Black-hole concept of a point-like nucleus with supercritical charge

TL;DR

This paper introduces a novel black-hole-inspired approach to solve the Dirac equation for supercritical charge nuclei, providing cut-off-free results and analyzing electron absorption and pair production.

Contribution

It presents a new method based on a black-hole concept to address the singular Coulomb problem with supercritical charge, avoiding finite-size regularization.

Findings

Analytical expression for barrier transmission coefficient

Complete orthogonal eigenfunctions with singular measure

Description of electron absorption and pair production effects

Abstract

The Dirac equation for an electron in the central Coulomb field of a point-like nucleus with the charge greater than 137 is considered. This singular problem, to which the fall-down onto the centre is inherent, is addressed using a new approach, based on a black-hole concept of the singular centre and capable of producing cut-off-free results. To this end the Dirac equation is presented as a generalized eigenvalue boundary problem of a self-adjoint operator. The eigenfunctions make complete sets, orthogonal with a singular measure, and describe particles, asymptotically free and delta-function-normalizable both at infinity and near the singular centre $r=0$. The barrier transmission coefficient for these particles responsible for the effects of electron absorption and spontaneous electron-positron pair production is found analytically as a function of electron energy and charge of the nucleus. The singular threshold behaviour of the corresponding amplitudes substitutes for the resonance behaviour, typical of the conventional theory, which appeals to a finite-size nucleus.