Propagating edge states for a magnetic Hamiltonian

Abstract

We study the quantum mechanical motion of a charged particle moving in a half plane (x>0) subject to a uniform constant magnetic field B directed along the z-axis and to an arbitrary impurity potential W_B, assumed to be weak in the sense that ||W_B||_\infty < δB, for some δsmall enough. We show rigorously a phenomenon pointed out by Halperin in his work on the quantum Hall effect, namely the existence of current carrying and extended edge states in such a situation. More precisely, we show that there exist states propagating with a speed of size B^{1/2} in the y-direction, no matter how fast W_B fluctuates. As a result of this, we obtain that the spectrum of the Hamiltonian is purely absolutely continuous in a spectral interval of size γB (for some γ<1) between the Landau levels of the unperturbed system (i.e. the system without edge or potential), so that the corresponding eigenstates are extended.