Pólya's conjecture up to $ε$-loss and quantitative estimates for the remainder of Weyl's law

Abstract

Let $Ω\subset\mathbb{R}^n$ be a bounded Lipschitz domain. For any $ε\in (0,1)$ we show that for any Dirichlet eigenvalue $λ_k(Ω)>Λ(ε,Ω)$, it holds \begin{align*} k&\le (1+ε)\frac{|Ω|ω(n)}{(2π)^n}λ_k(Ω)^{n/2}, \end{align*} where $Λ(ε,Ω)$ is given explicitly. This reduces the $ε$-loss version of Pólya's conjecture to a computational problem. This estimate is based on quantitative estimates on the remainder of the Weyl law with explicit constants, which we give a new proof without using Neumann eigenvalues. Our arguments in deriving such uniform estimates yield also, in all dimensions $n\ge 2$, classes of domains that may even have rather irregular shapes or boundaries but satisfy Pólya's conjecture. Another key observation is that on strip-tiling domains (and therefore any triangles for instance) one actually has better eigenvalue estimates than Pólya conjectured.