On conditions for the multiple expandability of functions in the root elements of a pencil of ordinary differential operators
type: Item
instance of: scholarly article
title: On conditions for the multiple expandability of functions in the root elements of a pencil of ordinary differential operators
zbMATH Open document ID: 1270.34196
DOI: 10.1134/S0012266112080010
author: A. I. Vagabov
published in: Differential Equations
publication date: 2013-01-14T00:00:00Z
review text: Given the linear ordinary differential equation \[ l(y)\equiv \sum_{\scriptstyle k_0+k_1\leq n\atop \scriptstyle k_0<n}\lambda ^{k_0}A^{(k_0,k_1)}(x)\frac{d^{k_1}y}{dx^{k_1}}-\lambda ^ny=0,\quad a<x<b, \] with complex eigenvalue parameter \(\lambda \) and with boundary conditions \[ U(y)\equiv\sum_{\scriptstyle k_0+k_1\leq n\atop\scriptstyle k_0<n}\lambda ^{k_0}\left\{\alpha ^{(k_0,k_1)} \left.\frac{d^{k_1}y}{dx^{k_1}}\right|_{x=a}+\beta ^{(k_0,k_1)} \left.\frac{d^{k_1}y}{dx^{k_1}}\right|_{x=b}\right\}=0. \] The author assumes a regularity condition which is a generalization of the standard Birkhoff regularity. Then it is shown that \(n\) integrable functions, \(n-1\) of which are in the domain of the differential operator, can be expanded simultaneously in terms of eigenfunctions and associated functions of the boundary eigenvalue problem. An explicit integral formula involving the Green function describes this expansion. This result is related to expansion theorems obtained in [\textit{J. D. Tamarkin}, M. Z. 27, 1--54 (1927; JFM 53.0419.02); \textit{G. D. Birkhoff} and \textit{R. E. Langer}, American Acad. Proc. 58, No. 2, 49--128 (1923; JFM 49.0723.01); \textit{M. V. Keldysh}, Dokl. Akad. Nauk SSSR, n. Ser. 77, 11--14 (1951; Zbl 0045.39402); \textit{M. A. Naimark}, Linear differential operators. I. New York: Frederick Ungar Publishing (1967; Zbl 0219.34001); II. (1968; Zbl 0227.34020); \textit{R. Mennicken} and \textit{M. Möller}, Non-self-adjoint boundary eingenvalue problems. Amsterdam: North-Holland (2003; Zbl 1033.34001)].
Mathematics Subject Classification ID: 34L10
Mathematics Subject Classification ID: 34B07
Mathematics Subject Classification ID: 34B09
Mathematics Subject Classification ID: 34B27
zbMATH DE Number: 6124958
zbMATH Keywords: \(n\)-fold eigenfunction expansion
zbMATH Keywords: Birkhoff regularity
zbMATH Keywords: Green function
MaRDI profile type: Publication
cites work: Q5631640
cites work: Q5332797
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description: scientific article; zbMATH DE number 6124958