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 \(M = A(Z,N)m_{\textrm{p}}\), where \(m_{\textrm{p}}\) 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 \(m_{\textrm{e}}\), charge \(-\,e\), and anomalous magnetic moment \(\mu _a (\approx -\, \frac{1}{4\pi }\frac{e^3}{m_{\textrm{e}}c^2})\) 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 \((-\,\infty ,-\,m_{\textrm{e}}c^2]\cup [m_{\textrm{e}}c^2, \infty )\) and a discrete spectrum of infinitely many eigenvalues in the gap \((-\,m_{\textrm{e}}c^2,m_{\textrm{e}}c^2)\), having \(m_{\textrm{e}}c^2\) 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.