Pólya's conjecture for thin products

2024/02/19 by Xiang He, Zuoqin Wang, He, Xiang +1 · 3 citations
Mathematics · #35P15 #35P20 #58J50 #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Holomorphic and Operator Theory #Rings, Modules, and Algebras #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2402.12093

openalex publication_date 2024/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Ω⊂ \mathbb R^d be a bounded Euclidean domain. According to the famous Weyl law, both its Dirichlet eigenvalue λ_k(Ω) and its Neumann eigenvalue μ_k(Ω) have the same leading asymptotics w_k(Ω)=C(d,Ω)k^2/d as k → ∞. G. Pólya conjectured in 1954 that each Dirichlet eigenvalue λ_k(Ω) is greater than w_k(Ω), while each Neumann eigenvalue μ_k(Ω) is no more than w_k(Ω). In this paper we prove Pólya's conjecture for thin products, i.e. domains of the form (aΩ₁) × Ω₂, where Ω₁, Ω₂ are Euclidean domains, and a is small enough. We also prove that the same inequalities hold if Ω₂ is replaced by a Riemannian manifold, and thus get Pólya's conjecture for a class of ``thin" Riemannian manifolds with boundary.

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• Pólya's conjecture up to ε-loss and quantitative estimates for the remainder of Weyl's law
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