Skip to main content
Log in

Constructive proof of localization in the Anderson tight binding model

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We prove that, for large disorder or near the band tails, the spectrum of the Anderson tight binding Hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues. The corresponding eigenfunctions are exponentially well localized. These results hold in arbitrary dimension and with probability one. In one dimension, we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder. Our techniques extend to other physical systems which exhibit localization phenomena, such as infinite systems of coupled harmonic oscillators, or random Schrödinger operators in the continuum.

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Anderson, P.: Absence of diffusion in certain random lattices, Phys. Rev.109, 1492 (1958)

    Google Scholar 

  2. Kunz, H., Souillard, B.: Sur le spectre des opérateurs aux différences finies aléatoires. Commun. Math. Phys.78, 201 (1980)

    Google Scholar 

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

    Google Scholar 

  4. Fröhlich, J., Spencer, T.: A rigorous approach to Anderson localization. Phys. Reports103, 9 (1984); Fröhlich, J., Spencer, T.: Mathematical theory of Anderson localization, proceedings of the I.A.M.P. Conference at Boulder, August 1983. Physica124A, 303 (1984)

    Google Scholar 

  5. Martinelli, F., Scoppola, E.: Remark on the absence of absolutely continuous spectrum in the Anderson model for large disorder or low energy. Commun. Math. Phys.,97, 465 (1985)

    Google Scholar 

  6. Berezanskii, J. M.: Expansion in eigenfunctions of selfadjoint operators. Transl. Math. Monogr.17, A.M.S. 1968; Simon, B.: Schrödinger semigroups, Bull. Am. Math. Soc., New Ser.7, 447 (1983)

  7. Jona-Lasinio, G., Martinelli, F., Scoppola, E.: Multiple tunnelings ind-dimension: A quantum particle in a hierarchical potential, Ann. Inst. Henri Poincaré42, No. 1, 73–108 (1985); A quantum particle in a hierarchical potential with tunneling over arbitrarily large scales, J. Phys.A17, (1984)

    Google Scholar 

  8. Spencer, T.: The Schrödinger equation with a random potential—a mathematical review, to appear in the proceedings of the 1984 Les Houches summer school “Critical Phenomena...”, Osterwalder, K., Stora, R. eds.

  9. Wegner, F.: Bounds on the density of states in disordered systems. Z. Phys.B22, 9 (1981)

    Google Scholar 

  10. Wegner, F.: Z. Phys.B14, 273 (1975)

    Google Scholar 

  11. Gol'dsheid, Ya., Molchanov, S., Pastur, L.: Pure point spectrum of stochastic one-dimensional Schrödinger operators. Funct. Anal. Appl.11, 1 (1977).

    Google Scholar 

  12. Delyon, F., Kunz, H., Souillard, B.: One-dimensional wave equation in disordered media. J. Phys.A16, No. 1, 25 (1983)

    Google Scholar 

  13. Holden, H., Martinelli, F.: Absence of diffusion near the bottom of the spectrum for a Schrödinger operator onL 2(ℝv). Commun. Math. Phys.93, 197 (1984)

    Google Scholar 

  14. Edwards, S., Thouless, D.: Regularity of the density of states in Anderson's localized electron model, J. Phys.C4, 453 (1971); 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, 571 (1984)

    Google Scholar 

  15. Fröhlich, J., Spencer, T.: The Kosterlitz-Thouless transition in two-dimensional abelian spin systems and the Coulomb gas. Commun. Math. Phys.81, 527 (1981)

    Google Scholar 

  16. Ishii, K.: Localization of eigenstates and transport phenomena in the one-dimensional disordered systems. Suppl. Progr. Theor. Phys.53, 77 (1973)

    Google Scholar 

  17. Carmona, R.: Exponential localization in one-dimensional disordered systems. Duke Math. J.49, 191 (1982)

    Google Scholar 

  18. Fürstenberg, H.: Trans. Am. Math. Soc.108, 377 (1963)

    Google Scholar 

  19. Thouless, D.J.: J. Phys.C5, 77 (1972).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by A. Jaffe

Work supported in part by National Science Foundations grant MCS-8108814 (A03).

Work supported in part by National Science Foundation grant DMR 81-00417.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fröhlich, J., Martinelli, F., Scoppola, E. et al. Constructive proof of localization in the Anderson tight binding model. Commun.Math. Phys. 101, 21–46 (1985). https://doi.org/10.1007/BF01212355

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01212355

Keywords

Navigation