## Abstract

Devices exhibiting the integer quantum Hall effect can be modeled by one-electron Schrödinger 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.

Original language | English |
---|---|

Pages (from-to) | 71-115 |

Number of pages | 45 |

Journal | Reviews in Mathematical Physics |

Volume | 20 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2008 |

### Bibliographical note

Funding Information:We thank J.-M. Combes for many discussions on edge currents and their role in the IQHE. We also thank F. Germinet, G.-M. Graf, E. Mourre and H. Schulz-Baldes for fruitful discussions. Some of this work was done when ES was visiting the Mathematics Department at the University of Kentucky and he thanks the Department for its hospitality and support. We thank the referees for a careful reading of the manuscript and helpful comments. The first-named author was supported in part by NSF grant DMS-0503784.

### Funding

We thank J.-M. Combes for many discussions on edge currents and their role in the IQHE. We also thank F. Germinet, G.-M. Graf, E. Mourre and H. Schulz-Baldes for fruitful discussions. Some of this work was done when ES was visiting the Mathematics Department at the University of Kentucky and he thanks the Department for its hospitality and support. We thank the referees for a careful reading of the manuscript and helpful comments. The first-named author was supported in part by NSF grant DMS-0503784.

Funders | Funder number |
---|---|

National Science Foundation (NSF) | DMS-0503784 |

Directorate for Mathematical and Physical Sciences | 0503784 |

## Keywords

- Asymptotic velocity
- Edge states
- Landau Hamiltonians
- Perturbation theory
- Quantum Hall effect
- Spectral theory

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics