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

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