Pólya's conjecture for thin products

Abstract

Let $\hat{I}\text{c◯}\subset {\mathbb{R}}^d$ be a bounded Euclidean domain. According to the famous Weyl law, both its Dirichlet eigenvalue $\hat{I}{»}_k(\hat{I}\text{c◯})$ and its Neumann eigenvalue $\hat{I}{¼}_k(\hat{I}\text{c◯})$ have the same leading asymptotics $w_k(\hat{I}\text{c◯})=C(d,\hat{I}\text{c◯})k^{2/d}$ as $k\to \infty$. G. Pólya conjectured in 1954 that each Dirichlet eigenvalue $\hat{I}{»}_k(\hat{I}\text{c◯})$ is greater than $w_k(\hat{I}\text{c◯})$, while each Neumann eigenvalue $\hat{I}{¼}_k(\hat{I}\text{c◯})$ is no more than $w_k(\hat{I}\text{c◯})$. In this paper we prove Pólya's conjecture for thin products, i.e. domains of the form $(a\hat{I}{\text{c◯}}_1)\times \hat{I}{\text{c◯}}_2$, where $\hat{I}{\text{c◯}}_1,\hat{I}{\text{c◯}}_2$ are Euclidean domains, and $a$ is small enough. We also prove that the same inequalities hold if $\hat{I}{\text{c◯}}_2$ is replaced by a Riemannian manifold, and thus get Pólya's conjecture for a class of ``thin" Riemannian manifolds with boundary.