Analytic approximation of transmutation operators for one-dimensional stationary Dirac operators and applications to solution of 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 [16]. 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.