Dynamical Properties of Quantum Systems with Infinitely Many Degrees of Freedom

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。本研究计划的目标是获得有关具有无限多个自由度的量子系统的动力学性质的数学结果。更具体地说，我们的兴趣将集中在两个主要领域。1)非相对论量子力学点状原子核和电子与量子化辐射场相互作用系统的光谱学。由于量子光学的应用越来越广泛，物理学家对实验情况越来越感兴趣，在实验情况下，理论解释必须超越常规的QED（量子电动力学）微扰理论。这就要求对非相对论性QED的数学理论进行更精细的分析，以描述带电量子物质和与这些现象相关的低能尺度上的量子化辐射的相互作用。我们在这个项目中的主要目标是改善对亚稳态动力学的数学控制，其中需要进行非摄动分析。为此，我们计划在开放量子系统的pi 2)不可逆过程和输运方程的贡献下，推进近年来发展起来的光谱、散射和展开方法的分析。输运理论与多体系统的研究密切相关，对于多体系统，详细的动力学控制是超出解析和数值方法的。因此，我们不得不处理粗粒度物理量的有效方程，但有人想从潜在的量子动力学中推导出它们。这是一个具有数学挑战性的程序，它将一些基本的物理概念与数学证明中表达的严格结果联系起来，这些数学证明至少在具有非平凡微动力学的简单模型中成立。我们将扩展我们最近在量子扩散方面取得的一些初步结果。本研究项目涉及的概念和技术连接了分析和数学物理的不同分支。我们的总体策略是从不同的角度来解决数学物理问题，因为我们相信坚实的物理直觉可以在不同的数学背景中建立联系。相反，我们希望从解决物理问题开始，在数学结构和技术方面获得更多的洞察力。这些问题将为数学和物理专业的学生提供博士研究项目。这些项目对那些将从事数学物理学术研究的学生，以及那些将应用分析和概率工具来模拟物理应用中的复杂系统的学生来说，都是一次有价值的训练。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The goal of this research program is to obtain mathematical results concerning the dynamical properties of quantum systems with infinitely many degrees of freedom. More specifically, our interest will be focused on two main areas.1) Spectroscopy of systems of non-relativistic, quantum mechanical point-like nuclei and electrons interacting with the quantized radiation field. Because of the growing applications of quantum optics, physicists have become increasingly interested in experimental situations where the theoretical explanation must go beyond regular QED (quantum electrodynamics) perturbation theory. This calls for a more refined analysis of the mathematical theory of non-relativistic QED that describes the interaction of charged quantum matter and the quantized radiation at low energy scales related to those phenomena. Our main goal within this project is to improve the mathematical control on the dynamics of metastable states where a non-perturbative analysis is necessary. To this end, we plan to push forward the analysis of spectral, scattering, and expansion methods developed in recent years also with the contribution of the PI.2) Irreversible processes and transport equations in open quantum systems. Transport theory is intimately related to the study of many-body systems, for which a detailed control of the dynamics is beyond both analytic and numerical methods. Therefore we are forced to deal with effective equations for coarse-grained physical quantities, but one one would like to derive them from the underlying quantum dynamics. This is a mathematically challenging program that links some basic physical concepts to rigorous results expressed in mathematical proofs which hold, at least, in simple models with a non-trivial micro-dynamics. We will extend some first results that we have recently attained regarding quantum diffusion.The concepts and techniques involved in this research project connect different branches of analysis and mathematical physics. Our general strategy is to address mathematical physics problems from different perspectives, because we are confident that solid physical intuition can make links across different mathematical contexts. Conversely we want to gain more insight in the mathematical structures and techniques starting from the solution of physics problems. These problems will provide doctoral research projects for students in mathematics and in physics. The projects represent a valuable training for both students who will pursue academic research in mathematical physics as well as students who will apply tools from analysis and probability to model complex systems in physical applications.