What is the significance of the frequency dependent order parameter?
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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."