Nonadiabatic superconductivity: The role of van Hove singularities

We consider the effect of van Hove singularities in the density of states (DOS) on the generalized theory of superconductivity that includes the first contribution beyond Migdal's theorem (nonadiabatic). Most of our results are not specific to a van Hove singularity and can be extended to the generic situation in which the Fermi surface is close to a peak in the DOS. Often the effect of a peak in the DOS is discussed in terms of an enhancement of the electron-phonon coupling lambda using the standard theory. Here we point out that in the most interesting situations this peak structure unavoidably leads to a breakdown of Migdal's theorem because the effective range of electronic energies becomes very narrow. We include therefore the first diagrams beyond Migdal's theorem that lead to a change in the structure of the theory and not just on the value of the effective coupling. These nonadiabatic effects lead naturally to an enhancement of the value of T-c with respect to the adiabatic theory with the same coupling. This enhancement is mainly due to the predominant role of small momentum scattering that is a consequence of the peak in the DOS. These results provide therefore a perspective for the effects of DOS peaks in the theory of superconductivity.