Location of the complex eigenvalues of dissipative problems

We study the location of the eigenvalues of the generator $G$ of a contraction semigroup $V(t) = e^{t G}, \: t \geq 0,$ related to the wave equation with dissipative boundary conditions. The spectrum of $G$ in $\Re z < 0$ is formed by isolated eigenvalues with finite multiplicity and the solutions $u(t, x) = e^{\lambda t} f(x)$ with $Gf = \lambda f, \Re \lambda < 0,$ are called asymptotically disappearing. The location of the eigenvalues of $G$ is important for the inverse scattering problems. We show that the eigenvalues are located in parabolic neighborhoods of the negative real axis or the imaginary one. For strictly convex obstacles we obtain a sharper result.