Discrete one-dimensional quasi-periodic Schrödinger operators with pure point spectrum. (English) Zbl 0908.34072

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 \(\varepsilon \rightarrow 0\).


34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34L05 General spectral theory of ordinary differential operators
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81V10 Electromagnetic interaction; quantum electrodynamics
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