Bogoliubov corrections and trace norm convergence for the Hartree dynamics

TL;DR

This paper proves convergence results for the dynamics of large bosonic systems in the mean field limit, including trace norm convergence of reduced density matrices and energy, extending previous results with optimal rates and general initial states.

Contribution

It introduces an auxiliary Hamiltonian similar to Bogoliubov theory to establish norm convergence of the dynamics for Coulomb and other interactions, with optimal rates and broad initial conditions.

Findings

Convergence of auxiliary evolution to full dynamics in N-particle space

Trace norm convergence of reduced density matrices with optimal rate

Convergence to Bogoliubov Hamiltonian dynamics with expected rate

Abstract

We consider the dynamics of a large number N of nonrelativistic bosons in the mean field limit for a class of interaction potentials that includes Coulomb interaction. In order to describe the fluctuations around the mean field Hartree state, we introduce an auxiliary Hamiltonian on the N-particle space that is very similar to the one obtained from Bogoliubov theory. We show convergence of the auxiliary time evolution to the fully interacting dynamics in the norm of the N-particle space. This result allows us to prove several other results: convergence of reduced density matrices in trace norm with optimal rate, convergence in energy trace norm, and convergence to a time evolution obtained from the Bogoliubov Hamiltonian on Fock space with expected optimal rate. We thus extend and quantify several previous results, e.g., by providing the physically important convergence rates, including time-dependent external fields and singular interactions, and allowing for general initial states, e.g., those that are expected to be ground states of interacting systems.