RUI: Asymptotics of Determinants of Perturbations of Convolution Operators

RUI：卷积算子扰动行列式的渐近

• 批准号: 0500892
• 负责人: Estelle Basor
• 金额: $ 11.3万
• 依托单位: California Polytechnic State University Foundation
• 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: 2005-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500892&HistoricalAwards=false
• 关键词: RUI; Asymptotics; Determinants; Perturbations; Convolution

AbstractBasor The focus of this project is to investigate the asymptotics of determinants of perturbations of convolution operators. Our goal will be to extend the classical limit theorems to these operators, both for scalar and matrix-valued symbols, and for both smooth and singular symbols. For many of these operators, the constant term is the most difficult piece of the asymptotic expansion to describe. For matrix-valued symbols there are only a few cases where the constants can be explicitly described. In particular, we will investigate the asymptotics in the case of a perturbation of a Toeplitz determinant by a Hankel operator with possibly different symbol. Other classes of operators of interest are Wiener-Hopf plus Hankel operators and Bessel operators. Classical operator methods will be used to study these problems as well as newer developments. For example, using the BorodinGeronimo-Geronimo-Case identity to bridge between smooth and singular symbols has been highly successful.There is increasing interest in finding asymptotic expansions of determinants of convolution type operators because they have connections to many problems in mathematical physics, including the Ising model (a model of a two-dimensional (or very thin) magnets), the classical dimer model, the entanglement problem in spin chain model, random growth models, and to the general area of random matrix theory. In these physical problems one is often interested in the complicated, unpredictable behavior of the models. Often a quantity that describes some statistical property of a system can be reformulated as a determinant approximation problem. The physical systems give predictions as to the right form of the approximation and show that many of the answers should be quite universal. The universality is especially important since it shows that many complicated systems and models are actually quite similar. Hence the idea is not simply to prove theorems and then find applications for the theorems, but to use the ideas of mathematical physics to give predictions of the mathematics and then conversely, to use the mathematics to tell us something about physical systems.

摘要basor该项目的重点是研究卷积操作员扰动决定因素的渐近学。我们的目标是将经典限制定理扩展到标量和矩阵值符号以及平滑和单数符号的这些操作员。对于许多这些操作员来说，恒定术语是要描述的渐近扩展中最困难的一部分。对于矩阵值符号，只有少数情况可以明确描述常数。特别是，在汉克尔操作员对符号可能不同的符号决定因素的情况下，我们将研究渐近学。其他感兴趣的运营商是Wiener-Hopf以及Hankel运营商和Bessel运营商。古典操作员方法将用于研究这些问题以及新的发展。 For example, using the BorodinGeronimo-Geronimo-Case identity to bridge between smooth and singular symbols has been highly successful.There is increasing interest in finding asymptotic expansions of determinants of convolution type operators because they have connections to many problems in mathematical physics, including the Ising model (a model of a two-dimensional (or very thin) magnets), the classical dimer model, the entanglement problem in spin chain model, random growth模型，以及随机矩阵理论的一般领域。在这些物理问题中，人们通常对模型的复杂，不可预测的行为感兴趣。 Often a quantity that describes some statistical property of a system can be reformulated as a determinant approximation problem.物理系统对近似值的正确形式进行了预测，并表明许多答案应该非常普遍。普遍性尤其重要，因为它表明许多复杂的系统和模型实际上非常相似。因此，这个想法不仅仅是证明定理，然后找到定理的应用程序，而是使用数学物理学的思想来提供数学的预测，然后相反，使用数学来告诉我们有关物理系统的信息。