Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Spectral localization in the hierarchical Anderson model
HTML articles powered by AMS MathViewer

by Evgenij Kritchevski PDF
Proc. Amer. Math. Soc. 135 (2007), 1431-1440 Request permission

Abstract:

We prove that a large class of hierarchical Anderson models with spectral dimension $\textrm {d}\leq 2$ has only pure point spectrum.
References
  • Michael Aizenman and Stanislav Molchanov, Localization at large disorder and at extreme energies: an elementary derivation, Comm. Math. Phys. 157 (1993), no. 2, 245–278. MR 1244867, DOI 10.1007/BF02099760
  • Anton Bovier, 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, DOI 10.1007/BF01025849
  • 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).
  • R. Del Rio, N. Makarov, and B. Simon, Operators with singular continuous spectrum. II. Rank one operators, Comm. Math. Phys. 165 (1994), no. 1, 59–67. MR 1298942, DOI 10.1007/BF02099737
  • Freeman J. Dyson, Existence of a phase-transition in a one-dimensional Ising ferromagnet, Comm. Math. Phys. 12 (1969), no. 2, 91–107. MR 436850, DOI 10.1007/BF01645907
  • A. Ya. Gordon, Pure point spectrum under $1$-parameter perturbations and instability of Anderson localization, Comm. Math. Phys. 164 (1994), no. 3, 489–505. MR 1291242, DOI 10.1007/BF02101488
  • S. Molchanov, Lectures on random media, Lectures on probability theory (Saint-Flour, 1992) Lecture Notes in Math., vol. 1581, Springer, Berlin, 1994, pp. 242–411. MR 1307415, DOI 10.1007/BFb0073874
  • S. Molchanov, Hierarchical random matrices and operators. Application to Anderson model, Multidimensional statistical analysis and theory of random matrices (Bowling Green, OH, 1996) VSP, Utrecht, 1996, pp. 179–194. MR 1463464
  • Barry Simon and Tom Wolff, Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians, Comm. Pure Appl. Math. 39 (1986), no. 1, 75–90. MR 820340, DOI 10.1002/cpa.3160390105
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47B80, 47A55, 93A13
  • Retrieve articles in all journals with MSC (2000): 47B80, 47A55, 93A13
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
  • 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
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2276652