ON SUPERSYMMETRIC HARMONIC-OSCILLATORS AND THE GREEN-CUSSON ANSATZE

New properties on parabosonic and parafermionic Cusson's realizations of the Green's Anzatze are first obtained. Then paraproperties are shown to have a specific interest for our real world, where bosons and fermions have to be superposed if Nature is supersymmetric, these results being obtained on supersymmetric harmonic oscillators. In particular, from the Green-Cusson Ansatze, we construct a class of supersymmetric Hamiltonians that are more general than the ones issued from standard or spin-orbit coupling considerations. We explicitly consider the simplest (previously unknown) super-Hamiltonian constructed from paraproperties in quantum mechanics and establish its invariance Lie superalgebra.