Multi-Scale Analysis for Random Quantum Systems with Interaction

ISBN-13: 9781493939527 / Angielski / Miękka / 2016 / 238 str.

Victor Chulaevsky; Yuri Suhov

Opis

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 Schrodinger 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.

Spis treści

Preface.- Part I Single-particle Localisation.- A Brief History of Anderson Localization.- Single-Particle MSA Techniques.- Part II Multi-particle Localization.- Multi-particle Eigenvalue Concentration Bounds.- Multi-particle MSA Techniques.- References.- Index.

Blurb

The study of quantum disorder has generated considerable research activity in mathematics and physics over past 40 years. While single-particle models have been extensively studied at a rigorous mathematical level, little was known about systems of several interacting particles, let alone systems with positive spatial particle density. Creating a consistent theory of disorder in multi-particle quantum systems is an important and challenging problem that largely remains open. Multi-scale Analysis for Random Quantum Systems with Interaction  presents the progress that had been recently achieved in this area.

 

The main focus of the book is on a rigorous derivation of the multi-particle localization in a strong random external potential field. To make the presentation accessible to a wider audience, the authors restrict attention to a relatively simple tight-binding Anderson model on a cubic lattice Z^d.

 

This book includes the following cutting-edge features:

* an introduction to the state-of-the-art single-particle localization theory

* an extensive discussion of relevant technical aspects of the localization theory

* a thorough comparison of the multi-particle model with its single-particle counterpart

* a self-contained rigorous derivation of both spectral and dynamical localization in the multi-particle tight-binding Anderson model.

 

Required mathematical background for the book includes a knowledge of functional calculus, spectral theory (essentially reduced to the case of finite matrices) and basic probability theory. This is an excellent text for a year-long graduate course or seminar in mathematical physics. It also can serve as a standard reference for specialists.