Solutions of Eliashberg equations for an electron-phonon coupling with a cutoff

In this paper we discuss the Eliashberg equations for the case of an electron-phonon coupling with an energy cutoff. This cutoff is imposed either for the energy difference by means of a strip function, or for both energies, with a Cooper-like expression. The strip function cutoff requires explicit calculation of not only the frequency renormalization function Z but also the energy renormalization X. The physical origin of such cutoffs might lie in the very strong electron-electron interaction which seems typical for high T_c superconductivity. If such cutoffs are admitted, the hypothesis that T_c is caused at least in part by a strong electron-phonon interaction can be reconsidered. We find that the combination of strong coupling and low-energy cutoff could produce high T_c with only small isotope effect and with little damping or pulling of the phonon modes. Correlation with other physical properties, such as specific heat, is reexamined in view to estimate the coupling constant λ. Some objections to the model using strong electron phonon interaction are removed and better agreement with observed properties is obtained