Absence of absolutely continuous spectrum for random scattering zippers

Boumaza, Hakim; Marin, Laurent

doi:10.1063/1.4906809

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Theorem 52 (Wegner Estimate1

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2023

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Cited by 2 publications

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References 22 publications

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“…For this we will use the criterion of Zariski-denseness of the Fürstenberg group, which Gol'dsheid and Margulis used in [18] to show positivity of Lyapunov exponents for the Anderson model on strips. More recently, this criterion has also been used in the proof of localization properties for continuum Anderson-type models with matrix-valued potential, see [4] and references therein, and for a class of unitary random operators [5].…”

Section: Theorem 52 (Wegner Estimatementioning

confidence: 99%

Localization for Random Block Operators Related to the XY Spin Chain

Chapman

Stolz

2014

Ann. Henri Poincaré

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Section: Theorem 52 (Wegner Estimatementioning

confidence: 99%

Localization for Random Block Operators Related to the XY Spin Chain

Chapman

Stolz

2014

Ann. Henri Poincaré

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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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Absence of absolutely continuous spectrum for random scattering zippers

Abstract: Absolutely continuous spectrum implies ballistic transport for quantum particles in a random potential on tree graphs Local vibrational modes in crystal lattices with a simply connected region of the quasi-continuous phonon spectrum Low Temp.

Cited by 2 publications

References 22 publications

Localization for Random Block Operators Related to the XY Spin Chain

Localization for Random Block Operators Related to the XY Spin Chain

Localization for random quasi-one-dimensional models

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