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 $Bθ=c\hbar^2/e$, where $θ$ is a parameter of noncommutativity. In this point the system reduces to exactly-solvable one-dimensional system. When $κ=1-eBθ/c\hbar^2<0$ there is a finite number of states corresponding to the given value of the angular momentum. In another phase, i.e. when $κ>0$ the number of states is infinite. The perturbative spectrum near the critical point $κ=0$ is computed.

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