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

The standard Eliashberg theory of superconductivity is studied, in which the effective electron-electron interactions are mediated by generally dispersive phonons, with Eliashberg spectral function $\alpha^2 F(\omega)\geq 0$ that is $\propto\omega^2$ for small $\omega>0$ and vanishes for large $\omega$. The Eliashberg function also defines the electron-phonon coupling strength $\lambda:= 2 \int_0^\infty\frac{\alpha^2 F(\omega)}{\omega}d\omega$. Setting $\frac{2\alpha^2 F(\omega)}{\omega}d\omega =: \lambda P(d\omega)$, formally defining a probability measure $P(d\omega)$ with compact support, and assuming as usual that the phase transition between normal and superconductivity coincides with the linear stability boundary $\mathscr{S}_{\!c}$ of the normal region against perturbations toward the superconducting region, it is shown that $\mathscr{S}_{\!c}$ is a graph of a function $\Lambda(P,T)$ that is determined by a variational principle: if $(\lambda,P,T)\in\mathscr{S}_{\!c}$, then $\lambda = 1/\mathfrak{k}(P,T)$, where $\mathfrak{k}(P,T)>0$ is the largest eigenvalue of a compact self-adjoint operator $\mathfrak{K}(P,T)$ on $\ell^2$ sequences constructed in the paper. Given $P$, sufficient conditions on $T$ are stated under which the map $T\mapsto \lambda = \Lambda(P,T)$ is invertible. For sufficiently large $\lambda$ this yields: (i) the existence of a critical temperature $T_c$ as function of $\lambda$ and $P$; (ii) a sequence of lower bounds on $T_c(\lambda,P)$ that converges to $T_c(\lambda,P)$. Also obtained is an upper bound on $T_c(\lambda,P)$. It agrees with the asymptotic form $T_c(\lambda,P) \sim C \sqrt{\langle \omega^2\rangle} \sqrt{\lambda}$ valid for $\lambda\sim\infty$, given $P$, though with a constant $C$ that is a factor $\approx 2.034$ larger than the sharp constant. Here, $\langle\omega^2\rangle := \int_0^\infty \omega^2 P(d\omega)$.