We study the global-in-time Strichartz estimates for the Schrödinger equation on a class of scattering manifolds X∘. Let LV=Δg+V
Nearby in the stack
where
Δg
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 HZ on the requirement of
V(z)=O(⟨z⟩−3)
to
O(⟨z⟩−2)
and secondly generalize the result to the scattering manifold with a mild trapped set as well as Bouclet-MizutaniBM but with a potential. We also obtain a global-in-time local smoothing estimate on this geometry setting.