Advertisement

Mathematical Physics, Analysis and Geometry

, Volume 13, Issue 3, pp 205–217 | Cite as

Wegner-type Bounds for a Two-particle Lattice Model with a Generic “Rough” Quasi-periodic Potential

  • Martin GaumeEmail author
Article
  • 34 Downloads

Abstract

In this paper, we consider a class of two-particle tight-binding Hamiltonians, describing pairs of interacting quantum particles on the lattice ℤ d , d ≥ 1, subject to a common external potential V(x) which we assume quasi-periodic and depending on auxiliary parameters. Such parametric families of ergodic deterministic potentials (“grands ensembles”) have been introduced earlier in Chulaevsky (2007), in the framework of single-particle lattice systems, where it was proved that a non-uniform analog of the Wegner bound holds true for a class of quasi-periodic grands ensembles. Using the approach proposed in Chulaevsky and Suhov (Commun Math Phys 283(2):479–489, 2008), we establish volume-dependent Wegner-type bounds for a class of quasi-periodic two-particle lattice systems with a non-random short-range interaction.

Keywords

Schrödinger operator Wegner bounds Two-particle Quasi-periodic 

Mathematics Subject Classifications (2010)

82B44 82B20 81Q10 47B80 47N50 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bovier, A., Campanino, M., Klein, A., Perez, F.: Smoothness of the density of states in the Anderson model at high disorder. Commun. Math. Phys. 114(3), 439–461 (1988)zbMATHCrossRefMathSciNetADSGoogle Scholar
  2. 2.
    Bourgain, J., Goldstein, M.: On nonperturbative localization with quasiperiodic potentials. Ann. Math. 152(3), 835–879 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bourgain, J., Goldstein, M., Schlag, W.: Anderson localization for Schrödinger operators on ℤ with potential generated by the skew-shift. Commun. Math. Phys. 220(3), 583–621 (2001)zbMATHCrossRefMathSciNetADSGoogle Scholar
  4. 4.
    Bourgain, J., Schlag, W.: Anderson localization for Schrödinger operators on ℤ with strongly mixing potentials. Commun. Math. Phys. 215(1), 143–175 (2000)zbMATHCrossRefMathSciNetADSGoogle Scholar
  5. 5.
    Campanino, M., Klein, A.: A supersymmetric transfer matrix and differentiability of the density of states in the one-dimensional Anderson model. Commun. Math. Phys. 104(2), 227–241 (1986)zbMATHCrossRefMathSciNetADSGoogle Scholar
  6. 6.
    Carmona, R., Lacroix, J.: Spectral Theory of Random Schrödinger Operators. Birkhäuser, Boston (1990)zbMATHGoogle Scholar
  7. 7.
    Chan, J.: Method of variations of potential of quasi-periodic Schrödinger equations. Geom. Funct. Anal. 17(5), 1416–1478 (2008)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Chulaevsky, V.: Wegner–Stollmann estimates for some quantum lattice systems. In: Contemp. Math., vol. 447, pp. 17–28. Amer. Math. Soc., Providence (2007)Google Scholar
  9. 9.
    Chulaevsky, V., Suhov, Y.: Wegner bounds for a two-particle tight binding model. Commun. Math. Phys. 283(2), 479–489 (2008)zbMATHCrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Combes, J.-M., Hislop, P.D., Klopp, F.: An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators. Duke Math. J. 140(3), 469–498 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Constantinescu, F., Fröhlich, J., Spencer, T.: Analyticity of the density of states and replica method for random Schrödinger operators on a lattice. J. Stat. Phys. 34(3–4), 571–596 (1984)zbMATHCrossRefADSGoogle Scholar
  12. 12.
    Craig, W., Simon, B.: Log Hölder continuity of the integrated density of states for stochastic Jacobi matrices. Commun. Math. Phys. 90(2), 207–218 (1983)zbMATHCrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators. Springer, Berlin (1987)zbMATHGoogle Scholar
  14. 14.
    Delyon, F., Souillard, B.: Remark on the continuity of the density of states of ergodic finite difference operators. Commun. Math. Phys. 94(2), 289–291 (1984)zbMATHCrossRefMathSciNetADSGoogle Scholar
  15. 15.
    von Dreifus, H., Klein, A.: A new proof of localization in the Anderson tight binding model. Commun. Math. Phys. 124(2), 285–299 (1989)zbMATHCrossRefADSGoogle Scholar
  16. 16.
    Fröhlich, J., Martinelli, F., Scoppola, E., Spencer, T.: Constructive proof of localization in Anderson tight binding model. Commun. Math. Phys. 101(1), 21–46 (1985)zbMATHCrossRefADSGoogle Scholar
  17. 17.
    Fröhlich, J., Spencer, T.: Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Commun. Math. Phys. 88(2), 151–184 (1983)zbMATHCrossRefADSGoogle Scholar
  18. 18.
    Fröhlich, J., Spencer, T., Wittwer, P.: Localization for a class of one dimensional quasi-periodic Schrödinger operators. Commun. Math. Phys. 132(1), 5–25 (1990)zbMATHCrossRefADSGoogle Scholar
  19. 19.
    Pastur, L.A., Figotin, A.L.: Spectra of Random and Almost Periodic Operators. Springer, Berlin (1992)zbMATHGoogle Scholar
  20. 20.
    Simon, B., Taylor, M.: Harmonic analysis on SL(2, ℝ) and smoothness of the density of states in the one-dimensional Anderson model. Commun. Math. Phys. 101(1), 1–19 (1985)zbMATHCrossRefMathSciNetADSGoogle Scholar
  21. 21.
    Stollmann, P.: Wegner estimates and localization for continuous Anderson models with some singular distributions. Arch. Math. (Basel) 75(4), 307–311 (2000)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Stollmann, P.: Caught by Disorder. A Course on Bound States in Random Media. Birkhäuser, Boston (2001)Google Scholar
  23. 23.
    Wegner, F.: Bounds on the density of states in disordered systems. Z. Phys., B. Condens. Matter 44(1–2), 9–15 (1981)CrossRefMathSciNetADSGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Institut de Mathématiques de JussieuUniversité Paris DiderotParis Cedex 13France

Personalised recommendations