pith. sign in

arxiv: 2606.03996 · v1 · pith:LK54UGSGnew · submitted 2026-05-07 · 🧮 math.AP · math.SP

Corrigendum to "Ramification of multiple eigenvalues for the Dirichlet-Laplacian in perforated domains" published in Journal of Functional Analysis 283 (2022) 109718

Laura Abatangelo , Corentin L\'ena , Paolo Musolino This is my paper

Pith reviewed 2026-06-30 23:18 UTC · model grok-4.3

classification 🧮 math.AP math.SP
keywords Dirichlet Laplacianperforated domainseigenvalue asymptoticscorrigendummultiple eigenvaluesramification
0
0 comments X

The pith

The original proof of asymptotic coefficients for Dirichlet-Laplacian eigenvalues in perforated domains is incorrect and requires correction.

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

This corrigendum fixes an error in the proof of Theorem 1.17 from the 2022 reference on ramification of multiple eigenvalues. The theorem addressed the asymptotic behavior of eigenvalues when a small hole is removed from the domain, but the original derivation does not produce the correct coefficients in general. The corrigendum supplies the revised computation while keeping the same domain and hole assumptions. A reader would care because these coefficients determine the precise perturbation of eigenvalues under small perforations.

Core claim

The proof of Theorem 1.17 does not in general give the correct coefficients in the asymptotic behavior of eigenvalues for the Dirichlet Laplacian when a small hole is removed from the domain, and the corrigendum supplies the correct computation under the same assumptions.

What carries the argument

The asymptotic expansion of eigenvalues for the Dirichlet-Laplacian in a domain with a small perforation, specifically the coefficients multiplying the leading correction terms.

If this is right

  • The coefficients stated in the original Theorem 1.17 are not valid in general.
  • Any subsequent analysis relying on those coefficients for eigenvalue ramification must use the revised values instead.
  • The precise rate at which multiple eigenvalues split under perforation changes once the correct coefficients are inserted.

Where Pith is reading between the lines

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

  • Citations or extensions of the 2022 paper that used the original coefficients may need re-evaluation.
  • Numerical verification on concrete domains with small holes can test whether the corrected coefficients match computed eigenvalue shifts.
  • The same derivation flaw could appear in related asymptotic analyses of the Laplacian under other small perturbations.

Load-bearing premise

The steps used to derive the coefficients in the original proof contain no algebraic or conceptual errors.

What would settle it

An explicit calculation of the first few coefficients in the eigenvalue expansion for the unit disk minus a small centered disk, compared against both the original stated values and the corrected ones.

read the original abstract

We fix the proof of Theorem 1.17 in the quoted reference, which does not in general gives the correct coefficients in the asymptotic behavior of eigenvalues for the Dirichlet Laplacian when a small hole is removed from the domain.

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 / 1 minor

Summary. This short corrigendum identifies an error in the proof of Theorem 1.17 from the authors' 2022 Journal of Functional Analysis paper on ramification of multiple eigenvalues for the Dirichlet-Laplacian in perforated domains. It asserts that the original derivation does not in general produce the correct coefficients in the asymptotic expansion of eigenvalues when a small hole is removed, and states that a corrected computation is supplied under the same domain and hole assumptions.

Significance. Corrections to published asymptotic expansions in spectral theory are valuable for maintaining accuracy in the literature, particularly when the coefficients affect quantitative predictions about eigenvalue perturbations. If the revised coefficients are correctly derived, the corrigendum restores reliability to results that may be cited in subsequent work on perforated domains and eigenvalue asymptotics.

minor comments (1)
  1. [Abstract] Abstract: the clause 'which does not in general gives the correct coefficients' contains a subject-verb agreement error and should read 'give'.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the corrigendum and for recommending acceptance. The report accurately captures the purpose of the note.

Circularity Check

0 steps flagged

No significant circularity

full rationale

This corrigendum directly identifies a flaw in the proof of Theorem 1.17 from the authors' prior paper and supplies a corrected computation of the asymptotic coefficients for the Dirichlet-Laplacian eigenvalues with a small hole. The derivation chain consists of mathematical corrections under identical domain assumptions, with no reduction of predictions to fitted inputs, no self-definitional loops, and no load-bearing reliance on unverified self-citations. The central claim is an explicit fix rather than a renaming or ansatz smuggling, rendering the result self-contained against external mathematical verification.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so a complete ledger cannot be extracted; the work relies on standard background from the original paper.

axioms (1)
  • standard math Standard properties of the Dirichlet-Laplacian and small-hole asymptotic analysis in smooth domains
    These background facts are presupposed by the eigenvalue perturbation setting described in the abstract.

pith-pipeline@v0.9.1-grok · 5572 in / 1045 out tokens · 41814 ms · 2026-06-30T23:18:49.764434+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

9 extracted references · 6 canonical work pages

  1. [1]

    Abatangelo, Laura and L\'ena, Corentin and Musolino, Paolo , TITLE =. J. Funct. Anal. , FJOURNAL =. 2022 , NUMBER =. doi:10.1016/j.jfa.2022.109718 , URL =

    work page doi:10.1016/j.jfa.2022.109718 2022

  2. [2]

    Nonlinear Anal

    Abatangelo, Laura and L\'ena, Corentin and Musolino, Paolo , TITLE =. Nonlinear Anal. , FJOURNAL =. 2024 , PAGES =. doi:10.1016/j.na.2023.113391 , URL =

    work page doi:10.1016/j.na.2023.113391 2024

  3. [3]

    Abatangelo, Laura and Felli, Veronica and Hillairet, Luc and L\'ena, Corentin , TITLE =. J. Spectr. Theory , FJOURNAL =. 2019 , NUMBER =. doi:10.4171/JST/251 , URL =

    work page doi:10.4171/jst/251 2019

  4. [4]

    and Colin de Verdi\`ere, Y

    Colbois, B. and Colin de Verdi\`ere, Y. , TITLE =. Comment. Math. Helv. , FJOURNAL =. 1988 , NUMBER =. doi:10.1007/BF02566762 , URL =

    work page doi:10.1007/bf02566762 1988

  5. [5]

    Colin de Verdi\`ere, Yves , TITLE =. Comment. Math. Helv. , FJOURNAL =. 1986 , NUMBER =. doi:10.1007/BF02621914 , URL =

    work page doi:10.1007/bf02621914 1986

  6. [6]

    Courtois, Gilles , TITLE =. J. Funct. Anal. , FJOURNAL =. 1995 , NUMBER =. doi:10.1006/jfan.1995.1142 , URL =

    work page doi:10.1006/jfan.1995.1142 1995

  7. [7]

    2002 , publisher =

    Schatzman, Michelle , title =. 2002 , publisher =

    2002

  8. [8]

    , Date-Added =

    Folland, Gerald B. , Date-Added =. Introduction to partial differential equations , Url =. 1995 , Bdsk-Url-1 =

    1995

  9. [9]

    and Musolino, P

    Dalla Riva, M. and Musolino, P. and Rogosin, S. V. , Date-Added =. Series expansions for the solution of the. Asymptot. Anal. , Mrclass =. 2015 , Bdsk-Url-1 =

    2015