标题： Hartree动力学在Sobolev迹范数下的简单证明 显示英文标题

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

摘要： 从玻色子多体系统在平均场极限下推导出Hartree方程在过去几年中得到了非常深入的研究。 然而，很少有结果表明N体密度矩阵的k阶边缘收敛到Hartree方程解的k重张量积的投影，在如能量迹范数这样的更强迹范数中，见\cite{MS}，\cite{Lu}。 从物理观点来看，这个问题非常重要。 原因是这样就可以通过Hartree方程来近似N体系统的某些可观测量的期望值，而无需满足非常严格的假设，即这些可观测量是有界算子。 我们考虑非相对论情况。 我们证明，在仅假设初始数据具有H^1正则性的情况下，能量迹范数下收敛但没有速率，而在任何其他较弱的Sobolev迹范数下收敛并带有速率。 我们的证明简单，并使用了Pickl在\cite{Pi}中引入的泛函$a_N$。 显示英文摘要

摘要： 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}.