Upper bounds in quantum dynamics

Damanik, David; Tcheremchantsev, Serguei

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

Upper bounds in quantum dynamics
HTML articles powered by AMS MathViewer

by David Damanik and Serguei Tcheremchantsev

J. Amer. Math. Soc. 20 (2007), 799-827

DOI: https://doi.org/10.1090/S0894-0347-06-00554-6

Published electronically: November 3, 2006

PDF

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.

References

Phys. Rev. A

36

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, DOI 10.1142/S0129055X94000419
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, DOI 10.1007/s00222-005-0463-y
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, DOI 10.1007/BF02099760
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, DOI 10.1007/s002200100441
Michael Baake and Robert V. Moody (eds.), Directions in mathematical quasicrystals, CRM Monograph Series, vol. 13, American Mathematical Society, Providence, RI, 2000. MR 1798986, DOI 10.1090/crmm/013
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
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, DOI 10.1215/S0012-7094-01-11015-6
J. M. Barbaroux and S. Tcheremchantsev, Universal lower bounds for quantum diffusion, J. Funct. Anal. 168 (1999), no. 2, 327–354. MR 1719245, DOI 10.1006/jfan.1999.3471
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, DOI 10.1007/s002200200642
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, DOI 10.1007/BF01218415
J. Bourgain and M. Goldstein, On nonperturbative localization with quasi-periodic potential, Ann. of Math. (2) 152 (2000), no. 3, 835–879. MR 1815703, DOI 10.2307/2661356
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, DOI 10.1007/BFb0107208
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, DOI 10.1016/S0076-5392(08)62372-3
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, DOI 10.1142/9789812794598_{0}006
John B. Conway, Functions of one complex variable. II, Graduate Texts in Mathematics, vol. 159, Springer-Verlag, New York, 1995. MR 1344449, DOI 10.1007/978-1-4612-0817-4
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, DOI 10.1007/978-3-540-77522-5
David Damanik, $\alpha$-continuity properties of one-dimensional quasicrystals, Comm. Math. Phys. 192 (1998), no. 1, 169–182. MR 1612089, DOI 10.1007/s002200050295
David Damanik, Dynamical upper bounds for one-dimensional quasicrystals, J. Math. Anal. Appl. 303 (2005), no. 1, 327–341. MR 2113885, DOI 10.1016/j.jmaa.2004.08.038
David Damanik, Rowan Killip, and Daniel Lenz, Uniform spectral properties of one-dimensional quasicrystals. III. $\alpha$-continuity, Comm. Math. Phys. 212 (2000), no. 1, 191–204. MR 1764367, DOI 10.1007/s002200000203

Math. Ann.

D. Damanik and P. Stollmann, Multi-scale analysis implies strong dynamical localization, Geom. Funct. Anal. 11 (2001), no. 1, 11–29. MR 1829640, DOI 10.1007/PL00001666
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, DOI 10.1016/j.jfa.2004.05.007
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, DOI 10.1007/s00220-003-0824-6

J. d’Analyse Math.

97

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, DOI 10.1007/BF02787106
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, DOI 10.1007/BF01212355
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, DOI 10.1007/BF01209475
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
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, DOI 10.1023/A:1004533629182
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, DOI 10.1007/s002200050360
François Germinet and Svetlana Jitomirskaya, Strong dynamical localization for the almost Mathieu model, Rev. Math. Phys. 13 (2001), no. 6, 755–765. MR 1841745, DOI 10.1142/S0129055X01000855
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, DOI 10.5802/aif.2034
François Germinet and Abel Klein, Bootstrap multiscale analysis and localization in random media, Comm. Math. Phys. 222 (2001), no. 2, 415–448. MR 1859605, DOI 10.1007/s002200100518
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, DOI 10.1017/S0308210500018680
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, DOI 10.1016/0022-247X(87)90212-5
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, DOI 10.2307/3062114
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

Europhys. Lett.

10

Europhys. Lett.

21

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. MR 1663518
Italo Guarneri and Hermann Schulz-Baldes, Intermittent lower bound on quantum diffusion, Lett. Math. Phys. 49 (1999), no. 4, 317–324. MR 1749574, DOI 10.1023/A:1007610717491
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, DOI 10.1007/BF02564647

J. Phys. Soc. Japan

57

J. Phys. Soc. Japan

57

B. Iochum, L. Raymond, and D. Testard, Resistance of one-dimensional quasicrystals, Phys. A 187 (1992), no. 1-2, 353–368. MR 1178630, DOI 10.1016/0378-4371(92)90426-Q
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, DOI 10.1007/BF01053750
Svetlana Ya. Jitomirskaya, Metal-insulator transition for the almost Mathieu operator, Ann. of Math. (2) 150 (1999), no. 3, 1159–1175. MR 1740982, DOI 10.2307/121066
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, DOI 10.1023/A:1018635811535
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, DOI 10.1007/s002200050376
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, DOI 10.1007/BF02392827
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, DOI 10.1007/s002200050830
S. Jitomirskaya, H. Schulz-Baldes, and G. Stolz, Delocalization in random polymer models, Comm. Math. Phys. 233 (2003), no. 1, 27–48. MR 1957731, DOI 10.1007/s00220-002-0757-5
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, DOI 10.1007/BF02099743
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, DOI 10.1103/PhysRevLett.79.1959
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
Rowan Killip, Alexander Kiselev, and Yoram Last, Dynamical upper bounds on wavepacket spreading, Amer. J. Math. 125 (2003), no. 5, 1165–1198. MR 2004433, DOI 10.1353/ajm.2003.0031
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, DOI 10.1215/S0012-7094-00-10215-3
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, DOI 10.1103/PhysRevLett.50.1870
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, DOI 10.1016/S0924-6509(08)70395-7
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, DOI 10.1007/BF02096752
Yoram Last, Quantum dynamics and decompositions of singular continuous spectra, J. Funct. Anal. 142 (1996), no. 2, 406–445. MR 1423040, DOI 10.1006/jfan.1996.0155
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, DOI 10.1007/s002220050288
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, DOI 10.1103/PhysRevLett.50.1870

Phys. Rev. Lett.

53

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, DOI 10.1090/S0002-9939-96-03599-X
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, DOI 10.1007/BF02099100
András Sütő, The spectrum of a quasiperiodic Schrödinger operator, Comm. Math. Phys. 111 (1987), no. 3, 409–415. MR 900502, DOI 10.1007/BF01238906
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, DOI 10.1007/BF01044450
Serguei Tcheremchantsev, Mixed lower bounds for quantum transport, J. Funct. Anal. 197 (2003), no. 1, 247–282. MR 1957683, DOI 10.1016/S0022-1236(02)00066-6
Serguei Tcheremchantsev, Dynamical analysis of Schrödinger operators with growing sparse potentials, Comm. Math. Phys. 253 (2005), no. 1, 221–252. MR 2105642, DOI 10.1007/s00220-004-1153-0
Serguei Tcheremchantsev, How to prove dynamical localization, Comm. Math. Phys. 221 (2001), no. 1, 27–56. MR 1846900, DOI 10.1007/s002200100460
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, DOI 10.1007/BF01219198

Phys. Rev. B

50

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, DOI 10.1007/978-3-540-47928-4_{3}