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Exact solutions of the 2D Schrödinger equation with central potentials induced by the non-commutativity of space

We obtain exact solutions of the 2D Schrödinger equation with the central potentials $V(r)=ar^2+br^{-2}+cr^{-4}$ and $V(r)=ar^{-1}+br^{-2}$ in a non-commutative space up to the first order of non-commutativity parameter using the power-series expansion method similar to the 2D Schrödinger equation with the singular even-power and inverse-power potentials respectively in commutative space. We derive the exact non-commutative energy levels and show that the energy is shifted to $m$ levels, as in the Zeeman effect.