Constants of Motion of the Harmonic Oscillator

We prove that Weyl quantization preserves constant of motion of the Harmonic Oscillator. We also prove that if $f$ is a classical constant of motion and $\mathfrak{Op}(f)$ is the corresponding operator, then $\mathfrak{Op}(f)$ maps the Schwartz class into itself and it defines an essentially selfadjoint operator on $L^2(\mathbb R^n)$. As a consequence, we provide detailed spectral information of $\mathfrak{Op}(f)$. A complete characterization of the classical constants of motion of the Harmonic Oscillator is given and we also show that they form an algebra with the Moyal product. We give some interesting examples and we analyze Weinstein average method within our framework.