Localization of the interior transmission eigenvalues for a ball

We study the localization of the interior transmission eigenvalues (ITEs) in the case when the domain is the unit ball ${x \in {\mathbb R}^d:: |x| \leq 1}, : d\geq 2,$andthecoefficients$c_j(x), : j =1,2,$ and the indices of refraction $n_j(x),j=1,2,$ are constants near the boundary $\mathrm{\mid }x\mathrm{\mid }=1$. 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 $\mathrm{\mid }x\mathrm{\mid }\le 1$, we show that all (ITEs) lie in a strip ${ \lambda \in {\mathbb C}::|{\rm Im}: \lambda| \leq C}$.