2 research outputs found
Positivity of Lyapunov exponents for a continuous matrix-valued Anderson model
We study a continuous matrix-valued Anderson-type model. Both leading
Lyapunov exponents of this model are proved to be positive and distinct for all
ernergies in except those in a discrete set, which leads to
absence of absolutely continuous spectrum in . This result is an
improvement of a previous result with Stolz. The methods, based upon a result
by Breuillard and Gelander on dense subgroups in semisimple Lie groups, and a
criterion by Goldsheid and Margulis, allow for singular Bernoulli
distributions
A matrix-valued point interactions model
We study a matrix-valued Schr\"odinger operator with random point
interactions. We prove the absence of absolutely continuous spectrum for this
operator by proving that away from a discrete set its Lyapunov exponents do not
vanish. For this we use a criterion by Gol'dsheid and Margulis and we prove the
Zariski denseness, in the symplectic group, of the group generated by the
transfer matrices. Then we prove estimates on the transfer matrices which lead
to the H\"older continuity of the Lyapunov exponents. After proving the
existence of the integrated density of states of the operator, we also prove
its H\"older continuity by proving a Thouless formula which links the
integrated density of states to the sum of the positive Lyapunov exponents