Riemann-Hilbert Problems, Toeplitz Determinants and Applications

数学、物理和工程中的许多问题都可以用Toeplitz矩阵(其对角线与主对角线平行的矩阵都是常量)来表示。近年来，利用Riemann-Hilbert分析和算子论方法，Toeplitz行列式的渐近研究在随机矩阵理论和数学物理中得到了重要的应用。特别值得关注的是由光滑函数生成的Toeplitz行列式的所谓Szegö渐近性和由可能具有间断、零点、卷曲和(可积)奇异性的函数生成的Fisher-Hartwig渐近性。具有外部参数T的Toeplitz行列式的渐近研究证明了双尺度极限在数学和物理中的重要性。例如，对于二维伊辛模型和XY自旋链模型，Toeplitz行列式可以用来分析某些关联函数的双标度极限，当T达到临界值TC时，同时自旋之间的间隔变得无限大。作为具体应用，该提议包括对纠缠的研究，这是一种量子现象，为量子计算和信息处理提供了原始资源。作为“量子”的一个基本衡量标准，它显示了我们可以观察到多少量子效应，并使用它来控制一个子系统和另一个子系统。在过去的二十年里，人们在理解各种量子系统中的纠缠方面做出了很大的努力。特别令人感兴趣的是所谓的XY量子自旋链，它可以被视为量子计算机的玩具模型。该模型由排列成直线的粒子组成，并以只有最近邻相互作用的正整数为索引。由于XY自旋链允许对物理相关量的精确计算，它们提供了对现实生活中目前不可能进行此类计算的物理系统的有价值的洞察。拟议的研究方法包括Riemann-Hilbert问题(RHP)，该问题具有悠久而令人印象深刻的历史，可追溯到Riemann的论文(1851年)和Hilbert在20世纪初的相关结果。Riemann-Hilbert问题与一维奇异积分算子、卷积算子、Toeplitz算子和Wiener-Hopf算子密切相关。Riemann-Hilbert问题可以描述为在给定的曲线上以给定的跳跃在复平面上寻找一个解析函数的问题。除了Riemann-Hilbert方法之外，还将发展与具体算子和矩阵相关的新的算子理论方法。

A considerable variety of problems in mathematics, physics and engineering can be expressed in terms of Toeplitz matrices (matrices whose diagonals parallel to the main diagonal are all constants). Recently the asymptotic study of Toeplitz determinants has found important applications in random matrix theory and mathematical physics using the Riemann-Hilbert analysis and operator theoretic methods. Of particular interest are the so-called Szegö asymptotics of Toeplitz determinants generated by smooth functions and Fisher-Hartwig asymptotics generated by functions that may possess discontinuities, zeros, winding and (integrable) singularities.The asymptotic study of Toeplitz determinants with an external parameter T demonstrates the importance of double-scaling limits in mathematics and physics. For example, regarding the 2D Ising model and the XY spin chain model, the Toeplitz determinants can be used to analyze the double-scaling limit for certain correlation functions as T goes to the critical value Tc and simultaneously the separation between the spins goes to infinity.As a concrete application, the proposal includes the study of entanglement, which is a quantum phenomenon providing a primary resource for quantum computation and information processing. As a fundamental measure of "quantumness," it shows how much quantum effect we can observe and use to control one subsystem by another. In the last two decades, substantial effort has been made to understand entanglement in various quantum systems. Of particular interest is the so-called XY quantum spin chain, which can be viewed as a toy model for quantum computers. The model consists of particles ordered in a line and indexed by the positive integers with only nearest neighbor interactions. Because the XY spin chain allows for exact calculations for physically relevant quantities, they provide valuable insight into real-life physical systems where such calculations are currently impossible.The methods of the proposed research include the Riemann-Hilbert problem (RHP), which has a long and impressive history going back to Riemann's dissertation (1851) and Hilbert's related results at the beginning of the 20th century. The Riemann-Hilbert problem, which can be described as a problem of finding an analytic function in the complex plane with a prescribed jump across a given curve, is closely connected to one-dimensional singular integral operators, convolution operators, Toeplitz operators, and Wiener-Hopf operators. In addition to the Riemann-Hilbert approach, new operator-theoretic methods will be developed related to concrete operators and matrices.