An infinite family of solvable and integrable quantum systems on a plane

An infinite family of exactly-solvable and integrable potentials on a plane is introduced. It is shown that all already known rational potentials with the above properties allowing separation of variables in polar coordinates are particular cases of this family. The underlying algebraic structure of the new potentials is revealed as well as its hidden algebra. We conjecture that all members of the family are also superintegrable and demonstrate this for the first few cases. A quasi-exactly-solvable and integrable generalization of the family is found.

💡 Research Summary

The paper introduces a broad, infinite family of two‑dimensional quantum potentials that are simultaneously exactly solvable and integrable. The authors begin by reviewing all previously known rational potentials that allow separation of variables in polar coordinates—most notably the Smorodinsky‑Winternitz family, the Calogero‑Moser‑Sutherland models, and various Dunkl‑operator constructions. They demonstrate that each of these earlier examples appears as a special case of a more general potential of the form

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