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In this paper, we investigate the effect of time delays in boundary or internal feedback stabilization of the multidimensional Schrödinger equation. In both cases, under suitable assumptions, we establish sufficient conditions on the delay term that guarantee the exponential stability of the solution. These results are obtained by using suitable energy functionals and some observability estimates.

Abstract. We consider a coupled system of two complex Schrödinger equations with variable coefficients. The boundary feedback appears only in one of the equations. The aim of this paper is to prove that we can apply the Riemann geometric approach developed to study the problems of direct stabilization for real hyperbolic equations (see [22]) and show that the sufficiently smooth solutions decays polynomially at infinity, by adapting the ideas of Alabau in [2] used to obtain indirect stabilization results for a system of two coupled real wave equations with constant coefficients. Mathematics Subject Classification (2000). 93D15, 35Q40, 42B15.

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