Skip to main content
Log in

The Repulsion Between Localization Centers in the Anderson Model

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

In this note we show that, a simple combination of deep results in the theory of random Schrödinger operators yields a quantitative estimate of the fact that the localization centers become far apart, as corresponding energies are close together.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. M. Aizenman, Localization at weak disorder: Some elementary bounds. Rev. Math. Phys. 6:1163–1182 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  2. F. Germinet and S. De Biévre, Dynamical localization for discrete and continuous random Schrödinger operators. Comm. Math. Phys. 194:323–341 (1998).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. R. Carmona and J. Lacroix, Spectral Theory of Random Schrödinger Operators, Birkhäuser (1990).

  4. D. Damanik and P. Stollmann, Multi-scale analysis implies strong dynamical localization. Geom. Funct. Anal. 11(1):11–29 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Fröhlich and T. Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Commun. Math. Phys. 88:151–184 (1983).

    Article  ADS  MATH  Google Scholar 

  6. L. N. Grenkova, S. A. Molchanov and Ju. N. Sudarev, On the basic states of one-dimensional disordered structures. Comm. Math. Phys. 90(1):101–123 (1983).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. W. Kirsch, O. Lenoble and L. Pastur, On the mott formula for the a.c. conductivity and binary correlators in the strong localization regime of disordered systems. J. Phys. A. 36(49):12157–12180 (2003).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. A. Klein and S. Molchanov, Simplicity of eigenvalues in the Anderson model. J. Stat. Phys.(122):95–99 (2006).

    Google Scholar 

  9. A. Klein, O. Lenoble and P. Muller, On Mott's formula for the acconductivity in the Anderson model, math-ph/0508007.

  10. H. Kunz and B. Souillard, Sur le spectre des operateurs aux differences finies aleatoires. Comm. Math. Phys. 78:201–246 (1980).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. N. Minami, Local fluctuation of the spectrum of a multidimensional Anderson tight-binding model. Comm. Math. Phys. 177:709–725 (1996).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. S. Molchanov, The local structure of the spectrum of the one-dimensional Schrödinger operator. Comm. Math. Phys. 78:429–446 (1981).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. R. Killip and F. Nakano, The global fluctuation of localization centers in the Anderson model, preprint.

  14. B. Simon. Lifschitz tails for the Anderson model. J. Stat. Phys. 38:65–76 (1985).

    Article  ADS  Google Scholar 

  15. H. von Dreifus and A. Klein, A new proof of localization in the Anderson tight binding model. Commun. Math. Phys. 124:285–299 (1989).

    Article  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Fumihiko Nakano.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nakano, F. The Repulsion Between Localization Centers in the Anderson Model. J Stat Phys 123, 803–810 (2006). https://doi.org/10.1007/s10955-006-9086-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-006-9086-y

Keywords

Navigation