Semiclassical limit of the Bogoliubov–de Gennes equation

Abstract

In this paper, we rewrite the time-dependent Bogoliubov–de Gennes (BdG) 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. [Ann. Henri Poincaré (2024)], 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 weak topology.