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

Abstract

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.

Authors

Michael Goldstein

Department of Mathematics
University of Toronto
Toronto, Ontario
Canada M5S 2E4

Wilhelm Schlag

Department of Mathematics
The University of Chicago
Chicago IL 60637