On resonances and the formation of gaps in the spectrum of quasi-periodic Schrodinger equations

We consider one-dimensional difference Schrodinger equations [H(x, omega)phi] (n) phi(n - 1) phi(n + 1) + V(x + n omega)phi(n) = E phi(n), n is an element of Z, x,omega is an element of [0, 1] with real-analytic potential function V(x). If L(E, omega(0)) is greater than 0 for all E is an element of (E', E '') and some Diophantine omega(0), then the integrated density of states is absolutely continuous for almost every b.) close to omega(0), as shown by the authors in earlier work. In this paper we establish the formation of a dense set of gaps in spec(H(x, omega)) boolean AND (E', E ''). 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, one relates a pre-gap to pairs of complex zeros of the Dirichlet determinants off the unit circle. Amongst other things, we establish a nonperturbative 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 Lambda. 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 omega up to a set of Hausdorff dimension zero.