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 ${\alpha }^{2}F(\omega )\ge 0$ that is $\propto {\omega }^2$ for small $\omega >0$ and vanishes for large $\omega$. The Eliashberg function also defines the electron-phonon coupling strength $\lambda :=2{\int }_0^{\mathrm{\infty }}\frac{{\alpha }^{2}F(\omega )}{\omega }d\omega$. 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 ${\mathcal{S}}_{\text{\hspace{0.17em}⁣}c}$ of the normal region against perturbations toward the superconducting region, it is shown that ${\mathcal{S}}_{\text{\hspace{0.17em}⁣}c}$ is a graph of a function $\mathrm{\Lambda }(P,T)$ that is determined by a variational principle: if $(\lambda ,P,T)\in {\mathcal{S}}_{\text{\hspace{0.17em}⁣}c}$, then $\lambda =1\mathrm{/}\mathfrak{k}(P,T)$, where $\mathfrak{k}(P,T)>0$ is the largest eigenvalue of a compact self-adjoint operator $\mathfrak{K}(P,T)$ on ${\mathrm{\ell }}^2$ sequences constructed in the paper. Given $P$, sufficient conditions on $T$ are stated under which the map $T\mapsto \lambda =\mathrm{\Lambda }(P,T)$ 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 $T_c(\lambda ,P)\sim C\sqrt{⟨{\omega }^2⟩}\sqrt{\lambda }$ valid for $\lambda \sim \mathrm{\infty }$, given $P$, though with a constant $C$ that is a factor $\approx 2.034$ larger than the sharp constant. Here, $⟨{\omega }^2⟩:={\int }_0^{\mathrm{\infty }}{\omega }^{2}P(d\omega )$.