Phaseless inverse scattering in the one-dimensional case

TL;DR

This paper develops explicit formulas for reconstructing the complex reflection coefficient from phaseless scattering data in a one-dimensional Schrödinger equation, enabling unique potential recovery.

Contribution

It introduces a method to determine the full reflection coefficient from phaseless data, advancing inverse scattering theory in 1D.

Findings

Explicit formulas for reflection coefficient reconstruction

Global uniqueness results for phaseless inverse scattering

Potential for practical applications in wave analysis

Abstract

We consider the one-dimensional Schr\"odinger equation with a potential satisfying the standard assumptions of the inverse scattering theory and supported on the half-line $x\ge 0$. For this equation at fixed positive energy we give explicit formulas for finding the full complex valued reflection coefficient to the left from appropriate phaseless scattering data measured on the left, i.e. for $x<0$. Using these formulas and known inverse scattering results we obtain global uniqueness and reconstruction results for phaseless inverse scattering in dimension $d=1$.