Nonadiabatic superconductivity: Electron phonon interaction beyond Migdal's Theorem

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A common characteristic of all the high Tc materials, oxides and Fullerene compounds, is that the Fermi energy is much smaller than in usual metals and it is of the order of the Debye phonon frequencies. This requires a generalization of the usual BCS-Eliashberg scheme to include non adiabatic effects beyond Migdal's Theorem. We have developed the first steps of this generalization and here we discuss the main results, the possible lines of development and the main open problems. The key point is that the first order non adiabatic effects (vertex corrections and similar processes) depend crucially on the momentum (q) of the exchanged phonons. If small q scattering is predominant, one obtains a strong enhancement of Tc and various other effecs also for the normal state that should lead to precise experimental predictions. The predominance of small q scattering can be naturally obtained from electronic correlations but it could arise also from different properties like peaks in the density of states.