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 $\omega$.