Past exam of the theoretical physics II course of the University of Cambridge / 2015 / Paper 1 / 3 / c / Solution

The outgoing wave is singular at the origin, so its value there cannot be inserted without regularization. At a short radial distance $\delta >0$, with $k\delta \ll 1$,The incident factor tends to one independently of the chosen approach direction; the singular part is isotropic. The discrete-state equation becomes, to the retained short-distance order,Keeping only $1/\delta$ and discarding $ik$ would lose the outgoing-wave imaginary part and produce an amplitude inconsistent with scattering unitarity.