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

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

doi:10.48550/arxiv.2001.02197

Cited by 1 publication

(6 citation statements)

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“…Our approach, which is discussed in Section 6 below, can be easily adapted to the Anderson model providing an alternative proof of Theorem 2.3 under more general assumptions. Furtheremore, in [8], we proved the corresponding result for a continuum version of the model, a problem which was left open in [14] where the authors develop a continuum version of the KSM.…”

Section: Model and Resultsmentioning

confidence: 72%

“…for all ϕ 0 ∈ l 2 (Z) with bounded support. A proof of these simple facts can be found in [9] in a related discrete model or in [8] in the continuum.…”

Section: Model and Resultsmentioning

confidence: 83%

“…The analysis in [8] provides a control on the eigenfunctions of H ω,λ . Let x ω,E = (x ω,E,n ) n denote the eigenfunction of H ω corresponding to the eigenvalue E. In [33,Theorem 8.6], it is showed that lim…”

Section: Model and Resultsmentioning

confidence: 99%

“…Sub-critical case: Dynamical Localization Theorem 2.3 was proved in [38] using the Kunz-Souillard method. We briefly outline the approach used in [8] to prove dynamical localization for the analogue continuum model and which becomes particularly simple when adapted to the discrete case (see [9] for the related discrete Dirac model). We use the fractional moment method [1,3].…”

Section: Control Of Generalized Eigenfunctionsmentioning

confidence: 99%

“…Dynamical localization in the regime 0 < α < 1 2 was showed in [38] providing, as a by-product, a proof of point spectrum in this regime. In [8], we provided a different proof of this behaviour for a continuum version of the model which can be easily adapted to the discrete case.…”

Section: Introductionmentioning

confidence: 99%

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

Bourget¹,

Flores²,

Taarabt³

2020

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We consider a one-dimensional Anderson model where the potential decays in average like n −α , α > 0. This simple model is known to display a rich phase diagram with different kinds of spectrum arising as the decay rate α varies.We review an article of Kiselev, Last and Simon where the authors show a.c. spectrum in the super-critical case α > 1 2 , a transition from singular continuous to pure point spectrum in the critical case α = 1 2 , and dense pure point spectrum in the sub-critical case α < 1 2 . We present complete proofs of the cases α ≥ 1 2 and simplify some arguments along the way. We complement the above result by discussing the dynamical aspects of the model. We give a simple argument showing that, despite of the spectral transition, transport occurs for all energies for α = 1 2 . Finally, we discuss a theorem of Simon on dynamical localization in the sub-critical region α < 1 2 . This implies, in particular, that the spectrum is pure point in this regime. 5.4. Absence of dynamical localization 12 6. Sub-critical case: Dynamical Localization 13 Appendix A. Some technical lemmas 13 A.1. Two results on unimodular matrices 13 A.2. Analysis of the Prüfer phases 16 References 18

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Section: Model and Resultsmentioning

confidence: 72%

“…for all ϕ 0 ∈ l 2 (Z) with bounded support. A proof of these simple facts can be found in [9] in a related discrete model or in [8] in the continuum.…”

Section: Model and Resultsmentioning

confidence: 83%