  The evolution of a quantum oscillator, with periodically varying frequency
and damping, is studied in the two cases of parametric resonance (PR) producing
a limited, or unlimited stretching of the wave function. The different
asymptotic behaviors of the energy distribution, show the non trivial interplay
between PR and the initial quantum state. In the first case, the oscillator's
mean energy tends asymptotically to a fully classical value, independent of the
initial state, with vanishing relative quantum fluctuations. In the second
case, the memory of the initial state persists over arbitrary long time scales,
both in the mean value and in the large quantum fluctuations of the energy.
