Maxwell operator in a cylinder. Separation of variables

TL;DR

This paper analyzes the spectral structure of the Maxwell operator in a cylindrical domain with scalar coefficients depending on one variable, providing a detailed spectral description for specific coefficient behaviors.

Contribution

It introduces a representation of the Maxwell operator as a sum of matrix differential operators and describes its spectrum for stabilizing and periodic coefficients.

Findings

Spectral structure characterized for coefficients stabilizing at infinity.

Spectral structure characterized for periodic coefficients.

Representation as a sum of matrix differential operators.

Abstract

The Maxwell operator in a 3D cylinder is considered. The coefficients are assumed to be scalar functions depending on the longitudinal variable only. Such operator is represented as a sum of countable set of matrix differential operators of first order acting in $L_2({\mathbb R})$. Based on this representation we give a detailed description of the structure of the spectrum of the Maxwell operator in two particular cases: 1) in the case of coefficients stabilizing at infinity; and 2) in the case of periodic coefficients.