1999/03/16 by S. De Bievre, J. V. Pule
Physics and Astronomy · Mathematics · #math-ph #math.MP
published as Mathematical Physics Electronic Journal, vol. 5, 1999 · 14 pages, 1 Figure
arxiv created 1999/03/16 · arxiv updated 2009/11/30
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 WB, assumed to be weak in the sense that ||WB||_∞ < 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 B1/2 in the y-direction, no matter how fast WB 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.