Mathematical Analysis of Interacting Quantum Gases

该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。它们展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性，这使得它们从物理和数学的角度都很有趣。从数学物理的角度来看，在过去的几年里，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的进展，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，必须为此目的开发新的数学思想和方法。沿着这些路线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。 在这个项目中解决的问题是在低密度和低温下量子气体的自由能的界限，以及有关相应的热平衡态的定性和定量陈述。要考虑的系统包括齐次和陷波系统，无论是连续的或在一个格子上。关注的问题，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性。此外，我们还将研究带电粒子气体的jeldom模型，目的是进一步加深对具有库仑相互作用的带电系统的理解。该项目的目标是开发新的计算工具，用于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。这些方法将是有用的，并将用于物理研究生课程。等人

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 interesting both from a physical and a mathematical point of view.Intellectual Merit. From the point of view of mathematical physics, there has been substantial progress in the last few years in understanding some 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 thermalequilibrium 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 jellium model for gases of charged particles will be investigated, with the goal of further increasing the understanding of charged systems with Coulomb interaction.Broader Impact. The goal of this project is the development of new mathematicaltools 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 useful, and will be used in physics graduate courses of the P.I. and others.