Two-dimensional electron gas in a uniform magnetic field in the presence of a δ-impurity

The density of states and the Hall conductivity of a two-dimensional electron gas in a uniform magnetic field and in the presence of a delta-impurity are exactly calculated using elementary field theoretic techniques. The impurity creates one localized state per Landau level, but the Hall conductivity is unaffected. Our treatment is explicitly gauge invariant, and can be easily adapted to other problems involving zero-range potentials.