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arxiv: 2606.02428 · v1 · pith:J5NJOPLLnew · submitted 2026-06-01 · 🧮 math.SP

Exact L^p growth rates of Laplace eigenfunctions on the unit disk

Haoyu Cheng This is my paper

Pith reviewed 2026-06-28 11:42 UTC · model grok-4.3

classification 🧮 math.SP
keywords Laplace eigenfunctionsunit diskLp normsgrowth ratesDirichlet boundary conditionsNeumann boundary conditionsBessel functionsstationary phase
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The pith

The logarithmic growth exponents of Lp norms for L2-normalized Laplace eigenfunctions on the unit disk are determined exactly for all 1 ≤ p ≤ ∞ under both Dirichlet and Neumann conditions.

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

The paper determines the precise rates at which the Lp norms of these eigenfunctions grow logarithmically with the eigenvalue. It covers the full range of p and both common boundary conditions on the disk. It further establishes that the upper and lower bounds are sharp and hold uniformly for every qualifying eigenfunction. A reader would care because the rates control how much the functions concentrate or oscillate at high frequencies. The argument rests on the explicit Bessel-function form of the eigenfunctions together with stationary-phase control of their integrals.

Core claim

We determine the logarithmic growth exponents of the Lp norms, 1≤p≤∞, of L2-normalized Laplace eigenfunctions on the unit disk, for both Dirichlet and Neumann boundary conditions. We also prove sharp uniform Lp upper and lower bounds for every L2-normalized Dirichlet eigenfunction and every non-constant Neumann eigenfunction. The proof uses stationary phase estimates and integral estimates for Bessel functions.

What carries the argument

Explicit representations of the eigenfunctions in terms of Bessel functions, together with stationary phase estimates applied to their integrals.

If this is right

  • The same growth exponents apply under both Dirichlet and Neumann boundary conditions.
  • Sharp uniform upper and lower Lp bounds hold simultaneously for every Dirichlet eigenfunction.
  • The uniform bounds also hold for every non-constant Neumann eigenfunction.
  • The exponents are obtained for the entire interval 1 ≤ p ≤ ∞.

Where Pith is reading between the lines

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

  • The explicit Bessel control on the disk supplies a benchmark that could be compared with upper-bound results known on domains lacking closed-form eigenfunctions.
  • The stationary-phase method used here might extend to other radial problems whose solutions satisfy similar integral representations.
  • Numerical checks of the predicted exponents on moderate eigenvalues would give an immediate consistency test before asymptotic regimes are reached.

Load-bearing premise

The eigenfunctions admit explicit expressions in terms of Bessel functions whose asymptotic and integral properties can be controlled by stationary phase methods to yield the precise exponents.

What would settle it

A direct numerical evaluation, for a sequence of high eigenvalues, of the L^∞ norm of an L2-normalized eigenfunction whose growth rate differs from the predicted logarithmic exponent.

Figures

Figures reproduced from arXiv: 2606.02428 by Haoyu Cheng.

Figure 1
Figure 1. Figure 1: The upper and lower exponents in Theorem 1.1 as functions of 1/p. The red curve represents the upper exponent bp, and the blue curve represents the lower exponent ap. Dirichlet eigenfunctions and λ 1/12−ε for Neumann eigenfunctions, for every ε > 0. The present theorem improves their results to λ 1/12 for both boundary conditions. Thus, the new contribution is the uniform lower bounds for 4 < p ≤ ∞ and the… view at source ↗
Figure 2
Figure 2. Figure 2: Φp(γ) as a function of 1/p for three representative values of γ. The blue curve represents γ = 0, the green dashed curve represents γ = 1, and the red curve represents γ = ∞. over all L 2 -normalized eigenfunctions satisfying the boundary condition B. When p = ∞, this definition was introduced by Sarnak [Sar04] on page 41 of his letter to Morawetz. Theorem 1.2 gives the following exact form of EB,p(D). Cor… view at source ↗
read the original abstract

We determine the logarithmic growth exponents of the $L^p$ norms, $1\le p\le\infty$, of $L^2$-normalized Laplace eigenfunctions on the unit disk, for both Dirichlet and Neumann boundary conditions. We also prove sharp uniform $L^p$ upper and lower bounds for every $L^2$-normalized Dirichlet eigenfunction and every non-constant Neumann eigenfunction $u_{\lambda}$ on the disk. The proof uses stationary phase estimates and integral estimates for Bessel functions.

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. The paper determines the logarithmic growth exponents of the L^p norms (1 ≤ p ≤ ∞) of L²-normalized Laplace eigenfunctions on the unit disk, for both Dirichlet and Neumann boundary conditions. It also proves sharp uniform L^p upper and lower bounds for every L²-normalized Dirichlet eigenfunction and every non-constant Neumann eigenfunction. The proofs rely on explicit Bessel-function expressions for the eigenfunctions, reduction of the L^p norms to one-dimensional radial integrals, and application of stationary-phase and Airy-type estimates in the oscillatory, transition, and evanescent regimes.

Significance. If the results hold, the work supplies exact growth exponents and matching sharp bounds on a canonical domain, using only classical properties of Bessel functions. This strengthens the literature on eigenfunction L^p norms by providing parameter-free, explicit rates that are uniform across the spectrum and both boundary conditions.

minor comments (1)
  1. The abstract states that the proof uses 'stationary phase estimates and integral estimates for Bessel functions,' but the manuscript should include a brief outline of the three radial regimes (oscillatory, transition, evanescent) already in the introduction for reader orientation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their thorough reading, positive assessment of the significance, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained analytic estimate

full rationale

The paper derives L^p growth exponents directly from the explicit Bessel-function formulas for disk eigenfunctions (Dirichlet/Neumann), splitting the radial integrals into oscillatory/transition/evanescent regimes and applying classical stationary-phase and Airy estimates. No parameters are fitted to data, no self-citations are load-bearing for the central claim, and no step reduces the target exponents to a definition or prior result by the same authors. The argument uses only standard properties of Bessel functions and integral estimates external to the paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Paper rests on standard analytic estimates for Bessel functions and stationary phase; no free parameters, new entities, or ad-hoc axioms are visible from the abstract.

axioms (1)
  • standard math Standard asymptotic and integral properties of Bessel functions hold and can be applied via stationary phase to control L^p norms of disk eigenfunctions.
    Invoked in the proof method described in the abstract.

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Works this paper leans on

26 extracted references · 9 canonical work pages

  1. [1]

    The spectral function of an elliptic operator , JOURNAL =

    H. The spectral function of an elliptic operator , JOURNAL =. 1968 , PAGES =. doi:10.1007/BF02391913 , URL =

    work page doi:10.1007/bf02391913 1968

  2. [2]

    Landau, L. J. , TITLE =. J. London Math. Soc. (2) , FJOURNAL =. 2000 , NUMBER =. doi:10.1112/S0024610799008352 , URL =

    work page doi:10.1112/s0024610799008352 2000

  3. [3]

    and Poliquin, G

    Lavoie, G. and Poliquin, G. , TITLE =. Ann. Math. Qu\'e. , FJOURNAL =. 2024 , NUMBER =. doi:10.1007/s40316-023-00219-y , URL =

    work page doi:10.1007/s40316-023-00219-y 2024

  4. [4]

    Olver, F. W. J. , TITLE =. 1974 , PAGES =

    1974

  5. [5]

    Watson, G. N. , TITLE =. 1995 , PAGES =

    1995

  6. [6]

    NIST Digital Library of Mathematical Functions

  7. [7]

    1960 , publisher=

    Bessel Functions: Part 3: Zeros and Associated Values , author=. 1960 , publisher=

    1960

  8. [8]

    Qu, C. K. and Wong, R. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 1999 , NUMBER =. doi:10.1090/S0002-9947-99-02165-0 , URL =

    work page doi:10.1090/s0002-9947-99-02165-0 1999

  9. [9]

    VanderKam, J. M. , TITLE =. Internat. Math. Res. Notices , FJOURNAL =. 1997 , NUMBER =. doi:10.1155/S1073792897000238 , URL =

    work page doi:10.1155/s1073792897000238 1997

  10. [10]

    2025 , eprint=

    Spherical harmonics and point configurations on the sphere , author=. 2025 , eprint=

    2025

  11. [11]

    and Germain, P

    Demeter, C. and Germain, P. , TITLE =. Proc. Edinb. Math. Soc. (2) , FJOURNAL =. 2024 , NUMBER =. doi:10.1017/S0013091524000099 , URL =

    work page doi:10.1017/s0013091524000099 2024

  12. [12]

    Ashu, A. M. , title =. 2013 , url =

    2013

  13. [13]

    On the wave equation on a compact

    B. On the wave equation on a compact. Mathematische Zeitschrift , volume=. 1977 , publisher=

    1977

  14. [14]

    and Lebeau, G

    Burq, N. and Lebeau, G. , journal=. Injections de

  15. [15]

    , journal=

    Bourgain, J. , journal=. Eigenfunction bounds for the. 1993 , publisher=

    1993

  16. [16]

    Forum of Mathematics, Sigma , volume=

    Bounds for spectral projectors on tori , author=. Forum of Mathematics, Sigma , volume=. 2022 , publisher=

    2022

  17. [17]

    and Levitin, M

    Filonov, N. and Levitin, M. and Polterovich, I. and Sher, D. , journal=. P. 2023 , publisher=

    2023

  18. [18]

    2024 , eprint=

    Generic simplicity of ellipses , author=. 2024 , eprint=

    2024

  19. [19]

    and Colin de Verdi

    Chabert, A. and Colin de Verdi. On the. 2026 , eprint=

    2026

  20. [20]

    Sogge, C. D. , TITLE =. J. Funct. Anal. , FJOURNAL =. 1988 , NUMBER =. doi:10.1016/0022-1236(88)90081-X , URL =

    work page doi:10.1016/0022-1236(88)90081-x 1988

  21. [21]

    Smith, H. F. and Sogge, C. D. , TITLE =. Acta Math. , FJOURNAL =. 2007 , NUMBER =. doi:10.1007/s11511-007-0014-z , URL =

    work page doi:10.1007/s11511-007-0014-z 2007

  22. [22]

    and Khan, R

    Humphries, P. and Khan, R. , TITLE =. Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2025 , NUMBER =. doi:10.1112/plms.70061 , URL =

    work page doi:10.1112/plms.70061 2025

  23. [23]

    , TITLE =

    Ki, H. , TITLE =. 2023 , NOTE =. 2302.02625 , archivePrefix =

    arXiv 2023

  24. [24]

    , title =

    Sarnak, P. , title =. 2004 , page =

    2004

  25. [25]

    , title =

    Grieser, D. , title =

  26. [26]

    Blair, M. D. and Ford, G. A. and Marzuola, J. L. , title =. Rev. Mat. Iberoamericana , volume =. 2018 , doi =

    2018