arXiv subjects
Publications and source records attributed to I. I. Safronov.
Stable and reliable numerical integration of second-order radial equations in the fields of classical black holes can be performed by using the Prüfer transformation and by passing to the use of phase functions, which allows uniquely selecting the solutions with physical asymptotics in numerical calculations.
An impermeable barrier at $r=r_{cl}$ in the effective potential of the relativistic Schrödinger-type equation leads to exclusion of the range $0 \leq r < r_{cl}$ from the wave function domain. Based on duality of the Schrödinger-type equation and the Dirac equation, a similar exclusion should be made in the wave functions domain of the Dirac equation. As a result, we obtain new solutions to the Dirac equation in the Coulomb repulsive field. Calculations show that depending on working parameters, at distances of several fractions or units of the Compton wavelength of the fermion from $r=r_{cl}$ new solutions almost coincide with the standard Coulomb functions of the continuous spectrum. Practically, matrix elements with new solutions will coincide to a good accuracy with standard matrix elements with the Coulomb functions of the continuous spectrum. Our consideration is methodological and helpful for discussing further development of quantum theory.
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 second-order 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 $Z \geq 170$ in case of realization of spontaneous emission of vacuum electron-positron pairs.
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 $j,l$ and the value of the azimuthal quantum number $m_φ$. 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 $j,l$. In case of extreme RN fields, absence of stationary bound states of fermions with the energies of $E<mc^{2}$ 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.
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 $E=0$ is proved. For each of the values of quantum numbers $j,l$, the physically meaningful energy $E=0$ (the binding energy is $E_{b}=mc^{2}$) is implemented at the value of the gravitational coupling constant $α\geqα_{min}$. The particles with $E=0$ 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 $j,l$. 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.
After transition from the Dirac equation to the Schrödinger-type relativistic equation with effective potentials of the Schwarzschild and Reissner-Nordström (RN) fields, the existence of the stationary state of fermions with real square-integrable radial wave functions is proved. The fermions are localized near the event horizon within the range from zero to several fractions or a few units of the Compton wavelength of a fermion as a function of the gravitational and electromagnetic coupling constants and the angular and orbital momenta j,l. Electrically neutral atomic-type systems (Schwarzschild and RN collapsars with fermions in bound states) are proposed as particles of dark matter.
An effective, reliable and time saving numerical method with using of the Pruefer transformation is proposed to calculate eigenvalues of Chandrasekhar-Page angular equations.
The paper explores quantum mechanics of half-spin particle motion in the field of Reissner-Nordstroem (RN) naked singularity. It is shown that for any quantum mechanical Dirac particle, irrespective of availability and sign of its electrical charge, the RN naked singularity is separated by an infinitely high positive potential barrier. With like charges of a particle and the source of the RN naked singularity, near the origin there exists the second completely impenetrable potential barrier. It has been proved that in the field of the RN naked singularity, bound states of half-spin particles can exist. The conditions for appearance of such states were revealed and computations were performed to find energy eigenvalues and eigenfunctions.
The "catastrophe" in solving the Dirac equation for an electron in the field of a point electric charge, which emerges for the charge numbers Z > 137, is removed in this work by effective accounting of finite dimensions of nuclei. For this purpose, in numerical solutions of equations for Dirac radial wave functions, we introduce a boundary condition at the nucleus boundary such that the components of the electron current density is zero. As a result, for all nuclei of the periodic table the calculated energy levels practically coincide with the energy levels in standard solutions of the Dirac equation in the external field of the Coulomb potential of a point charge. Further, for Z > 105, the calculated energy level functions E(Z) are monotone and smooth.The lower energy level reaches the energy E=-mc^{2} (the electron "drop" on a nuclei) at Zc = 178. The proposed method of accounting of the finite size of nuclei can be easily used in numerical calculations of energy levels of many-electron atoms.
Nondecaying bound states of elementary spin-half particles are validated and calculated numerically for the Schwarzschild gravitational field using a self-conjugate Hamiltonian with a flat scalar product for any value of the gravitational coupling constant. 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 horizon". At small values of the coupling constant, the energy spectrum is close to the hydrogen-like spectrum. Based on the results of this study, we can assume that there exists a new type of collapsars, for which the Hawking radiation mechanism is not present. From the standpoint of cosmology, if the value of the gravitational coupling constant is small, alpha<<1, the new type of nonradiating relict collapsars can manifest itself only through gravitation. Thus, they are good candidates for the role of "dark matter" carriers. In the wide range of admissible masses, there can exist collapsars of a new type with the masses of hypothesized WIMP particles, which are treated as representatives of "dark matter" in numerous scenarios of the expansion of the universe. The results of this study can lead to revisiting some concepts of the standard cosmological model related to the evolution of the universe and interaction of collapsars with surrounding matter.