Pólya's conjecture for higher-dimensional Neumann balls
Abstract
We prove Pólya's conjecture for the Neumann eigenvalues of the Laplacian on Euclidean balls in dimensions three and higher. The proof further develops the approach introduced in our earlier work on the two-dimensional case and on Dirichlet eigenvalues in arbitrary dimensions. The main difficulty in the higher dimensional Neumann case is that one has to estimate zeros of the derivatives of ultraspherical Bessel functions, rather than of the usual Bessel functions. For low-lying eigenvalues, we use variational estimates involving dimension-dependent test functions, which is a novel ingredient allowing us to control a larger dimension-scaled frequency range. Other components of the proof include phase-function bounds, lattice-point counting techniques, and computer-assisted arguments.
Related
- New Qualitative and Quantitative Estimates for Neumann Eigenvalues toward Pólya's Conjecture
- Pólya's conjecture for Neumann eigenvalues of spherical shells
- Pólya's conjecture for Neumann eigenvalues of annuli
- Sharp Harmonic-Mean Bounds for Neumann Eigenvalues on Riemannian Surfaces and Counterexamples to Geodesic Ball Optimality on Higher-Dimensional Spheres
- Von Neumann Type Inequalities for NC Operator Balls
BibTeX
@misc{filonov2026pólyas,
title = {{Pólya's conjecture for higher-dimensional Neumann balls}},
author = {Nikolay Filonov and Michael Levitin and Iosif Polterovich and David A. Sher},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.29305},
eprint = {2607.29305},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.29305},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Pólya's conjecture for higher-dimensional Neumann balls
AU - Nikolay Filonov
AU - Michael Levitin
AU - Iosif Polterovich
AU - David A. Sher
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2607.29305
UR - https://arxiv.org/abs/2607.29305
AB - We prove Pólya's conjecture for the Neumann eigenvalues of the Laplacian on Euclidean balls in dimensions three and higher. The proof further develops the approach introduced in our earlier work on the two-dimensional case and on Dirichlet eigenvalues in arbitrary dimensions. The main difficulty in the higher dimensional Neumann case is that one has to estimate zeros of the derivatives of ultraspherical Bessel functions, rather than of the usual Bessel functions. For low-lying eigenvalues, we use variational estimates involving dimension-dependent test functions, which is a novel ingredient allowing us to control a larger dimension-scaled frequency range. Other components of the proof include phase-function bounds, lattice-point counting techniques, and computer-assisted arguments.
ER -