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

François Golse (1) , Thierry Paul (1)

1 CMLS - Centre de Mathématiques Laurent Schwartz

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.

Mots clés: Mean-field limit; Empirical measure; Quantum N-body problem; Semiclassical scaling

Domaines: Equations aux dérivées partielles [math.AP]

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