Fractal dimensions and the phenomenon of intermittency in quantum dynamics

Duke Mathematical Journal

Volumn 110, Issue 1, 2001, Pages 161-193

  Barbaroux, Jean Marie ^a   Germinet, François ^a   Tcheremchantsev, Serguei ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000938886     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/S0012-7094-01-11015-6     Document Type: Article

References (37)

1. J.-M. Barbaroux, J.-M. Combes, and R. Montcho, Remarks on the relation between quantum dynamics and fractal spectra, J. Math. Anal. Appl. 213 (1997), 698-722.
2. J.-M. Barbaroux, F. Germinet, and S. Tcheremchantsev, Nonlinear variation of diffusion exponents in quantum dynamics, C. R. Acad. Sci. Paris. Sér. I Math. 330 (2000), 409-414.
3. _, "Quantum diffusion and generalized Rényi dimensions of spectral measures" in Journées "Équations aux dérivées partielles" (La Chapelle sur Erdre, France, 2000), exp. no. I, Univ. Nantes, Nantes, 2000.
4. _, Generalized fractal dimensions: Basic properties and equivalences, to appear in J. Math. Pures Appl. (9).
5. J.-M. Barbaroux and H. Schulz-Baldes, Anomalous quantum transport in presence of self-similar spectra, Ann. Inst. H. Poincaré Phys. Théor. 71 (1999), 539-559.
6. J.-M. Barbaroux and S. Tcheremchantsev, Universal lower bounds for quantum diffusion, J. Funct. Anal. 168 (1999), 327-354.
7. J. Bellissard and H. Schulz-Baldes, Anomalous transport: A mathematical framework, Rev. Math. Phys. 10 (1998), 1-46.
8. J.-M. Combes, "Connections between quantum dynamics and spectral properties of time-evolution operators" in Differential Equations with Applications to Mathematical Physics, ed. W. F. Ames, E. M. Harrel, and J. V. Herod, Math. Sci. Engrg. 192, Academic Press, Boston, 1993, 59-68.
9. J.-M. Combes and C. Mantica, Fractal dimensions and quantum evolution associated with sparse potential Jacobi matrices, preprint, 2000, http://ma.utexas.edu/mp_arc-bin/mpa?vn=00-365
10. C. D. Cutler, Some results on the behaviour and estimation of the fractal dimensions of distributions on attractors, J. Statist. Phys. 62 (1991), 651-708.
11. S. De Bièvre and G. Forni, Transport properties of kicked and quasi-periodic Hamiltonians, J. Statist. Phys. 90 (1998), 1201-1223.
12. S. De Bièvre and F. Germinet, Dynamical Localization for the random dimer Schrödinger operator, J. Statist. Phys. 98 (2000), 1135-1148.
13. D. H. Dunlap, H.-L. Wu, and P. W. Phillips, Absence of localization in a random dimer model, Phys. Rev. Lett. 65 (1990), 88-91., http://prola.aps.org/PRL/v65/i1
14. K. J. Falconer, Techniques in Fractal Geometry, Wiley, Chichester, England, 1997.
15. I. Guarneri, Spectral properties of quantum diffusion on discrete lattices, Europhys. Lett. 10 (1989), 95-100.
16. _, On an estimate concerning quantum diffusion in the presence of a fractal spectrum, Europhys. Lett. 21 (1993), 729-733.
17. I. Guarneri and H. Schulz-Baldes, Intermittent lower bound on quantum diffusion, Lett. Math. Phys. 49 (1999), 317-324.
18. _, Lower bounds on wave packet propagation by packing dimensions of spectral measures, Math. Phys. Electron. J. 5 (1999), no. 1, http://ma.utexas.edu/mpei/MPEJ.html#v5.
19. _, Upper bounds for quantum dynamics governed by Jacobi matrices with self-similar spectra, Rev. Math. Phys. 11 (1999), 1249-1268.
20. T. C. Halsey, M. Jensen, L. Kadanov, I. Procaccia, and B. Schraiman, Fractal measures and their singularities: The characterization of strange sets, Phys. Rev. A (3) 33 (1986), 1141-1151.
21. H. G. E. Hentschel and I. Procaccia, The infinite number of generalized dimensions of fractals and strange attractors, Phys. D 8 (1983), 435-444.
22. S. Jitomirskaya and Y. Last, Power-law subordinacy and singular spectra, I: Half-line operators, Acta Math. 183 (1999), 171-189.
23. R. Ketzmerick, K. Kruse, S. Kraut, and T. Geisel, What determines the spreading of a wave packet?, Phys. Rev. Lett. 79 (1997), 1959-1963.
24. R. Ketzmerick, G. Petschel, and T. Geisel, Slow decay of temporal correlations in quantum systems with Cantor spectra, Phys. Rev. Lett. 69 (1992), 695-698., http://prola.aps.org/toc/PRL/v69/i5
25. A. Kiselev and Y. Last, Solutions, spectrum, and dynamics for Schrödinger operators on infinite domains, Duke Math. J. 102 (2000), 125-150.
26. H. Kunz and B. Souillard, Sur le spectre des opérateurs aux différences finies aléatoires, Comm. Math. Phys. 78 (1980/81), 201-246.
27. Y. Last, Quantum dynamics and decomposition of singular continuous spectra, J. Funct. Anal 142 (1996), 406-445.
28. G. Mantica, "Quantum intermittency in almost periodic systems derived from their spectral properties" in Lattice Dynamics (Paris, 1995), Phys. D 103, North-Holland, Amsterdam, 1997, 576-589.
29. _, "Wave propagation in almost-periodic structures" in Physics and Dynamics Between Chaos, Order, and Noise (Berlin, 1996), Phys. D 109, North-Holland, Amsterdam, 1997, 113-127.
30. L. Olsen, A multifractal formalism, Adv. Math. 116 (1995), 82-196.
31. Y. B. Pesin, Dimension Theory in Dynamicals Systems: Contemporary Views and Applications, Chicago Lectures in Mathematics, Univ. Chicago Press, Chicago, 1996.
32. F. Piéchon, Anomalous diffusion properties of wave packet on quasiperiodic chains, Phys. Rev. Lett. 76 (1996), 4372-4375., http://prola.aps.org/toc/PRL/v76/i23
33. M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press, New York, 1972.
34. R. del Rio, S. Jitomirskaya, Y. Last, and B. Simon, Operators with singular continuous spectrum, IV: Hausdorff dimensions, rank one pertubations, and localization, J. Anal. Math. 69 (1996), 153-200.
35. S. J. Taylor and C. Tricot, Packing measure, and its evaluation for a Brownian path, Trans. Amer. Math. Soc. 288 (1985), 679-699.
36. S. Tcheremchantsev, Mixed lower bounds in quantum dynamics, preprint, 2001, http://ma. utexas.edu/mp/_arc-bin/mpa?yn=01-227
37. _, in preparation.