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，自然被经典的牛顿力学很好地描述了。这个项目的主要研究对象之一是玻色-爱因斯坦凝聚体(BEC)，对它来说，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在1924年从理论上预测，并于1995年由康奈尔和威曼通过实验产生。冷却到接近绝对零度的气体粒子占据最低量子态，然后气体形成BEC。这种物质状态具有非常不寻常的性质：例如，BEC中的光可以被完全阻止，或者显著减慢到每秒17米的速度。这个项目从严格的数学观点出发，通过使用来自数学分析、应用数学和数学物理的各种方法，研究重要量子系统的动力学性质。需要从理论上研究的现象之一是，当粒子通过BEC时，会出现量子摩擦。这项工作的目的是在第一原理的基础上推进对物理现象的理解，从而为理论物理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组与数学物理交界处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Hartree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。这个模型特别令人感兴趣的是量子摩擦的出现。此外，BEC周围的热涨落动力学将基于Hartree-Fock-Bogoliubov方程，该方程是从原始系统的准自由约化得到的。其他正在进行的项目侧重于弱随机势中电子的动力学(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，Boltzmann方程的适定性问题将在其Wigner变换下得到研究。这使得该模型可用于分析非线性薛定谔方程和Gross-Pitaevskii族的方法，例如密度矩阵的Strichartz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.