PACS. 71.38+i -Polarons and electron-phonon interactions. PACS. 74.20M n-Nonconventional mechanisms. PACS. 74.62Dh -Effects of crystal defects, doping and substitution.Abstract. -We show that, in contrast with the usual electron-phonon Migdal-Eliashberg theory, the critical temperature Tc of an isotropic s-wave nonadiabatic superconductor is strongly reduced by the presence of diluted non-magnetic impurities. Our results suggest that the recently observed Tc-suppression driven by disorder in K3C60 (Phys. Rev. B 55, 3866 (1997)) and in Nd2−xCexCuO 4−δ (Phys. Rev. B 58, 8800 (1998)) could be explained in terms of a nonadiabatic electron-phonon coupling. Moreover, we predict that the isotope effect on Tc has an impurity dependence qualitatively different from the one expected for anisotropic superconductors.High-T c superconductors are narrow band systems with Fermi energies (E F ) one or two orders of magnitudes smaller than those of conventional superconductors [1]. In these materials therefore it appears unavoidable to address the validity of Migdal's theorem [2] applied to the electron-phonon (el-ph) interaction and eventually other mediators [3]. In fact, in fullerene compounds, bismuth oxides and high-T c cuprates, the typical phonon frequencies ω 0 can be comparable to E F , making the quantity λω 0 /E F no longer negligible also for moderate values of the el-ph coupling constant λ. Given this situation, the ordinary Migdal-Eliashberg (ME) theory, which is based on Migdal's theorem, hardly can appropriately describe the properties of such materials and a more general treatment of the problem should be employed.In general, by moving from the ME regime one could end to qualitatively different situations depending on the values of λ and ω 0 /E F . For example, very strong el-ph couplings favour the formation of polarons and eventually of bi-polarons. This picture is certainly beyond the ME regime, however, as recently discussed in ref. [4], it is implausible that it can be at the basis of the phenomenon of high-T c superconductivity. Actually, one could go beyond ME regime without ending to polaron (bi-polaron) states by considering quasi-free charge carriers (λ < ∼ 1) coupled nonadiabatically (0 < ω 0 /E F < 1) to the lattice vibrations. In the following, to Typeset using EURO-L a T E X

The electron-phonon vertex correction has a complex structure both in momentum and frequency. We explain this structure on the basis of physical considerations and we show how the vertex correction can be decomposed into two terms with different physical origins. In particular, the first term describes the lattice polarization induced by the electrons and it is essentially a single-electron process whereas the second term is governed by the particle-hole excitations due to the exchange part of the phonon-mediated electron-electron interaction. We show that by weakening the influence of the exchange interaction the vertex takes mostly positive values giving rise to an enhanced effective coupling in the scattering with phonons. This weakening of the exchange interaction can be obtained by lowering the density of the electrons, or by considering only long-ranged (small q) electron-phonon couplings. These findings permit to understand why in the High-Tc materials the small carrier density and the long ranged electron-phonon interaction may play a positive role in enhancing Tc.