Mathematical Analysis of the Dynamics of Complex Quantum Systems

通常，量子现象是在非常小的微观尺度上表现出来的，而在大的宏观尺度上，自然是由经典的牛顿力学很好地描述的。该项目的主要研究对象之一是玻色-爱因斯坦凝聚体（BEC），因为宏观量子现象变得明显。BEC是一种新的物质状态，最早由玻色和爱因斯坦于1924年在理论上预测，1995年由康奈尔和威曼在实验中产生。气体的粒子，在非常接近绝对零度时，占据最低的量子态；然后气体形成BEC。物质的这种状态具有非常不寻常的特性：例如，BEC中的光可以完全停止或显著减慢到每秒17米的速度。本项目采用数学分析、应用数学和数学物理等多种方法，从严格的数学角度研究重要量子系统的动力学特性。理论上要研究的现象之一是量子摩擦的出现，当粒子通过BEC时。这项工作的目的是促进对基于第一原理的物理现象的理解，从而为理论物理学提供严格的数学基础。玻色气体相互作用的数学研究是色散非线性偏微分方程与数学物理相结合的一个活跃的研究课题。BEC的平均场描述由非线性Schrödinger或非线性Hartree方程给出。在这个项目中，量子力学示踪粒子与BEC相互作用的动力学将被研究。这个模型中特别有趣的是量子摩擦的出现。此外，基于Hartree-Fock-Bogoliubov方程（由原始系统的准自由约化得到），将对BEC周围的热波动动力学进行研究。其他正在进行的项目集中在弱随机势中的电子动力学（描述诸如半导体之类的材料），以及热平衡附近无限多个费米子系统的动力学。作为一个新的研究方向，玻尔兹曼方程的适定性问题将在其维格纳变换表示下进行研究。这使得该模型可用于分析非线性Schrödinger方程和Gross-Pitaevskii层次结构的方法，例如密度矩阵的Strichartz估计。这些项目涉及与多位高级研究人员、博士后研究人员和研究生的合作。

Ordinarily, quantum phenomena are exhibited on very small micro-scales, while on large macro-scales nature is well described by classical, Newtonian mechanics. One of the principal subjects of investigation in this project is a Bose-Einstein condensate (BEC), for which macroscopic quantum phenomena become apparent. A BEC is a new state of matter that was first predicted theoretically by Bose and Einstein in 1924, and was produced experimentally in 1995 by Cornell and Wieman. Particles of a gas, cooled very close to absolute zero, occupy the lowest quantum state; then the gas forms a BEC. This state of matter has very unusual properties: for example, light in a BEC can be stopped entirely or slowed down significantly to the velocity of 17 meters per second. This project investigates dynamical properties of important quantum systems from a rigorous mathematical viewpoint, by using a wide variety of methods from mathematical analysis, applied mathematics, and mathematical physics. One of the phenomena to be studied theoretically is the emergence of quantum friction, when a particle passes through a BEC. The purpose of this work is to advance the understanding of physical phenomena based on first principles, whereby giving theoretical physics a rigorous mathematical foundation. The mathematical study of interacting Bose gases is an active research topic at the interface between dispersive nonlinear PDEs and Mathematical Physics. The mean-field description of a BEC is given by the nonlinear Schrödinger or nonlinear Hartree equations. In this project, the dynamics of a quantum mechanical tracer particle in interaction with a BEC will be investigated. Of particular interest in this model is the emergence of quantum friction. Furthermore, the dynamics of thermal fluctuations around a BEC will be examined based on the Hartree-Fock-Bogoliubov equation that is obtained from the quasifree reduction of the original system. Other continuing projects focus on the dynamics of electrons in a weak random potential (describing materials such as semiconductors), and dynamics of a system of infinitely many fermions in the vicinity of a thermal equilibrium. As a new research direction, the well-posedness problem for the Boltzmann equation will be studied in its Wigner-transformed representation. This makes the model accessible to methods developed for the analysis of nonlinear Schrödinger equations and Gross-Pitaevskii hierarchies, such as Strichartz estimates for density matrices. These projects involve collaborations with various leading senior researchers, with a postdoctoral researcher, and with graduate students.