1 research outputs found
Black-hole concept of a point-like nucleus with supercritical charge
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
. 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.Comment: 22 pages, 5 figures, LATEX requires IOPAR