An inverse problem for Sturm-Liouville operators on trees with partial information given on the potentials

TL;DR

This paper addresses the inverse problem of recovering singular potentials on a tree graph from partial spectral data, providing a uniqueness theorem and a constructive solution method.

Contribution

It introduces a novel approach to solve the partial inverse problem for Sturm-Liouville operators on trees with singular potentials, using spectral completeness and Riesz basis techniques.

Findings

Proved uniqueness of potential recovery on the entire tree from partial spectral data.

Developed a constructive algorithm for the inverse problem.

Reduced the partial inverse problem to a complete one on a subgraph.

Abstract

We consider Sturm-Liouville operators on geometrical graphs without cycles (trees) with singular potentials from the class $W_2^{-1}$. We suppose that the potentials are known on a part of the graph, and study the so-called partial inverse problem, which consists in recovering the potentials on the remaining part of the graph from some parts of several spectra. The main results of the paper are the uniqueness theorem and a constructive procedure for the solution of the partial inverse problem. Our method is based on the completeness and the Riesz-basis property of special systems of vector functions, and the reduction of the partial inverse problem to the complete one on a part of the graph.