Black-hole 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 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.

22 pages, 5 figures, LATEX requires IOPART

References in corpus (4)

• Black-Hole Approach to the Singular Problem of Quantum Mechanics. II
• Interaction of Electromagnetic Radiation with Supercritical Magnetic Field
• Singular Centre in Quantum Mechanics as a Black Hole
• Superradiation in scattering of a Dirac particle off a point-like nucleus with Z>137 with Z>137

Cited by in corpus (6)

• Electric field of a pointlike charge in a strong magnetic field and ground state of a hydrogenlike atom
• Positronium collapse and the maximum magnetic field in pure QED
• Bethe-Salpeter approach for relativistic positronium in a strong magnetic field
• The use of Slater-type spinor orbitals in algebraic solution of two-center Dirac equation
• JRAF: A Julia Package for Computation of the Relativistic Molecular Auxiliary Functions
• Positronium collapse in hypercritical magnetic field and restructuring of the vacuum in QED