Pólya's conjecture for Dirichlet eigenvalues of annuli

Filonov, N., Levitin, M. ORCID: https://orcid.org/0000-0003-0020-3265, Polterovich, I. and Sher, D. A. (2026) Pólya's conjecture for Dirichlet eigenvalues of annuli. Journal of the London Mathematical Society, 113 (2). e70425. ISSN 0024-6107 doi: 10.1112/jlms.70425

Abstract/Summary

We prove Pólya'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 Pólya's original estimate.