  In this work, we estimate the relativistic corrections to the energy spectrum
of hydrogen resulting from the full one-photon-exchange interaction using a
highly precise numerical method. In the frame of the effective Schrodinger-like
equation, which is derived exactly from the Bethe-Salpeter equation in quantum
electrodynamics, we express the effective potential corresponding to the full
one-photon-exchange interaction in terms of eight scalar functions. Unlike the
usual calculations performed in the references, where the effective potential
is expanded in terms of momenta order by order, we retain the exact momentum
dependence in the effective potential to estimate its corrections to the energy
spectrum of hydrogen using a highly precise numerical method. We also use the
same numerical method to double-check the results in the non-relativistic case
to ensure accuracy. We discuss the comparison of the numerical results with
those obtained using the usual bound-state perturbative theory. Our
calculations suggest that it is possible to accurately account for all the
relativistic contributions using this method. It would be interesting to extent
these calculations to positronium, muonic hydrogen, and cases involving nuclear
structure and radiative corrections.
