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Pólya's conjecture for Dirichlet eigenvalues of annuli

Nikolay Filonov, Michael Levitin, Iosif Polterovich, David A. Sher

TL;DR

The paper proves Pólya’s conjecture for the Dirichlet Laplacian on all planar annuli by combining five analytic regions—two governed by isoperimetric and cylinder comparisons, and three regions handled via refined Bessel phase-function analysis and advanced lattice-point counting. Central to the approach are new trapezoidal floor-sum bounds for Lipschitz concave/convex functions, precise control of the Bessel phase difference, and a reduction of eigenvalue counting to a lattice-counting problem that is tackled both analytically and through a computer-assisted verification for the remaining parameter space. The authors also derive a two-term upper bound for the disk’s Dirichlet eigenvalue counting function, improving Polya’s original estimate. This work extends previous disk/sector results to a non-simply connected domain and integrates isoperimetric, spectral, and computational techniques to establish a comprehensive verification of Polya’s conjecture in 2D.

Abstract

We prove Pólya's conjecture for the eigenvalues of the Dirichlet Laplacian on annular domains. Our approach builds upon and extends the methods we previously developed for disks and balls. It combines variational bounds, estimates of Bessel phase functions, refined lattice point counting techniques, and a rigorous computer-assisted analysis. As a by-product, we also derive a two-term upper bound for the Dirichlet eigenvalue counting function of the disk, improving upon Pólya's original estimate.

Pólya's conjecture for Dirichlet eigenvalues of annuli

TL;DR

The paper proves Pólya’s conjecture for the Dirichlet Laplacian on all planar annuli by combining five analytic regions—two governed by isoperimetric and cylinder comparisons, and three regions handled via refined Bessel phase-function analysis and advanced lattice-point counting. Central to the approach are new trapezoidal floor-sum bounds for Lipschitz concave/convex functions, precise control of the Bessel phase difference, and a reduction of eigenvalue counting to a lattice-counting problem that is tackled both analytically and through a computer-assisted verification for the remaining parameter space. The authors also derive a two-term upper bound for the disk’s Dirichlet eigenvalue counting function, improving Polya’s original estimate. This work extends previous disk/sector results to a non-simply connected domain and integrates isoperimetric, spectral, and computational techniques to establish a comprehensive verification of Polya’s conjecture in 2D.

Abstract

We prove Pólya's conjecture for the eigenvalues of the Dirichlet Laplacian on annular domains. Our approach builds upon and extends the methods we previously developed for disks and balls. It combines variational bounds, estimates of Bessel phase functions, refined lattice point counting techniques, and a rigorous computer-assisted analysis. As a by-product, we also derive a two-term upper bound for the Dirichlet eigenvalue counting function of the disk, improving upon Pólya's original estimate.

Paper Structure

This paper contains 24 sections, 27 theorems, 191 equations, 11 figures.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Main result
  4. Ideas of the proof
  5. Plan of the paper
  6. Acknowledgements.
  7. Regions I and II via isoperimetric inequalities and comparison with flat cylinders
  8. Trapezoidal floor sums for Lipschitz functions and their bounds
  9. The eigenvalue counting function and the Bessel phase function
  10. Bounds on the phase functions difference and reduction to a lattice counting problem
  11. Some properties of functions $G_\lambda$ and $\Phi_{\lambda,\mu}$
  12. Lattice point count and the trapezoidal floor sums
  13. The outline of the scheme
  14. Region III
  15. Sharper bounds: Region IV
  16. ...and 9 more sections

Key Result

Theorem 1.1

Pólya's conjecture for the Dirichlet Laplacian holds for all planar annuli. In other words, inequality eq:PolyaA holds for all $(r,\lambda)\in\mathrm{R}\Lambda$, or, equivalently, eq:PolyaB holds for all $(\lambda,\mu)\in\Lambda\mathrm{M}$.

Figures (11)

  • Figure 1: Summary of all regions in $(r,\lambda)$-plane for which Pólya's conjecture is analytically proven to be true, with two side figures zooming near the axes. The exact definitions of the regions can be found in Theorem \ref{['thm:regI']} for $\mathrm{R}\Lambda_\mathrm{I}$, Theorem \ref{['thm:vari']} for $\mathrm{R}\Lambda_\mathrm{II}$, Theorem \ref{['thm:III']} for $\mathrm{R}\Lambda_\mathrm{III}$, Theorem \ref{['thm:IV']} for $\mathrm{R}\Lambda_\mathrm{IV}$, and Theorem \ref{['thm:V']} for $\mathrm{R}\Lambda_\mathrm{V}$.
  • Figure 2: Summary of all regions in $(\lambda, \mu)$-plane for which Pólya's conjecture is analytically proven to be true. The bottom figure zooms near the $\lambda$-axis and the origin.
  • Figure 3: An annulus $A_r$ and a cylinder $\mathcal{C}_{1-r}$. Note that we are comparing the spectrum of the Dirichlet Laplacian in $A_r$ with the spectrum of the Dirichlet realisation of \ref{['eq:scale']} in $\mathcal{C}_{1-r}$, which in turn coincides with the spectrum of the Dirichlet Laplacian in $\mathcal{C}_{h_r}$.
  • Figure 4: The plots of $\eta_\mathrm{II}(r)$ and $\zeta_\mathrm{II}(\lambda)$. The red dots indicate the positions of singularities in definitions \ref{['eq:eta2']} and \ref{['eq:zeta2']}.
  • Figure 6: A typical behaviour of bounds \ref{['eq:deltabound2']} and \ref{['eq:deltabound3']}, and of the upper bound \ref{['eq:deltaboundNF']}, with the solid line showing bound \ref{['eq:deltabound1']}. The inset zooms near intersections of the curves.
  • ...and 6 more figures

Theorems & Definitions (56)

  • Theorem 1.1
  • Remark 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • ...and 46 more