Scattering theory is about properties of solutions of the eigenfunction equation, usually closely related to the Laplacian. There is typically a big solution space to such equations, and a basis for solutions is given by those having some such form in the asymptotic region.
In fact, there is a well-defined map acting on functions defined on the sphere at infinity. This map is called the scattering matrix (in spite of the fact that it is not a matrix) and characterizes the large-distance effect of the potential function. There is a lot known about the scattering matrix, and still plenty of things still to be understood. For example, the high energy limit of the scattering matrix is still not very well understood.