J. Keating, The cat map: quantum mechanics and classical motion, Nonlinearity, 4, (1991), 309-341.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... of the torus were much simpler than chaotic billiards and did I not use this claim to motivate their study As it turns out, those systems have a rather special spectrum in some respects (Exercise 14.7) and the (properly reformulated) Bohigas GiannoniSchmit conjecture is not veri ed by them [Ke]. Numerical evidence indicates that perturbed hyperbolic automorphisms, on the other hand, do have eigenvalue statistics compatible with the BGS conjecture [MO] Unfortunately, whereas the proof of the Schnirelman theorem given in the previous section goes through almost unaltered for those ....

Keating J., The cat maps: quantum mechanics and classical motion, Nonlinearity 4, 309341 (1991).

....) that is one requires that in the semiclassical limit, quantum evolution becomes classical evolution. The analogue of eigenmodes are then the eigenfunctions of the propagator UN . The main focus in the literature has so far been on hyperbolic transformations of the torus, the so called cat maps [7, 11, 4, 5], to which the proof of Schnirelman s theorem [16, 17, 1] can be adapted to prove quantum ergodicity, but not QUE [2, 18] Assuming the Generalized Riemann Hypothesis, Degli Esposti, Graffi and Isola [5] found an explicit infinite (though sparse) subsequence of values of N , for which they show ....

J.P. Keating, The cat maps: quantum mechanics and classical motion. Nonlinearity 4 (1991) 309-341.

....quantized momenta j m N (mod 1) i, with the transformation matrix between these bases D m N fi fi fi n N E = 1 p N e Gamma2 imn=N = FN ) mn (finite Fourier transformation of rank N) and with N j (2 h) Gamma1 . Examples of unitary quantum maps on this space are the cat maps [8, 9], their perturbations [10] and the baker s map [11] defined (for even N) by this unitary matrix in the mixed (q 7 p) representation, B mixed def = FN=2 0 0 FN=2 = FN=2 Omega 1 0 0 1 ; i:e: B def = F Gamma1 N B mixed in position representation: 4) Because these ....

J.P. Keating, The cat maps: quantum mechanics and classical motion, Nonlinearity 4, 309-341 (1991).

....their perturbations) as well as the aforementioned discontinuous maps. Note that among those only the geodesic flows and the (smoothly perturbed) toral automorphisms are smooth. Their behaviour is often determined by special number theoretic properties and therefore not expected to be generic [HB, Ke1, Ke2, S]. This has been a major motivation for analyzing systems with singularities. Much attention has been paid to chaotic maps on the torus, even though they seem rather unrealistic as models for physical systems despite a recent attempt to realize the quantum Baker map as a physically realistic system ....

J. Keating, The cat maps: quantum mechanics and classical motion, Nonlinearity 4 (1991), 309-341.

....in [DT92] the cat map is discussed in detail. The main idea there is to use that the cat map A is the square of 0 1 1 1 0 1 A . The iterates of this latter matrix generate the Fibonacci numbers; hence one can use known results on the distribution of Fibonacci numbers mod n. In [HB80] [Kea91], and [PV87] more general matrices A 2 SL(2; Z Z ) are discussed. However, the results on Per A (n) are restricted to the case of prime numbers n = p or prime powers n = p ff : In this connection it should also be noted that general theorems on p Sylow subgroups can be used, if n = p is prime, ....

....be proved in section i 1 below. We recall that the special case A = 0 2 1 1 1 1 A was treated in [DT92] Impressive and very insightful numerical evidence on Per A (n) has been accumulated in the context of quantum theoretical models and discrete approximations to quantum chaos; see [HB80] [Kea91], PV87] In conclusion, our theorems exhibit a curious interaction between linearity of the iteration and equidistant lattice discretisation, which sheds some light on the intricacies involved in the discretisation of continuous systems. For further discussion see section 4. 2 The ....

J. Keating. The cat maps: quantum mechanics and classical motion. Nonlinearity, (4):309--341, (1991).

....those of [Z.2,3] in the case of wave groups. To illustrate the method and ergodicity results we will also study in detail the Toeplitz quantization of symplectic torus automorphisms (x5) undoubtedly the most popular of maps to undergo quantization see [A.d P.W] B.N.S] B.H] dB.B] d e.g.I] K. P] K][Ke][We] for just a few among the many treatments. As the reader is surely aware, quantization is not a uniquely defined process and it is not apriori clear how the plethora of quantizations defined in these articles are related to each other or to the quantization presented here. In fact, although it ....

....the unitary BKS operator in the Kahler setting is simply to forget the change in complex structure 0 . Thus, the unitary matrix defined relative to the classical theta functions is precisely the quantization of g in the Kahler sense. It is also the quantization of [B.H] dB.B] dE.G. I][Ke], as the interested reader may confirm by comparing their formulae for the quantized cat maps with the expressions in the transformation formulae. Our purpose now is to show that the Toeplitz method leads to the same result. 5.9.1) Lemma. Let Pi N be the orthogonal projection onto H 2 ....

J. Keating, The cat maps: quantum mechanics and classical motion, Nonlinearity 4 (1991), 309-341.

.... Z) in a N Gammadimensional Hilbert space, N = h Gamma1 , was obtained by Berry and Hannay [BH] the canonical quantization of the observables valid for any A 2 SL(2; Z) has been obtained in [DE] The spectral, periodicity and arithmetic properties of the quantum propagator VA are studied in [BV,E1,Ke1,Ke2,Ke3,PV] in addition to [BH,DE] if representations and classical limits are ignored, and the torus is instead the configuration space, the corresponding quantum system can be studied as a pure Weyl algebra automorphism also for h irrational [BNS,N] We employ here the canonical quantization procedure of ....

J.Keating, The cat maps: quantum mechanics and classical motion, Nonlinearity 4 (1991), 309--341.

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J. Keating, The cat map: quantum mechanics and classical motion, Nonlinearity, 4, (1991), 309-341.

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J. Keating, The cat map: quantum mechanics and classical motion, Nonlinearity, 4, (1991), 309-341.

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J.P. Keating, The cat maps: quantum mechanics and classical motion, Nonlinearity 4 (1991), 309-341.

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Keating, J. P.: The cat maps: quantum mechanics and classical motion, Nonlinearity, 4, 309--341 (1990)

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J. Keating, The cat maps: quantum mechanics and classical motion, Nonlinearity 4: 309-341 (1991)

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