Quantum ergodicity and quantum limits for sub-Riemannian Laplacians

Yves Colin de Verdière[1]; Luc Hillairet[2]; Emmanuel Trélat[3]

  • [1] Université de Grenoble, Institut Fourier, Unité mixte de recherche CNRS-UJF 5582 BP 74 38402-Saint Martin d’Hères Cedex France
  • [2] Université d’Orléans, Fédération Denis Poisson, Laboratoire MAPMO route de Chartres 45067 Orléans Cedex 2 France
  • [3] Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Institut Universitaire de France F-75005, Paris France

Séminaire Laurent Schwartz — EDP et applications (2014-2015)

  • page 1-17
  • ISSN: 2266-0607

Abstract

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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.

How to cite

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Colin de Verdière, Yves, Hillairet, Luc, and Trélat, Emmanuel. "Quantum ergodicity and quantum limits for sub-Riemannian Laplacians." Séminaire Laurent Schwartz — EDP et applications (2014-2015): 1-17. <http://eudml.org/doc/275752>.

@article{ColindeVerdière2014-2015,
abstract = {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.},
affiliation = {Université de Grenoble, Institut Fourier, Unité mixte de recherche CNRS-UJF 5582 BP 74 38402-Saint Martin d’Hères Cedex France; Université d’Orléans, Fédération Denis Poisson, Laboratoire MAPMO route de Chartres 45067 Orléans Cedex 2 France; Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Institut Universitaire de France F-75005, Paris France},
author = {Colin de Verdière, Yves, Hillairet, Luc, Trélat, Emmanuel},
journal = {Séminaire Laurent Schwartz — EDP et applications},
language = {eng},
pages = {1-17},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Quantum ergodicity and quantum limits for sub-Riemannian Laplacians},
url = {http://eudml.org/doc/275752},
year = {2014-2015},
}

TY - JOUR
AU - Colin de Verdière, Yves
AU - Hillairet, Luc
AU - Trélat, Emmanuel
TI - Quantum ergodicity and quantum limits for sub-Riemannian Laplacians
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2014-2015
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
SP - 1
EP - 17
AB - 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.
LA - eng
UR - http://eudml.org/doc/275752
ER -

References

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