Spectral theory of discontinuous functions of self-adjoint operators: essential spectrum

Abstract

Let $H_0$ and $H$ be self-adjoint operators in a Hilbert space. In the scattering theory framework, we describe the essential spectrum of the difference $\varphi(H)-\varphi(H_0)$ for piecewise continuous functions $\varphi$. This description involves the scattering matrix for the pair $H$, $H_0$.