Superconducting transition temperature: Interacting Fermi gas and phonon mechanisms in the nonadiabatic regime

We analyze the mathematical structure of equations for temperature T_C of the superconductivity transition in a gas of interacting Fermi particles or at the phonon-mediated pairing in a metal in the case of nonadiabatic conditions ω₀≥E_F , i.e., when the characteristic phonon frequency ω₀ is comparable or larger than the Fermi energy E_F. As the methods of calculating T_C in common superconductors are not applicable in the nonadiabatic regime, the integral equations for T_C are derived in the logarithmic approximation. The new equations contain no divergent terms in the antiadiabatic limit. The results can be immediately generalized to anisotropic band superconductors.