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

We obtain exact solutions of the 2D Schr"odinger equation with the central potentials $V(r)=a{r}^2+b{r}^{-2}+c{r}^{-4}$ and $V(r)=a{r}^{-1}+b{r}^{-2}$ in a non-commutative space up to the first order of non-commutativity parameter using the power-series expansion method similar to the 2D Schr"odinger 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.