MPEJ Volume 5, No.3, 17pp Received: Mar 24, 1999, Revised: Jun 23, 1999, Accepted: Jun 24, 1999 Stephan De Bievre, Joseph V. Pule Propagating Edge States for a Magnetic Hamiltonian ABSTRACT: We study the quantum motion of a charged particle in a half plane, subject to a perpendicular constant magnetic field $B$ and to an arbitrary weak impurity potential $W_B$ (i.e. $||W_B||_\infty < \delta B$, for some $\delta$ small enough). We show that there exist states propagating with a speed of size $B^{1/2}$ along the edge, no matter how fast $W_B$ fluctuates. As a consequence, the spectrum of the Hamiltonian is purely absolutely continuous in a spectral interval of size $\gamma B$ ($0 < \gamma < 1$) between the Landau levels of the system without edge or potential, so that the corresponding eigenstates are extended. This then provides a rigorous proof of a phenomenon pointed out by Halperin in his work on the quantum Hall effect.