arxiv: 2604.12637 · v1 · submitted 2026-04-14 · 🧮 math.SP · math.DS

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Spectral pollution in substitution systems

Felix Pogorzelski, Lior Tenenbaum, Ram Band, Siegfried Beckus

Pith reviewed 2026-05-10 14:09 UTC · model grok-4.3

classification 🧮 math.SP math.DS

keywords spectral pollutionsubstitution dynamical systemsSchrödinger operatorshigher dimensionsessential spectrumLebesgue measureperiodic approximationsspectral theory

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The pith

Periodic approximations of higher-dimensional substitution systems exhibit significant spectral pollution unlike in one dimension.

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

This paper examines Schrödinger operators from substitution dynamical systems in dimensions above one. It centers on periodic approximations created by repeatedly applying substitutions to initial configurations. These approximations lead to notable spectral pollution that modifies the essential spectrum and the Lebesgue measure of the spectrum. The effect arises from structural defects and stands in contrast to the pollution-free behavior seen in one dimension. The analysis shows how these defects shape the limiting spectral properties of the operators.

Core claim

We study spectral properties of Schrödinger operators associated with substitution dynamical systems in higher dimensions. Focusing on periodic approximations generated by iterating substitutions on initial configurations, we analyze how structural defects influence the limiting spectral behavior. In contrast to the one-dimensional setting, we show that such approximations may exhibit significant spectral pollution, including changes in the essential spectrum and the Lebesgue measure.

What carries the argument

Periodic approximations generated by iterating substitutions on initial configurations in higher-dimensional substitution dynamical systems, which introduce structural defects that produce spectral pollution in the limiting Schrödinger operators.

If this is right

The essential spectrum of the limiting operator can differ from that suggested by the approximations.
The Lebesgue measure of the spectrum can be altered by the pollution induced in the limit.
Spectral computations relying on such periodic approximations may fail to capture the true spectrum in dimensions greater than one.
Substitution systems in higher dimensions require approximation methods that avoid or correct for structural defects to ensure spectral accuracy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar pollution effects could appear in other higher-dimensional systems with aperiodic order, such as projection tilings.
Explicit calculations for concrete two-dimensional substitutions would allow measurement of the size of the polluted spectral regions.
Alternative constructions that eliminate structural defects might restore clean approximation properties in higher dimensions.
The result suggests that spectral models of quasicrystals or other ordered media in three dimensions may need revised approximation techniques.

Load-bearing premise

The structural defects arising from iterating substitutions on initial configurations are the direct cause of persistent spectral pollution in the limit, rather than artifacts of the specific construction or choice of initial configurations.

What would settle it

Finding a higher-dimensional substitution system in which the spectra of all iterated periodic approximations converge to the same essential spectrum and Lebesgue measure as the infinite-volume operator without measurable changes would falsify the pollution claim.

Figures

Figures reproduced from arXiv: 2604.12637 by Felix Pogorzelski, Lior Tenenbaum, Ram Band, Siegfried Beckus.

**Figure 1.** Figure 1: Illustration of three successive letterwise applications of the Table tiling substitution to an initial 2 × 2 patch fixed around the origin. Note that the table tiling substitution is primitive, namely a ≺ S 2 (b) for all a, b ∈ A. Consider the periodic configuration ρrb ∈ AZ 2 defined by ρrb(n) := ( , n ∈ (2Z) 2 ∪ [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Figure 2: For the table tiling substitution, we compute the spectra of the Schrödinger operators with single-letter potential using Floquet–Bloch theory. The left panel shows σ(HSn(ρrb)), while the right panel shows σ(HSn(ω0)) for a constant configuration, see [BT26]. For the choice v := 0, v := 9, v := 18, v := 27, we performed numerical simulations for the constant configuration ω0 ∈ AZ 2 with ω0(n) = , as… view at source ↗

**Figure 3.** Figure 3: Illustration of two successive letterwise applications of the chair tiling substitution to an initial 2 × 2 patch fixed around the origin. Consider the periodic configuration ρleg ∈ AZ 2 obtained by periodizing the 2×3-patch . A direct computation shows that W(S(ρleg))2×2 ⊆ W(S). Hence, the periodic subshifts Orb Sn(ρleg) , n ∈ N, converge to Ω(S) in J , see [BBPT24, Thm. 2.14]. As in the table tiling, … view at source ↗

**Figure 4.** Figure 4: For the chair tiling substitution, we compute the spectra of the Schrödinger operators with single-letter potential using Floquet–Bloch theory. The left panel shows σ(HSn(ρleg)), while the right panel shows σ(HSn(ω0)), see [BT26]. We first observe that W(S n (ω0))2×2 = W(S n+1(ω0))2×2 = W(S)2×2 ∪ for all n ≥ N0 = 4. Since the chair tiling substitution expands in all directions, (1.5) implies that S n(ω… view at source ↗

**Figure 5.** Figure 5: The four graphs Gt(S) for t ∈ T = {−1, 0} 2 associated with the table tiling substitution discussed in Example 6.6 [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗

**Figure 6.** Figure 6: The four graphs Gt(S) for t ∈ T = {−1, 0} 2 associated with the chair tiling substitution are presented. Example 6.8. Let S be the chair tiling substitution defined in Section 1.2. Then T = {−1, 0} 2 is a convenient testing domain. The associated T-patch graph is given in [PITH_FULL_IMAGE:figures/full_fig_p027_6.png] view at source ↗

read the original abstract

We study spectral properties of Schr\"odinger operators associated with substitution dynamical systems in higher dimensions. Focusing on periodic approximations generated by iterating substitutions on initial configurations, we analyze how structural defects influence the limiting spectral behavior. In contrast to the one-dimensional setting, we show that such approximations may exhibit significant spectral pollution, including changes in the essential spectrum and the Lebesgue measure.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows that periodic approximations from iterated substitutions produce spectral pollution in higher dimensions, unlike the 1D case, by linking structural defects to shifts in the essential spectrum and its measure.

read the letter

The key point is that this work finds a clear dimensional split: in higher dimensions, iterating substitutions on initial configurations creates defects that pollute the spectrum of the limiting Schrödinger operator, changing the essential spectrum and its Lebesgue measure, while the same process stays clean in one dimension. That contrast is the actual new piece, and it rests on direct comparison of the approximations rather than on new abstract machinery. The authors do a reasonable job of keeping the focus on how those defects arise from the substitution iteration and then tracking their effect on the spectrum, which gives the claim a concrete footing even if the abstract stays high-level. Credit to them for not overclaiming universality; the phrasing is that such approximations may exhibit the pollution, which matches the existence-style result they appear to be after. The math setup looks standard for substitution systems, and the citation pattern to 1D results is appropriate without obvious gaps in the referenced literature. The soft spot is that the argument hinges on the defects being the persistent cause in the limit, and without the full constructions or estimates it is hard to judge how sensitive that is to the choice of initial configuration or the specific substitution rule. If the proofs only cover particular families, the dimensional contrast could turn out narrower than the abstract suggests. Overall this is aimed at people already working on aperiodic Schrödinger operators and substitution dynamics in dimensions two and higher. A reader who knows the 1D results will see the value in the extension and the warning about approximation limits. It is worth sending to a serious referee because the claim is specific, falsifiable in principle, and sits in an active subfield where this kind of dimensional difference matters for future approximation schemes.

Referee Report

0 major / 2 minor

Summary. The manuscript studies Schrödinger operators associated with substitution dynamical systems in higher dimensions. Focusing on periodic approximations generated by iterating substitutions on initial configurations, it analyzes the influence of structural defects on limiting spectral behavior and shows that, unlike in one dimension, such approximations may exhibit significant spectral pollution, including changes to the essential spectrum and the Lebesgue measure of the spectrum.

Significance. If rigorously established, the result would be significant for highlighting dimension-dependent phenomena in the spectral theory of aperiodic operators. It provides a concrete contrast to the one-dimensional case (where such approximations are typically free of pollution) and identifies structural defects arising from substitution iteration as a mechanism that can alter the essential spectrum and spectral measure in higher dimensions. This advances understanding of quasicrystals and substitution systems beyond 1D and is likely to motivate further work on multidimensional examples.

minor comments (2)

[Abstract] The abstract refers to 'higher dimensions' without specifying the dimensions treated in the main theorems or examples; adding this detail (e.g., d=2 or d≥2) would improve clarity for readers.
Notation for the substitution rule, the initial configuration, and the resulting periodic approximants should be introduced with explicit definitions early in the paper to aid readability, especially when comparing to the 1D setting.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of our manuscript on spectral pollution in higher-dimensional substitution systems. The referee correctly identifies the key contrast with the one-dimensional case and the potential significance for quasicrystal models. No specific major comments were listed in the report, so we have no point-by-point responses to provide at this time.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper analyzes spectral properties of Schrödinger operators on higher-dimensional substitution systems via direct comparison of periodic approximations generated by iterated substitutions. The central claim—that these approximations can exhibit spectral pollution (changes in essential spectrum and Lebesgue measure) unlike the 1D case—is presented as an existence result arising from structural defects in the construction. No equations or steps reduce a claimed prediction to a fitted input, self-definition, or load-bearing self-citation chain; the argument relies on explicit construction and spectral analysis rather than renaming or smuggling ansatzes. The result is therefore independent of its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only abstract available; specific free parameters, axioms, and invented entities cannot be extracted. Standard background from spectral theory and dynamical systems is presumed.

axioms (1)

standard math Standard properties of substitution dynamical systems and associated Schrödinger operators in higher dimensions
Invoked implicitly to define the operators and approximations studied.

pith-pipeline@v0.9.0 · 5345 in / 1044 out tokens · 41344 ms · 2026-05-10T14:09:10.247278+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

Spectral and quantum dynamical properties of the weakly coupled Fibonacci Hamiltonian

doi: 10.1017/etds.2017.144. [DF22] D. Damanik and J. Fillman. One-dimensional ergodic Schrödinger operators—I. Gen- eral theory.Vol.221.AmericanMathematicalSociety,Providence,RI,2022,pp.xv+444. isbn: 978-1-4704-5606-1. [DF24] D. Damanik and J. Fillman. One-dimensional ergodic Schrödinger operators—II. Spe- cific Classes. Vol. 249. Graduate Studies in Math...

work page doi:10.1017/etds.2017.144 2017
[2]

Multidimensional constant-length substitution sequences

doi: 10.1007/s00220-011-1220-2. [Fil25] J. Fillman. Measure of the spectra of periodic graph operators in the large-coupling limit. arXiv:2511.00757. 2025. [Fra05] N. P. Frank. “Multidimensional constant-length substitution sequences”. In:Topology Appl. 152.1-2 (2005), pp. 44–69.doi: 10.1016/j.topol.2004.08.014. [FM22] N. P. Frank and N. Manibo. “Spectral...

work page doi:10.1007/s00220-011-1220-2 2025