arXiv subjects

Hakim Boumaza

Publications and source records attributed to Hakim Boumaza.

12 recordsLinked to original sources

arXiv2024

Dynamical localization for random scattering zippers

This article establishes a proof of dynamical localization for a random scattering zipper model. The scattering zipper operator is the product of two unitary by blocks operators, multiplicatively perturbed on the left and right by random unitary phases. One of the operator is shifted so that this configuration produces a random 5-diagonal unitary operator per blocks. To prove the dynamical localization for this operator, we use the method of fractional moments. We first prove the continuity and strict positivity of the Lyapunov exponents in an annulus around the unit circle, which leads to the exponential decay of a power of the norm of the products of transfer matrices. We then establish an explicit formula of the coefficients of the finite resolvent in terms of the coefficients of the transfer matrices using Schur's complement. From this we deduce, through two reduction results, the exponential decay of the resolvent, from which we get the dynamical localization.

math-ph↗

arXiv2024

Localization for quasi-one-dimensional Dirac operators

We consider a random family of Dirac operators on $N$ parallel real lines, modelling for example a graphene nanoribbon. We establish a localization criterion involving properties on the group generated by transfer matrices. In particular, we consider not only the case where this group is the symplectic group but also a strict subgroup of it. We establish under quite general hypotheses that the sum of the Lyapunov exponents and the integrated density of states are Hölder continuous. Moreover, for a set of concrete cases where the potentials are on Pauli matrices, we compute the transfer matrices and prove either localization or delocalization, depending on the potential and on the parity of $N$.

math-ph↗

arXiv2024

Analytic expression of the DOS for a new model of 1d-potential and its random perturbation

In this article we present comparisons between the spectrum of a one-dimensional Schrödinger operator for a particular periodic potential and for its restriction to a finite number of sites. We deduce from this finite, but large, number of sites, the Integrated Density of States (IDS) associated to the Hamiltonian operator whose derivate is the DOS. The exact formula for the IDS is given and the expression of the DOS is analytical. All our calculations are done on the particular periodic Airy-potential, which is a new case for which one has an analytical expression of the DOS. It is a continuous, periodic potential, piecewise affine. As a periodic operator, the spectrum is a band spectrum.

math-ph↗

arXiv2023

Localization for random quasi-one-dimensional models

In this paper we review results of Anderson localization for different random families of operators which enter in the framework of random quasi-one-dimensional models. We first recall what is Anderson localization from both physical and mathematical point of views. From the Anderson-Bernoulli conjecture in dimension 2 we justify the introduction of quasi-one-dimensional models. Then we present different types of these models : the Schr{ö}dinger type in the discrete and continuous cases, the unitary type, the Dirac type and the point-interactions type. In a second part we present tools coming from the study of dynamical systems in dimension one : the transfer matrices formalism, the Lyapunov exponents and the F{ü}rstenberg group. We then prove a criterion of localization for quasi-one-dimensional models of Schr{ö}dinger type involving only geometric and algebraic properties of the F{ü}rstenberg group. Then, in the last two sections, we review results of localization, first for Schr{ö}dinger type models and then for unitary type models. Each time, we reduce the question of localization to the study of the F{ü}rstenberg group and show how to use more and more refined algebraic criterions to prove the needed properties of this group. All the presented results for quasi-one-dimensional models of Schr{ö}dinger type include the case of Bernoulli randomness.

math-ph↗

arXiv2013

Lifshitz tails for matrix-valued Anderson models

This paper is devoted to the study of Lifshitz tails for a continuous matrix-valued Anderson-type model $H_ω$ acting on $L^2(\R^d)\otimes \C^{D}$, for arbitrary $d\geq 1$ and $D\geq 1$. We prove that the integrated density of states of $H_ω$ has a Lifshitz behavior at the bottom of the spectrum. We obtain a Lifshitz exponent equal to $-d/2$ and this exponent is independent of $D$. It shows that the behaviour of the integrated density of states at the bottom of the spectrum of a quasi-d-dimensional Anderson model is the same as its behaviour for a d-dimensional Anderson model.

math-ph↗

arXiv2013

Absence of absolutely continuous spectrum for random scattering zippers

A scattering zipper is a system obtained by concatenation of scattering events with equal even number of incoming and out going channels. The associated scattering zipper operator is the unitary equivalent of Jacobi matrices with matrix entries. For infinite identical events and random phases, Lyapunov exponents positivity is proved and yields to the absence of absolutely continuous spectrum.

math-ph↗

arXiv2010

Localization for an Anderson-Bernoulli model with generic interaction potential

We present a result of localization for a matrix-valued Anderson-Bernoulli operator, acting on $L^2(\R)\otimes \R^N$, for an arbitrary $N\geq 1$, whose interaction potential is generic in the real symmetric matrices. For such a generic real symmetric matrix, we construct an explicit interval of energies on which we prove localization, in both spectral and dynamical senses, away from a finite set of critical energies. This construction is based upon the formalism of the Fürstenberg group to which we apply a general criterion of density in semisimple Lie groups. The algebraic nature of the objects we are considering allows us to prove a generic result on the interaction potential and the finiteness of the set of critical energies.

math.SP↗

arXiv2010

Absence de spectre absolument continu pour un opérateur d'Anderson à potentiel d'interaction générique

We present a result of absence of absolutely continuous spectrum in an interval of $\R$, for a matrix-valued random Schrödinger operator, acting on $L^2(\R)\otimes \R^N$ for an arbitrary $N\geq 1$, and whose interaction potential is generic in the real symmetric matrices. For this purpose, we prove the existence of an interval of energies on which we have separability and positivity of the $N$ non-negative Lyapunov exponents of the operator. The method, based upon the formalism of Fürstenberg and a result of Lie group theory due to Breuillard and Gelander, allows an explicit contruction of the wanted interval of energies.

math-ph↗

arXiv2009

Exposants de Lyapounov pour un modèle d'Anderson à valeurs matricielles

Nous présentons un résultat d'absence de spectre absolument continu dans un intervalle de $\R$ pour un opérateur de Schrödinger aléatoire continu et à valeurs matricielles agissant sur $L^2(\R)\otimes \C^N$ pour $N\geq 1$ arbitraire. Pour cela nous prouvons l'existence d'un intervalle d'énergies sur lequel a lieu la séparabilité et la stricte positivité des $N$ exposants de Lyapounov positifs de l'opérateur. La méthode suivie, basée sur le formalisme de Fürstenberg et un résultat de théorie des groupes dû à Breuillard et Gelander, permet une construction explicite de l'intervalle d'énergie recherché.

math-ph↗

arXiv2009

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 $L^2(\R)\otimes \C^N$, for arbitrary $N\geq 1$. We prove that, under suitable assumptions on the Fürstenberg group of these operators, valid on an interval $I\subset \R$, 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.

math-ph↗

arXiv2008

A matrix-valued point interactions model

We study a matrix-valued Schrödinger 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 Hölder continuity of the Lyapunov exponents. After proving the existence of the integrated density of states of the operator, we also prove its Hölder continuity by proving a Thouless formula which links the integrated density of states to the sum of the positive Lyapunov exponents.

math-ph↗

arXiv2006

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

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.

math-ph↗