arxiv: 2604.15442 · v1 · submitted 2026-04-16 · 🧮 math.CA · math.AP· math.SP

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Uncertainty principles and singular potentials

A. Iosevich, C. Park

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

classification 🧮 math.CA math.APmath.SP

keywords uncertainty principlesRiemannian manifoldsLaplace-Beltrami operatorssingular potentialsspectral conditionsWeyl lawseigenfunction restrictions

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The pith

A quantitative spectral condition stabilizes uncertainty inequalities on compact Riemannian manifolds with singular potentials.

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

The paper establishes uncertainty principles for the Laplace-Beltrami operator on compact manifolds without boundary, extending to cases with real-valued singular potentials. By replacing the classical homogeneity assumption with a quantitative spectral condition, it derives stability versions of the uncertainty inequalities. The key result is an inequality that includes terms for spectral deviation and is proven sharp. In one dimension, the condition holds automatically, providing a link between spectral complexity and spatial support. Higher-dimensional cases use pointwise Weyl laws and eigenfunction restriction estimates.

Core claim

We prove that (1-ε-ε')^2 ≤ |E|/|M| · #X_S · sup_{x∈E} A_S(x) / (#X_S / |M|), which recovers the classical bound in the homogeneous case, quantifies its deterioration in the presence of spectral inhomogeneity, and is shown to be sharp in general. In dimension one the homogeneity condition holds automatically with additional Fourier-ratio complexity bounds, while higher dimensions rely on pointwise Weyl laws and eigenfunction restriction estimates.

What carries the argument

The quantitative spectral condition on the spectrum of the Laplace-Beltrami operator, which replaces homogeneity and enables the stability estimate through pointwise Weyl laws and eigenfunction restriction estimates.

If this is right

The classical uncertainty bound is recovered exactly when the spectral condition holds with ε = ε' = 0.
The bound deteriorates in a controlled way as spectral inhomogeneity increases, as measured by ε and ε'.
The inequality is sharp in general, so the constants cannot be improved without stronger assumptions.
In dimension one, automatic homogeneity yields a quantitative relation between spectral complexity and spatial support via Fourier-ratio bounds.
In higher dimensions, the results apply through pointwise Weyl laws and eigenfunction restriction estimates on submanifolds.

Where Pith is reading between the lines

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

The approach could be tested numerically on standard manifolds such as the sphere to check sharpness for concrete sets.
Similar quantitative conditions might stabilize uncertainty principles for other elliptic operators on manifolds.
The sharpness result indicates that spectral inhomogeneity fundamentally limits how tight the bounds can be.

Load-bearing premise

The quantitative spectral condition holds with sufficiently small error parameters ε and ε' for the given manifold and sets.

What would settle it

An explicit compact Riemannian manifold, a positive-measure set E, and spectral set X_S where the left side of the main inequality exceeds the right side for the ε and ε' determined by the spectral condition.

read the original abstract

We establish uncertainty principles on compact Riemannian manifolds without boundary in the setting of Laplace-Beltrami operators, including the case of real-valued singular potentials. We replace the classical homogeneity assumption by a quantitative spectral condition and obtain corresponding stability versions of uncertainty inequalities. In particular, we prove that \[ (1-\epsilon-\epsilon')^2 \leq \frac{|E|}{|M|}\cdot \# X_S \cdot \sup_{x\in E} \frac{A_S(x)}{\frac{\# X_S}{|M|}}, \] which recovers the classical bound in the homogeneous case, quantifies its deterioration in the presence of spectral inhomogeneity, and is shown to be sharp in general. In {\it dimension one}, we show that the homogeneity condition holds automatically, and we complement this rigidity by incorporating Fourier-ratio complexity bounds, yielding a quantitative relationship between spectral complexity and spatial support. In higher dimensions, we derive analogous results using pointwise Weyl laws and the eigenfunction restriction estimates on submanifolds.

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 gives a stability form of uncertainty principles on manifolds by replacing homogeneity with a quantitative spectral condition, but the higher-dimensional error control for singular potentials is not shown to be uniform enough.

read the letter

This paper extends uncertainty principles to the Laplace-Beltrami operator on compact Riemannian manifolds, including real singular potentials. It replaces the usual homogeneity assumption with a quantitative spectral condition and derives a stability inequality that recovers the classical bound when the condition holds and measures the deterioration otherwise. The inequality is claimed to be sharp in general. In dimension one the spectral condition holds automatically, and the authors add Fourier-ratio complexity bounds that tie spectral complexity to spatial support size. In higher dimensions they obtain the condition from pointwise Weyl laws and eigenfunction restriction estimates on submanifolds.

Referee Report

1 major / 2 minor

Summary. The paper establishes uncertainty principles for Laplace-Beltrami operators on compact Riemannian manifolds without boundary, including with real-valued singular potentials. It replaces the classical homogeneity assumption by a quantitative spectral condition and derives the stability inequality (1−ε−ε′)² ≤ |E|/|M| ⋅ #X_S ⋅ sup_{x∈E} A_S(x) / (#X_S / |M|), which recovers the homogeneous case, quantifies deterioration due to spectral inhomogeneity, and is claimed to be sharp. In dimension one the homogeneity condition holds automatically and is complemented by Fourier-ratio complexity bounds relating spectral complexity to spatial support; in higher dimensions the results follow from pointwise Weyl laws and eigenfunction restriction estimates on submanifolds.

Significance. If the central derivations hold, the work supplies a useful quantitative stability version of uncertainty principles that accounts for spectral inhomogeneity and singular potentials. The explicit form of the inequality, the sharpness statement, the automatic validity in 1D, and the use of standard asymptotic tools in higher dimensions constitute clear strengths. The result could inform further work in spectral geometry and harmonic analysis on manifolds.

major comments (1)

[higher dimensions paragraph of abstract] Abstract and higher-dimensional derivation: the quantitative spectral condition that controls ε and ε' is obtained from pointwise Weyl laws and eigenfunction restriction estimates. The manuscript must supply explicit remainder bounds for these estimates that are uniform with respect to real-valued singular potentials (and independent of the sets E and M). Without such uniform control the parameters ε, ε' may depend on the potential, which would undermine both the claimed stability of the inequality and the sharpness statement in general.

minor comments (2)

The symbols A_S(x), X_S, M and E appearing in the central inequality are not defined in the abstract; a short notation paragraph or reference to their definitions in the main text would improve readability.
[dimension one paragraph of abstract] In the dimension-one section, the Fourier-ratio complexity bounds should be stated with explicit dependence on the constants appearing in the estimates.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comment below.

read point-by-point responses

Referee: Abstract and higher-dimensional derivation: the quantitative spectral condition that controls ε and ε' is obtained from pointwise Weyl laws and eigenfunction restriction estimates. The manuscript must supply explicit remainder bounds for these estimates that are uniform with respect to real-valued singular potentials (and independent of the sets E and M). Without such uniform control the parameters ε, ε' may depend on the potential, which would undermine both the claimed stability of the inequality and the sharpness statement in general.

Authors: We agree that explicit remainder bounds uniform in the potential are required to fully substantiate the claims. The pointwise Weyl laws and eigenfunction restriction estimates we invoke are standard results from the spectral geometry literature (with references provided in the manuscript). These hold with remainders that depend only on the geometry of the manifold and the regularity class of the potential, and are independent of the specific sets E and M. For real-valued singular potentials in the class considered, the perturbation does not affect the leading asymptotics or the uniformity of the error terms. To address the referee's request explicitly, we will revise the higher-dimensional section by recalling the precise statements of these estimates, including their remainder bounds, and verifying the uniformity with respect to V. This will ensure that ε and ε' are controlled independently of the potential, preserving the stability and sharpness statements. revision: yes

Circularity Check

0 steps flagged

No circularity: central inequality derived from external spectral estimates under explicit quantitative condition

full rationale

The paper replaces the classical homogeneity assumption with a quantitative spectral condition (controlling ε, ε') and proves the stated inequality directly from it, recovering the homogeneous case when ε=ε'=0. In dimension one the condition is shown to hold automatically via an internal argument, while higher-dimensional cases invoke pointwise Weyl laws and eigenfunction restriction estimates as standard external tools rather than self-derived or fitted quantities. No step reduces the claimed result to a tautology, renamed input, or self-citation chain; the derivation remains self-contained against known spectral geometry benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The results rest on standard spectral properties of the Laplace-Beltrami operator and advanced analytic estimates rather than new free parameters or postulated entities.

axioms (2)

standard math The Laplace-Beltrami operator on a compact Riemannian manifold without boundary is self-adjoint and possesses a discrete spectrum of eigenfunctions
Fundamental for the eigenfunction expansion and spectral decomposition used throughout.
domain assumption Pointwise Weyl laws and eigenfunction restriction estimates hold on the manifold and its submanifolds
Invoked explicitly for the higher-dimensional results as stated in the abstract.

pith-pipeline@v0.9.0 · 5465 in / 1436 out tokens · 72223 ms · 2026-05-10T09:05:16.017990+00:00 · methodology

discussion (0)

Reference graph

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