Inverse problem for an inhomogeneous Schr\"odinger equation

TL;DR

This paper addresses the inverse problem of uniquely recovering an inhomogeneous Schrödinger potential from boundary wave function data across all energies, providing both theoretical proof and an algorithmic solution.

Contribution

It establishes the uniqueness of potential recovery under certain conditions and introduces a practical algorithm for reconstructing the potential from boundary measurements.

Findings

Proved uniqueness of potential recovery from boundary wave data.

Developed an algorithm for potential reconstruction.

Validated the approach with theoretical and computational results.

Abstract

An inverse problem is considered for an inhomogeneous Schr\"odinger equation. Assuming that the potential vanishes outside a finite interval and satisfies some other technical assumptions, one proves the uniqueness of the recovery of this potential from the knowledge of the wave function at the ends of the above interval for all energies. An algorithm is given for the recovery of the potential from the above data.