Local conditions for the existence of the spectral shift function

Let $U_0$, $U_1$ be unitary operators in a Hilbert space. If the operator $U_1-U_0$ is nuclear, then (as M. G. Krein established) there exists a function $\eta$ on the unit circle $\mathbf T$, $\eta=\eta(U_1,U_0)$, $\eta\in L_1(\mathbf T)$ satisfying the equality
\begin{gather} tr(\varphi(U_1)-\varphi(U_0))=\int_{\mathbf T}\eta(\zeta)\varphi'(\zeta)d\zeta \end{gather}
for all functions $\varphi$ with derivative $\varphi'$ from the Wiener class. M. Sh. Rirman and M. G. Krein proved that the function $\varphi'$ is connected with the scattering matrix $S$ for the pair $U_0$, $U_1$ by
\begin{gather} \det S(\zeta)=\exp(-2\pi i\eta(\zeta)), \tag{2} \end{gather}

In this paper (1) and (2) are proved under more general (local) conditions on the pair $U_0$, $U_1$. Under these conditions we investigate some properties of the function n and describe the class of functions $\eta$, which are admissible in (1). Applications to differential operators are given.