Proof of the P\'{o}lya conjecture

TL;DR

This paper proves the Pólya conjecture, which predicts a lower bound for higher eigenvalues of the Laplacian on bounded domains, confirming a long-standing mathematical hypothesis.

Contribution

The paper provides a complete proof of the Pólya conjecture, improving upon previous partial results by Li and Yau.

Findings

Confirmed the Pólya conjecture for all eigenvalues

Established a lower bound matching the Weyl asymptotic formula

Extended the argument of Li and Yau to fully solve the conjecture

Abstract

In this paper, we study lower bounds for higher eigenvalues of the Dirichlet eigenvalue problem of the Laplacian on a bounded domain $\Omega$ in $\mathbb{R}^n$. It is well known that the $k$-th Dirichlet eigenvalue $\lambda_k$ obeys the Weyl asymptotic formula, that is, \[ \lambda_k\sim\frac{4\pi^2}{(\omega_n\mathrm{vol}\Omega)^\frac{2}{n}}k^\frac{2}{n}\qquad\hbox{as}\quad k\rightarrow\infty, \] where $\mathrm{vol}\Omega$ is the volume of $\Omega$. In view of the above formula, P\'{o}lya conjectured that \[ \lambda_k\gs\frac{4\pi^2}{(\omega_n\mathrm{vol}\Omega)^\frac{2}{n}}k^\frac{2}{n}\qquad\hbox{for}\quad k\in\mathbb{N}. \] This is the well-known conjecture of P\'{o}lya. Studies on this topic have a long history with much work.In particular, one of the more remarkable achievements in recent tens years has been achieved by Li and Yau [Comm. Math. Phys. 88 (1983), 309--318]. They solved partially the conjecture of P\'{o}lya with a slight difference by a factor $n/(n+2)$. Here, following the argument of Li and Yau on the whole, we shall thoroughly solve the above conjecture.