Pólya's conjecture and the Weyl–Berry conjecture for the Vladimirov–Taibleson operator

Let $\Omega\subset\mathbb{Q}_p^d$ be an admissible bounded domain, let $D_{\Omega}^{\alpha}$ denote the Dirichlet realization of the Vladimirov–Taibleson operator, and let $N_{\Omega,D}^{\alpha}(\lambda)$ be its eigenvalue-counting function. If $W_{\Omega}^{\alpha}(\lambda)=C_{\Omega,p,d,\alpha}\lambda^{d/\alpha}$ denotes the Weyl main term, Pólya's conjecture asks whether $N_{\Omega,D}^{\alpha}(\lambda)\leq W_{\Omega}^{\alpha}(\lambda)$ for every $\lambda>0$ and every such domain $\Omega$ ; equivalently, whether the corresponding Dirichlet eigenvalues satisfy the sharp lower bound prescribed by Weyl's law.

References

Primary source

arXiv

Additional references

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

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new unrefereed paper reports that the p-adic analogues fail in general, while identifying the cases where a Pólya-type inequality survives.

The problem concerns Pólya's conjecture and the Weyl–Berry conjecture for the Vladimirov–Taibleson operator, contrasting these p-adic questions with their Archimedean counterparts.

August 2026 preprint

A new arXiv paper reports remainder estimates, proves failure of the Weyl–Berry conjecture for this operator, and characterizes the cases in which Pólya-type inequalities hold. These claims amount to a resolution of the tracked conjectures in the stated p-adic setting, but the preprint is unrefereed and independently unconfirmed.

Current status (as of August 2026): The preprint claims to settle the Weyl–Berry question negatively and to characterize Pólya-type inequalities, but these results remain unverified.

Sources

No solutions have been posted yet.