This project investigates spectral properties of Schrödinger operators and orthogonal polynomials, especially in the regime of slowly decaying perturbations, which are less understood than fast decaying and non-decaying perturbations. One of the topics is the study of oscillatory decaying perturbations, whose spectral properties show strong interplay between the frequencies of oscillation and the rate of decay. This project will seek to describe, among other things, spectral properties for very slow decay and the asymptotic behavior of the spectral density at critical points. The proposer also intends to study potentials which obey bounded variation conditions, investigating an emerging theme that conditions on derivatives of the potential can have the same spectral consequence as conditions on the potential itself. This includes the study of higher order Szegõ theorems and higher order Baxter theorems. Finally, the project includes some problems involving oscillatory, non-decaying potentials and decaying multi-dimensional Schrödinger operators.
Schrödinger operators are central to quantum mechanics and their spectral properties are connected to transport properties of electrons in a given potential. Slowly decaying potentials show a mix of features of fast decaying potentials, such as those describing the field of a single particle, and oscillatory potentials, such as those corresponding to a crystal or quasicrystal. This project is expected to bridge the gap between those two regimes and have a potential impact in physics. Orthogonal polynomials are used in approximation theory and many fields of science; for instance, a recent development establishes them as a tool in the study of quantum walks, a topic in quantum computing. Graduate students participate in some of the research in this project, and this contributes to their education.
