Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold
[Demi-délocalisation des fonctions propres du laplacien sur une variété d’Anosov]
Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2465-2523.

Nous étudions la limite de haute énergie pour les fonctions propres du laplacien, sur une variété riemannienne compacte dont le flot géodésique est d’Anosov. La localisation d’une mesure semiclassique associée à une suite de fonctions propres peut être mesurée par son entropie de Kolmogorov-Sinai. Nous obtenons pour cette entropie une borne inférieure qui, dans le cas des variétés à courbure négative constante, vaut la moitié de l’entropie maximale. En ce sens, on peut dire que les fonctions propres de haute énergie sont au moins à demi délocalisées.

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.

DOI : 10.5802/aif.2340
Classification : 81Q50, 35Q40, 35P20, 37D40, 58J40, 28D20
Keywords: Quantum chaos, semiclassical measure, ergodic theory, entropy, Anosov flows
Mot clés : chaos quantique, mesure semiclassique, théorie ergodique, entropie, flots d’Anosov
Anantharaman, Nalini 1 ; Nonnenmacher, Stéphane 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)
@article{AIF_2007__57_7_2465_0,
     author = {Anantharaman, Nalini and Nonnenmacher, St\'ephane},
     title = {Half-delocalization of eigenfunctions for the {Laplacian} on an {Anosov} manifold},
     journal = {Annales de l'Institut Fourier},
     pages = {2465--2523},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {57},
     number = {7},
     year = {2007},
     doi = {10.5802/aif.2340},
     zbl = {1145.81033},
     mrnumber = {2394549},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2340/}
}
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
SP  - 2465
EP  - 2523
VL  - 57
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2340/
DO  - 10.5802/aif.2340
LA  - en
ID  - AIF_2007__57_7_2465_0
ER  - 
%0 Journal Article
%A Anantharaman, Nalini
%A Nonnenmacher, Stéphane
%T Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold
%J Annales de l'Institut Fourier
%D 2007
%P 2465-2523
%V 57
%N 7
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2340/
%R 10.5802/aif.2340
%G en
%F AIF_2007__57_7_2465_0
Anantharaman, Nalini; Nonnenmacher, Stéphane. Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold. Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2465-2523. doi : 10.5802/aif.2340. http://archive.numdam.org/articles/10.5802/aif.2340/

[1] Anantharaman, N.; Koch, H.; Nonnenmacher, S. Entropy of eigenfunctions (2006) (to appear in the Proceedings of the International Congress on Mathematical Physics - Rio de Janeiro, August 6 - 11) | Zbl

[2] Anantharaman, Nalini Entropy and the localization of eigenfunctions (2008) (to appear in Ann. of Math.) | Zbl

[3] Anantharaman, Nalini; Nonnenmacher, Stéphane Entropy of semiclassical measures of the Walsh-quantized baker’s map, Ann. Henri Poincaré, Volume 8 (2007) no. 1, pp. 37-74 | DOI | Zbl

[4] Bérard, Pierre H. On the wave equation on a compact Riemannian manifold without conjugate points, Math. Z., Volume 155 (1977) no. 3, pp. 249-276 | DOI | MR | Zbl

[5] Berry, M. V. Regular and irregular semiclassical wavefunctions, J. Phys. A, Volume 10 (1977) no. 12, pp. 2083-2091 | DOI | MR | Zbl

[6] Bohigas, Oriol Random matrix theories and chaotic dynamics, Chaos et physique quantique (Les Houches, 1989), North-Holland, Amsterdam, 1991, pp. 87-199 | MR

[7] Bouzouina, A.; Robert, D. Uniform semiclassical estimates for the propagation of quantum observables, Duke Math. J., Volume 111 (2002) no. 2, pp. 223-252 | DOI | MR | Zbl

[8] Colin de Verdière, Y. Ergodicité et fonctions propres du laplacien, Comm. Math. Phys., Volume 102 (1985) no. 3, pp. 497-502 | DOI | MR | Zbl

[9] Colin de Verdière, Yves; Parisse, Bernard Équilibre instable en régime semi-classique. I. Concentration microlocale, Comm. Partial Differential Equations, Volume 19 (1994) no. 9-10, pp. 1535-1563 | DOI | MR | Zbl

[10] Dimassi, Mouez; Sjöstrand, Johannes Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, 268, Cambridge University Press, Cambridge, 1999 | MR | Zbl

[11] Donnelly, Harold Quantum unique ergodicity, Proc. Amer. Math. Soc., Volume 131 (2003) no. 9, p. 2945-2951 (electronic) | DOI | MR | Zbl

[12] Dunford, Nelson; Schwartz, Jacob T. Linear operators. I. General theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York, 1958 | MR | Zbl

[13] Evans, L.C.; worski, M.Z Lectures on semiclassical analysis (version 0.2), http://math.berkeley.edu/~zworski

[14] Faure, Frédéric; Nonnenmacher, Stéphane On the maximal scarring for quantum cat map eigenstates, Comm. Math. Phys., Volume 245 (2004) no. 1, pp. 201-214 | DOI | MR | Zbl

[15] Faure, Frédéric; Nonnenmacher, Stéphane; De Bièvre, Stephan Scarred eigenstates for quantum cat maps of minimal periods, Comm. Math. Phys., Volume 239 (2003) no. 3, pp. 449-492 | DOI | MR | Zbl

[16] Hadamard, Jacques Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques, Hermann, Paris, 1932 | Zbl

[17] Hörmander, Lars The analysis of linear partial differential operators. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 256, Springer-Verlag, Berlin, 1983 (Distribution theory and Fourier analysis) | Zbl

[18] Katok, Anatole; Hasselblatt, Boris Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995 (With a supplementary chapter by Katok and Leonardo Mendoza) | MR | Zbl

[19] Kelmer, D. Arithmetic quantum unique ergodicity for symplectic linear maps of the multidimensional torus (to appear in Ann. of Math., math-ph/0510079)

[20] Kelmer, D. Scarring on invariant manifolds for perturbed quantized hyperbolic toral automorphisms (preprint, arXiv:math-ph/0607033)

[21] Klingenberg, Wilhelm Riemannian manifolds with geodesic flow of Anosov type, Ann. of Math. (2), Volume 99 (1974), pp. 1-13 | DOI | MR | Zbl

[22] Kraus, K. Complementary observables and uncertainty relations, Phys. Rev. D (3), Volume 35 (1987) no. 10, pp. 3070-3075 | DOI | MR

[23] Ledrappier, F.; Young, L.-S. The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula, Ann. of Math. (2), Volume 122 (1985) no. 3, pp. 509-539 | DOI | Zbl

[24] Lindenstrauss, Elon Invariant measures and arithmetic quantum unique ergodicity, Ann. of Math. (2), Volume 163 (2006) no. 1, pp. 165-219 | DOI | MR | Zbl

[25] Maassen, Hans; Uffink, J. B. M. Generalized entropic uncertainty relations, Phys. Rev. Lett., Volume 60 (1988) no. 12, pp. 1103-1106 | DOI | MR

[26] Nonnenmacher, Stéphane; Zworski, Maciej Quantum decay rates in chaotic scattering (2007) (preprint, arXiv:0706.3242)

[27] Rudnick, Zeév; Sarnak, Peter The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys., Volume 161 (1994) no. 1, pp. 195-213 | DOI | MR | Zbl

[28] Sjöstrand, Johannes; Zworski, Maciej Asymptotic distribution of resonances for convex obstacles, Acta Math., Volume 183 (1999) no. 2, pp. 191-253 | DOI | MR | Zbl

[29] Šnirelʼman, A. I. Ergodic properties of eigenfunctions, Uspehi Mat. Nauk, Volume 29 (1974) no. 6(180), pp. 181-182 | MR

[30] Voros, André Semiclassical ergodicity of quantum eigenstates in the Wigner representation, Stochastic behavior in classical and quantum Hamiltonian systems (Volta Memorial Conf., Como, 1977) (Lecture Notes in Phys.), Volume 93, Springer, Berlin, 1979, pp. 326-333 | MR | Zbl

[31] Wolpert, Scott A. The modulus of continuity for Γ 0 (m) semi-classical limits, Comm. Math. Phys., Volume 216 (2001) no. 2, pp. 313-323 | DOI | MR | Zbl

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

Cité par Sources :