Dependence of the spectrum of a quantum graph on vertex conditions and edge lengths

TL;DR

This paper investigates how the spectrum of a quantum graph varies with changes in vertex conditions and edge lengths, providing interlacing results and discussing their applications.

Contribution

It introduces new interlacing theorems for quantum graph spectra based on vertex conditions and edge lengths, expanding understanding of spectral dependence.

Findings

Spectral interlacing results for different vertex conditions

Dependence of spectrum on edge lengths analyzed

Applications of spectral bracketing discussed

Abstract

We study the dependence of the quantum graph Hamiltonian, its resolvent, and its spectrum on the vertex conditions and graph edge lengths. In particular, several results on the interlacing (bracketing) of the spectra of graphs with different vertex conditions are obtained and their applications are discussed.