We generalize the Eliashberg equations to include the first nonadiabatic effects beyond Migdal’s theorem. The resulting theory is nonperturbative with respect to λ and perturbative with respect to (λ${\mathrm{\omega }}_{\mathit{D}}$/${\mathit{E}}_{\mathit{F}}$). The main effects are due to the vertex corrections and the cross diagram that show a complex behavior with respect to the exchanged momentum (q) and frequency (ω). Positive corrections and a corresponding enhancement of ${\mathit{T}}_{\mathit{c}}$ arise naturally if the electron phonon scattering is characterized mainly by small q values. For this reason we discuss our results in terms of an upper cutoff ${\mathit{q}}_{\mathit{c}}$ for the scattering. The generalized Eliashberg equations are solved numerically and analytically and we also provide a generalization of the McMillan equation that includes the nonadiabatic effects. For relatively small values of ${\mathit{q}}_{\mathit{c}}$, normal values of the coupling (λ≃0.5–1.0) can lead to strong enhancements of ${\mathit{T}}_{\mathit{c}}$ in the range of the observed values. This situation leads also to a complex behavior of the isotope effect that can be anomalously large (α>1/2) in some region of parameters but it can also vanish for ${\mathrm{\omega }}_{\mathit{D}}$>${\mathit{E}}_{\mathit{F}}$. It is therefore important to identify which features of more realistic models can lead to a situation in which the nonadiabatic effects are mainly positive. One way to achieve this is to have an upper cutoff ${\mathit{q}}_{\mathit{c}}$ for the electron-phonon scattering and electronic correlations appear to be a natural candidate to produce this effect. The nonadiabatic effects are expected to play an important role also for the normal properties of the system that can deviate appreciably from a normal Fermi liquid. It is important to study these effects in the future because they should lead to specific predictions of new effects that can be tested experimentally.

• Received 8 December 1994

DOI:https://doi.org/10.1103/PhysRevB.52.10530