Spectral Theory for Decaying Oscillatory Schrodinger Operators

衰变振荡薛定谔算子的谱理论

• 批准号: 1301582
• 负责人: Milivoje Lukic
• 金额: $ 12.26万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-06-01 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1301582&HistoricalAwards=false
• 关键词: Spectral; Theory; Decaying; Oscillatory; Schrodinger

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.

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。其中一个主题是振荡衰减扰动的研究，其光谱特性显示振荡频率和衰减率之间存在很强的相互作用。除其他外，该项目将寻求描述非常缓慢的衰减的光谱特性以及临界点光谱密度的渐近行为。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产生相同的光谱结果。这包括高阶 Szegõ 定理和高阶 Baxter 定理的研究。最后，该项目包括一些涉及振荡、非衰变电势和衰变多维薛定谔算子的问题。薛定谔算子是量子力学的核心，它们的光谱特性与给定电势下电子的输运特性有关。缓慢衰减的势显示出快速衰减势的特征的混合，例如描述单个粒子场的势和振荡势，例如对应于晶体或准晶体的势。该项目预计将弥合这两种机制之间的差距，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确立为研究量子行走（量子计算的一个主题）的工具。研究生参与该项目的一些研究，这有助于他们的教育。