On absolute continuity of spectra of periodic elliptic operators

TL;DR

This paper discusses a simplified proof of the absolute continuity of spectra for periodic elliptic operators, extending Sobolev's approach to a broad class of operators relevant in mathematical physics.

Contribution

It introduces a unified, simplified method for proving spectral absolute continuity applicable to various periodic elliptic operators, including Maxwell's equations.

Findings

Applicable to all periodic elliptic operators of interest in mathematical physics

Provides a unified approach based on complex analysis

Simplifies Sobolev's original proof

Abstract

The paper contains a brief description of a simplified version of A. Sobolev's proof of absolute continuity of spectra of periodic magnetic Schr\"{o}dinger operators. This approach is applicable to all periodic elliptic operators known to be of interest for math physics (including Maxwell), and in all these cases leads to the same model problem of complex analysis. The full account of this approach will be provided elsewhere.