Hakim Boumaza

Localization for a Matrix-valued Anderson Model

We study localization properties for a class of one-dimensional, matrix-valued, continuous, random Schrödinger operators, acting on

HÖLDER CONTINUITY OF THE INTEGRATED DENSITY OF STATES FOR MATRIX-VALUED ANDERSON MODELS

L^2(\mathbb R)\otimes \mathbb C^N

Positivity of Lyapunov Exponents for a Continuous Matrix-Valued Anderson Model

, for arbitrary N ≥ 1. We prove that, under suitable assumptions on the Fürstenberg group of these operators, valid on an interval

An exactly solvable non C1 periodic potential

I\subset \mathbb R

Description of the spectral bands for some 2D periodic Schrödinger operators

, they exhibit localization properties on I, both in the spectral and dynamical sense. After looking at the regularity properties of the Lyapunov exponents and of the integrated density of states, we prove a Wegner estimate and apply a multiscale analysis scheme to prove localization for these operators. We also study an example in this class of operators, for which we can prove the required assumptions on the Fürstenberg group. This group being the one generated by the transfer matrices, we can use, to prove these assumptions, an algebraic result on generating dense Lie subgroups in semisimple real connected Lie groups, due to Breuillard and Gelander. The algebraic methods used here allow us to handle with singular distributions of the random parameters.

Absence of absolutely continuous spectrum for random scattering zippers

We study a class of continuous matrix-valued Anderson models acting on L2(Rd) ⊗ C . We prove the existence of their Integrated Density of States for any d ≥ 1 and N ≥ 1. Then for d = 1 and for arbitrary N , we prove the Hölder continuity of the Integrated Density of States under some assumption on the group GμE generated by the transfer matrices associated to our models. This regularity result is based upon the analoguous regularity of the Lyapounov exponents associated to our model, and a new Thouless formula which relates the sum of the positive Lyapounov exponents to the Integrated Density of States. In the final section, we present an example of matrix-valued Anderson model for which we have already proved, in a previous article, that the assumption on the group GμE is verified. Therefore the general results developed here can be applied to this model.We study a class of continuous matrix-valued Anderson models acting on L2(ℝd) ⊗ ℂN. We prove the existence of their Integrated Density of States for any d ≥ 1 and N ≥ 1. Then, for d = 1 and for arbitrary N, we prove the Holder continuity of the Integrated Density of States under some assumption on the group GμE generated by the transfer matrices associated to our models. This regularity result is based upon the analoguous regularity of the Lyapounov exponents associated to our model, and a new Thouless formula which relates the sum of the positive Lyapounov exponents to the Integrated Density of States. In the final section, we present an example of matrix-valued Anderson model for which we have already proved, in a previous article, that the assumption on the group GμE is verified. Therefore, the general results developed here can be applied to this model.

A Matrix-Valued Point Interactions 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 energies in (2, +∞) except those in a discrete set, which leads to absence of absolutely continuous spectrum in (2, +∞). 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.

POSITIVITY OF LYAPUNOV EXPONENTS FOR ANDERSON-TYPE MODELS ON TWO COUPLED STRINGS

In this paper, we are concerned with the study of the spectrum of a periodic potential in 1D, modelling the interactions of an electron with a regular lattice of ions. The classical Bloch theory asserts that the spectrum has a band structure. In the case of a sawtooth potential, we have a very precise description and estimates of all bands and of all gaps under the potential barrier and near the minimum of the potential.

Localization for an Anderson-Bernoulli model with generic interaction potential

In this paper, we are concerned with the study of the spectrum of operators with periodic potentials in 1D and 2D, modelling the interactions of an electron with regular lattices of ions. The classical Bloch theory asserts that the spectrum has a band structure and we are able to localize the spectral bands which are in the range of the potential and to prove estimates on their widths in the semiclassical limit.

The band spectrum of the periodic airy-schrodinger operator on the real line

A scattering zipper is a system obtained by concatenation of scattering events with equal even number of incoming and outgoing channels. The associated scattering zipper operator is the unitary analog of Jacobi matrices with matrix entries. For infinite identical events and independent and identically distributed random phases, Lyapunov exponents positivity is proved and yields absence of absolutely continuous spectrum by Kotani’s theory.