Some properties of an infinite family of deformations of the harmonic oscillator
Abstract
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 superCalogero model. Some connection between both extensions is also outlined.
 Publication:

Symmetries in Nature: Symposium in Memoriam Marcos Moshinsky
 Pub Date:
 December 2010
 DOI:
 10.1063/1.3537856
 arXiv:
 arXiv:1012.3357
 Bibcode:
 2010AIPC.1323..275Q
 Keywords:

 harmonic oscillators;
 supersymmetry;
 integral equations;
 partial differential equations;
 03.65.Ge;
 11.30.Pb;
 02.30.Rz;
 02.30.Jr;
 Solutions of wave equations: bound states;
 Supersymmetry;
 Integral equations;
 Partial differential equations;
 Mathematical Physics;
 Quantum Physics
 EPrint:
 10 pages, no figure