In relativistic scattering theory (e.g. in quantum electrodynamics) the existence of the $S$-matrix as well as of Moller operators is postulated as far as I understand (although at some stage it has to be modified due to existence of infrared divergences). On the other hand if one considers the non-relativistic scattering problem of several particles with Coulomb potential, it seems to be very different from the usual scattering theory with rapidly decaying potential and much of it does not work (again, this is my impression, and I am not an expert).
I am wondering why one should expect existence of $S$-matrix in quantum electrodynamics with properties similar to $S$-matrix for non-relativistic scattering theory with rapidly decaying potential (rather than Coulomb potential). Do I miss something on the non-relativistic Coulomb scattering?
I guess that this question might not be appropriate for this site as QFT is not yet a mathematical theory. But this question is a direct continuation of this one on this site where I did get some helpful comments. Moreover I imagine that an answer might be known to some mathematical physicists.