Analytic spectral perturbation theory for a high-contrast Maxwell operator
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| Format: | Preprint |
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2026
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| _version_ | 1866908775439400960 |
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| author | Kohn, Robert V. Venkatraman, Raghavendra |
| author_facet | Kohn, Robert V. Venkatraman, Raghavendra |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_13408 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Analytic spectral perturbation theory for a high-contrast Maxwell operator Kohn, Robert V. Venkatraman, Raghavendra Analysis of PDEs 35P05, 35Q61, 47A55, 35B25 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. |
| title | Analytic spectral perturbation theory for a high-contrast Maxwell operator |
| topic | Analysis of PDEs 35P05, 35Q61, 47A55, 35B25 |
| url | https://arxiv.org/abs/2601.13408 |