Non-Commutative Quantum Mechanics

A general non-commutative quantum mechanical system in a central potential $V=V(r)$ in two dimensions is considered. The spectrum is bounded from below and for large values of the anticommutative parameter $\theta$, we find an explicit expression for the eigenvalues. In fact, any quantum mechanical system with these characteristics is equivalent to a commutative one in such a way that the interaction $V(r)$ is replaced by $V=V({\widehat{H}}_{HO},{\widehat{L}}_z)$, where ${\widehat{H}}_{HO}$ is the hamiltonian of the two-dimensional harmonic oscillator and ${\widehat{L}}_z$ is z- component of the angular momentum. For other finite values of $\theta$ the model can be solved by using perturbation theory.