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Tohoku Mathematical Journal
2013
March
SECOND SERIES VOL. 65, NO. 1
Tohoku Math. J.
65 (2013), 5774

Title
LOCALIZATION FOR AN ANDERSONBERNOULLI MODEL WITH GENERIC INTERACTION POTENTIAL
Author
Hakim Boumaza
(Received May 12, 2010, revised April 24, 2012) 
Abstract.
We present a result of localization for a matrixvalued AndersonBernoulli operator acting on the space of $\boldsymbol{C}^N$valued squareintegrable 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.
2000 Mathematics Subject Classification.
Primary 47B80; Secondary 37H15.
Key words and phrases.
Anderson localization, Lyapunov exponents, Fürstenberg group.


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