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P\'olya's conjecture for thin products

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arxiv 2402.12093 v3 pith:JEEIAG6S submitted 2024-02-19 math.SP math.DG

Xiang He , Zuoqin Wang This is my paper · ORCID
classification math.SPmath.DG
keywords omegaeigenvalueolyaconjecturethindirichletdomainseuclidean
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abstract

Let $\Omega \subset \mathbb R^d$ be a bounded Euclidean domain. According to the famous Weyl law, both its Dirichlet eigenvalue $\lambda_k(\Omega)$ and its Neumann eigenvalue $\mu_k(\Omega)$ have the same leading asymptotics $w_k(\Omega)=C(d,\Omega)k^{2/d}$ as $k \to \infty$. G. P\'olya conjectured in 1954 that each Dirichlet eigenvalue $\lambda_k(\Omega)$ is greater than $w_k(\Omega)$, while each Neumann eigenvalue $\mu_k(\Omega)$ is no more than $w_k(\Omega)$. In this paper we prove P\'olya's conjecture for thin products, i.e. domains of the form $(a\Omega_1) \times \Omega_2$, where $\Omega_1, \Omega_2$ are Euclidean domains, and $a$ is small enough. We also prove that the same inequalities hold if $\Omega_2$ is replaced by a Riemannian manifold, and thus get P\'olya's conjecture for a class of ``thin" Riemannian manifolds with boundary.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. P\'olya's Conjecture for the Neumann Eigenvalues on Euclidean Balls

    math.SP 2026-07 conditional novelty 8.0 of 10

    Every Euclidean ball in dimension d≥2 satisfies Pólya's Neumann inequality N^<(E) ≥ (ω_d/(2π)^d)|B|E^{d/2} at all energies E≥0.

  2. P\'{o}lya's conjecture for higher-dimensional Neumann balls

    math.SP 2026-07 accept novelty 7.0 of 10

    For every d≥3 and every λ≥0, the Neumann counting function of the unit d-ball satisfies N(λ) ≥ w_d λ^d, so Pólya's conjecture holds for all Euclidean balls.

  3. P\'olya's conjecture up to $\epsilon$-loss and quantitative estimates for the remainder of Weyl's law

    math.SP 2025-07 conditional novelty 7.0 of 10

    For every bounded Lipschitz domain, Pólya's eigenvalue bound holds up to factor 1+ε for all eigenvalues above an explicit threshold, and exact Pólya bounds are proved for new irregular domain classes.

  4. P\'{o}lya's conjecture on $\mathbb{S}^1 \times \R$

    math.SP 2025-06 conditional novelty 7.0 of 10

    For cylindrical strips S^1×[0,h], Pólya's conjecture holds exactly for h in (0,≈3.048] ∪ [≈3.238,≈4.046] ∪ [≈4.082,π^2/2], failing only for the 8th and 13th eigenvalues in two narrow gaps.

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