Bounds on $T_c$ in the Eliashberg theory of Superconductivity. I: The $\gamma$-model

Using the recent reformulation for the Eliashberg theory of superconductivity in terms of a classical interacting Bloch spin chain model, rigorous upper and lower bounds on the critical temperature $T_c$ are obtained for the $\gamma$ model – a version of Eliashberg theory in which the effective electron-electron interaction is proportional to $(g\mathrm{/}\mathrm{\mid }{\omega }_n-{\omega }_m\mathrm{\mid }{)}^{\gamma }$, where ${\omega }_n-{\omega }_m$ is the transferred Matsubara frequency, $g>0$ a reference energy, and $\gamma >0$ a parameter. The rigorous lower bounds are based on a variational principle that identifies $(T_c\mathrm{/}g{)}^{\gamma }$ with the largest (positive) eigenvalue of an explicitly constructed compact, self-adjoint operator $\mathfrak{G}(\gamma )$. These lower bounds form an increasing sequence that converges to $T_c(g,\gamma )$. The upper bound on $T_c(g,\gamma )$ is based on fixed point theory, proving linear stability of the normal state for $T$ larger than the upper bound on $T_c(g,\gamma )$.