Dependence of Discrete Sturm-Liouville Eigenvalues on Problems

2015/05/28 by Hao Zhu, Zhu, Hao, Shurong Sun +5
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Matrix Theory and Algorithms #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.SP

paper · pdf · doi:10.48550/arxiv.1505.07531

32 Pages, 6 figures

arxiv created 2015/05/28 · openalex publication_date 2015/05/28 · arxiv updated 2015/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Continuous Dependence of the n-th Eigenvalue on Self-adjoint Discrete Sturm-Liouville Problem
• Inequalities among eigenvalues of different self-adjoint discrete Sturm-Liouville problems
• Discrete Sturm-Liouville problems: singularity of the n-th eigenvalue with application to Atkinson type
• Singularity of the n-th eigenvalue of high dimensional Sturm-Liouville problems
• Eigenvalues of Sturm Liouville problems with discontinuity conditions inside a finite interval