DOI: 10.1063/5.0346146 ISSN: 0022-2488

The 2D Smorodinsky–Winternitz II system and the Laguerre–Heun algebra

Vutha Vichhea Chea, Luc Vinet, Alexei Zhedanov
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We identify the quadratic symmetry algebra of the two-dimensional Smorodinsky–Winternitz II system with a Laguerre-type confluent Heun algebra. The system is separable in Cartesian and parabolic coordinates. The complementary Cartesian separation operator Y=∂y2−ω2y2+1/4−c2y2 is of Laguerre type, while the parabolic integral W = L2 is its algebraic Heun partner. With Z = [Y, W], the defining relations are [Y,Z]=16ω2W−2bY,[W,Z]=6Y2−4HY+2bW+8ω2(1−c2), where H is central. This gives a direct superintegrable realization of the Laguerre–Heun algebra.

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