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

We study the global-in-time Strichartz estimates for the Schrödinger equation on a class of scattering manifolds X^∘. Let L_V = ∆_g + V where Δ_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 [23] 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–Mizutani[4] but with a potential. We also obtain a global-in-time local smoothing estimate on this geometry setting.