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

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We obtain exact solutions of the 2D Schrödinger equation with the central potentials V(r)=ar²+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.