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

从量子多体动力学到能量关键的非线性薛定谔方程以及返回

基本信息

批准号：
2005469
负责人：
Xuwen Chen
金额：
$ 21.05万
依托单位：
University of Rochester
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-15 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005469&HistoricalAwards=false
关键词：
Quantum Many Body Dynamics Energy

项目摘要

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.

该项目解决了量子多体动力学平均场极限的严格证明问题。多体系统作为物理系统中的基本模型自然产生。对这类系统的模拟只能通过一些近似，即所谓的平均场极限，因为多体系统可能包含大量的粒子。因此，平均场极限的数学证明，从它们应该描述的多体系统开始，是一个具有根本科学意义的问题。这位研究人员将解决玻色-爱因斯坦凝聚(BEC)研究中出现的问题，BEC是一种被冷却到非常接近绝对零度的玻色子稀薄气体的物质状态。在BEC中，很大一部分玻色子占据相同的量子态，在这一点上，量子效应在宏观尺度上变得明显。自从1995年Cornell和Ketterle(因他们的发现而获得诺贝尔奖)首次发现BEC以来，对这种新的物质状态的研究已经成为当代研究中最活跃的领域之一。本研究的重点是研究与含时多体薛定谔方程解的精细性质有关的几个问题，当粒子数在能量临界水平趋于无穷大时，在极限情况下。这个项目包括三个大方向。第一个方向旨在证明能量临界的非线性薛定谔方程是在重要的三维五次情形下在Gross-Pitaevskii标度下量子多体动力学的平均场极限。第二个方向集中于具有聚焦相互作用的多体薛定谔方程解的时空正则性。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和合作者将使用调和分析、概率和频谱理论的技术来分析这些问题。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。