We establish a quantitative observation inequality for the Schrödinger and the Heisenberg equations on $R^d$, uniform in the Planck constant $\hslash \in [0,1]$. The proof is based on the pseudometric introduced in F. Golse and T. Paul, The Schrödinger equation in the mean-field and semiclassical regime, Arch. Ration. Mech. Anal. 223 (2017) 57–94. This inequality involves only effective constants which are computed explicitly in their dependence in $\hslash$ and all parameters involved.

中文翻译：

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薛定谔和海森堡方程的定量可观测性：最优传输方法

我们建立了薛定谔方程和海森堡方程的定量观察不等式$R^d$, 在普朗克常数中均匀$\hslash \in [0,1]$. 证明基于 F. Golse 和 T. Paul 在平均场和半经典方案中的薛定谔方程Arch 中引入的伪度量。配给。机甲。肛门。223 (2017) 57–94。这种不等式只涉及有效常数，这些常数是根据它们在$\hslash$以及所有涉及的参数。