# EMPIRICAL MEASURES AND QUANTUM MECHANICS: APPLICATION TO THE MEAN-FIELD LIMIT

Abstract : In this paper, we define a quantum analogue of the notion of empirical measure in the classical mechanics of $N$-particle systems. We establish an equation governing the evolution of our quantum analogue of the $N$-particle empirical measure, and we prove that this equation contains the Hartree equation as a special case. Our main application of this new notion to the mean-field limit of the $N$-particle Schrödinger equation is an $O(1/\sqrt{N})$ convergence rate in some appropriate dual Sobolev norm for the Wigner transform of the single-particle marginal of the $N$-particle density operator, uniform in $\hbar\in(0,1]$ (where $\hbar$ is the Planck constant) provided that $V$ and $(−∆)^{3+d/2}V$ have integrable Fourier transforms.
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Type de document :
Pré-publication, Document de travail
35 pages, no figure. 2017

Littérature citée [39 références]

https://hal-polytechnique.archives-ouvertes.fr/hal-01644950
Contributeur : François Golse <>
Soumis le : mercredi 22 novembre 2017 - 16:55:53
Dernière modification le : jeudi 11 janvier 2018 - 06:12:13

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• HAL Id : hal-01644950, version 1

### Citation

François Golse, Thierry Paul. EMPIRICAL MEASURES AND QUANTUM MECHANICS: APPLICATION TO THE MEAN-FIELD LIMIT. 35 pages, no figure. 2017. 〈hal-01644950〉

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