One of the first results seen in elementary quantum mechanics is the closed form solution to the bound states of the hydrogen atom. In the usual approach, scattering theory is placed on the opposite side of spectrum; exclusively though of as a perturbative process. I am interested in knowing about the scattering states of the hydrogen atom, but more specifically in terms of electron-electron scattering.
My question is regarding the scattering states of electron-eletron scattering, which has the (non-relativistic) Hamiltonian below.
Because the problem is so similar to that of the electron-proton system, I would expect a closed form solution for dealing with the scattering states non-perturbatively. However, this problem (Rutherford scattering) is usually treated in textbooks using the Born approximation, with divergences at low deflection angles where the approach breaks down. I am especially interested in this regime.
So my questions are the following:
In the non-relativistic limit, is there a non-perturbative solution to electron-electron scattering? By solution I (naively) am thinking of something akin to an expression for the scattering cross-section.
When relativity is introduced (either with the Dirac-equation, or the full machinery of QED), at what point is a non-perturbative solution out of reach? Are there any special cases (e.g. total energies are below that needed for pair-production) that are significantly easier to handle?