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Quantum ergodicity and quantum limits for sub-Riemannian Laplacians
Yves Colin de Verdière; Luc Hillairet; Emmanuel Trélat
Séminaire Laurent Schwartz — EDP et applications (2014-2015), Talk no. 20, 17 p.
  • 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.

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Published online: 2015-12-15
DOI: 10.5802/slsedp.78
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@article{SLSEDP_2014-2015____A20_0,
     author = {Yves Colin de Verdi\`ere and Luc Hillairet and Emmanuel Tr\'elat},
     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 = {https://slsedp.centre-mersenne.org/articles/10.5802/slsedp.78/}
}
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PB  - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - https://slsedp.centre-mersenne.org/articles/10.5802/slsedp.78/
UR  - https://doi.org/10.5802/slsedp.78
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%J Séminaire Laurent Schwartz — EDP et applications
%Z talk:20
%D 2014-2015
%I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
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%F SLSEDP_2014-2015____A20_0
Yves Colin de Verdière; Luc Hillairet; Emmanuel Trélat. Quantum ergodicity and quantum limits for sub-Riemannian Laplacians. Séminaire Laurent Schwartz — EDP et applications (2014-2015), Talk no. 20, 17 p. doi : 10.5802/slsedp.78. https://slsedp.centre-mersenne.org/articles/10.5802/slsedp.78/
  • References
  • Cited by

[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.

[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.

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

[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.

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

[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.

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

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

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

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

[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.

[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.

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

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

[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.

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

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

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

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

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

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