Spectrum of a dilated honeycomb network

TL;DR

This paper investigates the spectral properties of a quantum honeycomb lattice with variable edge lengths and delta couplings, revealing how edge length ratios influence spectral gaps and eigenfunctions.

Contribution

It provides new conditions for the continuous spectrum and gap existence based on edge length ratios and analyzes the special case of coincident edge lengths.

Findings

Point spectrum exists with compactly supported eigenfunctions when edge lengths are commensurate.

Spectral gaps depend on number-theoretic properties of edge length ratios.

Detailed analysis of the case where two edge lengths are equal.

Abstract

We analyze spectrum of Laplacian supported by a periodic honeycomb lattice with generally unequal edge lengths and a $\delta$ type coupling in the vertices. Such a quantum graph has nonempty point spectrum with compactly supported eigenfunctions provided all the edge lengths are commensurate. We derive conditions determining the continuous spectral component and show that existence of gaps may depend on number-theoretic properties of edge lengths ratios. The case when two of the three lengths coincide is discussed in detail.