Constants of motion and non-stationary wave functions for the damped, time-dependent harmonic oscillator

We study a quantal time-dependent oscillator in the presence of a loss mechanism. We generalize to the damped oscillator the classical exact invariant found by Lewis for the undamped one. The quantal operator Ĵ associated with this constant of motion has no constant expectation values when the fluctuations are not negligible. To investigate the role of the quantal fluctuations we study the spectral properties of Ĵ for the undamped oscillator and show that its eigenstates are critical since they keep stationary the generalized uncertainty products. The nonlinear Schrödinger equation of Kostin and Kan-Griffin is shown to be a particular realization of critical-state-preserving descriptions for the damped oscillator motion.