Mathematical Analysis of the Dynamics of Complex Quantum Systems

通常，量子现象表现在非常小的微观尺度上，而在大的宏观尺度上，自然是很好地描述了经典，牛顿力学。该项目的主要研究对象之一是玻色-爱因斯坦凝聚（BEC），宏观量子现象变得明显。BEC是一种新的物质状态，由玻色和爱因斯坦在1924年首次从理论上预测，并于1995年由康奈尔和威曼在实验上产生。气体的粒子，冷却到非常接近绝对零度，占据最低的量子态;然后气体形成BEC。这种物质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。本课题从严格的数学观点出发，运用数学分析、应用数学、数学物理等多种方法，研究重要量子系统的动力学性质。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。这项工作的目的是推进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。 相互作用玻色气体的数学研究是色散非线性偏微分方程和数学物理之间的一个活跃的研究课题。BEC的平均场描述由非线性薛定谔方程或非线性哈特里方程给出。在这个项目中，将研究量子力学示踪粒子与BEC相互作用的动力学。在这个模型中，特别令人感兴趣的是量子摩擦的出现。此外，BEC周围的热涨落的动力学将被检查的基础上的Hartree-Fock-Bogoliubov方程，这是从原始系统的准自由约化。其他持续的项目集中在弱随机势中的电子动力学（描述半导体等材料），以及热平衡附近无限多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Wigner变换表示下进行研究。这使得该模型可以用于分析非线性薛定谔方程和Gross-Pitaevskii层次结构的方法，例如密度矩阵的Eschenhartz估计。这些项目涉及与各种领先的高级研究人员，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.