  Let $H_0 = -\Delta + V_0(x)$ be a Schroedinger operator on
$L_2(\mathbb{R}^\nu),$ $\nu=1,2,$ or 3, where $V_0(x)$ is a bounded measurable
real-valued function on $\mathbb{R}^\nu.$ 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.
