Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold

Nalini Anantharaman[1]; Stéphane Nonnenmacher[2]

  • [1] tabacckludge ’Ecole Normale Supérieure Unité de Mathématiques Pures et Appliquées 6, allée d’Italie 69364 LYON Cedex 07 (France)
  • [2] CEA/DSM/PhT Service de Physique Théorique Unité de recherche associé CNRS CEA/Saclay 91191 Gif-sur-Yvette (France)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 7, page 2465-2523
  • ISSN: 0373-0956

Abstract

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We study the high-energy eigenfunctions of the Laplacian on a compact Riemannian manifold with Anosov geodesic flow. The localization of a semiclassical measure associated with a sequence of eigenfunctions is characterized by the Kolmogorov-Sinai entropy of this measure. We show that this entropy is necessarily bounded from below by a constant which, in the case of constant negative curvature, equals half the maximal entropy. In this sense, high-energy eigenfunctions are at least half-delocalized.

How to cite

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Anantharaman, Nalini, and Nonnenmacher, Stéphane. "Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold." Annales de l’institut Fourier 57.7 (2007): 2465-2523. <http://eudml.org/doc/10304>.

@article{Anantharaman2007,
abstract = {We study the high-energy eigenfunctions of the Laplacian on a compact Riemannian manifold with Anosov geodesic flow. The localization of a semiclassical measure associated with a sequence of eigenfunctions is characterized by the Kolmogorov-Sinai entropy of this measure. We show that this entropy is necessarily bounded from below by a constant which, in the case of constant negative curvature, equals half the maximal entropy. In this sense, high-energy eigenfunctions are at least half-delocalized.},
affiliation = {tabacckludge ’Ecole Normale Supérieure Unité de Mathématiques Pures et Appliquées 6, allée d’Italie 69364 LYON Cedex 07 (France); CEA/DSM/PhT Service de Physique Théorique Unité de recherche associé CNRS CEA/Saclay 91191 Gif-sur-Yvette (France)},
author = {Anantharaman, Nalini, Nonnenmacher, Stéphane},
journal = {Annales de l’institut Fourier},
keywords = {Quantum chaos; semiclassical measure; ergodic theory; entropy; Anosov flows; quantum chaos},
language = {eng},
number = {7},
pages = {2465-2523},
publisher = {Association des Annales de l’institut Fourier},
title = {Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold},
url = {http://eudml.org/doc/10304},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Anantharaman, Nalini
AU - Nonnenmacher, Stéphane
TI - Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 7
SP - 2465
EP - 2523
AB - We study the high-energy eigenfunctions of the Laplacian on a compact Riemannian manifold with Anosov geodesic flow. The localization of a semiclassical measure associated with a sequence of eigenfunctions is characterized by the Kolmogorov-Sinai entropy of this measure. We show that this entropy is necessarily bounded from below by a constant which, in the case of constant negative curvature, equals half the maximal entropy. In this sense, high-energy eigenfunctions are at least half-delocalized.
LA - eng
KW - Quantum chaos; semiclassical measure; ergodic theory; entropy; Anosov flows; quantum chaos
UR - http://eudml.org/doc/10304
ER -

References

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