Dispersive bounds for the three-dimensional Schrödinger equation with almost critical potentials

Abstract.

We prove a dispersive estimate for the time-independent Schrödinger operator H  =   − Δ  + V in three dimensions. The potential V(x) is assumed to lie in the intersection

$$ L^{p} ({\user2{\mathbb{R}}}^{3} ) \cap L^{q} ({\user2{\mathbb{R}}}^{3} ), $$

p < 3/2 < q, and also to satisfy a generic zero-energy spectral condition. This class, which includes potentials that have pointwise decay

$$ {\left| {V(x)} \right|} \leq C(1 + {\left| x \right|})^{{ - 2 - \varepsilon }} , $$

is nearly critical with respect to the natural scaling of the Laplacian. No additional regularity, decay, or positivity of V is assumed.