Tohoku Mathematical Journal

Localization for an Anderson-Bernoulli model with generic interaction potential

Hakim Boumaza

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We present a result of localization for a matrix-valued Anderson-Bernoulli operator acting on the space of $\boldsymbol{C}^N$-valued square-integrable functions, for an arbitrary $N$ larger than 1, whose interaction potential is generic in the real symmetric matrices. For such a generic real symmetric matrix, we construct an explicit interval of energies on which we prove localization, in both spectral and dynamical senses, away from a finite set of critical energies. This construction is based upon the formalism of the Fürstenberg group to which we apply a general criterion of density in semisimple Lie groups. The algebraic nature of the objects we are considering allows us to prove a generic result on the interaction potential and the finiteness of the set of critical energies.

Article information

Tohoku Math. J. (2), Volume 65, Number 1 (2013), 57-74.

First available in Project Euclid: 8 April 2013

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47B80: Random operators [See also 47H40, 60H25]
Secondary: 37H15: Multiplicative ergodic theory, Lyapunov exponents [See also 34D08, 37Axx, 37Cxx, 37Dxx]

Anderson localization Lyapunov exponents Fürstenberg group


Boumaza, Hakim. Localization for an Anderson-Bernoulli model with generic interaction potential. Tohoku Math. J. (2) 65 (2013), no. 1, 57--74. doi:10.2748/tmj/1365452625.

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