Eigenvalue problem of Sturm-Liouville systems with separated boundary conditions

TL;DR

This paper develops a Hill-type and Krein-type trace formula for Sturm-Liouville systems with separated boundary conditions, enabling finite matrix representations of eigenvalue products and sums, with applications in geometry and analysis.

Contribution

It introduces the first Hill-type formula for non-periodic boundary conditions and derives a Krein-type trace formula for eigenvalue sums in Sturm-Liouville systems.

Findings

Derived Hill-type formula representing eigenvalue products as determinants.

Established Krein-type trace formula expressing eigenvalue sums as traces.

Applied formulas to estimate conjugate points on geodesics in Riemannian manifolds.

Abstract

Let $\lambda_j$ be the $j$-th eigenvalue of Sturm-Liouville systems with separated boundary conditions, we build up the Hill-type formula, which represent $\prod\limits_{j}(1-\lambda_j^{-1})$ as a determinant of finite matrix. This is the first attack on such a formula under non-periodic type boundary conditions. Consequently, we get the Krein-type trace formula based on the Hill-type formula, which express $\sum\limits_{j}{1\over \lambda_j^m}$ as trace of finite matrices. The trace formula can be used to estimate the conjugate point alone a geodesic in Riemannian manifold and to get some infinite sum identities.