# 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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https://hal-polytechnique.archives-ouvertes.fr/hal-01644950
Contributor : François Golse <>
Submitted on : Wednesday, November 22, 2017 - 4:55:53 PM
Last modification on : Thursday, March 5, 2020 - 7:07:24 PM

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François Golse, Thierry Paul. EMPIRICAL MEASURES AND QUANTUM MECHANICS: APPLICATION TO THE MEAN-FIELD LIMIT. Communications in Mathematical Physics, Springer Verlag, 2019, 369 (3), pp.1021-1053. ⟨10.1007/s00220-019-03357-z⟩. ⟨hal-01644950⟩

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