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

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(·) ∈ L_t ^∞(L_x²(ℝ³)) ∩ L _t²(L_x⁶(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.