Pseudogaps: A third peak in the fermion spectral function

I present an exactly solvable model of a pseudogap with two zero-energy fermion modes coupled to each other by a classical source of frequency ${\omega }_0$ and strength $|\Delta |$. A suitably defined fermion propagator has an infinite number of poles at frequencies that are multiple integers of ${\omega }_0$. In the adiabatic limit, ${\omega }_0\ll |\Delta |$, the situation is qualitatively different from the static case ${\omega }_0=0\mathrm{}$: the residue of the pole at $\omega =0\mathrm{}$ (a remnant of the bare fermion) vanishes linearly with ${\omega }_0$, a result that could not be anticipated by perturbation theory; the multiple poles of the propagator coalesce into a continuum instead of forming two single poles at $\pm |\Delta |$, which should be interpreted as inhomogeneous broadening of the Bogoliubov quasiparticles.