Quantum integrability and differential equations.

在过去的十年中，在复平面上定义的常微分方程和量子可积模型的研究中，在这些先前分离的领域之间惊人的对应关系的帮助下取得了进展。这个链接现在被称为ODE/IM通信。功能关系是对应关系的核心，在两个主题之间形成桥梁，使一个岛屿的技术适用于其邻国，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限的解集，由于可积模型与共形场论的关系，这些解被称为属于一族。除了su（2）的情况下，所有的情况下，只有最高权的状态在每个家庭已被探索和匹配的常微分方程或伪微分方程。第一个任务是映射出对应于可积模型激发态的微分方程组。我们将开始与一个简单的情况下，旨在推导出一般图片。所提出的研究的第二个目的是揭示李代数对称性在微分方程方面所起的隐藏作用。从可积模型的角度来看，我们期望相关Dynkin图的每个节点对应于不同的微分方程，直到图的对称性。我们将讨论丢失的微分方程的问题，将已知的方程组扩大到大多数Dynkin图的第一个节点之外。Bethe变换及其相关技术在可积模型的各个领域都起着核心作用，在许多相关领域都很重要。本文所述的研究将扩展当前用于求解Bethe不等式方程的非线性积分方程工具箱。

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.