Upper bounds in quantum dynamics

Damanik, David; Tcheremchantsev, Serguei

doi:10.1090/S0894-0347-06-00554-6

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ISSN 1088-6834(online) ISSN 0894-0347(print)

Upper bounds in quantum dynamics

Authors: David Damanik and Serguei Tcheremchantsev
Journal: J. Amer. Math. Soc. 20 (2007), 799-827
MSC (2000): Primary 81Q10; Secondary 47B36
Posted: November 3, 2006
MathSciNet review: 2291919
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Abstract: We develop a general method to bound the spreading of an entire wavepacket under Schrödinger dynamics from above. This method derives upper bounds on time-averaged moments of the position operator from lower bounds on norms of transfer matrices at complex energies.

This general result is applied to the Fibonacci operator. We find that at sufficiently large coupling, all transport exponents take values strictly between zero and one. This is the first rigorous result on anomalous transport.

For quasi-periodic potentials associated with trigonometric polynomials, we prove that all lower transport exponents and, under a weak assumption on the frequency, all upper transport exponents vanish for all phases if the Lyapunov exponent is uniformly bounded away from zero. By a well-known result of Herman, this assumption always holds at sufficiently large coupling. For the particular case of the almost Mathieu operator, our result applies for coupling greater than two.

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. Michael Aizenman, Localization at weak disorder: some elementary bounds, Rev. Math. Phys. 6 (1994), no. 5A, 1163–1182. Special issue dedicated to Elliott H. Lieb. MR 1301371 (95k:82018), http://dx.doi.org/10.1142/S0129055X94000419
3. Michael Aizenman, Alexander Elgart, Serguei Naboko, Jeffrey H. Schenker, and Gunter Stolz, Moment analysis for localization in random Schrödinger operators, Invent. Math. 163 (2006), no. 2, 343–413. MR 2207021 (2006m:81097), http://dx.doi.org/10.1007/s00222-005-0463-y
4. Michael Aizenman and Stanislav Molchanov, Localization at large disorder and at extreme energies: an elementary derivation, Comm. Math. Phys. 157 (1993), no. 2, 245–278. MR 1244867 (95a:82052)
5. Michael Aizenman, Jeffrey H. Schenker, Roland M. Friedrich, and Dirk Hundertmark, Finite-volume fractional-moment criteria for Anderson localization, Comm. Math. Phys. 224 (2001), no. 1, 219–253. Dedicated to Joel L. Lebowitz. MR 1868998 (2002h:82051), http://dx.doi.org/10.1007/s002200100441
6. Michael Baake and Robert V. Moody (eds.), Directions in mathematical quasicrystals, CRM Monograph Series, vol. 13, American Mathematical Society, Providence, RI, 2000. MR 1798986 (2001f:52047)
7. Joseph Bak and Donald J. Newman, Complex analysis, 2nd ed., Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1997. First edition [Springer, New York, 1982; MR0671250 (84b:30001)]. MR 1423130
8. Jean-Marie Barbaroux, François Germinet, and Serguei Tcheremchantsev, Fractal dimensions and the phenomenon of intermittency in quantum dynamics, Duke Math. J. 110 (2001), no. 1, 161–193. MR 1861091 (2003a:81058), http://dx.doi.org/10.1215/S0012-7094-01-11015-6
9. J. M. Barbaroux and S. Tcheremchantsev, Universal lower bounds for quantum diffusion, J. Funct. Anal. 168 (1999), no. 2, 327–354. MR 1719245 (2000h:81060), http://dx.doi.org/10.1006/jfan.1999.3471
10. Jean Bellissard, Italo Guarneri, and Hermann Schulz-Baldes, Phase-averaged transport for quasi-periodic Hamiltonians, Comm. Math. Phys. 227 (2002), no. 3, 515–539. MR 1910829 (2003g:81052), http://dx.doi.org/10.1007/s002200200642
11. J. Bellissard, B. Iochum, E. Scoppola, and D. Testard, Spectral properties of one-dimensional quasi-crystals, Comm. Math. Phys. 125 (1989), no. 3, 527–543. MR 1022526 (90m:82043)
12. J. Bourgain and M. Goldstein, On nonperturbative localization with quasi-periodic potential, Ann. of Math. (2) 152 (2000), no. 3, 835–879. MR 1815703 (2002h:39028), http://dx.doi.org/10.2307/2661356
13. J. Bourgain and S. Jitomirskaya, Anderson localization for the band model, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1745, Springer, Berlin, 2000, pp. 67–79. MR 1796713 (2002d:81053), http://dx.doi.org/10.1007/BFb0107208
14. Jean-Michel Combes, Connections between quantum dynamics and spectral properties of time-evolution operators, Differential equations with applications to mathematical physics, Math. Sci. Engrg., vol. 192, Academic Press, Boston, MA, 1993, pp. 59–68. MR 1207148 (94g:81205), http://dx.doi.org/10.1016/S0076-5392(08)62372-3
15. Jean-Michel Combes and Giorgio 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. Publ., River Edge, NJ, 2001, pp. 107–123. MR 1852219 (2003j:81057), http://dx.doi.org/10.1142/9789812794598_0006
16. John B. Conway, Functions of one complex variable. II, Graduate Texts in Mathematics, vol. 159, Springer-Verlag, New York, 1995. MR 1344449 (96i:30001)
17. H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger operators with application to quantum mechanics and global geometry, Springer Study Edition, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987. MR 883643 (88g:35003)
18. David Damanik, 𝛼-continuity properties of one-dimensional quasicrystals, Comm. Math. Phys. 192 (1998), no. 1, 169–182. MR 1612089 (99a:81031), http://dx.doi.org/10.1007/s002200050295
19. David Damanik, Dynamical upper bounds for one-dimensional quasicrystals, J. Math. Anal. Appl. 303 (2005), no. 1, 327–341. MR 2113885 (2006c:81054), http://dx.doi.org/10.1016/j.jmaa.2004.08.038
20. David Damanik, Rowan Killip, and Daniel Lenz, Uniform spectral properties of one-dimensional quasicrystals. III. 𝛼-continuity, Comm. Math. Phys. 212 (2000), no. 1, 191–204. MR 1764367 (2001g:81061), http://dx.doi.org/10.1007/s002200000203
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), no. 1, 11–29. MR 1829640 (2002e:82035), http://dx.doi.org/10.1007/PL00001666
23. David Damanik, András Sütő, and Serguei Tcheremchantsev, Power-law bounds on transfer matrices and quantum dynamics in one dimension. II, J. Funct. Anal. 216 (2004), no. 2, 362–387. MR 2095687 (2005j:81056), http://dx.doi.org/10.1016/j.jfa.2004.05.007
24. David Damanik and Serguei Tcheremchantsev, Power-law bounds on transfer matrices and quantum dynamics in one dimension, Comm. Math. Phys. 236 (2003), no. 3, 513–534. MR 2021200 (2005b:81038), http://dx.doi.org/10.1007/s00220-003-0824-6
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.
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. MR 1428099 (97m:47002), http://dx.doi.org/10.1007/BF02787106
27. J. Fröhlich, F. Martinelli, E. Scoppola, and T. Spencer, Constructive proof of localization in the Anderson tight binding model, Comm. Math. Phys. 101 (1985), no. 1, 21–46. MR 814541 (87a:82047)
28. Jürg Fröhlich and Thomas Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Comm. Math. Phys. 88 (1983), no. 2, 151–184. MR 696803 (85c:82004)
29. T. Geisel, R. Ketzmerick, and G. Petschel, Unbounded quantum diffusion and a new class of level statistics, Quantum chaos—quantum measurement (Copenhagen, 1991) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 358, Kluwer Acad. Publ., Dordrecht, 1992, pp. 43–59. MR 1155870
30. François Germinet, Dynamical localization II with an application to the almost Mathieu operator, J. Statist. Phys. 95 (1999), no. 1-2, 273–286. MR 1705587 (2000f:82094), http://dx.doi.org/10.1023/A:1004533629182
31. F. Germinet and S. De Bièvre, Dynamical localization for discrete and continuous random Schrödinger operators, Comm. Math. Phys. 194 (1998), no. 2, 323–341. MR 1627657 (99d:82033), http://dx.doi.org/10.1007/s002200050360
32. François Germinet and Svetlana Jitomirskaya, Strong dynamical localization for the almost Mathieu model, Rev. Math. Phys. 13 (2001), no. 6, 755–765. MR 1841745 (2002j:81061), http://dx.doi.org/10.1142/S0129055X01000855
33. François Germinet, Alexander Kiselev, and Serguei Tcheremchantsev, Transfer matrices and transport for Schrödinger operators, Ann. Inst. Fourier (Grenoble) 54 (2004), no. 3, 787–830 (English, with English and French summaries). MR 2097423 (2005h:81123)
34. François Germinet and Abel Klein, Bootstrap multiscale analysis and localization in random media, Comm. Math. Phys. 222 (2001), no. 2, 415–448. MR 1859605 (2002m:82035), http://dx.doi.org/10.1007/s002200100518
35. D. J. Gilbert, On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints, Proc. Roy. Soc. Edinburgh Sect. A 112 (1989), no. 3-4, 213–229. MR 1014651 (90j:47061), http://dx.doi.org/10.1017/S0308210500018680
36. D. J. Gilbert and D. B. Pearson, On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators, J. Math. Anal. Appl. 128 (1987), no. 1, 30–56. MR 915965 (89a:34033), http://dx.doi.org/10.1016/0022-247X(87)90212-5
37. Michael Goldstein and Wilhelm Schlag, Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions, Ann. of Math. (2) 154 (2001), no. 1, 155–203. MR 1847592 (2002h:82055), http://dx.doi.org/10.2307/3062114
38. A. Ja. Gordon, The point spectrum of the one-dimensional Schrödinger operator, Uspehi Mat. Nauk 31 (1976), no. 4(190), 257–258 (Russian). MR 0458247 (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. (electronic). MR 1663518 (2000d:81033)
42. Italo Guarneri and Hermann Schulz-Baldes, Intermittent lower bound on quantum diffusion, Lett. Math. Phys. 49 (1999), no. 4, 317–324. MR 1749574 (2001c:81044), http://dx.doi.org/10.1023/A:1007610717491
43. Michael-R. Herman, Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnol′d et de Moser sur le tore de dimension 2, Comment. Math. Helv. 58 (1983), no. 3, 453–502 (French). MR 727713 (85g:58057), http://dx.doi.org/10.1007/BF02564647
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, Phys. A 187 (1992), no. 1-2, 353–368. MR 1178630 (93i:82076), http://dx.doi.org/10.1016/0378-4371(92)90426-Q
47. B. Iochum and D. Testard, Power law growth for the resistance in the Fibonacci model, J. Statist. Phys. 65 (1991), no. 3-4, 715–723. MR 1137430 (93f:82006), http://dx.doi.org/10.1007/BF01053750
48. Svetlana Ya. Jitomirskaya, Metal-insulator transition for the almost Mathieu operator, Ann. of Math. (2) 150 (1999), no. 3, 1159–1175. MR 1740982 (2000k:81084), http://dx.doi.org/10.2307/121066
49. S. Jitomirskaya and M. Landrigan, Zero-dimensional spectral measures for quasi-periodic operators with analytic potential, J. Statist. Phys. 100 (2000), no. 3-4, 791–796. MR 1788487 (2002a:47054), http://dx.doi.org/10.1023/A:1018635811535
50. Svetlana Ya. Jitomirskaya and Yoram Last, Anderson localization for the almost Mathieu equation. III. Semi-uniform localization, continuity of gaps, and measure of the spectrum, Comm. Math. Phys. 195 (1998), no. 1, 1–14. MR 1637389 (99j:81038), http://dx.doi.org/10.1007/s002200050376
51. Svetlana Jitomirskaya and Yoram Last, Power-law subordinacy and singular spectra. I. Half-line operators, Acta Math. 183 (1999), no. 2, 171–189. MR 1738043 (2001a:47033), http://dx.doi.org/10.1007/BF02392827
52. Svetlana Ya. Jitomirskaya and Yoram Last, Power law subordinacy and singular spectra. II. Line operators, Comm. Math. Phys. 211 (2000), no. 3, 643–658. MR 1773812 (2001i:81079), http://dx.doi.org/10.1007/s002200050830
53. S. Jitomirskaya, H. Schulz-Baldes, and G. Stolz, Delocalization in random polymer models, Comm. Math. Phys. 233 (2003), no. 1, 27–48. MR 1957731 (2004f:82032), http://dx.doi.org/10.1007/s00220-002-0757-5
54. S. Jitomirskaya and B. Simon, Operators with singular continuous spectrum. III. Almost periodic Schrödinger operators, Comm. Math. Phys. 165 (1994), no. 1, 201–205. MR 1298948 (97a:47003)
55. R. Ketzmerick, K. Kruse, S. Kraut, and T. Geisel, What determines the spreading of a wave packet?, Phys. Rev. Lett. 79 (1997), no. 11, 1959–1963. MR 1471447 (98g:81012), http://dx.doi.org/10.1103/PhysRevLett.79.1959
56. A. Ya. Khinchin, Continued fractions, Translated from the third (1961) Russian edition, Dover Publications Inc., Mineola, NY, 1997. With a preface by B. V. Gnedenko; Reprint of the 1964 translation. MR 1451873 (98c:11008)
57. Rowan Killip, Alexander Kiselev, and Yoram Last, Dynamical upper bounds on wavepacket spreading, Amer. J. Math. 125 (2003), no. 5, 1165–1198. MR 2004433 (2005a:81255)
58. A. Kiselev and Y. Last, Solutions, spectrum, and dynamics for Schrödinger operators on infinite domains, Duke Math. J. 102 (2000), no. 1, 125–150. MR 1741780 (2001i:35224), http://dx.doi.org/10.1215/S0012-7094-00-10215-3
59. Mahito Kohmoto, Leo P. Kadanoff, and Chao Tang, Localization problem in one dimension: mapping and escape, Phys. Rev. Lett. 50 (1983), no. 23, 1870–1872. MR 702474 (84m:58113a), http://dx.doi.org/10.1103/PhysRevLett.50.1870
60. Shinichi Kotani, Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators, Stochastic analysis (Katata/Kyoto, 1982) North-Holland Math. Library, vol. 32, North-Holland, Amsterdam, 1984, pp. 225–247. MR 780760 (86h:60117), http://dx.doi.org/10.1016/S0924-6509(08)70395-7
61. Y. Last, A relation between a.c. spectrum of ergodic Jacobi matrices and the spectra of periodic approximants, Comm. Math. Phys. 151 (1993), no. 1, 183–192. MR 1201659 (93j:47053)
62. Yoram Last, Quantum dynamics and decompositions of singular continuous spectra, J. Funct. Anal. 142 (1996), no. 2, 406–445. MR 1423040 (97k:81044), http://dx.doi.org/10.1006/jfan.1996.0155
63. Yoram Last and Barry Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math. 135 (1999), no. 2, 329–367. MR 1666767 (2000f:47060), http://dx.doi.org/10.1007/s002220050288
64. Stellan Ostlund, Rahul Pandit, David Rand, Hans Joachim Schellnhuber, and Eric D. Siggia, One-dimensional Schrödinger equation with an almost periodic potential, Phys. Rev. Lett. 50 (1983), no. 23, 1873–1876. MR 702475 (84m:58113b), http://dx.doi.org/10.1103/PhysRevLett.50.1873
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. Barry Simon, Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schrödinger operators, Proc. Amer. Math. Soc. 124 (1996), no. 11, 3361–3369. MR 1350963 (97a:34223), http://dx.doi.org/10.1090/S0002-9939-96-03599-X
68. Eugene Sorets and Thomas Spencer, Positive Lyapunov exponents for Schrödinger operators with quasi-periodic potentials, Comm. Math. Phys. 142 (1991), no. 3, 543–566. MR 1138050 (93b:81058)
69. András Sütő, The spectrum of a quasiperiodic Schrödinger operator, Comm. Math. Phys. 111 (1987), no. 3, 409–415. MR 900502 (88m:81032)
70. András Sütő, Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian, J. Statist. Phys. 56 (1989), no. 3-4, 525–531. MR 1009514 (90e:82046), http://dx.doi.org/10.1007/BF01044450
71. Serguei Tcheremchantsev, Mixed lower bounds for quantum transport, J. Funct. Anal. 197 (2003), no. 1, 247–282. MR 1957683 (2004a:81089), http://dx.doi.org/10.1016/S0022-1236(02)00066-6
72. Serguei Tcheremchantsev, Dynamical analysis of Schrödinger operators with growing sparse potentials, Comm. Math. Phys. 253 (2005), no. 1, 221–252. MR 2105642 (2005f:47074), http://dx.doi.org/10.1007/s00220-004-1153-0
73. Serguei Tcheremchantsev, How to prove dynamical localization, Comm. Math. Phys. 221 (2001), no. 1, 27–56. MR 1846900 (2002i:81082), http://dx.doi.org/10.1007/s002200100460
74. S. Tcheremchantsev, Spectral and dynamical analysis of Schrödinger operators with growing sparse potentials, in preparation.
75. Henrique von Dreifus and Abel Klein, A new proof of localization in the Anderson tight binding model, Comm. Math. Phys. 124 (1989), no. 2, 285–299. MR 1012868 (90k:82056)
76. M. Wilkinson and E. Austin, Spectral dimension and dynamics for Harper's equation, Phys. Rev. B 50 (1994), 1420-1429.
77. Jean-Christophe Yoccoz, Analytic linearization of circle diffeomorphisms, Dynamical systems and small divisors (Cetraro, 1998) Lecture Notes in Math., vol. 1784, Springer, Berlin, 2002, pp. 125–173. MR 1924912 (2004c:37073), http://dx.doi.org/10.1007/978-3-540-47928-4_3

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