Two phases of the noncommutative quantum mechanics

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 = ch(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 /ch(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. (C) 2001 Elsevier Science B.V. All rights reserved.