Bogoliubov corrections and trace norm convergence for the Hartree dynamics

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.

Higher Order Corrections to the Mean-Field Description of the Dynamics of Interacting Bosons

“…Here, H Bog (t) denotes the Bogoliubov Hamiltonian 2 , an effective Hamiltonian in Fock space which is quadratic in the number of creation and annihilation operators. For three dimensions and scaling parameter β = 0, a similar result was obtained in [42,43] via a first quantised approach. More precisely, denote p ϕ(t) j := |ϕ(t, x j ) ϕ(t, x j )| 2 Written in second quantized form, HBog(t) is defined as…”

“…Let us first recall from [39,42,43] the explicit decomposition of an N -body wave function ψ in terms of a condensate ϕ ⊗N and k-particle fluctuations around this condensate. To this end, recall the following projections introduced in [50]:…”

Mean-Field Limits of Particles in Interaction with Quantized Radiation Fields

“…Moreover, we present the first explicit bounds on the rate of convergence of the oneparticle reduced density matrix of the charges in Sobolev norm. It seems possible to obtain the convergence rate N −1 , if one regards (similar to [19]) fluctuations around the mean-field dynamics. 6 We suppose that Conjecture III.2 can be proven by a standard fixed-point argument.…”

“…Remark VI.2. The functional β a was already used in [21,22,23,11,12,18,19,2] and others to derive the Hartree and Gross-Pitaevskii equation.…”

“…Inequality (43) follows from the monotonicity of the square root and (46). The estimate (44) originates from [19]. One starts with the relation…”