Dependence of Discrete Sturm-Liouville Eigenvalues on Problems

TL;DR

This paper investigates how eigenvalues of discrete Sturm-Liouville problems depend on problem parameters, establishing the continuity, analyticity, and monotonicity of eigenvalue branches in relation to problem variations.

Contribution

It introduces topologies and geometric structures on problem spaces and analyzes the analytic and geometric multiplicities of eigenvalues, revealing continuous eigenvalue branches.

Findings

Eigenvalues depend continuously on problem parameters.

Simple eigenvalues form continuous branches over problem space.

Eigenvalue branches exhibit analyticity and monotonicity.

Abstract

This paper is concerned with dependence of discrete Sturm-Liouville eigenvalues on problems. Topologies and geometric structures on various spaces of such problems are firstly introduced. Then, relationships between the analytic and geometric multiplicities of an eigenvalue are discussed. It is shown that all problems sufficiently close to a given problem have eigenvalues near each eigenvalue of the given problem. So, all the simple eigenvalues live in so-called continuous simple eigenvalue branches over the space of problems, and all the eigenvalues live in continuous eigenvalue branches over the space of self-adjoint problems. The analyticity, differentiability and monotonicity of continuous eigenvalue branches are further studied.