Post-Newtonian equation for the energy levels of a Dirac particle in a static metric

We study first the Hamiltonian operator H corresponding to the Fock-Weyl extension of the Dirac equation to gravitation. When searching for stationary solutions to this equation, in a static metric, we show that just one invariant Hermitian product appears natural. In the case of a space-isotropic metric, H is Hermitian for that product. Then we investigate the asymptotic post-Newtonian approximation of the stationary Schroedinger equation associated with H, for a slow particle in a weak-field static metric. We rewrite the expanded equations as one equation for a two-component spinor field. This equation contains just the non-relativistic Schroedinger equation in the gravity potential, plus correction terms. Those "correction" terms are of the same order in the small parameter as the "main" terms, but are numerically negligible in the case of ultra-cold neutrons in the Earth's gravity.