Spectral Theory for Decaying Oscillatory Schrodinger Operators

该项目研究薛定谔算子和正交多项式的谱特性，特别是在缓慢衰减扰动的情况下，这比快速衰减和非衰减扰动更难理解。其中一个主题是研究振荡衰减扰动，其频谱特性显示振荡频率和衰减速率之间的强烈相互作用。这个项目将寻求描述，除其他事项外，非常缓慢的衰减和光谱密度在临界点的渐近行为的光谱特性。提议者还打算研究服从有界变分条件的势，调查一个新兴的主题，即势的导数条件可以具有与势本身条件相同的谱结果。这包括研究高阶Szegleman定理和高阶巴克斯特定理。最后，该项目包括一些问题，涉及振荡，非衰减势和衰减多维薛定谔算子。薛定谔算子是量子力学的核心，其光谱性质与给定势中电子的输运性质有关。缓慢衰减势表现出快速衰减势（如描述单个粒子场的势）和振荡势（如对应于晶体或准晶的势）的混合特征。预计该项目将弥合这两种制度之间的差距，并在物理学方面产生潜在影响。正交多项式被用于近似理论和许多科学领域;例如，最近的发展使它们成为研究量子行走的工具，这是量子计算的一个主题。研究生参与了这个项目的一些研究，这有助于他们的教育。

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.