  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.
