Homogenization and nonselfadjoint spectral optimization for dissipative Maxwell eigenproblems

Abstract

The homogenization of eigenvalues of non-Hermitian Maxwell operators is studied by the H-convergence method. It is assumed that the Maxwell systems are equipped with suitable m-dissipative boundary conditions, namely, with Leontovich or generalized impedance boundary conditions of the form $n\times E=Z[(n\times H)\times n]$. We show that, for a wide class of impedance operators $Z$, the nonzero spectrum of the corresponding Maxwell operator is discrete. To this end, a new continuous embedding theorem for domains of Maxwell operators is obtained. We prove the convergence of eigenvalues to an eigenvalue of a homogenized Maxwell operator under the assumption of the H-convergence of the material tensor-fields. This result is used then to prove the existence of optimizers for eigenvalue optimization problems and the existence of an eigenvalue-free region around zero. As applications, connections with the quantum optics problem of the design of high-Q resonators are discussed, and a new way of the quantification of the unique (and nonunique) continuation property is suggested.

31 pages, the discussion on connections with quantum optics and unique continuation is added, in particular, Example 2.2 (ii); 3 references are added; some of the proofs are shortened; typos are corrected

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Cited by in corpus (1)

• Random acoustic boundary conditions and Weyl's law for Laplace-Beltrami operators on non-smooth boundaries