Localization for an Anderson-Bernoulli model with generic interaction potential

Abstract

We present a result of localization for a matrix-valued Anderson-Bernoulli operator, acting on L 2 (R) ⊗ R N , for an arbitrary N ≥ 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.

Absence of absolutely continuous spectrum for random scattering zippers

“…13 for the discrete case and Ref. 6 for the continuous case). The Lie algebraic techniques used to prove positivity of the Lyapunov exponents for these models, and on which Theorem 1 is based upon, were developed in the 1980s and 1990s in several papers and in particular the work of Goldsheid and Margulis 14 and the work of Klein, Lacroix, and Speis.…”

Lifshitz Tails for Continuous Matrix-Valued Anderson Models

“…. , V D , are obtained in [2] and [3]. These quasi-one-dimensional models are of physical interest as they can be considered as partially discrete approximations of Anderson models on a two-dimensional continuous strip.…”

“…This assumptions are hard to verify for a general D (where D is the size of the matrix-valued potential), but we were able to verify them for a very particular example of H ω in dimension d = 1, where the periodic and random potentials W and V ω acts like constant functions. In [3], we studied the following Anderson operator: The Proposition I.3 is interesting in itself but doesn't give any information about the behaviour of the IDS at the bottom of spectrum and, until now, it was not clearly stated that it has a Lifshitz behaviour. One of the motivation of the present article is to fill this lack of information of the IDS for quasi-one-dimensional operators and in particular those like H ℓ (ω) we have studied before from the localization point of view.…”

“…This assumptions are hard to verify for a general D (where D is the size of the matrix-valued potential), but we were able to verify them for a very particular example of H ω in dimension d = 1, where the periodic and random potentials W and V ω acts like constant functions. In [3], we studied the following Anderson operator: ). As we can see, H ℓ (ω) is a particular example of H ω with W constant and…”