Asymptotic Expansions for the Weakly Interacting Bose Gas

• Applicant: Professor Dr. Sören Petrat
• Subject Area: Mathematics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 512258249

In this proposal, we study pair-interacting Bose gases in non-relativistic quantum mechanics from a mathematical physics perspective. In particular, we aim at applying and proving asymptotic expansions with the inverse particle number as a small parameter; these have recently been developed by the applicant and co-authors. Concretely, we consider the time-dependent and time-independent Schroedinger equations and study series expansions of the solutions, i.e., of the time-evolved wave function, and of the low-lying eigenvalues and eigenfunctions. In the first part of the proposed project, we aim at applying these expansions for the mean-field limit in order to prove an Edgeworth-type expansion for bounded operators, to study the interaction energy of quasi-particles, and to prove an expansion for the binding energy. In the second part of the proposal, we aim at proving an expansion for the mean-field dynamics with Coulomb interaction, and for the dynamics with a singular scaling limit. The first project part establishes connections to probability theory and physics, and demonstrates the usefulness of the expansions. The second project part aims at developing the new technique further in physically interesting settings.

• DFG Programme: Research Grants