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.

对某些正交系统和运营商的分析：一维和多维系统。拟议的ResearchSerguei Denissov的取消该项目的目标是研究广泛的正交系统和相应的操作员。 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，将研究具有离散索引（单位圆和真实线上的正交多项式）和连续索引（Kerin Systems）的正交系统。将考虑操作员系数对光谱度量（反之亦然）的微妙问题。近似理论的标准方法将与操作者理论思想的扩展应用一起使用。 对于角林系统，即使是基本理论也没有发展。因此，有必要对此案进行系统研究。正交系统与数学物理学中的一些重要操作员直接相关 - 即Schroedinger和Dirac Operators。将研究这些操作员的光谱分析中的不同问题。对正交系统的深入分析将用于调查某些完全可以集成的系统（TODA晶格，KDV，非线性Schroedinger方程）的可能新结果。将尽力更好地理解多维案例，特别是Schroedinger和Schroedinger和Dirac操作员，具有缓慢衰减或随机衰减的潜力。近似理论表明，绿色函数及其空间渐近学是要分析的正确对象。光谱参数可以在分解集中进行，这通常使问题可以治疗。有了这种渐进性，可以解决散射理论中的许多困难问题。该技术还将试图理解数学物理学中有趣的现象：安德森模型中的离域化。