A representation of solutions of the one-dimensional Dirac equation is obtained. The solutions are represented as Neumann series of Bessel functions. The representations are shown to be uniformly convergent with respect to the spectral parameter. Explicit formulas for the coefficients are obtained via a system of recursive integrals. The result is based on the Fourier-Legendre series expansion of the transmutation kernel. An efficient numerical method for solving initial-value and spectral problems based on this approach is presented with a numerical example. The method can compute large sets of eigendata with non-deteriorating accuracy.

翻译：本文获得了一维狄拉克方程解的一种表示形式。该解被表示为贝塞尔函数的诺依曼级数。研究表明，该表示关于谱参数是一致收敛的。通过一个递归积分系统，得到了系数的显式公式。该结果基于变换核的傅里叶-勒让德级数展开。基于此方法，本文提出了一种求解初值问题和谱问题的高效数值方法，并给出了数值算例。该方法能够以不衰减的精度计算大规模的本征数据集。