Visibility of quantum graph spectrum from the vertices

TL;DR

This paper explores how the eigenvalues of a quantum graph's Laplacian relate to the Titchmarsh-Weyl function, providing a complete characterization of resonances and eigenvalues visible through this spectral data.

Contribution

It offers a comprehensive description of real resonances and eigenvalues visibility in quantum graphs based on edge lengths and connectivity, advancing spectral analysis methods.

Findings

Complete description of real resonances including multiplicities

Characterization of eigenvalues visible via Titchmarsh-Weyl function

Relation between graph structure and spectral properties

Abstract

We investigate the relation between the eigenvalues of the Laplacian with Kirchhoff vertex conditions on a finite metric graph and a corresponding Titchmarsh-Weyl function (a parameter-dependent Neumann-to-Dirichlet map). We give a complete description of all real resonances, including multiplicities, in terms of the edge lengths and the connectivity of the graph, and apply it to characterize all eigenvalues which are visible for the Titchmarsh-Weyl function.