The Quantum Mechanical Many-body Problem and Statistical Mechanics

这项研究致力于量子力学、凝聚态物理、量子信息论和量子电动力学中的多体理论的各个方面，以及一些相关的数学问题。根本的统一是，不同问题领域的解决方案相互揭示。这一建议的学术价值包含在以下具体研究目标的部分清单中。1.低密度玻色气体目前在实验和理论上都处于研究的前沿，它意在继续以前关于基态能量以及玻色-爱因斯坦凝聚及其与超流的关系的成功工作。具体问题是Bogolubov第二项在基态能量展开式中的有效性和旋转气体的yrast线的计算。2.对关联电子Hubbard模型中有关磁化强度和长程有序的猜想进行了研究。凝聚态物理学的其他目标是理解极化子的结合，并建立可以严格证明具有条纹态的模型。3.物质的一个基本统计力学性质是自由能热力学极限的存在。到目前为止，当考虑到电磁辐射场时，这一点的证明只完成了一部分。它的目的是完成演示。4.Lieb-Thirring不等式在各种物理和数学问题中都很重要。PI打算锐化常数，并将不等式扩展到具有背景粒子密度而不是真空的新区域。5.将尝试验证一个长期存在的猜想，即一个大原子的最大负电离只有一个电子左右。大自然似乎在告诉我们一个关于费米子的普遍事实，人们希望从第一原理就能理解它。6.原子和分子的密度泛函理论在实践和概念上都具有重要意义。我们将研究一些关于穆勒理论的公开问题，以及对交换关联能量的Lieb-Oxford界限的可能的改进。7.在该领域以往专门知识的基础上，将继续探讨日益重要的量子信息理论领域的专题。更广泛的影响：学生和博士后将参与这些项目，这些项目将把指导与研究结合起来，帮助培养下一代具有数学知识的物理学家。它还将进一步加强数学家和物理学家群体之间的跨学科联系，并建立现代数学分析概念与凝聚态物理问题的相关性。

This research is devoted to various aspects of many-body theory in quantum mechanics, condensed matter physics, quantum information theory, and quantum electrodynamics, as well as some related mathematical problems. The underlying unity is that solutions in the various problem areas shed light on each other. The intellectual merit of this proposal is contained in the following partial list of specific research goals. 1. Low density Bose gases are now in the forefront of research experimentally and theoretically and it is intended to continue the previous successful work on the ground state energy, as well as the question of Bose-Einstein condensation and its relation to superfluidity. Specific questions are the validation of Bogolubovs second term in the expansion of the ground state energy and the calculation of the yrast line of a rotating gas. 2. Conjectures concerning magnetization and long-range order in the Hubbard model of correlated electrons will be investigated. Other goals in condensed matter physics are to understand the binding of polarons and to build models that can be rigorously demonstrated to have striped states. 3. A fundamental statistical-mechanical property of matter is the existence of the thermodynamic limit of the free energy. A proof of this has been only partially accomplished so far when the electromagnetic radiation field is taken into account. It is intended to complete the demonstration. 4. The Lieb-Thirring inequalities are important in a variety of physical and mathematical problems. The PI intends sharpening the constants and to extend the inequalities to a new domain with a background density of particles instead of a vacuum. 5. An attempt will be made to verify the long-standing conjecture that the maximum negative ionization of a large atom is only about one electron. Nature seems to be telling us about a universal fact about fermions that it would be desirable to understand from first principles. 6. Density functional theory for atoms and molecules is important practically and conceptually. Some open questions about the Muller theory will be investigated, as well as possible improvements to the Lieb-Oxford bound for the exchange-correlation energy. 7. Building on previous expertise in the area, topics in the increasingly important field of quantum information theory will be pursued. Broader impact: Students and postdocs will be involved in these projects, which will combine mentoring with research and help produce the next generation of mathematically knowledgeable physicists. It will also further the interdisciplinary bonds between the communities of mathematicians and physicists and establish the relevance of modern concepts of mathematical analysis to problems of condensed matter physics.