  A generalized propagation matrix method is used to study how scattering off
local Einstein phonons affects resonant electron transmission through quantum
wells. In particular, the parity and the number of the phonon mediated
satellite resonances are found to depend on the available scattering channels.
For a large number of phonon channels, the formation of low-energy impurity
bands is observed. Furthermore, an effective theory is developed which
accurately describes the phonon generated sidebands for sufficiently small
electron-phonon coupling. Finally, the current-voltage characteristics caused
by phonon assisted transmission satellites are discussed for a specific double
barrier geometry.
