Anderson localization for multi-frequency quasi-periodic operators on $\mathbb{Z}^d$

Svetlana Jitomirskaya; Wencai Liu; Yunfeng Shi

REVIEW 3 major objections 4 minor 1 cited by

Anderson localization for multi-frequency quasi-periodic operators on $\mathbb{Z}^d$

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For large coupling, every analytic multi-frequency quasi-periodic operator on $\mathbb{Z}^d$ with phase-space dimension at least $d$ has pure point spectrum with exponentially decaying eigenfunctions for most frequencies.

desk verdict A genuine extension of Bourgain's localization theorem to arbitrary k, d and long-range hopping, with the main soft spot being an imported multi-scale step whose block-frequency extension is asserted rather than proved. read the letter →

arxiv 1908.03805 v1 pith:HLQRPHIP submitted 2019-08-10 math.SP math-phmath.MP

Svetlana Jitomirskaya , Wencai Liu , Yunfeng Shi This is my paper · ORCID

classification math.SPmath-phmath.MP MSC 47B8082B4481Q10

keywords Andersonlocalizationquasi-periodicoperatorsmulti-frequencyGreen'sfunctionsmulti-scaleanalysissemi-algebraicsetslargedeviationtheorempurepointspectrum

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves Anderson localization for a broad class of quasi-periodic lattice operators: for any analytic potential depending on any number of frequency coordinates, on $\mathbb{Z}^d$ with any $d$, and for any hopping matrix with exponentially decaying entries, sufficiently large coupling forces pure point spectrum with exponentially decaying eigenfunctions for most frequencies, for every fixed phase. Such a statement was previously known only when the number of frequencies equals the dimension, or in a one-frequency, arbitrary-dimension setting, so the paper removes a long-standing restriction. The result matters because these operators are the natural multi-dimensional, multi-frequency generalizations of the basic one-frequency lattice model, and they appear in dual families relevant to absolutely continuous spectrum and in models of interacting particles.

What carries the argument

The engine is a deterministic multi-scale analysis of Green's functions on finite 'elementary regions' of $\mathbb{Z}^d$. The central object is property $P$ at scale $N$: outside a set of phases of measure at most $e^{-N^\gamma}$ in every coordinate block, the Green's function on every elementary region of size $N$ has norm at most $e^{\sqrt N}$ and off-diagonal entries bounded by $e^{-\bar\rho|n-n'|}$ for $|n-n'|\ge N/10$, with the exceptional sets constrained to be semi-algebraic, meaning finite unions of polynomial equalities and inequalities, of controlled degree. The proof's main step is a scale induction that passes from property $P$ at scales $N$ and $N^{2/c_1}$ directly to an interval of subexponentially larger scales, avoiding the chain of intermediate scales used in prior schemes; this is carried by a new several-variable, matrix-valued small-value estimate for analytic matrix functions and by measure-and-complexity bounds for semi-algebraic sets that control how often a rotation trajectory meets forbidden sets.

What would settle it

Exhibit a non-degenerate analytic $v$ and a large coupling $\lambda$ for which, on a positive-measure set of frequencies, some finite-volume Green's function on an elementary region violates the exponential decay estimate $|G_{Q_N}(E;x)(n,n')|\le e^{-\bar\rho|n-n'|}$ for $|n-n'|\ge N/10$ at infinitely many scales; that would refute the large-deviation theorem at the proof's core. A more targeted check is the imported measure bound for $b>d$: if a concrete forbidden set $S$ gives a larger measure than claimed unless an extra Diophantine condition is imposed, the induction cannot start.

Watch

Extended reading notes

Core claim

The paper's central theorem asserts that if $H(x)=S+\lambda v(x+n\omega)$ on $\ell^2(\mathbb{Z}^d)$ has a translation-invariant hopping term $S$ with $|S(n,n')|\le e^{-\rho|n-n'|}$ and a real-analytic potential $v$ on $\mathbb{T}^b$, $b=\sum b_i\ge d$, that is nonconstant in each block of variables, then for any $\delta>0$ and any phase $x$ there is a coupling threshold $\lambda_0$ such that for all $\lambda\ge\lambda_0$ there is a set $\Omega\subset\mathbb{T}^b$ of frequencies of measure at least $1-\delta$ on which the operator has only pure point spectrum and all eigenfunctions decay exponentially. This covers arbitrary frequency count and spatial dimension, and it extends earlier high-coupling localization from the equal case and from the nearest-neighbor Laplacian to general long-range hopping. The paper notes in particular that the result applies to the most general form of a $d$-dimensional quasi-periodic operator with a $k$-dimensional phase space.

Load-bearing premise

The proof takes as given that a combinatorial estimate on how often a rotating phase trajectory can hit a small forbidden region, proved when the number of frequencies equals the dimension, extends to all phase-space dimensions $b\ge d$; if that extension needs extra arithmetic conditions or fails, the main multi-scale induction collapses.

Editorial extensions

If this is right

The restriction that the number of frequencies equals the dimension is removed: any frequency count is allowed whenever the total phase-space dimension $b=\sum b_i$ is at least $d$.
Localization holds for long-range hopping matrices with exponential decay, not only for the nearest-neighbor Laplacian.
For any fixed phase $x$ and any tolerance $\delta$, all but $\delta$ measure of the frequency torus exhibits pure point spectrum with exponentially decaying eigenfunctions once $\lambda$ is large enough.
The theorem covers the most general form of a $d$-dimensional quasi-periodic operator with $k$-dimensional phase space, including operators whose frequency vector is given by a $d\times k$ matrix.
The paper notes that this localization statement is a building block toward proving absolutely continuous spectrum for the dual family of operators.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The proof's explicit reliance on $b\ge d$ suggests that the genuinely hard regime is $b<d$, which includes models of interacting quasiperiodic particles; the present mechanism is unlikely to transfer there without a new arithmetic input.
The direct scale step, if it stands independently of the imported estimate, may simplify other deterministic multi-scale proofs that currently chain many intermediate scales.
A testable consequence is that the measure of exceptional frequencies should shrink to zero as $\lambda\to\infty$ at a rate controlled by the initial scale $\log\log\lambda$; numerical finite-volume checks for a two-frequency, two-dimensional model could look for this rate.
The non-degeneracy condition in each block is probably close to necessary: if the potential is constant in one block of variables, the corresponding frequency is redundant and the problem effectively reduces to a lower-dimensional system.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This paper genuinely extends Bourgain's 2007 localization theorem: it handles arbitrary k and d, and exponentially decaying Toeplitz hopping instead of the nearest-neighbor Laplacian. That is a real theorem, not a routine generalization. The proof also carries a simplification over Bourgain's scheme: a single-scale gluing from N1 to N2 using a subexponential norm bound, skipping the chain of intermediate scales. The multi-scale part is mostly written out, and the appendix contains a several-variable matrix-valued Cartan estimate. The authors are transparent about what comes from Bourgain: the overall scheme is adapted, and two pieces are explicitly imported. That is honest.

The main soft spot is Theorem 2.7, the deterministic multi-scale step. The statement is the engine of the paper: Theorem 3.6 and the final LDT depend on it. Its proof is not given; the text says it follows from Lemmas 2.2 and 2.4 and defers to the proof of the Claim in Bourgain's paper, page 694, with a notation alignment. The stress-test concern is accurate: Bourgain's Claim was proved for b=d with each frequency component one-dimensional, and the present paper needs b≥d with block frequencies ω_i∈R^{b_i}. The notation alignment shows the parameters match, but it does not by itself prove the block version. That said, I read Lemma 2.4 as the natural block generalization of Bourgain's Lemma 1.20, and the same semi-algebraic elimination/decomposition arguments appear to carry over. I do not see a hidden arithmetic condition or a loss in measure bounds. But because this is the load-bearing step, a referee should either get a proof of Theorem 2.7 in the block setting or, at minimum, a page-long explanation of why Bourgain's proof applies verbatim. The second soft spot is the final localization deduction, which is deferred to [4, Section 3] or [7, Section 6]; that is standard and I do not consider it a problem.

No circularity, no fitted parameters, no invented entities. The citation pattern is appropriate: the paper builds on Bourgain, Goldstein–Schlag, and Phong–Stein–Sturm, and the authors say so. The new gluing argument and the generality are the contribution.

Who benefits: researchers in quasi-periodic operators, especially those using Aubry duality to get absolutely continuous spectrum for dual models, as the authors note. It deserves a serious referee; desk rejection would be wrong. I would send it to review, with the explicit instruction that the referee verify the extension of Bourgain's Claim to the block-frequency setting.

Referee Report

3 major / 4 minor

Summary. The manuscript studies quasi-periodic operators H(x)=S+λ v(x+nω)δ_{nn'} on ℓ^2(Z^d), where S is a Toeplitz matrix with exponential off-diagonal decay and v is real analytic on T^b with b=∑_{i=1}^d b_i ≥ d. The main result, Theorem 1.1, claims that for any δ>0 there is λ0 such that for all λ≥λ0 and all phases x, the operator H(x) satisfies Anderson localization for all frequencies ω outside a set of measure at most δ. The proof follows Bourgain's multi-scale scheme for the k=d case: it defines good boxes and property P, formulates a deterministic multi-scale step (Theorem 2.7), proves resolvent identities and a multi-variable matrix-valued Cartan estimate (Lemma 3.4), derives a large deviation theorem (Theorem 4.1), and then asserts localization. The central deterministic step is deferred to Bourgain [5, p.694] with only a notation alignment, and the final localization argument is deferred to [4, Section 3] or [7, Section 6].

Significance. If correct, Theorem 1.1 is a substantial advance: it extends Bourgain's arbitrary-dimension localization (GAFA 2007) from k=d and nearest-neighbor hopping to arbitrary k,d and general exponentially decaying Toeplitz operators, and it covers the rectangular-frequency-matrix model (1.4), which is the natural setting for Aubry duality. The paper also contains genuinely new technical contributions: a simplified multi-scale induction that jumps directly from scale N1 to N2 without an intermediate chain of scales, a several-variables matrix-valued Cartan estimate proved in the appendix using Goldstein-Schlag's high-dimensional Cartan sets, and a careful treatment of elementary regions in the geometric case analysis of Theorem 3.6. However, the proof of the key deterministic multi-scale theorem and the final elimination-of-energy step are not provided in the manuscript; in particular, the claimed extension from Bourgain's b=d setting to the block case b≥d is asserted without proof. The result is therefore conditional on an external proof whose adaptation is far from purely notational, and the paper as submitted is not self-contained for its main claim.

major comments (3)

[Section 2, Theorem 2.7] Theorem 2.7 is the engine of the whole paper: Theorem 3.6, Theorem 4.1, and ultimately Theorem 1.1 depend on it. Its proof is not given; the text says only that it follows from Lemmas 2.2 and 2.4 and refers to the Claim in [5, p.694], followed by a list of notation alignments. Bourgain's Claim is proved in [5] for b=d with all b_i=1, i.e., one frequency per lattice coordinate. The present theorem requires b=∑ b_i ≥ d with arbitrary block sizes, as in Example 1 (d=2, b_1=2, b_2=1) and Example 2 (b_i=k). The notation alignment does not address why the semi-algebraic set elimination, arithmetic separation, and measure estimates in the proof of the Claim remain valid when the trajectory has block structure n_i ω_i with ω_i∈R^{b_i}. This is a load-bearing step, not a presentational detail; an actual proof of the extension, or a precise statement of the Claim together with a verification that all its hypotheses hold in the block setting, is required before the main result can be considered established.
[Section 2.3, Lemma 2.4] Lemma 2.4 is stated for ω_i∈R^{l_i} (i=1,...,r) and nω=(n_1ω_1,...,n_rω_r), with l=∑ l_i. The proof consists of the sentence that it follows from Lemmas 2.2 and 2.3 'just as the proof of Lemma 1.20 in [5]'. But Lemma 1.20 in Bourgain [5] is formulated for the diagonal case b=d with b_i=1, where the trajectory is a line in T^d and the elimination/separation arguments use the scalar structure of each frequency. With block frequencies ω_i∈R^{l_i}, the semi-algebraic set lies in [0,1]^{lJ}, the sections after elimination have dimension l, and the transversality condition in Lemma 2.3 must be checked for block variables. This is not a purely notational change; if the block analogue requires an extra Diophantine condition or loses a factor in the measure bound, the proof of Theorem 2.7 and hence the LDT collapses. The manuscript should provide the full proof of Lemma 2.4 or a detailed reduction to [5, Lemma 1.20].
[Section 4, Proof of Theorem 1.1] The paper's stated goal is Anderson localization, but the final step is dispatched with 'With Theorem 4.1 at hand, the proof Theorem 1.1 is rather standard' and a reference to [4, Section 3] or [7, Section 6]. The standard argument must be adapted to the present setting: the property P in Definition 2.6 holds with parameters (c1, ρ_i) where ρ_i→ρ/2, the phase space has dimension b that may exceed d, and the operator S is a general Toeplitz matrix rather than the nearest-neighbor Laplacian. None of these adaptations is shown, and the text does not even state which intermediate proposition (e.g., a semi-algebraic complexity bound for the set of energies with a nonlocalized eigenfunction) is being invoked. Since the final localization statement is the main conclusion, this deferral leaves a load-bearing gap in the proof as written.

minor comments (4)

[Throughout] There are numerous typos and small errors that should be corrected: 'opetarors' in the abstract, 'ﬁst' and 'the ﬁst multi-dimensional localization' in the introduction, and in Theorem 2.7 the statement 'mes((ΩN1\Ω3) ≤ N3−c3' is missing a closing parenthesis and should read 'mes((ΩN1\Ω3) ≤ N3^{−c3}'.
[Section 1, Remark (2)] The reference [9] (Bourgain, Jitomirskaya, and Parnovski, 'In preparation') is used to motivate the general Toeplitz setting. Since the present result is explicitly intended as a building block for that work, the 'in preparation' status makes it difficult for the reader to verify the claimed motivation; the relevant statement should be described precisely or the reference should be completed.
[Section 2.3, Lemma 2.4] In the statement of Lemma 2.4, the constant C is written as C=C(J,l), but the preceding line asserts a separation condition 'min_{1≤s≤r} |n_s| > (B max_{1≤s≤r} |m_s|)^C' with n∈N^i, m∈N^{i−1}; the proof of the measure bound also involves the parameter δ whose exponent is δ^{-1}=min_{n∈N^1} min_s |n_s|. The notation should be clarified so that it is explicit how C and δ are chosen.
[Section 3, Theorem 3.6] The figure captions refer to 'Fig.1' and 'Fig.2', but the figures themselves are not included in the arXiv source; the case analysis of Cases 1–3 would be much easier to follow with the actual figures or with a more detailed verbal description of the elementary-region geometry.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is anchored to independent prior results, not to a fitted or self-referential input.

full rationale

The paper's central claim is a new localization theorem for multi-frequency quasi-periodic operators on Z^d with arbitrary k and d. The derivation chain is: import Bourgain's single-scale elimination estimate (Theorem 2.7, referred to the Claim in [5, p.694]), combine it with the paper's own multi-scale induction (Theorem 3.6) and a several-variables Cartan estimate (Lemma 3.4, proved in the appendix), obtain an LDT (Theorem 4.1), and then deduce Anderson localization by the standard argument from [4, Section 3] or [7, Section 6]. No parameter is fitted to the target result, no quantity called a prediction is defined in terms of Anderson localization, and no assumption is equivalent by construction to the conclusion. The proof explicitly defers Theorem 2.7 to Bourgain's Claim: 'For details, we refer the reader to the proof of the Claim in [5, p.694]' (Section 2, Theorem 2.7). That is an external dependency, not a circular reduction; it is a possible completeness or correctness risk, since the paper does not itself supply the proof of the b>=d extension, but the rules exclude unverified external support from the circularity score. The only self-citations (e.g., [8,9,14-16]) are motivational or concern other spectral questions and are not load-bearing for the localization proof. Accordingly, the paper exhibits no self-definitional step, no fitted input disguised as a prediction, and no load-bearing self-citation chain.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No parameters are fitted to data. The proof works with analytic potentials and Toeplitz matrices satisfying exponential decay; all constants (c1-c4, gamma, N0, lambda0) are constructed in the proof. The central claim rests on external standard results in semi-algebraic geometry and several-variable Cartan theory, plus Bourgain's multi-scale Claim imported as Theorem 2.7.

assumptions (5)

standard math Lemma 2.2: elimination of variables for semi-algebraic sets (Bourgain, Lemma 1.18 in [5]).
Used to bound the measure of frequency sets whose trajectories hit a semi-algebraic set; foundational for Theorem 2.7 and the LDT.
standard math Lemma 2.3: decomposition of semi-algebraic sets into small-project and transversality parts ([4,5]).
Used in the proof of Lemma 2.4 and the multi-scale measure estimates.
domain assumption Theorem 2.7: the deterministic multi-scale step from property P at N1 to good scales up to N3, stated as a theorem with proof deferred to [5, p.694].
This is the load-bearing induction step. The paper assumes the general-b version follows from Bourgain's Claim for b=d via notation alignment, without providing the proof.
standard math Lemma 4.2: Lojasiewicz-type estimate from Phong-Stein-Sturm [20]: sup over slices of mes{ |v-E|<δ } ≤ C δ^a.
Provides the initial scale estimate in Theorem 4.3, controlling the measure of phases near a level set of v.
standard math Lemma A.2: higher-dimensional Cartan set estimate of Goldstein-Schlag [13, Lemma 2.15].
Used in the appendix to prove the several-variable matrix-valued Cartan estimate (Lemma 3.4), which is essential in Theorem 3.5.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Anderson localization for multi-frequency quasi-periodic operators on $\mathbb{Z}^d$." pith.science (2026). https://pith.science/paper/HLQRPHIP

@misc{pith2026190803805,
  author       = {Pith},
  title        = {Pith review of: Anderson localization for multi-frequency quasi-periodic operators on $\mathbbZ^d$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HLQRPHIP}},
  note         = {Machine review of arXiv:1908.03805}
}

abstract

We establish Anderson localization for general analytic $k$-frequency quasi-periodic operators on $\mathbb{Z}^d$ for \textit{arbitrary} $k,d$.

Figures

Figures reproduced from arXiv: 1908.03805 by the authors.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Infinite energy quasi-periodic solutions to nonlinear Schr\"odinger equations on $\mathbb R$
math.AP 2019-08 conditional novelty 8.0 of 10

For non-integrable NLS on R, the paper proves existence of space-time quasi-periodic, infinite-energy, smooth solutions with two frequencies for a large Cantor set of parameters.

Reference graph

Works this paper leans on

21 extracted references · 20 canonical work pages · cited by 1 Pith paper

[5]

Bourgain

J. Bourgain. Anderson localization for quasi-periodic lattice Schr ¨ odinger operators onZd, d arbitrary. Geom. Funct. Anal. , 17(3):682–706, 2007

work page 2007
[1]

A. Avila. Global theory of one-frequency Schr¨ odinger operators. Acta Math., 215(1):1–54, 2015

work page 2015
[2]

Avila, J

A. Avila, J. You, and Q. Zhou. Sharp phase transitions for the alm ost Mathieu operator. Duke Math. J. , 166(14):2697–2718, 2017

2017
[3]

Bourgain

J. Bourgain. Estimates on Green’s functions, localization and the quantum kicked rotor model. Ann. of Math. (2) , 156(1):249–294, 2002

work page 2002
[4]

Bourgain

J. Bourgain. Green’s function estimates for lattice Schr¨ odinger operators and applications, volume 158 of Annals of Mathematics Studies . Princeton University Press, Princeton, NJ, 2005

work page 2005
[6]

Bourgain and M

J. Bourgain and M. Goldstein. On nonperturbative localization with quasi-periodic po- tential. Ann. of Math. (2) , 152(3):835–879, 2000

work page 2000
[7]

Bourgain, M

J. Bourgain, M. Goldstein, and W. Schlag. Anderson localization fo r Schr¨ odinger opera- tors on Z2 with quasi-periodic potential. Acta Math., 188(1):41–86, 2002

work page 2002
[8]

Bourgain and S

J. Bourgain and S. Jitomirskaya. Absolutely continuous spectru m for 1D quasiperiodic operators. Invent. Math. , 148(3):453–463, 2002

work page 2002

Show all 21 references

[9]

Bourgain, S

J. Bourgain, S. Jitomirskaya, and L. Parnovski. Absolutely cont inuous spectrum for multidimensional quasiperiodic operators. In preparation
[10]

Bourgain and I

J. Bourgain and I. Kachkovskiy. Anderson localization for two in teracting quasiperiodic particles. Geom. Funct. Anal. , 29(1):3–43, 2019

2019
[11]

V. A. Chulaevsky and E. I. Dinaburg. Methods of KAM-theory f or long-range quasi- periodic operators on Zν. Pure point spectrum. Comm. Math. Phys. , 153(3):559–577, 1993

1993
[12]

Fr¨ ohlich and T

J. Fr¨ ohlich and T. Spencer. Absence of diﬀusion in the Anderson tight binding model for large disorder or low energy. Comm. Math. Phys. , 88(2):151–184, 1983

1983
[13]

Goldstein and W

M. Goldstein and W. Schlag. Fine properties of the integrated de nsity of states and a quantitative separation property of the Dirichlet eigenvalues. Geom. Funct. Anal. , 18(3):755–869, 2008

2008
[14]

Jitomirskaya and W

S. Jitomirskaya and W. Liu. Universal hierarchical structure o f quasiperiodic eigenfunc- tions. Ann. of Math. (2) , 187(3):721–776, 2018

2018
[15]

Jitomirskaya and W

S. Jitomirskaya and W. Liu. Universal reﬂective-hierarchical s tructure of quasiperiodic eigenfunctions and sharp spectral transition in phase. arXiv preprint arXiv:1802.00781 , 2018. 20 SVETLANA JITOMIRSKAYA, WENCAI LIU, AND YUNFENG SHI

2018 arXiv
[16]

Jitomirskaya and S

S. Jitomirskaya and S. Zhang. Quantitative continuity of singula r continuous spectral measures and arithmetic criteria for quasiperiodicSchr¨ odinger operators. arXiv preprint arXiv:1510.07086, 2015

2015 arXiv
[17]

S. Y. Jitomirskaya. Anderson localization for the almost Mathieu equation: a nonpertur- bative proof. Comm. Math. Phys. , 165(1):49–57, 1994

1994
[18]

S. Y. Jitomirskaya. Metal-insulator transition for the almost Ma thieu operator. Ann. of Math. (2) , 150(3):1159–1175, 1999

1999
[19]

C. A. Marx and S. Jitomirskaya. Dynamics and spectral theory of quasi-periodic Schr¨ odinger-type operators.Ergodic Theory Dynam. Systems , 37(8):2353–2393, 2017

2017
[20]

D. H. Phong, E. M. Stein, and J. A. Sturm. On the growth and st ability of real-analytic functions. Amer. J. Math. , 121(3):519–554, 1999

1999
[21]

J. You. Quantitative almost reducibility and its applications. In International Congress of Mathematicians, Rio de Janeiro , page 1916, 2018. (S. Jitomirskaya) Department of Mathematics, University of California, Irvin e, CA, USA E-mail address : szhitomi@math.uci.edu (W. Liu)...

1916

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [5]

Bourgain

J. Bourgain. Anderson localization for quasi-periodic lattice Schr ¨ odinger operators onZd, d arbitrary. Geom. Funct. Anal. , 17(3):682–706, 2007

work page 2007

[2] [1]

A. Avila. Global theory of one-frequency Schr¨ odinger operators. Acta Math., 215(1):1–54, 2015

work page 2015

[3] [2]

Avila, J

A. Avila, J. You, and Q. Zhou. Sharp phase transitions for the alm ost Mathieu operator. Duke Math. J. , 166(14):2697–2718, 2017

2017

[4] [3]

Bourgain

J. Bourgain. Estimates on Green’s functions, localization and the quantum kicked rotor model. Ann. of Math. (2) , 156(1):249–294, 2002

work page 2002

[5] [4]

Bourgain

J. Bourgain. Green’s function estimates for lattice Schr¨ odinger operators and applications, volume 158 of Annals of Mathematics Studies . Princeton University Press, Princeton, NJ, 2005

work page 2005

[6] [6]

Bourgain and M

J. Bourgain and M. Goldstein. On nonperturbative localization with quasi-periodic po- tential. Ann. of Math. (2) , 152(3):835–879, 2000

work page 2000

[7] [7]

Bourgain, M

J. Bourgain, M. Goldstein, and W. Schlag. Anderson localization fo r Schr¨ odinger opera- tors on Z2 with quasi-periodic potential. Acta Math., 188(1):41–86, 2002

work page 2002

[8] [8]

Bourgain and S

J. Bourgain and S. Jitomirskaya. Absolutely continuous spectru m for 1D quasiperiodic operators. Invent. Math. , 148(3):453–463, 2002

work page 2002