Pólya’s conjecture for thin products

Abstract Let $\Omega \subset \mathbb R^{d}$ be a bounded domain. According to the famous Weyl law, both its Dirichlet eigenvalues $\lambda _{k}(\Omega )$ and its Neumann eigenvalues $\mu _{k}(\Omega )$ have the same leading asymptotics $$ \begin{align*} & \frac{4\pi^{2}}{(\omega_{d} |\Omega|)^{\frac 2d}} k^{\frac 2d},\qquad \textrm{as }k\to+\infty. \end{align*} $$ In 1954, G. Pólya conjectured that for each $k$, $$ \begin{align*} & \lambda_{k}(\Omega)\ge \frac{4\pi^{2}}{(\omega_{d} |\Omega|)^{\frac 2d}} k^{\frac 2d}\ge \mu_{k}(\Omega). \end{align*} $$ In this paper we prove Pólya’s conjecture for all thin products, that is, domains of the form $(a\Omega _{1}) \times \Omega _{2}$, where $\Omega _{1}$ and $\Omega _{2}$ are arbitrary Euclidean domains, and $a$ is sufficiently small.