  Proofs of two statements are provided in this paper. First, the authors prove
that the formalism of the pseudo-Hermitian quantum mechanics allows describing
the Dirac particles motion in arbitrary stationary gravitational fields.
Second, it is proved that using the Parker weight operator and the subsequent
transition to the \eta -representation gives the transformation of the
Schroedinger equation for nonstationary metric, when the evolution operator
becomes self-conjugate. The scalar products in the \eta -representation are
flat, which makes possible the use of a standard apparatus for the Hermitian
quantum mechanics. Based on the results of this paper the authors draw a
conclusion about solution of the problem of uniqueness and self-conjugacy of
Dirac Hamiltonians in arbitrary gravitational fields including those dependent
on time. The general approach is illustrated by the example of Dirac
Hamiltonians for several stationary metrics, as well as for the cosmologically
flat and the open Friedmann models.
