Spectral theory of relativistic quantum Hamiltonians

量子力学的发展是数学和物理如何促进相互进步的一个很好的例子。这个成功故事的核心是自伴随算子的谱理论，即所谓的哈密顿算子，它决定了量子系统的能量和时间演化。这种共同努力的结果在工业上得到了大量的应用。然而，结合相对论效应的尝试遇到了许多困难，无论是物理解释还是数学分析。目前，物理、数学和应用之间存在着知识缺口。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。本项目将重点研究相对论哈密顿算符的谱理论，特别是著名的狄拉克算符。特别重要的是由狄拉克算子的特征值给出的束缚态能量。即使对于一个由单个相对论性粒子组成的系统，也有几个数学概念很难理解。这在很大程度上是由于狄拉克算子与非相对论算子相比缺少了一个重要的性质：半有界性。受这一性质的最新推广进展的激励，本项目的目的将是将变分原理、特征值界和特征值渐近等概念从非相对论情况扩展到狄拉克算子。

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.