Mathematical Analysis of Interacting Bose Gases

R·塞林格的《相互作用玻色气体的数学分析》这个项目的主要焦点是作者在相互作用玻色气体低温性质的数学分析方面的继续研究，特别是对玻色-爱因斯坦凝聚现象的关注。由于最近在处理稀薄玻色气体方面的实验突破，导致了2001年的诺贝尔奖，在过去的几年里，这一领域取得了很大进展。在一定程度上，通过与E·H·利布和J·杨格瓦森的合作，可以从基本的薛定谔方程出发，严格证明玻色-爱因斯坦凝聚、超流和其他现象的存在，比如高度拉长的陷阱中囚禁气体的一维行为。在开发必要的数学工具的帮助下，有可能对这些现象有相当大的物理洞察力。然而，有许多未解决的问题计划在此项目中解决。其中包括无限(与囚禁相反)系统的玻色-爱因斯坦凝聚的证明，旋转系统和涡旋的行为，玻色和费米气体的混合等等。这些问题从数学和物理的角度都很有趣，具有复杂的性质，带来了新的数学思想，并且与目前实验中首次研究的系统的性质密切相关，有望在不久的将来产生更令人着迷的结果。在对这些现象进行数学分析方面的进展肯定会进一步洞察在极低温度下相互作用的玻色子系统中正在进行的复杂物理。作者想在这个项目中研究的其他领域包括非相对论量子电动力学的Pauli-Fierz模型，关于正矩阵迹的Bessis-Moussa-Villani猜想，以及经典的夸克禁闭模型。

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.