Abstract:

The scattering matrix and the resolvent of Schrödinger operators with fast decaying potentials are well known to be meromorphic operator valued functions of the spectral parameter defined on a suitable cover of the complex plane. Poles of the resolvent are either eigenvalues or scattering resonances. If the dimension of space is even the resolvent is defined on a logarithmic cover of the complex place with branching point at zero. One can introduce a class of functions which makes it possible to analyse the behaviour of the scattering matrix and the resolvent at the point 0 in a very simple and direct way. I will review some of the classical stationary scattering theory and will then explain some new results that follow from the existence a more general field of functions that allows for poles at the logarithmic branching point.

(based on joint work with J. Müller)