Mean field propagation of infinite dimensional Wigner measures with a singular two-body interaction potential

TL;DR

This paper proves that Wigner measures for many-boson quantum systems with singular interactions, such as Coulomb potential, evolve according to the nonlinear Hartree flow, extending previous results to more realistic singular potentials.

Contribution

It introduces a measure transportation method to establish the propagation of Wigner measures for singular potentials, including Coulomb interactions, in the mean field limit.

Findings

Wigner measures propagate along the nonlinear Hartree flow for singular potentials

Extension of previous bounded potential results to Coulombic interactions

Use of measure transportation technique for singular interactions

Abstract

We consider the quantum dynamics of many bosons systems in the mean field limit with a singular pair-interaction potential, including the attractive or repulsive Coulombic case in three dimensions. By using a measure transportation technique, we show that Wigner measures propagate along the nonlinear Hartree flow. Such property was previously proved only for bounded potentials in our previous works with a slightly different strategy.