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arXiv · 1610.00380

On localization and the spectrum of multi-frequency quasi-periodic operators

Michael Goldstein·Wilhelm Schlag·Mircea Voda
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Abstract

We study multi-frequency quasi-periodic Schr\"odinger operators on $\mathbb{Z}$ in the regime of positive Lyapunov exponent and for general analytic potentials. Combining Bourgain's semi-algebraic elimination of multiple resonances with the method of elimination of double resonances via resultants, we establish exponential finite-volume localization as well as the separation between the eigenvalues. In a follow-up paper we develop the method further to show that for potentials given by large generic trigonometric polynomials the spectrum consists of a single interval, as conjectured by Chulaevski and Sinai.

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Michael Goldstein, Wilhelm Schlag, Mircea Voda. 2016-10-03. On localization and the spectrum of multi-frequency quasi-periodic operators. https://arxiv.org/abs/1610.00380

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