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.