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Letters in Mathematical Physics

, Volume 38, Issue 4, pp 343–348 | Cite as

Towards localisation by Gaussian random potentials in multi-dimensional continuous space

  • Werner Fischer
  • Hajo Leschke
  • Peter Müller
Article
  • 37 Downloads

Abstract

A rigorous proof is outlined to exclude the absolutely continuous spectrum at sufficiently low energies for a quantum-mechanical particle moving in multi-dimensional Euclidean space under the influence of certain Gaussian random potentials, which are homogeneous with respect to Euclidean translations.

Mathematics subject classifications(1991)

81Q10 47B80 35J10 

Key words

Random Schrödinger operators Localisation 

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References

  1. 1.
    Anderson, P. W.: Absence of diffusion in certain random lattices,Phys. Rev. 109 (1958), 1492.Google Scholar
  2. 2.
    Shklovskii, B. I. and Efros, A. L.,Electronic Properties of Doped Semiconductors, Springer, Berlin, 1984.Google Scholar
  3. 3.
    Lifshits, I. M., Gredeskul, S. A. and Pastur, L. A.:Introduction to the Theory of Disordered Systems, Wiley, New York, 1988.Google Scholar
  4. 4.
    Carmona, R. and Lacroix, J.:Spectral Theory of Random Schrödinger Operators, Birkhäuser, Boston, 1990.Google Scholar
  5. 1.
    Pastur, L. and Figotin, A.:Spectra of Random and Almost-Periodic Operators, Springer, Berlin, 1992.Google Scholar
  6. 6.
    Martinelli, F. and Holden, H.: On absence of diffusion near the bottom of the spectrum for a random Schrödinger operator onL 2 (ℝv),Comm. Math. Phys. 93 (1984), 197.Google Scholar
  7. 7.
    Combes, J.-M. and Hislop, P. D.: Localisation for some continuous, random Hamiltonians ind-dimensions,J. Funct. Anal. 124 (1994), 149.Google Scholar
  8. 8.
    Klopp, F.: Localisation for some continuous random Schrödinger operators,Comm. Math. Phys. 167 (1995), 553.Google Scholar
  9. 9.
    Kirsch, W.: Wegner estimates and Anderson localisation for alloy-type potentials,Math. Z. 221 (1996), 507.Google Scholar
  10. 10.
    Martinelli, F. and Scoppola, E.: Introduction to the mathematical theory of Anderson localisation,Riv. Nuovo Cimento 10(10) (1987), 1.Google Scholar
  11. 11.
    Fischer, W., Leschke, H. and Müller, P.: in preparation, to be submitted toJ. Stat. Phys. Google Scholar
  12. 12.
    Doukhan, P.:Mixing, Properties and Examples, Springer, New York, 1994.Google Scholar
  13. 13.
    Fernique, X. M.: Regularité des trajectoires des fonctions aléatoires Gaussiennes, in: P.-L. Hennequin (ed.),Ecole d'Eté de Probabilités de Saint-Flour IV-1974, Lecture Notes in Mathematics 480, Springer, Berlin, 1975, p. 1.Google Scholar
  14. 14.
    Ibragimov, I. A. and Rozanov, Y. A.:Gaussian Random Processes, Springer, New York, 1978.Google Scholar
  15. 15.
    Kirsch, W.: Random Schrödinger operators: a course, in: H. Holden and A. Jensen (eds),Schrödinger Operators, Lecture Notes in Physics 345, Springer, Berlin, 1989, p. 264.Google Scholar
  16. 16.
    Fröhlich, J. and Spencer, T.: Absence of diffusion in the Anderson tight binding model for large disorder or low energy,Comm. Math. Phys. 88 (1983), 151.Google Scholar
  17. 17.
    von Dreifus, H. and Klein, A.: Localisation for random Schrödinger operators with correlated potentials,Comm. Math. Phys. 140 (1991), 133.Google Scholar
  18. 18.
    Wegner, F.: Bounds on the density of states in disordered systems,Z. Phys. B 44 (1981), 9.Google Scholar
  19. 19.
    Glimm, J. and Jaffe, A.:Quantum Physics: A Functional Integral Point of View, 2nd edn, Springer, New York, 1987.Google Scholar
  20. 20.
    Gihman, I. I. and Skorokhod, A. V.:The Theory of Stochastic Processes I, Springer, Berlin, 1974.Google Scholar
  21. 21.
    Combes, J.-M. and Hislop, P. D.: Landau Hamiltonians with random potentials: Localisation and the density of states,Comm. Math. Phys. 177 (1996), 603.Google Scholar
  22. 22.
    Dorlas, T. C., Macris, N. and Pulé, J. V.: Localisation in a single-band approximation to random Schrödinger operators in a magnetic field,Helv. Phys. Acta 68 (1995), 329.Google Scholar
  23. 23.
    Dorlas, T. C., Macris, N. and Pulé, J. V.: Localisation in single Landau bands,J. Math. Phys. 37 (1996), 1574.Google Scholar
  24. 24.
    Wang, W.-M.: Microlocalisation, percolation and Anderson localisation for the magnetic Schrödinger operatorswith a random potential, to appear inJ. Funct. Anal. Google Scholar
  25. 25.
    Matsumoto, H.: On the integrated density of states for the Schrödinger operators with certain random electromagnetic potentials,J. Math. Soc. Japan 45 (1993), 197.Google Scholar
  26. 26.
    Ueki, N.: On spectra of random Schrödinger operators with magnetic fields,Osaka J. Math. 31 (1994), 177.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Werner Fischer
    • 1
  • Hajo Leschke
    • 1
  • Peter Müller
    • 1
  1. 1.Institut für Theoretische PhysikUniversität Erlange-NürnbergErlangenGermany

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