Perturbative evaluation of the eigenvalues of the Herbst Hamiltonian

We reconsider the well-known and long-debated problem of the calculation of the eigenvalues of the Herbst Hamiltonian 2\sqrt{p^2 +m^2} - \kappa/r. We give a formulation of the problem that allows, for the first time, a perturbative evaluation of the eigenvalues for any n and l, and in principle up to any order in \kappa via standard Kato perturbation theory. We present the evaluation of the energy of the n=1 and n=2 states up to \kappa^6, confirming the result previously obtained by Le Yaouanc et al. with a completely different technique. Moreover we give the n=2, l=1 level, which is new. Discussion of the results and comparison with previous findings are given at the end.