Let \(H_0\) and \(H\) be selfadjoint operators in a Hilbert space with spectral functions \(E_0(\cdot)\) and \(E(\cdot)\), respectively, and let \(D(\lambda):=E(\lambda)-E_0(\lambda)\), \(\lambda\in\mathbb R\). Under the assumption that the pair \(\{H,H_0\}\) satisfies a local Kato smoothness type assumption, the scattering matrix \(S(\cdot)\) – a multiplication operator with unitary values in the spectral representation of the absolutely continuous part of \(H_0\) – exists, and in the present paper the following interesting relation is proved:
\[ \sigma_{\text{ess}}(D(\lambda))=[-a,a],\qquad a=\tfrac{1}{2}\| S(\lambda)-I_\lambda\|. \]
It follows that \(D(\lambda)\) is compact if and only if \(S(\lambda)=I_\lambda\). Under some stronger trace class type condition on \(\{H,H_0\}\), also the absolutely continuous spectrum of \(D(\cdot)\) in terms of the spectrum of the scattering matrix \(S(\cdot)\) is described. As an example, the Schrödinger operator \(-\Delta+V\), where \(V\) satisfies \(| V(x)|\leq C(1+| x|)^{-\rho}\), \(\rho>1\), is studied.