Analytic spectral perturbation theory for a high-contrast Maxwell operator

TL;DR

This paper develops a spectral perturbation theory for high-contrast Maxwell operators in cavities, revealing how eigenvalues depend on contrast and geometry, and explaining the robustness of resonances.

Contribution

It introduces a novel operator-theoretic approach to analyze spectral dependence on contrast in Maxwell systems with high contrast, including asymptotic expansions and geometry effects.

Findings

Spectral dependence on contrast parameter is complex-analytic near zero contrast.

Eigenvalue asymptotics can be geometry-independent under certain conditions.

Resonance behavior is sensitive to shell geometry, even in symmetric configurations.

Abstract

We study analytic spectral perturbation theory for the time-harmonic Maxwell operator in a perfectly electrically conducting cavity containing a high-contrast core--shell structure. The dielectric permittivity equals $1$ in a bounded inclusion and a small complex parameter $\delta$ in the surrounding shell. The limit $\delta \to 0$ corresponds to an infinite-contrast regime and leads to a degenerate Maxwell system. Despite this degeneracy, we develop a detailed spectral theory for the limiting problem for general Lipschitz inclusions and shells. Using a novel operator-theoretic reformulation, we prove complex-analytic dependence of the spectrum on $\delta$ in a neighborhood of $\delta = 0$. When the inclusion is a ball, we analyze the asymptotic expansion of eigenvalues and identify conditions under which the leading-order term is independent of the geometry of the surrounding shell. We also construct examples of resonances for which the leading-order asymptotics depend sensitively on the shell geometry, even in this symmetric setting. These results clarify the mechanisms underlying geometry-invariance of resonances in high-contrast Maxwell systems and explain their robustness under small complex perturbations.