Solution of the Eliashberg equations for a very strong electron-phonon coupling with a low-energy cutoff

We solve the Eliashberg equations for the case of an explicit k dependence of the interactions, and of the resulting self-energies Σ₁(k,ω), Σ₂(k,ω). We consider a strong energy-dependence of the electron-electron scattering-rate τ, which is associated with a strong energy-dependence of the electron-phonon matrix element g(k,k′). We characterize this energy-dependence by a cutoff ζ₁, which is of the order of the phonon frequency ω_ph. We find that we can account for a large number of unexpected features of the superconductivity of the cuprates by the BCS electron-phonon theory, if we consider very large values of the McMillan coupling constant λ_ph, and small values of the cutoff ζ₁. Specifically, the Coulomb interaction is found not to depress T_c; the isotope effect is strongly reduced when ζ₁ < ω_ph. We find solutions in which the gap function Δ(k, ω) has extended s-wave symmetry but is very anisotropic. These large anisotropies are in good agreement with various experiments. We suggest that the underlying cause of the strong energy-dependence is a very small electronic screening parameter at the Fermi surface; the electron-phonon matrix element g is abnormally large, and this accounts for the high transition temperatures of the cuprates. An order of magnitude estimate suggests that the electron-phonon mechanism can account for transition temperatures up to about 200 K. We thus propose a very-strong-coupling theory, in which the renormalization functions, in particular the energy-renormalization X, depend very strongly on the superconducting gap Δ, and thus display a very strong temperature-dependence between T_c and T = 0. An experimental manifestation of the very strong coupling with a small cutoff is a zero bias anomaly sometimes observed in tunneling experiments.