Inverse Sturm-Liouville problem with singular potential and spectral parameter in the boundary conditions

TL;DR

This paper investigates an inverse Sturm-Liouville problem with singular potentials and spectral parameter dependence in boundary conditions, establishing new solvability, stability, and uniqueness results for potential and polynomial recovery from spectral data.

Contribution

It provides the first proof of local solvability and stability for this class of inverse problems, along with necessary and sufficient conditions for spectral data uniqueness and a reconstruction method.

Findings

Proved local solvability and stability of the inverse problem.

Derived necessary and sufficient conditions for spectral data uniqueness.

Developed a reconstruction procedure for potential and polynomial from spectral data.

Abstract

This paper deals with the Sturm-Liouville problem that feature distribution potential, polynomial dependence on the spectral parameter in the first boundary condition, and analytical dependence, in the second one. We study an inverse spectral problem that consists in the recovery of the potential and the polynomials from some part of the spectrum. We for the first time prove local solvability and stability for this type of inverse problems. Furthermore, the necessary and sufficient conditions on the given subspectrum for the uniqueness of solution are found, and a reconstruction procedure is developed. Our main results can be applied to a variety of partial inverse problems. This is illustrated by an example of the Hochstadt-Lieberman-type problem with polynomial dependence on the spectral parameter in the both boundary conditions.