Spectral analysis on interior transmission eigenvalues

TL;DR

This paper investigates the spectral properties of interior transmission eigenvalues, establishing their discreteness, density of eigenfunctions, and providing estimates on eigenvalue distribution and resolvent behavior.

Contribution

It proves the discreteness and density of eigenfunctions for interior transmission eigenvalues using spectral theory of Hilbert-Schmidt operators, and derives eigenvalue and resolvent estimates.

Findings

Spectrum is discrete and countable.

Generalized eigenfunctions are dense in the resolvent range.

Eigenvalue count estimate for large modulus.

Abstract

In this paper we prove some results on interior transmission eigenvalues. First, under rea- sonable assumptions, we prove that the spectrum is a discrete countable set and the generalized eigenfunctions spanned a dense space in the range of resolvent. This is a consequence of spectral theory of Hilbert-Schmidt operators. The main ingredient is to prove a smoothing property of resolvent. This allows to prove that a power of the resolvent is Hilbert-Schmidt. We obtain an estimate of the number of eigenvalues, counting with multiplicities, with modulus less than t2 when t is large. We prove also some estimate on the resolvent near the real axe when the square of the index of refraction is not real. Under some assumptions we obtain lower bound on the resolvent using the results obtained by Dencker, Sj\"ostrand and Zworski on the pseudospectra.