M. Degli Espositi, S. Gra and S. Isola, Classical limit of the quantized hyperbolic toral automorphisms, Comm. Math. Phys., 167, (1995), 471-507.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....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 ....

M. Degli Esposti, S. Gra, and S. Isola. Classical limit of the quantized hyperbolic toral automorphisms. Comm. Math. Phys., 167(3):471-507, 1995.

....given in [We5] Note that the Theorem only makes a statement for finite times, i.e. it is not strong enough to allow the interchange the limits h 0, and the ergodic time average, or some other version of the limit t 1. This would be very interesting for applications to quantum chaos (see [DGI] for a result in this direction) 4 Evolution Theorem. Let H h 2 C 2 (A; j) such that H h = H h for every h. Define the time evolution for each h by fl t h (A) e itH h =h A e GammaitH h =h ; 2:3) for A 2 A h , and t 2 IR. Let A h be j convergent, and define A t h = fl t h (A h ....

M. Degli Esposti, S. Graffi, and S. Isola: "Classical limit of the quantized hyperbolic toral automorphisms", Commun.Math.Phys. 167(1995) 471--507

....) 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 ....

.... 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 that the expectation values for all eigenfunctions 2 HN converge to the phase space average. Here, we will study a parabolic map of the torus (also called a skew translation) which is specified by ....

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M. Degli Esposti, S. Graffi and S. Isola, Classical limit of the quantized hyperbolic toral automorphisms, Comm. Math. Phys. 167 (1995) 471-507.

....counterpart, there remains as yet no well accepted concept of quantum ergodicity. Several inequivalent yet very natural approaches have been introduced. On the one hand, a system is deemed quantum ergodic if it has a welldefined classical limit which is itself ergodic [14] 15] 13] 4] [7]. On the other hand, the original notion of quantum ergodicity proposed by von Neumann defines, roughly speaking, a system as quantum ergodic if any observable is eventually distributed over the eigenstates according to the weight of each eigenstate. Let us discuss this latter notion first. Let H ....

Degli Esposti, M., Graffi, S., and Isola, S.: Classical limit of the quantized hyperbolic toral automorphisms, Comm. Math. Phys., 167, 471--507 (1995)

.... is an immediate consequence of (III.5) III.D. At this point it is not quite clear that Gamma is indeed a quantization of the classical map fl, i.e. that its classical limit 0 indeed yields fl. The goal of this subsection is to show that it is so. A related analysis can be found in [DGI]. We let jj Delta jj denote the operator norm on the Hilbert space H 2 (C ; d ) Theorem III.2. Let f be a continuous Z 2 invariant function on C . Then: jjF T (f)F Gamma1 Gamma T (f ffi fl)jj 0; as 0: III:8) Proof. Let ffl 0. We are going to show that for all ....

Degli Esposti, M., Graffi, S., and Isola, S.: Classical limit of the quantized hyperbolic toral automorphisms, Comm. Math. Phys., to appear

....theory of quantum computation and quantum information theory, etc. We will work within the operator algebra framework, as this is the natural setup for addressing the structural issues of quantum dynamics. Other approaches abound in the physics and mathematics literature, see e.g. 2] 5] 7] [11], and references therein. We shall focus on a somewhat restricted class of quantum dynamical systems, namely those which arise as quantizations of classical maps. 2. Classical dynamics Classical mechanics is formulated in terms of a phase space M which is usually assumed to be a symplectic ....

Degli Esposti, M., Graffi, S., and Isola, S.: Classical limit of the quantized hyperbolic toral automorphisms, Comm. Math. Phys., 167, 471 -- 507 (1995)

....the quantum system become equidistributed with respect to the Liouville measure . Made precise (see below) this is a statement about the diagonal matrix elements of quantized observables between eigenstates and is commonly referred to as the Schnirelman Theorem. It has been proven in many cases [Z1, CdV, HMR, GL, Z3, Z4, DIG, Sa, BD, ZZ]. If the system is in addition mixing, more can be inferred: in that case (most) off diagonal matrix elements tend to zero [Z2, CR] We will be interested here in the classical limit of quantized, discontinuous, ergodic or mixing symplectic transformations of the two torus. Combining ideas of [Z4] ....

....on we will suppress the explicit ( N) dependence on the e j ; f k and on U . We are now able to define the Weyl quantization Op W h (f) of a given function f = X n;m2Z f nm e 2 i(nq Gammamp) 2 C 1 (T 2 ) 12) as Op W h (f) X n;m2Z f nmU( m N ; n N ) 13) We refer to [BD, DIG] for more details. Turning to the dynamics, assume now that T is a symplectic map on the torus. In certain cases, it is possible to define a corresponding quantum operator V T on HN . There is no general procedure avaible for doing this. The cases for which it has been done are: 1. When T is ....

M. Degli Esposti, S. Isola and S. Graffi, Classical limit of the quantized hyperbolic toral automorphisms, Comm.Math.Phys. 167 (1995), 471-507.

....scalar products hu n ; Au n i turn out to be independent of n. This property cannot obviously hold in any sensible quantum system: however it may be valid at the classical limit, at which hu n ; Au n i may tend to the phase average of the r.h.s. of (1. 1) as verified in many instances (see e.g.[Sc, CdV, HMR, Ze, DEGI]) provided H is the quantization of a Hamiltonian generating an ergodic flow. This fact has even led some authors to assume the validity of this limiting property as the very definition of quantum ergodicity. The persisting dependence on the initial datum after time average can be looked at as a ....

M.Degli Esposti, S.Graffi, S.Isola, Classical Limit of the Quantized Hyperbolic Toral Automorphisms Commun.Math.Phys. 167 (1995), 471-507

....affine hyperbolic maps. It will turn out that the unitary matrices describing the quantum evolution of each of those systems can be computed straightforwardly and with relatively little effort in this way. The toral automorphisms and the Baker transformation were quantized respectively in [HB,DE, DGI] and in [BV] and they have been studied intensely ever since, both numerically and analytically [ Ke1, Ke2, Ke3, DGI, Eck, Sa] The methods of quantization used in these papers look very different from each other. Our approach reproduces the same results in those cases. In order to get a more ....

....of each of those systems can be computed straightforwardly and with relatively little effort in this way. The toral automorphisms and the Baker transformation were quantized respectively in [HB,DE, DGI] and in [BV] and they have been studied intensely ever since, both numerically and analytically [ Ke1, Ke2, Ke3, DGI, Eck, Sa]. The methods of quantization used in these papers look very different from each other. Our approach reproduces the same results in those cases. In order to get a more precise flavour of the ideas to be developed, recall that in classical mechanics the dynamics of a system is obtained by ....

M. Degli Esposti, S. Graffi and S. Isola, Classical limit of the quantized hyperbolic toral automorphisms, to appear in Commun. Math. Phys. (1994).

....classical notions as h 0. As a final remark let us mention that if A is a pseudodifferential operator also the Von Neumann definition (1. 6) reproduces the classical one at the classical limit if hu n ; Au n i tends to the phase average of the symbol of A, as verified in many instances (see e.g.[Sc, CdV, HMR, Ze1, Ze2, DEGI]) in which H is the quantization of a Hamiltonian generating an ergodic flow. Some authors ( Sa, Ze2] assume this limiting property as the very definition of quantum ergodicity. Ergodic Properties of Quantum Harmonic Crystals 29 ....

M.Degli Esposti, S.Graffi, S.Isola, Classical Limit of the Quantized Hyperbolic Toral Automorphisms Commun.Math.Phys. 167 (1995), 471-507

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M. Degli Espositi, S. Gra and S. Isola, Classical limit of the quantized hyperbolic toral automorphisms, Comm. Math. Phys., 167, (1995), 471-507.

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M. Degli Espositi, S. Gra and S. Isola, Classical limit of the quantized hyperbolic toral automorphisms, Comm. Math. Phys., 167, (1995), 471-507.

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M.Degli Esposti, S.Graffi, S.Isola, Classical limit of the quantized hyperbolic toral automorphisms, Commun. Math. Phys 167: 471-509 (1995)

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