Comptes Rendus
no. 2
Partial differential equations/Mathematical physics
Wave packets and the quadratic Monge–Kantorovich distance in quantum mechanics
[Paquets d'ondes et distance quadratique de Monge–Kantorovich en mécanique quantique]

Présenté par : Jean-Michel Coron

François Golse1 ; Thierry Paul1
1 CMLS, École polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau cedex, France
Comptes Rendus. Mathématique, Volume 356 (2018) no. 2, pp. 177-197.

Nous considérons dans ce texte la « pseudo-distance » entre densités quantiques introduite dans [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016) 165–205], analogue à la distance quadratique de Monge–Kantorovich(–Vasershtein). Nous en étendons les bornes inférieures et supérieures aux quantifications positives définies en termes de la famille des espaces de phase translatés d'un opérateur de densité, pas nécessairement de rang 1 comme dans le cas de la quantification de Töplitz. Comme corollaire, nous démontrons que le taux de convergence uniforme, lorsque ħ tend vers 0, de la limite de champ moyen de l'équation de Heisenberg à N particules vaut pour une classe beaucoup plus large de données initiales que dans [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016) 165–205]. Nous discutons également la pertinence de la pseudo-distance, comparée aux normes de Schatten, dans le but de métriser l'ensemble des opérateurs de densité quantique en régime semi-classique.

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 1 as in the case of the Töplitz quantization. As a corollary, we prove that the uniform as ħ→0 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, Commun. Math. Phys. 343 (2016) 165–205]. 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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2017.12.007
François Golse 1 ; Thierry Paul 1

1 CMLS, École polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau cedex, France 
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François Golse; Thierry Paul. Wave packets and the quadratic Monge–Kantorovich distance in quantum mechanics. Comptes Rendus. Mathématique, Volume 356 (2018) no. 2, pp. 177-197. doi : 10.1016/j.crma.2017.12.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.12.007/

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