Recovering quantum graph spectrum from vertex data

TL;DR

This paper investigates how spectral data of quantum graphs can be reconstructed from boundary measurements, showing that high-multiplicity eigenvalues and certain conditions allow full or partial spectrum recovery.

Contribution

It demonstrates that the Titchmarsh-Weyl function can recover high-multiplicity eigenvalues and, under specific conditions, the entire spectrum of quantum graphs.

Findings

High-multiplicity eigenvalues can be recovered from the Titchmarsh-Weyl function.

Full spectral information is obtainable under additional conditions.

Knowledge of the Titchmarsh-Weyl function is generally insufficient for complete spectrum recovery.

Abstract

We study the question to what extent spectral information of a Schr\"odinger operator on a finite, compact metric graph subject to standard or $\delta$-type matching conditions can be recovered from a corresponding Titchmarsh-Weyl function on the boundary of the graph. In contrast to the case of ordinary or partial differential operators, the knowledge of the Titchmarsh-Weyl function is in general not sufficient for recovering the complete spectrum of the operator (or the potentials on the edges). However, it is shown that those eigenvalues with sufficiently high (depending on the cyclomatic number of the graph) multiplicities can be recovered. Moreover, we prove that under certain additional conditions the Titchmarsh-Weyl function contains even the full spectral information.