Uniform stability for the inverse Sturm-Liouville problem with eigenparameter-dependent boundary conditions

TL;DR

This paper establishes the first uniform stability results for inverse Sturm-Liouville problems with eigenparameter-dependent boundary conditions, providing practical stability estimates using Darboux-type transforms.

Contribution

It introduces a novel approach to prove uniform stability for Sturm-Liouville operator pencils with boundary conditions depending on the eigenparameter.

Findings

Proved uniform stability for direct and inverse spectral problems.

Derived stability estimates for finite data approximations.

Applied Darboux-type transforms to demonstrate Lipschitz continuity.

Abstract

We consider a class of self-adjoint Sturm-Liouville problems with rational functions of the spectral parameter in the boundary conditions. The uniform stability for direct and inverse spectral problems is proved for the first time for Sturm-Liouville operator pencils with boundary conditions depending on the eigenparameter. Furthermore, we obtain stability estimates for finite data approximations, which are important from the practical viewpoint. Our method is based on Darboux-type transforms and proving of their Lipschitz continuity.