Quantum ergodicity and quantum limits for sub-Riemannian Laplacians
Séminaire Laurent Schwartz — EDP et applications (2014-2015), Exposé no. 20, 17 p.

This paper is a proceedings version of [6], in which we state a Quantum Ergodicity (QE) theorem on a 3D contact manifold, and in which we establish some properties of the Quantum Limits (QL).

We consider a sub-Riemannian (sR) metric on a compact 3D manifold with an oriented contact distribution. There exists a privileged choice of the contact form, with an associated Reeb vector field and a canonical volume form that coincides with the Popp measure. We state a QE theorem for the eigenfunctions of any associated sR Laplacian, under the assumption that the Reeb flow is ergodic. The limit measure is given by the normalized canonical contact measure. To our knowledge, this is the first extension of the usual Schnirelman theorem to a hypoelliptic operator. We provide as well a decomposition result of QL’s, which is valid without any ergodicity assumption. We explain the main steps of the proof, and we discuss possible extensions to other sR geometries.

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author = {Colin de Verdi\ere, Yves and Hillairet, Luc and Tr\'elat, Emmanuel},
title = {Quantum ergodicity and quantum limits for {sub-Riemannian} {Laplacians}},
journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
note = {talk:20},
publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
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Colin de Verdière, Yves; Hillairet, Luc; Trélat, Emmanuel. Quantum ergodicity and quantum limits for sub-Riemannian Laplacians. Séminaire Laurent Schwartz — EDP et applications (2014-2015), Exposé no. 20, 17 p. doi : 10.5802/slsedp.78. http://archive.numdam.org/articles/10.5802/slsedp.78/`

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