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

TL;DR

This work rigorously characterizes the discrete Dirac spectrum of a point electron with a physically realistic anomalous magnetic moment in the Reissner--Weyl--Nordström spacetime of a point nucleus. By introducing dimensionless variables and employing the Prüfer transformation, the authors reformulate the eigenvalue problem as a dynamical system on a compact cylinder and prove that the discrete spectrum is indexed by the spin-orbit quantum number $k$ and a winding number $N$, with $n=N+|k|$ corresponding to hydrogenic orbitals. They establish a one-to-one correspondence between the relativistic spectrum in curved spacetime and the traditional Dirac hydrogen spectrum, and provide numerical results extending beyond the empirical Z limit, while discussing open problems such as finite-nucleus effects, the $G\to0$ limit, and level-crossing phenomena. The analysis demonstrates that, for physically reasonable parameters, gravity leaves spectral differences between the general-relativistic and Minkowski cases minimal, reinforcing the robustness of hydrogenic spectral structure under general-relativistic corrections. Overall, the paper advances a non-perturbative, mathematically principled understanding of hydrogenic spectra in a relativistic gravitational setting 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 $μ_a (\approx - \frac{1}{4π}\frac{e^3}{m_{\mathrm{e}} c^2})$ 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 $Z\leq 45$ 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.

Figures (1)

• Figure 1: Electron Orbital Energies vs. the Nuclear Charge, with $\mathcal{G} < \mathcal{G}_{\hbox{\tiny res}}$

Theorems & Definitions (23)

• Theorem 2.1: Belgiorno--Martellini--BaldicchiBMB
• Remark 2.2
• Remark 2.5
• Remark 2.6
• Remark 2.7
• Theorem 2.8
• Remark 2.9
• Proposition 2.10
• Proposition 2.11
• Proposition 2.12