Half-shell scattering by a screened Coulomb potential

We investigate the physical half-shell T matrix for scattering by a smoothly screened Coulomb potential. In particular, we study the Kowalski-Noyes half-shell extension function Flhs(p;k) in the limit that the screening radius a tends to infinity. For the Coulomb potential the half-shell scattering cross section is a discontinuous (step) function of the off-shell momentum p at the on-shell point p=k. For the screened Coulomb potential (with screening of the Hulthén form, for l=0) we find that the half-shell cross section exhibits almost a step behavior near p=k: Its magnitude varies dramatically in an interval, the width of which is of the order of the inverse screening radius a. We discuss physical implications of this result.