  We study the isospectrality problem for a relativistic free quantum particle,
described by the Dirac Hamiltonian, confined in a one-dimensional ring with a
junction. We analyze all the self-adjoint extensions of the Hamiltonian in
terms of the boundary conditions at the junction, characterizing the energy
spectrum by means of a spectral function. By determining the symmetries of the
latter, we are able to divide the self-adjoint extensions in two classes,
identifying all the families of isospectral Hamiltonians, and thus completely
characterizing the isospectrality problem.
