On the local solvability and stability of the partial inverse problems for the non-self-adjoint Sturm-Liouville operators with a discontinuity

TL;DR

This paper investigates the local solvability and stability of partial inverse spectral problems for non-self-adjoint Sturm-Liouville operators with a discontinuity, providing new uniqueness theorems and methods for potential recovery.

Contribution

It introduces new results on the local solvability and stability of partial inverse problems for complex Sturm-Liouville operators with discontinuities, including novel uniqueness theorems.

Findings

Established local solvability and stability results.

Derived two new uniqueness theorems.

Analyzed error effects in potential recovery.

Abstract

In this work, we study the inverse spectral problems for the Sturm-Liouville operators on [0,1] with complex coefficients and a discontinuity at $x=a\in(0,1)$. Assume that the potential on (a,1) and some parameters in the discontinuity and boundary conditions are given. We recover the potential on (0,a) and the other parameters from the eigenvalues. This is the so-called partial inverse problem. The local solvability and stability of the partial inverse problems are obtained for $a\in(0,1)$, in which the error caused by the given partial potential is considered. As a by-product, we also obtain two new uniqueness theorems for the partial inverse problem.