Third order trace formula (Q2253672)

Let \(A\) be a self-adjoint operator (possibly unbounded) on a separable Hilbert space and \(V\) be a self-adjoint Hilbert-Schmidt operator. \textit{K. Dykema} and \textit{A. Skripka} [J. Funct. Anal. 257, No. 4, 1092--1132 (2009; Zbl 1170.47007)] showed that there exists a unique finite real valued function \(\nu\) (the spectral shift function) on \({\mathbb R}\) such that the trace formula \[ \mathrm{Tr}[ \phi(A+V)-\phi(A)-D^{(1)}\phi(A)(V)-\frac{1}{2}D^{(2)}\phi(A)(V,V)]=\int_{-\infty}^{\infty}\phi^{\prime\prime\prime}(\lambda)\,d\nu(\lambda), \] where \(D^{(1)}\phi(A)\) and \(D^{(2)}\phi(A)\) are the first and the second order Fréchet derivatives of \(\phi\) at \(A\), holds for suitable functions \(\phi\). In this paper, the authors give a different proof for the existence of the spectral shift function when the operator \(A\) is bounded below.