P\'olya's conjecture for Euclidean balls

TL;DR

This paper proves Pólya's conjecture for the disk and certain planar sectors, and for balls in any dimension, using analytic methods and computer-assisted proofs, advancing understanding in spectral geometry.

Contribution

It verifies Pólya's conjecture for the first non-tiling planar domain (the disk) and extends results to arbitrary planar sectors and higher-dimensional balls.

Findings

Pólya's conjecture holds for the disk and planar sectors.

The conjecture is confirmed for balls of any dimension in the Dirichlet case.

A new link between spectral problems and lattice counting is established.

Abstract

The celebrated P\'{o}lya's conjecture (1954) in spectral geometry states that the eigenvalue counting functions of the Dirichlet and Neumann Laplacian on a bounded Euclidean domain can be estimated from above and below, respectively, by the leading term of Weyl's asymptotics. P\'{o}lya's conjecture is known to be true for domains which tile Euclidean space, and, in addition, for some special domains in higher dimensions. In this paper, we prove P\'{o}lya's conjecture for the disk, making it the first non-tiling planar domain for which the conjecture is verified. We also confirm P\'{o}lya's conjecture for arbitrary planar sectors, and, in the Dirichlet case, for balls of any dimension. Along the way, we develop the known links between the spectral problems in the disk and certain lattice counting problems. A key novel ingredient is the observation, made in recent work of the last named author, that the corresponding eigenvalue and lattice counting functions are related not only asymptotically, but in fact satisfy certain uniform bounds. Our proofs are purely analytic, except for a rigorous computer-assisted argument needed to cover the short interval of values of the spectral parameter in the case of the Neumann problem in the disk.