CORRECTIONS TO MIGDALS THEOREM FOR SPECTRAL FUNCTIONS - A CUMULANT TREATMENT OF THE TIME-DEPENDENT GREENS-FUNCTION

The electron spectral function is calculated for a model including ele ctron-phonon coupling to Einstein phonons. The spectrum is studied as a function of the electronic bandwidth and the energy epsilon(k) Of th e level from which the electron is removed. A cumulant expansion is us ed for the time-dependent Green's function, and the second- and fourth -order cumulants are studied. This approach is demonstrated to give ac curate results for an exactly solvable two-level model with two electr onic levels coupling to local phonons. For a one-band, infinite, three -dimensional model the cumulant expansion gives one satellite in the l arge-bandwidth limit. As the bandwidth is reduced, the spectrum calcul ated with the fourth-order cumulant develops multiple satellites, if e psilon(k) is close to the Fermi energy E(F), and as the bandwidth beco mes small, results similar to the two-level model are obtained. If eps ilon(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 clos er to E(F) and a phonon. If the spin degeneracy of the electrons is ta ken into account, the broadening due to the decay of a hole into a hol e 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 M igdal'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 a rgued that the effective bandwidth is large. It is shown that Migdal's theorem is, nevertheless, not valid for A(3)C(60).