Source: National Natural Science Foundation of China

Grant No: 11322104 

Project: Variational Problems Arising in Quantum Many-Body Systems

Chief Specialist: Prof. Yujin Guo

Introduction: The theory of quantum many-body systems turns out to be an effective theoretic structure and solvable approach of understanding the collective behavior in

interacting many-particle systems. Over the past few decades, the theory of quantum many-body systems is applied widely to studying physical particle systems, such as Bose gas, Fermi gas and so on. There appear a lot of challenging and difficult mathematical problems in the studies of these physical systems, including many interesting and new variational problems. This project is focussed on the following three types of typical variational problems arising in quantum many-body systems: (1) We analyze the functional analysis of attractive Bose-Einstein Condensates (BEC), and study the existence, symmetry-breaking as well as dynamical stability of constraint minimizers. (2) Variational problems, including the existence and collapse of constraint minimizers, and the existence of the vortices in constraint minimizers, of rotating BEC with attractive interactions are investigated. (3) We study constraint minimizers of the Hartree-Fock-Bogoliubov energy functional, and discuss the existence of the pairing, the existence and uniqueness of constraint minimizers. Investigating deeply these variational problems cannot only contribute to deepening our understanding of quantum many-body systems theory, but also contribute to expanding the application ranges of modern nonlinear functional analysis.