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 ofsuperconductivity that includes the first contribution beyond Migdal's theorem ~nonadiabatic!. Most of ourresults are not specific to a van Hove singularity and can be extended to the generic situation in which theFermi surface is close to a peak in the DOS. Often the effect of a peak in the DOS is discussed in terms of anenhancement of the electron-phonon coupling l using the standard theory. Here we point out that in the mostinteresting situations this peak structure unavoidably leads to a breakdown of Migdal's theorem because theeffective range of electronic energies becomes very narrow. We include therefore the first diagrams beyondMigdal's theorem that lead to a change in the structure of the theory and not just on the value of the effectivecoupling. These nonadiabatic effects lead naturally to an enhancement of the value of Tc with respect to theadiabatic theory with the same coupling. This enhancement is mainly due to the predominant role of smallmomentum scattering that is a consequence of the peak in the DOS. These results provide therefore a perspectivefor the effects of DOS peaks in the theory of superconductivity.