Analytic spectral perturbation theory for a high-contrast Maxwell operator
Robert V. Kohn, Raghavendra Venkatraman
TL;DR
This work develops a rigorous analytic spectral perturbation theory for the time-harmonic Maxwell operator in a PEC cavity containing a high-contrast core–shell inclusion, where the shell permittivity is a small complex parameter $oldsymbol{}$ and the limit $oldsymbol{} o 0$ yields an infinite-contrast degenerate system. By reformulating the problem via a chain of compact, self-adjoint operators that depend analytically on $oldsymbol{}$, the authors apply Rellich's perturbation theory to establish complex-analytic dependence of the Maxwell spectrum on $oldsymbol{}$ near $0$ and to construct convergent Taylor expansions for eigenvalues and eigenfunctions. They first illuminate the scalar analogue through a pair of compact operators $K_0$ and $K_$, then lift the approach to the vector Maxwell setting by employing $$-dependent Helmholtz projections and a vector analogue $bK_$, connecting limiting eigenpairs to Maxwell fields with PEC boundary conditions. In the ball-inclusion case, the paper identifies conditions under which the leading-order eigenvalue is geometry-invariant with respect to the shell, and also demonstrates examples where shell geometry decisively affects the leading asymptotics; electrostatic resonances for ball cores are shown to yield infinite families of highly degenerate eigenpairs that persist in the ENZ limit, giving rise to geometry-invariant resonant mechanisms that bifurcate into quasi-resonances when losses are introduced. Overall, the work clarifies how geometry-invariance of resonances arises in high-contrast Maxwell systems and why certain resonances remain robust under small complex perturbations.
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.