Properties of Eigenvalues and Generalized Eigenfunctions for Sturm–Liouville Problem with Eigenparameter-Dependent Boundary Conditions

This paper investigates the eigenvalues and generalized eigenfunctions of a Sturm–Liouville problem in which the eigenparameter appears in the boundary conditions. By employing operator pencil theory in a Hilbert space, combined with an appropriate integral transformation, we prove that the system o...

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Bibliographic Details
Published in:Mediterranean journal of mathematics Vol. 22; no. 8; p. 222
Main Authors: Li, Zhiyu, Zheng, Zhaowen, Zhang, Yinghan
Format: Journal Article
Language:English
Published: Heidelberg Springer Nature B.V 01.12.2025
Subjects:
Boundary conditions
Eigenvalues
Eigenvectors
Hilbert space
Integral transforms
Lower bounds
Operators (mathematics)
Ordinary differential equations
Sturm-Liouville theory
ISSN:1660-5446, 1660-5454
Online Access:Get full text
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Summary:This paper investigates the eigenvalues and generalized eigenfunctions of a Sturm–Liouville problem in which the eigenparameter appears in the boundary conditions. By employing operator pencil theory in a Hilbert space, combined with an appropriate integral transformation, we prove that the system of generalized eigenfunctions forms a Riesz basis. Furthermore, we demonstrate that the spectrum consists of a countably infinite set of real, discrete eigenvalues accumulating at infinity and provide a lower bound estimation for the eigenvalues.
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ISSN:1660-5446
1660-5454
DOI:10.1007/s00009-025-02987-z