Pickl's Proof of the Quantum Mean-Field Limit and Quantum Klimontovich Solutions - Centre de mathématiques Laurent Schwartz (CMLS) Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2022

Pickl's Proof of the Quantum Mean-Field Limit and Quantum Klimontovich Solutions

Immanuel Ben Porath
  • Fonction : Auteur
  • PersonId : 1130951
François Golse
  • Fonction : Auteur
  • PersonId : 964158

Résumé

This paper discusses the mean-field limit for the quantum dynamics of $N$ identical bosons in $mathbf R^3$ interacting via a binary potential with Coulomb type singularity. Our approach is based on the theory of quantum Klimontovich solutions defined in [F. Golse, T. Paul, Commun. Math. Phys. 369 (2019), 1021-1053]. Our first main result is a definition of the interaction nonlinearity in the equation governing the dynamics of quantum Klimontovich solutions for a class of interaction potentials slightly less general than those considered in [T. Kato, Trans. Amer. Math. Soc. 70 (1951), 195-211]. Our second main result is a new operator inequality satisfied by the quantum Klimontovich solution in the case of an interaction potential with Coulomb type singularity. When evaluated on an initial bosonic pure state, this operator inequality reduces to a Gronwall inequality for a functional introduced in [P. Pickl, Lett. Math. Phys. 97 (2011), 151-164], resulting in a convergence rate estimate for the quantum mean-field limit leading to the time-dependent Hartree equation.
Fichier principal
Vignette du fichier
IBPFGPickl.pdf (353.52 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-03619187 , version 1 (24-03-2022)

Identifiants

  • HAL Id : hal-03619187 , version 1

Citer

Immanuel Ben Porath, François Golse. Pickl's Proof of the Quantum Mean-Field Limit and Quantum Klimontovich Solutions. 2022. ⟨hal-03619187⟩
74 Consultations
34 Téléchargements

Partager

Gmail Facebook X LinkedIn More