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Propagation and localization phenomena in solid state physics and, more generally, in complex and disordered media, have attracted the attention of physicists and mathematicians for many years. One of the most powerful and rigorous methods of analysis of these phenomena is so-called multi-scale analysis.

This monograph gives a systematic exposition of this method for random lattice Schrödinger operators (LSO) with interaction. These results belong to the authors; many of them are presented for the first time in the literature. The authors focus on the Anderson tight binding model, which is most accessible to a broad audience. Also given is a detailed review of various relevant methods (generated, mainly, for the single-particle case), and a discussion of their strong and weak sides regarding their applicability to multi-particle systems.

The level of presentation is suitable for graduate students possessing basic knowledge of functional analysis, probability theory and quantum mechanics. The complete presentation of basic results and methods of spectral theory of self-adjoint operators required for understanding of localization theory makes the presentation self-contained and available for non-specialists in this area.