Uniform spectral properties of onedimensional quasicrystals, II. The Lyapunov exponent
Abstract
In this paper we introduce a method that allows one to prove uniform local results for onedimensional discrete Schrödinger 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.
 Publication:

arXiv eprints
 Pub Date:
 May 1999
 arXiv:
 arXiv:mathph/9905008
 Bibcode:
 1999math.ph...5008D
 Keywords:

 Mathematical Physics;
 Mathematics  Mathematical Physics;
 Mathematics  Spectral Theory;
 81Q10;
 47B80
 EPrint:
 10 pages