Abstract
The canonical quantization of any hyperbolic symplectomorphismA of the 2-torus yields a periodic unitary operator on aN-dimenional Hilbert space,N=1/h. We prove that this quantum system becomes ergodic and mixing at the classical limit (N→∞,N prime) which can be interchanged with the time-average limit. The recovery of the stochastic behaviour out of a periodic one is based on the same mechanism under which the uniform distribution of the classical periodic orbits reproduces the Lebesgue measure: the Wigner functions of the eigenstates, supported on the classical periodic orbits, are indeed proved to become uniformly speread in phase space.
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References
[AA] Arnold, V.I., Avez, A.: Ergodic Problems in Classical Mechanics. New York: Benjamin, 1968
[Ap] Apostol, T.: Introduction to Analytic Number theory. Berlin, Heidelber, New York: Springer, 1976
[BH] Hannay, J. H., Berry, M.V.: Quantization of linear maps on a torus-Fresnel diffraction by a periodic grating. Physica D1, 267–291 (1980)
[BV] Bartuccelli, F., Vivaldi, F.: Ideal Orbits of Toral Automorphisms. Physica D39, 194 (1989)
[BNS] Benatti, R., Narnhofer, H., Sewell, G. L.: A non-commutative version of the Arnold cat map. Lett. Math. Phys.21, 157–172 (1991)
[CdV] Colin de Verdiere, Y.: Ergodicité et functions propres du Laplacien. Commun. Math. Phys.102, 497–502 (1985)
[Ch] Chandrasekharan, K.: Introduction to Analytic Number Theory. Berlin, Heidelberg, New York: Springer, 1968
[Da] Davenport, H.: On certain exponential sums. Reine, J. Angewandte Mathematik78, 158–176 (1932)
[DE] Degli Esposti, M.: Quantization of the orientation preserving automorphisms of the torus. Ann. Inst. Pincaré, H.58, 323–341 (1993)
[De1] Deligne, P.: La conjecture de Weil I. Pub. Math. I.H.E.S.48, 273–308 (1974)
[De2] Deligne, P.: Cohomologie Etale. Lecture Notes in Mathematics569, (1974)
[E1] Eckhardt, B.: Exact eigenfunctions for a quantized map. J. Phys. A19, 1823–1833 (1986)
[E2] Eckhardt, B.: Quantum mechanics of classically non-integrable systems. Phys. Reports163, 205–297 (1988)
[F] Folland, G.: Harmonic Analysis in Phase Space, Princeton: Princeton University Press, 1988
[FRM] Ford, J., Mantica, G., Ristov, G.H.: The Arnol'd cat: Failure of the correspondence principle. Physica D25, 105–135 (1991)
[H] Hasse, H.: Number Theory, Berlin, Heidelberg, New York: Springer, 1980
[HMR] Helffer, B., Martinez, A., Robert, D.: Ergodicité et limite semi-classique. Commun. Math. Phys.09, 313–326 (1987)
[I] Isola, S.: ξ-function and distribution of periodic orbits of toral automorphisms. Europhys. Lett.11, 517–522 (1990)
[Ka1] Katz, N.: Gauss Sums, Kloosterman Sums, and Monodromy Groups. Princeton, NJ: Princeton University Press, 1988
[Ka2] Katz, N. Sommes Exponentielles. Asterisque79, 1980
[Ke1] Keating, J.: Ph. D. thesis, University of Bristol (1989)
[Ke2] Keating, J.: Asymptotic properties of the periodic orbits of the cat map. Nonlinearity4, 277–307 (1991)
[Ke3] Keating, J.: The cat maps: Quantum mechanics and classical motion. Nonlinearity4, 309–341 (199)
[Kn] Knabe, S.: On the quantization of the Arnold cat map. J. Phys. A23, 2013–2025 (1990)
[LV] Leboeuf, J., Voros, A.: Chaos revealing multiplicative represenation of quantum eigenstates. J. Phys. A23, 1765–1773 (1990)
[M] Mañe, R.: Ergodic Theory and Differentiable Dynamics. Berlin, Heidelberg, New York: Springer, 1987
[N] Narnhofer, H.: Quantized Arnold cats can be entropicK-systems. J. Math. Phys.33, 1502–1510 (1992)
[PV] Percival, I., Vivaldi, F.: Arithmetical properties of strongly chaotic motions. Physica D25, 105–130 (1987)
[PP] Parry, W., Pollicott, M.: Zea functions and the periodic orbit structure of hyperbolic dynamics, Astérisque187–188 (1990)
[RM] Ram Murthy, M.: Artin's conjecture for primitive roots. Mathematical Intelligencer10, 59–70 (1988)
[S] Schnirelman, A.: Ergodic properties of the eigenfunctions. Usp. Math. Nauk29, 181–182 (1974)
[Sa] Sarnak, P.: Arithmetic Quantum Chaos. Tel Aviv Lectures 1993 (to apper)
[Sc] Schmidt, W.: Equations over finite fields. An elementary approach. Vol.536, Lecture Notes in Mathematis, 1976
[VN] Von Neumann, J.: Beweis des Ergoidensatzes und desH-Theorems in der Neuen Mechanik. Zschr. f. Physik57, 30–70 (1929)
[Z] Zelditch, S.: Uniform distribution of eigenfunction on compact hyperbolic surfaces. Duke Math. J.55, 919–941 (1987)
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Communicated by Ya. G. Sinai
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Esposti, M.D., Graffi, S. & Isola, S. Classical limit of the quantized hyperbolic toral automorphisms. Commun.Math. Phys. 167, 471–507 (1995). https://doi.org/10.1007/BF02101532
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DOI: https://doi.org/10.1007/BF02101532