Absence of eigenvalues of dissipative operator for strictly convex obstacles

We study the wave equation in the exterior of a strictly convex bounded domain $K\subset {\mathbb{R}}^d,d\ge 3,$ odd, with dissipative boundary condition ${\mathrm{\partial }}_{\nu }u-\gamma (x){\mathrm{\partial }}_{t}u=0$ on the boundary $\mathrm{\Gamma }$ and $0<\gamma (x)<1,\mathrm{\forall }x\in \mathrm{\Gamma }\mathrm{.}$ The solutions are described by a contraction semigroup $V(t)=e^{tG},t\ge 0.$ In [10] we established that for $\gamma \equiv const$ and $K=\{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }\le 1\}$ the operator $G$ has no eigenvalues and we conjectured that the same result holds for every strictly convex obstacle. In this paper we prove this conjecture.