Hydrogen-like Schrodinger Operators at Low Energies

Abstract: Recently, some microlocal tools of Vasy have allowed a systematic study of low energy scattering off of fairly generic short range potentials. The case of a long-range, Coulomb-like potential differs substantially in terms of the asymptotics seen at low energy, both at fixed spatial scales and in the joint regime where $E\to 0^+$ and $r\to\infty$ simultaneously. In this talk, we will consider the case of an *attractive* Coulomb-like (i.e. Hydrogen-like) potential, for which the low energy asymptotics can be described rather completely. The results are apparently new even in the case of an exact Coulomb potential, being related to the asymptotics of the Whittaker W-function as the argument and one parameter are taken to infinity simultaneously.