The electron spectral function is calculated for a model including electron-phonon coupling to Einstein phonons. The spectrum is studied as a function of the electronic bandwidth and the energy epsilon(k) Of the level from which the electron is removed. A cumulant expansion is used for the time-dependent Green's function, and the second- and fourth-order cumulants are studied. This approach is demonstrated to give accurate results for an exactly solvable two-level model with two electronic levels coupling to local phonons. For a one-band, infinite, three-dimensional model the cumulant expansion gives one satellite in the large-bandwidth limit. As the bandwidth is reduced, the spectrum calculated with the fourth-order cumulant develops multiple satellites, if epsilon(k) is close to the Fermi energy E(F), and as the bandwidth becomes small, results similar to the two-level model are obtained. If epsilon(k) is more than a phonon energy below E(F), the spectrum instead shows a very broad peak, due to the decay of the hole into a hole closer to E(F) and a phonon. If the spin degeneracy of the electrons is taken into account, the broadening due to the decay of a hole into a hole closer to E(F) and an electron-hole pair becomes important, even if epsilon(k) is closer to E(F) than the phonon energy. The validity of Migdal's theorem for A(3)C(60) (A=K,Rb) is discussed. The intersubband electron-phonon coupling is appreciable for A(3)C(60), and it may be argued that the effective bandwidth is large. It is shown that Migdal's theorem is, nevertheless, not valid for A(3)C(60).
