Skip to main content
SpringerLink
Log in

Anomalies in the one-dimensional Anderson model at weak disorder

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

Abstract

We show that at the special energiesE=2cosπp/q, the invariant measure, the Lyapunov exponent, and the density of states can be extended to zero disorder as C∞ functions in the disorder parameter. In particular, we obtain asymptotic series in the disorder parameter. This gives a rigorous proof of the existence of the anomalies originally discovered by Kappus and Wegner and studied by Derrida and Gardner and by Bovier and Klein.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bougerol, P., Lacroix, J.: Products of random matrices with applications to Schrödinger operators. Boston: Birkhäuser 1985

    Google Scholar 

  2. Kappus, M., Wegner, Z.: Anomaly in the band centre of the one-dimensional Anderson model. Z. Phys.B45, 15–21 (1981)

    Google Scholar 

  3. Thouless, D.: In: Ill-Condensed Matter. Balian, R., Maynard, R., Toulouse, G. (eds.). Amsterdam: North Holland 1979

    Google Scholar 

  4. Derrida, B., Gardner, E.: Lyapunov exponent of the one dimensional Anderson model: Weak disorder expansions. J. Phys.45, 1285–1295 (1984)

    Google Scholar 

  5. Bovier, A., Klein, A.: Weak disorder expansion of the invariant measure for the one-dimensional Anderson model. J. Stat. Phys.51, 501–517 (1988)

    Google Scholar 

  6. Klein, A., Speis, A.: Regularity of the invariant measure and of the density of states in the one dimensional Anderson model. J. Funct. Anal.88, 211–227 (1990)

    Google Scholar 

  7. 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, 227–241 (1986)

    Google Scholar 

  8. March, P., Sznitman, A.: Some connections between excursion theory and the discrete random Schrödinger equation with applications to analyticity and smoothness properties of the density of states in one dimension (preprint)

  9. Reed, M., Simon, B.: Methods of modern mathematical physics, II. Fourier analysis, self-adjointness. New York: Academic Press 1975

    Google Scholar 

  10. Chernoff, P.: Product formulas, nonlinear semigroups, and addition of unbounded operators. Mem. Am. Math. Soc.140, Providence, RI: Am. Math. Soc. 1974

    Google Scholar 

  11. Klein, A., Speis, A.: Smoothness of density of states in the Anderson model on a one-dimensional strip. Ann. Phys.183, 352–398 (1988)

    Google Scholar 

Download references

Author information

Author notes

  1. Massimo Campanino

    Present address: Dipartimento di Matematica, Universita de Bologna, p.zz.a S. Donato, 5, I-40126, Bologna, Italy

Authors and Affiliations

  1. Department of Mathematics, University of California, Irvine, 92717, Irvine, CA, USA

    Massimo Campanino & Abel Klein

Authors
  1. Massimo Campanino
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abel Klein
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by T. Spencer

Partially supported by NSF grant DMS 87-02301

Rights and permissions

Reprints and permissions

About this article

Cite this article

Campanino, M., Klein, A. Anomalies in the one-dimensional Anderson model at weak disorder. Commun.Math. Phys. 130, 441–456 (1990). https://doi.org/10.1007/BF02096930

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02096930

Keywords

Search

Navigation