Sharp Dispersive Estimates for the Schrödinger Equation with an Attractive Coulomb Potential

Abstract

We prove sharp dispersive $L^1\to L^{\infty }$ estimates for the three-dimensional attractive Coulomb operator $H_Z=-\Delta -Z|x{|}^{-1}$, where $Z>0$. The absolutely continuous part of the Schrödinger evolution decays at the free rate for short times, whereas its leading contribution decays like $|t{|}^{-1}$ for long times, with amplitude proportional to $Z$. This slower decay is driven by the threshold and is sharp when $Z^2|t|\gg 1$.