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The spectral analysis of the interior transmission eigenvalue problem for Maxwell's equations

Houssem Haddar 1 Shixu Meng 1, 2, *
* Corresponding author
1 DeFI - Shape reconstruction and identification
Inria Saclay - Ile de France, CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
Abstract : In this paper we consider the transmission eigenvalue problem for Maxwell's equations corresponding to nonmagnetic inhomogeneities with contrast in electric permittivity that has fixed sign (only) in a neighborhood of the boundary. Following the analysis made by Robbiano in the scalar case we study this problem in the framework of semiclassical analysis and relate the transmission eigenvalues to the spectrum of a Hilbert-Schmidt operator. Under the additional assumption that the contrast is constant in a neighborhood of the boundary, we prove that the set of transmission eigenvalues is discrete, infinite and without finite accumulation points. A notion of generalized eigenfunctions is introduced and a denseness result is obtained in an appropriate solution space.
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Submitted on : Friday, April 2, 2021 - 10:10:48 PM
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Houssem Haddar, Shixu Meng. The spectral analysis of the interior transmission eigenvalue problem for Maxwell's equations. Journal de Mathématiques Pures et Appliquées, Elsevier, 2018, 120, pp.1-32. ⟨hal-01945650v2⟩

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