[Démonstration de la conjecture du taux de Kurlberg–Rudnick]
Nous proposons une démonstration de la conjecture d'unique ergodicité quantique d'Hecke pour le modèle de Berry–Hannay, un modèle de mécanique quantique sur un tore de dimension deux. Cette conjecture a été proposée par Z. Rudnick à MSRI, Berkeley, 1999 et à l'ECM, Barcelona, 2000.
In this Note we present a proof of the Hecke quantum unique ergodicity conjecture for the Berry–Hannay model, a model of quantum mechanics on a two dimensional torus. This conjecture was stated in Z. Rudnick's lectures at MSRI, Berkeley, 1999 and ECM, Barcelona, 2000.
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@article{CRMATH_2006__342_1_69_0,
author = {Gurevich, Shamgar and Hadani, Ronny},
title = {Proof of the {Kurlberg{\textendash}Rudnick} rate conjecture},
journal = {Comptes Rendus. Math\'ematique},
pages = {69--72},
publisher = {Elsevier},
volume = {342},
number = {1},
year = {2006},
doi = {10.1016/j.crma.2005.10.033},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2005.10.033/}
}
TY - JOUR AU - Gurevich, Shamgar AU - Hadani, Ronny TI - Proof of the Kurlberg–Rudnick rate conjecture JO - Comptes Rendus. Mathématique PY - 2006 SP - 69 EP - 72 VL - 342 IS - 1 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2005.10.033/ DO - 10.1016/j.crma.2005.10.033 LA - en ID - CRMATH_2006__342_1_69_0 ER -
%0 Journal Article %A Gurevich, Shamgar %A Hadani, Ronny %T Proof of the Kurlberg–Rudnick rate conjecture %J Comptes Rendus. Mathématique %D 2006 %P 69-72 %V 342 %N 1 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2005.10.033/ %R 10.1016/j.crma.2005.10.033 %G en %F CRMATH_2006__342_1_69_0
Gurevich, Shamgar; Hadani, Ronny. Proof of the Kurlberg–Rudnick rate conjecture. Comptes Rendus. Mathématique, Tome 342 (2006) no. 1, pp. 69-72. doi : 10.1016/j.crma.2005.10.033. http://www.numdam.org/articles/10.1016/j.crma.2005.10.033/
[1] Classical limit of the quantized hyperbolic toral automorphisms, Comm. Math. Phys., Volume 167 (1995) no. 3, pp. 471-507
[2] La conjecture de Weil II, Publ. Math. IHES, Volume 52 (1981), pp. 313-428
[3] P. Deligne, Metaplectique, A letter to Kazhdan, 1982
[4] Quantization of linear maps on the torus – Fresnel diffraction by a periodic grating, Physica D, Volume 1 (1980), pp. 267-291
[5] Hecke theory and equidistribution for the quantization of linear maps of the torus, Duke Math. J., Volume 103 (2000), pp. 47-78
[6] Non-commutative tori – a case study of non-commutative differentiable manifolds, Contemp. Math., Volume 105 (1990), pp. 191-211
[7] Z. Rudnick, The quantized cat map and quantum ergodicity, Lecture at the MSRI conference “Random Matrices and their Applications”, Berkeley, June 7–11, 1999
[8] On quantum unique ergodicity for linear maps of the torus, European Congress of Mathematics, vol. II, Barcelona, 2000, Progr. Math., vol. 202, Birkhäuser, Basel, 2001, pp. 429-437
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