Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 47, pp. 1-18.

Title: Positivity of Lyapunov exponents for Anderson-type 
       models on two coupled strings

Authors: Hakim Boumaza (Univ. Paris 7 Denis Diderot, Paris, France)
         Guenter Stolz (Univ. of Alabama, Birmingham, Al, USA)
 
Abstract:
 We study two models of Anderson-type random operators on
 two deterministically coupled continuous strings. Each model is
 associated with independent, identically distributed four-by-four
 symplectic transfer matrices, which describe the asymptotics of
 solutions. In each case we use a criterion by Gol'dsheid and
 Margulis (i.e. Zariski denseness of the group generated by the
 transfer matrices in the group of symplectic matrices) to prove
 positivity of both leading Lyapunov exponents for most energies. In
 each case this implies almost sure absence of absolutely continuous
 spectrum (at all energies in the first model and for sufficiently
 large energies in the second model). The methods used allow for
 singularly distributed random parameters, including Bernoulli
 distributions.
  
Submitted October 31, 2006. Published March 20, 2007.
Math Subject Classifications: 82B44, 47B80, 81Q10.
Key Words: Random operators; Anderson model; Lyapunov exponents.