Vertex-corrected perturbation theory for the electron-phonon problem with non-constant density of states

A series of weak-coupling perturbation theories which include the lowest-order vertex corrections are applied to the attractive Holstein model in infinite dimensions. The approximations are chosen to reproduce the iterated perturbation theory in the limit of half-filling and large phonon frequency (where the Holstein model maps onto the Hubbard model). Comparison is made with quantum Monte Carlo solutions to test the accuracy of different approximation schemes.