Dynamics of complex quantum systems with randomness and nonlinearities

该项目涉及分析、应用数学和数学物理边界的研究问题。在项目的第一部分，我们在非相对论量子电动力学（QED）的框架下研究随机介质（弱无序安德森模型）中单个电子的性质。在第二部分中，我们讨论了弱随机势（描述半导体等材料）中电子气体的动力学，其中电子之间的相互作用在动态Hartree-Fock理论中建模。在项目的第三部分，我们研究了Gross-Pitaevskii层次的柯西问题，这是一个描述玻色粒子相互作用气体的多体平均场理论。研究了一般形式解的动力学性质，并将其与因式分解解的非线性薛定谔方程所预测的动力学性质进行了比较。非相对论量子电动力学描述了电子、原子和分子以普通速度运动并与光的能量量子（光子）相互作用。它是描述分子物理和量子化学过程的基本理论。在这个研究项目中，我们使用非相对论性量子电动力学的框架来研究电子在随机介质（例如，包括杂质的材料）中暴露于光或晶格振动时的运动。此外，我们还研究了电子之间的相互作用对随机介质中电子气体动力学的影响。这些问题的分析对于理解半导体的电学性质是至关重要的。这个项目的另一个主要部分是研究在超流体现象中起作用的玻色粒子系统的动力学。在这项工作中，我们研究了玻色气体的非相对论量子电动力学描述与其他模型的预测相匹配的程度。

This project addresses research problems at the boundary of analysis, applied mathematics, and mathematical physics. In a first part of the project, we study properties of a single electron in a random medium (weakly disordered Anderson model) in the framework of nonrelativistic Quantum Electrodynamics (QED). In a second part, we address the dynamics of a gas of electrons in a weak random potential (describing materials such as semiconductors), where the interactions between the electrons are modeled in dynamical Hartree-Fock theory. In a third part of the project, we investigate the Cauchy problem for the Gross-Pitaevskii hierarchy, which is a many body mean-field theory describing a gas of interacting Bose particles. We study dynamical properties of solutions of general form, and compare them to dynamical properties as predicted by the nonlinear Schrodinger equation obtained for solutions of factorized type.Non-relativistic quantum electrodynamics describes electrons, atoms, and molecules moving at ordinary speeds and interacting with the energy quanta of light (photons). It is the fundamental theory for the description of processes in molecular physics and quantum chemistry. In this research project, we use the framework of non-relativistic quantum electrodynamics to study the motion of an electron in a random medium (e.g., materials including impurities) when exposed to light or lattice vibrations. Moreover, we study the effect of the interaction between electrons on the dynamics of an electron gas in a random medium. The analysis of these questions is crucial for the understanding of electric properties of semiconductors. Another main part of this project investigates the dynamics of systems of many Bose particles, which play a role in the phenomenon of superfluidity. In this work, we investigate the question of how well the non-relativistic quantum electrodynamic description of a Bose gas matches predictions of other models.