[Demi-délocalisation des fonctions propres du laplacien sur une variété d’Anosov]
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.
Keywords: Quantum chaos, semiclassical measure, ergodic theory, entropy, Anosov flows
Mot clés : chaos quantique, mesure semiclassique, théorie ergodique, entropie, flots d’Anosov
@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://www.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://www.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://www.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://www.numdam.org/articles/10.5802/aif.2340/
[1] Entropy of eigenfunctions (2006) (to appear in the Proceedings of the International Congress on Mathematical Physics - Rio de Janeiro, August 6 - 11) | Zbl
[2] Entropy and the localization of eigenfunctions (2008) (to appear in Ann. of Math.) | Zbl
[3] 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] 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] Regular and irregular semiclassical wavefunctions, J. Phys. A, Volume 10 (1977) no. 12, pp. 2083-2091 | DOI | MR | Zbl
[6] Random matrix theories and chaotic dynamics, Chaos et physique quantique (Les Houches, 1989), North-Holland, Amsterdam, 1991, pp. 87-199 | MR
[7] Uniform semiclassical estimates for the propagation of quantum observables, Duke Math. J., Volume 111 (2002) no. 2, pp. 223-252 | DOI | MR | Zbl
[8] Ergodicité et fonctions propres du laplacien, Comm. Math. Phys., Volume 102 (1985) no. 3, pp. 497-502 | DOI | MR | Zbl
[9] 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] Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, 268, Cambridge University Press, Cambridge, 1999 | MR | Zbl
[11] Quantum unique ergodicity, Proc. Amer. Math. Soc., Volume 131 (2003) no. 9, p. 2945-2951 (electronic) | DOI | MR | Zbl
[12] 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] Lectures on semiclassical analysis (version 0.2), http://math.berkeley.edu/~zworski
[14] On the maximal scarring for quantum cat map eigenstates, Comm. Math. Phys., Volume 245 (2004) no. 1, pp. 201-214 | DOI | MR | Zbl
[15] Scarred eigenstates for quantum cat maps of minimal periods, Comm. Math. Phys., Volume 239 (2003) no. 3, pp. 449-492 | DOI | MR | Zbl
[16] Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques, Hermann, Paris, 1932 | Zbl
[17] 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] 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] Arithmetic quantum unique ergodicity for symplectic linear maps of the multidimensional torus (to appear in Ann. of Math., math-ph/0510079)
[20] Scarring on invariant manifolds for perturbed quantized hyperbolic toral automorphisms (preprint, arXiv:math-ph/0607033)
[21] Riemannian manifolds with geodesic flow of Anosov type, Ann. of Math. (2), Volume 99 (1974), pp. 1-13 | DOI | MR | Zbl
[22] Complementary observables and uncertainty relations, Phys. Rev. D (3), Volume 35 (1987) no. 10, pp. 3070-3075 | DOI | MR
[23] 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] Invariant measures and arithmetic quantum unique ergodicity, Ann. of Math. (2), Volume 163 (2006) no. 1, pp. 165-219 | DOI | MR | Zbl
[25] Generalized entropic uncertainty relations, Phys. Rev. Lett., Volume 60 (1988) no. 12, pp. 1103-1106 | DOI | MR
[26] Quantum decay rates in chaotic scattering (2007) (preprint, arXiv:0706.3242)
[27] The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys., Volume 161 (1994) no. 1, pp. 195-213 | DOI | MR | Zbl
[28] Asymptotic distribution of resonances for convex obstacles, Acta Math., Volume 183 (1999) no. 2, pp. 191-253 | DOI | MR | Zbl
[29] Ergodic properties of eigenfunctions, Uspehi Mat. Nauk, Volume 29 (1974) no. 6(180), pp. 181-182 | MR
[30] 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] The modulus of continuity for semi-classical limits, Comm. Math. Phys., Volume 216 (2001) no. 2, pp. 313-323 | DOI | MR | Zbl
[32] Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J., Volume 55 (1987) no. 4, pp. 919-941 | DOI | MR | Zbl
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