Uniform stability of the inverse problem for the non-self-adjoint Sturm-Liouville operator

TL;DR

This paper establishes uniform stability results for inverse spectral problems of non-self-adjoint Sturm-Liouville operators, including cases with simple and multiple eigenvalues, and explores stability under various data and restrictions.

Contribution

It introduces a new approach to uniform stability analysis for inverse spectral problems of non-self-adjoint Sturm-Liouville operators, covering both simple and multiple eigenvalues.

Findings

Lipschitz continuity of inverse mapping on specific spectral data subsets

Unconditional uniform stability estimates for inverse problems

Uniform stability results using Cauchy data for numerical reconstruction

Abstract

In this paper, we develop a new approach to investigation of the uniform stability for inverse spectral problems. We consider the non-self-adjoint Sturm-Liouville problem that consists in the recovery of the potential and the parameters of the boundary conditions from the eigenvalues and the generalized weight numbers. The special case of simple eigenvalues, as well as the general case with multiple eigenvalues are studied. We find various subsets in the space of spectral data, on which the inverse mapping is Lipschitz continuous, and obtain the corresponding unconditional uniform stability estimates. Furthermore, the conditional uniform stability of the inverse problem under a priori restrictions on the potential is studied. In addition, we prove the uniform stability of the inverse problem by the Cauchy data, which are convenient for numerical reconstruction of the potential and for applications to partial inverse problems.