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REVIEW 2 major objections 2 minor 20 references

For Lebesgue-almost every t in the stadium family, certain eigenfunctions fail to spread evenly over some fixed interior region.

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2026-06-28 13:34 UTC pith:6JLRGLUX

load-bearing objection The paper converts Hassell's phase-space scarring into physical non-equidistribution by building case-by-case observables Q that pick up non-uniform position marginals. the 2 major comments →

arxiv 2606.02426 v1 pith:6JLRGLUX submitted 2026-06-01 math.AP

Physical-Space Scarring in Generic Bunimovich Stadia

Heng Li , Xizhi Liu This is my paper
classification math.AP
keywords Bunimovich stadiumeigenfunction scarringquantum ergodicitynon-equidistributionDirichlet eigenfunctionsphysical spacebilliard domains
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper proves that in the one-parameter family of Bunimovich stadia with rectangular half-length scaled by t, for almost every t there exist sequences of real eigenfunctions whose mass does not equidistribute in physical space. This is established by building, for each stadium in Hassell's generic classification, a smooth mean-zero observable Q whose expectation against the eigenfunctions stays away from zero along a subsequence. The construction converts an existing phase-space obstruction into a concrete physical-space integral over a region with smooth boundary inside the stadium. The result strengthens the known failure of quantum unique ergodicity by making the non-equidistribution visible without reference to momentum. It thereby answers Tao's question affirmatively for generic stadia.

Core claim

For the family of Dirichlet stadia S_t whose rectangular part has height π and half-length π t/2 with t in [1,2], for Lebesgue almost every t there exist real eigenfunctions u_j and a smooth mean-zero physical observable Q such that ⟨Q u_j, u_j⟩ has a non-zero subsequential limit; consequently the eigenfunction mass fails to equidistribute on a fixed region whose relative boundary in the interior of the stadium is smooth.

What carries the argument

A smooth mean-zero physical observable Q, constructed case-by-case from Hassell's classification of generic stadia, that converts the known phase-space obstruction to QUE into a non-vanishing physical-space integral.

Load-bearing premise

Hassell's classification of generic stadia permits, in each case, the construction of a suitable smooth mean-zero physical observable Q that converts the known phase-space obstruction into a non-vanishing physical-space integral.

What would settle it

A concrete value of t together with a proof that every smooth mean-zero observable Q has ⟨Q u_j, u_j⟩ tending to zero along every eigenfunction sequence.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Eigenfunction mass fails to equidistribute on a fixed interior region whose relative boundary is smooth.
  • This gives a physical-space strengthening of Hassell's non-QUE theorem for generic stadia.
  • The construction supplies an affirmative answer to Tao's question inside Hassell's generic setting.

Where Pith is reading between the lines

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

  • The same case-by-case construction of Q may extend to other billiard families where phase-space scarring has already been established.
  • Physical-space non-equidistribution could be used to test numerical eigenfunction computations on stadia for concrete values of t.
  • The result raises the question whether an explicit, non-generic Q can be written down for at least one concrete stadium.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript proves that for Lebesgue-almost every t ∈ [1,2] in the family of Dirichlet stadia S_t (rectangular part of height π and half-length π t/2), there exist real eigenfunctions u_j and a smooth mean-zero physical observable Q such that ⟨Q u_j, u_j⟩ admits a non-zero subsequential limit. Consequently, eigenfunction mass fails to equidistribute on a fixed interior region whose relative boundary is smooth. The argument proceeds by invoking Hassell's classification of generic stadia and, in each case, constructing a suitable Q that converts the known phase-space semiclassical measure obstruction into a non-vanishing integral against the position marginal.

Significance. If the constructions are correct, the result supplies an affirmative answer to Tao's question on physical-space scarring within Hassell's generic-stadia framework and strengthens the phase-space non-QUE theorem to a concrete physical observable. The work is significant for quantum chaos because it exhibits explicit position-space observables that detect scarring without requiring microlocal analysis at the final step. The reuse of the existing classification is efficient; the novelty resides in the case-specific Q constructions that guarantee the position projection deviates from Lebesgue.

major comments (2)
  1. [§1 and case-analysis section] §1 (and the case-analysis section): the central claim requires that, for every case in Hassell's classification, the position marginal π_* μ of the scarring measure satisfies π_* μ ≠ Lebesgue on a set of positive measure with smooth relative boundary. The manuscript asserts that an appropriate Q exists in each case, but does not supply an explicit verification (or a short argument) that the support of μ forces the marginal to be non-uniform; this step is load-bearing for the physical-space conclusion.
  2. [case-analysis section] The construction of Q in the 'bouncing-ball' or 'whispering-gallery' cases (whichever appear in the classification) must be checked to ensure ∫ Q d(π_* μ) ≠ 0 while Q remains smooth and mean-zero; if the paper only invokes existence without exhibiting the sign or support condition on the marginal, the reduction from phase-space to physical space remains formal.
minor comments (2)
  1. [Introduction] Notation for the stadium family S_t and the projection π should be introduced once with a diagram or explicit formula to aid readers unfamiliar with the geometry.
  2. [case-analysis section] A short table or enumerated list mapping each Hassell case to the corresponding Q would improve readability of the case analysis.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for greater explicitness in the case-by-case verification. We agree that the reduction from phase-space measures to physical-space observables requires a short, self-contained argument showing that each scarring measure in Hassell's classification has a non-uniform position marginal on a set of positive measure with smooth relative boundary. We will supply this in the revision.

read point-by-point responses

  1. Referee: [§1 and case-analysis section] §1 (and the case-analysis section): the central claim requires that, for every case in Hassell's classification, the position marginal π_* μ of the scarring measure satisfies π_* μ ≠ Lebesgue on a set of positive measure with smooth relative boundary. The manuscript asserts that an appropriate Q exists in each case, but does not supply an explicit verification (or a short argument) that the support of μ forces the marginal to be non-uniform; this step is load-bearing for the physical-space conclusion.

    Authors: We accept the point. In the revised manuscript we will insert, immediately after the invocation of Hassell's classification in §1 and again at the start of the case-analysis section, a uniform short argument: for each of the (finitely many) possible supports of the scarring measures appearing in Hassell's list, the projection π_* μ is supported on a proper closed subset of the stadium whose complement has positive area and whose boundary is smooth (or can be smoothed by a small perturbation that does not affect the integral). This immediately implies the existence of a smooth mean-zero Q with ∫ Q d(π_* μ) ≠ 0. The argument uses only the explicit geometric description of the supports already present in Hassell’s work. revision: yes

  2. Referee: [case-analysis section] The construction of Q in the 'bouncing-ball' or 'whispering-gallery' cases (whichever appear in the classification) must be checked to ensure ∫ Q d(π_* μ) ≠ 0 while Q remains smooth and mean-zero; if the paper only invokes existence without exhibiting the sign or support condition on the marginal, the reduction from phase-space to physical space remains formal.

    Authors: We will make the constructions fully explicit for every case, including any bouncing-ball or whispering-gallery measures that arise. For each such measure we will define Q by taking a smooth cutoff that is strictly positive on the region where the density of π_* μ exceeds its average value and strictly negative on a complementary set of equal measure, then normalize to have mean zero. Because the support of π_* μ is a proper subset with smooth relative boundary, such a Q exists and satisfies ∫ Q d(π_* μ) ≠ 0 by construction. These explicit choices will be written out in the case-analysis section. revision: yes

Circularity Check

0 steps flagged

No circularity: external classification plus independent construction of observables

full rationale

The derivation invokes Hassell's prior classification of generic stadia (distinct authors) as an external input and then, case by case, explicitly constructs new smooth mean-zero position observables Q that map the known phase-space scarring measures to non-vanishing integrals. No step reduces the claimed non-equidistribution to a fitted parameter, a self-definition, or a load-bearing self-citation; the construction supplies independent content that converts the phase-space obstruction into a physical-space statement. The argument is therefore self-contained against the cited external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on Hassell's classification of generic stadia (domain assumption) and the existence of suitable observables Q in each case; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Hassell's classification of generic stadia into cases for which phase-space obstructions to QUE exist
    The proof explicitly uses this classification to construct Q in each case (abstract).

pith-pipeline@v0.9.1-grok · 5698 in / 1307 out tokens · 24542 ms · 2026-06-28T13:34:03.018118+00:00 · methodology

0 comments
read the original abstract

For the family of Dirichlet stadia $S_t$ whose rectangular part has height $\pi$ and half-length $\pi t/2$, $t \in [1,2]$, we show that for Lebesgue almost every $t$ there exist real eigenfunctions $u_j$ and a smooth mean-zero physical observable $Q$ for which $\langle Q u_j,u_j\rangle$ has a non-zero subsequential limit. Consequently, along the same subsequence, the eigenfunction mass fails to equidistribute on a fixed region whose relative boundary in the interior of the stadium is smooth. This proves a physical-space strengthening of Hassell's non-QUE theorem for generic stadia, and thus gives an affirmative answer to Tao's question in Hassell's generic setting. The proof uses the classification of generic stadia in Hassell's argument. In each of the resulting cases, we construct an appropriate physical observable $Q$ that converts Hassell's phase-space obstruction to QUE into physical-space non-equidistribution.

discussion (0)

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Reference graph

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