  Our objective is to study resonant tunneling of an electron in the presence
of inelastic scattering by optical phonons. Using a recently developed
technique, based on exact mapping of a many-body problem onto a one-body
problem, we compute transmission through a single site at finite temperatures.
We also compute current through a single site at finite temperatures and an
arbitrary strength of the potential drop over the tunneling region.
Transmission vs. incident electron energy at finite temperatures displays
additional peaks due to phonon absorption processes. Current at a voltage bias
smaller than the phonon frequency is dominated by elastic processes. We apply
the method to an electron tunneling through the Aharonov-Bohm ring coupled to
optical phonons. Elastic part of electron-phonon scattering does not affect the
phase of the electron. Dephasing occurs only through inelastic processes.
