On some Schrödinger and wave equations with time dependent potentials

The existence and uniqueness of solutions in the initial value problem for Schrödinger and wave equations in the presence of a (large) time dependent potential is studied. The usual Strichartz estimates for such linear evolutions are shown to hold true with optimal assumptions on the potentials. As a byproduct, one obtains a counterexample to the two dimensional double endpoint inhomogeneous Strichartz estimate.