The Schr\"odinger Equation in the Mean-Field and Semiclassical Regime

TL;DR

This paper rigorously analyzes the classical and mean-field limits of the Schrödinger and Hartree equations, establishing convergence to classical equations like Vlasov and Liouville, using quantum Monge-Kantorovich distances.

Contribution

It provides new estimates and bounds for the quantum-to-classical transition in many-body quantum systems, extending previous frameworks with refined distance measures.

Findings

Classical limit of Hartree equation to Vlasov equation established

Uniform classical limit of N-body Schrödinger to Liouville equation shown

Simultaneous mean-field and classical limit derived with quantitative estimates

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 Schr\"{o}dinger 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 Schr\"{o}dinger 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.