The interior transmission problem and bounds on transmission eigenvalues

TL;DR

This paper investigates the interior transmission eigenvalue problem for certain Laplacian perturbations, demonstrating that most complex eigenvalues are located near the positive real axis within a parabolic region.

Contribution

It provides new bounds on the location of transmission eigenvalues for sign-definite perturbations of the Laplacian, showing their confinement to a specific parabolic neighborhood.

Findings

Most complex transmission eigenvalues lie near the positive real axis.

Finitely many eigenvalues are outside the parabolic neighborhood.

The eigenvalues are confined within a predictable geometric region.

Abstract

We study the interior transmission eigenvalue problem for sign-definite multiplicative perturbations of the Laplacian in a bounded domain. We show that all but finitely many complex transmission eigenvalues are confined to a parabolic neighborhood of the positive real axis.