Discretization of inverse scattering on a half line

TL;DR

This paper introduces a novel discretization method for inverse scattering problems on a half line, enabling unique potential recovery from discretized S-matrix data without relying on traditional integral equations.

Contribution

It presents a new explicit formula for potential reconstruction from discretized S-matrix data, bypassing the Gelfand-Levitan-Marchenko equation.

Findings

Unique potential recovery from discretized S-matrix data

Explicit reconstruction formula without integral equations

Effective discretization approach for inverse scattering

Abstract

We solve inverse scattering problem for Schr\"odinger operators with compactly supported potentials on the half line. We discretize S-matrix: we take the value of the S-matrix on some infinite sequence of positive real numbers. Using this sequence obtained from S-matrix we recover uniquely the potential by a new explicit formula, without the Gelfand-Levitan-Marchenko equation.