Spectral asymptotics for sub-Riemannian Laplacians - Archive ouverte HAL Access content directly
Preprints, Working Papers, ... (Preprint) Year :

Spectral asymptotics for sub-Riemannian Laplacians

(1) , (2) , (3, 4)
1
2
3
4
Yves Colin de Verdìère
Luc Hillairet

Abstract

We study spectral properties of sub-Riemannian Laplacians, which are hypoelliptic operators. The main objective is to obtain quantum ergodicity results, what we have achieved in the 3D contact case. In the general case we study the small-time asymptotics of sub-Riemannian heat kernels. We prove that they are given by the nilpotentized heat kernel. In the equiregular case, we infer the local and microlocal Weyl law, putting in light the Weyl measure in sR geometry. This measure coincides with the Popp measure in low dimension but differs from it in general. We prove that spectral concentration occurs on the shief generated by Lie brackets of length r-1, where r is the degree of nonholonomy. In the singular case, like Martinet or Grushin, the situation is more involved but we obtain small-time asymptotic expansions of the heat kernel and the Weyl law in some cases. Finally, we give the Weyl law in the general singular case, under the assumption that the singular set is stratifiable.
Fichier principal
Vignette du fichier
weyl_sR.pdf (1.28 Mo) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

hal-03885610 , version 1 (05-12-2022)

Identifiers

Cite

Yves Colin de Verdìère, Luc Hillairet, Emmanuel Trélat. Spectral asymptotics for sub-Riemannian Laplacians. 2022. ⟨hal-03885610⟩
0 View
0 Download

Altmetric

Share

Gmail Facebook Twitter LinkedIn More