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

TL;DR

This paper develops two methods to analyze bound states in a one-dimensional Dirac equation with multiple delta-function potentials, providing explicit calculations for systems with up to three centers and exploring the effects of their strength and separation.

Contribution

It introduces Green's function and transfer matrix approaches for solving multi-delta potential Dirac equations, offering new analytical tools and explicit bound state calculations.

Findings

Bound state energies depend on delta-center strength and separation.

Explicit solutions for systems with up to three delta-centers.

Principle of strength additivity analyzed in merging and diverging limits.

Abstract

Two approaches are developed for the study of the bound states of a one-dimensional Dirac equation with the potential consisting of $N$ $\delta$-function centers. One of these uses the Green's function method. This method is applicable to a finite number $N$ of $\delta$-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.