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.
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.
Accepted:
Published online:
@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 = {https://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 - https://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 https://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, Volume 342 (2006) no. 1, pp. 69-72. doi : 10.1016/j.crma.2005.10.033. https://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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