Anderson localization for one-frequency quasi-periodic block Jacobi operators

Klein, Silvius

doi:10.1016/j.jfa.2017.04.017

Cited by 11 publications

(13 citation statements)

References 7 publications

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“…The purpose of the present work is to show the operator H ǫ,ω (x) defined in (1.1) exhibits non-perturbative AL. This generalizes a result of Bourgain [4] as well as a result of Klein [27]. More precisely, we have Theorem 1.1.…”

Section: Introduction and Main Resultssupporting

confidence: 83%

“…and V (x) = diag (v 1 (x), · · · , v l (x)) , ψ = { ψ n } ∈ ℓ 2 (Z, C l ). In a recent paper by Klein [27], he studied the quasi-periodic block Jacobi operators:…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The proof of our main theorem employs techniques developed by Bourgain and Goldstein in [6]. We also use some tools in [4,9], and some convenient notations of Klein in [27].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Anderson localization for one-frequency quasi-periodic block operators with long-range interactions

Jian

Shi

Yuan

2019

Journal of Mathematical Physics

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Section: Introduction and Main Resultssupporting

confidence: 83%

“…and V (x) = diag (v 1 (x), · · · , v l (x)) , ψ = { ψ n } ∈ ℓ 2 (Z, C l ). In a recent paper by Klein [27], he studied the quasi-periodic block Jacobi operators:…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Anderson localization for one-frequency quasi-periodic block operators with long-range interactions

Jian

Shi

Yuan

2019

Journal of Mathematical Physics

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“…Jacobi matrices with quasi-periodic coefficient sequences have been studied in numerous recent papers; compare, for example, [2,25,27,37,38,39] and references therein. On the other hand, Jacobi matrices with limit-periodic sequences have not yet been studied in similar depth and it is therefore our goal in this paper to advance the understanding of such operators.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Thin Spectra and Singular Continuous Spectral Measures for Limit-Periodic Jacobi Matrices

Damanik

Fillman²,

Wang

2021

Preprint

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This paper investigates the spectral properties of Jacobi matrices with limitperiodic coefficients. We show that for a residual set of such matrices, the spectrum is a Cantor set of zero Lebesgue measure, and the spectral measures are purely singular continuous. For a dense set of limit-periodic Jacobi matrices we can strengthen the result and show that the spectrum is a Cantor set of zero lower box counting dimension, and hence in particular of zero Hausdorff dimension, while still retaining the singular continuity of the spectral type. We also show how results of this nature can be established by fixing the off-diagonal coefficients and varying only the diagonal coefficients, and, in a more restricted version, by fixing the diagonal coefficients to be zero and varying only the off-diagonal coefficients. We apply these results to produce examples of weighted Laplacians on the multidimensional integer lattice having purely singular continuous spectral type and zero-dimensional spectrum. Contents 1. Introduction and Results 1 2. Spectral Estimates for Periodic Jacobi Matrices 5 3. Singular Continuous Cantor Spectrum 12 Appendix A. The Gordon Lemma and Applications 14 Appendix B. The IDS for Periodic Jacobi Matrices 17 Acknowledgement 21 References 21 D.D.

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“…Mathematically rigorous studies of the Anderson Model and other models started in the 1970s and several powerful methods have been found to prove Anderson localization, such as multiscale analysis (MSA) introduced by J. Fröhlich and T. Spencer [9] , fractional moments method (FMM) developed by M. Aizenman and S. Molchanov [1], etc. Recently, the method developed by J. Bourgain, M. Goldstein and W. Schlag [3,4] for one-dimensional Schrödinger operators has been applied widely to other one-dimensional models [5,7,11,12,13], which motivated us to apply this method to CMV matrices.…”

Section: Introductionmentioning

confidence: 99%