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

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 $H_k$ on the plane, depending on a positive real parameter $k$. Two algebraic extensions of $H_k$ are described. The first one, based on the elements of the dihedral group $D_{2k}$ and a Dunkl operator formalism, provides a convenient tool to prove the superintegrability of $H_k$ for odd integer $k$. The second one, employing two pairs of fermionic operators, leads to a supersymmetric extension of $H_k$ of the same kind as the familiar Freedman and Mende super-Calogero model. Some connection between both extensions is also outlined.