Strichartz estimates for long range perturbations

We study local in time Strichartz estimates for the Schrödinger equation associated to long range perturbations of the flat Laplacian on the Euclidean space. We prove that in such a geometric situation, outside a large ball centered at the origin, the solutions of the Schrödinger equation enjoy the same Strichartz estimates as in the nonperturbed situation. The proof is based on the Isozaki-Kitada parametrix construction. If in addition the metric is nontrapping, we prove that the Strichartz estimates hold in the whole space.