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 lowestorder 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. Pacs:74.20.-z, 71.27.+a, and 71.38.+i Typeset using REVTEX