Spectral theory of relativistic quantum Hamiltonians

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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.