半相对论Hartree动力学的收敛速度
Rate of Convergence Towards Semi-Relativistic Hartree Dynamics
摘要：
本文研究了具有引力常数$G$的半相对论$N$引力玻色子系统。系统的时间演化由相对论色散定律描述，并假设$N \to \infty$和$G \to 0$在$GN = λ$固定时的相互作用为平均场标度.在大$λ$的超临界区，我们引入了截止为零的正则化相互作用$N \to \infty$。我们证明了多体半相对论薛定谔动力学和相应的半相对论Hartree动力学之间的差别对于所有$λ$至多是$N^{-1}$阶，即，该结果涵盖了亚临界状态和超临界状态。界的$N$依赖性是最优的。
英文摘要：
We consider the semi-relativistic system of $N$ gravitating Bosons with gravitation constant $G$. The time evolution of the system is described by the relativistic dispersion law, and we assume the mean-field scaling of the interaction where $N \to \infty$ and $G \to 0$ while $GN = λ$ fixed. In the super-critical regime of large $λ$, we introduce the regularized interaction where the cutoff vanishes as $N \to \infty$. We show that the difference between the many-body semi-relativistic Schrödinger dynamics and the corresponding semi-relativistic Hartree dynamics is at most of order $N^{-1}$ for all $λ$, i.e., the result covers the sub-critical regime and the super-critical regime. The $N$ dependence of the bound is optimal.
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