2026/07/31 by Nikolay Filonov, Michael Levitin, Iosif Polterovich +1
Mathematics · #math.SP #msc:35P15. #msc:35P20 #msc:33C10 #msc:11P21
39 pages, 7 figures. Note: an alternative proof of the main result, independently obtained by Yutian Li, has been posted in arXiv:2607.25958 on 28 July 2026
arxiv created 2026/07/31 · arxiv updated 2026/08/03
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.