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Recurrent versus diffusive dynamics for a kicked quantum oscillator. (English) Zbl 0766.58061
This work studies the large-time behavior of the kicked quantum oscillator, where the time-dependent kick force is of the form \(F(t)=\sum_ n\varepsilon_ n\delta(t-n)\) with the \(\varepsilon_ n\) alternating randomly in sign. Typically, there is a diffusive energy growth in the system corresponding to the escape to infinity of classical trajectories. But there is also an exceptional ‘resonant’ time evolution with partially recurrent behavior of both the classical and the quantum paths. Though the author believes that the model provides a reasonable scenario for quantum chaos, it would be more fair to say that the stochastic behavior of this model is clearly induced by the external random force.

MSC:
58Z05 Applications of global analysis to the sciences
11B85 Automata sequences
81Q50 Quantum chaos
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References:
[1] J.E. Bayfield and P.M. Koch , Phys. Rev. Lett. , Vol. 33 , 1974 , p. 258 .
[2] J.G. Leopold and I.C. Percival , Phys. Rev. Lett. , Vol. 41 , 1978 , p. 944 .
[3] G. Casati , B.V. Chirikov and D.L. Shepelyanski , Phys. Rev. Lett. , Vol. 53 , 1984 , p. 2525 .
[4] R. Jensen , Phys. Rev. Lett. , Vol. 49 , 1982 , p. 1365 .
[5] S. Washburn and R.A. Webb , Adv. Phys. , Vol. 35 , 1986 , p. 375 .
[6] J. Moser , Stable and random motions in dynamical systems , Annals of Math. Studies , Princeton Univ. Press , Princeton , 1973 . MR 442980 | Zbl 0271.70009 · Zbl 0271.70009
[7] G. Casati , B.V. Chirikov , D.L. Shepelyanski and I. Guarneri , Phys. Rep. , Vol. 154 , 1987 , p. 77 .
[8] G. Casati and L. Molinari , Progr. Theo. Phys. Suppl. , Vol. 98 , 1989 , p. 287 . MR 1033467
[9] S. Fishman , D.R. Grempel and R.E. Prange , Phys. Rev. Lett. , Vol. 49 , 1982 , p. 509 . MR 669169
[10] M. Combescure , J. Stat. Phys. , Vol. 59 , 1990 , p. 679 . MR 1063177 | Zbl 0713.58044 · Zbl 0713.58044
[11] I. Guarneri , Lett. Nuovo Cimento , Vol. 40 , 1984 , p. 171 . MR 755105
[12] M. Queffelec , Substitution Dynamical Systems. Spectral Analysis , Springer , Berlin , 1987 . MR 924156 | Zbl 0642.28013 · Zbl 0642.28013
[13] B. Dorizzi , B. Grammaticos and Y. Pomeau , J. Stat. Phys. , Vol. 37 , 1984 , p. 93 . MR 774886
[14] T. Hogg and B.A. Huberman , Phys. Rev. Lett. , Vol. 48 , 1982 , p. 711 . MR 647749
[15] P.W. Milonni , J.R. Ackerhalt and M.E. Goggin , Phys. Rev. A , Vol. 35 , 1987 , p. 1714 . MR 879246
[16] Y. Pomeau , B. Dorizzi and B. Grammaticos , Phys. Rev. Lett. , Vol. 56 , 1986 , p. 681 . MR 827933
[17] K. Rzazewski and J. Mostowski , Phys. Rev. A , Vol. 35 , 1987 , p. 4414 . MR 892454
[18] J.M. Luck , H. Orland and U. Smilansky , J. Stat. Phys. , Vol. 53 , 1988 , p. 551 . MR 980148
[19] B. Sutherland , Phys. Rev. Lett. , Vol. 57 , 1986 , p. 770 .
[20] M. Combescure , J. Stat. Phys. , Vol. 62 , 1991 , issue 3 - 4 . Zbl 0746.11014 · Zbl 0746.11014
[21] L. Bunimovich , H.R. Jauslin , J.J. Lebowitz , A. Pellegrinotti and P. Nielaba , J. Stat. Phys. , 1991 . Zbl 0741.70017 · Zbl 0741.70017
[22] G.A. Hagedorn , M. Loss and J. Slawny , J. Phys. A , Vol. 19 , 1986 , p. 521 . MR 833433 | Zbl 0601.70013 · Zbl 0601.70013
[23] T. Kato , J. Fac. Sci. Univ. Tokyo , Sect. I, Vol. 17 , 1970 , p. 241 . MR 279626 | Zbl 0222.47011 · Zbl 0222.47011
[24] J. Piepenbrink , Pulsed Quantum Mechanical Hamiltonians , West Georgia College preprint , 1990 .
[25] I.S. Gradshteyn and I.M. Ryzhik , Tables of Integrals, Series and Products , Academic Press Inc ., 1965 . · Zbl 0918.65002
[26] Y.S. Chow and H. Teicher , Probability Theory , Springer , Berlin - Heidelberg - New York , 1978 . MR 513230 | Zbl 0399.60001 · Zbl 0399.60001
[27] W. Stout , Almost Sure Convergence , Academic Press , New York , 1974 . MR 455094 | Zbl 0321.60022 · Zbl 0321.60022
[28] A.N. Shiryayev , Probability , Springer , Berlin - Heidelberg - New York , 1984 .
[29] R. Blümel , R. Meir and U. Smilansky , Phys. Lett. , Vol. 103 A, 1984 , p. 353 .
[30] B. Milek and P. Seba , Phys. Rev. A , Vol. 42 , 1990 , pp. 3213 - 3220 . MR 1072316
[31] B. Saffari , C. R. Acad. Sci. Paris , T. 304 , 1987 , pp. 127 - 130 . MR 880114 | Zbl 0608.10052 · Zbl 0608.10052
[32] B. Saffari , Barker Sequences and Littlewood’s ”Two-Sided Conjectures” on Polynomials with \pm 1 Coefficients , Orsay review article, 1990 . Zbl 0739.11030 · Zbl 0739.11030
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