Shnol’s theorem and the spectrum of long range operators

Han, Rui

doi:10.1090/proc/14388

Cited by 30 publications

(15 citation statements)

References 20 publications

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“…We only give an outline of proof below. By Schnol's theorem [8,14,23], to prove Anderson localization, it suffices to show generalized eigenfunction Ψ, namely |Ψ y | ≤ C|y| for some constant C, to W λ 1 ,λ 2 ,ω,θ Ψ = zΨ decays exponentially. Throughout this section, let…”

Section: The Key Lemmamentioning

confidence: 99%

Localization for magnetic quantum walks

Yang¹

2022

Preprint

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Section: The Key Lemmamentioning

confidence: 99%

Localization for magnetic quantum walks

Yang¹

2022

Preprint

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“…The proof or localisation is based on Schnol's lemma, which we now recall (see [15] for a version applicable in the current setting). A function ∶ ℤ → ℂ is called a generalised eigenfunction corresponding to a generalised eigenvalue ∈ ℝ if…”

Section: Spectral Localisation: Proof Of Theoremmentioning

confidence: 99%

Anderson localisation for quasi-one-dimensional random operators

Macera¹,

Sodin²

2021

Preprint

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In 1990, Klein, Lacroix, and Speis proved (spectral) Anderson localisation for the Anderson model on the strip of width ⩾ 1, allowing for singular distribution of the potential. Their proof employs multi-scale analysis, in addition to arguments from the theory of random matrix products (the case of regular distributions was handled earlier in the works of Goldsheid and Lacroix by other means). We give a proof of their result avoiding multi-scale analysis, and also extend it to the general quasi-one-dimensional model, allowing, in particular, random hopping. Furthermore, we prove a sharp bound on the eigenfunction correlator of the model, which implies exponential dynamical localisation and exponential decay of the Fermi projection.Our work generalises and complements the single-scale proofs of localisation in pure one dimension ( = 1), recently found by Bucaj-Damanik-Fillman-Gerbuz-VandenBoom-Wang-Zhang, Jitomirskaya-Zhu, Gorodetski-Kleptsyn, and Rangamani.

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“…given by the closure of the set of energies for which there exists a non-trivial polynomially bounded solution satisfying the boundary condition. Since we are dealing with an unbounded potential, let us mention that Shnol's theorem holds in this setting as well [25].…”

Section: Appendix a A Remark On Unbounded Background Potentialsmentioning

confidence: 99%

Must the Spectrum of a Random Schrödinger Operator Contain an Interval?

Damanik¹,

Gorodetski²

2021

Preprint

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We consider Schrödinger operators in 2 (Z) whose potentials are given by independent (not necessarily identically distributed) random variables. We ask whether it is true that almost surely its spectrum contains an interval. We provide an affirmative answer in the case of random potentials given by a sum of a perturbatively small quasi-periodic potential with analytic sampling function and Diophantine frequency vector and a term of Anderson type, given by independent identically distributed random variables. The proof proceeds by extending a result about the presence of ground states for atypical realizations of the classical Anderson model, which we prove here as well and which appears to be new.

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Shnol’s theorem and the spectrum of long range operators

Cited by 30 publications

References 20 publications

Localization for magnetic quantum walks

Localization for magnetic quantum walks

Anderson localisation for quasi-one-dimensional random operators

Must the Spectrum of a Random Schrödinger Operator Contain an Interval?

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