tailieunhanh - Đề tài " On the distribution of matrix elements for the quantum cat map "

For many classically chaotic systems it is believed that the quantum wave functions become uniformly distributed, that is the matrix elements of smooth observables tend to the phase space average of the observable. In this paper we study the fluctuations of the matrix elements for the desymmetrized quantum cat map. We present a conjecture for the distribution of the normalized matrix elements, namely that their distribution is that of a certain weighted sum of traces of independent matrices in SU(2). This is in contrast to generic chaotic systems where the distribution is expected to be Gaussian. . | Annals of Mathematics On the distribution of matrix elements for the quantum cat map By P ar Kurlberg and Ze rev Rudnick Annals of Mathematics 161 2005 489 507 On the distribution of matrix elements for the quantum cat map By Par Kurlberg and Zeev Rudnick Abstract For many classically chaotic systems it is believed that the quantum wave functions become uniformly distributed that is the matrix elements of smooth observables tend to the phase space average of the observable. In this paper we study the fluctuations of the matrix elements for the desymmetrized quantum cat map. We present a conjecture for the distribution of the normalized matrix elements namely that their distribution is that of a certain weighted sum of traces of independent matrices in SU 2 . This is in contrast to generic chaotic systems where the distribution is expected to be Gaussian. We compute the second and fourth moment of the normalized matrix elements and obtain agreement with our conjecture. 1. Introduction A fundamental feature of quantum wave functions of classically chaotic systems is that the matrix elements of smooth observables tend to the phase space average of the observable at least in the sense of convergence in the mean 15 2 17 or in the mean square 18 . In many systems it is believed that in fact all matrix elements converge to the micro-canonical average however this has only been demonstrated for a couple of arithmetic systems For quantum cat maps 10 and conditional on the Generalized Riemann Hypothesis1 also for the modular domain 16 in both cases assuming that the systems are desymmetrized by taking into account the action of Hecke operators. As for the approach to the limit it is expected that the fluctuations of the matrix elements about their limit are Gaussian with variance given by classical This work was supported in part by the EC TMR network Mathematical aspects of Quantum Chaos HPRN-CT-2000-00103 . . was also supported in part by the NSF DMS-0071503 the Royal .