Geometric Scattering Theory and Applications

Classical scattering theory, by which we mean the scattering of acoustic and electromagnetic waves and quantum particles, is a very old discipline with roots in mathematical physics. It has also become an important part of the modern theory of linear partial differential equations. Spectral geometry is a slightly more recent subject, the goal of which is to understand the connections between the behavior of eigenvalues of the Laplace-Beltrami operator ∆g on a compact Riemannian manifold (M, g) and various features of the geometry and topology of this manifold. Geometric scattering theory has developed over the past few decades as a unification and extension of these two fields, though certain aspects of the field go back much further. On the one hand, scattering theory is the natural replacement for the study of eigenvalues on complete, noncompact manifolds since the spectrum of the Laplace-Beltrami operator may often contain only continuous spectrum, whereas there is still a rich theory for some of the other objects in scattering theory described below. On the other hand, the study of scattering theory in the setting of Riemannian manifolds adds many new subtleties and problems over those encountered for traditional Schrödinger operators on Euclidean space by allowing for spaces with various more intricate types of asymptotic geometries. This broader perspective has turned out to be surprisingly revealing and to shed light on many of the classical problems in scattering on Euclidean spaces. This ‘unification’ of scattering theory and spectral geometry was proposed as a systematic area of study in a series of lectures given by R. Melrose at Stanford in 1994 [11]. Since that time the field has grown substantially, partly along some of the lines that Melrose had foreseen, but in many exciting and unexpected directions as well. Geometric scattering theory encompasses the study, in the broadest sense, of the spectrum of the LaplaceBeltrami operator ∆g and other natural elliptic operators on complete, noncompact Riemannian manifolds (M, g) with geometries which are ‘asymptotically regular’ at infinity. While there is no precise definition of this condition, it encompasses many natural settings and includes such cases as manifolds which are asymptotically Euclidean or conic, asymptotically cylindrical or periodic, asymptotically (either real or complex) hyperbolic, or modeled on other locally or globally symmetric spaces of noncompact type. The objects of study include the resolvent of the Laplace-Beltrami operator∆g given by R(λ) = (∆g − λ)−1,