Some properties of an infinite family of deformations of the harmonic oscillator

摘要

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 Hk of the same kind as the familiar Freedman and Mende super-Calogero model. Some connection between both extensions is also outlined.