Upper bounds in quantum dynamics

Journal of the American Mathematical Society

Volumn 20, Issue 3, 2007, Pages 799-827

  Damanik, David ^a,b   Tcheremchantsev, Serguei ^c  

^a CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

^b RICE UNIVERSITY   (United States)

^c UNIVERSITÉ D'ORLÉANS   (France)

Author keywords

Anomalous transport; Quantum dynamics; Schr dinger operators

Indexed keywords

EID: 42449089102     PISSN: 08940347     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0894-0347-06-00554-6     Document Type: Article

References (77)

1. S. Abe and H. Hiramoto, Fractal dynamics of electron wave packets in one-dimensional quasiperiodic systems, Phys. Rev. A 36 (1987), 5349-5352.
2. M. Aizenman, Localization at weak disorder: some elementary bounds, Rev. Math. Phys. 6 (1994), 1163-1182. MR1301371 (95k:82018)
3. M. Aizenman, A. Elgart, S. Naboko, J. Schenker, and G. Stolz, Moment analysis for localization in random Schrödinger operators, Invent. Math. 163 (2006), 343-413. MR2207021 (Pubitemid 43078521)
4. M. Aizenman and S. Molchanov, Localization at large disorder and at extreme energies: an elementary derivation, Commun. Math. Phys. 157 (1993), 245-278. MR1244867 (95a:82052)
5. M. Aizenman, J. Schenker, R. Friedrich, and D. Hundertmark, Finite-volume fractionalmoment criteria for Anderson localization, Commun. Math. Phys. 224 (2001), 219-253. MR1868998 (2002h:82051) (Pubitemid 33381461)
6. M. Baake and R. V. Moody (Eds.), Directions in Mathematical Quasicrystals, CRM Monograph Series 13, American Mathematical Society, Providence, RI, 2000. MR1798986 (2001f:52047)
7. J. Bak and D. J. Newman, Complex Analysis, Second Edition, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1997. MR1423130
8. 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. MR1861091 (2003a:81058) (Pubitemid 33796965)
9. J.-M. Barbaroux and S. Tcheremchantsev, Universal lower bounds for quantum diffusion, J. Funct. Anal. 168 (1999), 327-354. MR1719245 (2000h:81060)
10. J. Bellissard, I. Guarneri, and H. Schulz-Baldes, Phase-averaged transport for quasi-periodic Hamiltonians, Commun. Math. Phys. 227 (2002), 515-539. MR1910829 (2003g:81052) (Pubitemid 36195020)
11. J. Bellissard, B. Iochum, E. Scoppola, and D. Testard, Spectral properties of one-dimensional quasicrystals, Commun. Math. Phys. 125 (1989), 527-543. MR1022526 (90m:82043)
12. J. Bourgain and M. Goldstein, On nonperturbative localization with quasi-periodic potential, Ann. of Math. 152 (2000), 835-879. MR1815703 (2002h:39028) (Pubitemid 33244792)
13. J. Bourgain and S. Jitomirskaya, Anderson localization for the band model, in Geometric Aspects of Functional Analysis, 6779, Lecture Notes in Math. 1745, Springer, Berlin (2000). MR1796713 (2002d:81053)
14. J. M. Combes, Connections between quantum dynamics and spectral properties of timeevolution operators, in Differential Equations with Applications to Mathematical Physics, 59-68, Academic Press, Boston (1993). MR1207148 (94g:81205)
15. J.-M. Combes and G. Mantica, Fractal dimensions and quantum evolution associated with sparse potential Jacobi matrices, in Long Time Behaviour of Classical and Quantum Systems (Bologna, 1999), 107-123, Ser. Concr. Appl. Math. 1, World Sci. Publishing, River Edge (2001). MR1852219 (2003j:81057)
16. J. B. Conway, Functions of One Complex Variable. II, Graduate Texts in Mathematics 159, Springer-Verlag, New York, 1995. MR1344449 (96i:30001)
17. H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, Texts and Monographs in Physics, Springer- Verlag, Berlin, 1987. MR0883643 (88g:35003)
18. D. Damanik, α-continuity properties of one-dimensional quasicrystals, Commun. Math. Phys. 192 (1998), 169-182. MR1612089 (99a:81031)
19. D. Damanik, Dynamical upper bounds for one-dimensional quasicrystals, J. Math. Anal. Appl. 303 (2005), 327-341. MR2113885 (2006c:81054)
20. D. Damanik, R. Killip, and D. Lenz, Uniform spectral properties of one-dimensional quasicrystals. III. α-continuity, Commun. Math. Phys. 212 (2000), 191-204. MR1764367 (2001g:81061)
21. D. Damanik, D. Lenz, and G. Stolz, Lower transport bounds for one-dimensional continuum Schrödinger operators, to appear in Math. Ann.
22. D. Damanik and P. Stollmann, Multi-scale analysis implies strong dynamical localization, Geom. Funct. Anal. 11 (2001), 11-29. MR1829640 (2002e:82035) (Pubitemid 33263741)
23. D. Damanik, A. Süto, and S. Tcheremchantsev, Power-law bounds on transfer matrices and quantum dynamics in one dimension II, J. Funct. Anal. 216 (2004), 362-387. MR2095687 (2005j:81056) (Pubitemid 39150842)
24. D. Damanik and S. Tcheremchantsev, Power-law bounds on transfer matrices and quantum dynamics in one dimension, Commun. Math. Phys. 236 (2003), 513-534. MR2021200 (2005b:81038) (Pubitemid 36761196)
25. D. Damanik and S. Tcheremchantsev, Scaling estimates for solutions and dynamical lower bounds on wavepacket spreading, J. d'Analyse Math. 97 (2005), 103-131. (Pubitemid 43409983)
26. 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. MR1428099 (97m:47002) (Pubitemid 126312240)
27. J. Fröhlich, F. Martinelli, E. Scoppola, and T. Spencer, Constructive proof of localization in the Anderson tight binding model, Commun. Math. Phys. 101 (1985), 21-46. MR0814541 (87a:82047)
28. J. Fröhlich and T. Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Commun. Math. Phys. 88 (1983), 151-184. MR0696803 (85c:82004)
29. T. Geisel, R. Ketzmerick, and G. Petschel, Unbounded quantum diffusion and a new class of level statistics, in Quantum Chaos-Quantum Measurement (Copenhagen, 1991), 43-59, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 358, Kluwer Acad. Publ., Dordrecht, 1992. MR1155870
30. F. Germinet, Dynamical localization II with an application to the almost Mathieu operator, J. Statist. Phys. 95 (1999), 273-286. MR1705587 (2000f:82094)
31. F. Germinet and S. De Bièvre, Dynamical localization for discrete and continuous random Schrödinger operators, Commun. Math. Phys. 194 (1998), 323-341. MR1627657 (99d:82033) (Pubitemid 128359969)
32. F. Germinet and S. Jitomirskaya, Strong dynamical localization for the almost Mathieu model, Rev. Math. Phys. 13 (2001), 755-765. MR1841745 (2002j:81061) (Pubitemid 33573981)
33. F. Germinet, A. Kiselev, and S. Tcheremchantsev, Transfer matrices and transport for Schrödinger operators, Ann. Inst. Fourier (Grenoble), 54 (2004), 787-830. MR2097423 (2005h:81123)
34. F. Germinet and A. Klein, Bootstrap multiscale analysis and localization in random media, Commun. Math. Phys. 222 (2001), 415-448. MR1859605 (2002m:82035) (Pubitemid 33332176)
35. D. Gilbert, On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints, Proc. Roy. Soc. Edinburgh Sect. A 112 (1989), 213-229. MR1014651 (90j:47061)
36. D. Gilbert and D. Pearson, On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators, J. Math. Anal. Appl. 128 (1987), 30-56. MR0915965 (89a:34033)
37. M. Goldstein and W. Schlag, Hölder continuity of the integrated density of states for quasiperiodic Schrödinger equations and averages of shifts of subharmonic functions, Ann. of Math. 154 (2001), 155-203. MR1847592 (2002h:82055) (Pubitemid 33321514)
38. A. Gordon, The point spectrum of the one-dimensional Schrödinger operator, Uspehi Mat. Nauk 31 (1976), 257-258. MR0458247 (56:16450)
39. I. Guarneri, Spectral properties of quantum diffusion on discrete lattices, Europhys. Lett. 10 (1989), 95-100.
40. I. Guarneri, On an estimate concerning quantum diffusion in the presence of a fractal spectrum, Europhys. Lett. 21 (1993), 729-733.
41. I. Guarneri and H. Schulz-Baldes, Lower bounds on wave packet propagation by packing dimensions of spectral measures, Math. Phys. Electron. J. 5 (1999), Paper 1, 16 pp. MR1663518 (2000d:81033) (Pubitemid 39169370)
42. I. Guarneri and H. Schulz-Baldes, Intermittent lower bounds on quantum diffusion, Lett. Math. Phys. 49 (1999), 317-324. MR1749574 (2001c:81044)
43. M. Herman, Une méthode pour minorer les exposants de Lyapunov et quelques exemples montrant le caractère local d'un théorème d'Arnold et de Moser sur le tore de dimension 2, Commun. Math. Helv. 58 (1983), 453-502. MR0727713 (85g:58057)
44. H. Hiramoto and S. Abe, Dynamics of an electron in quasiperiodic systems. I. Fibonacci model, J. Phys. Soc. Japan 57 (1988), 230-240.
45. H. Hiramoto and S. Abe, Dynamics of an electron in quasiperiodic systems. II. Harper model, J. Phys. Soc. Japan 57 (1988), 1365-1372.
46. B. Iochum, L. Raymond, and D. Testard, Resistance of one-dimensional quasicrystals, Physica A 187 (1992), 353-368. MR1178630 (93i:82076)
47. B. Iochum and D. Testard, Power law growth for the resistance in the Fibonacci model, J. Stat. Phys. 65 (1991), 715-723. MR1137430 (93f:82006)
48. S. Jitomirskaya, Metal-insulator transition for the almost Mathieu operator, Ann. of Math. 150 (1999), 1159-1175. MR1740982 (2000k:81084)
49. S. Jitomirskaya and M. Landrigan, Zero-dimensional spectral measures for quasi-periodic operators with analytic potential, J. Statist. Phys. 100 (2000), 791-796. MR1788487 (2002a:47054)
50. S. Jitomirskaya and Y. Last, Anderson localization for the almost Mathieu equation. III. Semiuniform localization, continuity of gaps, and measure of the spectrum, Commun. Math. Phys. 195 (1998), 1-14. MR1637389 (99j:81038) (Pubitemid 128110631)
51. S. Jitomirskaya and Y. Last, Power-law subordinacy and singular spectra. I. Half-line operators, Acta Math. 183 (1999), 171-189. MR1738043 (2001a:47033)
52. S. Jitomirskaya and Y. Last, Power-law subordinacy and singular spectra. II. Line operators, Commun. Math. Phys. 211 (2000), 643-658. MR1773812 (2001i:81079)
53. S. Jitomirskaya, H. Schulz-Baldes, and G. Stolz, Delocalization in random polymer models, Commun. Math. Phys. 233 (2003), 27-48. MR1957731 (2004f:82032) (Pubitemid 36229187)
54. S. Jitomirskaya and B. Simon, Operators with singular continuous spectrum. III. Almost periodic Schrödinger operators, Commun. Math. Phys. 165 (1994), 201-205. MR1298948 (97a:47003) (Pubitemid 24827924)
55. R. Ketzmerick, K. Kruse, S. Kraut, and T. Geisel, What determines the spreading of a wave packet?, Phys. Rev. Lett. 79 (1997), 1959-1963. MR1471447 (98g:81012) (Pubitemid 127644823)
56. A. Ya. Khinchin, Continued Fractions, Dover Publications, Mineola (1997). MR1451873 (98c:11008)
57. R. Killip, A. Kiselev, and Y. Last, Dynamical upper bounds on wavepacket spreading, Amer. J. Math. 125 (2003), 1165-1198. MR2004433 (2005a:81255) (Pubitemid 37207053)
58. A. Kiselev and Y. Last, Solutions, spectrum and dynamics for Schrödinger operators on infinite domains, Duke Math. J. 102 (2000), 125-150. MR1741780 (2001i:35224)
59. M. Kohmoto, L. P. Kadanoff, and C. Tang, Localization problem in one dimension: Mapping and escape, Phys. Rev. Lett. 50 (1983), 1870-1872. MR0702474 (84m:58113a)
60. S. Kotani, Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators, Stochastic analysis (Katata/Kyoto, 1982), 225-247, North-Holland Math. Library, 32, North-Holland, Amsterdam, 1984. MR0780760 (86h:60117)
61. Y. Last, A relation between a.c. spectrum of ergodic Jacobi matrices and the spectra of periodic approximants, Commun. Math. Phys. 151 (1993), 183-192. MR1201659 (93j:47053)
62. Y. Last, Quantum dynamics and decompositions of singular continuous spectra, J. Funct. Anal. 142 (1996), 406-445. MR1423040 (97k:81044) (Pubitemid 126166057)
63. Y. Last and B. Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math. 135 (1999), 329-367. MR1666767 (2000f:47060)
64. S. Ostlund, R. Pandit, D. Rand, H. J. Schellnhuber, and E. D. Siggia, One-dimensional Schrödinger equation with an almost periodic potential, Phys. Rev. Lett. 50 (1983), 18731877. MR0702475 (84m:58113b)
65. L. Raymond, A constructive gap labelling for the discrete Schrödinger operator on a quasiperiodic chain, preprint (1997).
66. D. Shechtman, I. Blech, D. Gratias, and J. V. Cahn, Metallic phase with long range orientational order and no translational symmetry, Phys. Rev. Lett. 53 (1984), 1951-1953.
67. B. Simon, Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schrödinger operators, Proc. Amer. Math. Soc. 124 (1996), 3361-3369. MR1350963 (97a:34223) (Pubitemid 126749033)
68. E. Sorets and T. Spencer, Positive Lyapunov exponents for Schrödinger operators with quasiperiodic potentials, Commun. Math. Phys. 142 (1991), 543-566. MR1138050 (93b:81058)
69. A. Süto, The spectrum of a quasiperiodic Schrödinger operator, Commun. Math. Phys. 111 (1987), 409-415. MR0900502 (88m:81032)
70. A. Süto, Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian, J. Statist. Phys. 56 (1989), 525-531. MR1009514 (90e:82046)
71. S. Tcheremchantsev, Mixed lower bounds for quantum transport, J. Funct. Anal. 197 (2003), 247-282. MR1957683 (2004a:81089) (Pubitemid 36269933)
72. S. Tcheremchantsev, Dynamical analysis of Schrödinger operators with growing sparse potentials, Commun. Math. Phys. 253 (2005), 221-252. MR2105642 (2005f:47074) (Pubitemid 40067041)
73. S. Tcheremchantsev, How to prove dynamical localization, Commun. Math. Phys. 221 (2001), 27-56. MR1846900 (2002i:81082) (Pubitemid 33311070)
74. S. Tcheremchantsev, Spectral and dynamical analysis of Schrödinger operators with growing sparse potentials, in preparation.
75. H. von Dreifus and A. Klein, A new proof of localization in the Anderson tight binding model, Commun. Math. Phys. 124 (1989), 285-299. MR1012868 (90k:82056)
76. M. Wilkinson and E. Austin, Spectral dimension and dynamics for Harper's equation, Phys. Rev. B 50 (1994), 1420-1429.
77. J.-C. Yoccoz, Analytic linearization of circle diffeomorphisms, in Dynamical Systems and Small Divisors (Cetraro, 1998), 125-173, Lecture Notes in Math. 1784, Springer, Berlin, 2002. MR1924912 (2004c:37073)