Bound states of a one-dimensional Dirac equation with multiple delta-potentials

Abstract

Two approaches are developed for the study of the bound states of a one-dimensional Dirac equation with the potential consisting of $N$ $δ$-function centers. One of these uses the Green's function method. This method is applicable to a finite number $N$ of $δ$-point centers, reducing the bound state problem to finding the energy eigenvalues from the determinant of a $2N\times2N$ matrix. The second approach starts with the matrix for a single delta-center that connects the two-sided boundary conditions for this center. This connection matrix is obtained from the squeezing limit of a piecewise constant approximation of the delta-function. Having then the connection matrices for each center, the transmission matrix for the whole system is obtained by multiplying the one-center connection matrices and the free transfer matrices between neighbor centers. An equation for bound state energies is derived in terms of the elements of the total transfer matrix. Within both the approaches, the transcendental equations for bound state energies are derived, the solutions to which depend on the strength of delta-centers and the distance between them, and this dependence is illustrated by numerical calculations. The bound state energies for the potentials composed of one, two, and three delta-centers ($N=1,\,2,\,3$) are computed explicitly. The principle of strength additivity is analyzed in the limits as the delta-centers merge at a single point or diverge to infinity.