pith. sign in

arxiv: 2606.30120 · v1 · pith:5VK2BBSTnew · submitted 2026-06-29 · 🧮 math.SP · math.AP

Inequalities between Dirichlet and Neumann eigenvalues in large dimensions

N. Filonov This is my paper

Pith reviewed 2026-06-30 03:58 UTC · model grok-4.3

classification 🧮 math.SP math.AP
keywords Dirichlet eigenvaluesNeumann eigenvaluesLaplace operatoreigenvalue inequalitieshigh dimensionsconvex domainsspectral shift
0
0 comments X

The pith

Dirichlet eigenvalues bound Neumann eigenvalues only after an index shift of size at least C(e/2)^d in dimension d.

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

The paper compares the ordered spectra of the Dirichlet and Neumann Laplacians on a bounded domain Ω in R^d. It introduces Ψ(d,k,Ω) as the smallest integer shift making the (k+Ψ)th Neumann eigenvalue no larger than the kth Dirichlet eigenvalue. The authors prove that this shift grows at least exponentially in d, specifically Ψ(d,1,Ω) ≥ C(e/2)^d for a positive constant C independent of Ω, and that the same lower bound holds for every k when Ω is convex. This quantifies a dimension-driven separation: in large d the Neumann spectrum is delayed by an exponentially growing number of modes relative to the Dirichlet spectrum.

Core claim

Let Ω be a bounded domain in R^d. Denote by λ_k (resp. μ_k) the eigenvalues of the Laplace operator in Ω with Dirichlet (resp. Neumann) boundary conditions. Denote by Ψ = Ψ(d,k,Ω) the shift of indices in the inequality μ_{k+Ψ} ≤ λ_k. We prove that a) Ψ(d,1,Ω) ≥ C(e/2)^d for all domains Ω; and b) Ψ(d,k,Ω) ≥ C(e/2)^d for all k and all convex domains Ω.

What carries the argument

Ψ(d,k,Ω), the minimal index shift satisfying the comparison inequality μ_{k+Ψ} ≤ λ_k between the ordered Neumann and Dirichlet eigenvalues of the Laplacian.

Load-bearing premise

The domain Ω is bounded in Euclidean space R^d.

What would settle it

A sequence of bounded domains Ω_d in dimensions d→∞ for which Ψ(d,1,Ω_d) divided by (e/2)^d tends to zero.

read the original abstract

Let $\Omega$ be a bounded domain in $R^d$. Denote by $\lambda_k$ (resp. $\mu_k$) the eigenvalues of the Laplace operator in $\Omega$ with Dirichlet (resp. Neumann) boundary conditions. Denote by $\Psi = \Psi (d,k,\Omega)$ the shift of indices in the inequality $\mu_{k+\Psi} \le \lambda_k$. We are interested to describe the behaviour of $\Psi$ for large $d$. We prove that a) $\Psi (d,1,\Omega) \ge C (e/2)^d$ for all domains $\Omega$; and b) $\Psi (d,k,\Omega) \ge C (e/2)^d$ for all $k$ and all convex domains $\Omega$.

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 / 2 minor

Summary. The paper studies the index shift Ψ(d,k,Ω) such that the (k+Ψ)th Neumann eigenvalue μ_{k+Ψ} is at most the kth Dirichlet eigenvalue λ_k of the Laplacian on a bounded domain Ω ⊂ R^d. It claims to prove that Ψ(d,1,Ω) ≥ C (e/2)^d holds for some C > 0 independent of d and for every bounded Ω, and that the same exponential lower bound holds for Ψ(d,k,Ω) for every k when Ω is convex.

Significance. If the claimed lower bounds hold with C independent of d, the result quantifies an exponentially large discrepancy between the ordered Dirichlet and Neumann spectra in high dimensions. This is a strong, dimension-dependent statement with potential implications for high-dimensional spectral geometry. The absence of free parameters or fitted constants in the stated bound is a positive feature of the claim.

minor comments (2)
  1. The abstract does not specify the value or dependence of the constant C; the full text should make explicit whether C is universal or depends on other quantities.
  2. Notation for the shift Ψ(d,k,Ω) is introduced without an explicit formula or definition in the provided abstract; the manuscript should include a precise definition early on.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their report and for acknowledging the potential significance of the claimed exponential lower bounds on the index shift Ψ(d,k,Ω). No specific major comments were raised in the report, and the recommendation is listed as uncertain. We would welcome the opportunity to address any concrete questions or concerns the referee may have about the proofs or statements.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines Ψ(d,k,Ω) explicitly as the minimal index shift making μ_{k+Ψ} ≤ λ_k hold, then proves an explicit lower bound C(e/2)^d on this quantity (for k=1 any bounded Ω, and for all k when Ω is convex). This is a standard lower-bound argument on a well-defined combinatorial quantity; no parameter is fitted to data and then relabeled as a prediction, no self-citation chain is invoked to justify the central claim, and the definition of Ψ does not presuppose the bound being proved. The only assumption (boundedness of Ω) is required for discreteness of the spectra and is external to the result. No step reduces to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard existence and ordering of Dirichlet and Neumann eigenvalues for bounded domains; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • standard math The Dirichlet and Neumann Laplacians on a bounded domain in R^d possess discrete spectra that can be ordered as increasing sequences λ_k and μ_k.
    Implicit background fact required to define the shift Ψ.

pith-pipeline@v0.9.1-grok · 5652 in / 1111 out tokens · 56634 ms · 2026-06-30T03:58:20.496866+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

21 extracted references · 3 canonical work pages · 1 internal anchor

  1. [1]

    Aviles,Symmetry theorems related to Pompeiu’s problem, Amer

    P. Aviles,Symmetry theorems related to Pompeiu’s problem, Amer. J. Math. 108 (1986), no. 5, 1023–1036

    1986

  2. [2]

    Bezdek,Tarski’s plank problem revisited, Geometry — intuitive, discrete, and convex, 45–64

    K. Bezdek,Tarski’s plank problem revisited, Geometry — intuitive, discrete, and convex, 45–64. Bolyai Soc. Math. Stud., 24, J´ anos Bolyai Math. Society, Budapest, 2013

    2013

  3. [3]

    G. Cox, S. MacLachlan, L. Steeves,Isoperimetric relations between Dirichlet and Neumann eigenvalues, arXiv:1906.10061 (2019)

    work page internal anchor Pith review Pith/arXiv arXiv 1906

  4. [4]

    G. Faber,Beweis, dass unter allen homogenen Membranen von gleicher Fl¨ ache und gleicher Spannung die kreisf¨ ormige den tiefsten Grundton gibt, Sitzungberichte der mathematisch- physikalischen Klasse der Bayerischen Akademie der Wissenschaften zu M¨ unchen (1923), 169–172

    1923

  5. [5]

    Filonov,On an inequality for the eigenvalues of the Dirichlet and Neumann problems for the Laplace operator, Algebra i Analiz 16 (2004), no

    N. Filonov,On an inequality for the eigenvalues of the Dirichlet and Neumann problems for the Laplace operator, Algebra i Analiz 16 (2004), no. 2, 172–176 (Russian). Translation in St. Petersburg Math. J. 16 (2005), no. 2, 413–416

    2004

  6. [6]

    Filonov, Y

    N. Filonov, Y. Safarov,Asymptotic estimates for the difference between the Dirichlet and Neumann counting functions, Funktsional. Anal. i Prilozhen., 44 (2010), no. 4, 54–64 (Russian); Engl. translation in Funct. Anal. Appl., 44 (2010), no. 4, 286–294

    2010

  7. [7]

    W. J. Firey,Lower bounds for volumes of convex bodies, Arch. Math. 16 (1965), 69–74

    1965

  8. [8]

    Freitas, M

    P. Freitas, M. Gama,On the (growing) gap between Dirichlet and Neumann eigenvalues, arXiv:2405.18079 (2024)

    work page arXiv 2024

  9. [9]

    Friedlander,Some inequalities between Dirichlet and Neumann eigenvalues, Arch

    L. Friedlander,Some inequalities between Dirichlet and Neumann eigenvalues, Arch. Ra- tional Mech. Anal. 116 (1991), no. 2, 153–160

    1991

  10. [10]

    Hatcher,Geometric inequalities between Dirichlet and Neumann eigenvalues, arXiv:2504.18517 (2025)

    L. Hatcher,Geometric inequalities between Dirichlet and Neumann eigenvalues, arXiv:2504.18517 (2025). 23

    work page arXiv 2025

  11. [11]

    Krahn, ¨Uber eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math

    E. Krahn, ¨Uber eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann. 94 (1925), no. 1, 97–100

    1925

  12. [12]

    Krahn, ¨Uber Minimaleigenschaften der Kugel in drei un mehr Dimensionen, Acta Comm

    E. Krahn, ¨Uber Minimaleigenschaften der Kugel in drei un mehr Dimensionen, Acta Comm. Univ. Dorpat. A9 (1926), 1–44

    1926

  13. [13]

    Kr¨ oger,Upper bounds for the Neumann eigenvalues on a bounded domains in Euclidean space, J

    P. Kr¨ oger,Upper bounds for the Neumann eigenvalues on a bounded domains in Euclidean space, J. Funct. Anal. 106 (1992), 353–357

    1992

  14. [14]

    H. A. Levine, H. F. Weinberger,Inequalities between Dirichlet and Neumann eigenvalues, Arch. Rational Mech. Anal. 94 (1986), 193–208

    1986

  15. [15]

    Lorch, P

    L. Lorch, P. Szego,Bounds and monotonicities for the zeros of derivatives of ultraspherical Bessel functions, SIAM J.Math. Anal. 25 (1994), no. 2, 549–554

    1994

  16. [16]

    P´ al,Ein minimumproblem f¨ ur Ovale, Math

    J. P´ al,Ein minimumproblem f¨ ur Ovale, Math. Ann. 83 (1921), 311–319

    1921

  17. [17]

    L. E. Payne,Inequalities for eigenvalues of membranes and plates, J. Rational Mech. Anal. 4 (1955), 517–529

    1955

  18. [18]

    P´ olya,Remarks on the foregoing paper, J

    G. P´ olya,Remarks on the foregoing paper, J. Math. Physics 31 (1952), 55–57

    1952

  19. [19]

    Best possible

    C. K. Qu, R. Wong,“Best possible” upper and lower bounds for the zeros of the Bessel functionJ ν(x), Trans. Amer. Math. Soc. 351 (1999), no 7, 2833–2859

    1999

  20. [20]

    Rohleder,Inequalities between Neumann and Dirichlet Laplacian eigenvalues on planar domains, Math

    J. Rohleder,Inequalities between Neumann and Dirichlet Laplacian eigenvalues on planar domains, Math. Ann. 392 (2025), no. 4, 5553–5571

    2025

  21. [21]

    G. N. Watson,A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, 1995. 24

    1995