Spectral theory of discontinuous functions of self-adjoint operators and scattering theory

In the smooth scattering theory framework, we consider a pair of self-adjoint operators $H_0$, $H$ and discuss the spectral projections of these operators corresponding to the interval $(-\mathrm{\infty },\lambda )$. The purpose of the paper is to study the spectral properties of the difference $D(\lambda )$ of these spectral projections. We completely describe the absolutely continuous spectrum of the operator $D(\lambda )$ in terms of the eigenvalues of the scattering matrix $S(\lambda )$ for the operators $H_0$ and $H$. We also prove that the singular continuous spectrum of the operator $D(\lambda )$ is empty.