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arxiv: 1907.03669 · v1 · pith:GKG6LT5Enew · submitted 2019-07-08 · 🧮 math.SP · math.CA· math.NT

The Weyl formula for planar annuli

Jingwei Guo , Wolfgang M\"uller , Weiwei Wang , Zuoqin Wang This is my paper

Pith reviewed 2026-05-25 00:33 UTC · model grok-4.3

classification 🧮 math.SP math.CAmath.NT
keywords Weyl formulaeigenvalue counting functionplanar annulusBessel functionslattice point countingvan der Corput boundsGauss circle problem
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The pith

The Dirichlet eigenvalue counting function for a planar annulus satisfies a two-term Weyl formula with remainder O(μ^{2/3}).

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

The paper establishes that the eigenvalue counting function N(μ) for the Dirichlet Laplacian on a planar annulus admits the two-term expansion (area/4π)μ minus (perimeter/4√π)√μ plus an error of size O(μ^{2/3}). The proof proceeds by deriving accurate approximations for the zeros of the cross-product of Bessel functions, which converts the spectral counting problem into a lattice-point counting problem whose boundary has unbounded curvature and whose lattice points are subject to varying translations. A reader would care because the resulting remainder controls the precision with which the spectrum of an annular domain can be predicted from its geometry alone. The same reduction also yields an improved remainder O(μ^{131/208}(log μ)^{18627/8320}) when a tangent of rational slope is present, and recovers the Huxley-type bound for the disk as a by-product.

Core claim

We obtain a two-term Weyl formula for the eigenvalue counting function for the planar annulus with a remainder of size O(μ^{2/3}). If we additionally assume that certain tangent has rational slope, we obtain an improved remainder estimate of the same strength as Huxley's bound in the Gauss circle problem, namely O(μ^{131/208}(log μ)^{18627/8320}). As a by-product of our lattice point counting results, we readily obtain this Huxley-type remainder estimate in the two-term Weyl formula for planar disks.

What carries the argument

Approximations of the zeros of the cross-product of Bessel functions that reduce the eigenvalue counting problem to a lattice point counting problem for a special domain with unbounded boundary curvature and varying translations of lattice points.

If this is right

  • The two-term Weyl formula with error O(μ^{2/3}) holds for the Dirichlet spectrum of any planar annulus.
  • Under the additional assumption that a certain tangent line has rational slope, the remainder improves to O(μ^{131/208}(log μ)^{18627/8320}).
  • The same Huxley-type remainder applies directly to the two-term Weyl formula for the planar disk.
  • The lattice-point problem that arises after the Bessel-zero reduction can be transformed into a standard form to which van der Corput bounds apply.

Where Pith is reading between the lines

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

  • The appearance of varying translations and unbounded curvature in the associated lattice-point problem may force the use of van der Corput estimates rather than more modern exponential-sum methods.
  • The reduction technique suggests that rotationally symmetric domains in higher dimensions could likewise be turned into counting problems whose error terms are governed by curvature properties of auxiliary curves.
  • Direct high-precision computation of Bessel cross-product zeros for large arguments could serve as an independent check on whether the derived error term is sharp.

Load-bearing premise

The approximations obtained for the zeros of the cross-product of Bessel functions are accurate enough that the reduction of the eigenvalue counting problem to the lattice-point counting problem does not introduce errors larger than the target remainder O(μ^{2/3}).

What would settle it

A numerical computation of the exact eigenvalues of a concrete annulus for successively larger values of μ that shows the difference between N(μ) and the two-term main terms exceeding every multiple of μ^{2/3} would falsify the claimed remainder.

Figures

Figures reproduced from arXiv: 1907.03669 by Jingwei Guo, Weiwei Wang, Wolfgang M\"uller, Zuoqin Wang.

Figure 3
Figure 3. Figure 3: ). These points are relatively easier to handle. T [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: The graph of G with R = 2 and r = 1. Proof. This result follows from an application of the method of stationary phase to the Bessel functions. More precisely, we first apply to all four factors in (2.1) the asymptotics (A.1) and (A.2) of Bessel functions and then use the angle difference formula for the sine. Lemma 2.2. There exists a constant c ∈ (0, 1) such that for any ε > 0 and all sufficiently large… view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: The graph of hn with n = 30, R = 2 and r = 1. such that hn maps (ak, bk) to (k−3/8, k+ 1/8) bijectively. It is obvious that these intervals (ak, bk)’s are disjointly located one by one as k increases. We claim that if n is sufficiently large then for each 1 ≤ k ≤ s (2.22) fn(ak)fn(bk) < 0. If this is true, the intermediate value theorem ensures the existence of at least one zero of fn in each (ak, bk). R… view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: The symmetric domain D. Then one can transfer the spectrum counting problem to a lattice count￾ing problem via the following result. In its proof we essentially follow the treatment for “the boundary parts” in [6, Theorem 3.1]. Proposition 3.1. There exists a constant C > 0 such that (3.1) |ND (µ) − ND(µ)| ≤ ND Ä µ + Cµ−0.4 ä −ND Ä µ − Cµ−0.4 ä +O Ä µ 0.6 ä . Proof. To study ND(µ) we would like to use th… view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: A decomposition of D in the first quadrant. where L12 = µr ρ(µG(r) + c) describes the contribution of the line segment separating D1 from D2. While (4.3) is true in general, we prove (4.4) in the irrational case only with the weaker error term O(µ 2/3 ). In µD1 we count lattice points along lines parallel to the x-axis. Denote by H : [0, G(r)] → [r, R] the inverse function of G restricted to [r, R]. Sinc… view at source ↗
read the original abstract

We study the zeros of cross-product of Bessel functions and obtain their approximations, based on which we reduce the eigenvalue counting problem for the Dirichlet Laplacian associated with a planar annulus to a lattice point counting problem associated with a special domain in $\mathbb{R}^2$. Unlike other lattice point problems, the one arisen naturally here has interesting features that lattice points under consideration are translated by various amount and the curvature of the boundary is unbounded. By transforming this problem into a relatively standard form and using classical van der Corput's bounds, we obtain a two-term Weyl formula for the eigenvalue counting function for the planar annulus with a remainder of size $O(\mu^{2/3})$. If we additionally assume that certain tangent has rational slope, we obtain an improved remainder estimate of the same strength as Huxley's bound in the Gauss circle problem, namely $O(\mu^{131/208}(\log \mu)^{18627/8320})$. As a by-product of our lattice point counting results, we readily obtain this Huxley-type remainder estimate in the two-term Weyl formula for planar disks.

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

2 major / 2 minor

Summary. The manuscript approximates the zeros of the cross-product of Bessel functions J_ν(μa)Y_ν(μb)−J_ν(μb)Y_ν(μa), reduces the eigenvalue counting function N(μ) for the Dirichlet Laplacian on a planar annulus to a lattice-point counting problem whose boundary has unbounded curvature and whose points are subject to position-dependent translations, transforms the latter problem into standard form, and applies van der Corput bounds to obtain the two-term asymptotic N(μ)= (area)μ²/4π + (perimeter)μ/4 + O(μ^{2/3}). Under the additional hypothesis that a certain tangent has rational slope, the remainder improves to O(μ^{131/208}(log μ)^{18627/8320}), matching the strength of Huxley's bound; the same improved remainder is obtained for planar disks as a byproduct.

Significance. If the error control in the Bessel-zero approximations is rigorous, the result supplies an explicit remainder of size O(μ^{2/3}) for annular domains (improving on the generic O(μ) remainder) and, under a Diophantine condition, reaches the current best exponent known for the Gauss circle problem. The technical handling of a lattice-point problem with unbounded curvature and variable translations, together with the clean reduction to classical van der Corput estimates, is a genuine contribution; the byproduct for disks is also useful.

major comments (2)
  1. [reduction step (following the statement of the main theorem)] The central claim that the Bessel-zero approximation errors remain absorbed inside the target remainder O(μ^{2/3}) is load-bearing. An explicit lemma is required that bounds the total discrepancy between the true eigenvalue counting function and the lattice-point count arising from the zero approximations (after scaling by the local spacing), showing that this discrepancy is o(μ^{2/3}).
  2. [transformation to standard form] After the transformation that produces a domain with unbounded curvature, the manuscript must verify that the error introduced by the change of variables does not exceed the O(μ^{2/3}) allowance before the van der Corput bounds are applied; otherwise the final remainder estimate is not justified.
minor comments (2)
  1. [Introduction] The notation for the cross-product of Bessel functions should be introduced once and used consistently; the current alternation between J_νY_ν−J_νY_ν and the explicit four-term expression is slightly confusing on first reading.
  2. A short table or diagram illustrating the geometry of the translated lattice-point region (including the location of the tangent of rational slope) would help the reader follow the reduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for recognizing the contribution of the lattice-point analysis with unbounded curvature and variable translations. We address each major comment below and will revise the manuscript accordingly to make the error controls fully explicit.

read point-by-point responses

  1. Referee: [reduction step (following the statement of the main theorem)] The central claim that the Bessel-zero approximation errors remain absorbed inside the target remainder O(μ^{2/3}) is load-bearing. An explicit lemma is required that bounds the total discrepancy between the true eigenvalue counting function and the lattice-point count arising from the zero approximations (after scaling by the local spacing), showing that this discrepancy is o(μ^{2/3}).

    Authors: We agree that an explicit lemma is required for full rigor. In the revised manuscript we will insert a new Lemma (placed immediately after the statement of the main theorem) that aggregates the individual Bessel-zero approximation errors (from the asymptotic expansions in Section 3) and shows, after scaling by the local eigenvalue spacing ~ μ^{-1/2}, that the total discrepancy is O(μ^{1/2 + ε}) for any ε>0, which is o(μ^{2/3}). The proof will combine the uniform error bounds already obtained for the cross-product zeros with a standard summation argument over the relevant range of orders ν. revision: yes

  2. Referee: [transformation to standard form] After the transformation that produces a domain with unbounded curvature, the manuscript must verify that the error introduced by the change of variables does not exceed the O(μ^{2/3}) allowance before the van der Corput bounds are applied; otherwise the final remainder estimate is not justified.

    Authors: We accept the point. The change-of-variables map is introduced in Section 5 to bring the problem into a form where the boundary curvature satisfies the hypotheses of van der Corput’s lemma. In the revision we will add a short paragraph (immediately preceding the application of van der Corput) that computes the Jacobian distortion and the perturbation of the boundary; we show that the induced error in the lattice-point count is O(μ^{1/2} log μ), which is absorbed into the target O(μ^{2/3}) remainder. This estimate uses only the C^2 regularity of the map and the already-established bounds on the original domain. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on external classical estimates

full rationale

The paper approximates zeros of the Bessel cross-product to map the annulus eigenvalue problem to a translated lattice-point count with unbounded curvature, then invokes van der Corput bounds and Huxley-type results from the literature to obtain the O(μ^{2/3}) remainder (and the conditional improvement). These bounds are independent external results, not fitted parameters or self-referential expressions. The by-product for disks follows the same external estimates. No self-definitional steps, no fitted inputs renamed as predictions, and no load-bearing self-citations appear in the derivation chain. The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard results from analytic number theory (van der Corput bounds) and the theory of Bessel functions; no free parameters, ad-hoc axioms, or invented entities are indicated in the abstract.

axioms (1)
  • standard math van der Corput bounds on the discrepancy of lattice points in domains with smooth boundary
    Invoked after the problem is transformed into standard form to obtain the O(μ^{2/3}) remainder.

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Reference graph

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