The Steen-Ermakov equation itself is obtained when \(g=0\). The equation for \(\sigma\) is relevant only if the force term is present. Otherwise, the above invariants resemble the case of the oscillator with time-dependent frequency already addressed in the ref. \cite{Bertin_2012}.
Quantization of the oscillator
Now we wish to explore the fact that the first-order operators (\ref{eq:10}) and (\ref{eq:15}) are two dynamical invariants of the oscillator if \(\beta\) and \(\beta^*\) are two L.I. solutions of (\ref{eq:11}). The commutation relations are found to be