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 modelled as 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 \(\lambda := 2 \displaystyle \int _{\mathbb {R}_+}\!\! \frac{\alpha ^2\!F(\omega )}{\omega }d\omega \). Setting \({ \displaystyle \frac{2\alpha ^2\!F(\omega )}{\omega }}d\omega =: \lambda P(d\omega )\), 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 in the \((\lambda ,P,T)\) parameter space against perturbations toward the superconducting region, it is shown that this critical hypersurface \(\mathscr {S}_{\!c}\) is a graph of a function \(\Lambda (P,T)\). This proves that the normal and the superconducting regions are simply connected. Moreover, it is shown that \(\mathscr {S}_{\!c}\) 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 that is constructed explicitly in the paper, for all admissible P. Furthermore, given any such 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 the following: (i) the existence of a critical temperature \(T_c\) as function of \(\lambda \) and P; (ii) an ordered sequence of lower bounds on \(T_c(\lambda ,P)\) that converges to \(T_c(\lambda ,P)\). Also obtained is an upper bound on \(T_c(\lambda ,P)\). Although not optimal, 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, \(\langle \omega ^2\rangle := \int _{\mathbb {R}_+} \omega ^2 P(d\omega )\).