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Absolutely continuous spectrum for random operators on trees of finite cone type

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Abstract

We study the spectrum of random operators on a large class of trees. These trees have finitely many cone types and they can be constructed by a substitution rule. The random operators are perturbations of Laplace type operators either by random potentials or by random hopping terms, i.e., perturbations of the off-diagonal elements. We prove stability of arbitrary large parts of the absolutely continuous spectrum for sufficiently small but extensive disorder.

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References

  1. M. Aizenman, R. Sims, and S. Warzel, Stability of the absolutely continuous spectrum of random Schrödinger operators on tree graphs, Probab. Theory Related Fields, 136 (2006), 363–394.

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Aizenman and S. Warzel, Resonant delocalization for random Schrödinger operators on tree graphs, J. European Math. Soc. to appear. arXiv:1104.0969v1.

  3. P.W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109 (1958) 1492–1505.

    Article  Google Scholar 

  4. J. Breuer, Localization for the Anderson model on trees with finite dimensions, Ann. Henri Poincaré 8 (2007), 1507–1520.

    Article  MathSciNet  MATH  Google Scholar 

  5. H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger Operators, Springer, 1987.

  6. R. Carmona, A. Klein, and F. Martinelli, Anderson localization for Bernoulli and other singular potentials, Comm. Math. Phys. 108 (1987), 41–66.

    Article  MathSciNet  MATH  Google Scholar 

  7. R. Carmona and J. Lacroix, Spectral Theory of Random Schrödinger Operators, Birkhäuser, Boston, 1990.

    Book  MATH  Google Scholar 

  8. R. Froese, F. Halasan, and D. Hasler, Absolutely continuous spectrum for the Anderson model on a product of a tree with a finite graph, J. Funct. Anal. 262 (2012), 1011–1042.

    Article  MathSciNet  MATH  Google Scholar 

  9. R. Froese, D. Hasler, and W. Spitzer, Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete Schrödinger operators on graphs, J. Funct. Anal. 230 (2006), 184–221.

    Article  MathSciNet  MATH  Google Scholar 

  10. R. Froese, D. Hasler, and W. Spitzer, Absolutely continuous spectrum for the Anderson model on a tree: a geometric proof of Klein’s theorem, Comm. Math. Phys. 269 (2007), 239–257.

    Article  MathSciNet  MATH  Google Scholar 

  11. R. Froese, D. Hasler, and W. Spitzer. Absolutely continuous spectrum for a random potential on a tree with strong transverse correlations and large weighted loops, Rev. Math. Phys. 21 (2009), 709–733.

    Article  MathSciNet  MATH  Google Scholar 

  12. R. Froese, D. Hasler, and W. Spitzer, A geometric approach to absolutely continuous spectrum for discrete Schrödinger operators, Boundaries and Spectral Theory, Birkhäuser Verlag, Basel, 2011.

    Google Scholar 

  13. F. Halasan, Absolutely continuous spectrum for the Anderson model on some tree-like graphs, Ann. Henri Poincaré 13 (2012) 789–811.

    Article  MathSciNet  MATH  Google Scholar 

  14. F. Halasan, Note on the absolutely continuous spectrum for the Anderson model on Cayley trees of arbitrary degree, arXiv:1008.1519v1.

  15. I. Ya. Goldsheid, S. Molchanov, and L. Pastur. A pure point spectrum of the stochastic onedimensional Schrödinger operator. Funct. Anal. Appl. 11 (1977), 1–8.

    Article  Google Scholar 

  16. M. Keller, On the spectral theory of operators on trees, PhD Thesis, Friedrich Schiller Universit ät Jena, 2010.

  17. M. Keller, D. Lenz, and S. Warzel On the spectral theory of trees with finite cone type, Israel J. Math., to appear, arXiv:1001.3600v2.

  18. H. Kesten, Aspects of first passage percolation, École d’été de probabilités de Saint-Flour XIV-1954, Lecture Notes in Math. 1180, Springer, Berlin, 1986, pp. 125–264.

    Chapter  Google Scholar 

  19. A. Klein, Absolutely continuous spectrum in the Anderson model on the Bethe lattice, Math. Res. Lett. 1 (1994), 399–407.

    MathSciNet  MATH  Google Scholar 

  20. A. Klein, Spreading of wave packets in the Anderson model on the Bethe lattice, Comm. Math. Phys. 177 (1996), 755–773.

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Klein, Extended states in the Anderson model on the Bethe lattice, Adv. Math. 133 (1998), 163–184.

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Klein and C. Sadel, Absolutely continuous spectrum for random operators on the Bethe Strip, Math. Nach. 285 (2012), 5–26.

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Klein and C. Sadel, Ballistic behavior for random Schrödinger operators on the Bethe strip, J. Spectral Theory 1 (2011), 409–442.

    Article  MathSciNet  MATH  Google Scholar 

  24. H. Kunz and B. Souillard, Sur le spectre des opérateurs aux différences finies aléatoires, Comm. Math. Phys. 78 (1980/81), 201–246.

    Article  MathSciNet  MATH  Google Scholar 

  25. R. Lyons, Random walks and percolation on trees Ann. Probab. 18 (1990), 931–958.

    Article  MathSciNet  MATH  Google Scholar 

  26. T. Nagnibeda and W. Woess, Random walks on trees with finitely many cone types, J. Theoret. Probab. 15 (2002), 383–422.

    Article  MathSciNet  MATH  Google Scholar 

  27. P. Stollmann, Caught by Disorder — Bound States in Random Media, Birkhäuser, Boston, Boston MA, 2001.

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematisches Institut, Friedrich-Schiller-Universität Jena, D-07743, Jena, Germany

    Matthias Keller

  2. Friedrich-Schiller-Universität Jena, Mathematisches Institut, D-07743, Jena, Germany

    Daniel Lenz

  3. Technische Universitätmünchen, Zentrummathematik, Boltzmannstrasse 3, D-85747, Garching, Germany

    Simone Warzel

Authors
  1. Matthias Keller
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  2. Daniel Lenz
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  3. Simone Warzel
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Correspondence to Matthias Keller.

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Keller, M., Lenz, D. & Warzel, S. Absolutely continuous spectrum for random operators on trees of finite cone type. JAMA 118, 363–396 (2012). https://doi.org/10.1007/s11854-012-0040-4

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  • DOI: https://doi.org/10.1007/s11854-012-0040-4

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