标题： Bogoliubov-de Gennes方程的半经典极限 显示英文标题

标题： Semiclassical Limit of the Bogoliubov-de Gennes Equation

摘要： 在本文中，我们将时间依赖的 Bogoliubov$\unicode{x2013}$de Gennes 方程改写为适当的半经典形式，并建立了其半经典极限到一个具有有效平均场背景势的两粒子动力输运方程，该势满足单粒子 Vlasov 方程。 此外，在某些半经典情形下，我们得到了两粒子动力输运方程的高阶修正，捕捉到了非平凡的两体相互作用效应。 收敛性在$C^2$相互作用势的情况下，通过半经典最优传输伪度量进行了证明。 此外，将我们的当前结果与 Marcantoni 等人的 [arXiv:2310.15280] 的结果相结合，我们在某些负阶 Sobolev 拓扑中通过 Vlasov 方程建立了自旋$\frac{1}{2}$费米子系统动力学的联合半经典和平均场近似。 显示英文摘要

摘要： In this paper, we rewrite the time-dependent Bogoliubov$\unicode{x2013}$de Gennes equation in an appropriate semiclassical form and establish its semiclassical limit to a two-particle kinetic transport equation with an effective mean-field background potential satisfying the one-particle Vlasov equation. Moreover, for some semiclassical regimes, we obtain a higher-order correction to the two-particle kinetic transport equation, capturing a nontrivial two-body interaction effect. The convergence is proven for $C^2$ interaction potentials in terms of a semiclassical optimal transport pseudo-metric. Furthermore, combining our current results with the results of Marcantoni et al. [arXiv:2310.15280], we establish a joint semiclassical and mean-field approximation of the dynamics of a system of spin-$\frac{1}{2}$ Fermions by the Vlasov equation in some negative order Sobolev topology.