Improvement of P\'{o}lya's conjecture for balls and cylinders

TL;DR

This paper advances Pólya's conjecture for eigenvalues of the Laplacian on balls and cylinders by refining analytical methods, extending proven regimes, and confirming conjectures in specific geometries, including in three dimensions.

Contribution

It refines analytical proofs for Pólya's conjecture on disks, extends the conjecture's validity, and confirms the Neumann case for cylinders in three dimensions.

Findings

Improved bounds for Pólya's conjecture on disks and balls.

Confirmed Neumann Pólya's conjecture for cylinders in 3D.

Extended the spectral parameter regime without computer assistance.

Abstract

P\'{o}lya's conjecture on the eigenvalues of the Laplacian has been one of the core problems in spectral geometry. Building upon the recent breakthrough works on P\'{o}lya's conjecture for balls and annuli by Filonov, Levitin, Polterovich and Sher, we study several aspects of P\'{o}lya's conjecture for balls and cylinders: by refining the purely analytical portion of the proof in [2] for the Neumann P\'{o}lya's conjecture for the disk, we extend the regime of the spectral parameter that can be established without computer assistance; we obtain improvement of P\'{o}lya's conjecture for disks and balls; we obtain improvement of P\'{o}lya's conjecture for cylinders and confirm the Neumann P\'{o}lya's conjecture for cylinders in $\mathbb{R}^3$. As a supplementary effort, we study Weyl's law for cylinders.