Some Properties of an Infinite Family of Deformations of the Harmonic Oscillator

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In memory of Marcos Moshinsky, who promoted the algebraic study of the harmonic oscillator, some results recently obtained on an infinite family of deformations of such a system are reviewed. This set, which was introduced by Tremblay, Turbiner, and Winternitz, consists in some Hamiltonians Hₖ on the plane, depending on a positive real parameter k. Two algebraic extensions of Hₖ are described. The first one, based on the elements of the dihedral group D₂ₖ and a Dunkl operator formalism, provides a convenient tool to prove the superintegrability of Hₖ for odd integer k. The second one, employing two pairs of fermionic operators, leads to a supersymmetric extension of Hₖ of the same kind as the familiar Freedman and Mende super-Calogero model. Some connection between both extensions is also outlined.