On resonances and the formation of gaps in the spectrum of   quasi-periodic Schroedinger equations
  We consider one-dimensional difference Schroedinger equations on the discrete line with a potential generated by evaluating a real-analytic potential function V(x) on the one-dimensional torus along an orbit of the shift x-->x+nw. If the Lyapunov exponent is positive for all energies and w, then the integrated density of states is absolutely continuous for almost every w. In this work we establish the formation of a dense set of gaps in the spectrum. Our approach is based on an induction on scales argument, and is therefore both constructive as well as quantitative. Resonances between eigenfunctions of one scale lead to "pre-gaps" at a larger scale. To pass to actual gaps in the spectrum, we show that these pre-gaps cannot be filled more than a finite (and uniformly bounded) number of times. To accomplish this, we relate a pre-gap to pairs of complex zeros of the Dirichlet determinants off the unit circle using the techniques of an earlier paper by the authors. Amongst other things, we establish in this work a non-perturbative version of the co-variant parametrization of the eigenvalues and eigenfunctions via the phases in the spirit of Sinai's (perturbative) description of the spectrum via his function as well as an multi-scale/finite volume approach to Anderson localization in this context. This allows us to relate the gaps in the spectrum with the graphs of the eigenvalues parametrized by the phase. Our infinite volume theorems hold for all Diophantine frequencies w up to a set of Hausdorff dimension zero. This is in contrast to earlier technology which only yielded exceptional sets of measure zero. Our only assumption on the potential apart from analyticity is the positivity of the Lyapunov exponent. 
Michael Goldstein, Wilhelm Schlag