  Using the exact path integral solution for the damped harmonic oscillator it
is shown that in general there does not exist an exact dissipative Liouville
operator describing the dynamics of the oscillator for arbitrary initial bath
preparations. Exact non-stationary Liouville operators can be found only for
particular preparations. Three physically meaningful examples are examined. An
exact new master equation is derived for thermal initial conditions. Second,
the Liouville operator governing the time-evolution of equilibrium correlations
is obtained. Third, factorizing initial conditions are studied. Additionally,
one can show that there are approximate Liouville operators independent of the
initial preparation describing the long time dynamics under appropriate
conditions. The general form of these approximate master equations is derived
and the coefficients are determined for special cases of the bath spectral
density including the Ohmic, Drude and weak coupling cases. The connection with
earlier work is discussed.
