An inverse spectral problem for a star graph of Krein strings

TL;DR

This paper addresses an inverse spectral problem for star graphs of Krein strings, demonstrating unique determination of weights from spectral data and characterizing all feasible spectral configurations.

Contribution

It introduces a method to uniquely recover weights on a star graph of Krein strings from spectral data, including spectra and coupling matrices, and characterizes all such spectral data.

Findings

Spectral data uniquely determine the weights on the graph.

Characterization of all possible spectral data for the class of weights.

Establishment of a concise inverse spectral theory for star graphs of Krein strings.

Abstract

We solve an inverse spectral problem for a star graph of Krein strings, where the known spectral data comprises the spectrum associated with the whole graph, the spectra associated with the individual edges as well as so-called coupling matrices. In particular, we show that these spectral quantities uniquely determine the weight within the class of Borel measures on the graph, which give rise to trace class resolvents. Furthermore, we obtain a concise characterization of all possible spectral data for this class of weights.