Influence of spatially varying pseudo-magnetic field on a 2D electron gas in graphene

The effect of a varying pseudo-magnetic field, which falls as $1/x^2$, on a two dimensional electron gas in graphene is investigated. By considering the second order Dirac equation, we show that its correct general solution is that which might present singular wavefunctions since such field induced by elastic deformations diverges as $x\rightarrow0$. We show that only this consideration yields the known relativistic Landau levels when we remove such elastic field. We have observed that the zero Landau level fails to develop for certain values of it. We then speculate about the consequences of these facts to the quantum Hall effect on graphene. We also analyze the changes in the relativistic cyclotron frequency. We hope our work being probed in these contexts, since graphene has great potential for electronic applications.