On general-relativistic hydrogen and hydrogenic ions

TL;DR

This study investigates how negative bare mass in certain general-relativistic spacetimes affects the mathematical self-adjointness of the Dirac Hamiltonian for electrons, revealing conditions under which self-adjointness fails or holds.

Contribution

It extends previous analyses by examining the impact of negative bare mass and electromagnetic vacuum properties on the Dirac Hamiltonian's self-adjointness in curved spacetime.

Findings

Self-adjointness fails unless the electric field diverges sufficiently fast at the nucleus.

The Dirac Hamiltonian is not essentially self-adjoint on Hoffmann spacetime with negative bare mass.

Operators have self-adjoint extensions with a specific spectrum and an infinite discrete spectrum in the gap.

Abstract

This paper studies how the static non-linear electromagnetic-vacuum spacetime of a point nucleus with negative bare mass affects the self-adjointness of the general-relativistic Dirac Hamiltonian for a test electron, without and with an anomalous magnetic moment. The study interpolates between the previously studied extreme cases of a test electron in (a) the Reissner--Weyl--Nordstr\"om spacetime (Maxwell's electromagnetic vacuum), which supports a very strong curvature singularity with negative infinite bare mass, and (b) the Hoffmann spacetime (Born or Born--Infeld's electromagnetic vacuum) with vanishing bare mass, which features the mildest possible curvature singularity. The main conclusion reached is: {on electrostatic spacetimes of a point nucleus with a strictly negative bare mass} (which may be $-\infty$) essential self-adjointness fails unless the radial electric field diverges sufficiently fast at the nucleus and the anomalous magnetic moment of the electron is taken into account. Thus on the Hoffmann spacetime with (strictly) negative bare mass the Dirac Hamiltonian of a test electron, with or without anomalous magnetic moment, is not essentially self-adjoint. All these operators have self-adjoint extensions, though, with the usual essential spectrum $(-\infty,-\mEL c^2]\cup[\mEL c^2,\infty)$ and an infinite discrete spectrum located in the gap $(-\mEL c^2,\mEL c^2)$