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Name:

Miloslav Znojil

Reviewer number:

9689

Email:

znojil@ujf.cas.cz

Item's zbl-Number:

DE 018 024 62X

Author(s):

Veselic, Ivan:

Shorttitle:

Localization of random perturbations of periodic Schroedinger operators

Source:

Ann. Henri Poincare 3, No. 2, 389-409 (2002)

Classification:

47B80

Random operators

Primary Classification:

82B44

Disordered systems random Ising models, random Schroedinger operators, etc.

Secondary Classification:

60H15

Stochastic partial differential equations

Keywords:

periodic Schroedinger operators; regular Floquet eigenvalues; random perturbations; proof of Andrson-type localization;

Review:

The possibility of absence of diffusion in certain random lattices
(i.e., the famous Anderson localization) is formulated and proved as
a theorem on non-negative, omega-dependent random perturbations V [of
a periodic (or, more generally, quasi-periodic) continuous
Schroedinger operator H] which preserve a spectral band. The theorem
states that there exists an interval I (containing the lower edge of
the band) where the perturbed spectrum is pure point (for almost all
omega). The proof relies on exclusion of the absolutely continuous
spectrum via establishing the exponential decay of the non-ergodic,
finite-size (box) resolvents, followed by the standard (still briefly
outlined) recursive multi-scale analysis. As long as the key
technical result of the paper concerns the integrated densities of
states, certain specific properties of the Floquet eigenvalues of H
are assumed.

Remarks to the editors:

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