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 \((g/|\omega _n-\omega _m|)^{\gamma }\), 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 \((2\pi T_c/g)^\gamma \) with the largest (positive) eigenvalue \(\mathfrak {g}(\gamma )\) 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 \(T_c(g,\gamma )\).