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

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 $X_a=e_a^{\mu }(x){\mathrm{\partial }}_{\mu }=({\delta }_a^{\mu }+{\omega }_{ab}^{\mu }{x}^b){\mathrm{\partial }}_{\mu }$, which induce a dynamical star product. The usual multiplication law can be hence reproduced in the ${\omega }_{ab}^{\mu }$ 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.