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

N. Aizawa, F. Toppan


We investigate the dynamical symmetry superalgebras of the one-dimensional Matrix Superconformal Quantum Mechanics with Calogero 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 $2^n\times 2^n$ supermatrices; their corresponding spectrum-generating superalgebras are given by the $osp(2n|2)$ series.
For $n=1$ the addition of a Calogero potential does not break the $osp(2|2)$ spectrum-generating superalgebra.
For $n=2$ two cases of Calogero 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 $\alpha$ determining the Calogero coupling constants.
The second $n=2$ case corresponds to deformed quantum oscillators of non-Klein type. In this case the
$osp(4|2)$ spectrum-generating superalgebra of the undeformed theory is broken to $osp(2|2)$.
The choice of the Hilbert spaces corresponding to the admissible range of the Calogero coupling constants and
the possible direct sum of lowest weight representations of the spectrum-generating superalgebras is presented.

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