Holstein model in infinite dimensions at half-filling

The normal state of the Holstein model is studied at half-filling in infini te dimensions and in the adiabatic regime. The dynamical mean-field equatio ns are solved using perturbation expansions around the extremal paths of th e effective action for the atoms. We find that the Migdal-Eliashberg expans ion breaks dawn in the metallic state if the electron-phonon coupling lambd a exceeds a value of about 1.3 in spite of the fact that the formal expansi on parameter lambda omega(0)/E-F (omega(0) is the phonon frequency, E-F the Fermi energy) is much smaller than 1. The breakdown is due to the appearan ce of more than one extremal path of the action. We present numerical resul ts which illustrate in detail the evolution of the local Green's function, the self-energy, and the effective atomic potential as a function of lambda . [S0163-1829(98)01845-1].