On the spectrum-generating superalgebras of the deformed one-dimensional quantum oscillators

We investigate the dynamical symmetry superalgebras of the one-dimensional matrix superconformal quantum mechanics with inverse square potential. They act as spectrum-generating superalgebras for the systems with the addition of the de Alfaro-Fubini-Furlan oscillator term. The undeformed quantum oscillators are expressed by 2" x 2" supermatrices; their corresponding spectrum-generating superalgebras are given by the osp(2n vertical bar 2) series. For n = 1, the addition of an inverse-square potential does not break the osp(212) spectrum-generating superalgebra. For n = 2, two cases of inverse square potential deformations arise. The first one produces Klein deformed quantum oscillators; the corresponding spectrum generating superalgebras are given by the D(2, 1; alpha) class, with a determining the inverse square potential coupling constants. The second n = 2 case corresponds to deformed quantum oscillators of non-Klein type. In this case, the osp(4 vertical bar 2) spectrum-generating superalgebra of the undeformed theory is broken to osp(2 vertical bar 2). The choice of the Hilbert spaces corresponding to the admissible range of the inverse square potential coupling constants and the possible direct sum of lowest weight representations of the spectrum-generating superalgebras is presented. Published under license by AIP Publishing.