Riemann-Hilbert Problems, Toeplitz Determinants and Applications

黎曼-希尔伯特问题、托普利茨行列式及应用

• 批准号: EP/T008636/1
• 负责人: Jani A. Virtanen
• 金额: $ 8.02万
• 依托单位: University of Reading
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FT008636%2F1
• 关键词: Riemann; Hilbert; Problems; Toeplitz; Determinants

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.

数学、物理和工程中的许多问题都可以用Toeplitz矩阵（与主对角线平行的对角线都是常数的矩阵）来表示。近年来，利用Riemann-Hilbert分析和算子理论的方法，Toeplitz行列式的渐近性研究在随机矩阵理论和数学物理中有重要的应用。特别令人感兴趣的是由光滑函数生成的Toeplitz行列式的Szegö渐近性和由可能具有不连续性、零点、缠绕和（可积）奇点的函数生成的Fisher-Hartwig渐近性。例如，对于二维伊辛模型和XY自旋链模型，当T达到临界值Tc时，同时自旋之间的分离达到无穷大，可以使用Toeplitz行列式来分析某些相关函数的双标度极限。作为具体应用，该提议包括纠缠的研究，其是为量子计算和信息处理提供主要资源的量子现象。作为“量子性”的基本度量，它显示了我们可以观察到多少量子效应，并利用它来控制一个子系统。在过去的二十年里，人们已经做出了大量的努力来理解各种量子系统中的纠缠。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。该模型由排列成行的粒子组成，并由正整数索引，只有最近邻相互作用。由于XY自旋链可以精确计算物理相关量，因此它们为了解目前无法进行此类计算的现实物理系统提供了宝贵的见解。拟议研究的方法包括Riemann-Hilbert问题（RHP），该问题具有悠久而令人印象深刻的历史，可以追溯到Riemann的论文（1851）和Hilbert在世纪初的相关结果。黎曼-希尔伯特问题，可以描述为在复平面上找到一个解析函数的问题，该函数具有给定的曲线上的预定跳跃，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。除了黎曼-希尔伯特方法，新的运营商理论的方法将开发相关的具体运营商和矩阵。

项目成果

Fredholm Toeplitz operators with VMO symbols and the duality of generalized Fock spaces with small exponents

具有 VMO 符号的 Fredholm Toeplitz 算子和小指数广义 Fock 空间的对偶性

• DOI: 10.1017/prm.2019.65
• 发表时间: 2019-12
• 期刊: Proceedings of the Royal Society of Edinburgh: Section A Mathematics
• 作者: Hu Zhangjian;Virtanen Jani A.
• 通讯作者: Virtanen Jani A.

Compact Hankel operators with bounded symbols

• DOI: 10.7900/jot.2020apr27.2276
• 发表时间: 2019-06
• 期刊: Journal of Operator Theory
• 影响因子: 0.8
• 作者: Raffael Hagger;J. Virtanen

On the spectrum of the double-layer operator on locally-dilation-invariant Lipschitz domains

局部膨胀不变Lipschitz域上双层算子的谱

• DOI: 10.1007/s00211-023-01353-z
• 发表时间: 2023
• 期刊: Numerische Mathematik
• 影响因子: 2.1
• 作者: Chandler-Wilde S

Entanglement entropy of two disjoint intervals separated by one spin in a chain of free fermion*

自由费米子链中由一个自旋分隔的两个不相交区间的纠缠熵*

• DOI: 10.1088/1751-8121/ab9cf2
• 发表时间: 2020
• 期刊: Journal of Physics A: Mathematical and Theoretical
• 作者: Brightmore, L.;Gehér, G. P.;Its, A. R.;Korepin, V. E.;Mezzadri, F.;Mo, M. Y.;Virtanen, J. A.
• 通讯作者: Virtanen, J. A.

Toeplitz Operators on the Unit Ball with Locally Integrable Symbols

• DOI: 10.1007/s00020-022-02695-3
• 发表时间: 2020-11
• 期刊: Integral Equations and Operator Theory
• 影响因子: 0.8
• 作者: Raffael Hagger;Cong Liu;J. Taskinen;J. Virtanen