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

Rearrange the continuum equation asThe incident plane wave contributes zero to the left side. The outgoing Green function identity gives $-4\pi f_k{\delta }^{(3)}(\mathbf{r})$ for the scattered term, henceThis fixes both the sign and the three-dimensional $4\pi$ factor. The model is an isotropic point coupling, so the scattered amplitude has no angular dependence. Take $g$ real, as required by the displayed Hermitian pair of coupling equations; an arbitrary phase of the discrete state can be chosen accordingly.