2015/10/31 by François Golse, Thierry Paul · 1 citation
Mathematics · Physics and Astronomy · #Classical limit #Exponent #Gas Dynamics and Kinetic Theory #Limit (mathematics) #Lipschitz continuity #Planck constant #Quadratic equation #Quantum #Quantum Mechanics and Non-Hermitian Physics #Semiclassical physics #Spectral Theory in Mathematical Physics #Vlasov equation #math-ph #math.AP #math.MP #msc:35Q41 #msc:35Q55 #msc:35Q83 #msc:82C05 #msc:82C10
paper · pdf · doi:10.1007/s00205-016-1031-x
published as Arch. Rational Mech. Anal. 223 (2017), 57-94 · 33 pages
openalex created_date 2016/06/24 · arxiv created 2016/07/16 · openalex publication_date 2016/08/12 · arxiv updated 2017/07/18 · openalex updated_date 2026/08/05
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.