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.
The following results have been obtained: 1) In joint work with R. Frank, E. Lieb, and L. Thomas, we have shown that there is a critical value of the polaron coupling constant for binding to occur. The critical value is bounded by a fixed constant times the fine-structure constant. Moreover, we show that the same holds true for an arbitrary number of electrons. If the polaron coupling is two small, binding does not occur, and the ground state energy of N electrons is just N times the one for one electron. 2) In joint work with R. Frank, C. Hainzl and J.P. Solovej, we were able to show rigorously how the Ginzburg-Landau equation arises from BCS theory in a certain limit. This limit concerns temperatures which are very close to the critical one, and system sizes which are proportional to the square root of the inverse of the the difference of the temperature to the critical one. The proof uses many tools from semiclassical analysis. 3) Lieb-Thirring inequalities at non-zero density and temperature were derived in joint work with R. Frank, M. Lewin and E. Lieb. The inequalities state that the minimal energy required to make a local change to the density of a non-interacting Fermi gas can be bounded from below by a uniform constant times to corresponding semiclassical expression.
