Jacobi, Ellipsoidal Coordinates and Superintegrable Systems
Jacobi, Ellipsoidal Coordinates and Superintegrable Systems
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DOI：
10.2991/jnmp.2005.12.2.5
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发表时间：
2005-01
影响因子：
0.7
通讯作者：
E. Kalnins;J. Kress;W. Miller
中科院分区：
文献类型：
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作者：
E. Kalnins;J. Kress;W. Miller
本文介绍Jacobi积分Hamilton-Jacobi方程的方法和他发现的椭圆坐标系，即真实的和复常曲率空间的一般可分坐标系。这项工作是一个重要的先驱现代理论的二阶超可积系统，我们然后转向。黎曼空间上一个具有位势的薛定谔算子是二阶超可积的，如果有2n − 1个（经典）函数独立的二阶对称算子。(The 2n − 1是这种对称性的最大可能数目。这些系统在特殊函数理论中具有相当大的意义，因为它们是多可分的，即，变量在多个坐标集中分开，并且可以用特殊函数显式求解。可分离解之间的相互关系提供了关于系统的更多信息。我们给出了一个例子的超可积系统，然后提出了最近的结果展示超可积系统的一般结构在所有真实的或复杂的二维空间和三维共形平坦空间和一个完整的列表，这样的空间和潜力在二维。
Abstract We describe Jacobi’s method for integrating the Hamilton-Jacobi equation and his discovery of elliptic coordinates, the generic separable coordinate systems for real and complex constant curvature spaces. This work was an essential precursor for the modern theory of second-order superintegrable systems to which we then turn. A Schrödinger operator with potential on a Riemannian space is second-order superintegrable if there are 2n − 1 (classically) functionally independent second-order symmetry operators. (The 2n − 1 is the maximum possible number of such symmetries.) These systems are of considerable interest in the theory of special functions because they are multiseparable, i.e., variables separate in several coordinate sets and are explicitly solvable in terms of special functions. The interrelationships between separable solutions provides much additional information about the systems. We give an example of a superintegrable system and then present very recent results exhibiting the general structure of superintegrable systems in all real or complex two-dimensional spaces and three-dimensional conformally flat spaces and a complete list of such spaces and potentials in two dimensions.