  We study the Dirac-Maxwell system coupled with an external potential of
Coulomb type. We use the Foldy--Wouthuysen (unitary) transformation of the
Dirac operator and its realization as an elliptic problem in the 4-dim half
space $\mathbb{R}^4_{+}$ with Neumann boundary condition. Using this approach
we study the existence of a "ground state" solution.
