Singular continuous spectrum for palindromic Schrödinger operators
In this very interesting paper the authors study the discrete Schrödinger operator defined by \[ \bigl( L(x)u \bigr)_n = u_{n + 1} + u_{n - 1} + x_n u_n \quad \text{for } u \in \ell^2 (\mathbb{Z}), \] where \(x\) denotes an element of \(X\), a compact shift-invariant subset of \(A^\mathbb{Z}\), for \(A\) a finite set. This operator has been studied in particular when \(X\) is the closure of the orbit of a sequence generated by a finite automaton or by a morphism: for a general overview on the subject I would advise to read the paper by \textit{A. Sütö} [Schrödinger difference equation with deterministic ergodic potentials, in: Beyond quasicrystals, F. Axel (ed.) et al., Les Éditions de Physique, 481-549 (1995)]. The authors of the paper under review show that, if there exists a \(z\) in \(X\) that contains arbitrarily long palindromes, then there is a generic set in \(X\) for which the operator has purely singular continuous spectrum. The result holds in particular if \(X\) is the closure of the orbit of primitive substitutions in a large class (including Fibonacci, Thue-Morse, period doubling, binary and ternary non-Pisot). They address also the case of potentials defined by circle maps. Note that the results overlap results by \textit{A. Bovier} and \textit{J.-M. Ghez} [Commun. Math. Phys. 158, No. 1, 45-66 (1993; Zbl 0820.35099) and Erratum 166, No. 2, 431-432 (1994)] but no set of results is included in the other. The authors also conjecture that the Rudin-Shapiro sequence does not contain arbitrarily long palindromes and hence that their method cannot be applied for a Rudin-Shapiro sequence of potentials. This conjecture has been proved recently by the reviewer [Rapport de Recherche du LRI, No. 1035, 1996, Orsay].
scientific article; zbMATH DE number 821650
scientific article