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Devices exhibiting the integer quantum Hall effect can be modeled byone-electron Schroedinger operators describing the planar motion of an electronin a perpendicular, constant magnetic field, and under the influence of anelectrostatic potential. The electron motion is confined to unbounded subsetsof the plane by confining potential barriers. The edges of the confiningpotential barrier create edge currents. In this, the first of two papers, weprove explicit lower bounds on the edge currents associated with one-edge,unbounded geometries formed by various confining potentials. This work extendssome known results that we review. The edge currents are carried by states withenergy localized between any two Landau levels. These one-edge geometriesdescribe the electron confined to certain unbounded regions in the planeobtained by deforming half-plane regions. We prove that the currents are stableunder various potential perturbations, provided the perturbations are suitablysmall relative to the magnetic field strength, including perturbations byrandom potentials. For these cases of one-edge geometries, the existence of,and the estimates on, the edge currents imply that the correspondingHamiltonian has intervals of absolutely continuous spectrum. In the secondpaper of this series, we consider the edge currents associated with two-edgegeometries describing bounded, cylinder-like regions, and unbounded,strip-like, regions.Comment: 68 page

EDGE CURRENTS FOR QUANTUM HALL SYSTEMS I: ONE-EDGE, UNBOUNDED GEOMETRIES