96-565 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) Nov 11, 96
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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.

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