analysis / Spectral Geometry, Laplace Eigenvalues
Pólya conjectured in 1954 that the Weyl-law expression bounds the eigenvalue counting function of the Laplacian. The paper proves the Neumann case for Euclidean balls in dimensions three and higher, extending the authors' earlier planar and Dirichlet results. Key difficulty: estimating zeros of derivatives of ultraspherical Bessel functions rather than of Bessel functions themselves.
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Append-only history
The ball case. For arbitrary domains Pólya's conjecture remains open; this continues the authors' programme after the planar disk, circular sectors, and the Dirichlet case in arbitrary dimensions.
Research memory
Pólya conjectured in 1954 that the Weyl-law expression bounds the eigenvalue counting function of the Laplacian. The paper proves the Neumann case for Euclidean balls in dimensions three and higher, extending the authors' earlier planar and Dirichlet results. Key difficulty: estimating zeros of derivatives of ultraspherical Bessel functions rather than of Bessel functions themselves.
The ball case. For arbitrary domains Pólya's conjecture remains open; this continues the authors' programme after the planar disk, circular sectors, and the Dirichlet case in arbitrary dimensions.
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