SUMMARY
Within the framework of the macroscopic dielectric continuum model the longitudinal optical (LO) phonon modes are derived for a cylindrical semiconductor quantum wire made of semiconductor 1 (well material) embedded in another finite semiconductor 2 (barrier material). The phonon states of modes are given by solving the generalized Born-Huang equation. It is shown that there may exist four types of longitudinal optical phonon modes according to the concrete materials forming the wire. The dispersion equations for phonon frequencies with wave-vector components parallel to the wire are obtained. After having quantized the phonon field we derive the Fröhlich Hamiltonian describing the electron--LO-phonon interaction. The influence of the thickness of the barrier layer as well as the thin metallic shell on the phonon frequencies and their interaction with electrons is studied.