Eigenvalues for Maxwell's equations with dissipative boundary conditions

Abstract: Let $V(t) = e^{tG_b},: t \geq 0,$ be the semigroup generated by Maxwell's equations in an exterior domain $\Omega \subset {\mathbb R}^3$ with dissipative boundary condition $E_{tan}- \gamma(x) (\nu \wedge B_{tan}) = 0, \gamma(x) > 0, \forall x \in \Gamma = \partial \Omega.$ We prove that if $\gamma(x)$ is nowhere equal to 1, then for every $0 < \epsilon \ll 1$ and every $N \in {\mathbb N}$ the eigenvalues of $G_b$ lie in the region $\Lambda_{\epsilon} \cup {\mathcal R}N,$ where $\Lambda{\epsilon} = { z \in {\mathbb C}:: |\Re z | \leq C_{\epsilon} (|\Im z|^{\frac{1}{2} + \epsilon} + 1), : \Re z < 0},$ ${\mathcal R}_N = {z \in {\mathbb C}:: |\Im z| \leq C_N (|\Re z| + 1)^{-N},: \Re z < 0}.$