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

## TL;DR

This paper explores the spectrum-generating superalgebras of one-dimensional deformed quantum oscillators, revealing how inverse-square potentials influence their algebraic structures and symmetries.

## Contribution

It characterizes the superalgebras associated with deformed quantum oscillators, including Klein and non-Klein types, and analyzes how inverse-square potentials affect their symmetry structures.

## Key findings

- For n=1, inverse-square potential preserves osp(2|2) superalgebra.
- For n=2, Klein deformations lead to D(2,1;α) superalgebras.
- Non-Klein deformations break osp(4|2) to osp(2|2).

## Abstract

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 $2^n\times 2^n$ supermatrices; their corresponding spectrum-generating superalgebras are given by the $osp(2n|2)$ series. For $n=1$ the addition of a inverse-square potential does not break the $osp(2|2)$ 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 $\alpha$ 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 $osp(4|2)$ 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 inverse-square potential coupling constants and the possible direct sum of lowest weight representations of the spectrum-generating superalgebras is presented.