Advertisement

Anderson localization for two interacting quasiperiodic particles

  • Jean Bourgain
  • Ilya KachkovskiyEmail author
Article

Abstract

We consider a system of two discrete quasiperiodic 1D particles as an operator on \(\ell^2(\mathbb Z^2)\) and establish Anderson localization at large disorder, assuming the potential has no cosine-type symmetries. In the presence of symmetries, we show localization outside of a neighborhood of finitely many energies. One can also add a deterministic background potential of low complexity, which includes periodic backgrounds and finite range interaction potentials. Such background potentials can only take finitely many values, and the excluded energies in the symmetric case are associated to those values.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

References

  1. AW09.
    Aizenman, M., Warzel, S.: Localization bounds for multiparticle systems. Comm. Math. Phys 290(3), 903–934 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. BPR06.
    S. Basu, R. Pollack and Roy M. Algorithms in Real Algebraic geometry. Springer, Berlin (2006)Google Scholar
  3. BCR98.
    Bochnak, J., Coste, M., Roy, M.: Real Algebraic Geometry. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  4. Bou05.
    Bourgain, J.: Green's Function Estimates for Lattice Schrödinger Operators and Applications. Princeton University Press, Princeton, Annals of Mathematics Studies (2005)CrossRefzbMATHGoogle Scholar
  5. Bou02a.
    Bourgain, J.: Estimates on Green's functions, localization and the quantum kicked rotor model. Ann. Math 156(1), 249–294 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Bou02b.
    Bourgain, J.: On the spectrum of lattice Schrödinger operators with deterministic potential I. J. Anal. Math. 87, 37–75 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Bou07.
    J. Bourgain.: Anderson localization for quasi-periodic lattice Schrödinger operators on \(\mathbb{Z}^d, d\) arbitrary. Geom. Funct. Anal., (3)17 682–706 (2007)Google Scholar
  8. BG00.
    Bourgain, J., Goldstein, M.: On nonperturbative localization with quasi-periodic potential. Ann. of Math. 152, 835–879 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. BGS02.
    Bourgain, J., Goldstein, M., Schlag, W.: Anderson localization for Schrödinger operators on \({\mathbb{Z}}^2\) with quasi-periodic potential. Acta Math. 188(1), 41–86 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. BJ.
    J. Bourgain and S. Jitomirskaya. Anderson localization for band model. Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 1745, 67–79Google Scholar
  11. BJK.
    J. Bourgain, S. Jitomirskaya and I. Kachkovskiy. Localization and delocalization for strongly interacting quasiperiodic particles, in preparationGoogle Scholar
  12. Bur08.
    Burget, D.: A proof of Yomdin-Gromov's algebraic lemma. Israel J. Math. 168, 291–316 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. CS14.
    Chulaevsky, V., Suhov, Yu.: Multi-Scale Analysis for Random Quantum Systems with Interaction. Springer, Berlin (2014)CrossRefzbMATHGoogle Scholar
  14. DGY.
    D. Damanik, A. Gorodetski and W. Yessen. The Fibonacci Hamiltonian. Inv. Math., 206(3), 629–692Google Scholar
  15. ES17.
    Elgart, A., Sodin, S.: The trimmed Anderson model at strong disorder: localisation and its breakup. J. Spectr. Theory 7(1), 87–110 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. GS01.
    Goldstein, M., Schlag, W.: Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions. Ann. Math. 154(1), 155–203 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  17. FS15.
    Frahm, K., Shepelyansky, D.: Freed by interaction kinetic states in the Harper model. D. Eur. Phys. J. B 88, 337 (2015)MathSciNetCrossRefGoogle Scholar
  18. Gab68.
    Gabrielov, A.: Projections of semi-analytic sets. Funct. Anal. Appl. 2(4), 282–291 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Gro87.
    M. Gromov. Entropy, homology and semialgebraic geometry. In: Séminaire Bourbaki, vol. 1985/86, exp. no. 661. Astérique, 145 – 146 (1987), 5, 225–240Google Scholar
  20. JSS03.
    Jitomirskaya, S., Schultz-Baldes, H., Stoltz, G.: Delocalization in random polymer models. Comm. Math. Phys. 233(1), 27–48 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  21. KN13.
    A. Klein and S. Nguyen. The bootstrap multiscale analysis of the multi-particle Anderson model. J. Stat. Phys., 151(5) (2013), 938–973.Google Scholar
  22. Lev96.
    B. Ya. Levin. Lectures on entire functions, American Mathematical Society Translations (1996)Google Scholar
  23. PW06.
    Pila, J., Wilkie, A.: The rational points of a definable set. Duke Math. J. 133(3), 591–616 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Wol95.
    Wolff, T.: An improved bound for Kakeya type maximal functions. Rev. Mat. Iberoam. 11(3), 651–674 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  25. Yom87.
    Yomdin, Y.: \(C^k\)-resolution of semi-algebraic mappings. Israel J. Math. 57, 301–317 (1987)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of MathematicsInstitute for Advanced StudyPrincetonUSA
  2. 2.Department of MathematicsMichigan State UniversityEast LansingUSA

Personalised recommendations