Essential spectrum for Maxwell's equations

TL;DR

This paper characterizes the essential spectrum of Maxwell operator pencils in anisotropic, unbounded domains, linking it to simpler spectral components and providing insights for broader electromagnetic system analysis.

Contribution

It establishes a precise description of the essential spectrum for Maxwell's equations with anisotropic media on complex domains, connecting it to scalar and constant-coefficient spectra.

Findings

The essential spectrum is the union of the spectrum of a scalar divergence operator and the constant-coefficient Maxwell spectrum.

The analysis applies to anisotropic, unbounded domains with finitely connected boundaries.

Results facilitate further spectral analysis of Maxwell's and related systems.

Abstract

We study the essential spectrum of operator pencils associated with anisotropic Maxwell equations, with permittivity $\varepsilon$, permeability $\mu$ and conductivity $\sigma$, on finitely connected unbounded domains. The main result is that the essential spectrum of the Maxwell pencil is the union of two sets: namely, the spectrum of the pencil $\mathrm{div}((\omega\varepsilon + i \sigma) \nabla\,\cdot\,)$, and the essential spectrum of the Maxwell pencil with constant coefficients. We expect the analysis to be of more general interest and to open avenues to investigation of other questions concerning Maxwell's and related systems.