One-dimensional Discrete Anderson Model in a Decaying Random Potential: from A.C. Spectrum to Dynamical Localization

Bourget, Olivier; Flores, Gregorio R. Moreno; Taarabt, Amal

doi:10.1007/978-3-030-55556-6_3

Cited by 2 publications

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References 33 publications

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“…The three above regimes correspond to α > 1 2 , α < 1 2 and α = 1 2 respectively. A complete study of the spectral behaviour of the one-dimensional discrete and continuum models is given in [36] (see also [8]).…”

Section: Introductionmentioning

confidence: 99%

“…From the dynamical point of view, it is standard to show that the system propagates for α > 1 2 . For the critical case α = 1 2 , no transition occurs at the dynamical level, despite of the spectral transition: there are non-trivial transport exponents for all values of the coupling constant [28] for both the discrete and continuum model (see also [8,9] for elementary arguments showing delocalization). This provides yet another example of a model where spectral localization and transport coexist.…”

Section: Introductionmentioning

confidence: 99%

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Dynamical Localization for the One-dimensional Continuum Anderson Model in a Decaying Random Potential

Bourget,

Flores,

Taarabt

2020

Preprint

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We consider a one-dimensional continuum Anderson model where the potential decays in average like |x| −α , α > 0. We show dynamical localization for 0 < α < 1 2 and provide control on the decay of the eigenfunctions. Contents 1. Introduction 1 Structure of the article 2 2. Model and main results 2 3. Asymptotics of Transfer Matrices and Prüfer transform 5 4. Fractional moments estimates 6 4.1. From Green's function to transfer matrices 6 4.2. Estimates on transfer matrices 8 5. Proof of Dynamical Localization 10 6. Proof of Theorem 2.5 and 2.6 13 Appendix A. General estimates 15 Appendix B. The martingale decomposition 15 Appendix C. Pure point spectrum 16 References 17

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dynamical Localization for the One-dimensional Continuum Anderson Model in a Decaying Random Potential

Bourget,

Flores,

Taarabt

2020

Preprint

Self Cite

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Spectral type of a class of random Jacobi operators

2021

Journal of Mathematical Physics

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In this paper, we use the generalized Prüfer variables to study the spectral type of a class of random Jacobi operators (Hτ,ωλu)(n)=τnu(n+1)+τn−1u(n−1)+λanωnu(n), in which the decay speed of the parameters an is n−α for some α > 0. We will show that the operator has an absolutely continuous spectrum for α>12, a pure point spectrum for 0<α<12, and a transition from a singular continuous spectrum to a pure point spectrum in α=12.

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One-dimensional Discrete Anderson Model in a Decaying Random Potential: from A.C. Spectrum to Dynamical Localization

Cited by 2 publications

References 33 publications

Dynamical Localization for the One-dimensional Continuum Anderson Model in a Decaying Random Potential

Dynamical Localization for the One-dimensional Continuum Anderson Model in a Decaying Random Potential

Spectral type of a class of random Jacobi operators

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