Spectrum of the Dirac Hamiltonian for Hydrogenic atoms on spacetimes with mild singularities

TL;DR

This paper investigates the spectral properties of the Dirac Hamiltonian for hydrogenic atoms modeled in spacetimes with mild singularities, showing that their spectral features resemble those in flat space.

Contribution

It demonstrates the essential self-adjointness of the Dirac Hamiltonian in such spacetimes and characterizes its spectrum, extending understanding of quantum behavior in curved, singular backgrounds.

Findings

Dirac Hamiltonian is essentially self-adjoint regardless of atomic number

Spectrum matches that of flat Minkowski space with Coulomb potential

Infinitely many eigenvalues with clustering behavior are present

Abstract

We model a single-electron ion (hydrogenic atom) as a static, spherically symmetric electrovacuum spacetime in which the nucleus is treated as a timelike line-singularity and the electron is treated as a test particle following Dirac's equation. The spacetime is a solution of Einstein-Maxwell equations with a non-linear vacuum law. An example is Hoffmann's spacetime obtained using the Born-Infeld law. The Dirac Hamiltonian is shown to be essentially self-adjoint, independent of the atomic number. The essential spectrum and absolutely continuous spectrum are the same as in Dirac's Hamiltonian on Minkowski space with a Coulomb potential. Presence of infinitely many eigenvalues is shown and a clustering result is obtained.