Uniform spectral properties of one-dimensional quasicrystals, IV. Quasi-Sturmian potentials

Abstract

We consider discrete one-dimensional Schrödinger operators with quasi-Sturmian potentials. We present a new approach to the trace map dynamical system which is independent of the initial conditions and establish a characterization of the spectrum in terms of bounded trace map orbits. Using this, it is shown that the operators have purely singular continuous spectrum and their spectrum is a Cantor set of Lebesgue measure zero. We also exhibit a subclass having purely $\alpha$-continuous spectrum. All these results hold uniformly on the hull generated by a given potential.

20 pages

References in corpus (3)

• Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrodinger operators
• Gordon-type arguments in the spectral theory of one-dimensional quasicrystals
• Hierarchical structures in Sturmian dynamical systems

Cited by in corpus (1)

• Arnoux-Rauzy Subshifts: Linear Recurrence, Powers, and Palindromes