Theory of nonadiabatic superconductivity

All high-temperature superconductors (and especially the C60 compounds) are characterized by a small Fermi energy (EF), of the order of the maximum phonon frequency (?D). We generalize therefore the many-body theory of superconductivity to the nonadiabatic case (?D/EF???0). This implies the breakdown of Migdal's theorem and it requires the inclusion of vertex renormalization and higher-order diagrams in the self-consistent gap equation. We show that these nonadiabatic effects produce a strong enhancement of Tc with respect to the usual theory (McMillan). The observed values of Tc for the C60 compounds and for the cuprates can be naturally obtained within the present theory with normal values for the electron-phonon coupling ?. In addition we predict an anomalous behaviour for the isotope effect.