Localization for random perturbations of periodic Schroedinger operators with regular Floquet eigenvalues

Details Localization for random perturbations of periodic Schroedinger operators with regular Floquet eigenvalues Localization for random perturbations of periodic Schroedinger operators with regular Floquet eigenvalues

2005-10-17

arxiv.org/abs/math-ph/0510063v1

Physics

Mathematical Physics

21 pages

Scientific paper

We prove a localization theorem for continuous ergodic Schr\"odinger operators $ H_\omega := H_0 + V_\omega $, where the random potential $ V_\omega $ is a nonnegative Anderson-type perturbation of the periodic operator $ H_0$. We consider a lower spectral band edge of $ \sigma (H_0) $, say $ E= 0 $, at a gap which is preserved by the perturbation $ V_\omega $. Assuming that all Floquet eigenvalues of $ H_0$, which reach the spectral edge 0 as a minimum, have there a positive definite Hessian, we conclude that there exists an interval $ I $ containing 0 such that $ H_\omega $ has only pure point spectrum in $ I $ for almost all $ \omega $.

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