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


We consider one-dimensional difference Schrödinger equations $$ \bigl[H(x,\omega)\varphi\bigr](n)\equiv -\varphi(n-1)-\varphi(n+1) + V(x + n\omega)\varphi(n) = E\varphi(n) , $$ $n \in \mathbb{Z}$, $x,\omega \in [0, 1]$ with real-analytic potential function $V(x)$. If $L(E,\omega_0)$ is greater than $0$ for all $E\in(E’, E”)$ and some Diophantine $\omega_0$, then the integrated density of states is absolutely continuous for almost every $\omega$ 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 $\mathop{\rm spec}( H(x,\omega))\cap (E’,E”)$. 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.