This book is an overview of scattering theory. The author shows how this theory provides a parametrization of the continuous spectrum of an elliptic operator on a complete manifold with uniform structure at infinity. In the first two lectures the author describes the simple and fundamental case of the Laplacian on Euclidean space to introduce the theory's basic framework. In the next three lectures, he outlines various results on Euclidean scattering, and the methods used to prove them. In the last three lectures he extends these ideas to non-Euclidean settings.

This IMA Volume in Mathematics and its Applications MICROLOCAL ANALYSIS AND NONLINEAR WAVES is based on the proceedings of a workshop which was an integral part of the 1988- 1989 IMA program on "Nonlinear Waves". We thank Michael Beals, Richard Melrose and Jeffrey Rauch for organizing the meeting and editing this proceedings volume. We also take this opportunity to thank the National Science Foundation whose financial support made the workshop possible. A vner Friedman Willard Miller, Jr. PREFACE Microlocal analysis is natural and very successful in the study of the propagation of linear hyperbolic waves. For example consider the initial value problem Pu = f E e'(RHd), supp f C {t ;::: O} u = 0 for t < o. If P( t, x, Dt,x) is a strictly hyperbolic operator or system then the singular support of f gives an upper bound for the singular support of u (Courant-Lax, Lax, Ludwig), namely singsupp u C the union of forward rays passing through the singular support of f.

These lecture notes are intended as a non-technical overview of scattering theory. The point of view adopted throughout is that scattering theory provides a parameterization of the continuous spectrum of an elliptic operator on a complete manifold with uniform structure at infinity. The simple and fundamental case of the Laplacian or Euclidean space is described in the first two lectures to introduce the basic framework of scattering theory. In the next three lectures various results on Euclidean scattering, and the methods used to prove them, are outlined. In the last three lectures these ideas are extended to non-Euclidean settings. These lecture notes will be of interest to graduate students and researchers in the field of applied mathematics.