弗拉基米罗夫-泰布勒松算子的魏尔定律与波利亚猜想
Weyl's law and Pólya's conjecture for the Vladimirov-Taibleson operator
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中文总结 AI 辅助
本文针对p进情形下的弗拉基米罗夫-泰布勒松算子，证明狄利克雷算子的魏尔定律、推导其余项精确估计，证明魏尔-贝里猜想不成立，还给出波利亚猜想成立的几何充要条件。
中文摘要 AI 辅助
本文研究p进情形下谱几何的若干基础问题。将弗拉基米罗夫-泰布勒松算子$D^\alpha$视为阿基米德情形下分数阶拉普拉斯算子$(-\Delta)^{\frac{\alpha}{2}}$的p进对应物，我们证明了狄利克雷算子的魏尔定律，并在勒贝格渐近密度为零的例外集外建立了诺伊曼算子的魏尔定律。对狄利克雷算子，我们推导了余项的精确估计并证明魏尔-贝里猜想不成立；此外，我们证明波利亚猜想一般不成立，并给出其成立的几何充要条件。
英文摘要
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 $(-Δ)^{\fracα{2}}$ 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.
