A novel approach to estimate the stability of one-dimensional quantum inverse scattering

TL;DR

This paper introduces a new numerical method to assess the stability of the Marchenko equation in one-dimensional quantum inverse scattering, using recursion relations and Lyapunov exponents to ensure reliable data inversion.

Contribution

It presents a novel recursive approach for Fourier coefficients and stability analysis using Lyapunov exponents, enhancing inverse scattering techniques.

Findings

Derived a recursion relation for Fourier coefficients

Established conditions for stable inversion based on scattering data

Provided a numerical example demonstrating the method

Abstract

We present a novel method to estimate the stability of the Marchenko equation for finite data-sets. We show that we can derive a recursion relationship for the Fourier expansion coefficients of the kernel which is solved by the Marchenko equation. The method can easily be implemented numerically. Moreover, we discus the stability of the one-dimensional inverse scattering problem by using Lyapunov exponents. We give conditions on the scattering data to provide stable inversion results. A numerical example is given.