Quantum graphs and their spectra

TL;DR

This paper investigates the spectral properties of quantum graphs, establishing finiteness of isospectral families, spectral invariants, and how the Bloch spectrum uniquely characterizes certain classes of planar quantum graphs.

Contribution

It introduces the concept of the Bloch spectrum for quantum graphs and demonstrates its power in uniquely determining planar 3-connected quantum graphs.

Findings

Families of isospectral leafless quantum graphs are finite.

Minimum edge length is a spectral invariant.

The Bloch spectrum determines key graph properties and classifies planar 3-connected quantum graphs.

Abstract

We show that families of leafless quantum graphs that are isospectral for the standard Laplacian are finite. We show that the minimum edge length is a spectral invariant. We give an upper bound for the size of isospectral families in terms of the total edge length of the quantum graphs. We define the Bloch spectrum of a quantum graph to be the map that assigns to each element in the deRham cohomology the spectrum of an associated magnetic Schr\"odinger operator. We show that the Bloch spectrum determines the Albanese torus, the block structure and the planarity of the graph. It determines a geometric dual of a planar graph. This enables us to show that the Bloch spectrum identifies and completely determines planar 3-connected quantum graphs.