The semi-classical ergodic Theorem for discontinuous metrics
Séminaire de théorie spectrale et géométrie, Tome 31 (2012-2014), pp. 71-89.

In this paper, we present an extension of the classical Quantum ergodicity Theorem, due to Shnirelman, to the case of Laplacians with discontinous metrics along interfaces. The “geodesic flow” is then no more a flow, but a Markov process due to the fact that rays can by reflected or refracted at the interfaces. We give also an example build by gluing together two flat Euclidean disks.

DOI : 10.5802/tsg.295
Colin de Verdière, Yves 1

1 Université de Grenoble, Institut Fourier UMR CNRS-UJF 5582 BP 74 38402-Saint Martin d’Hères Cedex (France)
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Colin de Verdière, Yves. The semi-classical ergodic Theorem for discontinuous metrics. Séminaire de théorie spectrale et géométrie, Tome 31 (2012-2014), pp. 71-89. doi : 10.5802/tsg.295. http://www.numdam.org/articles/10.5802/tsg.295/

[1] Abraham, Ralph Transversality in manifolds of mappings, Bull. Amer. Math. Soc., Volume 69 (1963), pp. 470-474 | MR | Zbl

[2] Abraham, Ralph; Robbin, Joel Transversal mappings and flows, An appendix by Al Kelley, W. A. Benjamin, Inc., New York-Amsterdam, 1967, pp. x+161 | MR | Zbl

[3] Bal, Guillaume Kinetics of scalar wave fields in random media, Wave Motion, Volume 43 (2005) no. 2, pp. 132-157 | DOI | MR | Zbl

[4] Berkolaiko, Gregory; Keating, Jon; Winn, Brian No quantum ergodicity for star graphs, Comm. Math. Phys., Volume 250 (2004) no. 2, pp. 259-285 | DOI | MR | Zbl

[5] Chazarain, Jacques Construction de la paramétrix du problème mixte hyperbolique pour l’équation des ondes, C. R. Acad. Sci. Paris Sér. A-B, Volume 276 (1973), p. A1213-A1215 | MR | Zbl

[6] Duistermaat, Hans; Guillemin, Victor The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math., Volume 29 (1975) no. 1, pp. 39-79 | MR | Zbl

[7] Dunford, Nelson; Schwartz, Jacob T. Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988, pp. xiv+858 (General theory, With the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1958 original, A Wiley-Interscience Publication) | MR | Zbl

[8] Gérard, Patrick; Leichtnam, Éric Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Math. J., Volume 71 (1993) no. 2, pp. 559-607 | DOI | MR | Zbl

[9] Jakobson, Dmitry; Safarov, Yuri; Strohmaier, Alexander; Colin de Verdière (Appendix), Yves The semi-classical theory of discontinuous systems and ray-splitting billiards (2015) (To appear in Amer. J. Math.)

[10] Krengel, Ulrich Ergodic theorems, de Gruyter Studies in Mathematics, 6, Walter de Gruyter & Co., Berlin, 1985, pp. viii+357 (With a supplement by Antoine Brunel) | DOI | MR | Zbl

[11] Shnirelman, Alexander Ergodic properties of eigenfunctions, Uspehi Mat. Nauk, Volume 29 (1974) no. 6(180), pp. 181-182 | MR | Zbl

[12] Thom, René Un lemme sur les applications différentiables, Bol. Soc. Mat. Mexicana (2), Volume 1 (1956), pp. 59-71 | MR | Zbl

[13] Colin de Verdière, Yves Ergodicité et fonctions propres du laplacien, Comm. Math. Phys., Volume 102 (1985) no. 3, pp. 497-502 http://projecteuclid.org/euclid.cmp/1104114465 | Zbl

[14] Colin de Verdière, Yves Semi-classical measures on quantum graphs and the Gauß map of the determinant manifold, Ann. Henri Poincaré, Volume 16 (2015) no. 2, pp. 347-364 | DOI | MR

[15] Colin de Verdière, Yves; Hillairet, Luc; Trélat, Emmanuel Quantum ergodicity for sub-Riemannian Laplacians. I: the contact 3D case (2015) (http://arxiv.org/abs/1504.07112)

[16] Zelditch, Steven Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J., Volume 55 (1987) no. 4, pp. 919-941 | DOI | MR | Zbl

[17] Zelditch, Steven; Zworski, Maciej Ergodicity of eigenfunctions for ergodic billiards, Comm. Math. Phys., Volume 175 (1996) no. 3, pp. 673-682 http://projecteuclid.org/euclid.cmp/1104276097 | MR | Zbl

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