Continuity properties of the electronic spectrum of 1D quasicrystals

In this paper we consider operatorsH(α,x) defined onl2(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.