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Damanik, David; Sims, Robert; Stolz, Günter

Localization for one-dimensional, continuum, Bernoulli-Anderson models. (English) Zbl 1107.82025

Duke Math. J. 114, No. 1, 59-100 (2002).
Summary: We use scattering theoretic methods to prove strong dynamical and exponential localization for one-dimensional, continuum, Anderson-type models with singular distributions; in particular, the case of a Bernoulli distribution is covered. The operators we consider model alloys composed of at least two distinct types of randomly dispersed atoms. Our main tools are the reflection and transmission coefficients for compactly supported single-site perturbations of a periodic background which we use to verify the necessary hypotheses of multi-scale analysis. We show that nonreflectionless single sites lead to a discrete set of exceptional energies away from which localization occurs.

Cited in 49 Documents

MSC:

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
47B80 Random linear operators
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References:

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