We study the time dependent Schrödinger equation for large spinless fermions with the semiclassical scale \(\hbar = N^{-1/3}\) in three dimensions. By using the Husimi measure defined by coherent states, we rewrite the Schrödinger equation into a BBGKY type of hierarchy for the k particle Husimi measure. Further estimates are derived to obtain the weak compactness of the Husimi measure, and in addition uniform estimates for the remainder terms in the hierarchy are derived in order to show that in the semiclassical regime the weak limit of the Husimi measure is exactly the solution of the Vlasov equation.

中文翻译：

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大型铁离子系统的组合平均场和半经典极限

我们研究了三维半经典尺度\（\ hbar = N ^ {-1/3} \）的大型无旋费米子的时间相关Schrödinger方程。通过使用由相干态定义的Husimi测度，我们将Schrödinger方程重写为k个粒子Husimi测度的BBGKY类型层次结构。得出进一步的估计以获得Husimi测度的弱紧致性，此外，还推导了层次结构中其余项的统一估计，以表明在半经典状态下，Husimi测度的弱极限恰好是该测度的解。弗拉索夫方程式。