On the P\'olya conjecture for the Neumann problem in tiling sets

TL;DR

This paper proves Pólya's conjecture for the Neumann eigenvalue problem in tiling sets, confirming the spectral counting function's lower bound matches Weyl's asymptotics for these sets.

Contribution

The paper establishes the Pólya conjecture for Neumann boundary conditions specifically in all tiling sets, extending previous partial results.

Findings

Pólya's conjecture holds for Neumann problem in tiling sets.

The spectral counting function is bounded below by Weyl's constant times volume and eigenvalue.

The result applies to all tiling sets, broadening the class of sets where the conjecture is confirmed.

Abstract

In 1954, G. P\'olya conjectured that the counting function of the eigenvalues of the Laplace operator of Dirichlet (resp. Neumann) boundary value problem in a bounded set $\Omega\subset{\mathbb R}^d$ is lesser (resp. greater) than $C_W |\Omega| \lambda^{d/2}$. Here $\lambda$ is the spectral parameter, and $C_W$ is the constant in the Weyl asymptotics. In 1961, P\'olya 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 P\'olya conjecture in the Neumann case for all tiling sets.