Position-dependent noncommutative quantum models: Exact solution of the harmonic oscillator

This paper is devoted to find the exact solution of the harmonic oscillator in a position-dependent 4-dimensional noncommutative phase space. The noncommutative phase space that we consider is described by the commutation relations between coordinates and momenta: $[{\widehat{x}}^1,{\widehat{x}}^2]=i\theta (1+{\omega }_2{\widehat{x}}^2)$, $[{\widehat{p}}^1,{\widehat{p}}^2]=i\bar{\theta }$, $[{\widehat{x}}^i,{\widehat{p}}^j]=i{\mathrm{\hslash }}_{eff}{\delta }^{ij}$. We give an analytical method to solve the eigenvalue problem of the harmonic oscillator within this deformation algebra.