Mathematics of Many-Body Quantum Systems

• Applicant: Professor Dr. Phan Thanh Nam
• Subject Area: Mathematics
• Term: from 2019 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 426365943

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.

• DFG Programme: Research Grants
• International Connection: Poland
• Partner Organisation: Narodowe Centrum Nauki (NCN)
• Cooperation Partners: Professor Dr. Jan Derezinski; Dr. Marcin Napiórkowski