论文标题
两个可整合二维系统的动力对称代数
Dynamical symmetry algebra of two superintegrable two-dimensional systems
论文作者
论文摘要
多年来，已经进行了二维式扁平空间上的2D可促进系统的完整分类，并获得了58个模型，分为12个等效类别。我们将重新检查两个伪休ud-Hermitian量子系统$ e_ {8} $和$ e_ {10} $从这种新方法中，基于额外的梯子运算符的新方法。这些额外的梯子运算符被利用以获得生成频谱代数和动态对称性。我们将将动态对称性代数的发电机与哈密顿量联系起来，从而表明后者可以以代数形式写入。我们还将将它们链接到提供可加密性属性的运动积分。这证明了动态对称代数如何解释对称性。此外，我们将利用这些代数结构来生成扩展的状态集，并赋予阶梯操作员对其的行动。我们将介绍哈密顿量的多项式以及在其中一些国家消失的运动积分，然后证明各个国家的集合不仅包含特征状态，还包含广义状态。我们的方法为此类国家提供了自然框架。
A complete classification of 2D superintegrable systems on two-dimensional conformally flat spaces has been performed over the years and 58 models, divided into 12 equivalence classes, have been obtained. We will re-examine two pseudo-Hermitian quantum systems $E_{8}$ and $E_{10}$ from such a classification by a new approach based on extra sets of ladder operators. Those extra ladder operators are exploited to obtain the generating spectrum algebra and the dynamical symmetry one. We will relate the generators of the dynamical symmetry algebra to the Hamiltonian, thus demonstrating that the latter can be written in an algebraic form. We will also link them to the integrals of motion providing the superintegrability property. This demonstrates how the dynamical symmetry algebra explains the symmetries. Furthermore, we will exploit those algebraic constructions to generate extended sets of states and give the action of the ladder operators on them. We will present polynomials of the Hamiltonian and the integrals of motion that vanish on some of those states, then demonstrating that the sets of states not only contain eigenstates, but also generalized states. Our approach provides a natural framework for such states.
