=\mu_{n}\left(q,\pi,\beta\right)=\left(n+\dfrac{1}{2}\right)^{2}+O\left(1\right)$ ( . [9, 1 . 286 $\delta_{n}\left(\pi,\beta\right)$ . 292]) $a_{n}=a_{n}\left(q,\pi,\beta\right)=\dfrac{\pi}{2\left(n+\frac{1}{2}\right)^{2}}\left(1+o\left(\dfrac{1}{n}\right)\right)$ ( . [11, 1.1 ], . 9-10), $\phi_{N}\left(x\right)$ ( . (2.20)) :

[TABLE]

$q_{n}\left(x\right)=O\left(\dfrac{1}{n^{2}}\right)$ $\left[0,\pi\right].$

$\displaystyle\sum\limits_{n=0}^{\infty}\dfrac{\sin\left(n+\frac{1}{2}\right)x}{n+\frac{1}{2}}=\dfrac{\pi}{2},$ $0<x<2\pi$ ( . , [8, (37) . 578]), $\phi_{N}\left(x\right)$ $\phi\left(x\right)$ ( $N\to\infty$ ) $\left[a,\pi\right],$ $a\in\left(0,\pi\right).$

, , $\phi\left(x\right)\equiv 0,\,x\in\left(0,\pi\right],$ , $\phi=0$ . .

, :

[TABLE]

(2.21) (2.22)

[TABLE]

$\left\{\varphi_{m}\left(x\right)\right\}_{m=0}^{\infty}$ $L(q,\pi,\beta)$ $L^{2}\left(0,\pi\right)$ , $\phi=0$ . . (2.18), (2.19) $N\to\infty$ (2.19), (1.7). 1.3 . ∎

2.1.

, 1.2 , , $L\left(q,\alpha,\beta\right),$ $\alpha,\beta\in\left(0,\pi\right)$

$L\left(0,\dfrac{\pi}{2},\dfrac{\pi}{2}\right),$ * . ., $\left\{\cos nx\right\}_{n\geq 0}$ ( . [5, 2, 7]). - ( . [15, . 121–122]) 1.2 . , : $L\left(q,\pi,\beta\right),$ $\beta\in\left(0,\pi\right)$ $\sin\left(n+\dfrac{1}{2}\right)x,$ $n=0,1,2,\dots,$ . ., $L\left(0,\pi,\dfrac{\pi}{2}\right)$ ( . [5, . 304] [2, . 71]). , , - $\left\{\sin\left(n+\dfrac{1}{2}\right)x\right\}_{n\geq 0}$ ( . [16, 2.6]).*

3.

[10] (b) 1.5 $\alpha,\beta\in\left(0,\pi\right)$ $\alpha=\pi,$ $\beta\in\left(0,\pi\right).$ , 1.3 1.4, . $\sigma\left(x\right)=\displaystyle\int_{0}^{x}q\left(t\right)dt$ $l_{n}\left(q,\pi,\beta\right)$ ( . (1.11)) :

[TABLE]

$\sigma_{1}\left(x\right)\equiv\sigma\left(\dfrac{x}{2}\right)$ $\left[0,2\pi\right].$

$\delta_{n}\left(\alpha,\beta\right)$ (1.9) $\alpha=\pi$ $\beta\in\left[0,\pi\right)$ , $\arccos$ .

(1.9) ( . [9]), $\beta\in\left(0,\pi\right)$

[TABLE]

,

[TABLE]

$d_{n}=O\left(\dfrac{1}{n^{2}}\right),\;e_{n}=O\left(\dfrac{1}{n}\right).$

, $l\left(x,\beta\right)$ ( . (1.12))

[TABLE]

$f_{n}=\displaystyle\int_{0}^{2\pi}\sigma_{1}\left(t\right)\sin\left(n+\delta_{n}\left(\pi,\beta\right)\right)tdt.$

$f_{n}=\displaystyle\int_{0}^{\pi}\sigma_{1}\left(t\right)\sin\left(n+\delta_{n}\left(\pi,\beta\right)\right)tdt+\displaystyle\int_{\pi}^{2\pi}\sigma_{1}\left(t\right)\sin\left(n+\delta_{n}\left(\pi,\beta\right)\right)tdt$

[TABLE]

, $\delta_{n}\left(\alpha,\beta\right)$ $n\geq 2,$ $\lambda_{0}\left(0,\pi,\beta\right),$ $\lambda_{1}\left(0,\pi,\beta\right)$ $\lambda_{n}\left(0,\pi,\beta\right)=n+\delta_{n}\left(\pi,\beta\right)$ $n\geq 2.$ ,

[TABLE]

$L\left(0,\pi,\beta\right)$ 1.3,

[TABLE]

$\left[a,\pi\right]\subset\left(0,\pi\right]$

[TABLE]

(3.1) (3.2),

[TABLE]

(3.7), ,

[TABLE]

(3.6)

[TABLE]

$\left[a,\pi\right]\subset\left(0,\pi\right].$

(3.4), (3.5), (3.8) $x\in\left(0,\pi\right]$

[TABLE]

$d_{n}=O\left(\dfrac{1}{n^{2}}\right),$ $e_{n}\displaystyle\int_{0}^{\pi}\sigma\left(\pi-\dfrac{t}{2}\right)\cos\left(n+\delta_{n}\left(\pi,\beta\right)\right)tdt=O\left(\dfrac{1}{n^{2}}\right),$ $g_{n}=O\left(\dfrac{1}{n^{2}}\right),$ $l_{3}\in AC\left(0,\pi\right].$ , ( . (3.4) (3.2))

[TABLE]

$l_{3}\in AC\left[\pi,2\pi\right)$ $l_{3}\in AC\left(0,2\pi\right)$ . $\displaystyle\sum_{n=2}^{\infty}\dfrac{\sin\left(n+\delta_{n}\left(\pi,\beta\right)\right)x}{\left(n+\delta_{n}\left(\pi,\beta\right)\right)}$ $\left(0,2\pi\right)$ ( . [12, 10]), $l_{1}\in AC\left(0,2\pi\right)$ . $d_{n}=O\left(\dfrac{1}{n^{2}}\right),$ (3.3) $\left[0,2\pi\right]$ , $l_{2}\in AC\left[0,2\pi\right].$ (b) 1.5 $\alpha=\pi,$ $\beta\in\left(0,\pi\right)$ . $\Box$

. . .

Abstract. Uniform convergence of the expansion of an absolutely continuous function for eigenfunctions of the Sturm-Liouville problem $-y^{\prime\prime}+q\left(x\right)y=\mu y,$ $y\left(0\right)=0,$ $y\left(\pi\right)\cos\beta+y^{\prime}\left(\pi\right)\sin\beta=0,$ $\beta\in\left(0,\pi\right)$ with summable potential $q\in L_{\mathbb{R}}^{1}\left[0,\pi\right]$ is proved. This result is used to obtain more precise asymptotic formulae for eigenvalues and norming constants of this problem.

Список литературы

[1] . . , , , (1969).
[2] . . , . . , , , (1970).
[3] . . , - , , (1977).
[4] . . , . . , ‘‘ - ’’, , 1, no. 3, 59–74 (2005).
[5] E. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw Hill Book Company, New York (1955).
[6] . . , , , (2007).
[7] . . , . . , - , , (1988).
[8] I. N. Bronshtein, K. A. Semendyayev, Handbook of Mathematics, Springer-Verlag Berlin Heidelberg, New York (1998).
[9] T. N. Harutyunyan, ‘‘The dependence of the eigenvalues of the Sturm-Liouville problem on boundary conditions’’, Matematicki Vesnik, 60, no. 4, 285 – 294 (2008).
[10] . . , ‘‘ - ’’, , , 51, no. 4, 3 – 16 (2016).
[11] T. N. Harutyunyan, A. A. Pahlevanyan, ‘‘On the norming constants of the Sturm-Liouville problem’’, , no. 3(31), 7 – 26 (2016).
[12] . . , - , , (2010).
[13] J. Pöschel, E. Trubowitz, Inverse Spectral Theory, Academic Press, Inc., Boston, MA (1987).
[14] . . , , , (1985).
[15] . . , , , (1961).
[16] A. Iserles, S. P. Nørsett, ‘‘From high oscillation to rapid approximation I: Modified Fourier expansions’’, IMA Journal of Numerical analysis, 28, no. 4, 862 – 887 (2008).

10 2017

Bibliography16

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] . . , , , (1969).
2[2] . . , . . , , , (1970).
3[3] . . , - , , (1977).
4[4] . . , . . , ‘‘ - ’’, , 1 , no. 3, 59–74 (2005).
5[5] E. Coddington, N. Levinson, Theory of Ordinary Differential Equations, Mc Graw Hill Book Company, New York (1955).
6[6] . . , , , (2007).
7[7] . . , . . , - , , (1988).
8[8] I. N. Bronshtein, K. A. Semendyayev, Handbook of Mathematics, Springer-Verlag Berlin Heidelberg, New York (1998).

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