Stationary bound states of spin-half particles in the Reissner-Nordstroem gravitational field

We prove the possibility of existence of stationary bound states of spin-half particles in the Reissner-Nordstroem gravitational field using a self-conjugate Hamiltonian with a flat scalar product of wave functions. Bound states of Dirac particles with a real discrete energy spectrum can exist both for particles above the external "event horizon", and for particles under the internal "event horizon", or the Cauchy horizon. The Hilbert condition g_{00}>0 leads to a boundary condition such that components of the vector of current density of Dirac particles are zero near the "event horizons". Based on the results of this study, we can assume that there exists a new type of charged collapsars, for which the Hawking radiation is not present. The results of this study can lead to a revision of some concepts of the standard cosmological model related to the evolution of the universe and interaction of charged collapsars with surrounding matter.