On the ill-posed character of the Lorentz integral transform

TL;DR

This paper analyzes the ill-posed nature of the Lorentz integral transform (LIT), providing an exact inversion formula, explaining the source of instability, and discussing potential remedies with numerical examples.

Contribution

It offers an exact inversion formula for LIT, analyzes the spectral properties causing ill-posedness, and discusses possible solutions.

Findings

Inverse Fourier transform of LIT kernel increases exponentially

Continuous spectrum approaches zero, contributing to ill-posedness

Proposed cure for ill-posedness is numerically illustrated

Abstract

An exact inversion formula for the Lorentz integral transform (LIT) is provided together with the spectrum of the LIT kernel. The exponential increase of the inverse Fourier transform of the LIT kernel entering the inversion formula explains the ill-posed character of the LIT approach. Also the continuous spectrum of the LIT kernel, which approaches zero points necessarily to the same defect. A possible cure is discussed and numerically illustrated.