Mathematical Studies in Quantum Dynamics of Matter and Radiation

在地球上常见的能量尺度上，物质的基石是电子和原子核，相互作用由量子电磁场和库仑力调节。整个系统的动力学--粒子和场--由一个定义明确的演化方程--薛定谔方程给出。构建描述这一动态的数学工具是我们未来几年研究的主要方向。 我们将讨论广泛的问题，从单个电子的性质，或者更准确地说，被诱导的光子云包围的电子的性质，原子和分子对辐射的发射和吸收，到形成相干的宏观结构(比如，像玻色-爱因斯坦凝聚体或像固体晶体这样的扩展结构)的原子的大集合的行为，或者更广泛地说，一些具有量子化的集体振荡(机械或弹性波)的连续介质，称为声子。我们感兴趣的过程是不可逆的，伴随着无限的信息丢失。 上述问题是我们所经历的物质的结构和性质的基础。毫不奇怪，他们在物理和化学领域获得了不成比例的诺贝尔奖。然而，处理这些问题的数学仪器只是处于建造的初级阶段，特别是与过去几个世纪为解决经典物理问题而发展的数学技术相比。 然而，这一步伐正在迅速加快。我们预计，在未来几年内，这个项目中出现的那种类型的演化方程的分析以及相应的纳米、细观和宏观近似将发展成为一个相当大的纯数学和应用数学领域。 我相信，与我的合作者(包括我过去和现在的研究生和博士后)一起，我将能够为这一发展做出实质性的贡献。

At energy scales common on Earth, the building blocks of matter are electrons and nuclei, with interactions mediating by quantum electro-magnetic field, as well as Coulomb force. The dynamic of the total system - particles and fields - is given by a well-defined evolution equation - the Schroedinger equation. Constructing the mathematical apparatus describing this dynamic is the main thrust of our research in the next several years. We will address a wide gamut of issues, from properties of a single electron, or more precisely the electron surrounded by an induced cloud of photons, to emission and absorption of radiation by atoms and molecules, to behaviour of large aggregates of atoms forming coherent, macroscopic structures (say, localized structures like Bose-Einstein condensates or extended ones like solid crystals), or more generally some continuous medium, with quantized collective oscillations (mechanical or elastic waves), called phonons. The processes we are interested in are "irreversible", accompanied by information loss at infinity. The issues described above underlie the structure and properties of matter as we experience it. Not surprisingly, they have garnered a disproportionate number of Nobel prizes in Physics and Chemistry. Nevertheless the mathematical apparatus dealing with them is only in the initial stages of construction, especially compared with the mathematical techniques developed in the last few centuries to tackle problems of classical physics. However the pace is quickly picking up. We expect that in the next few years, the analysis of evolution equations of the type appearing in this project and the corresponding nano-, meso- and macroscopic approximations will grow into a sizable area of pure and applied mathematics. I am confident that jointly with my collaborators (including my past and present graduate students and postodcs) I will be able to make substantial contributions to this development.