  The domain of wave functions and effective potentials of the Dirac and
Klein-Gordon equations for quantum-mechanical particles in static centrally
symmetric gravitational fields are analyzed by taking into account the Hilbert
causality condition. For all the explored metrics, assuming existence of event
horizons, the conditions of a "fall" of a particle to the appropriate event
horizons are implemented. The exclusion is one of the solutions for the
Reissner-Nordstroem extreme field with the single event horizon. In this case,
while fulfilling the condition found by V.I.Dokuchaev, Yu.N.Yeroshenko, the
normalization integral is convergent and the wave functions become zero on the
event horizon. This corresponds to the Hilbert causality condition. In our
paper, due to the analysis of the effective potential for the
Reissner-Nordstroem extreme field with real radial wave functions of the Dirac
equation, the impossibility is demonstrated for the bound stationary state
existence of quantum-mechanical particles, with positive energy, off the event
horizon. The analysis of the effective potential of the Dirac equation for the
case of the naked singularity of the Reissner-Nordstroem field shown the
possible existence of stationary bound states of quantum-mechanical half-spin
particles.
