AI 中文总结
本文构造了二维正方形格点三角形区域上的离散Smorodinsky–Winternitz II超可积系统，证明其极大超可积且对称代数具Hahn代数表示，通过特定二元正交多项式精确求解谱问题，其连续极限对应连续系统，极限代数结构由Laguerre–Heun代数描述。
AI 中文摘要
我们在二维正方形格点的三角形区域上构造了一个离散Smorodinsky–Winternitz II超可积系统。该模型由一对具有有限谱的对易有限差分数值算符，以及相关的 ladder-operator（梯级算符）结构构建而成。我们证明该系统是极大超可积的，且其对称代数具有Hahn代数表示形式。谱问题通过与因式分解的A₂-Leonard对相关的混合Krawtchouk和对偶Hann型二元正交多项式得到精确求解。最后，我们证明其连续极限可恢复连续Smorodinsky–Winternitz II系统及其Hermite–Laguerre本征函数。我们进一步解释了离散对称代数的Hahn表示在该极限下如何变为奇异，而极限代数结构可由与笛卡尔和抛物变量分离相关的Laguerre–Heun代数自然描述。
英文摘要
We construct a discrete Smorodinsky--Winternitz II superintegrable system on a triangular region of the two-dimensional square lattice. The model is built from a pair of commuting finite-difference number operators with finite spectrum together with an associated ladder-operator structure. We show that it is maximally superintegrable and that its symmetry algebra admits a Hahn-algebra presentation. The spectral problem is solved exactly in terms of bivariate orthogonal polynomials of mixed Krawtchouk and dual Hahn type associated with the factorized $A_2$-Leonard pair. Finally, we show that the continuum limit recovers the continuous Smorodinsky--Winternitz II system together with its Hermite--Laguerre eigenfunctions. We further explain how the Hahn presentation of the discrete symmetry algebra becomes singular in this limit, while the limiting algebraic structure is naturally described by the Laguerre--Heun algebra associated with Cartesian and parabolic separation of variables.
Comments20 pages, small edit changes in v.2
