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

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.