On the Pólya conjecture for circular sectors and for balls

Abstract

In 1954, G. Polya conjectured that the counting function $N(\Omega ,\Lambda )$ of the eigenvalues of the Laplace operator of the Dirichlet (resp. Neumann) boundary value problem in a bounded set $\Omega \subset R^d$ is lesser (resp. greater) than $(2\pi {)}^{-d}{\omega }_d|\Omega |{\Lambda }^{d/2}$. Here $\Lambda$ is the spectral parameter, and ${\omega }_d$ is the volume of the unit ball. We prove this conjecture for both Dirichlet and Neumann boundary problems for any circular sector, and for the Dirichlet problem for a ball of arbitrary dimension. We heavily use the ideas from \cite{LPS}.

merged into "Pólya's conjecture for Euclidean balls" by N. Filonov, M. Levitin, I. Polterovich, D. Sher, arXiv:2203.07696v3