P\'{o}lya's conjecture for Dirichlet eigenvalues of annuli
Spectral Theory
2026-02-10 v2 Number Theory
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.
Cite
@article{arxiv.2505.21737,
title = {P\'{o}lya's conjecture for Dirichlet eigenvalues of annuli},
author = {Nikolay Filonov and Michael Levitin and Iosif Polterovich and David A. Sher},
journal= {arXiv preprint arXiv:2505.21737},
year = {2026}
}
Comments
28 pages, 12 figures; the final published version