Higher-order spectral shift function for resolvent comparable perturbations

Given a pair of self-adjoint operators \(H\) and \(V\) such that \(V\) is bounded and \((H+V-i)^{-1}-(H-i)^{-1}\) belongs to the Schatten-von Neumann ideal \(\mathcal{S}^n\), \(n\ge 2\), of operators on a separable Hilbert space, we establish higher order trace formulas for a broad set of functions \(f\) containing several major classes of test functions and also establish existence of the respective locally integrable real-valued spectral shift functions determined uniquely up to a low degree polynomial summand. Our result generalizes the result of \cite{PSS13} for Schatten-von Neumman perturbations \(V\) and settles earlier attempts to encompass general perturbations with Schatten-von Neumman difference of resolvents, which led to more complicated trace formulas for more restrictive sets of functions \(f\) and to analogs of spectral shift functions lacking real-valuedness and/or expected degree of uniqueness. Our proof builds on a general change of variables method derived in this paper and significantly refining those appearing in \cite{vNS21,PSS15,S17} with respect to several parameters at once.