P. Kurlberg and Z. Rudnick. Hecke theory and equidistribution for the quantization of linear maps of the torus. Duke Math. J., 103(1):47-77, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....toral automorphisms. One nevertheless expects the sequences that satisfy (1.3) to be rather exceptional: there do indeed exist two results in the direction of (1.5) valid for a particular but rather large class of hyperbolic toral automorphisms in d = 1. The rst one is this. Theorem 1.4. [KR1] If A 2 SL(2; Z) is hyperbolic and A I 2 mod 4, then there exists for each N a basis f 1 ; 2 ; N g of eigenfunctions of M(A) so that (1.4) holds with E(N) f1; Ng. This obviously constitutes a strengthening of the Schnirelman theorem for the particular class of A considered. ....

....exists for each N a basis f 1 ; 2 ; N g of eigenfunctions of M(A) so that (1.4) holds with E(N) f1; Ng. This obviously constitutes a strengthening of the Schnirelman theorem for the particular class of A considered. The basis for which the result holds is explicitly described in [KR1]. Note the di erence between Theorem 1.4 and (1.5) Indeed, the eigenvalues of M(A) may be degenerate so that it is possible that exceptional sequences of eigenfunctions not belonging to the above basis have a di erent semiclassical limit. This is all the more true since there exists a sequence N ....

[Article contains additional citation context not shown here]

Kurlberg P., Rudnick Z., Hecke theory and equidistribution for the quantization of linear maps of the torus, preprint 1999, chao-dyn/9901031, Duke Math. J., to appear.

....x 0 . In other words, is the natural A invariant measure supported on a periodic orbit. Such sequences, if they exist, are called (strong) scars . For the case at hand, various strengthenings of the Schnirelman theorem as stated above indicate strongly that such sequences should not exist [DEGI][KR1][KR2] The existence or absence of some (possibly weaker) form of scarring is in general an interesting question in quantum chaos, that can be asked also in the context of billiards, for example. For more information on the question of scarring, you may consult [S1] and references therein. 14.2. ....

Kurlberg, P., Rudnick, Z. Hecke theory and equidistribution for the quantization of linear maps of the torus, preprint 1999, chao-dyn/9901031, Duke Math. J., to appear.

.... at least in the sense of convergence in the mean [14, 2, 16] or in the mean square [17] 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 [11], and conditional on the Generalized Riemann Hypothesis also for the modular domain [15] 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 uctuations of the matrix ....

....in part by the US Israel Bi National Science Foundation. An unconditional proof was recently announced by Elon Lindenstrauss. AR KURLBERG AND ZE study these uctuations for the quantum cat map. Our nding is that for this system, the picture is very di erent. We recall the basic setup [8, 3, 4, 11] (see section 2 for further background and any unexplained notation) The classical mechanical system is the iteration of a linear hyperbolic map A 2 SL(2; Z) of the torus T = R =Z (a cat map ) The quantum system is given by specifying an integer N , which plays the role of the inverse ....

[Article contains additional citation context not shown here]

P. Kurlberg and Z. Rudnick. Hecke theory and equidistribution for the quantization of linear maps of the torus. Duke Math. J., 103(1):47-77, 2000.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC