Analysis of some Orthogonal Systems and Operators: One-Dimensional and Multidimensional Problems

一些正交系统和算子的分析：一维和多维问题

• 批准号: 0500177
• 负责人: Sergey Denisov
• 金额: --
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500177&HistoricalAwards=false
• 关键词: Analysis; some; Orthogonal; Systems; Operators

Analysis of some orthogonal systems and operators: one-dimensional and multidimensional systems.Abstract of proposed researchSerguei DenissovThe goal of this project is to study a broad spectrum of orthogonal systems and the corresponding operators. There are essentially two different cases: one-dimensional systems and multidimensional systems. In the one-dimensional case, the orthogonal systems with discrete index (orthogonal polynomials on the unit circle and on the real line) and continuous index (Krein systems) will be studied. The subtle problems of dependence of operator coefficients on the spectral measure (and vice versa) will be considered. The standard methods of approximation theory will be used together withextensive application of operator theory ideas. For Krein systems, even the basic theory is not developed. Thus, the systematic study of this case is warranted. The orthogonal systems are directly related to some important operators in mathematical physics - namely the Schroedinger and Dirac operators. Different questions in the spectral analysis of these operators will be studied. The deep analysis of orthogonal systems will be used to investigate possible new results for some completely integrable systems (Toda lattice, KdV, nonlinear Schroedinger equation).Considerable effort will be made to better understand the multidimensional case In particular, the Schroedinger and Dirac operators with slowly decaying or random decaying potentials will be studied. Approximation theory suggests that the Green function and its spatial asymptotics is the right object to analyze. The spectral parameter can be taken in the resolvent set, which often makes the problem treatable. Having this asymptotics, many difficult problems in scattering theory can be solved. This technique will also be tried to understand an interesting phenomenon in mathematical physics: the delocalization in Anderson model.

分析一些正交系和算子：一维系统和多维系统。建议研究的摘要Serguei Denissov这个项目的目标是研究广谱的正交系和相应的算子。本质上有两种不同的情况：一维系统和多维系统。在一维情形下，研究具有离散指标(单位圆上和实线上的正交多项式)和连续指标的正交系(Krein系)。将考虑算子系数依赖于谱测量(反之亦然)的微妙问题。近似理论的标准方法将与算子理论思想的广泛应用结合使用。对于Krein系统来说，甚至连基本理论都没有发展起来。因此，有必要对此案进行系统的研究。正交系与数学物理中的一些重要算符直接相关，即薛定谔算符和狄拉克算符。这些算子的频谱分析中的不同问题将被研究。对于一些完全可积系统(Toda格子，KdV，非线性薛定谔方程)，我们将利用正交系的深入分析来研究可能的新结果。为了更好地理解多维情形，我们将特别研究具有缓慢衰减势或随机衰减势的薛定谔算子和狄拉克算子。近似理论表明格林函数及其空间渐近性是正确的分析对象。谱参数可以取在预解集中，这使得问题往往是可处理的。有了这个渐近性，就可以解决散射理论中的许多难题。这项技术也将被用来解释数学物理中一个有趣的现象：安德森模型中的离域现象。