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Spectral localization in the hierarchical Anderson model


Author: Evgenij Kritchevski
Journal: Proc. Amer. Math. Soc. 135 (2007), 1431-1440
MSC (2000): Primary 47B80, 47A55, 93A13.
DOI: https://doi.org/10.1090/S0002-9939-06-08614-X
Published electronically: November 13, 2006
MathSciNet review: 2276652
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that a large class of hierarchical Anderson models with spectral dimension $ {\rm d}\leq 2$ has only pure point spectrum.


References [Enhancements On Off] (What's this?)

  • [AM] Aizenman, M., Molchanov, S.: Localization at large disorder and at extreme energies: an elementary derivation. Commun. Math. Phys. 157 (1993), no. 2, 245-278. MR 1244867 (95a:82052)
  • [Bo] Bovier, A.: The density of states in the Anderson model at weak disorder: a renormalization group analysis of the hierarchical model. J. Statist. Phys. 59 (1990), no. 3-4, 745-779. MR 1063180 (91m:82063)
  • [BS] Bleher, P. M., Sinai, Ya. G.: Investigation of the Critical Point in Models of the Type of Dyson's Hierarchical Models. Commun. Math. Phys. 33, (1973).
  • [DMS] del Rio, R., Makarov, N., Simon, B.: Operators with singular continuous spectrum: II. Rank one operators. Commun. Math. Phys. 165 (1994), 59. MR 1298942 (97a:47002)
  • [D] Dyson, F.J.: Existence of a phase-transition in a one dimensional Ising Ferromagnet. Comm. Math. Phys. 12, 91 (1969). MR 0436850 (55:9786)
  • [G] Gordon, A.: Pure point spectrum under 1-parameter perturbations and instability of Anderson localization. Commun. Math. Phys. 164 (1994), 489. MR 1291242 (95k:47019)
  • [M1] Molchanov, S.: Lectures on random media. Lectures on probability theory (Saint-Flour, 1992), 242-411, Lecture Notes in Math., 1581, Springer, Berlin, 1994. MR 1307415 (95m:60165)
  • [M2] Molchanov, S.: Hierarchical random matrices and operators. Application to Anderson model. Multidimensional statistical analysis and theory of random matrices (Bowling Green, OH, 1996), 179-194, VSP, Utrecht, 1996.MR 1463464 (99b:82054)
  • [SW] Simon, B., Wolff, T.: Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians. Communications in Pure and Applied Mathematics 49 (1986), 75. MR 0820340 (87k:47032)

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Additional Information

Evgenij Kritchevski
Affiliation: Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, Quebec, Canada H3A 2K6
Email: ekritc@math.mcgill.ca

DOI: https://doi.org/10.1090/S0002-9939-06-08614-X
Received by editor(s): December 8, 2005
Published electronically: November 13, 2006
Additional Notes: This work was supported in part by an FQRNT grant.
Communicated by: Mikhail Shubin
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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