Stationary solutions of second-order equations for point fermions in the Schwarzschild gravitational field

Abstract

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 $E_b=m{c}^2$) is implemented at the value of the gravitational coupling constant $\alpha \ge {\alpha }_{min}$. 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.

23 pages, 2 figures, 1 table