Lifshitz Tails for Continuous Matrix-Valued Anderson Models

Boumaza, Hakim; Najar, Hatem

doi:10.1007/s10955-015-1255-4

Cited by 3 publications

(2 citation statements)

References 19 publications

(41 reference statements)

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“…A step towards Dirac operators has been done in the case where the kinetic energy is given by a Laplacian on L 2 (R d ) ⊗ C ν , ν > 1 and the random potential is matrix-valued (see [4] and references therein). In [23,24] the authors considered discretized versions of Dirac operators on ℓ 2 (Z d , C ν ) (d = 1, 2, 3), with a simple mass potential, and a random potential given by a matrix-valued diagonal operator, and proved spectral and dynamical localization near band edges.…”

Section: Introductionmentioning

confidence: 99%

Localization for Gapped Dirac Hamiltonians with Random Perturbations: Application to Graphene Antidot Lattices

2019

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In this paper we study random perturbations of firstorder elliptic operators with periodic potentials. We are mostly interested in Hamiltonians modeling graphene antidot lattices with impurities. The unperturbed operator H 0 := D S + V 0 is the sum of a Dirac-like operator D S plus a periodic matrix-valued potential V 0 , and is assumed to have an open gap. The random potential V ω is of Anderson-type with independent, identically distributed coupling constants and moving centers, with absolutely continuous probability distributions. We prove band edge localization, namely that there exists an interval of energies in the unperturbed gap where the almost sure spectrum of the family H ω := H 0 + V ω is dense pure point, with exponentially decaying eigenfunctions, that give rise to dynamical localization.

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Section: Introductionmentioning

confidence: 99%

Localization for Gapped Dirac Hamiltonians with Random Perturbations: Application to Graphene Antidot Lattices

2019

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“…A step towards Dirac operators has been done in the case where the kinetic energy is given by a Laplacian on L 2 (R d ) ⊗ C ν , ν > 1 and the random potential is matrix valued (see [4] and references therein). In [22,23] the authors considered discretized versions of Dirac operators on ℓ 2 (Z d , C ν ) (d = 1, 2, 3), with a simple mass potential, and a random potential given by a matrix valued diagonal operator, and proved spectral and dynamical localization near band edges.…”

Section: Introductionmentioning

confidence: 99%

Localization for gapped Dirac Hamiltionians with random perturbations: Application to graphene antidot lattices

Barbaroux¹,

Cornean²,

Zalczer³

2018

Preprint

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In this paper we study random perturbations of first order elliptic operators with periodic potentials. We are mostly interested in Hamiltonians modeling graphene antidot lattices with impurities. The unperturbed operator H 0 := D S + V 0 is the sum of a Dirac-like operator D S plus a potential V 0 , and is assumed to have an open gap. The random potential Vω is of Anderson-type with independent, identically distributed coupling constants and moving centers, with absolutely continuous probability distributions. We prove band edge localization, namely that there exists an interval of energies in the unperturbed gap where the almost sure spectrum of the family Hω := H 0 + Vω is dense pure point, with exponentially decaying eigenfunctions, that give rise to dynamical localization.

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Localization for random quasi-one-dimensional models

Boumaza

2023

Journal of Mathematical Physics

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In this Review Article, we review the results of Anderson localization for different random families of operators that enter the framework of random quasi-one-dimensional models. We first recall what is Anderson localization from both physical and mathematical points of view. 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 Schrödinger type in the discrete and continuous cases, the unitary type, the Dirac type, and the point interaction type. We present tools coming from the study of dynamical systems in dimension one: the transfer matrix formalism, the Lyapunov exponents, and the Furstenberg group. We then prove a criterion of localization for quasi-one-dimensional models of Schrödinger type involving only geometric and algebraic properties of the Furstenberg group. Then, we review results of localization, first for Schrödinger-type models and then for unitary type models. Each time, we reduce the question of localization to the study of the Furstenberg group and show how to use more and more refined algebraic criteria to prove the needed properties of this group. All the presented results for quasi-one-dimensional models of Schrödinger type include the case of Bernoulli randomness.

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Lifshitz Tails for Continuous Matrix-Valued Anderson Models

Cited by 3 publications

References 19 publications

Localization for Gapped Dirac Hamiltonians with Random Perturbations: Application to Graphene Antidot Lattices

Localization for Gapped Dirac Hamiltonians with Random Perturbations: Application to Graphene Antidot Lattices

Localization for gapped Dirac Hamiltionians with random perturbations: Application to graphene antidot lattices

Localization for random quasi-one-dimensional models

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