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

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.