Absolutely continuous spectrum for the isotropic Maxwell operator with coefficients that are periodic in some directions and decay in others

TL;DR

This paper proves that the spectrum of an isotropic Maxwell operator with mixed periodic and decaying coefficients is purely absolutely continuous, using a novel operator identity and spectral analysis techniques.

Contribution

It introduces a new operator identity linking Maxwell and Schrödinger operators, enabling spectral analysis of Maxwell operators with mixed periodic and decay properties.

Findings

Spectrum is purely absolutely continuous under given conditions.

Develops a new operatorial identity for Maxwell operators.

Extends spectral analysis techniques to mixed periodic-decay coefficients.

Abstract

The purpose of this paper is to prove that the spectrum of an isotropic Maxwell operator with electric permittivity and magnetic permeability that are periodic along certain directions and tending to a constant super-exponentially fast in the remaining directions is purely absolutely continuous. The basic technical tools is a new ``operatorial'' identity relating the Maxwell operator to a vector-valued Schrodinger operator. The analysis of the spectrum of that operator is then handled using ideas developed by the same authors in a previous paper.