Strichartz estimates and local smoothing estimates for asymptotically flat Schr\"odinger equations

TL;DR

This paper establishes global-in-time Strichartz estimates for Schr"odinger equations with long-range perturbations of the Laplacian, extending previous results to large perturbations under various geometric and spectral conditions.

Contribution

It extends local smoothing to global Strichartz estimates for large perturbations of the Laplacian in different geometric scenarios.

Findings

Estimates outside a compact set without non-trapping assumptions.

Global estimates with non-trapping assumptions.

Global estimates for operators with no resonance or eigenvalue at spectrum bottom.

Abstract

In this article we study global-in-time Strichartz estimates for the Schr\"odinger evolution corresponding to long-range perturbations of the Euclidean Laplacian. This is a natural continuation of a recent article of the third author, where it is proved that local smoothing estimates imply Strichartz estimates. In the aforementioned paper, the third author proved the local smoothing estimates 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.