The Adiabatic Approximation and Frohlich Model in the Theory of Metals

Based on the adiabatic expansion for metals, a method is developed whereby if is possible to compute the nonadiabatic corrections to the energy of any order by standard perturbation theory and diagram techniques. It turns out that in addition to the Frohlich one-phonon Hamiltonian the many-phonon Hamiltonians also play a significant role in the theory of metals. Inasmuch as the ground state of the system corresponds to adiabatic perturbation theory, the largest correction to the energy and phonon frequency is of the order (m/M)(1/2), as opposed to the results deduced from the Frohlich Hamiltonian. The expression for the ordinary self-energy contribution differs substantially from its expression in the Frohlich model, and the equation for the pairing self-energy contribution coincides up to terms of order (m/M)(1/2) with the corresponding equation in the Frohlich model. The expression for the critical temperature is discussed.