Asymptotic Stability for the Damped Schrödinger Equation on Noncompact Riemannian Manifolds and Exterior Domains

The Schrödinger equation subject to a nonlinear and locally distributed damping, posed in a connected, complete, and noncompact n dimensional Riemannian manifold (ℳ, g) is considered. Assuming that (ℳ, g) is nontrapping and, in addition, that the damping term is effective in ℳ \ Ω, where Ω ⊂ ⊂ ℳ is an open bounded and connected subset with smooth boundary ∂Ω, such that is a compact set, exponential and uniform decay rates of the L 2 − level energy are established. The main ingredients in the proof of the exponential stability are: (A) an unique continuation property for the linear problem; and (B) a local smoothing effect for the linear and nonhomogeneous associated problem. Keywords: Exponential stabilityNon-compact manifoldsSchröedinger equationMathematics Subject Classification: 58Jxx Acknowledgments The authors would like to thank Professor Nicolas Burq for fruitful discussions during the preparation of this manuscript and the anonymous referee for careful reading the manuscript and for his(her) comments and suggestions which resulted in the present version of it.