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Spectral properties of one dimensional quasi-crystals

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Abstract

In this paper we prove that the one dimensional Schrödinger operator onl 2(ℤ) with potential given by:

$$\upsilon (n) = \lambda \chi _{[1 - \alpha , 1[} (x + n\alpha )\alpha \notin \mathbb{Q}$$

has a Cantor spectrum of zero Lebesgue measure for any irrationalα and any λ>0. We can thus extend the Kotani result on the absence of absolutely continuous spectrum for this model, to allEquationSource % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepGe9fr-xfr-x% frpeWZqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhaiiaacq% WFiiIZtuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbbiab% +nj8ubaa!4628!\[x \in \mathbb{T}$$ .

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Authors and Affiliations

  1. Université de Provence, Marseille

    J. Bellissard & B. Iochum

  2. Centre de Physique Theorique, CNRS, Luminy Case 907, F-13288, Marseille Cedex 9, France

    J. Bellissard, B. Iochum & D. Testard

  3. Dip. Fisica, Università di Roma “La Sapienza”, Ple A. Moro 2, I-00185, Roma, Italy

    E. Scoppola

  4. Université d'Aix, Marseille 2, Luminy

    D. Testard

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  1. J. Bellissard
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  2. B. Iochum
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  3. E. Scoppola
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  4. D. Testard
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Communicated by B. Simon

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Bellissard, J., Iochum, B., Scoppola, E. et al. Spectral properties of one dimensional quasi-crystals. Commun. Math. Phys. 125, 527–543 (1989). https://doi.org/10.1007/BF01218415

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  • DOI: https://doi.org/10.1007/BF01218415

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