ABSTRACT
This chapter considers two kinds of polaron models. The first type is the one in which an electron interacts with one or two phonon modes. The one-mode problem was first studied by Gross. The symmetric two-mode problem was solved by Devreese. Devreese later extended the Gross model to consider the interaction of the electron with two phonon modes that have wave vectors q and -q and showed that this extended model also admits an exact solution. In the second type of model, consider the problem of a localized electron interacting with phonons. The problem of a localized electron interacting with optical phonons is an exactly soluble problem. In this case, neither the electron position nor its momentum is a dynamical variable. The role of the electric field is essentially to displace the mean position of the oscillator, and the energy of the oscillator is lowered by the work done in shifting the mean position of the oscillator.
