Cantor and band spectra for periodic quantum graphs with magnetic fields

TL;DR

This paper analyzes the spectral properties of two-dimensional periodic quantum graphs with magnetic fields, revealing complex Cantor spectra for irrational flux and band structures for rational flux, with implications for the Bethe-Sommerfeld conjecture.

Contribution

It provides a comprehensive spectral analysis of magnetic quantum graphs, identifying conditions for spectral gaps and demonstrating the failure of the Bethe-Sommerfeld conjecture in this context.

Findings

Spectrum includes Dirichlet eigenvalues and preimages of a discrete operator spectrum.

Between Dirichlet eigenvalues, the spectrum forms a Cantor set for irrational flux.

For rational flux, the spectrum is absolutely continuous with a band structure.

Abstract

We provide an exhaustive spectral analysis of the two-dimensional periodic square graph lattice with a magnetic field. We show that the spectrum consists of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant (Lyapunov function) of a suitable Kronig-Penney Hamiltonian. In particular, between any two Dirichlet eigenvalues the spectrum is a Cantor set for an irrational flux, and is absolutely continuous and has a band structure for a rational flux. The Dirichlet eigenvalues can be isolated or embedded, subject to the choice of parameters. Conditions for both possibilities are given. We show that generically there are infinitely many gaps in the spectrum, and the Bethe-Sommerfeld conjecture fails in this case.