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Continuity properties of the electronic spectrum of 1D quasicrystals

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Abstract

In this paper we consider operatorsH(α,x) defined onl 2(ℤ) by

$$H(\alpha ,x)\psi (n) = \sum\limits_{m \in \mathbb{Z}} {t_m \circ \phi ^{ - n} (\alpha ,x)} \psi (n - m),$$

where ϕ(α,x)=(α,x−α),t m is in the algebra of bounded periodic functions on ℝ2 generated by the characteristic functions of the sets

$$\phi ^n \left\{ {(\alpha ,x) \in \mathbb{R}^2 \left| {\left. {1 - \alpha \leqq x< \alpha (\bmod 1)} \right\}.} \right.} \right.$$

This class of hamiltonian includes the Kohmoto model numerically computed by Ostlund and Kim, where the potential is given by

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

(see [B.I.S.T.]). We prove that the spectrum (as a set) ofH(α,x), varies continuously with respect to α near each irrational, for anyx. We also show that the various strong limits obtained as α converges to a rational numberp/q describe either a periodic medium or a periodic medium with a localized impurity. The corresponding spectrum has eigenvalues in the gaps and the right and left limits as α→p/q do not coincide, for the Kohmoto model. The results are obtained throughC *-algebra techniques.

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Author notes

  1. J. Bellissard & B. Iochum

    Present address: Université de Provence, Marseille

  2. D. Testard

    Present address: Université d'Aix-Marseille II, Luminy

Authors and Affiliations

  1. Centre de Physique Théorique, C.N.R.S. Luminy Case 907, F-13288, Marseille Cedex 9, France

    J. Bellissard, B. Iochum & D. Testard

Authors
  1. J. Bellissard
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  2. B. Iochum
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  3. D. Testard
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Additional information

Communicated by H. Araki

Laboratoire propre L.P. 7061 du Centre National de la Recherche Scientifique

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Bellissard, J., Iochum, B. & Testard, D. Continuity properties of the electronic spectrum of 1D quasicrystals. Commun.Math. Phys. 141, 353–380 (1991). https://doi.org/10.1007/BF02101510

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