Dispersive Bounds for the three-dimensional Schrodinger equation with almost critical potentials

Abstract

We prove a dispersive estimate for the time-independent Schrodinger operator H = -\Delta + V in three dimensions. The potential V(x) is assumed to lie in the intersection L^p(R^3) \cap L^q(R^3), p < 3/2 < q, and also to satisfy a generic zero-energy spectral condition. This class, which includes potentials that have pointwise decay |V(x)| < C(1+|x|)^{-2-\epsilon}, is nearly critical with respect to the natural scaling of the Laplacian. No additional regularity, decay, or positivity of V is assumed.