Let $H_0 = -Delta + V_0(x)$ be a Schroedinger operator on $L_2(mathbb{R}^ u),$ $ u=1,2,$ or 3, where $V_0(x)$ is a bounded measurable real-valued function on $mathbb{R}^ u.$ Let $V$ be an operator of multiplication by a bounded integrable real-valued
function $V(x)$ and put $H_r = H_0+rV$ for real $r.$ We show that the associated spectral shift function (SSF) $xi$ admits a natural decomposition into the sum of absolutely continuous $xi^{(a)}$ and singular $xi^{(s)}$ SSFs. This is a special case of an analogous result for resolvent comparable pairs of self-adjoint operators, which generalises the known case of a trace class perturbation while also simplifying its proof. We present two proofs -- one short and one long -- which we consider to have value of their own. The long proof along the way reframes some classical results from the perturbation theory of self-adjoint operators, including the existence and completeness of the wave operators and the Birman-Krein formula relating the scattering matrix and the SSF. The two proofs demonstrate the equality of the singular SSF with two a priori different but intrinsically integer-valued functions: the total resonance index and the singular $mu$-invariant.
In this paper we prove for rank one perturbations that negative two times reciprocal of the imaginary part of resonance point is equal to the rate of change of the scattering phase as a function of the coupling constant, where the coupling constant i
s equal to the real part of the resonance point. This equality is in agreement with Breit-Wigner formula from quantum scattering theory. For general relatively trace class perturbations, we also give a formula for the spectral shift function in terms of resonance points, non-real and real.
It is a well-known result of T.,Kato that given a continuous path of square matrices of a fixed dimension, the eigenvalues of the path can be chosen continuously. In this paper, we give an infinite-dimensional analogue of this result, which naturally
arises in the context of the so-called unitary spectral flow. This provides a new approach to spectral flow, which seems to be missing from the literature. It is the purpose of the present paper to fill in this gap.
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 s
ingular $mu$-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.
