On the spectrum of an "even" periodic Schroedinger operator with a rational magnetic flux

TL;DR

This paper proves that a Schrödinger operator with periodic metric, electric, and magnetic fields, and rational magnetic flux, has an absolutely continuous spectrum, extending known results from zero magnetic flux cases.

Contribution

It establishes absolute continuity of the spectrum for a class of Schrödinger operators with rational magnetic flux and reflection symmetry, generalizing previous zero-flux results.

Findings

Spectrum is absolutely continuous under given conditions.

Reflection symmetry and rational flux are key assumptions.

Extends known spectral results to non-zero magnetic flux cases.

Abstract

We study the Schr\"odinger operator on $L_2(\mathbb R^3)$ with periodic variable metric, and periodic electric and magnetic fields. It is assumed that the operator is reflection symmetric and the (appropriately defined) flux of the magnetic field is rational. Under these assumptions it is shown that the spectrum of the operator is absolutely continuous. Previously known results on absolute continuity for periodic operators were obtained for the zero magnetic flux.