Structure of the Excitation Spectrum for Many-Body Quantum Systems

The main focus of this research project is the mathematical analysis of many-body quantum systems, in particular, interacting quantum gases at low temperature. The recent experimental advances in studying ultra-cold atomic gases have led to renewed interest in these systems. They display a rich variety of quantum phenomena, including, e.g., Bose-Einstein condensation and superfluidity, which makes them interesting both from a physical and a mathematical point of view. The goal of this project is the development of new mathematical tools for dealing with complex problems in many-body quantum systems. New mathematical methods lead to different points of view and can thus increase the understanding of physical systems. From the point of view of mathematical physics, there has been substantial progress in the last few years in understanding some of the interesting phenomena occurring in quantum gases, and the goal of this project is to further investigate some of the relevant issues. Due to the complex nature of the problems, new mathematical ideas and methods will have to be developed for this purpose. Progress along these lines can be expected to yield valuable insight into the complex behavior of many-body quantum systems at low temperature. The main question addressed in this research proposal is the validity of the Bogoliubov approximation for the excitation spectrum of many-body quantum systems. While its accuracy has been successfully shown for the ground state energy of various models, its predictions concerning the excitation spectrum have so far only been verified in the Hartree limit, an extreme form of a mean-field limit where the interaction ranges over the whole system size. Among the questions that are addressed in this project are the extension of these results to the physically more relevant case of short-range interactions, to the case of rotating systems, and to the study of the structure of the excitation spectrum in the thermodynamic limit.

The results obtained in this research project all concern the mathematically rigorous analysis of many-body systems in quantum mechanics. One main focus lies on questions of stability of quantum systems with zero-range interactions. Quantum mechanics allows for an idealized description of inter-particle interactions that act point-like, i.e., have zero range. While the two-body problem for such point interactions is completely understood, the question of stability vs. collapse for systems containing three or more particles represents a hard and partly unsolved problem. In this project, this problem was solved for one special case, namely the one of a gas of fermionic particles interacting with one impurity particle via point interactions. It was shown that if the ratio of the mass of the impurity particle and the one of the fermions is larger than some critical value, such a system is stable irrespective of the number of particles involved. Stability here refers to the fact that the ground state energy, i.e., the lowest possible energy of the system, is bounded from below. As a consequence of these stability results, additional important properties of such systems, known as Tan relations in the physics literature and relating various physical quantities directly measurable in experiments, were also established in a mathematically rigorous fashion. A second key focus of this project concerns Bose gases and the phenomenon of Bose- Einstein condensation, a phase transition that occurs in these systems. Bose gases and Bose-Einstein condensates are currently of great interest due to the possibility of experimentally realizing such systems with cold atomic gases. An important theoretical contribution towards understanding such complicated quantum many-body systems is the 1947 work by Bogoliubov, in which he introduces an approximation scheme that leads to concrete predications of physical relevance, e.g., superfluidity of such systems. Investigating the accuracy and regime of validity of the approximation represents a hard mathematical problem, however. The goal of this research project was to investigate the validity of the Bogoliubov approximation for the excitation spectrum and dynamics of weakly interacting bosons. The results of this project contribute to our understanding of this question in an essential way. The validity of the Bogoliubov approximation could be established for the dynamics of Bose-Einstein condensates with weak interactions, and a version of Bogoliubov`s theory at non-zero temperature was used to predict the effect of weak interactions on the transition temperature for Bose-Einstein condensation.