论文标题
对2D高阶量子可促进系统进行分类的新方法
A new way to classify 2D higher order quantum superintegrable systems
论文作者
论文摘要
我们修订了Kalnins，Kress和Miller（2010）的方法，用于为Schrödingereigenvalue方程$Hψ\ equiv（Δ_2 +v）ψ= e光构建任意顺序的对称运算符的规范形式。我们将方法作为一个例子来重新审视Tremblay和Winternitz（2010）parelevéVI的潜在潜力的衍生物，这是一个3阶可整合的平坦空间系统，这些平面系统分离为极性坐标，并且作为新的结果，我们在2个球体上列出了在2个球体上的潜力列表，并在2个球体上分离出势和2个螺旋式的2-刺激性（两点），并在2个seetc上（两点）分开。特别是，我们表明，在2个球体上的3阶可取代系统也出现了PainlevéVI电位，该系统在球形坐标中分离，以及在2-倍曲Boid上分离的3个阶可超积极系统，该系统分离了无球坐标，并在horocyclic coordinate中分离。我们的目的是开发用于在任何2D Riemannian空间上的高阶可促进系统分析和分类的工具，而不仅仅是欧几里得空间。
We revise a method by Kalnins, Kress and Miller (2010) for constructing a canonical form for symmetry operators of arbitrary order for the Schrödinger eigenvalue equation $HΨ\equiv (Δ_2 +V)Ψ=EΨ$ on any 2D Riemannian manifold, real or complex, that admits a separation of variables in some orthogonal coordinate system. We apply the method, as an example, to revisit the Tremblay and Winternitz (2010) derivation of the Painlevé VI potential for a 3rd order superintegrable flat space system that separates in polar coordinates and, as new results, we give a listing of the possible potentials on the 2-sphere that separate in spherical coordinates and 2-hyperbolic (two-sheet) potentials separating in horocyclic coordinates. In particular, we show that the Painlevé VI potential also appears for a 3rd order superintegrable system on the 2-sphere that separates in spherical coordinates, as well as a 3rd order superintegrable system on the 2-hyperboloid that separates in spherical coordinates and one that separates in horocyclic coordinates. Our aim is to develop tools for analysis and classification of higher order superintegrable systems on any 2D Riemannian space, not just Euclidean space.
