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PreprintCC0arXiv

Pólya's conjecture for Neumann eigenvalues of annuli

  • Guowei Dai
  • Yingxin Sun
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Abstract

We prove strict Pólya inequalities for every positive Neumann eigenvalue of every Euclidean annulus. The proof preserves the exact Weyl area term through a radial action identity and combines variational bounds with uniform Bessel-phase and angular-sum estimates. Explicit polynomial positivity arguments and angular-layer comparisons cover the remaining finite spectral ranges. We also obtain a positive counting surplus uniform in the radius ratio, an explicit subtraction of order √k/(R2−r2) for all k≥1604, and a positive square-root correction over the entire spectrum with a universal, non-explicit coefficient. The square-root order is sharp, and the first positive pair and the low-energy count admit optimal constants expressed by the first positive zero of J1'. Neumann inequalities for annular products and congruent annular sectors follow by established transfer principles.

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Cite

BibTeXRIS
BibTeX
@misc{dai2026pólyas,
  title = {{Pólya's conjecture for Neumann eigenvalues of annuli}},
  author = {Guowei Dai and Yingxin Sun},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.05146},
  eprint = {2610.05146},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.05146},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - Pólya's conjecture for Neumann eigenvalues of annuli
AU  - Guowei Dai
AU  - Yingxin Sun
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.05146
UR  - https://arxiv.org/abs/2610.05146
AB  - We prove strict Pólya inequalities for every positive Neumann eigenvalue of every Euclidean annulus. The proof preserves the exact Weyl area term through a radial action identity and combines variational bounds with uniform Bessel-phase and angular-sum estimates. Explicit polynomial positivity arguments and angular-layer comparisons cover the remaining finite spectral ranges. We also obtain a positive counting surplus uniform in the radius ratio, an explicit subtraction of order √k/(R^2−r^2) for all k≥1604, and a positive square-root correction over the entire spectrum with a universal, non-explicit coefficient. The square-root order is sharp, and the first positive pair and the low-energy count admit optimal constants expressed by the first positive zero of J_1'. Neumann inequalities for annular products and congruent annular sectors follow by established transfer principles.
ER  -