# The spectral dichotomy for one-frequency Schrödinger operators

# The spectral dichotomy for one-frequency Schrödinger operators

In the theory of one-frequency Schrödinger operators, the best understood potentials have been those that can be somehow considered either small or large. Roughly, small potentials tend to inherit the behavior of the Laplacian and present absolutely continuous spectral measures (leading to good transport properties), while for large potentials it is Anderson localization that prevails. Dynamically, those two distinct local theories correspond to a good understanding of cocycles near constant ("the domain of KAM''), and nonuniformly hyperbolic cocycles. We have proposed to build a global theory by focusing on the description of the phase-transition, from KAM to non-uniform hyperbolicity, in the infinite-dimensional space of cocycles. Its goal was to prove the Spectral Dichotomy: typically, an operator is the direct sum of a "small-like'' and a "big-like'' operators, with disjoint spectra. We will describe the structure of the proof of the Spectral Dichotomy, which has two main parts. The firt one describes the locus of criticality from the point of view of Lyapunov exponents, i.e., the boundary of nonuniform hyperbolicity. The second one relates zero Lyapunov exponents and KAM.