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Electron--optical-phonon interactions are studied in single and double heterostructures. The Hamiltonian describing electron-phonon interactions is derived with use of orthonormal eigenmodes of phonons calculated in the dielectric continuum model. Effects of interactions are expressed in the form of an effective electron-electron interaction mediated by virtual exchange of phonons. It is given by the sum of products of coupling constants and form factors for different modes. The coupling constants correspond to the Fr\"ohlich coupling constant in bulk and the form factors describe effects of electron confinement. The resulting expression is convenient and useful in discussing relative importance of interface modes and bulklike confined modes and also in exploring the possibility of reduction of electron--optical-phonon interactions in heterostructures. It is applied to calculations of the polaron mobility and magnetophonon resonance spectra. The results clearly demonstrate roles of interface phonons.
Mori et al. (Fri,) studied this question.