AI 中文总结
该研究构造了二维方格三角形区域上Smorodinsky–Winternitz I超可积系统的有限离散实现，其对称代数具Hahn代数表示，谱问题由Tratnik型二元对偶Hahn多项式精确求解，连续极限可恢复原连续系统。
AI 中文摘要
我们在二维方格的三角形区域上构造了Smorodinsky–Winternitz I超可积系统的有限离散实现。该构造基于一对具有有限谱的对易数算符，以及与之关联的梯级算符结构。我们证明所得模型是极大超可积的，且其对称代数具有Hahn代数表示形式。其谱问题通过Tratnik型二元对偶Hahn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 I系统及其梯级算符、本征函数和对称代数，从而确立当前构造是该连续模型的真实有限离散实现。
英文摘要
We construct a finite discrete realization of the Smorodinsky--Winternitz I superintegrable system on a triangular region of the two-dimensional square lattice. The construction is based on a pair of commuting number operators with finite spectrum together with an associated ladder-operator structure. We show that the resulting model is maximally superintegrable and that its symmetry algebra admits a Hahn-algebra presentation. Its spectral problem is solved exactly in terms of the bivariate dual Hahn polynomials of Tratnik type. Finally, we show that the continuum limit recovers the continuous Smorodinsky--Winternitz I system together with its ladder operators, eigenfunctions and symmetry algebra, thereby establishing the present construction as a genuine finite discrete realization of the continuous model.
Comments20 pages, Minor edit changes in v.2
