PreprintCC BY-NC-ND 4.0arXiv
Spectral asymptotics of sub-Riemannian Laplacians on compact Heisenberg manifolds
Abstract
Let NM(λ) be the spectral counting function of the sub-Laplacian on the compact Heisenberg manifold M=Γbackslashℍd, where Γ is a lattice subgroup of the Heisenberg group ℍd. In 2016, Strichartz \cite[\textit{J. Geom. Anal.}]{Str16} proved the Weyl law with remainder RM(λ)=NM(λ)−Advol(M)λd+1 = OM(λdlogλ), and conjectured the optimal remainder to be OM(λd). In this work, we establish a new upper bound and the first two-sided lower bounds RM(λ)=OM (λd(logλ)2/3), RM(λ)=ΩM,± (λdloglogλ). As a result, this implies that the sharp polynomial order is d, and disproves Strichartz's conjecture.
Related
- Lp Convergence of AMV Laplacians on completions of Riemannian manifolds
- On Polynomial Complex Hessian Equations I: Non-Linear Non-Spectral Elliptic Hessian Equations on Riemannian Manifolds
- The pseudo-Riemannian spectral zeta function density on scattering manifolds
- Sub-Riemannian Selberg Trace Formulae for Compact Quotients of SL(2,R) and Determinants of Sub-Laplacians
- Robin Laplacians with point interactions on unbounded domains: discrete spectrum and coupling asymptotics
BibTeX
@misc{mao2026spectral,
title = {{Spectral asymptotics of sub-Riemannian Laplacians on compact Heisenberg manifolds}},
author = {Sheng-Chen Mao and Ye Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08309},
eprint = {2609.08309},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08309},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Spectral asymptotics of sub-Riemannian Laplacians on compact Heisenberg manifolds
AU - Sheng-Chen Mao
AU - Ye Zhang
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08309
UR - https://arxiv.org/abs/2609.08309
AB - Let N_M(λ) be the spectral counting function of the sub-Laplacian on the compact Heisenberg manifold M=Γbackslashℍ_d, where Γ is a lattice subgroup of the Heisenberg group ℍ_d. In 2016, Strichartz \cite[\textit{J. Geom. Anal.}]{Str16} proved the Weyl law with remainder R_M(λ)=N_M(λ)−A_dvol(M)λ^d+1 = O_M(λ^dlogλ), 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 (λ^d(logλ)^2/3), R_M(λ)=Ω_M,± (λ^dloglogλ). As a result, this implies that the sharp polynomial order is d, and disproves Strichartz's conjecture.
ER -