Peter D. Hislop, Eric Soccorsi
Edge Currents for Quantum Hall Systems, I. One-Edge, Unbounded Geometries
(159K, Latex 2e)
ABSTRACT. Devices exhibiting the integer quantum Hall effect can be modeled
by one-electron Schroedinger operators describing the planar motion of an
electron in a perpendicular, constant magnetic field, and under the
influence of an electrostatic
potential. The electron motion is confined to unbounded subsets of the
plane by confining potential barriers. The edges of the confining potential
barrier create edge currents.
In this, the first of two papers, we
prove explicit lower
bounds on the edge currents associated with one-edge, unbounded
geometries formed by various confining
potentials. This work extends some known results that we review.
The edge currents
are carried by states with energy localized between any two Landau
levels. These one-edge geometries
describe the electron confined to certain unbounded regions
in the plane obtained by deforming half-plane regions.
We prove that the currents are stable under various
potential perturbations, provided the perturbations
are suitably small relative to
the magnetic field strength, including perturbations by random potentials.
For these cases of one-edge geometries,
the existence of, and
the estimates on, the edge currents imply that the corresponding
Hamiltonian has intervals of absolutely continuous spectrum.
In the second paper of this series, we consider the edge currents
associated with two-edge geometries
describing bounded, cylinder-like
regions, and unbounded, strip-like, regions.