  In the paper we analyze the quantum-mechanical equivalence of the metrics of
a centrally symmetric uncharged gravitational field. We consider the
Schwarzschild metrics in the spherical, isotropic and harmonic coordinates, and
the Eddington-Finkelstein, Painleve-Gullstrand, Lemaitre-Finkelstein, Kruskal
metrics. The scope of the analysis includes domains of the wave functions of
Dirac's equation, hermiticity of Hamiltonians, and the possibility of existence
of stationary bound states of spin-half particles. The constraint on the domain
of the wave functions of the Hamiltonian in a Schwarzschild field in spherical
coordinates (r > r_{0}) resulting from the fulfillment of Hilbert's condition
g_{00} > 0 also holds in other coordinates for all the metrics considered. The
self-adjoint Hamiltonians for the Schwarzschild metrics in the spherical,
isotropic and harmonic coordinates and also for the Eddington-Finkelstein and
Painleve-Gullstrand metrics are Hermitian, and for them the existence of
stationary bound states of spin-half particles is possible. The self-adjoint
Hamiltonians for non-stationary Lemaitre-Finkelstein and Kruskal metrics have
the explicit dependence on the temporal coordinates and stationary bound states
of spin-half particles cannot be defined for these Hamiltonians. The results of
this study can be useful when addressing the issues related to the evolution of
the universe and interaction of collapsars with surrounding matter.
