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