# Quantum graphs with the Bethe-Sommerfeld property

## TL;DR

This paper investigates conditions under which periodic quantum graphs can have a finite, nonzero number of spectral gaps, highlighting the roles of vertex couplings and edge commensurability, and providing explicit examples.

## Contribution

It demonstrates that the existence of a finite nonzero number of spectral gaps depends on vertex couplings and edge ratios, and constructs examples with such properties.

## Key findings

- Graphs with scale invariant couplings have no finite nonzero gaps.
- Existence of graphs with a finite nonzero number of gaps is shown.
- A rectangular lattice with tuned $ ext{delta}$-coupling exemplifies this property.

## Abstract

In contrast to the usual quantum systems which have at most a finite number of open spectral gaps if they are periodic in more than one direction, periodic quantum graphs may have gaps arbitrarily high in the spectrum. This property of graph Hamiltonians, being generic in a sense, inspires the question about the existence of graphs with a finite and nonzero number of spectral gaps. We show that the answer depends on the vertex couplings together with commensurability of the graph edges. A finite and nonzero number of gaps is excluded for graphs with scale invariant couplings; on the other hand, we demonstrate that graphs featuring a finite nonzero number of gaps do exist, illustrating the claim on the example of a rectangular lattice with a suitably tuned $\delta$-coupling at the vertices.