Localization of the interior transmission eigenvalues for a ball

TL;DR

This paper investigates the localization of interior transmission eigenvalues within a unit ball, demonstrating that under certain constant coefficient conditions, all eigenvalues are confined to a specific vertical strip in the complex plane.

Contribution

It improves existing eigenvalue-free region results for the unit ball, showing all ITEs lie within a bounded imaginary part when coefficients are constant.

Findings

All ITEs are confined to a strip with bounded imaginary part.

The eigenvalue-free region can be significantly expanded for the ball case.

Results apply when coefficients are constant near the boundary.

Abstract

We study the localization of the interior transmission eigenvalues (ITEs) in the case when the domain is the unit ball $\{x \in {\mathbb R}^d:\: |x| \leq 1\}, \: d\geq 2,$ and the coefficients $c_j(x), \: j =1,2,$ and the indices of refraction $n_j(x), \: j =1,2,$ are constants near the boundary $|x| = 1$. We prove that in this case the eigenvalue-free region obtained in [16] for strictly concave domains can be significantly improved. In particular, if $c_j(x), n_j(x), j = 1,2$ are constants for $|x| \leq 1$, we show that all (ITEs) lie in a strip $\{ \lambda \in {\mathbb C}:\:|{\rm Im}\: \lambda| \leq C\}$.