Mean-field quantum dynamics with magnetic fields

TL;DR

This paper proves that the quantum dynamics of a large bosonic system with Coulomb interactions in a magnetic field can be approximated by a magnetic Hartree equation, with explicit convergence rates.

Contribution

It establishes the mean-field limit for bosons in a magnetic field, including convergence in trace norm and energy, with quantitative estimates.

Findings

Convergence of the one-particle density matrix to the magnetic Hartree solution.

Quantitative rates of convergence in trace norm and energy.

Validation of the magnetic Hartree equation as an effective description.

Abstract

We consider a system of $N$ bosons in three dimensions interacting through a mean-field Coulomb potential in an external magnetic field. For initially factorized states we show that the one-particle density matrix associated with the solution of the $N$-body Schr\"odinger equation converges to the projection onto the solution of the magnetic Hartree equation in trace norm and in energy as $N \rightarrow \infty$. Estimates on the rate of convergence are provided.