Homogenization and nonselfadjoint spectral optimization for dissipative Maxwell eigenproblems
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| Format: | Preprint |
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2024
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| _version_ | 1866909997177241600 |
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| author | Eller, Matthias Karabash, Illya M. |
| author_facet | Eller, Matthias Karabash, Illya M. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_01049 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Homogenization and nonselfadjoint spectral optimization for dissipative Maxwell eigenproblems Eller, Matthias Karabash, Illya M. Analysis of PDEs Optimization and Control Spectral Theory 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. |
| title | Homogenization and nonselfadjoint spectral optimization for dissipative Maxwell eigenproblems |
| topic | Analysis of PDEs Optimization and Control Spectral Theory |
| url | https://arxiv.org/abs/2401.01049 |