Two phases of the noncommutative quantum mechanics

TL;DR

This paper explores the phase structure of noncommutative quantum mechanics on a plane with magnetic field, revealing two distinct phases separated by a critical magnetic field value, with exact solutions at the critical point.

Contribution

It identifies and characterizes two different phases of noncommutative quantum mechanics distinguished by the magnetic field parameter, including an exactly solvable point.

Findings

Two phases separated by a critical magnetic field value

Finite states in one phase, infinite in the other

Perturbative spectrum near the critical point

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.