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Résumé |
All, so far known explicitly two-dimensional quantum superintegrable systems in flat space are exactly solvable with spectra linear in quantum numbers (in agreement with the Montreal Conjecture by Tempesta-Turbiner-Winternirz (2001)). Their eigenfunctions are bivariate orthogonal polynomials in invariants of the (discrete) symmetry group. It includes the Smorodinsky-Winternitz potentials I-IV (or the Holt potential), the Fokas-Lagerstrom model, the 3-body Calogero (equivalently, A2 rational) and the Wolfes (equivalently, G2 rational, or I6) models, and the Tremblay-Turbiner-Winternitz system with integer index k=1,2,3,4. They have hidden (Lie) algebraic structure g(k) with various k, and they possess a (finite order) polynomial algebras of integrals generated by the Hamiltonian, two algebraically-independent integrals and their commutator. |
