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: $[\hat{x}^1,\hat{x}^2]=iθ(1+ω_2 \hat x^2)$, $[\hat{p}^1,\hat{p}^2]=i\barθ$, $[\hat{x}^i,\hat{p}^j]=i\hbar_{eff}δ^{ij}$. We give an analytical method to solve the eigenvalue problem of the harmonic oscillator within this deformation algebra.

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