广义五维Kepler系统、Yang-Coulomb变换和Hurwitz变换
Generalized five-dimensional Kepler system, Yang-Coulomb monopole and Hurwitz transformation
摘要：
5D开普勒系统具有许多有趣的性质。这个系统是超可积的，并且具有$su(2)$非阿贝尔相互作用（杨-库仑相互作用）。通过Hurwitz变换，该系统也与8D各向同性谐振子相关。我们引入一个新的超可积哈密顿量，它包含在一个5D开普勒系统与Smorodinsky-Winternitz型的新条款。得到了该系统的运动积分。他们生成一个二次代数的结构常数涉及的Casimir算子的$so(4)$李代数。我们还表明，该系统仍然是超可积的$su(2)$ nonAbel的？（广义杨库仑？？）。我们利用抛物坐标系研究了该系统，并通过Hurwitz变换得到了它的对偶，即一个8D奇异振子。这个8D奇异振子也是一个新的超可积系统和多重可分系统。我们得到了它的包含$so(4)$李代数的两个Casimir算子的二次代数。利用这种对应关系，从代数上得到了广义杨库仑体系的能谱。
英文摘要：
The 5D Kepler system possesses many interesting properties. This system is superintegrable and also with a $su(2)$ nonAbelian monopole interaction (Yang-Coulomb monopole). This system is also related to a 8D isotropic harmonic oscillator by a Hurwitz transformation. We introduce a new superintegrable Hamiltonian that consists in a 5D Kepler system with new terms of Smorodinsky-Winternitz type. We obtain the integrals of motion of this systems. They generate a quadratic algebra with structure constants involving the Casimir operator of a $so(4)$ Lie algebra. We also show that this system remains superintegrable with a $su(2)$ nonAbelian monopole (generalized Yang-Coulomb monopole). We study this system using parabolic coordinates and obtain from Hurwitz transformation its dual that is a 8D singular oscillator. This 8D singular oscillator is also a new superintegrable system and multiseparable. We obtained its quadratic algebra that involves two Casimir operators of $so(4)$ Lie algebras. This correspondence is used to obtain algebraically the energy spectrum of the generalized Yang Coulomb monopole.
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