Discrete one-dimensional quasi-periodic Schrödinger operators with pure point spectrum (Q1389802)

The author considers properties of discrete quasiperiodic Schrödinger operators in one dimension, defined by \[ ({\mathcal L}_{\theta} u)_n= -\varepsilon (u_{n+1} + u_{n-1}) + E (\theta + n\omega) u_n, \] where \(\omega\) is a real number and \(E\) is a smooth function on the torus belonging to the Gevrey class. The main result is a proof of the following theorem: Assume that \(E\) and \(\omega\) are as stated above. Then there exists a constant \(\varepsilon_0\) such that if \(| \varepsilon|< \varepsilon_0\), then \({\mathcal L}_{\theta}\) has a pure point spectrum with a complete set of eigenfunctions in \(l^2({\mathbb{Z}})\) for a.e. \(\theta\). Moreover, the measure of the set \([ \inf E, \sup E]\;\sigma ({\mathcal L}_{\theta})\) goes to \(0\) as \(\varepsilon \rightarrow 0\).