Analytic approximation of transmutation operators for one-dimensional stationary Dirac operators and applications to solution of initial value and spectral problems

TL;DR

This paper introduces an analytic approximation method for transmutation operators to solve initial value and spectral problems for one-dimensional Dirac equations, achieving high accuracy in eigenvalue computations.

Contribution

It presents a novel approach that reduces the numerical approximation of solutions to potential matrix approximation using generalized formal powers, with proven convergence rates.

Findings

Method achieves high accuracy in eigenvalue computation.

Convergence rate depends on potential smoothness.

Applicable to both initial value and spectral problems.

Abstract

A method for approximate solution of initial value and spectral problems for one dimensional Dirac equation based on an analytic approximation of the transmutation operator is presented. In fact the problem of numerical approximation of solutions is reduced to approximation of the potential matrix by a finite linear combination of matrix valued functions related to generalized formal powers introduced in arXiv:1904.03361. Convergence rate estimates in terms of smoothness of the potential are proved. The method allows one to compute both lower and higher eigendata with an extreme accuracy.