  The Dirac equation for the Coulomb problem is restated by incorporating a
nonlinear effective interaction into the Dirac Hamiltonian: one keeps the $1/r$
dependence for the Coulomb field, but the coupling constant is modified by a
factor depending on the n (principal quantum number) power of the mean value of
the Hamiltonian. In this simple context we study the Lamb shift and the
hyperfine splitting of the s-levels of hydrogenic atoms. We discuss to what
extent the corresponding calculations fit the energy splittings to the
appropriate order in the fine structure constant.
