论文标题
在笛卡尔坐标中与高阶梯子阶梯运算符分离的可累加系统的多项式代数
Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators
论文作者
论文摘要
我们介绍了一类代表一类高阶可促进系统的一般多项式代数，这些系统在笛卡尔坐标中分开。该构建依赖于潜在的多项式海森堡代数及其定义的高级梯子操作员。这些代数的一个特征是它们通过构造$ \ mathfrak {gl}（n）$ lie代数的结构的某些方面来保存。在此框架中出现的哈密顿量的类别中，有各种谐波振荡器和与特殊的正交多项式相关的谐波和奇异振荡器的变形，甚至是Painlevé，以及高阶Painlevé类似物。作为一个明确的例子，我们研究了与III型的Hermite特殊正交多项式相关的新的三维可整合系统。主要结果之一是根据多项式代数的有限维不可减少表示，确定了模型的退化。
We introduce the general polynomial algebras characterizing a class of higher order superintegrable systems that separate in Cartesian coordinates. The construction relies on underlying polynomial Heisenberg algebras and their defining higher order ladder operators. One feature of these algebras is that they preserve by construction some aspects of the structure of the $\mathfrak{gl}(n)$ Lie algebra. Among the classes of Hamiltonians arising in this framework are various deformations of harmonic oscillator and singular oscillator related to exceptional orthogonal polynomials and even Painlevé and higher order Painlevé analogs. As an explicit example, we investigate a new three-dimensional superintegrable system related to Hermite exceptional orthogonal polynomials of type III. Among the main results is the determination of the degeneracies of the model in terms of the finite-dimensional irreducible representations of the polynomial algebra.
