Asymptotics of the number of the interior transmission eigenvalues

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

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