  With the essential spectrum of a self-adjoint operator given a relatively
trace class perturbation one can associate an integer-valued invariant which
admits different descriptions as the singular spectral shift function, total
resonance index, and singular $\mu$-invariant. In this paper we give a direct
proof of the equality of the total resonance index and singular $\mu$-invariant
assuming only the limiting absorption principle. The proof is based on an
application of the argument principle to the poles and zeros of the analytic
continuation of the scattering matrix considered as a function of the coupling
parameter.
