Some remarks on the Schroedinger equation with a potential in L^r_tL^s_x

摘要

We study the dispersive properties of the linear Schr¨odinger equation with a timedependentpotential V (t, x). We show that an appropriate integrability condition in space and timeon V , i.e. the boundedness of a suitable L^rL^sx norm, is sufficient to prove the full set of Strichartzestimates. 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 ∈ L^rL^s.