Dinaburg E.I.,Sinai Ya.,Soshnikov A.
Splitting of the Low Landau Levels into a Set of Positive Lebesgue Measure
under Small Periodic Perturbations.
(198K, PostScript)
ABSTRACT. We study two-dimensional Schr\"odinger operator with uniform magnetic field
and small periodic external field :
$$ \ L_{\varepsilon_0}(B) = - (\partial/\partial x -iBy)^2 \ - \partial^2/
\partial y^2 \ +\varepsilon_0 \ V(x,y) $$
where $ \ B \ $ is a magnetic field , and external potential $ \ V(x,y) \ $
has a special form
$$ V(x,y)=V_0(y) + \varepsilon_1 V_1(x,y) , $$
$ \ \varepsilon_0 \ , \varepsilon_1 \ $ are small parameters \ , the potential
$ \ V \ $ is smooth enough.\\
We restrict our attention to the case of typical $ \ B \ \ ( \
B/2\pi \ $ is Diophantine ) and the low Landau bands.
Representing $ \ L_{\varepsilon_0} \ $ as the direct integral of
one-dimensional quasi-periodic difference operators with long range potential
and employing recent results by E.Dinaburg about Anderson localization for
such operators, we construct for
$ \ L_{\varepsilon_0} \ $
the full set of generalized eigenfunctions.