A transmutation operator method for solving the inverse quantum scattering problem

TL;DR

This paper introduces a practical transmutation operator method for solving the inverse quantum scattering problem, reducing it to linear algebra and demonstrating high accuracy and stability through numerical tests.

Contribution

It presents a new approach using Fourier-Jacobi series and Gelfand-Levitan equations to efficiently solve the inverse scattering problem.

Findings

Method reduces inverse problem to linear algebraic equations.

Proven convergence, stability, and uniqueness of the solution.

Numerical tests show high accuracy and stability.

Abstract

The inverse quantum scattering problem for the perturbed Bessel equation is considered. A direct and practical method for solving the problem is proposed. It allows one to reduce the inverse problem to a system of linear algebraic equations, and the potential is recovered from the first component of the solution vector of the system. The approach is based on a special form Fourier-Jacobi series representation for the transmutation operator kernel and the Gelfand-Levitan equation which serves for obtaining the system of linear algebraic equations. The convergence and stability of the method are proved as well as the existence and uniqueness of the solution of the truncated system. Numerical realization of the method is discussed. Results of numerical tests are provided revealing a remarkable accuracy and stability of the method.