Year: 2026
Type: preprint
Abstract: Let \(N_M(λ)\) be the spectral counting function of the sub-Laplacian on the compact Heisenberg manifold \(M=Γ\backslash\mathbb H_d\), where $Γ$ is a lattice subgroup of the Heisenberg group $\mathbb H_d$. In 2016, Strichartz \cite[\textit{J. Geom. Anal.}]{Str16} proved the Weyl law with remainder \(R_M(λ)=N_M(λ)-A_d\operatorname{vol}(M)λ^{d+1} = O_M(λ^d\logλ)\), and conjectured the optimal remainder to be \(O_M(λ^d)\). In this work, we establish a new upper bound and the first two-sided lower bounds $$ R_M(λ)=O_M\!\left(λ^d(\logλ)^{2/3}\right), \qquad R_M(λ)=Ω_{M,\pm}\!\left(λ^d\log\logλ\right). $$ As a result, this implies that the sharp polynomial order is $d$, and disproves Strichartz's conjecture.
Source: arXiv (Cornell University)
Authors: Mao, Sheng-Chen, Zhang, Ye
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212087236