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arxiv: 2605.26264 · v1 · pith:SP5GJXFSnew · submitted 2026-05-25 · 🧮 math.AP · math.CA· math.SP

Uncertainty Principles, Spectral Localization, and Singular Schr\"odinger Operators on Compact Manifolds

Alex Iosevich , Chamsol Park This is my paper

Pith reviewed 2026-06-29 20:17 UTC · model grok-4.3

classification 🧮 math.AP math.CAmath.SP
keywords uncertainty principlescompact manifoldsspectral localizationSchrödinger operatorsspectral projectionsKato classincomplete data recovery
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The pith

Uncertainty principles on compact manifolds relate the support of spectrally localized functions to the size of their spectral clusters.

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

The authors establish uncertainty principles that connect the measure of the support of a function localized to a spectral cluster with the number of eigenvalues in that cluster and certain Fourier ratio quantities. This is achieved by merging restriction estimates for systems of orthonormal functions with bounds on spectral projections associated to the Laplace-Beltrami operator. Similar principles are derived for Schrödinger operators that include singular potentials from the Kato class or scaling-critical classes. The results also provide uniqueness theorems for reconstructing functions from partial spectral information on these manifolds. Under additional assumptions of nonpositive or negative sectional curvature, logarithmically improved bounds are obtained for shrinking spectral windows.

Core claim

The paper claims that by combining restriction estimates for orthonormal systems with spectral projection bounds, one obtains uncertainty principles on compact Riemannian manifolds without boundary. These principles bound the support size of spectrally localized functions in terms of the cardinality of the spectral cluster and Fourier-ratio type quantities. Analogues hold for Schrödinger operators with potentials in the Kato and scaling-critical classes, and the principles imply uniqueness for recovery from incomplete spectral data. Logarithmically improved versions follow under curvature assumptions such as nonpositive sectional curvature.

What carries the argument

Restriction estimates for orthonormal systems combined with spectral projection bounds for the Laplace-Beltrami and Schrödinger operators, which relate support size to spectral cluster cardinality.

If this is right

  • Analogues of the uncertainty principles hold for Schrödinger operators with singular potentials in Kato and scaling-critical classes.
  • Uniqueness results follow for recovery of functions from incomplete spectral data on compact manifolds.
  • Logarithmically improved uncertainty principles hold under assumptions of nonpositive or negative sectional curvature for shrinking spectral windows.

Where Pith is reading between the lines

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

  • These principles may extend to other types of operators on manifolds if similar projection bounds can be established.
  • Applications could include numerical recovery algorithms that exploit the uniqueness from partial data.
  • Testing on specific manifolds like the sphere or torus could verify the constants involved in the bounds.

Load-bearing premise

The results rely on the manifold being compact and without boundary, and on the potentials satisfying membership in the Kato or scaling-critical classes.

What would settle it

Finding a spectrally localized function on a compact manifold where the support size is larger than what the cluster cardinality and Fourier ratios would allow would disprove the uncertainty principle.

Figures

Figures reproduced from arXiv: 2605.26264 by Alex Iosevich, Chamsol Park.

Figure 1
Figure 1. Figure 1: Logarithmically shrinking spectral windows yield improved uncertainty princi￾ples on manifolds with nonpositive or negative sectional curvature. 4.2. V ∈ L n 2 (M) and n ≥ 3 for super-critical exponents. Let V ∈ L n 2 (M). The estimate (1.11) follows by combining the proof of (1.6) with the logarithmically improved spectral projection bounds ∥1[λ,λ+(log λ)−1]( p HV )f∥Lq(M) ≲ λ σ(q) (log(2 + λ))1/2 ∥f∥L2(M… view at source ↗
Figure 2
Figure 2. Figure 2: Recovery from incomplete spectral data. Agreement outside a missing spectral cluster forces the difference function to be spectrally localized. The uncertainty principle then prevents simultaneous spectral and spatial concentration. the spectral projection associated with an interval I ⊂ [0, ∞). If f = X j ⟨f, ej ⟩ej , we interpret the coefficients ⟨f, ej ⟩, λj ∈/ I, as the observed spectral data, while th… view at source ↗
read the original abstract

We establish uncertainty principles on compact Riemannian manifolds without boundary by combining restriction estimates for orthonormal systems with spectral projection bounds for Laplace-Beltrami and Schr\"odinger operators. Our results relate the size of the support of spectrally localized functions to the cardinality of the underlying spectral cluster and to Fourier-ratio type quantities. We obtain analogues for Schr\"odinger operators with singular potentials belonging to Kato and scaling-critical classes. As an application, we prove uniqueness results for recovery from incomplete spectral data on compact manifolds. Under curvature assumptions, including nonpositive and negative sectional curvatures, we also prove logarithmically improved uncertainty principles associated with shrinking spectral windows.

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.

Referee Report

0 major / 2 minor

Summary. The paper establishes uncertainty principles on compact Riemannian manifolds without boundary by combining restriction estimates for orthonormal systems with spectral projection bounds for Laplace-Beltrami and Schrödinger operators. The results relate the size of the support of spectrally localized functions to the cardinality of the underlying spectral cluster and to Fourier-ratio type quantities. Analogues are obtained for Schrödinger operators with singular potentials in Kato and scaling-critical classes. Applications include uniqueness results for recovery from incomplete spectral data on compact manifolds. Logarithmically improved uncertainty principles are proved under curvature assumptions such as nonpositive and negative sectional curvatures.

Significance. If the results hold, they extend uncertainty principles and spectral localization results to compact manifolds with singular potentials in standard classes, with applications to inverse spectral problems. The approach of combining existing restriction estimates and spectral projection bounds is standard in the field, but the extensions to Kato-class and scaling-critical potentials, together with the curvature-dependent logarithmic improvements, add to the literature on spectral geometry and PDEs on manifolds.

minor comments (2)
  1. [Abstract] The abstract is quite dense with technical terms (e.g., 'Fourier-ratio type quantities'); a brief parenthetical clarification or reference to the relevant section would improve readability for a broad audience.
  2. [Introduction] Notation for the support size and spectral cluster cardinality should be introduced consistently in the introduction before being used in the statements of the main theorems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper explicitly frames its results as obtained by combining pre-existing restriction estimates for orthonormal systems with spectral projection bounds for the Laplace-Beltrami and Schrödinger operators. These inputs are treated as external and are invoked at the points where the modeling assumptions (compactness, Kato-class potentials, curvature conditions) enter. No derivation step reduces a claimed prediction or uniqueness result to a fitted parameter, a self-defined quantity, or a load-bearing self-citation chain. The application to incomplete spectral data recovery follows directly from the combined estimates without internal redefinition. This matches the default expectation of a non-circular combination of known analytic tools.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard background results from spectral theory and harmonic analysis on manifolds. No free parameters or invented entities appear in the abstract.

axioms (2)
  • standard math The Laplace-Beltrami operator on a compact Riemannian manifold without boundary admits well-defined spectral projections and eigenvalue clusters.
    Invoked when spectral projection bounds are combined with restriction estimates.
  • domain assumption Restriction estimates for orthonormal systems hold on the given manifold.
    Used as one of the two main ingredients to derive the uncertainty principles.

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