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We also note that a two-body Dirac equation composed of a Dirac operator for each of the two point particles interacting via the Coulomb interaction can be exactly separated in the (relativistic) center of momentum frame and the resulting ground state eigenvalue has been obtained very accurately using the Finite element methods of J. Shertzer.
With Ezra Getzler, Michèle Vergne, heat kernels and Dirac operator, principles of mathematical sciences 298, Springer Verlag 1992, 2004