Electron - phonon coupling in Eliashberg - McMillan theory beyond adiabatic approximation

Eliashberg - McMillan theory of superconductivity is essentially based on the adiabatic approximation. Small parameter of perturbation theory is given by $\lambda\frac{\Omega_0}{E_F}\ll 1$, where $\lambda$ is the dimensionless electron - phonon coupling constant, $\Omega_0$ is characteristic phonon frequency, while $E_F$ is Fermi energy of electrons. Here we present an attempt to describe electron - phonon interaction within Eliashberg - McMillan approach in situation, when characteristic phonon frequency $\Omega_0$ becomes large enough (comparable or exceeding the Fermi energy $E_F$). We consider the general definition of electron - phonon pairing coupling constant $\lambda$, taking into account the finite value of phonon frequency. Also we obtain the simple expression for the generalized coupling constant $\tilde\lambda$, which determines the mass renormalization, with the account of finite width of conduction band, and describing the smooth transition from the adiabatic regime to the region of strong nonadiabaticity. In the case of strong nonadiabaticity, when $\Omega_0\gg E_F$, the new small parameter appears $\lambda\frac{E_F}{\Omega_0}\sim\lambda\frac{D}{\Omega_0}\ll 1$ ($D$ is conduction band half - width), and corrections to electronic spectrum become irrelevant. At the same time, the temperature of superconducting transition $T_c$ in antiadiabatic limit is still determined by Eliashberg - McMillan coupling constant $\lambda$, while the preexponential factor in the expression for $T_c$, conserving the form typical of weak - coupling theory, is determined by the bandwidth (Fermi energy). For the case of interaction with a single optical phonon we derive the single expression for $T_c$, valid both in adiabatic and antiadiabatic regimes and describing the continuous transition between these two limiting cases.