Special concept of a point-like nucleus with supercritical charge

Abstract

The Dirac equation for an electron in the central Coulomb field of a point-like and almost 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 special concept of the singular centre, capable of producing results independent of the nucleus size. 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 that depends on the cut-off procedure at the edge of a finite-size nucleus.