On Relativistic and Non-Relativistic Fermi Systems

点击翻译按钮获取中文摘要

In recent years, in condensed matter physics, significant attention has been devoted to the study of ultra-cold atomic gases. In some remarkable experiments, the experimentalists have learned to tune the inter-atomic interaction from very weak to very strong, observing thereby different phases of the gas. Within our first project we intend to study the weak-coupling regime, which is usually called the BCS (Bardeen-Cooper-Schrieffer)-regime. Mathematically, this regime can be quite accurately described by the famous BCS gap-equation. This equation describes the wavefunction of, so called, Cooper pairs. It is highly non-linear, nonetheless, in a recent work we were able to link the BCS gap-equation to the spectral analysis of a linear, pseudo-differential, operator. This allowed a precise characterization of the class of potentials giving rise to superfluidity. In further projects we plan to study the energy gap, continuity properties of the momentum distribution, and higher order calculations of the critical temperature, as well as systems with where the two spin states are unequally populated, all questions that are particularly important for applications, such as the classification of different types of superfluids. Our second project concerns the study of relativistic particles, described by Dirac's operator. In recent works with Lewin, Solovej and Sere we established a framework within mean-field approximation which allows us to obtain ground states via a minimization principle, which is a major departure from prior results. Although originally developed in the context of QED these methods can be applied to quite arbitrary infinite quantum systems. All our previous results hold in the case of zero temperature. In near future we plan to extendour results to arbitrary positive temperature. The second part of our project on Dirac particles is associated with gravitational forces. First, we consider classical Newtonian forces and study a time-evolution which is supposed to describe the dynamical collapse of white dwarfs. Second, we plan a more innovative approach, incorporating general relativity. Broader Impact: The goal is to develop new mathematical tools for superfluid systems and Dirac systems. Such methods lead to different points of views and increase the understanding of the underlying physical systems. For example, in the long run, such methods could help to understand the mechanism of high T_c-superconductors. These have all kind of applications such as ultrafast computers, magnetic levitation, loss-less powergrid, etc. The work will include multiple collaborative efforts and contribute to the training of PhD students.