论文标题
Pickl的量子平均场限制和量子Klimontovich解决方案的证明
Pickl's Proof of the Quantum Mean-Field Limit and Quantum Klimontovich Solutions
论文作者
论文摘要
本文讨论了$ n $相同玻色子在$ \ mathbf r^3 $中通过二进制电位与库仑型奇异性的二进制电位相互作用的量子动力学的平均场限制。我们的方法基于[F. Golse，T。Paul，Commun。数学。物理。 369（2019），1021-1053]。我们的第一个主要结果是对相互作用的相互作用非线性定义的定义，该方程式管理量子klimontovich解决方案的动力学对于一类相互作用势的动力学，而不是[t。t。加藤，译。阿米尔。数学。 Soc。 70（1951），195-211]。我们的第二个主要结果是在与库仑型奇异性的相互作用潜力的情况下，量子Klimontovich解决方案满足了新的操作员不平等。当对初始骨纯状态进行评估时，该操作员不平等将降低至Gronwall不等式，以便在[P.中引入的功能pickl，lett。数学。物理。 97（2011），151-164]，导致量子平均场极限的收敛速率估计，导致时间依赖于时间依赖的Hartree方程。
This paper discusses the mean-field limit for the quantum dynamics of $N$ identical bosons in $\mathbf R^3$ interacting via a binary potential with Coulomb type singularity. Our approach is based on the theory of quantum Klimontovich solutions defined in [F. Golse, T. Paul, Commun. Math. Phys. 369 (2019), 1021-1053]. Our first main result is a definition of the interaction nonlinearity in the equation governing the dynamics of quantum Klimontovich solutions for a class of interaction potentials slightly less general than those considered in [T. Kato, Trans. Amer. Math. Soc. 70 (1951), 195-211]. Our second main result is a new operator inequality satisfied by the quantum Klimontovich solution in the case of an interaction potential with Coulomb type singularity. When evaluated on an initial bosonic pure state, this operator inequality reduces to a Gronwall inequality for a functional introduced in [P. Pickl, Lett. Math. Phys. 97 (2011), 151-164], resulting in a convergence rate estimate for the quantum mean-field limit leading to the time-dependent Hartree equation.
