Approximation problems for Dirac and Schrödinger operators

In many applications in science and engineering it is not possible to solve the underlying mathematical models exactly. Hence, suitable parameters in these mathematical models are replaced by idealized ones. The parameters should be chosen in such a way that the idealized model is easier accessible from a mathematical point of view and such that it still reflects the physical reality up to a reasonable level of exactness. To verify that the idealized models have similar properties as the original ones coming from applications is a difficult mathematical problem which is unsolved in many cases. It is the main goal of this project to justify the usage of several types of idealized models in a mathematically rigorous way. In the first part of the project so called Schrödinger operators with singular potentials are investigated. They play an important role in solid state physics to describe the propagation of particles in certain nano structures and also in the description of photonic crystals, which are already in use in computer systems as faster replacements for semiconductors. For these models there exist, under elementary assumptions, results which justify the replacement of the realistic parameters by the idealized singular potentials. It is one of the main goals in this project to extend these results to situations that appear in realistic applications in science and engineering. The second part is on so called Dirac operators, which are used in problems, where effects of the special theory of relativity play an important role. For instance, this is the case in the description of elementary particles like quarks or in the analysis of graphen, which appear in research for batteries, water filters or photovoltaic cells. For these problems the mathematical investigations are still at the very beginning. It is one of the main goals in this project to find elementary results on how parameters should be chosen in certain models such that the mathematical models reflect the physical reality in the correct way.

Mathematical models based on the Dirac and Schrödinger equations are fundamental for describing quantum phenomena and they are used to study a wide range of physical systems. In many applications such as graphene, leaky quantum graphs, photonic crystals, and other nanostructures, interactions are highly localized on curves, surfaces, or interfaces. Such interactions are commonly represented by singular (idealized) mathematical models, which are significantly easier to analyze than models with realistic short-range potentials. However, the mathematical justification for replacing realistic interactions by these idealized models had been incomplete in many important situations. The main objective of this project was to establish rigorous approximation results for Dirac and Schrödinger operators with singular interactions. The project investigated whether operators with regular, strongly localized potentials converge to operators with singular interactions in the norm resolvent sense. Such convergence guarantees that the approximating and limiting models possess essentially the same spectral properties and therefore describe the same physical phenomena. The first part of the project focused on relativistic Dirac operators with singular interactions supported on curves and surfaces. A major goal achieved was the derivation of approximation results for Dirac operators with delta-shell interactions, as well as the determination of the non-relativistic limit of Dirac operators with boundary conditions. This established a rigorous connection between relativistic quantum models and Laplace operators, providing a physical interpretation of the parameters appearing in the boundary conditions. The second part addressed Schrödinger operators with singular interactions. The project successfully developed approximation results for magnetic Schrödinger operators with delta-interactions supported on general networks. The approximation problem for Schrödinger operators with more singular delta'-interactions was also treated within the project and partial results were obtained. Besides the approximation results also various other spectral phenomena for different classes of ordinary and partial differential operators were studied within the project. The main results obtained in this project on approximation problems for differential operators with singular potentials substantially extend existing theory from simple geometries to graph-like structures and higher-dimensional settings that are directly relevant for applications. The achieved results provide rigorous mathematical foundation for widely used idealized quantum models, strengthen the theoretical understanding of their spectral properties, and improved their applicability in the mathematical description of quantum systems arising in nanotechnology, graphene, and photonic crystals.