Half Inverse Sturm-Liouville Problem on Three-Star Graph with a Discontinuity

TL;DR

This paper addresses the inverse Sturm-Liouville problem on a three-star graph with a discontinuity, demonstrating that a single spectrum can uniquely determine the potential and jump conditions given partial potential information.

Contribution

It introduces a method to uniquely recover the potential and jump data on a three-star graph with a discontinuity from a single spectral measurement, under partial potential knowledge.

Findings

Eigenvalue distribution on the graph was derived.

Uniqueness of potential and jump determination from one spectrum was proved.

Partial potential knowledge on edges suffices for full reconstruction.

Abstract

We consider the inverse Sturm-Liouville problem with one discontinuous point on three-star graph, we deduced the distribution of the eigenvalues, and proved that one spectrum could uniquely determine the unknown potential and jump information when we know the whole potentials on two edges and half potential on another edge.