Dynamical upper bounds on wavepacket spreading

American Journal of Mathematics

Volumn 125, Issue 5, 2003, Pages 1165-1198

  Killip, Rowan ^a   Kiselev, Alexander ^b   Last, Yoram ^c  

^a CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

^b UNIVERSITY OF WISCONSIN MADISON   (United States)

^c HEBREW UNIVERSITY OF JERUSALEM   (Israel)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0141541804     PISSN: 00029327     EISSN: None     Source Type: Journal    
DOI: 10.1353/ajm.2003.0031     Document Type: Article

References (44)

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, Fractal dimensions and the phenomenon of intermittency in quantum dynamics, Duke Math. J. 110 (2001), 161-193.
3. J. M. Barbaroux and H. Schulz-Baldes, Anomalous quantum transport in presence of self-similar spectra, Ann. Inst. H. Poincaré 71 (1999), 1-21.
4. J. M. Barbaroux and S. Tcheremchantsev, Univeral lower bounds for quantum diffusion, J. Funct. Anal. 168 (1999), 327-354.
5. Y. Berezanskii, Expansions in Eigenfunctions of Self-Adjoint Operators, Transl. Math. Monogr., vol. 17, Amer. Math. Soc., Providence, RI, 1968.
6. J. Bellissard, B. Iochum, E. Scoppola, and D. Testard, Spectral properties of one-dimensional quasicrystals, Comm. Math. Phys. 125 (1989), 527-543.
7. M. Birman and M. Solomyak, Spectral Theory of Selfadjoint Operators in Hilbert Space (trans. by S. Khrushchëv and V. Peller), Math. Appl. (Soviet Ser.), D. Rédel Publishing Co., Dordrecht, 1987.
8. R. Carmona and J. Lacroix, Spectral Theory of Random Schrödinger Operators, Birkhauser, Boston, MA, 1990.
9. J. M. Combes, Connections between quantum mechanics and spectral properties of time-evolution operators, Differential Equations with Applications to Mathematical Physics (W. F. Ames, E. M. Harrel, and J. V. Herod, eds.), Academic Press, Boston, 1993, pp. 59-68.
10. J. M. Combes and G. Mantica, Fractal dimensions and quantum evolution associated with sparse potential Jacobi matrices, Long Time Behaviour of Classical and Quantum Systems (Bologna, 1999), Ser. Concr. Appl. Math., vol. 1, World Sci. Publishing, River Edge, NJ, 2001, pp. 107-123.
11. D. Damanik, α-continuity properties of one-dimensional quasicrystals, Comm. Math. Phys. 192 (1998), 169-182.
12. _, Gordon-type arguments in the spectral theory of one-dimensional quasicrystals, Directions in Mathematical Quasicrystals (M. Baake and R. V. Moody, eds.), CRM Monog. Ser., vol. 13, Amer. Math. Soc., Providence, RI, 2000, pp. 277-304.
13. D. Damanik, R. Killip, and D. Lenz, Uniform spectral properties of one-dimensional quasicrystals, III. α-continuity, Comm. Math. Phys. 212 (2000), 191-204.
14. R. del Rio, S. Jitomirskaya, Y. Last, and B. Simon, Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank-one perturbations, and localization, J. Anal. Math. 69 (1996), 153-200.
15. F. Germinet and S. de Bievre, Dynamical localization of discrete and continuous random Schrödinger operators, Comm. Math. Phys. 194 (1998), 323-341.
16. I. Guarneri, Spectral properties of quantum diffusion on discrete lattices, Europhys. Lett. 10 (1989), 95-100.
17. _, On an estimate concerning quantum diffusion in the presence of a fractal spectrum, Europhys. Lett. 21 (1993), 729-733.
18. I. Guarneri and G. Mantica, Multifractal energy spectra and their dynamical implications, Phys. Rev. Lett. 73 (1994), 3379-3383.
19. I. Guarneri and H. Schulz-Baldes, Intermittent lower bound on quantum diffusion, Lett. Math. Phys. 49 (1999), 317-324.
20. _, Lower bounds on wave packet propagation by packing dimensions of spectral measures, Electron J. Math. Phys. 5 (1999).
21. B. Iochum and D. Testard, Power law growth for the resistance in the Fibonacci model, J. Stat. Phys. 65 (1991), 715-723.
22. S. Jitomirskaya and Y. Last, Power law subordinacy and singular spectra. I. Half line operators, Acta Math. 183 (1999), 171-189.
23. _, Power law subordinacy and singular spectra. II. Line operators, Comm. Math. Phys. 211 (2000), 643-658.
24. R. Ketzmerick, K. Kruse, S. Kraut, and T. Geisel, What determines the spreading of a wave-packet, Phys. Rev. Lett. 79 (1997), 1959-1963.
25. A. Kiselev and Y. Last, Solutions, spectrum, and dynamics for Schrödinger operators on infinite domains, Duke Math. J. 102 (2000), 125-150.
26. M. Kohmoto, L. Kadanoff, and C. Tang, Localization problems in one dimension: mapping and escape, Phys. Rev. Lett. 50 (1983), 1870-1873.
27. Y. Last, Quantum dynamics and decompositions 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, Phys. D 103 (1997), 576-589.
29. _, Wave propagation in almost-periodic structures, Phys. D 109 (1997), 113-127.
30. F. Piechon, Anomalous diffusion properties of wave packets on quasiperiodic chains, Phys. Rev. Lett. 76 (1996), 4372-4375.
31. L. Raymond, A constructive gap labeling for the discrete Schrödinger operator on a quasiperiodic chain, preprint.
32. M. Reed and B. Simon, Methods of Modern Mathematical Physics, III. Scattering Theory, Academic Press, London, 1979.
33. B. Simon, Absence of ballistic motion, Comm. Math. Phys. 134 (1990), 209-212.
34. _, Operators with singular continuous spectrum, VII. Examples with borderline time decay, Comm. Math. Phys. 176 (1996), 713-722.
35. _, The classical moment problem as a self-adjoint finite difference operator, Adv. Math. 137 (1998), 82-203.
36. Y. G. Sinai, Anomalous transport in almost periodic media, Uspekhi Mat. Nauk 54 (1999), 181-208.
37. B. Sutherland and M. Kohmoto, Resistance of a one-dimensional quasicrystal: power-law growth, Phys. Rev. B 36 (1987), 5877-5886.
38. A. Sütö, The spectrum of a quasiperiodic Schrödinger operator, Comm. Math. Phys. 111 (1987), 409-415.
39. _, Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian, J. Stat. Phys. 56 (1989), 525-531.
40. _, Schrödinger difference equation with deterministic ergodic potentials, Beyond Quasicrystals (Les Hauches, 1994) (F. Axel and D. Gratias, eds.), Springer-Verlag, Berlin, 1995, pp. 481-549.
41. S. Tcheremchantsev, How to prove dynamical localization, Comm. Math. Phys. 221 (2001), 27-56.
42. _, Mixed lower bounds in Quantum dynamics, preprint.
43. M. Toda, Theory of Nonlinear Lattices, Springer-Verlag, Berlin, 1989.
44. M. Wilkinson and E. J. Austin, Spectral dimension and dynamics for Harper's equation, Phys. Rev. B 50 (1994), 1420-1429.