From Quantum Many-Body Dynamics to Energy-Critical Nonlinear Schrodinger Equations and Back

该项目解决了量子多体动力学的平均场极限的严格理由。多体系统作为物理系统的基本模型自然出现。这种系统的模拟只能通过一些近似，即所谓的平均场极限，因为多体系统可以包含大量的粒子。因此，从平均场极限应该描述的多体系统出发，对平均场极限进行数学论证，是一个具有根本科学意义的问题。研究人员将解决玻色-爱因斯坦凝聚（BEC）研究中出现的问题，玻色子稀释气体冷却到非常接近绝对零度的温度的物质状态。在BEC中，大部分玻色子占据相同的量子态，此时量子效应在宏观尺度上变得明显。自1995年康奈尔和凯特勒首次观察到BEC以来，这一新的物质状态的研究已成为当代研究中最活跃的领域之一。本研究项目的重点是研究与时间相关的多体薛定谔方程解的精细性质有关的几个问题，在能量临界水平，粒子数趋于无穷大。该项目包括三个大方向。第一个方向的目的是证明能量临界非线性薛定谔方程是在Gross-Pitaevskii标度下的量子多体动力学在重要的三维五次情形下的平均场极限。第二个方向的重点是多体薛定谔方程与聚焦相互作用的解决方案的时空规律性。第三个方向转向离散量子多体动力学的概率方面的研究。PI和合作者将使用谐波分析、概率和谱理论等技术来分析这些问题。该奖项反映了NSF的法定使命，并且通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The project addresses the rigorous justification of mean-field limits of quantum many-body dynamics. Many-body systems arise naturally as fundamental models in physical systems. The simulation of such systems is only possible via some approximations, the so-called mean-field limits, since many-body systems can contain an enormous number of particles. The mathematical justification of mean-field limits, starting from the many-body systems they are supposed to describe, is therefore an issue of fundamental scientific importance. The investigator will address questions arising from the study of Bose-Einstein condensation (BEC), the state of matter of a diluted gas of bosons cooled to temperatures very close to absolute zero. In BEC, a large fraction of the bosons occupies the same quantum state, at which point quantum effects become apparent at the macroscopic scale. Since the first observation of BEC in 1995 by Cornell and Ketterle (who earned the Nobel Prize for their discovery), the investigation of this new state of matter has become one of the most active areas of contemporary research.The focus of this research project is to investigate several problems concerning the fine properties of solutions of the time-dependent many-body Schrödinger equation, in the limit as the particle number tends to infinity at the energy-critical level. This project encompasses three broad directions. The first direction aims to prove that the energy-critical Nonlinear Schrödinger equation is the mean-field limit of quantum many-body dynamics under the Gross-Pitaevskii scaling in the important three-dimensional quintic case. The second direction focuses on space-time regularity of solutions to the many-body Schrödinger equation with focusing interactions. The third direction turns to the study of probability aspects of discrete quantum many-body dynamics. The PI and collaborators will use techniques from harmonic analysis, probability, and spectral theory to analyze these problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.