Oscillators in a (2+1)-dimensional noncommutative space

Abstract

We study the Harmonic and Dirac Oscillator problem extended to a three-dimensional noncommutative space where the noncommutativity is induced by the shift of the dynamical variables with generators of \documentclass[12pt]{minimal}\begin{document}$SL(2,\mathbb {R})$\end{document}SL(2,R) in a unitary irreducible representation considered in Falomir et al. [Phys. Rev. D 86, 105035 (2012)]. This redefinition is interpreted in the framework of the Levi's decomposition of the deformed algebra satisfied by the noncommutative variables. The Hilbert space gets the structure of a direct product with the representation space as a factor, where there exist operators which realize the algebra of Lorentz transformations. The spectrum of these models are considered in perturbation theory, both for small and large noncommutativity parameters, finding no constraints between coordinates and momenta noncommutativity parameters. Since the representation space of the unitary irreducible representations \documentclass[12pt]{minimal}\begin{document}$SL(2,\mathbb {R})$\end{document}SL(2,R) can be realized in terms of spaces of square-integrable functions, we conclude that these models are equivalent to quantum mechanical models of particles living in a space with an additional compact dimension.