Abstract
The standard Eliashberg theory is extended to incorporate the narrow-band effects, that is, the vertex corrections and frequency dependence of the effective interaction between electrons. From the extended Eliashberg equation, the Coulomb pseudo-potential μ * is extracted. It is shown that even for narrow-band superconductors, where the Fermi energy ϵ F is comparable with the phonon frequency ω ph, the Coulomb pseudo-potential is a pertinent parameter, and is still given by μ *= μ/[1+ μln( ϵ F/ ω ph)], provided ω ph is appropriately scaled.
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