Both physical and mathematical theories of quantum scattering seem to be well developed in the case when the potential (or a more general perturbation of the Laplacian) decays fast enough at infinity and satsifies some regularity assumptions. For example under such rather general assumptions existence of $S$-matrix is proved in Ch. XIV of Hormander's "The Analysis of Linear Partial Differential Operators II".

However the Coulomb potential does not satisfy these assumptions of fast decay at infinity. Moreover my impression is that this is not just a technical issue, but the whole theory should look different for the Coulomb potential.

I am wondering if there is any notion of $S$-matrix for the Coulomb potential and how it looks like.

A reference would be helpful. Apologies if the question is not advanced enough- I am not an expert.