  Starting from the coupling of a relativistic quantum particle to the curved
Schwarzschild space-time, we show that the Dirac--Schwarzschild problem has
bound states and calculate their energies including relativistic corrections.
Relativistic effects are shown to be suppressed by the gravitational
fine-structure constant alpha_G = G m_1 m_2/(hbar c), where G is Newton's
gravitational constant, c is the speed of light and m_1 and m_2 >> m_1 are the
masses of the two particles. The kinetic corrections due to space-time
curvature are shown to lift the familiar (n,j) degeneracy of the energy levels
of the hydrogen atom. We supplement the discussion by a consideration of an
attractive scalar potential, which, in the fully relativistic Dirac formalism,
modifies the mass of the particle according to the replacement m -> m (1 -
\lambda/r), where r is the radial coordinate. We conclude with a few comments
regarding the (n,j) degeneracy of the energy levels, where n is the principal
quantum number, and j is the total angular momentum, and illustrate the
calculations by way of a numerical example.
