Bounds on Tc in the Eliashberg theory of Superconductivity. II: Dispersive phonons

The standard Eliashberg theory of superconductivity is studied. The effective electron-electron interactions are mediated by dispersive phonons, with Eliashberg spectral function α² F(ω)≥ 0 that is ∝ω² for small ω>0 and vanishes for large ω. The Eliashberg function also defines the electron-phonon coupling strength λ:= 2 ∫₀^∞α² F(ω)/ω. Setting 2α² F(ω)/ω=: λ P⁽ω), with P⁽ω) the density of a probability measure P(dω) supported in [0,Ω̄(P)] with 0< Ω̄(P), and assuming as usual that the phase transition between normal and superconductivity coincides with the linear stability boundary S\!c of the normal region against perturbations toward the superconducting region, it is shown that S\!c is a graph of a function Λ(P,T) that is determined by a variational principle: if (λ,P,T)\!c, then λ = 1/k(P,T), where k(P,T)>0 is the largest eigenvalue of a compact self-adjoint operator K(P,T) on ² sequences. Given P, sufficient conditions on T are stated under which the map T↦ λ = Λ(P,T) is invertible. For sufficiently large λ this yields: (i) the existence of a critical temperature Tc as function of λ and P; (ii) a sequence of lower bounds on Tc(λ,P) that converges to Tc(λ,P). Also obtained is an upper bound on Tc(λ,P). It agrees with the asymptotic form Tc(λ,P) ~ C √ ω² √λ valid for λ~∞, given P, though with C≈ 2.034 C_∞ where C_∞ = 0.1827262477... is the sharp constant. Here, ω² := ∫₀^∞ ω² P(dω).