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 $V(t) = e^{tG},\: t \geq 0,$ related to the wave equation in an unbounded domain $Ω\subset {\mathbb R}^d$ with dissipative boundary condition $\partial_νu - γ(x) \partial_t u = 0$ on $Γ= \partial Ω.$ We study two cases: $(A): \: 0 < γ(x) < 1,\: \forall x \in Γ$ and $(B):\: 1 < γ(x), \: \forall x \in Γ.$ We prove that for every $0 < ε\ll 1,$ the eigenvalues of $G$ in the case $(A)$ lie in the region $Λ_ε = \{z \in {\mathbb C}:\: |\Re z | \leq C_ε (|\Im z|^{\frac{1}{2} + ε} + 1), \: \Re z < 0\},$ while in the case $(B)$ for every $0 < ε\ll 1$ and every $N \in {\mathbb N}$ the eigenvalues lie in $Λ_ε \cup {\mathcal R}_N,$ where ${\mathcal R}_N = \{z \in {\mathbb C}:\: |\Im z| \leq C_N (|\Re z| + 1)^{-N},\: \Re z < 0\}.$

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