A dispersive bound for three-dimensional Schroedinger operators with zero energy eigenvalues

Abstract

We prove a dispersive estimate for the evolution of Schroedinger operators $H=-\Delta +V(x)$ in ${\mathbb{R}}^3$. The potential is allowed to be a complex-valued function belonging to $L^p({\mathbb{R}}^3)\cap L^q({\mathbb{R}}^3)$, $p<\frac{3}{2}<q$, so that $H$ need not be self-adjoint or even symmetric. Some additional spectral conditions are imposed, namely that no resonances of $H$ exist anywhere within the interval $[0,\infty )$ and that eigenfunctions at zero (including generalized eigenfunctions) decay rapidly enough to be integrable.

25 pages