Exponential Stability of a Schrödinger Equation Through Boundary Coupling a Wave Equation

In this paper, we are concerned with the stability of a Schrödinger equation through boundary coupling with a wave equation, where an internal dissipative damping is designed at the wave equation. The energy decay of the Schrödinger equation is obtained by the boundary transmission between the Schrödinger and wave equations. By a detailed spectral analysis, we show that all the eigenvalues of both the Schrödinger and wave equations have negative real parts, and the whole system is exponentially stable. A numerical simulation is presented for the distributions of the spectrum of the whole system, and it is found that the spectrum of the Schrödinger equation depends largely on the boundary transmission parameter and the decay of the wave equation.