Singular spectral shift function for resolvent comparable operators

Let H0=−Δ+V0(x) be a Schrödinger operator on L2(Rν), ν=1,2, or 3, where V0(x) is a bounded measurable real‐valued function on Rν. Let V be an operator of multiplication by a bounded integrable real‐valued function V(x) and put Hr=H0+rV for real r. We show that the associated spectral shift function (SSF) ξ admits a natural decomposition into the sum of absolutely continuous and singular SSFs. In particular, the singular SSF is integer‐valued almost everywhere, even within the absolutely continuous spectrum where the same cannot be said of the SSF itself. This is a special case of an analogous result for resolvent comparable pairs of self‐adjoint operators, which generalises the case of a trace class perturbation appearing in [2] while also simplifying its proof. We present two proofs which demonstrate the equality of the singular SSF with two a priori different and intrinsically integer‐valued functions which can be associated with the pair H0, V: the total resonance index [3] and the singular μ‐invariant [2].