Bounds on $T_c$ 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 $\Omega>0$, equipped with electron-phonon coupling strength $\lambda$. This allows for a detailed evaluation of the general results on $T_c$ for phonons with non-trivial dispersion relation, obtained in a previous paper, (II), by the authors. The variational principle for the linear stability boundary $\mathscr{S}_{\!c}$ of the normal state region against perturbations toward the superconducting region, obtained in (II), simplifies as follows: If $(\lambda,\Omega,T)\in\mathscr{S}_{\!c}$, then $\lambda = 1/\mathfrak{h}(\varpi)$, where $\varpi:=\Omega/2\pi T$, and where $\mathfrak{h}(\varpi)>0$ is the largest eigenvalue of a compact self-adjoint operator $\mathfrak{H}(\varpi)$ on $\ell^2$ sequences; $\mathfrak{H}(\varpi)$ is the dispersionless limit $P(d\omega)\to\delta(\omega-\Omega)d\omega$ of the operator $\mathfrak{K}(P,T)$ of (II). It is shown that when $\varpi \leq \sqrt{2}$, then the map $\varpi\mapsto\mathfrak{h}(\varpi)$ is invertible. For $\lambda>0.77$ this yields: (i) the existence of a critical temperature $T_c(\lambda,\Omega) = \Omega f(\lambda)$; (ii) a sequence of lower bounds on $f(\lambda)$ that converges to $f(\lambda)$. Also obtained is an upper bound on $T_c(\lambda,\Omega)$, which agrees with the asymptotic behavior $T_c(\lambda,\Omega) \sim C \Omega\sqrt{\lambda}$ for $\lambda\sim\infty$, given $\Omega$, though with $C\approx 2.034 C_\infty$, where $C_\infty := \frac{1}{2\pi}\mathfrak{k}(2)^\frac12 =0.1827262477...$ is the optimal constant, and $\mathfrak{k}(\gamma)>0$ the largest eigenvalue of a compact self-adjoint operator for the $\gamma$ model, determined in the first paper, (I), on $T_c$ by the authors.