Localization of the interior transmission eigenvalues for a ball

Abstract

We study the localization of the interior transmission eigenvalues (ITEs) in the case when the domain is the unit ball $\{x\in {\mathbb{R}}^d:|x|\le 1\},d\ge 2,$ 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 $|x|\le 1$, we show that all (ITEs) lie in a strip $\{\lambda \in \mathbb{C}:|\mathrm{Im}\lambda |\le C\}$.

References in corpus (1)

• Transmission eigenvalue-free regions