Local limits of high energy eigenfunctions on integrable billiards

Alba Garcia-Ruiz; Alberto Enciso; Daniel Peralta-Salas

REVIEW 2 major objections 3 minor 77 references

Local limits of high energy eigenfunctions on integrable billiards

T0 review · 2 major / 3 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read High-energy eigenfunctions of generic integrable billiards fail to reproduce random monochromatic waves, while rational rectangles and integrable triangles can approximate any such wave.

desk verdict The polygon results are the real contribution; Theorem D's ellipse proof has a coordinate-change error that is not cosmetic, so the generic-ellipse claim is unsupported as written. read the letter →

arxiv 2502.01291 v1 pith:DXA3PCNC submitted 2025-02-03 math.SP

Alberto Enciso , Alba Garcia-Ruiz , Daniel Peralta-Salas This is my paper · ORCID

classification math.SP MSC 35P2081Q5058J50

keywords randomwaveconjectureintegrablebilliardsinverselocalizationPlanckscaleeigenfunctionlocallimitsnodalsetsMathieufunctionslatticepointequidistribution

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

Berry's random wave conjecture says that on a chaotic billiard, high-energy eigenfunctions, looked at in a window of size $\lambda^{-1/2}$ (the Planck scale), should resemble random monochromatic waves; one consequence, called inverse localization, is that any solution of the Helmholtz equation $\Delta\varphi+\varphi=0$ should be approximated by eigenfunctions at the Planck scale around many points. This paper tests that expectation on the opposite class of systems, integrable billiards, whose eigenfunctions are known explicitly. It proves a dichotomy: rational rectangles and the three integrable triangles (equilateral, isosceles right, and hemiequilateral) possess inverse localization, approximating any monochromatic wave around almost every point; irrational rectangles, balls, and almost all ellipses fail it so strongly that a dense open set of waves is never approximated, so their local limits cannot be random waves. The paper concludes that random-wave-like behavior at the Planck scale is exceptional among integrable systems rather than typical.

What carries the argument

Two mechanisms carry the argument. The positive mechanism splits a localized eigenfunction, built from the polygon's explicit trigonometric basis, into a translation-invariant kernel plus an error; equidistribution of lattice points on ellipsoids turns the kernel into an integral of $\cos(\xi\cdot(z-z_\gamma))$ over the unit circle, while an $L^2$ bound on the error (Lemma 3.3) plus a Lipschitz interpolation lemma (Lemma 3.4) shows the error is uniformly small on most base points. The negative mechanism is a jet-space obstruction: jets of order 3 of every localized eigenfunction are shown to lie in a fixed algebraic variety with empty interior inside the jet space of all monochromatic waves, and the open mapping theorem converts that empty interior into density and openness of the set of waves that are never approximated. For ellipses and disks the same obstruction is fed by Mathieu-function and Bessel-function eigenfunctions, whose restrictions are sent by a grid evaluation map into the zero set of an explicit polynomial system; a codimension count over the continuum of base points then shows the approximable waves have empty interior.

What would settle it

The decisive check is to compute the restriction $\tilde R^{\xi_0}_{\eta_0}(\tilde u^{z_0}_{mn})(\xi)=S^p_m((\xi_0+I_{\xi_0}(\xi))/\sqrt{\lambda_{mn}},q_{mn})E^p_n(\eta_0,q_{mn})$ directly for a non-circular ellipse and test whether the evaluation map still lands in a fixed proper algebraic variety. Concretely, expand $F(\xi,\eta)=(c\cosh\xi\cos\eta,\,c\sinh\xi\sin\eta)$ around $(\xi_0,\eta_0)$; since $F$ is not affine, the curve $z_0+F(I_{\xi_0}(\xi),\eta_0)/\sqrt{\lambda}$ does not have elliptic coordinates $(\xi_0+I_{\xi_0}(\xi)/\sqrt{\lambda},\,\eta_0)$, and the extra $O(|\xi|^2/\lambda)$ terms change the differential equation that the restriction satisfies. If those extra terms drive the jets out of the zero set of the polynomial $Q_M$, the density argument for Theorem D fails; if the terms vanish or stay inside the variety, the theorem stands.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is a dichotomy inside the class of integrable billiards. Using the complete trigonometric basis of eigenfunctions on rectangles and the three integrable triangles, the authors construct, for each monochromatic wave $\varphi$, a sequence of Dirichlet or Neumann eigenfunctions $u_{\lambda_n}$ and open sets $O_{\lambda_n,\varphi}$ of almost full measure such that $\|u_{\lambda_n}(z_0+\cdot/\sqrt{\lambda_n})-\varphi\|_{C^k(B)}<\varepsilon$ for every base point $z_0\in O_{\lambda_n,\varphi}$; if $\varphi$ has the reflection symmetries of the polygon, the approximation holds at every fixed point. The negative half shows the opposite for every irrational rectangle $Q_l$, for balls in any dimension, and for ellipses $E_b$ with $b$ in a full-measure set of parameters: the set $\mathcal{N}_{\Omega,BC}$ of monochromatic waves that stay at positive distance from every localized eigenfunction is dense and open in the space of waves with sharp decay, which rules out random waves as local limits of any eigenfunction sequence. The paper's stated conclusion is that in a generic integrable billiard, the local limits of Dirichlet and Neumann eigenfunctions do not match random waves, contrary to what Berry's conjecture might suggest.

Load-bearing premise

For the ellipse theorem, the proof assumes that localizing an eigenfunction at a point shifts the radial Mathieu factor exactly as if elliptic coordinates were affine; this identification ignores the curvature of the coordinate map, and the density argument collapses if the neglected curvature terms let the jets of localized eigenfunctions escape the algebraic variety used in the proof.

Editorial extensions

If this is right

Rational rectangles and the three integrable triangles have inverse localization: every monochromatic wave is reproduced at the Planck scale by eigenfunctions on almost all of the domain, and at every point once the wave obeys the polygon's reflection symmetries.
No eigenfunction sequence on an irrational rectangle, a ball, or a generic ellipse can have Berry's random waves as a local limit, because the approximable waves form a meager set.
The positive results extend to rational almost integrable polygons, with a fixed finite number of eigenfunction sequences jointly covering almost every point of the domain.
As applications, these polygons admit eigenfunctions with prescribed complicated nodal topology, arbitrary nesting of nodal domains, and arbitrarily many nondegenerate critical points inside a Planck-scale ball.

Reading between the lines

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

If the dichotomy is read as arithmetic rather than dynamical, the rational-versus-irrational split of rectangles suggests inverse localization is governed by resonance of side ratios; a testable prediction is that the exceptional ellipses in Theorem D are exactly those with rationally related Mathieu parameters.
Should the proofs hold up, this supplies the first rigorous integrable counterexamples to random-wave-like local limits, complementing the single known positive example of the flat torus.
The jet-space obstruction is a transferable template: for any separable system with explicit eigenfunctions (higher-dimensional ellipsoids, other coordinate separations), checking whether localized eigenfunction jets satisfy an algebraic relation of empty interior would decide inverse localization there.
A numerical experiment could sharpen the negative result: fix an irrational rectangle and a generic wave, and measure the distance from the wave to the localized eigenfunction jets as $\lambda\to\infty$; the theorem predicts a positive limiting distance, and its rate of stabilization indicates how the jet span fails to fill the Helmholtz jet space.

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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 polygon results are the real contribution; Theorem D's ellipse proof has a coordinate-change error that is not cosmetic, so the generic-ellipse claim is unsupported as written.

What is new and good: Theorem A's positive part gives inverse localization for rational rectangles and the three integrable triangles, with quantitative error bounds using lattice-point equidistribution. The negative part for irrational rectangles, via a jet-space argument, is a genuinely new strong-failure result. The nodal set applications in Section 7 are a nice payoff. As far as I can tell, these results are likely correct and form a meaningful step toward the integrable side of Berry's conjecture.

The soft spot is Section 5.3.2. The authors restrict a Planck-scale localized ellipse eigenfunction to an arc of a hyperbola L_{z0}, parametrized in Cartesian coordinates, and then assert that this restriction equals the Mathieu product evaluated at the shifted elliptic coordinate (ξ0 + I_{ξ0}(ξ)/√λ, η0). That would be true only if the Cartesian-to-elliptic map were affine. It is not. On the major axis, for example, the point z0 + (F(I, η0) - F(ξ0, η0))/√λ has elliptic coordinate arccosh(cosh ξ0 + cosh I/√λ) to first order, not ξ0 + I/√λ. The modified Mathieu ODE and the inclusion into the algebraic variety Q^{-1}({0}) both depend on this exact identity. Without it, the argument does not apply to the actual eigenfunctions. This is a load-bearing error, not a typo: it removes the full-measure generic-ellipse statement. The disk case, handled separately in Section 5.3.1 with the radial Bessel argument, seems fine.

The reader's weakest-assumption is exactly right, and the stress-test note holds up on reading the paper. This is a genuinely substantial paper with a real flaw in one of its four main theorems. I would not reject the whole paper, but I would send it back for major revision, expecting the authors either to fix Theorem D or to rescope the paper to the polygon results. The rectangle and triangle theorems are worth publishing on their own.

Recommendation: send to peer review. A serious editor should not desk-reject; the error is specific and the remaining content is strong enough that referee time is warranted, with the ellipse claim flagged as needing repair.

Referee Report

2 major / 3 minor

Summary. The paper studies the inverse localization property for high-energy Laplace eigenfunctions in integrable planar billiards: whether, around typical points and at Planck scale, eigenfunctions can approximate any given solution to the Helmholtz equation. The positive results show that rational rectangles and the three integrable triangles admit sequences of Dirichlet or Neumann eigenfunctions that approximate any monochromatic wave around a set of points of asymptotically full measure, and around all points under symmetry assumptions. The negative results claim strong failure of inverse localization for irrational rectangles, for generic ellipses and all balls, and for the square and equilateral triangle under small Robin boundary conditions. The paper also extends the positive results to rational almost integrable polygons and draws consequences for nodal sets and critical points. The central advertised conclusion is that, in a generic integrable billiard, local limits of eigenfunctions do not match random waves.

Significance. If the results were all correct, the paper would be a significant step in the rigorous understanding of Berry's random wave conjecture for integrable systems: it would provide the first broad family of non-chaotic billiards where the random-wave local limit provably fails, complementing the known positive example of the flat torus. The positive theorems for rational rectangles and integrable triangles are carefully structured, use explicit trigonometric eigenfunctions and lattice-point equidistribution, and appear sound; the negative theorem for irrational rectangles via jet spaces is also internally coherent and elegant. These parts are of independent value. However, the proof of the ellipse theorem (Theorem D) contains a concrete coordinate-change error that invalidates the argument as written, and because the paper's abstract and conclusion emphasize generic integrable billiards including ellipses, this is a load-bearing failure rather than a local blemish.

major comments (2)

[§5.3.2, Step II] The central identity in the ellipse proof is false. The manuscript defines the restriction operator R^{ξ0}_{η0} by composing a function with the elliptic-coordinate curve F(ξ,η)=(c cosh ξ cos η, c sinh ξ sin η) at fixed η0, and then asserts that for a Planck-scale localized eigenfunction one has \tilde R^{ξ0}_{η0}(\tilde u^{z0}_{mn})(ξ)=u_{mn}(ξ0+I_{ξ0}(ξ)/√λ, η0). But \tilde u^{z0}_{mn}(x)=u_{mn}(F(ξ0,η0)+x/√λ), so the left-hand side is u_{mn}(F(ξ0,η0)+F(I_{ξ0}(ξ),η0)/√λ). This equals u_{mn}(ξ0+I_{ξ0}(ξ)/√λ,η0) only if F(ξ0+ε,η0)=F(ξ0,η0)+F(ε,η0), which is not satisfied by the hyperbolic trigonometric parametrization. For example, on the major axis η0=0, the elliptic coordinate of F(ξ0,0)+F(I,0)/√λ is arccosh(cosh ξ0+cosh I/√λ), whose first-order shift is (cosh I)/(sinh ξ0)·λ^{-1/2}, not I·λ^{-1/2}. Consequently the sampled restriction is not the shifted modified-Mathieu product S_m^p((ξ0+I)/√λ,q)E_n^p(η0,q), the claimed ODE for f_{s,t,m,n} does not apply to the composed function, and the inclusion EM_M(\tilde R(\tilde u^{z0}_{mn}))∈Q_M^{-1}({0}) does not follow. Since the generic-ellipse conclusion of Theorem D rests on this inclusion, Theorem D is unsupported as written.
[§5.3.1 and §5.3.2, Step IV] The density step in both the ball and ellipse proofs uses the assertion that a union of codimension-M sets parametrized continuously by a d-dimensional parameter family has codimension at least M-d. This is not automatic: one needs a parametrized transversality or Sard-type statement, and the manuscript does not provide it. Without such a statement, the conclusion that the set of monochromatic waves outside N_{B,BC} (or N_{Eb,BC}) is dense does not follow from the fiberwise codimension estimates. This gap affects the negative results for balls and ellipses, not just the ellipse theorem.

minor comments (3)

[§1.3 and Abstract] The statement that inverse localization fails for 'most' integrable polygons and ellipses should be revised if Theorem D is removed or substantially weakened; the current abstract overstates what is proved without the ellipse result.
[§2.2.2] There are typographical issues such as 'mayor axes' for 'major axes' and inconsistent use of 'Ea,BC' versus 'Eb'; these should be corrected.
[§5.3.2, Step II] The definition of R^{ξ0}_{η0} uses the arc {F(ξ,η0)} centered near ξ0 but not at ξ0; this shift is confusing and is connected with the error in the main identity. The authors should clarify the relationship between the coordinate parameter and the physical localization center.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: proofs use explicit eigenfunction formulas, external equidistribution results, and independent prior theorems; self-citations are auxiliary tools, and the ellipse concern is a soundness gap, not a circular reduction.

full rationale

The paper's derivation chain does not reduce to its own inputs. In the positive inverse-localization theorem (Theorem 3.1), the coefficients are explicitly chosen from eigenfunction values at z0+zγ/√λ, and the trigonometric identities produce a translation-invariant kernel; the approximation of a prescribed Helmholtz solution φ follows from lattice-point equidistribution results of Iwaniec and Cilleruelo–Córdoba, not from assuming the conclusion. The negative results for irrational rectangles, Robin square/triangle, balls, and generic ellipses work by showing that all localized eigenfunctions lie in a small set (a finite-dimensional space AT or the algebraic variety Q_M^{-1}({0})), so typical monochromatic waves are at positive distance; no parameter is fitted to the target claim. The self-citations (e.g., EG RPS22, EPSTdL21, EPS16, GR23) provide prior theorems — ellipsoid quadrature, Bessel-function approximation, global Helmholtz approximation with decay, and support properties of the random-wave measure — that are parameter-free and do not include the results being proved here; they therefore constitute real independent support rather than load-bearing circularity. A separate, non-circular concern is that in Section 5.3.2, Step II, the restriction of a Planck-scaled ellipse eigenfunction to the hyperbolic arc L_z0 is identified with a shifted Mathieu product in elliptic coordinates, which would require the elliptic coordinate map to behave affinely under z0+z/√λ. That identity is not proved and is in fact false for cosh/sinh coordinates; this is a correctness risk in the ellipse argument, but it is not an equation that makes the theorem equivalent to its own input.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper introduces no free fitted parameters and no new physical entities. It relies on standard external theorems and on classifications or spectral facts from the cited literature. The only invented object is the technical property 'inverse localization fails strongly', which is a definition, not an entity.

assumptions (7)

domain assumption Classification of integrable planar polygons: the only integrable polygons are rectangles and the three triangles of Definition 1.1.
Used to enumerate the domains covered by Theorem A; cited from [Gut84] and not proven in this paper.
standard math There exist sequences λ_n along which integer lattice points on the ellipsoid Q_P(ξ)=1 are asymptotically equidistributed (Definition 3.2).
Invoked in Step 3 of Theorem 3.1 to replace lattice sums by surface integrals; cited from [Iwa97, CC94].
standard math Canzani's L2-to-L-infinity bound for families of Lipschitz functions (Lemma 3.4, [Can19, lemma 2.18]).
Used to turn mean-square bounds on the error term into uniform bounds off a small set.
standard math Surjectivity of the jet map J: MW_s to R^{dH} (Lemma 5.5 from [Dam97] plus global approximation [GF18]).
Enters the proof that the set of eigenfunction jets has empty interior, which drives the negative results.
domain assumption Simple spectrum for all ellipses E_b with b outside a countable exceptional set (Theorem 2.7, [HJ24]).
Used in Theorem 5.8 so that every eigenfunction is a single Mathieu product; the exceptional set is uncharacterized.
standard math The derandomization result that a full-density subsequence of square and torus eigenfunctions has a random wave local limit ([Ing21], [Bou14], [BW16]).
Used in Appendix A to connect Berry's property to inverse localization on Q1.
standard math Lame's Fundamental Theorem: eigenfunctions on the basic polygon patched by reflections give smooth eigenfunctions on almost integrable polygons ([McC08]).
Used in Theorem 6.3 to define eigenfunctions on the almost integrable polygon.

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Pith. "Pith review of Local limits of high energy eigenfunctions on integrable billiards." pith.science (2026). https://pith.science/paper/DXA3PCNC

@misc{pith2026250201291,
  author       = {Pith},
  title        = {Pith review of: Local limits of high energy eigenfunctions on integrable billiards},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DXA3PCNC}},
  note         = {Machine review of arXiv:2502.01291}
}

abstract

Berry's random wave conjecture posits that high energy eigenfunctions of chaotic systems resemble random monochromatic waves at the Planck scale. One important consequence is that, at the Planck scale around "many" points in the manifold, any solution to the Helmholtz equation $\Delta\varphi+\varphi =0$ can be approximated by high energy eigenfunctions. This property, sometimes called inverse localization, has useful applications to the study of the nodal sets of eigenfunctions. Alas, the only manifold for which the local limits of a sequence of high energy eigenfunctions are rigorously known to be given by random waves is the flat torus $(\mathbf{R}/\mathbf{Z})^2$, which is certainly not chaotic. Our objective in this paper is to study the validity of this "inverse localization" property in the class of integrable billiards, exploiting the fact that integrable polygonal billiards are classified and that Birkhoff conjectured that ellipses are the only smooth integrable billiards. Our main results show that, while there are infinitely many integrable polygons exhibiting good inverse localization properties, for "most" integrable polygons and ellipses, this property fails dramatically. We thus conclude that, in a generic integrable billiard, the local limits of Dirichlet and Neumann eigenfunctions do not match random waves, as one might expect in view of Berry's conjecture. Extensions to higher dimensions and nearly integrable polygons are discussed too.

Figures

Figures reproduced from arXiv: 2502.01291 by the authors.

**Figure 1.** The integrable polygonal billiards. with rational angles and p sides, it is known (see e.g. [Gut86]) that the genus of S(P) satisfies (1.6) g(S(P)) = 1 + N 2 Xp i=1 mi − 1 ni , where all the angles of P are denoted by πmi/ni , and where N is the least common multiple of the integers ni . The integrable polygonal billiards [Gut84] are those whose associated surfaces are flat tori. Their billiard flow has no singulari… view at source ↗

**Figure 2.** Examples of some almost integrable billiards drawn in lattices. two degrees of freedom, has an additional integral of motion that “almost commutes” with the Hamiltonian. Thus, almost integrable billiards are not integrable, but they are still “very far” from being chaotic. 1.2.2. Main results for integrable polygonal billiards. We shall next state our main theorem for integrable billiards. This theorem consists of … view at source ↗

**Figure 1.3.** Circular and elliptic domains is (locally) integrable, then the domain is elliptic. More precisely, the conjecture says as follows: Conjecture (Birkhoff’s conjecture). Consider a domain whose boundary Ω ⊂ R 2 is a C 2 -smooth convex bounded curve in the plane. If a neighborhood of the boundary ∂Ω is foliated by caustics, ∂Ω is an ellipse. Although the conjecture remains open, much work has been done recently, as sum… view at source ↗

Figures from the paper (8 more)

**Figure 2.1.** Figure 2.1: Triangular Coordinate System [PITH_FULL_IMAGE:figures/full_fig_p018_2_1.png]

**Figure 2.2.** Figure 2.2: Change of coordinates in the equilateral triangle given by z1 = z1 − 1 2 and z2 = √ 3 2 − z2 T α,β s (z1, z2) = − sin 2 √ 3π(α + β) 3 z2 ! cos 2π(α − β) 3 z1 + sin 2 √ 3πα 3 z2 ! cos 2π(2β + α) 3 z1 + sin 2 √ 3πβ 3 z2 ! cos 2π(2α + β) 3 z1 , T α,β a (z1, …

**Figure 2.3.** Figure 2.3: Elliptical coordinates system eigenfuntions that we have associated to the same eigenvalue differ on the value of m. We see first the Dirichlet case. This result was conjectured by Bourget and it follows (see for example [IKS84, p.464]) from a number theory result ob…

**Figure 4.1.** Figure 4.1: Some Dirichlet eigenfunctions on Q1 associated to the first eigenvalues. Here we can observe the symmetry condition discussed in Remark 4.1. Original images created with the software Mathematica. satisfies that either u(z1, z2) = X N∈Z2,|N|= √ λ c˜N sin N1π z1 √ λ …

**Figure 6.1.** Figure 6.1: Example of a rational almost integrable polygonal billiard. We can also see the affine lines with respect to which we reflex to create this polygon. where A1 = I is the identity, A2 is the affine reflection with respect to R − √ 3 (1,0) (i.e., the affine line that go…

**Figure 6.2.** Figure 6.2: Following Remark 6.4, we can also have the inverse localization property in some polytope (i.e. polygonal like domains in R d ) like the one of the figure. Since O˜j n ⊂ P, we can define a new set given by On := z ∈ O˜ n such that B z, R √ λn ⊂ P , where R is …

**Figure 7.1.** Figure 7.1: Example of a tree and a transversal cut of the corresponding nesting of nodal domains. that we added when defining t ′ ensures that there is a bounded convex domain in BR inside which we only have those nodal sets that allow us to construct t. We conclude the proof …

**Figure 6.** Figure 6: ). This does not happen in the case of [PITH_FULL_IMAGE:figures/full_fig_p091_6.png]

Discussion (0). Continue with ORCID to comment.

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