A. Bouzouina, S. De Bivre, Equipartition of the eigenfunctions of quantized ergodic maps on the torus, Commun. Math. Phys. 178, 83 105 (1996)  Home/Search   Document Details and Download   Summary   Related Articles   Check

....email: debievre agat.univ lillel.fr 1 Introduction One of the main problems in quantum chaos is the understanding of the semiclassical behaviour of the eigenfunctions of quantum dynamical systems having a chaotic classical limit. The main theorem in this context is the Schnirelman theorem [Sc, CdV, Z1, HMR, BouDB]. It roughly states that most eigenfunctions equidistribute on the available phase space in the classical limit. This leaves open the question of the existence of exceptional sequences of eigenfunctions with a different limit. In the case of hard chaos (uniformly hyperbolic systems) numerical ....

....with the property that, for all f lira n( k [ f [k )n = f(xj) 1 fi) f(x) dx. 8) k T j=0 2 Our result helps to complete the picture of the semiclassical eigenfunction behaviour of quantized toral automorphisms known to date. Indeed, beyond the general Schnirelman theorem for these models [BouDB] the following results are known. First, suppose M is of checkerboard form , meaning AB semiclassically equidistribute, provided one takes the limit along a density one subsequence of values of N [KR2] for which the quantum period is larger than . Note that this sequence excludes the values N ....

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A. Bouzouina, S. De Bivre, Equipartition of the eigenfunctions of quantized ergodic maps on the torus, Commun. Math. Phys. 178, 83 105 (1996)

.... This result has been proven in many di erent contexts: for the Laplace Beltrami operator on compact Riemannian manifolds with an ergodic geodesic ow in [Sc] Z1] CdV] for ergodic billiards in [GL] ZZ] for nonrelativistic quantum mechanics in the classical limit in [HMR] for quantum maps in [BDB] [Z2] A precise statement in the latter context will be given below (Theorem 1.3) The theorem raises obvious questions: do there exist exceptional sequences of eigenfunctions allowing no semi classical limiting measure or a limit di erent from Liouville measure It is well known that the limit ....

....automorphisms of T . Such measures are sometimes called quantum limits . As such these results are to be compared with previous ones for the two torus available in the literature. Let us rst recall the precise statement of the Schnirelman theorem for ergodic symplectic toral automorphisms [BDB]. Theorem 1.3. Let A, M(A) be as above. Let, for each N , f 1 ; 2 ; N dg be a basis of eigenfunctions of M(A) Then, for each N 2 N, there exists a subset E(N) f1; N g such that: i) lim ]E(N) 1; ii) For any f 2 C ) for any sequence (j N 2 E(N) N2N , one ....

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Bouzouina A., De Bievre S., Equipartition of the eigenfunctions of quantized ergodic maps on the torus, Commun.Math.Phys. 178, 83-105 (1996).

....If A is of the form above, then M(A)H = H and M(A) is unitary on H . The general case, when A is an arbitrary element of SL(2; Z) is dealt with easily as well, but at the cost of some additional notational complications using the spaces H ( introduced in Exercise 13.3. I refer to [BDB1] for details. Proof: For 2 H , compute U(n)M(A) M(A) M(A) U(n)M(A) M(A)U(A n) where I used (12.17) Since, in view of (13.2) U(An) e 2 [An]1 [An]2 ; the rst part of the statement follows from an easy calculation. The unitarity of M(A) can be checked for example by ....

....from an easy calculation. The unitarity of M(A) can be checked for example by computing the matrix of M(A) in the basis e j explicitly. The proof is however long and messy this way, since the de nition of M(A) in section 12 is not very explicit and I will omit it. An alternative proof is given in [BDB1]. The operator M(A) on H is often called the quantum map associated to A. They were rst constructed and studied in the context of quantum chaos in [HB] 14. The Schnirelman theorem In this section, I nally return to quantum chaos. The question I want to address is how in the semi classical ....

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Bouzouina A., De Bivre S., Equipartition of the eigenfunctions of quantized ergodic maps on the torus, Commun.Math.Phys. 178, 83-105 (1996).

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