Pólya's conjecture for Euclidean balls

Abstract

The celebrated Pólya's conjecture (1954) in spectral geometry states that the eigenvalue counting functions of the Dirichlet and Neumann Laplacian on a bounded Euclidean domain can be estimated from above and below, respectively, by the leading term of Weyl's asymptotics. Pólya's conjecture is known to be true for domains which tile Euclidean space, and, in addition, for some special domains in higher dimensions. In this paper, we prove Pólya's conjecture for the disk, making it the first non-tiling planar domain for which the conjecture is verified. We also confirm Pólya's conjecture for arbitrary planar sectors, and, in the Dirichlet case, for balls of any dimension. Along the way, we develop the known links between the spectral problems in the disk and certain lattice counting problems. A key novel ingredient is the observation, made in recent work of the last named author, that the corresponding eigenvalue and lattice counting functions are related not only asymptotically, but in fact satisfy certain uniform bounds. Our proofs are purely analytic, except for a rigorous computer-assisted argument needed to cover the short interval of values of the spectral parameter in the case of the Neumann problem in the disk.

Figures (6)

• Figure 1: The Dirichlet eigenvalue counting function $\mathcal{N}^{\mathrm{D}}_{\mathbb{D}}(\lambda)$ (blue), the Neumann eigenvalue counting function $\mathcal{N}^{\mathrm{N}}_{\mathbb{D}}(\lambda)$ (red), and the leading Weyl's term $W_d(\lambda)=\frac{\lambda^2}{4}$ (black) in dimension $d=2$. The plot is produced using the floating-point evaluation of zeros of the Bessel functions and their derivatives. If we were to assume (contrary to the philosophy of this paper) the validity of floating-point arithmetic, this plot would have presented a numerically assisted (as opposed to computer-assisted) "proof" of Pólya's conjecture for the disk for $\lambda\lessapprox 15$.
• Figure 2: The region $\mathbb{P}_\lambda$, and the sets of shifted lattice points $Q^{\mathrm{D}}_d(\lambda)$ (blue disks) and $Q^{\mathrm{N}}_d(\lambda)$ (red diamonds), shown here for $d=2$ and $\lambda=23$.
• Figure 3: An illustration of inequalities \ref{['eq:BminusAD']} and \ref{['eq:BminusAN']}. The plots of $B^{\mathrm{D}}_\nu(\lambda)$ and $B_\nu^{\mathrm{N}}(\lambda)$ are drawn using the recipe from Hor. We remark that $B^{\mathrm{N}}_\nu(\lambda)$ has a minimum at $\lambda=\nu$.
• Figure 4: A numerical experiment: the computed $\mathcal{P}_2^{\mathrm{N}}(\lambda)/W_2(\lambda)-1$ as a function of $\lambda$.
• Figure 5: The numbers $L_k$, see \ref{['eq:Lk']} and also Remark \ref{['rem:Lk']}.

Theorems & Definitions (61)

• Remark 1.1
• Theorem 1.2
• Lemma 1.3
• proof
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Remark 1.7
• Theorem 1.8
• proof