On some integral operators appearing in scattering theory, and their resolutions

TL;DR

This paper analyzes specific integral operators relevant to scattering theory, expressing them via smooth functions of self-adjoint operators, and clarifies their properties independent of perturbation methods.

Contribution

It provides explicit expressions for integral operators like the Hilbert and Hankel transforms in terms of self-adjoint operators, advancing understanding in scattering theory.

Findings

Expressions for integral operators in terms of smooth functions of self-adjoint operators

Clarification of the operators' properties independent of perturbation theory

Application to operators like Hilbert and Hankel transforms

Abstract

We discuss a few integral operators and provide expressions for them in terms of smooth functions of some natural self-adjoint operators. These operators appear in the context of scattering theory, but are independent of any perturbation theory. The Hilbert transform, the Hankel transform, and the finite interval Hilbert transform are among the operators considered.