Time decay for solutions of Schrodinger equations with rough and time-dependent potentials

In this paper we establish dispersive estimates for solutions to the linear Schrodinger equation in three dimensions (0.1) 1/ipartial derivative(t)psi - Deltapsi + Vpsi = 0 psi(s) = f where V(t,x) is a time-dependent potential that satisfies the conditions [GRAPHICS] Here c(0) is some small constant and V((tau) over cap, x) denotes the Fourier transform with respect to the first variable. We show that under these conditions (0.1) admits solutions psi(.) is an element of L-t(infinity)(L-x(2)(R-3)) boolean AND L-t(2)(L-x(6)(R-3)) for any f is an element of L-2(R-3) satisfying the dispersive inequality (0.2) parallel topsi(t)parallel to(infinity) less than or equal to C\t - s\(-3/2) parallel tofparallel to(1) for all times t,s. For the case of time independent potentials V(x), (0.2) remains true if [GRAPHICS] We also establish the dispersive estimate with an epsilon-loss for large energies provided parallel toVparallel to(K) + parallel toVparallel to(2) < infinity. Finally, we prove Strichartz estimates for the Schrodinger equations with potentials that decay like vertical bar x vertical bar(-2-epsilon) in dimensions n >= 3, thus solving an open problem posed by Journe, Soffer, and Sogge.