Spectral theory of random Schrödinger operators : a genetic introduction / Reinhard Lang

This is an introductory exposition of the theory of random Schrö-dinger operators. It is addressed to nonspecialists and can be read with a minimum knowledge of operator theory (although some acquaintance with probability theory might be useful). The author discusses two topics from the theory, namely the Lifschits singularity of the integrated density of states and the theorem of Kotani concerning the determination of absolutely continuous spectra from the Lyapunov exponents. In each case, the author gives sufficient heuristic arguments before formally proving theorems.
After some introductory remarks, in §2 the author discusses two simple examples which suggest the main theorems of these notes. In §3, heuristic pictures behind the theory are given, and some basic notions such as that of the representation of the integrated density of states by means of the Brownian motion expectation, and that of Lyapunov exponents for one-dimensional random Schrödinger operators. In §4, two main theorems are formulated, one for the Lifshits (Theorem 1) singularity and Kotani's theorem (Theorem 2). §5 is devoted to the proof of Theorem 1. This section may serve as an introduction to the theory of large deviations. Finally, in §6, a full discussion and proof of Kotani's theorem are given. (MathSciNet)