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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