Israel Journal of Mathematics

, Volume 107, Issue 1, pp 125–139 | Cite as

Localization for random perturbations of anisotropic periodic media

  • Peter Stollmann


We prove localization for random perturbations of periodic divergence form operators of the form ∇ · aω · ∇ near the band edges. Here aω is a matrix function which results from an Anderson type perturbation of a periodic matrix function.


Band Edge Random Perturbation Pure Point Spectrum Generalize Eigenfunctions Schr6dinger Operator 
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  1. [1]
    J. M. Barbaroux, J. M. Combes and P. D. Hislop,Localization near band edges for random Schrödinger operators, Helvetica Physica Acta70 (1997), 16–43.MATHMathSciNetGoogle Scholar
  2. [2]
    R. Carmona and J. Lacroix,Spectral Theory of Random Schrödinger Operators, Birkhäuser, Boston, Basel, Berlin, 1990.MATHGoogle Scholar
  3. [3]
    J. M. Combes and L. Thomas,Asymptotic behavior of eigenfunctions of multiparticle Schrödinger operators, Communications in Mathematical Physics34 (1973), 439–482.CrossRefMathSciNetGoogle Scholar
  4. [4]
    A. Figotin and A. Klein,Localization of classical waves I: Acoustic waves, Communications in Mathematical Physics180 (1996), 439–482.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    R. Hempel,Second order perturbations of divergence type operators with a spectral gap, inSymposium on Operator Calculus and Spectral Theory, Lambrecht 1991 (M. Demuth, B. Gramsch and B.-W. Schulze, eds.), Operator Theory Advances and Applications vol. 57, Birkhäuser, Basel, Boston, Berlin, 1992.Google Scholar
  6. [6]
    R. Hempel and I. Herbst,Bands and gaps for periodic magnetic Hamiltonians, inPartial Differential Operators and Mathematical Physics, International Conference in Holzhau, 1994 (M. Demuth and B.-W. Schulze, eds.), Operator Theory Advances and Applications Vol. 78, Birkhäuser, Basel, Boston, Berlin, 1995.Google Scholar
  7. [7]
    T. Kato,Perturbation Theory for Linear Operators, 2nd ed., Springer-Verlag, Berlin, 1976.MATHGoogle Scholar
  8. [8]
    W. Kirsch,Wegner estimates and Anderson localization for alloy-type potentials, Mathematische Zeitschrift221 (1996), 507–512.MATHMathSciNetGoogle Scholar
  9. [9]
    W. Kirsch, P. Stollmann and G. Stolz,Localization for random perturbations of periodic Schrödinger Operators, preprint.Google Scholar
  10. [10]
    L. Pastur and A. Figotin,Random Schrödinger operators, Springer-Verlag, Berlin, 1992.Google Scholar
  11. [11]
    M. Reed and B. Simon,Methods of Modern Mathematical Physics IV, Academic Press, New York, 1978.MATHGoogle Scholar
  12. [12]
    G. V. Rozenblum, M. A. Shubin and M. Z. Solomyak,Spectral theory of differential operators, inPartial Differential Equations (M. A. Shubin, ed.), Encyclopedia of Mathematical Sciences Vol. 64, Springer-Verlag, Berlin, Heidelberg, 1994.Google Scholar
  13. [13]
    P. Stollmann,A short proof of a Wegner estimate and localization, preprint.Google Scholar
  14. [14]
    W. Thirring,A Course in Mathematical Physics 3, Quantum Mechanics of Atoms and Molecules Springer-Verlag, New York, Wien, 1981.MATHGoogle Scholar
  15. [15]
    J. Weidmann,Linear Operators in Hilbert Spaces, Graduate Texts in Mathematics Vol. 68, Springer-Verlag, Berlin, 1980.MATHGoogle Scholar

Copyright information

© The Magnes Press 1998

Authors and Affiliations

  1. 1.Fachbereich MathematikJohann Wolfgang Goethe-UniversitätFrankfurt and MainGermany

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