Strichartz estimates and local smoothing estimates for asymptotically flat Schrödinger equations

In this article we study global-in-time Strichartz estimates for the Schrödinger evolution corresponding to long-range perturbations of the Euclidean Laplacian. This is a natural continuation of a recent article D. Tataru, Parametrices and dispersive estimates for Schrödinger operators with variable coefficients, Amer. J. Math. 130 (2008) 571-634] of the third author, where it is proved that local smoothing estimates imply Strichartz estimates. By D. Tataru, Parametrices and dispersive estimates for Schrödinger operators with variable coefficients, Amer. J. Math. 130 (2008) 571-634] the local smoothing estimates are known to hold for small perturbations of the Laplacian. Here we consider the case of large perturbations in three increasingly favorable scenarios: (i) without non-trapping assumptions we prove estimates outside a compact set modulo a lower order spatially localized error term, (ii) with non-trapping assumptions we prove global estimates modulo a lower order spatially localized error term, and (iii) for time independent operators with no resonance or eigenvalue at the bottom of the spectrum we prove global estimates for the projection onto the continuous spectrum.