## Abstract

We study purely magnetic Schrödinger operators in two dimensions (x, y) with magnetic fields b(x) that depend only on the x-coordinate. The magnetic field b(x) is assumed to be bounded, there are constants 0<b_{-}<b_{+}<∞ such that b_{-}≤b(x)≤b_{+}, and outside of a strip of small width -ε<x<ε, where 0<ε<b^{-1/2}_{-}, we have b(x)=b_{±}x for ±x>ε. The case of a jump in the magnetic field at x=0 corresponding to ε=0 is also studied. We prove that the magnetic field creates an effective barrier near x=0 that causes edge currents to flow along it consistent with the classical interpretation. We prove lower bounds on edge currents carried by states with energy localized inside the energy bands of the Hamiltonian. We prove that these edge current-carrying states are well-localized in x to a region of size b--1/2, also consistent with the classical interpretation. We demonstrate that the edge currents are stable with respect to various magnetic and electric perturbations. These lower bounds on the edge current hold for all time. For a family of perturbations compactly supported in the y-direction, we prove that the time asymptotic current exists and satisfies the same lower bound.

Original language | English |
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Pages (from-to) | 594-624 |

Number of pages | 31 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 422 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2015 |

### Bibliographical note

Funding Information:PDH thanks the Centre de Physique Théorique, CNRS, Luminy, Marseille, France, for its hospitality. PDH was partially supported by the Université de Toulon , La Garde, France, and National Science Foundation grant DMS 11-03104 during the time part of the work was done. ES thanks the University of Kentucky, Lexington, KY, USA, where part of this work was done, for its warm welcome.

Publisher Copyright:

© 2014 Elsevier Inc.

## Keywords

- Asymptotic velocity
- Magnetic barrier
- Magnetic edge states
- Magnetic field
- Schrödinger operators

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics