Recovery of the matrix potential of the one-dimensional Dirac equation from spectral data

TL;DR

This paper introduces a practical numerical method for solving the inverse spectral problem of the one-dimensional Dirac equation using Gelfand-Levitan theory and Fourier-Legendre series, enabling matrix potential recovery from spectral data.

Contribution

It presents the first practical numerical approach for reconstructing the matrix potential of the 1D Dirac equation on a finite interval from spectral data.

Findings

Developed a linear algebraic system for potential reconstruction

Implemented a numerical method based on Gelfand-Levitan and Fourier-Legendre series

Demonstrated the method's effectiveness for inverse spectral problems

Abstract

A method for solving an inverse spectral problem for the one-dimensional Dirac equation is developed. The method is based on the Gelfand-Levitan equation and the Fourier-Legendre series expansion of the transmutation kernel. A linear algebraic system of equations is obtained, which can be solved numerically. To the best of our knowledge, this is the first practical method for the solution of the inverse problem for the one-dimensional Dirac equation on a finite interval.