Spectral properties of one-dimensional Schrödinger operators with potentials generated by substitutions
In this very interesting paper the authors study the spectra of one- dimensional discrete Schrödinger operators whose potentials are fixed point of substitutions. The usual method is to study the dynamical system generated by the trace map: this map can be defined in terms of the substitution giving the sequence of potentials, but it is often rather complicated to study. The nice idea of the authors is to introduce a so-called ``reduced trace map'' and to associate in a natural way to it a (new) substitution. If this new substitution is ``semi-primitive'' (a notion proposed by the authors which generalizes the notion of primitivity), if the initial substitution is primitive, if furthermore the sequence of potentials contains a square (i.e. two identical consecutive blocks), then the main result of the authors asserts that the spectrum of the associated Schrödinger operator is singular and supported on a set of zero Lebesgue measure. They also give a condition to exclude point spectrum. Finally they give many examples (known or not) where their result can be applied. They show at the end of the paper that the Rudin-Shapiro sequence ``retains its mystery'' as it does not enter this framework.
spectra of one-dimensional discrete Schrödinger operators
scientific article; zbMATH DE number 490659