Spectral asymptotics of sub-Riemannian Laplacians on compact Heisenberg manifolds

Abstract

Let \(N_M(\lambda)\) be the spectral counting function of the sub-Laplacian on the compact Heisenberg manifold \(M=\Gamma\backslash\mathbb H_d\), 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 \(R_M(\lambda)=N_M(\lambda)-A_d\operatorname{vol}(M)\lambda^{d+1} = O_M(\lambda^d\log\lambda)\), and conjectured the optimal remainder to be \(O_M(\lambda^d)\). In this work, we establish a new upper bound and the first two-sided lower bounds $$ R_M(\lambda)=O_M\!\left(\lambda^d(\log\lambda)^{2/3}\right), \qquad R_M(\lambda)=\Omega_{M,\pm}\!\left(\lambda^d\log\log\lambda\right). $$ As a result, this implies that the sharp polynomial order is $d$, and disproves Strichartz's conjecture.