Szegö theorems for semibounded subsets of the real axis

Spectral theory is the study of the relation of physical objects and their spectral characteristics. As an example one may consider as a physical object a piece of the ocean or the surface of a drum and as spectral characteristics the frequencies of the waves or the drum. That is, on the one hand we have an object that can be described by a mathematical equation and on the other hand we have the spectral characteristics that can be measured. If we extract from the object information about the spectral data, we call it a direct spectral problem and an inverse spectral problem, if we extract information in the opposite direction. Barry Simon defines a gem of spectral theory a theorem that describes a class of spectral data and a class of objects so that an object is in the second class if and only if its spectral data lies in the first class. We are mainly interested in self-adjoint problems, where the spectral data is a subset of the real line. If the subset is bounded, many spectral results are formulated in terms of the Robin constant of the spectral set. The Robin constant has the following physical interpretation. Assume that you have one unit of electric charge and you ask how this charge will distribute on the set in order to find an equilibrium. From a physical point of view it is expectable, that the potential associated to this charge distribution will be constant on the set. This can in fact be mathematically proven and the corresponding constant is given by the so-called Robin constant. Having in mind this physical interpretation it is clear that the notion of Robin constant does not extend to unbounded sets, for in this case all particles would escape to infinity. On the other hand many spectral problems have unbounded spectral sets. In a previous work with Milivoje Lukic, we were able to find a renormalized Robin constant which extends the notion of the classical Robin constant to certain unbounded sets relevant for spectral theory. A crucial part of the project deals with using this notion of renormalized Robin constant to extend spectral results from the bounded to the unbounded setting.

Orthogonal polynomials appear in many areas of mathematics, such as approximation theory, the study of discrete Schrödinger operators and random matrices, or numerical mathematics. Their origins date back to the early 19th century. Particular importance is given to the zeros of orthogonal polynomials, which play different roles in various fields. In numerical analysis, for example, the zeros serve as nodes for integration formulas, while in mathematical physics, they often appear as eigenvalues of physical systems. To model complex systems, one must choose polynomials of very high degree. In this project, I studied the local behavior of the zeros as the degree of the polynomials tends to infinity. It turns out that, for many systems, the zeros exhibit similar behavior on a local scale. This phenomenon is known as universal limits, which are of great significance, especially in random matrix theory. It had been conjectured that the local behavior of the zeros depends only on the local spectral properties of the underlying system. The systems I investigated in this project are completely described by the so-called spectral measure. Together with my coauthors, I was able to show that universal limits indeed depend only on the local properties of the spectral measure. In partular this proves the above stated conjecture. The precise result states that universality in the bulk occurs if and only if the spectral measure locally looks like the Lebesgue measure in a very weak sense. Furthermore, we established corresponding characterizations for so-called hard edges of the spectrum.