Pólya’s conjecture for arbitrary bounded domains

Prove Pólya’s conjecture for arbitrary bounded open sets Ω ⊂ ℝ^n (n ≥ 2) with discrete Neumann spectrum: establish that, for every integer k ≥ 1, the inequalities (|Ω| ω(n) / (2π)^n) γ_{k+1}^{n/2} ≤ k ≤ (|Ω| ω(n) / (2π)^n) λ_k^{n/2} hold, where {λ_k} are the Dirichlet Laplacian eigenvalues and {γ_k} are the Neumann Laplacian eigenvalues on Ω, and ω(n) denotes the volume of the unit ball in ℝ^n.

Background

The paper recalls Pólya’s 1954 conjecture, which provides sharp lower bounds for Dirichlet eigenvalues and upper bounds for Neumann eigenvalues solely in terms of the domain volume. It remains open in general, though it is known for tiling domains, product domains under certain conditions, balls (Dirichlet, all n ≥ 2), Neumann in n = 2, and some annuli. The present work proves that infinitely many eigenvalues satisfy Pólya’s bounds in certain dimensions (Dirichlet for n ≥ 3 and Neumann for n ≥ 5), but does not settle the full conjecture for arbitrary domains.

In the introduction, the authors define the Dirichlet and Neumann spectra on bounded open domains and state Pólya’s conjectured inequalities precisely, marking it as a conjecture and noting that the conjecture remains widely open. Their improvements to Berezin-Li-Yau and Kröger inequalities yield partial progress toward the conjecture but do not resolve it in full.