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Weyl's law and Pólya's conjecture for the Vladimirov-Taibleson operator

  • Yaojia Sun
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Abstract

This paper studies some fundamental problems in spectral geometry in the p-adic setting. By viewing the Vladimirov-Taibleson operator Dα as the p-adic counterpart of the fractional Laplacian (−Δ)α/2 in the Archimedean setting, we prove Weyl's law for the Dirichlet operator and establish it for the Neumann operator outside an exceptional set with zero Lebesgue asymptotic density. For the Dirichlet operator, we derive the sharp estimate for the remainder and prove that the Weyl-Berry conjecture fails. Furthermore, we show that Pólya's conjecture fails in general, and we give geometric necessary and sufficient conditions for it to hold.

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Cite

BibTeXRIS
BibTeX
@misc{sun2026weyls,
  title = {{Weyl's law and Pólya's conjecture for the Vladimirov-Taibleson operator}},
  author = {Yaojia Sun},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.26726},
  eprint = {2608.26726},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.26726},
}
RIS
TY  - UNPB
TI  - Weyl's law and Pólya's conjecture for the Vladimirov-Taibleson operator
AU  - Yaojia Sun
PY  - 2026
DA  - 2026/08/30
PB  - arXiv
DO  - 10.48550/arxiv.2608.26726
UR  - https://arxiv.org/abs/2608.26726
AB  - This paper studies some fundamental problems in spectral geometry in the p-adic setting. By viewing the Vladimirov-Taibleson operator D^α as the p-adic counterpart of the fractional Laplacian (−Δ)^α/2 in the Archimedean setting, we prove Weyl's law for the Dirichlet operator and establish it for the Neumann operator outside an exceptional set with zero Lebesgue asymptotic density. For the Dirichlet operator, we derive the sharp estimate for the remainder and prove that the Weyl-Berry conjecture fails. Furthermore, we show that Pólya's conjecture fails in general, and we give geometric necessary and sufficient conditions for it to hold.
ER  -