Superconducting Transition Temperature for Very Strong Coupling in the Antiadiabatic Limit of Eliashberg Equations

It is shown that the famous Allen–Dynes asymptotic limit for the superconducting transition temperature in the very strong coupling region ${{T}_{{\text{c}}}} > \frac{1}{{2\pi }}\sqrt \lambda {{\Omega }_{0}}$ (where $\lambda \gg 1$ is the Eliashberg–McMillan electron–phonon coupling constant and ${{\Omega }_{0}}$ is the characteristic frequency of phonons) in the antiadiabatic limit of Eliashberg equations ${{\Omega }_{0}}{\text{/}}D \gg 1$ ($D \sim {{E}_{{\text{F}}}}$ is the half-width of the conduction band and E_F is the Fermi energy) is replaced by ${{T}_{{\text{c}}}} > {{(2{{\pi }^{4}})}^{{ - 1/3}}}{{(\lambda D\Omega _{0}^{2})}^{{1/3}}}$, with the upper limit ${{T}_{{\text{c}}}} < \frac{2}{{{{\pi }^{2}}}}\lambda D$.