Scattering matrix and functions of self-adjoint operators

Abstract

In the scattering theory framework, we consider a pair of operators $H_0$, $H$. For a continuous function $\phi$ vanishing at infinity, we set ${\phi }_{\delta }(\cdot )=\phi (\cdot /\delta )$ and study the spectrum of the difference ${\phi }_{\delta }(H-\lambda )-{\phi }_{\delta }(H_0-\lambda )$ for $\delta \to 0$. We prove that if $\lambda$ is in the absolutely continuous spectrum of $H_0$ and $H$, then the spectrum of this difference converges to a set that can be explicitly described in terms of (i) the eigenvalues of the scattering matrix $S(\lambda )$ for the pair $H_0$, $H$ and (ii) the singular values of the Hankel operator $H_{\phi }$ with the symbol $\phi$.