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 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 2 N × 2 N 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 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.

Bound states and point interactions of the one-dimensional pseudospin-one Hamiltonian

“…where κ is defined in (30). In one dimension, similar equations have been established in [34,35] for the non-relativistic Schrödinger equation and in [29] for the Dirac equation.…”

“…Using then the boundary values of the components ψ 1 (x) − ψ 3 (x) and ψ 2 (x) obtained from wave function (29) and excluding the constants D 1 and D 2 , we get the equation for the bound state energy E = E b given in terms of the connection matrix Λ that describes any potential profile inside the interval x 1 ⩽ x ⩽ x 2 :…”

“…For example, in this way, the exactly solvable model has been constructed for the non-relativistic Schrödinger equation with a delta derivative potential δ ′ (x) [27,28]. In other studies, performed for instance in [29], the squeezing limit may be applied separately to each layer, fixing the distances between the layers. Here, using the transfer matrix method, the bound states of a one-dimensional Dirac equation with multiple delta potentials have been studied.…”

“…Finally, it should be emphasized that the squeezed connection matrices, which define the corresponding point interactions, depend on the shape of the functions used for the squeezing limit realization. This non-uniqueness problem refers to both the non-relativistic Schrödinger equation with a δ ′ -like potential [31][32][33] and the relativistic Dirac equation with a δ-potential [29]. In this regard, the piecewise representation of the potential profile of layers seems to be motivated from a physical point of view because of satisfying the principle of strength additivity [29].…”

“…This non-uniqueness problem refers to both the non-relativistic Schrödinger equation with a δ ′ -like potential [31][32][33] and the relativistic Dirac equation with a δ-potential [29]. In this regard, the piecewise representation of the potential profile of layers seems to be motivated from a physical point of view because of satisfying the principle of strength additivity [29].…”