On the discrete Dirac spectrum of general-relativistic hydrogenic ions with anomalous magnetic moment

Abstract

The Reissner-Weyl-Nordström (RWN) spacetime of a point nucleus features a naked singularity for the empirically known nuclear charges $Ze$ and masses $M=A(Z,N)m_{\mathrm{p}}$, where $m_{\mathrm{p}}$ is the proton mass and $A(Z,N)\approx Z+N$ the atomic mass number, with $Z$ the number of protons and $N$ the number of neutrons in the nucleus. The Dirac Hamiltonian for a test electron with mass $m_{\mathrm{e}}$, charge $-e$, and anomalous magnetic moment ${\mu }_a(\approx -\frac{1}{4\pi }\frac{e^3}{m_{\mathrm{e}}{c}^2})$ in the electrostatic RWN spacetime of such a 'naked point nucleus' is known to be essentially self-adjoint, with a spectrum that consists of the union of the essential spectrum $(-\infty ,m_{\mathrm{e}}{c}^2]\cup [m_{\mathrm{e}}{c}^2,\infty )$ and a discrete spectrum of infinitely many eigenvalues in the gap $(-m_{\mathrm{e}}{c}^2,m_{\mathrm{e}}{c}^2)$, having $m_{\mathrm{e}}{c}^2$ as accumulation point. In this paper the discrete spectrum is characterized in detail for the first time, for all $Z\le 45$ and $A$ that cover all known isotopes. The eigenvalues are mapped one-to-one to those of the traditional Dirac Hydrogen spectrum. Numerical evaluations that go beyond $Z=45$ into the realm of not-yet-produced hydrogenic ions are presented, too. A list of challenging open problems concludes this publication.

40 pages, 1 figure, submitted for publication

References in corpus (3)

• Measurement of the Electron Magnetic Moment
• Fine-Structure Constant for Gravitational and Scalar Interactions
• On the discrete Dirac spectrum of a point electron in the zero-gravity Kerr-Newman spacetime