The Rigorous Derivation of the 2D Cubic Focusing NLS from Quantum Many-body Evolution
The Rigorous Derivation of the 2D Cubic Focusing NLS from Quantum Many-body Evolution
复制标题
量子多体演化中二维立方聚焦NLS的严格推导
DOI：
10.1093/imrn/rnw113
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
J. Holmer
中科院分区：
文献类型：
--
作者：
Xuwen Chen;J. Holmer
我们考虑一个由 $N$-玻色子组成的二维时间相关量子系统，在 Gross-Pitaevskii 标度中具有谐波外部限制和 \emph{有吸引力} 粒子间相互作用。我们推导了物质类型估计的稳定性，表明能量的 $k$ 次方控制着 $k$ 粒子解的 $H^{1}$ Sobolev 范数。这种估计是新的，并且对于吸引相互作用比排斥相互作用更困难。为了证明，我们使用[49]中的有限维量子 di Finetti 定理的一个版本。证明中发挥了高粒子数平均效应，这对于排斥情况下的相应估计来说是不需要的。这个先验界限使我们能够证明相应的 BBGKY 层次收敛到 GP 极限，就像之前许多处理排斥相互作用情况的工作中所做的那样。因此，我们得到\emph{focusing}非线性Schr\"{o}dinger方程是具有有吸引力的原子间相互作用和渐近分解初始数据的二维时间相关量子多体系统的平均场极限。需要对原子间相互作用势的$L^{1}$-范数的大小进行假设，该假设对应于二维Gagliardo-Nirenberg不等式中的锐常数，尽管该不等式并不直接相关，因为我们正在处理的是痕迹而不是功率。
We consider a 2D time-dependent quantum system of $N$-bosons with harmonic external confining and \emph{attractive} interparticle interaction in the Gross-Pitaevskii scaling. We derive stability of matter type estimates showing that the $k$-th power of the energy controls the $H^{1}$ Sobolev norm of the solution over $k$-particles. This estimate is new and more difficult for attractive interactions than repulsive interactions. For the proof, we use a version of the finite-dimensional quantum di Finetti theorem from [49]. A high particle-number averaging effect is at play in the proof, which is not needed for the corresponding estimate in the repulsive case. This a priori bound allows us to prove that the corresponding BBGKY hierarchy converges to the GP limit as was done in many previous works treating the case of repulsive interactions. As a result, we obtain that the \emph{focusing} nonlinear Schr\"{o}dinger equation is the mean-field limit of the 2D time-dependent quantum many-body system with attractive interatomic interaction and asymptotically factorized initial data. An assumption on the size of the $L^{1}$-norm of the interatomic interaction potential is needed that corresponds to the sharp constant in the 2D Gagliardo-Nirenberg inequality though the inequality is not directly relevant because we are dealing with a trace instead of a power.