Stability analysis for a coupled Schrodinger system with one boundary damping

In this paper, we study the stability of a Schrodinger system with one boundary damping, which consists of two constant coefficients Schrodinger equations coupled through zero-order terms. First, we show that when rho=1,$$ \varrho =1, $$ the one-dimensional Schrodinger system is not exponentially stable by the asymptotic expansions of eigenvalues. Then, by the frequency domain approach and the multiplier method, we show that the energy decay rate of the multidimensional Schrodinger system is t-1$$ {t}<^>{-1} $$ for sufficiently smooth initial data when rho=1,$$ \varrho =1, $$ |alpha|$$ \mid \alpha \mid $$ is sufficiently small, and the boundary of domain satisfies suitable geometric assumption. Next, by solving the characteristic equation of unbounded operator, we show that the strong stability of the one-dimensional Schrodinger system is completely determined by rho$$ \varrho $$ and alpha$$ \alpha $$ and give the necessary and sufficient condition that rho$$ \varrho $$ and alpha$$ \alpha $$ satisfy. Finally, by solving the resolvent equation of unbounded operator and using the frequency domain approach, we show that when rho not equal 1$$ \varrho \ne 1 $$ and |alpha|$$ \mid \alpha \mid $$ is small enough, the energy of the one-dimensional Schrodinger system decays polynomially and the decay rate depends on the arithmetic property of rho.$$ \varrho . $$