Low frequency asymptotics and local energy decay for the Schrödinger equation

Abstract

We prove low frequency resolvent estimates and local energy decay for the Schrödinger equation in an asymptotically Euclidean setting. More precisely, we go beyond the optimal estimates by comparing the resolvent of the perturbed Schrödinger operator with the resolvent of the free Laplacian. This gives the leading term for the development of this resolvent when the spectral parameter is close to 0. For this, we show in particular how we can apply the usual commutators method for generalized resolvents and simultaneously for different operators. Then we deduce similar results for the large time asymptotics of the corresponding evolution problem. Even if we are interested in this paper in the standard Schrödinger equation, we provide a method which can be applied to more general non-selfadjoint (dissipative) operators.