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

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

2001-05-23

arxiv.org/abs/math-ph/0105034v1

Physics

Mathematical Physics

20 pages

Scientific paper

We consider discrete one-dimensional Schr\"odinger 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.

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