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首页> 外文期刊>Journal of Mathematical Physics >Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory - art. no. 053509
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Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory - art. no. 053509

机译：保形平坦空间中的二阶超可积系统。 I.二维古典结构理论-艺术。没有。 053509

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This paper is the first in a series that lays the groundwork for a structure and classification theory of second-order superintegrable systems, both classical and quantum, in conformally flat spaces. Many examples of such systems are known, and lists of possible systems have been determined for constant curvature spaces in two and three dimensions, as well as few other spaces. Observed features of these systems are multiseparability, closure of the quadratic algebra of second-order symmetries at order 6, use of representation theory of the quadratic algebra to derive spectral properties of the quantum Schrodinger operator, and a close relationship with exactly solvable and quasi-exactly solvable systems. Our approach is, rather than focus on particular spaces and systems, to use a general theoretical method based on integrability conditions to derive structure common to all systems. In this first paper we consider classical superintegrable systems on a general two-dimensional Riemannian manifold and uncover their common structure. We show that for superintegrable systems with nondegenerate potentials there exists a standard structure based on the algebra of 2x2 symmetric matrices, that such systems are necessarily multiseparable and that the quadratic algebra closes at level 6. Superintegrable systems with degenerate potentials are also analyzed. This is all done without making use of lists of systems, so that generalization to higher dimensions, where relatively few examples are known, is much easier. (C) 2005 American Institute of Physics.
机译：本文是该系列的第一篇，为共形平坦空间中的经典和量子二阶超可积系统的结构和分类理论奠定了基础。此类系统的许多示例是已知的，并且已经确定了二维和三维恒定曲率空间以及少数其他空间的可能系统列表。这些系统的观察到的特征是多重可分性，二阶对称二次代数的阶数为6的闭合，使用二次代数的表示理论导出量子Schrodinger算符的光谱性质以及与可精确解和拟可逆的紧密关系。完全可解决的系统。我们的方法不是基于特定的空间和系统，而是基于可积性条件使用通用的理论方法来推导所有系统共有的结构。在第一篇论文中，我们考虑了一般二维黎曼流形上的经典超可积系统，并揭示了它们的共同结构。我们表明，对于具有非简并势的超可积系统，存在一个基于2x2对称矩阵的代数的标准结构，该系统必须是可分的，并且二次代数在第6级处关闭。还对具有简并势的超可积系统进行了分析。无需使用系统列表即可完成所有操作，因此可以轻松地将其推广到已知相对较少示例的更高维度。 （C）2005美国物理研究所。

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