Nonadiabatic Superconductivity: Electron-Phonon Interaction Beyond Migdal's Theorem

We generalize Eliashberg's equations to include nonadiabatic effects like vertex corrections, cross phonon scattering, and others arising from the breakdown of Migdal's theorem. This generalization is imposed by the fact that all high Tc superconductors (oxides, fullerene compounds, etc.) have a very small Fermi energy ( EF). Nonadiabatic effects show a complex structure as a function of the exchanged frequency and momenta. In particular, a predominance of small momentum scattering leads to positive contributions with respect to Tc. This situation is actually realized in a correlated Fermi liquid with small EF that naturally leads to an enhancement of Tc and to various other consequences for the phenomenology of both the superconductive and normal phases.