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.