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arXiv · 2601.13408

Analytic spectral perturbation theory for a high-contrast Maxwell operator

Robert V. Kohn·Raghavendra Venkatraman
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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 $δ$ in the surrounding shell. The limit $δ\to 0$ 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 $δ$ in a neighborhood of $δ= 0$. 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.

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Robert V. Kohn, Raghavendra Venkatraman. 2026-01-19. Analytic spectral perturbation theory for a high-contrast Maxwell operator. https://arxiv.org/abs/2601.13408

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