When using a second-order Schrödinger-type equation with the effective potential of the Schwarzschild field, existence of a stationary state of half-spin particles with energy 0 E is proved. For each of the values of quantum numbers , jl , the physically meaningful energy) is implemented at the value of the gravitational coupling constant min . The particles with 0 E are, with the overwhelming probability, at some distance from the event horizon within the range from zero to several fractions of Compton wavelength of a fermion depending on value of the gravitational coupling constants and values , jl.In this paper, similar solutions of the second-order equation are announced for bound states of fermions in the Reissner-Nordström, Kerr, Kerr-Newman fields.Atomic-type systems: collapsars with fermions in bound states are proposed as particles of dark matter.
When using the quantum mechanical second-order equation with the effective potential of the Kerr-Newman (KN) field for fermions, results were obtained that qualitatively differ from results obtained when using the Dirac equation. In presence of two event horizons, existence of degenerate stationary bound states was proved for charged and uncharged fermions with square integrable wave functions vanishing on event horizons. The fermions in such states are localized near the event horizons with the maxima of probability densities away from the event horizons by fractions of the Compton wave length of fermions versus the values of coupling constants, the values of angular and orbital momenta , jl and the value of the azimuthal quantum number m .In the case of extreme KN fields, absence of stationary bound states of fermions was shown for any values of coupling constants.Existence of discrete energy spectra was shown for charged and uncharged fermions in the field of naked KN singularity at definite values of physical parameters.The KN naked singularity poses no threat to cosmic censorship because of the regular behavior of the effective potentials of the KN field in quantum mechanics with the second-order equation.
Existence of degenerate stationary bound states with square integrable radial wave functions was proved when second-order equations are used with the effective potential of the Reissner-Nordström (RN) field with two event horizons for charged and uncharged fermions,. The fermions in such states are localized near event horizons within the ranges from zero to several fractions of Compton wave length of fermions versus the values of gravitational and electromagnetic coupling constants and the values of angular and orbital momenta , jl. In case of extreme RN fields, absence of stationary bound states of fermions with the energies of 2 E mc is shown for solutions of the second-order equation for any value of gravitational and electromagnetic coupling constants. Existence of the discrete energy spectrum is shown for the naked RN singularity due to solution of the second-order equation at definite values of physical parameters. The discrete spectrum exists for both charged and uncharged fermions. The naked RN singularity in quantum mechanics with the second-order equation for half-spin particles poses no threat to cosmic censorship since it is covered with an infinitely large potential barrier. Electrically neutral systems of atomic type (RN collapsars with the definite number of fermions in degenerate bound states) are proposed to consider as particles of dark matter.
We have studied self-conjugate second-order equations with spinor wavefunctions for fermions moving in an external Coulomb field. For stationary states, the equations are characterized by separated states with positive and negative energies, which render probabilistic interpretation possible. For the Coulomb field of attraction, the energy spectrum of the secondorder equation coincides with the spectrum of the Dirac equation, while the probability densities of states are slightly different. For a Coulomb field of repulsion, there exists an impermeable potential barrier with radius depending on the classical electron radius and on the electron energy. The existence of the impermeable barrier does not contradict the results of experiment for determining the inner electron structure and does not affect (in the lowest order of perturbation theory) the Coulomb electron scattering cross section. The existence of the impermeable barrier can lead to positron confinement in supercritical nuclei with 170 Z in case of realization of spontaneous emission of vacuum electron-positron pairs.
An effective, reliable and time saving numerical method with using of the Pruefer transformation is proposed to calculate eigenvalues of Chandrasekhar-Page angular equations.
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