Q:

Relativistic corrections to quantum mechanics of Coloumb potential

Systems of charged particles (such as atomic nuclei and electrons) can be described by nonrelativistic quantum mechanics with the Coloumb interaction potential.  A fully relativistic description is given by quantum electrodynamics which is much more complex. 
Is it possible to expand various quantities in QED as power series in 1/c i.e. around the nonrelativistic approximation?  Examples of relevant quantities are:

Ground state energy of a given set of charged particles
Excited state energies
Scattering cross sections of charged particles & their bound states (assuming we trace over the photons in the final state)

A:

Yes, it's possible.  Look Breit equation, for example: http://en.wikipedia.org/wiki/Breit_equation

