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We obtain exact solutions of the 2D Schr\"odinger equation with the centralpotentials $V(r)=ar^2+br^{-2}+cr^{-4}$ and $V(r)=ar^{-1}+br^{-2}$ in anon-commutative space up to the first order of non-commutativity parameterusing the power-series expansion method similar to the 2D Schr\"odingerequation with the singular even-power and inverse-power potentials respectivelyin commutative space. We derive the exact non-commutative energy levels andshow that the energy is shifted to $m$ levels, as in the Zeeman effect.

Exact solutions of the 2D Schr\"odinger equation with central potentials induced by the non-commutativity of space