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<p>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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2024_2908_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="178" /> </InlineMediaObject> <EquationSource Format="TEX">\({\textbf{n}}\times {\textbf{E}}= Z [({\textbf{n}}\times {\textbf{H}})\times {\textbf{n}}] \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">n</mi> <mo>×</mo> <mi mathvariant="bold">E</mi> <mo>=</mo> <mi>Z</mi> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi mathvariant="bold">n</mi> <mo>×</mo> <mi mathvariant="bold">H</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mi mathvariant="bold">n</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. Under accretivity and coercivity assumptions on the impedance operator <i>Z</i>, we obtain a new embedding theorem for the domain of the associated Maxwell operator <i>M</i>, which ensures, in particular, that the nonzero spectrum of the Maxwell operator <i>M</i> is discrete.</p>

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Homogenization and nonselfadjoint spectral optimization for dissipative Maxwell eigenproblems

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  • 发表日期：

    2025年3月5日

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  • Matthias Eller,
  • Illya M. Karabash

摘要

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.