Location of eigenvalues for the wave equation with dissipative boundary conditions

We examine the location of the eigenvalues of the generator $G$ of a semi-group $V(t)=e^{tG},t\ge 0,$ related to the wave equation in an unbounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$ with dissipative boundary condition ${\mathrm{\partial }}_{\nu }u-\gamma (x){\mathrm{\partial }}_{t}u=0$ on $\Gamma = \partial \Omega.$Westudytwocases:$(A): : 0 < \gamma(x) < 1,: \forall x \in \Gamma$and$(B):: 1 < \gamma(x), : \forall x \in \Gamma.$ We prove that for every $0<ϵ\ll 1,$ the eigenvalues of $G$ in the case $(A)$ lie in the region $\Lambda_{\epsilon} = {z \in {\mathbb C}:: |\Re z | \leq C_{\epsilon} (|\Im z|^{\frac{1}{2} + \epsilon} + 1), : \Re z < 0},$whileinthecase$(B)$forevery$0 < \epsilon \ll 1$andevery$N \in {\mathbb N}$ the eigenvalues lie in ${\mathrm{\Lambda }}_{ϵ}\cup {\mathcal{R}}_N,$ where ${\mathcal R}_N = {z \in {\mathbb C}:: |\Im z| \leq C_N (|\Re z| + 1)^{-N},: \Re z < 0}.$