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 selfadjoint
versions of the Dirac-Coulomb Hamiltonian. Such discrete spectra display
naturally a fibred structure, whose bundle covers the whole gap of the continuum
spectrum.
