P\'{o}lya's conjecture for Dirichlet eigenvalues of annuli

TL;DR

This paper proves Pólya's conjecture for Dirichlet eigenvalues on annuli, extending previous methods from disks and balls, and introduces improved bounds for eigenvalue counting functions.

Contribution

The authors extend Pólya's conjecture proof to annular domains using advanced analytical and computational techniques, including lattice point counting and Bessel function estimates.

Findings

Proved Pólya's conjecture for annuli.

Derived a two-term upper bound for the eigenvalue counting function of the disk.

Enhanced bounds for Dirichlet eigenvalues using computer-assisted analysis.

Abstract

We prove P\'olya's conjecture for the eigenvalues of the Dirichlet Laplacian on annular domains. Our approach builds upon and extends the methods we previously developed for disks and balls. It combines variational bounds, estimates of Bessel phase functions, refined lattice point counting techniques, and a rigorous computer-assisted analysis. As a by-product, we also derive a two-term upper bound for the Dirichlet eigenvalue counting function of the disk, improving upon P\'olya's original estimate.