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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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.
M. Aizenman and S. Warzel, Resonant delocalization for random Schrödinger operators on tree graphs, J. European Math. Soc. to appear. arXiv:1104.0969v1.
P.W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109 (1958) 1492–1505.
J. Breuer, Localization for the Anderson model on trees with finite dimensions, Ann. Henri Poincaré 8 (2007), 1507–1520.
H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger Operators, Springer, 1987.
R. Carmona, A. Klein, and F. Martinelli, Anderson localization for Bernoulli and other singular potentials, Comm. Math. Phys. 108 (1987), 41–66.
R. Carmona and J. Lacroix, Spectral Theory of Random Schrödinger Operators, Birkhäuser, Boston, 1990.
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.
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.
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.
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.
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.
F. Halasan, Absolutely continuous spectrum for the Anderson model on some tree-like graphs, Ann. Henri Poincaré 13 (2012) 789–811.
F. Halasan, Note on the absolutely continuous spectrum for the Anderson model on Cayley trees of arbitrary degree, arXiv:1008.1519v1.
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.
M. Keller, On the spectral theory of operators on trees, PhD Thesis, Friedrich Schiller Universit ät Jena, 2010.
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.
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.
A. Klein, Absolutely continuous spectrum in the Anderson model on the Bethe lattice, Math. Res. Lett. 1 (1994), 399–407.
A. Klein, Spreading of wave packets in the Anderson model on the Bethe lattice, Comm. Math. Phys. 177 (1996), 755–773.
A. Klein, Extended states in the Anderson model on the Bethe lattice, Adv. Math. 133 (1998), 163–184.
A. Klein and C. Sadel, Absolutely continuous spectrum for random operators on the Bethe Strip, Math. Nach. 285 (2012), 5–26.
A. Klein and C. Sadel, Ballistic behavior for random Schrödinger operators on the Bethe strip, J. Spectral Theory 1 (2011), 409–442.
H. Kunz and B. Souillard, Sur le spectre des opérateurs aux différences finies aléatoires, Comm. Math. Phys. 78 (1980/81), 201–246.
R. Lyons, Random walks and percolation on trees Ann. Probab. 18 (1990), 931–958.
T. Nagnibeda and W. Woess, Random walks on trees with finitely many cone types, J. Theoret. Probab. 15 (2002), 383–422.
P. Stollmann, Caught by Disorder — Bound States in Random Media, Birkhäuser, Boston, Boston MA, 2001.
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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