A partial inverse problem for the Sturm-Liouville operator on the graph with a loop

TL;DR

This paper addresses a partial inverse problem for the Sturm-Liouville operator on a lasso graph with a loop, demonstrating uniqueness and providing a constructive solution method based on spectral data and known boundary potential.

Contribution

It introduces a new partial inverse problem for Sturm-Liouville operators on graphs with loops, including a uniqueness theorem and a constructive recovery algorithm.

Findings

Proved uniqueness of potential recovery on the loop from spectral data.

Developed a constructive algorithm for solving the inverse problem.

Validated the approach with theoretical guarantees.

Abstract

The Sturm-Liouville operator with singular potentials on the lasso graph is considered. We suppose that the potential is known a priori on the boundary edge, and recover the potential on the loop from a part of the spectrum and some additional data. We prove the uniqueness theorem and provide a constructive algorithm for the solution of this partial inverse problem.