Spectral and Pseudospectral Approximation of Finite-Interaction-Range Operators in Doubling Metric Measure Spaces

Mattes Wittig

Spectral and Pseudospectral Approximation of Finite-Interaction-Range Operators in Doubling Metric Measure Spaces

T0 review · 0 major / 7 minor · reviewed 2026-08-05 · deepseek-v4-flash

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

Pith's one-line read Finite windows determine global spectra with a 1/L error

desk verdict The central result is solid and worth refereeing; the only real weakness is that the Banach-valued and ℓ^p extensions in Section 8 are sketched rather than proved, which shouldn't block the main theorem. read the letter →

arxiv 2608.03526 v1 pith:YCP26CQR submitted 2026-08-04 math.SP math.FA

Mattes Wittig This is my paper · ORCID

classification math.SPmath.FA MSC 47A1047A5847B2847B9347N40

keywords finite-interaction-rangeoperatorsspectralapproximationpseudospectrafinitesectionslowernormsdoublingmetricmeasurespacesbandlocalisationofquasi-modes

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 aims to show that for a bounded finite-interaction-range operator on ℓ²(Γ), where Γ is a uniformly discrete subset of a left-invariant doubling metric measure space, the global spectrum and pseudospectrum are fully determined — up to a controlled additive error — by the finite rectangular sections H_{L,x} supported on metric balls. The central estimate pins the infimum of the local lower norms ν(H_{L,x}−λ) between the global lower norm ν(H−λ) and ν(H−λ)+C₀/L, with C₀ explicit in the interaction radius, coupling bound, interaction degree, and doubling geometry. From that sandwich follow two-sided pseudospectral inclusions, Hausdorff convergence of window pseudospectra, spectral gap tests, and sampling schemes. Why care: the geometric mechanism works without assuming Γ is a lattice or an Abelian group, so it covers quasicrystal-like disordered sets and discrete nilpotent groups such as the Heisenberg group. The paper also isolates the sole source of the error — the commutator of the operator with a Lipschitz tent cutoff.

What carries the argument

The load-bearing object is the Lipschitz tent localisation W_{L,x}, multiplication by w_{L,x}(y)=max{0,1−d(x,y)/L} on ℓ²(Γ). It is supported in B_L(x), is 1/L-Lipschitz, and its commutator with a band operator H is the sole source of localisation error. Theorem 11 bounds the X-average of ∥[W_{L,x},H]ψ∥² by (γ² C_geom m² M² / L²)∥ψ∥²∥w_{L,e}∥²_{L²(X)} using doubling and the packing bound on Γ; Lemma 12's averaging identity then converts global quasi-modes into a pointwise local quasi-mode, producing the O(1/L) loss and Corollary 14.

What would settle it

Take a concrete band operator on the discrete Heisenberg group with a potential, fix λ on the spectrum, and compute α_L(λ)=inf_x ν(H_{L,x}−λ) for growing L: if α_L(λ) does not stay at or below C₀/L with the paper's explicit C₀, or if α_L(λ)−ν(H−λ) decays strictly slower than 1/L over a sequence of windows, Corollary 14 is false. A simpler numerical experiment on a non-relatively-dense uniformly discrete set, using the empty-window convention of Section 8, would test whether the extension claim holds as stated.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is a quantitative localisation theorem: every global ε-quasi-mode of H−λ can be localised to a window B_L(x) to produce an (ε+C₀/L)-quasi-mode, because the integrated square of the commutator [W_{L,x}, H] decays like C₀²/L² when averaged over centres. The consequence is the two-sided window-pseudospectrum inclusion γ_{L,ε}(H) ⊂ σ_ε(H) ⊂ γ_{L,ε+C₀/L}(H), with the analogous closed version, so the global ε-pseudospectrum is trapped by data from windows of size L up to tolerance shift C₀/L. In the normal case this yields rigorous gap tests and finite-grid spectral approximations; in the non-normal case it yields deterministic pseudospectral approximation w

Load-bearing premise

The argument assumes the ambient space has a translation-invariant geometry whose ball volumes grow at a controlled (doubling) rate, and that the discrete set is spread evenly enough to be relatively dense; if those fail, the uniform 1/L error estimates and center-independent constants are not established.

Editorial extensions

If this is right

For every fixed tolerance ε>0, the window pseudospectrum is sandwiched between the global pseudospectrum at ε and at ε+C₀/L, and the two sides converge in Hausdorff distance as L→∞.
A point λ is certified to be at distance at least δ from the spectrum whenever the minimized window lower norm exceeds C₀/L by δ; this gives a computable spectral gap test.
In the normal case, a finite grid of local lower norms yields a rigorous δ-Hausdorff approximation of the spectrum, and this becomes a finite-time procedure under finite local complexity.
In the non-normal case, refining the grid until successive window pseudospectral approximations stabilize produces a deterministic algorithm whose output is a rigorous δ-approximation of the global pseudospectrum.
The constants are explicit and uniform over the geometry, so the same error control applies to every window size L and every center x, including irregular geometries covered by the framework.

Reading between the lines

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

Because the paper shows a compactly supported Lipschitz cutoff cannot beat the O(1/L) rate, a natural testable extension is whether smoother or non-compactly supported filters yield faster convergence while preserving the clean geometric constants.
For quasicrystal tight-binding models with finite local complexity, the finite-time sampling procedure suggests a practical certified workflow: compute local smallest singular values on a small atlas of windows and read off spectral gaps and pseudospectral enclosures without any translation invariance assumption.
The Section 8 sketches for ℓᵖ spaces and Banach-valued sequences indicate that the same mechanism should control localisation of quasi-modes in non-Hilbert norms; if confirmed, the approximation results would extend to weighted spaces and non-self-adjoint operator families.
Using the empty-window convention proposed for non-relatively-dense Γ, the theory would cover uniformly discrete sets with arbitrarily large gaps, making it applicable to sparse Delone sets rather than only relatively dense ones.

Share X Bluesky LinkedIn Reddit HN

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

Short version: the main result is real and the proof chain holds up on close reading. The paper proves explicit two-sided pseudospectral inclusion bounds for finite-interaction-range operators on uniformly discrete subsets of left-invariant doubling metric measure spaces, covering quasicrystal-like geometries and non-Abelian nilpotent groups like the discrete Heisenberg group. This genuinely extends earlier work on R^n and countable Abelian groups; the averaged tent commutator bound is the right technical core.

What's good: the constants are explicit, the argument is self-contained, and the paper is careful about what follows from what. I traced Theorem 11 → Lemma 12 → Lemma 13 → Corollary 14 → Theorem 17 and found no circularity and no fitted parameters. The O(1/L) rate is argued for Lipschitz cutoffs, and the Hausdorff convergence and gap tests are legitimate consequences. The paper also credits prior work appropriately; the citations to [24] and [14] are accurate, and the discussion of the Globevnik property is correct.

Soft spots: Section 8.2 and 8.3 jump from 'we expect' to 'it carries over with essentially no change' for Banach-valued and ℓ^p settings. Those are genuinely different settings—the combined lower norm requires Banach adjoints, and the averaging and commutator estimates need re-proving with Hölder and operator-valued entries. The author sketches the modifications but doesn't complete them. That's a real gap, but only in the extension section; Theorem 17 and the main theorem are unaffected. Also the relative denseness assumption is handled only as a sketch in Section 8.1, though that one is less worrying because it can be fixed by the same empty-window convention or the embedding trick.

One small thing: I do not think the claimed 'optimality' of the 1/L rate in Remark 19 is established in full generality—it assumes the metric realizes distances at scale L, which is plausible but not argued. Minor, not load-bearing.

Bottom line: this is a competent, honest paper that does what it claims for the Hilbert-space scalar setting. The proofs look sound from my reading. It belongs in the refereeing pipeline. I'd ask the editor to send it to a referee who can either fill in or certify the Section 8 claims, but the core result deserves to see the light of day.

Referee Report

0 major / 7 minor

Summary. The paper establishes a quantitative link between global spectral data of a bounded finite-interaction-range operator on ℓ²(Γ) and local finite-window reductions, under the assumption that Γ is a uniformly discrete, relatively dense subset of a left-invariant doubling metric measure space. The technical core is an averaged commutator bound for Lipschitz tent localisations (Theorem 11), an averaging identity (Lemma 12), and a quasimode localisation lemma (Lemma 13). These yield Corollary 14, an O(1/L) upper bound on the infimum of local lower norms in terms of the global lower norm. The main result (Theorem 17) gives two-sided pseudospectral inclusions γ_{L,ε}⊂σ_ε⊂γ_{L,ε+C0/L} and Hausdorff convergence of window pseudospectra, with explicit C0=γmM√C_geom. Applications include gap tests and spectral sampling for normal operators and an adaptive algorithm with stopping criterion for non-normal pseudospectra. Section 8 sketches extensions to non-relatively-dense sets, Banach-valued spaces, and ℓ^p.

Significance. The significance is in the unification and explicit quantification. The proof chain Theorem 11 → Lemma 12 → Lemma 13 → Corollary 14 → Theorem 17 is internally consistent; I checked the commutator estimate, the averaging identity, and the two-sided inclusions and found no errors. The constants depend only on the interaction range m, the coupling bound M, the interaction degree γ, and the doubling constant, with no fitted parameters. This rigorously transfers localisation strategies from R^n and countable Abelian groups to irregular geometries, including the discrete Heisenberg group, which is a genuine extension beyond previous frameworks. The two-sided pseudospectral inclusions are strong: they give computable gap certificates (Corollaries 20–21), Hausdorff approximations with explicit δ (Corollary 22), and a deterministic termination criterion in the non-normal case (Theorem 24). The optimal O(1/L) rate is argued in Remark 19. The main contribution is solid; the only notable weakness is that several Section 8 extension claims are stated more strongly than they are proved.

minor comments (7)

[§8.2–8.3] The Banach-valued and ℓ^p extensions are presented as consequences but only sketched. For example, §8.2 says the results carry over 'with essentially no change' and §8.3 says the lower-norm part 'admits a direct ℓ^p-analogue'; the combined lower norm with Banach adjoint is only 'expected' to hold. These extensions are not needed for the central Theorem 17, but as written they overstate the support. Please either provide full proofs or explicitly label these claims as conjectures/outlook.
[Theorem 11 / Corollary 14] The constant C_geom is defined via k=ceil(log2(2(1+m/L))) and therefore depends on L. Thus C0=γmM√C_geom is not actually independent of L for all L>0 as claimed in Lemma 13 and Corollary 14. The dependence disappears uniformly for L≥m (where C_geom≤4C_dbl²), and all asymptotic claims use L→∞, so this is easily fixed: state the uniform-constant result for L≥m, or define C_geom by the uniform bound from the outset.
[§2.2, Eq. (7)] The equality Σ_ε(A)=σ_ε(A) cannot hold as sets since σ_ε is open and Σ_ε is closed. What is meant, and what is used later via Hausdorff distance, is that the closure of σ_ε(A) equals Σ_ε(A). Please correct this to avoid a formal inconsistency.
[Lemma 13 proof] There is a misreference: 'By Lemma (11)' should be 'By Lemma 12' when invoking the averaging identity for the positive measure of the set E. The surrounding argument is correct, but the reference should be fixed.
[Corollary 22] The definition h_L := √2C_0/L is ambiguous. The proof uses h_L/√2 = C_0/L, so the intended formula is h_L = √2·C_0/L. Please clarify the displayed definition.
[§3.2, Intermezzo] The finite-local-complexity equivalence relation uses 'rotation' in a general metric group without giving a definition. Since FLC is not used in the main proofs, the notion should either be defined precisely or explicitly left as an illustrative condition.
[Assumption 3] The argument uses finiteness of ball volumes V(r) (e.g., in the packing bound and in ∥w_{L,e}∥_{L²(X)}). If this is not automatically implied by the stated assumptions, it should be added explicitly; otherwise the inequalities in Theorem 11 and Lemma 12 need a small justification that V(r)<∞ for all r>0.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central pseudospectral inclusion is derived by an explicit proof from stated geometric/operator assumptions, with no fitted parameters or load-bearing self-citation.

full rationale

The derivation chain is self-contained. Theorem 11 proves a commutator bound from the Lipschitz property of the tent function, the finite-interaction-range condition, the uniform bound on matrix entries, and the doubling/packing geometry; C_geom and C_0 = gamma*m*M*sqrt(C_geom) are explicit and arise from the proof, not from any fit. Lemma 12 is an averaging identity using only left-invariance of metric and measure plus Tonelli. Lemma 13 combines Theorem 11 and Lemma 12 to convert global quasi-modes into local quasi-modes with an explicit O(1/L) loss. Corollary 14 is a direct infimum argument, and Theorem 17 is a sandwich using monotonicity of local lower norms plus the Globevnik property, which is cited as standard independent material. There is no parameter fitted to data and then renamed a prediction; no self-citation carries a load-bearing assumption; no uniqueness theorem from the authors is invoked; and the new framework is not a renaming of a known result, since the proof supplies the explicit constants and inclusions. The only caveats are in Section 8.2–8.3, where Banach-valued and ell^p extensions are sketched rather than fully proved; this is a completeness issue, not circularity, and those sketches do not support Theorem 17.

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

The paper introduces no fitted parameters and no new entities. The constants m, M, gamma, C_dbl are assumed inputs of the operator and geometry; C0 and C_geom are derived from them. The axioms are either standard analysis facts or the standing geometric/operator assumptions that define the scope of the theorems.

assumptions (5)

domain assumption Left-invariant doubling metric measure structure (Assumption 3)
Used throughout; supplies the averaging identity (Lemma 12) and packing bounds (Theorem 11).
domain assumption Gamma uniformly discrete and relatively dense, H band operator with finite range m, bound M, degree gamma (Assumption 4)
Defines the operator class; gamma finiteness follows from uniform discreteness.
standard math Hilbert-space Globevnik property: sigma_eps(A)=Sigma_eps(A) for eps>0 (from [20,28])
Needed for Hausdorff convergence of open/closed pseudospectra in Section 2.4.
standard math Identity ||A^{-1}||^{-1}=nu_comb(A) (from [26,18])
Links lower norms to resolvent pseudospectra, Equation (4).
standard math Tonelli/Fubini and Cauchy-Schwarz estimates
Used in proofs of Theorem 11 and Lemma 12.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Spectral and Pseudospectral Approximation of Finite-Interaction-Range Operators in Doubling Metric Measure Spaces." pith.science (2026). https://pith.science/paper/YCP26CQR

@misc{pith2026260803526,
  author       = {Pith},
  title        = {Pith review of: Spectral and Pseudospectral Approximation of Finite-Interaction-Range Operators in Doubling Metric Measure Spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YCP26CQR}},
  note         = {Machine review of arXiv:2608.03526}
}

abstract

We study bounded finite-interaction-range operators $H$ on $\ell^2(\Gamma)$, where $\Gamma$ is a uniformly discrete subset of a left-invariant doubling metric measure space $(X,d,\mu)$. Our goal is to approximate spectral information of $H$ from finite sections $H_{L,x}$ supported on balls $B_L(x)\cap\Gamma$. The main technical input is a commutator estimate for Lipschitz ''tent'' localisations $W_{L,x}$, which depends only on geometric properties (doubling) and a uniform interaction degree. As a consequence, we obtain explicit two-sided pseudospectral inclusion bounds of the form \[ \gamma_{L,\varepsilon}(H)\subset \sigma_\varepsilon(H)\subset \gamma_{L,\varepsilon+C_0/L}(H), \] and Hausdorff convergence of window pseudospectra to the (global) pseudospectrum as $L\to\infty$. In the self-adjoint/ normal case this yields computable gap tests and spectral sampling schemes with rigorous $O(1/L)$ error control, while in the non-normal case it leads to corresponding approximation results for pseudospectra. The framework isolates the geometric core behind earlier approaches on $\mathbb{R}^n$ and on countable Abelian groups, and covers irregular geometries arising in quasicrystal models, as well as new cases such as discrete (non-Abelian) nilpotent groups (including for example the discrete Heisenberg group).

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

31 extracted references · 30 canonical work pages

[24]

358, 833–866

,Computing the spectrum and pseudospectrum of infinite-volume operators from local patches, Mathematics of Computation95(2026), no. 358, 833–866. 25

work page 2026
[14]

Localisation of pseudospectra on discrete groups

Simon N. Chandler-Wilde, Marko Lindner, and Christian Seifert,Localisation of pseu- dospectra on discrete groups, 2026, arXiv:2607.29354

work page Pith review arXiv 2026
[1]

1, 303–342

Artur Avila and Svetlana Jitomirskaya,The Ten Martini Problem, Annals of Mathematics 170(2009), no. 1, 303–342

work page 2009
[2]

Joseph Avron, Pieter H. M. van Mouche, and Barry Simon,On the measure of the spectrum for the almost Mathieu operator, Communications in Mathematical Physics132(1990), no. 1, 103–118

work page 1990
[3]

M. Ya. Azbel,Energy spectrum of a conduction electron in a magnetic field, Soviet Physics JETP19(1964), no. 3, 634–645

work page 1964
[4]

volume 1: A mathematical invitation, Cambridge University Press, Cambridge, 2013

Michael Baake and Uwe Grimm,Aperiodic order. volume 1: A mathematical invitation, Cambridge University Press, Cambridge, 2013

work page 2013
[5]

Ram Band, Siegfried Beckus, Felix Pogorzelski, and Lior Tenenbaum,Spectral approximation for substitution systems, Journal d’Analyse Math´ ematique (2026), https://doi.org/10.1007/s11854-026-0445-0

work page doi:10.1007/s11854-026-0445-0 2026
[6]

11, 3603–3631

Siegfried Beckus, Jean Bellissard, and Horia Cornean,H¨ older continuity of the spectra for aperiodic hamiltonians, Annales Henri Poincar´ e20(2019), no. 11, 3603–3631. 24

work page 2019

Show all 31 references

[7]

general theory, Journal of Functional Analysis275(2018), no

Siegfried Beckus, Jean Bellissard, and Giuseppe De Nittis,Spectral continuity for aperiodic quantum systems I. general theory, Journal of Functional Analysis275(2018), no. 11, 2917–2977

2018
[8]

2, 563–610

Siegfried Beckus and Alberto Takase,Spectral estimates of dynamically-defined and amenable operator families, Journal of Spectral Theory15(2025), no. 2, 563–610

2025
[9]

2, 353–380

Jean Bellissard, Bruno Iochum, and Daniel Testard,Continuity properties of the electronic spectrum of 1D quasicrystals, Communications in Mathematical Physics141(1991), no. 2, 353–380

1991
[10]

Colbrook, Anders C

Jonathan Ben-Artzi, Matthew J. Colbrook, Anders C. Hansen, Olavi Nevanlinna, and Markus Seidel,Computing Spectra – On the Solvability Complexity Index Hierarchy and Towers of Algorithms, 2020, arXiv:1508.03280

2020 arXiv
[11]

Simon N. Chandler-Wilde, Ratchanikorn Chonchaiya, and Marko Lindner,Convergent spectral inclusion sets for banded matrices, Proceedings in Applied Mathematics and Me- chanics23(2023), e202300016

2023
[12]

2, 719–804

,On spectral inclusion sets and computing the spectra and pseudospectra of bounded linear operators, Journal of Spectral Theory14(2024), no. 2, 719–804

2024
[13]

Chandler-Wilde and Marko Lindner,Gershgorin-type spectral inclusions for ma- trices, Linear Algebra and its Applications732(2025), 33–73

Simon N. Chandler-Wilde and Marko Lindner,Gershgorin-type spectral inclusions for ma- trices, Linear Algebra and its Applications732(2025), 33–73

2025
[15]

Matthew J. Colbrook, Mark Embree, and Jake Fillman,Computing Spectral Size: Rigorous Algorithms and the Limits of Computation, Communications of the American Mathemati- cal Society (2026), To appear; arXiv:2407.20353

2026 arXiv
[16]

Colbrook and Anders C

Matthew J. Colbrook and Anders C. Hansen,The foundations of spectral computations via the Solvability Complexity Index hierarchy, J. Eur. Math. Soc.25(2023), no. 12, 4639–4718

2023
[17]

E. B. Davies and E. Shargorodsky,Level sets of the resolvent norm of a linear operator revisited, Mathematika62(2015), 243–265

2015
[18]

Brian Davies,Linear operators and their spectra, Cambridge Studies in Advanced Math- ematics, vol

E. Brian Davies,Linear operators and their spectra, Cambridge Studies in Advanced Math- ematics, vol. 106, Cambridge University Press, 2007

2007
[19]

Folland,A Course in Abstract Harmonic Analysis, CRC Press, 1995

Gerald B. Folland,A Course in Abstract Harmonic Analysis, CRC Press, 1995

1995
[20]

Globevnik,Norm-constant analytic functions and equivalent norms, Illinois Journal of Mathematics20(1976), 503–506

J. Globevnik,Norm-constant analytic functions and equivalent norms, Illinois Journal of Mathematics20(1976), 503–506

1976
[21]

Hansen,On the Solvability Complexity Index, then-pseudospectrum and Ap- proximations of Spectra of Operators, Journal of the American Mathematical Society24 (2011), no

Anders C. Hansen,On the Solvability Complexity Index, then-pseudospectrum and Ap- proximations of Spectra of Operators, Journal of the American Mathematical Society24 (2011), no. 1, 81–124

2011
[22]

thesis, University of T¨ ubingen, 2024

Paul Hege,Spectral gaps in systems of finite local complexity, Ph.d. thesis, University of T¨ ubingen, 2024

2024
[23]

Paul Hege, Massimo Moscolari, and Stefan Teufel,Finding spectral gaps in quasicrystals, Physical Review B106(2022), 155140

2022
[25]

Hofstadter,Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Physical Review B14(1976), no

Douglas R. Hofstadter,Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Physical Review B14(1976), no. 6, 2239–2249

1976
[26]

Marko Lindner,Infinite Matrices and their Finite Sections: An Introduction to the Limit Operator Method, Frontiers in Mathematics, Birkh¨ auser, Basel, 2006

2006
[27]

Rabinovich, Steffen Roch, and Bernd Silbermann,Limit Operators and Their Applications in Operator Theory, Operator Theory: Advances and Applications, vol

Vladimir S. Rabinovich, Steffen Roch, and Bernd Silbermann,Limit Operators and Their Applications in Operator Theory, Operator Theory: Advances and Applications, vol. 150, Birkh¨ auser, 2004

2004
[28]

3, 493–504

Eugene Shargorodsky,On the level sets of the resolvent norm of a linear operator, Bulletin of the London Mathematical Society40(2008), no. 3, 493–504

2008
[29]

Struble,Metrics in locally compact groups, Compositio Mathematica28 (1974), no

Raimond A. Struble,Metrics in locally compact groups, Compositio Mathematica28 (1974), no. 3, 217–222

1974
[30]

Andr´ as S¨ ut˝ o,The spectrum of a quasiperiodic Schr¨ odinger operator, Communications in Mathematical Physics111(1987), 409–415

1987
[31]

Trefethen and Mark Embree,Spectra and Pseudospectra: The Behavior of Non- normal Matrices and Operators, Princeton University Press, Princeton, NJ, 2005

Lloyd N. Trefethen and Mark Embree,Spectra and Pseudospectra: The Behavior of Non- normal Matrices and Operators, Princeton University Press, Princeton, NJ, 2005. 26

2005

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [24]

358, 833–866

,Computing the spectrum and pseudospectrum of infinite-volume operators from local patches, Mathematics of Computation95(2026), no. 358, 833–866. 25

work page 2026

[2] [14]

Localisation of pseudospectra on discrete groups

Simon N. Chandler-Wilde, Marko Lindner, and Christian Seifert,Localisation of pseu- dospectra on discrete groups, 2026, arXiv:2607.29354

work page Pith review arXiv 2026

[3] [1]

1, 303–342

Artur Avila and Svetlana Jitomirskaya,The Ten Martini Problem, Annals of Mathematics 170(2009), no. 1, 303–342

work page 2009

[4] [2]

Joseph Avron, Pieter H. M. van Mouche, and Barry Simon,On the measure of the spectrum for the almost Mathieu operator, Communications in Mathematical Physics132(1990), no. 1, 103–118

work page 1990

[5] [3]

M. Ya. Azbel,Energy spectrum of a conduction electron in a magnetic field, Soviet Physics JETP19(1964), no. 3, 634–645

work page 1964

[6] [4]

volume 1: A mathematical invitation, Cambridge University Press, Cambridge, 2013

Michael Baake and Uwe Grimm,Aperiodic order. volume 1: A mathematical invitation, Cambridge University Press, Cambridge, 2013

work page 2013

[7] [5]

Ram Band, Siegfried Beckus, Felix Pogorzelski, and Lior Tenenbaum,Spectral approximation for substitution systems, Journal d’Analyse Math´ ematique (2026), https://doi.org/10.1007/s11854-026-0445-0

work page doi:10.1007/s11854-026-0445-0 2026

[8] [6]

11, 3603–3631

Siegfried Beckus, Jean Bellissard, and Horia Cornean,H¨ older continuity of the spectra for aperiodic hamiltonians, Annales Henri Poincar´ e20(2019), no. 11, 3603–3631. 24

work page 2019