CONTINUITY PROPERTIES OF THE ELECTRONIC-SPECTRUM OF 1D QUASI-CRYSTALS

In this paper we consider operators H(alpha,x) defined on l2(Z) by [GRAPHICS] where phi(alpha,x) = (alpha, x-alpha), t(m) is in the algebra of bounded periodic functions on R2 generated by the characteristic functions of the sets phi-n{(alpha,x) is-an-element-of R2\1-alpha less-than-or-equal-to x < alpha (mod 1)}. 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 is-an-element-of Z, x, lambda, alpha is-an-element-of R (see [B.I.S.T.]). We prove that the spectrum (as a set) of H(alpha,x) varies continuously with respect to alpha near each irrational, for any x. We also show that the various strong limits obtained as alpha converges to a rational number p/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 alpha --> p/q do not coincide, for the Kohmoto model. The results are obtained through C*-algebra techniques.