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 $ϕ$ vanishing at infinity, we set $ϕ_δ(\cdot)=ϕ(\cdot/δ)$ and study the spectrum of the difference $ϕ_δ(H-λ)-ϕ_δ(H_0-λ)$ for $δ\to0$. We prove that if $λ$ 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 $S(λ)$ for the pair $H_0$, $H$ and (ii) the singular values of the Hankel operator $H_ϕ$ with the symbol $ϕ$.