Two phases of the noncommutative quantum mechanics

Abstract

We consider quantum mechanics on the noncommutative plane in the presence of magnetic field $B$. We show, that the model has two essentially different phases separated by the point $B\theta=c\hbar^2/e$, where $\theta$ is a parameter of noncommutativity. In this point the system reduces to exactly-solvable one-dimensional system. When $\kappa=1-eB\theta/c\hbar^2<0$ there is a finite number of states corresponding to the given value of the angular momentum. In another phase, i.e. when $\kappa>0$ the number of states is infinite. The perturbative spectrum near the critical point $\kappa=0$ is computed.