The number of Dirac-weighted eigenvalues of Sturm-Liouville equations with integrable potentials and an application to inverse problems

TL;DR

This paper analyzes the number of Dirac-weighted eigenvalues in Sturm-Liouville equations with integrable potentials, introduces new computational methods, and applies findings to inverse problems involving Dirac weights.

Contribution

It provides a novel characterization of eigenvalues for Sturm-Liouville problems with Dirac weights and develops a direct algorithm for eigenvalue computation.

Findings

Maximum of n Dirichlet eigenvalues when weight is a positive combination of n Dirac deltas

Eigenvalues can be infinite or zero under certain conditions

Application to inverse Dirichlet problems with Dirac weights

Abstract

In this paper, we further Meirong Zhang, et al.'s work by computing the number of weighted eigenvalues for Sturm-Liouville equations, equipped with general integrable potentials and Dirac weights, under Dirichlet boundary condition. We show that, for a Sturm-Liouville equation with a general integrable potential, if its weight is a positive linear combination of $n$ Dirac Delta functions, then it has at most $n$ (may be less than $n$, or even be $0$) distinct real Dirichlet eigenvalues, or every complex number is a Dirichlet eigenvalue; in particular, under some sharp condition, the number of Dirichlet eigenvalues is exactly $n$. Our main method is to introduce the concepts of characteristics matrix and characteristics polynomial for Sturm-Liouville problem with Dirac weights, and put forward a general and direct algorithm used for computing eigenvalues. As an application, a class of inverse Dirichelt problems for Sturm-Liouville equations involving single Dirac distribution weights is studied.