Perturbation Theory on the Non-Commutative Plane with a Singular Potential

In this article we study the problem of a non-relativistic particle in the presence of a singular potential in the noncommutative plane. The potential contains a term proportional to $1/R^2$, where $R^2$ is the squared distance to the origin in the noncommutative plane. We find that the spectrum of energies is non analytic in the noncommutativity parameter $theta$.

Valley polarized electronic beam splitting in graphene

We show how the trigonal warping effect in doped graphene can be used to produce fully valley polarized currents. We propose a device that acts both as a beam splitter and a collimator of these electronic currents. The result is demonstrated trough an optical analogy using two dimensional photonic crystals.