Noncommutative quantum mechanics

A general noncommutative 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((H) over cap (HO), (L) over cap (z)), where (H) over cap (HO) is the Hamiltonian of the two-dimensional harmonic oscillator and (L) over cap (z) is the z component of the angular momentum. For other finite values of theta the model can be solved by using perturbation theory.