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