A 2-edge partial inverse problem for the Sturm-Liouville operators with singular potentials on a star-shaped graph

TL;DR

This paper addresses a partial inverse problem for Sturm-Liouville operators with singular potentials on a star-shaped graph, demonstrating that potentials on unknown edges can be reconstructed from spectral data, with proven uniqueness and an algorithmic approach.

Contribution

It introduces a method to reconstruct unknown potentials on a star-shaped graph using spectral data, extending inverse problem theory to singular potentials with a constructive algorithm.

Findings

Potential reconstruction from spectral data is possible on all but two edges.

A uniqueness theorem guarantees the reconstruction's validity.

An explicit algorithm for solving the inverse problem is provided.

Abstract

Boundary value problems for Sturm-Liouville operators with potentials from the class $W_2^{-1}$ on a star-shaped graph are considered. We assume that the potentials are known on all the edges of the graph except two, and show that the potentials on the remaining edges can be constructed by fractional parts of two spectra. A uniqueness theorem is proved, and an algorithm for the constructive solution of the partial inverse problem is provided. The main ingredient of the proofs is the Riesz-basis property of specially constructed systems of functions.