Continuous Dependence of the n-th Eigenvalue on Self-adjoint Discrete Sturm-Liouville Problem

TL;DR

This paper investigates how the n-th eigenvalue of self-adjoint discrete Sturm-Liouville problems varies continuously with changes in the problem's parameters, providing conditions for continuity and analyzing asymptotic behaviors at discontinuities.

Contribution

It offers a complete characterization of the continuity and discontinuity of the n-th eigenvalue function in the space of problems, including asymptotic analysis near discontinuities.

Findings

Necessary and sufficient conditions for eigenvalue continuity.

Complete characterization of eigenvalue discontinuities.

Asymptotic behavior of eigenvalues near discontinuities.

Abstract

This paper is concerned with continuous dependence of the n-th eigenvalue on self-adjoint discrete Sturm-Liouville problems. The n-th eigenvalue is considered as a function in the space of the problems. A necessary and sufficient condition for all the eigenvalue functions to be continuous and several properties of the eigenvalue functions in a set of the space of the problems are given. They play an important role in the study of continuous dependence of the n-th eigenvalue function on the problems. Continuous dependence of the n-th eigenvalue function on the equations and on the boundary conditions is studied separately. Consequently, the continuity and discontinuity of the n-th eigenvalue function are completely characterized in the whole space of the problems. Especially, asymptotic behaviors of the n-th eigenvalue function near each discontinuity point are given.