Comptes Rendus
no. 1
Mathematical Physics
Proof of the Kurlberg–Rudnick rate conjecture
[Démonstration de la conjecture du taux de Kurlberg–Rudnick]

Présenté par : Michèle Vergne

Shamgar Gurevich1 ; Ronny Hadani1
1 School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel
Comptes Rendus. Mathématique, Volume 342 (2006) no. 1, pp. 69-72.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2005.10.033

Shamgar Gurevich 1 ; Ronny Hadani 1

1 School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel 
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Shamgar Gurevich; Ronny Hadani. 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://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.10.033/

[1] M. Degli Esposti; S. Graffi; S. Isola Classical limit of the quantized hyperbolic toral automorphisms, Comm. Math. Phys., Volume 167 (1995) no. 3, pp. 471-507

[2] P. Deligne La conjecture de Weil II, Publ. Math. IHES, Volume 52 (1981), pp. 313-428

[3] P. Deligne, Metaplectique, A letter to Kazhdan, 1982

[4] J.H. Hannay; M.V. Berry Quantization of linear maps on the torus – Fresnel diffraction by a periodic grating, Physica D, Volume 1 (1980), pp. 267-291

[5] P. Kurlberg; Z. Rudnick Hecke theory and equidistribution for the quantization of linear maps of the torus, Duke Math. J., Volume 103 (2000), pp. 47-78

[6] M.A. Rieffel 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] Z. Rudnick 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

  • Shamgar Gurevich; Ronny Hadani Notes on the Self-Reducibility of the Weil Representation and Higher-Dimensional Quantum Chaos, Arithmetic and Geometry Around Quantization (2010), p. 201 | DOI:10.1007/978-0-8176-4831-2_7
  • L. Rosenzweig On the Fluctuations of Matrix Elements of the Quantum Cat Map, International Mathematics Research Notices (2010) | DOI:10.1093/imrn/rnq262
  • Vincent Lafforgue; Sergey Lysenko Geometric Weil representation: local field case, Compositio Mathematica, Volume 145 (2009) no. 1, p. 56 | DOI:10.1112/s0010437x08003771
  • Pär Kurlberg; Lior Rosenzweig; Zeév Rudnick Matrix elements for the quantum cat map: fluctuations in short windows, Nonlinearity, Volume 20 (2007) no. 10, p. 2289 | DOI:10.1088/0951-7715/20/10/001
  • Zeév Rudnick THE ARITHMETIC THEORY OF QUANTUM MAPS, Equidistribution in Number Theory, An Introduction, Volume 237 (2006), p. 331 | DOI:10.1007/978-1-4020-5404-4_15

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