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. Applications to the mean-field limit of the N-particle Schrödinger equation include 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 \({\hbar\in(0,1]}\) provided that V and \({(-\Delta)^{3/2+d/4}V}\) have integrable Fourier transforms.
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Acknowledgements
The problem of finding a quantum analogue to the notion of empirical measures and Klimontovich solutions used in [10,12] to prove the mean field limit in classical dynamics has been posed to us by Mario Pulvirenti.
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Golse, F., Paul, T. Empirical Measures and Quantum Mechanics: Applications to the Mean-Field Limit. Commun. Math. Phys. 369, 1021–1053 (2019). https://doi.org/10.1007/s00220-019-03357-z
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DOI: https://doi.org/10.1007/s00220-019-03357-z