Quantum dynamics and decompositions of singular continuous spectra

Journal of Functional Analysis

Volumn 142, Issue 2, 1996, Pages 406-445

  Last, Yoram ^a  

^a California Institute of Technology   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030589769     PISSN: 00221236     EISSN: None     Source Type: Journal    
DOI: 10.1006/jfan.1996.0155     Document Type: Article

References (39)

1. 1. W. O. Amrein, "Non-Relativistic Quantum Dynamics," Reidel, Dordrecht, 1981.
2. 2. J. Avron and B. Simon, Transient and recurrent spectrum, J. Funct. Anal. 43 (1981), 1-31.
3. 3. J. Avron and B. Simon, Almost periodic Schrödinger operators, II. The integrated density of states, Duke Math. J. 50 (1983), 369-391.
4. 4. J. Avron, P. van Mouche, and B. Simon, On the measure of the spectrum for the almost Mathieu operator, Commun. Math. Phys. 132 (1990), 103-118.
5. 5. R. Carmona and J. Lacroix, "Spectral Theory of Random Schrödinger Operators," Birkhäuser, Boston, 1990.
6. 6. M. D. Choi, G. A. Elliott, and N. Yui, Gauss polynomials and the rotation algebra, Invent. Math. 99 (1990), 225-246.
7. 7. 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.
8. 8. H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, "Schrödinger Operators," Springer, Berlin/New York, 1987.
9. 9. R. del Rio, S. Jitomirskaya, Y. Last, and B. Simon, Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank-one perturbations, and localization, J. D'analyse Math., to appear.
10. 10. R. del Rio, S. Jitomirskaya, Y. Last, and B. Simon, What is localization?, Phys. Rev. Lett. 75 (1995), 117-119.
11. 11. R. del Rio, N. Makarov, and B. Simon, Operators with singular continuous spectrum, II. Rank-one operators, Commun. Math. Phys. 165 (1994), 59-67.
12. 12. K. J. Falconer, "Fractal Geometry," Wiley, Chichester, 1990.
13. 13. T. Geisel, R. Ketzmerick, and G. Petschel, New class of level statistics in quantum systems with unbounded diffusion, Phys. Rev. Lett. 66 (1991), 1651-1654.
14. 14. A. Gordon, Pure point spectrum under 1-parameter perturbations and instability of Anderson localization, Commun. Math. Phys. 164 (1994), 489-505.
15. 15. I. Guarneri, Spectral properties of quantum diffusion on discrete lattices, Europhys. Lett. 10 (1989), 95-100.
16. 16. I. Guarneri, On an estimate concerning quantum diffusion in the presence of a fractal spectrum, Europhys. Lett. 21 (1993), 729-733.
17. 17. I. Guarneri and G. Mantica, On the asymptotic properties of quantum dynamics in the presence of a fractal spectrum, Ann. Inst. H. Poincaré Phys. 61 (1994), 369-379.
18. 18. I. Guarneri and G. Mantica, Multifractal energy spectra and their dynamical implications, Phys. Rev. Lett. 73 (1994), 3379-3382.
19. 19. G. H. Hardy and E. M. Wright, "An Introduction to the Theory of Numbers," fifth ed., Oxford Univ. Press, Oxford, 1979.
20. 20. M. Holschneider, Fractal wavelet dimensions and localization, Commun. Math. Phys. 160 (1994), 457-473.
21. 21. S. Jitomirskaya and B. Simon, Operators with singular continuous spectrum, III. Almost periodic Schrödinger operators, Commun. Math. Phys. 165 (1994), 201-205.
22. 22. 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.
23. 23. Y. Last, Zero measure spectrum for the almost Mathieu operator, Commun. Math. Phys. 164 (1994), 421-432.
24. 24. M. Reed and B. Simon, "Methods of Modern Mathematical Physics, I. Functional Analysis," Academic Press, London/San Diego, 1980.
25. 25. M. Reed and B. Simon, "Methods of Modern Mathematical Physics, II. Fourier Analysis, Self-Adjointness," Academic Press, London/San Diego, 1975.
26. 26. M. Reed and B. Simon, "Methods of Modern Mathematical Physics, III. Scattering Theory," Academic Press, London/San Diego, 1979.
27. 27. C. A. Rogers, "Hausdorff Measures," Cambridge Univ. Press, London, 1970.
28. 28. C. A. Rogers and S. J. Taylor, The analysis of additive set functions in Euclidean space, Acta Math. Stock. 101 (1959), 273-302.
29. 29. C. A. Rogers and S. J. Taylor, Additive set functions in Euclidean space, II, Acta Math. Stock. 109 (1963), 207-240.
30. 30. B. Simon, "Trace Ideals and their Applications," Cambridge Univ. Press, London, 1979.
31. 31. B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (1982), 447-526.
32. 32. B. Simon, Almost periodic Schrödinger operators: A review, Adv. Appl. Math. 3 (1982), 563-490.
33. 33. B. Simon, Absence of ballistic motion, Commun. Math. Phys. 134 (1990), 209-212.
34. 34. B. Simon, Operators with singular continuous spectrum, I. General operators, Ann. Math. 141 (1995), 131-145.
35. 35. B. Simon, Operators with singular continuous spectrum, VII. Examples with borderline time decay, submitted.
36. 36. R. S. Strichartz, Fourier asymptotics of fractal measures, J. Funct. Anal. 89 (1990), 154-187.
37. 37. D. J. Thouless, Bandwidth for a quasiperiodic tight binding model, Phys. Rev. B 28 (1983), 4272-4276.
38. 38. M. Toda, "Theory of Nonlinear Lattices," 2nd ed., Chap. 4, Springer, Berlin/New York, 1989.
39. 39. M. Wilkinson and E. J. Austin, Spectral dimension and dynamics for Harper's equation, Phys. Rev. B 50 (1994), 1420-1429.