98-569 Veseli\'c, Ivan

Localisation for random perturbations of periodic Schr\"odinger operators with regular Floquet eigenvalues (314K, ps) Aug 21, 98

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Abstract. We prove a localisation theorem for the ergodic Schr\"odinger operator $ H_\omega := H_0 + V_\omega $ on $ L^2 (\RR^d)$. Here $ V_\omega := \sum_{k \in \ZZ^d} \omega_k \, u( \cdot - k)$ is a nonnegative Anderson type random 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 perserved 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 \ni 0 $ such that $ H_\omega $ has only pure point spectrum in $ I $ for almost all $ \omega $.

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