Mathematical Analysis of Interacting Bose Gases

这个项目的主要重点是作者在相互作用玻色气体的低温性质的数学分析方面的研究的延续，特别关注玻色-爱因斯坦凝聚现象。受最近在稀释玻色气体处理方面的实验突破的推动，这一突破在2001年获得了诺贝尔奖，在过去的几年里，这一领域取得了很多进展。在与E.H. Lieb和J. Yngvason的部分合作中，从基本的薛定谔方程开始，有可能严格证明玻色-爱因斯坦凝聚、超流动性和其他现象的存在，比如在高度细长的陷阱中被困气体的一维行为。在发展必要的数学工具的帮助下，有可能对这些现象获得相当大的物理洞察力。然而，计划在这个项目中解决许多尚未解决的问题。其中包括无限大（相对于被困）系统的玻色-爱因斯坦凝聚的证明、旋转系统和漩涡的行为、玻色和费米气体的混合等。从数学和物理的角度来看，这些问题都是有趣的，它们具有复杂的性质，需要新的数学思想，并且与当前实验中首次研究的系统特性密切相关，这些特性有望在不久的将来产生进一步的迷人结果。对这些现象的数学分析的进展肯定会使我们对极低温下相互作用的玻色子系统中发生的复杂物理现象有更深入的了解。在这个项目中，作者想要研究的其他研究领域包括非相对论量子电动力学的Pauli-Fierz模型，关于正矩阵轨迹的Bessis-Moussa-Villani猜想，以及夸克约束的经典模型。

Mathematical Analysis of Interacting Bose Gases" by R. Seiringer The main focus of this project is the continuation of the author's research in the mathematical analysis of the low-temperature properties of an interacting Bose gas, with special attention on the phenomenon of Bose-Einstein condensation. Motivated by recent experimental breakthroughs in the treatment of dilute Bose gases, which led to Nobel prize awards in 2001, there has been a lot of progress in this field in the last few years. Partly in joint work with E.H. Lieb and J. Yngvason it was possible to rigorously prove the existence of Bose-Einstein condensation, superfluidity and other phenomena like one-dimensional behavior of trapped gases in highly elongated traps, starting from the basic Schroedinger equation. With the help of developing the necessary mathematical tools it was possible to gain considerable physical insight into these phenomena. There are a lot of open problems, however, that are planned to be addressed within this project. Among them are the proof of Bose-Einstein condensation for infinite (in contrast to trapped) systems, behavior of rotating systems and vortices, mixtures of Bose and Fermi gases, etc. These problems are interesting both from a mathematical and physical point of view, being of a complex nature that brings about the need for new mathematical ideas, and being closely related to properties of systems studied for the first time in current experiments which can be expected to yield further fascinating results within the near future. Progress in the mathematical analysis of these phenomena will certainly yield further insight into the complex physics that is going on in interacting bosonic systems at very low temperature. Other fields of research that the author would like to study within this project include the Pauli-Fierz model of non-relativistic Quantum Electro-dynamics, the Bessis-Moussa-Villani conjecture about traces of positive matrices, and classical models of Quark confinement.