Non-BCS behavior of the gap edge in strong-coupling superconductors nearTc

Eliashberg theory leads to a temperature $T^{*}<T_c$ such that the real part of the gap at the gap edge, $\Delta \mathrm{}\left(T\right)\mathrm{}$, is double valued for $0<T<\mathrm{}{T}^{*}$. The two values coalesce at $T^{*}$; above $T^{*}$ there is no nonzero solution. This behavior is qualitatively different from that of the BCS gap ${\Delta }^{\mathrm{BCS}}\mathrm{}\left(T\right)\mathrm{}$, particularly near $T^{*}$ where a real, frequency-independent gap function such as $\Delta \mathrm{}\left(T\right)\mathrm{}$ or ${\Delta }^{\mathrm{BCS}}\mathrm{}\left(T\right)\mathrm{}$ is inadequate. Possible implications for Ginzburg-Landau theory and nonequilibrium superconductivity are very briefly discussed. It is argued that $\frac{(T_c-T^{*})}{T_c\sim 1.3{\left[\frac{{\omega }^{*}}{\Delta \mathrm{}\left(0\right)\mathrm{}}\right]}^2}$, where ${\omega }^{*}$ is the lowest frequency at which the real and imaginary parts of the renormalization function $Z(\omega ,T_c)$ are equal. This formula works very well for the case studied in detail and predicts that $T^{*}$ should be 1 or 2% below $T_c$ for several well-studied strong-coupling superconductors. For weak-coupling superconductors the difference between $T^{*}$ and $T_c$ is extremely small.