Spectral properties of boundary value problems for functional-differential equations with integrable coefficients
type: Item
instance of: scholarly article
title: Spectral properties of boundary value problems for functional-differential equations with integrable coefficients
zbMATH Open document ID: 1210.34087
DOI: 10.1134/S0012266110080033
published in: Differential Equations
publication date: 2010-11-29T00:00:00Z
review text: The author considers boundary value problems for a functional-differential equation of the form \[ -y^{\prime\prime}(x)+q(x)y(x)=\lambda a^2y(x)+\alpha y(b),\tag{1} \] \[ 0\leq x\leq \pi,\qquad 0<b<\pi,\qquad \alpha>0, \] where \(\alpha\in\mathbb{R}\) and \(\lambda\) is a spectral parameter, with boundary conditions of the following form: \[ y(0)=y(\pi)=0; \] \[ y^\prime(0)=hy(0);\quad y^\prime(\pi)=-Hy(\pi);\quad h,H\in\mathbb{R}; \] \[ y(0)=0;\quad y^\prime(\pi)=0; \] \[ y^\prime(0)=0;\quad y(\pi)=0. \] The potential \(q(x)\) belongs to \(\mathcal{L}_1[0,\pi]\), or, which is the same, satisfies \[ \left(\int_0^xq(t)dt\right)_x^\prime=q(x) \] for almost all \(x\in[0,\pi]\). The author extends the method for constructing asymptotic formulas of arbitrary-order of accuracy for the eigenvalues and eigenfunctions developed by \textit{V. A. Vinokurov} and \textit{V. A. Sadovnichij} [Izv. Math. 64, No.~4, 695--754 (2000); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 64, No.~4, 47--108 (2000; Zbl 1001.34021)] to equation (1).
Mathematics Subject Classification ID: 34K08
Mathematics Subject Classification ID: 34K10
Mathematics Subject Classification ID: 34K06
zbMATH DE Number: 5820357
zbMATH Keywords: eigenvalue
zbMATH Keywords: eigenfunction
zbMATH Keywords: boundary value problems
zbMATH Keywords: functional-differential equations
author: S. I. Mitrokhin
MaRDI profile type: Publication
cites work: The development of general differential and general differential-boundary systems
cites work: Q3981931
cites work: Q3947981
cites work: Q3752740
description: scientific article; zbMATH DE number 5820357