A partial inverse problem for the Sturm-Liouville operator on a star-shaped graph

TL;DR

This paper addresses a partial inverse problem for the Sturm-Liouville operator on a star-shaped graph, developing a constructive method to recover unknown potentials from spectral data, with proofs of local solvability and stability.

Contribution

It introduces a new constructive approach for solving the partial inverse problem on star-shaped graphs, leveraging Riesz-basicity and proving key properties like solvability and stability.

Findings

Constructive method for potential recovery developed

Proof of local solvability of the inverse problem

Stability of the solution established

Abstract

The Sturm-Liouville operator on a star-shaped graph is considered. We assume that the potential is known a priori on all the edges except one, and study the partial inverse problem, which consists in recovering the potential on the remaining edge from the part of the spectrum. A constructive method is developed for the solution of this problem, based on the Riesz-basicity of some sequence of vector functions. The local solvability of the inverse problem and the stability of its solution are proved.