ar X iv : 0 81 0 . 20 72 v 1 [ m at h . SP ] 1 2 O ct 2 00 8 ABSOLUTELY CONTINUOUS AND SINGULAR SPECTRAL SHIFT FUNCTIONS

Given a self-adjoint operator H 0 , a self-adjoint trace class operator V and a fixed Hilbert-Schmidt operator F with trivial kernel and co-kernel, using limiting absorption principle an explicit set of full Lebesgue measure Λ(H 0 , F) ⊂ R is defined, such that for all points of this set the wave and the scattering matrices can be defined unambiguously. Many well-known properties of the wave and scattering matrices and operators are proved, including the stationary formula for the scattering matrix. This new abstract scattering theory allows to prove that the singular part of the spectral shift function is an almost everywhere integer-valued function for trace class perturbations of a self-adjoint operator.