# Discrete Sturm-Liouville problems: singularity of the $n$-th eigenvalue with application to Atkinson type

## TL;DR

This paper characterizes the singularity of the $n$-th eigenvalue in self-adjoint discrete Sturm-Liouville problems across dimensions, revealing jump phenomena and providing methods to analyze and partition the parameter space.

## Contribution

It offers a complete characterization of eigenvalue singularities, including jump phenomena, and extends analysis methods to any dimension, applying them to Atkinson type problems.

## Key findings

- Eigenvalue singularities depend heavily on coefficients and boundary conditions.
- The paper introduces a Hermitian matrix to determine parameter space divisions.
- It generalizes asymptotic analysis methods to higher dimensions and applies them to Atkinson type problems.

## Abstract

In this paper, we characterize singularity of the $n$-th eigenvalue of self-adjoint discrete Sturm-Liouville problems in any dimension. For a fixed Sturm-Liouville equation, we completely characterize singularity of the $n$-th eigenvalue. For a fixed boundary condition, unlike in the continuous case, the $n$-th eigenvalue exhibits jump phenomena and we describe the singularity under a non-degenerate assumption. Compared with the continuous case in [8, 12], the singular set here is involved heavily with coefficients of the Sturm-Liouville equations. This, along with arbitrariness of the dimension, causes difficulty when dividing areas in layers of the considered space such that the $n$-th eigenvalue has the same singularity in any given area. We study the singularity by partitioning and analyzing the local coordinate systems, and provide a Hermitian matrix which can determine the areas' division. To prove the asymptotic behavior of the $n$-th eigenvalue, we generalize the method developed in [21] to any dimension. Finally, by transforming the Sturm-Liouville problem of Atkinson type in any dimension to a discrete one, we can not only determine the number of eigenvalues, but also apply our approach above to obtain the complete characterization of singularity of the $n$-th eigenvalue for the Atkinson type.