Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem

TL;DR

This paper applies elliptic operator theory to the isotropic interior transmission eigenvalue problem, establishing spectrum discreteness, eigenvalue localization, and existence results using the Dirichlet-to-Neumann map under specific conditions.

Contribution

It introduces a method to reduce the non-elliptic ITE problem to an elliptic one, enabling new spectral analysis results and extending to complex-valued refractive indices.

Findings

Spectrum is discrete under certain conditions.

Infinitely many positive ITEs exist when the index is real.

Weyl type lower bounds on the eigenvalue counting function.

Abstract

The paper concerns the isotropic interior transmission eigenvalue (ITE) problem. This problem is not elliptic, but we show that, using the Dirichlet-to-Neumann map, it can be reduced to an elliptic one. This leads to the discreteness of the spectrum as well as to certain results on possible location of the transmission eigenvalues. If the index of refraction $\sqrt{n(x)}$ is real, we get a result on the existence of infinitely many positive ITEs and the Weyl type lower bound on its counting function. All the results are obtained under the assumption that $n(x)-1$ does not vanish at the boundary of the obstacle or it vanishes identically, but its normal derivative does not vanish at the boundary. We consider the classical transmission problem as well as the case when the inhomogeneous medium contains an obstacle. Some results on the discreteness and localization of the spectrum are obtained for complex valued $n(x)$.