Homogenization and nonselfadjoint spectral optimization for dissipative Maxwell eigenproblems
Year: 2025
Type: article
Abstract: The optimization of eigenvalues of non-Hermitian Maxwell operators is studied by the homogenization method. We obtain 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 to prove the existence of optimizers for eigenvalue optimization problems and the existence of an eigenvalue-free region. As applications, a connection with the quantum optics problem of the design of high-Q resonators is discussed and a new way of the quantification of the unique (and nonunique) continuation property is suggested. It is assumed throughout the paper that Maxwell systems are equipped with suitable m-dissipative boundary conditions, namely, with Leontovich or generalized impedance boundary conditions $${\textbf{n}}\times {\textbf{E}}= Z [({\textbf{n}}\times {\textbf{H}})\times {\textbf{n}}] $$ n × E = Z [ ( n × H ) × n ] . Under accretivity and coercivity assumptions on the impedance operator Z, we obtain a new embedding theorem for the domain of the associated Maxwell operator M, which ensures, in particular, that the nonzero spectrum of the Maxwell operator M is discrete.
Authors: Matthias Eller, Illya M. Karabash
Institutions: Georgetown University, University of Bonn
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
OpenAlex ID: W4408160456