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.