A new technique is presented for evaluating the electron Green function Gk(t) in an insulating electron–phonon system. The technique is particularly suitable for those electron–phonon interactions in which the long wavelength phonons are of prime importance such as the polaron and piezoelectric interactions.An exact formal expression is obtained for the time and temperature dependent Green function in the form[Formula: see text]where S is a functional of the differential operator d/dk. The Green function is then expressed as [Formula: see text], and from the exact result a series expansion is derived for the action function Ak(t).This series expansion has a very special property: to whatever order Ak(t) is evaluated the resulting expression for Gk(t) includes, in some way, every perturbation theory diagram. By this we mean that as in conventional techniques (the Hartree–Fock approximation, for example) an infinite subset of diagrams are summed exactly and in contrast to these techniques the remaining infinity of diagrams are not discarded but are evaluated approximately.The technique is applied to the polaron interaction.

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The quantum-limit cyclotron-resonance linewidth is studied theoretically, assuming electron scattering by acoustic phonons and ionized impurities. A minimum is predicted for the linewidth as a function of magnetic field but only for a restricted range of electron and ionized-impurity densities. Near this minimum the linewidth is proportional to the square root of the impurity density and is independent of temperature.

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