### Abstract

We study the spectral properties of a two-dimensional Schrödinger operator with a uniform magnetic field and a small external periodic field: (formula presented) where V(x,y) = V_{0}(y) + ε_{1}V_{1}(x,y), and ε_{0}, ε_{1} are small parameters. Representing L_{ε0} as the direct integral of one-dimensional quasi-periodic difference operators with long-range potential and employing recent results of E.I.Dinaburg about Anderson localization for such operators (we assume 2π/B to be typical irrational) we construct the full set of generalised eigenfunctions for the low Landau bands. We also show that the Lebesgue measure of the low bands is positive and proportional in the main order to ε_{0}.

Original language | English (US) |
---|---|

Pages (from-to) | 559-575 |

Number of pages | 17 |

Journal | Communications In Mathematical Physics |

Volume | 189 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1997 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

## Fingerprint Dive into the research topics of 'Splitting of the low Landau levels into a set of positive Lebesgue measure under small periodic perturbations'. Together they form a unique fingerprint.

## Cite this

*Communications In Mathematical Physics*,

*189*(2), 559-575. https://doi.org/10.1007/s002200050217