Some remarks on the Schrödinger equation with a potential in $L_t^r{L}_x^s$

Abstract

We study the dispersive properties of the linear Schrödinger equation with a time-dependent potential $V(t,x)$. We show that an appropriate integrability condition in space and time on $V$, i.e. the boundedness of a suitable $L_t^r{L}_x^s$ norm, is sufficient to prove the full set of Strichartz estimates. We also construct several counterexamples which show that our assumptions are optimal, both for local and for global Strichartz estimates, in the class of large unsigned potentials $V\in L_t^r{L}_x^s$.

to appear on Math. Annalen