# Global-in-time Strichartz estimates for Schrodinger on scattering manifolds

## TL;DR

This paper extends and generalizes global-in-time Strichartz estimates for the Schrödinger equation on scattering manifolds, including cases with trapping and potentials, and also establishes local smoothing estimates in this geometric setting.

## Contribution

It improves existing Strichartz estimates by weakening decay conditions on the potential and extends results to manifolds with trapping, adding potential considerations.

## Key findings

- Extended Strichartz estimates to $O(raket{z}^{-2})$ potentials.
- Generalized estimates to scattering manifolds with mild trapping.
- Established global-in-time local smoothing estimates.

## Abstract

We study the global-in-time Strichartz estimates for the Schr\"odinger equation on a class of scattering manifolds $X^{\circ}$. Let $\mathcal{L}_V=\Delta_g+V$ where $\Delta_g$ is the Beltrami-Laplace operator on the scattering manifold and $V$ is a real potential function on this setting. We first extend the global-in-time Strichartz estimate in Hassell-Zhang \cite{HZ} on the requirement of $V(z)=O(\langle z\rangle^{-3})$ to $O(\langle z\rangle^{-2})$ and secondly generalize the result to the scattering manifold with a mild trapped set as well as Bouclet-Mizutani\cite{BM} but with a potential. We also obtain a global-in-time local smoothing estimate on this geometry setting.