Bounds on Tc in the Eliashberg theory of Superconductivity. III: Einstein phonons

The dispersionless limit of the standard Eliashberg theory of superconductivity is studied. The effective electron-electron interactions are mediated by Einstein phonons of frequency Ω>0, equipped with electron-phonon coupling strength λ. This allows for a detailed evaluation of the general results on Tc for phonons with non-trivial dispersion relation, obtained in a previous paper, (II), by the authors. The variational principle for the linear stability boundary S\!c of the normal state region against perturbations toward the superconducting region, obtained in (II), simplifies as follows: If (λ,Ω,T)\!c, then λ = 1/h(), where :=Ω/2π T, and where h()>0 is the largest eigenvalue of a compact self-adjoint operator H() on ² sequences; H() is the dispersionless limit P(dω)→δ(ω-Ω)dω of the operator K(P,T) of (II). It is shown that when ≤ √2, then the map () is invertible. For λ>0.77 this yields: (i) the existence of a critical temperature Tc(λ,Ω) = Ω f(λ); (ii) a sequence of lower bounds on f(λ) that converges to f(λ). Also obtained is an upper bound on Tc(λ,Ω), which agrees with the asymptotic behavior Tc(λ,Ω) ~ C Ω√λ for λ~∞, given Ω, though with C≈ 2.034 C_∞, where C_∞ := 1/2(2)¹2 =0.1827262477... is the optimal constant, and k(γ)>0 the largest eigenvalue of a compact self-adjoint operator for the γ model, determined in the first paper, (I), on Tc by the authors.