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arXiv · 2608.12899

A discrete Smorodinsky--Winternitz I superintegrable system

Vutha Vichhea Chea·Luc Vinet
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Abstract

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.

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Vutha Vichhea Chea, Luc Vinet. 2026-08-13. A discrete Smorodinsky--Winternitz I superintegrable system. https://arxiv.org/abs/2608.12899

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