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Mathematics of Many-Body Quantum Systems

Subject Area Mathematics
Term from 2019 to 2023
Project identifier Deutsche Forschungsgemeinschaft (DFG) - Project number 426365943
 
Final Report Year 2023

Final Report Abstract

The general objective of the project is to obtain the rigorous understanding of large quantum systems from first principles, namely from the Schrödinger equation. As the complexity of the many-body Schrödinger equation escalates with the increasing number of particles, the analysis of large systems usually requires the creation of various mathematical tools in order to derive suitable approximate theories in certain regimes. Motivated by some concrete problems in quantum physics, we have developed a variety of mathematical methods encompassing the theory of operators on Hilbert spaces, functional analysis, spectral theory, and partial differential equations, which are hopefully helpful to gain new insights in both mathematics and physics. The main focus of the project is the mathematical treatment of interacting Bose gases, where we aim at constructing a unified theory to systematically justify Hartree, Gross–Pitaevskii and Bogoliubov approximations in various models. We are interested in some critical cases where the particle correlation plays a prominent role, leading to subtle corrections to these approximations. The common challenge of the project is that to capture the correlation between particles to a high precision, we need to find suitable methods to implement Bogoliubov’s idea of describing the excitation spectrum. This research line is originated by the statistic problem, namely the ground state and Gibbs state of the relevant systems, but the techniques developed are often helpful for the dynamical problem as well. The main part of the project’s findings concerns the macroscopic properties of interacting Bose gases in the mean-field and Gross–Pitaevskii regimes. In particular, some of the findings are devoted to the study of the Hartree dynamics of the Bose–Einstein condensate and its derivation from the Schrödinger equation, while others are concerned with Bogoliubov’s description for the ground state and Gibbs states at positive temperatures. We expect that these mathematical findings will be helpful for further investigation, especially the understanding of symmetry breaking in the thermodynamic limit. Another part of the project’s findings concerns the correlation energy of interacting Fermi gases. It turns out that the knowledge of the bosonic Bogoliubov theory is very helpful to compute the correlation energy of interacting Fermi gases via bosonization methods. Although this concept was introduced in the physics literature in the 1950s through pioneering works by Bohm and Pines, the mathematical exploration of this idea has largely remained an ongoing inquiry. We have introduced a general mathematical formulation of the random phase approximation and apply it to understand the correlation of Fermi gases in the meanfield regime. In the long term, our method may pave the way towards a proof of the famous Gell-Mann–Brueckner formula on Jellium’s correlation energy. The project also results in various functional inequalities with singular potentials which are of general interest from the mathematical perspective and have potential applications to physical systems with critical singularities.

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