P. Kurlberg and Z. Rudnick. On quantum ergodicity for linear maps of the torus. Comm. Math. Phys., 222(1):201-227, 2001. Home/Search Document Details and Download
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Controlling Strong Scarring for Quantized Ergodic Toral.. - Bonechi (2002) (1 citation) (Correct)
....all the more true since there exists a sequence N k for which the eigenspaces have a N k = ln N k fold degeneracy [BonDB] It is precisely this sequence of N that is used to construct the sequence of eigenfunctions in (1.3) Another result in the direction of (1.5) is the following. Theorem 1.5. [KR2] If A 2 SL(2; Z) is hyperbolic and a 11 a 12 0 a 21 a 22 mod 2, then there exists a density one sequence of integers (N ) 2N along which (1.5) holds. Theorem 1.5 states that quantum unique ergodicity holds along a subsequence of values of N . It is furthermore shown in [KR2] that, along ....
....Theorem 1.5. KR2] If A 2 SL(2; Z) is hyperbolic and a 11 a 12 0 a 21 a 22 mod 2, then there exists a density one sequence of integers (N ) 2N along which (1.5) holds. Theorem 1.5 states that quantum unique ergodicity holds along a subsequence of values of N . It is furthermore shown in [KR2] that, along this sequence, the degeneracies of the eigenspaces grow suciently slowly so that it is disjoint from the sequence N k mentioned above, as it should be in order not to contradict (1.3) It is therefore seen in both theorems that the obstacle to the validity of (1.5) for all N is the ....
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Kurlberg P., Rudnick Z., On quantum ergodicity for linear maps of the torus, preprint 1999, math.NT/9910145.
Quantum Chaos: A Brief First Visit - De Bièvre (Correct)
....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. On quantum ergodicity for linear maps of the torus, preprint 1999, math.NT/9910145.
On the Order of Unimodular Matrices Modulo Integers - Kurlberg (1991) (1 citation) Self-citation (Kurlberg) (Correct)
....of unimodular matrices modulo N 2 Z. That is, if A 2 SL 2 (Z) what can be said about lower bounds for ord N (A) the order of A modulo N , that hold for most N It is a natural generalization of the previous questions, but our main motivation comes from mathematical physics (quantum chaos) In [7] Rudnick and I proved that if A is hyperbolic , then quantum ergodicity for toral automorphisms follows from ord N (A) being slightly larger than N , and we then showed that this condition holds for a full density subset of the integers. Again, we expect that the typical order is much ....
....N we have NB p z p e z . Letting z = log log x and y = log x we get that NG = NB for N x with at most O (x( log log x) log x) o(x) exceptions. Now, the following Proposition gives that, for most N , ord N (A) is essentially given by ord p (A) Proposition ([7], Proposition 11) Let DA = 4(tr(A) 4) For almost all N x, ord N (A) pjd0 where d 0 is given by writing N = ds , with d = d 0 gcd(d; DA ) squarefree. Finally, since ord p (A) log p p for pjNG and p suciently large, we nd that ord N (A) pjNG ....
P. Kurlberg and Z. Rudnick. On quantum ergodicity for linear maps of the torus. Comm. Math. Phys., 222(1):201-227, 2001.
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