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.

@article{SLSEDP_2014-2015____A20_0,
     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},
     year = {2014-2015},
     doi = {10.5802/slsedp.78},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/slsedp.78/}
}
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/

[1] A. Agrachev, D. Barilari, U. Boscain, On the Hausdorff volume in sub-Riemannian geometry, Calc. Var. Partial Differential Equations 43 (2012), no. 3-4, 355–388. | MR 2875644 | Zbl 1236.53030

[2] A. Agrachev, U. Boscain, G. Charlot, R. Ghezzi, M. Sigalotti, Two-dimensional almost-Riemannian structures with tangency points, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), no. 3, 793–807. | Numdam | MR 2629880 | Zbl 1192.53029

[3] U. Boscain, C. Laurent, The Laplace-Beltrami operator in almost-Riemannian geometry, Ann. Institut Fourier (Grenoble) 63 (2013), 1739–1770. | Numdam | MR 3186507

[4] Y. Colin de Verdière, Calcul du spectre de certaines nilvariétés compactes de dimension 3, (French) [Calculation of the spectrum of some three-dimensional compact nilmanifolds], Séminaire de Théorie Spectrale et Géométrie (Grenoble) (1983–1984), no. 2, 1–6. | Numdam | MR 1046043 | Zbl 0738.53029

[5] Y. Colin de Verdière, Ergodicité et fonctions propres du laplacien, Commun. Math. Phys. 102 (1985), 497–502. | MR 818831 | Zbl 0592.58050

[6] Y. Colin de Verdière, L. Hillairet, E. Trélat, Spectral asymptotics for sub-Riemannian Laplacians. I: quantum ergodicity and quantum limits in the 3D contact case., arXiv:1504.07112 (2015), 41 pages.

[7] Y. Colin de Verdière, L. Hillairet, E. Trélat, Spectral asymptotics for sub-Riemannian Laplacians. II: microlocal Weyl measures, Work in progress.

[8] H. Duistermaat, V. Guillemin. The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975), 39–79. | MR 405514 | Zbl 0307.35071

[9] P. Gérard, E. Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Math. J. 71 (1993), no. 2, 559–607. | MR 1233448 | Zbl 0788.35103

[10] B. Helffer, A. Martinez, and D. Robert. Ergodicité et limite semi-classique, Commun. Math. Phys. 109 (1987), 313–326. | MR 880418 | Zbl 0624.58039

[11] L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. | MR 222474 | Zbl 0156.10701

[12] L. Hörmander. The spectral function of an elliptic operator, Acta Math. 121 (1968), 193–218. | MR 609014 | Zbl 0164.13201

[13] D. Jakobson, Y. Safarov, A. Strohmaier & Y. Colin de Verdière (Appendix), The semi-classical theory of discontinuous systems and ray-splitting billiards, American J. Math. (to appear).

[14] R.B. Melrose, The wave equation for a hypoelliptic operator with symplectic characteristics of codimension two, J. Analyse Math. 44 (1984-1985), 134–182. | MR 801291 | Zbl 0599.35139

[15] G. Métivier, Fonction spectrale et valeurs propres d’une classe d’opérateurs non elliptiques, Comm. Partial Differential Equations 1 (1976), no. 5, 467–519. | MR 427858 | Zbl 0376.35012

[16] R. Montgomery, Hearing the zero locus of a magnetic field, Commun. Math. Phys. 168 (1995), 651–675. | MR 1328258 | Zbl 0827.58076

[17] R. Montgomery, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs 91, American Mathematical Society, Providence, RI, 2002. | MR 1867362 | Zbl 1044.53022

[18] K. Petersen, Ergodic theory, Corrected reprint of the 1983 original. Cambridge Studies in Advanced Mathematics, 2. Cambridge University Press, Cambridge, 1989. xii+329 pp. | MR 1073173 | Zbl 0676.28008

[19] L.P. Rothschild, E.M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), 247–320. | MR 436223 | Zbl 0346.35030

[20] A.I. Shnirelman, Ergodic properties of eigenfunctions, Uspehi Mat. Nauk 29 (1974), 181–182. | MR 402834 | Zbl 0324.58020

[21] S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55 (1987), 919–941. | MR 916129 | Zbl 0643.58029

[22] S. Zelditch, Recent developments in mathematical quantum chaos, Current developments in mathematics, 2009, 115–204, Int. Press, Somerville, MA (2010). | MR 2757360 | Zbl 1223.37113

[23] S. Zelditch, M. Zworski, Ergodicity of eigenfunctions for ergodic billiards, Commun. Math. Phys. 175 (1996), no. 3, 673-682. | MR 1372814 | Zbl 0840.58048