Condensation energy in Eliashberg theory -- from weak to strong coupling

We consider two issues related to the condensation energy in superconductors described by the Eliashberg theory for various forms of the pairing interaction, associated either with phonon or electronic mechanisms of superconductivity. First, we derive a leading correction to the BCS formula for the condensation energy to first order in the coupling $\lambda$. Second, we show that at a given $\lambda$, the value of the condensation energy strongly depends on the functional form of the effective pairing interaction $\Gamma (\omega)$.