Pólya's conjecture for products $(0,1)\times\Omega$ with an interval factor and arbitrary bounded cross-section $\Omega$

Pólya's 1954 conjecture asserts that the eigenvalue counting function of the Dirichlet (resp. Neumann) Laplacian on any bounded Euclidean domain is bounded from above (resp. below) by the leading term of Weyl's asymptotics, equivalently that every Dirichlet eigenvalue $\lambda_k$ dominates its Weyl value $4\pi^2 k^{2/d}/(\omega_d|\Omega|)^{2/d}$ and every positive Neumann eigenvalue is dominated by it. It is known for tiling domains, disks, balls and annuli, and — by He and Wang's thin-product theorem — for products $a\Omega_1\times\Omega_2$ with $d_1,d_2\ge2$ and $a$ small, and for $(0,a)\times\Omega$ with $a$ small. The open problem is the one He and Wang explicitly single out: a one-dimensional thin factor at fixed normalization, i.e. the products $(0,1)\times\Omega$ (equivalently $(0,L)\times\Omega$ for any $L>0$) with an arbitrary bounded cross-section $\Omega$, where Laptev's Riesz-mean product argument fails because it requires Riesz exponent $\gamma\ge1$; He and Wang state that this remains open for general $\Omega$, with only special cross-sections (balls, by recent cylinder results) settled.

Background

For a bounded domain $\Omega\subset\mathbb{R}^{d}$, let $0<\lambda_1(\Omega)\le\lambda_2(\Omega)\le\cdots$ be the eigenvalues of the Dirichlet Laplacian (functions vanishing on the boundary) and $0=\mu_0(\Omega)\le\mu_1(\Omega)\le\mu_2(\Omega)\le\cdots$ the eigenvalues of the Neumann Laplacian (zero normal derivative on the boundary); Neumann statements below always refer to the $k$-th positive Neumann eigenvalue $\mu_k$, $k\ge1$, following the authoritative formulation of He–Wang. The eigenvalue counting functions are $\mathcal{N}^{D}_\Omega(\lambda)=\#\{k\ge1:\lambda_k\le\lambda\}$ and, in He–Wang's convention, $\mathcal{N}^{N}_\Omega(\lambda)=\#\{n\ge0:\mu_n<\lambda\}$ (strict inequality, the zero eigenvalue included); under these conventions the eigenvalue form and the counting-function form of the conjecture below are equivalent. Weyl's law states that both counting functions have the same leading asymptotics $L_d|\Omega|\lambda^{d/2}$ with $L_d=\omega_d/(2\pi)^d$, where $\omega_d$ is the volume of the unit ball in $\mathbb{R}^d$.

Pólya's conjecture (made in 1954; published as Pólya 1961) predicts that this leading term is a one-sided bound at every finite $\lambda$: every Dirichlet eigenvalue satisfies

$$\lambda_k(\Omega)\ \ge\ \frac{4\pi^2\,k^{2/d}}{(\omega_d|\Omega|)^{2/d}}\ =:\ w_k(\Omega),$$

and every positive Neumann eigenvalue satisfies $\mu_k(\Omega)\le w_k(\Omega)$; equivalently $\mathcal{N}^{D}(\lambda)\le L_d|\Omega|\lambda^{d/2}\le\mathcal{N}^{N}(\lambda)$ for all $\lambda\ge0$. (Pólya's original published Neumann statement was slightly weaker; the modern form above follows formulations (1.2)–(1.3) of He–Wang.) Pólya proved it for plane domains that tile $\mathbb{R}^2$ by congruent copies, and the argument extends to tiling domains in all dimensions, by a Dirichlet–Neumann bracketing argument: sufficiently many rescaled copies of the domain tile a large box, Dirichlet eigenvalues only decrease and Neumann eigenvalues only increase when the domain is enlarged, and the spectrum of the box is explicit, which pins both counting functions on either side by the Weyl term of the box. The conjecture is open for general domains in every dimension $d\ge2$. Li and Yau (1983) proved the corresponding statement for Riesz means — weighted sums $\sum_{\lambda_k<\lambda}(\lambda-\lambda_k)^{\gamma}$ of the eigenvalues below $\lambda$, equivalently bounds on averages of the first $k$ eigenvalues — which holds for all domains but does not imply the pointwise counting bound.

Two lines of progress define the current frontier. First, Filonov, Levitin, Polterovich and Sher proved Pólya's conjecture for disks and balls (Invent. Math. 2023) and annuli (JLMS 2026) via lattice-point counting and rigorous computer-assisted estimates — the first non-tiling examples — and Filonov (CPAM 2025) improved the Neumann bound for planar convex domains. Second, for product domains Laptev (1997) proved that if $\Omega_1\subset\mathbb{R}^{d_1}$ satisfies Pólya's conjecture with $d_1\ge2$, then $\Omega_1\times\Omega_2$ satisfies it for any bounded $\Omega_2$; his proof rests on the Berezin–Li–Yau Riesz-mean inequality $\sum_{\lambda_k<\lambda}(\lambda-\lambda_k)^{\gamma}\le L_{\gamma,d}|\Omega|\lambda^{\gamma+d/2}$, valid for Riesz exponent $\gamma\ge1$. He and Wang (IMRN 2026) proved Pólya's conjecture for thin products: for $a\Omega_1\times\Omega_2$ with $d_1,d_2\ge2$, $\Omega_1$ Lipschitz (boundary locally the graph of a Lipschitz-continuous function), $\Omega_2$ piecewise smooth, and $a>0$ small enough; for $(0,a)\times\Omega$ with $a$ small; and for analogous Riemannian products. Their method compensates for the absent Pólya inequality on the factor $\Omega_2$ by combining Seeley's two-term Weyl bounds (two-term spectral asymptotics whose second term reflects the boundary contribution, with remainder of order $\lambda^{(d-1)/2}$) with two-term Riesz-mean inequalities of Frank–Larson type (sharp second-term upper bounds for Riesz means of the eigenvalues).

The one-dimensional-factor case is exceptional. When the Pólya-verified factor has dimension $d_1=1$, Laptev's argument breaks down completely because his Riesz-mean inequality requires $\gamma\ge1$, which is unavailable for the spectrum $\pi^2 l^2$ of an interval. He and Wang's Theorem 1.2 covers $(0,a)\times\Omega$ only for $a$ sufficiently small, which by scale invariance corresponds to $(0,1)\times A\Omega$ with $A$ large; the fixed normalization $(0,1)\times\Omega$ for an arbitrary bounded $\Omega$ — a generalized cylinder in $\mathbb{R}^{d+1}$ with cross-section of arbitrary geometry — is left open. This is striking because the interval itself tiles $\mathbb{R}$, so the obstruction is not tiling but the failure of the Riesz-mean machinery at the $d_1=1$ endpoint.

Problem Statement

Determine whether Pólya's conjecture holds for every product of a bounded domain with a fixed interval: for every integer $d\ge2$ and every bounded domain $\Omega\subset\mathbb{R}^d$ with piecewise smooth boundary, is it true that all Dirichlet eigenvalues of the cylinder $(0,1)\times\Omega\subset\mathbb{R}^{d+1}$ satisfy

$$\lambda_k\big((0,1)\times\Omega\big)\ \ge\ \frac{4\pi^2\,k^{2/(d+1)}}{(\omega_{d+1}|\Omega|)^{2/(d+1)}}\quad\text{for all }k\ge1,$$

and all positive Neumann eigenvalues satisfy the reverse inequality $\mu_k\big((0,1)\times\Omega\big)\le\frac{4\pi^2\,k^{2/(d+1)}}{(\omega_{d+1}|\Omega|)^{2/(d+1)}}$ for all $k\ge1$? Equivalently, by scale invariance, whether $\mathcal{N}^{D}_{(0,L)\times\Omega}(\lambda)\le L_{d+1}\,L\,|\Omega|\,\lambda^{(d+1)/2}\le\mathcal{N}^{N}_{(0,L)\times\Omega}(\lambda)$ — where $L\,|\Omega|$ is the volume of the product, $\mathcal{N}^{D}$ counts Dirichlet eigenvalues $\lambda_k\le\lambda$, and $\mathcal{N}^{N}(\lambda)=\#\{n\ge0:\mu_n<\lambda\}$ counts Neumann eigenvalues strictly below $\lambda$, zero included — holds for every $L>0$ and every such $\Omega$. This is exactly the $d_1=1$ case left open by He–Wang's thin-product theorem, where Laptev's Riesz-mean product argument is unavailable because it requires Riesz exponent $\gamma\ge1$.

References

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2. Peter Li and Shing-Tung Yau, On the Schrödinger equation and the eigenvalue problem, Communications in Mathematical Physics 88(3), 309–318, 1983. DOI: 10.1007/bf01213210, https://doi.org/10.1007/bf01213210
3. A. Laptev, Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces, Journal of Functional Analysis 151(2), 531–545, 1997. DOI: 10.1006/jfan.1997.3155, https://doi.org/10.1006/jfan.1997.3155
4. Nikolay Filonov, Michael Levitin, Iosif Polterovich and David A. Sher, Pólya's conjecture for Euclidean balls, Inventiones mathematicae 234, 129–169, 2023. DOI: 10.1007/s00222-023-01198-1, arXiv:2203.07696, https://arxiv.org/abs/2203.07696
5. N. Filonov, On the Pólya conjecture for the Neumann problem in planar convex domains, Communications on Pure and Applied Mathematics 78(3), 537–544, 2025. DOI: 10.1002/cpa.22231, https://doi.org/10.1002/cpa.22231
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7. Nikolay Filonov, Michael Levitin, Iosif Polterovich and David A. Sher, Pólya's conjecture for Dirichlet eigenvalues of annuli, Journal of the London Mathematical Society 113(2), 2026. DOI: 10.1112/jlms.70425, arXiv:2505.21737, https://arxiv.org/abs/2505.21737
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12. Zhengchao Ji and Yong Luo, Improved lower bounds for Dirichlet eigenvalues of the Laplacian and poly-Laplacian on bounded Euclidean domains, 2026. arXiv:2604.20367, https://arxiv.org/abs/2604.20367
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