Hakim Boumaza, Günter Stolz
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.
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Hakim Boumaza Institut de Mathématique de Jussieu Université Paris 7 Denis Diderot 2 place Jussieu, 75251 Paris, France email: boumaza@math.jussieu.fr |
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Günter Stolz Department of Mathematics CH 452 University of Alabama at Birmingham 1300 University Boulevard Birmingham, Al 35294-1170, USA email: stolz@math.uab.edu |
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