Singular continuous spectrum for a class of substitution Hamiltonians
Let \((\omega_n)_{n\in\mathbb{Z}}\) be a sequence in the orbit closure of a primitive substitutive sequence arbitrarily extended to a sequence indexed by \(\mathbb{Z}\). Let \((V_\omega(n))_n\) be the image of \((\omega_n)\) by a real-valued map. The one-dimensional Schrödinger operator associated with \((V_\omega(n))\) is defined by \[ (H_\omega\Phi)(n)=\Phi(n+1)+\Phi(n-1)+V_\omega(n)\Phi(n),\quad\text{for}\quad \Phi\in\ell^2(\mathbb{Z}). \] These operators are related to the study of (one-dimensional) models of quasicrystals. Combinatorial properties of the sequence \((V_\omega(n))_n\) imply properties of the spectrum of \(H_\omega\) [see \textit{A. Bovier} and \textit{J.-M. Ghez}, Commun. Math. Phys. 158, 45-66 (1993; Zbl 0820.35099) and Erratum ibid. 166, 431-432 (1994; Zbl 0841.35072) when the sequence begins with long squares; see \textit{A. Hof}, \textit{O. Knill} and \textit{B. Simon}, Commun Math. Phys. 174, 149-159 (1995; Zbl 0839.11009) and \textit{J.-P. Allouche}, J. Math. Phys. 38, No. 4, 1843-1848 (1997) when the sequence contains long palindromes)]. In the paper under review the author proves the following theorem: If \((V_\omega(n))\) is the image of a sequence in the orbit closure of a primitive substitutive sequence, and if this primitive substructure sequence contains a fourth power (i.e., a subword \(wwww)\), then, for almost all elements \(\omega\) in this orbit closure, \(H_\omega\) has no eigenvalues and its spectrum is singular continuous. Note that Ref. [18] has appeared [\textit{Y. Last} and \textit{B. Simon}, Invent. Math. 135, 329-367 (1999)] and that in Ref. [19] 1284 should be replaced by 1294.
scientific article; zbMATH DE number 1268005