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.

一些正交系统和算子的分析：一维和多维系统。拟议研究摘要Serguei Denissov 该项目的目标是研究广泛的正交系统和相应的算子。 本质上有两种不同的情况：一维系统和多维系统。在一维情况下，将研究具有离散指标的正交系统（单位圆和实线上的正交多项式）和连续指标的正交系统（Krein系统）。将考虑算子系数对谱测量的依赖性（反之亦然）的微妙问题。近似论的标准方法将与算子理论思想的广泛应用一起使用。 对于 Kerin 系统来说，连基础理论都没有发展出来。因此，有必要对本案进行系统研究。正交系统与数学物理中的一些重要算子直接相关，即薛定谔算子和狄拉克算子。将研究这些算子谱分析中的不同问题。正交系统的深入分析将用于研究一些完全可积系统（Toda 晶格、KdV、非线性薛定谔方程）可能的新结果。将付出相当大的努力来更好地理解多维情况，特别是薛定谔和狄拉克算子将研究缓慢衰减或随机衰减的电位。逼近理论表明格林函数及其空间渐进性是正确的分析对象。光谱参数可以取入解析集中，这通常使得问题可以处理。有了这种渐进性，散射理论中的许多难题就可以得到解决。这项技术还将尝试理解数学物理中一个有趣的现象：安德森模型中的离域。