Eliashberg Theory in the Weak Coupling Limit

Eliashberg theory provides a theoretical framework for understanding the phenomenon of superconductivity when pairing between two electrons is mediated by phonons, and retardation effects are fully accounted for. BCS theory is often viewed as the weak coupling limit of Eliashberg theory, in spite of a handful of papers that have pointed out that this is not so. Here we present very accurate numerical solutions in the weak coupling limit to complement the existing analytical results, and demonstrate more convincingly the validity of this limit by extending the analytical results to first order in the coupling constant.