Existence and Regularity Properties of the Integrated Density of States of Random Schrödinger Operators

ISBN-13: 9783540726890 / Angielski / Miękka / 2007 / 160 str.

Opis

This book describes in detail a quantity encoding spectral feature of random operators: the integrated density of states or spectral distribution function. It presents various approaches to the construction of the integrated density of states and the proof of its regularity properties. The book also includes references to and a discussion of other properties of the IDS as well as a variety of models beyond those treated in detail here.

Recenzje

From the reviews:

"The aim of this small book is to give an overview of old and new results and methods in the spectral theory of random Schrödinger operators. ... The book ends with a bibliography of 501 items. It will be very useful for researchers interested in the quantum mechanics of disordered solids. It is accessible to graduate students with some knowledge on operator theory." (Dominique Lepingle, Zentralblatt MATH, Vol. 1189, 2010)

"The present lecture notes give a well written introduction to continuity properties of the integrated density of states (IDS) of random Schrödinger operators. ... it is a pleasant contribution to the current literature which will be useful for both graduate students and researchers interested in this active area." (G. Teschl, Monatshefte für Mathematik, Vol. 157 (2), June, 2009)

Spis treści

Random Operators.- Existence of the Integrated Density of States.- Wegner Estimate.- Wegner’s Original Idea. Rigorous Implementation.- Lipschitz Continuity of the IDS.

Blurb

The theory of random Schrödinger operators is devoted to the mathematical analysis of quantum mechanical Hamiltonians modeling disordered solids. Apart from its importance in physics, it is a multifaceted subject in its own right, drawing on ideas and methods from various mathematical disciplines like functional analysis, selfadjoint operators, PDE, stochastic processes and multiscale methods.
The present text describes in detail a quantity encoding spectral features of random operators: the integrated density of states or spectral distribution function. Various approaches to the construction of the integrated density of states and the proof of its regularity properties are presented.

The setting is general enough to apply to random operators on Riemannian manifolds with a discrete group action. References to and a discussion of other properties of the IDS are included, as are a variety of models beyond those treated in detail here.