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We prove the local energy decay and the smoothing effect for the damped Schrodinger equation on R^d. The self-adjoint part is a Laplacian associated to a long-range perturbation of the flat metric. The proofs are based on uniform resolvent estimates obtained by the dissipative Mourre method. All the results depend on the strength of the dissipation which we consider.
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Khenissi et al. (2017) studied this question.
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