For a pair of selfadjoint operators \(H\) and \(H_0\) in a Hilbert space such that \(\text{dom}(H)=\text{dom}(H_0)\) and with the difference \(V=H-H_0\) in the trace class, the spectral shift function \(\xi(\lambda)\) was introduced by M.G.Krein in [Dokl.Akad.Nauk.SSSR 144, 268–271 (1962; Zbl 0191.15201)] with the help of the perturbation determinant \(D_{H/H_0}:=\det((H-z)(H_0-z)^{-1})\). Since \(\lim_{|\text{Im}(z)|\to\infty}D_{H/H_0}(z)=1\), a branch of \(z\mapsto\log (D_{H/H_0}(z))\) in the upper half-plane \(\mathbb{C}_+\) is fixed by the condition \(\log (D_{H/H_0}(z))\to 0\) as \(\text{Im}(z)\to\infty\) and the spectral shift function is then defined by \(\xi(\lambda)=\frac{1}{\pi}\text{Im}(\log(D_{H/H_0}(\lambda+i0)))\). M.G.Krein proved that \(\xi\in L_1(\mathbb{R},d\lambda)\), \(\|\xi\|_{L_1}\leq\| V\|_1\) (where the latter is the trace-class norm of \(V\)), and that the trace formula \(\text{tr}((H-z)^{-1}-(H_0-z)^{-1})=-\int_\mathbb{R}\frac{\xi(\lambda)}{(\lambda-z)^2}\,d\lambda\) holds for all \(z\in\rho(H)\cap\rho(H_0)\). It turns out that the scattering matrix \(S(H,H_0;\lambda)\) [see, e.g., the book D.R.Yafaev, “Mathematical scattering theory.General theory” (Translations of Mathematical Monographs 105; Providence, RI:AMS) (1992; Zbl 0761.47001)] and the spectral shift function \(\xi(\lambda)\) are related via Birman–Krein formula \(\det(S(H,H_0;\lambda))=\exp(2\pi i\xi(\lambda))\) for a.e.\(\lambda\in\mathbb{R}\).
In the paper under review, the trace formula and the Birman–Krein formula are obtained for pairs of selfadjoint extensions \(A_0\) and \(A_\Theta\) of a densely defined symmetric operator \(A\) with finite deficiency indices (in this case, only the assumption that the resolvent difference \((H-z)^{-1}-(H_0-z)^{-1}\) is of trace class is used). These results are generalized further to a maximal dissipative extension \(A_D\) of \(A\) and the corresponding pair \(\{A_D,A_0\}\).
For the entire collection see [Zbl 1151.47002].