Classical and quantum theory of finite-dimensional integrable systems

1.修整链是一个重要的非线性微分方程，它包括Painleve方程。这些方程具有由二阶方阵表示的Lax表示，并且相关联的谱曲线成为超椭圆的。这使我们能够应用分离变量的技术，并获得一个周期性边界条件下的敷料链的哈密顿表示。本文以环面上的等单点形变为例，讨论了SU（2）Calogero-Gaudin系统的非自治形式。我们应用分离变量的技术将该系统重写成哈密顿形式。作为一个副产品，我们可以找到这个矩阵系统与一个标量等单道系统的联系。Inozemtsev系统是Calogero-Moser系统的变形，保留了经典的可积性。我们考虑了它的量子理论，发现该量子系统在其中一个耦合常数的一系列离散的特殊值下具有部分可解性（准精确可解性）.已知一类有理函数的模空间具有可积系统的结构。用三角函数或椭圆函数代替有理函数，构造了一个新的可积系统，并指出这些可积系统提供了非常简单的分离变量模型。

1. The dressing chains are known to be an important nonlinear differential equation that includes the Painleve equations. These equations have a Lax representation by a second order square matrix, and the associated spectral curve becomes hyperelliptic. This enabled us to apply the technique of separation of variables, and to obtain a Hamiltonian representation of the dressing chains under a periodic boundary condition.2. A non-autonomous version of the SU(2) Calogero-Gaudin system was taken up an example of isomonodromic deformations on a torus. We applied the technique of separation of variables to rewrite this system into a Hamiltonian form. As a byproduct, we could find a connection of this matrix system with a scalar isomonodromic system.3. The Inozemtsev system is a deformation of the Calogero-Moser system retaining the classical integrability. We considered its quantum theory, and discovered that the quantum system has partial solvability (quasi-exact-solvability) at a discrete series of special values of one of the coupling constants.4. The moduli space of a class of rational functions is known to carry the structure of an integrable system. We constructed a new integrable system by replacing the rational functions by trigonometric or elliptic functions, and pointed out that these integrable systems provide very simple models of separation of variables.