Scattering Length of Positive Potentials

There is a notion of scattering length of a positive function v on Rn, analogous to the notion of capacity of a compact set K in Rn. Seminal work on this was done in papers of M. Kac and J. Luttinger. This work played an important role in several previous papers of the author on Schrodinger operators. Here we give a systematic presentation of the fundamentals of the subject, and extend its scope from Rn with n bigger than 2 to a natural class of complete Riemannian manifolds, including two-dimensional cases. A central theme is the relation of the scattering length of v to the spectral behavior of v minus the Laplace operator.