In this paper we establish dispersive estimates for solutions to the linear Schrödinger equation in three dimensions (0.1) 1/i∂ tψ - Δ + Vψ = 0, ψ(s) = f where V(t, x) is a time-dependent potential that satisfies the conditions sup t∥V(t,·)∥L3/2(ℝ3) + sup x∈ℝ3∫ℝ3∫-∞ ∞|V(τ̂, x)|/|x - y| dτ dy < c0. Here c0 is some small constant and V(τ̂, x) denotes the Fourier transform with respect to the first variable. We show that under these conditions (0.1) admits solutions ψ(·) ∈ Lt ∞(Lx2(ℝ3)) ∩ L t2(Lx6(ℝ3)) for any f ∈ L2(ℝ3) satisfying the dispersive inequality (0.2) ∥ψ(t)∥∞ ≤ C|t - s| -3/2∥f∥1 for all times t,s. For the case of time independent potentials V(x), (0.2) remains true if ∫ℝ6 |V(x)| |V(y)|/|x - y|2dxdy < (4π)2 and ∥V∥K : = sup x∈ℝ3∫ ℝ3 |V(y)|/|x - y|dy < 4π. We also establish the dispersive estimate with an ε-loss for large energies provided ∥V∥K + ∥V∥2 < ∞. Finally, we prove Strichartz estimates for the Schrödinger equations with potentials that decay like |x|-2-ε in dimensions n ≥ 3, thus solving an open problem posed by Journé, Soffer, and Sogge.
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