Spectral Theory for Decaying Oscillatory Schrodinger Operators

本项目研究Schrödinger算子和正交多项式的谱性质，特别是在缓慢衰减摄动的情况下，这比快速衰减和非衰减摄动更少被理解。其中一个主题是振荡衰减微扰的研究，其频谱特性显示振荡频率和衰减速率之间有很强的相互作用。除其他事项外，本项目将寻求描述非常慢衰减的光谱特性和临界点处光谱密度的渐近行为。该提议者还打算研究服从有界变化条件的势，研究一个新兴的主题，即势的导数条件可以与势本身的条件具有相同的谱结果。这包括对高阶Szegõ定理和高阶Baxter定理的研究。最后，该项目还包括一些涉及振荡、非衰减势和衰减多维Schrödinger算子的问题。Schrödinger算符是量子力学的核心，它们的光谱性质与给定电位下电子的输运性质有关。慢速衰减电位表现出快速衰减电位的混合特征，例如描述单个粒子场的那些，以及振荡电位，例如对应于晶体或准晶体的那些。该项目有望弥合这两种制度之间的差距，并对物理学产生潜在影响。正交多项式用于近似理论和许多科学领域；例如，最近的一项发展将它们建立为量子行走研究的工具，量子行走是量子计算的一个主题。研究生参与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.