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

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