  The Dirac equation for a massive spin-1/2 field in a central potential V in
three dimensions is studied without fixing a priori the functional form of V.
The second-order equations for the radial parts of the spinor wave function are
shown to involve a squared Dirac operator for the free case, whose essential
self-adjointness is proved by using the Weyl limit point-limit circle
criterion, and a `perturbation' resulting from the potential. One then finds
that a potential of Coulomb type in the Dirac equation leads to a potential
term in the above second-order equations which is not even infinitesimally
form-bounded with respect to the free operator. Moreover, the conditions
ensuring essential self-adjointness of the second-order operators in the
interacting case are changed with respect to the free case, i.e. they are
expressed by a majorization involving the parameter in the Coulomb potential
and the angular momentum quantum number. The same methods are applied to the
analysis of coupled eigenvalue equations when the anomalous magnetic moment of
the electron is not neglected.
