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

TL;DR

This paper characterizes the essential spectrum of differences of functions of self-adjoint operators using scattering matrices, specifically for piecewise continuous functions, within the framework of scattering theory.

Contribution

It provides a new description of the essential spectrum of operator differences involving piecewise continuous functions via scattering matrices.

Findings

Essential spectrum characterized by scattering matrices

Applicable to piecewise continuous functions

Advances understanding of spectral differences in scattering theory

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 $H$, $H_0$.