  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.
