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A metal-insulator transition for the almost Mathieu model. (English) Zbl 0542.35059

The authors examine the spectral properties of the almost Mathieu Hamiltonian \[ H_ x^{\mu}\psi(n)=\psi(n+1)+\psi(n-1)+2\mu \quad \cos(x-n\theta)\psi(n),\quad n\in {\mathbb{Z}} \] where \(\theta\) is an irrational number and \(x\in T\). For sufficiently small coupling constant \(\mu\) and any x they show that there is a closed energy set of non-zero measure in the absolutely continuous spectrum of H. As well for sufficiently high \(\mu\) and almost all x the existence is established of a set of eigenvalues whose closure has positive measure. These results are obtained for a subset of irrational numbers \(\theta\) of full Lebesgue measure.
The first result depends on a reformulation of Rüssmann of results of Dinaberg and Sinai for studying the spectrum of \(H_ x\) for small values of \(\mu\) and has its basis in Newton’s algorithm for computing roots of certain equations together with a small divisor analysis. For the second result the authors use André-Aubry duality to show the existence of an infinite number of eigenvalues with exponentially localised eigenfunctions and finally techniques of number theory to show that the closure of this set is strictly positive.
Reviewer: M.Thompson

MSC:

35P25 Scattering theory for PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
47A40 Scattering theory of linear operators
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