Band edge localization beyond regular Floquet eigenvalues

We prove that Anderson localization near band edges of ergodic random Schrodinger operators with periodic background potential in $L^2(\mathbb{R}^d)$ in dimension two and larger is universal. By this we mean that Anderson localization holds without extra assumptions on the random variables and independently of regularity or degeneracy of the Floquet eigenvalues of the background operator. Our approach is based on an initial scale estimate the proof of which avoids Floquet theory altogether and uses instead an interplay between quantitative unique continuation and large deviation estimates. Furthermore, our reasoning is sufficiently flexible to prove this initial scale estimate in a non-ergodic setting, which promises to be an ingredient for understanding band edge localization also in these situations.