  The exact form of quantum propagator of a quantum oscillator interacting with
a bosonic bath consisting of $N$ distinguished quantum oscillators with
different frequencies is obtained in the Heisenberg picture. Reduced density
matrix for oscillator is obtained. The kernel or Green's function connecting
the initial density matrix of the oscillator to the density matrix in an
arbitrary time is obtained and its connection to Feynman-Vernon influence
functional is discussed. Weak coupling regime and squared mean values for
position, momentum and energy of the oscillator are obtained in equilibrium.
