A new approach to inverse Sturm-Liouville problems based on point interaction

TL;DR

This paper introduces a novel explicit method for solving inverse Sturm-Liouville problems with point interactions, enabling potential reconstruction from eigenvalues and extending classical theorems to measure differential equations.

Contribution

It presents a new approach leveraging point interactions and eigenvalue functions to explicitly and uniquely determine the potential in inverse Sturm-Liouville problems.

Findings

Explicit potential reconstruction from eigenvalues.

Generalization of classical theorems to measure differential equations.

Development of a family of perturbation models based on point interactions.

Abstract

In the present paper, motivated by point interaction, we propose a new and explicit approach to inverse Sturm-Liouville eigenvalue problems under Dirichlet boundary. More precisely, when a given Sturm-Liouville eigenvalue problem with the unknown integrable potential interacts with $\delta$-function potentials, we obtain a family of perturbation problems, called point interaction models in quantum mechanics. Then, only depending on the first eigenvalues of these perturbed problems, we define and study the first eigenvalue function, by which the desired potential can be expressed explicitly and uniquely. As by-products, using the analytic function theoretic tools, we also generalize several fundamental theorems of classical Sturm-Liouville problems to measure differential equations.