Infinite-dimensional representations of cubic and quintic algebras and special functions
Infinite-dimensional representations of cubic and quintic algebras and special functions
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三次代数和五次代数以及特殊函数的无限维表示
DOI：
10.1140/epjp/s13360-023-04155-2
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发表时间：
2023
期刊：
影响因子：
--
通讯作者：
Yao
中科院分区：
文献类型：
--
作者：
I. Marquette;Junze Zhang;Yao
对称代数的有限维和无限维表示在确定物理哈密顿量的谱性质方面起着重要的作用。在本文中，我们引入并应用一种实用的方法来构造出现在量子超可积系统中的某些多项式代数的无穷维表示。由于多项式代数的非线性，这些表示的显式构造是一项重要的任务。我们的方法具有相似之处，诱导模块的建设方法在李代数的背景下，并允许建设的超可积系统的状态超出了分离变量的范围。我们的主要重点是表示的多项式代数的超可积系统在2D达布空间。因此，我们能够构造大量的状态的艾里，贝塞尔和惠特克函数的复杂表达式，这将是很难获得的其他方式。
Finite and Infinite-dimensional representations of symmetry algebras play a significant role in determining the spectral properties of physical Hamiltonians. In this paper, we introduce and apply a practical method to construct infinite dimensional representations of certain polynomial algebras which appear in the context of quantum superintegrable systems. Explicit construction of these representations is a non-trivial task due to the non-linearity of the polynomial algebras. Our method has similarities with the induced module construction approach in the context of Lie algebras and allows the construction of states of the superintegrable systems beyond the reach of the separation of variables. Our primary focus is the representations of the polynomial algebras underlying superintegrable systems in 2D Darboux spaces. As a result, we are able to construct a large number of states in terms of complicated expressions of Airy, Bessel and Whittaker functions which would be difficult to obtain in other ways.