WAVE PACKETS AND THE QUADRATIC MONGE-KANTOROVICH DISTANCE IN QUANTUM MECHANICS

François Golse (1) , Thierry Paul (1)

1 CMLS - Centre de Mathématiques Laurent Schwartz

In this paper, we extend the upper and lower bounds for the " pseudo-distance " on quantum densities analogous to the quadratic Monge-Kantorovich(-Vasershtein) distance introduced in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016) 165–205] to positive quantizations defined in terms of the family of phase space translates of a density operator, not necessarily of rank one as in the case of the Töplitz quantization. As a corollary , we prove that the uniform (for vanishing h) convergence rate for the mean-field limit of the N-particle Heisenberg equation holds for a much wider class of initial data than in [F. Golse, C. Mouhot, T. Paul, loc. cit.]. We also discuss the relevance of the pseudo-distance compared to the Schatten norms for the purpose of metrizing the set of quantum density operators in the semiclassical regime.

Keywords: Wasserstein distance; Husimi transform; Töplitz operators; Semiclassical limit; Mean-field limit; Schrödinger equation; Hartree equation

Domains: Analysis of PDEs [math.AP]

François Golse, Thierry Paul. WAVE PACKETS AND THE QUADRATIC MONGE-KANTOROVICH DISTANCE IN QUANTUM MECHANICS. Comptes Rendus. Mathématique, 2018, 356, pp.177-197. ⟨10.1016/j.crma.2017.12.007⟩. ⟨hal-01562203⟩