CAREER: Analysis of quantum many body systems

本研究项目的主要重点是多体量子系统的数学分析，特别是在低温下相互作用的量子气体。最近在研究超冷原子气体方面的实验进展使人们对这些系统重新产生了兴趣。它们显示了丰富多样的量子现象，包括，例如，玻色-爱因斯坦凝聚和超流体，这使得它们从物理和数学的角度来看都很重要。在数学物理方面，过去几年在理解量子气体中发生的一些有趣现象方面取得了实质性进展，本项目的目标是进一步研究一些相关问题。由于问题的复杂性，必须为此目的发展新的数学思想和方法。沿着这些方向的进展有望对低温下多体量子系统的复杂行为产生有价值的见解。本课题研究的问题包括量子气体在低密度和低温下的自由能边界，以及相应的热平衡态的定性和定量表述。要考虑的系统包括齐次系统和捕获系统，或者是连续的，或者是在晶格上的。感兴趣的问题涉及，例如，玻色/费米混合物，低维系统，快速旋转的气体，以及晶格系统的超流动性。此外，还将研究费米子对的Bardeen-Cooper-Schrieffer理论，以进一步提高对具有一般相互作用的费米子系统的低温行为的理解。更广泛的影响。该项目的目标是开发新的数学工具来处理多体量子系统中的复杂问题。新的数学方法导致不同的观点，从而可以增加对物理系统的理解。这些方法将被用于P.I.的物理研究生课程和其他课程。数学物理会议和学校的组织将传播这些强大方法的使用。

The main focus of this research project is the mathematical analysis of many-body quantum systems, in particular, interacting quantum gases at low temperature. The recent experimental advances in studying ultra-cold atomic gases have led to renewed interest in these systems. They display a rich variety of quantum phenomena, including, e.g., Bose-Einstein condensation and superfluidity, which makes them of importance both from a physical and a mathematical point of view. In mathematical physics, there has been substantial progress in the last few years in understanding some of interesting phenomena occurring in quantum gases, and the goal of this project is to further investigate some of the relevant issues. Due to the complex nature of the problems, new mathematical ideas and methods will have to be developed for this purpose. Progress along these lines can be expected to yield valuable insight into the complex behavior of many-body quantum systems at low temperature. Among the questions that are addressed in this project are bounds on the free energy of quantum gases at low density and low temperature, as well as qualitative and quantitative statements about the corresponding thermal equilibrium states. The systems to be considered include both homogeneous and trapped systems, either continuous or on a lattice. The questions of interest concern, e.g., Bose/Fermi mixtures, low dimensional systems, rapidly rotating gases, as well as superfluidity for lattice systems. Moreover, the Bardeen-Cooper-Schrieffer theory of fermion pairing will be investigated, with the goal of further increasing the understanding of the low temperature behavior of fermionic systems with general interactions. Broader Impact. The goal of this project is the development of new mathematical tools for dealing with complex problems in many-body quantum systems. New mathematical methods lead to different points of view and can thus increase the understanding of physical systems. These methods will be used in physics graduate courses of the P.I. and others. The organization of mathematical physics conferences and schools will disseminate the use of these powerful methods.