Local solvability and stability of inverse problems for Sturm-Liouville operators with a discontinuity

TL;DR

This paper investigates the local solvability and stability of inverse problems for Sturm-Liouville operators with discontinuities, introducing a constructive algorithm and a new uniqueness theorem based on spectral data.

Contribution

It presents a novel constructive algorithm for inverse problems with discontinuities and establishes a new uniqueness theorem using fractional spectral data.

Findings

Proves local solvability and stability of inverse problems for Sturm-Liouville operators with discontinuities.

Develops a constructive algorithm based on Riesz-basis properties.

Introduces a new uniqueness theorem using a fractional part of the spectrum.

Abstract

Partial inverse problems are studied for Sturm-Liouville operators with a discontinuity. The main results of the paper are local solvability and stability of the considered inverse problems. Our approach is based on a constructive algorithm for solving the inverse problems. The key role in our method is played by the Riesz-basis property of a special vector-functional system in a Hilbert space. In addition, we obtain a new uniqueness theorem for recovering the potential on a part of the interval, by using a fractional part of the spectrum.