Remarks on the relation between quantum dynamics and fractal spectra

Journal of Mathematical Analysis and Applications

Volumn 213, Issue 2, 1997, Pages 698-722

  Barbaroux, J M ^a,b   Combes, J M ^a,b   Montcho, R ^a,b  

^a CENTRE DE PHYSIQUE THÉORIQUE   (France)

^b UNIVERSITÉ DE TOULON   (France)

Author keywords

[No Author keywords available]

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; FRACTALS; NONLINEAR EQUATIONS; PERTURBATION TECHNIQUES;

FRACTAL SPECTRA; SCHRODINGER EQUATIONS;

QUANTUM THEORY;

EID: 0031220833     PISSN: 0022247X     EISSN: None     Source Type: Journal    
DOI: 10.1006/jmaa.1997.5571     Document Type: Article

References (23)

1. 1. J. Bellissard, B. Iochum, E. Scoppola, and D. Testard, Spectral properties of one dimensional quasicrystal, Comm. Math. Phys. 125 (1989), 527-543.
2. 2. J. Bellissard, R. Mosseri, and J. X. Zhong, Green function analysis of energy spectra scaling properties, J. Phys. Condens. Matter 7 (1995), 3507-3514.
3. 3. A. Bovier and J. M. Ghez, Spectral properties of one dimensional Schrödinger operators with potentials generated by substitutions, Comm. Math. Phys. 158 (1993), 45-66.
4. 4. J. M. Combes, Connections between quantum dynamics and spectral properties of time-evolution operators, in "Differential Equations with Applications to Mathematical Physics" (W. F. Ames, E. M. Harrell, and J. V. Herod, Eds.), Academic Press, Boston, 1993.
5. 5. R. Del Rio, S. Jitomirskaya, Y. Last, and B. Simon, Operators with singular continuous spectrum. IV. Hausdorff dimensions, rank one perturbations and localization, J. Analyse Math. 69 (1996), 153-200.
6. 6. K. J. Falconer, "The Geometry of Fractal Sets," Cambridge Tracts in Mathematics, Cambridge Univ. Press, Cambridge, UK, 1985.
7. 7. I. Guarneri, Spectral properties of quantum diffusion on discrete lattices, Europhys. Lett. 10 (1989), 95-100.
8. 8. I. Guarneri, Private communication and communication at the workshop "Singular Spectra" at the Irwin Schrödinger Institute of Vienna, October 1995.
9. 9. C. A. Guerin and M. Holschneider, Three equivalent definitions of the correlation dimension of a probability measure, J. Statist. Phys. 29 (1996), 7651-7667.
10. 10. 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), 369-379.
11. 11. M. Holschneider, Fractal wavelet dimensions and localization, Comm. Math. Phys. 160 (1994), 457-473.
12. 12. 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.
13. 13. Y. Last, Quantum dynamics and decompositions of singular continuous spectra, J. Funct. Anal. 142 (1996), 406-445.
14. 14. K.-S. Lau, Fractal measures and mean p-variations, J. Funct. Anal. 108 (1992), 427-457.
15. 15. Y. B. Pesin, On rigourous mathematical definitions of correlation dimension and generalized spectrum for dimensions, J. Statist. Phys. 71 (1993), 529-547.
16. 16. C. A. Rogers, "Hausdorff Measures," Cambridge Univ. Press, London, 1970.
17. 17. M. Reed and B. Simon, "Methods of Modern Mathematical Physics," Vol. 3, Academic Press, New York, 1978.
18. 18. R. S. Strichartz, Fourier asymptotics of fractal measures, J. Funct. Anal. 89 (1990), 154-187.
19. 19. A. Süto, Singular continuous spectrum on a Cantor set of zero Lebesgue measure, J. Statist. Phys. 56 (1989), 525-531.
20. 20. B. Simon and T. Wolff, Singular continuous spectrum under rank one perturbations and localization for random hamiltonians, Comm. Pure Appl. Math. 39 (1986), 75-90.
21. 21. N. Wiener, Generalized harmonic analysis, Acta Math. 55 (1930), 117-258.
22. 22. L. S. Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynamical Systems 2 (1982), 109-124.
23. 23. J. X. Zhong and R. Mosseri, Quantum dynamics in quasiperiodic systems, preprint, 1996.