Spectral theory of relativistic quantum Hamiltonians

量子力学的发展是数学和物理如何相互促进的一个很好的例子。这个成功故事的核心是自伴算符的谱理论--所谓的哈密顿算符--它决定了量子系统的能量和时间演化。这种综合努力的结果在工业上得到了大量的应用。然而，试图纳入相对论效应遇到了许多困难，无论是物理解释，以及数学分析。目前，物理、数学和应用之间存在知识鸿沟。这提供了一个独特的机会，将已经在应用中产生良好数值结果的想法置于严格的数学基础上，同时挑战物理学中公认的观点。这个项目将集中在相对论性哈密顿量的谱理论，特别是著名的狄拉克算子。特别重要的是由狄拉克算子的本征值给出的束缚态能量。即使对于一个由单个相对论粒子组成的系统，也有几个数学概念是非常不容易理解的。这在很大程度上是由于狄拉克算子缺少了一个重要的性质相比，它的非相对论对应：半有界性。在此属性的推广的最新进展的动机下，这个项目的目的将是扩展概念，如变分原理，本征值界和本征值渐近从非相对论的情况下狄拉克算子。

The development of quantum mechanics is a great example of how mathematics and physics can stimulate mutual progress. At the core of this success story stands the spectral theory of self-adjoint operators - so called Hamiltonians - which determine the energy and time evolution of quantum systems. The results of this combined effort have found a large number of applications in industry. However, attempts to incorporate relativistic effects have encountered many difficulties, both for the physical interpretation as well as for the mathematical analysis. At present, there are knowledge gaps between physics, mathematics and applications. This provides a unique opportunity to put ideas that already yield good numerical results in applications onto a rigorous mathematical footing and at the same time to challenge accepted viewpoints in physics. This project will focus on the spectral theory of relativistic Hamiltonians, specifically on the famous Dirac operators. Of particular importance are bound state energies given by the eigenvalues of Dirac operators. Even for a system comprised of a single relativistic particle, there are several mathematical concepts that are very poorly understood. This is in large part owed to the Dirac operator missing an important property compared to its non-relativistic counterpart: semiboundedness. Motivated by recent advances in the generalisation of this property, the aim of this project will be to extend concepts such as variational principles, eigenvalue bounds and eigenvalue asymptotics from the non-relativistic case to Dirac operators.