CAREER: Analysis of quantum many body systems

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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 of importance both from a physical and a mathematical point of view. In mathematical physics, there has been substantial progress in the last few years in understanding some of 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. Among the questions that are addressed in this project are bounds on the free energy of quantum gases at low density and low temperature, as well as qualitative and quantitative statements about the corresponding thermal equilibrium states. The systems to be considered include both homogeneous and trapped systems, either continuous or on a lattice. The questions of interest concern, e.g., Bose/Fermi mixtures, low dimensional systems, rapidly rotating gases, as well as superfluidity for lattice systems. Moreover, the Bardeen-Cooper-Schrieffer theory of fermion pairing will be investigated, with the goal of further increasing the understanding of the low temperature behavior of fermionic systems with general interactions. Broader Impact. 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. These methods will be used in physics graduate courses of the P.I. and others. The organization of mathematical physics conferences and schools will disseminate the use of these powerful methods.