Discrete spectra for critical Dirac-Coulomb Hamiltonians

TL;DR

This paper generalizes Sommerfeld's fine structure formula to determine the discrete spectra of all self-adjoint realizations of the critical Dirac-Coulomb Hamiltonian, revealing a fibred spectral structure across the continuum gap.

Contribution

It extends the classical eigenvalue formula to all self-adjoint extensions of the Dirac-Coulomb operator at critical coupling, providing a comprehensive spectral classification.

Findings

Derived a generalized spectral formula for all self-adjoint realizations.

Identified a fibred structure of the discrete spectrum covering the continuum gap.

Classified all self-adjoint extensions of the critical Dirac-Coulomb Hamiltonian.

Abstract

The one-particle Dirac Hamiltonian with Coulomb interaction is known to be realised, in a regime of large (critical) couplings, by an infinite multiplicity of distinct self-adjoint operators, including a distinguished, physically most natural one. For the latter, Sommerfeld's celebrated fine structure formula provides the well-known expression for the eigenvalues in the gap of the continuum spectrum. Exploiting our recent general classification of all other self-adjoint realisations, we generalise Sommerfeld's formula so as to determine the discrete spectrum of all other self-adjoint versions of the Dirac-Coulomb Hamiltonian. Such discrete spectra display naturally a fibred structure, whose bundle covers the whole gap of the continuum spectrum.