Hartree动力系统的Bogoliubov修正和迹模收敛
Bogoliubov corrections and trace norm convergence for the Hartree dynamics
摘要：
我们考虑了一类包含库仑相互作用的相互作用势在平均场极限下大量N个非相对论玻色子的动力学。为了描述平均场Hartree态周围的涨落，我们在N粒子空间上引入了一个辅助哈密顿量，它与Bogoliubov理论中得到的哈密顿量非常相似。我们显示收敛的辅助时间演化的N粒子空间的规范完全相互作用的动力学。这一结果使我们能够证明其他几个结果：收敛的减少密度矩阵的跟踪范数与最佳的速度，收敛的能量跟踪范数，并收敛到一个时间的发展得到的Bogoliubov哈密顿Fock空间与预期的最佳速度。因此，我们扩展和量化了几个以前的结果，例如，通过提供物理上重要的收敛速率，包括依赖于时间的外部场和奇异相互作用，并允许一般的初始状态，例如，那些被认为是相互作用系统的基态的。
英文摘要：
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.
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