Non-Commutativity effects in the Dirac equation in crossed electric and magnetic fields

In this paper we present exact solutions of the Dirac equation on the non-commutative plane in the presence of crossed electric and magnetic fields. In the standard commutative plane such a system is known to exhibit contraction of Landau levels when the electric field approaches a critical value. In the present case we find exact solutions in terms of the non-commutative parameters $\eta$ (momentum non-commutativity) and $\theta$ (coordinate non-commutativity) and provide an explicit expression for the Landau levels. We show that non-commutativity preserves the collapse of the spectrum. We provide a dual description of the system: (i) one in which at a given electric field the magnetic field is varied and the other (ii) in which at a given magnetic field the electric field is varied. In the former case we find that momentum non-commutativity ($\eta$) splits the critical magnetic field into two critical fields while coordinates non-commutativity ($\theta$) gives rise to two additional critical points not at all present in the commutative scenario.