Harmonic oscillator in twisted Moyal plane: eigenvalue problem and relevant properties

Abstract

The paper reports on a study of a harmonic oscillator (ho) in the twisted Moyal space, in a well defined matrix basis, generated by the vector fields $X_{a}=e_{a}^μ(x)\partial_μ=(δ_{a}^μ+ω_{ab}^μx^{b})\partial_μ$, which induce a dynamical star product. The usual multiplication law can be hence reproduced in the $ω_{ab}^μ$ null limit. The star actions of creation and annihilation functions are explicitly computed. The ho states are infinitely degenerate with energies depending on the coordinate functions.