The Lorentz Integral Transform and its Inversion

TL;DR

The paper reviews the Lorentz integral transform method, discusses its inversion challenges, clarifies misconceptions about ill-posedness, and evaluates regularization techniques to improve solution stability.

Contribution

It clarifies the nature of the Lorentz integral transform's inversion problem and critically assesses regularization methods, offering insights into controlled resolution techniques.

Findings

Standard regularization procedures are effective for the transform inversion.

The term 'ill-posed' is misleading for this problem.

The Lorentz integral transform allows controlled resolution in applications.

Abstract

The Lorentz integral transform method is briefly reviewed. The issue of the inversion of the transform, and in particular its ill-posedness, is addressed. It is pointed out that the mathematical term ill-posed is misleading and merely due to a historical misconception. In this connection standard regularization procedures for the solution of the integral transform problem are presented. In particular a recent one is considered in detail and critical comments on it are provided. In addition a general remark concerning the concept of the Lorentz integral transform as a method with a controlled resolution is made.