The Quantum Mechanical Many-body Problem and Statistical Mechanics

本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。根本的统一性是，在不同问题领域的解决办法相互启发。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.低密度玻色气体现在处于实验和理论研究的最前沿，它旨在继续以前在基态能量方面的成功工作，以及玻色-爱因斯坦凝聚及其与超流性的关系问题。具体的问题是Bogolubov第二项在基态能量的扩展和旋转气体的yraast线的计算的验证。2.在哈伯德模型的相关电子的磁化和长程有序的猜测将进行调查。凝聚态物理学的其他目标是理解极化子的束缚，并建立可以严格证明具有条纹态的模型。3.物质的一个基本的热力学-力学性质是自由能的热力学极限的存在。到目前为止，当考虑电磁辐射场时，这一点的证明只是部分完成。它旨在完成演示。4. Lieb-Thirring不等式在许多物理和数学问题中是重要的。PI打算锐化常数并将不等式扩展到具有粒子背景密度而不是真空的新域。5.大自然似乎在告诉我们一个关于费米子的普遍事实，我们希望从第一原理来理解这个事实。6.原子和分子的密度泛函理论具有重要的理论意义和实用价值。一些公开的问题，穆勒理论将进行调查，以及可能的改进，利布-牛津界的交换相关能量。7.在该领域以前的专业知识的基础上，将追求越来越重要的量子信息理论领域的主题。更广泛的影响：学生和博士后将参与这些项目，这些项目将联合收割机指导与研究相结合，并帮助培养下一代数学知识丰富的物理学家。它还将进一步加强数学家和物理学家社区之间的跨学科联系，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.