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Wave propagation in almost-periodic structures. (English) Zbl 0925.58041

Summary: The unusual phenomena occurring in the wave propagation in almost-periodic structures - both classical and quantum - are studied in this paper. Focusing our attention on structures with spectral measures in the family of disconnected iterated function systems, we describe and characterize the concept of “quantum intermittency”. This theory shows that the non-trivial renormalization properties of the set of orthogonal polynomials associated with these systems are the origin of such “intermittency”, and leads to a new determination of the exponents of the asymptotic growth of the moments of the position operator.

MSC:

37A99 Ergodic theory
28A80 Fractals
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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[1] Brillouin, L., Wave Propagation in Periodic Structures (1953), Dover: Dover New York, and references therein · Zbl 0050.45002
[2] Akhiezer, N. I., The Classical Moment Problem (1965), Hafner: Hafner New York · Zbl 0135.33803
[3] Dyson, F., Phys. Rev., 92, 1331 (1953)
[4] Cycon, H. L.; Froese, R. G.; Kirsch, W.; Simon, B., Schrödinger Operators (1987), Springer: Springer Berlin, and references therein · Zbl 0619.47005
[5] Simon, B., Almost-periodic Schrödinger operators: A review, Adv. Appl. Math., 3, 463-490 (1982), and references therein · Zbl 0545.34023
[6] Kohmoto, M.; Oono, Y., Cantor spectrum of an almost-periodic Schrödinger equation and a dynamical map, Phys. Lett. A, 102, 145-148 (1984)
[7] Sütö, A., Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian, J. Stat. Phys., 56, 525-531 (1989) · Zbl 0712.58046
[8] Guarneri, I.; Mantica, G., Multi-fractal energy spectra and their dynamical implications, Phys. Rev. Lett., 73, 3379-3382 (1994)
[9] Mantica, G., Quantum intermittency in almost-periodic systems derived from their spectral properties, Physica D, 103, 576-589 (1997) · Zbl 1194.81085
[10] Evangelou, S. N.; Katsanos, D. E., Multi-fractal quantum evolution at the mobility edge, J. Phys. A, 26, L1243-L1250 (1993)
[11] Wilkinson, M.; Austin, E. J., Spectral dimension and dynamics for Harper’s equation, Phys. Rev. B, 50, 1420 (1994)
[12] Hiramoto, H.; Abe, S., Dynamics of an electron in quasi-periodic systems. II. Harper Model, J. Phys. Soc. Japan, 57, 1365-1371 (1988)
[13] Abe, S.; Hiramoto, H., Fractal dynamics of electron Wave packets in one-dimensional quasi-periodic Systems, Phys. Rev. A, 36, 5349-5352 (1987)
[14] Geisel, T.; Ketzmerick, R.; Petschel, G., Metamorphosis of a Cantor spectrum due to classical chaos, Phys. Rev. Lett., 67, 3635 (1991)
[15] Hiramoto, H.; Kohmoto, M., Electronic spectral and wavefunction properties of one-dimensional quasi-periodic Systems: A Scaling Approach, Int. J. Mod. Phys. B, 6, 281-320 (1992)
[16] Hutchinson, J., Fractals and self-similarity, Indiana J. Math., 30, 713-747 (1981) · Zbl 0598.28011
[17] Barnsley, M. F.; Demko, S. G., Iterated function systems and the global construction of fractals, (Proc. Roy. Soc. London A, 399 (1985)), 243-275 · Zbl 0588.28002
[18] Barnsley, M. F., Fractals everywhere (1988), Academic Press: Academic Press New York · Zbl 0691.58001
[19] Mantica, G., A Stieltjes technique for computing Jacobi matrices associated with singular measures, Constr. Appr., 12, 509-530 (1996) · Zbl 0878.42014
[20] Barnsley, M. F.; Geronimo, J. S.; Harrington, A. N., Infinite-dimensional Jacobi matrices associated with Julia sets, (Proc. Am. Math. Soc., 88 (1983)), 625-630 · Zbl 0535.30025
[21] Bellissard, J.; Bessis, D.; Moussa, P., Chaotic states of almost-periodic Schrödinger operators, Phys. Rev. Lett., 49, 702-704 (1982)
[22] Servizi, G.; Turchetti, G.; Vaienti, S., Generalized dynamical variables and measures for the Julia Sets, II Nuovo Cimento B, 101, 285-307 (1988)
[23] Baker, G. A.; Bessis, D.; Moussa, P., A family of almost-periodic Schrödinger operators, Physica A, 124, 61-77 (1984) · Zbl 0598.47054
[24] Zygmund, A., Classical Trigonometric Series (1968), Cambridge University Press: Cambridge University Press Cambridge
[25] Bessis, D.; Fournier, J. D.; Servizi, G.; Turchetti, G.; Vaienti, S., Mellin transform of correlation integrals and generalized dimensions of strange sets, Phys. Rev. A, 36, 920-928 (1987)
[26] Bessis, D.; Geronimo, J. S.; Moussa, P., Mellin transform associated with Julia sets and physical applications, J. Stat. Phys., 34, 75-110 (1984) · Zbl 0602.58028
[27] Ketzmerick, R.; Petschel, G.; Geisel, T., Slow decay of temporal correlations in quantum systems with cantor spectra, Phys. Rev. Lett., 69, 695 (1992)
[28] Makarov, K. A., Asymptotic expansions for Fourier transform of singular self-affine measures, J. Math. Anal. Appl., 187, 259-286 (1994) · Zbl 0812.28004
[29] Bessis, D.; Mantica, G., Orthogonal polynomials associated to almost-periodic Schrödinger operators, J. Comput. Appl. Math., 48, 17-32 (1993) · Zbl 0793.35023
[30] Mantica, G.; Guzzetti, D., Fourier transform of orthogonal polynomials of singular measures (1996), preprint
[31] Feudel, U.; Pikovsky, A. S.; Zaks, M. A., Correlation properties of a quasi-periodically forced two-level system, Phys. Rev. E, 51, 1762-1769 (1995)
[32] Guarneri, I., On an estimate concerning quantum diffusion in the presence of a fractal spectrum, Europhys. Lett., 21, 729 (1993)
[33] Guarneri, I.; Mantica, G., On the asymptotic properties of quantum dynamics in the presence of a fractal spectrum, Ann. Inst. H. Poincaré, 61, 369 (1994) · Zbl 0817.47078
[34] Mantica, G.; Sloan, A., Chaotic optimization and the construction of fractals, Complex Systems, 3, 37-62 (1989) · Zbl 0729.58033
[35] Handy, C. R.; Mantica, G., Inverse problems in fractal construction: Moment method approach, Physica D, 43, 17-36 (1990) · Zbl 0704.58033
[36] Mantica, G.; Giraud, B., Non-linear Forecasting and Iterated Function Systems, Chaos, 2, 225-230 (1992) · Zbl 1055.37561
[37] Bessis, D.; Mantica, G., Construction of multifractal measures in dynamics from their invariance properties, Phys. Rev. Lett., 66, 2939-2942 (1991) · Zbl 1050.37500
[38] Piéchon, F., Phys. Rev. Lett., 76, 4372 (1996)
[39] Ford, J.; Mantica, G.; Ristow, G., The Arnol’d cat: Failure of the correspondence principle, Physica D, 50, 493-520 (1991) · Zbl 0742.58024
[40] Mantica, G.; Ford, J., On the completeness of the classical limit of quantum mechanics, (Cvitanović, P.; Percival, I.; Wirzba, S., Quantum Chaos — Quantum Measurement (1992), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht)
[41] Ford, J.; Mantica, G., Does quantum mechanics obey the correspondence principle? Is it complete?, Am. J. Phys., 60, 12 (1992)
[42] Bellissard, J.; Schulz-Baldes, H., (Janot, C.; Mosseri, R., Proc. 5th Int. Conf. on Quasicrystals (1995), World Scientific: World Scientific Singapore)
[43] Guarneri, I.; Di Meo, M., Fractal spectrum of a quasi-periodically driven spin system, J. Phys. A, 28, 2717-2728 (1995)
[44] Guarneri, I., Singular continuous spectra and discrete wave particle dynamics, J. Math. Phys., 37, 5195-5206 (1996) · Zbl 0894.46051
[45] Magnus, A., Towards quantitative results on fibonacci chain orthogonal polynomials, (Brezinsky, C.; Gori, L.; Ronveaux, A., Orthogonal Polynomials and Their Applications (1991), J.C. Baltzer AG, Scientific Publishing Co) · Zbl 0855.42014
[46] Niu, Q.; Nori, F., Renormalization group study of one-dimensional quasi-periodic systems, Phys. Rev. Lett., 57, 2057-2060 (1986)
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