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Quantum integrability and differential equations.

量子可积性和微分方程。

基本信息

批准号：
EP/G039526/1
负责人：
Tania Dunning
金额：
$ 31.38万
依托单位：
University of Kent
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2009
资助国家：
英国
起止时间：
2009 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FG039526%2F1
关键词：
Quantum integrability differential equations.

项目摘要

Over the last decade, progress in the study of ordinary differential equations defined in the complex plane and that of quantum integrable models has advanced with the help of a surprising correspondence between these previously-separate fields. The link is now called the ODE/IM correspondence. Functional relations lie at the heart of the correspondence, forming a bridge between the two subjects and allowing techniques from one island to be applied to its neighbour, and vice versa. This has led to significant applications, for example in PT-symmetric quantum mechanics and boundary integrable quantum field theory. Each of the functional relations has an infinite set of solutions, which are known to fall into families due to the integrable model relation with conformal field theory. For all cases except the su(2) case, only the highest-weight state in each family has been explored and matched with either an ordinary differential equation or a pseudo-differential equation. The first task is to map out the full set of differential equations which correspond to the excited states of the integrable model. We will begin with a simple case and aim to deduce the general picture. The second aim of the proposed research is to shed light on the hidden role the Lie algebra symmetry has to play in the differential equation side of the picture. From the integrable model side we expect each node of the associated Dynkin diagram to correspond to a different differential equation, up to the symmetry of the diagram. We shall address the issue of the missing differential equations, enlarging the known set of equations beyond the first node of most of the Dynkin diagrams. The Bethe ansatz and related techniques play a central part in all areas of integrable models and are important in many related fields. The research described here will expand the current toolbox of nonlinear integral equations used for solving Bethe ansatz equations.

在过去的十年中，在复平面中定义的常微分方程的研究和量子可积模型的研究取得了进展，这得益于这些先前独立的领域之间惊人的对应关系。这个链接现在称为ODE/IM通信。功能关系是通信的核心，形成了两个主题之间的桥梁，允许一个岛屿的技术应用于相邻岛屿，反之亦然。这导致了重要的应用，例如在pt对称量子力学和边界可积量子场论。每一个函数关系都有一个无穷多的解集，这些解集由于保形场论的可积模型关系而被称为族。对于除su(2)情况外的所有情况，每个族中只有权重最高的状态被探索并与常微分方程或伪微分方程匹配。第一个任务是绘制出与可积模型的激发态相对应的微分方程的完整集合。我们将从一个简单的例子开始，目的是推断出总的情况。提出的研究的第二个目的是阐明李代数对称性在微分方程方面所扮演的隐藏角色。从可积模型方面，我们期望相关的Dynkin图的每个节点对应于不同的微分方程，直至图的对称性。我们将讨论缺失的微分方程的问题，将已知的方程集扩大到大多数Dynkin图的第一个节点之外。贝特函数及其相关技术在可积模型的所有领域中都起着核心作用，在许多相关领域中都很重要。这里描述的研究将扩展当前用于求解贝特安萨茨方程的非线性积分方程工具箱。