A simple proof of convergence to the Hartree dynamics in Sobolev trace norms

Abstract

The derivation of the Hartree equation from many-body systems of Bosons in the mean field limit has been very intensively studied in the last couple of years. However, very few results exist showing convergence of the k-th marginal of the N-body density matrix to the projection to the k-fold tensor product of the solution of the Hartree equation in stronger trace norms like the energy trace norm, see \cite{MS}, \cite{Lu}. This issue is from a physical view point very important. The reason is that one can then approximate expectation values of certain observables of the N-body system by means of the Hartree equation, with relaxation of the very restrictive assumption that the observables are bounded operators. Here we consider the non-relativistic case. We prove, assuming only H^1-regularity of the initial data, convergence in the energy trace norm without rates, and convergence in any other weaker Sobolev trace norm with rates. Our proof is simple and uses the functional $a_N$ introduced by Pickl in \cite{Pi}.

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• Higher order corrections to the mean-field description of the dynamics of interacting bosons
• Beyond Bogoliubov Dynamics
• Derivation of the Time Dependent Two Dimensional Focusing NLS Equation
• Derivation of the Maxwell-Schrödinger Equations from the Pauli-Fierz Hamiltonian
• Central limit theorem for Bose gases interacting through singular potentials
• Mean-field limits of particles in interaction with quantized radiation fields
• Convergence rate towards the fractional Hartree-equation with singular potentials in higher Sobolev norms
• Semiclassical approximation and critical temperature shift for weakly interacting trapped bosons
• Derivation of the Bogoliubov Time Evolution for a Large Volume Mean-field Limit
• On the emergence of quantum Boltzmann fluctuation dynamics near a Bose-Einstein Condensate