Mathematical studies in quantum dynamics

非相对论性量子物理的演化方程是在世纪上半叶由薛定谔、海森堡、狄拉克、冯·诺依曼、泡利等人的诺贝尔获奖著作中建立起来的。在过去的20年里，这些方程的数学理论经历了革命性的发展。与此同时，部分由于电子和通信设备的小型化，基础物理现象的新的强大应用已经出现。突然间，物理学和工程学文献中出现了数千篇论文，将这些方程应用于传统的“经典”通信，电子和计算领域。这一发展为量子光学、量子信息理论和半导体理论等成熟领域注入了新的活力，并产生了量子器件和量子计算等新领域。 在我们的项目中，我们将把非相对论量子物理演化方程的数学研究引向一个新的方向，处理微观多体系统形成的相干量子宏观和纳米结构的动力学，以及我们将继续我们最近对非相对论粒子系统与量子化电磁场相互作用的动力学的研究。虽然这个项目的主旨是纯理论的，但人们希望对量子动力学更好的数学理解将有助于上述应用领域。

The evolution equations of nonrelativistic quantum physics were formulated in the first part of the 20th century in Nobel prize winning works of Schroedinger, Heisenberg, Dirac, von Neumann, Pauli and others. In the last 20 years mathematical theory of these equations has undergone revolutionary development. At the same time, partly forced by miniturization of electronic and communication devices,  new powerful applications of the underlying physical phenomena have arisen. Suddenly, thousands of papers have appeared in Physics and Engineering literature applying these equations in traditionally 'classical' areas of communication, electronics and computing. This development has breathed new life into the well established fields of quantum optics, quantum information theory and theory of semiconductors and has given rise to the new fields of quantum devices and quantum computing.    In our project we will take our mathematical study of evolution equations of nonrelativistic quantum physics in a new direction dealing with dynamics of coherent quantum macroscopic and nanoscopic structures formed by microscopic many-body systems, as well as we will continue our recent investigations into dynamics of nonrelativistic particle systems interacting with the quantized electromagnetic field. Though the main thrust of this project is purely theoretical, one hopes that better mathematical understanding of quantum dynamics will contribute to the applied fields mentioned above.