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_c > 1/2π√(λ)Ω_0 (where λ≫ 1 is the Eliashberg–McMillan electron–phonon coupling constant and Ω_0 is the characteristic frequency of phonons) in the antiadiabatic limit of Eliashberg equations Ω_0/D ≫ 1 ( D ∼E_F is the half-width of the conduction band and E F is the Fermi energy) is replaced by T_c > (2π^4)^ - 1/3(λ DΩ _0^2)^1/3 , with the upper limit T_c < 2/π^2λ D .