Transmission problem of Schrödinger and wave equation with viscous damping

In this paper, we consider the transmission problem of a Schrödinger equation with a viscous damped wave equation which acts as a controller of the system. We show that the system operator generates a C0C0-semigroup of contractions in the energy state space, and the system is well-posed. By giving the asymptotic expressions of the eigenvalues of the system, we know they all locate in the left hand side of the complex plane. It follows that the C0C0-semigroup generated by the system operator achieves strong stability when the feedback gain is real.