Mean-Field- and Classical Limit of Many-Body Schr\"odinger Dynamics for Bosons

TL;DR

This paper proves the convergence of many-body Schrödinger dynamics for bosons to the Hartree and Vlasov equations in the mean-field limit, providing uniform estimates and new proof techniques.

Contribution

It introduces a new proof of convergence for bosonic Schrödinger dynamics to mean-field equations, with uniform estimates for certain interactions.

Findings

Convergence of N-particle Schrödinger dynamics to Hartree equation.

Uniform convergence estimates for specific two-body interactions.

Classical limit described by the Vlasov equation when Planck's constant approaches zero.

Abstract

We present a new proof of the convergence of the N-particle Schroedinger dynamics for bosons towards the dynamics generated by the Hartree equation in the mean-field limit. For a restricted class of two-body interactions, we obtain convergence estimates uniform in the Planck constant , up to an exponentially small remainder. For h=0, the classical dynamics in the mean-field limit is given by the Vlasov equation.