On the Pólya conjecture for the Neumann problem in tiling sets

2020/06/18 by N. Filonov, Filonov, Nikolai
Materials Science · Mathematics · #35P15 #FOS: Mathematics #Graph theory and applications #Quasicrystal Structures and Properties #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2006.10663

openalex publication_date 2020/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In 1954, G. Pólya conjectured that the counting function of the eigenvalues of the Laplace operator of Dirichlet (resp. Neumann) boundary value problem in a bounded set Ω⊂\mathbb R^d is lesser (resp. greater) than C_W |Ω| λ^d/2. Here λ is the spectral parameter, and C_W is the constant in the Weyl asymptotics. In 1961, Pólya proved this conjecture for tiling sets in the Dirichlet case, and for tiling sets under some additional restrictions for the Neumann case. We prove the Pólya conjecture in the Neumann case for all tiling sets.

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