Uniqueness Criteria for the Vlasov–Poisson System and Applications to Semiclassical Analysis

We review some uniqueness criteria for the Vlasov–Poisson system, emerging as corollaries of stability estimates in strong or weak topologies, and show how they serve as a guideline to solve problems arising in semiclassical analysis. More specifically, we look at the Schatten norms and at the quantum analogue of the Wasserstein distance and prove the semiclassical limit from the Hartree equation with Coulomb interaction to the Vlasov–Poisson system in these topologies by mimicking weak-strong uniqueness principles for the Vlasov–Poisson equation in the respective classical topologies. Different topologies allow to treat different classes of quantum states.