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

We prove sharp dispersive $L^1\to L^{\mathrm{\infty }}$ estimates for the three-dimensional attractive Coulomb operator $H_Z=-\mathrm{\Delta }-Z\mathrm{\mid }x{\mathrm{\mid }}^{-1}$, 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 $\mathrm{\mid }t{\mathrm{\mid }}^{-1}$ for long times, with amplitude proportional to $Z$. This slower decay is driven by the threshold and is sharp when $Z^2\mathrm{\mid }t\mathrm{\mid }\gg 1$.