Spectral asymptotics of sub-Riemannian Laplacians on compact Heisenberg manifolds

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.