On Relativistic and Non-Relativistic Fermi Systems

近年来，在凝聚态物理中，对超冷原子气体的研究受到了极大的关注。在一些引人注目的实验中，实验者已经学会了将原子间的相互作用从很弱调整到很强，从而观察到气体的不同相。在我们的第一个项目中，我们打算研究弱耦合区域，它通常被称为BCS(Bardeen-Cooper-Schrieffer)区域。从数学上讲，这一机制可以用著名的BCS间隙方程来相当准确地描述。这个方程描述了所谓的库珀对的波函数。它是高度非线性的，尽管如此，在最近的一项工作中，我们能够将BCS间隙方程与线性伪微分算子的谱分析联系起来。这使得能够精确地描述引起超流的位势的类别。在进一步的项目中，我们计划研究能隙、动量分布的连续性、临界温度的高阶计算，以及两个自旋态不相等填充的系统，所有这些问题对应用都特别重要，例如不同类型的超流体的分类。我们的第二个项目是关于狄拉克算符所描述的相对论粒子的研究。在最近与Luin，Solovej和Sere的工作中，我们在平均场近似下建立了一个框架，允许我们通过最小化原理来获得基态，这与以前的结果有很大的不同。虽然这些方法最初是在量子电动力学的背景下发展起来的，但它们可以应用于非常任意的无限量子系统。我们以前的所有结果在零温度的情况下都成立。在不久的将来，我们计划将结果推广到任意正温度。我们关于狄拉克粒子的项目的第二部分与引力有关。首先，我们考虑了经典的牛顿力，研究了一个描述白矮星动力学崩塌的时间演化。其次，我们计划采用一种更具创新性的方法，将广义相对论纳入其中。更广泛的影响：目标是为超流体系统和狄拉克系统开发新的数学工具。这些方法导致了不同的观点，并增加了对底层物理系统的理解。例如，从长远来看，这些方法可以帮助理解高T_c超导体的机制。这些都有各种应用，如超高速计算机、磁悬浮、无损耗电网等。这项工作将包括多方面的合作努力，并为培养博士生做出贡献。

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.