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Given a self-adjoint operator H, a self-adjoint trace class operator V and afixed Hilbert-Schmidt operator F with trivial kernel and co-kernel, usinglimiting absorption principle an explicit set of full Lebesgue measure isdefined such that for all points of this set the wave and the scatteringmatrices can be defined and constructed unambiguously. Many well-knownproperties of the wave and scattering matrices and operators are proved,including the stationary formula for the scattering matrix. This new abstractscattering theory allows to prove that for any trace class perturbations ofarbitrary self-adjoint operators the singular part of the spectral shiftfunction is an almost everywhere integer-valued function.

Absolutely continuous and singular spectral shift functions