A nonprimitive substitution Schrödinger operator with generic singular continuous spectrum
The nature of the spectrum of a discrete one-dimensional Schrödinger operator with potentials generated by substitutions has been studied by several authors. We quote in particular the paper by \textit{A. Hof, O. Knill} and \textit{B. Simon}, where they prove that the spectrum is generically purely singular continuous when the sequence of potentials is generated by a primitive morphism (substitution), and contains arbitrarily long palindromes [Commun. Math. Phys. 174, 149-159 (1995; Zbl 0839.11009)]. Here the authors study the morphism \(a\to aabaa\), \(b\to b\) that is not primitive. They prove nevertheless that the spectrum of the corresponding Schrödinger operator is generically purely singular continuous: the main ingredient (in comparison with the paper of Hof, Knill and Simon) is to prove that the discrete dynamical system associated with \(a\to aabaa\), \(b\to b\) is minimal. Note that it would be interesting to try to obtain the result by using the work of \textit{A. Bovier} and \textit{J.-M. Ghez} [see Commun. Math. Phys. 158, 45-66 (1993; Zbl 0820.35099) and Errata in ibid. 166, 431-432 (1994; Zbl 0841.35072)].
discrete one-dimensional Schrödinger operator
scientific article; zbMATH DE number 1507663