The Response to a Perturbation in the Reflection Amplitude

TL;DR

This paper develops a method using inverse scattering theory to determine how the potential and wave function in a one-dimensional Schrödinger equation change when the reflection amplitude is varied.

Contribution

It introduces a novel approach to compute the functional derivatives of potential and wave function with respect to reflection amplitude in quantum scattering.

Findings

Derived explicit formulas for functional derivatives of potential and wave function

Enhanced understanding of the inverse scattering problem

Potential applications in quantum control and inverse problems

Abstract

We apply inverse scattering theory to calculate the functional derivative of the potential $V(x)$ and wave function $\psi(x,k)$ of a one-dimensional Schr\"odinger operator with respect to the reflection amplitude $r(k)$.