Partial inverse problems for Sturm-Liouville operators on trees

TL;DR

This paper investigates inverse spectral problems for Sturm-Liouville operators on trees, demonstrating unique determination of potentials with multiple spectral sets and providing constructive solutions using spectral mapping methods.

Contribution

It introduces a method to uniquely recover potentials on a tree from multiple spectral sets when the potential on one edge is known.

Findings

b - 1 spectral sets determine the potential functions

Constructive solutions are developed using spectral mappings

Unique recovery of potentials on trees with external edges

Abstract

In this paper, inverse spectral problems for Sturm-Liouville operators on a tree (a graph without cycles) are studied. We show that if the potential on an edge is known a priori, then b - 1 spectral sets uniquely determine the potential functions on a tree with b external edges. Constructive solutions, based on the method of spectral mappings, are provided for the considered inverse problems.