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

Ivan Veselić (auth.)3540726896, 9783540726890

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.

Table of contents :
Front Matter….Pages I-X
Random Operators….Pages 1-11
Existence of the Integrated Density of States….Pages 13-43
Wegner Estimate….Pages 45-56
Wegner’s Original Idea. Rigorous Implementation….Pages 57-77
Lipschitz Continuity of the IDS….Pages 79-97
Back Matter….Pages 99-146