Completeness of multiseparable superintegrability on the complex 2-sphere

E. G. Kalnins, W. Miller, G. S. Pogosyan

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The possibility that Schrödinger's equation with a given potential can separate in more than one coordinate system is intimately connected with the notion of superintegrability. Here we demonstrate, for nondegenerate potentials, how to establish a complete list of such potentials that are superintegrable on the complex 2-sphere, using essentially algebraic means. We classify all such potentials that admit a pair of second-order constants of motion. Here 'nondegenerate' means that the potentials depend on four independent parameters. The method of proof generalizes to other spaces and dimensions. We show for the 2-sphere that all these superintegrable systems possess the remarkable property that they correspond to quadratic algebras, and we work out the detailed structure relations and their quantum analogues.

Original languageEnglish (US)
Pages (from-to)6791-6806
Number of pages16
JournalJournal of Physics A: Mathematical and General
Issue number38
StatePublished - Sep 29 2000

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