Leonid Pastur

Surveys in Applied Mathematics is a series of volumes, each of which contains expo- of several topics in mathematics and their applications. They are written at a sitions level accessible to advanced graduate students and interested nonspecialists, but they also contain the results of recent research. Volume I consists of three articles. The first is the classic paper of J. B. Keller and R. M. Lewis, "Asymptotic Methods for Partial Differential Equations: The Reduced Wave Equation and Maxwell's Equations." The second is by D. W. McLaughlin and E. A. Overman on "Whiskered Tori for Integrable Pde's: Chaotic Behavior in Near Integrable Pde's." This is a systematic analytical and numerical study of near integrable wave equations, including the sine-Gordon equations and the perturbed nonlinear SchrOdinger equation. The third article is by G. Papanicolaou on "Diffusion in Random Media." It is an introductory survey of homogenization methods for the diffusion equation with random diffusivity.

In the last fifteen years the spectral properties of the Schrodinger equation and of other differential and finite-difference operators with random and almost-periodic coefficients have attracted considerable and ever increasing interest. This is so not only because of the subject's position at the in­ tersection of operator spectral theory, probability theory and mathematical physics, but also because of its importance to theoretical physics, and par­ ticularly to the theory of disordered condensed systems. It was the requirements of this theory that motivated the initial study of differential operators with random coefficients in the fifties and sixties, by the physicists Anderson, 1. Lifshitz and Mott; and today the same theory still exerts a strong influence on the discipline into which this study has evolved, and which will occupy us here. The theory of disordered condensed systems tries to describe, in the so-called one-particle approximation, the properties of condensed media whose atomic structure exhibits no long-range order. Examples of such media are crystals with chaotically distributed impurities, amorphous substances, biopolymers, and so on. It is natural to describe the location of atoms and other characteristics of such media probabilistically, in such a way that the characteristics of a region do not depend on the region's position, and the characteristics of regions far apart are correlated only very weakly. An appropriate model for such a medium is a homogeneous and ergodic, that is, metrically transitive, random field.