Singularity of the $n$-th eigenvalue of high dimensional Sturm-Liouville problems

TL;DR

This paper investigates the continuity and asymptotic behavior of the $n$-th eigenvalue in high-dimensional Sturm-Liouville problems, characterizing boundary conditions that cause discontinuities and analyzing eigenvalue multiplicities and derivatives.

Contribution

It provides a complete characterization of boundary conditions affecting eigenvalue continuity, divides boundary conditions into layers with continuous dependence, and establishes eigenvalue multiplicity and derivative formulas.

Findings

Identified boundary conditions causing eigenvalue discontinuities.

Divided boundary conditions into $2d+1$ layers with continuous eigenvalue dependence.

Proved equality of analytic and geometric multiplicities, and derived eigenvalue derivative formulas.

Abstract

It is natural to consider continuous dependence of the $n$-th eigenvalue on $d$-dimensional ($d\geq2$) Sturm-Liouville problems after the results on $1$-dimensional case by Kong, Wu and Zettl [14]. In this paper, we find all the boundary conditions such that the $n$-th eigenvalue is not continuous, and give complete characterization of asymptotic behavior of the $n$-th eigenvalue. This renders a precise description of the jump phenomena of the $n$-th eigenvalue near such a boundary condition. Furthermore, we divide the space of boundary conditions into $2d+1$ layers and show that the $n$-th eigenvalue is continuously dependent on Sturm-Liouville equations and on boundary conditions when restricted into each layer. In addition, we prove that the analytic and geometric multiplicities of an eigenvalue are equal. Finally, we obtain derivative formula and positive direction of eigenvalues with respect to boundary conditions.