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A metal-insulator transition for the almost Mathieu model

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Abstract

We study the spectrum of the almost Mathieu hamiltonian:

$$\left( {H_x \psi } \right)\left( n \right) = \psi \left( {n + 1} \right) + \psi \left( {n - 1} \right) + 2\mu \cos \left( {x - n\theta } \right)\psi \left( n \right),n \in \mathbb{Z}$$

where ϑ is an irrational number andx is in the circle\(\mathbb{T}\). For a small enough coupling constant μ and anyx there is a closed energy set of non-zero measure in the absolutely continuous spectrum ofH. For sufficiently high μ and almost allx we prove the existence of a set of eigenvalues whose closure has positive measure. The two results are obtained for a subset of irrational numbers ϑ of full Lebesgue measure.

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Authors and Affiliations

  1. Centre de Physique Théorique, CNRS, F-13288, Marseille Cedex 9, France

    J. Bellissard, R. Lima & D. Testard

  2. Université de Provence, France

    J. Bellissard

  3. Centre Universitaire d'Avignon, France

    D. Testard

Authors
  1. J. Bellissard
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  2. R. Lima
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  3. D. Testard
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Communicated by B. Simon

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Bellissard, J., Lima, R. & Testard, D. A metal-insulator transition for the almost Mathieu model. Commun.Math. Phys. 88, 207–234 (1983). https://doi.org/10.1007/BF01209477

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  • DOI: https://doi.org/10.1007/BF01209477

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