Quantum intermittency: Old or new phenomenon?

Journal De Physique. IV : JP

Volumn 8, Issue 6, 1998, Pages

  Mantica, G ^a  

^a UNIVERSITY OF MILAN   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

FRACTALS; FUNCTIONS; TURBULENCE; WAVE PROPAGATION;

NON-TRIVIAL SCALING FUNCTIONS; QUANTUM INTERMITTENCY;

QUANTUM THEORY;

EID: 11644297481     PISSN: 11554339     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1051/jp4:1998635     Document Type: Article

References (34)

1. Out of Giovanni's papers on these topics-which are many-I would just like to recall a few: R. Benzi, G. Paladin, G. Parisi, and A. Vulpiani, On the multifractal nature of fully developed turbulence and chaotic systems, J. Phys. A 17 (1984) 3521. M.H. Jensen, G. Paladin, and A. Vulpiani, Intermittency in a cascade model of three-dimensional turbulence, Phys. Rev. A 43 (1991) 798. G. Paladin, and A. Vulpiani, Anomalous scaling laws in multifractal objects, Phys. Rep. 156 (1987) 147.
2. Out of Giovanni's papers on these topics-which are many-I would just like to recall a few: R. Benzi, G. Paladin, G. Parisi, and A. Vulpiani, On the multifractal nature of fully developed turbulence and chaotic systems, J. Phys. A 17 (1984) 3521. M.H. Jensen, G. Paladin, and A. Vulpiani, Intermittency in a cascade model of three-dimensional turbulence, Phys. Rev. A 43 (1991) 798. G. Paladin, and A. Vulpiani, Anomalous scaling laws in multifractal objects, Phys. Rep. 156 (1987) 147.
3. Out of Giovanni's papers on these topics-which are many-I would just like to recall a few: R. Benzi, G. Paladin, G. Parisi, and A. Vulpiani, On the multifractal nature of fully developed turbulence and chaotic systems, J. Phys. A 17 (1984) 3521. M.H. Jensen, G. Paladin, and A. Vulpiani, Intermittency in a cascade model of three-dimensional turbulence, Phys. Rev. A 43 (1991) 798. G. Paladin, and A. Vulpiani, Anomalous scaling laws in multifractal objects, Phys. Rep. 156 (1987) 147.
4. G. Mantica, Algorithms for the Quantum Evolution Generated by Jacobi Matrices of Singular Continuous Measures, Preprint (1997).
5. I. Guarneri and G. Mantica, Multifractal Energy Spectra and their Dynamical Implications, Phys. Rev. Lett. 73 (1994) 3379-3382.
6. G. Mantica, Quantum Intermittency in Almost Periodic Systems Derived from their Spectral Properties, Physica D 103 (1997) 576.
7. G. Mantica, Wave propagation in almost-periodic structures, Physica D 109 (1997) 113.
8. S.N. Evangelou and D.E. Katsanos, Multifractal Quantum Evolution at the Mobility Edge, J. Phys. A: Math. Gen. 26 (1993) L1243-L1250.
9. M. Wilkinson and E.J. Austin, Spectral Dimension and Dynamics for Harper's Equation, Phys. Rev. B, 50 (1994) 1420.
10. H.Hiramoto and S.Abe, Dynamics of an Electron in Quasi-Periodic Systems. I. Fibonacci Model, J.Phys.Soc. Japan 57 (1988) 230-240; II. Harper Model, ibid., (1988) 1365-1371.
11. H.Hiramoto and S.Abe, Dynamics of an Electron in Quasi-Periodic Systems. I. Fibonacci Model, J.Phys.Soc. Japan 57 (1988) 230-240; II. Harper Model, ibid., (1988) 1365-1371.
12. S.Abe and H.Hiramoto, Fractal Dynamics of Electron Wave Packets in One-Dimensional Quasi Periodic Systems, Phys. Rev. A. 36 (1987) 5349-5352.
13. See R. Artuso, G. Casati, F. Borgonovi, L. Rebuzzini, and I. Guarneri, Fractal and dynamical properties of the kicked Harper model, Int. J. Mod. Phys. B 8 (1994) 207, and references therein.
14. A.N. Kolmogorov, The local structure of turbulence in an incompressible flow with very large Reynold numbers, Dokl. Akad. Nauk SSSR 30 (1941) 301.
15. J. Hutchinson, Fractals and Self-Similarity, Indiana J. Math. 30 (1981) 713-747.
16. M.F. Barnsley and S.G. Demko, Iterated Function Systems and the Global Construction of Fractals, Proc. R. Soc. London A 399 (1985) 243-275.
17. M.F. Barnsley, Fractals Everywhere, Academic Press, (New York 1988).
18. G. Mantica, A Stieltjes Technique for Computing Jacobi Matrices Associated With Singular Measures, Constr. Appr. 12 (1996) 509-530.
19. M.F. Barnsley, J.S. Geronimo, and A.N. Harrington, Infinite-Dimensional Jacobi Matrices Associated With Julia Sets, Proc. Am. Math. Soc. 88 (1983) 625-630.
20. J. Bellissard, P. Bessis, and P. Moussa, Chaotic States of Almost Periodic Schrödinger Operators, Phys. Rev. Lett. 49 (1982) 702-704.
21. G. Servizi, G. Turchetti, and S. Vaienti, Generalized Dynamical Variables and Measures for the Julia Sets, Il Nuovo Cim. 101 B (1988) 285-307.
22. G.A. Baker, D. Bessis, and P. Moussa, A family of Almost Periodic Schrödinger Operators, Physica A 124 (1984) 61-77.
23. One should notice that, when λ is close to two, the maps φ± are not everywhere contractive, and, in principle, some results of I.F.S. theory may not hold. Nonetheless, it is possible to show that none of interest to us fail.
24. The result for n = 0 was found in R.Ketzmerick, G.Petschel and T.Geisel, Slow Decay of Temporal Correlations in Quantum Systems with Cantor Spectra, Phys.Rev.Lett. 69 (1992) 695. The general result, proven with the application of the Mellin transform is in [4]. See also the more mathematical papers [22] (for n = 0) and [24] for the general case.
25. K. A. Makarov, Asymptotic Expansions for Fourier Transform of Singular Self-Affine Measures, J. of Math. An. and Appl. 187 259-286 (1994).
26. D. Bessis and G. Mantica, Orthogonal Polynomials Associated to Almost Periodic Schrödinger Operators, J. Comp. Appl. Math. 48 (1993) 17-32.
27. G. Mantica and D. Guzzetti, Fourier Transform of Orthogonal Polynomials of Singular Measures Preprint (1997).
28. I.Guarneri, On an Estimate Concerning Quantum Diffusion in the Presence of a Fractal Spectrum, Europhys.Lett. 10, 95(1989); ibid., 21, 729 (1993); I.Guarneri and G.Mantica, On the Asymptotic Properties of Quantum Dynamics in the Presence of a Fractal Spectrum, Ann. Inst. H. Poincaré 61 (1994) 369.
29. I.Guarneri, On an Estimate Concerning Quantum Diffusion in the Presence of a Fractal Spectrum, Europhys.Lett. 10, 95(1989); ibid., 21, 729 (1993); I.Guarneri and G.Mantica, On the Asymptotic Properties of Quantum Dynamics in the Presence of a Fractal Spectrum, Ann. Inst. H. Poincaré 61 (1994) 369.
30. I.Guarneri, On an Estimate Concerning Quantum Diffusion in the Presence of a Fractal Spectrum, Europhys.Lett. 10, 95(1989); ibid., 21, 729 (1993); I.Guarneri and G.Mantica, On the Asymptotic Properties of Quantum Dynamics in the Presence of a Fractal Spectrum, Ann. Inst. H. Poincaré 61 (1994) 369.
31. G. Mantica and A. Sloan, Chaotic Optimization and the Construction of Fractals, Complex Systems 3, (1989) 37-62; C.R. Handy and G. Mantica, Inverse Problems in Fractal Construction: Moment Method Approach, Physica D 43, (1990) 17-36.
32. G. Mantica and A. Sloan, Chaotic Optimization and the Construction of Fractals, Complex Systems 3, (1989) 37-62; C.R. Handy and G. Mantica, Inverse Problems in Fractal Construction: Moment Method Approach, Physica D 43, (1990) 17-36.
33. In F. Piéchon, Phys. Rev. Lett. 76(1996) 4372, a relation formally equivalent to eq. (29), but radically different in the meaning of the symbols is proposed. Since Ref. [4] was accepted for publication much before [27] had appeared, I commented on the important differences in [5]: in [27] the focus is on the global density of states, rather than the local, and on average moments over ensembles of initial conditions.
34. The renormalization properties of Fibonacci systems-even if only approximate-are analogous: Q. Niu and F. Nori, Renormalization Group Study of One-Dimensional Quasi-Periodic Systems, Phys. Rev. Lett. 57 (1986) 2057-2060.