Symmetries and supersymmetries of the quantum harmonic oscillator

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Abstract
The supersymmetric version of the one-dimensional harmonic oscillator is studied by taking into account its conformal properties. The largest superalgebra of symmetries and supersymmetries is derived as Osp(2/2) Square Operator Sh(1), the semidirect sum of Osp(2/2) and the Heisenberg superalgebra. Through a one-to-one correspondence between the nonrelativistic free case and the harmonic oscillator description, the authors deduce the (expected) supersymmetries of the Schrodinger equation. The above structure appears as the largest spectrum-generating superalgebra of the harmonic oscillator and its representation within an energy basis is given. The physical three-dimensional case is also considered when the maximal set of (super)symmetries is required and this case is compared with recent work.