Localization in \(\nu\)-dimensional incommensurate structures (Q795983)

The authors construct a class \({\mathcal P}\) of quasi-periodic functions having singularities for which the discrete Schrödinger operator, \[ \epsilon \sum_{| e| =1}\psi(n-e)+V(x-\omega n)\psi(n)=E\quad \psi(n), \] has only pure point spectrum at small \(\epsilon\) with exponentially localized states and a dense set of eigenvalues in the real line. Here \(V\in {\mathcal P}\) and is of period 1, \(x\in R\), \(n\in {\mathbb{Z}}^{\nu}\) and \(\omega\) is a \(\nu\)-dimensional vector satisfying the diophantine condition \(\exists \gamma>0\), \(\sigma>\nu\), \(| \omega n+m| \geq \gamma | n|^{-\sigma}, \forall m\in {\mathbb{Z}}\), \(n\in {\mathbb{Z}}^{\nu}\), where \(| n| =\sum^{\nu}_{\mu =1}| n_{\mu}|\). After describing the properties of the class \({\mathcal P}\) and introducing an algebra of holomorphic kernels, the result is obtained by techniques based on the Kolmogoroff, Arnold, Moser method which was used in some previous work of the first two authors and \textit{D. Testard} [Commun. Math. Phys. 88, 207-234 (1983; Zbl 0542.35059)].