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Communications in Mathematical Physics

, Volume 189, Issue 2, pp 559–575 | Cite as

Splitting of the Low Landau Levels into a Set of Positive Lebesgue Measure under Small Periodic Perturbations

  • E.I. Dinaburg
  • Ya.G. Sinai
  • A.B. Soshnikov

Abstract:

We study the spectral properties of a two-dimensional Schrödinger operator with a uniform magnetic field and a small external periodic field:
$$$$
where
$$$$
and \(\), \(\) are small parameters. Representing \(\) 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 \(\) 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 \(\).

Keywords

Lebesgue Measure Landau Level Periodic Perturbation Small Periodic Positive Lebesgue Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • E.I. Dinaburg
    • 1
  • Ya.G. Sinai
    • 2
  • A.B. Soshnikov
    • 2
  1. 1.United Institute of Earth Physics, Russian Academy of Sciences, Moscow, RussiaRU
  2. 2.Princeton University, Mathematics Department, Fine Hall, Washington Road, Princeton, NJ 08544, USAUS

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