Inverse scattering on the half-line for energy-dependent Schr\"{o}dinger equations

TL;DR

This paper develops a method to solve inverse scattering problems for energy-dependent Schrödinger equations on the half-line by transforming them into Dirac systems and applying known scattering theories, enabling reconstruction of the potential.

Contribution

It introduces a novel transformation of energy-dependent Schrödinger inverse problems into Dirac systems, expanding the applicability of existing scattering theories.

Findings

Complete description of scattering functions S for the class of problems.

Algorithm for reconstructing the potential from scattering data.

Extension of scattering theory to energy-dependent boundary conditions.

Abstract

In this paper, we study the inverse scattering problem for energy-dependent Schr\"{o}dinger equations on the half-line with energy-dependent boundary conditions at the origin. Under certain positivity and very mild regularity assumptions, we transform this scattering problem to the one for non-canonical Dirac systems and show that, in turn, the latter can be placed within the known scattering theory for ZS-AKNS systems. This allows us to give a complete description of the corresponding scattering functions S for the class of problems under consideration and justify an algorithm of reconstructing the problem from S