The Spectral Analysis of the Interior Transmission Eigenvalue Problem for Maxwell's Equations

TL;DR

This paper investigates the spectral properties of the interior transmission eigenvalue problem for Maxwell's equations with non-magnetic inhomogeneities, establishing discreteness and denseness results using semiclassical analysis.

Contribution

It introduces a spectral analysis framework for Maxwell's transmission eigenvalues, proving discreteness and denseness under specific boundary contrast conditions.

Findings

Transmission eigenvalues form a discrete, infinite set

Eigenvalues have no finite accumulation points

Generalized eigenfunctions are dense in the solution space

Abstract

In this paper we consider the transmission eigenvalue problem for Maxwell's equations corresponding to non-magnetic inhomogeneities with contrast in electric permittivity that has fixed sign (only) in a neighborhood of the boundary. 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.