A pr 2 00 4 Proof of the Rudnick - Kurlberg Rate

In this paper we give a proof of the Hecke quantum unique ergodic-ity rate conjecture for the Berry-Hannay model. A model of quantum mechanics on the 2-dimensional torus. In the paper " Quantization of linear maps on the torus-Fresnel diffrac-tion by a periodic grating " , published in 1980 (see [BH]), the physicists M.V. Berry and J. Hannay explore a model for quantum mechanics on the 2-dimensional symplectic torus (T, ω). Berry and Hannay suggested to quan-tize simultaneously the functions on the torus and the linear symplectic group G = SL(2, Z). One of the motivations was to study the phenomenon of quantum chaos in this model (see [R2] for a survey). 1 Considering a classical mechanical system, which is ergodic. In what sense the ergodicity property is reflected in the corresponding quantum system. Put it a little differently, is there a meaningful notion of Quantum Ergodicity. This is a fundamental meta-question in the area of quantum chaos. For the specific case of the Berry-Hannay model, This question was addressed in a paper by Rudnick and Kurlberg [KR]. In this paper they formulated a rigorous definition of quantum ergodicity. We call this notion Hecke Quantum Unique Ergodicity. A focal step in their work, was to introduce a group of hidden symmetries, they called the Hecke group. The statement of Hecke Quantum Unique Ergodicity say about semi-classical convergence of certain matrix coefficients which are defined in terms of the Hecke group. In their paper they proved a bound on the rate of convergence. In Rudnick's ECM lecture [R2], Barcelona 2000, and in his lecture [R1] at MSRI, Berkeley 1999 he conjectured about a stronger bound to be valid. In this paper we prove the correct bound on the semiclassical asymptotic of the Hecke matrix coefficients for the two dimensional torus, as stated in Rudnick's lectures [R1],[R2].