Uniform spectral properties of one-dimensional quasicrystals, II. The Lyapunov exponent

Details Uniform spectral properties of one-dimensional quasicrystals, II. The Lyapunov exponent Uniform spectral properties of one-dimensional quasicrystals, II. The Lyapunov exponent

1999-05-16

arxiv.org/abs/math-ph/9905008v1

Physics

Mathematical Physics

10 pages

Scientific paper

In this paper we introduce a method that allows one to prove uniform local results for one-dimensional discrete Schr\"odinger operators with Sturmian potentials. We apply this method to the transfer matrices in order to study the Lyapunov exponent and the growth rate of eigenfunctions. This gives uniform vanishing of the Lyapunov exponent on the spectrum for all irrational rotation numbers. For irrational rotation numbers with bounded continued fraction expansion, it gives uniform existence of the Lyapunov exponent on the whole complex plane. Moreover, it yields uniform polynomial upper bounds on the growth rate of transfer matrices for irrational rotation numbers with bounded density. In particular, all our results apply to the Fibonacci case.

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