Eigenvalue properties of Sturm-Liouville problems with transmission conditions dependent on the eigenparameter

This paper studies a discontinuous Sturm-Liouville problem in which the spectral parameter appears not only in the differential equation but also in the transmission conditions. By constructing an appropriate Hilbert space and inner product, the eigenvalue and eigenfunction problems of the Sturm-Lio...

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Bibliographic Details
Published in:	Electronic research archive Vol. 32; no. 3; pp. 1844 - 1863
Main Authors:	Zhang, Lanfang, Ao, Jijun, Zhang, Na
Format:	Journal Article
Language:	English
Published:	AIMS Press 01.01.2024
Subjects:	dependence of eigenvalues eigenparameter-dependent transmission conditions green's function sturm-liouville problems
ISSN:	2688-1594, 2688-1594
Online Access:	Get full text
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Summary:	This paper studies a discontinuous Sturm-Liouville problem in which the spectral parameter appears not only in the differential equation but also in the transmission conditions. By constructing an appropriate Hilbert space and inner product, the eigenvalue and eigenfunction problems of the Sturm-Liouville problem are transformed into an eigenvalue problem of a certain self-adjoint operator. Next, the eigenfunctions of the problem and some properties of the eigenvalues are given via construction of the basic solution. The Green's function for the Sturm-Liouville problem is also given. Finally, the continuity of the eigenvalues and eigenfunctions of the problem is discussed. Especially, the differential expressions of the eigenvalues for some parameters have been obtained, including the parameters in the eigenparameter-dependent transmission conditions.
ISSN:	2688-1594 2688-1594
DOI:	10.3934/era.2024084