Dynamics of complex quantum systems with randomness and nonlinearities

该项目解决了分析，应用数学和数学物理边界的研究问题。 在第一部分中，我们在非相对论量子电动力学（QED）框架下研究了随机介质（弱无序安德森模型）中单电子的性质。在第二部分中，我们解决了电子气体在弱随机势（描述材料，如半导体），其中电子之间的相互作用在动态Hartree-Fock理论建模的动态。 在该项目的第三部分中，我们研究了Gross-Pitaevskii层次的柯西问题，这是一个描述相互作用玻色粒子气体的多体平均场理论。我们研究了一般形式的解的动力学性质，并将它们与非线性薛定谔方程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，我们使用非相对论量子电动力学的框架来研究电子在随机介质中的运动（例如，包括杂质的材料）。 此外，我们还研究了随机介质中电子间相互作用对电子气动力学的影响。 这些问题的分析对于理解半导体的电学性质至关重要。 该项目的另一个主要部分是研究许多玻色粒子系统的动力学，这些粒子在超流现象中发挥作用。 在这项工作中，我们调查的问题，以及非相对论量子电动力学描述的玻色气体匹配预测的其他模型。

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.