03-396 Joaquim Puig

Cantor Spectrum for the Almost Mathieu Operator. Corollaries of localization,reducibility and duality. (348K, Postscript) Sep 1, 03

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Abstract. In this paper we use results on reducibility, localization and duality for the Almost Mathieu operator, \[ \left(H_{b,\phi} x\right)_n= x_{n+1} +x_{n-1} + b \cos\left(2 \pi n \omega + \phi\right)x_n \] on $l^2(\mathbb{Z})$ and its associated eigenvalue equation to deduce that for $b \ne 0,\pm 2$ and $\omega$ Diophantine the spectrum of the operator is a Cantor subset of the real line. This solves the so-called ``Ten Martini Problem'' for these values of $b$ and $\omega$. Moreover, we prove that for $|b|\ne 0$ small enough or large enough all spectral gaps predicted by the Gap Labelling theorem are open. (This is a revised version of preprint mp_arc 03-145).

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