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

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 .