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 $\Gamma$ 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 $$R_M(\lambda )=O_M\left({\lambda }^d(\log \lambda {)}^{2/3}\right),R_M(\lambda )={\Omega }_{M,\pm }\left({\lambda }^d\log \log \lambda \right).$$ As a result, this implies that the sharp polynomial order is $d$, and disproves Strichartz's conjecture.

19 pages