Discreteness of Transmission Eigenvalues via Upper Triangular Compact Operators

TL;DR

This paper proves that interior transmission eigenvalues are discrete by demonstrating that the associated operator has an upper triangular compact resolvent, leading to spectral properties similar to compact operators.

Contribution

It introduces a novel approach using upper triangular compact operators to establish discreteness of transmission eigenvalues under a boundary-localized coercivity condition.

Findings

Spectrum of the operator is discrete

Generalized eigenspaces are finite dimensional

Eigenvalues are isolated and countable

Abstract

We show that the interior transmission eigenvalues are discrete by proving that the interior transmission operator has upper triangular compact resolvent, and that the spectrum of these operators share many of the properties of operators with compact resolvent. In particular, the spectrum is discrete and the generalized eigenspaces are finite dimensional. Our main hypothesis is a coercivity condition on the contrast that must hold only in a neighborhood of the boundary.