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.
%0 Journal Article
%1 citeulike:4017282
%A Shabad, A. E.
%D 2005
%K black-holes, nucleus
%T Black-hole concept of a point-like nucleus with supercritical charge
%U http://arxiv.org/abs/hep-th/0502139
%X 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.
@article{citeulike:4017282,
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.},
added-at = {2009-02-18T14:54:47.000+0100},
author = {Shabad, A. E.},
biburl = {https://www.bibsonomy.org/bibtex/231c5932c36c1c18a987815d752a7c820/janpaniev},
citeulike-article-id = {4017282},
eprint = {hep-th/0502139},
interhash = {c57b81fb503d58f3d817b49eded6d320},
intrahash = {31c5932c36c1c18a987815d752a7c820},
keywords = {black-holes, nucleus},
month = Feb,
posted-at = {2009-02-06 19:16:39},
priority = {2},
timestamp = {2009-02-18T14:54:49.000+0100},
title = {Black-hole concept of a point-like nucleus with supercritical charge},
url = {http://arxiv.org/abs/hep-th/0502139},
year = 2005
}