THE SCHRÖDINGER EQUATION IN THE MEAN-FIELD AND SEMICLASSICAL REGIMEReport as inadecuate




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1 CMLS - Centre de Mathématiques Laurent Schwartz

Abstract : In this paper, we establish 1 the classical limit of the Hartree equation leading to the Vlasov equation, 2 the classical limit of the N-body linear Schrödinger equation uniformly in N leading to the N-body Liouville equation of classical mechanics and 3 the simultaneous mean-field and classical limit of the N-body linear Schrödinger equation leading to the Vlasov equation. In all these limits, we assume that the gradient of the interaction potential is Lipschitz continuous. All our results are formulated as estimates involving a quantum analogue of the Monge-Kantorovich distance of exponent 2 adapted to the classical limit, reminiscent of, but different from the one defined in F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 2016, 165-205. As a by-product, we also provide bounds on the quadratic Monge-Kantorovich distances between the classical densities and the Husimi functions of the quantum density matrices.

Keywords : Classical limit Vlasov equation Mean-field limit Schrödinger equation Hartree equation Liouville equation Monge-Kantorovich distance





Author: François Golse - Thierry Paul -

Source: https://hal.archives-ouvertes.fr/



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