Weyl's law and P\'olya's conjecture for the Vladimirov-Taibleson operator

Abstract

This paper studies some fundamental problems in spectral geometry in the $p$-adic setting. By viewing the Vladimirov-Taibleson operator $D^{\alpha}$ as the $p$-adic counterpart of the fractional Laplacian $(-\Delta)^{\frac{\alpha}{2}}$ 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 P\'olya's conjecture fails in general, and we give geometric necessary and sufficient conditions for it to hold.