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.

中文翻译：

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谐波振荡器的运动常数

我们证明外尔量化保持谐波振荡器的运动常数。我们还证明，如果 $f$ 是一个经典的运动常数，而 $\mathfrak{Op}(f)$ 是相应的算子，那么 $\mathfrak{Op}(f)$ 将 Schwartz 类映射到自身并定义$L^2(\mathbb R^n)$ 上的一个本质上的自伴随算子。因此，我们提供了 $\mathfrak{Op}(f)$ 的详细光谱信息。给出了谐波振荡器的经典运动常数的完整特征，并且我们还表明它们与 Moyal 乘积形成代数。我们给出了一些有趣的例子，并在我们的框架内分析了 Weinstein 平均方法。