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

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 $\omega$.Comment: 21 page

Spectral gaps for self-adjoint second order operators

We consider a second order self-adjoint operator in a domain which can be bounded or unbounded. The boundary is partitioned into two parts with Dirichlet boundary condition on one of them, and Neumann condition on the other. We assume that the potential part of this operator is non-negative. We add a localized perturbation assuming that it produces two negative isolated eigenvalues being the two lowest spectral values of the resulting perturbed operator. The main result is a lower bound on the gap between these two eigenvalues. It is given explicitly in terms of the geometric properties of the domain and the coefficients of the perturbed operator. We apply this estimate to several asymptotic regimes studying its dependence on various parameters. We discuss specific examples of operators to which the bounds can be applied.Comment: Accepted for publication in Journal of Analysis and its Applications (Journal of Analysis and its Applications). 33 pages, 3 figures. (The estimates are more precise than in the first version.