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 ${\omega }_0\ge E_F$, i.e., when the characteristic phonon frequency ${\omega }_0$ 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.