Asymptotics of the number of the interior transmission eigenvalues

Abstract

We prove Weyl asymptotics $N(r)=c{r}^d+{\mathcal{O}}_{ϵ}(r^{d-\kappa +ϵ})$, for all $0<ϵ\ll 1$, for the counting function $N(r)=♯\{{\lambda }_j\in \mathbb{C}\setminus \{0\}:|{\lambda }_j|\le r^2\}$, $r>1$, of the interior transmission eigenvalues (ITE), ${\lambda }_j$. Here $d\ge 2$ denotes the space dimension and $0<\kappa \le 1$ is such that there are no (ITE) in the region $\{\lambda \in \mathbb{C}:|\mathrm{Im}\lambda |\ge C(|\mathrm{Re}\lambda |+1{)}^{1-\frac{\kappa }{2}}\}$ for some $C>0$.