On the calculation of the discrete spectra of one‐dimensional Dirac operators

Abstract

In this work, we consider the one‐dimensional Dirac operator where is Pauli's matrix, is a ‐matrix representing a regular potential that includes the electrostatic and scalar interactions as well as the anomalous magnetic momentum, is a singular potential consisting of delta distributions ( ) are ‐matrices representing the strengths of Dirac deltas, and is a two‐spinor. We associate to the operator an unbounded in symmetric operator denoted by , where is the support of singular potential . The operator includes only the regular potential together with certain interaction conditions at each point . The paper presents a method for determining the discrete spectrum of the operator for arbitrary potential whose entries are given by ‐functions. The eigenvalues of the operator are the zeros of a dispersion equation , where the characteristic function is determined explicitly in terms of power series involving the spectral parameter . The construction of the characteristic function from a set of monodromy matrices and the interaction conditions is presented in the paper. Moreover, its power series representation leads to an efficient numerical method for calculating the eigenvalues of the Dirac operator from the zeros of certain approximate function which is obtained by truncating the series up to a finite number of terms. Several examples show the applicability and accuracy of the numerical method.