Temperature of superconducting transition for very strong coupling in antiadiabatic limit of Eliashberg equations

Abstract: It is shown that the famous Allen -- Dynes asymtotic limit for superconducting transition temperature in very strong coupling region $T_{c}>\frac{1}{2\pi}\sqrt{\lambda}\Omega_0$ (where $\lambda\gg 1$ - is Eliashberg - McMillan electron - phonon coupling constant and $\Omega_0$ - the characteristic frequency of phonons) in antiadiabatic limit of Eliashberg equations $\Omega_0/D\gg 1$ ($D\sim E_F$ is conduction band half-width and $E_F$ is Fermi energy) is replaced by $T_c>(2\pi^4)^{-1/3}(\lambda D\Omega_0^2)^{1/3}$, with the upper limit for $T_c$ given by $T_c<\frac{2}{\pi^2}\lambda D$.