Anti-resonances and sharp analysis of Maryland localization for all parameters

Han, Rui; Jitomirskaya, Svetlana; Yang, Fan

doi:10.48550/arxiv.2205.04021

Cited by 2 publications

(2 citation statements)

References 34 publications

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“…To the best of our knowledge, our result is the first localization result for all Diophantine frequencies in the CMV setting and the quantum walk setting, see [17,19,56]. The sharp asymptotic (1.5) also adds to the recent growing collection of exact characterisations of quasi-periodic eigenfunctions [30,38,39,58].…”

Section: Introductionsupporting

confidence: 52%

Anderson localization for the unitary almost Mathieu operator

Yang

2024

Nonlinearity

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We prove Anderson localization for all Diophantine frequencies and all non-resonant phases for a model that arises from a 2D quantum walk model subject to an external magnetic field, also known as the unitary almost Mathieu operator. Our work provides the first localization results for all Diophantine frequencies in quasi-periodic quantum walk and CMV matrix setting. We also obtain sharp asymptotics of the localized eigenfunctions.

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Section: Introductionsupporting

confidence: 52%

Anderson localization for the unitary almost Mathieu operator

Yang

2024

Nonlinearity

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“…As a starting point, we focus on a particular case, namely, completely resonant phases for the almost Mathieu operator, in which phase resonances and frequency resonances overlap. As we were finalizing this paper we learned of a preprint [26] where the study of the numerators also plays an important role, but in a completely different setting of unbounded potentials and the so-called anti-resonances.…”

Section: Introductionmentioning

confidence: 99%

Small denominators and large numerators of quasiperiodic Schrödinger operators

Liu¹

2022

Preprint

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We initiate an approach to simultaneously treat numerators and denominators of Green's functions arising from quasi-periodic Schrödinger operators, which in particular allows us to study completely resonant phases of the almost Mathieu operator.Let (H λ,α,θ u)(n) = u(n + 1) + u(n − 1) + 2λ cos 2π(θ + nα)u(n) be the almost Mathieu operator on ℓ 2 (Z), where λ, α, θ ∈ R. LetWe prove that for any θ with 2θ ∈ αZ + Z, H λ,α,θ satisfies Anderson localization if |λ| > e 2β(α) . This confirms a conjecture of Avila and Jitomirskaya [The

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Anti-resonances and sharp analysis of Maryland localization for all parameters

Abstract: We develop the technique to prove localization through the analysis of eigenfunctions in presence of both exponential frequency resonances and exponential phase barriers (antiresonances) and use it to prove localization for the Maryland model for all parameters.

Cited by 2 publications

References 34 publications

Anderson localization for the unitary almost Mathieu operator

Anderson localization for the unitary almost Mathieu operator

Small denominators and large numerators of quasiperiodic Schrödinger operators

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