Asymptotics of the number of the interior transmission eigenvalues

Abstract

We prove a Weyl asymptotics $N(r) = c r^d + {\mathcal O}_{\epsilon}(r^{d - \kappa + \epsilon})$, $\forall\, 0< \epsilon \ll 1$, for the counting function $N(r) = \sharp\{\lambda_j \in {\mathbb C} \setminus \{0\}:\: |\lambda_j| \leq r^2\}$, $r>1$, of the interior transmission eigenvalues (ITE), $\lambda_j$. Here $0<\kappa\leq 1$ is such that there are no (ITE) in the region $\{\lambda\in {\mathbb C}:\: |{\rm Im}\:\lambda|\geq C(| {\rm Re}\:\lambda|+1)^{1-\frac{\kappa}{2}}\}$ for some $C>0$.