On the Discrete Dirac Spectrum of General-Relativistic Hydrogenic Ions with Anomalous Magnetic Moment

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)mp, where mp is the proton mass and A(Z,N)≈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 me, charge -e, and anomalous magnetic moment μa(≈-14πe3mec2) 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 (-∞,-mec2]∪[mec2,∞) and a discrete spectrum of infinitely many eigenvalues in the gap (-mec2,mec2), having mec2 as accumulation point. In this paper, the discrete spectrum is characterized in detail for the first time, for all Z≤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.