The semi-classical ergodic Theorem for discontinuous metrics

Yves Colin de Verdière[1]

  • [1] Université de Grenoble, Institut Fourier UMR CNRS-UJF 5582 BP 74 38402-Saint Martin d’Hères Cedex (France)

Séminaire de théorie spectrale et géométrie (2012-2014)

  • Volume: 31, page 71-89
  • ISSN: 1624-5458

Abstract

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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.

How to cite

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Colin de Verdière, Yves. "The semi-classical ergodic Theorem for discontinuous metrics." Séminaire de théorie spectrale et géométrie 31 (2012-2014): 71-89. <http://eudml.org/doc/275763>.

@article{ColindeVerdière2012-2014,
abstract = {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.},
affiliation = {Université de Grenoble, Institut Fourier UMR CNRS-UJF 5582 BP 74 38402-Saint Martin d’Hères Cedex (France)},
author = {Colin de Verdière, Yves},
journal = {Séminaire de théorie spectrale et géométrie},
language = {eng},
pages = {71-89},
publisher = {Institut Fourier},
title = {The semi-classical ergodic Theorem for discontinuous metrics},
url = {http://eudml.org/doc/275763},
volume = {31},
year = {2012-2014},
}

TY - JOUR
AU - Colin de Verdière, Yves
TI - The semi-classical ergodic Theorem for discontinuous metrics
JO - Séminaire de théorie spectrale et géométrie
PY - 2012-2014
PB - Institut Fourier
VL - 31
SP - 71
EP - 89
AB - 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.
LA - eng
UR - http://eudml.org/doc/275763
ER -

References

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  12. René Thom, Un lemme sur les applications différentiables, Bol. Soc. Mat. Mexicana (2) 1 (1956), 59-71 Zbl0075.32201MR102115
  13. Yves Colin de Verdière, Ergodicité et fonctions propres du laplacien, Comm. Math. Phys. 102 (1985), 497-502 Zbl0592.58050
  14. Yves Colin de Verdière, Semi-classical measures on quantum graphs and the Gauß map of the determinant manifold, Ann. Henri Poincaré 16 (2015), 347-364 Zbl1306.81043MR3302601
  15. Yves Colin de Verdière, Luc Hillairet, Emmanuel Trélat, Quantum ergodicity for sub-Riemannian Laplacians. I: the contact 3D case, (2015) 
  16. Steven Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55 (1987), 919-941 Zbl0643.58029MR916129
  17. Steven Zelditch, Maciej Zworski, Ergodicity of eigenfunctions for ergodic billiards, Comm. Math. Phys. 175 (1996), 673-682 Zbl0840.58048MR1372814

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