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De Verdière, Yves Colin; Hillairet, Luc; Trélat, Emmanuel

Spectral asymptotics for sub-Riemannian Laplacians. I: Quantum ergodicity and quantum limits in the 3-dimensional contact case. (English) Zbl 1388.35137

Duke Math. J. 167, No. 1, 109-174 (2018).
The authors study the spectral asymptotics for a sub-Riemannian Laplacian on a closed 3-dimensional manifold with an oriented contact distribution. They prove a quantum ergodicity theorem for the eigenfunctions of any associated sub-Riemannian Laplacian assuming that the Reeb flow is ergodic. A second main result establishes a decomposition of any quantum limit in a sum of two mutually singular measures for which additional geometric properties are available. It is worth noticing that this second main result holds true without any ergodicity assumption.
Reviewer: Dumitru Motreanu (Juiz de Fora)

Cited in 29 Documents

MSC:

35P20 Asymptotic distributions of eigenvalues in context of PDEs
53D10 Contact manifolds (general theory)
53D25 Geodesic flows in symplectic geometry and contact geometry

Keywords:

sub-Riemannian Laplacian; contact manifolds; quantum ergodicity
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References:

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