Communications in Mathematical Physics

, Volume 273, Issue 3, pp 601–618 | Cite as

Upper Bounds On Wavepacket Spreading For Random Jacobi Matrices

  • Svetlana Jitomirskaya
  • Hermann Schulz-BaldesEmail author


A method is presented for proving upper bounds on the moments of the position operator when the dynamics of quantum wavepackets is governed by a random (possibly correlated) Jacobi matrix. As an application, one obtains sharp upper bounds on the diffusion exponents for random polymer models, coinciding with the lower bounds obtained in a prior work. The second application is an elementary argument (not using multiscale analysis or the Aizenman-Molchanov method) showing that under the condition of uniformly positive Lyapunov exponents, the moments of the position operator grow at most logarithmically in time.


Lyapunov Exponent Critical Energy Dynamical Localization Jacobi Matrice Anomalous Diffusion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. AM.
    Aizenman M., Molchanov S. (1993). Localization at Large Disorder and at Extreme Energies: an Elementary Derivation. Commun. Math. Phys. 157: 245–278 zbMATHCrossRefMathSciNetADSGoogle Scholar
  2. BGT.
    Barbaroux J.M., Germinet F., Tcheremchantsev S. (2001). Fractal Dimensions and the Phenomenon of Intermittency in Quantum Dynamics. Duke Math. J. 110: 161–193 zbMATHCrossRefMathSciNetGoogle Scholar
  3. BS.
    Barbaroux J.-M., Schulz-Baldes H. (1999). Anomalous transport in presence of self-similar spectra. Ann. I.H.P. Phys. Théo. 71: 539–559 zbMATHMathSciNetGoogle Scholar
  4. BeS.
    Bellissard J., Schulz-Baldes H. (2000). Subdiffusive quantum transport for 3d-Hamiltonians with absolutely continuous spectra. J. Stat. Phys. 99: 587–594 zbMATHCrossRefMathSciNetGoogle Scholar
  5. BG.
    Germinet F., Bièvre S. (2000). Dynamical Localization for the Random Dimer Schrödinger Operator. J. Stat. Phys. 98: 1135–1148 zbMATHCrossRefGoogle Scholar
  6. BL.
    Bougerol P., Lacroix J. (1985). Products of Random Matrices with Applications to Schrödinger Operators. Boston, Birkhäuser, 1985 zbMATHGoogle Scholar
  7. Bov.
    Bovier A. (1992). Perturbation theory for the random dimer model. J. Phys. A: Math. Gen. 25: 1021–1029 zbMATHCrossRefADSGoogle Scholar
  8. BLS.
    Breuer, J., Last, Y., Strauss, Y.: Upper bounds on the dynamical spreading of wavepackets. In: preparationGoogle Scholar
  9. CKM.
    Carmona R., Klein A., Martinelli F. (1987).,erson localization for Bernoulli and other singular potentials. Commun. Math. Phys. 108: 41–66 zbMATHCrossRefMathSciNetADSGoogle Scholar
  10. Com.
    Combes, J.-M.: Connections between quantum dynamics and spectral properties of time- evolution operators. In: Differential Equations with Applications to Mathematical Physics, Ames, W.F., Harell, E.M., Herod, J.V., eds. Boston: Academic Press, 1993Google Scholar
  11. CM.
    Combes, J.-M., Mantica, G.: Fractal dimensions and quantum evolution associated with sparse potential Jacobi matrices. In: Proceedings of the Bologna APTEX International Conference, World Scientific. Ser. Concr. Appl. Math. 1, River Edge, NJ: world Scientific, pp. 107–123 (2001)Google Scholar
  12. DSS.
    Damanik D., Sims R., Stolz G. (2004). Localization for discrete one-dimensional random word models. J. Funct. Anal. 208: 423–445 zbMATHCrossRefMathSciNetGoogle Scholar
  13. DT.
    Damanik, D., Tcheremchantsev, S.: Upper bounds in quantum dynamics. To appear in J. Amer. Math. Soc., electronically posted on Nov. 30, 2006Google Scholar
  14. DJLS.
    del Rio R., Jitomirskaya S., Last Y., Simon B. (1996). Operators with singular continuous spectrum: IV, Hausdorff dimension, rank-one perturbations and localization. J. d’Analyse Math. 69: 153–200 zbMATHMathSciNetCrossRefGoogle Scholar
  15. DMS.
    del Rio R., Makarov N., Simon B. (1994). Operators with singular continuous spectrum: ii, Rank one operators. Commun. Math. Phys. 165: 59–67zbMATHCrossRefMathSciNetADSGoogle Scholar
  16. DWP.
    Dunlap D.H., Wu H.-L., Phillips P.W. (1990). Absence of Localization in Random-Dimer Model. Phys. Rev. Lett. 65: 88–91 CrossRefADSGoogle Scholar
  17. Gor.
    Gordon A. (1994). Pure point spectrum under 1-parameter perturbations and instability of Anderson localization. Commun. Math. Phys. 164: 489–505 zbMATHCrossRefADSGoogle Scholar
  18. Gua.
    Guarneri, I.: Spectral properties of quantum diffusion on discrete lattices. Europhys. Lett. 10, 95–100 (1989); On an estimate concerning quantum diffusion in the presence of a fractal spectrum. Europhys. Lett. 21, 729–733 (1993)Google Scholar
  19. GS1.
    Guarneri I., Schulz-Baldes H. (1999). Upper bounds for quantum dynamics governed by Jacobi matrices with self-similar spectra. Rev. Math. Phys. 11: 1249–1268 zbMATHCrossRefMathSciNetGoogle Scholar
  20. GS2.
    Guarneri, I., Schulz-Baldes, H.: Lower bounds on wave packet propagation by packing dimensions of spectral measures. Math. Phys. Elect. J. 5(1), 16 pages (1999)Google Scholar
  21. GS3.
    Guarneri I., Schulz-Baldes H. (1999). Intermittent lower bound on quantum diffusion. Lett. Math. Phys. 49: 317–324 zbMATHCrossRefMathSciNetGoogle Scholar
  22. Jit.
    Jitomirskaya, S.: Ergodic Schrödinger operators (on one foot). Preprint 2006, to appear in Barry Simon FestschriftGoogle Scholar
  23. JL.
    Jitomirskaya S., Last Y. (1999). Power-Law subordinacy and singular spectra, I.Half line operators. Acta Math. 183(2): 171–189 zbMATHCrossRefMathSciNetGoogle Scholar
  24. JSS.
    Jitomirskaya S., Schulz-Baldes H., Stolz G. (2003). Delocalization in random polymer chains. Commun. Math. Phys. 233: 27–48 zbMATHCrossRefMathSciNetADSGoogle Scholar
  25. KKL.
    Killip R., Kiselev A., Last Y. (2003). Dynamical upper bounds on wavepacket spreading. Amer. J. Math. 125: 1165–1198 zbMATHCrossRefMathSciNetGoogle Scholar
  26. KL.
    Kiselev A., Last Y. (2000). Solutions, spectrum and dynamics for Schrödinger operators on infinite domains. Duke Math. J. 102: 125–150 zbMATHCrossRefMathSciNetGoogle Scholar
  27. Las.
    Last Y. (1996). Quantum Dynamics and decomposition of singular continuous spectra. J. Funct. Anal. 142: 402–445 CrossRefMathSciNetGoogle Scholar
  28. PF.
    Pastur, L., Figotin, A.: Spectra of Random and Almost-Periodic Operators. Berlin:Springer, 1992Google Scholar
  29. SSS.
    Schrader R., Schulz-Baldes H., Sedrakyan A. (2004). Perturbative test of single parameter scaling for 1d random media. Ann. H. Poincare 5: 1159–1180 zbMATHCrossRefMathSciNetGoogle Scholar
  30. Sch.
    Schulz-Baldes, H.: Lyapunov exponents at anomalies of SL(2, \({\mathbb{R}}\)) actions, to appear in Operator Theory: Advances and Applications. Basel:Birkhäuser, 2006Google Scholar
  31. Tch.
    Tcheremchantsev S. (2005). Dynamical analysis of Schrödinger operators with growing sparse potentials. Commun. Math. Phys. 253: 221–252 zbMATHCrossRefMathSciNetADSGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California at IrvineIrvineUSA
  2. 2.Mathematisches InstitutUniversität Erlangen-NürnbergErlangenGermany

Personalised recommendations