  The conventional Bardeen-Cooper-Schrieffer (BCS) model of superconductivity
assumes a frequency-independent order parameter, which allows a relatively
simple description of the superconducting state. In particular, its excitation
spectrum readily follows from the Bogoliubov-de-Gennes (BdG) equations. A more
realistic description of a superconductor is the Migdal-Eliashberg theory,
where the pairing interaction, the order parameter, and electronic self-energy
are strongly frequency dependent. This work combines these ingredients of
phonon-mediated superconductivity with the standard BdG approach. Surprisingly,
we find qualitatively new features such as the emergence of a shadow
superconducting gap in the quasiparticle spectrum at energies close to the
Debye energy. We show how these features reveal themselves in standard
tunneling experiments. Finally, we also predict the existence of additional
high-energy bound states, which we dub "dark Andreev states."
