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Pólya's conjecture for Dirichlet eigenvalues of annuli

This paper proves Pólya's conjecture for the Dirichlet Laplacian on annular domains by extending previous methods used for disks and balls through a combination of variational bounds, Bessel function estimates, lattice point counting, and computer-assisted analysis.

Original authors: Nikolay Filonov, Michael Levitin, Iosif Polterovich, David A. Sher

Published 2026-02-10

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CC BY 4.0

Original authors: Nikolay Filonov, Michael Levitin, Iosif Polterovich, David A. Sher

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Mystery of the Vibrating Ring: A Simple Guide to a Mathematical Breakthrough

Imagine you have a circular metal ring, like a wedding band or a donut. If you strike this ring, it vibrates. In physics and math, these vibrations happen at very specific "notes" or frequencies, which we call eigenvalues.

For over 70 years, mathematicians have been obsessed with a famous prediction called Pólya’s Conjecture. It’s essentially a rule about how many of these "notes" can exist below a certain pitch. Pólya predicted that for certain shapes, the number of notes is always less than a specific mathematical limit.

Think of it like a musical buffet: Pólya predicted that no matter how you shape your instrument, you will never find more "notes" than what a specific formula allows. For decades, we knew this was true for simple shapes like squares or triangles, but for more complex shapes—especially shapes with holes in them—it was a massive mystery.

This paper is the "Eureka!" moment where scientists finally proved that Pólya was right about rings (annuli).

The Three Tools of the Trade

To solve this, the authors didn't just use one method; they used a "mathematical toolkit" that combined three very different approaches.

1. The "Shape-Shifter" Strategy (Isoperimetric Inequalities)

Imagine you have a lump of clay. You can shape it into a long, thin snake or a perfect ball. The ball is the most "efficient" shape—it holds the most volume with the least surface area.

The researchers used this idea of "efficiency." They proved that for certain types of rings, the "notes" behave predictably because the ring is constrained by its geometry. They compared the ring to a "flat cylinder" (like a rolled-up piece of paper) to show that the ring couldn't possibly break Pólya's rule.

2. The "Lattice Point" Strategy (Counting Dots)

This is where the math gets clever. Instead of trying to calculate the vibrations directly (which is incredibly hard), they turned the problem into a game of "Connect the Dots."

Imagine a sheet of graph paper. The researchers found a way to translate the "notes" of the ring into "dots" on this grid. If they could prove that the number of dots under a certain curved line was always less than a specific amount, they would prove Pólya was right.

They used a technique called "Trapezoidal Floor Sums." Think of this like trying to estimate how much water is in a wavy swimming pool by looking at the floor tiles. By using very precise "tile-counting" math, they could bound the number of dots with extreme accuracy.

3. The "Digital Detective" (Computer-Assisted Proof)

Even with all that math, there was still a "gap"—a small, messy region of possibilities where the formulas weren't quite strong enough to give a definitive answer. It was like having a map of a forest that shows the mountains and the rivers, but leaves a small, foggy patch in the middle.

To clear the fog, they didn't just guess; they built a rigorous digital detective. They wrote a computer program that didn't just "estimate" numbers (which can be prone to tiny, rounding errors), but instead used "Verified Rational Approximations."

Think of it like this: instead of a calculator saying "the answer is roughly 3.14," the computer says, "I have checked every single microscopic grain of sand in this patch of fog, and I guarantee the answer is between 3.141 and 3.142." The computer scanned the "foggy" area, checking thousands of points, until it could confidently say, "Pólya was right here, too."

Why Does This Matter?

You might ask, "Who cares about the vibrations of a ring?"

While it sounds abstract, this kind of math is the foundation of how we understand the physical world. The same math used to study these "notes" is used to understand:

Quantum Mechanics: How subatomic particles behave in tiny spaces.
Acoustics: How sound travels through complex structures.
Structural Engineering: How vibrations affect the stability of buildings or parts of a machine.

By proving Pólya’s conjecture for rings, these mathematicians have added a vital piece to the puzzle of how geometry and vibration dance together. They have turned a 70-year-old "maybe" into a mathematical "certainty."

Technical Summary: Pólya’s Conjecture for Dirichlet Eigenvalues of Annuli

Authors: Nikolay Filonov, Michael Levitin, Iosif Polterovich, and David A. Sher
Publication: Journal of the London Mathematical Society (2026)

1. The Problem: Pólya’s Conjecture

The paper addresses a long-standing problem in spectral geometry: Pólya’s Conjecture. For a bounded domain $\Omega \subset \mathbb{R}^2$ , let $N_{\text{Dir}}^{\Omega}(\lambda)$ denote the eigenvalue counting function for the Dirichlet Laplacian. Weyl’s law provides the asymptotic behavior:
$N_{\text{Dir}}^{\Omega}(\lambda) \sim \frac{\text{Area}(\Omega)}{4\pi}\lambda^2 \quad \text{as } \lambda \to \infty$
Pólya conjectured that this asymptotic leading term serves as a universal upper bound for all $\lambda > 0$ :
$N_{\text{Dir}}^{\Omega}(\lambda) < \frac{\text{Area}(\Omega)}{4\pi}\lambda^2$
While proven for domains that tile the plane, the conjecture remains open for most other shapes. This paper specifically targets annular domains $A_r = \{x \in \mathbb{R}^2 : r < |x| < 1\}$ , where $r \in (0,1)$ is the inner radius.

2. Methodology

The authors employ a hybrid approach, combining classical spectral theory, advanced analytic estimates, and rigorous computer-assisted verification. The proof is partitioned into five distinct regions of the parameter space $\mathcal{R}_{\Lambda}$ (defined by $r$ and $\lambda$ ).

Spectral-Theoretic Comparison (Regions I & II):
- Region I: Uses the Faber–Krahn and Krahn–Szego isoperimetric inequalities to handle small values of $\lambda$ where the first two eigenvalues dominate.
- Region II: Employs a comparison between the annulus and flat cylinders. By showing that the eigenvalues of an annulus are bounded by those of a specific cylinder, the authors reduce the problem to an explicit counting problem for cylinders.
Analytic & Lattice Point Techniques (Regions III, IV, & V):
- The authors relate the eigenvalue count to the Bessel phase function $\theta_\nu(x)$ . The counting function is expressed as a sum involving the difference of phase functions: $\Theta_{r,m}(\lambda) = \theta_m(\lambda) - \theta_m(r\lambda)$ .
- This is transformed into a lattice point counting problem under specific curves.
- New Mathematical Tools: The authors develop new bounds for trapezoidal floor sums for both concave and convex Lipschitz functions. These bounds allow for much tighter control over the error terms in the lattice count than previous methods.
Computer-Assisted Verification (The "Void"):
- The analytic methods leave a bounded "void" in the parameter space (where $\lambda$ is relatively small).
- The authors implement a rigorous computer-assisted algorithm using verified rational approximations. This avoids the pitfalls of floating-point errors by using interval arithmetic and rational bounds to ensure that the inequality holds strictly across the remaining domain.

3. Key Contributions

Extension to Non-Simply Connected Domains: This is the first proof of Pólya’s conjecture for a non-simply connected planar domain (the annulus).
Refined Lattice Counting: The paper introduces new theorems regarding the estimation of trapezoidal floor sums for Lipschitz functions, which are of independent mathematical interest.
Improved Bounds for the Disk: As a byproduct, the authors derive a sharper two-term upper bound for the Dirichlet eigenvalue counting function of the unit disk, improving upon Pólya's original estimates.
Algorithmic Rigor: The development of a robust algorithm to "fill the void" provides a template for proving similar conjectures in spectral geometry.

4. Main Results

Theorem 1.1: Pólya’s conjecture holds for all planar annuli. For any $r \in (0,1)$ and $\lambda > 0$ :
$N_{A_r}^{\text{Dir}}(\lambda) < \frac{1-r^2}{4}\lambda^2$
Theorem 1.5: For a unit disk $D$ , the authors provide a refined bound:
$N_{\text{Dir}}^{D}(\lambda) < \frac{\lambda^2}{4} - \frac{\lfloor \omega_0 \lambda \rfloor}{2}$
(where $\omega_0 \approx 0.108998$ ).

5. Significance

The paper represents a major milestone in spectral geometry. By successfully tackling the annulus, the authors have moved beyond the "simply connected" barrier that has limited progress on Pólya's conjecture for decades. The integration of high-level analysis with rigorous computation sets a new standard for how complex conjectures in mathematical physics can be resolved.