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

We analyze the mathematical structure of equations for temperature TC 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≥EF, i.e., when the characteristic phonon frequency ω₀ is comparable or larger than the Fermi energy EF. As the methods of calculating TC in common superconductors are not applicable in the nonadiabatic regime, the integral equations for TC 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.