Scattering of electrons by a strong standing wave in adiabatic switching of interaction

A theoretical analysis is made of the scattering of electrons by a strong standing wave with a slowly varying amplitude. It is shown that a consistent transition to the adiabatic limit is impossible unless higher orders of the perturbation theory are used. General expressions are derived for the scattering probabilities, which can be used in special cases to find the results of the perturbation theory or to pass the adiabatic limit. An analysis is given of the resonant scattering of electrons which appears when a third wave of similar frequency is added to the standing wave. The WKB time method is used to find expressions which govern the line profile and the resonant scattering probability.