Mathematics of Many-Body Quantum Systems

该项目的目标是从数学角度严格理解多体量子物理中各种近似的有效性。根据量子力学的第一原理，量子系统的物理性质编码在薛定谔方程中。然而，随着系统中粒子数目的增加，多体薛定谔方程的复杂性急剧增加，因此，在实践中，依赖于更容易处理的近似理论是至关重要的。理解这些近似的有效性是数学物理的一项重要任务。在这个项目中，我们将重点研究玻色多体量子系统。在玻色气体相互作用的背景下，物理学文献中使用的主要有效理论被命名为Hartree、Gross-Pitaevskii和Bogoliubov。在过去的10年里，无论是从静态还是动态的角度来看，在严格证明这些近似的合理性方面都取得了实质性的进展。该项目的总体目标是在一些关键情况下证明这些近似是正确的，包括平均场/稀疏极限中的粒子关联、具有吸引相互作用的系统的稳定性/爆破行为、自发对称性破缺和相关的热力学性质。我们的工作将遵循数学推理的规则。结果将被表述为定理。我们将使用各种数学方法，包括希尔伯特空间上的算子理论、泛函分析和偏微分方程。在我们的分析过程中，我们将牢记激发我们数学研究的物理问题。我们相信，我们计划在项目中解决的问题属于数学物理的中心课题，积极的结果将导致更好地理解宏观玻色气体的一些关键性质。我们预计，我们的项目将导致各种数学技术的发展，这些技术将在该领域引发新的研究方向。潜在的双边合作将促进高级研究人员的合作，并通过定期访问、研讨会和讲习班促进参与该项目的年轻研究人员(学生和博士后)的发展。

The objective of the project is the rigorous understanding of the validity of various approximations in many-body quantum physics from a mathematical perspective. From the first principles of quantum mechanics, the physical properties of a quantum system are encoded in the Schroedinger equation. However, the complexity of the many-body Schroedinger equation grows dramatically with the number of particles in the system and thus, in practice, it is crucial to rely on approximate theories which are easier to deal with. Understanding the validity of these approximations is an important task of mathematical physics. In this project we will focus on bosonic many-body quantum systems. In the context of interacting Bose gases, the main effective theories that are used in the physics literature go under the names of Hartree, Gross-Pitaevskii and Bogoliubov. There has been substantial progress in rigorously justifying these approximations over the last 10 years, both from a static and dynamical point of view. The general goal of the project is to justify these approximations in some critical cases, including the particle correlations in a mean field/dilute limit, the stability/blow-up behavior of systems with attractive interactions, spontaneous symmetry breaking and related thermodynamical properties. Our work will follow the rules of mathematical reasoning. The results will be formulated as theorems. We are going to use a variety of mathematical methods including theory of operators on Hilbert spaces, functional analysis, and partial differential equations. Throughout our analysis we will keep in mind the physical problems motivating our mathematical investigations.We believe that the questions that we plan to address in our project belong to central subjects of mathematical physics and that positive results will lead to a better understanding of some crucial properties of macroscopic Bose gases. We expect that our project will lead to the development of various mathematical techniques that will trigger new research directions in the field.The potential bilateral cooperation will facilitate the collaboration of the senior investigators as well as contribute to the development of younger researchers (students and post-docs) involved in the project through regular visits, seminars and workshops.