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Using the exact path integral solution for the damped harmonic oscillator itis shown that in general there does not exist an exact dissipative Liouvilleoperator describing the dynamics of the oscillator for arbitrary initial bathpreparations. Exact non-stationary Liouville operators can be found only forparticular preparations. Three physically meaningful examples are examined. Anexact new master equation is derived for thermal initial conditions. Second,the Liouville operator governing the time-evolution of equilibrium correlationsis obtained. Third, factorizing initial conditions are studied. Additionally,one can show that there are approximate Liouville operators independent of theinitial preparation describing the long time dynamics under appropriateconditions. The general form of these approximate master equations is derivedand the coefficients are determined for special cases of the bath spectraldensity including the Ohmic, Drude and weak coupling cases. The connection withearlier work is discussed.Comment: to be published in Phys. Rev.

Exact time evolution and master equations for the damped harmonic oscillator