Spectral theory of discontinuous functions of self-adjoint operators and scattering theory

TL;DR

This paper investigates the spectral properties of the difference between spectral projections of a pair of self-adjoint operators within scattering theory, linking the absolutely continuous spectrum to the eigenvalues of the scattering matrix.

Contribution

It provides a complete description of the absolutely continuous spectrum of the difference operator in terms of the scattering matrix eigenvalues and proves the absence of singular continuous spectrum.

Findings

Absolutely continuous spectrum characterized by scattering matrix eigenvalues

Singular continuous spectrum shown to be empty

Spectral properties linked to scattering theory framework

Abstract

In the smooth scattering theory framework, we consider a pair of self-adjoint operators $H_0$, $H$ and discuss the spectral projections of these operators corresponding to the interval $(-\infty,\lambda)$. The purpose of the paper is to study the spectral properties of the difference $D(\lambda)$ of these spectral projections. We completely describe the absolutely continuous spectrum of the operator $D(\lambda)$ in terms of the eigenvalues of the scattering matrix $S(\lambda)$ for the operators $H_{0}$ and $H$. We also prove that the singular continuous spectrum of the operator $D(\lambda)$ is empty.